saqarTvelos teqnikuri universiteti beJan kotia eleqtronebis da ...

saqarTvelos teqnikuri universitetixelnaweris uflebiTbeJan kotiaeleqtronebis da polaronebis Zvradobis kvanturiTeoriis zogierTi sakiTxi naxevargamtarebsa da ionurkristalebSidoqtoris akademiuri xarisxis mosapovebladwardgenili disertaciisavtoreferatiTbilisi2010 weli1

samuSao Sesrulebulia saqarTvelos teqnikur universitetisinformatikisa da marTvis sistemebis fakultetisfizikisdepartamentismyari sxeulebis fizikis mimarTulebaze.samecniero xelmZRvaneli: -----------------------------------------------------recezentebi: -----------------------------------------------------------------------------------------------------------------------------------------------------dacva Sedgeba ------------ wlis `_____~ --------------------------- saaTzesaqarTvelos teqnikuri universitetis informatikisa da marTvissistemebis fakultetis sadisertacio sabWos kolegiissxdomaze, korpusi-------------, auditoria-----------------misamarTi: Tbilisi 0175, kostavas 77disertaciis gacnoba SeiZleba stu-s biblioTekaSi,xolo avtoreferatisa stu-s vebgverdzesadisertacio sabWos mdivani ---------------------------------2

naSromis zogadi daxasiaTebaSesavali kristalebSi eleqtronuli gadatanis movlenebis zonuriTeoria principSi dafuZnebulia sam ZiriTad koncefciaze: 1) denisgadamtanebi warmoadgenen kvazi nawilakebs gansazRvruli kvaziimpulsiTda dispersiis kanoniT. 2) denis gadamtanTa eleqtrogamtarobada Zvradoba ganisazRvreba maTi gabneviT kristalis idealuri mesrisstruqturis dinamiur da statikur damaxinjebebze (defeqtebze). 3) denisgadamtanis Tavisufali ganarbenis sigrZe warmoadgens sasrul sididesda is bevrad aRemateba Sesabamisad kvazi nawilakis de-broilistalRis sigrZes. am pirobebis gaTvaliswinebiT denis gadamtanTa gabnevaSeiZleba CaiTvalos rogorc “iSviaTi”. am debulebebidan gamomdinare,denis gadamtanTa yofaqceva kristalSi aRiwereba albaTuri ganawilebisfunqciiT kvazi impulsebis mixedviT, romelic ganisazRvreba,rogorc bolcman-bloxis kinetikuri gantolebis amoxsna. gadatanis kinetikuri(meqanikuri) koeficientebis (Zvradoba, eleqtrogamtaroba) gamosaTvleladgamoiyeneba denis gadamtanebis arawonasworuli ganawilebisfunqciisTvis kinetikuri gantoleba–bolcmanis gantoleba–romelic iTvaliswinebs denis (muxtis) matarebelTa urTierTqmedebas(gabnevas) kristaluri mesris rxevebze.ukanasknel wlebSi, eleqtronuli gadatanis movlenebis gamokvlevebSimyari sxeulebis fizikaSi, Zalian farTo gamoyeneba hpova ufrozogadma midgomam, romelic dafuZnebulia kubos wrfivi reaqciisTeoriaze. am TeoriaSi-romelsac gadamwyveti mniSvneloba aqvs wrfivarawonasworul TermodinamikaSi–gadatanis kinetikuri (meqanikuri)koeficientebi bunebrivad gamoisaxebian (aRiwerebian) droiTi korelaciurifunqciebiT. isini asaxaven sistemis reaqcias hamiltonianisSeSfoTebisas, romlis tipiur magaliTs warmoadgens eleqtrogamtaroba.amrigad, aRniSnulidan gamomdinare naTelia, rom rogorc samecniero,aseve akademiuri TvalsazrisiT did interess warmoadgens eleqtronulida polaronuli gadatanis movlenebis koreqtuli kvanturi3

Teoriis ageba da gadatanis meqanikuri koeficientebis gamoTvlanaxevargamtarebsa da ionur kristalebSi dafuZnebuli kubos wrfivireaqciis Teoriaze.Temis aqtualoba Tanamedrove pirobebSi farTod gamokvlevissagans warmoadgens eleqtronuli da polaronuli gadatanis movlenebisSeswavlis sakiTxi myari sxeulebisa da kondensirebul garemoTafizikaSi. ukanasknel wlebSi, rTuli molekuluri aRnagobis mqonenivTierebaTa Seqmnis tendencia da maTi eleqtronuli da polaronuliTvisebebis Seswavla stimuls aZlevda mravali Teoriuli gamokvlevebisSesrulebas avtolokalizebuli (polaronuli) mdgomareobebisaRsawerad dinamiurad mouwesrigebel sistemebSi. polaronis koncefcias,romelic warmoadgens martiv magaliTs arawrfivi kvazinawilakisa, Zalian didi mniSvneloba da gamoyeneba aqvs myari sxeulebis(kondensirebul garemoTa) fizikaSi, da kerZod igi mWidrod arisdakavSirebuli kvanturi dinamiuri sistemebis Teoriis funadementurproblemebTan da velis kvanturi Teoriis sakiTxebTan. eleqtronulida polaronuli movlenebis ganxilvisa da Seswavlisas myar sxeulebSi,Teoriuli gamokvlevebis Zalian didi raodenoba samecnieroliteraturaSi miZRvnili aris eleqtronebisa da polaronebis eleqtrogamtarobisada Zvradobis gamoTvlaze naxevargamtarebsa da ionurkristalebSi. es gamokvlevebi dafuZnebuli aris sxvadasxva TeoriulmeTodebze–grinis funqciis teqnikaze, bolcmanis kinetikuri gantolebisSeswavlaze, TviTSeTanxmebul meTodebze da sxva. Sedegebi miRebulisxvadasxva meTodebis gamoyenebiT da Sesabamisad sxvadasxvamiaxloebebze dayrdnobiT, Zireulad gansxvavdeba erTmaneTisagan. miuxedavadimisa, rom eleqtronebis da polaronebis eleqtrogamtarobisada Zvradobis gamoTvla warmoadgens erT-erT uZveles problemas myarisxeulebis fizikaSi, is mainc rCeba erT-erT urTules da Znel amocanadTeoriulad amoxsnis TvalsazrisiT.4

amgvarad, Tanamedrove pirobebSi kvlav aqtualurs warmoadgenssakiTxi eleqtronebis da polaronebis eleqtrogamtarobis da Zvradobiskoreqtuli gamoTvlisa naxevargamtarebsa da ionur kristalebSi.myari sxeulebis fizikisa da arawonasworuli statistikuri meqanikismravali amocanis ganxilvisas Seiswavleba mcire dinamiuri qvesistemisevolucia droSi, romelic imyofeba kontaqtSi didi Tavisuflebisricxvis mqone, Termodinamikur wonasworobaSi myof sistemasTan–TermostatTan.eleqtronuli da polaronuli gadatanis movlenebis gamokvlevebisasmyar sxeulebSi kubos wrfivi gamoZaxilis Teoriaze dayrdnobiT,ZiriTad amocanas warmoadgens zusti, ganzogadoebuli kvanturi evoluciuri(kinetikuri) gantolebebis miReba drois ormomentiani wonasworulikorelaciuri funqciebisTvis kvazinawilakebis aRmweri Sesabamisidinamiuri sidideebisTvis, rodesac xdeba am ukanasknelTaurTierTqmedeba (gabneva) kristaluri mesris rxevebze (fononebze), darodesac fononuri (bozonuri) veli ganixileba rogorc Termostati.samecniero literaturaSi, aseTi saxis gantolebebis misaRebad,rogorc wesi, gamoiyeneba aprioruli hipoTeza–sawyisi korelaciebisSesustebis principi, an msgavsi debulebebi–mag. SemTxveviTi fazebis miaxloeba(Sfm)–rodesac drois sawyisi momentisaTvis mTeli sistemis(qvesistema plus Termostati) statistikuri operatori moicema faqtorizebulisaxiT (mTeli sistemis statistikuri operatori ganixilebarogorc qvesistemisa da Termostatis statistikur operatorTa pirdapirinamravli). naTelia, rom aseTi daSvebebis Sedegad miRebuli ganzogadoebuli,kvanturi evoluciuri gantolebebi wonasworuli korelaciurifunqciebisTvis ar aris zusti.amrigad, eleqtronuli da polaronuli gadatanis movlenebiskoreqtuli kvanturi Teoriis asagebad da meqanikuri koeficientebis(mag. Zvradoba, eleqtrogamtaroba) gamosaTvlelad naxevargamtarebsada ionur kristalebSi da agreTve eleqtron-fononuri sistemis kinetikissakiTxebis gamosakvlevad kubos wrfivi reaqciis Teoriis CarCo-5

ebSi, aqtualurs warmoadgens amocana kvazinawilakebis dinamiuri sidideebisTvisdrois ormomentiani wonasworuli korelaciuri funqciebisTviszusti, ganzogadoebuli, kvanturi evoluciuri gantolebebismiReba–sawyisi korelaciebis Sesustebis principisa da Sfm-is gamoyenebisgareSe.samuSaos mizani da amocanebi. sadisertacio naSromis mizanswarmoadgens: myari sxeulebis fizikis zogierTi kvanturi dinamiurisistemisTvis, romelic urTierTqmedebs fononur (bozonur) velTan(eleqtron-fononuri sistema, frolixis polaronis modeli, akustikuripolaronis modeli susti eleqtron-fononuri urTierTqmedebisSemTxvevaSi, polaronis feinmanis ganzogadoebuli modeli (fgm)),mowesrigebul operatorTa formalizmsa da T-namravlTa teqnikazedayrdnobiT, agreTve liuvilis superoperatoruli formalizmisa daproeqciuli operatoris meTodis gamoyenebiT_zusti, ganzogadoebulikvanturi evoluciuri (kinetikuri) gantolebebis miReba da gamokvlevadrois ormomentiani wonasworuli korelaciuri funqciebisTvis Sfm-isgamoyenebis gareSe.− Aam modelebze dayrdnobiT, da am gantolebaTa gamoyenebiT,Tanmimdevruli, srulyofili eleqtronuli da polaronuli gamtarobisada dabaltemperaturuli dreifuli Zvradobis kvanturi Teoriisageba naxevargamtarebsa da ionur kristalebSi dafuZnebuli kuboswrfivi gamoZaxilisa da SeSfoTebis Teoriaze. gadatanis meqanikurikoeficientebis (eleqtrogamtaroba, Zvradoba) gamoTvla kvanturi dinamiurisistemebis zemoTmiTiTebuli modelebisTvis.sadisertacio naSromis ZiriTadi Sedegebi da mecnieruli siaxlewarmodgenil sadisertacio naSromSi gadawyvetilia Semdegi amocanebi:− mowesrigebul operatorTa formalizmisa da T-namravlTa teqnikisdaxmarebiT, sawyisi korelaciebis Sesustebis principisa da Sfm-isgamoyenebis gareSe, gamoyvanilia da gamokvleulia axali, zusti, ganzogadoebulikvanturi evoluciuri (kinetikuri) gantolebebi gamoricxulibozonuri (fononuri) amplitudebiT drois ormomentiani wona-6

sworuli korelaciuri da grinis funqciebisTvis dinamiuri qvesistemisTvis,romelic urTierTqmedebs bozonur TermostatTan. miRebuliaagreTve axali, zusti kvanturi kinetikuri gantolebebi korelaciurifunqciebisTvis_liuvilis superoperatoruli formalizmisa da proeqciulioperatoris meTodis daxmarebiT.− dinamiuri qvesistemis bozonur (fononur) TermostatTan urTierTqmedebishamiltonianis mixedviT, SeSfoTebis Teoriis meore miaxloebaSimiRebulia axali, ganzogadoebuli kvanturi kinetikuri gantolebebigamoricxuli bozonuri amplitudebiT, qvesistemis droisormomentiani wonasworuli korelaciuri funqciebisTvis–rogorc markoviseuli,ise aramarkoviseuli formiT–romelTa dajaxebiTi integralebiSeicaven cxadad gamoyofil sawyisi korelaciebis evoluciurwevrebs.− kubos wrfivi reaqciisa da SeSfoTebis Teoriis farglebSi,naxevargamtarebisa da ionuri kristalebisTvis agebulia eleqtronulida polaronuli dabalsixSiruli gamtarobisa da dabaltemperaturulidreifuli Zvradobis Tanmimdevruli, koreqtuli kvanturiTeoria, dafuZnebuli_kvanturi disipaciuri sistemebis zemoTmiTiTebulimodelebisTvis_ganzogadoebul kvantur kinetikur gantolebebzekorelaciuri funqciebisTvis “deni-denze” eleqtronisa da polaronisTvis,romlebic urTierTqmedeben fononebTan Sfm-is gamoyenebisgareSe.− miRebulia da gamokvleulia analizuri gamosaxulebebi eleqtronisada polaronis relaqsaciuri maxasiaTeblebisTvis (impulsisrelaqsaciis sixSire da sxv.); gamoTvlilia wonasworuli korelaciurifunqciebis–“deni-denze”–milevis dekrementebi da oscilirebadifaqtorebi. napovnia eleqtrogamtarobis tenzoris (disipaciuri nawilis)analizuri saxe eleqtron-fononuri sistemisTvis kristalisdabali temperaturebisa da dabalsixSiruli gareSe eleqtruli velebisSemTxvevaSi da gamoTvlilia gadatanis meqanikuri koeficientebi(Zvradoba, eleqtrogamtaroba) kvanturi disipaciuri sistemebis aR-7

3niSnul modelTaTvis; napovnia “2K BΤ-problemis” nawilobrivi gada-hω wyveta frolixis polaronis dabaltemperaturuli Zvradobis Teoria-Si.− miRebulia temperaturuli Sesworebebi eleqtronisa da polaronisdreiful Zvradobebze, romlebic ganpirobebulia sawyisi korelaciebisevoluciuri wevrebis arsebobiT kvantur kinetikur gantolebebSiwonasworuli korelaciuri funqciebisTvis “siCqare-siCqareze”(“impulsi-impulsze”) eleqtronisa da polaronisTvis. dadgenilia, romes Sesworebebi warmoadgenen mcire sidideebs Sesrulebuli miaxloebebisada ganxiluli Teoriis farglebSi.sadisertacio naSromis praqtikuli mniSvnelobanaSromSi dasmuli amocanebis gadawyvetam moiTxova arawonasworulistatistikuri meqanikis zogierTi meTodis Semdgomi ganvi-Tareba. sadisertacio naSromSi miRebuli ZiriTadi Teoriuli Sedegebispraqtikuli mniSvneloba (Rirebuleba) ganisazRvreba imiT, rommiRebuli zusti, ganzogadoebuli kvanturi evoluciuri gantolebebiwonasworuli korelaciuri funqciebisTvis SesaZlebelia gamoyenebuliiqnas gadatanis movlenebis gansaxilvelad da gamosakvlevad_kuboswrfivi reaqciis Teoriis farglebSi, sawyisi korelaciebis Sesustebisprincipisa da Sfm-is daSvebebis gareSe_myari sxeulebis da kondensirebulgaremoTa fizikis dinamiur qvesistemaTa sxva modelTaTvis(kvanturi disipaciuri sistemebisTvis), romlebic urTierTqmedebenbozonur velTan (TermostatTan). (mag. brounis kvanturi nawilakismoZraobis Sesaswavlad, romelic ganixileba rogorc wrfivi, milevadiharmoniuli oscilatori, da romlis dinamika aRiwereba kaldeiralegetismikroskopuli modeluri hamiltonianiT).naSromSi ganviTarebuli formalizmi, meTodebi da miRebuli kinetikurigantolebebi martivad SesaZlebelia ganvrcobil iqnas kinetikurimovlenebis Sesaswavlad da gadatanis meqanikuri koeficientebis(mag. Zvradoba, eleqtrogamtaroba) gamosaTvlelad: eleqtronebis urTi-08

erTqmedebisas (gabnevisas) arapolarul optikur fononebze, piezoeleqtrulfononebze, agreTve sxva didi radiusis mqone polaronTa modelTaTvis(mag. akustikuri polaronis modelisTvis-eleqtronis fononebTanZlieri urTierTqmedebis SemTxvevaSi). gamoyvanili ganzogadoebulikvanturi kinetikuri gantolebebi korelaciuri funqciebisTvis kvanturidinamiuri qvesistemisTvis, romelic urTierTqmedebs fononurvelTan, SesaZlebelia gamoyenebul iqnas normaluri (arazegamtari)metalebis eleqtrowinaRobis gamosaTvlelad eleqtronebis gabnevisasakustikur fononebze. naSromSi warmodgenili formalizmis daxmarebiTSesaZlebelia temperaturuli Sesworebebis povna metalTa eleqtrowinaRobisTvis(Sesworebebi blox_grunaizenis formulaSi), romlebicagreTve ganpirobebulia eleqtronebis fononebTan urTierTqmedebisassawyisi korelaciebis gaTvaliswinebiT.dasacavad gamotanilia Semdegi debulebebi:1. osakas Sedegis ganzogadoeba dabalsixSiruli kuTri eleqtroga-3mtarobisTvis da “2K BΤ-problemis” nawilobrivi gadawyveta frolixishω 0polaronis (eleqtronis) dabaltemperaturuli Zvradobis TeoriaSi.2. eleqtronis dabaltemperaturuli statikuri Zvradobis gansxvavebuli(2-jer naklebi) mniSvneloba `bolcmaniseul~ ZvradobasTanSedarebiT akustikuri polaronis modelSi susti eleqtron-fononuriurTierTqmedebis SemTxvevaSi.3. Zlieri eleqtron_fononuri urTierqmedebis SemTxvevaSi dabaltemperaturuliZvradobis gansxvavebuli yofaqceva eleqtron-fononuribmis mudmivas rigis mixedviT polaronis fgm_Si, pekarismodelTan SedarebiT.4. mcire sididis temperaturuli Sesworebebis arseboba eleqtronis dapolaronis dabaltemperaturul Zvradobebze ganxilul modelebSi.naSromis aprobacia disertaciis ZiriTadi Sinaarsi moxsenebuliiyo informatikisa da marTvis sistemebis fakultetis fizikis departa-9

mentisa da myari sxeulebis fizikis kolegiis samecniero seminarebissxdomebze.disertaciis Sinaarsi da ZiriTadi Sedegebi wardgenili iyo: 1986w. q.TbilisSi Catarebul 24-e sakavSiro TaTbirze_»24-? ?????? ???? ??????????? ?????? ?????? ??????????», Tbilisi, 1986w; 1991w q. xarkovSi (ukraina)Catarebul sakavSiro konferenciaze «??????????????????? ????????????????????», ???????, 14-17??? 1991?; 1992w. q. puSCinoSi (ruseTi) CatarebulsaerTaSoriso skolaSi_International Workshop `POLARONS and APPLICATIONS~May 23-31, 1992, Pushchino, Russia; 1992w. q. berlinSi (germania) CatarebulsaerTaSoriso konferenciaze `The 18 th IUPAP International Conference onStatistical Physics, Berlin, 2-8 August 1992~; saqarTvelos teqnikuri universitetisprofesor_maswavlebelTa samecniero_teqnikur konferenciaze,16-19 noemberi, Tbilisi, 1993w; 1993w. q. trondhaimSi (norvegia)Catarebul saerTaSoriso simpoziumze `The Lars Onsager Symposium. CoupledTransport Processes and Phase Transitions~, June 2-4 1993, Trondheim, Norway; 1993w. q.florenciaSi (italia) Catarebul saerTaSoriso konferenciaze `EPS9TRENDS IN PHYSICS~ Firenze, 14-17 September 1993; 1995w. q. qsiamenSi (CineTi)Catarebul saerTaSoriso konferenciaze `The 19 th UPAP InternationalConference on Statisitcal Physics~, Xiamen 31 July-4 August 1995; 1998w. q. parizSi(safrangeTi) Catarebul saerTaSoriso konferenciaze `XXth IUPAPINTERNATIONAL CONFERENCE ON STATISTICAL PHYSICS~, Paris, July 20-24,1998, UNESCO Sorbonne.publikaciebi: disertaciis ZiriTadi Sedegebi gamoqveyenebuliaCvidmet samecniero naSromSi, romelTa dasaxeleba moyvanilia avtoreferatisbolos.naSromis moculoba da struqtura: disertaciis sruli moculobaSeadgens 157 nabeWd gverds; disertacia Sedgeba reziumesagan (or enaze),sarCevisagan, naxazebis nusxisagan, Sesavlisagan, sami Tavisagan,ilustraciis saxiT moyvanili sami naxazisgan, daskvnebisa da 125dasaxelebis mqone gamoyenebuli literaturis siisgan, erTi danarTisa10

da avtoris mier gamoqveynebuli samecniero naSromebisgan, romlebSiacasaxulia disertaciis ZiriTadi Sedegebi.sadisertacio naSromis SinaarsiSesavalSi dasabuTebulia Temis aqtualoba, Camoyalibebulia naSromismiznebi da amocanebi, da gansazRvrulia kvlevis obieqtebi dameTodebi.pirveli Tavi ZiriTadad atarebs mimoxilviT xasiaTs$1.1 (1.1.1-1.1.6)_Si moyvanilia modeluri hamiltonianis saxe dinamiurisistemebisa, romlebic urTierTqmedeben bozonur TermostatTan da ganxiluliazogierTi aqtualuri magaliTi kvanturi disipaciuri da Riaarawonasworuli modeluri sistemebisa Tanamedrove fizikis sxadasxvadargidan, romlebic gaxdnen intensiuri kvlevisa da Seswavlis saganiukanaskneli 30-40 wlis ganmavlobaSi. am farTo gamokvlevaTa speqtrimoicavda metalTa eleqtrogamtarobisa da zegamtarobis Teoriis, metalTaSenadnobebisa da gadacivebuli `metaluri minebis~ eleqtronuliTeoriis sakiTxebs; susti da Zlieri lokalizaciisa da Zlier araerTgvarovannivTierebaTa eleqtrogamtarobis Teoriis sakiTxebs mouwesrigebelsistemaTa fizikaSi; lazeruli gamosxivebisa da zegamosxivebisTeoriis aspeqtebs kvantur radiofizikaSi; magnituri polaronebisda fluqtuonebis (fazonebis) Teoriis sakiTxebs magniturnivTierebebSi (garemoebSi) da sxva.$1.2 (1.2.1-1.2.2.)_Si ganxilulia dinamiurad mouwesrigebeli sistemaeleqtron-fononuri sistema da eleqtronis urTierTqmedeba akustikurda polarul optikur fononebTan; moyvanilia eleqtron-fononuri sistemishamiltonianis zogadi saxe (frolix-pekaris tipis hamiltoniani),eleqtronis akustikur da polarul optikuri fononebTan urTierTqmedebishamiltonianTa saxeebi da mokled mimoxilulia agreTve deformaciispotencialis meTodi.$1.3 (1.3.1-1.3.3)_Si moyvanilia didi radiusis mqone polaronTa modelebi.ganxilulia polaronis frolixisa da pekaris modelebi dapolaronis feinmanis erToscilatoriani da feinmanis ganzogadoebu-11

li modelebi. amave paragrafSi ganxilulia agreTve ukanasknel wlebSiganviTarebuli da gamoyenebuli axali midgoma polaronuli sistemebisTermodinamikisa da kinetikis sakiTxebis gamokvlevebisas _ mowesrigebulioperatorTa formalizmi, T_namravlTa meTodi da fononurioperatorebis gamoricxvis teqnika eleqtron-fononuri sistemis maxasiaTebelifizikuri sidideebis wonasworuli da arawonasworulisaSualo mniSvnelobebidan. aRniSnulia zogierT SemTxvevebSi am axalimidgomis upiratesoba, kontinualuri integri-rebis meTodTanSedarebiT, eleqtron-fononuri sistemis kinetikis sakiTxebisSeswavlisas.$ 1.4._Si ganxilulia fizikuri kinetikis zogierTi principuli sakiTxidinamiuri sistemebisa, romlebic urTierTqmedeben fononur(bozonur) velTan. mimoxilulia metad mniSvnelovani da principulisakiTxi, K_tipis dinamiur sistemebSi (rogorc klasikuris, aseve kvanturisTvis),evoluciuri (kinetikuri) gantolebebis gamoyvanis drosSemoklebuli aRweris SesaZleblobaze, romelic ar eyrdnoba hipo-Tezas_sawyisi korelaciebis Sesustebisa da Sfm-is gamoyenebis Sesaxeb.aRwerilia is ZiriTadi sqemebi da meTodebi, romlebsac mivyavarTbolcmanis saxis kinetikuri gantolebisa da ZiriTadi kinetikuri gantolebismiRebamde. ganxilulia is ZiriTadi principuli xasiaTissirTuleebi, romlebic warmoiSvebian eleqtron-fononuri sistemisa dapolaronTa zemoTmoyvanil modelTaTvis dreifuli Zvradobebis gamo-Tvlisas, gamomdinare rogorc bolcmanis kinetikuri gantolebidan dakubos wrfivi reaqciis Teoriidan, aseve gamtarobis wrfivi da arawrfiviTeoriebis zogierTi sxva meTodebis (arawonasworuli simkvrivismatricis meTodi, balansis gantolebis meTodi) gamoyenebisas.meore Tavi sadisertacio naSromSi originaluri xasiaTisaa.am TavSi dasmulia da gadawyvetilia zogadi saxis amocana_dinamiuriqvesistemisTvis, romelic urTierTqmedebs bozonur (fononur)TermostatTan_axali, zusti ganzogadoebuli kvanturi evoluciurigantolebebis miReba drois ormomentiani wonasworuli korelciuri12

da grinis funqciebisTvis sawyisi korelaciebis Sesustebis principisada Sfm_is gamoyenebis gareSe.$2.1_Si ganxilulia mowesrigebuli operatorTa formalizmi daT_namravlTa meTodi.$2.2_Si am formalizmze dayrdnobiT gamoyvanilia axali, zusti,araCaketili ganzogadoebuli kvanturi kinetikuri gantolebebi wonasworulikorelaciuri funqciebisTvis, rogorc markoviseuli, asevearamarkoviseuli saxis, saidanac gamoricxulia bozonuri amplitudebi.miRebuli zusti kvanturi kinetikuri gantolebebis dajaxebiTi intgralebiSeicaven damatebiT wevrebs, romlebic aRweren sawyisi korelaciebisevolucias droSi, ganpirobebuls qvesistemis urTierTqmedebiTbozonur TermostatTan drois sawyis momentSi. napovniaagreTve msgavsi saxis evoluciuri gantoleba grinis (dagvianebuli)funqciisTvis. sadisertacio naSromis danarTSi detalurad aris aRwerilibozonuri operatorebis (amplitudebis) gamoricxvis teqnikadrois ormomentiani wonasworuli korelaciuri funqciebis evoluciuri(kinetikuri) gantolebebidan.$2.3_Si ganxilulia markoviseuli miaxloeba qvesistemis dinamikisTvisda qvesistemis TermostatTan urTierTqmedebis hamiltonianismixedviT SeSfoTebis Teoriis meore miaxloebaSi gamoyvanilia axali,ganzogadoebili markoviseuli kvanturi kinetikuri gantoleba droisormomentiani wonasworuli korelaciuri funqciisTvis < B sA s( − t)>:13

∂∂tti1BA ( − t) = [ HB ] ⋅A ( −t) − ⎡2( 1 + ( ))×−hh∑∫ dξ⎣N βs s s s s kk 0( , ξ) ⎡ ( ),⎤ ( ) ( β)× e C s − ⋅ C s B A − t + N e ×iω( k) ξ + −iω( k)ξkH0⎣ k s⎦−s k1× C s − ⋅ C s B A − t + d + N ×t+kH01 ⎣ k s⎦−s⎦2h∑∫ ⎣ kk 0( ξ) ⎡ ( ), ⎤ ( ) ⎤ ξ ⎡( 1 ( β))( ), ⎤ ( , ξ) ( ) ( β)× e ⋅ ⎡C s B C s − A − t + N e ×− iω( k) ξ +iω( k)ξ⎣ k s⎦−kH0s khβ+× ⎡⎣C ( s), B ⎤⎦⋅C s,k s−kH ( ) ( ) ⎤ i−ξ A −t − ⎡( 1 + ( ))×0s⎦2 ∑h∫ dθ ⎣N βk( , θ) ⎡ ( ),⎤ ( ) ( β)× e e C s −t−i C s B ⋅ A − t + N e ×− θω( k) iω( k) t +θω( k)kH0⎣ k s⎦−s k( θ ) ⎡ ( ) ⎤ ( )− iω( k)t +× e C s, −− t i ⋅ , ⋅ − ⎤.kH0⎣C s Bk s⎦ A t− s⎦k0(1)sadac H s _qvesistemis hamiltoniania, A s , B s C k (s), C + k (s) –qvesistemisoperatorebia, N k (β)=[e β h ω(k) -1] -1 _warmoadgens bozonebis Sevsebis saSualoricxvs, xolo operatorebi C kH( S,) da C kH( S,) _ gansazRvruliatolobebiT:0Z0ZC kH( S,) = e0ZiH 0ZhC ( S)eki− HoZhC kH+ih( S,Z)= e0H0ZC+k( S)ei− HoZhH ;0= Hs+ H∑da H Σ - aris bozonuri Termostatis hamiltoniani. analogiurisaxis gantolebebia miRebuli agreTve sxva korelaciuri funqciebisTvis.$2.4_Si ganxilulia da ganviTarebulia meore (gansxvavebuli) midgomaigive problemisadmi, romelic eyrdnoba liuvilis superoperatorulformalizmsa da proeqciuli operatoris meTods. am formalizmisada meTodis daxmarebiT napovnia axali, zusti ganzogadoebulikvanturi evoluciuri (kinetikuri) gantolebebi wonasworuli korelaciurifunqciebisTvis Sfm_is gamoyenebis gareSe. gamoyvanil kinetikurgantolebas < A sB s( − t)>_korelaciuri funqciisTvis aqvs Semdegi saxe:14

∂∂t( ) [ ] ( ) { ( , β )AB − t = i ΡL ΡA B −t −i ΡI t ×s s s s s Q−1( ) ( β) ⎤} ( )× ⎡⎣1 + MQ t IQ t,⎦ MQQLiΡAs Bs− t + dτ×{ LQM ( t τ ) ( t, β) ⎡1 M ( t) ( t,β)× Ρi Q− IQ ⎣ +QIQ⎤⎦×( ) } ( τ ) τ ⎡ ( τ)× M t QLΡA B − − d ⎣ΡLQM t− Q×Q i s s i Q0] ( τ )× LΡA B −i s s.t∫t∫0−1(2)1sadac L = [ KSK HS, ]−_liuvilis superoperatoria, romelic SeesabamebaS–qvesistemis H S –hamiltonians, P-warmoadgens Termostatish(bozonuri velis) mdgomareobebis mixedviT gasaSualebis proeqciuloperators:2P =P daQ = 1 − P ; xolo `masuri~ M Q (t) da integraluriI ( t,β ) superoperatorebi gansazRvrulia Semdegi tolobebiT:QiLt λH λHM Q( t)= exp[ iQLQt ];Q ( t,β) dλe − e Hinte −QβI =∫miRebuli aramarkoviseuli saxis zusti, ganzogadoebuli (2)_kinetikuri gantolebis dajaxebiTi integrali Seicavs wevrebs, romlebicaRweren sawyisi korelaciebis evolucias droSi. (sawyisi korelaciebisevolucia drois mixedviT aRiwereba I ( t,β ) _ integralurisuperoperatoriT da es evolucia moicema rogorc markoviseuli, asevearamarkoviseuli formiT).$2.5_Si, iseve rogorc $2.3_Si, ganxilulia markoviseuli miaxloebaqvesistemis dinamikisTvis da qvesistemis TermostatTan urTierTqmedebisliuvilianis mixedviT _ SeSfoTebis Teoriis meore miaxloebaSi,proeqciuli operatoris meTodis daxmarebiT_napovnia axali,ganzogadoebuli markoviseuli kvanturi kinetikuri gantoleba wonasworulikorelaciuri funqciisTvis < A sB s( − t)>_gamoricxuli bozonuriamplitudebiT:0Q15

nTa modelebSi, polaronis fgm_Si. gamokvleva dafuZnebulia kuboswrfivi gamoZaxilisa da SeSfoTebis Teoriaze. gamoTvlilia eleqtronulida polaronuli gadatanis meqanikuri koeficientebi(dabaltemperaturuli Zvradoba, eleqtrogamtaroba) zemoT miTiTebulmodelebSi_korelaciuri funqciebisTvis miRebul ganzogadoebul kvanturkinetikur gantolebebze dayrdnobiT.$3.1_Si gamokvleulia eleqtron-fononuri sistema da gamoTvliliaeleqtronis dabaltemperaturuli Zvradoba da dabalsixSirulieleqtogamtaroba susti eleqtron-fononuri urTierTqmedebis SemTxvevaSi(1) da (3)_gantolebebze dayrdnobiT miRebulia miaxloebiTikinetikuri gantoleba eleqtronis `siCqare_siCqareze~ wonasworulikorelaciuri funqciisTvis. napovnia am bolcmanis tipis gantolebisamonaxsni kristalis dabali temperaturebis SemTxvevaSi erTi zonisada relaqsaciis drois miaxloebaSi (rdm) eleqtronisTvis, da fononebisdispersiis nebismieri izotropuli kanonis dros. dadgenilia,rom eleqtronis (qvesistemis) korelaciuri funqciebi miilevian oscilaciebiTdidi droebis asimptotur areSi, gamoTvlilia korelaciurifunqciebis milevis dekrementi da oscilirebadi faqtori. gamoyvaniliazogadi formulebi kuTri eleqtrogamtarobis disipaciuri nawilisTvisizotropul SemTxvevaSi da eleqtronis dabaltemperaturuli,eleqtruli velis ω_sixSireze damokidebuli ZvradobisTvis (acmobility). am formulebs aqvs Semdegi saxe:⎛1⎞th⎜βhω2⎟⎝ ⎠ r⎧ ⎡βh⎤= ∫ Ρ Ρ Ρ Ρ ⎨ ⎢Γ Ρ ×hω⎣ 2 ⎥⎩⎦s 2relRe σµνne d ρs( β, ) Vν ( ) Vµ ( ) cosν( β, )relrel( , )⎫Γ ( , )Γ Ρνβ Ρ ⎡βhrel ⎤ µβ ⎪× + cos Γ ( , Ρ) ,22 ⎢ µβ22⎬rel( , ) ⎣ 2 ⎥relω + ⎡Γ Ρ ⎤ ⎦⎣ ⎦ + ⎡⎣Γ ( , Ρ)⎤νβ ωµβ ⎦ ⎪⎭sadac: µ ( ω)= µ0(ω)− ∆µ( ω)(4)17

µ⎛ 1 ⎞eth⎜βhω⎟⎪⎧rrrelΓ ⎪⎫⎝ 2 ⎠r r r rrelΓ ( , )( , P)Vβ Pµβω)= ∫dPρs(β,P)VV( P)Vµ( P)⎨r +⎬;2 rel2rhω2 rel⎪⎩ ω + [ ΓV( β,P)] ω + [ Γµ( β,P)] ⎪⎭0( 2⎛ 1 ⎞2eth⎜βhω⎟∆ =⎝ 2 ⎠r r r rµ ( ω)∫ dPρS(β,P)VV( P)Vµ( P)×(5)hω⎪⎧rrrelrel2( , )2( , ) ⎪⎫⎡βhrrel ⎤ Γ⎡⎤ Γ⎨sin ( , )2sin ( , )2 ⎬;42⎢[ ( , )] 4 ⎥ ⋅ PVβ P βhrrelµβ−1× ⎢ ΓVβ P ⎥r + Γµ β Pr ( ω

lsis relaqsaiciis sixSire ar aris damokidebuli TviT eleqtronisimpulsze da warmoidgineba Semdegi saxiT:~relrel 2~ 2Γz ( γ , Ρ)≡ Γ0 ( γ ) = αω0Ν0( γ ); ( Ρ > 1) (6)3sadac: ω 0 _ fononebis rxevis sixSirea, α_ warmoadgens eleqtron-fononuriurTierTqmedebis (frolixis) bmis mudmivas, daN−10( γ ) = [ eγ −1]eleqtronis `siCqare-siCqareze~ korelaciuri funqciisTvis miRebulibolcmanis tipis miaxloebiTi kinetikuri gantoleba amoxsnilia rdm-Sikristalis dabali temperaturebis SemTxvevaSi, da napovnia eleqtrogamtarobisda Zvradobis mniSvnelobebi frolixis polaronis modelSi_racfaqtiurad warmoadgens osakas Sedegis ganzogadoebas (drudesformulas) dabal sixSiruli eleqtrogamtarobisTvis da dabaltemperaturuliZvradobisTvis, romlebic Seicaven temperaturul Sesworebebs,ganpirobebuls eleqtronis fononebTan arsebuli sawyisi korelaciebiT:2Re ~ ne 2 ⎛ 1 ~ ⎞⎡( ))2⎛γ⎞⎤Γ0γσ(ω = ~ th⎜γω ⎟ 1 2sin0() ~ ;22⎢ − ⎜ Γ γ ⎟mω2⎥20γω ⎝ ⎠⎣⎝ ⎠⎦ω + Γ0( γ)( α < 1, γ >> 1, ωγ ~ > 1, ω~γ

µ0=emω0Γ−10e( γ ) =mω03e2α2e2 ⎡γ⎤ −1e 1 2 −γ∆µ= sin⎢Γ0( γ ) Γ0( ) ≈;02 ⎥γ αγ e(9)mω⎣ ⎦ mω03( γ >> 1, α < 1).gamoTvlilia agreTve eleqtrogamtaroba da eleqtronis Zvradobaeleqtruli velis maRali ω ~ -sixSireebis SemTxvevaSi. am Sem-TxvevaSic temperaturuli Sesworeba eleqtronis Zvradobaze warmoadgensZalian mcire sidides.sadisertacio naSromis am paragrafSi avtoris mier miRebuli3 KSedegi (ix. (9) formula) warmoadgens ` BT2 hωγ;0-problemis~ nawilobrivgadawyvetas frolixis polaronis (eleqtronis) dabaltemperaturuliZvradobis TeoriaSi. Zvradobis miRebuli mniSvneloba 3-jer aRemateba`bolcmaniseul~ dabaltemperaturul statikur Zvradobas:1 γµ = eB. e ;mω2α( ω = 0, γ >> 1);da 1 -mamravliT gansxvavdeba fxip-isa2γ0(feinmani, xelvorsi, idingsi, platcmani) da tornberg_feinmanis Sede-e 3 1 γgisgan: µFXIP= µTF=e ; ( ω = 0, γ >> 1);rac Seexeba TviT fxip-isamω2γ2α0da tornberg_feinmanis Sedegebis Tanxvedras da1 _mamravls, maTi2γwarmoSobis buneba (Rrma mizezi) dRevandel dRemde bolomde dadgeniliar aris.$3.3_Si ganxilulia da gamokvleulia eleqtronis Zvradobisyofaqceva akustikuri polaronis modelSi susti eleqtron-fononuriurTierTqmdebis dros. am SemTxvevaSi adgili aqvs eleqtronis urTierTqmedebasdispersiis mqone akustikur fononebTan: ω ( k ) = V . k (Vr rs_bgeris siCqarea kristalSi), da iseve rogorc polaronis frolixismodelSi, eleqtronis energiisTvis gamtarobis zonaSi gvaqvs dispersiisparaboluri kanoni: T ( P)= P / 2m;(m-eleqtronis efeqturir r2masaa).am paragrafSic miRebuli Sedegebi eleqtrogamtarobisa da eleqtroniss20

Zvradobis yofaqcevis Sesaxeb eyrdnoba $3.1_Si gamoyvanil zogadTanafardobebsa da formulebs. napovnia eleqtronis `siCqare_siCqareze~_korelaciurifunqciebis milevis dekrementebi (eleqtronis impulsisrelaqsaciis sixSire) da oscilirebadi faqtorebi. ganxiluliaeleqtronis aradrekadi gabnevis procesebi akustikur fononebze da~dadgenilia, rom `mcire~ siCqariT ( P < 1)moZravi eleqtronisTvis, implsisrelaqsaciis sixSire (dro) kristalis Zalian dabali temperaturebisdros ar aris damokidebuli TviT eleqtronis impulsismniSvnelobebze da warmoidgineba Semdegi saxiT:ΓrelAc( γ ) = τ−1relAcmV( γ ) = sh264α4γ−[ e −1] 1~( Ρ >1;α < 1 ).(10)2~mVssadac: P = P / mV s: da γ = : xolo α_eleqtron-fononuri2KT2 2D murTierTqmedebis (bmis) mudmivaa: α = < 1.D_deformaciis pote-38πρhncialis mudmivaa, ρ_kristalis masuri simkvrive. napovnia `siCqaresiCqareze~korelaciuri funqciisTvis miRebuli bolcmanis tipismiaxloebiTi kinetikuri gantolebis amonaxsni rdm-Si kristalis dabalitemperaturebis dros, da gamoTvlilia dabalsixSiruli kuTrieleqtrogamtaroba da eleqtronis Zvradoba am modelSi izotropulSemTxvevaSi da erTi zonis miaxloebaSi:BV s2 relne Γ ⎡ ⎤Ac(γ)hγrelReσAc(ω)= cos⎢Γ ( γ)2 2rel2 Ac ⎥m ω +ΓAc( γ)⎣mVs⎦(11)( α < 1; γmV>> 1; ω

∆µAc=m V4γ[ αγe]he1 4γ2e sin 32−2 2s32α; ( α < 1; γ >> 1)naTelia, rom temperaturuli Sesworeba eleqtronis dabaltemperaturulZvradobaze, romelic ganpirobebulia sawyisi korelaciebisgaTvaliswinebiT, warmoadgens Zalian mcire sidides. miRebuli formulebiwarmoadgenen Tanmimdevrul da koreqtul Sedegs eleqtronisdabaltemperaturuli ZvradobisTvis akustikuri polaronis modelSisusti eleqtron_fononuri bmis SemTxvevaSi.am paragrafSi gamoTvlilia aseve elqtrogamtaroba da eleqtronisZvradoba akustikuri polaronis modelSi eleqtruli velismaRali ω_sixSireebis SemTxvevaSi, kristalis rogorc maRali, asevedabali temperaturebis dros, napovnia am sidideebis ω_sixSiresa daΤ_temperaturaze damokidebuleba (yofaqceva) am parametrebissxvadasxva mniSvnelobebis dros. am SemTxvevaSic temperaturuliSesworeba eleqtronis dabaltemperaturul Zvradobaze warmoadgensZalian mcire sidides.eleqtronis dabatemperaturuli Zvradobis gamosaTvlelad akustikuripolaronis modelSi susti eleqtron-fononuri bmis SemTxvevaSi(α> 1, α < 1, ω 0).2 2m Vs 32α0 BAC=xolo, fxip-is Teoriisa (miaxloebisa) da balansis gantolebis meTodis(tornberg-feinmanis Teoriis) gamoyenebiT napovni eleqtronisdabaltemperaturuli statikuri Zvradobis mniSvneloba tolia sidids:µ ~FXIPAC~ TF 3 1 4γhe= µ;~AC= ⋅ e µ = µ , ( γ >> 1, α < 1)2 24γ64αm Vsavtoris mier napovni dabaltemperaturuli statikuri ZvradobismniSvneloba Sfm_Si (ix. 12 formula). warmoadgens Tanmimdevrul dakoreqtul Sedegs akustikuri polaronis modelSi. is 2_jer naklebia(13)(14)22

eleqtronis `bolcmaniseul~ dabaltemperaturul Zvradobaze da3 -4γmamravliT gansxvavdeba fxip-isa da tornberg-feinmanis Sedegasgan. racSexeba fxip-isa da tornberg-feinmanis Sedegebis Tanxvedras da3 -4γmamravls_iseve, rogorc polaronis froli-xis modelSi, am modelSicmaTi warmoSobis Rrma mizezi (buneba) jer-jerobiT dadgenili ar aris.$3.4_Si ganxilulia polaronis fgm. gamoTvlilia da gamokvleuliakontinualuri optikuri polaronis drefuli Zvradoba da misiyofaqceva am modelSi dabali temperaturebis SemTxvevaSi. gamoyvaniliakvanturi kinetikuri gantolebebi polaronis `impulsi-_impulsze~(`deni_denze~). wonasworuli korelaciuri funqciebisTvis,ganxilulia markoviseuli miaxloeba polaronis dinamikisaTvis dadidi droebis asimpatotur areSi: t ~ τ >> t = 0max (t s , t Σ ); τ >> β;h = m = ω = 1). (t 0 s-polaronis dajaxebaTa maxasiaTebeli droa, t Σ -arisTermostatSi fluqtuaciebis korelaciebis maxasiaTebeli dro), miRebuliamiaxloebiTi gamosaxulebebi korelaciuri funqciebisTvis.gamoyvanilia agreTve kinetikuri gantolebebi korelaciuri funqciebisdiagonaluri matriculi elementebisTvis. napovnia korelaciurifunqciebis milevis dekrementebi (polaronis impulsis relaqsaciissixSire) da oscilirebadi faqtorebi.relrel−1dabali temperaturebis SemTxvevaSi ( β >> 1,β > εi− εj)-polaronis P P ( t)− korelaciuri funqciebissidide ZiriTadad gansazRvrulia impulsTa im mniSvnelobebiT,romelTaTvisac P r 2

polaronis P r impulsze (`mcire~ siCqariT moZravi polaroni) da ganisazRvrebaTanafardobiT:r)GFGF 2Γzrel( β , Ρ)≡ Γ0rel( β ) =3αN0( β ) M f ( 2Mr)2( β >> 1, Ρ / 2M |(16)napovni (dabalsixSiruli) kuTri eleqtrogamtarobisTvis da polaronisdabaltemperaturuli ZvradobisaTvis gvaqvs Semdegi saxisgamosaxulebebi:12th(βω)GFGF2 −1Γ⎡ ⎤=20rel( β)β GFRe σ ( ω)Ne β Mcos⎢Γ0( β)2 2GFrelω ω + Γ ( β)⎣ 2 ⎥(17)⎦0relGF−1( ω

gadadis polaronis frolixis modelSi, polaronis dabaltemperaturuliZvradobisTvis gvaqvs:GF 3eβµ0⇒ µ0= e ;1 GF2 −β∆µ⇒ ∆µ= eαβe ;( α < 1, β >> 1)2α3romelic emTxveva dabaltemperaturuli Zvradobis mniSvnelobaspolaronis frolixis modelSi.2. Zlieri eleqtron-fononuri bmis zRvrul SemTxvevaSi: (α>>1, M GF >>1,M GF →8), rodesac fgm aRadgens pekaris Teorias, polaronis dabaltemperaturuliZvradobis yofaqceva aRiwereba TanafardobiT:µ3e~ exp( β ) α2;GF13oΠ( >> 1, β >> 1)α (20)sadisertacio naSromSi avtoris mier ganviTarebuli formalizmida meTodebi, miRebuli ganzogadoebuli kvanturi kinetikurigantolebebi wonasworuli korelaciuri funqciebisTvis da mesame Tavis$3.1_$3.4_ebSi ganxilul modelebze dayrdnobiT miRebuli Sedegebi(gamoyvanili formulebi: rogorc zogadi, aseve miaxloebiTi) SesaZleblobasiZleva gadaugvarebel, farTozonian, erTgvarovan (polarul)naxevargamtarebSi, ionur da kovalentur kristalebSi_eleqtronulida polaronuli gamtarobisa da dabaltemperaturuli dreifuli Zvradobiswrfivi kvanturi Teoriis agebas_kvanturi dinamiuri sistemebissxva modelTaTvisac, romlebic urTierTqmedeben fononebTan (eleqtronebisgabneva arapolarul optikur fononebze, piezoeleqtuli gabneva,polaronis fm. da sxva)._eleqtronebisTvis erTi zonis miaxloebaSi,dispersiis rogorc zogadi, aseve paraboluri kanonis dros dafononebis dispersiis izotropuli kanonis SemTxvevaSi.daskvnebi1. sxvadasxva midgomebis-mowesrigebul operatorTa da liuvilissuperoperatoruli formalizmisa da proeqciuli operatoris meTodisgamoyenebiT, sawyisi korelaciebis gaTvaliswinebiT_gamoyvanilia axali,zusti, ganzogadoebuli kvanturi evoluciuri (kinetikuri) gantolebebidrois ormomentiani wonasworuli korelaciuri funqciebisTvis,dinamiuri qvesistemisTvis romelic urTierTqmedebs bozonurvelTan (TermostatTan). miRebul gantolebaTa dajaxebiTi integralebi(19)25

Seicaven rogorc wevrebs, romlebic aRweren namdvili korelaciebisevolucias droSi, aseve sawyisi korelaciebis evoluciur wevrebs,romlebic ganpirobebulia qvesistemis urTierTqmedebiT bozonur TermostatTandrois sawyis momentSi.2. SeSfoTebis Teoriis meore miaxloebaSi – qvesistemis TermostatTanurTierTqmedebis hamiltonianis mixedviT – napovnia ganzogadoebulikvanturi kinetikuri gantolebebi gamoricxuli bozonuriamplitudebiT korelaciuri funqciebisTvis, rogorc markoviseuli,ise aramarkoviseuli saxiT, romlebic Seicaven cxadad gamoyofilsawyisi korelaciebis evoluciur wevrebs.3. eleqtron-fononuri sistemisTvis, frolixisa da akustikuripolaronis modelTaTvis, SeSfoTebis Teoriis meore miaxloebaSi,susti eleqtron-fononuri urTierTqmedebis SemTxvevaSi da erTi zonismiaxloebaSi eleqtronisaTvis gamoyvanilia da gamokvleulia markovissaxis kinetikuri gantolebebi eleqtronis siCqaris operatoriskomponentebis saSualo mniSvnelobebis diagonaluri matriculi elementebisaTvis,romlebic warmoadgenen bolcmanis tipis gantolebebs,saidanac gamoricxulia fononuri amplitudebi. Gganxilulia eleqtronisaradrekadi gabnevis procesebi fononebze da dadgenilia, romganxilul modelebSi adgili aqvs relaqsaciur process korelaciurifunqciebis oscilaciebiT. Nnapovnia eleqtronis impulsis (siCqaris)relaqsaciis sixSireebis analizuri gamosaxulebebi kristalis dabalitemperaturebis SemTxvevaSi. gamoTvlilia eleqtronis “siCqaresiCqareze”korelaciuri funqciebis milevis dekrementebi da oscilirebadifaqtorebi.4. gamokvleulia da dadgenilia, rom eleqtronis siCqaris (impulsis)mcire mniSvnelobebisaTvis, siCqaris relaqsaciis droebi(sixSireebi) ganxilul modelebSi ar aris damokidebuli impulsissidideze. mcire siCqareebiTYmoZravi eleqtronebisTvis Zalian dabalitemperaturebis dros napovnia dabalsixSiruli eleqtrogamtarobisada eleqtronis dreifuli Zvradobis gamosaTvleli formulebi.5. frolixis polaronis modelSi miRebuli gamosaxulebebi eleqtronisdabaltemperaturuli dreifuli Zvradobisa da dinamiurigamtarobiTvis warmoadgens osakas mier napovni Sedegis ganzogadoebasmcire intensivobis mqone dabalsixSirul gareSe eleqtrul velSi,rac faqtiurad SesaZlebelia ganxiluli iqnas, rogorc drudes formulakuTri eleqtrogamtarobisTvis. napovnia agreTve statikuri( ω = 0) eleqtrogamtarobisa da dabaltemperaturuli dreifuli Zvradobisanalizuri gamosaxulebebi, rogorc frolixis, aseve akustikuripolaronis modelebSi.6. rogorc gamoTvlebi gviCvenebs, eleqtronebis gabnevisas polaruloptikur fononebze, dabaltemperaturuli dreifuli ZvradobisTvis(dcmobility; ω = 0 ) miRebuli mniSvneloba 3-jer aRematebaZvradobis im mniSvnelobas, romelic miiReba bolcmanis kinetikurigantolebis gamoyenebiT da amoxsniT relaqsaciis drois miaxloebaSi.26

3 KmiRebuli Sedegi warmoadgens – “ BΤproblemis” – nawilobriv gadawyvetasfrolixis polaronis dabaltemperaturuli Zvradobis Teo-2 hω0riaSi.7. eleqtronebis gabnevisas akustikur fononebze (akustikuripolaronis modeli) miRebuli dabaltemperaturuli dreifuli Zvradobis( ω = 0) mniSvneloba 2-jer naklebia Zvradobis im mniSvnelobaze,romelic aseve miiReba bolcmanis kinetikuri gantolebis amoxsnisasrelaqsaciis drois miaxloebaSi.8. ganxilul modelebSi napovnia agreTve eleqtronis dreifulZvradobaze temperaturuli Sesworebebi, romlebic ganpirobebulia sawyisikorelaciebis evoluciuri wevrebis arsebobiT gamoyvanilikinetikuri gantolebebis dajaxebiT integralebSi da naCvenebia, romes Sesworebebi warmoadgenen mcire sidideebs ganxiluli TeoriisfarglebSi.9. polaronis fgm-sTvis miRebuli kvanturi kinetikuri gantolebebieleqtruli denis operatoris komponentebis (polaronisimpulsis) drois ormomentiani wonasworuli korelaciuri funqciebisTvisgamoyenebulia polaronis dreifuli Zvradobisa daeleqtrogamtarobis tenzoris gamosaTvlelad. Gganxilul erTzonianizotropul SemTxvevaSi, markoviseul miaxloebaSi polaronis dinamikisTvis,napovnia miaxloebiTi gamosaxulebebi korelaciuri funqciebisTvis.10. kristalis Zalian dabali temperaturebis SemTxvevaSi gamoyvaniliabolcmanis tipis kinetikuri gantoleba korelaciuri funqciisdiagonaluri matriculi elementisTvis, romelic Seesabamebapolaronis ZiriTad mdgomareobas. gamokvleulia polaronis aradrekadigabnevis procesebi fononebze. napovnia impulsis relaqsaciissixSiris (drois) analizuri gamosaxuleba da dadgenilia, rom mciresiCqariT moZravi polaronisTvis impulsis relaqsaciis sixSire (dro)ar aris damokidebuli impulsis sidideze.11. kubos wrfivi reaqciis Teoriis gamoyenebiT miRebulia dabalsixSirulieleqtrogamtarobis tenzoris analizuri gamosaxulebaeleqtron-fononuri sistemisaTvis erTzonian miaxloebaSi da fononebisdispersiis zogadi (izotropuli) kanonis SemTxvevaSi. gamoTvliliapolaronis dabaltemperaturuli dreifuli Zvradoba fgm-Si. ammodelSi napovnia agreTve temperaturuli Sesworeba polaronis dreifulZvradobaze, romelic ganpirobebulia sawyisi korelaciebisevoluciuri wevrebis arsebobiT miRebuli kinetikuri gantolebebisdajaxebiT integralebSi, da dasabuTebulia, rom es temperaturuliSesworeba warmoadgens mcire sidides.12. ganxilulia da gaanalizebulia polaronis dabaltemperaturulidreifuli Zvradobis yofaqceva susti ( α < 1) da Zlieri(α >>1) eleqtron-fononuri urTierTqmedebis zRvrul SemTxvevebSi.susti eleqtron-fononuri urTierTqmedebis SemTxvevaSi ( MGF→ 0) , rodesacpolaronis fgm gadadis polaronis frolixis modelSi, pola-27

onis dabaltemperaturuli dreifuli ZvradobisTvis (γ >>1; ω =0)vRebulobT iseTive miSvnelobas, rogoric napovnia polaronis frolixismodelSi. Zlieri eleqtron-fononuri urTierTqmedebis SemTxvevaSi( MGF→∞), rodesac polaronis fgm aRadgens polaronis pekarisnaxevradklasikur Teorias, dabaltemperaturuli dreifuli Zvradobis313yofaqceva moicema Semdegi TanafardobiT: µGF~ e.exp(γ ) α ; (0;2h = m = ω = 1γ = β >> 1; ω =0); anu polaronis dabaltemperaturuli Zvradoba Zlierieleqtron-fononuri urTierTqmedebis SemTxvevaSi ( α >> 1)izrdeba α -bmis mudmivas mecamete rigis proporciulad am mudmivas didi mniSvnelobebisdros, maSin rodesac polaronis pekaris TeoriaSi dabaltemperaturuliZvradoba izrdeba misi mexuTe rigis proporciulad:-µ ~ α5 Π; rodesac α >>1;( h = m = ω0= 1; β >> 1 ; ω = 0).13. sadisertacio naSromSi Catarebuli gamokvlevebi gviCvenebs,rom ganviTarebul meTodebs, romlebic dafuZnebulia kinetikuri gantolebebismiRebaze wonasworuli korelaciuri funqciebisTvis da maTgamoTvlaze, gansxvavebiT sxva midgomebisgan, ar mivyavarT ganSladiwevrebisagan Sedgenili usasrulo mwkrivebis ajamvis aucileblobas-Tan kvazinawilakis (eleqtronis, polaronis) urTierTqmedebis mixedviTfononebTan, kristalze modebuli gareSe eleqtruli velis dabali( ω → 0) sixSireebis SemTxvevaSi.naSromSi dasabuTebulia, rom arsebuli sawyisi korelaciebisevolucia da korelaciuri funqciebis oscilaciebi drois mixedviT,romlebic ganpirobebulia kvazinawilakis (zogad SemTxvevaSi kvanturidinamiuri qvesistemis) urTierTqmedebiT fononur (bozonur) velTandrois sawyis momentSi, gavlenas ar axdenen relaqsaciur procesebzeda isini warmoadgenen Zvradobebze temperaturuli Sesworebebis ZiriTadmizezs (wyaros) ganxilul modelebSi.SUMMARYIn modern conditions a subject to research is a subject of electron and polarontransport phenomena study in solid states and condensed matter physics. Making electron andpolaron mobility and electrical conductivity quantum theory and quasi-particle kineticfeatures calculation remains one of the actual problem in modern theory of electron andpolaron. In the latest years a tendency of making materials of difficult molecular buildingand studying polaron features gave stimuli to implement a lot of theoretical research fordescribing autolocalized (polaron) matter. Polaron concepts, which represents a simpleexample of nonlinear quasi-particle, has great importance and is highly used in solid statesand condensed matter physics and especially it is closely connected to the fundamentalproblems of quantum dynamical systems theory and to the subjects of quantum theory of afield. In the latest period it became actual to research subjects of electron-phonon system andpolaron kinetic on the base of Kubo linear response theory and to build correct quantumtheory of electron and polaron transfer phenomena and calculation of mechanical coefficients(mobility, electrical conductivity) in semiconductors and ionic crystals.28

The aim of thesis work is to receive and research new, exact generalized quantumkinetic equations for time correlation functions for some quantum dynamical systems of solidphysics, which interacts with phonon field (electron-phonon system, Frohlich and acousticalpolaron models, polaron generalized model of Feynman) and on the base of such modelsbuilding of consecutive, correct electron and polaron low- frequency conductivity and lowtemperaturedrift mobility quantum theory for non degenerated wide-band semiconductorsand ionic crystals based on Kubo linear response and perturbation theory and calculation ofmechanical coefficients transport (mobility, electrical conductivity) for above mentionedquantum subsystems models.The thesis work discusses two method of approach for new, exact generalizedquantum kinetic equations for double-time equilibrium correlation functions for quantumdynamical systems, which interacts with boson (phonon) field (thermostat).The first method of approach is based on ordered operators formalism andchronological and antichronological T-product method; and the second method of approachwhich is based on Liouville superoperative formalism and projection operator method.The first chapter of the work generally gives a short brief. In the first paragraph of thischapter deals with model Hamiltonian kind of dynamical systems, which interact with boson(phonon) thermostat and there are discussed some actual examples of quantum dissipative andopen nonequilibrium modeling systems and from different fields of modern physics, whichbecame the subject of intensive research and learn in the latest 30-40 years. The spectrum ofthe wide research contained subjects of metals’ electrical conductivity and superconductivitytheories, subjects of metal alloy and cold “metal glasses” electronic theory; subjects of weekand strong localization and strong inhomogeneous substances electrical conductivity theoryin disordered systems physics; aspects of laser radiation and superradiation theory aspects inquantum radiophysics; subjects of magnetic polaron and fluctuon (phason) and etc inmagnetic substances (environment) and others. The second paragraph deals withdynamically disordered system - electron-phonon system and electron interaction withacoustical and polar optical phonons and gives general introduction of Hamiltonian ofelectron-phonon system (Frohlich - Pekar type Hamiltonian), Hamiltonians of electronacoustical and polar optical phonons interaction and short brief of deformation potentialmethod. The third part of the work deals with large radius polaron models. There arereviewed polaron Frohlich and Pekary models and polaron Feynman oscillator andgeneralized Feynman models there. The same paragraph deals with a new approach forpolaron systems thermodynamics and kinetic subjects, developed in latest years – orderedoperators formalism, T-product method and phonon operators elimination method fromequilibrium and nonequilibrium average value – physical quantity characteristic for electronphononsystem. The advantage of a new method of approach in some occasions to at studyingkinetic subjects of electron-phonon system. The fourth paragraph deals with several principalsubjects of physical kinetics of dynamical systems, which interact with phonon (boson) field.There is given review of very important and principal subject such as opportunity of shortendescription at evolution (kinetic) equation for K-type dynamical systems (for classic andquantum as well), which is not based of a hypothesis about usage of initial correlationsweakness and random phase approximation (RPA). There is described those basic schedulesand methods, which lead us to Boltzmann type kinetic equation and master equation there.The chapter deals with such basic principal difficulties, which are arisen at calculation ofdrift mobilities for above mentioned models, as according to Boltzmann kinetic equation andKubo linear response theory and also at using some methods of linear and nonlinearconductivity theories (nonequilibrium density matrix and balance equation methods).29

The second chapter of thesis work deals with general question – to receive exact,generalized, quantum evolutionary equations for equilibrium correlation and Green functionsof dynamical subsystems, which interacts with boson (phonon) thermostat. The firstparagraph deals with ordered operators formalism and T-product method. The secondparagraph deals with new and exact quantum evolutional (kinetic) initial correlationsweakness principles equations for without random phase approximation usage and doubletimeequilibrium correlation and Green retarded functions, with eliminated boson amplitudes.The fourth paragraph deals with new and exact generalized quantum kinetic equations fordouble-time equilibrium correlation and Green functions been found by using Liouvillesuperoperative formalism and projection operator method. Unlike kinetic equations forcorrelation functions, received by different authors in scientific literature, integrals ofevolutional equations presented in this work contain additional members, which describeinitial correlations evolution in the period of time and which are caused by subsysteminteraction with boson thermostat in initial moment of time. The third and fifth paragraphsdiscuss Markov method of approach for subsystem dynamics and accordingly by the help ofboth formalism and methods there has been found approximately quantum kinetic equationsfor correlation functions with eliminated boson amplitudes and initial correlation descriptionand additional members in collission integrals. Researches and results have been conducted inthis chapter of the work give opportunity for better and wider studying of kinetic phenomena,which take place in dynamical systems and which interact with boson field.Subjects of electron and polaron transport phenomena quantum theories in solid states– in semiconductors and ionic crystals – have been researched in the third chapter of thesiswork. All four paragraphs of the same work are dedicated to electron and polaron lowfrequencyconductivity and low-temperature drift mobility quantum theory, which is based onthe above mentioned models of quantum dynamical systems, Kubo linear response andperturbation theory and on quantum kinetic equations for equilibrium correlation functionspresented in the second chapter. The first paragraph researches Markovian type kineticequations for correlation functions of electron “velocity – on velocity” in relaxation timeapproximation and there has been found decrements of damping correlation functions andoscillation factors; for electron-phonon systems in the case of weak electron-phononinteraction in one band approximation there has been received analyze image of lowfrequencyelectrical conductivity tensor and low-temperature drift mobility of electron in thecase of anisotropy has been calculated there – conductivity band is of electron velocity. Thesecond and third paragraphs deals with several subjects of electronic transport phenomena inFrohlich and acoustical polaron models. There have been found formulas for calculating driftmobility and low-frequency conductivity at low-temperatures for electron in such models.Generalization of Osaka result for electron low-temperature mobility and low-frequencyconductivity in electric field are received in polaron Frohlich model. There is given partial3 K Tdecision for “ Bproblem” in Frohlich polaron low-temperature mobility theory and is2 hω0shown, that meaning of mobility given in this work excels three times those meaning ofmobility which is received by Boltzmann kinetic equation in relaxation time approximation.In acoustical polaron model (at scattered electron on acoustical phonons) meaning of lowtemperaturemobility is two times less than the meaning of mobility, which is also received byBoltzmann equation in relaxation time approximation. The fourth paragraph of the samechapter deals researches several subjects of polaron kinetic in generalized Feynman modeland kinetic equations are solved for polaron “momentum-on momentum” equilibrium30

correlation functions; there is calculated frequency of polaron momentum relaxation(relaxation time) at low temperature crystal and analyze image of low-frequency electricalconductivity tensor (dissipative part) is received; there is found polaron low-temperature driftmobility meaning there. There is analyzed polaron low-temperature mobility behavior in theevent of electron phonon interaction there and there is found mobility dependence oncoupling constant in the case of strong electron-phonon interaction and different behavior oflow-temperature mobility according to the degree of coupling constant by polaron Pekarmodel is established. The work also calculates temperature corrections on electron andpolaron low-temperature mobilitys in the discussed models and there is shown, that thesecorrections represent small quantities within discussed theory and approaches has been usedthere.disertaciis ZiriTadi Sedegebi gamoqveynebuliaSemdeg SromebSi:1. ???? ?.?., ????? ?.?. ??????????? ???????? ? ?????????? ?????? ??????????? ?????? ????????????. 24-? ?????????? ????????? ?? ?????? ????????????????. ?????? ????????: ????? II: ??????????? ??????? ??? ??????????????????. ???????, 8-10 ????????, 1986 ?., ???. 201-202.2. ????? ?.?., ???? ?.?. ? ?????? ???????? ????????. ?????????? ???????????:???-???????? ???????? ?????????????? ??????. ???????, 14-17 ??? 1991 ?.??????? ?????????, ???. 1-140. ?????? ????????, ???. 83.3. Kotiya B.A., Los’ V.F. Exact equations for subsystem correlation functions anddensity matrix. The 18th IUPAP International Conference on Statistical Physics.Berlin, 2-8 August, 1992, Programme and Abstracts. Exact and Rigorous Results, p.133.4. Kotiya B.A., Los’ V.F. Exact equations for subsystem correlation functions anddensity matrix. Application in Polaron mobility. International Workshop “Polaronsand Applications”, May 23-31, 1992, Puschino, Russia. Abstracts, p. 18.5. Kotiya B.A., Los’ V.F. Exact equations for subsystem correlation functions anddensity matrix. Application in Polaron mobility. Proceedings in Nonlinear Science.Polarons and Applications. Ed. Lakhno V.D. (John Wiley and Sons, Chichester,1994), pp. 407-418 (impaqt. datvirTvis mqone).6. Kotiya B.A., Los’ V.F. To the theory of transport Phenomena in open systems. TheNorwegian Academy of Technological Sciences. The Lars Onsager Symposium.Coupled Transport Processes and Phase Transitions. June 2-4, 1993,Trondheim,Norwey. Abstracts, p. 89.7. Kotiya B.A., Los’ V.F. Exact Equations for Correlation Functions and Density Matrixof a Subsystem Interacting with a Heat Bath and their Application in the PolaronMobility. European physical Society. Eps9. Trends in Physics. Firenze. Italy.September 14-17, 1993. Abstracts. Symposium 29. Statistical Machanics: RigorousResults. ED. Systems, p. 130.31

8. kotia b. gadatanis movlenebis Teoria eleqtron-fononuri sistemisaTvis.saqarTvelos teqnikuri universitetis profesor-maswavlebelTasamecniero-teqnikuri konferencia. programa, 16-19 noemberi,Tbilisi, 1993 w. I-Teoriuli fizika, gv. 32.9. kotia b., losi v. polaronis Zvradobis TeoriisTvis. saqarTvelosmecnierebaTa akademiis moambe. t. 149, #1, 1994 w., Tbilisi, fizika,gv. 61-68.10. ????? ?.?., ???? ?.?. ? ?????? ?????????????????? ??????????? ????????? ??????? ????????????? ????????. ?????????? ???????? ????. ???????????????? ????. ????? ??????????, ?. 59, ? 8, 1995 ?.: ? ???????? ??????????????????? ????????? «?????????????????? ????????? ? ????????????? ???????????????? ????????». ??????. ?????? 1995 ?., ???.133-138 (impaqt.datvirTvis mqone).11. Kotiya B.A., Los’ V.F. Low-Temperature Electron Mobility of Acoustical Polaron. InPerspectives of Polarons. Editors: G.N. Chuev and V.D. Lakhno. Russian Academy ofSciences. World Scientific. Singapore. New Jersey. London. Hong Kong. 1996, p.216-222 (impaqt. datvirTvis mqone).12. Kotiya B.A., Los’ V.F. On the theory on Transport phenomena in the Polaron’sSystems. The 19th IUPAP International Conference on Statistical Physics. Xiamen31 July-4 August, 1995. Programme and Abstracts. Transport and Relaxation, p. 42.13. Kotiya B.A., Los’ V.F. Gigilashvili T.G. On the theory of transport phenomena in.electron-phonons’systems. The 20 th IUPAP International Conference on StatisticalPhysics. Paris, UNESCO, Sorbonne. July 20-25, 1998. Book of Abstracts,Nonequlibrium systems, p. 7.14. Kotiya B.A. On the theory of exact equations for correlation functions of the systemInteracting with a thermostat. Georgian Engineering News, 2005, ? 1, pp. 7-13.15. Kotiya B.A. On the theory of low-temperature electron mobility in the electronphononsystem and Frohlich’s model of the Polaron. Georgian Engineering News,2005, ? 2, pp. 11-21.16. ????? ?.?. ? ???????? ????????? ???????????? ??????, ????????????????? ????????? ?????. ????? I. ????????? ????????????? ?????????? ? ?????????????????? ???????? ????????. Georgian Engineering News, 2005, ? 4, ???. 7-1717. ????? ?.?. ? ???????? ????????? ???????????? ??????, ????????????????? ????????? ?????. ????? II. ???????????? ????????? ??? ????????????????????? ? ??????? ?????. Georgian Engineering News, 2006, ? 1, ???. 7-17.32

????? ?? ????? ?????????? ? ? ??? ???????? ????????????????? ? ?????? ? ??????? ???????????????????????? ? ? ? ???? ??????? ????????????? ?? ???? ???? ???? ?????? ?????????????????? ????????????? ????????? ???????????????? ???, 0175, ????????? ?2010 ??? ?© ???????? ? ????? ????? ?????, 2010 ?.

??????????? ????????? ????????????????????????? ? ? ??????? ?????????? ?????????????, ??????? ??? ?????????? ??? ???????? , ??? ????????? ??????????? ???? ?????? ???? ??? ????????? ??????? ??????? ???? :“?? ??? ??????? ?? ??? ???????? ????????? ???????? ??????????????? ????? ?? ????????????????? ?? ????? ??????? ????” ?????? ??? ????????????? ?????????? ????????? ?????????????„????????????? ????????? ??????????“ ??? ????????? ??????? ????????? ??? ???? ???? ???? ?????????????? ?????????? ?? .2010 ??? ???? ???????? ?:??????????:??????????:??????????:ii

??????????? ????????? ????????????2010 ??? ???????: ????? ???????????? ???: “?? ?????????? ?? ??? ???????? ????????????????? ??????? ???????? ????? ??????????????????? ??????? ??????? ????”?????????: ????????????? ????????? ????????????? ?????????????: ???? ????????? ???????: 2010 ?.?????????? ??? ???????????? ?? ???????????? ???????????????? ? ??????? ???? ??????????? ???????? ?????? ???????????????????? ???? ???????????? ? ???????? ?????????? ????????? ?????????? ????????? ? ???? ?????? ????????????? ????????????.??????? ??? ???????????? ?????????? ???????? ??????????? ? ???????? ?? ????? ? ???? ???????? ????? ???? ??? ??? ??????????????? ??? ????? ?? ?????? ????? ??? ???? ??????????? ????????? ?? ??????? ??????????????????? ??????.?????? ?????????, ??? ???????? ??????????? ? ????????????????? ????? ? ????? ??????????? ?? ?????????? ???????? (?????? ? ????? ????? ??????????, ???? ???? ????????? ?????? ??????????????????? ? ?????????? ??????????, ?????? ?? ??????? ???????????? ?????????? ?????? ??????) ?? ???? ? ??? ????? ?????????????????????.iii

???????Eeleqtronebisa da polaronebis Zvradobis kvanturi Teoriisageba da kvazinawilakebis kinetikuri maxasiaTeblebis gamoTvlakvlav rCeba erT-erT aqtualur amocanad myari sxeulebisfizikaSi. ukanasknel periodSi did interss iwvevs kubos wrfivigamoZaxilis meTodze dayrdnobiT eleqtronuli da polaronuligadatanis movlenebis koreqtuli kvanturi Teoriis ageba dameqanikuri koeficientebisH(Zvradoba, eleqtrogamtaroba) gamoTvlanaxevargamtarebsa da ionur kristalebSi.sadisertacio naSromis mizans warmoadgens-eleqtron-fononurisistemisTvis kvazinawilakis susti (frolixisa da akustikuripolaronis modelebi) da Zlieri (polaronis feinmanis ganzogadoebulimodeli (fgm)) urTirTqmedebis SemTxvevaSi fononurvelTan - axali,zusti, ganzogadoebuli kvanturi kinetikurigantolebebis miReba da gamokvleva droiTi wonasworuli korelaciurifunqciebisaTvis; am gantolebaTa gamoyenebiT, am modelTaTvisTanmimdevruli, koreqtuli eleqtronuli da polaronulidabaltemperaturuli dreifuli Zvradobis kvanturi Teoriisageba naxevargamtarebsa da ionur kristalebSi.sadisertacio naSromSi ganxilulia axali, zusti ganzogadoebulikvanturi kinetikuri gantolebebis miRebis ori midgomadrois ormomentiani wonasworuli korelaciuri funqciebisTvis:pirveli midgoma dafuZnebulia mowesrigebul operatorTa formalizmzeda qronologiur da antiqronologiur T-namravlTa me-Todze, da meore midgoma eyrdnoba liuvilis superoperatorulformalizmsa da proeqciuli operatoris meTods.naSromis pirveli Tavis pirvel paragrafSi moyvaniliamodeluri hamiltonianis saxe dinamiuri sistemebisa, romlebicurTierTqmedeben bozonur (fononur) TermostatTan da ganxiluliazogierTi aqtualuri magaliTi kvanturi disipaciuri da Riaarawonasworuli modeluri sistemebisa Tanamedrove fizikissxvadasxva dargidan, romlebic gaxdnen intensiuri kvlevisa daSeswavlis sagani ukanaskneli 30-40 wlis ganmavlobaSi. meoreparagrafSi ganxilulia dinamiurad mouwesrigebeli sistema -eleqtron-fononuri sistema da eleqtronis urTierTqmedeba akustikurda polarul optikur fononebTan; moyvanilia eleqtronfononurisistemis hamiltonianis zogadi saxe (frolix-pekaristipis hamiltoniani), eleqtronis akustikur da polarul optikurfononebTan urTierTqmedebis hamiltonianTa saxeebi damokled mimoxilulia agreTve deformaciis potencialis meTodi.didi radiusis mqone polaronTa modelebi – polaronis frolixisada pekaris, feinmanis erToscilatoriani da fg modelebi –mimoxilulia mesame paragrafSi.amave paragrafSi ganxilulia agreTve ukanasknel wlebSiganviTarebuli da gamoyenebuli axali midgoma polaronuli sistemebisTermodinamikisa da kinetikis sakiTxebis gamokvlevebisasiv

– mowesrigebul operatorTa formalizmi, T-namravlTa meTodi dafononuri operatorebis gamoricxvis teqnika eleqtron-fononurisistemis maxasiaTebeli fizikuri sidideebis wonasworuli daarawonasworuli saSualo mniSvnelobebidan. aRniSnulia zogierTSemTxvevaSi am axali midgomis upiratesoba eleqtron-fononurisistemis kinetikis sakiTxebis Seswavlisas, kontinualuriintegrirebis meTodTan SedarebiT. meoTxe paragrafSi ganxiluliafizikuri kinetikis zogierTi principuli sakiTxi dinamiurisistemebisa, romlebic urTierTqmedeben fononur (bozonur) vel-Tan. mimoxilulia mniSvnelovani da principuli sakiTxi Κ -tipisdinamiur sistemebSi (rogorc klasikuris, aseve kvanturisTvis),evoluciuri gantolebebis gamoyvanis dros Semoklebuli aRwerisSesaZleblobaze, romelic ar eyrdnoba sawyisi korelaciebisSesustebisa da SemTxveviTi fazebis miaxloebis (Sfm) hipoTezas.aRwerilia ZiriTadi sqemebi da meTodebi, romlebsac mivyavarTbolcmanis saxis kinetikuri gantolebisa da ZiriTadi kinetikurigantolebis povnamde. ganxilulia is ZiriTadi principuli xasiaTissirTuleebi, romlebic warmoiSvebian eleqtron-fononurisistemisa da polaronis zemoTmoyvanil modelTaTvis dreifuliZvradobebis gamoTvlisas, gamomdinare rogorc bolcmanis kinetikurigantolebidan da kubos wrfivi reaqciis Teoriidan, aseveeleqtrogamtarobis wrfivi da arawrfivi Teoriis zogierTi sxvameTodis (arawonasworuli simkvrivis matricis meTodi, balansisgantolebis meTodi) gamoyenebisas.sadisertacio naSromis meore Tavi originaluri xasiaTisaa.am TavSi dasmulia zogadi amocana –dinamiuri qvesistemisTvis,romelic urTierTqmedebs bozonur (fononur) TermostatTanaxali,zusti, ganzogadoebuli kvanturi evoluciuri gantolebebismiReba wonasworuli korelaciuri da grinis funqciebisTvis.pirvel paragrafSi ganxilulia mowesrigebul operatorTa formalizmida T-namravlTa meTodi. meore da meoTxe paragrafebSigamoyvanilia axali, zusti ganzogadoebuli kvanturi evoluciurigantolebebi drois ormomentiani wonasworuli korelaciurida grinis dagvianebuli funqciebisTvis,liuvilis superoperatoruliformalizmisa da proeqciuli operatoris meTodisgamoyenebiT,sawyisi korelaciebis Sesustebis principisa da Sfm-s gamoyenebisgareSe. GgansxvavebiT samecniero literaturaSi sxvadasxvaavtorTa mier miRebuli kinetikuri gantolebebisagan korelaciurifunqciebisTvis, naSromSi gamoyvanili evoluciuri gantolebebisdajaxebiTi integralebi Seicaven damatebiT wevrebs,romlebic aRweren sawyisi korelaciebis evolucias droSi, daromlebic ganpirobebulia qvesistemis urTierTqmedebiT bozonurTermostatTan drois sawyis momentSi. mesame da mexuTe paragrafebSiganxilulia markoviseuli miaxloeba qvesistemis dinamikis-Tvis da Sesabamisad orive formalizmisa da meTodis daxmarebiTnapovnia SeSfoTebis Teoriis meore miaxloebaSi kvanturi kinetikurigantolebebi korelaciuri funqciebisTvis gamoricxuli bozonuriamplitudebiT da sawyisi korelaciebis evoluciis aRmweridamatebiTi wevrebiT dajaxebiT integralebSi.v

eleqtronuli da polaronuli gadatanis movlenebis kvanturiTeoriis sakiTxebi – naxevargamtarebsa da ionur kristalebSi– ganxilulia da gamokvleulia sadisertacio naSromismesame TavSi. am Tavis oTxive paragrafi eZRvneba eleqtronulida polaronuli dabalsixSiruli gamtarobisa da dabaltemperaturulidreifuli Zvradobis kvantur Teorias, romelic dafuZnebuliakubos wrfivi reaqciis Teoriaze, kvantur dinamiuri sistemebiszemoTmiTiTebul modelebze, da wina TavSi miRebul kvanturkinetikur gantolebebze. pirvel paragrafSi gamokvleuliada amoxsnilia relaqsaciis drois miaxloebaSi markovis saxiskinetikuri gantolebebi eleqtronis “siCqare-siCqareze” korelaciurifunqciebisTvis; miRebulia dabalsixSiruli gamtarobistenzoris analizuri gamosaxuleba da gamoTvlilia eleqtronisdabaltemperaturuli dreifuli Zvradoba eleqtron-fononurisistemisTvis erTi zonis miaxloebaSi izotropul SemTxvevaSi –eleqtronis siCqarisa da fononebis dispersiis nebismieri zogadi,izotropuli kanonis SemTxvevaSi. meore da mesame paragrafeb-Si napovnia dabalsixSiruli eleqtrogamtarobisa da dabaltemperaturulidreifuli Zvradobis gamosaTvleli formulebifrolixisa da akustikuri polaronis modelebSi. polaronisfrolixis modelSi miRebulia osakas Sedegis ganzogadoebaeleqtronis dabaltemperaturuli dreifuli ZvradobisTvis3 K TdabalsixSirul gareSe eleqtrul velSi; moyvanilia “ B2 hω0problemis” nawilobrivi gadawyveta frolixis polaronis dabaltemperaturuliZvradobis TeoriaSi da naCvenebia, rom naSromSimiRebuli Zvradobis mniSvneloba 3-jer aRemateba Zvradobisim mniSvnelobas, romelic miiReba bolcmanis kinetikuri gantolebisamoxsniT relaqsaciis drois miaxloebaSi. akustikuri polaronismodelSi naSromSi napovni eleqtronis dabaltemperaturulidreifuli Zvradobis mniSvneloba 2-jer naklebiaZvradobis im mniSvnelobaze, romelic aseve miiReba bolcmanisgantolebis amoxsniT relaqsaciis drois miaxloebaSi. meoTxe paragrafSiGgamoyvanilia da amoxsnilia (relaqsaciis droismiaxloebaSi) kinetikuri gantolebebi polaronis “impulsiimpulsze”wonasworuli korelaciuri funqciebisTvis fgm-Si. mi-Rebulia dabalsixSiruli eleqtrogamtarobis tenzoris analizurigamosaxuleba da gamoTvlilia polaronis dabaltemperaturulidreifuli Zvradoba. napovnia Zvradobis damokidebulebaeleqtron-fononuri urTierTqmedebis (bmis) mudmivaze Zlierieleqtron-fononuri urTierTqmedebis SemTxvevaSi, da dadgeniliadabaltemperaturuli Zvradobis gansxvavebuli yofaqceva (bmismudmivas rigis mixedviT) polaronis pekaris modelTan SedarebiT.naSromSi gamoTvlilia aseve temperaturuli Sesworebebieleqtronisa da polaronis dabaltemperaturul dreiful Zvradobebzeganxilul modelebSi, da naCvenebia, rom es Sesworebebiwarmoadgenen mcire sidideebs ganxiluli Teoriisa da gamoyenebulimiaxloebebis farglebSi.vi

AbstractIn modern conditions a subject to research is a subject of electron and polarontransport phenomena study in solid states and condensed matter physics. Makingelectron and polaron mobility and electrical conductivity quantum theory and quasiparticlekinetic features calculation remains one of the actual problem in moderntheory of electron and polaron. In the latest years a tendency of making materials ofdifficult molecular building and studying polaron features gave stimuli to implement alot of theoretical research for describing autolocalized (polaron) matter. Polaronconcepts, which represents a simple example of nonlinear quasi-particle, has greatimportance and is highly used in solid states and condensed matter physics andespecially it is closely connected to the fundamental problems of quantum dynamicalsystems theory and to the subjects of quantum theory of a field. In the latest period itbecame actual to research subjects of electron-phonon system and polaron kinetic onthe base of Kubo linear response theory and to build correct quantum theory ofelectron and polaron transfer phenomena and calculation of mechanical coefficients(mobility, electrical conductivity) in semiconductors and ionic crystals.The aim of thesis work is to receive and research new, exact generalizedquantum kinetic equations for time correlation functions for some quantum dynamicalsystems of solid physics, which interacts with phonon field (electron-phonon system,Frohlich and acoustical polaron models, polaron generalized model of Feynman) andon the base of such models building of consecutive, correct electron and polaron lowfrequencyconductivity and low-temperature drift mobility quantum theory for nondegenerated wide-band semiconductors and ionic crystals based on Kubo linearresponse and perturbation theory and calculation of mechanical coefficients transport(mobility, electrical conductivity) for above mentioned quantum subsystems models.The thesis work discusses two method of approach for new, exact generalizedquantum kinetic equations for double-time equilibrium correlation functions forquantum dynamical systems, which interacts with boson (phonon) field (thermostat).The first method of approach is based on ordered operators formalism andchronological and antichronological T-product method; and the second method ofapproach which is based on Liouville superoperative formalism and projectionoperator method.The first chapter of the work generally gives a short brief. In the firstparagraph of this chapter deals with model Hamiltonian kind of dynamical systems,which interact with boson (phonon) thermostat and there are discussed some actualexamples of quantum dissipative and open nonequilibrium modeling systems andfrom different fields of modern physics, which became the subject of intensiveresearch and learn in the latest 30-40 years. The spectrum of the wide researchcontained subjects of metals’ electrical conductivity and superconductivity theories,subjects of metal alloy and cold “metal glasses” electronic theory; subjects of weekand strong localization and strong inhomogeneous substances electrical conductivitytheory in disordered systems physics; aspects of laser radiation and superradiationtheory aspects in quantum radiophysics; subjects of magnetic polaron and fluctuon(phason) and etc in magnetic substances (environment) and others. The secondparagraph deals with dynamically disordered system - electron-phonon system andelectron interaction with acoustical and polar optical phonons and gives generalintroduction of Hamiltonian of electron-phonon system (Frohlich - Pekar typeHamiltonian), Hamiltonians of electron acoustical and polar optical phononsinteraction and short brief of deformation potential method. The third part of the workvii

deals with large radius polaron models. There are reviewed polaron Frohlich andPekary models and polaron Feynman oscillator and generalized Feynman modelsthere. The same paragraph deals with a new approach for polaron systemsthermodynamics and kinetic subjects, developed in latest years – ordered operatorsformalism, T-product method and phonon operators elimination method fromequilibrium and nonequilibrium average value – physical quantity characteristic forelectron-phonon system. The advantage of a new method of approach in someoccasions to at studying kinetic subjects of electron-phonon system. The fourthparagraph deals with several principal subjects of physical kinetics of dynamicalsystems, which interact with phonon (boson) field. There is given review of veryimportant and principal subject such as opportunity of shorten description atevolution (kinetic) equation for K-type dynamical systems (for classic and quantumas well), which is not based of a hypothesis about usage of initial correlationsweakness and random phase approximation (RPA). There is described those basicschedules and methods, which lead us to Boltzmann type kinetic equation and masterequation there. The chapter deals with such basic principal difficulties, which arearisen at calculation of drift mobilities for above mentioned models, as according toBoltzmann kinetic equation and Kubo linear response theory and also at using somemethods of linear and nonlinear conductivity theories (nonequilibrium density matrixand balance equation methods).The second chapter of thesis work deals with general question – to receiveexact, generalized, quantum evolutionary equations for equilibrium correlation andGreen functions of dynamical subsystems, which interacts with boson (phonon)thermostat. The first paragraph deals with ordered operators formalism and T-productmethod. The second paragraph deals with new and exact quantum evolutional(kinetic) initial correlations weakness principles equations for without random phaseapproximation usage and double-time equilibrium correlation and Green retardedfunctions, with eliminated boson amplitudes. The fourth paragraph deals with new andexact generalized quantum kinetic equations for double-time equilibrium correlationand Green functions been found by using Liouville superoperative formalism andprojection operator method. Unlike kinetic equations for correlation functions,received by different authors in scientific literature, integrals of evolutional equationspresented in this work contain additional members, which describe initial correlationsevolution in the period of time and which are caused by subsystem interaction withboson thermostat in initial moment of time. The third and fifth paragraphs discussMarkov method of approach for subsystem dynamics and accordingly by the help ofboth formalism and methods there has been found approximately quantum kineticequations for correlation functions with eliminated boson amplitudes and initialcorrelation description and additional members in collission integrals. Researches andresults have been conducted in this chapter of the work give opportunity for better andwider studying of kinetic phenomena, which take place in dynamical systems andwhich interact with boson field.Subjects of electron and polaron transport phenomena quantum theories insolid states – in semiconductors and ionic crystals – have been researched in thethird chapter of thesis work. All four paragraphs of the same work are dedicated toelectron and polaron low-frequency conductivity and low-temperature drift mobilityquantum theory, which is based on the above mentioned models of quantumdynamical systems, Kubo linear response and perturbation theory and on quantumkinetic equations for equilibrium correlation functions presented in the secondchapter. The first paragraph researches Markovian type kinetic equations forviii

correlation functions of electron “velocity – on velocity” in relaxation timeapproximation and there has been found decrements of damping correlation functionsand oscillation factors; for electron-phonon systems in the case of weak electronphononinteraction in one band approximation there has been received analyze imageof low-frequency electrical conductivity tensor and low-temperature drift mobility ofelectron in the case of anisotropy has been calculated there – conductivity band is ofelectron velocity. The second and third paragraphs deals with several subjects ofelectronic transport phenomena in Frohlich and acoustical polaron models. There havebeen found formulas for calculating drift mobility and low-frequency conductivity atlow-temperatures for electron in such models. Generalization of Osaka result forelectron low-temperature mobility and low-frequency conductivity in electric field are3 K Treceived in polaron Frohlich model. There is given partial decision for “ B2 hω0problem” in Frohlich polaron low-temperature mobility theory and is shown, thatmeaning of mobility given in this work excels three times those meaning of mobilitywhich is received by Boltzmann kinetic equation in relaxation time approximation. Inacoustical polaron model (at scattered electron on acoustical phonons) meaning oflow-temperature mobility is two times less than the meaning of mobility, which is alsoreceived by Boltzmann equation in relaxation time approximation. The fourthparagraph of the same chapter deals researches several subjects of polaron kinetic ingeneralized Feynman model and kinetic equations are solved for polaron “momentumonmomentum” equilibrium correlation functions; there is calculated frequency ofpolaron momentum relaxation (relaxation time) at low temperature crystal and analyzeimage of low-frequency electrical conductivity tensor (dissipative part) is received;there is found polaron low-temperature drift mobility meaning there. There isanalyzed polaron low-temperature mobility behavior in the event of electron phononinteraction there and there is found mobility dependence on coupling constant in thecase of strong electron-phonon interaction and different behavior of low-temperaturemobility according to the degree of coupling constant by polaron Pekar model isestablished. The work also calculates temperature corrections on electron and polaronlow-temperature mobilitys in the discussed models and there is shown, that thesecorrections represent small quantities within discussed theory and approaches hasbeen used there.ix

SinaarsiSinaarsi.......................................................................................................................xnaxazebis nusxa .....................................................................................................xiiiSesavali.........................................................................................................................................................xivTavi I. dinamiuri sistemebi, romlebic urTierTqmedebenTermostatTan (bozonur velTan)........................................................................................231.1. kvanturi disipaciuri da Ria arawonasworuli modelurisistemebi (zogierTi magaliTi) ..................................................................... 251.1.1. fermionuli sistema .............................................................................. 251.1.2. Txevadi metalis an Senadnobis modeli.......................................... 251.1.3. nawilaki, romelic urTierTqmedebs garemosTan......................... 271.1.4. eleqtron-minarevuli sistema............................................................... 301.1.5. dikes modelebi gamosxivebis TeoriaSi........................................... 311.1.6. eleqtronebi magnetikSi, romlebic urTierTqmedebenkristalis magnitur qvesistemasTan ........................................................... 331.2. eleqtron-fononuri sistema. eleqtronis urTierTqmedebaakustikur da polarul optikur fononebTan....................................... 361.2.1. eleqtronis urTierTqmedeba kristaluri mesris akustikurrxevebTan. deformaciis potencialis meTodi........................................ 371.2.2. eleqtronis urTierTqmedeba polarul optikur fononebTan...................................................................................................................................... 391.3. polaronis amocana. didi radiusis mqone polaronismodelebi.................................................................................................................. 411.3.1. polaronis frolixisa da pekaris modelebi................................ 421.3.2. polaronis feinmanis modeli............................................................... 52x

1.3.3. polaronis feinmanis ganzogadoebuli modeli........................... 60(latinjer-lus modeli)................................................................................... 601.4. fizikuri kinetikis zogierTi sakiTxi dinamiuri sistemebisa,romlebic urTierTqmedeben fononur (bozonur) velTan................. 66Tavi II. ganzogadoebuli kvanturi evoluciuri gantolebebidrois ormomentiani wonasworuli korelaciuri funqciebisa dagrinis funqciebisTvis dinamiuri qvesistemisa, romelicurTierTqmedebs TermostatTan (bozonur velTan).........................................822.1. mowesrigebul operatorTa formalizmi da T-namravlTameTodi....................................................................................................................... 822.2. zusti ganzogadoebuli kvanturi kinetikuri gantolebebiwonasworuli korelaciuri funqciebisa da grinisfunqciebisTvis...................................................................................................... 912.3. markoviseuli miaxloeba qvesistemis dinamikisTvismiaxloebiTi gantolebebi korelaciuri funqciebisTvis................ 952.4. ganzogadoebuli kvanturi evoluciuri gantolebebikorelaciuri funqciebisTvis Sfm-is gamoyenebis gareSe.proeqciuli operatoris meTodi.................................................................. 992.5. Termostatis bozonuri amplitudebis gamoricxva evoluciurigantolebidan korelaciuri funqciisaTvis. markoviseulimiaxloeba qvesistemis dinamikisTvis........................................................ 103Tavi III. naxevargamtarebsa da ionuri kristalebSi eleqtronulida polaronuli gamtarobisa da dabaltemperaturuli Zvradobiskvanturi Teoria dafuZnebuli kubos wrfivi gamoZaxilisTeoriaze........................................................................................................................................................1083.1.eleqtron-fononuri sistema. eleqtronis dabalsixSirulieleqtrogamtarobisa da dabaltemperaturuli Zvradobis gamoTvlasusti eleqtron-fononuri urTierTqmedebis SemTxvevaSi114xi

3.2. eleqtronis Zvradoba frolixis polaronis modelSi............ 1173.3. eleqtronis Zvradoba akustikuri polaronis modelSisusti eleqtron –fononuri urTierTqmedebis SemTxvevaSi........ 1273.4 polaronis dabaltemperaturuli Zvradoba feinmanisganzogadoebul modelSi............................................................................... 134daskvna..............................................................................................................................................................143gamoyenebuli literatura............................................................................. 146danarTibozonuri (fononuri) amplitudebis gamoricxva qronologiurda antiqronologiur T-namravlTa daxmarebiT................................... 153disertaciis TemasTan dakavSirebiT gamoqveynebulia SemdeginaSromebi............................................................................................................... 156xii

naxazebis nusxanax.1. potencialuri ormo, energetikuli doneebida eleqtronis Ψ -talRuri funqcia pekarispolaronis modelSi---------------------------------------------------------------51nax.2. polaronis feinmanis modeli. eleqtroni,romelic dakavSirebulia “zambaris” meSveobiTmeore nawilakTan masiT M F ---------------------------------------------------56nax.3. V da W – variaciuli parametrebis yofaqceva,rogorc α-urTierTqmedebis (bmis) mudmivas funqciebi,polaronis feinmanis TeoriaSi-------------------------------------------- 58xiii

Sesavaliukanaskneli 50-60 wlis ganmavlobaSi didi yuradReba eTmobodadenis (muxtis) gadamtanebis dinamiuri da kinetikuri TvisebebisSeswavlas myar sxeulebSi. zogadad denis aseT gadamtanebswarmoadgens sxvadasxva saxis polaronebi; TiToeuli maTgani moicavsgamtarobis eleqtrons (an xvrels) da maT mier gamowveulmesris struqturis deformacias. aseTi TvalsazrisiT gamtarobiseleqtroni (an xvreli) faqtiurad warmoadgens polaronszRvruli, susti deformaciis SemTxvevaSi. eleqtronebis da polaronebisdinamiuri da kinetikuri yofaqcevis Seswavla stimulirebuliiyo mravali eqsperimentuli masaliT, romelic exebodagadatanis movlenebis gamokvlevas myar sxeulebSi, romelTainterpretacia mraval SemTxvevaSi standartuli zonuri TeoriisCarCoebSi Zalze rTuli iyo, gansakuTrebiT ionur kristalebSida polarul naxevargamtarebSi. garda amisa, gadatanis movlenebiszogadi debulebebi kristalebSi arsebiTad iqna ganviTarebuliim fundamentaluri gamokvlevebis safuZvelze, romelicasaxavs disipaciuri sistemebis dinamiur da kinetikur Tvisebebsda romlebSic mimdinareobs Seuqcevadi gadatanis movlenebi. eleqtronuligadatanis movlenebis Teoriis ganviTarebis aseTmapirobebma gamoiwvia mniSvnelovani zegavlena iseTi kvlevis areebisCamoyalibebaSi, rogoric aris – polaronuli gadatanis movlenebi,eleqtronuli gadatanis movlenebi Zlier (kvantur) magniturvelSi, eleqtronuli gadatanis movlenebis Teoria magniturkristalebSi da mouwesrigebel garemoebSi (sistemebSi) da sxva.rogorc cnobilia, kristalebSi eleqtronuli gadatanismovlenebis zonuri Teoria principSi dafuZnebulia sam ZiriTadkoncefciaze: 1) denis gadamtanebi warmoadgens kvazinawilakebsgansazRvruli kvaziimpulsiT da dispersiis kanoniT. 2) denisgadamtanTa eleqtrogamtaroba da Zvradoba ganisazRvreba maTigabneviT kristalis idealuri mesris struqturis dinamiur dastatikur damaxinjebebze (defeqtebze). 3) denis gadamtanis Tavi-xiv

sufali ganarbenis sigrZe warmoadgens sasrul sidides da isbevrad aRemateba Sesabamisad kvazinawilakis de-broilis talRissigrZes. am pirobebis gaTvaliswinebiT denis gadamtanTa gabnevaSeiZleba CaiTvalos rogorc “iSviaTi”. am debulebebidangamomdinare, denis gadamtanTa yofaqceva kristalSi aRiwerebaalbaTuri ganawilebis funqciiT kvaziimpulsebis mixedviT,romelic ganisazRvreba, rogorc bolcman-bloxis kinetikurigantolebis amoxsna. ukanaskneli SeiZleba ganxiluli iqnas,rogorc balansis (uwyvetobis) gantoleba, romelSic gaTvaliswinebuliadenis gadamtanTa ganawilebis funqciis cvlilebakristalSi, gamowveuli rogorc gareSe modebuli ZalebiszegavleniT, agreTve denis gadamtanTa kvaziimpulsis gadanawilebiT(cvlilebiT) gabnevis procesSi. aseTi gagebiT (Tu mxedvelobaSiar miviRebT dispersiis kanonis kvantur-meqanikur bunebasda denis gadamtanebis gabnevis albaTurobas), gadatanis movlenebisTeoria, romelsac xSirad uwodeben bolcmaniseuls, arsebi-Tad warmoadgens klasikurs da igi ar moicavs specifiur kvanturefeqtebs. amrigad, klasikur areebSi gadatanis kinetikuri(meqanikuri) koeficientebis (Zvradoba, eleqtrogamtaroba) gamosa-Tvlelad gamoiyeneba denis gadamtanebis arawonasworuli ganawilebisfunqciisaTvis kinetikuri gantoleba – bolcmanis gantoleba– romelic iTvaliswinebs denis (muxtis) matarebelTaurTierTqmedebas (gabnevas) kristaluri mesris rxevebze.ukanasknel wlebSi, eleqtronuli gadatanis movlenebis gamokvlevebSimyari sxeulebis da naxevargamtarebis fizikaSi, ZalianfarTo gamoyeneba hpova ufro zogadma, alternatiulma midgomam,romelic dafuZnebulia kubos wrfivi reaqciis Teoriaze.am TeoriaSi - romelsac gadamwyveti mniSvneloba aqvs wrfiv arawonasworulTermodinamikaSi – gadatanis kinetikuri (meqanikuri)koeficientebi bunebrivad gamoisaxebian (aRiwerebian) droiTi korelaciurifunqciebiT. isini asaxaven sistemis reaqcias hamiltonianisSeSfoTebisas, romlis tipiur magaliTs warmoadgens eleqtrogamtaroba.kubo-grinis formulebi gadatanis koeficientebi-xv

saTvis warmoadgenen kerZo SemTxvevas TanafardobaTa farToklasisa, romelic cnobilia fluqtuaciur-disipaciuri TeoremebissaxelwodebiT. am Teoremebis amsaxveli maTematikuri formulebiamyareben kavSirs mikroskopul da makroskopul damzeradsidideebs Soris. wrfivi gadatanis movlenebis ganxilvisas kubosformula “deni-denze” droiTi korelaciuri funqciisaTvis warmoadgensarsebiTad zust gamosaxulebas kvantur mravalnawilakovanformalizmSi, im gansxvavebiT bolcmanis kinetikuri gantolebisagan,rom araviTari daSveba ar keTdeba gabnevis kveTis sididisSesaxeb denis gadamtanebis (eleqtronebis) gabnevisas kristalurmesris rxevebze. unda aRiniSnos, rom tradiciulad eqvivalentobaskubos formulasa da bolcmanis gantolebas Sorismkacrad ikvleven mxolod denis gadamtanebis susti gabneviszRvrul SemTxvevaSi kristaluri mesris rxevebze. rogorc cnobilia,kubos formula amyarebs kavSirs eleqtrogamtarobasa dadroiT wonasworul korelaciur funqcia “deni-dens” Soris, romelicaris ornawilakovani (oTxwertilovani) grinis funqciismsgavsi, maSin, rodesac bolcmanis gantoleba warmoadgens – kinetikurgantolebas arawonasworuli ganawilebis funqciisaTvis,da romelic aris erTnawilakovani (orwertilovani) grinis funqciissaxis.Teoriuli gamokvlevebis Zalian didi raodenoba samecnieroliteraturaSi miZRvnili aris eleqtronebisa da polaronebiseleqtrogamtarobisa da Zvradobis gamoTvlaze naxevargamtarebsada ionur kristalebSi. es gamokvlevebi dafuZnebuli arissxvadasxva Teoriul meTodebze – grinis funqciis teqnikaze,bolcmanis kinetikuri gantolebis Seswavlaze, TviTSeTanxmebulmeTodebze da sxv. rezultatebi miRebuli sxvadasxva meTodebisgamoyenebiT da Sesabamisad sxvadasxva miaxloebebze dayrdnobiT,Zireulad gansxvavdeba erTmaneTisagan. miuxedavad imisa, romeleqtronebis da polaronebis eleqtrogamtarobisa da ZvradobisgamoTvla warmoadgens erT-erT uZveles problemas myari sxeule-xvi

is fizikaSi, is mainc rCeba erT-erT urTules da Znel amocanadTeoriulad amoxsnis TvalsazrisiT.amgvarad, Tanamedrove pirobebSi kvlav aqtualurs warmoadgenssakiTxi eleqtronebis da polaronebis eleqtrogamtarobisda Zvradobis koreqtuli gamoTvlisa naxevargamtarebsa da ionurkristalebSi.rogorc cnobilia, efeqtur meTods urTierTqmed nawilakTasistemis kinetikuri maxasiaTeblebis gamoTvlisa warmoadgens wonasworulkorelaciur funqciaTa da grinis funqciebis meTodi.kubos wrfivi gamoZaxilis Teoriis upiratesoba arawonasworulistatistikuri meqanikis sxva midgomebTan SedarebiT mdgomareobsimaSi, rom es Teoria saSualebas iZleva uSualod gamoyenebuliiqnas wonasworuli mravalnawilakovani meTodebi gadatanis kinetikuri(meqanikuri) koeficientebis gamosaTvlelad myar sxeuleb-Si. am midgomis dros ZiriTad problemas warmoadgens wonasworulikorelaciuri funqciebis gamoTvla, romlebisTvisac iwerebakvanturi evoluciuri (kinetikuri) gantolebebi.myari sxeulebis fizikisa da arawonasworuli statistikurimeqanikis mravali amocanis ganxilvisas Seiswavleba mcire dinamiuriqvesistemis evolucia droSi, romelic imyofeba kontaqt-Si didi Tavisuflebis ricxvis mqone, Termodinamikur wonasworobaSimyof sistemasTan – TermostatTan.eleqtronuli da polaronuli gadatanis movlenebis gamokvlevisasmyar sxeulebSi kubos wrfivi gamoZaxilis TeoriazedayrdnobiT, ZiriTad amocanas warmoadgens zusti ganzogadoebuli.kvanturi evoluciuri (kinetikuri) gantolebebis miReba droisormomentiani wonasworuli korelaciuri funqciebisTvis kvazinawilakebisaRmweri Sesabamisi dinamiuri sidideebisaTvis, rodesacxdeba am ukanasknelTa urTierTqmedeba (gabneva) kristalurimesris rxevebze (fononebze), da rodesac fononuri (bozonuri)veli ganixileba rogorc Termostati. samecniero literaturaSi,aseTi saxis gantolebebis misaRebad, rogorc wesi gamoiyenebaaprioruli hipoTeza – sawyisi korelaciebis Sesustebisxvii

principi, an msgavsi debulebebi – mag. SemTxveviTi fazebis miaxloeba(Sfm) – rodesac drois sawyisi momentisaTvis mTelisistemis (qvesistema plus Termostati) statistikuri operatorimoicema faqtorizebuli saxiT (mTeli sistemis statistikurioperatori ganixileba rogorc qvesistemisa da Termostatis statistikuroperatorTa pirdapiri namravli). naTelia, rom aseTidaSvebebis Sedegad miRebuli ganzogadoebuli, kvanturi evoluciurigantolebebi wonasworuli korelaciuri funqciebisaTvisar aris zusti. amitom cxadia, rom zogadad kvanturi disipaciurisistemebis ganxilvisas da kinetikuri movlenebis Seswavlisasmcire dinamiur sistemebSi, romlebic urTierTqmedeben bozonur(fononur) TermostatTan, kubos wrfivi gamoZaxilis TeoriisfarglebSi mkacri midgomis gansaviTareblad saWiroa zusti, ganzogadoebuli,kvanturi evoluciuri gantolebebis gamoyvana qvesistemisdinamiuri sidideebis wonasworuli korelaciuri funqciebisTvis.amrigad, eleqtronuli da polaronuli gadatanis movlenebiskoreqtuli Teoriis asagebad da meqanikuri koeficientebis(mag. Zvradoba, eleqtrogamtaroba) gamosaTvlelad naxevargamtarebsada ionur kristalebSi da agreTve eleqtron-fononuri sistemiskinetikis sakiTxebis gamosakvlevad kubos wrfivi reaqciisTeoriis CarCoebSi, aqtualurs warmoadgens amocana kvazinawilakebisdinamiuri sidideebisTvis drois ormomentiani wonasworulikorelaciuri funqciebisTvis zusti, ganzogadoebuli, kvanturievoluciuri gantolebebis miReba – sawyisi korelaciebis Sesustebisprincipisa da Sfm-is gamoyenebis gareSe.sadisertacio naSromis mizans warmoadgens:− myari sxeulebis fizikis zogierTi kvanturi dinamiuri sistemisTvis,romelic urTierTqmedebs fononur (bozonur) velTan(eleqtron-fononuri sistema, frolixis polaronis modeli,akustikuri polaronis modeli susti eleqtron-fononuri urTierTqmedebisSemTxvevaSi, polaronis fgm), mowesrigebul opera-xviii

torTa formalizmsa da T-namravlTa teqnikaze dayrdnobiT, agreTveliuvilis superoperatoruli formalizmisa da proeqciulioperatoris meTodis gamoyenebiT zusti, ganzogadoebulikvanturi evoluciuri (kinetikuri) gantolebebis miReba da gamokvlevadrois ormomentiani wonasworuli korelaciuri funqciebisTvisSfm-is gamoyenebis gareSe.− Aam modelebze dayrdnobiT Tanmimdevruli, srulyofilieleqtronuli da polaronuli gamtarobisa da dabaltemperaturulidreifuli Zvradobiskvanturi Teoriis ageba naxevargamtarebsada ionur kristalebSi dafuZnebuli kubos wrfivi gamo-Zaxilisa da SeSfoTebis Teoriaze. gadatanis meqanikuri (kinetikuri)koeficientebis (eleqtrogamtaroba, Zvradoba) gamoTvlakvanturi disipaciuri sistemebis zemoTmiTiTebuli modelebisTvis.ZiriTadi Sedegebi da mecnieruli siaxlewarmodgenil sadisertacio naSromSi gadawyvetilia Semdegiamocanebi:− mowesrigebul operatorTa formalizmisa da T-namravlTateqnikis daxmarebiT, sawyisi korelaciebis Sesustebis principisada Sfm-is gamoyenebis gareSe, gamoyvanilia da gamokvleulia axali,zusti, ganzogadoebuli kvanturi evoluciuri (kinetikuri)gantolebebi gamoricxuli bozonuri (fononuri) amplitudebiTdrois ormomentiani wonasworuli korelaciuri da grinis funqciebisTvisdinamiuri qvesistemisTvis, romelic urTierTqmedebsbozonur TermostatTan. aseTive saxis kvanturi kinetikuri gantolebebikorelaciuri funqciebisaTvis miRebulia liuvilissuperoperatoruli formalizmisa da proeqciuli operatorismeTodis daxmarebiT.− dinamiuri qvesistemis bozonur (fononur) TermostatTanurTierTqmedebis hamiltonianis mixedviT, SeSfoTebis Teoriismeore miaxloebaSi miRebulia axali, ganzogadoebuli kvanturikinetikuri gantolebebi gamoricxuli bozonuri amplitudebiT,xix

qvesistemis drois ormomentiani wonasworuli korelaciuri funqciebisTvis– rogorc markoviseuli, ise aramarkoviseuli formiT– romelTa dajaxebiTi integralebi Seicaven cxadad gamoyofilsawyisi korelaciebis evoluciur wevrebs.− kubos wrfivi reaqciisa da SeSfoTebis Teoriis farglebSi,naxevargamtarebisa da ionuri kristalebisTvis agebulia eleqtronulida polaronuli dabalsixSiruli gamtarobisa da dabaltemperaturulidreifuli Zvradobis Tanmimdevruli, koreqtulikvanturi Teoria, dafuZnebuli kvanturi disipaciuri sistemebiszemoTmiTiTebuli modelebisTvis ganzogadoebul kvanturkinetikur gantolebebze korelaciuri funqciebisaTvis “deni-denze”eleqtronisa da polaronisaTvis, romlebic urTierTqmedebenfononebTan Sfm-is gamoyenebis gareSe.− miRebulia da gamokvleulia analizuri gamosaxulebebieleqtronisa da polaronis relaqsaciuri maxasiaTeblebisTvis(impulsis relaqsaciis sixSire da sxv.); gamoTvlilia wonasworulikorelaciuri funqciebis – “deni-denze” – milevis dekrementebi.napovnia eleqtrogamtarobis tenzoris analizuri saxeeleqtron-fononuri sistemisTvis kristalis dabali temperaturebisada dabalsixSiruli gareSe eleqtruli velebis Sem-TxvevaSi da gamoTvlilia gadatanis meqanikuri koeficientebi(Zvradoba, eleqtrogamtaroba) kvanturi disipaciuri sistemebis3aRniSnuli modelebisTvis; napovnia agreTve “2K BΤhω -problemis”nawilobrivi gadawyveta frolixis polaronis dabaltemperaturuliZvradobis TeoriaSi.− miRebulia temperaturuli Sesworebebi eleqtronisa da polaronisdreiful Zvradobebze, romlebic ganpirobebulia sawyisikorelaciebis evoluciuri wevrebis arsebobiT kvantur kinetikurgantolebebSi wonasworuli korelaciuri funqciebisTvis“siCqare-siCqareze” (“impulsi-impulsze”) eleqtronisa da polaronisTvis.dadgenilia, rom es Sesworebebi warmoadgenen mcire0xx

sidideebs Sesrulebuli miaxloebebisa da ganxiluli TeoriisfarglebSi.sadisertacio naSromis praqtikuli mniSvneloba:naSromSi dasmuli amocanebis gadawyvetam moiTxova arawonasworulistatistikuri meqanikis zogierTi meTodis SemdgomiganviTareba. miRebuli zusti, ganzogadoebuli kvanturievoluciuri gantolebebi wonasworuli korelaciuri funqciebisTvisSesaZlebelia gamoyenebuli iqnas gadatanis movlenebisgansaxilvelad da gamosakvlevad - kubos wrfivi reaqciis TeoriisfarglebSi, sawyisi korelaciebis Sesustebis principisa daSfm-is daSvebebis gareSe - myari sxeulebis da kondensirebulgaremoTa fizikis dinamiur qvesistemaTa sxva modelTaTvis(kvanturi disipaciuri sistemebisTvis), romlebic urTierTqmedebenbozonur velTan (TermostatTan). (mag. brounis kvanturi nawilakismoZraobis Sesaswavlad, romelic ganixileba rogorcwrfivi, milevadi harmoniuli oscilatori, da romlis dinamikaaRiwereba kaldeira-legetis mikroskopuli modeluri hamiltonianiT).naSromSi ganviTarebuli formalizmi, meTodebi da miRebulikinetikuri gantolebebi martivad SesaZlebelia ganvrcobiliiqnas kinetikuri movlenebis Sesaswavlad da gadatanis meqanikurikoeficientebis (mag. Zvradoba, eleqtrogamtaroba) gamosa-Tvlelad: eleqtronebis urTierTqmedebisas (gabnevisas) arapolaruloptikur fononebze, piezoeleqtrul fononebze, agreTvesxva didi radiusis mqone polaronis modelTaTvis (mag. akustikuripolaronis modelisTvis - eleqtronis fononebTan ZlieriurTierTqmedebis SemTxvevaSi). gamoyvanili ganzogadoebuli kvanturikinetikuri gantolebebi korelaciuri funqciebisTvis kvanturidinamiuri qvesistemisTvis, romelic urTierTqmedebs fononurvelTan, SesaZlebelia gamoyenebul iqnas normaluri (arazegamtari)metalebis eleqtrowinaRobis gamosaTvlelad eleqtronebisgabnevisas akustikur fononebze. naSromSi warmodgeniliformalizmis daxmarebiT SesaZlebelia temperaturuli Seswore-xxi

ebis povna metalTa eleqtrowinaRobisTvis (Sesworebebi bloxgrunaizenisformulaSi), romlebic agreTve ganpirobebuliaeleqtronebis fononebTan urTierTqmedebisas sawyisi korelaciebisgaTvaliswinebiT.sadisertacio naSromSi miRebuli ZiriTadi Teoriuli Sedegebispraqtikuli mniSvneloba (Rirebuleba) ganisazRvrebaagreTve imiT, rom ukanasknel wlebSi samecniero literaturaSigamoqveynda mravali Teoriuli naSromi, romelic Seexeba bozonurvelTan (TermostatTan) urTierTqmedebaSi myofi kvanturidinamiuri sistemis kinetikis sakiTxebs, sadac ganixileba msgavsiformalizmi da meTodebi, romelic ganviTarebulia avtoris mierwarmodgenil naSromSi.xxii

Tavi I. dinamiuri sistemebi, romlebic urTierTqmedebenTermostatTan (bozonur velTan)ukanaskneli aTwleulebis ganmavlobaSi arsebiTad gaizarda interesiim dinamiuri sistemebis SeswavlaSi, romlebic urTierTqmedebaSiimyofebian TermostatTan (bozonur velTan) [1]. amasTan erTad didi yuradRebaeTmoboda im procesebis detalur Seswavlas, romlebic mimdinareobdnenim mcire sistemebSi, romlebic urTierTqmedebdnen rogorcsustad, agreTve Zlierad Termodinamikur wonasworobaSi myof didiTavisuflebis xarisxis mqone sistemasTan – TermostatTan. ZiriTadi me-Todi, romlis saSualebiTac xdeboda am procesebis gamokvleva eyrdnobodamidgomebs, romlebic ganviTarebuli iyo arawonasworul statistikurmeqanikaSi da stohastikuri (SemTxveviTi) procesebis TeoriaSi [1-2].ukanasknel periodSi, mkacri midgomebis ganviTarebam arawonasworulstatistikur meqanikaSi SesaZlebeli gaxada Camoyalibebuliyo sakmaodsruli da mkacri maTematikuri Teoria arawonasworuli modelurisistemebisa [3]. Ria arawonasworuli modeluri sistemebis (Ria sistemawarmoadgens mcire qvesistemas, romelic urTierTqmedebs TermostatTan)srulyofili maTematikuri Teoriis Seqmna warmoadgens SedarebiT ufromartiv da perspeqtiul problemas (amocanas) arawonasworuli statistikurimeqanikisa [3-4].arawonasworuli statistikuri meqanikis da fizikuri kinetikis problemaTafarTo wre dakavSirebulia im modeluri hamiltonianebis SeswavlasTan,rodesac dinamiuri qvesistema s urTierTqmedebs S TermostatTan,romelic aRiwereba bozonuri veliT; Tanac, rogorc wesi, ur-TierTqmedeba aiReba wrfivi Termostatis (boze) operatorebis (amplitudebis)mixedviT. zogad SemTxvevaSi, mTliani (s+S) sistemis hamiltonianiSesaZlebelia warmovadginoT Semdegi saxiT:Hts (, , Σ ) =Γ (, ts) + H + H () t(1.1)sadac: Γ (,) ts aris s qvesistemis sakuTari hamiltoniani;H Σ- Termostatis (bozonuri velis) hamiltoniania, xoloΣHintwarmoadgens urTierTqmedebis hamiltonians s qvesistemisa STermostatTan.int23

saxe:s qvesistemis urTierTqmedebisas bozonur velTanH int-s aqvs Semdegi+ +Hint () t = ∑ ⎡⎣Ck(, tsb )k+ Ck(,)tsb ⎤k ⎦. (1.2)koperatorebi, romlebic figurireben (1.1) da (1.2) gamosaxulebebSiarian Cveulebriv Sredingeriseul warmodgenaSi Cawerili dinamiuricvladebi. b kda b +kwarmoadgenen Termostatis boze amplitudebs – gaqrobisada dabadebis operatorebs Cawerils meoradi dakvantvis warmodgenaSi.bozonur velTan urTierTqmedebis SemTxvevaSi: H = +Σ ∑ h ω( kbb ) warmoadgensk kbozonuri velis sakuTar hamiltonians. k – aris im kvanturi ricxvebis er-Toblioba, romlebic aRweren s qvesistemis urTierTqmedebas STermostatTan da TviT Termostatis mdgomareobas. unda aRiniSnos, rom sdinamiuri qvesistemis operatorebis cxadi saxe arsad ar konkretdeba.gamokvlevaTa Zalian didi raodenoba miZRvnilia kinetikur movlenebisadmida procesebisadmi, romlebic mimdinareoben kvantur dinamiursistemebSi, romlebic urTierTqmedeben TermostatTan (bozonur velTan) daaRiwerebian (1.1-1.2) hamiltonianiT [1-21].ganvixiloT axla Tanamedrove fizikis TvalsazrisiT zogierTi aqtualurida mniSvnelovani magaliTebi, romlebic farTo Seswavlis saganigaxda statistikuri fizikis, myari sxeulebisa da kondensirebul garemoTafizikis, gamosxivebis Teoriis da sxv. - kvanturi dinamiuri sistemebisa,romlebic urTierTqmedeben TermostatTan.k24

1.1. kvanturi disipaciuri da Ria arawonasworuli modeluri sistemebi(zogierTi magaliTi)1.1.1. fermionuli sistemaqvesistema s warmoadgens araurTierTqmed fermionTa sistemas,romlebic xasiaTdebian fermi amplitudebiT da adgili aqvs SemdegTanafardobebs:eΓ (,) ts = Λ ( f) a a ; C (, ts)= L a aεt+ +∑ f f k k∑f + k f( f ) V ( f )eeC ts L aa L a aVVεtεt+ * + * +k(, ) =k∑f f + k=k∑f −k f,( f ) ( f )(1.3)sadac L k, L * kwarmoadgenen “C-sidideebs”. vinaidan fermionebs gaaCniaT spini,(1.3) gamosaxulebebSi f ( f,σ )≡ r da f r miekuTvneba kvazidiskretul speqtrs(σ - warmoadgens spinis aRmniSvnel maCvenebels). simbolo ( f k)rogorc ( f + k) ≡ ( f + k,σ )rr+ waikiTxeba; SesaZlebelia agreTve ganxiluli iqnas SemTxveva,rodesac fermionebi urTierTqmedeben erTmaneTTan. maSin Γ (,) ts hamiltonianSigaTvaliswinebuli unda iqnas agreTve damatebiTi wevrebi,romlebic aRweren fermionebs Soris urTierTqmedebas da maT urTierTqmedebasgareSe velebTan. unda aRiniSnos, rom aseTi saxis dinamiurisistemebis ganxilvamde mivyavarT amocanebs, romlebic moiTxoven metalTaeleqtrogamtarobis gansazRvras, zegamtarobis movlenis axsnas da sxv. [14-15].1.1.2. Txevadi metalis an Senadnobis modelimravalkomponentiani Txevadi metalis an Senadnobis hamiltonianiSesaZlebelia CavweroT Semdegi saxiT [9]:MH ts , H : H C tsδρ ,( ) ( )=Γ + =∑∑ r rintintl=1klklkNl+ −1−ikRlj( , ) =,( , ) : δρ = δlk l −k lk l ∑rrr r r − rko ,j=1C ts C ts N eδρ= δρ*rrlk l,− k.(1.4)aq s warmoadgens eleqtronuli sistemis maCvenebels, Γ (,) ts aris eleqtronulisistemis sakuTari hamiltoniani, romelic zogad SemTxvevaSi Seicavs25

eleqtron-eleqtronuli urTierTqmedebis wevrebsac; Hintwarmoadgenseleqtronul da ionur sistemebs Soris urTierTqmedebis hamiltonians, laris ionuri sistemis maCvenebeli, 1≤l ≤M ; M M- komponentebis ricxvia; R ljwarmoadgens l-uri komponentis j-uri ionis koordinatas;δρ r aris l-urilkkomponentis koleqtiuri koordinati, xolo N laris l-uri komponentis ion-Ta ricxvi: ajamva k rsimboloTi xorcieldeba kvazidiskretuli speqtrismixedviT. V-ki warmoadgens sistemis moculobas. magaliTad,araurTierTqmedi eleqtronebis modelisTvis, romlebic imyofebian gareSereleqtronul velSi Et () daZabulobiT, erTeleqtronian hamiltonians aqvsSemdegi saxe:rrH =Τ p −eE() t ⋅ r+ V r : V r = r −R,M Nr r r lr( ) ( ) ( ) ∑∑υl ( lj )l= 1 j=1sadac: e - eleqtruli muxtia, Τ( p)- warmoadgens energias (magaliTad,r2r pΤ ( p)= , m - efeqturi masaa), rr da2m impulsi. V ( r )( r Rlj)p aris eleqtronis koordinati da– eleqtronis ionebTan urTierTqmedebis potencialia, xoloυ − rwarmoadgens eleqtronis urTierTqmedebis psevdopotencials l-urilkomponentis j-ur ionTan. am SemTxvevaSi Cven gveqneba:r rr r Γ ts , = ε p −eE() t ⋅ r;ε p =Τ p + V r( ) ( ) ( ) ( ) ( )Mr r r r−ikrV r = Cυ% (0); υ%( k) = n dre υ ( r)( )∑l=1l l l lNC C r r nMl*l= : ∑ l= 1; υl ( ) = υl( ); =Nl=1(,) = r r rrikrC ts e Cυ % ( k).lkTxevadi metalis an Senadnobis modelSi, romelic aRiwereba (1.4)hamiltonianiT, ionuri sistema ganixileba rogorc klasikuri da aRiwerebakoleqtiuri cvladiTll∫NVδρ r . Txevadi metalis modelisaTvis ionuri sistemislkkonfiguraciebis mixedviT gasaSualoeba dafuZnebulia f { ρ rek}% koleqtiuricvladebiT gausis ganawilebis funqciiT gasaSualoebaze, romlisTvisacsamarTliania Semdegi formula:26

Maq% ≡ ∏∏df%lk ({ lk})⎧⎫⎧1MM*exp⎨∑α rδρ r⎬ = exp ⎨ ∑∑α rα r δρ r ⋅δρr .lk lk , −% ⎬r l k mk lk mk f⎩ i= 1 ⎭ % ⎩2fem , = 1 k⎭r r∫ rl=1 k... δρ δρ ...; sadac α r warmoadgens lda k r -s nebismierffunqcias da adgili aqvs Semdeg Tanafardobebs:* 1 1rrr−ikrδρ r ⋅ δρ r = s ( ) = ⎡1 + ( ( ) −1 ) ⎤.%llk Cnlk lk f⎣l ∫ dre gllrCN CN⎦ll* 1 n rrr−ikrδρ r ⋅ δρ r = s ( ) = ∫ ( () −1 ) : ≠ ,lk mk f%lmk dr glmr e l mN Nsadac g () r warmoadgenen radialuri ganawilebis funqciebs:lmlk( )g () r = g R −R,lm lm li mjlmxolo s ( k ) sidideebi warmoadgenen ionuri sistemis struqturul faqtorebs:s ( k) = s ( k ), s * ( k) = s ( k ). struqturuli faqtorebi iZlevian srullmmllmlminformacias Txevadi metalis an Senadnobis ionuri sistemis Sesaxeb,romlebic aiReba eqsperimentuli monacemebidan. unda aRiniSnos, rommodelis sruli formulirebisaTvis aucilebelia ionebis⎫R ljSemTxveviTikoordinatebis ganawilebis funqciis miTiTeba, romlis mixedviTac xdebagasaSualoeba. Txevadi metalis an Senadnobis aRniSnuli modeli xSiradgamoiyeneba metalebis, Senadnobebisa da gadacivebuli “metaluri minebis”kvantur eleqtronul TeoriaSi, romlebic dafuZnebulia am nivTierebaTaatomur da struqturul Taviseburebebze [9].1.1.3. nawilaki, romelic urTierTqmedebs garemosTanam modelSi ganixileba qvesistema s (gamoyofili nawilaki), romelicurTierTqmedebaSi imyofeba sxva nawilakebisagan Sedgenil garemosTan.aseTi sistemis hamiltonians aqvs saxe:r r2 N 2Ρ ΡiH = + ∑ + U( r1, r2, r3,... rN).2m2Mi=1iaq: m da p r warmoadgens masasa da impulss gamoyofili nawilakisa; MidaΡ riarian masebi da impulsebi garemos nawilakebisa. r i– garemos nawilakTaradius-veqtorebia, U warmoadgens mTeli sistemis nawilakTa ur-27

TierTqmedebis potenciur energias, xolo r r – gamoyofili nawilakisradius-veqtoria.sadac:sawyisi hamiltoniani SesaZlebelia warmovadginoT Semdegi formiT:( ) int,H =Γ s + H + HΣr2rrΡr r rΓ ≡s= + −1 22mrN 2ΡiHΣ= ∑ + U( r0, r1, r2...rN)2MU∇( s) H ( e 1 ) U ( r, r, r...r )i=1irrU∇( ) ⎡r r r r r r r r( N) ( N)Hint = e −1 0, 1, 2... −0, 1, 2... ⎤⎣U r r r r U r r r r .Σ ⎦hamiltonianis aseTi saxiT warmodgenisas igulisxmeba, rom gansaxilvelgamoyofil nawilaks gaaCnia wonasworuli mdebareoba radiusveqtoriTr 0raRac saSualo velSi, romelic ganpirobebulia garemosnawilakebTan urTierTqmedebiT da U - potencialuri energia gaSliliausasrulo mwkrivad nawilakis r u wanacvlebis mixedviT wonasworulir r rmdebareobidan: =0+rrr r r r r rr r u; ( , , ... ) U ∇U r r r r e U( r , r , r ... r )1 2 N= rr0 1 2rNΣN(1.5)(∇ r - warmoadgensgradientis operators gamoyofili nawilakis koordinatebis mixedviT).amrigad, gamoyofili nawilaki, romelic imyofeba saSualo velSi ( Hhamiltoniani), ganicdis am saSualo velis fluqtuaciebis zemoqmedebas( Hinthamiltoniani). simbolo ... (1.5) formulaSi aRniSnavs gasaSualoebasgaremos mdgomareobaTa mixedviT. Tu gavSliT u r wanacvlebas normaluri* +koordinatebis mixedviT: u r = ∑α r a +rχ χαχaχda gamoviyenebT formulas:rr∇ ≈ ∇=rrU* +e u ∑( χaχ χaχ)χχr r r= α + α ∇ (rac Seesabameba nawilakis mxolod wrfivisurTierTqmedebis gaTvaliswinebas garemosTan, romelic aRiwerebaHint-hamiltonianiT), da daviyvanT diagonalur saxezeHshamiltonianis imnawils, romelic kvadratulia r u wanacvlebis mixedviT, maSin SesaZlebelia(1.5) hamiltoniani warmovadginoT (1.1) formiT. simboloebi χ aRniSnavsgamoyofil oscilatorebs, xolo a χda a χ+warmoadgenen gaqrobisa dadabadebis operatorebs χ oscilatorebisa. amrigad s qvesistemis rolSigveqneba oscilatorebis erToblioba hamiltonianiTHs= ∑ ω aa+ , sadacχχ χ χ28

ωχaris χ oscilatoris sixSire. Tu gavSliT U( 0, 1, 2... N ) potenciurr r renergias wanacvlebebis - u1, u2,...uNmixedviT mwkrivad, maSin garemoshamiltoniani H ΣSesaZlebelia warmovadginoT agreTve rogorcoscilatorebis erToblioba (rasakvirvelia, am SemTxvevaSic undagamoviyenoT Sesabamisi miaxloebebi). garemos nawilakTa radius-rveqtorebisTvis gveqneba: r1 = ri0 + ui( i=1,2,... N ), sadac, r i0-warmoadgensgaremos nawilakTa wonasworuli mdebareobebis Sesabamis radius-veqtorebs.samecniero literaturaSi xSirad ganixileba magaliTi kvanturi oscilatorisa,romelic urTierTqmedebs TermostatTan (bozonur velTan) [6].aseTi sistemis hamiltonianisTvis gvaqvs gamosaxuleba:( , )sadac:+ * * +k k k( ) Σ intH =Γ ts , + H + H ,+*+hω0aa + h[f ( t)a + f ( t ]; H = +Σ ∑ h ω ( kbb ) k k;k? ( t,s)= ) a+ +∑ ⎡k( ) k k ( ) ⎤k( )Hint = ⎣C tsb , + C tsb , ⎦;k+C ts , = hλa + h υ a;k k kC ts = hλa+D υ a da kvanturi harmoniuli oscilatori imyofeba gareSe,cvladi klasikuri Zalis moqmedebis qveS, romelic wrfivadaadakavSirebuli (urTierTqmedebs) TermostatTan – araurTierTqmedi kvanturioscilatorebis sistemasTan. aa , , b , b+ +k karian gaqrobis da dabadebisbozeoperatorebi,λ , λ , υ , υ warmoadgenen urTierTqmedebis mudmivebs,* *k k k kromlebic proporciuli arian1Vsididisa, sadac: V - sistemis moculobaa,r k talRuri veqtoria, romelic iRebs kvazidiskretul mniSvnelobebs,f () t aris cvladi klasikuri gareSe Zala, xoloω , 0ωk– oscilatorebissixSireebia.aseTi saxis dinamiur sistemebs vxvdebiT zogierTi amocanebis ganxilvisaskvanturi radiofizikidan, myari sxeulebisa da kondensirebulgaremoTa fizikidan da sxv. magaliTebis saxiT SesaZlebelia moviyvanoTsistemebi, romlebic Sedgebian atomebisa da molekulebisagan da imyofebianlazeris talRis velSi myar sxeulebSi an airebSi, romlebic TamaSobenTermostatis rols [7-8, 10-12].29

1.1.4. eleqtron-minarevuli sistemaes modeli warmoadgens sistemas, romelic Sedgeba eleqtronebisagan,romlebic urTierTqmedeben garemoSi SemTxveviT ganlagebul minarevulircentrebTan da gareSe eleqtrul velTan, romlis daZabulobaa - Et (). aseTisaxis modeli aRiwereba Semdegi hamiltonianiT:rH =Γ ( ts , ) + Hint; Γ ( ts , ) = Hs−eE( t)⋅r;r2NΠNΠΡ(1.6)H = ; H = ≡int ∑ϕ∑ϕ( r−r),s e l2ml= 1 l=1r rsadac: e, m, , p Sesabamisad warmoadgenen eleqtronis muxts, masas, radiusveqtorsda impulss. ϕ ( r − r l ) aris eleqtronis urTierTqmedebis energiaminarevul centrTan, romelic imyofeba wertilSi, romlis radius-veqtoriarl, xolo N Π– minarevuli centrebis ricxvia. xSirad ganixilaven ufrozogad models eleqtron-minarevuli sistemisa, romlis hamiltonianiCaiwereba Semdegi saxiT [22]:sadac: Hint = Hie+ HeΣ;H( ) int,H =Γ s + H + H + H(1.7)eΣee= 1 0 i, j=1 i jΣNr2 2 Npie 1= ∑ + ∑ r r warmoadgens N eleqtronebis2m 8πε−hamiltonians, romlebic urTierTqmedeben erTmaneTTan kulonuri ZaliT, r ida p riaris i-uri eleqtronis koordinata da impulsi masaTa centrismimarT da Γ ( s), Hie,HeΣmoicema Semdegi formulebiT:r2Ρ rrr rr−iqRaΓ ( s) = − NER;Hie= ∑U( q)e2Nmrqa ,rHΣ= ( , ) ΦrreM q λq,λρq.∑rq,λ(1.8)(1.8) formulebSi H iedaH eΣ- eleqtronis urTierTqmedebis hamiltonianebiaminarevTan da fononur velTan; Γ ( s)warmoadgens eleqtronulisistemis masaTa centris hamiltonians (Ρ r da R r – masaTa centris impulsida koordinataa Sesabamisad), xoloρNiqriq= ∑ er aris eleqtronebisi=1+simkvrivis operatoris furie saxe (warmodgena); Φ rq, λ= b rq, λ+b r− q,λfononuri+velis operatoria; br q, λ,b r−q,λ- fononebis gaqrobis da dabadebisoperatorebia talRuri veqtoriT q r , λ warmoadgens fononuri speqtris30

Stos, M( q,λ )- eleqtron-fononuri urTierTqmedebis matriculi elementebia,U( q r ) - minarevTa potencialis furie saxea, ε 0- dieleqtrikuliSeRwevadobaa, xoloR ra- a-uri minarevuli centris koordinatia. (1.7) da(1.8) formulebi faqtiurad warmoadgenen N-urTierTqmedi eleqtronebisa daN Π- SemTxveviT ganlagebuli minarevuli centrebis modelur hamiltonians,romlebic imyofebian E rdaZabulobis mqone gareSe eleqtrul velSi daurTierTqmedeben TermostatTan, romelic aRiwereba fononuri (bozonuri)velis H ΣhamiltonianiT.modeluri hamiltonianebi (1.6)–(1.8) xSirad gamoiyeneba mouwesrigebelisistemebis speqtraluri da kinetikuri maxasiaTeblebis Seswavlisas dagamokvlevisas. zemoTmiTiTebuli hamiltonianebis SeswavlaSi mivyavarTeleqtronuli gadatanis movlenebis Teoriismraval amocanas (rogorcwrfivs, aseve arawrfivs) myar sxeulebSi; zogierT sakiTxs susti daZlieri lokalizaciis Teoriisa, mouwesrigebeli sistemebis eleqtrogamtarobisTeoriis sxvadasxva amocanebs da sxv. [22-23].1.1.5. dikes modelebi gamosxivebis TeoriaSiam sistemebSi umartivesi modeluri hamiltoniani aRwers qvesistemisurTierTqmedebas fotonebTan (fotonur velTan). urTierTqmedebishamiltoniani aiReba wrfivi fotonebis (boze) operatorebis mixedviT. ammodelis farglebSi ganixileba N raodenobis ori energetikuli donismqone (ordoniani) gamomsxiveblebi, romelTa sixSirea O. uSveben, romgamomsxiveblebi urTierTqmedeben erTmaneTTan mxolod velis saSualebiT.rogorc cnobilia, aseTi saxis hamiltoniani pirvelad SemoRebuli iyoatomTa spontanuri koherentuli gamosxivebis movlenis aRsawerad.Tu ugulvebelvyofT kinetikur efeqtebs, romlebic ganpirobebuliagamomsxivebelTa moZraobiT, gacvliTi da eleqtrostatikuri urTierTqmedebebiTda aRvwerT gamomsxivebelTa dipolur urTierTqmedebaseleqromagnitur velTan “mbrunavi talRis” miaxloebaSi, maSin miviRebTmodels, romelic aRiwereba hamiltonianiT [10-11]:31

( Σ ) =Γ ( ) + + ( Σ )H t,, s s H H ts , , ,zsadac: ( ) hΣh ( )Γ s = Ω S H r r=∑ kbbΣint: +0rk kkεtεte− + e * +Hint ( ts , , Σ ) = gSb r r r + gSb r r r∑rk k k ∑rk k kVVkω (1.9)da gamomsxivebeli sistemis spinuri operatorebi S, dakavSirebulia paulismatricebTan standartuli TanafardobebiT:kS± ±= ∑ ±r rikrjkSjej;Szkz= ∑ S e±jjr rikr j;± x y ( x,y,z)1 ( x,y,z)j= SjiS ; Sj= σjS ±aq: r jaris j -uri gamomsxiveblis radius-veqtori, brkda2b + r - warmoa-kdgenen fotonebis gaqrobis da dabadebis boze operatorebs (amplitudebs)talRuri veqtoriT k r , urTierTqmedebis (bmis) mudmivag r dakavSirebuliakd±gadasvlis denis dadebiT-sixSiriani operatoris furie-komponentasTanSemdegi TanafardobiT:( k )32πh ω r rr = r±⋅ rk k dr( )g d e eυsadac: υ warmoadgens nivTierebis kuTr moculobas, h - plankis mudmivaa,xolo ω ( kr ) – fotonis sixSirea. aq: Σ Termostatis rols TamaSobsfotonuri veli, romelSic “moTavsebulia” gamomsxivebelTa ssistema.calke aRebuli, erTeulovani urTierTqmedebis aqtis Sedegad, romelicxorcieldeba eleqtromagnitur velsa da gamomsxivebels Soris, adgiliaqvs fotonuri velis kvantis gamosxivebas (STanTqmas), ris gamocgamomsxivebeli gadadis ori SesaZlo energetikul mdgomareobidan erTerTSi.unda aRiniSnos, rom mocemuli modelisTvis SesaZlebelia agreTvesxva midgoma, romlis drosac Termostatis rols TamaSobs gamomsxivebelTasistema, xolo dinamiuri qvesistemis saxiT gvevlineba bozonuriveli [10]. sakiTxi Termostatisa da dinamiuri sistemis SerCevis Sesaxebwydeba imisda mixedviT, Tu rogori procesebi ganixileba. ase magaliTad,midgoma romelic ganxilulia [7]-Si Seesabameba eleqtromagnituri velisrelaqsaciis procesebis Seswavlas rezonatorSi, xolo midgoma, romelicgaSuqebulia [8]-Si aRwers Tavisufal sivrceSi gamosxivebisas spinuri sis-32

temis relaqsaciis procesebs, romlis drosac sivrce ganixilebaTermostatis rolSi. xSirad ganixilaven dikes iseT modelebs, romlebicaRiwerebian ufro zogadi hamiltonianiT da romlebic Seicaven agreTveordoniani obieqtebis kinetikuri energiis operators [ix. mag. [11,24]].Seswavlili iyo agreTve dikes modelebi, rodesac dinamiuri qvesistema s(gamosxiveba) urTierTqmedebda erTdroulad S atomur da S 1 fononursistemebTan [12].aqve unda aRiniSnos, rom aseTi tipis dinamiur sistemebamde (dikesmodelebamde) mivyavarT mravali problemis ganxilvas kvanturi radiofizikidanda lazeruli gamosxivebis fizikidan, zegamosxivebis Teoriidanda sxv. amitom dikes tipis modeluri sistemebis Seswavla dagamokvleva, rogorc Ria arawonasworuli kvanturi dinamiuri sistemebisa,warmoadgens did samecniero interess. cxadia, rom dikes modelebisaRmweri hamiltonianebi miekuTvnebian (1.1) da (1.2) formulebiT gamoxatulhamiltonianTa klass.1.1.6. eleqtronebi magnetikSi, romlebic urTierTqmedeben kristalis magniturqvesistemasTanaseTi saxis dinamiur sistemebSi ganixileba eleqtronebis urTier-Tqmedeba magnetikis atomTa spinebTan S-d modelis farglebSi [25-26]. er-Tzonian miaxloebaSi sistemis hamiltonians aqvs Semdegi saxe:aqH = He+ HM+ HAH = Ea a ; Hr2ArrRnα= a SS σσ ' a e(1.10)∑∑r ri( χ−χ')( ) '+ +r r r r re χ χσ χσ A χσ e nα χσrNr rχσχ, χ'σσ , 'He– gamtarobis eleqtronebis hamiltoniania,H A- warmoadgens hamiltonians,romelic aRwers eleqtronebis urTierTqmedebas magnetikis lokalizebulspinebTan (A – spin-eleqtronuli urTierTqmedebis mudmiva, N –magnitur atomTa ricxvia, S rre- zonuri eleqtronis spinis operatoria, S nα-warmoadgens magnituri atomis spinis operators, romelic moTavsebuliarkvanZSi, romlis radius-veqtoria ; n - kvanZis nomeria, α - qvemesrisnomeri).RnαHM– magnituri qvesistemis hamiltoniania (Cveulebriv, rogorcwesi iReben haizenbergis gacvliTi urTierTqmedebis hamiltonians33

lokalizebuli spinebisaTvis).H A– hamiltonianSi eleqtronis spinissaSualo spinebTan urTierTqmedebis gancalkevebis Semdeg, (1.10)hamiltoniani SesaZlebelia Caiweros Semdegi saxiT:H = H0+ H1+HMr rH Ea a a S a S er2Ar ri − R0= ∑ + ∑n αχ χσ χσ χσ e χσ ' ' αrN r rσσ 'χσχ, χ'σσ , ', nα+ +( χ χ')sadac: r r r( )rr r rH a S a S S er r( χ χ')( )re ( n )r2A+i − R1= r∑' '−n αχσ χσ α αN r rσσ 'χχ , 'σσ , ', nαSnα(1.11)Tu CavatarebT kanonikur gardaqmnas da gadavalTa + r χσ,a r χσoperatorebidanaxalC + g,Cgoperatorebze kristalis mocemuli gansazRvrulimagnituri struqturis dros, maSin SesaZlebelia H 0hamiltonianis dayvanadiagonalur saxeze (magaliTad: feromagnetiki-ormesriani antiferomagnetikida a.S.). magnituri mowesrigebisas magnituri struqturiswarmoqmnas mivyavarT eleqtronuli zonis gaxleCamde or zonad (qvezonad)[26], ise rom axali C ( + )gda C geleqtronuli operatorebi xasiaTdebian kvanturiricxvebis erTobliobiT: g r = ( χσυ r , , ) da energetikuli speqtriT εgeleqtronebisTvis (υ warmoadgens qvezonis nomers). magaliTad, aseTSemTxvevas aqvs adgili calRerZa helikoidaluri struqturebisSemTxvevaSi, romelTa kerZo SemTxvevas warmoadgenen feromagnetiki daantiferomagnetiki). holstein-primakovis gardaqmnis gamoyenebiTSesaZlebelia Srn,αspinuri operatorebidan gadavideT spinuri talRebisoperatorebze da miviRoT hamiltoniani, romelic aRwers eleqtronebisurTierTqmedebas magnonebTan da romelic zogadad Seicavs mravalmagnonian(erTmagnoniani, ormagnoniani da a.S.) wevrebs. (holstein-primakovisgardaqmnis samarTlianoba SesaZlebelia martivad davasabuToTmagnetikebisaTvis, romlebic imyofebian mowesrigebul fazaSi Zaliandabali temperaturebis dros: Τ> 1). zogierT SemTxvevebSi, rogorc wesi, SesaZlebeliaSemovisazRvroT mxolod wrfivi operatoruli wevrebiT magnonebis34

mixedviT. aseT SemTxvevebSi (1.10)-(1.11) hamiltonianebi erTmagnonianiprocesebisaTvis miiReben Semdeg saxes:sadac:∑H = H + H + H( )sΣ+ + +int ⎣ k k k k ⎦ k ggk , ' g g'kN gg , 'int+ +H = ε g C C ; H = h Ω dd(1.12)s g g Σk k kgk1H = ⎡B () sd + B () sd ⎤; B () s = A CC ,∑Ωk- k magnonuri modis sixSirea; k ( k,j)∑∑( )= r , j - magnonuri Stos nomeria,d , d + - magnonebis gaqrobis da dabadebis boze operatorebia, xolokkAgg'kwarmoadgens “C-sidides”.magaliTad, antiferomagnetikebSi, sadac eleqtronebis energetikulidoneebi gadagvarebulia spinis mixedviT, SesaZlebelia SemovisazRvroTerTmagnoniani procesebis ganxilviT dabali temperaturebis dros.(ormagnoniani procesebi, romlebic aRiwereba H1wevriT (1.11) hamiltonianSi,romelic SeicavsSznαzα− S wevrebs, warmoadgenen mcire sidideebs,rodesac 2 S >> 1 da temperatura aris dabali.( S − Sr

eleqtron-fononur sistemaze da didi radiusis mqone polaronis modelebze.1.2. eleqtron-fononuri sistema. eleqtronis urTierTqmedeba akustikur dapolarul optikur fononebTankristalur myar sxeulebSi, magaliTad farTozonian naxevargamtarebSi,eleqtronis fizikuri Tvisebebi, romelic gadaadgildeba kristalSi,ZiriTadad ganisazRvreba perioduli potencialiT, romelsac qmniankristaluri mesris kvanZebSi ganlagebuli ionebi.ionTa rxevebs wonasworuli mdebareobebis maxloblobaSi mivyavarTeleqtronebis energiebis praqtikulad myisier cvlilebamde dakristalisaTvis Sredingeris adiabaturi gantolebis hamiltonianSi, romelicaRwers eleqtronebis mdgomareobebs, damatebiTi wevris -Hint-iswarmoqmnamde. es damatebiTi wevri aRwers eleqtronebis urTierTqmedebasfononebTan. ionebis mcire amplitudebiT rxevebisSemTxvevaSi, fononuri(bozonuri) sistema Σ warmoadgens araurTierTqmedi harmoniuli oscilatorebiserTobliobas. am sistemis energias aqvs Semdegi saxe:kH = +Σ ∑h ω ( kbb ) k k, da igi warmoadgens fononuri velis sakuTriv hamiltoniansmeoradi dakvantvis warmodgenaSi (nulovani rxevebimxedvelobaSi ar miiReba). rac Seexeba urTierTqmedebis energias –H int– igiwrfivad aris damokidebuli calkeuli, aRebuli harmoniuli oscilatorisnormalur koordinatebze.im sistemis Tvisebebis Seswavla, romelic Sedgeba kvazinawilakisagan(eleqtronisagan) da fononuri (bozonuri) velisagan, romelTa SorisurTierTqmedeba aris wrfivi fononuri operatorebis mixedviT, warmoadgensmyari sxeulebis da kondensirebul garemoTa fizikis da agreTve veliskvanturi Teoriis metad mniSvnelovan amocanas.amgvarad, eleqtron-fononuri sistema warmoadgens magaliTs eleqtronisurTierTqmedebisa dakvantul fononur velTan kristalSi. am SemTxvevaSiqvesistema S Sedgeba erTi kvazinawilakisagan (eleqtronisagan),xolo Σ Termostatis rolSi gvevlineba fononuri veli.cnobili gamartivebebis Semdeg, ararelativisturi kvazinawilaki(eleqtroni), romelic urTierTqmedebs kristaluri mesris rxevebis dakva-36

ntul skalarul velTan, aRiwereba frolix-pekaris tipis hamiltonianiT[29-31]:r2Ρ r+ ikr * − ikr +H = + ∑ ω ( kbb ) + Ve b Ve bk k ∑⎡r r+r rh r r r r r ⎤rr k k k k2m⎣⎦kkrω > 0,( k )sadac r – eleqtronis radius-veqtoria, Ρ r - kvaziimpulsis operatori, br,b + r(1.13)- fononebis gaqrobis da dabadebis operatorebia talRuri veqtoriT k r , m- eleqtronis efeqturi masaa kristaluri mesris periodul potencialurvelSi, xolo ω ( kr ) – warmoadgens fononuri velis kvantis sixSires,romelic aris k r -talRuri veqtoris radialurad simetriuli funqcia:r rω − k = ω k .( ) ( )kk1.2.1. eleqtronis urTierTqmedeba kristaluri mesris akustikur rxevebTan.deformaciis potencialis meTodirogorc cnobilia, eleqtronebis gabnevisas akustikur fononebzegrZeltalRovan miaxloebaSi ka «1 (a - mesris mudmivaa) fononebis dispersiismiaxloebiT kanons aqvs Semdegi saxe [32-33]: ω ( k) = Vsk , sadac V sr r–bgeris siCqarea kristalSi, xolo (1.13) formulis Vrk– mudmivebi, romlebicwarmoadgenen eleqtronis fononebTan urTierTqmedebis energiis furie-komponentebs,moicemian Semdegi TanafardobiT:Vrk⎛4πα⎞= ⎜V ⎟⎝ ⎠12 2h 1 2km,(1.14)sadac: V – kristalis moculobaa, xoloα=Dm2 238πρhVs– warmoadgenseleqtron-fononuri urTierTqmedebis uganzomilebo mudmivas; D aris deformaciispotencialis mudmiva da ρ - kristalis masuri simkvrive. undaaRiniSnos, rom zemoTmoyvanili gamosaxulebebi samarTlianiaIII VA B daII VIA B naxevargamtarebisaTvis, romelTaTvisac adgili aqvs dispersiisizotropul kanons da romlebSic zonis eqstremumi imyofeba brilueniszonis centrSi.37

mokled SevexoT deformaciis potencialis meTods. am meTodis ideamdgomareobs SemdegSi: rodesac kristalSi vrceldeba drekadi talRaelementaruli ujredi ganicdis deformacias, icvleba misi moculoba(icvleba kristaluri mesris mudmiva), rac Tavis mxriv iwvevs gamtarobiszonis fskerisa da savalento zonis Weris mdebareobis cvlilebas, vinaidanzonis sigane mgrZnobiarea mesris mudmivas sididis cvlilebis mimarT. gamtarobiszonis fskeris cvlileba ki faqtiurad warmoadgens gamtarobiseleqtronis urTierTqmedebis energias kristaluri mesris rxevebTan.akustikuri rxevebis SemTxvevaSi, rodesac k r→ 0 yvela atomi elementarulr rujredSi irxeva sinfazurad. rogorc cnobilia, r wertilis U( r)wanacvleba kristalSi grZeltalRovan miaxloebaSi (ka «1) SesaZlebeliaCavweroT Semdegi saxiT [32]:r rsadac er( k)kjr 3rrr 1 r rrrikr + −ikr( ) = r ⎡ r∑+ rr kj jk jkN ⎣k, j=1U r e ( k) b e b e ⎤av⎦,(1.15)aris polarizaciis veqtori, romlis absoluturi mniSvnelobaakustikuri rxevebisaTvis grZeltalRovan miaxloebaSi erTis tolia, xoloNN – elementarul ujredTa raodenobaa kristalSi.vinaidan grZeltalRovan miaxloebaSi akustikuri rxevebisas irxevamxolod elementaruli ujredis masaTa centri da TviT ujredi ki arganicdis deformacias, amitom urTierTqmedebis energia warmoadgensr rmxolod U ( av ) wanacvlebis koordinatebis mixedviT pirveli warmoebulebiswrfiv funqcias da ar aris damokidebuli TviT wanacvlebissidideze. amitom urTierTqmedebis energia Caiwereba Semdegi saxiT:r rH = DdivUint ( ).(1.16)avHintsidides uwodeben deformaciis potencials, xolo D proporciulobiskoeficients deformaciis potencialis mudmivas, romelicZiriTadad ganisazRvreba eqsperimentidan. (1.15) da (1.16) formulebisdaxmarebiT martivad miviRebT:3iD rrrr rrrikr + −ikrHint= ( ker ( k)) ⎡br ∑e −b re⎤.kj jk jkN⎣⎦rk, j=1(1.17)ukanaskneli formulidan Cans, rom im SemTxvevaSi, rodesac gvaqvsizotropuli gabneva gamtarobis eleqtroni urTierTqmedebs mxolod grZiv38

grZeltalRovan akustikur fononebTan ( k // er ( k )) . zogad SemTxvevaSi,rodesac gabneva anizotropulia da deformaciis potencialis mudmivakjwarmoadgens tenzorul sidides D αβ. ( αβ= , 1,2,3)- deformaciis potencialisTvisgveqneba gamosaxuleba:3int=∑( ) ( )j=1H DαβU, sadac ( j)αβU αβwarmoadgensdeformaciis tenzors. am SemTxvevaSi urTierTqmedebis procesSimonawileobas Rebuloben agreTve ganivi fononebic [32-33].1.2.2. eleqtronis urTierTqmedeba polarul optikur fononebTanpolarul nivTierebebSi (polarul naxevargamtarebsa da ionur kristalebSi)ionTa rxevebisas warmoiqmneba ara marto deformaciis potenciali,aramed garda amisa, adgili aqvs Sorsmqmedi makroskopulieleqtruli velis warmoSobas da amitom eleqtronis urTierTqmedebas amvelTan mivyavarT damatebiTi gabnevis meqanizmis warmoSobamde. esurTierTqmedeba, romelsac uwodeben eleqtronis gabnevas polarul optikurfononebze, bevr kristalebSi ufro arsebiTia, vidre denis (muxtis)gadamtanis gabneva akustikur fononebze. am saxis gabnevac grZeltalRovanmiaxloebaSi aRiwereba (1.13) saxis hamiltonianiT. Vr – sidideebskeleqtronis gabnevisas polarul optikur fononebze aqvs Semdegi saxe [29-31]:r12rhω( k) ⎛4πα⎞⎛2 mω( k)⎞Vr=− i ;k r 12 ⎜ ⎟ U =⎜ ⎟12k U ⎝ V ⎠ ⎝ h ⎠21⎛1 1 ⎞ eUα = ⎜ − ⎟ r .2 ⎝ε∞ε0⎠hω( k)12(1.18)aq V aris sistemis (kristalis) moculoba, e da m – eleqtronis muxtida efeqturi masaa, Sesabamisad; α - eleqtron-fononuri urTierTqmedebisfrolixis uganzomilebo bmis mudmivaa, xolo ε 0da ε ∞- warmoadgenenmocemuli nivTierebis statikuri da optikuri dieleqtrikuliSeRwevadobis mudmivebs. eleqtronis urTierTqmedebisas grZiv optikurrfononebTan: max ω( k)≡ ω0- warmoadgens dispersiis armqone, kristalisgrZivi optikuri rxevebis sixSires. zogierT SemTxvevaSi, eleqtron-39

fononurisistemisTvis:Hsr2p=2m gamosaxulebis nacvlad aucilebeliagamoyenebuli iqnas ufro zogadi saxe eleqtronis energiisa gamtarobiszonidan- H =Τ( p r ). iseve rogorc (1.2) formulaSi, ajamva k r -talRurisveqtoriT (1.13). formulaSi xorcieldeba kvazidiskretuli speqtrismixedviT:r ⎛2π 2π 2π⎞k = ⎜ h h h ⎟ L = V⎝ L L L ⎠31, 2, 3; ,(1.19)sadac h 1, h 2, h 3– mTeli ricxvebia (rogorc dadebiTi, aseve uaryofiTi).amrigad, eleqtron-fononuri sistemisaTvis (1.1), (1.2), (1.13) formulebis+Tanaxmad, Ck()s da Ck() soperatorebs aqvs Semdegi saxe:ikrC () s = Ve r ;kkr rrr+ * −ikrk()=kC s Ve(simbolo * aRniSnavs kompleqsurad SeuRlebuls).(1.20)rodesac eleqtron-fononuri sistema imyofeba gareSe, erTgvarovanreleqtrul velSi, romlis daZabuloba Et (), maSin eleqtronis gareSeeleqtrul velTan urTierTqmedebis hamiltonians aqvs saxe:Hsext=−eE r r(). tr(1.21)im SemTxvevaSi, rodesac gareSe eleqtruli veli icvleba harmoniulikanoniT eleqtruli velis daZabulobisTvis gvaqvs:r −() =r εEt Ee t0cos( ω t),(1.22)sadac: ω warmoadgens cvladi eleqtruli velis sixSires, xolo E r0- arisvelisamplituda,te −εTanamamravli aRwers gareSe urTierTqmedebis+adiabatur CarTvas (gamorTvas), rodesac t →m ∞ da ε → 0( ε > 0) .eleqtronis polarul optikur fononebze gabnevisas, eleqtronfononuriurTierTqmedebis (ufro zogadad polaronis amocanis) specifikamdgomareobs imaSi, rom saerTod, uganzomilebo parametri α , romelicaRwers eleqtron-fononuri urTierTqmedebis siZlieres, ar SeiZlebaCaiTvalos rogorc mcire sidide. III-V jgufis naxevargamtarebSi eleqtronfononuriurTierTqmedeba SesaZlebelia iyos susti ( α < 1); II-VI jgufispolarul naxevargamtarebsa da dieleqtrikebSi saSualo siZlieris ( α ≈ 1)da ionur kristalebSi Zlieri ( α >> 1). amitom, zogad SemTxvevaSi,uaryofili unda iqnas daSveba imis Sesaxeb, rom muxtis gadamtanTa40

(eleqtronebis) energetikuli speqtri ganisazRvreba maTi sustiurTierTqmedebiT idealuri kristaluri mesris potencialur velTan dakristaluri mesris rxevebis roli mdgomareobs imaSi, rom maT mivyavarTSedarebiT iSviaT gadasvlebTan eleqtronis mdgomareobebs Soris idealurkristalSi. polarul kristalebSi eleqtron-fononuri urTierTqmedeba iwvevseleqtronuli speqtris arsebiT cvlilebas (polaronuli efeqti),denis gadamtanebis SedarebiT Zlier gabnevas optikur fononebze da amukanasknelTa dispersiis kanonis cvlilebas. yvelaferi es ki arTulebspolaruli kristalebis eleqtruli da agreTve optikuri TvisebebisSeswavlas [29-31, 36-39].amrigad, Zlieri (arasusti) eleqtron-fononuri urTierTqmedebisSemTxvevaSi, zogadad eleqtron-fononuri sistemis Termodinamikis sakiTxebisganxilvisas da kinetikuri movlenebis Seswavlisas Cven bunebrivaddavdivarT polaronis amocanis ganxilvamde [29-30, 37-39].1.3. polaronis amocana. didi radiusis mqonepolaronis modelebiukanasknel wlebSi aqtualuri gaxda sakiTxi struqturuladSedgenili nawilakebis dinamiuri modelebis agebisa. warmodgenebi nawilakisrTuli struqturisa da am nawilakis agznebuli mdgomareobebisarsebobis Sesaxeb yovelTvis warmoadgenda Zlieri urTierTqmedebis (bmis)Teoriis ZiriTad sakiTxs. unda aRiniSnos rom Zlieri bmis TeoriebissiZneleebi Tavidanve ganpirobebuli iyo iseTi cnebebis SemoRebasa daoperirebasTan, romlebic arsebiTad gansxvavebodnen Tavisufali velebisTeoriis warmodgenebisagan. n.n. bogolubovis SromebSi ganxiluli dagamokvleuli iyo amocana ararelativisturi nawilakis urTierTqmedebisadakvantul skalarul velTan adiabaturi miaxloebis zRvrul SemTxvevaSi[35,40].41

1.3.1. polaronis frolixisa da pekaris modelebipirvelad amocana ararelativisturi nawilakis urTierTqmedebisadakvantul skalarul velTan dasmuli iyo frolixis mier egreTwodebulipolaronebis problemis gamosakvlevad. polaronebis problema mdgomareobsgamtarobis eleqtronebis yofaqcevis SeswavlaSi polarul kristalebSi.eleqtroni, romelic imyofeba kristalSi, Tavisi kulonuri veliT waanacvlebsionebs TavianTi wonasworuli mdebareobebidan; Tavis mxriv warmoqmniliionuri polarizacia moqmedebs eleqtronze da iwvevs misi energiisSemcirebas. kristalSi gadaadgilebisas, eleqtrons Tan gadaaqvskristaluri mesris damaxinjebis are. eleqtroni da masTan erTad aRebuliTviTSeTanxmebuli polarizaciuli veli SesaZlebelia ganxiluli iqnesrogorc kvazinawilaki, romelsac uwodeben polarons [29,38]. undaaRiniSnos, rom TviT polaronis koncepcia SemoRebuli iqna pekaris mier,romelic ganixilavda zRvrul SemTxvevas eleqtronisa da kristalurimesris rxevebis Zlieri urTierTqmedebisa da mis aRsawerad iyenebdaadiabatur miaxloebas, romelic warmoadgens naxevradklasikur meTods [29-30].istoriulad pirveli da yvelaze gavrcelebuli midgoma polaronuliamocanis amoxsnisaTvis dafuZnebuli iyo or ZiriTad miaxloebaze,romlebic arsebiTad gansazRvraven polaronuli mdgomareobebis Tvisebebs:1) efeqturi masis miaxloeba, romlis Tanaxmadac eleqtronisurTierTqmedeba xisti kristaluri mesris periodul potencialTangaTvaliswinebulia m efeqturi masiT, rogorc parametriT. aseT SemTxvevaSieleqtronis kinetikuri energia Τ ( p)= gamtarobis zonaSi da TviTr2r Ρ2mzonis sigane formalurad warmoadgenen SemousazRvrel sidideebs daamitom ar TamaSoben parametrebis rols polaronis amocanisaTvis; 2)kontinualuri miaxloeba, romlis drosac ar gaiTvaliswineba kristalurimesris diskretuli struqtura. am miaxloebaTa CarCoebSi, polaronisamocana warmoadgens velis kvanturi Teoriis tipiur magaliTs ferminawilakisurTierTqmedebisa dakvantul fononur velTan, da amitom,rogorc aseTi, igi gaxda erT-erTi magaliTi velis Teoriis meTodebisgamoyenebisa myari sxeulebis fizikaSi.42

polaronis ZiriTadi mdgomareobis Tvisebebis Seswavla arsebiTadmartivdeba susti da Zlieri eleqtron-fononuri urTierTqmedebis zRvrulSemTxvevebSi. susti eleqtron-fononuri urTierTqmedebis SemTxvevaSipolaronis fizikuri maxasiaTeblebi kargad aRiwereba CveulebriviSeSfoTebis Teoriis gamoyenebiT, xolo Zlieri eleqtron-fononuriurTierTqmedebis SemTxvevaSi polaronuli mdgomareoba aRiwerebaTviTSeTanxmebuli midgomiT, romelic pirvelad ganxiluli iyo pekarismier da romelic eyrdnoboda adiabatur miaxloebas [30-31, 39].velis Teoriis lagranJiseuli formulirebisa da standartuli dakvantvisproceduris gamoyenebiT, frolixisa da pekaris mier miRebuli daganxiluli iyo polaronuli sistemis hamiltoniani (1.13-1.18) saxiT. vinaidanSredingeris gantolebis zusti amoxsna (1.13) saxis hamiltonianiT ver iqnanapovni, amitom sxvadasxva dros mravali avtoris mier gamoyenebuli iyomiaxloebiTi meTodebi, romlebsac susti eleqtron-fononuriurTierTqmedebis SemTxvevaSi mivyavarT an standartuli tipis SeSfoTebisTeoriasTan, an sxvadasxva saxis variaciul midgomebTan, romlebicfaqtiurad warmoadgenen hartri-fokis miaxloebis ama Tu im modifikacias[30, 37-38]. am gamokvlevebis Sedegebi gviCvenebs, rom uZravi eleqtronisenergia mcirdeba sididiT: ∆ E =−hωα, 0romelic SesaZlebelia ganxiluliqnas, rogorc polaronis sakuTari energia. garda amisa icvlebaeleqtronis efeqturi masa:* −1m = m(1−α 6) .susti eleqtron-fononuri urTierTqmedebis SemTxvevaSi polaronipirvel miaxloebaSi SesaZlebelia warmovidginoT rogorc kvazinawilaki(eleqtroni), romelic garemoculia fononuri velis arakorelirebulikvantebis RrubliT, romelTa saSualo ricxvi proporciulia bmismudmivasi (polaronis frolixis modeli) [29].fizikis TvalsazrisiT bevrad ufro saintereso da aqtualuria ZlierurTierTqmedebis SemTxveva nawilakisa fononur velTan. Zlier eleqtronfononururTierTqmedebas mivyavarT fononuri vakumis polarizaciaze, raciwvevs sabolood Sedgenili kvazinawilakis struqturis Camoyalibebas.cxadia, rom nebismier gamoTvliT sqemas(algoriTms) unda Seswevdes unari am Camoyalibebuli struqturismodelirebisa ukve pirvel miaxloebaSi [38-40].43

pekaris Tanaxmad, Zlieri eleqtron-fononuri urTierTqmedebisSemTxvevaSi, kristaluri mesris deformacias, romelic gamowveuliagamtarobis eleqtroniT, mivyavarT efeqturi potencialuri ormosCamoyalibebamde da eleqtronis CaWeramde diskretul energetikul doneze,romelsac Seesabameba ukanasknelis finituri moZraoba. GaerTianebulisistema (eleqtroni plus deformaciis are), rogorc mTliani gadaadgilebakristalSi efeqturi masiT, romelic sagrZnoblad aRemateba eleqtronismasas (efeqturs) gamtarobis zonaSi [30,37,39].amgvarad, Zlieri eleqtron-fononuri urTierTqmedebis SemTxvevaSipekaris mier ganviTarebuli Teoria iTvaliswinebs ionuri kristalisdieleqtrikul polarizacias gamowveuls gamtarobis eleqtroniseleqtruli veliT. warmoqmnili lokaluri polarizacia dakavSirebuliaionTa wanacvlebasTan da amitom is aris inerciuli; mas ar SeuZlia “mxariaubas” SedarebiT ufro swrafad moZrav eleqtrons da amitom es lokaluriinerciuli polarizacia, rogorc ukve iTqva, eleqtronisTvis qmnispotencialur ormos. sakmarisad Rrma ormoSi xdeba eleqtronisavtolokalizacia da am warmoqmnili potencialuri ormos siRrmesakmarisia, raTa masSi arsebobdes diskretuli energetikuli doneebieleqtronisTvis. kristalTa didi umravlesobisTvis ormos siRrme 0,5-1eleqtron-voltis rigisaa. eleqtroni, romelic imyofebaavtolokalizebul mdgomareobaSi erT-erT aseT energetikul donezeTavisi eleqtruli veliT “iWers” kristaluri mesris lokalurpolarizacias. ionebi TavianTi inerciulobis gamo “aRiqvamen” eleqtronisara myisier, aramed saSualo eleqtrul vels. eleqtronis aseTiavtolokalizebuli mdgomareoba anu kristalis aseTi mdgomareobebipolarizaciuli potencialuri ormoTi, romelSiac lokalize-buliaeleqtroni, iwodeba polaronul mdgomareobebad (polaronis pekarismodeli) [30,39-40].erT-erT mniSvnelovan da ZiriTad parametrs, romelic axasiaTebspolaronul mdgomareobas, warmoadgens polaronis radiusi, romelicaRwers maxasiaTebel zomebs kristaluri garemos deformaciis arisa. Tueleqtron-fononuri urTierTqmedeba aris susti, maSin polaronis radiusiSesaZlebelia Sefasebuli iqnas ganusazRvrelobis Tanafardobidan:12 12r ≈ h Ρ( mω0) , rogorc maxasiaTebeli zoma eleqtronis fluqtuaciebisa44

sivrceSi, romelic ganpirobebulia virtualuri fononebis gamosxivebisada STanTqmis procesebiT. Zlieri eleqtron-fononuri urTierTqmedebisSemTxvevaSi polaronis radiusis sidide arsebiTadaa damokidebuli amurTierTqmedebis xasiaTze. frolix-pekaris polaronis modelSi, rodesacα >>1, martivi Tvisobrivi Sefaseba polaronis radiusisaTvis iZlevaSemdeg mniSvnelobas:12 12r ≈ h Ρ( mω0) α , rac Seesabameba polaronis radiusisSemcirebas bmis mudmivas zrdasTan erTad [37,41].didi radiusis mqone polaronis adiabatur TeoriaSi polaronis zomaganisazRvreba optimaluri balansis pirobidan gamomdinare eleqtroniskinetikuri energiis dadebiTi wvlilisa, romelic lokalizebuliaSemosazRvruli sivrceSi, dadebiTi wvlilisa deformirebuli kristalurimesris energiisa da uaryofiTi wvlilisa eleqtron-fononuriurTierTqmedebis energiisa. yvela es wvlili erTi da igive rigisaasididis mixedviT, rac jamSi iZleva mogebas polaronis energiisaTvis,SedarebiT Tavisufali eleqtronis mdgomareobisaTvis aradeformirebulikristaluri mesrisaTvis (polaronuli wanacvleba) [37,42-43]. undaaRiniSnos, rom principSi polaronuli mdgomareobebis Tvisebebi arsebiTadganisazRvreba sami masStaburi energetikuli parametriT: D, hω , E , sadacE Ρwarmoadgens polaronul wanacvlebas – energiis Semcirebas eleqtronfononurisistemisaTvis polarizaciuli saxis urTierTqmedebis gamo dawarmoadgens im energetikul masStabs, romelic axasiaTebs amurTierTqmedebis intensivobas. D aris eleqtronebis gadaunomrvi zonissigane, xolo ω warmoadgens fononebis maxasiaTebel rxevebis saSualo six-Sires. didi radiusis mqone polaronis piroba moicema Semdegi TanafardobiT:rΡ >a, sadac a aris kristaluri mesris mudmiva. SesaZlebelianaCveneb iqnas, rom didi radiusis mqone polaronisaTvis adgili aqvsutolobas: E D< 1, romelic faqtiurad ekvivalenturia zemoT moyvaniliutolobisa [37,41].Ρpolaronuli mdgomareobebis mdgradobisTvis aucilebelia, romeleqtronis bmis energia potencialur ormoSi aRematebodes ionTa siTburimoZraobis saSualo energias kristalSi. variaciuli meTodis daxmarebiTpekarma ganaviTara mkacri Teoria eleqtronis Zlieri urTierTqmedebisaΡ45

izotropul ionur dieleqtrikTan, rodesac kristali ganixilebodarogorc uwyveti garemo.pekaris Tanaxmad eleqtronis energia polarizebul kristalSi(dieleqtrikSi), romlis polarizaciis veqtoria - Ρ r r( ) ,warmoidginebaSemdegi saxiT [29-30] (nulovan adiabatur miaxloebaSi, rodesac argaiTvaliswineba ionTa moZraobis kinetikuri energia).r2r r* * 2h r r r r r r rΕ⎡⎣Ψ, Ρ ⎤⎦=− Ψ ( ) ∆Ψ ( ) + 2 Ρ ( ) − Ρ( ) ( ) ,2∫ r rdr πε ∫ rdr ∫ rDrdr(1.23)mr rr r 2 1−rsadac: Dr ( ) = e∫Ψ( 1)r r dr3 1warmoadgens eleqtruli velis induqciis−1veqtors dieleqtrikis wertilSi, romlis radius-veqtoria r r , xolo*1 − −1 −1ε = ε∞− ε 0aris polarizebuli garemos efeqturi dieleqtrikuliSeRwevadoba (ix. (1.18) formula). (1.23) formulis pirveli ori wevri aRwerseleqtronisa da polarizaciuli velis energiebs, xolo ukanaskneli wevri- maT Soris urTierTqmedebis energias. (1.23) gamosaxuleba ganixilebarogorc funqcionali – Ψ( r ) da Ρ r r( ) funqciebis mimarT. rodesac kristaliimyofeba ZiriTad mdgomareobaSi, maSin am funqcionals unda gaaCndesabsoluturi minimumi, Ψ( r ) da Ρ r r( ) funqciebis damoukidebeli variaciebisdros, im pirobiT rom adgili eqneba talRuri funqciis normirebisSenaxvis pirobas:∫r 2 rΨ ( ) dr = 1.(1.24)Tu gavutolebT nuls (1.23) funqcionalis variacias, romelic ganpirobebuliaΡ r r( ) polarizaciis cvlilebiT, rodesac Ψ( r ) talRurifunqcia fiqsirebulia, maSin martivad davadgenT kavSirs Ρ r r( )sidideebs Soris:r r 1 rΡ ( ) = Dr ( ),*4πεda Dr ( r )(1.25)sadac: Dr ( r ) - induqcia (1.23) formulis Tanaxmad ganisazRvreba Ψ( r1)talRuri funqciis saSualebiT. (1.25) gamosaxulebis daxmarebiT, martivadvpoulobT (1.23) funqcionalis saxes, romelic damokidebulia mxolod Ψ( r )talRur funqciaze:22 1 2I[ Ψ ] = h ( ) dr D ( rdr ) .*2m∫ ∇Ψ r r −8πε∫ r r r(1.26)46

IΨ [ ] funqcionali gansazRvravs polarizebuli kristalis (dieleqtrikis)energias, romlisTvisac polarizacia SeTanxmebulia eleqtronismdgomareobasTan, romelic aRiwereba Ψ( r )-funqciiT.polarizebulikristalis aseTi mdgomareobisaTvis, eleqtronisTvis ufro “xelsayrelia”energetikuli TvalsazrisiT imyofebodes ara gamtarobis zonaSi, aramedmdgomareobaSi, romelic Seesabameba eleqtronis moZraobas kristalisSemosazRvrul (finitur) areSi da romelic ganisazRvreba - Ψ( r ) talRurifunqciiT. eleqtronisTvis es energetikulad ufro xelsayreli mdgomareobaSesaZlebelia ganisazRvros (1.26) funqcionalis minimumispirobidan, (1.24) Tanafardobis gaTvaliswinebiT, Ψ( r ) funqciis mixedviT.(1.26) funqcionalis minimizaciisaTvis pekarma gamoiyena pirdapirivariaciuli meTodi [29-30]. ZiriTadi mdgomareobis (polaronulimdgomareobis) funqciis - Ψ ( r r) =Ψ0( ) . aproqsimacia xdeboda ramodenimeparametris saSualebiT. am funqcias hqonda Semdegi saxe:r2 −α′rΨ () = A[1+ α′r + βr] e0(1.27)sadac: A mudmiva ganisazRvreboda normirebis pirobidan (talRurifunqciisTvis) da igi toli iyo sididis:A232α=π γ γ2( 14+ 168 + 720 );1β1− x2 2γ = ; Ψ ( ) (1 ) ;4α′2 0x = A + x + γ x e x = 2α ′ γ , (1.28)2sadac x warmoadgenda uganzomilebo cvlads. variaciuli gaTvlebi α′ da2meγ parametrebisaTvis iZleoda Semdeg mniSvnelobebs: α′ = 0 ,6585 ,,2 *h εγ = 0,1129; (1.28) formulebis daxmarebiT polaronis ZiriTadi mdgomareobisenergiisaTvis miRebuli iyo Semdegi gamosaxuleba [30]:2 2 2 2 43 πhA2 π eAΕ0( αγ , ) = ( 1+ 4γ + 24γ ) −* 5( 10,494+ 209,88γ+2 mαεα+ + +2 3 41884γ 8507,8γ 16727 γ .)(1.29)SedarebiT ufro uxeSi gaTvlebisTvis, pekaris mier gamoyenebuli iyoagreTve talRuri funqcia, romelic Seicavda erT variaciul parametrs.saerTod polaronis ZiriTadi mdgomareobis energia ar iyo Zalian“mgZnobiare” SerCeuli variaciuli parametrebis raodenobis mimarT.47

magaliTad, erTi damoukidebeli meore parametris damateba talRurfunqciaSi saSualebas iZleoda ZiriTadi mdgomareobis energiis mxolod2%-iT Semcirebas. erTi variaciuli parametris mqone normirebul talRurfunqcias hqonda Semdegi saxe:r α 'Ψ = +7π32− ' r0( r) ( 1 α ' r)e α ,(1.30)sadac21 meα ' ≡ = , α′ parametris (1.30) mniSvneloba ganisazRvreboda (1.23),2 *r 2 h ε0(1.26) da (1.30) formulebis daxmarebiT – (1.26) funqcionalis minimumispirobidan. TviT (1.30) talRur funqcias Seesabameboda (1.26) funqcionalisSemdegi mniSvneloba:meIΨ [ ] =− ≡− ⋅ hhε4200,054 20,054 ωα*20,(1.31)sadac: α - aris eleqtron-fononuri urTierTqmedebis frolixisuganzomilebo mudmiva (ix. (1.18) formula). (1.31) gamosaxuleba faqtiuradgansazRvravs polarizebuli kristalis srul energias. (1.23) da (1.25)gamosaxulebebis daxmarebiT SesaZlebelia ganisazRvros kristalispolarizacia, romelic Seesabameba (1.30) mdgomareobas.rr rr e r 21−rΡ0( ) = Ψ0( 1) dr3 1,x4πε∫ r r (1.32)−rdivΡ0( r)rxolo (1.32) formulisa da V0()r =−∫dr1r−r'1Tanafardobis daxmarebiTvipoviT sferuli polarizaciuli potencialuri ormos formas (saxes),romelSiac moZraobs eleqtroni (ix. nax.1). eleqtronis talRuri funqcia(ix. 1.28) napovni α′ variaciuli parametris saSualebiT Caiwereba SemdegisaxiT:( )Ψ = + + ⋅32 2 2() r 0,1229α 1 αr 0,4516 α r e −αr.0TviT V 0() r potencials aqvs Semdegi analizuri saxe:22αe ⎡1 −x⎛ 120() =− − + 0,7605+ 0,2605 + 0,05087 +*V r e x xε⎢x⎜⎣ ⎝ x3 4+ 0,005703x+ 0,0003024 x ) ⎤⎦,(1.33)sadac, x ganisazRvreba (1.28) formuliT. rodesac x >> 1, maSin eqsponencialuriTanamamravli aris Zalian mcire da V 0() r potencialis saxe aris48

2ekulonuri: V0( r)→ − ; ( r >>1); xolo r = 0 wertilSi potencials gaaCnia*ε rparaboluri tipis minimumi. polaronis ZiriTadi mdgomareobis energia (1.33)fiqsirebul potencialur ormoSi gamoiTvleba (1.23), (1.30) da (1.32)formulebis daxmarebiT. gamoTvlebi iZleva eleqtronis energiis SemdegmniSvnelobas:4meΕ0=− 0,163 =−20,163⋅ hωα2 *20h ε2; Tu polarizaciuli potencialuriormo aris sakmaod Rrma, maSin masSi SesaZlebelia agreTve sxvadiskretuli energetikuli doneebis arseboba eleqtronisaTvis (ix. nax.1).nax.1. potencialuri ormo, energetikuli doneebi da eleqtronis Y -talRuri funqcia pekaris polaronis modelSiamrigad, pekaris polaronuli mdgomareobebi SesaZlebelia ganxiluliiqnes rogorc eleqtronisa da kristalis lokaluri polarizaciisbmuli (avtolokalizebuli) mdgomareobebi. es mdgomareobebi xasiaTdebianerTi an ramodenime diskretuli Sinagani energetikuli doneebis arsebobiTavtolokalizebuli eleqtronisTvis. gansxvavebiT polaronis frolixismodelisagan, romelic aRiwereba susti eleqtron-fononuriurTierTqmedebis α mudmivaTi ( α < 1), pekaris polaronuli mdgomareobebixorcieldeba mxolod α bmis mudmivas didi mniSvnelobebis dros ( α >> 1).aseTi avtolokalizebuli mdgomareobebi – polaronebi SesaZlebelia49

gadaadgildnen kristalSi, rogorc erTiani mTliani struqtura(kvazinawilaki), raRac efeqturi masiT – polaronis masiT.pekaris mier gamoTvlili iyo agreTve polaronis masa, rodesac polaroniasrulebda gadataniT moZraobas kristalSi mcire siCqareebiT:V

Wrrkk , '=e2 0324πh 112Mω⎛1 Z⎞⎜ +7⎟⎝ ⎠N;( + Z )−1r r 2hω0⎡ ⎤ k'− kK BΤN = ⎢e − 1 ⎥ ; Z = ,2⎢⎣⎥⎦4 α '82(1.37)sadac: α ' - sidide moicema (1.30) formuliT. mcire siCqareebiT moZravir2 2h kpolaronisTvis, rodesac

ganvixiloT axla dinamiuri modeli struqturulad Sedgenilikvazinawilakis–polaronis, romelic aRiwereba eleqtron-fononuri bmis α -mudmivas nebismieri mniSvnelobis dros, e.w. polaronis feinmanis modeli.1.3.2. polaronis feinmanis modelieleqtron-fononuri sistemisTvis da polaronis amocanaSi farTogamoyeneba hpova feinmanis kontinualuri integrebis meTodma (integralebitraeqtoriebis gaswvriv) [44-45]. am meTodis warmateba polaronis TermodinamikissakiTxebis ganxilvisas dakavSirebulia Semdeg mTavar teqnikurmomentebTan: 1) es meTodi SesaZlebels xdis, raTa moxdes zusti gamoricxvafononuri amplitudebisa polaronis amocanidan (problemidan), ris Sedegadacmravalnawilakovani amocana daiyvaneba erTnawilakovani amocanisganxilvamde, romelic aRiwereba aralokaluri funqcionaliT da romelicdamokidebulia mxolod eleqtronis traeqtoriebze. polaronis feinmanismodelis formulirebisas fononuri operatorebi (amplitudebi)gamoiricxeba zustad eleqtron-fononuri sistemis qmedebidan. amgvarad,miRebuli qmedeba, romelic ar aris lokaluri fononuri operatorebisgamoricxvis Semdeg eleqtron-fononuri sistemidan, aRwers eleqtronisiseT moZraobas, rodesac amocana erTi nawilakis urTierTqmedebisa nawilakTausasrulo ricxvTan, daiyvaneba erTi nawilakis (eleqtronis) ur-TierTqmedebaze Tavis TavTan (TviTqmedebaze). 2) amocanis aseTi arsebiTigamartivebis Sedegad advili xdeba amoxsnebisTvis variaciuli meTodebisformulireba, dafuZnebuli iensenis utolobaze, romelic warmoadgensfunqcionalur analogs n.n. bogolubovis variaciuli principisa kvanturstatistikurisistemebis Tavisufali energiisTvis [25]. 3) miRebul funqcionalTadaxmarebiT, SesaZlebeli xdeba zustad amoxsnad modelTa klasisageba, romlebic Seesabamebian kvadratul funqcionalebs da romlebicgamoiyenebian rogorc miaxloebiTi modelebi variaciuli gamoTvlebisaTvis.feinmanis modelis (miaxloebis) arsi polaronis amocanisTvismdgomareobs imaSi, rom ganixileba zusti, aralokaluri qmedebis aprosimirebaaralokaluri kvadratuli qmedebiT. aralokalurobis gaTvaliswinebaanu “maxsovrobis” efeqtis CarTva qmedebaSi, aZlevs feinma-nismidgomas polaronis amocanisadmi unikalobis Tvisebas, polaronis sxva52

Teoriebisagan gansxvavebiT [46]. feinmanis sacdeli qmedeba aRwerseleqtrons, romelic urTierTqmedebs meore fiqtiur nawilakTan, romliskoordinatebis gamoricxva xdeba qmedebidan–mas Semdeg, rac ganixilebaeleqtronis moZraoba ionur kristalSi da kristaluri me-sris rxevebis(fononebis) gavlena eleqtronze modelirdeba meore nawilakiT–romlismasaa MF da romelic urTierTqmedebs kvadratulad, k-urTierTqmedebismudmivaTi eleqtronTan. aseTi saxis sistemas, romelic aRwers eleqtronisurTierTqmedebas polarul optikur fononebTan, urTierTqmedebis (bmis) αmudmivas nebismieri mniSvnelobis dros, uwodeben polaronis feinmanis(erToscilatorian) models, romelic aris translaciurad invariantuli[44,46] (ix. nax. 2).nax.2. polaronis feinmanis modeli. eleqtroni, romelic dakavSirebulia“zambaris” meSveobiT meore nawilakTan masiT M Famrigad, polaronis feinmanis modelSi eleqtronisa da masTandakavSirebuli virtualuri, korelirebuli fononebis “Rrublis” moZraobaaRiwereba eleqtronis urTierTqmedebiT fiqtiur nawilakTan. am modelishamiltonians aqvs Semdegi saxe [44,46]:r r2 2F Ρ ΡF1 r( ),2Hs= + + kr −rF(1.41)2m2MF2sadac: Ρ r da Ρ rF- warmoadgenen eleqtronisa da fiqtiuri nawilakisimpulsis operatorebs, Sesabamisad; xolo r da r Farian eleqtronisa dafiqtiuri nawilakis radius-veqtorebi, m aris eleqtronis efeqturi masagamtarobis zonidan, xolo M F – fiqtiuri nawilakis masa. (1.41) polaronisfeinmanis modeluri hamiltoniani aRwers nulovan miaxloebaSi polaronisaradisipaciur yofaqcevas eleqtron-fononuri urTierTqmedebis (bmis) αmudmivas nebismieri mniSvnelobis dros. Tu SemoviRebT axal kanonikurcvladebs, (1.41) hamiltoniani SesaZlebelia daviyvanoT diagonalur saxeze.marTlac gveqneba:53

aq:FHrPM + m r k r2FF 2 2s= + Ρos+ ros.2( m+MF) 2mMF2(1.42)r r rP =Ρ+Ρ - warmoadgens mTliani impulsis operators sistemisa, romelickanonikurad SeuRlebulia masaTa centris radius-veqtorTan:rrmr + MFrFR = ;m+MFxolo r os= r −rF- warmoadgens fardobiT koordinatas.r rr MFΡ−mΡFΡos=aris r oskoordinatis kanonikurad SeuRlebuli impulsi. (1.42)m+MFhamiltonianis energetikul speqtrs aqvs Semdegi saxe:r2PErr = + hν( h + h + h + 32);P, hx y z2( m+M )F( h , h , h = 0,1,2,...),x y z(1.43)xolo:M + mFν = k warmoadgens harmoniuli oscilatoris sixSires damMFmMFM + mFaris sistemis dayvanili masa. MF+ m - sidide mocemul modelSi warmoadgenspolaronis efeqtur masas.Zlieri eleqtron-fononuri urTierTqmedebis SemTxvevaSi, rodesacα >>1, adgili aqvs utolobebs: M F>> mda ν >> ω0. feinmanis mier SemoRebuliiyo uganzomilebo parametrebi V da W, romlebic dakavSirebuliaM F da ν sidideebTan Semdegi tolobebiT:2⎡⎛V⎞ ⎤MF= m⎢⎜⎟ − 1; ⎥ ν = Vω 0;V da W⎢⎣⎝W⎠ ⎥⎦parametrebi ganisazRvrebodnen polaronis Tavisufali energiisminimizaciiT [44,46]. modelis yvela sxva danarCeni parametri, rogorebicarian: k, M F + m,mMFm+M da sxv. SesaZlebelia ganisazRvron ( m,ω0) da ( VW) ,Fparametrebis daxmarebiT. susti da saSualo siZlieris (intensivobis) eleqtron-fononuriurTierTqmedebis SemTxvevaSi ( α < 1): V >> W; MF→ 0; ν ≥ ω0;V ≥ 1; V≈ 1+α12[47]. (V da W variaciuli parametrebis yofaqcevis Sesaxebpolaronis feinmanis modelSi ix. nax. 3).polaronis problemisadmi variaciuli meTodis gamoyenebiT, feinmanismier napovni iyo polaronis ZiriTadi mdgomareobis energiis zeda54

sazRvari, kristaluri mesris (fononebis) nulovani temperaturis (T = 0)dros:32 αVE ≤ ( V −W)−4Vπ∞∫02{⎡2 2−UVWU + ⎡( V −W ) V⎤( 1−e) ⎤}⎣⎣( h = ω = m=1)0dUe−U⎦⎦12.(1.44)fononuri operatorebis (amplitudebis) gamoricxvis Sedegad miRebulzust qmedebas hqonda Semdegi saxe (gamoiyeneboda feinmanis meTodiintegralebi traeqtoriebis gaswvriv kristaluri mesris nulovanitemperaturis pirobebSi):1S =2∫r⎛ dr⎜⎝ dt2⎞⎟⎠dt −α8∞ ∞∫∫0 0− t−ser r dtdS.( t)−( s)(1.45)nax.3. V da W – variaciuli parametrebis yofaqceva,rogorc a-urTierTqmedebis(bmis) mudmivas funqciebi, polaronis feinmanis TeoriaSi(1.44) da (1.45) Tanafardobebis ganzogadoeba kristaluri mesrisnebismieri temperaturebis dros Sesrulebuli iyo [48] naSromSi. V da Wparametrebis varirebis Sedegad (1.44) formulidan miiReboda polaronisZiriTadi mdgomareobis energiis yvelaze saukeTeso umciresi zedasazRvris Sefaseba α -bmis mudmivas sxvadasxva mniSvnelobebis dros:55

1) α -s mcire mniSvnelobebis dros ( α < 1), V da W-s saukeTesomniSvnelobebia: V = 31 [ + 2 α (1 −Ρ)3W], sadacW-s am mniSvnelobebisaTvis polaronisSefaseba:12 ⎡(1)12W 1Ρ= ⎣ − −W⎤⎦da W = 3. V daE-energiisaTvis gveqneba Semdegi2 3 2 3E ≤−α − α +Ο ( α ) =−α − 0.0123 α +Ο( α ).(1.46)81xolo SeSfoTebis Teoriis Sedegad miRebuli rezultati tolia sididis2E =−α − 0.0126 α + ...(1.47)2) α -s didi mniSvnelobebis SemTxvevaSi, rodesac ( α >> 1), V da W-ssaukeTeso mniSvnelobebisTvis gvaqvs:V2= α + +4 ⎛ C ⎞4⎜ln2 ⎟ 1, sadac C - eiler-9π⎝ 2 ⎠maskeronis mudmivaa: C = 0,5772... da W = 1. am SemTxvevaSi polaronisenergiisTvis gveqneba Sefaseba:2α 3 3 ⎛ 1 ⎞E ≤− − ( 2ln2 + C)− +Ο + ...23π2 4⎜ ⎟⎝α⎠(1.48)rogorc me-3 nax. naTlad Cans, V da W variaciuli parametrebi,polaronis feinmanis modelSi, warmoadgenen bmis α -mudmivas uwyvetfunqciebs. ricxviTi gamoTvlebis Sedegad miRebuli iyo polaronisZiriTadi mdgomareobis energiis damokidebuleba bmis mudmivaze. GgamoTvlebmaaCvena, rom polaronis ZiriTadi mdgomareobis energiismniSvneloba pekaris TeoriaSi:2E0 =−0.326α (0hω -erTeulebSi) – ufro mciresididisaa, vidre polaronis feinmanis TeoriaSi, rodesac α > 34.29.saerTod ki unda aRiniSnos, rom polaronis ZiriTadi mdgomareobisenergiis Sefaseba feinmanis TeoriaSi bevrad ufro mcirea yvela sxvaSefasebebTan SedarebiT, romlebic miiReba polaronis sxva Teoriebis saSualebiT;garda amisa, unda aRiniSnos agreTve is garemoebac, rom feinmanismeTods gaaCnia didi upiratesoba sxva meTodebTan SedarebiT polaronisTeoriebSi, vinaidan es meTodi (integralebi traeqtoriebis gaswvriv)iZleva saSualebas rom vipovoT polaronis energiis Sefasebebi bmis α -mudmivas rogorc mcire, aseve saSualedo da didi mniSvnelobebis dros;Tanac es meTodi iZleva erTaderT saimedo Sedegebs α -bmis mudmivas saSualedo(5< α < 10 ) mniSvnelobebisaTvis [46,49].56

funqcionalur-variaciuli meTodis daxmarebiT feinmanis mier gamoTvliliiyo agreTve polaronis efeqturi masis sidide [44,46]. integralebiTtraeqtoriebis gaswvriv miaxloebiT gamoTvlas – polaronis feinmanismodelis qmedebisTvis _ mivyavarT polaronis efeqturi masis Semdeggamosaxulebamde:−323 ∞2* αV−τ2⎡V −1−VWτ⎤mF= 1+ ⎢ + ( 1 − ) ⎥ ( = 1).3∫ dτe τ τe mπ0 ⎣ VW ⎦(1.49)(1.49) formulaSi V da W variaciuli parametrebis optimaluri mni-Svnelobebis CasmiT SesaZlebelia Sefasebuli iqnasm * Fefeqturi masa α -bmis parametris mniSvnelobaTa mTel intervalSi. mcire siCqareebiT moZravifeinmanis polaronis efeqturi masis (1.49) mniSvneloba susti da Zlierieleqtron-fononuri urTierTqmedebis zRvrul SemTxvevebSi warmoidginebaSemdegi saxiT:2* α 2αmF = 1 + + ; α >1.rasakvirvelia, (1.49-1.50) gamosaxulebebi samarTliania kristalisnulovani temperaturis (T=0) SemTxvevaSi. (1.50) gamosaxulebebi efeqturimasisaTvis ZiriTadi rigiT α -parametris mixedviT emTxveva, Sedegebsgamomdinare rogorc SeSfoTebis Teoriidan ( α > 1).sainteresoa aRiniSnos is garemoeba, romm * Fefeqturi masis mniSvnelobebi,romlebic napovnia (1.49) gamosaxulebidan mxolod umniSvnelod(ramodenime procentiT) gansxvavdeba sacdeli modelis sruli masismniSvnelobisagan:MF1 ⎛ V+ = ⎞⎜ ⎟⎝W⎠2(m – eleqtronis efeqturi masiserTeulebSi). sasruli temperaturebis dros, polaronis feinmanis Teoriisformulirebisas gamoiTvleboda gibsis operatoris kvali, traeqtoriebisgaswvriv integralebis meTodis gamoyenebiT, Zlieri eleqtron-fononuriurTierTqmedebis SemTxvevaSi [48]. eleqtron-fononuri sistemis mdgomareobaTaerToblioba dakavSirebulia helmholcis Tavisufali energiis57

mniSvnelobebTan, romlis minimizacia xdeboda V da W parametrebismixedviT mocemuli, fiqsirebuli temperaturis pirobebSi; amgvaradganisazRvreboda polaronis saSualo energia, rogorc temperaturisfunqcia. temperaturis zrdasTan erTad (α -parametris fiqsirebulimniSvnelobis dros), polaronis saSualo energiisa da misi efeqtur masismniSvnelobebi mcirdeba. polaronis saSualo energiisa da efeqturi masisaseTi yofaqceva, rodesac Τ→∞ , E → 0 da m → 0 [48] naSromSi aixsnebaimiT, rom vinaidan Zalian dabali temperaturebis dros fononebi arian“Tavisufali”, maTi entropia izrdeba im fononebis entropiasTanSedarebiT, romlebic dakavSirebuli arian eleqtronTan.eleqtronuli da polaronuli gadatanis movlenebis TeoriaSiarsebul im mravalricxovan gamokvlevaTa Soris, romelic miZRvniliapolaronis feinmanis modelis kinetikis sakiTxebisadmi, gansakuTrebulimniSvneloba eniWeba naSromebs, romlebSiac gamokvleulia polaronisZvradoba da eleqtrogamtaroba. es gamokvlevebi damyarebulia grinisfunqciaTa meTodze [50] bolcmanis kinetikuri gantolebis Seswavlaze [51-52], TviTSeTanxmebuli da feinmanis kontinualuri integrebis meTodebisgamoyenebaze [53-54], kubos wrfivi gamoZaxilis Teoriaze [55-56] da sxv.rogorc wesi, rezultatebi miRebuli am sxvadasxva meTodebis daxmarebiT,romlebSiac gamoiyeneba sxvadasxva miaxloebebi, aris sxvadasxva (zogjerarsebiTadac gansxvavdeba erTmaneTisagan). unda aRiniSnos, rom zogadadpolaronis kinetikis ganxilvisas yvela zemoT CamoTvlili meTodiiTvaliswinebs ZiriTadi mdgomareobis energiis Semcirebas dakvazinawilakis maxis gazrdas fononebTan urTierTqmedebis Sedegad,romelic gamowveulia eleqtron-fononuri urTierTqmedebis didi nawilisgaTvaliswinebiT. darCenili eleqtron-fononuri urTierTqmedeba (e.w. “nar-Ceni” urTierTqmedeba) aRwers polaronis gabnevas realur (siTbur)fononebze da iZleva Sesworebebs polaronis ZiriTadi mdgomareobisenergiisaTvis da efeqturi masisTvis.amgvarad, TviT sakuTriv polaronis feinmanis modelSi, eleqtronisurTierTqmedebas polarizaciul velTan mivyavarT arsebiTad or,erTmaneTisagan gansxvavebul movlenasTan (efeqtTan):1) eleqtroni garSemortymulia virtualuri fononebis “RrubliT”.sakmarisad Zlieri eleqtron-fononuri urTierTqmedebis SemTxvevaSi,58F

struqturulad Sedgenil kvazinawilaks – polarons gaaCnia Sinaganimdgomareoba. es “Cacmis” efeqti, romelic aris Sedegi virtualurifononebis gamosxivebisa da STanTqmisa, nulovan miaxloebaSi aRiwerebafeinmanis modeluri hamiltonianiT (1.41).2) disipaciis efeqti. es efeqti, romelic arsebiTad gansxvavdeba pirvelisagan,aRwers polaronis disipaciis movlenas; eleqtroni, fononurvelTan urTierTqmedebis Sedegad - asxivebs da STanTqavs realur (siTbur)fononebs, ris gamoc gadadis erTi stacionaruli mdgomareobidan meoreSi,magaliTad gareSe velis zemoqmedebis Sedegad. TviT polaronisurTierTqmedeba realur fononebTan am modelSi moicema frolix-pekaris(1.13) urTierTqmedebis hamiltonianiT, sadac V rksidideebi ganisazRvreba(1.18) TanafardobebiT.amgvarad, feinmanis kontinualuri integrebis formalizmisa dafunqcionaluri-variaciuli meTodis gamoyenebiT SesaZlebeli xdeba er-Tiani midgomis farglebSi aRweril iqnas polaronis amocana α -bmismudmivas nebismieri mniSvnelobis dros, Tanac α -mudmivas mniSvnelobaTafarTo intervalSi vRebulobT polaronis ZiriTadi mdgomareobis energiisSefasebas bevrad ufro zusts, vidre sxva cnobili meTodebis damiaxloebebis gamoyenebisas; am formalizmis daxmarebiT SesaZlebeliagamoTvlil iqnas TiTqmis yvela statikuri (Termodinamikuri) da dinamiurisidide, romelic ki warmoadgens interess polaronis amocanisTvis didiradiusis mqone polaronis TeoriaSi. rasakvirvelia, ar arsebobs winaswargarantia imisa, rom kargi miaxloeba polaronis Tavisufali energiisTvisaseve kargad aRwers polaronis sxva fizikur maxasiaTeblebs (zogierTSemTxvevaSi, marTlac es ase ar aris), magram zogadad unda iTqvas, romfunqcionalur-variaciuli midgoma SesaZlebels xdis polaronulisistemis yofaqcevis erTiani suraTis Camoyalibebas α -bmis parametris mniSvnelobaTafarTo intervalSi, kristalis sxvadasxva temperaturis dros,gareSe velebis sxvadasxva daZabulobebisa da sixSireebis SemTxvevaSi dasxv.magram miuxedavad polaronis feinmanis modelis unikalobisa, am modelsacgaaCnia erTgvari SemosazRvruloba. rogorc ukve iyo aRniSnuli,feinmanis polaronis TeoriaSi xdeba zusti, aralokaluri qmedebismiaxloeba aralokaluri kvadratuli qmedebiT, rac adebs erTgvarad Sez-59

Rudvas urTierTqmedebis potencialis formas, romelic Semoifarglebamxolod harmoniuli urTierTqmedebiT. Ggarda amisa, Zlieri eleqtronfononuriurTierTqmedebis SemTxvevaSi naTlad ar Cans kavSiri polaronisfeinmanis Teoriasa da polaronis pekaris models Soris.polaronis feinmanis modelis ganzogadoeba Sesrulebuli iyo [57-58]naSromebSi. gansxvavebiT polaronis feinmanis modelisagan, ganzogadoebulmodelSi ar xdeboda qmedebis aproqsimireba aralokaluri kvadratuliqmedebiT, anu urTierTqmedebis potenciali fiqtiur nawilakTan ar iyoharmoniuli tipis; ufro metic, eleqtronis urTierTqmedebis potencialifiqtiur nawilakTan ar iyo fiqsi-rebuli saxis - am potencialis povnaxdeboda variaciuli meTodis gamoyenebiT [57-58]. mimovixiloT axlaSedarebiT ufro dawvrilebiT polaronis feinmanis ganzogadoebulimodeli (fgm).1.3.3. polaronis feinmanis ganzogadoebuli modeli(latinjer-lus modeli)polaronis feinmanis modelisagan gansxvavebiT fgm-Si, eleqtronisada masTan dakavSirebuli virtualuri, korelirebuli, optikuri fononebis“Rrublis” moZraobis aproqsimireba xdeba eleqtronis traeqtoriiT,romlis drosac eleqtroni urTierTqmedebsMGFmasis mqone fiqtiur nawilakTanVGFpotencialis meSveobiT. SedarebiT yvelaze ufro martivSemTxvevaSi aseTi sistemis (eleqtroni+fiqtiuri nawilaki) hamiltonianiaiReba Semdegi saxiT [57]:r r2 2s Ρ ΡGFr rHGF = + + VGF ( − RGF),(1.51)2 2MGFr rsadac: Ρ , R da MGFwarmoadgenen impulsis operators, radius-veqtorsGFGFda masas fiqtiuri nawilakisa, Sesabamisad. gamoTvlebis gamartivebismizniT, Cven SemovisazRvrebiT erTeulTa sistemiT, romelSiac: h = m = ω0 = 1. .(1.51) formulaSi, daSvebulia, rom eleqtroni urTierTqmedebs fiqtiurrV − Rr- centraluri, Zaluri potencialiT. Tu SemoviRebTnawilakTanGF( GF )axal kanonikur cvladebs, SesaZlebelia (1.51) hamiltonianis dayvanadiagonalur saxeze da misi warmodgena Semdegi formiT:60

HrPM+ 1 r2sGF 2GF= + Ρ1GF + VGF2( MGF+ 1)2MGFr( ξ ).(1.52)r r raq: P= Ρ+ΡGFaris sistemis sruli impulsis operatori, romelic kanonikuradSeuRlebulia masaTa centris radius-veqtorTan: R =;rrr−RGFMGFMGF+ 1r rr r rr MGFΡ−ΡGFξ =−R GFwarmoadgens fardobiT koordinatas, xolo Ρ 1 GF=arisM + 1Sesabamisi impulsi. unda aRiniSnos, rom sidide MGF+ 1 mocemulmiaxloebaSi warmoadgens polaronis efeqtur masas fgm-Si; TviT V GFpotencializogadad ar aris fiqsirebuli, misi forma (saxe) moiZebnebavariaciuli principidan mocemul modelSi (SevniSnavT, rom polaronisfeinmanis modelSi Teoriis variaciul parametrebs warmoadgenen k dasidideebi).[57-58] naSromebSi ganzogadoebuli iqna feinmanis kontinualuri integrebismeTodi (feinmanis formalizmi – integralebi traeqtoriebisgaswvriv), romlis daxmarebiTac da iensenis utolobis gaTvaliswinebiT,GFMFmiRebuli iyo optikuri polaronisTvis energiis mniSvnelobaGFΕ 0, rogorcqveda sazRvari variaciuli energiisaGFΕV, ise rom ar iyo fiqsirebuliforma (saxe) V GFvariaciuli potencialisTvis.MGFsadac: µ = ;M + 1r2∞Ρr rGF GF 1GFαΕ0 ≤ΕV= u0 u0− ∑ ' ×2∫∫dξdξµ 2µn=0r r r rr r12* * ⎧1 exp 2 (1 ) ' ⎫u0( ξ ') u0( ξ ) u ( ) ( ') − ⎡− +∆ − ⎤⎪ ⎣nnξ unξC ε ξ ξ⎦⎪× r r ⎨ ⎬,ξ −ξ' ⎪ 1+∆εn⎩⎪⎭GF∆ ε = ε − εnn0 ;C =MGF2( M + 1)GFrda: u ( ξ ) dan(1.53)εnwarmoadgenensakuTar funqciebs da sakuTar mniSvnelobebs Sredingeris gantolebisa,arafiqsirebuli (ganusazRvreli) variaciuli potencialiT V GF:⎡ 1⎤− ∆ r⎢ + ( ) ( ) = ( );2⎥⎣V r r rGFξ⎦u nξ ε u n nξξµ (1.54)( n = 0,1,2,...).(1.52) hamiltonianis cxadi saxe gviCvenebs, rom Sredingeris gantolebasistemisaTvis (eleqtroni+fiqtiuri nawilaki) SesaZlebelia ganvacalkevoT61

da CavweroT rogorc (1.54) Sredingeris gantoleba VGF( ξ r ) potencialiT, daragreTve Semdegi saxis Sredingeris gantoleba Ψ r ( R)talRuriPfunqciisTvis:⎡ 1 1 ⎤ r r⎢− ∆ Ψ ( R) =ΕΨ( R),2M1 Rr r r r⎥(1.55)P P P⎣ GF+ ⎦sadac: sistemis sruli energia Ε GF r moicema TanafardobiTP , nr2GFPrrr1 iPRΕ r =Ε r + ε,, nn≡ + εPPnxolo Ψ r ( R)= e warmoadgens brtyel talRas,P2( + 1)VM GFromelic normirebulia sistemis V moculobaze.vinaidan (1.53) gantolebis yvela wevri, romelic figurirebs ajamvissimbolos qveS, warmoadgens dadebiT sidides, amitom cxadia, rom Tu CvenSemovisazRvrebiT mxolod ZiriTadi wevriT (h=0) (1.53) gantolebis marjvenanawilSi, maSin am gantolebis marjvena mxare gaxdeba zeda sazRvaripolaronis energiisTvis. ZiriTadi mdgomareobis miaxloebaSi Cven gveqneba:rr r2* Ρ r r rGF GF 1GFαΕ0 ≤ΕV= ∫dξu0( ξ) u0( ξ) − ' ⋅2 2∫∫dξdξµ µGF 0GFr r 2Eu0( ξ ) u0( ξ ')v= Ev(1.56)r r⋅ r r { 1−exp⎡−2 C ξ −ξ' ⎤}.ξ − ξ ' ⎣ ⎦rvinaidan (1.56) funqcionali Seicavs µ da u0( ξ ) sidideebs, SesaZlebeliaam sidideebis varireba, imisaTvis, rom vipovoT Ε 0GF V( µ , u0)energiisminimaluri mniSvneloba. Tu gamoviyenebT ritcis variaciul princips(meTods) pekaris tipis sacdeli talRuri funqciebiT, romelsac aqvsSemdegi saxe:r3 32 2 2 −bµξ 2 2bµu0( ξ ) = N( 1 + bµξ + ab µξ ) e ; N =,2π(14+ 42a+45 a )(1.57)sadac a da b warmoadgenen variaciul parametrebs, maSin SesaZlebelia gamoTvliliqnasE0GFV( µ , ab , ) - polaronis ZiriTadi mdgomareobis energiisminimaluri mniSvneloba. SesaZlebelia agreTve gamoyenebuli iqnasvariaciuli meTodi pirdapiri integrirebisa. vinaidan E 0GFVwarmoadgens µr(dayvanili masis) parametris funqcias da u 0( ξ ) funqciis funqcionals, darSeicavs damatebiTi pirobis saxiT mxolod u0( ξ ) funqciis normirebas:62

∫r r 2dξ u0( ξ ) = 1, amitom moTxovna saukeTeso (optimaluri) urTierTqmedebispotencialis SerCevisa (romelic ganapirobebs polaronis ZiriTadi mdgomareobis-E0GFV( µ , ab , ) energiis minimaluri mniSvnelobis miRebas)ekvivalenturia Semdeg gantolebaTa sistemisa:δrδu( ξ)0δEδµ0GFVr r0GF{ E ∫ }2Vλ dξ u0ξ= 0.− ' ( ') = 0;(1.58)amrigad, amocana faqtiurad daiyvaneba µ sididis TiToeulimniSvnelobisaTvis, hartris saxis TviTSeTanxmebuli Sredingerisgantolebis amoxsnamde:rr 22Ρ r12 r 0( ')r rGFα u ξu ( ξ ) − ' ⎡1−exp( −2 − ') ⎤×02∫dξ r rC ξ ξµ µ ξ −ξ' ⎣⎦r r× u ( ξ) = ε u ( ξ).0 0 0amitom, aseTi miaxloebis drospotencialisTvis gveqneba Semdegi saxis gamosaxuleba:r 2r0 α 2 r u0( ξ ')r rV ( ) =− ∫ ' ⎡1−exp( −2 − ') ⎤GFξ dξ r r.− ' ⎣C ξ ξµ ξ ξ⎦(1.59)V GFTviTSeTanxmebuli variaciuli(1.60)(1.56-1.57) da (1.59) gantolebebis daxmarebiT SesaZlebelia aRdgenili(miRebuli) iqnas pekaris naxevradklasikuri Teoria [30,58]. Zlierieleqtron-fononuri urTierTqmedebis SemTxvevaSi ( α >> 1), rodesacrM >> 1, ( ε0 >> hω0) da C →∞. (1.60) gantolebis Tanaxmad V0 ( ξ )GFpotencialisTvis gveqneba gamosaxuleba:r 2r0 α 2 r u0( ξ ')VGF( ξ) =− ∫dξ' r r .µ ξ −ξ'(1.61)susti eleqtron-fononuri urTierTqmedebis SemTxvevaSi ( α < 1), rodesacM → 0 , µ → 0, C → 0 , Tu (1.56), (1.60) formulebis, integralqveSaGFgamosaxulebebSi gavSliT eqsponentas mwkrivad, maSin martivad davrwmundebiT,rom am zRvrul SemTxvevaSi polaronis energia fgm-Si Semosaz-Rvrulia mniSvnelobiT: −α , anu E ≤− 0α . zogad SemTxvevaSi, polaronisGF63

E0GFV( µ , ab , ) energiis mniSvneloba SesaZlebelia gamoTvlil iqnasanalizurad Tu gamoviyenebT Semdeg cnobil formulebs:r rexp ( ik ξ −ξ') ∞ + e(1) *r r = ik∑Ie( kξ < ) hl ( kξ > ) ∑Ylm( θϕ , ) Ylm( θϕ , );4 πξ −ξ'l= 1m=−ln! n!ZZnkn −bt −bZ∫te dt = −en+ 1 ∑ n− k+1b0k=0 k!b∞∫Zte dt = en −bt −bZ∞∑k = 0n!Zk!bkn− k+1.;(1.62)polaronis energiisE0GFV( µ , ab , ) mniSvneloba gamoTvlili iyo [58]naSromSi. Zlieri eleqtron-fononuri urTierTqmedebis SemTxvevaSi ( α >> 1)polaronis energiis zeda sazRvris mniSvneloba tolia sididis:−0,108504α2[57-58].ricxviTi meTodebis gamoyenebiT, TviTSeTanxmebuli (1.60) potencialirrogorc sawyisi potenciali, saSualebas iZleva rom gamovTvaloT u ( ξ )aRgznebuli talRuri funqciebi, Sredingeris (1.59) gantolebis daxmarebiT.polaronis fgm-Si, TviTSeTanxmebul potencialis yofaqceva did manZilzearis kulonuri tipis, xolo mcire manZilze (areSi, sadac eleqtronistalRuri funqcia aris didi) potenciali aris paraboluri saxis(harmoniuli oscilatoris potencialis tipis) [30,57-58]. gansxvavebiT polaronisfeinmanis modelisagan, am modelSi martivad xerxdeba miRebuliSedegebis (magaliTad: polaronis ZiriTadi mdgomareobis energiis)dakavSireba SedegebTan, romlebic gamomdinareoben polaronis pekarisTeoriidan, Zlieri eleqtron-fononuri urTierTqmedebis SemTxvevaSi. [58]naSromSi naCvenebia, rom midgoma, romelic eyrdnoba variaciuli meTodisgamoyenebas optimaluri, TviTSeTanxmebuli potencialis sapovnelad, uke-Tes Sedegs iZleva polaronis ZiriTadi mdgomareobis energiis gamoTvlisas,vidre harmoniuli aproqsimacia urTierTqmedebis potencialisTvis.iseve rogorc polaronis feinmanis modelSi, polaronis fgm-Si (1.51)modeluri hamiltoniani aRwers nulovan miaxloebaSi polaronisaradisipaciur yofaqcevas, α-parametris nebismieri mniSvnelobis dros.amrigad, polaronis fgm-Sic, eleqtronis urTierTqmedebas polaruloptikur fononebTan mivyavarT or gansxvavebul efeqtTan. erTi efeqtiganapirobebs polaronis - rogorc kvazinawilakis struqturis Camoya-n64

libebas, romelic aris Sedegi virtualuri fononebis gamosxivebisa daSTanTqmisa da romelsac adgili aqvs Zlieri eleqtron-fononuri urTierTqmedebisdros da meore efeqti, romelic gansxvavdeba pirvelisagan daromelic aRwers polaronis disipaciis movlenas, ganpirobebuls eleqtronisurTierTqmedebiT siTbur polarul optikur fononebTan. esukanaskneli efeqti, iseve rogorc polaronis feinmanis modelSi, aRiwerebafrolix-pekaris tipis - (1.13) urTierTqmedebis hamiltonianiT.miuxedavad kontinualuri integrirebis (integralebi traeqtoriebisgaswvriv) meTodis upiratesobisa sxva meTodebTan SedarebiT da funqcionalur-variaciulimidgomis dadebiTi mxareebisa, romlebsac CvenSevexeT polaronis feinmanis modelisa da fgm-is ganxilvis dros,wonasworul da arawonasworul statistikur meqanikaSi da, kerZodpolaronuli sistemebis Termodinamikisa da kinetikis sakiTxebis SeswavlisasfarTo gamoyeneba hpova mowesrigebul operatorTa formalizmmada T-namravlTa meTodma, romelic agreTve emyareba fononuri amplitudebisgamoricxvis teqnikas [59-62]. ufro metic, SeiZleba iTqvas, rom zogierTiSedegi polaronis TeoriaSi pirvelad miRebul iqna swored am midgomisgamoyenebiT. garda amisa, mowesrigebul operatorTa formalizmsa da T-namravlTa meTods eniWeba upiratesoba zogierTi sakiTxis ganxilvisas,gansakuTrebiT maSin, rodesac saWiroa rezultatebis marTebulobissafuZvlianobisa da zogierTi Teoremebis damtkiceba statistikur meqanikaSi,da kerZod, polaronis TeoriaSi [60-62].gasuli saukunis 80-ian wlebSi n.n. bogolubovisa da n.n. bogolubov(umc.) mier ganviTarebul iqna axali midgoma eleqtron-fononurisistemisaTvis da polaronis wonasworul TeoriaSi (ganixilebodaeleqtronis urTierTqmedeba polarul optikur fononebTan), romelicsamarTliania nebismieri temperaturebisa da eleqtron-fononuriurTierTqmedebis α-parametris nebismieri mniSvnelobis dros. am midgomassafuZvlad edo T-namravlTa teqnikis gamoyeneba fononebis Tavisuflebisxarisxis gamosaricxad eleqtron-fononuri sistemidan, da Sesabamisadeleqtron-fononuri sistemis maxasiaTebeli fizikuri sidideebis wonasworulisaSualo mniSvnelobebis gamoTvla [62]. T-namravlTa teqnikasaSualebas iZleva mkacri maTematikuri sizustiT da SedarebiT martivdoneze daasabuTos kontinualuri integrebis meTodis safuZvlianobis idea65

[14-15]. zustad amoxsnadi hamiltonianisa da agreTve variaciuli principisgamoyenebam polaronis TeoriaSi, SesaZlebeli gaxada polaronis Tavisufalienergiis zeda sazRvris Sefaseba sasruli temperaturebisa da α-bmismudmivas nebismieri mniSvnelobis dros [40,62]. unda aRiniSnos, rommowesrigebul operatorTa formalizmi da T-namravlTa teqnika gansakuTrebiTefeqturi aRmoCnda polaronis Tavisufali energiisgamosaTvlelad SeSfoTebis Teoriis meTodebis gamoyenebisas sasrulitemperaturebis SemTxvevaSi da agreTve eleqtron-fononuri sistemis(polaronis) kinetikis sakiTxebis ganxilvisas [5,14-16,62].axla gadavideT fizikuri kinetikis zogierTi principuli sakiTxismimoxilvaze dinamiuri sistemebisa, romlebic urTierTqmedeben bozonur(fononur) TermostatTan.1.4. fizikuri kinetikis zogierTi sakiTxi dinamiuri sistemebisa, romlebicurTierTqmedeben fononur(bozonur) velTanrogorc cnobilia, kinetikuri gantolebebis miRebis Cveulebriviprocedura dakavSirebulia korelaciebis Sesustebis hipoTezasTan anekvivalentur daSvebebTan, magaliTad Sfm-Tan. es hipoTeza an miaxloebasaSualebas iZleva rom moxdes sistemis Semoklebuli aRwera kinetikurigantolebis saxiT. magram, rogorc cnobilia, dinamiur sistemebSi mimdinarestohastikuri procesebis Teoriidan, Tu dinamiuri sistema warmoadgens Ktipis sistemas (a.n. kolmogorovis saxis sistemebi) [20,63], maSin araviTarihipoTeza ar aris saWiro kinetikuri gantolebis misaRebad, SemoklebuliaRwera warmoiqmneba avtomaturad dinamiuri sistemis evoluciis procesSifazur sivrceSi Serevis procesebis arsebobis gamo erT-erTi an ramdenimedinamiuri cvladis mixedviT. swored am cvladis an cvladebis mixedviTxdeba korelaciebis swrafi Sesusteba. analogiuri debuleba samarTlianiaagreTve kvanturi - K sistemebisaTvis. tradiciulad, rogorc wesi,kinetikuri gantolebis gamoyvanas Tan axlavs specifiuri tipis apriorulihipoTeza. kargad aris cnobili, rom kinetikuri gantolebis forma dastruqtura ganisazRvreba im procesebis albaTuri bunebiT da TvisebebiT,romlebsac aRwers TviT es gantoleba. Tu gamovalT liuvilisgantolebidan ganawilebis funqciisTvis (kvantur-meqanikuri midgomisas66

fon-neimanis gantolebidan statistikuri operatorisTvis), Cven SegviZliavipovoT kinetikuri gantolebis saxe, Tu ugulvebelvyofT wevrebs romel-Ta gamo moZraobis dinamiuri xasiaTi arsebiTad gansxvavdeba SemTxveviTiprocesebisagan. amis gamo kinetikuri gantolebis miRebas ama Tu im formiTTan axlavs raime principis an aprioruli hipoTezis formulireba, daamitom aseTi hipoTezis arsi (datvirTva) mdgomareobs imaSi, rom mocemulidinamiuri sistemisTvis SemoaqvT SemTxveviTobis (qaosis) esa Tu is elementi[3,20].im meTodebs Soris, romlebsac mivyavarT kinetikuri gantolebismiRebamde, yvelaze ufro srulyofili ganviTareba da gamoyeneba hpova n.n.bogolubovis meTodma, romelic gadabmuli gantolebebis ierarqiulijaWvis daxmarebiT saSualebas iZleva ganawilebis funqciis agebisa, daromelic gamomdinareobs liuvilis gantolebidan. es meTodi cnobilia,rogorc – bbgki-is (bogolubovi-borni-grini-kirkvudi-ivoni) saxelwodebiT.bbgki-is jaWvi sabolood mTavrdeba bolcmanis tipis gantolebis miRebiT.am meTodSi gamoiyeneba sivrculi korelaciebis Sesustebis principi,romelic mdgomareobs imaSi, rom nawilakebi romlebic sakmao manZiliTarian dacilebuli erTmaneTisagan sivrceSi, asruleben arakorelirebadmoZraobas [4,34,64-65].unda aRiniSnos, rom fundamenturi miRwevebi bolcmanis tipiskinetikuri gantolebis (rogorc klasikuris, aseve kvanturis) da ZiriTadikinetikuri gantolebis gamoyvanaSi, miuZRvis n.n. bogolubovs. bolcmanissaxis kvantur-kinetikuri gantoleba mkacri saxiT miRebuli iqna n.n.bogolubovis mier, romelic eyrdnoboda korelaciebis SesustebishipoTezas [36,66]. mis mier damuSavebuli iyo formaluri sqemebi zogadadkinetikuri gantolebebis miRebisa da formulirebuli iyo mkacrimaTematikuri formiT is daSvebebi, romlebic avseben damatebiT amgamoyvanebis meTodebs. kvanturi kinetikuri gantolebis sxva formac –ZiriTadi kinetikuri gantoleba (master equation), romelic pirvelad SemoTavazebuliiyo paulis mier – gamoyvanili iyo n.n. bogolubovis mier Sfm-Si,romlis miRebis drosac gamoyenebuli iyo SeSfoTebis operatorisspeqtraluri Tvisebebi [36,66].K-tipis dinamiuri sistemebisaTvis (rogorc klasikuris, asevekvanturisTvis), SesaZlebelia paulis saxis kvanturi kinetikuri gantole-67

is miReba Sfm-is gamoyenebis gareSe. aseTi saxis gantolebis miReba demonstrirebuliiyo [20] naSromSi, kvanturi arawrfivi oscilatorismagaliTze, romelzedac moqmedebda gareSe perioduli Zala. meTodisZiriTadi idea, romelic gamoyenebuli iyo [20] naSromSi mdgomareobdaimaSi, rom kvanturi K sistemebis ergodiul Tvisebebs klasikur zRvrulSemTxvevaSi mivyavarT statistikuri operatoris aradiagonalurimatriculi elementebis swraf milevamde. Sedegi miRebuli iyo kvaziklasikurmiaxloebaSi, romelsac mivyavarT kvanturi dinamiuri sistemisSemoklebuli aRweris SesaZleblobaze (fazur sivrceSi) da rogorc Sedegi- napovni iyo paulis saxis kinetikuri gantoleba.ukanasknel wlebSi, dinamiuri sistemebis TeoriaSi, farTo ganviTarebahpova mkacrma meTodebma kvanturi dinamiuri qvesistemis dinamikisaRwerisa, romelic urTierTqmedebs dakvantul bozonur velTan(TermostatTan). es meTodebi gamoirCeva im TaviseburebiT, rom saSualebasiZleva, raTa miRebuli iqnas Tanafardobebi, romlebic aRweren qvesistemisdinamikas, romelic samarTliania bozonur velTan urTierTqmedebis (bmis)mudmivas nebismieri mniSvnelobebis dros, nebismieri temperaturebisa da nebismierigareSe moqmedi Zalebis SemTxvevaSi. am midgomebis dros argamoiyeneba daSveba mTeli dinamiuri sistemis statistikuri operatorisgamartivebis Sesaxeb didi droebis SemTxvevaSi: t >>τ0(sadac: τ0-warmoadgens qaotizaciis maxasiaTebel dros), rodesac am daSvebisTanaxmad, mTeli kvanturi dinamiuri sistemis statistikuri operatorixdeba qvesistemis dayvanili (reducirebuli) statistikuri operatorisfunqcionali.pirvelad aseTi saxis evoluciuri gantoleba qvesistemis statistikurioperatorisaTvis, rodesac qvesistema urTierTqmedebs bozonurvelTan (fononebTan), romlebic imyofebian statistikuri wonasworobismdgomareobaSi T temperaturiT, miRebuli iyo [15-16] naSromebSi, bozonuriamplitudebis zusti gamoricxviT am evoluciuri gantolebidan.evoluciuri gantoleba Sfm-is gamoyenebiT qvesistemis statistikurioperatorisTvis, romelic urTierTqmedebs TermostatTan, miRebuli iyo [67-71] SromebSi, romelic gansxvavebiT [5-6, 14-16] naSromebSi miRebuligantolebebisagan aris Caketili. sivrculi erTgvarovnebis SemTxvevaSi,qvesistemis (eleqtronis) TermostatTan susti urTierTqmedebis dros68

SeSfoTebis Teoriis meore miaxloebaSi am evoluciuri gantolebidangamomdinareobs bolcmanis gantoleba eleqtronisaTvis [57,69-71].ganzogadoebuli kvanturi kinetikuri gantolebebis mkacri gamoyvanaeleqtron-fononuri sistemisaTvis Sesrulebuli iyo [6,14-15] SromebSi,sadac ganxiluli iyo agreTve am gantolebebis gamoyeneba sxvadasxvamodelisTvis. Seqmnili iyo meTodi kinetikuri gantolebebis miRebisa,romelic eyrdnoboda ganviTarebul specifiur teqnikas – velis kvanturiTeoriisa da T-namravlTa formalizmis daxmarebiT, – bozonuri amplitudebisgamoricxvisa Sesabamisi kinetikuri gantolebebidan [5-6]. Sesabamisimiaxloebebis SesrulebiT, gamoyvanili kinetikuri gantolebebidanmiRebuli iyo konkretuli fizikuri Sedegebi, romlebic aRweren eleqtronisada polaronis kinetikas [27,29-31,38]. amrigad, fundamenturi midgoma,romelic ganviTarebuli da gamoyenebuli iyo [14-15] SromebSi, saSualebasiZleoda Sfm-Si ganzogadoebuli kvanturi evoluciuri gantolebebismiRebas sistemis statistikuri operatorisTvis, gamoricxuli bozonuriamplitudebiT.ukanasknel periodSi, arawonasworuli da Seuqcevadi procesebisaRsawerad da Sesabamisi adekvaturi Teoriis Sesaqmnelad, myari sxeulebisfizikaSi farTo gamoyeneba da ganviTareba hpova arawonasworuli statistikurimeqanikis sxvadasxva meTodebma. arawonasworul sistemebSirelaqsaciuri procesebis Sesaswavlad didi mniSvneloba eniWebakorelaciuri da grinis funqciebis meTods [4,34,64-65,72-73]. ukanasknelwlebSi, arawonasworuli movlenebis gamosakvlevad xSirad gamoiyenebaagreTve proeqciuli operatoris meTodi [74]. Gganzogadoebuli kvanturikinetikuri gantoleba statistikuri operatorisaTvis qvesistemisa,romelic urTierTqmedebs bozonur TermostatTan, SesaZlebelia miRebuliqnas aseve arawonasworuli statistikuri operatoris meTodis gamoyenebiT[4,34].[68] naSromSi, proeqciuli operatorisa da grinis superoperatorismeTodi gamoiyeneboda Caketili evoluciuri gantolebis misaRebadqvesistemis statistikuri operatorisaTvis, romelic urTierTqmedebdabozonur velTan, saidanac gamoricxuli iyo bozonuri operatorebi. amnaSromSi, liuvilis superoperatoruli formalizmisa da proeqciulioperatoris meTodis daxmarebiT, gamoyvanili iyo moZraobis ganzo-69

gadoebuli, kvanturi gantoleba grinis dagvianebuli superoperatorisTvis.grinis superoperatorebis meTodi gamoiyeneboda agreTve [75-76] naSromebSi,sadac miRebuli iyo kvanturi evoluciuri gantolebebi qvesistemisoperatorebis drois ormomentiani wonasworuli korelaciurifunqciebisTvis. Sesabamisi miaxloebebis ganxilvisas qvesistemis(eleqtronis) dinamikisTvis, am gantolebebidan martivad miiRebodacnobili Sedegebi eleqtron-fononuri sistemisTvis da, kerZod polaroniskinetikaSi [29,77]; magram, miuxedavad amisa, xazi unda gaesvas im garemoebas,rom yvela zemoT moyvanil naSromSi, qvesistemis statistikuri operatoris-Tvis ganzogadoebuli kvanturi evoluciuri gantolebis gamoyvanisas, dagrinis superoperatorebisTvis moZraobis gantolebebis miRebisas, gamoiyenebodaaproqsimacia – Sfm, da amitom am naSromebSi yvela miRebuligantoleba – rogorc qvesistemis statistikuri operatorisTvis, iseqvesistemis operatorebis wonasworuli korelaciuri funqciebisTvis – araris zusti.rogorc cnobilia, wrfivi gadatanis movlenebis Sesaswavlad myarsxeulebSi, erT-erT efeqtur xerxs warmoadgens meTodi, romelicdamyarebulia bolcmanis kinetikuri gantolebis gamoyenebaze. Ggadatanismovlenebis kvanturi Teoria, romlis aRsawerad gamoiyeneba bolcmanisgantoleba, faqtiurad warmoadgens kvaziklasikur Teorias, romelicdafuZnebulia adiabatur miaxloebasa da SeSfoTebis Teoriaze.eleqtronuli da polaronuli gadatanis movlenebis kvantur TeoriaSibolcmaniseuli midgomisas igulisxmeba, rom gamtarobis eleqtronebi (anpolaronebi) imyofebian stacionarul da TiTqmis “Tavisufal” mdgomareobebSikvaziimpulsebis gansazRvruli mniSvnelobebiT. kristaluri mesrisperiodulobis darRveva ganpirobebuli, magaliTad fononebiT, iwvevseleqtronebis mdgomareobaTa arastacionarobas. eleqtronebis fononebzegabnevisas igulisxmeba, rom gabnevis procesebis aqtebi kargad aris gancalkevebulisivrcesa da droSi, da amitom, faqtiurad, adgili ar aqvsinterferenciis movlenas eleqtronebis erTi mdgomareobidan meoreSigadasvlis dros. kristalze modebuli gareSe eleqtruli veli ganixilebarogorc susti intensivobisa (mcire amplitudebis mqone) da mdored cvladisivrcesa da droSi. rogorc Sedegi, aseTi gareSe eleqtruli velismoqmedeba iwvevs mxolod muxtis matarebelTa aCqarebas da maT gadasvlebs70

sxvadasxva mdgomareobebSi ise, rom ar xdeba TviT am mdgomareobebisarsebiTi cvlileba. aseT SemTxvevebSi eleqtronis gabnevis kveTebi ariReben did mniSvnelobebs. amgvarad, bolcmaniseuli aRwera SeiZleba CaiTvalosrogorc kvaziklasikuri, specifiuri kvantur-meqanikuri efeqtebisgareSe.eleqtron-fononuri sistemisaTvis bolcmanis ganzogadoebuligantoleba da polaronis frolixis modelisaTvis bolcmanis gantolebamiRebuli iyo susti eleqtron-fononuri urTierTqmedebis SemTxvevaSi [14-15] naSromebSi. bolcmanis gantolebis miRebisa da gamoyenebissafuZvlianoba eleqtronebis drekadi gabnevis SemTxvevaSi minarevulcentrebze, rodesac es ukanasknelni ganlagebulni arian qaosuradsivrceSi, ganxiluli iyo [78-79] naSromebSi. rogorc ukve aRvniSneT,bolcmanis gantoleba SesaZlebelia miRebuli iqnas agreTve qvesistemis(eleqtronis) simkvrivis matricis zusti, Caketili kinetikurigantolebidan, rodesac adgili aqvs eleqtronis sust urTierTqmedebaspolarul optikur fononebTan, romelic ganxiluli iyo [67,69-71]naSromebSi. magram miuxedavad amisa unda aRiniSnos, rom gadatanismovlenebis wrfiv TeoriaSi, gadatanis meqanikuri koeficientebisgamosaTvlelad bolcmanis kinetikuri gantolebis gamoyenebis areSezRudulia. ase magaliTad, eleqtron-fononuri sistemis SemTxvevaSi, Tueleqtronebis fononebze gabnevis sxvadasxva aqtebi erTmaneTTan ganicdianinterferencias, maSin cxadia, rom bolcmanis kinetikuri gantoleba ukvear gamoiyeneba [80].gasuli saukunis 70-ian wlebSi, gadatanis arawrfivi movlenebis aRsaweradda meqanikuri koeficientebis (mag. Zvradoba) gamosaTvlelad,tornbergisa da feinmanis mier ganviTarebuli iyo axali midgoma, romelicTavisufali iyo im SezRudvebisagan rasac moicavda gadatanis movlenebisbolcmaniseuli aRwera. gamoyenebul iqna ra kontinualuri integrebismeTodi, tornbergisa da feinmanis mier dadgenili iyo arawrfividamokidebuleba kristalze modebul gareSe eleqtrul velsa da muxtismatareblis (eleqtronis) damyarebul saSualo siCqares Soris polarulnivTierebebSi [84]. tornberg-feinmanis Teoriis sqemaSi eleqtron-fononurisistemis statistikuri operatorisTvis, romelic ganisazRvreboda liuvilfonneimanis evoluciuri gantolebidan, gamoyenebuli iyo Zalian rTuli71

maTematikuri formalizmi dafuZnebuli feinmanis integralebzetraeqtoriebis gaswvriv [53,84]. unda aRiniSnos, rom feinmanistraeqtoriebiT integralebis gamoTvla ar xerxdeboda zustad da amitomzemoTmiTiTebuli avtorebi iyenebdnen sxvadasxva miaxloebebs fizikuriSedegebis misaRebad. am midgomis farglebSi, eleqtron-fononuri sistemissimkvrivis matricidan xdeboda fononuri operatorebis gamoricxva.amgvarad, tornberg-feinmanis midgoma da meTodi amyarebda arawrfivTanafardobas gareSe sasrul (Zlier) eleqtrul velebsa da eleqtronissaSualo siCqareebs Soris polarul kristalSi, rac faqtiuradsaSualebas iZleoda myar sxeulebSi gadatanis arawrfivi movlenebisSeswavlis SesaZleblobas [84-85]. xazi unda gaesvas im garemoebas, romtornberg-feinmanis mier miRebuli zemoTnaxsenebi Sedegebi samarTlianiarogorc Zlieri, aseve susti eleqtron-fononuri urTierTqmedebisSemTxvevaSi. samecniero literaturaSi tornberg-feinmanis midgoma cnobiliarogorc balansis gantolebis meTodi. aqve unda aRvniSnoT, rombalansis gantolebis meTodi awydeba siZneleebs eleqtronis (an polaronis)Zvradobis gamoTvlisas polarul nivTierebebSi Zalian dabalitemperaturebis dros. am meTodSi miCneulia, rom arc Tu ise Zlierieleqtruli velebisas miiRweva stacionaruli mdgomareoba, rodesaceleqtroni moZraobs mudmivi siCqariT. [84] naSromSi daxatuli fizikurisuraTi, eleqtronis damyarebuli siCqaris damokidebulebisa Zlierstatikur eleqtrul velze, principSi samarTliania α, T da E-s nebismierimniSvnelobebisaTvis, magram avtorebi aRniSnaven or ZiriTad sirTules,romelic jer kidev ar aris gadalaxuli dRevandel dRemde:1) impulsis balansis gantolebaze dayrdnobiT miRebuli Sedegebi aremTxveva im Sedegebs, romlebic miiReba energiis balansis gantolebidan.2) susti intesivobis eleqtruli velebis SemTxvevaSi, ZvradobagamoTvlili balansis gantolebis daxmarebiT ar emTxveva Zvradobis immniSvnelobas, romelic miiReba bolcmanis gantolebis amoxsniT (gansxvavdebamisgan3 KBT-faqtoriT).2 hω0rogorc mogvianebiT gairkva [15-16], tornbergisa da feinmanis TeoriisSedegi (mxolod susti eleqtron-fononuri urTierTqmedebis zRvrulSemTxvevaSi) SeiZleba miviRoT bolcmanis gantolebidan, Tu vivaraudebT,72

om mis stacionarul amoxsnas aqvs kvazimaqsvelis ganawilebis funqciissaxe. marTlac, erTis mxriv eleqtron-fononuri sistemis qmedebisaproqsimacias kvadratuli funqcionaliT (feinmanis meTodSi – integralebitraeqtoriebis gaswvriv) yovelTvis mivyavarT kvazimaqsvelis tipisimpulsis ganawilebis funqciasTan [53,85] da amave dros, meores mxrivbolcmanis gantolebis amonaxsnTa gamokvlevebi [86] da eleqtronis dreifulisiCqaris gamoTvlis eqsperimentaluri monacemebi Zlier eleqtrulvelebSi dabali temperaturebisas [87] cxadad gviCveneben, rom impulsisganawilebis funqcia mkveTrad gansxvavdeba maqsveliseulisagan.Tvisobrivad es dakavSirebulia im faqtTan, rom dabali temperaturebisada Zlieri velebis SemTxvevaSi, relaqsaciis dominirebul meqanizmadgvevlineba polaruli optikuri fononebis gamosxiveba, romelic arisZlier aradrekadi da anizotropuli xasiaTis procesi da romelsacmivyavarT denis matareblebis specifiur yofaqcevamde, romelic cnobilia,rogorc “Streaming motion” movlena. miuxedavad zemoT aRniSnuli sir-Tuleebisa, [84-85] naSromebSi mocemuli Teoria warmoadgens jerjerobiTerTaderT mikroskopul Teorias eleqtronisa, nebismieri eleqtronfononuriurTierTqmedebis SemTxvevaSi Zlier gareSe eleqtrul velSi.imisaTvis, rom daeZliaT eleqtronuli da polaronuli gadatanismovlenebis siZneleebi naxevargamtarebsa da ionur kristalebSi, romlebicdafuZnebuli iyo bolcmanis gantolebis gamoyenebaze, da im mizniT, romaegoT impendansis erTiani wrfivi Teoria polaronisTvis (eleqtronisTvis)gareSe eleqtruli velis nebismieri sixSireebis, kristalis nebismieritemperaturebisa da eleqtron-fononuri bmis mudmivas nebismieri mniSvnelobisdros, feinmanis, xelvorsis, idingsisa da platcmanis (fxip) mierganviTarebuli iyo midgoma, romelic dafuZnebuli iyo eleqtron-fononurisistemisTvis arawonasworuli simkvrivis matricis gamoTvlaze feinmanismeTodiT – integralebi traeqtoriebis gaswvriv. fxip-is mier miRebuli iyowrfivi gamoZaxilis funqciis zusti mniSvneloba ormagi kontinualuriintegralis saxiT traeqtoriebis gaswvriv, romelic Semdeg aproqsimirebuliiyo feinmanis erToscilatoriani modelis farglebSi [88]. fxip-ismidgomam Semdgomi ganviTareba hpova [47,89-90] SromebSi. fxip-is miermiRebuli impendansis zogadi gamosaxulebidan gamoyvanili iyo polaronisefeqturi masisTvis zogadi formula, napovni iyo polaronis optikuri73

STanTqmis koeficienti da sxv. miRebuli iyo agreTve polaronis Zvradobiszogadi formula bmis mudmivas nebismieri mniSvnelobisa da nebismieritemperaturis dros. unda aRiniSnos, rom fxip-is mier miRebuli yvelaSedegi samarTliania agreTve eleqtronisTvisac, bmis mudmivas mcire (α < 1)mniSvnelobebisas. magram miuxedavad fxip-is midgomis universalobisa, ammidgomasac gaaCnia sirTuleebi, romlebic dRemde ar aris gadawyvetili.erT-erTi ZiriTadi sirTule mdgomareobs imaSi, rom iseve rogorctornberg-feinmanis TeoriaSi, aqac napovni dreifuli ZvradobismniSvneloba kristalis dabali temperaturebis dros3 KBT-TanamamravliT2 hωgansxvavdeba Zvradobis im mniSvnelobisagan, romelic miiReba bolcmanisgantolebis amoxsniT dabali temperaturebisas da bmis mudmivas rogorcmcire, aseve didi mniSvnelobis dros. Aarawonasworuli simkvrivis matricismeTodis sirTule mdgomareobs agreTve imaSi, rom arsebobs garkveuliganusazRvreloba impendansis miaxloebiTi mniSvnelobis SerCevaSi,romelic aiReba maTematikurad aramkacri mosazrebebisa da fizikisTvalsazrisiT gonivruli aproqsimaciis ganxilvaSi. saerTo jamSi undaiTqvas, rom fxip-is mier napovni eleqtrogamtarobis tenzorisgamosaxuleba ZiriTadad sworad da erTiani TvalsazrisiT misaRebadaRwers polaronul efeqtebs optikaSi, galvanomagnitur movlenebs daciklotronul rezonans myari sxeulebis fizikaSi da sxv.rogorc ukve aRniSnuli iyo sadisertacio naSromis SesavalSi,eleqtronuli da polaronuli gadatanis movlenebis gamosakvlevadnaxevargamtarebsa da ionur kristalebSi, garda bolcmanis kinetikurigantolebisa da zemoTmoyvanili midgomebisa da meTodebisa, SesaZlebeliagamoyenebuli iqnas sruliad gansxvavebuli (alternatiuli) midgoma,romelic dafuZnebulia kubos wrfivi gamoZaxilis Teoriaze [4,64,72-73]. ammidgomis farglebSi araviTari SezRudva ar edeba eleqtronis gabnevisaqtebs (gabnevis kveTebs) fononebze, gansxvavebiT bolcmaniseuliaRwerisagan. magram miuxedavad amisa, eleqtrogamtarobis gamoTvlisaskubos meTodiT warmoiSveba sirTuleebi, romlebic dakavSirebulia ganSladobebTanσ(? )-eleqtrogamtarobis gaSlisas mwkrivad eleqtronisurTierTqmedebis hamiltonianis mixedviT fononebTan (gambnevebTan), gareSeeleqtruli velis dabali ? →0 – sixSireebis SemTxvevaSi. susti eleqtron-074

fononuri urTierTqmedebis SemTxvevaSic, σ(? )-eleqtrogamtarobisTvis sworiSedegis misaRebad, rodesac ? →0, aucilebelia ganSladi wevrebisganSedgenili usasrulo mwkrivis ajamva [80]. kubos wrfivi reaqciis Teoriam –eleqtronuli da polaronuli gadatanis movlenebis aRsawerad – SemdgomiganviTareba hpova [91,93-95] SromebSi. [91] naSromSi liuvilis gantolebazedayrdnobiT da proeqciuli operatoris meTodis gamoyenebiT, eleqtronfononurisistemisTvis miRebuli iyo eleqtrowinaRobis zogadi formula.SeSfoTebis Teoriis gamoyenebiTa da Sesabamisi miaxloebebis ganxilviT(e.w. kenkre-drezdenis aproqsimacia) am zogadi gamosaxulebidan napovni iyoeleqtrowinaRobis (impendansis) miaxloebiTi formula. mogvianebiT [101]-naSromSi naCvenebi iyo, rom kenkre-drezdenis aproqsimacia impendansisTvisda kubos formulisTvis ekvivalenturia fxip-is mier gamoyvaniliimpendansis (eleqtrowinaRobis) miaxloebiTi mniSvnelobisa sustieleqtron-fononuri urTierTqmedebis SemTxvevaSi. kubos wrfivigamoZaxilis Teoria da proeqciuli operatoris meTodi [94]-naSromSicgamoyenebuli iyo eleqtruli winaRobis Teoriis asagebad. miRebuli iyozogadi gamosaxuleba eleqtrowinaRobisTvis da naCvenebi iyo, rom iseverogorc eleqtrogamtarobisTvis, eleqtrowinaRobis swori mniSvnelobismisaRebad aucilebelia calkeuli wevrebisgan Sedgenili usasrulomwvrivis ajamva. zogadi saxiT dadgenili iyo martivi kavSiri kompleqsureleqtrogamtarobasa da impendans Soris. [95]-naSromSi kubos Teoriisa, fonneimanis gantolebisa da cvancigis tipis proeqciuli operatoris daxmarebiTmiRebuli iyo araerTgvarovani ZiriTadi kinetikuri gantoleba,romelic saSualebas iZleoda gamoZaxilis funqciis povnisa kubos formalizmSi.gamoyvanili iyo agreTve formulebi ganzogadoebuli amTviseblobisada eleqtrogamtarobisTvis. ZiriTadi kinetikuri gantolebisdaxmarebiT napovni iyo bolcmanis tipis gantoleba pirveli momentisTvis,romelic Seicavda rogorc disipaciur, aseve nakadisebr wevrebs.kubos formalizmi da grinis funqciaTa meTodi [83], [93] naSromSigamoyenebuli iyo polaronis dabaltemperaturuli Zvradobis gamosaTvleladSeSfoTebis Teoriis meoTxe miaxloebaSi. avtorTa mier amnaSromSi, eleqtronis siCqaris drois ormomentiani korelaciuri funqciagamosaxuli iqna grinis ornawilakovani funqciis saSualebiT, xologrinis ornawilakovani funqcia warmodgenili iqna grinis erTnawilakovani75

funqciisa da eleqtron-fononuri urTierTqmedebis mixedviT mwkrivissaxiT. sabolood, maT mier napovni dabaltemperaturuli Zvradobis mniSvnelobaiZleoda karg miaxloebas [55]-naSromSi miRebul polaronisdabaltemperaturuli Zvradobis sididesTan, rodesac α

fononebze, e.w. polaronis rezonansuli urTierTqmedeba (gabneva)fononebTan, da am procesis gaTvaliswinebiTa da bolcmanis gantolebisamoxsniT, napovni iyo dabaltemperaturuli Zvradobis mniSvnelobapolaronisTvis. [52]- naSromSi, polaronis Zvradoba feinmanis modelSigamoTvlili iyo sasruli temperaturebisa da arasusti eleqtron-fononuribmis SemTxvevaSi. [52]- naSromi faqtiurad warmoadgenda [51,55-56]Sromebis (osakas Sedegebis) ganzogadoebas – oRond gansxvavebiT yvelasxva zemoT miTiTebuli naSromebisgan, am naSromSi bolcmanis gantolebaamoxsnili iyo variaciuli meTodis gamoyenebiT (zemoTaRniSnul SromebSibolcmanis gantolebis amosaxsnelad ZiriTadad gamoiyeneboda relaqsaciisdrois miaxloeba, an am meTodis esa Tu is modificirebuli forma).ricxviTi gamoTvlebi Catarebuli iyo TIBr-kristalebis nimuSTa monacemebismixedviT da karg TanxvedraSi iyo eqsperimentis SedegebTan. aqve undaaRiniSnos, rom zemoTmoyvanil naSromebSi da saerTod kinetikurigantolebis meTodSi, bolcmanis gantolebis amoxsnisas (saubariagawrfivebul gantolebaze) gamoiyeneba sxvadasxva saxis miaxloebebi (ix.mag. [106]), da amitom eleqtronis da polaronis dabaltemperaturuliZvradobebisTvis miRebuli mniSvnelobebi ar aris Tanmimdevruli da zusti.zogadad Zalian rTulia mkafiod dadgindes bolcmaniseuli midgomisgamoyenebis sazRvrebi saSualo da Zlieri eleqtron-fononuri bmisSemTxvevebSi. mosalodnelia, rom es midgoma samarTliani iqneba kristalisdabali temperaturebis dros, rodesac dajaxebaTa xangrZlivobis dro(romelic aris hβ= h KBT-s rigis) bevrad naklebia t-relaqsaciis droze.variaciuli meTodi [106]- bolcmanis kinetikuri gantolebis amosaxsneladpolaronis frolixis modelSi ganixileboda agreTve [99]- naSromSi.gamoTvlili iyo frolixis polaronis dabaltemperaturuli dreifuliZvradobis sidide, romelic emTxveoda relaqsaciis drois miaxloebaSifrolixis mier napovn dabaltemperaturuli statikuri ZvradobismniSvnelobas polaronisTvis [43]. amrigad, rogorc zemoTmoyvanilinaSromebis mimoxilva gviCvenebs, arsebobs arsebiTi gansxvaveba polaronisZvradobis mniSvnelobebs Soris, romelic erTis mxriv miiReba bolcmanisgantolebis amoxsnisas relaqsaciis drois miaxloebaSi da meores mxrivfeinmanis meTodis – integralebi traeqtoriebis gaswvriv gamoyenebisaspolaronis frolixisa da feinmanis modelebSi [100]. susti gareSe77

eleqtruli velebisa da susti eleqtron-fononuri bmis SemTxvevaSi,polaronis Zvradobis gamoTvlisas relaqsaciis drois miaxloebaSedarebiT ufro realuri Cans, vidre balansis gantolebis meTodi [84-85,100]. [84,88] -naSromebSi polaronis Zvradoba gamoTvlili iyo feinmanismeTodiT – integralebi traeqtoriebis gaswvriv, xolo [15,85,101] SromebSiZvradobis gamoTvlisas gamoiyeneboda eleqtronis impulsebis mixedviT maqsveliswanacvlebuli ganawilebis funqcia. [101] - naSromSi naCvenebi iyo,rom meTodebi romlebic ganviTarebuli da gamoyenebuli iyo [91-92,84,88]SromebSi, susti eleqtron-fononuri bmisa da susti gareSe eleqtrulivelebis SemTxvevaSi, polaronis dabaltemperaturuli ZvradobisTvis iZleodnenfaqtiurad erTnair mniSvnelobebs; rac Seexeba gamoTvlebs, romlebicdafuZnebulia bolcmanis kinetikuri gantolebis gamoyenebaze,optikuri polaronis dabaltemperaturuli dreifuli ZvradobisTvismivyavarT sxva mniSvnelobebTan. [97,100]. kritikuli analizi, polaronisZvradobisTvis miRebuli sxvadasxva mniSvnelobebis Tanxvdenis Sesaxeb,Catarebuli iyo [102,103]- SromebSi, da dadgenili iyo, rom optikuri polaronisZvradobis sidide damokidebulia zRvruli gadasvlebis α→0, ? →0operaciebis Tanmimdevrobaze (am SromebSi naCvenebi iyo, rom zRvruligadasvlebis swori Tanmimdevrobaa:limlim ).α→0ω→ 0polaronis fgm-Si[57-58]-bolcmanis tipis kinetikuri gantolebaZlieri eleqtron-fononuri bmisa da kristalis dabali temperaturebisSemTxvevaSi, miRebuli iyo [70-71]- SromebSi. bolcmanis gantolebismiRebisas, avtorebi eyrdnobodnen maT mier [68-69]- naSromebSi gamoyvanilCaketil kinetikur gantolebas qvesistemis (polaronis) statistikurioperatorisaTvis, romelic napovni iyo Sfm-Si. Gganixileboda mciresiCqariT moZravi polaroni dabali temperaturebis dros da miRebuli iyokinetikuri gantoleba qvesistemis statistikuri operatoris, rogorcdiagonaluri matriculi elementisaTvis (polaronis ganawilebisfunqciisTvis, polaronis ZiriTadi mdgomareobisTvis), asevearadiagonaluri matriculi elementebisaTvis (polaronis agznebuli,gadagvarebuli mdgomareobebisTvis). rac Seexeba TviT sakuTriv polaronisZvradobas, igi gamoTvlili ar iyo am modelSi. aqve, erTxel kidev undaaRiniSnos da xazi gaesvas im garemoebas, rom TviT bolcmanis kinetikurigantolebis sxvadasxva meTodebiT gamoyvanisas da misi amoxsnisas gamo-78

iyeneba sxvadasxva saxis miaxloebebi, da amitom eleqtronis da polaronisZvradobis Tanmimdevruli da koreqtuli gamoTvlis problema sxvadasxvamodelSi moiTxovs Semdgomi damatebiTi gamokvlevebis Catarebas.kovalentur (araionur) kristalebSi, magaliTad germaniumisa dasiliciumis tipis naxevargamtarebSi, eleqtronebis urTierTqmedebafononebTan aRiwereba deformaciis potencialis meTodiT, romelicmoqmedebs zonur eleqtronze da romelic warmoadgens axlomqmedSeSfoTebas kristalis perioduli potencialisTvis [39,42,107]. eleqtronismoZraobis SeSfoTeba, gamowveuli am urTierTqmedebiT, warmoadgenskvazinawilakis gabnevas deformaciis potencialze. Tavisi bunebidangamomdinare, es urTierTqmedeba damaxasiaTebelia yvela naxevargamtarebis-Tvis da saerTod myari sxeulebisTvis. rogorc cnobilia, eleqtronisurTierTqmedeba (gabneva) akustikur fononebTan SesaZlebelia iyos rogorcsusti, aseve Zlieri [41,104]. eleqtronis susti urTierTqmedebisasakustikur fononebTan, eleqtronis gadaadgilebas kovalenturi kristalebisgamtarobis zonaSi Tan axlavs kristalis lokaluri deformaciisgadanacvleba, romelic aRiwereba SeSfoTebis Teoriis enaze, rogorceleqtronis mier virtualuri fononebis gamosxivebisa da STanTqmisprocesi; xolo kristalebSi, romlebsac gaaCniaT mcire drekadobismodulebi da gamtarobis zonaSi eleqtronebis didi efeqturi masebi –eleqtronis urTierTqmedeba grZiv akustikur fononebTan aris Zlieri,rasac mivyavarT kristalis lokalur deformaciasTan, romelic sakmarisiapotencialuri ormos warmosaqmnelad da romelSiac eleqtroni asrulebsstacionarul moZraobas diskretuli energiiT (e.w. akustikuri polaronismodeli) [39,41-42,104-105]. Aakustikuri polaronis modelTan, da saerTodkvazinawilakis lokalizaciis problemasTan myar sxeulebSi, mWidrod arisdakavSirebuli sakiTxi eleqtronis e.w. TviTCaWerisa, romelic gamowveuliaZlieri eleqtron-fononuri urTierTqmedebiT [41-42,104-105]. akustikuripolaronis modeli – feinmanis meTodiT, integralebi traeqtoriebis gaswvriv– Seswavlili iyo [41,104-105] SromebSi. dadgenili iyo, rom Zlierieleqtron-fononuri bmis SemTxvevaSi adgili aqvs eleqtronis TviTCaWeras,romelic ganpirobebulia deformaciis potencialiT gamowveuli axlomqmediurTierTqmedebiT. saerTod unda aRiniSnos, rom eleqtronebisTviTCaWeris amocanis Seswavlas didi mniSvneloba aqvs eleqtronuli79

(polaronuli) gadatanis movlenebis aRsawerad myar sxeulebSi, vinaidankristaluri mesris deformaciis Sedegad TviTCaWerili kvazinawilaki ariZleva wvlils eleqtrogamtarobaSi (ZvradobaSi).susti eleqtron-fononuri bmis SemTxvevaSi, eleqtronuli gadatanismovlenebis Sesaswavlad akustikuri polaronis modelSi kristalisdabali temperaturebis dros, gamoiyeneba kinetikuri (bolcmanis) gantolebismeTodi. dabali temperaturebisas eleqtronis Zvradoba ZiriTadadganisazRvreba misi gabneviT kristalSi arsebul minarevebze da akustikurfononebze. sufTa kristalebSi dabali temperaturebisas- rodesacoptikuri fononebi sustad arian aRgznebuli – dominirebs eleqtronisgabneva akustikur fononebze. eleqtronis urTierTqmedebis energia grZeltalRovangrZiv akustikur fononebTan mcire sididisaa, vidre misiurTierTqmedebis energia polarul optikur fononebTan da, garda amisa,eleqtronis energiis cvlileba akustikur fononebze gabnevisas,warmoadgens mcire sidides (kvazidrekadi gabneva). amis gamo, eleqtronisgabneva akustikur fononebze SesaZlebelia ganxilul iqnas relaqsaciisdrois miaxloebaSi [29,32-33,107]. vinaidan am SemTxvevaSic bolcmaniskinetikuri gantolebis amoxsnisas gamoiyeneba sxvadasxva saxis miaxloeba(magaliTad, Sfm), eleqtronis Zvradobis gamoTvla mis akustikurfononebze gabnevisas, moiTxovs damatebiTi gamokvlevebis Catarebas [29,108].sadisertacio naSromSi ganviTarebulia da gamoyenebulia midgoma,eleqtronuli da polaronuli gadatanis movlenebis gamosakvlevadkvanturi dinamiuri sistemebis zemoTmoyvanili modelebisTvis (romlebicurTierTqmedeben fononur velTan), romelic damyarebulia kubos wrfivigamoZaxilisa da SeSfoTebis Teoriaze [109-122, 125].mowesrigebul operatorTa formalizmi da proeqciuli operatorismeTodi, T-namravlTa meTodi (teqnika) da fononuri (bozonuri) operatorebisgamoricxvis procedura wonasworuli, droiTi korelaciurifunqciebisaTvis aRwerili da gamoyenebuli iqneba sadisertacio naSromisII da III TavSi. III TavSi ganxilulia eleqtronuli da polaronuligadatanis movlenebis sakiTxebi kvantur disipaciur sistemebSi –eleqtron-fononur sistemaSi, polaronis frolixis modelSi, akustikuripolaronis modelSi, polaronis fgm-Si– dafuZnebuli kubos wrfivi gamoZaxilisTeoriaze; kerZod, III TavSi gamoTvlilia eleqtronuli da po-80

laronuli gadatanis meqanikuri (kinetikuri) koeficientebi (Zvradoba,eleqtrogamtaroba) zemoT miTiTebul modelebSi, zemoT naxseneb formalizmsada meTodze, da korelaciuri funqciebisaTvis ganzogadoebulkvantur kinetikur gantolebebze dayrdnobiT.81

Tavi II. ganzogadoebuli kvanturi evoluciuri gantolebebidroisormomentiani wonasworuli korelaciuri funqciebisa da grinisfunqciebisTvis dinamiuri qvesistemisa, romelic urTierTqmedebsTermostatTan (bozonur velTan)2.1. mowesrigebul operatorTa formalizmida T- namravlTa meTodiganvixiloT mcire dinamiuri qvesistema s, romelic urTierTqmedebsbozonur (fononur) velTan S. mTliani (s + S) sistemis hamiltoniani aviRoTSemdegi saxiT:sadacH = H + H + H , (2.1)sΣint.Hsaris s qvesistemis sakuTari hamiltoniani; H Σ- warmoadgensbozonuri (fononuri) velis hamiltonians; xolourTierTqmedebis hamiltoniani – bozonur velTan.H+C ( s ) da C ( s)k= +Σ ∑ h ω ( kbb ) ; k k = ⎡ ( ) + ( )kk+ +∑ ⎣ k k k kkHint.- aris s qvesistemisH ⎤int.C s b C s b ⎦, (2.2)– warmoadgenen operatorebs, romlebic miekuTvnebian sqvesistemas. mTliani (2.1) – hamiltoniani ar aris damokidebuli droze.A , B .... aRvniSnoT operatorebi Sredingeris warmodgenaSi, romlebicssdamokidebuli arian mxolod s qvesistemis dinamiur cvladebze daromlebic ar arian damokidebuli droze. operatorebi, romlebicdamokidebuli arian mxolod s-is an S-s cvladebze, komutirebenerTmaneTTan. bozonuri (fononuri) sistema S ganixileba rogorcTermostati. davuSvaT, rom As() t da Bs() t – warmoadgenen s qvesistemisoperatorebs haizenbergis warmodgenaSi:A ( t)= eSiHthA eS−iHth;SiHτhS−iHτhB ( τ ) = e B e ;am operatorebis sawyisi mniSvnelobebisaTvis gveqnebaAs = As() t , B ( )t=0 s= Bsττ = 0drois ormomentiani wonasworuli korelaciuri funqciebi da grinisfunqciebi (dagvianebuli, winmswrebi da mizezobrivi) ganisazRvrebianSemdegi tolobebiT (ix. mag. [25,34,64]).F ( t− τ) =< A () t B ( τ)>.AB s ss s82

a− τ = θ − τ < [ τ ] >; ( τ) θτ ( ) [ () ( τ )]rG ( t ) ( t ) As() t Bs( )ηG t− =− − t < Ast Bs>ηsadac:⎧1; x > 0θ( x)= ⎨⎩0; x < 0c 1G ( t− τ) = < Tη{ AtBs() s( τ)} > , (2.3)ih{ }; [ ]A () t B ( τ) = A () t B ( τ) − ηB ( τ) A () t ;s s η s s s sTη As() t Bs( τ) = θ ( t− τ) As() t Bs() τ + ηθ( τ −tB )s( τ ) As()t .η = 1 - boze operatorebisaTvis da η =−1Ffermi operatorebisaTvis, da< ... >= Z( β ) SP s( e−1 −βH∑...);β kT, (2.4)− 1= B−βHsadac: Z(β ) = SP s( e ) - warmoadgens statjams mTeli (s + S) sistemisaTvis; k B∑- aris bolcmanis mudmiva; T - absoluturi temperatura, xolo h –plankismudmiva; gasaSualoeba (2.4) formulaSi xorcieldeba gibsis kanonikuriansamblis wonasworuli statistikuri operatoris mixedviT.ganixileba Semdegi saxis zogadi amocana [109-113] – agebuli undaiqnas Teoria, romlis saSualebiTac SesaZlebeli iqneba miviRoT zusti,ganzogadoebuli (albaT araCaketili), kvanturi evoluciuri (kinetikuri)gantolebebi drois ormomentiani wonaworuli korelaciuri da grinisfunqciebisaTvisAsdaBs– operatorebisaTvis, s – dinamiuri qvesistemisaTvis,romelic urTierTqmedebs S sistemasTan – bozonur velTan. amTeoriis agebisas, gamoyenebuli unda iqnas mowesrigebul operatorTameTodi [5-6,59-61], (s + S) -sistemis (2.1) hamiltonianis cxadi saxe da gibsiskanonikuri ansamblis statistikuri operatori. gamoviyvanoT jerganzogadoebuli kvanturi, evoluciuri gantoleba wonasworulikorelaciuri funqciisaTvis:F () t =< B () t A >=< BA ( − t)>. (2.5)BA s ss s s sdasmuli amocanis gadasawyvetad CavweroT (2.5) korelaciuri funqciaSemdegi saxiT:−1 −βH−1< B () t A >= Z ( β) SΡ ⎡ () ⎤ = ( ) Ρ⎡ (, ) ⎤s s sΣ⎣e B t As s⎦ Z β S⎣AG t βs Bs⎦(2.6)(2.6) formulaSi, Cven SemoviReT damxmare operatori G (, t β ), romelicganisazRvreba tolobiT:−βHGs(, t β ) = SΡ ⎡ () ⎤BΣ ⎣e B ts ⎦. ( 2.7)B s83

Tu gavawarmoebT (2.5) – korelaciur funqcias t-droiTi cvladismixedviT, maSin Cven miviRebT gantolebas, romelic gansazRvravs misevolucias droSi:∂ iFBA( t)= < [ H , Bs( t)]s s∂th−⋅ As> , (2.8)sadac [...,...] – warmoadgens komutatoris agebuls ori operatorisagan.+ganvsazRvroT, agreTve evoluciis wrfivi operatorebi: W ( t,0),W( t ,0), W +( β,0), W ( β ,0)- Semdegi TanafardobebiT:i i i iHt Ht Ht Ht0 − −+0( ) ( )−βH0 βH−βH0( β,0 ) , ( β,0)h h h he = W t,0 e , e = e W t,0,, (2.9)− βH+ −e = W e e = e Wsadac: H0 = Hs+ H Σ.Tu SemoviRebT T-namravlTa cnebas [5-6], maSin es operatorebiformalurad SesaZlebelia warmovadginoT Semdegi saxiT:W+W t( t,0)( ,0)t⎧ ⎡i⎤⎪Τ exp ⎢%a ∫dτHint( τ) ⎥ t > 0⎪ ⎣h0 ⎦= ⎨0⎪ ⎡ i ⎤⎪ Τ exp ⎢ − %∫dτHint( τ) ⎥ t < 0⎩ ⎣ ht ⎦t⎧ ⎡ i ⎤⎪Τexp ⎢− ∫dτH%int( τ) ⎥ t > 0⎪ ⎣ h0 ⎦= ⎨0⎪ ⎡i⎤⎪ Τ exp ⎢%a ∫ dτHint( τ) ⎥ t < 0,⎩ ⎣ht ⎦(2.10)sadac:% i i0 − 0int( ) = h H τinth H τH τ e H e da:WW+β⎡⎤( β,0)=Τ exp ⎢− %a ∫ dλHint( λ) ⎥; β > 0⎣ 0 ⎦%−λH0 λH0Hint( λ)= e Hinte. (2.11)β⎡⎤β,0 =Τexp ⎢− %∫ dλHint( λ) ⎥; β > 0⎣ 0 ⎦H%( ) H e λ( )intλH0 − H0λ = eintam operatorebis sawyisi mniSvnelobebisaTvis gveqneba:( ) ( ) ( β ) ( β )W + t,0 = W t,0 = 1; W +,0 = W ,0 = 1.t= 0 t= 0 β = 0β = 0SemoRebul operatorebs gaaCniaT agreTve jgufuri Tvisebebi:W+−( t,0)= W (0. t)= W1 ( t,0);W ( t,τ ) W ( τ ,0) = W ( t,0)(2.12)84

+ −da W ( β,0) = W( 0, β) = W 1( β,0; ) W( βλ , ) W( λ,0 ) = W( β,0)nebismieri t,τ da β,λ ( β > 0, λ >0) mniSvnelobebisaTvis.(2.10) formulebSi T - qronologiurad mowesrigebis simboloa (operatorTamowesrigeba xdeba drois mixedviT marjvenidan marcxniv droismomentis zrdasTan erTad, rodesac t>0), xolo Τa- warmoadgensantiqronologiurad mowesrigebis simbolos drois mixedviT (operatorTamowesrigeba xdeba drois mixedviT marcxnidan marjvniv drois momentiszrdasTan erTad, rodesac t>0). analogiurad, (2.11) formulebSi T dawarmoadgens qronologiurad da antiqronologiurad mowesrigebissimboloebs operatorebisaTvis λ temperaturuli cvladis mixedviT. unda+aRiniSnos, rom W ( t,0), W( t ,0), W +( β,0)Τ a, W ( β ,0)operatorebisaTvis droiTida temperaturuli cvladebi H%( τ ) da H % ( λ)intoperatorebs aniWeben araintmarto raRac mniSvnelobebs, aramed gansazRvraven agreTve maTmowesrigebul Tanmimdevrobas T namravlebSi.Tu gamoviyenebT W( t ,0), W +( t,0)maSin G ( , )Bsda W ( β ,0)evoluciis operatorebs,t β operatori SesaZlebelia warmovadginoT Semdegi saxiT:−βH0( , β) ⎡ ( β,0 ) ( ,0) ( ) ( ,0)G ⎤Bst = SΡ Σ ⎣e W W t BSHt W t =s ⎦−βHs− β HΣ+= e SP ⎡Σ ( ,0) ( ,0) ( ) ( ,0)⎤⎣e W β W t BSHt W t ⎦=, (2.13)s− β Hs+( β) ( β) ⎡ ( β,0 ) ( ,0) ( ) ( ,0)= ZΣe ΡΣ⎣W W t BSHt W ts−1sadac: ( ) ( )P β ... = Z β S Ρ ⎡ −e βH Σ... ⎤Σ ΣΣ ⎣ ⎦⎤⎦- warmoadgens bozonuri velis mdgomareobebismixedviT gasaSualoebis operators (proeqciul operators);( β )ZΣ - bozonuri velis statjamia da B () t operatori ganisazRvrebaSemdegi formuliT:sH sHt s Ht sB () t = e h Be − h . (2.14)sHscxadia, rom adgili aqvs Tanafardobas:B () t = B () t , sadacsH0 sH sisii i() Ht 0 Ht 0BsHt = e h Be −0 sh .SemoviRoT axla specialuri operacia, romelic aRiniSneba simboloTiΤ (ix. mag. [5-6,109]).LRs , ,Fs() tLRs , ,Fs() tΤ operacia garkveuli wesiT awesrigebs operatorTanamravls, romelic Sedgeba erTi Fs()t - operatorisagan da mis85

gverdiT mdgomi Aj( tj, s ) da A ( β , s)operatorebisagan. tt 1, 2,.... t nwarmoadgenenkk(0,t)-droiTi intervalis sxvadasxva wertilebs. ( j = 1,2,... n), da ββ 1, 2,... βmarian(0, β ) – temperaturuli intervalis sxvadasxva wertilebi ( k = 1,2,... m).vTqvaT simbolo aRniSnavs erTi Fs()t operatoris namravls sxvaA ( t, s ) ( j = 1,2,... n) da A ( β , s)( k = 1,2,... m) operatorebze, romlebicjjkaRniSnuli (moniSnuli) arian indeqsebiT: L (left) da R (right).kaRniSnuli operatorebis Tanmimdevruli ganlageba -Si nebismieria.operaciaLRs , ,Fs() ttj( ot , )∈ ; j 1,2,..., n∈ ; β ( β)k∈ o, ; k ∈ 1,2,... m.LΤ ganisazRvreba Semdegnairad: yvela A ( , sj j )β –operatoris movaTavsebT Fs()t – operatoris marcxniv da ganvalagebT anti-Lqronologiurad droiTi momentebis mixedviT; yvela Ak ( βk,s)operatorsLmovaTavsebT ( ,j j )A t s operatorTa marcxniv da ganvalagebT qronologiuriwesis mixedviT da bolos yvela R'( , j j' )A t s - ( j ' = 1,2,... ) operators movaTavsebTmarjvniv Fs()t operatoridan da ganvalagebT qronologiurad droiTimomentebis mixedviT. am proceduris Sesrulebis Semdeg CamovacilebT L daR indeqsebs.T Fsamrigad, am gansazRvris Tanaxmad Cven gveqneba= ⎡⎡ ⎤ ⎡ ⎤L ⎤( ⎥ ×L⎢⎢ ⎥ × ⎢ ∏RT ∏ Ak( β k , s)Ta∏ Aj( t j , s)Fs( t)T Aj′′( t j′, s)⎥ (2.15)⎣ k ⎦ ⎣ j ⎦ ⎣ j′⎦L,R,st)cxadia, romΤ operaciis niSnis qveS, Semavali yvela operatoriLRs , ,Fs() tSegviZlia gadavanacvloT nebismierad ise, rom ar SevcvaloT Sedegi –TiTqos es operatorebi warmoadgendnen C sidideebs – vinaidan TviT esoperacia avtomaturad amyarebs wesrigs sabolood operatorTa ganlagebaSi.Cvens mier SemoRebuli operacia bunebrivad SegviZlia agreTveganvazogadod operatorebisagan Sedgenil transcendentulfunqcionalebzec. unda aRiniSnos, rom Cvens mier SemoRebuli specialurioperaciaT FsF(warmoadgens adre SemoRebuli Τs ( L ) s ( R )operaciis bunebrivL,R,st)ganzogadoebas droiT da temperaturul intervalebze [5-6,109,111-113].86

sadac:axla ganvixiloT (2.13) formula da CavweroT is Semdegi saxiT:( ) ( )G t, = Z e G%t, : t > 0Bs−βHs( β) ( β) ( β)+G t, β =Ρ β ⎡ ( ,0) (,0) () (,0) ⎤=BsΣ ⎣W β W t B tWtsHs⎦βt⎧⎪⎡ ⎤ ⎡ ⎤=Σ ( ) ⎨Τexp − % iP β ⎢ ∫dλHint( λ) ⎥⋅Τ exp %a ⎢ ∫dτHint() τ ⎥⋅ BsH() t ×s⎪⎩⎣ 0 ⎦ ⎣h0 ⎦t⎧β⎡ i ⎤⎪⎫ , , ⎪ ⎡ ⎡ ⎤×Τexp − % LRSint() ⎬=Τ () ⎨ΡΣ( ) ⎢ΤΣexp − % L⎢ ∫dτH τ ⎥ Bt β ⎢ int( ) ⎥×sHs⎣ h0 ⎦⎪⎭⎪⎩⎢∫dλHλ⎣ ⎣ 0 ⎦×ΤaΣtt⎡i⎤⎫exp % L⎤ ⎡ i ⎤ ⎪int( ) ⋅ () ΤΣexp − % R⎢ ∫dτH τ ⎥ BsHt ⎢ ∫dτHint( τ) ⎥⎥⎬; t>0s⎣h0 ⎦ ⎣ h0 ⎦ ⎥ ⎦⎭⎪ΣBs(2.16)(2.16) formulaSi H%( λ)da H % ( τ )intoperatorebi gansazRvrulia (2.10-int2.11) tolobebiT: Τ Σda ΤaΣsimboloebi awesrigeben mxolod im operatorebs,romlebic moqmedeben bozonur (fononur) velis cvladebze. TuLgaviTvaliswinebT (2.10-2.11) tolobebs H% int( λ)dagveqneba:∑H%( λ) = ⎡C% (, s λ) b ( λ) + C%(, s λ) b ( λ)⎤⎦L L + L+int ⎣ kH0 kH0 kH0 kH0k∑H%() τ = ⎡C% ( s, τ) b ( τ) + C%( s, τ) b ( τ)⎤⎦LR , LR , + LR ,+int ⎣ kH0 kH 0 kH 0 kH 0kH% operatorebisaTvisLR ,int( τ )(2.17)sadac:b e be e bλH0 −λH0−λhω( k )kH( λ) = ;0k=kb e be e b+ λH0 + −λH0λhω( k ) +kH( λ) = ;0k=ki iH0τH0τ−iω( k)τ() τ =h h= ;kH0k kb e be e biiH0τ− H0τ+ h + h iω( k)τ +kH() τ = .0k=kb e be e b(2.18)i i i iH0τ − H0τ Hsτ − Hsτh h h hkH( , τ) = ( , ) () ()0kHτ =sk=kC% s C%s e C se e C sei i i iH0τ − H0τ Hsτ − Hsτ+ + h + h h + hkH( , τ) = ( , ) () () .0kHτ =sk=kC% s C%s e C se e C segamosaxuleba, romelic mocemulia (2.16) formuliT P ( β )gasaSualoebis simbolos qveS G ( t,β )% operatorisaTvis, gamoiTvlebaBsdanarTSi miRebul formulis daxmarebiT (ix. danarTi, (1.4) formula). amyvelafris gaTvaliswinebiT gveqneba:Σ87

sadac ( t β )LRS , ,,{ , ,⎤}LRs , ,( ) = ⎡Φ( )G%Bt, β T () ()exp ⎣ , ⎦ , t>0,s BHt Bs s sHtsLRSt β (2.19)Φ funqcionali ganisazRvreba Semdegi gamosaxulebiT:t τ1iω()( k τ−ξ), , ( , )2 ∫ ∫ ∑⎡⎣( 1 ( ))% LLRSt β dτ dξ Nk β e CkH(,) sξ0h0 0 kΦ =− + ×t+ − ()( − ) + + 1× % L iωk τ ξ(,) + ( ) % L(, )%LC (,) ⎤kHsτ N0 kβ e CkH sξ C − ×0 kHsτ 0 ⎦ 2 ∫dτh0τ−iω()( k τ− ξ)+× ∫ ∑⎡⎣( 1 + ( ))% R(,)%Rdξ Nk β e CkH sτ C (, ) + ( ) ×0 kHsξ N0kβ0 k× eiω()( k τ−ξ)t% + R1CkH(,)%Rsτ C (,) ⎤+ ( 1 + ( ))×0 kHsξ 0 ⎦ 2 ∫dτ∫dξ∑⎡ h ⎣ Nkβτ0 0k× e C% ( s, τ) C% (, s ξ) + N ( β) e C%( s, τ)×−iω( k)( τ− ξ) L + R iω( k)( τ− ξ)+ LkH0 kH0 k kH0t βR iω( k)( iτ−Dλ)L⎤kH⎡( )0 ⎦ ∫ ∫ ∑ ⎣ k kH0h0 0 k× C% (, s ξ) − dτ dλ 1 + N ( β) e C%(, s λ)×t+ − ( )( − ) +× % L ω k iτ Dλ( , ) + ( ) % L(, ) % L iC ( , ) ⎤kHsτ N + ×0 kβ e CkH s λ C0 kHsτ 0 ⎦ h ∫ dτβω( k)( iτ− Dλ)+× d ⎡ 1+N ( ) % L⎣(, ) % Rkβ e CkH s λ C ( , ) + ( ) ×0ksτ Nkβ∫0λ ( )β γ−ω( k)( iτ− Dλ)% + L % R⎤kH0 kH0⎦ ∫ ∫ ∑ ⎡⎣ k0 0 k−hω( k)( γ− λ) L + L hω( k)( γ− λ)+ LkH0 kH0 k kH 0LkH00( )× e C (, s λ) C ( s, τ) + dγ dλ 1 + N ( β)×× e C% (, s γ) C% (, s λ) + N ( β) e C%(, s γ)×× C%∑k(, s λ ) ⎤⎦; t > 0.+aq:k( β) ( β)( k k)h ( k )−1Σ⎡ 1⎤N = P bb =⎣e β ω −⎦(2.20)warmoadgens bozonebis (fononebis)Sevsebis ricxvebis saSualo mniSvnelobas k mdgomareobaSi; (2.18)formulebis msgavsad C% (, )kHs λ da C % +(, )0 kHs λ – operatorebisaTvis gvaqvs0Semdegi gamosaxulebebi:C% s C% s e C se e C se ;( , ) (, )λH0 () −λH0λHs()−λHskHλ =0 kHλ =sk=kC% s C% s e C se e C se .+ ( , ) + (, ) λH0 + () −λH0λHs+()−λHskHλ =0 kHλ =sk=kTu gavawarmoebT (2.19) gamosaxulebas t droiTi cvladis mixedviT dagamoviyenebT (2.16) formulas, maSin martivad vipoviT G%(, t β)operatorisaTvis evoluciur gantolebas (moZraobis gantolebas):B s88

da{ ⎡ ⎤ ⎡ β ⎤}∂i LRS , ,G%B(, t β) = T(), ()exp, ,(, )s BH s s tHs BsH t ΦsLRSt +∂th ⎣ ⎦ ⎣ ⎦LRS , , ⎧ ∂⎫+ TBH ()(), ,(, )exp, ,(, ) ; 0s s t ⎨BsH t ΦsLRSt β ⋅ ⎡LRSt tt⎣Φ β ⎤⎦⎬>⎩ ∂⎭∂iG%+B(, t β) = Ρ ( β) ⎡W( β,0 ) W ( t,0 ) ⎡H , () ( ,0)sΣsBsHt ⎤W t ⎤+s∂th ⎣⎣ ⎦ ⎦i++ ΡΣ( β) ⎡W( β,0 ) W ( t,0 ) ⎡H%int()t , BsH () ( ,0); 0hst ⎤ W t ⎤⎣⎣⎦ ⎦t >(2.21)xolo TviT Φ, ,(, t β ) funqcionalis warmoebulisaTvis miviRebTLRSSemdeg gamosaxulebas:∂1Φ (, t β) = dξ 1 + N ( β) e C ( s, ξ)×∂ttiω( k )( t−ξ), , {( ) ⎡2 ∫ ∑% LLRS k ⎣ kH0h0 k+ + − ( )( − ) +× % R(,) − % L(, ) % L iωk t ξ(,) ⎤+ ( ) ⎡ % LCkH st C ( , ) ×0 kHs ξ C0 kHst0 ⎦ Nk β e ⎣CkHs ξ0t1} 2 ∑{( )× C% (,) st − C% (, s ξ) C%(,) st ⎤ + dξ 1 + N ( β)×× eR + L LkH0 kH0 kH0⎦kh0 k−iω( k )( t−ξ){( )% L + +⎡ (,) % R(, ) − % R(,) % R⎣C ( , ) ⎤kHstC + ( ) ×0 kHs ξ C0 kHstC0 kHs ξ0 ⎦ Nkββiω( k )( t− ξ ) + +× ⎡ % L(,) % R( , ) − % R⎣(,) % R ie C ( , ) ⎤kHstC + ×0 kHs ξ C0 kHstC0 kHs ξ0 ⎦ h∫dλ∑ω( k)( it− hλ)+ +× 1 + ( ) ⎡ % L(, ) % R⎣(,) − % L( , ) % LN (,) ⎤kβ e CkH s λ C0 kHst C0 kHs λ C0 kHst0 ⎦+k− ( k)( it− h ) + ++ ( ) ⎡ % L(, )% R(,) − % L⎣(, ) % LN (,) ⎤ke ω λβ CkH s λ C ; > 00 kHst C0 kHsλC0 kHst0 ⎦t∫}0}(2.22)Tu visargeblebT (2.16, 2.22) formulebiT, da agreTve W( t ,0), W +( t,0)W ( β ,0)evoluciis wrfivi operatorebis (2.9) gansazRvrebiT da (2.12)jgufuri TvisebebiT, maSin ( , )G t β operatorisaTvis evoluciurigantoleba SesaZlebelia warmovadginoT Semdegi saxiT [109]:B s,89

ii t∂i ⎧Ht− β H⎫ 1G (, t β ) = SP [ , ]2{( 1 ( )Σ ⎨e ehH B eh⎬ − + ) ×∂∫ dξ∑N βBss s kth ⎩ ⎭ h0 kiiiω ( k ) ξ⎡− βH Ht+− Ht ⎤hhiω ( k ) ξ× e SPΣ⎢e e Ck ( s,− ξ ) Ck ( s) e Bs( t) ⎥+ Nk ( β)e ×⎣⎦iit⎡Ht− Ht ⎫− β H +⎤⎪ 1hh× SPΣ⎢e e Ck ( s, −ξ ) Ck( s) e Bs( t) ⎥⎬−2 {( 1 + ( ))×⎣⎦⎭⎪∫ dξ∑Nkβh0 k−iω ( k)ξ⎡−βH× e SPΣ⎢e Bs( t)e⎣iHth⎤C s C s e ⎥ N e⎦i+− Hthiω( k ) ξk ( ) k ( , − ξ ) +k ( β ) × (2.23)iit⎡Ht− Ht ⎫− β H1h +h⎤⎪× SPΣ⎢e Bs ( t) e Ck ( s) Ck ( s, − ξ ) e ⎥⎬+2 {( 1 + ( ))×⎣⎦⎭⎪∫ dξ∑Nkβh0 kii−iω ( k) ξ⎡− β HHt+− Ht ⎤hhiω( k)ξ× e SPΣ⎢e Ck ( s, t) Bs ( t) e Ck ( s,− ξ) e ⎥ + Nk ( β ) e ×⎣⎦⎡− β H +× SPΣ⎢e Ck ( s,t) Bs( t)e⎣iHthit− Ht ⎫ 1h⎤⎪Ck ( s, − ξ) e ⎥⎬+2 {( 1 + ( ))×⎦⎭⎪∫ dξ∑Nkβh0 kiii ( k) ⎡− H− Ht− Ht+⎤hh−i ( k)⋅e ω ξ SPΣ⎢e β e Ck ( s1− ξ) e Bs( t) Ck ( st , ) ⎥ + Nk ( β)eω ξ×⎣⎦iiβ⎡ − Ht− Ht⎫− β H +⎤⎪ 1× SP ( , − ) ( ) ( , ) − {( 1 + ( )Σ ⎢e ehC s ξ ehB t C st ⎥⎬) ×k s k⎣⎦⎪∫dλ∑N βk⎭ h0 k( , λ) ( , ) ( ) ⎤ ( β)− ω( k)( it+ hλ) + − βH ω( k)( it+hλ)× e SPΣ⎡⎣Ck s − e Ck stBs t⎦+ Nke ×β− βH + i− ω( k)( it+hλ)× SP ⎡Σ ⎣C ( , − ) ( , ) ( ) ⎤ks λ e Ck stBs t ⎦} + ∫dλ∑{( 1 + Nk( β)) e ×h0 kβ Hω( k)( it+hλ)( , λ ) ( ) ( , t) ⎦ Nk( β) e SPΣ⎡⎣ Ck( s,λ)( ) ( , ) ⎤k }; 0+ −× SP ⎡Σ ⎣Ck s − e Bs t Cks ⎤+ − ×− β H+× e Bst C st t >⎦++(2.23) gantolebaSi Semavali: C ( s,±ξ ) , C ( s,± ξ ) , C ( s,−λ ) da C ( s,−λ)operatorebi ganisazRvrebian Semdegi tolobebiT:± i Hξ ± i Hξ ±i Hξ mi Hξh h h hk k k k( ξ) ( ξ)( λ) ( λ)+ +C s, ± = e C () se ; C s, ± = e C () se− λH λH + − λH + λHC s, − = e C () se ; C s, − = e C () se .k k k kamrigad, rogorc vxedavT G ( , )B skkkk(2.24)t β operatorisTvis (2.23) evoluciurgantolebaSi bozonuri amplitudebi ar gvaqvs. bozonuri amplitudebisgamoricxva moxerxdagasaSualoebis ( β )TL,R,SB H t)ss(specialuri operaciisa da bozonuri veliTΡ operatoris gamoyenebis Sedegad (naTelia, romΣbozonuri cvladebi (amplitudebi) gveqnebai Hte ± hevoluciis operatorebSi90

daB se −βH( , )gibsis faqtorSi). Semdeg paragrafSi (2.23) evoluciuri gantolebaG t β operatorisaTvis gamoyenebuli iqneba zusti, ganzogadoebulikvanturi evoluciuri (kinetikuri) gantolebebis misaRebad F ( t )korelaciuri funqciisTvis da grinis funqciebisTvis [110].BA s s2.2. zusti ganzogadoebuli kvanturi kinetikuri gantolebebi wonasworulikorelaciuri funqciebisa da grinis funqciebisTvisgamoviyvanoT axla zusti ganzogadoebuli kvanturi kinetikurigantolebebi korelaciuri funqciebisTvis. amisTvis ganvixiloT (2.6)formula F () t korelaciuri funqciisTvis da (2.5)-is daxmarebiT CavweroTBA s sis Semdegi saxiT:FBsAs= Zsadac: G ( t,β )A sgamosaxuleba:( t)=−1( β ) SPB ( t)Ass[ B G ( −t,β)]ssAs=B A ( −t)ss= Z−1( β)SPs[ A G ( t,β)]sBs=(2.25)− operatorisTvis gansazRvris Tanaxmad gvaqvs Semdegi( − , ) = ( − )⎡− HG t SP A t e βAsΣ sβ⎣⎤ ⎦. (2.26)Tu gavawarmoebT (2.25) Tanafardobas t-droiTi cvladis mixedviT,maSin miviRebT:∂F∂t= ZBsAs∂( t)=∂tB A ( −t)1 ⎡ ∂( β ) SPs⎢BsG⎣ ∂t−ssAs= Z⎤( −t,β ) ⎥⎦−1⎡ ∂( β ) SPs⎢AsG⎣ ∂tTu CavsvamT (2.27) gantolebaSi G ( t,β )(2.23) formulidan da visargeblebT ( )∂∂tB sBs⎤( t,β )⎥=⎦(2.27)operatoris gamosaxulebasSPs, Σ... operaciis qveS operatorTacikliuri gadanacvlebis SesaZleblobiT, maSin martivi gamoTvlebis SemdegvipoviT saZiebel zust, ganzogadoebul kvantur kinetikur gantolebasBAss( t)− korelaciuri funqciisTvis [110-111,113]:91

∂∂tti1BA − t = HB ⋅A −t − d + N ⋅( ) [ ] ( ) ( 12( )− ∑ ∫ ξ ⎡⎣β )s s s s s khh k 0( , ξ) ⎡ ( ),⎤ ( ) ( β)iω( k) ξ + −iω( k)ξ⋅e Ck s − ⋅⎣Ck s Bs⎦⋅A − + ⋅−st Nke1C s − ⋅ C s B ⋅ A − t + d + N ⋅t+k ⎣ k s⎦−s⎦ 2 ∑h∫ ⎣ kk 0( , ξ) ⎡ ( ), ⎤ ( ) ⎤ ξ ⎡( 1 ( β))( ),⎤ ( ξ) ( ) ( β)⋅e ⋅ ⎡C s B ⋅C s − ⋅ A − t + N e ⋅− iω( k) ξ +iω( k)ξ⎣ k s⎦k 1s k−( ), ( , ξ ) ( t) ⎤ dθ⎡( 1 N ( β ))+⎡⎣C s B ⎤⎦⋅C s − ⋅ A −k s−k s( θ ) ⎡ ( ) ⎤ ( )− θω( k) iω( k)t+⋅ − − ⋅⎣ ⎦⋅ −k k s s−hβi− + ⋅⎦2 ∑h∫ ⎣ ke e C s, t i C s , B A t +θω( k) − iω( k)t +( β) ( θ) ⎡ ( ) ⎤ ( )+ N , , ⎤ke e Ck s −t −i ⋅⎣Ck s Bs⎦ ⋅A − ; > 0.−st t⎦k0(2.28)analogiurad SesaZlebelia miviRoT zusti, ganzogadoebuli kvanturikinetikuri gantoleba ( ) ( ) ( )AB s ss s s sF − t = AB t = A − t B korelaciuri funqciisTvis.am gantolebis gamoyvana principulad ar gansxvavdebazemoTmoyvanili gamoTvlebisagan da amitom moviyvanT mxolod sabolooSedegs:∂∂tti1A − t B = A −t ⋅ HB − d + N ⋅( ) ( ) [ ] ( 12( )− ∑∫ ξ ⎡⎣β )s s s s s khh k 0( ) ( , ξ) ⎡ ( ),⎤ ( β)iω( k) ξ + −iω( k)ξ⋅e As −t ⋅Ck s − ⋅ ⎣Ck s Bs⎦+ N ⋅−ke1A −t ⋅C s − ⋅ C s B + d + N ⋅∑∫( )+( ) ( , ξ) ⎡ ( ), ⎤ ⎤ ξ ⎡ 1 ( β)s k ⎣ k s⎦−⎦ 2h ⎣ kk 0( ) ⎡ ( ), ⎤ ( , ξ) ( β)− iω( k) ξ +iω( k)ξ⋅e ⋅ As −t ⋅⎣Ck s Bs⎦⋅Ck s − + Nke ⋅−+A ( ) ⎡ ( ), ⎤s−t ⋅⎣Ck s Bs⎦⋅C , −−k( s ) ⎤ iξ − dθ ⎡2 ⎣( 1+ Nk( β)) ⋅⎦ h+( ) ⎡ ( ), ⎤ ( , θ )θω( k) iω( k)t⋅e e As −t ⋅ Ck s Bs Cks − t+ i +⎣θω( k) iω( k)t+( β) ( ) ⎡ ( ) ⎤ ( θ)⎦+ N , , ⎤ke e As −t ⋅⎣Ck s Bs⎦ C − + ; > 0.−ks t i t⎦t∑ ∫ hkβ0(2.29)martivi SesamCnevia, rom (2.28) da (2.29) evoluciuri gantolebebi korelaciurifunqciebisTvis arian araCaketili da markoviseuli. t0 da t

zemoTwarmodgenili gamoyvanisagan da amitomac aq ar moiyvaneba. aRvniSnavTmxolod imas, rom am gantolebaTa miRebisas saWiroaΤ operaciisLRS , ,BsAs()t+xelaxali gansazRvra Wt (,0) da W (,0) toperatorebis (2.10) warmodgenisLRS , ,Sesabamisad. ( Τ operaciis xelaxali gansazRvra aucilebelia agreTveFAB s sBsAs()t( − t)korelaciuri funqciisTvis zusti, ganzogadoebuli kinetikuri gantolebismisaRebadac, rodesac t>0). naTelia, rom ganzogadoebuli kvanturikinetikuri gantolebebi BA () t da AtB () - korelaciuri funqciebisTvisssmiiReba (2.28) da (2.29) gantolebebidan t droiTi cvladis formaluri cvlilebiT:t → –t. kinetikuri gantolebebi: BA () t da AtB () korelaciurifunqciebisTvis SesaZlebelia warmovadginoT agreTve aramarkoviseulisaxiT. Tu SevasrulebT (2.28) da (2.29) gantolebebSi t sididis cvlilebas: tHξ− Hξ→ –t, visargeblebT operatoruli tolobiT: As() t = e As( t−ξ) e ; ( ξ ∈[ 0, t])ssssssiih h,da SPs∑(...) operaciis qveS operatorTa cikliuri gadanacvlebis SesaZleblobiT,maSin miviRebT Semdeg zust, ganzogadoebul kvantur kinetikurgantolebebs BA () t da AtB () korelaciuri funqciebisTvis [110]:ssss∂∂tti1BA ( t) =− [ HB ] A ( t) − ⎡2 ∑∫ dξ⎣( 1 + N ( β)) ×hhs s s s s kk 0( ) ⎡ ( , ξ) , ( ξ) ⎤ ( ξ) ( β)− iω( k) ξ +iω( k)ξ× e Ck s ⋅⎣Ck s − Bs − ⎦ A − + ×−st Nke1× C s C s − B − A t− + d + N ×t+k ⎣ k s ⎦−s 2h∑∫ ⎣ kk 0( ) ⎡ ( , ξ) , ( ξ) ⎤ ( ξ) ξ ⎡( 1 ( β))( , ξ) , ( ξ) ⎤ ( ) ( ξ) ( β)iω( k) ξ + −iω( k)ξ× e ⋅ ⎡⎣Ck s − Bs − ⎦ C − + ×− ks As t Nke( , ξ) , ( ξ ) ⎤ ( ) ( t ξ) ⎤ dθ ⎡( 1 N ( β))+× ⎡⎣− − ⋅k s ⎦−kC s B C s A( , θ) ⎡ ( ),⎤ ( ) ( β)−θω( k) − iω( k) t +θω( k)× e e Ck st−i ⋅ ⎣Ck s Bs⎦A + ×−st Nke( θ ) ⎡ ( ) ⎤ ( )e C st , i , ⎤⎣C s B ⎦ A t ;⎦iω( k)t +× − ⋅ ⋅k k s − sshβi− + + ×⎦2 ∑h∫ ⎣ kk0(2.30)93

∂∂tti1A ( t) B =− A ( t) ⋅[ HB ] − ⎡2 ( 1+ ( )− ∑∫ dξ⎣N β ) ×hhs s s s s kk 0( ξ) ( ) ⎡ ( , ξ) , ( ξ) ⎤ ( β)− iω( k) ξ +iω( k)ξ× e As t− ⋅Ck s ⎣Ck s − Bs − ⎦ + N ×−ket1× A t− ⋅C s C s − B − + d + N ×+( ξ) ( ) ⎡ ( , ξ) , ( ξ) ⎤ξ ⎡( 1 ( β))s k ⎣ k s ⎦−2h ⎣ kk 0( ξ) ⎡ ( , ξ) , ( ξ) ⎤ ( ) ( β)iω( k) ξ + −iω( k)ξ× e ⋅ As t− ⎣Ck s − Bs − ⎦ C + ×−ks Nke+( ξ) ⎡ ( , ξ) , ( ξ ) C ( ) ⎤⎡ks dθ( 1 Nk( β))× As t− ⎣Ck s − Bs−∑∫i⎤⎦ ⋅ + + ×− ⎦2 ∑h ∫ ⎣( ) ⎡ ( ), ⎤ ( , θ) ( β)−θω( k) − iω( k) t +θω( k)× e e As t ⎣Ck s Bs⎦⋅ C + + ×− kst i Nkekhβ0(2.31)( ) ⎡ ( ) ⎤ ( θ )e A t , , ⎤⎣C s B ⎦ C st i ;⎦− iω( kt )+×s⋅k s⋅ +−k(2.28) da (2.29) gantolebebisagan gansxvavebiT, (2.30) da (2.31) evoluciuri(kinetikuri) gantolebebi arian araCaketili da aramarkoviseuliformis. (2.3) gamosaxulebebi da (2.28)-(2.29) evoluciuri gantolebebi gansazRvravengantolebebs grinis funqciebisTvis. ase magaliTad, gantolebasrdagvianebuli grinis funqciisTvis: ( ) ( ) ( ),∂ rG ( t)= δ ( t)∂tG t = θ t ⎡⎣Ast Bs⎤⎦ηaqvs Semdegi saxe:∂[ As,Bs] η+ θ ( t)[ As( t),Bs] η∂t. (2.32)amdagvarad SesaZlebelia ganisazRvros agreTve evoluciurigantolebebi sxva danarCeni grinis funqciebisTvis (winmswrebisa damizezobrivisTvis). (2.30)–(2.31) formulebis daxmarebiT evoluciurir(kinetikuri) gantoleba G ( t ) – grinis dagvianebuli funqciisTvisSesaZlebelia warmovadginoT Semdegi saxiT [110-111]:94

∂ riG ( t) = δ ( t) [ A , ] − ( ) ⎡⎣ ( ),[ , ] ⎤sBs θ t As t Bs Hs+η−∂th⎦η−iω( k)ξ( ) ( ){( 1k ( ))s( ) k ( )+∞1+ − + − ×2 ∑ ∫ dξθ ξ θ t ξ N β eh⎡⎣ A t ξ C sk−∞+ iω( k ) ξ+( ξ) ( , ξ ) ⎤ ( β) ( ξ) ( )× ⎡⎣B − C s − ⎤⎦ + ⎡⎦N e ⎣A t − C s ×s k−k s kη+∞1× ⎡⎣B ( −ξ) , C ( s, −ξ) ⎤⎦ ⎤⎦ } − ∑ ∫ dξθ ( ξ ) θ ( t − ξ ){( 1 + N ( β)) × (2.33)s k−2kη h k −∞e ⎡⎣A t ⎣Biω( k ) ξ×s−+ +( ξ ) ⎡ ( ξ ), C ( s,ξ) ⎤ C ( s) ⎤ N ( β )shβ− iω( k)ξ+× ⎡ ( − ) ⎡ ( − ), ( , − ) ⎤ ( ) ⎤ie⎣As t ξ ⎣Bs ξ Ck s ξ ⎦ C − ×−ks⎦2 ∑h ∫ dθηk 0− θω ( k) iω( k ) t+{( 1 Nk ( β ))e θ ( t){ e As( t) ⎡Bs, Ck ( s) ⎤ Ck( st , iθ)θω( , ) ⎡ , ( ) ⎤ ( ) } ( ) ( )− iω( k) t +( k )−ηe Ckst − θ Bs Ck s A + ⋅ ×−st Nkβ e θ t− iω( k ) t+{ ( ) ⎡⎣, ( )⎣Bs, Ck ( s) ⎦ A ( ) }}.−sts s k−k− − ⎦ ⎦+ ×k−k kη× + ⎣ ⎦ + −×⎣⎦( ) ( )iω( k ) t +e A t B C s ⎤⎦C s t + iθ −ηe C st , − iθ×1 k× ⎡ ⎤miRebuli (2.28–2.31) da (2.33) kvanturi evoluciuri (kinetikuri) gantolebebiwarmoadgenen zust, ganzogadoebul gantolebebs drois ormomentiani wonasworuli korelaciuri funqciebisa da grinis funqciisTvis[25,34,64]. gansxvavebiT kinetikuri gantolebebisagan, romlebic miRebulia [5-6,75-77] SromebSi, gamoyvanili evoluciuri gantolebebis dajaxebiTiintegralebi Seicaven rogorc maRali rigis korelaciur da grinisfunqciebs, aramed aseve sawyisi korelaciebis evoluciis amsaxvel wevrebs(ukanaskneli wevrebi (2.28–2.31 da 2.33) gantolebebis marjvena mxareSi). SevniSnavTagreTve, rom (2.28–2.31) gantolebebis marjvena mxareebis pirveliwevrebi aRweren korelaciuri funqciebis Tavisufal (aradajaxebiT)dinamikas.−2.3. markoviseuli miaxloeba qvesistemis dinamikisTvis.miaxloebiTi gantolebebi korelaciuri funqciebisTvisdavuSvaT, rom (2.2) urTierTqmedebis hamiltoniani s qvesistemasa da S+bozonur Termostats Soris Seicavs mcire parametrs ( Ck( s ) da Ck( s)urTierTqmedebis operatorebi Seicaven mcire parametrs). aseT SemTxvevaSi,95

s qvesistemasa da S bozonur vels Soris susti urTierTqmedebis gamo,adgili eqneba droTa ierarqias:τree( t t )>> t0 = maxs,Σ, (2.34)sadac:τ ree- warmoadgens s qvesistemis relaqsaciis maxasiaTebel dros,xolot ⋅ hβaris s qvesistemis dajaxebiTi (gabnevis) dro;s1tΣ⋅ω ~- ariskorelaciebis fluqtuaciebis milevis dro TermostatSi (ω% - bozonebisrxevebis maxasiaTebeli sixSirea). (2.34) utoloba saSualebas iZlevaSemovifargloT SeSfoTebis Teoriis meore miaxloebiT ( Hint- ur-TierTqmedebis hamiltonianis mixedviT) (2.28–2.31) evoluciur gantolebebSikorelaciuri funqciebisTvis. ase magaliTad, Tu SevasrulebT (2.28)evoluciur gantolebaSi Semdegi saxis miaxloebebs:+ +( , −ξ) ⇒ ( , −ξ) ; ( , −ξ) ⇒ ( , −ξ);0 0+ +( − − θ) ⇒ ( − − θ) ( − − θ) ⇒ ( − − θ)C s C s C s C sk kH k kHC s, t i C s, t i ; C s, t i C s, t i .k kH0 k kH 0(2.35)maSin miviRebT, drois ormomentiani wonasworuli korelaciuri funqciisTvis:( − )BA t _ ganzogadoebul, markoviseul kvantur kinetikurssgantolebas SeSfoTebis Teoriis meore miaxloebaSi [110-111]:∂∂tti1BA ( − t) = [ HB ] ⋅A ( −t) − ⎡2( 1 + ( ))×−hh∑∫ dξ⎣N βs s s s s kk 0( , ξ) ⎡ ( ),⎤ ( ) ( β)× e C s − ⋅ C s B A − t + N e ×iω( k) ξ + −iω( k)ξkH0⎣ k s⎦−s k1× C s − ⋅ C s B A − t + d + N ×t+kH01 ⎣ k s⎦−s⎦2h∑∫ ⎣ kk 0( ξ) ⎡ ( ), ⎤ ( ) ⎤ ξ ⎡( 1 ( β))( ), ⎤ ( , ξ) ( ) ( β)× e ⋅ ⎡C s B C s − A − t + N e ×− iω( k) ξ +iω( k)ξ⎣ k s⎦−kH0s khβ+× ⎡⎣C ( s), B ⎤⎦⋅C s,k s−kH ( ) ( ) ⎤ i−ξ A −t − ⎡( 1 + ( ))×0s⎦2 ∑h∫ dθ ⎣N βk( , θ) ⎡ ( ),⎤ ( ) ( β)× e e C s −t−i C s B ⋅ A − t + N e ×− θω( k) iω( k) t +θω( k)kH0⎣ k s⎦−s k( θ ) ⎡ ( ) ⎤ ( )− iω( k)t +× e C s, −− t i ⋅ , ⋅ − ⎤.kH0⎣C s Bk s⎦ A t− s⎦k0(2.36)analogiurad miiReba evoluciuri gantolebebi sxva korelaciurifunqciebisa da grinis funqciisTvis (ix. (2.29), (2.30–2.31) da (2.33) formulebi).rogorc cnobilia, xelsayrel da efeqtur meTods sustadarawonasworulimdgomareobebisa, relaqsaciuri procesebisa da kinetikuri movlenebisSeswavlisa – romlebic mimdinareoben mcire dinamiur qvesistemaSi,96

omelic urTierTqmedebs TermostatTan da gareSe velTan, - warmoadgenskorelaciuri funqciebisa da grinis funqciis meTodi.drois ormomentiani wonasworuli korelaciuri funqciebi da grinisfunqciebi warmoadgenen ZiriTad sidideebs kubos wrfivi gamoZaxilis(reaqciis) TeoriaSi [4,34,64,72-73]. (2.28–2.29) gantolebebi gansxvavdebianmsgavsi gantolebebisagan, romlebic gamoyvanili iyvnen SemTxveviTi fazebismiaxloebaSi [5-6,68,75-77], ukanaskneli wevrebiT gantolebebis marjvenamxareebSi, romlebic aRweren sawyisi korelaciebis evolucias dagavlenas dajaxebiT procesebze gabnevis (dajaxebiT) integralebSi (ix.agreTve (2.36) gantoleba). unda aRiniSnos agreTve is garemoeba, rom sawyisikorelaciebis evolucia droSi moicema markoviseuli formiT ((2.28–2.29),(2.30–2.31)) kinetikur gantolebebSi korelaciuri funqciebisTvis, da isiniar Seicaven maxsovrobis efeqtebs. Sfm-Si, rodesac t=0 drois sawyismomentSi mTeli ( s +Σ ) sistemis wonasworuli statistikuri operatori(gibsis kanonikuri ganawileba) aiReba faqtorizebuli saxiT:−1 −βH−1( ) ( ) ( ) Σ ( ) : ( ) ( )−1−βHΣρ ( β) = Z ( β)⋅e.ρ β = Z β ⋅e ⇒ ρ β ρ β ρ β = Z β ⋅eeq s s s− Hsadac: Z ( ) S ⎡e β s− Σβ ⎤ da Z ( ) S e βHs= Ρs⎣ ⎦ΣΣ−βH s(2.37)β = Ρ ⎡ ⎤Σ Σ ⎣ ⎦– warmoadgenen statjamebs s-qvesistemisa da Σ bozonuri velisa, Sesabamisad; (anu ugulvebelvyofTurTierTqmedebis hamiltoniansHinte −βHgibsis faqtorSi da Z ( β ) statjamSi,magram vinarCunebTi Hte ± h +- operatorebSi, ris gamoc W( β ) W ( β ),0 = ,0 = 1 (ix.(2.9); (2.11) formulebi)); _ sawyisi korelaciebis evoluciis wevrebi (2.28)–(2.29) da (2.30)–(2.31); (2.36) gantolebebSi xdeba nulis toli da, rogorcSedegi am miaxloebaSi vRebulobT ganzogadoebul kvantur-kinetikurgantolebebs korelaciuri funqciebisTvis, romlebic miRebuli iyo [6,68,75-77] SromebSi.sawyisi korelaciebis Sesustebis principis (postulatis) Tanaxmad“xelovnurad” Seqmnili sawyisi korelaciebi, romlebic ganpirobebulia t=0drois sawyisi momentisTvis s-qvesistemis urTierTqmedebiT S TermostatTan,unda miilion drois mixedviT, rodesac t →∞, Tu isini ar miekuTvnebianSenaxvad sidideebs, da unda moxdes dajaxebiTi (namdvili) korelaciebisaRdgena, romlebic aRiwerebian maRali rigis korelaciuri funqciebiTa da97

grinis funqciebiT (2.28), (2.29), (2.30), (2.31), (2.36) – gantolebebis dajaxebiTintegralebSi.amrigad, ( ) − 1( )−eqZ e β Hρ β = β ⋅ - sawyisi wonasworuli ganawilebis “detalebi”TiTqos ar unda iyos arsebiTi [4]; magram, miuxedavad amisa cxadia,rom principSi sawyisi korelaciebis evoluciis wevrebis arseboba ganzogadoebulkvantur kinetikur gantolebebSi korelaciuri funqciebisa dagrinis funqciebisTvis gavlenas axdenen evoluciur procesze, romelicmimdinareobs s qvesistemaSi. s qvesistemis da S Termostatis (bozonurivelis) susti urTierTqmedebis SemTxvevaSic, sawyisi korelaciebisevoluciuri wevrebi korelaciuri funqciebisTvis kinetikur gantolebebSiar miilevian nulisaken didi droebis asimptotur areSi, rodesac( t ∼ τ >> t0;t → ∞)da iZlevian TavianT wvlils gadatanis kinetikurrel(meqanikur) koeficientebSi (Zvradoba, eleqtrogamtaroba) s qvesistemisTvis(ix. III Tavi). Kkinetikuri koeficientebis [34], gamoZaxilis funqciis(impedansi, admitansi) [72-73] da dinamiuri amTviseblobis [4] gamoTvlisas;garda amisa, relaqsaciuri procesebis Seswavlisas, romlebicmimdinareoben Ria modelur sistemebSi, da myari sxeulebis fizikiskvanturi disipaciuri sistemebis ganxilvisas, ganzogadoebul kvanturkinetikur gantolebebSi sawyisi korelaciebis evoluciuri wevrebisgaTvaliswinebas aqvs principuli mniSvneloba da maTi ugulvebelyofa arSeiZleba. miuxedavad amisa unda aRiniSnos, rom kinetikuri movlenebisSeswavlisas am sistemebSi “namdvili” (dajaxebiTi) korelaciebi TamaSobendominirebul rols sawyis korelaciebTan SedarebiT [4].axla ganvixiloT gansxvavebuli midgoma igive amocanisadmi, romeliceyrdnoba liuvilis superoperatorul formalizmsa da proeqciulioperatoris meTods [114-122].98

s2.4. ganzogadoebuli kvanturi evoluciuri gantolebebi korelaciurifunqciebisTvis Sfm-is gamoyenebis gareSe.proeqciuli operatoris meTodiganvixiloT drois ormomentiani wonasworuli korelaciuri funqcia( )A t B da warmovadginoT is Semdegi saxiT:sA ( t)BSS−1= Z ( β)SPs∑−βHiLt[ B e e A ]ss(2.38)(2.38) formulaSi Cven SemoviReT liuvilis superoperatoris (L) cneba,romelic moqmedebs nebismier D-operatorze Semdegi wesiT:1hii± −±= [ , ] : = h Hth HtiLt± iLtLD H D e D e De , anu e D = D( ± t)da =−( )t=0sD D t .liuvilis sruli L superoperatori Caiwereba Semdegi saxiT:L L L L= + + , sadac L s= [ H ,...], L ∑= [ H ,...]da ... [ ,...]Σi1...sh1...∑h1= ; (2.1)–(2.2)hLiHint−hamiltonianis TiToeuli wevris Sesabamisad. (2.38) formulidan gamomdinarecxadia, rom korelaciuri funqciebis dinamika xelsayrelia ganvixiloTSemdegi superoperatorebis daxmarebiT, romlebic moqmedeben D D-operatorze Semdegi wesiT:−1( ) ( β )−βH iLtR t D = Z SΡ ⎡ ⎤Σ ⎣e e ΡD⎦−1( ) ( β )−βH iLtI t D = Z SΡ ⎡ ⎤Σ ⎣ e e QD ⎦ ,(2.39)sadac: Ρ warmoadgens Termostatis (bozonuri velis) mdgomareobebismixedviT gasaSualebis proeqciul operators:−( β) ( ρ ) ρ ( β)Ρ =Ρ; Ρ=Ρ ; Ρ D= SΡ D = D ; = Z e2 1Σ Σ Σ Σ Σ Σ(2.38) gamosaxuleba ( )−βHΣ( β) S ( e ) Q Q ( β)−βHΣZ = Ρ ; = = 1 −Ρ.Σ Σ Σss(2.40)A t B korelaciuri funqciisaTvis CavweroTSemdegi formiT:A t)( = A B ( t)= SP [ B R(t)A ]SB SSS− . (2.41)sssP da Q – proeqciuli operatorebis daxmarebiT martivad miviRebTSemdeg zust moZraobis gantolebebs R(t) da I(t) superoperatorebisaTvis:∂R( t) = iR( t) ΡLΡ+ iI( t)QLΡ;∂t∂I( t) = iI( t) QLQ+ iR( t)ΡLQ.∂t(2.42)99

Tu CavatarebT (2.42) gantolebis integrebas I(t) superoperatorisaTvis,maSin Cven miviRebT Semdeg gamosaxulebas:sadac:t∫0( ) = ( ) ( ) + ( ) Ρ ( − )I t I 0 M t i dτR τ LQM t τ ,(2.43)Q−1( 0) ( )−βHI Z S e QQ= β Ρ Σ(2.44)aris I(t)-superoperatoris sawyisi mniSvneloba da M ( t) = exp[ iQLQt]warmoadgens “masur” superoperators.Tu CavsvamT I(t)-superoperatoris (2.43) gamosaxulebas (2.42) formulispirvel gantolebaSi, maSin Cven vipoviT zust, araerTgvarovan evoluciurgantolebas R(t)-superoperatorisaTvis:∂tR( t) = iR( t) ΡLΡ+ iI( 0) MQ( t) QLΡ− dτR( t) ΡLQMQ( t−τ)QLΡ∂t∫ . (2.45)0miRebuli gantoleba gansazRvravs evoluciur gantolebas (2.41)korelaciuri funqciisaTvis. marTlac, Tu visargeblebT (2.39) gansaz-RvrebiTa da S Ρ s ,Σoperaciis qveS operatorTa cikliuri gadanacvlebis SesaZleblobiTada (2.41)–(2.45) formulebiT, maSin martivad vipoviT Semdegevoluciur gantolebas korelaciuri funqciisTvis [119]:∂∂t( ) [ ] ( ) ⎡ ( )AB − = Ρ Ρ − + ⎣ Ρ ⎤s st i L As Bs t i QMQ t QL As⎦Bs−t∫0( ) ⎤ ( )− dτ ⎡⎣ΡLQMQ t−τ QLΡAs⎦Bs−τ.Q(2.46)(2.45) da (2.46) gantolebebi warmoadgenen zust, aramarkoviseul evoluciurgantolebebs R(t) superoperatorisa da AB ( t)ss− korelaciurifunqciisTvis. am gantolebaTa araerTgvarovani wevrebi (meore wevrebi (2.45),(2.46) ganatolebaTa marjvena nawilSi) aRweren sawyisi korelaciebisevolucias drois mixedviT, romlebic ganapirobebulia s qvesistemisurTierTqmedebiT S TermostatTan (bozonur velTan) drois sawyis moment-Si: t=0. amis gamo, sawyisi amocanis (koSis amocana) amoxsnisas (2.45) gantolebisaTvis(da korelaciuri funqciis povnisaTvis) saWiroa codna ara martosuperoperatorisa – R(0), aramed agreTve I(0) sididisa. Tu maxsovrobasawyisi korelaciebisa drois mixedviT miileva, maSin evoluciis procesSidominirebs dajaxebaTa gavlena. magram, miuxedavad amisa, principSi sawyisi100

korelaciebis wevrebis arseboba evoluciur gantolebebSi, romelicganapirobebulia I(0)-is arsebobiT, gavlenas moaxdens relaqsaciur procesze,romelic aRiwereba korelaciuri funqciiT, qvesistemis susti urTierTqmedebisSemTxvevaSiac TermostatTan. mogvianebiT Cven vaCvenebT, rom sawyiskorelaciebs SeaqvT TavianTi wvlili s qvesistemis (mag. eleqtronis) gadataniskinetikur koeficientSi (eleqtrogamtaroba) [115-122].bevrad ufro efeqturi da xelsayrelia, Tu Cven ganvixilavT evoluciiszust da erTgvarovan gantolebas, romelSiac sawyisi korelaciebimoicema aracxadi saxiT. imisaTvis, rom vipovoT aseTi saxis gantoleba,visargebloT cnobili integraluri operatoruli igiveobiT:β−βH −βH0 −βH λH−λH0λint0e = e −∫ d e e H e , (2.47)sadac H0moicema (2.9) formuliT. am igiveobis gamoyenebiT, adviliaimis Cveneba, rom adgili aqvs Semdeg gantolebaTa sistemas I(t) da I(0)-superoperatorebisaTvis, romlebic gamomdinareoben (2.43) da qvemoTmoyvanili gantolebidan:( 0 ) ( ) ( , β) ( ) ( , β)I =−R t I t −I t I t(2.48)Qda romelic miiReba (2.47) gamosaxulebisa da I(t)-superoperatorisTvis (2.44)sawyisi pirobidan. (2.48) gantolebaSi Cven SemoviReT integralurisuperoperatori, romelic ganisazRvreba Semdegi tolobiT:βiLt λH λHQ ( t,β) dλe − e Hinte −Q0QI =∫ . (2.49)Tu amovxsniT (2.43) da (2.48) gantolebebs I(0)-is mimarT da CavsvamT(2.45) gantolebaSi, maSin Cven miviRebT saZiebel zust, erTgvarovan Caketilgantolebas R(t)-superoperatorisaTvis:∂−1R( t) = iR( t) ΡLΡ−iR( t) ΡI ( , ) ⎡⎣1 + ( ) I ( , ) ⎤Qt β MQ tQt β ⎦ ×∂tt( ) τ ( τ) ( τ ) ( , β) ⎡1( )∫× M t QLΡ+ d R ΡLQM t− I t ⎣ + M t ×Q Q Q Q0−1( β) ⎤ ( ) τ ( τ ) ( τ)×I t, ⎦ M t QLΡ− d R ΡLQM t− QLΡ.Q Q Q0t∫(2.50)unda aRvniSnoT, rom miRebuli gantolebebi SesaZlebelia gavamartivoT,Tu visargeblebT Semdegi TanafardobebiT:L Ρ= 0; Ρ LQ = QL Ρ= 0; ΡLΡ= 0. (2.51)Σs s i101

dabolos, Tu gamoviyenebT (2.50)–(2.51) formulebs, (2.46) gantolebisnacvlad miviRebT Semdeg zust kvantur evoluciur (kinetikur) gantolebaskorelaciuri funqciisTvis [119]:∂∂t( ) [ ] ( ) { ( , β )AB − t = i ΡL ΡA B −t −i ΡI t ×s s s s s Q−1( ) ( β) ⎤} ( )× ⎡⎣1 + MQ t IQ t,⎦ MQQLiΡAs Bs− t + dτ×{ LQM ( t τ ) ( t, β) ⎡1 M ( t) ( t,β)× Ρi Q− IQ ⎣ +QIQ⎤⎦×( ) } ( τ ) τ ⎡ ( τ)× M t QLΡA B − − d ⎣ΡLQM t− Q×Q i s s i Q0] ( τ )× LΡA B −i s s.t∫t∫0−1(2.52)(2.50)–(2.52) evoluciur gantolebebs gaaCniaT sakmarisad rTulioperatoruli struqtura. sawyisi korelaciebis evolucia droSi aRiwereba(2.49) integraluri superoperatoriT; igi Sedis evoluciur gantolebebSirogorc markoviseuli, ise aramarkoviseuli formiT (meore da mesamewevrebi (2.50) da (2.52) gantolebebis marjvena nawilSi). Sfm-is dros,− H0rodesac (2.38) korelaciuri funqciis gansazRvrebaSi Cven viRebT e β –operators, nacvladHe β-gibsis statistikuri operatorisa, vRebulobT, romI(0)=0, rogorc es naTlad Cans (2.39) da (2.40) formulebidan (vinaidan PQ=0).am garemoebas mivyavarT im Sedegamde, rom integraluri superoperatorixdeba nulis toli ( ( t β )I , ≡ 0), rac Tavis mxriv iwvevs sawyisiQkorelaciebis evoluciis amsaxveli wevrebis gaqrobas (isini xdebian nulistoli) (2.50) da (2.52) evoluciur gantolebebSi. rogorc Sedegi da kerZoSemTxveva Cven vRebulobT evoluciur gantolebas korelaciuri funqciis-Tvis, romelic gamoyvanilia [75-77] SromebSi.evoluciuri gantolebebi (2.3)-grinis funqciebisTvis gamomdinareoben(2.52)-gantolebidan. ase magaliTad, grinis dagvianebuli funqciisTvisgveqneba gantoleba:∂∂t∂= + − −∂t∂−ηθ( t) BAs s( t) .∂t( ) δ ( ) [ ] θ( ) ( )rG t t ABs st ABs stηnaTelia, rom evoluciuri gantoleba ( )(2.53)AB t korelaciuri funqciisTvismiiReba (2.52) gantolebidan, Tu gamoviyenebT Casmas t → –t [119, 121-122].ss102

2.5. Termostatis bozonuri amplitudebis gamoricxva evoluciurigantolebidan korelaciuri funqciisaTvis. markoviseuli miaxloebaqvesistemis dinamikisTvisganvixiloT axla ufro detalurad (2.52) evoluciuri gantolebakorelaciuri funqciisTvis. advili misaxvedria, rom (2.52) gantolebaSeicavs Termostatis (bozonuri velis) amplitudebs, romlebic ar ariangamoricxuli am gantolebidan. davuSvaT, rom hamiltoniani Hint(liuviliani L i ) Seicavs mcire parametrs (susti urTierTqmedebis SemTxvevaqvesistemasa da Termostats Soris); maSin (2.52)-gantolebidan principSiSesaZlebelia bozonuri amplitudebis gamoricxva. am amocanisgadasawyvetad gavSaloT mwkrivebad superoperatorebi, romlebicfigurireben (2.52)-gantolebaSi ( )mixedviT. gveqneba:L H liuvilianis (hamiltonianis)iint−1k( ) ( β) ( ) ( ) ( β)⎡⎣1 + M t I t, ⎤⎦ = −1 ⎡⎣M t I t,⎤⎦Q Q Q Qk=0∞∑( n) (0)0( ) = ( );( ) =Q Q Qk = 0∞∑M t M t M t eiLtkt t1tn−1( n)iL0( t−t1) iL0( t1−t2)Q ( ) ∫ 1∫ 2... ∫ n i i...0 0 0M t = dt dt dt e iQLe iQL ×iL0( tn −1−tn)iLt 0 n× e iQLein=( ); 1,2,...t−iLt−iLt0e = e Τexp⎢−i∫Li( ξ)d0λλHλH0e = e Τ'exp⎢∫Hint( γ ) d0−( ξ) ; ( γ)⎡⎣⎤ξ⎥⎦⎤γ⎥⎦L = e Le H = e H eiiL 0 ξ iL 0 ξ −γH int0 γHiint00⎡⎣L = L + LsΣ,(2.54)sadac T da T’ simboloebi aRniSnaven operatorTa mowesrigebas Sesabamisadξ da γ cvladebis mixedviT.s qvesistemis S TermostatTan susti urTierTqmedebis SemTxvevaSi(2.54) gaSlebi formalurad SegviZlia ganvixiloT rogorc SeSfoTebisTeoriis mwkrivebi da gamovTvaloT wevrebi (2.50)–(2.52) gantolebebismarjvena nawilebSi mocemuli sizustiT. martivi saCvenebelia, rom aseTigaSlebisas urTierTqmedebis mixedviT im SemTxvevaSi, rodesac H ΣdaaqvT (2.2) saxe, nulisgan gansxvavebuli iqnebian mxolod iseTi saSualoHint103

sidideebi – ... Σ, romlebic Seicaven bozonebis dabadebisa da gaqrobisoperatorebis erTnair raodenobas. amitom aseTi saxis saSualoebi toliiqneba sidideebis, romlebic proporciuli iqnebian bozonebis SevsebissaSualo ricxvebis namravlebis. amrigad, bozonuri (fononuri)operatorebi (amplitudebi) mTlianad gamoiricxeba (2.52) evoluciurigantolebidan wonasworuli korelaciuri funqciisaTvis.davuSvaT, rom susti urTierTqmedebis gamo, romelsac adgili aqvs sqvesistemasa da S Termostats Soris, gvaqvs droTa ierarqia mTel ( s +Σ )sistemaSi, romelic aRiwereba (2.34) utolobiT. (2.34) utolobissamarTlianoba gvrTavs nebas CavataroT markoviseuli miaxloeba (2.50)–(2.52)gantolebebSi; Tu SemovisazRvrebiT SeSfoTebis Teoriis meore miaxloebiTurTierTqmedebis ( )intiH L hamiltonianis (liuvilianis) mixedviT, visargeblebT(2.54) gaSlebiT da SevasrulebT Semdeg miaxloebebs:Q( ) ( βQ Q )(0)( ) ( )0( , ) ( , )−1⎡⎣1 + M t I t, ⎤⎦⇒1iL0tM t ⇒ M t = esQQQI t β ⇒I t β = dλe e H e QiL0( t−τ)( −τ) ⇒ ( − )B e B tβ∫0−iL0t λH0 −λH0ints.(2.55)maSin (2.52) evoluciuri gantoleba martivdeba da igi iRebs saxesmarkoviseuli, ganzogadoebuli kvanturi kinetikuri (evoluciuri)gantolebisa wonasworuli korelaciuri funqciisTvis AB ( t)∂∂tβ−iLt0 λH0( − ) = [ ] ( − ) − ⎡s s s s sλ Ρ ×AB t i LA B t i d⎣e et−λH0 iLt 0 iL0( t−τ)× H ⎤inte e LA ⎦ ( − ) −∫⎡⎣Ρ ⎤i sBs t d LeiLAi s⎦×0( )iL0( t−τ)×s−e B t.∫0ss− [119]:τ (2.56)miRebuli (2.56) gantoleba korelaciuri funqciisTvis jer kidevSeicavs gamouricxav bozonur amplitudebs. Tu visargeblebT H Σdahamiltonianebis – (2.2) cxadi saxiT da gamovricxavT bozonur operatorebszemoT aRwerili proceduris daxmarebiT, maSin (2.56) evoluciurigantoleba korelaciuri funqciisTvis Caiwereba Semdegi saboloo saxiT[119]:H int104

∂∂t⎡⎡ti1iω( k)ξs s ( − ) =− [ s,s] ( − ) −2 { ( ) ×− s ∫ ξ∑kβhh0 kAB t A H B t d N e( )( ) ⎤ ( ξ) ⎤ ( ) ( β)+ −iω( k)ξ×⎣⎣As, Ck s ⎦ ⋅C , − − + 1+ ×−ks⎦Bs t Nke−ω( k)β+ − ( )( + )⎡ , ( ) ( , ) ⎤iiωk t ihλ×⎣⎡⎣As Ck s ⎤⎦⋅C − ( − ) + { ( ) ×− ks ξ⎦Bs t ∑( ) h ∫dλ Nkβ eω k0 k( )( , hλ) ⎡⎣, ( ) ⎤⎦( ) 1 ( β)−+( , hλ) ⎡ , ( ) ⎤ ( ) },× C s −t −i ⋅ A C s B − t + + N e+ iω( k )( t + ihλ)k s k s k× C s −t−i ⋅⎣A C s⎦B −tk s k−s×(2.57)sadac: Nk( β )( k)e β h= ⎡ω −saSualo ricxvs, da⎣1⎤⎦−1- warmoadgens bozonebis (fononebis) Sevsebis± iLZ s+ ± iLZ s +( 1± ) = ( );( 1± ) = ( )( )[ , ] = − m β h ω kED ED e DE.C s Z e C s C s Z e C sk k k kxolo± ω( k)(2.58)nebismieri E da D operatorebisaTvis.(2.57) gantolebis gamoyvanisas, Cven visargebleT agreTve SemdegiTanafardobebiT:e b = e b ; e b = e b± iLt Σ m i ω( k ) t ± iLt Σ + ± i ω( k ) t +k k k k+ +( b , b ) N ( ) ; ( b , b ) ( 1 N ( ))+ +Ρ ( bk, bk') =Ρ ( bk, bk') = 0.Ρ = β δ Ρ = + β δk k' k kk , ' k k' k kk , '(2.59)Tu SevasrulebT t → –t Casmas (2.57) evoluciur gantolebaSi, maSinmartivad miviRebT ganzogadoebul kvantur kinetikur gantolebas AB ( t )korelaciuri funqciisTvis:∂∂ti1AB ( t) = [ A , H ] B ( t) − ⋅2{ ( ) ×− ∫ dξ∑ N βhhs s s s s k0 k⎡⎣⎣( ) ⎤ ( ) ⎤⎦ ξ ( ) ( ( β))− ⎦−ω( k)β+i( ) ⎤⎦( ξ) ⎤ ( )− ⎦ω( k)} ∫ λ− iω( k)ξ+× e ⎡As, Ck s ⋅ Ck s, Bs t + 1+ Nk×⎡iω( k)ξ× e⎣⎡⎣As, Ck s ⋅Ck s,Bst − d ×h0∑iω( k )( t − ihλ)+{ N ( β)e C ( st , ihλ) ⎡A , C ( s) ⎤ B ( t)× − ⋅ ⎣ ⎦ +kk k s k−s−iω( k )( t −ihλ)+( 1 N ( β ))e C st , ikk( hλ) ⎡A C ( s) ⎤ Bs k s( t) }+ + −t⋅ ⎣, ⎦.−s(2.60)(2.57)–(2.60) gantolebebi gansxvavdeba Sfm-Si miRebuli gantolebebisagan[75-77]. mesame wevrebis arsebobiT gantolebaTa marjvena nawilSi,romlebic aRweren sawyisi korelaciebis evolucias [114, 119,121-122].s105

dabolos, unda aRiniSnos erTi principuli sakiTxi, romelic exebasawyisi korelaciebis evoluciis wevrebis gaTvaliswinebas ganzogadoebulikvanturi kinetikuri gantolebebis dajaxebiT integralebSi,wonasworuli korelaciuri da grinis funqciebisTvis. Sfm – mTeli (s + S)-kvanturi dinamiuri sistemis statistikuri operatorisTvis – dafuZnebuliaim daSvebaze da aRwers iseT situacias, rodesac drois sawyisi t = 0momentisTvis adgili ar aqvs korelaciebs s-kvanturi dinamiuri qvesistemisada S-Termostatis mdgomareobebs Soris, da drois am momentisTvisxdeba “CarTva” urTierTqmedebisa, rogorc qvesistemasa da TermostatsSoris, aseve qvesistemasa da gareSe velebs Soris. cxadia, rom aseTSemTxvevaSi mTliani (s + S)-kvanturi dinamiuri sistemis sawyisistatistikuri ganawileba (statistikuri operatori drois sawyis momentSi)aiReba faqtorizebuli saxiT (ix. (2.37) formula). amitom, rogorc ukve iyoaRniSnuli, rogorc Sedegi vRebulobT ganzogadoebul kinetikur gantolebebskorelaciuri da grinis funqciebisTvis, romelTa dajaxebiTiintegralebi ar Seicaven sawyisi korelaciebis evoluciis wevrebs [75-77].yvela im realur modelebSi fizikuri sistemebisa (kvanturidisipaciuri sistemebi, Ria arawonasworuli modeluri sistemebi da sxv.),romlebsac ganixilavs statistikuri fizika (meqanika) da Termodinamika,fizikuri kinetika da a.S., qvesistemis urTierTqmedebis gamo TermostatTan,s dinamiuri qvesistemac da S Termostatic erTad Seadgenen ganuyofelnawilebs erTiani mTliani dinamiuri sistemisa - (s + S) (qvesistema plusTermostati). ufro metic, SeiZleba iTqvas, rom qvesistemasa da TermostatsSoris urTierTqmedebis gamo, eqsperimentebze faqtiurad SeuZlebelia maTierTmaneTisagan gancalkeveba. aseT situaciaSi, kinetikuri movlenebis srulda adekvatur aRwerasa da gamokvlevas, romlebic mimdinareoben mcirekvantur dinamiur qvesistemaSi, Seesabameba – mTeli (s + S) dinamiurisistemis statistikuri operatoris sawyisi mniSvneloba – ara Sfm-is tipis,aramed iseTi saxis sawyisi mniSvnelobebi mTeli sistemis statistikurioperatorisa, romlebic iTvaliswineben korelaciebs kvanturi dinamiuriqvesistemisa da Termostatis mdgomareobebs Soris drois sawyisi momentisaTvis(t = 0) (arafaqtorizebuli sawyisi statistikuri ganawilebebimTeli (s + S)-sistemisa) [19,21].106

Cvens mier dasmuli amocanis Tanaxmad, mTeli (s + S) dinamiurisistema drois sawyisi t = 0 momentisaTvis imyofeba statistikuri wonasworobismdgomareobaSi, romelic aRiwereba gibsis kvanturi kanonikuri (andidi kanonikuri) ganawilebiT mTeli (s + S) sistemisTvis:ρeq β Z β e β H−1−( ) = ( ) ,da drois amave t = 0 momentSi xdeba urTierTqmedebis “CarTva” mTel (s + S)sistemasa da gareSe wyaroebs (velebs) Soris. amis gamo, kinetikuriprocesebis mkacrad, zustad da koreqtulad aRsawerad da Sesaswavlad,romlebic mimdinareoben mcire kvantur dinamiur qvesistemaSi, romelicurTierTqmedebs TermostatTan (mag. bozonur (fononur) velTan), sawyisikorelaciebisa da sawyisi korelaciebis evoluciis wevrebisgaTvaliswinebas - ganzogadoebuli kvanturi kinetikuri gantolebebisdajaxebiT integralebSi, wonasworuli korelaciuri da grinisfunqciebisTvis s-qvesistemis dinamiuri sidideebisaTvis– aqvs principulimniSvneloba, da maTi apriori ugulvebelyofa (ignorireba) dauSvebelia [19,21, 110, 114-115, 119-122].miRebuli ganzogadoebuli kvanturi kinetikuri gantolebebi droisormomentiani wonasworuli korelaciuri funqciebisTvis, sadisertacionaSromis III TavSi gamoyenebuli iqneba eleqtronuli da polaronuligadatanis movlenebis gamosakvlevad myar sxeulebSi – naxevargamtarebsa daionur kristalebSi [114-118, 120-122,125].107

Tavi III. naxevargamtarebsa da ionuri kristalebSi eleqtronuli dapolaronuli gamtarobisa da dabaltemperaturuli Zvradobis kvanturiTeoria dafuZnebuli kubos wrfivi gamoZaxilis Teoriazerogorc ukve aRniSnuli iyo Sesavalsa da literaturis mimoxilvaSi,myar sxeulebSi gadatanis wrfivi movlenebis Sesaswavlad da denismatareblebis gadatanis kinetikuri (meqanikuri) koeficientebis gamosaTvlelad,SesaZlebelia gamoyenebuli iqnas ori gansxvavebuli midgoma: 1)dafuZnebuli kubos wrfivi gamoZaxilis (reaqciis) Teoriaze, da 2)damyarebuli kinetikur gantolebaze denis gadamtanebis ganawilebisfunqciisTvis, romelic cnobilia rogorc bolcmanis gantoleba.rogorc ukve araerTxel iyo xazgasmuli, eleqtronuli da polaronuligadatanis movlenebis Teoriis Seswavla myar sxeulebSi da maTikinetikuri maxasiaTeblebis gamoTvla, warmoadgens erT-erT aqtualuramocanas eleqtronisa da didi radiusis mqone polaronis kinetikisTanamedrove TeoriaSi [29-30,37-38]. gamoviyenoT axla wina TavSi gamoyvaniliganzogadoebuli kvanturi kinetikuri gantolebebi eleqtronuli dapolaronuli gadatanis movlenebis Sesaswavlad naxevargamtarebsa daionur kristalebSi; kerZod – ganvixiloT konkretuli magaliTebi kvanturidinamiuri qvesistemebisa, romlebic urTierTqmedeben fononur velTan:eleqtron-fononuri sistema (eleqtronis urTierTqmedeba (gabneva)polarul optikur da akustikur fononebTan), polaronis latinjer-lus(fg) modeli da maTze dayrdnobiT avagoT denis gadamtanebisTviseleqtrogamtarobisa da Zvradobis wrfivi kvanturi Teoria da gamov-TvaloT kinetikuri maxasiaTeblebi (korelaciuri funqciebis milevis devrementebi,relaqsaciis droebi (sixSireebi) da sxv.) da gadatanis meqanikurikoeficientebi (kuTri eleqtrogamtaroba, dabaltemperaturuli Zvradoba,da sxv.) kvanturi disipaciuri (qve) sistemebis am kerZo modelebisTvis. ganixilebamidgoma, romelic eyrdnoba kubos wrfivi reaqciisa da SeSfoTebisTeorias.108

3.1. eleqtron-fononuri sistema. eleqtronis dabalsixSirulieleqtrogamtarobisa da dabaltemperaturuli Zvradobis gamoTvla sustieleqtron-fononuri urTierTqmedebis SemTxvevaSiganvixiloT SemTxveva nawilakis (eleqtronis) urTierTqmedebisadakvantul fononur velTan (eleqtron-fononuri sistema), rodesac mTelisistemis hamiltoniani moicema (1.13) formuliT da eleqtronis energiasgamtarobis zonidan aqvs Semdegi zogadi saxe: Hs=ΤΡ ( ).rogorc cnobilia, kubos wrfivi reaqciis Teoriis Tanaxmadsistemaze moqmedi susti intensivobisa da ω-sixSiris mqone gareSe eleqtrulivelis SemTxvevaSi, eleqtrogamtarobis tenzori SesaZlebeliagamovsaxoT “deni-denze” korelaciuri funqciis saSualebiT. kubosformulis Tanaxmad eleqtrogamtarobis tenzori Caiwereba Semdegi saxiT[4,25,64,72,83]:⎛ 1 ⎞+∞ ⎜thβ hΩi⎟21σ ( ω ) =− Ω⎝ ⎠∫ dI{ µν , }( Ω)µνπhΩ Ω−ω −iε−∞+ε > 0; ε → 0 ( µν , = xyz , , ),+∞{2∫−∞ν µµ ν1 iΩtsadac: I µ , ν}(Ω)= dte [ j (0) J ( t)+ j ( t)J (0) ](3.1) formulis Tanaxmad eleqtrogamtarobis tenzoris disipaciurinawilisTvis gveqneba:.(3.1)12 th( βhω)Re σ ( ω) dtcos( ωt) (), t∞s 2sµν≡Ψµνhω∫0(3.2)sssadac: σ ( ω ) da Ψ () t warmoadgenen gamtarobis σ ( ω ) tenzorisa daµνµνΨ () t korelaciuri funqciis simetriul nawilebs:µνs 1σµν( ω) = ⎡ ( ) ( ) ;2⎣σµν ω + σνµω ⎤⎦s 1 1Ψµν() t = ⎡ () t () t () t ( t) ;2⎣Ψ µν+Ψνµ⎤⎦ = ⎡2⎣Ψ µν+Ψµν− ⎤⎦(3.3)1 1Ψµν() t = ⎡ jν (0) jµ () t + jµ () t jν(0) ⎤= ⎡ jν (0) jµ () t + jµ (0) jν( −t)⎤2⎣ ⎦ 2⎣ ⎦µ µiLtda j () t = e j (0) aris r j -eleqtruli denis operatoris µ komponenti haizenbergiswarmodgenaSi. amrigad, eleqtrogamtarobis gamosaTvleladµν109

2esaWiroa vipovoT: Ψ () = (0) ()2 ⎡µνt⎣Vν Vµ t + + V (0) ( − ) ⎤µVν t⎦korelaciuri funqciismniSvneloba. V rwarmoadgens eleqtronis siCqaris operators gamtarobiszonaSi. imisaTvis, rom vipovoT Ψ () t korelaciuri funqciis sidide,µνgamoviyenoT naSromis II-TavSi miRebuli (2.57) da (2.60) miaxloebiTi evoluciurigantolebebi wonasworuli korelaciuri funqciebisTvis eleqtronfononurisistemisTvis, da visargebloT mTeli sistemis hamiltonianis(1.13) gamosaxulebiT. miaxloebiT evoluciur (kinetikur) gantolebaseleqtronis drois ormomentiani wonasworuli korelaciuri funqciisTvis“siCqare-siCqare” eqneba Semdegi saxe ( H A V B Vs s ν s µ )∂∂t−( k ) ν µ }−=ΤΡ ( ); = ; = .[120-122]:i1= ⎡ ΤΡ⎤ − +t2−iω( k ) ξV (0) () , ( ) () ⎡νVµ t ⎣Vν ⎦ V ( )− µt2 ∫ dξ∑V⎣e Nkβkhh0 krriω( k ) ξ ikr −ikr( ξ ) iω( k ) ξ −iω( k)ξ( 1 ( ))⎤ ⎡ , ⎤ () ⎡kβ⎦ ⎣ ν ⎦ µ( )−⎣kβ+ e + N V e e V t − e N + e ×βνkr ( ξ) ikr i2iω( k )( t −ihλ)⎡ ⎤ ∑ ⎡h∫k0 k× 1 + N ( β) ⎤⎦ e⎣V , e⎦V () t − dλ V⎣e Nk( β)+( β )+ e 1 + N ( ) ⎤ e ⎡V , e ⎤ V (), t− iω ( k )( t − i h λ ) − ( − h )⎦ ikr t i λ⎣ ikrkν ⎦ µ−{(3.4)iLtsadac: V () t = e V (0) ,µ µiiΤΡ ( ) ξ − Τ ( Ρ)ξ rh h iLsξr( ξ ) = e re ≡e r da r ( ξ ) aris eleqtronis TavisufalimoZraobis “traeqtoria”. analogiurad Caiwereba kinetikuriµ νgantoleba V (0) V ( − t)korelaciuri funqciisTvis. (3.4) gantolebis gamoyvanisasCven visargeblebT agreTve Semdegi TanafardobebiT: ω( − k) = ω( k);V= V .− k kCven SemovifarglebiT mxolod erTi zonis miaxloebiT eleqtronisTvis,rodesac eleqtronis siCqaris operatori diagonaluriaimpulsur (kvaziimpulsis) warmodgenaSi da amitom adgili aqvs Semdegtolobebs:∂( Ρ ) = ΤΡ ( );∂ΡV µµ⎡⎣V µ( Ρ), ΤΡ ( ) ⎤⎦ = 0, ( µ = xyz , , ).Cven viyenebT agreTve aRniSvnas V µ( Ρ ) eleqtronis siCqaris operatorismatriculi elementisTvis ΡV µΡ , romelic gamoiTvleba eleqtroniskvaziimpulsis operatoris sakuTari funqciebis Ρ meSveobiT.−110

Tu SemoviRebT Semdeg relevantur (damxmare) operators (SedarebisTvisix. (2.39)) Gµ (, t β ); sawyisi mniSvnelobiT, rodesac t = 0;Gµ ( β) = Gµ(, t β )t =0da ganvsazRvravT tolobebiT:G t = Z S Ρ ⎣V te β H⎦−1(, β) ( β) ⎡ −() ⎤ ;µ Σ µsZ ( β)Gµ ( β) = Vµ,(3.5)S ZsΡ ⎡ ⎣( β)⎤ ⎦ssHsadac: Z ( β ) = S Ρ ⎡ −e β ⎤Σ ⎣ ⎦- dayvanili (reducirebuli) statjamia eleqtronfononurisistemisTvis; maSin SemoRebuli (3.5) operatoris daxmarebiTSesaZlebelia V (0) V ( ± t)- korelaciuri funqciebi da maTi Sesabamisiν µkinetikuri gantolebebi gamovsaxoT Semdegi formiT:∂∂tVν (0) Vµ ( ± t) = SΡ s⎡⎣Vν (0) Gµ( ± t, β ) ⎤⎦⎡ ∂ ⎤V (0) ( ± ) ≡ Ρ ⎢ (0) ( ± , ) ⎥=± Ρ ⎡⎣ (0) Γ ( ± , ) ( ± , ) ⎤νVµ t SsVν Gµ t β SsVν νt β Gµt β ⎦,⎣ ∂t⎦(3.6)sadac: operatori Γ (, t β)uSualod ganisazRvreba (3.4) gantolebidan da aqvsSemdegi saxe:0ν{t12 −iω( k ) ξ iω( k) ξ−1Γ (, ) =− ⎡ ( ) +2 ∫ ∑( 1 + ( ))⎤νt β dξ V⎣e N β e N β⎦V ×k k kνh( β )⎡ ⎤ ⎡ ⎤ikr −ikr ( ξ) iω( k ) ξ −iω( k ) ξ −1 −ikr( ξ )×⎣V , e⎦e − ( ) + 1 + ( ) ×− ⎣e N β e Nkk ⎦V eννβi2 ω( )( − hλ) −νω( )( − hλ)} ∫ λ∑βk k( βk)ikr i k t i k t i× ⎡ , ⎤ − ⎡ ( ) + 1 + ( ) ⎤⎣Vνe⎦d V ×− h⎣e N e N⎦× Vikre ⎡ , ⎤⎣Vνe⎦,−1 − ikr ( t − i hλ)ν0kk−(3.7)sadac:V ⋅ V = V ⋅ V = 1.−1 −1ν ν ν ν(3.6) kinetikur gantolebebSi kvali eleqtronis (qvesistemis) mdgomareobebismixedviT gamovTvaloT eleqtronis kvaziimpulsis operatorisΡ sakuTari funqciebis saSualebiT. amis gaTvaliswinebiT (4.6)Tanafardobebi SesaZlebelia CavweroT Semdegi saxiT:V (0) V ( ± t) = dV Ρ ( Ρ ) G ( ± t, β, Ρ)∫ν µ ν µ∫∂dV Ρ ( Ρ ) G ( ± t, β, Ρ ) =± dV Ρ ( ΡΓ ) ( ± t, β, Ρ ) G( ± t, β, Ρ).ν µ ν ν µ∂t∫(3.8) gantolebebis miRebisas Cven gaviTvaliswineT, rom eleqtronissiCqaris operatori diagonaluria kvaziimpulsis warmodgenaSi da, vinaidanganixileba sivrculad erTgvarovani sistema, amitom mxolod diagonaluri(3.8)111

matriculi elementebi – ΡΓ (, t β) Ρ ≡Γ (, t β , Ρ); Γ (, t β)- operatoris arisνgansxvavebuli nulisagan. Tu visargeblebT Semdegi TanafardobebiT:ν± ikre δ ( hk)ikrikr( Ρ ) = ( Ρmh) ;± ikr ± ikr( Ρ ) = ( Ρ± h ),Ρ Ρ = Ρ ± −Ρ1 2 2 1 ;e f f k e± ±f e e f ksadac: f ( Ρ ) warmoadgens Ρ kvaziimpulsis (impulsis) nebismier funqcias davisargeblebT (3.7) gantolebiT, maSin Γν (, t β ) operatoris Γν(, t β, Ρ ) matriculielementisaTvis miviRebT Semdeg gantolebas [120]:ν⎧ ⎡t− ⎤sin ( , )2 2 ( ) ( )⎢∆ k ΡV ⎥νΡ+ hk −VνΡ ⎪(, , ) ( )⎣hΓ⎦νt β Ρ = ∑ V ⎨Nβ+kk−h k Vν( Ρ) ⎪ ∆ ( k, Ρ)⎪⎩⎡t+ ⎤sin ( , )−⎢ ∆ k Ρ ⎥ i⎡− −β∆( k , Ρ)− ∆ ( , Ρ)1( 1 ( )⎣h⎦ it k e −h+ + N β )+ ( )++ ⎢N β ekk−∆ ( k, Ρ) 2 ⎢⎣∆ ( k, Ρ)+i + −β∆( k , Ρ)− t∆ ( k, Ρ)e −1⎤⎫⎪+ ( 1 + N ( β )h) e,k+ ⎥⎬∆ ( k , Ρ)⎥⎪ ⎦⎭(3.9)±sadac: ∆ ( k, Ρ ) =ΤΡ+ ( hk) −ΤΡ± ( ) h ω( k).ganvixiloT axla izotropuli SemTxveva, rodesac eleqtrogamtarobistenzors da simetrizebul korelaciur funqcias aqvT Semdegnairisaxe:ssssσ ( ω) = σ ( ωδ ) ; Ψ () t =Ψ () t δµν µνµν µνda visargebloT miaxloebiTi gantolebiT, romelic gamomdinareobs (3.8)gantolebidan:∂Gµ ( ± t, β, Ρ ) =±Γµ( ± t, β, Ρ ) Gµ( ± t, β, Ρ).(3.10)∂tcxadia, rom (3.10) kinetikuri gantoleba warmoadgens relaqsaciisdrois miaxloebis (rdm) saxis aproqsimacias. naTelia, rom (3.10) miaxloebiTigantoleba xdeba zusti, rodesac Γ ( ± t, , Ρ ) funqcia ar arisµβdamokidebuli Ρ kvaziimpulsze. magaliTad, aseT SemTxvevas aqvs adgilifrolixis polaronisTvis dabali temperaturebis areSi [116-118, 120].Tu CavatarebT (3.10) gantolebis integracias da gaviTvaliswinebT (3.5)sawyis pirobebs, maSin Cven miviRebT:112

~ r r[ Γ ( t,β , P] G ( β,);rG ( ± t,, P)= expP Γ~ tr( t,, P)= ∫dτΓ( τ , β,Pr)µβµ±µµβµ;0sV ( Ρ) Z ( β, Ρ)µssGµ ( β, Ρ ) = Ρ Gµ( β) Ρ = ; Z ( β, Ρ ) = Ρ Z ( β) Ρ .sdΡZ( β, Ρ)∫(3.11)Tu CavsvamT Γ (, t , Ρ)-s (3.9) mniSvnelobas (3.11) formulaSi, maSinµβintegraciis Sedegad Γ% (, t , Ρ)-Tvis Cven miviRebT gamosaxulebas:µβΓ% (, t β, Ρ ) = Re Γ% (, t β, Ρ ) + iIm Γ%(, t β, Ρ);µ µ µV ( Ρ+ hk) −V( Ρ)Re Γ (, t β, Ρ ) = ∑ V N ( ) ⎡1+ e ⎤×µk⎦k2 −µ µ −β∆( k , Ρ)βkV ( Ρ)⎣µ{⎡t− ⎤ ⎡t+ ⎤⎫1−cos ⎢ ∆ ( k , Ρ) ⎥ 1− cos ⎢ ∆ ( k, Ρ)+⎣h⎦⎥− ∆ ( , Ρ)2 ( 1 ( ))⎡β k⎪1 ⎤ ⎣h× + + N β +⎦;2 ⎬− ⎣e⎦(3.12)k+⎡⎣∆ ( k , Ρ) ⎤⎦⎡⎣∆ ( k, Ρ)⎤⎦ ⎪⎪⎭( , )2 ( ) ( ) ⎧−−β∆k ΡΡ+ − Ρ ⎪ 1−Im Γ%V hk Vµ µe(, t β, Ρ ) =− ∑ V ⎨ N ( β )×µkk2( ) −kVµΡ ⎪ ⎡ ( , ) ⎤⎩ ⎣∆k Ρ ⎦+−β∆( k , Ρ)− 1−⎫+ ⎪× sin⎡t ∆ ( , Ρ )⎤ e+ ( 1 + ( ))sin⎡t∆ ( , Ρ) ⎤⎢k⎥N β.k+2 ⎢k⎥⎬⎣h⎦ ⎡⎣∆ ( k , Ρ)⎤⎦⎣h⎦⎪⎭(3.11) da (3.12) gamosaxulebebi asaxaven (3.6) korelaciuri funqciebisdroze damokidebulebas da principSi saSualebas iZlevian, rom gamoTvliliqnes eleqtrogamtarobis tenzori (3.2)-(3.3) formulebis daxmarebiT.vinaidan Cven SemovifarglebiT SeSfoTebis Teoriis mxolod meoremiaxloebiT (3.4) kinetikur gantolebaSi eleqtron-fononuriurTierTqmedebis ( )H V hamiltonianis mixedviT, amitom Tanmimdevrobisik−βH−β 0 sdacvis mizniT SevasruloT miaxloeba: e e H −β e H −βeH Σ⇒ = - korelaciurifunqciebis (3.5)-(3.11) – sawyisi mniSvnelobebisaTvis. Aamrigad Cven gvaqvs:−βHs−βΤ( Ρ)eeGµ ( β) = V ; G ( , ) V ( ).H µ µβ−βs−βΤ( Ρ)µSΡ⎡e ⎤Ρ = sdΡeΡ(3.13)⎣ ⎦ ∫(3.8), (3.11) da (3.13) gantolebebs mivyavarT Semdeg gamosaxulebamdeΨ () t korelaciuri funqciisaTvis:µν∫{ ⎡% β ⎤ ⎡%β ⎤}2−βΤ( Ρ)e dΡe Vν ( Ρ) Vµ ( Ρ) exp ⎣Γν (, t , Ρ ) ⎦+ exp ⎣Γµ( −t, , Ρ)⎦Ψµν() t =,−βΤ( Ρ)2dΡe∫(3.14)113

sadac: Γ % (, t β, Ρ)da Γ% ( −t, , Ρ ) ganisazRvrebian (3.12) formulebiT.νµβ(3.12) da (3.14) tolobebis Tanaxmad, eleqtronis (qvesistemis) korelaciurifunqciebi miilevian oscilaciebiT imis gamo, rom relaqsaciisΓ % (, t β, Ρ)-faqtori warmoadgens kompleqsur sidides, da rogorc CvenνmogvianebiT vaCvenebT, es oscilaciebi “sicocxlis unariani” arian dididroebis asimptotur areSi t >> t0da iZlevian wvlils gadatanis kinetikurkoeficientebSi (eleqtrogamtaroba, Zvradoba) [117-118, 120-122]. es daskvnagamomdinareobs im martivi garemoebidan, rom wonasworuli droiTikorelaciuri funqciebis gansazRvrebaSi (ix. mag. (2.6) formula) figurirebsgibsis ganawileba, romelic Seicavs sawyisi korelaciebis wevrs ( Hint ) daromlis zusti gaTvaliswinebac xdeba evoluciur (kinetikur) gantolebebSiam funqciebisTvis. Sfm-Si Γ (, t β, Ρ ) _ relaqsaciis faqtoris warmosaxviTiνnawili nulis tolia, da, amitom zemoT miTiTebul oscilaciebs adgili areqneba [75,77]. unda aRiniSnos, rom ganxiluli msjeloba samarTliania imSemTxvevaSi, rodesac adgili aqvs droTa ierarqias (ix. (2.34) formula);rogorc (3.12) formulidan naTlad Cans, im SemTxvevaSi, rodesac ganixilebaeleqtronis urTierTqmedeba fononebTan: t ~ hβdastΣ~1ϖ, sadac ω - warmoadgensfononTa maxasiaTebel sixSires. ufro metic, (3.12)gamosaxulebidan gamomdinareobs, rom energiis Senaxvis kanoni eleqtronisfononebTan gabnevis procesis dros samarTliania didi droebisSemTxvevaSi: t0t >> , t ( )0= max ts,t Σ; Tu ganvixilavT droTa did intervalst >> t 0, maSin Cven SegviZlia SevasruloT zRvruli gadasvla t →∞ (3.12)gamosaxulebaSi da, rogorc Sedegi miviRebT [120]:1limRe Γ%relν(, t β, Ρ ) =−Γν( β, Ρ ) t =− t ;t→∞relτ ( β, Ρ)rel 2π2 Vν( Ρ+ hk) −Vν( Ρ)−Γ ( , Ρ ) =− ∑( ) ⎡∆ ( , Ρ ) ⎤νβV N βδ +kkh( Ρ)⎣ kV⎦+( N β ) δ ⎡ k ⎤k}+ 1 + ( ) ⎣∆ ( , Ρ) ⎦ ;kν2 ( Ρ+ ) − ( Ρ)−limIm Γ%Vνhk Vν(, , ) ( ) ⎡ ( , ) ⎤νt β Ρ =−βπ∑V N βδ ∆ Ρ +t→∞kk( Ρ)⎣ k ⎦k Vν++ ( 1 + N ( β)k ) δ ⎡⎣∆( k, Ρ) ⎤⎦}Signt.{{(3.15)(3.15) formulebis gamoyvanisas Cven visargebleT TanafardobebiT:114

1−cos( ωt) sin( ωt)lim = π t δω ( ); lim = πδ( ω) Signt.t→∞2ωt→∞ωaRvniSnavT, rom (3.15) gamosaxulebebSi limt→∞Sesruldes Termodinamikuri zRvruli gadasvla (Τ -zRvari):V... dk (...).(2 π)∑( ) =3 ∫kzRvrul gadasvlamde undacxadia, rom Zalian didi droebis asimptotur areSi, rodesac( t >> t , t → ) (3.15) Tanafardobebi Cven gvaZlevs Semdeg tolobas:0∞1limIm Γ % (, , Ρ ) = Γ ( , Ρ) .→∞2h relνt β βνβ Signt(3.16)tνrelamrigad, Γ ( β, Ρ ) - sidide, romelic faqtiurad warmoadgens eleqtronis“siCqare-siCqare” – korelaciuri funqciis milevis dekrements –SesaZlebelia ganxilul iqnas rogorc eleqtronis V ν( Ρ ) siCqaris ? –relkomponentis relaqsaciis sixSire ( τ ( β, Ρ)- warmoadgens Sesabamisrelaqsaciis dros).dabolos, Tu visargeblebT (3.2)-(3.3) da (3.14)-(3.15) TanafardobebiT,sabolood vpoulobT gamosaxulebas kuTri eleqtrogamtarobisdisipaciuri nawilisTvis [120]:⎛1⎞th⎜βhω2⎟⎝ ⎠ r⎧ ⎡βh⎤= ∫ Ρ Ρ Ρ Ρ ⎨ ⎢Γ Ρ ×hω⎣ 2 ⎥⎩⎦s 2relRe σµνne d ρs( β, ) Vν ( ) Vµ ( ) cosν( β, )relrel( , )⎫Γ ( , )Γ Ρνβ Ρ ⎡βhrel ⎤ µβ ⎪× + cos Γ ( , Ρ) ,22 ⎢ µβ22⎬rel( , ) ⎣ 2 ⎥relω + ⎡Γ Ρ ⎤ ⎦⎣ ⎦ + ⎡⎣Γ ( , Ρ)⎤νβ ωµβ ⎦ ⎪⎭ν(3.17)sadac: n - warmoadgens eleqtronebis koncentracias gamtarobis zonaSi,−βΤ( Ρ) −βΤ( Ρ)xolo ρ ( β, Ρ ) = e dΡe.s ∫ releqtronis dabaltemperaturuli Zvradoba, romelic damokidebuliagareSe eleqtruli velis ? sixSireze (ac-mobility), SesaZlebeliawarmodgenili iqnas Semdegi saxiT (saubaria dreiful Zvradobaze):sadac:µω ( ) = µ ( ω ) −∆ µω ( ),0µ⎛ 1 ⎞eth⎜βhω⎟⎪⎧rrrelΓ ⎪⎫⎝ 2 ⎠r r r rrelΓ ( , )( , P)Vβ Pµβω)= ∫dPρs(β,P)VV( P)Vµ( P)⎨r +⎬;2 rel2rhω2 rel⎪⎩ ω + [ ΓV( β,P)] ω + [ Γµ( β,P)] ⎪⎭0( 2115

⎛ 1 ⎞2eth⎜βhω⎟∆ =⎝ 2 ⎠r r r rµ ( ω)∫ dPρS(β,P)VV( P)Vµ( P)×(3.18)hω⎪⎧rrrelrel2( , )2( , ) ⎪⎫⎡βhrrel ⎤ Γ⎡⎤ Γ⎨sin ( , )2sin ( , )2 ⎬;42⎢[ ( , )] 4 ⎥ ⋅ PVβ P βhrrelµβ−1× ⎢ ΓVβ P ⎥r + Γµ β Pr ( ω

3.2. eleqtronis Zvradoba frolixis polaronis modelSi3(“2βhω0problema” polaronis dabaltemperaturuliZvradobis TeoriaSi)ganvixiloT axla eleqtroni, romelic moZraobs polarul naxevargamtarSian ionur kristalSi, romlis moZraoba aRiwereba (1.13)-(1.18)hamiltonianiT da romelic sustad urTierTqmedebs polarul optikurfononebTan: ( α < 1) (e.w. frolixis polaronis modeli) [29-30]. ganixilebaeleqtronis urTierTqmedeba dispersiis armqone optikur fononebTan,rodesac ω( k ) ≡ ω0. vinaidan eleqtronis energiisTvis gamtarobis zonidanrr2Ρgvaqvs gamosaxuleba ΤΡ ( ) = , amitom eleqtronis siCqaris µ -2mkomponentisTvis gveqneba Semdegi formula:r ΡµVµ( Ρ ) =m(3.19)rogorc (3.14), (3.15) da (3.17) gamosaxulebebidan Cans, sakiTxi daiyvanebareleleqtronis relaqsaciis sixSiris: Γ ( , Ρ r )- gamoTvlaze, romelic zogadµβSemTxvevaSi damokidebulia eleqtronis Ρ r impulsze.davuSvaT, rom eleqtruli veli mimarTulia z RerZis dadebiTi mimar-relTulebis gaswvriv, da, gansazRvrulobisaTvis ganvixiloT Γ ( β, Ρ r ). TuSevasrulebT Termodinamikur zRvrul gadasvlas da CavatarebTintegracias k rcvladis mixedviT, maSin (1.18) da (3.15) gamosaxulebebidanmiviRebT:2ππ~ αω ⎪⎧0( β,Ρ)= ~ 1⎨Ν0(γ ) dϕdθsin θ cosθ2πΡz⎪⎩ 0 0relΓ0∫ ∫~Ρ cos Φ~+2 2Ρ cos Φ + 12ππ~Ρ cosΦ⎫ ~ 2 2+ ( 1+ Ν ( γ )) ∫ d ϕ∫ dθsin θ cosθ⎬(Ρ cos Φ ≥ 1 ).0 ~(3.20)2 20 0Ρ cos Φ −1⎭aq Cven SemoviReT eleqtronis uganzomilebo impulsi:~ ΡiΡi = ; i =2mhω0( x,y,z)da10e γ −γ = ⎡ − ⎤ γ = βhω0N ( )⎣1⎦; .Φ - warmoadgens kuTxes k r da Ρ ~ veqtorebs Soris, romelic dakavSirebuliaθ da ϕ kuTxeebTan (θ da ϕ kuTxeebi gansazRvraven k r veqtoriszmimarTulebas) Semdegi TanafardobiT:117

~Ρ cosϕ~= sin θ cosϕΡx~+ sin θ sin ϕΡy~+ cosθΡ.z(3.20) gamosaxulebis gamosaTvlelad SemovisazRvroT dabalitemperaturebis SemTxveviT, rodesacγ >> 1(3.21)da (3.14) korelaciuri funqciebi gansazRvrulia eleqtronis impulsismcire mniSvnelobebiT:~Ρ 2> ).(3.24)ω 0(3.24) gamosaxulebis gamoTvlisas Cven visargebleT SemdegiformulebiT:∫r~dPe3~r2~ r2 ~r~2−γP2 1 ⎛π⎞−γPPz⎜ ⎟ ; dPe=2γ∫⎝ γ ⎠⎛ π= ⎜⎝ γ32⎞⎟⎠(3.25)amrigad, rogorc (3.24) gamosaxulebidan Cans, eleqtronis “siCqaresiCqare”korelaciuri funqciebi eqsponencialurad miilevian drois mixedviTdabali temperaturebis ganxilul SemTxvevaSi. (3.24)-is Tanaxmadrelrelrelaqsaciis droisTvis gveqneba Semdegi gamosaxuleba: τ ( γ) = ⎡ ⎣Γ0( γ) ⎤ ⎦.−1118

cxadia, rom Cvens mier miRebuli rezultati napovnia Sfm-isgamoyenebis gareSe [120]. (3.24) gamosaxuleba gansxvavdeba Sedegisagan, romel-⎡ iγ rel⎤ic napovnia Sfm-Si [77] eqsponencialuri faqtoriT exp ⎢± Γ0( γ ) .2ω⎥⎣ 0 ⎦gamovTvaloT axla eleqtrogamtaroba da dabaltemperaturuliZvradoba eleqtronisTvis frolixis polaronis modelSi susti eleqtronfononuriurTierTqmedebis dros: ( α < 1). Tu gaviTvaliswinebT (3.19) da(3.23)–(3.25) gamosaxulebebs, (3.17) formulidan kuTri eleqtrogamtarobisTvismiviRebT [120]:2~ ne 2 ⎛ 1 ~ ⎞⎡( ))2⎛γ⎞⎤Γ0γRe σ ( ω = ~ th⎜γω⎟1 2sin0() ~ ;2 202⎢ − ⎜ Γ γ ⎟mωγω2⎥⎝ ⎠⎣⎝ ⎠⎦ω + Γ0( γ)( α < 1, γ >> 1, ωγ ~ > 1, ω~γ t0pirobidan) da γ . Γ0 ( γ )

mcire sidideebs ( γ >> 1). (3.27) gamosaxulebebidan eleqtronisdabaltemperaturuli statikuri ZvradobisTvis (dc-mobility) frolixispolaronis modelSi gveqneba [120]:µ0e=mω0Γ−10( γ ) =emω32α2e2⎡γ⎤∆µ= sin⎢Γ0( γ ) Γmω0⎣ 2 ⎥⎦( γ >> 1, α < 1).0−10eγ;e( γ ) =mω01 2αγ e3−γ;(3.28)napovni (3.28) gamosaxulebebi warmoadgens Tanamimdevrul da sworrezultats eleqtronis dabaltemperaturuli Zvradobisa, frolixispolaronis modelSi, mcire temperaturuli SesworebiT susti eleqtronfononuriurTierTqmedebis SemTxvevaSi [114-118,120].ganvixiloT axla eleqtronis gamtaroba da Zvradoba polaronisfrolixis modelSi eleqtruli velis maRali sixSireebis SemTxvevaSi,rodesac adgili aqvs Semdeg Tanafardobas:rrel−1Γ ( β,Ρ)

gamovTvaloT axla (3.32)-is daxmarebiT eleqtronebis kuTri eleqtrogamtarobada Zvradoba eleqtruli velis maRali %ω -sixSireebis Sem-TxvevaSi. vipovoT jer σ ( %)0ω -kuTri eleqtrogamtarobis mniSvneloba. TurelCavsvamT Γ ( β, Ρ r )-s (3.32) gamosaxulebas (3.30)-is pirvel formulaSi,CavatarebT martiv gardaqmnebs da gamoviyenebT (3.19) da (3.31) formulebs,maSin Ρ r -impulsuri cvladiT integraciisa da Termodinamikuri zRvruligadasvlis Semdeg, σ 0( ω%)-eleqtrogamtarobisTvis miviRebT Semdeg gamosaxulebas:2neσ ( ω% ) = σ%( ω%);0 0mω0∞2 2 1 2 3210( )⎛ ⎞ ⎡ γωσ % % ω% = th0( )⎛ ⎞⎜ γω%2⎟ αγ N γ3∫dk⎢e ⎜k+ ×γω%⎟⎝ ⎠ π0 ⎣ ⎝ k ⎠2 2γ ⎛ 1 ⎞ γ ⎛ 1 ⎞− ⎜k+ ⎟4 1− ⎜k−⎤⎟⎝ k ⎠ ⎛ ⎞ 4⎝ k ⎠× e + ⎜k− ⎟e⎥; Re σ0( ω% ) ≡σ0( ω%),⎝ k ⎠ ⎥ ⎦(3.33)sadac: σ% 0( ω% ) warmoadgens uganzomilebo dinamiur (sixSireze damokidebul)gamtarobas, xolo uganzomilebo parametrebi: γ , ω% da N 0( γ ) moicema (3.20) da(3.26) formulebiT. SevniSnoT, rom sixSireze damokidebul eleqtroniseZvradobisaTvis (3.33)-is msgavsad gveqneba: µ0( ω %) = µ %0( ω%) da µ % 0( ω% ) ≡ σ%0( ω %).mωeleqtronis dinamiuri gamtarobisTvis miRebuli (3.33) gamosaxulebafrolixis polaronis modelSi SesaZlebelia CavweroT kompaqturi formiT,Tu CavatarebT integracias k -cvladiT (3.33) gamosaxulebaSi. gveqneba:02 1 2γ2 32 2ωσ % %10( ω% ⎛ ⎞ ⎛ ⎞) = th γω αγ eN0( γ) K1γ ,γω⎜ %% 2⎟ ⎜3 π2⎟⎝ ⎠ ⎝ ⎠sadac: K 1() z warmoadgens pirveli gvaris makdonaldis funqcias.(3.34)(3.32) formula saSualebas iZleva, rom gamoTvlili iqnas eleqtronisimpulsis relaqsaciis sixSire, k r -cvladiT integraciis Semdeg. (3.32)relgamosaxuleba: Γ ( β, Ρ r )-sididisTvis miiRebs Semdeg saxes:⎧~ ⎡ ~~ ⎤ ⎡ ~~ ⎤⎫Γ Ρ = ~ 1⎨ + Ν Ρ −⎢Ρ − + ~ 1 Ρ⎥+ Ν⎢Ρ + − ~ 1rel ~220( γ , ) ω0α(10(γ ) θ(1) 1 arcch0(γ)1 arcshΡ2⎥⎬Ρ ⎩⎣ Ρ ⎦ ⎣ Ρ ⎦⎭(3.35)121

cxadia, rom eleqtronis impulsis mcire mniSvnelobebisaTvis, rodesac:Ρ

~Γ1). SedarebiT dabal sixSiruli are, rodesac:~1( γ , Ρ)

32−2it−ittS t)= D ( t)[(1+ N ( γ )) e + N ( γ ) e ]; da D(t)= − it;(3.41)γ(00Ffxip-is TeoriaSi (miaxloebaSi) eleqtronis dreifuli ZvradobaMmoicema (ganisazRvreba) Semdegi TanafardobiT:1µ ~FXIP~ 2~= lim Re ~ ( ~ ) lim Re ( ~~ ω σ ω ~ Z ω );(3.42)ω → 0=ω →0sadac: (3.40)-formulis Tanaxmad, eleqtronis impendansis realuri nawiliganisazRvreba Semdegi tolobiT:3 1~ ~ 2αsin( ω~ t412 2=~ )γRe Z ( ω)dt Im[ S(t)] = αγ e N0(γ)K1(γ ). ( ω ~ → 0)(3.43)3 π ω3 π2∞∫0(3.43)-gamosaxulebidan martivad miiReba eleqtronis dabaltemperaturulistatikuri ( ω = 0)dreifuli Zvradobis mniSvneloba ( ix. Sedarebi-saTvis (3.37)-formula) fxip-is miaxloebaSi:3 1 γµ = FXIPe ;2γ2α( γ >> 1, γ → ∞)(3.44)Bbalansis gantolebis meTodze dayrdnobiT tornberg-feinmanis miermiRebuli eleqtronis statikuri dreifuli Zvradobis mniSvneloba moicemaSemdegi gamosaxulebiT: [ 53, 84-85, 101-103 ];31241~ 1 α∞ γ2 2= Im[ ( )] αγ0(γ )1(γ)µ 3 π∫dt⋅t⋅ S t = e N K(3.45)3 π2TF0romelic emTxveva fxip-is mier miRebul dreifuli Zvradobis (3.43)-mniSvnelobas.Aamitom cxadia, rom frolixis polaronis (eleqtronis)dabaltemperaturuli ZvradobisTvis gvaqvs igive saxis gamosaxuleba,rogoric fxip-is TeoriaSi:~ 3 1 γµ = e ;( γ >> 1, γ → )2γ2α∞TF(3.46)[ 15 ] - naSromSi gamokvleuli da naCvenebi iyo, rom tornberg-feinmanismidgomaSi eleqtronis dreifuli Zvradobis gamoTvlisas gamoiyenebodamaqsvelis wanacvlebuli ganawilebis funqcia eleqtronisaTvis; xolofxip-is TeoriaSi eleqtronis Zvradobis gamosaTvlelad gamoiyenebaeleqtrogamtarobis (eleqtrowinaRobis) mniSvneloba SeSfoTebis Teoriismeore miaxloebaSi. (α -bmis mudmivas rigis mixedviT).rogorc aRniSnuli iyo literaturis mimoxilvaSi, frolixis polaronisdabaltemperaturuli Zvradobis gamosaTvlelad mravali gamokvleva124

iqna Catarebuli bolcmanis kinetikur gantolebaze dayrdnobiT [ 55, 91, 93,98 ]. aRniSnuli gamokvlevebis Sedegad miRebuli dabaltemperaturuliZvradobis mniSvneloba moicema Semdegi gamosaxulebiT:~ 1 γµ = Be ; ( ω ~ =2α0, γ >> 1; γ → ∞ )(3.47)(3.47)- formulis miRebis dros, bolcmanis gawrfivebuli (gareSe eleqtrulivelis mixedviT) kinetikuri gantolebis amoxsnisas gamoiyenebodarelaqsaciis drois miaxloeba da Tanac dajaxebiT integralebSi gadasvlissixSireebis da TviT integralebis gamoTvlisas gaTvaliswinebuliiyo mxolod “danaklisis” wevrebi. Aam miaxloebebis farglebSi -eleqtronis ganawilebis funqciis relaqsaciis sixSirisaTvis (droisTvis)miiReboda Semdegi saxis gamosaxuleba:Γr+ rr− r{(1+ N ( β )) δ [ ∆ ( k , p)],+ N ( β ) δ [ ∆ ( k,p ]}β r 1 2π2, p ) = =rel r ∑ V)2 K0τ ( β , p)h(3.48)relB(0BK(SedarebisTvis ix. (3.31) formula).~saidanac, mcire siCqareebiT moZravi eleqtronisaTvis Ρ 2> 1),Tu CavatarebT martiv ga-moTvlebsdgindeba, rom relaqsaciis sixSire Γ Ρ~ rrelB( γ , ) − warmoidgineba Semdegi formiT:~rrel~Γ γ , Ρ)= Γ ( γ ) 2αN( ); ( γ >> 1, Ρ 2

~ ~1sazRvrebs: Γ rel ( , ) ~0γ Ρ

Vν( p + hk) − VνrV ( p)νr( p)(ix. (3.9),(3.12), (3.15)), romelic aRwers eleqtronissiCqaris cvlilebas misi gabnevisas fononebze, maSin rodesacrel releqtronis ganawilebis funqciis relaqsaciis sixSire: Γ ( β,p)(ix. (3.48))-aseTi saxis Tanamamravls ar Seicavs. swored es faqtori ganapirobebs amdamatebiTi mamravlis warmoqmnas [120]. rac Seexeba fxip-isa da tornbergfeinmanisSedegebis Tanxvedras ~ 1 3 γµFXIP= µTF= e ; ( γ >> 1) da2γ2αmamravls, maTi warmoSobis buneba dRevandel dRemde bolomde garkveuliar aris.B12γ-3.3. eleqtronis Zvradoba akustikuri polaronismodelSi susti eleqtron –fononuri urTierTqmedebis SemTxvevaSiganvixiloT eleqtroni, romelic moZraobs kovalentur (araionur)kristalSi an naxevargamtarSi, romlis moZraoba aRiwereba (1.13)-(1.14) hamiltonianiT.davuSvaT, rom eleqtroni sustad urTierTqmedebs akustikur2 2D mfononebTan; α = < 1. Aam SemTxvevaSi adgili aqvs eleqtronis38πρhV srrurTierTqmedebas dispersiis mqone akustikur fononebTan: ω ( k ) = V k;k ≡ k .eleqtronis energiisaTvis gamtarobis zonidan gvaqvs dispersiisstandartuli paraboluri kanoni da eleqtronis siCqariss⋅µ −komponentimoicema (3.19) formuliT. eleqtronis Zvradobis gamosaTvleladakustikuri polaronis modelSi visargebloT (3.17)–(3.18) formulebiT,rel rxolo Γ ( β,p)- relaqsaciis sixSiris gamosaTvlelad ki – (3.15)vacgamosaxulebiT. Tu gamoviyenebT (1.14) da (3.15) – formulebs, mivmarTavTeleqtrul velsΖ − RerZis dadebiTi mimarTulebis gaswvriv, SevasrulebTTermodinamikur zRvrul gadasvlas da CavatarebT integrebas K-cvladiT,rrelmaSin Γzac( γ , Ρ)− relaqsaciis sixSirisaTvis miviRebT gamosaxulebas:Γrelzacr~ mV( γ,Ρ)= −h2s16α~Ρz∫dΩrk~cosθ(1 −ΡcosΦ)3exp1~[ 4γ(1−ΡcosΦ)]2mV− 16 sα~1~ ∫ d Ω r cosθ(1+ ΡcosΦ)3;kh Ρexp−1z~[ 4γ(1 +ΡcosΦ)]( α < 1.)−−1(3.50)127

sadac:2mVsγ = - warmoadgens modelis maxasiaTebel uganzomilebo fi-2KT −zikur parametrs,B~rΡ −uganzomilebo impulsia:~ rΡ = Ρ / mVs~~~ ~dΩ r k= sin θdθdϕ;Ρ cosΦ = sin θ cosϕΡ+ sin θ sin ϕΡ+ cosθΡ,sadac θ da ϕ - sferuli kuTxeebia, xolo Φ - warmoadgens kuTxesK r da Ρ rveqtorebs Soris. gamovTvaloT (3.50) – formulis daxmarebiTeleqtronis impulsis relaqsaciis sixSire kristalis dabali temperaturebisSemTxvevaSi, rodesac γ >>1.amisaTvis ganvixiloT “mcire“ siC-~qariT moZravi eleqtroni: Ρ = exp[ − Γ ( ) | | ]Acγ t exp( ) ;22⎢±i ΓAcγγ⎥⎣ mVs⎦(3.52)( α < 1; γrel>> 1; t ≥ τ Ac)(3.52)–korelaciuri funqciebis asimptoturi gamosaxulebebis gamoyva-nisasCven visargebleT (3.25) – TanafardobebiT.Aamgvarad, rogorc (3.52) –formulebidan Cans, “siCqare – siCqareze” –korelaciuri funqciebi eqsponencialurad miilevian drois mixedviT128

dabali temperaturebis drosrel( γ >> 1), Γ ( γ ) − milevis dekrementiT. naTe-lia,rom miRebuli Sedegi napovnia Sfm-is gamoyenebis gareSe, da gan-sxvavdeba.Sedegisagan, romelic miiReba Sfm-Si eqsponencialuri Tana-mamravliT:⎡ ihγrel⎤exp ⎢± Γ ( γ ) .2 Ac ⎥ [121-122].⎣ mV s ⎦vipovoT axla eleqtrogamtaroba da dabaltemperaturuli ZvradobaeleqtronisTvis erTzonian miaxloebaSi da izotropul SemTxvevaSi akustikuripolaronis modelSi susti eleqtron- fononuri bmis dros. TuvisargeblebT (3.17), (3.19) da (3.25) – formulebiT, maSin kuTri ele-qtrogatarobisaTvismiviRebT gamosaxulebas [121-122]:Ac2 relne Γ ⎡ ⎤Ac(γ)hγrelReσAc(ω)= cos⎢Γ ( γ)2 2rel2 Ac ⎥m ω +ΓAc( γ)⎣mVs⎦(3.53)( α < 1; γmV>> 1; ω > 1)(3.54)rogorc (3.54) – formulebidan Cans, temperaturuli Sesworeba ele-qtronisdabaltemperaturul, dreiful, statikur (dc)−Zvradobaze, romelicganpiribebulia sawyisi korelaciebis gaTvaliswinebiT am modelSiwarmoadgens Zalian mcire sidides:∆µAche≈2m V2s32αγe2 −4γ;( α < 1; γ >> 1)∆ µ Ac 2 2 8γ≈ 2048 ⋅αγ e− ;(3.55)µAcmiRebuli (3.54- 3.55)–formulebi warmoadgenen Tanmimdevrul da koreqtulSedegs eleqtronis dabaltemperaturuli Zvradobisa akustikuri polaronismodelSi susti eleqtron – fononuri bmis SemTxvevaSi [121-122].gamovTvaloT axla eleqtrogamtaroba da dreifuli Zvradobaakustikuri polaronis modelSi eleqtruli velis maRali sixSireebis129

SemTxvevaSi, rodesac sruldeba (3.29)–Tanafardoba romelic akusti-kuripolaronis modelSi miiRebs Semdeg saxes:2~ ~ ~ 1Γ relAc( γ , Ρ)

2 1γ >> 1;( ~~ th γω) ⇒ 1. amitom (3.57) formulidan gamomdinare (Tu CavatarebTγω 2martiv gardaqmnebsa da gamoTvlebs) dinamiuri gamtarobisTvis gveqnebaSemdegnairi (asimptoturi) yofaqceva:ω~2Re σ~0128( ω~) ≈ α3γe−4γAC ; ( >> 1, γ → ∞);α < 1γ (3.59)amrigad, am areSi eleqtrogamtaroba ukuproporciulia sixSiris kvadratisa(maRali sixSireebis SemTxvevaSi) da miileva eqsponencia-lurad Ttemperaturis SemcirebasTan erTad.II) maRal temperaturaTa are, rodesac~ΓrelAC1( γ , ~ p )

ogorc ukve aRniSnuli iyo naSromis I TavSi, eleqtronis daba-ltemperaturuliZvradobis gamosaTvlelad akustikuri polaronis modelSisusti eleqtron-fononuri bmis SemTxvevaSi ( α < 1), gamoiyeneba kinetikuri(bolcmanis) gantolebis meTodi [99,108]. gareSe eleqtruli velis mixedviTgawrfivebuli bolcmanis gantolebis amoxsnisas relaqsaciis droismiaxloebaSi, dajaxebiT integralebSi gadasvlis sixSireebisa da TviT amintegralebis gamoTvlisas, gaiTvaliswineba mxolod `danaklisis” wevrebi.aseTi saxis miaxloebis Sesruleba (araferi rom ar vTqvaT Sfm-ze)warmoadgens sakmaod uxeSs da zogadad arasrulyofilad asaxavseleqtronuli gadatanis movlenebs eleqtron-fononur sistemaSi.warmodgenil modelSi, zemoT Tqmuli miaxloebebis farglebSi, eleqtronisganawilebis funqciis relaqsaciis sixSires aqvs Semdegi saxe:rel r 2π2ΓBAC(β , p)=2 ∑ Vkh kr2⎛ hkkp ⎞x δ ⎜ Vsk⎟+ + ; Γ2mm⎝⎠~ rrel ~ 8αrΓBAC(γ , p)= ∫ dKKπr2⎛ hkδ ⎜ +⎝ 2mr2( , ~ mVp )sγ =h[ Ν ( β )krelBACkpm~ΓrelBAC⎞−V k ⎟s+ (1 + N⎠r( γ , ~ p );( β ))xr2rr ~ rr2rr ~[(1+ N (4γK ) δ ( K + Kp + K ) + N (4γK)δ ( K + Kp − K )]sadac: uganzomilebo talRuri veqtori K rformulebSi) ganisazRvreba TanafardobiT:k(3.61)(iseve rogorc (3.57) da(3.58)r h rK = ⋅ k ;2 mVsmartivad dgindeba, rom dabali temperaturebis SemTxvevaSi (γ >>1) da~`mcire~ siCqariT moZravi eleqtronisTvis ( P 2 1, α < 1).(3.62)(3.62)-is daxmarebiT ki martivad vpoulobT eleqtronis dabaltemperaturulZvradobas (Sfm-Si) akustikuri polaronis modelSi:µhe1 4γ= ⋅ e ;( γ >> 1, α 1).2 2m Vs 32α0 BAC

amrigad, sadisertacio naSromSi avtoris mier miRebuli Sedegiakustikuri polaronis modelSi eleqtronis dabaltemperaturuli(statikuri) dreifuli ZvradobisTvis (ix. (3.54) formula) Sfm-Si 1/2-jernaklebia eleqtronis `bolcmaniseul~ dabaltemperaturul dreifulZvradobaze (ix. 3.63)). mamravli 1/2-is warmoSoba ganpirobebulia imrel relgaremoebiT, rom Γ ( γ )/ Γ0 ( γ ) = 2,AC BACda TviT eleqtronis siCqaris (impu-lsis)relaqsaciis sixSiris (3.58)-gamosaxulebis integralqveSa wevrebi Seicaven~ r r~( K ⋅ P)/P 2_Tanamamravls, maSin rodesac (3.61) relaqsaciis sixSirisgamosaxulebis integralqveSa wevrebi aseT Tanamamravls ar Seicaven; xoloTviT am Tanamamravlis warmoSoba ganpirobebulia korelaciurifunqciebisTvis kinetikuri gantolebebis dajaxebiT integralebSieleqtronis siCqaris cvlilebis wevris arsebobiT misi gabnevisasfononebze (ix. (3.9), (3.12, (3.15) da 3.2 paragrafi)). swored am garemoebis gamomiiReba eleqtronis dabaltemperaturuli Zvrado-bisTvis 2-jer naklebimniSvneloba am modelSi, romelic warmoadgens Tanamimdevrul da sworSedegs gansxvavebiT kinetikuri gantolebis gamoyenebiT miRebuliSedegisagan (`bolcmaniseuli~ midgomisagan) [121-122].rac Seexeba eleqtronis dabaltemperaturul (statikur) Zvradobebsam modelSi romlebic miiRebian fxip-is Teoriisa (miaxloebisa) dabalansis gantolebis meTodis (tornberg-feinmanis Teoriis) gamoyenebiT,avtoris mier Catarebuli gamokvlevebis safuZvelze napovnia amdabaltemperaturuli Zvradobebis mniSvnelobebi da dadgenilia, rom isiniemTxvevian erTmaneTs:~ FXIP ~ TF 3 1 4γµAC= µAC= ⋅ e ; ( γ >> 1, α < 1)(3.64)4γ64αiseve rogorc polaronis frolixis modelSi, am modelSic3 -4γmamravlis warmoSobis Rrma mizezi (buneba) jer-jerobiT dadgenili araris.amrigad sadisertacio naSromis am Tavis (3.1_3.3) paragrafebSi avtorismier ganxilul modelebze dayrdnobiT miRebuli Sedegebi (gamoyvaniliformulebi: rogorc zogadi, aseve miaxloebiTi) SesaZleblobas iZlevagadaugvarebel, farTozonian, erTgvarovan (polaruli) naxevargamtarebSi,ionur da kovalentur kristalebSi eleqtronuli gadatanis meqanikuri koe-133

ficientebis (dreifuli Zvradoba, dinamiuri gamtaroba) gamoTvlisa erTizonis miaxloebaSi kvazinawilakebis (eleqtronebis) dispersiis rogorczogadi, aseve paraboluri kanonis dros, fononebis dispersiisizotropuli kanonisa da susti eleqtron-fononuri urTierTqmedebisSemTxvevaSi.3.4 polaronis dabaltemperaturuli Zvradobafeinmanis ganzogadoebul modelSipolaronuli gadatanis meqanikuri koeficientebis (dreifuli Zvradoba,eleqtrogamtaroba) gamosaTvlelad polarul naxevargamtarebsa anionur kristalebSi SemovifargloT polaronis fgm-iT, romelic aRweriliiyo naSromis I Tavis 1.3.3 paragrafSi da romlis modeluri hamiltonianiSH GF moicema (1.52)-tolobiT. polaronis dreifuli Zvradobis sapovneladvisargebloT kubos wrfivi gamoZaxilis TeoriiT da gamoTvlebisgasamartiveblad SemovifargloT 1.33 paragrafSi warmodgenil erTeulTasistemiT: ( h = m = ω 1). CavTvaloT, rom deni romelic figurirebs (3.2)-0 =formulaSi, ganpirobebulia eleqtronisa, da fiqtiuri nawilakis(polaronis) masaTa centris gadaadgilebiT, vinaidan isini dakavSirebuliarian erTmaneTTan. amgvarad, ganixileba sistema, romelic aRiwerebahamiltonianiT [57-58,125]:HsGFHGF+ H + H int=∑(3.65)sadac: H∑-fononuri velis hamiltoniania, xolo H inturTierTqmedebishamiltoniani ganisazRvreba (1.2), (1.18) da (1.20)-tolobebiT. eleqtronfononuriurTierTqmedeba ganixileba rogorc mcire SeSfoTeba. naTelia,rom eleqtronis r -koordinati dakavSirebulia axal kanoni-kurcvladebTan TanafardobiT (ix. 1.33. paragrafi):rMGF= Rr+ µeξr; µe= , µ ≡ µe(3.66)M + 1GFSemovifargloT erTi zonis miaxloebiT da izotropuli SemTxveviTda ganvixiloT, magaliTad polaronis `impulsi-impulsze~-korelaciurifunqciis z-komponenti.134

)ΡzS Ρ)ΡsGFz( t)GF= Z−1GF( β ) S ΡGF ∑⎡⎢ e⎣− βHGFiH GF thi− H GF th)'' )GFGF−1t L GF t −βH GF[ Ρ z G z ( t,β )] ; G z ( t,β ) = Z GF ( β ) S Ρ [(e Ρ z ) e ])Ρze∑)Ρze⎤⎥⎦=(3.67)ZGF(GF−βHβ)= SΡ( e ) Pˆ ≡GF∑zsadac: Cven SemoviReT damxmare (relevanturi) operatori G GF ( t,β )msgavsi gamosaxuleba gveqneba agreTve korelaciuri funqciisTvis:)Ρ)z ( t) Ρ(t)GFz.. rogorc (3.67)-tolobebidan Cans kvali korelaciuri funqciis)gamoTvlisas aiReba Ρ > n > − . orTonormirebul sakuTar funqciebzeSredingeris (1.54) da (1.55)-gantolebebisa;u n(ξ v )v) ) v)s GFGFn ≡ ); . S Ρ [ Ρ G ( t)] = ∑∫dΡΡG ( Ρ,n,t,β)daGGFzzhzzv) v)v)GF( Ρ,n,t,β ) = Ρ < n G ( t,β n Ρ(3.68)GFz z)) )martivad SesaZlebelia naCvenebi iqnas, rom korelaciuri funqcia Ρ z(t) Ρ(t)da G GF ( t,β ) -operatori akmayofileben Semdegi saxis kinetikur gantolebasz(ix. SedarebisaTvis (3.4)-gantoleba):−t∂∂t) )⎡)∂ ⎤ i ⎧⎡ˆ)sGFss= Ρ ⎤ GFΡ⎫z Ρz( t)= S ΡGF⎢ΡzGz( t,β ) ⎥ S GF ⎨ Ρ z,H GF . Gz( t,β ) ⎬ −⎣ ∂t⎦ h ⎩⎢⎣⎥⎦ ⎭) rr rr−iτiτS ikr −ikr(τ ) GF[ e N ( β)+ e (1 + N ( β)] SP {[ P , e ] ⋅ e G ( t,β)}2∫dτ∑ V rGF Z −⋅Z+r k000sadac:k+t−iβ∫0dz∑rkV2rkrr ) rriz−izS −ikr( z)ikr GF[ e N ( β)+ e (1 + N ( β)] ⋅ SP { e [ P , e ] ⋅G( t,β)};0r − ganisazRvreba (3.66) – tolobiT da0GFZr )rZ(3.69)siH tS(gf −iHGF tt = e rewarmoa-dgenspolaronis Tavisufali moZraobis “traeqtorias” fgm-Si. analogiurad) ) ) )Caiwereba kinetikuri gantoleba < PZ( t)PZ>GF=< PZPZ( −t)>GFkorelaciurifunqciisaTvis. Tu gaviTvaliswinebT (3.66) – tolobas, CavatarebTintegrebas τ da z - cvladebiT, maSin martivi gardaqmnebisa da gamoTvlebisSemdeg miviRebT kinetikur (moZraobis) gantolebas G GF Z(t, β )- operatorisr)diagonaluri matriculi elementebisaTvis- G GF ( P,n,t,β ) :∂G∂tGFZr)( P,n,t,β ) = iZ2 KZ∑ V r ) ×kPrk , n1,n2z135

⎧r⎡r)− ∆− ⎤− ⎡ ∆+ v r )1 exp it ( k,P,n⎤⎫⎪1n2)1 exp it ( k,P,n1n2)⎢⎣⎥⎦⎢⎣⎥⎦ ⎪× ⎨N0(β ) r r) + (1 + N0( β ))⎬ ×−+ r r )⎪ ∆ ( k , P,n1,n2)∆ ( k , P,n1,n2)⎪⎩⎭r rrr r)iµk−ikGF2 Z< | |1> ( ,2,, , ) − ∑ | |ke ξµ e ξn e n n e n GZP n n t β i Vs) ×r kk , n1 , nP2Z⎧⎡+ r rr))⎤⎫−1 − exp − i(t − iβ) ∆ ( k,P,n , )⎪ 1 − exp[ − ( − ) ∆ ( , , , )]1ni t iβk P n1n⎢⎣⎥⎦ ⎪× ⎨N0( β )r r)+ (1 + N0( β ))⎬−+ r r )P⎪∆ ( k,P,n1,n)∆ ( k,P,n1,n)⎪⎩⎭rrr rr)−iµekξiµekξGF× < n e | n >< n | e | n > G ( P,n , n,t,)aq|1 12 Z 2βr rs2 )r r) k kP∆ ± ( k,P,n1,n2)= + ± 1+εn−ε1 n22MM(3.70).−sidide warmoadgens fononebze gabnevisaspolaronis energiis cvlilebas misi erTi aRgznebuli energe-tikulimdgomareobidan meore aRgznebul mdgomareobaSi gadasvlis dros;M ≡ MGF+1-ki polaronis masaa. Tu (3.70) – gantolebaSi Sevasru-lebT Casmasr)t → −t martivad vpoulobT gantolebas G GF ( P,n,−t,β ) sididisaTvis. cxadia,rom (3.70)- kinetikuri gantoleba ar warmoadgens bolcmanis tipisgantolebas, vinaidan am gantolebis dajaxebiTi inte-gralebi Seicavenaradiagonalur matricul elementebs G GFZ( t,β)operatorisa:r)r)r)GFGFG P,n , n,t,β ) =< P | < n | G ( t,β ) | n > | P . naTelia, rom ajamva (3.70) –Z(2 2 Z>gantolebis marjvena mxareSi xorcieldeba Sredingeris (1.54)–gantolebisyvela | n >≡ ( ξ r) −orTonormirebuli ZiriTadi da aRgznebuli (zogadadu ngadagvarebuli) mdgomareobebis mixedviT. (Cven vTvliT, rom ZiriTadimdgomareobis| 0 > −energetikuli done ε − ar aris gadagvarebuli).r)bolcmanis tipis kinetikuri gantolebis misaRebad G GFZ( P,n,t,β )v)sididisaTvis warmovadginoTG GF ( P,n2,n,t,β)− aradiagonaluri matriculielementebi Semdegi formiT:r)r)GFGFG P,n , n,t,β)= G ( P,n,t,β)δ + exp it(εZsadac~G GFZZ0Z~ GF[ −ε)] G ( P,n , n,t,β)1[ −δ]:( 2 Zn2nn2n Z 2n 2 nδn 2 n− warmoadgens kronekeris simbolos da SemoRebuli sidider)( P,n2,n,t,β ) −SedarebiT mdored icvlebar)(3.71)t −didi droebis asimptoturareSi t ≥ τ ). ganvixiloT did droTa mniSvnelobebi, rodesac(rel−1t >> | ε −ε| ;( n , n = 01, ,2,...)(3.72)ninjij136

Tu CavsvamT (3.71)–tolobas (3.70)–kinetikuri gantolebis marjvena mxareSida gaviTvaliswinebT (3.72)–utolobas, maSin miviRebT bolcmanis saxisv)kinetikur gantolebas G GF ( P,n,t,β ) −sididisaTvis. (Cven ugulvebelyaviTZdajaxebiTi integralebi aradiagonaluri matriculi elementebiT, vinaidanisini warmoadgenen swrafad oscilirebad funqciebs didi droebisSemTxvevaSi) [125]:⎧r⎡r)− ∆− ⎤− ⎡ ∆+ r r )⎤⎫∂ v)1 exp ( , ,1,)1 exp ( , ,1GF r2 kit k P n nit k P nnZ ⎪ ⎢⎣⎥⎦⎢⎣ ⎥⎦ ⎪GZ( P,n,t,β)= i∑ | Vr | ) ⎨N0(β)r + (1 +⎬×−0()kv)N β rr∂+ )tk,n P1 Z ⎪ ∆ ( k,P,n1, n)∆ ( k,P,n1, n)⎪⎩⎭r rr r r)iµk−ikGFke ξµ e ξ2Z× < n | e | n >< n e n > ⋅GZP n t −i∑Vr)1 1| | ( , , , β)| | ×r kP⎧r⎡r)− ⎤⎡+ r r )1−exp − i(t − iβ) ∆ ( k,P,n− − − ∆ ⎤⎫⎪1,n)1 exp i(t iβ) ( k,P,n1, n)⎢⎣⎥⎦⎢⎣⎥⎦ ⎪× ⎨N0(β )r)+ (1 + N0(β ))⎬×−+ r r )⎪∆ ( k,P,n1,n)∆ ( k , P,n1,n)⎪⎩⎭rrr r r)−iµekξiµekξGF−1× < n e | n > × < n | e | n > G ( P,n,t,); t >> | ε −ε|(3.73)k,n|1 1Zβ1Zn i n j(3.73) gantolebaSi ajamva n 1− simboloTi moicavs agreTve yvela im mdgomareobebs,romelTaTvisac ε = ε , vinaidan zogadad aRgznebulin2 n 1| n > − mdgomareobebis energetikuli speqtri gadagvarebulia. msgavsi saxisr)v)gantolebebi gveqneba agreTve G GF ( P,n,−t,β ) da ( P,0,t,β)− ZiriTadiZmdgomareobis ε | 0 ) matriculi elementebisaTvis. Tu CavatarebT (3.73)-(0 ,>r)gantolebisa da misi msgavsi gantolebis G GF ( P,n,− t,β ) − sididisaTvisintegrebast −droiTi cvladis mixedviT, gaviTvaliswinebTG GFZ( ± t,β ) − operatorebisaTvis sawyis pirobebs t = 0 − drois momenti-saTvis:r) ) )GF−1 −βH) )GFG (0, β ) = Z GF ( β)P SP ( e ); < P P ( t)> da < PZPZ( −t)>GFkorelaciuriZZΣZZGFfunqciebisTvis (3.68) formulebs, maSin am korelaciuri funqciebisTvismiviRebT Semdegi saxis gamosaxulebebs:β 3/2( )⎡r)) )2⎤± > = MPnPZPZ( t)2πr)βε 2 βGF∑∑e∫dPP⎢ − ⎥ ×−Zexpβεne n⎢⎣2M⎥⎦

− ⎡ ∆− ⎤r) 1 cos ( , ,⎪1,)−GF2sadac, Γ= −⎢⎣⎥⎦ −⎨[ + ]∆ ( , , 1,)Re ( , , , )kt k P n nr r)Zβ k P n nZ t β P n ) | V | N0(β )1 e∑rk , n1PZ⎧r r )rv rk2−)⎪ ∆ ( k,P⎩, n , n)r⎡r)− ∆+1 cos t ( k,P,n n ⎤1,)+ ( 1+N⎢⎣⎥⎦0(β )) r v) ×12+∆ ( k,P,n , n)11r+r)− ∆ /−[ + e ]} Pr vr rβ ( k,, n1 , n)iµki ek1< n | ee ξµ ξ| n >< n | e |1n >r v)⎧−v)⎪ sin [ ∆ ( , , ]GFkt k P nZ 21ndaIm ΓZ( t,β , P,n)=∑vk , n1r⎡r)∆ ⎤r)2+∆ ( k,P,n1,n)kr ξn >(3.75))PZ| V r | ⎨N0( β )kr v2−)⎪⎩ ∆ ( k,P,n1,n)+r−r)sin t ( k,P,n1nr+r)− ∆ ( , , 1,)[ − ] + +⎢⎣⎥⎦ − ∆ ( , , 1,1 e(1 N0())[ 1−e ] Pr vβ k P n nβ k n niµekξβ r} < n | e | n1> ×× < n1| e−iµ re|+⎫⎪× ⎬⎪⎭xolo, TviT korelaciuri funqciebis milevis dekrementisaTvis(polaronisimpulsisZ − komponentis relaqsaciis sixSirisTvis) da oscilirebadifaqtorisaTvis gveqneba Semdegi Tanafardobebi:r)GFr)−1 GFRe Γz( t,β,Ρ,n)kz 2((t >> | εn i− εn j| ) ; ΓZrel( β , Ρ,n)= lim| t|→∞= −2π∑ ) | Vr| ×rk| t |rk , n Ρ1 zr r)r v)rrrr−+iµekξ−iµekξ× N β)⋅δ∆ ( k,Ρ,n , n)+ (1 + N ( β))δ ∆ ( k,Ρ,n , n)< n | e | n >< n | e | n{0( [1]0[1]} 1 1> ;r)r)GFGFβr)GFIm ΓZ ( β , Ρ,n)= lim| t|→∞Im ΓZ( t,β , Ρ,n)= ΓZrel( β , Ρ,n)Signt2napovni (3.74) formulisa da (3.2)–(3.3) Tanafardobebis daxmarebiT(3.76)martivad vRebulibT polaronisTvis eleqtrogamtarobis tenzoris disipaciurinawilis mniSvnelobas ganxilul izotropul SemTxvevaSi [125]:Re σGFSzzNe( ω)=Msadac:× exp2212th(βω)2ω∞∫0β( )dt cos( ωt)2πMe∑)r)GFGF[ Re ΓZ( t,, Ρ,n)] cos[ ImΓZ( t,β,Ρ,n)]n3r)r) β2⎡ 2)⎤−βεΡn 2⋅− ∑e∫dΡΡZ⋅exp⎢−⎥ ×βε nn2Mβ (3.77)N −warmoadgens polaronebis koncentracias gamtarobis zonaSi.polaronis dabaltemperaturuli dreifuli Zvradobis gamosa-Tvleladfgm-Si ganvixiloT kristalis Zalian dabali temperaturebis zRvruliSemTxveva, rodesac:⎢⎣⎥⎦β−1>> 1;β

sadacε 1− warmoadgens polaronis pirveli aRgznebuli mdgomareobisenergias; (3.78) – pirobebis dacvis SemTxvevaSi ZiriTad wvlils (3.67)-(3.74)korelaciur funqciebSi (ix. agreTve (3.77)) iZlevian polaronisGF 1mdgomareobebi, romelTaTvisac: Er) ≈ β− = ⎜ ⎟ ∫ ΡΡ exp⎢−⎥⋅exp[ −Γ ( , Ρ)⋅|| ]⎡ βz ztGFdzzrelβ t ×GFr) ⎤exp⎝2πM⎠ ⎢⎣2M⎢± i Γzrel( β,Ρ)⎥(3.79)⎥⎦⎣ 2 ⎦r)GFkr r)r r)z−+ΓzrelΡ = − ∑ V r2( β , ) 2π) | | { N ( ) [ ∆ ( k,Ρ)] + (1 + N ( )) [ ∆ ( k,Ρ)]}×r k 0β δ0β δΡkzµ 2× | < 0 |r ξrr)r)i ekGF β GFe | 0 > | ; Im⋅Γz( β , Ρ)= Γzrel( β,Ρ)2r rrr)2 r) k k ⋅Ρsadac: ∆ ± ( k,Ρ)= + ± 1.2MMr)2−1Ρ( β >> 1; β 1, β

mokidebuli TviT polaronis Ρ v )−impulsis sidideze da ganisazRvrebaSemdegi TanafardobiT:r)GFGF 2Γzrel( β , Ρ)≡ Γ0rel( β ) =3αN0( β ) M f ( 2M)r)2( β >> 1, Ρ / 2M |(3.81)N−[ e − ] 10(β ) = β 1polaronis dabalsixSiruli eleqtrogamtarobisa da dabaltemperaturuliZvradobis sapovnelad fgm-Si visargebloT (3.77), (3.79) da (3.80) –formulebiT. martivi gamoTvlebis Sesrulebis Semdeg Cven vRebulobTSemdegi saxis gamosaxulebebs polaronis kuTri eleqtrogamtarobisa daZvradobisTvis [125]:12th(βω)GFGF2 −1Γ⎡ ⎤=20rel( β)β GFRe σ ( ω)Ne β Mcos⎢Γ0( β)2 2GFrelω ω + Γ ( β)⎣ 2 ⎥(3.82)⎦0relGF−1( ω

f ( k)≡< 0 | ekr ξ 2F1( k)2 2 2 12| 0 > | ; sadac F2(k)= B0( a)(k + 4b) ;F ( k)iµ re=28 4 2 12 610 8 28da F ( k ) = 16384b[ b B ( a)k + 2bB ( a)B ( a)k + b ( B ( a)+ 2B( a)B ( a k +1 11 221 3)106 12 2+ 2b( B ( a)B ( a)+ B ( a)B ( a))k + b ( B ( a)+ 2B( a)B ( a))k142332142 16 2+ 2b B ( a)B ( a)k + b B ( a)](3.84)34444+aq:B0( a)+ 45a2;= 14 + 42a; B1 ( a)= 1−3a;B2( a)= 36 −132a+ 90a222B ( a)= 240 −144a−1200; B a)= 448 + 1344a+ 1440a= 32B( )3a4(0axolo, a da b - sidideebi (mudmivebi) warmoadgenen variaciul parametrebsfgm-Si. ganvixiloT polaronis fgm-is zRvruli SemTxvevebi:I. Zlieri eleqtron-fononuri urTierTqmedeba:Zlieri eleqtron-fononuri bmis zRvrul SemTxvevaSi, rodesacα >> 1, M >> 1;( M → ∞,C → ∞),µ → 1GFGFtalRuri funqcia gadadis pekaris talRur funqciaSi:2 2 −bΠξ[ 1+b ξ a b ξ ] e ;u ( ξv ') ⇒ Nα = b Π; β ≡ β = a b 2Π Π Π;0Π Π+ΠΠe(ix. 1.33 – paragrafi ) fgm-isda manormirebeli mamravli tolia:NΠ2b=π(14+ 42a3ΠΠ+ 45a2Π− sididis,)xoloaΠdabΠ-pekaris variaciuli parametrebia, romlebic SerCeulerTeulTa sistemaSi moicemian Semdegi saxiT: b = 0,65852α; a = 0, 4516. (ix.1.3.1 paragrafi ). vinaidan f −Πfunqcia warmoadgens ori mravalwevrisSefardebas da fgm-Si da M M + 1- aris polaronis efeqturi masa,=GFΠamitom Tu SevinarCunebT wamyvan wevrsmaSin Cven miviRebT:α −s rigis mixedviT f −funqciaSi,12 2 1216384bΠB1( aΠ)kf ( 2M) ≈; k = 2M ; ( α >> 1)(3.85).2 2 12( k + 4b)Πvinaidan pekaris TeoriaSi4MΠ≈ α da = M + 1 → MΠ,MGFamitom xarisxTa12rigis gamoTvla gviCvenebs, rom: f ( 2M) ≈ α− ;( α >> 1).amrigad, dabaltemperaturuliZvradobisTvis polaronis fgm-Si vRebulobT Semdegi saxisyofaqcevasα −bmis mudmivas xarisxis rigis mixedviT [125]:µ3 GF13oΠ ; ( >> 1, β >> 1)e= exp( β )2αf (M 3e≈ exp( β ) α2M) 2α (3.86)141

maSin rodesac polaronis pekaris TeoriaSi (ix. 1.3.1- paragrafi) dabaltemperaturuliZvradobis yofaqceva α −bmis mudmivas xarisxis rigismixedviT moicema Semdegi damokidebulebiT:5µ0≡ µ0Π ≈ α ;( α >> 1, β >> 1) (SevniSnavT, rom Zlieri eleqtron-fononuri bmisSemTxvevaSi ( α >>1)polaronis fgm-i gadadis polaronis pekaris modelSi).II.susti eleqtron-fononuri urTierTqmedeba:susti eleqtron-fononuri bmis zRvrul SemTxvevaSi, rodesacMGFα < 1,MGF> 1)2α3GF(3.87)amrigad, polaronis dabaltemperaturuli dreifuli ZvradobisTvisvRebulobT iseTive mniSvnelobas, rogoric napovnia polaronis frolixismodelSi.dabolos unda aRiniSnos, rom sadisertacio naSromSi avtorismier ganviTarebuli formalizmi (meTodebi) da miRebuli ganzogadoebulikvanturi kinetikuri gantolebebi korelaciuri funqciebisaTvisSesaZlebelia gamoyenebuli iqnas wrfivi eleqtronuli da polaronuligadatanis meqanikuri koeficientebis (dreifuli Zvradoba, dinamiuri gamtaroba)gamosaTvlelad urTierTqmedebis rigis mixedviT SeSfoTebisTeoriis maRal miaxloebebSi zemoTganxiluli modelebisTvis, da myarisxeulebis fizikis ((polaruli) naxevargamtarebi, ionuri da kovalenturikristalebi da sxva). kvantur dinamiuri sistemebis sxva modelTaTvisac,romlebic urTierTqmedeben fononebTan (eleqtronebis gabnevaarapolarul optikur fononebze, piezoeleqtruli gabneva, polaronisfeinmanis modeli da sxva).142

daskvna1. sxvadasxva midgomebis-mowesrigebul operatorTa da liuvilissuperoperatoruli formalizmisa da proeqciuli operatoris meTodisgamoyenebiT, sawyisi korelaciebis gaTvaliswinebiT- gamoyvanilia axali,zusti, ganzogadoebuli kvanturi evoluciuri (kinetikuri) gantolebebidrois ormomentiani wonasworuli korelaciuri funqciebisTvis, dinamiuriqvesistemisTvis romelic urTierTqmedebs bozonur velTan (TermostatTan).miRebul gantolebaTa dajaxebiTi integralebi Seicaven rogorc wevrebs,romlebic aRweren namdvili korelaciebis evolucias droSi, aseve sawyisikorelaciebis evoluciur wevrebs, romlebic ganpirobebulia qvesistemisurTierTqmedebiT bozonur TermostatTan drois sawyis momentSi.2. SeSfoTebis Teoriis meore miaxloebaSi – qvesistemis TermostatTanurTierTqmedebis hamiltonianis mixedviT – napovnia ganzogadoebulikvanturi kinetikuri gantolebebi gamoricxuli bozonuriamplitudebiT korelaciuri funqciebisTvis, rogorc markoviseuli, isearamarkoviseuli saxiT, romlebic Seicaven cxadad gamoyofil sawyisikorelaciebis evoluciur wevrebs.3. ganviTarebuli midgoma da formalizmi gamoyenebulia gadaugvarebelfarTozonian, erTgvarovan naxevargamtarebsa da ionur kristalebSieleqtronuli da polaronuli gamtarobisa da dabaltemperaturulidreifuli Zvradobis wrfivi kvanturi Teoriis asagebad. ganxilul kvanturdinamiur qvesistemebis modelTaTvis, romlebic urTierTqmedeben fononurvelTan - eleqtron-fononuri sistemisTvis, frolixisa da akustikuripolaronis modelTaTvis, polaronis fgm–sTvis gamoyvanilia ganzogadoebulikvanturi evoluciuri gantolebebi wonasworuli korelaciuri funqciebisTvis– “deni-denze” (“siCqare-siCqareze”) – eleqtronisa da polaronisTvisSfm-is gamoyenebis gareSe.4. eleqtron-fononuri sistemisTvis, frolixisa da akustikuripolaronis modelTaTvis, SeSfoTebis Teoriis meore miaxloebaSi, sustieleqtron-fononuri urTierTqmedebis SemTxvevaSi da erTi zonismiaxloebaSi eleqtronisaTvis gamoyvanilia da gamokvleulia markovissaxis kinetikuri gantolebebi eleqtronis siCqaris operatoriskomponentebis saSualo mniSvnelobebis diagonaluri matriculielementebisaTvis, romlebic warmoadgenen bolcmanis tipis gantolebebs,saidanac gamoricxulia fononuri amplitudebi. Gganxilulia eleqtronisaradrekadi gabnevis procesebi fononebze da dadgenilia, rom ganxilulmodelebSi adgili aqvs relaqsaciur process korelaciuri funqciebisoscilaciebiT. Nnapovnia eleqtronis impulsis (siCqaris) relaqsaciis six-Sireebis analizuri gamosaxulebebi kristalis dabali temperaturebis Sem-TxvevaSi. gamoTvlilia eleqtronis “siCqare-siCqareze” korelaciurifunqciebis milevis dekrementebi da oscilirebadi faqtorebi.5. gamokvleulia da dadgenilia, rom eleqtronis siCqaris (impulsis)mcire mniSvnelobebisaTvis, siCqaris relaqsaciis droebi (sixSireebi)ganxilul modelebSi ar aris damokidebuli impulsis sidideze. mciresiCqareebiTYmoZravi eleqtronebisTvis Zalian dabali temperaturebis drosnapovnia dabalsixSiruli eleqtrogamtarobisa da eleqtronis dreifuliZvradobis gamosaTvleli formulebi.6. frolixis polaronis modelSi miRebuli gamosaxulebebi eleqtronisdabaltemperaturuli dreifuli Zvradobisa da dinamiurigamtarobiTvis warmoadgens osakas mier napovni Sedegis ganzogadoebasmcire intensivobis mqone dabalsixSirul gareSe eleqtrul velSi, rac143

faqtiurad SesaZlebelia ganxiluli iqnas, rogorc drudes formulakuTri eleqtrogamtarobisTvis. napovnia agreTve statikuri ( ω = 0)eleqtrogamtarobisa da dabaltemperaturuli dreifuli Zvra-dobisanalizuri gamosaxulebebi, rogorc frolixis, aseve akustikuri polaronismodelebSi.7. rogorc gamoTvlebi gviCvenebs, eleqtronebis gabnevisas polaruloptikur fononebze, dabaltemperaturuli dreifuli ZvradobisTvis(dcmobility; ω = 0 ) miRebuli mniSvneloba 3-jer aRemateba Zvradobis immniSvnelobas, romelic miiReba bolcmanis kinetikuri gantolebisgamoyenebiT da amoxsniT relaqsaciis drois miaxloebaSi. miRebuli Sedegi3 Kwarmoadgens – “ BΤproblemis” – nawilobriv gadawyvetas frolixis2 hω0polaronis dabaltemperaturuli Zvradobis TeoriaSi.8. eleqtronebis gabnevisas akustikur fononebze (akustikuripolaronis modeli) miRebuli dabaltemperaturuli dreifuli Zvradobis( ω = 0) mniSvneloba 2-jer naklebia Zvradobis im mniSvnelobaze, romelicaseve miiReba bolcmanis kinetikuri gantolebis amoxsnisas relaqsaciisdrois miaxloebaSi.9. ganxilul modelebSi napovnia agreTve eleqtronis dreifulZvradobaze temperaturuli Sesworebebi, romlebic ganpirobebulia sawyisikorelaciebis evoluciuri wevrebis arsebobiT gamoyvanili kinetikurigantolebebis dajaxebiT integralebSi da naCvenebia, rom es Sesworebebiwarmoadgenen mcire sidideebs ganxiluli Teoriis farglebSi.10. polaronis fgm-sTvis miRebuli kvanturi kinetikuri gantolebebieleqtruli denis operatoris komponentebis (polaronis impulsis) droisormomentiani wonasworuli korelaciuri funqciebisTvis gamoyenebuliapolaronis dreifuli Zvradobisa da eleqtrogamtarobis tenzorisgamosaTvlelad. Gganxilul erTzonian izotropul SemTxvevaSi,markoviseul miaxloebaSi polaronis dinamikisTvis, napovnia miaxloebiTigamosaxulebebi korelaciuri funqciebisTvis.11. kristalis Zalian dabali temperaturebis SemTxvevaSi gamoyvaniliabolcmanis tipis kinetikuri gantoleba korelaciuri funqciisdiagonaluri matriculi elementisTvis, romelic Seesabameba polaronisZiriTad mdgomareobas. gamokvleulia polaronis aradrekadi gabnevisprocesebi fononebze. napovnia impulsis relaqsaciis sixSiris (drois)analizuri gamosaxuleba da dadgenilia, rom mcire siCqariT moZravipolaronisTvis impulsis relaqsaciis sixSire (dro) ar aris damokidebuliimpulsis sidideze.12. kubos wrfivi reaqciis Teoriis gamoyenebiT miRebuliadabalsixSiruli eleqtrogamtarobis tenzoris analizuri gamosaxulebaeleqtron-fononuri sistemisaTvis erTzonian miaxloebaSi da fononebisdispersiis zogadi (izotropuli) kanonis SemTxvevaSi. gamoTvliliapolaronis dabaltemperaturuli dreifuli Zvradoba fgm-Si. am modelSinapovnia agreTve temperaturuli Sesworeba polaronis dreifulZvradobaze, romelic ganpirobebulia sawyisi korelaciebis evoluciuriwevrebis arsebobiT miRebuli kinetikuri gantolebebis dajaxebiT integralebSi,da dasabuTebulia, rom es temperaturuli Sesworeba warmoadgensmcire sidides.13. ganxilulia da gaanalizebulia polaronis dabaltemperaturulidreifuli Zvradobis yofaqceva susti ( α < 1) da Zlieri (α >>1) eleqtron-144

fononuri urTierTqmedebis zRvrul SemTxvevebSi. susti eleqtronfononuriurTierTqmedebis SemTxvevaSi ( MGF→ 0) , rodesac polaronis fgmgadadis polaronis frolixis modelSi, polaronis dabaltemperaturulidreifuli ZvradobisTvis (γ >>1; ω =0) vRebulobT iseTive miSvnelobas,rogoric napovnia polaronis frolixis modelSi. Zlieri eleqtronfononuriurTierTqmedebis SemTxvevaSi ( MGF→∞), rodesac polaronis fgmaRadgens polaronis pekaris naxevradklasikur Teorias, dabaltemperaturulidreifuli Zvradobis yofaqceva moicema Semdegi313TanafardobiT: µGF~ e.exp(γ ) α ; (0;2γ = β >> 1;ω =0); anu polaronisdabaltemperaturuli Zvradoba Zlieri eleqtron-fononuriurTierTqmedebis SemTxvevaSi ( α >> 1)izrdeba α -bmis mudmivas mecameterigis proporciulad am mudmivas didi mniSvnelobebis dros, maSinrodesac polaronis pekaris TeoriaSi dabaltemperaturuli Zvradobaizrdeba misi mexuTe rigis proporciulad: µ ~ α5Π; rodesac α >>1;( h = m = ω0= 1; β >> 1 ; ω = 0).14. sadisertacio naSromSi Catarebuli gamokvlevebi gviCvenebs, romganviTarebul meTodebs, romlebic dafuZnebulia kinetikuri gantolebebismiRebaze wonasworuli korelaciuri funqciebisTvis da maT gamoTvlaze,gansxvavebiT sxva midgomebisgan, ar mivyavarT ganSladi wevrebisaganSedgenili usasrulo mwkrivebis ajamvis aucileblobasTan kvazinawilakis(eleqtronis, polaronis) urTierTqmedebis mixedviT fononebTan, kristalzemodebuli gareSe eleqtruli velis dabali ( ω → 0) sixSireebis Sem-TxvevaSi.naSromSi dasabuTebulia, rom arsebuli sawyisi korelaciebisevolucia da korelaciuri funqciebis oscilaciebi drois mixedviT,romlebic ganpirobebulia kvazinawilakis (zogad SemTxvevaSi kvanturidinamiuri qvesistemis) urTierTqmedebiT fononur (bozonur) velTan droissawyis momentSi, gavlenas ar axdenen relaqsaciur procesebze da isiniwarmoadgenen Zvradobebze temperaturuli Sesworebebis ZiriTad mizezs(wyaros) ganxilul modelebSi.145

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danarTibozonuri (fononuri) amplitudebis gamoricxva qronologiur daantiqronologiur T-namravlTa daxmarebiTganvixiloT axla ufro dawvrilebiT bozonuri (fononuri) cvladebisgamoricxvis meTodi (teqnika) qronologiur da antiqronologiur T-namravlTa daxmarebiT Cvens mier warmodgenili im klasis sistemebisaTvis,romlebic aRiwerebian (2.1), (2.2) saxis hamiltonianiT [109].fononuri cvladebis (amplitudebis) gamoricxva CavataroT HinthamiltonianisTvis, romelic warmoadgens wrfiv formas boze-amplitudebisgan.cxadia, rom komutatorebi: ⎡H% ( ξ) , H % ( τ)⎤ , ⎡H( ) , H ( )( ),H ( )γintλ ⎦ −⎣intint⎦−% %⎣⎤int,ξintλ ⎦ −⎡H% %⎣⎤int- warmoadgenen C sidideebs (ricxvebs) bozonuri (fononuri)velis mimarT.Tu gamoviyenebT magnusis formulas [6,81] da veilis igiveobas [6,82]Cven gveqneba:β β γ⎡ ⎤ ⎡1⎤ΤΣexp⎢− % Lint( ) ⎥ = exp ⎢ ⎡ % Lint ( ),% L∫dλH λint ( )2∫dγ∫dλ ⎣H γ H λ ⎤⎦ ⎥×−⎣ 0 ⎦ ⎣ 0 0⎦β⎡⎤× exp ⎢−% L∫ dλHint( λ)⎥;⎣ 0 ⎦tt ξ⎡i⎤ ⎡ 1⎤ΤΣexp⎢ % L∫ int ( ) ⎥ = exp ⎢⎡ % L2int ( ),% LadτH τint ( ) ⎤ ⎥×2∫dξ∫dτ ⎣H ξ H τ⎦−⎣h0 ⎦ ⎣ h0 0⎦t⎡ i⎤× exp ⎢−% L∫ dτHint ( τ)⎥;⎣ h0 ⎦tt ξ⎡ i⎤ ⎡ 1⎤ΤΣexp⎢− % Rint ( ) ⎥ = exp ⎢⎡ % R∫ 2int ( ),% RdτH τ int ( ) ⎤ ⎥×2∫dξ∫dτ ⎣H τ H ξ⎦−⎣ h0 ⎦ ⎣ h0 0⎦⎡ i× exp ⎢−⎣ ht∫0⎤% RdτHint ( τ ) ⎥.⎦ganvixiloT axla Semdegi gamosaxuleba:(1.1)153

βt⎡ ⎡ ⎤ ⎡ ⎤ΡΣ( ) ⎢ΤΣexp⎢− % Liint( ) ⎥⋅Τ Σexp⎢ % Lβ ∫dλHλa ∫dτHint( τ)⎥×⎢⎣⎣ 0 ⎦ ⎣h0 ⎦tβ γ⎡i⎤⎤ ⎡1⎤×ΤΣexp⎢ % Lint( ) ⎥⎥ = exp ⎢ ⎡ % Lint( ),% L∫dλH τint ( ) ⎤ ⎥×2∫dγ∫dλ⎣H γ H λ ⎦−⎣h0 ⎦⎥⎦⎣ 0 0⎦t ξt ξ⎡ 1 ⎤ ⎡ 1× exp ⎢ ⎡ % L2 int( ), % Lint ( ) ⎤ ⎥⋅ exp⎢×22∫dγ∫dτ ⎣H ξ H τ ⎦−⎣ h2∫dξ∫dτ0 0 ⎦ ⎣ h0 0× % R⎡ ( ), % Rint int ( ) ⎤⎣H τ H ξ ⎤⎦⋅I( t, β),−⎦sadac, gansazRvris Tanaxmad, I( t,β ) funqcionalisTvis gvaqvs SemdegimniSvneloba:βt⎡ ⎤%⎡i%⎤⎢ ∫ int ⎥ ⎢ ∫ int ⎥h0 0LL( , β) ( β){ exp λ ( λ) exp τ ( τ )I t =ΡΣ− d H ⋅ d H ⋅⎣⎦ ⎣⎦t⎡ iR⎤⎪⎫⋅exp − dτH%⎢int ( τ ) ⎥⎬.⎣ h∫0 ⎦⎭⎪da bolos, Tu gamoviyenebT Tanafardobebs:β γ β β⎡1 ⎤L L 1L L⎢ dγ dλ⎡H% int( γ) , H% int ( λ) ⎤ ⎥+ dλ dγH% int( λ) , H%int ( γ ) =2∫ ∫ ⎣⎦−⎣2∫ ∫0 0 ⎦ 0 0=β∫γdγ dλH% γ , H%λ ;∫0 0LintL( ) ( )1 1ξ τ ( ),( ) ( ) ( )2 ⎡ ξ τh⎤ − τ ξ τ ξ2=t ξt t% L % L % L % L2∫d ∫d ⎣Hint Hint ⎦−2∫d ∫d Hint Hinth0 0 0 01=−2ht∫τ∫0 0LintintL( ) ( )dτ dξH% ξ H%τ1 1ξ τ ( τ ),( ξ) τ ξ ( τ ) ( ξ)2h⎡ ⎤ − 2=t ξt tR R R Rd d H% 2 intH% intd d H% 2intH%∫ ∫ ⎣⎦−∫ ∫inth0 0 0 01=−2ht∫τ0 0RintintR( ) ( )dτ dξH% τ H%ξ1 1τ ξ ( τ) , ( ξ) τ ξ ( τ) ( ξ)2h⎡ ⎤ + 2+t t t tL R L Rd d H% 2 intH% intd d H% 2intH%∫ ∫ ⎣⎦int− ∫ ∫0 0h0 01 1+ τ ξ ξ τ = τ ξ τ ξ2ht t t τR L L Rd d H% 2 intH% intd d H% 2intH%∫ ∫ ∫ ∫inth0 0 0 0( ) ( ) ( ) ( )β tβ t⎡ % L % L i % L % L∫d ∫d ⎣Hint Hint ⎦int int− ∫d ∫d H H0 0h0 0i− λ τ ( λ) , ( τ) ⎤ − λ τ ( λ) ( τ ) −2h2β tβ ti− % L % L iint int=− % L % Lint int2h∫dλ∫d τH τ H λh∫dλ∫dτH λ H τ( ) ( ) ( ) ( )0 0 0 0∫int;;;;(1.2)154

β tβ t% L % R i % L % R∫d ∫d ⎣Hint Hint ⎦int int− ∫d ∫d H H0 0h0 0iλ τ ( ),( ) ( ) ( )2 ⎡ λ τh⎤ + λ τ λ τ2+β iβ ti+ % R % L iint int= % L % Rint int2h∫dλ∫dτH τ H λh∫dλ∫dτH λ H τ( ) ( ) ( ) ( )0 0 0 0da SevasrulebT gasaSualoebas bozonuri (fononuri) velismdgomareobebis mixedviT, Cven miviRebT [109]:∑ΡΣ( β) ⎡⎣H % ( γ) H%( λ) ⎤⎦ = ⎡⎣(1 + N ( β )) e C%( s , γ ) ×L L −hω( k)( γ −λ) Lint intkkH 0k+ ( )( − ) +× % L hω k γ λ( , ) + ( ) % L( , ) % LC ( , ) ⎤kHs λ N ;0 kβ e CkH s γ C0 kHs λ0 ⎦ΡΣ( β) ⎡ %⎣H ( ξ) H%( τ ) ⎤⎦ = ∑ ⎡⎣(1 + N ( β)) e C%( s , ξ ) ×L L −iω( k )( ξ−τ)Lint intkkH 0k+× % Liω ( k)( ξ − τ )CkH( s, τ ) + ( ) % + L( , ) % LN ( , ) ⎤ ;0kβ e CkH s ξ C0 kHs τ0 ⎦− ( )( − )ΡΣ( ) ⎡ % R⎣ int( ) % R iω k τ ξint( ) ⎤⎦ = ∑ ⎡⎣(1 + ( )) % Rβ H τ H ξ Nkβ e CkH( s, τ ) ×0% ( , ) ( ) % ( , ) % ( , ) ⎤ ;+ R iω( k)( τ− ξ ) + R R× CkH s ξ + N0 k β e CkH s τ C0 kH s ξ0 ⎦kΡΣ( β) ⎡⎣H % ( τ) H%( ξ) ⎤⎦ = ⎡⎣(1 + N ( β)) e C%( s, τ ) ×L R −iω( k )( τ −ξ) Lint intkkH 0k% ( , ) ( ) % ( , ) % ( , ) ⎤ ;+ R iω( k)( τ− ξ ) + L R× CkH s ξ + N0 kβ e CkH s τ C0 kHs ξ0 ⎦ΡΣ( β) ⎡⎣H % ( λ) H%( τ ) ⎤⎦ = ⎡⎣ (1 + N ( β )) e C%( s, λ)×× C%+ LkH 0∑∑L L ω ( k)( iτ −hλ)Lint intkkH 0k( s, τ) N ( β) e C% ( s, λ) C%( s, τ ) ⎤ ;−ω( k)( iτ − hλ)+ L L+ k kH0 kH 0 ⎦∑ΡΣ( β) ⎡ ( ) ( ) ⎤ ⎡⎣H% λ H%τ⎦=⎣(1 + N ( β )) e C%( s, λ)×L R ω ( k)( iτ −hλ)Lint intkkH 0k.(1.3)+ − ( )( − ) +× % R ω k iτ hλ( , ) + ( ) % L( , ) % RC ( , ) ⎤kHs τ N .0 kβ e CkH s λ C0 kHs τ0 ⎦martivi gamoTvlebis Catarebis Semdeg advilad vipoviT( t β )exp ⎡Φ,⎤⎣ LRS , , ⎦funqcionalis saboloo gamosaxulebas [109]:β⎡t⎡ ⎤ ⎡ ⎤ΡΣ( ) ⎢ΤΣ exp⎢− % Liint( ) ⎥⋅Τ Σexp⎢% Lβ ∫dλHλa ∫dτHint( τ)⎥×⎢⎣⎣ 0 ⎦ ⎣h0 ⎦(1.4)t⎡ i ⎤⎤×Τ exp −int ( ) = exp ⎡Φ , , ( , ) ⎤; > 0, > 0Σ ⎢% R∫ dτH τ ⎥⎥⎣ t βLRS ⎦ t β⎣ h0 ⎦⎦⎥da (1.4) formuliT gamoxatuli ( t β )ganisazRvreba (2.20) gamosaxulebiT.Φ funqcionalis mniSvnelobaLRS , ,,155

disertaciis TemasTan dakavSirebiT gamoqveynebuliaSemdegi naSromebi:1. ???? ?.?., ????? ?.?. ??????????? ???????? ? ?????????? ?????? ???????? ????????? ????????????. 24-? ?????????? ????????? ?? ?????? ?????? ??????????.?????? ????????: ????? II: ??????????? ??????? ??? ?????? ????????????. ???????,8-10 ????????, 1986 ?., ???. 201-202.2. ????? ?.?., ???? ?.?. ? ?????? ???????? ????????. ?????????? ???????????: ???-???????? ???????? ?????????????? ??????. ???????, 14-17 ??? 1991 ?. ????????????????, ???. 1-140. ?????? ????????, ???. 83.3. Kotiya B.A., Los’ V.F. Exact equations for subsystem correlation functions and densitymatrix. The 18th IUPAP International Conference on Statistical Physics. Berlin, 2-8August, 1992, Programme and Abstracts. Exact and Rigorous Results, p. 133.4. Kotiya B.A., Los’ V.F. Exact equations for subsystem correlation functions and densitymatrix. Application in Polaron mobility. International Workshop “Polarons andApplications”, May 23-31, 1992, Puschino, Russia. Abstracts, p. 18.5. Kotiya B.A., Los’ V.F. Exact equations for subsystem correlation functions and densitymatrix. Application in Polaron mobility. Proceedings in Nonlinear Science. Polarons andApplications. Ed. Lakhno V.D. (John Wiley and Sons, Chichester, 1994), pp. 407-418(impaqt. datvirTvis mqone).6. Kotiya B.A., Los’ V.F. To the theory of transport Phenomena in open systems. TheNorwegian Academy of Technological Sciences. The Lars Onsager Symposium. CoupledTransport Processes and Phase Transitions. June 2-4, 1993,Trondheim, Norwey. Abstracts,p. 89.7. Kotiya B.A., Los’ V.F. Exact Equations for Correlation Functions and Density Matrix of aSubsystem Interacting with a Heat Bath and their Application in the Polaron Mobility.European physical Society. Eps9. Trends in Physics. Firenze. Italy. September 14-17, 1993.Abstracts. Symposium 29. Statistical Machanics: Rigorous Results. ED. Systems, p. 130.8. kotia b. gadatanis movlenebis Teoria eleqtron-fononuri sistemisaTvis.saqarTvelos teqnikuri universitetis profesor-maswavlebelTasamecniero-teqnikuri konferencia. programa, 16-19 noemberi, Tbilisi,1993 w. I-Teoriuli fizika, gv. 32.9. kotia b., losi v. polaronis Zvradobis TeoriisTvis. saqarTvelos mecnierebaTaakademiis moambe. t. 149, #1, 1994 w., Tbilisi, fizika, gv. 61-68.10. ????? ?.?., ???? ?.?. ? ?????? ?????????????????? ??????????? ????????? ??????? ????????????? ????????. ?????????? ???????? ????. ???????? ????????????. ????? ??????????, ?. 59, ? 8, 1995 ?.: ? ???????? ??????????? ????????????????? «?????????????????? ????????? ? ????????????? ? ???????????????????????». ??????. ?????? 1995 ?., ???.133-138 (impaqt. datvirTvis mqone).11. Kotiya B.A., Los’ V.F. Low-Temperature Electron Mobility of Acoustical Polaron. InPerspectives of Polarons. Editors: G.N. Chuev and V.D. Lakhno. Russian Academy ofSciences. World Scientific. Singapore. New Jersey. London. Hong Kong. 1996, p. 216-222(impaqt. datvirTvis mqone).156

12. Kotiya B.A., Los’ V.F. On the theory on Transport phenomena in the Polaron’s Systems.The 19th IUPAP International Conference on Statistical Physics. Xiamen 31 July-4August, 1995. Programme and Abstracts. Transport and Relaxation, p. 42.13. Kotiya B.A., Los’ V.F. Gigilashvili T.G. On the theory of transport phenomena in.electron-phonons’systems. The 20 th IUPAP International Conference on StatisticalPhysics. Paris, UNESCO, Sorbonne. July 20-25, 1998. Book of Abstracts, Nonequlibriumsystems, p. 7.14. Kotiya B.A. On the theory of exact equations for correlation functions of the systemInteracting with a thermostat. Georgian Engineering News, 2005, ? 1, pp. 7-13.15. Kotiya B.A. On the theory of low-temperature electron mobility in the electron-phononsystem and Frohlich’s model of the Polaron. Georgian Engineering News, 2005, ? 2, pp.11-21.16. ????? ?.?. ? ???????? ????????? ???????????? ??????, ????????????????? ????????? ?????. ????? I. ????????? ????????????? ?????????? ? ?????????????????? ???????? ????????. Georgian Engineering News, 2005, ? 4, ???. 7-17.17. ????? ?.?. ? ???????? ????????? ???????????? ??????, ????????????????? ????????? ?????. ????? II. ???????????? ????????? ??? ?????????????? ???????? ??????? ?????. Georgian Engineering News, 2006, ? 1, ???. 7-17.157