Influence of Coulomb scattering of electrons and holes between ...
Influence of Coulomb scattering of electrons and holes between ... Influence of Coulomb scattering of electrons and holes between ...
146ARTICLE IN PRESSS.A. Moskalenko et al. / Physica E 39 (2007) 137–149results illustrated in Fig. 1 with the ones obtained in thissection. Putting the aim into practice we need to calculatethe wave functions and the energy spectrum of the lowestexciton band. We will confine ourselves into the frame offour exciton levels model. By this restriction we willdetermine the influence of nearly situated exciton levelswith the same wave vector on the lowest exciton level inwhich the BEC of magnetoexcitons takes place. Thesecalculations are needed to obtain more exact expressionsfor the exciton wave function as well as of the excitoncreation and annihilation operators. They will be representedas coherent superpositions of their zero orderexpressions and will lead us to the necessity to investigatethe BEC in the exciton superposition state.For the beginning we will define the magnetoexcitoncreation operator [25–27] characterized by the number n ofthe electron LL and by the number m of the hole LL, aswell as by the 2D wave vector k with two components k xand k yX y n;m;k ¼ p1 ffiffiffiffiX exp½ ik y tl 2 Ša y n;kNx =2þt by m;k x =2 t . (43)tThe state with the pair of numbers ðn; mÞ equals to (0,0) willfor simplicity be denoted as the state 1, the pair of numbers(1,1) gives rise to the magnetoexciton state 2, the pairs (1,0)and (0,1) will be mentioned as the states 3 and 4,correspondingly. The exciton wave function c i ðkÞ isobtained as action of this operator on the vacuum statej0i determined as a n;p j0i ¼0; b m;p j0i ¼0c i;k ¼ X y n;m;kj0i, (44)where i labels the quantum numbers i !ðn; mÞ. Thus, onecan straightforwardly prove the orthogonality and normalizationproperties of eigenstates:hc i;k jc j;k 0i¼d i;j d k;k 0. (45)The matrix elements of the Hamiltonian Eq. (2), where thechemical potentials m e and m h are omitted as we areconcerned in single exciton properties, on the wavefunctions Eq. (44) are denoted asH ij ðkÞ ¼hc i;k j ^Hjc j;k i¼d i;j hc i;k j ^H 0 jc i;k iþV i;j ðkÞ, (46)where V i;j ðkÞ ¼hc i;k j ^H Coul jc j;k i. Analytical expression forV i;j ðkÞ one can derive from Eqs. (6) to (7) (for details seee.g. in Ref. [43]). The magnetoexciton energy bands will bedetermined at first in the zeroth order of the perturbationtheory, when only the diagonal matrix elements H ii andtwo off-diagonal matrix elements V 12 ðkÞ and V 21 ðkÞ aretaken into account. The reason for the choice of the zeroorder approximation will be given below. All other offdiagonalmatrix elements, which are smaller than thevalues V 12 and V 21 will be introduced in higher orders ofthe perturbation theory. We consider first the nondegeneratecase when o ce ao ch ,so that the four zero ordermagnetoexciton bands accounted from the LLLs areE 1 exðkÞ ¼Eð0;0Þ ex ðkÞ ¼H 1;1 ðkÞ ¼V 1;1 ðkÞ ¼ I ð0;0Þex ðkÞ, ð47aÞE 2 exðkÞ ¼Eð1;1Þ ex ðkÞ ¼H 2;2 ðkÞ ¼_o ce þ _o ch I ð1;1Þex ðkÞ, ð47bÞE 3 exðkÞ ¼Eð1;0Þ ex ðkÞ ¼H 3;3 ðkÞ ¼_o ce I ð1;0Þex ðkÞ, ð47cÞE 4 exðkÞ ¼Eð0;1Þ ex ðkÞ ¼H 4;4 ðkÞ ¼_o ch I ex ð0;1Þ ðkÞ. ð47dÞThe ionization potentials of four bands were determined asI ðn;mÞex ðkÞ ¼ X sF e2h ðp; n; k x p; m; p s; n; k x p þ s; mÞ exp½ik y sl 2 Š¼ e2e 0 lx 2 exp 12Z 1dx0 nþmx 22J 0 ðklxÞ; n; m ¼ 0; 1. ð48ÞAn explicit expression for I ðn;mÞex ðkÞ can be found e.g. in Ref.[44].A more exact expression of the magnetoexciton wavefunction is the linear combinationc n;k ¼ X4i¼1a in c i;k . (49)The new functions c n;k are denoted by Greek symbolsn ¼ a; b; g; d, whereas the initial zero order functions c ik byLatin letter i ¼ 1; 2; 3; 4. They obey to the Schro¨dingerequation^Hc n;k ¼ E n ðkÞc n;k (50)what leads to four linear equations which determine theenergy spectrum E n ðkÞ and the coefficients a in . Theequations contain the matrix elements Eq. (46) and havethe formX 4i¼1a in H ij ðkÞ ¼E n ðkÞa jn ; j ¼ 1; 2; 3; 4. (51)The energy spectrum E n ðkÞ can be found solving secularequationH 11 ðkÞ E n ðkÞ V 12 ðkÞ V 13 ðkÞ V 14 ðkÞV 21 ðkÞ H 22 ðkÞ E n ðkÞ V 23 ðkÞ V 24 ðkÞ¼ 0.V 31 ðkÞ V 32 ðkÞ H 33 ðkÞ E n ðkÞ V 34 ðkÞ V 41 ðkÞ V 42 ðkÞ V 43 ðkÞ H 44 ðkÞ E n ðkÞ (52)Neglecting 10 off-diagonal matrix elements denoted as afirst order infinitesimals ejV 13 jjV 14 jjV 23 jjV 24 jjV 34 je (53)we will obtain the zero order solutions of Eqs. (51) and (52)E 0 a;b ðkÞ ¼ H 11ðkÞþH 22 ðkÞ2 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH 11 ðkÞ H 22 ðkÞÞ 2 þ 4V 2 122ðkÞ ,E 0 g ðkÞ ¼H 33ðkÞ; E 0 d ðkÞ ¼H 44ðkÞ. ð54Þ
ARTICLE IN PRESSS.A. Moskalenko et al. / Physica E 39 (2007) 137–149 147In the limit V 12 ðkÞo_o ce þ _o ch two magnetoexcitonbands look as follows:E 0 a ðkÞ ¼E1 ex ðkÞ V 2 12 ðkÞ ,_o ce þ _o chE 0 b ðkÞ ¼E2 ex ðkÞþ V 2 12 ðkÞ .ð55Þ_o ce þ _o chThe coefficients a 0 in can be also determined in the zerothorderja 0 1a j2 ¼ja 0 ðH 22 ðkÞ H 11 ðkÞÞ 22b j2 ¼ðH 22 ðkÞ H 11 ðkÞÞ 2 þ V 2 12 ðkÞV 2 12 1ðkÞð_o ce þ _o ch Þ 2 ,ja 0 2a j2 ¼ja 0 V 21b j2 12¼ðkÞðH 22 ðkÞ H 11 ðkÞÞ 2 þ V 2 12 ðkÞV 2 12ðkÞð_o ce þ _o ch Þ 2 ,a 0 2a V 12 ðkÞ; a 0 1b_o ce þ _o V 12ðkÞ,ch _o ce þ _o cha 0 3n ¼ a0 4n ¼ 0 for n ¼ a; b. ð56ÞThe different signs of the coefficients a 0 1b and a0 2aare takento obey Eqs. (51) and will be important in the discussionsbelow. We are interested in a more exact expressions foronly the lowest exciton band taking into account theinfluence of three exciton bands situated upper on theenergy scale. For these three bands involved in this schemethe starting wave function (49) is not sufficient becausethere are another exciton bands which do not affect toomuch the lowest exciton bands, but can stronger influenceon the upper exciton bands. It can be easily shown that firstorder corrections give zero contribution to the energy andthe following contribution to the coefficients a in :a 0 1n ¼ a0 2n ¼ 0,a 0 3n ¼ðV 31a 0 1n þ V 32a 0 2n ÞðH 33 E 0 n Þ ; n ¼ a; b,a 0 4n ¼ ðV 41a 0 1n þ V 42a 0 2n ÞðH 44 E 0 n Þ . ð57ÞThe perturbation theory used here gives rise to the secondorder correction to the lowest exciton band E 00aE 00a ðkÞ ¼ jV 13 ðkÞj 2E 0 a ðkÞ H 33ðkÞ þ jV 14ðkÞj 2 E 0 a ðkÞ H ja 0 1a44ðkÞj2jV 23 ðkÞj 2þE 0 a ðkÞ H 33ðkÞ þ jV 24 ðkÞj 2 E 0 a ðkÞ H ja 0 2a44ðkÞj2þ a 01a a0 2aþ a 02a a0 1aV 13 ðkÞV 32 ðkÞE 0 a ðkÞ H 33ðkÞ þ V 14ðkÞV 42 ðkÞE 0 a ðkÞ H 44ðkÞV 23 ðkÞV 31 ðkÞE 0 a ðkÞ H 33ðkÞ þ V 24ðkÞV 41 ðkÞE 0 a ðkÞ H . ð58Þ44ðkÞSubstituting the coefficients a 0 1a and a0 2a, given by (56) andthe matrix elements V ij ðkÞ from (46) into (58) we obtain thedependence of E 00aðkÞ on the dimensionless wave vector kl.The dependence of the ionization potential of the lowestexciton band on the wave vector k with the account of thesecond order corrections is represented in Fig. 1 by thedotted and dash-dotted lines. This corrections are 40%smaller as compared with the influence of all ELL.In the frame of four level models the magnetoexcitoncreation operator has the formX y ak ¼ a0 1a X y 0;0;k þ a0 2a X y 1;1;k þ a0 3a X y 1;0;k þ a0 4a X y 0;1;k . (59)On its base it is possible to construct a new displacementoperator ^DðN ex Þ and to discuss the phenomenon of BECtaking into account explicitly the LLLs and FELLs.Consider the BEC in the lowest exciton superpositionstate in a simplified variant, when only two main terms inexpression (59) are taken into account and the coefficientsof superposition are a 0 1a a 1 and a 0 2a a 2, with thecondition a 2 1 þ a2 2 ¼ 1. The expression Eq. (31) now canbe rewritten as:X y k ¼ a 1X y 0;0;k þ a 2X y 1;1;k . (60)pffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiThe unitary transformation ^Dð N exÞ¼exp½N exðXy ðkÞXðkÞÞŠ leads do generalized Bogoliubov’s u-v transformationa p ¼ ^Da p ^D y¼ cosðga 1 Þa p sinðga 1 Þ exp½ ik y ðp k x =2ÞŠb y k x p ,b p ¼ ^Da p ^D y¼ cosðga 1 Þb p þ sinðga 1 Þ exp½ ik y ðk x =2 pÞŠa y k x p ,g p ¼ ^Dc p ^D y¼ cosðga 2 Þa p sinðga 2 Þ exp½ ik y ðp k x =2ÞŠd y k x p ,d p ¼ ^Dd p ^D y¼ cosðga 2 Þd p sinðga 2 Þ exp½ ik y ðk =2 pÞŠc y k x p . ð61ÞpHere g ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pl 2 n ex ; n ex ¼ N ex =S; Fermi operators c p ; d pand c y p ; dy p describe the annihilation and creation of anelectron and a hole in the FELL, respectively. Then thenew ground state wave function acquires the followingform:pffiffiffiffiffiffiffiffijC g ðkÞi ¼ ^Dð N exÞj0i¼ Y ðcosðgÞþsinðgÞ expð ik y tl 2 Þ½a 1 a y k x =2þt by k x =2 ttþ a 2 c y k x =2þt dy k x =2 tŠÞ, ð62Þwhich plays the role of vacuum state for the operatorsa p ; b p ; g p ; d p , i.e.a p jC g ðkÞi ¼ b p jC g ðkÞi ¼ g p jC g ðkÞi ¼ d p jC g ðkÞi ¼ 0.With the above definitions one can calculate the averagenumber of excitons (which is equal to the average number
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146ARTICLE IN PRESSS.A. Moskalenko et al. / Physica E 39 (2007) 137–149results illustrated in Fig. 1 with the ones obtained in thissection. Putting the aim into practice we need to calculatethe wave functions <strong>and</strong> the energy spectrum <strong>of</strong> the lowestexciton b<strong>and</strong>. We will confine ourselves into the frame <strong>of</strong>four exciton levels model. By this restriction we willdetermine the influence <strong>of</strong> nearly situated exciton levelswith the same wave vector on the lowest exciton level inwhich the BEC <strong>of</strong> magnetoexcitons takes place. Thesecalculations are needed to obtain more exact expressionsfor the exciton wave function as well as <strong>of</strong> the excitoncreation <strong>and</strong> annihilation operators. They will be representedas coherent superpositions <strong>of</strong> their zero orderexpressions <strong>and</strong> will lead us to the necessity to investigatethe BEC in the exciton superposition state.For the beginning we will define the magnetoexcitoncreation operator [25–27] characterized by the number n <strong>of</strong>the electron LL <strong>and</strong> by the number m <strong>of</strong> the hole LL, aswell as by the 2D wave vector k with two components k x<strong>and</strong> k yX y n;m;k ¼ p1 ffiffiffiffiX exp½ ik y tl 2 Ša y n;kNx =2þt by m;k x =2 t . (43)tThe state with the pair <strong>of</strong> numbers ðn; mÞ equals to (0,0) willfor simplicity be denoted as the state 1, the pair <strong>of</strong> numbers(1,1) gives rise to the magnetoexciton state 2, the pairs (1,0)<strong>and</strong> (0,1) will be mentioned as the states 3 <strong>and</strong> 4,correspondingly. The exciton wave function c i ðkÞ isobtained as action <strong>of</strong> this operator on the vacuum statej0i determined as a n;p j0i ¼0; b m;p j0i ¼0c i;k ¼ X y n;m;kj0i, (44)where i labels the quantum numbers i !ðn; mÞ. Thus, onecan straightforwardly prove the orthogonality <strong>and</strong> normalizationproperties <strong>of</strong> eigenstates:hc i;k jc j;k 0i¼d i;j d k;k 0. (45)The matrix elements <strong>of</strong> the Hamiltonian Eq. (2), where thechemical potentials m e <strong>and</strong> m h are omitted as we areconcerned in single exciton properties, on the wavefunctions Eq. (44) are denoted asH ij ðkÞ ¼hc i;k j ^Hjc j;k i¼d i;j hc i;k j ^H 0 jc i;k iþV i;j ðkÞ, (46)where V i;j ðkÞ ¼hc i;k j ^H Coul jc j;k i. Analytical expression forV i;j ðkÞ one can derive from Eqs. (6) to (7) (for details seee.g. in Ref. [43]). The magnetoexciton energy b<strong>and</strong>s will bedetermined at first in the zeroth order <strong>of</strong> the perturbationtheory, when only the diagonal matrix elements H ii <strong>and</strong>two <strong>of</strong>f-diagonal matrix elements V 12 ðkÞ <strong>and</strong> V 21 ðkÞ aretaken into account. The reason for the choice <strong>of</strong> the zeroorder approximation will be given below. All other <strong>of</strong>fdiagonalmatrix elements, which are smaller than thevalues V 12 <strong>and</strong> V 21 will be introduced in higher orders <strong>of</strong>the perturbation theory. We consider first the nondegeneratecase when o ce ao ch ,so that the four zero ordermagnetoexciton b<strong>and</strong>s accounted from the LLLs areE 1 exðkÞ ¼Eð0;0Þ ex ðkÞ ¼H 1;1 ðkÞ ¼V 1;1 ðkÞ ¼ I ð0;0Þex ðkÞ, ð47aÞE 2 exðkÞ ¼Eð1;1Þ ex ðkÞ ¼H 2;2 ðkÞ ¼_o ce þ _o ch I ð1;1Þex ðkÞ, ð47bÞE 3 exðkÞ ¼Eð1;0Þ ex ðkÞ ¼H 3;3 ðkÞ ¼_o ce I ð1;0Þex ðkÞ, ð47cÞE 4 exðkÞ ¼Eð0;1Þ ex ðkÞ ¼H 4;4 ðkÞ ¼_o ch I ex ð0;1Þ ðkÞ. ð47dÞThe ionization potentials <strong>of</strong> four b<strong>and</strong>s were determined asI ðn;mÞex ðkÞ ¼ X sF e2h ðp; n; k x p; m; p s; n; k x p þ s; mÞ exp½ik y sl 2 Š¼ e2e 0 lx 2 exp 12Z 1dx0 nþmx 22J 0 ðklxÞ; n; m ¼ 0; 1. ð48ÞAn explicit expression for I ðn;mÞex ðkÞ can be found e.g. in Ref.[44].A more exact expression <strong>of</strong> the magnetoexciton wavefunction is the linear combinationc n;k ¼ X4i¼1a in c i;k . (49)The new functions c n;k are denoted by Greek symbolsn ¼ a; b; g; d, whereas the initial zero order functions c ik byLatin letter i ¼ 1; 2; 3; 4. They obey to the Schro¨dingerequation^Hc n;k ¼ E n ðkÞc n;k (50)what leads to four linear equations which determine theenergy spectrum E n ðkÞ <strong>and</strong> the coefficients a in . Theequations contain the matrix elements Eq. (46) <strong>and</strong> havethe formX 4i¼1a in H ij ðkÞ ¼E n ðkÞa jn ; j ¼ 1; 2; 3; 4. (51)The energy spectrum E n ðkÞ can be found solving secularequationH 11 ðkÞ E n ðkÞ V 12 ðkÞ V 13 ðkÞ V 14 ðkÞV 21 ðkÞ H 22 ðkÞ E n ðkÞ V 23 ðkÞ V 24 ðkÞ¼ 0.V 31 ðkÞ V 32 ðkÞ H 33 ðkÞ E n ðkÞ V 34 ðkÞ V 41 ðkÞ V 42 ðkÞ V 43 ðkÞ H 44 ðkÞ E n ðkÞ (52)Neglecting 10 <strong>of</strong>f-diagonal matrix elements denoted as afirst order infinitesimals ejV 13 jjV 14 jjV 23 jjV 24 jjV 34 je (53)we will obtain the zero order solutions <strong>of</strong> Eqs. (51) <strong>and</strong> (52)E 0 a;b ðkÞ ¼ H 11ðkÞþH 22 ðkÞ2 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH 11 ðkÞ H 22 ðkÞÞ 2 þ 4V 2 122ðkÞ ,E 0 g ðkÞ ¼H 33ðkÞ; E 0 d ðkÞ ¼H 44ðkÞ. ð54Þ
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