Christophe Bichara CNRS and Aix Marseille University

Outline Some technical aspects : Faster Tight Binding energy calculationValidation• Carbon solubility• Melting temperature of NickelUse of Grand Canonical Monte Carlo algorithm to simulate growthGrowth of nanotubes State of catalystIdentify growth conditions:• Tube/Nanoparticle size matching• Parameters : temperature, carbon chemical potential (µ C ), flux– Low T, large µ Cand flux : encapsulation– Too high temperature : tubes detaches from metal nanoparticle (NP)– Intermediate regime : growth !Growth mechanisms, role of carbon chainsGrowth of graphene on NiConclusions3/30

Tight Binding model• Minimal basis set :– C s and p electrons– Ni d electrons• Total energy :E =⎛∑ ⎜ ∫⎝E⎟ ⎞Eni(E)dE⎠fi −∞Band structure termLocal densities of states+12∑V(r ij )i,jEmpirical repulsive termHopping integrals :- C-C : ssσ, spσ, ppσ, ppπ- Ni-Ni : ddσ, ddπ, ddδ- Ni-C : sdσ, pdσ, pdπ• Moments :Local DOS on red atom depends on- 1 st neighbors (2 nd moment); cut off = 2.7 Å- 1+2 nd neighbors (4 th moment)4 th moment and beyond : directional bonding (p)• Parameters :– Energy levels, hopping integrals, repulsion, cut off dist.Amara et al. Phys. Rev. B 73, 113404 (2006)Phys. Rev. B 79, 014109 (2009).4/30

Original algorithm Moments calculations: Use recursion algorithm• stable, efficient beyond 4 th moment At each MC step calculate only moments that have changed• very fast, possible only for limited nb. of moments Continued fraction to calculate DOS and total energy with 3 choices CF truncated at 4 th moment level : (b 3=0)• simple but not efficient Constant a iand b icoefs up to M th level + diagonalizing (MxM) tridiagonal matrix• efficient and stable, but slow (older version) Asymptotic expansion (M∞) + numerical integration• very fast but not very stable.5/30

Faster algorithm (thanks to Jan H. Los) Moments calculations: Use recursion algorithm• stable, efficient beyond 4 th moment At each MC step calculate only moments that have changed• very fast, possible only for limited nb. of moments Continued fraction to calculate DOS and total energy with 3 choices CF truncated at 4 th moment level : (b 3=0)• simple but not efficient Constant a iand b icoefs up to M th level + diagonalizing (MxM) tridiagonal matrix• efficient and stable, but slow (older version) Asymptotic expansion (M∞) + numerical integration• very fast but not very stable.6/30

Tight binding model : important features Pure Carbon : Carbon linear chain about 1 eV/ atom less stable thansp 2 carbon (DFT-GGA calculation) Pure Ni : melting temperature is 2040 K (model) instead of 1728 K (exp t ) ≈ 15 % too high∆ LDA∆ GGA Solubility of C in bulk Ni Heat of solution = + 0.5 eV / C (experimental value) Tendency to favor C or C 2species in subsurface sites. Surface Ni layer distorted by adsorbed C atoms ‘Clock’ reconstruction of (100) surfaceAmara et al. Phys. Rev. B 73, 113404 (2006)Phys. Rev. B 79, 014109 (2009)Klink PRL 1993M. Moors et al., ACS Nano, 2009, 3 (3), 511-516Our TB µ4 model7/30

Melting of small Ni clustersInternal energy (eV/ at.)1400 KTemperature (K)Extrapolating melting temperatureof clusters (Gibbs-Thompson)Pure Ni clusters with more than 55 atoms arestill solid up to 1400 K in our modelCalculatedExperimentalT m= 2360 KT m= 1728 KJ. H. Los et al. PRB 81, 064112 (2010)8/30

Melting temperature of bulk NiBetter estimate by calculating Gibbs energies of bulkliquid and solid phases using thermodynamicintegration on 864 atoms boxesCalculated T m = 2040 ± 50 KExperimental T m = 1728 KWe will consider rescaled temperatures :T* = T x (T m exp /T m calc ) = 0.85 x T T = 1000 K corresponds to 843 K = 570 CT = 1200 K corresponds to 1013 K = 740 CT = 1400 K corresponds to 1180 K = 910 Cspans the experimentaltemperature range9/30

Grand canonical Monte Carlo calculations (1) Carbon chemical potential (µ) is an essential control parameter Idea is to use GCMC algorithm to control growth number of Ni atoms fixed, C atoms incorporatedThermodynamic probability of a configurationNVPi ≈ exp( −β ( E − µ N))3NΛ N!Random“move” ofatomsRandomly alternate canonical displacement moves+ attempts to insert a particle with acceptance probability:⎪⎧V⎪⎫P acc( i → j)= min⎨1,exp( β ( µ − ∆E))3 ⎬⎪⎩ Λ ( N + 1)⎪⎭++ attempts to remove a particle with acceptance probability:⎪⎧3Λ N⎪⎫P acc ( i → j)= min⎨1,exp( −β( µ − ∆E))⎬V ⎪⎩⎪⎭insertionremoval10/30

Grand canonical Monte Carlo calculations (2)Ingredients of GCMC calculations• Temperature• µ Carbon: tells you how often a randomattempted insertion/removal will beaccepted≈ sticking coefficient• Number of relaxation steps betweeninsertions/destructions of C atoms≈ 1 / (« flux » or « feeding rate »)… but no time scale in MC••Real time scale (< 100 ns)Do not control µ carbonC atoms are tentatively inserted close to Ni cluster surface,with a given chemical potential, to simulate CVD reaction100000 attempted insertion/cycle, return after first acceptedAll C atoms are tentatively removedEach C atom is tentatively removed once / cycle• Molecular Dynamics simulations:displacementAll atoms allowed to relaxC insertionNi clusterC removalMC_disp = 10000 to 50000 / at. / cycle11/30

Just a toy example : chemical potential controls C incorporation …T = 1200 K ; 10 relaxation steps/atom = unphysical !Mu_C = - 7.0 eV / CMu_C = -4.5 eV / CLow carbon chemical potential : only favorable incorporation sites areaccepted Chains are growing on surfaceHigher carbon chemical potential : Less selective incorporation More disordered structures12/30

Our first goal : try and grow a defectless tube from initial cap Play with GCMC calculations varying Structures : (6,5), (6,6), (9,1) and (10,0) tube butts sitting on Ni clusters (55, 85 and 147 atoms) Temperature : 1000 to 1400 K 10000 to 50000 relaxation MC steps between attempted insertions/removal Carbon chemical potential (range -7.0 to -5.0 eV/at.) Try and identify growth or no-growth conditions Particle/tube matching Role of temperature / flux / µ carbon conditions Growth mechanisms Comparison with graphene / Ni Problems to face : Small systems, hence large fluctuations … statistical analysis Time scale … always too fast as compared to experiment :• 1 ring of C hexagons added/millisecond• Local thermodynamic equilibrium approximation is valid13/30

Yes, we can … grow tubes !T100NI85.C40Helweg et al., Nature (2004)1200 K; 50000 MC disp./at.Note Ni atoms, first invading the cap, then expelled !Similar « stick and slip » growth process alreadyobserved by in situ TEM14/30

First analysis …Example :(9, 1) butt on Ni85 clusterAB-5.468 eV / atom -5.475 eV / atomB is slightly more stable than A (10-30 meV/atom) This depends on number of Ni atoms … Ni is expelled from carbon nanotube during successful growth, Can be qualitatively explained by surface tension arguments Cf. :Schebarchov & Hendy Nano Lett., 8, 8, 2253-7 (2008)15/30

Tube does not grow (1) : low temperature, fast flux ⇒ encapsulationT91_NI85.CT2T100NI85.C101000 K; 10000 MC disp./at.1000 K; 50000 MC disp./at. At T= 1000 K, we almost systematically obtain a encapsulation of Ni cluster, speciallywhen flux is fast (small number of relaxation steps between insertion/removal)16/30

Tube does not grow (1) : low temperature, fast flux ⇒ encapsulationT91_NI85.CT2T100NI85.C101000 K; 10000 MC disp./at.1000 K; 50000 MC disp./at. At T= 1000 K, we almost systematically obtain a encapsulation of Ni cluster, speciallywhen flux is fast (small number of relaxation steps between insertion/removal)17/30

Tube does not grow (2) : too hot ⇒ tube detaches from NPT100NI85.C50T66N147.C301400 K; 50000 MC disp./at.1400 K; 50000 MC disp./at. At T = 1400 K, tube almost always detaches from Ni cluster, before or after closing … Note that our model tends to give too small an adhesion energy of tubes on Ni cluster whena carbide is formed close to surface… temperature for tube detachment might be higher18/30

Tube does not grow (2) : too hot ⇒ tube detaches from NPT100NI85.C50T66N147.C301400 K; 50000 MC disp./at.1400 K; 50000 MC disp./at. At T = 1400 K, tube almost always detaches from Ni cluster, before or after closing … Note that our model tends to give too small an adhesion energy of tubes on Ni cluster whena carbide is formed close to surface… temperature for tube detachment might be higher19/30

Growth mechanisms (1) : saturation of NP (sub-)surface is requiredAlready observed for nucleationNucleation of a capWhen starting from tube butt on pure Nicluster, saturation of subsurface sites isobserved prior to growth Carbon solubility around 20%,depends (±5%) on : size of NP (55 to 201 Ni atoms),Temperature and µ CAmara et al., PRL 100, 056105 (2008)Video : C within Ni cluster are blueC at Ni surface are lightblueC outside are blackT66_NI85.x2020/30

Growth mechanisms (2) : growth of sp 2 layers is tangential to particleT65_NI85.C101000 K; 50000 MC disp./at. sp 2 structure tends to formFinal configurationon left side, tangential to Ni surface,while right side (perpendicular) hardly grows. Better growth conditions when metal NP is just slightly larger than tube With (6,5) (6,6) (9,1) and (10,0) tube butts, 85 Ni atoms are better than 55, 147 or 201 In real life metal NP size imposes tube diameter …21/30

Growth mechanisms (3) : µ C controls C incorporation … butT100_Ni85.x77T100_Ni85.x40Mu_C = - 6.5 eV / C; 25000 relax stepsMu_C = - 5.5 eV / C; 50000 relax stepsStart from (10, 0) tube butt on Ni 85 cluster; T = 1200 KMu_C and number of relaxation steps are not completely decoupled22/30

Growth mechanisms (4) : µ C controls C incorporation … but Start : 6 pentagons + 30 hexagons Final : 9 pentagons + 80 hexagons + 8 heptagons ?? can be improved ?? Higher (-5.50 vs -6.50 eV/at) carbon chemical potential leads to faster growth More relaxation steps (50000 vs. 25000) leads to structures of ≈ same quality More investigations are required23/30

Growth mechanisms (5) : carbon chains are essential for growthT65_NI85.C10T66_NI85.C20T66_NI85.C301000 K; 50000 MC disp./at.1200 K; 10000 MC disp./at.1400 K; 10000 MC disp./at. Carbon sp 2 network is growing through carbon chains that attach to existing structure Also seen by Page, Irle, Morokuma (e.g. : J. Phys Chem. C 2010) Chains are present at all temperatures and carbon chemical potential Not enough statistics yet to study chain length as a function of T and µ C No direct experimental evidence ? G. Eres : deduced from kinetics analysis24/30

Defect healing … poster by Mamadou Diarra Idea is to study defect healing mechanisms with and without catalyst to see whetherthis might induce a chiral selectivity … For the moment answer is NO !25/30

Conclusions / open questions for nanotube growth : Before growth, Ni catalyst gets saturated with subsurface C atoms It would be reverse in real situation (saturation / nucleation / growth) The size ratios are important, We never see growth perpendicular to the surface, always tangential… With caps tested, no growth on Ni_55, difficult on Ni_147, best on Ni_85 We always see + or – long C chains forming Controlling this length might be important:• Too long chains have lower probability to get attached to tube lip• cf. role of etching agent for chiral selection (NH 3in JACS 2011 Kauppinen) Chiral selectivity ? Come from thermodynamically favored cap/NP matching ? Favored by defect healing mechanisms ? (see Mamadou DIARRA’s poster) Might originate from differences in growth kinetics and C incorporation mechanisms ? Role of substrate not considered here… too difficult for the moment26/30

Graphene on Ni : 1200 K, flat Ni (111) surfaceSide viewTop viewdepth Thick slab 12 x 12 x 12 = 1728 Ni atoms Mu_C = -4.5 eV/at Fast growth Subsurface layer stuffed with C Topmost Ni layers strongly perturbateddepth27/30

Graphene on Ni : 1200 K, flat Ni (111) surfaceSide viewTop viewdepth Thick slab Mu_C = -6.0 eV/ at. Slower growth more site selective Less C atoms in subsurface layers Ni layers less perturbateddepth28/30

Graphene on Ni : 1200 K, stepped Ni(111) surfaceSLABSTEP_MU4.5__WG_topSLABSTEP_MU4.5__WG_side Thick slab Mu_C = -4.5 eV/at Fast growth Subsurface layer stuffed with C Topmost Ni layers and step edges strongly perturbated29/30

Graphene on Ni : 1200 K, stepped Ni (111) surfaceSLABSTEP_MU6.0_WG_topSLABSTEP_MU6.0_WG_side Thick slab Mu_C = -6.0 eV/ at. Slower growth, step edges sites are favored for C adsorption Less C atoms in subsurface layers Ni layers and steps less perturbated Growth of graphene via C chains formation30/30

Conclusion graphene growth Growing graphene seems easier than growing tubes No need to find window between encapsulation of NP and tube detachment Longer CPU time required (≈ 1 month) because more atoms Setting simulation parameters to match experimental conditions is not trivial Some preliminary conclusions : C concentration in subsurface layers depends on chemical potential / flux conditions As for tubes, small carbon chains play important role C incorporation is favored close to step edges in mild (realistic ?) growth conditions …31/30

Thanks toAlexandre ZappelliJan H. LosCINaM - CNRS and Aix Marseille UniversityMamadou DiarraHakim AmaraFrançois DucastelleLEM - ONERA/CNRS Chatillon FranceKim BoltonAnders BörjessonUniv. Gothenburgh + Borås SwedenSOS_NanotubesANR-09-Nano-02832/30