Large scale electronic structure calculations for nanosystems - PowerPoint PPT Presentation

Large scale electronic structure calculations for nanosystems Lin-Wang Wang Computational Research Division Lawrence Berkeley National Laboratory US Department of Energy BES, ASCR, Office of Science

The challenge for nanoscience simulations � Different electronic structures than bulk materials � 1,000 ~ 100,000 atom systems are too large for direct O(N 3 ) ab initio calculations � O(N) computational methods are required � Parallel supercomputers critical for the solution of these systems

Outline � Charge patching method: electronic structures for nanodots and wires � Electron phonon coupling: carrier transport in disordered polymers � LS3DF calculation: ZnTe:O alloy for solar cell � Quantum transport: a molecular switch � LCBB: million atom CMOS simulations � Some thoughts for future simulations

Motif based charge patching method ρ − ( ) r R ρ = ρ × 0 ( ) ( ) atom r r ∑ ρ − motif graphite ( ) r R atom R ρ ρ ( LDA ) motif graphite = ∑ ρ ρ − patch aligned ( ) ( ) r r R nanotube motif R Error: 1%, ~20 meV eigen energy error. Phys. Rev. B 65, 153410 (2002).

Charge patching: free standing quantum dots In 675 P 652 LDA quality calculations (eigen energy error ~ 20 meV) 64 processors (IBM SP3) for ~ 1 hour Total charge density CBM VBM motifs The band edge eigenstates are calculated using linear scaling folded spectrum method (FSM), which allows for 10,000 atom calculations.

The accuracy for the small Si quantum dot 8 (22)

Folded Spectrum Method (FSM) and Post Processing ψ = ε ψ − ε ψ = ε − ε ψ 2 2 ( ) ( ) H H ref i i ref i i i i ν h ν h Using { Ψ i , ε i } and Coulomb/exchange integral for limited CI calc. ---- many-body effects, optical fine struct., Auger effects, etc.

CdSe quantum dot results

CdTe nanowire

Quantum dot and wire calculations for semiconductor materials IV-IV: Si III-V: GaAs, InAs, InP, GaN, AlN, InN II-VI: CdSe, CdS, CdTe, ZnSe, ZnS, ZnTe, ZnO

Polarization of quantum rods (CdSe) Stock shift (meV) Calc. Expt. Energy (eV) Aspect ratio of the quantum rods Calc: Expt:

A CdSe core in CdS rod Red: hole Green: electron

Solar cell using stable, abundant, and env. benign mat ZnO/ZnS core/shell wire The absorption length is similar to bulk Si, Band gap lowers down further from superlattices. thus similar among f material for solar cell. 23% theoretical efficiency for solar cell

Charge patching method for organic systems LUMO HOMO HOMO-1 LUMO+1 An amorphous P3HT blend A 3 generation PAMAM dendrimer

Explicit calculation of localized states and their transition rates � Classical force field MD for P3HT blend atomic structure � Take a snapshot of the atomic structure � CPM and FSM to calculate the electronic states ψ i . � Classical force field calculation for all the phonon modes � Quick CPM calculation for electron-phonon coupling constants υ = ψ ∂ ∂ υ ψ ( ) | / | C H , i j i j � transition rate W ij from C ij ( ν ): = ∑ ν + δ ε − ε − ω + 2 h | ( ) | [ 1 / 2 ] ( ) .. W C n ν ν ij ij i j ν � using W ij and multiscale approach to simulate carrier transport

How good is the current phenomenological models The Miller model for weak electron-phonon coupling exp( -( ε j - ε i )/kT) for ε j > ε i W ij =C exp(- α R ij ) for ε j < ε i 1 Marcus like formula for strong electron-phonon coupling, polaron π = − λ + ε − ε λ 2 2 exp[ ( ) / 4 ] W V kT λ ij ij j i h kT Some electron density of state (e.g., Gaussian) is assumed These formulas have been used for decades without checking their validities. Our case is the weak coupling case

The test for W ij models A better model: Miller’s model = ∫ 2 ψ ψ ε − ε | || | ( ) W C dr D ij i j phonon j i

Multiscale model for electron transport in random polymer 10x10x10 box 30nm 3nm Exp Refs: PRL 91 216601 (2003) PRL 100 056601 (2008) 0.14nm 300nm 10x10x10 box

Why are quantum mechanical calculations so expensive? Density functional theory calculations 2 ∇ 2 + V tot ( r ) + ] ψ i ( r ) = ε i ψ i ( r ) [ − 1 � If the size of the system is N : ψ i ( r ) � N coefficients to describe one wavefunction ψ i ( r ) � i = 1,…, M wavefunctions , M is proportional to N . ∫ � Orthogonalization: , M 2 wavefunction ψ ( ψ * 3 ) ( ) r r d r i j pairs, each with N coefficients: N*M 2 , i.e N 3 scaling. The repeated calculation of these orthogonal wavefunctions make the computation expensive, O(N 3 ). For large systems, an O(N) method is needed

Previous divide-and-conquer methods ∑ = • Using partition function P F (r): ( ) 1 P r F F ∑ ρ = ρ ( ) ( ) ( ) r P r r tot F F F Center region = ( ) 1 P r F P F (r) Overlap region < ( ) 1 P r F Buffer region = ( ) 0 P r F • No unique way to divide kinetic energy ∑∫ ∑∫ ∇ ψ ψ ∇ ψ or 2 3 2 3 ( ) | ( ) | ( ) ( ) ( ) P r r d r P r r r d r F F F F F F F • Require ρ = ρ in the overlap region ( ) ( ) r r F tot • Introduce additional correction terms: the result is not the same as the original full system LDA W. Yang, Phys. Rev. Lett., 66:1438, 1991. F. Shimojo, R.K. Kalia, A. Nakano and P. Vashishta, Comp. Phys. Commun., 167:151, 2005.

Divide-and-conquer scheme for LS3DF LS3DF: linear scaling three dimensional fragments method ρ (r) } Total = Σ F { F F Phys. Rev. B 77, 165113 (2008); J. Phys: Cond. Matt. 20, 294203 (2008)

The patching scheme in 2D and 3D Interior area Buffer area Artificial surface (i,j,k) passivation Fragment (2x1) } Total = Σ F { F F F F Boundary effects are (nearly) cancelled out between the fragments { } ∑ = + + + − − − − System F F F F F F F F 222 211 121 112 221 212 122 111 , , i j k

Accuracy of LS3DF method compared with direct DFT � The accuracy of LS3DF is determined by the fragment size. � A comparison to LDA on 339 atom silicon (Si 235 H 104 ) quantum dot The total energy error: 3meV/atom ~ 0.1 kcal/mol • Charge density difference: 0.2% • Atomic force difference: 10 -5 a.u • � A test on 178 atom CdSe nano-rod The dipole moment difference: 1.3x10 -3 Debye/atom • For most practical purposes, LS3DF and direct LDA method can be considered the same.

Schematics for LS3DF computation

LS3DF scaling (440 Tflops on 150,000 processors) LS3DF 500 450 Performance [ Tflop/s] 400 (XT5) Jaguar 350 Intrepid 300 Franklin TFlop/s . (quad-core) 250 200 150 100 50 0 0 50,000 100,000 150,000 200,000 Number of cores Cores 2008 Gordon Bell prize, 440 Tflops/s on 150,000 cores

Dipole moment of a realistic ZnO nanorod

Use intermediate state to improve solar cell efficiency � Single band material theoretical PV efficiency: 30% � With an intermediate state, the PV efficiency can be 60% � ZnTe:O could be one example � Is there really a gap? Are there oscillator strength? Yes, there is a gap, and O induced states are very localized. LS3DF calculation for 3500 atom 3% O alloy [one hour on 17,000 processors] INCITE project, NERSC, NCCS]. Highest O induced state ZnTe bottom of cond. band state

Density of states J. Li and S.-H. Wei, Phys. Rev. B 73, 041201 ZnTe Density of States 3% Oxygen 6% Oxygen 9% Oxygen

Density of states 3% O 6% O 9% O Density of States

Oscillator strength 3% O 6% O Valence band – oxygen band and oxygen band -conduction band optical transition is possible.

Sunlight Solar cell efficiency photon-flux � Quasi-Fermi level Absorption ε 2 within IB VB-CB � Same absorbed IB-CB photon flux for IB-CB and VB-IB � Total number of VB-IB IB-CB carrier pair equals photon flux VB-CB +IB-CB � Output energy Absorption length (nm) equal to carrier number x VB-CB band gap. Original ZnTe efficiency: 24% IB solar cell efficiency: 63% ! Photon energy (eV)

Electron quantum transport problem ψ = • ( ) exp( ( ) ) r ik E r + β − • exp( ( ) ) ik E r ψ = α • ( ) exp( ( ) ) r ik E r In coming wave 1 − ∇ + ψ = ψ 2 { ( )} ( ) ( ) V r r E r 2 Transmitted wave Reflected wave It is a Schrodinger’s equation with an open boundary condition

A new algorithm for elastic transport calc. (1) Make an auxiliary periodic boundary condition (2) For a given E, solve a few Ψ l (r) [for diff. W l (r)] from the linear equation ψ = . − ψ = ik r ( ) ( ) , r e u r ( ) ( ) ( ) H E r W r l l l l (3) Take a linear combination of ψ l (r), to construct the scattering state Ψ sc (r). M ∑ ∑ R ψ = ψ = φ + φ ∈ Ω R * R for ( ) ( ) ( ) ( ) r C r r B r r ≠ sc l l m n R n m n = 1 l ∑ * r ∈ Ω = n φ L L r for ( ) A L n n Allow the use of PW ground state codes to calculate scattering states

Au electrode tunneling at Fermi surface A B cross cross section section at A at B distance

A molecular switch F S E S F

Switching under electric field Planar Perpendicular Electric field Electron transmission