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

Different electronic structures than bulk materials 1,000 ~ 100,000 atom systems are too large for direct O(N3) ab

initio calculations

O(N) computational methods are required Parallel supercomputers critical for the solution of these systems

The challenge for nanoscience simulations

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

) ( LDA

graphite

ρ

motif

ρ

Motif based charge patching method

) ( ) ( R r r

R aligned motif patch nanotube

− =∑ ρ ρ

Error: 1%, ~20 meV eigen energy error.

∑

− − × =

R atom atom graphite motif

R r R r r r ) ( ) ( ) ( ) ( ρ ρ ρ ρ

• Phys. Rev. B 65, 153410 (2002).

Charge patching: free standing quantum dots In675P652 LDA quality calculations (eigen energy error ~ 20 meV) CBM VBM 64 processors (IBM SP3) for ~ 1 hour Total charge density 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

i i i

H ψ ε ψ =

i ref i i ref

H ψ ε ε ψ ε

2 2

) ( ) ( − = −

Using {Ψi, εi} and Coulomb/exchange integral for limited CI calc.

• --- many-body effects, optical fine struct., Auger effects, etc.

ν h

ν h

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)

Energy (eV)

Stock shift (meV) Aspect ratio of the quantum rods

Calc. Expt. 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

Band gap lowers down further from superlattices. The absorption length is similar to bulk Si, thus similar among f material for solar cell.

23% theoretical efficiency for solar cell

Charge patching method for organic systems

HOMO-1 HOMO LUMO LUMO+1

A 3 generation PAMAM dendrimer

An amorphous P3HT blend

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 ψi. Classical force field calculation for all the phonon modes Quick CPM calculation for electron-phonon coupling constants transition rate Wij from Cij(ν): using Wij and multiscale approach to simulate carrier transport

j i j i

H C ψ υ ψ υ | / | ) (

,

∂ ∂ = .. ) ( ] 2 / 1 [ | ) ( |

2

+ − − + = ∑

ν ν ν

ω ε ε δ ν h

j i ij ij

n C W

How good is the current phenomenological models

The Miller model for weak electron-phonon coupling Wij=C exp(-αRij) exp( -(εj-εi)/kT) for εj > εi 1 for εj < εi Marcus like formula for strong electron-phonon coupling, polaron

] 4 / ) ( exp[

2 2

kT kT V W

i j ij ij

λ ε ε λ λ π − + − = h

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 Wij models Miller’s model A better model:

) ( | || |

2 i j phonon j i ij

D dr C W ε ε ψ ψ − = ∫

10x10x10 box 3nm

30nm

300nm 10x10x10 box

0.14nm

Multiscale model for electron transport in random polymer

Exp Refs: PRL 91 216601 (2003) PRL 100 056601 (2008)

[− 1 2 ∇2 + Vtot(r)+]ψi(r) = εiψi(r)

If the size of the system is N: N coefficients to describe one wavefunction i = 1,…, M wavefunctions , M is proportional to N. Orthogonalization: , M2 wavefunction

pairs, each with N coefficients: N*M2, i.e N3 scaling.

r d r r

j i 3 *

) ( ) ( ψ ψ

∫

The repeated calculation of these orthogonal wavefunctions make the computation expensive, O(N3). For large systems, an O(N) method is needed

ψi(r)

ψi(r)

Why are quantum mechanical calculations so expensive? Density functional theory calculations

Previous divide-and-conquer methods

1 ) ( =

∑

r P

F F

Center region Overlap region Buffer region

1 ) ( = r P

F

1 ) ( < r P

F

) ( = r P

F

) ( ) ( ) ( r r P r

F F F tot

ρ ρ

∑

=

• No unique way to divide kinetic energy

r d r r r P

F F F F 3 2

) ( ) ( ) ( ψ ψ ∇

∑∫

r d r r P

F F F 3 2

| ) ( | ) (

∑∫

∇ψ

• r
• Require

in the overlap region

) ( ) ( r r

tot F

ρ ρ =

• Introduce additional correction terms:

the result is not the same as the original full system LDA

• Using partition function PF(r):
• W. Yang, Phys. Rev. Lett., 66:1438, 1991.
• F. Shimojo, R.K. Kalia, A. Nakano and P. Vashishta, Comp. Phys. Commun., 167:151, 2005.

PF(r)

F F

Total = ΣF {

}

• Phys. Rev. B 77, 165113 (2008); J. Phys: Cond. Matt. 20, 294203 (2008)

ρ(r) Divide-and-conquer scheme for LS3DF LS3DF: linear scaling three dimensional fragments method

(i,j,k) Fragment (2x1) Interior area Artificial surface passivation Buffer area Boundary effects are (nearly) cancelled out between the fragments

Total = ΣF {

F F

}

F F

{ }

∑

− − − − + + + =

k j i

F F F F F F F F System

, , 111 122 212 221 112 121 211 222

The patching scheme in 2D and 3D

The accuracy of LS3DF is determined by the fragment size. A comparison to LDA on 339 atom silicon (Si235H104) 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

Accuracy of LS3DF method compared with direct DFT For most practical purposes, LS3DF and direct LDA method can be considered the same.

Schematics for LS3DF computation

Performance [ Tflop/s]

50 100 150 200 250 300 350 400 450 500 50,000 100,000 150,000 200,000 TFlop/s . Cores

LS3DF

Jaguar Intrepid Franklin

Number of cores

(XT5)

(quad-core)

LS3DF scaling (440 Tflops on 150,000 processors) 2008 Gordon Bell prize, 440 Tflops/s on 150,000 cores

Dipole moment of a realistic ZnO nanorod

ZnTe bottom of cond. band state Highest O induced state

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? LS3DF calculation for 3500 atom 3% O alloy [one hour on 17,000 processors] INCITE project, NERSC, NCCS]. Yes, there is a gap, and O induced states are very localized.

Density of States 3% Oxygen 6% Oxygen 9% Oxygen ZnTe

• J. Li and S.-H. Wei, Phys. Rev. B 73, 041201

Density of states

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

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

Solar cell efficiency Photon energy (eV) Absorption length (nm)

Sunlight photon-flux

Absorption ε2

IB-CB VB-IB VB-CB IB-CB

Quasi-Fermi level within IB Same absorbed photon flux for IB-CB and VB-IB Total number of carrier pair equals photon flux VB-CB +IB-CB Output energy equal to carrier number x VB-CB band gap. Original ZnTe efficiency: 24% IB solar cell efficiency: 63% !

Electron quantum transport problem

) ( ) ( )} ( 2 1 {

2

r E r r V ψ ψ = + ∇ −

) ) ( exp( ) ( r E ik r

• = α

ψ

) ) ( exp( ) ( r E ik r

• =

ψ

) ) ( exp( r E ik

• −

+ β

In coming wave Reflected wave Transmitted wave It is a Schrodinger’s equation with an open boundary condition

A new algorithm for elastic transport calc.

) ( ) ( ) ( r W r E H

l l

= − ψ

(2) For a given E, solve a few Ψl(r) [for diff. Wl(r)] from the linear equation (3) Take a linear combination of ψl(r), to construct the scattering state Ψsc(r).

(1) Make an auxiliary periodic boundary condition

) ( ) (

.

r u e r

l r ik l

= ψ

) ( ) ( ) ( ) (

* 1

r B r r C r

R n R n m n R m l M l l sc

φ φ ψ ψ

∑ ∑

≠ =

+ = = ) (

* r

A

L n n L nφ

∑

=

R

r Ω ∈

L

r Ω ∈

for for , Allow the use of PW ground state codes to calculate scattering states

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

A molecular switch E

F F S S

Planar Perpendicular

Electric field

Switching under electric field

Electron transmission

Linear Combination of Bulk Band (LCBB) method Number of plane wave basis ~ 100 million

∑

= Ψ

q iqr

e q C ) (

Can not truncate q

] ) ( [

, , , ikr k n k n k n

e r u C

∑

= Ψ

The bulk Bloch state could be a better basis Can truncate (n,k) Empirical pseudopotential H

) ( 2 1

, 2

R r v H

R

− + ∇ − =

∑

α α

unk(r) nth Bloch state at k point

• Truncate the basis {n,k} a few thousand
• Fast algorithm to evaluate

unk eik.r | HEPM | un’k’eik’.r

• Being able to treat strains in system (using VFF)

speed of LCBB 10 hours on a work station error of LCBB 10-20 meV for band gap, 3 meV for intraband splitting LCBB method for million atom calculations ψ(r) = Σ Cnk unk(r) exp(i k.r)

nk

Can be used to replace the effective mass k.p methods

e3 e2 e1 e0 e3 e5 e4 e6 e7

Electronic states in embedded InAs quantum dot

0 GaAs CBM InAs 1ML Wetting

• 60 meV

e0 e1,2 e3,4,5 h0 h1 h2 0 GaAs VBM 82 meV 249 (235) 1.044 eV (1.098) e6,7 59 (50) 59 (48) 55 2 (2) 8

• 228 (280)

168 (180)

Black: Calculation Red: Petrof, UCSB Blue: Schmidt, PRB 54, 11346 (96) green: Itskevich, PRB 58, R4250(98)

Energy levels, comparison with experiment

N-type CMOS device

The basic formalism

Empirical pseudopotential Hamiltonian for the carrier wavefunctions Ψ: Approximation for wavefunction occupation under nonequilibrium to get n(r): Poisson equation for electrostatic potential Φ:

How to calculate the occupied charge density ?

1 ) / ) exp(( 1 ) (

) ( ) (

+ − = kT E E E f

R L F i i R L

x L ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − =

∫

x i CBM L i

dx E x E f x W ) ) ( ( 2 exp ) (

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − =

∫

x L i CBM R i

dx E x E f x W ) ) ( ( 2 exp ) (

Fermi-Dirac distribution using and

L F

E

R F

E Effective mass WKB approx. for

and

) (x W L

i

) (x W R

i

Carrier charge density and gate threshold potential

Discrete random dopant fluctuation

Threshold lowering due to discrete dopant positions

Threshold fluctuation

Some thoughts about the future challenges The surface atomic structure and electronic states are still poorly understood Ab initio MD simulation for Pt on CdSe surface LS3DF simulation for the dipole moment of a 5,000 atom ZnO rod

Some thoughts about future challenges ZnO ZnS VBM CBM

Carrier dynamics is important to understand the escape rate

Conclusion Much remains to be done and understood Large scale high fidelity numerical simulation can play a critical role.

Thanks for Thanks for your attention ! your attention !

Acknowledgement:

BES, ASCR/SC INCITE project NERSC, NCCS, ALCF Jingbo Li Byounghak Lee Maia Garcia Vergniory Nenad Vukmirovic Zhengji Zhao Shuzhi Wang Sefa Dag Denis Demchenko Joshua Shcrier Aran Garcia-Lekue Jun Wei Luo