Efficient solution algorithm of non-equilibrium Green’s functions

in atomistic tight binding representation

Yu He, Lang Zeng, Tillmann Kubis, Michael Povolotskyi, Gerhard Klimeck

Network of Computational NanotechnologyPurdue University, West Lafayette, Indiana

May, 2012

Why atomistic tight binding?

Nature Nanotechnology 7, 242 (2012)Ryu et al., Wednesday, 9.40am

Single atom transistor

Countable device atoms suggest atomistic descriptionsModern device concepts, e.g.

• Band to band tunneling• Topological insulators (gap less materials)• Band/Valley mixing etc.

require multi band representations

Topological insulators

Nature Physics 6, 584 (2010)Sengupta et al., Thursday, 12.40pm

Band-to-band tunneling

IEEE Elec. Dev. Lett. 30, 602 (2009)Jiang et al., Wednesday, Poster P82

Why non-equilibrium Green’s functions?

http://newsroom.intel.com/docs/DOC-2035

This requires a consistent description of coherent quantum effects (tunneling, confinement, interferences,…)

andincoherent scattering (phonons, impurities, rough interfaces,…)

Device dimensions

State of the art semiconductor devices

utilize or suffer from quantum effects (tunneling, confinement, interference,…)

are run in real world conditions (finite temperatures, varying device quality…)

Numerical load of atomic NEGF

NEGF requires for the solution of four coupled differential equations

GR = (E – H0 – ΣR)-1 ΣR = GRDR + GRD< + G<DR

G< = GRΣ<GA

Σ< = G<D<

G‘s and Σ‘s are matrices in discretized propagation space (RAM ~N2, Time ~N3)

Atomic device resolutions can yield very large N (e.g. N = 107)

Motivation – transport in reality

Full TB NEGF:Atomistic tight binding represents electrons by N*No states (N atoms, No

orbitals)

sp3d5s* TB band structure of 3nm (111) GaAs quantum well

This work:1. Find the n relevant states and form a n-dimensional basis2. Transform the NEGF equations into the n-dimensional basis and solve therein

(low rank approximation)

G(z,z’,k|| = 2/nm,E=0)

Physics:Electrons with given (k|| ,E) do not couple with all N*N0 statesA few states should suffice to describe the physics at (k|| ,E)

Incomplete basis transformation: LRA

In NEGF: “Low rank approximation”

Given NEGF equation in an N-dimensional basis, i.e. matrix MNxN

n < N orthonormal functions { Ψi } in the N-dimensional basis

Approximate NEGF equation P is given by:

P = T† M T, with T = ( Ψ1 , Ψ2 , … Ψn )

T is a matrix of N x nP is a matrix of n x n

Wikipedia.org:“In mathematics, low-rank approximation is a minimization problem, in which the cost function measures the fit between a given matrix (the data) and an approximating matrix (the optimization variable), subject to a constraint that the approximating matrix has reduced rank.”

Key: How to find “good” Ψi ?

Method – Find propagating states

One method: For electrons at energy E and momentum k …

1. Solve the eigenproblem of nonhermitian Hamiltonian of the open system( H(k) + ƩR(k,E) ) Φi = εi Φi

2. Choose the n states { Φ i } with Re(εi) closest to the considered particle energy E

3. Orthonormalize the n states{ Φi } → { Ψi }

4. LRA: P = T† M T, with T = ( Ψ1 , Ψ2 , … Ψn )

5. Solve the approximate NEGF equations P in the reduced basis { Ψ }

6. Transform the results back into the original basis representationGNxN = TNxn gnxn (TNxn)†

5nm

5nm

Result:Benchmark: Density in Si nanowire

Original matrix rank was reduced down to 10% No loss in accuracy of the electron density Preliminary implementation shows already a speed up of 8

times (effectively limited by the solution of the basis functions)

5x5nm squared Si nanowire in sp3d5s* empirical tight binding model

5nm

5nm

Benchmark: Transmission in Si nanowire5x5nm squared Si nanowire in sp3d5s* empirical tight binding model

Original matrix rank can be reduced down to 10% Too strong matrix rank reduction results in increasing deviations LRA works for electrons and holes

electronsholes

L-shape GaSb-InAs tunneling FET Broken gap bandstructure – mixture

electrons/holes Periodic direction – momentum dependent

basis functions 2D transport (nonlinear geometry)

Low rank approximation is Applicable to arbitrary geometries and periodicities Applicable to band to band tunneling

Benchmark: Periodicity & band-to-band tunneling

10 nm 10 nm

InAs

GaSb

contact

contact

10 nm

10 nm

periodic

TFET concept (taken from MIND)

NEGF with LRA: 12nm diameter Si nanowire in sp3d5s* TB With 10% of original matrix rank Calculation done on ~100 CPUs

With LRA, NEGF is easily applicable to larger device dimensions than everSee also Lang et al., Poster Thursday (LRA + inelastic scattering in eff. mass)

LRA in NEGF: Feasibility of large devices

Atomistic NEGF without LRA on Supercomputers:Typical maximum Si wire diameter ~ 8 nm (1000s CPUs)

Approximate basis functions

eig(H+Σ(-1.2eV))eig(H+Σ(-0.6eV))

eig(H+Σ(1.9eV))eig(H+Σ(2.5eV))

Transmission in resonant tunneling diode GaAs

GaA

s

GaA

s

AlA

s

AlA

s

Challenge for further speed improvement:Solving a set of basis functions for every energy takes too much timeSolution:Reuse basis functions of at (E,k) for different (E’,k’)

9% matrix rankexact

9% matrix rankexact

Approximate solutions of basis functions are feasible

Conclusion & Outlook

Ongoing workLRA for NEGF with phononsLRA for NEGF with inelastic scattering in tight bindingLRA for multiscaling transport problems

Conclusion Developed systematic efficiency improvement of NEGF in atomistic tight binding for LRA in effective mass + inelastic scattering see Lang et al., Wed., Poster P11 Freely tunable approximation of NEGF Applicable to electrons, holes and mixed particles, arbitrary device geometries,… Enables efficient NEGF solutions in large devices

Implemented in free software tool NEMO5

Thank you!