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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 representation
GNxN = 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 nanowire
5x5nm 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 rank
exact
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!