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 Nanotechnology
Purdue University, West Lafayette, Indiana
May, 2012
Why atomistic tight binding?
Single atom transistor
Nature Nanotechnology 7, 242 (2012)
Ryu et al., Wednesday, 9.40am
Band-to-band tunneling
IEEE Elec. Dev. Lett. 30, 602 (2009)
Jiang et al., Wednesday, Poster P82
Topological insulators
Nature Physics 6, 584 (2010)
Sengupta et al., Thursday,
12.40pm
Countable device atoms suggest atomistic descriptions
Modern device concepts, e.g.
• Band to band tunneling
• Topological insulators (gap less materials)
• Band/Valley mixing etc.
require multi band representations
Why non-equilibrium Green’s functions?
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…)
http://newsroom.intel.com/docs/DOC-2035
This requires a consistent description of
coherent quantum effects (tunneling, confinement, interferences,…)
and
incoherent scattering (phonons, impurities, rough interfaces,…)
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
sp3d5s* TB band structure of
3nm (111) GaAs quantum well
Full TB NEGF:
Atomistic tight binding represents
electrons by N*No states (N atoms, No
orbitals)
G(z,z’,k|| = 2/nm,E=0)
Physics:
Electrons with given (k|| ,E) do not couple
with all N*N0 states
A few states should suffice to describe the
physics at (k|| ,E)
This work:
1. Find the n relevant states and form a n-dimensional basis
2. Transform the NEGF equations into the n-dimensional basis and solve therein
(low rank approximation)
Incomplete basis transformation: LRA
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.”
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 n
P is a matrix of n x n
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)†
Benchmark: Density in Si nanowire
5x5nm squared Si nanowire in sp3d5s* empirical tight binding model
Result:
5nm
 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)
Benchmark: Transmission in Si nanowire
5x5nm squared Si nanowire in sp3d5s* empirical tight binding model
5nm
holes
electrons
 Original matrix rank can be reduced down to 10%
 Too strong matrix rank reduction results in increasing deviations
 LRA works for electrons and holes
Benchmark: Periodicity & band-to-band tunneling
L-shape GaSb-InAs tunneling FET
 Broken gap bandstructure – mixture
electrons/holes
 Periodic direction – momentum dependent
basis functions
 2D transport (nonlinear geometry)
TFET concept (taken from MIND)
10 nm
10 nm
Low rank approximation is
 Applicable to arbitrary geometries and periodicities
 Applicable to band to band tunneling
LRA in NEGF: Feasibility of large devices
Atomistic NEGF without LRA on Supercomputers:
Typical maximum Si wire diameter ~ 8 nm (1000s CPUs)
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 ever
See also Lang et al., Poster Thursday (LRA + inelastic scattering in eff. mass)
Approximate basis functions
exact
9% matrix rank
eig(H+Σ(-1.2eV))
GaAs
AlAs
GaAs
Transmission in resonant tunneling diode
AlAs
Challenge for further speed improvement:
Solving a set of basis functions for every energy takes too much time
Solution:
Reuse basis functions of at (E,k) for different (E’,k’)
exact
9% matrix rank
eig(H+Σ(-0.6eV))
eig(H+Σ(2.5eV))
eig(H+Σ(1.9eV))
Approximate solutions of basis functions are feasible
Conclusion & Outlook
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
Ongoing work
LRA for NEGF with phonons
LRA for NEGF with inelastic scattering in tight binding
LRA for multiscaling transport problems
Thank you!