Denis Mamaluy public.asu/~dmamaluy/research.htm

1. Efficient Simulation of Nanoelectronic Deviceswith Contact Block Reduction method Denis Mamaluy http://www.public.asu.edu/~dmamaluy/research.htm

2. What we will discuss • A brief overview of general approaches to quantum transport simulations in nano-devices • Major computational methods for the quasi-ballistic quantum transport in nano-devices • Contact block reduction (CBR) method • #1 point: reduction to contact block(s) • #2 point: use of an incomplete set of eigenstates (generalized von Neumann boundary conditions) • #3 point: mode space reduction • Numerical efficiency estimation

3. Quantum transport in nanostructures Commonly used schemes: • Non-equilibrium Green’s function approach • Wigner-function approach • Pauli master equation • Landauer-Buttiker formalism

4. NEGF theory and applications • Microscopic theory for quantum transport that may include all interactions • Known as Keldysh formalism (sometimes also referred to as “generalized Kadanoff-Baym approach”) • Uses second quantization language • Equivalent to Green’s functions formalism in the case of the ballistic (coherent) transport

5. The lowest order perturbation terms in NEGF include • Electron-electron interaction through the Hartree-Fock approximation (correlation & exchange integrals) • Does not affect coherence (“ballisticity”) of the transport [Datta] • Electron-phonon interaction through a self-consistent solution for the self-energy term • No rigorous self-consistent scheme has been implemented so far even in 1D!

6. NEGF applications • NEMO 1D (R. Lake, G. Klimecket al.) • Application: resonant tunneling diodes • Datta’s approximation “Buttiker probes” (similar to the relaxation time approximation) • Non self-consistent forms of phonon self-energy (e.g. relaxation time, deformational potential approximation, etc.) • The rest of “NEGF” publications is nothing else than standard Green’s function formalism with no scattering!

7. ‘True NEGF’ problem • The computational costs: • From the numerical point of view this approach is almost hopeless for realistic 2D-3D devices…

8. Wigner functions • Transport is rigorously quantum-mechanical • Similarities to the quasi-classical transport theory (Boltzmann equation) • Scattering can be taken into account in a convenient (standard to MC) way • Integral equations can be solved using EMC technique • So far, the method is rather slow (days to obtain a converged solution). Known simulations are restricted to quasi 1D systems and RTDs.

9. Pauli master equation • A simplified form of quantum master equations used in optics • Assumes that • PME scales as O(N) • Violates continuity equation (Frensley)

10. Landauer approach & ballistic transport • Applicabilityballistic or quasi-ballistic quantum transport • Main assumptionapplied voltage drops at the interfaces with the device (contact resistance) or inside the device: no voltage drop in the leads • Modelleads have to be “infinitely” more conducting than the device, and have known distribution functions and potentials

11. Transfer matrix and QTBM • Transfer matrix methods • Standard (1969) • Usuki method (1995), Ferry’s “recursive scattering matrix” (J.Appl.Phys, 2004) • Boundary conditions are given by the Quantum Transmitting Boundary Method (QTBM) • Frensley (1990) • Lent and Kirkner (1992) • Ting (1995) • Laux & Fischetti method (PRB,2004)

12. Transfer matrix in 2D & 3D • The size of the linear system to be solved is determined by the area or the volume of entire device for every energy step

13. Recursive Green’s functions • Widely used due to the popularity of NEMO 1D • Works with NEGF • Efficient (in 1D and 2D) • Flexible: can be applied to different geometries

14. Recursive Green’s functions • Scales as • Unfortunately works onlyfor two contacts!! • Cannot be applied to calculate (at least self-consistently) the gate leakage current, gate charge, and, generally, multi-terminal devices.

15. Open system problem • We consider general n-dimensional system consisting of • The “active device” region: may be under applied bias and contain spatially varying potential • External leads assumed to have known Hamiltonians (far enough from the active device) “contacts” Number of leads: L

16. System’s Hamiltonian • “Big” (infinite) system’s Hamiltonian:

17. What do we want to solve?

18. Green’s functions formalism

19. Self-energy Σ

20. Important notations • Ddenote the (internal) device region • Cdenote contact (boundary) region C “contacts”

21. Indexing (labeling) scheme

22. Contact block reduction

23. Contact block reduction: A-matrix

24. The left upper block fully determine the transmission function1 • The left lower block determines density of states, charge density, etc2. Contact block reduction: the first key point All elements of GR can be determined from inversion of small matrixAC 1 D. Mamaluy et al.,J. App. Phys. 93, 4628 (2003). 2 D. Mamaluy et al., Phys. Rev. B 71, 245321 (2005).

25. CBR: the transmission function • The transmission function is given by

26. CBR: local DOS

27. CBR: density matrix

28. CBR: the particle density and the density matrix

29. The second CBR key-point (boundary conditions)

30. The boundary conditions • We want to find such boundary conditions for G0, which would “mimic” the open boundary conditions, but still form a Hermitian linear eigenvalue problem… • Then, the elements of GR can be obtained using even an incomplete set of such eigenstates of the closed system… C “closed” boundary conditions(for the “decoupled” G0) “open” boundary conditions (for the retarded Green’s function) Dirichlet 1D example: von Neumann Robbins

31. Decomposition of the self-energy in CBR Close to the band edge we can expand the exponential term:

32. Von Neumann boundary conditions in CBR

33. Incomplete set of eigenstates and von Neumann boundary conditions • Using the von Neumann boundary conditions we can use incomplete set of eigenstates. • Typically it is enough to find only <1% for 3D or 5-7% for 2D of all eigenstates to obtain quite accurate results! • The explanation is simple: • As one can easily check that von Neumann boundary conditions are zero-th order approximation to the open boundary conditions (close to the band edge or for a very small real-space grid step).

34. Von Neumann boundary conditions in the single band case Simple 2D example: Aharonov-Bohm ring GaAs, 1-band N=6000

35. Generalized von Neumann boundary conditions (multi-band case) 2D-example: T-junction p-GaAs, 3 leads 4-band k∙p Hamiltonian N = dim(H0) = 25000 D. Mamaluy, D. Vasileska, M. Sabathil, T. Zibold, P. Vogl, Phys. Rev. B, 2005.

36. The third CBR key-point: lead mode reduction • The size of the CBR matrices is of the size of the contacts, however we can • use lead modes Nm<=Nc (eigenstates) as a basis. • Only the propagating modes (often only a few) should be included into the calculation! • Additional reduction of computational costs (especially important for 3D calculations, when contacts are 2D objects)

37. The third CBR key-point: lead mode reduction Transmission function for an asymmetric, highly resonant wave-guide structure <=

38. Computational costs

39. Features presently incorporated in the CBR simulator • Fully self-consistent quantum mechanical transport in 2D and 3D structures (any geometry, any number of leads) • Effective mass approximation, 6 silicon valleys (working on extending to k-dot-p Hamiltonians) • Arbitrary crystallographic orientation • Electron-electron interaction (via the LDA) • Surface/interface roughness effects • Scattering on the local impurities (3D only) • Scattering on phonons via relaxation time approximation(plans to include more rigorous models) • Order of magnitude faster than most of the available 2D-3D quantum transport simulators