Stanford 11/10/11 Modeling the electronic structure of semiconductor devices M. Stopa Harvard University Thanks to Blanka Magyari-Kope, Zhiyong Zhang and Roger Howe

Introduction Self-consistent electronic structure for nanoscale semiconductor devices requires calculation of charge density Conceptually simple solutions (Solve the Schrödinger equation!) not practical in most cases (too many eigenstates). Thomas-Fermi approaches can be developed in some cases, but even these are limited. Nano by Numbers

Outline I will describe self-consistent electronic structure code SETE for density functional theory calculation of electronic structure for semiconductor devices. Highlight the role of density calculation for increasingly complex systems. Present various results for different systems. Case of “exact diagonalization” and using Poisson’s equation to calculate Coulomb matrix elements.

SETE: Density functional calculation for heterostructures Approximations (1) effective mass (2) effective single particle (3) exchange and correlation via a local spin density approximation Allows full incorporation of (1) wafer profile (2) geometry and voltages of surfaces gates voltages (3) temperature and magnetic field Self-consistent electronic structure of semiconductor heterostructures including quantum dots, quantum wires and nano-wires, quantum point contacts. Outputs: 1.electrostatic potential  (r) 2.charge density  (r) 3.for a confined region (i.e. a dot) eigenvalues E i, eigenfunctions  I tunneling coefficients  i 4.total free energy F(N,V g,T,B)

define a mesh discretize Poisson equation guess initial  (r), V xc (r) solve Poisson equation Compute  (r) 1.Schrödinger equation 2.Thomas-Fermi regions  in =  out ? no yes adjust V xc (r) V xc same ? yes DONE no Mesh must be inhomogeneous, encompassing wide simulation region so that boundary conditions are simple Jacobian Thomas-Fermi appx. wave functions N or  fixed ? Bank-Rose damping convex Optimize t by calculating  several times for different t. R. E. Bank and D. J. Rose, SIAM J. Numer. Anal., 17, 806 (1980) 2D Schrödinger equation classically isolated region provided by gate potentials fix either N or  cut off wave function in barrier regions (Dirichlet B.C.’s) dot nearly circular  expand in eigenfunctions (Bessel fns.); otherwise discretize on mesh (Arnoldi method) use perturbation theory details Adiabatic treatment of z Newton-Raphson

Density from potential 3D Thomas-Fermi zero temperature: Quasi-2D Thomas-Fermi zero temperature: Quasi-2D Thomas-Fermi T≠0: Sandia, NM 10/11/11 Only true under the assumption of parabolic bands

Examples of SDFT results Triple dot rectifier M. Stopa, PRL 88, (2002)

Blue dots are donors, red circles are ions donor layer disorder/order M. Stopa, Phys. Rev. B, 53, 9595 (1996) M. Stopa, Superlattices and Microstructures, 21, 493 (1997) Statistics of quantum dot level spacings Transition from Poisson statistics to Wigner statistics as disorder increases

Degenerate 2D electron gas (quantum Hall regime) Density of states

Single photon detector Evolution of magnetic field induced compressible and incompressible strips in a quantum dot Magnetic terraces Quantum dot Radial potential profile as B is increased Komiyama et al. PRB 1998 Stopa et al. PRL 1996

Charge density in two parts: (i) Thomas-Fermi density from adiabatic subband energies: (ii) Schrödinger density, eigenvalue problem in restricted 2D region: Eigenvalues in Quantum dots Frequently divide 2DEG region into “dot” and “leads. Dot = small number of isolated electrons.

E=-0.5 Ry* Coulomb interaction of scars

Schematic of wire simulation Metallic leads InP barriers Wire length 100 nm (smaller than expt.) Back gate 40 nm from wire InAs wire simulation (SETEwire) SPM tip

Complex band structure TF – in progress Luttinger Hamiltonian for valence band (light holes and heavy holes) replaces the Laplacian No analytic relation between Fermi momentum and Fermi energy. Numerical relation has to be determined at each position in space! Tough problem.

Going beyond mean field theory – using Kohn-Sham states as a basis for Configuration Interaction calculation

Exact diagonalization in quantum dots Typical case: double dot potential with N=2 Coulomb interaction Simple single particle basis states:

Two-particle basis states Singlet, S=0 Triplet, S=1 Singlet energy = single ptcls. + interdot Coulomb + exchange - delocalization Triplet energy = single ptcls. + interdot Coulomb - exchange

Kohn-Sham equationsexact diagonalization Dirichlet boundary conditions on gates DFT basis for exact diagonalization

Summary: exact diagonalization N=2 1. Solve DFT problem for spinless electrons with full device fidelity. 2.Remove Coulomb interaction and exchange-correlation effects from Kohn-Sham levels. 3.Truncate basis to something manageable. 4.Compute Coulomb matrix elements using Poisson’s equation. 5.Diagonalize Hamiltonian. Form all symmetric and anti-symmetric combinations of basis states for singlet and triplet two electron states, resp. Symmetric states Anti-symmetric states SETE solves Kohn-Sham problem, i.e. mean field

Modeling of electronic structure by configuration interaction (CI) with a basis of states from density functional theory (DFT) 1.Use DFT and realistic geometry (gate configuration, wafer profile, wide leads, magnetic field B) with N=2. 2.Resulting “Kohn-Sham” states used as basis for “exact diagonalization” (configuration interaction) of Coulomb interaction. MAIN MESSAGE: capture both geometric effects and many-body correlation. ADVANTAGES: 1.Fewer basis states needed because basis already includes potential profile and B. 2.Coulomb matrix elements calculated with Poisson’s equation  screening of gates included automatically plus no 3D quadratures required. 3.No artificial introduction of tunneling coefficient. Basis states are states of full double dot.

NSEC CECAM08 Dirichlet boundary conditions on gates Calculating Coulomb matrix elements POINT: calculated matrix element without ever knowing V(r 1,r 2 ) ! POINT: inhomogeneous screening automatically included.

LR Exact diagonalization calculation for realistic geometry double dot. We calculate the N=2 (many-body) spectrum, lowest two singlet and triplet states, near the transition from (1,1) to (0,2). For ε<0 singlet and triplet ground states have one electron in each dot, singlet and triplet excited states have both electrons in right dot. T 1 must have occupancy of higher orbital in R NSEC M. Stopa and C. M. Marcus, NanoLetters 2008

GATE nanoparticle/dots dot 1 dot 2 Exciton transfer via Förster process motivation Similar to quantum dot, we can calculate electronic structure of confined excitons taking gate into account via boundary conditions on Poisson equation.

Conclusions In contrast to molecular systems, number of eigenstates in semiconductor systems is too great to calculate all states. Thomas-Fermi is valuable, both 3D and effective 2D, in some cases For complex band structure of inhomogeneous systems there is no systematic way to implement TF. Finally, for isolated, small N systems, can go beyond even standard Kohn-Sham method to incorporate many-body correlation into self-consistent calculation in realistic environment.