Quantum Transport

Outline: What is Computational Electronics? Semi-Classical Transport Theory  Drift-Diffusion Simulations  Hydrodynamic Simulations  Particle-Based Device Simulations Inclusion of Tunneling and Size-Quantization Effects in Semi-Classical Simulators  Tunneling Effect: WKB Approximation and Transfer Matrix Approach  Quantum-Mechanical Size Quantization Effect Drift-Diffusion and Hydrodynamics: Quantum Correction and Quantum Moment Methods Particle-Based Device Simulations: Effective Potential Approach Quantum Transport  Direct Solution of the Schrodinger Equation (Usuki Method) and Theoretical Basis of the Green’s Functions Approach (NEGF)  NEGF: Recursive Green’s Function Technique and CBR Approach  Atomistic Simulations – The Future Prologue

Quantum Transport  Direct Solution of the Schrodinger Equation: Usuki Method (equivalent to Recursive Green’s Functions Approach in the ballistic limit)  NEGF (Scattering): Recursive Green’s Function Technique, and CBR approach  Atomistic Simulations – The Future of Nano Devices

i=0 i=N j=0 j=M+1 y x incident waves transmitted waves reflected waves Wavefunction and potential defined on discrete grid points i,j i th slice in x direction - discrete problem involves translating from one slice to the next. Grid spacing: a<< F Description of the Usuki Method Usuki Method slides provided by Richard Akis.

Obtaining transfer matrices from the discrete SE apply Dirichlet boundary conditions on upper and lower boundary: Wave function on ith slice can be expressed as a vector j=0 j=M+1 j=1 j=M i Discrete SE now becomes a matrix equation relating the wavefunction on adjacent slices: where: (1b)

(1b) can be rewritten as: Combining this with the trivial equation one obtains: Modification for a perpendicular magnetic field (0,0,B) : B enters into phase factors important quantity: flux per unit cell (2) where Is the transfer matrix relating adjacent slices

Mode eigenvectors have the generic form: redundant There will be M modes that propagates to the right (+) with eigenvalues: propagating evanescent There will be M modes that propagates to the left (+) with eigenvalues: propagating evanescent anddefining Complete matrix of eigenvectors: Solving the eigenvalue problem: yields the modes on the left side of the system

Transfer matrix equation for translation across entire system Transmission matrix Zero matrix no waves incident from right Unit matrix waves incident from left have unit amplitude reflection matrix Converts from mode basis to site basis Converts back to mode basis Recall: In general, the velocities must be determined numerically

Boundary condition- waves of unit amplitude incident from right Variation on the cascading scattering matrix technique method Usuki et al. Phys. Rev. B 52, 8244 (1995) plays an analogous role to Dyson’s equation in Recursive Greens Function approach Iteration scheme for interior slices Final transmission matrix for entire structure is given by A similar iteration gives the reflection matrix

After the transmission problem has been solved, the wave function can be reconstructed wave function on column N resulting from the kth mode The electron density at each point is then given by: One can then iterate backwards through the structure: It can be shown that:

First propagating mode for an irregular potential confining potential u 1 (+) for B=0.7 T j u 1 (+) for B=0 T u 1j Mode functions no longer simple sine functions general formula for velocity of mode m obtained by taking the expectation value of the velocity operator with respect to the basis vector.

V g = -1.0 VV g = -0.9 VV g = -0.7 V Potential felt by 2DEG- maximum of electron distribution ~7nm below interface Potential evolves smoothly- calculate a few as a function of V g, and create the rest by interpolation Conduction band [eV] z-axis [  m] Fermi level E F Conduction band profile E c Energy of the ground subband Simulation gives comparable 2D electron density to that measured experimentally Example – Quantum Dot Conductance as a Function of Gate voltage

V V V Subtracting out a background that removes the underlying steps you get periodic fluctuations as a function of gate voltage. Theory and experiment agree very well Same simulations also reveal that certain scars may RECUR as gate voltage is varied. The resulting periodicity agrees WELL with that of the conductance oscillations * Persistence of the scarring at zero magnetic field indicates its INTRINSIC nature  The scarring is NOT induced by the application of the magnetic field

Magnetoconductance Conductance as a function of magnetic field also shows fluctuations that are virtually periodic- why? B field is perpendicular to plane of dot classically, the electron trajectories are bent by the Lorentz force

Green’s Function Approach: Fundamentals The Non-Equilibrium Green’s function approach for device modeling is due to Keldysh, Kadanoff and Baym It is a formalism that uses second quantization and a concept of Field Operators It is best described in the so-called interaction representation In the calculation of the self-energies (where the scattering comes into the picture) it uses the concept of the partial summation method according to which dominant self-energy terms are accounted for up to infinite order For the generation of the perturbation series of the time evolution operator it utilizes Wick’s theorem and the concepts of time ordered operators, normal ordered operators and contractions

Relevant Literature A Guide to Feynman Diagrams in the Many-Body Problem, 2 nd Ed. R. D. Mattuck, Dover (1992). Quantum Theory of Many-Particle Systems, A. L. Fetter and J. D. Walecka, Dover (2003). Many-Body Theory of Solids: An Introduction, J. C. Inkson, Plenum Press (1984). Green’s Functions and Condensed Matter, G. Rickaysen, Academic Press (1991). Many-Body Theory G. D. Mahan (2007, third edition). L. V. Keldysh, Sov. Phys. JETP (1962).

Schrödinger, Heisenberg and Interaction Representation Schrödinger picture Interaction picture Heisenberg picture

Time Evolution Operator Time evolution operator representation as a time-ordered product

Contractions and Normal Ordered Products

Wick’s Theorem Contraction (contracted product) of operators For more operators (F 83) all possible pairwise contractions of operators Uncontracted, all singly contracted, all doubly contracted, … Take matrix element over Fermi vacuum All terms zero except fully contracted products

Propagator

Partial Summation Method

Example: Ground State Calculation

GW Results for the Band Gap

Correlation functions Direct access to observable expectation values Retarded, Advanced Simple analitycal structure and spectral analysis Time ordered Allows perturbation theory (Wick’s theorem) * 1 = x 1,t 1 Definitions of Green’s Functions

 Just one indipendent GF General identities Spectral function Fluctuation-dissipation th. G r, G a, G are enough to evaluate all the GF’s and are connected by physical relations See eg: H. Haug, A.-P. Jauho A.L. Fetter, J.D. Walecka Equilibrium Properties of the System

 Contour-ordered perturbation theory: No fluctuation dissipation theorem G r, G a, G are all involved in the PT Time dep. phenomena Electric fields Coupling to contacts at different chemical potentials 2 of them are indipendent Contour ordering See eg: D. Ferry, S.M. Goodnick H.Haug, A.-P. Jauho J. Hammer, H. Smith, RMP (1986) G. Stefanucci, C.-O. Almbladh, PRB (2004) Non-Equilibrium Green’s Functions

Dyson Equation  Two Equations of Motion Keldysh Equation Computing the (coupled) G r, G < functions allows for the evaluation of transport properties In the time-indipendent limit G r, G < coupled via the self-energies Constitutive Equations

Summary This section first outlined the Usuki method as a direct way of solving the Schrodinger equation in real space In subsequent slides the Green’s function approach was outlined with emphasis on the partial summation method and the self-energy calculation and what are the appropriate Green’s functions to be solved for in equilibrium, near equilibrium (linear response) and high-field transport conditions