Quantum Transport

1. Quantum Transport

2. 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

3. 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

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

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

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

7. yields the modes on the left side of the system Solving the eigenvalue problem: 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 defining and Complete matrix of eigenvectors:

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

9. Variation on the cascading scattering matrix technique method Usuki et al. Phys. Rev. B 52, 8244 (1995) Boundary condition- waves of unit amplitude incident from right 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

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

11. u1(+) for B=0 T First propagating mode for an irregular potential u1(+) for B=0.7 T u1j Mode functions no longer simple sine functions confining potential 40 80 0 j general formula for velocity of mode m obtained by taking the expectation value of the velocity operator with respect to the basis vector.

12. Conduction band profile Ec 0.8 Energy of the ground subband 0.6 Conduction band [eV] 0.4 Vg= -1.0 V Vg= -0.9 V Vg= -0.7 V 0.2 Fermi level EF 0.0 z-axis [mm] -0.2 0.00 0.02 0.04 0.06 0.08 0.10 Example – Quantum Dot Conductance as a Function of Gate voltage Simulation gives comparable 2D electron density to that measured experimentally Potential felt by 2DEG- maximum of electron distribution ~7nm below interface Potential evolves smoothly- calculate a few as a function of Vg, and create the rest by interpolation

13. -0.897 V -0.923 V -0.951 V 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 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

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

15. 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

16. Relevant Literature • A Guide to Feynman Diagrams in the Many-Body Problem, 2nd 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).

17. Schrödinger, Heisenberg and Interaction Representation • Schrödinger picture • Interaction picture • Heisenberg picture

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

19. Contractions and Normal Ordered Products

20. 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

21. Propagator

22. Partial Summation Method

23. Example: Ground State Calculation

24. GW Results for the Band Gap

25. Allows perturbation theory (Wick’s theorem) Time ordered Simple analitycal structure and spectral analysis Retarded, Advanced Correlation functions Direct access to observable expectation values Definitions of Green’s Functions * 1 = x1,t1

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

27. Non-Equilibrium Green’s Functions 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) • Time dep. phenomena • Electric fields • Coupling to contacts at different chemical potentials • Contour-ordered perturbation theory: Gr, Ga, G<, G> are all involved in the PT 2 of them are indipendent Contour ordering No fluctuation dissipation theorem

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

29. 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