Quantum Transport Outline: What is Computational Electronics?

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Quantum Transport


  • 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

Transport Properties of system/device using Green’s functions formalism

  • Low field transport

    • Linear response theory (ASU)
  • High field transport

    • Bulk systems – Airy approach (Rita Bertoncini, ASU, PhD Thesis)
    • Devices:
      • Recursive Green’s Functions Approach (ASU, Purdue)
      • CBR Approach (ASU, WSI, Purdue)

Linear Response Theory

  • Only the retarded Green’s function is needed as it includes the collisional broadening of the states

  • In the ASU’s simulator for low-field mobility calculation in silicon inversion layers, strained-Si layers and InGaAs/InAlAs heterostructures the following features have been implemented:

    • Realistic treatmet of scattering within the self-consistent Born approximation
    • Modification of the density of states function is accounted for due to the collisional broadening of the states and the intersubband scattering
    • Random phase approximation in its full implementation is included to properly treat static screening of Coulomb and Interface-Roughness scattering
    • Bethe-Salpether integral equation is solved in the calculation of the conductivity
    • Excellent agreement is obtained with measured low-field mobility data in silicon inversion layers and predictions were made for the mobility behavior in Strained-Si layers and InGaAs/InAlAs heterostructures that were later confirmed with experimental measurements

Relevant Literature

  • D. Vasileska, P. Bordone, T. Eldridge and D.K. Ferry, “Calculation of the average interface field in inversion layers using zero-temperature Green’s functions formalism”, J. Vac. Sci. Technol. B 13, 1841-7 (1995).

  • P. Bordone, D. Vasileska and D.K. Ferry, “Collision duration time for optical phonon emission in semiconductors”, Physical Review B 53, 3846-55 (1996).

  • D. Vasileska, T. Eldridge and D.K. Ferry, “Quantum transport: Silicon inversion layers and InAlAs-InGaAs heterostructures”, J. Vac. Sci. Technol. B 14, 2780-5 (1996).

  • D. Vasileska, P. Bordone, T. Eldridge and D. K. Ferry, “Quantum transport calculations for silicon inversion layers in MOS structures”, Physica B 227, 333-5 (1996).

  • D. Vasileska and D. K. Ferry, “Scaled silicon MOSFET’s: Part I - Universal mobility behavior”, IEEE Trans. Electron Devices 44, 577-83 (1997).

  • G. Formicone, D. Vasileska and D.K. Ferry, “Transport in the surface channel of strained Si on a relaxed Si1-xGex substrate”, Solid State Electronics 41, 879-886 (1997).

Proposed Strained-Si and Strained-SiGe Devices

Is Strain Beneficial in Nanoscale MOSFETs With High Channel Doping Densities?

High Field Transport in Devices: Recursive Green’s Functions Approach

  • The most complete 1D transport in resonant tunneling diodes (RTDs) that operate on purely quantum mechanical principles was accomplished with the NEMO1D Code

  • The NEMO 1D Code was developed by Roger Lake, Gerhard Klimeck, Chris Bowen and Dejan Jovanovich while working at Texas Instruments/Raytion

  • It solves the retarded Green’s function (spectral function) in conjuction with less-than Green’s function (occupation function) self-consistently

  • References for NEMO1D:

    • Roger. K. Lake, Gerhard Klimeck, R. Chris Bowen, Dejan Jovanovic, Paul Sotirelis and William R. Frensley, "A Generalized Tunneling Formula for Quantum Device Modeling",VLSI Design, Vol. 6, pg 9 (1998).
    • Roger Lake, Gerhard Klimeck, R. Chris Bowen and Dejan Jovanovic, "Single and multiband modeling of quantum electron transport through layered semiconductor devices", J. of Appl. Phys. 81, 7845 (1997).

The Philosophy Behind the Recursive Green’s Function Approach

Representative Simulation Results

High Field Transport in Devices: Contact Block Reduction Method

Retarded Green’s Function

Transmission Function and Local Density of States Calculation

  • Transmission Function

  • Local Density of States Function

    • CBR Formalism

Properties of Widely Acceptable 2D Simulators

Need for 3D Device Simulations

DG vs. TG FinFET

Atomistic Simulations – The Future of Nano-Devices

  • Examples of devices for which atomistic simulations will be necessary include:

    • Devices in which local Strain exists
    • Alloy Disorder has to be properly described

Why Tight-Binding ?

Scalability of TB approaches

The sp3s* Hamiltonian [Vogl et al. J. Phys. Chem Sol. 44, 365 (1983)]

The sp3d5s* Hamiltonian [Jancu et al. PRB 57 (1998)]

Tight-Binding sp3d5s* model for nitrides

Boundary conditions

Where do we put the atoms ?

Example: Strain and Pseudomorphic growth

Strain in a AlGaN/GaN Nanocolumn

AlGaN/GaN Nanocolumns

How do we describe alloys ?

Self-Consistent Tight-Binding

Self-Consistent Tight-Binding


  • Linear response and solution of the Beth-Salpether equation in conjunction with the Dyson equation for the retarded Green’s function is useful when modeling low-field mobility of inversion layers

  • When modeling high field transport both Dyson equation for the retarded Green’s function and the kinetic equation for the less-than Green’s function have to be solved self-consistently

  • CBR approach and recursive Green’s function method have both their advantages and their disadvantages

  • When local strains and stresses have to be accounted for in ultra-nano-scale devices then atomistic approaches become crucial


What are the lessons that we have learned?

  • Semi-classical simulation is still a very important part of Today’s semiconductor device modeling as power devices and solar cells (traditional ones) operate on semi-classical principles

  • Quantum corrections can quite accurately account for the quantum-mechanical size quantization effect which gives about 10% correction to the gate capacitance

  • For modeling ultra-nano scale devices one can successfully utilize both Poisson-Monte Carlo-Schrodinger solvers and fully quantum-mechanical approaches based on NEGF (tunelling + size quantization)

  • Full NEGF is a MUST when quantum interference effects need to be captured and play crucial role in the overall device behavior

  • For a subset of ultra-nano scale devices that are in the focus of the scientific community now, in which band-structure, local strain and stresses, play significant role, atomistic simulations are necessary.

Simulation Strategy for Ultra-Nano-Scale Devices

Atomistic Simulations Selected Literature

  • Mathieu Luisier and Gerhard Klimeck, "A multi-level parallel simulation approach to electron transport in nano-scale transistors", Supercomputing 2008, Austin TX, Nov. 15-21 2008. Regular paper - 59 accepted papers, 277.

  • Mathieu Luisier, Neophytos Neophytou, Neerav Kharche, and Gerhard Klimeck, "Full-Band and Atomistic Simulation of Realistic 40 nm InAs HEMT", IEEE IEDM, San Francisco, USA, Dec. 15-17, 2008, DOI : 10.1109/IEDM.2008.4796842,

  • Mathieu Luisier, and Gerhard Klimeck, "Performance analysis of statistical samples of graphene nanoribbon tunneling transistors with line edge roughness", Applied Physics Letters, Vol. 94, 223505 (2009), DOI:10.1063/1.3140505,

  • Mathieu Luisier, and Gerhard Klimeck, "Atomistic, Full-Band Design Study of InAs Band-to-Band Tunneling Field-Effect Transistors ", IEEE Electron Device Letters, Vol. 30, pp. 602-604 (2009), DOI:10.1109/LED.2009.2020442.

Questions or Comments?

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