Computational Single-Electronics
by Chris Wasshuber

For the first time, this ebook brings together a review of the theory behind single-electronics, a state-of-the-art description of simulation methods for single-electron devices and circuits, and a comprehensive discussion of a wide selection of applications ranging from logic and memory to metrology. The author shares his experience in single-electronics. Attention is given to open challenges such as the random background charge problem, where old and new ideas are presented to overcome it. Many simulation methods described in this ebook are not particular to single-electronics. The efficient calculation of capacitances, tunnel rates, and bound energy states has applications in many other fields. This ebook is written with the practical side in mind. Simulation methods are described in a manner that makes their implementation straightforward.

Single-electronics is a fascinating technology that reveals new physical effects of charge transport. It has many benefits and great figures of merits but also several open challenges waiting for elegant solutions. In order to collect, categorize, and summarize a good part of this body of knowledge as well as to introduce some new points of view, variations, and extensions, I set out to write this book. A book targeted at the student eager to delve into single-electronics as well as the expert who needs a reference for theory, circuits, and algorithms for system analyses. This book addresses three areas: the theory which goes beyond the orthodox theory, the computational methods necessary to analyze sing-electron circuits, and applications and manufacturing methods, the practical side of single-electronics. The theory was kept short and concise, suitable for people seeking a compact introduction or reference. For in-depth coverage one has to consult cited articles and books. The computational part is very complete and can be considered state-of-the-art for single-electronics. Almost all algorithms which are necessary for a successful and efficient implementation are stated. Not all of them are exhaustively explained but at least a recipe for their successful implementation is given.

Several of the described algorithms have been implemented in SIMON a single-electron device and circuit simulator.

• 1 Introduction
  • 1.1 Single-Electronics - Made Easy
  • 1.2 A Historical Look Back
• 2 Theory
  • 2.1 Orthodox Single-Electron Theory
    • 2.1.1 Thermodynamic Formulation
  • 2.2 Time and Space Correlations
  • 2.3 Master Equation of Electron Transport
  • 2.4 Extensions to the Orthodox Theory
    • 2.4.1 Cotunneling
    • 2.4.2 Influence of the Electromagnetic Environment; Quantum Langevin Theory; Phase Correlation Theory
    • 2.4.3 Different Materials - Different Density of States; Discrete Energy Levels
    • 2.4.4 Superconducting Tunnel Junctions; Quasiparticle Tunneling; Parity Effect; Andreev Reflections; Coherent Cooper Pair Tunneling
    • 2.4.5 Self-Heating
    • 2.4.6 Image Charge
• 3 Simulation Methods and Numerical Algorithms
  • 3.1 Monte Carlo Method
    • 3.1.1 Time-Dependent Tunnel Rates
    • 3.1.2 Deterministic Model
    • 3.1.3 Random Numbers
      • Linear Congruential Generators
      • Lagged Fibonacci or Shift Register Generators
      • Inverse Congruential Generators
      • Resolution Limit for Rare Tunnel Events
  • 3.2 Solution of the Master Equation
    • 3.2.1 Krylov Subspace Approximate of the Matrix Exponential Operator
    • 3.2.2 Schur-Frechet Algorithm
  • 3.3 Coupling with SPICE
  • 3.4 Free Energy
  • 3.5 Tunnel Transmission Coefficient
    • 3.5.1 Analytic Solutions
    • 3.5.2 Wentzel-Kramers-Brillouin Approximation
    • 3.5.3 Piecewise Potential Approximation
      • Three Dimensions
      • Transfer Matrix versus Scattering Matrix
      • Piecewise-Constant Potential Approximation
      • Piecewise-Linear Potential Approximation
    • 3.5.4 Finite Differences with Continued Fraction
    • 3.5.5 Finite Elements
    • 3.5.6 Detour via the Time-Dependent Schrödinger Equation
      • Finite Differences
      • Spectral Method
  • 3.6 Energy Levels
    • 3.6.1 Analytic Solutions
    • 3.6.2 Bohr-Sommerfeld Quantization Rule
    • 3.6.3 Piecewise Potential Approximation; Transmission Line Analogy
    • 3.6.4 Finite Differences with Continued Fraction
    • 3.6.5 Time-Dependent Solutions
  • 3.7 Evaluation Schemes for Cotunneling
  • 3.8 Rate Calculation Including Electromagnetic Environment
    • 3.8.1 Network Impedance Calculation
  • 3.9 Numerical Integration of Tunnel Rates
  • 3.10 Time-Dependent Node Voltages and Node Charges
  • 3.11 Stability Diagram and Stable States
  • 3.12 Capacitance Calculations
    • 3.12.1 Analytic Formulas
    • 3.12.2 Capacitance of Ellipsoid, Elliptic Disc, and Circular Disc
    • 3.12.3 Image Charge Method for Spheres
      • Capacitance of Two Equal Spheres
      • Capacitance of an Arbitrary Arrangement of Spheres
      • Capacitance of Two Intersecting Spheres - Capacitance by Inversion
    • 3.12.4 Source Point Collocation Method
    • 3.12.5 Stochastic Capacitance Calculation for Rectangular Geometries
• 4 Circuits and Applications
  • 4.1 Fundamental Circuits
    • 4.1.1 Single-Electron Transistor
    • 4.1.2 Single-Electron Turnstile; Asymmetric Turnstile
    • 4.1.3 Single-Electron Pump
    • 4.1.4 Linear Array of Junctions
    • 4.1.5 Two-Dimensional Array of Junctions
  • 4.2 Metrology Applications
    • 4.2.1 The Quantum Metrology Triangle
    • 4.2.2 Electron Pump - Current Standard
    • 4.2.3 Supersensitive Electrometer
    • 4.2.4 Single-Electron Proximity Probe
    • 4.2.5 Coulomb Blockade Thermometer
  • 4.3 Memory
    • 4.3.1 Single-Electron Flip-Flop
    • 4.3.2 Electron Trap Memory
    • 4.3.3 Ring Memory
    • 4.3.4 Background-Charge-Independent Memory
    • 4.3.5 Single-Island Memory
    • 4.3.6 Multiple-Island Memory
    • 4.3.7 T-Memory Cell
    • 4.3.8 Combinatorial Access Memory
    • 4.3.9 Switch-Source-Sink Memory
    • 4.3.10 Negative Differential Resistance Flip-Flop
    • 4.3.11 Multivalued Memory from Asymmetric Tunnel Junctions
    • 4.3.12 Nanocrystal Memory
  • 4.4 Logic
    • 4.4.1 Transistor-like Design - Voltage State Logic
    • 4.4.2 Bits by Single Electrons - Charge State Logic
      • Binary-Decision Diagram
      • Lattice Gas Machines
      • Systolic Processors
    • 4.4.3 Quantum Cellular Automata
    • 4.4.4 Wireless Logic
    • 4.4.5 Tunneling Phase Logic
    • 4.4.6 Parametron Logic
  • 4.5 Interfacing to CMOS
  • 4.6 Exotic Circuits
    • 4.6.1 Neuronal Networks
    • 4.6.2 Boltzmann Machines
    • 4.6.3 Mixed Bag
      • Negative Differential Resistance
      • Digital-to-Analog Converter
      • Asymmetric Tunnel Barriers
  • 4.7 Evolutionary Circuit Design
• 5 Random Background Charges
  • 5.1 The Good Side of High Charge Sensitivity
  • 5.2 Solutions on the Material Level
  • 5.3 Solutions on the Device Level
    • 5.3.1 Refresh for Single-Electron Logic
    • 5.3.2 Coulomb Oscillations
    • 5.3.3 Resistive Elements
    • 5.3.4 One- and Two-Dimensional Island Structures
  • 5.4 Solutions on the Circuit and System Level
• 6 Manufacturing Methods and Material Systems
  • 6.1 Shadow Evaporation
  • 6.2 Step-Edge Cutoff
  • 6.3 Nanoimprint
  • 6.4 Planar Quantum Dots
  • 6.5 Scanning Probe Microscopy
  • 6.6 Granular Films
  • 6.7 Self-Assembled Structures
  • 6.8 Outlook
• Appendixes
  • A Fermi's Golden Rule
  • B Capacitance and Resistance Extraction from Measured Data
  • C Analytic Solutions of the Cotunneling Rate
  • D Algorithms from Number Theory
  • E Convex Hull of Point Set
  • F Analytic Capacitance Calculation
• References
• Subject Index

1st edition 2001, PDF 278 pages.
word count: 91345 which is equivalent to 365 standard pages of text