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Sharp Switching in Tunnel Transistors and Physics- Based Machines for Optimization

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Sharp Switching in Tunnel Transistors and Physics- Based Machines for Optimization Sharp Switching in Tunnel Transistors and Physics- based Machines for Optimization Sri Krishna Vadlamani Electrical Engineering and Computer Sciences University of California, Berkeley Technical Report No. UCB/EECS-2021-97 http://www2.eecs.berkeley.edu/Pubs/TechRpts/2021/EECS-2021-97.html May 14, 2021 Copyright © 2021, by the author(s). All rights reserved. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission. Sharp Switching in Tunnel Transistors and Physics-based Machines for Optimization by Sri Krishna Vadlamani A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Electrical Engineering and Computer Sciences in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Eli Yablonovitch, Chair Professor David Limmer Professor Alistair Sinclair Spring 2021 Sharp Switching in Tunnel Transistors and Physics-based Machines for Optimization Copyright 2021 by Sri Krishna Vadlamani 1 Abstract Sharp Switching in Tunnel Transistors and Physics-based Machines for Optimization by Sri Krishna Vadlamani Doctor of Philosophy in Electrical Engineering and Computer Sciences University of California, Berkeley Professor Eli Yablonovitch, Chair In this thesis, I report my work on two different projects in the field of energy-efficient computation: Part 1 (Device Physics): Tunnel Field-Effect Transistors (tFETs) are one of the candidate devices being studied as energy-efficient alternatives to the present-day MOSFETs. In these devices, the preferred switching mechanism is the alignment (ON) or misalignment (OFF) of two energy levels or band edges. Unfortunately, energy levels are never perfectly sharp. When a quantum dot interacts with a wire, its energy level is broadened. Its actual spectral shape controls the current/voltage response of such transistor switches, from on (aligned) to off (misaligned). The most common model of spectral line shape is the Lorentzian, which falls off as reciprocal energy offset squared. Unfortunately, this is too slow a turnoff, algebraically, to be useful as a transistor switch. Electronic switches generally demand an ON/OFF ratio of at least a million. Steep exponentially falling spectral tails would be needed for rapid off-state switching. This requires a new electronic feature, not previously recognized: narrowband, heavy-effective mass, quantum wire electrical contacts, to the tunneling quantum states. Part 2 (Systems Physics): Optimization is built into the fundamentals of physics. For exam- ple, physics has the principle of least action, the principle of minimum power dissipation, also called minimum entropy generation, and the adiabatic principle, which, in its quantum form, is called quantum annealing. Machines built on these principles can solve the mathemati- cal problem of optimization, even when constraints are included. Further, these machines become digital in the same sense that a flip–flop is digital when binary constraints are in- cluded. A wide variety of machines have had recent success at approximately optimizing the Ising magnetic energy. We demonstrate that almost all those machines perform opti- mization according to the principle of minimum power dissipation as put forth by Onsager. Moreover, we show that this optimization is equivalent to Lagrange multiplier optimization for constrained problems. i To my wonderful Mummy, Daddy, and Madhav ii Contents Contents ii List of Figures v 1 Introduction 1 1.1 Part 1: MOSFETs, subthreshold slope, tunnel transistors . 1 1.2 Part 2: Physics-based optimization|Ising solvers . 4 I Spectral lineshape theory for tunnel transistors and optical absorption 8 2 Tunnel Transistors and Spectroscopy 9 2.1 The performance of energy-filtering transistors depends on lineshape . 11 2.2 Standard derivation of Lorentzian lineshapes; Fermi's Golden Rule . 15 2.3 Id − Vg characteristics from lineshape . 20 2.4 Deriving non-Lorentzian lineshapes | Exact solution using Laplace transforms 23 2.5 Deriving non-Lorentzian lineshapes | Tunneling from a dot into a wire that has a Lorentzian density of states . 25 ~ 2.6 Smoothness of d1(t) and spectral lineshape tails . 28 2.7 Proposed tunnel transistor design and its analysis . 34 2.8 Final structure and Id − Vg characteristics . 39 2.9 Conclusion . 43 3 Optical Spectroscopy and the Quest for the Urbach Tail 44 3.1 Absorption coefficient α(!) in terms of the susceptibility χ(!) . 45 3.2 Lorentz-Lorenz model . 47 3.3 Statistical models of lineshape based on dipole moment fluctuations; Brownian motion; Kubo's lineshape . 48 3.4 Linear response and autocorrelations . 50 3.5 Warm-up: Absorption in a harmonic potential well . 53 iii 3.6 Absorption by an electron in a harmonic potential well coupled to a phonon reservoir . 57 3.7 Optical absorption in a 2-level system coupled to a phonon reservoir . 62 3.8 Optical absorption in a general crystal . 70 3.9 Conclusion . 72 II Physics-based Optimization 73 4 Physics principles as Optimization tools 74 4.1 Optimization in Physics: Principles and Algorithms . 75 4.2 The Ising problem . 80 5 Coupled LC Oscillator Ising Solver and Lagrange Multipliers 82 5.1 Coupled LC oscillator Ising solver . 82 5.2 The circuit follows the principle of minimum power dissipation . 89 5.3 Lagrange multipliers . 91 5.4 Exact equivalence between coupled LC oscillators and Lagrange multipliers . 95 5.5 Augmented Lagrange function . 100 5.6 Conclusion . 104 6 Other Physical Ising Solvers 105 6.1 Other Physical Ising Solvers . 105 6.2 Other methods in the literature . 114 6.3 Applications in Linear Algebra and Statistics . 117 6.4 Conclusion . 119 7 Conclusion 120 7.1 Part 1: MOSFETs, subthreshold slope, tunnel transistors . 120 7.2 Part 2: Physics-based optimization|Ising solvers . 121 Bibliography 122 A Optical absorption in a damped harmonic potential well|No rotating wave approximation 136 B Motional Narrowing 142 B.1 Jumping within a Gaussian distribution of frequencies . 142 B.2 Impossibility of getting asymptotically exponential lineshapes . 144 C Minimum Power Dissipation 145 C.1 Minimum power dissipation implies circuit laws . 145 C.2 Onsager's dissipation and the standard power dissipation . 148 iv C.3 Minimum power dissipation in nonlinear circuits . 150 D Equations of motion derivation 152 D.1 Signal circuit . 152 D.2 Power dissipation . 155 D.3 Pump circuit . 156 D.4 Augmented Lagrange equations of motion . 159 D.5 Circuit for arbitrary real values in the Ising J matrix . 160 E Lagrange Multipliers 162 E.1 Lagrange multipliers theory . 162 E.2 Lagrange multipliers algorithms: Augmented Lagrangian . 164 E.3 Application of Lagrange multipliers to the Ising problem; Cubic terms . 165 E.4 Iterative Analog Matrix Multipliers . 165 v List of Figures 1.1 A standard MOSFET has three terminals, the source, the drain, and the gate. When a voltage is applied between the source and the drain terminals, current flows between them through an intermediary channel. The conductance of the channel is modulated using the gate terminal. 2 1.2 (a) When the gate voltage is low, the channel barrier is high, allowing only a few source electrons to the drain. (b) A higher gate voltage pushes the barrier down and allows an exponentially greater number of electrons to flow to the drain. 2 1.3 The Ising problem setup. A collection of N spins, which can each individually only point up (xi = 1 if the i-th spin is up) or down (xi = −1), interact via pairwise interaction energies −Jijxixj. The problem is to find the vector of orientations of the N spins, x, that minimizes the sum of all the pairwise energies. 5 2.1 Tunnel distance modulating tunnel transistors. The barrier between the source and the channel is large in the absence of gate voltage and the device is OFF (left). A positive gate voltage narrows the barrier and turns the device ON (right). 10 2.2 Na¨ıve picture of energy-filtering tunnel transistors. A slight offset in the drain and source quantum dots leads to the current between them immediately going to zero so that energy conservation remains satisfied. 10 2.3 More realistic picture of energy-filtering tunnel transistors. Current keeps flowing between the source and drain dots even after they are offset by a gate voltage because of spectral broadening of the energy levels. Current flows between the energy tails and contributes to off-state leakage. 13 2.4 (a) The pure exponential decay of amplitude from a quantum dot level into a wire with a broad density of states. (b) The lineshape is proportional to the Fourier transform of the time domain decay. The Fourier transform of a pure exponential time decay is a Lorentzian function in frequency. Lorentzians have slow decay 1=!2 at large !..................................... 20 2.5 The full tunnel transistor. The source wire is coupled to the source quantum dot via the interaction X as is the drain wire to its own quantum dot. This interaction X led to the broadening of the quantum dot's energy levels. The quantum dots themselves interact via the coupling M and it is this interaction that leads to current in the device. 21 vi 2.6 From the point of view of an electron leaking from a quantum dot into a wire, these two systems are equivalent: (a) The quantum dot is coupled to a wire with a Lorentzian density of states, and (b) The quantum dot is coupled to an intermediary quantum dot that is then coupled to a wire with a broad flat density of states through an appropriate interaction jX00j.
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