Quantum Statistical Transport Phenomena in Memristive Computing Architectures

Christopher N. Singh, Brian A. Crafton, Mathew P. West, Alex S. Weidenbach, Keith T. Butler, Allan H. MacDonald, Arjit Raychowdury, Eric M. Vogel, W. Alan Doolittle, L.F.J. Piper, and Wei-Cheng Lee

Phys. Rev. Applied 15, 054030 – Published 14 May 2021

Abstract

The advent of reliable nanoscale memristive components is promising for next-generation compute-in-memory paradigms; however, the intrinsic variability in these devices has prevented widespread adoption. Here, we show coherent electron wave functions play a pivotal role in the nanoscale transport properties of these emerging nonvolatile memories. By characterizing both filamentary and nonfilamentary memristive devices as disordered Anderson systems, the switching characteristics and intrinsic variability arise directly from the universality of electron transport in disordered media. Our framework suggests that localization phenomena in nanoscale solid-state memristive systems are directly linked to circuit-level performance. We discuss how quantum conductance fluctuations in the active layer set a lower bound on device variability. This finding implies that there is a fundamental quantum limit on the reliability of memristive devices and that electron coherence will play a decisive role in surpassing or maintaining Moore’s law with these systems.

Received 31 August 2020
Revised 8 April 2021
Accepted 9 April 2021

DOI:https://doi.org/10.1103/PhysRevApplied.15.054030

Physics Subject Headings (PhySH)

Anderson localization Quantum transport

Amorphous semiconductors Artificial neural networks Devices Transition metal oxides

Density functional theory Linear response theory

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Christopher N. Singh ^1,2, Brian A. Crafton³, Mathew P. West³, Alex S. Weidenbach³, Keith T. Butler ⁴, Allan H. MacDonald⁵, Arjit Raychowdury³, Eric M. Vogel³, W. Alan Doolittle³, L.F.J. Piper ^1,6, and Wei-Cheng Lee^1,*

¹Department of Physics, Applied Physics, and Astronomy, Binghamton University, Binghamton, New York 13902, USA
²Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
³Department of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
⁴SciML, Scientific Computing Department, Rutherford Appleton Laboratory, Didcot OX110QX, United Kingdom
⁵Department of Physics, University of Texas at Austin, Austin, Texas 78712-1081, USA
⁶WMG, University of Warwick, Coventry CV4 7AL, United Kingdom

^*wlee@binghamton.edu

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Vol. 15, Iss. 5 — May 2021

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Images

Figure 1
The relation between device-level variance and circuit-level performance. (a) The circuit-level crossbar architecture with 1-bit word-line drivers and higher-precision ADCs with a NMOS transistor and memristor. In this circuit, the word line (WL) and select line (SL) are set to a high voltage and the resulting current along the bit line (BL) is the result of the read operation. (b) The stack architecture of filamentary memristors set length scale at the device level. (c) A quantum-level description of the mechanism. The set and reset operations generate dynamic potentials for electrons. When the filament is fully formed, a high-transmission-probability path exists. (d) The measured and simulated conductance fluctuations for ${Hf O}_{x}$ , showing a log-normal character indicative of phase-coherent localization. (e) The effect of variance on the readout performance. The left and right panels show the total normalized conductance of four (left) and eight (right) devices with 5%, 10%, and 20% variance. The black vertical lines represent ADC regions set by conductance difference between on and off state devices ( $G_{on} - G_{off}$ ).
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Figure 2
Experimentally measured distributions of ${Hf O}_{x}$ memristors. The plot shows conductance distributions on different substrates and in both high- and low-resistance states: (a) glass LRS; (b) glass HRS; (c) silicon LRS; (d) silicon HRS. The data show that over many devices and multiple measurements, a clear statistical distribution emerges that shows log-normal behavior in the conductance, a hallmark of phase-coherent transport.
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Figure 3
Mobility edge switching and hysteresis with dynamic disorder. (a) The electronic structure: the density of states for amorphous stoichiometric ( $a$ - ${Nb}_{2} O_{5}$ ) and off-stoichiometric ( ${Nb}_{2} O_{4.875}$ ) niobium oxide. Oxygen deficiency draws out the conduction band and shifts the chemical potential by approximately 1 eV. The shading indicates a region of strong localization and the region integrated to produce the isodensity surface, shown as an inset in part (c) in the rightmost panel. (b) Simulated conductance with dynamic disorder. By assuming a greater vacancy transport (disorder potential redistribution), a single operating bias can have multiple conductance states. Each point is averaged over five disorder realizations with $W = 3$ eV. The dotted line is a guide to the eye. (c) The annular memdiode-device $I$ - $V$ character. The inset shows the $0.1 e^{-} / Å^{3}$ isodensity surface, demonstrating the stochastic landscape for electrons in establishing a current.
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Figure 4
The characterization of a dynamic distribution of disorder. The plot shows that the probability distribution of having a defect is dependent on the ionic velocity $v$ .
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Figure 5
The quantification of the in-gap contribution to the conductivity. The dotted lines plot the finite-size Kubo conductivity as a function of the energy at 300 K in (a) a single-band model, (b) the two-band model, and (c) the first-principles DFT ${Nb}_{2} O_{5}$ . The solid lines show the density of states on the same dependent axis with a globally arbitrary, but internally relative, vertical scale.
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Figure 6
Modeling the population of in-gap states and their localization. The lines plot the density of states as a function of disorder strength $W$ and the dots represent the Gini coefficient: (a) a single-band model; (b) **HERE** an arbitrarily gapped two-band model; (c) the density-functional result for a 50-band 197-unit supercell of ${Nb}_{2} O_{5}$ . $W$ specifies the width of the box distribution in electronvolts.
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