Quantum-statistical transport phenomena in memristive computing architectures

Christopher N. Singh,1, 2 Brian A. Crafton,3 Mathew P. West,3 Alex S. Weidenbach,3 Keith T. Butler,4 Allan H.

MacDonald,5 Arjit Raychowdury,3 Eric M. Vogel,3 W. Alan Doolittle,3 L. F. J. Piper,1, 6 and Wei-Cheng Lee1

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, NM 87545, USA3Department of Electrical and Computer Engineering,

Georgia Institute of Technology, Atlanta, GA, 30332, USA4SciML, Scientific Computing Department, Rutherford Appleton Laboratory, Didcot, OX110QX, UK

5Department of Physics, University of Texas at Austin, Austin Texas 78712-1081, USA6WMG, University of Warwick, Coventry CV4 7AL, UK

(Dated: June 2, 2021)

The advent of reliable, nanoscale memristive components is promising for next generationcompute-in-memory paradigms, however, the intrinsic variability in these devices has preventedwidespread adoption. Here we show coherent electron wave functions play a pivotal role in thenanoscale transport properties of these emerging, non-volatile memories. By characterizing bothfilamentary and non-filamentary memristive devices as disordered Anderson systems, the switchingcharacteristics and intrinsic variability arise directly from the universality of electron transport indisordered media. Our framework suggests localization phenomena in nanoscale, solid-state memris-tive systems are directly linked to circuit level performance. We discuss how quantum conductancefluctuations in the active layer set a lower bound on device variability. This finding implies there isa fundamental quantum limit on the reliability of memristive devices, and electron coherence willplay a decisive role in surpassing or maintaining Moore’s Law with these systems.

I. INTRODUCTION

The von Neumann model of computing is the founda-tion of modern digital technologies, but this paradigmharbors an intrinsic bottleneck. This bottleneck arisesbecause data cannot be operated on in the same placethat it is stored, and therefore, computation speed is lim-ited by the rate data can be transferred between memoryand compute locations [1]. Because of this, solid-state,bio-mimetic computing that avoids this inefficient datatransfer and enables a ‘compute-in-memory’ paradigm ishighly desired [2]. However, mimicking biological systemsthat typically have billions of neurons [3] suggests a solid-state analog would require a similar number of logicalunits. With transistors already hitting the quantum lim-its of performance [4], a different class of nanoscale, solid-state components is needed, and adaptive-oxide mem-ristors are considered among the most promising candi-dates [5, 6]. Although viable solid-state memristors havealready been reported [7], the variability inherent in theiroperation has been severely detrimental to circuit-levelperformance [8]. What’s more, their underlying switch-ing mechanisms have been hotly debated [9–12].

In solid-state devices, the migration of atomic defectsis generally considered sufficient for memristive switch-ing [13], but whether it is necessary–or even optimal–is unclear [14], as purely-electronic switching mecha-nisms have also been proposed [10, 15]. In eithercase, quantum transport effects such as strong elec-tronic correlations, interface scattering, tunneling, andinterference may all additionally contribute to the to-tal transport characteristics. Thus, to improve the per-formance of nanoscale, compute-in-memory devices, a

quantum-theoretical framework is necessary. This is es-pecially pressing given that predicting device propertiespost fabrication remains the single greatest challenge towidespread adoption [16, 17]. Treating all the relevantvariables in a non-equilibrium, quantum framework willtherefore accelerate the development pipeline of memris-tive materials [18].

This work fills that gap in the microscopic de-scription of memristive transport by developing acomputationally-tractable, first-principles approach topredict transport properties of electrons in environmentswith stochastic disorder potentials. We apply it to fila-mentary (a-HfOx) and non-filamentary (a-Nb2O5−x) sys-tems. In filamentary devices, we find that electron-phaseeffects are a significant contribution of variability. Innon-filamentary systems, we find that dynamic disor-der potentials can give rise to hysteretic conductancecurves. Together, these results indicate that electron-phase effects can significantly influence the performanceof compute-in-memory devices. The stochastic nature ofthe conductance in filamentary systems [19] is shown tobe analogous to that of quantum wires near an Andersonlocalization transition [20]. One of the most astoundingproperties of quantum disordered systems is that the log-arithm of the conductance, not the resistance, stabilizesin the thermodynamic limit [21]. In this way, the conduc-tance becomes a key circuit-level design element becausenear an Anderson transition, the resistance is not self-averaging. This immediately suggests interference phe-nomena could play a central role in the rational design ofbio-mimetic hardware. We anticipate an understandingof the origin of variability will enable circuit engineers todesign more robust compute-in-memory architectures.

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(a) Circuit Level

A prototypical compute-in-memory crossbar using cross point devices in 1 Transistor 1 Resistor (1T1R) configuration

(b) Device Level (c) Atomic Level

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(d) Conductance distributions (e) Effect of variance on read-out performance

Memristor stack geometry

~1 mmSiO2/Si or Glass

Ti 20 nm

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FIG. 1. Relation between device level variance and circuit-level performance. a) Crossbar architecture with 1-bitword line drivers and higher precision ADC’s 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 readoperation. b) Stack architecture of filamentary memristors set length scale at the device level. c) Quantum level descriptionof mechanism. The set and reset operations generate dynamic potentials for electrons. When the filament is fully formed, ahigh transmission probability path exists. d) Measured and simulated conductance fluctuations for HfOx showing a log-normalcharacter indicative of phase coherent localization. e) Left and Right show total normalized conductance of 4 (Left) and 8(Right) devices with 5, 10, and 20 percent variance. The black vertical lines represent ADC regions set by conductance differentof on and on state devices (Gon −Goff)

II. OVERVIEW OF COMPUTE-IN-MEMORYARCHITECTURE

Modern computing in general, but deep learning tech-niques especially, contain a workload of almost entirelymatrix multiplication (~y = W~x) [22]. Neuromorphichardware is poised to perform this much more efficiently.In traditional von Neumann machines, both the featuredata ~x and matrix weights Wij are transported frommain memory to the compute units, where the multiply-and-accumulate (MAC) operations are performed. Afterwhich, the result ~y is transported and written back intomain memory. In this procedure, the energy cost of read-ing and transporting data from memory to logic greatlyoutweighs the cost of the MAC [23, 24], thus motivatingin-memory computing.

A compute-in-memory paradigm performs the MACoperations in a crossbar structure using Ohm’s law anda non-volatile conductance state. This is often known asresistive random access memory (RRAM) [25], but thereare other memories that behave similarly. A circuit-to-atomic level schematic of the typical (HfOx) RRAM isgiven in Figure 1a-c. Each element of the matrix Wij

is programmed as a conductance in the crossbar (Fig-

ure 1a), and each value of the vector xi is converted tovoltage. The conductance of the active layer (Figure 1b),tuned with external bias, is ultimately set by control-ling the electric potential cross section seen by electrons.These electrons carry currents along the filaments formedby mobile oxygen vacancies in disordered HfOx [26], asshown in Figure 1c. By Ohm’s law, the current througheach RRAM device is proportional to the product of theprogrammed conductanceWij and applied voltage ~xi. ByKirchhoff’s current law, the resulting currents summedalong the columns of the crossbar are proportional to theproduct of the matrix and vector, ~y. Using this archi-tecture, the only data transport required is the featurevector ~x and result ~y, whereas a typical von Neumannarchitecture would move ~x and Wij . Assuming ~x andWij have the same dimension d, moving only ~x and ~yamounts to a significant savings of d(d− 1) elements.

Although compute-in-memory using the crossbar ar-chitecture can greatly reduce data transport, it faces itsown limitations at the device and circuit level. The twolimitations are: (1) the number of states the circuit canread at once and (2) the number of distinguishable statesthat can be accurately read from a column of the cross-bar. To read states from the crossbar, an analog-to-digital converter (ADC) converts the analog current value

3

from the crossbar to a digital value. In Figure 1a, the endof each column feeds into an ADC. Modern ADC’s arethe result of decades of research and can be used to readthousands of states at relatively high speed [27]. How-ever, device variance limits the number of distinguish-able states that can be accurately read. Figure 1d showsthat HfOx RRAM stacks do have significant variance.It shows that both the measured and predicted dimen-sionless conductance (g = G/Go) of electro-formed HfOx

devices is distributed log-normally. We will expand insection V on the details of the quantum framework thatenable this prediction, but at this point, it is sufficient tosay that this is a quantitative model that also explainsthe origin of the variability in these devices.

A log-normally distributed conductance means thatproblem 2 (clearly distinguishing states) is exacerbated,as a large proportion of the available states lie close toeach other in value. In Figure 1e, we plot the cumula-tive distribution function for the total conductance re-sulting from four (left) and eight (right) devices. Thisfigure demonstrates that given enough variance, the cur-rent summed along a column will result in an error witha probability given by the distribution of the variance ofeach device. In this circuit simulation, we use a conduc-tance ratio on par with experiment (see Figure 1d) andsweep five, ten, and twenty percent variance. In bothcases, ten and twenty percent normalized variance resultsin erroneous compute-in-memory results. Although it isa simple analysis, it clearly demonstrates that increasingthe number of devices read will increase the probabilityof errors due to accumulated variance, and that any in-trinsic quantum variability is directly tied to circuit levelperformance. Therefore, it is desirable to understandthe physical origin of the variance in order to mitigateit. Before laying out the theory however, we will firstestablish the well-known conductance properties of fila-mentary RRAMs.

III. CONDUCTANCE VARIATIONS INFILAMENTARY DEVICES

The conductance of nanoscale, filamentary memristorsgenerally does not follow a normal distribution. Instead,the log of the conductance is distributed normally. Thisis a general feature of these devices, but its origin hasbeen debated [28–31]. There are phenomenological mod-els that have considered the statistical properties of theconductance [32], and heuristic models such as trap as-sisted tunneling [33] that can recover it, but none affordit an origin. In other words, significant effort has beendevoted to understanding the log-normal conductance inwith the goal of mitigating uncertainty from an engineer-ing perspective, but none have considered the possibilitythat a fundamental quantum limit would manifest. Thereis, however, this possibility due to the length scales ofthese devices, and the stochastic nature of electron trans-port in disordered systems.

For example, the many possible configurations of a fil-ament in a thermal environment are one source of con-ductance fluctuations [34]. In fact, this has been studiedquite extensively already, but we do not focus on thatin this work. Another source of variability though is themany possible paths for electrons to take within any par-ticular filament configuration. Phase-coherence effectsbetween these different electron paths are known to giverise to log-normal conductance fluctuations [21]. To seethis schematically, we revisit figure 1c. It depicts a fil-amentary device in two arbitrary resistance states. Inone resistance state, the defects form a filament, and inanother resistance state, the defects do not form a con-tinuous filament. In either state, there are many possibledefect configurations, each with a different disorder po-tential for electrons (indicated schematically by the heatmaps). As a result, these devices can be considered as dy-namically disordered, and transport will be heavily influ-enced by phase coherence and localization effects [35, 36].More importantly, if the transport length L approachesthe phase coherence length Lφ, the transmission proba-bility for electrons will approach a universal distributiongiven only by the mean conductance–its character beinglog-normal with a normalized variance inversely propor-tional to its mean [37].

Beyond just a heuristic argument, however, there isadditional theoretical and experimental evidence to sug-gest quantum effects may play an important role. Theevidence mainly takes the form of thermal and dimen-sional arguments. For example, Basnet et al. have shownthat thermally insulating substrates can improve the per-formance of HfOx stacks by decreasing the variance [38].Other comprehensive investigations of HfOx RRAMs alsoshowed that increasing the temperature of the entire de-vice from 25◦C to 150◦C reduced the log-normal tail-ing [39]. Beilliard et al. demonstrated that differentstack compositions such as Al2O3/TiO2−x display an in-creased variability at cryogenic temperatures [40], sug-gesting these effects are not even unique to HfOx. Allthese works are commensurate with the idea that in-creased inelastic scattering at higher temperature coulddrive Lφ � L, restoring uncorrelated, diffusive transportand normally distributed conductance. With regards todimensionality, Bradley et al. have shown experimen-tally that electron injection tends to drive aggregationof oxygen defects into a quasi-1D filament [41, 42]. Thepopular trap-assisted tunneling model [33] sets the cur-rent proportional to a barrier-tunneling probability, orequivalently, to the rate of traversing the slowest bridgealong a classical percolation path. Therefore, existingtheory and experiment both tend to view the transportin nanoscale-filamentary memristors as quasi-1D. In spiteof this, existing models are generally semi-classical, eventhough Funck et al. have shown that the quantum behav-ior might be fundamentally different from semi-classicalpredictions [43]. Thus, both current theoretical modelsand experimental evidence suggest phase-coherence ef-fects can play an important role. Our results indicate

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FIG. 2. Experimentally measured distributions ofHfOx memristors. The plot shows conductance distribu-tions on different substrates and in both high and low re-sistance states. The data shows that over many devices,and multiple measurements, a clear statistical distributionemerges that shows log-normal behavior in the conductance,a hallmark of phase coherent transport.

that the log-normality of the conductance can in fact beattributed to quantum disorder effects, and although weuse HfOx as a model system, these results likely apply toother materials systems.

Figure 2 characterizes the conductance variability inour devices. The experimental data was obtained byswitching the devices between the low resistance state(LRS) and the high resistance state (HRS) multiple timeson many devices. To switch to the LRS, a positive volt-age sweep was applied to the top electrode from 0 to 1.2V. After the device was set to the LRS, a voltage sweepfrom 0 to 0.1 V was conducted. In post processing, theinverse slope of this low voltage sweep was calculated todetermine the resistance of the device. A negative volt-age sweep from 0 to -1.5 V was used to reset the deviceto the HRS. Then another 0 to 0.1 V sweep was usedto determine the resistance of the HRS. Each device wasswitched between the LRS and HRS ten times to ob-tain the cycle-to-cycle variation. This was repeated on37 devices on a glass substrate and 36 devices on a thinSiO2/Si substrate. In each panel, a histogram of conduc-tance measurements is overlaid with either a log-normalor Gaussian distribution defined as

p(g) =

β

g√

2πσ2exp

[− (ln[g]− µ)2

2σ2

]Log-normal

β√2πσ2

exp

[− (g − µ)2

2σ2

]Gaussian.

We find that in both the HRS and the LRS with a sili-con substrate, the conductance distribution remains log-

normal (panels C&D of Figure 2). In the HRS on glass(panel B of Figure 2), we also find log-normal. In the LRSon glass however (panel A of Figure 2), we find Gaussianor multi-modal, and attribute this to the difference inthermal conductivity between substrates. These resultsclearly show that a log-normal conductance manifests ex-perimentally, and furthermore, that it can be altered byreducing the heat flow away from the active layer.

To summarize the filamentary section, our data, alongwith the generally observed log-normal features of fil-amentary memristors, as well as the reduction of log-normal tailing with increased temperature, strongly sug-gests the origin and behavior of device variance can be de-scribed as a quantum wire with dynamic disorder. Whatthis means for filamentary RRAM design is that in ad-dition to variability from stochastic filament dynamics,electron transport dynamics induce additional variabilityfrom interference phenomena. If dominated by electronphase effects, the conductance variance σ is bounded bythe analytical limit of σ ≈ 2/(3〈g〉) [37]. This has sev-eral meaningful consequences. The first is that to re-duce variability and improve RRAM performance, onemust reduce Lφ to be less than the transport lengthLφ � L. The second is that the statistical behaviorof a single device with many, bias-induced random fila-ments is the same as the statistical behavior of many de-vices with equally random filaments. This means ergodicconsiderations1 can be taken into account at the circuitlevel design, i.e., that device-to-device variance is a man-ifestation of different disorder distributions, and will bedistributed in the same way as cycle-to-cycle variance.The third is that the read-to-read variance would be lim-ited by universal conductance fluctuations for cryogenicapplications if Lφ is larger than the inelastic scatteringlength. Finally, because the variance of synaptic weightsis closely tied to the learning capability of biological andartificial neural networks [48, 49], the existence of a quan-tum source of stochasticity suggests one could leverage itto mimic the limited precision computation model of bi-ological systems [48].

IV. CONDUCTANCE VARIATIONS INNON-FILAMENTARY SYSTEMS

This section considers the possibility that hystereticswitching in non-filamentary systems can be equally de-scribed within a framework of dynamic disorder. Weare motivated by recent advancements demonstratingthat Anderson localization can be leveraged as a mech-anism of purely electronic switching in silicon [36], and

1 The ergodic hypothesis is that after a long enough time, all themicrostates of the system with the same energy will be accessedequiprobably. It has been widely employed in the study of dis-order effects [44], and is one of the most basic assumptions inequilibrium quantum statistics [45–47].

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FIG. 3. Mobility edge switching and hysteresis with dynamic disorder. a) The density of states for amorphousstoichiometric (a-Nb2O5) and off-stoichiometric (Nb2O4.875) niobium oxide. Oxygen deficiency draws out the conduction bandand shifts the chemical potential by ∼ 1 eV. The shaded region indicates a region of strong localization, and the region integratedto produce the isodensity surface shown as an inset in part in the rightmost panel. b) Simulated conductance with dynamicdisorder. By assuming a greater vacancy transport (disorder potential redistribution), a single operating bias can have multipleconductance states. Each point is averaged over five disorder realizations with W = 3 eV. The dotted line is a guide to theeye. c) Annular memdiode device IV character. The inset shows the 0.1 e−/A3 isodensity surface demonstrating the stochasticlandscape for electrons in establishing a current.

by amorphous, electroforming-free, niobium oxide mem-diodes recently developed by Shank et al [50]. We usea-Nb2O5−x as a model system to test this hypothesis.The first notable difference between filamentary and non-filamentary systems is that by definition, non-filamentarytransport cannot be assumed to be quasi-1D. Therefore,the way quantum conductance variations present them-selves could be totally different. In addition, disorder dy-namics are known to be different in the crystalline phaseand the amorphous phase [51], so this can have an effecttoo. In fact, the introduction of disorder can increase theionic conductivity by up to four orders of magnitude insome materials [52, 53]. Considering this, and the uniquepropensity of the niobium oxide series to accommodatedynamic defects [54–56], it is reasonable to assume thatdisorder potentials will evolve in three dimensions underhigh field. By extension, if the potential felt by elec-trons in the presence of a changing ionic environment isdynamic, it is possible that there can be multiple elec-tron transmission probabilities at a given bias. In thisway, dynamic disorder allows a unified description of fila-mentary HfOx and non-filamentary a-Nb2O5−x–the maindifference being the dimensionality of transport. We willshow in section V that the dimensionality, however, isinherently encoded in the matrix elements of the velocityoperator, so a single framework can treat both cases.

The non-filamentary devices studied here are annularNi/a-Nb2O5−x/Ni memdiode stacks fabricated on Al2O3

substrates. They are three-hundred nanometers thickwith a radius of one-hundred micrometers. The normal-ized conductance in the high-resistance state is approx-imately one-hundred times smaller than HfOx, suggest-ing stronger localization and 3D transport. By contrast,the transport direction in the HfOx RRAMs is approx-imately five nanometers, and the transverse direction is

sub-nanometer, leading to quasi-1D transport. Two dif-ferent memdiode devices were fabricated and tested. Apositive voltage sweep from zero to three volts was ap-plied to the top electrode of each device, and the sub-sequent current generated was measured. This voltagesweep was repeated one-hundred times with a two sec-ond delay in between voltage cycles.

Figure 3 summarizes the experimental and theoreti-cal results pertaining to non-filamentary devices. Panela gives the first-principles electronic structure of the a-Nb2O5−x active layer of the memdiode Computationaldetails are given in the Supplementary Material [57].Panel b and c show the theoretical and experimentaltransport characteristics respectively. In part a, oxygendeficiency draws out the conduction band and shifts thechemical potential to a region of finite density of states.Traditionally, this would be an indication of metallictransport, but the measurements clearly show diode-likebehavior. Therefore, the naive, band-theoretical descrip-tion has already fallen short. If, however, we considerthese states are localized due to disorder, then they wouldgive zero contribution to the conductance until an addi-tional energy is applied to cross the mobility edge. Giventhe experimental turn on voltage is near two volts (Fig-ure 3c), the localization picture seems more appropriatethan the band picture. Part b of Figure 3 shows the simu-lated conductance as a function of energy.2 The differentdata points represent conductance values parameterizedby different normalized defect velocities v/vf (effectively,how much of the active layer is disordered). We will pro-

2 The abscissa is shifted by one electronvolt to connect with thefact that the electronic structure of oxygen-deficient a-Nb2O5−x

is shifted relative to the stoichiometric counterpart.

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vide the complete theoretical details in section V, but themain result at this stage is that dynamic disorder givesrise to multiple conductance states at a given energy, evenwhile averaged over several specific realizations of disor-der. More concretely, at E −Ef − 1 = 3 eV, we find theconductance is greater with v = 0.9vf than v = 0.6vf ,but the opposite is true at E − Ef − 1 = 2.5 eV. Thismeans an interplay between the density of states and lo-calization of the wave functions is contributing to thefinal conductance. Actually, wave function localizationplays an important role in defining the critical turn-onvoltage (details in section V B). This variation in the con-ductance at a given energy can also be interpreted as aresult of the different disorder dynamics in the forwardand reverse bias directions. Thus, dynamic disorder isa quantum analogue of the dynamic boundary betweenregions of high and low dopant concentrations originallyenvisioned by Strukov et al [58]. Part c gives the mea-sured IV curve. Comparing the theoretical result in partb and the experimental result in part c, the similar fea-tures indicate dynamic disorder and localization effectscan in fact enable memdiode-like IV behavior.

To summarize the non-filamentary section, dynamicdisorder potentials in 3D have manifested conductancevariations that are quite different than in quasi-1D struc-tures. Nevertheless, both can be attributed to electron-phase effects in highly disordered systems. This resultsuggests that the critical voltage is dictated by localiza-tion, and that the spread in the hysteresis is related tothe dynamic disorder potential. Because the statisticaldistribution and IV characteristics of two very differentsystems can be reproduced in a single approach, we re-solve dynamic disorder as a viable quantum frameworkto treat memristive switching. Given the evidence pre-sented thus far, we now turn to detailing the theoreticalframework, and providing a complete theoretical justifi-cation for its use in this circumstance.

V. THEORETICAL TREATMENT OFDYNAMIC DISORDER

A. Construction of the Hamiltonian

A quantum treatment begins with the Hamiltonian.We define it as the sum of the kinetic and stochasticcomponents as

H = Ht +Hs +Hh. (1)

Here Ht is the kinetic term, Hs is the substitutional dis-order induced by vacancies or replacements, and Hh isthe hopping disorder arising from deformation of atomicpositions, interstitials, etc. The kinetic term can be ex-pressed as

Ht =1

2

∑ijαβ

(tαβij − 2µδijδαβ)c†iαcjβ . (2)

The fermion operators c†iα create (destroy) particles atsite i (j), with orbital and spin character denoted byα (β), and µ denotes the chemical potential. In the

limit of Hs = Hh = 0, the kinetic integrals tαβij aresufficient to determine bulk transport properties of long-range, ordered solids. They can be calculated using stan-dard density functional theory techniques and a Fouriertransformation into a real-space, localized basis.3 Thechallenge, however, is that to study the statistical be-havior of many disorder configurations, many, expen-sive, molecular-dynamic simulations of large supercells,followed by density-functional calculations would be re-quired. It is therefore worthwhile to pursue a cheaperway to model statistical properties, yet retain (to theextent that it’s possible) a completely first-principles de-scription of the electronic structure. In the spirit of this,the two types of disorder can easily be parameterized asa perturbation once Ht is known. Atomic vacancies orsubstitutions are diagonal elements in the total Hamil-tonian, and act purely as a local potential as describedby

Hs =∑ijαβ

Θαβij (W )δijδαβc

†iαcjβ . (3)

Deformation of atomic positions or interstitials manifestas off-diagonal modulation of the kinetic energy den-sity. We make the approximation that in the amorphousphase, the atomic positions are randomly perturbed fromtheir equilibrium positions in the ordered state, and thiseffect is written as

Hh =∑ijαβ

(1− δijδαβ)Θαβij (W )c†iαcjβ . (4)

The matrix elements Θαβij (W ) appearing in Hs and Hh

are random numbers drawn from a uniform distributionof width W centered at zero. With this distribution, thechemical potential of the disordered system is the sameas the clean system. This is not strictly required for ourargument, but simplifies the analysis.

With the kinetic term Ht taken from first principles,we need only justify the disorder potential (Hs +Hh) isphysical. We achieve this by performing twenty ab initiomolecular-dynamic quenches, and calculating the distri-bution of the Hartree potential using density functionaltheory. We then set the maximum disorder strength Wto be less than the full width at half maximum of thisdistribution [57]. The advantage of this approach is thatwe can generate any number of physically-justified dis-order realizations, and gather statistical information onimportant metrics such as transport coefficients, the mo-bility edge, and wave function localization, for far less

3 To obtain the kinetic integrals of HfOx and a-Nb2O5−x, we haveperformed ab initio molecular dynamic quenches, and subsequentdensity functional theory calculations. Further computationaldetails can be found in Supplementary Materials [57].

7

computational expense than in traditional approach. Todetermine the distribution of the conductance in many,disordered filaments, this is sufficient. Each filament re-alizes a random disorder potential. The final part how-ever, is to incorporate the dynamic aspect of a disorderpotential.

To model a dynamic disorder potential in 3D, we maketwo reasonable assumptions. First, we assume electronsequilibrate instantaneously compared to ionic timescales.Second, we assume defects drift in an applied field. Thisis an approximation because diffusion and thermophore-sis also contribute to defect transport [59, 60]. However,this simple approximation allows us to treat dynamic dis-order through a spatially-dependent probability density.The probability that any given site in the lattice has adefect, Pk, becomes a function of time and applied biasPk → Pk(V, t). In the linear drift approximation, we canassume this probability is parameterized at small timesby the drift velocity v. Thus, in the absence of an exter-nal field, v = 0, and every site in the lattice is equallylikely to have a defect. But as the external bias is ap-plied, defects drift, and are less likely reside in a regionof vt away from (or closer to) an electrode (dependingon the defect charge and bias polarity). Motivated byStrukov et al’s boundary idea, we assume Pk(V, t) = 0if the lattice site is in a region of reduced defects, andPk(V, t) = 1 in the other region. In principle, the natureof the probability function can be adapted to include anynumber of effects, and if it were exact, then the descrip-tion would be exact. This approach effectively redefinesEquation 1 to become bias and time dependent. Thecoupled, system-environment HamiltonianH(V, t) can bewritten (ignoring Hh) as

H(V, t) = Ht +∑ijαβ

Θαβij (W )Pαβij (V, t)δijδαβc

†iαcjβ . (5)

Figure 4 shows the linear probability distribution im-plemented as a function of Hamiltonian super-index k(covers site and orbital indices). This is used in simu-lating the conductance of a-Nb2O5−x (Figure 3b). Thecentral quantity is the defect velocity v. If v = 0, thereis no motion of defects, and every site is equally likelyto have a defect. If v 6= 0, then the probability for asite to have a defect changes proportional to v. We cali-brate a limiting velocity vf by the equilibration times forIV curves in our a-Nb2O5−x devices. For example, if ittakes one-hundred seconds for the current to stabilize ina three-hundred nanometer thick sample, we may assumethe slowest moving ions are traveling three nanometersper second. This is reasonable for good ionic conduc-tors [61]. This defines the shallowest slope of our proba-bility curve (v = vf ), but if ions have a smaller propensityto move, we can set the slope to a fraction of vf .

To implement this approach, one chooses an arbitrarymeasurement time and a velocity. This defines the dis-order potential. Then, the Hamiltonian is constructed,exactly diagonalized, and the wave functions are used todetermine the transport coefficients (linear response the-

FIG. 4. Characterizing a dynamic distribution of dis-order. The plot shows the probability distribution of havinga defect is dependent on the ionic velocity v.

ory details in section V B). It is worth mentioning thisapproach is not a simulation of the ionic motion itself,rather, it enables a quantum-statistical description of thememristive transport. The time dynamics of the poten-tial we have employed here is only an approximation.Nevertheless, by taking a linear one, we have found thespread in the conductance at a given bias is in line withthe experimental result.

B. Transport in the presence of disorder: Validityof Kubo expression

Thus far, we have not established a theoretical ap-proach to determine the transport coefficients in dynamicdisorder environments. We use the finite-size implemen-tation of the Kubo conductivity tensor defined as [62]

σαβ(µ, T ) ≡ −ihe2

N

∑nn′

∆fnn′

εn − εn′

〈n|vα|n′〉〈n′|vβ |n〉εn − εn′ + iη

. (6)

In Equation 6, N is the number of unit cells, |n〉 is aneigenstate of the total Hamiltonian H with eigenenergyεn, the velocity operator is v, and ∆fnn′ represents theoccupation difference between the state n and n′. ∆fnn′

carries implicit dependence on temperature and chemi-cal potential through the Fermi function. The matrixelements of the velocity operator are defined by Equa-tion 7 where ri is the real space position of state i andeα is the unit vector along direction α.

ihvαij = (ri − rj) · eαHij (7)

To establish the validity of the finite-size Kubo for-mula for applications to real material systems near an

8

−10 −5 0 5 10

W = 1

W = 5

W = 20

−10 −5 0 5 10

W = 1

W = 5

W = 20

−2 −1 0 1 2 3 4 5

W = 1

W = 3

W = 5

Energy (eV)

One Band Model

Density of statesKubo conductivity

Energy (eV)

Two Band Model

Density of statesKubo conductivity

Energy (eV)

DFT Nb2O5

Density of statesKubo conductivity

FIG. 5. Quantifying in-gap contribution to the conductivity. The dotted lines plot the finite size Kubo conductivity asa function of energy at 300K in a single band model (left), the two-band model (center) and the first principles Nb2O5 (right).The solid lines show the density of states on the same dependent axis with globally arbitrary, but internally relative verticalscale.

Anderson transition, we benchmark against model sys-tems with a known localization transition. We choosea one- and two-band, nearest-neighbor, tight-bindingmodel with cubic symmetry and 1331 unit cells. Thesingle-band Hamiltonian has a known localization tran-sition at W = 16.5t [63], but because we are primar-ily interested in gapped systems, we also benchmark thetwo-band model. The dispersion in the one band case issimply defined as

E(k) = −2t(cos(kx) + cos(ky) + cos(kz)). (8)

It is common to set the hopping parameter t in this one-band model to unity. The bandwidth is then 12t. In thetwo-band model, there are two on-site degrees of freedomwith energy ±ε. In the Hamiltonian the only non-zero

hopping elements tαβij are defined as

ti=jα,α = −ε

ti=jα,β = e−2|ri−rj|

ti=jβ,α = e−2|ri−rj|

ti=jβ,β = +ε.

(9)

Using these expressions, we can investigate the electronicstructure and wavefunction localization as a function ofdisorder W across the transition in known systems, andcompare that to the density functional theory (DFT) re-sult for Nb2O5−x. The details for obtaining the DFTresult are given in the Supplementary Materials [57].

Figure 5 shows the Kubo response as a function ofenergy for various strengths of disorder in the varioussystems overlaid with the density of states. The solidlines show the density of states, and the dotted linesshow the Kubo conductivity. As expected, we see the

one-band and two-band models undergo a transition to acompletely transportless phase with increasing disorderstrength. This demonstrates the finite size Kubo expres-sion is in fact capable of capturing Anderson localization.The evidence is that we see a finite density of states,yet zero conductivity inside the Anderson localized phase(W = 20). The implications for oxygen-poor a-Nb2O5−xare shown in the right panel of Figure 5, where at W = 5eV, the highly localized in gap states give zero contribu-tion to the linear response. The importance of the matrixelement effects are readily apparent across all systemsbecause the finite density of states across the chemicalpotential means the occupation factor ∆fnn′/(εn − εn′)in the Kubo expression is positive, but the phase coher-ence forms a transport gap. This is only possible if thevelocity matrix elements go to zero. This analysis showsthat for pairs of wave functions with similar thermal oc-cupation, or small eigenstate coupling, the contributionto the finite size Kubo conductivity will be small, andthat interference effects are captured by the matrix ele-ments of the velocity operator. Because the time scale forthe ionic drift is usually much slower than that for theelectronic response, the electronic structure will be es-tablished immediately after the oxygen vacancies reachnew locations. Given these pieces of evidence, the Kuboformula is a valid approach to compute the conductivityat each snapshot in time.

C. Characterization of localized wave functions

To determine if the mid-gap states contribute to trans-port in the presence of localization effects, we handcrafta metric of localization focused on transport. In some

9

0.4

0.5

0.6

0.7

0.8

0.9

1.0

−10 −5 0 5 10

Localized states

Conductivestates

−10 −5 0 5 10 −5 0 5 100.0

0.1

0.2

0.3

0.4

0.5

gapless butlocalizedat W = 5

Ginic

oefficient(abs.un

its)→

Energy (eV)

One Band Model

W = 0W = 10W = 20

Energy (eV)

Two Band Model

W = 0W = 5

W = 20

DOS(arb.

units)→

Energy (eV)

DFT Nb2O5

W = 0W = 3W = 5

FIG. 6. Modeling the population of in-gap states and their localization. The lines plot the density of states as afunction of disorder strength W , and the dots represent the Gini coefficient. The left panel is a single-band model. The centerpanel is an arbitrarily gapped, two-band model. The right panel is the density functional result for a 50-band 197-unit super-cellof Nb2O5. W specifies the width of the box distribution in electron volts

sense it is analogous to the well known inverse partici-pation ratio [64], but considers not only wave functionextent, but also virtual coupling of all states in the sys-tem. To do so, we first define the object φmn as

φmn =∑r

∣∣〈m|r〉〈r|n〉∣∣2. (10)

If {|n〉} and {|m〉} represent eigenvectors of the totalHamiltonian H in Equation 1, then Eqn 10 can inter-preted as a collection of the probabilities for states tocouple through the eigenspace. We would like to charac-terize the dispersion of φmn, so we can define the Ginicoefficient [65] of the nth eigenstate with energy εn as

g(ε) =∑n

χnδ(ε− εn), (11)

where

χn ≡[2N∑m

φmn]−1 ∑

mm′

∣∣φmn − φm′n∣∣. (12)

In this formulation, g ∈ [0, 1], and will describe thedistribution of an electron’s propensity to change state.If this state is characterized by g = 0, then it has equalpropensity to transition to any other state in the system.This is not the case in real systems, so in practice the Ginicoefficient of active states lies in the middle of the mea-sure. On the other hand, if g = 1, the state cannot transi-tion to any other state, and this is a physically realizablesituation by many mechanisms. Therefore, states with aGini coefficient of one, will effectively be silent, as theydo not couple through the eigenspace to any other state.This approach may prove to generalize beyond the inverseparticipation ratio in characterizing transport properties

because there are many transport mechanisms facilitatedby spatially local wave functions [66]. While we have onlyconsidered electronically coupled states here, in princi-ple this could be extended to include states coupled byother mediators, for example phonons in variable rangehopping situations. When connected to ab initio simula-tions of disordered systems, this is a simple yet powerfulmethod to gauge the mobility edge in real materials.

The density of states and Gini coefficient as a func-tion of disorder are shown in the panels of Figure 6. Thedensity of states is shown by the solid lines and the Ginicoefficient by the points. In all three panels, the ab-scissa is absolute and physical. The ordinate for the Ginicoefficient is absolute, but the ordinate for the densityof states is only internally relative and arbitrary. Theleftmost panel shows the single band model; the centerpanel the two-band model; and the right panel shows theDFT result for Nb2O5. The single and two band mod-els shown in Figure 6 are again used as a benchmark.For zero disorder, the Gini coefficient is uniform, and theinsulating gap is clearly visible. As the disorder is in-creased to W = 3, the Gini coefficient spikes for statesnear the chemical potential, then quickly decays at higherenergy. This demonstrates that in principle, we can re-solve a mobility edge. The right panel shows the firstprinciples result for Nb2O5. At W = 0 eV, there is aninsulating gap in the density of states and the Gini co-efficient. As the disorder is increased to W = 3 eV, thegap begins to close but remains finite, and the Gini co-efficient of the in-gap states spikes. At W = 5 eV, statescompletely span the gap. In clean systems, this would bean indication of metallic behavior. However, the extentto which states inside the gap couple to all other states isgreatly reduced, indicated by a tendency of the Gini co-

10

efficient towards unity. This situation greatly reduces thenumber of conducting channels available, and indicatesphase-coherent localization of these mid gap states.

To summarize the theoretical section, development ofthe dynamic disorder potential, a viable implementationof the linear response, and a metric of localization rep-resents a framework to understand quantum-statisticaleffects in memristive computing architectures. Predic-tions in filamentary and non-filamentary systems havebeen presented in Figures 1, 2, and 3–they compare wellwith experiment. Figures 6 and 5 demonstrate that wecan resolve localized wavefunctions in our approach, andthat the Kubo formula is a viable tool to model the lin-ear response. One framework gives a quantum-statisticalorigin to the log-normal conductance fluctuations in fil-amentary RRAMs, and enables quantitative predictions.It also exposes the role of localization in the critical volt-age and hysteresis in non-filamentary systems. With thisin mind, we now discuss the general implications of thisframework for in-memory computing.

VI. GENERAL IMPACT FORCOMPUTE-IN-MEMORY ARCHITECTURES

With the evidence presented thus far, some rather gen-eral and striking aspects of the agency of neuromorphicsto transcend Moore’s law emerge. These general aspectsoriginate in the behavior of nanoscale memristors. Theyare:

1. Basic quantum properties of disordered systems atthe nanoscale are an immutable source of variabil-ity.

2. The ergodic hypothesis ensures that device-to-device variability is the same as cycle-to-cycle vari-ability, as each are just different instantiations ofdisorder.

3. The logarithm of the conductance, not the resis-tance, obeys the central limit theorem.

So while the classical conception of neuromorphics is toaccelerate matrix operations using a well-defined state,there is an additional, and possibly more important abil-ity of a truly neuromorphic system; that is the abilityto fall into non-deterministic states. Any machine, nomatter how complex or multi-variate, can hardly be con-sidered intelligent if its products, possibly ad infinitum,are yet deterministic.

Therefore, the promise of neuromorphics lies not onlyin the ability to circumvent the von Neumann bottle-neck, but also in the ability to leverage quantum non-determinism for truly intelligence hardware designs. Fu-ture efforts can now be directed towards learning al-gorithms incorporating the statistical nature of logicalnodes. For example, one may envision a simple neuralnetwork where each node’s state is not known precisely,but rather, is drawn from a log-normal distribution withvariance inversely proportional to the running average of

its activations. This is how the brain itself works [48], andas such, a machine of this type may have greater general-izability due to the finite probability of each of its nodesto take on values far from their means. In this sense,what has thus far been considered a hindrance, is real-ized to actually be a utility, and is rooted in the quantumtransport properties of nanoscale disordered systems.

VII. CONCLUSION

We have presented a unified description of the switch-ing characteristics and intrinsic variability of two verydifferent classes of memristive devices, suggesting designparadigms and connections to the fundamental processesof bio-mimetic learning. Our framework’s ability to re-produce salient experimental signatures shows that quan-tum interference phenomena are directly linked to cir-cuit level performance with implications for endurance,reliability, and scaling of neuromorphic hardware. Al-though we have demonstrated the close relation betweenoxygen deficiency, disorder, and memristive response inelectro-forming free a-Nb2O5−x and filamentary a-HfOx,the minimal ingredients for this physics can be found inmany systems, requiring only dynamic disorder. In fact,we suspect that memristive hysteresis should be foundin practically all disordered systems as long as there arenearby meta-stable configurations that can be accessedwith an impulse. The understanding gained from circuitrealizations of intelligence may even provide insight intothe functioning of human brain.

ACKNOWLEDGMENTS

This work was supported by the Air Force Office ofScientific Research Multi-Disciplinary Research Initiative(MURI) entitled, “Cross-disciplinary Electronic-ionicResearch Enabling Biologically Realistic AutonomousLearning (CEREBRAL)” under Award No. FA9550-18-1-0024 administered by Dr. Ali Sayir.

VIII. AUTHOR CONTRIBUTIONS

C.N. Singh, L.F.J. Piper, and W.-C. Lee conceived thestudy of conductance distributions. C.N. Singh designedand performed the theoretical analysis of the conduc-tance fluctuations. K. T. Butler performed the moleculardynamic simulations. B. A. Crafton and A. Raychowduryconducted the circuit analysis. A. S. Weidenbach and W.Alan Doolittle fabricated and performed experiments forNb2O5−x devices, M. P. West and E. M. Vogel for HfOx.C.N. Singh, L.F.J. Piper, A. H. MacDonald, and W.-C. Lee analyzed the experimental data and theoreticalsimulations. C.N. Singh, L.F.J. Piper, and W.-C. Leewrote the manuscript with inputs from all authors. C.N.

11

Singh, L.F.J. Piper, and W.-C. Lee ensured the clarityof the manuscript.

IX. COMPETING INTERESTS

The authors declare no competing interests.

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