Observation, Characterization and Modeling of Memristor Current Spikes
aa r X i v : . [ c ond - m a t . m t r l - s c i ] F e b Abstract
Memristors have been compared to neurons (usually specifically the synapses)since 1976 but no experimental evidence has been offered for support for this po-sition. Here we highlight that memristors naturally form fast-response, highly re-producible and repeatable current spikes which can be used in voltage-driven neu-romorphic architecture. Ease of fitting current spikes with memristor theories bothsuggests that the spikes are part of the memristive effect and provides modelingcapability for the design of neuromorphic circuits.
Keywords: memristors, d.c., current transients, mem-con theory bservation, Characterization andModeling of Memristor Current Spikes Ella Gale ∗ , Ben de Lacy Costello and Andrew Adamatzky
1. Unconventional Computing Group, University of the West of England,Frenchay Campus, Coldharbour Lane, Bristol, BS16 5SR, UKJune 4, 2018
Neuromorphic computing is the concept of using computer components to mimicbiological neural architectures, primarily the mammalian brain. Although an areaof current and active research, we do not know exactly how the brain works, how-ever it is believed that the brain is a neural net. Signals travel along neurons viavoltage spikes known as action potentials which are caused by the movement ofions across the neuron’s cell membrane, and the signals pass between neurons viachemical neurotransmitters (the gap crossed between neurons is the synapse) [1].The interaction of these spikes is thought to be a cause of brain waves, thought,learning and cognition. The long-term potentiation of neurons is related to achange in structure of the synaptic cleft, which is thought to result from the SpikeTime Dependent Plasticity (STDP) of these synapses and result in Hebbian (asso-ciative) learning [2].The memristor is the 4 th fundamental circuit element as predicted by LeonChua [3]. First reported experimentally using this terminology in 2008 [4], mem-ristors have been an object of scientific study for at least 200 years [5]. Memristortheory was first demonstrated in a model of the action of nerve axon membranesin 1976 [6], which was proposed as an alternative to the Hodgkin-Huxley circuitmodel) and this has led to the suggestion that they would be appropriate compo-nents for a computer built using a neuromorphic architecture [4]. Several simu-lations of neural nets containing memristors have been performed (see for exam-ple [7]). Recently, it was reported that circuits combining two memristors withtwo capacitors could produce self-initiating repeating phenomena similar in formto brain waves [8].Perhaps it is not merely the case that memristor models fit neuron behavior,but that neurons themselves are memristive. Thus, we would expect that advancesin the study of memristors would explain neurological phenomena (as happenedwith computer science and STDP). A circuit theoretic analysis of an updated ver-sion of Hodgkin-Huxley’s model of the neuron has been undertaken [9, 10]. TheHodgkin-Huxley model is often used to explain the transmission of voltage spikesalong the neuron. However, this model predicts huge inductances which are notexperimentally observed in biology and it has been demonstrated [9] that updat-ing the Hodgkin-Huxley model with memristors avoids this requirement. A recent ∗ e-mail: [email protected] aper suggested that memristance could explain the STDP in neural synapses [2].The authors used memristor equations to adjust simulated spikes, found a similar-ity to experimentally measured biological synapse action [11] and concluded thata memristive mechanism was behind the biological STDP phenomenon.In this paper we will show experimentally that memristors spike naturally anddo not require a spiking input to cause them to spike in a manner qualitatively sim-ilar to neurons. We shall attempt to quantify the spikes. We will then demonstratethat these spikes are also present in theoretical models of memristors and discussthe cause of them. We think that utilizing these naturally-occurring spikes will bethe most fruitful way to create neuromorphic memristor architectures. Memristors come in two flavours, charge-controlled (left) and flux-controlled (right)as shown below in Equation 2 where q is the charge, ϕ , is the magnetic flux, M isthe memristance and W is the memductance (inverse memristance) [3] V ( t ) = M ( q ( t )) I ( t ) , I ( t ) = W ( ϕ ( t )) V ( t ) . For a charge-controlled memristor we would input a current, I , and measurethe voltage, V . Biological neurons may be described as charge-controlled becauseit is the movement of ions that causes the change in voltage giving rise to a voltagespike. Our memristors are flux-controlled and a change in voltage causes a spikein the current. Thus, creation of a neuromorphic computer with memristors willbe using the complimentary effect to the one utilized by nature, in that memris-tors have voltage-change-caused current spikes and neurons have current-change-caused voltage spikes. That both types of spikes have a similar form arises fromthe similarity in the underlying electromagnetics, in that circuits can considered asbeing constructed with either a voltage source or a current source. −8 Time / s C u rr en t / A Figure 1: Current spikes recorded from a memristor subjected to the voltage squarewave in figure 2. The spike heights are highly repeatable and qualitatively resembleneuronal spikes. 31
10 20 30 40 50 60 70−0.1−0.0500.050.1 Time / s V o l t age / V Figure 2: Voltage square wave that the memristor measured in figure 1 was subjectedto.
Our memristors are flexible sol-gel titanium dioxide gel layers sandwiched be-tween aluminium electrodes [12, 13] and they show a distinctive large spike thatoccurs when the voltage is changed. The experiments reported here were carriedout with a Keithley 2400 sourcemeter sourcing voltage. There are no spikes inthe voltage profiles, (see Figure 2) and no current spikes are seen when the sameexperiment is done across a resistor. It has been suggested that these spikes arecapacitance; however the timescale is too long. The spikes have been reported byother groups in their memristors (see for example, [14]), however they are usuallyoverlooked or attributed to artefacts arising from the experimental set-up or notreported at all (many researchers only report the I − V curves to demonstrate thatthey have a memristor). However, the current spike is an equilibrating process thatis responsible for the frequency dependence of the I − V curves. In Figure 2 eachvoltage step had 40 timesteps ( ≈ I − V loop. This effect increases with frequency until it reaches the limitwhere the voltage frequency is too fast for the memristor to relax at all and the I − V curve just traces out the maximal spike currents for each voltage.These current spikes can be seen whenever a voltage change occurs acrossthe memristor. Unlike some neuronal spikes, the voltage does not need to spike.The current spikes are highly reproducible. For the experiment shown in Figure 1(10 pairs of positive to negative switches), the standard deviation was 0.0729% ofthe mean for the negative voltages (where n = 10 ) and 0.1192% of the positivevoltages (where n = 9 , due to incomplete recording of the first spike)). For therepeated spikes in figure 3 (3 repeats each of both positive and negative ramps,as shown in figure 4) the largest difference between the spike current repeats wasonly 3.06 × − A and only 2.33 × − A for the equilibrated current - both takenfrom the positive side as it has a larger hysteresis than the negative side.The direction of the current spikes is related to the change in voltage, not itssign, so a change from a positive voltage to zero (turning the voltage source off)gives a negative spike and vice versa for a negative voltage to zero. The spikecurrent still flows for a short while after the voltage source has been turned off.This lag is a general thing and has been recorded in several different devices. Indifferent devices the spikes are the same shape and seem to be following similardynamics. The spike current is proportional to the equilibrated current. Intrigu-ingly, spike shape closely resembles that of Bi and Poo’s experimentally observed TDP function [11] and thus could be used to perform a similar function.
Figure 3: The spikes for 5 successive runs up and then down the voltage staircase shownin figure 4. The runs are coloured and overlap. The spikes are highly reproducible onsuccessive runs V o l t age / V Figure 4: Voltage ramps for figure 3. 5 sets of positive voltage ramps-negative voltageramps were run, to give the spike response in figure 3.
Figure 6 shows the I − t response of a single spike to a voltage step like thatshown in figure 5. The current spikes are roughly the same shape, and thus we canmake some statements about the nature of the current spikes in memristors, whichshould also relate to the voltage spike in neurons. As shown in figure 6, there is asteady-state current, i ∞ , a spike current i and a transition between the two whichis a time-dependent transient i ( t ) . We don’t currently know if the i ( t ) is dependent V o l t age / V Figure 5: An example of a voltage step as applied to a memristor. −9 Time / s C u rr en t / A Figure 6: An example Spike. Red dashed line: τ ; orange dotted line τ ; greendot-dashed τ ; blue dotted τ . Horizonal purple dot-dashed line is i ∞ and the spikeheight is i . 34 Time / s R ( t ) / Ω Figure 7: The resistance profile for the memristor subjected to the voltage in figure 5.Note that the ‘zero’ resistance is due to zero measured resistance as no voltage is ap-plied, not a true zero resistance. on i or not. We do know that i is related to i ∞ . Until a thorough experimentalstudy is undertaken, we shall assume that i ( t ) is not dependent on i as this is whatthe experimental evidence seems to suggest.Thus, the time-dependent current response, I ( t ) is assumed to be of the form: I ( t ) = i ∞ + i ( t ) where i < i ( t ) < .The current response to the voltage is thus: ∆ I = VR ( T ) The time taken to get to i ( t ) = 0 the equilibration lifetime which we shallcall τ , and this lifetime is the short-term memory of the memristor and relatesto its dynamical properties; from longer time spike studies with our devices, weknow that τ is approximately 3.3s. We shall define the concept of the equilibra-tion frequency as the ‘frequency’ associated with changing a descretised triangularvoltage waveform such that each voltage step n lasts for τ seconds.We know that q e = Z I ( t ) dt. thus, the total measured charge in a memristor spike is ∆ q spike = Z τt =0 = i ( t ) dt + i ∞ τ. This number includes all the charge carrying species in the system. Knowledge ofthis number may help us elucidate the mechanism of the spikes. For our examplesystem shown in figures 6, we have an i of 1.37 × − A, an i ∞ of . × − A,with the τ of 0.56s and an τ of 0.84s, which shows how quick the fall off is(and τ of 1.13s and τ of 2.34s, as drawn in figure 6). The resistance profile for he memristor subject to a voltage step as shown in figure 5 is shown in figure 7.This is approximately a straight-line which is interesting as it is not required tobe by memristor theory and tells us that the spike current response depends on aquantity in the system that is varying with linearly time. The mem-con model of memristance [16] is a recently announced theoretical modelthat relates real world q and ϕ to Chua’s constitutive equations and has been suc-cessful in modeling our memristors [17]. The mem-con theory has the concept ofa memory property, the physical or chemical attribute of the device that holds thememory of the device. In titanium dioxide (and many others) it is related to thenumber of the oxygen vacancies. The presence of oxygen vacancies allows thecreation of a doped form of titanium dioxide TiO − x which is more conductingthan the undoped (TiO ) form. The mem-con theory requires that we calculate thememristance from the point of view of the memory property, i.e. the ions.Theoretically, the voltage step is a discontinuous function and the voltagechanges from voltage A, V A to voltage B, V B in an infinitesimal, i.e. ∆ V = V B → V A t , t → δt . Experimentally this is not the case of course, but the responsetimescale of the memristor is long enough that we needn’t worry about this ap-proximation.Thus to elucidate what happens to the memristor during a current spike, andhow the final current i ∞ is determined, we take differences of the mem-con theory.We shall assume our device is a TiO memristor, with oxygen vacancies acting asthe memory property [16].As a reaction to the voltage step, we get a current spike, ∆ i , which can beexpressed as a volume current within the device as ∆ J as given by: ∆ ~J = { ∆ q v µ v ~Lvol , , } for vacancies moving in the + x direction where q v is the charge in that volumedue to the vacancies, µ v is the ion mobility of vacancies and L is the averageelectric field causing the movement of the vacancies and vol is the volume full ofmoving ions. The change in the magnetic field at point p , ∆ ~B ( p ) would then be: ∆ ~B ( p ) = µ π Z ∆ J ~ ˆ J ~ × ~ ˆ rr dτ (1)where µ is the permittivity of a vacuum, ~ ˆ J and ~ ˆ r are the unit vectors for ~J and ~r where ~r is the vector of length r from the volume infinitesimal dτ to point p , given by ~r = { r x ˆ ı, r y ˆ , r z ˆ k } .Thus, to get a measure of the effect of the spikes, we need to solve this inte-gral over a time-interval covering from the start of the spike to the tail-off of thememristor’s response. The voltage input is non-integrable, but we can integratefrom the start of the step, which we shall take as t ( n ) where n is the number ofthe voltage step, which is zero for this case if it is understood that this is not thezero at the start of an experiment with many steps (i.e. we are considering a caseas in figure 6) to when the memristor has responded, which we shall take as T .Dependent on the situation T can be one of many values, for a staircase we wouldpresumably want T = t ( n + 1) where t ( n + 1) is the time that the voltage stepis input. For a response to a single step function we could take the integral out to (which is what we shall do here). For experimental purposes we might be moreinterested in integrating to τ or τ .Solving the integral gives: ∆ ~B ( p ) = µ π Lµ v ∆ q { , − xzP y , xyP z } with P y = F w + E + F ) − wEF (cid:16) ∆ wE (cid:16) F (cid:0) E + F (cid:1) + a + b (cid:17)(cid:17) c + F arctan (cid:18) ∆ wEF √ ∆ w + E + F (cid:19) , and P z = E w + E + F ) − wEF (cid:16) ∆ wF (cid:16) E (cid:0) E + F (cid:1) + a + b (cid:17)(cid:17) c + E arctan (cid:18) ∆ wFE √ ∆ w + E + F (cid:19) , where a = ∆ w (cid:0) E + F (cid:1) b = ∆ w (cid:0) E + 5 E F + 2 F (cid:1) c = (cid:0) ∆ w + F (cid:1) (cid:0) E + F (cid:1) (cid:0) ∆ w + E + F (cid:1) . Where the effect on the magnetic field is due to both the influx of charge andthe resulting movement of the boundary between doped and undoped TiO .To calculate the change in magnetic flux through a surface associated with thisfield, ϕ , we need to take the surface integral ∆ ϕ = Z ∆ ~B · d ~A where d ~A is the normal vector from the surface infinitesimal dA .As it is a surface integral, to calculate the magnetic flux we need to pick asurface to evaluate over. It makes sense to choose a surface that correlates to one ofthe surfaces of the device. Picking the surface just above the device ( < x < D , < y < E , z = F ), we use the surface normal area infinitesimal, ~dA , which isgiven by ~dA = { , , ˆ ı ˆ } . As is standard in electromagnetism, we integrate overthe entire area. The limits of the surface are taken to be the dimensions of thedevice.Thus we derive the general form of the magnetic flux passing through a surface i - j : where, because ϕ is entirely dependent on q , which is time-varying, we caninclude the time varying effects by taking the differentials thus δϕ = µ π Lµ v ijP k δq v , (2)And, as in mem-con theory [ ? ], by using Chua’s constitutive relation for thememristor, we can then arrive at the change in the Chua memristance as experi-enced by the ions: M q (∆ q v ( t )) = UXµ v ∆ P k (∆ q v ( t )) , (3)where we have gathered up the constants and explicitly included P k ’s dependenceon q v .Equation 3 can be considered as three separate parts:1. U , the universal constants: µ π .2. X , the experimental constants: DEL .3. the material variable: µ v P k (called β is ref [!!Mem-Con]), this includes thephysical dimensions of the doped part of the device and the drift speed ofthe dopants.Writing out the differences explicitly of equation 3 we end up with: M ( B ) = M ( A ) + UXµ v [ P k ( q B ) − P k ( q A )] , which allows us to calculate how the final Chua memristance from knowledgeof the peak and final currents. The Chua memristance is written for the vacancycharge, so to put it into the standard format for the electronic current we need toscale it thus: R M = C M M, where R M is the electronic resistance of the doped part of the memristor and C M is a fitting coefficient. The memory part of the function only describes the effect of the memristancechange on the doped part of the memristor. To cover the other one we use theconservation function, this is most easily expressed in terms of w ( t ) , but w ( t ) isrelated to q ( t ) by w ( t ) = µ v Lq ( t ) EF v d . Thus, the difference in conservation function, ∆ R con , written as a differenceequation is: R con ( B ) = R con ( A ) + ( D − [ w ( B ) − w ( A )]) ρ Off EF which based on the definition of resistivity and where ρ off is the resistivity ofthe undoped part of TiO .The mem-con model describes a memristor by being the sum of the memoryand conservation functions (both written for the electrons) and this then gives usthe following expression for the change in time-varying resistance, R ( t ) , as mea-sured after a change from V A → V B as: ∆ R ( t ) = c m M ( A ) + R con ( A ) + ρ off DEF + c M UXµ v [ p k ( q B ( t )) − p k ( q A ( t T ))] − Lρ off µ v [ q B ( t ) − q A ( t T )] E F v d , where we have substituted for w . This equation has two parts: . S , the time-invarient part, which is: c m M ( A ) + R con ( A ) + ρ off DEF Y , the time variant part: c M UXµ v [ p k ( q B ( t )) − p k ( q A ( t T ))] − Lρ off µ v [ q B ( t ) − q A ( t T )] E F v d , the last two terms which are both dependent on q (remember p k is dependent on w but can also be written in terms of q .In the above equation 4 highlights a few subtleties of the model. p k and q aretime-dependent and thus change after the voltage step from V A → V B . If we askthe question of what the difference will be between the equilibrated current at V A and that at V B , ∆ R A ∞ → B ∞ equation 4 collapses to: ∆ R = c m M ( A ) + R con ( A ) + ρ off DEF + c M UXµ v [ p k ( q B ( τ )) − p k ( q A ( τ ))] − Lρ off µ v [ q B ( τ ) − q A ( τ )] E F v d , which is time invariant and allows us to predict the value of the equilibrated currentafter a voltage step from the equilibrated current from the step before.What if there was previous step in which the device did not equilibrate to i ∞ ?This would happen if the voltage was changed quicker than τ , i.e. T where T < τ .The q A ( t T ) is not q A ( τ ) and thus needs to be shifted by its value as a proportionof τ . As an example, if we sped the voltage ramps up to 90% of the equilibrationfrequency, q A would be q A ( τ ) and the length of a time step would be τ . Atfirst glance it might appear that this would merely modulate the starting point for q B ( t ) , which, at times under t < τ , this would be time dependent. But there is theinteraction between q B ( t ) and q A ( t T ) , the memristor hasn’t finished responding to V A and that response should be mixed in with V B , further complicating predictiveefforts. The mem-con model consists of sum of two components: the memory function, M e , and Conservation function, R c . The memory function has a fitting parameter c m within the model to account for the conversion between the material’s resis-tance as for an oxygen vacancy and as for an electron. The conservation functionhas the fitting parameter c c which accounts for the resistivity of the undoped ma-terial, ρ off , which may not be the same as the bulk titanium dioxide. R on is thefinal fitting parameter and relates to the resistivity of the doped material, which isthe memristor in the equilibrated state and any resistance in the wires. The fittedequation is I ( t ) = VR on − Vc c R c ( t ) − c m M e ( t ) . As figures 8 and 9 shows, the mem-con model fits these spikes quite well andmuch better than an exponential fit. For the positive spike, c M − . × , c c = 1 . × and V /R on = 2 . × − , with a summed square of residualsof 1.61 × − . For the negative spike, c M − . × , c c = 1 . × − and V /R on = − . × − , with a summed square of residuals of 1.63 × − .For the exponential fit, I ( t ) = Ae λt , and A = 3 . , λ = − . with a summedsquare of residuals of 2.43 × − . The exponential fit could be fit to either theshort time spike or the long time tail but not both, the short term spike fit goes rroneously to zero and the long-term spike fit grossly over-estimates the size of thespike. Furthermore, there is no experimental justification for using an exponentialfit, unlike the mem-con fit. This model can be utilized to perform simulations ofmemristor spiking networks to test out possible neuromorphic architectures. −9 Time / s C u rr en t / A Experimental dataMem−con theory fitExponetial fit
Figure 8: A longer-term spike response fit by the mem-con theory. The mem-contheory fits the experimental data well, the best result fitting the data with an exponentialis added as a comparison. Blue dots: experimental data, red line: mem-con fit, greenline: exponential fit to the spike.
The memory property of these memristors is the oxygen ions, usually viewed aspositive holes in a semi-conducting material. We suspect that the motion of theseions is behind both the spikes and the memristance as we postulate that the two arethe same phenomena. The current that flows at t = 0 s may be the ionic current,which would have a greater inertia, and thus takes longer to stop compared to theelectrons, which may explain the cause of the devices hysteresis. This current flowcan also explain the open-loop memristors (suggested by Pershin and di Ventra [18]to explain experimental results such as [19] which are similar to ones seen in ourlabs and others’). The spike shape would then be the result of the equilibrating ofthe ionic current to a change in voltage. We expect that the timescale and dynamicsof the spikes will relate to the frequency effects seen in memristors. However, thereis much further experimental work to be done to prove this mechanism. Chua’s definitions of his two types of memristors, flux and charge controlled, wasgiven above. The mem-con model has the concept of a two-level system wherewe have two charge carriers, q , our memory property and e − the electronic currentwhich is what is measured in an experiment. Level 0 is the relationship between thevacancy charge, q and vacancy flux, ϕ . This is experienced at level 1 by resistance
50 100 150 200 250−1.4−1.2−1−0.8−0.6−0.4−0.20 x 10 −8 Time / s C u rr en t / A Figure 9: A longer-term negative spike, demonstrating that the negative spikes are fitequally well by the mem-con theory. Blue dots: experimental data, red line: mem-confit. changes ( R ( t ) ) which effect the electronic current, I e − . The circuit measurablesare the voltage, V and the total current I where I = dqdt + I e − .For our memristors, driven by a voltage, the right hand side of figure 10 sum-marizes the operation. There is a change in voltage, which acts on the electronsand the vacancies, causing a change in the number of charge carriers ( ∆ e − and ∆ q . The change in q causes a change in the magnetic flux associated with q andthus a change in the Chua memristance. This, due to the conservation of space,causes a change in the amount of material described by the conservation function R c , which then changes the total resistance ∆ R . This change in resistance willdraw more current, e − and thus the change in the number of electrons is influ-enced by both the change in voltage and the change in resistance that change hascaused. The change in total current is due to both the electrons and the vacancies.A neuron is the opposite way round, see the left hand side side of figure 10.The cell is always pumping ions back and forth, so we have a change current dueto an influx of charge carrier. This causes a change in magnetic flux and affectsthe total resistance (the values of the memory and conservation functions for thissystem have not yet been worked out). This change in resistance causes a voltagespike. Thus, similarities can been seen between neuronal voltage spikes and mem-ristive current spikes, in that they are the opposite way round with respect to thecircuit measurables, in that the memristor as operated here is a current response toa voltage-sourced circuit, and the neuron is a voltage response to a current-sourcedcircuit. Essentially the shape of the circuit variable, i.e. that which is being mea-sured, is qualitatively similar. It has been suggested since 1976 that neurons are memristive, but experimentalevidence for neuron-like spiking in memristors had not been collated or analyzed inthis way before. If this spiking behavior is an integral result of memristance then itis evidence for the suggestion that neurons may be memristive in action and furtherunderstanding of memristor theory may further the neurological understanding. eurons Memristors I Vq Φ V Iq Φ
Figure 10: Scheme
This work shows that to make neuromorphic computers that compute withspikes memristors are an obvious choice for this task as they spike naturally. Inter-ruption to the equilibrating current curves as shown in figure [!3], by, for example,changing voltage, would potentiate the connection by modifying the memristanceand could thus be used to do STDP with memristors without requiring CMOSneurons to generate the spikes.
Memristors, when subject to a change in voltage, undergo a current spike. Thisspike has been shown to be reproducible and repeatable. The mem-con theoryhave been shown to fit the time-dependent current behaviour with only two fittingparameters (which come from the missing material values in the theory) suggest-ing that this I − t spike behaviour is an aspect of memristance. Rewriting themem-con theory as a difference equation allows the formulation of a predictiveequation to related the equilibrated currents at different (and successive) voltages.Application of the equilibration lifetime ( τ ) to this equation highlights where thetime-responsive interactions might arise in a memristor switched faster than theequilibration frequency. Acknowledgement
This project has been funded by EPSRC grant No. EP/H014381/1.
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