Abstract
In 1963 Ridley postulated that under certain bias conditions circuit elements exhibiting a current or voltagecontrolled negative differential resistance will separate into coexisting domains with different current densities or electric fields, respectively, in a process similar to spinodal decomposition of a homogeneous liquid or disproportionation of a metastable chemical compound. The ensuing debate, however, failed to agree on the existence or causes of such electronic decomposition. Using thermal and chemical spectromicroscopy, we directly imaged signatures of currentdensity and electricfield domains in several metal oxides. The concept of local activity successfully predicts initiation and occurrence of spontaneous electronic decomposition, accompanied by a reduction in internal energy, despite unchanged power input and heat output. This reveals a thermodynamic constraint required to properly model nonlinear circuit elements. Our results explain the electroforming process that initiates information storage via resistance switching in metal oxides and has significant implications for improving neuromorphic computing based on nonlinear dynamical devices.
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Introduction
Electronic instabilities in materials, such as negative differential resistance (NDR), have seen a surge of recent interest because of their potential applications in storageclass memory^{1,2,3} and bioinspired neuromorphic computing^{4,5,6,7}. By invoking symmetrybreaking via maximum entropy creation, Ridley^{8} predicted the formation of two different coexisting domains of currentdensity channels when a material that exhibits currentcontrolled NDR is voltage biased in an unstable operating regime. The proposed mechanism was questioned by Landauer^{9,10}, who suggested that minimum heat generation and interaction dynamics between coexisting stable states were key to understanding electronic domain decomposition. Schöll demonstrated that domain decomposition and chaos were possible for the specific case of hot electrons interacting with impurity levels in semiconductors^{11,12}. However, no consensus on a general mechanism has arisen in this debate, and in particular the question of how to distinguish between different, and possibly coexisting, steady states in a nonlinear electronic system remains unanswered^{12,13,14,15,16,17}. There has been a parallel debate on whether state transitions, such as resistance switching in metal oxides, are initiated by electronic or thermal instabilities^{12,15,16,18,19}. With a few exceptions^{20}, most recent models of NDR and related instabilities have ignored the possibility of decomposition into coexisting domains^{1,6,21,22,23,24,25,26,27}.
Here we show from experimental imaging and an extended nonlinear dynamics model that an electronic system biased within an unstable operating region undergoes spontaneous decomposition into coexisting stable and distinct domains, even when the power input and heat generated by the decomposed states are identical to those of the uniform unstable state. However, the increase of the internal energy of the nonlinear material from ambient conditions is lower for the decomposed compared to the uniform unstable system. We illustrate these concepts via currentdensity decomposition during currentcontrolled NDR, and the coexistence of currentdensity and electricfield decomposition during the temperaturecontrolled instability of a Mott transition. We utilize the concept of local activity^{28,29} as the framework to understand the electronic instabilities and the amplification mechanism of local thermal fluctuations that drives these systems to the lowest internal energy steady state.
Results
Currentcontrolled NDR from temperatureactivated transport
A currentcontrolled NDR in materials such as the oxides of Ti, Hf, Nb, Ta, etc. results from a highly nonlinear and activated transport mechanism that enables positive feedback by Joule heating, leading to an increase in current accompanied by a decrease in device voltage^{1,21,30}. When the power dissipated and heat removed are not at steady state, such a system can be modeled by a memristor formalism with the temperature T as the state variable^{6,31}. We reprise one such model using a modified 3D Poole–Frenkel equation^{1} as the quasistatic equation for NbO_{2} connecting the memristor current, i_{m}, and voltage, v_{m}:
where A is the lateral device area; d the thickness of the oxide; k_{B} the Boltzmann constant (8.617 × 10^{−5} eV); T the absolute temperature; and ω and σ_{ 0 } are material constants, described elsewhere^{1}. The equation determining the device dynamics is just Newton’s law of cooling:
where T_{amb} is the ambient temperature (300 K); \({\it{C}}_{{\mathrm{th}}}^{{\mathrm{sys}}}\) is the thermal capacitance of the system comprising the active NbO_{2} layer, the device structure, and its thermal environment^{32}; and R_{th} is the effective thermal resistance of the entire device embedded in its ambient (1.4 × 10^{6} KW^{−1}). Estimates for the thermal constants are discussed elsewhere^{32,33}. For an applied external voltage V_{ext}, the steady state i_{m}–V_{ext} relationship (Fig. 1a) obtained by solving Eqs. (1 and 2) with \(\frac{{{\mathrm{d}}{\it{T}}}}{{{\mathrm{d}}{\it{t}}}} = 0\) and V_{ext} = v_{m} + i_{m}R_{s} exhibits NDR when the series resistance (R_{S}) is less than R_{NDR}, where R_{NDR} is defined as the absolute magnitude of the largest negative slope of the i_{m}–v_{m} curve within the NDR region (about 350 Ω). However, when R_{S} = 360 Ω is added to the circuit, the electrical behavior is stabilized^{8,34} (green curve). The origins of electronic decomposition and subsequent stabilization within the circuit by adding R_{S} are revealed by plotting the dynamical route of the system (\(\frac{{{\mathrm{d}}{\it{T}}}}{{{\mathrm{d}}{\it{t}}}}\) against T by parametrically sweeping T, after combining Eqs. (1 and 2))^{35,36}. In the case of R_{S} less than R_{NDR}, at an applied voltage within the NDR region, there are three zerocrossings of the dynamical route (Fig. 1b), corresponding to three possible steadystate temperatures \(\left( {\frac{{{\mathrm{d}}{\it{T}}}}{{{\mathrm{d}}{\it{t}}}} = 0} \right)\). The arrows on the dynamical route are determined by whether \(\frac{{{\mathrm{d}}{\it{T}}}}{{{\mathrm{d}}{\it{t}}}} > 0\) (T increases) or \(\frac{{{\mathrm{d}}{\it{T}}}}{{{\mathrm{d}}{\it{t}}}}\) < 0 (T decreases). Upon a small perturbation in T, stable steadystates will return to the original state (arrows pointing inwards), while unstable steadystates do not return (arrows pointing outwards). A possible state evolution when the system is biased at an unstable steady state is to decompose into coexisting physical domains corresponding to the two stable steadystates on either side of the unstable state. The unstable current levels within the NDR region can be accessed through transients, but are typically not accessed through voltage sweeps (Supplementary Fig. 1). For the circuit with R_{S} greater than R_{NDR} (Fig. 1c), there is only one stable steadystate (blue sphere), and thus no regions of instability or NDR (also see Supplementary Fig. 2). In other words, operation by a current source, or a voltage source with a sufficiently large R_{S}, will lead to stabilization and no decompositions. Thus the necessary conditions for electronic domain decomposition under a given bias are: (i) the presence of at least one unstable and two stable steady states, and (ii) experimental access to an unstable steady state. We determine the lower and higher stable (T_{L} and/or T_{H}, respectively) and unstable (T_{U}), if any, steadystate value(s) of T through the dynamical route maps corresponding to every V_{ext} (Fig. 1d,e). Using this ‘phase plot’ of T vs. V_{ext}, and the functional relationship between T and i_{m} (Supplementary Fig. 3), we calculated the current densities (j_{m}) corresponding to every V_{ext}, which reveals that for a range of V_{ext}, there are a pair of stable current densities (j_{H} and j_{L}) along with an unstable current density (j_{U}) (Fig. 1f, g, Supplementary Fig. 4). The decomposition occurs via the formation of two adjacent current density domains,
where A is the lateral device area and x is the fraction of A carrying current density j_{H}. The values of x and 1 − x can be obtained from Eq. (3) for different V_{ext} (Fig. 1h) in the NDR region. In order to estimate the relative internal energies of the decomposed (D) and unstable (U) configurations, we calculated the change in enthalpy of the active oxide film (ΔH) from ambient conditions to the steady states (also see Supplementary Note 1):
and
respectively, where \({\it{C}}_{{\mathrm{th}}}^{{\mathrm{act}}}\) is the thermal capacitance of only the active NbO_{2} layer within the device structure, ΔH_{L} and ΔH_{H} are the enthalpy changes from ambient corresponding to regions with temperatures T_{L} and T_{H}, respectively, and T_{amb} is the ambient temperature (300 K). Figure 1i shows that the decomposed configuration has a lower enthalpy at steady state. Criteria based on maximization of entropy or minimizing total heat generated fail to distinguish between the two configurations^{8,10,15}, because the input power and heat released by the two possibilities are identical.
We now demonstrate that currentdensity decompositions can be the precursor to nonvolatile resistance switching in a TaO_{x} memristor, which initially exhibited reversible currentcontrolled NDR in its i_{m}–v_{m} behavior when powered by a quasistatic current source (blue curve, Fig. 2a, see Methods)^{37}. However, when subsequently operated with a quasistatic voltage source, it displayed an abrupt jump in current followed by nonvolatile resistive switching (red curve, Fig. 2a). This behavior was caused by a decomposition that created a highcurrent density and thus hightemperature region in the oxide. The large temperature gradient induced thermophoresis and subsequent Fick diffusion of oxygen, creating radial chemical and electrical conduction gradients in the oxide that persisted even after the power was withdrawn (Fig. 2b, c)^{38,39}, thus providing a channel for nonvolatile resistance switching. Minimization of the interface energy between the two coexisting domains yields a cylindrical channel of one current density embedded in the other, explaining the circular feature in the oxygen map of Fig. 2b. To further support this interpretation, we used the measured NDR and subsequent resistive switching behavior of a memristor^{38} to identify the stable and unstable current levels (i_{H}, i_{L} and i_{U} in Fig. 2a, with i_{U} limited by the device’s steadystate internal resistance). From these, we calculated the expected area of the high current density domain using Eq. (3) (dashed circle, Fig. 2c), which agrees with the observed channel in Fig. 2b. Thus, we conclude that the voltagesourced electroforming process for transition metal oxides is consistent with a currentdensity decomposition (illustrated in Supplementary Fig. 5) that produces a much higher temperature channel than its surroundings, which in turn induces the oxygen stoichiometry changes responsible for nonvolatile electrical switching. The formation of a channel can be avoided by using a current source to initially power the sample, which will result in uniform current flow and heating of the device^{32}.
Temperaturecontrolled instability of a Mott transition
We next consider a temperaturecontrolled instability in which both the current and voltage decrease as the temperature of the system is increased (Fig. 3a), which is caused by a Mott transition, observed for example in VO_{2} and NbO_{2}^{4,32,40}. In order to model this behavior, we still used Eq. (1) to describe electrical transport, but we introduced an abrupt increase^{32,41} in R_{th} at the transition temperature T_{MIT} (chosen to be 340 K to represent VO_{2}) in Eq. (2), which produces the i_{m}–v_{m} plot in Fig. 3a. Further, to account for the latent heat of the Mott transition, we utilized a spikedincrease in \({\it{C}}_{{\mathrm{th}}}^{{\mathrm{act}}}\) as a function of T at T_{MIT} (see Methods, Supplementary Fig. 7). The dynamical routes of the system corresponding to operation separately by a current source or a voltage source within the region of instability are each characterized by an unstable steadystate flanked by stable steadystates (Fig. 3b, c), revealing that both bias conditions will produce electronic decompositions. We repeated the dynamical analysis presented in Fig. 1 to obtain a plot of j_{m} vs. V_{ext} (Fig. 3d, Supplementary Fig. 8), for the case of using an external voltage source with R_{S} = 0. For the case of using an external current source, Fig. 3e shows the plot of the electric field (ε_{m}) vs. applied current (I_{ext}). In each case, there are three steadystate current densities or electric fields (two stable and one unstable). We represent the decomposition in the electric field as
where ε_{U} is the unstable uniform electric field, ε_{H} and ε_{L} are stable high and low electric fields, respectively, d is the thickness of the oxide film, and z is the fraction of d containing the electric field ε_{H}. Using Eqs. (3 and 5) on the data in Fig. 3d, e, we calculate the area fractions (A_{f}) and the thickness fractions (Z_{d}) for the currentdensity and the electricfield decompositions, respectively (Fig. 3f,g). For the case of electricfield decompositions, we calculate ΔH using Eq. (4) replacing x with z, while T_{H} and T_{L} correspond to ε_{H} and ε_{L}, respectively. ΔH for the unstable state is calculated as \({\mathrm{\Delta }}{\it{H}}_{\mathrm{U}} = \mathop {\smallint }\limits_{{\it{T}}_{{\mathrm{amb}}}}^{{\it{T}}_{\mathrm{U}}} {\it{C}}_{{\mathrm{th}}}^{{\mathrm{act}}}({\it{T}}){\mathrm{d}}{\it{T}}\), where T_{U} corresponds to ε_{U}. Enthalpy calculations for both decompositions (Fig. 3h, i) show that the decomposed states have the lower ΔH. A temperaturecontrolled instability caused by a Mott transition will exhibit decomposed currentdensity channels during voltage source operation and electricfield fragments during current source operation.
We experimentally demonstrate the above using VO_{2} (see Methods), a Mott insulator with a temperaturecontrolled instability^{40}, for current flow in the plane of a thin film to visualize the field domains. A i_{m}–v_{m} curve measured using a current source exhibits a pinched hysteresis (Fig. 4a) due to the Mott transition, which appears as a pair of sharp NDR transitions. Figure 4c–h displays inoperando blackbody emission temperature maps of a lateral VO_{2} device with metallic electrodes deposited on top of a thin VO_{2} film (device fabrication and measurements are detailed elsewhere)^{40,42}. When powered by a voltage source near the unstable region, a high currentdensity channel, as revealed by the hightemperature map, connects the two electrodes (Fig. 4c–e). However, when operated with a current source within the unstable region, hightemperature domains appear (Fig. 4f–h) that do not connect the electrodes. This data qualitatively supports the model presented in Fig. 3. Notably, the hightemperature (above T_{MIT}) and lowtemperature (below T_{MIT}) domains correspond to the regions of lower and higher power dissipation, respectively, in the i_{m}–v_{m} curve in Fig. 4a. This counterintuitive result arises because of the increase^{32,41} in the thermal resistance R_{th} observed at T_{MIT} (Supplementary Fig. 9).
Dual current and temperaturecontrolled instabilities
Having separately analyzed currentcontrolled NDR (similar to that in TaO_{x}, TiO_{x}, etc.) and a Mott transition (e.g., that in VO_{2}), we now present the full example of NbO_{2}, which exhibits both instabilities. This was modeled by using Eq. (1) to describe transport and Eq. (2) for the Mott transition^{43,44} with T_{MIT} = 1070 K. The resultant i_{m}–v_{m} curve for R_{S} = 0 Ω is shown in Fig. 5a, and a voltage bias within the range containing both instabilities intersects the curve at five distinct points (α–λ). A plot of the dynamical route (Fig. 5b) at this bias reveals that two of the steadystates are unstable, with each flanked by a pair of stable steadystates. A plot of \(\frac{{\mathrm{d}}}{{{\mathrm{d}}{\it{T}}}}\left( {\frac{{{\mathrm{d}}{\it{T}}}}{{{\mathrm{d}}{\it{t}}}}} \right)\) vs. T reveals the change in state equation of \(\frac{{{\mathrm{d}}{\it{T}}}}{{{\mathrm{d}}{\it{t}}}}\) for a small perturbation in T (Fig. 5c). If this quantity is positive, it indicates amplification of the rate of temperature change for small perturbations in T, which is a signature of local activity^{28,45}. This quantity also serves as a criterion for verifying local activity and the presence of associated NDR and/or other instabilities.
As demonstrated in Fig. 5c, both the currentcontrolled NDR and the Motttransition instability are regions of local activity. Further, in order to analyze the possible currentdensity decompositions occurring within different operating regions, we repeat the exercise detailed in Fig. 1 to obtain a j_{m} vs. T plot (Fig. 5d). Here we see three regions (I–III) of bias where region II contains three stable steadystates while regions I and II contain two each. A potential ternary decomposition involving a combination of all three stable steadystates within region II has a higher enthalpy than the component decompositions and is thus unstable; see Methods. We consider decompositions from state δ into α and λ, and separately into γ and λ. The enthalpy increases of the two possible decompositions and that for the unstable state δ (Fig. 5e, obtained using a procedure similar to that established in Fig. 1 and Eqs (3 and 4)), reveals that in region I, the α–λ decomposition has the lowest ΔH, while in regions II and III, the γ–λ decomposition has the lowest ΔH. A transition from one decomposed state to another would in general involve switching between different channels.
We experimentally imaged micrometersized NbO_{2} crosspoint devices (see Methods, Supplementary Figs. 13–15), in which current flow is normal to the plane of the device, using in operando timemultiplexed thermoreflectance (device structures, the technique and measurements are detailed elsewhere)^{32,46}. The temperature maps presented in Fig. 6 are averages of more than 1000 sequential maps to improve the signaltonoise ratio and to establish repeatability^{46}. The bias voltages reported in Figs 5 and 6 are not exactly equivalent, since the data for each were collected using different samples and measurement apparatus. When the devices were operated using a voltage source within the regions of instabilities, we observed a diffuse hightemperature channel (Fig. 6a, 0.75 V) that appeared to transform into two adjacent channels at higher voltage (Fig. 6c, 0.80 V). A further voltage increase was accompanied by extinguishing the original channel and a temperature increase in the second channel (Fig. 6f, 0.85 V). This crossover from one channel to another is consistent with that expected from the results in Fig. 5e (a transition from the α–λ decomposition to the γ–λ decomposition, causing a new channel to form).
Discussion
The experimental observations of (a) voltagesourced electroforming of a nonvolatile conducting channel in TaO_{x} memristors (Fig. 2), (b) currentdensity channel and electricfield fragment formation in VO_{2} (Fig. 4), and (c) currentdensity channel formation in NbO_{2} (Fig. 6) are all the result of instabilityinduced decompositions. The temperature image in Fig. 6a at 0.75 V clearly contains more noise or higher amplitude fluctuations than the other images, despite the lower applied voltage and identical measurement procedures. Since local activity can amplify thermal fluctuations (Fig. 5c), this could be a direct image of amplified local temperature variations upon the system entering a region of instability represented by region I in Fig. 5d. Thus, our proposed resolution of the debate on the cause of state transitions^{12,15,16,18,19} such as resistance switching and electronic domain decomposition is that nonlinear activated transport and instabilities provide a feedback mechanism to amplify random thermal fluctuations that trigger spontaneous symmetrybreaking and drive the system to its lowest internal energy state, which would occur even for structurally uniform and defectfree systems. Inhomogeneities (in the material composition, film thickness, etc.) may provide nucleation sites for a high current density channel. For example, the more prominent hot region observed in Fig. 6a may have nucleated at a defect. Another interesting observation is that oxides such as TaO_{x} and HfO_{x} undergo electroforming to create chemically differentiated conduction channels that support nonvolatile resistance switching, but oxides such as NbO_{2} crystallize under significant Joule heating but otherwise only display reversible NDR even when voltagedriven. We attribute this difference to the activation energies for creation of oxygen defects^{38,39,47}, which are relatively low in oxides such as HfO_{x} but are much larger in NbO_{x}.
This model of how an electronic device can minimize its internal energy via amplified fluctuations, symmetry breaking and decomposition is completely missing from conventional numerical multiphysics simulators. The best that they can do is approximate the unstable state in a region of NDR or other electronic instability. For nanometer scale devices with increasing nonlinearity and thermal fluctuation amplitudes^{4}, especially those that exhibit local activity, the results obtained by simply integrating coupled field equations can be qualitatively incorrect. Inhomogeneous and asymmetric currentdensity and electricfield configurations can be stabilized by spacecharge regions^{8}, as long as the energy required to form them is less than the energy gained through decomposition (Supplementary Note 1). Not understanding the consequences of currentdensity decomposition in the design of an electronic device can lead to unanticipated faults and poor reliability in an integrated circuit. However, introducing temperature fluctuations into full scale Monte Carlo or moleculardynamicsstyle simulations with a uniform initial state and iterating until a lowest internal energy steady state of an electronic device has been found will likely require very long computation times, especially to generate current–voltage characteristics with high fidelity. One way to significantly cut execution times and computation costs would be to use the methods described above to map out the parameter space and use a decomposed state as the initial condition for a full physics simulation. For the interested reader, we provide an extended but preliminary analysis of the free energy of the system in Supplementary Note 2 and Supplementary Figs 16–18.
In conclusion, we performed nonlinear dynamical analysis of decompositions arising from electronic instabilities such as currentcontrolled NDR and Motttransition instabilities, and showed that both currentdensity and electricfield decompositions are possible depending on the operating conditions and the type of instability. The decomposed states were shown to have a lower internal energy than corresponding uniform unstable steady states. As noted by Landauer^{10,15}, thermodynamic minimization constraints can have a significant impact on the numerical simulations of any electronic device and are especially important for accurate modeling of nonlinear devices. We also provided experimentally obtained thermal images of such decompositions, directly revealing the formation of currentdensity channels and electricfield domains. We further make the connection among local activity, amplification of thermal fluctuations, and electronic instabilities, which together cause decompositions to occur.
Methods
Film growth and device fabrication
The TaO_{x} was deposited using ion beam induced reactive sputter deposition^{38} (using an Oxford Instruments Ionfab 300Plus machine) from a target of Ta_{2}O_{5}. VO_{2} was grown by hightemperature annealing of an evaporated V film in an oxygen environment^{40}. NbO_{2} was grown by reactive sputter deposition^{32} from a target of NbO_{2}. The compositions and film structure were studied using a variety of techniques including xray absorption spectroscopy (Supplementary Fig. 13), transmission electron microscopy and electron diffraction (Supplementary Fig. 15), etc. The electrodes in all cases were lithographically defined and deposited using evaporation of Pt. Detailed analyses are presented elsewhere (see the preceding references).
Electrical measurements
Quasistatic current–voltage behavior (Fig. 2) was measured using an Agilent B1500 parameter analyzer and a Cascade probe station. The parameter analyzer was controlled through a General Purpose Interface Bus (GPIB) using software programs written in Igor. Dynamical electrical measurements (to obtain data in Fig. 6) were carried out using a custommade circuit board capable of producing controlled and synchronized pulsed voltages that is described elsewhere^{48}.
Physical measurements
The oxygen concentration map in Fig. 2b was obtained using a scanning transmission xray microscope at the synchrotron at the Advanced Light Source at Lawrence Berkeley National Laboratory, beamline 11.0.2 at the oxygen Kedge. The devices for this experiment were fabricated on top of 200 nm of freely suspended silicon nitride membranes, to enable xray transmission in the oxygen Kedge. The details of the measurement can be found elsewhere^{38,49}. The temperature maps in Fig. 4 were obtained using blackbody emission microscopy in the infrared wavelength, using InfraScope^{50}, emissivitycalibrated to a temperature resolution of about 1 K and spatial resolution of about 1.5 µm. The temperature maps in Fig. 6 were obtained using thermoreflectance microscopy^{46}, with a temperature resolution of about 1 K and spatial resolution of less than 0.5 µm. To enable a higher signaltonoise ratio, we utilized a timemultiplexed technique to average the signal over synchronously repeated electrical operation^{48}.
Latent heat for the Mott transition
The latent heat for the Mott transition was accounted for by introducing a smoothed increase in \({\it{C}}_{{\mathrm{th}}}^{{\mathrm{act}}}\) over the transition temperature range, such that the area within the increase in \({\it{C}}_{{\mathrm{th}}}^{{\mathrm{act}}}\) in a plot of \({\it{C}}_{{\mathrm{th}}}^{{\mathrm{act}}}\) vs. T is equal to the latent heat. \({\it{C}}_{{\mathrm{th}}}^{{\mathrm{act}}}\) was calculated by using literature values for the intrinsic heat capacity and the volume of the oxide within the device. Since the rest of the device structure is unaltered during the Mott transition, we consider the thermal capacitance of only the Mott insulator within the device. The magnitude of increase in \({\it{C}}_{{\mathrm{th}}}^{{\mathrm{act}}}\) during the Mott transition depends on the width of the transition (in temperature), which in turn depends on several factors including purity of the material, crystal quality, etc. There is a wide range of transition widths and corresponding changes in heat capacities reported in the literature. The choice of the transition width does not affect our conclusions^{51,52,53}.
For VO_{2}, (in WsK^{−1})
For NbO_{2}, (in WsK^{−1})
Eliminating the possibility of ternary decomposition
Let us suppose from Fig. 5 that a total current in unstable steadystate β is decomposed among three stable steadystates α, γ and λ. This can be represented as
where x and y are the fractions of the area carrying current densities \({\it{j}}_{\it{\lambda }}\) and \({\it{j}}_{\it{\gamma }}\), respectively. The enthalpy change from the ambient for the hypothetical ternary decomposition can be calculated as
where \({\it{T}}_{\it{\lambda }}\), \({\it{T}}_{\it{\gamma }}\) and \({\it{T}}_{\it{\alpha }}\) represent the temperatures corresponding to \({\it{j}}_{\it{\lambda }}\), \({\it{j}}_{\it{\gamma }}\) and \({\it{j}}_{\it{\alpha }}\), respectively.
Substituting Eq. (8c) in Eq. (9),
which is a linear function of x and therefore the minimum value of \({\mathrm{\Delta }}{{H}}_{{\mathrm{tri}}}\) will appear at x = 0 or x = 1. Both cases eliminate one of the three components of the ternary decomposition, and hence the most stable configuration is a binary decomposition.
Data availability
Additional data, if any, which support the findings of this study are available from the authors upon reasonable request.
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Acknowledgements
The authors gratefully thank Leon Chua, Stefan Slesazeck, and the anonymous reviewers for providing very useful reviews that helped in significantly improving the manuscript. The research is in part based upon work supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via contract number 201717013000002. This research used resources of the Advanced Light Source at Lawrence Berkeley National Laboratory, which is a DOE Office of Science User Facility under contract no. DEAC0205CH11231.
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S.K. and R.S.W. together conceived the ideas presented here, analyzed the experimental data, performed the calculations and modeling, and wrote the manuscript. S.K. designed and performed the experiments.
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Kumar, S., Williams, R.S. Separation of current density and electric field domains caused by nonlinear electronic instabilities. Nat Commun 9, 2030 (2018). https://doi.org/10.1038/s4146701804452w
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DOI: https://doi.org/10.1038/s4146701804452w
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