Spontaneous Symmetry‐Breaking of Nonequilibrium Steady–States Caused by Nonlinear Electrical Transport

Negative differential resistance (NDR) in certain materials has been attributed to spontaneous emergence of symmetry‐breaking electrical current density localization from a previously homogeneous distribution, which is postulated to occur due to the nonequilibrium thermodynamic force of minimization of entropy production. However, this phenomenon has not been quantitatively predicted based on intrinsic material properties and an applied electrical stimulus. Herein an instability criterion is derived for localization of current density and temperature from a thermal fluctuation in a parallel conductor model of a thin film that is subject to Newton's law of cooling. The conditions for steady–state electro‐thermal localization is predicted, verifying a decrease in entropy production upon localization. Electro‐thermal localization accompanied by a decrease of entropy production is confirmed in a multiphysics simulation of current flow in a thin film. The instability criterion predicts conditions for spontaneous current density localization, relating symmetry breaking fundamentally to dynamical instability via Local Activity theory.


DOI: 10.1002/aelm.202300265
systems, [1] remains a critical topic of interest across many disciplines from the differentiation of clusters of cells [2] to galaxy formation during the early universe. [3] Within the field of materials science, spontaneous pattern formation occurs from atomic to macroscopic length scales, as exemplified by spinodal decomposition [4] and buckling in curved elastic media. [5] In each of these examples, symmetry breaking results when an energetically metastable system is perturbed, resulting in spontaneous formation of spatial heterogeneity. Similar spontaneous heterogeneities are predicted to occur in current-controlled semiconductor systems [6] and have recently been experimentally realized, [7,8] resulting in the formation of domains of high current density parallel to the direction of current flow (Figure 1a). This phenomenon has been generally rationalized in terms of steady-state entropy minimization principles. [6,9] However, the intrinsic properties and parametric conditions that result in the spontaneous localization of current, and the properties of the resulting high current density channel, remain largely unresolved.
Negative differential resistance is a non-Ohmic response for which the steady-state-current-voltage curve of an electronic circuit element exhibits non-monotonicity ( dV dI < 0) [6] due to the electrical transport being a sufficiently nonlinear function of some internal variable, e.g., temperature. Devices operated in a region of NDR are locally active, meaning that they are capable of utilizing DC power to couple to external circuitry and thereby amplify small fluctuations in their internal state variable or external electrical biasing, which can be used in circuits to implement amplifiers, oscillators, signal multiplexers and other useful functions. [8,[10][11][12][13] NDR can manifest from a variety of nonlinear physical mechanisms and associated state variables, including temperature in thermally-activated transport, tunneling distance in ion electro-migration, and reaction coordinate in phase transitions and chemical reactions, and each may either cause or result from current density localization in the device (Figure 1a). [14][15][16][17][18][19][20] In this work, we restrict our analysis to current-controlled, or S-type, NDR resulting from the feedback between thermallyactivated nonlinear electrical conductivity, internal Joule heating, and heat dissipation via Newton's law of cooling, such as that observed in VO x , NbO x , or TaO x . [7,[21][22][23][24][25][26] In these materials, the Conditions for electro-thermal localization in a lateral thin film device were predicted based on intrinsic material properties, device parameters, and the applied electrical stimulus. a.) Current density localization was predicted to result from negative differential resistance (NDR) in certain materials. [6] b) In lateral devices under an applied current, thermal localizations form in VO 2 after a critical current is exceeded, bridging electrodes on the left and right sides of a film (scale bar = 2 μm). To model this phenomenon, a thin film is treated as N identical electrically coupled but thermally uncoupled parallel conductors. Schematics of c) the thermal model in cross-section and d) the lateral electrical model in plain view. One of the parallel conductors, G h , is initially perturbed to a higher temperature, T 0 h = T ∞ + T seed , where T ∞ is the steady-state temperature of the homogeneous system, under the same parametric conditions. The other channels have initial conductance of G i at temperature T 0 i . e) Utilizing the dynamical equation describing this system (Equation 1) with parameters predicted to either dissipate (top) or magnify (bottom) a seeded thermal perturbation, the model reproduces spontaneous current density localization depending on the thermal conductance th and applied current I app ( th = 5 × 10 −6 W · K −1 , N = 10, I app = 1.0 mA (top), I app = 1.4 mA (bottom)). f) Depending on temperature (color scale in b) and the electrical conductivity, the instability criterion Ω (Equation 7) varies nonlinearly with I app and th . There exists for a set of parameters a critical value of applied current, I crit , beyond which localization occurs.
appearance of a dramatic and abrupt NDR has been observed to coincide with the appearance of thermal inhomogeneities that occur simultaneously with current density localizations, referred to collectively as electro-thermal localizations (Figure 1b).
NDR is intrinsic to oscillatory and spiking behavior for implementing neuromorphic computing [27] , but symmetry breaking, and current density localization are not included in any of the lumped models for biological or artificial neurons. A quantitative understanding of this phenomenon is critical to developing ideal materials for such applications. Previous attempts to explain the co-occurrence of NDR and current density localization focused on empirically-derived results that cannot be easily generalized to other materials without extensive data collection, [28] or on ex-plicit inclusion of a high current density channel to produce specific electrical responses. [20,29] To this end, we investigated the underlying mechanism that produces the spontaneous formation of current density localization, with the goal of deriving criteria to predict the parametric boundaries of these localizations in thermally-activated materials and their final steady-state physical attributes.

Results and Discussion
To quantify conditions that facilitate localization, an analytic parallel electrical conductor model of a two-terminal thin film device was developed from Kirchhoff's current law and Newton's www.advancedsciencenews.com www.advelectronicmat.de law of cooling (Figure 1c,d). [25] Lateral heat transfer among the N conductors and heat transfer through convection from the top surface of the device were assumed to be negligible compared to heat conduction through the bottom interface of the channels to an isothermal substrate to simplify the thin film device model ( Figure 1c). Each parallel conductor was treated as a homogeneous, thermally lumped volume with uniform current density. This simplification necessarily sacrifices some fidelity, while maintaining the core underlying nature of the problem being investigated.
The dynamical equation for the temperature (T i ) of the i th conductor (with a conductance G i ) under a constant applied current (I app ) evolves according to Newton's Law of Cooling in the form: where th is the effective thermal capacitance of each conductor, G tot is the total electrical conductance of the set of conductors, and T sub is the temperature of the underlying substrate. The net interfacial thermal boundary conductance net [W · K −1 ] combines the thermal conductances of any relevant thermal boundaries. [30][31][32] The functionally continuous (i.e., excluding an abrupt conductivity change such as an insulator-to-metal transition) temperaturedependent electrical and thermal conductance functions were chosen for a generic rather than specific system to illustrate the generality of the model (see Supporting Information for additional detail). This both avoids biasing the predicted output discontinuity (localization) by assuming a discontinuity in the input and shows that that the localization phenomenon manifests in a much more general class of materials. Electro-thermal localization occurs when a thermally perturbed "hot" conductor has a higher temperature (T ∞ h ) than that of any of the N − 1 other reference conductors (T ∞ i ) at steadystate, Allowing for the possibility that the thermal conductance is temperature-dependent, the relationships between thermal and electrical parameters for a current density and temperature localization at steady-state follow as: where the parameters at steady-state are abbreviated as ∞ = (T ∞ ); G∞=GT∞.
A metastable system will remain in its metastable state until perturbed, in the present case by an electrical or thermal fluctuation, [33] or alternatively passively through a material defect that produces a local temperature heterogeneity, which can initiate its transition to a more stable state. Assuming an initially homogeneous temperature distribution T 0 i = T ∞ for all N conductors, we evaluate their stability by incrementing, or "seeding", the temperature in the hot conductor by a finite temperature perturbation: T 0 h = T ∞ + T seed . We then evaluate the time evolution of the set of conductors following (Equation 1) and observe whether they return to their original homogeneous state, meaning that their initial state is stable, or if instead they evolve to a new, more stable state, meaning the initial state was metastable (see Supporting Information, Section 2 for more details) (Figure 1e). After the perturbation, if the temperature increases in the seeded hot conductor relative to the others, then a feed-forward relationship between the increasing current density and temperature caused by Joule heating allows electro-thermal localization in the parallel conductors. This instability occurs when the time-dependent temperature rise in the "hot" conductor exceeds that in the other conductors: The total electrical conductance of the system of N parallel conductors is expressed in terms of the electrical conductance of the thermally perturbed conductor, and the unperturbed conductors as: For an infinitesimal thermal fluctuation T seed → 0 such that where: This expression can be used to assess at what ranges of I app the initial homogeneous state of the set of conductors becomes unstable (Figure 1f). Equation 7 can also be rewritten to emphasize the electrical conductance function: showing that the destabilization of the set of conductors upon a thermal fluctuation is the result of the interplay between extrinsic device and circuit parameters and the conductors' intrinsic, nonlinear electrical conductivity. Finally, this criterion can be expressed in terms of the steadystate temperature (T ∞ i ) of the device and the temperature of the underlying substrate (T sub ): where GT is the change of electrical conductance with respect to a change in temperature of the thermally perturbed conductor, and G i is the electrical conductance of one conductor [See Supporting Information for derivation]. This expression is the same as the criterion derived for edge of chaos (EOC) and negative differential resistance (NDR) properties in a constrained system in which the thermal conductance is constant, showing the interdependence and simultaneous appearance of EOC with electrothermal localizations. [10] To provide insight into the thermodynamic driving forces behind temperature and current density localizations in thermallyactivated material systems, we used the conductor-based model to verify whether the electro-thermal localizations shown throughout this work are consistent with the principle of minimum entropy production. [6] This principle states that the most stable configuration of a steady-state system has the lowest rate of entropy production to be as close as possible to the equilibrium state, which by definition has an entropy production rate ofṠ eq = 0. Using a numerical implementation of the conductorbased model (Equation 1), the steady-state rate of entropy production for the set of conductors,Ṡ ss , can be calculated as the summation of entropy production over all conductors [33,34] : where I i and T i are the current and temperature of the i th conductor, and V is the voltage drop across the parallel conductors.
To calculate the entropy production rateṠ ss for the conductorbased model, a numerical implementation of the parallel conductor model (Equation 1), and two conductors (N = 2) was used to obtain the steady-state temperatures and current through each conductor, and the voltage drop V across the two conductors. Holding the total width of the conducting film constant (W = 10 μm), the width of the thermally perturbed conductor G h was varied between W h = 0.1 − 2 μm (Figure 2a). Two sets of parametric sweeps of W h were computed, one in which the initial temperatures of the two conductors were homogeneous and one in which the G h was initially thermally perturbed. The differential entropy production rate between the thermally localized and thermally homogeneous cases can be calculated from the equation: Based on the principle of minimum entropy production, the most stable electro-thermal localization width W min ΔṠ = 0.88 m corresponds to the minimum of ΔṠ(W h ) (Figure 2a). For a set of given parameters {W, N, th }, there exists a critical current beyond that localization occurs (I crit ). By varying I app , minimization of entropy production predicts that as I app is increased beyond I crit , the width of the localization increases (Figure 2b). When the current density J is held constant, there is a linear relationship between the optimal channel width and the total device width (Figure 2c).
To relax some of the limiting assumptions of the parallel conductor model, we performed multiphysics-based finite element (FE) modeling, which facilitates studies of interactions among temperature, voltage and current in complex material structures and devices. This method allowed us to include elements neglected in the parallel conductor model, such as lateral thermal coupling throughout the film and in-plane thermal gradients due to the competing trends of thermal conduction and heat generation. Electro-thermal localization was demonstrated in the x − y plane of a 3D FE model of the thin film device ( Figure S2,   Figure 2. Numerical simulations implementing Equation 1 were used to calculate the electro-thermal localization widths that minimize the differential entropy production, ΔṠ. a) While holding the total width of the device constant, the width of the thermally perturbed conductor (W h ) was varied to find a width corresponding to the minimum in the differential entropy production rate ( W minΔṠ = 0.88 m) using Equation 11 (constant th = 5 × 10 5 W K −1 unless otherwise indicated). b) As I app increases beyond I crit , the W minΔṠ predicted by entropy minimization increases. c) The current density J (A · m −2 ) was held constant as the modeled device's width W was varied between W = 2 − 10 μm and the width of the hot conductor was varied between W h = 0.01 − 2 μm. Using minimization of entropy production as a metric of optimal width, W minΔṠ increases linearly as a function of increasing W.
Supporting Information) in the form of temperature and current density differential maps at simulated times of t sim = 0 μs and t sim = 10 ms (Figure 3). The initial temperature perturbation of T seed = 10 K in a 0.1 μm 3 volume (Figure 3a) offset from the center of the film did not bridge the two electrodes. Using a smaller temperature perturbation will produce the localization effect, albeit with less control over the steady-state position of Figure 3. Spontaneous electro-thermal localization was exhibited in a 3D finite element (FE) model of a lateral thin film device. a) The differential temperature distribution of a temperature localization in a 3D FE model of the generic electro-thermal device. The lateral device is oriented with electrodes at the top and bottom of each temperature map. At t sim = 0 μmus, a rectangular prism of V = 10 −3 μm 3 (dimensions of L, W = 1 μm, t = 0.1 μm) is perturbed Tseed = 10 K above the steady-state temperature of the surrounding thin film (size of the perturbed volume is exaggerated). Below I app = 2.6 mA, the temperature of the simulation equilibrates homogeneously throughout the device volume. At and beyond I app = 2.6 mA, there is an abrupt localization of temperature into a channel ( W FWHM = 1.37 ± 0.02 μm) parallel to the direction of current flow. b) The differential current density map reflects the temperature map, as a narrow high current density channel forms from a previously homogeneous distribution at and beyond I app = 2.6 mA. c) Negative differential resistance corresponding to the spontaneous localization is seen in the simulated steady-state V(I) curve. the electro-thermal localization that is subject to the relative rates of heat and charge transfer within and outside of the electrothermal localization. For example, smaller temperature perturbation of Tseed = 1 K under these same conditions will still destabilize the system but could result in an electro-thermal localization closer to the central axis of the device. For the chosen material properties (see, Section 1, Supporting Information) below 2.6 mA, the steady-state temperature of the device is nearly homogeneous. For I app ≥ 2.6 mA, a steady-state thermal localization bridging the electrodes forms parallel to the direction of current flow even with the initial perturbation in a small cubic volume of material. This indicates that the state of uniform current flow is extremely unstable above I crit . The localized channel has a maximum temperature of T max > T min + 500 K that is hottest in its center and decreases along the perpendicular direction to the formation of the localization, with a width of W FWHM = 1.37 ± 0.02 μm ( Figure S3, Supporting Information). The width of the localization in the FE model is somewhat greater than that predicted by entropy minimization in the parallel conductor model. However, in the FE model, 3D heat transfer produces temperature and current density gradients that contribute to the width of the localization and are not explicitly included in the parallel conductor model. The current density differential maps confirm homogeneous current density throughout the device until I app = 2.6 mA, at which a high current density channel forms, corresponding to the temperature localization ( Figure 3b). As an additional degree of symmetry-breaking, the space charge distributes asymmetrically with respect to the temperature and current density localization ( Figure S4, Supporting Information). In the simulated steady-state V(I) curve of the device, NDR emerges simultaneously with the localization at I app = 2.6 mA (Figure 3c). These simulations demonstrate electro-thermal localizations correlating to NDR that are initiated by a small temperature perturbation in a system with otherwise spatially homogeneous material and device properties, confirming the results of the parallel electrical conductor model.
The temperature and current density solutions of the FE model were used to calculate the differential rate of entropy production between a homogeneous temperature distribution and a localized temperature distribution (See Section 3, Supporting Information for details). Using Equation 11, the differential entropy production rate was found to be ΔṠ = −7.71 × 10 −8 ± 0.26 × 10 −8 J ⋅ K −1 s −1 for the localized distribution relative to the homogeneous one. This decrease in entropy production is consistent with electro-thermal localization being favored thermodynamically by steady-state minimization of entropy production.
The multiphysics simulations in combination with the relatively simple but intuitive treatment of a thin film as a set of parallel conductors helps to elucidate the critical link between nonlinear dynamical conductivity, which is a key property in the implementation of neuromorphic functions such as action potentials and spiking, and current density localization, which has been neglected in lumped models of neuromorphic behavior. Our findings show that current density spontaneously localizes above a critical threshold current in response to a thermal perturbation, due to an instability in the dynamical equations that describe temperaturedependent electrical transport and Newton's law of cooling, www.advancedsciencenews.com www.advelectronicmat.de which also produce NDR and Local Activity. Experimental studies have thus far relied on indirect methods to observe current density localization, e.g. surface temperature mapping, emergence of NDR, or spectroscopic detection of phase transitions. We directly calculate the spontaneous localization of current density triggered by a thermal fluctuation in a model of a homogeneous material without defects or additional assumptions that is consistent with minimization of entropy production. The methods developed here define a critical current threshold for the prediction of current density localization based on a small set of material and device parameters, as well as quantitative methods for predicting the localization properties, such as channel width and temperature.
This research has consequences far beyond the results presented here. We see that a full quantitative description of the behavior of a nonlinear dynamical system may require input from both the theory of Local Activity and nonequilibrium (or stochastic) thermodynamics beyond lumped models. This provides a platform for further analyses of Turing instabilities and Prigogine's "instability of the homogeneous". Given the relatively simple requirements for current density localization, it is plausible that current localization occurs more broadly than is currently recorded, perhaps even in biological systems. These results have immediate implications for the design of artificial neurons. For instance, if a single neuron is multiplied addressable, can a single artificial neuron communicate different signals simultaneously in localized and nonlocalized regions of the neuron? We look forward to the examination and resolution of these and many other related questions.

Experimental Section
Thermoreflectance Experiments: Quasistatic temperatures of a twoterminal VO 2 thin film device under an applied current (Figure 1b) were measured using a Microsanj differential Thermal Image Analyzer (TIA) as described elsewhere. [7] The dimensions of the imaged devices were L = 5 μm, W = 10 μm, and t = 0.1 μm.
Multiphysics Simulations: The 3D finite element (FE) model of a lateral thin film device was developed in the multiphysics modeling platform COMSOL using the Joule Heating multiphysics interface, combining electrical current and heat transfer modules. The model had an average element size of 7 × 10 −4 μm 3 , or ≈8000 elements in the thin film domain. Simulations were run under a free time-stepping condition, which generally took ≈100 steps. The simulated time of the model was set between 1 − 10 ms per simulation, which was sufficient to achieve a steady-state temperature and current distribution (see Section 3, Supporting Information).

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.