Time-Resolved In Situ Joule Heating

• Time-resolved in situ Joule heating is a technique that uses controlled electrical dissipation to dynamically drive and measure thermal transitions in materials in real time.
• Its methodologies incorporate pulsed measurements, quasi-static bias sweeps, and advanced thermometry to differentiate electron heating from lattice heating across various platforms.
• The approach informs applications from superconducting spectroscopy to phase transformation studies by decoupling thermal artifacts from intrinsic nonequilibrium phenomena.

Time-resolved in situ Joule heating denotes experimental and theoretical approaches in which electrical dissipation generated within a specimen, or in a directly coupled conductor, is used to drive thermal evolution while the resulting transport, magnetic, structural, chemical, or thermodynamic response is monitored during bias application, during a heating pulse, or during subsequent relaxation. In the literature, the phrase covers both direct time-domain measurements with controlled delays and quasi-static bias sweeps that expose a bias-controlled thermal transition in real time. Across topological insulators, hybrid superconducting nanodevices, correlated oxides, mesoscopic metals, transmission electron microscopy platforms, and diamond anvil cells, a recurring result is that Joule heating is often not a parasitic side effect but the dominant control parameter, and failure to separate electron heating, lattice heating, and thermal diffusion can misidentify thermal artifacts as intrinsic nonequilibrium states (Kölling et al., 16 May 2025, Wang et al., 2017, Kumar et al., 2015, Fang et al., 16 Feb 2025).

1. Experimental scope and meanings of “time-resolved” and “in situ”

In this context, “in situ” refers to measurements performed while the sample remains under electrical drive and within the operative environment of the instrument, such as a PPMS transport stage, a transmission electron microscope, a blackbody spectromicroscope, scanning transmission x-ray microscopy, magnetic-force microscopy, or a diamond anvil cell. “Time-resolved” has at least two established meanings. In mesoscopic thermometry and pulse transport, it means direct sampling of a thermal transient with controlled delay times. In Joule spectroscopy of hybrid superconductor-semiconductor devices, it instead denotes an in situ, bias-controlled dynamical thermal transition observed during a transport sweep rather than an ultrafast pump-probe transient (Wang et al., 2017, Ibabe et al., 2022).

Platform	Joule-heating protocol	In situ observable
Normal metal film with proximity SNS thermometer	Rectangular heating pulse plus delayed probe pulse	Electron temperature from switching statistics
BST Hall bars	DC source-drain bias analyzed through measured $V_{xx}$	Differential conductivity, WAL cusp, ZBRP, coercive-field proxy in VBST
VO$_2$ planar device	Current ramp through Pt-contacted film	Optical filament, blackbody temperature map, STXM phase map
Mn$_3$Si$_2$Te$_6$	Triangular-wave current and 500 $\mu$s rectangular pulses	Real-time resistance and MFM domain evolution
GO suspended in TEM	Repeated current sweeps with increasing $I_{\max}$	Conductivity, TEM, ED, EELS
DAC microassembly	Pulsed Joule heating in four-point geometry	Simultaneous resistance and spectroradiometric temperature

The breadth of these implementations shows that Joule heating is used both as a perturbation and as a measurement modality. In some cases it is the quantity to be eliminated or decoupled, as in electrocaloric measurements and low-temperature quantum transport. In others it is the signal itself, as in hybrid-device Joule spectroscopy, modulation calorimetry, or electrically driven phase transformations (Quintero et al., 2011, Geballe et al., 2017, Geballe et al., 2023).

2. Electrothermal principles, thermal balances, and scaling relations

The basic power relations used throughout are the standard Joule-heating identities

$P = I^2R,\qquad P = VI,\qquad P = \frac{V^2}{R},$

with the experimentally relevant control parameter depending on geometry and context. In BST Hall bars, the analysis is performed in terms of the measured bias voltage $V \equiv V_{xx}$, because the authors emphasize that Joule heating depends on electric field or bias voltage rather than on current alone (Kölling et al., 16 May 2025). In nanowires, the local volumetric source term is written as

$$Q = \mathbf j\cdot \mathbf E = \frac{1}{\sigma}j^2,$$

which makes constrictions and current crowding dominant hot spots because the heating scales as $_2$0 (Fangohr et al., 2010).

A central distinction is between electron temperature and lattice or substrate temperature. In a Joule-heated normal metal film coupled to an SNS thermometer, the steady-state thermal balance is written as

$_2$1

with electron-phonon cooling modeled by

$_2$2

For small delays after a heating pulse begins, the temperature rise is approximately

$_2$3

whereas relaxation after the pulse obeys

$_2$4

These relations explicitly separate injected Joule power, heat capacity, and thermal conductance (Wang et al., 2017).

Low-temperature quantum transport introduces an additional energy-scale mapping. In disordered BST, the relevant scale is written as

$_2$5

and the central empirical correspondence is

$_2$6

with data consistent with

$_2$7

Here voltage acts as an effective temperature scale through electron heating rather than through direct bias-induced suppression of weak antilocalization or electron-electron interaction corrections (Kölling et al., 16 May 2025).

Time-domain transport in Mn$_2$8Si$_2$9Te$_3$0 uses an explicit lumped electrothermal equation,

$_3$1

with $_3$2 approximated by a second-order polynomial. The measured equilibrium $_3$3 is inserted directly into the simulation, so the apparent current-induced switching is recast as heating-driven motion through the equilibrium resistance curve (Fang et al., 16 Feb 2025).

3. Time-domain instrumentation and thermometry

A defining achievement of time-resolved in situ Joule-heating work is the development of thermometry that minimally perturbs the thermal state being measured. In a proximity Josephson-junction thermometer, an SNS weak link is used only as a probe, not as a heater. The junction remains superconducting until the current reaches the switching current $_3$4, and the crucial point is that dissipation starts only after the junction switches. Consequently, the switching statistics reflect the unperturbed electron temperature of the normal region before the junction overheats itself. With a timed sequence of heating and probe pulses, the thermometer samples nonequilibrium electron temperature with effectively zero back-action; the switching probability is measured as

$_3$5

and the readout current $_3$6 is defined at 50% switching probability (Wang et al., 2017).

Another time-domain strategy is to exploit distinct thermal time scales to separate concurrent processes. In a BaTiO$_3$7-based multilayer capacitor, direct temperature-vs-time measurements under a voltage step reveal a very fast initial electrocaloric jump, an intermediate sample-holder equilibration, and a slower relaxation to the bath. The thermal model treats the sample and holder as two coupled bodies with

$_3$8

$_3$9

under the designed condition $_2$0. The reported characteristic times are $_2$1, $_2$2, and $_2$3. Because $_2$4, electrocaloric and Joule-heating fingerprints can be decoupled in the transient (Quintero et al., 2011).

High-frequency modulation calorimetry extends the same logic to extreme confinement. In diamond anvil cell mock-ups, AC Joule heating produces a temperature oscillation at $_2$5, which modulates the sample resistance and generates a measurable third-harmonic voltage at $_2$6. A coupled electrical-thermal model predicts that the specific heat of metals inside diamond cells can be measured directly at about $_2$7 MHz with better than $_2$8 accuracy, whereas a thin-film-heater geometry already enables thermal-effusivity measurements with $_2$9 accuracy using 10 Hz to 300 kHz (Geballe et al., 2017).

At the hardware level, a three-layer DAC microassembly shows how pulsed Joule heating and four-point resistance measurements can be stabilized simultaneously under extreme pressure. The stack combines two KCl thermal-insulation layers, four electrical leads, a sample, and a polycrystalline alumina buttressing layer, with the sample and inner electrodes fabricated by focused-ion-beam milling. Three Fe proof-of-concept samples were successfully compressed and pulsed Joule heated while maintaining a four-point configuration, reaching approximately $_6$0 GPa and $_6$1 K (Geballe et al., 2023).

4. Bias-driven transport, quantum corrections, and superconducting spectroscopy

Low-temperature transport is a major domain in which time-resolved or bias-resolved Joule heating alters interpretation. In $_6$2, the apparent non-Ohmic differential resistance consists of a zero-bias resistance peak and a high-bias background. The zero-bias feature is robust in magnetic field and follows the standard 2D EEI form

$_6$3

while the bias dependence at sufficiently large bias is written as

$_6$4

The high-bias background is suppressed by out-of-plane magnetic field and is associated with weak antilocalization. Transport and VBST coercive-field measurements show that at the highest bias the electron temperature inferred from transport can reach about $_6$5 K while the lattice temperature rises only to about $_6$6 K from a $_6$7 K base, with only about a $_6$8 K substrate increase inferred from a neighboring Hall bar. The consequence is that the bias dependence is largely explained by Joule-heating-induced electron heating, not by direct bias suppression of quantum corrections (Kölling et al., 16 May 2025).

Hybrid superconductor-semiconductor nanodevices convert the same effect into a spectroscopic tool. In full-shell Al-InAs nanowire Josephson junctions, finite-bias transport injects quasiparticles into superconducting leads whose heat diffusion is poor at low temperature. When the local lead temperature near the junction reaches the lead-specific $_6$9, the excess current is suppressed and a sharp dip appears in $\mu$0. These dips, $\mu$1 and $\mu$2, provide lead-resolved information on $\mu$3, $\mu$4, coherence length, shell continuity, and inverse proximity effects. The work explicitly states that the measurements are quasi-static transport traces rather than ultrafast pump-probe experiments, but the observed transition is nonetheless an in situ dynamical thermal transition driven by bias increase (Ibabe et al., 2022).

A broader interpretive pattern emerges from these transport studies. Bias-dependent anomalies in quantum materials do not automatically imply direct electric-field control of quantum coherence or phase structure. In both BST and Al-InAs hybrid devices, transport nonlinearities become intelligible only after the electrothermal pathway is modeled explicitly. A plausible implication is that time resolution must be combined with an independent thermal observable whenever finite-bias transport is used to claim a novel nonequilibrium state.

5. Phase transitions, filament formation, and operando structural or chemical transformation

Joule heating frequently drives first-order-like or threshold-like transformations that are highly nonuniform in space. In planar VO$\mu$5 devices with Pt electrodes, optical microscopy shows a filament that appears only in the low-resistance state, grows with current, and disappears on returning below the OFF threshold. Blackbody microscopy provides emissivity-calibrated temperature maps with approximately $\mu$6 K temperature resolution and $\mu$7m spatial resolution, and STXM on membrane-supported devices adds $\mu$8 nm spatial and $\mu$9 meV energy resolution. The thermal maps show a discontinuous jump in local maximum temperature at switching, with some regions cooling by about $I_{\max}$0 K even while total current increases because current redistributes into a narrow filament. The local temperature jump reaches up to about $I_{\max}$1 K. STXM confirms formation of the rutile metallic phase in the same region, and the paper concludes that switching is a Joule-heating-driven local transition in which the insulator-metal transition and structural phase transition occur at essentially the same applied current, even if the structural transition can lag locally by a few kelvin (Kumar et al., 2015).

Ferrimagnetic Mn$I_{\max}$2Si$I_{\max}$3Te$I_{\max}$4 provides a closely related but magnetically resolved example. MFM images show ferrimagnetic domains at $I_{\max}$5 K and $I_{\max}$6, and under current bias the domain wall disappears abruptly at $I_{\max}$7 mA on the up-sweep and reappears at $I_{\max}$8 mA on the down-sweep. Triangular-wave I–V traces exhibit hysteresis, negative differential resistance, and nonreciprocity that are strongest around $I_{\max}$9 Hz and strongly suppressed at frequencies of order $P = I^2R,\qquad P = VI,\qquad P = \frac{V^2}{R},$0 Hz. With 500 $P = I^2R,\qquad P = VI,\qquad P = \frac{V^2}{R},$1s rectangular current pulses and a repetition period of about $P = I^2R,\qquad P = VI,\qquad P = \frac{V^2}{R},$2 s, the measured $P = I^2R,\qquad P = VI,\qquad P = \frac{V^2}{R},$3 closely mirrors the equilibrium $P = I^2R,\qquad P = VI,\qquad P = \frac{V^2}{R},$4, and simulations based on the heat-balance equation reproduce the onset times of peaks and dips while showing that $P = I^2R,\qquad P = VI,\qquad P = \frac{V^2}{R},$5 can rise by more than $P = I^2R,\qquad P = VI,\qquad P = \frac{V^2}{R},$6 K within $P = I^2R,\qquad P = VI,\qquad P = \frac{V^2}{R},$7s. When the I–V curve is sampled at $P = I^2R,\qquad P = VI,\qquad P = \frac{V^2}{R},$8s, before substantial heating develops, it is linear and Ohmic across different temperatures and magnetic fields (Fang et al., 16 Feb 2025).

In situ TEM extends Joule heating from phase switching to chemical reduction. A suspended GO film contacted inside a TEM is subjected to 47 individual current sweeps with increasing $P = I^2R,\qquad P = VI,\qquad P = \frac{V^2}{R},$9 up to $V \equiv V_{xx}$0 mA, and after each sweep the same region is characterized by TEM, electron diffraction, and EELS. The experiment separates into a low-current regime, attributed to diffusion processes, and a true reduction regime beginning at an applied power density of about $V \equiv V_{xx}$1. Over the full sequence, conductivity rises from about $V \equiv V_{xx}$2–$V \equiv V_{xx}$3 to $V \equiv V_{xx}$4, oxygen content falls from more than $V \equiv V_{xx}$5 at.% to less than $V \equiv V_{xx}$6 at.%, thickness decreases from about $V \equiv V_{xx}$7 nm to $V \equiv V_{xx}$8–$V \equiv V_{xx}$9 nm, and the sp$$Q = \mathbf j\cdot \mathbf E = \frac{1}{\sigma}j^2,$$0 fraction rises from about $$Q = \mathbf j\cdot \mathbf E = \frac{1}{\sigma}j^2,$$1 to about $$Q = \mathbf j\cdot \mathbf E = \frac{1}{\sigma}j^2,$$2. The paper emphasizes that the reduction is highly uniform and localized when the film is contacted symmetrically (Hettler et al., 2021).

A manufacturing-scale analogue appears in graphene-nanoplatelet/epoxy composites, where a percolated GNP network acts as distributed nanoheaters. Above the percolation threshold, the composite can be post-cured out of autoclave by adjusting current to maintain $$Q = \mathbf j\cdot \mathbf E = \frac{1}{\sigma}j^2,$$3C for 4 h. Infrared thermography shows an initial temperature field that is very uniform, highly homogeneous up to about $$Q = \mathbf j\cdot \mathbf E = \frac{1}{\sigma}j^2,$$4C, and the heating rate is about $$Q = \mathbf j\cdot \mathbf E = \frac{1}{\sigma}j^2,$$5C/min compared with a typical oven heating rate of $$Q = \mathbf j\cdot \mathbf E = \frac{1}{\sigma}j^2,$$6–$$Q = \mathbf j\cdot \mathbf E = \frac{1}{\sigma}j^2,$$7C/min. The resulting composites are fully cured, more compact, and less microvoided than their oven-cured counterparts (Xia et al., 2019).

6. Modeling frameworks, extreme regimes, and interpretive controversies

Time-resolved in situ Joule heating depends strongly on geometry, thermal dimensionality, and coupled multiphysics, so analytical and numerical models are not optional add-ons. For nanowires on thick substrates, the heat-flow problem separates into three regimes: an early-time regime in which the temperature rises approximately logarithmically and is well described by the You model, an intermediate regime in which the nanowire temperature stays constant while a hemispherical heat front carries heat into the substrate, and a boundary-limited regime in which both wire and substrate warm rapidly after the heat front reaches the substrate boundary. The crossover time is estimated as

$$Q = \mathbf j\cdot \mathbf E = \frac{1}{\sigma}j^2,$$8

For thin membranes, by contrast, there is only one long-time logarithmic regime before the heat front reaches the boundary, and a separate analytical expression is derived for effectively two-dimensional substrates (Fangohr et al., 2010).

Fully coupled electro-thermo-mechanical simulation is developed for thermoviscoelastic Joule heating with the unknowns temperature $$Q = \mathbf j\cdot \mathbf E = \frac{1}{\sigma}j^2,$$9, electric potential $_2$00, and displacement $_2$01. The governing system,

$_2$02

$_2$03

$_2$04

is discretized by standard finite elements in space and a semi-implicit Euler-type method in time. The proved convergence rates are optimal: second order in space, $_2$05 in $_2$06, and first order in time, $_2$07 in $_2$08, with the expected one-order loss in $_2$09-type norms. This provides a rigorous basis for time-resolved simulations of thermistor- or MEMS-like devices in which Joule heating induces both temperature change and deformation (Målqvist et al., 2016).

The literature also contains important controversies and cautionary cases. In the normal-superconductor phase transition of a type-I cylinder in a magnetic field, one paper argues that the total Joule heat

$_2$10

is fixed by thermodynamics, independent of the normal-state resistivity, and generated only in the normal region. The same work further argues that the additional dissipation in the superconducting boundary layer predicted by conventional theory would contradict this thermodynamic result (Hirsch, 2020). That claim is presented in the source as a challenge to conventional superconductivity theory rather than as a settled consensus.

At the opposite scale, Joule heating also appears as a theoretical mechanism in hot and dense matter. In a magnetized plasma with nonzero electron chiral chemical potential, the CME-induced electric field produces a heating rate

$_2$11

The paper argues that even modest $_2$12 can deposit QCD-scale energy densities on millisecond or second timescales under strong magnetic fields, making CME-driven Joule heating potentially relevant to proto-neutron stars and neutron-star mergers (Sen et al., 30 Sep 2025).

Across these disparate settings, the principal methodological lesson is stable. Time-resolved in situ Joule heating is most informative when it is treated as a coupled electrothermal problem with explicit attention to heat generation, thermal escape channels, and the distinction between electron, lattice, and environmental temperatures. Conversely, when those distinctions are neglected, Joule heating can masquerade as bias-induced quantum suppression, current-induced phase stabilization, or other ostensibly intrinsic nonequilibrium phenomena (Kölling et al., 16 May 2025, Fang et al., 16 Feb 2025).