Introduction

Phonon transport, governing the heat conduction in nonmetallic crystalline materials, has garnered widespread attention in various advanced technologies, including thermal management1,2, energy conversion3, and the design of materials for extreme environments4. Effectively manipulating phonon transport is highly desirable for enhancing the functionalities and performance of these applications, yet its realization poses significant challenges5,6. Strategies to manipulate phonon transport typically involve altering specific phonon scattering processes, consequently leading to changes in phonon thermal conductivity6,7. In crystalline materials, the primary scattering processes for phonons include phonon-boundary scattering, phonon-phonon scattering, phonon-impurity/defect scattering, and phonon interactions with electrons8,9. Extensive prior research has attempted to alter phonon-boundary/phonon/impurity/defect scattering by modifying lattice structures, introducing approaches such as nanostructure construction10,11, pressure tuning12, strain engineering13, and impurity/defect engineering14,15. These approaches have achieved alterations in phonon thermal conductivity. However, due to changes in intrinsic material properties6,16,17 and the typically irreversible nature of lattice structural modifications18, active and reversible phonon transport manipulation proves exceptionally challenging.

A more promising approach for achieving reversible phonon transport manipulation is by adjusting the phonon-electron scattering processes19,20, as electrons exhibit a rapid response to the external field and the lattice structure remains nearly unchanged21. Prior theoretical studies have confirmed the significant impact of phonon-electron scattering on phonon thermal conductivity in semiconductor materials22,23. Especially in two-dimensional (2D) semiconductor materials which have broad prospective applications in electronic devices, optoelectronic devices, and energy conversion24,25,26,27, the influence of electron-phonon scattering on phonon thermal conductivity can be comparable to that of phonon-phonon scattering even at room temperature28 and moderate carrier concentrations29,30. However, to date, effective experimental approaches for utilizing the electrical response characteristics of electrons to manipulate phonon transport remain unexplored.

Here, we achieve active and reversible manipulation of phonon transport in a 2D semiconductor MoSe2-WSe2 heterojunction by modulating the electron-phonon scattering processes through tuning bias voltages. We construct the heterojunction with electrical rectification effects using atmospheric pressure chemical vapor deposition method. Phonon thermal conductivity values of the heterojunction under electrical forward bias and reverse cutoff are measured using the H-type sensor method. The results indicate that the thermal conductivity under forward bias is lower than that under reverse cutoff. This difference amplifies with decreasing temperature, reaching a 13.8% decrease in thermal conductivity under forward bias compared to reverse cutoff at an ambient temperature of 258 K. The thermal conductivity is actively and reversibly manipulated by changing the bias voltage magnitude and direction over the applied voltage range of −20 to +15 V in the experiment. By combining device simulation with an effective first-principles-based model, we find this manipulation can be theoretically explained as varying strengths of phonon-electron scattering under different bias voltages.

Results

Fabrication and thermal conductivity measurement of MoSe2-WSe2 lateral heterojunctions

As shown in Fig. 1a, b, the monolayer MoSe2-WSe2 lateral heterostructure samples were synthesized by atmospheric pressure chemical vapor deposition (AP-CVD) route as reported in our previous work31,32, and a higher current density was achieved by interface doping. We deposited two gold sensors with a thickness of 100 nm on top of the samples by physical vapor deposition (PVD), and the sensors were designed in an H-shape to facilitate the measurement of their thermal conductivities33. As a comparison, the monolayer pure MoSe2 and monolayer pure WSe2 were synthesized via the chemical vapor deposition (CVD) technique34,35, shown in Fig. 1c, d. Subsequently, a 370-nm-thick polymethyl methacrylate (PMMA) film was spin-coated on top of the samples for transfer to the suspended H-type gold electrodes. To avoid potential damage caused by the limited mechanical strength of transition metal dichalcogenide (TMD) materials, the PMMA layers atop the pure MoSe2 and pure WSe2 were preserved to serve as a supportive matrix, with their triangular profiles clearly shown in the SEM images (Fig. 1c, d). Similarly, a 300-nm-thick SiO2 layer film was underneath the MoSe2-WSe2 heterostructure to provide necessary support. Moreover, the silicon substrate underlying all the samples was deeply etched to a depth exceeding 2.5 μm, ensuring the sensor remained suspended and avoiding heat leakage from the silicon substrate. This design feature enhanced the gold sensor’s responsiveness to heat flow passing through the sample, thereby optimizing its sensitivity for thermal detection applications.

Fig. 1: Experimental measurement circuit and thermal conductivity changes of MoSe2-WSe2 lateral heterojunctions under bias currents.
Fig. 1: Experimental measurement circuit and thermal conductivity changes of MoSe2-WSe2 lateral heterojunctions under bias currents.
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a Schematic illustration for measuring the thermal conductivity. V1, V2, and V3 are used to measure the voltage applied to the two gold sensors and the bias voltage across the sample respectively. V1′, V2′, and V3′ are used to measure the voltage of the three standard resistors respectively in order to obtain current in each circuit. By controlling the value of V3, zero voltage bias, forward voltage bias and reverse voltage bias can be achieved. b Scanning electron microscope (SEM) image of the monolayer MoSe2-WSe2 lateral heterostructure sample (Sample #1). SEM images of monolayer pure MoSe2 (Sample #2) (c) and monolayer pure WSe2 (Sample #3) (d), respectively. The measured thermal conductivities of MoSe2-WSe2 lateral heterojunction (e), pure MoSe2 (f), and pure WSe2 (g) at different bias voltages, within a temperature range of 258–318 K, respectively. In each figure, the circle, upper triangle, and lower triangle patterns represent zero bias voltage, reverse bias voltage, and forward bias voltage respectively. κ0 corresponds to the thermal conductivity under reverse bias at 288 K. Error bars show experimental uncertainties of ±5%. The shaded region in (e) highlights the difference in thermal conductivity under forward and reverse bias conditions. Part (b) reprinted with the permission from ref. 31, AAAS.

The measurement methodology is predicated on the utilization of two gold sensors, which serve as heaters while concurrently delivering precise temperature measurements (temperature resolution reaches 0.01 K). In this experimental setup, a large current passed through one gold sensor to heat the sample, while a comparatively weaker current was applied to the other sensor to monitor the attendant temperature increase. The thermal conductivity of the sample was subsequently determined through the meticulous analysis of the temperature signal using finite element simulations of the heat conduction equation. Furthermore, the thermal transport properties of the PMMA and SiO2 layers are also included in the numerical simulations, and these considerations did not adversely affect the thermal conductivity analysis conducted in this study (“Methods” and Supplementary Note 1). Notably, since the electron thermal conductivity in the heterojunction is minimal31, the measured lattice thermal conductivity is attributed to phonon thermal conductivity.

In order to investigate the thermal conductivity of the whole heterojunction under different bias conditions, an additional voltage (V3 in Fig. 1a) was introduced across the sample to inject carriers. The thermal conductivity was evaluated under applied bias voltages of 0 V, −9.2 V, and +13.3 V within a high vacuum chamber maintaining a pressure of 10−4 Pa, as illustrated from e to g in Fig. 1. The thermal conductivity at 288 K under reverse bias condition was determined for the MoSe2-WSe2 heterojunction, as well as for the pure MoSe2 and pure WSe2, to be 15.7 W/mK, 42.2 W/mK, and 45.0 W/mK, respectively. These values were adopted as a benchmark for normalizing the thermal conductivity measurements under varying conditions. The experimental data revealed that for both pure MoSe2 and pure WSe2, the alteration in thermal conductivity due to the application of reverse and forward bias voltages relative to the zero-bias condition does not exceed 3%. This finding indicates that the thermal conductivity of these two pure materials is essentially unchanged under the influence of an electric field applied in this study and is insensitive to the direction of the electric field.

Upon the formation of a heterojunction between MoSe2 and WSe2, a notable transformation in their behavior is observed. Prior studies have documented that the MoSe2-WSe2 heterojunction forms a natural p-n junction with a pronounced electrical rectification behavior31. Thermal conductivity measurements indicate that the discrepancy in thermal conductivity between the material under reverse bias (electrical cutoff state) and under zero bias conditions is minimal, remaining within a 3% threshold, as anticipated. Conversely, under forward bias conditions (electrical conduction state), the thermal conductivity of the sample significantly decreases compared to that under zero bias and reverse bias. This reduction becomes more pronounced with decreasing temperature. At 258 K, the thermal conductivity under forward bias is 13.8% lower than that under reverse bias. Such findings suggest that the direction of the applied bias voltage effectively alters the thermal conductivity of the MoSe2-WSe2 heterojunction. To further substantiate the active and reversible nature of our manipulation method, another Sample #4 is used to investigate the effects of different bias voltages on the thermal conductivity of the heterojunction. The dynamic phonon transport modulation is achieved over a wide range of bias voltages, from −20 V to +15 V (Supplementary Note 2).

Device simulation for MoSe2-WSe2 heterojunctions

To elucidate the experimental phenomenon of reduced thermal conductivity at electrical forward bias, we propose a first-principles-based computational model. For 2D semiconductor materials under moderate doping conditions, lattice thermal conductivity is primarily influenced by phonon-phonon and phonon-carrier (i.e., electron and hole) scattering22,36. The expression for lattice thermal conductivity is given by \({\kappa }_{{lat}}={\sum }_{\lambda }{c}_{V,\lambda }{v}_{\lambda }^{2}{\tau }_{\lambda }\), where \({c}_{V,\lambda }\) and \({v}_{\lambda }\) are the volumetric heat capacity and group velocity of a specific phonon mode λ. \({\tau }_{{{{\rm{\lambda }}}}}\) is the phonon relaxation time, related to the phonon-phonon scattering rate \(1/{\tau }_{\lambda }^{{pp}}\), phonon-electron scattering rate \(1/{\tau }_{\lambda }^{{pe}}\) and phonon-hole scattering rate \(1/{\tau }_{\lambda }^{{ph}}\). Based on Matthiessen's rule, the phonon relaxation time is shown as \(1/{\tau }_{\lambda }=1/{\tau }_{\lambda }^{{pp}}+1/{\tau }_{\lambda }^{{pe}}+1/{\tau }_{\lambda }^{{ph}}\). The phonon-phonon scattering rate is primarily associated with the lattice vibration, while the phonon-electron (hole) scattering rate also depends on the electron (hole) states. The expression for the phonon-electron (or phonon-hole) scattering rate, based on first-principles Boltzmann transport theory, is as follows9,

$$\frac{1}{{\tau }_{\lambda }^{{pe}/{ph}}}=\frac{2{{{\rm{\pi }}}}}{{{\hslash }}}{\sum }_{{{{\bf{k}}}},m,n}{\left|{g}_{m{{{\bf{k}}}}+{{{\bf{q}}}},n{{{\bf{k}}}}}^{\lambda }\right|}^{2}\left({f}_{n{{{\bf{k}}}}}-{f}_{m{{{\bf{k}}}}+{{{\bf{q}}}}}\right)\times {{{\rm{\delta }}}}\left({{{{\rm{\varepsilon }}}}}_{n{{{\bf{k}}}}}-{{{{\rm{\varepsilon }}}}}_{m{{{\bf{k}}}}+{{{\bf{q}}}}}+{{\hslash }}{\omega }_{\lambda }\right),$$
(1)

where \(\left|{g}_{m{{{\bf{k}}}}+{{{\bf{q}}}},n{{{\bf{k}}}}}^{\lambda }\right|\) is the electron-phonon coupling matrix element, calculated by the density functional perturbation theory (Methods). ε and ℏω are the energy for the electrons and phonons in a scattering event. \({f}_{n{{{\bf{k}}}}}\) and \({f}_{m{{{\bf{k}}}}+{{{\bf{q}}}}}\) are the electron distributions related to temperature, carrier concentration, and external field in the heterojunction system.

To determine the carrier concentration and electric field distribution, we first performed the device simulation on the 2D MoSe2-WSe2 heterojunction system based on the experimental structure (Fig. 1b), as illustrated in Fig. 2a (Detailed settings discussed in Methods). The n-type MoSe2 and p-type WSe2 form a natural p-n junction with type-II band alignment at the interface (Supplementary Fig. S3), and the simulated current-voltage (I–V) curves at room temperature quantitatively match the experimental measurements (Supplementary Fig. S4). The carrier concentration in the MoSe2-WSe2 heterojunction at zero bias (Fig. 2c), reverse bias (Fig. 2b), and low forward bias (at Va = 0.2 V, Supplementary Fig. S5) approximately follow the carrier concentration distribution characteristics of an ideal PN junction37. However, under larger forward bias conditions (Fig. 2d), the electron and hole concentrations in the MoSe2 and WSe2 regions increase significantly, exceeding the evenly doping concentration (2 × 1010/cm2), indicating a high injection regime38. The average concentrations of electrons and holes in the MoSe2 region are 5.16 × 1010/cm2 and 4.07 × 1010/cm2, respectively, while in the WSe2 region, they are 3.16 × 1010/cm2 and 4.06 × 1010/cm2, significantly higher than at zero bias and reverse bias (Supplementary Table. S2).

Fig. 2: Device simulation for MoSe2-WSe2 lateral heterojunction.
Fig. 2: Device simulation for MoSe2-WSe2 lateral heterojunction.
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a Schematic illustration of the simulated device structure. The MoSe2 and WSe2 regions are both 2 μm long and 6 μm wide. Gold electrodes are applied to both sides of MoSe2 and WSe2. A bias voltage Va is applied. Spatial distribution of electron and hole concentrations along the x direction (cross-section at y = 0) under reverse bias (b, Va = −9.2 V), zero bias (c, Va = 0 V), and forward bias (d, Va = 13.3 V) conditions at 258 K. e Schematic diagram of the equivalent electron temperature model. Under forward bias conditions, electrons gain energy in the electric field and transfer the energy to phonons. The electron temperature Te is determined based on the power density of Joule heating and the phonon-electron scattering rate can be calculated. The phonon-phonon scattering rate is calculated at the ambient temperature.

Furthermore, under large forward bias conditions, there exists an electric field (~104 V/cm, supplementary Fig. S6) within the heterojunction, where electrons acquire energy and generate a current in response to the external electric field. Due to the complexity in accurately describing electron behavior in heterojunctions21, we adopt the equivalent electron temperature model, which is well-accepted in semiconductor device simulations37,39, illustrated in Fig. 2e. In this model, the electrons (or holes) are assumed to be in equilibrium at an equivalent electron temperature Te, representing the energy acquired under an electric field. At zero bias or reverse cutoff, where minimal current is generated, electrons (or holes) and phonons are essentially in thermal equilibrium. Hence, the electron and phonon temperatures (Tlat) are nearly equal to the ambient temperature (Te = Tlat). However, under forward bias conditions, electrons gain energy and occupy higher energy state, while phonons can be considered to remain at the ambient temperature, resulting in Te > Tlat.

Thermal conductivity calculations for MoSe2 and WSe2 regions in the heterojunction

To quantitatively analyze the impact of carrier concentrations and electron temperatures on the thermal conductivity, we first divide the heterojunction into two regions: n-type MoSe2 and p-type WSe2, and separately consider their thermal conductivities at an ambient temperature of 258 K. Notably, the carrier concentrations and electron temperatures are obtained based on device simulation results of the heterojunction. In the reverse bias condition, the primary charge carriers are electrons in MoSe2, and holes in WSe2 (Fig. 2b). The phonon-electron scattering rates in MoSe2 and the phonon-hole scattering rates in WSe2 can be computed using density-functional perturbation theory and maximally localized Wannier functions40,41,42. The anharmonic lattice dynamics are employed for phonon-phonon scattering rate calculations43 (“Methods” and Supplementary Note 4). The phonon-phonon and phonon-electron (hole) scattering rates under reverse bias conditions are shown in Fig. 3a, b. The phonon-phonon scattering rates in both materials exhibit an increase with phonon frequency, reaching magnitude on the order of 10−1 ps−1. Conversely, the phonon-electron (hole) scattering rates are relatively weak, being lower by more than two orders of magnitude compared to the phonon-phonon scattering rates, thus contributing less to the lattice thermal conductivity. Factoring in both phonon-phonon and phonon-electron (hole) scattering, the lattice thermal conductivity under reverse bias conditions is determined to be 73.2 W/mK for MoSe2 and 66.9 W/mK for WSe2. The experimentally measured thermal conductivities are lower than the simulation values due to the increased phonon scattering channels provided by the supporting SiO2 layer44.

Fig. 3: First-principles calculation for MoSe2 and WSe2 regions in the heterojunction.
Fig. 3: First-principles calculation for MoSe2 and WSe2 regions in the heterojunction.
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Phonon-phonon scattering rates (3ph), phonon-electron scattering rates (Ph-el), and phonon-hole scattering rates (Ph-hole) at 258 K under forward bias and reverse bias conditions (a, MoSe2, b, WSe2). Each point in the figures represents a phonon mode. The x-axis represents the frequency of the phonon mode relative to the maximum frequency. c Electron temperature changes with Joule-heating power density under forward bias condition (Va = 13.3 V) at 258 K for MoSe2 and WSe2. The gray dashed line represents the experimentally measured average power density, 2.70×107 W/m2. d Thermal conductivities for MoSe2 and WSe2 regions calculated by first-principles methods change with ambient temperature. κf /κr represents the ratio of forward bias thermal conductivity to reverse bias thermal conductivity at the same ambient temperature.

In the forward bias condition, a substantial current density (48.7 μA) is achieved by applying a positive voltage of 13.3 V (Supplementary Fig. S7), and the electron and hole concentrations in both MoSe2 and WSe2 are significantly higher than those in the reverse bias condition (Fig. 2d). Phonon-phonon, phonon-electron, and phonon-hole scattering need to be simultaneously considered. To calculate the phonon-electron (hole) scattering rates, it is also necessary to determine the equivalent electron temperature (Te). In this heterojunction, where no luminescent effects or electromagnetic radiation were present, the energy supplied by the voltage is ultimately converted into Joule heat through electron (hole)-phonon interactions (Average power density: 2.70 × 107 W/m2, Supplementary Note 5). By utilizing first-principles method to compute the electron-phonon energy generation process, the relationship between Te and the Joule heating power density per unit area of the heterojunction is established (Methods), as shown in Fig. 3c. At an ambient temperature of 258 K and Va = 13.3 V, Te in MoSe2 is 330 K, while in WSe2 it is 563 K. The corresponding phonon-electron (hole) scattering rates for both materials are shown in Fig. 3a, b. Under the forward bias condition, there is a significant increase in the phonon-electron scattering rates compared to the reverse cutoff. For some phonon modes (near the Γ point, or around the normalized frequencies 0.4 and 0.8), the phonon-electron (hole) scattering rates are essentially equivalent to the phonon-phonon scattering rates. The total lattice thermal conductivity of MoSe2 is 60.7 W/mK, and that of WSe2 is 46.9 W/mK. Compared to reverse bias, the thermal conductivity under forward bias decreases by 17.1% in MoSe2 and 29.9% in WSe2, primarily due to the increased carrier concentrations and elevated electron temperatures (Supplementary Tables S2, S3 and S5).

Next, we consider the temperature dependence of the lattice thermal conductivity of MoSe2 and WSe2 regions in the heterojunction. With increasing ambient temperature from 258 K to 318 K, the difference between the forward bias and reverse bias thermal conductivities gradually diminishes Fig. 3d, \({\kappa }_{f}/{\kappa }_{r}\) increases and approaches 1.0, consistent with experimental trends. This can be partially attributed to the strong temperature dependence of phonon-phonon scattering45. Moreover, the rise in ambient temperature results in a decrease in carrier concentration at high forward bias conditions (Supplementary Tables S2 and S3). Consequently, as the ambient temperature rises, phonon-phonon scattering gradually dominates over phonon-electron (hole) scattering, and the values of thermal conductivity at forward bias and reverse bias tend to consistent. Notably, due to the lower carrier concentration (~108/cm2, Supplementary Fig. S10) and negligible electron temperature rise under forward bias conditions (Supplementary Note 6), the effect of heterojunction-induced reduction in thermal conductivity is difficult to directly observe in the pure material samples (Fig. 1f, g).

Temperature-dependent thermal conductivity calculations for MoSe2-WSe2 heterojunction

Next, we consider the thermal conductivity of the complete heterojunction. Due to interfacial doping, an interfacial region forms at the junction of MoSe2 and WSe2, contributing to interfacial thermal resistance (Rg), thereby influencing the overall lattice thermal conductivity of the heterojunction, as depicted in Fig. 4a. The direct measurement of interfacial thermal resistance in 2D lateral heterojunctions is challenging46. Therefore, we estimate that the interfacial thermal resistance is approximately 1 × 10−7 m2K/W at 258 K, based on experimental measurements of the thermal conductivity of pure MoSe2 and WSe2, as well as the heterojunction (Supplementary Note 7). With increasing temperature, the enhanced anharmonicity at the interface results in higher phonon transmission thereby reducing interfacial thermal resistance47,48. Based on the thermal conductivity measurements and molecular dynamics (MD) simulations49,50 (Supplementary Note 7), it is found that as the ambient temperature increased from 258 K to 318 K, the interfacial thermal resistance decreased by around 20%, consistent with previously reported temperature dependence trends for 2D heterojunctions47,51.

Fig. 4: Temperature-dependent thermal conductivity calculations for the heterojunction.
Fig. 4: Temperature-dependent thermal conductivity calculations for the heterojunction.
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a Thermal resistance-in-series model for the thermal conductivity of the heterojunction, with thermal resistances corresponding to MoSe2, WSe2, and the interface denoted as R1, R2, and Rg, respectively. b Comparison of calculated (Sim) and experimental (Exp) thermal conductivities under forward and reverse bias conditions at different ambient temperatures. The upper triangle and lower triangles represent the reverse bias (I-) and forward bias (I+) thermal conductivities, respectively. Error bars show experimental uncertainties of ±5%. The thermal conductivities in the experiment and simulation are normalized to the corresponding reverse bias thermal conductivities at 288 K (κ0).

Subsequently, using the thermal resistance-in-series model (Fig. 4a), we incorporate the calculated thermal conductivities of the MoSe2 and WSe2 regions, and the estimated temperature-dependent interfacial thermal resistance values to determine the total thermal conductivities of the heterojunction at different ambient temperatures and bias conditions. These results are compared with the experimental measurements, as shown in Fig. 4b. Both computational and experimental results demonstrate that at lower ambient temperatures (258 K), the thermal conductivity at forward bias in the heterojunction is lower than at reverse bias. As temperature increases, this difference between the two thermal conductivities gradually diminishes. Furthermore, the reverse bias thermal conductivity shows a slight increase with temperature. This phenomenon is attributed to the decrease in interfacial thermal resistance as the ambient temperature rises, thereby mitigating the inhibitory effect of temperature-dependent phonon-phonon scattering on thermal conductivity. These findings underscore the significant impact of interfacial effect on the thermal conductivity of the heterojunction. Due to the complexity of the interfacial structure, we only provide a preliminary quantitative exploration of its impact mechanism on thermal conductivity. Recent studies have proposed interfacial transport models for electron-phonon interactions48,52,53 and various methods for tuning interfacial thermal transport, including doping and surface modification54,55,56. Further research is required to gain a more comprehensive understanding of interfacial electron and phonon transport and to achieve more pronounced phonon transport manipulation in heterostructures based on advanced interfacial thermal tuning strategies.

In summary, the active modulation of phonon transport based on bias voltage is experimentally achieved and successfully applied in the MoSe2-WSe2 heterojunction. Under large forward bias, the lattice thermal conductivity of the heterojunction is significantly lower than that under zero and reverse bias. This effect becomes more pronounced as the temperature decreases. At 258 K, the thermal conductivity under forward bias decreases by 13.8% compared to that under reverse cutoff. Subsequently, based on device simulation in combination with first-principles method, calculations are performed for MoSe2 region, WSe2 region, and the complete heterojunction. We speculate that the bias-dependent thermal conductivity originates from a significant increase in the phonon-electron (or phonon-hole) scattering rate of the heterojunction under forward bias conditions. The increased carrier concentration and elevated equivalent electron temperature under high forward bias injection both contribute to the phonon-electron (or phonon-hole) scattering rate. Furthermore, our calculation illustrates the significant interfacial effects on the heat conduction of the heterojunction. The bias voltage-based manipulation method holds promise for achieving active and reversible modulation of phonon transport without affecting the material structure and has the potential to be extended to a wider range of materials or structural systems.

Methods

Thermal conductivity measurements

Prior to the measurement of the thermal conductivity of 2D materials, a meticulous calibration of the resistance temperature coefficient of two gold sensors was conducted. The ambient temperature of the experimental apparatus was stringently managed by a vacuum thermoelectric heating and cooling stage (INSTEC, TP102SV-PM-F8, ±0.001 K). By adjusting the electrical power supplied to the gold sensors and fitting the resistance-power curve, the resistance of the gold sensors at zero power was determined through extrapolation, which is also the resistance of the gold sensors at ambient temperature. Subsequently, the resistance temperature coefficient was determined by altering the ambient temperature within the controlled environment and by fitting the resistance-temperature curve.

In the experimental configuration, one gold sensor is actuated as a “heater” by applying a substantial current across its terminals, while another gold sensor is utilized as a “detector” by subjecting it to a minimal current. The Joule heat produced by the “heater” is propagated through the sample to the “detector,” resulting in a measurable temperature increase. The magnitude of this temperature increase is meticulously measured by monitoring the resistance changes of the gold sensor acting as the “detector”. By changing the Joule heating power on the “heater” and fitting the temperature curves of the detector and the heater, the temperature response coefficient β can be obtained. The physical significance of β is the value of the temperature increase of the “detector” for every 1 Kelvin increase in the temperature of the “heater”.

$$\beta=\frac{\Delta {T}_{{\rm{detector}}}}{\Delta {T}_{{\rm{heater}}}}$$
(2)

Eventually, a steady-state thermal conductivity model was developed within the COMSOL Multiphysics (Supplementary Note 1). The geometric dimensions of the model were meticulously extracted from a scanning electron micrograph of the test specimen. The thermal conductivity values for gold and silicon were incorporated into the model, having been previously measured and calibrated utilizing the methodologies described in a prior scholarly investigation. Experimental data for heating power served as the input for the model. In this configuration, the thermal conductivity of the sample itself represented the sole unknown variable. A consistent internal heat source was applied within the gold sensor to emulate the Joule heat generated by electric current passage. The thermal conductivity of the 2D material can be determined through the process of curve fitting the temperature response coefficient, β.

Device simulations

The device simulation is conducted using a modified drift-diffusion model and calibrated with the experimental measurement at room temperature (303 K). The MoSe2 and WSe2 regions are both 2 μm long and 6 μm wide based on the experimental SEM figure (Fig. 1b), doped as n-type and p-type, respectively, with a doping concentration of 2 × 1010 cm2. Gold electrodes are applied to both sides of MoSe2 and WSe2, each with a length of 0.5 μm. Metallic electrode contacts have minimal impact on the heterojunction’s electrical rectification effect31. Therefore, in the device simulation, the metal-semiconductor contacts are simplified to ideal Ohmic contacts. The electron affinities, dielectric constants, bandgaps, and carrier density of states for MoSe2 and WSe2 are obtained from experimental values or calculated values in relevant literature31,57,58,59,60 (Supplementary Table S4). The high-field saturation model is adopted for mobility61. Carrier generation and recombination processes are considered using the Shockley-Read-Hall (SRH) model and Auger recombination in the device simulation37.

First-principles calculations

The ground states of electrons and phonons in MoSe2 and WSe2 are computed with Quantum ESPRESSO package40. We employ scalar relativistic Optimized Norm-Conservinng Vanderbilt (ONCV) pseudopotentials with the Perdew–Burke–Ernzerhof form of the exchange-correlation functional62. DFT-D2 of the Grimme method is applied to describe the van der Waals (vdW) interactions63. The lattice constants for MoSe2 and WSe2 are both 3.32 Å, which is consistent with previous studies57. Approximately 15 Å of vacuum is set for the out-of-plane direction in our ground-state calculation. The cutoff energy of the plane wave is set to be 100 Ry and the convergence threshold for the electron energy is set to be 10−11 Ry for self-consistent calculation with a Monkhorst Pack k-point grid of 24 × 24 × 1. The harmonic force constants are obtained using density functional perturbation theory (DFPT) with a 6 × 6 × 1 q-point grid and a threshold of 10−16 Ry to guarantee the convergence.

The thermal transport properties are calculated with the anharmonic lattice dynamics approach using ShengBTE package43. We use 5 × 5 × 1 supercells with a cutoff of the sixth-nearest atom to calculate the third-order force constants using the finite-difference method. The mode-level phonon-phonon scattering rates are calculated under the relaxation time approximation (RTA) with 100 × 100 × 1 q-points grids. Although the iterative method has been demonstrated to be a more accurate approach for calculating phonon thermal conductivity, RTA ensures the effective extraction of scattering rates. The phonon-electron and phonon-hole scattering rates are calculated by the Electron-Phonon Wannier (EPW) package, which is based on the maximally localized Wannier functions (MLWFs) and generalized Fourier interpolation41,42. The initial coarse electron-phonon coupling matrix element is calculated with a 24 × 24 × 1 k-point grids and 6 × 6 × 1 q-point grids. Then the initial coarse matrix element is interpolated into dense 100 × 100 × 1 k/q-points grids to obtain the mode-level phonon-electron scattering rate. Note that the q-point grids for phonon-phonon scattering rate and phonon-electron scattering rate must be consistent for the total lattice thermal conductivity calculation \({\kappa }_{{lat}}={\sum }_{\lambda }{c}_{V,\lambda }{v}_{\lambda }^{2}{\tau }_{\lambda },\) where \(1/{\tau }_{\lambda }=1/{\tau }_{\lambda }^{{pp}}+1/{\tau }_{\lambda }^{{pe}}+1/{\tau }_{\lambda }^{{ph}}\). The details of first-principles calculations can be found in Supplementary Note 4.

Determination of equivalent electron temperature

The equivalent electron temperature (Te) can be determined from the electron-phonon energy generation process. Applying a forward bias voltage to the heterojunction results in a significant conduction current. The average power density (P) of the device is calculated as \(P={V}_{a}\times I/A\), where Va and I are the applied voltage and current, respectively (Supplementary Note 5). And A is the surface area of the heterojunction. Due to the interaction between electrons and phonons, the power density is eventually converted into Joule heat. Based on first-principles Boltzmann transport theory in combination with equivalent electron temperature model, the average power density, i.e., electron-phonon energy generation rate, can be calculated as39,64

$$P= \frac{4{{{\rm{\pi }}}}}{{{\hslash }}V} {\sum}_{{{{\bf{k}}}},m,n,{{{\rm{\lambda }}}}}{{\hslash }}{\omega }_{\lambda }{\left|{g}_{m{{{\bf{k}}}}+{{{\bf{q}}}},n{{{\bf{k}}}}}^{\lambda }\right|}^{2}\{{n}_{\lambda }\left({f}_{m{{{\bf{k}}}}+{{{\bf{q}}}}}\left({T}_{e}\right)-{f}_{n{{{\bf{k}}}}}\left({T}_{e}\right)\right)\\ +{f}_{m{{{\bf{k}}}}+{{{\bf{q}}}}}\left({T}_{e}\right)\left(1-{f}_{n{{{\bf{k}}}}}\left({T}_{e}\right)\right)\}{{{\rm{\delta }}}}({E}_{n{{{\bf{k}}}}}-{E}_{m{{{\bf{k}}}}+{{{\bf{q}}}}}+{{\hslash }}{\omega }_{\lambda }),$$
(3)

where \({n}_{{{{\rm{\lambda }}}}}\) is the phonon distribution, assumed at the ambient temperature. Note that for simplicity, in each material region, it is assumed that the electron and hole temperatures are the same, and only a single electron temperature is used to represent the average Joule heating effect. As shown in Fig. 3c, when the electron temperature is equal to the phonon temperature (ambient temperature), the electron-phonon energy generation rate is zero, indicating no net energy exchange between the electron and phonon systems. With an increase in electron temperature, the electron-phonon energy generation rate gradually increases, maintaining a monotonically increasing trend. As a result, there is a one-to-one correspondence between the electron temperature (Te) and the electron-phonon energy generation rate (P). Therefore, by calculating the average power density based on current-voltage relationship, the corresponding electron temperature can be determined. This calculation was implemented by modifying the source code of EPW package39.