Direct observation of hot-electron-enhanced thermoelectric effects in silicon nanodevices

The continuous downscaling of silicon (Si) technology in modern electronics leads to a situation where electrons in nano-sized transistors are far from thermal equilibrium with the lattice. This results in a significantly higher effective electron temperature (Te) compared to the lattice temperature (Tl)1,2. The energy relaxation of these high-density electron hotspots creates a substantial heat load on chips, limiting their functionality and performance3. Effective thermal management is thus crucial for post-Moore nanoelectronics4. Current approaches primarily rely on passive cooling, which depends on the intrinsic thermal conductance of materials. However, the limited thermal conductance of materials (e.g., Ks ~150 Wm−1K−1 at 300 K)8,9 and the presence of interfacial thermal resistance significantly hinder efficient heat dissipation, making external cooling less effective. Therefore, innovative cooling solutions are urgently needed. Semiconductor thermoelectric (TE) cooling offers an active approach, where heat flow is generated along with the electrical current through the Peltier effect (QPeltier = πI, where π is the Peltier coefficient)10–18. This method isn't limited by material properties. State-of-the-art thermoelectrics typically use two dissimilar materials with different Seebeck coefficients to generate a large temperature gradient and substantial cooling power19–21. While successful at a macroscopic scale, this approach may not directly translate to the nanoscale because electron temperature can significantly exceed lattice temperature (Te ≠ Tl), and highly localized hotspots create extreme temperature gradients and current densities beyond what's seen macroscopically. For microscopic TE applications, single-material TEs offer advantages in nanodevice design and material compatibility with Si technology. The theoretical understanding of TE Peltier effects under nonequilibrium conditions (Te ≠ Tl) has been explored23,25,31–33, predicting a nonlinear dependence of π on current density25,31–33, potentially enhancing cooling performance under high electric fields. Experimental verification has been limited due to the challenges in nanothermometry34,35, although nonlinear phenomena were observed in graphene nanostructures and attributed to electron wind effects or van der Waals barrier effects34. Direct experimental evidence of hot electrons and quantitative studies of the Peltier coefficient under nonequilibrium conditions remain scarce.

Previous research has explored thermoelectric effects in miniatured transistors, particularly focusing on nanothermometry to study microscopic temperature profiles in various materials such as metals, semiconductors, two-dimensional materials, and molecular junctions. However, direct observations of thermoelectric effects (like Peltier and Thomson effects) in silicon—crucial for on-chip refrigeration using silicon itself—had not been addressed prior to this study. Theoretical work had predicted nonlinear thermoelectric effects under nonequilibrium conditions, where the electron temperature significantly differs from the lattice temperature. These theories suggested a redefinition of the Peltier coefficient and a nonlinear current dependence, potentially leading to enhanced cooling. Limited experimental evidence existed, mainly from studies in materials other than silicon and often relying on indirect observations or interpretations.

This study employed a nanoconstriction structure fabricated in phosphorus-doped silicon films to generate highly localized electron hotspots. The device consists of a narrow channel (approximately 400 nm wide) designed to concentrate the electric field. Atomic force microscopy (AFM) was used to characterize the device's structure. The research used a combined approach of scanning noise microscopy (SNoiM) and scanning thermal microscopy (SThM) to separately measure the electron temperature (Te) and lattice temperature (Tl) distributions. SNoiM, a non-contact radiative electronic nanothermometry technique, detects near-field fluctuating electromagnetic fields (around 20.7 ± 1.2 THz) from hot electrons, providing Te information. SThM, a contact nanothermometry technique, uses a scanning thermocouple tip to measure Tl. Measurements were conducted at room temperature and atmospheric pressure. Both AC square-wave and sinusoidal wave bias voltages were applied to the device, enabling the decoupling of Joule heating and thermoelectric signals using a lock-in technique with harmonic analysis. The first and second harmonic responses were used to separate thermoelectric and Joule signals, respectively. The study also involved numerical simulations using a two-temperature model to analyze thermal transport in the device, considering the decoupling of electron and lattice subsystems, classical transport laws (accounting for non-Fourier heat transport), and isotropic heat diffusion. The model solves the heat transport equation for the electron subsystem, incorporating the Peltier effect. The nonequilibrium Peltier coefficient was evaluated considering the local average kinetic energy of electrons, incorporating drift and diffusion components. Finite element method simulations were used to solve the two-temperature model equation to extract electron and lattice temperature profiles, which were then used to compute Joule heating and thermoelectric cooling/heating. The simulations were validated against experimental results, comparing the Joule and thermoelectric signal profiles and their dependence on bias voltage.

The experimental results revealed a significant temperature difference between the electron and lattice subsystems within the nanoconstriction, with Te reaching ~1500 K while Tl remained around 320 K. The spatial mapping of Joule heating and thermoelectric effects showed that Joule heating was concentrated within the constriction, while thermoelectric cooling/heating occurred on either side of the constriction, with a clear change in sign at the center. This sign change reversed when the current direction was reversed, demonstrating a current-direction-dependent thermoelectric effect. Quantitative analysis revealed that Joule heating exhibited a quadratic dependence on current (consistent with Joule's law), while the thermoelectric cooling/heating showed a distinctive cubic dependence on current. This cubic dependence is significantly different from the linear dependence expected under local thermal equilibrium conditions. Numerical simulations, employing a two-temperature model and considering the non-equilibrium Peltier coefficient, were in excellent agreement with the experimental data. Both simulated and experimental results showed a linear increase in the ratio of thermoelectric to Joule signals with increasing bias voltage. Further analysis using sinusoidal wave modulation confirmed the cubic current dependence of the thermoelectric effect and revealed the presence of both first and third harmonic components in the thermoelectric signal, with the first harmonic being three to four times stronger than the third harmonic. The study ruled out alternative explanations, such as energy filtering, thermionic emission cooling, boundary scattering, and the conventional Thomson effect, demonstrating the dominant role of the nonequilibrium thermoelectric effect.

The observed cubic dependence of the thermoelectric effect on current, along with the excellent agreement between experimental and simulation results, provides strong evidence for the dominance of nonequilibrium thermoelectric effects driven by hot electrons. The results indicate that the Peltier coefficient under these nonequilibrium conditions is significantly modulated by the electron temperature, leading to the nonlinear behavior observed. This study directly demonstrated the potential for enhancing thermoelectric performance through hot-electron effects. The exceptionally high electron temperature gradients achieved in this study are not achievable in macroscale devices, suggesting that scaling down to the nanoscale plays a critical role in this enhanced effect. This is also why several other potential mechanisms were ruled out during the analysis. The observed phenomena open up new possibilities for on-chip thermal management strategies in advanced nanoelectronics.

This research provides direct experimental and computational evidence for hot-electron-enhanced thermoelectric effects in silicon nanodevices. The observed cubic current dependence of the thermoelectric effect, significantly deviating from conventional TE behavior, highlights the potential of exploiting nonequilibrium hot carriers for enhanced on-chip cooling. Future research could explore different methods to further enhance the thermoelectric properties, such as using laser-induced temperature gradients or exploring intervalley-assisted energy transport. This work contributes significantly to the understanding and advancement of nanoscale thermal management.

The study focused on a specific type of silicon nanodevice with a particular geometry and doping level. The generalizability of these findings to other Si nanostructures or different materials needs further investigation. The two-temperature model used in the simulations simplified some aspects of the complex energy transport processes. A more detailed model incorporating phonon dynamics could provide deeper insights into the underlying mechanisms. The experimental techniques used, while advanced, have inherent limitations in spatial resolution, which could influence the precise determination of temperature gradients.