{'fraction_non_alphanumeric': 0.028717948717948718, 'fraction_numerical': 0.01064102564102564, 'mean_word_length': 4.748710390567428, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Thermal Conductance of a metallic dot Single-Electron Transistor

Dots are ideal systems to study fundamentals on heat transfer at the nanoscale and promising nanoscale heat-engines and thermal devices. Here, we report on the validation of our theoretical model on the thermal conductance of a metallic dot single-electron transistor (md-SET) by a recent experiment on the low-T thermal conductance. We compare with the experiment, we emphasize the physics interpretation and characteristic values and we apply the model to evaluate the operation the md-SET as heat-switch. Perfect agreement is shown between the calculated and the measured charge conductance G, heat conductance \k{appa} and the ratio \k{appa}/GT. The experimental findings confirm the theoretical predictions on the periodicity of the Coulomb oscillations in the classical regime, the low-T extreme values and the high-T limits of G and \k{appa}. The calculated conductances of the md-SET are presented in universal curves from low-T to high-T. It is shown that the md-SET can efficiently operate as a heat switch at temperatures , Ec being the charging energy of the dot.

2 Dots have been extensively studied for many years. They still attract much scientific research interest because they are model systems to study quantum physics effects, interference effects in carriers' states, fundamentals of transport theory, effects of interactions, novel mechanisms and applications [1][2][3]  The charge and thermal transport properties of dots have been at the center of scientific research for more than three decades. The cooperative progress between theory and experiment in dot-based nanostructures has been proved essential for understanding fundamentals on combined charge and heat transfer and designing applications and devices [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] . Here, we report on the experimental validation of our previously published theoretical work on the thermal conductance of a dot coupled with two electrodes with tunneling barriers 14,21 . The thermal conductance of a metallic dot SET (md-SET) has been recently measured 22 . We compare the calculations using our theoretical formalism in the classical regime that is appropriate for the md-SET, with the experimental findings. Perfect agreement is shown between the calculated and the measured charge conductance G, heat conductance κ and the ratio κ/GT. The experimental findings confirm the theoretical predictions on the periodicity of the Coulomb oscillations in the classical regime, the low-T extreme values and the high-T limits of G and κ. The theoretical model is applied to evaluate the operation of the md-SET as heat-switch.

3
The transport properties of dots are closely related with discrete states of carriers.
Interactions between charge carriers can result in discretization of their energy states.
A characteristic example is the transfer of an electron into an uncharged dot. In this case, the electrostatic energy of the dot with capacitance C increases by e 2 /2C due to the Coulomb interaction. Transport is blocked unless the required energy is provided by the thermal energy, a voltage bias applied between the electrodes or an external field as that of a gate electrode. This is the well-known Coulomb blockade effect resulting from the Coulomb interaction between electrons 23 . Due to this effect the charge conductance of a dot shows oscillations 4 . The spectrum of the oscillations depends on the charging energy E c =e 2 /C and the energy spectrum of the dot. In 'quantum dots', the energy spectrum is discrete and the separation of subsequent Coulomb blockade oscillations is determined by ΔΕ+e 2 /C, ΔΕ being the energy level separation 4,24-27 . In 'metallic' dots, the discreteness of the energy spectrum is screened and the charge conductance shows periodic Coulomb blockade oscillations with period equal to the charging energy E c [28][29] . The thermal conductance of a dot in the presence of the Coulomb effect was calculated in Refs. [14,21] from the quantum limit Distinct behavior was found for the different transport regimes. In a quantum dot, the thermal conductance is more sensitive to the dot energy spectrum than the charge conductance: the separation between the peaks and the magnitude of the peaks of the Coulomb oscillations depend on the separation of neighboring energy states. In the case of a metallic dot, the thermal conductance oscillations show the same features as the charge conductance: the periodicity of the oscillations is determined by the charging energy and the magnitude of the peak depends on the density of states.
The thermal conductance κ of a dot coupled to two electron reservoirs (electrodes) is defined as: where Q is the heat flux and ΔT is the temperature difference between the two electrodes. In the second part of equation (1), the thermal conductance is related to the transport coefficients that are calculated from the linearized expressions for the electric current 4 and the heat flux Q 14 . For a metallic dot, the energy levels separation is smaller than the thermal energy (ΔΕ<<k B T) and the 'classical regime' formalism describes the charge conductance G 4 the thermopower S 5 and the thermal coefficient where and In the above expressions, μ is the chemical potential of the dot in equilibrium, ρ is the density of states of the dot, γ is the tunneling probability and , where C is the total capacitance of the 5 dot and ext  is for the contribution of external fields such as the gate in the case of the SET.
In a metallic dot, the discreteness of the energy spectrum is screened by the thermal broadening and the transport properties depend on the density of states ρ of the dot and on the tunneling probability γ through the barriers separating the dot from the electrodes (the source and the drain). This is explicitly shown in equations (2) and for the thermal conductance 14 In this regime, the thermal energy provides enough kinetic energy to the carriers so As temperature decreases and the thermal energy becomes comparable to the charging energy, ( Bc k T E ), the kinetic energy of the carriers also decreases and they start to 'see' the Coulomb barrier and their transport properties start fluctuating with E F (Figure 1). Now, the energy required to overcome the Coulomb barrier is provided by the external field/gate that shifts E F relative to the charging states of the dot. In  At low-T, the thermal conductance oscillates between a vanishing value at the conduction valleys and a maximum at the conduction peaks. In this regime, the md-SET operates as a heat-switch. The switching ability of the md-SET, can be evaluated by the switching ratio R SW defined as The calculated ratio R SW is shown in Figure 3 as a function of the thermal energy k B T with respect to the charging energy E c . R SW is ~1 at low-T where the thermal energy is small compared with the charging energy. The switching efficiency of the md-SET 11 decreases rapidly with increasing thermal energy. This is because the switching operation lies on a thermally activated process: heat conduction is thermally activated above the Coulomb barrier. The thermal activation is controlled by the capacitance C of the dot that determines the charging energy E c and therefore the maximum height of the energy barrier. In Figure 3, it can be seen that for k B T<0.1E C , R th-SW >0.5, which corresponds to switching efficiency of 50%. The temperature 0.1 /  for the charge and the thermal conductances provide guidance for applications of the md-SET. The thermal conductance curves have been used to evaluate the md-SET as heat-switch. Efficient operation has been shown below a threshold temperature that is determined by the metallic dot capacitance.