*3.5. Time-Dependent Thermal Transport Theory*

Modeling thermal transport and thermoelectric effects at the nanoscale is difficult, because besides the intricacies of certain materials, we lack a working definition of temperature at those scales. Indeed, the concepts of equilibrium and thermodynamic limits are difficult to extend to the case of a few electrons or phonons strongly localized in a nanodevice. Clearly, also the idea that the leads/reservoirs have a well-established temperature (distribution function) down to the contact with the device is an abstraction. Many attempts have been tried to remedy this situation and general extensions of some thermodynamics concepts to the nanoscale have been attempted [107,108].

One of such attempts is to completely remove the needs for macroscopic reservoirs and leads by replacing them with an effective radiation of known properties. One might think for example, of replacing the electrostatic bias with the time-dependent vector potential that produces the same electric field. A similar attempt can be made with the temperature gradient. We replace the coupling to the energy reservoirs by two black bodies each radiating according to their own temperature and spectral properties: the electromagnetic field of the blackbody radiation is then included in the system dynamics and establish in this way the needed temperature gradient and transport dynamics. To maintain a steady state and avoid that the energy stored in the device saturates, we assume that the system is weakly coupled to an external environment at a given temperature. This corresponds to an experimental set-up; the quantum systems gets excited out of its equilibrium by thermal radiation and at the same time can release energy to its surrounding. (such as a metal heated up by a laser and starts to glow). Recent progress has been made to include these quantum-electro-dynamical effects into an ab-initio formalism [109]. Being able to study and detect the radiation produced by the

transport dynamics could possibly allow the definition of an effective "local temperature" based on the electrodynamics theory.

We assume each blackbody to be located far away from the devices, therefore for all our purposes we will consider the radiation they produce as composed solely of plane waves propagating along the line connecting the device and the blackbody. In Figure 7 the considered nanodevice is shown and the left and right blackbody radiation is indicated by **A***L* and **A***R* respectively. While the blackbodies radiate according to their respective temperature *TL* and *TR*, the system is interacting with an environment at temperature *TE*. The amplitude of each plane wave of the blackbody field is determined by the temperature of the blackbody itself. We assume therefore the thermal radiation of the form

$$\mathbf{A}\_{L(R)}(\mathbf{r},t) = \mathbf{E}\_0 \int d\Omega \sqrt{\Omega n \left(\Omega\_r T\_{L(R)}\right)} \sin(kx \pm \Omega t + \varphi(\Omega)),\tag{57}$$

where *<sup>n</sup>*(<sup>Ω</sup>, *T*) is the Bose-Einstein distribution, **E**0 = *<sup>E</sup>*0**e***y*. Here, *E*0 is the strength of the corresponding electric field, where **<sup>e</sup>***y* is the unit vector in the y direction and *k* is the momentum of the photon with frequency Ω. We have considered a random phase, *ϕ*(Ω) ∈ [0, <sup>2</sup>*π*) that ensures the plane waves are incoherently reaching the device. In principle, there are more realistic, but numerically more expensive, ways to model the blackbody thermal radiation [110,111]. A comparison of these methods with the results of Equation (57) is still missing. Due to the coupling with an external environment, we expect that the system reaches a steady state. This coupling is effectively described through a master equation where the environmental power spectrum for |*ω*| < *ωc* is

$$\mathcal{K}(\omega) = 4|\omega|^3 n(|\omega|, T\_\mathcal{E}) + \theta(-\omega),\tag{58}$$

where *<sup>θ</sup>*(*ω*) is the Heaviside step function, and *ωc* is a cutoff frequency determined by the scale of the system. For |*ω*| > *ωc*, *<sup>C</sup>*(*ω*) = 0. We apply this model to the case of a device described in the tight-binding formalism where the electrons are not interacting. In this case, the Hamiltonian for the system is given by

$$H\_{\sf s} = \sum\_{\langle i,j \rangle} T\_{i,j} \mathbf{c}\_i^{\dagger} \mathbf{c}\_{j,i} \tag{59}$$

where the operator *cj* destroys an electron in the site *j* and *Ti*,*<sup>j</sup>* is the hopping parameters. The blackbody radiation enters here as a Peierls transformation into the Hamiltonian

$$T\_{i,j} = t\_{i,j} \exp\left[-i \int\_{\mathbf{R}\_i}^{\mathbf{R}\_j} \text{d}\mathbf{r}\left(\mathbf{A}\_L(\mathbf{r}, t) + \mathbf{A}\_R(r, t)\right)\right],\tag{60}$$

where *ti*,*j* is the hopping parameter in the absence of the radiation. We align the device along the direction joining the two blackbodies. If one considers initially the case where the blackbodies and the environment have the same temperature, we see that the dynamics of the system points towards a steady state with no net energy transport in the long-time limit [109].

**Figure 7.** Set-up under consideration. Six tight-binding sites are connected locally to the radiation of two blackbody at temperature, *TL* and *TR*. Furthermore, the system is connected to an environment at temperature *TE*. Reprinted (adapted) from [109]. Copyright (2015) American Physical Society.

In the case of a finite temperature gradient we observe in Figure 8 a turnover of the energy current, *jT*, with respect to the flux of energy from the blackbodies. Here, *S* is the time-averaged Poynting vector indicating the coupling strength to the blackbodies. Indeed, for small fluxes, the energy current increases when increasing the flux, then reaches a maximum and later vanishes for intense radiations. The reason for this turnover lies in the finite number of states available to carry energy. For small fluxes almost all states are empty and there is the possibility to allocate for a larger number of carriers. When the flux increases, the states fill up, and less and less will be available for conduction, until essentially all the states will by full and no dynamics is possible [109]. We want to point out that this theory can be applied to large temperature gradients: the coupling with the black body radiation is not restricted to linear response, while the system is coupled weakly to the environment [109].

**Figure 8.** Thermal current along the *x* direction for different temperature gradients as a function of the strength of the coupling *S*. For large coupling, the current decreases since the energy level are all filled, and the dynamics is quenched. Reprinted (adapted) [109]. Copyright (2015) American Physical Society.
