Theory of shift heat current and its application to electron-phonon coupled systems

We propose a heat current analog of the shift current, “shift heat current.” We study nonlinear heat current responses to an applied ac electric field by a diagrammatic method and derive a microscopic expression for the second-order dc heat current response. As a result, we find that the shift heat current is related to the shift vector, a geometric quantity that also appears in the expression for the shift current. The shift heat current directly depends on and can be controlled through the chemical potential. In addition, we apply the diagrammatic method to electron-phonon coupled systems, and we find that even if only the phonons are excited by an external field, the amplitude of the shift heat current is determined by the energy scale of electrons, not of phonons.

Figure 1

Schematic pictures of shift current and shift heat current. (a) Illuminating light of frequency $\omega$ induces dc electric current (shift current) and dc heat current (shift heat current). (b) In the momentum space, an incident photon excites an electron from the valence band (with dispersion ${\varepsilon }_v\left(\mathit{k}\right)$) to the conduction band [with dispersion ${\varepsilon }_c\left(\mathit{k}\right)$]. (c) In the real space, the center of a wave packet is shifted during the interband transition. This shift of the electron wave packet (indicated by the black arrow) induces an electric current, the shift current, and simultaneously induces the shift of energy of $({\varepsilon }_c+{\varepsilon }_v)/2-\mu$ (indicated by the red arrow), which results in the shift heat current.

Figure 2

The Feynman diagrams which contribute to shift heat current. Solid lines with an arrow and wavy lines represent propagators of electrons and photons, respectively, and black dots with $n$ photons denote vertices for $h^{{\alpha }_1\cdots {\alpha }_n}$ and cross dots with $n$ photons are the vertex for ${\tilde{g}}^{{\alpha }_1\cdots {\alpha }_n}$.

Figure 3

An illustration of the Rice-Mele model [Eq. (64)]. In order to break the particle-hole symmetry, next-nearest-neighbor hoppings $t_{AA},t_{BB}$ are included in the model.

Figure 4

The shift heat current and the shift current for the Rice-Mele model. We used the parameters: $t_{AB}=0.9t,t_{BA}=1.1t,t_{AA}=0.1t,t_{BB}=0.1t,r_A=0,r_B=0.5a,\mathrm{\Delta }=2t,{\varepsilon }_0=-0.20$ with lattice constant $a$ and an energy scale $t$. ${\varepsilon }_0$ is determined so that the middle of the minimum of the conduction band and the maximum of the valence band is zero. The temperature $T$ is set to be zero ($T=0$). The total number of the unit cells $N$ is set to be 10 001, and we set $\eta =0.01t$ so that $t/N\ll \eta \ll t$ with the energy scale $t$. (a) The band structure of the Rice-Mele model for the given parameters. The black, red, and blue dashed lines correspond to the chemical potential $\mu =0,-0.1t,0.1t$, respectively. (b) The response coefficients for shift heat current ${\alpha }^{xxx}(2i\eta ;\mathrm{\Omega }+i\eta ,-\mathrm{\Omega }+i\eta )$ and shift current ${\sigma }^{xxx}(2i\eta ;\mathrm{\Omega }+i\eta ,-\mathrm{\Omega }+i\eta )$ as a function of the photon energy $\hslash \mathrm{\Omega }$. ${\alpha }^{xxx}$ is calculated for $\mu =0,-0.1t,0.1t$ and ${\sigma }^{xxx}$ is calculated for $\mu =0$.

Figure 5

The Feynman diagrams which describe the phonon-induced shift heat current. Here, curly lines are phonon propagators and open dots are the electron-phonon interaction. Solid lines represent propagators of electrons. Black dots and crossdots represent vertices for $h$ and $\tilde{g}$ as in Fig. 2.