Quantitative relationship between polarization differences and the zone-averaged shift photocurrent

A relationship is derived between differences in electric polarization between bands and the “shift vector” that controls part of a material's bulk photocurrent, then demonstrated in several models. Electric polarization has a quantized gauge ambiguity and is normally observed at surfaces via the surface charge density, while shift current is a bulk property and is described by shift vector gauge invariant at each point in momentum space. They are connected because the same optical transitions that are described in shift currents pick out a relative gauge between valence and conduction bands. We also discuss subtleties arising when there are points at the Brillouin zone where optical transitions are absent. We conclude that two-dimensional materials with significant interband polarization differences should have high bulk photocurrent, meaning that the modern theory of polarization can be used as a straightforward way to search for bulk photovoltaic material candidates.

Figure 1

(a) Top panel: Photoexcitation induces shift of the electron wave packet in real space. (a) Bottom panel: Rice-Mele (RM) tight-binding model. The unit cell of size $a$ has two sites and alternating hoppings $t_1=t/2+\delta /2$ and $t_2=t/2-\delta /2$. The distance between conduction and valence band centers is ${\overline{R}}_{cv}$. For $\delta =0$, ${\overline{R}}_{cv}=\pm a/2$ is ambiguous because the system does not break inversion symmetry. For $\delta >0$, the centers of charge move towards one another by a distance $d$. The polarization is $P_v\left(\delta \right)-P_v\left(0\right)=-ed=({\overline{R}}_{cv}-a/2)/2$. When a photon is absorbed the electron jumps to another atom a distance ${\overline{R}}_{cv}$ away. (b) Integral of the shift vector over the BZ and polarization difference. ${\overline{R}}_{cv}$ has an integer discontinuity at $\delta =0$. (c) Stream plot of the vector field ${\mathbf{R}}_{cv}=(R_{cv}^{kk},R_{cv}^{k\delta })$ which has vortex of charge $+1$ in this gauge-independent vector field (see main text). The discontinuity in ${\overline{R}}_{cv}$ is the charge of the optical zero. In the numerical examples $\mathrm{\Delta }>0$ and $t=e=a=1$.

Figure 2

Polarization and integrated shift vector in a three-band model, Eq. (13). We find jumps in $W_{12}$ indicating that no single gauge choice gives vanishing winding numbers over the parameter $\alpha$. However, $e{\overline{R}}_{12}=a(P_1-P_2)+W_{12}ea$ holds for all $\alpha$. We used parameters $B/A=0.5,e=a=1$ and $0<\varepsilon \ll 1$. In evaluating $P_1-P_2$, we adopted the gauge given in Eq. (E4) with ${\phi }_n=0$.

Figure 3

(a) polarizations of each band of model Eq. (E1) as a function of $\alpha$. (b) $W_{12}$ changes at optical zeros $\alpha =0,4\pi /3$ and at the inversion symmetric point $\alpha =5\pi /3$. (c) Gauge-invariant field $(R_{12}^{kk},R_{12}^{\delta k})$ showing the vorticity of the optical zeros giving the discontinuities of ${\overline{R}}_{12}$. The loop $\gamma ={\sum }_n{\gamma }_n$ encloses a vortex of charge +1 (see main text). One can check that ${\mathbf{R}}_{12}=0$ at $ka=0$, and hence it does not contribute to the path integral. We chose units such that $e=a=1$.