Geometric Photon-Drag Effect and Nonlinear Shift Current in Centrosymmetric Crystals

The nonlinear shift current, also known as the bulk photovoltaic current generated by linearly polarized light, has long been known to be absent in crystals with inversion symmetry. Here we argue that a nonzero shift current in centrosymmetric crystals can be activated by a photon-drag effect. Photon-drag shift current proceeds from a “shift current dipole” (a geometric quantity characterizing interband transitions) and manifests a purely transverse response in centrosymmetric crystals. This transverse nature proceeds directly from the shift-vector’s pseudovector nature under mirror operation and underscores its intrinsic geometric origin. Photon-drag shift current can be greatly enhanced by coupling to polaritons and provides a new and sensitive tool to interrogate the subtle interband coherences of materials with inversion symmetry previously thought to be inaccessible via photocurrent probes.

Figure 1

Comparison between (a) vertical transitions and (b) nonvertical transitions (e.g., from photon drag) between two bands. Dashed black contours denotes the Fermi surface (FS) in the valence band (blue pockets). Green contours are energy-momentum-conserving contours (ECs) for (a) vertical transitions ${\omega }_{c,\mathbf{p}}-{\omega }_{v,\mathbf{p}}-\omega =0$ and (b) nonvertical transitions ${\omega }_{c,\mathbf{p}+\mathbf{k}/2}-{\omega }_{v,\mathbf{p}-\mathbf{k}/2}-\omega =0$. For nonvertical transitions, only part of the EC that lies below the FS is optically allowed (right green arrow). Top: arrows from light to dark circles in upper panels represent transitions of electrons in energy-momentum space; in the lower panels, they illustrate real space displacements. Yellow and purple circles denote electron Bloch states with opposite wave vectors.

Figure 2

Schematic of graphene plasmon enhanced photon-drag effect. A centrosymmetric 2D or thin-film target material (blue layer) is stacked on top of a graphene monolayer (thin gray layer), with an insulating layer (purple layer) in between. A propagating graphene plasmon (GP, red curve) can induce nonvertical optical transitions and generates a shift current in the adjacent target material (that can be perpendicular to the GP propagation direction, i.e., a transverse photon-drag shift current). Fermi surface of the target crystal is tunable with top and bottom gates (thick gray layers).

Figure 3

(a) Weighted shift vector ${\mathbf{R}}^{\uparrow }(\mathbf{p})$ for the BHZ model at $T=0$ where the black contours denote EC contributing to the shift current dipole. The relative sizes and colors denote the relative magnitude of the vector field. Shift current dipole (picoamperes) (b) as a function of chemical potential $\mu$ at a fixed plasmon energy $\hslash \omega =200\mathrm{meV}$ [38], and (c) as a function of plasmon energy $\hslash \omega$ with a fixed $\mu =-\hslash \omega /2$ at different temperatures ($T=10$, 20, 40 K for blue, green, red curves). (d) Shift current calculated directly from Eq. (3) (solid curves) versus linear approximation from shift current dipole (dashed lines). Illustrative parameters used for the two-band model: $m_0=0.2\mathrm{eV}$, $c_x=6\mathrm{eV}{\AA }^2$, $c_y=3\mathrm{eV}{\AA }^2$, $v_x=1\mathrm{eV}\AA$, $v_y=0.2\mathrm{eV}\AA$, of a system with an inversion center and a mirror plane, e.g., a monolayer 1T’-${\mathrm{MX}}_2$ [44, 45, 46]. Here we used a plasmon electric field with $E=2000\text{V}{\text{cm}}^{-1}$ [40].