Photon drag effect in $\left({\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x\right){}_2{\mathrm{Te}}_3$ three-dimensional topological insulators

We report on the observation of a terahertz radiation-induced photon drag effect in epitaxially grown $n$- and $p$-type $\left({\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x\right){}_2{\mathrm{Te}}_3$ three-dimensional topological insulators with different antimony concentrations $x$ varying from 0 to 1. We demonstrate that the excitation with polarized terahertz radiation results in a dc electric photocurrent. While at normal incidence a current arises due to the photogalvanic effect in the surface states, at oblique incidence it is outweighed by the trigonal photon drag effect. The developed microscopic model and theory show that the photon drag photocurrent can be generated in surface states. It arises due to the dynamical momentum alignment by time- and space-dependent radiation electric field and implies the radiation-induced asymmetric scattering in the electron momentum space. We show that the photon drag current may also be generated in the bulk. Both surface states and bulk photon drag currents behave identically upon variation of such macroscopic parameters as radiation polarization and photocurrent direction with respect to the radiation propagation. This fact complicates the assignment of the trigonal photon drag effect to a specific electronic system.

Figure 1

ARPES investigation of the surface electronic structure of different TI samples, measured at low temperature $(T\approx 25$ K) using photon energy of $8.4$ eV. The spectra unambiguously prove the existence of topological surface states in each material. (a) Results for pure $n$-type ${\mathrm{Bi}}_2{\mathrm{Te}}_3$. The corresponding results for samples (b) BST306 and (c) BST641, i.e., the $n$-type $\left({\mathrm{Bi}}_{0.6}{\mathrm{Sb}}_{0.4}\right){}_2{\mathrm{Te}}_3$ and $\left({\mathrm{Bi}}_{0.57}{\mathrm{Sb}}_{0.43}\right){}_2{\mathrm{Te}}_3$ ternary TIs, (d) for sample BST508, i.e., the $n$-type 10-nm ${\mathrm{Bi}}_2{\mathrm{Te}}_3$/7.5-nm ${\mathrm{Sb}}_2{\mathrm{Te}}_3$ TI heterostructure, and (e) for sample BST305, i.e., the $p$-type ${\mathrm{Sb}}_2{\mathrm{Te}}_3$. The top panels depict the constant-energy contour at ${\varepsilon }_B=0$ with indicated crystallographic directions showing the hexagonal warping of the energy spectrum of the topological surface states. The middle panel illustrates the binding-energy-dispersion spectra $\varepsilon \left(k\right)$ map at $k_y=0$ along the $\overline{\mathrm{\Gamma }}M$ direction, where the upper part of the topological surface states is revealed while the Dirac point is buried in the bulk valence-band maximum. Additionally, the energy distribution curves, integrated over the entire image, are shown beside the middle panels. The bottom panels depict the respective momentum distribution curves. The ones for ${\varepsilon }_B=0$ are highlighted in red.

Figure 2

(a) X-ray-diffraction pole figure scan around the $(1,0,5)$ reflection of the ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ sample BST127 showing that one domain orientation (highlighted by the solid red line) dominates. It also reveals that the crystallographic axes lie parallel to the sample edges. Insets sketch domain orientations illustrated by the solid red line connecting the top Bi atoms in the top right panel and the side view of one quintuple layer (see the bottom right panel). (b) Photocurrent $J_x\left(\alpha \right)/I$ in ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ sample BST127 measured for front and back illumination at $T=296$ K. Solid lines show fits after Eq. (1). Note that the same dependencies are obtained after phenomenological [see Eq. (4)] and microscopic theory [see Eq. (19)] for the photogalvanic effect.

Figure 3

(a) Photocurrent $J_x/I$ measured in ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ sample BST127. (b) Photocurrent $J_y/I$ measured in $\left({\mathrm{Bi}}_{0.57}{\mathrm{Sb}}_{0.43}\right){}_2{\mathrm{Te}}_3$ sample BST641. Plots show the dependence of the photocurrent excited by normal incident radiation with $f=2.0$ THz on the azimuth angle $\alpha$. Angles of incidence $\theta =0$ and ${180}^{\circ }$ correspond to the front and back excitations, respectively. Solid lines show fits after Eq. (1). Note that the same dependencies are obtained after phenomenological [see Eq. (4)] and microscopic theory [see Eqs. (19) and (26) for the photogalvanic and the photon drag effect, respectively]. Insets sketch the setup and the orientation of the electric field. Note that the photocurrent is probed in the directions coinciding with the principal axes of the trigonal system. (c) Temperature dependence of the photocurrent measured in the ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ sample BST127 and ${\mathrm{Sb}}_2{\mathrm{Te}}_3$ sample BST305. All data are normalized to the value for $T=4.2$ K.

Figure 4

(a) $J_y/I$ measured in ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ sample BST127 for $f=3.3$ THz. (b) $J_y/I$ measured in $\left({\mathrm{Bi}}_{0.57}{\mathrm{Sb}}_{0.43}\right){}_2{\mathrm{Te}}_3$ sample BST641 for $f=1.1$ THz. The data show the dependence of the photocurrent on the azimuth angle $\alpha$ excited by normal incident radiation. Angles of incidence $\theta =0$ and ${180}^{\circ }$ correspond to the front and back excitations, respectively. Solid lines show fits after Eq. (1). Note that the same dependencies are obtained after phenomenological [see Eq. (4)] and microscopic theory [see Eqs. (19) and (25) for the photogalvanic and the $q_z$-photon drag effects, respectively].

Figure 5

Azimuth angle dependencies of the photocurrent $J_y/I$ excited in $\left({\mathrm{Bi}}_{0.57}{\mathrm{Sb}}_{0.43}\right){}_2{\mathrm{Te}}_3$ sample BST641 by normal and oblique incident radiation. The data demonstrate that the polarization dependence does not change upon variation of the angle of incidence either for (a) the case in which the current decreases at oblique incidence $(f=2.0$ THz) or (b) the case in which it increases with increasing $\left|\theta \right|$ $(f=3.3$ THz). Solid lines show fits after Eq. (1). Note that the same dependencies are obtained after phenomenological [see Eq. (4)] and microscopic theory [see Eqs. (17), (19), and (25) for the photogalvanic and both photon drag effects].

Figure 6

Dependencies of the photocurrent amplitudes $A_{x,y}$ on the angle of incidence $\theta$ obtained for different frequencies and three samples: ${\mathrm{Sb}}_2{\mathrm{Te}}_3$ (BST305), $\left({\mathrm{Bi}}_{0.57}{\mathrm{Sb}}_{0.43}\right){}_2{\mathrm{Te}}_3$ (BST641), and ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ (BST127). Solid lines show fits after Eq. (4) and calculations after Eqs. (19), (25), and (17). Dotted and dashed lines show contributions of the photogalvanic effect and photon drag effect caused by the $q_x$ component of the photon wave vector. The curves are calculated after Eqs. (19) and (17), respectively. Note that solid curves in (e) and (f) are obtained by also taking into account the contribution of the photon drag effects caused by the $q_z$ component of the photon wave vector. The latter effect (not shown) mainly contributes to the signal at normal incidence and is the cause of the difference between the value of the calculated total photocurrent (solid line) and the photogalvanic effect contribution (dotted line). It has a negative sign and decreases with increasing $\theta$. The relative contributions of the photogalvanic and photon drag effects are obtained from the measurements applying front and back illuminations; see Figs. 3 and 4 as well as Table 1. The inset in (a) sketches the setup.

Figure 7

Azimuth angle dependencies of the photocurrent $J_y/I$ excited in ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ sample BST127. The data are obtained at oblique incidence radiation $(f=2.0$ THz) for $\theta ={20}^{\circ }$ and different orientations of the plane of incidence with respect to the $y$ direction given by the angle $\psi$; see the inset. The data are shown for the angles $\psi =0^{\circ },{40}^{\circ }$, and ${90}^{\circ }$. The solid lines are calculated after Eq. (5).

Figure 8

(a) Azimuth angle dependencies of the photocurrent $J_x/I$ excited in ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ sample BST127. The data are obtained at oblique incidence radiation $(f=2.0$ THz) for $\theta ={20}^{\circ }$ and different orientations of the plane of incidence with respect to the $y$ direction given by the angle $\psi$. The data are shown for the angle $\psi$ changing from $0^{\circ }$ to ${130}^{\circ }$ with steps of $\mathrm{\Delta }\psi ={10}^{\circ }$. Solid lines show fits after Eqs. (5). The measured phase shift as a function of $\psi$ for (b) the data shown in (a) and (c) samples BST641, BST127, and BST305 and two radiation frequencies. Here solid and open symbols correspond to the frequencies $f=2.0$ and 3.3 THz, respectively. The solid line shows a fit with $\varphi =2\psi$.

Figure 9

Helicity dependence of the photocurrent, $J_x/I$, measured in ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ sample BST127 at normal as well as oblique incidence in the direction normal to the plane of incidence. The ellipses on top illustrate the polarization states for various angles $\phi$. Solid lines show fits after Eqs. (4), where the polarization-dependent terms take, for this geometry, the form $J_x=A_x\left(f\right)(cos4\phi +1)/2$.

Figure 10

Model of the photogalvanic effect, excited in surface states of $\left({\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x\right){}_2{\mathrm{Te}}_3$ due to the asymmetry of elastic scattering of holes by wedges.

Figure 11

Model of the trigonal photon drag effect caused by the in-plane wave vector $q_x$. To be specific, we discuss the hole gas in the surface states excited by oblique radiation with the incidence plane $\left(xz\right)$; see (c). The triangle in the left picture in (d) shows the top view of the considered scattering potential. The side view of the scatters is sketched in the right picture.

Figure 12

Frequency dependencies for photogalvanic and photon drag photocurrents calculated after Eq. (19) (dashed curve) and Eq. (17) (solid curve), respectively. The curves are normalized by the corresponding photocurrent maximum. The photon drag current is multiplied by $-1$ because these terms always have, in experiments, the opposite sign.