Nonlinear intensity dependence of edge photocurrents in graphene induced by terahertz radiation

We report on the observation of terahertz-radiation-induced edge photogalvanic currents in graphene, which are nonlinear in intensity. The increase of the radiation intensities up to MW/${\mathrm{cm}}^2$ results in a complex nonlinear intensity dependence of the photocurrent. The nonlinearity is controlled by the back gate voltage, temperature, and radiation frequency. A microscopic theory of the nonlinear edge photocurrent is developed. Comparison of the experimental data and theory demonstrates that the nonlinearity of the photocurrent is caused by the interplay of two mechanisms, i.e., by direct interband optical transitions and Drude-like absorption. Both photocurrents saturate at high intensities but have different intensity dependencies and saturation intensities. The total photocurrent shows a complex sign-alternating intensity dependence. The functional behavior of the saturation intensities and amplitudes of both kinds of photogalvanic currents, depending on gate voltages, temperature, radiation frequency, and polarization, is in good agreement with the developed theory.

Figure 1

(a) Cross section of the graphene device displaying the exfoliated graphene sandwiched between boron nitride. (b) Sketch of the Hall-bar-shaped samples #A and #B with corresponding contact labels. Panel (c) shows the longitudinal resistivity ${\rho }_{xx}$ as a function of the effective gate voltage, obtained for room and liquid helium temperatures. Vertical up arrow indicates the charge neutrality point. (d) Schematic representation of the experimental setup. The photocurrent is measured as a voltage drop over 50 $\mathrm{\Omega }$ load resistors.

Figure 2

(a) Intensity dependencies of the photocurrents obtained in samples #A and #B for a pair of contacts at the left and right sample edges. All curves are obtained for an angle $\alpha ={45}^{\circ }$. Curves are fits to Eq. (42). (b) Polarization dependence of the edge photocurrent $J_{\mathrm{edge}}=(J_{\mathrm{AC}}-J_{\mathrm{GE}})/2$ obtained for different radiation intensities. Note that a small polarization-independent offset was subtracted for visibility. Curves are fits to Eq. (1). The inset defines the angle $\alpha$ describing the relative orientation of the radiation electric field vector $\mathit{E}$ in respect to the samples long side. Arrows on top illustrate the polarization plane orientation for several angles $\alpha$.

Figure 3

(a) Intensity dependencies of the edge current excited in sample #A at liquid helium temperature by radiation with $f=2.02$ THz. The curves obtained for different values of the effective gate voltage are vertically shifted by $2\mu \mathrm{A}$ (except the ones for $U_{\mathrm{g}}^{\mathrm{eff}}=1$ and 3 V). The solid lines are fits to Eq. (42) with Drude and interband amplitudes $J_1^{\mathrm{Dr}}/I, J_1^{\mathrm{i}-\mathrm{b}}/I$, and saturation intensities $I_s^{\mathrm{Dr}}, I_s^{\mathrm{i}-\mathrm{b}}$ as fit parameters. (b) The magnitudes of the photocurrent contributions caused by Drude ($J_1^{\mathrm{Dr}}/I$, full circles) and interband ($J_1^{\mathrm{i}-\mathrm{b}}/I$, open circles) absorption as a function of the gate voltage. The dashed curve is calculated using Eq. (40). The best fit is obtained assuming an electron temperature $T=230$ K. Note that the substantial electron-gas heating enables the interband absorption at large gate voltages, despite the fact that the final state of such transitions lies below the Fermi energy. (c) Gate dependencies of the saturation intensities of these contributions: $I_s^{\mathrm{Dr}}$ (full circles) and $I_s^{\mathrm{i}-\mathrm{b}}$ (open circles).

Figure 4

(a) Intensity dependencies of the edge current excited in sample #A at room temperature with $f=2.02$ THz. The curves obtained for different values of the effective gate voltage are vertically shifted by $2\mu \mathrm{A}$ (except the ones for $U_{\mathrm{g}}^{\mathrm{eff}}=1$ and 3 V). Curves are fits using Eq. (42) with Drude and interband amplitudes $J_1^{\mathrm{Dr}}/I, J_1^{\mathrm{i}-\mathrm{b}}/I$, and saturation intensities $I_s^{\mathrm{Dr}}, I_s^{\mathrm{i}-\mathrm{b}}$ as fit parameters. (b) Photocurrent contributions caused by Drude ($J_1^{\mathrm{Dr}}/I$, full circles) and interband ($J_1^{\mathrm{i}-\mathrm{b}}/I$, open circles) absorption as a function of the gate voltage. The dashed curve is calculated after Eq. (40). The best fit is obtained assuming an electron temperature $T=370$ K. (c) Gate dependencies of the saturation intensities of these contributions: $I_s^{\mathrm{Dr}}$ (full circles) and $I_s^{\mathrm{i}-\mathrm{b}}$ (open circles).

Figure 5

Intensity dependencies of the edge current excited in sample #A at liquid helium temperature and high positive gate voltage. Panel (a) shows the data for three radiation frequencies, $f=3.33$, 2.02, and 1.07 THz, and $\alpha ={45}^{\circ }$. Curves are fits using Eq. (2). Inset shows the frequency dependencies of the magnitude $J^{\mathrm{Dr}}$ and saturation intensities of the Drude absorption related photocurrent. Solid and dashed curves show $J^{\mathrm{Dr}}\propto 1/{\omega }^2$ and $I_s^{\mathrm{Dr}}\propto {\omega }^2$, respectively. Panel (b) presents a zoom in of the data for $f=1.07$ THz. The dashed green line is a linear fit of the low-power photocurrent without saturation.

Figure 6

Intensity dependencies of the edge current excited in sample #A at room (panel a) and liquid helium (panel b) temperatures obtained for small positive gate voltages. The data are presented for the radiation with $f=2.02$ THz and $\alpha ={45}^{\circ }$. Solid lines are fits using Eq. (2).

Figure 7

(a) Intensity dependence of the signal measured in the photoconductivity setup in which a dc bias voltage $V_{dc}$ is applied to the sample. The data are obtained for $f=2.02$ THz, $U_{\mathrm{g}}^{\mathrm{eff}}=-4$ V, and angle $\alpha ={45}^{\circ }$. Fits are after $U_{\mathrm{GE}}\propto I/(1+I/I_s^{\mathrm{Dr}})$. Panel (b) displays the radiation-induced change of the sample conductivity $\mathrm{\Delta }\sigma$ normalized by the dark conductivity $\sigma$. The full line is a fit after Eq. (2). The inset shows the setup used for the photoconductivity measurements.

Figure 8

Model of the edge current formation. The current originates from the alignment of the free-carrier momenta by the linearly polarized THz electric field. Blue and red arrows illustrate the anisotropy in the distribution of carrier momenta $\mathit{p}$ and, consequently, the velocities $\mathit{v}$. (a) The carrier fluxes (red and blue tilted arrows) induced by the optical alignment in stripes of the mean free path width $\delta$ at the sample edge ($\mathrm{\Delta }{x}_{\mathrm{edge}}$) and in the sample bulk. In the edge vicinity, the uncompensated bottom right flux drives a net electric current $J$ flowing along the edge, illustrated by a vertical red arrow. In the bulk the fluxes compensate each other and a bulk current is absent. After Ref. [17]. (b, c) Optical alignment of carrier momenta induced by linearly polarized terahertz radiation, resulting in an anisotropy of the carrier movement. The alignment is plotted for Drude absorption (b) and direct interband optical transitions (c). Panel (d) shows the momentum distribution at two-photon interband transitions.

Figure 9

(a) The dependence of the edge photocurrent on the light intensity at $\alpha =\pi /4$. The dashed line is the low-intensity current $J_1$. Inset shows the ratio $J/J_1$ as a function of ${\mathcal{E}}^2$ on a semilogarithmic scale. (b) Deviation of the edge current from the $sin2\alpha$ dependence given by the ratio $J/J_1$ at different values of the electric field amplitude $\mathcal{E}$ indicated in the figure. The ratios $J/J_1$ are normalized to their values at $\alpha =0$.

Figure 10

Model illustrating the edge photocurrent formation for Drude optical transitions (bent arrow) and direct interband transitions (upward arrow). Open and closed circles illustrate carriers in the initial and final state of the optical transitions, respectively. Right panel illustrates modification of the Fermi-Dirac distribution due to the electron-gas heating. The carrier redistribution is illustrated for the Fermi level lying in the conduction band and three different temperatures. Despite that the final state of the indicated direct transition is below the Fermi level, a substantial heating resulting in a high temperature leads to a depopulation of this state, making allowance for the interband transition.