Electric field induced injection and shift currents in zigzag graphene nanoribbons

We theoretically investigate the one-color injection currents and shift currents in zigzag graphene nanoribbons by applying a static electric field across the ribbon, which breaks the inversion symmetry to generate nonzero second-order optical responses by dipole interaction. These two types of currents can be separately excited by specific light polarization, circularly polarized lights for injection currents and linearly polarized lights for shift currents. Based on a tight binding model formed by carbon $2p_z$ orbitals, we numerically calculate the spectra of injection coefficients and shift conductivities, as well as their dependence on the static field strength and ribbon width. The spectra show many peaks associated with the optical transition between different subbands, and the positions and amplitudes of these peaks can be effectively controlled by the static electric field. By constructing a simple two band model, the static electric fields are found to modify the edge states in a nonperturbative way, and their associated optical transitions dominate the current generation at low photon energies. For typical parameters, such as a static field ${10}^6$ V/m and light intensity 0.1 GW/${\mathrm{cm}}^2$, the magnitude of the injection and shift currents for a ribbon with width of 5 nm can be as large as the order of 1 $\mu \mathrm{A}$. Our results provide a physical basis for realizing passive optoelectronic devices based on graphene nanoribbons.

Figure 1

Illustration of an $N$-zGNR. Red and gray dots correspond to carbon atoms at the $A$ and $B$ sites, respectively. The unit cell of the ribbon is indicated by the parallelogram. An external static electric field $E_d$ is applied across the ribbon.

Figure 2

(a), (b) Band structures of 24-zGNR for gate field $E_d=0, {10}^8$ V/m, $3\times {10}^8$ V/m, and $5\times {10}^8$ V/m. At zero field, the dashed and solid curves correspond to different parity. The matrix elements of (c) $r_{s_1{s}_{2}k}^{0;c}$ and (d) ${\mathcal{R}}_{s_1{s}_{2}k}^{0;cx}$ at zero gate field, with solid (dashed) curves for imaginary (real) parts.

Figure 3

(a) The energy difference ${\varepsilon }_{sk}-{\varepsilon }_{s^{\prime }}$ for different $(s,s^{\prime })$ pairs at zero gate field. (b) The gate-field dependence of the energy gaps $E_{g;s{s}^{\prime }}$ between different bands $(s,s^{\prime })$. The line color indicates the $k$ values of these gaps. The black dotted line gives the energy difference ${\varepsilon }_{\left(1\right)g/2}-{\varepsilon }_{(-1)g/2}$.

Figure 4

(a) Spectra of injection coefficient ${\tilde{\eta }}^{xxyy}\left(\omega \right)$ for a 24-zGNR for gates fields $E_d={10}^2, {10}^4, {10}^6$, and ${10}^7$ V/m at room temperature; the curves at the right of the vertical dashed line are scaled by ten times. (b) The spectra of ${\tilde{\eta }}_{(+1)(-1)}^{xxyy}$ and ${\tilde{\eta }}_l^{xxyy}$ for $l=2,3,...,8$ for $E_d={10}^2$ V/m. Two specific labels mark the separated contribution of transition from band $\pm 1$ to the band $l=3$. (c) The injection coefficient ${\eta }^{xxy}$ of zGNR under large electric field $E_d={10}^7, 5\times {10}^7$, and ${10}^8$ V/m at room temperature. (d) The spectra of ${\tilde{\eta }}_l^{xxy}$ for $l=2,3,...,8$ for $E_d=5\times {10}^7$ V/m.

Figure 5

Spectra of shift conductivity ${\sigma }^{xxy}\left(\omega \right)$ for an undoped 24-zGNR at different gate fields. (a) Transition-resolved contribution of ${\sigma }^{xxy}\left(\omega \right)$ at $E_d={10}^4$ V/m. The shadowed region gives the total conductivity. The plotted contribution from different band pairs are ${\sigma }_{(+1)(-1)}^{xxy}\left(\omega \right), {\sigma }_{(+3)(\pm 1)}^{xxy}\left(\omega \right)$, as well as ${\sigma }_l^{xxy}\left(\omega \right)={\sum }_{\pm }{\sigma }_{(+l)(\pm 1)}^{xxy}\left(\omega \right)+{\sigma }_{(\pm 1)(-l)}^{xxy}\left(\omega \right)$ for $l=2,3,4,5$. (b) Spectra of ${\sigma }^{xxy}\left(\omega \right)$ at $E_d={10}^2, {10}^3, {10}^4, {10}^5$, and ${10}^6$ V/m at room temperature. (c) Spectra of ${\sigma }^{xxy}\left(\omega \right)$ at gate fields up to ${10}^8$ V/m.

Figure 6

The spectra of injection coefficients and shift conductivities for different ribbon width $W=5$, 10, 15, and 20 nm. (a) ${\tilde{\eta }}^{xxy}\left(\omega \right)$ at $E_d={10}^4$ V/m, (b) ${\sigma }^{xxy}\left(\omega \right)$ at $E_d={10}^4$ V/m, (c) ${\tilde{\eta }}^{xxy}\left(\omega \right)$ at $E_d=5\times {10}^7$ V/m, (d) ${\sigma }^{xxy}\left(\omega \right)$ at $E_d=5\times {10}^7$ V/m.