Franz-Keldysh effect and electric field-induced second harmonic generation in graphene: From one-dimensional nanoribbons to two-dimensional sheet

It is well established that external electrostatic fields modify the electronic structure and optical response of materials. Modifications of the optical response of quasi-one-dimensional graphene nanoribbons (GNRs) depend strongly on the direction of the electrostatic field. While transverse fields primarily lift degeneracies in the band structure, longitudinal fields are responsible for considerable nonperturbative Franz-Keldysh effects. Also, electric fields break the inversion symmetry of GNRs and result in strong electric field-induced second harmonic generation. In this work, we study field-induced modifications of the linear and nonlinear optical response of narrow and wide semiconducting armchair GNRs (AGNRs). Both finite and infinite AGNRs with and without electrostatic fields are studied and length convergence is analyzed. Similarly, the width convergence of wide AGNRs to the two-dimensional graphene limit with and without longitudinal fields is investigated.

Figure 1

Atomic structure of AGNRs with varying $n_d$ number of rows of carbon-carbon dimer bonds. Longitudinal and transversal axes are along the $x$ and $y$ directions, respectively. The black dashed lines show the AGNR unit cell with the length $a$ and the red diamond is the two-dimensional graphene unit cell with $\mathit{b}$ and ${\mathit{b}}^{\prime }$ as unit vectors. The green hexagon and yellow triangle show full and irreducible Brillouin zone of graphene, respectively, and $\mathrm{\Gamma }$, M, and K are high symmetry points in $k$ space.

Figure 2

Band structure of (a) AGNR-3 and (b) AGNR-9 in the nearest neighbor tight-binding model. The red curves show the band structures under $F_{\mathrm{dc}}^y=0$ and the blue curves under $F_{\mathrm{dc}}^y=0.36\mathrm{V}/\AA$ for AGNR-3 and $F_{\mathrm{dc}}^y=0.12\mathrm{V}/\AA$ for AGNR-9.

Figure 3

Real part of linear optical conductivity of finite and infinite (a) AGNR-3 and (b) AGNR-9.

Figure 4

Real part of linear optical conductivity of finite and infinite (a) AGNR-3 and (b) AGNR-9 under a transversal electrostatic field $F_{\mathrm{dc}}^y=0.36\mathrm{V}/\AA$ for AGNR-3 and $F_{\mathrm{dc}}^y=0.12\mathrm{V}/\AA$ for AGNR-9.

Figure 5

Real part of linear optical conductivity of infinite (a) AGNR-3 and (b) AGNR-9 with and without transversal electrostatic field $F_{\mathrm{dc}}^y=0.36\mathrm{V}/\AA$ for AGNR-3 and $F_{\mathrm{dc}}^y=0.12\mathrm{V}/\AA$ for AGNR-9.

Figure 6

Real part of linear optical conductivity for finite and infinite (a) AGNR-3 and (b) AGNR-9 under the effect of a longitudinal electrostatic field $F_{\mathrm{dc}}^x=0.01\mathrm{V}/\AA$. Note that the low-energy parts of the spectra have been down-scaled.

Figure 7

Real part of linear optical conductivity for finite and infinite AGNR-3 under the effect of the longitudinal electrostatic field $F_{\mathrm{dc}}^x=0.003\mathrm{V}/\AA$. Note that above 5 eV, the spectra are down-scaled.

Figure 8

Real part of second-order nonlinear optical conductivity for finite and infinite (a) AGNR-3 and (b) AGNR-9 under the effect of the longitudinal electrostatic field $F_{\mathrm{dc}}^x=0.01\mathrm{V}/\AA$. Note that the low-energy parts of the spectra have been down-scaled.

Figure 9

Real part of linear optical conductivity for AGNR-45, AGNR-90, AGNR-240, AGNR-390, and two-dimensional graphene $n_d=\infty$.

Figure 10

Real part of linear optical conductivity for AGNR-45, AGNR-90, AGNR-240, AGNR-390, and two-dimensional graphene $n_d=\infty$ under the effect of $F_{\mathrm{dc}}^x=0.01\mathrm{V}/\AA$.

Figure 11

Real part of second-order nonlinear optical conductivity for AGNR-45, AGNR-90, AGNR-240, AGNR-390, and two-dimensional graphene $n_d=\infty$ under the effect of $F_{\mathrm{dc}}^x=0.01\mathrm{V}/\AA$.