Current injection by coherent one- and two-photon excitation in graphene and its bilayer

Coherent control of optically injected carrier distributions in single and bilayer graphene allows the injection of electrical currents. Using a tight-binding model and Fermi’s golden rule, we derive the carrier and photocurrent densities achieved via interference of the quantum amplitudes for two-photon absorption at a fundamental frequency, $\omega$, and one-photon absorption at the second harmonic, $2\omega$. Strong currents are injected under co-circular and linear polarizations. In contrast, opposite-circular polarization yields no net current. For single-layer graphene, the magnitude of the current is unaffected by the rotation of linear-polarization axes, in contrast with the bilayer and with conventional semiconductors. The dependence of the photocurrent on the linear-polarization axes is a clear and measurable signature of interlayer coupling in AB-stacked multilayer graphene. We also find that single and bilayer graphene exhibit a strong, distinct linear-circular dichroism in two-photon absorption.

Figure 1

Crystal structure of (a) single-layer graphene and (b) bilayer graphene. The basis vectors ${\mathit{a}}_1$ and ${\mathit{a}}_2$ define the unit cell, ${\gamma }_0$ and ${\gamma }_1$ are the intralayer and interlayer coupling strengths, and the sublattices are denoted by A (A${}^{\prime }$) and B (B${}^{\prime }$).

Figure 2

Reciprocal space and linear energy-crystal momentum dispersion of graphene near $K$. The basis vectors ${\mathit{b}}_1$ and ${\mathit{b}}_2$ form the reciprocal unit cell, enclosing one $K$ and one $K^{\prime }$ valley. The dispersion shows the initially empty conduction band $c$ and occupied valence band $v$ touching at the $K$ point. The excitation scheme employs interference between two-photon absorption at $\omega$ (red arrows) and one-photon absorption at $2\omega$ (blue arrows), leading to generation of charge and current.

Figure 3

Distribution $\mathrm{ṅ}(\mathit{k})$ of the carrier injection through reciprocal space under irradiation by an optical field with components $E(\omega )$ and $E(2\omega )$ satisfying $\Delta \phi =\frac{\pi }{2}$. (a) Opposite-circular polarization ($-\delta {\phi }_{\omega }=\delta {\phi }_{2\omega }=\pm \frac{\pi }{2}$, $\theta =0$). (b) Co-circular polarization (${\sigma }^{\pm }$ light, $\delta {\phi }_{\omega }=\delta {\phi }_{2\omega }=\pm \frac{\pi }{2}$, $\theta =0$). (c), (d) Linear polarization ($\delta {\phi }_{\omega }=\delta {\phi }_{2\omega }=0$) with ${\stackrel{\wedge }{\mathit{e}}}_{2\omega }=\stackrel{\wedge }{\mathit{x}}$ and ${\stackrel{\wedge }{\mathit{e}}}_{\omega }=\stackrel{\wedge }{\mathit{x}}\cos \theta +\stackrel{\wedge }{\mathit{y}}\sin \theta$; (c) $\theta =0$ and (d) $\theta =\frac{\pi }{4}$. The distribution in (a) results in no net current; the asymmetric distributions [(b)–(d)] result in net electrical currents injected in the graphene plane along $\stackrel{\wedge }{\mathit{x}}$ [(b), (c)] or $\stackrel{\wedge }{\mathit{y}}$ (d).

Figure 4

Band dispersion of bilayer graphene and breakdown of the transition amplitudes for two-photon absorption. Bands $v_1$ and $v_2$ are valence bands and initially filled; $c_1$ and $c_2$ are initially empty conduction bands. Bands $v_2$ and $c_1$ form a gapless doublet touching at the $K$ point; $c_2$ and $v_1$ are split-off bands shifted by an energy ${\gamma }_1$ above and below the gapless doublet, respectively. All bands are quadratic near $K$ and linear at larger $k$. Transition amplitudes appear in four variants: (i) the gapless term (GLT) between bands $v_2$ and $c_1$, (ii) two- and (iii) three-band terms involving exactly one split-off band (2BT and 3BT, respectively), and (iv) the split-off term (SOT) between bands $v_1$ and $c_2$. The notation $\{...\}$ next to a virtual state indicates that the sum in Eq. (9) is restricted to $m\in \{...\}$. Not shown are the 2BT and 3BT between bands $v_1$ and $c_1$.

Figure 5

Quantities describing two-photon carrier injection in bilayer graphene as a function of photon energy for an intermediate-state linewidth $\Gamma /{\gamma }_1=0.35$. (a) The individual contributions ${\mathrm{\xi ̄}}_{2a-d}$ ($a$, plain red; $b$, long-dashed blue; $c$, short-dashed green; $d$, dotted orange) from Eq. (20). (b) The independent nonzero tensor components ${\xi }_2^{\mathit{xxxx}}$ (plain black) and ${\xi }_2^{\mathit{xxyy}}$ (dashed red) from Eq. (19). (c) The linear-circular dichroism $\delta ={\xi }_2^{\mathit{xxyy}}/{\xi }_2^{\mathit{xxxx}}$.

Figure 6

Diagrams of the four contributions ${\mathrm{\eta ̄}}_{\mathit{Ia}-d}$ to the current injection in bilayer graphene [4b].

Figure 7

The current-injection tensor ${\eta }_I$ in bilayer graphene for an intermediate-state linewidth $\Gamma /{\gamma }_1=0.25$. (a) The four contributions ${\mathrm{\eta ̄}}_{\mathit{Ia}-d}$ from Eq. (22): Plain red, long-dashed blue, short-dashed green, and dotted orange curves correspond to components $a$, $b$, $c$, and $d$, respectively. (b) The tensor components ${\eta }_I^{\mathit{xxxx}}$ (plain black) and ${\eta }_I^{\mathit{xyyx}}$ (dashed red).

Figure 8

Results for the current injection in bilayer graphene with linearly polarized $\omega$ and $2\omega$ beams. Left-hand side: The disparity parameter $d={\eta }_I^{\mathit{xyyx}}/{\eta }_I^{\mathit{xxxx}}$ describing the asymmetry between parallel and perpendicular polarization axes, with $\Gamma /{\gamma }_1=0.15$ (dotted orange), $0.25$ (short-dashed green), $0.35$ (long-dashed blue), and $0.45$ (plain red). Right-hand side: Polar plots of $f(\theta ,d)$ and $g(\theta ,d)$, the angular distributions of the projections of $\stackrel{·}{\mathit{J}}$ parallel and perpendicular to ${\stackrel{\wedge }{\mathit{e}}}_{2\omega }$, as a function of the angle $\theta$ between the polarization vectors for $d=-2$, $-1$, $-0.5$, $0$, and $0.5$. The shaded circles represent unit amplitude and dashed lines represent negative projections. The graphene prediction, $d=-1$, is highlighted.