Wannier-Stark states of graphene in strong electric field

We find theoretically the energy spectrum of a graphene monolayer in a strong constant electric field using a tight-binding model. Within a single band, we find quantized equidistant energy levels (Wannier-Stark ladder), separated by the Bloch frequency. Singular interband coupling results in mixing of the states of different bands and anticrossing of corresponding levels, which is described analytically near Dirac points and is related to the Pancharatnam-Berry phase. The rate of interband tunneling, which is proportional to the anticrossing gaps in the spectrum, is only inversely proportional to the tunneling distance, in a sharp contrast to conventional solids where this dependence is exponential. This singularity will have major consequences for graphene behavior in strong ultrafast optical fields, in particular, leading to nonadiabaticity of electron excitation dynamics.

Figure 1

(a) Honeycomb lattice structure of two-dimensional (2D) graphene, which consists of two sublattices with atoms labeled by $A$ (open circles) and $B$ (filled circles), respectively. The nearest-neighbor coupling with hopping integral $\gamma$ is also shown. (b) The first Brillouin zone of graphene. Points $K$ and $K^{\prime }$ are two inequivalent Dirac points, which correspond to two valleys of low-energy spectrum of graphene. The direction of electric field is shown by the blue line and is characterized by angle $\theta$ relative to the $x$ axis. (c) Energy dispersion of graphene within the nearest-neighbor tight-binding model. The $K$ and $K^{\prime }$ Dirac points are labeled. The conduction and the valence bands correspond to positive and negative energies, respectively.

Figure 2

Lines of coupled states in reciprocal space. The electron states of the reciprocal space, which are coupled by a constant electric field parallel to the $x$ axis, are shown by solid lines of two different colors (red and blue), where different colors correspond to two different values of $k_y$. (a) The coupled states are shown in the first Brillouin zone. The equivalent points (at the edges of the Brillouin zone) are shown by the same type of points, i.e., solid red points or open red points. The equivalent points are connected by a vector of reciprocal lattice. (b) The coupled states are shown in the whole reciprocal space. The first Brillouin zones, localized at different points of the reciprocal lattice, are also shown. The equivalent points, which are connected by a vector of reciprocal lattice, are shown by the same type of points, e.g., two red points are equivalent.

Figure 3

Dipole matrix element $D_x$ as a function of $k_x$ for different values of $k_y$. The wave vector $k_x$ is measured in units of $2k_0$, where $k_0=2\pi /a\sqrt{3}$. The Dirac point is at $k_x=k_0$ and $k_y=k_0/\sqrt{3}=(1/3)(2\pi /a)$. The numbers near the lines are the values of $k_y$ in units of $(2\pi /a)$. Panels (a) and (b) differ by the vertical scale.

Figure 4

(a) Dimensionless energies ${\varepsilon }_n^{(\pm )}$ of the WS states, calculated from Eq. (32), as a function of dimensionless parameter ${\varepsilon }_{c,0}$ for different values of integer number $n$. Parameter ${\Lambda }_0$ is ${\Lambda }_0=0.6$. Different types of anticrossing points are labeled by integer parameter $l$. With increasing electric field, the last anticrossing point corresponds to $l=1$ and occurs at ${\varepsilon }_{c,0}=\pi$. (b) The energies $E_n^{(\pm )}$ of the WS states, calculated from Eq. (31), as a function of Bloch frequency $\hslash {\omega }_B$, which is proportional to electric field. The anticrossing points, corresponding to $l=1$ and 2, are marked by red lines. The parameter ${\Lambda }_0$ is ${\Lambda }_0=0.6$, and $E_{c,0}=1$ eV.

Figure 5

Electron densities ${\rho }_v\left(x\right)$ and ${\rho }_c\left(x\right)$ in the conduction and valence bands of a given WS state. The electric field is (a) $F=1.8$ V/Å, (b) $F=2.4$ V/Å, (c) $F=3.6$ V/Å. The fields 1.8 and 3.6 V/Å corresponds to $l=2$ and 1 anticrossing points. The $y$ component of the wave vector is $k_y=k_{y,0}$.

Figure 6

Total electron density $\rho \left(x\right)={\rho }_v\left(x\right)+{\rho }_c\left(x\right)$ of three WS states. The electric field is $F=3.6$ V/Å, corresponding to the $l=1$ anticrossing point. The $y$ component of the wave vector is $k_y=k_{y,0}$. The curves are displaced vertically for clarity.

Figure 7

Energy spectra of graphene in a constant electric field, parallel to the $x$ axis. The spectra are calculated numerically for a finite-size system for two values of $k_y$: (a) $k_y=0$ and (b) $k_y=0.32(2\pi /a)$. The number of states in each band is 100. The anticrossing points corresponding to $l=1$ and 2 are marked by red lines.

Figure 8

(a) Anticrossing gaps, calculated for the $l=2$ and 1 anticrossing points, are shown as a function of the $y$ component of the wave vector $k_y$. (b) The positions of $l=1$ and 2 anticrossing points are shown as a function of $k_y$. The electric field is parallel to the $x$ axis.

Figure 9

(a) Line of coupled states in the reciprocal space is shown by blue solid line. Along this line there are two inequivalent Dirac points $K$ and $K^{\prime }$. The direction of electric field is also shown. (b) Dimensionless energies ${\varepsilon }_n^{(\pm )}$ of WS states, calculated from Eq. (58), are shown as a function of dimensionless parameter ${\tilde{\varepsilon }}_{c,0}$ for different values of integer number $n$. The parameters ${\Lambda }_1={\Lambda }_2={\Lambda }_0$ and $\alpha$ are ${\Lambda }_0=0.6$ and $\alpha =0.7$. (c) The energies $E_n^{(\pm )}$ of the WS states, calculated from Eq. (55), are shown as a function of Bloch frequency $\hslash {\omega }_B$, which is proportional to electric field. The anticrossing points, corresponding to $l=1$ and 2, are marked by red lines. The parameters are ${\Lambda }_0=0.6,\alpha =0.7$, and $E_{c,0}=1$ eV.