Magneto-optical and optical probes of gapped ground states of bilayer graphene

In the two-band model (3) in the Landau gauge $\mathbf{A}=(0,Bx)$, the GF $G(\mathbf{r},\mathbf{r}^{\prime},\omega)$ in the mixed coordinate-frequency representation has a form

where the translation invariant part of the GF is represented as a sum over the Landau levels[17]

Here $z=\mathbf{r}^{2}/(2l^{2})$, $L_{n}^{\alpha}(z)$ are associated Laguerre polynomials (by the definition $L^{\alpha}_{-2}(z)=L^{\alpha}_{-1}(z)\equiv 0$), and $\mu$ is the chemical potential. For brevity of notation the subscript $\xi$ in $M_{n\xi}$ is omitted in what follows, i.e. $M_{n}=M_{n\xi}$. Since the $2\times 2$ Hamiltonian is quadratic in $\hat{\pi}$ and $\hat{\pi}^{\dagger}$, the electric current operator contains the derivatives

The phase factors in Eq. (5) cancel and we obtain at finite temperature

where $\omega_{n}=\pi(2n+1)/\beta$ and

Noting that $\tilde{j_{i}}({\bf r},{\bf r}^{\prime})=\tilde{j_{i}}({\bf r}-{\bf r}^{\prime})$ depend on the difference of the coordinates ${\bf r}-{\bf r}^{\prime}$, one can integrate over ${\bf r}+{\bf r}^{\prime}$ in Eq. (10) canceling the volume factor $V$ in the denominator

Writing Eq. (12) we also used that $\tilde{G}(\mathbf{r},i\omega_{n})$ is even under the transformation $\mathbf{r}\rightarrow-\mathbf{r}$, while $\tilde{j_{i}}({\bf r})$ is odd.

The calculation of the optical conductivity follows closely the corresponding calculation done for monolayer graphene,[32, 33] so we directly proceed to the final expression. The only difference is that we introduce a finite Landau level width $\Gamma$ at the very end of the calculation in the final expressions for the conductivities. The final $\Gamma=0$ expression for the polarization operator $\Pi_{\pm}^{R}(\Omega)\equiv\Pi_{xx}^{R}(\Omega)\pm i\Pi_{xy}^{R}(\Omega)$ takes the form

where $n_{F}(\omega)=1/[\exp((\omega-\mu)/T)+1]$ is the Fermi distribution function. It is clear that the last $\sim\Omega$ term in $\Pi_{\pm}$ may only be present if the time-reversal symmetry breaking gap, $\Delta_{T}\neq 0$.

It is convenient to consider the optical conductivities $\sigma_{\pm}(\Omega)=\sigma_{xx}(\Omega)\pm i\sigma_{xy}(\Omega)=\hbar[\Pi_{\pm}^{R}(\Omega)-\Pi_{\pm}^{R}(0)]/(i\Omega)$ which correspond to the opposite circular polarizations of light. After introducing a finite width of Landau levels in Eq. (13), the final result for the complex conductivities reads as

The expressions for conductivities in the case of more general gaps $\Delta_{\xi s}$ are obtained from Eq. (14) by replacing $\Delta_{\xi}$ with $\Delta_{\xi s}$ and inserting the overall factor $1/2$ and summing over the spin variable. Equation (14) is one of the main results of this paper which generalizes the results of Refs. 37, 38 to a finite $\Delta_{\xi}$ case. For $\Delta_{\xi}=0$, there is a practically overall agreement between Eq. (14) and the corresponding expressions from Ref. 37, 38 except to the intensity of the $0$ to $1$ transition. To verify this issue, in Appendix A we consider the behavior of the spectral weight. We show both that our result agrees with the behavior of the spectral weight for $B=0$ and that the weight is conserved when the chemical potential moves from the region $M_{0}<\mu<M_{2}$ to the region $\mu>M_{2}$.

It has to be stressed that rather simple analytical result (14) was possible to obtain because we used the effective $2\times 2$ Hamiltonian. The $4\times 4$ consideration is much more involved [40, 41]. Moreover, the neglected tight-binding terms, including the trigonal warping, can only be treated numerically[54] which is beyond the scope of the present work.

The scheme of Landau levels and allowed transitions for bilayer can be found, for example, in the review,[19] so that we can go directly to the discussion of the behavior of magneto-conductivity. In Figs. 1 and 2 we show the results based on the computation of Eq. (14) for the real part of $\sigma_{\pm}(\Omega)$ in units of $\sigma_{0}=e^{2}/(4\hbar)$ as a function of $\Omega$ measured in meV. The highest value of $\Omega={\text{\unboldmath$\mathrm{80}$}}\,\text{\unboldmath$\mathrm{mev}$}$ is less than $\gamma_{1}/4$. In both figures we set $B={\text{\unboldmath$\mathrm{3}$}}\,\text{\unboldmath$\mathrm{T}$}$, $T={\text{\unboldmath$\mathrm{5}$}}\,\text{\unboldmath$\mathrm{K}$}$ and $\Gamma={\text{\unboldmath$\mathrm{2}$}}\,\text{\unboldmath$\mathrm{meV}$}$.

The reference case[37, 38] $\Delta_{\xi}=0$ is shown in Fig. 1. To scrutinize the curves in this figure, we provide the energies of the Landau levels, viz. for the $n=2$ level $|E_{\pm 2}|=M_{2}\approx{\text{\unboldmath$\mathrm{15{.}3}$}}\,\text{\unboldmath$\mathrm{meV}$}$, for the $n=3$ level $|E_{\pm 3}|=M_{3}\approx{\text{\unboldmath$\mathrm{26{.}6}$}}\,\text{\unboldmath$\mathrm{meV}$}$ and for the $n=4$ level $|E_{\pm 4}|=M_{4}\approx{\text{\unboldmath$\mathrm{37{.}6}$}}\,\text{\unboldmath$\mathrm{meV}$}$. The long dashed (red) curve is for $\sigma_{+}(\Omega)$ and the dash-dotted (black) curve is for $\sigma_{-}(\Omega)$. Both curves are for $\mu={\text{\unboldmath$\mathrm{10}$}}\,\text{\unboldmath$\mathrm{meV}$}$ which is less than $E_{2}$. Accordingly, in the long dashed (red) curve we observe the lines with the energy $M_{2}-M_{0,1}\approx{\text{\unboldmath$\mathrm{15{.}3}$}}\,\text{\unboldmath$\mathrm{meV}$}$, $M_{2}+M_{3}\approx{\text{\unboldmath$\mathrm{41{.}9}$}}\,\text{\unboldmath$\mathrm{meV}$}$ and $M_{3}+M_{4}\approx{\text{\unboldmath$\mathrm{64{.}2}$}}\,\text{\unboldmath$\mathrm{meV}$}$. They correspond to the transitions from the Landau levels with the energy $E_{0,1}$ to $E_{2}$, from $E_{-2}$ to $E_{3}$, and from $E_{-3}$ to $E_{4}$, respectively. Since for the opposite polarization of light presented by the dash-dotted (black) curve there is no transition between the 0th ($n=0,1$) and $E_{2}$ level, the first observed line is for the transition from $E_{-3}$ to $E_{2}$ levels with the energy $M_{2}+M_{3}$ coinciding with the second line on the previous curve. The second observed line corresponds to the transition from $E_{-4}$ to $E_{3}$ levels with the energy $M_{3}+M_{4}$. The solid (blue) curve is for $\sigma_{+}(\Omega)$ and the short dashed (green) curve is for $\sigma_{-}(\Omega)$, and both curves are for $\mu={\text{\unboldmath$\mathrm{20}$}}\,\text{\unboldmath$\mathrm{meV}$}$, so that $E_{2}<\mu<E_{3}$. Accordingly, in the solid (blue) curve the transition with the energy $M_{2}-M_{1}$ is impossible, but instead the transition with the energy $M_{3}-M_{2}\approx{\text{\unboldmath$\mathrm{11{.}3}$}}\,\text{\unboldmath$\mathrm{meV}$}$ develops. Two remaining transitions from $E_{-2}$ to $E_{3}$, and from $E_{-3}$ to $E_{4}$ which coincide with the transitions on the considered above long dashed (red) and dash-dotted (black) curves are also present. For the opposite polarization of light presented by the short dashed (green) curve the only possible transition is from $E_{-4}$ to $E_{3}$, so that the peak at the energy $M_{3}+M_{4}$ is the only peak shared by all curves.

Now we are at the position to discuss how the opening of either time-reversal symmetry breaking gap $\Delta_{T}$ or preserving this symmetry gap $U$ changes the presented above picture (see Fig. 2).

Again long dashed (red) curve is for $\sigma_{+}(\Omega)$ and the dash-dotted (black) curve is for $\sigma_{-}(\Omega)$. Both curves are for $\mu={\text{\unboldmath$\mathrm{10}$}}\,\text{\unboldmath$\mathrm{meV}$}$, $U={\text{\unboldmath$\mathrm{5}$}}\,\text{\unboldmath$\mathrm{meV}$}$ and $\Delta_{T}=0$. The solid (blue) curve is for $\sigma_{+}(\Omega)$ and the short dashed (green) curve is for $\sigma_{-}(\Omega)$, and while the value of the chemical potential is kept the same, $\mu={\text{\unboldmath$\mathrm{10}$}}\,\text{\unboldmath$\mathrm{meV}$}$, the gaps are now $U=0$ and $\Delta_{T}={\text{\unboldmath$\mathrm{5}$}}\,\text{\unboldmath$\mathrm{meV}$}$. For relatively small values of the gaps, $\mathrm{5}$ $\mathrm{meV}$, the energies of the levels with $n\neq 0,1$ remain practically intact, viz. $|E_{\pm 2}|=M_{2}\approx{\text{\unboldmath$\mathrm{16{.}1}$}}\,\text{\unboldmath$\mathrm{meV}$}$, $|E_{\pm 3}|=M_{3}\approx{\text{\unboldmath$\mathrm{27{.}1}$}}\,\text{\unboldmath$\mathrm{meV}$}$ and $|E_{\pm 4}|=M_{4}\approx{\text{\unboldmath$\mathrm{37{.}9}$}}\,\text{\unboldmath$\mathrm{meV}$}$. Accordingly, the last two peaks from the right remain practically unchanged, the first from the right peak with the energy $M_{3}+M_{4}$ shifts in energy from $\mathrm{64{.}2}$ $\mathrm{meV}$ to $\mathrm{65}$ $\mathrm{meV}$, and the second from the right peak with the energy $M_{2}+M_{3}$ shifts in energy from $\mathrm{41{.}9}$ $\mathrm{meV}$ to $\mathrm{43{.}2}$ $\mathrm{meV}$. The most essential changes occur due to the partial removal of the degeneracy of the lowest Landau level.

Since we took $\mu={\text{\unboldmath$\mathrm{10}$}}\,\text{\unboldmath$\mathrm{meV}$}$ all curves in Fig. 2 have to be compared with the two curves in Fig. 1 plotted for the same value of the chemical potential, viz. the long dashed $\sigma_{+}(\Omega)$ curve and the dash-dotted $\sigma_{-}(\Omega)$ curve.

Let us begin with the $\sigma_{+}(\Omega)$ polarization. We observe that for the gap $U={\text{\unboldmath$\mathrm{5}$}}\,\text{\unboldmath$\mathrm{meV}$}$ the peak at $\mathrm{15{.}3}$ $\mathrm{meV}$ corresponding to the transition from $E_{0,1}$ to $E_{2}$ levels is split into two peaks at $\mathrm{11{.}1}$ $\mathrm{meV}$ and $\mathrm{21{.}1}$ $\mathrm{meV}$. These new peaks correspond to the transitions from the levels $E_{0,1}=U$ and $E_{0,-1}=-U$ to $E_{2}$ level, respectively. For the time-reversal symmetry breaking gap, $\Delta_{T}={\text{\unboldmath$\mathrm{5}$}}\,\text{\unboldmath$\mathrm{meV}$}$, we observe that there is only one peak at $\mathrm{21{.}1}$ $\mathrm{meV}$ which corresponds to the transition from the only level $E_{0,-1}=-\Delta_{T}$ to $E_{2}$. One can verify that the position of the peak is sensitive to the sign of the gap, i.e. for $\Delta_{T}=-{\text{\unboldmath$\mathrm{5}$}}\,\text{\unboldmath$\mathrm{meV}$}$ the position of the peak is at $\mathrm{11{.}1}$ $\mathrm{meV}$, because there is the only level at the positive energy, $E_{0,1}=-\Delta_{T}>0$.

For the $\sigma_{-}(\Omega)$ polarization, the discussed above transitions are forbidden. The only small difference is observed between the height of the second from the right peaks in the solid (blue) and short dashed (green) curves. These curves interchange when the sign of the gap $\Delta_{T}$ is reversed to $-\Delta_{T}$.

It should be noted that although a measurement of the optical Hall conductivity $\sigma_{xy}(\Omega)$ allows one to distinguish holes and electrons,[55] some essential features such as splitting of the absorption peak into two peaks depending on the type of the gap remain present even in the diagonal conductivity $\sigma_{xx}(\Omega)$.

The obtained results allow us to conclude that the investigation of the optical magneto-conductivity may provide an additional insight into the nature of the gaps in the bilayer graphene. The features discussed here could be observed if the condition $\Gamma<|\Delta_{\xi}|$ is satisfied. Since our value $\Gamma={\text{\unboldmath$\mathrm{2}$}}\,\text{\unboldmath$\mathrm{meV}$}$ is almost 5 times smaller than that used in Ref. 45 for fitting the data on the gated bilayer at $B=0$, the observation of the discussed here effects requires samples of a much better quality.

Starting from Eq. (14) one can reproduce correctly $B\to 0$ limit. Similarly to the consideration done for monolayer graphene,[33] introducing a continuum variable $\omega$ instead of $M_{n}$ given by Eq. (6) and replacing the sum over $n$ by the integral, in the limit $\Gamma\to 0$ for the real part of the diagonal optical conductivity $\sigma_{xx}(\Omega)=(\sigma_{+}(\Omega)+\sigma_{-}(\Omega))/2$ one obtains

Here we also included the dependence of the gap $\Delta_{\xi s}$ on the spin variable $s$. The first term in curved brackets corresponds to the Drude peak while the second one describes interband electron-photon scattering processes.

Finally, at $T=0$ we obtain

This expression for longitudinal optical conductivity is similar to that obtained for single layer graphene[20] and for topological insulators.[56] It is worth noting that there is a more deep analogy between the band structures and optical conductivities of the single layer graphene with Rashba term and biased bilayer graphene.[57]

Similarly, one can consider the $B\to 0$ limit for the optical Hall conductivity, $\sigma_{xy}(\Omega)=(\sigma_{+}(\Omega)-\sigma_{-}(\Omega))/(2i)$ which takes the form

Note that the last expression is an even function of chemical potential $\mu$ in contrast to the Hall conductivity in a magnetic field which is an odd function of $\mu$. It is clear that $\sigma_{xy}(\Omega)\neq 0$ only if the time-reversal symmetry breaking gaps $U_{T}$ or $\Delta_{T}$ are nonzero. For zero temperature we obtain

Equation (18) resembles the result obtained for topological insulators. [56] Making the change of the variable, $s\rightarrow\xi s$, in Eqs.(16) and (18), one can see that both conductivities are invariant under the interchange $U_{T}\leftrightarrow\Delta.$ The real part of the dc Hall conductivity takes the form

which at $\mu=0$, $U=U_{T}=\Delta=0$, and $\Delta_{T}\neq 0$ is in agreement with the corresponding expression in Ref. 58. At the neutral point, $\mu=0$, we have,

Clearly, the factor $\nu$ can take the values $\nu=0,\pm 2,\pm 4$ depending on the relations between the gaps $U,U_{T},\Delta_{T},\Delta$ and their signs. The QAH gap $\Delta_{T}$ and LAF gap $U_{T}$ break time-reversal symmetry and this is the necessary condition for observation of nonzero Hall conductivity. For example, $\nu=4$ is realized in the case $\Delta_{T}>0,U=U_{T}=\Delta=0$ (for $\Delta_{T}<0$, obviously, $\nu=-4$), thus QAH phase has a zero-field quantized charge Hall conductivity. In the QSH phase with $\Delta\neq 0$ and $U=U_{T}=\Delta_{T}=0$, two spin components have opposite Hall conductivity, hence zero charge Hall conductivity. On the other hand, in this phase a spin Hall conductivity $\sigma_{xy}(s=+)-\sigma_{xy}(s=-)$ is nonzero and quantized. For $\Delta_{T}=0$ and taking without the loss of generality all other gaps positive, we obtain $\nu=-2$ if the following conditions are satisfied

[$\nu=2$ is obtained if we invert the signs of all three gaps and replace the gap values by their absolute values]. The second inequality in (21) arises as a consequence of the symmetry $\sigma_{xy}(\Omega)$ under the interchange $U_{T}\leftrightarrow{\Delta}$.

In Fig. 3 we plotted the real part of frequency dependent Hall conductivity at zero temperature and scattering rate $2\Gamma={\text{\unboldmath$\mathrm{5}$}}\,\text{\unboldmath$\mathrm{meV}$}$. The scattering rate $\Gamma$ is introduced by replacing $i0$ in Eq.(18) by $2i\Gamma$. In Fig. 3 two different cases for time-reversal breaking gaps are presented: 1) Long dashed (red) line is for a finite QAH gap $\Delta_{T}={\text{\unboldmath$\mathrm{5}$}}\,\text{\unboldmath$\mathrm{mev}$}$, all the other gaps are zero, $U=U_{T}=\Delta=0$, and 2) For solid (blue) line the gaps are taken $\Delta_{T}=0$, $U={\text{\unboldmath$\mathrm{5}$}}\,\text{\unboldmath$\mathrm{mev}$}$, $U_{T}={\text{\unboldmath$\mathrm{4}$}}\,\text{\unboldmath$\mathrm{mev}$},\Delta={\text{\unboldmath$\mathrm{1{.}5}$}}\,\text{\unboldmath$\mathrm{mev}$}$. Since the two-band model is valid for $\Omega\leq\gamma_{1}/4\sim{\text{\unboldmath$\mathrm{0{.}1}$}}\,\text{\unboldmath$\mathrm{eV}$}$, only this range of energies is shown in the figure.

These two types of time-reversal symmetry breaking gaps in bilayer graphene are expected to manifest themselves through a nonzero dc Hall response in the absence of a magnetic field. At finite frequencies, the behavior of the Hall conductivities differs essentially for two types of gaps: the curve for nonzero gap $\Delta$ crosses zero, while the curve for $\Delta_{T}\neq 0$ does not. The crossing takes place in infrared range of frequencies, while at near-infrared and optical frequencies, $\Omega\sim\gamma_{1}/4\gg|\Delta_{\xi s}|$, the Hall conductivity is small and of order ${\rm Re}\sigma_{xy}\sim\sum_{\xi,s}\Delta_{\xi s}|\Delta_{\xi s}|/\Omega^{2}$ [see, Eq. (18)]. Nevertheless, it is possible to observe such gaps in a Hall response at optical frequencies as was suggested in Ref. 48. The point is that the Hall conductivity at optical frequencies is dominated by transitions to the high-energy bands and can be manifested in the polar Kerr or Faraday rotation. However, to make reliable calculations, it is necessary to move up to the four-band model. We study the optical Hall conductivity in the four-band model in Sec. IV.