Effects of electron-phonon coupling on Landau levels in graphene

We calculate the density of states (DOS) in graphene for electrons coupled to a phonon in an external magnetic field. We find that coupling to an Einstein mode of frequency ${\omega }_E$ not only shifts and broadens the Landau levels (LLs), but radically alters the DOS by introducing a new set of peaks at energies $E_n\pm {\omega }_E$, where $E_n$ is the energy of the $n$th LL. If one of these new peaks lies sufficiently close to an LL, it causes the LL to split in two; if the system contains an energy gap, an LL may be split in three. The new peaks occur outside the interval $(-{\omega }_E,{\omega }_E)$, leaving the LLs in that interval largely unaffected. If the chemical potential is greater than the phonon frequency, the zeroth LL lies outside the interval and can be split, eliminating its association with a single Dirac point. We find that coupling to an extended phonon distribution such as a Lorentzian or Debye spectrum does not qualitatively alter these results.

Figure 1

The bare DOS with a magnetic field (solid black) and without (dashed red), along with the corresponding real and imaginary parts of the self-energy calculated from those bare densities. Integers label the bare Landau levels. Here $A=50$ meV, ${\omega }_E=200$ meV, $\Gamma =5$ meV, and the black curves are calculated for $B=15.2$T. Note that $N_0$ has dimensions of inverse energy squared, so the quantity $N/N_0$ has dimensions of energy rather than the density of states’ usual dimensions of inverse energy.

Figure 2

Density of states and self-energy as a function of energy, for $A=50$ meV, ${\omega }_E=200$ meV, $B=15.2$ T, and $\Gamma =5$ meV. The dashed red curve shows the bare DOS, while the solid black curve shows the DOS renormalized by $\Sigma$. Integers $n$ label the dressed Landau levels. Integers $m_p$ label the phonon peaks, pointed to by blue arrows.

Figure 3

The sum of states ${\int }_0^{\omega }N\left({\omega }^{\prime }\right)d{\omega }^{\prime }$ as a function of energy for $B=15.2$ T (solid black curves) and $B=0$ (dashed red curves) in both the bare (left) and dressed (right) cases. The remaining parameters are $A=250$ meV, ${\omega }_E=200$ meV, and $\Gamma =5$ meV. The number of states is written in units of ${10}^5$ meV${}^2{N}_0$.

Figure 4

Locations of peaks in the density of states, as determined by $\omega -{\Sigma }_1\left(\omega \right)$, for $A=250$ meV, ${\omega }_E=50$ meV, $B=12.2$ T, and $\Gamma =4$ meV. Straight lines indicate the bare Landau-level energies $M_n$ (dotted black lines), the energies of renormalized Landau levels $E_n$ (dot-dashed blue), and the positions $P_m$ of phonon peaks (dashed red). Blue circles mark the intersection of $\omega -{\Sigma }_1$ with $M_n$, which determines the energies $E_n$; since these intersections occur twice for $M_3$ and $M_5$, the third and fifth LLs are split. Red circles mark the center points of oscillations in $\Sigma$, which determine the positions of phonon peaks. Those red circles outlined in blue also satisfy the equations $\omega -{\Sigma }_1=M_n$ (for $n=3$ and $n=5$).

Figure 5

The effect of $\Gamma$ on the number of peaks in the density of states, showing a small portion of Fig. 4, where the dashed black curves here are identical to the solid black curves therein, and the solid green curves are for a halved value of $\Gamma$. The curve with $\Gamma =2$ meV intersects two more $M_n$'s than does the curve with $\Gamma =4$ meV, creating two more split Landau levels.

Figure 6

A schematic depiction of the locations of energy levels on the Dirac cone for three increasing values of $B$. Solid colored lines labeled with integers $n$ or $n^{\pm }$ indicate Landau levels. Dashed black lines labeled with integers $m_p$ or $m_p^{\pm }$ indicate energies of phonon peaks.

Figure 7

(Top) Density of states as a function of energy for a sequence of magnetic field strengths, with curves offset by 500 meV for clarity. Black arrows point to the first two sets of phonon peaks, at $\pm {\omega }_E$ and $\pm ({\omega }_E+M_1)$. Integers label the first five sets of Landau levels. In all cases, ${\omega }_E=200$ meV, $A=250$ meV, and $\Gamma =5$ meV. (Bottom) Locations of peaks in the DOS as a function of $\sqrt{B}$, with the same parameters. The solid curves labeled $E_{n^{\pm }}$ show the positions of the renormalized Landau levels. The thin dashed lines show $M_n/(1+{\lambda }_0)$, where ${\lambda }_0\equiv {\lambda }^{\mathrm{eff}}(B=0)$. The thick black dashed lines show the first two phonon peaks. (Inset) The fractional change in ${\lambda }^{\mathrm{eff}}$ away from ${\lambda }_0$ as a function of $B$, for ${\omega }_E=150$ meV (dashed red curve), ${\omega }_E=200$ meV (solid black), and ${\omega }_E=250$ meV (double-dot dashed blue).

Figure 8

Density of states and self-energy in the Einstein model (dashed red curves) and in the truncated Lorentzian model (solid black curves) with a magnetic field $B=15.2$T. The Landau levels that are split about ${\omega }_E$ in the Einstein model are flattened by the broadening of the phonon in the Lorentzian model. In the Einstein model we use $A=250$ meV and ${\omega }_E=200$ meV; in the Lorentzian, ${\omega }_0=200$ meV, $A^{\prime }=555$ meV, $\delta =15$ meV, and ${\delta }_c=30$ meV. The value of $A^{\prime }$ is chosen to ensure that ${\lambda }^{\mathrm{eff}}=0.19$ in both models. In both cases, $\Gamma =5$meV.

Figure 9

Density of states and self-energy after one iteration (dashed red curves) and after iterating to convergence (solid black curves), with parameters $B=15.2$ T, $A=250$ meV, ${\omega }_E=200$ meV, and $\Gamma =5$ meV.

Figure 10

(Top) Density of states as a function of energy for three finite values of $\mu$: 150 meV (top), 50 meV (middle), and $-$150 meV (bottom), chosen to lie in the intervals ${\mu }_0>\sqrt{2B}$, $-\sqrt{2B}<{\mu }_0<\sqrt{2B}$, and ${\mu }_0<-\sqrt{2B}$, respectively. Arrows point to the locations of the phonon peaks. In all cases, $B=15.2$ T, ${\omega }_E=200$ meV, $A=250$ meV, and $\Gamma =5$ meV. (Bottom) The locations of the phonon peaks for ${\mu }_0=150$ meV (left) and ${\mu }_0=-150$ meV (right). The colors of the curves match those of the arrows in the top set of plots. Dotted lines indicate $\pm {\omega }_E$.

Figure 11

The arrangement of energy levels on the Dirac cone in the cases $\mu >0$ (top) and $\mu <0$ (bottom) for three increasing values of magnetic field. The shaded region indicates filling up to the Fermi energy. Landau levels are indicated by solid lines labeled with integers $n$ or $n^{\pm }$, and phonon peaks by dashed lines labeled with integers $m_p$ or $m_p^{\pm }$.

Figure 12

Bare (thin dashed blue) and renormalized (solid black) density of states and the real part of the self-energy in the case of $\mu >{\omega }_E$. Arrows point to the (negative of the) bare and renormalized chemical potentials ${\mu }_0=340$ meV and $\mu =291$ meV, respectively. The remaining parameters are ${\omega }_E=200$ meV, $A=250$ meV, $B=15.2$ T, and $\Gamma =5$ meV. We also display the bare (dot-dashed magenta) and renormalized (thick dashed red) curves in the $B=0$ case.

Figure 13

A small portion of Fig. 12 focusing on the region around $-\mu$. Here we see that the three peaks near $-\mu$ consist of a phonon peak and the split $n=0$ Landau level. None of the three lies at $-\mu$.

Figure 14

Color maps of the bare (top) and dressed (bottom) spectral density $\mathrm{tr}\left[{\gamma }^{0}A(\vec{k},\omega )\right]$ in the $(k,\omega )$ plane for ${\mu }_0=-340$ meV, ${\omega }_E=200$ meV, $A=250$ meV, $B=15.2$ T, and $\Gamma =3$ meV. The plane is extended to negative values of $k$ in order to show the conelike structure.

Figure 15

Density of states with a finite energy gap $\Delta =10$ meV for three values of ${\mu }_0$: 150 meV (top), 0 meV (middle), and $-$150 meV (bottom), chosen to lie in the intervals ${\mu }_0>\Delta$, $-\Delta <{\mu }_0<\Delta$, and ${\mu }_0<-\Delta$, respectively. Red arrows point to the locations of the phonon peaks around $\pm {\omega }_E$, shown in the insets. For $-\Delta <{\mu }_0<\Delta$, these peaks lie at $\pm ({\omega }_E+\Delta )-{\mu }_0$; for ${\mu }_0>\Delta$, at $-{\omega }_E\pm \Delta -{\mu }_0$; for ${\mu }_0<-\Delta$, at ${\omega }_E\pm \Delta -{\mu }_0$. In all cases, ${\omega }_E=200$ meV, $A=250$ meV, $B=15.2$ T, and $\Gamma =5$ meV.

Figure 16

A density of states curve displaying a Landau level ($n=-3$) split in three by a phonon peak that is itself split in two by an energy gap. Here ${\mu }_0=15$ meV, $\Delta =10$ meV, $A=250$ meV, ${\omega }_E=200$ meV, $\Gamma =3$ meV, and $B=15.2$ T.

Figure 17

The bare density of states (dashed red curves) and the density of states renormalized by an acoustic phonon (solid black curves), for three values of ${\mu }_0$: 15 meV (top), 0 meV (middle), and $-$15 meV (bottom). In all cases, $B=15.2$ T, $\Gamma =5$ meV, and the parameters of the Debye distribution are ${\omega }_D=20$ meV and $A^{\prime }=0.03125$ meV${}^{-2}$.