Excitation gap of a graphene channel with superconducting boundaries

We calculate the density of states of electron-hole excitations in a superconductor–normal-metal–superconductor (SNS) junction in graphene, in the long-junction regime that the superconducting gap ${\Delta }_0$ is much larger than the Thouless energy $E_T=\hslash v∕d$ (with $v$ the carrier velocity in graphene and $d$ the separation of the NS boundaries). If the normal region is undoped, the excitation spectrum consists of neutral modes that propagate along the boundaries—transporting energy but no charge. These “Andreev modes” are a coherent superposition of electron states from the conduction band and hole states from the valence band, coupled by specular Andreev reflection at the superconductor. The lowest Andreev mode has an excitation gap of $E_0=\frac{1}{2}(\pi -\mid \varphi \mid )E_T$, with $\varphi ∊(-\pi ,\pi )$ the superconducting phase difference. At high doping (Fermi energy $\mu ⪢E_T$) the excitation gap vanishes $\propto E_0{(E_T∕\mu )}^2$, and the usual gapless density of states of Andreev levels is recovered. We use our results to calculate the $\varphi$ dependence of the thermal conductance of the graphene channel.

Figure 1

Three types of states in an SNS channel in graphene. The solid and dashed lines show the classical trajectories of an electron (filled circle) and a hole (open circle), converted into each other upon Andreev reflection at the superconductor. The transition from a localized Andreev level (a) to a propagating Andreev mode (b) occurs when the excitation energy $\epsilon$ becomes larger than the Fermi energy $\mu$ in the normal region. These two types of states are both charge neutral. Purely electronic states (c) exist near grazing incidence. While the states of type (a) and (b) are sensitive to the phase difference $\varphi$ of the two superconductors, the states of type (c) are not.

Figure 2

Dispersion relation of the SNS junction, calculated numerically from Eq. (7) for three values of the superconducting phase difference $\varphi$ at $\mu ∕E_T=0.1$. The lowest modes ${\epsilon }_n^{\pm }\left(q\right)$ with $n=0,1,2$ are nearly degenerate for $\varphi =0$ and nondegenerate for $\varphi =\pi ∕2$ (thicker lines correspond to ${\epsilon }_n^{+}$). For $\varphi =\pi$ all modes are nearly degenerate except the lowest one ${\epsilon }_0^{-}$.

Figure 3

Same as in Fig. 2 for $\mu ∕E_T=100$. The smoothed dispersion relation (14) is indicated by dashed lines.

Figure 4

(Color online) Dispersion relation of the SNS junction, calculated numerically from Eq. (7) for $\varphi =0$ and $\mu ∕E_T=100$. The curves show ${\epsilon }_n^{+}\left(q\right)$ with $n=5,10,15,\dots 60$. The three types of states from Fig. 1 are color coded; $\text{red}=\text{type}$ $a$, $\text{blue}=\text{type}$ $b$, $\text{green}=\text{type}$ $c$.

Figure 5

(Color online) Double-logarithmic plot of the energy dependence of the excitation gap, calculated numerically from Eq. (7), for three different values of the superconducting phase difference $\varphi$ (colored lines). The straight black line is the asymptote $E_{\text{gap}}∕E_0\left(\varphi \right)\propto {(\pi E_T∕\mu )}^2$.

Figure 6

Thermodynamic limit $E_T=\hslash v∕d\to 0$ of the density of states $\rho$ of the SNS junction, according to Eq. (24) with ${\rho }_0=4\mathcal{L}∕\pi \hslash v$. The total density of states $\rho$ (solid) is the sum of the density ${\rho }_e\propto \mu +\epsilon$ of electron states (dashed) and the density ${\rho }_h\propto \mid \mu -\epsilon \mid$ of hole states (dotted).

Figure 7

Density of states of the SNS junction in the low-doping regime, for superconducting phase difference $\varphi =0$ (solid curves) and $\varphi =\pi$ (dashed curves). The curves are calculated from Eq. (26), normalized by ${\rho }_0=4\mathcal{L}∕\pi \hslash v$. The excitation gap for $\varphi =0$ is at $E_0=\pi E_T∕2$.

Figure 8

Smoothed density of states of the SNS junction in the high-doping regime for $\varphi =0$, calculated from Eq. (31).

Figure 9

A temperature difference $\Delta T=T_L-T_R$ between the two ends of the graphene channel drives a heat current $I_Q$, carried by Andreev modes in the normal region at temperatures below the gap ${\Delta }_0$ in the superconductors.

Figure 10

Thermal conductance of the SNS junction in the low-doping, low-temperature regime $(\mu ,k_B{T}_0⪡E_T)$, calculated as a function of the superconducting phase difference $\varphi$ (modulo $2\pi$) from Eq. (37). The peak value equals $G_{\text{peak}}=\pi k_B^2{T}_0∕3\hslash$, which is half of the thermal conductance quantum (Refs. 22, 23, 24) per spin and valley degree of freedom.