Geometric and Conventional Contribution to the Superfluid Weight in Twisted Bilayer Graphene

By tuning the angle between graphene layers to specific “magic angles” the lowest energy bands of twisted bilayer graphene (TBLG) can be made flat. The flat nature of the bands favors the formation of collective ground states and, in particular, TBLG has been shown to support superconductivity. When the energy bands participating in the superconductivity are well isolated, the superfluid weight scales inversely with the effective mass of such bands. For flat band systems one would therefore conclude that even if superconducting pairing is present, most of the signatures of the superconducting state should be absent. This conclusion is at odds with the experimental observations for TBLG. We calculate the superfluid weight for TBLG taking into account both the conventional contribution and the contribution arising from the quantum geometry of the bands. We find that both contributions are larger than one would expect treating the bands as well isolated, that at the magic angle the geometric contribution is larger than the conventional one, and that for small deviations away from the magic angle the conventional contribution is larger than the geometric one. Our results show that, despite the flatness of the bands the superfluid weight in TBLG is finite and consistent with experimental observations. We also show how the superfluid weight can be tuned by varying the chemical potential and the twist angle opening the possibility to tune the nature of the superconducting transition between the standard BCS transition and the Berezinskii-Kosterlitz-Thouless transition.

Figure 1

The dispersion of the lower flat band of TBLG for (a) $\theta =1.05°$ and (b) $\theta =1.00°$. The high symmetric points in the moiré BZ are also shown.

Figure 2

The dispersion of superconducting band and superfluid weight integrand. Left-hand column, $\theta =1.05°$; right-hand column, $\theta =1.00°$. (a),(b) The lowest quasiparticle bands with superconducting gap. (c),(d) $d_{xx}^{\mathrm{s},\mathrm{conv}}(\mathbf{k})$. (e),(f) $d_{xx}^{\mathrm{s},\mathrm{geom}}(\mathbf{k})$. (g),(h) $d_{xx}^{s,\text{total}}(\mathbf{k})$. All the figures are obtained with $\mu =-0.30\mathrm{meV}$ and $T_c=1.6305\mathrm{K}$, $T_c=1.2119\mathrm{K}$ for $\theta =1.05°$, $\theta =1.00°$, respectively.

Figure 3

$D_s$ as a function of the chemical potential for (a) $\theta =1.05°$ and (b) $\theta =1.00°$ at $\mathrm{\Delta }=\mathrm{\Delta }(\mu )$.

Figure 4

$D_s$ as a function of $\mathrm{\Delta }$ for $\theta =1.05°$ (a) and $\theta =1.00°$ (b). $\mu =-0.3\mathrm{meV}$.

Figure 5

(a) $D^s$ versus $T$ for $\theta =1.05°$ and different values of $\mu$: $\mu$ goes from 0 to $-0.5\mathrm{meV}$ along the direction of the arrow. (b) $T_{\mathrm{KT}}/T_c$ as function of $\mu$ for $\theta =1.05°$. (c) Same as (a) for $\theta =1.00°$. (d) $T_{\mathrm{KT}}/T_c$ as function of $\mu$ for $\theta =1.00°$.

Figure 6

(a) $D_s$ and (b) $T_c$ and $T_{\mathrm{KT}}$ as functions of twist angle. $\mu =-0.3\mathrm{meV}$.