Critical magnetic fields and electron pairing in magic-angle twisted bilayer graphene

The velocities of the quasiparticles that form Cooper pairs in a superconductor are revealed by the upper critical magnetic field. Here we use this property to assess superconductivity in magic-angle twisted bilayer graphene (MATBG), which has been observed over a range of moiré band-filling, twist angle, and screening environment conditions. An average Fermi velocity can be defined as $v_F^{*}\equiv k_B{T}_c{\ell }_c/\hslash$, where $T_c$ and ${\ell }_c$ are the critical temperature and magnetic length, respectively. An advantage of this definition is that $v_F^{*}$ can be directly extracted from the existing experimental data. Mean-field theory calculations of upper critical fields in model superconductors are consistent with the expectation that Fermi velocities defined in this way are nearly independent of the strength of pairing interaction. Moreover, for fixed strength pairing interaction, minima in $v_F^{*}$ as a function of band filling are coincident with maxima in $T_c$, as expected from the McMillan formula. Since no association between $T_c$ maxima and $v_F^{*}$ minima is present in MATBG experimental data, we argue that the pairing interaction in MATBG is strongly filling-factor dependent. Any theory of MATBG superconductivity must explain this dependence, which is apparently primarily responsible for the observed superconducting domes.

Figure 1

(a) Schematic diagram of a moiré Brillouin zone (MBZ) and four shells of reciprocal lattice vectors. (b) Band structures of MATBG for different values of flat-band filling $\nu$ after including the HF self-energy corrections. These results are calculated by choosing twist angle $\theta =1.{05}^{\circ }$, interlayer tunneling ratio $\eta =w_0/w_1=0.7$, nonlocal tunneling coefficient $\xi =0.1$, gate-sample distance $d_s=30$ nm, and dielectric constant $\epsilon =25$.

Figure 2

Finite-wave-vector superconductivity calculated within the bare band structure model of MATBG for twisting angle $\theta =1.{07}^{\circ }$. (a) Fourier coefficients of the real-space pairing potential expanded up to four shells of the reciprocal lattice vectors, which are generated by acting symmetry operations $\widehat{P}$ from point group ${\mathcal{C}}_6$ on ${\mathit{Q}}_i$ depicted in Fig. 1 with ${\mathit{Q}}_0=0$. The inset shows ${\mathrm{\Delta }}_{\widehat{P}{\mathit{Q}}_1}$ for flat-band filling $\nu =-1$ at small pairing wave vector $q_x$, where the dashed curve corresponds to ${\mathrm{\Delta }}_{{\mathit{Q}}_1}$. (b) Superconducting condensation energy normalized per moiré supercell versus $q_x$ for different band fillings. The circles denote numerical results and solid curves are fits to Eq. (13). (c) Color scale plot of $-\delta F\left(\mathit{q}\right)$ within the MBZ (dashed hexagon) at $\nu =-1$. These results are calculated by choosing $\eta =0.7, \xi =0$, and $u=40\mathrm{meV}{\mathrm{nm}}^2$ that yields $T_c\sim 1.7$ K.

Figure 3

(a)–(c) Zero-$q$ critical temperature $T_c$, perpendicular critical magnetic field $H_{c2}$, and average Fermi velocity $v_F^{*}$ as functions of band filling $\nu$ for several strengths of reduced Coulomb repulsion $u$ in units of $\mathrm{meV}{\mathrm{nm}}^2$. These results are calculated for a bare MATBG band structure model with $\theta =1.{07}^{\circ }, \eta =0.7$, and $\xi =0$ (see main text). The dotted lines indicate the values of $\nu$ at which VHSs occur, as plotted in (c). $v_F^{*}$ in (c) exhibits a minimum where $T_c$ in (a) is maximized.

Figure 4

(a)–(c) Zero-$q$ critical temperature $T_c$, perpendicular critical magnetic field $H_{c2}$, and average Fermi velocity $v_F^{*}$ as functions of band filling $\nu$ for several strengths of Coulomb repulsion $u$ in units of $\mathrm{meV}{\mathrm{nm}}^2$. These results are calculated for a MATBG bands that take account of HF self-energy corrections with $\theta =1.{05}^{\circ }, \eta =0.7$, and $\xi =0.1$. The corresponding band-filling-dependent band structures are depicted in Fig. 1. No flavor symmetries are broken. The dotted lines indicate the values of $\nu$ at the dressed band VHSs occur. The dashed line in (c) is the dressed band DOS. $v_F^{*}$ in (c) exhibits minimum versus $\nu$ near each maximum in $T_c$ versus $\nu$.

Figure 5

Critical magnetic fields calculated at zero pairing wave vector by only including Zeeman coupling ($H_P$), and extracted from the critical pairing wave vector $q_c$ with (w/t) and without (w/o) Zeeman coupling. In the presence of Zeeman coupling, $H_{c2}=({\mathrm{\Phi }}_s/2\pi )q_{x,c}{q}_{y,c}$, where $q_{x,c}$ and $q_{y,c}$ are the critical pairing wave vectors along $x$ and $y$ directions. These results are obtained by choosing $\theta =1.{07}^{\circ }, \eta =0.7, \xi =0$, and $u=30\mathrm{meV}{\mathrm{nm}}^2$ without including the HF self-energy correction.

Figure 6

(a) Fermi-surface averaged electron-acoustic phonon spectral function ${\alpha }^{2}F\left(\omega \right)$ at the valence band VHS. The dotted and solid lines are results obtained without (w/o) umklapp processes and with (w/t) umklapp processes up to the fourth shell characterized by ${\mathit{Q}}_4$ in Fig. 1. (b) Electron-acoustic phonon coupling strength $\lambda$ and the estimated $T_c$ for MATBG with $\theta =1.{07}^{\circ }$ and $\eta =0.7$. The dotted vertical lines in (b) show the positions of the VHSs.

Figure 7

The effects of (a) Hartree, (b) Fock, and (c) Hartree-Fock self-energies on the flat bands of MATBG with $\theta =1.{05}^{\circ }$ at band filling $\nu =\pm 4$. The dashed curves show the noninteracting flat bands, where the particle-hole asymmetry is attributed to the nonlocal interlayer tunneling described by Eq. (3). These calculations are performed by choosing $\eta =0.7, \xi =0.1, d_s=30$ nm, and the effective dielectric constant $\epsilon =25$.

Figure 8

(a)–(c) $T_c, H_{c2}$, and $v_F^{*}$ as functions of chemical potential $\mu$ measured from charge neutrality for each value of $\eta$. In these calculations, we choose $\theta =1.{07}^{\circ }$ and $u=30\mathrm{meV}{\mathrm{nm}}^2$. (d)–(f) $T_c, H_{c2}$, and $v_F^{*}$ as functions of $\mu$ for several different twist angles. In these calculations, we choose $\eta =0.7$ and $u=20\mathrm{meV}{\mathrm{nm}}^2$. Vertical arrows in (c) and (f) highlight the positions of flat-band VHSs.