Unconventional superconducting states of interlayer pairing in bilayer and trilayer graphene

We develop a theory for interlayer pairing of chiral electrons in graphene materials which results in an unconventional superconducting state with an $s$-wave spin-triplet order parameter. In a pure bilayer graphene, this superconductivity exhibits a gapless property with an exotic effect of temperature-induced condensation causing an increase of the pairing amplitude with increasing temperature. We find that a finite doping opens a gap in the excitation spectrum and weakens this anomalous temperature dependence. We further explore the possibility of realizing a variety of pairing patterns with different topologies of the Fermi surface, by tuning the difference in the doping of the two layers. In trilayer graphene, the interlayer superconductivity is characterized by a two-component order parameter which can be used to define two distinct phases in which only one of the components is nonvanishing. For ABA stacking the stable state is determined by a competition between these two phases. On variation of the relative amplitude of the corresponding coupling strength, a first-order phase transition can occur between these two phases. For ABC stacking, we find that the two phases coexist with the possibility of a similar phase transition, which turns out to be second order.

Figure 1

Left panel: Lattice structure of bilayer graphene. Interlayer S correlations in real space are shown by wavy lines which couple electrons of different layers with opposite pseudospin degrees of freedom. Right panel: Interlayer superconductivity in momentum space is shown by coupling of two time-reversed momentum states $|-k\rangle$ and $|k\rangle$ which are located in valleys $K_1$ and $K_2^{\prime }$ and belong to different layers. Since interlayer pairing is antisymmetric with respect to the pseudospin degree of freedom, pairing between components $|A_1\rangle$ and $|B_2\rangle$ is possible.

Figure 2

The quasiparticle excitation spectrum $E_{k\mathit{l}}^{+}$ versus ${\epsilon }_k$ in the S state. The dispersions correspond to the regimes of (a) BCS pairing with no effective Fermi surface for both branches $\mathit{l}=\pm 1$, (b) Sarma pairing with two effective Fermi surfaces for the branch $\mathit{l}=-1$ but no Fermi surface for the other branch $\mathit{l}=1$, (c) pairing with a single Fermi surface in $\mathit{l}=-1$ but no Fermi surface for the other branch $\mathit{l}=1$, and (d) $pn$ pairing with a single Fermi surface in each branch $\mathit{l}=\pm 1$.

Figure 3

For symmetric doping of bilayer graphene, $h=0$: (a) The gap solution ${\Delta }_{\perp 1}$ versus $T$ for different levels of the mean doping $\mu$ at $t_{\perp }=0$. (b) The mean-field phase diagram of the chiral superconductivity showing the dependence on the interlayer hopping energy $t_{\perp }$ and the temperature $T$ for $\mu /g_{\perp 1}=0,0.03,0.06,0.09$ (from bottom to top, respectively). The line of transition from the N to the S phase is shown, where the dashed and solid parts indicate the first- and second-order transitions, respectively.

Figure 4

For asymmetric doping of bilayer graphene, $h\ne 0$: (a) TNonzero solutions of the gap equation ${\Delta }_{\perp 1}$ versus $T$ for different levels of the mean doping $\mu$ at $t_{\perp }=0$. (b) The mean-field phase diagram of the chiral superconductivity showing dependence on the doping difference $h$ and the temperature $T$ for $\mu /g_{\perp 1}=0,0.03,0.06,0.09$ (from top to bottom, respectively). The line of transition from the N to the S phase is shown, where the dashed and solid parts indicate the first- and second-order transitions, respectively.

Figure 5

For asymmetric doping of bilayer graphene, $h\ne 0$, the phase diagram of S states in the $\mu -h$ plane is shown at zero temperature. Solid thin lines indicate the boundaries of different phases and the thick dashed line represents the first-order phase transition from the S to the N state.

Figure 6

Lattice structure of trilayer graphene. (a) Lattice structure of ABA-stacked trilayer graphene in which the sublattices $A_1,B_2$, and $A_3$ from different layers are overlapped (left panel). In the right panel, interlayer correlations between electrons of sublattices $A_1$ and $B_2$ ($B_2$ and $A_3$) from the first and second (second and third) layers are indicated by wavy lines. (b) Lattice structure of ABC-stacked trilayer graphene in which the sublattices $A_1$ and $B_2$ from the first and second layers and also $A_2$ and $B_3$ from the second and third layers are overlapped (left panel). The S correlations between interlayer nearest-neighbor sublattices are shown in the right panel.

Figure 7

Solutions of gap equations ${\Delta }_{\perp (1,2)}$ versus temperature for ABA and ABC stacking at $\mu =0$ and $t_{\perp }=0$.

Figure 8

Finite-temperature thermodynamic potential contours in the ${\Delta }_{\perp 1}$-${\Delta }_{\perp 2}$ plane showing stable points. In the case of ABA stacking, steps (a) $g_{\perp 1}>g_{\perp 2}$ ($g_{\perp 1}/g_{\perp 2}=1.02$), (b) $g_{\perp 1}=g_{\perp 2}$, and (c) $g_{\perp 1}<g_{\perp 2}$ ($g_{\perp 1}/g_{\perp 2}=0.98$) indicate first-order phase transitions between ${\Delta }_{\perp 1}$ and ${\Delta }_{\perp 2}$, whereas in the case of ABC stacking, steps (d) $g_{\perp 1}>g_{\perp 2}$ ($g_{\perp 1}/g_{\perp 2}=3.11$), (e) $g_{\perp 1}=g_{\perp 2}$, and (f) $g_{\perp 1}<g_{\perp 2}$ ($g_{\perp 1}/g_{\perp 2}=0.32$) imply that second-order phase transitions occur.