Properties of superconductivity on a density wave background with small ungapped Fermi surface parts

We investigate the properties and the microscopic structure of superconductivity (SC) coexisting and sharing the common conducting band with density wave (DW). Such coexistence may take place when the nesting of the Fermi surface (FS) is not perfect, and in the DW state some quasiparticle states remain on the Fermi level and lead to the Cooper instability. The dispersion of such quasiparticle states strongly differs from that without DW, and so do the properties of SC on the DW background. The upper critical field $H_{c2}$ in such a SC state increases as the system approaches the critical pressure, where the ungapped quasiparticles and superconductivity just appear, and it may considerably exceed the usual $H_{c2}$ value without DW. The spin-density wave (SDW) background strongly suppresses the singlet SC pairing, while it does not affect so much the triplet SC transition temperature. The results obtained explain the experimental observations in layered organic metals ${\left(\text{TMTSF}\right)}_2{\text{PF}}_6$ and $\alpha \text{−}{\left(\text{BEDT-TTF}\right)}_2\text{KHg}{\left(\text{SCN}\right)}_4$, where SC appears in the DW states under pressure and shows many unusual properties.

Figure 1

The schematic picture of the phase diagram in ${\left(\text{TMTSF}\right)}_2{\text{PF}}_6$, where superconductivity coexists with DW in some pressure interval above $P_{c1}$ but below $P_c$. ${\text{DW}}_0$ stands for the uniform fully gapped DW. ${\text{DW}}_1$ denotes the DW state when the imperfect nesting term $t_b^{\prime }$ is strong enough as compared to the SDW energy gap ${\Delta }_0$, so that the ungapped FS pockets or nonuniform DW structure appear.

Figure 2

(Color online) The schematic picture of small open pockets on one Fermi surface sheet, which get formed when the antinesting term ${\epsilon }_{+}$ in Eq. (16) exceeds the DW energy gap ${\Delta }_0$. The blue dashed line shows the Fermi surface sheet with imperfect nesting, i.e., with $2t_b>{\Delta }_0$. The green dashed-dotted line shows the other Fermi surface sheet shifted by the nesting vector. If the nesting was perfect, these two lines would coincide. The dotted brown line shows the perfectly nested Fermi surface sheet. The red solid ellipses are the small Fermi surface pockets that appear in the DW state when $2t_b>{\Delta }_0$, i.e., when the pressure exceeds $P_{c1}$.

Figure 3

The schematic picture of the density of states $\rho \left(\epsilon \right)$ near the Fermi level in the DW phase with ungapped FS pockets. On the figure, ${\rho }_0$ is the DoS in the metallic phase, ${\Delta }_0$ is the DW energy gap, and $\delta$ is the size of new ungapped pockets [see Eq. (22)].

Figure 4

The diagram equations for the Gor’kov functions ${\widehat{f}}^{LR}$ and ${\widehat{f}}^{RL}$ without DW. The solid lines represent the electron Green’s functions: $g^{RR\left(LL\right)}\left(\mathit{k},\omega \right)$ in the metallic state. The dashed lines represent the short-range e-e interaction $U_c^f$ or $U_c^b$.

Figure 5

The diagram equations for the Gor’kov functions ${\widehat{f}}^{LR}$ and ${\widehat{f}}^{RL}$ in the presence of the DW ordering with two coupling constants. The solid lines represent the electron Green’s functions $g_n^{R\left(L\right)R\left(L\right)}\left(k,\omega \right)$ in the DW state. The dashed lines represent the short-range e-e interaction, i.e., backward $U_c^b$ or forward $U_c^f$ scattering.

Figure 6

The SC transition temperature $T_{c\text{DW}}^{\text{SC}}$ on the DW background in the pocket scenario. The solid line shows the ratio $T_{c\text{DW}}^{\text{SC}}/T_{c0}^{\text{SC}}$, calculated numerically from Eq. (42), as function of the size $\delta$ of the ungapped Fermi surface pockets. The dashed-dotted line represents analytical formula (47) with the numerically calculated constant $C\approx 1.86$. The approximate experimental value ${\delta }_{\text{max}}$ for ${\left(\text{TMTSF}\right)}_2{\text{PF}}_6$ is denoted by the vertical dashed line.