Pauli-limited upper critical field in dirty $d$-wave superconductors

We calculate the Pauli-limited upper critical field and the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) instability for dirty $d$-wave superconductors within the quasiclassical theory using the self-consistent $\widehat{t}$-matrix approximation for impurities. We find that the phase diagram depends sensitively on the scattering rate and phase shift of nonmagnetic impurities. The transition into the superconducting state is always second order for weak (Born) scattering, while in the unitarity (strong) scattering limit a first-order transition into both uniform and spatially modulated superconducting states is stabilized. Contrary to general belief, we find that the FFLO phase is robust against disorder and survives impurity scattering equivalent to a $T_c$ suppression of roughly 40%. Our results bear on the search of FFLO states in heavy-fermion and layered organic superconductors.

Figure 1

(Color online) The critical fields of Pauli-limited dirty $d$-wave superconductors for transitions to USC (dashed lines, first order) and FFLO states (solid lines, second order) for $\mathbf{q}$ along nodes (squares) and antinodes (circles). Superconductivity sets in at the highest $B_{c2}$. (a) Pure case: Below $T_P=T_{\mathbf{q}\ne 0}$ the second-order N-FFLO transition is above the N-USC transition. (b) Impurities quickly suppress $B_{c2}\left(\mathbf{q}\parallel \text{antinode}\right)$ relative to $B_{c2}\left(\mathbf{q}\parallel \text{node}\right)$, and split $T_P$ and $T_{\mathbf{q}\ne 0}$. (c),(d) Born and unitarity impurities move $T_P$ in opposite directions, thus $T_P^B<T_{\mathbf{q}\ne 0}<T_P^u$. For Born impurities the second-order FFLO transition is above the N-USC line at all temperatures, as for the clean case. In the unitarity limit the first-order N-USC transition preempts a N-LO transition.

Figure 2

(Color online) The phase diagram for $\Gamma /\pi T_{c0}=0.14$. Left panel: transition into the LO state at low $T$ becomes first order above $T_{\text{LO}}^{\text{I-II}}$ for unitarity impurities. There is also a region of a first-order transition into the USC state below $T_P$. Right panel: Born impurities result in second-order transitions. The LO state is favored in large parts of the phase diagram over the FF state, except near $T_{\mathbf{q}\ne 0}$ (Refs. 7, 9). The dotted line denotes the expected transition between the LO and USC states; its exact location is beyond the GL theory.

Figure 3

(Color online) Phase transitions along the $B_{c2}$ line in the $\Gamma \text{−}{T}_c$ plane. The (black) solid line is $T_c\left(\Gamma \right)$ at $B=0$. For Born impurities the LO state exists for $T<T_{\mathbf{q}\ne 0}$, with a small region occupied by the FF state. For unitarity scattering a first-order transition appears in the USC and LO states. At low $T$ a square lattice FFLO state (SQ) (Ref. 19) is rapidly suppressed with increasing $\Gamma$ (Ref. 23).