Impact of disorder on unconventional superconductors with competing ground states

Nonmagnetic impurities are known as strong pair breakers in superconductors with pure $d$-wave pairing symmetry. Here we discuss $d$-wave states under the combined influence of impurities and competing instabilities, such as pairing in a secondary channel as well as lattice symmetry breaking. Using the self-consistent $T$-matrix formalism, we show that disorder can strongly modify the competition between different pairing states. For a $d$-wave superconductor in the presence of a subdominant local attraction, Anderson’s theorem implies that disorder always generates an $s$-wave component in the gap at sufficiently low temperature, even if a pure $d_{x^2-y^2}$ order parameter characterizes the clean system. In contrast, disorder is always detrimental to an additional $d_{xy}$ component. This qualitative difference suggests that disorder can be used to discriminate among different mixed-gap structures in high-temperature superconductors. We also investigate superconducting phases with lattice symmetry breaking in the form of bond order and show that the addition of impurities quickly leads to the restoration of translation invariance. Our results highlight the importance of controlling disorder for the observation of competing order parameters in cuprates.

Figure 1

(Color online) Superconducting gaps ${\Delta }_d$ (continuous lines) and ${\Delta }_s$ (broken lines) as a function of $n_{\mathrm{imp}}$ for various values, from left to right, of the $s$-wave pairing $V_s=-0.45$ (mixed region), $V_s=-0.40$ ($s\to s+id$ critical point), $V_s=-0.3$ (clean $d$ phase with $s$ component induced by disorder), and $V_s=-0.1$ (pure $d$ phase). Here $V_d=-1.0$ is fixed and the inverse temperature is taken as $\beta =1000$.

Figure 2

(Color online) Phase diagram $(\mid V_s\mid ,n_{\mathrm{imp}})$ of the dirty $s+i{d}_{x^2-y^2}$ superconductor for $\beta =1000$. The thin dashed lines refer to the gaps plotted in Fig. 1. The inset illustrates the evolution of the phase boudaries with temperature, here shown for $\beta =100$ (broken lines) and $\beta =1000$ (continuous lines).

Figure 3

(Color online) Zero-temperature phase diagram $(\mid V_s\mid ,n_{\mathrm{imp}})$ in the dirty $s+i{d}_{x^2-y^2}$ case. The thick line stands for the limited pure $d_{x^2-y^2}$ region. Compare with the finite temperature plot displayed in the inset of Fig. 2.

Figure 4

Superconducting gaps ${\Delta }_d$ (continuous line) and ${\Delta }_{xy}$ (broken line) as a function of $n_{\mathrm{imp}}$ for the $d_{xy}$-wave pairing $V_{xy}=-0.47$ (mixed phase in the clean limit), with $V_d=-0.5$ and $\beta =100$.

Figure 5

Phase diagram $(\mid V_{xy}\mid ,n_{\mathrm{imp}})$ of the dirty $d_{xy}+i{d}_{x^2-y^2}$ superconductor, with $V_d=-0.5$ and $\beta =100$. The thin dashed line refers to the gaps drawn in Fig. 4.

Figure 6

Gap structure of a bond-ordered superconductor (dubbed spin Peierls in the following). In general, gaps take complex values, so that time reversal symmetry is also broken.

Figure 7

Superconducting phase diagram of the clean $t\text{−}J$ model for $\beta =100$ (in units of $J=1$), with a two-site unit cell. The transition from $s^{\mathrm{\star }}$ to $d_{x^2-y^2}$ is discontinuous, whereas all other phase boundaries are second order. These results are consistent with Ref. 8.

Figure 8

(Color online) Superconducting gaps as a function of impurity concentration, for $t∕J=1$, $\beta =100$, and $\delta =0.375$. Single (double) primed quantities denote real (imaginary) parts, respectively. Here we chose a gauge where ${\Delta }_1^{\prime }=0$. The initial state at $n_{\mathrm{imp}}=0$ is the bond-ordered superconductor described in the text.

Figure 9

(Color online) Superconducting gaps as a function of impurity concentration, same conditions as in Fig. 8, with $\delta =0.45$ here. The initial state at $n_{\mathrm{imp}}=0$ is $s^{\mathrm{\star }}+i{d}_{x^2-y^2}$.

Figure 10

Superconducting phase diagram of the dirty $t\text{−}J$ model as a function of impurity concentration $n_{\mathrm{imp}}$ and doping $\delta$, for $t∕J=1$ and $\beta =100$. The two dashed lines refer to the cuts in Figs. 8, 9, respectively.