Spontaneous symmetry breaking at surfaces of $d$-wave superconductors: Influence of geometry and surface ruggedness

Surfaces of $d$-wave superconductors may host a substantial density of zero-energy Andreev states. The zero-energy flat band appears due to a topological constraint, but comes with a cost in free energy. We have recently found that an adjustment of the surface states can drive a phase transition into a phase with finite superflow that breaks time-reversal symmetry and translational symmetry along the surface. The associated Doppler shifts of Andreev states to finite energies lower the free energy. Direct experimental verification of such a phase is still technically difficult and controversial, however. To aid further experimental efforts, we use the quasiclassical theory of superconductivity to investigate how the realization and the observability of such a phase are influenced by sample geometry and surface ruggedness. Phase diagrams are produced for relevant geometric parameters. In particular, critical sizes and shapes are identified, providing quantitative guidelines for sample fabrication in the experimental hunt for symmetry-breaking phases.

Figure 1

A $d$-wave superconducting grain at temperature $T=0.1T_{\mathrm{c}}$ with spontaneous superflow (colors) that spontaneously breaks translational (along individual surface segments) and time-reversal symmetries. The Andreev states exist only at the pair-breaking edges, which for this sample the geometry occurs along the nodal directions. There is no superflow at surfaces along the lobe directions, since those surfaces have no Andreev states. The inset shows the vector field ${\mathbf{p}}_{\mathrm{s}}$ (superfluid momentum) with a periodic structure of topological defects [25] in the form of edge sources and sinks.

Figure 2

The surface density of states averaged over one side of a $60{\xi }_0\times 60{\xi }_0$ square grain with $Imz=\delta =0.02k_{\mathrm{B}}{T}_{\mathrm{c}}$. Different curves correspond to different values of $\theta$, as indicated by colors. (a) Above the transition, the peak is narrow (note the logarithmic scale on the ordinate). The steps at $\epsilon \approx 1k_{\mathrm{B}}{T}_{\mathrm{c}}\approx {\mathrm{\Delta }}_0/2$ come from the features in the DOS at the square corners, with ${\mathrm{\Delta }}_0\approx 2.14k_{\mathrm{B}}{T}_{\mathrm{c}}$ being the bulk gap. (b, c) As the temperature is lowered, the midgap states are broadened due to the presence of spontaneous superflow.

Figure 3

(a)–(c) Free-energy difference between phases with and without spontaneous superflow, and (d)–(f) sample-averaged magnitude of the spontaneous superfluid momentum. These quantities are plotted versus the angle $\theta$ between the grain edges and the crystal $\widehat{\mathbf{a}}$ axis in a square grain with area $\mathcal{A}={\left(60{\xi }_0\right)}^2$, as illustrated in the inset in (a). Here, the units are ${\mathrm{\Omega }}_0\equiv \mathcal{A}{N}_{\mathrm{F}}{k}_{\mathrm{B}}^2{T}_{\mathrm{c}}^2$ and $p_0\equiv 2\pi k_{\mathrm{B}}{T}_{\mathrm{c}}/v_{\mathrm{F}}$. Note the varying scales in (a)–(c).

Figure 4

Phase diagram showing the transition temperature $T^{*}$ of the symmetry-broken phase, as a function of the angle $\theta$ between a vacuum-superconductor interface and the $d$-wave crystal $\widehat{\mathbf{a}}$ -axis, in a grain of area $\mathcal{A}={\left(60{\xi }_0\right)}^2$ (see inset). Error bars denote the uncertainty due to the discrete angular resolution. $T^{*}\left(\theta \right)$ is roughly described by the angular dependence of the MGS peak $N_{\mathrm{MGS}}$, denoted $\mathcal{F}\left(\theta \right)$ (dashed line), defined in Eq. (27).

Figure 5

Effect of sample size $\mathcal{L}\times \mathcal{L}$ with maximal pair-breaking edges (see inset) on (a) the transition temperature into the spontaneous superflow phase (circles and squares, left axis) and the superconducting transition temperature of the grain (dashed line, right axis), (b) the heat capacity jump of the spontaneous superflow phase transition, and (c) the sample-averaged current magnitude at $T=0.1T_{\mathrm{c}}$. Error bars denote uncertainty due to (a) discrete resolution in temperature and (b) numerical uncertainty in the heat capacity.

Figure 6

(a) Mesoscopic surface roughness, where the disorder is on the coherence length scale or larger. The roughness is modeled as mesoscopic facets that scatter incoming quasiparticle states specularly. (b) In contrast, atomic surface roughness, where the disorder is on the atomic scale, i.e., generally much smaller than the superconducting coherence length, leads to diffuse scattering of any incoming quasiparticle state. This type of roughness is not considered in this work.

Figure 7

$D$-wave grains with side lengths (a)–(d) $150{\xi }_0$ and (e)–(h) $60{\xi }_0$, and with different degrees of mesoscopic surface roughness (gray boundaries) of the type introduced in Fig. 6. Colors indicate the magnitude of magnetic fields induced by spontaneous currents, with a maximum and minimum flux density of roughly $\pm {10}^{-5}{\mathrm{\Phi }}_0/{\xi }_0^2$, where ${\mathrm{\Phi }}_0\equiv hc/2\left|e\right|$ is the magnetic flux quantum.