Andreev-Bragg Reflection from an Amperian Superconductor

We show how an electrical measurement can detect the pairing of electrons on the same side of the Fermi surface (Amperian pairing), recently proposed by Patrick Lee for the pseudogap phase of high-$T_c$ cuprate superconductors. Bragg scattering from the pair-density wave introduces odd multiples of $2k_F$ momentum shifts when an electron incident from a normal metal is Andreev reflected as a hole. These Andreev-Bragg reflections can be detected in a three-terminal device, containing a ballistic $Y$ junction between normal leads (1, 2) and the superconductor. The cross-conductance $d{I}_1/d{V}_2$ has the opposite sign for Amperian pairing than it has either in the normal state or for the usual BCS pairing.

Figure 1

(a) Density of states $\rho (E,\mathit{k})$ as a function of energy $E$ and momentum $k_y$ integrated over $k_x\in (-\pi ,\pi )$. Only the electron contribution is shown (in units of $1/t$), the full density of states also includes the hole contribution $\rho (-E,-\mathit{k})$ to ensure particle-hole symmetry. (b) Density of states as a function of $k_x$ and $k_y$, integrated over a narrow energy interval around the Fermi level $E=0$. The Amperian pairing takes place near the momenta $\pm {\mathit{K}}_x$ and $\pm {\mathit{K}}_y$ where the free-electron Fermi surface crosses the boundary of the first Brillouin zone (black arc and arrows). The PDW with wave vector $\pm {\mathit{Q}}_x$ or $\pm {\mathit{Q}}_y$ has a periodicity of 8 unit cells. This folds the Brillouin zone, but for clarity the figure shows the bands unfolded (extended zone scheme). Inset: Magnified region of the Brillouin zone near $k_y=-3\pi /8$, showing a minigap.

Figure 2

(a) Total Andreev reflection probability [Eq. (4)] as a function of energy $E$ and incoming transverse momentum $k_y^{\text{in}}$, integrated over the outgoing transverse momenta $k_y^{\text{out}}$. The regions of nonzero $R_{\mathrm{tot}}$ correspond to the gapped regions in Fig. 1. (b) Andreev reflection probability at $E=0$ as a function of incoming and outgoing transverse momenta. The diagonal black lines correspond to momentum shifts $k_y^{\text{out}}=k_y^{\text{in}}+n{Q}_0$, $n\in \mathbb{Z}$. Region I around $E=0$ and $k_y^{\text{in}}=0$ exhibits Andreev-Bragg reflection with transverse momentum shift $Q_0$ between the incoming electron and the outgoing hole. Region II shows transverse momentum shifts of $3Q_0$. [The resonance lines labeled II in panel (a) are interrupted by the finite $k_y$ resolution.] Region III supports Andreev retroreflection (without transverse momentum shift), because the periodic modulation of the order parameter is perpendicular to the normal-superconductor interface. Region IV (white) has no incoming electron modes from the normal metal.

Figure 3

Right panel: Same as Fig. 2, but now as a function of angles $\theta$ of velocities (measured in degrees) rather than transverse momentum $k_y$. The left panel indicates the geometry of the normal-superconductor (NS) interface, with retroreflection corresponding to ${\theta }_{\text{out}}=-{\theta }_{\text{in}}$ and specular reflection to ${\theta }_{\text{out}}={\theta }_{\text{in}}$.

Figure 4

Computer simulation of the differential cross-conductance for the $Y$ junction geometry shown in the inset. The sign of the cross-conductance $d{I}_1/d{V}_2\propto T_{12}-A_{12}$ distinguishes Amperian pairing ($A_{12}>T_{12}$) from BCS pairing ($T_{12}>A_{12}$).

Figure 5

Effect of disorder on the cross-conductance of Fig. 4, for Amperian pairing at $V_2=0$. The data are averaged over four disorder realizations (error bars show standard error of the mean). Data collapse for different system widths $W$ is achieved by rescaling the conductance by a factor $W_0/W$, with $W_0=256$. The negative cross-conductance persists for $W/{\ell }_{\text{mf}}\lesssim 2$.

Figure 6

Effect of an interface barrier (height $E_B$, length $\delta L=3$, width $W=256$) on the cross-conductance of Fig. 4, for Amperian pairing at $V_2=0$. The sign changes occur when the average normal-state transmission probabilities of the barrier are $T_B=0.68$ ($E_B=-0.63t$) and $T_B=0.50$ ($E_B=1.5t$). The $T_B$’s are calculated by averaging over all the incoming modes at different energies, weighted with the Fermi distribution.