Andreev Reflection in Scanning Tunneling Spectroscopy of Unconventional Superconductors

We evaluate the differential conductance measured in an STM setting at arbitrary electron transmission between STM tip and a 2D superconductor with arbitrary gap structure. Our analytical scattering theory accounts for Andreev reflections, which become prominent at larger transmissions. We show that this provides complementary information about the superconducting gap structure beyond the tunneling density of states, strongly facilitating the ability to extract the gap symmetry and its relation to the underlying crystalline lattice. We use the developed theory to discuss recent experimental results on superconductivity in twisted bilayer graphene.

Figure 1

Electron transport in a setup where an STM tip is placed over a high-symmetry point of a 2D superconductor. Symmetric blue arrows: the particle wave spreading from the tip carries the symmetry of crystalline lattice. Asymmetric green arrows: the Andreev-reflected hole wave [Eq. (4)] carries information about the superconducting gap symmetry, which may differ from the crystalline one.

Figure 2

Dependence of the normalized differential conductance $G/G_n$ on bias $V$ for weak ($s_0\to 1$, red) and strong ($s_0=0$, blue) tunneling at a high-symmetry point. The conductance is evaluated with the help of Eqs. (9)–(14) for a 2D superconductor with a circular Fermi surface (parameterized by the angle $\phi$) and gap $\mathrm{\Delta }(\mathbf{k})=\mathrm{\Delta }(\phi )$. (a) $s$-wave superconductor; (b) $d$-wave superconductor preserving time-reversal symmetry; $G(V)$ remains linear in the limit $V\to 0$ at any tunneling strength; the van Hove singularity at $s_0\to 1$ is replaced by a Fano resonance (inset) at strong tunneling; (c) $s+d$ gap preserving time-reversal symmetry, but breaking the lattice point symmetry; parameters chosen to preserve the $G(0)=2G_Q$, but significantly shrink the plateau $G(V)<2G_Q$ at $V>0$ compared to the case of $s$-wave superconductor [cf. (a)]; (d) $d+id$ gap preserving point-group symmetry, but breaking time-reversal symmetry; $G(V)$ remains zero below the gap at any $s_0$; (e) $s+d+id$ gap breaking point-group and time-reversal symmetries; a prominent Fano resonance develops at $eV=\min \{|\mathrm{\Delta }(\phi )|\}$ in the strong tunneling limit.