Determining the superconducting order parameter of UPt$_3$ using scanning tunneling microscopy

TL;DR

This work uses ultra-low-temperature scanning tunneling microscopy to determine the superconducting order parameter of UPt3. By observing a zero-energy Andreev bound state at a surface normal to the c-axis and correlating it with vortex-lattice imaging, the authors discriminate between competing pairing symmetries. Through tight-binding BdG modelling and DFT-based Rashba-splitting calculations, they conclude that UPt3 hosts a spin-singlet E1g (chiral) order parameter, with the bound state robust against surface Rashba splitting, thereby constraining the pairing mechanism in this heavy-fermion superconductor. The results have implications for the broader understanding of unconventional superconductivity in uranium-based materials and suggest further avenues to explore surface-phase behavior and neutron-resonance signatures.

Abstract

Superconductivity, a state in which electrical currents can flow without resistance, occurs because of pairing of electrons into quasiparticles with integer spin $S$. In practically all known superconducting materials, these pairs form a singlet with $S=0$. Finding a material that has triplet pairing, $S=1$, would have profound fundamental and technological implications. UPt$_3$ has been a key candidate material for spin-triplet superconductivity. Because of a lack of direct evidence for the pairing symmetry, the nature of the superconducting pairing remains under debate. Here, we use ultra-low temperature scanning tunneling microscopy to resolve this question. Our data reveals a zero-bias Andreev bound state within the gap for a surface normal to the $c$-axis of UPt$_3$. The superconducting origin of the features is confirmed through vortex imaging. For triplet pairing, such an Andreev state is fragile against Rashba spin-splitting, whereas for singlet pairing it remains robust, classifying UPt$_3$ as a spin-singlet superconductor with a chiral order parameter.

Figures (8)

• Figure 1: Superconducting order parameters of UPt3. (a) Crystal structure of UPt3, showing planes of UPt3 stacked with a hexagonally-close-packed (hcp) alignment along the $c$-axis. (b, c) Graphical representations of the leading contenders for the order parameter in the low temperature, low field phase in UPt3. (b) Spin-singlet order parameter with $E_{1g}$ symmetry, (c) spin-triplet order parameter with $E_{2u}$ symmetry. (d) Andreev reflection at a normal-superconducting interface, with electron paths indicated by solid arrows and hole paths shown by dashed arrows. The trajectories are specularly reflected at the vacuum interface, whereas at the $N-S$ interface, an electron with momentum $\mathbf{k}$ is reflected as a hole with momentum $-\mathbf{k}$ and vice-versa. (e, f) Local density of states (LDOS) at the surface of superconductors with singlet order parameter $E_{1g}$ and the triplet order parameter $E_{2u}$, respectively, including a non-negligible spin-orbit coupling (see suppl. sect. \ref{['andreevsection']} for details).
• Figure 2: Imaging and spectroscopy of UPt3. (a) Topography of the surface of the UPt3 crystal ($T = 10\mathrm{K}$, $V_\mathrm{s} = 100\mathrm{mV}$, $I_\mathrm{s} = 300 \mathrm{pA}$). The inset shows a close up with the crystal structure superimposed. (b) Tunneling spectrum, $g(V)$, recorded in the normal state, showing a huge asymmetry of the density of states across the Fermi energy ($T = 10\mathrm{K}$, $V_\mathrm{s} = 50\mathrm{mV}$, $I_\mathrm{s} = 400 \mathrm{pA}$, $V_\mathrm{mod} = 1\mathrm{mV}$). (c) Tunneling spectrum $g(V)$ recorded in the superconducting state taken at $T = 40\mathrm{mK}$, showing a clear gap-like structure. The size of the of superconducting gap $2\Delta = 280 \mu V$ is indicated ($V_\mathrm{s} = 2\mathrm{mV}$, $I_\mathrm{s} = 800\mathrm{pA}$, $V_\mathrm{mod} = 10 \mathrm{\mu V}$).
• Figure 3: Vortex lattice and coherence length. (a) Real space map of tunneling conductance, $g(V)$, for $V = -40\mathrm{\mu V}$ taken in a magnetic field $B = 0.75\mathrm{T}$ at $T = 40\mathrm{mK}$, showing the vortex lattice. In the vortex cores, $g(V)$ reverts to the normal state density of states ($V_\mathrm{s} = 2\mathrm{mV}$, $I_\mathrm{s} = 800\mathrm{pA}$, $V_\mathrm{mod} = 50 \mathrm{\mu V}$). $3\times3$ pixel averaging applied to an $86\times86$ pixel map. (b) $g(V)$ spectra taken from left to right along the white dashed line in (a), showing suppression of the gap within vortices ($V_\mathrm{s} = 2\mathrm{mV}$, $I_\mathrm{s} = 800 \mathrm{pA}$, $V_L = 10 \mathrm{\mu V}$). (c) Radially averaged $g_r(V)$, with a constant background value subtracted and then normalised, measured from the centre of a vortex core outwards. Data fit to $g_r(V)=1-\tanh(r/\xi)$ where $\xi=126$Å, shown by the solid line. (d) Linecut of the step edge along the pink line in the inset; inset is a topography of a step edge ($V_\mathrm{s} = 10 \mathrm{mV}$, $I_\mathrm{s} = 400\mathrm{pA}$), showing a step height of 1.3 nm. Tunneling spectra along the pink line (e) show suppression of the gap at the step edge ($T = 40 \mathrm{mK}$, $V_\mathrm{s} = 2\mathrm{mV}$, $I_\mathrm{s} = 800 \mathrm{pA}$, $V_L = 10 \mathrm{\mu V}$).
• Figure 4: Temperature and field dependence of the gap. (a) Low temperature phase diagram of UPt3, with $\mu_0 H \parallel c$ showing the three superconducting phases (based on Adenwalla1990Suderow1997Huxley2000). (b) Magnetic field dependence ($B \parallel c$) of $g(V)$ spectra recorded at $T = 45 \mathrm{mK}$. The gap is completely suppressed at fields larger than $\mu_0 H=1\mathrm{T}$ ($V_\mathrm{s} = 2\mathrm{mV}$, $I_\mathrm{s} = 800\mathrm{pA}$, $V_\mathrm{mod} = 10 \mathrm{\mu V}$). (c) Temperature dependence of $g(V)$ spectra, showing that the gap structure disappears above $550\mathrm{mK}$ ($V_\mathrm{s} = 2\mathrm{mV}$, $I_\mathrm{s} = 800\mathrm{pA}$, $V_\mathrm{mod} = 10 \mathrm{\mu V}$). (d) Tunneling spectrum taken at the lowest temperature but with simulated thermal broadening, but otherwise constant shape of the gap. The simulation shows that indeed in the experiment the gap is suppressed at $T=550 \mathrm{mK}$, and does not just become indiscernable. The blue, purple, grey and pink colours in b-d indicate the B-phase, C-phase, A-phase and normal state as in panel a, respectively.
• Figure S1: Lowest harmonic form factors for the real space superconducting order parameter on a hexagonal lattice with $D_{6h}$ point group symmetry for (a) an E$_{1g}$ order parameter with symmetry $xz+iyz$ and (b) an $E_{2u}$ order parameter with symmetry $(x^2-y^2)z + 2ixyz$. The solid (dashed) lines indicate the sign of the gap connecting nearest neighbour unit cells.