Large superconducting diode effect indicates nonuniform triplet pairing in UTe2

Introduction

Since the prediction of nonuniform superconducting states in 1964 by Fulde, Ferrell, Larkin and Ovchinnikov[1, 2] – broadly referred to as FFLO states – evidence for their existence has been found in multiple candidate materials. These include organic superconductors (SCs),[3, 4] cuprates,[5] noncentrosymmetric SCs with helical states,[6] SCs coexisting with magnetic order including heavy fermion SCs such as $CeCoIn_5$,[7, 8, 9, 10] as well as in cold atomic systems.[11, 12] With more recent imaging techniques, e.g., scanning Josephson tunneling microscopy (SJTM), pair density waves (PDWs) – nonuniform SCs that either do not break time reversal symmetry or break it spontaneously,[5] as opposed to FFLO states that occur with explicit time reversal symmetry breaking (TRSB) – have been reported on surfaces of multiple SCs,[13] including cuprates,[14] kagome SCs,[15] and heavy-fermion $UTe_2$.[16, 17] Related nonuniform superconductivity with intra-unit-cell modulation that does not break translation symmetries has also been observed in Fe-chalcogenides.[18, 19]

Despite the multitude of indirect experimental observations supporting the existence of FFLO states, conclusive identification of bulk nonuniform superconductivity – characterized by finite-momentum Cooper pairs – remains challenging due to the difficulty of resolving the bulk modulation of the SC order parameter. While recent advances in imaging techniques have partly addressed this issue, such measurements are constrained to probe only the material surface, making it challenging to rule out surface reconstruction effects as reported for example in the case of $UTe_2$.[20, 21, 22, 23] A promising alternative bulk transport probe for identifying FFLO states with inversion symmetry breaking (ISB), the so-called FF-type FFLO and PDW states,[5] is provided by the superconducting diode effect (SDE).[24, 25, 26, 27] SDE, defined by unequal critical currents $J_{c+}$ and $J_{c-}$ for opposite current flow directions, has initially attracted much attention precisely as a possible signature of a helical FFLO state in a noncentrosymmetric heterostructure;[28] related effects in Josephson junctions have also been proposed[29, 30, 31] and used[32] as methods of characterizing FFLO states. However, it has since been pointed out that extrinsic vortex diode effects (VDE), another form of SDE known but unnamed for a long time,[33, 34] can account for the critical current nonreciprocity without invoking finite-momentum Cooper pairs.[35, 36, 37, 38] Ruling out such extrinsic effects is, therefore, crucial for robustly identifying an FFLO state.

Here we report SDE in a bulk UTe$_2$ sample, with the diode efficiency $\eta= \frac{J_{\mathrm{c}}^{+}+J_{\mathrm{c}}^{-}}{J_{\mathrm{c}}^{+}-J_{\mathrm{c}}^{-}}$ maximized at 33% near a field-induced first order metamagnetic phase transition into a polarized paramagnetic (PPM) state at $\mu_{0}H \approx33$ T, making it the highest-field observation of SDE to date. This is also a rare example of SDE in a bulk 3D sample besides e.g.,[34, 39] most other reports being in 2D or thin-film samples. Most significantly, the absence of SDE away from a narrow region close to the metamagnetic transition strongly disfavors a purely extrinsic vortex effect due to either bulk or surface pinning asymmetry, instead implying spontaneous ISB with accompanying nonuniform superconductivity. Moreover, we establish for the first time the coexistence of superconductivity within the PPM phase over a narrow field region in which the SDE is observed, suggesting the mixing of superconducting and magnetic orders as a likely source of the ISB. This ISB may be either due to some domain structure and/or an intrinsic FFLO state due to the coexistence of the two orders, which we model phenomenologically using a Ginzburg–Landau (GL) theory below.

In addition to other signatures of nonuniform SC in UTe$_2$, in both surface SJTM/STM experiments in the low field (SC1) phase[16, 17] and anomalous vortex transport in a mixed region between SC1 and the field-induced SC2 phase,[40, 41, 42, 43] the observation of SDE provides the strongest evidence yet for a bulk (FF-like) nonuniform SC state in UTe$_2$. Given the abundant evidence for triplet pairing in UTe$_2$,[44, 45, 46, 47, 48, 49] this is a rare candidate for a triplet FFLO state besides CeCoIn$_5$,[7, 8, 9] adding to the long list of possible unconventional phases in this material.[48, 49, 50]

SDE marks the overlap of SC + magnetism

We plot the electrical resistivity $\rho$ of UTe$_2$ versus applied magnetic field strength $\mu_{0}H$ in Figure 1a at various temperatures, with $\mathbf{H}$ aligned along the hard magnetic $b$-axis and $\mathbf{J}$ along the $a$-axis. At low temperatures $T \lesssim1.1$ K, $\rho(H) = 0$ up until around 34 T, where a steep upturn occurs at a metamagnetic phase transition as the material enters the high-spin PPM state, characterized by an abrupt increase in the bulk magnetization of 0.5 Bohr magnetons per uranium site [51, 52, 53, 54]. At higher temperatures re-entrant superconductivity is observed, distinguishing two separate superconducting phases (numbered SC1 and SC2) [49]. In Figure 1b we compare the $J$–$V$ characteristics of the SC2 state at various fixed $H$ values for positive- and negative-polarity $J$-sweeps. At 33 T we observe $V(J^{+}) = V(J^{-})$, within experimental resolution. By contrast, a large nonreciprocity in $\pm J_{\mathrm{c}}$ develops upon increasing $H$, yielding a sizeable SDE. This is maximal at $\mu_{0}H = 33.6$ T, exhibiting a SDE efficiency of 33%. Then, by 34 T, SC2 is fully suppressed and Ohmic, polarity-symmetric magnetoconductivity is observed.

In Figure 2b we replot the 1.11 K and 1.72 K $\rho(H)$ curves from Figure 1a, to evaluate how the manifestation of a large SDE for $\mu_{0}H \ge33$ T relates to the metamagnetic phase boundary. At 1.72 K the sample transitions from SC1 to the normal state at 5.8 T, and then does not enter SC2 at any higher $H$. At 33.0 T, $\rho$ bends up sharply (marked by dashed line), indicating that the metamagnetic transition has been crossed into the field polarized state. However, interestingly, upon taking a closer look in the zoomed inset it is clear that at 1.11 K $\rho$ remains zero up until 33.8 T. This implies that there is a small region of overlap between the SC2 and PPM states from 33.0 to 33.8 T – corresponding to the field domain over which the SDE occurs (Figure 1). This strongly suggests that the interplay of superconductivity and magnetism drives the manifestation of the SDE in UTe$_2$.

In Figure 3 we examine the profile of $V(J)$ upon switching the polarity of $H$. In Figure 3a, it is clear that for positive polarity magnetic field (purple curve), $J_c^- > J_c^+$. By contrast, for negative polarity field (ochre curve), the opposite is true with $J_c^- < J_c^+$. In panels c and d sit right at

the edge of the nonreciprocal $J - V$ curves, and in ±33.5 T switching $J$ between ±15 A cm$^{-2}$ alternates $V$ between 0 and ±1 µV.

Phenomenological Model of Spontaneous Intrinsic SDE

The experimental data shows that the SDE occurs in a narrow region of the phase diagram in which the SC2 and PPM states coexist, indicating spontaneous inversion symmetry breaking in the mixed state. To model this phenomenologically, we use an extension of the Ginzburg–Landau (GL) theory considered in [55]. Taking $M$ to be the magnetization field and $\Delta$ to be the (possibly multicomponent) SC2 order parameter, the free energy $\mathcal{F}[M,\Delta] = \mathcal{F}_{M}[M] + \mathcal{F}_{\Delta}[\Delta] +

$\mathcal{F}_{\mathrm{mix}}[M,\Delta]$ has three terms:

Here $\mathcal{F}_{M}$ and $\mathcal{F}_{\Delta}$ model the pure magnetic and superconducting orders, with parameters chosen to approximately fit the metamagnetic phase transition and the critical temperature of SC2, respectively (see Supplementary Information). Here $q$ is the Cooper pair momentum (the corresponding GL terms originate from Fourier transforms of gradients of $\Delta$). In the case of a multicomponent order parameter considered in the following section, the components can be parametrized as $\Delta_{m}=|\Delta|d_{m}$, in which case the coefficients in front of $|\Delta|^{2}$ terms in Eq. (1) should be understood as including cross terms of the form $\Delta_{m}^{*}\Delta_{n}$, etc.

The two orders in general interact via the mixed terms in $\mathcal{F}_{\mathrm{mix}}$. This includes the standard diamagnetic response of the superconductor through the $\chi_{\Delta}^{-1}\gg0$ term, which strongly suppresses any coexistence. The crucial observation, however, is that an additional mixing term $\gamma(q)=\gamma_{2}q^{2}$ is allowed by symmetry depending on the form of the order parameters, and may favor coexistence, provided $q\ne0$ in the ground state and $\gamma_{2}$ is sufficiently negative. Such a state corresponds to $\Delta\sim e^{i\mathbf{q}\mathbf{r}}$, i.e., an FF-type FFLO state, which breaks inversion symmetry. We note that on symmetry grounds, ISB implies finite-momentum pairing in absence of fine-tuning; typically translation symmetry is also broken, unless $\mathbf{q}$ is an element of the reciprocal lattice, in which case a so-called pair density modulation (PDM) state is realized.[18, 19] We comment on the possible microscopic origin of $\gamma$ in the following section. Figure 4 shows the numerically computed ground state values of the order parameters $\Delta$ and $M$, and $q$; the inset shows the diode efficiency $\eta$; see Supplementary Information for parameter values. The metamagnetic phase transition occurs at $\mu_{0}H=33\mathrm{T}$ in the normal state, but a phase characterized by non-zero $q$ emerges just below this field in the SC state. Here we look at a fixed temperature $T$, but $T_{\mathrm{c}}$ as a function of $H$ is strongly correlated with $\Delta(H)$.

Possible Origins of SDE

Having presented the data and GL model for the SDE, we now discuss its possible origin and implications for the symmetry of the order parameter. We note that free energies similar to the one considered above have been considered in the context of uniform SC (i.e., $q = 0$) in studies of TRSB in $\mathrm{SC}_1$[56, 57, 58] as well as phenomenological studies of $H_{\mathrm{c}2}$ in $\mathrm{SC}_2$ and $\mathrm{SC}_3$ with non-unitary pairing.[59, 60, 61] As noted in those works, at zero applied field a coupling of the form $\gamma(q)M|\Delta|^2$ generally requires an SC order parameter that breaks TRS assuming $\gamma(q)$ is an even function of $q$. Since the $\mathrm{UTe}_2$ point group, $D_{2h}$, only contains one dimensional irreducible representations (irreps), this generally requires $\mathrm{SC}_2$ to be a multicomponent phase with at least two SC components $\Delta_n,\Delta_m \propto\Delta$ belonging to two distinct irreps with an imaginary phase between them (e.g., $B_{1u}+iB_{3u}$ pairing, $B_{2u}+iB_{3u}$ pairing, etc.). For a single-band triplet superconductor, this generally implies non-unitary pairing with a gap function $\Delta=i\sigma_y\boldsymbol{\sigma}\cdot\mathbf{d}$ with $\mathbf{d}\ne\mathbf{d}^{*}$, furthermore implying Cooper pairs possessing a non-vanishing Cooper pair spin $\mathbf{s}=i\mathbf{d}\times\mathbf{d}^{*}\ne0$.[61] However, we note that in a multicomponent SC, which is plausible in UTe$_2$, non-unitary pairing is not necessary to produce a non-zero $\gamma$.[62]

There are several possible microscopic mechanisms for generating a coupling of the form $\gamma M|\Delta|^2$, the simplest being an exchange-like interaction $\gamma\mathbf{M}\cdot\mathbf{s}=i\gamma\mathbf{M}\cdot\mathbf{d}\times\mathbf{d}^{*}\sim\gamma M\Delta_n^{*}\Delta_m$. Noting that UTe$_2$ is a locally noncentrosymmetric material with two sublattices $s=A,B$ with locally broken inversion symmetry,[63, 64] another possibility is a (interfacial) Dzialoshinskii-Moriya like interaction (DMI) of the form $\mathbf{D}_s(\mathbf{q})\cdot\mathbf{M}_s(-\mathbf{q})\times\mathbf{s}_s(\mathbf{q})$ with $\mathbf{D}_A=-\mathbf{D}_B$ and $\mathbf{D}_s(\mathbf{q})=-\mathbf{D}_s(-\mathbf{q})\propto\hat{\mathbf{c}}\times\mathbf{q}$.[65, 66] In principle, the triplet nonuniform SC order could be characterized either by a nonuniform phase, $\Delta\sim e^{i\mathbf{q}\mathbf{r}}$, with fixed Cooper pair spin $\mathbf{s}$, or by a nonuniform $\mathbf{s}$ (and thus $\mathbf{d}$) rotating in space, for example forming a helical or conical pattern.[67, 68] The latter could be favored by DMI, which generally favors the formation of magnetic spirals. On general symmetry grounds, we anticipate that the magnetic order in the mixed state also acquires a nonuniform character with corresponding SDW momentum $\mathbf{q}_M$, which could be probed by experiments sensitive to magnetic order. We note that a ground state with mixed magnetic and SC order with $q=0$ but $q_M\ne0$ has been considered in the vicinity of an SC to FM phase transition, e.g., in.[69]

Finally, we comment on possible extrinsic factors. It should be emphasized that some extrinsic factors necessarily do contribute to the observed SDE: at the relevant high magnetic fields, the $J-V$ characteristics clearly show vortex flow behavior above $J_c$, which must therefore be set by vortex depinning. The observed SDE is therefore a form of a VDE. Moreover, extrinsic effects certainly determine the polarity of the diode effect (i.e., the sign of $\eta$), which is never seen to reverse in the experiment between subsequent sweeps (as one would expect if inversion symmetry were to be broken spontaneously each time). This is in contrast to field-free SDE with spontaneous TRSB, for which the polarity can typically be switched by training with a small magnetic field. The polarity of SDE with spontaneous ISB, on the other hand, is likely selected by either the sample geometry or contact placement, which is not as easily manipulated. Experiments with more complicated contact geometry may potentially find such polarity switching, which we hope to carry out in future studies.

Some known extrinsic effects may be at play in $\mathrm{UTe}_{2}$. For example, VDE without any finite momentum pairing is known to occur in superconductor/ferromagnetic thin bilayer heterostructures with in-plane magnetization, possibly due to stray fields near the edges of the bilayer.[35, 38, 70] From a symmetry point of view, this is an illustration of a possible SDE mechanism due to coexistence of (inhomogeneous) magnetic order with superconductivity, which draws some parallels to the SDE we observe in the SC2+PPM region. However, the particular mechanism identified in[35, 38, 70] is only applicable to thin films and not bulk 3D samples that we studied. It may be relevant, however, if a layered structure (characterized by momentum $q$) emerges microscopically in the mixed SC2+PPM state. A purely 3D VDE, on the other hand, is known to occur due to sample geometry affecting the length of the flux tubes.[34, 39] This mechanism may play a role in the SDE we observe if domains with broken inversion symmetry form in the SC2+PPM, providing a source of asymmetric vortex pinning (cf.[42]). In this case too, however, the formation of a domain structure with ISB and some characteristic length scale $1/q$ must take place. Whether a nonuniform state in such a scenario can be considered an FFLO state depends to a large degree on that length scale and the microscopic origin of the domains, which we cannot resolve in the present experiment.

We also stress that the presence of extrinsic effects does not by itself imply the absence of intrinsic effects. The source of the asymmetry of the vortex pinning potential can itself be of intrinsic origin. It is known theoretically, as one would expect on symmetry grounds, that in FFLO states the vortex lattices themselves exhibit translation symmetry breaking reflecting that of the underlying superconducting order parameter.[71, 72, 73] On the other hand, orbital effects responsible for the vortex lattice formation are also known theoretically to lead to d-vector rotation[74, 75] and the stabilization of finite-momentum pairing.[36, 76] Calculations in helical SC states additionally show that the shape of individual vortices lacks inversion symmetry.[77, 78]

We note that translation symmetry breaking vortex stripes[79] and asymmetric vortex profiles lacking inversion symmetry[80, 81, 82] have been seen on the surface of $UTe_2$, but at fields far below the metamagnetic transition. Vortex imaging in the SC2+PPM region – while experimentally challenging – could be very informative in confirming or ruling out an intrinsic origin of the SDE, as would other imaging probes sensitive to translation symmetry breaking.

Discussion and Outlook

We reported an SDE in the SC2+PPM coexistence region of the $UTe_2$ phase diagram above 33T with a large diode efficiency $\eta\approx33\%$, and the existence of superconductivity inside the PPM phase was seen for the first time. The absence of SDE outside of this narrow coexistence region of the phase landscape implies that inversion symmetry is only (spontaneously) broken inside the region, which strongly suggests an intrinsic origin of the SDE from a nonuniform SC state. This makes $UTe_2$ a rare candidate for realizing a triplet FFLO. Directly sensitive imaging experiments are necessary to conclusively establish the existence of such a state; nevertheless, the discovery of SDE in $UTe_2$ is an important first step towards its possible later identification.

The potential mechanism we have identified for producing SDE from coexistence between magnetic order and superconductivity may be relevant to other materials where FFLO states have been suggested, for example $CeCoIn_5$ or organic superconductors in which SDE has also been observed recently.[83] In general, our results call for revisiting past FFLO candidates in SDE experiments. As in $UTe_2$, a combination of intrinsic and extrinsic factors are likely to play a role in such systems, but the presence of spontaneous ISB within the SC phase may be helpful in isolating the intrinsic mechanisms.

Several other systems in which spontaneous symmetry breaking occurs, including ISB, have been found to exhibit SDE. These include Ru-substituted $Sr_2RuO_4$, [84, 85] the kagome $CsV_3Sb_5$, [86, 87, 88] $FeSe$, [89] and $Mo_2C$. [90] In all of these, ISB has been attributed to the formation of asymmetric domain networks, but without a clear explanation of the origin of the asymmetry. It is possible that in some of these systems the ISB takes place due to intrinsic material properties. Finally, we mention that given observations of signatures of PDW and FFLO states – as well as theoretical predictions of SDE[91, 92, 93] – in other parts of the $UTe_2$ phase diagram, multiple FFLO states of distinct origin may yet be observed in $UTe_2$ itself.

Acknowledgements

This project was supported by the EPSRC of the UK through grants EP/X011992/1 EP/Z533695/1 and EP/R513180/1. We acknowledge financial support by the Czech Science Foundation GAČR under the Junior Star Grant No. 26-21795M (STiUS). We acknowledge the support of HFML-RU/NWO-I, member of the European Magnetic Field Laboratory (EMFL) and the Engineering and Physical Sciences Research Council (EPSRC, UK) via its membership to the EMFL (grant EP/N01085X/1). A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement No. DMR-2128556 and the State of Florida. The work of D. S. was financially supported by the NSF Quantum Leap Challenge Institute for Hybrid Quantum Architectures and Networks Grant No. OMA-2016136. D.V.C. acknowledges financial support from Washington University in St. Louis through the Edwin Thompson Jaynes Postdoctoral Fellowship. A.G.E. acknowledges support from Sidney Sussex College (University of Cambridge).

Data availability

The datasets supporting the findings of this study will be uploaded to the University of Cambridge Apollo Repository prior to publication.

Competing interests statement

The authors declare no competing interests.