Long-lived non-equilibrium superconductivity in a non-centrosymmetric Rashba semiconductor

TL;DR

This study uncovers long-lived non-equilibrium superconducting states in Rashba semiconductor GeTe, revealing asymmetric magnetoresponse and slow dynamics that challenge conventional understanding of superconductivity in non-centrosymmetric materials.

Contribution

It demonstrates the existence of long-lived non-equilibrium superconductivity and develops a model explaining slow relaxation dynamics in Rashba systems.

Findings

Non-equilibrium states decay on timescales much longer than electronic scattering times.

The magnetoresponse is asymmetric under magnetic field reversal.

Evidence of slow dynamics in another Rashba system.

Abstract

We report non-equilibrium magnetodynamics in the Rashba-superconductor GeTe, which lacks inversion symmetry in the bulk. We find that at low temperature the system exhibits a non-equilibrium state, which decays on time scales that exceed conventional electronic scattering times by many orders of magnitude. This reveals a non-equilibrium magnetoresponse that is asymmetric under magnetic field reversal and, strikingly, induces a non-equilibrium superconducting state distinct from the equilibrium one. We develop a model of a Rashba system where non-equilibrium configurations relax on a finite timescale which captures the qualitative features of the data. We also obtain evidence for the slow dynamics in another non-superconducting Rashba system. Our work provides novel insights into the dynamics of non-centrosymmetric superconductors and Rashba systems in general.

Figures

Equations

$$\frac{\mathrm{d}{ϵ}_{\mathrm{F}}^{\pm }}{\mathrm{d}t}=\frac{\partial {ϵ}_{\mathrm{F}}^{\pm }}{\partial B}\frac{\partial B}{\partial t}+\frac{\partial {ϵ}_{\mathrm{F}}^{\pm }}{\partial n^{\pm }}\frac{\partial n^{\pm }}{\partial t},$$

Full text

Rashba systems are a class of spin-orbit coupled materials in which spatial inversion symmetry is absent and whose band structure, therefore, lacks spin degeneracy. The dispersion of Rashba systems features two concentric Fermi surfaces with opposing helical spin structures that are separated in momentum space by twice the Rashba wavevector $k_{\mathrm{R}}$. Systems displaying a large Rashba effect are desirable for all-electrical spin-based logic schemes, and Rashba superconductors are expected to harbour topological superconducting phases Sato and Fujimoto (2009), much sought-after towards fault-tolerant quantum computation. While the breaking of spatial inversion is most readily achieved in low-dimensional systems, recently three-dimensional materials such as bismuth tellurohalides Saeed and Naoki (2017) and GeTe Sante et al. (2012) have been shown to have a giant bulk Rashba effect.

It is known that the presence or absence of specific symmetries in a system has a telling effect on the allowed dynamical processes Narayan et al. (2007). In the specific case of Rashba systems, transitions between the two Rashba bands are constrained by the finite momentum split $k_{\mathrm{R}}$ and the helical spin-structure. As we will show in this Letter, this has important consequences for equilibration. The spin-structure of the Rashba bands also has important consequences for superconducting systems and, in particular, the nature of Cooper pairs Bauer (2012); Gor’kov and Rashba (2001). Thus, Rashba superconductors can harbour unconventional superconducting phases including Fulde-Ferrell-Larkin-Ovchinnikov-type Fulde and Ferrell (1964); Larkin and Ovchinnikov (1965) phases in which the Cooper pair has a finite momentum, and/or topological superconductor phases Sato and Fujimoto (2009); Tafti et al. (2013); Nakajima et al. (2015a); Sato and Ando (2017); Xiao et al. (2018).

We report here low temperature ($T$), magnetic field ($B$)-induced dynamics in molecular-beam-epitaxy (MBE)-grown (Fig. S1 SOM (rial)) ultra-thin films ($18\text{\,}\mathrm{n}\mathrm{m}$-thick) of GeTe. GeTe is a narrow band-gap semiconductor with giant bulk and surface Rashba couplings Sante et al. (2012); Liebmann et al. (2016), and is inherently superconducting Hein et al. (1964); Narayan et al. (2016). We present data from two Hall bar samples patterned from the same wafer (for details see SOM (rial)), which go superconducting below $0.2\text{\,}\mathrm{K}$. Strikingly, we find that a second non-equilibrium superconducting state with a higher critical temperature ($T_{\mathrm{c}}$) and critical field ($B_{\mathrm{c}}$) is accessed when the system is subjected to a continuously-ramped magnetic field $B(t)$. This non-equilibrium state is extremely long-lived, relaxing on macroscopic timescales of several minutes. By ruling out other well-known sources of slow dynamics, we demonstrate that the mechanism underlying the observed dynamics is novel. We have shown that such long-lived non-equilbrium behaviour can generically be expected in clean Rashba systems Verpoort and Narayan (shed): in materials with strong Rashba coupling, where $k_{\mathrm{R}}$ is much larger than the thermal phonon momentum scale, there is a suppression of all relaxation processes involving real phonon modes. Furthermore, the spin texture at the Fermi surface serves to significantly reduce the scattering events due to inter-carrier interactions, ultimately resulting in non-equilibrium states with finite lifetimes. Based on this we formulate a model of a Rashba superconductor with suppressed inter-band transitions, within which the non-equilibrium superconducting behaviour arises from the enhancement of the density of states. Our model suggests similar dynamics in normal Rashba materials, evidence of which we observe in a topological insulator heterostructure with dominantly bulk-type transport.

Fig. 1 characterises the equilibrium electrical properties of GeTe in the normal and superconducting states. Between room $T$ and $\approx$100\text{,}\mathrm{K} GeTe shows activated behaviour indicative of a band gap of $0.1\text{\,}\mathrm{eV}$. Below $100\text{\,}\mathrm{K}$ the transport becomes $T$ independent, suggesting the role of two-dimensional (2D) modes. Correspondingly, the electrical characteristics are plotted as ‘sheet resistances’ $R_{\mathrm{s}}=R\times W\,/\,L$, where $R$ is the electrical resistance measured using a constant-current, four-terminal setup, and $W$ and $L$ are the width and length of the Hall bar. The 2D nature of transport is corroborated by the low-$T$ magnetotransport, where positive quantum corrections to the electrical conductivity $\sigma_{\mathrm{xx}}$ or weak anti-localisation (WAL) characteristics are seen to be 2D-like (Fig. [1](#S0.F1)b, Fig. S2 SOM ([rial](#bib.bib14))). Here $\sigma_{\mathrm{xx}}\equiv(L/W)\,R_{\mathrm{xx}}/(R_{\mathrm{xx}}^{2}+R_{\mathrm{xy}}^{2})$, where $R_{\mathrm{xx}}$ and $R_{\mathrm{xy}}$ are the longitudinal and Hall components of resistance, respectively. The observation of 2D modes is consistent with recent spectroscopic measurements Liebmann *et al.* ([2016](#bib.bib15)), although our results are not affected by the dimensionality of transport. Fig. [1](#S0.F1)c shows the onset of the superconductivity at $T=$0.2\text{\,}\mathrm{K} and its suppression under the influence of a perpendicular magnetic field ($B$).

Evidence of a second non-equilibrium superconducting state is shown in Fig. 2. Under the influence of a slowly-ramped $B$ field perpendicular to the plane of the sample, it is found that (Fig. 2a-c): (1) superconductivity is no longer observed at $B=$0\text{,}\mathrm{T}, but instead appears at a finite $B$; (2) the magnetoresistance is asymmetric about the new superconducting state, depending explicitly on the sign of $\nicefrac{{\mathrm{d}B}}{{\mathrm{d}t}}$, and (3) the non-equilibrium state is highly persistent, relaxing on the timescale of minutes (Fig. [2](#S0.F2)d and Fig. S4 SOM ([rial](#bib.bib14))). The occurrence of this state relies on a finite $\nicefrac{{\mathrm{d}B}}{{\mathrm{d}t}}$ without which one obtains the ‘equilibrium’ magnetoresistance (Fig. [1](#S0.F1)c inset). Strikingly, however, Fig. [1](#S0.F1)e shows that sweeping at a slower rate serves to enhance the finite-$B$ minimum ($R_{\mathrm{min}}$), implicating an optimum sweep rate at which the non-equilibrium superconducting state manifests most clearly. The enhancement of $B_{\mathrm{c}}$ evidenced in Fig. [2](#S0.F2)a-c in conjunction with the behaviour in Fig. [2](#S0.F2)d strongly suggests that the dynamical superconducting state is distinct from the initial equilibrium superconducting state, and results from a long-lived transient configuration. This is supported by Fig. [2](#S0.F2)f-i (and Fig. S6 SOM ([rial](#bib.bib14))), where we find that its existence is not contingent on the equilibrium superconducting state, occurring even above $T_{\mathrm{c}}$ and remaining perceptible up to $T=$0.4\text{\,}\mathrm{K}. The absence of complete loss of resistance could be interpreted as a competition of the timescales for the transition into the superconducting phase, and the decay of the non-equilibrium state (see Fig. S4 SOM (rial)).

There are various mechanisms that are known to result in slow relaxation in solid-state systems. In the Supplement SOM (rial) we discuss and rule out contributions due to magnetocaoric effects, superconducting vortices, nuclear spins, trapped flux, and inhomogeneities in the GeTe film. We also note that the recent findings of ‘non-reciprocal transport’ in Rashba systems Wakatsuki et al. (2017); Hoshino et al. (2018) cannot explain our experimental observations as these are equilibrium effects. In particular, we find no violation of reciprocity in the equilibrium transport (Fig. 1c inset).

In Ref. Verpoort and Narayan (shed), we explicitly consider the relaxation dynamics in Rashba-coupled systems and establish that in the absence of charged impurities and below a characteristic $T$, non-equilibrium configurations relax on timescales that can be many orders of magnitude greater than the conventionally-observed picosecond relaxation timescales. In our MBE-grown samples, we expect the role of charged-impurity scattering to be negligible (the spontaneously-formed Ge vacancies are uncharged Edwards et al. (2005, 2006), and do not contribute to Coulomb scattering), and we show that the characteristic $T$ scale can be as large as $1\text{\,}\mathrm{K}$ for GeTe. In the following we verify whether the existence of such a timescale is a sufficient condition to induce the observed novel magnetoresponse by considering a model Rashba superconductor and introducing, by hand, a finite timescale $\tau$ for inter-band transitions. We estimate $\tau\approx$100\text{,}\mathrm{s}$$ for the GeTe films from Fig. S5a SOM (rial).

The Rashba dispersion is given by $\epsilon_{k}^{\pm}=\nicefrac{{\hbar^{2}\mathbf{k}^{2}}}{{2m}}\pm\sqrt{(g\mu_{\mathrm{B}}B)^{2}+(\alpha_{\mathrm{R}}\mathbf{r}_{\mathrm{SO}}\times\mathbf{k})^{2}}$, where the $+$ ($-$) superscript refers to the inner (outer) Rashba band. Here, $\hbar$ is Planck’s constant, $\mathbf{k}$ is the wavevector, $m$ is the effective mass of carriers, $g$ the Lande g-factor, $\mu_{\mathrm{B}}$ the Bohr magneton, $\alpha_{\mathrm{R}}$ the Rashba parameter, and $\mathbf{r}_{\mathrm{SO}}$ is the direction of the spin-orbit coupling along which inversion symmetry is broken. This assumes the direction of the $B$ field to lie parallel to $\mathbf{r}_{\mathrm{SO}}$ (however this assumption does not affect the qualitative results as in-plane fields cause a redistribution of carriers between bands similar to the out-of-plane one). To study the consequences of the time-varying magnetic field we compute the dynamical Fermi energies $\epsilon^{\pm}_{\mathrm{F}}$ as a function of time $t$, whose time-dependence is governed by the differential equation:

[TABLE]

where $n^{\pm}$ are the carrier densities of the two bands. The first term on the right describes how the Fermi surfaces change with $B$ and the second term describes carriers relaxing so as to equilibrate $\epsilon_{\mathrm{F}}^{+}$ and $\epsilon_{\mathrm{F}}^{-}$. We model $\nicefrac{{\partial n^{\pm}}}{{\partial t}}$ using a relaxation-time approximation with time constant $\tau$. In order to describe the superconductivity, we make the following generic assumptions: (1) superconductivity is assumed to emerge through pairing of opposite spin carriers within a band (this is the simplest prescription based on Cooper pairs with zero net momentum Smidman et al. (2017), and while it is assumed for simplicity, the model is readily extended to include inter-band Cooper pairing); (2) the transition temperature $T_{c}$ is assumed to depend exponentially on the density-of-states $\nu$: $T_{c\pm}\sim\mathrm{e}^{(-1/\Gamma\nu_{\pm})}$, where the $\pm$ indicate the two Rashba bands, $\nu_{\pm}=\nicefrac{{\mathrm{d}k}}{{\mathrm{d}\epsilon_{\pm}}}$, and $\Gamma$ is the strength of the contact interaction that mediates superconductivity. $\Gamma$ is drawn from a uniform probability distribution to reflect local variations in dopant concentration, interaction strength etc. The magnetoresistance is modelled by $R=R_{\mathrm{N}}\min(T/T_{\mathrm{C-}},1)$, which arises only from the parts of the sample that are normal. The results of our model calculation are shown in Fig. 3a, and capture the essential features of the experimental observations in Fig. 2a – c and f – i. Notably, this simple model does not capture the $\nicefrac{{\mathrm{d}B}}{{\mathrm{d}t}}$-dependence of the depth of the dip (Fig. 2e), which is likely a higher-order effect requiring a more microscopic treatment. One plausible scenario is that the non-equilibrium superconducting state has a more complex history dependence (Fig. S9 SOM (rial)) or involves inter-band Cooper pairing and thus requires a threshold occupancy in both Rashba bands.

It is noteworthy that the arguments of long relaxation times presented in Ref. Verpoort and Narayan (shed) are generic to Rashba systems as opposed to only Rashba superconductors. In this context, we first point out that the novel asymmetric and rate-dependent magnetodynamics are observed in GeTe even in samples that do not go superconducting Narayan et al. (2016). Fig. 3b demonstrates similar, albeit less pronounced qualitative features in a vertical topological insulator (TI) $p$-$n$ junction Eschbach et al. (2015); Backes et al. (2017) in which a $6\text{\,}\mathrm{nm}$ layer of $\mathrm{Bi}_{2}\mathrm{Te}_{3}$ is capped with a $15\text{\,}\mathrm{nm}$ layer of $\mathrm{Sb}_{2}\mathrm{Te}_{3}$. Here transport is in the plane of the film and $B$ is out of plane. TIs are well-known spin-orbit materials, and the in-built potential of the $p$-$n$ junction Eschbach et al. (2015) provides a clear mechanism for the breaking of inversion symmetry. Importantly, the specific layer configuration is known to show significant bulk transport Backes et al. (2017) rather than the surface-dominated transport in ultra-thin TIs, where this effect is not expected due to the single Dirac cone. We stress that this behaviour should not be confused with the so-called ‘butterfly hysteresis’ observed in magnetic Dirac materials Wolgast et al. (2015); Brinkman et al. (2007); Nakajima et al. (2015b); Daptary et al. (2017); Tiwari et al. (2017) since 1) the materials reported here are manifestly non-magnetic, and 2) the data reflect non-equilibrium states in the samples.

Fig. 3c shows the qualitative behaviour of $\rho(B)\approx\nicefrac{{1}}{{(\sigma^{+}+\sigma^{-})}}$ as derived from the conductivities $\sigma^{\pm}\propto\int\operatorname{d}^{3}\!{k}\,(\nicefrac{{\partial\epsilon_{\mathbf{k}}^{\pm}}}{{\partial k}})^{2}\,\delta(\epsilon_{\mathbf{k}}^{\pm}-\epsilon^{\pm}_{\mathrm{F}})$ of the bands and the dynamics of $\epsilon^{\pm}_{\mathrm{F}}(t)$ arising due to Eq. (1), evaluated for a parabolic band dispersion. The figure clearly captures the salient features of the experimental data (although for a quantitative comparison, in addition to realistic band structure, one also needs to account for WAL corrections to the conductivity). The physical mechanism is the exact same as before: the two Rashba bands develop unequal Fermi levels when subjected to a $B$ sweep. Since changes in conductivity of the two bands do not cancel, i.e. $\frac{\partial\sigma^{+}}{\partial n^{+}}\neq\frac{\partial\sigma^{-}}{\partial n^{-}}$, there is a net change in the total conductivity $\sigma^{+}+\sigma^{-}$. Lastly, we stress that while GeTe shows a dramatic magnetoresponse, this can be much more subtle as shown in Fig. 3b. We question whether similar observations may have been overlooked in the past.

In conclusion, we have reported ultra-slow relaxation and rich non-equilibrium magnetodynamics in the non-centrosymmetric Rashba superconductor GeTe. These novel dynamics reveal a second non-equilibrium superconducting state with a higher $T_{\mathrm{c}}$ and $B_{\mathrm{c}}$. Importantly, the observed slow dynamics are inconsistent with more common sources of slow dynamics such as nuclear spin relaxation and vortex creep. They are also inconsistent with magnetocaloric-driven cooling and/or Eddy current-induced heating. We develop a model that successfully captures the salient features of the experimental data, and also predicts a specific response in normal Rashba systems, qualitative evidence of which we observe in a TI-based heterostructure. We suggest, therefore, that our observations might be generic to Rashba systems and discuss the conditions under which they might be observed in experiment. Our work has provided striking new experimental insights into the behaviour of Rashba superconductors and possibly indicates a non-equilibrium behaviour unique to Rashba systems in general.