1 Introduction↩︎

Recent experimental advances in fabrication and exfoliation techniques have enabled the isolation of atomically thin layers of transition metal dichalcogenides (TMDs), ranging from monolayers to few-layer stacked structures. A particularly striking discovery was that many of these systems remain superconducting even in the two-dimensional limit, with critical temperatures often comparable to those of their bulk counterparts [1]–[9]. This unexpected robustness of superconductivity has brought TMD materials to the forefront of condensed matter research. Early prominent examples include superconductivity in few-layer NbSe\(_2\) and electrostatically gated MoS\(_2\). Subsequent studies revealed an even broader landscape of low-temperature superconducting TMD platforms, including gate-tunable superconductivity in WTe\(_2\) with \(T_c\sim 1\) K [10], [11], superconductivity in multilayer \(T_d\)-MoTe\(_2\) under ambient pressure with \(T_c\sim 0.1\) K [12], and pressure-induced superconductivity in WTe\(_2\), reaching \(T_c\sim 3\) K [13]. More recently, superconductivity has also been established in moiré-engineered twisted TMD systems, such as twisted WSe\(_2\) [14], opening new opportunities for studying strongly correlated and topological superconducting phases in highly tunable two-dimensional materials.

Samples with an odd number of layers, including monolayer systems such as exfoliated NbSe\(_2\) and gated MoS\(_2\), lack inversion center. In metals with broken inversion symmetry, dc transport can exhibit intrinsically nonreciprocal nonlinear responses. One prominent example is magnetochiral anisotropy (MCA) proposed by Rikken and co-authors [15], [16], in which the electric current contains a contribution quadratic in the applied electric field and linear in the external magnetic field, thus changing sign upon reversal of the field direction. Canonical MCA requires a pseudoscalar constructed from the current and magnetic field, usually \(\boldsymbol{I}\cdot\boldsymbol{B}\). In what follows, we consider more general tensor structures appropriate for nonlinear second-order conductivity.

Experimental studies demonstrated that MCA becomes strongly enhanced in the vicinity of the superconducting transition in noncentrosymmetric superconductors such as MoS\(_2\) [17]. This enhancement was attributed to the proliferation of superconducting fluctuations above the critical temperature \(T_c\). Qualitatively, the mechanism can be understood from the hierarchy of energy scales in the problem. In the normal metallic state, inversion- and time-reversal-symmetry-breaking effects enter through comparatively small parameters associated with the spin-orbit \(\Delta_{\text{SO}}\) and Zeeman energies \(\Delta_{\text{Z}}\) measured relative to the Fermi energy \(E_{\text{F}}\). By contrast, for preformed Cooper pairs in the fluctuation regime, the relevant energy scale is set by the inverse Ginzburg-Landau relaxation time, \(\tau^{-1}_{\text{GL}}\propto T-T_c\), which is parametrically smaller near the transition. As a result, symmetry-breaking perturbations have a much stronger influence on the fluctuating superconducting condensate than on normal-state electrons. These and other related findings of superconducting diode effects [18]–[20] sparked tremendous interest in nonreciprocal charge transport phenomena in noncentrosymmetric superconducting 2D quantum materials (see Refs. [21], [22] for recent reviews on this topic). Beyond MCA, recent theoretical studies have explored effects of fluctuation on the photogalvanic responses [23]–[25], second-harmonic generation and the nonlinear Hall effect [26]–[28]. Additional related studies of fluctuation effects include investigation of the observed in-plane magnetic anisotropy [29] and planar Hall effect [30].

The reported analyses of the enhancement of MCA by superconducting fluctuations have been predominantly based on the phenomenological framework of time-dependent Ginzburg-Landau (TDGL) theory [17], [31]–[33]. It is well established that TDGL correctly captures the leading fluctuation-induced conductivity enhancement due to preformed Cooper pairs, the so-called Aslamazov-Larkin (AL) contribution [34], but it does not include quantum interference effects, most notably the Maki-Thompson (MT) term [35], [36]. From a more microscopic perspective, for instance within nonlinear sigma-model approaches formulated in the Keldysh technique, it can be shown that the MT contribution emerges naturally as a correction to the current rather than as a modification of the TDGL equation itself [37]. This observation allows the MT term to be incorporated into the theory of MCA on an equal footing with the AL contribution, which in part provides one of the main motivations for the present work.

At the level of linear conductivity, the key distinction between the AL and MT contributions lies in their sensitivity to dephasing processes. The MT term is formally divergent in the absence of pair-breaking mechanisms and therefore requires regularization by processes such as spin-flip scattering from magnetic impurities or other dephasing channels, closely analogous to the regularization procedures familiar from weak-localization physics [38]. In two-dimensional systems this leads to an additional logarithmic enhancement of the MT correction, while both AL and MT contributions share the same leading scaling with \(T-T_c\), with the MT term being a factor of two larger.

In contrast, the situation becomes qualitatively richer in the nonlinear response regime. The AL contribution corresponds to transport carried by the collective momentum of fluctuating Cooper pairs, with the nonlinearity arising from the modification of the condensate dynamics in external fields. By contrast, in the MT channel the current couples more directly to the external electric field through quasiparticle interference processes. Importantly, these two objects – the Cooper pair momentum in AL and the electric field coupling in MT – transform differently under point-group symmetries. Since magnetochiral anisotropy is fundamentally constrained by symmetry and relies on simultaneous breaking of time-reversal and inversion symmetries, one can expect the AL and MT channels to contribute in distinct ways.

We find that this expectation is indeed borne out in the concrete case of MoS\(_2\). This material belongs to the \(D_{3h}\) point group, exhibits pronounced trigonal warping, and hosts Ising-type superconductivity. When a perpendicular magnetic field is applied, symmetry considerations allow for a cubic Lifshitz invariant in the free energy, which is sufficient to generate a finite MCA response in the AL channel. However, this symmetry-allowed structure does not contribute to the MT part of the nonlinear response. We further find that lowering the symmetry perturbatively, for example via strain-induced tensor distortions, allowes additional invariant structures in the free energy that do contribute to MCA in the MT channel.

In other crystallographic classes the interplay between AL and MT contributions can be markedly different. For instance, our preliminary considerations suggest that in Rashba-type superconductors with \(C_{3v}\) symmetry subjected to an in-plane magnetic field, the MT channel contributes to MCA already at the unstrained level. However, its contribution exhibits a distinct tensorial structure compared to the AL term, reflecting the different symmetry origins and microscopic mechanisms underlying the two channels.

These considerations motivate us to begin the paper with a brief discussion of magnetochiral anisotropy in normal conductors with appropriately broken symmetries and in the presence of strain. This pedagogical introduction is primarily intended to explicitly verify the allowed vector structures contributing to the MCA current when strain is present – structures that could otherwise be established purely on symmetry grounds. This material is presented in Sec. 2.

In Sec. 3, we turn to fluctuation-induced corrections to the MCA. Building on earlier microscopic approaches, we in particular incorporate the Maki-Thompson contribution alongside the Aslamazov-Larkin term. Special attention is devoted to symmetry analysis and to the construction of the relevant cubic Lifshitz invariants that govern the effective energy landscape of fluctuating Cooper pairs and ultimately give rise to a pronounced MCA response.

We conclude in Sec. 4 with a summary of the main results and a discussion of their broader implications.

2 MCA from symmetry considerations↩︎

Magnetochiral anisotropy is fundamentally a second-order nonlinear transport response that is odd under both inversion \(\mathcal{P}\) and time reversal \(\mathcal{T}\). The current must therefore be linear in the magnetic field \(\boldsymbol{B}\) and quadratic in the electric field \(\boldsymbol{E}\). The current density \(\boldsymbol{j}\) induced in the medium can be expanded in powers of electric field \(\boldsymbol{E}\) as follows \[j_i=\sigma_{ij}E_j+\chi_{ijk}E_jE_k+\gamma_{ijkl}B_jE_kE_l+\ldots\] The expansion coefficients are tensors of the second, third, and fourth ranks, respectively. We restrict our considerations only to a response to static field, as more terms can be added to the current density for the case of oscillating fields or responses at finite wave vector [39]. The linear conductivity tensor \(\sigma_{ij}\) describes the Ohm’s law and \(\chi_{ijk}\) is the ordinary noncentrosymmetric quadratic conductivity tensor (allowed when \(\mathcal{P}\) is broken ¹).The MCA tensor \(\gamma_{ijkl}\) is additionally odd in magnetic field and it can be viewed as the magnetic-field-linear part of the quadratic conductivity. It is symmetric in the last two indices \(\gamma_{ijkl}=\gamma_{ijlk}\) because \(E_kE_l\) is symmetric.

For a fully isotropic chiral medium, rotational symmetry constrains the tensor strongly. The only allowed structure is \(\boldsymbol{j}_{\text{MCA}}=\gamma(\boldsymbol{B}\cdot\boldsymbol{E})\boldsymbol{E}\). This is the familiar form used in discussions of nonreciprocal resistivity, namely \(\Delta\rho\propto \boldsymbol{B}\cdot\boldsymbol{E}\) [15], [16].

For a 2D crystal with out-of-plane field \(B_z\) the structure reduces to \(\varsigma_{ikl}B_zE_kE_l\). The remaining tensor \(\varsigma_{ikl}\) is determined entirely by the crystal point group. For example, for the point group \(D_{3h}\), the tensor structure is extremely constrained by: threefold rotation \(C_3\), vertical mirrors, the horizontal mirror \(\sigma_h\). Because the MCA tensor is symmetric in \(k,l\) the relevant object to consider from symmetry is the symmetric product \(E_kE_l\). In 2D this decomposes into irreducible pieces: \(E_kE_l=\frac{E^2}{2}\delta_{kl}+Q_{kl}\), where \[\label{eq:Q} Q_{kl}=\left(\begin{array}{cc}E^2_x-E^2_y & 2E_xE_y \\ 2E_xE_y & -(E^2_x-E^2_y)\end{array}\right)\tag{1}\] is the traceless quadrupolar part. Under \(D_{3h}\), \(E^2\) transforms as the trivial irreducible representation (irrep) \(A'_1\) and \((E^2_x-E^2_y,2E_xE_y)\) transforms as the two-dimensional irrep \(E'\). The current \((j_x,j_y)\) also transforms as \(E'\), thus the MCA coupling must map \(E'\otimes E'\to E'\). The isotropic contribution proportional to \(\delta_{kl}\) is forbidden because it would require an invariant vector. Therefore only the traceless \(E'\) sector contributes. There is only one independent \(D_3\)-invariant symmetric rank-3 tensor in 2D. It can be written compactly as \[\label{eq:t} t_{ikl}=\Re\left[(e_x+ie_y)_i(e_x+ie_y)_k(e_x+ie_y)_l\right]\tag{2}\] This is exactly the same tensor that appears in trigonal warping and cubic anisotropy. Explicitly, the nonzero components are \(t_{xxx}=1\), \(t_{xyy}=t_{yxy}=t_{yyx}=-1\) up to an overall normalization/sign convention. Therefore the MCA current in \(D_{3h}\) has the universal form with \(\varsigma_{ikl}=\gamma t_{ikl}\). Contracting indices this gives \[\label{eq:j-MCA} \boldsymbol{j}_{\text{MCA}}=\gamma B_z\boldsymbol{F}(\boldsymbol{E}), \quad \boldsymbol{F}(\boldsymbol{E})=\left(\begin{array}{c}E^2_x-E^2_y \\ -2E_xE_y\end{array}\right).\tag{3}\] This is the unique quadratic MCA structure allowed by \(D_{3h}\). As a result, the main task of the microscopic theory is reduced to the calculation of the coefficient \(\gamma\).

We proceed to extend these considerations. Let the strain-induced symmetry breaking be described by the symmetric traceless tensor \[\varepsilon_{ij}=\left(\begin{array}{cc}(\varepsilon_{xx}-\varepsilon_{yy}) & 2\varepsilon_{xy} \\ 2\varepsilon_{xy} & -(\varepsilon_{xx}-\varepsilon_{yy})\end{array}\right)\] or equivalently by the corresponding two-component vector (doublet) \[\boldsymbol{\varepsilon}=(\varepsilon_{xx}-\varepsilon_{yy},-2\varepsilon_{xy})\] We seek all contributions to the MCA current that are: quadratic in \(\boldsymbol{E}\), linear in \(B_z\), linear in strain \(\boldsymbol{\varepsilon}\). The most general response is \[j_i=\lambda_{iakl}B_z\varepsilon_aE_kE_l\] where \(a=1,2\) labels the strain doublet and \(\lambda_{iakl}\) is symmetric in \(k,l\). Going through the list of irreducible representations under the symmetry operation of \(D_{3h}\) that constrain the form of \(\lambda_{iakl}\), we can identify two additional vector combinations in the MCA current induced by strain. In particular we find \[\label{eq:j-MCA-strain} \boldsymbol{j}_{\text{MCA}}=B_z\left[\gamma_1E^2\boldsymbol{\varepsilon}+\gamma_2(\boldsymbol{\varepsilon}\cdot\boldsymbol{E})\boldsymbol{E}\right]\tag{4}\] Indeed, \(B_z \in A_2'\) \(\varepsilon \in E'\) so \(B_z \varepsilon \in E'\), \((E_x,E_y)^2 = A'_1 + E'\). so one vector \((E')\) emerges out of \(A_1' \times E'\) and the other from \(E' \times E'\).

3 MCA from superconducting fluctuations↩︎

In this section, we consider magnetochiral anisotropy in a band model relevant to MoS\(_2\). Our analysis closely follows earlier calculations reported in Ref. [17] while providing several further extensions. The minimal model includes a trigonal-warping term together with spin-splitting terms in the Hamiltonian arising from the Zeeman effect and spin-orbit interaction.

The calculation of the normal-state transport properties reveals that MCA vanishes to leading order in the symmetry-breaking perturbations as a result of mutual cancellations between current contributions from different valleys.

The experimentally observed giant MCA signal near the superconducting transition temperature, \(T_c\sim 9\)K, therefore motivates consideration of fluctuation effects. The total current can be written as \[\boldsymbol{j}=\sigma\boldsymbol{E}+\boldsymbol{j}_{\text{MCA}}\] The linear conductivity \(\sigma=\sigma_{\text{D}}+\sigma_{\text{AL}}+\sigma_{\text{MT}}+\sigma_{\text{DOS}}\) consists of the normal state Drude term \((\sigma_{\text{D}})\), as well as fluctuation-induced corrections including Aslamazov-Larkin \((\sigma_{\text{AL}})\), Maki-Thompson \((\sigma_{\text{MT}})\), and density of states \((\sigma_{\text{DOS}})\) contributions [40]. The MCA part of the current can be classified by the same type of terms. Specifically we find \[\boldsymbol{j}_{\text{MCA}}=\delta\boldsymbol{j}_{\text{AL}}+\delta\boldsymbol{j}_{\text{MT}}.\] The density of states effect is weak and will be omitted in the following.

The part of the current corresponding to the AL-contribution, can be extracted from the TDGL formalism. In this framework it was first derived by Schmid [41] and later used by Dorsey [42] to address the nonlinear response in the centrosymmetric case. Adapting these results to the disordered limit the current can be found in the form \[\begin{align} \label{eq:j-AL} \boldsymbol{j}_{\text{AL}}&=4eT\int_{\boldsymbol{q}}\partial_{\boldsymbol{q}}\alpha(\boldsymbol{q})\nonumber \\ &\times\int^{0}_{-\infty}dt_1 \exp\left[-2\int^{0}_{t_1}dt_2\alpha(\boldsymbol{q}-2e\boldsymbol{A}(t_2+t)+2e\boldsymbol{A}(t))\right]. \end{align}\tag{5}\] We work in the gauge with the vector potential \(\boldsymbol{A}(t)=-\boldsymbol{E}t\) corresponding to the static uniform electric field. The short-hand notation \(\int_{\boldsymbol{q}}=\int\frac{d^2\boldsymbol{q}}{(2\pi)^2}\) denotes \(2D\) momentum integration. Up to an overall factor of density of states, \(\alpha(\boldsymbol{q})\) corresponds to the Fourier transform of the Ginzburg-Landau free energy of preformed Cooper pairs, it reads \[\alpha(\boldsymbol{q})=Dq^2+\tau^{-1}_{\text{GL}}+\delta\alpha(\boldsymbol{q})\] where \(D\) is the coefficient of diffusion, \(\tau^{-1}_{\text{GL}}=\frac{8}{\pi}(T-T_c)\) is the GL time. The correction terms odd in the collective Cooper pair momentum \(\boldsymbol{q}\) are captured by \(\delta\alpha(\boldsymbol{q})\) that are precisely Lifshitz invariants [43]–[45]. It can be shown from the gradient expansion that magnetic-field-induced nonreciprocal dispersion specific to \(D_{3h}\) reads \[\label{eq:alpha} \delta\alpha_{D_{3h}}(\boldsymbol{q})=\kappa B_z (q^3_x-3q_xq^2_y)\tag{6}\] The microscopic coefficient of the model \(\kappa\propto\lambda g\mu_\text{B}\Delta_{\text{SO}}/T^2_c\) is proportional to the warping parameter in the band structure \(\lambda\) and strength of spin-orbit splitting \(\Delta_{\text{SO}}\), where \(g\) and \(\mu_{\text{B}}\) are the \(g\)-factor and Bohr magneton respectively. Expanding Eq. 5 to the linear order in \(\boldsymbol{E}\) gives \(\boldsymbol{j}_{\text{AL}}=\sigma_{\text{AL}}\boldsymbol{E}\) with \(\sigma_{\text{AL}}=\frac{e^2}{2\pi}(T_c\tau_{\text{GL}})\). Expanding it further to the second order gives \[\delta\boldsymbol{j}_{\text{AL}}=e^3T\int_{\boldsymbol{q}} \frac{\partial_{\boldsymbol{q}}\alpha(\boldsymbol{q})\partial^2_{q_jq_k}\alpha(\boldsymbol{q})}{\alpha^4(\boldsymbol{q})}E_jE_k\] This term results in the nonvanishing contributions only in the presence of \(\delta\alpha(\boldsymbol{q})\). Working to the leading order in \(B_z\) we find after the averaging over the direction of \(\boldsymbol{q}\) the following expression \[\delta\boldsymbol{j}_{\text{AL}}=6 e^3T\kappa B_z\boldsymbol{F}(\boldsymbol{E}) \int_{\boldsymbol{q}}\frac{Dq^2}{(Dq^2+\tau^{-1}_{\text{GL}})^4}.\] Performing the remaining momentum integral, the final result is \[\label{eq:AL-MCA} \delta\boldsymbol{j}_{\text{AL}}=\frac{e^3\kappa B_z}{4\pi DT_c}(T_c\tau_{\text{GL}})^2\boldsymbol{F}(\boldsymbol{E}).\tag{7}\] The vector form of the current matches the expectations based on the symmetry arguments. In particular, matching this result to Eq. 3 gives us MCA coefficient in the form \(\gamma=\frac{e^3\kappa}{4\pi DT_c}(T_c\tau_{\text{GL}})^2\).

The quantum-interference term in the Cooper channel described by the MT-process does not appear in the TDGL equation but arises as a correction term to the current. It has to be derived from the microscopic theory as shown in Ref. [37]. It can be written in the form \[\begin{align} \label{eq:j-MT} \boldsymbol{j}_{\text{MT}}&=16e^2TD\boldsymbol{E}\nonumber \\ &\times \int_{\boldsymbol{q}}\int^{t}_{-\infty}dt_1\exp\left[-2\int^{t}_{t_1}dt'\beta(\boldsymbol{q}-e\boldsymbol{A}(t')-e\boldsymbol{A}(2t_1-t'))\right]\nonumber \\ &\times \int^{t_1}_{-\infty}dt_2\exp\left[-2\int^{t_1}_{t_2}dt''\alpha(\boldsymbol{q}-2e\boldsymbol{A}(t'')) \right] \end{align}\tag{8}\] where \[\beta(\boldsymbol{q})=Dq^2+\tau^{-1}_\phi\] is related to the two-Cooperon part of the anomalous MT-diagram and \(\tau_\phi\) is the dephasing time. To the linear order one restores the known result \(\boldsymbol{j}_{\text{MT}}=\sigma_{\text{MT}}\boldsymbol{E}\) with \(\sigma_{\text{MT}}=\frac{e^2}{\pi}(T_c\tau_{\text{GL}})\ln(\tau_\phi/\tau_{\text{GL}})\). As usually done, in the prefactor of the logarithm, we absorbed dephasing term into the shift of the critical temperature, \(T_c\to T_c-\pi/8\tau_\phi\), thus effectively renormalizing the GL time. As a next step, working to the second order in the field, we extract from Eq. 8 the leading MCA part of the current \[\label{eq:j-MT-MCA} \delta\boldsymbol{j}_{\text{MT}}=4e^3TD\boldsymbol{E}\int_{\boldsymbol{q}}\frac{\boldsymbol{E}\cdot\partial_{\boldsymbol{q}}\alpha(\boldsymbol{q})}{\beta(\boldsymbol{q})\alpha^3(\boldsymbol{q})}.\tag{9}\] However, upon substituting \(\delta\alpha(\boldsymbol{q})\) from Eq. 6 into this expression, we find that the current vanishes because the integrand is odd (\(d\)-wave–like) and therefore disappears upon angular averaging over the Cooper-pair momentum \(\boldsymbol{q}\). A closer inspection of this result suggests that the cancellation is not generic, but rather a specific consequence of the \(D_{3h}\) form of the Lifshitz invariant.

These considerations prompted us to examine physically relevant perturbations that lower the symmetry, such as strain. A group-theoretical analysis then suggests the possibility of a mixed magnetic field-strain terms in the Ginzburg-Landau functional. The first symmetry allowed term emerges in the first order \(B_z(\boldsymbol{\varepsilon}\cdot\boldsymbol{q})\). This term alone can’t give rise to the MCA current as it can be gauged away. In other words, it simply picks a particular vector \(\boldsymbol{q}_0\propto\boldsymbol{\varepsilon}\) near which a pair-propagator becomes soft, therefore shifting the free energy minimum to that point removes this term. This is a well-known nuance which was discussed extensively in the context of superconducting diode effect in helical superconductors [45], [46]. The next in complexity would be quadratic in \(\boldsymbol{q}\) terms that are symmetry allowed but do not contribute to the current as they vanish in Eq. 9 upon averaging over the directions of \(\boldsymbol{q}\). Therefore, we have to consider cubic terms and the simplest we identify is the following \[\delta\alpha_{D_{3h}}(\boldsymbol{q})=\eta B_z(\boldsymbol{\varepsilon}\cdot\boldsymbol{q})q^2,\] which captures two distinct microscopic strain channels, namely uniaxial anisotropic bond stretching and shear (bond-angle distortion). From a tight-binding model perspective the coefficient \(\eta\) can be related to the change of the hopping term, \(\eta\propto\partial\ln t/\partial\ln a\), with \(t\) being the nearest neighbor hopping integral and \(a\) the bond length. Other more complex terms are possible that also resolve trigonal anisotropy.

Then, in complete analogy, expanding to the linear order in \(\delta\alpha(\boldsymbol{q})\) and averaging over the directions of \(\boldsymbol{q}\), e.g. \(\langle(\boldsymbol{E}\cdot\boldsymbol{q})(\boldsymbol{\varepsilon}\cdot\boldsymbol{q})\rangle=\frac{1}{2}q^2(\boldsymbol{\varepsilon}\cdot\boldsymbol{E})\), the MT term transforms into \[\delta\boldsymbol{j}_{\text{MT}}=4e^3T\eta B_z\boldsymbol{E}(\boldsymbol{\varepsilon}\cdot\boldsymbol{E}) \int_{\boldsymbol{q}}\frac{2Dq^2(Dq^2+\tau^{-1}_{\text{GL}})-3(Dq^2)^2}{(Dq^2+\tau^{-1}_\phi)(Dq^2+\tau^{-1}_{\text{GL}})^4}\] After the final integration we find \[\label{eq:MT-MCA} \delta\boldsymbol{j}_{\text{MT}}=\frac{e^3\eta B_z}{\pi T_cD}(T_c\tau_{\text{GL}})^2f(\tau_{\text{GL}}/\tau_\phi)\boldsymbol{E}(\boldsymbol{\varepsilon}\cdot\boldsymbol{E}),\tag{10}\] which conforms to the vector structure of Eq. 4 . The dimensionless function is found in the form \[\label{eq:f} f(x)=\frac{2x(2+x)\ln x+x(4-5x)+1}{2(x-1)^4}.\tag{11}\] We note that, although in this model \(\delta\boldsymbol{j}_{\text{AL}}\) and \(\delta\boldsymbol{j}_{\text{MT}}\) originate from physically distinct symmetry-breaking terms, they exhibit the same temperature dependence near \(T_c\), at least in the limit of weak pair breaking.

4 Summary and Discussion↩︎

The separation of energy scales between fermionic and bosonic degrees of freedom enables the onset of the nonlinear transport regime to occur near \(T_c\) at sufficiently lower electric fields than in the normal state transport regime. Indeed, one can estimate the threshold by comparing the work done by the electric field over the size of a Cooper pair, \(\sim eE\xi\), to the characteristic energy of fluctuation-driven pairs, \(\sim T-T_c\). In the GL regime, \(\xi=\sqrt{D\tau_{\text{GL}}}\). Therefore, the critical field required to drive fluctuating pairs into the nonlinear response regime scales as \(eE_c\propto (T-T_c)^{3/2}\).

For a centrosymmetric system the current can be described by the nonlinear conductivity, \(\boldsymbol{j}=\sigma(E)\boldsymbol{E}\), where \(\sigma(E)=\sigma_{\text{D}}+\sigma_{\text{AL}}(E)+\sigma_{\text{MT}}(E)\), with both the nonlinear AL and nonlinear MT conductivities being even functions of \(E\). For example, in the limit \(E\gg E_c\) one can deduce from Eq. 5 \[\sigma_{\text{AL}}(E)=\frac{1}{3}\Gamma(1/3)\sigma_{\text{fl}}\left(\frac{E_c}{E}\right)^{2/3},\] with \(\sigma_{\text{fl}}=\frac{e^2}{2\pi}(T_c\tau_{\text{GL}})\), while the similar expression for MT-term from Eq. 8 reads with the logarithmic accuracy \[\sigma_{\text{MT}}(E)=2^{1/3}\Gamma(4/3)\sigma_{\text{fl}}\left(\frac{E_c}{E}\right)^{2/3}\ln\left(\frac{(E/E_c)^{2/3}}{\tau_{\text{GL}}/\tau_\phi}\right).\] These asymptotic expression are only valid within the Gaussian fluctuation theory. The general scaling form for the nonlinear conductivity can be written in the form [47] \[\sigma(E)\sim \xi^{2-d+z}\Sigma_{\pm}(\xi^{1+z}E)\] where \(\Sigma_\pm(x)\) are the universal scaling functions above \((+)\) and the \((-)\) the critical temperature \(T_c\); and \(z\) is the dynamic critical exponent. At \(T=T_c\) the correlation length diverges, and the nonlinear conductivity scales as a power-law function of electric field \(\sigma(E)\sim E^{-(2-d+z)/(1+z)}\). The mean-field expressions derived above agree with this scaling law for the mean-field exponent \(z=2\).

For a noncentrosymmetric system, the expansion of the current density starts at quadratic order in the electric field. Our main results for the MCA are given by Eq. 7 and Eq. 10 . The key distinction between them lies in their vector structure. For the D\(_{3h}\) point group, the MT term requires an additional symmetry-lowering perturbation, such as strain.

In the limit of weak dephasing, both the AL and MT contributions to the MCA effect exhibit the same temperature singularity near \(T_c\). However, in the strong depairing regime, the MT term is suppressed by a factor of \((T_c\tau_\phi)^2\ll1\), as can be extracted from Eq. 11 in the limit \(x\gg1\).

It should be noted that, at the level of diagrammatic technique, there are additional contributions to the AL and MT currents [Eqs. 5 and 8 ] that arise from the renormalization of the current vertices. These terms can be obtained via the Peierls substitution and physically originate from the warping terms in the Hamiltonian. However, these terms contain extra powers of the bosonic momentum and are therefore less singular. In other words, Eqs. 5 and 8 capture the leading singularity in \(T-T_c\).

It is important to discuss the role of disorder scattering in this problem. A well-known feature of fluctuation-induced transport described by the AL contribution is its apparent independence of the elastic scattering time, (\(\tau\)). Specifically, calculations performed within the TDGL framework yield the same result in both the diffusive limit, (\(T_c\tau \ll 1\)), and the clean limit, (\(T_c\tau \gg 1\)). At first sight, this conclusion is problematic. Superconducting fluctuations originate from electron-electron interactions in the Cooper channel, and such interactions conserve the total electronic momentum. Therefore, in the absence of impurity scattering, which provides a mechanism for momentum relaxation, one would not expect fluctuations to generate any correction to the resistivity. This apparent paradox between the finite AL conductivity obtained from phenomenological calculations and the physical expectation that interaction effects alone cannot produce dissipation was resolved by Aronov et al. [48]. They showed that, in the absence of impurity scattering, the coupling of an external electric field to normal electrons does not generate an effective force acting on superconducting fluctuations. A proper microscopic treatment of fluctuations in the nonlocal ballistic regime consequently reveals that the AL conductivity vanishes in the limit (\(\tau \to \infty\)). Moreover, the result is sensitive to the order in which the limits are taken: the dc limit, (\(\omega \to 0\)), and the clean limit, (\(\tau \to \infty\)), do not generally commute. For these reasons, we expect similar issues to arise in the context of nonlinear fluctuation-induced transport, including MCA, in the ballistic regime. Existing studies of fluctuation-induced MCA [17], [31]–[33] have been carried out exclusively within the clean-limit GL theory, and the role of momentum relaxation has not been explicitly addressed. To the best of our knowledge, no extension of the microscopic analysis of Ref. [48] to the nonlinear regime currently exists. Consequently, the problem of fluctuation-induced MCA in clean superconductors remains unresolved.

Our results for the MCA apply to the diffusive limit. The parameter \(\kappa\) of the warping term that enters Eq. 7 is strongly renormalized by disorder scattering. This occurs due to the broadening of the single-particle Green’s functions and vertex corrections. The latter enter through the inclusion of a disorder ladder diagram in the Cooper pairing channel. We find that in the diffusive limit, \(T_c\tau\ll1\), the overall suppression factor scales parametrically as \(\kappa_{\text{dis}}\simeq\kappa (T_c\tau)^3\). Despite this strong suppression, the overall magnitude of the MCA induced by fluctuations remains large.

Indeed, the typical magnitude of the MCA current in the normal state can be estimated for one of the valleys using the Boltzmann equation. It is found to be \[\delta\boldsymbol{j}_{\text{N}}\simeq e^3\lambda\nu\tau^2\Delta_Z\boldsymbol{F}(\boldsymbol{E}),\] where \(\Delta_Z=g\mu_{\text{B}}B_z\) and \(\nu\) is the single-particle density of states. The applicability of the Gaussian fluctuation theory is controlled by the Ginzburg-Levanyuk criterion, which constrains the proximity to \(T_c\). Specifically, one must satisfy \(T-T_c>\text{Gi}\). For a disordered conductor, the Ginzburg number is related to the dimensionless conductance as \(\text{Gi}\sim 1/(E_F\tau)\). Putting these arguments together, one arrives at the following estimate for the superconducting contribution to the MCA current: \[\delta\boldsymbol{j}_{\text{S}}\simeq e^3\lambda\nu\frac{\Delta_{\text{SO}}\Delta_{\text{Z}}}{T^3_c}(E_{\text{F}}\tau)^2(T_c\tau)^3 \boldsymbol{F}(\boldsymbol{E}).\] The ratio between the two expressions is determined by three parameters, \(\delta\boldsymbol{j}_{\text{S}}/\delta\boldsymbol{j}_{\text{N}}\sim (\Delta_{\text{SO}}/T_c)(T_c\tau)(E_{\text{F}}\tau)^2\). According to ab initio calculations, \(\Delta_{\text{SO}}\approx 7.5\) meV. Therefore, for \(T_{c}\approx 8.8\) K, one finds \(\Delta_{\text{SO}}/T_c\sim10\). The experimentally reported normal-state sheet resistivity in MoS\(_2\), \(\rho_{\text{N}}\approx140\) Ohms, translates into the dimensionless conductance \(E_{\text{F}}\tau\sim10^2\). Based on the separation between the characteristic energy scales of the Fermi energy and the superconducting gap, one estimates \(T_c\tau\sim0.1\). Taken together, these numbers imply an enhancement by approximately four orders of magnitude.

In agreement with earlier conclusions [17], [31]–[33], our analysis suggests that this enhancement mechanism is fairly generic and should also occur in other superconductors, for example in systems with Rashba-type spin-orbit coupling subject to an in-plane magnetic field, and the superconducting surface state of a topological insulator.

Acknowledgements↩︎

The work of J. T. M. was supported in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001 and by the NSF Quantum Leap Challenge Institute for Hybrid Quantum Architectures and Networks Grant No. OMA-2016136. M. K. acknowledges the support of the grant NSF-BSF DMR2023693. The work of A. L. was supported by NSF Grant No. DMR-2452658 and H. I. Romnes Faculty Fellowship provided by the University of Wisconsin-Madison Office of the Vice Chancellor for Research and Graduate Education with funding from the Wisconsin Alumni Research Foundation.

References↩︎