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Sci Adv. 2017 April; 3(4): e1602390.

Published online 2017 April 21. doi: 10.1126/sciadv.1602390

PMCID: PMC5400453

Ryohei Wakatsuki,^{1,}^{*} Yu Saito,^{1,}^{*} Shintaro Hoshino,^{2} Yuki M. Itahashi,^{1} Toshiya Ideue,^{1} Motohiko Ezawa,^{1} Yoshihiro Iwasa,^{1,}^{2} and Naoto Nagaosa^{1,}^{2,}^{†}

Received 2016 September 28; Accepted 2017 February 22.

Copyright © 2017, The Authors

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is **not** for commercial advantage and provided the original work is properly cited.

Lack of spatial inversion symmetry in crystals offers a rich variety of physical phenomena, such as ferroelectricity and nonlinear optical effects (for example, second harmonic generation). One such phenomenon is magnetochiral anisotropy, where the electrical resistance depends on the current direction under the external magnetic field. We demonstrate both experimentally and theoretically that this magnetochiral anisotropy is markedly enhanced by orders of magnitude once the materials enter into a superconducting state. To exemplify this enhancement, we study the magnetotransport properties of the two-dimensional noncentrosymmetric superconducting state induced by gating of MoS_{2}. These results indicate that electrons feel the noncentrosymmetric crystal potential much coherently and sensitively over the correlation length when they form Cooper pairs, and show open a new route to enhance the nonreciprocal response toward novel functionalities, including superconducting diodes.

Rectification is a fundamental nonreciprocal effect or phenomenon, where the current-voltage (*I*-*V*) characteristics of right and left directions are distinct. As naturally expected, it requires systems without inversion symmetry, for example, p-n junction. In p-n junctions, the width of the depletion layer changes differently as a function of voltage between right and left directions. In contrast with these junction systems, in bulk crystals without interfaces, the crystal momentum **k** is a good quantum number, and the situation is much less trivial. The lack of inversion symmetry manifests itself as the spin splitting of bands when the relativistic spin-orbit interaction is present. However, time-reversal symmetry relates the energy dispersion at **k** and **−k** with the spin direction reversed. Therefore, the nonreciprocal current responses usually appear when both inversion and time-reversal symmetries are broken. In the case of dc transport in metals with broken inversion symmetry, the nonreciprocal nature necessarily appears in nonlinear responses, such as the current proportional to the square of the electric field **E** under the external magnetic field **B**, as expressed by the current-dependent resistivity *R*

$$\mathit{R}={\mathit{R}}_{0}(1+\mathrm{\gamma}\mathit{B}\mathit{I})$$

(1)

where *I* = |**I**| is the current and γ is the coefficient representing the strength of this effect called magnetochiral anisotropy (MCA) (*1*–*6*). The magnitude of γ is usually small in metals typically of the order of ~10^{−3} to 10^{−2} T^{−1} A^{−1}. The reason is that the spin-orbit interaction and magnetic energy are much smaller than the Fermi energy ε_{F} and hence are weak perturbations (see Fig. 1A).

When the superconductivity sets in, a new energy scale enters, that is, the transition temperature *T*_{c} or the superconducting gap Δ, which are much smaller than ε_{F}. Therefore, it is possible that the effects of spin-orbit interaction and magnetic field are enhanced in the superconducting state. It is known that the noncentrosymmetric superconductors have several unique features, including the mixed singlet and triplet pairings and the associated large upper critical field beyond the Pauli limit, super spin current, and supercurrent induced by the Zeeman field (*7*–*9*). These superconducting properties have been intensively studied thus far, but the nature of the superconducting fluctuation (*10*–*15*) in the resistive state has not been well explored, especially under a magnetic field.

Here, we report the MCA in the resistive superconducting fluctuation in noncentrosymmetric superconductors. In particular, with the example of electric field–induced superconductivity in MoS_{2}, the signal of the MCA in the superconducting state was found to be markedly enhanced compared to that in the normal state (see Fig. 1, A and B). This experimental result is consistent with the theoretical prediction, which suggests that intrinsic properties of noncentrosymmetric systems can be accessed by the coherence of Cooper pairs.

MoS_{2} is an archetypal layered semiconducting transition metal dichalcogenides (TMDs), which have been attracting significant attentions as two-dimensional (2D) materials for various functionalities (*16*–*18*). MoS_{2} is composed of the stacked layers with weak van der Waals interaction. In the monolayer, one Mo atom is sandwiched by two S atoms, forming a 2D network with *D*_{3h} symmetry (Fig. 2A). Reflecting this threefold symmetry, the semiconducting TMDs have trigonal warping in the Fermi surfaces (*19*). In the isolated monolayer, in-plane inversion symmetry is broken, causing out-of-plane spin polarization together with effective Zeeman fields, which results in spin splitting at the *K* points (*20*, *21*). Because of the sign difference in the effective Zeeman fields at the *K* and −*K* valleys, the spin polarization is oppositely locked depending on the valley. However, in this state, only spatial inversion symmetry is broken, and time-reversal symmetry still holds. Time-reversal symmetry is broken by the application of an external magnetic field. A schematic structure of the conduction band bottom under an external out-of-plane magnetic field is illustrated in Fig. 2B. Recently, MoS_{2} was reported to show superconductivity by electrostatic doping in an electric double-layer transistor (EDLT) configuration (*22*).

The EDLT forms an electrochemical interface, in which a high-density 2D electron system is created without introducing extrinsic disorder. Furthermore, because superconductivity occurs in chemically undoped crystals, highly crystalline 2D superconductors can be realized, (*23*, *24*), showing quantum metallic states in the low temperature (*25*). In addition, a signature of the noncentrosymmetric superconductor was observed by an enhanced in-plane upper critical fields owing to the spin-valley locking (*26*–*28*). Thus, the gate-induced superconductivity in MoS_{2} is an ideal platform for investigating the properties intrinsic to the crystal structure, such as broken spatial inversion symmetry.

We prepared an EDLT structure with a 20-nm-thick 2*H*-MoS_{2} thin flake with *N*,*N*-diethyl-*N*-(2-methoxyethyl)-*N*-methylammonium bis(trifluoromethylsulphonyl) imide (DEME-TFSI) as a gate medium (Fig. 2C). The electrodes (Cr/Au, 5 nm/90 nm) in a Hall bar configuration were covered with SiO_{2} thin film for each electrode to define the channel shape (see Materials and Methods). Following the recent literature (*29*), the device was fabricated so that the source-drain current (*I*_{DS}) flow was parallel to the zigzag edge. Figure 3A shows the temperature dependence of sheet resistance *R*_{sheet} of the MoS_{2}-EDLT at a gate voltage *V*_{G} of 5.0 V and a sheet carrier density *n*_{2D} of 1.2 × 10^{14} cm^{−2}. Superconducting transition was observed at the critical temperature of 8.8 K, as defined at the midpoint of the transition with *R*_{sheet} being 50% of the normal state. We identify this temperature as the mean-field transition temperature *T*_{c}, whereas that determined by the zero resistance is regarded as the Kosterlitz-Thouless (KT) transition temperature *T*_{KT} (*25*). Here, note that the sample we measured was a MoS_{2} multilayer flake with the 2*H* polytype, in which the two adjacent layers are rotated by π with respect to each other, therefore making the whole structure centrosymmetric in contrast to the noncentrosymmetric monolayer (Fig. 2A). However, once the gate voltage is applied, the electric field breaks the out-of-plane inversion symmetry, making the neighboring monolayers inequivalent. At the same time, the conduction band minimum shifts to the ±*K* points, where interlayer coupling is very weak, and thus electrons are quite localized within each layer. Consequently, the in-plane inversion asymmetry in each monolayer reappears. Therefore, the gate-induced conducting and superconducting states are effectively regarded as a monolayer (*30*). Note that the point group symmetry is reduced down to *C*_{3v} in the presence of the strong electric field.

To extract the intrinsic MCA in a 2*H*-MoS_{2}–EDLT, we measured both the first (*R*^{ω}) and second harmonic signals (*R*^{2ω}) of the longitudinal and transverse magnetoresistance in the out-of-plane magnetic field geometry by the application of the ac excitation bias voltage with the constant frequency of 13 Hz. Figure 3B shows the antisymmetric component of *R*^{2ω} for the longitudinal and transverse magnetoresistance (${\mathit{R}}_{\text{sheet}}^{2\mathrm{\omega}}$ and ${\mathit{R}}_{\text{Hall}}^{2\mathrm{\omega}}$) at 2 K at a *I*_{DS} of 17 μA. Once the magnetic field is switched on, ${\mathit{R}}_{\text{sheet}}^{2\mathrm{\omega}}$ first increases, then peaks around 0.5 T, and finally decreases. On the other hand, ${\mathit{R}}_{\text{Hall}}^{2\mathrm{\omega}}$ was extremely small compared with ${\mathit{R}}_{\text{sheet}}^{2\mathrm{\omega}}$, being consistent with the expected selection rule, as expressed in Eq. 4. Figure 3 (C and D) shows *R*_{sheet} and ${\mathit{R}}_{\text{sheet}}^{2\mathrm{\omega}}$, respectively, as a function of the magnetic field between 2 and 15 K. The ${\mathit{R}}_{\text{sheet}}^{2\mathrm{\omega}}$ signal was significantly enhanced in the superconducting state with decreasing temperature. In the normal state above 10 K, ${\mathit{R}}_{\text{sheet}}^{2\mathrm{\omega}}$ was too small to be detected by the lock-in amplifier. The maximum value of ${\mathit{R}}_{\text{sheet}}^{2\mathrm{\omega}}$ increased with decreasing temperature, reaching 0.65 ohms at 2 K, which corresponds to 0.46% of the normal state resistance (Fig. 3E). In Fig. 3F, we plotted the γ value (see Eq. 1), which is calculated as γ = $\frac{2{\mathit{R}}_{\text{sheet}}^{2\mathrm{\omega}}}{{\mathit{R}}_{\text{sheet}}\mathit{B}{\mathit{I}}_{\text{DS}}}$, as a function of temperature. Here, the values of ${\mathit{R}}_{\text{sheet}}^{2\mathrm{\omega}}$ were taken from the maximum point of the ${\mathit{R}}_{\text{sheet}}^{2\mathrm{\omega}}$ (Fig. 3D) for each temperature, and γ was calculated by the corresponding values of *B* and *R*_{sheet}. In the normal state above the transition temperature, the γ value shows almost zero, whereas it suddenly increases up to ~1200 T^{−1} A^{−1} at the mean-field transition temperature *T*_{c} = 8.8 K. In the superconducting state, all the γ values are order of 10^{3} T^{−1} A^{−1}, and the maximum value is nearly 8000 T^{−1} A^{−1}, both of which are approximately five orders of magnitude larger than those of conventional materials (*1*–*3*). These results provide unambiguous evidence that the MCA signals are significantly enhanced in the superconducting phase at the same current.

Now, we describe the theoretical analysis of MCA near and above *T*_{c} for MoS_{2} under an out-of-plane magnetic field. 2D MoS_{2} is well described by the Hamiltonian given by (*31*, *32*)

$${\mathit{H}}_{\mathit{k}\mathrm{\sigma}\mathrm{\tau}}=\frac{{\mathit{\u0127}}^{2}{\mathbf{k}}^{2}}{2\mathit{m}}+{\mathrm{\tau}}_{\mathit{z}}\mathrm{\lambda}{\mathit{k}}_{\mathit{x}}({\mathit{k}}_{\mathit{x}}^{2}-3{\mathit{k}}_{\mathit{y}}^{2})-{\mathrm{\Delta}}_{\mathrm{Z}}{\mathrm{\sigma}}_{\mathit{z}}-{\mathrm{\Delta}}_{\text{SO}}{\mathrm{\sigma}}_{\mathit{z}}{\mathrm{\tau}}_{\mathit{z}}$$

(2)

around the *K* and −*K* points. Here, λ is the parameter of the trigonal warping (*21*), Δ_{Z} and Δ_{SO} are the spin splittings due to the Zeeman effect and the spin-orbit interaction, respectively, and σ_{z} = ±1 (or ↑, ↓) and τ_{z} = ±1 (or +, −) represent the spin and valley degrees of freedom, respectively. For simplicity, we neglected the Rashba spin-orbit interaction because of the electric field in the EDLT device because the Rashba term does not contribute to the nonreciprocal response (see Supplementary Materials).

We first derive the Ginzburg-Landau (GL) free energy as (see Materials and Methods)

$$\mathit{F}={\displaystyle \int}\mathit{d}\mathit{r}\mathrm{\Psi}*[\mathit{a}+\frac{{\mathit{p}}^{2}}{4\mathit{m}}+\frac{\mathrm{\Lambda}\mathit{B}}{{\mathit{\u0127}}^{3}}({\mathit{p}}_{\mathit{x}}^{3}-3{\mathit{p}}_{\mathit{x}}{\mathit{p}}_{\mathit{y}}^{2}\left)\right]\mathrm{\Psi}+\frac{\mathit{b}}{2}{\displaystyle \int}\mathit{d}\mathit{r}{|\mathrm{\Psi}(\mathit{r})|}^{4}$$

(3)

where ** p** = −i

$$\mathit{j}=\frac{{\mathit{e}}^{2}}{16\mathit{\u0127}}{\mathrm{\u03f5}}^{-1}\mathit{E}-\frac{\mathrm{\pi}{\mathit{e}}^{3}\mathit{m}\mathrm{\Lambda}\mathit{B}}{64{\mathit{\u0127}}^{3}{\mathit{k}}_{\mathrm{B}}{\mathit{T}}_{\mathrm{c}}}{\mathrm{\u03f5}}^{-2}\mathit{F}(\mathit{E})$$

(4)

where $\mathrm{\u03f5}=\frac{\mathit{T}-{\mathit{T}}_{\mathrm{c}}}{{\mathit{T}}_{\mathrm{c}}}$ and $\mathit{F}(\mathit{E})=({\mathit{E}}_{\mathit{x}}^{2}-{\mathit{E}}_{\mathit{y}}^{2},-2{\mathit{E}}_{\mathit{x}}{\mathit{E}}_{\mathit{y}})$, which satisfies the transformation rule of the *C*_{3v} point group. We have finite nonreciprocal current, which develops around the critical region.

On the other hand, the normal phase does not carry the nonreciprocal current in this model because the contributions from the two pairs of the bands cancel out in the lowest order (see Materials and Methods). Hence, neglecting the nonreciprocal current in the normal state, one can define the linear and nonlinear transport coefficients as **j** = (σ_{1} + σ_{n})** E** + σ

$$\mathrm{\gamma}=\frac{{\mathrm{\sigma}}_{2}}{{\mathrm{\sigma}}_{1}^{2}\mathit{W}\mathit{B}}=\frac{4\mathrm{\pi}\mathit{m}\mathrm{\Lambda}}{\mathit{e}\mathit{W}\mathit{\u0127}{\mathit{k}}_{\mathrm{B}}{\mathit{T}}_{\mathrm{c}}}$$

(5)

where *W* is the channel width. We will estimate the amplitude of the γ value and show below that it is markedly enhanced in the superconducting state.

Although the normal state does not carry nonreciprocal current in this model, one can estimate its typical order of magnitude using the current from only one of the band pairs. Thus, the ratio of the γ values in the normal region (γ_{N}) and the resistive superconducting region (γ_{S}) is

$$\frac{{\mathrm{\gamma}}_{\mathrm{S}}}{{\mathrm{\gamma}}_{\mathrm{N}}}~{\left(\frac{{\mathrm{\epsilon}}_{\mathrm{F}}}{{\mathit{k}}_{\mathrm{B}}{\mathit{T}}_{\mathrm{c}}}\right)}^{3}$$

(6)

which is quite large. We emphasize that this anomalous enhancement of the γ parameter should be a universal feature in noncentrosymmetric superconductors because it derives from the energy scale difference between the Fermi energy and the superconducting gap.

In the resistive state at low temperature *T* *T*_{c}, which is continued smoothly to the normal state, the amplitude develops, whereas the phase fluctuation remains, and, hence, the nonlinear interaction term in Eq. 3 becomes relevant. The dynamics of vortices under the current flow and its nonreciprocal nature becomes relevant in this case, which is left for future investigations.

Here, we estimate the physical quantities of the clean MoS_{2}. According to ab initio calculations, 2*m*λ/*ħ*^{2} = −0.49 Å and Δ_{SO} ≈ 7.5 meV around the Fermi energy (*26*, *31*, *32*). We take the experimental parameters *W* 3 μm, ${\mathrm{\sigma}}_{\mathrm{n}}^{-1}=140\text{ohms}$, and *T*_{c} = 8.8 K. The MCA is reflected in the physical quantity γ = (σ_{2}/*WB*)/(σ_{1} + σ_{n})^{2}. The γ value is largely enhanced by the superconducting fluctuation and reaches γ ≈ 250 T^{−1} A^{−1}, which should be compared with γ ≈ 1200 T^{−1} A^{−1} that is experimentally observed at *T*_{c} = 8.8 K (Fig. 3F). Although there is a discrepancy of a factor of ~5, the rapid increase in γ is captured by theoretical calculation.

In conclusion, we have demonstrated that the MCA in the gate-induced superconductivity in MoS_{2} is enhanced by five orders of magnitude compared with the normal state. It is ascribed to the difference of the characteristic energy scale between the Fermi energy (~1 eV) and the superconducting gap (~1 meV). Our argument is quite general, and we believe that the anomalous enhancement of the γ value occurs in any noncentrosymmetric superconductors, although we have shown explicitly so for MoS_{2}.

Bulk 2*H*-polytype MoS_{2} single crystals were cleaved into thin flakes with dozens of nanometers in thickness using the Scotch tape method, and then flakes were transferred onto a Si/SiO_{2} substrate. Au (90 nm)/Cr (5 nm) electrodes were patterned onto an isolated thin flake in a Hall bar configuration, and a side-gate electrode was patterned onto the substrate. We covered the device with the polymethylmethacrylate (used as the resist for electron beam lithography), except for the channel surface, to avoid chemical intercalation from the edge of the flake and define the channel. We used DEME-TFSI as a gate medium.

The resistance measurements (including both the first and second harmonic measurements) of a MoS_{2}-EDLT under zero and finite magnetic fields were made with the four-point probe 2D MoS_{2} is well described by the Hamiltonian method in a Quantum Design Physical Property Measurement System combined with two ac lock-in amplifiers (Stanford Research Systems Model SR830 DSP and Signal Recovery Model 5210). We confirmed that the phase of the first and second harmonic signals in the ac resistance measurements kept showing around 0° and 90°, respectively, which indicates the validity of our ac measurements. The gate voltage was supplied by a Keithley 2400 SourceMeter. We applied gate voltages to the device at 240 K, just above the glass transition temperature of DEME-TFSI, under high vacuum (less than 10^{−4} torr), and then lowered temperatures. The magnetoresistance measurements were performed under a helium atmosphere (5 to 10 torr).

We note the relations derived from the definition of γ in Eq. 1. The current density is described as *j* = σ_{1}*E* + σ_{2}*E*^{2}, where we considered only one direction. If we solve in terms of *E*, then we obtain

$$\mathit{E}=\frac{-{\mathrm{\sigma}}_{1}\pm \sqrt{{\mathrm{\sigma}}_{1}^{2}+4{\mathrm{\sigma}}_{2}\mathit{j}}}{2{\mathrm{\sigma}}_{2}}\approx \frac{1}{{\mathrm{\sigma}}_{1}}\mathit{j}+\frac{{\mathrm{\sigma}}_{2}}{{\mathrm{\sigma}}_{1}^{3}}{\mathit{j}}^{2}$$

(7)

Here, we chose the plus sign to satisfy the linear order relation.

With the sample width *W* and length *L*, we had the current *I* = *Wj* and the voltage drop *V* = −*EL*. Then, we obtained

$$\mathit{V}=-\frac{\mathit{L}}{{\mathrm{\sigma}}_{1}\mathit{W}}\mathit{I}-\frac{{\mathrm{\sigma}}_{2}\mathit{L}}{{\mathrm{\sigma}}_{1}^{3}{\mathit{W}}^{2}}{\mathit{I}}^{2}$$

(8)

With the ac *I* = *I*_{DS}cos ω*t*, the first and second harmonic signals are

$$\begin{array}{c}{\mathit{R}}^{\mathrm{\omega}}={\mathit{R}}_{\text{sheet}}=\mathit{R}\\ {\mathit{R}}^{2\mathrm{\omega}}=\frac{\mathit{R}\mathrm{\gamma}\mathit{B}{\mathit{I}}_{\text{DS}}}{2}\end{array}$$

(9)

Therefore, γ can be calculated as

$$\mathrm{\gamma}=\frac{2{\mathit{R}}^{2\mathrm{\omega}}}{{\mathit{R}}^{\mathrm{\omega}}\mathit{B}{\mathit{I}}_{\text{DS}}}$$

(10)

We reviewed how to calculate the paraconductivity due to the superconducting fluctuation following Schmid (*33*). The GL free energy quadratic in the order parameter reads

$$\mathit{F}={\displaystyle \int}\mathit{d}\mathit{r}\mathrm{\Psi}*(\mathit{r})\mathrm{\eta}(\mathit{r})\mathrm{\Psi}(\mathit{r})={\displaystyle \sum _{\mathit{q}}}{\mathrm{\eta}}_{\mathit{q}}{|{\mathrm{\Psi}}_{\mathit{q}}|}^{2}$$

(11)

where we assumed the uniform system and expressed the free energy in the momentum space, and η_{q} will be given as Eq. 30. The expectation value of the excess current density was evaluated by calculating the time dependence of Ψ explicitly on the basis of the time-dependent GL equation with a stochastic force, namely

$$\mathit{\u0127}\mathrm{\Gamma}{\partial}_{\mathit{t}}\mathrm{\Psi}(\mathit{r},\mathit{t})=-\mathrm{\eta}(\mathit{r},\mathit{t})\mathrm{\Psi}(\mathit{r},\mathit{t})+\mathit{f}(\mathit{r},\mathit{t})$$

(12)

where Γ is the damping coefficient for Ψ, and *f* is the uncorrelated random force satisfying

$$<\mathit{f}*(\mathit{r},\mathit{t})\mathit{f}({\mathit{r}}^{\prime},{\mathit{t}}^{\prime})>=2\mathit{\u0127}\mathrm{\Gamma}{\mathit{k}}_{\mathrm{B}}\mathit{T}\mathrm{\delta}(\mathit{r}-{\mathit{r}}^{\prime})\mathrm{\delta}(\mathit{t}-{\mathit{t}}^{\prime})$$

(13)

or

$$<{\mathit{f}*}_{\mathit{q}}(\mathit{t}){\mathit{f}}_{{\mathit{q}}^{\prime}}({\mathit{t}}^{\prime})>=2\mathit{\u0127}\mathrm{\Gamma}{\mathit{k}}_{\mathrm{B}}\mathit{T}{\mathrm{\delta}}_{\mathit{q},{\mathit{q}}^{\prime}}\mathrm{\delta}(\mathit{t}-{\mathit{t}}^{\prime})$$

(14)

One can check that the stochastic force reproduces the appropriate value |Ψ_{q}|^{2} = *k*_{B}*T*/η_{q}. Next, we calculated the time evolution of Ψ_{q}(*t*), and then the expectation value of the current density. The formal solution of Eq. 12 is given by

$${\mathrm{\Psi}}_{\mathit{q}}(\mathit{t})=\frac{1}{\mathit{\u0127}\mathrm{\Gamma}}{\displaystyle {\int}_{-\infty}^{\mathit{t}}}\mathit{d}{\mathit{t}}^{\prime}{\mathit{f}}_{\mathit{q}}({\mathit{t}}^{\prime})\text{exp}[-\frac{1}{\mathit{\u0127}\mathrm{\Gamma}}{\displaystyle {\int}_{{\mathit{t}}^{\prime}}^{\mathit{t}}}\mathit{d}{\mathit{t}}^{\prime \prime}{\mathrm{\eta}}_{\mathit{q}}({\mathit{t}}^{\prime \prime})]$$

(15)

where we assumed Ψ_{q}(−∞) = 0. Therefore, the expectation value of the order parameter is

$$\langle {|{\mathrm{\Psi}}_{\mathit{q}}(\mathit{t})|}^{2}\rangle =\frac{2{\mathit{k}}_{\mathrm{B}}\mathit{T}}{\mathit{\u0127}\mathrm{\Gamma}}{\displaystyle {\int}_{-\infty}^{\mathit{t}}}\mathit{d}{\mathit{t}}^{\prime}\text{exp}[-\frac{2}{\mathit{\u0127}\mathrm{\Gamma}}{\displaystyle {\int}_{{\mathit{t}}^{\prime}}^{\mathit{t}}}\mathit{d}{\mathit{t}}^{\prime \prime}{\mathrm{\eta}}_{\mathit{q}}({\mathit{t}}^{\prime \prime})]$$

(16)

On the other hand, the current density is expressed as

$$\widehat{\mathit{j}}=-\frac{\mathit{c}\mathrm{\delta}\mathit{F}}{\mathit{V}\mathrm{\delta}\mathit{A}}=\frac{1}{\mathit{V}}{\displaystyle \sum _{\mathit{q}}}(-\frac{\mathit{c}\partial {\mathrm{\eta}}_{\mathit{q}}}{\partial \mathit{A}}){|{\mathrm{\Psi}}_{\mathit{q}}|}^{2}=\frac{1}{\mathit{V}}{\displaystyle \sum _{\mathit{q}}}{\widehat{\mathit{j}}}_{\mathit{q}}{|{\mathrm{\Psi}}_{\mathit{q}}|}^{2}$$

(17)

whose expectation value leads to

$$\mathit{j}(\mathit{t})=\frac{2{\mathit{k}}_{\mathrm{B}}{\mathit{T}}_{\mathrm{c}}}{\mathit{V}\mathit{\u0127}\mathrm{\Gamma}}{\displaystyle \sum _{\mathit{q}}}{\displaystyle {\int}_{-\infty}^{\mathit{t}}}\mathit{d}{\mathit{t}}^{\prime}{\widehat{\mathit{j}}}_{\mathit{q}}({\mathit{t}}^{\prime})\text{exp}[-\frac{1}{\mathit{\u0127}\mathrm{\Gamma}}{\displaystyle {\int}_{{\mathit{t}}^{\prime}}^{\mathit{t}}}\mathit{d}{\mathit{t}}^{\prime \prime}{\mathrm{\eta}}_{\mathit{q}}({\mathit{t}}^{\prime \prime})]$$

(18)

where we set the vector potential ** A** = −

We derived the GL theory for the TMDs and showed that the nonreciprocal current emerges because of the third-order term in the momentum. The Hamiltonian is

$${\mathit{H}}_{\mathit{k}\mathrm{\sigma}\mathrm{\tau}}={\mathrm{\epsilon}}_{\mathit{k}\mathrm{\tau}}-\mathrm{\mu}-{\mathrm{\sigma}\mathrm{\Delta}}_{\mathrm{Z}}-{\mathrm{\sigma}\mathrm{\tau}\mathrm{\Delta}}_{\text{SO}}$$

(19)

$${\mathrm{\epsilon}}_{\mathit{k}\mathrm{\tau}}=\frac{{\mathit{\u0127}}^{2}{\mathit{k}}^{2}}{2\mathit{m}}+\mathrm{\tau}\mathrm{\lambda}{\mathit{k}}_{\mathit{x}}({\mathit{k}}_{\mathit{x}}^{2}-3{\mathit{k}}_{\mathit{y}}^{2})\equiv {\mathrm{\epsilon}}_{\mathit{k}}^{0}+\mathrm{\tau}\mathrm{\lambda}{\mathit{f}}_{\mathit{k}}$$

(20)

If we assume a contact-attractive interaction, then s-wave singlet pairing occurs between the (↑, +) and (↓, −) pair and between the (↑, −) and (↓, +) pair. The free energy for the order parameter Δ is calculated as

$$\mathit{F}={\displaystyle \sum _{\mathit{q}}}[\frac{1}{\mathit{g}}+\frac{\mathit{T}}{\mathit{V}}{\displaystyle \sum _{\mathit{k},\mathrm{\tau}=\pm}}{\mathit{G}}_{\mathit{k}+\mathit{q}/2,\uparrow ,\mathrm{\tau}}^{\mathit{e}}{\mathit{G}}_{\mathit{k}-\mathit{q}/2,\downarrow ,-\mathrm{\tau}}^{\mathit{h}}]{|{\mathrm{\Delta}}_{\mathit{q}}|}^{2}$$

(21)

where *g* is the strength of the attractive interaction, and

$${\mathit{G}}_{\mathit{k}\mathrm{\sigma}\mathrm{\tau}}^{\mathit{e}}=\frac{1}{\mathit{i}{\mathrm{\omega}}_{\mathit{n}}-{\mathit{H}}_{\mathit{k}\mathrm{\sigma}\mathrm{\tau}}}=\frac{1}{\mathit{i}{\mathrm{\omega}}_{\mathit{n}}-{\mathrm{\epsilon}}_{\mathit{k}\mathrm{\tau}}+\mathrm{\mu}+{\mathrm{\sigma}\mathrm{\Delta}}_{\mathit{Z}}+{\mathrm{\sigma}\mathrm{\tau}\mathrm{\Delta}}_{\text{SO}}}$$

(22)

$${\mathit{G}}_{\mathit{k}\mathrm{\sigma}\mathrm{\tau}}^{\mathit{h}}=\frac{1}{\mathit{i}{\mathrm{\omega}}_{\mathit{n}}+{\mathit{H}*}_{-\mathit{k}\mathrm{\sigma}\mathrm{\tau}}}=\frac{1}{\mathit{i}{\mathrm{\omega}}_{\mathit{n}}+{\mathrm{\epsilon}}_{\mathit{k}\mathrm{\tau}}-\mathrm{\mu}-{\mathrm{\sigma}\mathrm{\Delta}}_{\mathit{Z}}-{\mathrm{\sigma}\mathrm{\tau}\mathrm{\Delta}}_{\text{SO}}}$$

(23)

are the electron and hole Green’s functions with the Matsubara frequency ω_{n} = (2*n* + 1)π*k*_{B}*T*. We also define the Green’s functions without the trigonal warping term as ${\mathit{g}}_{\mathit{k}}^{\mathit{e}}$ and ${\mathit{g}}_{\mathit{k}}^{\mathit{h}}$.

We expanded Eq. 21 up to the third order of ** q** and the first order of λ and Δ

We first expanded Eq. 21 with respect to ** q**. The zeroth-order term in

$${\mathit{G}}_{\mathit{k}+\mathit{q}}^{\mathit{e}}{\mathit{G}}_{\mathit{k}}^{\mathit{h}}={\mathit{G}}_{\mathit{k}}^{\mathit{h}}[1+\mathit{q}\cdot {\nabla}_{\mathit{k}}+\frac{1}{2}{(\mathit{q}\cdot {\nabla}_{\mathit{k}})}^{2}+\frac{1}{6}{(\mathit{q}\cdot {\nabla}_{\mathit{k}})}^{3}]{\mathit{G}}_{\mathit{k}}^{\mathit{e}}$$

(24)

where _{k} is the derivative of the wave number. The first-order term is

$$\frac{\mathit{T}}{\mathit{V}}{\displaystyle \sum _{\mathit{k}}}{\mathit{G}}_{\mathit{k}}^{\mathit{h}}\mathit{q}\cdot {\nabla}_{\mathit{k}}{\mathit{G}}_{\mathit{k}}^{\mathit{e}}=\frac{\mathit{T}}{\mathit{V}}{\displaystyle \sum _{\mathit{k}}}{\mathit{q}}_{\mathit{i}}\frac{\partial {\mathrm{\epsilon}}_{\mathit{k}}}{\partial {\mathit{k}}_{\mathit{i}}}{\mathit{G}}_{\mathit{k}}^{\mathit{h}}{({\mathit{G}}_{\mathit{k}}^{\mathit{e}})}^{2}=0$$

(25)

The second-order term is

$$\begin{array}{c}\frac{\mathit{T}}{\mathit{V}}{\displaystyle \sum _{\mathit{k}}}\frac{1}{2}{\mathit{G}}_{\mathit{k}}^{\mathit{h}}{(\mathit{q}\cdot {\nabla}_{\mathit{k}})}^{2}{\mathit{G}}_{\mathit{k}}^{\mathit{e}}=-\frac{\mathit{T}}{2\mathit{V}}{\displaystyle \sum _{\mathit{k}}}\mathit{q}\cdot {\nabla}_{\mathit{k}}{\mathit{G}}_{\mathit{k}}^{\mathit{h}}\mathit{q}\cdot {\nabla}_{\mathit{k}}{\mathit{G}}_{\mathit{k}}^{\mathit{e}}\\ =\frac{\mathit{T}}{2\mathit{V}}{\displaystyle \sum _{\mathit{k}}}{\mathit{q}}_{\mathit{i}}{\mathit{q}}_{\mathit{j}}\frac{\partial {\mathrm{\epsilon}}_{\mathit{k}}}{\partial {\mathit{k}}_{\mathit{i}}}\frac{\partial {\mathrm{\epsilon}}_{\mathit{k}}}{\partial {\mathit{k}}_{\mathit{j}}}{({\mathit{G}}_{\mathit{k}}^{\mathit{h}})}^{2}{({\mathit{G}}_{\mathit{k}}^{\mathit{e}})}^{2}\\ =\frac{\mathit{T}}{\mathit{V}}{\mathrm{\epsilon}}_{\mathit{q}}^{0}{\displaystyle \sum _{\mathit{k}}}({\mathrm{\epsilon}}_{\mathit{k}}^{0}+3\mathrm{\lambda}{\mathit{f}}_{\mathit{k}}){({\mathit{G}}_{\mathit{k}}^{\mathit{h}})}^{2}{({\mathit{G}}_{\mathit{k}}^{\mathit{e}})}^{2}\\ =\frac{\mathit{T}}{\mathit{V}}{\mathrm{\epsilon}}_{\mathit{q}}^{0}{\displaystyle \sum _{\mathit{k}}}{\mathrm{\epsilon}}_{\mathit{k}}^{0}{({\mathit{g}}_{\mathit{k}}^{\mathit{e}})}^{2}{({\mathit{g}}_{\mathit{k}}^{\mathit{h}})}^{2}\\ =\frac{\mathrm{\pi}\mathrm{\nu}\mathit{T}}{2}{\mathrm{\epsilon}}_{\mathit{q}}^{0}{\displaystyle \sum _{\mathit{n}}}\frac{\mathrm{\mu}+{\mathrm{\Delta}}_{\text{SO}}}{{|{\mathrm{\omega}}_{\mathit{n}}|}^{3}}\\ =\frac{7\mathrm{\zeta}(3)}{16}\frac{{\mathit{\u0127}}^{2}\mathrm{\nu}(\mathrm{\mu}+{\mathrm{\Delta}}_{\text{SO}})}{\mathit{m}{(\mathrm{\pi}\mathit{T})}^{2}}{\mathit{q}}^{2}\end{array}$$

(26)

The third-order term is

$$\begin{array}{c}\frac{\mathit{T}}{\mathit{V}}{\displaystyle \sum _{\mathit{k}}}\frac{1}{6}{\mathit{G}}_{\mathit{k}}^{\mathit{h}}{(\mathit{q}\cdot {\nabla}_{\mathit{k}})}^{3}{\mathit{G}}_{\mathit{k}}^{\mathit{e}}\\ =\frac{\mathit{T}}{\mathit{V}}{\displaystyle \sum _{\mathit{k}}}{\mathit{G}}_{\mathit{k}}^{\mathit{h}}{\mathit{q}}_{\mathit{i}}{\mathit{q}}_{\mathit{j}}{\mathit{q}}_{\mathit{k}}[\frac{1}{6}\frac{{\partial}^{3}{\mathrm{\epsilon}}_{\mathit{k}}}{\partial {\mathit{k}}_{\mathit{i}}\partial {\mathit{k}}_{\mathit{j}}\partial {\mathit{k}}_{\mathit{k}}}{({\mathit{G}}_{\mathit{k}}^{\mathit{e}})}^{2}+\frac{{\partial}^{2}{\mathrm{\epsilon}}_{\mathit{k}}}{\partial {\mathit{k}}_{\mathit{i}}\partial {\mathit{k}}_{\mathit{j}}}\frac{\partial {\mathrm{\epsilon}}_{\mathit{k}}}{\partial {\mathit{k}}_{\mathit{k}}}{({\mathit{G}}_{\mathit{k}}^{\mathit{e}})}^{3}+\frac{\partial {\mathrm{\epsilon}}_{\mathit{k}}}{\partial {\mathit{k}}_{\mathit{i}}}\frac{\partial {\mathrm{\epsilon}}_{\mathit{k}}}{\partial {\mathit{k}}_{\mathit{j}}}\frac{\partial {\mathrm{\epsilon}}_{\mathit{k}}}{\partial {\mathit{k}}_{\mathit{k}}}{({\mathit{G}}_{\mathit{k}}^{\mathit{e}})}^{4}]\\ =\frac{\mathit{T}}{\mathit{V}}{\mathit{f}}_{\mathit{q}}{\displaystyle \sum _{\mathit{k}}}{\mathit{G}}_{\mathit{k}}^{\mathit{h}}{({\mathit{G}}_{\mathit{k}}^{\mathit{e}})}^{2}[\mathrm{\lambda}+6{\mathrm{\lambda}\mathrm{\epsilon}}_{\mathit{k}}^{0}{\mathit{G}}_{\mathit{k}}^{\mathit{e}}+(\frac{{\hslash}^{6}}{4{\mathit{m}}^{3}}{\mathit{f}}_{\mathit{k}}+9\mathrm{\lambda}{({\mathrm{\epsilon}}_{\mathit{k}}^{0})}^{2}\left){({\mathit{G}}_{\mathit{k}}^{\mathit{e}})}^{2}\right]\\ =\frac{\mathit{T}}{\mathit{V}}\mathrm{\lambda}{\mathit{f}}_{\mathit{q}}{\displaystyle \sum _{\mathit{k}}}{\mathit{g}}_{\mathit{k}}^{\mathit{h}}{({\mathit{g}}_{\mathit{k}}^{\mathit{e}})}^{2}[1+6{\mathrm{\epsilon}}_{\mathit{k}}^{0}{\mathit{g}}_{\mathit{k}}^{\mathit{e}}+9{({\mathrm{\epsilon}}_{\mathit{k}}^{0})}^{2}{({\mathit{g}}_{\mathit{k}}^{\mathit{e}})}^{2}+{({\mathrm{\epsilon}}_{\mathit{k}}^{0})}^{3}{({\mathit{g}}_{\mathit{k}}^{\mathit{e}})}^{2}(4{\mathit{g}}_{\mathit{k}}^{\mathit{e}}-{\mathit{g}}_{\mathit{k}}^{\mathit{h}})]\\ =\mathrm{\pi}\mathit{T}{\mathrm{\lambda}\mathrm{\Delta}}_{\mathrm{Z}}\mathrm{\nu}{\mathit{f}}_{\mathit{q}}{\displaystyle \sum _{\mathit{n}}}(\frac{1}{2{|{\mathrm{\omega}}_{\mathit{n}}|}^{3}}+\frac{3{(\mathrm{\mu}+{\mathrm{\Delta}}_{\text{SO}})}^{2}}{{|{\mathrm{\omega}}_{\mathit{n}}|}^{5}})\\ =\mathrm{\pi}\mathit{T}{\mathrm{\lambda}\mathrm{\Delta}}_{\mathrm{Z}}\mathrm{\nu}{\mathit{f}}_{\mathit{q}}[\frac{7\mathrm{\zeta}(3)}{8}\frac{1}{{(\mathrm{\pi}\mathit{T})}^{3}}+\frac{93\mathrm{\zeta}(5)}{16}\frac{{(\mathrm{\mu}+{\mathrm{\Delta}}_{\text{SO}})}^{2}}{{(\mathrm{\pi}\mathit{T})}^{5}}]\\ \approx \frac{93\mathrm{\zeta}(5)}{16}\frac{\mathrm{\nu}{(\mathrm{\mu}+{\mathrm{\Delta}}_{\text{SO}})}^{2}{\mathrm{\Delta}}_{\mathit{Z}}\mathrm{\lambda}}{{(\mathrm{\pi}\mathit{T})}^{4}}{\mathit{f}}_{\mathit{q}}\end{array}$$

(27)

Finally, by adding the two contributions in Eq. 21 and defining the wave function ${\mathrm{\Psi}}_{\mathit{q}}=\sqrt{\frac{7\mathrm{\zeta}(3)\mathit{n}}{2{(\mathrm{\pi}{\mathit{T}}_{\mathrm{c}})}^{2}}}{\mathrm{\Delta}}_{\mathit{q}}$ with the electron number *n*, we obtained the GL free energy as

$$\mathit{F}={\displaystyle \sum _{\mathit{q}}}[\mathit{a}+\frac{{\mathit{\u0127}}^{2}{\mathit{q}}^{2}}{4\mathit{m}}+\mathrm{\Lambda}\mathit{B}{\mathit{q}}_{\mathit{x}}({\mathit{q}}_{\mathit{x}}^{2}-3{\mathit{q}}_{\mathit{y}}^{2}\left)\right]{|{\mathrm{\Psi}}_{\mathit{q}}|}^{2}+\frac{\mathit{b}}{2}{|{\mathrm{\Psi}}_{\mathit{q}}|}^{4}$$

(28)

where

$$\mathrm{\Lambda}=\frac{93\mathrm{\zeta}(5)\mathit{g}{\mathrm{\mu}}_{\mathrm{B}}{\mathrm{\Delta}}_{\text{SO}}\mathrm{\lambda}}{28\mathrm{\zeta}(3){(\mathrm{\pi}{\mathit{k}}_{\mathrm{B}}{\mathit{T}}_{\mathrm{c}})}^{2}}$$

(29)

*a* = 4(π*k*_{B}*T*_{c})^{2}ϵ/7ζ(3)ε_{F} and *b* = *a*/8*n*ϵ are the conventional coefficients in the GL theory.

Then, if we neglect the interaction term for *T*_{c} *T*, then we have

$${\mathrm{\eta}}_{\mathit{q}}(\mathit{t})=\mathit{a}+\frac{{\mathit{\u0127}}^{2}}{4\mathit{m}}{(\mathit{q}-\frac{2\mathit{e}}{\mathit{\u0127}}\mathit{E}\mathit{t})}^{2}+\mathrm{\Lambda}\mathit{B}({\mathit{q}}_{\mathit{x}}-\frac{2\mathit{e}}{\mathit{\u0127}}{\mathit{E}}_{\mathit{x}}\mathit{t})\times [{({\mathit{q}}_{\mathit{x}}-\frac{2\mathit{e}}{\mathit{\u0127}}{\mathit{E}}_{\mathit{x}}\mathit{t})}^{2}-3{({\mathit{q}}_{\mathit{y}}-\frac{2\mathit{e}}{\mathit{\u0127}}{\mathit{E}}_{\mathit{y}}\mathit{t})}^{2}]$$

(30)

$${\mathit{j}}_{\mathit{q}}(\mathit{t})=-\frac{\mathit{e}\mathit{\u0127}}{\mathit{m}}(\mathit{q}-\frac{2\mathit{e}}{\mathit{\u0127}}\mathit{E}\mathit{t})-\frac{6\mathit{e}\mathrm{\Lambda}\mathit{B}}{\mathit{\u0127}}\mathit{F}(\mathit{q}-\frac{2\mathit{e}}{\mathit{\u0127}}\mathit{E}\mathit{t})$$

(31)

where we introduced $\mathit{F}(\mathit{E})=({\mathit{E}}_{\mathit{x}}^{2}-{\mathit{E}}_{\mathit{y}}^{2},-2{\mathit{E}}_{\mathit{x}}{\mathit{E}}_{\mathit{y}})$. Note that the model is valid only in a small momentum region. By applying Eq. 18 and expanding the excess current with respect to both of λ and *E*, we obtained

$$\mathit{j}=\frac{{\mathit{e}}^{2}}{16\mathit{\u0127}}{\mathrm{\u03f5}}^{-1}\mathit{E}-\frac{\mathrm{\pi}{\mathit{e}}^{3}\mathit{m}\mathrm{\Lambda}\mathit{B}}{64{\mathit{\u0127}}^{3}{\mathit{k}}_{\mathrm{B}}{\mathit{T}}_{\mathrm{c}}}{\mathrm{\u03f5}}^{-2}\mathit{F}(\mathit{E})$$

(32)

which is Eq. 4.

We calculated the normal state conductivity on the basis of the Boltzmann theory with the relaxation time approximation. The Boltzmann equation is

$${\partial}_{\mathit{t}}\mathit{f}+\stackrel{.}{\mathit{r}}\cdot {\partial}_{\mathit{r}}\mathit{f}+\stackrel{.}{\mathit{k}}\cdot {\partial}_{\mathit{k}}\mathit{f}=-\frac{\mathit{f}-{\mathit{f}}^{0}}{\mathrm{\tau}}$$

(33)

where τ is the relaxation time and *f* and *f*^{0} are the nonequilibrium and equilibrium distribution functions, respectively. For the uniform and stationary system, we have

$$\stackrel{.}{\mathit{k}}\cdot {\partial}_{\mathit{k}}\mathit{f}=-\frac{\mathit{f}-{\mathit{f}}^{0}}{\mathrm{\tau}}$$

(34)

The equation of motion of the wave packet is

$$\mathit{\u0127}\stackrel{.}{\mathit{k}}=-\mathit{e}\mathit{E}$$

(35)

where we ignored the magnetic field and Berry curvature contributions. By solving the equation iteratively, we obtained

$$\mathit{f}\approx {\mathit{f}}^{0}+\frac{\mathrm{\tau}\mathit{e}}{\mathit{\u0127}}\mathit{E}\cdot {\partial}_{\mathit{k}}({\mathit{f}}^{0}+\frac{\mathrm{\tau}\mathit{e}}{\mathit{\u0127}}\mathit{E}\cdot {\partial}_{\mathit{k}}{\mathit{f}}^{0})$$

(36)

Then, the current density was calculated as

$$\mathit{j}=-\mathit{e}{\displaystyle \int}\frac{\mathit{d}\mathit{k}}{{(2\mathrm{\pi})}^{2}}{\mathit{v}}_{\mathit{k}}{\mathit{f}}_{\mathit{k}}$$

(37)

where **${\mathbf{v}}_{\mathit{k}}=\frac{\partial {\mathrm{\epsilon}}_{\mathit{k}}}{\mathit{\u0127}\partial \mathit{k}}$** is the group velocity. The (↑,+) band contribution is

$$\mathit{j}=\frac{\mathrm{\tau}{\mathit{e}}^{2}\mathit{n}}{\mathit{m}}\mathit{E}-\frac{6{\mathrm{\lambda}\mathrm{\tau}}^{2}{\mathit{e}}^{3}\mathit{n}}{{\mathit{\u0127}}^{3}}\mathit{F}(\mathit{E})$$

(38)

The contribution from the (↑,+) and (↓,−) bands is

$$\mathit{j}=\frac{2\mathrm{\tau}{\mathit{e}}^{2}\mathit{n}}{\mathit{m}}\mathit{E}-\frac{12{\mathrm{\lambda}\mathrm{\tau}}^{2}{\mathit{e}}^{3}{\mathrm{\Delta}}_{\mathit{Z}}\mathrm{\nu}}{{\mathit{\u0127}}^{3}}\mathit{F}(\mathit{E})$$

(39)

Then, the total current density from all the four bands is

$$\mathit{j}=\frac{4\mathrm{\tau}{\mathit{e}}^{2}\mathit{n}}{\mathit{m}}\mathit{E}$$

(40)

which shows that the system has no nonreciprocal response in the lowest order of ** q**.

We thank T. Morimoto for fruitful discussions. **Funding:** This work was supported by Ministry of Education, Culture, Sports, Science, and Technology (nos. JP25000003 to Y.I., JP24224009, JP26103006 to N.N., JP15H06133 to T.I., JP25400317, and JP15H05854 to M.E.). It was also supported by the Grants-in-Aid for Japan Society for the Promotion of Science (nos. JP15J09045 to R.W. and JP15J07681 to Y.S.), as well as by the Impulsing Paradigm Change through Disruptive Technologies Program of Council for Science, Technology and Innovation (Cabinet Office, Government of Japan) (to N.N.), and Core Research for Evolutionary Science and Technology (CREST) (no. JPMJCR16F1) (to M.E. and N.N.). **Author contributions:** R.W. and S.H. carried out the theoretical calculations. Y.S. designed and conducted the experiments. Y.S. and Y.M.I. fabricated the samples and analyzed the data. T.I., Y.I., and N.N. conceived and guided the project. R.W., Y.S., S.H., Y.I., and N.N. wrote the manuscript. All authors contributed to the interpretation of the results and commenting on the manuscript. **Competing interests:** The authors declare that they have no competing interests. **Data and materials availability:** All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/4/e1602390/DC1

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