Giant second harmonic transport under time-reversal symmetry in a trigonal superconductor

Nonreciprocal or even-order nonlinear responses in symmetry-broken systems are powerful probes of emergent properties in quantum materials, including superconductors, magnets, and topological materials. Recently, vortex matter has been recognized as a key ingredient of giant nonlinear responses in superconductors with broken inversion symmetry. However, nonlinear effects have been probed as excess voltage only under broken time-reversal symmetry. In this study, we report second harmonic transport under time-reversal symmetry in the noncentrosymmetric trigonal superconductor PbTaSe2. The magnitude of anomalous nonlinear transport is two orders of magnitude larger than those in the normal state, and the directional dependence of nonlinear signals are fully consistent with crystal symmetry. The enhanced nonlinearity is semiquantitatively explained by the asymmetric Hall effect of vortex-antivortex string pairs in noncentrosymmetric systems. This study enriches the literature on nonlinear phenomena by elucidating quantum transport in noncentrosymmetric superconductors.


Linear magnetotransport of PbTaSe2
In Supplementary   shows the fitting curve for ( ) by the two-carrier model.

Vortex Hall effect probed by linear Hall effect in PbTaSe2
In the layered superconductors, the difference in chemical potentials in the normal core and the other superconducting region leads to the charging of vortices, which causes transverse motion of vortices and resultant anomalous Hall resistance near the superconducting transition [4][5][6][7][8][9][10] . Supplementary Figures 2a and 2b show the temperature dependence of symmetrized ω ( sym , Supplementary Fig. 2a , antisymmetrized ω ( asym , red and ω ( asym , blue (Supplementary Fig. 2b at B = ±0.003 T and I = 10 A, respectively, for sample 5. During the superconducting transition, asym shows the large negative peak. The sign of the Hall effect is opposite to asym in the normal state, which is much smaller than the negative component as seen in Supplementary Fig. 1b, because we applied the magnetic field of only 0.003 T in order not to suppress superconductivity. Such a peak is absent in asym .
This anomalous sign reversal in the Hall resistance between the normal state and the superconducting state strongly suggests the occurrence of the vortex Hall effect in this layered superconductor. Supplementary Figures 2c and 2d display the magnetic field dependence of sym and asym , respectively, at T = 3.1 K (purple and 3.4 K (orange for sample 6. The current value was set at I = 50 A in this measurement. The negative peak is clearly observed around the superconducting transition for both cases, which are consistent with the above scenario of the vortex Hall effect. We also calculated the Hall angle of vortex Hall effect as  According to the previous studies 11,12 , these energies are described as where rv is the diameter of vortex loop, and ⊥ are the intralayer and interlayer coupling constant, respectively, is the in-plane coherence length, d is the thickness of one layer, n is the number of layers and c is the energy to create one vortex in one layer. and ⊥ is related to anisotropy parameter as / ⊥ = 2 and we ignore c because it is small. Thus, we conclude that vortex string pairs are excited in PbTaSe2 flake near the superconducting transition and serve as free vortices and antivortices at zero magnetic field to cause the nonlinear superconducting transport.

Theoretical description of vortex-induced linear and nonlinear transport
We describe the transport coefficients in terms of vortex contributions. With the weak force = ( , ) acting on a vortex, the velocity for the vortex (vorticity = + or antivortex ( = − is given by The second term in the right-hand side originates from the trigonal symmetry with the where 0 * = ℎ/2| | is the superconducting flux quantum, and is the number density of vortices or antivortices. The sample lengths along x and y directions are and , respectively. Thus, the voltage is proportional to the vorticity current.
Next, we determine the concrete form of the force . For the linear response, the driving force 0 * on a vortex, where is the current density, is balanced by the force from environment as where ∥ is the friction coefficient and ⊥ is responsible for the vortex Hall effect. The hat symbol represents a unit vector. Assuming that the Hall effect is small, i.e., ≡ ⊥ ∥ ≪ 1, we obtain the relations 1 = 1 ∥ , = 0 * , and = .
As for the ratchet effect, we employ the information from the Brownian motion of point particle in the one-dimensional asymmetric potential for simplicity [15][16][17] . In order to determine the response coefficients, we focus on the case with the external field along the y-direction ( = 0, ≠ 0 . According to Supplementary Eq. (5 , only the motion in the ydirection is involved, and then we consider the one-dimensional kinetic equation: where X is the coordinate of the particle, represents the force from a thermal noise which as a function of the force. 1 and 2 are then obtained as the coefficients of F and 2 , respectively.
We take the potential form shown in Supplementary Fig. 4a, where ℓ v is the periodicity of the potential for vortices, and the length ℓ v is the size of each pinning center.
The potential height is given by , and we will take = 0.1 in the numerical evaluation.
From the Fokker-Planck equation, the response coefficients are given by is the friction coefficient without the pinning potentials. In the Bardeen-Stephen model 20 , it is given by 0 = ℏ 2 n 2 2 2 with the normal conductivity n and the coherence length . While we have introduced the simplified ratchet potential phenomenologically by regarding the vortex as a point particle, it is difficult to derive the potential from the microscopic point of view. It may originate from the impurities and lattice defects, which pin the vortex in a collective manner for the weak pinning and the vortex may be pinned by a single center for the strong pinning. Our purpose here is to confirm the validity of the ratchet vortex scenario and hence we deal with the effective models introduced above. We also note that the randomness of the potential is neglected for simplicity, since the pinning effect and asymmetric feature, which are needed for the nonreciprocal transport, are accounted by our model.

Without magnetic field
First, we consider the case without magnetic field. In this case we have the relation where v is the total number density of the vortices. We then obtain the linear and nonlinear transport coefficients as 14 which satisfies the selection rule in the trigonal symmetry. Here we use the electrical current instead of the current density. Note that the vortex Hall effect is essential for the nonreciprocal transport signal as seen from the presence of the factor = ⊥ / ∥ . The ratio is written by the simple quantity which is not influenced by the number of vortices and friction coefficient. Taking the low- The value at low temperature is estimated by using pin,0 ≃ 400 μA (see Supplementary Fig.   4b , and we get 0 ≃ 40 meV. If we considered a temperature-independent potential height, the transport signal with the magnitude observed experimentally could not be reproduced due to the exponential factor − 0 . Hence it is necessary and natural to consider the temperature dependence of which goes to zero as approaches to the mean-field critical temperature c ≃ 3.8 K. In order to estimate the temperature dependence of potential height, we employ the condensation energy gain at the vortex core in the presence of the normal state at the impurity site as ≃ c 2 2 0 2 ∝ c − where c is the thermodynamic critical magnetic field, the fraction of pinning points 21,22 , 0 the permeability in vacuum, and t the sample thickness.
In addition, we assume the temperature dependence of ℓ v as ℓ v ∝ √ c − , which corresponds to the fact that the size of vortices increases as → c and the potential periodicity for vortices becomes effectively shorter together with the increasing coherence length . That is to say, the temperature-dependent variation of the vortex size, which cannot be treated in our model of Brownian motion of the point particle, was incorporated into ℓ v . By Next, we consider the number of vortices which is necessary for the direct evaluation of 2ω . We assume that the description of two-dimensional superconductors applies to the present system. Below the Kosterlitz-Thouless transition temperature KT , which is very close to the mean-field transition temperature c away from dirty limit 22 show ω at I = 60 A and 100 A, respectively. Errorbars indicate the uncertainty of the signals estimated from the current dependence of 2ω at each temperature.

Comparison with signals in normal state
We compare the transport signals in the superconducting state with those of the normal state 25  and we take the low-temperature limit for the estimation of its magnitude. The ratio between normal and superconducting states is where ⊥ / ∥ and /̃ are related to the Hall angles of the vortex and skew scattering of electron, respectively. Assuming that these factors are comparable, the difference in nonlinear transport signals is explained by the two huge factors F ℓ v and F / B . Namely, the signal is much enhanced in the superconducting state because the nearly atomic length ( F −1 and Fermi energy are replaced by the much larger characteristic length for the vortices and the temperature, respectively, in the superconducting regime. This enhancement below superconducting transition temperature is consistent with the experimental observation discussed in Fig. 4 of the main text.

Under magnetic field
In the presence of the external magnetic field, the number of vortices is determined by the total magnetic flux. Then the number densities are given by − ( ) = 0 and + ( ) = 0 * .
Using Supplementary Eq. (6 we obtain the transport coefficients as It is notable that the nonlinear Ohmic signal 2ω ( ) is now finite and is not dependent on originating from the vortex Hall effect. Here again the ratio between linear and nonlinear coefficients is given by the simple quantity Assuming that the properties of the pinning remain unchanged from the zero-field case, the ratio 2ω ( )/ ω ( ) in the magnetic field can be larger by the factor −1 = ∥ / ⊥ compared with Supplementary Eq. (15 for the zero-field case.

Nonlinear anomalous transport in other samples
Nonlinear anomalous transverse response and the rectification effect have been observed in other samples. In Supplementary Table 1,

Nonlinear superconducting transport under the magnetic field
To achieve the comprehensive understanding of the nonlinear superconducting transport in trigonal PbTaSe2, we measured the second-harmonic resistance under the magnetic field 27 . Supplementary Figures 7a and 7b show the magnetic field dependence of ω ( Supplementary Fig. 7a , 2ω (red and 2ω (blue ( Supplementary Fig. 7b , in sample 6. In this measurement, the current flows along the zigzag direction (configuration A . We observed clear peak structure in 2ω during the superconducting transition, whose sign is reversed by

Comparison between nonlinear superconducting transport with and without magnetic field
We now compare the magnitude of nonlinear transport with and without magnetic field. By combining Supplementary Eqs. (13 , (15 , (19 , and (21 ,

Effect of Joule heating
We consider the heating effect on the transport coefficients caused by the external current 29,30 . The current-voltage relation is in general written as where is the temperature of the sample. The Joule heating is accounted by the power = . The energy transfer from the sample to the environment with the temperature 0 is given by = where is the thermal boundary conductance and = − 0 represents the temperature variation. Then, the temperature change is expressed as a function of the current, and the leading-order term is proportional to the square of the current. We thereby obtain = (1) ( 0 ) + (2) where (1) ′ = (1) is the temperature derivative of the resistance. It is notable that there is a qualitative difference between the second-order and third-order terms. Namely, the thirdorder contribution can be generated through the combination of the first-order resistance and the temperature-varying effect, but the second-order coefficient is not generated by such an effect. Hence the second-order signals studied in this work captures only the intrinsic effect from the nonreciprocal response in the noncentrosymmetric superconductors.
In order to further clarify that second harmonic response comes from the noncentrosymmetric crystal symmetry, we investigated the centrosymmetric superconductor as a control experiment. As a comparable centrosymmetric superconductor, we investigated the second-order nonlinear response in 2H-NbSe2. In PbTaSe2 (Supplementary Fig. 9a Fig. 9c and measured the first and second harmonic resistance similarly to PbTaSe2 (Fig. 3a in the main text . 2H-NbSe2 shows metallic behavior and superconductivity around T = 7 K ( Supplementary Fig. 9d , which is consistent with previous studies 31,32 .
First, we focus on the nonlinear signals in the normal state. Supplementary Figure 10a (b shows 2ω and 2ω as a function of the current in PbTaSe2 (2H-NbSe2 at T = 20 K.
PbTaSe2 shows the large second harmonic signal in 2ω while 2ω is small (nonlinear transverse response . With contrast, in 2H-NbSe2 both 2ω and 2ω are indicernible. This indicates that the intrinsic second harmonic signal is absent in centrosymmetric system.
Next, we discuss the nonlinear signals in the superconducting state. Supplementary   Figure 10c (d shows the current dependence of 2ω and 2ω in PbTaSe2 (2H-NbSe2 at T = 2 K. 2ω shows the peak sturucture during the superconducting transition while the signal in 2ω is indiscernible as discussed in the main text. With contrast, both 2ω and 2ω in 2H- NbSe2 are negligibly small, which is the expected behavior of the centrosymmetric crystals.
We note a small peak structure in 2ω in the transition, which also appears in 2ω of PbTaSe2.
This might come from the asymmetry of the contacts or inhomogeniety of the superconductivity. However, its magnitude is much smaller than the intrinsic signals of the nonlinear transverse response in PbTaSe2 (Supplementary Fig. 10c .