Stripe order and diode effect in two-dimensional Rashba superconductors

In superconductors and Fermionic superfluids, Cooper pairs with their center-of-mass momenta ${\bf q}$’s are usually condensed into a uniform state of ${\bf q}=0$, as ${\bf q}\neq 0$ modes and associated spatial variations generally require additional energy cost. When the Cooper pair is subject to pair-breaking effects due to an external magnetic field FF ; LO ; FFLO_Matsuda_review_07 and surface scatterings Vorontsov_SF ; Vorontsov_SC ; Hachiya ; Aoyama_cylinder ; Aoyama_film ; Wiman_film , it can happen that the Cooper-pair condensate chooses to have a ${\bf q}\neq 0$ rather than to be uniformly suppressed. A prime example of such a ${\bf q}\neq 0$ superconducting (SC) state is the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state induced by the Pauli-paramagnetic pair-breaking effect FF ; LO ; FFLO_Matsuda_review_07 where the SC gap function exhibits a phase modulation $e^{i{\bf q}\cdot{\bf r}}$ (FF state) or an amplitude modulation $\cos({\bf q}\cdot{\bf r})$ (LO state). The latter can be viewed as a superposition of $\pm{\bf q}$ modes of equal weight. In this work, we theoretically investigate the stability of a weight-biased superposition state called a stripe order Stripe_Agterberg_prb_07 ; 2layer-Stripe_Yoshida_jpsj_13 in two-dimensional non-centrosymmetric superconductors possessing a Rashba-type spin-orbit coupling (RSOC) where the degeneracy between ${\bf q}$ and $-{\bf q}$ is lifted by a combined effect of the RSOC and an in-plane Zeeman field Kaur ; Springer and resultantly, a supercurrent becomes nonreciprocal diode_Wakatsuki_prl_18 ; diode_Wakatsuki_sadv_17 , giving rise to a SC diode effect diode_Ando_nature_20 ; diode_Daido_prl_22 ; diode_Yuan_pnas_22 ; diode_He_njp_22 ; diode_Daido_prb_22 .

The RSOC of the form $\alpha_{\rm R}\left({\bf k}\times\hat{z}\right)\cdot{\mbox{\boldmath$\sigma$}}$ is generic to non-centrosymmetric systems lacking mirror symmetry with respect to a crystalline plane such as the heavy-fermion bulk superconductors CePt₃Si CePt3Si and CeTSi₃ (T=Rh, Ir) CeRhSi3 ; CeIrSi3 and two-dimensional superconductors realized on artificial superlattices superlattice_Goh_prl_12 ; superlattice_Shimozawa_prl_14 ; superlattice_Naritsuka_prb_17 , at interfaces between two different materials interface-SC ; tunable-SC ; tunable-RSO , and in perpendicular electrical gate fields gateE-SC_Ueno_natmat_08 ; gateE-SC_Ueno_prb_14 . In a magnetic field ${\bf H}$ parallel to the mirror plane, a field-induced in-plane spin polarization is connected to supercurrents via the RSOC SC_jM ; SC_Bj ; Dimitrova ; Samokhin ; Kaur ; Fujimoto ; Helical_AS_12 ; Helical_ASS_14 . Such a magneto-electric effect can induce a so-called helical SC state with a phase modulation $e^{i{\bf q}\cdot{\bf r}}$ Kaur , where in contrast to the conventional FF state, the direction of ${\bf q}$ is fixed to be parallel to $\alpha_{\rm R}\left({\bf H}\times\hat{z}\right)$ due to the RSOC and thus, ${\bf q}$ and $-{\bf q}$ are not equivalent. Noting that the phase gradient yields a SC current, it turns out that the nonequivalence between ${\bf q}$ and $-{\bf q}$ should be reflected as a nonreciprocity between the supercurrents flowing in the ${\bf q}$ and $-{\bf q}$ directions. Actually, a nonreciprocal SC transport of this kind has been observed as the SC diode effect diode_Ando_nature_20 which can be understood as the nonreciprocity of the SC critical current diode_Daido_prl_22 ; diode_Yuan_pnas_22 ; diode_He_njp_22 ; diode_Daido_prb_22 . Here, we emphasize again that the key ingredient for the helical phase and the associated SC diode effect is the RSOC.

In the case without the RSOC, on the other hand, it is well known that the Zeeman field or the Pauli-paramagnetic pair-breaking effect leads to the occurrence of the amplitude modulated LO state in a high-field and low-temperature region FFLO_Matsuda_review_07 . Since as mentioned above, the RSOC favors the FF-like helical state, how robust the LO state is against the RSOC would be an interesting question. This issue has already been discussed in three-dimensional Rashba superconductors with a dominant spin-singlet $s$-wave pairing interaction Stripe_Agterberg_prb_07 , although in three dimensions, the orbital pair-breaking effect, which is not incorporated in Ref. Stripe_Agterberg_prb_07 , is non-negligible Ikeda_0 . It has been reported that an intermediate state between the FF and LO states taking the form of $\Delta_{+}e^{i{\bf q}\cdot{\bf r}}+\Delta_{-}e^{-i{\bf q}\cdot{\bf r}}$ with $|\Delta_{+}|\neq|\Delta_{-}|\neq 0$ can be stabilized Stripe_Agterberg_prb_07 . Since this intermediate state breaks the translational symmetry similarly to the LO state of $|\Delta_{+}|=|\Delta_{-}|$, it is called the stripe order, being distinguished from the conventional LO state. In contrast to the helical phase where only one modulation vector ${\bf q}$ determines the preferred flow-direction of the supercurrent, the stripe phase involves both ${\bf q}$ and $-{\bf q}$, so that how the SC diode effect looks like in the stripe phase would be an interesting question. Furthermore, when we consider a $d$-wave pairing instead of the isotropic $s$-wave one, we notice that another fundamental question arises.

In $d$-wave superconductors, there exists a spatial anisotropy originating from of the SC gap nodes. In the $d_{x^{2}-y^{2}}$ case without the RSOC, it has been theoretically shown that for a cylindrical Fermi surface, the LO state with ${\bf q}$ parallel to the nodal [110] direction is stabilized in the high-field and low-temperature region and that at further high fields, the FF state with ${\bf q}$ parallel to the anti-nodal [100] direction becomes more stable FFLO_d_Maki_physicaB_02 ; FFLO_Vorontsov_prb_05 . We note that the direction of the applied field has nothing to do with the ${\bf q}$ direction as the spin and orbital sectors are completely decoupled in the absence of a spin-orbit coupling. Then, the question is what happens when the RSOC is introduced. Since the RSOC favors the phase-modulated helical state with ${\bf q}\parallel\alpha_{\rm R}\left({\bf H}\times\hat{z}\right)$, the field direction relative to the nodal direction should be important for the stability of the LO-like stipe phase involving both ${\bf q}$ and $-{\bf q}$. To our knowledge, this issue has not yet been discussed. In addition, a recent experiment on the globally-noncentrosymmetric tricolor superlattice of YbRhIn₅/CeCoIn₅/YbCoIn₅ superlattice_Naritsuka_prb_17 containing CeCoIn₅ whose pairing symmetry in the bulk is $d_{x^{2}-y^{2}}$ VLFF ; Hiasa ; Magreso_Stock_prl_08 ; angledep_Vorontsov ; d-wave ; node-imaging_Allan_natphys_13 ; QPI_Zhou_natphys_13 has shown that a nonreciprocal SC transport exhibits an in-plane anisotropy when the in-plane field is rotated from the nodal ([110]) direction to the anti-nodal ([100]) direction superlattice_Matsuda_private . In view of such theoretical and experimental situations, we theoretically investigate the stripe instability in the two-dimensional Rashba superconductors for the $d$-wave pairing and the $s$-wave one as well for reference, and examine the SC diode effect in the stripe phase.

Based on the Ginzburg-Landau (GL) analysis, we will show that for ${\bf H}$ parallel to the nodal [110] direction, the stripe phase can be stabilized in a relatively wide range of the high-field and low-temperature region, while not for ${\bf H}$ parallel to the anti-nodal [100] direction [see Figs. 1 (b) and (c)]. In both cases, with increasing $\delta N$ measuring the strength of the RSOC (see Sec. II), the stripe phase gets unstable against the helical phase. It is also found that the nonreciprocity of the critical SC current, i.e., the intrinsic SC diode effect diode_Daido_prl_22 ; diode_Yuan_pnas_22 ; diode_He_njp_22 ; diode_Daido_prb_22 , emerges in the stripe phase as well as in the helical phase and that the second order transition between the two phases does not leave a footprint in the temperature dependence of the critical-current nonreciprocity.

This paper is organized as follows: In Sec. II, we derive the GL free energy functional from the microscopic BCS Hamiltonian with the RSOC and explain how to examine the stability of the stripe order and the nonreciplocity of the critical current. Results on the former and latter issues are discussed in Secs. III and IV, respectively. We end the paper with summary and discussions in Sec. V.

In this section, we derive the GL free energy functional from the microscopic Hamiltonian, and explain how to examine the stability of the stripe order and the nonreciplocity of the critical current.

Figure 1 shows the temperature and magnetic field phase diagrams in the two-dimensional superconductors with the weak RSOC of $\delta N=0.01$ (upper panels) and the moderate RSOC of $\delta N=0.05$ (lower panels). Although our interest is mainly in the $d$-wave case shown in Figs. 1 (b) and (c), we show the result for the $s$-wave case in Fig. 1 (a) for completeness. In all the cases shown in Fig. 1, the helical SC state with its modulation ${\bf Q}\parallel(\hat{H}\times\hat{z})$ is realized just below the $H_{c2}$ curve. For the $\delta N$ values used here, the direction of ${\bf Q}$ remains unchanged at temperatures lower than but not so far from the $H_{c2}$ transition where the GL theory should work well. Figure 2 shows the field dependence of the helical modulation $\tilde{Q}$ (black curves) and the coefficient of the $|\Delta_{+}|^{4}$ term $\beta({\bf Q})$ (cyan curves) along the $H_{c2}$ curve. One can see that around $h=1.6$ slightly above the Pauli limiting field of $h=1.25$, $\tilde{Q}$ exhibits a steep increase and at the same time, $\beta({\bf Q})$ tends to become small. Although the latter tendency is remarkable in the $d$-wave case with ${\bf H}\parallel$ node shown in Fig. 2 (b), $\beta({\bf Q})$ is always positive, so that the transition between the helical SC and normal phases is of second order. We note in passing that in Fig. 2 (b), due to the sharp drop in $\beta({\bf Q})$ near $h=1.6$, the present GL expansion becomes invalid at further low temperatures (for details, see Appendix A).

In the reference case of $s$-wave shown in Fig. 1 (a), one can see that the stripe order is realized in the high-field and low-temperature region. The stripe region for $\delta N=0.05$ is almost quantitatively the same as that in the associated three dimensional system Stripe_Agterberg_prb_07 in spite of the difference in the dimensionality. The top panels in Fig. 3 show the temperature dependence of the gap amplitudes for the $\pm{\bf Q}$ modes $\Delta_{\pm}$ obtained at $h=2.6$ for $\delta N=0.05$. One can see from Fig. 3 (a) that with decreasing temperature, $\Delta_{+}$ first develops and then, $\Delta_{-}$ starts developing at the helical-stripe transition which is determined by the condition $\alpha^{\rm eff}(-{\bf Q})=0$ and turns out to be of second order as $\beta(-{\bf Q})>0$ (see the lower panel). Since the RSOC prefers a single-${\bf Q}$ helical state, the stripe order involving both ${\bf Q}$ and $-{\bf Q}$ gets unstable with increasing the RSOC [compare the upper and lower panels in Fig. 1 (a)]. In the associated three dimensional system, it has been reported that for $\delta N=0.25$, the stripe phase cannot exit any more Stripe_Agterberg_prb_07 .

In contrast to the $s$-wave case where the SC gap is isotropic and thus, the stability of the stripe order does not depend on the field direction, the stripe ordering in the $d_{x^{2}-y^{2}}$ case strongly depends on the field direction due to the existence of the line node running along the [110] and [1$\overline{1}$0] directions. As exemplified by the lower panels of Figs. 3 (b) and (c), $\alpha^{\rm eff}(-{\bf Q})$ at the fixed field strength of $h=2.6$ becomes negative at low temperatures for ${\bf H}\parallel$ node, whereas it remains positive for ${\bf H}\parallel$ anti-node, each suggesting the presence and absence of the stripe phase of $\Delta_{-}\neq 0$. The results of this kind at different field strengths are summarized in Fig. 1. As readily seen in Figs. 1 (b) and (c), although the stripe phase can be stable in the high-field and low-temperature region for both the two field configurations, its stability region is significantly suppressed in the ${\bf H}\parallel$ anti-node case. Such a difference between the ${\bf H}\parallel$ node and ${\bf H}\parallel$ anti-node cases becomes more remarkable for stronger RSOC. This field-angle dependence can easily be understood from the FFLO instability in the absence of the RSOC where the LO state with the modulation parallel to the nodal direction is widely stable. Since the RSOC favors the helical modulation perpendicular to the field, the directions of the LO and helical modulations becomes compatible for ${\bf H}\parallel$ node, while not for ${\bf H}\parallel$ anti-node, which results in the angle-dependent stability of the stripe phase.

In both cases of ${\bf H}\parallel$ node and ${\bf H}\parallel$ anti-node, we have confirmed that the transition between the helical and stripe phases is of second order [for example, see the sign of $\beta(-{\bf Q})$ in Fig. 3 (b)], except the low-field phase boundary for ${\bf H}\parallel$ node [see the dashed line in Fig. 1 (b)]. Near this low-field boundary, the present GL approach taking account of the terms up to the fourth order in the SC gap does not work well (for details, see Appendix A). Nevertheless, considering that in the absence of the RSOC, the low-field boundary is of second order FFLO_Vorontsov_prb_05 , it is likely to be of second order as well in the presence of the moderate RSOC.

In the previous section, we show that the stripe order of the from $\Delta_{+}e^{i{\bf Q}\cdot{\bf r}}+\Delta_{-}e^{-i{\bf Q}\cdot{\bf r}}$ with $|\Delta_{+}|>|\Delta_{-}|\neq 0$ can appear via the second order transition from the higher-temperature helical state of the form $\Delta_{+}e^{i{\bf Q}\cdot{\bf r}}$. For the $d$-wave pairing, the stripe ordering gets suppressed when the in-plane field is rotated from the nodal direction to the anti-nodal direction. In this section, we will discuss the nonreciprocity of the critical current called the intrinsic SC diode effect diode_Daido_prl_22 ; diode_Yuan_pnas_22 ; diode_He_njp_22 ; diode_Daido_prb_22 , putting particular emphasis on how the $-{\bf Q}$ mode additionally emerging in the stripe phase affects the nonreciprocity.

We shall start from the $s$-wave case. Figure 4 (a) shows the temperature dependence of the nonreciprocity $R$ in the $s$-wave case with $\delta N=0.05$ and $h=2.6$, the same parameter set as that in Fig. 3 (a). In Fig. 4 (a), in addition to the main result represented by the solid black curve, we show, for comparison, the result obtained under the constraint of $\Delta_{-}=0$ for which only the helical state is allowed (the gray dashed curve). One can see that $R$ is nonzero not only in the helical phase but also in the stripe phase and that a signature of the second order transition into the stripe phase cannot be found in $R$. At further low temperatures, the nonreciprocity $R$ in the stripe phase becomes slightly smaller than the one for the helical state [compare the black solid and gray dashed curves in Fig. 4 (a)]. To understand these behaviors of $R$, we shall look into the details of the supercurrent $\tilde{j}$ as a function of $\tilde{q}_{\rm ex}$ corresponding to the external current.

Figures 4 (b) and (c) show the $\tilde{q}_{\rm ex}$ dependence of $\tilde{j}$ (upper panels) and the gap amplitudes $|\tilde{\Delta}_{\pm}|$ (lower panels) at $t=0.27$ and $t=0.21$, respectively, where in the upper panels, the negative part of $\tilde{j}$ is folded back (see the pink curve) such that the nonreciprocity, which corresponds to the peak-height difference indicated by blue dotted lines, can easily be confirmed. In Figs. 4 (b) and (c), the notation of the gray dashed curve is the same as that in Fig. 4 (a); it represents the result obtained by assuming that only the helical phase is realized. One can see from Fig. 4 (b) that just below the helical-stripe transition, the $-{\bf Q}$ component $\tilde{\Delta}_{-}$, which is very small due to the second-order nature of the transition, is rapidly suppressed by $\tilde{q}_{\rm ex}$ (see the lower panel) and does not affect the peak heights of $\tilde{j}$, or equivalently, the critical currents (see the upper panel). Therefore, the nonreciprocity $R$ in the stripe phase just below the transition is the same as that for the helical phase, and does not show any clear anomaly at the transition. Further below the transtion, on the other hand, as shown in Fig. 4 (c), $|\tilde{\Delta}_{-}|$ is relatively robust against $\tilde{q}_{\rm ex}$, so that it can affect the critical current. The peak hight of the positive part of $\tilde{j}$ is slightly suppressed in the stripe phase [compare the red solid and gray dashed curves in Fig. 4 (c)], and correspondingly, the nonreciprocity $R$ takes a smaller value [see Fig. 4(a)].

We next discuss the $d$-wave case. Figure 5 shows the result on the nonreciprocity in the $d$-wave case with ${\bf H}\parallel$ node, where the notations are the same as those in the $s$-wave case of Fig. 4. One can see from Fig. 5 (a) that the critical-current nonreciprocity emerges in the stripe phase without showing an anomaly at the helical-stripe transition. The reason for the absence of the signature of the transition is the same as that in the $s$-wave case: just below the transition, the $-{\bf Q}$ component $\tilde{\Delta}_{-}$ is fragile against the external current $\tilde{q}_{\rm ex}$ [see Fig. 5 (b)]. At further low temperatures, there is a difference between the $s$-wave and $d$-wave cases. In Fig. 5 (c), although it is common that $\tilde{\Delta}_{-}$ is relatively robust against $\tilde{q}_{\rm ex}$, the positive peak of $\tilde{j}$ is not suppressed but rather enhanced by the contribution from $\tilde{\Delta}_{-}$, which results in a slight increase in $R$ [compare the black solid and gray dashed curves in Fig. 5 (a)]. In experiments, however, such a slight deviation from the helical value could not be captured, since as will be discussed below, even in the helical phase, the temperature dependence of $R$ is not so simple.

Figure 6 (a) shows the temperature dependence of $R$ for ${\bf H}\parallel$ anti-node, where only the helical phase is realized over the temperature range of this figure. The nonreciprocity $R$ exhibits a non-monotonic temperature dependence. Note that $\tilde{\Delta}_{-}$ remains zero even after $\tilde{q}_{\rm ex}$ is introduced [see Figs. 6 (b) and (c)] and thus, this behavior is purely of helical origin. In addition, the value of $R$ for ${\bf H}\parallel$ anti-node is much smaller than $R$ for ${\bf H}\parallel$ node and takes negative sign at the lowest temperature, reflecting the fact that, as will be explained below, $R$ changes its sign at lower fields.

Figure 7 shows the temperature and magnetic-field dependence of $R$ for the same parameter set as that for Fig. 1, where the reddish and bluish colors represent positive and negative values of $R$, respectively, and the stability region of the stripe phase in Fig. 1 is indicated by translucent gray. In Fig. 7 (b), data blanks near $h=1.6$ are due to the invalidity of the GL approach used here (see Sec. II and Appendix A). As already reported elsewhere diode_Daido_prl_22 ; diode_Daido_prb_22 , the nonreciprocity $R$ tends to changes its sign near the field (in the present case, $h\sim 1.6$) at which the helical modulation $Q$ rapidly develops (see Fig. 2). In the $d$-wave case with the moderate RSOC of $\delta N=0.05$, however, the sign change is suppressed when the magnetic field is applied along the nodal direction [compare the lower panels of Figs. 7 (b) and (c)]. Although the sign of $R$ seems to depend on the details of specific systems such as the Fermi surface shape controlled by electron fillings diode_Daido_prl_22 ; diode_Daido_prb_22 , the above result obtained for the isotropic cylindrical Fermi surface should capture the essential part of roles of the gap anisotropy. Concerning the main focus of this work, i.e., how the nonreciprocity looks like in the stripe phase, it continuously changes on cooling across the transition from the helical phase into the stripe phase, and any characteristic feature such as a sign change in $R$ cannot be found at the transition, being irrespective of the orbital pairing symmetry, the strength of the RSOC, and the field direction.

In this work, we have theoretically investigated the stability of the stripe order, a LO-like state described as a superposition of the $+{\bf Q}$ and $-{\bf Q}$ modes of different weight, in two-dimensional superconductors with the Rashba spin-obrit coupling (RSOC) and an in-plane magnetic field where the helical SC state with only the $+{\bf Q}$ mode is widely stabilized. Based on the GL analysis, we show that the stripe phase can be stabilized in the high-field and low-temperature region for both the $s$-wave and $d$-wave pairing symmetries, as originally pointed out for the three-dimensional $s$-wave case Stripe_Agterberg_prb_07 . Interestingly, in the $d$-wave case, the stability region of the stripe phase shrinks when the in-plane field is merely rotated from the nodal direction to the anti-nodal direction. It is also found that the nonreciprocity of the critical current, the so-called the SC diode effect, emerges not only in the helical phase but also in the stripe phase. The transition between the helical and stripe phases does not leave a footprint in the temperature dependence of the critical-current nonreciprocity due to the second-order nature of the transition.

In experiments on film superconductors, it is usually difficult to perform bulk measurements such as the specific heat, so that transport measurements have widely been used to study SC states. Although a signature of the stripe phase has not been observed so far in relevant two-dimensional Rashba systems diode_Ando_nature_20 ; superlattice_Naritsuka_prb_17 ; interface-SC ; tunable-SC ; tunable-RSO ; gateE-SC_Ueno_natmat_08 ; gateE-SC_Ueno_prb_14 , this might be simply because the RSOC is too strong for the stripe order to survive or because as discussed above, the critical current is insensitive to the second order transition into the stripe phase. Also, a current-driven dynamics of vortices could contribute to the SC transport. First, in two dimensions, thermally activated Kosterlitz-Thouless vortices with their axis parallel to the out-of-plane direction can appear, but they are active basically at higher temperatures near the SC transition diode_Hoshino_prb_18 and thus, should be irrelevant to the low-temperature transport. Second, in real experimental systems with finite thickness in the out-of-plane direction, the in-plane field may yield SC vortices extending along the in-plane field direction and a current-driven out-of-plane motion of these vortices could be relevant. If the vortex motion is affected by the stripe modulation perpendicular to the vortex line, we may have a chance to detect a signature of the stripe order in the vortex dynamics.

Even in the presence of the vortex dynamics, the nonreciprocity of the critical current should more or less exist. In contrast to the isotropic $s$-wave case, in the anisotropic $d$-wave case, the sign of the nonreciprocity is dependent on the in-plane field direction relative to the nodal direction (see Fig. 7). A recent transport measurement on the tricolor superlattice of YbRhIn₅/CeCoIn₅/YbCoIn₅, the two-dimensional Rashba surperconductor with the $d_{x^{2}-y^{2}}$ pairing symmetry, has shown that the field dependence of the nonreciprocity exhibits an anomaly for ${\bf H}\parallel$ [100], while not for ${\bf H}\parallel$ [110] superlattice_Matsuda_private , which might be a manifestation of the field-angle-dependent sign change in the critical-current nonreciprocity originating from the $d$-wave anisotropy. Although the Fermi surface shape, which is assumed, for simplicity, to be cylindrical in this work, could quantitatively affect the results and thus, should carefully be considered in our future work, we believe that our result obtained for the simplified model captures the essential part of the stripe ordering and the associated nonreciprocal phenomena in two-dimensional Rashba superconductors.