Abstract
Spin current is a central theme in spintronics, and its generation is a keen issue. The spinpolarized current injection from the ferromagnet, spin battery, and spin Hall effect have been used to generate spin current, but Ohmic currents in the normal state are involved in all of these methods. On the other hand, the spin and spin current manipulation by the supercurrent in superconductors is a promising route for dissipationless spintronics. Here we show theoretically that, in twodimensional superconductors with Rashba spinorbit interaction, the generation of dissipationless bulk spin current by charge supercurrent becomes highly efficient, exceeding that in normal states in the dilute limit, i.e. when the chemical potential is close to the band edge, although the spin density becomes small there. This result manifests the possibility of creating new spintronic devices with longrange coherence.
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Introduction
In spintronics, spin current plays an essential role to transfer the information associated with the spin degrees of freedom. Therefore the generation of spin current is an important issue, and several methods have been proposed and experimentally verified^{1}. Various methods, such as the spinpolarized current injection from the ferromagnet^{2,3,4}, spin battery^{5,6,7,8,9}, and spin Hall effect^{10,11,12,13,14,15,16}, have been employed to generate the spin current. It has been proposed also that the spin current second order in the electric field can be generated in noncentrosymmetric systems with spinorbit interaction^{17,18}. The spin currents mentioned above are either carried by itinerant electrons through dissipative electric currents, or by localized magnetic moments through exchange interactions. However, there can also be nondissipative spin current in itinerantelectron systems. In this case, it is an equilibrium spin current without dissipation^{19}. Although such a spin current is detectable according to Sonin^{20}, it does not contribute to the transport property in a set up where the spin current can flow in and out.
On the other hand, superconducting spintronics is an emerging field attracting recent interest^{21,22,23,24,25,26,27,28,29,30,31,32,33,34}. Although the spin degrees of freedom are usually quenched in singlet superconductors, the triplet component can be finite in noncentrosymmetric superconductors, ferromagnetsuperconductor hybrids and Josephson junctions, or oddparity superconductors, where the spins become (partially) active. Therefore, it is an intriguing issue if one can generate a spin current in superconductors with zero or small dissipation. Actually, the spin supercurrent has been discussed in He3 related to the internal degrees of freedom^{39}. Recently, Leurs et al.^{40} have reexamined the spin supercurrent in spinorbit coupled systems, and classified it to the coherent and noncoherent parts, only the latter of which contributes to the continuity equation of the spin density and its generation or manipulation is the focus of superconducting spintronics.
The superconducting magnetoelectric effect^{35,36,37,38} is also a result of the spins of Cooper pairs being active. For example, the spin density (or magnetization) induced by a supercurrent has been discussed by Edelstein^{35} in the case where the Fermi energy is large compared to the spin splitting by the Rashba interaction. Although this is justified in many situations, there are also systems where the spinorbit splitting is comparable or even larger than the Fermi energy, e.g., the interface between LaAlO\({}_{3}\) and SrTiO\({}_{3}\)^{41}, where the electron density can be tuned by gating. Therefore, it is desirable to cover the wide range of parameters, e.g., chemical potential, the strength of Rashba interaction, and temperature.
Here, we study spin density and spin current produced by a superconducting current in a twodimensional superconductor with Rashba spinorbit interaction for wide parameter regions. We investigate in detail the properties of spin current in this system and find that the spin current generation efficiency is comparable to or even larger than that of normal state when normalized with the charge current density. The spin current we discuss here corresponds to the noncoherent part in the classification by Leurs et al.^{40} and hence contributes to the spin accumulation, unlike the equilibrium spin current in normal systems^{19}. Furthermore, it is different from those in other superconducting spintronic setups involving ferromagnetsuperconductor hybrids where the spin degrees of freedom of Cooper pairs become active due to the presence of ferromagnets. In contrast, what we obtain here is a bulk spin supercurrent induced by charge supercurrent without timereversal symmetry breaking of the ground state. The spin degrees of freedom become active because of the spinorbit coupling and the flow of a Cooper pair condensate. Similar to normal spin currents, such a spin supercurrent may be detected by connecting to a material that shows inverse spin Hall effect and converts the injected spin currents into a transverse voltage. The study of the bulk spin supercurrent carried by Cooper pairs is a significant step towards nondissipative spintronics.
Results
Model
A twodimensional superconductor (SC) with Rashba spinorbit interaction (SOI) can be described by the following Hamiltonian,
where \(\alpha\) is the Rashba SOI strength, \(\Delta\) is the SC order parameter and \({\boldsymbol{\sigma }}=\hat{{\bf{x}}}{\sigma }^{x}+\hat{{\bf{y}}}{\sigma }^{y}+\hat{{\bf{z}}}{\sigma }^{z}\) is a spatial vector of Pauli matrices. The constants \(m\) and \(\hslash\) are the electron mass and the Planck constant respectively. Summation over implicit indexes is assumed.
Without superconductivity, the normalstate band structure is schematically shown in Fig. 1a. When \(\mu \gg {E}_{{\rm{R}}}\), \({E}_{{\rm{R}}}=m{\alpha }^{2}/2\) being the Rasbha band splitting energy, the Fermi level is far above the band touching point. If \(\mu =0\), the Fermi level cuts through this point and the inner Fermi surface shrinks to a point. The band bottom is reached when \(\mu ={E}_{{\rm{R}}}\).
The superconductivity order parameter considered here is spinsinglet swave. In systems with Rashba spinorbit interaction and superconductivity, swave may be mixed with pwave and a general theory should include all of them. It has been shown that the actual amplitudes of the two different pairing potential depend on the form of the interaction^{42,43}. When only evenparity interaction is considered, the pwave order parameter vanishes. We assume pure swave order parameter in our calculation for simplicity.
Although we do not assume any spintriplet order parameters in the Hamiltonian, the anomalous Green’s function \({F}(\omega ,{\bf{k}})=({\psi }_{s}+{\bf{d}}\cdot {\boldsymbol{\sigma }})i{\sigma }_{y}\), which carries the full information about the Cooper pairs, has both spinsinglet part \({\psi }_{s}\) and spintriplet components \({\bf{d}}\) when spinorbit interaction is present^{43}. As a result, the Cooper pairs become spinactive due to SOI even though the pairing potential is purely swave. When timereversal symmetry is broken, such as by the supercurrent \({J}_{x} \sim {q}_{x}\) in our case, the spin of the Cooper pairs become polarized and the polarization is given by \({\bf{S}} \sim i{\bf{d}}\times {{\bf{d}}}^{* }\) (more discussion in Supplementary Note 1). Since the pairs carry both polarized spin and momentum, they generate spin currents. This is illustrated schematically in Fig. 1b.
To investigate the spin and charge properties, it is convenient to introduce an SU(2) gauge field \({{\bf{W}}}^{\nu }{\sigma }^{\nu }\)^{40,44,45} along with the electromagnetic field \({\bf{A}}\). Electrons coupled with both fields are described by the Hamiltonian
Apparently, the Rashba term in \({\hat{H}}_{0}\) breaks SU(2) symmetry. It is actually equivalent to a constant gauge field (neglecting a numeric constant) \(\bar{{\bf{W}}}=({\bar{W}}_{y}^{x}{\sigma }^{x},{\bar{W}}_{x}^{y}{\sigma }^{y},0)\), with \({\bar{W}}_{y}^{x}={\bar{W}}_{x}^{y}=2m\alpha /\hslash\)^{46}. In this formulism, the spin current operator is conveniently defined as \({\hat{{\bf{J}}}}^{\nu }=\partial \hat{H}/\partial {{\bf{W}}}^{\nu }\). Or, starting from the free energy, the spin current can be obtained as \({{\bf{J}}}^{\nu }=\partial {\mathcal{F}}/\partial {{\bf{W}}}^{\nu }\).
When a uniform supercurrent is flowing (say along \(x\)direction), the SC order parameter acquires a constant phase gradient, i.e. \(\Delta ={\Delta }_{0}\exp [2i{q}_{x}x]\). Or, by a gauge transformation, it is equivalent to a constant vector potential \({A}_{x}=\hslash {q}_{x}/e\), which is effective only for SCs. Before showing the results, it is helpful to discuss the symmetry properties of the involved physical quantities. The spin current (spin density) is odd (even) under spatial inversion but even (odd) under timereversal operation, while charge current, or \({q}_{x}\), is odd under both of them. Consequently, the lowest order of the spin current is \({J}^{\nu } \sim {q}_{x}^{2}\) and that of spin density is \({M}_{\nu } \sim {q}_{x}\). Both of them must be odd functions of \(\alpha\).
Spin density and spin current
At zero temperature, the free energy \({\mathcal{F}}\) is just the ground state energy. For arbitrary Rashba SOI strength and chemical potential, the spin polarization is along \(y\)direction and the density is (derivation in the Methods section)
where we have defined the Rashba splitting energy \({E}_{{\rm{R}}}=m{\alpha }^{2}/2\). This is a generalized SC Edelstein effect^{35} in largeRashba systems. The \(\mu\)dependence of the spin density \({M}_{y}\) is shown in Fig. 2a with various values of the pairing potential. When \(\mu \gg {E}_{{\rm{R}}}\gg \Delta\), we get \({M}_{y}(\mu \gg {E}_{{\rm{R}}})=m\alpha {q}_{x}/4\pi\), in which the dependence on \(\mu\) is absent. When \(\mu < 0\), the spin density gradually decreases as \(\mu\) goes down.
In Fig. 2a, one readily notes that the spin density remains finite when \(\Delta\) approaches zero. On the other hand, if \(\Delta =0\), i.e. in a normal state, spin density due to the vector potential must vanish since one can trivially gauge out \({q}_{x}\). Thus, there is a discontinuity in \({M}_{y}/{q}_{x}\) at \(\Delta \to 0\). However, as we will see later, \({M}_{y}\) actually vanishes when \(\Delta\) decreases to zero since the supercurrent \({J}_{x}\) (and thus \({q}_{x}\)) also approaches zero, as discussed in Supplementary Note 2. Also, this discontinuity is absent at finite temperature, as shown in Supplementary Note 3. The \(\Delta\)dependence of the function \(g\) is shown in the inset of Fig. 2a, where a finite value is obtained at \(\Delta \to 0\) for \(\mu /{E}_{{\rm{R}}}> \!\!1\). In the limit \(\Delta /{E}_{{\rm{R}}}\to 0\), Eq. (3) simply becomes
Due to the form of the Rashba SOI, and as indicated by the direction of the spin density obtained above, the only nonvanishing spin current induced by the supercurrent in \(x\)direction is \({{\bf{J}}}^{y}\). In the limit where the chemical potential is high and the order parameter is small, i.e. \(\mu \gg {E}_{{\rm{R}}}\gg \Delta\), the spin current density is (derivation in the Methods section)
and all other spin current components vanish. This expression is independent of \(\mu\) and \(\Delta\), similar to the spin density discussed previously.
When \({E}_{{\rm{R}}} \, < \, \mu \, < \, 0\) and \(\Delta\) is small, the spin current density can be written as
Particularly, when \(d=\Delta /{E}_{{\rm{R}}}\to 0\), the function \(f(d,u)\) simplifies to \(f(d\to 0,u)=3/\sqrt{u}\sqrt{u}\). Combined with Eq. (7), this leads to the same spin current density as Eq. (6) when \(u=1\) (corresponding to the band crossing point \(\mu =0\)). Also, the spin current increases because of the increase of the density of states when the chemical potential goes down until it reaches the band edge at \(\mu \to {E}_{{\rm{R}}}\) where it diverges, as shown by the dashed curve in Fig. 2b. The two Fermi surfaces have the same (opposite) spin textures and the contributions from the two subbands add up (tend to cancel) when \(\mu\) is negative (positive). While the spin current is similar to spin density for large \(\mu\) in the sense that they both keep constant, it shows different behavior when \(\mu \, < \, 0\). Note that the \(\mu\)dependence of spin current resembles the density of states of the normal Rashba band, which also diverges at the band edge.
For general parameters, the spin current of the SC is calculated numerically and shown in Fig. 2b. Because of finite \(\Delta\), the aforementioned divergence of the spin current at the band edge is smoothed out while the spin current at \(\mu \, > \, 0\) is hardly changed even the pairing potential reaches \(0.5{E}_{{\rm{R}}}\). The change of the peak position (\({\mu }_{\text{p}eak}\)) and peak height (\({j}_{\,\text{m}\,ax}^{y}/{q}_{x}^{2}\)) as functions of \(\Delta\) are shown in the inset of Fig. 2b. It turns out that the peak position shifts almost linearly from the band edge when the order parameter increases while the height of the peak decreases in a nonlinearly.
BoseEinstein condensation regime
When the chemical potential is below the band edge, i.e. \(\mu \, < \!{E}_{{\rm{R}}}\), the normal electronic state has no Fermi surface and SC cannot happen in the weak coupling limit of the BardeenCooperSchrieffer (BCS) theory. Assuming that the SC exists, it must be in BEC regime where electrons are tightly bond and the critical temperature is determined by the BoseEinstein condensation (BEC) temperature^{47,48}. As shown in Fig. 2, spin density and spin current quickly drops to zero when \(\mu \, < \!{E}_{{\rm{R}}}\) if \(\Delta\) is small. However, when the pairing potential is large and a BEC superconductor is achieved, they become finite. Especially, when \(u\gg d\) and \(u\gg 1\), direct expansions of Eq. (3) and Eq. (7) respect to \(d\) and \({u}^{1}\) give \(M_y \approx \frac{m \alpha q_x}{32\pi}{d}^{2}/{u}^{2}\) and \({{\mathbf{j}}}_{\text{BEC}}^{y}=\alpha\hslash{q}_{x}^{2}{d}^{2}{(12\pi)}^{1}u{}^{3}\hat{{\mathbf{x}}}\). Both the spin density and spin current show powerlaw decay as \(\mu\) decreases. Note that the discontinuity at \(\Delta \to 0\) does not appear when \(\mu \, < \!{E}_{{\rm{R}}}\).
Efficiency
To relate our calculation to experiments, it is useful to convert the wave vector \({q}_{x}\) to the supercurrent density \({j}_{x}\). To calculate \({j}_{x}\), we note that (charge) current operator in a Rashba system is \({\hat{J}}_{x}=e\int {d}^{2}{\bf{k}}{[\frac{\hslash }{m}({k}_{x}+{q}_{x})\alpha {\sigma }^{y}]}_{ss^{\prime} }{c}_{s,{\bf{k}}}^{\dagger }{c}_{s^{\prime} ,{\bf{k}}}.\) At \(T=0\), the usual paramagnetic term vanishes while the diamagnetic term remains. The last term proportional to \(\alpha\) contributes a supercurrent of \(e\alpha {M}_{y}/\hslash\). So the zero temperature supercurrent density can be written as
The quantity \({n}_{e}\) is the electron density. Eq. (9) may be regarded as the generalized London equation for Rashba systems.
Since \({j}_{x} \sim {q}_{x}\), we define two coefficients
which denote the efficiency of spin density and spincurrent generation, respectively, for given charge current density. They are shown in Fig. 3. For large \(\mu\), both the spin density and spin current decrease in powerlaw, \(\gamma \sim {(\mu +{E}_{{\rm{R}}})}^{1}\), \(\eta \sim {(\mu +{E}_{{\rm{R}}})}^{2}\). When \(\mu\) goes below zero, both \(\gamma\) and \(\eta\) increase. Interestingly, if \(\Delta\) is small compared to Rashba splitting, \(\gamma\) stays constant for \(\mu \, < \, 0\) until it reaches the band edge, below which it decreases again. For spin current, the ratio \(\eta\) keep increasing before \(\mu\) reaches the band edge. Below that, \(\eta\) decreases very slowly. That means the efficiency of spin current generation in the BEC regime is very high.
Relation between spin density and spin current
As shown above, the functional behaviors of the spin density and spin current are similar in some parameter regions but very different in others. To further clarify the mechanism of the spincurrent generation, we investigate its relation to the spin polarization.
When spin polarization is induced by supercurrent, we may intuitively expect a spin current
which is just the particle current times the spin polarization of each particle. Using Eqs. (9)–(11) and Eq. (3), we get \({\eta }_{0}=\gamma e/{n}_{e}\) and \(\gamma =\frac{m\alpha \hslash }{4e{E}_{{\rm{R}}}}\frac{g(d,u)}{\pi {\hslash }^{2}{n}_{e}(u)/(m{E}_{{\rm{R}}})g(d,u)}.\) In Fig. 3b, by comparing \(\eta\) (solid curves) with \({\eta }_{0}\) (dashed curves in corresponding colors), we find that the differences between them are not small in general. Thus, Eq. (11) does not fully describe the origin of the spin current because the spin and momentum degrees of freedom cannot be separated in systems with SOI.
Comparison to normal states
In normal dissipative Rashba systems, spin density^{49} and spin current^{17,18} can also be generated when an external electric field \({\bf{E}}\) is applied. In such a case, the coefficients \(\gamma\) and \(\eta\) are shown by the black dashed curves in Fig. 3. It turns out that the spin polarization efficiency for SC state and that for normal state are almost the same if \(\Delta\) is small. For spin current generation, \(\eta\) for the SC state is smaller than the normal state in most of the parameter regime. Only when \(\mu\) is very close to the band edge and \(\Delta\) is small, the SC state shows a higher efficiency. The comparison of the normal state and the SC state becomes clearer in the loglog plots as shown in the insets of Fig. 3. When \(\mu \, < \!\!{E}_{{\rm{R}}}\), it is not possible to pass a current in the normal state at \(T=0\). However, there could be BEC SCs in this limit, which can carry supercurrent and gives nonzero spin densities and spin currents.
Although a dissipative normal system seems actually better at generating spin current than its SC counterpart for most of the parameter space, we should note that, in the normal state, the spin current, as well as the charge current, is generated by an electric field and suffers from dissipation and heating, making it impossible to maintain longrange coherence. On the other hand, the spin current in the SC state has very low dissipation (if any) because it is purely due to the flow of a supercurrent. In other words, it is the property of the Cooper pair condensate and longrange coherence is guaranteed, making it a spin supercurrent^{27,50,51,52}.
It should be emphasized that the spin current in a coherent system does not necessarily correspond to the coherent spin current as defined by Leurs et al.^{40} as mentioned previously. In the formalism of nonAbelian gauge fields, the spin current operator can be decomposed into two parts, \({\hat{{\mathbf{J}}}}^{a}={\hat{{\mathbf{J}}}}_{\,\text{NC}}^{a}+{\hat{{\mathbf{J}}}}_{\,\text{C}\,}^{a}\). The first term \({\hat{{\mathbf{J}}}}_{\,\text{NC}}^{a}=\hat{{\mathbf{J}}}{\hat{S}}^{a}\) is a direct product of the charge current operator and the spin operator and is called noncoherent. The second term \({\hat{{\mathbf{J}}}}_{\,\text{C}\,}^{a}\) involves spin entanglement, according to Leurs at al., and is called coherent. In our calculation, the spin current operator is actually \({\hat{{\mathbf{J}}}}_{\,\text{NC}}^{a}\). Such spin supercurrent carried by Cooper pairs does induce spin transport and may be used to create new spintronic devices with longrange coherence.
Discussion
We have shown that a supercurrent in an superconductor with Rashba SOI can induce spin density and spin current. Let’s take the interface between a LaAlO\({}_{3}\) and a SrTiO\({}_{3}\) as an example and estimate the realistic magnitude of these effects. The critical current density (2D) is \(\sim \!\!1{0}^{3}{\mathrm{A/cm}}\), effective mass \({m}^{* }=1.5{m}_{e}\) and the superfluid density \({n}_{s} \sim 1{0}^{13} \, {\mathrm{cm}}^{2}\)^{53}. Assuming \({E}_{{\rm{R}}} \sim 10 \, {\mathrm{meV}}\) (\(\alpha \sim 1{0}^{5} \, {\mathrm{m}}/ \!{\mathrm{s}}\)) and replacing the electron density \({n}_{e}\) by the superfluid density \({n}_{s}\), the second term in Eq. (9) has the same order of magnitude as the first term when the Fermi level is high, i.e. or \(u\gg 1\). For a supercurrent density of \({j}_{x} \sim 1{0}^{5} \, {\mathrm{A/cm}}\), which is about \(1/100\) of the critical current density, the corresponding \({q}_{x} \sim 1{0}^{3} \, {\mathrm{m}}^{1}\). Then, according to Eqs. (6) and (3), the spin current density \({j}^{y} \sim 1{0}^{8}\hslash \cdot {\mathrm{s}}^{1}{\mathrm{cm}}^{1}\) and spin density \({M}_{y} \sim 1{0}^{7}\hslash \cdot {\mathrm{cm}}^{2}\). Converted to charge current by replacing \(\hslash /2\) by \(e\), such a spin current corresponds to an charge current density of \(\sim \!\!0.01 \, {\mathrm{nA/cm}}\). This is rather small. However, If the Fermi level is decreased, say by gating^{41,54,55,56}, and it is still superconducting, the spin current can be enhanced by several orders of magnitude as shown in Fig. 3b. In that case, it should be detectable by inverse spin Hall effect^{57}, by connecting the superconductor with a lightemitting diode^{58}, or by other methods. Our results may be generalized to other spinorbit coupled systems where general properties of the Edelstein effect have been recently studied^{38}.
Another important issue is the effect of finite temperature which affects the results through both the Fermi distribution of the quasiparticles and change of \(\Delta\). In the following, we assume that the function \(\Delta (T)\) is determined by the BCS theory. Fig. 4a shows the spin density as a function of temperature for different values of \(\mu\). Generally, the spin density starts to grow linearly when \(T \, < \, {T}_{c}\), and it saturates when \(T\to 0\). It is exactly the same feature as the superfluid density \({\rho }_{s}\) in standard BCS theory. This is no surprise since Eq. (9) indicates that the spin density is part of the supercurrent. When the Fermi level moves downward, the temperature dependence remains the same. In Fig. 4b, we show the temperature dependence of the spin current. Further discussion about temperature effect can be found in Supplementary Note 3. The temperature dependence in the case of normal state is shown in Supplementary Note 4 for comparison. The spin current generation efficiency as a function of chemical potential is obtained for finite temperature in Supplementary Note 5 which shows that our conclusions are still valid.
One may also note that this system is a paradigm for the study of topological superconductors. In fact, an outofplane Zeeman field can drive it into a topological phase with Majorana edge modes if \(\mu \sim 0\). Then the following questions become interesting: (1) What is the spin current contributed by the edge modes, if any? (2) How are topological phase transitions affected by the supercurrent? In fact, the second question has recently been discussed but in a quasionedimensional system where it is found that a supercurrent shifts the topological phase transitions^{59}. Similar things may happen for twodimensional systems as well. In this way, one may control topological phases by a supercurrent, achieving new methods for coherent quantumdevice manipulation.
To conclude, we have studied the spin and spin current generation by the supercurrent in twodimensional noncentrosymmetric superconductor with Rashba spinorbit interaction. The spin degrees of freedom are partially activated due to the noncentrosymmetry even when onsite pairing between up and down spins is considered. When the chemical potential is below the band crossing point and approaching to the Band edge, i.e., in dilute electron density limit, the large enhancement of the spin supercurrent generation occurs although the spin density is small. The efficiency of the spin supercurrent does not decrease so much even when the chemical potential is below the band edge, i.e., in the BEC limit of the superconductivity. Furthermore, the carrier density can be controlled there by gating, and the chemical potential dependence can be studied experimentally. These studies will pave a route toward the dissipationless superconducting spintronics, where the transfer of spin information without the energy loss is possible through the longrange quantum coherence of the system.
Methods
NonAbelian gauge fields and Rashba spinorbit interaction
The Hamiltonian with SU(2) gauge symmetry of free electrons in magnetic field is
where \({\bf{A}}\) is the U(1) gauge potential of electromagnetic field and \({{\bf{W}}}^{\nu }\) is the SU(2) gauge potential. The constant \({\mu }_{{\rm{B}}}\) denotes the Bohr magneton and \({\sigma }^{\nu =1,2,3}\) are Pauli matrices. The coefficient \(\bar{g}\) is a coupling constant to be assigned later. Rashba SOI corresponds to the existence of the following SU(2) gauge field^{46},
Note that we have introduced the speed of light \(c\) and the LeviCivita symbol \({\epsilon }_{ij\nu }\). The Hamiltonian with this gauge field can be rewritten as
The second term describes the Rashba SOI. So, the Rashba SOI will be included by assuming a constant SU(2) gauge field \({\tilde{{\bf{W}}}}^{\nu }\) and the SU(2) gauge symmetric Hamiltonian with certain Rashba strength becomes
The last two terms are just constant energy shift due to the electric field of the Rashba SOI, which we ignore hereafter. Then we have
In order for the nonAbelian gauge field \({{\bf{W}}}^{\nu }\) to couple with the spin, we set
Derivation of spin density
We calculate the spin density induced by the supercurrent, or by the vector potential \({\bf{A}}=(\hslash {q}_{x}/e)\hat{{\bf{x}}}\) in Eq. (2), using perturbation method. The perturbation Hamiltonian \(\hat{H}^{\prime}\) including both the test fields (\({{\bf{W}}}^{\nu }\) and \({\bf{B}}\)) and the external field (\({\bf{A}}\)) is
with
which is composed of terms linear and quadratic in \({\bf{A}}\) respectively, a term proportional to nonAbelian gauge field \({{\bf{W}}}^{\nu }\), a term bilinear in \({\bf{A}}\) and \({{\bf{W}}}^{\nu }\), and the term due to the field \({\bf{B}}\). As mentioned previously, the spin density \({M}_{y}\) is linear in \({q}_{x}\) (or \({A}_{x}\)) to the lowest order. Consequently, we should calculate the contribution to the free energy by \(\hat{H}^{\prime}\) up to the second order.
At zero temperature, the free energy is simply the ground state energy.
The subscript \(h\) labels the hole bands, i.e. \({E}_{h\pm }(k) \, < \, 0\). Here we ignored the energy due to Cooper pairs, assuming that the order parameter is not affected by the perturbation term. The unperturbed energy spectrum of the system is
\({\xi }_{\pm }=\frac{{\hslash }^{2}{k}^{2}}{2m}\mu \pm \alpha \hslash k\) are the energy spectrum of the normal Rashba bands. The lowest order term of the spin density is from the bilinear term \(\sim {q}_{x}{B}_{i}\). Then, the perturbation correction to the free energy is
The spin density is
After substituting the expressions of \({\xi }_{\pm }\) into the above integral and defining \({k}_{\pm }=k\pm m\alpha /\hslash\), it becomes
with \(u=\mu /{E}_{{\rm{R}}}+1\) and \(d=\Delta /{E}_{{\rm{R}}}\).
Derivation of spin current
Similar to the spin density derivation, we use perturbation theory to obtain the correction to the free energy \(\delta {F}_{J}\). The difference here is that we need to go to the third order perturbation due to the fact that the spin current is quadratic in \({q}_{x}\). After lengthy but straightforward calculation, the free energy correction is
The general form of the function \({F}_{}\) is complicated. However, in the limit \(\Delta \to 0\), it becomes
By change of integral variable from \(k\) to \({k}_{}\) to \(x\) similar to the previous calculation, the integral can be written as
The spin current density is
We have defined two wave vectors \({k}_{2}\) and \({k}_{1}\) which denotes the Fermi wave number of the outer and inner Fermi surfaces respectively.
When \(\mu\) is large, the integral of the second term is negligible and the first and third terms lead to Eq. (6) of the main text. When \({E}_{{\rm{R}}} \, < \,\mu \, < \, 0\), the third term becomes negligible and Eqs. (7)–(8) are obtained.
Data availability
All essential data are available in the manuscript. Additional data are given in Supplementary Figs. 2–8. Further supporting data can be provided from the corresponding author upon request.
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Acknowledgements
N.N. was supported by Ministry of Education, Culture, Sports, Science, and Technology Nos. JP24224009 and JP26103006, the Impulsing Paradigm Change through Disruptive Technologies Program of Council for Science, Technology and Innovation (Cabinet Office, Government of Japan), and Core Research for Evolutionary Science and Technology (CREST) No. JPMJCR16F1 and No. JPMJCR1874, Japan.
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N.N. conceived the idea and supervised this project. J.J.H. and K.Hiroki carried out the calculations for superconducting states while K.Hamamoto did the calculations for normal states. J.J.H. and N.N. prepared the manuscript with the help from all other authors.
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He, J.J., Hiroki, K., Hamamoto, K. et al. Spin supercurrent in twodimensional superconductors with Rashba spinorbit interaction. Commun Phys 2, 128 (2019). https://doi.org/10.1038/s4200501902309
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DOI: https://doi.org/10.1038/s4200501902309
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