Kramers Fulde-Ferrell state and superconducting spin diode effect

Recent experimental observations of the diode effect in superconductors [1, 2, 3, 4] and Josephson junctions (JJ) [5, 6, 7, 8, 9] have stimulated the research of nonreciprocal transport properties in superconducting (SC) systems. Following the proposal of SC diode effect [10], the so-called $\phi_{0}$ Josephson state has been extensively studied [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] as a possible mechanism to realize nonreciprocal transport in JJs. More recently, many theoretical proposals [24, 25, 26, 27, 28, 29, 30, 31] have been put forward for SC diode in bulk superconductors. In particular, the finite-momentum pairing Flude-Ferrell-Larkin-Ovchinnikov state [32, 33] is believed to provide a physical mechanism, since the order parameter of Fulde-Ferrell (FF) state $\Delta(\textbf{r})=\Delta e^{i\textbf{q}\cdot\textbf{r}}$ can directly generate a difference in the critical current along and against the direction of q, leading to SC diode effect [28, 27]. The FF order, also known as helical superconductivity, can be realized in noncentrosymmetric superconductors with spin-orbit coupling (SOC) and time reversal symmetry breaking fields [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47].

So far, the study of SC diode effect focused on the nonreciprocity of charge transport. One may wonder if there exists a similar nonreciprocal property in spin transport in certain SC systems. From the symmetry point of view, SC diode effect in charge transport, where critical currents in opposite directions have different magnitudes, requires the system to break both inversion and time-reversal (${\cal T}$) symmetry, since the charge current operator changes sign under either inversion or ${\cal T}$. While the spin current operator changes sign under inversion, it is invariant under ${\cal T}$. Thus, the nonreciprocity in spin transport only requires breaking inversion symmetry and can be realized in ${\cal T}$ invariant superconductors.

In this article, we propose a novel SC state that can realize nonreciprocal spin transport. This state has equal-spin pairing and a FF type of order parameter $\Delta_{\sigma}(\textbf{r})=\Delta e^{i\sigma\textbf{Q}\cdot\textbf{r}}$, with opposite Cooper pair center of mass momentum for opposite spin polarizations as shown schematically in Fig. 1(a). We term this SC state as a Kramers FF (KFF) state since ${\cal T}$ symmetry is maintained. Such FF state has pairing field with only one Q vector in each pairing channel, which is translational invariant unlike the Larkin-Ovchinnikov state also known as the pair density wave state, where the pairing field has both Q and $-\textbf{Q}$ vectors in each channel so that the pairing order parameters varies in space. We demonstrate that the KFF state can be realized in a meanfield theory of a concrete model describing a spin-orbit coupled chain with nearest neighbor attractions as illustrated in Fig. 1(a,b). The nonzero Q pairing across the Fermi points in Fig. 1(a) is enabled by the SOC split bands. We study the condition to realize the nonreciprocal spin transport where the critical spin current along positive and negative directions are unequal in magnitude for both bulk SC state and Josephson junction structures. Moreover, we find intriguing properties and rich phases in the charge transport across JJs of the KFF state, which can be realized by simply changing the length of the SC chain. Similar pairing state was also studied in the two dimensional honeycomb system where the valley degree of freedom plays the role of spin here [48].

This article is organized as follows. We start with the introduction of our model Hamiltonian and meanfield formulation for the KFF state in Sec. II. In Sec. III, we discuss the nonreciprocal spin transport for the bulk SC with KFF order, which is followed by the discussion of the hidden inversion symmetry that is related to the nonreciprocal spin transport in Sec. IV Then we study the transport properties of the Josephson junction structure constructed from the KFF state in Sec. V and discuss various phases realized in the charge transport across JJs of the KFF state in Sec. VI. We finalize the discussion in Sec. VII.

The meanfield Hamiltonian in KFF state becomes

which can be written in the Nambu basis $\psi_{k\sigma}^{\dagger}=\left(c_{k+\frac{\sigma Q}{2},\sigma}^{\dagger},c_{-k+\frac{\sigma Q}{2},\sigma}\right)$ as

where,

is block diagonal in spin space, leading to the eigenenergy as

and we have $E_{k\sigma,\pm}=E_{-k\bar{\sigma},\pm}$ due to the ${\cal T}$ symmetry. Then the free energy density at zero temperature $\Omega(\Delta_{\parallel},Q)$ can be calculated as

where $\Theta(x)$ the Heaviside step function. The order parameter $\Delta_{\parallel}$ for a given $Q$ can be determined self-consistently by minimizing $\Omega(\Delta_{\parallel},Q)$ with respect to $\Delta_{\parallel}$, leading to the self-consistency equation

The optimal $Q$ value can be further determined by minimizing $\Omega(\Delta_{\parallel},Q)$ with respect to $Q$, i.e., $\partial_{Q}\Omega(\Delta_{\parallel},Q)=0$. Because the latter is directly related to the spin current carried by the KFF state (see Appendix. B) $j_{s}(Q)=\sum_{\sigma}\frac{\sigma}{2}j_{\sigma}=\partial_{Q}\Omega(\Delta_{\parallel},Q,T)$, the ground state with optimized $Q=Q_{0}$ does not carry any net spin current since $\partial_{Q}\Omega(\Delta_{\parallel},Q_{0},T)=0$. When the KFF state is driven out of equilibrium into a state with $Q\neq Q_{0}$, a nonzero applied spin current $j_{s}(Q\neq Q_{0})=j_{s}\neq 0$ is realized. Throughout the remaining text, we define the charge and spin currents in units of $\frac{e}{\hbar}$ and unity, respectively. Since the spin current carrying state has equal but opposite center of mass momentum $\pm Q$ for Cooper pairs in opposite spin channels, the charge current always vanishes due to $\cal T$ symmetry. The critical spin currents in $+$ and $-$ directions are determined by the maximum and minimum values of $j_{s}(Q)$ sustained by the SC state according to $j_{s,c+}=\max_{Q}[j_{s}(Q)]$ and $j_{s,c-}=\min_{Q}[j_{s}(Q)]$. When $\left|j_{s,c+}\right|\neq\left|j_{s,c-}\right|$, the SC state enables nonreciprocal spin transport as shown in Fig. 1(c).

We performed meanfield calculations using two sets of parameters. In the first case, we set $t_{2}=0$ and obtain analytically that the ground state has $Q_{0}=2\theta_{\alpha}=\sigma(k_{f\sigma,+}+k_{f\sigma,-})$, consistent with the SOC split bands where electrons pair across the Fermi points $k_{f\sigma,\pm}$ in the same spin sector, giving rise to finite Cooper pair momenta $\sigma Q_{0}$ with more details shown in Appendix. C. The zero temperature free energy in Eq. (17) and the current density from its momentum derivative are calculated numerically and plotted in Figs. 2(a, c) as a function of $Q$. The critical spin currents $j_{s,c\pm}$ are determined by the maximum and minimum values of the spin current density. Fig. 2(c) shows that the two critical momenta $Q_{\pm}$ at which the spin current reaches critical values coincide with the two momenta where the SC order parameter $\Delta_{\parallel}(Q_{\pm})$ vanishes. In this case with $t_{2}=0$, we find that the critical spin currents $j_{s,c+}=-j_{s,c-}$, as shown in Fig. 2(a), and the spin transport is reciprocal. The absence of nonreciprocal spin transport turns out to be due to a hidden inversion symmetry when $t_{2}=0$.

To demonstrate the hidden inversion symmetry, we can perform a local gauge transformation $c_{i\sigma}^{\dagger}\rightarrow e^{-\frac{i}{2}\sigma Qx_{i}}d_{i\sigma}^{\dagger}$, corresponding to $c_{k\sigma}^{\dagger}\rightarrow d_{k-\frac{\sigma Q}{2},\sigma}^{\dagger}$ in momentum space. Then the meanfield Hamiltonian Eq. 13 becomes

Since in the absence of $t_{2}$, $\varepsilon_{k+\frac{\sigma Q}{2},\sigma}=-2t_{\alpha}\cos(k+\frac{\sigma Q}{2}-\sigma\theta_{\alpha})$, we can see that in the new basis, the only inversion breaking term $\theta_{\alpha}$ owing to the spin-orbit coupling $\alpha$, is cancelled if $Q=Q_{0}=2\theta_{\alpha}$. In other words, when $Q=Q_{0}=2\theta_{\alpha}$, the inversion symmetry can be recovered in the new basis, where the meanfield Hamiltonian becomes

The transformed Hamiltonian describes two spin-degenerate $p$-wave Kitaev chains [51] with inversion symmetry in each spin sector. It is precisely this hidden inversion symmetry that forbids the nonreciprocal property of spin current, since this hidden inversion symmetry changes the sign of the current operator $\hat{j}_{\sigma}$, while keeping the total Hamiltonian invariant, which guarantees a one to one correspondence between the positive and negative current, such that $j_{s,c+}$ and $j_{s,c-}$ have to have the same magnitude.

In the case with finite $t_{2}$, the dispersion of the noninteracting Hamiltonian reads $\varepsilon_{k\sigma}=-2t_{\alpha}\cos(k-\sigma\theta_{\alpha})-2t_{2}\cos(2k)$, and then the Fermi momentum $k_{f\sigma,\pm}$ no longer have a closed form and the sum of the two Fermi momentum belonging to the same spin polarization is incommensurate in general. Here, we can immediately see that the gauge transformation above can not recover the inversion symmetry as above, since now $\varepsilon_{k+\frac{\sigma Q}{2},\sigma}=-2t_{\alpha}\cos(k+\frac{\sigma Q}{2}-\sigma\theta_{\alpha})-2t_{2}\cos(2k+\sigma Q)$, and the inversion breaking phase of the two cosine function $\frac{\sigma Q}{2}-\sigma\theta_{\alpha}$ and $\sigma Q$ cannot be cancelled by $Q$ simultaneously, so that there is no hidden inversion symmetry that forbids the presence of the nonreciprocal spin transport. Therefore, the transformed model describes the two p-wave Kitaev chains with complex hoppings, which are time-reversal counterparts of each other but not identical and can be written in the real space as

Indeed, the results shown in Fig. 2(b) confirms that critical spin currents are nonreciprocal with $j_{s,c+}=0.11$ and $j_{s,c-}=-0.20$ along $+$ and $-$ directions, respectively. As a result, a spin current $j_{s}$ satisfying $0.11<|j_{s}|<0.20$ flows as dissipationless supercurrent in the negative direction since $|j_{s}|<|j_{s,c-}|$, but can only be transported as a dissipative normal current in the positive direction since $|j_{s}|>|j_{s,c+}|$. This SC spin diode effect is shown schematically in Fig. 1(c).

We next study the transport properties of Josephson chains consisting of a normal metal sandwiched between two KFF superconductors depicted in Fig. 3(a) with more detailed setups shown in Appendix. D. We consider the general case with nonzero $t_{2}$ in the KFF state.

An intriguing feature in Fig. 3(b) is the phase difference between Josephson currents in two spin sectors, which is clearly revealed in the energy spectrum plotted in Fig. 3(c). Apart from the continuum states outside the SC gap, there are eight in-gap states. Among them, four are at exactly zero energy, corresponding to two pairs of Majorana zero modes (one for each spin) located at the two ends of the chain due to $p$-wave nature of the KFF state [51], which do not contribute to Josephson current. The other four in-gap states are Andreev bound states of the S-N-S junction. For junctions made of conventional superconductors, the energies of the bound states cross at $\phi=\pi$ [54], corresponding to zero modes trapped by $\pi$-junction [55]. In the current KFF JJs, the energy spectrum of the bound states for two spin species shifts in opposite directions as shown in Fig. 3(c). As a result, the phase bias where the bound states cross zero shifts from $\pm\pi$ to $\pm\phi_{0\sigma}$, where the Josephson current jumps due to branch switching as shown in Fig. 3(b).

Such phase differences have a great impact on charge transport. The charge current $I_{e}=I_{\uparrow}+I_{\downarrow}$ crosses zero at both $\phi=0$ and $\phi=\pi$ with positive slopes, indicating the free energy $\Omega_{Jc}(\phi)$ of the junction reaches a local minimum at both $\phi=0$ and $\pi$. This state is called a $\mathbf{0}^{\prime}$ or $\pmb{\pi}^{\prime}$ state depending on the momentum of the global minimum. It was studied previously in JJs where two superconductors are coupled through an Anderson impurity [56] or a magnetic quantum dot [57]. Here, these remarkable states are realized in ${\cal T}$-invariant systems due to the novel KFF SC order. Remarkably, we find that distinct Josephson junction states ($\mathbf{0}$, $\pmb{\pi}$, $\mathbf{0}^{\prime}$ and $\pmb{\pi}^{\prime}$) can all be realized by tuning the phase shift, which can be easily achieve by changing the length of the SC region. The definition of these states are listed in Table. 1.

The dependence of the phase shift on the length $N_{s}$ of the SC region can be understood by performing a local gauge transformation that maps the KFF JJ onto a Kitaev JJ consisting of two spin-degenerate $p$-wave Kitaev chains subject to spin-dependent phase bias $\phi_{\sigma}=\phi+\sigma Q(N_{s}-1)$ as shown in Appendix. D. If we further assume the Josephson current for the Josephson chain consisting of two spin degenerate $p$-wave Kitaev chains with phase bias $\phi$ as $I_{0}(\phi)$ which is identical for the two spin species due to the spin degeneracy, we can then immediately get the Josephson current for each spin species as $I_{\sigma}(\phi)=I_{0}(\phi+\sigma Q(N_{s}-1))$, i.e., the Josephson current $I_{\sigma}(\phi)$ is shifted from the current of the transformed junction $I_{0}(\phi)$ by a phase $\sigma Q(N_{s}-1)$ (mod $2\pi$) so that the relative phase difference of the current between the two spin species is $\delta\phi=2\sigma Q(N_{s}-1)$ (mod $2\pi$). Various Josephson junction states can be realized by tuning $\delta\phi$ from 0 to $2\pi$ through the length $N_{s}$. This relation is verified numerically in Figs. 5(a-e) for $Q=\frac{\pi}{10}$ and $N_{s}\in[321,332]$. Any combination of $Q$ and $N_{s}$ can produce similar results as long as $\delta\phi$ covers the range $[0,2\pi]$. We also calculate the total free energy of the system for different $N_{s}$ and indeed observe transitions between all these states controlled by $N_{s}$ as shown in Fig. 5(f). Specifically, for $N_{s}$=321 (331), there is only one global minimum located at $\phi=0\ (\pi)$ and the system is in $\mathbf{0}\ (\pmb{\pi})$ state. For $N_{s}$=323 (329), the free energy has a global minimum at $\phi=0\ (\pi)$ and a local minimum at $\phi=\pi\ (0)$ so that the system is in $\mathbf{0}^{\prime}$ ($\pmb{\pi}^{\prime}$) state. The transition between the two states is reached at $N_{s}=326$, where $\delta\phi=\pi$ and $\Omega_{Jc}(0)=\Omega_{Jc}(\pi)$.

Interestingly, in this case with only nearest neighbor hopping $t_{1}$, the charge current $I_{e}(\phi)$ acquires a period of $\pi$ in phase $\phi$ instead of $2\pi$ as in conventional Josephson current at the critical point ($N_{s}=326$), which can also be understood in the gauge transformed basis. In this case with $N_{s}=326$, the phase bias becomes $\phi_{\sigma}=\phi+\frac{\sigma\pi}{2}$ and we thus have $I_{\sigma}(\phi)=I_{0}(\phi+\frac{\sigma\pi}{2})$, from which we can get $I_{\uparrow}(\phi+\pi)=I_{0}(\phi+\frac{3\pi}{2})=I_{0}(\phi-\frac{\pi}{2})=I_{\downarrow}(\phi)$. Then we can derive a new relation for the total charge current $I_{e}$ as

so that the charge current $I_{e}(\phi)$ acquires a period of $\pi$ instead of $2\pi$. The spin current $I_{s}(\phi)$ then acquires a minus sign when progressing $\pi$ phase, so that its period is still $2\pi$.

The realization of diverse Josephson junction states by simply controlling the length of the superconductor is an intriguing property. It originates from the opposite nonzero momentum of Cooper pairs in each spin sector in the novel KFF state. The quantum interference of relative phase shifted pairing functions with opposite spin polarization leads to rich phases and physical phenomena.

We reported the theoretical discovery of a novel time-reversal invariant, finite momentum pairing Fulde-Ferrell state – the KFF state. The concrete effective 1D model we used to realize the KFF state and its many unprecedented properties is intimately connected to the novel physics observed at the atomic line defect (ALD) in monolayer iron-based superconductor Fe(Te,Se), where zero-energy bound states emerge at both ends of the ALD with no signatures of ${\cal T}$ symmetry breaking [50]. The missing atoms cause inversion symmetry breaking and induces Rashba SOC. It was shown that significant equal-spin triplet pairing can be induced by coherent quantum mechanical processes along such a Rashba ALD embedded in 2D unconventional superconductors  [49]. This makes it plausible for materializing the effective 1D model with significant equal-spin triplet pairing to generate the KFF state. More recently, evidence for finite momentum pair density wave order has been observed in monolayer Fe(Te,Se) along one-dimensional domain walls [58]. The experimental evidence suggests time-reversal symmetry is preserved, which makes the KFF state a plausible candidate in addition to the Larkin-Ovchinnikov state.

The most remarkable of the KFF state is that, in the presence of broken inversion symmetry, it supports nonreciprocal spin supercurrent in both bulk superconductor and JJ. In contrast to nonreciprocal charge transport in SC systems which requires breaking both inversion and ${\cal T}$ symmetry, nonreciprocal SC spin transport only requires breaking inversion symmetry. This is because the spin current operator is invariant under time-reversal, such that systems with positive and negative spin current are unrelated by the ${\cal T}$ operation. This is true regardless of whether the system respects the ${\cal T}$ symmetry or not, making it free of the constraint by the Onsager relation. We thus propose a novel SC spin diode effect as a potential new frontier for using spins to make dissipationless electronic devices in SC spintronics. While future work is clearly needed which is outside the scope of the current paper, we point out that the unique properties of the KFF state make it plausible for possible realizations in JJs consisting of a ferromagnetic barrier. The exchange field of the barrier favors spin-triplet pairing and has very little effect on the critical current of equal-spin triplet pairing such as in the KFF state, resulting in the slow decay of the critical current with increasing barrier length. Both effects have been demonstrated experimentally [59, 60]. In turn, detecting the nonreciprocal spin transport together with the slow decay of the critical current with the barrier length can serve as the smoking gun evidence for the KFF state.