Introduction

The invention of diodes marked a pivotal breakthrough in solid-state electronics, with their distinctive non-reciprocal current flow making them fundamental to semiconductor devices1,2. In quantum materials, inversion and time-reversal symmetry crucially determine charge current nonreciprocity. Magnetochiral anisotropy (MCA), arising in systems with broken inversion symmetry3, is a dominant mechanism for this effect. MCA leads to non-reciprocity in the magnetoresistance with respect to the current direction and can be greatly enhanced by superconducting correlations4,5. In superconductors, MCA can lead to an extreme manifestation of nonreciprocity known as the superconducting diode effect (SDE), where resistance is zero in one current direction and non-zero in the other6,7,8. SDE thus realizes an ideal diode with perfect rectification. Such a scenario arises in the regime where the critical current that causes superconductor to normal metal transition differ in the opposite directions (\(| {J}_{c}^{+}| \ne | {J}_{c}^{-}|\))6,7. Recent experiments in engineered superlattice systems8,9,10 and bulk materials/ thin films5,6,11,12 have spurred research interests into more efficient superconducting diode devices, including in twisted multilayer graphene13,14 and transition metal dichalcogenides15,16.

The inversion symmetry breaking required for critical current non-reciprocity can be achieved through extrinsic mechanisms (e.g., asymmetric geometry or artificial superlattice potential8,10,17) or intrinsic mechanisms like Rashba spin-orbit coupling (SOC)9,18,19,20,21,22,23,24,25. Correspondingly, SDE can be broadly classified into two types: extrinsic and intrinsic. Although predicted decades ago26,27, the mechanism of SDE is beginning to be understood only recently, leading to extensive investigations into its potential for dissipationless electronic circuits6,18,19,21,28,29,30,31,32. Recent theoretical studies18,19,28,29, using phenomenological Ginzburg-Landau theory and mean-field analysis of the two-dimensional Rashba-Zeeman Hubbard model, have shown that intrinsic SDE can arise in systems with broken inversion and time-reversal symmetry. In the helical superconducting ground state, this occurs due to asymmetry in Cooper pair momentum in the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state33,34, resulting in Kramers non-degenerate s-wave Cooper pairs6,18,28.

A Rashba nanowire device proximitized to an s-wave superconductor and subjected to external magnetic fields serves as a prototypical platform for realizing helical superconductivity25,31,35,36,37,38. This device also shows interesting magnetoelectric effects depending on the direction of the applied fields39. While previous phenomenological/ non-self-consistent studies25 demonstrated the potential for SDE in such systems, they reported low diode efficiencies ( ~2%). To enhance the relatively low diode efficiency of this device and better understand its properties, it’s crucial to systematically investigate the superconducting order parameter using self-consistent methods19,28,31. Recent studies19,28,31 have shown that this approach can potentially uncover pathways for improvement across various parameters. In this paper, we address this gap through a comprehensive, self-consistent mean-field investigation of the interplay between various Rashba SOC terms and external magnetic fields in determining emergent FFLO pairing and highly efficient superconducting diodes in a Rashba nanowire device. We show that even with only linear Rashba SOC we can achieve a very large diode efficiency ( ≳45%) by suitably tuning the parameters in this system. We extend beyond linear Rashba SOC to include higher-order (cubic) Rashba SOC terms, which are not only naturally present in semiconducting nanowires40,41,42 but also can dominate the normal state properties43,44,45,46,47,48,49,50. Moreover, we demonstrate that higher-order Rashba SOC can induce finite diode efficiency ( ≳15%) even without longitudinal magnetic fields. This behavior differs qualitatively from the linear SOC case and can be used to detect the presence of higher-order SOC terms in the nanowire. Our study establishes the Rashba nanowire device as a promising platform for highly efficient superconducting diodes, crucial for advancing nanoscale superconducting electronics.

Method

Model hamiltonian

We consider a single channel one-dimensional semiconducting nanowire along the z-direction with both linear and higher order Rashba SOC terms40,41,42,51 placed in close proximity to a three dimensional bulk s-wave superconductor as schematically shown in Fig. 1a. In the presence of externally applied magnetic fields, the normal state physics of the nanowire is governed by the one-dimensional Rashba-Zeeman Hamiltonian52

$${{{\mathcal{H}}}}_{0}=\mathop{\sum}\limits_{s,{s}^{{\prime} }}\int\,dz\,{\psi }_{s}^{{\dagger} }(z){h}_{s{s}^{{\prime} }}(z){\psi }_{{s}^{{\prime} }}(z),$$
$$\hat{h}(z)=\frac{{p}_{z}^{2}}{2m}-\mu +\left(\frac{\alpha }{\hslash }{p}_{z}+\frac{\delta }{{\hslash }^{3}}{p}_{z}^{3}\right){\sigma }_{x}+\frac{\gamma }{{\hslash }^{3}}{p}_{z}^{3}{\sigma }_{y}+{{\bf{B}}}\cdot {{\mathbf{\sigma }}}\,.$$
(1)

Here, the spin degree of freedom is labeled by \(s,{s}^{{\prime} }\in \{\uparrow ,\downarrow \}\). \({\psi }_{s}^{{\dagger} }(z)\) creates a fermion at the position z with spin s, while σ = (σx, σy, σz) is the vector of the Pauli matrices in spin space and μ is the chemical potential. \({p}_{z}=\frac{\hslash }{i}{\partial }_{z}\) is the momentum operator in the direction of current flow (z-direction) within the nanowire. The strength of the usual linear Rashba SOC is denoted by α and the cubic Rashba SOC of strengths are δ and γ. The presence of Rashba SOC breaks the inversion symmetry of the system. The externally applied magnetic fields lead to the Zeeman term characterized by B = (Bx, By, Bz) that breaks time-reversal symmetry. We assume the radius of the nanowire to be small such that there is only one occupied mode, i.e., single channel. This nanowire Hamiltonian [Eq. (1)] in momentum space can be written as

$${{{\mathcal{H}}}}_{0}=\mathop{\sum}\limits_{s,{s}^{{\prime} }} \mathop{\sum}\limits_{k}\,{c}_{k,s}^{{\dagger} }{h}_{s{s}^{{\prime} }}(k){c}_{k,{s}^{{\prime} }},$$
$$\hat{h}(k)={\xi }_{k}+\left[\left(\alpha k+\delta {k}^{3}\right){\sigma }_{x}+\gamma {k}^{3}{\sigma }_{y}\right]+{{\bf{B}}}\cdot {{\mathbf{\sigma }}}\,.$$
(2)

Here, k represents the momentum along the nanowire (z-direction) and \({\xi }_{k}=\frac{{\hslash }^{2}{k}^{2}}{2m}-\mu\). The bulk energy spectrum of the nanowire obtained by diagonalizing \({\hat{h}}_{k}\) is given by

$${\epsilon }_{\pm }(k)={\xi }_{k}\pm \sqrt{{(\alpha k+\delta {k}^{3}+{B}_{x})}^{2}+{(\gamma {k}^{3}+{B}_{y})}^{2}+{B}_{z}^{2}}$$
(3)

where,  ± label the two helicity bands. Note that, the Zeeman field Bz (longitudinal field) opens a Zeeman gap while Bx (transverse field) creates asymmetry between the two bands as schematically shown in Fig. 1b. On the other hand, By opens up a gap as well as creates band asymmetry (not shown). Throughout the majority of this paper, we will discuss the results with By = 0 for simplicity and will separately discuss the effects of nonzero By for completeness. We choose m = 0.015 me, where me is the bare electron mass, and α = 100 meV-nm, values representative of semiconductor nanowires with strong Rashba SOC53.

Fig. 1: Schematic of the proximity-induced Rashba nanowire device.
figure 1

a The schematic of our proposed device consisting of a Rashba nanowire in close proximity to an s-wave superconductor and in presence of external Zeeman fields: Bx and Bz to investigate SDE. The strength of the linear Rashba SOC is denoted by α, while the cubic Rashba SOC strengths are represented by δ and γ as illustrated in Eq. (2). b, c The dispersions of the normal state and the Bogoliubov quasi particles in the superconducting state are qualitatively shown in b and c respectively considering both linear and higher order SOC.

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Self-consistent BdG mean-field formalism

We seek to elucidate the nature of the superconducting ground state in the Rashba nanowire in the presence of s-wave superconducting correlations that can be induced via the proximity effect in presence of a bulk s-wave superconductor, as well as supercurrents (see Fig. 1a). In general, the presence of supercurrents can lead to a superconducting state with finite momentum Cooper pairs termed as the FFLO state. To establish that, we consider an attractive Hubbard type interaction in the bulk s-wave superconductor described by the Hamiltonian

$${{{\mathcal{H}}}}_{I}=-\frac{U}{2}\mathop{\sum}\limits_{s,{s}^{{\prime} }}\int\,{d}^{3}{{\bf{r}}}\,{\psi }_{s}^{{\dagger} }({{\bf{r}}}){\psi }_{{s}^{{\prime} }}^{{\dagger} }({{\bf{r}}}){\psi }_{{s}^{{\prime} }}({{\bf{r}}}){\psi }_{s}({{\bf{r}}})\,,$$
(4)

where, U is the attractive Hubbard interaction strength. We now decouple the Hubbard interaction term (\({{{\mathcal{H}}}}_{I}\)) in the finite momentum s-wave channel within the mean-field approximation23,28,31 and consider these correlations to be proximity induced in the nanowire. Thus, our self-consistent mean-field treatment effectively mimics an intrinsic superconducting one-dimensional wire with SOC. Note that, superconducting correlations leak from the bulk superconductor into the nanowire, with the magnitude of the proximity-induced gap depending on system-specific factors such as interface quality and external magnetic fields. In general, these “leaked” correlations in the nanowire are not constrained by the same self-consistency conditions as in intrinsically superconducting systems. However, as a zeroth-order approximation, we assume that the self-consistency condition applies similarly, apart from a renormalization of the pairing gap. This assumption is justified by the fact that the proximity-induced pairing originates from a high-symmetry three-dimensional superconductor, which effectively suppresses fluctuations31,54. Then, using the Bogoliubov de-Gennes (BdG) mean-field formalism the effective Hamiltonian of the nanowire takes the form

$${{\mathcal{H}}}= \frac{1}{2}\int\,dz\,{\Psi }^{{\dagger} }(z){{{\mathcal{H}}}}_{BdG}(z)\Psi (z)+{{{\mathcal{E}}}}_{0}\,,\\ {{{\mathcal{H}}}}_{BdG}(z)= \left[\begin{array}{cc}\hat{h}(z)&\hat{\Delta }(z)\\ -{\hat{\Delta }}^{* }(z)&-{\hat{h}}^{* }(z)\end{array}\right];\,\hat{\Delta }(z)=-i{\sigma }_{y}\Delta {e}^{iqz}\,,$$
(5)

where, \(\Psi (z)={[{\psi }_{\uparrow }(z),{\psi }_{\downarrow }(z),{\psi }_{\uparrow }^{{\dagger} }(z),{\psi }_{\downarrow }^{{\dagger} }(z)]}^{T}\) is the Nambu spinor inside the nanowire, \({{{\mathcal{E}}}}_{0}=\frac{L}{U}| \Delta {| }^{2}\) with L denoting the length of the nanowire. The s-wave FFLO order parameter is given by Δ(z) = Δ( − iσy)eiqz, where q is the Cooper pair momentum. The Hamiltonian in Eq. (5) effectively describes the physics of the Rashba nanowire device schematically shown in Fig. 1a. To proceed further, we write the mean-field Hamiltonian [Eq. (5)] in the momentum space as

$${{\mathcal{H}}}= \frac{1}{2}{\sum}_{k}{\Psi }_{k}^{{\dagger} }{{{\mathcal{H}}}}_{BdG}(k){\Psi }_{k}+{{{\mathcal{E}}}}_{0}\,,\\ = {\sum}_{n,k}\left({E}_{n,k}{\gamma }_{n,k}^{{\dagger} }{\gamma }_{n,k}-\frac{{E}_{n,k}}{2}\right)+{{{\mathcal{E}}}}_{0}\,,$$
$$\,{\mbox{with}}\,,\,\,{{{\mathcal{H}}}}_{BdG}(k)=\left[\begin{array}{cc}\hat{h}(k+\frac{q}{2})&-i{\sigma }_{y}\Delta \\ i{\sigma }_{y}\Delta &-\hat{h}{(-k+\frac{q}{2})}^{* }\end{array}\right]\,.$$
(6)

Here, \({\Psi }_{{{\bf{k}}}}\equiv {[{c}_{k+q/2,\uparrow },{c}_{k+q/2,\downarrow },{c}_{-k+q/2,\uparrow }^{{\dagger} },{c}_{-k+q/2,\downarrow }^{{\dagger} }]}^{T}\) and En,k(q) denotes the Bololiubov quasiparticle energies. The behavior of the Bogoliubov quasiparticle spectrum is schematically shown in Fig. 1c. This depicts that the introduction of the pairing potential Δ opens a gap in the spectrum. As in the case of the nanowire spectrum, the asymmetry in BdG bands occurs due to the application of the Zeeman field Bx, and the Zeeman field Bz makes the bands distinctly gapped.

The condensation energy Ω(q) of the system, defined as the difference of free energy per unit length between the superconducting and normal state28,55, is

$$\Omega (q,\Delta )=F(q,\Delta )-F(q,0)\,,$$
(7)

where, F(q, Δ) is the free energy density of the nanowire given by

$$F(q,\Delta )=-\frac{1}{L\beta }{\sum}_{n,k}\ln \left[1+{e}^{-\beta {E}_{n,k}(q)}\right]+\frac{| \Delta (q){| }^{2}}{U}\,,$$
(8)

where, \(\beta ={({k}_{B}T)}^{-1}\) with kB being the Boltzmann constant and T is the temperature.

We determine the order parameter Δ(q) for a given value of q by self-consistently solving the gap equation, obtained via minimizing the free energy F(q, Δ) in Eq. (8), given by

$$\Delta (q)=-\frac{U}{L}{\sum}_{n,k}\frac{\partial {E}_{n,k}}{\partial {\Delta }^{* }}\,{n}_{F}({E}_{n,k})$$
(9)

where \({n}_{F}({E}_{n,k})=1/(1+{e}^{\beta {E}_{n,k}})\) is the Fermi function. Using the self-consistent solution of Δ(q) in Eq. (7), we optimize the condensation energy with respect to q to obtain the Cooper pair momentum (q0) of the true FFLO superconducting ground state. In this equilibrium state, there is no charge current through the system. The BCS gap in the absence of any external magnetic fields (B = 0) is denoted by Δ0 and we choose the value of U in such a way that we are in the weak coupling BCS regime with Δ0 ~1 meV. In the rest of the paper, we present results in the low temperature limit choosing a fixed optimized value of β−1= 0.1 meV much smaller than Δ0.

The variation of the self-consistent solutions of the superconducting order parameter Δ(q) and the momentum of the Cooper pairs q0 in the FFLO ground state with the external Zeeman fields are shown in Figs. 2 and 3 with only linear SOC case and, with both the linear and higher order SOC case respectively. We note from Figs. 2a and 3a that there is only a marginal reduction in the superconducting gap Δ(q0) ≈ Δ(q=0) from Δ0 with the external Zeeman fields. We also note that for both the cases the superconducting gap vanishes for a critical value of the Cooper pair momentum either above a positive value of \(q={q}_{c}^{+}\) or below a negative value of \(q={q}_{c}^{-}\). Figures 2b and 3b show that q0 varies linearly with Bx for Bx ≪ Δ0 for a given value of Bz consistent with previous studies18. We note Fig. 2b that in the case of only linear SOC q0 is finite only when Bz ≠ 0 and Bx ≠ 0. This is because when only linear SOC is present the effect of the SOC can be gauged away by using a spin-dependent gauge transformation56 and then when only Bx ≠ 0 the band structure is symmetric leading to zero momentum Cooper pairs. However, when both Bx, Bz ≠ 0, then in the transformed basis a spiral magnetic field texture is realized leading to momentum dependent asymmetry in the band structure and hence leading to finite momentum Cooper pairs. In contrast, when both the linear and cubic SOC terms are present, the effect of SOC can not be gauged away and as a result only Bx ≠ 0 is sufficient to give rise to q0 ≠ 0 in this case. Consequently, this feature also gets manifested in realizing finite SDE in this Rashba nanowire device, as extensively discussed in the next section.

Fig. 2: FFLO ground state with linear SOC.
figure 2

a The self-consistent superconducting gap Δ(q) is shown as a function of the Cooper-pair momentum q(nm-1) for Bx/Δ0 = 0.3 with only linear Rashba SOC (α = 100 meV-nm). Δ0 is the BCS gap in the absence of any external magnetic field (B = 0). b The Cooper pair momentum of the FFLO ground state q0 as a function of Bx. The presence of the FFLO ground state (q0 ≠ 0) is established only when both Bx and Bz are finite. Other model parameters are chosen as: (By, μ/Δ0, U, β−1)=(0, 0.5, 25.55 meV, 0.1 meV).

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Fig. 3: FFLO ground state with both linear and higher order SOC.
figure 3

a The self-consistent superconducting gap Δ(q) is depicted as a function of Cooper-pair momentum q for Bz = 0 with both the linear Rashba SOC term (α = 100 meV-nm) and the cubic Rashba SOC terms (δ = γ = 10 eV-nm3). b The Cooper pair momentum of the FFLO ground state q0 as a function of Bx is shown. Note that the FFLO state is stable even when Bz = 0 in the presence of a finite Bx. Other system parameters are chosen as: (By, μ/Δ0, U, β−1)=(0, 0.5, 16.45 meV, 0.1 meV).

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Results

The supercurrent density J(q) is computed from the condensation energy Ω(q, Δ) in Eq. (7) as18,25,28,

$$J(q)=-2e\,\frac{\partial \Omega (q,\Delta )}{\partial q}.$$
(10)

The sign of J(q) indicates the direction of the supercurrent flow. The critical (optimal) values of the supercurrent along and opposite to the flow direction within the range \({q}_{c}^{-}\le q\le {q}_{c}^{+}\) for which the superconducting phase is stable, are denoted as \({J}_{c}^{+}\) and \({J}_{c}^{-}\) respectively. When \(| {J}_{c}^{+}| \ne | {J}_{c}^{-}|\), we have a finite SDE in the system and the efficiency or the quality factor of the SDE is defined as18

$$\eta =\frac{| {J}_{c}^{+}| -| {J}_{c}^{-}| }{| {J}_{c}^{+}| +| {J}_{c}^{-}| }\,.$$
(11)

We first compute J(q) over the range \({q}_{c}^{-}\le q\le {q}_{c}^{+}\) using the self-consistent solutions Δ(q) for each value of q (shown in Figs. 2a and 3a for example). Then finding its optimal values \({J}_{c}^{+}\) and \({J}_{c}^{-}\), we evaluate the diode efficiency (η) numerically. We systematically analyze the SDE in our Rashba nanowire device considering two broadly distinct scenarios: with only linear SOC and, with both linear and higher order SOC for various system parameters as discussed in the following two subsections.

Case-I: Rashba nanowire with only linear SOC

In the presence of only linear Rashba SOC, we uncover the SDE by investigating the behavior of the supercurrent density (J(q)) and the diode efficiency (η) for different system parameters as presented in Fig. 4. Figure 4a shows that the supercurrent density is symmetric with respect to q for Bz = 0 even with Bx ≠ 025. We note that as we start increasing the magnitude of Bz with Bx ≠ 0, J(q) becomes more and more asymmetric, leading to nonreciprocal critical supercurrents, \(| {J}_{c}^{+}| \ne | {J}_{c}^{-}|\) and hence to finite SDE. This is a direct consequence of the fact that the presence of FFLO state and the asymmetry in critical momenta: \(| {q}_{c}^{+}| \ne | {q}_{c}^{-}|\) is distinctly realized only in presence of both the fields Bx, Bz as noted in Fig. 2a.

Fig. 4: Superconducting diode effect with linear SOC.
figure 4

a The supercurrent density (J) is shown as a function of q(nm-1) considering solely linear Rashba SOC and Bx/Δ0 = 0.3. The nonreciprocity in the supercurrent, J(q) ≠ − J( − q), is achieved in this case only when both Bx and Bz are finite. It then leads to nonreciprocity in the critical supercurrents, \({J}_{c}^{+}\ne | {J}_{c}^{-}|\), hence exhibiting a finite diode efficiency (η ≠ 0). Here, \({J}_{0}\equiv {J}_{c}({B}_{z}=0)=| {J}_{c}^{+}({B}_{z}=0)| =| {J}_{c}^{-}({B}_{z}=0)|\). b and c: The corresponding diode efficiency (η) is shown as a function of Bx (with fixed Bz) and Bz (with fixed Bx) with only linear Rashba SOC in the panels b and c respectively. The system parameters are chosen as: (By, α, μ/Δ0, U, β−1)=(0, 100 meV-nm, 0.5, 25.55 meV, 0.1 meV).

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The diode efficiency as a function of Bx for fixed values of Bz and as a function of Bz for fixed values of Bx are shown in Fig. 4b, c respectively. In both cases, η increases linearly at low field strengths, reaches a maximum at moderate values, and then decreases to zero as the field strengths continue to intensify. This behavior can be understood as a consequence of a competition between the asymmetry in the band structure of the system and changes in the superconducting pairing susceptibility as the magnetic field strengths are varied, consistent with previous studies18,20. Note that, the self-consistent BCS gap vanishes for very large values of the Zeeman fields Bx and Bz consequently leading to zero efficiency, however the resultant nanowire spectrum might still be gapped like an ordinary insulator.

Preceding investigations on a Rashba nanowire featuring only linear Rashba SOC25 reported maximum diode efficiency of  ~2%, where the superconducting order parameter has been fixed to a particular value. Contrastingly, our study reveals that the self-consistently derived superconducting order parameter within the full parameters space has significant influence on the performance of the diode where the highest η value is found to be ≳45%. This is evident from calculation presented in Fig. 4b, c. The vanishing of the superconducting gap within our self-consistent treatment necessitates restricting the maximum values of the magnetic fields used in our model. We note that in this case By also plays a similar role as Bz due to the geometry of the setup (see Fig. 1a) under consideration and hence further analysis considering By ≠ 0 is not presented. However, By plays a significant and distinct role when we consider the effect of cubic Rashba SOC terms as discussed in the next subsection.

Case-II: Rashba nanowire with both linear and higher order SOC

The variation of J(q) and η in the case when both the linear (α≠0) and cubic Rashba SOC terms (δ, γ ≠ 0) are present in the nanowire are shown in Fig. 5. First of all, comparing Fig. 5a with Fig. 4a, we note that the cubic Rashba SOC terms interestingly lead to asymmetry in J(q) even with Bz = 0 as long as Bx≠0. This leads to nonreciprocal critical supercurrents, \(| {J}_{c}^{+}| \ne | {J}_{c}^{-}|\) with a single magnetic field Bx ≠ 0 in contrast to only linear SOC case discussed earlier. The variation of η with the magnetic fields Bx and Bz with By=0 are shown in Fig. 5b, c. We note from Fig. 5b that the diode efficiency at a fixed Bz exhibits a linear increase as a function of Bx for small values of Bx ≪ Δ0, eventually attaining a maxima around Bx/Δ0 = 0.4 similar to the case with only linear Rashba SOC described earlier. This behavior, as in the earlier linear SOC case18,20, results from two competing effects: the system’s asymmetric band structure versus changes in superconducting pairing susceptibility as magnetic field strength varies. The maximum efficiency increases as one increases the magnitude of Bz. However, η shows only marginal enhancement at a fixed value of Bx when Bz is varied as depicted in Fig. 5c.

Fig. 5: Superconducting diode effect with both linear and higher order SOC.
figure 5

a The supercurrent density (J) as a function of q(nm-1) with both cubic Rashba SOC (δ = γ = 10 eV nm3) and linear Rashba SOC (α = 100 meV-nm) choosing Bz=0. The nonreciprocity in the critical supercurrents: \(| {J}_{c}^{+}| \ne | {J}_{c}^{-}|\) (also implying a finite diode efficiency η ≠ 0) is achieved for any finite value of Bx. Here, \(J{{\prime} }_{0}\equiv {J}_{c}({B}_{x}={B}_{z}=0)=| {J}_{c}^{+}({B}_{x}={B}_{z}=0)| =| {J}_{c}^{-}({B}_{x}={B}_{z}=0)|\). b, c The corresponding diode efficiency (η) is depicted as a function of Bx (with fixed Bz) and Bz (with fixed Bx) in the panels b and c respectively. We have chosen the other model parameters as: (By, μ/Δ0, U, β−1)=(0, 0.5, 16.45 meV, 0.1 meV).

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Modifying the values of δ and γ evidently changes the band dispersions in the normal and superconducting state of the system and consequently the efficiency of the superconducting diode will also change. We depict the diode efficiency phase diagrams in the δ − γ plane in Fig. 6. We obtain the phase diagrams by varying U for every configuration of (δ, γ) such that the self-consistent BCS gap Δ0 remains fixed. For the Bz = 0 and Bx ≠ 0 case, we can achieve a peak diode efficiency of approximately 15% for an optimal choice of higher-order Rashba SOC parameters as shown in Fig. 6a. We emphasize that this behavior is qualitatively different from the linear SOC case, where the efficiency is zero with only the Bx field. This distinctive response, arising solely from higher-order SOC terms, provides a method to detect and quantify the strength of these terms in a nanowire device. The application of an additional magnetic field Bz readily elevates η to approximately 20%, as illustrated in Fig. 6b since effects of both the linear and cubic SOC terms are relevant.

Fig. 6: Diode efficiency phase diagram with both linear and higher order SOC.
figure 6

The variation of the diode efficiency η is shown in the plane of cubic Rashba SOC parameters δ − γ choosing “optimal” values of the Zeeman fields (Bx/Δ0, Bz/Δ0) = (0.4, 0.0) in panel a and (Bx/Δ0, Bz/Δ0) = (0.4, 0.5) in panel b, respectively. We note a small enhancement in the maximum value of η for finite Bz. The other system parameters are chosen as: (By, α, μ/Δ0, β−1) = (0, 100 meV-nm, 0.5, 0.1 meV).

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The results discussed so far were obtained by fixing By = 0. The effect of finite By in η is shown in the Fig. 7. In this case, By serves the role of both Bx and Bz in realizing a finite SDE by creating both gap and asymmetry in the band structure. Figure 7a and b show that the application of By produces systematic increase in η with increasing By for different values of Bz with Bx = 0 and Bx ≠ 0 respectively. However, By can not increase the efficiency beyond a certain limit because the superconducting state becomes unstable after some value of By when the other magnetic fields are present as well.

Fig. 7: By dependence of diode efficiency.
figure 7

The diode efficiency η as a function of By with both linear and cubic Rashba SOC terms for Bx = 0 and finite Bx/Δ0 = 0.3 are shown in a and b respectively. The chosen model parameters are (α, δ, γ, μ/Δ0, β−1) = (100 meV-nm, 10 eV-nm3, 10 eV-nm3, 0.5, 0.1 meV).

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For completeness, we consider a special case with a different type of the higher order Rashba SOC terms51 when only δ ≠ 0 and γ = 0 (with α ≠ 0 as usual). The variation of the corresponding diode efficiencies with the magnetic fields are shown in Fig. 8. A finite diode efficiency with a maximum of  ~12% even in presence of a finite Bx only is obtained as shown in Fig. 8a which is also in contrast to the linear SOC case. Thus the presence of even a single cubic Rashba parameter δ is sufficient to manifest SDE. By further application of Bz the diode efficiency η is readily enhanced upto  ~21% as a function of Bx shown in Fig. 8a. There is only a marginal increase in efficiency by increasing Bz for fixed Bx as shown in the Fig. 8b.

Fig. 8: Diode efficiency for the case δ ≠ 0, γ = 0.
figure 8

The diode efficiency (η) with linear SOC and with cubic SOC of different type where δ = 10 eV nm3 but γ = 0 as a function of Bx (with fixed Bz) and Bz (with fixed Bx) are shown in a and b respectively. Other parameters are: (α, μ/Δ0, By, β−1)=(100 meV-nm, 0.5, 0, 0.1 meV).

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It is important to note that the chemical potential was fixed at μ/Δ0 = 0.5 in the discussion thus far. The variation of the SDE efficiency with respect to μ is presented in Fig. 9. As illustrated in Fig. 9a, we find that the SDE efficiency in the nanowire device reaches its maximum near the subband crossing point at μ/Δ0 ≈ 0 when considering linear SOC alone consistent with previous reports25,29. Thus by suitably choosing the value of the chemical potential, it is possible further increase the SDE efficiency  ~ 50% in this case. Similarly, when both linear and cubic SOC are taken into account, the SDE efficiency also peaks at μ/Δ0 ≈ 0, as shown in Fig. 9b.

Fig. 9: μ-dependence of efficiency.
figure 9

Diode efficiency (η) versus chemical potential (μ) for different values of Bx, considering a solely linear Rashba spin-orbit coupling (SOC) (α = 100 meV-nm) for Bz/Δ0 = 0.4 with U = 25.55  meV and b combined linear and cubic Rashba SOC (δ = γ = 10 eV-nm3) for Bz/Δ0 = 0.5 with U = 16.45 meV. Other model parameters are chosen as (By, β−1)=(0, 0.1 meV).

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Conclusions

This article analyzes how the interplay between linear and cubic Rashba SOC terms, combined with external Zeeman fields, can optimize the efficiency of SDE in a proximitized one-dimensional nanowire device. Using self-consistent mean-field analysis, we demonstrate that SDE, characterized by nonreciprocal critical currents (\(| {J}_{c}^{+}| \ne | {J}_{c}^{-}|\)), arises from Cooper pair momentum asymmetry in the helical superconducting ground state of this time-reversal and inversion symmetry broken system. Note that, we achieve a diode efficiency η≳45% with only linear Rashba SOC significantly higher than the previously reported efficiency of η~2% from non-self-consistent analyses25. In this case with only linear SOC, at least two magnetic fields are required to obtain a finite diode efficiency: one along the SOC direction and one perpendicular to it. This requirement stems from the fact that linear SOC effects can be eliminated through a spin-dependent gauge transformation56, resulting in a conventional BCS superconducting state with zero Cooper pair momentum when only one Zeeman field is present. However, the presence of two orthogonal Zeeman fields creates a spiral magnetic field texture via the gauge transformation, leading to momentum-dependent band asymmetry and asymmetric FFLO Cooper pairs56. In contrast, higher-order SOC effects cannot be eliminated via gauge transformations, allowing for finite diode efficiency even with only one Zeeman field unlike the linear SOC case and this behavior can be used to detect the presence of higher-order SOC in nanowires. However, including higher-order SOC doesn’t improve the maximum diode efficiency (only η~20%), this value remains significant compared to previously reported diode efficiencies6,25. Our results underscore the importance of self-consistently determining the order parameter of the superconducting ground state to fully understand the capability and optimize Rashba nanowire devices as superconducting diodes. Note that the effect of topology on the diode effect in the nanowire device is not addressed in our study, as we have not considered any contributions from the ends of the nanowire.

Our findings can be readily experimentally tested in spin orbit coupled Rashba nanowires52,57,58,59 such as zinc-blende InSb and wurtzite InAs, with lengths of approximately 100 nm, where cubic Rashba SOC interactions are significant depending on the growth direction40. Superconductivity can be proximity induced in these nanowires by common bulk s-wave superconductors like Nb or Al. Future research directions include investigating SDE in proximity-induced multichannel Rashba nanowires with overlapping channels52,60,61,62, extending beyond the single-channel case examined here. Additionally, the influence of topological superconductivity realized in the nanowire and the nanowire device geometry on SDE warrants further exploration63,64,65,66,67,68. An intriguing prospect is the potential replication of magnetic field effects using electric fields via the Rashba-Edelstein effect6,69,70,71, which could lead to the realization of field-free SDE.