Nonreciprocal transport effects can occur in the normal state of conductors and in superconductors when both inversion and time-reversal symmetry are broken. Here, we consider systems where magnetochiral anisotropy of the energy spectrum due to an externally applied magnetic field results in a rectification effect in the normal state and a superconducting (SC) diode effect when the system is proximitized by a superconductor. Focusing on nanowire systems, we obtain analytic expressions for both normal state rectification and SC diode effects that reveal the commonalities—as well as differences—between these two phenomena. Furthermore, we consider the nanowire brought into an (almost) helical state in the normal phase or a topological SC phase when proximitized. In both cases, this reveals that the topology of the system considerably modifies its nonreciprocal transport properties. Our results provide insights into how to determine the origin of nonreciprocal effects and further evince the strong connection of nonreciprocal transport with the topological properties of a system.

A diode is a device in which the resistance depends on the direction of current flow and is a fundamental element of most modern electronics. A diode effect requires a nonreciprocal resistance, R, due to a current, I, such that R ( + I ) ≠ R ( − I )⁠; in other words, current is rectified. To achieve such an effect, it is necessary that inversion symmetry and time-reversal symmetry are broken simultaneously. In most cases, time-reversal symmetry is broken by dissipation and inversion symmetry breaking is achieved extrinsically, e.g., a pn-junction.

In contrast, however, it is also possible for a nonreciprocal resistance to arise as an intrinsic property of a material, e.g., due to the band structure. For instance, applying an external magnetic field can result in a magnetochiral anisotropy (MCA) of the band structure that results in a diode effect.1–10 In particular, MCA results in a resistance that is proportional to the current, I, itself, such that R = R 0 ( 1 + γ B I )⁠, where R0 is the reciprocal resistance, B is the magnetic field strength, and γ the MCA rectification coefficient.4 

It has also recently been discovered that it is possible to have a diode effect in superconductors.11–13 In this case, a superconductor exhibits a critical current, I c ±⁠, that is dependent on the direction of current flow, such that, e.g., I c + > | I c − |⁠, where ± indicates the direction of current flow.12–42 This effect is known as the superconducting (SC) diode effect, because there exists a range of currents—in the above example | I c − | < | I | < I c +—that experiences the zero-resistance of a superconductor in one direction but has a finite resistance for current flow in the opposite direction.12,13

Several mechanisms have been proposed to produce SC diode effects. As in normal state diodes, broken inversion symmetry—either explicitly or intrinsically—and broken time-reversal symmetry—either explicitly by a magnetic field21,22,24 or by some other mechanism17–19,37–40—are required. In particular, it has been shown that MCA of the underlying energy spectrum in a superconductor can also result in a SC diode effect.13,24,25,30

Although normal state rectification and the SC diode effect can both stem from MCA of the energy spectrum,9,24–26 the relationships between the two effects have not been explored. Since MCA rectification can be measured in the normal state, understanding how MCA of the normal state and SC phase relate to each other could provide a pathway to produce hybrid SC diode devices with high efficiencies, which has significant potential technological applications.28,32,35,42,43 Furthermore, such a relationship could provide a measure of the reduction of spin–orbit interaction (SOI) due to metalization effects44,45 in hybrid SC devices.46,47 Finally, it has been shown that SC diode effects can be altered dramatically in the topological SC phase,24,30,48 which is still not fully understood, and further insights into the connections with topology could be gained from the normal state rectification effect.

In this paper, we investigate systems with MCA of the energy spectrum and consider relations between the normal state rectification effect and SC diode effect when proximitized by a superconductor. Focusing on nanowire systems,46,49 we obtain analytic expressions for both normal state rectification and SC diode effects that reveal the links between both phenomena. We also investigate the case where the nanowire is brought into an (almost) helical state in the normal phase or a topological SC phase when proximitized. This reveals that the topology of the system considerably modifies the nonreciprocal transport response. We conclude with a discussion of the relations and differences between these two effects.

First, we investigate normal state nonreciprocal transport in a (quasi) one-dimensional nanowire. In particular, we focus on the diffusive regime with a scattering rate 1 / τ as, e.g., considered in recent experiments on topological insulator nanowires.9 For simplicity, we consider only a single subband of a one-dimensional band structure. As such, we assume that the system is shorter than the localization length and do not take into account other conductivity channels, e.g., in the bulk of the TI nanowire. Although sufficient for our purpose, these contributions can still be important in real experimental systems since they affect γ, the rectification coefficient.

The normal state of a nanowire in the presence of a magnetic field is described by the following Hamiltonian:24 
(1)
where ξ k = ξ − k (to be defined later), Δz (Δx) is the Zeeman energy due to a magnetic field parallel (perpendicular) to the SOI vector, i.e., direction, of strength α k = − α − k⁠, defining the quantization axis, and σi denote the Pauli matrices. The eigenenergies are described by an energy dispersion ε k ( s )⁠, where k is the momentum along the nanowire and s the band index. We will first consider the case of a magnetic field that is parallel to the SOI vector [Fig. 1(a)] and then later consider the case where an additional component of magnetic field perpendicular to the SOI vector results in an (almost) helical pair of states at Fermi levels within the resulting gap [Fig. 1(b)].
FIG. 1.
Energy spectrum: schematic normal state band structure, ε ( k ), of a one-dimensional system with spin–orbit interaction (SOI), i.e., a nanowire. Inversion symmetry breaking plays a crucial role in both the case of normal state rectification and the SC diode effect. In particular, a difference in Fermi velocity for the inner and outer branches of the band structure ( v in ≠ v out) directly results in an SC diode effect and stems from the difference in band curvature for the inner and outer Fermi points, which results in a normal state rectification. Both the normal state rectification and SC diode effect are substantially modified when a magnetic field component opens a gap at k = 0, as in (b), resulting in an (almost) helical state of the normal state and a topological SC state in the proximitized system.
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Energy spectrum: schematic normal state band structure, ε ( k )⁠, of a one-dimensional system with spin–orbit interaction (SOI), i.e., a nanowire. Inversion symmetry breaking plays a crucial role in both the case of normal state rectification and the SC diode effect. In particular, a difference in Fermi velocity for the inner and outer branches of the band structure (⁠ v in ≠ v out⁠) directly results in an SC diode effect and stems from the difference in band curvature for the inner and outer Fermi points, which results in a normal state rectification. Both the normal state rectification and SC diode effect are substantially modified when a magnetic field component opens a gap at k = 0, as in (b), resulting in an (almost) helical state of the normal state and a topological SC state in the proximitized system.

FIG. 1.
Energy spectrum: schematic normal state band structure, ε ( k ), of a one-dimensional system with spin–orbit interaction (SOI), i.e., a nanowire. Inversion symmetry breaking plays a crucial role in both the case of normal state rectification and the SC diode effect. In particular, a difference in Fermi velocity for the inner and outer branches of the band structure ( v in ≠ v out) directly results in an SC diode effect and stems from the difference in band curvature for the inner and outer Fermi points, which results in a normal state rectification. Both the normal state rectification and SC diode effect are substantially modified when a magnetic field component opens a gap at k = 0, as in (b), resulting in an (almost) helical state of the normal state and a topological SC state in the proximitized system.
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Energy spectrum: schematic normal state band structure, ε ( k )⁠, of a one-dimensional system with spin–orbit interaction (SOI), i.e., a nanowire. Inversion symmetry breaking plays a crucial role in both the case of normal state rectification and the SC diode effect. In particular, a difference in Fermi velocity for the inner and outer branches of the band structure (⁠ v in ≠ v out⁠) directly results in an SC diode effect and stems from the difference in band curvature for the inner and outer Fermi points, which results in a normal state rectification. Both the normal state rectification and SC diode effect are substantially modified when a magnetic field component opens a gap at k = 0, as in (b), resulting in an (almost) helical state of the normal state and a topological SC state in the proximitized system.

Close modal
To calculate the current, we expand the Fermi distribution function in powers of electric field, E, applied along the nanowire, such that f l ( k ) ∝ E l⁠, and then solve the Boltzmann equation in the constant relaxation time approximation (as in, e.g., Refs. 4, 9, and 10) such that the lth order current contribution is given by
(2)
where e < 0 is the electron charge, ℏ V l , k s = ∂ k l ε k ( s ) ,   ℏ v k s = ∂ k ε k ( s )⁠, and k i s is the right (R) or left (L) Fermi wavevector associated with the band s = ± [blue/red in Fig. 1(a)]. The full current at all orders of electric field is given by j = ∑ l j l⁠.
Consider the case of a magnetic field parallel to the SOI vector (see Fig. 1) and expand around each band minimum to cubic order in δ k s = k − s k SO with s = ± the band index, where s k SO is the momentum corresponding to the band minima in the absence of magnetic fields [see Fig. 1(a)], such that
(3)
where the quadratic coefficient α m > 0 is related to the band mass and the coefficient β is related to cubic SOI. Here, μ is the chemical potential, and Δ z = g μ B B z / 2 is the Zeeman energy due to the magnetic field Bz along the SOI axis with μB the Bohr magneton and g the g-factor. Note the chemical potential, μ, is measured from the band crossing point at k = 0 and − ε so is the energy of the band minima, both in the absence of a magnetic fields. We will see that this is the minimal order expansion required to obtain a finite nonreciprocal effect. We also note that we assume that β is not too large (⁠ β k SO ≪ α m⁠) such that the quadratic expansion is already a good approximation of the bands and cubic corrections are small. In this case, we could ignore the spurious Fermi points at large momentum arising from the cubic order or, alternatively, add an additional quartic term.50 
Using Eq. (2) to calculate the second-order conductivity, i.e., the leading order nonreciprocal component, at temperature T = 0, we obtain
(4)
where v in (⁠ v out⁠) are the Fermi velocities of the inner (outer) branches of the spectrum in the absence of a magnetic field [see Fig. 1(a)]. In the second line, the approximation is to leading order in β and Δz, which are treated perturbatively, such that k R + + k L − ≈ 2 Δ z / ℏ v out and k R − + k L + ≈ − 2 Δ z / ℏ v in⁠. We note that, in general, v out ≠ v in if β ≠ 0⁠, which is required to achieve a finite σ 2 n⁠. A significant cubic contribution, β, arises, for instance, in germanium nanowires10 or topological insulator (TI) nanowires.9 We note that in the absence of the cubic term in our expansion of the energy spectrum, β = 0, the nonreciprocal coefficient will vanish unless higher orders of momentum in Eq. (3) are included. In other words, a purely quadratic dispersion, even with broken inversion symmetry, is not sufficient to produce a finite nonreciprocal conductivity.9,10 As expected, in the absence of a magnetic field, σ 2 n vanishes.
We now consider an additional component of magnetic field parallel to the nanowire (perpendicular to the SOI vector) that opens a partial gap at small momentum. The chemical potential μ can be placed inside this partial gap, and the system hosts a pair of (almost) helical states51–54 [see Fig. 1(b)]. The presence of Δx has two effects: (1) it modifies the inner Fermi points in Eq. (4) and (2) it modifies the scattering time τ. The interplay of these two effects will determine the behavior of σ 2 n⁠; however, when the inner points no longer contribute, we can again use Eq. (2) to calculate the second order conductivity given by
(5)
where τh is the scattering time in the helical state. As such, we find the simple relationship
(6)
where v in and v out are, as above, the Fermi velocities without an applied magnetic field.

The nonreciprocal conductivity can be either reduced or enhanced, depending on the relative ratios of v in / v out and τ h / τ⁠. However, we assume that the main source of scattering is due to nonmagnetic impurities, and scattering processes that enable spin-flip scattering are small. As a result, backscattering is reduced in the (almost) helical state due to the small scattering matrix element between opposite spin branches. As such, we generically expect that the scattering time will be much longer in the helical state, τ h ≫ τ⁠, and given the fact that it depends on the square of the τ h / τ⁠, there is likely an increase in σ2 in the helical state.

We now turn to the SC diode effect in nanowires.24,33,34,41 We will assume that the superconductivity in the nanowire is induced by the proximity effect, and therefore, no self-consistent treatment of the pairing potential is required. Such a setup was considered numerically in Ref. 24, where it was shown that a change of sign and dramatic reduction in the SC diode efficiency occurred for values of Zeeman energy, Δx, when the topological SC phase transition had occurred. Here, we establish analytically the origin of the SC diode effect and the connection to the topological SC phase transition.

When brought into proximity with a superconductor, a pairing potential, Δ, is induced in the nanowire. Furthermore, the presence of a current through the nanowire can be described by a finite Cooper pair momentum, q, in the nanowire.20–22,24 The full Hamiltonian in the presence of a supercurrent and pairing potential is given by H q = 1 2 ∫ d k ψ k † H k , q ψ k⁠, where
(7)
and, as in Eq. (1), hk is the normal state Hamiltonian. Here, ψ k † = ( c k + q 2 ↑ † ,   c k + q 2 ↓ † ,   c − k + q 2 ↑ ,   c − k + q 2 ↓ )⁠, where the creation (annihilation) operator c k σ † (⁠ c k σ⁠) acts on an electron with momentum k and spin σ. Throughout we assume Δ ≥ 0 is real.
First, we consider only a magnetic field component parallel to the SOI vector. To make analytic progress in calculating the current, we linearize the normal state spectrum in the absence of magnetic fields, which will later be taken into account perturbatively. Furthermore, we assume that SC pairing only occurs between equivalent Fermi points, i.e., only between outer and only between inner points, such that they can be treated separately.55 In this case, the Hamiltonian of each pair of Fermi points can be described by
(8)
where η = + 1 ( − 1 ) indicates the outer (inner) Fermi points, such that k − 1 0 (⁠ k 1 0⁠) is the Fermi wave number of the inner (outer) point of the + band at zero magnetic field and velocities v − 1 = v in and v 1 = v out are as defined above. In Fig. 2, we show that this approximation captures well the SC diode behavior at low temperatures. For each pair of Fermi points, η, the two eigenenergies are given by Legg et al.24 (see the supplementary material),
(9)
where n = ±⁠. Here, we chose two branches that correspond to positive energies in the absence of supercurrents and magnetic fields. The remaining two branches are related to E n , η ( Δ , q , k ) by particle-hole symmetry and are given by − E − n , η ( Δ , q , − k )⁠. The current can then be obtained from the free energy density21,24 (see the supplementary material for details), such that
(10)
where β = ( k B T ) − 1⁠, with T being the temperature and kB the Boltzmann constant.
FIG. 2.
Superconducting diode efficiency: the diode efficiency initially increases linearly with Zeeman energy before saturating at a Zeeman field that is set by the difference in Fermi velocities of the inner and outer branches, see Eq. (15). The linearized approximation utilized for analytic calculations (dark blue line) is a good approximation at low temperatures. For the numerical solution, the full energy spectrum of H k , q [see Eq. (7)] is used to find the maxima and minima of the current [see Eq. (10)]. We use hk, as defined in Eq. (1), with ξ k = ℏ 2 m 0 k 2 + C k 4 and α k = α SO k. Here, m0 is the band mass, C the quartic contribution that results in v out ≠ v in necessary for an SC diode effect, and α SO the linear spin–orbit coefficient. Parameters: m 0 = 0.015   m e ,   C = 2 × 10 5   meV   nm 4 ,   α S O = 0.65   eV  Å, μ = 0.4   meV ,   Δ = 0.2   meV ,   Δ x = 0.
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Superconducting diode efficiency: the diode efficiency initially increases linearly with Zeeman energy before saturating at a Zeeman field that is set by the difference in Fermi velocities of the inner and outer branches, see Eq. (15). The linearized approximation utilized for analytic calculations (dark blue line) is a good approximation at low temperatures. For the numerical solution, the full energy spectrum of H k , q [see Eq. (7)] is used to find the maxima and minima of the current [see Eq. (10)]. We use hk, as defined in Eq. (1), with ξ k = ℏ 2 m 0 k 2 + C k 4 and α k = α SO k⁠. Here, m0 is the band mass, C the quartic contribution that results in v out ≠ v in necessary for an SC diode effect, and α SO the linear spin–orbit coefficient. Parameters: m 0 = 0.015   m e ,   C = 2 × 10 5   meV   nm 4 ,   α S O = 0.65   eV  Å, μ = 0.4   meV ,   Δ = 0.2   meV ,   Δ x = 0⁠.

FIG. 2.
Superconducting diode efficiency: the diode efficiency initially increases linearly with Zeeman energy before saturating at a Zeeman field that is set by the difference in Fermi velocities of the inner and outer branches, see Eq. (15). The linearized approximation utilized for analytic calculations (dark blue line) is a good approximation at low temperatures. For the numerical solution, the full energy spectrum of H k , q [see Eq. (7)] is used to find the maxima and minima of the current [see Eq. (10)]. We use hk, as defined in Eq. (1), with ξ k = ℏ 2 m 0 k 2 + C k 4 and α k = α SO k. Here, m0 is the band mass, C the quartic contribution that results in v out ≠ v in necessary for an SC diode effect, and α SO the linear spin–orbit coefficient. Parameters: m 0 = 0.015   m e ,   C = 2 × 10 5   meV   nm 4 ,   α S O = 0.65   eV  Å, μ = 0.4   meV ,   Δ = 0.2   meV ,   Δ x = 0.
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Superconducting diode efficiency: the diode efficiency initially increases linearly with Zeeman energy before saturating at a Zeeman field that is set by the difference in Fermi velocities of the inner and outer branches, see Eq. (15). The linearized approximation utilized for analytic calculations (dark blue line) is a good approximation at low temperatures. For the numerical solution, the full energy spectrum of H k , q [see Eq. (7)] is used to find the maxima and minima of the current [see Eq. (10)]. We use hk, as defined in Eq. (1), with ξ k = ℏ 2 m 0 k 2 + C k 4 and α k = α SO k⁠. Here, m0 is the band mass, C the quartic contribution that results in v out ≠ v in necessary for an SC diode effect, and α SO the linear spin–orbit coefficient. Parameters: m 0 = 0.015   m e ,   C = 2 × 10 5   meV   nm 4 ,   α S O = 0.65   eV  Å, μ = 0.4   meV ,   Δ = 0.2   meV ,   Δ x = 0⁠.

Close modal
When subject to a magnetic field, the ground state SC condensate could develop a finite Cooper pair momentum.21,22,56 If there is a finite gap and the energies E n , η ( Δ , q , k ) > 0 for all k, in the zero-temperature limit, the current is given by
(11)
where the first line follows from the fact that, for T = 0, the terms in Eq. (10) will cancel for all k except where sgn { E n , η ( 0 , q , k ) } ≠ sgn { E n , η ( Δ , q , k ) }⁠. In the second line, we note that the Δz contribution cancels between inner and outer branches because the product of the Cooper pair velocity of each branch, v η⁠, and the shift in momentum, η Δ z ℏ v η⁠, results in equal magnitude contributions with opposite signs (see the supplementary material for details). Importantly, this means that the condition j ( q 0 ) = 0 is only satisfied by q 0 = 0⁠, and there is no finite pairing momentum in the ground state. We will see below that this is not the case in the topological phase. Although analytically only valid at zero-temperature, we find this is a good approximation also at finite temperature (see Fig. 3).
FIG. 3.
Superconducting diode efficiency of the topological phase: numerical calculation of (a) the diode coefficient δ and (b) the ground-state Cooper pair momentum q0 as a function of the Zeeman energy Δx resulting from a magnetic field parallel to the nanowire. Deep within the topological regime ( Δ x ≫ Δ), we find that the diode efficiency is considerably suppressed by the presence of a finite pairing momentum q0 in the ground state of the superconductor. We find q0 is not strongly affected by small finite temperature. Normal state Hamiltonian and parameters are the same as Fig. 2. Other than μ = 0   meV ,   Δ x ≠ 0, and Δ z = 0.006   meV.
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Superconducting diode efficiency of the topological phase: numerical calculation of (a) the diode coefficient δ and (b) the ground-state Cooper pair momentum q0 as a function of the Zeeman energy Δx resulting from a magnetic field parallel to the nanowire. Deep within the topological regime (⁠ Δ x ≫ Δ⁠), we find that the diode efficiency is considerably suppressed by the presence of a finite pairing momentum q0 in the ground state of the superconductor. We find q0 is not strongly affected by small finite temperature. Normal state Hamiltonian and parameters are the same as Fig. 2. Other than μ = 0   meV ,   Δ x ≠ 0⁠, and Δ z = 0.006   meV⁠.

FIG. 3.
Superconducting diode efficiency of the topological phase: numerical calculation of (a) the diode coefficient δ and (b) the ground-state Cooper pair momentum q0 as a function of the Zeeman energy Δx resulting from a magnetic field parallel to the nanowire. Deep within the topological regime ( Δ x ≫ Δ), we find that the diode efficiency is considerably suppressed by the presence of a finite pairing momentum q0 in the ground state of the superconductor. We find q0 is not strongly affected by small finite temperature. Normal state Hamiltonian and parameters are the same as Fig. 2. Other than μ = 0   meV ,   Δ x ≠ 0, and Δ z = 0.006   meV.
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Superconducting diode efficiency of the topological phase: numerical calculation of (a) the diode coefficient δ and (b) the ground-state Cooper pair momentum q0 as a function of the Zeeman energy Δx resulting from a magnetic field parallel to the nanowire. Deep within the topological regime (⁠ Δ x ≫ Δ⁠), we find that the diode efficiency is considerably suppressed by the presence of a finite pairing momentum q0 in the ground state of the superconductor. We find q0 is not strongly affected by small finite temperature. Normal state Hamiltonian and parameters are the same as Fig. 2. Other than μ = 0   meV ,   Δ x ≠ 0⁠, and Δ z = 0.006   meV⁠.

Close modal
We define the diode efficiency as19,
(12)
where j c ± are the critical currents to the left and right, such that j c − = − min { j ( q ) } and j c + = max { j ( q ) }⁠. At zero temperature, we find that the extremal current densities occur precisely at the values of q, where the system becomes gapless as one of the energies E n , η ( Δ , q , k ) in Eq. (9) becomes zero, i.e., when the condition E n , η ( Δ , q , k ) > 0 for Eq. (11) is no longer satisfied for all k (see the supplementary material for details).24 Note that | Δ z | ≥ Δ results in a gapless system already for q = 0, such that δ = 0 and we assume throughout 0 ≤ | Δ z | < Δ⁠. From Eq. (9), for each n and η, we find the critical values,
(13)
Using Eq. (11), the diode efficiency can be written as24 
(14)
and in other words, the diode efficiency is set by the largest negative and smallest positive q η n⁠. In particular, in the regime of small | Δ z |⁠, both critical momenta q c ± correspond to the same η branch, namely, that with the largest v η⁠, and the diode efficiency grows linearly with Zeeman energy, Δz. For larger Zeeman strengths, q c ± corresponds to different η branches and the diode efficiency saturates to a constant. For instance, taking v out > v in⁠, we find the diode efficiency is given by
(15)
We emphasize that a finite diode efficiency is only possible when there is a difference in Fermi velocities v in ≠ v out⁠; otherwise, the contributions of two branches cancel each other even in the presence of a magnetic field. The resulting diode efficiency is shown in Fig. 2 both using this analytic formula based on the linearized approximation and using the numerical maxima and minima of Eq. (10) without the linearized approximation (see the supplementary material for details).
We now turn to the topological phase that can exist for a region of chemical potential when a parallel Zeeman field component, Δx, is sufficiently large.49 We note that for a SC diode effect to occur, we still require a component of magnetic field such that Δz is also non-zero, ensuring that the outer branches are no longer Kramers partners. In general, Eq. (10) must now be calculated numerically, as in Ref. 24. However, in this regime, a finite pairing momentum q0 can develop. In particular, if temperature is sufficiently low and Δx sufficiently large, solving Eq. (11) for j ( q 0 ) = 0 in the absence of v in contributions gives a finite pairing momentum q 0 ≈ 2 Δ z / ℏ v out⁠. As a result, deep in the topological phase, the critical currents are
(16)
Since q c ± = q 1 ± = 2 ( Δ + Δ z ) / ℏ v out⁠. Note that q0 is precisely the finite momentum due to the shift of the outer branches by the Δz component of the magnetic field and q 0 = 0 for Δ z = 0⁠. As such, we see that the critical currents deep in the topological SC phase are (approximately) reciprocal and the diode efficiency δ ≈ 0⁠. The full numerical results of δ and q0 in this regime, for various temperatures T, are shown in Fig. 3 and confirm this behavior (see the supplementary material for details of numerics).

We have investigated the relations between normal state nonreciprocal transport and the SC diode effect in proximitized nanowire systems. In both cases, MCA can result in a difference in the inner and outer Fermi velocities and this is necessary to cause a finite diode effect. This difference in Fermi velocities generically arises either due to cubic SOI or due to a momentum dependent mass term in systems with SOI. We also found that the topology of the system strongly affects its nonreciprocal transport properties for both normal and SC systems. In the (almost) helical normal state, we found that the rectification is generically enhanced due to the expected increase in the scattering time. In contrast, the SC diode effect is substantially reduced due to the emergence of a finite Cooper pair momentum, q0, deep in the topological phase. Our results indicate that systems with large nonreciprocal transport coefficients in the normal state can be prime candidates for SC diodes when proximitized by a superconductor. However, we find that the impact of a magnetic field parallel to the nanowire can be strikingly different for normal state rectification in comparison with the SC diode effect.

See the supplementary material for calculations of the full superconducting energy spectrum, the current density, and details of the numerics used to obtain critical currents

This work was supported by the Georg H. Endress Foundation and the Swiss National Science Foundation. This project received funding from the European Union's Horizon 2020 Research and Innovation Program (ERC Starting Grant, Grant No 757725).

The authors have no conflicts to disclose.

Georg Angehrn and Henry F. Legg contributed equally to this work.

Georg Angehrn: Formal analysis (equal); Validation (equal); Writing – review & editing (equal). Henry F. Legg: Conceptualization (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Daniel Loss: Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Jelena Klinovaja: Conceptualization (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article and its supplementary material.

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