# Giant nonreciprocity of current-voltage characteristics of
noncentrosymmetric supercondctor-normal metal-superconductor junctions

T. Liu  Department of Physics, University of Washington, Seattle, WA 98195,
USA  M. Smith  Materials Science Division, Argonne National Laboratory,
Lemont, Illinois 60439, USA  A. V. Andreev  Department of Physics, University
of Washington, Seattle, WA 98195, USA  B. Z. Spivak  Department of Physics,
University of Washington, Seattle, WA 98195, USA

###### Abstract

We develop a theory of nonreciprocal current-voltage (I-U) characteristics in
noncentrosymmetric superconductor-normal metal-superconductor junctions. We
show that at small voltages the nonreciprocal features of the I-U
characteristics can be expressed entirely in terms of the dependence of the
nonreciprocal part of the quasiparticle density of states in the normal metal
part of the junction on the order parameter phase difference $\chi$ across the
junction. The amplitude of the nonreciprocity in this regime is proportional
to the inelastic quasiparticle relaxation time $\tau_{in}$, and can be much
larger than that in normal materials, where it is proportional to the elastic
relaxation time $\tau_{el}$. At low bias the I-U characteristics possess
additional symmetry, not present in normal conductors; they remain invariant
under simultaneous reversal of current, voltage and the magnetic field.

Due to Onsager’s principle, the linear two-terminal conductance of time-
reversal invariant systems, $G_{l}(\bf{H})$, must be an even function of the
magnetic field ${\bf H}$ Onsager ; Onsager1

$G_{l}({\bf H})=G_{l}(-{\bf H}).$ (1)

Beyond the linear in $U$ regime, the current-voltage ($I-U$) characteristics
in noncentrosymmetric systems can be nonreciprocal: $J(U,{\bf H})\neq
J(U,-{\bf H})$.

In noncentrosymmetric normal conductors this phenomenon has been investigated
in several articles, see for example Refs. Nagaosa1 ; IvchenkoSpivak ; Krstic
; Cobden ; Ideue ; Rikken ; Rikken1 ; Loss . In this case, at small ${\bf H}$
and $U$, the degree of nonreciprocity is proportional to the elastic
relaxation time $\tau_{el}$,

$\delta J=J({\bf H})-J({\bf-H})\sim\eta\tau_{el}U^{2}{\bf H}.$ (2)

Here $\eta$ is a material dependent parameter.

The shapes of the current-voltage (I-U) characteristics in superconductor-
normal metal-superconductor (SNS) junctions are more complicated than those of
normal metals. The reciprocal part of the I-U characteristics of SNS junctions
have been studied in many articles, see for example Refs. LarkinOvchinnikov1 ;
Tinkham ; Volk ; Averin1 ; Liu . It was shown in Ref. Liu that the shape of
I-U characteristics at small and large bias are qualitatively different, as
illustrated in Figs. 1 and 2. At relatively large bias, the I-U
characteristics are controlled by the elastic mean free time $\tau_{el}$. As a
result, the degree of nonreciprocity in this regime is expected to be of order
of that of the normal part of the junction, which is relatively small and
featureless. These parts of the I-U characteristics are shown in Figs. 1,2 by
dashed black lines. At small bias the I-U characteristics are controlled by
the long quasiparticle inelastic relaxation time $\tau_{in}\gg\tau_{el}$,
which is typically much longer than the elastic relaxation time. Therefore,
the nonlinear conductance of the junctions turns out to be much larger than
the normal state conductance. These parts of the I-U characteristics are shown
in Figs. 1 and 2 by green and blue lines.

There is extended literature about the nonreciprocity of supercurrent in
superconductors in general, and the nonreciprocity of the critical current of
Josephson junctions in particular (see for example Refs. Hasan ; Ando ; Fu1 ;
Lotfizadeh ; Efetov ; Edelstein ; Baumgartner ; Baumartner1 ; Strunk ; Turini
; Lin ; Nagaosa1 ; Fu2 ; LiyandaGeller ; Zutich ; Ilie ; Li , and references
herein). On the other hand, the nonreciprocity of the dissipative part of
nonlinear I-U characteristics has attracted much less attention. The goal of
this article is to develop a theory of nonreciprocity of $I-U$ characteristics
of noncentrosymmetric SNS junctions in this regime. We will show that at small
voltages the degree of the nonreciprocity of the I-U characteristics of SNS
junctions turns out to be much larger than in normal metals.

At small voltages $eU\ll E_{T}$, the quasi-particle spectrum, and the I-U
characteristics of the junction can be calculated in the adiabatic
approximation, treating the phase difference $\chi(t)$ as a slowly varying
parameter Liu . Here $E_{T}$ is the Thouless energy, which is inversely
proportional to the characteristic time required for quasiparticles to travel
between the two superconducting banks of the junction. For simplicity we
assume that the transmission coefficient of superconductor-normal metal
boundary is of order one. In this case the charge transport through SNS
junction can be expressed entirely in terms of the dependence of the density
of states $\nu(\epsilon,\chi)$ on $\chi$ and the quasiparticle energy
$\epsilon$.

In the presence of the voltage, the phase difference across the junction
$\chi(t)$ evolves in time according to the Josephson relation,

$\dot{\chi}\equiv\frac{d\chi}{dt}=2eU(t).$ (3)

Due to Andreev reflection at the normal metal-superconductor boundaries the
low energy quasiparticles ($\epsilon<\Delta$) are trapped inside the normal
region, and the spectrum of these quasiparticles depends on the phase
difference $\chi$. At nonzero temperature $T$, the quasiparticles occupying
these levels move in energy space together with the levels. This motion
creates a nonequilibrium quasiparticle distribution, which relaxes via
inelastic scattering producing a dissipative contribution to the current. The
physical mechanism of this contribution is similar to the Debye mechanism of
microwave absorption in gases Debye , Mandelstam-Leontovich mechanism of the
second viscosity in liquids Landau , the Pollak-Geballe mechanism of microwave
absorption in the hopping conductivity regime Pollak , and the mechanism of
low frequency microwave absorption in superconductors Smith1 ; Smith2 ; Mike .

A quantitative description of the current in the low bias regime can be
obtained as follows (see for example Liu and references therein). In the
adiabatic approximation, contributions to the current related to transitions
between energy levels can be neglected, and the current may be expressed as

$\displaystyle J(t)=$
$\displaystyle-2e\int_{0}^{\infty}d\epsilon\nu\big{(}\epsilon,\chi(t)\big{)}V_{\nu}\big{(}\epsilon,\chi(t)\big{)}[n(\epsilon,t)-n_{F}\big{(}\epsilon\big{)}]$
$\displaystyle+J_{s}\big{(}\chi(t)\big{)}.$ (4)

Here $\nu(\epsilon,\chi)$ is the total density of states which includes spin
degree of freedom, $n(\epsilon,t)$ is the nonequilibrium occupancy of
quasiparticle levels with energy $\epsilon$, $n_{F}(\epsilon)$ is the Fermi
distribution function at temperature $T$ (which we will assume to be much
smaller than the superconducting gap in the banks of the junction,
$T\ll\Delta$), and $V_{\nu}\big{(}\epsilon,\chi(t)\big{)}$ is the sensitivity
of quasiparticle energy levels to changes in $\chi$. The latter can be
expressed in terms of the density of states as,

$V_{\nu}\big{(}\epsilon,\chi\big{)}\equiv-\frac{1}{\nu(\epsilon,\chi)}\int_{0}^{\epsilon}d\tilde{\epsilon}\frac{\partial\nu(\tilde{\epsilon},\chi)}{\partial\chi}.$
(5)

Finally, $J_{s}(\chi)$ is the supercurrent can be written as

$J_{s}(\chi)=-2e\int^{\infty}_{0}d\epsilon\tanh\left(\frac{\epsilon}{2T}\right)\nu(\epsilon,\chi)V_{\nu}(\epsilon,\chi),$
(6)

(see for example Ref. Binn ).

The time evolution of the distribution function is described by the kinetic
equation,

$\partial_{t}n(\epsilon,t)+\dot{\chi}\cdot
V_{\nu}(\epsilon,\chi(t))\,\partial_{\epsilon}n(\epsilon,t)=\frac{n_{F}(\epsilon)-n(\epsilon,t)}{\tau_{in}},$
(7)

where $\tau_{in}$ is the inelastic relaxation time. Equation (7) can be
derived both phenomenologically and more rigorously using Green’s functions
Liu .

Equations (3)-(7) together with the expression for the density of states
provide a complete description of charge transport through SNS junctions
provided the phase evolution rate is sufficiently slow. Thus, in this regime
the I-U characteristics of the junctions are determined by $\epsilon$ and
$\chi$ dependence of the density of states.

In general, both the current and voltage across junctions exhibit oscillations
in time. We will be interested in the shape of the I-U characteristics
averaged over the period of oscillations, and will indicate the time-averaged
quantities by overline, e.g. $\bar{J}$ , and $\bar{U}$.

In nonmagnetic systems the density of states is invariant under the change of
sign of both $\chi$ and $\bm{H}$, $\nu(\epsilon,\chi,{\bf
H})=\nu(\epsilon,-\chi,-{\bf H})$, which implies

$V_{\nu}(\epsilon,\chi,{\bf H})=-V_{\nu}(\epsilon,-\chi,-{\bf H})$ (8)

for the level sensitivity.

It follows from Eqs. (3)-(8) that in the low bias regime considered here the
I-U characteristics in SNS junctions possesses a symmetry, which is not
present in devices based on normal conductors (see e.g. Ref. Glazman ), and is
independent of the device geometry. Namely, it is invariant under the
simultaneous reversal of the magnetic field, current and voltage,

$\displaystyle\bar{J}(-\bar{U},-{\bf H})=-\bar{J}(\bar{U},{\bf H}).$ (9)

Thus, at low bias one can equivalently define nonreciprocity as part of the
I-U characteristic, which is odd under the reversal of current and voltage for
a fixed magnetic field, $\delta\bar{J}=\bar{J}(\bar{U},{\bf
H})-J(\bar{U},-{\bf H})=\bar{J}(\bar{U},{\bf H})+J(-\bar{U},{\bf H})$.

The nonreciprocity of the I-U characteristics in the low bias regime described
by Eqs. (3)-(8) arises from the odd in $\bf{H}$ dependence of the density of
states, $\delta\nu(\epsilon,\chi,{\bf H})=\nu(\epsilon,\chi,{\bf
H})-\nu(\epsilon,\chi,-{\bf H})$. The latter depends on the orientation of the
${\bf H}$ relative to the current. It is important to note that in the special
case where the influence of ${\bf H}$ on $\nu(\epsilon,\chi,{\bf H})$ reduces
a constant phase shift $\phi({\bf H})$,

$\nu(\epsilon,\chi,{\bf H})=\nu_{0}(\epsilon,\chi+\phi({\bf H})),$ (10)

the I-U characteristics remain reciprocal. The reason is that neither the
average voltage $\bar{U}$ nor the average current $\bar{J}$ are affected by
the phase shift. Furthermore, it follows from Eq. (6) that in this case the
critical current of the junction is also reciprocal Hasan . We note however,
that if different junctions obeying Eq. (10) are connected in parallel
K.C.Fong ; Fominov , the critical current and resistance at current bias are
nonreciprocal.

Figure 1: The I-U characteristics of a nonreciprocal SNS junction at low
voltage bias for opposite signs of the magnetic field are sketched in blue and
green. The dashed lines correspond to the high voltage regime.

The shapes of I-U characteristics of SNS junctions depends on the external
circuits, we will consider below two limiting cases of voltage and current
biased junctions.

_Voltage bias:_ We begin the consideration of nonreciprocity with the voltage
bias setup. The I-U characteristic in this case has an $N$ shape Liu ,
sketched in Fig. 1. At low bias, where the rate of phase evolution obeys the
inequality $\dot{\chi}(t)\tau_{in}\ll 1$ the solution to Eq. (7) may be
expressed as a series in the small parameter $\dot{\chi}(t)\tau_{in}$. To
second order we obtain for the instantaneous current

$\displaystyle J(t,T,{\bf H})=$ $\displaystyle\,J_{s}\big{(}\chi(t),T,{\bf
H}\big{)}+g_{1}\big{(}\chi(t),{\bf
H}\big{)}\big{(}\dot{\chi}(t)\tau_{in}\big{)}$
$\displaystyle+g_{2}\big{(}\chi(t),{\bf
H}\big{)}\big{(}\dot{\chi}(t)\tau_{in}\big{)}^{2},$ (11)

where

$\displaystyle g_{1}(\chi,{\bf H})=$
$\displaystyle\,-2e\int_{0}^{\infty}d\epsilon\partial_{\epsilon}n_{F}(\epsilon)\nu(\epsilon,\chi,{\bf
H})V^{2}_{\nu}(\epsilon,\chi,{\bf H}),$ (12a) $\displaystyle g_{2}(\chi,{\bf
H})=$
$\displaystyle\,e\int^{\infty}_{0}d\epsilon\bigg{\\{}\partial_{\epsilon}n_{F}(\epsilon)\partial_{\chi}\left[\nu(\epsilon,\chi,{\bf
H})V^{2}_{\nu}(\epsilon,\chi,{\bf H})\right]$
$\displaystyle+\partial_{\epsilon}^{2}n_{F}(\epsilon)\nu(\epsilon,\chi,{\bf
H})V_{\nu}^{3}(\epsilon,\chi,{\bf H})\bigg{\\}}.$ (12b)

At $\dot{\chi}(t)=2eU=\mathrm{const}$, the average current can be obtained by
averaging Eq. (Giant nonreciprocity of current-voltage characteristics of
noncentrosymmetric supercondctor-normal metal-superconductor junctions) over
the phase $\chi$. The supercurrent, averages to zero. The linear conductance
arises from the second term on the RHS, and has the form

$\displaystyle G_{l}({\bf H})=2e\tau_{in}\langle g_{1}(\chi,{\bf H})\rangle,$
(13)

where $\langle\ldots\rangle$ denotes averaging over the phase $\chi$. Using
Eqs. (8) and (12a) it is easy to see that the linear conductance is an even
function of ${\bf H}$ in accordance with the Onsager symmetry principle.

The nonreciprocal part of the _dc_ current $\delta\bar{J}(U,{\bf
H})=\bar{J}(U,{\bf H})-\bar{J}(U,-{\bf H})$ arises from the third term on the
RHS of Eq. (Giant nonreciprocity of current-voltage characteristics of
noncentrosymmetric supercondctor-normal metal-superconductor junctions), and
has the form

$\displaystyle\delta\bar{J}(U,{\bf H})$ $\displaystyle=2\,\langle
g_{2}(\chi,{\bf H})\rangle\big{(}2eU\tau_{in}\big{)}^{2}.$ (14)

It follows from Eq. (8) that

$\langle g_{2}(\chi,-{\bf H})\rangle=-\langle g_{2}(\chi,{\bf H})\rangle.$

is an odd function of ${\bf H}$, in agreement with Eq. (9). It is interesting
that the linear conductance in Eq. (13) is proportional to $\tau_{in}$, while
the nonreciprocal current in Eq. (14) is proportional to $\tau^{2}_{in}$.

In the voltage bias setup it is possible to obtain closed-form expressions for
the I-U characteristic for arbitrary values of $2eU\tau_{in}$. The results are
presented in the Appendix. Here we summarize their main features. The
magnitude of the current reaches its maximum at $eU\tau_{in}\sim 1$, and the
values of the maximum current $J_{max}({\bf H})$ and its nonreciprocity
$\delta J_{max}({\bf H})=J_{max}({\bf H})-J_{max}(-{\bf H})$ may be estimated
as

$\displaystyle J_{max}({\bf H})\sim\langle g_{1}(\chi,{\bf
H})\rangle,\quad\delta J_{max}({\bf H})\sim\langle g_{2}(\chi,{\bf
H})\rangle.$ (15)

We note that at sufficiently large temperatures the ciritical current becomes
exponentially small in $T$, while according to Eqs. (15) and (12a), the
maximal current $J_{max}$ does not have such strong temperature dependence.
Therefore, $J_{max}$ can be much larger than $J_{c}$ Liu .

At relatively large voltages $eU\tau_{in}\gg 1$ the solutions of Eqs. (3) -(7)
yield an expression for the current which decreases with voltage,

$\bar{J}(U,{\bf H})\sim\frac{C_{1}({\bf H})}{eU\tau_{in}}+\frac{C_{2}({\bf
H})}{(eU\tau_{in})^{2}}.$ (16)

Here to within a factor of order unity $C_{1}\sim\langle g_{1}(\chi,{\bf
H})\rangle$ and $C_{2}\sim\langle g_{2}(\chi,{\bf H})\rangle$.

Eventually, at large voltages the I-U characteristics are controlled by the
contribution to the current which is beyond the adiabatic approximation and is
of order $G_{N}U$. Here $G_{N}$ is the conductance of the normal metal part of
the junction. As the voltage approaches this regime, the current reaches a
minimum at $U=U_{min}$. At this point, the contribution to the conductance
proportional to $\tau_{in}$ and $\tau_{el}$ are of the same order. The
position of the minimum is approximately given by,

$U_{min}({\bf H})\sim\frac{1}{\tau_{in}}\sqrt{\frac{G_{l}({\bf
H})}{G_{N}}}+\frac{\langle g_{2}({\bf H})\rangle}{\langle g_{1}({\bf
H}))\rangle}{\frac{G_{l}({\bf H})}{G_{N}}}.$ (17)

At $U\gg U_{min}$ the I-U characteristics of the system (including its
nonreciprocity) is roughly the same as that of the normal part of the SNS
junction.

Equations (14)-(17) show that the general key features of nonreciprocity of
the I-U characteristics of voltage-biased junctions are determined by the
parameters $\langle g_{1}(\chi,{\bf H})\rangle$ and $\langle g_{2}(\chi,{\bf
H},)\rangle$. Note that the first term on the HRS of Eq. (12) is a total
derivative and averages to zero. In many physical situations the relevant
quasiparticle energies are of order of the Thouless energy $E_{T}$. In this
case, for $T\gg E_{T}$ Eqs. (12) simplify to

$\displaystyle\langle g_{1}(\chi,{\bf H})\rangle=$
$\displaystyle\,\frac{e}{2T}\int_{0}^{\infty}d\epsilon\langle\nu(\epsilon,\chi,{\bf
H})V^{2}_{\nu}(\epsilon,\chi,{\bf H})\rangle,$ (18a) $\displaystyle\langle
g_{2}(\chi,{\bf H})\rangle=$ $\displaystyle\,\frac{e}{32\pi
T^{3}}\int^{\infty}_{0}d\epsilon\,\epsilon\,\langle\nu(\epsilon,\chi,{\bf
H})V_{\nu}^{3}(\epsilon,\chi,{\bf H})\rangle.$ (18b)

and we get

$\frac{\langle g_{2}(\chi,{\bf H},T)\rangle}{\langle
g_{1}(\chi,T)\rangle}=\gamma({\bf H})\frac{E_{T}^{2}}{T^{2}},$ (19)

where $\gamma({\bf H})$ is an odd function of ${\bf H}$, which is determined
by the sensitivity of the phase-dependent quasiparticle spectrum on the
magnetic field. The different temperature dependence of $\langle
g_{1}(\chi,{\bf H})\rangle$ and $\langle g_{2}(\chi,{\bf H})\rangle$ in Eq.
(18) arises because the second term in Eq. (12) requires going to higher order
in the Sommerfeld expansion.

_Current bias:_ Let us now turn to the consideration of nonreciprocity in the
current-bias setup. The shape of the I-U characteristic in this case Liu is
illustrated in Fig. 2. The nonreciprocity of the critical current $J_{c}$ of
SNS junctions has been studied in several articles Hasan ; Baumgartner ; Ando
; Fu1 ; Lotfizadeh ; Efetov ; Edelstein ; Baumartner1 ; Turini ; Lin ;
Nagaosa1 ; Fu2 ; LiyandaGeller ; Zutich ; Ilie . Here we study the I-U
characteristics $\bar{U}(J)$ at currents larger than the critical current.

Similarly to the voltage-biased case, the dissipative part of the I-U
characteristics can be divided into two parts. At $\bar{U}>1/\tau_{in}$ and
$\bar{U}<1/\tau_{in}$ it is controlled by the elastic and inelastic relaxation
times respectively. The description of the I-U characteristics in the
adiabatic approximation is valid at relatively small average voltages
$\bar{U}<1/\tau_{in}$ when $J<J_{jump}$ Liu . At higher currents,
$J>J_{jump}$, the voltage increases dramatically. In this regime the
dissipation is determined by the elastic relaxation time, and the shape of
this part of the I-U characteristics is, roughly speaking, the same as in the
normal metal. Therefore, similar to the case of current-biased junctions, the
degree of nonreciprocity in this region is expected to be relatively small.

Figure 2: A qualitative picture of the I-U characteristics of a nonreciprocal
SNS junction at fixed current. The blue and green curves correspond to the low
voltage regime regime of the I-U characteristics for opposite signs of
magnetic field. The dashed lines correspond to the high voltage regime.

In the current bias setup the phase difference $\chi(t)$ increases
monotonically with time, but at a varying rate. Using the Josephson relation
Eq. (3) we can relate the average voltage across the junction to the duration
$t_{p}$ of the time interval during which $\chi$ increases by $2\pi$,

$\bar{U}=\frac{\pi}{et_{p}}.$ (20)

To evaluate $t_{p}$, we must determine the dynamics of $\chi(t)$.

When the current is close to the critical current, $J-J_{c}({\bf H})\ll
J_{c}({\bf H})$, the dominant contribution to the period $t_{P}$ comes from
the interval of phase where $\chi(t)$ is near the phase $\chi_{m}({\bf H})$,
at which the supercurrent reaches its maximum value (see Liu ). In this
interval $\dot{\chi}(t)\ \ll 1/\tau_{in}$, and Eqs. (Giant nonreciprocity of
current-voltage characteristics of noncentrosymmetric supercondctor-normal
metal-superconductor junctions)-(12) can be used, leading to

$t_{p}\approx\tau_{in}\int^{\pi}_{-\pi}\frac{g_{1}(\chi_{m}({\bf H}),{\bf
H})d\chi}{J-J_{s}(\chi,{\bf H},T)}.$ (21)

Using the quadratic phase dependence of $J_{s}(\chi,{\bf H},T)$ near
$\chi=\chi_{m}({\bf H})$, $J_{s}(\chi,{\bf H},T)\approx J_{c}({\bf
H},T)+\frac{(\chi-\chi_{m}({\bf
H}))^{2}}{2}\partial^{2}_{\chi}\left.J_{s}(\chi,{\bf
H},T)\right|_{\chi=\chi_{m}({\bf H})}$ we can express the average voltage as

$\bar{U}(J,{\bf H})\approx A({\bf H})\sqrt{(J-J_{c}({\bf H},T))},$ (22)

where

$\displaystyle A({\bf H})=$
$\displaystyle\,\frac{\sqrt{-\partial^{2}_{\chi}J_{s}(\chi,{\bf
H},T)|_{\chi=\chi_{m}({\bf H})}}}{2^{5/2}e\tau_{in}g_{1}(\chi_{m}({\bf
H}),{\bf H})}.$ (23)

The voltage in Eq. (22) has the standard square root dependence on the excess
current $J-J_{c}({\bf H},T)$. The nonreciprocity in this regime is
characterized by the nonreciprocity of the critical current $J_{c}({\bf H},T)$
and the coefficient $A({\bf H})$. Equation (23) shows that nonreciprocity of
$A({\bf H})$ is determined not only by the sensitivity of the phase-dependent
supercurrent to the magnetic field but also by the dissipative parameter
$g_{1}(\chi)$. Indeed, the phase $\chi_{m}({\bf H})$ at which the supercurrent
attains the maximal value in the direction of the bias shifts linearly with
${\bf H}$. Since $g_{1}(\chi)$ does not have an extremum at $\chi_{m}({\bf
H})$ the denominator in Eq. (22) contributes to nonreciprocity of $A({\bf
H})$.

It was shown in Ref. Liu , that

$J_{jump}\sim J_{max},$ (24)

where $J_{max}$ was determined for the voltage bias case. Interestingly, it
follows from our equations that at $T>E_{T}$ the degree of nonreciprocity for
current bias, $\delta J_{jump}=J_{jump}({\bf H})-J_{jump}(-{\bf H})$, is
enhanced with respect to the voltage bias case, Eqs. (15),(19), by the factor
$T^{2}/E_{T}^{2}$

$\displaystyle\frac{\delta J_{jump}}{\delta
J_{max}}\sim\left\\{\begin{array}[]{cc}1,&T\lesssim E_{T},\\\
\frac{T^{2}}{E_{T}^{2}},&T>E_{T}.\end{array}\right.$ (27)

The reason for this can be traced to the non-constant phase evolution rate
$\dot{\chi}(t)$. Because of this the leading term in the Sommerfeld expansion
of $g_{2}(\chi,{\bf H})$ (first term in Eq. (12)) can no longer be written as
a total derivative of some function of $\chi$, and its contribution to the
average voltage does not vanish.

The results presented above assume that the low-energy quasiparticles are
trapped inside the normal region of the junction by insulating boundaries and
Andreev reflection from the S-N boundaries, and the only channel of
quasiparticle relaxation is inelastic scattering. In situations where escape
of quasiparticles from the normal region is possible, $\tau_{in}$ should be
replaced by the characteristic time of quasiparticle escape from the normal
region.

The above consideration can also be extended to the cases where the
nonreciprocity of the I-U characteristics is associated with existence of a
spontaneous magnetization ${\bf M}$ (or spontaneous valley symmetry breaking
in twisted graphene Efetov ; Lin ; Scammell ) in the normal region of the
junction at ${\bf H}=0$. In this case the nonreciprocal part of the I-U
characteristics can be expressed in terms of $\delta\nu=\nu(\epsilon,\chi,{\bf
M})-\nu(\epsilon,\chi,-{\bf M})$.

To estimate the magnitude of the effect, below we apply the general results
obtained above to a planar junction of length $L$ and width $L_{1}$(shown in
Fig. 3), in which the normal region is described by the following Hamiltonian,

$\displaystyle H$ $\displaystyle={\bf p}^{2}/2m-E_{F}+\beta^{\alpha
i}p^{i}\sigma^{\alpha}+V_{imp}({\bf r})+g\mu_{0}{\bf H}\cdot{\bm{\sigma}}.$
(28)

Here $E_{F}$ is the Fermi energy, $m$ is the electron mass, $\sigma_{i}$ are
the Pauli matrices in spin space, $g$ is the g-factor, $\mu_{0}$ is the Bohr
magneton, and $V_{imp}({\bf r})$ is the random impurity potential. For Rashba
spin-orbit coupling $\beta^{\alpha i}=\alpha_{R}\epsilon^{\alpha
ij}\hat{n}^{j}$, where $\hat{n}$ is a unit polar vector, and for Dresselhaus
spin-orbit coupling $\beta^{\alpha i}=\alpha_{D}\delta^{\alpha i}$.

The direction of the magnetic field is chosen to be parallel to the film, as
depicted in Fig. 3. Therefore, it enters the Hamiltonian, Eq. (28), only via
the Zeeman term.

Below we will focus on linear in ${\bf H}$ contribution to the nonreciprocity
of the I-U characteristics. We consider a case of weak spin-orbit coupling
$\beta p_{F}\ll\tau_{el}^{-1}$ and focus on the diffusive regime,
$L\gg\sqrt{D\tau_{so}}$. Here $D=v_{F}^{2}\tau_{el}/2$ is the typical value of
the electron diffusion coefficient in the normal region, $\tau_{so}$ is the
spin relaxation time, and $v_{F}$ is the Fermi-velocity. We also assume that
the distance between the superconductors, $L$, is much larger than the
coherence length in the superconductors, and therefore the order parameter
$\Delta({\bf r})$ has a constant modulus $\Delta$ in the superconducting
leads, and vanishes in the normal region, see Fig. 3.

Figure 3: Top down view of a planar SNS junction. The junction is aligned
along the $\hat{\bf x}$ direction, there is a parallel magnetic field ${\bf
H}$ directed in the $\hat{\bf y}$ direction, and there is an out of plane
vector $\hat{\bf n}$ pointing in the $\hat{\bf z}$ direction which breaks
inversion symmetry.

After averaging over the random impurity potential, the density of states in
the SNS junction $\nu(\epsilon,\chi,{\bf H})$ is obtained by solving Usadel’s
equation in the presence of spin-orbit coupling and a magnetic field Loss ;
tokaltly ; Ilie . Here we present the main results leaving the details of the
calculations to Appendix B .

The main feature of the density of states of a diffusive SNS junction is the
existence of a mini-gap at $\chi=0$ of order $E_{T}\sim D/L^{2}$ Atland ;
Melsen ; Frahm ; ZhouSpivak (for simplicity we restrict ourselves to the case
where the S-N boundaries of the junction are transparent). The density of
states $\nu(\epsilon,\chi,{\bf H})$ exhibits a significant $\chi$-dependence
only for energies of the order of the mini-gap. This means that the level
sensitivity, Eq. (5), is peaked in the energy interval $\epsilon\gtrsim
E_{T}$.

We show in Appendix B that if the diffusion coefficient $D(x)$ and the
strength of spin-orbit coupling $\beta^{\alpha x}(x)$ depend only on the $x$
coordinate, the density of states at ${\bf H}\neq 0$ can be written in the
form Eq. (10) with

$\phi({\bf H})=\int^{L/2}_{-L/2}dx\frac{2\tau_{so}g\mu_{0}\beta^{\alpha
x}(x){\bf H}^{\alpha}}{D(x)}.$ (29)

Therefore, in this idealized 1D model the I-U characteristics are reciprocal.
However, in the general case where $D({\bf r})$ and $\beta({\bf r})$ are
functions of the two coordinates, $x$ and $y$, or the shape of the normal
metal part of the junction is not rectangular, the I-U characteristics of the
junction are non-reciprocal.

Below we estimate the degree of non-reciprocity of the I-U characteristics in
the case where $L\lesssim L_{1}$, the amplitude of fluctuations the diffusion
coefficient in the y-direction is of order $\delta D$, and the correlation
length of such fluctuations is of order $L_{1}$. In this case we get the
following estimates for $\langle g_{1}\rangle$ and $\langle g_{2}\rangle$,

$\displaystyle\langle g_{1}\rangle$ $\displaystyle\sim
e\nu_{N}\frac{E_{T}^{3}}{T},\quad\langle g_{2}\rangle\sim\langle
g_{1}\rangle\frac{\beta(g\mu_{0}H)\tau_{so}E_{T}}{LT^{2}}\bigg{(}\frac{\delta
D}{D}\bigg{)}^{2},$ (30)

where $\tau_{so}^{-1}=4p_{F}^{2}\beta^{2}\tau_{el}$. For the nonreciprocal
part $\delta A\equiv[A({\bf H})-A(-{\bf H})]$ of the coefficient $A({\bf H})$
in Eq. (23) we find

$\begin{split}\delta A\sim\frac{\sqrt{J_{c}(0,T)}}{e\tau_{in}\langle
g_{1}\rangle}\frac{\tau_{so}\beta(g\mu_{0}H)}{LE_{T}}\bigg{(}\frac{\delta
D}{D}\bigg{)}^{2}.\end{split}$ (31)

The parameters in Eqs. (30)–(31) together with Eqs. (14) and (22) characterize
the degree of nonreciprocity of I-U characteristics of SNS junctions with weak
spin-orbit coupling in both the voltage and current bias cases.

We note that transport in diffusive junctions in which the normal region is
formed by a surface of a topological insulator can also be analysed using
Usadel’s equation. This is done in Appendix B. The magnitude of nonreciprocity
in this case is obtained by setting in Eqs. (30) and (31)
$\tau_{so}\to\tau_{el}$ and $\beta\to v$, where $v$ is the velocity of the
relativistic dispersion.

We conclude by summarizing our main results. We have shown that at low bias
the nonreciprocal features of the I-U characteristics in SNS junctions can be
expressed in terms of the nonreciprocal part of the density of states,
$\delta\nu=\nu(\epsilon,\chi,{\bf H})-\nu(\epsilon,\chi,-{\bf H})$. This leads
to an a symmetry not present for normal conductors; the I-U characteristics
remain invariant under a simultaneous reversal of curr ent, voltage and
magnetic field. At low bias all features of I-U characteristic are controlled
by the inelastic relaxation time. This leads to a much stronger nonreciprocity
in comparison to normal conductors. Although the maximal current $J_{max}$ in
the voltage bias setup and the current $J_{jump}$ at which the voltage rapidly
rises in the current bias setup are of the same order, their nonreciprocity
turn out to be parametrically different at $T>E_{T}$.

This work of T. L. and A. A. was supported by the US National Science
Foundation through the MRSEC Grant No. DMR-1719797, the Thouless Institute for
Quantum Matter (TIQM), and the College of Arts and Sciences at the University
of Washington.

## References

  * (1) L. Onsager,  Reciprocal relations in irreversible processes. I,  Phys. Rev. 37, 405 – Published 15 February 1931
  * (2) L. Onsager,  Reciprocal relations in irreversible processes. II,  Phys. Rev. 38, 2265 – Published 15 December 1931
  * (3) G. L. J. A. Rikken, J. Fölling, and P. Wyder,  Electrical Magnetochiral Anisotropy, Phys. Rev. Lett. 87, 236602 – Published 14 November 2001
  * (4) E. L. Ivchenko and B. Spivak,  Chirality effects in carbon nanotubes, Phys. Rev. B 66, 155404 – Published 2 October 2002.
  * (5) V. Krstic, S. Roth, M. Burghard, K. Kern, and G. L. J. A. Rikken,  Magneto-chiral anisotropy in charge transport through single-walled carbon nanotubes,  The Journal of Chemical Physics 117, 11315 (2002).
  * (6) J. Wei, M. Shimogawa, Z. Wang, I. Radu, R. Dormaier, and D. H. Cobden,  Magnetic-field asymmetry of nonlinear transport in carbon nanotubes,  Phys. Rev. Lett. 95, 256601 (2005).
  * (7) T. Ideue, K. Hamamoto, S. Koshikawa, M. Ezawa, S. Shimizu, Y. Kaneko, Y. Tokura, N. Nagaosa, and Y. Iwasa,  Bulk rectification effect in a polar semiconductor,  Nature Physics 13, 578 (2017).
  * (8) T. Morimoto, N. Nagaosa,  Nonreciprocal current from electron interactions in noncentrosymmetric crystals: roles of time reversal symmetry and dissipation,  Sci Rep 8, 2973 (2018).
  * (9) G. L. J. A. Rikken and N. Avarvari,  Strong electrical magnetochiral anisotropy in tellurium,  Phys. Rev. B 99, 245153 (2019).
  * (10) H.F. Legg, M. Roßler, F. Munning, Dingxun Fan, O. Breunig, A. Bliesener, G. Lippertz, A. Uday, A.A. Taskin, D. Loss, J. Klinovaja, and Y. Ando,  Giant magnetochiral anisotropy from quantum-confined surface states of topological insulator nanowires,  Nature Nanotechnology 17, 696 (2022).
  * (11) A. I. Larkin and Yu. N. Ovchinnikov,  Tunnel effect between superconductors in an alternating field.  Sov. Phys. JETP 24, 1035 (1967).
  * (12) S. N. Artemenko, A. F. Volkov, and A. V. Zaitsev,  Theory of nonstationary Josephson effect in short superconducting contacts.  Sov. Phys. JETP 49, 924, 1979.
  * (13) M. Tinkham,  _Introduction to superconductivity_ ,  Courier Corporation, 1986.
  * (14) D. Averin and A. Bardas,  Adiabatic dynamics of super- conducting quantum point contacts,  Phys. Rev. B 53, R1705 (1996).
  * (15) T. Liu, A.V. Andreev, B.Z. Spivak,  Current–voltage characteristics of superconductor-normal metal-superconductor junctions,  Annals of Physics, 2023, 169327, ISSN 0003-4916.
  * (16) V. M. Edelstein,  Characteristics of the Cooper pairing in two-dimensional noncentrosymmetric electron systems,  J. Exp. Theor. Phys. 68, 1244 (1989).
  * (17) F. Ando, Y. Miyasaka, T. Li, J. Ishizuka, T. Arakawa, Y. Shiota, T. Moriyama, Y. Yanase, and T. Ono,  Observation of superconducting diode effect,  Nature 584, 373 (2020).
  * (18) J. Diez-Merida, A. Diez-Carlon, S. Y. Yang, Y. M. Xie, X. J. Gao, K. Watanabe, T. Taniguchi, X. Lu, K. T. Law, and D. K. Efetov,  Symmetry-broken Josephson junctions and superconducting diodes in magic-angle twisted bilayer graphene,  _Nat Commun_ 14, 2396 (2023).
  * (19) C. Baumgartner, L. Fuchs, A. Costa, J. Pico-Cortes, S. Reinhardt, S. Gronin, G. C. Gardner, T. Lindemann, M. J. Manfra, P. E. F. Junior, D. Kochan, J. Fabian, N. Paradiso, and C. Strunk,  Effect of Rashba and Dresselhaus spin-orbit coupling on supercurrent rectification and magnetochiral anisotropy of ballistic Josephson junc-tions,  J. Phys.: Condens. Matter 34, 154005 (2022).
  * (20) L. Bauriedl, C. Bauml, L. Fuchs, C. Baumgartner, N. Paulik, J. M. Bauer, K.-Q. Lin, J. M. Lupton, T. Taniguchi, K. Watanabe, C. Strunk, and N. Paradiso,  Supercurrent diode effect and magnetochiral anisotropy in few-layer NbSe2,  Nat. Commun. 13, 4266 (2022).
  * (21) B. Turini, S. Salimian, M. Carrega, A. Iorio, E. Stram- bini, F. Giazotto, V. Zannier, L. Sorba, and S. Heun,  Josephson Diode Effect in High-Mobility InSb Nanoflags,  Nano Lett. 22, 8502 (2022).
  * (22) J.-X. Lin, P. Siriviboon, H. D. Scammell, S. Liu, D. Rhodes, K. Watanabe, T. Taniguchi, J. Hone, M. S. Scheurer, and J. I. A. Li,  Zero-field superconduct- ing diode effect in small-twist-angle trilayer graphene,  Nat. Phys. 18, 1221 (2022).
  * (23) S. Ilić and F. S. Bergeret,  Theory of the Supercurrent Diode Effect in Rashba Superconductors with Arbitrary Disorder,  Phys. Rev. Lett. 128, 177001 – Published 27 April 2022
  * (24) N. F. Q. Yuan and L. Fu,  Supercurrent diode ef- fect and finite-momentum superconductors,  PNAS 119, e2119548119 (2022).
  * (25) B. Pal, A. Chakraborty, P. K. Sivakumar, M. Davydova, A. K. Gopi, A. K. Pandeya, J. A. Krieger, Y. Zhang, M. Date, S. Ju, N. Yuan, N. B. M. Schroter, L. Fu, and S. S. P. Parkin,  Josephson diode effect from Cooper pair momentum in a topological semimetal,  Nat. Phys. 18, 1228 (2022).
  * (26) Jaglul Hasan, Konstantin N. Nesterov, Songci Li, Manuel Houzet, Julia S. Meyer, and Alex Levchenko,  Anomalous Josephson effect in planar noncentrosymmetric superconducting devices,  Phys. Rev. B 106, 214518 – Published 20 December 2022
  * (27) Baumgartner, C., Fuchs, L., Costa, A. et al. ,  Supercurrent rectification and magnetochiral effects in symmetric Josephson junctions,  Nat. Nanotechnol. 17, 39–44 (2022).
  * (28) N. Lotfizadeh, B. Pekerten, P. Yu, W. Strickland, A. Matos-Abiague, and J. Shabani,  Superconducting Diode Effect Sign Change in Epitaxial Al-InAs Josepshon Junctions,  arXiv:2303.01902 (2023).
  * (29) D. Kochan, A. Costa, I. Zhumagulov, I. Žutić ,  Phenomenological Theory of the Supercurrent Diode Effect: The Lifshitz Invariant,  arXiv:2303.11975 (2023).
  * (30) T. de Picoli, Z. Blood, Y. Lyanda-Geller, and J. I. Vayrynen,  Superconducting diode effect in quasi-one- dimensional systems,  arXiv:2302.04277 (2023).
  * (31) Naiyuan James Zhang1, Jiang-Xiazi Lin1, Yibang Wang1, Kenji Watanabe2, Takashi Taniguchi3, Liang Fu4, and J.I.A. Li1  Momentum-polarized superconductivity in twisted trilayer graphene,  arXiv:2209.12964v4 (2023).
  * (32) Peter Debye,  _Polar molecules_.  Dover Publ., 1970.  Google-Books-ID: f70ingEACAAJ.
  * (33) L. D. Landau and E. M. Lifshitz,  _Fluid Mechanics_.  Elsevier, October 2013.  ISBN 978-1-4831-4050-6.  Google-Books-ID: CeBbAwAAQBAJ.
  * (34) M. Pollak and T. H. Geballe,  Low-Frequency Conductivity Due to Hopping Processes in Silicon,  Physical Review, 122(6):1742–1753, June 1961.
  * (35) M. Smith, A. V. Andreev, and B. Z. Spivak,  Debye mechanism of giant microwave absorption in superconductors.  Phys. Rev. B 101, 134508 – Published 20 April 2020
  * (36) M. Smith, A. V. Andreev, and B. Z. Spivak,  Giant magnetoconductivity in noncentrosymmetric superconductors.  Phys. Rev. B 104, L220504 – Published 6 December 2021
  * (37) M. Smith, A.V. Andreev, B.Z. Spivak, Giant microwave absorption in s- and d- wave superconductors, Annals of Physics, Volume 417, 2020, 168105, ISSN 0003-4916.
  * (38) Beenakker, C.W.J. (1992).  Three “Universal” Mesoscopic Josephson Effects. In: Fukuyama, H., Ando, T. (eds) Transport Phenomena in Mesoscopic Systems. Springer Series in Solid-State Sciences, vol 109. Springer, Berlin, Heidelberg.
  * (39) A. V. Andreev, and L. I. Glazman,  Nonlinear magnetoconductance of a classical ballistic system,  Phys. Rev. Lett. 97, 266806 (2006).
  * (40) Chui-Zhen Chen, James Jun He, Mazhar N. Ali, Gil-Ho Lee, Kin Chung Fong, and K. T. Law,  Asymmetric Josephson effect in inversion symmetry breaking topological materials,  Phys. Rev. B 98, 075430 – Published 27 August 2018
  * (41) Ya. V. Fominov and D. S. Mikhailov,  Asymmetric higher-harmonic SQUID as a Josephson diode,  Phys. Rev. B 106, 134514 – Published 27 October 2022
  * (42) Harley D Scammell, J I A Li, and Mathias S Scheurer,  Theory of zero-field superconducting diode effect in twisted trilayer graphene,  2D Mater. 9 (2022) 025027
  * (43) I. V. Tokatly,  Usadel equation in the presence of intrinsic spin-orbit coupling: A unified theory of magnetoelectric effects in normal and superconducting systems,  Phys. Rev. B 96, 060502(R) – Published 2 August 2017
  * (44) A. Altland and M. R. Zirnbauer,  Random Matrix Theory of a Chaotic Andreev Quantum Dot,  Phys. Rev. Lett. 76, 3420 (1996).
  * (45) K. M. Frahm, P. W. Brouwer, J. A. Melsen, and C. W. J. Beenakker,  Effect of the Coupling to a Superconductor on the Level Statistics of a Metal Grain in a Magnetic Field,  Phys. Rev. Lett. 76, 2981 – Published 15 April 1996
  * (46) J. A. Melsen, P. W. Brouwer, K. M. Frahm, and C. W. J. Beenakker,  Superconductor-proximity effect in chaotic and integrable billiards,  Physica Scripta T 69,223-225(1997).
  * (47) Zhou, F., Charlat, P., Spivak, B. et al. ,  Density of States in Superconductor-Normal Metal-Superconductor Junctions,  Journal of Low Temperature Physics 110, 841–850 (1998).
  * (48) I. V. Tokatly,  _Usadel equation in the presence of intrinsic spin-orbit coupling: A unified theory of magnetoelectric effects in normal and superconducting systems,_ Phys. Rev. B 96, 060502(R)
  * (49) G. Eilenberger, _Transformation of Gorkov’s equation for type II superconductors into transport-like equations_ , Z. Physik 214, 195 (1968).

Supplemental Material: Giant nonreciprocity of current-voltage characteristics
of noncentrosymmetric supercondctor-normal metal-superconductor junctions

##  Appendix A Appendix A: Derivation of general expression for the current of
voltage-biased junctions

To obtain a general expression for the current in the fixed voltage case, it
is useful introduce the variable N, defined as the density of states
integrated up to energy $\epsilon$,

$N=\int^{\epsilon(N,\chi(t),{\bf
H})}_{0}d\tilde{\epsilon}\nu(\tilde{\epsilon},\chi(t),{\bf H}),$ (A1)

and make a change of variables from $\epsilon$ to $N$ in Eqs. (4)–(7).
Following this change of variables, the function
$\epsilon\big{(}N,\chi(t),{\bf H}\big{)}$ is to be interpreted as the energy
of the Nth level counting from $\epsilon=0$. This change of variables is
similar to changing from Euler to Lagrange variables in hydrodynamics.
Equation (7) written with the variables $\big{(}N,\chi(t),{\bf H}\big{)}$ has
the form,

$\partial_{t}n(N,t)=\frac{n_{F}\big{(}\epsilon(N,\chi(t),{\bf
H})\big{)}-n(N,\chi(t))}{\tau_{in}}.$ (A2)

At $t\gg\tau_{in}$ the solution to Eq. (A2) is insensitive to initial
conditions,

$n(N,t)=\int^{\infty}_{0}\frac{d\tau}{\tau_{in}}e^{-\frac{\tau}{\tau_{in}}}n_{F}\big{(}\epsilon\big{(}N,\chi(t-\tau),{\bf
H}\big{)}\big{)}.$ (A3)

Substituting it into Eq. (4) and averaging over the period of oscillations, we
get the following expression for the average current,

$\bar{J}(U,{\bf
H})=\frac{e}{\pi}\int^{\infty}_{0}dN\int^{\pi}_{-\pi}d\chi\partial_{\chi}\epsilon(N,\chi,{\bf
H})\int^{\infty}_{0}\frac{d\phi}{\lambda}e^{-\phi/\lambda}n_{F}\big{(}\epsilon\big{(}N,\chi-\mathrm{sgn}(U)\phi,{\bf
H}\big{)}\big{)}.$ (A4)

Here $\lambda=|2eU\tau_{in}|$, and we changed the integral of $\tau$ to an
integral over $\phi=|2eU\tau|$. Since we have converted the integrals over
time to integrals over phase, we omit the time dependence of $\chi(t)$ for the
rest of the appendix.

In the regime $E_{T}/T\ll 1$, where the typical distance of level motion is
small compared to the temperature, we can expand the Fermi function about the
average energy of the Nth energy level $\bar{\epsilon}(N,{\bf
H})=1/2\pi\int^{\pi}_{-\pi}d\chi\epsilon(N,\chi,{\bf H})$. Keeping terms up to
second order in deviations from the average value, we substitute this
expansion into Eq. (A4) and obtain the following expression for the current,

$\displaystyle\bar{J}(U,{\bf H})=$
$\displaystyle\,\frac{e}{\pi}\int^{\infty}_{0}dN\int^{\pi}_{-\pi}d\chi\partial_{\chi}\epsilon(N,\chi,{\bf
H})\int^{\infty}_{0}\frac{d\phi}{\lambda}e^{-\phi/\lambda}\bigg{(}n_{F}\big{(}N,\bar{\epsilon}(N,{\bf
H})\big{)}$
$\displaystyle+\partial_{\epsilon}n_{F}\big{(}\bar{\epsilon}(N,{\bf
H})\big{)}\big{(}\epsilon(N,\chi-\mathrm{sgn}(U)\phi,{\bf
H})-\bar{\epsilon}(N,{\bf H})\big{)}$
$\displaystyle+\frac{1}{2}\partial^{2}_{\epsilon}n_{F}\big{(}\bar{\epsilon}(N,{\bf
H})\big{)}\big{(}\epsilon(N,\chi-\mathrm{sgn}(U)\phi,{\bf
H})-\bar{\epsilon}(N,{\bf H})\big{)}^{2}\bigg{)}.$ (A5)

To simplify this expression, it is useful to define the Fourier expansion of
$\epsilon(N,\chi)$,

$\epsilon(N,\chi,{\bf H})=\bar{\epsilon}(N,{\bf H})+\sum_{k}C_{k}(N,{\bf
H})e^{ik\chi}.$ (A6)

The coefficients must satisfy $C_{k}(N,{\bf H})=C^{*}_{-k}(N,{\bf H})$ since
the LHS is real. Furthermore, time reversal symmetry ensures that
$\epsilon(N,\chi,{\bf H})=\epsilon(N,-\chi,-{\bf H})$. As a result, we also
have $C_{k}(N,{\bf H})=C_{-k}(N,-{\bf H})$.

Substituting this into the expression for the current and evaluating the
integrals of $\phi$, we get

$\displaystyle\bar{J}(U,{\bf H})=$
$\displaystyle(2e)\mathrm{sgn}(U)\int^{\infty}_{0}dN\partial_{\epsilon}n_{F}\big{(}\bar{\epsilon}(N,{\bf
H})\big{)}\sum_{k}|C_{k}(N,{\bf
H})|^{2}\bigg{(}\frac{-k^{2}\lambda}{1+(k\lambda)^{2}}\bigg{)}$
$\displaystyle-e\int^{\infty}_{0}dN\partial^{2}_{\epsilon}n_{F}\big{(}\bar{\epsilon}(N,{\bf
H})\big{)}\sum_{k_{1},k_{2}}C_{k_{1}}(N,{\bf H})C_{k_{2}}(N,{\bf
H})C_{-k_{1}-k_{2}}(N,{\bf H})$
$\displaystyle\times(k_{1}+k_{2})\bigg{(}\frac{i+\mathrm{sgn}(U)\lambda(k_{1}+k_{2})}{1+(k_{1}+k_{2})^{2}\lambda^{2}}\bigg{)}.$
(A7)

Let us separate the average nonlinear current into the reciprocal,
$\bar{J}_{rec}(U,{\bf H})$, and nonreciprocal, $\delta\bar{J}(U,{\bf H})$
parts,

$\displaystyle\bar{J}_{\mathrm{rec}}(U,{\bf H})=$
$\displaystyle\,\frac{1}{2}[\bar{J}(U,{\bf H})+\bar{J}(U,-{\bf
H})],\quad\delta\bar{J}(U,{\bf H})=\bar{J}(U,{\bf H})-\bar{J}(U,-{\bf H}).$
(A8)

To lowest order in $E_{T}/T$, the reciprocal part is given by,

$\bar{J}_{\mathrm{rec}}(U,{\bf
H})=2e\int^{\infty}_{0}dN\partial_{\epsilon}n_{F}\big{(}\bar{\epsilon}(N,{\bf
H})\big{)}\sum_{k}|C_{k}(N,{\bf
H})|^{2}\bigg{(}\frac{-k^{2}\lambda}{1+(k\lambda)^{2}}\bigg{)}.$ (A9)

The nonreciprocal part of the current is given by,

$\displaystyle\delta\bar{J}(U,{\bf H})=$ $\displaystyle
2e\int^{\infty}_{0}dN\partial^{2}_{\epsilon}n_{F}\big{(}\bar{\epsilon}(N,{\bf
H})\big{)}\sum_{k_{1},k_{2}}C_{k_{1}}(N,{\bf H})C_{k_{2}}(N,{\bf
H})C_{-k_{1}-k_{2}}(N,{\bf
H})\bigg{(}\frac{-i(k_{1}+k_{2})^{3}\lambda^{2}}{1+(k_{1}+k_{2})^{2}\lambda^{2}}\bigg{)}.$
(A10)

Note that the nonreciprocal current contains 2 derivatives of the Fermi
function and is proportional to $(E_{T}/T)^{2}$, as opposed to the reciprocal
current which is proportional to $E_{T}/T$.

At small voltages $\lambda\ll 1$, the expressions for the reciprocal and
nonreciprocal current can be simplified. To lowest order in $\lambda$, the
reciprocal current is given by ,

$\displaystyle\bar{J}_{\mathrm{rec}}(U,{\bf H})$
$\displaystyle=(2e)\mathrm{sgn}(U)\int^{\infty}_{0}dN\partial_{\epsilon}n_{F}\big{(}\bar{\epsilon}(N,{\bf
H})\big{)}\sum_{k}|C_{k}(N,{\bf H})|^{2}\big{(}-k^{2}\lambda\big{)}$
$\displaystyle=2eU\tau_{in}\int^{\infty}_{0}dN\partial_{\epsilon}n_{F}\big{(}\bar{\epsilon}(N,{\bf
H})\big{)}\frac{1}{2\pi}\int^{\pi}_{-\pi}d\chi\big{(}\partial_{\chi}\epsilon(N,\chi,{\bf
H})\big{)}^{2}.$ (A11)

Similarly, the nonreciprocal current is given by

$\displaystyle\delta\bar{J}(U,{\bf H})=$ $\displaystyle
2e\int^{\infty}_{0}dN\partial^{2}_{\epsilon}n_{F}\big{(}\bar{\epsilon}(N,{\bf
H})\big{)}\sum_{k_{1},k_{2}}C_{k_{1}}(N,{\bf H})C_{k_{2}}(N,{\bf
H})C_{-k_{1}-k_{2}}(N,{\bf H})\bigg{(}(i(k_{1}+k_{2})^{3}\lambda^{2})\bigg{)}$
$\displaystyle=$ $\displaystyle
2e(2eU\tau_{in})^{2}\int^{\infty}_{0}dN\partial^{2}_{\epsilon}n_{F}\big{(}\bar{\epsilon}(N,{\bf
H})\big{)}\frac{1}{2\pi}\int^{\pi}_{-\pi}d\chi\big{(}\partial_{\chi}\epsilon(N,\chi,{\bf
H})\big{)}^{3}.$ (A12)

Changing variables back to $(\epsilon,\chi)$, we arrive at Eqs. (12) and (13).
To estimate the reciprocal and nonreciprocal current at relatively large
voltages, $1/\tau_{in}\ll|2eU|\ll E_{T}$, we use the fact that for a typical
phase-dependence of the quasiparticle spectrum, the sums over $k$ in Eqs. (A9)
and (A10) are dominated by $k$ of order unity. With this in mind, we obtain
the following estimates

$\displaystyle\bar{J}_{\mathrm{rec}}(U,{\bf H})\sim$
$\displaystyle\frac{1}{U\tau_{in}}\int^{\infty}_{0}dN\partial_{\epsilon}n_{F}\big{(}\bar{\epsilon}(N,{\bf
H})\big{)}\sum_{k}|C_{k}(N,{\bf H})|^{2}(-k^{2})\sim\frac{\langle
g_{1}(\chi,{\bf H})\rangle}{eU\tau_{in}},$ (A13)
$\displaystyle\delta\bar{J}(U,{\bf H})\sim$
$\displaystyle\frac{-ie}{(2eU\tau_{in})^{2}}\int^{\infty}_{0}dN\partial^{2}_{\epsilon}n_{F}\big{(}\bar{\epsilon}(N,{\bf
H})\big{)}\sum_{k_{1},k_{2}}C_{k_{1}}(N,{\bf H})C_{k_{2}}(N,{\bf
H})C_{-k_{1}-k_{2}}(N,{\bf H})\sim\frac{\langle g_{2}(\chi,{\bf
H})\rangle}{(eU\tau_{in})^{2}}.$ (A15)

Combining Eqs. (A13) and (A15) we get Eq. (15) in the maintext.

The current as a function of the voltage is increasing at small voltages and
decreasing at large voltages, reaching a maximum $J_{max}({\bf H})$ in the
intermediate regime. The location of the maximum is found by setting to zero
the derivative $\partial_{\lambda}\bar{J}$,

$\displaystyle\partial_{\lambda}\bar{J}(U,{\bf H})=$
$\displaystyle-e\int^{\infty}_{0}dN\bigg{[}\mathrm{sgn}(U)\partial_{\epsilon}n_{F}\big{(}\bar{\epsilon}(N,{\bf
H})\big{)}\sum_{k}|C_{k}(N,{\bf
H})|^{2}\bigg{(}\frac{2k^{2}-2k^{4}\lambda^{2}}{\big{(}1+k^{2}\lambda^{2}\big{)}^{2}}\bigg{)}$
$\displaystyle+\partial^{2}_{\epsilon}n_{F}\big{(}\bar{\epsilon}(N,{\bf
H})\big{)}\sum_{k_{1},k_{2}}C_{k_{1}}(N,{\bf H})C_{k_{2}}(N,{\bf
H})C_{-k_{1}-k_{2}}(N,{\bf H})(k_{1}+k_{2})^{2}$
$\displaystyle\times\bigg{(}\frac{\mathrm{sgn}(U)\big{(}1+\lambda^{2}(k_{1}+k_{2})^{2}\big{)}-2\lambda(k_{1}+k_{2})\big{(}\mathrm{sgn}(U)\lambda(k_{1}+k_{2})+i\big{)}}{\big{(}1+\lambda^{2}(k_{1}+k_{2})^{2}\big{)}^{2}}\bigg{)}\bigg{]}.$
(A16)

Since the sums are dominated by $k\sim 1$, $\partial_{\lambda}\bar{J}$
vanishes at $\lambda\sim 1$. We can estimate the reciprocal and nonreciprocal
part of $J_{max}({\bf H})$ by substituting $\lambda=1$ into Eqs. (A) and (A),

$\displaystyle J_{max}({\bf H})$ $\displaystyle\sim
e\int^{\infty}_{0}dN\partial_{\epsilon}n_{F}\big{(}\bar{\epsilon}(N,{\bf
H})\big{)}\sum_{k}|C_{k}(N,{\bf
H})|^{2}\bigg{(}\frac{-k^{2}}{1+k^{2}}\bigg{)}\sim\langle g_{1}(\chi,{\bf
H})\rangle,$ (A17) $\displaystyle\delta J_{max}({\bf H})$ $\displaystyle\sim-
ie\int^{\infty}_{0}dN\partial^{2}_{\epsilon}n_{F}\big{(}\bar{\epsilon}(N,{\bf
H})\big{)}\sum_{k_{1},k_{2}}C_{k_{1}}(N,{\bf H})C_{k_{2}}(N,{\bf
H})C_{-k_{1}-k_{2}}(N,{\bf H})\sim\langle g_{2}(\chi,{\bf H})\rangle.$ (A18)

This reproduces Eq. (15) in the main text.

##  Appendix B Appendix B: Derivation of Usadel’s equations in diffusive
junctions with spin-orbit coupling and a Zeeman field

In this section we derive Usadel’s equations for the electron Green’s
functions in the diffusive approximation in the presence of spin-orbit
coupling and a Zeeman field. We then we show that in the special case of
quasi-one dimensional geometry the I-U characteristics of the junctions turn
out to be reciprocal. Thus, the non-reciprocity of the junctions is related to
either a general character of position-dependence of the microscopic
parameters describing the normal metal, such as $D({\bf r})$ and $\beta({\bf
r})$, or geometry of the normal region.

###  B.1 Weak spin-orbit coupling

We will first consider the case of weak spin-orbit coupling, where $\beta
p_{F}\ll\tau_{el}^{-1}$ . The Hamiltonian describing a 2D SNS junction in the
presence of weak spin-orbit coupling and an in plane magnetic field has the
form

$H=\left(\xi_{p}+\beta^{ij}\sigma^{i}p^{j}+V_{imp}({\bf
r})\right)\tau_{3}+g\mu_{0}{\bf H}\cdot{\bm{\sigma}}+\Delta({\bf r})\tau_{1}.$
(B1)

Here $\xi_{p}=p^{2}/2m-E_{F}$, $\tau_{i}$ are the Pauli matrices in Nambu
space, and $\Delta({\bf r})$ is the superconducting order parameter. We will
focus on the case where the length of the normal metal region is much larger
than the superconducting coherence length of the nodes $L\gg\xi$, and
$\Delta({\bf r})$ has the form

$\Delta({\bf r})=\begin{cases}\Delta,&\quad x<-\frac{L}{2},\\\
0,&\quad-\frac{L}{2}<x<\frac{L}{2},\\\ \Delta e^{-i\chi},&\quad
x>\frac{L}{2}.\end{cases}$ (B2)

We start with the Eilenberger equation eilenberger corresponding to the
Hamiltonian in Eq. (B1)

$\displaystyle\left[\epsilon\underline{\tau}_{3}-\Delta({\bf
r})\tau_{1}+A_{0}\underline{\tau}_{3},\underline{g}\right]+iv_{F}\hat{p}^{k}\tilde{\nabla}_{\bf
r}^{k}\underline{g}-{\frac{i}{2m}}\left\\{A_{k},\nabla_{\bf
r}^{k}\underline{g}\right\\}=$
$\displaystyle\,\frac{i}{2\tau_{el}}\left[\underline{g}_{0},\underline{g}\right],$
$\displaystyle\underline{g}^{2}=1.$ (B3)

Here $\underline{g}({\bf r},\epsilon,{\bf\hat{p}})$ is the quasi-classical
Green’s function, which is a matrix in Nambu-spin space, $\\{\cdot,\cdot\\}$
and $[\cdot,\cdot]$ denote the anti-commutator and commutator respectively,
$\hat{p}$ is the unit vector pointing in the direction of ${\bf p}$, and
$g_{0}$ is the Green’s function averaged over the direction of $\hat{p}$.
Finally, ${\bf h}=g\mu_{0}{\bf H}$ is the Zeeman energy,

$\displaystyle A_{0}=$ $\displaystyle\,-h^{\alpha}\sigma^{\alpha},\qquad
A_{k}={-m\beta^{\alpha k}\sigma^{\alpha}},$ (B4)

and $\tilde{\nabla}_{\bf r}^{k}=\nabla^{k}_{\bf r}-i[A_{k},\cdot]$.

We note that the third Eq. (B3) differs from the Eilenberger equation used in
Ref. tokatly . The reason for this is because in Ref. tokatly the author
chooses to perform a unitary transformation of the semiclassical Green’s
functions which eliminates terms which are linear in momentum and spin-orbit
coupling, we choose not to do this. The density of sates in the normal region
can be written in terms of the Green’s function in the form,

$\nu(\epsilon)=\frac{\nu_{N}}{4L_{1}L}\int d{\bf
r}\Re\big{[}\mathrm{Tr}_{\tau,\sigma}\big{(}\tau_{3}\underline{g}_{0}(\epsilon,{\bf
r})\big{)}\big{]}.$ (B5)

Here $\nu_{N}$ is the total density of states of the normal metal part of the
junction in the absence of the superconductors, $\mathrm{Tr}_{\tau,\sigma}$
denotes the trace over Nambu-spin space, and the integral is taken over the
area of the normal region.

In the diffusive regime where the Green’s functions are nearly isotropic, we
can expand the Green’s function into its zeroth and first angular harmonics,

$\displaystyle\underline{g}=$
$\displaystyle\underline{g}_{0}+\underline{g}_{1}^{k}\hat{p}^{k},\quad|\underline{\bm{g}}_{1}|\ll|\underline{g}_{0}|.$
(B6)

With these assumptions, the normalization conditions are then

$\displaystyle\underline{g}_{0}^{2}$
$\displaystyle=1,\quad\left\\{\underline{g}_{0},\underline{g}_{1}^{k}\right\\}=0.$
(B7)

Substituting Eq. (B6) into Eq. (B3), and using Eq. (B7) we get the equations
for $g_{0}$ and $g_{1}^{k}$

$\displaystyle\left[\epsilon\underline{\tau}_{3}-\Delta({\bf
r})\tau_{1}+A_{0}\underline{\tau}_{3},\underline{g}_{0}\right]+\frac{iv_{F}}{2}\tilde{\nabla}_{\bf
r}^{k}\underline{g}_{1}^{k}-{\frac{i}{2m}}\left\\{A_{k},\nabla_{\bf
r}^{k}\underline{g}_{0}\right\\}=$ $\displaystyle\,0,$ (B8a) $\displaystyle
v_{F}\tilde{\nabla}_{\bf
r}^{k}\underline{g}_{0}-\frac{1}{2{\tau_{\mathrm{el}}}}\left[g_{0},\underline{g}_{1}^{k}\right]=$
$\displaystyle\,0.$ (B8b)

Multiplying both sides of Eq. (B8b) by $\underline{g}_{0}$ and using Eq. (B7),
we get

$\displaystyle\underline{g}_{1}^{k}$
$\displaystyle=-v_{F}{\tau_{\mathrm{el}}}\underline{g}_{0}\tilde{\nabla}_{\bf
r}^{k}\underline{g}_{0}.$ (B9)

Substituting Eq. (B9) back into Eq. (B8a), we arrive at Usadel’s equation,

$\displaystyle\left[\epsilon\underline{\tau}_{3}-\Delta({\bf
r})\tau_{1}+A_{0}\underline{\tau}_{3},\underline{g}_{0}\right]-i\tilde{\nabla}_{\bf
r}^{k}\left(D({\bf r})\underline{g}_{0}\tilde{\nabla}_{\bf
r}^{k}\underline{g}_{0}\right)-{\frac{i}{2m}}\left\\{A_{k}({\bf
r}),\nabla_{\bf r}^{k}\underline{g}_{0}\right\\}$ $\displaystyle=0.$ (B10)

To account for long range variations of the microscopic parameters, we allow
$D({\bf r})$ and $A_{k}({\bf r})$ to depend on position. We note that Eq.
(B10) is valid provided that $D({\bf r})$ and $A_{k}({\bf r})$ change slowly
on the scale of the spin relaxation length.

Equation (B10) is written in Nambu-spin space. If $h\ll\tau_{so}^{-1}$ , it is
useful to decompose to the Green’s functions into a singlet and triplet
components,

$\displaystyle\underline{g}_{0}=\underline{g}_{s}+\underline{g}_{t}^{\alpha}\sigma^{\alpha},\quad|\underline{g}_{t}^{\alpha}|\ll|\underline{g}_{s}|.$
(B11)

Here $\underline{g}_{s}$, $\underline{g}_{t}^{\alpha}$ are matrices in Nambu
space. In this approximation, the normalization conditions Eq. B7 yield

$\displaystyle\underline{g}_{s}^{2}$
$\displaystyle=1,\quad\left\\{\underline{g}_{s},\underline{g}_{t}^{\alpha}\right\\}=0.$
(B12)

Substituting Eq. (B11) into (B10) and using Eq. (B12), we get

$\displaystyle\left[\epsilon\underline{\tau}_{3}-\Delta({\bf
r})\tau_{1},\underline{g}_{s}\right]-i\nabla_{\bf r}^{k}\left(D({\bf
r})\underline{g}_{s}\nabla_{\bf
r}^{k}\underline{g}_{s}\right)-h^{\alpha}\left[\underline{\tau}_{3},\underline{g}_{t}^{\alpha}\right]+i\beta^{\alpha
k}({\bf r})\nabla_{\bf r}^{k}\underline{g}_{t}^{\alpha}$ $\displaystyle\,=0,$
(B13a)
$\displaystyle\frac{i}{\tau_{so}}\underline{g}_{s}\underline{g}_{t}^{\alpha}+\left[\epsilon\underline{\tau}_{3}-\Delta({\bf
r})\tau_{1},\underline{g}_{t}^{i}\right]-i\nabla_{\bf r}^{k}\left(D({\bf
r})\underline{g}_{s}\nabla_{\bf
r}^{k}\underline{g}_{t}^{i}+\underline{g}_{t}^{i}\nabla_{\bf r}^{k}(D({\bf
r})\underline{g}_{s})\right)$ $\displaystyle+2imD({\bf
r})\epsilon^{i\alpha\beta}\beta^{\alpha k}({\bf
r})\left(\underline{g}_{s}\nabla_{\bf
r}^{k}\underline{g}_{t}^{\beta}+\underline{g}_{t}^{\beta}\nabla_{\bf
r}^{k}\underline{g}_{s}+\nabla_{\bf
r}^{k}\left(\underline{g}_{s}\underline{g}_{t}^{\beta}\right)\right)$
$\displaystyle\,=h^{i}\left[\underline{\tau}_{3},\underline{g}_{s}\right]-i\beta^{ik}({\bf
r})\nabla_{\bf r}^{k}\underline{g}_{s}.$ (B13b)

Note that we have dropped terms
$\mathcal{O}\left(\underline{g}_{t}^{2}\right)$, and that only ordinary
derivatives are left in the equations. The terms on the RHS of Eq. (B13) act
as source terms for the triplet Green’s functions. Thus, the triplet Green’s
function is generated either by the Zeeman field or the linear in gradient
terms arising from spin orbit coupling.

Since we are interested in solutions for $\epsilon\sim E_{T}$ and $\Delta({\bf
r})=0$ in the normal metal, the second term on the LHS of Eqs. (B13a) is on
the order of $E_{T}$. The typical length scale on which the Green’s functions
change is on the order of $L$ (see Ref. ZhouSpivak ), so the gradients in Eqs.
(B13a) and (B13) are of order $\frac{1}{L}$. With this in mind, we note that
the third term on the LHS of Eq. (B13) is also on the order of $E_{T}$. In the
case $E_{T}\ll\tau_{so}^{-1}$ the first term in Eq. (B13) is the largest term
on the LHS, and we get

$\displaystyle\underline{g}_{t}^{\alpha}$
$\displaystyle=-i\tau_{so}\left(\underline{g}_{s}\left[h^{\alpha}\underline{\tau}_{3},\underline{g}_{s}\right]-i\beta^{\alpha
k}({\bf r})\underline{g}_{s}\nabla_{\bf r}^{k}\underline{g}_{s}\right).$ (B14)

Finally, substituting Eq. (B14) into Eq. (B13a), we get the Usadel equation
for $\underline{g}_{s}$

$\displaystyle\left[\epsilon\underline{\tau}_{3},\underline{g}_{s}\right]-i\nabla_{\bf
r}^{k}\left(D({\bf r})\underline{g}_{s}\nabla_{\bf
r}^{k}\underline{g}_{s}\right)+i\tau_{so}h^{2}\left[\underline{\tau}_{3},\underline{g}_{s}[\underline{\tau}_{3},\underline{g}_{s}]\right]$
$\displaystyle+\tau_{so}\beta^{\alpha k}({\bf
r})h^{\alpha}\left(\left[\underline{\tau}_{3},\underline{g}_{s}\nabla_{\bf
r}^{k}\underline{g}_{s}\right]+\nabla_{\bf
r}^{k}\left(\underline{g}_{s}\left[\underline{\tau}_{3},\underline{g}_{s}\right]\right)\right)$
$\displaystyle=0.$ (B15)

In the subsequent equations we only keep terms which are linear in $h$, as we
will be interested only in the part of the density of states which is linear
in ${\bf H}$.

Let us first consider junctions with a 1D geometry, where $D({\bf r})$ as well
as $\beta({\bf r})$ depend only on the $x$ coordinate. In this case, Eq. (B.1)
and Eq. (B5) are given by

$\displaystyle\left[\epsilon\tau_{3},g_{s}\right]-i\partial_{x}\left(D(x)g_{s}\partial_{x}g_{s}\right)+\tau_{so}\beta^{\alpha
x}(xh^{\alpha}\left(\left[\tau_{3},g_{s}\partial_{x}g_{s}\right]+\partial_{x}\left(g_{s}\left[\tau_{3},g_{s}\right]\right)\right)=0,$
(B16)
$\displaystyle\nu(\epsilon)=\frac{\nu_{N}}{2L}\int^{L/2}_{-L/2}dx\Re\big{[}\mathrm{Tr}_{\tau}\big{(}\tau_{3}\underline{g}_{s}(\epsilon,x)\big{)}\big{]}$
, (B17)

where and $\mathrm{Tr}_{\tau}$ is the trace over Nambu space.

It is convenient to parameterize the Green’s functions in the following form,

$\displaystyle
g_{s}(\epsilon,x)=\left(\begin{array}[]{cc}\cos\theta(\epsilon,x)&\sin\theta(\epsilon,x)e^{i\tilde{\chi}(\epsilon,x)}\\\
\sin\theta(\epsilon,x)e^{-i\tilde{\chi}(\epsilon,x)}&-\cos\theta(\epsilon,x)\end{array}\right),$
(B20)

where $\theta(\epsilon,x)$ and $\tilde{\chi}(\epsilon,x)$ are complex
variables. In this parametrization the Usadel equation reduces to the
following two equations,

$\displaystyle\partial_{x}\left[\big{(}D(x)\partial_{x}\tilde{\chi}+2\tau_{so}\beta^{\alpha
x}(x)h^{\alpha}\big{)}\sin^{2}\theta\right]$ $\displaystyle=0,$ (B21)
$\displaystyle\frac{1}{2}\partial_{x}\bigg{(}D(x)\partial_{x}\theta\bigg{)}+i\epsilon\sin\theta-\frac{D(x)}{4}\sin
2\theta\bigg{(}\partial_{x}\tilde{\chi}+\frac{2\tau_{so}\beta^{\alpha
x}(x)h^{\alpha}}{D(x)}\bigg{)}^{2}$ $\displaystyle=0,$ (B22)

and the expression for the density of states becomes

$\displaystyle\nu(\epsilon,\chi)$
$\displaystyle=\frac{\nu_{N}}{L}\int^{L/2}_{-L/2}d{x}\Re\cos\theta(\epsilon,x).$
(B23)

In the case of perfectly transmitting NS interfaces, the boundary conditions
for Eqs. (B21), (B22) are given by

$\displaystyle\theta(\epsilon,\pm L/2)$
$\displaystyle=\frac{\pi}{2},\quad\tilde{\chi}(\epsilon,\pm
L/2)=\pm\frac{\chi}{2}.$ (B24)

We can further simplify these equations by making the change of variables
$(\theta,\tilde{\chi})\rightarrow(\theta,\hat{\chi})$, where the shifted phase
$\hat{\chi}$ is given by

$\hat{\chi}=\tilde{\chi}+\int_{-L/2}^{x}dq\frac{2\tau_{so}\beta^{\alpha
x}(q)h^{\alpha}}{D(q)}.$ (B25)

In the new variables Eqs. (B21), (B22), and (B24) become

$\displaystyle\partial_{x}\left[D(x)\partial_{x}\hat{\chi}\sin^{2}\theta\right]$
$\displaystyle=0,$ (B26)
$\displaystyle\frac{1}{2}\partial_{x}\bigg{(}D(x)\partial_{x}\theta\bigg{)}+i\epsilon\sin\theta-\frac{D(x)}{4}\sin
2\theta\big{(}\partial_{x}\hat{\chi}\big{)}^{2}$ $\displaystyle=0,$ (B27)
$\displaystyle\theta(\epsilon,\pm
L/2)=\frac{\pi}{2},\qquad\tilde{\chi}(\epsilon,\pm
L/2)=\pm\bigg{(}\frac{\chi+\phi}{2}\bigg{)}$ . (B28)

We note that Eqs. (B26)–(B28) are identical to Usadel’s equations and their
boundary conditions at $h=0$, with a phase difference of $\chi+\phi$ across
the junction. Thus, the solution for $\theta$ has the form

$\displaystyle\theta(\epsilon,x,\chi)=\theta_{0}(\epsilon,x,\chi+\phi)$ (B29)

Here $\theta_{0}(\epsilon,x,\chi)$ is the solution for zero magnetic field and
phase difference $\chi$, and $\phi$ is given by Eq. (27). Thus, we see that
the phase dependence of the density of states has the form of Eq. (10). As
discussed in the main text, in this case the I-U characteristics are
reciprocal.

We would like to stress however that this result arose from the 1D character
of the idealized model of the the SNS junction. In the general situation,
where the parameters $D({\bf r})$ and $\beta({\bf r})$ are functions of $x$
and $y$, or the shape on the N-region is non-rectangular, the density of
states $\nu(\chi)$ can not be expressed in the form of Eq. (25), and the I-U
characteristics are nonreciprocal.

Consider for example a simple model where the diffusion coefficient is a
function of $y$, a coordinate parallel to the SN interface,

$D(x,y)=\begin{cases}D+\frac{\delta D}{2},&0<y<L_{1}/2,\\\ D-\frac{\delta
D}{2},&-L_{1}/2<y<0,\end{cases}$ (B30)

where $\delta D\ll D$. If $L_{1}\gg L$ then the local density of states in the
regions with $y>0$ and $y<0$ can be approximated ${\bf H}=0$ solutions shifted
by phases $\phi_{\pm}=\frac{2\tau_{so}\beta^{\alpha x}h^{\alpha}L}{D\pm\delta
D/2}$ respectively. Then the total density of states can be written as

$\nu(\epsilon,\chi,D)\approx\frac{1}{2}\bigg{(}\nu_{0}(\epsilon,\chi+\phi_{+},D+\delta
D/2)+\nu_{0}(\epsilon,\chi+\phi_{-},D-\delta D/2)\bigg{)},$ (B31)

where $\nu_{0}(\epsilon,\chi,D)$ is density of states of a junction with
diffusion coefficient $D$ and dimensions $L_{1}\times L$, at ${\bf H}=0$. In
this case the total density of states cannot be expressed in the form of
equation of Eq. (25), and the I-U characteristics are non-reciprocal.
Substituting Eq. (B31) into Eq. (17b) in the main text and keeping terms
linear in ${\bf H}$, we get the following estimate for $\langle g_{2}\rangle$,

$\displaystyle\langle g_{2}(\chi,{\bf H})\rangle$
$\displaystyle\approx\bigg{(}\frac{\tau_{so}\beta^{\alpha x}(g\mu_{0}{\bf
H}^{\alpha})}{LE_{T}}\bigg{)}\bigg{(}\frac{\delta
D}{D}\bigg{)}^{2}\frac{E_{T}^{2}}{T^{2}}\langle g_{1}(\chi,{\bf H})\rangle.$
(B32)

Similarly, we substitute Eq. (B31) into Eq. (22) to obtain the following
estimate for $\delta A$,

$\displaystyle\delta A$
$\displaystyle\approx\frac{\sqrt{J_{c}(0,T)}}{e\tau_{in}\langle
g_{1}\rangle}\frac{\beta(\tau_{so}g\mu_{0}H)}{LE_{T}}\bigg{(}\frac{\delta
D}{D}\bigg{)}^{2}.$ (B33)

Here we have used the fact that both the critical current and the phase
$\chi_{m}({\bf H})$, at which the supercurrent reaches its maximum, are
functions of $\nu(\epsilon,\chi,D)$.

In the more general case where $L_{1}\gtrsim L$, the scale of spacial
fluctuations of $D({\bf r})$ and $\beta({\bf r})$ are of order $L$, and
$\delta D\lesssim D$, we expect these estimates to be accurate to order unity.

###  B.2 Superconductor-topological insulator-superconductor (S-TI-S) junction

In this section we consider a model in which the normal region of the junction
is comprised of a conducting surface of a topological insulator described by
the Hamiltonian

$\displaystyle\mathcal{H}(k)=\left(\beta^{\alpha
k}k^{i}\sigma^{\alpha}-\mu+V_{dis}\right)\tau_{3}+h^{\alpha}\sigma^{\alpha}+\Delta\tau_{1}.$
(B34)

This model can be viewed a limiting case of strong spin-orbit coupling in
which the conduction band has a definite helicity.

The derivation of the Usadel equation in S-TI-S junction was done by Ref. Loss
; here we will sketch the main steps. The Eilenberger equation for the
Hamiltonian in Eq. (B34) is given by,

$\displaystyle\left[\epsilon\underline{\tau}_{3}-\Delta({\bf
r})\tau_{1}-h^{\alpha}\sigma^{\alpha}\underline{\tau}_{3},\underline{g}(\bm{n},\epsilon,{\bf
r})\right]+p_{F}\beta^{\alpha
k}\hat{p}^{k}\left[\sigma^{\alpha}\underline{g}({\bf\hat{p}},\epsilon,\bm{r})\right]+\frac{i}{2}\beta^{\alpha
k}\left\\{\sigma^{\alpha},\nabla_{\bf
r}^{k}\underline{g}({\bf\hat{p}},\epsilon,\bm{R})\right\\}=$
$\displaystyle\,-\frac{i}{2{\tau_{\mathrm{el}}}}\left[\langle\underline{g}\rangle,\underline{g}({\bf\hat{p}},\epsilon,\bm{R})\right],$
(B35)

In the regime of strong spin orbit coupling where the system has complete
spin-momentum locking, the Green’s function has the following helical spin
structure,

$\displaystyle\underline{g}=\frac{1}{2}\underline{g}^{\prime}\left(1+\hat{\beta}^{\alpha
k}\hat{p}^{k}\sigma^{\alpha}\right),$ (B36)

where $\underline{g}^{\prime}$ is a matrix only in Nambu space. We then insert
Eq. (B36) into Eq. (B35),

$\displaystyle\left[\epsilon\underline{\tau}_{3},-\Delta({\bf
r})\tau_{1}-h^{\alpha}\sigma^{\alpha},\frac{1}{2}\underline{g}^{\prime}\left(1+\hat{\beta}^{\gamma
k}{\sigma^{\gamma}}\hat{p}^{k}\right)\right]+ip_{F}\epsilon^{\alpha\beta\gamma}\beta^{\alpha
k}\hat{\beta}^{\beta i}\hat{p}^{k}\hat{p}^{i}g^{\prime}\sigma^{\gamma}$
$\displaystyle+\frac{i\beta^{\alpha k}}{2}\nabla_{\bf
r}^{k}g^{\prime}\left(\sigma^{\alpha}+\hat{\beta}^{\alpha
i}\hat{p}^{i}\right)+\frac{i}{2{\tau_{\mathrm{el}}}}\left[\left\langle\frac{1}{2}\underline{g}^{\prime}\left(1+\hat{\beta}^{\alpha
k}\hat{p}^{k}\right)\right\rangle,\frac{1}{2}\underline{g}^{\prime}\left(1+\hat{\beta}^{\alpha
k}\hat{p}^{k}\right)\right]$ $\displaystyle=0.$ (B37)

In the diffusive limit where $v_{F}\tau_{el}\ll L$, we expand the Green’s
function into its zeroth and first moment
$\underline{g}^{\prime}=\underline{g}_{0}+\underline{g}_{1}^{k}\hat{p}^{k}$,
insert the expansion into Eq. (B37), project the equation onto its zeroth and
first angular harmonics, and obtain two equations for
$\underline{g}_{0},\underline{g}_{1}$. After taking the trace of the resulting
equations over spin indices, we arrive at the following equations,

$\displaystyle\left[\epsilon\underline{\tau}_{3}-\Delta({\bf
r})\tau_{1},\underline{g}_{0}\right]+i\frac{\beta}{2}\hat{\nabla}_{\bf
r}^{i}\underline{g}_{1}^{i}=0,$ (B38) $\displaystyle\underline{g}_{1}^{i}$
$\displaystyle=-2\beta{\tau_{\mathrm{el}}}\underline{g}_{0}\hat{\nabla}_{\bf
r}^{i}\underline{g}_{0},$ (B39)

where we have defined,

$\displaystyle\hat{\nabla}_{\bf r}^{i}\cdot=\nabla_{\bf
r}^{i}\cdot+i\frac{\hat{\beta}^{\alpha
i}h^{\alpha}}{\beta}\left[\underline{\tau}_{3},\cdot\right].$ (B40)

Substituting Eq. (B39) in (B38), we have Usadel’s equation for the strong spin
orbit case,

$\displaystyle\left[\epsilon\underline{\tau}_{3}-\Delta({\bf
r})\tau_{1},\underline{g}_{0}\right]-i\nabla_{\bf r}^{i}\left(D({\bf
r})\underline{g}_{0}\nabla_{\bf
r}^{i}\underline{g}_{0}\right)+\tau_{el}\beta^{\alpha i}({\bf
r})h^{\alpha}\left(\nabla_{\bf
r}^{i}\left(\underline{g}_{0}\left[\underline{\tau}_{3},\underline{g}_{0}\right]\right)+\left[\underline{\tau}_{3},\underline{g}_{0}\nabla_{\bf
r}^{i}\underline{g}_{0}\right]\right)+i\tau_{el}h^{2}\left[\underline{\tau}_{3},\underline{g}_{0}\left[\underline{\tau}_{3},\underline{g}_{0}\right]\right]=$
$\displaystyle\,0,$ (B41)

which is the same as Eq. (B.1), the equation for $g_{0}$ in the case of weak
spin-orbit coupling, with the replacement of $\tau_{so}$ with $\tau_{el}$ .
This can be understood by the fact that in the case where spin-momentum
locking is strong, spin-relaxation is limited only by the rate of elastic
scattering. This suggests that the form of Eq. (B41) is universal within the
diffusive regime, where the parameters in the equation depend on the strength
of the spin-orbit coupling. As a result, we can apply the estimates for
$\langle g_{2}\rangle$ and $\delta A$ given in Eqs. (B32) and (B33) to the
case of a S-TI-S junction with the appropriate substitution of
$\tau_{so}={\tau_{\mathrm{el}}}$.