Rashba proximity states in superconducting tunnel junctions
We consider a new kind of superconducting proximity effect created by the tunneling of “spin split” Cooper pairs between two conventional superconductors connected by a normal conductor containing a quantum dot. The difference compared to the usual superconducting proximity effect is that the spin s...
Збережено в:
| Опубліковано в: : | Физика низких температур |
|---|---|
| Дата: | 2018 |
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2018
|
| Теми: | |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/176152 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Rashba proximity states in superconducting tunnel junctions / O. Entin-Wohlman, R.I. Shekhter, M. Jonson, A. Aharony // Физика низких температур. — 2018. — Т. 44, № 6. — С. 701-710. — Бібліогр.: 31 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860097300116799488 |
|---|---|
| author | Entin-Wohlman, O. Shekhter, R.I. Jonson, M. Aharony, A. |
| author_facet | Entin-Wohlman, O. Shekhter, R.I. Jonson, M. Aharony, A. |
| citation_txt | Rashba proximity states in superconducting tunnel junctions / O. Entin-Wohlman, R.I. Shekhter, M. Jonson, A. Aharony // Физика низких температур. — 2018. — Т. 44, № 6. — С. 701-710. — Бібліогр.: 31 назв. — англ. |
| collection | DSpace DC |
| container_title | Физика низких температур |
| description | We consider a new kind of superconducting proximity effect created by the tunneling of “spin split” Cooper pairs between two conventional superconductors connected by a normal conductor containing a quantum dot. The difference compared to the usual superconducting proximity effect is that the spin states of the tunneling Cooper pairs are split into singlet and triplet components by the electron spin-orbit coupling, which is assumed to be active in the normal conductor only. We demonstrate that the supercurrent carried by the spin-split Cooper pairs can be manipulated both mechanically and electrically for strengths of the spin-orbit coupling that can realistically be achieved by electrostatic gates.
|
| first_indexed | 2025-12-07T17:27:08Z |
| format | Article |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6, pp. 701–710
Rashba proximity states in superconducting tunnel
junctions
O. Entin-Wohlman1,2, R.I. Shekhter3, M. Jonson3, and A. Aharony1,2
1Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel
2Physics Department, Ben Gurion University, Beer Sheva 84105, Israel
E-mail: oraentin@bgu.ac.il
3Department of Physics, University of Gothenburg, SE-412 96 Göteborg, Sweden
Received January 15, 2018, published online April 25, 2018
We consider a new kind of superconducting proximity effect created by the tunneling of “spin split” Cooper
pairs between two conventional superconductors connected by a normal conductor containing a quantum dot.
The difference compared to the usual superconducting proximity effect is that the spin states of the tunneling
Cooper pairs are split into singlet and triplet components by the electron spin-orbit coupling, which is assumed
to be active in the normal conductor only. We demonstrate that the supercurrent carried by the spin-split Cooper
pairs can be manipulated both mechanically and electrically for strengths of the spin-orbit coupling that can real-
istically be achieved by electrostatic gates.
PACS: 72.25.Hg Electrical injection of spin polarized carriers;
72.25.Rb Spin relaxation and scattering.
Keywords: spin-orbit interaction, Rashba spin splitter, Josephson effect, electric weak link.
1. Introduction
The prominent role that the electronic spin plays in de-
termining the properties of solid-state devices has been at
the forefront of experimental and theoretical research during
the last decade. The topological surface electronic states,
that are formed due to a strong spin-orbit coupling, with
their vast potential for quantum computations [1,2] and
spintronic applications [3] of spin-polarized currents, are just
a few conspicuous examples. Conducting nanostructures,
e.g., quantum dots, nanowires and nanorings, where the
mesoscopic behavior of the electrons is dominated by Cou-
lomb correlations and quantum-phase coherence, are by
now the tools of choice for studying spin-related phenom-
ena, in particular effects induced by the spin-orbit cou-
pling. Composite mesoscopic structures comprising such
nanometer-sized elements are currently of considerable
interest due to their applicability in quantum communica-
tion systems [4–6]. The hope is to provide a coherent plat-
form for flying qubits: moving two-state spinors, which may
represent the electronic spin [7] or any other pseudo-spin
state, e.g., of particles moving in two coupled wires [8].
Clearly, spin-state decoherence is detrimental to spin-
tronics applications involving, e.g., flying qubits. Reducing
the scattering rate of spin-polarized electrons in order to pre-
serve spin coherence is therefore essential, and is the rea-
son why using superconducting materials have been con-
sidered. However, while electron transport in a supercon-
ductor is indeed fully coherent, the supercurrent carried by
spin-singlet Cooper pairs in a conventional superconductor
conducts charge but not spin. If, on the other hand, the
Cooper pairs could be spin polarized it would mean that a
coherent, dissipationless spin current could be generated.
Hence, it is highly desirable to find methods for generating
spin-polarized Cooper pairs. Recently, such a method —
involving the creation of spin-polarized Cooper pairs in
superconducting weak links made of materials with a strong
spin-orbit interaction (SOI) — was proposed. It was shown
that the spin-structure of the Cooper pairs, injected into a
non-superconducting material in which the spin-orbit inter-
action is significant, can be “predesigned” in such a way that
a net electronic spin-polarization is carried through an SOI-
active weak link that connects the superconducting leads.
The physics behind this phenomenon is the splitting of the
transferred electronic states within the weak link with re-
spect to spin — the so-called “Rashba spin-splitting” [9]. As
a consequence of this spin splitting, the electronic spin expe-
riences quantum fluctuations that lead to a “triplet-channel”
for Cooper-pair transport through the link.
© O. Entin-Wohlman, R.I. Shekhter, M. Jonson, and A. Aharony, 2018
O. Entin-Wohlman, R.I. Shekhter, M. Jonson, and A. Aharony
The ability to inject electrons paired in a spin-triplet
state into a conventional BCS superconductor from an
SOI-active superconducting weak link, opens a route to all
kinds of spintronics applications that can be implemented
by using a dissipationless spin current. However, the ap-
pearance of spin-triplet Cooper pairs in a conventional
BCS superconductor is a so-called proximity effect, and
spin-polarized Cooper pairs are present only in the vicinity
of the weak link. In addition, the triplet states are vulnera-
ble to any spin-relaxation mechanism in the superconduc-
tor. A clever composite device-design is therefore required
to allow for the accumulation of paired electrons with a
non-zero net spin, while significantly blocking their spin
relaxation.
In the present paper we suggest such a design, and pro-
pose a new type of a superconducting weak link in which
Rashba spin-split states involving pairs of time-reversed
(“Cooper pair”) states can be established through the prox-
imity effect. The generic component of the device is a
quantum dot coupled to two superconductors through SOI-
active weak links in the form of nanowires, as illustrated in
Fig. 1. A significant advantage of the device is its relative-
ly low spin relaxation rate — a well-known property of
quantum dots [10–14]. The extent to which spin is accu-
mulated on the dot can be controlled by electric fields that
modify the SOI strength [15–18]. Another handle on the
device is the possibility to tune mechanically the physical
location of the dot, and thus affect the amount of tunneling
between the two reservoirs [19,20].
The paper is organized as follows. Section 2 introduces
the Hamiltonian of our model and details the calculation of
the transmission of Cooper pairs between two supercon-
ductors connected by a weak link on which the transferred
electrons are subjected to a spin-orbit interaction. We in-
clude in the calculation two important effects. (i) The SOI
on the left wire can differ from the one on the right wire
(see Fig. 1), both in strength and in the direction of the
effective magnetic field that characterizes this interaction.
(ii) The passage of a Cooper pair through the weak link can
take place either by sequential tunneling, during which the
two electrons tunnel one by one and the dot is at most sin-
gly occupied, or by events in which the two electrons hap-
pen to reside simultaneously on the dot during the tunnel-
ing. In the latter case, one has to account for the Coulomb
interaction on the dot. Obviously these two processes con-
tribute disparately to the transport. We dwell on the two
separate contributions and their dependence on the geome-
try of the junction in Sec. 3. Since we consider in Sec. 2
the transfer of Cooper pairs, in which the two electrons are
in time-reversed states, we need to construct the relation
between the (spin-dependent) tunneling amplitudes in the-
se two states. This task is accomplished in Appendix A.
The transmission of Cooper pairs is analyzed by studying
the equilibrium Josephson current between the two reser-
voirs. In particular, we analyze the manner by which the
spins of the tunneling electrons precess as they pass
through the weak link. Technical details of this calculation
are relegated to Appendix B.
Section 3 presents the results. We derive there explicit
expressions for the spin-precession factor of each of the
two processes alluded to above, and explain the way the
disparity between the two reflects the coherence of the
sequential single-electron tunneling process, and the inco-
herence of the double-electron one, during which the dot is
doubly occupied. We also analyze there the dependence of
the Cooper pairs’ transmission on the relative angle be-
tween the two directions of the SOI’s on the two wires.
Our conclusions are discussed in Sec. 4.
2. Tunneling of Cooper pairs
2.1. Description of the model
The spin splitting of electrons that flow through a weak
link in which the Rashba [21,22] spin-orbit interaction is
active, can be understood within a semiclassical picture.
As the electrons pass through the weak link, their spins
precess around an effective magnetic field associated with
the SOI. This spin dynamics splits the electron wave func-
tion into different spin states and yields a certain probabil-
ity, which can be controlled externally, for the spins to be
flipped as they emerge from the weak link [23]. The spin-
splitting phenomenon becomes more complicated when the
transmission of a pair of electrons in two time-reversed
states through an SOI-active weak link is considered. In
that case there are two types of tunneling events, those
where the electrons are transferred one by one sequentially,
and those in which the dot is doubly occupied during the
tunneling. This is taken into account in our model by rep-
resenting the quantum dot as a single localized level of
energy ε which can accommodate two electrons in two
spin states (“up” and “down”). In our simple model the
reservoirs that supply the electrons are two bulk BCS su-
perconductors; these are coupled together by a nanowire
on which the quantum dot is located. When on the dot, the
spin state of one electron of the Cooper pair is projected on
the spin-up state of the dot, and that of the other on the
spin-down state [24]. We find that the projection breaks
the coherent evolution of the spin states. The Pauli princi-
Fig. 1. (Color online) Spin-orbit active superconducting weak-
link. The quantum dot is represented by a single localized level of
energy ε , and the superconducting reservoirs are denoted LS and
RS . The tunneling amplitude (a matrix in spin space) between the
dot and the left (right) lead is denoted kt ( pt ).
702 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6
Rashba proximity states in superconducting tunnel junctions
ple is assumed to be effective only on the quantum dot;
elsewhere the passage of the electrons in and out of the dot
is viewed as a single-electron tunneling event, whose am-
plitude includes the electronic spin precession [25].
The Hamiltonian of the junction illustrated in Fig. 1
reads
0 tun= .+ (1)
The Hamiltonian 0 pertains to the decoupled system, and
includes the Hamiltonian of the quantum dot and that of
the superconducting leads,
† ††
0 lead
= ,
= .
L R
d d Ud d d d α
σ σ ↑ ↓↑ ↓
σ α
ε + +∑ ∑ (2)
The operator dσ ( †dσ) annihilates (creates) an electron in
the spin state | σ〉 on the dot, and U denotes the Coulomb
repulsion. The BCS leads are described by the annihilation
(creation) operators of the electrons there, ( )c σk p ( †
( )c σk p ).
[k (p) enumerates the single-particle orbital states on the
left (right) lead.] Denoting by ( )k pε the single-electron
energy measured relative to the common chemical poten-
tial of the device, the Hamiltonian of the leads is
= ( ) †
( ) ( )lead ( )
( ),
=L R
k p c cα
σσ
σ
ε −∑ k pk p
k p
† †( )
( ) ( ) ( )
( )
(e H.c.),
i L R
L R c c
ϕ
↑ − − ↓
−∆ +∑ k p k p
k p
(3)
where ( )L R∆ and ( )L Rϕ are the amplitude and the phase of
the superconducting order parameters.
The tunneling Hamiltonian is the key component of our
model,
tun = H.c. ,LD RD+ + (4)
where
( )†
( ) ( ) ( )
( ), ,
= [t ] .L R D
L R D c d′ ′σσ σσ
′σ σ
∑ k p k p
k p
(5)
The probability amplitude for the transfer of an electron
from the spin state | ′σ 〉 on the dot to the state | ( ),σ〉k p in
the left (right) reservoir is ( )
( )[t ]L R D
′σσk p , which allows for
spin flips during the tunneling. This amplitude is conven-
iently separated into a (scalar) orbital amplitude, and a
unitary matrix (in spin space), denoted W, that contains the
effects of the SOI (whether of the Rashba [21,22] or the
Dresselhaus [26] type), and also the dependence on the
spatial direction of the SOI-active wire. The spin-orbit in-
teraction associated with strains is briefly mentioned in
Sec. 3. For the linear SOI [9], the tunneling amplitude can
be presented in the form
( ) ( )( )
( )t = e ,
ikL R D L R DF L R
L R
d
it
−
k W (6)
where Fk is the Fermi wave vector in the leads, and
( )L Rd is the length of the bond between the left (right)
lead and the dot. The generic form of W is [27,28]
( )
( ) ( )= ,L R D
L R L Ra i+ ⋅W bσ (7)
where ( )L Ra is a real scalar and ( )L Rb is a real vector (de-
termined by the symmetry axis of the SOI and the spatial
direction of the weak link), with 2
( ) ( ) ( ) = 1L R L R L Ra + ⋅b b
(σ is the vector of the Pauli matrices). In the absence of the
spin-orbit interaction ( )L RW is just the unit matrix, name-
ly, ( ) = 1L Ra and ( ) = 0L Rb . Explicit expressions for the
spin-orbit coupling are discussed in Sec. 3.
Since the two electrons of a Cooper pair are in two
states connected by the time-reversal transformation, one
has to consider also the tunneling amplitude between the
time-reversed states of | ′σ 〉 and | ( ),σ〉k p , which we denote
by an overline, i.e., | ( ) |yi′ ′σ 〉 ≡ σ σ 〉 and | ( ), .− − σ〉k p The
corresponding amplitude, ( )
( )[ ]L R D
′σσk pt , describes the trans-
fer of an electron from the spin state | ′σ 〉 on the dot to the
state | ( ),− − σ〉k p in the left (right) lead. It is given by
( ) ( ) 1
( ) ( )
ˆ ˆ; = ( ) ,L R D L R D
yK i−≡ σk p k pt Tt T T (8)
( yσ is a Pauli matrix, and K is the complex conjugation
operator). We derive in Appendix A the relation
( ) ( )
( ) ( )[ ] = [( ) ] ,L R D L R D ∗
′ ′σσ σσk p k pt t (9)
which is of paramount importance in our considerations.
2.2. The particle current
The flow of electrons between the two superconductors
is analyzed by studying the equilibrium Josephson current,
i.e., the rate by which electrons leave the left (or the right)
superconductor, when the chemical potentials of the two
reservoirs are identical (and therefore a flow of quasi-
particles is prohibited by the gap in the quasiparticle densi-
ty of states in the superconducting leads).
Using units in which = 1 , this rate is
†
,
= ( / ) = 2Im ( ) ,LDJ d dt c c tσσ
σ
− 〈 〉 − 〈 〉∑ kk
k
(10)
where the angular brackets denote quantum averaging. It is
evaluated using the S-matrix,
1( ) = ( , ) ( ) ( , ) ,LD LDt S t t S t−〈 〉 〈 −∞ −∞ 〉 (11)
with 0 0( ) = exp[ ] exp[ ]LD LDt i t i t− , and the quan-
tum average is with respect to 0 . As the leading-order
contribution to J is fourth-order in the tunneling Hamilto-
nian, it is found from the expansion up to third order of the
S-matrix [29],
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6 703
O. Entin-Wohlman, R.I. Shekhter, M. Jonson, and A. Aharony
1
1 2
1 tun 1 1 2 tun 1 tun 2
1 2 3 tun 1 tun 2 tun 3
( , ) =1 ( ) ( ) ( )
( ) ( ) ( ) .
tt t
t tt
S t i dt t dt dt t t
i dt dt dt t t t
−∞ −∞ −∞
−∞ −∞ −∞
−∞ − − +
+
∫ ∫ ∫
∫ ∫ ∫
(12)
The energy level on the dot is assumed to lie well above
the chemical potential of the leads, and thus the small pa-
rameter of the expansion is /Γ ε , where = L RΓ Γ +Γ is the
total width of the resonance level created on the dot due to
the coupling with the bulk reservoirs (each coupling induc-
es a partial width ( )L RΓ , see Appendix B). This implies
that the perturbation expansion is carried out on a dot
which is empty when decoupled [30]. We list and discuss
the relevant terms in the expansion of J in Appendix B.
As mentioned, the total Josephson current in our junc-
tion is due to the two processes available for the tunneling
pairs. In the first, whose contribution is denoted sJ , double
occupancy on the dot does not occur, and the transfer of
the electron pair is accomplished by a sequential tunneling
of the paired electrons one by one. In the second process
that contributes dJ , the dot is doubly occupied during the
tunneling. The (lengthy but straightforward) calculation
presented in Appendix B yields
,s dJ J J= + (13)
where
0
0
( / ) ,
2 ( / , / ) .
s s s
d d d
J I F
J I F U
= ε ∆
= ε ∆ ∆
(14)
These results are obtained, for simplicity, in the zero-
temperature limit, and for =L R∆ ∆ ≡ ∆. The common fac-
tor 0I in the expressions for sJ and dJ is the Josephson
amplitude of the interface between the two superconduc-
tors (i.e., when the localized level on the dot as well as the
SOI are ignored), 0 = 2sin( )[ / ]R L L RI ϕ −ϕ Γ Γ ∆ . The dis-
parity of the two tunneling processes comes into play in
the other two factors in Eqs. (14). The functions sF and
dF [31], given in Eqs. (B5) and (B9) and reproduced here
for convenience,
1( )
cosh
1 1 , ,
cosh cosh cosh
ps k
k
k p p
dd
F
∞ ∞
−∞ −∞
ζζ
ε = ×
π π ζ + ε
ε
× ε =
ζ + ζ ζ + ε ∆
∫ ∫
(15)
and
1( , )
cosh
1 1 , , ,
cosh2
pd k
k
p
dd
F U
UU
U
∞ ∞
−∞ −∞
ζζ
ε = ×
π π ζ + ε
ε
× ε = =
ζ + ε ∆ ∆ε +
∫ ∫
(16)
convey the effect of the resonance on the dot, and the Cou-
lomb repulsion there. As seen, the Pauli principle tends to
reduce the contribution of the second tunneling process.
The last factors, s and d , describe the amount of spin
precession. Their detailed analysis is the topic of the next
subsection. These two factors become 1 in the absence of
the SOI. It follows that in the absence of the SOI, the Jo-
sephson current of our junction, 0J , is
0 0= [ ( / ) 2 ( / , / )] .s dJ I F F Uε ∆ + ε ∆ ∆ (17)
The normalized Josephson current, i.e., J of Eq. (13) di-
vided by 0J , is
0
( / ) 2 ( / , / ) .
( / ) 2 ( / , / )
s s d d
s d
J F F U
J F F U
ε ∆ + ε ∆ ∆
=
ε ∆ + ε ∆ ∆
(18)
2.3. Spin precession
In the sequential tunneling process, where the pair of
electrons tunnel one by one, their spin-precession factor is
, ,
1= sgn ( )sgn ( )
2
[( ) ] [ ] [( ) ] [ ] ,
s
L R
L R
LD LD DR DR
L L R R
′σ σ σ σ
∗ ∗
′ ′σ σ σ σ σ σ σσ
σ σ ×
×
∑ ∑
W W W W
(19)
while the spin-precession factor of the tunneling process
during which the dot is doubly occupied is
,
1= sgn( )sgn ( )
2
[( ) ] [ ] [( ) ] [ ] .
d
L R
L R
LD LD DR DR
L L R R
σ σ σ
∗ ∗
σ σ σ σ σσ σσ
σ σ ×
×
∑ ∑
W W W W
(20)
[See the derivation in Appendix B that leads to Eqs. (B4)
and (B8), reproduced in Eqs. (19) and (20) for convenience.]
It is illuminating to scrutinize the summations over the
spin indices. The factor s can be written as
,
2 2
1= sgn( )sgn( )[( ) ] [ ]
2
= |[ ] | |[ ] | ,
s LR LR
L R L R L R
L R
LR LR
∗
σ σ σ σ
σ σ
↑↑ ↑↓
σ σ =
−
∑ W W
W W
(21)
where the direct spin part of the tunneling amplitude from
the right lead to left one is
[ ] = [ ] [ ] .LR LD DR
L R L Rσ σ σ σ σσ
σ
∑W W W (22)
In contrast, the spin-precession factor of the double-
occupancy channel cannot be expressed in terms of the
direct amplitudes; we find
2 2
2 2
= (| [ ] | | [ ] | )
(| [ ] | | [ ] | ) .
d LD LD
DR DR
↑↑ ↑↓
↑↑ ↑↓
− ×
× −
W W
W W
(23)
704 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6
Rashba proximity states in superconducting tunnel junctions
One notes that d , Eq. (23), is a product of two factors of
the same structure as the single factor in s , Eq. (21). Our
interpretation is that s describes the coherent transfer of
a Cooper pair from the right to the left lead, while d de-
scribes first a coherent Cooper pair transfer from the right
lead to the dot, where coherence is lost, then a second co-
herent transfer from the dot to the left lead. Both contribu-
tions to the Josephson current, sJ and dJ [see Eqs. (13)
and (14)], are suppressed to a certain extent by the spin-
precession (as compared with their respective values in the
absence of the SOI). Which of them is more severely af-
fected is determined by the geometry of the junction and
the symmetry direction of the SOI. This feature is dis-
cussed in Sec. 3.
The effect of the spin-orbit coupling on the super-
current is embedded in the precession of the spins it in-
duces. It is therefore natural to express the current, in
particular the precession factors, in terms of orientations
of the effective magnetic fields responsible for the pre-
cession in the left and right SOI-active nanowires. In-
deed, upon carrying out the spin summations in Eqs. (18)
and (20) within each reservoir [i.e., on Lσ and Rσ using
Eq. (7) for W], one finds that these factors can be ex-
pressed in terms of the vectors LV and RV , that represent
the effective magnetic fields,
sgn ( )[( ) ] [ ] [ ] ,
sgn ( )[ ] [( ) ] [ ] .
LD LD
L LL L
L
DR DR
R RR R
R
∗
′ ′σ σ σ σ σ σ
σ
∗
′ ′σσ σ σ σσ
σ
σ ≡ ⋅
σ ≡ ⋅
∑
∑
W W V
W W V
σ
σ
(24)
The vectors ( )L RV are determined by the detailed form of
the SOI, Eq. (7),
( ) ( ) ( )
2
( ) ( ) ( )
= 2
ˆ ˆ2 (1 2 ).
L R L R z L R
L R L R L R
b
a b
+
+ × + −
V b
b z z
(25)
Inserting Eqs. (24) into the spin-precession factors
Eqs. (19) and (20), one finds
= ,
= .
s
L R
d
Lz RzV V
⋅V V
(26)
It is thus seen that due to the Pauli exclusion principle,
only the components of the vectors LV and RV that are
parallel to the quantization axis on the dot contribute to
the spin-precession factor arising from the tunneling pro-
cess in which the dot is doubly occupied, whereas all
components of these vectors participate in the spin-
precession factor of the sequential transmission. We also
note that the difference between the two spin-precession
factors disappears when either of the vectors ( )L RV or
both are directed along z-axis (which is the situation in
the absence of the SOI coupling).
3. Results
Although it is possible in principle to calculate an effec-
tive SOI ab initio, it is rather ubiquitous to adopt the phe-
nomenological Hamiltonian proposed in Ref. 22. This
Hamiltonian (named after Rashba) is valid for systems
with a single high-symmetry axis that lack spatial inver-
sion symmetry. For an electron of an effective mass m∗
and momentum p propagating along a wire where the SOI
is active, it reads
Rashba
ˆ= ( ).so
so
k
m∗
σ ⋅ ×p n (27)
Here n̂ is a unit vector along the symmetry axis (the c-axis
in hexagonal wurtzite crystals, the growth direction in a
semiconductor heterostructure, the direction of an external
electric field), and Rashba
sok is the strength of the SOI in
units of inverse length, which is usually taken from exper-
iments [16,18]. By exploiting the Hamiltonian (27) to find
the propagator along a one-dimensional wire, one obtains
that the tunneling amplitude is [19,27]
Rashba ˆ= e exp[ ],ikF LL so L
rit ik × ⋅kt r n σ (28)
with an analogous form for pt . Here ( )L Rr is the radius
vector pointing from the dot to the left (right) reservoir
along the wire, and Fk is the Fermi wave vector of an elec-
tron traversing this nanowire. The linear Dresselhaus SOI
[26] gives rise to a comparable form for so [28]. Another
source for SOI’s are strains, created for instance when a
single flat graphene ribbon is rolled up to form a tube. This
type of SOI was modeled by the Hamiltonian
strain
strain ˆ= ,F so
so
k k
m∗
⋅n σ (29)
where strain
sok is a phenomenological parameter that gives
the strength of the SOI in units of inverse length and the
unit vector n̂ points along the nanotube [14,17].
In the case of the Rashba SOI, the spin-dependent fac-
tor in the tunneling amplitude Eq. (28) can be written in
the form
ˆ ˆ= exp[ ] = cos ( ) sin ( ) ,i iα ⋅ α + α ⋅W v v σ σ (30)
where α is proportional to the strength of the spin-orbit
coupling and both α and the unit vector v̂ are determined
by the symmetry direction of the SOI and the geometrical
configuration of the junction: Rashba= sinsok dα ϕ , where d
is the length of the SOI-active bridge, and ϕ is the angle
between the wire direction and n̂, the symmetry axis of the
interaction.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6 705
O. Entin-Wohlman, R.I. Shekhter, M. Jonson, and A. Aharony
The parameters that characterize the spin-orbit coupling
can be controlled experimentally. The capability to
tune the strength of the spin-orbit interaction electrostati-
cally was first demonstrated in Ref. 15 on inverted
In0.53Ga0.47As/In0.52Al0.48As heterostructures. A more
recent experimental evidence is found in Ref. 16, that
describes measurements on the inversion layer of a
In0.75Ga0.25As/In0.75Al0.25As semiconductor heterostruc-
ture, and in Ref. 18, which reports on a dual-gated
InAs/GaSb quantum well. The electrodes in these experi-
ments are two-dimensional electron (or hole) gas bulk
conductors; two gate electrodes are used to tune both the
carriers’ concentration and the strength of the SOI. The
spin-orbit coupling is characterized by the “Rashba param-
eter” Rα , whose relation to Rashba
sok is
Rashba 2= / ,so Rk m∗α (31)
where Rα is measured in [meV⋅Å] (we keep in the ex-
perimental estimations). In Ref. 16, the spin-orbit coupling
constant α varied with the gate voltage between roughly
150 and 300 meV⋅Å. Attributing α mainly to the Rashba
SOI parameter Rα , and using = 0.04m m∗ (m is the free-
electron mass), one concludes that if a weak link were to
be electrostatically defined in such a system, then
Rashba
sok d could be varied from ~ 8 to ~ 16 for d = 1 µm.
The ratio of the effective mass to the free-electron mass
quoted in Ref. 18 is comparable; in this quantum well the
Dresselhous SOI was kept constant while Rα could be
varied between 53 and 75 meV Å, leading to Rashba
sok d that
varies from ~ 8 to ~ 16 for d = 1 µm.
As an explicit example, we consider a weak link of the
form of two straight segments, whose SOI’s parameters are
( )L Rα and ( )ˆ L Rv . One then finds that the two spin-
precession factors, Eqs. (26), are [see Eqs. (25) and (30)]
2 2= [cos (2 ) 2 ][cos(2 ) 2 ],d
L Lz R Rz′ ′α + α + v v (32)
and
= 4sin( )sin( )
ˆ ˆ{[ ] [cos( ) cos( ) ]
[cos( ) cos( ) ]} ,
s d
L R
L R z R Lz L Rz
L R L R Lz Rz⊥ ⊥
+ α α ×
′ ′× × α − α +
′ ′+ ⋅ α α +
v v
v v
v v
v v
(33)
where
( ) ( ) ( )= sin( ).L R z L R z L R′ αv v (34)
(The notation ⊥ indicates the part of the vector normal to
ˆ.)z For instance, when both ˆ Lv and ˆ Rv are along ẑ , then
1s d= = , and the effect of the SOI disappears. On the
other hand, when the spin-orbit coupling is due to the
Rashba interaction with an electric field directed along ẑ
and the junction is lying in the XY plane (see Fig. 1), then
ˆ Lv and ˆ Rv are in the XY plane. In that case the spin-
precession factors are
cos (2 )cos (2 ) ,
sin (2 )sin (2 )cos ( ) ,
d
L R
s d
L R
= α α
= + α α θ
(35)
where θ is the angle between ˆ Lv and ˆ Rv . We plot in Fig. 2
the normalized Josephson current, Eq. (18), for = =R Lα α α
and the spin-precession factors as given in Eqs. (35), as a
function of the angle θ, for various values of the spin-
orbit strength α . One notes the change of sign of the
normalized Josephson current, as a function of the angle
θ between ˆ Lv and ˆ Rv .
Figure 3 displays the normalized Josephson current as a
function of both θ and the spin-orbit coupling constant α .
Fig. 2. (Color online) The normalized Josephson current, Eq. (18), as
a function of the angle θ between ˆ Lv and ˆ Rv [see Eqs. (35)] for
various values of = =R Lα α α . Straight (black) line — = 0α (1),
tiny-dashed (blue) curve — = 0.2α (2), medium-dashed (magenta)
curve — = 0.4α (3), large-dashed (red) curve — = 0.6α (4), dot-
ted (brown) curve — = 0.8α (5), dot-dashed (black) curve —
= 1α (6), dot-dashed (orange) curve — = 1.2α (7). The parame-
ters that determine Eqs. (15) and (16) are / = 0ε ∆ and / = 5U ∆ .
Fig. 3. (Color online) A density plot of the normalized Josephson
current, Eq. (18), as a function of the angle θ between ˆ Lv and ˆ Rv
and the spin-orbit coupling constant, α [see Eqs. (35)]. The parame-
ters that determine Eqs. (15) and (16) are / = 0ε ∆ and / = 5U ∆ .
706 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6
Rashba proximity states in superconducting tunnel junctions
Here one notes the conspicuous oscillations as a func-
tion of ,α which arise from the trigonometric functions
in Eqs. (35). Figures 2 and 3 exemplify the possibility to
vary the supercurrent in our device mechanically, by
bending the bridge (and thus changing θ) connecting the
superconductors.
As mentioned, the coupling constant of the SOI can
be manipulated experimentally by varying gate voltages.
Additional functionality is obtained when the orienta-
tions of these electric fields (induced by the gates) in the
two arms of the bridge are made to be different. The
simplest example is when the two arms of the bridge lie
along the x-axis, with the electric field on the left nan-
owire directed along z, while that on the right one is in
the YZ-plane, making an angle γ with the electric field
on the left wire. In this configuration, ˆ ˆ=L −v y and
ˆ ˆ ˆ= cos ( ) sin ( ).R − γ + γv y z Using these expressions in
Eqs. (32), (33), and (34) gives
2 2
2
cos(2 )[cos (2 ) 2sin ( )sin ( )] ,
sin (2 )cos ( ) ,
d
s
= α α + α γ
= + α γ
(36)
where for simplicity we have assumed that Rashba=L so Lk dα
and Rashba=R so Rk dα coincide, i.e., =L Rα α ≡ α . The nor-
malized Josephson current pertaining to this configuration,
as a function of the angle γ between the electric fields, is
shown in Fig. 4 for various values of α . Its oscillation with
respect to both γ and α is displayed in Fig. 5. Importantly,
all our illustrations are based on gate-controlled SOI
strengths that are amenable in experiment.
4. Discussion
We have considered the spin splitting of Cooper pairs
that carry a supercurrent through a weak-link Josephson
junction. Our main result, illustrated in Figs. 2–5, is the
rich oscillatory dependence of the normalized Josephson
current, 0/J J [see Eq. (18)], on both the spin-orbit cou-
pling constant α and the geometrical properties of the
junction. In the example illustrated in Fig. 1, the latter var-
iation is manifested in the dependence of 0/J J on the
bending angle θ between ˆ Lv and ˆ Rv , which are normal
to the wires connecting the dot with the left and right res-
ervoirs, respectively, and are lying in the the plane of the
junction. As seen in Fig. 2, for certain specific values of θ
and the spin-orbit coupling strength α the current vanish-
es. Another possibility to manipulate the geometry is to
‘mis-orient’ the electric fields that give rise to the spin-
orbit interactions on the weak link. Figures 4 and 5 display
the dependence of the supercurrent on the angle in-
between these two fields.
The oscillatory dependence of the supercurrent on the
SOI strength (i.e., the dependence on α in Figs. 2–5) re-
sults from a rather complex interference between different
transmission events: the single-electron transmission one,
that yields sJ , and the double-electron transmission that
gives dJ , Eqs. (14). In the single-electron transmission
channel the two electrons are transferred sequentially one
by one, so that at any time during the tunneling there is
only one electron on the bridge. By contrast, in the other
transmission channel both electrons appear in the link for
some period of time, which means that in the Coulomb-
blockade limit the transfer of Cooper pairs in this channel
Fig. 4. (Color online) The normalized Josephson current, Eq. (18),
as a function of the angle γ , the ‘mis-orientation’ of the electric fields
on the left and right nanowires [see Fig. 1 and Eqs. (36)] for various
values of = =R Lα α α . Straight (black) line — = 0α (1), tiny-
dashed (blue) curve — = 0.2α (2), medium-dashed (magenta)
curve — = 0.4α (3), large-dashed (red) curve — = 0.6α (4), dotted
(brown) curve — = 0.8α (5), dot-dashed (black) curve — = 1α (6),
dot-dashed (orange) curve — = 1.2α (7). The parameters that deter-
mine Eqs. (15) and (16) are / = 0ε ∆ and / = 5U ∆ .
Fig. 5. (Color online) A density plot of the normalized Joseph-
son current, Eq. (18), as a function of the angle γ between in-
dicating the ‘mis-orientation’ of the electric fields on the two
nanowires, and the spin-orbit coupling constant, α [see Eqs. (36)].
The parameters that determine Eqs. (15) and (16) are / = 0ε ∆
and / = 5U ∆ .
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6 707
O. Entin-Wohlman, R.I. Shekhter, M. Jonson, and A. Aharony
is completely suppressed. As the Coulomb blockade is
lifted, the probability of pairs to be transferred in the dou-
ble-electron tunneling increases. As seen from Eqs. (21),
(22), and (23), the Pauli principle operating on the dot
breaks the coherence of the pair transfer in the double-
electron tunneling process, but does not ruin completely
the contribution to the Josephson current.
The pronounced oscillations of the supercurrent and the
sign reversal can be observed for plausible lengths of the
weak link, of the order of a micron, supposedly achievable
by suitably-designed geometries of the gates. The magni-
tude of the Josephson current through a quantum dot is set
by the functions sF and dF , Eqs. (15) and (16), that are
derived for short weak links [31]. However, whereas the
restriction on the length d of the bridge might be strict,
d << ξ for the orbital part (ξ is the superconducting coher-
ence length), it is far weaker for the spin-dependent part:
Rashba
so Fk d k<< ξ, since the spin-precession factors s and
d are not sensitive to the energy dependence of the
transmission amplitude [9]. Our results indicate interesting
phenomena caused by SOI-induced spin polarization of
Cooper pairs.
An intriguing feature of our result concerns the spin-
polarization created on the dot due to the superconducting
proximity effect in conjunction with the spin-orbit cou-
pling. Calculating this polarization may require higher-
orders in the tunneling, which are beyond the scope of the
present analysis.
Acknowledgment
We thank the Computational Science Research Center
in Beijing for the hospitality that allowed for the accom-
plishment of this project. RIS and MJ thank the IBS Cen-
ter for Theoretical Physics of Complex Systems, Daejeon,
Rep. of Korea, and OEW and AA thank the Dept. of
Physics, Univ. of Gothenburg, for hospitality. This work
was partially supported by the Swedish Research Council
(VR), by the Israel Science Foundation (ISF), by the in-
frastructure program of Israel’s Ministry of Science and
Technology under contract 3-11173, by the Pazi Founda-
tion, and by the Institute for Basic Science, Rep. of Korea
(IBS-R024-D1).
Appendix A: Time-reversal symmetry and the
tunneling amplitudes
Here we discuss the effect of the time-reversal trans-
formation on the tunneling amplitudes of Eq. (5) as given
in Eqs. (6) and (7), and prove Eq. (9). Consider for in-
stance [ ]LD
′σσkt , the probability amplitude for an electron
to go from the the state | ′σ 〉 on the dot to the state | ,σ〉k on
the left lead. We denote by an overline the quantities relat-
ed to the time-reversed process. Thus, [ ]LD
′σσkt is the
probability amplitude for the time-reversed process which
takes an electron from the time-reversed state of | ′σ 〉 on the
dot, — i.e., from | ′σ 〉 — to the time-reversed state of | ,σ〉k
in the left lead, that is, to | ,− σ〉k . The time-reversal trans-
formation is given in Eq. (8), and is reproduced here for
clarity,
1ˆ ˆ= ,LD LD −
k kt Tt T (A1)
where ˆ = ( )yK iσT is the time-reversal operator; K is the
complex conjugation operator, and yσ is the Pauli matrix.
Hence,
| ( ) | = | ,
| ( ) | =| .
y
y
i
i
↑〉 ≡ σ ↑〉 − ↓〉
↓〉 ≡ σ ↓〉 ↑〉
(A2)
The spin-orbit interaction by itself is time-reversal sym-
metric, i.e., its matrix part W [see Eqs. (6) and (7)] is in-
variant under the time-reversal transformation, =W W,
while the scalar factor (i.e., ( ) ( )exp[ ]L R F L Rit ik d− is com-
plex-conjugated. It remains to find the tunneling amplitude
in the basis of the time-reversed states. To this end we use
the generic form of the linear SOI, W [see Eq. (7)]. Using
Eqs. (A2), one finds
( ) ( )[ ] = [( ) ] ,
L R D L R D ∗
′ ′σσ σσW W (A3)
which leads to the relation Eq. (9).
Appendix B: Expansion of the particle current
Upon using the expansion Eq. (12) in the expression
(10) for the particle current, one finds quite a number of
terms. However, only four of them describe the transfer of
Cooper pairs at thermal equilibrium,
____________________________________________________
1 2
2 1
1 2 3 1 2 3 1 2 3
1 2 3 1 2 3 1 2 3 2 3
[ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ]
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) .
t tt
LD DR LD DR LD LD DR DR
t tt t t t
LD LD DR DR LD DR LD DR
i dt dt dt t t t t t t t t
i dt dt dt t t t t i dt dt dt t t t t
−∞ −∞ −∞
−∞ −∞ −∞ −∞ −∞ −∞
〈 〉 + 〈 〉 −
− 〈 〉 + 〈 〉
∫ ∫ ∫
∫ ∫ ∫ ∫ ∫ ∫
(B1)
_______________________________________________
Recall that the dot is empty in the decoupled state of
the junction [30]. Examining the expressions in Eq. (B1)
in conjunction with Eq. (5) shows the following fea-
tures. (i) Each of the terms corresponds to the annihila-
tion of a pair of electrons in the right reservoir and the
creation of a pair in the left reservoir. [Note that the
708 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6
Rashba proximity states in superconducting tunnel junctions
particle current, Eq. (10), requires the imaginary part of
(B1), which means that it includes also analogous terms
corresponding to a pair creation in the right reservoir,
and a pair annihilation in the left one.] As the electrons
in each pair are in two time-reversed states, the two tun-
neling amplitudes are related according to Eq. (9). For in-
stance, the first term in Eq. (B1) is
____________________________________________________
1 2
, , , ,
† †
1 2 3 1 1 2 2 3 3
[ ] [ ] [ ] [ ]
† †( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) .
L R L R
RL
LD DR LD DR
L R
t tt
L
i
dt dt dt c t d t d t c t c t d t d t c t
′ ′ ′σ σ σ σ σ σ σ σ
′σ σ σ σ
′ ′σ − σ σ σ′ ′σ σ− σ σ
−∞ −∞ −∞
×
× 〈 〉
∑ ∑
∫ ∫ ∫
k p k p
k p
p pk k
t t t t
(B2)
_______________________________________________
(ii) Two of the terms in Eq. (B1), the first and the fourth,
correspond to sequential tunneling, in which the dot is only
singly occupied in the intermediate state. In the other two
terms, the dot is doubly occupied in the intermediate state,
and therefore the evolution of the spin states of the tunneling
pair is disrupted. (iii) The quantum averages of the operators
of the reservoirs are nonzero only in the superconducting
state, i.e., when both leads are superconducting.
The remaining part of the calculation is routine: using the
Bogoliubov transformation, one derives the time-dependent
quantum average of the operators of the reservoirs. Those on
the dot are calculated using the Hamiltonian of the decoupled
dot [the first two terms on the right hand-side of Eq. (2)].
In this way, the expression in Eq. (B2) becomes
( 2 2
,
)2e | | | |
1 1 1 .
2 2
i R L
sL R
k p k k p p
t t
E E E E E E
ϕ −ϕ− ×
∆ ∆
×
+ ε + + ε
∑ k p
k p
(B3)
where 2 2 2
( ) ( ) ( )=k p k p L RE ε + ∆ . For simplicity, the tempera-
ture is set to zero. In deriving this expression, we have
made use of Eq. (9), that relates the tunneling amplitudes
of two time-reversed events. The factor s describes the
spin precession in the sequential tunneling processes [i.e.,
the first and the fourth terms in Eq. (B2)]. Explicitly,
,
, ,
1= sgn ( )sgn ( )[( ) ]
2
[ ] [( ) ] [ ] .
L
L R
L
s LD
L R
LD DR DR
R R
∗
′σ σ
′σ σ σ σ
∗
′σ σ σ σ σσ
σ σ ×
×
∑ ∑ W
W W W
(B4)
As in the absence of the SOI the matrices W are all just the
unit matrix, the spin-precession factor s becomes then 1.
The sums over k and p are carried out assuming that
2
( )| |tk p and the single-particle density of states of the
leads can be approximated by their respective values at the
Fermi energy, 2
( )| |L Rt and ( )L RN . For a short weak link
with =L R∆ ∆ ≡ ∆ [31], these sums then give
s( / ) ( / )L R FΓ Γ ∆ ε ∆ , where 2
( ) ( ) ( )= | |L R L R L RN tΓ π , and
the function sF is
1
( ) =
[(cosh )(cosh cosh )(cosh )] ,
= / .
s k
p
k k p p
d
F
d
∞
−∞
∞
−
−∞
ζ
ε ×
π
ζ
× ζ + ε ζ + ζ ζ + ε
π
ε ε ∆
∫
∫
(B5)
An identical result is obtained for the fourth term in the
expansion (B1). The imaginary part of the expression in
Eq. (B3) consists of three factors, the Josephson amplitude
of the interface between the two superconductors (i,e., in
the absence of the resonant level on the dot and the SOI),
0 = 2sin( )[ / ]R L L RI ϕ −ϕ Γ Γ ∆ , the function sF that con-
veys the effect of the localized level on the dot, and the
spin-precession factor, s . The two latter factors are dis-
cussed in Sec. 3.
The second term in Eq. (B1), which pertains to the situ-
ation where during the tunneling process the dot is doubly
occupied, reads
1 2
, , ,
†
1 2 3 1 1 2
†
2 3 3
[ ] [ ] [ ] [ ]
† †( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ,
L R L R
R
L L
R R
LD DR LD DR
L
t tt
i
dt dt dt c t d t c t d t d t
c t d t c t
σ σ σσ σ σ σσ
σ σ σ
σ σ σ− σ σ
−∞ −∞ −∞
− σ σ σ
×
× 〈 ×
× 〉
∑ ∑
∫ ∫ ∫
k p k p
k p
k k
p p
t t t t
(B6)
where we have taken into account the Pauli principle, and
therefore there are only three summations over the spin
indices [c.f. Eq. (B2)]. In this case we obtain
( 2 2
,
1 1 1)4e | | | | ,
2 2 2
i dL RR L
k p k p
t t
E E E U E
ϕ ∆ ∆−ϕ−
+ε ε+ +ε∑ k p
k p
(B7)
where d describes the spin precession in the tunneling
processes in which the two electrons reside simultaneously
on the dot in the intermediate sate [i.e., the second and the
third terms in Eq. (B2)]. Its explicit form is
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6 709
O. Entin-Wohlman, R.I. Shekhter, M. Jonson, and A. Aharony
,
1= sgn ( )sgn ( )[( ) ]
2
[ ] [( ) ] [ ] .
L
L R
L R R
d LD
L R
LD DR DR
∗
σ σ
σ σ σ
∗
σ σ σσ σσ
σ σ ×
×
∑ ∑ W
W W W
(B8)
Similar to s , this factor also becomes 1 in the absence of
the SOI. The sums over k and p are carried out as ex-
plained above. Because of the double occupancy of the dot,
the energy denominators in Eq. (B7) differ from those in
Eq. (B3). These summations give rise to another function,
( / , / )dF Uε ∆ ∆ , of the energies on the dot [31]
1
( , ) =
[(cosh )(2 )(cosh )] ,
= / , = / .
d
pk
k p
F U
dd
U
U U
∞ ∞
−
−∞ −∞
ε
ζζ
= ζ + ε ε + ζ + ε
π π
ε ε ∆ ∆
∫ ∫
(B9)
Since the third term in the expansion (B1) turns out to be
identical to Eq. (B7), it follows that the contribution from
these tunneling processes to the Josephson current is again
a product of three factors, 0I , dF , and d .
_______
1. J. Lehmann, A. Gaita-Arino, E. Coronado, and D. Loss,
Nanotechnology 2, 312 (2007).
2. C. Kloeffel and D. Loss, Annu. Rev. Condens. Matter Phys.
4, 51 (2013).
3. I. Vobornik, U. Manju, J. Fujii, F. Borgatti, P. Torelli, D.
Krizmancic, Y.S. Hor, R.J. Cava, and G. Panaccione, Nano
Lett. 11, 4079 (2011).
4. S. Hermelin, S. Takada, M. Yamamoto, S. Tarucha, A.D.
Wieck, L. Saminadayar, C. Bäuerle, and T. Meunier, Nature
477, 435 (2011).
5. R.P.G. McNeil, M. Kataoka, C.J.B. Ford, C.H.W. Barnes, D.
Anderson, G.A.C. Jones, I. Farrer, and D.A. Ritchie, Nature
477, 439 (2011).
6. B. Bertrand, S. Hermelin, P.-A. Mortemousque, S. Takada,
M. Yamamoto, S. Tarucha, A. Ludwig, A.D. Wieck, C.
Bäuerle, and T. Meunier, Nanotechnology 27, 214001
(2016).
7. A.E. Popescu and R. Ionicioiu, Phys. Rev. B 69, 245422
(2004).
8. M. Yamamoto, S. Takada, C. Bäuerle, K. Watanabe, A.D.
Wieck, and S. Tarucha, Nanotechnology 7, 247 (2011). See
also A. Aharony, S. Takada, O. Entin-Wohlman, M.
Yamamoto, and S. Tarucha, New J. Phys. 16, 083015 (2014).
9. See, e.g., R.I. Shekhter, O. Entin-Wohlman, M. Jonson, and
A. Aharony, Fiz. Niz. Temp. 43, 368 (2017) [Low Temp.
Phys. 43, 303 (2017)].
10. A.V. Khaetskii and Y. Nazarov, Phys Rev. B 61, 12639
(2000).
11. K. Yakushiji, F. Ernult, H. Imamura, K. Yamane, S. Mitani,
K. Takanashi, S. Takahashi, S. Maekawa, and H. Fujimori,
Nature Mater. 4, 57 (2004).
12. S. Amasha, K. MacLean, I.P. Radu, D.M. Zumbühl, M.A.
Kastner, M.P. Hanson, and A.C. Gossard, Phys. Rev. Lett.
100, 046803 (2008).
13. P.N. Hai, S. Ohya, and M. Tanaka, Nature Nanotechnol. 5,
593 (2010).
14. M.S. Rudner and E.I. Rashba, Phys. Rev. B 81, 125426
(2010).
15. J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys.
Rev. Lett. 78, 1335 (1997).
16. Y. Sato, T. Kita, S. Gozu, and S. Yamada, J. Appl. Phys. 89,
8017 (2001).
17. K. Flensberg and C.M. Marcus, Phys. Rev. B 81, 195418
(2010).
18. A.J.A. Beukman, F.K. de Vries, J. van Veen, R. Skolasinski,
M. Wimmer, F. Qu, D.T. de Vries, B.-M. Nguyen, W. Yi,
A.A. Kiselev, M. Sokolich, M.J. Manfra, F. Nichele, C.M.
Marcus, and L.P. Kouwenhoven, Phys. Rev. B 96, 241401
(2017).
19. R.I. Shekhter, O. Entin-Wohlman, and A. Aharony, Phys.
Rev. Lett. 111, 176602 (2013).
20. R.I. Shekhter, O. Entin-Wohlman, and A. Aharony, Phys.
Rev. B 90, 045401 (2014).
21. E.I. Rashba, Fiz. Tverd. Tela 2, 1224 (1960) [Sov. Phys.
Solid State 2, 1109 (1960)].
22. Y.A. Bychkov and E.I. Rashba, P. Zh. Eksp. Teor. Fiz. 39,
66 (1984) [JETP Lett. 39, 78 (1984)].
23. R.I. Shekhter, O. Entin-Wohlman, M. Jonson, and A.
Aharony, Phys. Rev. Lett. 116, 217001 (2016).
24. The quantization axis of the spins is taken to be the same,
along the z-direction, in both the superconducting leads and
the dot (it can be set, e.g., by a small external magnetic field
along that direction).
25. This statement can be justified in the limit where the energy
of the electrons passing simultaneously through the wires
(that bridge the dot with the leads) exceeds significantly the
superconducting energy gap, see Eqs. (15) and (16), and also
R.I. Shekhter, O. Entin-Wohlman, M. Jonson, and A.
Aharony, Phys. Rev. B 96, 241412(R) (2017).
26. G. Dresselhaus, Phys. Rev. 100, 580 (1955).
27. T.V. Shahbazyan and M.E. Raikh, Phys. Rev. Lett. 73, 1408
(1994).
28. O. Entin-Wohlman, A. Aharony, Y.M. Galperin, V.I. Kozub,
and V. Vinokur, Phys. Rev. Lett. 95, 086603 (2005).
29. M. Gisselfält, Physica Scripta 54, 397 (1996).
30. The other possibility, of a dot which is initially occupied, is
also of interest. We hope to pursue the spin-splitting
phenomenon for this configuration in the future.
31. L.I. Glazman and K.A. Matveev, P. Zh. Eksp. Teor. Fiz. 49,
570 (1989) [JEPT Lett. 49, 660 (1989)].
___________________________
710 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6
https://doi.org/10.1146/annurev-conmatphys-030212-184248
https://doi.org/10.1021/nl201275q
https://doi.org/10.1021/nl201275q
https://doi.org/10.1038/nature10416
https://doi.org/10.1038/nature10444
https://doi.org/10.1088/0957-4484/27/21/214001
https://doi.org/10.1103/PhysRevB.69.245422
https://doi.org/10.1088/1367-2630/16/8/083015
https://doi.org/10.1063/1.4976637
https://doi.org/10.1063/1.4976637
https://doi.org/10.1103/PhysRevB.61.12639
https://doi.org/10.1038/nmat1278
https://doi.org/10.1103/PhysRevLett.100.046803
https://doi.org/10.1038/nnano.2010.130
https://doi.org/10.1103/PhysRevB.81.125426
https://doi.org/10.1103/PhysRevLett.78.1335
https://doi.org/10.1103/PhysRevLett.78.1335
https://doi.org/10.1063/1.1362356
https://doi.org/10.1103/PhysRevB.81.195418
https://doi.org/10.1103/PhysRevB.96.241401
https://doi.org/10.1103/PhysRevLett.111.176602
https://doi.org/10.1103/PhysRevLett.111.176602
https://doi.org/10.1103/PhysRevB.90.045401
https://doi.org/10.1103/PhysRevB.90.045401
https://doi.org/10.1103/PhysRevLett.116.217001
https://doi.org/10.1103/PhysRev.100.580
https://doi.org/10.1103/PhysRevLett.73.1408
https://doi.org/10.1103/PhysRevLett.95.086603
https://doi.org/10.1088/0031-8949/54/4/018
1. Introduction
2. Tunneling of Cooper pairs
2.1. Description of the model
2.2. The particle current
2.3. Spin precession
3. Results
4. Discussion
Acknowledgment
Appendix A: Time-reversal symmetry and the tunneling amplitudes
Appendix B: Expansion of the particle current
|
| id | nasplib_isofts_kiev_ua-123456789-176152 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T17:27:08Z |
| publishDate | 2018 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Entin-Wohlman, O. Shekhter, R.I. Jonson, M. Aharony, A. 2021-02-03T19:05:44Z 2021-02-03T19:05:44Z 2018 Rashba proximity states in superconducting tunnel junctions / O. Entin-Wohlman, R.I. Shekhter, M. Jonson, A. Aharony // Физика низких температур. — 2018. — Т. 44, № 6. — С. 701-710. — Бібліогр.: 31 назв. — англ. 0132-6414 PACS: 72.25.Hg, 72.25.Rb https://nasplib.isofts.kiev.ua/handle/123456789/176152 We consider a new kind of superconducting proximity effect created by the tunneling of “spin split” Cooper pairs between two conventional superconductors connected by a normal conductor containing a quantum dot. The difference compared to the usual superconducting proximity effect is that the spin states of the tunneling Cooper pairs are split into singlet and triplet components by the electron spin-orbit coupling, which is assumed to be active in the normal conductor only. We demonstrate that the supercurrent carried by the spin-split Cooper pairs can be manipulated both mechanically and electrically for strengths of the spin-orbit coupling that can realistically be achieved by electrostatic gates. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Специальный выпуск К 90-летию со дня рождения A.A. Абрикосова Rashba proximity states in superconducting tunnel junctions Article published earlier |
| spellingShingle | Rashba proximity states in superconducting tunnel junctions Entin-Wohlman, O. Shekhter, R.I. Jonson, M. Aharony, A. Специальный выпуск К 90-летию со дня рождения A.A. Абрикосова |
| title | Rashba proximity states in superconducting tunnel junctions |
| title_full | Rashba proximity states in superconducting tunnel junctions |
| title_fullStr | Rashba proximity states in superconducting tunnel junctions |
| title_full_unstemmed | Rashba proximity states in superconducting tunnel junctions |
| title_short | Rashba proximity states in superconducting tunnel junctions |
| title_sort | rashba proximity states in superconducting tunnel junctions |
| topic | Специальный выпуск К 90-летию со дня рождения A.A. Абрикосова |
| topic_facet | Специальный выпуск К 90-летию со дня рождения A.A. Абрикосова |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/176152 |
| work_keys_str_mv | AT entinwohlmano rashbaproximitystatesinsuperconductingtunneljunctions AT shekhterri rashbaproximitystatesinsuperconductingtunneljunctions AT jonsonm rashbaproximitystatesinsuperconductingtunneljunctions AT aharonya rashbaproximitystatesinsuperconductingtunneljunctions |