Spin Hall effect in a Josephson contact
A. G. Mal’shukov1,2,3and C. S. Chu2,31Institute of Spectroscopy, Russian Academy of Sciences, 142190 Troitsk, Moscow oblast, Russia 2Department of Electrophysics, National Chiao Tung University, Hsinchu 30010, Taiwan
3Physics Division, National Center for Theoretical Sciences, Hsinchu 30043, Taiwan 共Received 30 May 2008; published 9 September 2008兲
The spin Hall effect on the Josephson tunneling through a two-dimensional normal contact with a spin-orbit split conduction band has been studied in the diffusive regime and at the zero electric bias. Linearized Usadel equations for triplet components of the pairing function in the presence of intrinsic spin-orbit interaction have been derived. These equations have been employed for analysis of the spin Hall effect induced by the super-current. We predict that a nondissipative out-of-plane spin Hall polarization accumulates at lateral edges and an in-plane polarization is induced throughout the entire normal region. At the same time, in contrast to the spin Hall effect in normal systems, the spin current is absent in the considered case of the stationary Josephson effect.
DOI:10.1103/PhysRevB.78.104503 PACS number共s兲: 72.25.Dc, 71.70.Ej, 73.40.Lq, 74.50.⫹r
I. INTRODUCTION
Various spintronic applications have attracted much recent interest in the spin-orbit interaction共SOI兲 effects on electron transport in normal metals and semiconductors. This interac-tion gives rise to fundamental transport phenomena, such as the spin Hall effect 共SHE兲 共for a review, see Ref. 1兲, and electric spin orientation.1,2 The former shows up in a spin-polarization flux flowing perpendicular to the electric cur-rent. This flux, in its turn, gives rise to the spin polarization at the sample boundaries. The spin polarization can also be directly induced by the electric current in the sample bulk, that is, the electric spin orientation effect. These effects dem-onstrate delicate coupling of spin and charge degrees of free-dom in electron transport. On the other hand, we have the interplay between SOI and superconductivity. Indeed, vari-ous systems including superconductor-ferromagnet super-conductor junctions3 共F stands for ferromagnet兲, superconductor-normal-superconductor 共SNS兲4,5 and superconductor-normal 共SN兲 共Refs. 6兲 contacts, or bulk superconductors,6,7 have been considered. In Refs. 4 and 5 the effect of the SOI onto the Josephson current has been studied in the case when the normal共N兲 part of a SNS junc-tion is a normal metal with a noticeable SOI. We will, how-ever, focus on a different problem. Namely, we are going to consider the spin Hall current and spin accumulation associ-ated with the supercurrent across the junction. As was pointed out in Refs. 6 and7, SOI causes the admixture of triplet components in the pairing function. This sort of singlet-triplet coupling finds similarity in the spin-charge coupling in normal systems. Thus, one might expect that phenomena closely related to SHE could manifest them-selves also in superconducting structures, such as SNS junc-tions. But, after a moments’ thought, it becomes clear that this analogy cannot go too far. For example, at least in the case of zero voltage across the junction, the spin Hall current cannot be a linear response to the supercurrent. It is because, on the one hand, these two currents are, by nature, of oppo-site parities in time; but, on the other hand, the parities must be the same in the case of stationary nondissipative
super-conducting transport. Yet, it is still of great interest to look for another signature of SHE: the accumulation of magneti-zation in response to the supercurrent in superconducting structures. It should be noted that, despite a formal similarity, such a magnetization is fundamentally distinct from that in-duced by the normal SHE since it is determined by a coher-ent many-particle quantum state and, hence, is not subject to dissipative processes of spin diffusion and relaxation that take place in normal systems.
We will consider SHE and the electric spin orientation in the case of the Josephson tunneling through a two-dimensional 共2D兲 N contact 共see Fig. 1兲. The SOI in the normal contact may be caused by impurities or it can be of an intrinsic origin, due to the crystal field in noncentrosym-metric lattices. We will consider the intrinsic SOI represented by the Hamiltonian Hso=· hk, where is the Pauli spin
vector. The spin-orbit field hk, which is an odd function of
the electron wave vector k, can be given, for example, by Rashba8 or Dresselhaus9 SOI, as well as by their combina-tion. In this case the vector hk lies in the plane of the 2D
system. The electron transport through the contact will be treated within the diffusion approximation so that the length of the junction L, the condensate wave-function penetration depth into N region Lc, and the spin precession length Lso =vF/2h with vF as the Fermi velocity and h as the angular-averaged spin-orbit field are all assumed to be much larger than the electron mean-free path l. For our Josephson setting,
FIG. 1. Josephson contact with S and N denoting superconduct-ing and normal regions, respectively.
the bias voltage across the junction is zero and the supercur-rent is determined by the phase difference between the two S electrodes. Our analysis involves a standard semiclassical treatment of Gor’kov’s equations in the diffusion approxima-tion共for a review, see Ref.10兲. Our goal is to derive Usadel-type equations and to calculate the spin density induced by the SHE.
II. USADEL EQUATIONS
In the considered case of the thermal equilibrium, all ob-servables of interest can be expressed via retarded and ad-vanced Green’s functions. The corresponding Gor’kov’s equations in the Nambu representation have the form
冉
it− Hˇ − ⌺ˇ
r/a
冊
Gˇr/a共X,X
⬘
兲 =␦共X − X⬘
兲, 共1兲where r and a denote retarded and advanced functions, re-spectively, X =共t,r兲 and
Hˇ = 3
2mⴱkˆ 2−
3+· hkˆ, 共2兲
with the momentum operator kˆ = −i/r, and the chemical potential. After averaging the initial Green’s functions over random positions of short-range impurities, the self-energy in Eq. 共1兲 takes the form11
⌺ˇr/a共X,X
⬘
兲 = 3 2NFGˇr/a共t,t
⬘
,r,r兲3␦共r − r⬘
兲, 共3兲 where is the elastic-scattering time. Unperturbed Green’s functions are easily obtained from Eq.共1兲. In the momentum representation and after the time Fourier transform, they can be written asGˇ0r/a共,k兲 = 共−3Ek−· hk⫾ i⌫兲−1, 共4兲
where Ek=共k2/2mⴱ兲−. Below we will perform calculations for retarded functions and drop the labels r and a.
Proximity to superconducting contacts results in appear-ance of anomalous共proportional to 1and2兲 Green’s func-tions. These functions are inhomogeneous in space. In order to calculate them, we will follow a well-known procedure in the framework of the semiclassical approximation.12 First, we perform the Fourier transform with respect to X − X
⬘
in-troducing, accordingly, the frequency and wave-vector vari-ables and k. The center-of-mass variables will remain in-tact and will be denoted as r. Since the problem is stationary, the corresponding center-of-time variable is absent. Taking into account that variations of G on the scale of the Fermi wavelength are small, Eq.共1兲 should be expanded in terms of gradients /r. The next step is to simplify the self-energy part of Eq. 共1兲, keeping only the terms linear in the anoma-lous part. Such a linearization can be done if the transpar-ency of the SN contact is small or the leads are close to the superconducting critical temperature. By combining Eq. 共1兲 and its conjugate one and by making use of the fact that hkisan odd function of k for the anomalous part G12, we obtain the equation
冉
2−v · qˆ +i 冊
G12−兵hk·,G12其 − 1 2关␦hk,qˆ·,G12兴 = Isc, 共5兲 where␦hk,qˆ=共qˆ·ⵜk兲hkwith qˆ = −i/r andIsc= − 1 2NF
共G110g
12+ g12G220 兲. 共6兲 The subscripts of G0 denote the matrix elements in the Nambu space and g12=兺kG12. The 2⫻2 matrix G12,␣ can be transformed to the conventional pairing function F␣¯ ⬅G12,␣, where¯ denotes the spin projection opposite to. Further, it is convenient to decompose F into triplet F1,F−1, and F0 and singlet Fscomponents as
F0=F12
冑
+ F21 2 , Fs=F12− F21
冑
2F1= F11, F−1= F22. 共7兲
After this transformation, it is easy to see that the last term in the left-hand side of Eq. 共5兲 is responsible for a coupling between the singlet and triplet components of the pairing function. Besides, the singlet-triplet coupling also originates from the spin-dependent parts of G110 and G220 in Eq.共6兲. Due to such coupling, the triplet component of F is generated within the junction between two singlet S electrodes.
The Usadel diffusion equation can be obtained from Eq. 共5兲 by iterating it with respect to small,共v·qˆ兲, and hk
up to the second order in the last two parameters. By this way, components of Fmare expressed in terms of Iscand the last term in the left-hand side of Eq.共5兲. Further summation over k leads to the closed diffusion equation for fm=兺kFm.
The derivation of the diffusion equation and the following analysis will be restricted to a limiting case of the strong SOI so that LsoⰆLc. From the theory of SNS contacts,10 it fol-lows that Lc= min共L,LT兲 where LT=
冑D
/kBT is the thermal diffusion length. It is assumed that the energy gap in super-conducting contacts 兩⌬兩ⰇkBT and D/L2. If the N region is represented by a narrow gap semiconductor quantum well,Lsomay vary from less than a micron to several microns. For example, the Dresselhaus interaction in a GaAs/AlGaAs quantum well provides the spin splitting 2h = 0.1 meV at n = 5⫻1011 cm−2共Ref.13兲 that gives Lso⬇2 m. This length strongly decreases, if a strong Rashba SOI takes place in addition to the Dresselhaus interaction. Hence, at low enough temperatures and the junction lengths of severalm, the requirement LsoⰆLc can be realized in practice. Taking the leading terms we arrive at the following diffusion equa-tion for the triplet pairing funcequa-tion fm=兺kFm where 共m = 0 , 1 , −1兲:
2if =
冓
冉
− iv · r+ 2J · hk
冊
2冔
f + Mfs, 共8兲
where J is the vector of 3⫻3 angular moment operators and 具...典 denotes the angular averaging over the Fermi surface. The triplet-singlet coupling is given by
M0= 0, M⫾1= 42
冑
2具hk ⫿共h k⫻␦hk,qˆ兲典, 共9兲 with hk⫿= hk x⫿ih k y. The singlet fs satisfies the similar equa-tion, where the second term in angular brackets is absent and the mixing with triplets is represented by the Hermitian con-jugate to M. Since triplets, in their turn, are expressed through fs, such a mixing gives rise to a correction term in the closed equation for fs. From Eqs.共8兲 and 共9兲 it is easy to evaluate this correction as ⬃h22/共kF
2
Lso2兲fsⰆ2fs. There-fore, the effect of SOI on fs and, hence, on the Josephson current is weak in the semiclassical case, which is in agree-ment with Ref. 5. We also neglect a depairing effect on fs due to exchange Zeeman field associated with the finite spin Hall polarization. This effect is weak as evaluated in the discussion in Sec. IV. Hence, the singlet function fsis given by the well-known unperturbed solution in the SNS contact. Without the last term in the right-hand side in Eq. 共8兲, it 关Eq. 共8兲兴 formally coincides with the spin-diffusion equation for two-dimensional electron gas 共2DEG兲 in a zero electric field.14 The spin-diffusion equation in the presence of the electric field has been derived in Ref.15for the case of the Rashba SOI and for a general SOI in Ref. 16. After a linear transformation to spin-density variables,14 Equation共8兲 will coincide with these more general equations, if, apart from a constant factor, fsis formally identified with the electric-field potential. Hence, a coupling of the spin to the electric field in normal spin transport appears to be very similar to the singlet-triplet coupling in Eq.共8兲. We note, however, a prin-cipal distinction from the normal transport. Equation 共8兲 is written not for an observable spin density but for the anoma-lous Green’s function, which plays the role of the Cooper pair wave function. Hence, the observables, such as spin den-sities, cannot be directly represented by a solution of the diffusion equation but must be calculated from bilinear com-binations of fmas will be done below.
III. SPIN DENSITY
Let us consider an example of the Rashba SOI. In this case hkx=␣ky and hk
y
= −␣kx. For a homogeneous in y direc-tion case, all funcdirec-tions depend only on x and we get f0= 0 and f1= f−1 with f1satisfying the equation
D 2 x2f1−⌫sof1= i ␣⌫so
冑
2 xfs, 共10兲 where ⌫so= 2␣2k F2 is the D’yakonov-Perel’ spin-relaxation time.17The small left hand side of Eq.共8兲 has been neglected in Eq.共10兲. Neglecting the third and higher derivatives of fs, the solution of Eq. 共10兲 can be written as
f1= − i
␣
冑
2
xfs+共x兲, 共11兲
where, in order to satisfy appropriate boundary conditions18 at x =⫾L/2, 共x兲 is taken as a linear combination of exp共⫾kx兲 with k=
冑D
/⌫so= 1/Lso. Since LⰇLso, this func-tion is important only close to the boundaries. We, however, are interested in the bulk solution given by the first term in Eq. 共11兲.Our next step is to calculate the spin-polarization density associated with triplet components of the pairing function. This polarization is given by
Si共r兲 = i 2
兺
k冕
d 2nF共兲 ⫻ Tr兵 i关G k11 r 共 ,r兲 − Gk11 a 共 ,r兲兴其, 共12兲 where nF is the equilibrium Fermi distribution function. It is easy to see that the nonzero value of Eq.共12兲 is provided by triplet components of anomalous Green’s functions, which contribute to G11with a correction term⬀f2. Up to the lead-ing second order with respect to fs and keeping only the linear terms of the triplet fm 共m=1,−1,0兲, for the retarded function we obtain from Eqs.共1兲–共4兲,兺
k Tr关i Gk11r/a兴 = ⫿1 NF冋
i␦iz 2 共f0 r/a fs+r/a− fs r/a f0+r/a兲 +冑
1 2共fi r/a fs+r/a+ fs r/a fi+r/a兲册
, 共13兲 where fy=共f1+ f−1兲/2 and fx= −i共f1− f−1兲/2. The conjugate functions f+共兲=−fⴱ共−兲.In the case of Rashba SOI fx= f0= 0 and fy= f1. The latter is given by Eq. 共11兲. Then, from Eq. 共13兲 it immediately follows that only the y projection of the spin density is finite. Using the relations fs
a共兲= f s r共−兲 and f m a共兲=−f m r共−兲 共m = 1 , −1 , 0兲, we arrive to the spin polarization,
Sy共x兲 = eNF␣
J dc
, 共14兲
wheredcis the dc conductivity of the normal metal and J is the Josephson current density,
J = eD 42NF
冕
dnF共兲冋
冉
fs rfs+r x − fs r xfs +r冊
−共r a兲册
. 共15兲 The spin polarization in Eq.共14兲 coincides with polarization induced in normal metals by the electric field E,2 if the Jo-sephson current is substituted for the normal dissipative dc current Jdc=dcE. Similar effect has been predicted by Edelstein6 for bulk superconductors and at NS boundary, providing the supercurrent flows along the SN interface.Let us now check, if the analogy with the electric spin orientation extends to the spin Hall effect. Hence, our goal is to calculate Jy
z
, which is the y projection of a spin flux po-larized in the z direction. The corresponding spin current operator can be written as Jy
z
=兵z, vy其/2 where the velocity
vy= ky/mⴱ+共· hk兲/ky. Since it has been assumed that hz = 0, one gets Jy
z
=zky/mⴱ. The spin Hall current JsH, in its turn, can be derived from Eq.共12兲 withisubstituted for J
y z . Keeping the same leading terms as in calculation of the spin density, we arrive at JsH= 0. This result does not depend on whether hk is given by the Rashba or Dresselhaus
interac-tions. That is very distinct from the normal spin Hall effect, where in the diffusive regime the spin Hall conductance is zero for the Rashba SOI, but finite for the cubic Dresselhaus interaction.19 In general, as it was discussed above, the zero
value of JsH in superconducting transport follows from the time inversion symmetry.
Besides JsH, in normal systems the dc current together with SOI gives rise to the accumulation of the z component of spin at the lateral edges of the sample.16,20–22In the case of the Josephson junction, the z projection of the spin density is given by Eq.共12兲 and the first term in Eq. 共13兲. Hence, it is proportional to the f0 component of the pairing function which, in its turn, can be found from Eq. 共8兲, which near lateral edges of the sample, takes the form
0 = D 2f 0 y2 + 4i具vyhk典
冉
J01 f1 y + J0,−1 f−1 y冊
− 2⌫sof0, 0 = D 2f i y2+ 4i具vyhk典Ji0 f0 y −⌫so共fi− fi b兲. 共16兲 where the subscript i =⫾1 and fib
= −Mifs/⌫so. For simplicity we assumed that具hk
x
hk
y典=0, which does not take place if SOI is, for example, the sum of Rashba and Dresselhaus interac-tions. The boundary conditions to this equation depend on the type of the boundary as discussed in Refs.16,21, and22. Let us consider hard wall boundaries of 2DEG at y =⫾Ly/2. In this case one can borrow the boundary condi-tions for Eq. 共8兲 from Refs. 16 and21. In normal systems these conditions correspond to the vanishing spin current at
y =⫾Ly/2. In our case similar equations can be written for triplet “currents” j =兺kvF. We thus have jy兩y=⫾Ly/2= 0 where the zero triplet component is given by
j0y= − Df0
y − 2i具兵vy关hk⫻ 共
冑
2f +␦hk,qˆfs兲兴其典, 共17兲where f =共fx, fy兲. The first term in this equation is the diffu-sive current, the first term in the brackets is determined by the spin precession in the effective spin-orbit field, and the last term is associated with the singlet-triplet coupling. The following analysis of Eq.共16兲 with the above boundary con-ditions is the same as for SHE in normal systems. Indeed, in the case of Rashba SOI, taking fx= 0 and fy= f1= f−1⬅ f⫾1b in the form of the first term of Eq.共11兲, one can easily see that the second term in Eq.共17兲 vanishes. Hence, the solution of Eq. 共16兲 is simply f0= 0 and fi= fi
b
. This, according to Eqs. 共12兲 and 共13兲, leads to Sz= 0. For other type of boundaries the resulting z polarization may be finite.22It is also finite in the case of the hard wall boundary and cubic Dresselhaus SOI
hkx=␥kx共ky
2−2兲 and h
k
y
= −␥ky共kx
2−2兲. It is easy to see that for this interaction the second term in Eq.共17兲 does not turn to zero. Calculating the spin-charge coupling in Eq.共9兲 and
f⫾1b , the solution of Eq.共8兲 can be expressed in the form f0 =共y兲共fs/x兲 where is a real function of y. Then, Eqs. 共12兲, 共13兲, and 共15兲 give
Sz= − eNF共y兲共J/dc兲. 共18兲 The numerical plots for the function , in their turn, can be taken from Fig. 1 of Ref. 16.
IV. DISCUSSION
For a numerical evaluation of the spin polarization, we write an expression for the critical Josephson current through a long SNS junction in the form23 J
c= adc⑀T/eL where⑀T = D/L2 is the Thouless energy and a is a dimensionless pa-rameter. The latter increases up to ⬃10 at kBT⬍⑀Tand de-creases at higher temperatures. In its turn, the spin polariza-tion in Eq. 共14兲 can be written in the form Sy= N
F⌬0/2, where⌬0is an effective spin splitting, which has been mea-sured in Ref.24by the Faraday rotation method. The z com-ponent of the spin polarization is of the same order of mag-nitude as Sy because Eqs. 共16兲 and 共17兲 acquire a dimensionless form when y is measured in units of Lso and
f0, f⫾1are rescaled as f0/ f1b, f⫾1/ f1b. Substituting the above expression for the Josephson current into Eq.共14兲 and taking into account that D= l2/2, we get
⌬0=␣al
2
L3. 共19兲
Taking the Rashba parameter of an asymmetric InAs-based quantum well25 ␣= 5⫻10−12 eV m, for L = 1 m, l = 0.1 m, and a⬃5 at ⌬/⑀T= 100, kBT/⑀T= 5共see Ref.23兲, we arrive at⌬0= 0.25 eV. For comparison, we note that the Faraday rotation method24 allows to measure even smaller values of ⌬0. It should be noted that the above number is evaluated on the verge of the diffusion approximation appli-cability since Lsois only three times larger than l. Also Lsois not sufficiently small compared to L and LT so that the ap-proximation of the strong SOI is not very accurate. An obvi-ous trend, however, is that the spin density increases with higher l, stronger SOI, and larger a. Therefore, a theory valid in the case of short ballistic junctions carrying high Joseph-son currents is necessary to study the regime where SHE is able to induce high spin densities. It should be taken into account that at the higher magnetization, the depairing effect due to an exchange interaction ignored in Eqs. 共5兲 and 共8兲 might become important. The exchange Zeeman energy pro-duced by polarized electrons can be evaluated as Eex ⬃e2S/共k
F⑀0兲. Taking ⌬0= 0.25 eV and kFaBⴱ⬃1, where the effective Bohr radius is aBⴱ=ប2⑀0/mⴱe2, with ⑀0 ⬃10, mⴱ/m=0.23 we obtain Eex⬃4·10−2 eV. This en-ergy is much less than⑀Tand, hence, the Cooper pair energy 2 in the Usadel Eq. 共8兲. Therefore, for the chosen param-eters this depairing effect can be ignored.
In conclusion, the stationary spin Hall effect induced by a supercurrent across an SNS junction has been studied in the diffusive regime for a relatively strong SOI in the 2D junc-tion. We found out that the spin Hall current is forbidden by the time inversion symmetry. On the other hand, the out-of-plane magnetization accumulates at lateral edges in a very close analogy to SHE in normal systems. Also, similar to the electric spin orientation, the spin polarization parallel to 2DEG is finite throughout the entire N region. On the other hand, such a close analogy takes place only in the considered above limiting case of a long junction. We also expect a behavior quite distinct from SHE in normal systems in the case of the Josephson effect driven by a dc electric bias.
ACKNOWLEDGMENTS
This work was supported by Russian RFBR with Grant
No. 060216699, by Taiwan NSC with Grant No. 96-2811-M-009-038–MY3 and NCTS with Grant No. 93-2119-M-007-002, and by MOE-ATU Grant.
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