Andreev Reflection and Spin Injection into $s-$ and $d-$wave Superconductors

4 Pages·0.12 MB·English

by  Robert L. Merrill

Andreev Reflection and Spin Injection into s and d wave Superconductors
− −
Robert L. Merrill and Qimiao Si
Department of Physics, Rice University, Houston, TX 77251-1892
(February 1, 2008)
Westudytheeffectofspininjectionintos−andd−wavesuperconductors,withanemphasisonthe
9 interplay between boundary and bulk spin transport properties. The quantities of interest include
9 the amount of non-equilibrium magnetization (m), as well as the induced spin-dependent current
9 (Is)andboundaryvoltage(Vs). Ingeneral,theAndreevreflectionmakeseachofthethreequantities
1
dependonadifferentcombination oftheboundaryandbulkcontributions. Thesituation simplifies

n either for half-metallic ferromagnets or in the strong barrier limit, where both Vs and m depend
a solely on the bulk spin transport/relaxation properties. The implications of our results for the
J on-going spin injection experimentsin high Tc cuprates are discussed.
7
1 PACS numbers: 71.10.Hf, 73.40. -c,71.27. +a, 72.15.Gd

] A number of spin-injection experiments have recently zation. We will show below that, in general these quan-
l
e been carried out for the high T cuprates [1–5]. These tities reflect very different combinations of the bulk spin
c
- experiments are of interest for a variety of reasons. In transport and interface transport contributions.
r
t particular, they in principle allow us to extract the bulk TherelevantexperimentalsetupsareillustratedinFig.
s
. spin transportproperties of the high T cuprates. In the 1. Fig. 1a) is representative of those used in the on-
t c
a normal state, such spin transport properties have been going spin-injection experiments in the high T cuprates
c
m
proposed as a probe of spin-charge separation [6,7]. In [1–4]. An injection current (I, per unit area) from a
- the superconducting state, they can provide important ferromagnetic metal (FM1) to the superconductor (S) is
d
clues to the nature of the quasiparticles. In addition, applied,andthe criticalcurrentofthe superconductoris
n
o the anisotropy of the spin transport properties should then measured. The suppression of the critical current,
c shed new light on the nature of the c axis transport. ∆J (J (I = 0) J (I))/J (I = 0), is expected to be
c c c c
[ − ≡ −
Finally,spin-injectionintosuperconductorsalsoprovides a measure of the amount of injected magnetization (m).

2 a setting to study the phenomenon of spatial separation Fig. 1b) illustrates the spin-injection-detection setup of
v of charge and spin currents [8,9]. Johnson and Silsbee [16,5]. Here a superconductor (S)
4 For the purpose of extracting the bulk spin transport is in contact with two itinerant ferromagnets, FM1 and
0
properties, it is essential that their contributions to the FM2. ThemagnetizationofFM1iseitherparallel(σ = )
0 ↑
measuredphysicalquantitiesareseparatedfromthoseof orantiparallel(σ = )tothatofFM2. Foragivenσ,I is
1 σ
↓
0 the boundary transport processes. The boundary of in- theinducedcurrentacrosstheS-FM2interfaceinaclosed
9 terest here is formed between the high T cuprates and circuit. Likewise, V is the induced boundary voltage
c σ
9 ferromagnetic metals. One transport process through (V ) inanopen circuit. The spin-dependent currentand
σ
/
t such a boundary is the Andreev reflection [10], which boundary voltage are defined as Is = I↑ I↓ and Vs =
a −
carries pair-current into the superconductor. The novel V V , respectively.
m ↑ ↓
−
featuresoftheAndreevreflectioninvolvingaferromagnet To highlight the interplay between the boundary and
-
d haverecentlybeenaddressed[11–13]. Anotherboundary bulk transportprocesses, we will make a number of sim-
n processisthesingle-particletransport. While thecharge plifyingassumptions. Thesuperconductor(S)isassumed
o transportinvolvesbothprocesses,thespintransportpro- to be a BCS superconductor, with either an s wave
c −
: ceedsthroughthesingle-particleprocessonly. Theinter- or a dx2−y2−wave order parameter. The ferromagnetic
v play between the two processes is therefore expected to metal will be simply modeled by an exchange energy h
i
X playanimportantroleinthe spininjection experiments. [11,17]. The Hamiltonians for FM1 and FM2 are, 1 =
H
r The consequences of such an interplay are explored in k,τ(ǫk−hτ)c†k,τck,τ and H2 = k,τ(ǫk−hτσ)c†k,τck,τ,
a this paper. The s wave case is simpler, which we will Prespectively. The unshifted enerPgy dispersions for both
address first. (Thi−s part of our analysis is also relevant ferromagnets are assumed to be ǫ =h¯2k2/2m, as is the
k
to the spin injection experiments in the non-cuprate su- normal state energy dispersion for the superconductor.
perconductors[14,15].) We will then extendour analysis In addition, the Fermi wavevectors of the superconduc-
to the d wave case. tor and the ferromagnets in the absence of polarization
−
Severalphysicalquantitiesareofinterestinspininjec- are assumed to be equal to k . For FM1, this implies
f
tion experiments. One is the amount of non-equilibrium k =k t and a Fermi velocity v =v t where
fτ f τ fτ f τ
magnetization (m) injected into the superconductor.
The others include the spin-dependent current (I ) and
s t 1+τh/ǫ (1)
τ f
boundary voltage (Vs) induced by the injected magneti- ≡q
1
I =ηµ I /e (6)
I spin B sp
where η is the single-particle current polarization,which
itself also needs to be determined. In the following, we
firstcalculatethethreecurrents,I ,I ,andI ,for
pair sp spin
S
a given voltage,V, across the FM1-S barrier. The latter
FM1 can then be expressed in terms of I, through Eq. (5),
a) leading to an expression for I as a function of I.
spin
For a fixed V, the values of the three currents depend
on the nature of the interface. Here, we model this in-
I terfacebyadelta-functionpotential, = δ(x).
interface
Vσ Iσ The Bogoliubov-deGennes equation [H18] is, V
σ H0−hτ ∆ uτ =E uτ (7)
S (cid:18)∆∗ −(H0+hτ)(cid:19)(cid:18)vτ¯ (cid:19) (cid:18)vτ¯ (cid:19)
2
FM1 FM2 where 0 =p /2m µ+ interface. Here, h and ∆ are
H − H
b) non-zero only for x < 0 and x > 0, respectively. In-
side FM1, the solution is the sum of three plane waves,
describing the incident electron of spin τ and wavevec-
tor~k , the Andreev-reflected hole of spin τ¯, wavevector
FIG. 1. a) A spin injection setup involving a fer- iτ
~k and amplitude a , and the normally-reflected elec-
romagnetic metal (FM1) and a superconductor (S). b) A aτ τ
spin-injection-detection setupinvolvingasuperconductor(S) tron of spin τ, wavevector~kbτ and amplitude bτ, respec-
and two ferromagnetic metals (FM1 and FM2). tively. Insidethesuperconductor,thesolutionisthesum
of two plane waves, one without branch-crossing and of
wavevector~k andamplitudec ,theotherwithbranch-
cτ τ
For FM2, kfτ =kftστ and vfτ =vftστ. crossing and of wavevector ~k and amplitude d . The
dτ τ
We introduce m(x) to denote the steady state spin
wavevectorsaredeterminedbytherequirementthat,the
magnetizationdensityinthesuperconductor. Sincem(x)
components parallel to the interface are equal. The am-
is entirely carried by the quasiparticles, it satisfies
plitudes are then calculated by solving Eq. (7), together
2 2 with the boundary conditions [18] appropriate to this
Ds∂ m/∂x = m(x)/T1 (2)
− − equation. These amplitudes, in turn, allow us to deter-
where T1 is the longitudinalspin relaxationtime andDs mine the various currents.
is the spin diffusion constant. In the linear response regime, the pair, single-
Wenowspecifytheboundaryconditions. Forthesetup particle, and spin currents can be written as Iλ/V =
of Fig. 1a), x = 0 and x = d correspond to the FM1-S dE( ∂f/∂E) δ(ξτ E)vxiτ(E,~k). Here, ξτ =
− ~kτF k − k λ k
interaceandtheopenendofS,respectively. ForFig. 1b), Rǫk ǫf hτ, f(PE) is the Fermi-Dirac distribution func-
− −
they instead describe the FM1-S and S-FM2 interfaces, tion, isafilteringfactorwhichspecifiesthedistribution
F
respectively. At x=0, of incident angles. In addition,
−Ds∂m/∂x|x=0 =Ispin (3) iτpair=2e2Aτ
ForthesetupofFig. 1a),andtotheleadingorderinFig. iτsp=e2(Cτ +Dτ)
1b) as well, the boundary condition at x=d is iτ =µ e(C +D )τ (8)
spin B τ τ
−Ds∂m/∂x|x=d =0 (4) where Cτ = |cτ|2(u(kcτ)2 − v(kcτ)2)vfx/vkxτ, Dτ =
d 2(u(k )2 v(k )2)vx/vx , and A = a 2vx /vx .
The crucial question then is, what is the spin-current | τ| dτ − dτ f kτ τ | τ| kτ¯ kτ
(Here kτ labels the wavevectorssatisfying ξτ =E.)
I for a fixed injection electrical-current I. At the k
spin The potential barrier can be represented in terms of
FM1-S boundary the electrical current is the sum of a 2
a dimensionless quantity, z = m /¯h k . In the follow-
pair current (Ipair), carried through Andreev reflection, ing, we will focus on the two extrVeme lfimits, z 0 and
and a single-particle current (Isp), z . In the latter case we expand in terms→of 1/z.
→ ∞
I =I +I (5) Inaddition,wewillconsideronlytemperatureslowcom-
pair sp
paredtothesuperconductinggap∆. Thesecasesaresuf-
Thespincurrent,ontheotherhand,hasnocontribution ficient to illustrate the main points of this paper. More
from the Andreev process given that the Cooper pairs general cases will be discussed elsewhere.
are spin singlets. We then expect,
2
s wave, vanishing barrier: We address first the case current that would have been induced by the effective
−
of an isotropic s wave superconductor, in the limit of a magnetic field drop. The total induced current is,
−
vanishing barrier, z =0. For k T ∆, we found,
B ≪ Iσopen =σg0(µB/e)Ms,0Ys,0(T)(m/χ)+g0Ps,0Vσ (14)
Ipair/V=g0Ps,0
ComparedtoEq. (12),theadditionaltermisthecurrent
Isp/V=g0Ss,0Ys,0(T) induced by the voltage drop; here only the pair current
Ispin/V=(µB/e)g0Ms,0Ys,0(T) (9) has been kept as it dominates over the corresponding
single-particle current. Setting Iopen =0 determines V ,
Hereg0 =N0vfe2whereN0 isthedensityofstatesatthe leadingtoaspin-dependentbounσdaryvoltage,V =V σ
unshifted Fermi energy, Ps,0 = 16αpairt↑t↓/(t↑t↓+1)2, V , as follows, s ↑−
Ss,0 = 2√2παsp[t↑(t2↓+1)+t↓(t2↑+1)]/(t↑t↓+1)2, and ↓
Ms,0 = 2√2παspin(t↑−t↓)(1−t↑t↓)/(t↑t↓+1)2. The Vs/I =2(µB/e)2(Ms,0/Ps,0)2[Ys,0(T)]2 T1 (15)
(polarization-dependent)factorsαpair,αsp,andαspinare χδssinh(d/δs)
of order unity; they describe the averaging over the an-
Eqs. (11, 13,15)revealone key point: In generaleach
gle of incident electrons, and are all equal to unity when
of the three quantities, m, V , and I , reflects a different
forward incidence dominates. s s
combination of the boundary and bulk contributions.
Both I and I have an exponential temperature
sp spin Consider next the special case of a half-metallic ferro-
dependence, as specified by the function Ys,0(T) = magnet. I vanishes in this case, as can be seen from
k T/∆e−∆/kBT. This reflects the simple physics that pair
B the expressionfor Ps,0 inEq. (9); the Andreev reflection
pthetransportofboththesingle-particlecurrentandspin
is completely suppressedin this case [11]. The total cur-
current involve the quasiparticles of the superconductor
rent in Eq. (5) is then entirely given by I . This, com-
sp
and hence only energies above the superconducting gap.
bined with the fact that Ms,0 = Ss,0 for a half-metallic
We are now in a position to determine the non-
ferromagnet, leads to a simple relationship between the
equilibrium magnetization (m) and the induced spin-
spin and charge current,
dependent boundary voltage (V ) and current (I ) as
s s
a function of I. Consider first the general case, when I /I =µ /e (16)
spin B
the ferromagnet has a less than 100% polarization. For
k T ∆, I then dominates the total current in Eq. In addition, the second term in Eq. (14) is now replaced
B pair
(5). A≪s a result, by the correspondingsingle-particlecurrent. The results
then become,
Ispin/I =(µB/e)(Ms,0/Ps,0)Ys,0(T) (10) T1
m(x=d)/I=(µ /e)
B
Combining Eq. (10) with Eqs. (2, 3, 4), we have, δssinh(d/δs)
2 T1
T1 Is/I=2g0(µB/e) Ms,0Ys,0(T)
m(x=d)/I =(µB/e)(Ms,0/Ps,0)Ys,0(T) (11) χδssinh(d/δs)
δ sinh(d/δ )
s s 2 T1
V /I=2(µ /e) (17)
s B
where δs = √DsT1 is the spin-diffusion length of the χδssinh(d/δs)
superconductor.
Eq. (17)revealsanothermainconclusionofthispaper:
This magnetization accumulation leads to a drop
The boundary-transport-contributions are absent in the
across the interface of an effective magnetic field [16],
expressions for the non-equilibrium magnetization, m,
∆H =m/χ where χ is the uniform spin susceptibility of
and the induced spin-dependent boundary voltage, V ,
s
thesuperconductor. Inthespin-injection-detectionsetup
when the ferromagnetis half-metallic. For m, this is the
(Fig. 1b),foraclosedcircuit,thiseffectivefielddropwill
resultofthesimplerelationshipbetweenthespincurrent
induce acurrentacrosstheS-FM2interface. Followinga
and total current (Eq. (16)), which arises whenever the
proceduresimilartotheoneleadingtoEq. (8),wefound
Andreevreflectionisabsent. (Notethatthetotalcurrent
I is fixed.) For V , the reasoning leading to this conclu-
Iσclosed =σg0(µB/e)Ms,0Ys,0(T)(m/χ) (12) sionisslightlymosresubtle. WhentheAndreevreflection
The induced spin-dependent current, I Iclosed is absent,the boundary-transport-contributionsgiverise
s ≡ ↑ − to exactly the same prefactorsin the two currentsin Eq.
Iclosed, is then equal to
↓ (14). Since the boundary voltage V is determined by
σ
2 2 2 T1 balancing these two terms, the boundary-transport fac-
Is/I =2g0(µB/e) (Ms,0/Ps,0)[Ys,0(T)] χδ sinh(d/δ ) (13) tors cancel out exactly in the expression for Vs.
s s
The above cancellation argument does not apply to
Similarly, in the open circuit case, a boundary voltage the induced current I . Indeed, I does contain the
σ s
will develop across the S-FM2 interface to balance the boundary-transportfactors.
3
s wave, large barrier: Consider now the limit of To summarize, we have studied the effect of spin in-
−
a large barrier. The results for the pair, single-particle, jection into s and d wave superconductors. Through
− −
andspincurrentsparallelEq. (9),exceptthatthefactors the explicit results in the small and larger barrier lim-
Ps,0, Ss,0, Ms,0, and Ys,0 are replaced by its,weconcludethatAndreev-reflectionmakesthediffer-
entphysicalpropertiesmeasuredinspininjectionexperi-
4
Ps,∞=(1/z )g0αpairt↑t↓ mentsreflectverydifferentcombinationsoftheboundary
Ss,∞= π/2(1/z2)g0αsp(t↑+t↓) and bulk spin transport properties. For the purpose of
p 2 isolating bulk spin transport properties, it is desirable
Ms,∞= π/2(1/z )g0αsp(t↑−t↓) to make the Andreev contribution as small as possible.
Y (T)=p∆/k Te−∆/kBT (18) In this case, the non-equilibrium magnetization and the
s,∞ B
p spin-dependent boundary voltage depends solely on the
respectively. The resulting expressions for m, Is and Vs bulkspintransportproperties. Thisoccursintwolimits.
are also given by Eqs. (11, 13, 15), with an appropriate One is the limit of a strong barrier in tunneling geome-
substitution by the factors given in Eq. (18). triessuchthattheAndreevboundstatesareabsent. The
The pair current is of order 1/z4. Both the single- other is for the half-metallic ferromagnet.
particle currentand spin current, on the other hand, are This work has been supported in part by the NSF
of order 1/z2. The latter is to be expected, since in the GrantNo. DMR-9712626,aRobertA.Welchgrant,and
large barrier limit the single-particle transport can be an A. P. Sloan Fellowship. While this paper was under
described in terms of a tunneling picture. For the same preparation we learned of a preprint [20] of Kashiwaya
reason, the temperature dependence of Isp and Ispin in et al. who calculated the non-linear spin current for the
this case should reflect simply the thermal smearing of case of a d wave superconductor.
the single-particle density of states in the superconduc- −
tor. Indeed,Y (T)correspondstothelowtemperature
s,∞
limit of the Yosida function.
Whenever the Andreev reflection is non-negligible,
eachofthethreequantities,m,V andI ,againdepends
s s
onadifferentcombinationoftheboundarytransportand
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When the potential barrier is so strong that the [2] Z. W. Dong et al., Appl. Phys.Lett. 71, 1718 (1997).
4 [3] V. A.Vas’ko et al, Appl. Phys.Lett. 73, 844 (1998).
temperature-independent 1/z contribution is negligible
comparedto the temperature-dependent1/z2 terms,the [4] N.-C. Yeh et al., preprint(1998).
[5] N. Hass et al., Physica C235-240, 1905 (1994).
Andreev processis absent. Here again,while the bound-
[6] Q. Si, PhysRev.Lett. 78, 1767 (1997).
ary transport terms still appear in I , they are cancelled
s [7] Q. Si, Phys. Rev.Lett. 81, 3191 (1998).
out in m and Vs. [8] H. L. Zhao and S. Hershfield, Phys. Rev. B52, 3632
d wave case: We now turn to the case of a dx2−y2 (1995).
−
superconductor. Here we will focus on the case when [9] S. A. Kivelson and D. Rokhsar, Phys. Rev. B41, 11693
both the FM1-S and S-FM2 interfaces involve the 110 (1990).
{ }
surface of the cuprates. Consider firstthe limit of a van- [10] A. F. Andreev,Sov.Phys. JETP 19 1228 (1964).
ishing barrier. The resultfor the currentsagainparallels [11] M. J. M. de Jong and C. W. J. Beenakker, Phys. Rev.
Lett. 74 1657 (1995).
Eq. (9), with the prefactors replaced, respectively, by
[12] J.-X. Zhu,B. Friedman, and C. S. Ting, preprint(1998).
Pd,0=16g0αpairt↑t↓/(t↑+t↓)2 [13] I. Zutic and O. T. Valls, cond-mat/9808285.
[14] R. J. Soulen et al., Science 282, 85 (1998).
2
Sd,0=(ln2)g0αsp[C↑t↑+(↑→↓)]/(t↑t↓+1) [15] S. K.Upadhyayet al., Phys.Rev.Lett. 81, 3247 (1998).
2
Md,0=(ln2)g0αspin[C↑t↑ ( )]/(t↑t↓+1) [16] M. Johnson, Appl. Phys. Lett. 65, 1460 (1994) and ref-
− ↑→↓
erences therein.
Yd,0(T)=kBT/∆ (19)
[17] Thespinwaveshaveasuper-ohmicspectralfunctionand
where C =(t +1)2+4(3 π)t (19/3 2π)(t 1)2. are henceunimportant for ouranalysis.
↑ ↓ ↓ ↓
− − − − [18] G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Phys
For the large barrier limit, we found Pd,∞ = Pd,0, Rev. B 25, 4515 (1982) and references therein.
Yd,∞(T)=Yd,0(T), and [19] Chia-Ren Hu,Phys. Rev.Lett. 72, 1526 (1994).
2 [20] Y. Tanaka and S. Kashiwaya, Phys Rev. Lett., 74, 3451
S =(4 π)(ln2/2)(1/z )α (t +t )
d,∞ − sp ↑ ↓ (1995);S.Kashiwayaetal.,PhysRev.B53,2667(1996).
2
Md,∞=(4 π)(ln2/2)(1/z )αspin(t↑ t↓) (20) [21] J. H.Xu et al., Phys.Rev.B 53, 3604 (1996).
− −
[22] M. Fogelstrom et al., Phys.Rev.Lett. 79, 281 (1997).
Notethat,whileIsp andIspin arestilloforder1/z2,Ipair [23] S.Kashiwaya,Y.Tanaka,N.Yoshida,andM.R.Beasley,
is of order 1/z0. The latter reflects the formation of the cond-mat/9812160.
Andreev-bound states [19–22].
4