Subharmonic Gap Structure in Superconductor/Ferromagnet/Superconductor Junctions

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by  I. V. Bobkova

Subharmonic Gap Structure in Superconductor/Ferromagnet/Superconductor
Junctions
I. V. Bobkova∗
Institute of Solid State Physics, Chernogolovka, Moscow reg., 142432 Russia
(Dated: February 2, 2008)
6
0 The behavior of dc subgap current in magnetic quantum point contact is discussed for the case
0
of low-transparency junction with different tunnel probabilities for spin-up (D↑) and spin-down
2
(D↓) electrons. Due to the presence of Andreev bound states ±ε0 in the system the positions of

n subgap electric current steps eVn =(∆±ε0)/n are split at temperature T 6=0 with respect to the
a nonmagneticresulteVn=2∆/n. ItisfoundthatundertheconditionD↑ 6=D↓ thespincurrentalso
J manifests subgap structure, but only for odd values of n. The split steps corresponding to n=1,2

7 insubgapelectricandspincurrentsareanalyticallycalculatedandthefollowingstepsaredescribed
1 qualitatively.

] PACSnumbers: 74.45.+c,74.50.+r
n
o
c It is wellestablishednow thatsubharmonic gapstruc- in qualitative agreement with the experimental data17.
-
r ture in superconducting weak links can be described in The incoherentMAR regime with eV,∆ ETh has also
p termsofmultipleAndreevreflection(MAR).Theconcept beenstudied18,19 anditwasfoundthatsu≫bharmonicgap
u
of MAR was first introduced1 for SNS (superconductor structure shows qualitatively different behavior for even
s
. - normalmetal - superconductor)junctions and then ex- and odd subharmonic gap structures.
t
a tendedtoincludethe effectofresistanceofSNinterface2 Magnetic interfaces can not be characterized by
m (OTBK theory). Then it was shown3 that this mech- the only parameter D and Andreev bound states are
- anism is the reason for subharmonic gap structures in known to take place on the magnetic surface of s-wave
d junctions of any type, and specifically in superconduct- superconductor20. So it is natural that new qualitative
n
ingpointcontacts. Aslongasthejunctionisshortonthe features with respect to nonmagnetic case appear in dc
o
c scaleofthecoherencelengthξ0 themicroscopicdetailsof current even in the simplest models of a short magnetic
[ thejunctionareirrelevantandforthenonmagneticjunc- junction21,22. In Ref.22 the dc-current-voltage charac-

tion the only parameter is the transmission probability teristic of magnetic quantum point contact between two
2
ofthebarrierD. AnexacttheoryofMARforshortnon- superconductors is studied theoretically for the model
v
magnetic junctions between clean superconductors has wheremagneticquantumdotischaracterizedbytwopa-
3
3 beendevelopedinRefs.4,5onthe basisofscatteringam- rameters: transparency D and the spin-mixing angle Θ.
7 plitudes method and in Ref.6 making use of nonequilib- It is found that subgap current steps take place at volt-
0 rium Green function techniques. It was obtained that ages eV =∆(1+cosΘ/2)/n.
n
1 for this case MAR manifests itself in dc current as cur- In this paper the dc subgap current due to MAR pro-
5
rent steps at voltages eVn = 2∆/n, n = 1,2,.... The cesses in short ballistic quantum point contact for the
0
/ magnitude of the current onset at Vn is proportional to mostgeneralcaseofsymmetricmagneticinterfaceinclud-
t Dn. Thetheoreticalresultsareconsistentwiththeexper- ing different transparencies for spin-up and spin-down
a
m imental situation in atomic-size superconducting point quasiparticles (D↑ = D↓) is studied theoretically. I con-
contacts7,8,9,10. The subharmonic gap structure due to sider the case of sm6 all D and D of the same order of
- ↑ ↓
d photon-assistedMAR in the presence of microwaveradi- magnitude and find that at T = 0 current steps occur
n ation has also been studied and theoreticalpredictions11 at voltages eV = ∆(1 cosΘ/62)/n. This splitting dis-
n
o are confirmed by the experimental observations12. appears in the limit T± 0 and the result obtained in
c Arnold3 was the first to take into account the phase Ref.22 for the current s→tep positions is recovered. Also
:
v coherence of subsequent Andreev reflections, which were I found that the difference between D and D is very
↑ ↓
Xi absent in the OTBK theory. The quantum coherence in important. It leads to non-zero spin current which has
short ballistic junctions lead to ac Josephson effect4,5,6. subharmonicgapstructuredifferentfromthatoneforthe
r
a In addition, in long ballistic SNS junctions coherence ef- dc electric current.
fects give rise to resonant structures in the dc current The theoretical analysis is based on the non-
due to Andreev quantization13. equilibrium quasiclassical theory of superconductivity
MARindiffusive nonmagneticjunctions hasalsobeen in terms of Riccati amplitudes developed by Eschrig23
investigated in details. Short diffusive junctions with and generalized for the case of magnetic interfaces in
Thouless energy E ∆ can be described by coher- Refs.20,24. Now I briefly outline this formalism. The
Th
≫
ent MAR theory and show subharmonic gap structures fundamental quantity in non-equilibrium quasiclassical
at eV = 2∆/n14,15. In long diffusive junctions with theory of superconductivity is the quasiclassical Green’s
n
E .∆additionalpeaksinthedifferentialconductance function gˇ = gˇ(p ,R,ǫ,t). It is a 8 8 matrix form
Th f
at eV 2(∆ E )/2n are predicted16. This result is in the product space of Keldysh, parti×cle-hole and spin
n Th
≈ ±
2
variables. In generalthe quasiclassicalGreen’s functions paper. gˇK(p ,x,ǫ,t) is a 4 4 KeldyshGreen’s function
f
×
are depend onspace R, time t variables, the directionof in the product space of particle-hole and spin variables.
quasiparticleFermimomentump andtheexcitationen- p stands for incoming quasiparticle trajectories and p
f f f
ergy ǫ. In our case of one-mode quantum point contact for the outgoing ones. τˆ and σˆ are Pauli matrices in
i i
the problem is effectively one-dimensional and R x, particle-holeandspin spaces,respectively. The spincur-
≡
where x - is the coordinate measured along the normal rent jsp/se can be calculated making use of Eq.(1) with
to the junction. The momentum pf has only two values, the substitution σˆ3 for σˆ0. se =1/2 is the electron spin.
which correspondto incoming andoutgoing trajectories.
Quasiclassical Green’s function gˇ can be expressed in
Theelectricandspincurrentsshouldbecalculatedvia
terms of Riccati coherence functions γˆR,A and γˆ˜R,A,
Keldysh part of the quasiclassical Green’s function. For
which measure the relative amplitudes for normal-state
the one-mode quantum point contact the corresponding
quasiparticle and quasihole excitations, and distribution
expression for the charge current reads as follows
functions xˆK and xˆ˜K. All these functions are 2 2 ma-
×
+∞ trices in spin space and depend on (pf,x,ǫ,t).
jch = sgnpf dǫ InthispaperIconsiderashortjunctionwiththechar-
e 2π 4πi ×
acteristicsize d ξ . Butit is assumedthat despite the
−Z∞ ≪ 0
small size of the grain charging effects can be neglected.
Tr4 τˆ3σˆ0 gˇK(pf,x,ǫ,t) gˇK(p ,x,ǫ,t) , (1) For definiteness the current is calculated on the left side
− f
of the interface. Keldysh Green’s function for incoming
h (cid:16) (cid:17)i
where e is the electron charge and ~=1 throughout the trajectory is parameterized by
(xˆK γˆR Xˆ˜K γˆ˜A) (γˆR Xˆ˜K xˆK ΓˆA)
gˇ1K(pf)=−2iπNˇR⊗ (Γˆ˜1R− xˆ1K⊗ Xˆ˜1K⊗ γˆ˜1A) −(Xˆ˜K1 ⊗Γˆ˜R1 −xK1 ⊗ΓˆA1) !⊗NˇA , (2)
− 1 ⊗ 1 − 1 ⊗ 1 1 − 1 ⊗ 1 ⊗ 1
1 γˆR(ΓˆA) Γˆ˜R(γˆ˜A) −1 0
NˇR(A) = − 1 1 ⊗ 1 1 . (3)
(cid:16) 0 (cid:17) 1 Γˆ˜R(γˆ˜A) γˆR(ΓˆA) −1 
− 1 1 ⊗ 1 1
 
 (cid:16) (cid:17) 
Heresubscript1meansthatthe correspondingfunctions
sfuhnocutlidonbsetisakoemniatttexd=fo−r 0b,reavrgituym. eTnhtepfproofdaullctth⊗eRoifcctwatoi xˆKl,r(ǫ)= 1−|γˆlR,r((ǫ−eVl,r),t)|2 tanhǫ−2eTVl,r , (5)
functionsofenergyandtimeisdefinedbythenoncommu-
tative convolution A B = ei(∂ǫA∂tB−∂tA∂ǫB)A(ǫ,t)B(ǫ,t). wherethes(cid:0)ubscriptl,rdenotesthat(cid:1)theappropriateRic-
⊗ catifunctionbelongstothebulkoftheleft(right)super-
Keldysh Green’s function gˇ (p ) for the outgoing tra-
1 f conductor. ∆ is the bulk absolute value of supercon-
jectory can be obtained from Eqs. (2), (3) by the ducting order parameter for a given temperature, which
substitution (γˆR,γˆ˜A,xˆK)(p ) (ΓˆR,Γˆ˜A,XˆK)(p ) and is assumed to be the same in the both superconductors.
1 1 1 f → 1 1 1 f
V is the electric potential in the bulk of left (right)
(Γˆ˜R,ΓˆA,Xˆ˜K)(p ) (γˆ˜R,γˆA,xˆ˜K)(p ). l,r
1 1 1 f → 1 1 1 f superconductor, so V = Vr Vl is the voltage bias ap-
Riccati coherence and distribution functions obey plied to the junction. Quan−tities γˆ˜R,A and xˆ˜K are ob-
l,r l,r
Riccati-type transport equations. The equations are tained from Eqs. (4), (5) respectively by the operation
given in Refs.23,24, so I do not write them here. The a˜(ǫ,t)=a( ǫ,t)∗.
quantities (γˆR,A,γˆ˜R,A,xˆK,xˆ˜K), denoted by lower case Foranaly−ticalconsiderationsuperconductingorderpa-
1 1 1 1
symbols, are obtained by solving the Riccati equations rameter and electric potential are assumed to be spa-
for the appropriate trajectory with the asymptotic con- tially constant in the superconductors. Under this as-
ditions,whicharefor spin-singlets-wavesuperconductor sumption the voltage drop only occurs at the junc-
as follows tion region. These simplifications seem to be rea-
sonable for quantum point contact. Under the as-
∆e−2ieVl,rt sumptions above the solutions of Riccati equations for
iσˆ , ǫ <∆
γˆlR,r,A(ǫ,t)= ǫǫ±+∆ispgen−∆ǫ22ie−ǫV2l,ǫr2t ∆22iσˆ2, ||ǫ||>∆ , (4) st(γipˆoaRnc,sAei(nvxa)tr,hiγˆ˜aeRbc,lAeor(arxen)s,dpxˆocKno(idxnicn),igdxˆ˜esKuw(pxie)tr)hcodtnhodeunacotstoyrdm.eppteontdicocnontdhie-
−
 p
3
The quantities (ΓˆR1,A,Γˆ˜R1,A,Xˆ1K,Xˆ˜1K), de- where R↑,↓ +D↑,↓ = 1, α = ±1 depending on the par-
noted by upper case symbols, are expressed via ticular model25. Sˆ˜ = Sˆ in the considered problem.
ij ij
(γˆ1R,A,γˆ˜1R,A,xˆK1 ,xˆ˜K1 ) = (γˆlR,A,γˆ˜lR,A,xˆKl ,xˆ˜Kl ) and the The particular expressions for (ΓˆR,A,Γˆ˜R,A,XˆK,Xˆ˜K) in
elements of the interface scattering matrix for the 1 1 1 1
S termsof(γˆR,A,γˆ˜R,A,xˆK,xˆ˜K)and -matrixelementscan
normal-state electrons and holes with the energies at 1 1 1 1 S
be found in Ref.24, so I do not write them explicitly.
the Fermi surface. The interface -matrix is a unitary
S
8 8 matrix in the combined spin, particle-hole and Now we can proceed with the current. Substituting
×
directional spaces. The explicit structure of -matrix in Keldysh Green’s function (2) into Eq.(1) it can be seen
S
directional space is that the current is expressed as a sum over harmonics
j(t) = J e2iNeVt. I focus on the dc component and
= Sˇ11 Sˇ12 , (6) N N
S Sˇ21 Sˇ22 only caPlculate the term corresponding to N = 0. I-V
(cid:18) (cid:19) characteristics in charge and spin currents exhibit steps.
where matrix Sˇ contains spin-dependent reflection am- In the tunnel limit one can extract the elementary pro-
ii
cessesgivingrisetoeachofthesesteps. Inthispaperthe
plitudes of normal-state quasiparticles from the inter-
face in i-th half-space, while Sˇ with i= j incorporates positions, height and shape of several first steps in sub-
ij
6 gap I-V characteristic are calculated analytically. The
spin-dependent transmission amplitudes of normal-state
quasiparticlesfromsidej. EachelementSˇ isadiagonal linear in transparency thermally-activated term in the
ij
matrixinparticle-holespaceSˇ =Sˆ (1+τˆ )/2+Sˆ˜ (1 dc current, which is not related to the steps, has also
ij ij 3 ij − been calculated, but it is not pronounced in the subgap
τˆ )/2. The most generalform of S-matrix for a symmet-
3 region for the considered problem and is not discussed
ric magnetic interface without spin-orbit interaction can
below. The part of the dc current corresponding to the
be written as25:
steps takes the form:
R eiΘ/2 0
Sˆ =Sˆ = ↑ , (7)
11 22 0 R e−iΘ/2
(cid:18)p ↓ (cid:19) jch(jsp)=jch(sp)+jch(sp)+jch(sp)+jch(sp) , (9)
p e se 1+ 1− 2+ 2−
D eiΘ/2 0
Sˆ =Sˆ = i ↑ , (8)
12 21 ± 0 α D↓e−iΘ/2
(cid:18)p − (cid:19)
p
∆ sinΘ/2 ε eV (ε eV)2 ∆2 ε eV ε
j1c+h(sp) = | 4 |(D↑±D↓)|(0ε− eV| )2 0∆−2cos2Θ−/2 tanh 02−T −tanh2T0 , (10)
0− p − (cid:18) (cid:19)
∆ sinΘ/2 ε 2eV (ε 2eV)2 ∆2 1+ γ(ε eV)2 ε 2eV ε
j2c+h = | 2 |D↑D↓| 0(ε− 2e|V)2 0−∆2cos2Θ−/2 1 γ2|(ε 0−eV)ei|Θ 2 tanh 0−2T −tanh2T0 , (11)
0− p − | − 0− | (cid:18) (cid:19)
jch(sp) = jch(sp)(V V), jch = jch(V V), forthecaseofnonmagneticjunctionbetweentwod-wave
j1s−p = 0.−I1n+Eqs.(11→),(1−0) ε 2=− ∆c−os2Θ+/2. →γ(−ǫ) = superconductors26.
2± 0
(ǫ i√∆2 ǫ2)/∆, where for ǫ > ∆ the square root
− − | | Let us describe all the steps for n > 2 qualitatively.
√∆2 ǫ2 should be substituted by isgnǫ√ǫ2 ∆2.
Theb−ehaviorofjchandjspdescribedby−Eq. (9)iss−hown The estimate for the magnitude of jnch(sp) is easily ob-
tained if one considers physical processes of MAR lead-
in Fig.1 for Θ=π/2.
ing to subgap current steps. For the spin-up band the
It is seen from Fig.1 that at zero temperature there bound state energy is ε = +ε and at T = 0 the fol-
B 0
is a shift of well-known step positions in electric current lowing process gives spin-up current at eV+: a spin-up
n
for a non-magnetic interlayer: eV = 2∆/n eV+ = electron tunnels into the bound state from the contin-
n → n
(∆+ε )/n. Thisresultiscoincidewiththatonefoundin uum states below ∆ undergoing n 1 Andreev reflec-
0
− −
Ref.22. However,at non-zerotemperature non-magnetic tionsinsidethegap. Anelectronconvertsintoaholeand
step positions are split: eV = 2∆/n eV± = (∆ vice versa under each Andreev reflection event. In every
ε )/n. Itiseasilyobtainedtnhattheaddi→tionalnstepseV±− passing of the electron (hole) through the boundary the
0 n
disappearrapidly( e−ε0/T)whenthetemperaturegoes amplitude of its wave function should be multiplied by
∝
down. InthefirstorderofDtheanalyticalexpressionfor D ( D ). For the spin-down band with ε = ε
↑ ↓ B 0
−
theelectriccurrentsimilartoEq. (10)hasbeenobtained the same process takes place with the modification that
p p
4
jch/e jsp/se
| |
(cid:12)(cid:12) 0.1(cid:12)(cid:12) (a) 0.03(b)
V1+ 0.025 V1+
0.08
0.06 V2+ 0.02 V1−
0.015
T =0 T =0
0.04 −
0.02 V2− V1 T =0.5Tc 0.00.0051 T =0.5Tc
0.5 1 1.5 2 0.5 1 1.5 2
eV/∆0 eV/∆0
FIG.1: (a)Thesubgapstructureofdcelectriccurrent(inunitsof∆0e/~)isplottedinaccordancewithEq.(9)forT =0,0.5Tc.
(b) The same for thespin current. Θ=π/2, D↑ =0.2, D↓ =0.1. ∆0 ≡∆(T =0).
a spin-down electron starts in the bound state and goes in the junction region, exceeds the average time of in-
up into the continuum states at +∆. So the current elastic scattering τ . So the steps with n more than
in
j is (D D )k and j (D D )kD . Fi- τ ∆ (ξ /d ) can not be observed. For the case under
↑,↓2k ↑ ↓ ↑,↓(2k+1) ↑ ↓ ↑,↓ in 0 0 eff
∝ ∝
nally, we obtain for jch(sp) = j j the following re- considerationthisestimateseemsnottoberestrictivebe-
n n↑ n↓
± cause ξ /d 1 for a short junction and τ ∆ due to
sju(s2plkt): =j(c2h0k(.+sp1)) ∝(D↑D↓)k(D↑±D↓)andj(c2hk) ∝2(D↑D↓)k, esildecetrraobn0ly-pehexofcnfeoe∼ndsscuantitteyrifnogr t∝em(ωpeDr/aTtu)r2e(s∆u0p/Titno) Ta0nc.d con-
ForT =0thereisanon-zeroprobabilitytofindaspin-
6
upelectronatε =ε andavacantplaceforaspin-down In conclusion, I have presented a theoretical study of
B 0
electron at ε . So the processes leading to the addi- thedcelectricandspincurrentsubgapstructureinmag-
0
tional steps−at eV− arise: (i) a spin-up electron tunnel- netic point contact. At T = 0 electric current step posi-
n 6
ing from ε =ε to the continuum spectrumat +∆ and tions eV =(ε ∆)/n aresplit comparedto the case of
B 0 n 0
±
(ii) a spin-down electron tunneling from the continuum nonmagneticjunctionandatT =0theyarejustshifted:
spectrum below ∆ into the bound state ε . Conse- eV =(ε +∆)/n. Thesubgapspincurrent,takingplace
− − 0 n 0
quently, for T =0 the splitting of current step positions atD =D manifestsstepsatthesamevoltagesbutonly
↑ ↓
occurs and the6re are all the steps at eV± = (∆ ε )/n for odd6 values of n. The current for steps with n = 1,2
n ± 0
inthechargecurrentandonlyoddsplitstepsinthespin is calculatedanalyticallyanda qualitativedescriptionof
current. the following steps is presented.
Of course, the processes of inelastic scattering lead
to impossibility of observing the subgap structures cor- Acknowledgments. The author thanks A.M. Bobkov
responding to large n due to the loss of coherence if for enlightening discussions and acknowledges the sup-
the time τ = nd /v , which a quasiparticle spends port by grant RFBR 05-02-17175.
eff f
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