Nonequilibrium Josephson current in ballistic multiterminal SNS junctions

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Nonequilibrium Josephson current in ballistic

multiterminal SNS-junctions

P. Samuelsson, A. Ingerman, V.S. Shumeiko, and G. Wendin

Department of Microelectronics and Nanoscience, Chalmers University of

Technology, S-412 96 Goteborg, Sweden

Abstract

We study the nonequilibrium Josephson current in a long two-dimensional ballisticSNS-junction with a normal reservoir coupled to the normal part of the junction.The current for a given superconducting phase difference φ oscillates as a functionof voltage applied between the normal reservoir and the SNS-junction. The periodof the oscillations is πhvF /L, with L the length of the junction, and the amplitudeof the oscillations decays as V −3/2 for eV ≫ hvF /L and zero temperature. Thecritical current Ic shows a similar oscillating, decaying behavior as a function ofvoltage, changing sign every oscillation. Normal specular or diffusive scattering atthe NS-interfaces does not qualitatively change the picture.

pacs[74.50.+r, 74.20.Fg, 74.80.Fp]

1 Introduction

In recent years there has been an increased interest in the nonequilibriumJosephson current in mesoscopic multiterminal SNS-junctions. Quasiparticleinjection from one or several normal reservoirs coupled to the SNS-junctionleads to a nonequilibrium population of the current carrying Andreev levels,and thus to a modification of the Josephson current. As predicted by theory,[1–6] experiments show that suppression [7,8], switching [9] and even enhance-ment [10] of the Josephson current under nonequilibrium is possible.

The theory for the nonequilibrium Josephson effect has been developed formainly two types of junctions, quantum ballistic [1,3,4] and diffusive [2,6,5].A growing experimental interest is however shown for multiterminal SNS-junctions where the normal part is a ballistic semiconductor 2DEG.[8,11] Inthis paper, a theory for these type of structures is presented, and it is applied

Preprint submitted to Elsevier Preprint 1 February 2008

to junctions with and without normal reflection at the NS interfaces. Forjunctions with normal reflection, both specular and diffusive scattering is takeninto account.

2 Model and theory

A model of the junction is presented in Fig. 1. A ballistic two-dimensionalnormal region, width W , is connected to two superconducting electrodes, withelectrode separation L. The phase difference between the superconductors isφ. A normal electron reservoir is connected to the normal part of the junctionvia a quantum point contact, with width d ≪ W , and a voltage V is appliedbetween the normal reservoir and the SNS-junction.

L

SS dW

NV

Fig. 1. A schematic picture of the junction.

The resistance of the point contact is assumed to be the dominating resis-tance of the junction, and the applied voltage thus drops completely over theinjection point. Zero magnetic field is assumed.

A natural framework for studying multiterminal ballsitic two-dimensional SNS-junctions is the Landauer-Buttiker scattering approach. The junction is de-scribed by the Bogoliubov-de Gennes-equation (BdG), where it is assumedthat the superconducting gap ∆ is constant in the superconductors and zeroin the normal metal. Hard wall boundary conditions for the 2DEG are alsoassumed. Solutions to the BdG-equation are matched at the NS-interfaces andat the three lead connection. It is assumed that all transverse modes coupleequally to the modes in the injection point contact.[12]

Boundary conditions are incoming electron and hole quasiparticles from thenormal reservoir and incoming electron- and hole-like quasiparticles from the

2

superconductors at energies above the gap. Knowing the wave function coef-ficients the current density is straightforwardly calculated.

Since the normal reservoir is weakly coupled to the SNS-junction (d ≪ W ), theinjected current is small and the current flowing between the superconductorsis only the nonequilibrium Josephson current. [13] However, the coupling mustbe large enough so that the injected quasiparticles do not scatter inelasticallybefore leaving the junction. Under these assumptions, the distribution of theinjected electrons and holes are governed by the normal reservoir.

The total nonequilibrium Josephson current is naturally parted into the equi-librium current Ieq (at eV = 0) and the current due to nonequilibrium, Ineq.It has been shown [14] that the nonequilibrium current can be split into twocomponents, one associated with the nonequilibrium population of the An-dreev states and the other with nonequilibrium mesoscopic fluctuations of thecurrent. Here, this mesoscopic fluctuation term is neglected, since it is smallcompared to term from the nonequilibrium population of the Andreev levels.The total nonequilibrium current then becomes

I ≡ Ieq + Ineq =

∞∫

−∞

dE i(E) nF +

∆∫

−∆

dE

[

i(E)

2(ne + nh − 2nF )

]

(1)

with ne(h) = nF (E ∓ eV ) being the distribution functions of electrons (holes)in the normal reservoir, where nF = [1+exp(E/kT )]−1. Clearly, the propertiesof the current density i(E) directly determines the nonequilibrium Josephsoncurrent, and the current density will thus be the staring point for the discus-sions below. For energies outside the gap, E > ∆, i(E) is given by [15,16],

i(E) =4e

h

d

dφIm

(

tr ln[

1 − α2S(E)rAS∗(−E)r∗A])

, (2)

where S(E)[S∗(−E)] is the electron(hole) scattering matrix for the normalregion and α = exp[−acosh(E/∆)] and rA = diag[exp(iφ/2), exp(−iφ/2)] de-scribe the Andreev reflection at the NS-interfaces. The dimensions of the scat-tering matrices S(E) and rA are 2N × 2N , with N = 2W/λF the number oftransverse transport modes in the normal region between the superconduc-tors, with λF the Fermi wavelength. The current density for energies withinthe gap is given by adding a small, positive imaginary part to the energyE → E + i0.[17,16] The expression (2) then reduces to the well known result[18]

i(E) =2e

h

∑

m

dEm

dφδ(E − Em), (3)

3

where the index m is labeling the bound states given by the equation det[1−α2S(E)rAS∗(−E)r∗A] = 0. This form of the current density is useful when thebound state energy as a function of phase difference φ is explicitly known. Thecurrent density has the energy symmetry i(−E) = −i(E).

The junctions studied are in the long limit, L ≫ ξ0, ξ0 = hvF/∆. Thenonequilibrium Josephson current in junctions in the opposite, short limit, hasbeen studied in Ref.[14]. Only classically wide junctions, with many transportmodes N ≫ 1, are considered below. The opposite quantum limit, N = 1, wasstudied in Refs. [1,3,4].

3 Perfectly transmitting NS-interfaces

We first consider the case when there is perfect Andreev reflection at the NS-interfaces, i.e no normal reflection. For the low energy levels, E ≪ ∆, thebound state energies are given by [19]

E±

p,n =hvFn

2L[(2p + 1)π ± φ] , (4)

where the index n denotes the transverse mode and p,± labels the Andreevlevels for a given mode. Due to the hard wall conditions, the Fermi velocity for

each mode n is vFn = vF

√

1 − (n/N)2. The current density for each Andreev

level is given by inserting the expression (4) into (3). The total current densityis then obtained by first summing over the modes n, equivalent to integratingover angles, and then summing over p,±, giving

i(E) = Ne

h

hvF

L

∑

p,±

±E2θ(E±

p0 − E)

(E±

p0)2√

(E±

p0)2 − E2

, (5)

where θ is the Heavyside stepfunction and E±

p0 is given from Eq. (4) withn = 0. The current density for phase differences φ = π/4 and 3π/4 is plottedin Fig. 2. The current density consists of alternating positive and negativepeaks at energies E±

p0. The peaks arise from the square root singularities ofthe current density in Eq. (5) and the amplitude of the peaks is decreasing forincreasing energy.

The expression for the equilibrium part, Ieq, of the Josephson current in Eq. (1)is known.[19–21] The nonequilibrium part, Ineq, is straightforwardly calculatedfrom Eqns. (5) and (1). It is clear from the form of the current density in Eq.(5) that the total current I will be an oscillating function of voltage, withalternating maxima and minima, at voltages eV = E±

p0 for zero temperature

4

0 50 100 150 200 250 300 350 400−6

−5

−4

−3

−2

−1

0

1

2

3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.06

−0.04

−0.02

0

0.02

0.04

0.06

I (N

ev /

L)

i(E

) (N

e/h)

FeV/(hv /L)

0

F

21

-0.4

4

0.4

F

0

3

-4

2

0

2

0 21 3

E/(hv /L)

Fig. 2. Upper: The current density as a function of energy. Lower: The total cur-rent as a function of voltage for zero temperature. The phase difference φ = π/4(dashed)and 3π/4 (solid) and L = 10ξ0

(see Fig. 2). The period of oscillation is thus πhvF /L. The amplitude of theoscillations, ∆Ip = I(eV = E+

p0) − I(eV = E−

p0), in the limit eV ≫ hvF/L,decays with voltage as

∆Ip ≃ NevF

L

(

|φ|hvF

LeV

)3/2

. (6)

For finite temperatures the amplitude of the voltage oscillations decreases,and the oscillations are completely washed out for kT ≫ hvF /L.

5

The equilibrium current Ieq is always positive for phase differences 0 < φ < πand negative for −π < φ < 0. In nonequilibrium, the total current for agiven phase difference I(φ) changes sign as a function of applied voltage, i.ethe junction becomes a so called π-junction [2,9]. This can bee seen from thecurrent to phase relationship I(φ), shown in Fig. 3 for different voltages. Itcan be noted that for certain voltages, the current to phase relationship hasseveral local current minima and maxima.

The critical current is defined as the maximum possible current for phasedifference −π < φ < π. To study the π-junction behavior, the critical currentmultiplied by the sign of the critical phase difference φc, sgn(φc)Ic, is shown inFig. 3 for different temperatures. The critical current multiplied by the criticalphase difference sgn(φc)Ic oscillates between positive and negative values asa function of voltage, with a period πhvF /L, i.e it shows a typical π-junctionbehavior. The amplitude of the oscillations decreases with increasing voltage.It can be noted that for zero temperature, there are, for certain voltages, jumpsbetween the branches of positive and negative critical current, i.e the criticalcurrent Ic never becomes zero. This can be understood from the current-phaserelationship in Fig. 3, where for certain voltages, the critical phase differenceφc jumps between positive and negative values, changing the sign of sgn(φc)Ic.

4 Normal reflection at the NS-interfaces

Normal scattering at the NS-interfaces is taken into account by introducingeffective interface barriers with the transmission probability Γ. The low lyingbound states energies, E ≪ ∆, are given by[22]

E±

p,n =hvFn

2L

[

2pπ ± acos

(

4(1 − Γ) cos(2kFnL) − Γ2 cos φ

(2 − Γ)2

)]

. (7)

The bound state energies oscillate rapidly as a function of length of the junc-tion, due to the term cos(2kFnL) in Eq. (7). The corresponding rapid oscilla-tions of the current density are averaged out when summing over the transversemodes.[23] The total current density, summed over n and p,±, is given by

i =e

h

2Γ2 sin φ

π

N∫

0

dnsgn[sin(2EL/hvFn

)]√

16(1 − Γ)2 − [(2 − Γ)2 cos(2EL/hvFn) + Γ2 cos φ]2

.(8)

The current density as a function of energy is plotted for different barriertransparencies Γ in Fig. 4. The current density has a similar shape, withalternating positive and negative peaks, as the current density for the junction

6

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

eV/(hv /L)F

I (N

ev /

L)

FI

(N

ev /

L)

cF

-0.4

86420

0.40.2

0

−π π0φ

-0.2

0.2

0

0.4

-0.4-0.2

Fig. 3. Upper: The current phase relationship for different voltages eV = 0 (dotted),eV = hvF /L (dashed) and eV = 2hvF /L (solid), temperature T = 0. Lower: Thecritical current multiplied by the sign of the critical phase difference sgn(φc)Ic as afunction of applied voltage for different temperatures, kT = 0, 0.3, 0.5hvF /L, withdecaying amplitude for increasing temperature.

without barriers at the NS-interfaces, shown in Fig. 2. The effect of the NS-barriers, apart from the overall decreased current density amplitude, is thateach current density peak is shifted towards lower energies, as is seen in Fig. 4.The current as a function of voltage, for a fixed phase difference, thus oscillateswith the same period πhvF /L as in the case without barriers at the NS-interfaces, and with a decreasing amplitude of the oscillations for increasingvoltage (see Fig. 4).

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Γ2Γ2

0 0.5 1 1.5 2 2.5 3 3.5 4−1.5

−1

−0.5

0

0.5

1

i(E

) (N

e

/h)

I (

N

ev

/L)

F

FE/(hv /L)

0 1 2 3

0

-1.0

-0.5

0.5

0

0.2

0.4

2 4310-0.2

eV/(hv /L)F

Fig. 4. Upper: The current density as a function of energy. Lower: The total cur-rent as a function of appled voltage. The barrier barrier transparencies Γ = 0.1(solid), 0.5 (dashed) and 0.9 (dotted). The phase difference φ = 3π/4 and thelength L = 10ξ0

5 Dirty NS-interfaces

Usually, there is dirt at the NS-interfaces from the junction processing, leadingto diffusive scattering at the interfaces. To simulate this diffusive scattering,we introduce random scattering matrices S1 and S2 to model the interfaces.The matrices are written in the polar decomposition[24]

8

Sj =

rj tTj

tj r′j

=

Vj

√

1 − ΓjVTj Vj

√

ΓjUTj

Uj

√

ΓjVTj −Uj

√

1 − ΓjUTj

, (9)

where the barrier transmittances Γj are taken to be mode independent andequal for both barriers. The unitary matrices Uj , Vj are taken to be indepen-dent members of the ensemble of unitary, symmetric matrices (COE)[25]. Thetotal electron scattering matrix for the normal region is given by [26]

S = r + tS0(1 − r′S0)−1tT , S0 =

0 P

P 0

, (10)

where r = diag(r1, r2) and similarily for r′ and t. The matrix S0 is the scat-tering matrix for the normal region without NS-barriers, with the diagonalmatrices P with elements Pn = exp(i[kFn + EL/(hvFn)]), kFn beeing theFermi wave vector. This is inserted in Eq. (2) to give the current density. Itcan be pointed out that for only forward mode mixing scattering (Γ = 1), themode mixing at each interface is completely reversed by the Andreev reflec-tion, giving the same result as in the absence of barriers at the NS-interfaces.

We are interested in the current density averaged over the random matricesUj , Vj, which is calculated numerically by generating a large number of matri-ces. The current density is plotted in Fig. 5 for different barrier transparenciesΓ.

0 0.5 1 1.5 2 2.5 3 3.5 4−4

−3

−2

−1

0

1

2

Γ2i(

E)

(N

e/h

)

E/(hv /L)F

0 1 2 3

0

0.5

-0.5

-1.0

Fig. 5. The current density as a function of energy for different barrier transparen-cies Γ = 0.1 (solid), 0.5(dashed) and 0.9(dotted). The current density has beencalculated by generating 1000 random matrices of dimension N = 15. The phasedifference φ = 3π/4 and the length L = 10ξ0

9

The main result is that the peak-like structure of the current density is notsubstantially changed compared the corresponding specular barrier case in Fig.4. From this one can draw the conclusion that the effect of diffusive scatteringat the NS-interfaces does not modify the nonequilibrium Josephson currentqualitatively. The result that the current for a fixed phase difference oscillatesperiodically with applied voltage is still valid. Whether rough 2DEG-sidewalls,giving rise to diffusive boundary scattering, have a more profound effect onthe current, remains to be investigated. It is known that in the case with thenormal region being a chaotic cavity, the peak-like current density structureis completely washed out, and a gap opens up in the spectrum.[16]

6 Conclusions

In conclusion, we have studied the nonequilibrium Josephson current in longtwo-dimensional ballistic SNS-junctions weakly coupled to a normal metalreservoir. The total current is given by a convolution of a single current densityi(E) with the quasiparticle distribution functions [See Eq. (1)]. Junctions withand without specular normal scattering at the NS-interfaces and also junctionswith diffusive NS-interfaces are studied. It is found that the current densityin all cases has a peak-like structure, with alternating signs of the peaks.The resulting nonequilibrium Josephson current for a given phase differencethus oscillates as a function of applied voltage, with a period πhvF /L, andan amplitude decreasing with increasing voltage. This behaviour also carriesover to the critical current, which changes sign as a function a voltage, i.e thejunctions displays so called π-behavior.

7 Acknowledgements

This work has been supported by research grants from NFR, TFR, NUTEK(Sweden) and NEDO International Joint Research Grant (Japan).

References

[1] B.J.van Wees, K.M.H. Lenssen, and C.J.P.M. Harmans, Phys.Rev.B 44, 470(1991).

[2] A.F. Volkov, Phys. Rev. Lett. 74, 4730 (1995).

[3] L.F. Chang, and P.F. Bagwell, Phys. Rev. B 55, 12678 (1997).

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[4] P. Samuelsson, V.S. Shumeiko, and G. Wendin, Phys. Rev. B 56, R5763 (1997).

[5] S.K. Yip, Phys. Rev. B 58, 5803 (1998)

[6] F.K. Wilhelm, G. Schon, and A. Zaikin, Phys. Rev. Lett. 81, 1682 (1998).

[7] A.F.Morpurgo, T.M.Klapwijk, and B.J. van Wees, Appl. Phys. Lett. 72, 966(1998).

[8] Th. Schapers, J. Malindretos, K. Neurohr, S. Lachenmann, A. van der Hart, G,Crecelius, H. Hardtdegen, H. Luth, and A.A. Golubov, Appl. Phys. Lett. 73,2348 (1998).

[9] J.J.A. Baselmans, A.F. Morpurgo, B.J. van Wees, and T.M. Klapwijk, Nature397, 43 (1999).

[10] The experiment by J. Kutchinsky, R. Taboryski, C. B. Sørensen, J. BindslevHansen, and P. E. Lindelof, Phys. Rev. Lett. 83, 4856 (1999), was performedwith a superconducting injector, but qualitatively similar results have beenpredicted for a normal injector in Ref. [13].

[11] See contributions from B.J. van Wees et. al and P. Delsing et. al in this issue.

[12] It is shown in experiments with a wide injection lead, K. Neurohr, Th. Schapers,J. Malindretos, S. Lachenmann, A. I. Braginski, H. Luth, M. Behet, G. Borghsand A.A. Golubov, Phys.Rev.B. 59, 11197 (1999), that this holds for junctionwidths up to several microns.

[13] R. Seviour and A. F. Volkov, Phys. Rev. B. 61, R9273 (2000).

[14] P. Samuelsson and H. Schomerus, cond-mat/0004197.

[15] E. Akkermans, A. Auerbach, J. E. Avron, and B. Shapiro, Phys. Rev. Lett. 66,76 (1991).

[16] P. W. Brouwer and C. W. J. Beenakker, Chaos, Solitons and Fractals 8, 1249(1997).

[17] E. Doron and U. Smilansky, Phys. Rev. Lett 68, 1255 (1992).

[18] C. W. J. Beenakker, Phys. Rev. Lett. 67, 3836 (1991).

[19] I.O. Kulik, Sov. Phys. JETP 30, 944 (1970)

[20] C. Ishii, Prog. Theor. Phys. 44, 1525 (1970).

[21] A.V. Svidzinsky, T.N. Antsygina, and E.N. Bratus’, Sov. Phys. JETP 34, 860(1972).

[22] H. Blom, A. Kadigrobov, A.M. Zagoskin, R.I. Schekter, and M. Johnson, Phys.Rev. B. 57, 9995 (1998).

[23] In shorter junctions, these oscillations show up as oscillations of the criticalcurrent, as discussed in U. Schussler and R. Kummel, Phys. Rev. B. 47, 2754(1993) and A. Chrestin, T. Matsuyama and U. Merkt, Phys. Rev. B. 49, 498(1994).

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[24] P.A. Mello, P. Pereyra and N.Kumar, Ann. Phys. NY, 181, 290 (1988).

[25] C.W.J Beenakker, Rev. Mod. Phys. 69, 808 (1997).

[26] P. Brouwer, Phys. Rev. B. 51, 16878 (1995).

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