ON THE THEORY OF QUANTIZED RESISTANCES IN SUPERCONDUCTING-NORMAL-SUPERCONDUCTING FILM STRUCTURES

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ON THE THEORY OF QUANTIZED RESISTANCES

IN

SUPERCONDUCTING-NORMAL-SUPERCONDUCTING FILM STRUCTURES

P. Hansack, R. Kümmel

To cite this version:

P. Hansack,

R. Kümmel.

ON THE THEORY OF QUANTIZED RESISTANCES IN

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JOURNAL DE PHYSIQUE

Colloque C6, srrppliment au

no

8, Tome 39, aotit

1978,

page

C6-563

ON THE THEORY OF Q U A N T I Z E D RESISTANCES I N SUPERCONDUCTING-NORMAL-SUPERCONDUCTING

P. Hansack and R. ~iimmel

PhysikaZisches I n s t i t u t der Universittit Wiirzburg, 87 Wii~zburg, Germany

R6sumd.- Nous prdsentons le calcul des caractdristiques IV de film S.N.S. avec un courant parallGle etabli

P

partir de la dynamique de diffusion des quasi-particules. I1 permet de retrouver les carac- tdristiques essentielles des expdriences de rssistance quantifise.

Abstract.- We present a calculation of current-voltage-characteristics of SNS films with current flow parallel to the phase-boundaries which, on the basis of quasiparticle scattering dynamics, re- produces the main features of the quantized resistances experiments.

1.I;:TODUCTION.- Recently, current-voltage-characte- ristics (CVC) of SNS films with current flow

paral-

lel to the phase- boundaries have been measured /1,

-

41. They exhibit steps and linear current branches with quantized resistances. One of us (R.K.) worked out a microscopic theory / 5 / for current driven

SNS systems which at T = 0 K yields CVC resembling the returning sections of the experimental CVC 11,

41. The underlying physical picture is the follo- wing /5/. In thermal equilibrium with the lattice, a certain class of "current excited" quasiparticles /6/(q.p.), mainly localized in the N region (bound states), with a well defined Fermi momentum compo- nent kZF normal to the phase-boundaries, decay in the S-regions, merge into the ground state and transfer their momenta, which are essentially

oppo-

site to the current-flow, to all ground state elec-

-

trons. In order to balance the negative momentum ifput from these particle-hole (Andreev)scattering processes and maintain a stationary current,a voltage must appear between the film ends. It is rela- ted to the (virtual) phase-shift of the order para- meter, caused by the q.p. momentum transfer, by a Josephson-like relation 151. In order to account for the observed linear current branches/l,4/ a model of inhomogeneous ground state currents in the N and S regions has been proposed /5/ and subse- quently explored at finite temperatures. In this paper we are reporting the results obtained with this model.

2.PROCEDURE.- As before /5/ we use the Bogoliubov- de Gennes Equations (BdGE) with a step-like pair-

+work supported

in

part by the Deutsche Forschungs- gemeinschaft

.

potential (see insert figure 1) and without a ma- gnetic field H. (Experiments were done with H being either zero/3/ or parallel to the film surface / I ,4/)

Fig.] : currentvoltage-characteristic of an SNS- film with returning section/l/(solid line, qS =7 x 10-5 kF) and linear branches of increasing current (dashed lines, qS-qN = const.). The film parameters are about the same as in the experiments /1,4/: lenggh L = 5 m , width W = 0.3m,thickness 2 D = 5000 A , ns= 1 . 2 ~

However, we do no longer assume a homogeneous ground state current 151, but allow for a variation of the average net momentum e q (z) of ground sta-

-Y

te electrons from the

N

to the S regions, accor- ding to a step-function ansatz :

q (2) = qN B(-IzI+ a) +q@(~-Izl)e (1.1-a) ( 1 )

with d q /dz = 0. ~ h u s , the model of the pair-

, N,S

potential in the BdGE is

and for the q.p. wavefunctions in the N and S regions we have

(3)

where u0 (r) and v0 (r) are the solutions of the

N,S N,S

BdGE without current flow'/5/,i.e.q (z) = 0. The mis- match of the phases of the wave-functions (3) at

the N-S phase boundaries in z = a is caused by

theansatz ( 1 ) which, however, is consistent with, and should not introduce more serious errors than the widely used /5,6/ step-function approximation (2) for

A(z)

.

3.RESULTS. Matching of the absolute values of the

wavefunctions and of their derivatives at z = a

and the condition that the functions vanish at the

outer films surfaces at z = D yield the eigenva-

lue equation for the energies E l of the q.p. states These states are superpositions of electrons and holes travelling in i z-direction, having plane wave character in x- and y-directions parallel to the phase-boundaries/5/. In the set of quantum numbers

1 Z (kx, ky, n, spin), characterizing one q.p. state, n labels the quantized energy levels for fixed

kx and k The eigenvalue equation, even more com-

Y'

plicated than that of refdrence /5/,has to be sol- ved numerically. Its solutions E l determine the vol-

tage U between the film ends and the current I parallel to the phase-boundaries in y-direction accor- ding to the equations 151.

Here, e and m are the electronic charge and mass, V

and L are the volume of the film and its length in y-direction, ns = ki / 3 lr2, is the q.p. momen-

kY

+

- ₂

tum parallel ta the current, k; =Lk:F

2

2 (mEI

/H

-

2 2

k q f11/2, kzp =

Ep

-

kx

-

kGJi/r

,

and f ( E l ) Y N-

is the Fermi distribution function. The probability

P (k ) of particle-hole scattering in bound states

zF

is approximated by a weighted delta function /5/ 8(kZF-ko) , 2 ~ ~ < ko <2.5 K ~ , Ko=(m A) '/2/

H.

Numerical integration of eqs. (4) and (5) yields the current- voltage-characteristic of figure 1.

4.DISCUSSION.- A q.p. state is bound and leads to voltage induction for kZF = ko, if its energy E l

1 5 1 and E l % E; +

n2

k a/D +qS(~-a)/$

1

/m repre- sents a rather good fit to the numerically calculated eigenvalues. In the returning section of the CVC at finite temperaturesthe voltage steps occur, whenever with decreasing qN a class of q.p. states

with kZF = k and fixed quantum number n ceases

(ky< 0) or starts (k > 0) to satisfy the binding Y

criterion (6). The condition : "q fixed, qNdecrea- S

sing" is the simplest one for a most rapid reduc- tion of energy dissipation with decreasing current. On the other hand, in the linear current branches where the current rises so that the difference qS- q remains fixed at the values it has in the res-

N

pective starting points on the returning section,

an observer moving either with the velocity

H

qN/

m with the velocity $ qS/ m of the ground state

flow in the N-or in the S-region would observe no change in the q.p. energy and thus in q.p.nature (i.e. bound vs.unbound, amplitudes, etc.)

The width of the calculated voltage-steps in figure 1 has about the magnitude of the measured ones /1,4/. The net current is the small difference between the large ground state flow and the large q.p. counter-current 161. It is extreme- ly sensitive to the accuracy of the numerical cal- culations, within the errors of which agreement between the theoretical and the experimental/l,4/ current data may be considered as satisfactory.

References

/I/ Chen,J.T., Hayler,L.G.and Kim,Y.W., Phys.Rev. Lett.

30

(1973) 645

../2/ Hayler,L.G., Geppert,L.M., Chen,J.T.' and Kim Y.W.,Phys.Rev. B

2

(1975) 1924

131 Geppert,L.M., Hayler,L.G.,Chen,J.T., and Kim Y.W., Phys.Rev. B

14

(1976) 1062

/4/ Geppert,L.M.,Thomas,R.L. and Chen,J.T., Phys. Rev. B

16

(1977) 1032

/5/ ~Gnunel,~. Phys. Rev. B

16

(1977) 1979

161 Bardeen,J. and Johnson,J.L., Phys.Rev. B

5

(1972) 72

satisfies the criterion