VOLUME 80, NUMBER 17
PHYSICAL REVIEW LETTERS
27 APRIL 1998
Hall Effect in SN and SNS Junctions
F. Zhou* and B. Spivak
Physics Department, University of Washington, Seattle, Washington 98195
(Received 17 March 1997)
Hall effect in SN and SNS junctions is considered. It is shown that at low temperatures the Hall
voltage is significantly suppressed as compared to its normal metal value. The time dependence of the
Hall voltage in SNS junctions has a form of narrow pulses with the Josephson frequency. [S00319007(98)05980-8]
PACS numbers: 74.50. + r, 05.20. – y, 82.20. – w
Recent progress in microfabrication technology has
rekindled the interest to low temperature proximity effect
in superconductor–normal-metal (SN) junctions [1–10].
In this paper we consider the Hall effect in superconductor–normal-metal and superconductor–normal-metal–
superconductor (SNS) junctions. At low temperatures
T ø D, quasiparticles with energies e ø D cannot
tunnel from the metal into the superconductor. Here D is
the value of the gap in the superconductors. On the other
hand the tunneling of electron pairs, which is known as
Andreev reflection, is possible. This gives rise to a coherence between electrons and holes inside the metal,
which
p
extends over distances of the order of Le ­ Dye ¿ l.
1
Here D ­ 3 yF l is the electron diffusion coefficient, l is
the elastic mean free path, and yF is the Fermi velocity.
Consequently, the wave packets which carry the current in
the metal are coherent superpositions of electron and hole
wave functions. At
p T ø 1yt ­ yF yl the packets’ size
is of order LT ­ DyT ¿ l and their effective charge
is much smaller than the electron charge e. The above
mentioned electron-hole coherence manifests itself in experimentally observable effects. For example, the density
of states in the metal near the SN boundary nse, r$ d is
significantly suppressed as compared with the bulk metal
value n0 ­ m2 yF [8–10]; the low temperature resistance
of the SN junction and the electric field distribution near
the SN boundary turn out to be very sensitive to the phase
breaking rate in the metal [4–7].
The suppression of the effective charge should also
lead to a suppression of the Hall effect. We show
below that the Hall voltage in SN junctions measured
with the help of leads 1,2 shown in Fig. 1(a) is indeed
significantly suppressed as compared to its bulk metal
value. It is worth mentioning here that at zero temperature
the resistance of a long SN junction is approximately
the same as the resistance of the normal segment [4–7].
Therefore the suppression of the Hall voltage is not
directly connected to the suppression of the electron
density of states near the SN boundary [8–10].
We also show that the Hall voltage in SNS junctions
exhibits oscillations of its amplitude with Josephson
frequency and with no oscillations of the sign. It is
different from the behavior of supercurrent through the
junction which at a given voltage oscillates both in
amplitude and in sign.
In this article we neglect the electron-electron interaction in metal and are interested only in effects linear in
voltage applied to the junction. In the “clean limit” when
Dt, Tt ¿ 1, the kinetic scheme describing Hall effect in
superconductors was developed in [11,12]. In this limit,
however, the above mentioned proximity effect leads only
to small corrections to the value of the Hall voltage in
metal. To describe the Hall effect in a metal near the SN
boundary in the “dirty limit” when T ø D, 1yt we use
the following set of equations analogous to that in [13,14]:
J$ ­ J$ s 1 J$ n 1 J$ H ,
D Z
D Z
J$ s ­
n0 def0 sed Trssz fG R =G R 2 G A =G A gd,
n0 de=f1 sed Trs1 2 sz G R sz G A d ,
J$ n ­
4
4
Z
vc t
J$ H ­
Dn0 de e$ b 3 =f1 3 Trhs1 1 sz G R sz G R 2 G A G R 2 sz G A sz G R d sG R sz 2 G A sz dj ;
16
eFsr$ d ­
Z
def1 se, r$ dnse, r$ d,
nse, r$ d ­
1
Trssz G R 2 G A sz d ;
2
R
ehG R sz 2 sz G R j 1 D≠ ? sG R ≠G R d ­ Iph
;
(1)
(2)
(3)
=hTrs1 2 sz G R sz G A d=f1 1 Trfsz sG R ≠G R 2 G A ≠G A dgf0 1 vc t e$ b 3
=f1 Trfs1 1 sz G R sz G R 2 G A G R 2 sz G A sz G R d sG R sz 2 sz G A dgj ­ Trssz Iph d .
0031-9007y98y80(17)y3847(4)$15.00
© 1998 The American Physical Society
(4)
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VOLUME 80, NUMBER 17
PHYSICAL REVIEW LETTERS
Here J$ s , J$ n , J$ H are the supercurrent density, normal current density, and the Hall current density, respectively,
f0 ­ tanhsey2kT d is the Fermi distribution function,
vc ­ eHymc is the cyclotron frequency; H is the magnetic field, and e$ b is the unit vector in the direction
of magnetic field; F ­ 1y2≠t x 1 f, f, and x are the
gauge invariant scalar potential, electrical potential, and
phase of anomalous Green’s function, respectively; sz is
the Pauli matrix, ≠X ­ =X 2 issz X 2 Xsz dA$ is the covariant derivative, A$ is the vector potential of the magnetic
field; G R,A are retarded and advanced Green’s functions
which are matrices in the Nambu space
µ R,A
∂
g
F R,A
R,A
G se, r$ d ­
,
(5)
2F R,A gR,A
gR,A se, r$ d and F R,A se, r$ d are normal and anomalous
Green’s functions, respectively. Thus the set of equations
consists of electroneutrality condition Eq. (2), the Usadel equation for retarded and advanced Green’s functions
G R,A Eq. (3), and the kinetic equation Eq. (4) for the distribution function f1 se, r$ d which describes the imbalance
of populations between the electron and hole branches of
R
ø isz tin
spectrum [13,14]. The scattering integrals Iph
and Trssz Iph d describe the broadening of the spectrum
and the imbalance charge relaxation due to inelastic processes. In Eqs. (3),(4), we take into account that in noninteracting metal the order parameter and the supercurrent
are zero.
In the zeroth order approximation in the parameter vc t,
these equations were derived in [13]. However, to describe the Hall effect one has to keep terms linear in vc t.
The problem of the Hall effect in the mixed state of superconductors was addressed recently in [12,15,16]. We
derive Eqs. (1),(3) which include the contributions linear
in vc t using the same procedure as in [11,12,15,16].
Let us now consider the Hall effect in the SN junctions shown in Fig. 1(a). We assume that the magnetic
field is smaller than the critical one and does not penetrate
into the superconductor. Taking into account the normalization condition fgR se, r$ dg2 ­ F2R se, r$ dF1R se, r$ d 1 1 one
27 APRIL 1998
can express gR , F R , and Eqs. (1)–(4) in the form
cos use, r$ d ­ gR se, r$ d,
sin use, r$ d ­ iF1R se, r$ d ,
µ
∂
D 2
1
sin used 2
= used 1 ie 2
2
tin
µ
∂
D eH 2 2
x sin 2u ­ 0 ,
4 h̄c
(6)
(7)
= ? hcosh2 u2 =f1 1 tvc e$ b 3 =f1 cos u1 cosh3 u2 j ­ 0 ,
Z
J$ ­ Dn0 dehcosh2 u2 =f1 1 tvc e$ b
3 =f1 cos u1 cosh3 u2 j ,
(8)
where u ­ u1 1 iu2 is the complex variable. We have
chosen A$ ­ e$ y Hx.
The boundary conditions for Eqs. (7),(8) have the
forms [17]
$ ­ t sinfuse, 01 d 2 use, 02 dg ,
D n$ ? =use, Rd
D cosh u22 ? =f1 ­ tyF hf1 s02 d 2 f1 s01 dj
3 cosh u2 s01 d cosh u2 s02 d
3 cosfu1 s01 d 2 u1 s02 dg ,
f1 s0d ­ 0 ,
f1 se, x ­ `d ­ 2eV ≠e f0 sed .
(9)
Here n$ and t are the unit vector perpendicular to the
boundary and the transmission coefficient of the boundary, respectively. Indexes 1, 2 indicate two sides of the
SN or NNp interface [Fig. 1(a)]. Np represents lead 1,2.
Inside the superconductor use ø Dd ­ py2 while inside
the bulk metals u ­ 0.
Solving Eqs. (8),(9) in zeroth order approximation in
vc t we get the expressions for the conductance GSN ­
I0 yU0 of the junction shown in Fig. 1(a) [4–7] measured
by the two probe method and the conductance Gp measured between lead 1 and lead 2. (Here I0 is the current
L
L
D
D
FIG. 1. (a) Temperature dependence of Hall voltage in SN junction. a p ­ maxh L0t , L0 j, T p ­ minh L20 , L20 j. (b) Time dependence
t
of Hall voltage in SNS junctions with geometry shown in the inset, t p ­ t0 T yEc ø t0 ­ 1y2eU0 . U0 is applied along X direction
and UH is measured along Y direction in normal metals.
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VOLUME 80, NUMBER 17
PHYSICAL REVIEW LETTERS
27 APRIL 1998
through the junction and U0 is the voltage across the junction.) That is
Z
Z
1
1
de≠e f0
GSN sTd ­ S0 s0 de≠e f0
,
Gp sT d ­ Sp sp
,
LS sed
2Lp sed
(10)
where
LS sed ­
Lp sed ­
Z L0
1
Lt
dx
,
1
2
cosh u2 se, 0d sin u1 se, 0d
cosh
u
0
2 se, xd
Z Lp
Ltp
1
,
dy
1
2
cosh u2 se, Ld cosh u2p se, 0d cosfu1 se, Ld 2 u1p se, 0dg
se, yd
cosh u2p
0
(11)
S0,p are the cross sections of the sample and the leads, s0,p , D0,p are the Drude conductivities and diffusion constants in
the sample and leads; Lt ­ D0 yt0 yF , Ltp ­ Dp ytp yF are the length characterizing SN and NNp boundaries, t0 and tp
are the transmission coefficients for SN and NNp boundaries, up se, yd is the solution of Eq. (7) inside the leads.
In the first order approximation in vc t we get an expression relating the Hall current IH and Hall voltage UH
measured by leads 1,2
Z
S p sp 1
cos u1 se, Ld
d
de
IH ­ Gp sT dUH sT d ­ tvc I0
(12)
≠e f0 .
GSN sT d
Lp sed LS sed cosh u2 se, Ld
Here d is the width of the junction (see Fig. 1). The values of u1 , u2 , and UH (as well as GSN and Gp ) depend on the
processes breaking the electron-hole coherence inside the sample and in the leads. Therefore, they are very sensitive to
the ratio between parameters L0 , LT , Lt , Ltp [4,5,7].
The solution of Eq. (7) in the metal for SN junctions when L0 ¿ Lt with the corresponding boundary conditions in
Eq. (9) is [7]
8p
p L 1L
>
2 2 tL0 ,
Lt , L ø L0 ø Le ;
>
2
<p
Lt 1L
L t , L ø L e ø L0 ;
(13)
use, Ld ­ p2 2 s1 2 id Le ,
>
>
2
ps11idL
x
e
:
p
expf2s1 2 id
g, L ø L ø L .
4Lt
The most interesting results appear in the cases when
the processes which break the electron-hole coherence are
not effective and the value of u1 sT , Ld is close to py2.
At small T and large t
8
< maxh Lt , Lt j, L ø L ø L ;
UH
t
T
L 0 LT
~
(14)
: maxh L , L j, Lt ø L ø LT .
UHN
L0 LT
Here UHN ­ vc tdI0 yS0 s0 is the Hall voltage measured
by the leads 1,2 in the absence of the proximity effect.
The main feature of Eq. (14) is that UH is significantly
suppressed as compared with UHN [see Fig. 1(a)].
At high temperatures when LT ø Lt or L the proximity effect gives only small corrections to UHN
Ω 2 2æ
UHN 2 UH
LT LT
(15)
~ min 2 , 2 ø 1 .
UHN
Lt L
The above results are obtained in the limit
p of low
magnetic field when LH ø LT . Here LH ­ F0 yH is
the magnetic length, F0 is flux quantum. In the opposite
limit one should substitute LT for LH in Eqs. (14),(15).
As we have mentioned the value of UH is very sensitive
to the nature of the leads. The requirement that the leads
do not contribute to the breaking of the electron-hole
coherence is d ø Ltp , Lt ø L0 .
2 Le
e
t
0
Let us now turn to the discussion of the Hall effect in
SNS junctions. In this case the ac Josephson effect causes
the values of usr$ , x0 d and nsr$ , x0 d to be time dependent.
At small U0 these dependences adiabatically follow the
corresponding time dependence of the order parameter
phase difference x0 across the junction: ≠t x0 std ­ 2eU0 .
For simplicity let us consider the case of a thin junction
when d ø LH when one can neglect any y dependence
of x0 along the junction. It is well known that at x0 ­ 0,
the energy dependence of nsed ­ cos u1 sed cosh u2 sed exhibits a gap Eg sx0 ­ 0d , Ec ­ DyL02 [3]. The x0 dependences of u, n, and Eg were calculated in [10,18].
It was shown that the value of the gap Eg monotonically decreases with x0 and closes at x0 ­ p. Hence
cos u1 sed ­ 0 when e # Eg sx0 d. In two limiting cases
when x0 is close to p or 0 Eg sx0 d is determined by [18]
Ω
s1 2 Cx02 d,
x0 ø p ;
Eg sx0 d ­ Eg s0d
(16)
C1 sp 2 x0 d, p 2 x0 ø p .
Here C and C1 are of the order of unity. It follows from
Eq. (12) that the Hall voltage UH is a periodical function
of time with a period 1y2eU0 . At low temperatures
T ø Ec the value of UH std is exponentially small except
when the gap is small Eg fx0 stdg , T . As a result UH std
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VOLUME 80, NUMBER 17
PHYSICAL REVIEW LETTERS
consists of short pulses of duration t p , TyEc eU0 with
maximums of the order of UHN [see Fig. 1(b)]. Let tn be
the time when the nth maximum of UH std occurs. Then
at t p ø jt 2 tn j ø 1y2eU0 ,
µ
∂
jt 2 tn j
.
(17)
UH std , UHN exp 2
tp
The main contribution to the Hall current averaged over
the period of oscillations comes from the time intervals
jt 2 tn j , t p ø 1yU0 and as a result
T
UHN .
(18)
Ec
We would like to mention the difference in the temperature dependences
in the SN and SNS cases. In the first
p
case UH , T, while in the second case UH , T. Let
us note that the time dependence of the resistance of the
junction exhibit peaks which are similar to the above considered peaks of the Hall voltage.
In addition to the quasiparticle’s contribution to the
Hall current in SNS junctions there is another contribution
which can be associated with the supercurrent [15,16].
We believe, however, that in the limit when d ø LH , the
supercurrent part of the Hall current does not contribute
to the Hall voltage. In the case of SN junctions the
supercurrent contribution to the Hall current does not exist.
In this paper we considered the limit eU0 ø 1yte
when it is possible to neglect nonequilibrium corrections
to the quasiparticle distribution function inside the normal
metal. In the opposite limit the quasiparticle distribution
in the metal at e , Ec becomes nonequilibrium and as a
result the duration of the pulses t p becomes of the order
of seU0 d21 .
The above results were obtained for the quantities
averaged over realizations of random potential. We are
planning to consider mesoscopic contributions in UH
elsewhere.
We acknowledge useful discussions with B. Altshuler,
A. I. Larkin, and B. Pannetier. This work was supported
kUH l ~
3850
27 APRIL 1998
by the Division of Materials Sciences, U.S. National
Science Foundation under Contract No. DMR-9625370.
*Permanent address: Physics Department, Princeton
University, Princeton, NJ 08540.
[1] A. Kastalsky, A. W. Kleinsasser, L. H. Greene, R. Bhat,
F. P. Milliken, and J. P. Harbison, Phys. Rev. Lett. 67,
3026 (1991).
[2] H. Pothier, S. Gueron, D. Esteve, and M. H. Devoret,
Phys. Rev. Lett. 73, 2488 (1994).
[3] H. Courtois, Ph. Gandit, D. Mailly, and B. Pannetier,
Phys. Rev. Lett. 76, 130 (1996).
[4] C. W. J. Beenakker, Phys. Rev. B 46, 12 841 (1992).
[5] A. F. Volkov, Physica (Amsterdam) 203B, 267 (1994).
[6] Yu. V. Nazarov and T. H. Stoof, Phys. Rev. Lett. 76, 823
(1996).
[7] F. Zhou, B. Spivak, and A. Zyuzin, Phys. Rev. B 52, 4467
(1995).
[8] A. A. Golubov and M. Y. Kupriyanov, JETP Lett. 61, 851
(1995).
[9] W. Belzig, C. Bruder, and G. Schon, Phys. Rev. B 54,
9443 (1996).
[10] J. A. Melsen, P. W. Brower, K. M. Fraham, and C. W. J.
Beenakker, Phys. Scr. T69, 223 (1997).
[11] U. Eckern and A. Schmid, J. Low Temp. Phys. 45, 137
(1981).
[12] N. B. Kopnin, J. Low Temp. Phys. 97, 157 (1994).
[13] A. I. Larkin and Yu. N. Ovchinnikov, Zh. Eksp. Teor. Fiz.
73, 299 (1977) [Sov. Phys. JETP 41, 960 (1977)].
[14] A. Schmidt and G. Schon, J. Low Temp. Phys. 20, 207
(1975).
[15] A. I. Larkin and Yu. N. Ovchinnikov, Phys. Rev. B 51,
5965 (1995).
[16] N. B. Kopnin and A. V. Lopatin, Phys. Rev. B 51, 15 291
(1995).
[17] M. Yu. Kupriyanov and V. F. Lukichev, Zh. Eksp. Teor.
Fiz. 94, 139 (1988) [Sov. Phys. JETP 67, 1163 (1988)].
[18] F. Zhou, P. Charlet, B. Pannetier, and B. Spivak, J. Low
Temp. Phys. 110, 841 (1998).