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Josephson equations for an electron in the presence of a barrier
J. Riess, Y. Grandati
To cite this version:
J. Riess, Y. Grandati. Josephson equations for an electron in the presence of a barrier. Journal de
Physique I, EDP Sciences, 1991, 1 (2), pp.261-268. �10.1051/jp1:1991129�. �jpa-00246319�
J.
Phys.
I1(1991)
261-268 F#VRIER 1991, PAGE 261Classification
Physics
Abslracts03.65G 73.40G 74.50
Josephson equations for
anelectron in the presence of
abarrier
J. Riess and Y. Grandafi
(*)
Centre de Recherches sun les Trds Basses
Tempbratures,
C-N-R-S-, B-P-166X,
38042 Grenoble Cedex, France(Received14 September
1990,accepted
29Oclober1990)
Rksunlk. A
partir
del'bquation
deSchroedinger
onexprime
lesbquations
deJosephson
pourun blectron en
prksence
d'une barridre depotentiel.
Pour cela on btudie, en fonction de la hauteur de la barridre, l'bvolutiontemporelle
de la phase de la fonction d'onde. Cette bvolution(qui
estdbterminbe par le mouvement des vortex associbs au
gradient
de laphase)
conduit, dans le cas de barridres trds hautes, I une oscillationmonochromatique
du courant(effet Josephson altematio.
Abstract. The
Josephson equations
for asingle
electron in the presence of apotential
barrierare derived from the
Schroedinger equation.
To this end the time evolution of thephase
of the wavefunction as a function of the barrierheight
isinvestigated.
The evolution of thephase
function(which
is determinedby
the motion ofphase gradient
vortices) leads to a monochromaticcurrent oscillation
(ac-Josephson effect)
in the limit of veryhigh
barriers.Inboduction.
The
dynamical
behaviour ofparticles
in quantum mechanics may bequite
different from classical behaviour. This is related to the fact that in quantum mechanics theposition
and thevelocity
cannot be chosenindependent
of each other. The former is related to the modulusand the latter to the
gradient
of thephase
of acomplex space-dependent
function#r(x)
=
#r(x) bxp [i~ (x)].
Thephase
and the modulus are related in a characteristic way,and each state is
represented by
a characteristicphase
structure[Ii.
These structures arecomposed
of one or severalphase gradient
vortices aroundphase singularities
situated on(n
2)-dimensional
nodal(hyper-)lines
of the modulus of the wave function#r(x),
wherex denotes the
position
in the n-dimensionalconfiguration
space.In the presence of a
homogeneous
electric field the wave functionsusually change
withtime.
Recently general
laws have been established for the adiabaticchange
of thecharacteristic
phase
structures of the wave functions of alarge
class of systems[I].
These lawsare at the
origin
of non-classicaldynamical behaviour,
which inparticular
occurs in systems with reducedspatial symmetry [1-3].
This breakdown of classical behaviour in the presence of
phase singularities
is consistentwith a result of Bohm and
Hiley [4],
who introduced a quantumpotential
(*)
Present address-. Institut Laue-Langevin, B-P- 156X, F-38012 Grenoble Cedex, France.262 JOURNAL DE PHYSIQUE I M 2
Q
~
(fi~/2m)(grad)~ R(x)/R(x), R(x)
=
[#r(x) [,
andshowed,
that non-classical behaviour results from theSchroedinger equation,
ifQ
is notnegligeable. According
to this criterium classical behaviour cannot beexpected
in the presence ofphase singularities,
sincethey
arelocated at nodes of the wave
function,
whereQ diverges.
Typical
non-classicaldynamical
effects areJosephson
type oscillations.They
appear in smalljunctions
and insuperconducting
and normal micro-circuits and have been thesubject
of intense research in recent years
[5].
In this article we will discuss theac-Josephson
effect ofa
single
electron in thelight
of thegeneral
lawsill
mentioned above. We will see that this effect is thelimiting
case of a verygeneral behaviour,
which is common to alarge
class ofsystems, which can be described
by Schroedinger type equations.
We restrict our considerations to motion in one
spatial
direction. To bespecific
we considera
single particle
withcharge
q in the presence of asingle potential
barrier(Fig. I)
andsubject
to a
homogeneous
electric fieldE~.
This field can beexpressed by
atime-dependent
vectorpotential
Ax(x)
=
~ (t)/L
,
(i)
~§
(t)
=
cE~
Lt + const.(2)
Vixi
o xi x~ x
Fig.
I. One-dimensional system oflength
L with apotential
barrier ofheight
V.We assume the
boundary
conditionS'(xi
= $r
(x
+Li (3)
Condition
(3)
is fulfilled on a realloop
of circumferenceL,
but it is often also the correctphysical
condition on an opensystem.
This is e-g- the case for asuperconducting
orderparameter (which obeys
theGinzburg-Landau equation,
which is ofSchroedinger type),
which finds bulk conditions on both sides of an
insulating junction (corresponding
to ahigh
barrier in ourone-particle problem),
or for asingle-particle
solution withplane
wavebehaviour at a sufficient distance away from the barrier.
Together
with(3)
thephase
difference between x and x +L, A~
= ~
(xi
~(x
+L) (modulo
2vi,
determines the self-adjoint representation
of the Hamiltonian[6].
On a realloop,
Which is embedded in the real two- or three-dimensionalphysical
space, thisphase
difference is zero(modulo
2vi [2].
Thiscorresponds
to wave functions which aresingle-valued
on theloop.
On an open line all othervalues of
A~
would also beacceptable
from aphysical point
of view. In ourfollowing
discussion these values would appear in the form of different choices of the
origin
of~§ at t
=
0
(see
e-g- Refs.[1, 2].
Since thisorigin
is irrelevant for thefollowing,
we set itequal
to zero
together
with the constant in(2).
We consider the Hamiltonian
H
=
(1/2 m)j (fill)(d/dx) q~ (i)/(cL)j
+v(x)
,
(4)
with
#r(0)=#r(0+L),
whereV(x)=
V for xi «x«x~ and zero otherwise(x,
xi,M 2 JOSEPHSON
EQUATIONS
FOR AN ELECTRON 263J~ e
[0, L]).
Weinvestigate
thephase
structure of the adiabatic solutions of the time-dependent Schroedinger equation
forincreasing
barrierheights
V.(Adiabatic
solutions areeigenfunctions
ofH(t)
where t is considered as a parameter. This is thelimiting
case forsufficiently
smallE~.)
Inparticular
we are interested in theground
state. The time evolution of an adiabatic state is visiblethrough
its characteristicphase
structure, which for theground
state of our system is
particularly simple.
The time evolution of thephase
function illustrates thegeneral
laws discussed in referenceill,
which lead to an oddperiodic
current withfrequency [qE~L/h[
if the symmetry issufficiently
low. In the system considered in this article an increase in the barrierheight
V will let theperiodic
current tend towards anexactly
monochromatic current. We will obtain this limit
analytically. Further,
the intermediate situations will be illustratednumerically.
The
following general
result is trueill
: ifV(x)
hasonly
the trivialperiodicity V(x
+L)
=
V(x),
as is the case in oursystem,
theeigenfunctions
ofH, S'(x;~§)
=
j$r(x; ~§)j expii~(x; ~§)1, (5)
show the
following
characteristic#-dependence
:S'(X
~b +hciq )
= $r
(x
~§)
expi-
iF(x)ifii
,
(6)
where
F(x)
=
hx/L (7)
This means that the
phase winding
numberw(~
=
(i/2 ar) j~ (o/ox)
~(x ~ ) (8)
obeys
thegeneral
relationw(~
+hc/q )
=
w(~
+hc/q (9)
While the wave function itself is continuous as a function of ~§, the
winding
number Wchanges discontinuously
injumps
ofinteger
values[1, 2].
This process is associated with the appearance ofphase singularities,
situated at cores ofphase gradient
vortices. In thegeneral
case of a three-dimensionalsystem (threaded by
a flux ~§) thesesingularities
arelocated on nodal lines of
#r.
If theconfiguration
space istwo-dimensional, they
reduce to nodalpoints.
When ~§increases,
these lines~points) periodically
move across thephysical
domain into the hole threaded
by
the flux ~§. In our case, where the domainmathematically corresponds
to a one-dimensionalloop,
this process can be visualizedby extrapolating
thephase
~(x)
from theloop
into theplane
where theloop
can be imbedded. Thisextrapolation
is notunique,
but thetopology
of thephase
structure, I-e- theposition
of the vortex cores(the phase singularities)
withrespect
to theloop
isuniquely represented [2].
The current of an elecbon.
The current
j(~§ ) along
theloop
reflects thisphase
structure, sincej
has thegeneral
fornJJ (~ )
=
i-
q~/(mc )i
#r(x
;~ )
2iAx(x ~ ) (tic/q) (o/ox)
~(x
;~ ) (io)
The
eigenfunctions
of the Hamiltonian(4)
can be obtained in the usual wayby
asuperposition
of twoplane
waves in each part of theloop
with constantV(x)
and then2M JOURNAL DE
PHYSIQUE
I M 2connecting
thepiecewise
solutions with the correctbranching
conditions. The solution has thegeneral
fornJ[2, 7]
#r~(s~ t)
= exp
[- ir~(s~)] (#rj
sinh(0~
s~0~/L~)
+ #r~exp[iy~]
sinh(s~ 0~/L~)) /sinh 0~
(11)
#ib(Sb
t)
" ~XP
[~ ll~b(Sb)I
#i2 Slllfib
Sb°b/l~b)
+ #ii ~XP II
Ybi
Slll(Sb °b/l~b)) /Slll fib
,
where s~ = x xi, s~ = x x~,
L~
= x~ xi,L~
= LL~.
Here#r~(#~) represents
thesolution in
(outside)
the barrier.r~,~(x)
=j~ (q/tic) A~dx (12)
ya "
ra(X2)
, yb "
rbl'i
+Lb) (13)
0
~,~ =
L~,
~(2
m E V~,~)"~/fi (14)
with
V~
=V, V~
= 0.Here E is the
corresponding
energyeigenvalue
at thegiven parameter
value ~§(t) (only
thecase E ~ V is
considered),
and #r1, #r~ are the values of the solution at xi, x~respectively (see
e-g-
Eq. (15)
of Ref.[7]).
With(11)
the current(10)
can beexpressed
as :Ja(t)
"
#i1( #i2' (°a/l~a
Slll~l°a)(~ ~~l'~)
Slll[~ (Xl
~(~h) Yai
~~~~
Jb(t)
=
$i1( #i2( (0b/l~b
SInll0b)(~ Q~/m)
Sin[~ (J§)
~ (X2 +Lb) Ybi
where
j~, j~
denote theexpression
for the current on the intervalsL~, L~ respectively.
Here#r and
j
arefully
determinedby E(~§ ), #r1(~§),
#r~(~§),
which are obtainedby solving
thecorresponding
linearsystem [2]
for the unknown values#ri
and #r~ at agiven
~§= ~§
(t).
Let us calculate the adiabatic current as a function of t for different barrier
heights
V.
Equation (15) suggests,
that a monochromatic behaviour is obtained if ~(xi
~(J~) changes linearly
with time(y~
isalready
linear in taccording
to(1), (2), (12)
and(13)).
We will nowshow,
that such a monochromatic behaviour is indeed obtainedmathematically
as alimiting
case.From the
Schroedinger equation
itfollows,
that the current isdivergence free,
hence theexpression (15)
for the current in the barrier interval oflength L~
= x~- xiequils
theexpression
for the current on thecomplementary
interval oflength L~ (which
due to theperiodic boundary
conditions is also boundedby
x, andJ~).
From theequality
of these two currentexpressions
one obtainsA
w ~
(xi )
~(x~)
= arctan
(-
X sin y~ + sin
y~)/(Xcos
y~ + cosy~), (16)
where X
=
0~
L~
sinh0~/( 0~ L~
sin 0~) (17)
If V is
sufficiently high, 0E(~§ )/0~§
can be setequal
to zero(for
theground
state asimple
estimation
yields
the condition Vmh~/(2~'~mL~L)
; similar limits could begiven
for the excitedstates).
One obtains then(expressing
~§ in units oftic/q)
i
X~ L~ L~
+X(L~ L~)
cos ~§°~~°'~
L I +
X~
+ 2 Xcos ~§~~~~
0~A/0~§
~ =~~ ~j
~~~~
~
(19) (1+
X+
2 Xcos ~§
)
M 2 JOSEPHSON EQUATIONS FOR AN ELECTRON 265
Further,
from(17)
we obtainsinh
[L~(2
m E V) "~/fi]
[X[
m~~~
(20) L~(2
m E V) /fi
Therefore,
for veryhigh
V theexpression ix
tends towardsinfinity,
hence0~A/0~§~
tendstowards zero and
0A/0~§
tends towardsL~/L.
This means that in this case we haveA=-L~~§/L, (21)
°~
A~ya"~ (La+Lb)fb/L"~fb. (22)
Hence
j
sin ~§(23)
according
to the firstequation
of(15).
Equation (23)
leads to thepredicted monochromaticity,
since ~§obeys (2)
and theprefactor
in front of sin ~§ can be considered as
#-independent (see appendix).
One obtains(expressing
~§
again
inordinary units)
j (t)
sin w t,
w = q A
V/fi (24)
Here AV
=
E~L
is thepotential drop along
L(or
the electromotive force around theloop)
and w is the
Josephson frequency
for asingle particle
withcharge
q.Numerical results.
The
preceeding analytical
results are illustratedby
thefollowing
numerical calculations.Figure
2 shows thephase ~(x)
of theground
state wave function for different values of~§ and three different barrier
heights
V. One sees, that~(x)
becomes linear outside the barrier forhigh
barrierheight.
It is linear in the wholespatial
domain for ~§=
2 w times an
integer,
but itjumps
in the middle of the barrier at~§ = w times an odd
integer.
Thesejumps correspond
to thefact,
that if the system is considered as aloop,
at these#-values
the centre of aphase gradient
vortex(I.e.
aphase singularity)
with individualwinding
number one issituated on the
perimeter
of theloop, exactly
in the middle of the barrier. Hence thephase
~
(x) jumps by
an amount of w if thispoint
is crossed in x-direction.(For
the motion of thevortex pattem in the
extrapolation plane
of theloop
see Ref.[2].)
Figure
3 shows thephase
difference A across the barrier as a function of ~§= tc AV for
different barrier
heights
V. It illustrates thelinearity
of A with t forhigh
barriers (« weak links»).
Discussion.
It has been shown
previously [1, 2],
that for systems describedby Schroedinger
typeequations (with
theboundary
conditions mentionedabove),
which describecharged particles
in the presence of an electricfield,
theperodic
passage ofquantized
vortices(whose
centres arephase singularities,
which also are calledphase slip centres)
is a consequence of the mathematical structure of theequation.
If thespatial
translational symmetry isonly
the trivial one,I-e-, V(x+L)= V(x),
thefrequency
of passage of the vortices isequal
to qAV/h [Ii.
This isjust
the timeperiodicity
of the adiabatic current in thesystem.
In the
present
paper alimiting
case of thisgeneral
behaviour has beeninvestigated.
We considered an electron in theground
state drivenby
an electric field across asingle potential
barrier. We
showed,
that forsufficiently high
barriers theperiodic
current tends to a266 JOURNAL DE PHYSIQUE I M 2
2~
~~~~ ~~~ ~'~
0~,__,,, j fi,
--~
-~/2
~
Xi X2
~~""
~
~ 5fl/2 ( =3~-0.3
( ~
~
52
/" ,"
0
~
~
'
/
_qj2
~
( =~+0.3~
~
~ j
~
, /
,
t °
(= 3fl -2~
~r3fl/2 / ~z 2~
~
(
= 3~$~~~
a)
-&~b)
Fig.
2. Numerical values of thephase ~(x)
of theground
state wavefunction as a function of j=
cE~
Ll) for different barrierheights
V. Full lines : V= 3.75 eV
pointed
lines : V= 0.375 eV dashed fines V
= 0.0375 eV. L
=
100
h,
L~= x~ xj = 10
h.
A
'
"
"
",
Fig.
3. Phase difference A across the barrier as a function of j for the same values ofV as in
figure
2.monochromatic current with the
frequency
v=
w/(2
w)
= q
AV/h (Eq. (24)). Equations (15)
and(22), (2) together
constitute the same type ofequations
as theJosephson equations
inthe
superconducting
case(apart
from a constantprefactor depending
on theparameters
of thesuperconductor
andapart
from the factor two in thecharge),
seeequations (42)
and(38), (28)
of reference
[8].
M 2 JOSEPHSON EQUATIONS FOR AN ELECTRON 267
In the
present
case of asingle
electron the differentialequation (the Schroedinger equation)
is known in the entirespatial
range of the system,I-e-, including
the barrierregion
and its
edges.
The behaviour of theparticle
can therefore be derived from thisequation
without any additionalassumptions
on the structure of the wave function outside the barrier(once
theboundary
conditions of the Hamiltonian aregiven).
Our results show inparticular,
that the crucial linear relation of the
phase
difference A with respect to time and to thepotential
difference AV(Eq. (21),
where~§ is
expressed by (2))
holdsonly
in the limit of a veryhigh
barrier(corresponding
to a weak link in thesuperconducting
andsuperfluid cases).
In the derivation of this limit the nature of the barrier appears
explicitely through (18).
On the otherhand,
in thesuperconducting
andsuperfluid
cases thecorresponding linearity (Eqs. (I)
and
(38)
of Ref.[8]
andEq. (19)
of Ref.[9],
where the chemicalpotential
enters instead of the electrostaticpotential)
is derivedindependently
of the nature of thebarrier, by
a separateargument for the
phase
of the order parameter in the bulk outside the weak link. This is because ingeneral
nomacroscopic
waveequation
exists inside the weaklink,
whose width is of the order of the coherencelength
or smaller.(An exception
may be the case of thickjunctions
indirty superconductors
close to the transitiontemperature.
Here similar methodsas in the present
Schroedinger
case could beapplied,
with the orderparameter
fittedtogether
with
corresponding boundary
conditionsii Ii.)
Our
analysis shows,
that thesingle-particle ac-Josephson
effect is aquantum
effect due toquantized
vortex motion. This isanalogous
to theac-Josephson
effect insuperfluids [9,10]
and
superconductors [12].
Weemphasize,
that theperiodic
passage(with
timeperiod
r =
hi [q
AVi )
of thephase gradient
vortex centres~phase singularities, phase slip centres) perpendicular
to the direction of thepotential drop,
which leads to thephase jumps
across the middle of the barrier with timeperiod
r, is crucial for thefrequency equation (24) (which corresponds
toEq. (42)
of Ref.[8]
in thesuperconducting case).
This fact is not obvious from the usual[8] macroscopic
derivation of theJosephson equations.
It means that(21) (and (38)
of Ref.[8])
isactually only
true within disconnected time intervalsslightly
shorter than r, sinceperiodically (with period r)
thephase ~(x)
on one side of the barrierglobally
increases
by
2w, see
figures
2 and 3. But thisjump
occursjust
atvalues,
where theargument
of sin (~§)
in(23)
is itselfequal
to an(odd)integer multiple
of w. Hence thisjump
has nomathematical consequences for the
equation (23)
of the current and may pass unnoticed.The derivation in the
present
paper has been amissing
link in theanalogy
betweenmacroscopic systems
describedby quantum
wave functions ofGinzburg-Landau type
and the behaviour of asingle particle
inSchroedinger quantum
mechanics. Here the influence of the barrier on thephase
function and its consequences forac-Josephson
behaviour could beanalyzed
in detail. Wehope
that ouranalysis
will contribute to theunderstanding
of this effect.An
experimental
verification of thepredicted
behaviour of the electron current appears tobe rather difficult.
Presently
one canhope
to measure a current ofroughly
2nA in amesoscopic ring (without
abarrier)
of 2 ~Lm in diameter(by detecting
the createdmagnetic
flux with asquid).
This currentcorresponds
to the sum of about thousandsingle-electron
currents, and its
amplitude
turns out to be of the order of the current of asingle
electron at the Fermi level in such aring. However,
in the presence of a barrier theamplitudes
of the individual and hence of the total current are reduced. Therefore the monochromatic limit ofthe current seems not
yet
to be detectable withpresent experimental
methods.Acknowledgement.
We should like to thank A. Benoit for information on the present
experimental possibilities.
JOURNAL DE PHYSiQUE I T I, M 2, F#VRIER 199i
268 JOURNAL DE
PHYSIQUE
I M 2Appendix.
We
study
theprefactor
of sin~
in(23),
whichdepends
on the wavefunction #r(x) given by (11).
Here[#rj[
=[#r~[,
which isarbitrarily
setequal
to one. TMs determines the£
normalization factor
N~
=
o #r(x)
[~ dx of the normalized function#r(x)/N.
After somealgebra
we find from(11)
N~=A+Bcos~§
where
A
=
L~[1/(0~
th0~) 1/sh~ 0~]
+1/(a sh~ 0~)[0~/th 0~ Ii
++
L~[I/sin~ 0~ 1/(0~
tg0~)]
+1/(a sin~ 0~)[1 0~/tg 0~]
B
=
[L~ 0~/(aL~
sh0~
sin0~)][0~/th
0~Ii
+[L~ 0~/(aL~ 0~
sh0~
sin0~)] II 0~/tg 0~]
a =
I/(L~ 0~th 0~)
+1/(L~ 0~tg 0~).
If V is very
high, 0E/0#
tends to zero, hence0~
and0~
are constant withrespect
to~§. In addition
0~
tends towardsinfinity (at
least in the caseL~
« L consideredhere),
and onecan see that B is
negligeable compared
to A for all values of0~. Therefore,
after normalization of thewavefunction,
theproduct [#ri11
#r~[ mI/N~ (which
represents therelevant
prefactor
in(23))
is constant with respect to#.
References
[Ii
RIESS J.,Phys.
Rev. B 38(1988)
3133.[2] RIESS J., J.
Phys.
A 20 (1987) 5179.[3] RIESS J.,
Phys.
Rev. B 39(1989)
4978.[4] BOHM D, and HILEY B.,
Phys.
Rev. Left. 55(1985)
2511.[5] For a review see e.g. LIKHAREV K. K.,
Dynamics
of Josephson Junctions and Circuits(Gordon
and Breach Science Publishers, New York) 1986 ;WASHBURN S. and WEBB R. A., Adv. Phys. 35 (1986) 375.
[6] see e-g- SCHULMAN L. S., J. Math.
Phys.
12 (1971) 304.[7] RIESS J., J. Phys. tell. France 43
(1982)
L-277.[8] JOSEPHSON B. D., in
Superconductivity
Vol. I, Ed. R. D. Parks(1969)
p. 423.[9] ANDERSON P. W., Rev. Mod. Phys. 38
(1966)
298.[10] For a review of the experimental situation see e-g- VAROQUAUX E., Phys. World1
(1988)
32.[I
I] DE GENNES, P. G., Rev. Mod.Phys.
36(1964)
225.[12] ANDERSON P. W. and DAYEM A. H.,