Macroscopic quantum tunneling in "small" Josephson junctions in magnetic field

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by  Yu. N. Ovchinnikov

Macroscopic Quantum Tunneling in “Small” Josephson Junctions in magnetic field
Yu.N.Ovchinnikov,1 A. Barone,2 and A.A.Varlamov3
1Max-Plank Institute for Physics of Complex Systems, Dresden,
D-01187 Germany and Landau Institute for Theoretical Physics,
RAS, Chernogolovka, Moscow District, 142432 Russia
7
2Dipartimento Scienze Fisiche, Universita’ di Napoli “Federico II”
0
and Coherentia-INFM, CNR, piazzale Tecchio, 80, 80125, Napoli, Italy
0
2 3Coherentia-INFM, CNR, Viale del Politecnico 1,
I-00133 Rome, Italy and Materials Science Division,
n
Argonne National Laboratory, 9700 S.Cass Avenue, Argonne Il 60439, USA
a
(Dated: February 6, 2008)
J

1 Westudythephenomenonofmacroscopicquantumtunneling(MQT)insmallJosephsonjunctions
3 (JJ)withanexternallyappliedmagneticfield. ThelatterresultsintheappearanceoftheFraunhofer
typemodulation of the current density along the barrier. The problem of MQT for a point-like JJ

] isreducedtothemotion ofthequantumparticle inthewashboard potential. Inthecaseofafinite
n
size JJ under consideration, this problem corresponds to a MQT in potential which itself, besides
o
thephase,dependsonspacevariables. Finally,thegeneralexpressionforthecrossovertemperature
-c T0 betweenthermally activatedandmacroscopicquantumtunnelingregimesandtheescapingtime
r τesc havebeen calculated.
p
u PACSnumbers: 74.50.+r
s
.
t
a The paramount importance of macroscopic quantum tunneling lies in the role of such a process in a variety of
m
physical phenomena, which span from the cosmological implications of the false ”vacuum” to recent studies on BEC
- systems. GreatattentiondeservesinthiscontexttheoccurrenceofMQTinJosephsonjunctions1 observedbothinlow
d
T 2 and high T 3 structures. Usually the phenomenon of MQT is considered in a “small” JJ, i.e. in the assumption
n c c
o that the characteristic size L of the junction is much smaller than all other related parameters of the problem (some
c exceptions can be found in experimental4 and theoretical papers5 ). Such parameters are the Josephson penetration
[ depth λJ = ~c2/8πej1ℓeff 1/2 and the characteristic length LH = ℓ2H/ℓeff, with ℓH = (~c/eHext)1/2 as the standard
1 quantum mechanicalmagnetic length. The externalmagnetic field H is supposedto be directed alongthe plane of
ext
(cid:0) (cid:1)
v junction. The critical current density j of the JJ will be defined explicitly below. The effective length ℓ depends
1 eff
4 on the relation between the thickness d (i=L,R) of the superconductive electrodes and the magnetic penetration
(i)
8
depth λ of the bulk superconductor materials, which the electrodes are made of. In the limiting cases one can
7 (i)
find6,7,8,9
1
0
λ +λ +d , λ ≪d
7 ℓ = (1) (2) ox (i) (i) . (1)
eff d +d +d , d ≪λ
0 (1) (2) ox (i) (i)
(cid:26)
/
t ThefinitevalueofthejunctionlengthLbecomesimportantevenforL<λ .Suchasituationtakesplacewhenthe
a J
m magnetic field is applied to the junction what results in appearance of the Fraunhofer diffraction pattern Ic vs Hext.
In our analysis we will suppose that L≪λ but will not impose any restrictions on the relation between L and L .
- J H
d The effect of magnetic field on the properties of JJ leads to three different manifestations. First of all magnetic
n applied filed suppresses the value of the order parameter ∆ and changes the frequency dependence of the single-
o particledensityofstatesρ(ω)(formally,thepresenceofmagneticfieldchangesthefrequencydependenceoftheGreen
c functions)10 . Then, its presence leads to the spacial modulation of the current flowing through the junction6,7,8,9,11.
:
v Finally,the magneticfieldaffects ontheprobabilityofthe electrontunneling throughthebarrier. Since the thickness
i of insulating barriers in the case of JJ is very small (about 10A˚) this effect we will neglect in our discussion.
X
Let us start our analysis from the suppression of the order parameter ∆ and changes of the frequency dependence
r
a oftheGreenfunctions bythe magneticfield. The Greenfunctions integratedoverthe energyvariable(α,β) forboth
electrodes can be found from the equation10:
D ∂ 2 ∂2
α∆−βω+ α −2ieA β−β α =αβΓ, (2)
2 ∂r ∂r2
( (cid:18) (cid:19) )
with the normalization condition α2(r,ω)+β2(r,ω) = 1. We choose the vector potential gauge in each electrode in
theformA=(Hz,0,0),wherethedirectionof z axisissupposedtobeperpendiculartothejunctionsurfaceandthe
magnetic field H is directed along the y-axis. The coordinate z in each electrode changes in the symmetric interval
z ∈[−d/2,d/2],while the coordinate along the barrier x∈[−L/2,L/2].In view of the smallness of the JJ length one
b
b
2
can see that the spacialdependence of ∆ and Green functions can be neglected. Averagingthe Eq.(2) over the space
variable one can determine the frequency dependence of the averagedGreen functions (α(ω),β(ω))
α ∆ −β ω =α β Γ +D e2H2 d2 /6 , (3)
(i) (i) (i) (i) (i) (i) (i) ext (i)
(cid:16) (cid:17)
where Γ =τ−1 is the inverse electron spin-flip scattering time.
(i) s(i)
The critical current density J (H) can be determined from the Ambegaokar-Baratofftype formula11:
1
π 1
J (H)= T β (ω)β (−ω), ω ≡ω =2πT n+ , (4)
1 eR S (1) (2) n 2
N ω (cid:18) (cid:19)
X
whereR isthejunctionresistanceinnormalstateandS isthejunctionarea. Thustheeffectofmagneticfieldresults
N
just in the renormalizationof the critical current density. The latter becomes essential when D e2H2 d2 ∼∆.
(i) ext (i)
Let us discuss of the second external magnetic field manifestation on the properties of a JJ, related to the space
modulationofcurrentdensityinit. Inthecaseofa“point-like”JJthevalueJ determinestheheightofthepotential
1
barriersand hence the escaping time for the MQT. In the case of a small JJ of finite size the currentdensity is space
modulatedbytheexternalmagneticfieldandtheescapingtimecalculationbecomesmorecomplicated. Indeed,while
forthe pointcontactitisreducedtothe motionofthe quantumparticleinthe washboardpotentialinthe caseunder
consideration the problem corresponds to the tunneling of a lengthy object in the potential which itself, besides the
phase, depends on space variables.
To calculatethe escapingtime τ weneedto write downthe functionalofeffective actionA[ϕ], whatcanbe done
esc
in this case basing on the results of12,13:
1 1/2T C ∂ϕ(r,τ) 2 ~ π~ 1/2T ϕ(r,τ)−ϕ(r,τ )
A[ϕ]= dτ d2r − Iϕ(r,τ)− dτ 2sin2 1
S Z−1/2T Z (2e2 (cid:18) ∂τ (cid:19) e 2RNe2 Z−1/2T 1(cid:20) (cid:18) 2 (cid:19)
× α (τ −τ )α (τ −τ )+cos(ϕ(r,τ)+ϕ(r,τ ))β (τ −τ )β (τ −τ )]
L 1 R 1 1 L 1 R 1
+ π~T2 1/2T dτ sin2 ϕ(r,τ)−2ϕ(r,τ1) + ~2c2S ∂ϕ(r,τ) − eℓeff [H ×n] 2 . (5)
Rshe2 Z−1/2T 1 sin(cid:16)2[πT (τ −τ1)](cid:17) 8πe2ℓeff (cid:18) ∂r ~c ext (cid:19) 
Here C is the junction capacity, I is the full current flowing through the junction, Rshis the shunt resistance.
The integrals are carried out over the imaginary time and the junction area. The first term corresponds to the
kinetic energy of the junction. The second and fourth terms describe the contribution of the potential energy to the
action. Let us stress that the third term, corresponding to the capacity renormalization appears only beyond the
quasi-classical approximation. The action accounting for all these terms was already found in12,13 in the case of
microscopic study of the Josephson current decay in point-like junction. The last term in Eq.( 5) accounts for the
finite size of the junction and the magnetic field contribution to the effective action. The motion in the classically
permitted region is described by the same functional Eq.( 5) in the assumption τ = τ what eliminates the third
1
term. The further variation procedure of Eq.( 5) over ϕ results in the getting of the quasi-classical equations of
motion6,7,8,9.
Wearelookingfortheescapetimeτ withtheaccuracyuptothepre-exponentialfactor. Theexponentialfactorin
esc
itis determined by the extremalvalue ofthe functionalEq.(5) foundatthe class ofsolutionsperiodicalin imaginary
time τ with the period 1/T.The function ϕ(x,τ) at which the extremum of functional Eq.( 5) is realized(the saddle
point of functional ) we will search in the form
x 1
ϕ(x,τ)=ϕ − +ϕ(x,τ)= Φ(x)+ϕ(x,τ), (6)
0
L 2
H
where ϕ0 =const, is the phase difference at the junctionein the absence of meagnetic field, while the function ϕ(x,τ)
satisfies the boundary condition9
e
∂ϕ ∂ϕ
= =0. (7)
∂x ∂x
(cid:12)x=−L/2 (cid:12)x=L/2
e(cid:12) e(cid:12)
(cid:12) (cid:12)
The experimentally realistic case corresponds(cid:12) to the curren(cid:12)t close to its critical value Ic. This is why we restrict
our consideration namely by the condition I → I . Let us note that in this case the third term in Eq.( 5) can be
c
3
∗
excluded from consideration by the simple renormalizationof the junction capacity C →C , where
π~ ∞ dω ∂α ∂α
∗ L R
C =C+ . (8)
2RN Z−∞ 2π (cid:18) ∂ω (cid:19)(cid:18) ∂ω (cid:19)
The substitution of Eq.( 6) to the Eq.( 5) with the further expansion of the cosine over ϕ(x,τ) in the fourth term
leads to the cancellation of the linear in ϕ second term. Since the function ϕ has to be periodical in imaginary time
τ with the period 1/T we present it in the form of the Fourier series. This makes the effective action ( 5) to become
e
the functional of the Fourier coefficients ben (let us note that bn(x)=b−n(x)e):
~J ∞
A[b ]= 1 d2r cos[Φ(x)]+ζ n2+Q |n| b2
n eT sh n
Z n=−∞
X (cid:8)(cid:2) (cid:0) (cid:1)(cid:3)
∞ 2
2 ∂b
− sin[Φ(x)] b b b +λ2 n , (9)
3 k k+n n J ∂x
k=−∞ ! (cid:18) (cid:19) )
X
where we have introduced the notations ζ = 2π2T2C∗/(e~J S) and the Q-factor Q−1 = 2πTR C∗/~.The further
1 sh sh
variation of the functional (9) over b results in the system of equations:
n
∂2b ∞ n−1
−λ2J ∂x2n + cos[Φ(x)]+ζ n2+Qsh|n| bn−sin[Φ(x)] (2−δn,0)b0bn+2 bkbn+k+ bkbn−k =0 (10)
" #
k=1 k=1
(cid:2) (cid:0) (cid:1)(cid:3) X X
Letusrecallthatwewilllookforthesolutionofthissystemwiththeboundaryconditions(7). Theanalogoussystem
of equations for the point junction was solved for the cases of low and high shunt resistance12 and also close to the
temperatureT correspondingtothecrossoverbetweenthe thermallyactivating(Arreniuslaw)andquantumregimes
0
in tunneling through the barrier.
In order to calculate the escape time we need to know the value of the effective action ( 5) at the saddle point, i.e.
at the optimal trajectories b (x), being solutions of the system ( 10). The related expression can be considerably
n
simplified, subtracting from the functional Eq. ( 9) the Eq. ( 10) multiplied by corresponding b (x):
n
~J ∞ ∞ ∞ ∞ n−1
A[bn]= 3eT1 d2rsin[Φ(x)] b30+6b0 b2n+4 bn bkbn+k+2 bn bkbn−k . (11)
" #
Z n=1 n=1 k=1 n=1 k=1
X X X X X
e
We consider the junction of a small enough size: L≪λ . Thus the system Eq. (10) acquires the large parameter
J
λ2/L2 ≫ 1. Consequently the functions b (x) weakly vary with the change of its argument and can be represented
J n
in the form:
b =hb i+b , |b |≪|hb i|, (12)
n n n n n
where h...i indicates the average over the coordinate ex along tehe junction. The corresponding equation for bn takes
the form
e
∂2b 2x
λ2 n =hb isin(2ϕ )sin . (13)
J ∂x2 n 0 L
(cid:18) H(cid:19)
e
Its solution satisfying the boundary conditions ( 7) is
L2 2x L 2x
b = H hb isin(2ϕ ) cos −sin . (14)
n 4λ2 n 0 L 2L L
J (cid:20)(cid:18) H(cid:19) (cid:18) H(cid:19) (cid:18) H(cid:19)(cid:21)
e
What concerns the equations for the average values hb i they can be obtained directly from the Eq.( 10) by the
n
method of subsequent approximations.
Now we cancalculate the escape time τ near the crossovertemperature T , taking into accountthe effects of the
esc 0
current density coordinate dependence, due to the applied magnetic field. All coefficients hb i with n ≥ 1 are small
n
and quickly decrease as n increases. It is why at temperatures close to T in Eq. ( 11) one can restrict consideration
0
by the contribution of b , b and b ∼b2 .
0 1 2 1
(cid:0) (cid:1)
4
The crossover temperature T is characterized by the fact that below it the quantum tunneling through barrier
0
becomes more probable than the thermally activated escape process. Formally this temperature is the bifurcation
pointofthe system(10)whenthefirstnon-zerosolutionsforn≥1appear. Aboveit existstheonly,trivial,solution
is b = const, what results in the action ( 5) A[ϕ] = ∆U/T corresponding to the Arrenius law for the escape time.
0
BelowT the non-zerosolutionsb appear,whatleadsto the appearanceofthe imaginarytime dependence ofϕ, the
0 n
action( 5) changesits form and,consequently,teheetemperature dependence ofthe escape time τesc ∼exp[−A(ϕ0,L)]
changes qualitatively. After some cumbersome but straightforward calculations one can find the quadratic equation
e
determining the crossover temperature T :
0
2π2C∗T2 πT L L
0 + 0 =I sin(2ϕ ) H sin Υ(ϕ ,L), (15)
~e R e c 0 L L 0
sh (cid:18) (cid:19) (cid:18) H(cid:19)
where I =J (H)S is the value of junction’s critical current, while
c 1
2 1+ 1cos 2L
LH 3 L L 2 LH
Υ(ϕ ,L)=cot(2ϕ )+sin(2ϕ ) cos − . (16)
0 0 0 (cid:18)2λJ(cid:19) 2 (cid:18)LH(cid:19) (cid:18)LH(cid:19) sin L(cid:16) (cid:17)
LH
 (cid:16) (cid:17) 
In the case of a point junction the right-hand-side of Eq. ( 15) reduces to I2−I2.
c
InvicinityofT theessentialcontributiontoEq. (11) giveonly thetermswithb ,b andb .Theiraveragevalues
0 0 1 2
p
can be found from Eq.( 12)-( 14). We present here only
Υ(ϕ ,L)ζsin(2ϕ ) LH sin L −ζ2(Q +1)
hb i2 = 0 0 L LH sh . (17)
1 2sin2(2ϕ ) LH 2(cid:0)sin2(cid:1) L (cid:16) (cid:17)2 − 1
0 L LH Qsh+1 Qsh+3
(cid:0) (cid:1) (cid:16) (cid:17)h i
what finally permits to find
~J S L L 6hb i4 L L 2 1
A[ϕ]= 1 sin(2ϕ ) H sin Υ3(ϕ ,L)− 1 sin(2ϕ ) H sin − .
0 0 0
3eT L L ζ L L Q +1 Q +3
(cid:18) (cid:19) (cid:18) H(cid:19)" (cid:18) (cid:19) (cid:18) H(cid:19)(cid:18) sh sh (cid:19)#
e
One can see that the effect of magnetic field in a finite size junction, even in the case L ≪ λ , does not reduce
J
simply to the renormalization of the value of I . Indeed, both characteristic parameters of the problem, T and the
c 0
escape time τ ,turn outto be the functions oftwoparameters: I/I andL/L .Letus stress thatwhenthe current
esc c H
I isveryclosetoitscriticalvalueI theapproximation(12)forthefunctionsb (x)becomesincorrect,andthisregion
c n
has to be studied separately. These results are of great interest for the whole scenario of quantum coherence, as for
instance in the context of MQT in Josephson structures based on d-wave superconductors3,4
Yu.N.Ovchinnikov acknowledgesthe financial support of the grantRFBR-06-01-90507-BNTS-aand Russian Foun-
dation of Basic Research.
1 A.J.Leggett, Suppl. Prog. Theor. Phys. 69, 80 (1980) and references reported therein.
2 For an excellent review see: J.Clarke, A.N.Cleland, M.H.Devoret, D.Esteve, and J.H.Martinis, Science 29, 992 (1988).
3 T.Bauch,F.Lombardi,F.Tafuri,A.Barone,G.Rotoli,P.DelsingandT.Claeson, Phys.Rev. Lett. 94,87003(2005);T.Bauch,
T.Lindstrom, F.Tafuri, G.Rotoli, P.Delsing, T.Claeson and F.Lombardi, Science 311, 57 (2006).
4 A.L.Fetter, M.J.Stephen, Phys. Rev. 168, 475 (1968).
5 M.G. Castellano, G. Torrioli, C. Cosmelli, A. Costantini, F. Chiarello, P. Carelli, G. Rotoli, M. Cirillo and R. L. Kautz
Phys. Rev.B 54,15417(1996);A.Walraff,A.Lukashenko,J.Lisenfeld,A.Kemp,M.V.Fistul,Y.KovalandA.V.Ustinov,
Nature 425, 155 ( 2003); B.Ivlev arXiv: quant-ph/0511052v (7 Nov 2005);
6 R.E.Eck, D.J.Scalapino, B.N.Taylor, Phys. Rev. Lett. 13, 15 (1964).
7 I.O.Kulik, Soviet Physics - JETP Letters 2, 84 (1965).
8 Yu.M.Ivanchenko,A.V.Svidzinskii and V.A.Slyusarev, Soviet Physics - JETP 24, 131 (1967).
9 A.I.Larkin, Yu.N.Ovchinnikov,Soviet Physics - JETP 26, 1219 (1968).
10 A.I.Larkin, Yu.N.Ovchinnikov,Soviet Physics - JETP 34, 1144 (1972).
11 A.Barone, G.Paterno, Physics and Applications of the Josephson Effect, John Willey, NY (1982).
12 A.I.Larkin, Yu.N.Ovchinnikov,Soviet Physics - JETP 57, 876 (1983).
13 A.I.Larkin, Yu.N.Ovchinnikov,Phys. Rev. B 28, 6281 (1983).