Voltage fluctuations in Josephson tunnel junctions

Physica 108B (1981) 1293-1294 North-Holland Publishing Company

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VOLTAGE FLUCTUATIONS IN JOSEPHSON TUNNEL JUNCTIONS

A.B.Zorin

Department of Physics, Moscow State University Moscow I17234, U.S.S.R. Starting from microscopic theory of the Josephson tunnel junctions, the general expession forthe power spectrum of small voltage fluctuations SV(~) is obtained. According to this expression, SV(~) could be considered as a result of parametric conversion of "original fluctuations" at composite frequencieS~n=~+n~ v (~v=2eV/h is the Josephson frequency) to the observation frequency,. The low-frequency spectral density Sv(o) of voltage fluctuations is numerically calculated as a function of dc voltage V for the case of relatively small junction capacitance ( ~ l ) . This dependence is compared with the results obtained for generalized RSJ model and for normal tunnel junction.

Frequency spectrum of fluctuations in Josephson tunnel junctions has been calculated earlier for the case of relatively large capacitance of the junction (see, e.g., [I]), = 2elcR2C/h~l,

(1)

On the other hand, tunnel junctions with small capacitance ( ~ l), which are preferred for a number of app~cations, have been fabricated recently [2]. The objective of this paper is to present the result of an analysis of the fluctuations for the case of finite junction capacitance ( ~ I ) within the framework of the microscopic theory 13,4]. In contrast with the case (1), in our case the capacitance does not short out the ac Josephson current perfectly, and ac components of the voltage across the junction cannot be neglected, as it has been done earlier [l]. In order to take these ac components into account, we have utilized the method similar to that developed by Tucker for quasiparticle component of the tunnel current [5]. At the first stage of calculation, we have solved the auxiliary problem of the Josephson junction biased with some determinate (noiseless) voltage V(t). For this situation, the spectral density of small fluctuations of the current I through the junction can be expressed in terms of zeroth element of correlation matrix SI(~) = Soo,

(2)

where the arbitrary matrix element Snma' = < [~(-~m ) 'I ~m' ) ] +> is the expectation value of anticom~nutator for appropriate frequency components of the current fluctuation operator I = I -(I>. The frequency ~m = ~ + m @ v is the linear combination of the observation frequency ~ and m-th harmonic of the Josephson oscillation f r e q u e n c Y ~ v = 2eV/h. A direct calculation using the tunneling-Hamiltonian formalism and Kubo relationship yields Strum,= e--E[(W ~ +W -n-l+m ~ W -n-l+~ )Imlq(~n+~ 2~ n n-m W n-m'

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, imlp(~n+ ~ +(Wn_mW_n_l+nf+W_n_l+mWn_ff) ~coth[h~On+(Ov/2)/2kBT] ,

)1 (3)

and according to Eq,(2) one obtains

n +2Re (WnW- 1- n ) I m l p ( t O n ~ - ) ] xcoth[~(~n+%/2)/2kBT]

.

(4)

The f u n c t i o n s Iqlp(tO) a r e w e l l - k n o w n c o m p l e x amplitudes of quaslparticle and p a i r c o m p o n e n t s o f tunnel current,respectively [3,4], and Fourier coefficients Wn a r e d e t e r m i n e d by t h e d o s e p h s o n phase difference dynamics~(t), expEi~(t)/2]

= ~, W n e x p [ i ( n + l / 2 ) f ~ v t ] ~ n w h e r e V ( t ) h a s b e e n a s s u m e d t o be a 2 ~ / ~ v - p e r i o dic function of time. Equations (3)-(4) is the generalization o f t h e r e s u l t by T u c k e r [51 f o r the case of the presence of pair component of the current. In the particular c a s e o f dc a p p l i ed v o l t a g e ( V ( t ) = V = c o n s t ) t h e r e s u l t (4) c o i n sides with the earlier one [l]. If, moreover, V=O, and t h u s n o n e q u i l i b r i u m p r o c e s s o f J o s e p h son oscillations is absent, Eq.(4) reduces to the fluctuation-dissipation theorem [6]. At t h e s e c o n d s t a g e o f t h e c a l c u l a t i o n the real situation h a s b e e n c o n s i d e r e d when t h e J o s e p h s o n junction is attached to external circuit of high impedance. In the present case it is the current through the junction which has the determinate value, while the voltage can fluctuate. A simple linear analysis yields the fo~lowing connection of these ~otage fluctuations V and current fluetuations I of the above auxiliary problem: ~(~)

= ~m Zom(~)~(~m)'

(5)

IlZmm,l = IIYmm#i/OmC~mm,ll-1 , where llYmm,ll is admittance matrix of the junction

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t ....

ti

Sv(O)

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I/x/ ~,

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with zero capacitance [51. Due to Eq.(5), power spectrum of voltage fluctuations SV(~ ) can be also expressed in terms of parametric conversion as

SV(~) : Z

Z~mZo~Smm ,

with Smm, determined by Eq.(3). culate the value of SV(~) , it first to find coefficients W n and then the coefficients Zom

Vc"

VA .

.

Y

0

f

T .0

i

o

~o)

.~

I

(6) In order to calis neccessary at (see, e.g.,J7,8]) [9,10].

As an illustration of our general result, the low-frequency spectral density Sv(O) ! has been calculated for the zero-capacitance junction (~=0). Figure 1 shows the plot of Sv(O) versus V for several values of reduced temperature T/T c (solid lines). For comparison, the result of calculation with simplified expressions for lq, P (~) I = i~/eR, I = I , (7) q p c which corresponds to the RSJ model, is plotted by the dashed lines. For every plot, the voltage values Vc=IcR , V~=2A(T)/e, and VT=T/e , are also shown. At temperatures close to critical (Fig. la), SV(O ) is close to that of generalized RSJ model except small singularities in the vicinity of odd subharmonics of the gap voltage VA. At lower tempe-

I eV/2AIT)

2

Figure I : The low-frequency spectral density Sv(O) of voltage fluctuations vs average voltage across the junction ~ for several values of reduced temperature T/T c. The spectral density is normalized to So=2~(o)R~/R~where R d is the dc differential resistance, and R is a normal resistance. Solid lines correspond to calculated dependence for the small-capacitance tunnel junction, ~ I ; unstable regions of its I-V curve, (di/dV¢O) are indicated on the top. Dashed lines show the same dependence calculated from "generalized RSJ model" (3),(6),(7); dash-dot lines show Sv(O) for the case of normal tunnel junction ~=O). Label "N" indicates the level of Nyquist noise, and the thin lines show shot noise in the normal tunnel junction. ratures (Fig.|b,c) the voltage fluctuation density is, however, close to that of the normal junction, while gap singularities become stronger. The thermal noise level lowers, and at T=O all voltage fluctuations are of shot-noise and quantum-noise origin. The author would like to thank Dr.K.K.Likharev and Dr.V.K.Semenov for helpful discussions. I

Sv(O) is directly related to the linewidth of the Josephson oscillation:2F=2¢(2C/@o)2Sv(o). [l]Rogovin,D.and Scalapino,D.J.,Ann. Phys.66~74) l. [2]Hu,E.L.et al.,IEEE Tra~£D-27(1980)2030. [31Werthamer,N.R.,Phys.Rev. 147(1966)255. [4]Larkin,A.l.and Ovchinnikov,Yu.N.,Zh.Exp.Teor. Fiz.51 (1966)1535 lSov.Phys.-JETP 240967)1035]. rS]Tucker,J.R.,IEEE J. QE-15(1979)1234. [6]Callen,H.BmndWelton,T.A.,Phys.Rev]83(1951)34. [7]McDonald,D.C.et al,Phys.Rev. B J2(1976) I028 . [8]Zorin,A.B.and Likharev,K.K.,Fiz.Nizk.Temp.3 (1977) 148[Sov.J.Low Temp.Phys.3 (1977)70]. [9]Zorin,A.B.and Likharev,K.K.,Radiotekhnika i Elektronika 26(1981)834. [10]Zorin,A.B.,Fiz.Nizk.Temp.7(1981)N6[Sov.J.Low Temp. Phys.].