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Physica B 227 (1996)241-244

Noise and adiabatic dynamics of superconducting quantum point contacts D. Averin Department of Physics, SUNY at Stony Brook, Stony Brook, NY 11794, USA Abstract

Dynamics of superconducting quantum point contacts at small bias voltages is governed by a simple kinetic equation for occupation probabilities of the two current-carrying states localized in the contact region. Starting from this equation, I show that superconducting quantum point contacts should have unusual noise properties. In the stationary case, flow of the DC supercurrent at non-vanishing temperatures gives rise to a noise with very large spectral density at zero frequency. At finite voltages the contact exhibits giant shot noise. Surprisingly, the origin of the noise is the supercurrent coherence, because of which the charge is transferred through the contact in large quanta. In contact with small reflection the noise can be calculated in terms of Landau-Zener transitions between the current-carrying states. Keywords: Josephson junctions; Shot noise; Point contacts

One of the most direct manifestations of the correlations between electrons caused by Pauli principle is the suppression of the shot noise in ballistic point contacts between normal metals. First predicted for classical point contacts [1, 2], such a suppression was later rediscovered for quantum point contacts [3, 4], where it was also observed in experiments [5, 6]. In point contacts between normal metals and superconductors, or in fully superconducting point contacts at large voltages, Andreev scattering leads to a partial reflection of electrons in the contact region giving rise to a finite shot noise [-2, 7]. This noise, however, is associated only with the excess current and therefore always represents only a small fraction of the full classical shot noise• The aim of this work is to study the current noise in a quantum point contact between two identical superconductors at small bias voltages V ~ Ale, including the DC case V = 0. There is a common opinion (supported by old calculations of the current noise in Josephson tunnel junctions; for review, see, e.g., Ref. [8]) that the flow of supercurrent

in Josephson junctions is a coherent continuous process and does not lead to any current noise. In what follows, I show that this opinion is only a prejudice. In fact, the coherent nature of the supercurrent flow in superconducting quantum point contacts leads to a giant current noise. The reason for this is that because of the supercurrent coherence the charge is transferred through the constriction in large quanta, much larger than individual Cooper pairs, and fluctuations of these charge quanta cause giant noise. The model I consider is a short single-mode quantum point contact with characteristic dimensions that are much smaller than both the elastic scattering length and coherence length of the superconducting electrodes. The current in such a contact can be described in terms of the non-equilibrium occupation probabilities p± of the two states carrying the currents +(Io/2)sin ~/2 [9]:

•

~o(t)

I(t) = Io sm ~ -

0921-4526/96/$15.00 g~ 1996 ElsevierScienceB.V. All rights reserved PII S0921-4526(96)00410-3

p(q~(t)),

eA Io = ~ .

(1)

D. Averin/Physica B 227 (1996) 241-244

242

Here q~ is the Josephson phase difference across the contact, p is the difference of the two occupation probabilities, p(q)(t))==-p_ - p + , and Io has the meaning of the zero-temperature critical current. The factor of two due to summation over spin is included in Eq. (1). The rate equation for p(q)) is jO(~o(t)) = 7(e)En(e) - p(~0(t))],

f'

(g -- d ) 3 c o s h ( e / 2 T )

, ~9(g '2 - - A 2)

r-i i I i i I t_._A

--1

[-i i J i I i

t

(2)

where n(~) = tanh(e/2T) is the equilibrium value of p(e(q~)), 7(e) is the rate of quasiparticle exchange between the bulk electrodes and two discreet states in the constriction, and e = e(q~)-= A cos ~o/2. The rate 7 is roughly proportional to the subgap density of states in the superconducting electrodes; it vanishes in the ideal BCS case; if the gap is slightly smeared by finite electron-phonon interaction, 7 is given by the following expression [10]:

y(e):~Jde ~

I

s i n h ( - ~ ~ / 2

T)'

(3) were ~ is a constant determined by the parameters of electron-phonon interaction. The rate equation (2) should be supplemented with a "boundary condition" which states that the level occupation reaches equilibrium as soon as the levels hit the gap edges, ~ = __+A (Fig. 1), that is

Fig 2. Typical realization of the "DC" supercurrent in a short constriction between two superconductors as a function of time at non-vanishing temperatures. The current sign changes randomly with the characteristic time interval Y 1 between the jumps.

ization of the occupation probabilities is a dynamic process. In particular, even in the stationary case V = 0 the system undergoes transitions between the two current-carrying states. This means that what is considered to be a stationary supercurrent is in fact a stochastic process with a typical realization illustrated in Fig. 2. The current randomly changes sign with the rate ~, so that the probabilities of the positive/negative current are [1 _+

n(e(q~))]/2. Quantitatively, the solution p(t) of Eq. (2) with the boundary condition (4) enables one to find the current correlation function

~(t)

K(t, t + z) = 21o2 sin - ~ - sin

p+(t)p_(t)

p(q~) = ( - 1)ran(A) for q~ = 2rcm, m = 0, ___1, ...

(4) Eq. (2) implies that at non-vanishing temperatures (when y -~ 0 inside the energy gap) thermal-

where p± = (1 _+ p)/2. In the stationary case K is independent of t and the spectral density of the current fluctuations is [11]

lfo

A

S1(e)) = ~

dz K(z) cos(o)z)

I2sin2(~0/2) 0

~. . . . . .

= 2~ cosh2(e(qg)/2T)7

--h-"

m =

1

2

3

Fig 1. Energies e± = + A cos(~o/2) of the two Andreev-bound states in a short constriction between two superconductors as functions of the Josephson phase difference qx Solid dots represent "thermalization points" where occupation of the two level always reaches equilibrium see Eq.(4).

7 2 -~-

092,

(6)

where ~ = y(e(~0)). We see that although the total noise intensity decreases as T -~ 0, the spectral density at zero frequency may actually increase at small temperatures because of the rapid decrease of 7 (3). At finite voltages the correlation function (5) oscillates in time t with the period lrh/eV of the

243

D. Averin/Physica B 227 (1996) 241--244

Josephson oscillations. By averaging over t and Fourier transforming with respect to z as in the stationary case, the spectral density of current fluctuations can be found in various regimes. In particular, at T = 0 and small voltages V ~ 7(O)/e (assuming that 7(0) is finite even at T = 0) the spectral density is Igev

Sl((O ) = n2h(72(0 ) -[- (D2).

(7)

At finite temperatures and voltages large on the scale of 7, V >> 7/e, the occupation probabilities p+ stay constant throughout the period of the Josephson oscillations and change only due to thermalization described by the boundary condition (4). In this case hi 2 1 + cos(Tth~o/eV ) $i(~o) = 2rt2eV c o s h 2 ( A / 2 T ) (1 - (hog/eV)2) 2 " (8)

Eq. (8) shows that the noise has unusual voltage dependence: it decreases with growing voltage despite the fact that the average current saturates in this voltage range. This reflects an unusual mechanism of the noise: random transfer of charge through the point contact in each period of the Josephson oscillations. The quantum of the transferred charge (i.e. the charge in one period) is 2 A / V . At small voltages it is much larger than the charge of individual Cooper pairs and decreases with growing voltage. The noise (8) is a shot noise of these large charge quanta. So far we discussed ballistic contacts (R = 0). In contacts with small but finite reflection coefficient R ,~ 1 there is another noise-generating mechanism in addition to the one discussed above. It is associated with the Landau-Zener transitions between the two current-carrying states which occur at ~p = it mod(2~) with the probability 2 -e x p { - n R A / e V } [12]. The noise in this case is caused by the randomness of the Landau-Zener transitions. The current correlation function can be found either from the rate equation (2) modified to account for these transitions or from fully quantum mechanical approach producing the same result. In the latter approach one starts with the current operator in the basis of the two current-carrying states: l(t) = (Io/2)sin(~o(t)/2) U t ( t , - ~ ) GzU(t, - ~), where the evolution operator U describing

the Landau-Zener transition is U(t, t') = S(t, O) SLzS(0, t'). Here the time is chosen such that the transition moment is t = 0; SLZ is the scattering matrix of the Landau-Zener transition, and

0 ) e - iz~t,t,)

,

with X being the phase accumulated due to the time evolution of the Josephson phase q~. Explicit expression for the current operator together with the unitarity of SLZ give directly the zero-temperature current correlation function for time arguments within the same period of the Josephson oscillations: K(t, t') = 2I~)~(1 - ,~)sin ~0(t)/2 sin ~o(t')/2, for t, t' > 0. The currents in different periods are not correlated because of the thermalization (4). This gives the following spectral density of current fluctuations: SI(e)) -

hlo22(1 - 2) 1 - 2v sin (~v/2) + y2 rc2eV (1 - v2) 2 '

v - he)/eV.

(9)

The qualitative features of this spectral density are the same as those of Eq. (8). It is much larger that the spectral density of the classical shot noise; it grows at small and decreases at large voltages. The maximum is reached at V ~- ~ R A / e l n 2. In conclusion, it was shown that the superconducting quantum point contacts should have very unusual noise properties. In the DC regime the flow of supercurrent leads to a two-state noise, while in the AC regime the contact exhibits giant shot noise. The spectral density of current fluctuations was calculated in both regimes. The author gratefully acknowledges the suggestion of K.K. Likharev which stimulated this work. The work reported here was partly done in collaboration with H. Imam, and was supported by D O D URI through AFOSR Grant # F49620-92-J0508.

References

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D. Averin /Physica B 227 (1996) 241-244

[3] G.B. Lesovik, JETP Lett. 49 (1989) 592. I-4] M. Biittiker, Phys. Rev. Lett. 65 (1990) 2901. 1-5] Y.P. Li, D.C. Tsui, J.J. Heremans, J.A. Simmons and G.W. Weimann, Appl. Phys. Lett. 57 (1990) 774. [6] M. Reznikov, M. Heiblum, H. Shtrikman and D. Mahalu, Phys. Rev. Lett. 75 (1995) 3340. [-7] J.P. Hessling, V.S. Shumeiko, Yu.M. Galperin and G. Wendin, cond-mat 9509018.

[8] D. Rogovin and D.J. Scalapino, Ann. Phys. 86 (1974) 1. [-9] D. Averin and A. Bardas, Phys. Rev. B 53 (1996) R1705. 1-10] A.I. Larkin and Yu.N. Ovchinnikov, Sov. Phys.-JETP 41 (1976) 960. [-11] D. Averin and H. Imam, Phys. Rev. Lett. 76 (1996) 3814. [,12] D. Averin and A. Bardas, Phys. Rev. Lett. 75 (1995) 1831.

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