Coulomb blockade in superconducting
47 (1999) 385-387
D.V. Averin Department of Physics and Astronomy, SUNY at Stony Brook, Stony Brook NY 11794
Amplitude of the Coulomb blockade oscillations is calculated for a single-mode Josephson junction with large electron transparency D. It is shown that the Coulomb blockade is suppressed as D + 1. The suppression is described quantitatively in terms of the Landau-Zener transition in imaginary time. 1. INTRODUCTION
Quantization of electric charge Q of an isolated conductor gives rise to a wide range of Coulomb blockade phenomena in mesoscopic systems [1,2]. However, if the conductor is connected to an external circuit by a ballistic junction, the charge is free to move in and out of the conductor and both the charge quantization and Coulomb blockade should be suppressed. Indeed, it was shown [3,4] that for a single-mode junction between normal conductors with quasi-continuous energy spectrum, amplitude of the Coulomb blockade oscillations vanishes together with the reflection coefficient R = 1 - D of the junction. The aim of this work was to study this problem for superconducting junctions, where the situation appears to be different. Coulomb blockade oscillations arise in this case  from the formation of the Bloch bands in the Josephson potential U(cp) which is periodic in the Josephson phase difference cp. Since the ballistic junctions also have periodic Josephson potential, one could expect that Coulomb blockade exists even in the ballistic regime. It is shown below that this expectation is incorrect and, similarly to the normal case, the Coulomb blockade disappears when R + 0. 2. MODEL
The Hamiltonian of a nearly ballistic single-mode Josephson junction consits of the charging energy of the junction capacitance C and internal energy of electron motion in the junction. As in the normal case , the latter can be written as the sum of energies HL,R of electrons with momenta fkF moving forward and backward through the junction, and potential V responsible for scattering between these two directions of propagation. The energy of the forward-moving electrons can be written in the standard matrix form: HL =
-iv~alax A(4 *L(X),4) = A*(4 ha/ax
where SL = (&+,&I) is the creation operator for quasiparticles with momentum kp, A is the superconducting energy gap, and ?&?is the Fermi velocity. HR is given by the same expression with VF + -vF. 0167-$1317/99/$ - SIXfrontmatter PIt: SO167-9317(99)00240-3
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D.K Averin 1 Microelectronic
Engineering 47 (1999) 385-387
All relevant energies, including characteristic charging energy EC = (2e)2/2C and temperature T, are assumed to be much smaller than A. In this “adiabatic” regime the junction Hamiltonian can be simplified considerably. Most importantly, since the variations of cpare slow and can not cause transitions between different quasiparticle states, the Hamiltonian (1) reduces to a sum of the quasiparticle energies Ed of the occupied states. Another simplification is that in this regime only Cooper pairs transfer charge across the junction (no quasiparticles can be created), and the operator of the transferred charge Q can be expressed directly through the Josephson phase difference : Q = -2eid/&. Introducing the charge Q injected into the junction from external circuit, we finally write the total Hamiltonian as: H=
3. l3BSULTS AND DISCUSSION The spectrum of eigenenergies ek(cp) is found by solving the Bogolyubov-de Gennes equations with the pair-potential A(z) (1). The spectrum consists of the continuum of states at energies outside the gap ]E]> A, and two discrete states in the gap:
E*(cp)= FA cm 42,
where t = (A/tiv~) sin (p/2. In all these expressions cp E [0,27r], and they should be continued periodically in cp beyond this interval. At cp = 0 the discrete states (3) merge with the continuum and at 2’<< A all the states with e < -A are filled, while those with 6 > A are empty. However, when cp deviates from 0, the spectrum shifts by one level per period of cp: the states with +/i+ shift up in energy, while the -kF states shift down. The change in energies in this process is infinitesimal for all states besides the two states (3) which move across the gap and change their energy by 2A. Such a motion of the energy spectrum gives rise to the effective Josephson potential in the Hamiltonian (2): V(p) = Cek(cp) = A(2m + (-l)m+1coscp/2),
m E int(] cp ( /27r).
The increase of the potential (4) with cp means that the phase can increase beyond the points cp = Omod(2n) only at the expense of creating qua&particles in the junction electrodes. In the case of classical Josephson dynamics, this process generates real quasiparticles and creates dissipative component of the Josephson current . The energy relaxation restores then the 2x-periodicity of all the junction characteristics. It should be noted that the potential (4) for quantum phase dynamics can not be obtained if one takes into account only the subgap states . An important consequence of aperiodicity of the potential (4) is the complete suppression of the Coulomb blockade oscillations in the ballistic junction. The periodic nature of the potential and Coulomb blockade are restored by finite reflection in the junction. At cp = r, where the energies of the two subgap states (3) coincide, the backscattering term V in the Hamiltonian (2) couples the branch (4) of the Josephson potential with no quasiparticles at cp = 0 to the one with no quasiparticles at cp = 27r (the same as (4)
D.VOAverin I Microelectronic Engineering 47 (1999) 385-387
but shifted along the cp axis by 27r), thus creating the periodic low-energy branch of the potential. Quantitatively, the backscattering potential is V = J dzU(z)p(z), where p is the operator of electron density, and V(z) is the potential profile along the junction. Evaluating it in the basis of subgap states (3), we find that the coupling matrix element in the relevant region around cp = 7r is: (q- 1V 1CO+) = rA, where T is the reflection amplitude of the junction. Dynamics of cp in this region reduces then to the standard two-state dynamics. In particular, the probability amplitude 20 of staying on the low-energy branch of the potential is controlled by the usual Landau-Zener transition, the same as in the case of classical phase dynamics . The only difference with the classical case is that now the transition should take place in the course of cp motion under the potential barrier, i.e. in “imaginary time”. Using the quasiclassical approximation in the Schrodinger equation for cp near the point cpN r we find: ‘1~= &
X = (R/2)(A/E&/2.
Equation (5) shows that w vanishes as (27rrX) ‘I2 at small reflection probabilities R < (EC/A) l/2 * If the system stays on the periodic branch of the Josephson potential, the Bloch bands of states are formed and the junction free energy F oscillates as a function of the injected charge Q, F(q) = A cos(vrq/e + Q), where 8 is the phase of the reflection amplitude r. The amplitude A of these “Coulomb blockade” oscillations at zero temperature is:
The oscillations of the free energy lead to oscillations of the voltage across the junction: V(q) = dF(q)l&! as a function of q. In summary, we have shown that the Coulomb blockade in a single-mode Josephson junction is suppressed in the ballistic limit and the suppression is governed by the same Landau-Zener transition that determines the magnitude of the dissipative component of the current in the regime of classical Josephson dynamics. REFE-NCES 1. D.V. Averin and K. K. Likharev, in: Mesoscopic Phenomena in Solids, ed. by B. L. Altshuler, P. A. Lee, and R. A. Webb (Elsevier, Amsterdam, 1991). 2. Single Charge Zknneling, ed. by H. Grabert and M.H. Devoret (Plenum, 1992). 3. K. Matveev, Phys. Rev. B 51 (1995) 1743. 4. I.L. Aleiner and L.I. Glazman, Phys. Rev. B 57 (1998) 9608. 5. D.V. Averin, A.B. Zorin, and K.K. Likharev, Zh. Eksp. Teor. Fiz. 88 (1985) 692 [Sov. Phys. JETP 61, 4071. 6. P.W. Anderson, in: The many-body problem, ed. by E.R. Caianiello (Academic Press, New York, 1964), p. 113. 7. D.V. Aver-in and A. Bardas, Phys. Rev. B 53 (1996) R1705. 8. D.A. Ivanov and M.V. Feigel’man, cond-mat/9808029. 9. D.V. Averin and A. Bardas, Phys. Rev. Lett. 75 (1995) 1831.