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Damping of current-carrying quasiparticle states in superconducting junctions and superlattices
Physica C 235-240 (1994)3249-3250 North-Holland
PHfS~CA
Damping of current-carrying quasiparticle states in superconducting junctions and superlattices S. V. Kuplevakhsky ~ and I. I. Fal'ko b ~Department of Physics, Kharkov State University, 310077 Kharkov, Ukraine ~Physico-Technical Department, Kharkov Polytechnical Institute, 310002, Ukraine
Within the framework of a self-consistent microscopic theory, we study the effect of weak structural inhomogeneities of tunnel barriers on recently predicted supercurrent-induced subgap states in Josephson structures. We show that tunneling with the non-conservation of the momentum component parallel to the barrier leads to the damping of these states. Recent theoretical studies reveal that supercurrents in Josephson structures with tunnel barriers induce quasiparticle states within the superconducting gap. The energies of these states are parameterized by the coherent phase difference, ¢, and the cosine of the angle of incidence on the barrier, cosO. For example, in an isolated junction with an ideally flat tunnel barrier a single discrete level
[1,2] occurs: E°(t;¢)=A~I1
1
• 2~
7 T(t)sln 71'
invariant. In this case, one should expect a shift and damping of the level (1). Below we give an explicit mathematical derivation of these results on the basis of a simple but rather general microscopic model of a Josephson junction. In terms of the retardered Green's function, the problem is formulated as follows:
IE'to+I2-~V2+EF)~3-½(~I+~2)A(r) -½/~1 - xel A*(r)- W(r) 7
(1)
Here A~is the energy gap in the bulk of superconductors, t---cos~, T(t) is the tunneling probability for given t (T(1) << 1), EF is the Fermi energy. In a superlattice, the level (1) is transformed into a Bloch-type band [3]. The consideration of the papers [1-3] is based essentially on the assumption of translational ivariance in the directions parallel to the interfaces. This assumption guarantees the conservation of the corresponding momentum components, yielding a quasi-one-dimentional problem to solve. On the other hand, it is quite clear that owing to the inevitable presence of structural inhomogeneities (at least on the atomic scale) real tunnel barriers are not translationally
Here E is the energy parameter with an infinitesimal positive imaginary part; m is the mass of an electron; Xk ( k = 1 , 3 ) is a Pauli matrix in Gor'kov-Nambu space, and t0 is the corresponding unit matrix. The potential of the barrier is described by ^
W(r) = Ev+ u(p)~5(x)x3,
where V is a positiv constant, p : (y, z), U(p) is a random function of coordinates with the properties
( u ( p ) ) : 0,
0921-4534/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved. SSDI 0921-4534(94)02190-2 -
(3)
(4)
S. E Kupleuakhs~'v, I.I. Fal'ko/Pt~vsica C 235~ 240 (1994) 3249 3250
3250
(u(p)u(p/)) = ~aIp
p/~,
(5)
The subgap part of the energy spectrum of our system is determined by the /
where P0 -- mvo (Vo is the Fermi velocity), u 2 is a constant, and C..> means averaging. Discarding the self-consistency condition, we adopt for the pair potential the approximation (6)
A(0 = A~ exp [/sgn(x)d~].
This approximation is valid for the evaluation of the subgap states in the limit of a low barrier transmission (V>> v0) [2]. Additionally, we restrict ourselves to the limit of small structural fluctuations: s -- u2/V 2 << 1.
(7)
We are interested in the solution of (2), (6) averaged over the structural fluctuations. Making use of (4), (5), (7) and mathematical methods developed for the treatment of superconducting alloys [4], we derive
where we have taken Fourier transforms in the coordinates x and y, with ~ being the momentum component parallel to the barrier. /
The Green's function
0;
det
GE~Io, o;~!
~ ~0.
(9)
Solving (8), (9) in the lowest order in tunneling probabilities yields
i3~ T~(t)ltl3 sin2{ ~
(lO)
where T(t)=[l+d2=~jv20t21V2+T~(O is the averaged total tunneling probability, while T,(t)-sv201tl/V 2 is the averaged probability of tunneling with the non-conservation of K; }tl >> max{(A=lEF) 112,s}. As expected, in the absence of structural inhomogeneities (10) reduces to (1). Physically, T(t) determines the resistance of the junction in the normal state [5]. It is interesting to observe that TK(t) not only gives rise to damping, but also contributes to the shift of ReEo. The authors thank A. N. Omelyanchuk for fruitful discussions.
gives us the
-1
E
which is equivalent to the condition
\
GEIX, X;,~~
solution of (2), (6) for U(p)-=0. Diagonal elements of GE with one or both coordinates in the plane of the barrier have a pole at the energy of the current-carrying level (1). The self-energy part, T., satisfies the self-consistent equation
z=~3f~2-~L
\
poles of diagonal elements of ( d ; E ) [ 0 , 0 ; ~ ' ,
-
The solution of (9) is expressed in terms of the matrices to,-c~, 1:3 with complex coefficients. Note that the emergence of ImE indicates damping caused by tunneling with the non-conservation of K --- [~t = P0 I sin 8[.
REFERENCES 1. A. Furusaki and M. Tsukada, Phys. Rev. B43 (1991) 10164; 2. S. V. Kuplevakhsky and I. I. Fal'ko, Sov. J. Low Temp. Phys. 17 (1991) 501. 3. S. V. Kuplevakhsky and I. I. Fal'ko, Sov. J. Low Temp. Phys. 18 (1992) 776. 4. A. A. Abrikosov, L P. Gorkov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Englewood Cliffs, 1963. 5. I. O. Kulik, Yu. N. Mitsay and A. N. Omelyanchuk, Zh. Eksp. Teor. Fiz. 66 (1974) 1051.
PHfS~CA
Damping of current-carrying quasiparticle states in superconducting junctions and superlattices S. V. Kuplevakhsky ~ and I. I. Fal'ko b ~Department of Physics, Kharkov State University, 310077 Kharkov, Ukraine ~Physico-Technical Department, Kharkov Polytechnical Institute, 310002, Ukraine
Within the framework of a self-consistent microscopic theory, we study the effect of weak structural inhomogeneities of tunnel barriers on recently predicted supercurrent-induced subgap states in Josephson structures. We show that tunneling with the non-conservation of the momentum component parallel to the barrier leads to the damping of these states. Recent theoretical studies reveal that supercurrents in Josephson structures with tunnel barriers induce quasiparticle states within the superconducting gap. The energies of these states are parameterized by the coherent phase difference, ¢, and the cosine of the angle of incidence on the barrier, cosO. For example, in an isolated junction with an ideally flat tunnel barrier a single discrete level
[1,2] occurs: E°(t;¢)=A~I1
1
• 2~
7 T(t)sln 71'
invariant. In this case, one should expect a shift and damping of the level (1). Below we give an explicit mathematical derivation of these results on the basis of a simple but rather general microscopic model of a Josephson junction. In terms of the retardered Green's function, the problem is formulated as follows:
IE'to+I2-~V2+EF)~3-½(~I+~2)A(r) -½/~1 - xel A*(r)- W(r) 7
(1)
Here A~is the energy gap in the bulk of superconductors, t---cos~, T(t) is the tunneling probability for given t (T(1) << 1), EF is the Fermi energy. In a superlattice, the level (1) is transformed into a Bloch-type band [3]. The consideration of the papers [1-3] is based essentially on the assumption of translational ivariance in the directions parallel to the interfaces. This assumption guarantees the conservation of the corresponding momentum components, yielding a quasi-one-dimentional problem to solve. On the other hand, it is quite clear that owing to the inevitable presence of structural inhomogeneities (at least on the atomic scale) real tunnel barriers are not translationally
Here E is the energy parameter with an infinitesimal positive imaginary part; m is the mass of an electron; Xk ( k = 1 , 3 ) is a Pauli matrix in Gor'kov-Nambu space, and t0 is the corresponding unit matrix. The potential of the barrier is described by ^
W(r) = Ev+ u(p)~5(x)x3,
where V is a positiv constant, p : (y, z), U(p) is a random function of coordinates with the properties
( u ( p ) ) : 0,
0921-4534/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved. SSDI 0921-4534(94)02190-2 -
(3)
(4)
S. E Kupleuakhs~'v, I.I. Fal'ko/Pt~vsica C 235~ 240 (1994) 3249 3250
3250
(u(p)u(p/)) = ~aIp
p/~,
(5)
The subgap part of the energy spectrum of our system is determined by the /
where P0 -- mvo (Vo is the Fermi velocity), u 2 is a constant, and C..> means averaging. Discarding the self-consistency condition, we adopt for the pair potential the approximation (6)
A(0 = A~ exp [/sgn(x)d~].
This approximation is valid for the evaluation of the subgap states in the limit of a low barrier transmission (V>> v0) [2]. Additionally, we restrict ourselves to the limit of small structural fluctuations: s -- u2/V 2 << 1.
(7)
We are interested in the solution of (2), (6) averaged over the structural fluctuations. Making use of (4), (5), (7) and mathematical methods developed for the treatment of superconducting alloys [4], we derive
where we have taken Fourier transforms in the coordinates x and y, with ~ being the momentum component parallel to the barrier. /
The Green's function
0;
det
GE~Io, o;~!
~ ~0.
(9)
Solving (8), (9) in the lowest order in tunneling probabilities yields
i3~ T~(t)ltl3 sin2{ ~
(lO)
where T(t)=[l+d2=~jv20t21V2+T~(O is the averaged total tunneling probability, while T,(t)-sv201tl/V 2 is the averaged probability of tunneling with the non-conservation of K; }tl >> max{(A=lEF) 112,s}. As expected, in the absence of structural inhomogeneities (10) reduces to (1). Physically, T(t) determines the resistance of the junction in the normal state [5]. It is interesting to observe that TK(t) not only gives rise to damping, but also contributes to the shift of ReEo. The authors thank A. N. Omelyanchuk for fruitful discussions.
gives us the
-1
E
which is equivalent to the condition
\
GEIX, X;,~~
solution of (2), (6) for U(p)-=0. Diagonal elements of GE with one or both coordinates in the plane of the barrier have a pole at the energy of the current-carrying level (1). The self-energy part, T., satisfies the self-consistent equation
z=~3f~2-~L
\
poles of diagonal elements of ( d ; E ) [ 0 , 0 ; ~ ' ,
-
The solution of (9) is expressed in terms of the matrices to,-c~, 1:3 with complex coefficients. Note that the emergence of ImE indicates damping caused by tunneling with the non-conservation of K --- [~t = P0 I sin 8[.
REFERENCES 1. A. Furusaki and M. Tsukada, Phys. Rev. B43 (1991) 10164; 2. S. V. Kuplevakhsky and I. I. Fal'ko, Sov. J. Low Temp. Phys. 17 (1991) 501. 3. S. V. Kuplevakhsky and I. I. Fal'ko, Sov. J. Low Temp. Phys. 18 (1992) 776. 4. A. A. Abrikosov, L P. Gorkov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Englewood Cliffs, 1963. 5. I. O. Kulik, Yu. N. Mitsay and A. N. Omelyanchuk, Zh. Eksp. Teor. Fiz. 66 (1974) 1051.