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Resonance phenomenon in current-voltage characteristic of the Josephson junction
PHYSICA
Physica B 194-196 (1994) 1473-1474 North-Holland
RESONANCE PHENOMENON JOSEPHSON JUNCTION
IN C U R R E N T - V O L T A G E
CHARACTERISTIC
OF THE
Yu. N. Ovchinnikov L. D. Landau Institute for Theoretical Physics, Academy of Sciences of Russia, Kosygin St. 2, 117334 Moscow, Russia A. Schmid Institut fur Theorie der Kondensierten Materie, Universitat Karlsruhe Kaiserstrasse 12, 7500 Karlsruhe, Germany Sharp peaks appear in I-V characteristic of the Josephson Junction when levels in neighboring wells are crossing. The shape and size of these peaks strongly depend on the parameters of the junction as well as on temperature and number of levels lying below the resonance one. 1. MODEL It is well known from quantum mechanics that a resonant phenomenon takes place, when levels in double well potential are close together [1]. In Josephson junctions such situations take place at the special values of the external current j. On the I-V characteristic at these special values of current there appear sharp asymmetric peaks [2]. In the resonance region are essential as excitations of small energy so also resonance excitations with energy equal to the distance between levels in the well. To find the form of the resonance peak, we will use the tunneling Hamiltonian, where viscosity will be introduced by help of the interaction of the "particle" with an infinite number of oscillators [3]. The tunneling amplitude will be considered as small
lltNk being the same for the oscillators. The energy E of this state, counted from the ground state is E = N ~ + Z NkC,Ok
k
(3)
Transition probability per unit time W can be represented in the form
Sdt ITNTNI C~ ( N ) C R (N + e ) . --cx~
\
\
t
k h
)t
k
tl
n = I210 + f l l
HI=~
l
^
+
(4)
"~
TkC L ( k ) C R (k + Q) + ^. + k ~T k C R ( k + Q ) C L ( k )
(1)
Here we take into account only transitions between resonant states, that are shifted by the whole number Q. In zero approximation we can introduce normal coordinates of oscillators and particles when "particle" is in left or right well, respectively. The eigenstates of the Hamiltonian [-2I0 now can be written in the form
The expression (4) for the transition probability can be written in the form [4]
(
/-1
NN]
• N L ~ - N R ~ * + Y~N~mk - Y~N~mk +A k
(5) Here { N L , N L } are the occupation numbers in the
initial state and {NR,N R} is the same for final state, A- is energy shift between bottoms of left and right wells. The sum over all final states is taken.
where { N , N k }
are the occupation numbers and
~ N is the wave function of the particle in N-state,
For small value of viscosity quantity ~
[4]
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved S S D I 0921-4526(93)E1317-F
is equal to
1474
~:£2-i7,
(6)
7=4MX~
where £"2, M , X 0 - me paramelers of polenlial ([2 -
where 8 = E L - E R , 0 ) c - is cutoff energy.
The
dependence of W from energy difference is given in Fig. 1.
distance between neighboring levels, M - mass of the "particle," 2X 0 - distance between bottoms at two wells), ~ - is the interaction constant of particle with "thermal bath" [3], connected with the constants of coupling of the particle O¢k with the oscillalors by relation '~ 2 o~(x~ Z~{,Olk k mkO)k
.
- o1) : o~
(7)
Between the normal coordinates Z k of the oscillators, when the "particle" is in the left or right well, respectively, there exists a simple conneclion
:
+
2qt-2x OOCk
F ~ -6
Relation (8) is essenlial fl~r calculation of overlapping integrals kind of (2) in formula (5). The detailed calculation of transition probability W in full temperature region 0
I[ r O,c ;
/
W = 21~i"0 Re e ir~x Y + 1~
1
LY
7
1
2 (9)
./Y
i
10
,1.=0,2
2
E/0,
Fig. 1 REFERENCES
[1] Quantum Mechanics, L. D. Landau, E. M. Lifshitz. Pergamon Press, London-Paris, 1959. [2] A. 1. Larkin, Yu. N. Ovchinnikov, A. Schmid. Physica B152 (1988) 266. [3] A. O. Calderia, A. J. Leggett Phys. Rev. Left. 46 (1981) 211. [4] Yu. N. Ovchmnikov, A. Schmid, Phys. Rev. B (1993).
Physica B 194-196 (1994) 1473-1474 North-Holland
RESONANCE PHENOMENON JOSEPHSON JUNCTION
IN C U R R E N T - V O L T A G E
CHARACTERISTIC
OF THE
Yu. N. Ovchinnikov L. D. Landau Institute for Theoretical Physics, Academy of Sciences of Russia, Kosygin St. 2, 117334 Moscow, Russia A. Schmid Institut fur Theorie der Kondensierten Materie, Universitat Karlsruhe Kaiserstrasse 12, 7500 Karlsruhe, Germany Sharp peaks appear in I-V characteristic of the Josephson Junction when levels in neighboring wells are crossing. The shape and size of these peaks strongly depend on the parameters of the junction as well as on temperature and number of levels lying below the resonance one. 1. MODEL It is well known from quantum mechanics that a resonant phenomenon takes place, when levels in double well potential are close together [1]. In Josephson junctions such situations take place at the special values of the external current j. On the I-V characteristic at these special values of current there appear sharp asymmetric peaks [2]. In the resonance region are essential as excitations of small energy so also resonance excitations with energy equal to the distance between levels in the well. To find the form of the resonance peak, we will use the tunneling Hamiltonian, where viscosity will be introduced by help of the interaction of the "particle" with an infinite number of oscillators [3]. The tunneling amplitude will be considered as small
lltNk being the same for the oscillators. The energy E of this state, counted from the ground state is E = N ~ + Z NkC,Ok
k
(3)
Transition probability per unit time W can be represented in the form
Sdt ITNTNI C~ ( N ) C R (N + e ) . --cx~
\
\
t
k h
)t
k
tl
n = I210 + f l l
HI=~
l
^
+
(4)
"~
TkC L ( k ) C R (k + Q) + ^. + k ~T k C R ( k + Q ) C L ( k )
(1)
Here we take into account only transitions between resonant states, that are shifted by the whole number Q. In zero approximation we can introduce normal coordinates of oscillators and particles when "particle" is in left or right well, respectively. The eigenstates of the Hamiltonian [-2I0 now can be written in the form
The expression (4) for the transition probability can be written in the form [4]
(
/-1
NN]
• N L ~ - N R ~ * + Y~N~mk - Y~N~mk +A k
(5) Here { N L , N L } are the occupation numbers in the
initial state and {NR,N R} is the same for final state, A- is energy shift between bottoms of left and right wells. The sum over all final states is taken.
where { N , N k }
are the occupation numbers and
~ N is the wave function of the particle in N-state,
For small value of viscosity quantity ~
[4]
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved S S D I 0921-4526(93)E1317-F
is equal to
1474
~:£2-i7,
(6)
7=4MX~
where £"2, M , X 0 - me paramelers of polenlial ([2 -
where 8 = E L - E R , 0 ) c - is cutoff energy.
The
dependence of W from energy difference is given in Fig. 1.
distance between neighboring levels, M - mass of the "particle," 2X 0 - distance between bottoms at two wells), ~ - is the interaction constant of particle with "thermal bath" [3], connected with the constants of coupling of the particle O¢k with the oscillalors by relation '~ 2 o~(x~ Z~{,Olk k mkO)k
.
- o1) : o~
(7)
Between the normal coordinates Z k of the oscillators, when the "particle" is in the left or right well, respectively, there exists a simple conneclion
:
+
2qt-2x OOCk
F ~ -6
Relation (8) is essenlial fl~r calculation of overlapping integrals kind of (2) in formula (5). The detailed calculation of transition probability W in full temperature region 0
I[ r O,c ;
/
W = 21~i"0 Re e ir~x Y + 1~
1
LY
7
1
2 (9)
./Y
i
10
,1.=0,2
2
E/0,
Fig. 1 REFERENCES
[1] Quantum Mechanics, L. D. Landau, E. M. Lifshitz. Pergamon Press, London-Paris, 1959. [2] A. 1. Larkin, Yu. N. Ovchinnikov, A. Schmid. Physica B152 (1988) 266. [3] A. O. Calderia, A. J. Leggett Phys. Rev. Left. 46 (1981) 211. [4] Yu. N. Ovchmnikov, A. Schmid, Phys. Rev. B (1993).