Plasma waves in a Josephson chain

Physica C 168 (1990) 279-281 North-Holland

P L A S M A WAVES I N A J O S E P H S O N

CHAIN

M.V. S I M K I N Laboratory of Neutron Physics. Joint Institute for Nuclear Research, Head Post Office P.0. Box 79, Moscow, USSR Received 5 March 1990

Plasma waves in a periodic Josephson chain are predicted. The wave equation is obtained and solved for the small amplitude case. The dispersion relation and its long-wave asymptotic are obtained. Plasma waves in thin superconducting wire are also considered. These one dimensional (1D) plasmons are analogous to plasmons in 1D semiconductor heterostructures.

1. Introduction and model

neously. In its turn this voltage changes the phase difference Jn. This means that oscillations are possible in the system.

It is well known that in long Josephson j u n c t i o n s the p r o p a g a t i o n o f waves is possible. See, for example, the textbook by A b r i k o s o v [ 1 ]. However, there exists a n o t h e r system where one can observe waves - the Josephson arrays. Recently the technology o f p r o d u c i n g highly periodical Josephson networks has been d e v e l o p e d [2,3]. In a p a p e r by M i s h o n o v a n d G r o s h e v [4 ], p l a s m a waves in t w o - d i m e n s i o n a l arrays are predicted. The aim o f the present paper is the investigation o f plasma waves in a o n e - d i m e n s i o n a l Josephson chain. Let us consider an infinite periodical chain o f sup e r c o n d u c t i n g grains connected by identical Josephson j u n c t i o n s (see fig. 1 ). The phase o f the wave function o f the C o o p e r pairs On d e p e n d s on the position n u m b e r n. The phase difference ~ . = 0 n - 0 . + ~ causes a Josephson current between neighbouring grains. The current changes the charges o f grains and a voltage between grains On--On+~ arises simulta-

~r'L-- t

Ir~-t

2. Wave equation The d y n a m i c s o f Josephson j u n c t i o n s are described by the well-known equations [ 1 ] I , = I c s i n Jn + (On - ~ , + ~)/R ,

(1)

dJ, = 2 e / h ( O , -0~+ l ) dt

(2)

where Ic is the critical current, R the o h m i c resistance o f a Josephson junction, 2e the charge o f a C o o p e r pair a n d h is Planck's constant. The potentials o f the grains are connected with their charges by the equation +or

(~n =

~

enmqm ,

(3)

m = --oo

69

I

I't

T't

Ctr~- I ~rt-1

8~+ i qr~+t ¢ ' n + l

¢3g <

)

Fig. 1. Periodic Josephson chain. Here 0. denotes the order parameter phase, qn the charge and 0, the electric potential of the nth grain, and a is the lattice period. 0921-4534/90/$03.50 © Elsevier Science Publishers B.V. ( North-Holland )

M. V. Simkin / Plasma wavesin a Josephson chain

280

where P~m=P( In - m l ) is the potential matrix of the system [5 ]. The change of the grain charges is connected with the currents between them by the equation

X [12 -121 ,/a p(k).},/2 2sin(ka/2), +oo

P(k)=P(O)+2 ~ P(m)coskma.

(7)

m=l

dqm =I.,- l -Ira. dt

(4)

By differentiating eq. (2) with respect to time and expressing with the help of eqs. ( 1 - 4 ) all necessary variables through ~ we obtain the wave equation

d2~n

2e +~ dt 2 - h lc m ~ ~--oo P ( L n - m l )

X ~ (~,,-1 +6m+~ - 2 ~ m )

Here P ( k ) is the Fourier transformation of P(n) and Coo(k) is the dispersion relation obtained neglecting ohmic dissipation. The necessary condition for the existence of a wave is that the imaginary part of the frequency should be much less than its real part,

hCoo(k) << 4eR(I 2 - I 2) 1/2

X {sin 8m_ ~+sin 8m+l - 2sin

(~m d l - h / ( 2 e l c R

)

•

(5)

If the chain is placed on a metallic substrate then P ( I n l ) rapidly decreases when n increases. In this case we can take into account P ( 0 ) only in eq. (5). Neglecting also terms with first time derivatives describing the ohmic dissipation we obtain d2~n 2e dt 2 IcP(O)[sin3~+t+sin6~_~-2sin3~].(6) This equation is similar to the completely integrable equation for the Toda lattice [ 6 ]. The question arises of whether eq. (6) is integrable.

3. Small amplitude waves Let us linearize eq. (5), considering

C~n(t)=60+dn(t),

Coo(k) = { ( 2 e / h )

3n << 1 .

30 is determined by the constant current Io=Ic sin 80 flowing through the chain. The linearized equation has running wave solutions

An =3 exp{ i ( kna - Cot) } , where Co denotes frequency, k the wave vector and a the distance between grain centers. The dispersion relation for these waves is

co(k) =coo(k) [ 1 - ihcoo ( k ) / ( 4 e g [I~ - I 2 ] l/E) ] ,

(8)

P ( n ) is the potential created by the charge of the nth grain on the Oth grain. When n >> 1, polarization of grains may be neglected because the dipole potential is proportional to 1/n 2 and P is the Coulomb potential,

P(n)=l/na.

(9)

When ka is small ( l n ( l / k a ) > > 1 ) then the P ( n ) with n >> 1 are most important in eq. (7). Substituting eq. (9) into eq. (7) we obtain Co(k) = (2/L)1/2 k[ln( 1/ka)

1 '/2 ,

(10)

where L = (h/2ea) / (12 - 1 2 )1/2 is the Josephson inductance per unit length. The dispersion relation of long wave plasmons in a thin superconducting wire or a narrow superconducting strip has the same form. In this case L = 4n22/(c2S) is the kinetic inductance per unit length, 2 is the London penetration depth, c is the light velocity and S is the wire (strip) section area. Das Sarma and Lai [ 7 ] obtained a similar dispersion relation for l D plasmons in a semiconductor heterostructure. In the last case L=m/e2n~D, where t/1D is the number of electrons per unit length and m is the electron effective mass. If the chain is placed on a metallic substrate then P ( n ) is proportional to 1/n3 because of screening. At small ka the Fourier transformation of this potential is approximately constant and, as we can see from eq. (7), the dispersion relation is linear. However, there are more limitations than eq. (8). Obtaining the group velocity vgr=dCo/dk from eq. (l 0), we see that Vgr increases ad infinitum when k goes to 0. This is because in eqs. (1) and (3) the retardation of the electromagnetic field is neglected.

M. V. Simkdn / Plasma waves in a Josephson chain

So we can use eq. (l 0) only if the group velocity is much lower than the light velocity c: d t o / d k << c .

( 11 )

Notice that oscillations with a frequency comparable to A / h , where d is the coupling energy of Cooper pairs, destroy them. This causes wave attenuation. So we have another limitation, og<
281

Acknowledgements The author wishes to thank T.M. Mishonov for critical reading of the manuscript and his interest in present work; A.V. Vagov for useful discussions and A.T. Filippov for pointing out the similarity between eq. (6) and the Toda lattice.

(12)

Note that for tunneling contacts the A m b e g a o k a r Baratov equation [ 1 ], Ic = n J / ( 2 e R )

exists. In this case limitations (8) and ( 12 ) coincide. The calculations show that in real systems the fulfillment o f the criteria o f eqs. (8), ( 11 ) and (12) may be achieved. For example, for the array o f N b tunnel junctions described by Webb and Woss [ 3 ] with a = 1 0 lam and I c = l nA: h o g / z l < l / 3 0 at all k and eq. ( 11 ) is fulfilled at I n ( l / k a ) < 105. To excite these waves we can use the well-known method for low-dimensional plasmon excitation (see Allen et al. [ 8 ] ).

References [ 1] A.A. Abrikosov, Fundamentals of the Theory of Metals (North-Holland, Amsterdam, 1988) chap. 22. 12 ] J.M. Mooij and G.B.J. Sch6n (eds.), Proc. on Coherence in Superconducting Networks, Physica B 152 ( 1988). [3] R.F. Voss and R.A. Webb, Phys. Rev. B25 (1982) 3446. [4 ] T.M. Mishonov and A.G. Groshev, JINR Report El 7-89-752. [5] L.D. Landau and E.M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York). [6] M. Toda, Theory of Nonlinear Lattices (Springer, Berlin, 1981). [7] S. Das Sarma and W.Y. Lai, Phys. Rev. B32 (1985) 1401. [8] S.J. Allen Jr., D.C. Tsui and R.A. Logan, Phys. Rev. Lett. 35 (1977) 980.