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The characteristic behavior of the inductance and capacitance of
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multi junction ac Josephson effect in superconductor has been presented. Few parameters characterizing the behavior of Josephson junctions are needed to evaluate for technological applications. In this paper, the inductance and capacitance of the multi Josephson junction connected in parallel are evaluated, using simple classical argument. The numerical results for inductance and capacitance have also been included, indicating few technological applications.

The Josephson effect [

A model of the multi junction ac Josephson effect in superconductor has been proposed [

Josephson [

I = I 0 sin ϕ ( t ) , (1)

where I 0 is the maximum supercurrent and ϕ ( t ) is the time dependent phase difference between the superconductors and follow the relation

d ϕ / d t = 2 e V / ℏ = ω J .

Let us first consider two Josephson junctions connected in parallel with an applied voltage V, as shown

Now, the mathematical expressions for current I_{1} and I_{2} flowing through the Josephson junctions 1 and 2, can be written as [

I 1 = I 01 e i ( ω J t + ϕ ) (2)

and

I 2 = I 02 e i ( ω J t + 2 ϕ ) (3)

where, I_{01} and I_{02} be the maximum current flowing through the junctions 1 and 2, respectively; ϕ and 2 ϕ are arbitrarily chosen for the time independent phase differences across the junctions 1 and 2, respectively.

The resultant current can be written as

I = I 01 e i ( ω J t + ϕ ) + I 02 e i ( ω J t + 2 ϕ ) = ( I 01 e i ϕ + I 02 e 2 i ϕ ) e i ω J t (4)

Considering identical junction ( I 01 = I 02 = I 0 ) and after some mathematical exercises, the resultant current due to ac Josephson effect for two junctions can be expressed as:

I = I 0 e i ω J t ( e i ϕ + e 2 i ϕ ) = 2 I 0 cos ( ϕ / 2 ) sin ( ω J t + 3 ϕ / 2 ) (5)

For convenience, we have taken imaginary part of the resultant current. The above formulation can be extended for more than two junctions. Let us proceed for N number of identical Josephson junctions; the resultant current can be written as

I = I 0 [ e i ( ω J t + ϕ ) + e i ( ω J t + 2 ϕ ) + ⋯ + e i ( ω J t + N ϕ ) ] (6)

Using some mathematical calculations, we have

I = I 0 sin ( N ϕ / 2 ) sin ( ϕ / 2 ) sin ( ω J t + ( N + 1 ) ϕ / 2 ) (7)

This is the resultant current for multi junction ac Josephson effect in superconductor. It can be noted that for N = 2, the Equation (7) resembles to Equation (5) for two identical Josephson junctions in parallel connection, which supports our assumption.

Let us start with a single junction in presence of applied voltage V, an alternating supercurrent of frequency ω J flows through the junction which is given by

I = I 0 e i ( ω J t + ϕ ) , (8)

where the frequency of alternating supercurrent is ω J = ( 2 e V / ℏ ) and ϕ is the phase difference, independent of time. Differentiating above current equation with respect to time and using the classical argument V = L ( d I / d t ) ; we have

V = L ( i ω J ) I 0 e i ( ω J t + ϕ ) (9)

Putting ω J = 2 e V / ℏ for single junction and taking imaginary parts as before, we have

L J = L 0 cos ( ω J t + ϕ ) , (10)

where L 0 = ( ℏ / 2 e I 0 ) is called the parametric inductance.

Therefore, the inductance of an ac Josephson junction is a periodic function of phase and frequency.

Now we proceed for the calculation of inductance due to 2 Josephson junctions connected in parallel (as shown in

The corresponding inductance is found to be

L J = L 0 2 cos ( ϕ / 2 ) cos ( ω J t + 3 ϕ / 2 ) . (11)

In a similar way, we proceed for N number of identical Josephson junction, the result is as follows

L J = L 0 cos ( ω J t + ( N + 1 ) ϕ / 2 ) sin ( N ϕ / 2 ) sin ( ϕ / 2 ) . (12)

This is the mathematical expression for inductance of the multi Josephson junction connected in parallel.

The concept of capacitance for Josephson junction is analogously defined as the capacitance of a parallel plate capacitor. A Josephson junction consists of two superconductors separated by an insulating layer, the layer may be oxide or dielectric material. Any two adjacent conductors can function as a capacitor if the charges on the conductors are +q (for hole pair) and −q (for electron pair). The applied voltage between the conductors gives capacitance, written as

C = I ( d V d t ) − 1 , (13)

where the supercurrent flowing through the Josephson junction is given in Equation (8).

Compare to the ordinary frequency of oscillation ω = 1 / L C and using the inductance L 0 = L ≡ ( ℏ / 2 e I 0 ) , we have

V = ( I 0 ℏ 2 e C ) . (14)

Using ( C ) − 1 ∫ I d t = V and I = I 0 e i ( ω J t + ϕ ) for single Josephson junction, the capacitance is found to be

C J = 2 e I 0 ℏ ω J 2 cos 2 ( ω J t + ϕ ) . (15)

For 2 junctions connected in parallel (shown in

C J = 8 e I 0 ℏ ω J 2 cos 2 ϕ 2 cos 2 ( ω J t + 3 ϕ 2 ) . (16)

In a similar way, for N identical Josephson junction connected in parallel, the resultant capacitance is found to be

C J = 2 e I 0 ℏ ω J 2 sin 2 ( N ϕ / 2 ) sin 2 ( ϕ / 2 ) cos 2 ( ω J t + ( N + 1 ) ϕ / 2 ) . (17)

This is the expression for capacitance of the multi junction ac Josephson effect in superconductor.

The numerical work has been carried out for the sample HgTe based Josephson junction [

2, 3, 4). Here, we see that the capacitance increases with the increasing number of junctions and decreases with frequency. The result is comparable with the experimental results discussed in references [

with phase for the different number of junctions (N=1, 2, 3, 4). Here, we see that the amplitude of the capacitance increases with the increasing number of junction and vary periodically. The periodic variations of the capacitance indicate that the multi junction can be used as a local oscillator.

We have studied the model of multi Josephson junction in superconductivity. The theoretical investigation based on numerical analysis of the characteristic parameters of multi Josephson junction indicates few technological advantages, like high frequency oscillator, interferometer, and qubit etc. already mentioned before. This is the first time we have evaluated the inductance and capacitance of the multi Josephson junction using simple classical argument. Addition of the multi Josephson junction model gives kinetic inductance which is highly nonlinear. The periodic variation of the kinetic inductance, i.e., positive and negative inductance indicates the simultaneous energy received and released. It is well known that the nonlinearity of the Josephson junction breaks the degeneracy of the energy states, and picks up only the two qubit states simultaneously [

The authors are grateful to the authority of Chittagong University of Engineering and Technology (CUET), Chittagong 4349, Bangladesh for all sorts of support during this work.

The authors declare no conflicts of interest regarding the publication of this paper.

Maruf, H.M.A.R., Islam, M.R. and Chowdhury, F.-U.-Z. (2018) Inductance and Capacitance of the Multi Josephson Junction in Superconductor. Journal of Applied Mathematics and Physics, 6, 2544-2552. https://doi.org/10.4236/jamp.2018.612212