Influence of Frustration Effects on the Critical Current of DC SQUID

Abstract

In this paper, we conducted the calculation of the critical current of DC SQUID based on the Josephson junction on a multi-band superconductor with frustration effect. It is shown that the critical current of DC SQUID on the frustrated multi-band superconductor with a small geometrical inductance of the loop is determined by the supercurrent amplitude in different channels and by the external magnetic field. In the case of a DC SQUID with high inductance, frustration effects can be ignored.

1. Introduction

A Direct Current (DC) Superconducting Quantum Interference Device (SQUID) consists of two Josephson junctions including a superconducting loop in parallel (Figure 1). The foundation of DC SQUID on low-temperature superconductor-based Josephson junctions is presented in Refs. [1,2]. In the calculation of DC SQUID characteristics and dynamical effects, the sinusoidal current–phase relation of Josephson junctions [1,2] was used. For low-temperature superconductor-based junctions, the relationship $I=I_{c0}\sin \varphi$ was fulfilled with high accuracy [3]. In the case of Josephson junctions between single- and multi-band compounds, the dynamics were influenced by the frustration effects. The result of frustration effects on the characteristics of systems with Josephson junctions should be taken into account [4,5]. In particular, the presence of a frustrated ground state in multi-band superconductors causes $\varphi$-junction peculiarity [4]. The frustration effects in many-band superconductors and Josephson junctions based on them are described in several papers [6,7,8,9,10,11]. The influence of frustration effects in multiband superconductors on the escape rate in Josephson junctions was considered in Refs. [12,13]. The theoretical analysis of the escape rate for the AC SQUID based on the junction with a non-sinusoidal current–phase relation was analyzed in Ref. [14].

In this study, we carried out a calculation of the critical current of the DC SQUID on the junction based on many-band superconductors with frustration effects.

2. Basic Equations

It is well known that the dynamics of DC SQUID in general cases can be described by the system of equations.

$$I_1+I_2=I_e,$$

(1)

$$\mathsf{\Phi }={\mathsf{\Phi }}_e-L_1{I}_1+L_2{I}_2,$$

(2)

$$\varphi ={\varphi }_e-\frac{2\pi L_{+}{I}_L}{{\mathsf{\Phi }}_0}; I_L=(L_1{I}_1-L_{2_{}}{I}_2)/(L_1+L_2).$$

(3)

In Equation (1), I₁ and I₂ are the currents on the left and right sides of the ring, where Φ₀ is the quantum of magnetic flux and Φ_e is external magnetic flux. ϕ_e = 2π Φ_e/Φ₀ is the normalized external flux. It is also well known that [1], in the case of DC SQUID, the small total inductance of the loop is $L_1{I}_1,L_2{I}_2<<{\mathsf{\Phi }}_0$, $l=2\pi \frac{L{I}_c}{{\mathsf{\Phi }}_0}<<1$ (total inductance is the sum of left and right inductances L = L₁ + L₂) and with the sinusoidal current–phase relation is equivalent to a single Josephson junction with effective critical current

$$I_e=I_m\sin \chi$$

(4)

and with effective phase

$$\chi ={\varphi }_1+\eta ={\varphi }_2+\eta -{\varphi }_e,$$

(5)

where ϕ₁ and ϕ₂ are the phases of junctions on the left and right sides of the ring.

$$\eta =-\frac{2I_{c1}\tan ({\varphi }_e/2)}{I_{c1}+I_{c2}+(I_{c1}-I_{c2}){\tan }^2({\varphi }_e/2)}.$$

(6)

In Equation (3), the effective critical current can be calculated as

$$I_m{}^2=I_{c1}^2+I_{c2}^2+2I_{c1}{I}_{c2}\cos {\varphi }_e.$$

(7)

where I_c1 and I_c2 are the critical currents of the junctions in DC SQUID (Figure 1).

For the study of frustration effects in DC SQUID, we considered that the left junction had a single-/single-band and the right junction a single-/multi-band character (see Figure 1). For the Josephson junctions between single- and multi-band superconductors (in single-band/single-band case, $I=I_c\sin \chi$), the supercurrent was the sum of different tunneling channel currents [14,15]:

$$I=I_{c1}\sin \chi +I_{c2}^{}\sin (\chi +\varphi )+I_{c3}\sin (\chi +\theta )+\dots ,$$

(8)

where $I_{c1,2,3},\dots$ are the different channel critical currents and $\varphi$, $\theta$,…. are the phase differences between order parameters in a frustrated state of the multi-band superconductor. In the single-band superconductor with the zeroes phase, we have ${\mathsf{\Psi }}_0=\left|{\mathsf{\Psi }}_0\right|\exp (0)$. For the multi-band superconductor, the following expressions are true: ${\mathsf{\Psi }}_1=\left|{\mathsf{\Psi }}_1\right|\exp (\chi )$, ${\mathsf{\Psi }}_2=\left|{\mathsf{\Psi }}_2\right|\exp (\chi +\varphi )$, ${\mathsf{\Psi }}_3=\left|{\mathsf{\Psi }}_3\right|\exp (\chi +\theta )$, …… The critical currents in the different channels $I_{c1,2,3},\dots$ in Equation (8) are proportional to the product of amplitude of wave functions above, i.e., I_c.n = Ψ₀Ψ_n. The Ginzburg–Landau free energy functional of the multiband character of the superconducting state can be written as [16,17,18]:

$$F={\displaystyle \int d^{3}r({\displaystyle \sum _{ij}(F_{ii}-F_{ij}+\frac{H^2}{8\pi }}}),$$

(9)

where

$$F_{ii}=\frac{{\hslash }^2}{4m_i}{\left|\left(\nabla -\frac{2\pi i\stackrel{\rightharpoonup }{A}}{{\mathsf{\Phi }}_0}\right){\mathsf{\Psi }}_i\right|}^2+{\alpha }_i(T){\left|{\mathsf{\Psi }}_i^{}\right|}^2+{\beta }_i{\left|{\mathsf{\Psi }}_i^{}\right|}^4/2,$$

(10)

$$F_{ij}={\epsilon }_{ij}({\mathsf{\Psi }}_i^{*}{\mathsf{\Psi }}_j+c.c.)+{\epsilon }_1^{ij}\left\{\left.\left(\nabla +\frac{2\pi i\overrightarrow{A}}{{\mathsf{\Phi }}_0}\right){\mathsf{\Psi }}_i^{*}\left(\nabla -\frac{2\pi i\overrightarrow{A}}{{\mathsf{\Phi }}_0}\right){\mathsf{\Psi }}_j+c.c.\right\}\right.$$

(11)

where Ψ_i is the superconducting order parameter in the superconductor Ψ∞ exp(iϕ), A is the vector potential of magnetic field H = rotA, m_i are the effective masses of the electrons in different bands, (i = 1–3); α_i= γ_i(T − T_ci) are the quantities linearly dependent on temperature T; β_i and γ_i are constants; and ε_ij = ε_ji and ε₁^ij = ε₁^ji describe the interaction between order parameters and their gradients in different bands, respectively. H is the magnetic-field-applied superconductor and Φ₀ is the magnetic flux quantum. In the case of single- and two-band junctions, for the phase differences $\varphi$ of order parameters, we can find the effective critical current as [15]:

$$I_{ceff}=(I_{c1}+I_{c2})\hspace{1em}\hspace{1em}\mathrm{for}\hspace{1em}\hspace{1em}\varphi =0,$$

(12a)

$$I_{ceff}=(I_{c1}-I_{c2})\hspace{1em}\hspace{1em}\mathrm{for}\hspace{1em}\hspace{1em}\varphi =\pi .$$

(12b)

For single-/three-band junctions, in the case of an identical and positive interband interaction term ε_ij = ε_ji = ε > 0, one of the phase differences will be zero and other phase differences in frustration states are given as $\left(\begin{array}{l}\varphi \\ \theta \end{array}\right)=\left(\begin{array}{l}2\pi /3\\ -2\pi /3\end{array}\right)$ and $\left(\begin{array}{l}\varphi \\ \theta \end{array}\right)=\left(\begin{array}{l}-2\pi /3\\ 2\pi /3\end{array}\right)$ [15]. Another frustration state corresponds to phase differences $\left(\begin{array}{l}\varphi \\ \theta \end{array}\right)=\left(\begin{array}{l}0\\ \pi \end{array}\right);\left(\begin{array}{l}\pi \\ 0\end{array}\right)$ and $\left(\begin{array}{l}\varphi \\ \theta \end{array}\right)=\left(\begin{array}{l}\pi \\ \pi \end{array}\right)$. From the expression for potential energy for single-/three-band junctions under an external current $I_e=I_{e1}+I_{e2}+I_{e3}$, we can obtain the following for effective critical current:

$$I_{ceff}=I_{c1}{\left({(1-\frac{I_{c2}}{2I_{c1}}-\frac{I_{c3}}{2I_{c1}})}^2+{(\frac{I_{c3}}{I_{c1}}-\frac{I_{c2}}{I_{c1}})}^2\right)}^{1/2}$$

(13)

In the derivation of Equation (13), we use the Josephson junction reveal $\phi$-junction peculiarity $I=I_{ceff}\sin (\varphi -\phi )$, with $\phi =\arctan \frac{I_{c3}-I_{c2}}{I_{c1}-\frac{I_{c2}}{2}-\frac{I_{c3}}{2}}$. In the other frustration state $\left(\begin{array}{l}\varphi \\ \theta \end{array}\right)=\left(\begin{array}{l}-2\pi /3\\ 2\pi /3\end{array}\right)$, the terms I_c2 and I_c3 in Equation (12) were replaced by the places. The frustration case $\left(\begin{array}{l}\varphi \\ \theta \end{array}\right)=\left(\begin{array}{l}0\\ \pi \end{array}\right)$ corresponded to the effective critical current:

$$I_{ceff}=(I_{c1}+I_{c2}-I_{c3})$$

(14a)

In the state $\left(\begin{array}{l}\varphi \\ \theta \end{array}\right)=\left(\begin{array}{l}\pi \\ \pi \end{array}\right)$ for effective critical current, the following expression is true:

$$I_{ceff}=(I_{c1}-I_{c2}-I_{c3})$$

(14b)

3. Results

The inclusion of frustration effects in the current–phase relation (See Equations (12) and (13)) leads to the renormalization of the critical current of DC SQUID I_m in comparison without similar effects. The normalized critical current of a DC SQUID with a frustrated Josephson junction is a maximum value of the superconducting current and can be written as:

$$i_m={(1+i_{c2}^2)}^{1/2}{(1+A\cos {\varphi }_e)}^{1/2}$$

(15a)

where the modulation function A is calculated as:

$$A=\frac{2i_{c2}}{(1+i_{c2}{}^2)}$$

(15b)

The calculated critical current $I_m$ of DC SQUID on the SB/two-band (SB-single band) Josephson junction with a frustrated state of 0 and $\pi$ using Equation (12) is presented in Figure 2. In calculations without a restriction of generality, we considered the critical current of the left junction to be I_c1 = 1. For the corresponding normalization of critical currents in different channels in SB/MB in Equations (12) and (13), we also used the same scale. The monotonic increasing and decreasing character of the critical current in 0 and $\pi$ cases is correspondingly clear. In Figure 3, we plotted the results of calculations of the modulation coefficient A in DC SQUID based on the SB/two-band Josephson junction. The changing of this parameter is clearly crucial in the frustrated $\pi$ case. When the high ratio parameter i_c2 is close to 1, the sensitivity of DC SQUID to the external magnetic field becomes very small.

In Figure 4, we present the results of calculations of the critical current of DC SQUID based on SB/three-band junctions for different ratios, i_c2 versus i_c3. The minimum in the dependence of i_m(i_c3) at a small i_c2 in contrast to the SB/two-band case is clear. Another important moment is related to the changing of the critical current in restricted regions close to 1. In Figure 5, the modulation coefficient A in DC SQUID on the SB/three-band Josephson junction is presented. It means that the sensitivity in the case of the DC SQUID on the SB/three-band Josephson junction is higher than in the SB/two-band case.

For the small values of geometrical inductance $l<<1$, the calculations show that in all cases of the current–phase relation the changing of the inductance of DC SQUID had a small impact on the presented results; see Figure 2, Figure 3, Figure 4 and Figure 5. For the high values of inductance of DC SQUID l >> 1, the Josephson inductance of junctions can be ignored in consideration of the dynamical effects [1]. As a result, the phase of Josephson junctions on the superconducting loop of DC SQUID (Figure 1) changes independently and in this limit is a true system of linear equations for currents [1]:

$$i_1=\frac{i_e}{2}+\frac{{\varphi }_e}{l}$$

(16a)

$$i_2=\frac{i_e}{2}-\frac{{\varphi }_e}{l}$$

(16b)

This means that in a DC SQUID with high geometrical inductance l >> 1, the frustrated effects in the current–phase relation can be neglected.

4. Conclusions

Finally, in this paper, the influence of the frustration effect in the many-band superconductor on the critical current I_m of DC SQUID was investigated. The renormalization of the critical current in such frustrated junctions under an external magnetic field was taken into account in the limit of the small geometrical inductance of DC SQUID. In the opposite case of high geometrical inductance (l >> 1), the influence of frustration effects in the current–phase relation was negligibly small.