# Influence of Frustration Effects on the Critical Current of DC SQUID

^{1}

^{2}

^{3}

## Abstract

**:**

## 1. Introduction

## 2. Basic Equations

_{1}and I

_{2}are the currents on the left and right sides of the ring, where Φ

_{0}is the quantum of magnetic flux and Φ

_{e}is external magnetic flux. ϕ

_{e}= 2π Φ

_{e}/Φ

_{0}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 = L

_{1}+ L

_{2}) and with the sinusoidal current–phase relation is equivalent to a single Josephson junction with effective critical current

_{1}and ϕ

_{2}are the phases of junctions on the left and right sides of the ring.

_{c}

_{1}and I

_{c}

_{2}are the critical currents of the junctions in DC SQUID (Figure 1).

_{c.n}= Ψ

_{0}Ψ

_{n}. The Ginzburg–Landau free energy functional of the multiband character of the superconducting state can be written as [16,17,18]:

_{i}is the superconducting order parameter in the superconductor Ψ∞ exp(iϕ),

**A**is the vector potential of magnetic field

**H**= rot

**A**, 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 ε

_{1}

^{ij}= ε

_{1}

^{ji}describe the interaction between order parameters and their gradients in different bands, respectively. H is the magnetic-field-applied superconductor and Φ

_{0}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]:

_{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:

_{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:

## 3. Results

_{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:

_{c}

_{1}= 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

_{c}

_{2}is close to 1, the sensitivity of DC SQUID to the external magnetic field becomes very small.

_{c}

_{2}versus i

_{c}

_{3}. The minimum in the dependence of i

_{m}(i

_{c}

_{3}) at a small i

_{c}

_{2}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.

## 4. Conclusions

_{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.

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Critical current of the DC SQUID with small inductance of l for SB/two-band junction as function of normalized current in second channel of SB/two-band junction.

**Figure 3.**Modulation coefficient of the DC SQUID with small inductance of l for SB/two-band junction as function of normalized current in second channel of SB/two-band junction.

**Figure 4.**Critical current of DC SQUID with small inductance of l for SB/three band junction as function of normalized current in the third channel of SB/three-band junction for different values of i

_{c}

_{2}.

**Figure 5.**Modulation coefficient of the DC SQUID with small inductance of l for SB/three band junction case as function of normalized current in the third channel of SB/three-band junction for different values of i

_{c}

_{2}.

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Askerzade, I.N.
Influence of Frustration Effects on the Critical Current of DC SQUID. *Condens. Matter* **2023**, *8*, 65.
https://doi.org/10.3390/condmat8030065

**AMA Style**

Askerzade IN.
Influence of Frustration Effects on the Critical Current of DC SQUID. *Condensed Matter*. 2023; 8(3):65.
https://doi.org/10.3390/condmat8030065

**Chicago/Turabian Style**

Askerzade, Iman N.
2023. "Influence of Frustration Effects on the Critical Current of DC SQUID" *Condensed Matter* 8, no. 3: 65.
https://doi.org/10.3390/condmat8030065