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where ˜ n = n + 1 $_{2}$, T$_{A}$ = ℏ · v$_{F}$ 2 πd · k$_{B}$ is Andreev temperature, M$_{0}$ = − c$_{3}$ · σ 2 · ( T T$_{A}$ ) · Φ ( T, h) · ∂ Φ( T, h) ∂ h , the ”phase” Φ ( T, h) is a solution of the differential equation (8), n$_{0}$ is the number of Andreev levels in the potential well ( n$_{0}$ ≈ Φ( T, h) π tan ( α$_{c}$ ) ). Eq. (19) shows that the mag- netic moment is diamagnetic in the range of small fields ( Φ (h) = const · h, const > 0 ) and allows for the contri- butions of ”particles” and ”holes”. 

Let us compare two approaches described above for the calculation of the magnetic moment of the NS structure. Fig.5 shows the function M $^{∗}$(Φ) and dependency of the current on the magnetic flux obtained in the Green’s function approach. For comparison, we obtained the de- pendence M $^{∗}$(Φ) at the same value α$_{c}$ = π/ 2 as was used in the derivation of the formula j (Φ) in [18]. In the initial part (linear in the Φ) both curves coincide. In this approximation (Φ (h) = const · h) the self-consistent equation (9) turns into Eq. (10). Thus, at small values of the magnetic field we would obtain the same field de- pendence of the magnetic moment M ( T, h) for the NS structure in both approaches. However, in large fields the behavior M ( T, h) is quite different. To calculate M ( T, h) from Eq. (9), we have used the following physi- cal values of the NS structure: R = 8 . 3 µ m, d = 3 . 2 µ m, ( α$_{c}$ = 36 $^{0}$), v$_{F}$ (Au) = 1 . 4 · 10 8 cm/s, Δ (Nb) = 1 . 42 meV ( σ = 0 . 644, η · c$_{3}$ = 5 . 3 10 $^{3}$, H$_{0}$ = 51A/m = 0 . 64 Oe ). The selected parameters are close to those used in the experiment [3, 4, 22]. 

The results of calculation according to formulas (15) and (18) are illustrated in Fig. 3, 4 . While plotting Fig. 3, the nonzero quantity Ω (Φ = 0) was omitted. The dependence M $^{∗}$(Φ) = − d Ω d Φ ( Fig. 4 ) crosses the ab- scissa thereby determining singular points of the differ- ential equation (8). The dependence Φ (h , T) calculated through numerical solution of the self-consistency equa- tion (8) exhibits jumps and is illustrated in Fig.6a for the branches corresponding to the minimum of the Gibbs free energy [23]: 

<img src='content_image/401377.jpg'>

where ⃗ B = rot ( ⃗ A ) , ⃗ H = ⃗ H (0 , 0 , H). The magnetic mo- ment M (h) and the free energy Ω (h) as functions of the magnetic field are shown in Fig. 7a, Fig. 6b. Each jump ΔΩ of the free energy (see Fig. 6b ) is accompa- nied by the jump of the magnetic moment Δ M (see Fig. 7a ) in such a way that the Gibbs free energy (20) is a continuous function of versus the magnetic field h. We have not performed an analysis of behavior Gibbs free energy (20) near the points where the magnetic moment has jumps because it is beyond the semi-classical approx- imation adopted in this article. 

## D. Numerical results 

The goal of our study was to interpret the experiments performed by A. C. Mota et al., [3, 4], who detected an anomalous behavior of the magnetic susceptibility of the NS structure in a weak magnetic field at millikelvin tem- peratures. Previously [8, 9], the anomalous behavior of the NS structure was attributed to the properties of the quantized Andreev levels depending on the magnetic flux that varies with the temperature and magnetic field. We used the ”thermodynamic” approach for the calculation of the magnetic moment of normal region of NS structure. Within the framework of the self-consistent equation (8), we have managed to trace the role of the parameter α$_{c}$ in thermodynamics of NS of structures ( Fig. 4 and Fig. 5 ). Failure connected with the use of the quasi-classical 

<img src='content_image/401370.jpg'>

## III. CONCLUSIONS 

