Magnetic Properties of Diluted Magnetic Nanowire

Abstract

A magnetic diluted nanowire with cylindrical structure described by the Ising model is investigated. Using the effective field theory with a probability distribution technique, the influence of the dilution on the phase diagrams, susceptibility and the hysteresis loops are discussed in detail. Novel features are obtained for the thermal variations of longitudinal susceptibility and longitudinal magnetization. We have investigated the magnetic reversal of the system and have found the existence of triple hysteresis loops patterns, affected by the concentration of magnetic atoms, the temperature, and the exchange interaction between the core and the surface shell.

Appendix

For the nanowire center spin c ₁:

(23)

for the spins around the center:

(24)

for spins of type-1 surface shell nanowire:

(25)

for spins of type-2 surface shell nanowire:

(26)

Susceptibility of the nanowire:

$$ \left. \frac{\partial m_{c_{1}}^{z}}{\partial h} \right)_{h=0}=A_{1,1}\left. \frac{\partial m_{c_{1}}^{z}}{\partial h} \right)_{h=0}+A_{1,2}\left. \frac{\partial m_{c_{2}}^{z}}{\partial h} \right)_{h=0}+B_{1} $$

(27)

$$ \left. \frac{\partial m_{c_{2}}^{z}}{\partial h} \right)_{h=0}=A_{2,1}\left. \frac{\partial m_{c_{1}}^{z}}{\partial h} \right)_{h=0}+A_{2,2}\left. \frac{\partial m_{c_{2}}^{z}}{\partial h} \right)_{h=0}+A_{2,3}\left. \frac{\partial m_{s_{1}}^{z}}{\partial h} \right)_{h=0}+A_{2,4}\left. \frac{\partial m_{s_{2}}^{z}}{\partial h} \right)_{h=0}+B_{2} $$

(28)

$$ \left. \frac{\partial m_{s_{1}}^{z}}{\partial h} \right)_{h=0}=A_{3,2}\left. \frac{\partial m_{c_{2}}^{z}}{\partial h} \right)_{h=0}+A_{3,3}\left. \frac{\partial m_{s_{1}}^{z}}{\partial h} \right)_{h=0}+A_{3,4}\left. \frac{\partial m_{s_{2}}^{z}}{\partial h} \right)_{h=0}+B_{3} $$

(29)

$$ \left. \frac{\partial m_{s_{2}}^{z}}{\partial h} \right)_{h=0}=A_{4,2}\left. \frac{\partial m_{c_{2}}^{z}}{\partial h} \right)_{h=0}+A_{4,3}\left. \frac{\partial m_{s_{1}}^{z}}{\partial h} \right)_{h=0}+A_{4,4}\left. \frac{\partial m_{s_{2}}^{z}}{\partial h} \right)_{h=0}+B_{4}. $$

(30)

With N ₁=1, N ₂=2, N ₃=4, and N ₄=6 denote respectively the coordination numbers and $C_{k}^{l}$ are the binomial coefficient $C_{k}^{l}=\frac{l!}{k!(l-k)!}$.

The elements A _i,j and B _i are given by:

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$$ \frac{\partial f_{z}}{\partial h}\bigg)_{h=0}=\beta \bigl(1-\tanh^{2}(\beta y) \bigr). $$

(47)