Unique Phenomena in Transverse Ising Nanoislands

Abstract

The phase diagrams and magnetizations in four transverse Ising nanoislands with an antiferromagnetic configuration and a ferromagnetic spin configuration are examined by the use of the effective field theory with correlations. They exhibit the same behaviors for the magnetic properties. Some characteristic phenomena, such as the reentrant phenomena, are found in the magnetic properties. The phenomena come from the frustration induced by the finite size of the systems.

Appendices

Appendix 1

The longitudinal magnetizations of two nanoparticles shown in Fig. 1 are given by

$$ m=[\cos h(A)+m\sin h(A)]^{\mathrm{2}}F(x)\|{~}_{x = 0.0} $$

(13)

and

$$ \begin{array}{l} m_{1} =\left[ {\cos h\left( A \right)-m_{2} \sin h\left( A \right)} \right]^{2}F\left( x \right)\vert_{x = 0.0} \\ m_{2} =\left[ {\cos h\left( A \right)-m_{1} \sin h\left( A \right)} \right]^{2}F\left( x \right)\vert_{x = 0.0} \end{array} $$

(14)

These equations exactly reduce to

$$ m[1.0-F(2.0J)]= 0.0 $$

(15)

and

$$ m_{\mathrm{1}}m_{\mathrm{2}}[1.0-F(2.0J)][1.0+F(2.0J)]= 0.0 $$

(16)

When h ≠ 0.0, 1.0 > F(2.0J). Accordingly, the magnetizations must be m = 0.0 and m₁ = m₂ = 0.0, when T > 0.0. When h = 0.0, the function is given by F(2.0) = tanh(2.0t), so that the transition temperature (T_C) must be given by T_C = 0.0 K, in order to satisfy the relation of 1.0 = F(2.0J). Thus, the two-dimensional nanoparticles in Fig. 1 do not exhibit a finite magnetization at a finite temperature (m = 1.0 at T = 0.0 K and m = 0.0 for T > 0.0 K), as noted in Section 1.

Appendix 2

The longitudinal magnetizations of two nanoislands shown in Fig. 10 are given by

$$ \begin{array}{l} m_{1} =[\cos h\left( A \right)+m_{1} \sin h\left( A \right)\left] {{~}^{2}} \right[\cos h\left( B \right)-m_{1} \sin h\left( B \right)]F\left( x \right)\vert_{x = 0} \\ \text{and} \\ m_{2} =[\cos h\left( A \right)+m_{2} \sin h\left( A \right)\left] {{~}^{2}} \right[\cos h\left( B \right)-m_{1} \sin h\left( B \right)]F\left( x \right)\vert_{x = 0} \end{array} $$

(17)

for system C and

$$ \begin{array}{l} m_{1} =[\cos h\left( A \right)-m_{2} \sin h \left( A \right)\left] {{~}^{2}} \right[\cos h \left( B \right)-m_{2} \sin h \left( B \right)]F\left( x \right)\vert_{x = 0} \\ \text{and} \\ m_{2} =[\cos h \left( A \right)-m_{1} \sin h \left( A \right)\left] {{~}^{2}} \right[\cos h \left( B \right)-m_{1} \sin h \left( B \right)]F\left( x\right)\vert_{x = 0} \end{array} $$

(18)

for system D. Expanding the right hand side of coupled equations, we get

$$ \begin{array}{l} m_{1} = 2.0m_{1} K_{1} -m_{2} K_{2} -\left( {m_{1} } \right)^{2}m_{2} K_{3}\\ \text{and} \\ m_{2} = 2.0m_{2} K_{1} -m_{1} K_{2} -\left( {m_{1} } \right)^{2}m_{2} K_{3} \end{array} $$

(19)

for system C and

$$ \begin{array}{l} m_{1} =-\,m_{2} \left[ {2.0K_{1} +K_{2} } \right]-\left( {m_{2} } \right)^{3}K_{3} \\ \text{and} \\ m_{2} =-\,m_{1} \left[ {2.0K_{1} +K_{2} } \right]-\left( {m_{1} } \right)^{3}K_{3} \end{array} $$

(20)

for system D. Following the discussions in Appendix 1, the phase diagrams (or transition temperature) in the two systems (C and D) can be determined from the following relation:

$$ 2.0K_{\mathrm{1}}+K_{\mathrm{2}}= 1.0 $$

(21)

Equation (21) is nothing but (6) for systems A and B. In this way, the phase diagrams of systems C and D depicted in Fig. 10 are completely equivalent to those given in Section 3.

On the magnetizations of the two (C and D) systems, the results of system B given in Section 4 (Fig. 8) are also valid for these two systems, although, at first sight, the coupled equations ((19) and (20)) seem to be different from the coupled (7). In Fig. 11, the thermal variations of m_T (solid line), m₁ (dashed curve), and m₂ (dashed curve) are plotted for system C with h = 0.0 and r = 5.0, by solving the coupled (20) numerically. In fact, these results can be also obtained by solving the coupled (19) and (7). Furthermore, the behavior of m₁ in the figure is completely equivalent to that of m in Fig. 7 (or the curve labeled r = 5.0) for system A.