Spin Ordering in Low-Dimensional Magnetic System Induced by Model Interaction

Abstract

The magnetic susceptibility has been studied by using a new trial function for the model interaction between spins arranged in a low-dimensional square pyramidal system. We suggest that the exchange interaction depends on a power exponent which depends on the pyramidal structure in addition to the magnetic energy and thermal energy. We have calculated the magnetization as a function of the temperature and magnetic field keeping the exponent constant. Dependence of the susceptibility on a characteristic length related to the low-dimensional system exhibits novel features. Magnetic field dependence of the susceptibility has a maximum which shifts toward higher field as the temperature is increased. Even at higher fields, the susceptibility can be very small. The variations of the susceptibility with temperature exhibit maximum which depends on the external perturbing field.

Appendix 1

The Hamiltonian of the system can be written as follows:

$$ \mathcal{H}=-\frac{1}{2}\left[\sum\limits_{i=1}^{4}{J_{ij}\left( \textbf{S}_{i}.\textbf{S}_{j}\right)}+\sum\limits_{j_{i}=1}^{4}{J_{j_{i}j_{j}}.\left( \textbf{S}_{j_{i}}.\textbf{S}_{j_{j}}\right)}\right]-\textbf{H}.\left( \textbf{S}_{i}+{\sum}_{j=1}^{4}{\textbf{S}_{j}}\right) $$

(6)

Here, S_i is the apex spin. S_j’s are the spins on the plane.

$$ J_{ij}=J_{0}\left[1+{\Gamma}\left( \frac{a}{r_{ij}}\right)^{n}\coth{\left( \frac{k_{B}T}{\mu_{B}H}\right)}\right] $$

(7)

For $J_{j_{i}j_{j}}$, d = 0 that is r_ij = a. From the geometry of the model system, the expression of r_ij is:

$$ r_{ij}=\frac{1}{\sqrt{2}}\sqrt{a^{2}+2d^{2}} $$

(8)

Magnetization of the model system can be written as:

$$ M=\left<S_{i}\right>+4\left<S_{j}\right> $$

(9)

Applying mean field theory, we can write:

$$ \left<S_{i}\right>=\frac{tr S_{i}e^{\upbeta S_{i}\mathcal{H}_{eff}^{i}}}{tr e^{\upbeta S_{i}\mathcal{H}_{eff}^{i}}}=\frac{1}{2}\tanh{\left( \frac{1}{2k_{B}T}\mathcal{H}_{eff}^{i}\right)} $$

(10)

Where, ${\mathscr{H}}_{eff}^{i}=2J_{ij}\left <S_{j}\right >+H$. Putting the value of ${\mathscr{H}}_{eff}^{i}$ into the expression, we can obtain the expression of $\left <S_{i}\right >$, which is given as follows:

$$ \left<S_{i}\right>=\frac{1}{2}\tanh{\left[\frac{1}{2k_{B}T}\left( 2J_{ij}\left<S_{j}\right>+H\right)\right]} $$

(11)

Similarly, the expression of $\left <S_{j}\right >$ is:

$$ \left<S_{j}\right>=\tanh{\left[\frac{1}{k_{B}T}\left( 2J_{ij}\left<S_{i}\right>+J_{j_{i}j_{j}}\left<S_{j}\right>+4H\right)\right]} $$

(12)

Adding $\left <S_{i}\right >$ and $4\left <S_{j}\right >$, we have the expression of magnetization M:

$$ \begin{array}{@{}rcl@{}} M&=&\frac{1}{2}\tanh{\left[\frac{1}{2k_{B}T}\left( 2J_{ij}\left<S_{j}\right>+H\right)\right]}\\ &&+4\tanh{\left[\frac{1}{k_{B}T}\left( 2J_{ij}\left<S_{i}\right>+J_{j_{i}j_{j}}\left<S_{j}\right>+4H\right)\right]} \end{array} $$

(13)

The susceptibility is defined as follows:

$$ \chi=\left( \frac{\partial M}{\partial H}\right)_{T} $$

(14)

Now putting the expression of M following equation, we have:

$$ \chi=\frac{\partial\left<S_{i}\right>}{\partial H}+4\frac{\partial\left<S_{j}\right>}{\partial H} $$

(15)

Using derivative $\frac {\partial \left <S_{i}\right >}{\partial H}$, we have:

$$ \frac{\partial\left<S_{i}\right>}{\partial H}=\frac{1}{4T}\textup{sech}^{2}{\left[\frac{2J_{ij}\left<S_{j}\right>+H}{2T}\right]}\left( 1+2J_{ij}^{\prime}\left<S_{j}\right>+2J_{ij}\frac{\partial\left<S_{j}\right>}{\partial H}\right) $$

(16)

Similarly,

$$ \frac{\partial\left<S_{j}\right>}{\partial H}=\frac{4+J_{j_{i}j_{j}}^{\prime}\left<S_{j}\right>+2J_{ij}^{\prime}\left<S_{i}\right>+2J_{ij}\frac{\partial\left<S_{i}\right>}{\partial H}}{T\cosh^{2}{\left( \frac{4H+J_{j_{i}j_{j}}\left<S_{j}\right>+2J_{ij}\left<S_{i}\right>}{T}\right)-J_{j_{i}j_{j}}}} $$

(17)

Here,

$$ J_{j_{i}j_{j}}=J_{0}\left[1+{\Gamma}\coth{\left( \frac{k_{B}T}{\mu_{B}H}\right)}\right] $$

(18)

$$ J_{ij}^{\prime}=\left( \frac{a}{r_{ij}}\right)^{n}\frac{k_{B}T}{\mu_{B}H^{2}}\textup{cosech}^{2}{\left[\frac{k_{B}T}{\mu_{B}H}\right]} $$

(19)

$$ J_{j_{i}j_{j}}^{\prime}=\frac{k_{B}T}{\mu_{B}H^{2}}\textup{cosech}^{2}{\left[\frac{k_{B}T}{\mu_{B}H}\right]} $$

(20)

The final expression of the susceptibility $\left (\chi \right )$ is given as follows:

$$ \begin{array}{@{}rcl@{}} \chi=\frac{1}{4T}\textup{sech}^{2}{\left[\!\frac{2J_{ij}\left<S_{j}\right>+H}{2T}\!\right]}\!\left( \!1+2J_{ij}^{\prime}\left<S_{j}\right>+2J_{ij}\frac{\partial\left<S_{j}\right>}{\partial H}\!\right)\\ +4\frac{4+J_{j_{i}j_{j}}^{\prime}\left<S_{j}\right>+2J_{ij}^{\prime}\left<S_{i}\right>+2J_{ij}\frac{\partial\left<S_{i}\right>}{\partial H}}{T\cosh^{2}{\left( \frac{4H+J_{j_{i}j_{j}}\left<S_{j}\right>+2J_{ij}\left<S_{i}\right>}{T}\right)-J_{j_{i}j_{j}}}} \end{array} $$

(21)