Perturbation of exchange interaction in Nd–Ni spin subsystem of nickelate

Abstract

A spin subsystem of a nano-sized NdNi$_8$ of a nickelate superconductor is chosen to calculate its contribution in controlling magnetic phase transition. The role of the spin distribution of NdNi$_8$ in controlling magnetization (M) is studied within the framework of the mean field theory (MFT). Two different levels of perturbation in the exchange interaction between spins of Ni and Nd have been used. M (H, T) has been calculated numerically for perturbations of 0, 0.15 and 0.30 with respect to an unperturbed $J_0$. Changes in M (H) in the range of T = 10.0–300.0 K have been found to be very sensitive to perturbation. Possibility of a magnetic phase transition has been found which is sensitive to the perturbation of the exchange integral. Sensitivity of the broadening of the phase transition is weaker for the level of perturbation in comparison with that for the magnetic field. Susceptibility is also calculated and analysed to understand how perturbed exchange integral affects magnetic phase transition in spin distribution of NdNi$_8$.

Appendix

Mean value of a thermodynamic quantity (f) can be written with the help of the partition function as follows: $\left<f\right>=\frac{{{\,\textrm{tr}\,}}{fe^{\beta {E}}}}{{{\,\textrm{tr}\,}}{e^{\beta {E}}}}$. Applying the MFT, we get $\left<S\right>$ as

$$\begin{aligned} \left<S\right>=\frac{{{\,\textrm{tr}\,}}{Se^{\beta S{\mathcal {H}}_{eff}}}}{{{\,\textrm{tr}\,}}{e^{\beta S{\mathcal {H}}_{eff}}}}=\vert S\vert \tanh {\beta S {\mathcal {H}}_{eff}}. \end{aligned}$$

(5)

We have ${\mathcal {H}}_{eff}$ given as follows:

$$\begin{aligned}{} & {} {\mathcal {H}}^{j}_{eff} = 2J \big ( \left<S_i\right> + \left<S_k\right>\big )+\mu _{B}H \\{} & {} {\mathcal {H}}^{k}_{eff} = 2J \left<S_j\right> +4\mu _{B}H \\{} & {} {\mathcal {H}}^{i}_{eff} = 2J \left<S_j\right> +4\mu _{B}H. \end{aligned}$$

Now substituting ${\mathcal {H}}_{eff}$ values in Eq. (5), we obtain the following expressions for $\left<S\right>$ :

$$\begin{aligned}{} & {} \left<S_j\right> = 2 \tanh \left[ \frac{2}{k_{B}T}\left( 2J\left<S_k\right>+2J\left<S_i\right>+\mu _{\text {B}} H\right) \right] \\{} & {} \left<S_k\right> = \tanh \left[ \frac{1}{k_{B}T}\left( 2J\left<S_j\right>+4\mu _{\text {B}} H\right) \right] \\{} & {} \left<S_i\right> = \tanh \left[ \frac{1}{k_{B}T}\left( 2J\left<S_j\right>+4\mu _{\text {B}} H\right) \right]. \end{aligned}$$

Total average spin ($\left<S\right>_{Tot}$) is taken as,

$$\begin{aligned} \left<S\right>_{Tot} = \left<S_j\right> + 4\left<S_k\right> + 4\left<S_i\right>. \end{aligned}$$

Magnetization (M) of the system is, $M = ng\mu _{\text {B}}\left<S\right>_{Tot}$

$$\begin{aligned} M= & {} (1/V)g\mu _{\text {B}} \left[ 2\tanh \left[ \frac{2}{k_{B}T}\left( 2J\left<S_k\right>+2J\left<S_i\right>+\mu _{\text {B}} H \right) \right] + 4 \tanh \left[ \frac{1}{k_{B}T}\left( 2J\left<S_j\right>+4\mu _{\text {B}} H \right) \right] + 4 \tanh \left[ \frac{1}{k_{B}T}\left( 2J\left<S_j\right>+4\mu _{\text {B}} H\right) \right] \right] \nonumber \\= & {} (1/V)g\mu _{\text {B}} \left[ 2\tanh \left[ \frac{2}{k_{B}T}\left( 2J\left<S_k\right>+2J\left<S_i\right>+\mu _{\text {B}} H \right) \right] + 8\tanh \left[ \frac{1}{k_{B}T}\left( 2J\left<S_j\right>+4\mu _{\text {B}} H \right) \right] \right] \end{aligned}$$

(6)

$n = N/V$, number of spins per unit volume, $g=2$ and $\mu _{\text {B}} = 9.27\times 10^{-24}$ J/T.

Susceptibility ($\chi$) has been calculated by the derivative of M(H).

$$\begin{aligned} \chi = \mu _{\text {0}}\left( \frac{\partial M}{\partial H} \right) _T = \mu _{\text {0}}(1/V)g\mu _{\text {B}}\left( \frac{\partial \left<S\right>_{Tot} }{\partial H}\right) _T. \end{aligned}$$

(7)

We obtain $\left<S\right>^{'}$, by taking the derivative with respect to H,

$$\begin{aligned} \frac{\partial \left<S_j\right>}{\partial H}= & {} \frac{4.0}{k_{B}T} \frac{\left[ 2J^{'}\big (\left<S_{k}\right>+\left<S_{i}\right>\big ) + 2J\big ( \left<S_{k}\right>^{'}+ \left<S_{i}\right>^{'}\big ) + \mu _{B} \right] }{\cosh ^2\left[ \frac{2}{k_{B}T}\big (2j\left<S_{k}\right>+2j\left<S_{i}\right> +\mu _{B}H \big )\right] } \end{aligned}$$

(8)

$$\begin{aligned} \frac{\partial \left<S_k\right>}{\partial H}= & {} \frac{\partial \left<S_i\right>}{\partial H}=\frac{1}{k_{B}T} \frac{\left[ 2J^{'}\left<S_j\right> + 2J\left<S_j\right>^{'} + 4\mu _{B} \right] }{\cosh ^{2} \left[ \frac{1}{k_{\text {B}}T} \left( 2J\left<S_{j}\right>+4\mu _{\text {B}}B \right) \right] }. \end{aligned}$$

(9)

Substituting (8) and (9) in (7), we have an expression for the susceptibility as follows:

$$\begin{aligned} \chi = \mu _{\text {0}}(1/V)g\mu _{\text {B}} \left[ \frac{2}{k_{B}T} \frac{\left[ 2J^{'}\left<S_j\right> + 2J\left<S_j\right>^{'} + 4\mu _{B} \right] }{\cosh ^{2} \left[ \frac{1}{k_{\text {B}}T} \left( 2J\left<S_{j}\right>+4\mu _{\text {B}}H \right) \right] } + \frac{4.0}{k_{B}T} \frac{\left[ 2J^{'}\big (\left<S_{k}\right>+\left<S_{i}\right>\big ) + 2J\big ( \left<S_{k}\right>^{'}+ \left<S_{i}\right>^{'}\big ) + \mu _{B} \right] }{\cosh ^2\left[ \frac{2}{k_{B}T}\big (2j\left<S_{k}\right>+2j\left<S_{i}\right>+\mu _{B}H \big )\right] } \right]. \end{aligned}$$

(10)