Magnetic and Magnetocaloric Effects in Systems with Reverse First-Order Transitions

Abstract

Reverse first-order magnetostructural phase transitions have been theoretically analyzed within the model of interacting parameters of magnetic and structural orders. A characteristic feature of these transitions is stepwise occurrence of the magnetic order upon cooling (as in the case of the first-order phase transition) and its smooth disappearance upon heating (as in the conventional second-order phase transition). These transitions are observed in some alloys of the Mn_1 – xCr_xNiGe magnetocaloric systems under pressure (x = 0.11) and without it (x = 0.18) and are accompanied by specific magnetic and magnetocaloric features. These specific features are phenomenologically described within the concept of soft mode for a structural subsystem undergoing the first-order structural phase transition (P6₃/mmc–Pnma) and the Heisenberg model for a spin subsystem. It is shown for systems with magnetostructural instability within the molecular-field approximation for a spin subsystem and the approximation of biased harmonic oscillator for a lattice subsystem that reverse phase transitions occur when the magnetic disordering temperature is in the range of temperature hysteresis of the P6₃/mmc–Pnma first-order structural phase transition. It is also shown that the two-peak form of isothermal entropy (characteristic of reverse transitions) is due to separation of contributions from the structural and magnetic entropies.

APPENDIX

The Hamiltonian of a spin subsystem in the presence of external magnetic field H₀ can be written as

$${{{\mathbf{\hat {H}}}}_{s}} = - \sum\limits_{nk,n'k'} {J_{{nn'}}^{{kk'}}{\mathbf{\hat {s}}}_{n}^{k}{\mathbf{\hat {s}}}_{{n'}}^{{k'}}} - 2{{\mu }_{0}}{{{\mathbf{H}}}_{0}}\sum\limits_{ni} {{\mathbf{\hat {s}}}_{n}^{k},} $$

(A.1)

where ${\mathbf{\hat {s}}}_{n}^{k}$ is the spin operator of atom k in hexagonal unit cell n; ${\mathbf{J}}_{{nn'}}^{{kk'}}$ ≡ $J(\left| {\Delta {\mathbf{R}}_{{nn'}}^{{kk'}}} \right|)$ are the corresponding integrals of exchange interaction between magnetoactive atoms at distances $\left| {\Delta {\mathbf{R}}_{{nn'}}^{{kk'}}} \right|$; $\sum\nolimits_n {{{N}_{0}}}$ and $\sum\nolimits_{n,k}$ = N = 2N₀(1 – x) are the number of unit cells and the number of magnetic active atoms (Mn) per unit volume; H₀ = (0,  0, H₀) is the vector of external uniform magnetic field; and μ₀ is the Bohr magneton.

For a simple helimagnetic (HM) with the wave vector k = (0, 0, k_a), angle $\Psi _{{nn'}}^{{kk'}}$ between the directions of spin means of atoms spaced by distances $\left| {\Delta {\mathbf{R}}_{{nn'}}^{{kk'}}} \right|$ is determined by the expression $\Psi _{{nn'}}^{{kk'}}$ = ${\mathbf{k}}({\mathbf{R}}_{{n'}}^{{k'}}$ – ${\mathbf{R}}_{n}^{k})$ ≡ ${\mathbf{k}}\Delta {\mathbf{R}}_{{nn'}}^{{kk'}}$ = ${{k}_{a}}c_{{nn}}^{{kk'}}$ ($c_{{nn}}^{{kk'}}$ is the distance between atoms kn and k'n' along the wave vector k). To describe HM ordering within the mean-field approximation, we use Hamiltonian (A.2), in which the direction of the spatially inhomogeneous mean field ${\mathbf{h}}_{n}^{k}$ = $h{\mathbf{U}}_{n}^{k}$ for atom at the site ${\mathbf{R}}_{n}^{k}$ is determined by the unit vector ${\mathbf{U}}_{n}^{k}$ = (cos(${\mathbf{kR}}_{n}^{k}$))sin(ϑ)sin(${\mathbf{kR}}_{n}^{k}$)sin(ϑ), cos(ϑ)) and coincides with the direction of the local quantization axis. It is assumed that the local quantization axis at H₀ = 0 lies in the basal plane oriented perpendicular to k (ϑ = π/2). The model Hamiltonian ${{{\mathbf{\hat {H}}}}_{{hs}}}$ and the model free energy Ω_h ≡ Ω(h) are determined by the expressions

$${{{\mathbf{\hat {H}}}}_{{hs}}} = - \sum\limits_{nk} {h{\mathbf{U}}_{n}^{k}{\mathbf{\hat {s}}}_{n}^{k}} = - \sum\limits_{nk} {h\hat {m}_{n}^{k},} $$

(A.2)

$${{\Omega }_{h}} = {{\left\langle {{{{{\mathbf{\hat {H}}}}}_{s}} - {{{{\mathbf{\hat {H}}}}}_{{hs}}}} \right\rangle }_{h}} - N{{k}_{{\text{B}}}}T\ln [z(X)],$$

(A.3)

where $\hat {m}_{n}^{k}$ = ${\mathbf{U}}_{n}^{k}{\mathbf{\hat {s}}}_{n}^{k}$ is the operator of spin projection on the quantization axis and

$${{\left\langle {{{{{\mathbf{\hat {H}}}}}_{s}} - {{{{\mathbf{\hat {H}}}}}_{{hs}}}} \right\rangle }_{h}} = - \sum\limits_{n,k} {\left[ {\left[ {\sum\limits_{n',k'} {J(\Delta {\mathbf{R}}_{{nn'}}^{{kk'}}){{{\left\langle {{\mathbf{\hat {s}}}_{n}^{k}} \right\rangle }}_{h}}{{{\left\langle {{\mathbf{\hat {s}}}_{{n'}}^{{k'}}} \right\rangle }}_{h}}} } \right]} \right.} $$

$$\left. {^{{^{{^{{^{{}}}}}}}} + \;2{{\mu }_{0}}{{{\mathbf{H}}}_{0}}\left\langle {{\mathbf{\hat {s}}}_{n}^{k}} \right\rangle - {{{\left\langle {\hat {m}} \right\rangle }}_{h}}h} \right]$$

$$\begin{gathered} = - \sum\limits_{n,k} {\left[ {\left[ {\left\langle {\hat {m}} \right\rangle _{h}^{2}\sum\limits_{n',k'} {J(\Delta {\mathbf{R}}_{{nn'}}^{{kk'}}){\mathbf{U}}_{n}^{k}{\mathbf{U}}_{{n'}}^{{k'}}} } \right]} \right.} \\ \left. {^{{^{{^{{^{{}}}}}}}} + \;2{{\mu }_{0}}{{{\mathbf{H}}}_{0}}{\mathbf{U}}_{n}^{k}{{{\left\langle {\hat {m}} \right\rangle }}_{h}} - {{{\left\langle {\hat {m}} \right\rangle }}_{h}}h} \right] \\ \end{gathered} $$

(A.4a)

$$ = - N\left\langle {\hat {m}} \right\rangle _{h}^{2}[J(k){{\sin }^{2}}(\vartheta ) + J(0){{\cos }^{2}}(\vartheta )]$$

$$ - \;N2{{\mu }_{0}}{{H}_{0}}\cos (\vartheta ){{\left\langle {\hat {m}} \right\rangle }_{h}} + N{{\left\langle {\hat {m}} \right\rangle }_{h}}h.$$

It is taken into account in (A.4a) that ${{{\mathbf{H}}}_{0}}{\mathbf{U}}_{n}^{k}$ = ${{H}_{0}}\cos (\vartheta )$ and only components $\hat {m}_{n}^{k}$ directed along the mean field contribute to the diagonal eigenvalues ${\mathbf{\hat {s}}}_{n}^{k}$; therefore, the relation between the mean values of spin ${{\left\langle {{\mathbf{\hat {s}}}_{n}^{k}} \right\rangle }_{h}}$ and ${{\left\langle {\hat {m}_{n}^{k}} \right\rangle }_{h}}$ within the mean-field approximation is determined by the expression

$${{\left\langle {{\mathbf{\hat {s}}}_{n}^{k}} \right\rangle }_{h}} = {\mathbf{U}}_{n}^{k}{{\left\langle {\hat {m}_{n}^{k}} \right\rangle }_{h}} = {\mathbf{U}}_{n}^{k}{{\left\langle {\hat {m}} \right\rangle }_{h}} = {\mathbf{U}}_{n}^{k}ys.$$

(A.4b)

Averaging within the mean-field approximation is performed according to the scheme

$$ys = {{\left\langle {\hat {m}} \right\rangle }_{h}} = \operatorname{Sp} \hat {m}{{e}^{{\beta h\hat {m}}}}{\text{/}}z(x) = \sum\limits_{m = - s}^s {m{{e}^{{\beta hm}}}{\text{/}}z(X)} $$

$$ \equiv {{\beta }^{{ - 1}}}\partial \ln z(X){\text{/}}\partial h = s{{B}_{s}}(X)$$

(A.5)

$$ = s\left[ {\left( {\frac{1}{{2s + 1}}} \right)\coth \frac{1}{{2s + 1}}X - \left( {\frac{1}{{2s}}} \right)\coth \frac{1}{{2s}}X} \right],$$

$$\begin{gathered} z(X) = \operatorname{Sp} {{e}^{{\beta h\hat {m}_{n}^{k}}}} = \sum\limits_{m = - s}^s {{{e}^{{\beta hm_{n}^{k}}}}} \\ = \sinh [(1 + {{(2s)}^{{ - 1}}})X]{\text{/}}\sinh [{{(2s)}^{{ - 1}}}X], \\ \end{gathered} $$

(A.6a)

$$X = \beta sh.$$

(A.6b)

Within the approximation of nearest and next Mn atoms in the plane oriented perpendicular to the wave vector k, J(k_a) has the form

$$\begin{gathered} J({{k}_{a}}) = \sum\limits_{\Delta R} {J(\left| {\Delta {\mathbf{R}}} \right|)\cos ({\mathbf{k}}\Delta {\mathbf{R}})} \\ \approx {{J}_{0}} + {{J}_{1}}\cos (\Psi ) + {{J}_{2}}\cos (2\Psi ), \\ \end{gathered} $$

(A.6c)

$$J(0) \equiv J({{k}_{a}} = 0) \equiv J(\Psi = 0) = {{J}_{0}} + {{J}_{1}} + {{J}_{2}}.$$

(A.6d)

Taking into account (A.4–A.6), Ω_h can be written in the form

$$\begin{gathered} {{\Omega }_{h}} = - N{{s}^{2}}{{y}^{2}}[J({{k}_{a}}){{\sin }^{2}}(\vartheta ) + J(0){{\cos }^{2}}(\vartheta )] \\ - \;Nsy2{{\mu }_{0}}{{H}_{0}}\cos (\vartheta ) + Nsyh - N{{k}_{{\text{B}}}}T\ln [z(X)]. \\ \end{gathered} $$

(A.7)

The expression for the mean-field magnitude h is determined from the condition dΩ_h/dh = $\partial {{\Omega }_{h}}{\text{/}}\partial h$ + $(\partial {{\Omega }_{h}}{\text{/}}\partial y)dy{\text{/}}dh$ = 0, which, with allowance for (A.5), is reduced to the form

$$\begin{gathered} (dy{\text{/}}dh)\{ - 2sy[J({{k}_{a}}){{\sin }^{2}}(\vartheta ) + J(0){{\cos }^{2}}(\vartheta )] \\ - \;2{{\mu }_{0}}{{H}_{0}}\cos (\vartheta )s + h\} = 0. \\ \end{gathered} $$

(A.8)

Hence,

$$\begin{gathered} h = 2sy[J({{k}_{a}}){{\sin }^{2}}(\vartheta ) + J(0){{\cos }^{2}}(\vartheta )] \\ + \;2{{\mu }_{0}}{{H}_{0}}\cos (\vartheta )s. \\ \end{gathered} $$

(A.9)

After substituting the expression for h into (A.7), Ω_h ≡ Ω(h) takes the form of Ω₂(y) in (3c).