Statistical and Analytical Approaches
to Finite Temperature Magnetic Properties of SmFe₁₂ compound

The Hamiltonian for $4f$ shell in $j$-th $R$ ion in Eq. (1) is

with

The first and second terms in Eq. (2) represent the single electron contribution and the electron-electron repulsion in $4f$ shell, respectively, where $n_{4f}$ is the number of $4f$ electrons, $\varepsilon_{0}$ and $e$ are the vacuum permittivity and the elementary charge, respectively. $\hat{h}_{j}(i)$ in Eq. (3) is the Hamiltonian for $i$-th $4f$ electron on $j$-th $R$ site, where the first term in Eq. (3) is the spin-orbit interaction (SOI) between spin ($\hat{\bm{s}}_{i}$) and orbital ($\hat{{\bm{l}}_{i}}$) angular momenta, with a coupling constant $\xi$. The second term represents the exchange interaction between spin moment and temperature dependent exchange field ${\bm{B}}_{{\rm ex},j}(T)=-{\bm{e}}^{TM}B_{{\rm ex},j}(T)$ on $j$-th $R$ site, where $\mu_{B}$ is the Bohr magneton. The third and fourth terms are the CF and Zeeman terms, respectively. In the expression of CF, $V_{j}({\bm{r}}_{i})$ and $R_{4f}(r_{i})$ are Coulomb potential and radial parts of the $4f$ wave function on $j$-th $R$ site, respectively. Note that the kinetic energy and screened central potential terms are effectively taken into account in the formation of $4f$ orbital.

To obtain the electronic properties at $T=0$, we apply the first-principles and determine the parameters in the Hamiltonians in Eq. (3). We use the full-potential linearized augmented plane wave plus local orbitals (APW+lo) method implemented in the WIEN2k code wien2k . The Kohn-Sham equations are solved within the generalized-gradient approximation (GGA). To simulate localized $4f$ states, we treat $4f$ states as atomic-like core states, which is so called opencore method Richter1 ; Novak ; Hummler0 ; Richter2 ; Divis1 ; Divis2 .

We calculate the ground state properties of SmFe₁₂ such as Coulomb potential, charge distribution, and sublattice magnetizations. In accord with the previous theoretical studies for SmFe₁₂ Harashima ; Koerner ; Delange , we assume that Sm ion has trivalent-like electronic structure. The exchange fields $B_{{\rm ex},j}(0)$ at $T=0$ are determined from an energy increase caused by spin flip of $4f$ electrons Brooks ; Yoshioka , and CFs acting on $i$-th $4f$ electron are directly estimated from Coulomb potential $V_{j}({\bm{r}}_{i})$ acting on $j$-th $R$ site. It is noted that the single ion Hamiltonian $\hat{\cal H}_{R,j}$ thus determined for $j$-th $R$ ions includes effects of $TM$ atoms surrounding the $R$ ions as a mean field.

Practically, the CF term is rewritten as the following formula Novak ; Richter2 :

where $A_{l,j}^{m}\langle r^{l}\rangle$ is CF parameter on $j$-th $R$ site, $a_{l,m}$ is a numerical factor Hutchings , $t_{l}^{m}(\hat{\theta}_{i},\hat{\varphi}_{i})$ is tesseral harmonic function of a solid angle $\Omega=(\hat{\theta}_{i},\hat{\varphi}_{i})$, and $r_{\rm c}$ is a cut-off radius.

Values of CF parameters $A_{l,j}^{m}\langle r^{l}\rangle$ in Eq. (5), exchange field $B_{{\rm ex},j}(0)$ in Eq. (3), $TM$-sublattice magnetization $M^{TM}(0)$ in Eq. (1) in SmFe₁₂ are shown in TABLE 1. The lattice constants used in this calculations are the experimental values $a=b=8.35$ Å and $c=4.8$ Å Hirayama2 . For Wycoff positions, we apply the theoretically optimized ones given in Ref. Harashima . The crystal structure of SmFe₁₂ is shown in Fig. 1.

We here apply the concept of $LS$ coupling to the single electron Hamiltonian of Eq. (3) with Russell Saunders states $|L,S;J,M\rangle$, due to the strong Coulomb interaction between $4f$ electrons. According to the Hund’s rule, we specify the quantum number of total orbital (spin) moment $L(S)$. Total angular momentum $J$ is varied from $|L-S|$ to $L+S$, and $M$ is the magnetic quantum number. Thus the single ion Hamiltonian in Eq. (2) can be reduced to:

Hereafter, the site index $j$ is omitted for single-ion quantity. $\hat{\bm{L}}$ and $\hat{\bm{S}}$ are total orbital and spin momenta of $4f$ electrons, respectively, $B_{l}^{0}=\sqrt{(2l+1)/4\pi}A_{l}^{0}\langle r^{l}\rangle/a_{l,0}$ and $B_{l}^{\pm|m|}=(\mp 1)^{m}\sqrt{(2l+1)/8\pi}\left[A_{l}^{|m|}\langle r^{l}\rangle\mp iA_{l}^{-|m|}\langle r^{l}\rangle\right]/a_{l,m}$ for $m\neq 0$, and $\Theta_{l}^{L}=\langle L\parallel\sum_{i}C^{(l)}_{m}(\hat{\theta}_{i},\hat{\varphi}_{i})\parallel L\rangle/\langle L\parallel\sum_{i}C^{(l)}_{m}(\hat{\bm{L}})\parallel L\rangle$. In the treatment of SOI, we should note that the eigenstates of $LS$ coupling are specified by the quantum number of $J$. In general, the term $\hat{\cal H}_{\rm so}$ is dominating in Eq. (6). Thus $J$ is a good quantum number in most of the $R$-$4f$ systems. Because the $LS$ coupling in Sm compounds is weak compared with other $R$ ones, it is necessary to include excited $J$-multiplets. Hereafter, we abbreviate the states $|L,S;J,M\rangle$ as $|J,M\rangle$.

The energy levels of Sm-$4f$ states in SmFe₁₂ depend on ${\bm{B}}_{\rm ex}(T)$ and applied field ${\bm{B}}$. Fig. 2 shows the $B_{\rm ex}(T)/B_{\rm ex}(0)$ dependence of the energy levels for ${\bm{e}}^{TM}$=${\bm{n}}_{c}$, which is a unit vector parallel to $c$-axis, and for ${\bm{B}}={\bm{0}}$. The data needed are given in TABLE 1. As for SOI constant, we use experimental value of $\lambda/k_{B}=\xi/5k_{B}=411$ K Elliott . At $B_{\rm ex}(T)=0$ the $S$ is strongly coupled with $L$ to form a Kramers doublet with a total angular momentum $J$ due to the large $LS$ coupling with fine CF splitting. With increasing $B_{\rm ex}(T)/B_{\rm ex}(0)$, the exchange field breaks the time-reversal symmetry and lift the degeneracy.

For finite temperature magnetic properties of TM, we apply a phenomenological formula assuming uniform $M^{TM}(T)$ and $K_{1}^{TM}(T)$. For $M^{TM}(T)$, we apply the Kuz’min formula Kuzminmag :

where, $T_{\rm C}$ is Curie temperature and $s$ is a fitting parameter. The temperature dependence of $K_{1}^{TM}(T)$ has been expressed by an extended power law Miura-pl :

where $C_{1}$ and $C_{2}$ are fitting parameters.

In present study for SmFe₁₂ compound, we use values of $s=0.01$ and $T_{\rm C}=555$ K in Eq. (12), as used by Hirayama $et$ $al$. Hirayama2 . They showed that the magnetization agrees well with experimental measurement for SmFe₁₂. The values of $C_{1}$, $C_{2}$ and $V_{0}K_{1}^{TM}(0)$ in Eq. (13) are determined as $-0.263$, $-0.237$ and $47.7$ K, respectively, by fitting the expression to observed data for YFe₁₁Ti in Ref. Nikitin .

To calculate the finite temperature magnetic properties, we use the model Hamiltonian and calculate MA and magnetic moment for Sm $4f$ electrons using the statistical method for the partial system. Using the eigenvalues of the Hamiltonian Eq. (6), we express the free energy density as,

where $g_{j}({\bm{e}}^{TM},T,{\bm{B}})$ is Gibbs free energy for $R$-$4f$ partial system, $E_{n,j}({\bm{e}}^{TM},T,{\bm{B}})$ and $Z_{j}({\bm{e}}^{TM},T,{\bm{B}})$ are the eigenvalue and the partition function of $j$-th $R$ Hamiltonian $\hat{\cal H}_{R,j}$ [Eq. (6)] for given ${\bm{e}}^{TM}$, respectively. The direction of the $TM$ magnetization ${\bm{e}}^{TM}$ is treated as an external parameter. The equilibrium condition of the system for given $T$ and $\bm{B}$ is:

where ${\bm{e}}_{0}^{TM}$ is the direction of $TM$ sublattice magnetization in the equilibrium. In practice, we determine the minimal $G({\bm{e}}^{TM},T,{\bm{B}})$ numerically by changing ${\bm{e}}^{TM}$.

The MA energy is given by the free energy $G({\bm{e}}^{TM},T,{\bm{0}})$ with different directional vector ${\bm{e}}^{TM}$. In the tetragonal symmetry, $g_{j}({\bm{e}}^{TM},T,{\bm{0}})$ in $G({\bm{e}}^{TM},T,{\bm{0}})$ is formally expressed as Kuzminlinear ; Miura-pl :

where $\theta^{TM}$ and $\varphi^{TM}$ are polar and azimuthal angle of ${\bm{e}}^{TM}$, respectively, $\lfloor p/2\rfloor$ indicates the greatest integer of $p/2$, and $k_{p,j}(T)$ and $k_{p,j}^{q}(T)$ are out-of-plane and in-plane MA constant for $j$-th $R$ ion. The $C(T)$ is an angle independent constant. The series expansion does not guarantee the convergence Kuzmin2nd ; Kuzminlimit , however, for finite $p$, $k_{p,j}^{q}(T)$ can be obtained from the comparison between Taylor series of $g_{j}({\bm{e}}^{TM},T,{\bm{0}})$ of Eqs. (15) and (18) with respect to $\theta^{TM}$ for a fixed $\varphi^{TM}$ Sasaki ; Miura-direct as:

and

respectively, which are resulting in

etc. Using MA energy on the single $R$ ion in Eq. (18), the total MA constants are obtained as

where $K_{p}(T)$ and $K_{p}^{q}(T)$ are out-of-plane and in-plane MA constant in whole system.

The orbital and spin components of the magnetic moment of a single $R$ ion in the equilibrium, can be calculated by:

respectively, where $\rho_{n,j}(T,{\bm{B}})=\exp\left[-\beta E_{n,j}({\bm{e}}^{TM}_{0},{\bm{B}})\right]/Z_{j}({\bm{e}}^{TM}_{0},T,{\bm{B}})$ and $|n,j\rangle$ is the $n$-th eigenstate for $E_{n,j}({\bm{e}}_{0}^{TM},{\bm{B}})$, and the total magnetization $M_{\rm s}(T,{\bm{B}})$ is given as,

with ${\bm{m}}_{j}(T,{\bm{B}})={\bm{m}}_{L,j}(T,{\bm{B}})+{\bm{m}}_{S,j}(T,{\bm{B}})$.

Finally, to confirm the convergence of the probability weights for excited-$J$ multiplet states at ${\bm{B}}={\bm{0}}$, we define a following weight function:

In the case of SmFe₁₂ crystal, the value of $W_{J}(T)$ is independent of site index $j$.

The results are shown in TABLE 2, which indicates good convergence of weight for the number of the excited $J$-multiplets even at $T=T_{\rm C}=555$ K. Thus in the calculation using statistical method for SmFe₁₂, we take the excited $J$-multiplets up to $J=9/2$. In the analytical calculation, the $J$-mixing effects are approximately treated only for the lowest-$J$ multiplet by using unitary transformation.

According to hierarchy of energy scale in $R$ intermetallic compounds: $\hat{\cal H}_{\rm so}\gg\hat{\cal H}_{\rm ex}\gg\hat{\cal H}_{\rm CF}\sim\hat{\cal H}_{\rm Z}$, we develop an analytical method for finite temperature magnetic properties, which enables us to connect the thermodynamic properties directly to our model parameters based on electronic states. Practically, we generalize the analytical expression of Gibbs free energy Yamada ; Kuzminlinear to include the effects of $J$-mixing using a first-order perturbation for the CF potential and Zeeman energy. We also derive an analytical expression for the magnetization curve, which enables us to estimate the CF potential using the observed results. The procedure of the formalism consists of (i) construction of starting Hamiltonian for single $R$ ion, (ii) approximation for diagonal matrix element of an effective Hamiltonian, (iii) finite temperature perturbation for single $R$ ion, and (iv) thermodynamic analysis.