Abstract Generated abstract
This paper develops a spin-wave treatment of magnetic anisotropy in ferromagnetic single crystals with hexagonal symmetry, extending Dyson’s method for interacting spin waves to an anisotropic exchange Hamiltonian. The analysis formulates the partition function in terms of spin-wave states, separates kinematic and dynamic interactions, and evaluates the free energy using a graph expansion while neglecting exponentially small kinematic corrections in the low-temperature series. The resulting anisotropy free energy includes the Bloch temperature term, corrections from lattice discreteness and anisotropic spin-wave energy, and a contribution from dynamical spin-wave interaction. The dynamical interaction is estimated to remain small up to about one quarter of the Curie temperature, supporting the applicability of spin-wave theory in that range.
Full Text
PHYSICS
N. A. POTAPKOV
ON THE THEORY OF ANISOTROPY OF FERROMAGNETIC SINGLE CRYSTALS
(Presented by Academician N. N. Bogolyubov on 8 VIII 1957)
Magnetic anisotropy is characterized by the so-called anisotropy constants, which are defined as the coefficients in the expansion of the free energy in a series in powers of the direction cosines (with respect to the crystallographic axes). From the theoretical point of view, the temperature dependence of the anisotropy constants was investigated qualitatively in \((^{1,2})\). The dependence of the anisotropy constants on temperature and field was considered in \((^{3,4})\), by means of the method of approximate secondary quantization.
The results \((^{3,4})\) were obtained in the linear approximation of spin-wave theory without taking into account the interaction of spin waves. The theory of spin-wave interaction for an isotropic ferromagnet was considered by Dyson \((^{5})\). He determined spin-wave states in which the Heisenberg Hamiltonian is approximately diagonal, while the corrections due to the dynamical interaction do not exceed several percent even for \(T = \frac{1}{2}T_c\).
In the present work Dyson’s method is applied to an anisotropic ferromagnet with hexagonal symmetry. The Hamiltonian of the system is
\[ \mathcal{H} = \mu \sum_j (\mathbf{H}\mathbf{S}_j) - \frac{1}{2} I \sum_{j\delta} \mathbf{S}_j \mathbf{S}_{j+\delta} - \frac{1}{2}\delta I \sum_{j\delta} S_j^{z'} S_{j+\delta}^{z'}, \tag{1} \]
where \(H\) is the external magnetic field; \(\mu\) is the Bohr magneton; \(\delta I\) characterizes the magnetic interaction; \(I\) is the ordinary exchange integral; \(S_j\) is the electron spin operator belonging to site \(j\); \(z'\) coincides with the direction of the crystallographic axis; \(\vec{\delta}\) is the vector connecting nearest-neighbor sites.
Let us pass to another coordinate system, with the direction of the \(z\)-axis along the magnetization vector, and define the operators
\[ S_\lambda = N^{-1/2}\sum_\lambda \exp(i\vec{\lambda}\mathbf{j}) S_j, \]
\[ S_\lambda^{\pm} = S_\lambda^x \pm iS_\lambda^y . \tag{2} \]
Here \(\vec{\lambda}\) is a reciprocal-lattice vector. The operators \(S_\lambda\) satisfy the relations
\[ [S_\lambda^z S_\mu^+] = N^{-1/2}S_{\lambda+\mu}^+, \qquad [S_\lambda^z S_\mu^-] = - N^{-1/2}S_{\lambda+\mu}^-, \qquad [S_\lambda^+ S_\mu^-] = 2N^{-1/2}S_{\lambda-\mu}^z . \tag{3} \]
The Hamiltonian of the system is reduced to the form
\[ \begin{aligned} \mathcal H={}&\mu N^{-1/2}(\mathbf H\mathbf S_0)-\frac12 I\sum_\lambda \gamma_\lambda \mathbf S_\lambda\mathbf S_{-\lambda} -\frac12 \delta I\cos^2\theta \sum_\lambda S_\lambda^z S_{-\lambda}^z\gamma_\lambda \\ &-\frac18 \delta I\sin^2\theta \sum_\lambda \gamma_\lambda \left(S_\lambda^+S_{-\lambda}^+ + S_\lambda^-S_{-\lambda}^- + S_\lambda^+S_{-\lambda}^- + S_\lambda^-S_{-\lambda}^+\right) \\ &+\frac12 \delta I\sin\theta\cos\theta \sum_\lambda \gamma_\lambda \left(S_\lambda^+S_{-\lambda}^z+S_\lambda^-S_{-\lambda}^z\right), \end{aligned} \tag{4} \]
where
\[ \gamma_\lambda=\sum_\delta e^{i(\vec\lambda\vec\delta)} . \]
The ground state of the system is determined by the formulas:
\[ S_j^-|0\rangle=0,\qquad S_j^z|0\rangle=-S|0\rangle \tag{5} \]
for all \(j\), or, equivalently,
\[ S_\lambda^-|0\rangle=0,\qquad S_\lambda^z|0\rangle=-N^{1/2}S\delta_{\lambda 0}|0\rangle \tag{6} \]
for all \(\vec\lambda\).
Following Dyson, let us define the spin-wave states
\[ |a\rangle=\prod_\lambda \left[(2S)^{-1/2 a_\lambda}(a_\lambda!)^{-1/2}(S_\lambda^+)^{a_\lambda}\right]|0\rangle, \tag{7} \]
where \(a_\lambda\) is the number of spin waves with wave vector \(\vec\lambda\).
States with \(\sum a_\lambda>1\) are neither normalized nor orthogonal. The nonorthogonality of the states (7) gives rise to an interaction between spin waves, called the kinematic interaction. Its physical cause is that the spin of an individual atom can assume only \(2S+1\) values. Another interaction, called dynamic, arises from the fact that the Hamiltonian (4) is not diagonal in the states (7).
Let us find the action of the Hamiltonian \(\mathcal H\) on the states \(|a\rangle\):
\[ \mathcal H|a\rangle= \left[E_0+\sum_\lambda a_\lambda(L+\varepsilon_\lambda)\right]|a\rangle +\sum_b Q_{ba}|b\rangle, \tag{8} \]
where
\[ E_0=-\mu H^zNS-\frac12(I+\delta I\cos^2\theta)\gamma_0NS^2, \tag{9} \]
\[ L_0=\mu H^z+\delta IS\gamma_0\left(\cos^2\theta-\frac12\sin^2\theta\right), \tag{10} \]
\[ \varepsilon_\lambda=\left(I+\frac12\delta I\sin^2\theta\right)S(\gamma_0-\gamma_\lambda) =I'S(\gamma_0-\gamma_\lambda), \tag{11} \]
and \(Q_{ba}\) is determined from the expressions for the commutators of \(\mathcal H\) with \(S_\lambda^+\).
The sum of states of the system, according to (5), is expressed in the form
\[ Z=\sum_u F_u^{-2}\sum_a \langle a|u\rangle\langle u|\exp(-\beta\mathcal H)|a\rangle,\qquad \beta=\frac1{kT}. \tag{12} \]
The states \(|u\rangle\) are defined as follows:
\[ |u\rangle=\prod_j\left[(2S)^{-1/2u_j}(u_j!)^{-1/2}(S_j^+)^{u_j}\right]|0\rangle, \tag{13} \]
where \((u)\) is a system of numbers \(u_j\) taking the values \(0,1,\ldots,2S\),
\[ F_u=\langle v|u\rangle. \tag{14} \]
Following Dyson, we express \(Z\) in the ideal spin-wave model, in which a harmonic oscillator is associated with each lattice site \(j\).
The oscillators are described by means of creation operators \(\eta_j^*\) and annihilation operators \(\eta_j\), satisfying the relations
\[ [\eta_j\eta_k^*]=\delta_{jk}, \qquad \eta_j^*\eta_j=u_j . \tag{15} \]
The complete set of orthogonal and normalized states of the ideal system is
\[ |u)=\prod_j \left[(u_j!)^{-1/2}(\eta_j^*)^{u_j}\right]|0). \tag{16} \]
In the ideal model one can define another orthogonal system
\[ |a)=\prod_\lambda \left[(a_\lambda!)^{-1/2}(\alpha_\lambda^*)^{a_\lambda}\right]|0), \tag{17} \]
where \(\alpha_\lambda^*\) are defined as follows:
\[ \alpha_\lambda^*=N^{-1/2}\sum_j \exp(i\lambda j)\eta_j^* \tag{18} \]
and satisfy the relations
\[ [\alpha_\lambda \alpha_\mu^*]=\delta_{\lambda\mu}, \qquad \alpha_\lambda^*\alpha_\lambda=a_\lambda . \tag{19} \]
The sum of states is expressed in the ideal model as
\[ Z=e^{-\beta E_0}\sum_{ab} U_{ba}V_{ab}, \tag{20} \]
where
\[ U_{ba}=(b|\exp[-\beta(\mathcal H-E_0)]|a), \]
\[ V_{ab}=\sum_u(a|u)E_u(u|b); \tag{21} \]
\[ E_u=\prod_j E(u_j); \tag{22} \]
\[ E(u)=1 \text{ for } u=0,1,\ldots,2S; \qquad E(u)=0 \text{ for } u>2S. \]
The expression \(V_{ab}\) represents the effect of the kinematic interaction. The physical states of our system correspond to a set \((u)\) of numbers \(u_j\), in which the \(u_j\) take the values \(0,1,\ldots,2S\). The sum of states of our system can be represented in the form
\[ Z=Z_T-Z_1, \tag{23} \]
where \(Z_T\) extends over states with arbitrary \(u_j\), while \(Z_1\) only over states with \(u_j>2S\). The sum \(Z_1\) contains the factor \(e^{-T_c/T}\) (\(T_c\) is the Curie temperature). Therefore, in the expansion of the free energy in powers of \(T\), the terms taking account of the kinematic interaction may be omitted. To compute \(U_{ba}\), we represent \(\exp[-\beta(\mathcal H-E_0)]\) in the form of a series:
\[ \exp[-\beta(\mathcal H-E_0)] = \]
\[ =\sum_{m=0}^{\infty}(-1)^m\int_0^\beta d\beta_1\ldots \int_0^{\beta_{m-1}} d\beta_m\, \exp[(\beta_1-\beta)\mathcal H_1]\mathcal H_2\ldots \mathcal H_2\exp(-\beta_m\mathcal H_1), \tag{24} \]
where
\[ \mathcal H_1=\sum_\lambda (L+\varepsilon_\lambda)\alpha_\lambda^*\alpha_\lambda, \qquad \mathcal H_2=\mathcal H-E_0-\mathcal H_1 . \tag{25} \]
To compute the sums we shall apply Feynman’s graph method. For the free energy per atom we obtain
\[ F=-\mu H^z S-\frac{1}{2}(I+\delta I\cos^2\theta)\gamma_0S^2-kT\{Z_{1/2}(\beta L)\theta'^{3/2} +{}^3/4\pi\nu Z_{7/2}(\beta L)\theta'^{5/2} +\left(\frac{3\pi\nu}{4S}\right)[Z_{5/2}(\beta L)]^2\theta'^4\} -\delta I\gamma_0\left({}^3/2\cos^2\theta-{}^1/2\right)[Z_{3/2}(\beta L)]^2\theta'^3, \tag{26} \]
where
\[ \theta'=\frac{3kT}{2\pi I'\gamma_0\nu},\qquad \nu=\frac{a^2}{v^{2/3}},\qquad Z_n(x)=\sum_{j=1}^{\infty}j^{-n}e^{-jx}. \]
Expanding \(Z_n(\beta L)\) in powers of \(\delta I/kT\) and \(\theta'\) in powers of \(\delta I/I\), we obtain for the free energy
\[ F=F_0+F_A, \]
where \(F_A\) is the free energy of anisotropy:
\[ F_A=-{}^1/2\delta I\gamma_0\cos^2\theta\{S^2-3SZ_{3/2}(\beta L')\theta^{3/2} -5SZ_{5/2}(\beta L')\theta^{5/2} +3[Z_{3/2}(\beta L')]^2\theta^3\}, \tag{27} \]
\[ L'=\mu H^z,\qquad \theta=\frac{3kT}{2\pi IS\gamma_0\nu}=\frac{T}{2\pi T_c}. \]
The term with \(\theta^{3/2}\) corresponds to Bloch’s approximation and agrees with the results of \((^3)\); the term with \(\theta^{5/2}\) appears because of the discreteness of the crystal lattice and the dependence of the spin-wave energy on \(\delta I\). The term with \(\theta^3\) represents the dynamical interaction of spin waves. In the temperature range up to \(T={}^1/4T_c\), the term representing the dynamical interaction does not exceed \(4\%\) (for \(S={}^1/2\)) and \(2\%\) (for \(S=1\)) of the term corresponding to Bloch’s approximation. The small magnitude of the dynamical interaction shows that spin-wave theory is applicable to this temperature range.
In conclusion, I express my gratitude to S. V. Tyablikov and V. V. Tolmachev for a valuable discussion.
Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
29 VII 1957
REFERENCES
- S. V. Vonsovskii, ZhETF, 8, 1104 (1938).
- J. Van Vleck, Phys. Rev., 52, 1104 (1938).
- S. V. Tyablikov, ZhETF, 20, 661 (1950).
- S. V. Tyablikov, A. A. Gusev, Fiz. met. i metalloved., 2, no. 3 (1956).
- F. J. Dyson, Phys. Rev., 102, 1217 (1956).