Numeric Calculation of Antiferromagnetic Resonance Frequencies for the Noncollinear Antiferromagnet

Abstract

We present an algorithm for the numeric calculation of antiferromagnetic resonance frequencies for the noncollinear antiferromagnets of general type. This algorithm uses general exchange symmetry approach (Andreev and Marchenko, Sov. Phys. Usp. 130:39, 1980) and is applicable for description of low-energy dynamics of an arbitrary noncollinear spin structure in weak fields. Algorithm is implemented as a MatLab and C$++$ program codes, available for download. Program codes are tested against some representative analytically solvable cases.

Appendix: Analytically Solvable Models Used as Test Cases

We recall here some of the known examples of application of exchange symmetry theory to low-energy dynamics of noncollinear antiferromagnets. These analytical solutions were used as test cases to ascertain correctness of numeric algorithms.

The first test example is an antiferromagnet on a triangular lattice CsNiCl$_3$ [3]. In the ordered phase of this magnet, spins form a planar 120$^\circ$ structure. High symmetry of triangular lattice leaves single invariant in the anisotropy energy $U_\mathrm{A}=\beta ({l_{3}^z})^2$, here z-axis is normal to hexagonal plane and vector ${\mathbf {l}}_3$ is the normal to the plane of the planar spin structure, $\beta >0$ as at zero-field spin plane is orthogonal to the hexagonal crystallographic plane. Magnetic susceptibility normal to the spin plane dominates: $\chi _3>\chi _2=\chi _1$ (i.e., $I_3<I_1=I_2$). Two of the zero-field frequencies are zero, non-zero zero-field frequency is $\omega _{0}=\gamma \sqrt{\frac{I_1-I_3}{I_1+I_3}\beta }=\gamma \sqrt{\frac{\chi _3-\chi _1}{\chi _1}\beta }$. As the field is applied along z-axis spin plane reorients at the field $H_0=\sqrt{\frac{\beta }{\gamma ^2 (I_1-I_3)}}=\sqrt{\frac{\beta }{\chi _3-\chi _1}}$. Magnetic resonance frequencies at $\mathbf {H}||z$ are given by equations:

$$\begin{aligned} H<H_0&:\omega _1^2=\omega _{0}^2+\left( \gamma H\right) ^2\\& \quad\omega _2=\omega _3=0\\ H>H_0&:&\omega _{1,2}=\sqrt{\left( \frac{I_1}{I_1+I_3} \gamma H \right) ^2-\omega _{10}^2}\pm \frac{I_3}{I_1+I_3}\gamma H\\&=\sqrt{\left( \frac{\chi _3}{2\chi _1}\gamma H \right) ^2-\omega _{0}^2}\pm \frac{2\chi _1-\chi _3}{2\chi _1}\gamma H\\&\omega _3=0 \end{aligned}$$

Because of simplicity of anisotropy energy, this problem can be solved analytically at arbitrary field orientation, see Ref. [3] for details.

To reproduce experimental results of Ref. [3], we take for our modeling $\beta =1$ kOe$^2$, $\gamma =18.8 \frac{10^9 \mathrm{rad}~ \mathrm{s}^{-1}}{\mathrm{kOe}}$ (3.0 GHz kOe$^{-1}$ in frequency units), $I_1=I_2=8.77\times 10^{-6} \frac{ \mathrm{kOe}^2}{(10^9 \mathrm{rad}~\mathrm{s}^{-1})^2}$ and $I_3=9.75\times 10^{-7}\frac{ \mathrm{kOe}^2}{(10^9 \mathrm{rad}~\mathrm{s}^{-1})^2}$.

Second, we consider 12-sublattice antiferromagnet Mn$_3$Al$_2$Ge$_3$O$_{12}$ [5]. Here, $I_1=I_2$ because of the cubic symmetry, anisotropy energy $U_\mathrm{A}=\lambda [l_{2z}^2-l_{1z}^2+\frac{2}{\sqrt{3}}(l_{1x}l_{2x}-l_{1y}l_{2y})]$ ($\lambda >0$) (we use notations of Ref. [22]). At zero-field plane of the spiral structure is orthogonal to one of the $\langle 111\rangle$ directions. Oscillation eigenfrequencies can be found at $\mathbf {H}||[111]$:

$$\begin{aligned} \omega _{1,2}= & {} \sqrt{ \left( \frac{I_1}{I_1+I_3}\gamma H\right) ^2+\frac{4}{3}\frac{\lambda }{(I_1+I_3)}} \pm \frac{I_3}{I_1+I_3}\gamma H\\= & {} \sqrt{ \left( \frac{\chi _3}{2\chi _1}\gamma H\right) ^2+\frac{4}{3}\frac{\lambda }{\chi _1}\gamma ^2} \pm \frac{2\chi _1-\chi _3}{2\chi _1}\gamma H \\ \omega _3= & {} \sqrt{\frac{8}{3}\frac{\lambda }{I_1}}=\gamma \sqrt{\frac{8}{3}\frac{2\lambda }{\chi _3}} \end{aligned}$$

To reproduce experimental results of Ref. [5], we take for our modeling $\lambda =1$ kOe$^2$, $\gamma =17.6\frac{10^9 \mathrm{rad}~\mathrm{s}^{-1}}{\mathrm{kOe}}$ (2.80 GHz kOe$^{-1}$), $I_1=I_2=1.42\times 10^{-5}\frac{ \mathrm{kOe}^2}{(10^9~\mathrm{rad}~~ \mathrm{s}^{-1})^2}$, $I_3=7.99\times 10^{-6}\frac{ \mathrm{kOe}^2}{(10^9 \mathrm{rad}~\mathrm{s}^{-1})^2}$. Results of the modeling for this case are shown at the Fig. 1.

Finally, it is a spiral magnet LiCu$_2$O$_2$ [8]. Despite the orthorhombic symmetry $I_1=I_2$ as there is no anisotropy in the plane of the spiral structure, $U_\mathrm{A}=\frac{A}{2}l_{3z}^2+\frac{B}{2} l_{3y}^2$ ($A\le B \le 0$). It turns out that in the case of LiCu$_2$O$_2$ A and B constants in anisotropy energy are close within 1 %. Thus, normal to the spin plane $\mathbf {l}_3$ rotates almost freely in the (yz) plane. One of the oscillation frequencies corresponds to the rotation in the plane of spiral structure and is always zero since phase of the helix can be changed at no energy cost. Two other modes have non-zero zero-field frequencies

$$\begin{aligned} \omega _{10}^2= & {} -\frac{A}{I_1+I_3}=-\gamma ^2 \frac{A}{\chi _1}\\ \omega _{20}^2= & {} \frac{B-A}{I_1+I_3}=\gamma ^2\frac{B-A}{\chi _1}<\omega _{10}^2\\ \end{aligned}$$

For LiCu$_2$O$_2$ $\chi _3>\chi _1$, in this case at $\mathbf {H}||z$ vector $\mathbf {l}_3$ always remains aligned along z and non-zero oscillation frequencies are

$$\begin{aligned} \omega _{1,2}^2= & {} \frac{\omega _{10}^2+\omega _{20}^2}{2}+ \gamma ^2 H^2\frac{I_3^2+I_1^2}{\left( I_3+I_1\right) ^2}\\&\pm \sqrt{\left( \frac{\omega _{10}^2-\omega _{20}^2}{2}\right) ^2+4\frac{\gamma ^4H^4 I_1^2I_3^2}{\left( I_1+I_3\right) ^4}+2\frac{\gamma ^2 H^2 (\omega _{10}^2+ \omega _{20}^2)I_3^2}{\left( I_1+I_3\right) ^2}} \end{aligned}$$

At $\mathbf {H}||x$ spin plane rotates orthogonally to the magnetic field at some critical field. Critical field $H_{\mathrm{cx}}=\frac{\omega _{10}}{\gamma }\sqrt{\frac{I_1+I_3}{I_1-I_3}}=\frac{\omega _{10}}{\gamma }\sqrt{\frac{\chi _1}{\chi _3-\chi _1}}$ and oscillation frequencies are

$$\begin{aligned}H<H_{\mathrm{cx}}:\\\omega _1^2=\, & {} \omega _{10}^2+\gamma ^2 H^2\\ \omega _2^2=\, & {} \omega _{20}^2\\H>H_{\mathrm{cx}}:\\ \omega _{1,2}^2=\, & {} \frac{\omega _{20}^2-2\omega _{10}^2}{2} +\frac{I_1^2+I_3^2}{\left( I_1+I_3\right) ^2}\gamma ^2 H^2 \\&\pm \sqrt{\frac{\omega _{20}^4}{4}+ 2\gamma ^2 H^2\frac{\left( \omega _{20}^2-2\omega _{10}^2\right) I_3^2}{\left( I_1+I_3\right) ^2} +4\frac{\gamma ^4 H^4 I_1^2 I_3^2}{\left( I_1+I_3\right) ^4}} \end{aligned}$$

To reproduce experimental results of Ref. [8], we take for our modeling $\gamma =17.59\frac{10^9 \mathrm{rad}~ \mathrm{s}^{-1}}{\mathrm{kOe}}$ (corresponds to 2.80 GHz kOe$^{-1}$), $A=-1$ kOe$^2$, $B=-0.99$ kOe$^2$, $I_1=I_2=1.85\times 10^{-7}\frac{ \mathrm{kOe}^2}{(10^9~\mathrm{rad}~\mathrm{s}^{-1})^2}$, $I_3=6.18\times 10^{-8}\frac{ \mathrm{kOe}^2}{(10^9 \mathrm{rad}~\mathrm{s}^{-1})^2}$

The detailed comparison of the modeled curves with analytical predictions is given in a supplementary material.