Semiclassical model of magnons in double-layered antiferromagnets

TL;DR

The paper addresses how double-layered antiferromagnetic order can be stabilized and how its magnon spectrum evolves in realistic materials. It develops two 1D models (linear chain and railroad trestle) to obtain analytic magnon dispersions and stability criteria, and then bridges to CrN by computing ab initio exchange parameters. For CrN, four NN exchanges emerge with strong inter-sublattice couplings that vary with distance due to competition between AFM Cr–Cr direct exchange and FM Cr–N–Cr superexchange, captured by a distance-power law, yielding a magnon gap at the X point and a Néel temperature near TN ≈ 284 K. The results show that orthorhombic distortion lifts magnetic frustration and validates a first-order magnetostructural transition, providing a semiclassical framework for magnetism and magnons in double-layered AFMs.

Abstract

The stability and magnonic properties of double-layered antiferromagnets are investigated using two model systems, a linear chain (LC) and a more complex chain of railroad trestle (RT) geometry, and the results are confronted with properties of the real material CrN. The spin-paired order ($\cdots{+}{+}{-}{-}\cdots$) in LC requires alternating ferromagnetic and antiferromagnetic (AFM) exchanges, whereas in RT, an analogous order remains stable even when all interactions are AFM within certain analytical constraints. The rock-salt structure of CrN evokes clear magnetic frustration since Cr atoms in a face-centered cubic lattice form links to twelve nearest neighbors (NNs) all equivalent and AFM. Nonetheless, the magnetostructural transition to an orthorhombically distorted phase below $T_\text{N} = 287~\text{K}$ leads to four different NN Cr-Cr distances and consequently, to a large diversification of the exchange strength, which suppresses the frustration and allows for stable double-layered AFM order of CrN. This behavior originates from a competition at each NN link between Cr-Cr direct exchange and 90$^\circ$ Cr-N-Cr superexchange, both exhibiting specific power-law dependences on the interatomic distance. Finally, based on the $\textit{ab initio}$ calculated exchange parameters, the magnon spectrum and temperature evolution of ordered magnetic moments are derived.

Figures (9)

• Figure 1: (a) One-dimensional (1D) model of double-layered antiferromagnets in the linear chain structure. The spin arrangement is represented by the arrows (circles) which are upwards ($\bullet$) for $S^z = +S$ and downwards ($\circ$) for $-S$. The exchange parameters $J_1$ and $J_2$ are for the intra- ($\multimapdotboth$, $\multimapboth$) and inter-sublattice ($\multimapdotbothB$) interactions, respectively. All sites have the same spacing $d$ and are indexed by $4p + n$, where $n = 0$, 1, 2, and 3 in the $p$th unit cell. (b) 1D model of the railroad trestle type, possessing an additional exchange parameter $J_3$. Each site has one nearest neighbor (NN) of same spin (related by $J_1$) and three NNs of opposite spin (related by $J_2$ and $J_3$).
• Figure 2: Selected magnon dispersion relations of double-layered antiferromagnets in the models of (a) linear chain and (b) railroad trestle (RT). Four types of lines indicate $\gamma = J_1/J_2 = -1$ (red), $-1/3$ (black dashed), 0 (black solid), and $1/3$ (blue). The characteristic frequencies are $\omega_0 = \sqrt{2|\omega_1\omega_2|}$ and $\omega_0' = \sqrt{2}|\omega_2|$. Here positive $\gamma$ of $1/3$ means all magnetic interactions in RT being antiferromagnetic.
• Figure 3: (a) The crystal structure and magnetic ordering of CrN projected on the orthorhombic $ab$ plane. The symbols "$\bullet$ ($\circ$)" and "$+$" indicate the Cr sites of $S^z = +S$ ($-S$) and N sites, respectively. Four kinds of magnetic exchange parameters $J_1$ (red), $J_2$ (blue), $J_3$ (green), and $J_4$ (grey) are marked. For more detailed structural description, see Ref. Ahn2024. (b) Schematic two-dimensional picture of exchange links. The full lines denote the motif running along the $a$ axis that closely reminds the railroad trestle in Fig. \ref{['fig:struct']}(b) with magnetic exchange parameters $J_1$, $J_2$, and $J_3$. The dashed lines show links toward Cr atoms in neighboring units that include in addition to $J_1$ and $J_2$, also a new parameter $J_4$. The projection to the $a$ axis, seen on the right side, refers to the spin-paired linear chain of Fig. \ref{['fig:struct']}(a).
• Figure 4: Calculated magnetic exchange parameters for three variants of the CrN structure: the hypothetical cubic phase of rock-salt (RS) type (purple squares), the intermediate structure upon orthorhombic distortion but without (w/o) local atomic shifts (AS) (empty triangles), and the real fully optimized orthorhombic phase with (w/) AS (black dots with $J_1$, …, $J_4$ denoted). The dependence of the last data on the distance between nearest-neighbor (NN) Cr pairs of $S = 3/2$ is fitted as an interplay of Cr--Cr direct exchange and 90° Cr--N--Cr superexchange (dashed line, see the text).
• Figure 5: (a) Magnon dispersion relation of antiferromagnetic CrN. The presence of a magnon gap at the X point shows good agreement with the model analysis in Fig. \ref{['fig:model']}. (b) Calculated temperature evolution of local magnetic moments from the classical Monte-Carlo simulations. The ordering temperature of $T_\text{N} = 284$ K was obtained by fitting with $(1 - T/T_\text{N})^{1/3}$.