Magnetic exchange interactions in ${\mathrm{BaMn}}_2{\mathrm{As}}_2$: A case study of the $J_1$-$J_2$-$J_c$ Heisenberg model

${\mathrm{BaMn}}_2{\mathrm{As}}_2$ is unique among Ba$T_2$As${}_2$ compounds crystallizing in the body-centered-tetragonal ${\mathrm{ThCr}}_2{\mathrm{Si}}_2$ structure, which contain stacked square lattices of $3d$ transition metal $T$ atoms, since it has an insulating large-moment ($3.9{\mu }_{\mathrm{B}}/$Mn) G-type (checkerboard) antiferromagnetic (AF) ground state. We report measurements of the anisotropic magnetic susceptibility $\chi$ versus temperature $T$ from 300 to 1000 K of single crystals of ${\mathrm{BaMn}}_2{\mathrm{As}}_2$, and magnetic inelastic neutron scattering measurements at 8 K and ${}^{75}$As nuclear magnetic resonance (NMR) measurements from 4 to 300 K of polycrystalline samples. The Néel temperature determined from the $\chi \left(T\right)$ measurements is $T_{\mathrm{N}}=618$(3) K. The measurements are analyzed using the $J_1$-$J_2$-$J_c$ Heisenberg model for the stacked square lattice, where $J_1$ and $J_2$ are, respectively, the nearest-neighbor (NN) and next-nearest-neighbor intraplane exchange interactions and $J_c$ is the NN interplane interaction. Linear spin wave theory for G-type AF ordering and classical and quantum Monte Carlo simulations and molecular field theory calculations of $\chi \left(T\right)$ and of the magnetic heat capacity $C_{\mathrm{mag}}\left(T\right)$ are presented versus $J_1$, $J_2$, and $J_c$. We also obtain band-theoretical estimates of the exchange couplings in ${\mathrm{BaMn}}_2{\mathrm{As}}_2$. From analyses of our $\chi \left(T\right)$, NMR, neutron scattering, and previously published heat capacity data for ${\mathrm{BaMn}}_2{\mathrm{As}}_2$ on the basis of the above theories for the $J_1$-$J_2$-$J_c$ Heisenberg model and our band-theoretical results, our best estimates of the exchange constants in ${\mathrm{BaMn}}_2{\mathrm{As}}_2$ are $J_1\approx 13$ meV, $J_2/J_1\approx 0.3$, and $J_c/J_1\approx 0.1$, which are all antiferromagnetic. From our classical Monte Carlo simulations of the G-type AF ordering transition, these exchange parameters predict $T_{\mathrm{N}}\approx 640$ K for spin $S=5/2$, in close agreement with experiment. Using spin wave theory, we also utilize these exchange constants to estimate the suppression of the ordered moment due to quantum fluctuations for comparison with the observed value and again obtain $S=5/2$ for the Mn spin.

Figure 1

Crystal and magnetic structures of ${\mathrm{BaMn}}_2{\mathrm{As}}_2$ (Ref.13). The crystal structure is body-centered-tetragonal ${\mathrm{ThCr}}_2{\mathrm{Si}}_2$-type in which the Mn atoms form a square lattice within the $ab$ plane, the axes of which are rotated by 45${}^{\circ }$ with respect to the a and b unit cell axes. The Mn atoms in adjacent layers are directly above or below each other along the $c$ axis. The magnetic structure is a G-type antiferromagnetic structure in which the magnetic moments of nearest-neighbor spins are antiparallel both in the $ab$ plane and along the $c$ axis. The ordered moment at $T=0$ is $3.9\left(1\right){\mu }_{\mathrm{B}}/$Mn.[13]

Figure 2

Collinear commensurate in-plane magnetic structures in the $J_1$-$J_2$-$J_c$ model for the square lattice antiferromagnet. The top panel shows the G-type (Néel or checkerboard) AF structure where nearest-neighbor spins are aligned antiparallel. The bottom panel shows stripe-type ordering, along with the definitions of the in-plane exchange constants $J_1$ and $J_2$. A $J_2$ interaction is present for both diagonals of each square. According to Eqs. (4), the G-type in-plane ordering is favored if $J_2<J_1/2$, whereas the stripe-type ordering is favored if $J_2>J_1/2$. By examining the bottom panel, one sees that the stripe magnetic structure consists of two interpenetrating G-type magnetic structures, each respectively consisting of next-nearest-neighbor spins.

Figure 3

Inelastic neutron scattering data from a powder sample of BaMn${}_2$As${}_2$ as measured on the Pharos spectrometer with incident energies (a) $E_{\mathrm{i}}=150$ meV and (b) $E_{\mathrm{i}}=200$ meV. The white lines delineate scattering angles of 7${}^{\circ }$ and 30${}^{\circ }$ where the magnetic scattering was estimated. Panels (c) and (d) show calculations of the polycrystalline averaged spin wave scattering using a Heisenberg model. The calculations in (c) and (d) are identical; however, panel (c) shows the trajectories of the angle summation limits for $E_{\mathrm{i}}=150$ meV and (d) for $E_{\mathrm{i}}=200$ meV.

Figure 4

$Q$ dependence of the inelastic neutron scattering intensity averaged over several energy ranges with incident energies (a) $E_{\mathrm{i}}=150$ meV and (b) $E_{\mathrm{i}}=200$ meV. The red lines show calculations of the polycrystalline averaged spin wave scattering using the $J_1$-$J_2$-$J_c$ Heisenberg model.

Figure 5

Energy dependence of the inelastic neutron scattering intensity averaged over a scattering angle range from 7${}^{\circ }$ to 30${}^{\circ }$ with incident energies (a) $E_{\mathrm{i}}=150$ meV and (b) $E_{\mathrm{i}}=200$ meV. The magnetic intensity was extracted from the total scattering as described in the text. The red lines show calculations of the polycrystalline averaged spin wave scattering using the $J_1$-$J_2$-$J_c$ Heisenberg model.

Figure 6

The spin wave dispersion of the $J_1$-$J_2$-$J_c$ model along various symmetry directions for three different combinations of the exchange parameters: $J_2/J_1=0$ and $J_c/J_1=1$ (top black curve); $J_2/J_1=0$ and $J_c/J_1=0.1$ (middle red curve); and $J_2/J_1=0.25$ and $J_c/J_1=0.1$ (bottom blue curve). Energies are reported in units of $S{J}_1$. The $\Gamma$ point in the Brillouin zone corresponds to wave vector q $=$ 0. The q values corresponding to the high-symmetry M, X, and N points are given in Table . These wave-vector directions are written with respect to the lattice translation vectors of the $I$4/$mmm$ chemical unit cell. The labels $\Sigma$, F, U, Y, $\Delta$, and $\Lambda$ correspond to high-symmetry lines in reciprocal space. In the far right-hand panel, the energy versus the spin wave density of states is shown for $J_2/J_1=0$ and $J_c/J_1=1$ (top gray region) and $J_2/J_1=0.25$ and $J_c/J_1=0.1$ (bottom blue region).

Figure 7

(a) Magnetic susceptibility $\chi$ versus temperature $T$ of single crystals of ${\mathrm{BaMn}}_2{\mathrm{As}}_2$ with the applied magnetic field parallel to the $c$ axis (${\chi }_c$) and to the $ab$ plane (${\chi }_{ab}$). The individual symbols are the data previously reported in Ref.11. The solid curves are the present data obtained in an applied magnetic field $H=3$ T. The Néel temperature $T_{\mathrm{N}}$ is indicated. (b) Expanded plot of ${\chi }_c\left(T\right)$ for temperatures around $T_{\mathrm{N}}$. The temperature of the maximum slope of ${\chi }_c\left(T\right)$ gives $T_{\mathrm{N}}=618$(3) K.

Figure 8

Parallel susceptibility ${\chi }_{\parallel }$ versus temperature $T$ through the Néel temperature $T_{\mathrm{N}}$ for G-type AF ordering using the $J_1$-$J_2$-$J_c$ model in MFT for various values of $f=\theta /T_{\mathrm{N}}$, as listed, for spins (a) $S=1/2$ and (b) $S=5/2$. The order of the curves from top to bottom is the same as in the figure legends. At temperatures $T>T_{\mathrm{N}}$, $\chi$ is isotropic. For $T<T_{\mathrm{N}}$, the perpendicular susceptibility is constant, ${\chi }_{\perp }={\chi }_{\perp }\left(T_{\mathrm{N}}\right)$ (not shown). The G-type AF state is unstable against the stripe-AF state for $f>3$.

Figure 9

Parallel susceptibility ${\chi }_{\parallel }$ versus temperature $T$ through the Néel temperature $T_{\mathrm{N}}$ for the $J_1$-$J_2$-$J_c$ model in MFT for spins $S=1/2,$ 5/2, and 10 and $f=\theta /T_{\mathrm{N}}$ values of (a) 0, (b) 1, and (c) 3. The order of the curves in each panel from top to bottom is the same as in the figure legends. The value $f=1$ corresponds to the conventional bipartite lattice with $J_2=0$.

Figure 10

Ordered moment ${\mu }_z\equiv {\mu }_z^{†}$ versus temperature $T$ from molecular field theory of the $J_1$-$J_2$-$J_c$ model for a collinear antiferromagnet for classical spins and for several quantum spins as listed, where the saturation moment is ${\mu }_{\mathrm{sat}}=gS{\mu }_{\mathrm{B}}$. The order of the curves from top to bottom is the same as in the figure legend. Remarkably, the results are independent of $J_2$.

Figure 11

Magnetic component of the heat capacity $C_{\mathrm{mag}}$, divided by the molar gas constant $R$, versus the ratio of the temperature $T$ to the Néel temperature $T_{\mathrm{N}}$ according to the molecular field theory [Eq. (56)] of the $J_1$-$J_2$-$J_c$ model for a collinear antiferromagnet for classical spins and for several quantum spins as listed. The order of the curves from top to bottom is the same as in the figure legend. As in Fig. 10, the results are independent of $J_2$.

Figure 12

Magnetic entropy $S_{\mathrm{mag}}/R$ for quantum spins versus reduced temperature $T/T_{\mathrm{N}}$ according to molecular field theory for the quantum spins $S$ indicated. The order of the curves from top to bottom is the same as in the figure legend. In the disordered state at $T>T_{\mathrm{N}}$, the magnetic entropy is the constant value $S_{\mathrm{mag}}=R\ln (2S+1)$ for each $S$, as indicated along the right-hand ordinate.

Figure 13

Comparison of the MFT prediction below $T_{\mathrm{N}}$ of the anisotropic susceptibilities in Eqs. (55) and (B18) versus temperature with the experimental data from Fig. 7a.

Figure 14

Ordered moment ${\mu }_z$ versus temperature $T$ measured for ${\mathrm{BaMn}}_2{\mathrm{As}}_2$ from Ref.13 (solid red circles). The Néel temperature $T_{\mathrm{N}}$ and saturation moment ${\mu }_{\mathrm{sat}}$ are given in the figure. Also shown are the MFT predictions for quantum spins $S=1/2$, 2, and 5/2 from Eq. (C1) (solid and dashed curves), where the order of the curves from top to bottom is the same as in the figure legend.

Figure 15

Heat capacity $C$ versus temperature $T$. The experimental $C_{\mathrm{p}}\left(T\right)$ data obtained for a single crystal of ${\mathrm{BaMn}}_2{\mathrm{As}}_2$ (Ref.11) are shown as the solid black circles. The lattice heat capacity for a Debye temperature ${\Theta }_{\mathrm{D}}=300$ K is plotted versus $T$ as the solid red curve. The sum of the lattice heat capacity and the magnetic heat capacity for spin $S=2$ is shown as the dashed blue curve.

Figure 16

Spin wave dispersion relation $\hslash \omega /4J_{1}S$ of the isotropic two-dimensional square lattice over the Brillouin zone of the primitive tetragonal space lattice. The dispersion relation is doubly degenerate everywhere. At low temperatures, there are two distinct doubly degenerate spin wave branches that are relevant, one at the $\Gamma$ point at (0,0) and the other at $(\frac{\pi }{a},\frac{\pi }{a})$ (and equivalent points).

Figure 17

(a) Normalized magnetic spin susceptibility $\chi J_1/\left(N{g}^2{\mu }_{\mathrm{B}}^2\right)$ versus normalized temperature $k_{\mathrm{B}}T/\left[J_{1}S(S+1)\right]$ for the classical spin $S$ Heisenberg square lattice with $J_c=0$ and $J_2/J_1=-0.4$ to 0.4, with $S(S+1)$ replacing $S^2$. The lattice size is $80\times 80$ in each case. (b) Expanded plot at low temperatures (open circles) of the data for $J_2=0$ in (a). The error bars are shown and are inside the open circles. A linear fit to the data up to a reduced temperature of 0.14 is shown as the solid straight line, and the dashed line is an extrapolation of the fit. The coefficients of the fit are listed in Eq. (89).

Figure 18

Normalized magnetic spin susceptibility $\chi (J_1+J_1)/\left(N{g}^2{\mu }_{\mathrm{B}}^2\right)$ versus normalized temperature $k_{\mathrm{B}}T/\left[(J_1+J_2)S(S+1)\right]$ determined from classical Monte Carlo simulations for the spin $S$ Heisenberg square lattice with $J_c=0$ and $J_2/J_1=-0.4$ to 0.4, as indicated, with $S(S+1)$ replacing the classical $S^2$. The lattice size is $80\times 80$ in each case. The axis scaling is superior at high temperatures to that in Fig. 17a. A plot of the Curie-Weiss law in Eq. (86) is shown by the black dashed curve.

Figure 19

Classical Monte Carlo (CMC) simulations on $80\times 80$ spin lattices of the magnetic heat capacity $C_{\mathrm{mag}}$ divided by the molar gas constant $R$ versus the scaled temperature $k_{\mathrm{B}}T/J_{1}S(S+1)$ for $J_c=0$ and $J_2/J_1=-0.4$ to 0.4 as shown.

Figure 20

(a) Magnetic heat capacity and (b) magnetic susceptibility $\chi$ versus temperature $T$ from classical Monte Carlo simulations for the spin $S$ Heisenberg square lattice with $J_2/J_1=0.1$ and with $J_c/J_1=0$, 0.02, 0.05, and 0.1, as indicated. The order of the curves from top to bottom is the same as in the respective figure legends. The lattice size in each case is $20\times 20\times 10$.

Figure 21

(a) Magnetic heat capacity $C_{\mathrm{mag}}$ and (b) spherically averaged magnetic susceptibility $\chi$ versus temperature $T$ from classical Monte Carlo simulations for the spin $S$ Heisenberg square lattice with $J_2/J_1=0$ to 0.4, as indicated, and with fixed $J_c/J_1=0.1$. The order of the curves from right to left is given in the figure (a) legend. The lattice size in each case is $20\times 20\times 10$.

Figure 22

Antiferromagnetic ordering temperature $T_{\mathrm{N}}$ versus (a) the interlayer coupling $J_c$ for fixed values of $J_2/J_1$ and (b) $J_2/J_1$ at fixed $J_c/J_1$. In (b), the order of the curves from top to bottom is the same as in the figure legend. The solid curves are a global fit to all the data by Eq. (96), using the parameters in Eqs. (93) and (95).

Figure 23

Normalized magnetic spin susceptibility $\chi J_1/\left(N{g}^2{\mu }_{\mathrm{B}}^2\right)$ versus normalized temperature $k_{\mathrm{B}}T/\left[J_{1}S(S+1)\right]$ determined from quantum Monte Carlo (QMC) simulations for quantum Heisenberg square lattices with (a) $J_2/J_1=0$ and spins $S=5/2,$ 2, 3/2, 1, and 1/2, and (b) $J_2/J_1=0.1$ and spins $S=5/2$ and 2. The data for the semiclassical model (green) with $J_2/J_1=0$ and 0.1 are included in the two panels, respectively. The lattice size for each simulation is indicated in the respective figure.

Figure 24

Normalized QMC magnetic spin susceptibility $\chi J_1/\left(N{g}^2{\mu }_{\mathrm{B}}^2\right)$ versus normalized temperature $k_{\mathrm{B}}T/\left[J_{1}S(S+1)\right]$ data (symbols) at low temperatures from Fig. 23a for spins $S=1/2$ (bottom), 1, 3/2, 2, and 5/2 (top). The error bars for the QMC data are also plotted. The corresponding Hasenfratz-$\mathrm{Niedermayer}+\mathrm{spin}$ wave theory predictions ($\mathrm{HN}+\mathrm{SWT}$) for the low-temperature behaviors (Refs. 52 and 53) are also shown for these spin values as solid curves.

Figure 25

Magnetic heat capacity $C_{\mathrm{mag}}$ divided by the molar gas constant $R$ versus normalized temperature $k_{\mathrm{B}}T/\left[J_{1}S(S+1)\right]$ for the square lattice with only nearest-neighbor couplings $J_1$ (exchange constants $J_2=J_c=0$). Quantum Monte Carlo (QMC) data (solid circles) for spins $S=1/2$ (bottom), 1, 3/2, 2, and 5/2 (top) are shown, together with classical Monte Carlo (CMC) data from Fig. 20a (solid black curve at the top) and the first term in the high-temperature series expansion HTSE for the magnetic heat capacity from Eq. (A9) using $z=4$ (dashed black curve). The error bars for the QMC data are also plotted, and the lines connecting the data points are guides for the eye. The order of the Monte Carlo data from top to bottom is the same as in the figure legend.

Figure 26

Low-temperature magnetic heat capacity $C_{\mathrm{mag}}$ times $S^2$ divided by the molar gas constant $R$ and $T^2$ versus normalized temperature squared, ${(k_{\mathrm{B}}T/J_1)}^2$, for the square lattice with only nearest-neighbor couplings $J_1$ (exchange constants $J_2=J_c=0$). Quantum Monte Carlo (QMC) data (solid symbols) for spins $S=1/2$, 1, and 3/2 are shown, together with error bars. The dotted lines are fits of the data by Eq. (103) assuming the specific $A\left(S\right)$ $y$ intercepts given in the text below Eq. (103). Only the slopes $D\left(S\right)$ were fitted. The fitting ranges of $T^2$ were 0–0.06 for $S=1/2$, 0–0.25 for $S=1$, and 0–0.5 for $S=3/2$.

Figure 27

Product of the maximum susceptibility ${\chi }^{\max }$ per mole of spins and the temperature $T^{\max }$ at which it occurs for the $J_1$-$J_2$ model on a Heisenberg square spin lattice for spins (a) $S=2$ and (b) $S=5/2$, versus the ratio $J_2/J_1$ of the diagonal next-nearest-neighbor coupling to the nearest-neighbor coupling. Here a $g$ factor $g=2$ was assumed. The data were obtained using quantum Monte Carlo (blue, QMC) and classical Monte Carlo (red, CMC) simulations on the lattice sizes indicated. The ${\chi }^{\max }{T}^{\max }$ values obtained from CMC for the three-dimensional lattices are very insensitive to the coupling $J_c$ between layers up to at least $J_c/J_1=0.1$, as shown in Table . The solid black curves in (a) and (b) are least-squares fits to the respective data by the second-order polynomials in Eqs. (105).

Figure 28

Two fits (Fit 1 and Fit 2, dashed curves) of the high-temperature magnetic susceptibility $\chi$ of ${\mathrm{BaMn}}_2{\mathrm{As}}_2$ from Fig. 7a by CMC simulations. The fits are only valid above $T_{\mathrm{N}}$ but are extrapolated to lower temperatures. The fit parameters are given in Table . The temperatures of the breaks in slope of the fits are discernable and denote the predicted Néel temperatures in Eq. (96) for the respective parameters, which are somewhat above the observed value.

Figure 29

${}^{75}$As NMR spectra for polycrystalline ${\mathrm{BaMn}}_2{\mathrm{As}}_2$ at different temperatures. The solid red line is a fit to the spectrum at 4.2 K.

Figure 30

Nuclear spin-lattice relaxation rate ($1/T_1$) measured at the ${}^{75}$As site versus temperature $T$. The solid and dashed lines represent $T^3$ and $T^5$ behaviors, respectively. The slope of the former line fitted to the 50–300 K data is $2.51\times {10}^{-6}{\mathrm{s}}^{-1}{\mathrm{K}}^{-3}$. (Inset) ${\left(T_{1}T\right)}^{-1}$ versus $T$.

Figure 31

Total energy $E$ of a system when the polar angle ${\theta }_{ij}$ between two local magnetic moments $i$ and $j$ is varied in band theory. The energies of the ferromagnetic $E_{\mathrm{FM}}$ and antiferromagnetic $E_{\mathrm{AF}}$ magnetic structures and their difference are indicated. Here the antiferromagnetic state is the ground state. For a ferromagnetic ground state, the minimum in energy would be at ${\theta }_{ij}=0$.

Figure 32

Reduction $S-\langle S\rangle$ in the ordered spin $\langle S\rangle$ from its value $S$ in the absence of quantum fluctuations versus $J_2/J_1$ according to linear spin wave theory to first order (solid black curves) and second-order (solid circles) in $1/S$ for (a) $J_c=0$ and (b) $J_c/J_1=0.1$.

Figure 33

Influence of a perpendicular magnetic field ${\mathbf{H}}_{\perp }$ on the sublattice magnetizations of an ordered antiferromagnet. The ${\mathbf{H}}_{\perp }$ tilts the ordered sublattice magnetizations ${\mathbf{M}}_1$ and ${\mathbf{M}}_2$ that are initially pointed along the $z$ axis by an angle $\gamma$ toward the applied field ${\mathbf{H}}_{\perp }$ along the $x$ axis. The angle $\gamma$ is greatly exaggerated for clarity.