⇦Chinese Physics Letters, 2022, Vol. 39, No. 6, Article code 067501 Microscopic Magnetic Origin of Rhombohedral Distortion in NiO Guangmeng He (何光萌)1,2†, Huimin Zhang (张慧敏)1,2†, Jinyang Ni (倪斤阳)1,2, Boyu Liu (刘博宇)1,2, Changsong Xu (徐长松)1,2*, and Hongjun Xiang (向红军)1,2* Affiliations 1Key Laboratory of Computational Physical Sciences (Ministry of Education), Institute of Computational Physical Sciences, and Department of Physics, Fudan University, Shanghai 200433, China 2Shanghai Qi Zhi Institute, Shanghai 200030, China Received 21 March 2022; accepted 22 April 2022; published online 29 May 2022 †Guangmeng He and Huimin Zhang contributed equally to this work.
*Corresponding authors. Email: csxu@fudan.edu.cn; hxiang@fudan.edu.cn Citation Text: He G M, Zhang H M, Ni J Y et al. 2022 Chin. Phys. Lett. 39 067501    Abstract Numerous investigations have been conducted to explore the structural phase transition in antiferromagnetic 3$d$ transition metal monoxides accompanied by appearance of magnetic phase transition. However, how the spins induce distortion in the high symmetric structure has not yet been fully understood. In this study, the monoxide NiO is used as an example to investigate what lowers the structural symmetry. By comparing two different magnetic structures, our results reveal that the spin–lattice coupling is responsible for such a structural distortion. Then, a spin–lattice model, including the strain component, is constructed to simulate the transition procedure. Moreover, the results from the first-principles calculations are used to compare with our model results. Both first-principles calculations and model simulations clarify the structural phase transition caused by a unique magnetic arrangement.
cpl-39-6-067501-fig1.png
cpl-39-6-067501-fig2.png
 DOI:10.1088/0256-307X/39/6/067501 © 2022 Chinese Physics Society Article Text Antiferromagnetic (AFM) Mott insulators not only are fundamentally crucial, but also have technical applications.[1–3] Among such Mott antiferromagnets, NiO is critical in studying electronic, magnetic, and structural properties of strong correlated 3$d$ electron materials. In NiO, the partially occupied Ni $d$ shell and strong superexchange interactions result in a high magnetic transition temperature ($T_{\rm N} = 523$ K),[4–6] making NiO a promising candidate for room-temperature spintronics applications.[7–9] Above $T_{\rm N}$, NiO crystallizes into the face-centered-cubic structure (space group $Fm\bar{3}m$) [Fig. 1(a)] in the paramagnetic state.[10] As the temperature decreases, a slight trigonal structural distortion appears and NiO crystallizes into the rhombohedral structure (space group $R\bar{3}m$) accompanying the existence of the AFM phase.[10–12] Such a distortion also happens in other antiferromagnets, such as MnO,[13–15] which may also have the same mechanism. The AFM properties of NiO have been reported in the literature,[2] and its ground magnetic state is discovered to be a ferromagnetic (FM) (111) layer stacked antiferromagnetically along the [111] direction [Fig. 1(b)]. The [111] direction is set as the $z$ axis in this study. The magnetic and structural phase transitions occur simultaneously due to the competition of nearest-neighbor (NN) and next-nearest-neighbor (NNN) exchange interactions.[16] The crystal structure contracts along the [111] direction from a cubic structure to a rhombohedral structure as the specific AFM arrangement emerges in NiO. The x-ray diffraction study by Rooksby and Slack showed that such a distortion causes the rhombohedral angle $\alpha$ [Fig. 1(a)] from 60$^{\circ}$ at high temperature to be 60.07$^{\circ}$ below the Neel temperature.[5] The distance of NNN Ni ions was measured to be 4.176 Å at 298 K and 4.171 Å near 0 K.[10] Several attempts have been made to investigate the transition mechanism based on the structural distortion that accompanies the magnetic phase transition. Greenwald and Smart first proposed that the ordering of an antiferromagnet was responsible for the observed lattice distortions. They also highlighted that such distortion was induced by the exchange striction.[1] However, Li[17] proposed that NiO must have the type-B arrangement, as shown in Fig. 2(d), and considered that the distortion was induced by anisotropy magnetostriction. However, Roth[2] reported that magnetic dipole–dipole interactions predominated and type-A arrangement [Fig. 2(c)] should be the magnetic ground state. Further, Rodbell and Owen[15] derived an expression for the distortion with a molecular-field approach and proposed that the anisotropic strain plays a significant role in NiO and MnO. Other efforts have also been made to focus on the individual aspects of the structural distortions, such as single-ion magnetostriction,[18] Heisenberg model with distance-dependent interactions,[19] lattice dynamical property calculations,[20] and Heisenberg exchange analysis.[21] However, the relationship between the magnetic and structural phase transitions remains unclear and requires a more detailed study for interpretation.
cpl-39-6-067501-fig1.png
Fig. 1. (a)–(d) Grey sphere: Ni; red sphere: O. The red and yellow arrows represent the spin-up and spin-down states of Ni ions. Here, $\alpha$, $\beta$, $\alpha'$, and $\beta'$ denote the Ni–Ni–Ni angles, whereas $\gamma$, $\theta$, $\gamma'$, and $\theta'$ denote the Ni–O–Ni angles. The bond lengths are labeled $l_{1}$, $l_{1}'$, and $l_{1}''$. (a) The relaxed structure in cubic symmetry, where $J_{1}$ and $J_{2}$ represent NN and NNN, respectively, exchange interactions. (b) The distorted crystal structure is optimized from the cubic structure with experimental parameters in case (iii). $J_{1}$ in high symmetry structure turns into different groups, $J_{1}'$ and $J_{1}''$, due to the distortion. The bonds corresponding to $J_{1}'$ and $J_{1}''$ are shown as green and blue lines, respectively. (c) $J_{1}'$ represents the six-fold inter-plane exchange interactions. (d) $J_{1}''$ represents the six-fold intra-plane exchange interactions.
cpl-39-6-067501-fig2.png
Fig. 2. (a) The simple-cubic sublattice in which two adjacent Ni ions are AFM. (b) The combination of the four sublattices marked in different colors. Each sublattice satisfies the AFM relationship shown in (a). (c) The type-A magnetic structure. (d) The type-B magnetic structure. The red arrows in (c) and (d) represent up-spins, and the yellow arrows represent down-spins. Note that only Ni ions are shown in spheres with green, blue, purple, and grey.
Both density functional theory (DFT) calculations and spin Hamiltonian model simulations were performed in this study. An effective spin Hamiltonian model is constructed to describe the energy of NiO within a cubic structure. Monte Carlo (MC) simulations with such Hamiltonian are performed to determine the magnetic ground states, i.e., types A and B. Further, symmetry analyses are performed to deduce that only type A with rhombohedral distortion is the magnetic and structural ground state. Notably, a spin–lattice model is proposed to analyze the microscopic mechanism of NiO's rhombohedral distortion. Meanwhile, the DFT calculations were performed as a means of comparison to verify the accuracy of our model study. The DFT calculations are performed using the Vienna ab initio simulation package (VASP)[22,23] with the projector-augmented wave method. The Perdew–Burke–Ernzerhof functional is used to describe electron interactions,[24] and the generalized gradient approximation with Hubbard $U$ (GGA+$U$) method is used to treat electron exchange and correlation. The Hubbard effective $U$ is imposed on the 3$d$ orbitals of the Ni ion using the Dudarev approach.[25] According to the previous studies,[3,26] $U_{\rm eff}$ ($U_{\rm eff}=U - J$) is set to be 4.5 eV in our calculations. In addition, different $U_{\rm eff}$ values ranging from 4 to 6 eV are tested, and we found that a reasonable $U_{\rm eff}$ choice does not change our main results. The cut-off energy is 550 eV and Ni 3$d$, O 2$s$, and O 2$p$ are considered to be valence states. The internal degrees of freedom are optimized by relaxation of shape of the unit cell and the atomic positions until the Hellmann–Feynman force is less than 1 meV/Å. For NiO, the unit cell contains one Ni ion and one O ion. A $2 \times 2 \times 2$ supercell is adopted to calculate the energies of different spin configurations, and the $k$-mesh is set to be $13 \times 13 \times 13$ for optimization and $19 \times 19 \times 19$ for static calculation. A $3 \times 3 \times 3$ supercell is constructed to calculate the parameters of magnetic interaction, with a $k$-mesh of $5 \times 5 \times 5$. The four-state method is used to calculate the quadratic exchange interaction parameters in this study,[27] using the formula $$ J_{ij}=\frac{E_{\mathrm{\uparrow \uparrow }}+E_{\downarrow \downarrow }-E_{\uparrow \downarrow }-E_{\downarrow \uparrow }}{4S^{2}},~~ \tag {1} $$ where $E_{gh}$ ($g,h = \uparrow, \downarrow$) represents the energies of different spin configurations, in which the sites $i$ and $j$ can change the spin direction. The other spins are set being perpendicular to that of sites $i$ and $j$, and they remain unchanged during the calculation. The parameters of biquadratic exchange interactions, $K_{ij}$, can be obtained similarly by mapping the spins of sites $i$ and $j$.[27] In addition, the coefficient $\partial J_{ij}/\partial \eta_{p}$ can be achieved using the four-state method[28] with the following formula: $$\begin{align} \frac{\partial J_{ij}}{\partial \eta_{p}}={}&\frac{1}{4}\Big(\frac{\partial E_{\uparrow \uparrow }}{\partial \eta_{p}}+\frac{\partial E_{\downarrow \downarrow }}{\partial \eta_{p}}-\frac{\partial E_{\uparrow \downarrow }}{\partial \eta_{p}}-\frac{\partial E_{\downarrow \uparrow }}{\partial \eta_{p}}\Big)\\ ={}&-\frac{1}{4}(\sigma_{p}^{\mathrm{\uparrow \uparrow }}+\sigma_{p}^{\downarrow \downarrow }-\sigma_{p}^{\uparrow \downarrow }-\sigma_{p}^{\downarrow \uparrow}),~~ \tag {2} \end{align} $$ where $\eta_{p}$ ($p = 1, \ldots, 6$) is the homogeneous strain in Voigt notation and $\sigma_{p}$ denotes the corresponding stress. To describe the magnetic energy of undistorted cubic NiO, we adopt the effective spin Hamiltonian model expressed as $$\begin{align} E({\boldsymbol{S}})={}&\sum\limits_{i,j>i} {J_{1}{\boldsymbol{S}}_{i}\cdot } {\boldsymbol{S}}_{j} +\sum\limits_{l,k>l} {J_{2}{\boldsymbol{S}}_{l}\cdot } {\boldsymbol{S}}_{k} +\sum\limits_{i,j>i} {K_{1}({{\boldsymbol{S}}_{i}\cdot {\boldsymbol{S}}_{j})}^{2}}\\ &+\sum\limits_{l,k>l} {K_{2}({{\boldsymbol{S}}_{l}\cdot {\boldsymbol{S}}_{k})}^{2}} ,~~ \tag {3} \end{align} $$ where $J_{1}$ and $J_{2}$ are the NN and NNN Heisenberg exchange interaction parameters, respectively; $K_{1}$ and $K_{2}$ are the NN and NNN biquadratic interactions, respectively. The biquadratic term is considered here because it is previously highlighted to be important for the $d^{8}$ configuration.[29] The spin value ${\boldsymbol S}=1$ is adopted.
Table 1. The magnetic parameters of NiO at different symmetries. The values in parenthesis are measured experimentally in previous works. The unit is meV.
Cubic Rhombohedral
$J_{1} = -1.34$ ($-1.37$, Ref. [30]) $J_{1}'=-1.35$ ($-$1.36, Ref. [30])
$J_{1}''=-1.38$ ($-$1.39, Ref. [30])
$J_{2} = 22.29$ (17, 19.8, Refs. [30,31]) $J_{2}'=23.22$
$K_{1} = -0.13$ $K_{1}'=-0.13$
$K_{1}'' =-0.14$
$K_{2} = -0.60$ $K_{2}' =-0.60$
The aforementioned exchange interaction parameters can be obtained from the DFT calculations with a relaxed structure in cubic symmetry[10] using the four-state energy mapping method, and our results are listed in Table 1. For the relaxed cubic structure [Fig. 1(a)], the NN superexchange interactions ($J_{1}$) are 12-fold degenerate with Ni–Ni coupled through a 90$^{\circ}$ Ni$^{2+}$–O$^{2-}$–Ni$^{2+}$ path. The Ni–Ni–Ni angles marked as $\alpha$ and $\beta$ are both 60$^{\circ}$. The bond length of all NN Ni sites, $l_{1}$, is 2.987 Å within the cubic structure. $J_{1}$ is calculated to be $-$1.34 meV, implying the FM interactions due to the negative sign. However, the NNN superexchange interactions ($J_{2}$) are six-fold degenerate with Ni–Ni coupling through a 180$^{\circ}$ Ni$^{2+}$–O$^{2-}$–Ni$^{2+}$ path. $J_{2}$ is simulated to be 22.29 meV, indicating AFM couplings. Our results for $J_{1}$ and $J_{2}$ are consistent with Goodenough's rule.[32] The FM nature of $J_{1}$ and the AFM of $J_{2}$ indicate exchange competition among NNs and NNNs. Notably, the derived $J_{1}$ and $J_{2}$ in our calculations are comparable with those of previous studies, as presented in Table 1, which proves the accuracy of our results. Moreover, the biquadratic parameters are also determined by the four-state method, yielding $K_{1}=-0.13$ meV and $K_{2}=-0.60$ meV, indicating the non-negligible fourth-order interactions. The negative biquadratic parameter favors collinear alignment, either parallel or antiparallel. Further, MC simulations with such Hamiltonian are performed using differently sized and shaped supercells in $Fm\bar{3}m$ symmetry. The ground state is determined to be an AFM pattern [Fig. 2(c)]. Such a pattern is consistent with the reported configuration[17,33] and named type-A structure. Besides type-A, type-B is another low-energy (or degenerate) state that satisfies the NNN AFM relationship [Fig. 2(d)]. Note that type-A and type-B states are ordered states and have a similar contribution to the configurational entropy. Types A and B can be considered as the different coupling of four simple-cubic sublattices, with each sublattice satisfying a 180$^{\circ}$ Ni$^{2+}$–O$^{2-}$–Ni$^{2+}$ AFM relationship. Figure 2(a) shows the simple-cubic sublattice. Figure 2(b) shows the combination of the four sublattices. Any two NN Ni ions are in different sublattices. In type A, the (111) planes are in the FM state, whereas two adjacent planes are in the AFM state, i.e., the magnetic ground state. However, the special spin arrangement is absent in type B. The DFT energies of types A and B in the cell, compared with MC simulations, yield a negligible difference of 0.001 meV/Ni, confirming our model simulation results. Further analyses show that each type-A or type-B configuration satisfies six pairs of the 12-fold FM NN interactions, six pairs of the 12-fold AFM NN interactions, and six pairs of the six-fold AFM NNN couplings. This suggests that type-A and B configurations would have the same energy if the effective spin Hamiltonian model shown in Eq. (3) is adopted. Such analyses explain the fact that types A and B are close in energy.
Table 2. The DFT energies of type-A and type-B structures under different optimization conditions. Note that $E$ (iii) of type-A is set to be an energy reference.
Case Type $E$ (meV/Ni) The change of group symmetry
(i) Lattice vectors are frozen while atoms relaxed A 8.51 Maintain $Fm\bar{3}m$
B 8.51 Maintain $Fm\bar{3}m$
(ii) Lattice vectors are relaxed while atoms frozen A 0 Transform to $R\bar{3}m$
B 0.08 Maintain $Fm\bar{3}m$
(iii) Lattice vectors and atoms are both relaxed A 0 Transform to $R\bar{3}m$
B 0.08 Maintain $Fm\bar{3}m$
Now, we investigate why the type-A structure widely exists in nature, whereas type-B does not. Symmetry analyses are further performed considering types A and B when the optimization is applied. Three different approaches are used to optimize the cubic structure with experimental parameters using DFT calculations: (i) Lattice vectors are frozen while atoms are allowed to relax. (ii) Atoms are frozen while lattice vectors are allowed to change. (iii) Atoms and lattice vectors are both relaxed. Table 2 summarizes the results of types A and B in different optimization methods. In case (i), types A and B have the same energy and both maintain the cubic symmetry due to the frozen lattice vectors. We discover that cases (ii) and (iii) produce the same results. For type A, the DFT structural optimization results in a distorted rhombohedral $R\bar{3}m$ structure, whereas type B maintains the cubic $Fm\bar{3}m$. Therefore, lattice distortion, rather than relative atomic displacements, causes rhombohedral distortion. The energy of type A with a distorted structure is 0.08 meV/Ni lower than that of type B in an undistorted structure, indicating that the distorted structure is the ground state structure. Regarding the structural distortions, the NN interactions also lower their degeneracy. The distorted structure [Fig. 1(b)] is optimized from the cubic structure using experimental parameters in case (iii) and used to calculate the exchange interaction parameters. The NN pairs are divided into two groups, with $J_{1}' = -1.35$ meV being inter-(111) plane interactions and $J_{1}'' = -1.38$ meV being intra-(111) layer couplings. Figures 1(c) and 1(d) show the six-fold NN inter-plane $J_{1}'$ and six-fold NN intra-plane $J_{1}''$, with the Ni–Ni–Ni angles, $\alpha'$ and $\beta'$, and Ni–Ni bond lengths, $l_{1}'$ and $l_{1}''$. Let us analyze the split of $J_{1}$ based on the changes of $\alpha$, $\beta$, and $l_{1}$. The changes in angle and bond indicate a contraction along [111] direction. We can also conclude the structural contraction along the $z$-axis from the changes in the superexchange angle of Ni$^{2+}$–O$^{2-}$–Ni$^{2+}$. The angle of Ni$^{2+}$–O$^{2-}$–Ni$^{2+}$ with Ni ions in adjacent (111) planes changes from 90$^{\circ}$ ($\gamma$) to 89.91$^{\circ}$ ($\gamma'$) when an angle with Ni ions in the same (111) plane changes from 90$^{\circ}$ ($\theta$) to 90.09$^{\circ}$ ($\theta'$), which implies this contraction. Table 1 summarizes the parameters in a distorted structure due to the contraction, showing the previous results for comparison. Similar results from other studies confirm the accuracy of our calculation. Accordingly, the biquadratic interaction, $K_{1}$, also split into $K_{1}'= -0.13$ meV and $K_{1}'' =-0.14$ meV. In contrast, the NNN interaction maintains its degeneracy since all NNN interactions are inter-(111)-typed. The rhombohedral distortion associated with the type-A configuration can be understood as follows. In the type-A configuration, the neighboring spins between the (111) layers are antiparallel to each other, whereas neighboring spins within each (111) layer are parallel to each other. To lower the spin interaction energy of the type-A configuration, the interlayer NN FM-exchange interaction should be weaker than the intralayer NN FM-exchange interaction. This is achieved in the rhombohedral structure with interlayer $J_{1}'=-1.35$ meV and intralayer $J_{1}''=-1.38$ meV. The NiO lattice being contracted rather than elongated along the [111] direction is consistent with the modulation of the NN FM-exchange interactions. Two competing spin-exchange mechanisms exist for the NN exchange interaction with the 90$^{\circ}$ Ni–O–Ni angle: the AFM direct Ni–Ni exchange and the FM superexchange through the bridging of O-ion. By comparing interlayer and intralayer exchange interactions in the rhombohedral structure, the FM superexchange can be regarded as the same since the magnitude ($\propto |\cos(\angle$Ni–O–Ni)$|$) of the effective $e_{\rm g}-t_{\rm 2g}$ hopping is the same (note that $\theta'=180^{\circ}-\gamma'$). The AFM direct Ni–Ni exchange for the interlayer $J_{1}'$ is stronger than that of the intralayer $J_{1}''$ as the interlayer Ni–Ni bond distance is shorter than that of the intralayer bond.
Table 3. The symmetrized elastic moduli $C_{pq}$ of NiO obtained from cubic structure (kbar). The coordinate system is the same as that in Fig. 1.
$xx$ $yy$ $zz$ $xy$ $yz$ $zx$
$xx$ 3169 1247 1344 0 0 $-138$
$yy$ 1247 3169 1344 0 0 138
$zz$ 1344 1344 3071 0 0 0
$xy$ 0 0 0 961 138 0
$yz$ 0 0 0 138 1059 0
$zx$ $-138$ 138 0 0 0 1059
Table 4. The magnetic parameters of the cubic structure of NiO. The coordinate system is the same as that in Fig. 1.
$xx$ $yy$ $zz$ $xy$ $yz$ $zx$
$\frac{\partial J_{1}}{\partial \eta_{p}}$ (eV) $-0.004$ 0.008 $-0.008$ 0 0 $-0.004$
$\frac{\partial J_{2}}{\partial \eta_{p}}$ (eV) $-0.049$ $-0.063$ $-0.055$ $-0.012$ $-0.017$ $-0.010$
We now establish an effective model to further understand the microscopic mechanism of the rhombohedral distortion. Based on the preceding discussions, it is reasonable to consider elastic energy and spin–lattice couplings (rather than spin–phonon interactions) in such a model,[34] resulting in the following expressions: $$\begin{align} E(\eta_{p},{\boldsymbol{S}})={}&E({\boldsymbol{S}})+\sum\limits_{p,q} {\frac{1}{2} C_{pq}\eta_{p}\eta_{q }V_{0}} \\ &+\sum\limits_{i,j} {\Big(\frac{\partial J_{ij}}{\partial \eta_{p}}{\boldsymbol{S}}_{i}\cdot {\boldsymbol{S}}_{j}\Big)\eta_{p}},~~ \tag {4} \end{align} $$ where the first part $E({\boldsymbol S})$ is defined in Eq. (3), and the second part represents the elastic energy, with $C_{pq}$ being the elastic moduli and $\eta_{p}$ ($p = 1, \ldots,6$) being the homogeneous strain in Voigt notion. The third part denotes the Taylor expansion of spin–lattice couplings concerning $\eta_{p}$, and only the NN and NNN Heisenberg exchange interactions are considered. The spin value of $| {\boldsymbol S}|$ is also set to 1. The relaxed structure in cubic symmetry is applied to obtain $C_{pq}$ and $\partial J_{ij}/\partial \eta_{p}$ in DFT calculations and to perform the model simulations [Fig. 1(a)], and has a volume $V_{0}$. $C_{pq}$ is given in Table 3, and the coefficients $\partial J_{ij}/\partial \eta_{p}$ are presented in Table 4, which are calculated using Eq. (2), indicating how the strain affects the exchange parameters of the ordered NN pair [close to label $l_{1}$ in Fig. 1(a)] and NNN pair [shown in black solid line in Fig. 1(a)]. Any NN or NNN pair can be used for calculation due to the degenerate relationship in the cubic symmetry. Note that the coordinate system shown in Fig. 1 is applied here.
Table 5. The strain tensor induced by spin order in types A and B. The coordinate system is the same as that in Fig. 1.
Strain by $xx$ $yy$ $zz$ $xy$ $yz$ $zx$
spin order
Type A $-0.00173$ $-0.00173$ $-0.00391$ 0 0 0
Type B $-0.00246$ $-0.00246$ $-0.00246$ 0 0 0
We estimate the lattice distortion using the spin–lattice model and compare it with DFT calculations. The different magnetic structures, types A and B, in the cubic phase are both considered in the model. The strain $\eta_{p}$ is the only variable in the model. We minimize the total energy $E (\eta_{p}, {\boldsymbol S})$ concerning $\eta_{p}$ to obtain the structural distortion by spin order. Table 5 shows the strain tensor. For types A and B, the tensor only has nonzero diagonal elements, indicating that the strain merely occurs in the $x$, $y$, and $z$ directions. For type A, the negative sign of the diagonal element indicates a shortened strain in a certain direction, and the absolute value of the $z$ component is almost twice the value of the $x$ or $y$ component. Thus, a contraction along the $z$ direction, i.e., the [111] direction, occurs and lowers the cubic symmetry. The diagonal elements for type B are the same, shrinking the overall volume but maintaining the cubic symmetry. Table 6 shows the structure parameters and symmetry at the beginning and final states, with the strain obtained from the spin–lattice model. The results of the aforementioned DFT calculations are also presented for comparison. For type A, the model results show that the structural ground state is $R\bar{3}m$, and the angle $\alpha$ (60$^{\circ}$) changes to $\alpha'$ (60.097$^{\circ}$), which is extremely close to the 60.100$^{\circ}$ obtained from DFT calculations. The bond length $l$ is split into $l'$ and $l''$ in both the methods. The changes in angle and bond length indicate a contraction along [111] direction. For type B, the structural ground state is $Fm\bar{3}m$, while $\alpha$ maintains 60$^{\circ}$ and $l$ does not change into two groups in the final state from the model, which is consistent with the DFT calculations. Similar results from DFT calculations confirm the accuracy of the spin–lattice model.
Table 6. The NiNi–Ni angle, $\alpha$, and bond length $l$ of NN Ni ions, and symmetry at the beginning and final states obtained from both the spin–lattice model simulations and DFT calculations for types A and B.
Type Parameters Model results DFT calculations
Begin state Final state Begin state Final state
A $\alpha$ 60.000$^{\circ}$ 60.097$^{\circ}$ 60.000$^{\circ}$ 60.100$^{\circ}$
$l$ 2.987 Å 2.977 Å 2.953 Å 2.969 Å
2.982 Å 2.974 Å
Symmetry $Fm\bar{3}m$ $R\bar{3}m$ $Fm\bar{3}m$ $R\bar{3}m$
B $\alpha$ 60.000$^{\circ}$ 60.000$^{\circ}$ 60.000$^{\circ}$ 60.000$^{\circ}$
$l$ 2.987 Å 2.980 Å 2.953 Å 2.971 Å
Symmetry $Fm\bar{3}m$ $Fm\bar{3}m$ $Fm\bar{3}m$ $Fm\bar{3}m$
To further examine whether it is $\partial J_{1}/\partial \eta_{p}$ or $\partial J_{2}/\partial \eta_{p}$ that causes the contraction along the [111] direction, we arbitrarily turn on and off each of the two terms in the spin–lattice model of Eq. (4). When starting with type A, it is discovered that only $\partial J_{1}/\partial \eta_{p}$ increases $\alpha$, whereas $\partial J_{2}/\partial \eta_{p}$ keeps $\alpha$ unchanged. These results indicate that the distortion originates from a change in $J_{1}$, rather than a change in $J_{2}$, which is consistent with a previous study.[1] In summary, we have explicitly studied the structural phase transition using NiO as an example. First, the energies of different types of magnetic order, types A and B, are found to degenerate in the cubic structure with both DFT calculations and MC simulations using a spin Hamiltonian model. However, type A is more stable in the distorted NiO structure. Further, we conclude that the spin–lattice coupling induces the rhombohedral contraction in NiO, according to symmetry analysis and DFT calculations. Therefore, the spin–lattice model is constructed to further understand the microscopic mechanism of the rhombohedral distortion with the strain item included. We apply both types A and B in our model, and the results show that the rhombohedral distortion only appears when the type-A magnetic structure is used, and it maintains the cubic symmetry when the type-B magnetic structure is considered. The results of DFT calculations are used as a comparison to ensure the accuracy of our model, and they show good consistency. Moreover, our model reveals that $J_{1}$ is responsible for the structural distortion, whereas the change in $J_{2}$ is irrelevant. Therefore, the mechanism of structural phase transition is demonstrated in this study, which also guides the understanding of similar phase transition phenomena in other monoxides. Acknowledgments. The work at Fudan was supported by the National Natural Science Foundation of China (Grant Nos. 11825403, 12188101, and 11804138). References Deformations in the Crystal Structures of Anti-ferromagnetic CompoundsMagnetic Structures of MnO, FeO, CoO, and NiOCrystalline and magnetic anisotropy of the 3 d -transition metal monoxides MnO, FeO, CoO, and NiOMagnitude and Origin of the Band Gap in NiOA note on the structure of nickel oxide at subnormal and elevated temperaturesThe space group corepresentations of antiferromagnetic NiOFerromagnetic dislocations in antiferromagnetic NiOEnhancement of Thermally Injected Spin Current through an Antiferromagnetic InsulatorSpin-flop transition in the easy-plane antiferromagnet nickel oxideExchange Striction in NiOA high-temperature X-ray diffraction study of the NiO–Li2 O systemStructural Distortion Stabilizing the Antiferromagnetic and Insulating Ground State of NiOExchange Striction Effects in MnO and MnSMagnetic ordering and exchange effects in the antiferromagnetic solid solutions Mn x Ni 1 − x O Sublattice Magnetization and Lattice Distortions in MnO and NiOMagnetostructural phase transitions in NiO and MnO: Neutron diffraction dataMagnetic Moment Arrangements and Magnetocrystalline Deformations in Antiferromagnetic CompoundsSingle-Ion Magnetostriction in the Iron Group Monoxides from the Strain Dependence of Electron-Paramagnetic-Resonance SpectraStructural, electronic, and magnetic properties of MnOLattice dynamical properties of antiferromagnetic MnO, CoO, and NiO, and the lattice thermal conductivity of NiOHeisenberg exchange in the magnetic monoxidesEfficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis setFrom ultrasoft pseudopotentials to the projector augmented-wave methodGeneralized Gradient Approximation Made SimpleElectron-energy-loss spectra and the structural stability of nickel oxide: An LSDA+U studyAntiferromagnetic structures and electronic energy levels at reconstructed NiO(111) surfaces: A DFT + U studyCrystalline Peroxosolvates: Nature of the Coformer, Hydrogen-Bonded Networks and Clusters, Intermolecular InteractionsGeneral microscopic model of magnetoelastic coupling from first principlesGiant Biquadratic Exchange in 2D Magnets and Its Role in Stabilizing Ferromagnetism of NiCl 2 MonolayersMeasurement of Spin-Wave Dispersion in NiO by Inelastic Neutron Scattering and Its Relation to Magnetic PropertiesAnalysis of the Exchange Parameters and Magnetic Properties of NiONeutron Diffraction by Paramagnetic and Antiferromagnetic SubstancesNature of spin-lattice coupling in two-dimensional CrI3 and CrGeTe3
[1] Greenwald S and Smart J S 1950 Nature 166 523
[2] Roth W L 1958 Phys. Rev. 110 1333
[3] Schrön A, Rödl C, and Bechstedt F 2012 Phys. Rev. B 86 115134
[4] Sawatzky G A and Allen J W 1984 Phys. Rev. Lett. 53 2339
[5] Rooksby H 1948 Acta Crystallogr. 1 226
[6] Cracknell A P and Joshua S J 1969 Mathematical Proceedings of the Cambridge Philosophical Society 66 493
[7] Sugiyama I, Shibata N, Wang Z, Kobayashi S, Yamamoto T, and Ikuhara Y 2013 Nat. Nanotechnol. 8 266
[8] Lin W, Chen K, Zhang S, and Chien C L 2016 Phys. Rev. Lett. 116 186601
[9] Machado F L A, Ribeiro P R T, Holanda J, Rodríguez-Suárez R L, Azevedo A, and Rezende S M 2017 Phys. Rev. B 95 104418
[10] Bartel L C and Morosin B 1971 Phys. Rev. B 3 1039
[11] Toussaint C 1971 J. Appl. Crystallogr. 4 293
[12] Krüger E 2020 Symmetry 12 56
[13] Morosin B 1970 Phys. Rev. B 1 236
[14] Cheetham A K and Hope D A O 1983 Phys. Rev. B 27 6964
[15] Rodbell D S and Owen J 1964 J. Appl. Phys. 35 1002
[16] Balagurov A M, Bobrikov I A, Sumnikov S V, Yushankhai V Y, and Mironova-Ulmane N 2016 JETP Lett. 104 88
[17] Li Y Y 1955 Phys. Rev. 100 627
[18] Phillips T G and White R L 1967 Phys. Rev. 153 616
[19] Pask J E, Singh D J, Mazin I I, Hellberg C S, and Kortus J 2001 Phys. Rev. B 64 024403
[20] Linnera J and Karttunen A J 2019 Phys. Rev. B 100 144307
[21] Harrison W A 2007 Phys. Rev. B 76 054417
[22] Kresse G and Furthmüller J 1996 Comput. Mater. Sci. 6 15
[23] Kresse G and Joubert D 1999 Phys. Rev. B 59 1758
[24] Perdew J P, Burke K, and Ernzerhof M 1996 Phys. Rev. Lett. 77 3865
[25] Dudarev S L, Botton G A, Savrasov S Y, Humphreys C J, and Sutton A P 1998 Phys. Rev. B 57 1505
[26] Li L and Kanai Y 2015 Phys. Rev. B 91 235304
[27] Li X, Yu H, Lou F, Feng J, Whangbo M H, and Xiang H 2021 Molecules 26 26
[28] Lu X Z, Wu X, and Xiang H J 2015 Phys. Rev. B 91 100405
[29] Ni J Y, Li X Y, Amoroso D, He X, Feng J S, Kan E J, Picozzi S, and Xiang H J 2021 Phys. Rev. Lett. 127 247204
[30] Hutchings M T and Samuelsen E J 1972 Phys. Rev. B 6 3447
[31] Shanker R and Singh R A 1973 Phys. Rev. B 7 5000
[32]Coey J M D 2009 Magnetism and Magnetic Materials (New York: Cambridge University Press)
[33] Shull C G, Strauser W A, and Wollan E O 1951 Phys. Rev. 83 333
[34] Li J, Feng J, Wang P, Kan E, and Xiang H 2021 Sci. Chin. Phys. Mech. & Astron. 64 286811