Broken ${\mathrm{C}}_4$ symmetry in the tetragonal state of uniaxial strained ${\mathrm{BaCo}}_{0.9}{\mathrm{Ni}}_{0.1}{\mathrm{S}}_{1.9}$

A compound with large anions is known to show large compressibility, whereby a novel response may emerge from the degenerated state by a uniaxial pressure. Neutron scattering study of ${\mathrm{BaCo}}_{0.9}{\mathrm{Ni}}_{0.1}{\mathrm{S}}_{1.9}$ crystal reveals that the tetragonal insulating state has two magnetic domains with in-plane anisotropic antiferromagnetic wave vectors ${\mathbf{Q}}_{\mathbf{1}}=(\pi ,0)$ and ${\mathbf{Q}}_{\mathbf{2}}=(0,\pi )$. The magnetic order with in-plane broken $C_4$ symmetry is realized as a twin of these two domains in the tetragonal state without any strain. One magnetic domain with ${\mathbf{Q}}_{\mathbf{2}}$ becomes dominant under a weak strain without any appreciable structural distortion. Correspondingly, the in-plane broken $C_4$ symmetry is also observed in the in-plane magnetic excitation of the tetragonal state.

Figure 1

Two types of stripe order structures in a two-dimensional ${\mathrm{Co}}_{0.9}{\mathrm{Ni}}_{0.1}{\mathrm{S}}_{1.9}$ layer and magnetic exchange couplings. (a) ${\mathbf{Q}}_{\mathbf{1}}=(\pi ,0)$ observed in iron-based superconductor parent materials. The magnetic moments (red arrows) align along the $a$-axis with the AF wave vector of ${\mathbf{Q}}_{\mathbf{1}}$ in the two-dimensional square lattice. Three exchange coupling parameters, $J_{1a}$, $J_{1b}$, and $J_2$ are major parts for the spin-wave calculation. (b) ${\mathbf{Q}}_{\mathbf{2}}=(0,\pi )$. The magnetic moments align along the $a$-axis with the AF wave vector of ${\mathbf{Q}}_{\mathbf{2}}$. (c) Three-dimensional view of the magnetic structure in panel (b). ${\mathrm{Co}}_{0.9}{\mathrm{Ni}}_{0.1}$ (blue) is coordinated by a pyramid of five sulfur ions (yellow). ${\mathrm{Co}}_{0.9}{\mathrm{Ni}}_{0.1}$ ions alternately buckle along the $c$-axis in the layer due to the alternating buckling of apical sulfur ions. Uniaxial pressure is applied along $b$ axis. The exchange coupling parameter $S{J}_c$ is between two Fe magnetic moments at equivalent sites in the neighboring unit cells along $c$ axis.

Figure 2

Temperature dependence of magnetic susceptibility, in-plane lattice parameters, and magnetic peak intensity. (a) Temperature dependencies of the field-cooled and zero-field-cooled DC magnetic susceptibility ($\chi =M/B$) curves at an ambient pressure. The peak appears at the Néel temperature $T_N$. A hysteresis is observed in the mid-temperature range. (b) In-plane lattice parameters estimated from 400 and 040 peaks at a uniaxial pressure of 1 MPa along the $b$-axis. The tetragonal-to-orthorhombic phase transition $T_s$ can be observed by the splitting of the peaks. Filled circles and open circles denote cooling and warming processes, respectively. (c) Temperature dependence of 102 magnetic Bragg peak intensity obtained by the neutron diffraction measurement upon cooling at a uniaxial pressure of 1 MPa along the $b$-axis. Below the structural phase transition $T_s^c$ upon cooling, the intensity suddenly drops. The solid line is a guide to the eye. The arrows denote the thermal cycle directions. PM, AFI, and PI are the paramagnetic metal phase, antiferromagnetic insulator phase, and paramagnetic insulator phase, respectively.

Figure 3

Observed nuclear Bragg peak intensities as a function of calculated intensity. The measurement of BCNS crystal was carried out at a uniaxial pressure of 1 MPa.

Figure 4

Observed magnetic Bragg peak intensities at uniaxial pressures of 0.1 and 1 MPa as a function of calculated intensity upon cooling. (a) A twin model of two magnetic structures of ${\mathbf{Q}}_{\mathbf{1}}$ with $m//a$ and ${\mathbf{Q}}_{\mathbf{2}}$ with $m//b$ at 0.1 MPa. The twin ratio of these was refined as 0.44(10) to 0.56(10), respectively. The magnetic moment per ${\mathrm{Co}}_{0.9}{\mathrm{Ni}}_{0.1}$ was $2.14\pm 0.13{\mu }_B$. The conventional reliability factor $R_F$ was 18.3%. (b) A twin model of two magnetic structures of ${\mathbf{Q}}_{\mathbf{1}}$ with $m//b$ and ${\mathbf{Q}}_{\mathbf{2}}$ with $m//a$ at 0.1 MPa. The twin ratio of these was refined as 0.48(5) to 0.52(5), respectively. The magnetic moment per ${\mathrm{Co}}_{0.9}{\mathrm{Ni}}_{0.1}$ was $2.13\pm 0.06{\mu }_B$. The conventional reliability factor $R_F$ was 9.38%. (c) A twin model of two magnetic structures of ${\mathbf{Q}}_{\mathbf{1}}$ with $m//a$ and ${\mathbf{Q}}_{\mathbf{2}}$ with $m//b$ at 1 MPa. The twin ratio of these was refined as 0.38(10) to 0.62(10), respectively. The magnetic moment per ${\mathrm{Co}}_{0.9}{\mathrm{Ni}}_{0.1}$ was $2.34\pm 0.15{\mu }_B$. The conventional reliability factor $R_F$ was 19.7%. (d) A twin model of two magnetic structures of ${\mathbf{Q}}_{\mathbf{1}}$ with $m//b$ and ${\mathbf{Q}}_{\mathbf{2}}$ with $m//a$ at 1 MPa. The twin ratio of these was refined as 0.41(4) to 0.59(4), respectively. The magnetic moment per ${\mathrm{Co}}_{0.9}{\mathrm{Ni}}_{0.1}$ was $2.33\pm 0.06{\mu }_B$. The conventional reliability factor $R_F$ was 8.54%.

Figure 5

Temperature dependence of nonmagnetic domain ratio under uniaxial pressures $P=0.1$ (black open and closed circles) and 1 MPa (red open and closed circles). The temperature dependences are fitted by a tanh function and a cubic polynomial to estimate the transition temperatures. The transition temperature increased at 1 MPa from that of 0.1 MPa. Inset shows the observed relationship between the orthorhombic nonmagnetic domain ratio and the tetragonal magnetic domain ratio of ${\mathbf{Q}}_{\mathbf{2}}$ with $m//a$ at the same uniaxial pressure.

Figure 6

Constant-energy cuts of dynamical structure factor $S(\mathbf{Q},E)$ along $H$ and $K$ at a uniaxial pressure of 1 MPa. Two equivalent data sets in the $(H,K)$ plane were averaged and symmetrized, based on the $C_2$ symmetry. The intensities are integrated in $\pm 0.3$ r.l.u. along the normal direction to the corresponding scan direction of $H$ or $K$. $E_i$ is 54.1 meV for $E=10$, 20, and 30 meV, whereas $E_i$ is 120.0 meV for the other energies above 30 meV.

Figure 7

Constant-Q cuts of dynamical structure factor multiplied by $E$, $S(\mathbf{Q},E)\times E$, at zone boundaries at a uniaxial pressure of 1 MPa. Intensities were subtracted by those at (1, 1) as background. (a) Zone boundary energies at (0.5, 0) were estimated as $53.0\pm 0.7$ and $58.4\pm 0.3$ meV from the fit with a fixed full-width at half maximum of 3 meV. (b) Zone boundary energies at (0.5, 1) were $59.5\pm 0.8$, and $69.5\pm 1.1$ meV. (c) Zone boundary energies at (1, 0.5) were $58.6\pm 0.5$, and $70.5\pm 1.4$ meV. Because of the twin model with $C_2$ symmetry, these peak energies at (1, 0.5) are expected to be the same as those at (0.5, 1). Based on these energies, basic spin-wave parameters $S(J_{1a}+J_{1b})/2$ and $S{J}_2$ were determined. The solid lines are guides to the eye with expected intensity ratio from the spin-wave calculation.

Figure 8

$L$-dependence of magnetic excitation with an incident energy of 19.7 meV at a uniaxial pressure of $1\mathrm{MPa}$. (a) In this measurement configuration, $E$ increases from 0 to 8 meV with increasing $L$ from 0 to 1 r.l.u. (b) The intensity was integrated in the reciprocal region of ($H,K)=(0\pm 0.2,-1\pm 0.2$). The continuous intensity along $L$ suggests that the two-dimensional nature of a magnetic rod in this energy range.

Figure 9

Constant-energy slices of dynamical structure factor $S(\mathbf{Q},E)$ in the ($H$, $K$) plane at a uniaxial pressure of 1 MPa. (a–c) $S(\mathbf{Q},E)$ in the ($H$, $K$) plane at $E=40$ (a), 50 (b), and 60 (c) meV. Each slice is integrated in the energy width of 10 meV. The incident energy is $E_i=120.0\mathrm{meV}$. The color bars are in units of mbarn ${\mathrm{sr}}^{-1}{\mathrm{meV}}^{-1}{\mathrm{Co}}^{-1}$. The quadrant intensity is averaged based on the $C_2$ symmetry. (d–f) Theoretical simulations of a twin model with a ratio of 60:40 at $E=40$ (d), 50 (e), and 60 (f) meV for $\left|H\right|\le$ 2 and $\left|K\right|\le$ 2 r.l.u. The used exchange coupling parameters are $S{J}_{1a}=-4.3\mathrm{meV}$, $S{J}_{1b}=-2.7\mathrm{meV}$, $S{J}_2=14.0\mathrm{meV}$, and $S{J}_c=-0.06\mathrm{meV}$.

Figure 10

Observed dispersion (filled circles) and calculated spin-wave dispersion with $S{J}_{1a}$=$-4.3\mathrm{meV}$, $S{J}_{1b}=-2.7\mathrm{meV}$, $S{J}_2=14.0\mathrm{meV}$, and $S{J}_c=-0.06$ meV. The maximum energy of the dispersion from Y to Y+Z points is limited below 5 meV. Because of the quasi-two-dimensionality, $L$ is neglected for the other spectra. The calculated dispersions are drawn with a full-width at half-maximum of 8 meV, except for the dispersion along $L$ in (0, $-1$, $L$), where the width is 2 meV to show the low energy dispersion.

Figure 11

Momentum-integrated dynamical spin susceptibility ${\chi }^{\prime \prime }\left(E\right)$ at a uniaxial pressure of $1\mathrm{MPa}$. The closed circles are integrated along $H$ in $(H,1)$ within $|K-1|\le 0.3$ r.l.u. of ${\chi }^{\prime \prime }(Q,E)$ at 200 K on cooling. The open circles are along $K$ in $(1,K)$ within $|H-1|\le 0.3$ r.l.u. The solid and dashed lines were calculated dynamical spin susceptibilities of the twin model integrated along $H$ in $(H,1)$ and $K$ in $(1,K)$, respectively. They were calculated based on the spin-wave model with a twin ratio of 60:40. The used exchange parameters were $S{J}_{1a}=-4.3\mathrm{meV}$, $S{J}_{1b}=-2.7\mathrm{meV}$, $S{J}_2=14.0\mathrm{meV}$, and $S{J}_c=-0.06\mathrm{meV}$. The horizontal error bar is the integrated region in energy.