Complex magnetic order in the kagomé staircase compound ${\mathrm{Co}}_3{\mathrm{V}}_2{\mathrm{O}}_8$

${\mathrm{Co}}_3{\mathrm{V}}_2{\mathrm{O}}_8$ (CVO) has a different type of geometrically frustrated magnetic lattice, a kagomé staircase, where the full frustration of a conventional kagomé lattice is partially relieved. The crystal structure consists of two inequivalent (magnetic) Co sites, one-dimensional chains of $\mathrm{Co}\left(2\right)$ spine sites, linked by $\mathrm{Co}\left(1\right)$ cross-tie sites. Neutron powder diffraction has been used to solve the basic magnetic and crystal structures of this system, while polarized and unpolarized single crystal diffraction measurements have been used to reveal a rich variety of incommensurate phases, interspersed with lock-in transitions to commensurate phases. CVO initially orders magnetically at $11.3\mathrm{K}$ into an incommensurate, transversely polarized, spin density wave state, with wave vector $k=(0,\delta ,0)$ with $\delta =0.55$ and the spin direction along the $a$ axis. $\delta$ is found to decrease monotonically with decreasing temperature and then locks into a commensurate antiferromagnetic structure with $\delta =\frac{1}{2}$ for $6.9<T<8.6\mathrm{K}$. In this phase, there is a ferromagnetic layer where the spine site and cross-tie sites have ordered moments of $1.39{\mu }_B$ and $1.17{\mu }_B$, respectively, and an antiferromagnetic layer where the spine-site has an ordered moment of $2.55{\mu }_B$, while the cross-tie sites are fully frustrated and have no observable ordered moment. Below $6.9\mathrm{K}$, the magnetic structure becomes incommensurate again, and the presence of higher-order satellite peaks indicates that the magnetic structure deviates from a simple sinusoid. $\delta$ continues to decrease with decreasing temperature and locks in again at $\delta =\frac{1}{3}$ over a narrow temperature range $(6.2<T<6.5\mathrm{K})$. The system then undergoes a strongly first-order transition to the ferromagnetic ground state $(\delta =0)$ at $T_c=6.2\mathrm{K}$. The ferromagnetism partially relieves the cross-tie site frustration, with ordered moments on the spine-site and cross-tie sites of $2.73{\mu }_B$ and $1.54{\mu }_B$, respectively. The spin direction for all spins is along the $a$ axis (Ising-like behavior). A dielectric anomaly is observed around the ferromagnetic transition temperature of $6.2\mathrm{K}$, demonstrating that there is significant spin-charge coupling present in CVO. A theory based on group theory analysis and a minimal Ising model with competing exchange interactions can explain the basic features of the magnetic ordering.

Figure 1

(Color online) Crystal structure of ${\mathrm{Co}}_3{\mathrm{V}}_2{\mathrm{O}}_8$. The nonmagnetic $\mathrm{V}\text{−}\mathrm{O}$ tetrahedra provide isolation between the magnetic Co-O layers.

Figure 2

(Color online) (a) The crystal structure of CVO viewed down the crystallographic $c$ axis. There are two Co sites: the $\mathrm{Co}\left(1\right)$ cross-tie (at the apex of the triangles) and $\mathrm{Co}\left(2\right)$ spine sites. (b) Edge-on view of the kagomé staircase. (c) Kagomé staircase lattice, showing only the Co sites. Note that when the $\mathrm{Co}\left(2\right)$ sites are ordered antiferromagnetically, the $\mathrm{Co}\left(1\right)$ spins are fully frustrated.

Figure 3

The specific heat of a single crystal sample of CVO as a function of temperature is plotted as $C∕T$ vs $T$, at zero magnetic field.

Figure 4

(Color online) Observed (plus signs), calculated (solid lines), and difference (bottom curve) low angle portions of the neutron powder diffraction intensity. (a) $T=15\mathrm{K}$. Fit to a nuclear structure. (b) $T=8.4\mathrm{K}$. Fit includes an incommensurate magnetic structure. (c) $T=3.1\mathrm{K}$. Fit includes a ferromagnetic structure. The vertical lines indicate the calculated Bragg peak positions. The lower row of vertical lines in (b) and (c) indicates the nuclear Bragg peak positions.

Figure 5

(Color online) The integrated intensity (left axis) and the position (right axis) of the (0,δ,0) peak as a function of temperature measured on the powder sample of CVO.

Figure 6

(Color online) Incommensurate magnetic structure model. The transversely polarized spin density wave shown in the figure corresponds to the initial ordering wave vector of $\delta \approx 0.55$ for CVO. The wave vector is along the $b$ axis, and the direction of the moments is along the $a$ axis.

Figure 7

(Color online) Antiferromagnetic structure model of CVO with $\delta =0.5$. The adjacent kagomé layers alternate between ferromagnetic and antiferromagnetic coupling of the ferromagnetic $\mathrm{Co}\left(2\right)$ chains. The $\mathrm{Co}\left(1\right)$ sites fail to exhibit an ordered moment in the antiferromagnetic layer due to frustration as a result of trying to satisfy interactions with the two antiparallel $\mathrm{Co}\left(2\right)$ chains, which $\mathrm{Co}\left(1\right)$ sites link. The direction of the Co moments is along the crystallographic $a$ axis.

Figure 8

(Color online) Ferromagnetic structure model with $\delta =0$ of CVO. All the Co moments are along the crystallographic $a$ axis.

Figure 9

(Color online) (a) $a$ (left axis) and $b$ (right axis) lattice parameters of CVO as a function of temperature. (b) $c$ (left axis) lattice parameter and volume of the unit cell (right axis) as a function of temperature. (c) $\mathrm{Co}\left(1\right)\text{−}\mathrm{Co}\left(2\right)$ distance (left axis) and $\mathrm{Co}\left(2\right)\text{−}\mathrm{Co}\left(2\right)$ chain distance (right axis) as a function of temperature.

Figure 10

$k$ scans at four different temperatures, showing that the incommensurability $\delta$ is strongly temperature dependent.

Figure 11

(Color online) Evolution of the fundamental magnetic Bragg peak $(0,\delta ,0)$ as a function of temperature, exhibiting a rich variety of incommensurate phases, interspersed with lock-in transitions to commensurate phases. The initial structure that forms is incommensurate, which locks-in to the simple frustrated antiferromagnetic structure at $\delta =\frac{1}{2}$. The fundamental peak then becomes incommensurate and moves to smaller $\delta$, while a weak higher order satellite peak moves to higher $\delta$. The structure locks in again at $\delta =\frac{1}{3}$, and finally undergoes a first-order transition to the ferromagnetic ground state $(\delta =0)$.

Figure 12

(a) Integrated intensity of the fundamental incommensurate magnetic Bragg peak $(0,\delta ,0)$ as a function of temperature on cooling (circles) and warming (squares). (b) Incommensurability $\delta$ as a function of temperature on cooling (circles) and warming (squares). The solid (cooling) and dotted (warming) lines in (a) and (b) are guides to the eye.

Figure 13

(a) Integrated intensity of the (0,$\delta$,0) peak (open symbols), and 20 times the integrated intensity of the (0,${\delta }^{\prime }$,0) satellite peak (filled symbols) as a function of temperature on cooling (circles) and warming (squares). (b) Incommensurability $\delta$ (open symbols) and ${\delta }^{\prime }$ (filled symbols) as a function of temperature on cooling (circles) and warming (squares). (c) Ratio $\equiv \frac{q^{\prime }-0.5}{0.5-q}$ (see text) as a function of temperature. The solid (cooling) and dashed (warming) lines in (a)–(c) are guides to the eye.

Figure 14

Temperature dependence of the peak intensity of (a) (0,2,1), (b) (0,0,2), and (c) (0,2,0) on warming (filled circles) and on cooling (open circles).

Figure 15

Temperature dependence of the ratio between spin-flip scattering and non-spin-flip scattering at $\mathbf{Q}=(0,2,1)$, with polarization vector $\mathbf{P}\perp \mathbf{Q}$, on cooling and warming.

Figure 16

The integrated intensity of the (0,3,0) peak as a function of temperature on cooling and on warming. The line is a guide to the eye.

Figure 17

Temperature dependence of the (0,1,0) peak on warming (squares) and cooling (circles) for spin-flip (filled symbols) and non-spin-flip (open symbols) channels. The solid (flipper on) and the dashed (flipper off) lines are guides to the eye.

Figure 18

Dielectric constant as a function of temperature of a powder sample of CVO measured at $100\mathrm{K}\mathrm{Hz}$. The dashed lines are guides to the eye.