Magnetic order and fluctuations in the presence of quenched disorder in the kagome staircase system (Co${}_{1-x}$Mg${}_x$)${}_3$V${}_2$O${}_8$

Co${}_3$V${}_2$O${}_8$ is an orthorhombic magnet in which $S=3/2$ magnetic moments reside on two crystallographically inequivalent Co${}^{2+}$ sites, which decorate a stacked, buckled version of the two-dimensional kagome lattice, the stacked kagome staircase. The magnetic interactions between the Co${}^{2+}$ moments in this structure lead to a complex magnetic phase diagram at low temperature, wherein it exhibits a series of five transitions below 11 K that ultimately culminate in a ferromagnetic ground state below $T\sim 6.2$ K. Here we report magnetization measurements on single- and polycrystalline samples of (Co${}_{1-x}$Mg${}_x$)${}_3$V${}_2$O${}_8$ for $x<0.23$, as well as elastic and inelastic neutron scattering measurements on single crystals of magnetically dilute (Co${}_{1-x}$Mg${}_x$)${}_3$V${}_2$O${}_8$ for $x=0.029$ and $x=0.194$, in which nonmagnetic Mg${}^{2+}$ ions substitute for magnetic Co${}^{2+}$. We find that a dilution of 2.9$\%$ leads to a suppression of the ferromagnetic transition temperature by $\sim$15$\%$ while a dilution level of 19.4$\%$ is sufficient to destroy ferromagnetic long-range order in this material down to a temperature of at least 1.5 K. The magnetic excitation spectrum is characterized by two spin wave branches in the ordered phase for (Co${}_{1-x}$Mg${}_x$)${}_3$V${}_2$O${}_8$ ($x=0.029$), similar to that of the pure $x=0$ material, and by broad diffuse scattering at temperatures below 10 K in (Co${}_{1-x}$Mg${}_x$)${}_3$V${}_2$O${}_8$ ($x=0.194$). Such a strong dependence of the transition temperatures on long-range order in the presence of quenched nonmagnetic impurities is consistent with two-dimensional physics driving the transitions. We further provide a simple percolation model that semiquantitatively explains the inability of this system to establish long-range magnetic order at the unusually low dilution levels which we observe in our experiments.

Figure 1

The crystal structure of Co${}_3$V${}_2$O${}_8$. (a) View of the kagome staircase considering only the Co${}^{2+}$ ions (red and blue) which are stacked along the $b$ axis. (b) View of the kagome plane projected on the $a$-$c$ plane with the crystallographically inequivalent cross-tie (red) and spine sites (blue). The magnetic exchange interactions discussed in the text are indicated.

Figure 2

The phase diagram of (Co${}_{1-x}$Mg${}_x$)${}_3$V${}_2$O${}_8$ as a function of $x$ obtained from magnetization measurements described in the text. The error bars result from the determination of the transition temperatures using data in Fig. 3. The phase boundaries (dashed lines) are guides to the eye and are extrapolated to $T=0$ K. The theoretically expected site percolation limit for the perfect 2D kagome lattice ($x_c$ $\sim$ 0.35)[28] is shown as red star. The critical doping concentration $x_c$ above which any magnetic ordering gets suppressed at $T=0$ corresponds to the determined percolation limit discussed later in the text and is shown as blue diamond. The inset shows representative scans of the field-cooled molar susceptibility $\chi \left(T\right)$ for an applied field of ${\mu }_{0}H=0.005$ T, and the phase transition temperatures $T_{\mathrm{C}}$ and $T_{\mathrm{N}}$ for data points in the phase diagram.

Figure 3

Representative scans of the susceptibility $\chi \left(T\right)·T$ in a field of ${\mu }_{0}H=0.005$ T reveal anomalies at the phase transition temperatures $T_{\mathrm{C}}$ and $T_{\mathrm{N}}$ (indicated by arrows). These transition temperatures are used to construct the phase diagram shown in Fig. 2. Closed symbols refer to data obtained from polycrystalline samples, while open symbols represent measurements on single crystals with ${\mu }_{0}H$ aligned parallel to the $a$ axis of the system. Note that the maximum in the $\chi \left(T\right)·T$ curve in the $x=0.194$ doped sample is indicated as a phase transition with $T_{\mathrm{N}}\sim 5$ K in Fig. 2; however, elastic neutron scattering data show no clear evidence for long-range magnetic correlations at any temperature above 1.5 K in this sample.

Figure 4

Contour plots showing the temperature evolution of the elastic magnetic scattering $S$(Q, $E=0$) around the $\mathbf{Q}=(0,0,2)$ Bragg peak position for (a) $x$ $=$ 0.029 and (c) $x=0.194$. A background data set within the paramagnetic state at $T=20$ K has been subtracted from the individual $(0,0,L)$ scans so as to eliminate the weak nuclear component to the $(0,0,2)$ Bragg scattering. Representative scans making up the contour plots are shown in panels (b) and (d). The instrumental resolution of SPINS is indicated in panels (b) and (d) as the black horizontal bar. The error bars represent one standard deviation in this and the following figures.

Figure 5

The order parameter and diffuse scattering as a function of temperature measured near the $(0,0,2)$ magnetic Bragg peak in the $x=0.029$ sample. Note the different intensity scales and counting times.

Figure 6

Elastic scattering in the (a) $x=0.029$ and (b) $x=0.194$ doped (Co${}_{1-x}$Mg${}_x$)${}_3$V${}_2$O${}_8$ samples. (a) The integrated intensities of the Gaussian (black) and Lorentzian (blue) components to the line shape used to describe the magnetic scattering in the $x=0.029$ sample discussed in the text are shown. A representative scan with corresponding fits in the Griffiths-like phase at $T=5.8$ K is shown in the inset. (b) The integrated intensity of the Lorentzian line shape used to describe the magnetic scattering in the $x=0.194$ doped sample is shown, while the inset displays representative scans in $L$ along the $(0,0,2)$ direction for different temperatures with fits to the Lorentzian line shape as described in the text. For both panels (a) and (b) a $T=20$ K high-temperature nonmagnetic background has been subtracted to isolate the magnetic scattering. Note that the peaks for the $x=0.194$ doped sample in the inset of panel (b) are much wider than the instrumental resolution at SPINS, which is shown as a black horizontal bar.

Figure 7

FWHM of the Lorentzian components shown for both doped single-crystal samples. The dashed lines are guides to the eye. While for $x=0.029$, the FWHM of the scattering near $(0,0,2)$ drops to 0 at 5.2 K as the sample develops long-range order, for the higher doping of $x=0.194$, the scattering maintains a finite Q-width to the lowest temperatures measured. The trend for the evolution of the FWHM for $x=0.194$, however, extrapolates approximately to 0 at $T=0$.

Figure 8

Maps of the elastic scattering intensity $S$(Q, $E=0$) for (Co${}_{1-x}$Mg${}_x$)${}_3$V${}_2$O${}_8$ with $x=0.029$ and $x=0.194$ and for two different temperatures, $T=1.5$ K (top row) and $T=5.5$ K (bottom row).

Figure 9

Cuts through the color contour maps maps shown in Fig. 8 along $0,0,L$. The observed $(0,0,1)$ peak at $T=1.5$ K in the $x=0.194$ sample is a clear signature of incommensurate fluctuations of the form $(0,\delta ,L)$ that are observed in the $x=0.029$ sample at $T=5.5$ K and $T=7$ K. This is consistent with the phase diagram presented in Fig. 2.

Figure 10

$S(\mathbf{Q},E)$ in $x=0$ (top), $x=0.029$ (middle), and $x=0.194$ (bottom row) samples along two perpendicular directions in reciprocal space. All data shown have had their respective high-temperature $T=20$ K background data sets subtracted. Note that the data have been taken at the base temperatures appropriate to each experiment, which was 200 mK for $x=0$ and $T=1.5$ K for both $x=0.029$ and $x=0.194$.

Figure 11

Cuts through scattering intensity around $[1.5,0,0]$ and integrating over a narrow range in $H=[1.4,1.6]$ and $L=[-0.2,0.2]$ on a logarithmic intensity scale. To facilitate the comparison between these two sets of measurements, the scattered intensity has been normalized to the incoherent elastic scattering of the pure sample.

Figure 12

The temperature evolution of the inelastic scattering $S(\mathbf{Q},E)$ along $[0,0,L]$, contrasting the $x=0.029$ and $x=0.194$ samples. Panels (a), (b), and (c) correspond to the $x=0.029$ sample at the temperatures indicated in phase diagram in the top right panel. Panels (e) and (f) correspond to the $x=0.194$ sample with temperatures as shown in the phase diagram.

Figure 13

Percolation calculations showing the evolution of the largest cluster size in a lattice containing 100 $\times$ 100 unit cells as a function of site dilution $x$. The inset shows the length of the shortest path through the system connecting the edges of the lattice, if it exists. Above the percolation threshold, the lattice edges cannot be connected, and there is no shortest path. Note that the evolution of the largest cluster size is normalized to its value at zero site dilution ($x=0$). Error bars in the normalized largest cluster sizes are determined from the standard error. The determination of the shortest path through the system was based on only those data sets that showed connectivity of the lattice, and the error bars are given by the standard error of these data sets.