Field-induced magnetic behavior in quasi-one-dimensional Ising-like antiferromagnet BaCo${}_2$V${}_2$O${}_8$: A single-crystal neutron diffraction study

BaCo${}_2$V${}_2$O${}_8$ is a nice example of a quasi-one-dimensional quantum spin system that can be described in terms of Tomonaga-Luttinger liquid physics. This is explored in the present study where the magnetic field-temperature phase diagram is thoroughly established up to 12 T using single-crystal neutron diffraction. The transition from the Néel phase to the incommensurate longitudinal spin density wave (LSDW) phase through a first-order transition, as well as the critical exponents associated with the paramagnetic-Néel phase transition, and the magnetic order both in the Néel and in the LSDW phase are determined, thus providing a stringent test for the theory.

Figure 1

Crystal structure of BaCo${}_2$V${}_2$O${}_8$. (a) Projection in the $(b,c)$ plane: For sake of clarity, only the two chains with $0\le x\le 0.5$ and $-0.25\le y\le 0.75$ are represented for $-0.62<z<1.62$. (b) Projection in the $(a,b)$ plane of the four chains of the unit cell ($0\le x\le 1$, $-0.25\le y\le 0.75$, $-0.15<z<1.15$): The arrows indicate the sense of rotation of the screw chains on increasing $z$.

Figure 2

Temperature dependence of the specific heat measured in BaCo${}_2$V${}_2$O${}_8$ in different magnetic fields from 0 to 7 T applied along the $c$ axis.

Figure 3

Graphical representation of the magnetic structure refinement in the Néel phase of BaCo${}_2$V${}_2$O${}_8$ at $T=1.8$ K and $H=0$: calculated versus measured integrated intensities.

Figure 4

Magnetic structure of BaCo${}_2$V${}_2$O${}_8$ in the two magnetic domains of the Néel phase ($H=0$ and $T=1.8$ K): perspective view (left panels) and projection along the $c$ axis (right panels). The cobalt atoms and their magnetic moment are labeled as in Fig. 1. The two kinds of chains are plotted in different colors: An axis $4_1$ runs through the red chains while an axis $4_3$ runs through the blue ones. Looking at the figures on the right allows an easy visualization of the connection between the two magnetic domains: The application, for example, of the $4_1$ screw axis running along the ${\mathrm{Co}}_1,...,{\mathrm{Co}}_4$ chain transforms domain #1 into domain #2 (Co${}_1$ transforms into Co${}_2$, while Co${}_{7^{\prime }}$ transforms into Co${}_8$).

Figure 5

(a) Temperature dependence of the magnetic signal: neutron counts on top of the strong 2 $\overline{3}$ 0 reflection of the Néel phase for various magnetic field values between 0 and 3.25 T and on top of the 2 $\overline{3}$ 0.127 reflection of the LSDW phase at $H=5$ T. The latter curve was obtained from $Q_L$ scans performed across the 2 $\overline{3}$ 0.127 position counting 1 min per point (and then renormalized to 10 s for the figure). The black solid lines are power-law fits performed on a temperature range $\delta T=T_c-T\sim 1$ K as explained in the text. (b),(c) The set of the $\beta$ and $T_c$ values as obtained for various fitting temperature ranges $\delta T$ at $H=0$ together with a linear extrapolation of both parameters to $\delta T=0$ for a precise determination of their values (see text). Inset of panel (a): log-log plot of the magnetic intensity with subtracted background, $I-I_0$, as a function of the reduced temperature, $1-T/T_c$, at $H=0$. The data (open circles) and the power-law fit (solid line) are presented using the extrapolated values for $\delta T\to 0$: $T_c=5.556$ K (for the data and the fit) and $\beta =0.307$ (for the fit).

Figure 6

Field dependence of the magnetic signal at various temperatures between 1.85 K and 5 K: neutron counts on top of the strong $\overline{2}$ 3 0 reflection of the Néel phase. The black solid lines are fit to a power law, as explained in the text. (Inset) ${\beta }^{\prime }$ values as a function of the temperature (see text). The dashed black line is a linear fit of these data.

Figure 7

Field dependence of the magnetic signal at $T=50$ mK, measured on top of $\overline{2}$ $\overline{3}$ 0 for two different field sweep rates, 0.2 T/min (a) and 3 mT/min (b), with increasing and decreasing field. A sizable field hysteresis is observed for the Néel-LSDW phase transition, whose magnitude increases with the sweep rate ($\Delta H\sim 0.2$ T at 3 mT/min, $\sim$$0.32$ T at 0.2 T/min).

Figure 8

$Q_L$ scans performed across the $\mathbf{Q}=(\overline{2},\overline{3},0)$ scattering vector at $T=50$ mK for different increasing magnetic field values. (a) Néel phase; (b) coexistence of the Néel and LSDW phases; (c),(d) LSDW phase. For a better visibility, the scans measured at 4 and 3.98 T in (b) were shifted vertically by 2000 and 4000, respectively.

Figure 9

Field dependence of the position along $c^{☆}$ of the magnetic reflections (red and blue circles for increasing and decreasing field, respectively), obtained from the fit of the $Q_L$ scans: antiferromagnetic peak at $Q_L=0$ in the Néel phase, incommensurate satellites at $Q_L=\pm \xi$ in the LSDW phase with $\xi$ the incommensurate modulation. The inset presents an enlargement of these curves around the transition. The solid black line in the main panel corresponds to $4M_z^{\mathrm{cor}}/g$, where $M_z^{\mathrm{cor}}$ is the magnetization curve $M_z\left(H\right)$ of Ref. 14 corrected for a residual Van Vleck contribution (Ref. 33) and $g=6.0$ as described in the text.

Figure 10

Field dependence of the integrated intensity of the magnetic peaks, obtained from the fit of the $Q_L$ scans for increasing (solid circles) and decreasing (open circles) field. The inset presents an enlargement of these data around the transition. The scattering of the red and blue data sets reflects the typical error bars that should be considered.

Figure 11

Magnetic $H$-$T$ phase diagram of BaCo${}_2$V${}_2$O${}_8$ obtained from the specific heat measurements presented in Sec. 3a (black squares) and from the present neutron diffraction measurements (red circles). As concerns the Néel-LSDW phase transition (see enlargement in the inset), the solid and open red circles correspond to increasing and decreasing field scans, respectively. The blue open triangles in the main panel are the data of Kimura et al.[14] obtained from macroscopic measurements and the blue dashed line is the theoretical LSDW-paramagnetic transition line predicted by Okunishi and Suzuki (Ref. 15).

Figure 12

Graphical representation of the magnetic structure refinement in the LSDW phase of BaCo${}_2$V${}_2$O${}_8$ at $T=1.8$ K and $H=4.2$ T in decreasing field (coming from 6 T): calculated versus measured integrated intensities.

Figure 13

Magnetic structure of BaCo${}_2$V${}_2$O${}_8$ in the LSDW phase at $H=4.2$ T (decreased from 6 T) for the magnetic domain  #1, shown over about one half period ($\sim$$5c$), using the same labeling and the same colors as in Fig. 4 for the two types of cobalt chains. This projection on the $(b,c)$ plane is to be compared to Fig. 4a, sketching the magnetic structure of domain  #1 in zero field. The magnetic domain #2 of the LSDW phase can simply be obtained from domain #1 by reverting all spins plotted in red (like for the structure in zero magnetic field).

Figure 14

Count on top of the 2 0 0 reflection as a function of the magnetic field evidencing the ferromagnetic magnetization $M_z$ arising when the field increases. The dashed horizontal black line corresponds to the intensity $I_0$ on top of the pure nuclear reflection. The solid black curve corresponds to the square of the magnetization curve $M_z$ of Ref. 14 corrected for a residual Van Vleck contribution (Ref. 33), scaled to our data, and with an offset $I_0$. (Inset) $\omega$ scan on the 2 0 0 reflection at $H=0$ (pure nuclear reflection) and $H=12$ T (nuclear and ferromagnetic reflection), allowing a precise determination of the ferromagnetic moment at 12 T.

Figure 15

(a) Enlargement on the Néel phase of the $H$-$T$ phase diagram. The same symbols as in Fig. 11 are used for the experimental data. Phase boundary is very well approximated by the shape of the superellipse (blue line) as described in the text. (b) Dependence of the “critical” power-law exponent obtained in a fitting procedure as a function of the width of the fitting interval, simulated for field sweeps at various constant temperatures. As the values close to zero are experimentally inaccessible, a linear extrapolation (red lines) of the available points (open symbols) to zero clearly underestimates the critical value, with progressively stronger underestimate as the temperature is lowered. (c) Temperature dependence of the obtained “critical” exponent ${\beta }^{\prime }$, to be compared with the inset of Fig. 6.