Magnetic phase diagram of the frustrated $S=\frac{1}{2}$ chain magnet LiCu${}_2$O${}_2$

We present the results of the magnetization and dielectric constant measurements on untwinned single-crystal samples of the frustrated $S=1/2$ chain cuprate LiCu${}_2$O${}_2$. Magnetic phase transitions were observed. A spin-flop transition of the spiral spin plane was observed for the field orientations $\mathit{H}\parallel \mathit{a}$, $\mathit{b}$. The second magnetic transition was observed at $H\approx 15$ T for all three principal field directions. This high-field magnetic phase is discussed as a collinear spin-modulated phase, which is expected for an $S=1/2$ nearest-neighbor ferromagnetic and next-nearest-neighbor antiferromagnetic chain system.

Figure 1

(a) Schematic representation of the arrangement of Cu${}^{2+}$ moments in LiCu${}_2$O${}_2$. Only one of four Cu${}^{2+}$ ($\alpha$, $\beta$, $\gamma$, $\delta$) positions is shown. Arrows correspond to the spin directions below $T_{c2}$. $J_1,J_2$, and $J_3$ are the main exchange integrals.[19] (b) Scheme of the orientation of the spin plane for $\mathit{H}$ in the $bc$ plane. $H>H_{c1}$. Spin plane is shadowed with gray. The direction of spin rotation by displacement along the chain ($b$ axes) is shown by the circular arrow. $\mathit{n}$ is the vector normal to the spin plane.

Figure 2

Upper panel: Polarized optical microscope images of the $ab$ plane of LiCu${}_2$O${}_2$ samples (a) without twinning, (b) with two large twinned domains, and (c) with complicated twinning structure. Lower panel: X-ray diffraction pattern observed in $\theta –2\theta$ geometry on an untwinned single crystal. The spectra were obtained from the $bc$ and $ac$ planes of the crystal.

Figure 3

Upper panel: Temperature dependence of the low-field magnetization curves $M\left(T\right)/{\mu }_{0}H$ for three field directions $\mathit{H}\parallel \mathit{a},\mathit{b},\mathit{c}$ and ${\mu }_{0}H=0.1$ T. Lower panel: $M\left(T\right)/{\mu }_{0}H$ in the vicinity of the transition temperatures $T_{c1}$ and $T_{c2}$ at ${\mu }_{0}H=1$, 3, 5, and 7 T for $\mathit{H}\parallel \mathit{b}$.

Figure 4

Temperature dependences of the magnetization derivative $dM/dT$ for three field directions in the vicinity of $T_{c1}$ and $T_{c2}$ obtained from $M\left(T\right)$ data at ${\mu }_{0}H=1$, 5, and 7 T.

Figure 5

Upper panel: Magnetization curves $M\left(H\right)$ at 4.2 K for three field directions $\mathit{H}\parallel \mathit{a},\mathit{b},\mathit{c}$ measured in pulse magnetic fields of up to 52 T. The inset shows the field dependence of $dM/dH$ at 4.5 K for $\mathit{H}\parallel \mathit{b}$ measured with a SQUID magnetometer. Lower panels: Measured magnetic field dependences of $dM/dH$ for the principal field directions $\mathit{H}\parallel \mathit{a},\mathit{b},\mathit{c}$. The inset shows the enlarged dependency in the range of transition field $H_{c2}$ for $\mathit{H}\parallel \mathit{b}$.

Figure 6

Upper panel: Temperature dependences of dynamic dielectric constants of three samples at zero magnetic field for alternating electric field $\mathit{E}\parallel \mathit{a},\mathit{c}$. Dependences $\varepsilon \left(T\right)$ presented in the main panel were obtained by subtraction of the linear temperature drifts. Curves are shifted along the $y$ direction for better presentation. Inset shows the raw temperature dependence of the capacitance $C\left(T\right)$ for $\mathit{E}\parallel \mathit{a}$ at $H=0$, with the linear extrapolation of $C\left(T\right)$ at $T>T_c$ to the range of $T\lesssim T_{c2}$ shown by the dotted line. Lower panel: The field dependences of the capacitance for $\mathit{E}\parallel \mathit{a}$ and the magnetic field $\mathit{H}\parallel \mathit{b}$ at different temperatures. The data in the lower panel are plotted without shift along the $y$ axis.

Figure 7

Magnetic phase diagrams of the LiCu${}_2$O${}_2$ compound for $\mathit{H}\parallel \mathit{b}$ (upper panel) and $\mathit{H}\parallel \mathit{c}$ (lower panel). The different symbols in the figure correspond to data from different experimental methods. Solid triangles and circles show the phase boundaries obtained from ${\varepsilon }_c\left(T\right)$ and ${\varepsilon }_a\left(H\right)$ measurements, respectively. Open red circles and squares show the phase boundaries obtained from $M\left(H\right)$ and $M\left(T\right)$ measurements, respectively. The solid lines are guides to the eye. The grey shadowed area in the upper panel marks the sample-dependent position of the phase boundary between phases I and II. High-field spin structure III schematically shown in the phase diagram is merely speculative. The dashed line is a schematic phase boundary between the paramagnetic polarized and ordered phases.