Charge density wave and lock-in transitions of ${\mathrm{CuV}}_2{\mathrm{S}}_4$

The three-dimensional charge density wave (CDW) compound ${\mathrm{CuV}}_2{\mathrm{S}}_4$ is known to undergo phase transitions at $\sim 91$ and $\sim 50\mathrm{K}$. Employing single-crystal x-ray diffraction on an annealed crystal, we confirm the formation of an incommensurate CDW at $T_{\text{CDW}}\approx 91\mathrm{K}$, and we establish the nature of the transition at $T_{\text{lock-in}}\approx 50\mathrm{K}$ as a lock-in transition toward a threefold superstructure. As-grown crystals develop the same incommensurate CDW as the annealed crystal does, but they fail to go through the lock-in transition. Instead, the length of the modulation wave vector continues to decrease down to low temperatures in as-grown crystals. These findings are corroborated by distinct temperature dependencies of the electrical resistivity, magnetic susceptibility, and specific heat measured on as-grown and annealed crystals. A superspace model for the crystal structure of the incommensurate CDW suggests that the formation of extended vanadium clusters is at the origin of the CDW. In the lock-in phase, short and long V-V distances persist, but clusters now percolate the entire crystal. The lowering toward orthorhombic symmetry appears to be responsible for the precise pattern of short and long V-V distances. However, the orthorhombic lattice distortion is nearly zero for the annealed crystal, while it is visible for the as-grown material, again suggesting the role of lattice defects in the latter.

Figure 1

Temperature dependence (2–300 K) of the electrical resistivity ($\rho$) of ${\mathrm{CuV}}_2{\mathrm{S}}_4$, as measured on an as-grown crystal (blue open circles) and on the same crystal after annealing (filled red dots).

Figure 2

Temperature dependence (5–250 K) of the specific heat ($C_p$) of ${\mathrm{CuV}}_2{\mathrm{S}}_4$, as measured on an as-grown crystal (blue open circles) and on the same crystal after annealing (red filled dots). Data for the as-grown crystal have been offset by $35\mathrm{J}{\mathrm{mol}}^{-1}{\mathrm{K}}^{-1}$. The inset shows $C_p/T$ at low temperatures. The linear fit resulted in ${\gamma }_{\text{asgr}}=37.6\left(9\right)$ and ${\gamma }_{\text{ann}}=28.0\left(4\right)\mathrm{mJ}{\mathrm{mol}}^{-1}{\mathrm{K}}^{-2}$ for the as-grown and annealed samples, respectively.

Figure 3

Temperature dependence (5–300 K) of the magnetic susceptibility ($\chi$) of ${\mathrm{CuV}}_2{\mathrm{S}}_4$. Measured on an annealed crystal upon cooling (red open circles) and heating (orange filled dots), and measured on an as-grown crystal upon cooling (blue open diamonds) and heating (violet filled diamonds).

Figure 4

Temperature dependence near the CDW transition of the intensity ($I$) of the satellite Bragg reflection $(10,12,0,-1)$ (open circles; left scale), and square of the magnitude ($u$) of the modulation amplitude of atom V1 (filled circles; right scale), as determined for crystal B. Notice the perfect scaling between these two quantities. The satellite intensity and modulation are zero for 85 K and higher temperatures.

Figure 5

Temperature dependence of the incommensurate component $\sigma$ of the modulation wave vector. Filled circles for crystal A (annealed). Open symbols for as-grown crystals, B (diamonds), C (up triangles), D (squares), and E (down triangle).

Figure 6

Temperature dependence of the volume of the $F$-centered unit cell. Filled circles for crystal A (annealed). Open symbols for as-grown crystals, B (diamonds), C (up triangles), D (squares), and E (down triangle).

Figure 7

Schematic representation of the $({\mathbf{a}}_F^{*},{\mathbf{b}}_F^{*})$ plane of the diffraction pattern of incommensurate ${\mathrm{CuV}}_2{\mathrm{S}}_4$ below $T_{\text{CDW}}$. Open circles indicate the positions of reflections that are extinct due to the $F$ centering. Positions of second-order satellites are indicated, but these reflections have not been observed.

Figure 8

Relation between the $F$-centered (blue) and $I$-centered (orange) unit cells for the pseudocubic $F$-centered lattice.

Figure 9

Temperature dependence of (a) ratio $c_F/a_F$ of the lattice parameters $a_F$ and $c_F$, and (b) the lattice parameter ${\gamma }_F$. Both refer to the $F$-centered setting of the $I$-centered tetragonal or orthorhombic lattices. Filled circles for crystal A (annealed). Open symbols for as-grown crystals, B (diamonds), C (up triangles), D (squares), and E (down triangle).

Figure 10

Perspective view of the crystal structure of ${\mathrm{CuV}}_2{\mathrm{S}}_4$. The basis vectors of the $F$-centered unit cell are indicated as ${\mathbf{a}}_F, {\mathbf{b}}_F$, and ${\mathbf{c}}_F={\mathbf{c}}_I$. Labels V1-1, V2-1, and other labels indicate atoms equivalent by symmetry to V1 and V2, respectively. Blue is vanadium, gray is copper, and yellow is sulfur.

Figure 11

$t$-plot of interatomic distances between one V atom and its six surrounding V atoms for the incommensurate CDW state of crystal A at 80 K. Horizontal dashed lines indicate distances for the basic structure. (a),(b) The central atom is V1 at ($1/8,-1/8,3/8$), which is part of V-atom chains along $-{\mathbf{a}}_F+{\mathbf{b}}_F, {\mathbf{b}}_F+{\mathbf{c}}_F$, and $-{\mathbf{a}}_F+{\mathbf{c}}_F$. (c),(d) The central atom is V2 at ($1/8,1/8,5/8$), which is part of V-atom chains along ${\mathbf{a}}_F+{\mathbf{b}}_F, {\mathbf{b}}_F+{\mathbf{c}}_F$, and ${\mathbf{a}}_F+{\mathbf{c}}_F$.

Figure 12

$t$-plot of interatomic distances between one V atom and its six surrounding V atoms for the commensurately modulated structure of the lock-in state of crystal A at 40 K. The threefold superstructure involves distances and atomic positions at $t$-values of $\frac{1}{6}, \frac{1}{2}$, and $\frac{5}{6}$, indicated by vertical dashed lines. Horizontal dashed lines indicate distances for the basic structure. (a),(b) The central atom is V1 at ($1/8,-1/8,3/8$). (c),(d) The central atom is V2 at ($1/8,1/8,5/8$). Distances are shown between the same atoms as have been used for Fig. 11.