Charge density wave and crystalline electric field effects in ${\mathrm{TmNiC}}_2$

Single crystals of ${\mathrm{TmNiC}}_2$ were grown by the optical floating-zone technique and were investigated by x-ray diffraction (XRD), thermal expansion, electrical resistivity, specific heat, and magnetic susceptibility measurements. Single-crystal XRD reveals the formation of a commensurate charge density wave (CDW) characterized by a CDW modulation vector ${\mathbf{q}}_{\mathbf{2}\mathbf{c}}=(0.5,0.5,0.5)$, which is accompanied by a symmetry change from the orthorhombic space group $Amm2$ to the monoclinic space group $Cm$, i.e., to a CDW superstructure which is isostructural with that of ${\mathrm{LuNiC}}_2$. For all transport and thermodynamic properties, anomalies related to a second order-type thermodynamic CDW phase transition are observed at around $T_{\mathrm{CDW}}\simeq 375\mathrm{K}$. The large specific heat anomaly at $T_{\mathrm{CDW}}, \mathrm{\Delta }C\simeq 6.2\mathrm{J}{\mathrm{mol}}^{-1}{\mathrm{K}}^{-1}$, together with noticeable changes in entropy and enthalpy related to the CDW transition, suggests that this point group symmetry breaking CDW phase transition affects more significant parts of the Fermi surface as compared to the incommensurate CDW transition of, e.g., ${\mathrm{SmNiC}}_2$ with no change in point group symmetry. The results on the antiferromagnetic and paramagnetic state of ${\mathrm{TmNiC}}_2$ obtained by the above macroscopic techniques were complemented by microscopic studies via inelastic neutron scattering. A crystalline electric field modeling of macroscopic susceptibility and magnetic specific heat and entropy contributions as well as microscopic neutron scattering data, reveal crystal field eigenstates and eigenvalues with a ground-state doublet of the Tm-$4f$ electrons, which is well separated by about 25 meV from exited states of the $J=6$ ground-state multiplet.

Figure 1

(a) The high-temperature phase of ${\mathrm{TmNiC}}_2$ at 400 K viewed down [100]. Tm (yellow), Ni (blue) and C (gray) atoms are represented by ellipsoids drawn at the 75% probability levels. Symmetry operations of the $Amm2$ space group are indicated using the usual graphical symbols [32]. A gray backdrop indicates the unit cell of the low-temperature superstructure. (b) The low-temperature phase of ${\mathrm{TmNiC}}_2$ at 280 K viewed down [001] (corresponding to the $\left[01\overline{1}\right]$ direction of the high-temperature phase). The indicated unit-cell corresponds to the conventional setting of the low-temperature phase. A $p$-subscript indicates a vector pointing out of the drawing plane. The lattice basis vector $\mathbf{a}$ of the aristotype is indicated. (c) $h=0$ and (d) $h=\frac{1}{2}$ sections of reciprocal space of a ${\mathrm{TmNiC}}_2$ crystal at 280 K reconstructed from single-crystal XRD frame data. The reciprocal basis of the basic structure and the modulation vectors of both twin domains are indicated. A subscript “p” designates a vector projected onto the drawing plane.

Figure 2

(a) Temperature-dependent thermal expansion along orthorhombic $a$ (green), $b$ (red), and $c$ (blue) orientations of ${\mathrm{TmNiC}}_2$. Inset zooms towards the AF transition. Both, CDW and AF transition temperatures are indicated by dotted lines; (b) thermal expansion coefficients and volume expansion coefficient vs temperature. Solid lines are a guide to the eye.

Figure 3

Temperature-dependent electrical resistivity measured with applied current along orthorhombic $a$ (green), $b$ (red), and $c$ (blue) orientations of ${\mathrm{TmNiC}}_2$ single-crystal bars; solid lines are guides to the eye. Inset zooms towards the AF transition. Both CDW and AF transitions are indicated by the black arrows and dotted lines.

Figure 4

(a) Temperature-dependent specific heat of ${\mathrm{TmNiC}}_2$ (blue) and ${\mathrm{LuNiC}}_2$ (pink) where lines connecting data points are just guides to the eye; the inset zooms in on the AF transition; (b) $\mathrm{\Delta }{C}_p$ in the vicinity of the CDW transition where the straight solid line serves to extrapolate the CEF Schottky anomaly; the inset shows our estimate of the contribution $C_{\mathrm{CDW}}/T$ (see text); (c) temperature-dependent magnetic specific heat and entropy contributions where red dotted lines mark $S=Rln(2J+1)$ for $J=1/2$ and $J=6$, respectively.

Figure 5

(a) The temperature-dependent dc magnetic susceptibility of single-crystalline ${\mathrm{TmNiC}}_2$ measured along the orthorhombic $a$ axis orientation with an applied magnetic field of 63 mT and along the $b$- and $c$-orientations with a field of 0.1 T; (b) corresponding inverse dc susceptibility, $1/\chi$, shown for principal orientations, all measured at 1 T. The original data are shown as symbols and data of $1/{\chi }_b$ and $1/{\chi }_c$ corrected with respect to an obvious misalignment of the measured crystal (see text) are presented as dashed lines.

Figure 6

Color-coded INS intensity, $I(Q,\omega )$, i.e., depending on wave vector $Q$ and energy $\hslash \omega$ transfer, measured at $T\simeq 16\mathrm{K}$ for (a) ${\mathrm{TmNiC}}_2$ and (b) ${\mathrm{LuNiC}}_2$. For the sake of better visibility of phonon intensities, high intensities are cutoff in the color coding, i.e., the actual CEF intensity of ${\mathrm{TmNiC}}_2$ at around 28 meV reaches about 120 mbarn ${\mathrm{sr}}^{-1}{\mathrm{meV}}^{-1}$ per f.u. at low $Q$.

Figure 7

(a) CEF fitting of alignment corrected inverse magnetic susceptibility data of single-crystalline ${\mathrm{TmNiC}}_2$; (b) of magnetic heat capacity and magnetic entropy contributions (solid lines are fits); and (c)–(f) of magnetic scattering functions $I_{\mathrm{mag}}\left(\omega \right)$ measured at four different temperatures, where error bars of data points representing the quasielastic line are artificially enlarged by orders of magnitude to effectively exclude the quasielastic line from the fitting process; blue solid lines are calculated $I_{\mathrm{mag}}\left(\omega \right)$ functions obtained by Gaussian convolutions of the calculated magnetic scattering intensities (see text for details of the temperature variation of the applied half widths); (g) resulting CEF energy level splitting.