Low-temperature properties of single-crystal ${\mathrm{CrB}}_2$

We report the low-temperature properties of ${}^{11}\mathrm{B}$-enriched single-crystal ${\mathrm{CrB}}_2$ as prepared from high-purity Cr and B powder by a solid-state reaction and optical float zoning. The electrical resistivity, ${\rho }_{\mathrm{xx}}$, Hall effect, ${\rho }_{\mathrm{xy}}$, and specific heat, $C$, are characteristic of an exchange-enhanced Fermi liquid ground state, which develops a slightly anisotropic spin gap $\Delta \approx 220\mathrm{K}$ below $T_{\mathrm{N}}=88\mathrm{K}$. This observation is corroborated by the absence of a Curie dependence in the magnetization for $T\to 0$ reported in the literature. Comparison of $C$ with $d{\rho }_{\mathrm{xx}}/dT$, where we infer lattice contributions from measurements of ${\mathrm{VB}}_2$, reveals strong antiferromagnetic spin fluctuations with a characteristic spin fluctuation temperature $T_{\mathrm{sf}}\approx 257\mathrm{K}$ in the paramagnetic state, followed by a pronounced second-order mean-field transition at $T_{\mathrm{N}}$, and unusual excitations around $\approx T_{\mathrm{N}}/2$. The pronounced anisotropy of ${\rho }_{\mathrm{xx}}$ above $T_{\mathrm{N}}$ is characteristic of an easy-plane anisotropy of the spin fluctuations consistent with the magnetization. The ratio of the Curie-Weiss to the $\mathrm{N}\acute{\mathrm{e}}\mathrm{el}$ temperatures, $f=-{\Theta }_{\mathrm{CW}}/T_{\mathrm{N}}\approx 8.5$, inferred from the magnetization, implies strong geometric frustration. All physical properties are remarkably invariant under applied magnetic fields up to 14 T, the highest field studied. In contrast to earlier suggestions of local-moment magnetism our study identifies ${\mathrm{CrB}}_2$ as a weak itinerant antiferromagnet par excellence with strong geometric frustration.

Figure 1

Schematic picture of the crystal structure of the C32 diborides for the case of ${\mathrm{CrB}}_2$.

Figure 2

(a) ${\mathrm{CrB}}_2$ rod after optical float zoning. Large facets and a clean metallic surface indicate high sample quality. The growth direction was from the bottom to the top. (b) Laue x-ray diffraction pattern along the $c$ axis showing the characteristic sixfold symmetry. (c) Laue x-ray diffraction pattern along the $a$ axis of the same part of the ingot. The characteristic twofold symmetry demonstrates that the single-crystal grain extends across the ingot.

Figure 3

Temperature dependence of the electrical resistivity of ${\mathrm{CrB}}_2$. (a) Electrical resistivity for current parallel to the $a$ axis and the $c$ axis, where a pronounced kink marks the antiferromagnetic transition. (b) Derivative of the resistivity with respect to the temperature. (c) Close-up view of the resistivity below $T_{\mathrm{N}}$ for both current directions as fitted with a model according to Eq. (1) combining Fermi liquid behavior and scattering by spin waves with a slightly anisotropic spin wave gap.

Figure 4

Magnetoresistance and Hall effect of ${\mathrm{CrB}}_2$. (a) Magnetoresistance of ${\mathrm{CrB}}_2$ for various temperatures. (b) Hall resistivity for various temperatures. (c) The normal Hall coefficient as a function of temperature for current parallel to the $a$ axis and the $c$ axis. For both directions a pronounced maximum is observed at the $\mathrm{N}\acute{\mathrm{e}}\mathrm{el}$ temperature $T_{\mathrm{N}}=88.5$ K. Data extracted from measurements as a function of temperature at fixed fields of $\pm 6$ T (solid symbols) and from measurements as a function of field at fixed temperatures (open symbols) are in good agreement. (d) Effective carrier density as calculated from the data in panel (c).

Figure 5

Magnetization of ${\mathrm{CrB}}_2$ as a function of temperature. (a) Magnetization for fields up to 9 T displaying a clear kink at the $\mathrm{N}\acute{\mathrm{e}}\mathrm{el}$ transition. The absolute values are small. The crystalline $c$ direction corresponds to the magnetic hard axis. No qualitative changes are observed up to the highest fields studied. (b) Normalized magnetization, $M/H$, for ${\mu }_{0}H=1$ T for both field directions. At low temperatures there is no sign of a Curie tail. (c) Inverse normalized magnetization, $H/M$, for both field directions as well as for ${\mu }_{0}H=1$ T and ${\mu }_{0}H=9$ T. The solid green lines are Curie-Weiss fits for temperatures $T>90$ K.

Figure 6

Specific heat of ${\mathrm{CrB}}_2$ as a function of temperature. (a) A pronounced anomaly is present at the $\mathrm{N}\acute{\mathrm{e}}\mathrm{el}$ transition. It hardly changes in fields up to 9 T. The Dulong-Petit limit corresponds to $9R=74.83$ J ${\mathrm{mol}}^{-1}$ ${\mathrm{K}}^{-1}$. (b) Specific heat divided by temperature, $C/T$. Data are shown for ${\mathrm{CrB}}_2$ and ${\mathrm{VB}}_2$, where the latter is used for an estimate of the lattice contribution in ${\mathrm{CrB}}_2$. (c) Specific heat divided by temperature around the $\mathrm{N}\acute{\mathrm{e}}\mathrm{el}$ transition in more detail. An entropy-conserving construction yields the $\mathrm{N}\acute{\mathrm{e}}\mathrm{el}$ temperature $T_{\mathrm{N}}=88.5$ K. Measurements with heat pulses of 0.1% of the current temperature (open symbols) reproduce data measured with 0.5% heat pulses.

Figure 7

Magnetic contribution to the entropy of ${\mathrm{CrB}}_2$. The lattice contribution was inferred from the entropy of ${\mathrm{VB}}_2$ as described in the text.

Figure 8

Kadowaki-Woods plots of selected $d$- and $f$-electron compounds as presented in Ref. [74]. The Kadowaki-Woods ratio of ${\mathrm{CrB}}_2$ is close to the value $a_{\mathrm{HF}}=10\mu \Omega \mathrm{cm}{\mathrm{mol}}^2{\mathrm{K}}^2{\mathrm{J}}^{-2}$ observed in heavy-fermion compounds (upper dashed line).

Figure 9

Estimate of the average fluctuation rate for nonmagnetic ${\mathrm{CrB}}_2$. (a) Correlation between the Sommerfeld coefficient, $\gamma$, and the average fluctuation rate, ${\Gamma }_{\mathrm{ave}}$, for various compounds. The figure has been reproduced from Ref. [75]. An estimate for nonmagnetic ${\mathrm{CrB}}_2$ is inferred from the specific heat in the following panel. (b) Magnetic contribution to the specific heat of ${\mathrm{CrB}}_2$ at high temperatures. A fit corresponding to $C_{\mathrm{mag}}/T={\gamma }_{\mathrm{norm}}ln(T_{\mathrm{SF}}/T)$ (solid line) yields an estimate of the Sommerfeld coefficient expected for nonmagnetic ${\mathrm{CrB}}_2$, ${\gamma }_{\mathrm{norm}}=70\mathrm{mJ}{\mathrm{mol}}^{-1}{\mathrm{K}}^{-2}$, and a spin fluctuation temperature, $T_{\mathrm{SF}}=257$ K.