Magnetoelastic coupling and Grüneisen scaling in ${\mathrm{NdB}}_4$

We report high-resolution capacitance dilatometry studies on the uniaxial length changes in a ${\mathrm{NdB}}_4$ single crystal. The evolution of magnetically ordered phases below $T_{\mathrm{N}}=17.2$ K [commensurate antiferromagnetic phase (cAFM)], $T_{\mathrm{IT}}=6.8$ K [intermediate incommensurate phase (IT)], and $T_{\mathrm{LT}}=4.8$ K [low-temperature phase (LT)] is associated with pronounced anomalies in the thermal expansion coefficients. The data imply significant magnetoelastic coupling and evidence of a structural phase transition at $T_{\mathrm{LT}}$. While both cAFM and LT favor structural anisotropy $\delta$ between in-plane and out-of-plane length changes, it competes with the IT type of order, i.e., $\delta$ is suppressed in that phase. Notably, finite anisotropy well above $T_{\mathrm{N}}$ indicates short-range correlations which are, however, of neither cAFM, IT, nor LT type. Grüneisen analysis of the ratio of thermal expansion coefficient and specific heat enables the derivation of uniaxial as well as hydrostatic pressure dependencies. While $\alpha /c_{\mathrm{p}}$ evidences a single dominant energy scale in LT, our data imply precursory fluctuations of a competing phase in IT and cAFM, respectively. Our results suggest the presence of orbital degrees of freedom competing with cAFM, and successive evolution of a magnetically and orbitally ordered ground state.

Figure 1

Relative length changes $d{L}_i/L_i$ along the crystallographic [001]- (black circles) and [110]-direction (red circles) vs. temperature. Insets show enlargement of the low temperature region. Left and right ordinates are to scale but shifted in order to account for the difference at 200 K. Vertical dashed lines indicate the transition temperatures $T_{\mathrm{N}}, T_{\mathrm{IT}}$, and $T_{\mathrm{LT}}$ toward the commensurate antiferromagnetic phase (cAFM), the incommensurate IT phase, and the low-temperature phase LT.

Figure 2

Uniaxial thermal expansion coefficients ${\alpha }_i$ along the crystallographic (a) [001] and (b) [110] direction. Dashed lines indicate the transition temperatures $T_{\mathrm{LT}}, T_{\mathrm{IT}}$, and $T_{\mathrm{N}}$ (see the text).

Figure 3

(a) Volumetric thermal expansion coefficient corrected by the phonon contribution. Inset: Uncorrected volumetric thermal expansion coefficient and phonon fit to the high temperature data (see the text). (b) Corresponding background corrected specific heat, and (c) Fisher's specific heat $\partial \left({\chi }_{i}T\right)/\partial T$ for $B=0.1\mathrm{T}$ applied along the crystallographic [100] (black circles) and [110] direction (red circles), respectively. Filled (open) circles indicate measurements upon heating (cooling). Inset: Corresponding magnetic susceptibility ${\chi }_{\left[001\right]}$ highlighting hysteresis at $T_{\mathrm{LT}}$. Vertical dashed lines indicate $T_{\mathrm{LT}}, T_{\mathrm{IT}}$, and $T_{\mathrm{N}}$.

Figure 4

Grüneisen parameters ${\gamma }_i={\alpha }_i^{\prime }/c_{\mathrm{p}}^{\prime }$, where $i$ denotes the two directions (a) [001] and (b) [110], as well as the volume (c). Dashed lines show the transition temperatures $T_{\mathrm{N}}, T_{\mathrm{IT}}$, and $T_{\mathrm{LT}}$.

Figure 5

Temperature dependence of the relative distortion of the uniaxial length changes $\delta =(d{L}_{\left[110\right]}-d{L}_{\left[001\right]})/(d{L}_{\left[110\right]}+d{L}_{\left[001\right]})$. The inset shows an enlargement of the low-temperature behavior. Arrows indicate the transition temperatures of competing magnetic phases.