Specific heat study of the magnetocaloric effect in the Haldane-gap $S=1$ spin-chain material $\left[\mathrm{Ni}{\left({\mathrm{C}}_2{\mathrm{H}}_8{\mathrm{N}}_2\right)}_2{\mathrm{NO}}_2\right]({\mathrm{BF}}_4)$

Magnetocaloric effect in the Haldane-gap $S=1$ spin-chain $\left[\mathrm{Ni}{\left({\mathrm{C}}_2{\mathrm{H}}_8{\mathrm{N}}_2\right)}_2{\mathrm{NO}}_2\right]({\mathrm{BF}}_4)$ was investigated at low temperatures $\left(k_{B}T\ll J\right)$ and magnetic fields up to nominally $0.1B_{\mathrm{sat}}$, where $J$ and $B_{\mathrm{sat}}$ denote an exchange interaction and saturation magnetic field, respectively. Magnetic entropy was found to scale as $exp(\mathrm{\Delta }/k_{B}T)$ in zero magnetic field and $T^{0.67}$ for $B=B_c$, in which the Haldane gap should be closed, in reasonable agreement with the theoretical predictions. The existence of the inverse magnetocaloric effect for $B<B_c$ was confirmed. The occurrence of normal magnetocaloric response at lowest temperatures and magnetic fields was attributed to $S=1/2$ degrees of freedom arising from the ends of chain segments.

Figure 1

Temperature dependence of specific heat of NENB studied at various magnetic fields oriented along the $b$ axis of the crystal below 5 K. The dashed line denotes the extrapolation of the specific heat data at 9 T. See text for a more detailed discussion. Inset: Selected specific heat data of NENB from 0.4 to 25 K.

Figure 2

Low-temperature part of the magnetic specific heat of NENB in zero magnetic field (full circles) analyzed using Eq. (2) (dashed line). The contribution of ferromagnetic dimers with $J_d/k_B=0.6$ K and $x_d=0.003$, respectively, is represented by a solid line. Inset: The analysis of the lattice contribution (open circles) using Eq. (1) (solid line). For clarity, only selected points are plotted.

Figure 3

Temperature dependence of magnetic entropy of NENB in magnetic fields $B=0$ (open circles), and $B=B_c\approx 9$ T (open squares) in the corresponding coordinates. Solid lines represent fits using the predictions $S\left(T\right)\approx exp(\mathrm{\Delta }/k_{B}T)$ T and $S\left(T\right)\approx T^{\beta }$ for $B=9\mathrm{T}$, respectively.

Figure 4

Adiabatic demagnetization curves for NENB calculated from magnetic entropy (solid lines). The contribution of ferromagnetic dimers with $J_d/k_B=0.6$ K and $x_d=0.003$, respectively, is represented by dashed lines.

Figure 5

Adiabatic demagnetization curves calculated for the simplified energy level scheme for a Haldane system, considering only a singlet ground state and the first excited triplet with $D/k=7.5$ K and $E/k=0.7$ K for magnetic fields oriented parallel (a) and perpendicular (b) to the axis of quantization. Magnetic field dependences of the energy levels are presented in the insets.