Low-temperature thermodynamic properties near the field-induced quantum critical point in NiCl${}_2$-4SC(NH${}_2$)${}_2$

We present a comprehensive experimental and theoretical investigation of the thermodynamic properties: specific heat, magnetization, and thermal expansion in the vicinity of the field-induced quantum critical point (QCP) around the lower critical field $H_{c1}\approx 2$ T in NiCl${}_2$-4SC(NH${}_2$)${}_2$. A $T^{3/2}$ behavior in the specific heat and magnetization is observed at very low temperatures at $H=H_{c1}$, which is consistent with the universality class of Bose-Einstein condensation of magnons. The temperature dependence of the thermal expansion coefficient at $H_{c1}$ shows minor deviations from the expected $T^{1/2}$ behavior. Our experimental study is complemented by analytical calculations and quantum Monte Carlo simulations, which reproduce nicely the measured quantities. We analyze the thermal and the magnetic Grüneisen parameters, which are ideal quantities to identify QCPs. Both parameters diverge at $H_{c1}$ with the expected $T^{-1}$ power law. By using the Ehrenfest relations at the second-order phase transition, we are able to estimate the pressure dependencies of the characteristic temperature and field scales.

Figure 1

Magnetization $M$ as a function of the magnetic field $H$ of DTN at 16 mK for measurements $H$ parallel (solid line, Ref. 21) and perpendicular to the crystallographic $c$ direction at 500 mK (dashed line, Ref. 20). The inset shows new results of the magnetization at 50 mK in low fields up to 2.5 T and a linear fit (dashed line) to the data.

Figure 2

Corrected magnetization $\Delta M$ versus $T^{3/2}$ for magnetic fields $H\parallel c$ between 1  and 3 T. The magnetization at the critical field ${\mu }_0{H}_{c2}=2.08$ T is labeled with stars. Data above 1 T are shifted vertically by $0.02{\mu }_B$ per Ni atom for better visualization. Arrows indicate the phase transition into the $XY$ AFM state.

Figure 3

Main panel: total specific heat $C/T$ as a function of temperature $T$ at the critical field $H_{c1}=2.06$ T (filled dots) and the resulting $C_{\mathrm{mag}}/T$ (open squares) after subtraction of the nuclear Schottky (NS) contribution at low temperatures. Inset (a) shows the data of the main panel as $C/T$ versus $T^{-3}$ to illustrate the NS specific heat as indicated by a dashed line. The prefactor of the NS specific heat $C_{\text{NS}}/T=a{T}^{-3}$ is plotted in inset (b) versus $H^2$. It does not follow the expected $H^2$ field dependence.

Figure 4

Magnetic specific heat $C_{\mathrm{mag}}/T$ as a function of temperature $T$ for magnetic fields between 0 and 2.2 T in a double-log scale. The arrows indicate the AFM phase transition for fields $H>H_{c1}$. The critical field obtained from MCE experiments is 2.06 T. The inset shows the energy gap between 0 and 1.7 T, estimated from the exponential temperature increase of the magnetic specific heat at low temperatures.

Figure 5

Linear thermal expansion coefficient ${\alpha }_c$ measured along the crystallographic $c$ axis ($H\parallel c\parallel \Delta L_c$) as a function of the temperature $T$ between 0 and 5.5 T, including the critical field $H_{c1}=2.02$ T. Arrows mark the phase transition into the AFM ordered state. The inset displays in addition to ${\alpha }_c$ the coefficient ${\alpha }_a$ ($H\parallel c,\Delta L_a$) and the calculated volumetric coefficient ${\alpha }_V=2{\alpha }_a+{\alpha }_c$ for 0 T.

Figure 6

Experimental magnetization $\Delta M\left(T\right)$ versus temperature $T$ (symbols) at the critical field ${\mu }_{0}H=2.08$ T applied along the crystallographic $c$ direction of DTN. The dashed line represents QMC results, whereas the solid line indicates the analytic calculations. The symbols in the inset show the magnetic Grüneisen parameter ${\Gamma }_{\mathrm{mag}}$ estimated from the data of the main panel and the specific-heat values shown in Fig. 4. Dashed and solid lines represent QMC and analytical results, respectively.

Figure 7

Magnetic specific heat as $C_{\mathrm{mag}}/T$ as a function of temperature $T$ at the critical field $H_{c1}=2.06$ T (stars), at 2.04 T (open circles), 2 T (filled triangles), and 1.9 T (open diamonds) compared with QMC results at $H_{c1}$ (solid line) and with analytical calculations (dashed, dotted lines) at $H_{c1}$ and for fields below the critical field.

Figure 8

The main panel shows the normalized length change $\Delta L/L$ (symbols) as a function of the temperature $T$ of DTN for $H\parallel c\parallel L_c$ at the critical field $H_{c1}=2.02$ T in a semilogarithmic display. The experimental data are compared with QMC calculations, where the spin-spin correlator (SSC) (dashed line) and the SSC together with the ${\left(S^z\right)}^2$ term (dotted line) are taken into account. The inset (a) shows the experimental data at zero field (solid), the critical field $H_{c1}=2.02$ T (dotted line), and the difference $\Delta L_c/L\left(H_{c1}\right)-\Delta L_c/L\left(0\right)$ (symbols). In inset (b), the absolute values $\Delta L/L$ (symbols) are plotted as a function of $T$ in comparison with $\sim T^{1/2}$ (solid), $\sim T^{3/2}$ (dashed), and $\sim T^2$ (dotted line) temperature dependence.

Figure 9

The volume thermal expansion coefficient divided by $T$ is presented for fields close to the critical fields as a function of temperature $T$ in a semilogarithmic plot for DTN with field direction $H\parallel c$. The inset shows the absolute value of the thermal Grüneisen parameter ${\Gamma }_{\mathrm{th}}={\alpha }_V/C$ at the critical field $H_{c1}$. The solid line in the inset illustrates $T^{-1}$ behavior.

Figure 10

Volume magnetostriction coefficient ${\lambda }_V$ and susceptibility $\chi ={\mu }_0^{-1}\partial M/\partial H$ at 0.1 K of DTN. The susceptibility is scaled by the hydrostatic pressure dependence of the critical field $\Omega$ estimated from the Ehrenfest relations.