Enhanced spin correlations in the Bose-Einstein condensate compound ${\mathrm{Sr}}_3{\mathrm{Cr}}_2{\mathrm{O}}_8$

Combined experimental and modeling studies of the magnetocaloric effect, ultrasound, and magnetostriction were performed on single-crystal samples of the spin-dimer system ${\mathrm{Sr}}_3{\mathrm{Cr}}_2{\mathrm{O}}_8$ in large magnetic fields to probe the spin-correlated regime in the proximity of the field-induced $XY$-type antiferromagnetic order also referred to as a Bose-Einstein condensate of magnons. The magnetocaloric effect, measured under adiabatic conditions, reveals details of the field-temperature $(H,T)$ phase diagram, a dome characterized by critical magnetic-fields $H_{c1}=30.4,H_{c2}=62\mathrm{T}$, and a single maximum ordering temperature $T_{\max }\left(45\mathrm{T}\right)\simeq 8\mathrm{K}$. The sample temperature was observed to drop significantly as the magnetic field is increased, even for initial temperatures above $T_{\max }$, indicating a significant magnetic entropy associated with the field-induced closure of the spin gap. The ultrasound and magnetostriction experiments probe the coupling between the lattice degrees of freedom and the magnetism in ${\mathrm{Sr}}_3{\mathrm{Cr}}_2{\mathrm{O}}_8$. Our experimental results are qualitatively reproduced by a minimalistic phenomenological model of the exchange striction by which sound waves renormalize the effective exchange couplings.

Figure 1

Adiabatic magnetocaloric-effect data obtained for ${\mathrm{Sr}}_3{\mathrm{Cr}}_2{\mathrm{O}}_8$ in a 36-ms short-pulse magnet with the magnetic field applied along the crystallographic $c$ axis. Individual curves were obtained for various initial temperatures between 4 and 20 K. The lowest-temperature curves show initial heating, possibly caused by the presence of a small number of paramagnetic impurities [3]. The sample temperature for all curves shows an overall cooling effect with increasing magnetic field, which can be attributed to the combined effect of enhanced excitations and correlations as the spin energy gap is suppressed. The temperature uncertainty is estimated to be $\le 0.2\mathrm{K}$.

Figure 2

Relative change in the sound velocity $\mathrm{\Delta }v/v$ vs temperature for the longitudinal $c_{11}$ and $c_{33}$ acoustic modes in ${\mathrm{Sr}}_3{\mathrm{Cr}}_2{\mathrm{O}}_8$. The inset shows the vicinity of the Jahn-Teller phase transition. The ultrasound frequency was 60 MHz for the $c_{11}$ mode (red line) and 47 MHz for the $c_{33}$ mode (blue line).

Figure 3

Relative change in the sound velocity $\mathrm{\Delta }v/v$ vs temperature for (a) the longitudinal $c_{11}$ mode and (b) the transverse $c_{44}$ and $\left(c_{11}\text{−}{c}_{12}\right)/2$ acoustic modes in ${\mathrm{Sr}}_3{\mathrm{Cr}}_2{\mathrm{O}}_8$ at selected magnetic fields.

Figure 4

(a) Thermal expansion measured on the $ab$ plane at various DC magnetic fields up to 34.5 T. (b) Thermal expansion measured along the $c$ axis direction at 45 T. The inset: Coefficient of thermal expansion vs temperature $\alpha \left(T\right)$ computed at 45 T.

Figure 5

Magnetic-field dependence of the axial strain measured in continuous magnetic fields (a) along the $ab$-plan direction and (b) along the $c$-axis direction at various temperatures. The small oscillations in the paramagnetic sate $H<H_{c1}$ are an artifact originated in Faraday rotation of light in the optical fiber caused by the large 45-T magnet fringe field [14].

Figure 6

Relative change in (a) the sound velocity and (b) sound attenuation of the $c_{44}$ mode vs magnetic-field ($H\parallel c$) measured at selected temperatures in ${\mathrm{Sr}}_3{\mathrm{Cr}}_2{\mathrm{O}}_8$.

Figure 7

Relative change in the sound velocity of (a) the longitudinal $c_{11}$ and (b) the transverse $\left(c_{11}\text{−}{c}_{12}\right)/2$ mode vs magnetic field measured at selected temperatures in ${\mathrm{Sr}}_3{\mathrm{Cr}}_2{\mathrm{O}}_8$.

Figure 8

($H,T$) phase diagram from our dilatometry and sound-velocity data for DC magnetic fields along the $ab$ plane, along the $c$ axis, and magnetocaloric effect data in pulsed magnetic fields. Although excess entropy seems to have an important effect at temperatures $T>T_{\max }$, our diagram largely agrees with that of Aczel et al., [3] obtained on samples from a different source. The temperature uncertainty is $<0.2\mathrm{K}$, the field uncertainty is negligible for DC fields and 0.5 T for pulsed fields.

Figure 9

Calculated magnetic-field dependence for the relative changes in the sound velocity related to the shear mode $c_{44}$ for ${\mathrm{Sr}}_3{\mathrm{Cr}}_2{\mathrm{O}}_8$ at $T=4.2\mathrm{K}$ using the known magnetization vs field. These results reproduce well the experiments [7, 8].

Figure 10

Calculated temperature dependence of the magnetization per spin for magnetic-fields $H\ge H_{c1}$ in the framework of the phenomenological model presented here. The shown curves were computed at the magnetic fields indicated.

Figure 11

Calculated temperature dependence of the magnetic susceptibility per spin at $H=0$ for the phenomenological model (black) and the simpler mean field [33] (dashed red). The parameters for the latter are taken from Aczel et al. [3] $N_A$ is Avogadro's number.

Figure 12

Calculated temperature and magnetic-field dependence of the magnetic susceptibility per spin. The intersection with a plane (gray) reproduces the critical ($H_c,T_c$) line.

Figure 13

(a) Calculated magnetic-field dependence of the relative change in the velocity of the shear acoustic mode $c_{44}$ for various temperatures indicated. (b) Calculated magnetic field dependence of the sound attenuation of the shear mode $c_{44}$ at the same temperatures. Compare to Fig. 6. The gradual broadening of the transition is likely the results of higher temperatures.

Figure 14

Calculated temperature dependence of the relative change in the velocity of the longitudinal acoustic mode $c_{11}$ for various values of the magnetic field. Compare to Fig. 3.