Sound Attenuation Study on the Bose-Einstein Condensation of Magnons in ${\mathrm{T}\mathrm{l}\mathrm{C}\mathrm{u}\mathrm{C}\mathrm{l}}_3$

We investigate experimentally and theoretically sound attenuation in the quantum spin system $\mathrm{T}\mathrm{l}\mathrm{C}\mathrm{u}\mathrm{C}{\mathrm{l}}_{\mathrm{3}}$ in magnetic fields at low temperatures. Near the point of Bose-Einstein condensation of magnons a sharp peak in the sound attenuation is observed. The peak demonstrates a hysteresis as a function of the magnetic field pointing to a first-order contribution to the transition. The sound damping has a Drude-like form arising as a result of hard-core magnon-magnon collisions. The strength of the coupling between lattice and magnons is estimated from the experimental data. The puzzling relationship between the transition temperature and the concentration of magnons is explained by their “relativistic” dispersion.

Figure 1

(a) Ultrasonic attenuation $\alpha (H)$ with increasing (full circles) and decreasing magnetic field (open circles) and a fitting of the broad maximum by a Gaussian (dashed line). Inset (b): transition regime after subtracting the Gaussian and a constant. Inset (c): phase diagram with $g{H}_c/2$ (thick line) from thermodynamic experiments [12]. The $g$ factor of $S_z=-1$ excitations $g=2.16$.

Figure 2

(a) Dimensionless concentration of magnons ${\overline{n}}_B(T_B)$ corresponding to the BEC temperature $T_B$. Squares present experimental data [12]. (b) $[{\overline{n}}_B(T_B)/{\overline{n}}_B(1\mathrm{K})]T_B^{-2}$. With increasing $\Delta$ the curves are closer to $T_B^{-1/2}$ (the dashed line), corresponding to $\varphi =3/2$.

Figure 3

Calculated ${\mu }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ (a) and damping (b),(c) for different spin-lattice couplings as a function of field $H$ at $T=3.5\mathrm{K}$. For $H<2.5\mathrm{T}$ the behavior of damping is almost flat, in agreement with the experiment. $\Delta =6.5\mathrm{K}$, $H_g=4.5\mathrm{T}$. The results depend on $V_{\mathrm{m}\mathrm{m}}$ which determines the $H_c$.