Bose-Einstein Condensation of Magnons in ${\mathrm{Cs}}_2{\mathrm{CuCl}}_4$

We report on results of specific heat measurements on single crystals of the frustrated quasi-2D spin-$1/2$ antiferromagnet ${\mathrm{Cs}}_2{\mathrm{CuCl}}_4$ ($T_N=0.595\mathrm{K}$) in external magnetic fields $B<12\mathrm{T}$ and for temperatures $T>30\mathrm{mK}$. Decreasing $B$ from high fields leads to the closure of the field-induced gap in the magnon spectrum at a critical field $B_c\simeq 8.51\mathrm{T}$ and a magnetic phase transition is clearly seen below $B_c$. In the vicinity of $B_c$, the phase transition boundary is well described by the power law $T_c(B)\propto (B_c-B{)}^{1/\varphi }$, with the measured critical exponent $\varphi \simeq 1.5$. These findings are interpreted as a Bose-Einstein condensation of magnons.

Figure 1

Specific heat of ${\mathrm{Cs}}_2{\mathrm{CuCl}}_4$ in zero magnetic field. From the total specific heat $C_{\mathrm{tot}}$ (open symbols), the phonon contribution $C_{\mathrm{ph}}$ has been subtracted to obtain the magnetic part $C_{\mathrm{mag}}$ (solid symbols). The solid and dashed lines represent the calculated $C(T)$ (see text). Inset: Temperature dependence of the magnetic entropy, $S_{\mathrm{mag}}(T)$.

Figure 2

Magnetic specific heat $C_{\mathrm{mag}}$ vs $T$ of ${\mathrm{Cs}}_2{\mathrm{CuCl}}_4$ close to the critical field. Inset: $C_{\mathrm{mag}}/T$ vs $T$ in a semilogarithmic plot. The data at $B=8.44\mathrm{T}$ reveal a small jump at $T_c=76\mathrm{mK}$, indicating the vicinity of the critical field.

Figure 3

Semilogarithmic plot of $C_{\mathrm{mag}}T$ vs $1/T$ of ${\mathrm{Cs}}_2{\mathrm{CuCl}}_4$ for fields above $B_c$. The slope of the data (solid lines) yields the value of the gap $\Delta$ present in the magnon excitation spectrum. The data shown in the inset were obtained at $B=8.6\mathrm{T}$ and are plotted as $C_{\mathrm{mag}}\sqrt{T}$ vs $1/T$.

Figure 4

$(T,B)$ phase diagram of ${\mathrm{Cs}}_2{\mathrm{CuCl}}_4$ for $B\parallel a$. The ordering temperature $T_c$ decreases as the magnetic field approaches the critical field $B_c=8.51\mathrm{T}$. A field-polarized ferromagnetic (FM) state is entered above $B_c$ and the gap $\Delta$ in the spin excitation spectrum opens. Inset: The experimental $T_c(B)$ data points and the calculated phase boundary of the BEC of magnons (solid line) agree very well.