Quasi-two-dimensional Bose-Einstein condensation of spin triplets in the dimerized quantum magnet ${\mathrm{Ba}}_2{\mathrm{CuSi}}_2{\mathrm{O}}_6{\mathrm{Cl}}_2$

We synthesized single crystals of composition ${\mathrm{Ba}}_2{\mathrm{CuSi}}_2{\mathrm{O}}_6{\mathrm{Cl}}_2$ and investigated their quantum magnetic properties. The crystal structure is closely related to that of the quasi-two-dimensional (2D) dimerized magnet ${\mathrm{BaCuSi}}_2{\mathrm{O}}_6$ also known as Han purple. ${\mathrm{Ba}}_2{\mathrm{CuSi}}_2{\mathrm{O}}_6{\mathrm{Cl}}_2$ has a singlet ground state with an excitation gap of $\mathrm{\Delta }/k_{\mathrm{B}}=20.8$ K. The magnetization curves for two different field directions almost perfectly coincide when normalized by the $g$ factor except for a small jump anomaly for a magnetic field perpendicular to the $c$ axis. The magnetization curve with a nonlinear slope above the critical field is in excellent agreement with exact-diagonalization calculations based on a 2D coupled spin-dimer model. Individual exchange constants are also evaluated using density functional theory (DFT). The DFT results demonstrate a 2D exchange network and weak frustration between interdimer exchange interactions, supported by weak spin-lattice coupling implied from our magnetostriction data. The magnetic-field-induced spin ordering in ${\mathrm{Ba}}_2{\mathrm{CuSi}}_2{\mathrm{O}}_6{\mathrm{Cl}}_2$ is described as the quasi-2D Bose-Einstein condensation of triplets.

Figure 1

Crystal structure of ${\mathrm{Ba}}_2{\mathrm{CuSi}}_2{\mathrm{O}}_6{\mathrm{Cl}}_2$ viewed along the (a) $b$, (b) $a$, and (c) $c$ axes.

Figure 2

(a) Temperature dependence of magnetic susceptibilities of BCuSOC measured for $H\parallel c$ and $H\perp c$, which are normalized by the $g$ factors of $g_{\parallel }=2.32$ and $g_{\perp }=2.06$. The solid line denotes the fit obtained using Eq. (3). (b) Temperature dependence of total specific heat divided by the temperature measured at zero magnetic field.

Figure 3

2D model of the exchange network in ${\mathrm{Ba}}_2{\mathrm{CuSi}}_2{\mathrm{O}}_6{\mathrm{Cl}}_2$. Thick solid lines represent the intradimer exchange interaction $J$, and thin solid and dashed lines, respectively, represent the interdimer exchange interactions $J_{\alpha \beta }^a$ and $J_{\alpha \beta }^b(\alpha ,\beta =1,2)$.

Figure 4

(a) Magnetic-field dependence of magnetization $M$ and its field derivative $dM/dH$ for BCuSOC measured at 1.4 K for $H\parallel c$ and $H\perp c$, which are normalized by the $g$ factors. An arrow indicates an anomaly in $dM/dH$ for $H\perp c$ owing to a phase transition accompanied with a small magnetization jump. (b) Axial magnetostriction ${\varepsilon }_a\left(H\right)$, with $H\perp c$ and $H\parallel c$, at $T=1.3$ K after subtraction of the smooth background of similar magnitude measured at $T=10$ K. The magnetostriction along and perpendicular to the $c$ axis [at $(g/2)H=25$ T] show magnitudes and signs which, in a magnetically isotropic system, would indicate conservation of the unit cell volume. (c) Comparison between experimental and theoretical magnetization curves for $H\parallel c$. The theoretical result was obtained using exact diagonalization (ED) for a 32-site spin cluster.

Figure 5

Detail of the Cu-$3d_{x^2-y^2}$ band manifold around the Fermi level in ${\mathrm{Ba}}_2{\mathrm{CuSi}}_2{\mathrm{O}}_6{\mathrm{Cl}}_2$ calculated using paramagnetic GGA-PBE and interpolated with MLWFs.

Figure 6

MLWF centered on a Cu site. Large antibonding O-$2p$ tails are clearly visible on the Cu coordinating oxygen atoms.

Figure 7

Graphical representation of the results obtained by the least-squares fitting procedure employed to evaluate the magnetic couplings: optimized energies calculated with the Ising Hamiltonian (2) are represented as a function of the DFT + $U_{\mathrm{scf}}$ relative energies. Positive couplings correspond to AFM interactions according to Eq. (1).