Spin relaxation in ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_{4-x}{\mathrm{Br}}_x$

The quantum-spin $S=1/2$ chain system ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_4$ is of high interest due to competing antiferromagnetic intrachain $J$ and interchain exchange $J^{\prime }$ interactions and represents a paramount example for Bose-Einstein condensation of magnons [R. Coldea et al., Phys. Rev. Lett. 88, 137203 (2002)]. Substitution of chlorine by bromine allows tuning the competing exchange interactions and corresponding magnetic frustration. Thereby, anisotropic exchange contributions may be decisive for the resulting ground state. Here we report on electron spin resonance (ESR) in single crystals of ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_{4-x}{\mathrm{Br}}_x$ with the aim to analyze the evolution of these anisotropic exchange contributions. The main source of the ESR linewidth is attributed to the uniform Dzyaloshinskii-Moriya interaction. The vector components of the Dzyaloshinskii-Moriya interaction are determined from the angular dependence of the ESR spectra using a high-temperature approximation. The obtained results support the site selectivity of the Br substitution suggested from the evolution of lattice parameters and magnetic susceptibility dependent on the Br concentration.

Figure 1

Crystal structure of ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_{4-x}{\mathrm{Br}}_x$ built up from $\mathrm{Cu}{\mathrm{Cl}}_4$ tetrahedra with Cu (small orange spheres), Cl1 (dark green spheres), Cl2 (green spheres), and Cl3 (light green spheres). The Cs ions are omitted for clarity. The relevant intra/interchain exchange $(J/J^{\prime })$ and DM interactions ($\mathbf{D}/{\mathbf{D}}^{\prime }$) are indicated by dashed/dotted lines. The numbers 1–4 mark the four inequivalent chains. The copper $\mathbf{g}$ tensors are equivalent ${\mathbf{g}}_{\widehat{I}}$ on chain 1 and 3, and differ from ${\mathbf{g}}_{\widehat{I}I}$ on chain 2 and 4. The intrachain DM interaction $\mathbf{D}$ attains only 2 nonzero components $D_a$ and $D_c$ within the light blue $ac$ plane, uniform within each chain but alternating with all possible combinations of plus and minus signs between inequivalent chains, while the interchain DM interaction ${\mathbf{D}}^{\prime }$ is staggered in all three components.

Figure 2

ESR spectra of ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_{4-x}{\mathrm{Br}}_x$ at selected Br concentrations $x$ for the magnetic field $H$ applied along the crystallographic $b$ axis. The red solid lines indicate the fit by a Lorentzian line shape.

Figure 3

Temperature dependence of the linewidth $\mathrm{\Delta }H$ of ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_{4-x}{\mathrm{Br}}_x$ with different Br concentration $x$ for the magnetic field $H$ applied along the crystallographic $c$ axis. The solid lines indicate fits by Eq. (1) containing an Arrhenius law and a critical divergence.

Figure 4

Temperature dependence of the $g$ value of ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_{4-x}{\mathrm{Br}}_x$ with different $x$ for the magnetic field $H$ applied along the crystallographic $c$ axis.

Figure 5

Concentration dependence of (from top to bottom) low-temperature critical exponent $p$ and corresponding divergent linewidth contribution $\mathrm{\Delta }{H}_{\mathrm{div}}$ as well as constant contribution $\mathrm{\Delta }{H}_{\infty }$ and activation gap $\mathrm{\Delta }$ with values $\mathrm{\Delta }{H}_{\text{act}}\left(a\right)=67$ kOe, $\mathrm{\Delta }{H}_{\text{act}}\left(b\right)=59$ kOe, $\mathrm{\Delta }{H}_{\text{act}}\left(c\right)=68$ kOe. The parameters are obtained from fitting the linewidth of ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_{4-x}{\mathrm{Br}}_x$ by Eq. (1) for the magnetic field $H$ applied along the crystallographic $a$ (blue squares), $b$ (green circles) and $c$ axes (red triangles).

Figure 6

Angular dependence of $g$ value (upper frame) and linewidth $\mathrm{\Delta }H$ (lower frame) in ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_4$ at $T=100$ K for the magnetic field applied within the $ac$ plane and a perpendicular plane containing the $b$ axis. Black squares and red circles: X-band, green triangles: Q-band. Dashed and dotted lines indicate the two inequivalent $\mathbf{g}$ tensors, solid lines are fits as described in the text.

Figure 7

Angular dependence of $g$ value (upper frame) and linewidth $\mathrm{\Delta }H$ (lower frame) in ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_{3.6}{\mathrm{Br}}_{0.4}$ at $T=100$ K and X-band frequency for the magnetic field applied within the $ac$ plane and a perpendicular plane containing the $b$ axis. Dashed and dotted lines indicate the two inequivalent $\mathbf{g}$ tensors, solid lines are fits as described in the text.

Figure 8

Angular dependence of $g$ value (upper frame) and linewidth $\mathrm{\Delta }H$ (lower frame) in ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_{2.6}{\mathrm{Br}}_{1.4}$ at $T=100$ K for the magnetic field applied within the $ac$ plane and a perpendicular plane containing the $b$ axis. Black squares and red circles: X-band, green stars: Q-band. The Q-band data have been measured on a different crystal with slightly larger linewidth. Dashed and dotted lines indicate the two inequivalent $\mathbf{g}$ tensors, solid lines are fits as described in the text.

Figure 9

Angular dependence of $g$ value (upper frame) and linewidth $\mathrm{\Delta }H$ (lower frame) in ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_{1.6}{\mathrm{Br}}_{2.4}$ at $T=100$ K and X-band frequency for the magnetic field applied within the $ac$ plane and a perpendicular plane containing the $b$ axis. Dashed and dotted lines indicate the two inequivalent $\mathbf{g}$ tensors, solid lines are fits as described in the text.

Figure 10

Top frame: bromine-concentration dependence of the diagonal components of the $\mathbf{g}$ tensor in local coordinates. Middle frame: DM-vector components and ratio of DM components (inset). Bottom frame: Intrachain $J$, interchain $J^{\prime }$, and resulting effective exchange parameter $J_{\mathrm{eff}}=\sqrt{\langle J^2\rangle }$ of ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_{4-x}{\mathrm{Br}}_x$. The Curie-Weiss temperature $\mathrm{\Theta }$ obtained from the temperature dependence of the spin susceptibility is shown for comparison. The absolute values of the exchange interactions and squared DM vectors may vary within a factor of about 2 dependent on the chosen cutoff condition [40].

Figure 11

View of the crystal structure of ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_{4-x}{\mathrm{Br}}_x$ along the $b$ direction (Cu large orange, Cl middle green, and Cs small grey spheres) to illustrate possible superexchange paths between copper ions Cu(1) and Cu(2) of neighboring chains of adjacent $bc$ layers. Note that the numbering of the Cl ions just counts the possible exchange bridges independent from the notation of nonequivalent Cl sites used in Fig. 1.

Figure 12

Temperature dependence of the linewidth $\mathrm{\Delta }H$ of ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_{4-x}{\mathrm{Br}}_x$ with different Br concentration $x$ for the magnetic field $H$ applied along the crystallographic $c$ axis. Solid and dashed lines indicate fits by the quantum approach and by the classical approximation for the spin relaxation via uniform DM interaction in a spin 1/2 chain, respectively.

Figure 13

Concentration dependence of (from top to bottom) residual linewidth $\mathrm{\Delta }{H}_0$, uniform DM linewidth contribution $\mathrm{\Delta }{H}_{\mathrm{DM}}$, and intrachain exchange $J$ of ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_{4-x}{\mathrm{Br}}_x$ obtained from the temperature dependence of the ESR linewidth using the approach derived in Ref [28].