Muon spin relaxation studies of the frustrated quasi-two-dimensional square-lattice spin system $\text{Cu}\left(\text{Cl},\text{Br}\right)\text{La}{\left(\text{Nb},\text{Ta}\right)}_2{\text{O}}_7$: Evolution from spin-gap to antiferromagnetic state

We report muon spin relaxation $\left(\mu \text{SR}\right)$ and magnetic-susceptibility measurements on $\text{Cu}\left(\text{Cl},\text{Br}\right)\text{La}{\left(\text{Nb},\text{Ta}\right)}_2{\text{O}}_7$, which demonstrate: (a) the absence of static magnetism in $\left(\text{CuCl}\right){\text{LaNb}}_2{\text{O}}_7$ down to 15 mK confirming a spin-gapped ground state; (b) phase separation between partial volumes with a spin-gap and static magnetism in $\left(\text{CuCl}\right)\text{La}{\left(\text{Nb},\text{Ta}\right)}_2{\text{O}}_7$; (c) history-dependent magnetization in the (Nb,Ta) and (Cl,Br) substitution systems; (d) a uniform long-range collinear antiferromagnetic state in $\left(\text{CuBr}\right){\text{LaNb}}_2{\text{O}}_7$; and (e) a decrease in Néel temperature with decreasing Br concentration $x$ in $\text{Cu}\left({\text{Cl}}_{1-x}{\text{Br}}_x\right){\text{LaNb}}_2{\text{O}}_7$ with no change in the ordered Cu moment size for $0.33\le x\le 1$. Together with several other $\mu \text{SR}$ studies of quantum phase transitions in geometrically frustrated spin systems, the present results reveal that the evolution from a spin-gap to a magnetically ordered state is often associated with phase separation and/or a first-order phase transition.

Figure 1

(Color online) (a) Crystal structure of $\left(\text{CuCl}\right){\text{LaNb}}_2{\text{O}}_7$ (Ref. 11). (b) Exchange interactions $J_1$ and $J_2$ on the 2D square lattice. (c) Conceptual phase diagram of the spin-1/2 square-lattice $J_1\text{−}{J}_2$ model (Refs. 12, 13) as a function of $J_1$ and $J_2$ with regions of collinear antiferromagnetic (CLAF), ferromagnetic (FM) and Néel AF order. A spin-gap or spin-liquid state is expected in the shaded region. The present work elucidates the evolution illustrated by the purple arrow between the FM and CLAF phases.

Figure 2

(Color online) Magnetic susceptibility $\chi$ of $\left(\text{CuCl}\right){\text{LaNb}}_2{\text{O}}_7$, $\left(\text{CuCl}\right){\text{LaTa}}_2{\text{O}}_7$ and $\chi$ and $1/\chi$ of $\left(\text{CuBr}\right){\text{LaNb}}_2{\text{O}}_7$ in an external field of 1 kG.

Figure 3

(Color online) (a) ZF-$\mu \text{SR}$ time spectra in $\left(\text{CuCl}\right){\text{LaNb}}_2{\text{O}}_7$ demonstrating absence of static magnetism, and in $\left(\text{CuCl}\right)\text{La}{\left({\text{Nb}}_{1-y}{\text{Ta}}_y\right)}_2{\text{O}}_7$ showing magnetic order in a partial volume fraction. (b) ZF-$\mu \text{SR}$ spectra in $\left({\text{CuCl}}_{1-x}{\text{Br}}_x\right){\text{LaNb}}_2{\text{O}}_7$ at low temperatures.

Figure 4

(Color online) ZF-$\mu \text{SR}$ time spectra in (a) $\left(\text{CuBr}\right){\text{LaNb}}_2{\text{O}}_7$ and (b) $\left(\text{CuCl}\right){\text{LaTa}}_2{\text{O}}_7$ showing the onset of long-range antiferromagnetic order. The solid lines represent Eq. (2) in (a) and Eq. (3) in (b).

Figure 5

(Color online) Muon spin precession frequency $\nu$ observed in $\text{Cu}\left({\text{Cl}}_{1-x}{\text{Br}}_x\right){\text{LaNb}}_2{\text{O}}_7$ with $x=1.0$, 0.67, and 0.33, and in $\left(\text{CuCl}\right){\text{LaTa}}_2{\text{O}}_7$

Figure 6

(Color online) [(a) and (b)]: Amplitude of persisting muon spin precession in a WTF of 100 G in (a) $\left(\text{CuCl}\right)\text{La}{\left(\text{Nb},\text{Ta}\right)}_2{\text{O}}_7$ and (b) $\text{Cu}\left(\text{Cl},\text{Br}\right){\text{LaNb}}_2{\text{O}}_7$. This represents the fraction of muons landing in paramagnetic or nonmagnetic environment. In each specimen, about 25% of the amplitude persists at any temperature and composition as denoted by the dashed lines. The dotted lines show the level of the background originating from instruments (Ref. 30).

Figure 7

(Color online) (a) System and (b) temperature dependence of the $\mu \text{SR}$ time spectra in zero field in $\left(\text{CuCl}\right)\text{La}{\left(\text{Nb},\text{Ta}\right)}_2{\text{O}}_7$, which exhibit signals of fast- and slow-decay components, with composition- and $T$-dependent amplitudes. (c) The exponential muon spin relaxation rate $\Lambda$ of the fast decay component in (a) and (b).

Figure 8

(Color online) [(a) and (b)] Magnetic susceptibility for both FC and ZFC in 100 G obtained in (a) $\left(\text{CuCl}\right)\text{La}{\left(\text{Nb},\text{Ta}\right)}_2{\text{O}}_7$ and (b) $\text{Cu}\left(\text{Cl},\text{Br}\right){\text{LaNb}}_2{\text{O}}_7$. [(c) and (d)] The irreversible magnetization $M_{irr}\equiv M_{\text{FC}}-M_{\text{ZFC}}$ at $T=2\text{K}$. For $\left(\text{CuCl}\right){\text{LaNb}}_2{\text{O}}_7$ (i.e., $y=0$ and $x=0$), results from two different specimens are shown: sample (I) (open circles) which was used in the high-field magnetization studies (Ref. 20), and sample (II) (closed circles) which was used in the present $\mu \text{SR}$ measurements. The inset of (b) shows that $M_{\text{ZFC}}$ and $M_{\text{FC}}$ of sample (II) increase below $T\sim 6\text{K}$ without significant irreversibility effect. In (d), we also added a point of $M_{irr}$ for the $x=0.05$ (Cl,Br) system obtained in $H=200\text{G}$. Nearly identical values of $M_{irr}$ for $H=100$ and 200 G indicate a static origin of the irreversible magnetization.

Figure 9

(Color online) Phase diagram of $\text{Cu}\left(\text{Cl},\text{Br}\right)\text{La}{\left(\text{Nb},\text{Ta}\right)}_2{\text{O}}_7$ obtained in the present study, with spin-gap (SG), glassy antiferromagnetic (GAF), and collinear antiferromagnetic (CLAF) states. The striped region indicates phase separation. The broken and solid lines denote, respectively, first- and second-order thermal transitions. The arrows attached to the horizontal axis show the region with history dependence (HD) of the magnetic susceptibility and the region where static magnetism exists in a partial volume fraction (PV).

Figure 10

(Color online) A schematic view of free-energy profile as a function of magnetic order parameter $m$. The curves (a) and (b) represent the case of standard second-order phase transitions, above and below $T_c$, respectively, while (d) and (c) for the first-order transitions with (d) above $T_c$ and (c) at $T_c$.