Heat capacity of the site-diluted spin dimer system Ba${}_3$(Mn${}_{1-x}$V${}_x$)${}_2$O${}_8$

Heat-capacity and susceptibility measurements have been performed on the diluted spin dimer compound Ba${}_3$(Mn${}_{1-x}$V${}_x$)${}_2$O${}_8$. The parent compound Ba${}_3$Mn${}_2$O${}_8$ is a spin dimer system based on pairs of antiferromagnetically coupled $S=1$, 3$d^2$ Mn${}^{5+}$ ions such that the zero-field ground state is a product of singlets. Substitution of nonmagnetic $S=0$, 3$d^0$ V${}^{5+}$ ions leads to an interacting network of unpaired Mn moments, the low-temperature properties of which are explored in the limit of small concentrations $0⩽x⩽0.05$. The zero-field heat capacity of this diluted system reveals a progressive removal of magnetic entropy over an extended range of temperatures, with no evidence for a phase transition. The concentration dependence does not conform to expectations for a spin-glass state. Rather, the data suggest a low-temperature random singlet phase, reflecting the hierarchy of exchange energies found in this system.

Figure 1

(a) Schematic diagram showing the transition-metal sublattice of the dimer compound Ba${}_3$(Mn${}_{1-x}$V${}_x$)${}_2$O${}_8$. Mn ions are shown in red. Dominant antiferromagnetic exchange $J_0$ between pairs of Mn moments in the undiluted compound ($x$ $=$ 0) leads to a singlet ground state. In this diagram, a single Mn ion has been substituted by a nonmagnetic V ion (green). The Mn ion vertically below the V ion is now no longer paired and, hence, contributes a local magnetic moment. Distances between the V ion and its nearest neighbors $r_0$–$r_4$ are labeled. (b) Probability, assuming no clustering, of finding at least one V ion within a distance $r$ from the substituted ion, shown for the specific vanadium concentrations ($x$) studied, and for the specific distances $r_0-r_4$ (Ref. 20). (c) Probability of finding the nearest-neighboring V ion a distance $r$ from the substituted ion (Ref. 21).

Figure 2

(a) Susceptibility as a function of temperature for single crystals of Ba${}_3$(Mn${}_{1-x}$V${}_x$)${}_2$O${}_8$ (compositions listed in legend). Successive curves are offset by 0.004 emu/mol for clarity. Data are fit (red lines) by a dimer model, described in the main text, including a mean-field correction to account for interdimer interactions, a temperature-independent term, and a Curie-Weiss term to account for the unpaired magnetic impurities introduced by V substitution. Values for the exchange constants were held fixed at the values obtained for the stoichiometric compound (black circles). (b) Curie constant C (left axis), and the associated effective moment per V, ${\mu }_{\text{eff}}$/ V (right axis). Note that V${}^{5+}$ is nonmagnetic, and that the moment arises from the unpaired Mn${}^{5+}$ $S=1$ spin on the broken dimers. The dotted horizontal line shows the anticipated effective moment ${\mu }_{\text{eff}}=g\sqrt{S(S+1)}{\mu }_B=2.83$ ${\mu }_B$ for Mn${}^{5+}$. (c) Measured versus nominal composition of the single crystals used in this study.

Figure 3

(a) Low-temperature heat capacity of Ba${}_3$(Mn${}_{1-x}$V${}_x$)${}_2$O${}_8$ in zero field. Substitution of vanadium leads to a substantial increase in the total heat capacity. (b) Left axis: The same data, plotted as $C_p/T$ and scaled by V concentration, revealing the evolution in the functional form of the heat capacity. Right axis: Change in entropy $\Delta S$ between 50 and 500 mK associated with the total heat capacity. Despite subtle changes in the functional form of the heat capacity as the vanadium concentration is increased, the integrated entropy per mole of V over this temperature window remains remarkably similar.

Figure 4

Heat capacity of Ba${}_3$(Mn${}_{0.980}$V${}_{0.020}$)${}_2$O${}_8$ in magnetic fields of 0, 0.25, and 0.5 T, shown as red, blue, and green points, respectively. Lines show the theoretical heat capacity for the Schottky anomaly associated with an isolated $S=1$ spin with single-ion anisotropy $D=$ $-0.024$ meV in the same fields. Inset shows the appropriate energy spectrum for this calculation. Vertical lines indicate the three different fields for which measurements were performed.

Figure 5

(a) Heat capacity of Ba${}_3$(Mn${}_{0.980}$V${}_{0.020}$)${}_2$O${}_8$, Ba${}_3$Mn${}_2$O${}_8$, and the difference at 0 T shown in black circles, red squares, and the blue solid line, respectively. The change in entropy calculated from the difference is plotted as a black dotted line along the right axis. (b) Heat capacity of Ba${}_3$(Mn${}_{0.020}$V${}_{0.980}$)${}_2$O${}_8$, Ba${}_3$V${}_2$O${}_8$, and the difference at 0 T shown in black circles, red squares, and the blue solid line, respectively. The change in entropy calculated from the difference is plotted as a black dotted line along the right axis.

Figure 6

Comparison of the experimentally observed heat capacity of Ba${}_3$Mn(${}_{1-x}$V${}_x$)${}_2$O${}_8$ with that of the random singlet model. Panels (a), (c), and (e) show $C$, $C/T$, and the integrated entropy $S$, respectively, for the random singlet model, as described in the main text. Values have been scaled by the vanadium concentration. Panels (b), (d), and (e) show the experimental results for the same quantities. For $x=$ 0.980, the data have been normalized by the amount of Mn (i.e., 0.020) rather than the V concentration.

Figure 7

(a) Experimental susceptibility of Ba${}_3$(Mn${}_{0.980}$V${}_{0.020}$)${}_2$O${}_8$ for fields both parallel (up triangles) and perpendicular (down triangles) to the $c$ axis. The solid black line shows Curie contribution to the susceptibility determined from a fit of the entire data to Eq. (A1). (b) Log-log plot comparison of susceptibility associated with concentration of unpaired Mn moments derived from the random singlet model (dashed red line) and Curie model (solid black line). The vertical dashed line at 1.8 K marks the lowest temperature to which experimental measurements reached. For the measured temperature range, the susceptibility of the random singlet model does not deviate substantially from Curie’s law.

Figure 8

Energy spectrum of a pair of moments with interactions defined by Eq. (A1) versus $J$, normalized by the single-ion anisotropy $\left|D\right|$. Large $J/\left|D\right|$ range and small $J/\left|D\right|$ range shown in (a) and (b), respectively.

Figure 9

Heat-capacity curves versus normalized temperature for specific $J/\left|D\right|$ values plotted on a log scale. Mol refers to one mole of spins.