Quantum spin liquid and electric quadrupolar states of single crystal Tb\(_{2+x}\)Ti\(_{2-x}\)O\(_{7+y}\)

Abstract

The ground states of the frustrated pyrochlore oxide Tb\(_{2+x}\)Ti\(_{2-x}\)O\(_{7+y}\), sensitively depending on the small off-stoichiometry parameter \(x\), have been studied by specific heat measurements using well characterized samples. Single crystal Tb\(_{2+x}\)Ti\(_{2-x}\)O\(_{7+y}\) boules grown by the standard floating zone technique are shown to exhibit concentration (\(x\)) gradient. This off-stoichiometry parameter is determined by precisely measuring the lattice constant of small samples cut from a crystal boule. Specific heat shows that the phase boundary of the electric quadrupolar state has a dome structure in the \(x\)-\(T\) phase diagram with the highest \(T_{\text{c}} \simeq 0.5\) K at about \(x = 0.01\). This phase diagram suggests that the putative U(1) quantum spin-liquid state of Tb\(_{2+x}\)Ti\(_{2-x}\)O\(_{7+y}\) exists in the range \(x<x_{\text{c}} \simeq -0.0025 \), which is separated from the quadrupolar state via a first-order phase-transition line \(x=x_{\text{c}}\).