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

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 $x=x_{\text{c}}$.


Introduction
Magnetic systems with geometric frustration have been intensively studied experimentally and theoretically for decades [1]. Spin systems on networks of triangles or tetrahedra, such as triangular [2], kagomé [3], and pyrochlore [4] lattices, play major roles in these studies. Subjects fascinating many investigators in recent years are quantum spin liquid (QSL) states [5,6], where conventional long-range orders (LRO) are suppressed to very low temperatures.
Among frustrated magnetic pyrochlore oxides [4], Tb 2 Ti 2 O 7 (TTO) has attracted much attention because it does not show any conventional LRO down to 50 mK [7], suggesting that it is a candidate for a QSL state. Although many experimental studies of TTO have been performed to date, the problem why TTO does not show any magnetic LRO remains very difficult [8,9]. This is partly because TTO shows strong sample dependence [10], extremely strong for single crystals. And accordingly, simple interpretation of experimental data is precluded.
Recently, we investigated polycrystalline samples of off-stoichiometric Tb 2+x Ti 2−x O 7+y , and showed that a very small change of x induces a quantum phase transition between a spin liquid state (x < −0.0025 = x c ) and a LRO state with a hidden order parameter (x c < x) [11]. The x-T phase diagram of Tb 2+x Ti 2−x O 7+y suggested in Ref. [11] has a dome-shape LRO phase boundary. More recently, we study the hidden LRO using an x-controlled single crystal, which shows a very sharp peak in specific heat at T c = 0.53 K (x ≃ 0.005) [12]. By using semi-quantitative analyses, we propose [12][13][14] that the LRO of Tb 2+x Ti 2−x O 7+y is an electric multipolar (or quadrupolar) state. This LRO state was theoretically predicted [15] using electronic superexchange interactions for non-Kramers ions, including Tb 3+ , which have both magnetic dipole and electric quadrupole (16-pole, and 64-pole) moments. In addition, quite intriguingly, the estimated parameter set [12] of the effective pseudospin-1/2 Hamiltonian is located very close to a theoretical phase boundary between the electric quadrupolar and U(1) quantum spin-liquid states [15,16], which could naturally explain the spin liquid state of TTO.
The purpose of this investigation is to extend our study of polycrystalline Tb 2+x Ti 2−x O 7+y [11] to single crystals in the hope that the above scenario for the TTO problem is reinforced. We grow single crystals of Tb 2+x Ti 2−x O 7+y by the standard floating zone (FZ) technique [17] and have found that very precise measurements of the lattice constant are useful to characterize the single crystals. Specific heat of these samples with different off-stoichiometry parameters (x) have been measured down to 0.1 K to obtain an x-T phase diagram.

Experimental methods and results
Polycrystalline samples of Tb 2+x Ti 2−x O 7+y were prepared by the standard solid-state reaction as described in Ref. [11]. The two starting materials, Tb 4 O 7 and TiO 2 , were heated in air at 1350 • C for several days with periodic grindings to ensure a complete reaction. The value of x was adjusted by changing the mass ratio of the two materials, and is nominal with an offset about ±0.002. The resulting Tb 2+x Ti 2−x O 7+y powder samples were used for single crystal growth by the standard FZ technique [17]. Crystal growth was carried out in an Ar gas flow atmosphere using a double ellipsoidal image furnace (NEC SC-N35HD).
X-ray powder-diffraction experiments were carried out using a RIGAKU-SmartLab diffractometer equipped with a Cu K α1 monochromator. To precisely measure the lattice constant we performed θ-2θ scans on powder mixtures of polycrystalline or crushed-crystalline Tb 2+x Ti 2−x O 7+y and Si [11,18]. Absolute values of lattice constants are normalized by using the certified lattice parameter for a temperature of 22.5 • C of the SRM-640d Si powder, a = 5.43123 A [19], being further corrected for the temperature dependence [20].
Temperature dependence of the lattice constant a(T, x) of Tb 2+x Ti 2−x O 7+y was measured using a polycrystalline sample with x = −0.0075, and the result is shown in Fig. 1(a). The x dependence of a(T = 26.0 • C, x) of polycrystalline samples is plotted in Fig. 1(b), where we converted the published lattice constants ( Fig. 1 in Ref. [11]) to those at 26.0 • C [18].    of small crystals cut from this boule were measured at 26.0 • C and are plotted as a function of the distance along the growth direction L shown in Fig. 3. We assume that a(T = 26.0 • C, x) of polycrystalline samples ( Fig. 1(b)) and its linear extension to the range x > 0.01 can be used to estimate the off-stoichiometry parameter (x) of the small crystals. These x values are shown on the right vertical axis of Fig. 3. One can see that the boule has a systematic x gradient. During the crystal growth the off-stoichiometry parameter starts from x ≃ 0.04 (L = 1 -5 mm), then decreases linearly as a function of L, and finally varies more slowly (L > 40 mm).
To characterize crystal samples we also measured specific heat C P (T ) at low temperatures using a 3 He or an adiabatic demagnetization refrigerator. In Fig. 4(a) we show specific heat as a function of temperature for the several crystals cut from the boule (Fig. 2) and a few from another boule. Based on these C P (T ) data we draw a tentative x-T phase diagram for the single crystals in Fig. 4(b). We note that these C P (T ) data and the x-T phase diagram for the single crystals are quite consistent with those of polycrystalline Tb 2+x Ti 2−x O 7+y [11]. This indicates that our trial method of estimating small x (|x| < 0.01) for single crystals using the precise measurement of the lattice constant is probably reliable.
The x-T phase diagram (Fig. 4(b)) implies that one has to take a special care of very small change of the off-stoichiometry existing even in a single crystal boule to investigate . Previous experimental investigations using small TTO crystals will have to be reinterpreted as investigations on different Tb 2+x Ti 2−x O 7+y crystals. In particular, previous experiments using large crystals, especially inelastic neutron scattering for example Refs. [21][22][23][24], require special caution in their interpretation, because the crystals may not be sufficiently homogeneous.

Discussion and summary
The x-T phase diagram shows that around x = x c ≃ −0.0025 the transition temperature T c of the quadrupolar state [12] disappears abruptly in a small x range. This suggests that the neighboring putative QSL state is separated by a first-order phase-transition line x = x c [11,12]. It is interesting that this type of first-order phase transition between U(1) QSL and quadrupolar states is predicted by a gauge mean-field theory [16], presumably relevant to TTO [12]. One may naturally expect that Tb 2+x Ti 2−x O 7+y with x = x c is on the theoretical border of U(1) QSL and quadrupolar states [16], and that the spin liquid state of Tb 2+x Ti 2−x O 7+y with x < x c is U(1) QSL of Ref. [16]. This is a very intriguing hypothesis for further studies.
On the other hand, in a larger x range of x > 0.01 the transition temperature of the quadrupolar state seems to decrease gradually and the specific heat peak gradually becomes smaller as x is increased. These suggest that an effect of randomness controls the system. A possible scenario of the randomness effect may be as follows. Most of excess Tb atoms reside on the Ti 4+ site and become Tb 4+ ions. These magnetic Tb 4+ ions behave as magnetic impurities in the system, where local magnetic short-range order is restored around each Tb 4+ ion. The quadrupolar state is completely suppressed in x > 0.04.
In summary, we have investigated single-crystalline samples of the frustrated pyrochlore oxide Tb 2+x Ti 2−x O 7+y by growing single crystals using the standard floating zone technique and by characterizing them using X-ray diffraction techniques and specific heat measurements down to 0.1 K. We show that a precise determination of the lattice constant is useful for estimating the small off-stoichiometry parameter x. Small crystals cut from single crystal rods, exhibiting x gradient, show three different low temperature behaviors: a paramagnetic QSL (x < x c ), a long range quadrupolar, and possibly a randomness dominating state. The phase boundary of the quadrupolar state shows a dome structure in the x-T phase diagram with the highest T c ≃ 0.5 K at x = 0.01 and suggests existence of a first-order phase-transition line separating the QSL and quadrupolar states.