Quantum spin liquid and electric quadrupolar states of single crystal Tb 2+x Ti 2−x O 7+y
M. Wakita, T. Taniguchi, H. Edamoto, H. Takatsu, H. Kadowaki
aa r X i v : . [ c ond - m a t . s t r- e l ] F e b Quantum spin liquid and electric quadrupolar statesof single crystal Tb x Ti − x O y M Wakita, T Taniguchi, H Edamoto, H Takatsu and H Kadowaki
Department of Physics, Tokyo Metropolitan University, Hachioji-shi, Tokyo 192-0397, Japan
Abstract.
The ground states of the frustrated pyrochlore oxide Tb x Ti − x O y , sensitivelydepending on the small off-stoichiometry parameter x , have been studied by specific heatmeasurements using well characterized samples. Single crystal Tb x Ti − x O y boules grownby the standard floating zone technique are shown to exhibit concentration ( x ) gradient. Thisoff-stoichiometry parameter is determined by precisely measuring the lattice constant of smallsamples cut from a crystal boule. Specific heat shows that the phase boundary of the electricquadrupolar state has a dome structure in the x - T phase diagram with the highest T c ≃ . x = 0 .
01. This phase diagram suggests that the putative U(1) quantum spin-liquidstate of Tb x Ti − x O y exists in the range x < x c ≃ − . x = x c .
1. Introduction
Magnetic systems with geometric frustration have been intensively studied experimentally andtheoretically for decades [1]. Spin systems on networks of triangles or tetrahedra, such astriangular [2], kagom´e [3], and pyrochlore [4] lattices, play major roles in these studies. Subjectsfascinating many investigators in recent years are quantum spin liquid (QSL) states [5, 6], whereconventional long-range orders (LRO) are suppressed to very low temperatures.Among frustrated magnetic pyrochlore oxides [4], Tb Ti O (TTO) has attracted muchattention because it does not show any conventional LRO down to 50 mK [7], suggesting that it isa candidate for a QSL state. Although many experimental studies of TTO have been performedto 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 singlecrystals. And accordingly, simple interpretation of experimental data is precluded.Recently, we investigated polycrystalline samples of off-stoichiometric Tb x Ti − x O y , andshowed that a very small change of x induces a quantum phase transition between a spinliquid state ( x < − . x c ) and a LRO state with a hidden order parameter ( x c < x )[11]. The x - T phase diagram of Tb x Ti − x O y suggested in Ref. [11] has a dome-shapeLRO phase boundary. More recently, we study the hidden LRO using an x -controlled singlecrystal, which shows a very sharp peak in specific heat at T c = 0 .
53 K ( x ≃ . x Ti − x O y is anelectric multipolar (or quadrupolar) state. This LRO state was theoretically predicted [15] usingelectronic superexchange interactions for non-Kramers ions, including Tb , which have bothmagnetic dipole and electric quadrupole (16-pole, and 64-pole) moments. In addition, quiteintriguingly, the estimated parameter set [12] of the effective pseudospin-1/2 Hamiltonian isocated 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 x Ti − x O y [11] to single crystals in the hope that the above scenario for the TTO problem is reinforced.We grow single crystals of Tb x Ti − x O y by the standard floating zone (FZ) technique [17]and have found that very precise measurements of the lattice constant are useful to characterizethe 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.
2. Experimental methods and results
Polycrystalline samples of Tb x Ti − x O y were prepared by the standard solid-state reactionas described in Ref. [11]. The two starting materials, Tb O and TiO , were heated in air at1350 ◦ 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 ± . x Ti − x O y powder samples were used for single crystal growth bythe standard FZ technique [17]. Crystal growth was carried out in an Ar gas flow atmosphereusing a double ellipsoidal image furnace (NEC SC-N35HD).X-ray powder-diffraction experiments were carried out using a RIGAKU-SmartLabdiffractometer equipped with a Cu K α monochromator. To precisely measure the latticeconstant we performed θ -2 θ scans on powder mixtures of polycrystalline or crushed-crystallineTb x Ti − x O y and Si [11, 18]. Absolute values of lattice constants are normalized by using thecertified lattice parameter for a temperature of 22.5 ◦ C of the SRM-640d Si powder, a = 5 . a ( T, x ) of Tb x Ti − x O y was measuredusing a polycrystalline sample with x = − . x dependence of a ( T = 26 . ◦ C , x ) of polycrystalline samples is plotted in Fig. 1(b), where weconverted the published lattice constants (Fig. 1 in Ref. [11]) to those at 26.0 ◦ C [18].Figure 2 shows a single crystal Tb x Ti − x O y boule that was grown from a feed rod of x = − .
005 powder and was post-annealed for about 7 days at 1000 ◦ C in air. Lattice constants a ( Å ) T ( ˚ C)(a) x = ! ! x (b) T = 26.0 ˚ C Figure 1.
Lattice constant a ( T, x ) of polycrystalline Tb x Ti − x O y samples. (a)Temperature dependence of a ( T, x = − . a ( T = 26 . ◦ C , x ) = 0 . x + 10 .
5 12 15 19 23 27 31 35 40 52 L (mm) Figure 2.
Single crystal Tb x Ti − x O y boule grown by the FZ method, where the missingpart (40 < L <
48 mm) was cut before taking this photograph. The numbers represent distances L along the growth direction, where small crystals are cut at these L values. Lattice constantand specific heat of these crystals are shown in Figs. 3 and 4. a ( Å ) x L (mm) T = 26 ˚ C Figure 3.
Lattice constants of small Tb x Ti − x O y crystals cut from the boule shownin Fig. 2. These lattice constants are converted to x using the polycrystalline curve, i.e., a ( T = 26 . ◦ C , x ) of Fig. 1(b) and are shown on the right vertical axis.of small crystals cut from this boule were measured at 26.0 ◦ C and are plotted as a function ofthe distance along the growth direction L shown in Fig. 3. We assume that a ( T = 26 . ◦ C , x ) ofpolycrystalline samples (Fig. 1(b)) and its linear extension to the range x > .
01 can be used toestimate the off-stoichiometry parameter ( x ) of the small crystals. These x values are shown onthe right vertical axis of Fig. 3. One can see that the boule has a systematic x gradient. Duringthe crystal growth the off-stoichiometry parameter starts from x ≃ .
04 ( L = 1 – 5 mm), thendecreases 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 temperaturesusing a He or an adiabatic demagnetization refrigerator. In Fig. 4(a) we show specific heat asa function of temperature for the several crystals cut from the boule (Fig. 2) and a few fromanother boule. Based on these C P ( T ) data we draw a tentative x - T phase diagram for the singlecrystals in Fig. 4(b). We note that these C P ( T ) data and the x - T phase diagram for the singlecrystals are quite consistent with those of polycrystalline Tb x Ti − x O y [11]. This indicatesthat our trial method of estimating small x ( | x | < .
01) for single crystals using the precisemeasurement 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 verysmall change of the off-stoichiometry existing even in a single crystal boule to investigate SL quadrupolar state paramagnetic state !"!!" ! !"!% !"!! !"!% !"!& T ’ ( ) * x (a) (b) C P ( J / K m o l - T b ) T (K) x = ! x = 0.003 x = 0.006 x = 0.016 x = 0.026 x = 0.032 Figure 4. (a) Temperature dependence of specific heat of several single crystals. The x valuesare estimated by the method shown in Fig. 3. (b) x - T phase diagram determined from the specificheat measurements of single crystals. Temperature ranges of the specific heat measurements areshown by vertical blue dashed lines.Tb x Ti − x O y (or nominal Tb Ti O ). Previous experimental investigations using small TTOcrystals will have to be reinterpreted as investigations on different Tb x Ti − x O y crystals. Inparticular, previous experiments using large crystals, especially inelastic neutron scattering forexample Refs. [21–24], require special caution in their interpretation, because the crystals maynot be sufficiently homogeneous.
3. Discussion and summary
The x - T phase diagram shows that around x = x c ≃ − . T c of the quadrupolar state [12] disappears abruptly in a small x range. This suggests that theneighboring 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 quadrupolarstates is predicted by a gauge mean-field theory [16], presumably relevant to TTO [12]. Onemay naturally expect that Tb x Ti − x O 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 x Ti − x O 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 > .
01 the transition temperature of thequadrupolar state seems to decrease gradually and the specific heat peak gradually becomessmaller as x is increased. These suggest that an effect of randomness controls the system. Apossible scenario of the randomness effect may be as follows. Most of excess Tb atoms reside onthe Ti site and become Tb ions. These magnetic Tb ions behave as magnetic impuritiesin the system, where local magnetic short-range order is restored around each Tb ion. Thequadrupolar state is completely suppressed in x > . x Ti − x O y by growing single crystals using the standard floating zone technique and bycharacterizing them using X-ray diffraction techniques and specific heat measurements down to0.1 K. We show that a precise determination of the lattice constant is useful for estimating thesmall 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 longrange quadrupolar, and possibly a randomness dominating state. The phase boundary of thequadrupolar state shows a dome structure in the x - T phase diagram with the highest T c ≃ . x = 0 .
01 and suggests existence of a first-order phase-transition line separating the QSLand quadrupolar states.
Acknowledgments
We thank S. Onoda and Y. Kato for useful discussions. This work was supported by JSPSKAKENHI grant numbers 25400345 and 26400336.
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