Investigation of the anisotropic thermal expansion of PbIn6Te10 crystal by high temperature x-ray diffraction measurements

The PbIn6Te10 crystal, an IR laser frequency conversion material, grown via the Bridgman method with dimensions ϕ35 mm × 90 mm in a four-zone furnace, was subject to the investigation of its thermal expansion behavior using high-temperature x-ray diffraction in the range of 25–450 °C. Based on the obtained data, the average thermal expansion coefficients of 15.21 × 10−6 K−1 for α a and 6.44 × 10−6 K−1 for α c were determined utilizing the least square method. The study revealed that the linear and volume thermal expansion coefficients of PbIn6Te10 crystals satisfy the relationships α a > α c > 0, α V = 2α a + α c , and α a increased while α c decreased with increasing temperature, accentuating a substantial anisotropy in thermal expansion between the crystal’s principal axes. A detailed exploration pinpointed that the variations in the PbTe6 octahedron primarily governed the changes in the PbIn6Te10 unit cells, with further investigation uncovering its association with variations in the nearest neighboring bonds, which is mainly related to Pb-Te4 and Pb-Te2 bonds. Additionally, the determination of temperature-dependent anisotropic thermal expansion coefficient α was complemented by calculating Grüneisen parameter γ using Quasi-harmonic Debye model. Remarkably, these parameters also exhibited anisotropic behaviors ( γ⊥ increases with temperature whereas γ// decreases, α ⊥ > α ∥), contributing additional insights into the crystal thermal characteristics.


Introduction
In the contemporary scientific and academic scene, there has been a discernible surge in the significance attributed to nonlinear optical (NLO) crystals, a trend underscored by their pivotal role in facilitating the continuous tuning of mid-to-far infrared and even terahertz lasers.Among them, the medium infrared and far infrared have a high degree of overlap with the 'fingerprint area' of molecular, which have significant application value in the military and civil field (medical laser, remote sensing, early warning, etc).This augmentation positions these crystals as fundamental constituents within the laser frequency conversion systems [1][2][3][4][5][6].It is currently available to achieve high-power laser output in the range of 3 ~5 μm and 8 ~14 μm [7].However, the laser output of 14 mm and above is rarely reported for lack of nonlinear crystals with the wide-band transmission.Compared with phosphorus, sulfur, and selenide crystals, telluride crystals have lower phonon energy and a wider transmission range in long wave bands [8].Ternary telluride PbIn 6 Te 10 (PIT) is a novel nonlinear optical material with a wide infrared transmission range (1.3 ~31 μm) [9], belonging to the trigonal system with space group R32 [10,11].Theoretically, PbIn 6 Te 10 is derived from the binary telluride In 7 Te 10 , in which Te atoms surround In atoms forming a tetrahedral structure with Pb atoms replacing part of In atoms and occupying 2/3 of the Tetrahedral interstice [12].Pb atoms reduce the crystal structure symmetry and increase the nonlinear effect of the crystal, resulting in a high nonlinear optical coefficient (d11 = 51 pm/V) [13].
In 1996, German scientists Müller and Deiseroth introduced PbIn 6 Te 10 into the study of β-Mn structure for the first time [11].More than a decade later, in 2011, Russian researcher Avanesov investigated ternary telluride and after attempting to grow high-quality In 7 Te 10 and failing under conditions where the only structural properties were known, he focused on growing ternary PbIn 6 Te 10 [14].High-quality single crystal with a size of f12 mm × 30 mm was grown by Czech researcher Reshak in 2016 [10].However, it is still challenging to obtain high-quality, crack-free, and large-size single crystals.On the one hand, PbIn 6 Te 10 crystal is a heterogeneous melt eutectic, so the components are difficult to control during crystal growth [15].On the other hand, it is due to the thermal expansion anisotropy between the a-axis and the c-axis.As a prospective avenue for future research, a pivotal area of focus resides in the comprehensive examination of the thermal expansion behavior exhibited by this crystal.To date, a dedicated inquiry into the thermal expansion characteristics specific to PIT crystals remains conspicuously absent within existing literature.Such an investigation is deemed indispensable for fostering an enhanced comprehension, not only of the impact exerted on the crystallization process by thermodynamic considerations but also of the temperature-dependent nuances influencing diverse physical characteristics, including optical properties.
This paper presents a new method based on the crystallization data of spontaneous nucleation as well as the selection of a small high-quality single crystal to grow a PbIn 6 Te 10 single crystal with the size of f 35 mm × 90 mm following the seed-oriented Vertical Bridgman method.Subsequently, PIT was characterized by x-ray diffraction (XRD), energy dispersive spectroscopy (EDS), and simultaneous thermal analysis (TG-DSC) measurements.The present work is primarily aimed at studying thermal properties and structural deformation using high-temperature x-ray diffraction.In general, the relatively modest expansion experienced by most materials usually arises from enhanced anharmonic vibrational amplitudes of the composition groups, including atoms, ions, or molecules [16].It is well acknowledged that strong anisotropy of positive thermal expansion is the main factor causing cracking during the process of crystal growth.This work is promising to help develop the growth process.In addition, the results may provide a valid reference model verifying the reliability of other parameters related to thermal expansion.

Experimental details
2.1.Synthesis of PbIn 6 Te 10 PbIn 6 Te 10 ingot was synthesized by the use of high-purity simple raw materials with purity 99.9999% including tellurium, indium and lead, mixed in stoichiometric ratio with an excess of 10% In 2 Te 3 in mole.The quartz ampoule containing raw materials was vacuumed to under 10 −3 pa by molecular pump, sealed off and placed into a two-zone furnace (figure 1).More detailed description of the synthesis process utilized in this work can be found in our previous paper [12].

Single crystal growth
PIT polycrystalline materials were ground into powder through corundum mortar, which was passed through by 300-mesh sieve.Subsequently, the well-sieved powder as well as selected high-quality oriented seed crystals was loaded into a quartz ampoule with pyrolytic boron nitride (PBN) crucible inside.Then the quartz ampoule was vacuumed and sealed under 10 −3 pa and placed into a four-zone furnace.The furnace is composed of two independent heating systems, each of them has two temperature zones, and the crucible is placed at the junction of two parts.According to the phase diagram of PIT and previous growth experience, the temperatures of the upper furnace and lower furnace were raised to 715 °C/690 °C and 520 °C/480 °C, respectively.Then, it was necessary to hold the temperature for 24 h to ensure the parts above the crystal seed were completely melted.Both the upper and lower furnaces were cooled down by 100 °C to make them supercooled.After ten hours of heat preservation, they were heated up again to 695 °C/670 °C in the upper furnace and 500 °C/460 °C in the lower preparing for crystal growth, as shown in figure 2.
The above process is aimed at reducing the possibility of polycrystalline nucleation crystallization, based on which the fusion of seed crystal has a good effect.The ampule was descended at a rate of 0.16-0.19mm h −1 through a thermal gradient of 10-12 °C.After the crystal growth process was finished, we cooled down the furnace slowly to room temperature.Figure 3 is the grown crystal with size of f35 mm × 90 mm, which is much larger than that grown by Reshak [10].

X-ray powder diffraction and energy x-ray dispersive spectroscopy
To confirm the composition and structure of the growth crystal, small pieces were selected from the middle and top parts of the boule and ground into a powder.as shown in figure 3. The powder samples were tested by XRD (Dandong DX-2700) measurements with Cu Kα radiation from 10°to 130°at a scanning speed of 0.013°/s.We  performed the x-ray energy dispersive spectroscopy (EDS) analysis using an Aztec X-Max80 high-resolution scanning electron microscopy (SEM) device attached to a dual-probe spectrometer.

Continuous scanning and single crystal XRD rocking curve
In order to detect the crystal quality of the grown single crystal, we cut the sample from a cross-section of the main body and carried out the continuous scanning method using a DX-2700 x-ray diffractometer (Dandong Haoyuan Instrument, China) with a speed of 4°/min from 10°to 90°at an interval of 0.02°.Subsequently, the scanning mode was switched to omega scanning to conduct a rocking curve test.

Simultaneous thermal analysis and constant pressure heat capacity
To study the thermal stability of PIT crystal in high temperatures, we employed a thermogravimetric and differential scanning calorimetry (TG-DSC) synchronous analyzer before the high-temperature XRD test, which was carried out by DSC1, TGA (Mettler Toledo, Switzerland).The results are shown in figure 4. The melting point of PbIn 6 Te 10 is about 627 °C, and it is consistent with other reports [8,17] .We noticed that there is rare decomposition before the temperature rises to the melting point and that the transformation of PIT crystal to other structures did not occur.In addition, the heat capacity at constant pressure with temperature was measured by an SDT Q600 apparatus combined with a TG-DSC analysis (figure 5).The constant-pressure heat capacity of the PIT was only evaluated in a smaller temperature range (0 °C−100 °C) due to the instrument's limitations to guarantee the accuracy of the test results.

High temperature in situ XRD study of PbIn 6 Te 10
The high-temperature x-ray diffraction was performed by an EMPYREAN x-ray diffractometer with Cu Kα radiation (λ = 1.54Å), which can reach up to 900 °C.The scanning range was determined by 10°−130°with a step of 0.0065°, and the temperature points of measurement were determined as 25, 50, 100, 150, 200, 250, 300, 350, 400 and 450 °C with heating rate of 25 °C/min.Before measurement, the samples from the main body of a single crystal were ground into powder with a size of about 75 mm.An elevated temperature environment was built by an HTK 16N strip heater chamber in a shielding inert atmosphere.The ceramic ampoule-containing specimen was heated by heating strips (graphite) mounted inside the chamber.In addition, before each test, the temperature of the environment was preheated for 5 min to avoid temperature fluctuation.
The collected high-temperature variable x-ray diffraction data were refined by the Rietveld method [18,19] to further study the relevance between thermal expansion properties and the crystal microstructure.The refinement work was performed by GSAS [20] software with EXPGUI [21] graphical toolkit package.The initial model was determined by the structure reported in [11].The background was matched by the shifted Chebyschev polynomial function, and the parameter of peak shape was rectified by the pseudo-Voigt function.The refined results include cell parameters, coordinates of atoms preferred orientation, etc According to all the refinement results of different temperatures, the goodness of fit (χ 2 ) is less than 2(1.6-1.9) and the R wp is lower than 10%, which confirms that the results of the work are credible and useable.

Results and discussion
3.1.Analysis of test from growth result Subsequently, we refined the XRD data also using the Rietveld method [18,19] and compared it to the standard spectra of PbIn 6 Te 10 , In 2 Te 3 and PbTe (seen in figure 6).The peak position of PbIn6Te10 was completely consistent with the standard PDF, and there are no In 2 Te 3 and PbTe and other peaks appeared, with sharp peak and high crystallinity.The crystal parameters calculated from the Rietveld refined method are close to the precious report from Xiong [12].The result of EDS (seen in figure 7) indicates that the proportion of Te, In, and Pb elements is close to the ideal stoichiometry, and the elements are evenly distributed.The continuous scanning and rocking curves are shown in figure 8, and the peaks are sharp with high intensity as well as good symmetry.The margin of error of 2θ between the continuous scanning curve and the rocking curve is within 0.4 degrees.The full width at half of the maximum (FWHM) is about 0.573°.All the pieces of evidence indicate that the crystalline quality is good.

XRD patterns at different temperature
XRD patterns at specified temperatures were recorded and compared to standard data (PDF# 87-0849) [11], as shown in figure 9.Each XRD reflection curve corresponds well to the standard PDF.In alignment with the results of TG-DSC measurements, the sound thermal stability contributes to the absence of structural transformation and phase transition throughout the measurement process.The patterns indicate that the intensities of most peaks uniformly decrease with increasing temperature.However, for specific peaks like (321) at 2θ of 30.41°, the intensities are enhanced with growing temperature.Additionally, some characteristic peaks slightly shift to the lower angles with increasing temperature.The partial details of XRD patterns are shown in figure 10, from which the shift of peaks is directly observed.The most significant variation is that the degree of the peak shift varies with the angle, indicating the thermal expansion anisotropy.

Temperature dependence of PbIn 6 Te 10 unit cell parameters
Based on the Cohen least square method as well as refined diffraction data, the cell parameters are obtained, as shown in table 1.The cell parameters at 25 °C are fairly close to those from the up-to-date PDF data, indicating that the reliability of measured data is credible.The variation of cell parameters as a function of temperature can be seen in figure 11.As it can be found from the curves, the parameters a, c, and V all increase nonlinearly with growing temperature, and their increments are evaluated to be 0.65, 0.28, and 1.44%, respectively.It is likely to imply the behavior of thermal expansion along a axis, which tends to play a more important role, is more notable than that of the c axis.We successfully accomplished the approximations of temperature dependence by nonlinear curve fitting and the polynomial functions of temperature dependence parameters are listed in table 2.

Non-axial thermal expansion coefficient
As for the thermal expansion coefficients in arbitrary direction, it can be defined as [22] ( ) where j is the angle between the direction [hkl] and the c-axis.The thermal expansion coefficients at different temperatures are substituted into the above formula.The results of α hkl at different temperatures are presented in figure 12.The values of α hkl are positive in any direction.In particular, when j = 0°, cos 2 j = 1, α hkl corresponds to α c ; whereas when j = 90°, cos 2 j = 0, α hkl corresponds to α a .The slope of the curves represents the degree of thermal expansion anisotropy, which is equal to α k .The absolute values of the slope, namely |α k |,     increase with elevated temperature.As it can be seen from the plot, all the curves tend to intersect at one point approximately cos 2 j ≈ 0.6(j ≈ 39°).In this case, there are specific directions where the thermal expansion coefficients have no significant changes or even tend to be stable despite the temperature variations.

Axial thermal expansion coefficient
Each crystal axis experiences a distinct expansion effect during thermal expansion due to the asymmetry of the crystal structure.It is capable of further investigating the thermal expansion of PIT crystals via temperature dependence of cell parameters.Based on the least square approximation [23][24][25], the temperature dependence of a cell parameter R can be described by the polynomial function of temperature in the following form [24]  ( ) where T represents the temperature and A 0 denotes the lattice parameter at 0 °C.As for A n , the coefficient of the polynomial is determined by the effect of temperature via the power n.Therefore, the expansion coefficient of lattice parameters α R is defined as By combining equations (1) and (3), the thermal expansion coefficient can be described as In addition, the average thermal expansion coefficient is defined as In figure 11 is the variation of cell parameters as a function of temperature, and it is well-fitted with the secondorder polynomial curves, which means the thermal expansion coefficients are in a linear relation with temperature.The results of thermal expansion coefficients are presented in table 3. α a represents the linear thermal expansion perpendicular to the c-axis, α c represents that parallel to the caxis, α V characterizes the thermal expansion of volume and α k implies the anisotropy of thermal expansion.The values of α a , α c, and α V are positive, thus, the PIT crystal tends to expand synchronously in all directions.The values of α a and α V increase while α c decreases with growing temperature, and the absolute values of α k also surge with temperature.All thermal expansion coefficients meet the following approximate relation:α In addition, to make a comparison and to study the thermal expansion behavior of PbIn 6 Te 10 more intuitively, the three-dimensional ellipsoidal diagram was depicted by calculating the cell parameters using the software PASCal [26] (version v2.1.0),as shown in figure 13.The degree of thermal expansion is determined by the shade of the color.From the graphic diagram, it is evident that the thermal expansivity perpendicular to the c axis is dominant.The thermal expansion coefficients related to crystallographic axes were also calculated and they are listed in table 4. The parameter σα is a possible error in the linear coefficient of thermal expansion.Based on the algorithm, the lattice parameters were employed by first-order function fitting, thus the error in coefficient exists reasonably and the coefficients of thermal expansion in three axes are constants.
Under this circumstance, the coefficient perpendicular to c axis, namely expansivity in a and b directions, is about 2 times larger than α c , indicating prominent anisotropy of thermal expansion.An approximate relationship of α V ≈ 2α a + α c was observed as well.The two methods share the similar conclusions.

Anisotropic thermal expansion correlated with structure
As shown in figure 14(a), PbIn 6 Te 10 crystal is in a trigonal structure where Pb atoms are in an incomplete occupation state (2/3), which is likely to be regarded as the substitution from In by Pb, namely Pb atoms replace part of In atoms from the InTe 4 coordination tetrahedra in the original In 7 Te 10 phase.Based on the distinct atom environment, we confirmed that the structure contains three kinds of basis, which are In 1 tetrahedra, In 2   Thermal expansion, driven by atom anharmonicity, is intricately tied to the atomic-molecular composition of the material [27].For anisotropic structures like PbIn 6 Te 10 , variations in expansivity and crystallographic unit cell dimension prompt speculation on two phenomena: changes in crystal framework geometry, specifically bond lengths and angles, and continuous oscillating motion in specific atoms about their equilibrium positions.High-temperature x-ray measurements on the PIT crystal demonstrate simultaneous expansion in all directions when the crystal is heated.Analyzing the volumes of three coordination polyhedra (PbTe 6 , In 1 Te 4 , In 2 Te 4 ) with temperature variation (figure 15), the chart reveals an enlargement in the PbTe 6 octahedron volume with rising temperature.In contrast, the volume of the InTe4 tetrahedron with two distinct central atoms (In 1 , In 2 ) remains largely unchanged.This suggests that the anisotropic thermal expansion in the PIT crystal is correlated with the variation in the PbTe 6 octahedron.The volume variation with temperature is linearly fitted as 38.5832 + 2.99 × 10 −3 T, yielding a thermal expansion coefficient of 77.49 × 10 −6 K −1 for the PbTe6 octahedron volume.Notably larger than the unit cell volume in PIT crystals, this underscores the pivotal role of the PbTe 6 coordination polyhedron's thermal expansion behavior in correlational structure expansion.
Bond lengths and angles were determined using the Rietveld refinement method with GSAS DISAGL.In figure16(a), temperature-dependent variations in bond length and angle are presented.The Pb-Te 4 and Pb-Te 2 bond lengths in PbTe 6 exhibit undulatory increases with temperature, while the Pb-Te 3 bond length remains

Anisotropic thermal expansion correlated with Grüneisen parameters
Understanding the Grüneisen parameter is a key to predicting and controlling the thermal expansion properties of solids in response to temperature changes.The anharmonicity of lattice vibrations, reflecting nonlinear interatomic interactions concerning atomic displacements, significantly influences crystal qualities [28].White et al conducted a comprehensive investigation into the coefficient of thermal expansion of axisymmetric crystals [29][30][31].Under the quasi-harmonic Debye model, the linear expansion coefficients for hexagonal crystals that are parallel to the c-axis and perpendicular to it are as follows c and c ^are the linear compressibility along and perpendicular to c axis, respectively.Therefore, equations ( 7)-( 8) can be described as g are the Grüneisen parameters in the direction perpendicular to the c-axis and parallel to the caxis, respectively.S ij is the elastic flexibility whose value is the reciprocal of the stiffness tensor and C V is the molar specific heat at constant volume.According to Barron et al [29][30][31][32][33], the two Grüneisen parameters are ascertained as follows.( ) ^Ŵhere V m represents the molar volume, C p denotes the molar specific heat at constant pressure, α a and α c are the thermal expansion coefficients along the a and c axes, respectively, and C ij is the elastic stiffness.The linear combination of // g and , g ^weighted by the corresponding linear compressibility // c and c ^along with the volume compressibility, yields the volume Grüneisen parameter .

V g
The elastic constants of stiffness tensor (C ij ) at high temperatures are currently unavailable in the previous report.To calculate the elastic constants, the spin-polarized density functional theory (DFT) calculations [34,35] were carried out in the Vienna ab initio simulation package (VASP) software based on the plane-wave basis sets with the projector augmented-wave method [36,37].The exchange-correlation potential was treated by using a generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) parametrization [38].The van der Waals correction of Grimme's DFT-D3 model was also adopted [39].The energy cutoff was set to be 450 eV.The Brillouin-zone integration was sampled with a Γ-centered Monkhorst-Pack mesh [40] of 2 × 2 × 2. The structures were fully relaxed until the maximum force on each atom was less than 0.03 eV Å, and the energy convergent standard was 10 −5 eV.Applying the proper set of distortions, with the distortion parameter d ranging from −0.02 to + 0.02, allowed for the calculation of the elastic constants.Finally, we could obtain the heat capacity at constant pressure C p by simultaneous thermal analysis method, as shown in figure 5. Since it is a temperature-dependent parameter and the density of PbIn 6 Te 10 is known as 5.957 g cm −3 , the molar volume V m could be easily computed using the lattice parameters from table 1.Ultimately, using equations ( 14)-( 16), the Grüneisen parameters , g ^, // g and V g are determined.As can be observed, , g ^, // g and V g are all positive values when the Grüneisen parameters are plotted versus temperature in figure 17.Furthermore, both g ^and V g increase with increasing temperature while // g decreases, with average value of 2.7 for g ^and 2.4 for .

//
g The plotted line of , g ^// g and // g approximately intersect at a temperature of 150 °C.Due to the relatively large value of the molar volume of PIT crystal, the numerical value of Grüneisen parameters is considerable.The previously deduced behavior of anisotropic thermal expansion agrees with the trends of three Grüneisen parameters with temperature.Based on the equations (12)-( 13) and calculated Grüneisen parameters, the anisotropy thermal expansion coefficients α ⊥ and α ∥ are also determined as temperaturedependent variables(seen in figure 18), which are all positive quantities and α ⊥ is always greater than α ∥ .The average value for α ⊥ and α ∥ are 18.07 × 10 −6 K −1 and 11.66 × 10 −6 K −1 , respectively.In addition, we also found that the thermal expansion coefficient parallel to the c-axis surges with temperature while that perpendicular to the c-axis decreases.Above all, not only is the Grüneisen parameter correlated with the degree of anisotropy of the coefficients of thermal expansion, but it is also associated with the different anisotropies of elasticity.

Conclusion
In summary, the successful growth of a PbIn 6 Te 10 single crystal was provided, with TG-DSC, XRD, and EDS characterizations collectively affirming its quality.Thermal expansion coefficients were systematically investigated using high-temperature x-ray diffraction and PASCal.Both the coefficients along the c-axis and perpendicular to it exhibit positive values, with the expansion behavior along the a-axis playing a predominant role, notably marked by a more pronounced expansivity (α a ).This underscores the structural influence of the volume of PbTe 6 octahedra on thermal expansion, with the coefficient along the c-axis intricately tied to the variations in Pb-Te 3 bonds.In addition, the temperature-dependent Grüneisen parameters V g and g ^increase with increasing temperature whereas // g decreases, which also exhibits anisotropic behavior related to crystallographic axes.The study is likely to be of great help in improving the growth technics, and machining nonlinear optical devices such as differential frequency devices and optical parametric occillator in future research.

Figure 1 .
Figure 1.Sketch map of two-zone synthesis furnace.

Figure 2 .
Figure 2. Temperature control program for crystal growth.

Figure 6 .
Figure 6.Partial plots of Rietveld refinement result from the main body of PIT crystal from 10°to 80°.

Figure 7 .
Figure 7. Results of EDS with weight/atomic percentage and distribution of three elements.

Figure 10 .Figure 11 .
Figure 10.Partial details of high temperature XRD patterns at different angles.

Figure 12 .
Figure 12.Non-axial thermal expansion coefficient α hkl of PIT crystal at different temperature.
tetrahedra, and Pb octahedra (figure 14(b)).These three kinds of coordination polyhedrons are connected by common atoms and edges.The connections of identical and different bases are presented in figures 14(c) and (d), respectively.

Figure 14 .
Figure 14.The details of PbIn 6 Te 10 structure.(a) Unit cell of PIT.(b) Different coordination polyhedrons of PIT.(c) Threedimensional connection of identical basis and (d) different basis.

Figure 16 .
Figure 16.Variation of bond lengths and bond angles in PbTe 6 octahedron with temperature.

Figure 17 .
Figure 17.Variation of the Grüneisen parameters with temperature.

Table 1 .
Refined cell parameters for PbIn 6 Te 10 crystal and standard PDF data.

Table 2 .
Temperature dependence of cell parameters of PbIn 6 Te 10 .

Table 4 .
Thermal expansivities related to crystallographic axes, as calculated by PASCal.