Anisotropic Thermal Expansion of Zirconium Diboride: An Energy-Dispersive X-Ray Diffraction Study

Zirconium diboride (ZrB 2 ) is an attractive material due to its thermal and electrical properties. In recent years, ZrB 2 has been investigated as a superior replacement for sapphire when used as a substrate for gallium nitride devices. Like sapphire, ZrB 2 has an anisotropic hexagonal structure which defines its directionally dependent properties. However, the anisotropic behavior of ZrB 2 is not well understood. In this paper, we use energy-dispersive synchrotron X-ray diffraction to measure the thermal expansion of polycrystalline ZrB 2 powder from 300 to 1150K. Nine Bragg reflections are fit using Pseudo-Voigt peak profiles and used to compute the a and c lattice parameters using a nonlinear least-squares approximation. The temperature-dependent instantaneous thermal expansion coefficients are determined for each a-axis and c-axis direction and are described by the following equations:


Introduction
Zirconium diboride (ZrB 2 ) is a nonoxide ceramic material with a melting temperature of 3245 ∘ C and high electrical conductivity (10 7 S⋅m −1 ) [1].The ultrahigh melting temperature and resistance to oxidation of ZrB 2 make the material attractive for applications in ultrasonic flight, continuous steel processing, and atmospheric reentry [1][2][3].ZrB 2 is also unique, however, for its electronic properties.Recently, single crystal ZrB 2 has been used as a substrate for gallium nitride (GaN) semiconductor devices [4,5].ZrB 2 has intrinsic advantages over sapphire, the industry standard.Notably, ZrB 2 has a smaller lattice mismatch to GaN and superior electronic and thermal conductivity.
Proper use of ZrB 2 in any application requires an indepth understanding of the thermal behavior.The crystal arrangement of ZrB 2 is hexagonal with alternating layers of Zr and B atoms [1].This hexagonal nature of ZrB 2 leads to anisotropic behavior and properties which are crucial to understand, especially in applications where the crystal orientation of ZrB 2 is controlled.
The thermal behavior of ZrB 2 has been studied in the past.The pioneering work by Kinoshita and coworkers was able to show that ZrB 2 is a viable substrate for GaN electronics.They grew single crystals of ZrB 2 using a float zone method and measured the coefficient of thermal expansion along the  direction [4,5].Thermal expansion was later studied by Okamoto et al.where they grew single crystals in a similar manner and measured expansion in both the  and  directions using a pushrod-type dilatometer from room temperature to 1073 K [6].Their results revealed the anisotropic nature of thermal expansion with average coefficient of thermal expansion (CTE) values of 6.66 × 10 −6 and 6.93 × 10 −6 K −1 in the  and  directions, respectively.They also showed that their values carry a temperature dependence, however.Zimmermann et al. used a densified polycrystalline sample to study the thermal expansion of ZrB 2 [7].Their study employed the use of dilatometer and measured expansion from 300 K to 1675 K.They were able to measure the temperature-dependent CTE but their polycrystalline sample means that they were insensitive to directional effects.The CTE of ZrB 2 can also be calculated from first principles.Milman et al. performed a molecular dynamics study and calculated values of thermal expansion that were within range of previous reports.While they mention that Okamoto's results show anisotropy and temperature dependence, they were unable to address these features within their experimental framework [8].
Scattering techniques such as X-ray diffraction can measure the distance between planes of atoms in a crystal.Because of this, X-ray diffraction has been used extensively to measure thermal expansion [9][10][11][12][13][14]. Energy-dispersive X-ray diffraction is particularly well-suited for measuring changes in interatomic distances [15][16][17][18][19].This is because unlike the more-common angular-dispersive X-ray diffraction, EDXRD satisfies the scattering condition with polychromatic radiation at a fixed angle.This enables one to make very precise lattice parameter measurements [10,20].Additionally, the use of an energy-discriminating detector means that the entire spectrum is collected in parallel which is ideal for timeresolved measurements [21,22].
In this paper, we take advantage of the penetrative power of synchrotron radiation and measure the thermal expansion of polycrystalline ZrB 2 in situ up to 1100 K. Particular focus is given to the anisotropic effects of ZrB 2 's hexagonal lattice and the temperature dependence of thermal expansion.

Experimental
Polycrystalline ZrB 2 powder was obtained from H. C. Stark Corporation (Grade B Hf min 0.2% 90% <6 m).The powder was loosely packed in an alumina vessel and a type-K thermocouple was placed in the center of the powder.The vessel was then placed in a furnace built for high temperature in situ diffraction work.The details of this furnace have been covered previously in the literature [21].
Energy-dispersive X-ray diffraction (EDXRD) was carried out at the National Synchrotron Light Source X17B1 beamline at Brookhaven National Laboratory.EDXRD uses a fixed angle and polychromatic radiation in order to satisfy the Bragg diffraction condition.As a result, the diffraction measurement works directly in reciprocal space and enables precise determination of the lattice parameters.The measurement volume is fixed in space and the spectra are collected in a parallel fashion.These factors of EDXRD make it a viable technique for in situ measurement of thermal expansion.Figure 1 shows the experimental configuration.
The sample was heated from room temperature to approximately 1150 K at a heating rate of 5 K per minute in an argon environment to prevent oxidation.Throughout the experiment, spectra were collected every 30 seconds.Additionally, the temperature was logged by the thermocouple every one second.The clocks on both the temperature logger and the EDXRD computer were synchronized so that the temperature and EDXRD data could be correlated.
Standards for calibration were collected from Ag, Au, CeO 2 , Cu, Ge, LaB 6 , W, and Y 2 O 3 and used to determine the channel number to energy relationship and the exact angle of diffraction.Additionally, laboratory powder diffraction was carried out to characterize the powder and included in the calibration.
The 001, 100, 101, 002, 110, 102, 111, 200, and 201 peaks were identified in the spectra and were fit with Pseudo-Voigt peak-shape function [23].The -spacing for the nine peaks were then used to calculate the  and  lattice parameters using a nonlinear least-squares determination incorporating the Levenberg-Marquardt algorithm using the fsolve function in MATLAB.

Results and Discussion
Figure 2 shows the lattice parameter  as a function of temperature and Figure 3 shows the lattice parameter  as a function of temperature.The data show a minimal degree of scatter which should be considered a representation of the error in both the peak fitting and the temperature recording.Additionally, throughout the experiment we observe uniform thermal expansion and no structural changes.
The CTE can be expressed as where  is a length and / is the rate of change of that length with temperature.Rearranging gives us assuming that the expansion coefficient does not change with temperature.For small changes in temperature, this provides a good approximation.However, this experiment deals with a temperature range of 850 K, rendering the above approximation useless.The behavior of the lattice parameter over the given range of temperatures can be approximated using the following second order polynomial equation: where  is the lattice parameter,  is the measured temperature, Θ is a reference temperature,   is the lattice parameter at the reference temperature, and   and   are the first-and second-order expansion coefficients, respectively.The quadratic behavior of the lattice parameter with temperature results in a CTE that is linearly dependent on temperature.By taking the derivative of the fit equation and dividing by the lattice parameter, one can calculate the temperature-dependent instantaneous expansion coefficient [24,25].This is shown in the following equation: In both Figures 2 and 3, the lattice parameter data are fit with the second order polynomial equation and a reference temperature of 293.15 K.The fitting results are given in Table 1.The fit is excellent with a coefficient of determination ( 2 ) value greater than 0.998 for both lattice parameters.
The CTE dependence on temperature for both the  and  directions is shown in Figure 4. Additionally, selected values are given in Table 2.At room temperature, we observe anisotropic behavior with a larger CTE in the  direction.The two lines converge around 780 K, however, where at this temperature the CTE is isotropic.At temperatures greater than 780 K, we observe a higher expansion in the  direction.4. For CTE values given with a range, they are plotted at their upper limit of the range.Above 650 K, there is good agreement between our values and those reported by others.However, below 650 K our values are lower than those of Okamoto et al.Because this is within operating temperature range of a typical GaN LED, it is important to understand the results.It is well known for thermal expansion that lattice parameter measurements differ from macroscopic measurements.Particularly, our diffraction is not sensitive to the increases in thermodynamic defects associated with higher temperatures.Additionally, Okamoto et al. do not mention anything about the quality of their crystal, that is, whether or not there is significant twinning or other defects.These things could explain the significantly higher values they observe.
Figure 5 shows both the EDXRD and ADXRD spectra at room temperature.We see a good agreement between the two methods and observe no secondary phases.Whole pattern fitting was carried out on the ADXRD measurement and the lattice parameters were found to be within range of the literature values.

Conclusion
We performed energy-dispersive synchrotron X-ray diffraction measurements while heating a polycrystalline ZrB 2 powder from 300 to 1150 K.The temperature-dependent instantaneous thermal expansion coefficients were determined for the -axis and -axis directions.The results show that   and   converge to the same value at around 780 K, below which   is higher and above which   is higher.

Figure 2 : 2 Adj
Figure 2: Lattice constant  as a function of temperature.

Figure 3 :
Figure 3: Lattice constant  as a function of temperature.

Table 1 :
Constants solved using the polynomial fit of lattice parameter versus temperature.

Table 2 :
Instantaneous values of expansion coefficient at selected temperatures.There are some interesting differences when comparing our results with other values reported in the literature.CTE values from Okamoto et al., Zimmermann et al., and Milman et al. are plotted in Figure