Thermal and Mechanical Properties

Material preparation, device fabrication, and device functionality depend on a variety of material properties, including thermal and mechanical properties thereof. To derive such properties, a diversity of experimental techniques are employed, which are more recently aided by theoretical calculations. The availability of bulk ZnO single crystals enabled to measure their thermal and mechanical properties, which are quite consistent, wherein minor differences may arise from the crystal structural quality and purity. This section discusses fundamental thermal and mechanical properties of bulk ZnO single crystals with some inclusion of theoretical results.

Melting Point

Only the wurtzite phase of ZnO is stable up to melting point, MP = 1975°C = 2248 К [77], where it melts with a melting enthalpy AH_{ = 54392 J/mol. For this solid phase and the liquid phase above

MP, thermal properties are compiled, e.g., in the FactSage database [245] and by Barin [246] as follows:

Figure 3.45 compares the Gibbs free energy functions G(7) for solid and liquid ZnO, which can be calculated from these experimental data with that of gaseous ZnO for total pressures p = 1 atm (solid) and p = 3 atm (dashed), respectively. Above MP = 1975°C, G for the liquid phase is lower than for the solid phase, almost independent on pressure. At 1 atm, however, G(7) for the gas becomes even lower below MP. This is the reason why ZnO cannot be molten at ambient pressure. Already a slight overpressure (as shown for 3 atm) is sufficient to stabilize the condensed phases beyond MP.

Figure 3.45 Gibbs free energy G(T) for the three states of aggregation of ZnO. Only for the gas phase, G depends significantly also on the pressure p.

Thermal Conductivity and Specific Heat Capacity

Thermal conductivity of hexagonal ZnO is described by the second-order tensor with two independent thermal conductivity coefficients perpendicular Aj__c and parallel ^ilc to the c-axis. Wolf and

Martin [247] reported thermal conductivities (with a steady-state longitudinal heat-flow apparatus) of bulk ZnO crystals (obtained by the hydrothermal and vapor phase methods) in a temperature range of3.5-300 К with a maximum value of about 1000 W/mK below 50 K.

At 300 K, the thermal conductivity is just below 100 W/mK. Also, the ratio of A||_C/A_lc was found to be 1.15, indicating a minor anisotropy of the thermal conductivity. The thermal conductivity is limited by phonon-phonon scattering at elevated and high temperatures.

Bulk ZnO single crystals obtained from the gas phase and measured by Florescu et al. [248] with scanning thermal microscopy showed at RT values of 102-116 ± 0.07(8) and 98-110 ± 0.08(9) W/mK for Zn-terminated and О-terminated faces of a c-oriented sample, respectively. For ZnO bulk crystals obtained by the cold crucible method, Ozgiir et al. [249] obtained thermal conductivities of 100 ± 0.08 and 95 ± 0.06 W/mK at RT for Zn-terminated and О-terminated faces, respectively. A thermal conductivity ratio for Zn- terminated and О-terminated faces is, therefore, between 1.04 and 1.05. Samples annealed in air (1050°C, 3 h) and N-plasma (750°C,

1 min) revealed the values of 135 ± 0.08 and 147 ± 0.08 W/mK, respectively. Higher values resulted likely from a change in surface electrical conductivity and surface morphology. Indeed, Gadzhiev

[250] reported that thermal conductivities of ZnO ceramics with a porosity of 0-21% decrease from 35 to 14.3 W/mK at RT, which is less than half the value of bulk single crystals.

Specific heat capacity reported by Klimm et al. [77] increases from 41.05 at 298 К to 61.576 J/molK at the melting point of ZnO at 2248 К (see Section 3.4.1). Similar values were reported by Passler

[251] in this temperature range, with a fast drop thereof below RT, to a value of 12 J/molK at 80 K. Serrano et al. [252] concluded from experimental and theoretical studies that heat capacity depends on isotope masses of Zn and О atoms, which affect mainly acoustic phonons that are active at low temperatures, and optical phonons that are active at higher temperatures, respectively.

Thermal Expansion

The linear thermal expansion coefficients of bulk ZnO crystals were determined, e.g., by Iwanaga et al. [253], Albertsson et al. [254], and Khan [255].

Iwanaga et al. [253] found by X-ray powder diffractometry of a bulk ZnO crystal the following relation at a temperature range of 300-1373 K:

which yield at RT а₀^ет = 4.31 x 10'⁶ K'¹ and a_c^RT = 2.49 x 10~⁶ K'¹ with a ratio a_a^RT/a_c^RT = 1.73.

The с/a ratio decreases with temperature from 1.602 at 300 К to 1.596 at 1373 K. This means that the wurtzite-type structure of ZnO becomes more stable with an increase in temperature with respect to zincblende-type structure.

Along the a- and c-axes, the thermal expansion coefficient relations according to dilatometric measurements by Albertsson et al. [254] are as follows:

where AT = T- 298 K, a„^RT = 6.511 x 10'⁶ K"¹ and а_с^ет = 3.017 x IQ-6 k~i for a- and c-axes, respectively. The ratio а₀^ет/а_с^кт = 2.16 is higher as compared with the above-mentioned study and may arise from the measurement technique.

For comparison, Khan [255] measured on powders in a temperature range of 27-619°C the following thermal expansion coefficients:

which result in а₀^ет = 6.11 x 10~⁶ K'¹ and а_с^ет = 3.59 x 10^-6 K'¹ at RT for a- and c-axes, respectively.

Stiffness and Hardness

The elastic properties of ZnO are described by the fourth-rank tensors, with the stiffness coefficients c,y forming a 6 x 6 matrix with five independent components: c_n, c₁₂, c₁₃, c₃₃, and c₄₄, wherein c₆₆ = (c_n - c₁₂)/2. Mechanical properties of ZnO were studied both experimentally and theoretically. Experimental results on elastic properties of bulk ZnO crystals can be found, e.g., in the works of Bateman [256], Solbrig [2], Kobiakov [257], and Azuhata et al. [258]. Theoretical study of mechanical properties was carried out, e.g., by Ahuja et al. [259], Catti et al. [260], Wu et al. [261], Tu and Hu [262], Gopal and Spaldin [263], Shein et al. [264], and Li et al. [265]. Selected experiments, determined mainly by ultrasonic techniques, and theoretical stiffness coefficients c,yare listed in Table 3.5.

Using the Voigt approach and symmetry for the hexagonal system (^ci l = c₂₂, c₁₃ = c₂₃, c₄₄ = c₅₅), the bulk modulus В and shear modulus Gcan be calculated from the following relations:

and both Young's modulus and Poisson ratio can be calculated as follows:

The calculated moduli and Poisson ratio on the basis of the stiffness coefficients along with the anisotropy ratio [266] defined by A = c₄₄/c₆₆ are also included in Table 3.5.

The anisotropy ratio is unity or close to unity, which indicates an almost isotropic elastic behavior of ZnO. This means Young's modulus is nearly independent of the orientation. Bulk, shear, and Young's moduli derived from experimental stiffness coefficients lie in the range of 128-144 GPa, 43-46 GPa, and 115-126 GPa, while theoretical values are a bit overestimated in a majority of cases.

Table 3.5 Experimental and theoretical values of stiffness coefficients of bulk ZnO crystals. Exp - experimental, Theor.- theoretical, A - anisotropy ratio, B- bulk modulus, G- shear modulus, E- Young's modulus, and v- Possions ratio

Cij [GPa]	Exp. Ref. [256]	Exp. Ref. [2]	Exp. Ref. [257]	Exp. Ref. [258]	Theor. Ref. [260]	Theor. Ref. [262]	Theor. Ref. [263]	Theor. Ref. [265]
	209.7	194	207	190	246	218	217	191.5
	121.1	102	117.7	110	127	137	117	108.7
	105.1	94	106.1	90	105	121	121	95
	210.9	217	209.5	196	246	229	225	206.7
	42.47	—	44.8	39	56	38	50	41.4
	44.29	46	44.6	40	59.5	40.5	50	38

With respect to plastic deformation, ZnO is a comparably soft material. As a mineral, "zincite" has the hardness of 4 [267] on the Mohs scale, which means that it can be scratched by a knife. The Mohs hardness of bulk ZnO single crystals was measured at a bit higher value of 4.5 [80]. Microhardness of bulk ZnO crystals was studied by numerous researches. Yonenaga [268] found the microhardness of ZnO by the Vickers indentation method of 4.7 GPa, while Kucheyev et al. [269] by nanoindentation and atomic force microscopy of 5 ± 0.1 GPa at RT with Young's modulus of 111.26 ±4.7 GPa. Multiple discontinuities (pop-ins) in force-displacement curves were observed as a result of a plastic slip on both the basal and pyramidal planes [269, 270]. Such pop-ins were also observed by Jian [271] with nanoindentation-induced deformation studies of bulk c-oriented ZnO crystals with microhardness and Young's modulus of 5.4 ± 0.7 GPa and 112.5 ± 8.4 GPa, respectively. The plastic slip occurs on the (1011) plane [271,272], which may lead to a non-pronounced cleaving parallel to {1010}. Coleman et al. [273] measured the microhardness and Young's modulus of bulk ZnO single crystals by nanoindentation of 2 ±0.2 and 163 ±6 (o-oriented) and 4.8 ± 0.2 and 143 ± 6 GPa (c-oriented), respectively.

Demyanets et al. [274] studied micro hardness ofhydrothermally grown bulk ZnO single crystals by the Vickers indentation method in a temperature range of-196 to 900°C on the planes of monohedra, hexagonal, prisms, and the pyramid. Indentation prints on the (0001) and (0001) monohedra did not show any cracks, indicating a maximum hardness and plasticity over the temperature range. On the other hand, around the indentation prints on the {1010} and {1120} prisms and {1011} pyramids, radial cracks were formed, which were aligned in the glide <1120> direction. The microhardness was found to be relatively isotropic with a value of 1.7 GPa, while Young's modulus was 47.7 GPa.

The experimental density p_s of bulk ZnO crystals is 5.676 [256] or 5.642 g/cm³ [275]. Having values of the solid density as well as bulk and shear moduli, one can estimate transverse and longitudinal sound velocity in ZnO crystals [264]:

Taking experimental values of stiffness coefficients [256-258] and density [256], the semi-empirical sound velocities for bulk ZnO single crystals are v_T = 2742-2860 and v_L = 5317-5600 m/s. Bateman [256] measured sound velocities with a high-frequency ultrasonic buffer technique and found the values of 2735-2793 and 5939-6096 m/s for shear and longitudinal waves, respectively.

The experimental Debye temperature of ZnO mentioned by Abrahams and Bernstein [275] is 0_D = 355-416 K. An example of the theoretical value 0_D by Shein et al. [264] is 405 K.

More details on elastic properties of the wurtzite structure in general can be found in books of Adachi [276] and Gil [277].

Conclusion

Melting point of ZnO is high, 1975°C, but to obtain the liquid phase, an overpressure is required. Thermal conductivity of bulk ZnO single crystals is very high, about 100 W/mK with a minor anisotropy parallel and perpendicular to the c-axis of about 1.15, as well as for Zn- and О-terminated c-faces of 1.04-1.05. Also, the linear thermal expansion of ZnO single crystals shows anisotropy with values at RT of 4.31-6.511 x 10'⁶ K'¹ along the а-axis and 2.49-3.017 x 10'⁶ K'¹ along the c-axis. The ratio of the linear thermal conductivity along the а-axis to c-axis is in the range of 1.73-2.16. The experimental density is 5.642-5.676 g/cm³. Basing on measured stiffness coefficients (five independent components) on bulk ZnO single crystals, bulk, shear, and Young's moduli are in the range of 128-144 GPa, 43-46 GPa, and 115-126 GPa, respectively. Measured microhardness was found in the range of 4.7-5.4 GPa. Plastic slip occurs on both the basal and pyramidal planes. Semi-empirical transverse and longitudinal sound velocities in bulk ZnO single crystals are 2742-2860 and 5317-5600 m/s, respectively, while the Debye temperature is between 355 and 416 K.

Basic thermal and mechanical properties of bulk ZnO single crystals are collected in Table 3.6.

Table 3.6 Basic thermal and mechanical properties of bulk ZnO single crystals

MSemi-empirical values based on measured stiffness coefficients from corresponding references and Eqs. 3.39-3.42; ^semi-empirical values based on measured stiffness coefficients and density from corresponding references, and Eq. 3.43