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 secondorder tensor with two independent thermal conductivity coefficients perpendicular Aj__{c} and parallel ^ilc to the caxis. Wolf and
Martin [247] reported thermal conductivities (with a steadystate longitudinal heatflow apparatus) of bulk ZnO crystals (obtained by the hydrothermal and vapor phase methods) in a temperature range of3.5300 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 phononphonon 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 102116 ± 0.07(8) and 98110 ± 0.08(9) W/mK for Znterminated and Оterminated faces of a coriented 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 Znterminated and Оterminated faces, respectively. A thermal conductivity ratio for Zn terminated and Оterminated faces is, therefore, between 1.04 and 1.05. Samples annealed in air (1050°C, 3 h) and Nplasma (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 021% 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 Xray powder diffractometry of a bulk ZnO crystal the following relation at a temperature range of 3001373 K:
which yield at RT а_{0}^{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 wurtzitetype structure of ZnO becomes more stable with an increase in temperature with respect to zincblendetype structure.
Along the a and caxes, 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 IQ6 k~i for a and caxes, respectively. The ratio а_{0}^{ет}/а_{с}^{кт} = 2.16 is higher as compared with the abovementioned study and may arise from the measurement technique.
For comparison, Khan [255] measured on powders in a temperature range of 27619°C the following thermal expansion coefficients:
which result in а_{0}^{ет} = 6.11 x 10~^{6} K'^{1} and а_{с}^{ет} = 3.59 x 10^{6} K'^{1} at RT for a and caxes, respectively.
Stiffness and Hardness
The elastic properties of ZnO are described by the fourthrank tensors, with the stiffness coefficients c,y forming a 6 x 6 matrix with five independent components: c_{n}, c_{12}, c_{13}, c_{33}, and c_{44}, wherein c_{66 }= (c_{n}  c_{12})/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 (^{c}i l = c_{22}, c_{13} = c_{23}, c_{44} = c_{55}), 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 = c_{44}/c_{66} 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 128144 GPa, 4346 GPa, and 115126 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 
A 
0.96 
— 
1.00 
0.98 
0.94 
0.94 
1.00 
1.09 
В 
144 
132 
143 
128 
157 
158 
153 
132 
G 
46 
— 
46 
43 
61 
42 
50 
44 
E 
124 
— 
126 
115 
162 
117 
135 
118 
V 
0.356 
— 
0.353 
0.350 
0.328 
0.377 
0.353 
0.351 
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 (popins) in forcedisplacement curves were observed as a result of a plastic slip on both the basal and pyramidal planes [269, 270]. Such popins were also observed by Jian [271] with nanoindentationinduced deformation studies of bulk coriented 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 nonpronounced 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 (ooriented) and 4.8 ± 0.2 and 143 ± 6 GPa (coriented), respectively.
Demyanets et al. [274] studied micro hardness ofhydrothermally grown bulk ZnO single crystals by the Vickers indentation method in a temperature range of196 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/cm^{3} [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 [256258] and density [256], the semiempirical sound velocities for bulk ZnO single crystals are v_{T} = 27422860 and v_{L} = 53175600 m/s. Bateman [256] measured sound velocities with a highfrequency ultrasonic buffer technique and found the values of 27352793 and 59396096 m/s for shear and longitudinal waves, respectively.
The experimental Debye temperature of ZnO mentioned by Abrahams and Bernstein [275] is 0_{D} = 355416 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 caxis of about 1.15, as well as for Zn and Оterminated cfaces of 1.041.05. Also, the linear thermal expansion of ZnO single crystals shows anisotropy with values at RT of 4.316.511 x 10'^{6} K'^{1} along the аaxis and 2.493.017 x 10'^{6} K'^{1 }along the caxis. The ratio of the linear thermal conductivity along the аaxis to caxis is in the range of 1.732.16. The experimental density is 5.6425.676 g/cm^{3}. Basing on measured stiffness coefficients (five independent components) on bulk ZnO single crystals, bulk, shear, and Young's moduli are in the range of 128144 GPa, 4346 GPa, and 115126 GPa, respectively. Measured microhardness was found in the range of 4.75.4 GPa. Plastic slip occurs on both the basal and pyramidal planes. Semiempirical transverse and longitudinal sound velocities in bulk ZnO single crystals are 27422860 and 53175600 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
Property 
Unit 
Value 
Reference 
Experimental solid density 
g/cm^{3} 
5.6425.676 
256,275 
Melting point 
°C 
1975 
77 
Thermal expansion coefficient of the lattice parameter at RT 
к^{1} 
a: 4.316.511 x 10'^{6 }c. 2.493.59 xlO^{6} 
253, 254, 255 
Thermal conductivity at RT 
W/mK 
95116 
247, 248, 249 
Molar heat capacity at RT 
J/molK 
41.05 
77 
Experimental stiffness coefficients 
GPa 
c_{u}= 210/194/207/190 c_{12}= 121/102/118/110 c_{13}= 105/94/106/90 c_{33} = 211/217/210/196 c_{44} = 42/ — /45/39 
256/2/257/258 
Bulk modulus^ 
GPa 
144/132/143/128 
256/2/257/258 
Shear modulus^ 
GPa 
46/ — /46/43 
256/2/257/258 
Young modulus^ 
GPa 
124/ —/126/115 
256/2/257/258 
Poisson ratioW 
— 
0.356/ —/0.353/0.350 
256/2/257/258 
Microhardness 
GPa 
csurface: 4.75.4 asurface: 2 
268, 269, 270, 271, 272, 273 
Mohs hardness 
— 
4.5 
80 
Transverse sound velocity^ Longitudinal sound velocity^ 
m/s 

256, 257, 258 
Debye temperature 
К 
355416 
275 
MSemiempirical values based on measured stiffness coefficients from corresponding references and Eqs. 3.393.42; ^semiempirical values based on measured stiffness coefficients and density from corresponding references, and Eq. 3.43