Thermal Expansion of GaN and AIN
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I.I'
X (ot1). (ot e (8/I)1]
(1)
Where X, and 0•are fitting parameters. This method provides a reliable analytical expression for the thermal expansion when accurate data is available at cryogenic and intermediate temperatures. For situations where only room temperature and intermediate temperature data is available, there is no way to approximate Blackman's average GrIneisen parameters in this model and more approximate approaches must be used. Graneisen introduced the Mie-Graneisen equation of state and his thermal expansion model [7], Eq. 2, in the early part of this century.
Y(T) - a~Q HW_ K E,) 0 -
(L-a)av
(2)
Here, the expansivity,Y(T), equals (v-vT)/VT, E,1,e is the vibrational energy, Qo 0 - K0oVW, and x (-rd, - 1)/2. Ko is the bulk modulus, Vo is'thepmolar volume, y is the Grilneisen parameter at zero K, av . VT 1v0 and T, is the reference temperature. The thermal expansion is obtained by differentiati*g (2) with respect to temperature. The vibrational energy is calculated by approximating the unknown vibration spectrum with a Debye model. Later, Suzuki et al. derived a more accurate expression for expansivity [8] 863 Mat. Res. Soc. Symp. Proc. Vol. 482 01998 Materials Research Society
Y(T) - 1t2K-(1c4ay1/Qo)"2
(3)
Unfortunately, the approximations in Eq. (2) and (3), limit the accuracy of the results to about twice the Debye temperature. Eq. (2) and (3) are not applicable at cryogenic temperature for Group IV elements, II-VI, III-V and other Grimm-Sommerfeld compounds which exhibit negative thermal expansion at very low temperatures. Linear expansion along different axis for anisotropic materials [8] are easily calculated with this approach. The parameters in Eq. (1) through (3) have physical significance. They are obtained by leastsquare fitting of the experimental data. The thermal expansion of the wurtzite structure GaN and AIN have been evaluated and predicted at high temperatures where data is unavailable. For AIN where there is cryogenic temperature data, Eq.(1) is appropriate. The high temperature thermal expansion
of GaN was estimated with Eq.(2) and the available experimental data. RESULTS Lattice Parameters Table 1 lists the 298 0K lattice parameters along the a- and c-axes for AIN and GaN. These are normalized to a Cu-Kc 1 x-ray wavelength of 1.540579A. AIN lattice parameters are provided by many authors with most results reviewed earlier by Slack [9]. Slack's measurements are adopted as a standard for AIN and Leszczynski et al.'s [10] for GaN. Table 1. Lattice Parameters at 298°K, X=1.540597 A Compound a-axis c-axis References AIN GaN
(A)
(A)
3.1129 3.1878
4.9819 5.1850
[9] [101
Thermal Expansion Hexagonal materials thermal expansions perpendicular and parallel to the c-axis are usually different. They are reviewed and predicted separately. The linear thermal expansion of the polycrystal can then be calculated as a.(2 a,)/3. AIN The lattice parameters and limited thermal expansion AIN results have been measured to intermediate temperatures earlier [11-20]. Single crystal AIN m
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