Lattice Parameters and Thermal Expansion of Important Semiconductors and Their Substrates
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Lattice Parameters and Thermal Expansion of Important Semiconductors and Their Substrates Robert R. Reeber and Kai Wang Department of Materials Science and Engineering, North Carolina State University Raleigh, NC 27695-7907 ABSTRACT Thermal expansion and lattice parameters are useful thermophysical properties of materials. A knowledge of their temperature dependence is essential for optimizing device design and crystal growth conditions while minimizing the residual stress in epitaxial films and electronic devices. Currently first principles theory can predict lattice parameters to 0.01%. Information required for device design and thin film crystal growth requires improvements two to three orders of magnitude better than this. Available experimental results from our work and the literature will be reviewed in terms of a high temperature predictive model. Tables of thermal expansion and lattice parameters for AlN, GaN, 6H-SiC, $-SiC, MgO, Al2O3, ZnO and GaAs are provided. INTRODUCTION AND METHOD Thermal expansion can be determined with a variety of techniques at low and intermediate temperatures. Elsewhere, we have reviewed the methods for evaluating and predicting the thermal expansion of materials[1]. These involve empirical, semiempirical, and theoretical modeling. Recently we utilized two semiempirical methods to predict the thermal expansion of group IV elements an well as AlN and GaN at high temperatures[2,3]. The simplest, developed by Reeber[4] and illustrated in Eq.(1), represents the thermal expansion of a solid within the constraints of a semiempirically determined multi-frequency Einstein model. 2
θi θi exp n T T αV = ∑ X i 2 i=1 θi exp T - 1
(1)
Here both Xi and θi are fitting parameters. When data are available at low and intermediate temperatures and defect energies are relatively high, this method provides a reliable evaluation and estimator of high thermal expansion. In our earlier work the available data was limited for GaN and a more approximate method was utilized for calculating and predicting thermal expansion. Our recent experimental results [5] and those of others permit use of our more robust model. As we describe in another symposium here[6] accurate thermophysical property data is essential for carrying out quantitative finite element calculations.
T6.35.1
Table I. Model Parameters for selected materials θ1 (Κ)
θ2 (Κ)
θ3 (Κ)
125 100 135 135 18.75 75 75 150 125 327.5 100 96.25
600 528.75 565.625 598.438 96.875 581.25 590.625 700 600 1350 585 575
1852.5 1723.75 1231.25 1468.75 375.625 1684.375 1675.0 1952.5 1412.5
Compound AlN Al2O3 GaAs GaN α−SiC β-SiC ZnO
a c a c a a c a c a a c
θ4 (Κ)
5468.75 5198.438
1667.5 1287.5
X1 (10-7/K)
X2 (10-7/K)
X3 (10-7/K)
-4.348 -5.174 1.2176 2.856 0.117 0.487 0.621 2.726 -12.725 12.873 -7.234 -11.510
44.074 29.857 53.401 72.079 -4.336 52.152 47.312 40.116 59.148 45.809 70.535 47.847
35.056 39.565 35.613 23.202 70.55 4.210 1.125 6.963 3.532
X4 (10-7/K)
23.661 29.087
35.674 20.074
Lattice Parameter
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