Lattice parameters and thermal expansion of GaN
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Lattice parameters and thermal expansion of GaN Robert R. Reeber and Kai Wang Department of Materials Science and Engineering, North Carolina State University, Raleigh, North Carolina 27695-7907 (Received 8 February 1999; accepted 26 October 1999)
Neutron powder diffraction methods with Rietveld analysis are utilized to determine GaN lattice parameters from 15 to 298.1 K. Using these measurements and literature data, we calculated the thermal expansion of gallium nitride (GaN) and predicted its higher temperature thermal expansion. The results are compared with available experimental data and earlier work.
I. INTRODUCTION AND METHOD
Gallium nitride (GaN) is an actively studied semiconductor that has promise for the construction of blue-lightemitting diodes and high-temperature electronic devices. Most of the methods for growing GaN rely on epitaxial layer deposition on a variety of substrates at higher temperatures. Thermal expansion differences between GaN and its substrate introduce residual stresses during and after cooling to room temperature. Such stress can affect device lifetime and performance. An accurate knowledge of the thermal expansion over an extended temperature range is critical for calculating the residual stress in such devices. This information is also relevant to an understanding of the interatomic potential and the equation of state. Lattice parameter data at low temperatures are limited for GaN. Several room-temperature values have been summarized elsewhere.1 Limited elastic and lattice parameter data exist for this wide-band-gap material. Thermal expansion can be determined with a variety of techniques at low and intermediate temperatures. Elsewhere, we have reviewed2–7 the methods for evaluating and predicting the thermal expansion of materials. These involve empirical, semiempirical, and theoretical modeling. Recently8 we used two semiempirical methods to predict the thermal expansion of AlN and GaN at high temperatures. The simplest, developed by Reeber9 and illustrated in Eq. (1), represents the thermal expansion of a solid within the constraints of a semiempirically determined multifrequency Einstein model:
␣V =
冱X i=1
冉冊 冉冊 冋 冉冊 册 i T
n
i
2
exp
i T
i exp −1 T
2
.
(1)
Here both Xi and i are fitting parameters. When data is available at low and intermediate temperatures and defect energies are relatively high, this method provides a 40
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J. Mater. Res., Vol. 15, No. 1, Jan 2000 Downloaded: 13 Jan 2015
reliable evaluation and estimator of thermal expansion. In our earlier work, the available data were limited for GaN and a more approximate method was used for calculating and predicting thermal expansion. This model, illustrated in Eq. (2), was developed by Gru¨neisen early in this century.10 It approximates the low-frequency vibration spectrum with a Debye model:
Y共T 兲 =
Evib 共1 − aV兲 + . aV 共Q0 − Evib兲 aV
(2)
Where Y(T) ⳱ (V − VTr )/VTr , Evib is the vibrational energy, Q0 ⳱ K0V0 /␥, and
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