Extrapolation of critical thickness of GaN thin films from lattice constant data using synchrotron X-ray
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557 Mat. Res. Soc. Symp. Proc. Vol. 423 ©1996 Materials Research Society
CL1
Fig. I Unit cells of sapphire (0001) and AIN on top of it. Solid line is for sapphire and dashed line for AIN. Open circles and hatched ones represent 0 in sapphire and Al in AIN, respectively. EXPERIMENT To test those ideas which will be explained in the next section we examined measurements of lattice constants of GaN thin film grown on AIN buffer layers on sapphire (0001).3 The AIN buffer layer between the sapphire and the GaN was used because its in-plane lattice constant is intermediate, as used previously 4, which redistributes the substantial mismatch between two interfaces. GaN samples with thicknesses in the range from 50 A to 1 gm were grown by MBE (Molecular Beam Epitaxy) on sapphire (0001) using a 32 A AIN buffer layer. In order to have a workable signal from the thinnest films, X-ray beamline X16C at the National Synchrotron Light Source (NSLS) was employed to determine the lattice constant a of the GaN films using a least-squares fit method. 3 Fig. I shows the known epitaxial arrangement of AIN on sapphire (0001)5.6 and table 17 lists the known bulk hexagonal lattice constants of the relevant materials. If there were no lattice mismatch, the in-plane lattice constant of AIN would be expected to be 4.758/N,3 = 2.747 A, considerably less than the bulk value of 3.112 A. Instead, we found a = 3.084 A indicating a partial compression of the buffer layer. THEORY First we consider the relationship between the film thickness and the lattice constant a using Van der Merwe's energy minimization theory. 8 From elementary elasticity theory the strain energy per unit area is proportional to h(ao- a)2, where h is a film thickness, a, is the lattice constant of the (completely relaxed) material of the film and a is the actual lattice constant of the film. Assuming that the effective range of the dislocation field is constant, the energy due to Table I Physical parameters of GaN, AIN and Sapphire GaN AIN Sapphire
a (A)
c (A)
3.189 3.112 4.758
5.185 4.982 12.991
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dislocations depend on their density alone, which is proportional to (a - as). energy of the strained film is given by E =cih(a - a, )2 + 3(a-a)
Then, the total
(1)
where u. and P are constants. For the system to be in equilibrium, this energy should be a minimum: aE/aa=O. Then, to obtain the value of h,, the critical thickness, we simply need to consider a = a,. Setting aEi/a=0 also gives a relation between h and a so long as a > as At a = as, h is equal to hc below which the film is pseudomorphic and a will have the same lattice constant as the substrate, a,. The derived relation between a and h from Eq.(I) is h a = a. +--h(as- a)
(2)
The method we have used to determine h, experimentally is to fit lattice constant data for thicker films, and use Eq.(2) as an extrapolation formula. Now we consider the total energy more rigorously. The elastic strain £ depends on the mismatch between a film and its substrate, m, as well as the average number of dislocations present
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