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(d) Deviations from the ideal c/a ratio may be observed in some materials, e.g., in ZnO c/a = 1.602. Taking into account the alternating Zn²+ and O²− planes in the hexagonal "Wurtzite" structure of ZnO (see Figure 2), what consequences may such a c/a deviation result in? C Zn Figure 2: Wurtzite ZnO unit cell.
a₁ 2 # 1 22 3 4 5 6 Figure 3: Triclinic crystal with (hkl) planes 1-6 indicated. (a) What are the Miller indices of the planes 1-6? (b) Make a drawing indicating the reciprocal lattice points corresponding to the planes in Figure 3. (c) Consider the reciprocal lattice vector G corresponding to the (010) planes in the drawing produced when solving (b) and an x-ray wave having wavevector k. Prove that diffraction does not occur for any kG G, where kG is a projection of vector k on the G direction. < (d) Generalize the argument to the rest of the nearest-to-origo reciprocal points and introduce the Brillouin zone concept on the same drawing. 5. Construct the three first Brillouin zones for a 1D, as well as for 2D and 3D simple cubic lattices. 6. What are limitations to observe crystallographic planes in real x-ray experiments? 7. Consider a crystal having N sites and a certain number of vacancies n. (a) Derive an expression for the equilibrium concentration of vacancies c = n/N as a function of temperature T and applied stress o. Show that the minimum free energy is reached only for a certain concentration of vacancies. (b) Explain the roles of the formation energy and the activation volume.
8. Consider a Si₁-xGex alloy with Ge mole fraction x. Si and Ge exhibit similar lattice structures, but different lattice parameters, as shown in Figure 4 below. A Si₁-xGex film, say Sio.8 Geo.2, can be grown on a Si substrate so that the in-plane atomic arrangement in the film is matching that in the substrate, i.e., epitaxy is occurring. The film would then be biaxially compressed as illustrated in Figure 4. (a) Estimate the stress occurring effectively in an epitaxial Sio.8 Geo.2 film on a Si (100) oriented substrate. Use 52 GPa for the shear modulus of elasticity μ and 0.28 for the poisson ratio v. (b) Assume an x-ray diffraction measurement (A 1.54 Å) of Si and Sio.8 Geo.2 crystals, as well as the Sio.8 Geo.2/Si heterostructure. Consider the (004) diffraction peak and make a qualitative plot showing the changes in the diffraction peak for the three samples. = (c) Figure 5 below represents Sb diffusion data in normal and biaxially compressed Sio.8 Geo.2. As- suming that Sb diffuses via a vacancy mechanism, estimate the ratio between the vacancy con- centration in normal and strained Sio.8 Geo.2 by using the data in Figure 5.
Cubic Si Cubic Ge 0.565 SiGe 0.560 0.555 0550 0.545 8 SioGeo film with «normal» lattice SiGe epitaxy Si mismatch Si-Ge-4.1% "Si Si crystall O is Ge atom and is Si atom 0.2 0.8 1.0 0.4 0.6 Ge content x Figure 4: Left: Lattice parameter of the Si₁-xGex alloy. Right: epitaxial Sio.8 Geo.2 film on a Si substrate. Temperature (°C) 1000 900 800 700 (cm /s) coefficient Sb diffusion 10 10.15 10¹ 10:17 Biaxially compressed Sio Geo film on Si substrate O Normal lattice Biaxially compressed lattice Sio.Ge 0.2 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 1/(KT) (eV)1¹ Figure 5: Sb diffusion data in normal and biaxially compressed Sio.8 Geo.2. Note that the stress in a biaxially compressed film may be given as¹ v +1 %=-2μm (x) fm(x) = 0.418x, v-1 where fm(x) is the deformation due to the difference in lattice parameters.