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Growth Physics

Growth Physics. The first MBE Experiments: Equilibrium vapor pressure of Ga over GaAs at 500 °C is low x Ga = c Ga F Ga c Ga = 1 (sticking or incorporation coefficient). Growth Physics. The first MBE Experiments: Sticking coefficient of As 2 depends on presence of Ga. As 2 on GaAs.

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Growth Physics

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  1. Growth Physics • The first MBE Experiments: • Equilibrium vapor pressure of Ga over GaAs at 500 °C is low • xGa = cGaFGa • cGa = 1 (sticking or incorporation coefficient)

  2. Growth Physics • The first MBE Experiments: • Sticking coefficient of As2 depends on presence of Ga As2 on GaAs As2 on Ga-covered GaAs From Ohring, Fig. 7-23, p. 341 • cAs ~ 0 without Ga • cAs > 0 with Ga

  3. Growth Physics • As2 requires presence of Ga to stick • Grow with excess flux of As (V/III > 1) • Excess As2 is desorbed • Do not require 1:1 flux ratio of As:Ga to achieve film stoichiometry • Growth rate is set by Ga flux

  4. Growth Physics • Physisorption • Adatoms • No electron transfer (van der Waals bonding) • E < ~ 0.4 eV • Chemisorption • Involves electron transfer (chemical binding occurs with the substrate atoms) • E > ~ 1 – 10 eV

  5. Growth Physics From Mahan, Fig. III.2, p. 47

  6. Nucleation Rate • Adatoms will migrate until they find a binding site • If adatoms do not find a binding site within the time t they will desorb • t = n-1 exp (Edes/kT) • = surface lifetime before desorption or incorporation • = attempt frequency ~ 1012 s-1 Edes = barrier to desorption • Growth temperature must allow for sufficient surface diffusion for atoms to find binding sites and to overcome potential barrier (As2 → 2As)

  7. Growth Physics • Rate equation for adatoms : • dN/dt = rate of chemisorption • – rate of desorption • Rate of adsorption = d (1 - q) z • Rate of desorption = N/t • d = trapping probability • q = surface coverage = fraction of filled surface sites • z = impingement rate • N = surface adatom density • t = surface lifetime before desorption or incorporation Sticking coefficient, c

  8. Growth Physics • At equilibrium, dN/dt = 0 : • q = QP / (1 + QP) • Q = d(2pmkT)-½ / [Nn exp(-Edes / kT)] • (Langmuir adsorption isotherm) From Mahan, Fig. III.1, p. 51

  9. Alloy Sticking Coefficients • e.g., In1-xGaxAsyP1-y • x / (1- x) = FGa / Fin • Assumes equilibrium VP at the growth temperature are not too high

  10. Alloy Sticking Coefficients • e.g., In1-xGaxAsyP1-y • V/V’ ratios ratios in the solid depend on the relative sticking coefficients of As2 and P2 which depend on the relative equilibrium vapor pressures • y / (1-y) = c Y / (1-Y) • c = cAs/cP • Rearranging, y = cY / [ 1 + (c-1)Y ]

  11. Alloy Sticking Coefficients • Experimentally, c > 1 • Sticking coefficient of As2 is greater than P2 • Due to higher equilibrium VP of P2 compared to As2 LaPierre, Ph.D. thesis

  12. Film Structure Amorphous Crystalline Polycrystalline

  13. Growth Modes Growth Modes Volmer- Weber Franke- van der Merwe Stranski- Krastanov Step-flow

  14. Growth Modes • Island (or Volmer-Weber) • Atoms are more strongly bound to each other than to the substrate From Ohring, Fig. 5-2, p. 197

  15. Growth Modes • Layer (or Franke-van der Merwe) • Atoms are more strongly bound to the substrate than to each other From Ohring, Fig. 5-2, p. 197

  16. Growth Modes • Layer Plus Island (Stranski-Krastanov) • Layer-by-layer initially favored followed by islanding • e.g., strained-layer epitaxy (above critical thickness/strain) From Ohring, Fig. 5-2, p. 197

  17. Growth Modes • Step-flow • Atoms migrate on surface terraces and stick preferentially to step edges

  18. Growth Modes • Vicinal surfaces From Tsao, Fig. 6.5, p. 208

  19. Surface Energy • g = Interfacial energy per unit area • = Work required to create unit area of interface DG = gdA = FdA • g = F • Surface energy (Jm-2) = surface tension (Nm-1) F = Force per unit length From Porter & Easterling, Fig. 3.1, p. 112

  20. Surface Energy • Broken bond model of surface energy From Porter & Easterling, Fig. 3.3, p. 114

  21. Surface Energy • e.g., (111) surface • 12 nearest neighbors in bulk • 3 missing neighbors at surface • e/2 energy per broken bond • Surface energy = 3e/2 per surface atom • Surface density, Ns • Surface energy, g ~ 3Nse/2 • ~ 1015 eVcm-2 • ~ 1 Jm-2 From Porter & Easterling, Fig. 3.2, p. 112

  22. Homogeneous Nucleation Change in G due to material formation: DG = VDGv + Avfgvf = (4/3 pr3) DGv + (4pr2) gvf Volume, V Surface Area, Avf Radius, r

  23. Homogeneous Nucleation • Trade-off between decrease in G due to volume formation but increase in G due to surface energy • dDG/dr = 0 • DG* = 16pgvf3/ 3(DGv)2 • r* = -2g/DGv From Ohring, Fig. 1-19, p. 42

  24. Homogeneous Nucleation • Critical nucleus, r* = - 2gvf / DGv • Clusters smaller than r* are unstable (DG > 0) and will evaporate (shrink by losing atoms) • Clusters larger than r* are stable (DG < 0) and will grow by accumulating atoms • DG* is the energy barrier to nucleation From Ohring, Fig. 1-19, p. 42

  25. Heterogeneous Nucleation Change in G due to island formation: DG = VDGv + Avfgvf- Afsgsv + Afsgfs Volume, V Surface Area, Avf Area of substrate covered = Afs

  26. Heterogeneous Nucleation • DG = VDGv + Avfgvf + Afsgfs- Afsgsv • = a3r3DGv + a1r2gvf + a2r2gfs – a2r2gsv • a1 = 2p ( 1 – cosq ) • a2 = p sin2q • a3 = p ( 2 – 3cosq + cos3q) / 3 From Ohring, Fig. 5-3, p. 199

  27. Heterogeneous Nucleation • Equilibrium among interfacial tensions : • gsv = gfs + gvf cosq (Young’s eqn.) From Ohring, Fig. 5-3, p. 199

  28. Heterogeneous Nucleation • gsv = gfs + gvf cosq (Young’s eqn.) • Island growth: • q > 0 • gsv < gfs + gvf From Ohring, Fig. 5-3, p. 199

  29. Heterogeneous Nucleation • gsv = gfs + gvf cosq (Young’s eqn.) • Layer growth: • q = 0 • gsv = gfs + gvf From Ohring, Fig. 5-3, p. 199

  30. Heterogeneous Nucleation • DG = VDGv + Avfgvf + Afsgfs- Afsgsv • = a3r3DGv + a1r2gvf + a2r2gfs – a2r2gsv • At equilibrium, dDG/dr = 0 • Gives maximum in DG(r) curve • Energy barrier • DG* = (16pgvf3 / 3DGv2) [ (2-3cosq + cos3q) / 4 ] • = (homogeneous barrier)*(shape factor) • 0 < shape factor < 0.5 • DG*(heterogeneous) < DG*(homogeneous) • Critical nucleus • r* = -2gvf / DGv • Same as homogeneous nucleation

  31. Heterogeneous Nucleation Critical nucleus, r* = - 2gvf / DGv Clusters smaller than r* are unstable and will evaporate (shrink by losing atoms) • Clusters larger than r* are stable and will grow by accumulating atoms From Ohring, Fig. 5-3, p. 199

  32. Growth Process 1. Growth Flux 2. Island Formation (Nucleation) 3. Island Growth and Coalescence

  33. Film Formation • Film formation occurs by nucleation, growth and coalescence of islands From Ohring, Fig. 5-1, p. 196

  34. Growth Process Number of islands, N* Many, small islands (N* large)  Amorphous or fine-grained polycrystalline film Few, large islands (N* small)  Single crystal or coarse-grained polycrystalline film

  35. Growth Process What is the effect of growth conditions? Temperature, T Growth Rate, R Change the rate of nuclei formation, dN*/dt

  36. dN*/dt ~ exp [ Edes – DG* – Edif) / kT)] • DG* is the energy barrier to nucleation Nucleation Barrier As T increases More adatoms have sufficient energy to overcome energy barrier to island formation (more nuclei are formed) From Ohring, Fig. 1-19, p. 42

  37. Diffusion Barrier dN*/dt ~ exp [ Edes – DG* – Edif) / kT)] Adatoms diffuse a distance X ~  Dt during the surface lifetime, t As T increases  Diffusion coefficient increases D = Doexp(-Edif/kT)  Adatom diffusion length is longer  Adatoms more likely to diffuse and be captured by existing island edges (fewer nuclei are formed)

  38. Desorption Barrier dN*/dt ~ exp [ Edes – DG* – Edif) / kT)] • As T increases • Surface lifetime is reduced due to desorption • t = (1/n) exp(Edes/kT) • Adatom diffusion length is shorter •  Adatoms less likely to diffuse and be captured by existing island edges (more nuclei are formed)

  39. Nucleation Rate • Higher growth rate and lower T • Smaller and greater number of nuclei • Polycrystalline or amorphous films • Lower growth rate and higher T • Larger and fewer nuclei • Large crystallites or monocrystalline films

  40. Nucleation Rate From Ohring, Fig. 5-13, p. 227

  41. Nucleation Rate From Ohring, Fig. 5-4, p. 205

  42. Atomistic Models • Surface dangling bonds energetically unfavorable • Surface atoms and bonds rearrange from their bulk positions to minimize surface energy • Results in surface reconstruction

  43. Atomistic Models • Surface reconstruction determined using combination of LEED, RHEED, STM From Ohring, Fig. 7-4, p. 312

  44. Atomistic Models • In 2-D, 5 unit meshes or nets are possible From Ohring, Fig. 7-5, p. 313

  45. Atomistic Models • Surface reconstruction phase diagram L. Daweritz and K. Ploog, Semicond. Sci. & Technol. 9 (1994) 123.

  46. Atomistic Models (2x4) III-V surface, e.g., GaAs at growth conditions (V/III > 1, 550 °C) Farrell et al, JVST B 5 (1987) 1482

  47. Atomistic Models • Electron counting model : • Bonding orbitals are completely filled (2 electrons) • Non-bonding (dangling) orbitals are either completely empty (above the Fermi level) or completely filled (below the Fermi level) Filled dangling orbital Farrell et al, JVST B 5 (1987) 1482

  48. Atomistic Models • Ga atom has three valence electrons • and three (sp2) orbitals From Oxtoby & Nachtrieb, Fig. 13-17, p. 477

  49. Atomistic Models • Ga inserted at As dangling bond has only one covalent bond and one half-filled dangling orbital Filled As dangling orbital Farrell et al, JVST B 5 (1987) 1482 From Oxtoby & Nachtrieb, Fig. 13-17, p. 477

  50. Atomistic Models • Formation of Ga dimers is more stable configuration (lower energy) Farrell et al, JVST B 5 (1987) 1482

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