1. Theory of critical thickness estimation B89202009 彭成毅

2. Outline • SiGe Alloys • Pseudomorphic Growth and Film Relaxation • Theory of critical thickness estimation Models : Van der Merwe (1962) Matthhew – Blakeslee (1974) People – Bean (1985) Huang (2000)

3. SiGe Alloys • Si and Ge - Group Ⅳ elemental semiconductors - Diamond lattice structure • Vegard’s rule - a(Si1-XGeX)＝aSi＋x(a Ge－a Si) a – lattice constant x - Ge fraction

4. SiGe Allloys Unit cell of the diamond lattice Theoretical and experimental lattice constant of a Si1-xGex alloy as a function of Ge fraction

5. Pseudomorphic Growth and Film Relaxation • Lattice mismatch between Si (a＝5.431A) and Ge (a＝5.658A) - 4.17% at 300k • SiGe film on thick Si substrate - Initial growth - Pseudomorphic - SiGe film is forced to adopt Si smaller lattice constant - Desired result for most device application - Reach “critical thithiness” - Relax - Strain energy too large to maintain local equilibrium - SiGe film relaxes via misfit dislocation formation

6. Pseudomorphic Growth and Film Relaxation Schematic 2-D representation of both strained and relaxed SiGe on a Si substrate Schematic representation of misfit dislocation formed at the Si／SiGe growth interface

7. Theory of critical thickness estimation • The existence of critical thickness was first detailed by Vand der Merwe. 「1」 • Theoretically, many different models have been established to predict the critical thickness for strained layers. • The most celebrated ones are Matthhew – Blakeslee (1974) 「2」 People – Bean (1985) 「3」 Huang (2000) 「4」

8. Theory of critical thickness estimation • The models of Matthhew – Blakeslee • Principle : Minimize the total energy to get the thermal equilibrium state • Result : tc=5.5/x ln(tc)

9. Theory of critical thickness estimation • The models of People – Bean • Principle : The critical thickness is determined by the condition that the strain energy is equal to the minimum of dislocation energy.

10. Theory of critical thickness estimation • The dislocation energy is given by • The stress energy is

11. Theory of critical thickness estimation • Results

12. Theory of critical thickness estimation • The large lattice mismatch of about 4% between germanium and silicon has limited the growth of high-quality SiGe alloys to within a certain thickness, the so-called critical thickness, beyond which misfit dislocations start to generate. • To circumvent this limitation, a novel approach via substrate engineering (i.e., tailoring the substrate to form a finite dimension in the vertical and/or lateral directions) has been proposed to transfer or dilute the misfit strain.

13. Theory of critical thickness estimation • The dislocation energy in the case of an epilayer situated on a bulk substrate is • However the dislocation energy in the case of a compliant structure must be reconsidered.

14. Theory of critical thickness estimation • A compliant structure : • The dislocation is refined by the multiple image dislocation :

15. Theory of critical thickness estimation • where • When only the first-order image dislocations are considered, the total dislocation energy becomes

16. Theory of critical thickness estimation • For the elastic strain energy - in the case of a compliant structure where

17. Theory of critical thickness estimation • Using the PB model condition :

18. References • [1]J. H. Van der Merwe, J. Appl. Phys, 34, 123 (1962) • [2]J. W. Mattehews and A. E. Blakeslee, J. Cryst. Growth 27,118(1974) • [3]R. People and J. Bean, Appl. Phys. Lett. 47, 322 (1985) • [4]F. Y. Huang, Phys. Rev. Lett. 85, 787 (2000)