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(Si 1-X Ge X ) ＝ a Si ＋ x(a Ge － a Si ) a – lattice constant x - Ge fraction

4. SiGe Allloys Unit cell of the diamond latticeTheoretical and experimental lattice constant of a Si 1-x Ge x 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)