Strain analysis of nanostructures using Raman spectroscopy: theory and experiment

1. Strain analysis of nanostructures using Raman spectroscopy: theory and experiment José Menéndez Department of Physics Arizona State University Tempe, AZ 85287-USA

2. Just a second...

3. Mexico and us • 30,000 Spanish refugees from Spanish War • 75,000 Americans moved to Mexico fleeing McCarthysm • 50,000 Chileans fleeing dictator Pinochet granted asylum • 200,000 Salvadorans fleeing civil war granted Mexican residency • 225,000 Guatemalans fleeing counter-insurgency granted refuge

4. In March of 1938 Mexico was the only country which officially protested at the League of Nations for the violent anexation of Austria to Nazi Germany. In honor of this act the city of Vienna named this park Mexicoplatz. Mexicoplatz in Vienna

5. Mexican heroes Advisor: But shouldn’t we make a prior selection of refugees to be admitted into Mexico? Cárdenas: Those who fought in their countries for their legal governments cannot be offended with an interrogation. We must take them all. President Lázaro Cárdenas

6. Mexican heroes II Manuel Azaña, Spain’s last constitutional president, buried with the Mexican flag in France. “For us it represents an honor, for the republicans, hope, and for you [the French government], a painful lesson” Ambassador Luis Ignacio Rodríguez Taboada

7. Other countries had a different approach: Dealing with dictators Mexico never recognized Spain’s illegitimate government. It restored diplomatic relations with Spain in 1977, two years after the dictator’s death.

8. Semiconductor nanowires G. Liang et al., Nano Lett. 7, 642 (2007) • Dense packing • FET performance close to balistic limit • Wrap gates • Core-shell: manipulation of optical and electronic properties via index mismatch, confinement, and strain. J. Drucker

9. Acknowledgements Rachna Singh Eric Dailey Prashant Madras Jeff Drucker

10. “Seedless” nanowire growth • Grown in CVD chamber on < 1ML Au / Si (111) • Si wires (Si2H6) and Ge wires (Ge2H6) • Tunable diameter and density • <011> nanowire axis direction Eric Dailey and Jeff Drucker, J. App. Phys. 105, 064317 (2009)

11. Core-shell Ge-Si nanowires Ge NWs were grown at 300 °C with 10 mTorr Ge2H6. Si shells were deposited at 540 °C, 3 mTorr Si2H6.

12. Critical thickness I.A. Goldthorpe et al.,Nano Lett. v. 8, 4081 (2008)

13. Strain characterization with Raman spectroscopy • Simple (No synchrotron needed) • Non-destructive (unlike electron microscopy) • Good spatial resolution.

14. Raman scattering

15. Why is Raman affected by strain? Triple-degeneracy Anharmonic coefficients. Same symmetry as elastic tensor 3 phonon branches here!

16. Strain lifts the degeneracy... p = K11, q = K12, and r = K44 are anharmonic coefficients . E. Anastassakis, A. Pinczuk, E. Burstein, F. Pollak, and M. Cardona, Sol. St. Comm. 8, 133 (1970). F. Cerdeira, C.J. Buchenauer, F.H. Pollak, and M. Cardona, Phys. Rev. B 5, 580 (1972)

17. Raman selection rules

18. Raman tensor

19. Raman tensors Eigenvector of un-perturbed phonon j Eigenvector of perturbed phonon i Raman tensor of unperturbed phonon j Raman tensor of perturbed phonon i

20. Can we get the strain tensor from Raman measurements? • In principle possible. In practice very hard.G. Loechelt et al., APL 66, 3639 (1995), JAP 86, 6164 (1999). • In nanostructures virtually impossible due to antenna effects. G. Chen et al Nano Lett. 85, 1341 (2008)

21. Basic equilibrium equations

22. Strain in cylindrical coordinates Cartesian Cylindrical

23. Stress-strain relations in isotropic materials

24. The core-shell nanowire

25. Strain in core-shell nanowires

26. Equilibrium equations

27. Boundary conditions Same stress at interface No net cross-section force No stress @ outer surface Shrink-fit Shrink-fit

28. Pressing rings around shafts

29. Strain solution-core

30. Strain solution-shell

31. Core-shell strain

32. Optical phonons with strain Phonons are the solution of the eigenvalue problem: 1,2,3 are the crystal’s cubic cartesian coordinates

33. <011> oriented nanowires 2 x 2 problem can be easily diagonalized:

34. <111> oriented nanowires 3 x 3 problem can be easily diagonalized:

35. Predictions for Ge-Si <011>

36. Predictions for Ge-Si <111>

37. Experiment: <011> nanowires Eric Dailey and Jeff Drucker

38. Raman from pure Ge <011> nanowires • Diameters 10-30 nm. • Twice the Lorentzian linewidth as bulk Ge • 1.2-1.4 cm-1 downshift. Not heating!

39. Raman from <011> Ge-Si core-shell nanowires 45 nm Ge / 9.1 nm Si 11 nm Ge / 5 nm Si 44 nm Ge / 3,7 nm Si 11 nm Ge / 3.4 nm Si

40. Core spectrum interpretation • Broader, downshifted peak is Ge-Ge mode from Si-Ge interface. • Narrow, upshifted peak is from Ge-core. • Since no mode splitting obvious, peak is assigned to LL mode.

41. Theoretical vs. experimental upshifts

42. Critical thickness I.A. Goldthorpe et al.,Nano Lett. v. 8, 4081 (2008)

43. Conclusions core Raman spectrum • Predicted shifts VERY sensitive to shell thickness. • Observed shifts can be explained with 2 nm roughness/oxide and 40% strain relaxation. • Strain level much higher than expected from critical thickness theory.

44. Shell Raman spectrum Theory

45. Conclusions • Core -shell Raman spectra predicted and measured. • Observed strain larger than expected from equilibrium theory. • Good news! Trisilane?

46. Arizona welcomes you (really) Write to us: José Menéndezjose.menendez@asu.edu Jeff Drucker jeff.drucker@asu.edu Fernando Ponce fernando.ponce@asu.edu