1. PSU – ErieComputational Materials Science2001 Properties of Point Defects in Semiconductors Dr. Blair R. Tuttle Assistant Professor of Physics Penn State University at Erie, The Behrend College

2. PSU – ErieComputational Materials Science2001 © Blair Tuttle 20012 Outline Semiconductor review and motivation Point defect calculations using ab initio DFT Applications from recent research: –Donor and acceptor levels for atomic H in c-Si –Paramagnetic defects –Energies of H in Si environments –Hydrogen in amorphous silicon –Hydrogen at Si-SiO 2 interface

3. PSU – ErieComputational Materials Science2001 © Blair Tuttle 20013 Properties of solids E Band Gap< 2 eV N Band Gap > 2 eV N E E N WiresSwitches Barriers occupied Conductors Semiconductors Insulators

4. PSU – ErieComputational Materials Science2001 © Blair Tuttle 20014 Silicon as prototype semiconductor Tetrahedral Coordination Semiconductor: E g = 1.1 eV : 4 bonds per SiDiamond Structure: E N E-Fermi

5. PSU – ErieComputational Materials Science2001 © Blair Tuttle 20015 Doping in c-Si P-type Boron acceptors h+ E N N-type Phosphorous donors +1 e - E

6. PSU – ErieComputational Materials Science2001 © Blair Tuttle 20016 Metal Oxide Semiconductor Field Effect Transistor (MOSFET) Gate Source Drain L ds ~ 90 nm t ox ~ 2.0 nm V sd ~ 2.0 V L ds ~ 90 nm t ox ~ 2.0 nm V sd ~ 2.0 V

7. PSU – ErieComputational Materials Science2001 © Blair Tuttle 20017 Hydrogen in Silicon Systems Compensates both p-type and n-type doping Passivates dangling bonds at surfaces and interfaces Hydrogen related charge traps in MOSFETs Participates in metastable defect formation in poly- and amorphous silicon Forms very mobile H 2 molecules in bulk Si Forms large platelets used for cleaving silicon For more details see reference below and references therein: C. Van de Walle and B. Tuttle, “Theory of hydrogen in silicon devices” IEEE Transactions on Electron Devices, vol. 47 pg. 1779 (2000)

8. PSU – ErieComputational Materials Science2001 © Blair Tuttle 20018 Concentration of defects: H in Si E tot = total energy for bulk cell with N si silicon atoms and N H hydrogen atoms.    Si = the chemical potential for hydrogen, Si The charge q and the Fermi energy (E F ).

9. PSU – ErieComputational Materials Science2001 © Blair Tuttle 20019 Acceptor and Donor levels for atomic hydrogen in crystalline silicon Donor level is the Fermi Energy when: Calculate E form for H at its local minima for each charge state q = +1,0,-1 Calculate valence band maximum to compare charge states

10. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200110 Choose Method Semi-empirical –Tight binding (TB) –Classical Potentials Ab intio –Quantum Monte Carlo (QMC) –Hartree-Fock methods (HF) –Density Function Theory (DFT) For more details on a state-of-the-art implimentation of DFT: Kresse and Furthmuller,”Efficient iterative schemes for ab intio total-energy calculations using a plane wave basis set” Phys. Rev. B vol. 54 pg. 11169 (1996). http://cms.mpi.univie.ac.at/vasp/vasp/vasp.html

11. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200111 Review of DFT Solve the Kohn-Sham equations: For more details see review articles below: W. E. Pickett, “Pseudopotential methods in condensed matter applications” Computer Physics Reports, vol. 9 pg. 115 (1989). M. C. Payne et al. “Iterative minimization techniques for ab initio total-energy calculations” Review of Modern Physics vol. 64 pg. 1045 (1992).

12. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200112 Choose {ex, cor} functional Local density approximation (LDA) –Calculates exhange-correlation energy (E ex,cor ) based only on the local charge density –Rigorous for slowly varying charge density General gradient approximations (GGA) –Calculates E ex,cor using density and gradients –Improves many shortcoming of LDA For more details see reference below: Kurth, Perdew, and Blaha “Molecular and solid-state tests of density functional approximations: LSD, GGAs, and meta-GGAs” Int. J. of Quantum Chem. Vol. 75 pg. 889 ( 1999).

13. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200113 Results of DFT-LDA Bond lengths, lattice constants ~ 1 – 5 % (low) Binding and cohesive energies ~ 10 % (high) Vibrational frequencies ~ 5 – 10 % (low) Valence bands good –valence band offsets for semiconductors Wavefunctions good –Hyperfine parameters

14. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200114 Shortcomings of DFT-LDA Poor when charge gradients vary significantly (better in GGA) –Atomic energies too low: E H = -13.0 eV –Barriers to molecular dissociation often low, Example: H + H 2 = H 3 –Energy of Phases, Ex: Stishovite vs Quartz Semiconductor band gaps poor ~ 50 % low

15. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200115 Choose boundary conditions Cluster models (20 – 1000 atoms) –Defect-surface interactions –Passivate cluster surface with hydrogen –Wavefunctions localized Periodic supercell (20 – 1000 atoms) –Defect-defect interactions –Wavefunctions de-localized –Bands well defined

16. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200116 Choose basis for wavefunctions Localized pseudo-atomic orbitals –Efficient but not easy to use or improve results Plane Waves –Easy to use and improve results:

17. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200117 Testing Convergence Convergence calculation –Total energy for defect at minima –Relative energies for defect in various positions Accuracy vs. Computational Cost Variables to converge –Basis set size –Supercell size –Reciprocal space integration –Spin polarization (include or not)

18. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200118 Convergence: Basis size Plane waves are a complete basis so crank up the G vectors until convergence is reached. E PW [Ryd.] D E [eV]

19. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200119 Convergence: Supercell size Prevent defect-defect interactions. –Electronic localization of defect level as determined by k-point integration –Steric relaxations: di-vancancy in silicon –Coulombic interaction of charged defects For more details see reference below and references therein: 1. C. Van de Walle and B. Tuttle, “Theory of hydrogen in silicon devices” IEEE Transactions on Electron Devices, vol. 47 pg. 1779 (2000) 2. http://cms.mpi.univie.ac.at/vasp/vasp/vasp.html

20. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200120 Convergence: k-point sampling Reciprocal space integration –For each supercell size, converge the number of “special” k-points –Data for 8 atom supercell: K pointsE per Si (eV) for c-Si  (eV) for H + BC in c-Si 2x2x25.88267.581 3x3x35.95497.514 4x4x45.96917.485 5x5x55.97057.484

21. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200121 Convergence data at E pw = 15 Ryd. N atomsK pointsE per Si (eV) in c-Si  (eV) for H + BC in c-Si 85x5x55.97057.484 642x2x25.96937.311 643x3x35.97007.309 644x4x45.97117.308 2162x2x25.97087.240 N=64, Kpt=2x2x2 results converged to within 0.1 eV

22. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200122 Bandstructure of 64 atom supercell Bulk c-Si Bulk c-Si + H + BC L G X Bulk bands retained even with defect in calculation

23. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200123 Results for H in c-Si H 0 and H +1 at global minimum H -1 at stationary point or saddle point –Will lower its energy by moving to Td site H -1 H +1 H0H0 E Fermi E Form 0.5 eV 1.0 eV E g lda For more info see: C. G. Van de Walle, “Hydrogen in crystalline semiconductors” Deep Centers I Semiconductors, Ed. by S. T. Pantelides, pg. 899 (1992).

24. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200124 Hydrogen in Silicon E in eVE(0,-) E(+,0)E(+,-)U-corr Exp. 0.51 0.92 0.72-0.41 LDA0.46 1.07 0.77-0.61 Solid = LDA, Dashed =LDA + rigid scissor shift

25. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200125 H 0 defect level chemistry Si 3sp 3 H 1s 110 001 For more info see: C. G. Van de Walle, “Hydrogen in crystalline semiconductors” Deep Centers I Semiconductors, Ed. by S. T. Pantelides, pg. 899 (1992). Defect level derived from Si-Si anti-bonding states

26. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200126 Metal Oxide Semiconductor Field Effect Transistor (MOSFET) Gate Source Drain L ds ~ 90 nm t ox ~ 2.0 nm V sd ~ 2.0 V L ds ~ 90 nm t ox ~ 2.0 nm V sd ~ 2.0 V

27. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200127 H 0 in silicon = paramagnetic defect Si 3sp 3 H 1s 110 001 For more info see: C. G. Van de Walle and P. Blochl, “First principles calculations of hyperfine parameters” Phys. Rev. B vol. 47 pg. 4244 (1993).

28. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200128 Paramagnetic Defects 1.Atomic H o in c-Si 2.D center defects in a-Si 3.P b centers at Si-SiO 2 interfaces 4.E ’ centers in SiO 2 5.Atomic H o in SiO 2

29. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200129 Hyperfine parameters All electron wavefunctions are needed !!!! For more info see: C. G. Van de Walle and P. Blochl, “First principles calculations of hyperfine parameters” Phys. Rev. B vol. 47 pg. 4244 (1993).

30. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200130 Hyperfine parameters for Si db Isotropic Parameters For more details see: B. Tuttle, “Hydrogen and P b defects at the Si(111)-SiO 2 interface” Phys. Rev. B vol. 60 pg. 2631 (1999).

31. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200131 H passivation of defects Binding energy for hydrogen passivation –Related to the desorption energy –Compare to vacuum annealing experiments

32. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200132 Atomic hydrogen in Silicon Si 3sp 3 H 1s H 0 min. energy at BC site, E B ~ 0.5 --1.1 eV In disordered Si, strain lowers E B ~ 0.25 eV per 0.1 Ang H + (BC) and H - (T): Negative U impurity Neutral hydrogen in Si is a paramagnetic defect 110 001

33. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200133 H 2 min. at T site E B ~ 1.9 eV per H atom 0.6 eV less than free space H 2 * along direction E B ~ 1.6 eV per H atom H + (BC) + H _ (T) H 2 complexes in Silicon

34. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200134 Si 3sp 3 2 H 1s 2 (Si-H) Hydrogen atoms remove electronic band tail states in a-Si E B ~ 2.3 eV per H atom (roughly the same as H 2 in free space) Negative U complex (equilibrium state includes only 0 or 2 H) H passivation of strained bonds

35. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200135 5-fold Si defects are paramagnetic: D center in a-Si & P b center at Si-SiO 2 interface E B ~ 2.45 eV per H for Si-H at Si-interstitials in c-Si E B ~ 2.55 eV per H for Si-H at a 5-fold defect in a-Si Si-H BondFrustrated Bond Passivation of a 5-fold Si defect

36. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200136 Si 3sp 3 H 1s Si dangling bonds paramagnetic E B ~ 4.1 eV for H-SiH 3 E B ~ 3.6 eV for pre-existing isolated Si db in c-Si E B ~ 3.1 - 3.6 eV for pre-existing isolated Si db in a-Si H passivation of dangling bonds

37. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200137 Hydrogen in SiO 2 H 0 favors open void E B ~ 0.1 eV Very little experimental info on charge states Defect is paramagnetic H 2 free to rotate E B ~ 2.3 eV per H atom

38. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200138 0.0 1.0 2.0 3.0 4.0 Binding Energy per H (eV) H 0 (free) & SiO 2 H in c-Si H 2 in c-Si H 2 * in c-Si (Si-H H-Si) in a-Si H 2 (free) & SiO 2 H at pre-existing isolated silicon dangling bond (db) H at pre-existing “frustrated” Si bond H at pre-existing db with Si-H in a cluster e.g. a Si vacancy

39. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200139 Hydrogenated Amorphous Silicon Electronic Band Tails  Strained Si-Si bonds Intrinsic paramagnetic defects: [D] ~ 10 16 cm -3 5-15 % Hydrogen  [H]~ 10 21 cm -3 E gap ~1.8 eV ln(DOS) Energy

40. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200140 Si-H behavior in a-Si:H [D] concentration thermally activated with E d ~ 0.3 eV Hydrogen diffusion thermally activated E a ~ 1.5 eV Spin Density [cm -3 ] 10 19 10 20 10 21 H Evolved [cm -3 ] 10 17 10 18 10 19 S. Zafar and A. Schiff, “Hydrogen and defects in amorphous silicon” Phys. Rev. Lett. Vol. 66 pg. 1493 (1991). Spin Density [cm -3 ] 1.2 1.6 2.0 1000/T [ k -1 ] 10 16 10 17 10 18 Hydrogen in (Si-H H-Si) clusters evolves first Dilute Si-H bonds stronger

41. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200141 Modelling a-Si:H Simulated annealing –Monte Carlo: bond switching –Molecular Dynamics: add defects Compare results to experiments q V

42. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200142 E B (eV) 0.0 1.0 2.0 3.0 4.0 Clustered Si-H H at frustrated bonds Isolated Si-H bonds Energy of H in a-Si H E a ~1.5 eV E d ~.3 eV B. Tuttle and J. B. Adams, “Ab initio study of H in amorphous silicon” Phys. Rev. B, vol. 57 pg. 12859 (1998).

43. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200143 Si-SiO 2 Interface M. Staedele, B. R. Tuttle and K. Hess, 'Tunneling through unltrathin SiO 2 gate oxide from microscopic models', J. Appl.Phys. {\bf 89}, 348 (2001).

44. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200144 Si-H dissociation at Si-SiO 2 interface Thermal vacuum annealing measurements –[P B ] versus time, pressure and temperature –Data fit by first-order kinetics –Rate limiting step: E B = 2.6 eV (Si-H) Si db H SiO 2 Si E B =2.6 eV [Si-H ] [ Si db + H ] E R K. Brower and Meyers, Appl. Phys. Lett. Vol. 57, pg. 162 (1990)..

45. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200145 E B (eV) 0.0 1.0 2.0 3.0 4.0 Isolated Si-H bonds Energy of H at Si(111)-SiO 2 interface H in Si E B ~2.6 eV H in SiO 2

46. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200146 Si-H Desorption Paths

47. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200147 P b H2H2 (P b H) H SiO 2 Si E B =1.6 eV [P b + H 2 ] [ (P b H) + H ] Possible ReactionsTheory 1. Si db + H 2 (SiO 2 ) => Si-H + H(SiO 2 ) E R = 1.0 eV 2. Si db + H 2 (SiO 2 ) => Si-H + H(Si) E R = 0.0 eV E R H 2 passivation of Si db (or P b ) Thermal Annealing Experiments

48. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200148 Path for H 2 dissociation and for H-D exchange Exchange of deeply trapped H and transport H is low ~ 0.2 eV B. Tuttle and C. Van de Walle, “Exchange of deeply trapped and interstitial H in Si” Phys. Rev. B vol. 59 pg. 5493 (1999).

49. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200149 H 2 dissociation in SiO 2 H 2 dissociation in SiO 2 H2H2 SiO 2 Si E B = 4.1 eV [H 2 ] [ H + H ] Reactions Theory 1. H 2 (SiO 2 ) => 2 H(SiO 2 ) E R = 4.4 eV 2. H 2 (SiO 2 ) => 2 H(Si) E R = 2.4 eV E R H H Thermal Annealing Experiments

50. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200150 H 0 diffusion in SiO 2 Experiments  E a = 0.05 – 1.0 eV Classical Potentials  E a = 0.6 -- 0.9 eV LDA & CTS Theory: E a = 0.2 eV D o = 8.1x 10 -4 cm 2 /sec B. Tuttle, “Energetics and diffusion of hydrogen in SiO 2 ” Phys. Rev. B vol. 61 pg. 4417 (2000). X position (Ang.) Y position (Ang.) Energy Contours (0.1 eV)

51. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200151 Good Classical Potentials Need insight into chemical processes Force Matching Method –J. B. Adams et al. (1990s) –Fit cubic spline potentials to a database of high level ab initio calculations Q(silicon coordination) V(Q..) 1 4 6

52. PSU – ErieComputational Materials Science2001 © Blair Tuttle 200152 Summary Computational methods based on DFT have been widely applied to important problems in materials science including point defects in semiconductors. DFT methods provide a powerful tool for calculating properties of interest including: –Static properties (potential energy surfaces, formation energies, donor/acceptor levels) –Dynamical properties (vibrational frequencies, diffusivities) –Electrical and structural properties (defect levels, defect localization, hyperfine parameters)