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

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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

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

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

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

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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 (2000)

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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 ).

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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 (1996).

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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 (1992).

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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).

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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

PSU – ErieComputational Materials Science2001 © Blair Tuttle Shortcomings of DFT-LDA Poor when charge gradients vary significantly (better in GGA) –Atomic energies too low: E H = 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

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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:

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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)

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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]

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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 (2000) 2.

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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 2x2x x3x x4x x5x

PSU – ErieComputational Materials Science2001 © Blair Tuttle Convergence data at E pw = 15 Ryd. N atomsK pointsE per Si (eV) in c-Si  (eV) for H + BC in c-Si 85x5x x2x x3x x4x x2x N=64, Kpt=2x2x2 results converged to within 0.1 eV

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

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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).

PSU – ErieComputational Materials Science2001 © Blair Tuttle Hydrogen in Silicon E in eVE(0,-) E(+,0)E(+,-)U-corr Exp LDA Solid = LDA, Dashed =LDA + rigid scissor shift

PSU – ErieComputational Materials Science2001 © Blair Tuttle H 0 defect level chemistry Si 3sp 3 H 1s 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

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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

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

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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 (1993).

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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 (1999).

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

PSU – ErieComputational Materials Science2001 © Blair Tuttle Atomic hydrogen in Silicon Si 3sp 3 H 1s H 0 min. energy at BC site, E B ~ 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

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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 ~ eV for pre-existing isolated Si db in a-Si H passivation of dangling bonds

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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

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

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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 ] H Evolved [cm -3 ] S. Zafar and A. Schiff, “Hydrogen and defects in amorphous silicon” Phys. Rev. Lett. Vol. 66 pg (1991). Spin Density [cm -3 ] /T [ k -1 ] Hydrogen in (Si-H H-Si) clusters evolves first Dilute Si-H bonds stronger

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

PSU – ErieComputational Materials Science2001 © Blair Tuttle E B (eV) 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 (1998).

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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).

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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)..

PSU – ErieComputational Materials Science2001 © Blair Tuttle E B (eV) 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

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

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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 (1999).

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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

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

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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

PSU – ErieComputational Materials Science2001 © Blair Tuttle 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)