PSU – Erie Computational Materials Science 2001
Properties of Point Defects in Properties of Point Defects in SemiconductorsSemiconductors
Dr. Blair R. Tuttle
Assistant Professor of Physics
Penn State University at Erie,The Behrend College
PSU – Erie Computational Materials Science 2001
© Blair Tuttle 2001 2
OutlineOutline
• 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-SiO2 interface
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Properties of solidsProperties of solids
E
Band Gap< 2 eV
N
Band Gap > 2 eV
N
EE
N
• Wires • Switches • Barriers
occupied
Conductors Semiconductors Insulators
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Silicon as prototype semiconductorSilicon as prototype semiconductorTetrahedral Coordination
Semiconductor: Eg = 1.1 eV :
4 bonds per Si Diamond Structure:
E
NE-Fermi
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Doping in c-SiDoping in c-SiP-typeBoronacceptors
-1 h+
E
N
N-typePhosphorousdonors
+1 e -
E
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Metal Oxide Semiconductor Metal Oxide Semiconductor Field Effect Transistor (MOSFET)Field Effect Transistor (MOSFET)
Gate Gate SourceSource DrainDrain
Lds ~ 90 nmtox ~ 2.0 nmVsd ~ 2.0 V
Lds ~ 90 nmtox ~ 2.0 nmVsd ~ 2.0 V
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Hydrogen in Silicon SystemsHydrogen 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 H2 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)
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Concentration of defects: H in SiConcentration of defects: H in Si
• Etot = total energy for bulk cell with Nsi silicon atoms and NH hydrogen atoms.
• Si = the chemical potential for hydrogen, Si
• The charge q and the Fermi energy (EF).
ZPFHHSiSitotform
formformformform
kTGsites
EqENNqEqE
PVTSEG
eNC form
)()(
/
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Acceptor and Donor levels for Acceptor and Donor levels for atomic hydrogen in crystalline siliconatomic hydrogen in crystalline silicon
• Donor level is the Fermi Energy when:
• Calculate Eform for H at its local minima for each charge state q = +1,0,-1
• Calculate valence band maximum to compare charge states
)()( 01 qq formform EE
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Choose MethodChoose 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
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Review of DFTReview of DFT
• Solve the Kohn-Sham equations:
N
iieff
effcorexeffHartexteff
iiieffm
rn
nVnVrVrV
rrrV
2
,
22
)(
][][)()(
)()()(
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).
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Choose {ex, cor} functionalChoose {ex, cor} functional
• Local density approximation (LDA) – Calculates exhange-correlation energy (Eex,cor)
based only on the local charge density– Rigorous for slowly varying charge density
• General gradient approximations (GGA)– Calculates Eex,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).
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Results of DFT-LDAResults 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
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Shortcomings of DFT-LDAShortcomings of DFT-LDA
• Poor when charge gradients vary significantly (better in GGA)
– Atomic energies too low: EH = -13.0 eV
– Barriers to molecular dissociation often low, Example: H + H2 = H3
– Energy of Phases, Ex: Stishovite vs Quartz
• Semiconductor band gaps poor ~ 50 % low
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Choose boundary conditionsChoose 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
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Choose basis for wavefunctionsChoose basis for wavefunctions• Localized pseudo-atomic orbitals
– Efficient but not easy to use or improve results
• Plane Waves– Easy to use and improve results:
corexHartioneff
GkiG
iGkieffGGm
mPW
GGkiki
VVVGGV
CCGGVGk
GE
rGkiCr
,
,,
2
2
2
max2
,,
)(
)(
)(exp)(
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Testing Convergence 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)
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Convergence: Basis sizeConvergence: Basis size• Plane waves are a complete basis so crank up
the G vectors until convergence is reached.
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
10 15 20 25 30 35 40
DE ( eV )
)()()( SictotBCtotPW EHEE E
EPW [Ryd.]
E [
eV]
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Convergence: Supercell sizeConvergence: 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
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Convergence: k-point samplingConvergence: k-point sampling• Reciprocal space integration
– For each supercell size, converge the number of “special” k-points
– Data for 8 atom supercell:
K points E per Si (eV) for c-Si
(eV) for H+
BC in c-Si
2x2x2 5.8826 7.581
3x3x3 5.9549 7.514
4x4x4 5.9691 7.485
5x5x5 5.9705 7.484
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Convergence data at EConvergence data at Epwpw = 15 Ryd. = 15 Ryd.
N atoms K points E per Si (eV) in c-Si
(eV) for H+
BC in c-Si
8 5x5x5 5.9705 7.484
64 2x2x2 5.9693 7.311
64 3x3x3 5.9700 7.309
64 4x4x4 5.9711 7.308
216 2x2x2 5.9708 7.240
•N=64, Kpt=2x2x2 results converged to within 0.1 eV
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Bandstructure of 64 atom supercellBandstructure of 64 atom supercell
0
0.5
1
1.5
2
2.5
3
1 2 3 4 5 6 7 8 9 10 11
VBM
CBM
0
0.5
1
1.5
2
2.5
3
1 2 3 4 5 6 7 8 9 10 11
VBM
Defect
CBM
Bulk c-Si Bulk c-Si + H+BC
L X L X
•Bulk bands retained even with defect in calculation
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Results for H in c-SiResults for H in c-Si
• H0 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
H0
EFermi
EForm
0.5 eV 1.0 eV
Eglda
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).
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Hydrogen in SiliconHydrogen in Silicon
E in eV E(0,-) E(+,0) E(+,-) U-corrExp. 0.51 0.92 0.72 -0.41LDA 0.46 1.07 0.77 -0.61
Solid = LDA, Dashed =LDA + rigid scissor shift
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HH00 defect level chemistry defect level chemistry
Si 3sp3
H 1s110
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
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Metal Oxide Semiconductor Metal Oxide Semiconductor Field Effect Transistor (MOSFET)Field Effect Transistor (MOSFET)
Gate Gate SourceSource DrainDrain
Lds ~ 90 nmtox ~ 2.0 nmVsd ~ 2.0 V
Lds ~ 90 nmtox ~ 2.0 nmVsd ~ 2.0 V
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HH00 in silicon = paramagnetic defect in silicon = paramagnetic defect
Si 3sp3
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).
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Paramagnetic DefectsParamagnetic Defects
1. Atomic Ho in c-Si
2. D center defects in a-Si
3. Pb centers at Si-SiO2 interfaces
4. E’ centers in SiO2
5. Atomic Ho in SiO2
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Hyperfine parametersHyperfine parameters
3
23
4
3
2
2
1cos3)(
)(
3)(
rsrdggb
Rgga
SSSSH
SASH
spinI
Ie
eo
spinI
Ie
eo
ez
Iz
Iesym
eI
bba
�
•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).
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Defect Exp. Tuttle (LDA-DPZ)
SiH3 in gas 190 173 (09 % low)
Pb at the Si(111)-SiO2
110 99 (10 % low)
D in a-Si 75
99 (32 % high)
Hyperfine parameters for SiHyperfine parameters for Sidbdb
Isotropic Parameters
For more details see: B. Tuttle, “Hydrogen and Pb defects at the Si(111)-SiO2
interface” Phys. Rev. B vol. 60 pg. 2631 (1999).
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H passivation of defectsH passivation of defects
• Binding energy for hydrogen passivation– Related to the desorption energy– Compare to vacuum annealing experiments
)()]()([ // HEHEHEE refnototwtotBind
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Atomic hydrogen in SiliconAtomic hydrogen in Silicon
Si 3sp3
H 1s
• H0 min. energy at BC site, EB ~ 0.5 --1.1 eV
• In disordered Si, strain lowers EB ~ 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
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• H2 min. at T site
• EB ~ 1.9 eV per H atom
• 0.6 eV less than free space
• H2* along <111> direction
• EB ~ 1.6 eV per H atom
• H+ (BC) + H_(T)
HH22 complexes in Silicon complexes in Silicon
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Si 3sp3
2 H 1s
2 (Si-H)
• Hydrogen atoms remove electronic band tail states in a-Si
• EB ~ 2.3 eV per H atom (roughly the same as H2 in free space)
• Negative U complex (equilibrium state includes only 0 or 2 H)
H passivation of strained bondsH passivation of strained bonds
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• 5-fold Si defects are paramagnetic:
• D center in a-Si & Pb center at Si-SiO2 interface
• EB ~ 2.45 eV per H for Si-H at Si-interstitials in c-Si
• EB ~ 2.55 eV per H for Si-H at a 5-fold defect in a-Si
Si-H Bond Frustrated Bond
Passivation of a 5-fold Si defect Passivation of a 5-fold Si defect
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Si 3sp3
H 1s
• Si dangling bonds paramagnetic
• EB ~ 4.1 eV for H-SiH3
• EB ~ 3.6 eV for pre-existing isolated Sidb in c-Si
• EB ~ 3.1 - 3.6 eV for pre-existing isolated Sidb in a-Si
H passivation of dangling bondsH passivation of dangling bonds
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Hydrogen in SiOHydrogen in SiO22
• H0 favors open void
• EB ~ 0.1 eV
• Very little experimental info on charge states
• Defect is paramagnetic
• H2 free to rotate
• EB ~ 2.3 eV per H atom
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0.0 1.0 2.0 3.0 4.0
Binding Energy per H (eV)
H0 (free)&SiO2
H in c-Si
H2 in c-Si
H2* in c-Si
(Si-H H-Si) in a-Si
H2 (free)&SiO2
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
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Hydrogenated Amorphous SiliconHydrogenated Amorphous Silicon
• Electronic Band Tails Strained Si-Si bonds
• Intrinsic paramagnetic defects: [D] ~ 1016 cm-3
• 5-15 % Hydrogen [H]~ 1021 cm-3
Egap ~1.8 eVln(DOS)
Energy
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Si-H behavior in a-Si:HSi-H behavior in a-Si:H• [D] concentration thermally
activated with Ed ~ 0.3 eV
• Hydrogen diffusion thermally activated Ea ~ 1.5 eV
Sp
in D
ensi
ty [
cm-3]
1019 1020 1021
H Evolved [cm-3]
1017
1018
1019
S. Zafar and A. Schiff, “Hydrogen and defects in amorphous silicon” Phys. Rev. Lett. Vol. 66 pg. 1493 (1991).
Sp
in D
ensi
ty [
cm-3]
1.2 1.6 2.01000/T [ k-1]
1016
1017
1018
• Hydrogen in (Si-H H-Si) clusters evolves first
• Dilute Si-H bonds stronger
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Modelling a-Si:HModelling a-Si:H
• Simulated annealing– Monte Carlo: bond switching
– Molecular Dynamics: add defects
• Compare results to experiments q
V
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EB (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-SiEnergy of H in a-Si
HEa~1.5 eV
Ed~.3 eV
B. Tuttle and J. B. Adams, “Ab initio study of H in amorphous silicon” Phys. Rev. B, vol. 57 pg. 12859 (1998).
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Si-SiOSi-SiO22 Interface Interface
M. Staedele, B. R. Tuttle and K. Hess, 'Tunneling through unltrathin SiO2 gate oxide from microscopic models', J. Appl.Phys. {\bf 89}, 348 (2001).
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Si-H dissociation at Si-SiOSi-H dissociation at Si-SiO22 interface interface• Thermal vacuum annealing measurements
– [PB] versus time, pressure and temperature
– Data fit by first-order kinetics
– Rate limiting step: EB = 2.6 eV
(Si-H) Sidb
H
SiO2
Si
EB=2.6 eV
[Si-H ] [ Sidb + H ]
ER
K. Brower and Meyers, Appl. Phys. Lett. Vol. 57, pg. 162 (1990)..
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EB (eV)0.0
1.0
2.0
3.0
4.0
Isolated Si-H bonds
Energy of H at Si(111)-SiOEnergy of H at Si(111)-SiO2 2 interfaceinterface
H in SiEB~2.6 eV
H in SiO2
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Si-H Desorption PathsSi-H Desorption Paths
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Pb
H2
(PbH)
H
SiO2
Si
EB=1.6 eV
[Pb + H2 ] [ (PbH) + H ]
Possible Reactions Theory 1. Sidb + H2(SiO2) => Si-H + H(SiO2) ER = 1.0 eV 2. Sidb + H2(SiO2) => Si-H + H(Si) ER = 0.0 eV
ER
HH22 passivation of Si passivation of Sidb db (or P(or Pb b ))
Thermal Annealing Experiments
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Path for HPath for H2 2 dissociation dissociation
and for H-D exchangeand 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).
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HH22 dissociation in SiO dissociation in SiO22
H2
SiO2
Si
EB= 4.1 eV
[H 2] [ H + H ]
Reactions Theory 1. H2 (SiO2) => 2 H(SiO2) ER = 4.4 eV 2. H2 (SiO2) => 2 H(Si) ER = 2.4 eV
ER H
H
Thermal Annealing Experiments
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HH00 diffusion in SiO diffusion in SiO22
•Experiments Ea = 0.05 – 1.0 eV
•Classical Potentials Ea = 0.6 -- 0.9 eV
•LDA & CTS Theory:
• Ea = 0.2 eV
• Do = 8.1x 10-4 cm2/sec
B. Tuttle, “Energetics and diffusion of hydrogen in SiO2” Phys. Rev. B vol. 61 pg. 4417 (2000).
X position (Ang.)
Y p
osit
ion
(Ang
.)
Energy Contours (0.1 eV)
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Good Classical PotentialsGood 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
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Summary 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)
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