Transcript Theory of Materials

PSU – Erie
Computational Materials Science
Properties of Point Defects in
Semiconductors
Dr. Blair R. Tuttle
Assistant Professor of Physics
Penn State University at Erie,
The Behrend College
2001
PSU – Erie
Computational Materials Science
2001
Outline
• Semiconductor review and motivation
• Point defect calculations using ab initio DFT
• Applications from recent research:
–
–
–
–
–
© Blair Tuttle 2001
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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Computational Materials Science
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Properties of solids
Semiconductors
Conductors
E
E
Insulators
E
Band Gap< 2 eV
Band Gap
> 2 eV
occupied
N
• Wires
© Blair Tuttle 2001
N
• Switches
N
• Barriers
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Computational Materials Science
2001
Silicon as prototype semiconductor
Tetrahedral Coordination
4 bonds per Si
Diamond Structure:
N
Semiconductor: Eg = 1.1 eV :
© Blair Tuttle 2001
E-Fermi
E
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Computational Materials Science
P-type
Boron
acceptors
-1
2001
Doping in c-Si
N-type
Phosphorous
donors
h+
+1
e-
N
© Blair Tuttle 2001
E
E
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Computational Materials Science
2001
Metal Oxide Semiconductor
Field Effect Transistor (MOSFET)
Source
© Blair Tuttle 2001
Gate
Drain
Lds ~ 90 nm
tox ~ 2.0 nm
Vsd ~ 2.0 V
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Computational Materials Science
2001
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 polyand 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)
© Blair Tuttle 2001
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Computational Materials Science
2001
Concentration of defects: H in Si
C  N sitese
 G form / kT
G form  E form  TS form  PVform
E form (q)  Etot (q)  N Si m Si  N H m H  qEF  EZP
• Etot = total energy for bulk cell with Nsi silicon
atoms and NH hydrogen atoms.
• mH , mSi = the chemical potential for hydrogen, Si
• The charge q and the Fermi energy (EF).
© Blair Tuttle 2001
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Computational Materials Science
2001
Acceptor and Donor levels for
atomic hydrogen in crystalline silicon
• Donor level is the Fermi Energy when:
E form (q  1)  E form (q  0)
• Calculate Eform for H at its local minima for
each charge state q = +1,0,-1
• Calculate valence band maximum to
compare charge states
© Blair Tuttle 2001
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Computational Materials Science
2001
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
© Blair Tuttle 2001
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PSU – Erie
Computational Materials Science
2001
Review of DFT
• Solve the Kohn-Sham equations:





  Veff (r ) i (r )   i i (r )


Veff (r )  Vext (r )  VHart [neff ]  Vex,cor [neff ]


2m
2
 2
  i (r )
N
neff
i
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”
© Blair Tuttle 2001
Review of Modern Physics vol. 64 pg. 1045 (1992).
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Choose {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).
© Blair Tuttle 2001
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Computational Materials Science
2001
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
© Blair Tuttle 2001
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Computational Materials Science
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Shortcomings 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
© Blair Tuttle 2001
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Computational Materials Science
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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
© Blair Tuttle 2001
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Computational Materials Science
2001
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:
i ,k (r )   Ci ,k G expi (k  G )  r 
G
EPW 

G

2m

2m
Gmax
2

k  G  GG  Veff (G  G) Ci ,k G   i Ci ,k G
2
Veff (G  G)  Vion  VHart  Vex,cor
© Blair Tuttle 2001
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Testing Convergence
• Convergence calculation
– Total energy for defect at minima
– Relative energies for defect in various positions
• Accuracy vs. Computational Cost
• Variables to converge
–
–
–
–
© Blair Tuttle 2001
Basis set size
Supercell size
Reciprocal space integration
Spin polarization (include or not)
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Computational Materials Science
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Convergence: Basis size
• Plane waves are a complete basis so crank up
the G vectors until convergence is reached.

EPW
c

Si
tot
BC
tot
E(


)  E (H )  E (
)
0.16
DE [eV]
0.14
DE ( eV )
0.12
0.1
0.08
0.06
0.04
0.02
0
10
© Blair Tuttle 2001
15
20
25
30
EPW [Ryd.]
35
40
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Computational Materials Science
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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
© Blair Tuttle 2001
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Computational Materials Science
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Convergence: 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
E (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
© Blair Tuttle 2001
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PSU – Erie
Computational Materials Science
2001
Convergence data at Epw = 15 Ryd.
N atoms
K points
8
5x5x5
E per Si (eV) E (eV) for
in c-Si
H+BC in c-Si
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
© Blair Tuttle 2001
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PSU – Erie
Computational Materials Science
2001
Bandstructure of 64 atom supercell
Bulk c-Si + H+BC
Bulk c-Si
3
3
VBM
CBM
2.5
2
2.5
2
1.5
1.5
1
1
0.5
0.5
0
0
1
2
3
4
5
6
G
L
7
8
VBM
Defect
CBM
9 10 11
X
1 2 3 4 5 6 7 8 9 10 11
L
G
•Bulk bands retained even with defect in calculation
© Blair Tuttle 2001
X
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Computational Materials Science
2001
Results for H in c-Si
EForm
Eglda
H+1
H0
H-1
0.5 eV
1.0 eV EFermi
• H0 and H+1 at global minimum
• H-1 at stationary point or saddle point
– Will lower its energy by moving to Td site
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).
© Blair Tuttle 2001
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Computational Materials Science
2001
Hydrogen in Silicon
Solid = LDA, Dashed =LDA + rigid scissor shift
E in eV
Exp.
LDA
© Blair Tuttle 2001
E(0,-)
0.51
0.46
E(+,0)
0.92
1.07
E(+,-)
0.72
0.77
U-corr
-0.41
-0.61
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Computational Materials Science
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H0 defect level chemistry
Si
3sp3
001
110
H 1s
•Defect level derived from Si-Si anti-bonding states
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).
© Blair Tuttle 2001
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Computational Materials Science
2001
Metal Oxide Semiconductor
Field Effect Transistor (MOSFET)
Source
© Blair Tuttle 2001
Gate
Drain
Lds ~ 90 nm
tox ~ 2.0 nm
Vsd ~ 2.0 V
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0
H
Computational Materials Science
2001
in silicon = paramagnetic defect
Si
3sp3
001
110
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. 4244 (1993).
© Blair Tuttle 2001
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Computational Materials Science
2001
Paramagnetic Defects
1.
2.
3.
4.
5.
Atomic Ho in c-Si
D center defects in a-Si
Pb centers at Si-SiO2 interfaces
E’ centers in SiO2
Atomic Ho in SiO2
© Blair Tuttle 2001
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Computational Materials Science
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Hyperfine parameters

I
e
H  S  A S
H sym  ( a  b) S  S  3bS S
e
a
b
2 mo
3
mo
4
I
I
z
e
z
g e m g I m  spin ( R)
e
I
3 cos   1
g e m g I m  d r  spin ( s )
3
2r
2
e
I
3
•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).
© Blair Tuttle 2001
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PSU – Erie
Computational Materials Science
2001
Hyperfine parameters for Sidb
Isotropic Parameters
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)
For more details see: B. Tuttle, “Hydrogen and Pb defects at the Si(111)-SiO2
interface” Phys. Rev. B vol. 60 pg. 2631 (1999).
© Blair Tuttle 2001
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PSU – Erie
Computational Materials Science
2001
H passivation of defects
• Binding energy for hydrogen passivation
– Related to the desorption energy
– Compare to vacuum annealing experiments
 EBind  [Etot (w / H )  Etot (no / H )]  Eref (H )
© Blair Tuttle 2001
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Computational Materials Science
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Atomic hydrogen in Silicon
001
Si
3sp3
H 1s
•
•
•
•
110E ~ 0.5 --1.1 eV
H0 min. energy at BC site,
B
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
© Blair Tuttle 2001
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Computational Materials Science
2001
H2 complexes in Silicon
• H2 min. at T site
• EB ~ 1.9 eV per H atom
• 0.6 eV less than free space
© Blair Tuttle 2001
• H2* along <111> direction
• EB ~ 1.6 eV per H atom
_
+
• H (BC) + H (T)
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Computational Materials Science
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H passivation of strained bonds
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)
© Blair Tuttle 2001
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PSU – Erie
Computational Materials Science
2001
Passivation of a 5-fold Si defect
Si-H Bond
Frustrated Bond
• 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
© Blair Tuttle 2001
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PSU – Erie
Computational Materials Science
2001
H passivation of dangling bonds
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
• Tuttle
EB 2001
~ 3.1 - 3.6 eV for pre-existing isolated Sidb in a-Si
© Blair
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Computational Materials Science
2001
Hydrogen in SiO2
• H0 favors open void
• EB ~ 0.1 eV
• Very little experimental
info on charge states
• Defect is paramagnetic
© Blair Tuttle 2001
• H2 free to rotate
• EB ~ 2.3 eV per H atom
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PSU – Erie
Computational Materials Science
2001
Binding Energy per H (eV)
0.0
H0 (free)
&
SiO2
1.0
2.0
3.0
H at pre-existing
isolated silicon
dangling bond (db)
H in c-Si
H2* in c-Si
H2 in c-Si
H2 (free)
&
SiO2
(Si-H H-Si) in a-Si
© Blair Tuttle 2001
4.0
H at pre-existing db
with Si-H in a cluster
e.g. a Si vacancy
H at pre-existing
“frustrated” Si bond
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Computational Materials Science
2001
Hydrogenated Amorphous Silicon
ln(DOS)
Egap ~1.8 eV
Energy
• Electronic Band Tails  Strained Si-Si bonds
• Intrinsic paramagnetic defects: [D] ~ 1016 cm-3
• 5-15 % Hydrogen  [H]~ 1021 cm-3
© Blair Tuttle 2001
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Computational Materials Science
2001
• [D] concentration thermally
activated with Ed ~ 0.3 eV
• Hydrogen diffusion thermally
activated Ea ~ 1.5 eV
1019
1018
1017
1019
1020
1021
H Evolved [cm-3]
• Hydrogen in (Si-H H-Si)
clusters evolves first
• Dilute Si-H bonds stronger
Spin Density [cm-3]
Spin Density [cm-3]
Si-H behavior in a-Si:H
1018
1017
1016
S. Zafar and A. Schiff, “Hydrogen and defects in amorphous
© Blair Tuttle 2001
silicon” Phys. Rev. Lett. Vol. 66 pg. 1493 (1991).
1.2
1.6
2.0
1000/T [ k-1]
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Computational Materials Science
2001
Modelling a-Si:H
• Simulated annealing
V
– Monte Carlo: bond switching
– Molecular Dynamics: add defects
• Compare results to experiments
© Blair Tuttle 2001
q
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Computational Materials Science
2001
Energy of H in a-Si
EB (eV)
0.0
1.0
Ea~1.5 eV
2.0
Ed~.3 eV
H
Clustered Si-H
H at frustrated bonds
3.0
Isolated Si-H bonds
4.0
B. Tuttle and J. B. Adams, “Ab initio study of H in amorphous silicon” Phys. Rev. B, vol. 57 pg. 12859 (1998).
© Blair Tuttle 2001
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PSU – Erie
Computational Materials Science
2001
Si-SiO2 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).
© Blair Tuttle 2001
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Computational Materials Science
2001
Si-H dissociation at Si-SiO2 interface
• Thermal vacuum annealing measurements
– [PB] versus time, pressure and temperature
– Data fit by first-order kinetics
– Rate limiting step: EB = 2.6 eV
H
ER
Sidb
[Si-H ]
EB=2.6 eV
SiO2
(Si-H)
Si
© Blair Tuttle 2001
K. Brower and Meyers, Appl. Phys. Lett. Vol. 57, pg. 162 (1990)..
[ Sidb + H ]
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Computational Materials Science
2001
Energy of H at Si(111)-SiO2 interface
EB (eV)
0.0
1.0
H in SiO2
H in Si
EB~2.6 eV
2.0
3.0
Isolated Si-H bonds
4.0
© Blair Tuttle 2001
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Si-H Desorption Paths
© Blair Tuttle 2001
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Computational Materials Science
2001
H2 passivation of Sidb (or Pb )
Thermal Annealing Experiments
H2
H
ER
EB=1.6 eV
SiO2
Pb
Si
1. Sidb
2. Sidb
(PbH)
[Pb + H2 ]
Possible Reactions
+ H2(SiO2) => Si-H + H(SiO2)
+ H2(SiO2) => Si-H + H(Si)
© Blair Tuttle 2001
[ (PbH) + H ]
Theory
ER = 1.0 eV
ER = 0.0 eV
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Computational Materials Science
2001
Path for H2 dissociation
and for H-D exchange
•Exchange of deeply trapped H and transport H is low ~ 0.2 eV
©B.Blair
Tuttle
Tuttle
and 2001
C. Van de Walle, “Exchange of deeply trapped and interstitial H in Si” Phys. Rev. B vol. 59 pg. 5493 (1999).48
PSU – Erie
Computational Materials Science
2001
H2 dissociation in SiO2
Thermal Annealing Experiments
H
H2
H
SiO2
ER
[H 2]
Si
Reactions
1. H2 (SiO2) => 2 H(SiO2)
2. H2 (SiO2) => 2 H(Si)
© Blair Tuttle 2001
EB= 4.1 eV
[H+H]
Theory
ER = 4.4 eV
ER = 2.4 eV
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Computational Materials Science
2001
H0 diffusion in SiO2
Y position (Ang.)
Energy Contours (0.1 eV)
X position (Ang.)
•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
© Blair Tuttle 2001
B. Tuttle, “Energetics and diffusion of hydrogen in SiO2” Phys. Rev. B vol. 61 pg. 4417 (2000).
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Computational Materials Science
2001
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
V(Q..)
Q(silicon coordination)
© Blair Tuttle 2001
1
4
6
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Computational Materials Science
2001
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)
© Blair Tuttle 2001
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