PSU – ErieComputational Materials Science2001 Properties of Point Defects in Semiconductors Dr....

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


© 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



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



Silicon as prototype semiconductorSilicon as prototype semiconductorTetrahedral Coordination

Semiconductor: Eg = 1.1 eV :

4 bonds per Si Diamond Structure:

E

NE-Fermi



Doping in c-SiDoping in c-SiP-typeBoronacceptors

-1 h+

E

N

N-typePhosphorousdonors

+1 e -

E



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




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)



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

)()(

/



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



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



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).



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).



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



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



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



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)(



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)



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]



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



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



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



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



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).



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



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



Metal Oxide Semiconductor Metal Oxide Semiconductor Field Effect Transistor (MOSFET)Field Effect Transistor (MOSFET)

Gate Gate SourceSource DrainDrain





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).



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



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).



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).



H passivation of defectsH passivation of defects

• Binding energy for hydrogen passivation– Related to the desorption energy– Compare to vacuum annealing experiments

)()]()([ // HEHEHEE refnototwtotBind



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



• H2 min. at T site

• EB ~ 1.9 eV per H atom

• 0.6 eV less than free space

• H2* along <111> direction


• H+ (BC) + H_(T)

HH22 complexes in Silicon complexes in Silicon



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



• 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



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



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




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



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



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



Modelling a-Si:HModelling a-Si:H

• Simulated annealing– Monte Carlo: bond switching

– Molecular Dynamics: add defects

• Compare results to experiments q

V



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).



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).



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)..



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



Si-H Desorption PathsSi-H Desorption Paths



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



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).



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



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)



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



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)

PSU – ErieComputational Materials Science2001 Properties of Point Defects in Semiconductors Dr....

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