Crystal Defects

Crystal Defects
Peter Rudolph
Crystal Technology Consultation (CTC)
Helga-Hahnemann-Str. 57, D-12529 Schönefeld
[email protected]
15th International Summer School on Crytsal Growth – ISSCG-15
Abstract
The quality of crystals is very sensitively influenced by structural and
atomistic deficiencies generated during crystal growth. Such imperfections comprise point defects, impurity and dopant inhomogeneities,
dislocations, grain boundaries, second-phase particles, twins. While
point defects are in thermodynamic equilibrium and, therefore, always
presented all another types of imperfections are in non-equilibrium and,
thus, in principle preventable. However, for that nearly ideal, mostly
unprofitable growth conditions are required. Additionally, each growing
crystal exhibits a propagating fluid-solid interface showing distinct
phase boundary characteristics. Such facts do not allow to obtain totally
perfect crystals. In praxi, only optimal crystals are achievable.
Today, most of defect-forming mechanisms have become well understood. There exists an enormous knowledge about the defect genesis and control supported by proper theoretical fundamentals and
technological know how. However, there are still problems to be solved,
especially for new high-temperature, high-dissociative substances and
epitaxial sequences. It is the aim of present lecture to combine defect
fundamentals with suggestions for improved defect engineering.
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Outline
1.
2.
Introduction - defect classification
Point defects
2.1 Native point defects
2.2 Extrinsic point defects
2.3 Segregation phenomena
3.
Dislocations
3.1
3.2
3.3
3.4
4.
Dislocation types and analysis
Dislocation dynamics
Low-angle grain boundaries - substructuring
Dislocation engineering
Second-phase particles
4.1 Precipitates
4.2 Inclusions
5.
6.
7.
Faceting
Twinning
Summary and outlook
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1. Introduction
a – interstitial impurity atom
c – self interstitial atom
e – precipitate of impurity atoms
g – interstitial type dislocation loop
Defect types
b – edge dislocation
d – vacancy
f – vacancy type dislocation loop
h – substitutional impurity atom
after H. Föll: http://www.tf.uni-kiel.de/matwis/amat/
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1. Introduction
Defect classification
Structural crystal defects are classified according to their dimensions.
in thermodynamic
equilibrium
0-dimensional defects
1-dimensional defects
in thermodynamic
nonequilibrium
2-dimensional defects
3-dimensional defects
15th International Summer School on Crystal Growth – ISSCG-15
atomic size („point“) defects
intrinsic (vacancies, interstitials)
and extrinsic (dopants) defects
dislocations
(edge, screw, 60°, 30°, mixed,
mobile, sessile, bunched, ordered...)
stacking faults, twins
grain and phase boundaries,
facets ? (expressing perfection !)
precipitates, inclusions,
voids (vacancy agglomerates),
bubbles, dislocation clusters
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1. Introduction
Point defects
Defect diagnostics
Dislocations
Grain boundaries
Inclusions
Hähnert, Rudolph 1993
Gebauer 2000
Fujiwara 2006
Schröder 1967
- unoccupied state -
- Dash necking -
- casting -
Scanning Tunneling
Microscopy (STM)
X-ray diffraction
(Lang) topography
Photo image;
Electron Back
Scattering (EBS)
Laser Scattering
Tomography (LST);
Transmission Electron
Microscopy (TEM)
(110) (1x1) GaAs
(110) FZ Si
PV Si
(100) VB CdTe
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- nonstoichiometry
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2. Point defects
Intrinsic defect minimum
2.1 Native point defects
interstitial
vacancy
antisite
AB compound
A certain point defect content
increases the entropy and, hence,
decreases the Gibbs potential !
G
N  n*
 Ed  kT ln
0
n
n*
G  H d  S d T
Hd = n Ed - defect enthalpy ( n - number of defects)
Sd = k lnW - configurational entropy, W = N ! /n !(N-n) !
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2. Point defects
Diffusivity and non-stoichiometry
2.1 Native point defects
stoich.
x
n* = N exp (- Ed / kT)
Ed = Eform + Evib + ES
Existence region of a compound
deviation from stoichiometry (netto defects in each sublattice):
x = A - B = (CiA - CvA + 2CA/B- 2CB/A )
formation, vibration, configuration
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– (CiB - CvB + 2CB/A- 2CA/B )
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2. Point defects
Segregation and condensation
2.1 Native point defects
stoich. growth
Tcong
non-stoich. growth
cmp
rejected
excess
component
(B)
segregation
IF
*
*
*
homogeneous
precipitation
diffusion area
~ 100 nm
dislocation
heterogeneous
XB
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2. Point defects
2.1 Native point defects
During crystal growth from melt
the native point defects undergo
various types of transport kinetics such as capture at the interface and diffusion by jumping via
interstitials and vacancies. Variations of the growth rate shifts
the point defect transport between incorporation and diffusion
dominated. Whereas in dislocation-free silicon crystals at high
rates the flux of vacancies dominates that of self-interstitials at
low rates or high temperature
gradients interstitials are in excess. This fact is of high significance for in situ control of native point defect type and content.
Generation and incorporation kinetics
antisite
pair in
thermal
equilibrium
vacancy
overgrowth
vst
Frenkel
pair
formation
by thermal
oscillation
crystal
vacancy
capture
from the
melt
melt
velocity of flowing step < > back diffusion
vst  i T < > DIF / hst
i - kinetic coefficient, T - supercooling,
DIF - interdiffusion coefficient, hst - step height
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2. Point defects
Point defect dynamics in silicon
2.1 Native point defects
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2. Point defects
Point defect engineering
2.1 Native point defects
Compound growth
Czochralski Silicon
Vacany-interstitial annihilation
Stoichiometry control by vapor source
source
seed
crystal melt
boat
container
HB
low temperature
furnace
temperature gradient
region
high temperature
furnace
V/G* = 1.34 x 10-3 cm2/K min
VCz
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VGF
2. Point defects
Impurities and dopants
2.2 Extrinsic point defects
Each real growing crystal contains
impurities or dopants. When their
concentrations are below the solubility
limits, the matrix is regarded as contributing one component in a phase
diagram and the solute another. The
equilibrium between the chemical potentials of the adding species i in the
liquid and solid phases µiL (x,T) = µiS
(x,T) yields:
Si - C
ios  kTln xis is  iol  kTln xil il
µoiL - µoiS = µoi = hoi - soiT and sio = hio/Tmi , with hio, sBo
intensive standard enthalpy and entropy, Tmi - melting point
of the dopant, hoMiS,L = kT lniS,L - mixing enthalpy
 hio  1 1  hMiL  hMiS 
xiS
 
 
 ko  exp 

xiL
k
T
T
kT
mi 



ko - equilibrium distribution coefficient
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2. Point defects
Extrinsic-intrinsic defect interaction
2.2 Extrinsic point defects
Concentration in the solid, cm-3
1020
CdTe
1018
1016
Segregation coefficient
koAg = CSAg/CLAg = 0.3
Cd
Incorporation coefficient of
Ag in substitutional AgCd
TeL excess
position
1018 cm-3
kAgCd = CSAgCd/CLAg
1017 cm-3
1016cm-3
1014
Vacancies provided by the
interface are occupied by
extrinsic impurity atoms:
Ag
Te
Cd
VCd
With increasing deviation from stoichiometry the growing number of vacancies
is occupied by silver atoms.
stoich
1020
1018
1014
1016
Concentration in the melt, cm-3
Rudolph, Rinas, Jacobs JCG 138 (1994) 249
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Note, electrically charged
intrinsic defects (vacancies)
tend to form complexes with
extrinsic atoms,
e.g. [VGa - ON] in GaN:O.
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2. Point defects
Diffusion boundary layer
2.3 Segregation phenomena
x
x
(mole fraction)
xBLo
x
jD
ko
L
xB
ko
S
xL
s
V>0
V=0
xBS
z
equilibrium segregation
coefficient:
xBLo
ko  S
xB
z
effective segregation
coefficient:
keff
ko
xBS
 L 
x ko  (1  ko ) exp(  R s / D)
Burton, Prim, Slichter, J. Chem. Phys. 21 (1953) 1987
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2. Point defects
Axial distributions
concentration x
2.3 Segregation phenomena
T
ko = xS / xL
liquid
koSi = xS /
xSxL= k0.4xL (1
- g)
o
k-1
solid
xS
xL
x
xL
I - no melt mixing
II - partial melt mixing
III - complete melt mixing
xS = koxL
xS = koxL (1-g) ko -1
Solidified fraction z/L = g
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E. Scheil (1952)
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2. Point defects
Constitutional supercooling
2.3 Segregation phenomena
GL (1  k ) C m

v
DL k
Phase diagram
W. A. Tiller et al., Acta Metalurgica 1 (1953) 428
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2. Point defects
Reduction of diffusion boundary layer
2.3 Segregation phenomena
mc-Si ingot
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3. Dislocations
Dislocation types
3.1 Types and analysis
b
edge
•
core with
dislocation
line
glide
planes
glide
plane
b
screw
60°
mixed
b
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3. Dislocations
Stress field of dislocations
3.1 Types and analysis
Each dislocation acts as a
source of elastic stress.
y
The stress value of screw dislocation:
xy
τs 
compression
x
expansion
Gb
2 r
G - shear modulus
b - Burgers vector
The elastic energy of screw ( =1)
and edge ( = 1 - ) dislocation:
Es = (Gb2/4) ln (R/ro)
copper:
 = 104 cm-2 Es = 4.52 eV
 = 106 cm-2 Es = 3.76 eV
 = 1010 cm-2 Es = 2.26 eV
screening effect !
R
Interaction energy
between dislocations
Ei ( ρ)  
2
2

f
(
r
,

)(
Gb
/ 2 ) ln( R / ro )dr
 u
 1 / 2
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R - crystal radius,  - dislocation density,
fu - dislocation interaction function (+/- b)
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3. Dislocations
Partial (Shokley) dislocations
3.1 Types and analysis
The Burgers vector may decompose into two Shockley partials
zinkblende:
1
1
1
[101]  [112]  [211]
2
6
6
Ecompl > Epart
b
ao
[100]
ao
1
b  110 
2
2
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Pohl 2013
dSh ~ 1/SF
SF - stacking fault energy
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3. Dislocations
3.1 Types and analysis
Basic considerations
Theoretically, for generation of dislocations in
a perfect crystal an extremly high stress of
~ 10-2 - 10-1 G
is required (G - shear modulus = 10 - 50 GPa).
1 cm
Much lower stress is necessary to move and
multiply already presented dislocations.
Near to the melting point the critical resolved
shear stress (CRSS) C to move (multiply)
dislocations yields:
Cu
C 0.02
1 cm
cm/cm3
pit/cm-2
1
= 1 etch
> cm/cm3 = 3 etch pits/cm-2
mean Dislocation distance: d
= -2
Si
Ge
GaAs
CdTe
9
1.5
0.5
0.2
MPa
However, dislocations can be generated by:
- intrinsic point defect condensation
- on precipitates and inclusions
- at the crystal surface (high local load)
- lattice misfit at heteroepitaxial systems
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3. Dislocations
Dislocation generation
3.1 Types and analysis
point
defect
condensation
(kPa – MPa)
vacancy condensation
lattice folding up
TEM of interstitial loops in Si
b
inclusioninduced
dislocations
(MPa - GPa)
lattice planes growing
around an inclusion
high EPD around inclusion
in GaAs
epitaxial
layer
epitaxial
misfit
Dislocations
(500 MPa - GPa)
dislocations in KDP
dislocation
cross-structure
in (Al,Ga)As layer
substrate
misfit dislocations
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3. Dislocations
Misfit and threading dislocations
3.1 Types and analysis
threading disloc.
misfit disloc.
Pohl 2013
misfit disloc.
threading disloc.
Kang 1997
Misfit dislocation network in GaN on sapphire
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misfit = 13.8 %
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3. Dislocations
3.1 Types and analysis
Laser scattering tomography
integrated depth: 2mm
Dislocation patterns are arranged honeycomb-like
consisting of globularly shaped cells with nearly
dislocation-free interiors.
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integrated depth: 0.5mm
M. Naumann, P. Rudolph, …
J. Crystal Growth 231 (2001) 22
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3. Dislocations
X-ray synchrotron tomography
3.1 Types and analysis
Burgers vector
analysis
g
511
Criterion
of disappearance:
g•b=0
g
151
b II [101]
cos (g • b) = 0
g – diffraction vector
b – Burgers vector
Dislocation cells in
GaAs :
- mainly 60°
dislocations with
b = ½ <110>
e.g. HASYLAB-DESY Hamburg
T. Tuomi, L. Knuuttila, P. Rudolph
J. Crystal Growth 237 (2002) 350
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1 mm
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3. Dislocations
Transmission electron microscopy
3.1 Types and analysis
CdTe
GaN
small-angle
grain
boundaries
etching
AlN
parallel dislocations
of identical b
GaN
Sapphire
Wang, Appl. Phys. Lett. 89, 152105 (2006)
TEM
GaN
1 µm
Hossain 2012
Durose (1988)
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3. Dislocations
Dislocation movement
3.2 Dislocation dynamics
b
vacancy
jog
glide
plane
interstitial
3D
2D
climb
glide
(of edge dislocation)
(of edge dislocation)
Nonconservative process of point defect diffusion
High-temperature process !
Velocity:
vg = vo (eff )m exp (-Ea/kT)
Ea – activation energy (Peierls potential)
(eff =  -Ao), o - mobile disloc. density,
vo - material constant, A - strain hardening
factor,  - strain
vcl = vo (eff )Nc exp (-ESD/kT)
(Di/b) cj(SF/Gb)2 (/G)
ESD - activation energy for self-diffusion,
Nc - climb exponent (~ 3), G - shear modulus
Di - point defect diffusion coeff.,  - strain,
Cj - concentration of jogs, SF - stacking fault energy
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3. Dislocations
Glide plane arrangements
3.2 Dislocation dynamics
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3. Dislocations
Thermomechanical stress
3.2 Dislocation dynamics
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3. Dislocations
Plastic relaxation
3.2 Dislocation dynamics
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3. Dislocations
Plastic relaxation
3.2 Dislocation dynamics
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3. Dislocations
Dislocation distribution
3.2 Dislocation dynamics
- reality -
- simulation 1.5 MPa
5 x 104 cm-2
7 x 103
0.5 MPa
0.8 MPa
[100]
[110]
radial stress distribution
GaAs
undoped
radial dislocation distribution
characteristic
dislocation cellular
structure
Frank-Rotsch, Rudolph (2006)
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3. Dislocations
Dislocation cell patterning
3.3 Substructuring
deformed samples
as-grown crystals
a
2 µm
b
1 µm
c
300 µm
a - Mo 12% deformed at 493 K
b - Cu-Mn deformed at 68.2 MPa
c - GaAs grown by LEC
100 µm
d
e
500 µm
200 µm
f
d - CdTe grown by VB
e - mc-Si grown by VGF
f - SiC grown by sublimation
g - Cd0.96Zn0.04Te grown by VB
h - NaCl deformed by 150 MPa
g 200 µm
h
500 µm
i
1000 µm
i - CaF2 grown by Cz
P. Rudolph, Crystal Res. Technol. 40 (2005) 7
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3. Dislocations
Origins of cellular substructures
3.3 Substructuring
1. Dynamic polygonization (DP)
in the course of plastic relaxation due to thermomechanical stress.
2. High-temperature dislocation dynamics (DD)
combining glide with point-defect assisted claim.
3. Morphological instability of the propagating crystallization front
in the form of cellular interface shape.
1. and 2. are close correlating. However, whereas DP requires in any case stress-related
driving force DD implies along with screening effects also evidences of self-organized
(dissipative) structuring in the course of irreversible thermodynamics (de facto, each directional crystallization system is an “open” one steadily importing and exporting energy).
DD takes place at high temperatures where the point defect diffusivity is still high enough.
It is noteworthy that the formation of spatial cellular patterns is only possible when threedimensional dislocation movements like climb and cross glide can take place. Glide alone
could be not responsible for.
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3. Dislocations
Dislocation interactions
3.3 Substructuring
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3. Dislocations
Dynamic polygonization
3.3 Substructuring
t=0
random
dislocation
distribution
Growing crystal under thermoelastic stress with excess
defect enthalpy Hd
t>0
elastic stress
+ annihilation
polygonized KCl
d
Hd = min
Hd minimization by dislocation
annihilation und lining up of the
excess dislocations in low-angle
grain boundaries
Hwall  ¼ Hd
Amelincks (1956)
RSS

simulation of
dislocation glide in
ensemble
Gulluoglou (1989)
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small-angle grain boundary etching
tilt angle sin    = b/d [rad]
1 rad = 180°/  57.3 °
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3. Dislocations
3.3 Substructuring
Numeric modeling of impact of
climb and cross glide
climb
cross glide
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3. Dislocations
Rules of correspondense
3.3 Substructuring
There are scaling relations fullfilled over a wide range of materials and
deformation conditions. Zaiser 2004
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3. Dislocations
Dislocation bunching
3.3 Substructuring
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3. Dislocations
Favourable growth conditions
3.4 Dislocation engineering
Generally, for a dislocation-reduced
growth the following conditions are
required:
• dislocation-free seed
• uniaxial heat flow at small T-grad
• detached growth conditions
• in-situ stoichiometry control
• no constitutional supercooling
• no fluid pressure fluctuations
stoich
As-rich
fluid
IF
  min
 > 90°
solid
Kiessling, Rudolph (2004)
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3. Dislocations
Reduction during heteroepitaxy
3.4 Dislocation engineering
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4. Second Phase Particles
4.1 Precipitates
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Point defect
condensations
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4. Second Phase Particles
4.2 Inclusions
Incorporation
at growing interface
Si : C
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4. Second Phase Particles
Correlation
4.2 Inclusions
CdTe
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4. Second Phase Particles
After-growth
treatment
CdTe
Franc (2010)
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5. Faceting
Examples
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5. Faceting
Correlation with kinetics
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6. Twinning
Correlation with stacking fault
V
InP
S
L
2D nucleation
with stacked fault
{111} facets with twins in InP
Shibata et al. (1990)
Concept of Hurle (1995):
(using Voronkov‘s facet growth theory)
A* = Tc (h H /Tm)
twins in InP (IKZ)
stacking fault energies (x 10-7 J cm-2)
Si: 100, GaAs: 55, InP: 18, CdTe: 10
A* - reduced work of twinned nucleus at
VLS boundary ~ supercooling Tc
 - twin plane energy~  !
SF
Tm - melting temperature
h - nucleus height,
H - latent heat
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6. Twinning
T instability
k/h
at anheater,
T/t dT/dt
HH [K/h]
40
Supercooling at the facet 
0.8
InP
0.6
30
20
10
0
-10
-20
-30
-40
0
10
20
30
40
50
60
70
80
Kristalllänge [mm]
crystal length, mm
0.4
often twinning
40
0.2
0
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Facet length, mm
k/h
at anheater,
T/t dT/dt
HH [K/h]
Relative frequency of twins
1.0
30
20
10
0
-10
-20
-30
-40
0
10
20
30
40
50
60
70
Kristalllänge [mm]
Neubert (2006)
crystal length, mm
seldom twinning
Twinning probability correlates with growth rate fluctuations !
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80
7. Summary
Defects vs. temperature
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7. Summary and outlook





Most of the defect-forming mechanisms have become well understood. Their avoidance, however, is still problematically. For instance,
it is not possible to reduce the thermal stresses to a sufficiently low
level to prevent dislocation multiplication and substructuring.
Although the conditions of morphological stability are well known it is
still not possible to grow large homogeneous mixed single crystals.
Twinning remains still a serious limiter of yield in the growth of single
crystals with low stacking fault energy, such as CdTe and InP.
One of the prior tasks is the heteroepitaxy of low-dislocation crackfree layers, especially GaN on sapphire or Si.
So what of the future?
- much better understanding of the thermodynamics and kinetics of native point defects and their interactions with dopants
during growth and post annealing;
- industrial scaling up to achieve cost reduction by modellingassisted prober hot-zone engineering and magnetic field control.
- find out a stress-free dislocation reduction method for heteroepitaxial processes.
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