View
7
Download
0
Category
Preview:
Citation preview
15th International Summer School on Crytsal Growth – ISSCG-15
Crystal Defects
Peter Rudolph Crystal Technology Consultation (CTC)
Helga-Hahnemann-Str. 57, D-12529 Schönefeld
rudolph@ctc-berlin.de
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
Abstract
The quality of crystals is very sensitively influenced by structural and atomistic deficiencies generated during crystal growth. Such imperfec-tions 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 un-derstood. There exists an enormous knowledge about the defect ge-nesis 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.
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
Outline
1. Introduction - defect classification
2. Point defects
2.1 Native point defects
2.2 Extrinsic point defects
2.3 Segregation phenomena
3. Dislocations
3.1 Dislocation types and analysis
3.2 Dislocation dynamics
3.3 Low-angle grain boundaries - substructuring
3.4 Dislocation engineering
4. Second-phase particles
4.1 Precipitates
4.2 Inclusions
5. Faceting
6. Twinning
7. Summary and outlook
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
1. Introduction Defect types
a – interstitial impurity atom b – edge dislocation
c – self interstitial atom d – vacancy
e – precipitate of impurity atoms f – vacancy type dislocation loop
g – interstitial type dislocation loop h – substitutional impurity atom
after H. Föll: http://www.tf.uni-kiel.de/matwis/amat/
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
1. Introduction Defect classification
Structural crystal defects are classified according to their dimensions.
precipitates, inclusions,
voids (vacancy agglomerates),
bubbles, dislocation clusters
3-dimensional defects
stacking faults, twins
grain and phase boundaries,
facets ? (expressing perfection !)
2-dimensional defects
dislocations
(edge, screw, 60°, 30°, mixed,
mobile, sessile, bunched, ordered...)
1-dimensional defects
atomic size („point“) defects
intrinsic (vacancies, interstitials)
and extrinsic (dopants) defects
0-dimensional defects
in thermo-
dynamic
equilibrium
in thermo-
dynamic
non-
equilibrium
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
1. Introduction Defect diagnostics
- unoccupied state -
Scanning Tunneling
Microscopy (STM)
(110) (1x1) GaAs
- Dash necking -
X-ray diffraction
(Lang) topography
(110) FZ Si
- casting -
Photo image;
Electron Back
Scattering (EBS)
PV Si
- nonstoichiometry
Laser Scattering
Tomography (LST);
Transmission Electron
Microscopy (TEM)
(100) VB CdTe
Point defects Dislocations Grain boundaries Inclusions
Schröder 1967 Gebauer 2000 Fujiwara 2006
Hähnert, Rudolph 1993
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
2. Point defects 2.1 Native point defects
A certain point defect content
increases the entropy and, hence,
decreases the Gibbs potential !
TSHG dd Hd = n Ed - defect enthalpy ( n - number of defects)
Sd = k lnW - configurational entropy, W = N ! /n !(N-n) !
interstitial
vacancy
antisite
AB compound
0*
*ln
n
nNkTE
n
Gd
Intrinsic defect minimum
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
2. Point defects 2.1 Native point defects stoich.
n* = N exp (- Ed / kT)
Ed = Eform + Evib + ES
Existence region of a compound
x = A - B = (CiA - Cv
A + 2CA/B- 2CB/A )
– (CiB - Cv
B + 2CB/A- 2CA/B )
x
deviation from stoichiometry (netto defects in each sublattice):
formation, vibration, configuration
Diffusivity and non-stoichiometry
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
2. Point defects 2.1 Native point defects
* *
*
XB
Tcong cmp
stoich. growth
segregation
precipitation
IF
rejected excess component (B)
dislocation
homogeneous
heterogeneous
diffusion area
~ 100 nm
non-stoich. growth
Segregation and condensation
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
During crystal growth from melt
the native point defects undergo
various types of transport kine-
tics such as capture at the inter-
face and diffusion by jumping via
interstitials and vacancies. Vari-
ations of the growth rate shifts
the point defect transport bet-
ween incorporation and diffusion
dominated. Whereas in disloca-
tion-free silicon crystals at high
rates the flux of vacancies domi-
nates that of self-interstitials at
low rates or high temperature
gradients interstitials are in ex-
cess. This fact is of high signifi-
cance for in situ control of nati-
ve point defect type and content.
2. Point defects Generation and incorporation kinetics 2.1 Native point defects
Frenkel
pair
formation
by thermal
oscillation
vacancy
over-
growth
antisite
pair in
thermal
equilibrium
vacancy
capture
from the
melt
crystal melt
vst
velocity of flowing step < > back diffusion
vst i T < > DIF / hst
i - kinetic coefficient, T - supercooling,
DIF - interdiffusion coefficient, hst - step height
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
2. Point defects Point defect dynamics in silicon 2.1 Native point defects
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
2. Point defects Point defect engineering 2.1 Native point defects
Czochralski Silicon Compound growth
V/G* = 1.34 x 10-3 cm2/K min
low temperature
furnace
source seed crystal melt boat container
high temperature
furnace
temperature gradient
region
Vacany-interstitial annihilation Stoichiometry control by vapor source
HB
VGF VCz
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
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 contri-
buting one component in a phase
diagram and the solute another. The
equilibrium between the chemical po-
tentials of the adding species i in the
liquid and solid phases µiL (x,T) = µiS
(x,T) yields:
liliolisisi
osi xkTxkT lnln
kT
hh
TTk
hk
x
x MiSMiL
mi
o
io
iL
iS 11exp
µoiL - µ
oiS = µo
i = hoi - so
iT and sio = hi
o/Tmi , with hio, sB
o
intensive standard enthalpy and entropy, Tmi - melting point
of the dopant, hoMiS,L = kT lniS,L - mixing enthalpy
ko - equilibrium distribution coefficient
Si - C
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
Rudolph, Rinas, Jacobs JCG 138 (1994) 249
Cd
VCd
Te Cd Ag
1014 1016 1018 1020
1014
1016
1018
1020
Concentration in the melt, cm-3
Concentr
ation in t
he s
olid
, cm
-3
stoich
TeL excess
1018 cm-3
1017 cm-3
1016cm-3
Segregation coefficient
koAg = CS
Ag/CLAg = 0.3
Incorporation coefficient of
Ag in substitutional AgCd
position
kAgCd = CSAgCd/CL
Ag
CdTe
Vacancies provided by the
interface are occupied by
extrinsic impurity atoms:
2. Point defects Extrinsic-intrinsic defect interaction
2.2 Extrinsic point defects
Note, electrically charged
intrinsic defects (vacancies)
tend to form complexes with
extrinsic atoms,
e.g. [VGa - ON] in GaN:O.
With increasing deviation from stoichio-
metry the growing number of vacancies
is occupied by silver atoms.
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
2. Point defects Diffusion boundary layer
2.3 Segregation phenomena
xBLo
x
z
xL
xBS
ko
V > 0
x (mole fraction)
z
xL
xBS
ko
V = 0
S
B
L
Bo
x
xk
o
equilibrium segregation
coefficient:
)/exp()1( DRkk
k
x
xk
soo
o
L
S
Beff
s
effective segregation
coefficient:
jD
Burton, Prim, Slichter, J. Chem. Phys. 21 (1953) 1987
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
2. Point defects Axial distributions
2.3 Segregation phenomena
xL
xS = koxL
koSi = xS /
xL 0.4 xS = ko xL (1
- g) k-1
liquid
solid
xL xS
T
x
Solidified fraction z/L = g
concentr
atio
n x
ko = xS / xL
I - no melt mixing
II - partial melt mixing
III - complete melt mixing
xS = koxL (1-g) ko -1
E. Scheil (1952)
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
2. Point defects Constitutional supercooling
2.3 Segregation phenomena
W. A. Tiller et al., Acta Metalurgica 1 (1953) 428
kD
mCkG
L
L
)1(
v
Phase diagram
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
2. Point defects Reduction of diffusion boundary layer
2.3 Segregation phenomena
mc-Si ingot
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
3. Dislocations 3.1 Types and analysis
glide
plane
b
core with dislocation
line
•
glide
planes
b
edge
screw
b
mixed 60°
Dislocation types
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
3. Dislocations 3.1 Types and analysis
Stress field of dislocations
r
Gbτs
2
Each dislocation acts as a
source of elastic stress.
The stress value of screw dislocation:
Es = (Gb2/4) ln (R/ro)
The elastic energy of screw ( =1)
and edge ( = 1 - ) dislocation:
G - shear modulus
b - Burgers vector
drrRGbrfρE o
R
ui )/ln()2/)(,(2)( 2
2/1
Interaction energy
between dislocations
R - crystal radius, - dislocation density,
fu - dislocation interaction function (+/- b)
copper: = 104 cm-2 Es = 4.52 eV
= 106 cm-2 Es = 3.76 eV
= 1010 cm-2 Es = 2.26 eV
screening effect !
expansion
y
x
xy compression
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
3. Dislocations 3.1 Types and analysis
Partial (Shokley) dislocations
The Burgers vector may decompose into two Shockley partials
]112[6
1]211[
6
1]110[
2
1 Ecompl > Epart
dSh ~ 1/SF
SF - stacking fault energy
ao
[100]
b
2110
2
1 oab
zinkblende:
Pohl 2013
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
3. Dislocations 3.1 Types and analysis
Basic considerations
1 cm
1 c
m
1 cm/cm3 = 1 etch pit/cm-2
> cm/cm3 = 3 etch pits/cm-2
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).
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 Si Ge GaAs CdTe
0.02 9 1.5 0.5 0.2
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
C MPa
mean Dislocation distance: d = -2
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
3. Dislocations 3.1 Types and analysis
Dislocation generation
lattice folding up vacancy condensation TEM of interstitial loops in Si
point
defect
condensation
(kPa – MPa)
epitaxial
layer
substrate
misfit dislocations
dislocation
cross-structure
in (Al,Ga)As layer
epitaxial
misfit
Dislocations
(500 MPa - GPa)
lattice planes growing
around an inclusion
b
high EPD around inclusion
in GaAs
dislocations in KDP
inclusion-
induced
dislocations
(MPa - GPa)
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
3. Dislocations 3.1 Types and analysis
Misfit and threading dislocations
threading disloc. misfit disloc.
Misfit dislocation network in GaN on sapphire
threading disloc.
misfit disloc.
misfit = 13.8 %
Kang 1997
Pohl 2013
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
3. Dislocations 3.1 Types and analysis
Laser scattering tomography
integrated depth: 2mm
integrated depth: 0.5mm Dislocation patterns are arranged honeycomb-like
consisting of globularly shaped cells with nearly
dislocation-free interiors. M. Naumann, P. Rudolph, …
J. Crystal Growth 231 (2001) 22
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
3. Dislocations 3.1 Types and analysis
X-ray synchrotron tomography
e.g. HASYLAB-DESY Hamburg
T. Tuomi, L. Knuuttila, P. Rudolph
J. Crystal Growth 237 (2002) 350 1 mm
g
511
g
151
Burgers vector
analysis
Criterion
of disappearance:
g • b = 0
cos (g • b) = 0 g – diffraction vector b – Burgers vector b II [101]
Dislocation cells in
GaAs :
- mainly 60°
dislocations with
b = ½ <110>
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
3. Dislocations 3.1 Types and analysis
Transmission electron microscopy
parallel dislocations
of identical b
Durose (1988)
CdTe
etching
small-angle
grain
boundaries
1 µm
TEM Wang, Appl. Phys. Lett. 89, 152105 (2006)
GaN
AlN
GaN
Sapphire
GaN
Hossain 2012
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
3. Dislocations 3.2 Dislocation dynamics
Dislocation movement
glide (of edge dislocation)
b
glide
plane
climb (of edge dislocation)
Nonconservative process of point defect diffusion
High-temperature process !
interstitial
vacancy
vg = vo (eff )m exp (-Ea/kT)
Velocity:
Ea – activation energy (Peierls potential)
(eff = -Ao), 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
2 D 3 D
jog
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
3. Dislocations 3.2 Dislocation dynamics
Glide plane arrangements
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
3. Dislocations 3.2 Dislocation dynamics
Thermomechanical stress
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
3. Dislocations 3.2 Dislocation dynamics
Plastic relaxation
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
3. Dislocations 3.2 Dislocation dynamics
Plastic relaxation
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
3. Dislocations 3.2 Dislocation dynamics
Dislocation distribution
radial stress distribution
[100]
5 x 104 cm-2
7 x 103
radial dislocation distribution
- simulation - - reality -
GaAs characteristic
dislocation cellular
structure Frank-Rotsch, Rudolph (2006)
[110]
undoped
1.5 MPa
0.5 MPa
0.8 MPa
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
3. Dislocations 3.3 Substructuring
Dislocation cell patterning
2 µm 1 µm 300 µm
500 µm 100 µm
200 µm 1000 µm 500 µm
a b c
d e f
g h i
200 µm
a - Mo 12% deformed at 493 K
b - Cu-Mn deformed at 68.2 MPa
c - GaAs grown by LEC
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
i - CaF2 grown by Cz
P. Rudolph, Crystal Res. Technol. 40 (2005) 7
deformed samples
as-grown crystals
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
3. Dislocations 3.3 Substructuring
Origins of cellular substructures
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 di-
rectional 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 three-
dimensional dislocation movements like climb and cross glide can take place. Glide alone
could be not responsible for.
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
3. Dislocations 3.3 Substructuring
Dislocation interactions
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
3. Dislocations 3.3 Substructuring
Dynamic polygonization
Hd minimization by dislocation annihilation und lining up of the excess dislocations in low-angle grain boundaries
Growing crystal under thermo-elastic stress with excess defect enthalpy Hd
simulation of
dislocation glide in
ensemble Gulluoglou (1989)
t > 0
elastic stress
+ annihilation
Hd = min
RSS
t = 0
random
dislocation
distribution
Hwall ¼ Hd
polygonized KCl Amelincks (1956)
d
small-angle grain boundary etching
tilt angle sin = b/d [rad]
1 rad = 180°/ 57.3 °
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
3. Dislocations 3.3 Substructuring
Numeric modeling of impact of
climb and cross glide
climb
cross glide
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
3. Dislocations 3.3 Substructuring
Rules of correspondense
There are scaling relations fullfilled over a wide range of materials and
deformation conditions. Zaiser 2004
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
3. Dislocations 3.3 Substructuring
Dislocation bunching
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
3. Dislocations 3.4 Dislocation engineering
Favourable growth conditions
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
fluid
solid
IF min
> 90°
stoich
As-rich
Kiessling, Rudolph (2004)
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
3. Dislocations 3.4 Dislocation engineering
Reduction during heteroepitaxy
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
4. Second Phase Particles 4.1 Precipitates
Point defect
condensations
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
4. Second Phase Particles 4.2 Inclusions
Incorporation
at growing interface
Si : C
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
4. Second Phase Particles 4.2 Inclusions
Correlation
CdTe
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
Franc (2010)
CdTe
4. Second Phase Particles
After-growth
treatment
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
5. Faceting
Examples
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
5. Faceting
Correlation with kinetics
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
6. Twinning
{111} facets with twins in InP
Shibata et al. (1990)
twins in InP (IKZ)
Concept of Hurle (1995): (using Voronkov‘s facet growth theory)
A* = Tc (h H /Tm)
A* - reduced work of twinned nucleus at
VLS boundary ~ supercooling Tc
- twin plane energy
Tm - melting temperature
h - nucleus height,
H - latent heat
stacking fault energies (x 10-7 J cm-2)
Si: 100, GaAs: 55, InP: 18, CdTe: 10
~ SF !
2D nucleation
with stacked fault
S
V
L
InP
Correlation with stacking fault
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
6. Twinning R
ela
tive
fre
qu
en
cy o
f tw
ins
0
0.2
0.4
0.6
0.8
1.0
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Facet length, mm
Supercooling at the facet
Neubert (2006)
0 10 20 30 40 50 60 70 80-40
-30
-20
-10
0
10
20
30
40
dT
/dt a
n H
H [K
/h]
Kristalllänge [mm]
T
/t a
t h
ea
ter,
k/h
crystal length, mm
often twinning
0 10 20 30 40 50 60 70 80-40
-30
-20
-10
0
10
20
30
40
dT
/dt a
n H
H [K
/h]
Kristalllänge [mm]
T
/t a
t h
ea
ter,
k/h
crystal length, mm
seldom twinning
Twinning probability correlates with growth rate fluctuations !
InP
T instability
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
7. Summary Defects vs. temperature
15th International Summer School on Crystal Growth – ISSCG-15 LAST NAME, First Name – talk id
7. Summary and outlook Most of the defect-forming mechanisms have become well under-
stood. 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 crack-free layers, especially GaN on sapphire or Si.
So what of the future?
- much better understanding of the thermodynamics and kine- tics of native point defects and their interactions with dopants during growth and post annealing;
- industrial scaling up to achieve cost reduction by modelling- assisted prober hot-zone engineering and magnetic field control.
- find out a stress-free dislocation reduction method for hetero- epitaxial processes.
Recommended