crystal defects - gdańsk university of technologya certain point defect content increases the...

15th International Summer School on Crytsal Growth – ISSCG-15

Crystal Defects

Peter Rudolph Crystal Technology Consultation (CTC)

Helga-Hahnemann-Str. 57, D-12529 Schönefeld

[email protected]

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.


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


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/


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 !)


dislocations

(edge, screw, 60°, 30°, mixed,

mobile, sessile, bunched, ordered...)


atomic size („point“) defects

intrinsic (vacancies, interstitials)

and extrinsic (dopants) defects


in thermo-

dynamic

equilibrium

in thermo-

dynamic

non-

equilibrium


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


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


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


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


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


2. Point defects Point defect dynamics in silicon 2.1 Native point defects


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


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


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.


2. Point defects Diffusion boundary layer


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


2. Point defects Axial distributions


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)


2. Point defects Constitutional supercooling


W. A. Tiller et al., Acta Metalurgica 1 (1953) 428

kD

mCkG

L

L

)1(

v

Phase diagram


2. Point defects Reduction of diffusion boundary layer


mc-Si ingot


3. Dislocations 3.1 Types and analysis

glide

plane

b

core with dislocation

line

•

glide

planes

b

edge

screw

b

mixed 60°

Dislocation types



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



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



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



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)



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



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



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>



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


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



Glide plane arrangements



Thermomechanical stress



Plastic relaxation



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


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



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.



Dislocation interactions



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 °



Numeric modeling of impact of

climb and cross glide

climb

cross glide



Rules of correspondense

There are scaling relations fullfilled over a wide range of materials and

deformation conditions. Zaiser 2004



Dislocation bunching


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)


3. Dislocations 3.4 Dislocation engineering

Reduction during heteroepitaxy


4. Second Phase Particles 4.1 Precipitates

Point defect

condensations


4. Second Phase Particles 4.2 Inclusions

Incorporation

at growing interface

Si : C


4. Second Phase Particles 4.2 Inclusions

Correlation

CdTe


Franc (2010)

CdTe

4. Second Phase Particles

After-growth

treatment


5. Faceting

Examples


5. Faceting

Correlation with kinetics


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


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


7. Summary Defects vs. temperature


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 kinetics 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 heteroepitaxial processes.

crystal defects - gdańsk university of technologya certain point defect content increases the...

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