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Introduction to Dislocation Mechanics
What is Dislocation Mechanics? (as meant here)
The study of the stress state and deformation of continua whose elastic response is mediated by the nucleation, presence, motion, and interaction of distributions of crystal defects called dislocations
Dislocation?For the moment, An imaginary curve in an elastic continuum
• that induces a stress field in the elastic body• is capable of moving and altering its shape
kinetics driven by stress
Multiple dislocations interact through their stress fields• The stress field of one dislocation modifies the stress
acting on another thus affecting the latter’s placement
Dislocation loops in Si http://www.tf.uni-kiel.de/matwis/amat/def_en/kap_5/illustr/i5_4_1.html
Cell walls in OFHC Cu, fatigue loadingZhang, Jiang, ‘07
What is a Dislocation?
Edge dislocation (after G. I. Taylor, 1934)
Line direction
Burgers vector
www.roosterteeth.com
2012books.lardbucket.org
coefs.uncc.edu/hzhang3/w-o-m/
Also Polanyi ’34; Orowan, ‘34
What is a Dislocation? Screw dislocation (after J. M. Burgers, 1939)
Burgers vector
Line directioncoefs.uncc.edu/hzhang3/w-o-m/
What is a Dislocation? Edge-Screw dislocation (after J. M. Burgers)
Burgers vector forentire dislocation
Dislocation line
coefs.uncc.edu/hzhang3/w-o-m/
7
Why should Dislocation Mechanics be studied?
Dislocations in crystalline materials are an inevitable consequence of the storage of elastic energy
Once formed, they critically affect mechanical electrical electronic optical
performance of devices and structures built from crystalline materials
8
Electronic Materials Mechanical stress is a central factor in
Fabrication Performance Reliability
Source of generation and motion of undesirable threading dislocations Easy diffusion paths for dopants short circuits across
layers Electron-hole recombination centers Sites for defect nucleation, growth and multiplication
Strain engineering Altering electronic band-gaps of devices by suitably positioning
defects
9
Semiconductor Technology:Thin film-Substrate Heterostructure
Interface misfit dislocation
Slip plane
Threading segment
Interface misfit segment
Bulk substrate(Si, SiC)
Semiconductor(GaInAs, SiGe, GaN)thin film ~ 10 nmthick film ~ 600 nm
10
Semiconductor Technology:Thin Film-Compliant Substrate
System
Twist-bonded Compliant substrateGaAs ~3-10 nm
InSb, InGaP film~300-600 nm
Bulk substrate
X-grid of screw dislocations
11
IC, MEMS Technology
Interconnects: Thermal and residual stress
—Voids and cracks in metallization layers leading to failure - “stress migration”
Dislocation motion induces stress relaxation Important to understand magnitude of relaxation for quantitative
failure estimates (reliability)
Fatigue failure of MEMS components Experimental results indicate evolving dissipative mechanisms Residual dislocation stress + relaxation by dislocation motion need
to be modeled to understand plasticity at micron length-scales.
12
Interconnects
grain
Passivation layerSiO2
Al metallization~500nm X 1000 nm
Si substrate
13
MEMS
Macroscopic plasticity does not work for structural dimensions of ~10 m - 0.1 m
Gradient plasticity inadequate for detailed analysis of local stress concentrations that drive failure processes
Dislocation mechanics required to understand stress concentration and relaxation in MEMS structures
freq
uenc
y
cycles
14
Structural Components
Inhomogeneous deformation - precursor to failure
Strength Formability - Ductility Residual Stress Fatigue
15
Target Predictions of Dislocation Mechanics Capability:
Fine Features Dislocation nucleation due to elastic
instability
Simple cubic lattice
s
b
c cohesive reaction
shear traction
slip0 b
c c
ohes
ive
reac
tion
b/4
Instability leadingto nucleation
max
16
Nucleation vs. Motion
Figure 1. Schematic illustration of dislocation motion and nucleation; a) motion of an existing edgedislocation resulting in an advance of the slipped region; b) nucleation of an edge dislocation; c)nucleation of an edge dislocation dipole. Red lines indicate slipped regions of the crystal; green linesrepresent unslipped (but possibly deformed) regions; and black lines represent dislocations as theboundary between slipped and unslipped regions.
a)
b)
c)
AA, Beaudoin,Miller, 08
17
Nucleation vs. Motion
Figure 6: Nucleation and motion of a dislocation dipole during nano-indentation, with contours showing relative magnitudes of atomic motion (Å). (a) the undefected cystal. (b) nucleation (c) growth to a full Burgers vector and (d)-(f) motion.
AA, Beaudoin,Miller, 08
18
Target Predictions of Dislocation Mechanics Capability:
Fine Features Dislocation multiplication - Frank-Read
source
Screw dislocation
19
Target Predictions of Dislocation Mechanics Capability:
Fine Features Hardening due to interactions
Dissociation energeticallyfavorable
Stacking faultin crystal
Partial dislocation
Long range stress fieldLomer-Cottrell lock
Short range stress field
Forest hardening
Orientation dependence of work hardening
20
Clarebrough and Hargreaves, AJP, 1960
21
Target Predictions of Dislocation Mechanics Capability:
Coarse Features Stress
Arising from lattice stretching—Due to presence of dislocations (residual or internal stress)—Due to applied loads
Hardening Retardation of dislocation motion due to modification of local stress field
acting on dislocation due to stress field of others
Deformation Elastic stretching Slip due to dislocation motion (permanent deformation) –
deformation microstructure Time dependence of mechanical response
22
Patchy Slip
Piercy, Cahn & Cottrell, 1955
brass
23
Slip bands, localization
Chang & Asaro, 1980
Al Cu
:q q q+ -= -
Related kinematical question
1
Characterize the possible jumps in
field on such that
on \
where is a given vector field
on satisfying .
S
grad S
C
curl
q
q W
W
*
*
=
=
A
A
A 0
W
S
+-
{ }\ :OW W*=
O
for any closed curve surrounding
, and this is constant on .
Cd
C
O S
q = - ⋅ò A x
C
a
b c
d
e
Discontinuity of a Discontinuity
Terminating curve of Displacement discontinuity = DISLOCATION
Polar angle of director discontinuity = DISCLINATION
The classical question(Volterra - dislocations, Frank – nematic disclinations)
2Minimize
or
solve 0 on \
subject to : 2 on
0 on and
say 0 on
grad dv
div grad S
K S
grad S
grad
Wq
q W
q q q p
qq W
+ -
üïïïïïïýïïï= ïïïþ- = =
⋅ =⋅ = ¶
ò
nn
2
1# has to blow up like as
1# energy density like
total energy in is unbounded
grad Or
r
q
W
x
W
S
+-
{ }\ :OW W*=
O
Classical field of a screw dislocation/nematicwedge disclination
x2
x1x3
23
1
13 1
23 2
arctan
1 sin,
2 41 cos
,2 4
xu
x
b
rb
r
q
qe q
pq
e qp
æ ö÷ç ÷= = ç ÷ç ÷çè ø
= = -
= =
DiscontinuousDisplacement(even apart from origin)
Except origin,smooth strain field !!!!
So, dislocation strain fields arenot really the ones from takinga deriv. of the displacement field
BUTderivative fields obtained on the
Simply-connected domainInduced by the cut
Moral
A ‘slightly’ different, partial, alternate formulation
As an alternate problem for
, ask to find s.t.
2 on
0
0 on
O z
grad
curl K
div
q
d pW
W
ü= - ïïýï= ïþ⋅ = ¶
A
A eA
A n
However, this does not say
anything about determining
with the required properties.q
How to do this?
Punctured domain etc.not physical and
Impossible for practicalcomputation
Need formulation that producesfinite energy AND an associated
qfield without requiring
cuts, holes etc.
Dislocation-Eigendeformation Formulation# Infinite TOTAL energy classical solution
is troublesome. Can the problem be forced
so as to give finite energy, keeping most
global features intact?
nq+
q-
core
lt
lS
(but not only )# Regularize jump across S q
( )( )
( )
( ) ( ) ( )
# in ; otherwise.
and constant in layer outside core
and decays to zero inside core
So, is only non-zero component
0 and , ,
l
n
t n t t n
g t l S
t g
q q
l
l l l
+ -= -
= ⋅
\ = ¹
l n 0
x t
# : and localized in core,
and 2
for any area patch containing corezA
curl
curl da K
A
q p
= ¹
⋅ = =òb l
l
0
e
Dislocation-Eigendeformation formulation
# Replace in classical defect theory by
:
# Replace 0 on \ by
a) 0 on
# Replace 0 on by
b) 0 on
# 2 smeared over core
d
grad
grad
div grad S
div
grad
K curl
q
qq W
Wq W
Wp
= -=
=⋅ = ¶
⋅ = ¶= - =
l
b
E
En
E nE
Recall, alternate problem for
field in classical theory
2 on
0
0 on
O z
grad
curl K
div
qd p
W
W
ü= - ïïýï= ïþ⋅ = ¶
A eA
A n
(where is a classical construct).
but has finite energy.
=morally E A AE
# since we also have that outside by construction, we
have managed to define a potential field whose gradient field
matches in most of the domain.
dl
d d
S grad
grad
q
q q
\ =E
A
Main utility of eigendeformation formulation is it provides a new field for specification of dynamics of dislocation lines.
Connection of eigendeformation formulation with classical picture
# Write
on
0
0 on
grad z
curl curl
div
W
W
üï= + ïïï= = ýïïï= ïþ⋅ = ¶
l gg b l
gg n
SH
nq+
q-
core
lt
lS
# classical question: solve
0 on \
subject to 2 on
0 on and
say 0 on
div grad S
K S
grad S
grad
q W
q q q p
qq W
+ -
é ù =ê úë û- = =
⋅ =⋅ = ¶
C
C nC n
values of 0, 0
is a smooth field except
at origin for 0.
l c
c
" ³ ³
=g
c
as 0z z lq q+ - + - - = ⋅ = - ò lp
dx
( )1 1
1
#
s.t.
(say with 0 on ) with smooth for 0
d
d
div div grad z div
z div grad div
c
q
q q q
q W q
é ùé ù é ù= - =ê ú ê ú ë ûë û ë ûé ù é ù - = =ê ú ë ûë û
= ¶ ¹
g
g
C E 0 C C
C C
d d zq q q+ - - = =
0
0
: 0,12
ly x yl
x c
æ ö÷çé ù ' + - + ÷ç ÷ë û ç ÷è ø>
p t n
for 0
:
across any surface
d
d
c ¹
=
=
T CE
T 0
without assumption
=gradqA
Connection of eigendeformation formulation and classical picture
( )
as 0
In \ ;
, but in \
In \ ;
;
c c
d dl
d d dld d
l
S grad div
grad S
S grad div
W q
q W
W q
q q
= =
= - =
\ = =
= =
l l
T C T 0
T C 0
T C T 0
T 0
But did not require cut surface punctures etc.