Introduction to Dislocation Mechanicsjz203/research/aarg/lecturenotes/... · 2015. 11. 3. · 21 Target Predictions of Dislocation Mechanics Capability: Coarse Features Stress Arising

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.