! ! ! 3. Dislocations

Hartmut S. Leipner: Structure of imperfect materials

!"#"$%!"#"$"%&'"(')*+$",-."/+&'()*+,-.+/$,+/,0123$456706)$80,2+63$2910)3$'0/)(29:

!";"$$<*-).,(29$2=0+69$+>$'()*+,-.+/)

!"!"$$?+60$)265,2560$+>$'()*+,-.+/)$(/$1-6.,5*-6$,69)2-*)

!"@"$$A()*+,-.+/$B+.+/$-/'$70/06-.+/

&A()*+,-.+/$,+60$)265,2560:C**$6(7=2)$60)0680'$D$;E#E$FGHIHJK$L-**0

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hsl 2010 – Structure of imperfect crystals – 3.1 Dislocation topology

Navaho

“Symmetrie wird durch kleine Abweichungen von exakter Regelmäßigkeit erst faszinierend: Kein Kristall ohne Baufehler, kein Mosaik ohne kleine Störungen, und selbst die Navaho-Indianer bauten in ihre Teppiche einen weißen Faden ein, damit die bösen Geister entweichen können.”

[Hahn 2000]

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hsl 2010 – Structure of imperfect crystals – 3.1 Dislocation topology

! Existence of dislocations as line defects deduced as early as in the 1930ies (Orowan, Polanyi, Taylor independently in 1934)

! Strong evidence for dislocations: comparison of theoretical and experimental shear stress (Frenkel)

Plastic deformation of crystals

b

a

StressPeriodic shearing force to move the top atomic row:

For small shear x/b ! Hooke’s law,

Max. ! is the theoretical critical shear stress:

i. e. !max ~ G

Experimentally:

10-4 to 10-8 G

Stress

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hsl 2010 – Structure of imperfect crystals – 3.1 Dislocation topology

[E. Weber]

Dislocation concept

[Russ 1996]

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hsl 2010 – Structure of imperfect crystals – 3.1 Dislocation topology

Gedankenexperiment for the generation of an edge-type dislocation by inserting an extra half-plane of atoms in a simple cubic structure

[Hull, Bacon 1992]

Geometry of dislocations

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hsl 2010 – Structure of imperfect crystals – 3.1 Dislocation topology

Edge dislocation in the sc structure

S03-90

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hsl 2010 – Structure of imperfect crystals – 3.1 Dislocation topology

Shiites

!

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hsl 2010 – Structure of imperfect crystals – 3.1 Dislocation topology

Echinicactus grusonii

“Chair for the mother in law“

!

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hsl 2010 – Structure of imperfect crystals – 3.1 Dislocation topology

Screw dislocation

S04-90

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hsl 2010 – Structure of imperfect crystals – 3.1 Dislocation topology

Surface spiral

Spiral growth on a silicon carbide facette. The material is grown from the vapor phase on a seed crystal.

[Phys. uns. Zeit 2002]

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hsl 2010 – Structure of imperfect crystals – 3.1 Dislocation topology

Burgers circuit

Burgers vector b: 1 ! 19

" !

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hsl 2010 – Structure of imperfect crystals – 3.1 Dislocation topology

Transmission electron microscopy: [110] lattice image of a dissociated 60° dislocation in GaAs[Gerthsen, Carter 1993]

High-resolution TEM

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hsl 2010 – Structure of imperfect crystals – 3.1 Dislocation topology

! Sign of the Burgers vector depends on the sense of a dislocation line; not unambiguous

! FS/RH rule by Hirth, Lothe

! b ! ! for edge dislocation, b || ! for screw.

! b = 0 for point defects.

! Dislocations with b and -b are different(alternatively with equal b and opposite !).

Properties of the Burgers vector

FS/RH circuits in a real and a perfect reference crystal. ! points into the drawing plane. [Hirth, Lothe 1982]

Burgers circuit

" !

b

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hsl 2010 – Structure of imperfect crystals – 3.1 Dislocation topology

Annihilation of dislocations

b2

b1

b1 + b2 = 0

S06-90

Opposite sign of the Burgers vectors give:

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b =12�111�

hsl 2010 – Structure of imperfect crystals – 3.1 Dislocation topology

! b are shortest lattice translation vectors (perfect dislocations),

e. g. bcc , fcc ! b and ! define the slip plane of the dislocation.

! b || ! for screws, all planes containing ! are slip planes

! b conserves along a dislocation (important if the dislocation line ! changes).

! Existence of mixed dislocations with arbitrary angle between b and !! Dislocations cannot end within a crystal

(only at surfaces, grain boundaries, or other dislocations).! Dislocation loops

Geometric properties of dislocations

b =12�110�

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hsl 2010 – Structure of imperfect crystals – 3.1 Dislocation topology

Mixed dislocation

! b

! b

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hsl 2010 – Structure of imperfect crystals – 3.1 Dislocation topology

Perfect dislocation loop

b

b " !"

b " !#

Perfect dislocation loop consisting of screw and 60° segments

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hsl 2010 – Structure of imperfect crystals – 3.1 Dislocation topology

Dislocation node

b3b2 b3b2

b1 b1

b1 = b2 + b3 b1 + b2 + b3 = 0

!2 !3 !3!2

!1 !1

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hsl 2010 – Structure of imperfect crystals – 3.1 Dislocation topology

Arrangement of dislocations in a well annealed crystal [Hull, Bacon 1992]

! Definition of the dislocation density: " = l/V(total dislocation length per unit volume)

! Given in units of cm–2

! Typical numbers in well annealed metals 106 to 108 cm–2,in semiconductors 10 to 105 cm–2

! After plastic deformation 1012 cm–2 and above

Dislocation density in cm–2

102 104 106 108 1010 1012 1014

Total length in km/cm3

10–3 0.1 10 103 105 107 109

Average distance 1/2 in m

10–3 10–4 10–5 10–6 10–7 10–8 10–9

Relative distance in 5 10–10 m

2 106 2 105 2 104 2 103 200 20 2

Dislocation density

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hsl 2010 – Structure of imperfect crystals – 3.1 Dislocation topology

<2,=$1-M06/$+/$-$&###:$7-**(5B$1=+)1=('0$)56>-,0$&+1.,-*$B(,6+),+19$(B-70:

Etch pit density

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hsl 2010 – Structure of imperfect crystals – 3.1 Dislocation topology

Dislocation etch pits

Formation of etch pits at the intersection of dislocations with the surface.

(a) The cylinder around an edge dislocation represents the region with different chemical and physical properties.

(b) A conical pit forms due to preferential removal of atoms from the imperfect region. Emergent site of a screw dislocation. The pit forms due to the chemical resolution as a reverse process to crystal growth.

[Hull:93]

Dislocation etch pits

(a)

(b)

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hsl 2010 – Structure of imperfect crystals – 3.1 Dislocation topology

Visualization of dislocations by TEM

N6-/)B())(+/$0*0,26+/$B(,6+),+19$+>$1*-).,-**9$'0>+6B0'$O-C)$&'(P6-,.+/$,+/26-)2$(B-70:

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hsl 2010 – Structure of imperfect crystals – 3.1 Dislocation topology

Dislocation imaging in TEM

! Dislocations are visible in diffraction contrast through their strain field,

giving rise to a local bending of the lattice plane.! Parameters to be determined: magnitude and direction of b and !, slip plane

! Imaging of dislocation configurations: interactions, helix structures, tangles

etc.! Determination of the dislocation density! Imaging of defects on the dislocation line, such as jogs and kinks,

determination of dislocation dissociation

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hsl 2010 – Structure of imperfect crystals – 3.1 Dislocation topology

Dislocation imaging in TEM

For a samples slightly tilted out of the Bragg position, only the lattice planes bended near the dislocations fulfil the Bragg condition.

(hkl) planes reflect (hkl) planes reflect---

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hsl 2010 – Structure of imperfect crystals – 3.1 Dislocation topology

Extinction rule in TEM

g1

g2

g3

b

Demonstration of the extinction rule for an edge dislocation. Only the net planes used for the imaging with the diffraction vector g1 are strongly bent.

g1"b ! 0, g2"b = 0, g3"b = 0

g"b = 0

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