Crystal-Air surface
Interphase
boundary
Crystal Boundary
Crystal-Crystal
Grain
boundary
2D DEFECTS
(Surface / Interface)
Stacking Faults
Anti-phase Boundary
Twin Boundary
Low
angle
High
angle
Low angle
High angle
Based on angle of rotation
Homophase
Based on axis
Based on Lattice Models
Twist
Tilt
Mixed
Special
Based on Geometry
of the Boundary plane
Random
Curved
Faceted
Mixed
CSL/Other
Curtesy S. Van Tenderloo
Picture ARM-UC Berkeley4
Low angle
Semicoherent
High angle Incoherent
Based on angle of rotation
Interphase
Based on axis
Based on Lattice Models
Twist
Tilt
Mixed
Special
Based on Geometry
of the Boundary plane
Epitaxial/Coherent
Random
Curved
Faceted
Mixed
Wulff-type
constructions
Coherence at interfaces
• Coherent interface means an interface in which
the atoms match up on a 1-to-1 basis (even if
some elastic strain is present).
• Incoherent interface means an interface in which
the atomic structure is disordered.
• Semi-coherent interface means an interface in
which the atoms match up, but only on a local
basis, with defects (dislocations) in between.
Coherent interfaces
• Coherent interface means an interface in
which the atoms match up on a 1-to-1
basis (even if some elastic strain is
present).
• Near identical lattice parameters, often
thin layers of A on B
Incoherent interfaces
• Incoherent interface means an interface in
which the atomic structure is disordered.
• General case, analogous to a general
high-angle grain boundary (roughly)
Semi-coherent interfaces
• Semi-coherent interface means an interface in
which the atoms match up, but only on a local
basis, with defects (dislocations) in between.
• Comparable to a low-angle grain boundary with
a dislocation array (now called misfit
dislocations)
Epitaxy
Britannica Concise Encyclopedia: epitaxy
Process of growing a crystal of a particular orientation on
top of another crystal. If both crystals are of the same
material, the process is known as homoepitaxy; if the
materials are different, it is known as heteroepitaxy.
Common types of epitaxy include vapour phase, liquid
phase, and solid phase, according to the source of the
atoms being arranged on the substrate.
Comment 1: “growth” is not needed here…
Comment 2: often used more generally than this
Main Types of Epitaxy
• Homoepitaxy
– Growth of material on the same substrate (Si on Si)
• Pseudomorphic growth
– Material adopts the lattice of substrate/matrix
• Coincidence
– Material has certain spacings common with
substrate/matrix
– Similar to CSL
• Cube-Cube
– Major orientations are parallel, e.g. [001]A//[001]
substrate
Heteroepitaxial growth modes
Frank-van der
Merwe
Volmer-Weber
Stranski-Krastanov
trade surface for interface
relieve stress
2
1
layer-by-layer
Pseudomorphic Growth
Pseudomorphic Growth
• Consider a layer of “A” on “B”, of thickness
t
• Take z normal to film, x in plane
• Suppose that lattice of A is larger than that
of B, and would match that of B is strained
by exx along x
• Strain energy scales as texx2 (I leave to
you to work this out in detail…) per unit
area
Interface Energy
• If A matches the lattice of B, the “bonding”
will be good
• Energy of interface per unit area is gAB
• Total energy of system
– E = t*exx2 + gAB
Alternative
• Hetero epitaxial growth (“lattice-mismatched” growth) permits
the fabrication of dissimilar materials on the same substrate
• Strain in the growing film depends on thickness and mismatch
Thin layer - the film will
elastically deform to match
the in-plane lattice
parameter of the substrate
Thick layer - film will revert to its unstrained
lattice parameter, with misfit dislocations at
the interface with the substrate
Alternative, dislocations
• Put dislocations at the interfaces of Burgers
vector b, separation L
• Assume that these remove all the strain
– b/L = exx
• Energy of dislocations per unit area will scale as
b2/L (better, use Read-Shockley model or similar,
Frank-Van Der Merwe)
– Note: no t dependence
T
T
T
T
Energy Balance
• For “phase transition” pseudomorphic to
dislocations
DE = -C1*t*exx2 + C2(b2/L)
2.5
Strained
Relaxed
2
1.5
1
0.5
0
0
0.5
1
1.5
2
Dislocation Standoff
Dislocation energy scales with
shear modulus
T
m1
T
T
m2
T
T
T
T
T
T
m1 > m2
T
T
m1 = m2
m1 < m2
T
Energy Balance
• Better, consider a half dislocation loop growing
in (kinetics)
• Energy of loop = pRC2b2
• Strain energy relieved = C1pR2/2exx2
• For transition (remove p & 1/2)
DE = -C1R2exx2 + RC2b2
4.5
Strained
4
Relaxed
3.5
3
R
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
Classic Nucleation Problem
0.5
0
0
-0.5
-1
-1.5
-2
-2.5
0.5
1
1.5
2
Islands
Similar Cases
• Thin films/precipitates can have different
structures
– Energy for phase change < interface energy
DE = C1*V + C2V2/3(gA-gB)
B1-AlN
VN
3 nm
Epitaxial Stabilization of B1-AlN in AlN/VN
Al Superlattices
N
a VN
w-AlN
zb-AlN
B1-AlN
5
a
relative to wurtzite AlN
Energy of B1-AlN and zb-AlN vs.
underlayer lattice constant (not
including the interfacial energy).
[Madan et al.]
Total energy per unitcell (eV)
TiN
4
B1
3
2
1
zinc-blende
0
4.0
4.2
4.4
4.6
4.8
Underlayer Lattice Constant (Å)
Similar Cases
• Nanoparticles can have different
structures
– Energy for elastic strain < surface energy
DE = C1*V + (CA-CB) V2/3gA
Stranski-Krastanow Growth
• Formation of 3D
structures (q-dots)
preceded by wetting
layer
• Relieve strain energy,
increase surface
energy
DE = C1V2/3+C2V
Comments
• Similar to CSL boundaries, one can have
dislocations of the coherency between the
two materials at an interface
• A step at the interface is normally a
different type of dislocation – sessile
(immobile)
• There is more….
Low angle
Semicoherent
High angle Incoherent
Based on angle of rotation
Interphase
Based on axis
Based on Lattice Models
Twist
Tilt
Mixed
Special
Based on Geometry
of the Boundary plane
Epitaxial/Coherent
Random
Curved
Faceted
Mixed
Wulff-type
constructions
Curtesy S. Van Tenderloo
Picture ARM-UC Berkeley30