semi conducting materials misfit dislocations march 6 2012
Post on 05-Apr-2018
221 Views
Preview:
TRANSCRIPT
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
1/58
Semi-conducting & Magnetic Materials
Prof S. B. Sant
Department of Metallurgical & Materials EngineeringIIT Kharagpur
MT41016
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
2/58
Semi-conducting & Magnetic Materials
Misfit Dislocations
Two cubic crystals with lattice constants a1 = 1 units and a2 = 1.05 units
(i.e. a misfit of5%) form a phase boundary interface.
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
3/58
Semi-conducting & Magnetic Materials
Misfit Dislocations
Two cubic crystals with lattice constants a1 = 1 units and a2 = 1.05 units(i.e. a misfit of5%) form a phase boundary interface.
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
4/58
Semi-conducting & Magnetic Materials
Misfit Dislocations
Misfit dislocations compensate for differences in the lattice constants by concentrating
the misfit in one-dimensional regions - the dislocation lines.
Between the dislocation lines the interface is coherent;
a phase boundary with misfit dislocations is called semi-coherent.
Misfit dislocations - in contrast to general grain boundary dislocations
must have an edge componentthat accounts for the lattice constant mismatch
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
5/58
Semi-conducting & Magnetic Materials
Misfit Dislocations
Phase boundary dislocations -misfit dislocations are only a subset),
"simple" misfit dislocations are the dominant defects in technologically importantman-made phase boundaries.
Misfit dislocations are not restricted to boundaries between two chemically different
types of materials.
Silicon heavily doped with, e.g., Boron, has a slightly changed lattice constant and
thus formally can be seen as a different phase.
The rather ill defined interface between a heavily doped region and an undoped
region thus may and does have misfit dislocations, an example is given in the illustration.
Phase boundaries
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
6/58
Semi-conducting & Magnetic Materials
Misfit DislocationsMisfit Dislocations in the Interface between Heavily and Normally Doped Silicon
The TEM micrograph shows a loose network of dislocations between "regular" &
Heavily B-doped Si. The expected square network has not yet fully developed. Many
dislocations are "on their way" from the surface to their proper place in the interface.
The geometry is also not too well defined, because there is no abrupt change of lattice
constants as in the case of phase boundaries between chemically different phases.
The lattice constant changes continuously following the B-concentration whichobeys some diffusion profile.
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
7/58
Semi-conducting & Magnetic Materials
Misfit Dislocations
The mere existence of misfit dislocations coupled with their usually detrimental
influence on electronic properties is the reason why many "obvious" devices do not
exist at all (e.g. optoelectronic GaAs structures integrated on a Si chip), and othershave problems.
The aging of Laser diodes, e.g., may be coupled to the behavior of misfit dislocations
in the many phase boundaries of the device.
Optoelectronics in general practically always involves having phase boundaries, e.g.
devices like Lasers, LEDs, as well as all multi quantum well structures. A very
careful consideration of misfit and misfit dislocations is always needed and some
special process steps are often necessary to avoid these defects.
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
8/58
Semi-conducting & Magnetic Materials
Misfit DislocationsOptoelectronics
Optoelectronics includes all semiconductor devices which emit light through
recombination of electrons and holes. Prime materials are GaAs, GaAlAs, GaP, InSb
and generally all III - V semiconductors, but also GaN orSiC.
In making optoelectronic devices, defect engineering is needed.
Diffusion plays a major role; the precise atomic mechanisms are not too well
understood at present.
Defects in interfaces (= phase boundaries between different optoelectronic materials)play a major role; they essentially limit or prohibit applications in many cases.
In contrast to Si microelectronics, defects may also play a role in thefinished device
while it is in operation. Dislocations, not wholly unavoidable in most III - V materials,
may start to climb and degrade the function.
Early Lasers diodes, e.g., stopped working after few hours of operation because
defects evolved that served as recombination centers impeding radiant recombination.
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
9/58
Semi-conducting & Magnetic Materials
Misfit Dislocations
This is shown in the schematic three-dimensional view of an edge dislocations
in a cubic primitive lattice. This beautiful picture (from Read?) shows the inserted
half-plane very clearly; it serves as the quintessential illustration of what anedge dislocation looks like.
Look at the picture and try to grasp the concept. But don't forget
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
10/58
Semi-conducting & Magnetic Materials
Misfit Dislocations
1. There is no such crystal in nature: All real lattices are more complicated.
2. The exact structure of the dislocation will be more complicated.Edgedislocations are just an extreme form of the possible dislocation structures,
and in most real crystals would be split into "partial" dislocations and look
much more complicated.
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
11/58
Semi-conducting & Magnetic Materials
Misfit Dislocations
We therefore must introduce a more general and necessarily more abstract
definition of what constitutes a dislocation.Before we do that, however, we will continue to look at some properties of
(edge) dislocations in the simplified atomistic view, so we can appreciate
some elementary properties.
First, we look at a simplified but principally correct rendering of the connection
between dislocation movement and plastic deformation - the elementary
process of metal working which contains all the ingredients for a complete
solution of all the riddles and magic of the smiths art.
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
12/58
Semi-conducting & Magnetic Materials
Misfit Dislocations
Generation of an edgeDislocation by a shear
stress
Movement of thedislocation
through the crystal
Shift of the upperhalf of the crystalafter the dislocation
emerged
Dislocations move in response to an external stress .
As soon as a critical shear stress is reached, the dislocation starts movingand deformation is no longer elastic but plastic, because the dislocation will
not move back when the stress is removed.
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
13/58
Semi-conducting & Magnetic MaterialsMisfit Dislocations
The dislocation line moves on its glide plane and produces, upon leaving
the crystal (and thus disappearing), an elementary step on the crystal surface.
Note that after the dislocation disappeared, the crystal is completely stress-free.
Formacroscopic deformation in three dimensions, many dislocations have
to move through the crystal. The elementary process shown above thus has
to be repeated literally billions of times on many (at least 5) different
planes of the lattice.
Plastic deformation proceeds - atomic step by atomic step - by the
generation and movement of dislocations
Without dislocations, there can be no elastic stresses whatsoever in a single crystal!
"Discovery" of dislocations as source of plastic deformation = answer to one of thebiggest and oldest scientific puzzles in 1934 (Taylor, Orowan and Polyani).
No Noble prize!
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
14/58
Semi-conducting & Magnetic Materials
Misfit DislocationsWe already know enough by now, to deduce some elementary properties ofdislocations which must be generally valid.
1. A dislocation is one-dimensional defectbecause the lattice is only disturbed
along the dislocation line (apart from small elastic deformations which we do
not count as defects farther away from the core). The dislocation line thus canbe described at any point by a line vector t(x,y,z).
2. In the dislocation core the bonds between atoms are notin an equilibrium
configuration, i.e. at their minimum enthalpy value; they are heavily distorted.
The dislocation thus must possess energy (per unit of length) and entropy.3. Dislocations move under the influence of external forces which cause internal
stress in a crystal. The area swept by the movement defines a plane,
the glide plane, which always (by definition) contains the dislocation line vector.
4. The movement of a dislocation moves the whole crystal on one side of the
glide plane relative to the other side. (Edge) dislocations could (in principle) be generated by the agglomeration
of point defects: self-interstitial on the extra half-plane, or vacancies on the
missing half-plane. The Burgers vector is perpendicular to the dislocation
line direction.
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
15/58
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
16/58
Semi-conducting & Magnetic Materials
Misfit Dislocations
The special vector needed for closing the circuit in the reference crystal
is by definition theBurgers vectorb.But beware! As always with conventions, you may pick the sign of the
Burgers vector at will.
In the version given here (which is the usual definition), the closed circuit
is around the dislocation, the Burgers vector then appears in the reference crystal.
You could, of course, use a closed circuit in the reference crystal and define
the Burgers vector around the dislocation. You also have to define if you go
clock-wise or counter clock-wise around your circle. You will always get the
same vector, but the sign will be different! And the sign is very important for
calculations! So whatever you do, stay consistent!. In the picture above wewent clock-wise In both cases.
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
17/58
Semi-conducting & Magnetic Materials
Misfit DislocationsNow we go on and learn a new thing: There is a second basic type of dislocation,
called screw dislocation. Its atomistic representation is somewhat more difficult
to draw - but a Burgers circuit is still possible:
You notice that here we chose to go clock-wise - for no particularly good reason.The Burgers vector is parallel to the dislocation line direction.
If you imagine a walk along the non-closed Burgers circuit, which you keep
continuing round and round, it becomes obvious how a screw dislocation got its name.
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
18/58
Semi-conducting & Magnetic Materials
Misfit DislocationsDislocations are characterized by
Their Burgers vectorb =
Vector describing the step obtained after a dislocation passed through the crystal.
Vector obtained by a Burgers circuit around a dislocation.
All definitions ofb give identical results for a given dislocations; but watch out
for sign conventions!
By definition, b is always a translation vectorTof the lattice.
For energetic reasons b is usually the shortest translation vector of the lattice;e.g. b = a/2 for the fcc lattice.
Their line vectort(x,y,z) describing the direction of the dislocation line in the lattice.
t(x,y,z) is an arbitrary (unit) vector in principle but often a prominent latticedirection in reality.
While the dislocation can be curved in any way, it tends to be straight
(= shortest possible distance) for energetic reasons.
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
19/58
Semi-conducting & Magnetic Materials
Misfit Dislocations
The glide plane by necessity must contain t(x,y,z) and b and is thus defined by the
two vectors .
The angle between t(x,y,z) and b determines the character or kind of dislocation:Note that any plane containing tis a glide plane for a screw dislocation.
= 90: Edge dislocation. = 0: Screw dislocation. = 60: "Sixty degree" dislocation. = arbitrary : "Mixed" dislocation.
Dislocations have a large line energyEdisper length and therefore are
never thermal equilibrium defects
//
5
b
eV
disE >>
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
20/58
Semi-conducting & Magnetic Materials
Misfit Dislocations
Misfit dislocations compensate for differences in the lattice constants by concentrating
the misfit in one-dimensional regions - the dislocation lines.
Between the dislocation lines the interface is coherent;
a phase boundary with misfit dislocations is called semi-coherent.
Misfit dislocations - in contrast to general grain boundary dislocations
must have an edge componentthat accounts for the lattice constant mismatch
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
21/58
Semi-conducting & Magnetic Materials
Misfit Dislocations
The total elastic energy contained in the "strained layer" scales with the thickness
of the layer and the expenditure in elastic energy below a critical thickness for an
epitaxial layer may be smaller than the energy needed to introduce misfit dislocations.
However, not every ( = 1) phase boundary with some misfit between the partners
contains misfit dislocations - provided one of the phases consists of a thin layeron
top of the other phase.
Only if the thickness of the thin-layer phase exceeds a critical value, misfit
dislocations will be observed.
It is easy to understand why this is so:For thin layers, it may be energetically more favorable to deform the layerelastically,
so that a perfect match to the substrate layer is achieved.
No Misfit Dislocations
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
22/58
Semi-conducting & Magnetic Materials
Misfit Dislocations
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
23/58
Semi-conducting & Magnetic Materials
Misfit Dislocations
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
24/58
Semi-conducting & Magnetic Materials
Strained-Layers
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
25/58
Semi-conducting & Magnetic Materials
Misfit DislocationsEnergy of Misfit Dislocations and Critical Thickness
The critical thickness for the introduction of misfit dislocations can be obtained by
equating the energy contained in a misfit dislocation network with the elastic
energy contained in a strained layer of thickness h.Since the elastic energy increases directly with h, whereas the energy contained
in the dislocation network increases only very weakly with h, the thickness for
which both energies are equal is the critical thickness hc.
Thicker layers are energetically better off with a dislocation network, thinner
layers prefer elastic distortion.This computation was first done by Frankand van der Merwe in 1963
The resulting Frank and van der Merwe formulabecame quite famous.
Somewhat later in 1974 Matthews and Blakeslee reconsidered the situation
and looked at the forces needed to move a few pre-existing dislocations into the
interface in order to form the misfit dislocation network.
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
26/58
Semi-conducting & Magnetic Materials
Misfit Dislocations
They obtain the same formula for the critical thickness as van der Merwe
(i.e. the equilibrium situation), but their treatment also allows to consider the
kinetics of the process to some extent (i.e. how the network is formed)
and is therefore widely used.
We are looking at the situation retrospectivelyby studying an article of the
possibly most famous TEM and defect expert, Peter Hirsch from
Oxford University, or, to be precise, Sir Peter as he must be called
after his nobilitation by Elizabeth II, Queen of England.
This is to show that honor-wise - a defect expert can go just as far as a rock
Star (several of whom have been knighted by the Queen).
Money-wise, however, it is a completely different matter.
We use parts ofhis articleprinted in the Proceedings of the
2nd International Conf on Polycrystalline Semiconductors
(Schwbisch Hall, Germany, 1990, p. 470).
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
27/58
Semi-conducting & Magnetic Materials
Misfit Dislocations
IntroductionIn 1949 Frank and van der Merwe discussed theoretically the stresses and the
energies at the interface of an epitaxial layer grown on a matrix with a slightly
different lattice parameter. Their one-dimensional model was extended to two
dimensions by Jesser et al. These studies show that if the lattice mismatch issmall, and/or the thickness of the overlayer is not large, the growth of the epilayer is
pseudomorphic (commensurate) with the matrix, with the atomic planes on the two
sides of the interface being in perfect register with each other. The mismatch is
accommodated by an elastic strain in the epilayer giving a biaxial stress of
Nucleation and Propagation of Misfit Dislocations in
Strained Epitaxial Layer Systems
P.B. Hirsch (Sir Peter)
Department of Materials, University of Oxford
( )( )
+=
1
1...2 f [1]
where is the shear modulus, is the Poisson's ratio andf the misfit parameter.Elastic isotropy is assumed.
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
28/58
Semi-conducting & Magnetic Materials
Misfit Dislocations
The misfit parameter is given byf= (ae -am)/am, where ae , am are thelattice parameters of the unconstrained epilayer and matrix in the plane
parallel to the Interface.
For the case ofGexSi1-x alloys Vegard's law is approximately obeyed
i.e. aGeSi = aSi + (aGe - aSi).x, where the a's are the lattice parameters.
This means that forGexSi1-x epilayers on an Si(001) surface,fis a linearfunction ofx; since the lattice parameter ofGe (0.5657nm) is greater than
that ofSi (0.5431nm), fincreases with x (f(x) =0.042x), and the epilayeris in compression.
Beyond a critical strain and/or thickness it becomes energetically
favourable for the misfit to be accommodated by a network of
interface dislocations.
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
29/58
Semi-conducting & Magnetic Materials
Misfit Dislocations
In view of the importance ofstrained epilayers or superlattices for device
applications and the development of methods of growing them, much research
has been devoted in recent years to the conditions which control the relaxation
of elastic strain by the introduction of misfit dislocations, and to the mechanisms
by which they are formed.
This paper presents a brief review of this field of research; it does not pretend tobe exhaustive.
The energy of the system involving a strained epilayer and an array ofmisfit dislocations is generally discussed for the case of the layer and
matrix material being cubic, and the interface parallel to a cube plane.
Energetic Considerations
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
30/58
Semi-conducting & Magnetic Materials
Misfit Dislocations
where h is the film thickness, r0 the core radius, and where the factor 2
arises because of the presence of two orthogonal sets of edge dislocations.The elastic strain remaining is given by:
The energy per unit area of a square grid of edge dislocations, with
Burgers vectorb, with dislocation spacing p is given approximately by
( )
0
2
ln14
.2reh
pb
[2]
p
bf=
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
31/58
Semi-conducting & Magnetic Materials
Misfit DislocationsThe total energy per unit area E is then given by
( )
( )
( )
( )
+
+=
0
2 ln
121
12
r
ehfbhE
where the first term is the elastic strain energy. For a given thickness the minimum
energy occurs for a value 0 given by:
( )
+
=
0
0 ln
18 r
eh
h
b
If0 > f, then the layer is ideally commensurate with the substrate, and the elastic
strain is equal to f.
If0 < fthen some misfit will be relaxed by dislocations, the spacing being given by (5)
with 0 = f - b/p. The critical film thickness, hc , at which it becomes energeticallyfavourable for the first dislocation to be introduced is obtained with 0 = f, i.e.
[3]
[4]
[5]
( )
+
=
0
ln
18 r
eh
f
bh c
c
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
32/58
Semi-conducting & Magnetic Materials
Misfit Dislocations
Two points should be made about this relation.
First, hc depends on the core radius r0 ( b), and the uncertain value of this parameter
introduces some uncertainty into this relation, particularly for small h/r0.
Secondly, hc depends on the assumed dislocation arrangement; for example themisfit might be relieved by dislocations with different b; eqn. (2) shows that the
dislocation strain field energy is smaller for edge dislocations of smaller energy,even though for the same relief of strain (f - ), the spacing p will be smaller.
[5]
( )
0
2
ln
14
.2
r
eh
p
b
[2]
( )
+
=
0
ln
18 r
eh
f
bh c
c
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
33/58
Semi-conducting & Magnetic Materials
Misfit Dislocations
Thus it is necessary to take care in making comparisons between theory and
experiment. In practice, however, it is generally found that the observed
values ofhc are larger than those predicted over most of the range of misfits(see for example People and Bean [6] forGe-Si layers on (100) Si).
The reasons for this discrepancy are partly due to insensitivity of the
experimental techniques used, and partly kinetic in origin.
In order to introduce dislocations, there have to be mechanisms for doing
so, and for most practical cases, except these with very large misfits, the
strain relief is limited by kinetic considerations.
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
34/58
Semi-conducting & Magnetic Materials
Misfit Dislocations - Critical Thickness
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
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
35/58
Semi-conducting & Magnetic Materials
Misfit DislocationsAs you saw, great minds sometimes make great steps and are not immune to
small errors!
If you didn't see that, consider:
How exactly do you get eq. 1?
Why is the strain for minimum energy calculated in eq. 3 equalto the unrelaxed elastic strain at the point of the introduction of dislocations?
What is h, the thickness of the layer, doing in an equation for thecritical thickness hc (eq. 5)? After all, the critical thickness can not possibly
depend on the thickness itself.
Well, if you want to know, turn to the annotated version of Sir Peters paper.
( )
( )
+=
1
1...2 f
( )( )
( )( )
+
+
=
0
2 ln121
12
r
ehfbhE
( )
+
=
0
ln18 r
eh
f
bh c
c
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
36/58
Semi-conducting & Magnetic Materials
Misfit DislocationsThe annotated versionComments on light blue background
1. Introduction
In 1949 Frank and van der Merwe discussed theoretically the stresses and theenergies at the interface of an epitaxial layer grown on a matrix with
a slightly different lattice parameter.
Their one-dimensional model was extended to two dimensions by Jesser et al.
These studies show that if the lattice mismatch is small, and/or the thickness of theoverlayer is not large, the growth of the epilayer is pseudomorphic commensurate)
with the matrix, with the atomic planes on the two sides of the interface being in
perfect register with each other.
The mismatch is accommodated by an elastic strain in the epilayer
giving a biaxial stress of:
( )
+=
1
1.2 f
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
37/58
Semi-conducting & Magnetic Materials
Misfit Dislocations
The misfit parameter is given byf = (ae -am)/am, where ae , am are the latticeparameters of the unconstrained epilayer and matrix in the plane parallelto the interface.
How did he get this formula? Well, this is easy, but deriving the starting
formula also illustrates a certain problem one might encounter from lookingat simple pictures all the time. Let's see:
( )
+=
1
1.2 f
where is the shear modulus, is the Poisson's ratio andf the misfit parameter.Elastic isotropy is assumed.
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
38/58
Semi-conducting & Magnetic Materials
Misfit DislocationsIf you strain the lattice of the epitaxial layer in one dimension, so that it
fits the matrix perfectly, you have the following situation:
The strain needed for perfect fit is = (ae am)/ae; but since ae
and am are nearly equal, dividing by ae (as is correct) or by am (asSir Peter does) makes no difference.
So we can equate withf= (ae am)/am, the misfit parameter
according to Sir Peter.
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
39/58
Semi-conducting & Magnetic Materials
Misfit Dislocations The strain needed for perfect fit is = (ae am)/ae; but since ae and am arenearly equal, dividing by ae (as is correct) or by am (as Sir Peter does)makes no difference.
So we can equate withf= (ae am)/am, the misfit parameter according to Sir Peter.
Strain and stress are usually related by:= E. withE= modulus of elasticity (Youngs modulus).
But sinceEcan be expressed in terms of the shear modulus and Poissons ratio byE=2(1+),
we can write = 2(1+). = 2(1+). fApart from the factor(1 ), this is Sir Peters starting formula.Where does the (1 ) term come from?Let's look at the problem carefully. We actually have a two-dimensionalproblem
and must considerbiaxial stress. Our simple figure was one-dimensional - and that
is where we might have missed something.
Did we miss something? Well, yes - we did.
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
40/58
Semi-conducting & Magnetic Materials
Misfit DislocationsIn the picture above, we have applied a suitable stress to strain the blue lattice to thedesired value in thex-direction. If we now apply the same stress in they-direction
perpendicular to the first one, we will have to use a larger strain because the
y-dimension of the crystal layer will now be smaller as expressed by Poissons modulus.
This is illustrated below.
S i d i M i M i l
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
41/58
Semi-conducting & Magnetic Materials
Misfit DislocationsAfter we pulled the blue crystalsheet to the desired dimension by a
strain 1, its lateral dimension iny
direction decreased by 2q
as
shown. We now must apply a strain
of2 = 1 (1 ) to make thematch in y-direction.
That's it. For reasons of symmetry,this must be the strain corrected for
biaxial stress in both directions, i.e.
we have
1 = 1,2 (1 )This is the expression used by Sir
Peter.
S i d i & M i M i l
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
42/58
Semi-conducting & Magnetic Materials
Misfit DislocationsAfter we pulled the blue crystal sheet to the desired dimension by a strain 1,
its lateral dimension iny direction decreased by 2q as shown. We now must
apply a strain of2
= 1
(1 ) to make the match in y-direction.
That's it. For reasons of symmetry, this must be the strain corrected for
biaxial stress in both directions, i.e. we have
1 = 1,2 (1 )
This is the expression used by Sir Peter.
Of course, we made a little mistake. The deformation iny-direction will lead to a
shrinkage inx-direction which me must compensate, which in turn will lead to
a shrinkage in y-direction, which will lead to a shrinkage - and so on ad infinitum.
But all this does is to add higher order terms in which we commonly
neglect in linear elasticity theory.
S i d ti & M ti M t i l
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
43/58
Semi-conducting & Magnetic Materials
Misfit DislocationsFor the case ofGexSi1-x alloys, Vegard's law is approximately obeyed,
i.e. aGeSi = aSi+(aGe - aSi)x, where the a's are the lattice parameters.
This means that forGexSi1-x epilayers on an Si(001) surface, fis a linearfunction ofx; since the lattice parameter ofGe (0.5657nm) is greater than
that ofSi (0.5431nm), fincreases with x (f(x) =0.042x), and the epilayeris in compression.
Beyond a critical strain and/or thickness it becomes energetically favourable
for the misfit to be accommodated by a network of interface dislocations.
In view of the importance of strained epilayers or superlattices for device
applications and the development of methods of growing them, much research
has been devoted in recent years to the conditions which control the relaxation
of elastic strain by the introduction of misfit dislocations, and to the
mechanisms by which they are formed.
S i d ti & M ti M t i l
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
44/58
Semi-conducting & Magnetic Materials
Misfit DislocationsEnergetic ConsiderationsThe energy of the system involving a strained epilayer and an array of misfit
dislocations is generally discussed for the case of the layer and matrix material
being cubic, and the interface parallel to a cube plane. The energy per unit area
of a square grid of edge dislocations, Burgers vectorb, with dislocation spacing pis given approximately by
where h is the film thickness, r0 the core radius, and where the factor2arises because of the presence of two orthogonal sets of edge dislocations.
First, there is a slight correction in eqn (2):
The "x" after the "2" as shown in the original has been replaced by a dot
- because the "", written as "x" as a sign for multiplication is no longer
allowed; it has been used up as denoting the amount ofGe in the alloy GexSi1 x.
( )
0
2
ln
14
.2
r
eh
p
b
Semi cond cting & Magnetic Materials
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
45/58
Semi-conducting & Magnetic Materials
Misfit Dislocations
Now the formula contains an unexplained"e".
Since we know that the energy of a dislocation contains the term ln(R/r0) with
R being some outer radius, it is clear thatR cannot be larger than h, thethickness of the layer.
But by simply equatingR with h, we make some numerical mistake which wemight correct by introducing a unspecified (but probably not very large)
correction factore?
That was the first thought. Well - wrong! Sir Peter simply takes one of themany formulas for the total energy of a dislocation that float around, it isthe same formula as shown before (for the purpose of recalling it here),ande ise indeed - the base for natural logarithms.
Semi conducting & Magnetic Materials
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
46/58
Semi-conducting & Magnetic Materials
Misfit DislocationsWe then have the correct formula except for the 2/p.
But this is easy and pointed out (albeit somewhat obliquely) by Sir Peter:
The general formula for the dislocation energy gives the energy per unit length
of the dislocation. If we want the energy per unit area, we have to multiply by
the length of the dislocations in a unit area and then divide by the unit area.
If we take the unit area to bep2, the areas of one cell of the (square)dislocation network, it contains dislocations with the length 2p
we have the factor2/p.
The elastic strain remaining is given by = f b/p.
This is something easy to figure out for yourself.
Just take into account that every dislocation with a Burgers vectorb relaxes
the total deformation by one b (provided it is fully contained in the plane
of the boundary).
Semi conducting & Magnetic Materials
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
47/58
Semi-conducting & Magnetic Materials
Misfit DislocationsFull relaxation thus would occur if a misfit dislocation network with a spacingp = b/f = b/(ae am)/am is introduced which partially relaxes the epitaxial layer.
The essential trick is to generate a variable e which is the residual straincontained in a partially relaxed epitaxial layer. So some of the strain and its
energy is gone, but at a cost: Dislocations, carrying their own energy penalty,
are introduced.
Since is a variable, it can now be used to optimize the system as we will see.The total energy per unit area E is then given by
( )( )
( )( )
+
+=
0
2 ln121
12rehfbhE
(3)
where the first term is the elastic strain energy.
Semi conducting & Magnetic Materials
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
48/58
Semi-conducting & Magnetic Materials
Misfit Dislocations
However, p has been replacedusing the relation = f b/p, or
p = b/ (f ).
The next part is straight forward. The total energy per unit area is a functionof, the strain still present in the epitaxial layer even after some dislocations
have been introduced.So we can find the minimum energy of the system with respect to bycalculating dE/d = 0.
The calculation is straight forward:For a given thickness the minimum energy occurs for a value
0 given by
Here Sir Peters gets a bit tricky once more. The elastic energy Eelast of a
uniaxially elastically deformed material is simple ./2. For biaxial strainit is twice that, and that gives the first term.
The second term is the energy of the dislocation network; it is almosttheformula from above.
Semi conducting & Magnetic Materials
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
49/58
Semi-conducting & Magnetic Materials
Misfit Dislocations
If0 = f, then the layer is ideally commensurate with the substrate, and the
elastic strain is equal to f.
If0 < f, then some misfit will be relaxed by
dislocations, the spacing being given by (5) with 0 = f b/p.
The critical film thickness, hc , at which it becomes energetically favourable
for the first dislocation to be introduced is obtained with 0 = f, i.e.
( )
+
=
0
ln18 r
eh
f
bh c
c
Got it? Well, lets look at the argumentation in detail?
( )
+
=
0
0 ln
18 r
eh
h
b
(4)
For a given thickness the minimum energy occurs for a value 0 given by
(5)
Semi-conducting & Magnetic Materials
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
50/58
Semi-conducting & Magnetic Materials
Misfit Dislocations
If the remaining strain 0 = f, then it is the strain of the unrelaxed layer andthe formula defining yields b/p = 0 which, since b has a defined value, can
only meanp is infinite - in other words there is no dislocation network.
If0 > f, the layer is not ideally commensurate with the substrate as stated inthe original, but a dislocation network must be present (b/p < 0 is required)
which increases the strain - the sign ofb is the wrong way around.This is of course a totally unphysical high energy situation, and we can safely
exclude 0 > f,from the possible range ofe values.
If0 < f, again a dislocation network must be present, but this time with theright sign ofb - it decreases the total elastic strain.
Semi-conducting & Magnetic Materials
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
51/58
Semi-conducting & Magnetic Materials
Misfit DislocationsNow, 0, the optimal residual strain is a function of the layer thickness h.
It decreases with increasing h and this means that dislocations must be
introduced at some critical thickness hc.
If the thickness is below hc we have no dislocations and = fobtains.If the thickness is above h
c
, we have dislocations and = fobtains.
This leaves us with 0 = fat the point of critical thickness.
All we have to do now is to express the equation above forh;
it will then give hc by substitutingffor0.Sir Peter now writes
( )
+
=
0
ln18 r
eh
f
bh c
c
Semi-conducting & Magnetic Materials
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
52/58
Semi conducting & Magnetic Materials
Misfit DislocationsTwo points should be made about this relation:
First, the above version of equation (5) corrects the little mistake of the
original - forgetting the index "c" at the h in the argument of the logarithm, andSecond; this equation is now a transcendent equation forhc;we cannot write it down in closed form.
hc depends on the core radius r0 ( b), and the uncertain value of thisparameter introduces some uncertainty into this relation, particularly for small h/r0.
hc depends on the assumed dislocation arrangement; for example the
misfit might be relieved by dislocations with different b; eqn. (2) shows that thedislocation strain field energy is smaller for edge dislocations of smaller energy,
even though for the same relief of strain (f ), the spacing p will be smaller.
Semi-conducting & Magnetic Materials
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
53/58
Semi conducting & Magnetic Materials
Misfit DislocationsThus it is necessary to take care in making comparisons between theory
and experiment.
In practice, however, it is generally found that the observed values ofhcare larger than those predicted over most of the range of misfits
(see for example People and Bean [6] forGe-Si layers on (100) Si).
The reasons for this discrepancy are partly due to insensitivity of the
experimental techniques used, and partly kinetic in origin. In order to
introduce dislocations, there have to be mechanisms for doing so, and
for most practical cases, except these with very large misfits, the strain
relief is limited by kinetic considerations.
Now, what order of magnitude do we get forhc, forgetting about thesmall detail of unclear core radii and so on?
Semi-conducting & Magnetic Materials
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
54/58
Semi conducting & Magnetic Materials
Misfit Dislocations
Well, if we rewrite the above equation, with 8(1 + ) 30,andln hc/r0 = y, we have
hc b/f (ln y)/30
How large is ln y? If we take r0 to be about0.3 nm, andhc to be anywhere
between 10 nm and1000 nm, we have a range of values from ln(10/0.3) = 3.51to ln(1000/0.3) = 8.11.
In other words, it doesn't matter much for orders of magnitude.
Lets take an intermediate value of5 and we gethc b/6 f .
Now here is a simple formula!
But how good is it?
Semi-conducting & Magnetic Materials
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
55/58
Semi conducting & Magnetic Materials
Misfit DislocationsWell, we live in the age of easy accessible PCs with tremendous computingpower, so solving the transcendental equation from above is actually no
problem at all.
The best approximation is actually obtained forln y = 3.03 leading to
So if your misfit is 1% (f = 0.01),your critical thickness will be roughly
around10 b. Burger vectors usually are b = a/2 orb = a/22which is around0.3 nm.
This gives
hc 3 nm - which is not all that much!
f
bhc
9.9
Fortunately, as Sir Peter points out in the remainder of the article, the
critical thicknesses observed are usually considerably larger than the
calculated ones.
Semi-conducting & Magnetic Materials
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
56/58
Semi conducting & Magnetic Materials
Still, Sir Peter got it right in principle, and his derivation of the critical
thickness is short and most elegant. The final formula for the critical
thickness hc is
Misfit Dislocations
( )
+=
0
ln18 r
eh
f
bh cc
With b = Burgers vector of the misfit dislocations (actually only their edge
component in the plane of the interface), f= misfit parameter, i.e a/a,e = 2.7183... =base of natural logarithms, and
r0 = core radius of the dislocations.
This transcendental equation may be roughly approximated by
f
bhc
9.9
Semi-conducting & Magnetic Materials
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
57/58
g gMisfit Dislocations
Lets see what the calculations tell us for real phase boundaries (for a b value of
0.376 nm (which applies to Si)). We note that misfit dislocations are only to be
expected if the layer thickness h exceeds the critical value hc.
For a misfit of 1% the critical thickness is about 4 nm - not much at all!
Semi-conducting & Magnetic Materials
-
8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012
58/58
g gMisfit Dislocations
Experiments confirm the theory.
Very thin epitaxial layers do not show dislocations in the interface, but with
increasing thickness misfit dislocations will appear.
Considering that misfit dislocations are usually unwanted but that they mustappear with increasing layer thickness - however not out of thin air
we ask an important question:
Exactly how are misfit dislocations produced and incorporated into theinterface if the critical thickness is reached. More to the point: How can I
prevent this nucleation and migrationprocess?
Suffice it to say that while this question has not been fully answered,there are many ways and tricks to keep misfit dislocations from appearing
at the earliest possible moment.
top related