Semi Conducting Materials Misfit Dislocations March 6 2012

8/2/2019 Semi Conducting Materials Misfit Dislocations March 6 2012

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Semi-conducting & Magnetic Materials

Prof S. B. Sant

Department of Metallurgical & Materials EngineeringIIT Kharagpur

MT41016


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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.


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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.


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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


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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


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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.


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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.


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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.


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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


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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.


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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.


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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.


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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!


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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.


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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.


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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.


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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.


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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 >>


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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


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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


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Misfit Dislocations


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Misfit Dislocations


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Strained-Layers


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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.


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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).


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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.


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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.


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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


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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=


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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


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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


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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.


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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


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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


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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


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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.


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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.


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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.


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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


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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


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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


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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


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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


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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


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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).



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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.



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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.



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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)



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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.



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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



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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.



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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?



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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?



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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.



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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



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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!



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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.

Semi Conducting Materials Misfit Dislocations March 6 2012

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