Empirical Potential Methods and Strain in Atomistic Simulations

agreementkittensSemiconductor

Nov 1, 2013 (3 years and 11 months ago)

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TMCS
III, Leeds 18
th

Jan
2012

Slide
1

Outline

1.
Epitaxy

in Semiconductor Crystal Growth

2.
Elastic Description of Strain in Cubic Semiconductor Crystals

3.
Atomistic Description of Strain

4.
Molecular Statics and Force Fields

5.
Keating's Valence Force Field

6.
Stillinger
-
Weber Potential

7.
Tersoff

Potential

8.
Simulation of
Nanostructures

9.
Piezoelectricity in
Zincblende

and
W
urtzite

crystals


TMCS
III, Leeds 18
th

Jan
2012

Slide
2

Semiconductors

Group IV: Si, Ge, C

Group III
-
V: GaAs, InAs, AlAs, GaP, InP, AlP, GaN, InN, AlN, GaSb, InSb, AlSb

Group II
-
VI: CdSe, ZnSe, ZnS, CdS, MgSe, ZnTe

InN

TMCS
III, Leeds 18
th

Jan
2012

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3

Epitaxy

Epitaxy
: (Greek; epi "above" and taxis "in ordered manner")
describes an ordered crystalline growth on a monocrystalline
substrate.


Homo
-
epitaxy

(same layer and substrate material)


Hetero
-
epitaxy

(different layer and substrate material).


In the Hetero
-
epitaxy case growth can be:


Lattice Matched
: same, or very close, lattice constant of layer and
substrate

e.g. GaSb/InAs or AlAs/GaAs


Lattice Mismatched
: different lattice constant of layer and substrate
material e.g. InP/GaAs or InN/GaN.

TMCS
III, Leeds 18
th

Jan
2012

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4

Lattice Matched and Mismatched Epitaxy

Lattice Matched

Lattice Mismatched

TMCS
III, Leeds 18
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Jan
2012

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5

Lattice Mismatched Epitaxy

Lattice Mismatched

In lattice mismatched heteroepitaxy the
layer material can be made to “adapt”
(can become smaller or larger) its in
plane lattice constant to match that of
the substrate (pseudomorphic growth).

Consequently volume conservation
(though volume is not perfectly
conserved) dictates that the lattice
constant in the growth direction needs
to become larger/smaller.

In this way the lattice periodicity is
maintained in the growth plane, but lost
in the growth direction.

TMCS
III, Leeds 18
th

Jan
2012

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6

Quantum Mechanics in action

1D Quantum Well

In
0.52
Al
0.48
As

In
0.52
Al
0.48
As

In
0.84
Ga
0.16
As

AlAs

AlAs

Growth direction

Nanostructures: 2D Growth, 2D Growth + etching, 3D Growth

Green: Free Carrier
,
Red: Confinement

2D Multi Quantum Wires

Taurino et al Mat Sci and Eng B
,
67 (1999) 39


Scanning
Tunneling
Microscopy

TMCS
III, Leeds 18
th

Jan
2012

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7

Outline

1.
Epitaxy

in Semiconductor Crystal Growth

2.
Elastic Description of Strain in Cubic Semiconductor Crystals

3.
Atomistic Description of Strain

4.
Molecular Statics and Force Fields

5.
Keating's Valence Force Field

6.
Stillinger
-
Weber Potential

7.
Tersoff

Potential

8.
Simulation of
Nanostructures

9.
Piezoelectricity in
Zincblende

and
Wurtzite

crystals



TMCS
III, Leeds 18
th

Jan
2012

Slide
8

Elastic Strain

Semiconductors are produced by depositing liquid or gasses that
when coalesce and solidify follow the crystal structure of the “seed”,
usually a substrate of high crystalline quality.

During this deposition, often done in very small amounts (low growth
rate), as small as depositing one atomic layer at the time, if the layer
material has a bulk lattice constant larger than the substrate, then
the crystal will appear slightly deformed from its equilibrium state.

We refer to this material as “strained”.

Unstrained

Strained

We chose the axes vectors
x,y,z arbitrarily, but need to
be linearly independent.

Note that while the axes
vectors are chosen to be
unitary (in units of the lattice
constant) in the unstrained
case, the strained axes are
not necessarily unitary.

TMCS
III, Leeds 18
th

Jan
2012

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9

Elastic Strain

Unstrained and strained axis can be easily related:

The numerical coefficient
ε
ij
define the deformation of all the
atoms in the Crystal.

The diagonal terms
ε
ii
control
the length of the axis, while the
off diagonal terms
ε
ij

control the
angles between the axis.

This picture is general
and valid for all types
of crystals, not just
simple cubic.

Unstrained

Strained

TMCS
III, Leeds 18
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Jan
2012

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10

Elastic Strain

Unstrained

Strained

This set of equations are in the
form of a mathematical entity
called Tensor.


The equations define the strained
position of any atom within the
crystal that upon strain moves
from
R
to

R’
.

Unstrained and strained
positions are written in terms of
the old and new axis:

Important: notice how the
coefficients
α
,
β
,
γ

are the same in the
unstrained and strained system

TMCS
III, Leeds 18
th

Jan
2012

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11

Elastic Strain

We now substitute the new axis with the expressions for the distortion:

After a little manipulation and taking into account the expression for
R:


Provided the original position and the distortion tensor are known, this
expression gives a practical way of calculating the position of any atom
inside a strained unit cell.

TMCS
III, Leeds 18
th

Jan
2012

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12

Strain Components

Often there is confusion between the terms strain and distortion. In
this lectures we follow the notation used in Jasprit Singh’s book, for
which the strain components e
ij

are different from the distortion
components but related to them by:

The final expressions for the off
-
diagonal terms e
ij

are an
approximation in the limit of small strain.

Dilation
: expresses how much the volume of the unit cell changes, and
in the limit of small strain is given by:

Biaxial Strain
: expresses how much the unit cell is strained in the z
direction compared to the x and y:

Uniaxial Strain
: strain in one direction only, e.g. if e
ij
= constant and
e
ii
=0 then the strain is uniaxial in the [111]

TMCS
III, Leeds 18
th

Jan
2012

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13

Stress Components

Stress components
: the force components (per unit area) that
causes the distortion of the unit cell.

There are 9 components:

X
x
, X
y
, X
z
, Y
x
, Y
y
, Y
z
, Z
x
, Z
y
, Z
z


Capital letters: direction of the force

Subscript: direction normal to the plane on which the stress is
applied (x is normal to yz, y is normal to xz, z is normal to xy, )

X
x

x
y

The number of independent
components reduces when we consider
that in cubic systems (like diamond or
zincblende) there is no torque on the
system (stress does not produce
angular acceleration).

Therefore X
y
= Y
x
, Y
z
= Z
y
, Z
x
= X
z

And we are only left with 6:

X
x
, Y
y
, Z
z

; Y
z
, Z
x
, X
y


TMCS
III, Leeds 18
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Jan
2012

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14

Elastic constants

The stress components are connected to the strain components via
the small strain elastic constants:

In practice we never have to deal with all 36 elastic constants.

First of all it is always the case that c
ij
=c
ji
which reduced the total to 21.

Second in real crystals, particularly cubic, the lattice symmetry reduces
the number even more.

Therefore in ZB we only have 3 independent constants: c
11
,c
12
,c
44

In WZ there are 5: c
11
,c
12
,c
13
, c
33
, c
44

TMCS
III, Leeds 18
th

Jan
2012

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15

Some more definitions

Elastic strain energy density for ZB
:

Bulk Modulus for ZB
:

Shear Constant for ZB
:

TMCS
III, Leeds 18
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Jan
2012

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16

Properties of Semiconductors

ZB


a

B


C’


c11


c12


c44



(Ǻ)


(Mbar)


(Mbar) (Mbar) (Mbar)

(Mbar)


Si


5.431

0.980


0.502 1.660


0.640 0.796

Ge


5.658 0.713 0.410


1.260


0.440 0.677

C


3.567 0.442 0.478


10.79


1.24 5.78



Ga
-
As


5.653

0.757


0.364 1.242


0.514


0.634

In
-
As


6.058

0.617


0.229


0.922


0.465 0.444

Al
-
As


5.662

0.747


0.288


1.131


0.555 0.547

Ga
-
P


5.451

0.921


0.440


1.507


0.628 0.763

In
-
P


5.869

0.736


0.269


1.095


0.556 0.526

Al
-
P


5.463

0.886


0.329


1.325


0.667 0.627

Ga
-
N


4.500

2.060


0.825


3.159 1.510 1.976

In
-
N


4.980

1.476


0.424


2.040


1.190 1.141

Al
-
N


4.380

2.030


0.698


2.961


1.565 2.004

Ga
-
Sb


6.096

0.567


0.270


0.927


0.378 0.462

In
-
Sb


6.479

0.476


0.183


0.720


0.354 0.341

Al
-
Sb


6.135

0.855


0.414


1.407


0.579 0.399

TMCS
III, Leeds 18
th

Jan
2012

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17

Strain in Lattice Mismatched Epitaxy

Poisson ratio
: is a measure of the tendency of materials to
stretch in one direction when compressed in another. This ratio
depends on the substrate orientation and the type of crystal. For
cubic crystals including ZB:

Strain
: in pseudomorphic growth one can consider, independent of
the substrate orientation, strain to have only two components, one
parallel to the growth plane and one perpendicular.

Important: in [001] growth:
e


= e
xx
= e
yy

and
e

= e
zz


TMCS
III, Leeds 18
th

Jan
2012

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18

Strain in [111] pseudomorphically
grown layers

Important: in [111] growth the combination of
e


and
e


results in a
strain tensor with e
xx
= e
yy
= e
zz
and e
xy
= e
xz
= e
yz


The distortions in this case are:

[111]

(1,1,1)

x

z

y

Important: the distortions are expressed
in the basis system where x, y and z are
aligned with the [100], [010] and [001]
directions.

Instead
e


and
e


are defined so that they
relate to strain in the (111) plane and the
[111] direction, respectively.

TMCS
III, Leeds 18
th

Jan
2012

Slide
19

Outline

1.
Epitaxy

in Semiconductor Crystal Growth

2.
Elastic Description of Strain in Cubic Semiconductor Crystals

3.
Atomistic Description of Strain

4.
Molecular Statics and Force Fields

5.
Keating's Valence Force Field

6.
Stillinger
-
Weber Potential

7.
Tersoff

Potential

8.
Simulation of
Nanostructures

9.
Piezoelectricity in
Zincblende

and
Wurtzite

crystals



TMCS
III, Leeds 18
th

Jan
2012

Slide
20

Tetrahedral Bonding

In the Zincblende crystal, just like in the diamond one, atoms
bond together to form tetrahedrons.

Hence the individual atomic orbitals merge to form sp
3

hybrid orbitals

TMCS
III, Leeds 18
th

Jan
2012

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21

Wurtzite

While Zinblende is the preferred crystal structure of III
-
As, III
-
P and
III
-
Sb, III
-
N tend to crystallize preferentially in hexagonal form. The
hexagonal crystal with a two atom basis consisting of cations and
anions is called Wurtzite.

View from the top

Perspective View

Wurtzite

Zincblende

Two adjacent tetrahedrons overlap in the z direction in WZ but not in ZB.
Hence second nearest neighbours in WZ are actually closer than in ZB at
equilibrium. The modified inter
-
atomic forces result in a slight reduction of
the interatomic distance between the first nearest neighbours.

TMCS
III, Leeds 18
th

Jan
2012

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22

The 7
th

elastic parameter

Is a description based on 6 strain components enough to describe
all deformations in a ZB or WZ crystals?

The distance that the atom is
displaced by is characterized by
the
Kleinman parameter

With a the lattice constant and
γ

the shear strain.

This results in a crystal where the
atomic bonds are not all of the
same length.

Strain in the [111]

Only 3
identical
sp
3
orbitals

+

+

+

+

-

TMCS
III, Leeds 18
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Jan
2012

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23

Strain from atomic positions

Given the 5 coordinates of the atoms in a tetrahedron how do we
reverse engineer the strain?

This become a simple system of linear equations easily solvable.

The solution gives the 6 components of the strain tensor.

However the deformation on the position of the yellow atom, dependant
on the Keinman parameter, is still undetermined and requires a separate
calculation.

TMCS
III, Leeds 18
th

Jan
2012

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24

The issue of local/global composition

Microscopists

refer to strain as difference in the bond lengths
compared to the host.

Theorists

think of strain as deformation of a material from its bulk
state.

Everyone else

does not usually know what they are talking about!!

Furthermore strain is a relative property (variation of e.g. bond
length compared to an initial state).

If dealing with an alloy and if wanting to take the theorist approach,
one needs to know what the lattice constant of the alloy is.


But what does composition mean?

It makes sense for a large uniform block, not for non uniform.


We take the approach of counting atoms up to second nearest
neighbour form the centre of the tetrahedron

TMCS
III, Leeds 18
th

Jan
2012

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25

Outline

1.
Epitaxy

in Semiconductor Crystal Growth

2.
Elastic Description of Strain in Cubic Semiconductor Crystals

3.
Atomistic Description of Strain

4.
Molecular Statics and Force Fields

5.
Keating's Valence Force Field

6.
Stillinger
-
Weber Potential

7.
Tersoff

Potential

8.
Simulation of
Nanostructures

9.
Piezoelectricity in
Zincblende

and
Wurtzite

crystals



TMCS
III, Leeds 18
th

Jan
2012

Slide
26

Modelling Strain in Real Structures

Because of its impact on the electronic properties strain in
semiconductor nanostructures always needs to be evaluated with
the highest possible accuracy.


Measurements (usually involving electron microscopy analysis) are
not usually sufficiently accurate, so modelling is the only viable
alternative.


Simple elasticity formulas are acceptable when dealing with
standard cases where strains are uniform or approximating strains
as uniform is acceptable, e.g. a simple quantum well.


They become useless however in complex quantum well
structures, wires and dots where strains are non uniform.


In time several methods have been developed ranging from
continuum, finite element, analytic and atomistic.


Atomistic methods are now widely used for quantum dots while
continuum methods are the preferred methods for quantum wells.

TMCS
III, Leeds 18
th

Jan
2012

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27

Molecular Dynamics



Molecular Dynamics is a computer simulation in which a starting set
of atoms or molecules is made to interact for a period of time
following the laws of Physics (e.g Newton’s Laws).




In Semiconductor science one can build an atomistic model of a
strained crystal but if the strain is not known a priori then atoms are
not going to be in their equilibrium positions.




Then their motion paths are dictated by the “
force field
” generated
by the potential of the solid.

TMCS
III, Leeds 18
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Jan
2012

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28

Often the simulation does not require very large atomic motion.

For instance for calculating strain one might only want to allow small
atomic displacements from the crystal structure, without atom switching.

When Energy minimisation is the fundamental criterion and forces are
used to direct the geometry optimisation rather than predicting the final
positions, we are using a “Molecular Statics” simulation.

Molecular Dynamics

Initial Position of the atoms r
0
i

Evaluate the positions after
Δ
t

Potential V (r
0
i
)

Forces F=
-
grad

V (r
0
i
)

Velocities and acceleration

Repeat till Forces are low

TMCS
III, Leeds 18
th

Jan
2012

Slide
29

Outline

1.
Epitaxy

in Semiconductor Crystal Growth

2.
Elastic Description of Strain in Cubic Semiconductor Crystals

3.
Atomistic Description of Strain

4.
Molecular Statics and Force Fields

5.
Keating's Valence Force Field

6.
Stillinger
-
Weber Potential

7.
Tersoff

Potential

8.
Simulation of
Nanostructures

9.
Piezoelectricity in
Zincblende

and
Wurtzite

crystals



TMCS
III, Leeds 18
th

Jan
2012

Slide
30

Valence Force Field

The “force field” that is generated by the potential of the atoms in
the solid can be represented as a 3 body potential.

In the Keating's Valence Force Field:

The Potential is the sum of the
potential energy between the
pairs of atoms i and j (two
body), plus a term that depends
on the angle between i,j and a
third atom k (three body).

d
ij
0

is the unstrained bond length of atoms
i

and
j

and

0

is the
unstrained bond angle (e.g. for zinc
-
blend cos

0
=
-
1/3), and

ijk

is the
angle between atoms

i
,
j

and
k
.

The local chemistry is contained in the parameters


and


, which are
fitted to the elastic constants


i

j

k

k

k


jki

R
j

R
i

R
k

P.N. Keating, Phys. Rev. 145, 637 (1966)


TMCS
III, Leeds 18
th

Jan
2012

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31

Valence Force Field

The VFF is widely used for all types of nanostructures.

VFF is basically a parabolic approximation to the potential of solids

The main limitation is that there
are only 2 parameters (


and


) but
3 elastic constants even for
Zincblende!!!

R
0

R

V(R)

Binding Energy

Uniform: same
distortion in
x,y and z

Non Uniform: z
stretch, x,y compress
(by the same amount)
and viceversa

Ω

is the volume
occupied by one
atom

TMCS
III, Leeds 18
th

Jan
2012

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32

Progress in Valence Force Field

Anharmonicity

correction
:




Ability to reproduce
anharmonic

effects is linked to the quality of prediction
of the phonon spectrum.



Some progress has been presented (e.g.
Lazarenkova

et al,
Superlattices

and Microstructures 34, 553 (2003)).



Not clear why phonon frequencies, elastic constants and mode
Grüneisen

parameters are not correlated (Porter et al J. Appl. Phys. 81, 96 (1997
)).



For
Ionicity

in
Zincblende

to solve this problem check recent
P. Han and G.
Bester, Phys. Rev. B
83

174304 (2011)





Ionicity

and
Wurtzite
:



Empirical potentials were historically developed for Si and
Ge

(pure covalent
bonds)



III
-
V are mainly covalent, partially ionic. II
-
VI are both covalent and ionic



Only for infinite crystals or systems were the charge is uniformly distributed
this it’s not a big deal.



Important in III
-
N WZ (Grosse and
Neugebauer
, PRB 63, 085207 (2001)),
and can be incorporated following
Ewald

summation scheme (codes available).



Also check Camacho
et al (
Physica

E,
Vol. 42, p. 1361 (2010)
)
“application
of Keating’s valence force field to non

ideal
wurtzite

materials


Distance between 2
ions, one of which is in
the central cell

TMCS
III, Leeds 18
th

Jan
2012

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33

Outline

1.
Epitaxy

in Semiconductor Crystal Growth

2.
Elastic Description of Strain in Cubic Semiconductor Crystals

3.
Atomistic Description of Strain

4.
Molecular Statics and Force Fields

5.
Keating's Valence Force Field

6.
Stillinger
-
Weber Potential

7.
Tersoff

Potential

8.
Simulation of
Nanostructures

9.
Piezoelectricity in
Zincblende

and
Wurtzite

crystals



TMCS
III, Leeds 18
th

Jan
2012

Slide
34

Stillinger
-
Weber

F. Stillinger and T. A. Weber, Phys. Rev. B 31, 5262 (1985)


The “force field” that is generated by the potential of the atoms in
the solid can be represented as a 3 body potential.

In the Stillinger
-
Weber potential:

i

j

k

k

k


jki

R
j

R
i

R
k

The Potential is the sum of the
potential energy between the
pairs of atoms i and j (two
body), plus a term that depends
on the angle between i,j and a
third atom k (three body).

This in an adaptation of the well known Lennard
-
Jones
potential used for liquefied noble gasses.

This potential works very well for Si in

diamond structure where
the
bond
angle cos

0
=
-
1/3.

The local chemistry is contained in the parameters A, B , p, q , a,
λ

and
γ

which are fitted to various material properties.


TMCS
III, Leeds 18
th

Jan
2012

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35

Stillinger
-
Weber

The SW is not as widely used as VFF, but it has his niche
(thermodynamics of Si mainly).

In a way it should perform much better than VFF as it is not a
parabolic approximation to the potential of solids.

Parameterisations take into account the crystal phase diagram and
check that diamond is the lowest energy structure

Works

reasonably well for diamond
-
Si but not for other crystal structures.

TMCS
III, Leeds 18
th

Jan
2012

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36

Outline

1.
Epitaxy

in Semiconductor Crystal Growth

2.
Elastic Description of Strain in Cubic Semiconductor Crystals

3.
Atomistic Description of Strain

4.
Molecular Statics and Force Fields

5.
Keating Valence Force Field

6.
Stillinger
-
Weber Potential

7.
Tersoff

Potential

8.
Simulation of
Nanostructures

9.
Piezoelectricity in
Zincblende

and
Wurtzite

crystals



TMCS
III, Leeds 18
th

Jan
2012

Slide
37

Tersoff Potential

The “force field” that is generated by the potential of the atoms in
the solid can be represented as a 3 body potential.

In the Tersoff potential:

i

j

k

k

k


jki

R
j

R
i

R
k

The Potential is the sum of the
potential energy between the
pairs of atoms i and j (two body),
multiplied times a term (b
ij
) that
depends on the angle between i,j
and a third atom k (three body).

The expression for b
ij

(known as bond order) is written
as to emulate the atomic coordination number Z.

Hence
ζ

is sometimes called the pseudo
-
coordination.

g(
θ
) and
ω

describe the angular and radial forces dependence.

TMCS
III, Leeds 18
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Jan
2012

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38

i

j

k

k

k


jki

g

θ

θ
eq

ω

Tersoff Potential

angular forces: resistance to bend

radial forces: resistance to stretch



When fitting to Bulk Modulus g(
θ
) is always g(
θ
eq
)

and
ω
ijk
==1



When fitting to Shear Constant g(
θ
)≠ (
θ
eq)

but
ω
ijk
==1



When fitting c
44
then both g(
θ
) ≠ (
θ
eq)

and
ω
ijk



1




Hence the Kleinman parameter links angular and radial forces!!!

TMCS
III, Leeds 18
th

Jan
2012

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39

Tersoff Potential

This potential describes covalent bonding and works very
well for different crystal structures for group IV and despite
the partial ionicity of the bond, group III
-
V.


The local chemistry is contained in the parameters A, B , r
e
,
α
,
β

,
γ
, c, d, h, n and
λ
,
which are fitted to various material
properties.


J. Tersoff, Phys Rev Lett
56
, 632 (1986) & Phys Rev B
39
, 5566 (1989)

Sayed et al,
Nuclear Instruments and Methods in Physics Research
102
, 232 (1995)

TMCS
III, Leeds 18
th

Jan
2012

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40

Tersoff Potential

The TP is not as widely used as VFF, but its use is rapidly increasing
as parameterizations are improved.

Again it should perform much better than VFF as it is not a parabolic
approximation to the potential of solids.

As there are many parameters, parameterisations can take into
account many things, including the crystal phase diagram, all the
cohesive and elastic properties and many more.

Works

rather well for zincblende and diamond group IV and III
-
V but it is
not yet optimized for thermodynamic and vibrational properties.

D. Powell, M.A. Migliorato and A.G. Cullis, Phys. Rev. B 75, 115202 (2007)


TMCS
III, Leeds 18
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Jan
2012

Slide
41

Progress in Tersoff

The Kleinman parameter



The many parameters need putting to good use.



Kleinman deformation is critical because expresses the balance between
radial and angular forces (Powell et al PRB 75, 115202 (2007))

DFT

DFT

Range of physical shear strains

Tersoff


Tersoff

Ionicity and Phonons



Ionicity, like VFF, is missing.



Crystal growth only possible if ionic contribution is included (Nakamura et
al J. Cryst. Growth
209,
232 (2000)



Phonons are still independent of elastic constants (Powell et al, Physica E
32, 270 (2006)

TMCS
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Jan
2012

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42

Beyond Tersoff: bond order potentials

Π

versus
σ


bonding




Tersoff neglects Π

bonding. Is it of consequence?



Tersoff can to some extent reproduce surface
reconstruction energies (Hammerschmidt, PhD thesis)

Beyond
σ

-
bonding











It is generally possible to rewrite the b
ij
with expressions directly obtained
from tight binding. (D.G. Pettifor, “Many atom Interactions in Solids”,
Springer Proceedings in Physics 48, 1990, pag 64))




In this way the “bond order” can be explicitly obtained analytically to any
order (Murdick et al, PRB 73, 045206 (2006)).




The second moment approximation is essentially equivalent to Tersoff


(Conrad and Scheerschmidt, PRB 58, 4538 (1998))

TMCS
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Jan
2012

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43

Outline

1.
Epitaxy

in Semiconductor Crystal Growth

2.
Elastic Description of Strain in Cubic Semiconductor Crystals

3.
Atomistic Description of Strain

4.
Molecular Statics and Force Fields

5.
Keating's Valence Force Field

6.
Stillinger
-
Weber Potential

7.
Tersoff

Potential

8.
Simulation of
Nanostructures

9.
Piezoelectricity in
Zincblende

and
Wurtzite

crystals



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2012

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44

General Tips for MD

Building Models:


If possible try and use existing software


Try and guess final positions: it saves a lot of computational time

Empirical Potentials:


Codes that use VFF, SW and Tersoff are usually freely available!


IMD (Stuttgart), CPMD (IBM
-
Zurich) are parallel (for running on
clusters) and open source


Nemo
3
(Purdue)

uses VFF


Always check what version of the potentials are being used!!

Molecular Statics:


Make sure that the parameters that control the length of time the
simulation is running for are set to reasonable values


Build your simulation up in size to see what you can get away
with in terms of system sizes and check that results do not
depend on the size chosen

Strain:


Good strain algorithms exist and are freely available


If you write your own you need a nearest neighbour list. Usually
MD produces one

Gridding:


Strain is first obtained onto the atomic grid. Then to use it often
it needs converting to an ordered grid. One can use various
methods like Gaussian smoothing or weighted average.

TMCS
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Jan
2012

Slide
45

MD of QDs using Tersoff Potential

ε
xx

ε
z
z

ε
yy

Before MD

After MD

Fixed

PBC

Floating

TMCS
III, Leeds 18
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Jan
2012

Slide
46

Outline

1.
Epitaxy

in Semiconductor Crystal Growth

2.
Elastic Description of Strain in Cubic Semiconductor Crystals

3.
Atomistic Description of Strain

4.
Molecular Statics and Force Fields

5.
Keating's Valence Force Field

6.
Stillinger
-
Weber Potential

7.
Tersoff

Potential

8.
Simulation of
Nanostructures

9.
Piezoelectricity in
Zincblende

and
Wurtzite

crystals



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III, Leeds 18
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Jan
2012

Slide
47

Kleinman Parameter

The distance that the atom is displaced by is
characterized by the Kleinman parameter

With a the lattice constant and
γ

the shear strain.

This results in a crystal where the atomic bonds
are not all of the same length.

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Jan
2012

Slide
48

+

+

+

+

-

Strain in the [111]

4
identical
sp
3
orbitals

Only 3
identical
sp
3
orbitals

+

+

+

+

-

Piezoelectricity

In the case of a uniaxial distortion the displacement is in the [111]
direction, and can still be characterized by the Kleinman parameter

The displacement of cations relative to anions in III
-
V semiconductors
results in the creation of electric dipoles in the polar direction which in
ZB is the direction that lacks inversion symmetry.

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Jan
2012

Slide
49

Piezoelectricity in
Zincblende


The effect can be quantified by writing a general expression for the
polarization as a function of the so called “piezoelectric coefficients”
and the distortion components.

Convention is:

xx=1, yy=2, zz=3, yz=4, zx=5, xy=6

In ZB, for symmetry, the only non zero
coefficients are e
14
= e
25
= e
36

In
actual fact this picture is incomplete as only includes coefficients
linked to linear terms in the strain.

In the past 6 years the importance of including also
coeffiecients

linked
to quadratic terms in the strain (e.g.
𝛆
xx
2
or
𝛆
xy

𝛆
xz

)

has been
highlighted (so called non linear or second order
Piezo

effect).



M.A
.
Migliorato

et al
,
Phys. Rev. B 74, 245332 (2006)




L. C. Lew Yan
Voon

and M.
Willatzen
, J. Appl. Phys. 109, 031101 (2011) REVIEW



A.
Beya
-
Wakata

et al,
Phys. Rev. B 84, 195207 (2011)

TMCS
III, Leeds 18
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Jan
2012

Piezoelectricity in
Wurtzite


Wurtzite

Zincblende

Spontaneous
polarization

Strain induced polarization

Quadratic terms in the strain (e.g.
𝛆
xx
2
or
𝛆
xy

𝛆
xz

)

are also important.

There is still some controversy between the early accepted values of
mainly for the spontaneous polarization coefficients



L. C. Lew Yan
Voon

and M.
Willatzen
, J. Appl. Phys. 109, 031101 (2011) REVIEW



J. Pal
et al,
Phys. Rev. B
84,
085211 (2011)

TMCS
III, Leeds 18
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Jan
2012

Slide
51

Thank you!!!

Acknowledgments
:

Joydeep

Pal , Umberto
Monteverde
, Geoffrey
Tse
,
Vesel

Haxha
, Raman
Garg

(University of Manchester)

Dave
Powell (University of
Sheffield)

GP
Srivastava

(University of Exeter)