Upload
ferdinand-rose
View
221
Download
2
Embed Size (px)
Citation preview
TMCS III, Leeds 18th Jan 2012 Slide 1
Outline
1. Epitaxy in Semiconductor Crystal Growth2. Elastic Description of Strain in Cubic Semiconductor Crystals3. Atomistic Description of Strain4. Molecular Statics and Force Fields5. Keating's Valence Force Field6. Stillinger-Weber Potential7. Tersoff Potential8. Simulation of Nanostructures9. Piezoelectricity in Zincblende and Wurtzite crystals
TMCS III, Leeds 18th Jan 2012 Slide 2
Semiconductors
Group IV: Si, Ge, CGroup 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 18th Jan 2012 Slide 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 18th Jan 2012 Slide 4
Lattice Matched and Mismatched Epitaxy
Lattice Matched Lattice Mismatched
TMCS III, Leeds 18th Jan 2012 Slide 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 18th Jan 2012 Slide 6
Quantum Mechanics in action
1D Quantum Well
In0.52Al0.48As
In0.52Al0.48As
In0.84Ga0.16AsAlAs
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 18th Jan 2012 Slide 7
Outline
1. Epitaxy in Semiconductor Crystal Growth2. Elastic Description of Strain in Cubic Semiconductor Crystals3. Atomistic Description of Strain4. Molecular Statics and Force Fields5. Keating's Valence Force Field6. Stillinger-Weber Potential7. Tersoff Potential8. Simulation of Nanostructures9. Piezoelectricity in Zincblende and Wurtzite crystals
TMCS III, Leeds 18th 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”.
x
y
z
Unstrainedx
y
z
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 18th Jan 2012 Slide 9
Elastic Strain
Unstrained and strained axis can be easily related:
The numerical coefficient εij
define the deformation of all the atoms in the Crystal.The diagonal terms εii control the length of the axis, while the off diagonal terms εij control the angles between the axis.
ˆ ˆ ˆ1 xx xy xzx x y z
ˆ ˆ ˆ1yx yy yzy x y z
ˆ ˆ ˆ1zx zy zzz x y z
This picture is general and valid for all types of crystals, not just simple cubic.
x
y
z
Unstrainedx
y
z
Strained
TMCS III, Leeds 18th Jan 2012 Slide 10
Elastic Strain
x
y
z
Unstrainedx
y
z
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’.
ˆ ˆ ˆ1 xx xy xzx x y z
ˆ ˆ ˆ1yx yy yzy x y z
ˆ ˆ ˆ1zx zy zzz x y z
R
R ˆ ˆ ˆR x y z
ˆ ˆ ˆR x y z
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 18th Jan 2012 Slide 11
Elastic Strain
ˆ ˆ ˆ1 xx xy xzx x y z
ˆ ˆ ˆ1yx yy yzy x y z
ˆ ˆ ˆ1zx zy zzz x y z
We now substitute the new axis with the expressions for the distortion:
ˆ ˆ ˆR x y z
ˆ ˆ ˆxx yx zx xy yy xz xz yz zz
R R
x y z
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 18th Jan 2012 Slide 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 eij are different from the distortion components but related to them by:
xx xx yy yy zz zz
xy yx xy yz zy yz xz zx xz
e e e
e x y e y z e x z
The final expressions for the off-diagonal terms eij 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:
xx yy zze e e
Biaxial Strain: expresses how much the unit cell is strained in the z direction compared to the x and y:
12bx zz xx yye e e e
Uniaxial Strain: strain in one direction only, e.g. if eij= constant and eii=0 then the strain is uniaxial in the [111]
TMCS III, Leeds 18th Jan 2012 Slide 13
Stress Components
Stress components: the force components (per unit area) that causes the distortion of the unit cell. There are 9 components:Xx, Xy, Xz, Yx, Yy, Yz, Zx, Zy, Zz
Capital letters: direction of the forceSubscript: 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, )
z
x yyz xz
xy
Xxxy
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 Xy= Yx, Yz= Zy, Zx= Xz
And we are only left with 6:Xx, Yy, Zz ; Yz, Zx, Xy
TMCS III, Leeds 18th Jan 2012 Slide 14
Elastic constants
The stress components are connected to the strain components via the small strain elastic constants:
11 12 13 14 15 16
21 22 23 24 25 26
31 32 33 34 35 36
41 42 43 44 45 46
51 52 53
x xx yy zz yz zx xy
y xx yy zz yz zx xy
z xx yy zz yz zx xy
z xx yy zz yz zx xy
x xx yy zz
X c e c e c e c e c e c e
Y c e c e c e c e c e c e
Z c e c e c e c e c e c e
Y c e c e c e c e c e c e
Z c e c e c e
54 55 56
61 62 63 64 65 66
yz zx xy
y xx yy zz yz zx xy
c e c e c e
X c e c e c e c e c e c e
In practice we never have to deal with all 36 elastic constants.First of all it is always the case that cij=cji 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: c11,c12,c44
In WZ there are 5: c11,c12,c13, c33, c44
TMCS III, Leeds 18th Jan 2012 Slide 15
Some more definitions
Elastic strain energy density for ZB:
Bulk Modulus for ZB:
Shear Constant for ZB:
11 122
3
c cB
11 12
2
c cC
2 2 2 2 2 2111 12 442 xx yy zz xx yy xx zz zz yy xy yz xzU c e e e c e e e e e e c e e e
TMCS III, Leeds 18th Jan 2012 Slide 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 0.677C 3.567 0.442 0.478 10.79 1.24 5.78 Ga-As 5.653 0.757 0.364 1.242 0.514 0.634In-As 6.058 0.617 0.229 0.922 0.465 0.444Al-As 5.662 0.747 0.288 1.131 0.555 0.547Ga-P 5.451 0.921 0.440 1.507 0.628 0.763In-P 5.869 0.736 0.269 1.095 0.556 0.526Al-P 5.463 0.886 0.329 1.325 0.667 0.627Ga-N 4.500 2.060 0.825 3.159 1.510 1.976In-N 4.980 1.476 0.424 2.040 1.190 1.141Al-N 4.380 2.030 0.698 2.961 1.565 2.004Ga-Sb 6.096 0.567 0.270 0.927 0.378 0.462In-Sb 6.479 0.476 0.183 0.720 0.354 0.341Al-Sb 6.135 0.855 0.414 1.407 0.579 0.399
TMCS III, Leeds 18th Jan 2012 Slide 17
Strain in Lattice Mismatched Epitaxy
11 11 12 44
12 11 12 44
2[001] [111]
2 4
c c c c
c c c c
1Substrate
Layer
eae e
a
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 = exx= eyy and e= ezz
TMCS III, Leeds 18th Jan 2012 Slide 18
11 12 44//
11 12 44
11 12 44//
11 12 44
2 4 42 1
3 3 2 4
2 4 41 1
3 3 2 4
xx yy zz
xy yz xz
c c ce
c c c
c c ce
c c c
Strain in [111] pseudomorphically grown layers
Important: in [111] growth the combination of e and e results in a strain tensor with exx = eyy = ezz and exy = exz = eyz 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 18th Jan 2012 Slide 19
Outline
1. Epitaxy in Semiconductor Crystal Growth2. Elastic Description of Strain in Cubic Semiconductor Crystals3. Atomistic Description of Strain4. Molecular Statics and Force Fields5. Keating's Valence Force Field6. Stillinger-Weber Potential7. Tersoff Potential8. Simulation of Nanostructures9. Piezoelectricity in Zincblende and Wurtzite crystals
TMCS III, Leeds 18th 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 sp3 hybrid orbitals
TMCS III, Leeds 18th Jan 2012 Slide 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 WurtziteZincblende
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 18th Jan 2012 Slide 22
The 7th 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
4
az
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
sp3
orbitals
+
++
+
-
TMCS III, Leeds 18th Jan 2012 Slide 23
Strain from atomic positions
Given the 5 coordinates of the atoms in a tetrahedron how do we reverse engineer the strain?
'0 0
'0 0
'0 0
1
1
1
xx xy xz
yx yy yz
zx zy zz
x x
y y
z z
'1 1
'1 1
'1 1
1
1
1
xx xy xz
yx yy yz
zx zy zz
x x
y y
z z
'2 2
'2 2
'2 2
1
1
1
xx xy xz
yx yy yz
zx zy zz
x x
y y
z z
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 18th Jan 2012 Slide 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 18th Jan 2012 Slide 25
Outline
1. Epitaxy in Semiconductor Crystal Growth2. Elastic Description of Strain in Cubic Semiconductor Crystals3. Atomistic Description of Strain4. Molecular Statics and Force Fields5. Keating's Valence Force Field6. Stillinger-Weber Potential7. Tersoff Potential8. Simulation of Nanostructures9. Piezoelectricity in Zincblende and Wurtzite crystals
TMCS III, Leeds 18th 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 18th Jan 2012 Slide 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 18th Jan 2012 Slide 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 r0i
Evaluate the positions after Δt
Potential V (r0i)
Forces F=-grad V (r0i)
Velocities and acceleration
Repeat till Forces are low
TMCS III, Leeds 18th Jan 2012 Slide 29
Outline
1. Epitaxy in Semiconductor Crystal Growth2. Elastic Description of Strain in Cubic Semiconductor Crystals3. Atomistic Description of Strain4. Molecular Statics and Force Fields5. Keating's Valence Force Field6. Stillinger-Weber Potential7. Tersoff Potential8. Simulation of Nanostructures9. Piezoelectricity in Zincblende and Wurtzite crystals
TMCS III, Leeds 18th 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:
2 3ˆ( ) ( )VFF i j ijk
ij ij
V V R R V
2, 0 23 00 2
,
31( ) ( ) ( ) cos
2 8 ( )
nni jk
j i k i iji j k i ij
V R R R R dd
22 0 22 0 2
31( ) ( )
2 8 ( )
nnij
i j iji j ij
V R R dd
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).
dij0 is the unstrained bond length of atoms i and j and 0 is the
unstrained bond angle (e.g. for zinc-blend cos0=-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
kk
k
jki
Rj
Ri
Rk
P.N. Keating, Phys. Rev. 145, 637 (1966)
TMCS III, Leeds 18th Jan 2012 Slide 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!!!
R0 R
V(R)
Binding Energy
Uniform: same distortion in x,y and z
Bd E
dvE Ecoh
1 2
2
Non Uniform: z stretch, x,y compress (by the same amount) and viceversa
EcohEd
EdC
2
21'
Ω is the volume occupied by one
atom
R v R1 13
R R
R R
R R
x x
y y
z z
1
1
1
1
1
( )
/ ( )
11 12
44
1 13
4
c ca a
ca
TMCS III, Leeds 18th Jan 2012 Slide 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”
2
0,0
1
2c
ll s
Z Z eU
r
Distance between 2 ions, one of which is in
the central cell
TMCS III, Leeds 18th Jan 2012 Slide 33
Outline
1. Epitaxy in Semiconductor Crystal Growth2. Elastic Description of Strain in Cubic Semiconductor Crystals3. Atomistic Description of Strain4. Molecular Statics and Force Fields5. Keating's Valence Force Field6. Stillinger-Weber Potential7. Tersoff Potential8. Simulation of Nanostructures9. Piezoelectricity in Zincblende and Wurtzite crystals
TMCS III, Leeds 18th 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:
2
2 3
1( ) ( ) cos
3
nn nn
SW ij ij ijkij ijk
V V r V r
i
j
kk
k
jki
Rj
Ri
Rk
12
1
2ij
nnr ap q
ij iji j
V A Br r e
1 1 2
3,
1 1cos
2 3
ij iknn r a r a
ijki j k i
V e
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 cos0=-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 18th Jan 2012 Slide 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 18th Jan 2012 Slide 36
Outline
1. Epitaxy in Semiconductor Crystal Growth2. Elastic Description of Strain in Cubic Semiconductor Crystals3. Atomistic Description of Strain4. Molecular Statics and Force Fields5. Keating Valence Force Field6. Stillinger-Weber Potential7. Tersoff Potential8. Simulation of Nanostructures9. Piezoelectricity in Zincblende and Wurtzite crystals
TMCS III, Leeds 18th 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
kk
k
jki
Rj
Ri
Rk
The Potential is the sum of the potential energy between the pairs of atoms i and j (two body), multiplied times a term (bij) that depends on the angle between i,j and a third atom k (three body).
( ) ( )e eij ij e ij ij er r r r
ij ij ij ijV A b B
12
1
nini
ij ij ijb
The expression for bij (known as bond order) is written as to emulate the atomic coordination number Z. Hence ζ is sometimes called the pseudo-coordination.
,
( ) ( )ij c ik ijk ijkk i j
f r g
gcidi
c
d hijki
i i jki
( )( cos )
1
2 2
2 2
3 3( )e ij ikr r
ijk
g(θ) and ω describe the angular and radial forces dependence.
TMCS III, Leeds 18th Jan 2012 Slide 38
gcidi
c
d hijki
i i jki
( )( cos )
1
2 2
2 2
i
j
kk
k
jki
g
θθeq
3 3( )e ij ikr r
ijk
ω
3 3( )ij ikr r
Tersoff Potential
angular forces: resistance to bendradial forces: resistance to stretch
• When fitting to Bulk Modulus g(θ) is always g(θeq) and ωijk==1• When fitting to Shear Constant g(θ)≠ (θeq) but ωijk==1• When fitting c44 then both g(θ) ≠ (θeq) and ωijk ≠ 1
• Hence the Kleinman parameter links angular and radial forces!!!
TMCS III, Leeds 18th Jan 2012 Slide 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 , re, 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 18th Jan 2012 Slide 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)
cohEE
coh
d
Edc
2
2
44
1
zz
yy
yxx
RR
RR
RRR
TMCS III, Leeds 18th 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
0.00 0.02 0.04 0.06 0.08 0.10
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.00 0.02 0.04 0.06 0.08 0.10
hydrostatic distortion
shear distortion
su
bla
ttic
e d
isp
lac
em
en
t
circles: InAssquares: GaAs
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 III, Leeds 18th Jan 2012 Slide 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 bij 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))
12
1
nini
ij ij ijb
,
( ) ( )ij c ik ijk ijkk i j
f r g
gcidi
c
d hijki
i i jki
( )( cos )
1
2 2
2 2
3 3( )e ij ikr r
ijk
TMCS III, Leeds 18th Jan 2012 Slide 43
Outline
1. Epitaxy in Semiconductor Crystal Growth2. Elastic Description of Strain in Cubic Semiconductor Crystals3. Atomistic Description of Strain4. Molecular Statics and Force Fields5. Keating's Valence Force Field6. Stillinger-Weber Potential7. Tersoff Potential8. Simulation of Nanostructures9. Piezoelectricity in Zincblende and Wurtzite crystals
TMCS III, Leeds 18th Jan 2012 Slide 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 timeEmpirical 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• Nemo3 (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 oneGridding:• 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 III, Leeds 18th Jan 2012 Slide 45
MD of QDs using Tersoff Potential
0.005000
-0.005000
-0.01500
-0.02500 -0.03500
0.005000
-0.04500
0.01500
-0.05500-0.05500
-0.008500
-0.01850
0.001500
-0.02850
0.001500
-0.03850
-0.04850
0.01150
0.05825
0.05825
-0.006900-0.01690
-0.02690-0.03690
-0.04690-0.05690
0.002000
0.002000
0.01200
-0.008000
-0.008000
0.02200
0.03200
-0.01800
0.0020000.002000
-0.02800
-0.03800
0.04200 0.04200
0.002000
-0.01800
-0.006900
50 100 150 200 250 300 350
50
100
150
200
250
300
[001] Å
[00
1] Å
-0.06500
-0.05500
-0.04500
-0.03500
-0.02500
-0.01500
-0.005000
0.005000
0.01500
0.02500
0.03500
0.04500
50 100 150 200 250 300 350
50
100
150
200
250
300
[010] Å
[00
1] Å
-0.09850
-0.08850
-0.07850
-0.06850
-0.05850
-0.04850
-0.03850
-0.02850
-0.01850
-0.008500
0.001500
0.01150
0.02150
0.02900
50 100 150 200 250 300 350
50
100
150
200
250
300
[010] Å
[00
1] Å
-0.02175
-0.01175
-0.001750
0.008250
0.01825
0.02825
0.03825
0.04825
0.05825
0.06825
0.07825
0.08825
0.09825
0.1082
0.1182
0.1282
0.1382
0.1482
0.1500
50 100 150 200 250 300 350
50
100
150
200
250
300
[010] Å
[00
1] Å
-0.06690
-0.05690
-0.04690
-0.03690
-0.02690
-0.01690
-0.006900
2.000E-4
50 100 150 200 250 300 350
50
100
150
200
250
300
[010] Å
[00
1] Å
-0.09800
-0.08800
-0.07800
-0.06800
-0.05800
-0.04800
-0.03800
-0.02800
-0.01800
-0.008000
0.002000
0.01200
0.02200
0.03200
0.04200
0.05200
0.06200
0.07200
0.08200
0.09200
0.1020
0.1120
0.1220
0.1320
0.1420
0.1500
50 100 150 200 250 300 350
50
100
150
200
250
300
[010] Å
[00
1] Å
-0.06690
-0.05690
-0.04690
-0.03690
-0.02690
-0.01690
-0.006900
2.000E-4
εxx
εzz
εyy
Before MD After MD
Fixed
PBC
Floating
TMCS III, Leeds 18th Jan 2012 Slide 46
Outline
1. Epitaxy in Semiconductor Crystal Growth2. Elastic Description of Strain in Cubic Semiconductor Crystals3. Atomistic Description of Strain4. Molecular Statics and Force Fields5. Keating's Valence Force Field6. Stillinger-Weber Potential7. Tersoff Potential8. Simulation of Nanostructures9. Piezoelectricity in Zincblende and Wurtzite crystals
TMCS III, Leeds 18th Jan 2012 Slide 47
Kleinman Parameter
cohEE
coh
d
Edc
2
2
44
1
zz
yy
yxx
RR
RR
RRR
The distance that the atom is displaced by is characterized by the Kleinman parameter
4
az
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.
TMCS III, Leeds 18th Jan 2012 Slide 48
+
++
+-
Strain in the [111]
4 identical
sp3
orbitals
Only 3 identical
sp3
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
2 2
2 2
2 2
x x y z
y x y z
z x y z
R R R R
R R R R
R R R R
4
ax y z
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.
TMCS III, Leeds 18th 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.
, , ,
, , ,
, , ,
x xkl klk l x y z
y ykl klk l x y z
z zkl klk l x y z
P e
P e
P e
Convention is:xx=1, yy=2, zz=3, yz=4, zx=5, xy=6
In ZB, for symmetry, the only non zero coefficients are e14= e25= e36
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. 𝛆xx
2 or 𝛆xy 𝛆xz ) has been
highlighted (so called non linear or second order Piezo effect).
14 yz zy zx xz xy yxP e
• 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 18th Jan 2012
Piezoelectricity in Wurtzite
WurtziteZincblende
Spontaneous polarization
Strain induced polarization
Quadratic terms in the strain (e.g. 𝛆xx2 or 𝛆xy
𝛆xz ) 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 18th 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)