 
nextnano^{3}  Tutorial
next generation 3D nano device simulator
1D Tutorial
Strain: Band shifts and splittings due to conduction and valence band
deformation potentials
Author:
Stefan Birner
If you want to obtain the input files that are used within this tutorial, please contact stefan.birner@nextnano.de.
> 1Ddeformation_potentials_no_strain.in / *_nnp.in 
input file for the nextnano^{3} and nextnano++ software
> 1Ddeformation_potentials_strain.in / *_nnp.in 
input file for the nextnano^{3} and nextnano++ software
> 1Ddeformation_potentials_hydro.in
InAs/GaAs/InAs structure
 This input file simulates an InAs/GaAs/InAs structure.
The structure is grown pseudomorphically on InAs, i.e. the GaAs is
tensilely strained, the InAs is unstrained. The growth direction [001]
is along z, the interfaces are in the (x,y) plane.
 We artificially set the electrostatic potential to zero to avoid any
further shifting of
the band edges.
By default Varshni parameters are used to determine temperature dependent
band gaps
(i.e. temperature dependent conductionbandenergies ).
Here, the Varshni parameters are switched off, thus the conduction and valence
band edges from the database (or input file) are taken.
$numericcontrol
simulationdimension
= 1
zeropotential
= yes
varshniparameterson = no !
Band gaps independent of temperature. Absolute values from database are taken.
latticeconstantstempcoeffon = no !
Lattice constants independent of temperature.
Absolute values from database are taken.
$end_numericcontrol
No strain
> 1Ddeformation_potentials_no_strain.in
 The following figure shows the conduction and valence band edges of the
heterostructure when no strain is applied (
straincalculation =
nostrain ). The heavy and
light hole bands are degenerate.
 GaAs:
conductionbandenergies = 2.979d0 3.275d0
3.441d0 ! [eV] Gamma, L, X
valencebandenergies =
1.346d0
! [eV] E_{v,av}
6x6kpparameters =
... ...
... !
0.341d0 ! [eV] Delta_{splitoff}
Conduction bands:
> Gamma
band = 2.979
> L
band = 3.275
> X
band = 3.441
Valence bands:
> heavy hole band =
1.346 + 1/3 * 0.341 = 1.45967
(= E_{v,av} + 1/3 *
Delta_{splitoff} =
E_{v,max})
> light hole band =
1.346 + 1/3 * 0.341 = 1.45967
(= E_{v,av} + 1/3 *
Delta_{splitoff} =
E_{v,max})
> splitoff hole band = 1.346 
2/3 * 0.341 = 1.11867 (= E_{v,av}
 2/3 *
Delta_{splitoff} =
E_{v,max} 
Delta_{splitoff})
Note: We apply an overall bandshift to all bands in order to align the
topmost valence bands (heavy hole/light hole) to zero (0 eV).
!
! Shift all bands, so that GaAs (hh/lh) = 0 eV. ==>
E_{v,max} = 0
!
bandshift = 1.45967d0 ! [eV]
Conduction bands:
> Gamma
band = 2.979  1.45967 =
1.519
> L
band = 3.275  1.45967 =
1.815
> X
band = 3.441  1.45967 = 1.981
Valence bands:
> heavy hole band =
1.45967  1.45967 =
0 (=
E_{v,max})
> light hole band =
1.45967  1.45967 =
0 (=
E_{v,max})
> splitoff hole band = 1.11867 
1.45967 = 0.341
For details of the calculations of the band gap or band edges see
FAQ section.
Strain (biaxial strain)
> 1Ddeformation_potentials_strain.in
 Then we turn on strain (
straincalculation =
homogeneousstrain ). Here it is
interesting to see that the sixfold degenerate X bands in GaAs split.
 Valence bands
absolutedeformationpotentialvb =
1.21d0 ! a_{v} [eV]
(Ref. Zunger)
uniaxvbdeformationpotentials =
2.0d0 ...
! b,d [eV]
InAs has a larger lattice constant than GaAs.
The tensile strain for GaAs has the following three components (the
offdiagonal strain components are zero):
e_{ }= e_{xx }= e_{yy }= 0.07165
e_{__ }= e_{zz }=  0.06643
The hydrostatic strain is the trace of the strain tensor and
corresponds to the change in volume dV/V:
e_{hydro }= Tr(e_{ij}) = e_{xx }+ e_{yy }+ e_{zz
}= 2e_{ }+_{ }e_{__} = 0.07687
The three valence bands shift by a constant amount (hydrostatic or absolute
deformation potential):
E_{v,av}' = E_{v,av} + a_{v }e_{hydro}
= E_{v,av} + (1.21 e_{hydro}) =
E_{v,av}  0.0930127
In addition to this shift each hole band gets a further shift which depends on
the growth direction: For growth direction along [001] the additional valence
bands shift in the following way:
E_{v}^{*}(hh) =  1/2 E_{sh}^{001}
E_{v}^{*}(lh) =  1/2 Delta _{so} + 1/4 E_{sh}^{001}
+ 1/2 [ SQRT(Delta_{so}^{2} + Delta_{so }E_{sh}^{001}
+ 9/4 (E_{sh}^{001})^{2 }) ]
E_{v}^{*}(so) =  1/2 Delta_{so} + 1/4 E_{sh}^{001}
 1/2 [ SQRT(Delta_{so}^{2} + Delta_{so }E_{sh}^{001}
+ 9/4 (E_{sh}^{001})^{2 }) ]
These three equations include the spinorbit splitoff energy
Delta_{so}
and are thus given relative to the unstrained valence band edge
maximum
E_{v,max} =
E_{v}(hh) = E_{v}(lh) .
Note that without strain, these equations lead to:
E_{v}^{*}(hh) = 0
E_{v}^{*}(lh) = 0
E_{v}^{*}(so) =  Delta_{so}
E_{sh}^{001} = b(e_{xx }+ e_{yy } 2e_{zz})
= 2b(e_{__}e_{}) = 2(2.0)(e_{__}e_{})
= 0.55232
E_{v}(hh) =
E_{v,max} + a_{v }e_{hydro}
+ E_{v}^{*}(hh) =
E_{v,max}
 0.0930127 + ( 
0.27616 )
= 0.369
E_{v}(lh) =
E_{v,max} + a_{v }e_{hydro}
+ E_{v}^{*}(lh) =
E_{v,max}
 0.0930127 + ( 
0.1705 + 0.13808 + 0.497745 ) = 0.372
E_{v}(so) =
E_{v,max} + a_{v }e_{hydro}
+ E_{v}^{*}(so) =
E_{v,max}  0.0930127
+ (  0.1705 +
0.13808  0.497745 ) = 0.623
In our example, we have shifted the bands (bandshift = 1.45967d0 ),
so that it holds for unstrained GaAs:
E_{v,max}
= 0
 Conduction bands
absolutedeformationpotentialscbs = 9.36d0 4.91d0 0.16d0
! a_{c}(Gamma), a_{c}(L),
a_{c}(X) uniaxcbdeformationpotentials =
0d0 14.26d0 8.61d0 !
Xi(Gamma), Xi(L), Xi(X)
Each of the three conduction bands shifts by a constant amount
(hydrostatic or absolute deformation potential):
E_{c}'(Gamma) = E_{c}^{0}(Gamma) + a_{c}(Gamma)
e_{hydro} = E_{c}^{0}(G) + (9.36
e_{hydro}) = 1.519  0.7195 =
0.799
E_{c}'(L) = E_{c}^{0}(L)
+ a_{c}(L) e_{hydro} = E_{c}^{0}(L)
+ (4.91 e_{hydro}) =
1.815
 0.3774 = 1.437
E_{c}'(X) = E_{c}^{0}(X)
+ a_{c}(X) e_{hydro} = E_{c}^{0}(X)
+ (0.16 e_{hydro}) =
1.981
 0.0123 = 1.9687
The X conduction band of GaAs is also subjected to a splitting (upwards and
downwards relative to E_{c}'(X) ). This amount depends on
the growth direction. For [001] growth direction the splitting is given by
E_{c}^{001*}(X) = 2/3
Xi(X) (e_{__}e_{}) = 2/3 *
8.61 * (0.13808) =  0.7925792
E_{c}^{100,010*}(X) =  1/3 Xi(X) (e_{__}e_{})
=  1/3 * 8.61 * (0.13808) = 0.3962896
> E_{c}^{001}(X) = E_{c}'(X)
+ E_{c}^{001* }(X) = 1.9687  0.7925792 =
1.176 (2fold valley degeneracy)
> E_{c}^{100,010}(X) = E_{c}'(X) + E_{c}^{100,010*}(X)
= 1.9687 + 0.3962896 = 2.365
(4fold valley degeneracy)
 The above given equations for the valence band are not used
inside nextnano³, however. We get the band shifts by diagonalizing the
BirPikus strain Hamiltonian which is given in
Basics 2 (strain
effects). This is a more general approach as it gives the correct shifts
for arbitrary orientations (However, it is only for valence bands). The
equations above are the special case for growth along the [001] direction.
 Note: There are two different definitions of the valence band
deformation potential in the literature. What can be measured experimentally
is the deformation potential of the band gap which is composed of the
deformation potential of the valence and conduction band edge.
a_{gap} = a_{c} + a_{v}
(e.g. definition in
Vurgaftman1)
One can also argue that the conduction band deformation potential can be
constructed from the valence band edge and the band gap deformation potential
like this:
a_{c} = a_{v} + a_{gap}
(definition in nextnano³)
The equations differ in the sign of the valence band deformation potential.
Note that the Erratum of
Bahder1 is not in agreement with the definiton inside nextnano³.
Note also the comment of
Vurgaftman1: "... the band gap increases for a compressive strain. Under
positive hydrostatic pressure, i.e. negative strain, the change in energy
DeltaE_{gap }= a_{gap}(e_{xx }+ e_{yy }+ e_{zz})
must be positive. This implies a negative value for a_{gap} .
[...] It is generally believed that the conduction band edge moves upward in
energy while the valence band moves downward, with most of the change being in
the conduction band egde, although Wei and Zunger recently argued that this is
not always the case."
Note also the comment of Wei and Zunger (Wei1):
"We show that the volume deformation potentials a_{c} of the
conductionband minimum state Gamma_{6c} are usually large and always
negative (energy increases with pressure) while the volume deformation
potentials a_{v} of the VBM state Gamma_{8v} are usually small
and negative for zinc blende compounds containing occupied valence d state
(e.g. GaAs, InAs) but positive for compounds without occupied valence d
orbitals (e.g. AlAs)."
Hydrostatic strain
> 1Ddeformation_potentials_hydro.in
 Now we want to apply a hydrostatic strain to our 1D structure (
straincalculation =
hydrostaticstrain ). The difference to
pseudomorphic strain is simply that the pressure acts equally from each
direction. This leads per definition to:
e_{xx }= e_{yy }= e_{zz
} for cubic crystals._{
} Normally e_{zz } is calculated in the
following way: e_{zz }=  2 (c_{12}/c_{11})_{
}e _{xx } (only for [001] growth direction!)_{
} For option hydrostaticstrain
we artificially use inside the code the constant c_{12 }= c_{11}/2_{
} leading to e_{zz}=e _{xx}.
This allows us to study the effects of absolute deformation potentials
only (without splitting of the bands and without shifts due to uniaxial
deformation potentials).
In this case we have:
e_{xx }= e_{yy }=_{ }e_{zz }= 0.07165
e_{hydro }= Tr(e_{ij}) = e_{xx }+ e_{yy }+ e_{zz
}= 3e_{xx} = 0.21495
E_{v}' = E_{v}^{0} + a_{v }e_{hydro} =
E_{v}^{0} + (1.21*0.21495) = E_{v}^{0}
 0.26009


E_{v}(hh) = E_{v}^{0}(hh) + a_{v }e_{hydro}
= 0  0.26009 =
0.260
E_{v}(lh) = E_{v}^{0}(lh) + a_{v }e_{hydro}
= 0  0.26009 =
0.260
E_{v}(so) = E_{v}^{0}(so) + a_{v }e_{hydro}
= 0.341  0.26009 = 0.601
E_{c}'(Gamma) = E_{c}^{0}(Gamma) + a_{c}(Gamma)
e_{hydro} = E_{c}^{0}(G) + (9.36
e_{hydro}) = 1.519  2.0119 =
0.492
E_{c}'(L) = E_{c}^{0}(L)
+ a_{c}(L) e_{hydro} = E_{c}^{0}(L)
+ (4.91 e_{hydro}) =
1.815  1.0554 = 0.759
E_{c}'(X) = E_{c}^{0}(X)
+ a_{c}(X) e_{hydro} = E_{c}^{0}(X)
+ (0.16 e_{hydro}) =
1.981  0.0343 = 1.946
Here we have negative hydrostatic pressure = increase in volume = positive
hydrostatic strain.
The valence bands shift downwards (a_{v } is negative) and
the conduction bands shift downwards, too (a_{c } is
negative).
In this case, the Gamma conduction band is below the heavy/light hole band
edge.
 Please help us to improve our tutorial. Send comments to
support
[at] nextnano.com .
