Download Optical properties of cylindrical nanowires

Optical properties of cylindrical nanowires
L.A. Haverkate; L.F. Feiner
15th December 2006
Abstract
A theoretical analysis is presented of the optical absorption of III-V semiconductor cylindrical nanowires. The optical properties are described by means
of the dielectric function, calculated for band-to-band transitions close to
the band gap.
We have treated the electronic structure using effective mass theory,
taking the degeneracy of the valence band into account.
A strong polarization anisotropy is found which is due to quantum confinement, in agreement with atomistic methods. We show that the effective
mass approach provides a fast and flexible tool to analyze the diameter dependent properties of nanowires for a wide range of semiconductor materials.
In addition we discuss the effect of classical Mie scattering and show that
it is negligible in the quantum regime.
Contents
Introduction
6
I
8
Classical theory of light scattering by a cylinder
1 General solution
1.1 General theory . . . . . . . . . . . . . . . . . . . .
1.1.1 Maxwell equations . . . . . . . . . . . . . .
1.1.2 Boundary conditions . . . . . . . . . . . . .
1.2 Mie’s formal solution for circular cylinders . . . . .
1.3 Scattering problem . . . . . . . . . . . . . . . . . .
1.3.1 Scattering coefficients, general solution . . .
1.4 Far field theory . . . . . . . . . . . . . . . . . . . .
1.4.1 Far field approximation . . . . . . . . . . .
1.4.2 Poynting vector and electromagnetic energy
1.4.3 Cross sections and efficiencies . . . . . . . .
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2 Small dielectric cylinders
23
2.1 Coefficients in Rayleigh approximation . . . . . . . . . . . . . 23
2.2 Fields inside the wire . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Efficiency, polarization anisotropy and - contrast in Rayleigh
approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.1 Polarization anisotropy, polarization contrast . . . . . 28
2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4.1 Efficiencies and polarization anisotropy at oblique incidence . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.2 Efficiencies and polarization anisotropy as a function
of wavelength . . . . . . . . . . . . . . . . . . . . . . . 36
II
Absorption
40
3 Electronic properties
41
3.1 The k · p method . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1.1 Top valence bands in III-V semiconductors . . . . . . 44
3
Contents
3.2
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5 Dielectric function nanowire
5.1 General theory . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Atomic polarizability approach . . . . . . . . . . . . .
5.1.2 Transition rate method . . . . . . . . . . . . . . . . .
5.1.3 Dielectric function expressed in reduced effective mass
5.2 Dielectric function for finite group transitions . . . . . . . . .
5.3 Polarization anisotropy nanowire . . . . . . . . . . . . . . . .
5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Estimation kz dependence of |Tcv |2 . . . . . . . . . .
5.4.2 Polarization anisotropy and R dependence . . . . . . .
5.4.3 Material dependence . . . . . . . . . . . . . . . . . . .
5.4.4 Effect of the dielectric background . . . . . . . . . . .
91
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105
3.3
3.4
3.5
Effective mass approximation . . . . . . . . . . . . . . .
3.2.1 Crystal Hamiltonian in envelope representation .
3.2.2 Top valence bands in III-V semiconductors . . .
Envelope description for infinite cylinders . . . . . . . .
3.3.1 Hole in III-V semiconductor nanowires . . . . . .
3.3.2 Electron in III-V semiconductor nanowires . . .
Hole dispersion around kz = 0 . . . . . . . . . . . . . . .
3.4.1 Solutions at the wire zone center . . . . . . . . .
3.4.2 Hole dispersion around kz = 0 for |fz | = 12 , (−) .
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Hole energy bands of III-V material nanowires .
3.5.2 Hole wave functions of III-V material nanowires
3.5.3 Band gap in III-V material nanowires . . . . . .
4 EM transition matrix
4.1 General theory . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Radiation matter interaction . . . . . . . . . .
4.1.2 EM transition matrix . . . . . . . . . . . . . .
4.2 Bloch representation . . . . . . . . . . . . . . . . . . .
4.2.1 Total wavefunction in Bloch functions . . . . .
4.2.2 EM transition matrix in Bloch functions . . . .
4.3 Reformulation of transition matrix element . . . . . .
4.3.1 EM field in dipole approximation . . . . . . . .
4.3.2 EM field including Mie scattering . . . . . . . .
4.3.3 Polarization anisotropy of the transition matrix
4.4 Selection rules . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Polarization selection rules . . . . . . . . . . .
4.4.2 Selection rules on the envelope wavefunctions .
4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Dipole approximation . . . . . . . . . . . . . .
4.5.2 EM field including Mie scattering . . . . . . . .
4
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Contents
Conclusions
107
A Hole wavefunctions for different kz
112
B Polarization selection rules
116
C Interband matrix elements
119
D Reference articles
122
Bibliography
125
5
Introduction
In the last several years semiconductor nanowires have attracted much interest, due to the tunability of their fundamental optical and electronic
properties. Techniques for the growth of nanostructures have been developed and high quality III-V semiconductor nanowires with a length of several
microns and a lateral size of only a few nanometers have been obtained. Recent experiments have shown a large polarization anisotropy in such wires
[1][2]. For example, Figure 1 shows the photoluminescence and excitation
spectra of an InP nanowire on a flat gold surface [2]. The radius of this wire
was ∼ 15 nm and the measured polarization anisotropy is fully explained by
the dielectric mismatch between the wire and the surrounding.
Figure 1: Experimental results for optical absorption [2]. a) Photoluminescence image (CCD camera, incident laser light polarized parallel
to the wire axis) and b) Excitation spectra for parallel (k) and perpendicular (⊥) polarized incident light of an InP nanowire on a flat gold
surface. The length of the wire is ∼ 2 µm, its radius ∼ 15 nm and the
wavelength of the exciting laser beam is 457.9 nm. The emitted light
was unpolarized in order to take only the polarization anisotropy in the
absorption process into account.
However, next to this classical effect of dielectric contrast it is expected
that quantum confinement starts to contribute significantly for decreasing
6
wire radius. This quantum effect has already been observed in the shift in
the fundamental band gap [3], but based on atomistic theories[4][5] it is also
predicted that quantum confinement causes drastic changes in the polarization anisotropy of nanostructures.
In this paper we will analyze the optical absorption properties of III-V
semiconductor cylindrical nanowires using effective mass theory. Within this
approach it is possible to describe the optical and electronic properties for
varying wire thickness and for a wide range of semiconductor materials. Contrary, ab initio methods using a fully atomistic description (tight-binding,
pseudo-potential,...) are limited with respect to the dimensions of the nanosystem since with increasing number of atoms the calculations become more
and more complex, or even impossible. Although the effective mass approach generally is less accurate, it thus provides a relative fast and flexible tool
to simulate real nanowires, with dimensions which are technically feasible
at the present day.
Despite the large amount of papers on the subject of nanowires, or even
nanostructures in general, little attention has been paid to the effects of
classical scattering. Usually it is assumed that the wavelength of the incident
light is sufficiently larger than the wire radius in order to neglect the spatial
variance of the electromagnetic (EM) field within the wire, which justifies
considering the response of the nanowire to the incident light in the dipole
limit. For increasing wire radius, however, the wave behavior of the EM
field cannot simply be neglected any more. Therefore, it is one of the main
questions in this thesis if this so called Mie-scattering already starts to play
a significant role in the quantum confinement regime.
For this purpose we have to know the local response field inside the wire.
But historically most of the work in classical scattering theory was dedicated
to measurable quantities far from the scattering objects, driven by the large
interest from application fields as astronomy and meteorology. In Part I,
Chapters 1 and 2, we therefore start with a classical theory describing the
scattering of light by an infinite cylindrical structure. In particular we derive
explicit expressions for the EM field inside the wire by using a procedure
originally developed by Mie [6].
In Part II we subsequently focus on the effects of quantum confinement
by means of a corrected description of the dielectric function of cylindrical
nanowires. In Chapter 3 the electronic properties of nanowires made from
III-V compounds are discussed, Chapter 4 treats the EM matrix element for
band-to-band transitions between the top Γ8 valence bands and the lowest
lying Γ6 conduction band in III-V semiconductor nanowires and finally in
Chapter 5 the dielectric function and polarization anisotropy of a nanowire
are obtained including the quantum confinement corrections by the bandto-band transitions.
7
Part I
Classical theory of light
scattering by a cylinder
8
Chapter 1
General solution
In this chapter classical theory is treated which describes the scattering of
light by an infinite cylinder at arbitrary angle of incidence and wire radius.
In the first sections general theory is discussed and specified to the case of
an infinite cylinder by using a procedure originally developed by Mie [6]. In
section the theory will be put in an applicable form by deriving measurable
quantities (cross sections, efficiency factors) in the far field region.
1.1
General theory
1.1.1
Maxwell equations
The scattering of light at oblique incidence by an infinite cylinder needs a
full, formal treatment, in particular when the solution has to be expanded
in the wire radius.
The starting-point of the full problem is Maxwell’s theory. Assuming the
light waves to be periodic with time dependence e−iωt , the charge density
ρ equal to zero and the magnetic permeability µ equal to 1, the Maxwell
equations are:
∇ × H = −ik0 m2 E,
(1.1)
∇ × E = ik0 H,
(1.2)
∇ · H = 0,
(1.3)
2
∇ · (m E) = 0,
(1.4)
where
k0 =
ω
c
=
2π
,
λ0
(1.5)
is the wave number in vacuum and
m2 = ε +
4πiσ
.
ω
(1.6)
9
Chapter 1. General solution
The parameter m is the complex refractive index of the medium at the
frequency ω of the light waves and consists of an optic part and an electric
part. The former is associated with ε, the dielectric constant, the latter
with the conductivity σ, which is taken to be zero since the electrical part
is beyond the scope of this paper. Both parts are complex and depend on
the circular frequency ω of the light waves.
It should be noted that in general m is a tensor and moreover depends on
the position in the medium. For the applications considered in this paper the
medium is assumed to be homogeneous and in that case m is a constant. We
will also assume here that m is a scalar. As a consequence, from (1.1)-(1.4),
the field vectors E and H satisfy the vector wave equation:
∆A + k02 m2 A = 0.
(1.7)
As a consequence the rectangular components of E and H satisfy the scalar
wave equation
∆ψ + k02 m2 ψ = 0,
(1.8)
which has plane wave solutions with the propagation constant equal to k0 m.
This shows that the wave is damped if m has a negative imaginary part and
in that case absorption takes place.
1.1.2
Boundary conditions
In case of a sharp boundary between two homogeneous media (1 and 2) the
integral representation of the Maxwell equations (1.1) and (1.2) gives the
boundary conditions on the tangential components of the fields, after a well
known limiting process (Jackson [12], page 16):
n × (H2 − H1 ) = 0,
(1.9)
n × (E2 − E1 ) = 0,
(1.10)
where n is the normal to the boundary.
In the same way the Maxwell equations (1.3) and (1.4) lead to the boundary
conditions on the normal components:
n · (m2 2 E2 − m1 2 E1 ) = 0,
(1.11)
n · (H2 − H1 ) = 0.
(1.12)
The tangential and normal boundary conditions are not independent. For
instance, boundary condition (1.10) can be derived from (1.12), Maxwell
equation (1.3) and applying the limiting procedure. In the same way it can
be shown that (1.9) and (1.11) are dependent on each other. Therefore it is
sufficient to look only at the tangential components.
10
1.2. Mie’s formal solution for circular cylinders
1.2
Mie’s formal solution for circular cylinders
In order to solve the boundary value problem exactly the coordinate system
should be the one in which the scalar wave equation is separable in the
coordinates. In case of circular cylinders these coordinates are (ρ, φ, z),
where the cylinder axis coincides with the z -axis (see Figure 1.1). As a
condition for this separability the cylinder length L has to be assumed much
larger then its diameter:
L À 2R,
(1.13)
where R denotes the cylinder radius. In this case the cylinder can be seen
as infinitely long and then it is possible to use the following formal solution
developed by Mie [6]. If ψ satisfies the scalar wave equation (1.8), define
M ψ and N ψ as
M ψ = ∇ × (êz · ψ),
(1.14)
mk0 N ψ = ∇ × M ψ .
(1.15)
Then both M ψ and N ψ satisfy the vector wave equation (1.7), and the
elementary solutions of Maxwell’s equations can be expressed as
E = M v + iN u ,
(1.16)
H = mM u − imN v ,
(1.17)
where u and v are the two independent solutions of the scalar wave equation.
The scalar wave equation (1.8) in cylindrical coordinates for a homogeneous medium with complex refractive index m is
µ 2
¶
1 ∂2
∂
1 ∂2
∂2
2 2
+
(1.18)
+
+
+ m k0 ψ = 0,
∂ρ2 ρ ∂ρ ρ2 ∂φ2 ∂z 2
and its solutions can be found by separating the variables. The resulting
differential equation for the ρ coordinate is the Bessel equation, which has
two independent solutions: Jn , the integral order Bessel function and Nn ,
the integral order Neumann function. This means that the solutions of (1.8)
can be found by an appropriate superposition of:
q
(1.19)
ϕn = Zn (ρ m2 k02 − g 2 )ei(gz−ωt) einφ ,
with n an integer, Zn any Bessel function of order n and g arbitrary. In
cylindrical coordinates M ϕn and N ϕn are then derived as:

M ϕn = 
in
ρ
∂
− ∂ρ
0

 ϕn ,


mk0 N ϕn = 

∂
ig ∂ρ
−ng
ρ
m2 k02 −

 ϕn , (1.20)
g2
11
Chapter 1. General solution
on the basis of cylindrical unit vectors êρ , êφ , êz . Consequently, with un =
An ϕn and vn = Bn ϕn for certain An , Bn and taking the sum over all n, the
components of E and H are
∞
X
in
g ∂un
Eρ =
vn −
,
(1.21)
ρ
mk0 ∂ρ
n=−∞
Hρ =
∞
X
g ∂vn
inm
un +
,
ρ
k
∂ρ
0
n=−∞
normal to the cylinder surface and
∞
X
∂vn
ing
Eφ =
−
un ,
−
∂ρ
mk0 ρ
n=−∞
Ez =
Hφ =
Hz =
∞
X
i(m2 k02 − g 2 )
un ,
mk0
n=−∞
∞
X
n=−∞
∞
X
n=−∞
(1.23)
(1.24)
∂un ing
+
vn ,
∂ρ
k0 ρ
(1.25)
i(m2 k02 − g 2 )
vn
k0
(1.26)
−m
−
(1.22)
tangential to the cylinder surface.
1.3
Scattering problem
With the above formal solution it is now possible to solve the general scattering problem of an arbitrary polarized plane electromagnetic wave incident
obliquely on a circular cylinder of infinite length.
For oblique incidence the direction of propagation of the incident wave makes an angle θ with the normal to the z -axis, see Figure 1.1. Furthermore
the cylinder is assumed to be surrounded by vacuum and the refractive index of the cylinder is equal to m. In case of a surrounding homogeneous
medium with refractive index m1 the solutions are of the same form if m
is considered as the refractive index of the cylinder relative to the medim2
. With the above definitions the incident wave, depicted in
um: m = m
1
Figure 1.1 is represented by the scalar wave function
ψ0 = Ẽ0 e−i(k0 x cos θ+k0 z sin θ+ωt)
∞
X
= Ẽ0 e−i(hz+ωt)
(−i)n Jn (lρ)eınφ ,
(1.27)
n=−∞
where
h ≡ k0 sin θ,
l ≡ k0 cos θ =
12
(1.28)
q
k02 − h2 .
(1.29)
1.3. Scattering problem
H
O II
E O II
k
E sca
H
OI
EO I
k
X
Z
E
int
Y
Figure 1.1: Definition of the coordinates for scattering by a circular
cylinder. The incident waves are showed, including the corresponding
incident fields: E 0 I , H 0 I in Case I and E 0 II , H 0 II in Case II. The
angle of incidence is defined by θ .
Equation (1.27) represents a wave travelling in the −êx direction if θ equals
zero. Note that the last expression in (1.27) is an expansion in Bessel functions and has the required form of (1.19). In this way also the scattered
wave and internal wave (inside the cylinder) can be formed from a superpositionpof functions of the form (1.19). Finiteness at the origin requires that
Jn (ρ m2 k02 − h2 ) is the radial function describing the internal wave, where
h is given by (1.28) because of continuity at the boundary ( (1.9) and (1.10)).
The last argument also holds forpthe scattered wave, which is described by
(1)
the first Hankel function Hn (ρ k02 − h2 ), describing an outgoing wave at
large distances from the cylinder.
Following the procedure of Van de Hulst [7] and Kerker [8], the polarized
incident wave has to be resolved into two components:
• Case I: a Transverse Magnetic (TM) mode. The magnetic field of
the incident wave is perpendicular to the cylinder axis (Figure 1.1).
This mode is described by choosing un = il1 Ẽ0 (−i)n ϕn (with
g = −h, Zn = Jn ) and vn = 0 in (1.21)- (1.26). This choice also
fixes the orientation of the incident electric field: E 0 I = Ẽ0 (cos θêz −
sin θêx ) e−i(hz+lx+ωt) . The factor il1 is just a normalization constant,
for further details see Bohren and Huffman [11].
• Case II: a Transverse Electric (TE) mode.The electric field is perpendicular to the cylinder axis. Now un = 0 and vn = il1 Ẽ0 (−i)n ϕn and
the incident field is given by E 0 II = Ẽ0 êy e−i(hz+lx+ωt) .
For an arbitrary elliptically polarized incident wave the solutions can be
found by an appropriate superposition of Case I and Case II. The decompo13
Chapter 1. General solution
sition of the incident wave does not necessarily mean that the scattered and
internal waves resolve in the same way. This can be explained by looking
closely at the general expressions of the scalar fields inside and outside the
cylinder. These are:
Case I
ρ > R unI
vnI
= Ẽ0 Fn {Jn (lρ) − bnI Hn(1) (lρ)} ,
=
Ẽ0 Fn {anI Hn(1) (lρ)} ,
(1.30)
(1.31)
ρ < R unI
= Ẽ0 Fn {dnI Jn (jρ)} ,
(1.32)
vnI
= Ẽ0 Fn {cnI Jn (jρ)} ,
(1.33)
Case II
ρ > R unII
vnII
ρ < R unII
vnII
= Ẽ0 Fn {bnII Hn(1) (lρ)} ,
= Ẽ0 Fn {Jn (lρ) −
anII Hn(1) (lρ)} ,
(1.34)
(1.35)
= Ẽ0 Fn {dnII Jn (jρ)}
(1.36)
= Ẽ0 Fn {cnII Jn (jρ)} ,
(1.37)
where
Fn ≡
and
j ≡
1 −i(hz+ωt)
e
(−i)n einφ
il
q
m2 k02 − h2 .
(1.38)
(1.39)
Unlike the incident waves, which are chosen to be TM or TE, the solutions
for the scattered and internal scalar waves are in general decomposed into
two components:
• A solution with the same orientation as the incident wave (TM or TE),
contained in unI (Case I) and vnII (Case II) respectively.
• A ”cross mode” with an opposite orientation, TE (v1 ) in Case I and
TM (uII ) in Case II.
Only in case of normal incidence, θ = 0, the cross terms turn out to be zero
and the scattered and internal waves resolve in the same way as the incident
wave (see paragraph below).
As stated in section 1.2, the scalar wave expressions (1.30)-(1.37) also
determine the fields inside and outside the cylinder in the various cases. The
incident, scattered and internal fields are denoted with E 0 , E sca and E int ,
respectively.
As an example, combining the expression for the scattered scalar wave in
Case I (the second term in (1.30)) with the equations for the field components (1.21)-(1.26) one finds for the scattered electric field E sca in cylinder
coordinates:
14
1.3. Scattering problem

E sca I
= Ẽ0
∞
X


Fn anI 
in
ρ
∂
− ∂ρ


 Hn(1) (lρ) + i(−bnI ) 

0
n=−∞
−ih
∂
k0 ∂ρ
nh
k0 ρ
l2
k


 (1)

 Hn (lρ) .
(1.40)
The scattered electric field E sca II in Case II can be obtained from (1.40)
by replacing anI by −anII and −bnI by bnII .
1.3.1
Scattering coefficients, general solution
The coefficients anI , bnI , cnI and dnI (anII , bnII , cnII and dnII ) are in general
functions of the angle of incidence θ and the wire radius R. They can be
determined by the fact that the boundary conditions (1.11)-(1.12) require
continuity of the tangential components of E and H. As a consequence the
equations (1.23)-(1.26) have to be continuous at R = ρ. These conditions
lead in both cases to four linear algebraic equations which can be solved for
the four coefficients:
Case I
anI (R, θ) =
ı sin θ n(m2 − 1){Nn−1 − On−1 }
2 2
lR{( kj0 )2 Ln − (m2 + 1) kj0 Dn + L−1
n (Cn − m Dn )}
(1) 0
bnI (R, θ) =
cnI (R, θ) =
dnI (R, θ) =
Hn
,
(1)
(lR){( kj0 )2 Kn − m2 kj0 Dn } + Hn (lR){− kj0 Dn Kn + m2 Dn2 − Cn }
,
(1)
2 2
Hn (lR)Mn {( kj0 )2 Ln − (m2 + 1) kj0 Dn + L−1
n (Cn − m Dn )}
(
)
(1)
anI (R, θ)Hn (lR)
l2
,
j2
Jn (jR)
(
)
(1)
ml2
Jn (lR)
bnI (R, θ)Hn (lR)
−
,
(1.41)
j2
Jn (jR)
Jn (jR)
Case II:
(1) 0
anII (R, θ) =
Hn
(lR){( kj0 )2 Kn −
j
k0 Dn }
(1)
+ Hn (lR){m2 kj0 Dn Kn + m2 Dn2 − Cn }
(1)
2 2
Hn (lR)Mn {( kj0 )2 Ln − (m2 + 1) kj0 Dn + L−1
n (m Dn − Cn )}
,
bnII (R, θ) = −anI (R, θ),
cnII (R, θ) =
dnII (R, θ) =
l2
j2
(
(1)
Jn (lR)
bnII (R, θ)Hn (lR)
−
Jn (jR)
Jn (jR)
(
)
(1)
ml2 anII (R, θ)Hn (lR)
,
j2
Jn (jR)
)
,
(1.42)
15
Chapter 1. General solution
where
(m2 − 1)2 n2 tan2 θ
,
j 2 R2
0
J (jR)
,
≡ cos θ n
Jn (jR)
Cn ≡
(1.43)
Dn
(1.44)
(1) 0
0
Kn ≡
Jn (lR)
Hn (lR)
,
, Ln ≡ (1)
Jn (lR)
Hn (lR)
Mn ≡
Hn (lR)
Hn (lR)
, Nn ≡
,
Jn (lR)
Jn (lR)
On ≡
Hn (lR)
Jn0 (lR)
(1) 0
(1.45)
(1)
(1.46)
(1) 0
(1.47)
and the functions l (1.29) and j (1.39) depend on θ. Despite of the different
form, equations (1.30)-(1.37) are the same as derived by Bohren [11]. They
give the complete, formal solution for the scattering problem of a plane
electromagnetic wave incident obliquely on a circular cylinder of infinite
length. In principle the electromagnetic fields and the intensities can be
obtained by calculating the full expansion of (1.30)-(1.33) for Case I ((1.34)(1.37) for Case II) and subsequently use these expressions to calculate the
fields (1.21-1.26). However, in practice it is impossible to get an exact
analytic solution and a numerical procedure is the only way to solve the
full problem.
In the special case of a normal incident wave (θ = 0), the scattering
coefficients (1.30)-(1.37) reduce to:
Case I
anI (R, 0) = 0,
0
bnI (R, 0) =
0
mJn (k0 R)Jn (mk0 R) − Jn (k0 R)Jn (mk0 R)
(1) 0
(1)
mHn (k0 R)Jn0 (mk0 R) − Hn
cnI (R, 0) = 0,
(1) 0
dnI (R, 0) =
Hn
(1) 0
mHn
(k0 R)Jn (mk0 R)
(1)
,
0
(k0 R)Jn (k0 R) − Hn (k0 R)Jn (k0 R)
(1.48)
,
(1)
(k0 R)Jn (mk0 R) − m2 Hn (k0 R)Jn0 (mk0 R)
Case II
0
anII (R, 0) =
0
Jn (k0 R)Jn (mk0 R) − mJn (k0 R)Jn (mk0 R)
(1) 0
(1)
Hn (k0 R)Jn0 (mk0 R) − mHn
bnII (R, 0) = 0,
(1) 0
cnII (R, 0) =
Hn
(1) 0
m2 Hn
dnII (R, 0) = 0.
16
(k0 R)Jn (mk0 R)
(1)
,
0
(k0 R)Jn (k0 R) − Hn (k0 R)Jn (k0 R)
,
(1)
(k0 R)Jn (mk0 R) − mHn (k0 R)Jn0 (mk0 R)
(1.49)
1.4. Far field theory
As stated before, the cross terms disappear, which means that all waves in
Case I are TM and all waves in Case II TE.
1.4
Far field theory
In principle the scattering theory derived in the previous sections is complete and everything one wants to know can be derived from it. However, in
order to make predictions about measurable quantities, in this section the
theory will be put in an applicable form.
It is important to realize that usually the experimental measurements are done at a large distance from the scattering object(s), so in the first paragraph
general expressions for the fields in this region are derived. Subsequently measurable quantities (cross sections, efficiency factors) are defined and
applied to the situation of scattering by an infinite cylinder. The theory
depicted here is derived in a detailed form by Bohren and Huffman [11].
More intuitive approaches are found in [7],[8].
1.4.1
Far field approximation
As stated in section 1.3 the scattered wave is associated with the first Hankel
function, based on the fact that the wave has to be an outgoing wave. At large distances from the cylinder the first Hankel function can be approximated
by its asymptotic expression:
r
2 iz
e (−i)n e−iπ/4 ,
| z | À n2 .
(1.50)
Hn(1) (z) ∼
πz
This is the only ingredient needed to approximate the scattered part of the
fields at large distances from the wire.
Consider for this purpose equation (1.40) for the scattered electrical field.
In the far field approximation (lρ À 1) the Hankel functions in this expression are approximated by (1.50). After elaboration of the derivatives and
1
, which fall of much faster then the terms ∼ √1lρ ,
neglecting all terms ∼ lρ√
lρ
this results in:
r
E sca I ∼ −Ẽ0 e
−iπ/4
∞
2 i(lρ−hz−ωt) X
e
(−1)n einφ [anI êφ + bnI (sin θêρ + cos θ)êz ] .
πlρ
n=−∞
(1.51)
This is the result for an incident wave with the magnetic field perpendicular to the wire axis (Case I). For Case II, when the incident field is TE
(the electric field perpendicular to the wire axis) the asymptotic expression
of the scattered field has the same form, apart from changing anI into −anII
and −bnI into bnII .
17
Chapter 1. General solution
Equation (1.51) shows that the surfaces of constant phase, or wavefronts,
of the scattered wave obey
ρ cos θ − z sin θ = C,
C ∈ R,
(1.52)
which represents cones of half-angle θ and apexes at z = −C/ sin θ. Including the e−iωt factor, the scattered wave can be visualized as a cone sliding
down the cylinder [11].
1.4.2
Poynting vector and electromagnetic energy rates
One of the most important properties of electromagnetic (EM) waves is the
flux of EM energy through a certain area. In the case of light scattering at
a particle not only the magnitude of this flux has to be specified, but also
c
its direction. This is given by the Poynting vector S = 8π
Re{E × H ∗ },
which defines the time-averaged flux of energy crossing a unit area. As a
consequence the rate of EM
R energy crossing a plane surface A, with normal
unit vector n̂, is equal to S · n̂ dA.
For a surface A which encloses a volume V the net rate W at which EM
energy crosses the boundary A is defined as
I
(1.53)
W = −
S · n̂ dA,
n̂ ≡ unit normal outward to A.
A
This is a definition in the sense that the minus sign ensures that W is positive
if there is a net rate of EM energy flowing into the volume V (S · n̂ < 0),
so in the case of absorption of EM energy in the volume.
Denoting the incident and scattered EM fields in the same way as before,
the Poynting vector at any point outside the particle can be written in these
fields as:
c
S =
Re{(E 0 + E sca ) × (H ∗0 + H ∗sca )} = S 0 + S sca + S ext ,
8π
where
S0 =
S ext =
c
c
Re{E 0 × H ∗0 } , S sca =
Re{E sca × H ∗sca } , (1.54)
8π
8π
c
Re{E 0 × H ∗sca + E sca × H ∗0 )} .
8π
The decomposition in (1.54) nicely shows that, next to the expected Poynting vectors of the incident (S 0 ) and scattered (S sca ) fields, a term S ext
arises which describes the interaction between the incident and scattered
waves.
To be more precise, it turns out that S ext represents the removal of energy from the incident light waves, the extinction. Consider for this purpose
an imaginary sphere of radius a and surface A around a particle of finite
size . The rate of energy Wabs absorbed within the sphere equals the energy
18
1.4. Far field theory
rate absorbed by the particle because the surrounding medium is supposed
to be non-absorbing. Wabs is given by equation (1.53), now with êa the
outward unit normal to the sphere and may be decomposed in:
Wabs = W0 − Wsca + Wext ,
where
I
I
W0 = −
Wext = −
IA
A
S 0 · êa dA ,
Wsca =
A
S sca · êa dA ,
(1.55)
S ext · êa dA .
The choice of the minus signs here again ensures that all the energy rates
are positive, note in particular S sca · êa > 0. Furthermore the energy rate
W0 associated with the incident wave vanishes for a non-absorbing medium,
so
Wext = Wabs + Wsca ,
(1.56)
which shows that Wext indeed represents the extinction, namely the sum of
the energy scattering rate and energy absorbing rate.
In case of an infinite cylinder the imaginary sphere in the preceding
argumentation has to be replaced by an imaginary surrounding cylinder of
infinite length. Now it is convenient to look at the rate of EM energy flow per
unit length , since this quantity is finite. Furthermore, infinite cylinders don’t
exist except as an idealization, so the statements here have to be carefully
applied to the situation of a cylinder long compared with is diameter, as
used in the previous sections. This is possible if edge effects are negligible,
such that there is no net contribution to Wabs from the ends of the imaginary
cylinder.
In that case, denoting r as the radius of the constructed cylinder and a length
L equal to the length of the cylinder, the expressions for the scattering and
extinction rates of EM energy per unit length become
Z
Wsca /L =
(S sca )ρ ρ dφ |ρ = r ,
0
Z
Wext /L =
2π
0
2π
(S ext )ρ ρ dφ |ρ = r ,
(1.57)
with (S sca )ρ and (S ext )ρ the (positive) radial components of the expressions
derived in (1.54). On physical grounds the absorption energy rate has to
be independent of r if the medium outside the cylinder is non absorbing.
Indeed, with the far field solution of (1.51), the r dependence in (1.57) drops
out.
19
Chapter 1. General solution
1.4.3
Cross sections and efficiencies
In stead of using the energy rates it is more convenient to take the normalized
forms of them: cross sections, or, better, efficiency factors. The former are
surfaces, defined as
Wabs
,
I0
Cabs =
Csca =
Wsca
,
I0
Cext =
Wext
,
I0
(1.58)
c
|Ẽ0 |2 is the incident intensity. Dividing these optical cross
where I0 = 8π
sections by the geometrical cross section G, dimensionless efficiency factors
are found:
Cabs
,
G
Qabs =
Qsca =
Csca
,
G
Qext =
Cext
.
G
(1.59)
Note that equation (1.56) has a synonym in terms of efficiency factors:
Qext = Qsca + Qabs .
(1.60)
For a circular cylinder with radius R and length L the geometrical cross section equals 2RL. Note that the efficiency factors indeed are dimensionless.
With the far field scattered electric field (1.51) and a similar expression
for the scattered magnetic field now it is a matter of patience to derive:
Qsca I
=
=
Z 2π
1
(|T11 (π − φ)|2 + |T12 (π − φ)|2 ) dφ
πx 0
(
)
∞
X
2
|b0I |2 + 2
(|bnI |2 + |anI |2 ) ,
x
(1.61)
n=1
Qext I
=
=
2
Re{T11 (π = φ)}
x
(
)
∞
X
2
Re b0I + 2
(bnI ) ,
x
(1.62)
n=1
Qsca II
=
=
Z 2π
1
(|T22 (π − φ)|2 + |T21 (π − φ)|2 ) dφ
πx 0
)
(
∞
X
2
(|bnII |2 + |anII |2 ) ,
|a0II |2 + 2
x
(1.63)
n=1
Qext II
=
=
2
Re{T22 (π = φ)}
x
(
)
∞
X
2
Re a0II + 2
(anII ) ,
x
n=1
20
(1.64)
1.4. Far field theory
where
T11 (π − φ) ≡
T22 (π − φ) ≡
T12 (π − φ) ≡
T21 (π − φ) ≡
∞
X
n=−∞
∞
X
n=−∞
∞
X
n=−∞
∞
X
bnI e−in(π−φ) ,
anII e−in(π−φ) ,
anI e−in(π−φ) ,
bnII e−in(π−φ)
(1.65)
n=−∞
are the four components of the amplitude scattering matrix T, as defined by
Kerker [8] and Bohren & Huffman [11]. 1 2
Two general features can be mentioned from (1.61)-(1.64), the efficiency
factors for light falling obliquely on a cylinder long compared to its radius:
• The efficiencies are expansions in the size parameter kR of the particle
in question.
• The extinction quantities only depend on the scattering amplitudes in
the forward direction (φ = π), while it contains the effect of scattering
in all directions by the particle. This is a particular form of the optical
theorem and a intuitive explanation is given in [7], [11].
The efficiencies Qabs , Qsca and Qext are the main quantities which can be
measured in optical experiments. If the resolution in a particular experiment
is high enough, also the differential efficiencies dQsca /dφ can be estimated,
which are given by
dQsca I /dφ =
dQsca II /dφ =
1
(|T11 (π − φ)|2 + |T12 (π − φ)|2 ),
πx
1
(|T22 (π − φ)|2 + |T21 (π − φ)|2 ).
πx
(1.66)
They specify the angular distribution of the scattered light.
It is important to note that the efficiencies defined here in principle can
take values larger then unity, contrary to what one should expect from the
meaning of the word ”efficiencies”. In particular it can be shown that in the
geometrical limit, i.e. if all the dimensions of the scattering object are much
1
The expressions for Qext are derived after quite a lot of algebraic work [11], it can be
done faster by using the optical theorem in advance [7].
2
The transformation of φ to π − φ comes from the definition of the incident wave: as
in [11] the incident wave is in the −êx direction, while Van de Hulst [7] and Kerker [8]
use the opposite and no transformation is needed.
21
Chapter 1. General solution
larger then the wavelength, the extinction efficiency approach the limiting
value two. This is rather peculiar, because it suggests that the object removes twice the energy that is incident on it. This so called extinction paradox
is resolved by taking also diffraction into account: the edge deflects rays in
its neighborhood which from a geometrical view would have passed undisturbed. In this way the incident wave is influenced beyond the geometrical
size of the scattering particle.
22
Chapter 2
Small dielectric cylinders
As stated in chapter 1, it is not possible to express the general solution of the
scattering problem explicitly as a function of the material properties (dielectric constant, wire radius), geometric configuration (angle of incidence,
radial distance from cylinder) and the wave number of the incident light. In
this chapter the special case of cylindrical wires with radius small compared
to the wavelength of the incident light will be treated. It will be shown that
in this approximation it is possible to get an analytic solution. In section 2.4
numerical results are given for InP.
2.1
Coefficients in Rayleigh approximation
When the radius of the cylinder is sufficiently small compared to the wavelength of the incident light, the Bessel functions appearing in the scattering
coefficients can be expanded in terms of kR. To be precise, sufficiently small
means the following condition:
|m|x ¿ 1,
(2.1)
where
x ≡ k0 R
(2.2)
is defined as the size parameter of the circular cylinder. This condition is
physically based on the two Rayleigh assumptions:
• The wave behavior of the incident field can be neglected with respect
to the size of the particle: x ¿ 1. This implies the external field can
be considered as an homogeneous field.
• The applied field should penetrate so fast into the particle that the
static polarization is established in a time t short compared to the
period T , so t/T ¿ 1. Since the velocity inside the cylinder is c/m
and the wave period T = 1/ck this assumption is satisfied by (2.1).
23
Chapter 2. Small dielectric cylinders
With condition (2.1) the following expressions for the Bessel and first
Hankel functions can be used:
z
2
J0 (z) ' 1 − z4 ,
J00 (z) ' − ,
2
1
3
0
0
J1 (z) ' −J0 (z) ,
J1 (z) ' − z 2 ,
2 16
2i 1
0
2i
z
H0 (z) ' 1 + 2i
π γ + π log 2 , H0 (z) ' π z ,
2i 1
,
(2.3)
H1 (z) ' −H00 (z) ,
H10 (z) '
π z2
where | z | ¿ 1, H denotes the first Hankel function and γ is Euler’s constant.
The Hankel functions in 2.3 are expanded to zeroth order in z, because this
is sufficient for a second order approximation of the scattering coefficients.
With these expansions it can be easily shown that the scattering coefficients
(1.41) and (1.42) up to second order in the size parameter x are approximated by:
Case I
a0I (x, θ) = 0,
πx2 (m2 − 1)
a1I (x, θ) =
sin θ + O(x4 ),
4 (m2 + 1)
iπx2 2
(m − 1) cos2 θ + O(x4 ),
b0I (x, θ) = −
4
iπx2 (m2 − 1)
b1I (x, θ) = −
sin2 θ + O(x4 ),
4 (m2 + 1)
Case II
a0II (R, θ) = O(x4 ),
iπx2 (m2 − 1)
+ O(x4 ),
a1II (R, θ) = −
4 (m2 + 1)
b0II (R, θ) = 0,
πx2 (m2 − 1)
b1II (R, θ) = −
sin θ + O(x4 ),
4 (m2 + 1)
(2.4)
(2.5)
This result is in agreement with the earlier work of Wait [9], [10] and
also gives the expressions derived by Van de Hulst [7] and Kerker [8] for
normal incidence (θ = 0) . The internal coefficients for the fields inside the
wire, cnI , dnI , cnII and dnII are not explicitly shown here because they have
a complex form. They follow directly from equations (1.41) and (1.42).
In principle now it is possible to proceed further and use equations (2.4) and
(2.5) for the approximation of the fields (equations (1.21)-(1.26)). However,
it is really important to be careful, because an expansion of the fields to
second order in the size parameter needs more and further expanded coefficients then showed above.
24
2.2. Fields inside the wire
2.2
Fields inside the wire
To our best knowledge the expressions for the fields inside a dielectric cylinder have only been derived in the dipole limit x → 0 [7][13]. This is a
relatively small result compared to the the huge amount of research done
in the far field region outside the scattering object, where a lot of interest
in particular for applications in meteorology and astronomy worked as a
driving force.
In the dipole approximation, also used by Wang and Lieber [1], the
incident field is really taken to be homogeneous. It is a special, stronger
form of the Rayleigh approximation discussed above, since the second order
corrections now are completely neglected. In this way the expressions for
the fields are independent of the size parameter and are derived in terms of
the incident fields as [7][13]:
E int k = E 0 k ,
2
E int ⊥ =
E0 ⊥ .
1 + m2
(2.6)
(2.7)
Here we will extend this solution to finite values of the size parameter
x. Before doing this care has to be taken by expanding the coefficients, as
noted before. This is because the internal fields also depend implicitly on the
size parameter via ρ, apart from the explicit dependence via the coefficients.
This implicit dependence can be split up in two parts:
• The internal fields are expressed in terms of Jn (jρ), see (1.30)-(1.37).
In second order this results in a ρ dependence by (2.3).
• Some of the components of the fields (1.21)-(1.26) have an extra 1/ρ
dependence.
Taking this into account for a second order approximation of the internal
fields one needs the internal coefficients cnI , dnI , cnII and dnII up to the
following orders in x:
n
order
0
3
1
2
2
1
3
0
...
...
Now it is a matter of mathematics to get the solution of the fields inside the wire up to second order. Since the expressions for oblique incidence
are too complex to show in an illuminating way, only the results for normal
incidence are showed below.
Starting with Case I, where for normal incidence Ẽint I ρ (x, 0) and Ẽint I φ (x, 0)
are directly zero (see end of section 1.3), the z component of the electric field
25
Chapter 2. Small dielectric cylinders
becomes
Eint k z (x, 0) =
©
m2 k02 ρ2
cos2 φ +
E0 e−iωt 1 − imk0 ρ cos φ −
2
1 2
ρ2
(m − 1)(1 − 2 ) x2 +
(2.8)
4
R
¾
³
1 2
x´ 2
x + O(x3 ).
(m − 1) −2γ + iπ − 2 log
4
2
As required, this solution reduce to (2.6) in the dipole limit x → 0
(note: mk0 ρ = x Rρ → 0). Also the limit m → 1 provides the desired result
E int I = E 0 I : for m = 1 there is no optical difference between inside and
outside any more.
Furthermore, the part between brackets in equation (2.8) can be divided in
three parts, each written on a different line here and each with a different
physical background:
• The first part (including the Ẽ0 e−iωt term in front of the brackets)
expresses the original wave behavior of the incident field: it is the
expansion of e−imkρ cos φ up to second order.
• As the cylinder radius increases, the effect of optical focusing gets a
more important role. This is described by the second part: it has its
maximum in the middle of the cylinder and falls off quadratically to
zero at ρ = R. Note that this term is quadratic in the size parameter.
• Also the third part is quadratic in x, but in contrast to the second term
constant over the wire. It has its origin in the expansion of H0 (kρ), see
(2.3). It is a rather striking expression: the iπ part in it can be seen as
a constant phase shift of the field. Roughly speaking it is responsible
for a correction on the absorption: taking the absolute value squared
this iπ part mixes with the complex part of m and decreases the flux
of energy crossing the cylinder.
Note that the log x2 part together with the x2 after the brackets is
finite: limx→0 x2 log x = 0. It gives positive contribution to Iint =
|Ẽint I z (x, 0)|2 that can be large enough to get a value for Iint /I0
larger then unity. This is an example of the extinction paradox, as
will be explained further in the next sections where a link will be
made between internal quantities and the external efficiency factors.
The components of the electric field in Case II show mainly the same
features as mentioned above. Remember from section 1.3 that the internal
electric field is perpendicular to the wire axis (TE) at normal incidence, so
Ẽint II z (x, 0) = 0. The other two components become
26
2.3. Efficiency, polarization anisotropy and - contrast in Rayleigh approximation
Ẽint II ρ (x, 0) =
Ẽint II φ (x, 0) =
½
2
m2 k02 ρ2
−iωt
cos2 φ +
Ẽ
e
1 − imk0 ρ cos φ −
0
2
(m + 1)
2
1 2
ρ2
(2.9)
(m − 1)(1 − 2 ) x2 +
8
R
µ
¶
¾
1 (m2 − 1) 1
x
− 2γ + iπ − 2 log
x2 + O(x3 ),
2
4 (m + 1) 2
2
½
m2 k02 ρ2
2
−iωt
Ẽ0 e
1 − imk0 ρ cos φ −
cos φ 2
cos2 φ +
(m + 1)
2
ρ2
3 2
(m − 1)(1 − 2 ) x2 +
(2.10)
8
R
µ
¶
¾
1 (m2 − 1)
1
x
−(m2 + ) − 2γ + iπ − 2 log
x2 + O(x3 ).
2
4 (m + 1)
2
2
sin φ
Again, the terms on the first line describe the original wave behavior of
the incident field. The incident TE field was taken to be in the positive êy
direction and decomposing this in cylindrical coordinates one gets indeed
the first terms in the expressions (2.9) and (2.10).
2.3
Efficiency, polarization anisotropy and - contrast in Rayleigh approximation
Contrary to the quantities inside the wire, in the far field region it is possible to get simple analytic expressions in Rayleigh approximation at oblique
incidence. For expansion of the scattering and extinction efficiencies (1.61)(1.64) to third order in the size parameter x, the coefficients have to be
estimated to second and fourth order for Qsca and Qext , respectively.
The third order term for Qext I and Qext II are too extended to show
here, but are included in all calculations of the next section.
With this in mind the efficiencies (1.61)- (1.64) are approximated by:
½
¾
2 sin2 θ(1 + sin2 θ) π 2 x3
Qsca I (x, θ) = |m2 − 1|2 cos4 θ +
+ O(x5 ) ,
|m2 + 1|2
8
(2.11)
2
2
2
3
ªπ x
|m − 1| ©
Qsca II (x, θ) =
(2.12)
+ O(x5 ) ,
2 − cos2 θ
2
2
|m + 1|
4
¾
½
πx
4 sin2 θ
2
2
+ O(x3 ), (2.13)
Qext I (x, θ) = Im{m − 1} cos θ + 2
2
|m + 1|
2
Qext II (x, θ) =
Im{m2 − 1}
2πx + O(x3 ) .
|m2 + 1|2
(2.14)
27
Chapter 2. Small dielectric cylinders
As required, in the limit m → 1 all efficiencies become zero: the refractive
index is the same everywhere and there is no scattering any more. Also in
the dipole limit x → 0 the efficiencies become zero: in the far field region
the scattered field can be neglected with respect to the incident field.
The solutions (2.11)-(2.14) depend in a specific way on the angle of
incidence θ, which will be illustrated in the next paragraph. It has its origin
in the boundary conditions (1.9)-(1.12) on the fields.
Apart from this the limit θ → π2 gives an extra requirement: in this limit
the difference between Case I and Case II has to vanish as can be argued
with symmetry arguments. This can be seen by taking θ = π2 in Figure 1.1.
In both cases the incident fields E0 and H0 become perpendicular to the
wire axis and by rotational symmetry around this axis Case I and Case II
describe the same situation. Indeed, taking the limit θ → π2 the efficiencies
in Case I and Case II are equal:
π
π
|m2 − 1|2 π 2 x3
Qsca I (x, ) = Qsca II (x, ) =
,
2
2
|m2 + 1|2 2
(2.15)
π
Im{m2 − 1}
π
Qext I (x, ) = Qext II (x, ) =
2πx + O(x3 ) .
2
2
|m2 + 1|2
(2.16)
It is a limit in the sense that an incident wave in the same direction as the
wire axis (θ = π2 ) needs a special treatment. How to consider light incident
on the endpoints of an (relatively) infinite cylinder? In fact this is the situation of wave guiding, which will not be treated in this paper.
Furthermore, the solutions (2.11)-(2.14) can be compared to literature
for θ = 0. At normal incidence they are in agreement with the efficiency
factors derived by Van de Hulst [7] and Kerker [8]:
π 2 x3
+ O(x5 ) ,
8
(2.17)
|m2 − 1|2 π 2 x3
+ O(x5 ) ,
|m2 + 1|2 4
(2.18)
πx
+ O(x3 ),
2
(2.19)
Qsca I (x, 0) = |m2 − 1|2
Qsca II (x, 0) =
Qext I (x, 0) = Im{m2 − 1}
Qext II (x, 0) =
2.3.1
Im{m2 − 1}
2πx + O(x3 ) .
|m2 + 1|2
(2.20)
Polarization anisotropy, polarization contrast
In order to express the difference between incident TM (Case I) and TE
(Case II) waves properly, it is common to define a polarization anisotropy
ρ [1] [26]. Up to now, for dielectric cylinders this quantity has mainly been
28
2.3. Efficiency, polarization anisotropy and - contrast in Rayleigh approximation
estimated by looking at the internal fields at normal incidence and in the
dipole limit. Denoting the polarization anisotropy in this special case by
ρint , it is defined by:
ρint ≡
|Eint I |2 − |Eint II |2
,
|Eint I |2 + |Eint II |2
(2.21)
where |Eint |2 = Iint indicates the internal intensity in dipole approximation.
By using the fields in dipole approximation (2.6) and (2.7) this yields
ρint =
|m2 + 1|2 − 4
.
|m2 + 1|2 + 4
(2.22)
In principle one could proceed further by using the expressions (2.8), (2.9)
and (2.10) to get the expanded intensities and so an expansion of the polarization anisotropy ρint (x, 0) inside the wire , but in fact this last step requires
far too much calculations: also the direction of the field has to be taken in
to account properly. Even harder, the intensity (or Poynting vector) starts
to depend on the position in the wire.
Instead, it is much easier to calculate the polarization anisotropy by
making use of the efficiency factors in the far field region. In terms of the
extinction efficiencies, the extinction polarization anisotropy is defined by
ρext ≡
Qext I − Qext II
,
Qext I + Qext II
(2.23)
A formal prove that this ratio in general equals the polarization anisotropy
inside the wire, ρext = ρint , is complicated and will not be given here. Instead, the equality can be explained by the following argument: the internal
fields are modified with respect to the incident field both by scattering and
absorption, so ρint contains the relative difference of the total removal of
energy. This is nothing else than the relative difference in extinction between the two cases, which is described by ρext .
Contrary to the general case, it is easy to prove the equality at normal
incidence in the dipole limit. Using equations (2.19) and (2.20), ρext is
approximated by
ρext (x, 0) =
|m2 + 1|2 − 4
+ O(x2 ) = ρint + O(x2 ) ,
|m2 + 1|2 + 4
(2.24)
so taking the dipole limit x → 0 on both sides yields ρext = ρint .
Next to the extinction polarization anisotropy defined above, it is also
insightful to define scattering and absorption polarization anisotropies. Since scattering, absorption and extinction are related to each other by (1.60)
only the scattering polarization anisotropy will be treated here. It is defined
by
ρsca
≡
Qsca I − Qsca II
.
Qsca I + Qsca II
(2.25)
29
Chapter 2. Small dielectric cylinders
Using (2.17) and (2.18) the scattering polarization anisotropy at normal
incidence is expanded in the size parameter by
ρsca (x, 0) =
|m2 + 1|2 − 2
+ O(x2 ) ,
|m2 + 1|2 + 2
(2.26)
Although often used, the polarization anisotropy is not a quite useful
quantity to work with in practice. This will be illustrated in the next section, but at this stage it is anticipated by introducing a new quantity that
describes the difference between the case of incident TM waves and incident
TE waves. It is called the polarization contrast and defined by
Cext ≡
Qext I
,
Qext II
(2.27)
Csca ≡
Qsca I
,
Qsca II
(2.28)
for the total removal of EM energy and for scattering, respectively.
Again, using the expanded efficiencies (2.17)-(2.20) the approximations
at normal incidence are
Cext (x, 0) =
|m2 + 1|2
+ O(x2 ) ,
4
(2.29)
Csca (x, 0) =
|m2 + 1|2
+ O(x2 ) .
2
(2.30)
Note that the limit m → 1 does not work for the scattering polarization
anisotropies and contrasts any more.
30
2.4. Results
2.4
Results
As an illustration of the results in the previous sections it is insightful to
choose a particular material, InP for example. In particular it is interesting
for which radius (or size parameter) the Rayleigh approximation is valid.
Recall that the complex refractive index is the only material property appearing in the classical theory derived here, apart from R. It depends on
the circular frequency ω of the incident light, see section 1.1 : the material
responds to the incident periodic EM field and this response depends on the
frequency and so on the wavelength of the incident light.
For InP this is illustrated in Figure 2.1. Actually the figure shows the real
and imaginary parts ²0 and ²00 of the complex dielectric function ² [11], which
is related to the complex refractive index by
m2 ≡ ² ≡ ²0 + i ²00 .
(2.31)
Absorption and scattering are more simply described by these optical ”constants”, so from now on all quantities are discussed in terms of ².
Ε' , Ε''for InP
17.5
15
12.5
Ε'
10
7.5
Ε''
5
2.5
0
350
400
450
500
550
600
Λ
Figure 2.1: Bulk values of the real and imaginary part ²0 and ²00 of the
complex dielectric function ² for InP, as a function of λ0 . It is calculated
by interpolating between 32 (optic) experimental values in this interval.
From Figure 2.1 it becomes directly clear that showing the efficiencies
as a function of the dimensionless size parameter x is misleading: it really
matters if x is changed by varying the wave number or the radius. At different wave numbers also ² has changed, only over a narrow range at small
values of k the optical constants can be considered as constant.
However, in literature it is common to show the efficiency as a function
of x at a fixed ², mainly because it is the most convenient way. This is also
31
Chapter 2. Small dielectric cylinders
the starting point in this paper, see Figure 2.2. For six fixed values of λ0 , so
for six different values of ², the extinction efficiencies in Case I and Case II
up to third order in x at normal incidence are plotted as a function of x.
608
508
422
395
376
354
nm
nm
nm
nm
nm
nm
HbL: QIIext, Θ = 0
608
508
422
395
376
354
0.015
QIIext
QIext
HaL: QIext, Θ = 0
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.01
nm
nm
nm
nm
nm
nm
0.005
0
0
0.01 0.02 0.03 0.04 0.05
x
0
0.01 0.02 0.03 0.04 0.05
x
Figure 2.2: Extinction efficiencies Qext I and Qext II at normal incidence
as a function of the size parameter x = kR. In every plot k, and so ²,
is fixed. The corresponding values for the wavelength are showed in the
left corner. The efficiencies are expanded up to third order in x.
In the illustrated domain the linear terms (2.19) and (2.20) in the extinction efficiencies dominate and the total removal of EM energy from the
incident beam increases with the size parameter.
The slope of this relation depends on the wavelength and in a quite different
way for the two cases. It is explained by looking closely to the dependence
on the complex dielectric function:
• In Case I the slope increases to a maximum around λ0 = 400 nm
after which it decreases for increasing wavelength. This is caused by
the factor Im{m2 − 1} = ²00 in (2.19), see Figure 2.1.
• In Case II the slope decreases in the whole domain for increasing λ0 .
2 −1}
00
The increase of the denominator of Im{m
∝ (²0 )2²+(²00 )2 in (2.20)
|m2 +1|2
dominates the increase of the numerator to ∼ 400 nm. Afterwards
²00 falls off so fast that the slope remains decreasing for increasing
wavelength.
It even seems from the figure that the third order terms can be neglected
in the given domain. Nevertheless, a closer look gives the opposite: for
increasing x the third order terms start to give significant corrections. This
is illustrated in Figure 2.3: here the correction by the third order terms with
respect to (2.19) and (2.20) are shown in percentages.
The figure reveals that also the magnitude of the deviation depends on
the wavelength: in both cases the influence of the third order correction
becomes larger for increasing wavelength. This is explained with the same
32
2.4. Results
HaL: dev. QIext, Θ = 0
10
nm
nm
nm
nm
nm
nm
608
508
422
395
376
354
6
%
608
508
422
395
376
354
15
%
HbL: dev. QIIext, Θ = 0
8
20
4
nm
nm
nm
nm
nm
nm
2
5
0
0
0
0.01 0.02 0.03 0.04 0.05
x
0
0.01 0.02 0.03 0.04 0.05
x
Figure 2.3: Deviation of linear behavior in percentages of Qext I and
Qext II at normal incidence, as a function of x.
kind of arguments as for the extinction factors itself, but it is omitted here
since the third order corrections are not shown explicitly.
In the same line as for the extinction efficiencies also the scattering and
absorption efficiencies can be illustrated. Since these quantities are dependent of each other only Qsca I and Qsca II are showed here. Figure 2.4 shows
that for small x the extinction is completely dominated by the absorption:
the contribution of the scattering to the total extinction (Figure 2.2) is only
about 1%. This is due to the absence of a first order term in the expansion
HaL: QIsca , Θ = 0
608
508
422
395
376
354
QIsca
0.01
0.008
0.006
nm
nm
nm
nm
nm
nm
608
508
422
395
376
354
0.00008
QIIsca
0.012
HbL: QIIsca, Θ = 0
0.004
0.00006
0.00004
nm
nm
nm
nm
nm
nm
0.00002
0.002
0
0
0
0.01
0.02
x
0.03
0.04
0
0.01
0.02
x
Figure 2.4: Scattering efficiencies Qsca I and Qsca II at normal incidence
as a function of x. In every plot k, and so ², is fixed. The corresponding
values for the wavelength are showed in the left corner. The efficiencies
are expanded to third order in x.
of Qsca compared to Qext , see equations (2.17)-(2.20).
The correction to the third order terms in Figure 2.4 are depicted in
Figure 2.5. This is the deviation caused by the x5 terms in percentages.
The calculation of these x5 terms is only done for scattering, since in this
case the coefficients are needed to x4 while for extinction one needs also the
sixth order terms.
33
0.03
0.04
Chapter 2. Small dielectric cylinders
HaL: err
. QIsca , Θ = 0
608
508
422
395
376
354
%
3
2
nm
nm
nm
nm
nm
nm
608
508
422
395
376
354
4
3
%
4
HbL: err
. QIIsca, Θ = 0
5
2
1
nm
nm
nm
nm
nm
nm
1
0
0
0
0.01
0.02
x
0.03
0.04
0
0.05
0.1
x
0.15
Figure 2.5: Correction to the third order expansion of Qsca I and Qsca II
by the x5 terms in percentages; again at normal incidence, as a function
of x.
The above illustrations are useful for determining the range of the size
parameter in which a certain approximation is valid. For instance, Figure 2.5
shows that for a maximal deviation of 5% the third order approximation
holds to x ∼ 0.03 and x ∼ 0.2 for Qsca I and Qsca II , respectively.
However, this way of displaying is quite awkward for investigating the
wavelength dependence: the different curves belong to different values of k.
Actually, keeping the wire radius fixed requires looking at a smaller x value
by going to a curve at higher wavelength.
Also the determination of the limiting R values will not work properly.
A size parameter x ∼ 0.03 at λ0 = 400 nm gives R ∼ 2.2 nm, but it would be
much more convenient to get these values as a function of the wavelength.
Before doing this, the next paragraph will illustrate the dependence on the
angle of incidence.
2.4.1
Efficiencies and polarization anisotropy at oblique incidence
The most dominant feature that will appear by displaying the efficiencies
as a function of the angle of incidence θ is the symmetry requirement for
θ = π2 , as explained in section 2.3.
Starting with extinction, it is important to note that also the x3 terms
are encountered in Figure 2.6. This means for instance that the first order
term (2.14), which is independent of θ, is corrected a little bit by the third
order term. The scattering efficiencies are shown in Figure 2.7. In both
cases the cylinder radius is fixed, R = 2 nm. Figure 2.6 illustrates the θ
dependence of Qext I and Qext II as given in equations (2.13) and (2.14).
The difference between Qext I and Qext II at θ = 0 is explained by the large
denominator in (2.14): |m2 + 1|2 ' 200. For increasing θ, Qext I decreases to
a limiting value which is equal to Qext II at θ = π2 : the first term in (2.13)
34
0.2
2.4. Results
falls off to zero, while the second term increases leading to Qext I ' Qext II
at θ = π2 . Also the scattering efficiencies, shown in Figure 2.7, are the same
at θ = π2 . As discussed before this is due to the fact that the TM and TE
case describe the same physical situation at θ = π2 .
HaL: QIext, R = 2 nm
QIext
0.6
0.4
nm
nm
nm
nm
nm
nm
QIIext
608
508
422
395
376
354
0.8
HbL: QIIext, R = 2 nm
0.2
0
Π
€€€€€
8
0
Π
€€€€
4
Θ
Π
€€€€
2
3Π
€€€€€€€€
8
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
376 nm
354 nm
Π
€€€€€
8
0
422 nm
395 nm
608 nm
508 nm
Π
€€€€
4
Θ
3Π
€€€€€€€€
8
Π
€€€€
2
Figure 2.6: Extinction efficiencies Qext I and Qext II as a function of the
angle of incidence θ for a fixed cylinder radius R = 2 nm. In every plot
k, and so ², is fixed. The efficiencies are expanded to third order in x.
608
508
422
395
376
354
HbL: QIIsca, R = 2 nm
nm
nm
nm
nm
nm
nm
QIIsca
QIsca
HaL: QIsca , R = 2 nm
0.0175
0.015
0.0125
0.01
0.0075
0.005
0.0025
0
422
395
376
354
0.00015
nm
nm
nm
nm
608 nm
508 nm
0.0001
0.00005
0
Π
€€€€€
8
0
Π
€€€€
4
Θ
3Π
€€€€€€€€
8
Π
€€€€
2
0
Π
€€€€€
8
Π
€€€€
4
Θ
3Π
€€€€€€€€
8
Figure 2.7: Scattering efficiencies Qsca I and Qsca II as a function of θ
for R = 2 nm. In every plot k, and so ², is fixed. The efficiencies are
expanded to third order in x.
In order to look more closely to the difference between TM and TE
waves, the extinction polarization anisotropy (2.23) as well as the scattering
polarization anisotropy (2.25) corresponding to the expanded efficiencies at
R = 2 nm are depicted in Figure 2.8. Indeed, ρsca and ρext become zero in the
limit θ → π2 . At normal incidence the polarization anisotropies reach their
maximum value, around 0.985 for extinction as well as for scattering. But
the distinction between the curves for different wavelength is hard to extract
from the figures. Also the difference between extinction and scattering is
not illustrated clearly.
As stated in section 2.3 it is more convenient to look at the polarization
35
Π
€€€€
2
Chapter 2. Small dielectric cylinders
HaL: Ρ sca for R = 2 nm
HaL: Ρ extfor R = 2 nm
1
1
0.8
608
508
422
395
376
354
0.6
0.4
0.2
0
0
nm
nm
nm
nm
nm
nm
Π
€€€€€
8
Ρ ext
Ρ sca
0.8
608
508
422
395
376
354
0.6
0.4
0.2
0
Π
€€€€
4
Θ
Π
€€€€
2
3Π
€€€€€€€€
8
0
nm
nm
nm
nm
nm
nm
Π
€€€€€
8
Π
€€€€
4
Θ
Π
€€€€
2
3Π
€€€€€€€€
8
Figure 2.8: Polarization anisotropy ρext and scattering polarization anisotropy ρsca as a function of θ for R = 2 nm. In every plot k, and so ²,
is fixed. The factors are expanded up to second order in x.
contrast, equations (2.27) and (2.27). For R = 2 nm this is displayed in
Figure 2.9. Now the difference between scattering and extinction becomes
clear: at normal incidence the depolarization for scattering is significant
larger then for extinction. By rotating the angle of incidence to θ = π2 the
scattering polarization anisotropy also decreases faster to the limiting value
zero. This effect is also visible in Figure 2.8, but less clearly.
C sca for R = 2 nm
C sca
200
150
100
C extfor R = 2 nm
nm
nm
nm
nm
nm
nm
C ext
608
508
422
395
376
354
250
50
0
0
Π
€€€€€
8
Π
€€€€
4
Θ
3Π
€€€€€€€€
8
Π
€€€€
2
140
120
100
80
60
40
20
0
608
508
422
395
376
354
0
Π
€€€€€
8
Π
€€€€
4
Θ
3Π
€€€€€€€€
8
Figure 2.9: Polarization contrast Cext and scattering polarization contrast Csca as a function of θ for R = 2 nm. In every plot k, and so ², is
fixed. The factors are expanded up to second order in x.
2.4.2
Efficiencies and polarization anisotropy as a function
of wavelength
As stated in section 2.4, the most physically accurate picture of the scattering process is obtained by showing the efficiencies and polarization anisotropies as a function of the wavelength (or wave number).
Apart from the practical points made in the previous sections, this kind
36
nm
nm
nm
nm
nm
nm
Π
€€€€
2
2.4. Results
of figures also contain far more information: for every wavelength a set of
optical constants has to be used. This is done by interpolating between
measured data points for the complex dielectric function, see Figure 2.1. In
this way the response of the dielectric cylinder as a function of the frequency
of the incident EM field is obtained.
HaL: QIext, Θ = 0
HbL: QIIext, Θ = 0
2.5
5 nm
4 nm
3 nm
2 nm
1 nm
1.5
1
5 nm
4 nm
3 nm
2 nm
1 nm
0.03
QIIext
QIext
2
0.02
0.01
0.5
0
350
400
450
500
550
0
350
600
400
450
Λ
500
550
600
Λ
Figure 2.10: Extinction efficiencies Qext I and Qext II at normal incidence as a function of the wavelength at constant cylinder radii. The
corresponding five R values are showed in the right corner. The efficiencies are expanded to third order in x. The plots are calculated by
interpolating between thirty-two (optic) experimental values of ² in this
interval.
HaL: QIsca , Θ = 0
HbL: QIIsca, Θ = 0
0.3
0.2
0.15
5 nm
4 nm
3 nm
2 nm
1 nm
0.1
5 nm
4 nm
3 nm
2 nm
1 nm
0.001
0.00075
0.0005
0.05
0
0.0015
0.00125
QIIsca
QIsca
0.25
0.00025
350 375 400 425 450 475 500
Λ
0
350
400
450
500
Λ
Figure 2.11: Scattering efficiencies Qsca I and Qsca II at normal incidence
as a function of the wavelength at constant cylinder radii. The efficiencies
are expanded to third order in x.
For the extinction and scattering efficiencies at normal incidence this is
depicted in Figure 2.10 and Figure 2.11, respectively. The domain of the
wavelength is limited: the high and low frequency regions are omitted. The
figures show the dependence of the wavelength at five fixed values of R.
The dependence of the cylinder radius is visible, especially for extinction:
the extinction efficiencies increase linearly with increasing radius by going
from one curve to the next in Figure 2.10.
37
550
600
Chapter 2. Small dielectric cylinders
The shape of the curves has been explained above: they where already
visible in Figure 2.2 and Figure 2.4, but not so clearly. Now the particular
dependence of (2.19) and (2.20) on the complex dielectric function is really
visible. For instance, Figure 2.10 nicely shows that the behavior of the
imaginary part ²0 is completely reflected in the extinction efficiency.
HaL: err
. QIsca ,Θ=0
%
15
10
HbL: err
. QIIsca ,Θ=0
1.5
5 nm
4 nm
3 nm
2 nm
1 nm
5 nm
4 nm
3 nm
2 nm
1 nm
1.25
1
%
20
0.75
0.5
5
0.25
0
350
400
450
500
550
0
350
600
400
450
Λ
500
550
600
Λ
Figure 2.12: Correction to the third order expansion of Qsca I and Qsca II
by the x5 terms in percentages; again at normal incidence, as a function
of λ0 at constant R.
In the same way as in Figure 2.5, the correction to the third order terms
in Figure 2.11 is illustrated in Figure 2.12. This is the deviation caused by
the x5 terms given as a percentage. For a cylinder radius below 2 nm the
used expansion is accurate to 5%. For larger radii it really depends on the
wavelength if the approximation is acceptable.
HaL: Ρ sca , Θ = 0
0.992
HbL: Ρ ext, Θ = 0
0.985
1 - 5 nm
5 nm
4 nm
3 nm
2 nm
1 nm
0.98
0.988
Ρ ext
Ρ sca
0.99
0.986
0.975
0.97
0.984
0.965
0.982
0.98
350
0.96
400
450
500
Λ
550
600
350
400
450
500
Λ
Figure 2.13: Polarization anisotropy ρext and scattering polarization anisotropy ρsca at normal incidence as a function of the wavelength at constant cylinder radii. The corresponding five R values are showed in the
right corners. Figure (a) shows that ρsca is independent of the wire radius. The polarization anisotropy ρext is expanded including the second
order term, ρsca up to second order in x.
The polarization ratio and scattering polarization ratio corresponding to
Figure 2.10 and Figure 2.11 are shown in Figure 2.13. Figure 2.13 (a) shows
38
550
600
2.4. Results
that ρsca expanded up to second order in x is independent of the wire radius,
see (2.26). For extinction also the second order is included. Figure 2.13 (b)
shows that in that case ρext depends on the wire radius: in particular for
wavelengths larger then 400 nm the polarization ratio becomes larger. In
other words, increasing the wire radius implies a bigger difference between
the case of an incident wave with the magnetic field perpendicular to the
wire axis (TM) and the case where the electric field is perpendicular (TE).
Remember that this result only applies for small R, results for larger radius
or size parameter are showed in [7] [8] [11].
It is really interesting to compare the obtained polarization anisotropy
ρext with the depolarization in the dipole limit ρint (2.22), also used in [1]
[26].
This is shown in Figure 2.14. The blue curve shows the case when for ρint
in the dipole limit only the real part of the bulk ² is taken into account.
The difference with the the red curve, indicating ρint for the complete ²,
becomes dramatically large for λ below 400 nm, and remains significant for
the other wavelengths.
HaL: Ρ ext, Ρ int; , Θ = 0
HbL: C ext, C int; Θ = 0
C ext  C int
0.98
0.96
Ρ
0.94
0.92
0.9
0.88
0.86
350
Ρ int, dip.app., Ε'
Ρ int, dip.app.
Ρ ext,R = 5 nm
Ρ ext,R = 2 nm
400
450
500
Λ
550
600
140
120
100
80
60
40
20
0
350
C int, dip.app., Ε'
C int, dip.app.
C ext,R = 5 nm
C ext,R = 2 nm
400
450
500
Λ
Figure 2.14: Comparison of the internal polarization anisotropy/contrast
in the dipole limit with ρext / Cext . In blue curve, indicating ρint , only
the real part of ² is taken into account. The red curve shows ρint for the
complete ² .
The yellow and green curve show that increasing the wire radius to R =
5 nm already gives a significant difference between the solution of the dipole
limit and the expanded polarization anisotropy ρext . Figure 2.14 (b) shows
the same results in terms of the contrasts.
39
550
600
Part II
Absorption
40
Chapter 3
Electronic properties
In this chapter the electronic properties of nanowires made from III-V compounds are discussed. The results are based on more detailed studies which
can be found in basic semiconductor books [14] [15] and articles by Luttinger
[16], Sercel [17] and Marechal [18].
The electronic band structure and wave functions in a nanowire are calculated using the effective mass approximation. This method is in particular
convenient to study the optical properties of a semiconductor structure, because analytic expressions for the band dispersion, effective mass and electron/hole wavefunctions around high symmetry points can be obtained.
Before turning to the nanowire, in section 3.1 first the band dispersion in
bulk material will be derived. Next to general theory, the specific situation
of the degenerate top valence band in III-V semiconductor materials will be
treated. It has its specific importance in the next chapters and will therefore
also be the guideline in the other sections of this chapter: in section 3.2 the
effective mass theory for bulk systems is treated, sections 3.3 and 3.4 summarize the envelope description in case of an infinite nanowire and explicit
results for InP and InAs are found in section 3.5.
3.1
The k · p method
There are various ways to determine the electronic bands of a semiconductor.
Global dispersion relations of bulk materials are available (pseudo-potential
techniques, tight binding) but in a lot of cases they are unnecessary.
In particular, for describing the optical properties of a semiconductor structure it is often sufficient to know the band dispersion in a small range around
the band extremes. This is achieved by the k · p method, which differs from
the procedures mentioned above in the fact that, next to the band gaps, also
the oscillator strengths of the transitions are used as input. In the k · p method the band dispersion around any point ka is obtained by extrapolation
from the k = ka energy gaps and optical matrix elements, using either de41
Chapter 3. Electronic properties
generate or non-degenerate perturbation theory . The input data at k = ka
can be obtained from experimental results, typically at the high symmetry
points of the crystal.
Starting point is the one-electron Schrödinger equation describing the
motion of an electron in an averaged potential V (r), which is obtained from
the Hamiltonian of a perfect crystal containing N unit cells after usual
assumptions such as the Born-Oppenheimer and mean field approximation.
The potential V (r) is assumed to reflect the periodicity of the perfect crystal:
V (r + R) = V (r),
(3.1)
where R are the lattice vectors. Including the spin-orbit interaction the
Hamiltonian describing the unperturbed semiconductor becomes
H0 =
p2
~
+ V (r) + 2 2 (σ × ∇V ) · p,
2m0
4c m0
(3.2)
where m0 denotes the free electron mass and σ are the Pauli spin matrices.
The relativistic character of the spin-orbit term is reflected by the c12 dependence.
Note that the total Hamiltonian, including the spin-orbit interaction, is invariant under a translation by R. H0 thus commutes with the translation
operator of the crystal and has Bloch functions as solutions. After normalizing over the whole crystal, containing N unit cells, these are defined
as:
1
ψnk (r) = N − 2 eik·r unk (r),
(3.3)
where the unk ’s have the periodicity of the lattice, are normalized over one
unit cell and k lies in the first Brillouin zone.
The Bloch functions (3.3) form a complete and orthonormal set. Next
1
to this, the Bloch solutions ψn0 = N − 2 un0 at k = 0 are also periodic. Once
these, or to be more precise, the corresponding interband matrix elements
and energies ²n ≡ ²n (0) are known, the energy dispersion around the zone
center (k = 0) can be derived using perturbation theory.
In principle, this argumentation can be extended to any point k = ka ,
provided the transition matrix elements and energies at k = ka are known.
This result has been widely discussed in literature [14] [15] [16] [17] [18],
here only the results around the zone center are summarized.
Assuming the band structure has an extremum (almost) at the zone
center and taking k sufficiently small 1 , the dispersion relation for a nondegenerate band (apart from spin) is given by
²n (k) = ²n +
1
~2 k 2
,
2m∗n
(3.4)
Sufficiently small means that the corresponding energy difference ²n (k) − ²n remains
much smaller then the band edge differences ²n − ²n0 and that the terms linear k are small
enough to be neglected.
42
3.1. The k · p method
where m∗n is the effective mass of the band,
1
m∗n
=
1
2 X | π nn0 · k |2
+ 2 2
m0 m0 k 0
²n − ²n0
(3.5)
n 6=n
and π nn0 are the the interband matrix elements at the zone center:
π nn0
~
σ × ∇V | un0 0 i
≡ hun0 | p +
4m0 c2
µ
¶
Z
~
3
∗
=
d r un0 (r) p +
σ × ∇V un0 0 (r).
4m0 c2
(3.6)
(3.7)
With the same assumptions, for a degenerate band relation (3.4) is replaced
by
hj,j 0 (k) = ²j δj,j 0 +
~2 k 2 1
2 m∗jj 0
(3.8)
with
1
m∗jj 0
=
X (k · π jm )(k · π mj 0 )
1
2
δj,j 0 + 2 2
.
m0
²j − ²n0
m0 k
(3.9)
²n0 6=²j
Here j denotes the degeneracy and the summation over n0 describes the coupling between the group of degenerate states and the other bands. Contrary
to the case of a non-degenerate band, one is left with a matrix hj,j 0 (k) which
has to be diagonalized in order to get the dispersion relation(s).
It should be noted that within the notation used here the tensor behavior
of the effective mass is neglected. In general, the coupling between k and
the interband matrix elements causes the effective mass to be non isotropic
and inclusion of this effect is achieved by the substitution
X 1
1 2
k
−→
k k ,
(3.10)
∗ αβ α β
m∗n
αβ mn
1
m∗n αβ
=
β
1
2 X πnα0 n πnn0
δαβ + 2
m0
²n − ²n0
m0 0
(3.11)
n 6=n
in (3.4) and a similar one in case of a degenerate band. The surfaces of
constant energy belonging to this effective mass tensor are not spheres any
more, but warped in certain directions, depending on the symmetry properties of the band under consideration.
Furthermore, there are three remarks important to be made at this stage. First, in most of the cases the summation over the bands n0 in (3.5)
and (3.9) can be executed over a limited number of values. For large energy differences ²n − ²n0 the contribution of n0 to the effective mass becomes
43
Chapter 3. Electronic properties
relatively unimportant. Also the interband matrix elements in the numerator reduces the number of bands that contribute to the effective mass. As
will be explicitly shown in subsection 4.4.1, the matrix elements are subject
to selection rules which are determined by the symmetry properties of the
bands in question. Most of the matrix elements become zero by this kind of
symmetry arguments.
Secondly, including the spin-orbit interaction in case of a non-degenerate
band makes little practical difference, since its effect is absorbed in the interband matrix elements which are determined by experiment. For a degenerate
band this is different, because the spin-orbit interaction in general lifts the
degeneracy and will cause small splitting between the bands.
A last important remark has to be made concerning the limitations of the
k·p method as depicted here, up to second order in k. The above results rely
on the assumption that ²n (k) − ²n remains much smaller then the band edge
differences ²n − ²n0 (and a similar assumption in case of degenerate bands),
which is not necessarily satisfied, e.g. in semiconductor compounds with a
narrow band gap. Instead of expanding beyond second order in the same
framework, a commonly used approach [14] [15] [17] initiated by Kane [19]
solves this problem by diagonalizing the group of neighboring bands exactly
and afterwards treating the coupling with the well separated other bands in
a second order perturbation. However, in the remaining part of this paper
it is assumed that the bands under investigation are well separated from the
others, i.e. splitting terms as the band gap Eg and spin-orbit splitting ∆0
are assumed to be sufficiently large.
3.1.1
Top valence bands in III-V semiconductors
In principle the above theoretical statements now can be applied to any
band, or group of bands, once the energy and the interband matrix elements
are known. Here the band structure of the six fold degenerate (including
spin) top valence band at the Γ point (k = 0) in III-V semiconductor compounds will be summarized. However, the explicit diagonalization of the
matrix (3.8) will be performed further on in the envelope function framework since this is the most convenient way when the nanowire structure is
anticipated.
Starting with symmetry considerations, it is well known [14][15] that the
top valence bands in III-V materials have Γ4 like symmetry, apart from spin.
The corresponding spatial parts of the valence band wavefunctions at k = 0
are p-like, which means that they are triply degenerate and transform under
rotations like the three components of a vector. Including spin this leads to
six band edge Bloch functions, which are denoted by |Xi|σi, |Y i|σi, |Zi|σi,
with σ =↑,↓.
The one-electron Hamiltonian H0 is diagonalized by linear combinations
of these band edge Bloch functions. Rewriting the spin orbit term in (3.2)
44
3.1. The k · p method
as
Hs.o. = λL0 · σ,
(3.12)
with L0 the angular momentum of the atomic states and treating Hs.o. as a
small perturbation2 , this term is diagonalized by the eigenfunctions of the
total angular momentum J = L0 + σ of the atomic states. Subsequently
the total Hamiltonian H0 can be expressed in the transformed zeroth order
eigenfunctions |j, jz i, with j the eigenvalues of J and jz the eigenvalues of
its projection Jz along the z axis. These are defined as
¯3 3®
¯ ,
= − √12 |X + iY i| ↑ i,
2 2
q
¯3 1®
2
¯ ,
√1 |X + iY i| ↓ i +
=
−
2 2
3 |Zi| ↑ i,
6
q
¯3 1®
¯ ,−
= √16 |X − iY i| ↑ i + 23 |Zi| ↓ i,
2
2
¯3 3®
¯ ,−
= √12 |X − iY i| ↓ i
2
2
for the j =
3
2
(3.13)
(3.14)
(3.15)
(3.16)
quadruplet and
¯1 1®
¯ ,
= − √13 |X + iY i| ↓ i + |Zi| ↑ i,
2 2
¯1 1®
¯ ,−
= − √13 |X − iY i| ↑ i − |Zi| ↓ i,
2
2
(3.17)
(3.18)
for the two j = 12 states. The last ones are split from the j = 32 states by the
spin-orbit interaction, with a magnitude ∆0 = 23 λ. For ∆0 sufficiently large,
such that the matrix elements which couple the j = 32 and j = 12 bands are
negligible compared to ∆0 , the 6 × 6 Hamiltonian can be decoupled into a
4 × 4 and a 2 × 2 matrix.
In most III-V semiconductors, the 4 × 4 matrix of the j = 32 states corresponds to the top most valence band. Assuming the spin-orbit coupling
large enough, in this paper the valence band dispersion will be derived by
diagonalizing the Γ8 Hamiltonian of these j = 32 states. This is achieved in
the same framework as used by Sercel [17] and as in [18]; the explicit results
are given in section 3.2.
As stated above, corrections to this approach can be found by including the
split-off (Γ7 ) band of the j = 12 states and possibly also the lowest conduction band, which usually has Γ6 symmetry. The last one in general is
less important since in most III-V semiconductors the spin-orbit splitting is
much smaller than the band gap Eg . Including more bands will improve the
results, but makes the calculations harder. Focussing on the dispersion for
small k around the zone center, it is assumed that these corrections can be
neglected in first instance.
2
λ is small because of the relativistic character of the spin-orbit interaction
45
Chapter 3. Electronic properties
3.2
Effective mass approximation
Suppose an infinite system which is built from the perfect crystal, and a
disturbance δV which has to be restricted by specific properties, as will be
explained below. The Schrödinger equation (S.E.) of the system is given by
(H0 + δV ) |Ψi = E |Ψi.
(3.19)
In principle the solutions of the S.E. can be found by expanding Ψ in terms
of the complete orthonormal set of Bloch functions, but without making any
further approximation this requires an extensive job since the disturbance
δV breaks the translational symmetry of the crystal.
The problem is solved much easier by assuming δV to be slowly varying
over one unit cell and making use of the band parameters of the unperturbed
system, equations (3.4) and (3.8). This approach is known as the effective
mass approximation. It can be derived either by utilizing Bloch functions, or
in the context of the more localized Wannier functions. Here the Wannier
functions are used. They are related to the Bloch functions by Fourier
transformation and defined by
1 X
e−ik·R ψnk (r).
(3.20)
anR (r) = N − 2
k
Note that the Wannier functions are indexed by the lattice vector R, reflecting the localized character. They form a complete, orthonormal set
just as the Bloch functions and depend on the difference between r and R:
anR (r) = an (r − R).
Using the complete and orthonormal set of Wannier functions, the solution
Ψ(r) of (3.19) is expanded as:
X
Ψ(r) =
Fj (R)ajR (r),
(3.21)
jR
where j sums over the j degenerate bands and thus includes only one band
n in the non degenerate case. The functions Fj (R) are known as the envelope wave functions: as will be shown below, they describe wave packets,
extended over (a part of) the crystal and are the envelopes of the atomistic
variations caused by the Wannier functions.
In order to convert the total Hamiltonian H0 +δV in (3.19) into operators
acting on the Wannier functions, it is stated here that k and R are conjugate
operators in the sense that
R ←→ i∇k
and
k ←→ −i∇R ,
(3.22)
in the limit of large N . Note that R now is treated as a continuous variable,
which is justified by the large N limit, i.e. the size of the semiconductor
compound is much larger than the distance between the atoms.
46
3.2. Effective mass approximation
Using this result and assuming that δV is a slowly varying function with
respect to a lattice vector, it can be shown [14][18] that the S.E. (3.19)
reduces to a Schrödinger equation for the envelope functions:
{²n (−i∇R ) + δV (R)}Fn (R) = EFn (R)
(3.23)
in case of a non degenerate band n and
X
{hj,j 0 (−i∇R ) + δV (R)}Fj 0 (R) = EFj (R)
(3.24)
j0
for a degenerate band. For a given band, equation (3.23) ((3.24)) describes
the motion of a particle with effective mass m∗n (m∗jj 0 ) in a potential δV .
Note that the total wave function of this particle, moving in the perturbed
crystal, is obtained from the solutions of (3.23)/(3.24) by multiplying with
the Wannier functions as in (3.21).
3.2.1
Crystal Hamiltonian in envelope representation
The above envelope framework initially was derived in the context of impurity states, but as stated by Sercel [17], the procedure can also be used
to develop a representation of the unperturbed Hamiltonian H0 which anticipates a centrosymmetric or cylindrical heterostructure. Instead of the
Wannier representation (3.21), the solution is expanded in the zone center
Bloch functions |uj i by the ansatz
X
(3.25)
|Ψi =
|Fj i |uj i,
j
which is justified if the energy difference ²j − ²n0 between the degenerate
bands and al others is sufficiently large such that unk ' un0 .
The notation used in (3.25) stresses the fact that the envelope functions Fj
act in a different space as the zone center Bloch functions, this is shown
in more detail in chapter 4 considering the transition matrix element. The
Bloch functions are defined within a unit cell, while the envelopes are extended over a sufficiently large group of lattice points. It should be mentioned
again that the assumption unk ' un0 is essential in this context.
In addition to the assumptions in section 3.13 , an extra approximation
has to be made here concerning the anisotropy, in order to profit fully from
the envelope representation. Conform the situation in most of the III-V
semiconductor materials, it is assumed that the anisotropic terms in the
Hamiltonian can be neglected, at least as a first order approximation. In
this spherical approximation the lower cubic terms causing the warping of
3
I.e. k sufficiently small and energy gaps such as ∆0 and Eg large enough to neglect
the coupling of the band(s) under investigation with the others.
47
Chapter 3. Electronic properties
the bands are set to zero by a restriction on the involved Kohn-Luttinger
parameters: γ2 = γ3 [17][18].
As an upshot, adopting the spherical approximation amounts to replace
the space group Td of the crystal with the full rotational group. Now the
crystal Hamiltonian is invariant under rotations and additional operators
can be found which share the same basis of eigenstates. In a cylindrical
representation these operators are Pz and Fz , where Pz is the envelope
momentum along the z-axis and Fz denotes the total angular momentum
along the z axis:
Fz = Jz + Lz ,
(3.26)
with Lz the z component of the envelope angular momentum L. The z
component of the total angular momentum is a conserved operator, or in
other words, Fz commutes with the crystal Hamiltonian. Consequently,
the eigenvalue fz of Fz is a good quantum number and the Hamiltonian is
diagonal with respect to Fz .
3.2.2
Top valence bands in III-V semiconductors
This is illustrated in more detail by narrowing the focus again to the situation of the top Γ8 valence bands in III-V semiconductors.
Following the same approach as in section 3.1, j in (3.25) sums over the jz
values of the j = 32 quadruplet and |uj i = | 23 , jz i. The envelope functions
|Fj i now are represented as |kz ; k, mi, where kz is the eigenvalue of Pz , k denotes the radial wavenumber and m ² Z are the eigenvalues of the envelope
angular momentum Lz . Making use of Lz = Fz − Jz , the envelope functions
in the cylindrical representation are of the form Jfz −jz (kρ)ei(fz −jz )φ eikz z ,
where Jn (z) is a Bessel function. With hρ φ z|kz ; k, fz − jz i the envelope
functions in cylindrical coordinates, this results in
hρ φ z|kz ; k, fz − jz i | 32 , jz i ∝ Jfz −jz (kρ)ei(fz −jz )φ eikz z | 23 , jz i. (3.27)
as a basis for the solution (3.25), which is is orthogonal in fz , jz , k and kz .
The Hamiltonian HFΓz8 of the the top Γ8 valence band in III-V semiconductors
now is expressed in this basis by [17][18]

HFΓz8
48
p+q
 s
= 
 r
0
s
p−q
0
r
r
0
p−q
−s

0
r 
,
−s 
p+q
(3.28)
3.3. Envelope description for infinite cylinders
where the basis is ordered with respect to jz as { 23 , 12 , − 21 , − 32 } and p, q, r
and s are given by
~2
((γ1 + γ2 )k 2 + (γ1 − 2γ2 )kz2 ) ,
2m0
~2
((γ1 − γ2 )k 2 + (γ1 + 2γ2 )kz2 ) ,
p−q = −
2m0
~2 √
r =
3γ2 k 2 ,
2m0
~2 √
2 3γ2 kkz .
s =
2m0
p+q = −
(3.29)
(3.30)
(3.31)
(3.32)
This Hamiltonian has two different eigenvalues, corresponding to a heavy
hole (HH) and a light hole (LH) band which are degenerate at the zone
center:
~2
2
(γ1 − 2γ2 )(kHH
+ kz2 ),
2m0
~2
2
= −
(γ1 + 2γ2 )(kLH
+ kz2 ).
2m0
²HH = −
(3.33)
²LH
(3.34)
Both bands are doubly degenerate and the unnormalized eigenvectors are
given by
 k2


|HH1i = 


2
HH +4kz
√
2
3kHH

2kz

kHH  ,

1
0
 √ 
− 3
 2kz 
kLH 
|LH1i = 
 1 ,
0

0
1





|HH2i =  − 2kz  ,
 2 kHH 2 

(3.35)
kHH +4kz
√ 2
3kHH

0
 1 

|LH2i = 
− 2kz  ,
k√
LH
− 3
(3.36)
with respect to the basis given in (3.27), ordered as { 32 , 21 , − 21 , − 23 } with
respect to jz .
3.3
Envelope description for infinite cylinders
In principle it is possible to apply the effective mass approximation in the
context of the geometrical configuration of a nanowire. However, as pointed
out in [14][18], care has to be taken concerning the foundation of the theoretical framework developed in the previous sections.
49
Chapter 3. Electronic properties
In the first place, the effective mass approximation relies on the assumption that the potential is a slowly varying function over a unit cell. Imposing
the wire configuration by taking
δV (r) = −V0 Θ(R − ρ),
(3.37)
with Θ the Heaviside function, this requires the wire radius R to be sufficiently large. Intuitively this makes sense directly, for if there are just a
few atoms within the wire, the potential change at the boundary of the wire
cannot be neglected any more with respect to the interatomic distances a.
To be more precise, by rewriting the potential (3.37) in Fourier space, it can
be seen [14][18] that the entire concept of an effective mass is only useful if
a
R ¿ 1, the limit in which only the Fourier components δV (k) around the
zone center contribute significantly.
Secondly, in the theory of section 3.2 the atomic wavefunctions are assumed to be the same everywhere. If the effective mass approximation is not
treated in a suitable form, it thus fails to describe in a proper way the heterostructure situation with two or more completely different environments
(e.g. a semiconductor compound in vacuum) and consequently drastic changes in the atomic wavefunctions.
This problem is solved in a general way by assuming that every different environment can be described in a large part independently of the other(s)[18].
The bands in the different systems are subsequently related to each other
by matching bands with the same symmetry. Instead of equation (3.21), the
corresponding wavefunction is assumed to be of the form
X
(s)
(3.38)
Ψ(r) =
Fj (R)ajR (r),
jR
where s indicates that for the atomic functions, in this case in Wannier representation, the solutions are taken far in the corresponding system s.
Even if the environments are large enough to approximate them mainly
as bulk systems in this way, it still remains a problem to match the wavefunctions of the different systems near the boundaries. For example, it
cannot be expected that the atoms around the interface of different systems
simply are positioned at the lattice points of a perfect crystal. The atomic
functions on both sides of the boundary are not orthogonal to each other
and in case of a semiconductor structure in vacuum, there are even no atoms
outside the structure any more. Another practical point is the oxidation of
the structure, resulting in a system probably better described as a core shell
structure.
However, the neglect of these effects concerning changes in the atomic
wavefunctions and matching of the boundary can be justified by the spatial
50
3.3. Envelope description for infinite cylinders
extent of the envelope functions: by taking an infinite potential well for the
nanowire geometry, the envelope functions fall off to zero at the boundary.
In this model the atomic functions outside the wire are of no importance
and possible fluctuations around the boundary are neglected because the
overlapping envelope is almost zero. More problems are expected when V0
is finite. In this case the envelope function leaks with a certain extent into
the region outside the wire and the change in atomic wavefunctions plays a
more important role.
In the present paper the potential V0 in (3.37) is assumed to be large
enough to consider it as representing an infinite potential well. In case of
an infinite cylinder structure, this leads to a boundary condition on the
envelopes of bulk wavefunctions:
Fj (ρ = R, φ, z) = 0, ∀ j, φ, z.
(3.39)
The solutions for this boundary condition can be labeled with a set of quantum numbers, say λ, where λ will be specified for the valence band in paragraph 3.4.1 and for the conduction band in paragraph 3.3.2. In addition,
for a given band λ the wavefunctions depend on the wavenumber kz .
Denoting the complete labeling with λ kz one obtains a normalization condition forPthe envelope functions Fλ kz ,j if the total wavefunction
Ψλ kz (r) = Cλ kz jR Fλ kz ,j (R)ajR (r) is normalized to unity:
Z
Z
X X
dr |Ψλ kz (r)|2 = Cλ2 kz
Fλ∗ kz ,j 0 (R0 )Fλ kz ,j (R) dr a∗j 0 R0 (r)ajR (r)
= Cλ2 kz
X X
j, j 0 R, R0
Fλ∗ kz , j 0 (R0 )Fλ kz , j (R)δR0 ,R δj 0 ,j
j, j 0 R, R0
= Cλ2 kz
XX
j
|Fλ kz , j (R)|2 = 1,
(3.40)
R
from which the normalization constant Cλ kz is obtained. In the third step
in equation (3.40) the orthonormality of the Wannier functions is used. The
summation over R can be replaced by an integral in the same context as
equation (3.22). The same normalization condition is obtained using the
Bloch representation (3.25).
3.3.1
Hole in III-V semiconductor nanowires
With the above remarks in mind consider again the top valence bands of IIIV semiconductors. The wire geometry is imposed by the infinite potential
well:
½
∞, ρ > R,
δV (ρ) =
(3.41)
0, ρ ≤ R.
51
Chapter 3. Electronic properties
This leads to the boundary condition on the envelopes (3.39), with j =
jz = { 32 , 12 , − 12 , − 32 }. Since the bulk heavy- and light hole solutions (3.35)
and (3.36) have four components which cannot be zero simultaneously, this
requirement (3.39) can be satisfied only if the total wavefunction is a superposition of the four bulk heavy- and light hole eigenstates for a given
fz . Consequently, apart from normalization constant (3.40) the envelope
wavefunctions are determined by
Fλ kz ,jz (ρ, φ, z) = {(vHH1 |HH1ijz + vHH2 |HH2ijz )Jfz −jz (kHH ρ) +
(3.42)
(vLH1 |LH1ijz + vLH2 |LH2ijz )Jfz −jz (kLH ρ)} ei(fz −jz )φ eikz z ,
where |HH1i-|LH2i are the bulk eigenstates given in (3.35) and (3.36)
and vHH1 , vHH2 , vLH1 , vLH2 are the coefficients which satisfy Fλ kz ,jz (ρ =
R, φ, z) = 0. The boundary condition for jz = { 32 , 21 , − 21 , − 32 } results in the
determinant equation
n
0 =
Jf − 3 (kLH )Jf − 1 (kLH )Jf + 1 (kHH )Jf + 3 (kHH )
2
2
2
2
o
+ Jf − 3 (kHH )Jf − 1 (kHH )Jf + 1 (kLH )Jf + 3 (kLH )
2
2
2
2
+ 3Jf − 3 (kLH )Jf − 1 (kHH )Jf + 1 (kHH )Jf + 3 (kLH )
2
2
2
2
n
4kz2
+ kLH kHH Jf − 3 (kLH )Jf − 1 (kHH )Jf + 1 (kLH )Jf + 3 (kHH )
2
2
2
2
o
+Jf − 3 (kHH )Jf − 1 (kLH )Jf + 1 (kHH )Jf + 3 (kLH )
2
+
2
2
(3.43)
2
2 +4k 2 )(k 2
2
(kLH
z
HH +4kz )
Jf − 3 (kHH )Jf − 1 (kLH )Jf + 1 (kLH )Jf + 3 (kHH ),
2 k2
3kL
2
2
2
2
HH
which is a relation for the allowed energies. Here the wire radius R is absorbed in the wave numbers by kHH → kHH R, kLH → kLH R and kz → kz R.
From now on kHH , kLH and kz denote these dimensionless ”wavenumbers”,
unless stated otherwise.
Together with the constraint obtained from the equation for the energy,
²HH = ²LH = E,
(3.44)
where the bulk energies ²HH and ²LH are given by (3.33), the determinant
equation (3.43) fixes the radial wavenumbers kHH (kz ) and kLH (kz ) for a given kz . Using these solutions of kHH (kz ) and kLH (kz ), also the coefficients
vHH1 -vLH2 are obtained from the boundary equation (3.39).
Before turning to more explicit results in the next sections, the following
general remarks are important to keep in mind. First, the determinant
equation (3.43) is invariant under the inversion fz → −fz , which reflects the
time-reversal symmetry of the Hamiltonian HFΓz8 given in (3.28): the total
angular momentum reverses direction if t → −t. Consequently, the energy
solutions E are doubly degenerate in fz and the corresponding wavefunctions
turn into each other under fz → −fz .
52
3.3. Envelope description for infinite cylinders
Secondly it should be noted that the wavenumber kz , giving the dispersion in the z direction where the electron (hole) is still free to move, cannot
be separated from the lateral terms in the envelope wavefunction (3.42).
The radial wavenumbers kHH and kLH are functions of kz , so the dispersion in the z direction in general depends on the lateral distribution of the
wavefunction.
Furthermore, for a given kz the set of equations (3.43), (3.44) has to be
solved numerically. Only in special cases the energy E and hole wavefunction reduce to relative simple analytical expressions. In the next section, first
some analytical results at kz = 0 are summarized. Subsequently an expression for the effective mass of a hole in a III-V nanowires will be derived by
expansion around kz = 0, the wire zone center.
3.3.2
Electron in III-V semiconductor nanowires
Up till now only the situation of the degenerate top valence bands in III-V
semiconductors was discussed. Since also the conduction band properties
are needed in the remaining of this paper and because it is also illustrative
to consider a nondegenerate example which is much easier to handle, here the electron dispersion and wavefunctions of the lowest lying conduction
band in III-V semiconductors are treated shortly.
Again an infinite confinement is assumed, as given in equation (3.41).
Since the conduction band is non-degenerate, the S.E. for the electron is
given by (3.23) and the envelope function for ρ < R is given by
Fλ kz (ρ, φ, z) = Cλ Jlz (klz ρ)eilz φ eikz z ,
(3.45)
with Cλ the normalization constant.
Assuming also the warping sufficiently small, i.e. adopting the spherical
approximation by taking an uniform effective mass m∗c for the conduction
band, the energy dispersion becomes
~2
(k 2 + kz2 ) = E.
2m∗c lz
(3.46)
j
z ,n
, the alloThe boundary condition (3.39) now simply gives klz ,n = lR
wed values of klz which are independent of kz . Here jlz ,n is the nth zero of
the Bessel function Jlz (x)
It is convenient to introduce a notation which summarizes the labeling of
the conduction subbands, as derived in the current framework. In the present case, the total Hamiltonian already is diagonal in the envelope angular
momentum, so the conduction subbands are labeled with |lz |.
53
Chapter 3. Electronic properties
Following the notation of [4], the irreducible representation of the conduction subbands in cylinder configuration is characterized by
(±)
C|lz |, n ,
(3.47)
where (±) denotes the parity, n the nth solution at this parity and the
absolute value of the envelope angular momentum is taken because of the
degeneracy in lz .
3.4
Hole dispersion around kz = 0
As discussed in [17] [18], the top valence band Hamiltonian HFΓz8 , given by
(3.28), decouples into two 2 × 2 blocks at the wire zone center, kz = 0. Both
blocks have solutions which are characterized by parity: the corresponding
Bessel functions are only even or only odd under ρ → −ρ.
Apart from a general discussion, in this section the focus will be narrowed
to an exceptional case: the odd solutions for |fz | = 12 . In this case it is
possible to derive a transparent equation for the effective mass in the z
direction by Taylor expansion around kz = 0.
3.4.1
Solutions at the wire zone center
At the wire zone center, kz = 0, one obtains from the energy equation (3.44):
s
r
m∗HH
γ1 + 2γ2
kHH = βkLH , β ≡
=
,
(3.48)
γ1 − 2γ2
m∗LH
where m∗HH and m∗LH are the effective masses of the heavy and light hole
bulk bands, respectively. Also the boundary condition simplifies at kz = 0.
By block diagonalizing (3.28) it is found that the four heavy- and light hole
wavefunctions decouples into two groups: either
vHH2 = vLH2 = 0 , vLH1 = α1 vHH1 ,
(3.49)
vHH1 = vLH1 = 0 , vLH2 = α2 vHH2 .
(3.50)
or
Here α1 and α2 are determined by the determinant equation (3.43) at the
wire zone center, which decouple into two mutually excluding determinants
1 Jfz − 32 (kHH )
3 Jfz − 3 (kLH )
= −
1 Jfz + 32 (kHH )
3 Jfz + 3 (kLH )
= −
2
Jfz + 1 (kHH )
2
Jfz + 1 (kLH )
≡ α1 ,
(3.51)
≡ α2 ,
(3.52)
2
or
2
54
Jfz − 1 (kHH )
2
Jfz − 1 (kLH )
2
3.4. Hole dispersion around kz = 0
so the only possible solutions indeed are given by (3.49) and (3.50). Note
that the inversion symmetry of fz is revealed by the two determinants: using
J−n (z) = (−1)n Jn (z),
(3.53)
it easy to show that (3.49) turns into (3.50) under fz → −fz .
Energy equality (3.48), together with either (3.49) or (3.50) determines
the energy at the zone center. The different energy bands and corresponding
wavefunctions are characterized by parity at the wire zone center: for a
given fz ² Z + 21 the solution corresponding to (3.49)/(3.50) contains only
even/odd (odd/even) Bessel functions. Note that Bessel functions transform
under inversion in ρ in the same way as their label (i.e. under z → −z,
Jn (z) → Jn (z) if n is even, Jn (z) → −Jn (z) if n is odd).
As long as kz = 0, parity thus is a good quantum number and this is still
approximately the case for kz close to 0. Consequently, the energy bands
and corresponding valence subbands are labelled with +/− for respectively
even/odd solutions at kz = 0. Note that for a given parity there are different
solutions labelled by n.
At this point it is convenient to specify the labeling of the valence subbands further. As stated in paragraph 3.2.1, in a cylindrical representation
the total Hamiltonian H Γ8 is diagonal in Fz , which implies that the subbands are also labeled with |fz | (the absolute value is taken because of the
degeneracy in fz → −fz ). Consequently, the complete set of solutions for
the Γ8 valence band in the infinite cylinder configuration is characterized by
the quantum numbers fz , (±), nth solution at this parity. This irreducible
representation of the valence subbands is indicated with
(±)
(3.54)
E|fz |, n ,
where, contrary to the notation in [4], n denotes the nth solution at a particular parity.
In general, even at kz = 0 the original bulk heavy- and light hole solutions
are coupled to each other in a nanowire. However, it turns out that the odd
solutions at |fz | = 21 form an exceptional group. The determinant equation
(either (3.49) or (3.50)) in this case reduces to
J1 (kHH )J1 (kLH ) = 0,
(3.55)
with Bessel zeros j1,n as solutions:
kHH =
j1,n
j1,n
, kLH =
R
βR
(3.56)
55
Chapter 3. Electronic properties
or
kHH = β
j1,n
j1,n
, kLH =
,
R
R
(3.57)
where kHH , kLH are the original wavenumbers, so R is written out explicitely.
As can be seen from (3.51) or (3.52), the relevant coefficient α1 /α2 is zero
for these solutions. Since the other heavy -, light hole pair already is excluded (equation (3.49) or (3.50)) the odd wavefunctions at kz = 0, |fz | = 21
consequently are pure heavy - or light hole like. Note that for the light hole
solutions both α1 and α2 should be inverted.
3.4.2
Hole dispersion around kz = 0 for |fz | = 12 , (−)
The confinement by the infinite wire geometry, resulting in the determinant
equation (3.43), reduces the dimensions in which the electron (hole) is free
to move to one. The dispersion relation in this direction (z) becomes more
complex than the quadratic dispersion of the two original bulk bands, due to
the fact that kz cannot be separated from the lateral terms in the envelope
wavefunction in case of the degenerate III-V top valence band. However,
for the odd solutions at |fz | = 21 it is possible to approximate the dispersion
around the wire zone center with an effective mass.
For this purpose, the first step is to expand kHH (kz ) and kLH (kz ) up to
second order in kz . Recall that the Γ8 band minimum is assumed to be at
at the zone center, so
¯
∂E ¯¯
= 0.
(3.58)
∂kz ¯kz =0
Utilizing this assumption, one finds for the lateral wavenumbers, by Taylor
expansion around kz = 0,
2
2
kHH
(kz ) ' a2HH + b2HH kz2 , kLH
(kz ) ' a2LH + b2LH kz2 .
(3.59)
Consequently, expanding up to second order in kz , kHH and the corresponding Bessel function are approximated by
kHH (kz ) ' aHH +
b2HH 2
k ,
2aHH z
Jn (kHH (kz )) ' Jn (aHH ) +
(3.60)
b2HH 0
J (aHH )kz2
2aHH n
and the expressions for kLH are similar.
56
(3.61)
3.4. Hole dispersion around kz = 0
Before expanding (3.43) in this way, first it can be simplified for |fz | =
by using the Bessel function property (3.53):
1
2
4 2 2
k k J1 (kLH )J1 (kHH ) {J0 (kLH )J2 (kHH ) + 3J0 (kHH )J2 (kLH )} +
3 LH HH ©
ª
4kLH kHH J12 (kLH )J0 (kHH )J2 (kHH ) + J12 (kHH )J0 (kLH )J2 (kLH ) kz2 +
4 2
2
(3.62)
(k + kHH
)J1 (kLH )J1 (kHH )J0 (kLH )J2 (kHH )kz2 ,
3 LH
where it should be mentioned that kHH = 0 (or kLH = 0) is not a solution
[18]. Expanding the determinant equation up to second order with (3.59)(3.61), the zeroth order part of the first line in (3.62) gives the solutions at
kz = 0:
0 =
0 = J1 (aLH )J1 (aHH ) {J0 (aLH )J2 (aHH ) + 3J0 (aHH )J2 (aLH )} , (3.63)
where the term between the brackets in (3.63) corresponds to the even solutions.
In order to simplify the quadratic term in the expansion of (3.62), aHH
and aLH should be fixed by either the even or the odd solutions in (3.63).
For the odd solutions this results in a transparent equation for the dispersion
around the wire zone center. Imposing J1 (aHH )J1 (aLH ) = 0 and utilizing a
property of the Bessel functions,
n
(3.64)
Jn0 (z) = ∓Jn±1 (z) ± Jn (z),
z
the quadratic term becomes
n 2
h
i
b
1
0 = aLH aHH 13 2aLH
J
(a
)
∓J
(a
)
±
J
(a
)
J1 (aHH )J2 (aHH )+
0 LH
1±1
LH
aLH 1 LH
LH
h
i
2
1 bHH
1
3 2aHH J0 (aLH )J1 (aLH ) ∓J1±1 (aHH ) ± aHH J1 (aHH ) J2 (aHH ) +
h
i
b2HH
1
J
(a
)
∓J
(a
)
±
J
(a
)
J1 (aLH )J2 (aLH ) +
0
HH
1±1
HH
1
HH
2aHH
aHH
h
i
b2LH
1
J
(a
)J
(a
)
∓J
(a
)
±
J
(a
)
J2 (aLH ) +
(3.65)
0
HH
1
HH
1±1
LH
1
LH
2aLH
aLH
o
¡ 2
¢ 2
1
2
kz .
aLH aHH J1 (aLH )J0 (aHH ) J2 (aHH ) + J1 (aHH )J0 (aLH ) J2 (aLH )
The final step is to specify the odd solution further by choosing either
J1 (aLH ) = 0 or J1 (aHH ) = 0. Here the discussion is restricted to aHH =
j1,n , which corresponds to the lowest, heavy hole like energy bands.4 After
choosing the convenient signs in (3.65), the determinant equation in this
case reduces to an expression for the expansion factor bHH of the heavy hole
lateral wavenumber:
2
bHH = − 1
2
aLH J1 (aLH )
3 J0 (aLH )
4
− J2 (aLH )
=−
j1,n
2β
j1,n J1 ( β )
,
j1,n
j1,n
1
3 J0 ( β ) − J2 ( β )
(3.66)
Note again that for the odd solutions there is no heavy -, light hole mixing any more.
57
Chapter 3. Electronic properties
where the second expression follows from aLH = β1 aHH , see (3.48).
2 (k ) + k 2 )
Expanding also the energy equation UEh = −(γ1 − 2γ2 )(kHH
z
z
1
(equation (3.44)) using (3.59)-(3.61), the odd |fz | = 2 heavy hole bands are
approximately given by
E
Uh
= −(γ1 − 2γ2 )j1,n − (γ1 − 2γ2 )(1 + b2HH )(kz R)2 ,
(3.67)
where bHH is given in (3.66) and the dependence on the wire radius is explicitly shown by defining an energy unit Uh :
Uh ≡
~2
.
2m0 R2
(3.68)
This results in an expression for the effective mass of a heavy hole in the
odd |fz | = 12 energy bands of an infinite wire:
m∗HH , z
= m0 (γ1 − 2γ2 )−1 (1 + b2HH )−1 ,
where z is the only direction in which the hole is still free to move.
58
(3.69)
3.5. Results
3.5
Results
As an illustration of the above theory, in this section numerical results are
given on the basis of specific examples. In particular it is insightful to compare III-V materials with different properties, in this case different KohnLuttinger parameters. Hence, the III-V compounds InP and InAs are investigated, their Kohn-Luttinger parameters given in Table 3.1 are taken
from [20]. Note that the values of γ3 are not needed here: the theoretical
InP
InAs
γ1
5.08
20.0
γ2
1.60
8.5
Table 3.1: Kohn-Lutinger parameters for InP and InAs
framework rests on the assumption γ3 = γ2 .
3.5.1
Hole energy bands of III-V material nanowires
The first seven hole energy bands of InP and InAs nanowires are shown in
figure 3.1. They are calculated from equations (3.43) and (3.44). The black
line in the graphs corresponds to the reference band −(γ1 + 2γ2 )kz2 with
kLH = 0, where all bands end because there are no solutions if kLH ≤ 0. The
R dependence is absorbed in the units along the axes, kz R and E R 2 , with
R in nm, kz R dimensionless and E R2 in eV nm2 .
InP
-1
E R2 HeV nm2 L
InAs
fz = 12, H+L 1
fz = 12, H-L 1
fz = 32, H+L 1
fz = 32, H-L 1
fz = 32, H+L 2
fz = 12, H+L 2
fz = 12, H-L 2
-2
-3
-4
-5
-6
0
fz = 12, H+L 1
fz = 12, H-L 1
fz = 32, H+L 1
fz = 32, H-L 1
fz = 32, H+L 2
fz = 12, H+L 2
fz = 12, H-L 2
-1
E R2 HeV nm2 L
0
-2
-3
-4
-5
-6
0
1
2
4
3
kz R
5
6
0
0.5
1
1.5 2
kz R
Figure 3.1: Hole energy bands for InP and InAs. The bands are labeled
by absolute total angular momentum in the z-direction, |fz |, and parity,
denoted with (+) nth (nth even solution) and (−) nth (odd). The black
line in the graphs corresponds to the reference band (γ1 + 2γ2 )kz2 with
kLH = 0. The R dependence is absorbed in the units along the axes,
with wire radius R in nm.
Using the notation given in (3.54) and (3.47), the representation of the
first seven hole subbands of InP and InAs nanowires are shown in Table 3.2.
59
2.5
3
3.5
Chapter 3. Electronic properties
The subbands vi are ordered with respect to their zone center offset, see
Figure 3.1. For convenience, also the representation of first two electron
subbands c1 and c2 are given, within the framework of Subsection 3.3.2.
Subband
InP
v1
E 1 ,1
v2
E 1 ,1
v3
E 3 ,1
v4
E 3 ,1
v5
E 3 ,2
v6
E 1 ,2
v7
E 1 ,2
c1
C0, 1
c2
C1, 1
(+)
2
(−)
2
(+)
2
(−)
2
(+)
2
(+)
2
(−)
2
InAs
(−)
E 1 ,1
2
(+)
E 1 ,1
2
(+)
E 3 ,1
2
(+)
E 1 ,2
2
(−)
E 3 ,1
2
(−)
E 1 ,2
2
(+)
E 3 ,2
2
(+)
C0, 1
(+)
(−)
C1, 1
(−)
Table 3.2: Irreducible representation of the first seven hole subbands vi
and first two electron subbands cj for InP and InAs nanowires. The
characterization is also valid for conduction subbands calculated in a
finite potential well.
Around kz = 0 the hole band dispersion can be approximated by the
quadratic expressions given in Table 3.3. In general, for |fz | = 12 and odd
parity these numerical results are in good agreement with the analytical
expansion given in (3.67). For instance, for InP the effective mass in the z
direction m∗HH , z corresponding to the values in Table 3.3 are 3.45 m0 and
11.49 m0 for the first even and first odd subband, while the analytical expression (3.69) gives 3.39 m0 and 10.20 m0 , respectively.
Comparing the two materials, the following remarks are supported by
Figure 3.1, Table 3.2 and Table 3.3:
60
3.5. Results
hole state
|fz | = 12 , (+) 1
|fz | = 12 , (−) 1
|fz | = 12 , (+) 2
|fz | = 12 , (−) 2
|fz | = 32 , (+) 1
|fz | = 32 , (−) 1
|fz | = 32 , (+) 2
InP
−0.75 − 0.14(kz R)2
−1.05 − 0.29(kz R)2
−2.16 − 0.07(kz R)2
−3.53 − 0.087(kz R)2
−1.38 + 0.31(kz R)2
−1.79 − 0.59(kz R)2
−2.08 − 0.16(kz R)2
InAs
−2.14 − 1.87(kz R)2
−1.68 + 0.77(kz R)2
−3.91 − 0.72(kz R)2
−5.62 − 0.35(kz R)2
−2.95 + 0.44(kz R)2
−4.07 − 0.45(kz R)2
−6.52 + 1.29(kz R)2
Table 3.3: Numerical results for the hole energy E R2 (eV nm2 ), fitted
to kz2 R2 around the wire zone center
• The shape of a particular band, including its zone center energy, is
material dependent. It depends on the magnitude of the gamma’s by
γ1 − 2γ2 , but also their ratio γγ12 is a deciding quantity. Consequently,
the corrections to the band gap Eg caused by the confinement are
material dependent. For the present two examples the zone center
band gaps of InAs are more shifted by the infinite wire configuration.
• Next to the shape of the individual bands, also their mutual ordering
is material dependent. This means that the parity of the lowest lying band (and the others) can differ depending on the material. For
example, the lowest lying band is even for InP and odd for InAs.
Furthermore, it should be noted that the results are valid for all R, with
the only requirement that R should be sufficient large in order to consider the
confinement potential (3.41) as a slowly varying function with respect to the
unit cell dimensions. This means that in the limit R → ∞ the confinement
correction on the band gap has to disappear, which is indeed the case as can
be concluded from the 1/R2 dependence.
3.5.2
Hole wave functions of III-V material nanowires
As pointed out in section 3.3, the wavenumber along the cylinder axis kz is
not independent of the lateral part of the hole wavefunction. As a consequence, the total wavefunction for a particular band depends in a non trivial
way on kz and should be calculated from (3.42) for every value of kz separately. Here the results of this procedure are summarized by focussing, next to
the kz dependence, on three other subjects: the parity of the wavefunctions,
the invariance under total z-angular momentum reversion and the influence
of material properties (Kohn-Luttinger parameters).
Starting with parity and the invariance under fz → −fz , Figure 3.2
shows the φ = 0 radial part of the envelope wavefunction, decomposed
into the different jz components at the same value of kz and for the same
61
Chapter 3. Electronic properties
(+)
material. The graphs in the first row correspond to the two E 1 ,2 solutions,
2
(−)
those in the second row to the solutions with representation E 3 ,1 . The total
2
envelope function is normalized using equation (3.40) and the wire radius R
is absorbed in the dimensionless unit Rρ .
fz = 12 , kz R = 2.66667 , H+L 2
fz = -12 , kz R = 2.66667 , H+L 2
1.5
1
1
0.5
0.5
Χ jz
Χ jz
1.5
0
-0.5
0
-0.5
jz = 3 2
jz = 1 2
-1
0
0.2
0.4
0.6
jz = -1 2
jz = -3 2
0.8
jz = 3 2
jz = 1 2
-1
1
0
0.2
ѐR
0.8
1
1
0.5
0.5
0
-0.5
0
-0.5
jz = 3 2
jz = 1 2
-1
0
0.2
0.4
0.6
jz = -1 2
jz = -3 2
0.8
jz = 3 2
jz = 1 2
-1
1
0
0.2
ѐR
0.4
0.6
jz = -1 2
jz = -3 2
0.8
ѐR
Figure 3.2: Radial (φ = 0, eikz z omitted) part of the normalized hole
envelope functions, decomposed in the different jz components: the jz =
3
1
1
2 components are given in yellow, jz = 2 in green, jz = − 2 in blue and
3
jz = − 2 in red. The first row gives the solutions for |fz | = 12 , + (2),
the second row for |fz | = 32 , −(1). The Kohn-Luttinger parameters are
taken from InP as given in Table 3.1 and kz R = 2.67 is fixed.
(±)
Figure 3.2 illustrates that for a given subband E|fz |,n the solution at fz
for a particular jz is the same (apart from minus sign) as at −fz for −jz .
Moreover, as expected the two total wavefunctions turn into each other
under time reversal, because under fz = lz + jz → −fz = −lz − jz any
component Jl (kρ) |j, jz i → J−l (kρ) |j, −jz i, so besides jz → −jz the odd
solutions reverse sign, as shown in Figure 3.2.
The wavefunctions in Figure 3.2 are calculated away from the wire zone
center, at kz R = 1.5 10−9 with R in nm. As a consequence, next to the jz
components with the parity of the zone center, also other jz components
appear which have the opposite parity. For example, in the first graph the
dominant jz = 12 component (green curve) and the jz = − 23 component (red
curve) are the evolved even components which are present at the zone center, while the jz = 32 (orange) and jz = − 21 (blue) curves are odd in ρ → −ρ.
62
1
fz = -32 , kz R = 2.66667 , H-L 1
1.5
Χ jz
Χ jz
0.6
ѐR
fz = 32 , kz R = 2.66667 , H-L 1
1.5
0.4
jz = -1 2
jz = -3 2
1
3.5. Results
fz = 12 , kz R = 0. , H+L 1
fz = 12 , kz R = 0.125 , H+L 1
1.5
1
1
0.5
0.5
Χ jz
Χ jz
1.5
0
-0.5
0
-0.5
jz = 3 2
jz = 1 2
-1
0
0.2
0.4
0.6
jz = -1 2
jz = -3 2
0.8
jz = 3 2
jz = 1 2
-1
1
0
0.2
ѐR
1.5
1
1
0.5
0.5
Χ jz
Χ jz
0.8
1
fz = 12 , kz R = 0.5 , H+L 1
1.5
0
-0.5
0
-0.5
jz = 3 2
jz = 1 2
-1
0
0.2
0.4
0.6
jz = -1 2
jz = -3 2
0.8
jz = 3 2
jz = 1 2
-1
1
0
0.2
ѐR
0.4
0.6
jz = -1 2
jz = -3 2
0.8
1
ѐR
fz = 12 , kz R = 0.75 , H+L 1
1.5
fz = 12 , kz R = 1. , H+L 1
1.5
1
1
0.5
0.5
Χ jz
Χ jz
0.6
ѐR
fz = 12 , kz R = 0.25 , H+L 1
0
-0.5
0
-0.5
jz = 3 2
jz = 1 2
-1
0
0.2
0.4
0.6
jz = -1 2
jz = -3 2
0.8
jz = 3 2
jz = 1 2
-1
1
0
0.2
ѐR
0.4
0.6
jz = -1 2
jz = -3 2
0.8
1
ѐR
fz = 12 , kz R = 1.25 , H+L 1
1.5
fz = 12 , kz R = 1.5 , H+L 1
1.5
1
1
0.5
0.5
Χ jz
Χ jz
0.4
jz = -1 2
jz = -3 2
0
-0.5
0
-0.5
jz = 3 2
jz = 1 2
-1
0
0.2
0.4
0.6
ѐR
jz = -1 2
jz = -3 2
0.8
jz = 3 2
jz = 1 2
-1
1
0
0.2
0.4
0.6
jz = -1 2
jz = -3 2
0.8
ѐR
Figure 3.3: Radial part of the |fz | = 12 , + (1) hole envelope wavefunctions for InAs. The value of kz R changes from 0 in the first picture to the
maximum value 1.5 (at the end of the band) in the last graph.
This is a general property: at the wire zone center the total wavefunction
consist only of either even or odd components, while away from kz = 0 also
significant contributions with the other parity arise.
63
1
Chapter 3. Electronic properties
The variation of the wavefunction as a function of kz is illustrated in more
(+)
detail in Figure 3.3. It shows the radial part of the E 1 ,1 hole wavefunctions
2
for InAs for different values of kz , given at the top of each graph.
(−)
In Appendix A also the first odd solution, E 1 ,1 , for InAs is shown in
2
this way; the illustrations can be compared to the results in case of InP,
Figure 10 and Figure 11.
The following remarks are revealed by Figure 3.3 and the figures in
Appendix A:
• As noted before, at kz = 0 the total hole wavefunction is either even
or odd under inversion ρ → −ρ. Remarkable is that the shape of the
functions depends only a little on the choice of material.
• Increasing kz slowly, the shapes of the different jz components change
in a continuous way: an extra graph between the first and the second
would give a result in between.
• Comparing the results for InP and InAs, it can be seen that for a
given band the altering of the hole wavefunctions by increasing kz is
material dependent. Actually, the amount of change depends on the
shape of the corresponding band: a smaller effective mass corresponds
with a faster change in the hole wavefunctions around kz = 0, which
can be checked for the present examples with the help of Table 3.3.
• At the end of a particular band, one of the jz components disappears,
so in general there are three or fewer jz components which contribute
to the total wavefunction at the end of a band.
The above results have some important consequences, in particular concerning the calculation of the absorption matrix elements over the entire
band, see next chapter. Since the different jz components of the hole wavefunction change just slightly over a particular band, it suffices to choose
a suitable small number of kz points by which the hole wavefunctions at
neighboring points are approximated.
3.5.3
Band gap in III-V material nanowires
Finally, it is illustrative to estimate the effect of the infinite confinement in
the present model by comparing the bulk band gap Eg with the confinement
energy Evconf
1 →c1 of the fundamental transition v1 → c1 between the highest
lying hole state v1 and lowest electron state c1 at the wire zone center.
The values of effective mass of the Γ6 conduction band and the bulk
band gap Eg for InP and InAs are given in Table 3.4. Using the values
64
3.5. Results
m∗c /m0
Eg (eV )
V0 (eV )
InP
0.0795
1.4236
4.28
InAs
0.026
0.417
4.93
Table 3.4: The band gap Eg , potential well V0 and effective masses of
the Γ6 bulk conduction band for InP and InAs
of the effective mass, the first row in Table 3.5 gives the energy of the
lowest conduction subband c1 at the wire zone center. The second row
Ec1 (eV )
Evconf
1 →e1 (eV )
InP
InAs
2.77 R−2
3.52 R−2
8.48 R−2
10.16 R−2
Table 3.5: The energy in eV at kz = 0 of the lowest conduction subband
c1 and the confinement energy Evconf
of the fundamental transition
1 →c1
v1 → c1 for InP and InAs in an infinite wire confinement, as derived
with the model described in this paper. R denotes the wire radius.
in Table 3.5 shows the confinement energy Evconf
1 →c1 (R) of the fundamental
conf
transition v1 → c1 . Note that Ev1 →c1 (R) does not include the bulk Eg , it is
defined as
Evtrans
(R) = Eg + Eci (R) − Evi (R) ≡ Eg + Evconf
(R).
i →cj
i →cj
(3.70)
However, the assumption of an infinite potential well is too strong. The
difference between the vacuum level and the conduction band edge (electron
affinity) is in the order of electron volts for III-V materials and taking this
finiteness into account leads to significant corrections, in particular for the
conduction subbands[18][21].
Here the discussion will be restricted to correcting the conduction subband c1 for InP and InAs, given in Table 3.5, with a reduction factor due
to the finite potential well. For the valence subbands it is expected that the
correction is less crucial, in the first place because the correction is smaller
for bulk bands with a higher effective mass. Another reason is that, apart
from the extra energy difference by the band gap, the difference with the
vacuum level becomes larger for deeper lying subbands, in contrast to the
conduction band states.
In the finite potential well model, the dependence on the wire radius
becomes more complicated then the simple R−2 dependence in the infinite case. Actually, the electronic properties depend on the dimensionless
65
Chapter 3. Electronic properties
quantity
lV
R
lV
, where lV is defined as
r
~2
=
.
2m∗ V
(3.71)
For the conduction band, the potential V equals the vacuum level offset V0
which is given for InP and InAs in Table 3.4.
Ec1 (eV )
Evconf
1 →c1 (eV )
InP
InAs
0.07
0.11
0.13
0.20
Table 3.6: The energy in (eV ) at kz = 0 of the lowest conduction subband c1 and the confinement energy Evconf
of the fundamental transi1 →c1
tion v1 → c1 for InP and InAs at R = 4.83 nm and R = 4.85 nm, respectively. These are the corrected results of Table 3.5: the conduction
subbands are calculated in the finite potential wells given in Table 3.4.
The corresponding energies of the lowest conduction subband c1 in the
finite wire configuration are shown in Table 3.6. These values of Ec1 can be
compared to those for the infinite potential well case at R = 4.8 nm: 0.2 eV
and 0.36 eV for InP and InAs, respectively. Similarly, for the confinement
energy Evconf
1 →c1 , the values in the infinite potential well case are 0.15 eV and
0.43 eV for InP and InAs, respectively. The difference between the two
models is larger for InAs due to the smaller effective mass of the conduction
band.
66
Chapter 4
EM transition matrix
In this chapter the EM matrix element for band-to-band transitions between
the top Γ8 valence bands and the lowest lying Γ6 conduction band in III-V
semiconductor nanowires will be derived. Section 4.1 contains some general
theory, in section 4.2 the matrix element is developed further in the Bloch
representation, section 4.3 gives explicit expressions for the band-band matrix elements and section 4.4 treats the selection rules on the intersubband
transitions. Explicit results for InP and InAs are shown in section 4.5.
4.1
4.1.1
General theory
Radiation matter interaction
In this paper the interaction between the external EM field and the electrons
within the semiconductor system is described using a macroscopic, semiclassical approach. In this method the EM field is treated classically, while
the semiconductor material is described quantum mechanically in the spirit
of the previous chapter. Next to the assumption that this semi-classical picture approximates the more realistic QED model, it is also assumed that the
semiconductor heterostructure can be described using macroscopic Maxwell
equations, i.e. where the different parts in the system are characterized by
macroscopically averaged quantities, such as the dielectric function.
As will be discussed in more detail in Chapter 5, it is unclear if this
macroscopic framework still holds if the size of the system is reduced to nanoscale, when the dimensions of the system are not large any more compared
to the microscopic (atomic) distances. Then a microscopic semiclassical theory would be a more realistic approach [22].
However, proceeding with the macroscopic semiclassical approach, the
gauge freedom in choosing the scalar potential φ and vector field A is used
67
Chapter 4. EM transition matrix
by taking the Coulomb gauge,
∇ · A = 0.
(4.1)
Then the transverse part of the electric field equals − 1c ∂A
∂t . Note that the
longitudinal part −∇φ is absorbed in the matter part of the semiconductor
Hamiltonian: the averaged potential V (r) in equation (3.1) contains the full
Coulomb interaction between the particles. Further details are found in [22].
In this scheme the Hamiltonian representing the radiation-matter interaction is given by
e X
Hr−m = −
A(r i ) · pi ,
(4.2)
m0 c
i
where i labels the N electrons in the material and m0 the free electron
mass. Taking m0 instead of an effective mass m∗i is justified for interband
transitions [15]. Furthermore, since the EM field is typically is small, the
term of order |A|2 is neglected in (4.2).
4.1.2
EM transition matrix
Treating the time dependent EM interaction (4.2) as a small perturbation,
it induces a transition between initial state |Ψi i and final state |Ψf i, which
are eigenstates of the semiconductor system in the absence of the EM field.
The probability that the unperturbed system |Ψi i transforms under the
absorption/emission of light to |Ψf i is proportional to the transition matrix
element
Mf i = hΨf |Hr−m |Ψi i.
(4.3)
In case of a degenerate initial and/or final state, with |Ψim i and |Ψfn i the
m- and n-fold degenerate initial and final states, this degeneracy is taken
into account by
X
|Mf i |2 =
|hΨfn |Hr−m |Ψim i|2 ,
(4.4)
m,n
i.e. the different possible transitions at the same energy are summed as
probabilities. If the degeneracy is lifted by a perturbation which breaks a
particular symmetry, say an external magnetic field, then |Ψim i and |Ψim0 i
are separated in energy and their matrix elements can be distinguished as
belonging to different transitions |Mf i | and |Mf i0 |.
In this paper the absorption process is considered, in the specific case
of direct band-to-band transitions in nanowires. As initial state the many
68
4.1. General theory
body groundstate Ψ0 of the undoped semiconductor nanowire is taken, with
all valence bands completely filled and the conduction bands empty:
Ã
!
Y
Ψ0 (r 1 , r 2 , .., r N ) = A
Ψλi kzi (r i ), ,
(4.5)
i
where the product of one particle states is anti-symmetrized by the operator
A and λi denote all valence bands.
As final state an excited state is constructed by taking an electron out
of one of the top most valence bands,


Y
Ψexcited (r 1 , r 2 , .., r N ) = A Ψλj = c kzj (r j )
Ψλi kzi (r i ) , (4.6)
i 6=j
where j = c indicates that the electron is placed into one of the empty conduction bands. In the remainder this notation will be used to distinguish
wavefunctions belonging to conduction bands from valence band wavefunctions Ψλi kzi (r i ).
One can proceed further by forming the final exciton state from a linear
combination of the excited states (4.6). However, in this paper only the
band-to-band transitions from the groundstate () to a particular excited
state (4.6) are considered. By writing down the antisymmetrization
operator
P
explicitly in terms of the permutation operator P , A = √1N ! P (−1)P P , the
transition matrix (4.3) for these specific initial and final states becomes
Z
X
e 1 X
e
P +P 0
hΨexcited |
A(r i ) · pi |Ψ0 i = −
Mc v = −
(−1)
dr 1 . . . dr N ×
m0 c
m0 c N ! 0
i
PP
´
´P
³Q
³
Q
0
(r
)
.
A(r
)
·
p
P
Ψ
(4.7)
P Ψ∗λl = c kzl (r l ) k 6=l Ψ∗λk kzk (r k )
j
i
i
i
j λj kzj
In order to simplify this expression, we recall the orthonormality relations
Z
dr Ψ∗λi = c kzi (r i ) Ψλj kzj (r j ) = 0,
(4.8)
Z
dr Ψ∗λi kzi (r i ) Ψλj kzj (r j ) = δλi ,λj δkzi ,kzj ,
(4.9)
where (4.8) is due to the orthogonality of the zone center atomic functions,
s-like and p-like for conduction and valence band wavefunctions, respectively. Equation (4.9) is explained by the orthogonality of (3.27) and the
normalization (3.40) of the total wavefunction.
It can be shown [15] that, utilizing (4.8) and (4.9), the transition matrix
element (4.7) simplifies to
Z
e
Mc v = −
dr Ψ∗λc kzc (r) A(r) · p Ψλv kzv (r),
(4.10)
m0 c
69
Chapter 4. EM transition matrix
in other words, Mc v is the transition matrix for an electron in one particular state Ψλv kzv in a valence band λv which is excited to a conduction band
state Ψλc kzc by the EM radiation.
It is easy to check the simplification (4.10) of equation (4.7) if N = 2:
from the 8 terms only two identical terms are nonzero and they cancel the
2! in the denominator.
4.2
Bloch representation
The transition matrix (4.10) is developed further by making use of either the
Wannier or the Bloch representation for the wavefunctions. Here the Bloch
scheme is chosen, for the practical reason that zone center Bloch functions
are much better known. An other disadvantage of expanding in Wannier
functions lies in the fact that Wannier functions are extended over a region
larger then a unit cell, which makes it less defensible to approximate the
EM field as constant over the relevant intervals of integration [18].
4.2.1
Total wavefunction in Bloch functions
Instead of expanding the wavefunction in zone center Bloch functions by applying ansatz (3.25), the Bloch representation is generalized by expanding
the wavefunction in k space using the Fourier transform of the Wannier
functions, equation (3.20).
Defining the Fourier transform of the envelope function F̂λkz ,j (k0 ) as
1 P
0
F̂λkz ,j (k0 ) ≡ N − 2 R Fλkz ,j (R) e−ik ·R , the total wavefunction Ψλ kz (r)
in this way ecomes
X
0
1 XX
Ψλ kz (r) =
Fλkz ,j (R) ajR (r) = N − 2
Fλkz ,j (R) e−ik ·R ψnk0 (r)
jR
= N
− 21
X
jk
= N
− 21
0
X
jk
jR k0
(
N
− 12
X
Fλkz ,j (R) e
−ik0 ·R
)
eik
0
·r
ujk0 (r)
R
F̂λkz ,j (k0 ) eik
0
·r
ujk0 (r),
(4.11)
0
where in the second line the normalized Bloch functions (3.3) are used.
This expression is specified further in the infinite wire configuration by
decomposing the envelope Fλ kz , j in its radial part and the plane wave along
the cylinder axis,
eikz Z
Fλ kz , j (R) = χλ kz , j (R⊥ ) √ ,
M
70
(4.12)
4.2. Bloch representation
where the envelope is normalized in the Z direction by denoting the number
of atoms in this direction with M . Note that the not normalized lateral part
χλ kz , j (R⊥ ) in general depends on kz , as concluded in the previous section.
Utilizing equation (4.12), the Fourier transform of the envelope function
F̂λkz ,j (k0 ) is decomposed as
1
F̂λkz ,j (k0 ) = N − 2
=
X
0
χλ kz , j (R⊥ ) e−ik⊥ ·R⊥
R⊥
0
ξλkz ,j (k⊥
) δkz ,kz0 ,
X ei(kz −kz0 )Z
√
M
Z
(4.13)
where
µ
0
ξλkz ,j (k⊥
)
≡
N
M
¶− 1 X
2
0
χλ kz , j (R⊥ ) e−ik⊥ ·R⊥
(4.14)
R⊥
is the Fourier transform of the radial part χλ kz , j (R⊥ ).
Inserting (4.13) in equation (4.11), the total wavefunction expanded in
0
Bloch functions uj (k⊥
,kz ) reads
0
1 X
0
0
Ψλ kz (r) = N − 2
ξλkz ,j (k⊥
) ei(k⊥ ·r⊥ +kz z) uj (k⊥
(4.15)
,kz ) (r).
0
jk⊥
Within the framework developed in the previous chapter, this expression
is valid around the zone center of any bulk band. This general form thus
also describes the total wavefunction in a confined conduction band, where
the sum over j usually disappears because this band is non degenerate in
most of the III-V materials. Furthermore, equation (4.15) is also valid for
an arbitrary strength of the confinement V0 (as in equation (3.37)).
4.2.2
EM transition matrix in Bloch functions
Utilizing the general expression (4.15) both for the valence and conduction band wavefunction, the transition matrix (4.10) is expanded in Bloch
functions by
Z
e
Mc v = −
dr Ψλ∗c kzc (r) A(r) · p Ψλv kzv (r)
m0 c
e X X ∗
ξλc kzc ,jc (k0⊥c ) ξλv kzv ,jv (k0⊥v ) ×
= −
(4.16)
m0 c
jc jv k0⊥c k0⊥v
Z
0
0
1
dr e−i(k⊥c ·r⊥ +kzc z) ujc (k0⊥c ,kzc ) (r) A(r) · p ei(k⊥v ·r⊥ +kzv z) ujv (k0⊥v ,kzv ) (r),
N
where the labeling ⊥ indicates that the corresponding vector lies in the plane
perpendicular to the wire axis, as in section 4.2.1.
71
Chapter 4. EM transition matrix
For the moment, we narrow the focus to the last line in equation (4.16).
The integral
entire
r can be split up into N integrals over a
R over theP
R space
0
unit cell, drf (r) = R Ω0 dr f (R + r 0 ). Using the commutation relation
[p, eik·r ] = −~k eik·r , the third line in (4.16) becomes
1 X −i(k0⊥c −k0⊥v )·R⊥ −i(kzc −kzv )Z
e
e
×
(4.17)
N
Z R
0
0
dr e−i(k⊥c −k⊥v )·r⊥ e−i(kzc −kzv )z ujc (k0⊥c ,kzc ) (r) A(R + r) · (p + ~k) ujv (k0⊥v ,kzv ) (r).
Ω0
Further progress is made by assuming the variation of the EM waves to be
small over a unit cell, i.e.
A(R + r) ' A(R),
(4.18)
This approximation is justified by the fact that the wavelength λ0 of the
EM radiation typically is much larger then the interatomic distances a0 . As
in the case of the effective mass approximation, where the restriction on δV
(slowly varying over one unit cell) leads to a separation between atomic wavefunctions and ”macroscopic” envelope functions, the envelope and Bloch
parts of the transition matrix elements will factor into separate integrals due
to assumption (4.18).
In order to show this explicitly, reconsider expression (4.17) and include
the following:
• Assuming the EM field constant over a unit cell leads to the elimination
of the ~k term, because in this case the integral contains just two
orthogonal Bloch functions.
• In the case of an infinite cylinder the EM field is of the form
A(R) = A(R⊥ )eiqz Z
(4.19)
To be more precise, in Section 1.3 it was derived that qz = −k0 sin θ,
with θ the angle of incidence compared to a plane perpendicular to
the wire axis.
• It is more convenient to rewrite kc and kv in terms of the total momentum K of the system by defining
K ≡ kc − kv ;
k ≡ kv .
72
(4.20)
(4.21)
4.2. Bloch representation
Now (4.17) is simplified by
X
0
1 X
A(R⊥ )e−iK ⊥ ·R⊥
e−i(Kz −qz )Z ·
(4.22)
N
Z
R⊥
Z
0
dr e−iK ⊥ ·r⊥ e−iKz z ujc (K 0⊥ +k0⊥ ,Kz +kz ) (r) p ujv (k0⊥ ,kz ) (r),
Ω0
With the notion that under assumption (4.18) qz is much smaller then a
reciprocal lattice vector Giz and that Kz lies in the first Brillouin zone,
the summation over M lattice points results in a momentum conservation
relation in the Z direction:
X
e−i(Kz −qz )Z = M δqz ,Kz .
(4.23)
Z
This simplifies (4.22) further into
Z
0
δqz ,Kz Â(K 0⊥ ) ·
dr e−iK ⊥ ·r⊥ e−iKz z ujc (K 0⊥ +k0⊥ ,Kz +kz ) (r) p ujv (k0⊥ ,kz ) (r),
Ω0
where Â(K ⊥ ) is defined as the Fourier transform of A(R⊥ ),
MX
A(R⊥ )e−iK ⊥ ·R⊥ .
Â(K ⊥ ) ≡
N
(4.24)
R⊥
Furthermore, since the envelope functions are also assumed to be slowly varying over a unit cell, now it is possible to factor the envelope and Bloch
parts of the transition matrix elements into separate integrals. Any wavevector κ in (4.16) is much smaller then a reciprocal lattice vector Gi because
it is assumed that R À a0 and κ ∼ R1 and Gi ∼ a10 . Under this condition
0
ujκ ∼ uj0 and e−iK ⊥ ·r⊥ e−iKz z ∼ 1. Inserting (4.24) into (4.16) and using
P
P
0
0
0
0
∗
∗
∗
R⊥ χλc kzc ,jc (R⊥ )A(R⊥ )χλv kzv ,jv (R⊥ )
k0 K 0 ξλc kzc ,jc (k⊥ + K ⊥ )Â(K ⊥ )ξλv kzv ,jv (k⊥ ) =
⊥
⊥
the final form of the transition matrix element in Bloch functions is obtained:
XX
e
χ∗λc kzc ,jc (R⊥ )A(R⊥ )χ∗λv kzv ,jv (R⊥ ) ·
Mc v = −
δkzc ,kzv
m0 c
jc jv R⊥
Z
dr ujc 0 (r) p ujv 0 (r).
(4.25)
Ω0
This result is also obtained by using ansatz (3.25), a simplified expansion
of the wavefunction in Bloch states which is justified if the wire radius R
is sufficiently large, i.e. the Fourier related k values are so small that the
corresponding energy difference ²j (k) − ²j remains much smaller then the
band edge differences ²j − ²n0 .
It should be noted that the degeneracy of the hole and electron energy
bands is not taken into account in (4.25). It can be included in the same
way as by replacing (4.3) with (4.4), as will be done further on when the
matrix elements are calculated explicitly.
73
Chapter 4. EM transition matrix
4.3
Reformulation of transition matrix element
In this section the transition matrix element (4.25) is reconsidered by including the degeneracy of the conduction and valence subbands in case of
transitions between the top Γ8 valence bands and the lowest lying Γ6 conduction band in III-V semiconductors. Apart from the trivial degeneracy
in ± kz , the transitions are degenerate in the quantum numbers ± lz c and
± σ of the conduction subbands and in ± fz v of the valence subbands, see
Chapter 3.
The EM field is specified further by considering two approximations with
respect to the wire radius as derived in Part I. First, the spatial variation
of the EM field across the wire diameter is entirely neglected, i.e. the transition matrix element is formulated in the dipole limit, where λ0 À R so
A(R⊥ ) ' A. Secondly, a spatial variation of the EM field is taken into
account by applying the scattering fields (2.8), (2.9) and (2.10), which are
expansions up to second order in 2π
λ0 R for normal incident light.
Recall that the Coulomb gauge is chosen by deriving the transition matrix element, so the vector potential is related to the transverse electric field
by E = − 1c ∂A
∂t . In case of absorption this yields
E = −
iω
A,
c
(4.26)
with ω the frequency of the EM field.
4.3.1
EM field in dipole approximation
In Part I it was shown that the strength of the EM field inside the wire depends on the polarization of the incident light. In the dipole approximation
this resulted in (2.6) and (2.7) for polarization parallel and perpendicular to
the wire axis, respectively. In order to separate this polarization anisotropy
from the dielectric mismatch, a matrix element Tc v is defined by
|Mc v (kz )|2 ≡ (
e 2 E 2
) | | |Tc v (kz )|2 ,
m0 ω
2
(4.27)
where kz = kzc = kzv and E is the strength of the electric field inside the
wire in the dipole approximation, E ≡ E ε̂.
Including the degeneracy, from equation (4.4) and (4.25) one obtains for
the transition matrix Tc v between valence and conduction subband v and c:
¯
¯2
¯
X ¯¯X
¯
2
3
¯
¯ , (4.28)
hχ
|χ
(k
)
ihSσ|
ε̂
·
p
|
|Tc v (kz )| =
j
i
z
z
l
f
;j
zc
zv
z
2
¯
¯
¯
¯
jz
σ ld f d
zc zv
74
4.3. Reformulation of transition matrix element
d denotes l
where lzc
zc = {|lzc | , −|lzc |}, the degeneracy at a given |lzc |. A sid . The notation of the atomic part is explained
milar definition applies to fzv
R
∗
in (4.34). Furthermore, hχlzc |χfzv ;jz (kz ) i =
P dR⊥ χlzc (R⊥ )χfzv kz ;jz (R⊥ ),
where the replacement of the summation R⊥ by an integral is justified
since in the effective mass approximation the dimensions of the wire are assumed to be much larger then interatomic distances. The additional step size
by going from summation to integration is eliminated by the normalization
of the wavefunctions.
Note that indeed the strength of the internal field is separated from
Tcv . This classical penetration effect is absorbed in E, as shown in equation
(4.28).
4.3.2
EM field including Mie scattering
In case of a spatially varying EM field the separation in (4.27) is not possible
any more. Instead, the EM field has to be integrated between the conduction
and valence subband envelope functions over the wire cross section area
πR2 , as shown in (4.25). However, from equations (2.8), (2.9) and (2.10)
one finds that the reduction factor caused by the penetration into the wire
still can be divided out, just as in the dipole limit. Thus, by separating
out the factor E = E0 in case of polarization parallel to the wire axis and
2
E0 at perpendicular polarization, the matrix element Tc v contains
E = 1+ε
only the scattering (optical focusing) and expansion terms due to the wave
behavior of the EM field inside the wire. Denoting E 0 (R⊥ ) as the electric
field without the penetration strength, E E 0 (R⊥ ) ≡ E(R⊥ ), the right hand
side of (4.28) now is replaced with
¯2
¯
¯
¯
X ¯X
¯
2
0
3
¯ hχl ,n |E |χf ,n;j (kz ) i · hSσ| p | jz i¯ (4.29)
|Tc v (kz ; ε, R)| =
zc
zv
z
2
¯
¯ .
¯
¯
d
d
jz
σ l ,f
zc
zv
Note that, contrary to the dipole limit, the direction of the internal EM
field is in general different from that of the incident field . Furthermore,
due to the scattering field the matrix elements now depend on the dielectric
function and the wire radius.
4.3.3
Polarization anisotropy of the transition matrix
At this point it is instructive to define a polarization anisotropy purely originating from the matrix elements. As demonstrated by equation (4.28), in
the dipole limit it is possible to separate the anisotropy caused by the dielectric mismatch from that which is due to the polarization in the transition
matrix. In other words, in principle one is able to determine the polarization anisotropy caused by the transition matrix elements alone once it
is (experimentally) possible to eliminate the polarization anisotropy of the
75
Chapter 4. EM transition matrix
dielectric mismatch, for instance by a sufficient increase of the intensity of
the incident field in the perpendicular case or by changing the surrounding
(for instance if the nanowire is covered by an oxide). In analogy with (2.21),
the polarization anisosotropy ρTcv of the matrix element alone is defined as
ρTcv
≡
|Tcv, k |2 − |Tcv, ⊥ |2
,
|Tcv, k |2 + |Tcv, ⊥ |2
(4.30)
where Tcv, k = Tcv, z is the matrix element corresponding to a polarization
parallel to the wire and Tcv, ⊥ denotes the perpendicular case.
4.4
Selection rules
The interband transition matrix (4.25) is investigated in more detail by
considering the different kind of selection rules it imposes. A selection rule
originates from an underlying symmetry of the system under consideration
and generally disappears if the symmetry on which it relies is broken. By
a selection rule some transitions are ”selected” to be allowed, while others
are said to be forbidden.
As stated in section 4.3, a spatially varying EM field has to be treated
differently then the more common dipole approximation. The dipole approximation is crucial for the selection rules originating from the envelope
part of the transition matrix element (4.25). Away from this limit the variation of the EM field starts to break the symmetry of the matrix element
between the envelope parts of the electron and hole wavefunctions.
On the other hand, the selection rules originating from the atomic like
matrix element of the momentum operator p are independent of E, provided
that the field can be considered as constant over a unit cell, an approximation which was made earlier.
Leaving the discussion of a spatially varying EM field to paragraph 4.5.2,
the selection rules in the dipole approximation fall into three different classes:
• Polarization selection rules
• Selection rules on the lz -angular momentum of the envelope wavefunction
• Parity selection rules.
The polarization rules are caused by the atomic like matrix element of the
momentum operator p, while the selection rules on the lz -angular momentum of the envelope wavefunction and the related parity selection rules are
76
4.4. Selection rules
due to the envelope part of the transition matrix (4.25).
Explicit investigation of the transition matrix requires a restriction to a
more specific case. The selection rules depend on which particular system is
considered and subsequently which symmetry properties are valid. Therefore
the focus will be on the band-to-band transitions between the top most
valence bands and the lowest lying conduction band in III-V materials. In
this case the matrix element is given by (4.28), or by (4.29) in case of a
spatially varying EM field.
4.4.1
Polarization selection rules
In this paragraph the polarization selection rules in case of transitions between the top Γ8 valence bands and the lowest lying Γ6 conduction band in
III-V semiconductors are derived. However, since the theory strongly relies
on the results of atomic physics, it is instructive to summarize these shortly
in advance.
Suppose an atomic system which is built from the orthonormal base
{|η j mi}, where the quantum numbers j and m come from an angular momentum operator J and η refers to other possible quantum numbers which
complete the basis of the system in consideration. Then for any vector
operator V applies
hη j m0 |V+ |η j mi = 0
if m0 − m 6= 1,
hη j m0 |V− |η j mi = 0
if m0 − m 6= −1,
hη j m0 |Vz |η j mi = 0
if m0 − m 6= 0,
(4.31)
where V+ ≡ Vx + iVy and V− ≡ Vx − iVy , as usual.
In case of the absorption of light, where p is the vector operator of interest, this result gets its physical interpretation if one realizes that a photon
carries spin 1, so m = {1, 0, −1}: in this case (4.31) is just a consequence
of the conservation of angular momentum.
Turning to the optical transitions in III-V semiconductors, remember
that an analogy was made between the band edge Bloch states and atomic
functions, see paragraph 3.1.1. Now it becomes more clear what is actually
meant with ”atomic-like”: in the k · p method the optical matrix elements
are used as input. Without specifying the precise form of a particular band
edge Bloch function, it can be argued that its symmetry properties are the
same as a particular atomic function. All what remains is to determine
(experimentally) the optical matrix elements.
Concerning transitions between the Γ8 valence bands and the Γ6 con77
Chapter 4. EM transition matrix
duction band, the only nonzero matrix elements are given by
−
i
i
i
hS|px |Xi = −
hS|py |Y i = −
hS|pz |Zi = P,
m0
m0
m0
(4.32)
where |Si denotes the band edge Bloch function of the conduction band
which is s-like. The magnitude P of the matrix elements in (4.32) is related
to the Kane matrix element Ep by
EP
= 2m0 P 2 ,
(4.33)
which can be determined experimentally for each particular III-V bulk semiconductor material.
As an upshot, the polarization selection rules in semiconductor materials
result from restrictions imposed on the matrix element of the momentum
operator by the symmetry properties of the atomic-like Bloch states. Generally, from group theory it is known that a matrix element hψ1 |Ô|ψ2 i is
only nonzero if the symmetry S1 of ψ1 is the same as one of the irreducible
representations of the direct product O ⊗ S2 , where O denotes the symmetry of the operator Ô. Indeed, analyzing the symmetry properties of the
momentum operator, one finds that p-like and s-like states lead to the only
nonzero matrix elements given in (4.32)[14].
Narrowing the focus to band-to-band transitions between the top Γ8 valence bands and the lowest lying Γ6 conduction band in III-V semiconductors, the the atomic part of the transition matrix (4.25) is specified further
by
Z
(4.34)
dr ujc 0 (r) p ujv 0 (r) = hS σ | p | 32 jz i,
Ω0
again with σ = ↑, ↓ and jz ² { 32 , 12 , − 12 , − 32 }. This matrix can be calculated in
terms of the only nonzero matrix elements (4.32) by using the decomposition
of | 23 jz i in the states |Xi,|Y i and |Zi, as given in (3.13).
Table 4.1 shows the result for the unpolarized interband matrix element
hS σ | px + py + pz | 32 jz i.
Here the matrix elements Pu are introduced for convenience,
hS|pu |U i ≡ Pu ,
u = {x, y, z}, U = {X, Y, Z}.
(4.35)
In terms of the Kane matrix elements, equations (4.32) and (4.33), Table 4.2 gives the quantitative results of the polarization dependence of the
1
3
matrix element
1 hS σ | pu | 2 jz i. This result plays a dominant role
(m0 Ep ) 2
by determining the polarization anisotropy in the EM transition matrix elements, see paragraph 4.3.3.
78
4.4. Selection rules
| 32
px + py + pz
hS ↑ |
3
2
− √12 Px −
hS ↓ |
| 32
i
1
2
| 32 − 21 i
i
√
√2 P z
3
√i Py
2
− √16 Px −
0
√1 Px
6
√i Py
6
−
| 23 − 32 i
√i Py
6
√
√2 Pz
3
0
√1 Px
2
−
√i Py
2
Table 4.1: Result for the unpolarized interband matrix elements
hS σ | px + py + pz | 32 jz i in terms of Pu ≡ hS|pu |U i. For a particular transition, the spin σ of the conduction band electron is shown in
the left column and the Bloch angular momentum jz belonging to the
valence band in the first row.
P
For instance,
Table 4.2 clearly demonstrates that the ratio of σ |hS σ | pz | 32 jz =
P
± 12 i|2 and σ |hS σ | px | 23 jz = ± 12 i|2 equals 4. The polarization selection
rule has an even stronger effect on the valence subband states which are
dominated by terms with jz = ± 32 : in this case the matrix element of pz
is strictly zero. A strictly zero matrix element of a particular transition
and direction of the momentum operator is said to be polarization forbidden, or polF in short notation. This qualitative result is summarized in
Table 4.3, which shows the allowed polarizations. Here x, y denotes the allowed polarizations perpendicular to the wire axis and z the allowed parallel
1
1
(m0 Ep ) 2
hS ↑ |
hS ↓ |
pu
| 32
3
2
i
| 32
1
2
i
| 32 − 12 i
| 23 − 23 i
px
− 2i
0
i
√
2 3
0
py
1
2
0
1
√
2 3
0
pz
0
√i
3
0
0
px
0
− 2√i 3
0
i
2
py
0
1
√
2 3
0
1
2
pz
0
0
√i
3
0
Table 4.2: Selection rules on the atomic-like interband matrix elements
1
3
1 hS σ | pu | 2 jz i.
(m0 Ep ) 2
79
Chapter 4. EM transition matrix
polarization. Another feature which will be extracted from Table 4.2 further
ε̂ · p
| 32
3
2
i
| 32
1
2
i
| 32 − 12 i
| 23 − 23 i
hS ↑ |
x, y
z
x, y
polF
hS ↓ |
polF
x, y
z
x, y
Table 4.3: Selection rules on the atomic-like interband matrix elements
hS σ | ε · p | 32 jz i. The allowed polarizations are denoted with x, y and
z, where x, y are perpendicular to the wire axis and z gives the parallel
polarization. Polarization forbidden transitions are denoted with polF .
on is that the matrix elements corresponding to polarizations perpendicular
to the wire axis are the same, i.e. |Tcv, x |2 = |Tcv, y |2 , which is a consequence
of the rotational symmetry around the wire axis in the dipole approximation.
Finally, it is instructive to come back to the selection rules in the atomic
case. In fact, the results in Table 4.2 lead to the same selection rules as given
in (4.31). For this purpose, first note that Table 4.2 can be formulated in
an algabraic way by
r
1
1
hSσ|px | 32 jz i = i(m0 Ep ) 2
|jz | (δσ−jz ,1 − δσ−jz ,−1 ),
6
r
1
1
hSσ|py | 23 jz i = (m0 Ep ) 2
|jz | (δσ−jz ,1 + δσ−jz ,−1 ),
6
r
1
2
3
hSσ|pz | 2 jz i = i(m0 Ep ) 2
|jz | δσ,jz .
(4.36)
3
In terms of p+ ≡ px + ipy and p− ≡ px − ipy the matrix elements of px and
py yield
r
1
2
hSσ|p+ | 23 jz i = i(m0 Ep ) 2
|jz | δσ−jz ,1 ,
3
r
1
2
hSσ|p− | 23 jz i = (m0 Ep ) 2
|jz | δσ−jz ,−1 .
(4.37)
3
Together with the matrix element of pz in (4.36), these are the same selection
rules as in the atomic case, now originating from the conservation of angular
momentum on the Bloch part of the transition matrix.
4.4.2
Selection rules on the envelope wavefunctions
The selection rules on the envelope part of the transition matrix are not as
general as the polarization selection rules derived in paragraph 4.4.1. While
80
4.4. Selection rules
the polarization selection rules originate from the bulk, atomic like matrix
element of the momentum operator, the selection rules on the envelope part
are dependent on the configuration of the system (wire radius, length) and
also depend on whether the EM field can be considered in the dipole limit.
T ransition
P olarization
Class
|jz |
C0, 1 → E 1 ,n
x, y, z
MP
1
2
C0, 1 → E 3 ,n
x, y
SP
3
2
C0, 1 → E 5 ,n
−
lF
−
C1, 1 → E 1 ,n
x, y, z
MP
1
2
C1, 1 → E 3 ,n
x, y, z
MP
1
2
C1, 1 → E 5 ,n
x, y
SP
3
2
2
2
2
2
2
2
Table 4.4: Summary of the polarization and class for the lowest band-toband transitions in C∞ nanowires with a constant EM field. Polarization
perpendicular to the wire axis is denoted with x, y, parallel polarization
with z. The envelope angular momentum forbidden transitions are denoted with lF while SP and M P denote the polarization class, single
and mixed polarization respectively. The last column gives the allowed
values of |jz |.
Proceeding with the transitions between the Γ8 valence bands and the
Γ6 conduction band in III-V semiconductors in the dipole limit, equation
(4.28), the φ part of hχlzc ,n |χfzv ,n;jz (kz ) i gives
1
2π
Z
0
2π
dφ e−i(lzc −(fzv −jz ))φ = δ lzc ,fzv −jz ,
(4.38)
which can be considered as a selection rule on the envelope angular momentum, since lzv = fzv − jz . As a direct consequence, transitions for which
lzc − fzv 6= {− 32 , − 12 , 12 , 32 } are l-angular momentum forbidden (lF ). This
result can be found in Table 4.4, which shows a combination of the polarization and l-angular momentum selection rules. The last row gives the
allowed values of |jz |.
81
Chapter 4. EM transition matrix
The constraint (4.38) also leads to a selection rule which is a consequence
of the parity of the subbands at the wire zone center. An even (odd) wavefunction corresponds to an even (odd) Bessel function Jlz and by (4.38) lzc
and lzv = fzv − jz has to be the same, i.e. the parity of the valence subband
wavefunction has to match with the parity of the conduction subband. In
other words, transitions
E (±) → C (∓)
(4.39)
are parity forbidden (pF) at kz = 0. Away from the zone center this selection rule generally is broken: the valence subband wavefunctions are not
characterized by parity any more.
82
4.5. Results
4.5
Results
In this section the above theoretical framework is applied to specific examples.
Again InP and InAs are chosen since those III-V materials are two kind of
extremes regarding their electronic properties. As a matter of fact, although this is also the case for the Kane matrix element Ep , the effect of this
difference will be small since Ep is hardly material dependent, see Table 4.5.
Ep (eV )
InP
20.7
InAs
21.5
Table 4.5: The Kane matrix element Ep for InP and InAs
Since a spatially varying EM field requires a different approach compared
to the theory derived in the dipole limit, this case is treated separately in
paragraph 4.3.2.
4.5.1
Dipole approximation
The topmost illustration in Figure 4.1 (a) and Table 4.6 show the nume-
ÈΡ T v c È
v7 -> c1
v6 -> c1
v4 -> c1
v5 -> c1
v7 -> c1
v3 -> c1
0.80
v2 -> c1
1.00
0.80
v1 -> c1
1.00
v4 -> c1
0.65 0.68 0.72 0.75 0.78 0.82 0.62 0.65 0.68 0.72 0.75 0.78 0.82
0.14
aL : kzR = 0.
bL : kzR = 0.45
0.12
0.10
0.08
z - pol.
0.06
y - pol.
0.04
0.02
v3 -> c1
0.14
0.12
0.10
0.08
0.06
0.04
0.02
v2 -> c1
ÈTv c È2 Harb. unitsL
ETrans HeVL, R = 4.85 nm
0.60
0.60
Ρ>0
Ρ<0
0.40
0.20
0.20
0
0.40
0.49 0.5
0
0.5 0.51 0.52 0.53 0.48 0.49 0.5 0.5 0.51 0.52 0.53
ETrans HeVL, R = 9.96 nm
Figure 4.1: Matrix elements |Tcv,k |2 (first row, white bars) and |Tcv,⊥ |2
(black) in arbitrary units and corresponding polarization anisotropy ρTcv
(second row) of the first 7 transitions vi → c1 for InAs at a) : kz R = 0
and b) : kz R = 0.45. The colors are the same as in Figure 3.1; for the
representations see Table 4.6. The energy scale for all graphs is given at
two values of R, indicated at the top and bottom of the figure.
83
Chapter 4. EM transition matrix
rical results at kz = 0 for InAs of the matrix elements perpendicular and
parallel to the wire axis, |Tcv, y |2 and |Tcv, z |2 respectively. The upper graph
Figure 4.1 (b) gives the same results away from the wire zone center, at
kz R = 0.45. The matrix elements are calculated for wavefunctions valid
in an infinite potential well both for the valence and the conduction subbands and so the values are independent of R. Contrary to the energies, the
correction on the subband wavefunctions by taking a finite potential into
account is expected to be negligible, provided the dimensionless quantity lV
in (3.71) is not too small.
The last two columns in Table 4.6 give the transition energy and corresponding wavelength. In this case the conduction subbands are calculated in
the finite potential wells given in Table 3.4, at R = 4.85 nm.
T ransition
Class
|Tcv, y |2
|Tcv, z |2
Etrans (eV )
λtrans (nm)
(+)
pF
−
−
0.616
2011
(+)
MP
2.94 10−2
11.77 10−2
0.636
1949
(+)
SP
1.23 10−2
−
0.670
1850
(+)
MP
6.15 10−3
24.6 10−3
0.712
1754
(+)
pF
−
−
0.718
1726
(+)
pF
−
−
0.784
1581
(+)
SP
8.39 10−2
−
0.818
1515
Representation
(−)
v1 → c1
E 1 ,1 → C0, 1
v2 → c1
E 1 ,1 → C0, 1
v3 → c1
E 3 ,1 → C0, 1
v4 → c1
E 1 ,2 → C0, 1
v5 → c1
E 3 ,1 → C0, 1
v6 → c1
E 1 ,2 → C0, 1
v7 → c1
E 3 ,2 → C0, 1
2
(+)
2
(+)
2
(+)
2
(−)
2
(−)
2
(+)
2
Table 4.6: Interband matrix elements |Tcv, y |2 and |Tcv, z |2 of the first
seven transitions vi → c1 for InAs at kz = 0. The parity forbidden
transitions are denoted with pF , SP and M P refer to single and mixed
polarization, respectively. The last two columns show the transition
energy and corresponding wavelength calculated at R = 4.85 nm with
the conduction subband in the finite potential well model.
Furthermore, Table 4.6 summarizes the representation and class of the
first seven transitions vi → c1 as derived in the previous sections. In Figure 4.1 the transitions are indicated in the same colors as used in Figure 3.1.
The second row in Figure 4.1 gives the polarization anisotropy ρTcv corresponding to the matrix elements alone, as defined by equation (4.30).
84
4.5. Results
A closer investigation and comparison with the theory leads to the following conclusions:
• At the zone center only transitions between subbands with the same
parity are allowed. Away from the zone center, the parity selection is
broken since the hole states are not characterized by parity any more.
• The polarization anisotropy is completely determined by the polariza(+)
(+)
tion rules. All E 1
→ C0 transitions show a polarization of 0.6,
2
which is in agreement with the analytical result of paragraph 4.4.1,
|T
|2
cv, k
1
where a polarization contrast |Tcv,
2 of 4 was determined for |jz | = 2 .
⊥|
Indeed only these states contribute to the matrix elements, since in the
dipole limit it is required that |jz | = |fzv | by the l-angular momentum
(+)
(+)
selection rule on E 1 → C0 . Also the results of E 3 → C0 are as
2
expected: for the |jz | =
(polF).
3
2
2
states a parallel polarization is forbidden
• The matrix elements are independent of the wire radius. It should be
noted that the finite confinement for the conduction subbands is not
taken into account, but this correction is expected to be small. On the
other hand, the transition energies strongly depend on the wire radius.
Therefore, the energy scale is given at two values of R in Figure 4.1,
R = 4.85 nm and R = 9.96 nm indicated at the top and bottom
of the figure, respectively. Since the correction is substantial for the
transition energies, these are calculated with the conduction subband
in the finite potential well model.
• The present effective mass approach can be compared successfully with
the results derived in an atomistic approach. In Appendix D, Figure 14
the band-to-band matrix elements for an R = 4.8 nm InAs wire are
shown based on an atomistic, empirical pseudo-potential plane-wave
method [4]. It is noted that the C∞ v representations given in Table II
of the article differ from those derived in the present paper and are in
conflict with the basic symmetry considerations of Chapter 4. With
this in mind, comparing Figure 4.1 with FIG. 3 in [4] it is concluded
(+)
(+)
that also in the more accurate atomistic approach the E 3 → C0, 1
2
transitions are completely y-polarized. Considering the polarization
(+)
(+)
anisotropy of the E 1 → C0, 1 transitions, the deviations from 0.6 in
2
FIG. 3 in [4] are explained by the possible corrections of including the
split-off band, or even diagonalizing the full 8 × 8 Hamiltonian of the
three Γ8 valence bands and Γ6 conduction band in the present effective
mass approximation. This also explains that some pF transitions in
85
Chapter 4. EM transition matrix
[4] still have a small contribution at kz = 0. For the shifts in the
transition energies the additional argument holds that taking also finite
confinement for the valence subbands into account may change the
valence energies slightly.
ÈΡ T v c È
1.55 1.58
1.62 1.65
bL : kzR = 0.93
v7 -> c1
z - pol.
y - pol.
v5 -> c1
v6 -> c1
aL : kzR = 0.
1.6
v4 -> c1
v5 -> c1
v6 -> c1
1.58
v1 -> c1
v3 -> c1
v2 -> c1
1.55
v3 -> c1
1.53
0.14
0.12
0.10
0.08
0.06
0.04
0.02
v1 -> c1
ÈTv c È2 Harb. unitsL
ETrans HeVL, R = 4.83 nm
0.14
0.12
0.10
0.08
0.06
0.04
0.02
1.00
1.00
0.80
0.80
0.60
0.60
Ρ>0
Ρ<0
0.40
0.40
0.20
0.20
0
1.45
1.46
1.46
1.46 1.46 1.47 1.47 1.48
0
ETrans HeVL, R = 10. nm
Figure 4.2: Matrix elements |Tcv,k |2 and |Tcv,⊥ |2 and corresponding polarization anisotropy ρTcv of the first 7 transitions vi → c1 for InP at
a) : kz R = 0 and b) : kz R = 0.93.
The colors are the same as in FIgure 3.1, for the representations see
Table 4.7. The energy scale for all graphs is given at two values of R,
indicated at the top and bottom of the figure.
The above results of InAs can be compared with InP, see Figure 4.2 and
Table 4.7. This leads to the following conclusions:
• The large difference in the transition energies, for instance the tran(+)
(+)
sition E 1
→ C0, 1 corresponds to Etrans = 0.636 eV for InAs and
2
Etrans = 1.529 eV for InP, is caused by the difference in the bulk band
gap Eg , see Table 3.4. It is slightly reduced by the difference in confinement energies for the electron subband, see Table 3.6: since InP has
a larger conduction band effective mass m∗c than InAs, its confinement
energy of the conduction subbands is lower.
• As already stated in Chapter 3, the ordering of the valence subbands is
material dependent, due to the differences in the heavy- and light hole
(−)
(+)
effective masses. For instance the first two transitions E 1 → C0, 1
2
86
4.5. Results
T ransition
Class
|Tcv, y |2
|Tcv, z |2
Etrans (eV )
λtrans (nm)
(+)
MP
3.4 10−2
13.7 10−2
1.529
811
(+)
pF
−
−
1.542
804
(+)
SP
9.1 10−2
−
1.556
797
(+)
pF
−
−
1.574
788
(+)
SP
1.2 10−2
−
1.586
782
(+)
MP
3. 10−4
12. 10−4
1.590
780
(+)
pF
−
−
1.648
752
Representation
(+)
v1 → c1
E 1 ,1 → C0, 1
v2 → c1
E 1 ,1 → C0, 1
v3 → c1
E 3 ,1 → C0, 1
v4 → c1
E 3 ,1 → C0, 1
v5 → c1
E 3 ,2 → C0, 1
v6 → c1
E 1 ,2 → C0, 1
v7 → c1
E 1 ,2 → C0, 1
2
(−)
2
(+)
2
(−)
2
(+)
2
(+)
2
(−)
2
Table 4.7: Interband matrix elements |Tcv, y |2 and |Tvc, z |2 of the first
seven transitions vi → c1 for InP at kz = 0. The parity forbidden
transitions are denoted with pF , SP and M P refer to single and mixed
polarization, respectively. The last two columns show the transition
energy and corresponding wavelength calculated at R = 4.83 nm with
the conduction subband in the finite potential well model.
(+)
and E 1
2
(+)
→ C0, 1 are ordered the other way around in case of InP.
As an important consequence, the parity selection rule works on the
lowest possible transition in case of InAs, while for InP this selection
rule at kz = 0 works on the second possible transition.
• Next to the energies, also the transition strengths are material dependent. This is caused by the different heavy- and light hole effective
masses which result in different valence subband wavefunctions.
4.5.2
EM field including Mie scattering
In Figure 4.3 the numerical results are shown of the matrix elements |Tcv,k |2
and |Tcv,⊥ |2 and corresponding polarization anisotropy ρTcv for InAs, including the effects of spatial variation of the EM field up to second order. The
penetration strength, which is also present in the dipole limit, is not taken
into account and the results are obtained by using equation (4.29). Since
the matrix elements now depend on the wire radius by the scattering field,
87
Chapter 4. EM transition matrix
R = 4.85 nm is fixed. Contrary to the expressions for the expanded electric
field at normal incidence in Part I, which were given in cylindrical coordinates, here there components in cartesian coordinates are used. As can
be concluded from (2.9) and (2.10), for polarization perpendicular to the
wire axis the internal field does not have the same direction as the incident
field. However, contributions from the other direction (say x) to the matrix
elements are negligible.
v7 -> c1
v6 -> c1
v4 -> c1
v5 -> c1
v3 -> c1
v1 -> c1
v2 -> c1
v7 -> c1
v6 -> c1
v4 -> c1
v5 -> c1
v3 -> c1
v2 -> c1
0.62 0.65 0.68 0.72 0.75 0.78 0.82 0.62 0.65 0.68 0.72 0.75 0.78 0.82
0.14
0.14
aL : kzR = 0.
bL : kzR = 0.45
0.12
0.12
z - pol.
0.10
0.10
y - pol.
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
v1 -> c1
ÈTv c È2 Harb. unitsL
ETrans HeVL, R = 4.85 nm
1.00
ÈΡ T v c È
0.80
0.60
1.00
Ρ>0
Ρ<0
0.80
0.60
0.40
0.40
0.20
0.20
0
0
0.62 0.65 0.68 0.72 0.75 0.78 0.82 0.62 0.65 0.68 0.72 0.75 0.78 0.82
ETrans HeVL, R = 4.85 nm
Figure 4.3: Matrix elements |Tcv,k |2 and |Tcv,⊥ |2 and corresponding polarization anisotropy ρTcv of the first 7 transitions vi → c1 for InAs, calculated including the scattering terms in the EM field at R = 4.85 nm.
The corresponding energy scale is given at the top and bottom of the
figure.
Before proceeding further by analyzing the results and comparing them
with the dipole limit, at this point it is of particular importance to note that
the bulk value of the dielectric function is used by calculating the matrix
elements. As will be discussed in more detail in Chapter 5, this can only give
a first estimation since a proper calculation of the matrix elements requires
a self-consistent determination of the dielectric function.
Nevertheless, starting with a qualitative approach, at first sight Figure 4.3 seems to be not very different from the results in the dipole limit,
Figure 4.1. The selection rules in the dipole limit still determine almost
completely the strength of the matrix elements and away from the zone center the corresponding polarization anisotropy ρTcv . At kz = 0, however,
88
4.5. Results
the parity selection rule is broken and the resulting transition strengths,
regardless of their magnitude, are subject to a selection rule which requires
a strictly zero matrix element |Tcv,k |2 in case of the |fzv | = 12 subbands.
The explanation is subtle. First recall from paragraph 3.4.1 that the
(−)
E 1 valence subbands are heavy hole like. At the wire zone center, for
2
fzv = 12 the corresponding lateral part of the envelope wavefunction (the jz
component) is given by
χ 1 ,jz (ρ, φ) = |HH1ijz J 1 −jz (j1,n
2
2
1
ρ
)} ei( 2 −jz )φ ,
R
(4.40)
apart from a normalization constant and with j1,n the nth zero of J1 (x) = 0.
Here |HH1i is given in (3.35), with kz = 0.
(−)
(+)
Turning to the band-to-band transitions E 1 ,1 → C0, 1 , the transition
2
matrix for the ∼ ρ cos φ term in the internal field, see (2.9) and (2.10), is of
the form
¯2
¯
¯
¯
X ¯X
¯
¯ hχl =0 | ρ cos φ |χ 1 ihSσ|py | 3 jz i¯ ,
(4.41)
zc
,j
2
¯
¯
2 z
¯
σ ¯ jz
again only considering fz = + 12 and focussing on the perpendicular (y)
polarization. The integral over φ in the envelope part of (4.41) is only
nonzero if jz = 32 or jz = − 12 and the integral over ρ gives the same value
in these two cases, apart from a minus sign in case of jz = 23 (coming from
J−1 (x) = −J1 (x)). Absorbing the value of the integral over ρ in a constant
c, which also includes the overall normalization of the wavefunctions, and
using (4.40), equation (4.41) equals
¯2
X ¯¯
¯
¯−c 2π|HH1i 3 hSσ|py | 32 32 i + c 2π|HH1i− 1 hSσ|py | 23 − 21 i¯
σ
2
2
¯2
X ¯¯ 1
¯
2
2
3
3
3
1
¯
¯
= |c| (2π)
¯− √3 hSσ|py | 2 2 i + hSσ|py | 2 − 2 i¯
σ
¯
¯
¯
1 1
1 ¯¯2
2
2 ¯
+ √
= |c| (2π) ¯− √
3 2 2 3¯
= 0,
(4.42)
where in the first equality the |HH1ijz are inserted utilizing (3.35) at kz =
0, while for the second equality the polarization selection rules, given in
Table 4.2, are used.
(−)
Since the only nonzero contributions for transitions from E 1 to the
2
first conduction subband are coming from the ρ cos φ term (all others are
parity forbidden), equation (4.42) clearly demonstrates that for perpendicular (y) polarization the fzv = + 21 part of this transition is strictly forbidden.
89
Chapter 4. EM transition matrix
In a similar way it can be shown that this is also the case for fzv = − 12 , so
(−)
(+)
we conclude that the transitions E 1 ,1 → C0, 1 at perpendicular incidence
2
are polarization forbidden, if the envelope angular momentum selection rule
4.38 is changed by the ρ cos φ term in the scattering field.
Turning to the quantitative aspects of taking the spatial variation into
account, it can be concluded that the corresponding corrections are too
small to overcome the parity selection rule at kz = 0 significantly. This
is a general result: in Appendix C the effect of the Mie scattering is also
investigated for InP at different R. The maximal contribution at kz = 0 of
(−)
(+)
|Tcv,k |2 for the E 1 ,1 → C0, 1 transitions is of the order 10−6 and a similar
2
(−)
(+)
order was found for the matrix elements of the E 3 ,1 → C0, 1 transitions.
2
For the transitions already present in the dipole approximation it can
be concluded from Figure 4.2, Figure 12 and Figure 13, that the scattering
reduces the strength slightly, with a comparable amount for both |Tcv,k |2
and |Tcv,⊥ |2 . While the polarization anisotropy of the matrix elements more
or less remains the same, the strength of the transition thus reduces at larger
R. As an example, for ε = 12 and a transition wavelength of 900 nm, the
1
reduction is estimated with the expansion parameter |ε| 2 k0 R to be about
1 % at R = 5 nm and 6 % at R = 10 nm, which is indeed confirmed by the
results in Appendix C.
90
Chapter 5
Dielectric function nanowire
Finally, in this chapter the dielectric function and polarization anisotropy
of III-V semiconductor nanowires are obtained including the quantum confinement corrections by the band-to-band transitions between the top Γ8
valence bands and the lowest lying Γ6 conduction band. Section 5.1.3 considers general theory about the dielectric response of a particular group of
interband transitions. In section 5.2 a total dielectric function is formulated
including the background response of all transitions which are not connected with the quantum confinement. Subsequently, in section 5.3 the total
polarization anisotropy of a nanowire is derived and in section 5.4 explicit
results are given for InP and InAs.
5.1
General theory
The transition matrix elements, derived in the previous chapter for bandto-band transitions, are of crucial importance for determining the optical
response of a system to an incident EM field. There are mainly two approaches for determining the macroscopic dielectric function from the quantum
mechanical oscillator strengths, one based on the atomic polarizability, the
other using the absorption transition rate. Both methods rely on the assumption that the EM field can be considered in the dipole limit, i.e. E(R⊥ ) ' E.
Next to this it is required that the semiconductor heterostructure can be described using macroscopic Maxwell equations, i.e. where the different parts
in the system are characterized by macroscopically averaged quantities.
In the following both approaches are applied to the nanowire case, in
particular concerning the band-to-band transitions.
5.1.1
Atomic polarizability approach
In order to derive the dielectric function of the nanowire by using the atomic polarizability, the transition matrix has to be expressed in the position
91
Chapter 5. Dielectric function nanowire
operator x instead of the momentum operator p.
Already at this stage the dipole approximation is of crucial importance.
If E(R⊥ ) = E the electric field can be taken out of the integral as in
(4.27). Omitting the subband characterization for convenience by denoting
the conduction and valence subband wavefunctions with |Ψc i and |Ψv i now
it suffices to find the relation between hΨc | p |Ψv i and hΨc | x |Ψv i. Utilizing
the commutation relation
i~
p = [x, H],
m0
(5.1)
where H is the one-electron crystal Hamiltonian in the wire configuration
derived in Chapter 3, one obtains
hΨc | p |Ψv i =
=
m0
hΨc |[x, H]|Ψv i
i~
trans (k )
im0 Ecv
z
hΨc | x |Ψv i,
~
(5.2)
where
trans
Ecv
(kz ) ≡ ~ωcv (kz ) ≡ Eg + Ec (kz ) − Ev (kz ),
(5.3)
is the transition energy. The R dependence is omitted for the moment.
Following Ziman[23], the dipole moment is proportional to the local field,
h −ex(t) i = α0 (ω)E(t)
(5.4)
where α0 denotes the real part of the atomic polarizability. Also this only
makes sense in the dipole approximation, the proportionality is integrated
out in case of a spatially varying field. From equations (5.2), (4.27) and
(4.28) the atomic polarizability for the band-to-band transitions between
the Γ8 and Γ6 subbands considered in this paper is given by
α0 (ω) =
1 e2 X X
2
1
|Tc v (kz )|2 2
,
N m0 cv
m0 ~ωcv (kz )
ωcv (kz ) − ω 2
(5.5)
kz
where contrary to the one atom model in [23] here a factor N1 appears since
already N atoms are taken into account in (5.5) by the sum over the N
initial valence subband states.
The formal prove of equations (5.4) and (5.5) requires time-dependent
perturbation theory and in the present case this means that in the excited
state (4.6) the time dependence has to be included by solving the corresponding time-dependent Schrodinger equation. Here we make an analogy with
an atomic system, since the basic arguments are the same in the two pictures. Denote a ground-state orbital with |Φ0 i corresponding to a ground-state
92
5.1. General theory
energy ²0 , from which an electron can be excited to higher orbitals |Φj i with
energy ²j . Assuming the electric field in the x direction for simplicity1 , the
electron wavefunction |Ψ(t)i at a particular moment can be written as
X
|Ψ(t)i = |Φ0 ie−i²0 t/~ +
cj (t)|Φj ie−i²j t/~ ,
(5.6)
j
where the coefficients cj (t) ∝ eEx hΦj | x |Φ0 i are the solutions of the timedependent Schrödinger equation. The expectation value of the dipole moment −hΨ(t)| ex |Ψ(t)i consequently becomes proportional to the electric
field as in (5.4), where the atomic polarizability in the present case is given
by
α0 (ω) =
fj
e2 X
,
m
(²j − ²0 )2 − ω 2
(5.7)
j
with fj = 2m
~(²j − ²0 ) |hΦj |x|Φ0 i|2 the oscillator strength of these atomic
~2
transitions.
Turning back to the nanowire, using (5.2) it is easy to show that the
oscillator strength of a transition v → c at kz corresponding to (5.5) equals
fcv (kz ) =
2
|Tc v (kz )|2 ,
m0 ~ωcv (kz )
(5.8)
Classically, the oscillator strength is the number of oscillators with frequency
ωcv (kz ). Indeed, the quantity (5.8) is dimensionless. Quantum mechanically
it
P has
Pto satisfy the Thomas-Reiche-Kuhn sum rule, in the present case
cv
kz fcv (kz ) = N .
Once the real part of a linear response is known, the imaginary part is
uniquely determined by the Kramers-Kronig relations which are a consequence of the causality condition. This leads to
´
³
1 e2 X X
iπ
α(ω) =
fcv (kz ) ω2 (k1z )−ω2 + 2ω
δ(ω − ωcv ) . (5.9)
cv
N m0 cv
kz
Apart from the oscillator strength, this is basically the same result as obtained in the one atom model [4].
Since the atomic dipole moment (5.4) described in this way is the same for all the N electrons involved in transitions between the Γ8 and Γ6
subbands, the total dipole moment per unit volume P is given by
P
=
N
αE.
V
(5.10)
1
An electric field represented by Ex e−iωt corresponds to a real electric field with amplitude 2Ex , so take E(t) = Ex (eiωt + e−iωt )
93
Chapter 5. Dielectric function nanowire
All what remains is to determine the relation between this macroscopically
averaged quantity and the dielectric function. Again assuming a linear response, the dielectric displacement D equals
D = εE = E + 4πP ,
(5.11)
which leads to
ε−1 =
4πN
α.
V
(5.12)
Combining this with (5.9) the imaginary part of the dielectric function is
obtained:
2π 2 e2 X X
fcv (kz )δ(ω − ωcv (kz )),
(5.13)
ε00 (ω) =
m0 ωV cv
kz
where V is the volume πR2 L of the nanowire.
In the above derivation it is assumed that the local field which excites an
electron equals the macroscopic field obtained from the Maxwell equations.
It is well known [8][12][23] that a more realistic result requires inclusion of
the Lorentz correction, which has to be reconsidered when dealing with a
nanostructure. In the present paper this correction is not taken into account.
5.1.2
Transition rate method
In the second approach the quantum mechanically determined transition
probability is related to the macroscopic power loss in the wire due to the
absorption process. The method is explained here shortly since it is extended more easily to the case of a varying EM field.
Starting with the transition matrix Mc v , the probability Pcv for a transition between subbands c and v is obtained from from Fermi’s Golden Rule,
2π
trans
|Mc v (kz )|2 δ(~ω − Ecv
(kz )).
(5.14)
~
P P
The total transition rate P subsequently equals kz cv Pcv (kz ) and multiplying with the energy ~ω in each photon this equals the power loss in the
wire volume due to absorption. With equations (4.27) and (4.28), in the
dipole limit this results in
Pcv (kz ) =
P ower loss =
X
2π e2
2
trans
|E|
|Tc v (kz )|2 δ(~ω − Ecv
(kz )). (5.15)
ω m20
cv,kz
In Part 1 this quantity already was derived macroscopically for an infinite
cylinder by
c
c
Wext = Cext I0 = Cext |2E0 |2 = Qext |2E0 |2 2RL,
(5.16)
8π
8π
94
5.1. General theory
see equations (1.58) and (1.59). In the second step it is taken into account
that an electric field represented by E0 e−iωt corresponds to a real electric
field with amplitude 2E0 .
Utilizing the efficiency factors (2.17)-(2.20) in the dipole limit, with the
notion that the scattering contribution can be neglected in that case, the
total power loss Wabs due to absorption equals
ω
Wabs = ε00 V
|E|2 ,
(5.17)
2π
where E is defined in the same way as in (4.27).
Finally this gives
µ
¶
1 2πe 2 X X
00
trans
ε (ω) =
|Tc v (kz )|2 δ(~ω − Ecv
(kz )),
V m0 ω
cv
(5.18)
kz
which is the same result as in (5.13), here shown with fcv explicitly written
out.
As in the case of the polarizability approach, the current derivation relies
on the dipole approximation, but here it is more easy to see the consequences
of allowing a spatial variation of the EM field compared to the scale of
R. Starting with the classical absorption rate, equation (5.17) in fact is an
expansion in mk0 R up to first order, with an additional factor R coming from
the geometrical cross section. Increasing R and subsequently mk0 R requires
a further expansion than (5.17), here denoted with Wabs (ε, R). In case of
the quantum mechanically determined transition probability, in (5.14) now
equation (4.29) has to be used instead of (4.28).
Equalizing the two transition rates yields
XX
Pcv (kz ; ε, R)
Wabs (ε, R) = ~ω
(5.19)
=
kz cv
e2
X
2π
trans
|E|2
|Tc v (kz ; ε, R)|2 δ(~ω − Ecv
(kz , R)),
2
ω m0
cv,kz
trans . Together with the
with the R, ε dependence is indicated, also for Ecv
Kramers-Kronig relations, (5.19) has to be solved self-consistently in order
to find ε0 (ω) and ε00 (ω).
As a final remark, in the above expressions (5.13) and (5.18) for ε00 (ω)
in the dipole approximation and even more (5.19) as a relation for ε(ω) in
case of a spatial varying EM field, it is implicitly assumed that the dielectric
function is a constant, not depending on the wire radius. A correct description, however, requires allowing ε(R), also in the dipole limit: the inclusion
of the Lorentz correction has to be reconsidered within a microscopic, semiclassical approach, when dealing with nanoscale systems which are not
95
Chapter 5. Dielectric function nanowire
”microscopically large” any more: the resonant states are extended over the
whole volume by the envelope part of the wavefunctions and consequently
the induced polarization is position dependent.
As a first step, however, in this paper the dielectric function is approximated with a constant, which depends on the orientation of the incident field
but is calculated in a local approach, not including the spatial variation.
5.1.3
Dielectric function expressed in reduced effective mass
The dielectric function derived in the previous section is expressed in a more
easily applicable form by taking advantage of the effective mass approximation.
For this purpose, first note that the sum over kz in (5.18) is replaced
by an integral over the first Brillouin zone if L → ∞. Using L = M a and
1 2π
taking the reciprocal distance M
a into account, (5.18) is replaced with
00
ε (ω) =
Z π
2 e2 X a
trans
dkz |Tc v (kz )|2 δ(~ω − Ecv
(kz )), (5.20)
R2 m20 ω 2 cv − π
a
where the integral is taken over the first Brillouin zone.
Apart from the correction by the finite potential well on the conduction
trans (k ) derived in the effective
subbands, the simple kz2 dependence of Ecv
z
mass approximation results in
µ
dkz
trans (k )
dEcv
z
=
2µ∗cv z
~2
¶1
2
1
p
,
trans
trans
2 Ecv (kz ) − Ecv
(5.21)
where
1
µ∗cv, z
≡
1
1
+ ∗
m∗c
mv, z
(5.22)
is the reduced effective mass in the z-direction of the nanowire obtained
trans ≡ E trans (0).
from the results in Chapter 3 and Ecv
cv
With the density of states expression (5.21) the integral part in (5.20)
equals
Z
1
dE p
|Tc v (E)|2 δ(~ω − E)
(5.23)
trans
E − Ecv
and after evaluating the δ distribution finally one obtains
ε00 (ω) =
2 ³ 2µ∗cv z ´ 21 ³ e ´2 P
1
2 √
,
cv |Tc v (~ω)|
m0 ω
trans
~2
~ω−Ecv
R2
where |Tc v (~ω)|2 is given by (4.28).
96
(5.24)
5.2. Dielectric function for finite group transitions
In the last step of the above derivation it is more realistic to add a
broadening term, for instance by replacing the δ function with a Gaussian
distribution, since in practice never an infinite sharp line is seen. The quantum efficiency is always reduced by the radiative relaxation of the levels, or
by impurities for instance at the surface of the nanowire.
5.2
Dielectric function for finite group transitions
Although the results in the previous sections apply to band-to-band transitions between the Γ8 valence and Γ6 conduction subbands in a cylindrical
nanowire, up till now we did not specify precisely how to consider the sum
over v and c. In principle, the real part of the dielectric function at a frequency ω away from any absorption peak is built up from all responses with
frequencies ω 0 > ω, i.e. from all atomic oscillators which are able to follow
the oscillation of the EM field. In practice, however, it is hardly feasible to
calculate all contributions, for example those of bulk bands lying deep in a
particular system, or even all relevant symmetry points of the highest lying
bands.
Nevertheless, focussing again on the nanowire, there is a way to take
only a particular group of transitions into account explicitly without neglecting the others entirely, provided the bulk dielectric function εbulk is known
reasonably well. Denoting this group with c and v, referring to the notation
above, a background dielectric function εbg can be defined which is basically
the bulk dielectric function, from which the c, v group is projected out. In
other words, denoting εw
cv as the dielectric response of one particular transition c, v in the nanowire and εbulk
cv as its contribution at the same frequency
in the bulk material, the total dielectric function in the wire configuration
is given by
εw (ω) = εbg (ω) +
X
εw
cv (ω),
(5.25)
cv
with
εbg = εbulk −
X
εbulk
cv .
(5.26)
cv
As a first approximation, away from a bulk absorption peak εbg is approximated with εbulk , the bulk dielectric function. Contrary to common literature
[4][5][14][15], where εbg usually is not taken into account at all, this approach
will be retained carefully in the remaining of this paper.
97
Chapter 5. Dielectric function nanowire
5.3
Polarization anisotropy nanowire
Closely related to the discussion above, the common approach [4][5][14][15]
to derive the polarization anisotropy of an infinite cylinder in the dipole
approximation is insufficient when the quantum confinement becomes essential. In the usual procedure the polarization anisotropy is estimated by
considering the absorption coefficient of the nanowire. In contrast, by using
the efficiency factors, here an approach is followed in which the correct dielectric function εw (ω) appears in a natural way. Also the dipole limit can
be taken more precisely.
By defining the relative difference δ⊥ between the macroscopically determined internal field at parallel and perpendicular polarization by
2
δ⊥
(ω, x)
¯
¯2
¯
¯
2
¯ ,
≡ ¯¯
1 + ε⊥ w (ω, x) ¯
(5.27)
the efficiency factors derived in Part 1 are summarized for a nanowire at
normal incidence by
πx
+ O(x3 ),
2
πx 2
δ (ω, x) + O(x3 ),
Qext⊥ (ω, x) = ε00⊥ w (ω, x)
2 ⊥
Qextk (ω, x) = ε00k w (ω, x)
(5.28)
(5.29)
where the expansion parameter x is defined as
x ≡ k0 R,
(5.30)
while ε00k w (ω, x) and ε00⊥ w (ω, x) are the dielectric functions of the nanowire at parallel and perpendicular incidence, respectively. Consequently, the
polarization anisotropy due to the extinction is given by
ρext (ω, x) =
=
Qextk (ω, x) − Qext⊥ (ω, x)
Qextk (ω, x) + Qext⊥ (ω, x)
2 (ω, x) + O(x2 )
ε00k w (ω, x) − ε00⊥ w (ω, x) δ⊥
2 (ω, x) + O(x2 )
ε00k w (ω, x) + ε00⊥ w (ω, x) δ⊥
.
(5.31)
Taking the wire dipole limit by neglecting terms of higher order in x, this
leads to
ρdip (ω, R) =
2 (ω, R)
ε00k w (ω, R) − ε00⊥ w (ω, R) δ⊥
2 (ω, R)
ε00k w (ω, R) + ε00⊥ w (ω, R) δ⊥
,
(5.32)
where the R dependence is explicitly shown. Note that the scattering process
in the wire dipole limit is negligible compared to absorption. It should be
98
5.3. Polarization anisotropy nanowire
stressed again that for practical purpose it is more convenient to use a
polarization contrast, in the present case denoted with
Cdip (ω, R) =
ε00k w (ω, R)
2 (ω, R)
ε00⊥ w (ω, R) δ⊥
.
(5.33)
Utilizing the framework of section 5.2 it is of particular interest to consider these results in two special cases.
Starting with the one which leads in a natural way to the result in the
usual procedure, if the transitions are investigated at frequencies ω where
0
ε00bulk
P ∼ 0w and εbulk is large compared to both the real00 and imaginary part
of cv εcv (ω) , then the background contribution to εw (ω) is negligible and
δ⊥ is approximated with the bulk value. In this case
ρdip ∼
2
X |Tcv,k |2 − |Tcv,⊥ |2 δ⊥
2 ,
|Tcv,k |2 − |Tcv,⊥ |2 δ⊥
cv
(5.34)
which indeed equals the usual expression [14][4][5].
Secondly, take a macroscopically small, but microscopically large R which
allows to neglect the quantum corrections, but still satisfies the dipole limit λ0 À R. Considering
we conclude that in this case
P (5.25) andP(5.26),
bulk (ω) becomes negligible, which
ε
(ω)−
the quantum correction cv εw
cv
cv cv
leads back to the classical result (2.22), as required.
99
Chapter 5. Dielectric function nanowire
5.4
Results
Using the results of the previous chapters, in this section the above theoretical framework is applied to the specific examples InP and InAs. Initially
the focus will be on the dielectric response purely due to the band-to-band
transitions between the Γ8 valence and Γ6 conduction subbands in nanowires
by neglecting the imaginary part ε00bg of the background dielectric function
in (5.25). Hereby the following aspects will be discussed in more detail:
• The effect of the kz dependence of the transition matrix on the imaginary part ε00w of the nanowire dielectric function .
• The polarization anisotropy (4.30) of the matrix elements alone and including the polarization due to the dielectric mismatch only by taking
ε0bg into account, as in (5.34).
• The R dependence of ε00w and corresponding polarization anisotropy
(5.34).
• Material dependence and comparison with literature [4][5].
Finally, in paragraph 5.4.2, ε00bg is taken into account by calculating the
correct expression for the polarization contrast, corresponding to equation
(5.33) and equivalent to the polarization anisotropy given in (5.34). All
results are obtained in the dipole approximation.
5.4.1
Estimation kz dependence of |Tcv |2
In Chapter 4 it was shown that the transition matrix elements depend on
kz in a nontrivial way.
z-pol
y-pol.
4
aL: kz = 0.
4
bL: kz = 0.45
3
v4
2
v7
Ε''wire
Ε''wire
3
z-pol
y-pol.
v2
0.65
v7
v1
v3
1
v4
2
v2
0.7
0.75
0.8
Energy HeVL
v3
1
0.85
0.65
0.7
0.75
0.8
Energy HeVL
0.85
Figure 5.1: Complex part of the dielectric function εw at parallel (z) and
perpendicular (y) polarization for InAs and R = 4.85 nm, fixing |Tcv |2
at a) : kz R = 0 and b) : kz R = 0.45. Only the first 7 transitions vi → c1
are taken into account, the corresponding peaks are labeled in the same
colors as in Figure 4.1. The background response ε00bg is neglected.
100
5.4. Results
As a first simple trial, Figure 5.1 shows the imaginary part of the dielectric function εw for InAs and R = 4.85 nm , obtained by fixing |Tcv |2 at
its value at a) : kz R = 0 and b) : kz R = 0.45 in equation (5.24). The singutrans = 0 are broadened in a qualitative way by adding a
larities at ~ω − Ecv
displacement of 0.004 eV to ~ω in the denominator of (5.24). Although this
is not the common way to include the broadening, it qualitatively gives the
the same results as in the usual procedure, where the Dirac distribution is
replaced for instance with a Gaussian. Since the non-radiative decay which
causes the broadening is not estimated yet it makes little practical difference.
Focussing on the first peaks in Figure 5.1, it is concluded that simply
taking |Tcv (0)|2 in (5.24) not only quantitatively, but also qualitatively fails
since it does not include the contribution of the first peak, corresponding to
(−)
(+)
the transition E 1 ,1 → C0, 1 which is parity forbidden at kz R = 0, but not
2
at finite kz R.
In Figure 5.2 the kz dependence of |Tcv |2 is taken into account properly.
At each fixed value of ~ω equation (5.24) is calculated using the correct
value of |Tcv |2 , which means that for every point in the figure the correct
valence subband wavefunction is used in (4.28). Only the first two peaks
are shown.
4
z-pol
y-pol.
Ε''wire
3
2
v2
v1
1
0
0.62
0.64
0.66
0.68
Energy HeVL
0.7
0.72
Figure 5.2: Parallel (dots) and perpendicular (line) contributions to εw
of the first two transitions vi → c1 for InAs and R = 4.85 nm. At each
fixed value of ~ω equation (5.24) is calculated using the correct value of
|Tcv |2 , which means that for every point in the figure the correct valence
subband wavefunction is included in (4.28).
Since it is desirable to avoid such an extensive calculation, from now
on the simpler approach of taking |Tcv |2 constant is used, but at a finite
kz value such as in Figure 5.1 b). In this way the results of Figure 5.2
are reproduced qualitatively: the peaks which are parity forbidden (pF)
101
Chapter 5. Dielectric function nanowire
at kz = 0 are included. Quantitatively, the slope corresponding to one
particular transition is underestimated in most of the cases, since except for
the pF transitions, the matrix elements become smaller for larger kz . The
height of the peaks will be corrected with respect to the strength of the
transitions which are already present at kz = 0. As a consequence, the pF
transition peaks are slightly overestimated in the following paragraphs.
Furthermore, it is important to note that the tails of the different transition contributions to ε00w only represent a first rough qualitative estimation.
In the effective mass approach the bulk bands are assumed to be quadratic
and as stated in Chapter 3 this rests on the assumption that k is sufficiently
small. In fact, a particular transition stops contributing when it reaches the
boundary of the Brillouin zone, but in the present procedure this point is
not reached at all since the hole subbands derived in Chapter 3 have a finite
extent.
5.4.2
Polarization anisotropy and R dependence
Utilizing the above mentioned procedure, Figure 5.3 a) again illustrates the
imaginary part of the dielectric function εw for InAs and R = 4.85 nm ,
now obtained by fixing |Tcv |2 at kz R = 0.45 and correcting the height of
the peaks with respect to the strength of the transitions which are already
present at kz = 0. As stated before, the background response ε00bg will be
neglected up till paragraph 5.4.4. The remarks about broadening remain
the same.
1
v2
z-pol
y-pol.
0.75
v4
0.25
0.5
3
2
v7
v1
0
v3
1
Ρ
Ε''wire
4
Ρ bulk
Ρ Tvc
Ρ wire
-0.25
-0.5
0.65
0.7
0.75
0.8
Energy HeVL
0.85
0.65
0.7
0.75
0.8
Energy HeVL
0.85
Figure 5.3: a): Complex part of the dielectric function εw at parallel
(z) and perpendicular (y) polarization for InAs and R = 4.85 nm, fixing
|Tcv |2 at kz R = 0.45 and correcting the height of the peaks with respect
to the strength of the transitions which are already present at kz =
0. Only the first 7 transitions vi → c1 are taken into account, the
corresponding peaks are labeled in the same colors as in Figure 4.1. b):
Polarization anisotropy, calculated from a) alone (dotted), only from
bulk mismatch (line) and both together (red line).
In Chapter 4 it already was shown that v1 → c1 is the only pF transition
which contributes significantly. Indeed this is confirmed by Figure 5.3.
102
5.4. Results
Secondly, the influence of the valence subband effective mass on the
reduced effective mass µ∗cv z (5.22) is negligible: in all relevant situations
considered here, the effective mass of the conduction subband c1 , also in the
finite potential well model, is much smaller then m∗v, z .
As in Chapter 4, the transition energies include the correction by the finite potential well model on the conduction subbands. The correction on the
matrix elements is not taken into account, since it can be argued to be small.
Figure 5.3 b) gives the polarization anisotropy ρTcv due to the matrix
elements alone (dotted line), ρbulk only from bulk mismatch (black line) and
both together (red line). Starting from the left, up till the appearance of
the peak v3 , ρTcv has a constant value of 0.6, as expected (compare Figure 4.1 b)): for all kz both transitions are subject in the same way to the
envelope angular momentum and polarization selection rules described in
the previous chapter. The peak v3 changes ρTcv drastically: in the case
of the E 3 ,n → C0, 1 transitions the perpendicular component is polarizati2
on forbidden. This is also the case for the v7 peak, here the polarization
anisotropy of the matrix elements alone becomes even negative.
Compared to the bulk value ρbulk due to the dielectric mismatch, which
is almost a constant in the region of interest, it is concluded that ρTcv causes giant changes in the total polarization anisotropy of the wire. This will
become even more visible in paragraph 5.4.4 considering the polarization
contrast.
1
z-pol
y-pol.
0.6
v2
3
0.4
Ρ
Ε''wire
4
0.8
v4
2
1
v1
0.52
0.2
v7
Ρ bulk
Ρ Tvc
Ρ wire
0
v3
0.54
-0.2
0.56 0.58
Energy HeVL
0.6
0.62
-0.4
0.52
0.54
0.56 0.58
Energy HeVL
0.6
0.62
Figure 5.4: a): Complex part of the dielectric function εw at parallel
(z) and perpendicular (y) polarization for InAs and R = 7.5 nm. b):
Polarization anisotropy, calculated from a) alone (dotted), only from
bulk mismatch (line) and from both together (red line).
The statements about the R dependence of ε00w and corresponding polarization anisotropy (5.34) are summarized if one compares Figure 5.3 with the
same results at R = 7.5 nm, Figure 5.4. Without including the background
dielectric response εbg , the dielectric function of the wire behaves formally
as ε00w ∝ R12 while the polarization anisotropy ρTcv remains the same. Howe103
Chapter 5. Dielectric function nanowire
ver, the different transitions come closer to each other for increasing R, with
their mutual distances also behaving like R12 in the infinite-well approximation. Moreover, more and more higher transitions which are not taken into
account here enter the energy region of interest. Apart from the remarks
about the tails corresponding to the different transitions one should thus
keep in mind that a correct picture of the higher energy part includes more
peaks, for instance coming from vi → c2 transitions.
5.4.3
Material dependence
For InP the present effective mass approach can be compared with results
obtained from an tight-binding approach [5].
1
0.8 v1
v3
z-pol
y-pol.
0.6
0.4
0.2
0.6
v4
0.4
v5
Ρ
Ε''wire
v2
0.8
0.2
v6
0
Ρ bulk
Ρ Tvc
Ρ wire
-0.2
1.54 1.56 1.58 1.6
Energy HeVL
-0.4
1.62 1.64
1.54 1.56 1.58 1.6 1.62 1.64
Energy HeVL
Figure 5.5: a): Complex part of the dielectric function εw at parallel
(z) and perpendicular (y) polarization for InP and R = 4.83 nm, fixing
|Tcv |2 at kz R = 0.6. Only the first 7 transitions vi → c1 are taken into
account. b): Polarization anisotropy, calculated from a) alone (dotted),
only from bulk mismatch (line) and from both together (red line).
Comparing Figure 5.5 with Figure 15 in Appendix D and noting that
the present results are obtained with the parameters at T = 0 K only for
the first seven transitions vi → c1 , we conclude that the effective mass
approximation, without including the split off Γ8 band and assuming an
infinite potential well in case of the valence subbands, successfully describes
the overall features of ε00w . Remarkable is the difference in the lowest two
transition(s): in FIG 3 of [5] the band edge optical transition is fully zpolarized. This polarization selection cannot be explained from an effective
(+)
(+)
mass approach at all considering the E 1 → C0, 1 transitions, but we note
2
that also the more accurate atomistic approach [4] contradicts this strict
selection found in [5], see paragraph 4.5.1.
With the effective mass approach it is relatively easy to change the material parameters and wire radius. In Figure 5.6 the imaginary part of
the dielectric function εw is shown at R = 10 nm both for InAs and InP.
Comparing the two materials it is concluded that a particular band-to-band
transition has a significantly larger contribution in the dielectric function
104
1.75
1.5
1.25
1
0.75
0.5
0.25
aL: InAs
v2
bL: InP
1.75
z-pol
y-pol. v7
z-pol
y-pol.
1.5
1.25
Ε''wire
Ε''wire
5.4. Results
v3
v4
1
0.75
0.5
0.25
0.48 0.49 0.5 0.51 0.52 0.53 0.54
Energy HeVL
v1
v3
1.46
v4
1.47
1.48
Energy HeVL
1.49
Figure 5.6: Complex part of the dielectric function εw at parallel (z) and
perpendicular (y) polarization for a): InAs and b): InP at R = 10 nm.
The pF transitions are neglected by fixing |Tcv |2 at kz R = 0.
of InAs. This is directly caused by the larger band gap of InP since the
transition probability ∝ ω12 .
5.4.4
Effect of the dielectric background
Finally, in this paragraph the effect of the background response εbg is estimated by looking at the polarization contrast. For this purpose, compare
the results shown in Figure 5.7 and Figure 5.8.
200
175
160
aL: InAs
120
75
C
C
125
100
bL: InP
140
150
C bulk
C wire
100
C bulk
C wire
80
60
50
25
0.52 0.54 0.56 0.58 0.6
Energy HeVL
40
0.62
1.48
1.5
1.52
Energy HeVL
1.54
Figure 5.7: Polarization contrast Cbulk (black line), only due to the
dielectric mismatch, and Cwire (red line), including the polarization anisotropy caused by quantum confinement, for a:) InAs and b:) InP, at
R = 7.5 nm. The dielectric background is not taken into account properly: ε00bg is set to zero.
First of all, we recover the result that the effect of the quantum confinement is huge if εbg is not taken into account: both for InP and InAs Figure 5.7 shows a maximum enhancement by a factor 4 due to the quantum
confinement, as already predicted in paragraph 4.4.1. However, in general
this overestimates the polarization anisotropy substantially.
After including εbg in a proper way, see Figure 5.8, the effect of the
quantum confinement is still large for InAs (maximum enhancement ' 2),
105
Chapter 5. Dielectric function nanowire
100
C bulk
C wire
90
aL: InAs
48
bL: InP
46
80
70
C
C
C bulk
C wire
44
60
42
50
40
30
0.52
40
0.54
0.56 0.58
Energy HeVL
0.6
0.62
1.48
1.5
1.52
Energy HeVL
1.54
Figure 5.8: Polarization contrast Cwire compared to Cbulk for a): InAs
and b): InP, at R = 7.5 P
nm using the correct dielectric function of the
wire: εw (ω) = εbg (ω) + cv εw
cv (ω). It is assumed that εbg ' εbulk .
but considerably reduced for InP (maximum enhancement ' 1.2). These
results are far more conform reality and reflect the difference between InP
and InAs: as shown in the previous section the confinement correction to
εw is much smaller in the case of InP.
Furthermore, for both materials the effect of the quantum confinement
disappears if R is increased sufficiently. This is P
only achieved if εwire is
considered in the right way: εw (ω) = εbg (ω) + cv εw
cv (ω). To be more
precise, taking all transitions
c,
v
corresponding
to
one
particular
bulk tranP w
P bulk
sition into account, cv εcv (ω) '
cv εcv (ω) for sufficiently large R and
consequently
ε
(ω)
'
ε
(ω).
So
for
R → ∞ the quantum confinement
bulk
P ww
correction cv εcv (ω) equals the contribution of the same group of transitions projected in the bulk system.
106
Summary and Conclusions
In this paper we analyzed the optical absorption of III-V semiconductor cylindrical nanowires with the aim to get a theory which describes the optical
properties for arbitrary wire thickness and for a wide range of semiconductor
materials.
In Part I we started with a classical theory describing the scattering of
light by an infinite cylindrical structure. At arbitrary angle of incidence, in
Chapter 1 general expressions were found for the cross sections and corresponding efficiency factors, which are measurable quantities in the region far
from the cylindrical wire.
Subsequently, in Chapter 2 we focussed on the case of cylindrical wires
small compared to the wavelength of the incident light and derived analytic solutions explicitly as a function of the material properties (dielectric
constant, wire radius R), geometric configuration (angle of incidence) and
wavenumber (k0 ) of the incident light. Next to reproducing the well known
results in the dipole limit, we extended the theory by Mie expansion of the
EM field inside cylinder up to second order in the dimensionless parameter
k0 R. We concluded that for increasing cylinder radius, besides the wave
behavior of the EM field inside the wire, the effect of optical focussing gets
a more important role.
Furthermore, numerical results of the efficiencies and corresponding polarization anisotropy are given for InP in the region between 350 and 600 nm
and wire radii up till 5 nm. Hereby the nanowire is treated classically by
taking the bulk value of the dielectric function.
In Part II we have included the effects of quantum confinement by means
of a corrected description of the dielectric function of cylindrical nanowires.
For this purpose, in Chapter 3 we first derived the electronic structure
using effective mass theory. This method utilizes the already well known
bulk energy gaps and optical matrix elements at the band extreme. The resulting bulk dispersion obtained from the crystal Hamiltonian H0 is treated
as a kinetic energy term, which after including the cylindrical confinement
potential and assuming the wire radius R large compared to interatomic
distances, results in a one-particle Schrodinger equation acting on the enve107
lope of the nanowire wavefunction. Neglecting the small anisotropic terms
in H0 by applying the spherical approximation, the total Hamiltonian of the
nanowire becomes diagonal with respect to the total angular momentum
along the z axis Fz = Jz + Lz , with Lz and Jz the z-projection of the envelope angular momentum and the total angular momentum of the atomic
states, respectively. Consequently the eigenvalue fz of Fz is a good quantum
number.
In case of the Γ point valence band in III-V semiconductors we neglected
the split-off band and diagonalized the 4 × 4 Hamiltonian HFΓz8 of the heavy
and light hole band in the basis jz v = { 32 , 12 , − 12 , − 32 }. At kz = 0, the wire
zone center, HFΓz8 decouples into two 2 × 2 blocks with solutions which are
characterized by parity: the envelopes are even or odd under ρ → −ρ.
Assuming an infinite confinement potential, the complete set of solutions
for the Γ8 valence band in a cylindrical nanowire is thus characterized with
|fz |, the parity (±) and n, denoting the nth solution at this parity. Away
from the zone center, the valence subband wavefunctions depend in a non
trivial way on kz by the the lateral part of the envelope function
For the Γ6 conduction band the Hamiltonian is already diagonal in Lz
and consequently the irreducible representation of the conduction subbands
is given by the eigenvalue lz of Lz and again (±), n. We have taken the
finiteness of the potential well into account by including a reduction factor
to the conduction subband energies calculated in an infinite confinement
model, which results in a large correction since the bulk conduction band
has a relative small effective mass. Contrary to the energies, no correction
is made to the subband wavefunctions since in this case the difference with
the finite potential well model is expected to be negligible.
In Chapter 4 we analyzed the radiation-matter interaction − me0 c A · p
between the external electromagnetic (EM) field and the electrons within
the semiconductor system using a macroscopic, semiclassical approach: we
treated the EM field classically, while the semiconductor nanostructure is
described in the spirit of Chapter 3. Similar to the separation of the nanowire
wavefunction into an atomic part and ”macroscopic” envelope functions, the
EM transition matrix factors into separate integrals: a bulk, atomic like
matrix element of the momentum operator and the integral of the EM field
between the envelopes. This separation relies on the natural assumption
that the EM field varies slowly compared to atomic distances.
Both the envelope and the atomic part of the transition matrix are subject to selection rules. The last ones are the semiconductor variant of the
well known selection rules on p in atomic systems. In case of the p-like
Γ8 valence states and the s-like Γ6 conduction states we concluded for the
|jz | = 23 valence states that the atomic-like matrix elements corresponding
to components of p parallel to the wire axis are polarization forbidden. Con108
sidering the |jz | = 12 states we found a ratio of 4 between the parallel and
perpendicular p components, respectively. Physically, these polarization selection rules are based on the conservation of angular momentum on the
atomic part of the transition matrix.
While the polarization selection rules originate from the bulk, atomic
like matrix element of the momentum operator, the selection rules on the
envelope part depend on the configuration of the system (wire dimensions,
wavelength of incident EM field). Starting with the envelope integral over
the coordinate z parallel to the wire axis, in Part I we found for infinite
cylinders that the EM field always is of the form E(r ⊥ )eiqz z and together
with the ∝ eikz z dependence of the envelope functions this results in the
conservation of momentum equation along the z-direction.
Considering the lateral envelope part of the transition matrix, a spatially
varying EM field requires a different approach than the dipole limit.
In the common dipole approximation, the valence and conduction subband wavefunctions (∝ eilz φ ) must have the same envelope angular momentum. In addition, this envelope angular momentum conservation naturally
leads to parity selection at the wire zone center: transitions between states
with different envelope parity are not allowed. Away from the zone center
this is the only selection rule which is broken since the valence subband
wavefunctions are not characterized by parity any more. We calculated the
band-to-band transitions between the first 7 valence subbands and the lowest Γ6 conduction subband for InP and InAs at different wire radii R. First
of all, while the transition energies strongly depend on the wire radius, the
matrix elements are independent of R, approximatively even if the finite
confinement for the conduction subband wavefunctions would be taken into
account. Secondly, comparing the results of InAs with InP it can be concluded that next to the energy of a particular transition, also its strength
is material dependent. Finally we conclude that the polarization anisotropy
of the transition matrix is completely determined by the polarization selection rules. At all kz , the transitions from |fz | = 23 valence subbands to the
lowest lz = 0 conduction subband are strictly forbidden for parallel polarization and for |fz | = 12 the transitions to the lz = 0 subband have a fixed
polarization anisotropy of 0.6. These observations compare favorably with
the results derived in an atomistic, empirical pseudo-potential plane-wave
method[4]. The deviations can be explained by the possible corrections of
including the split-off band, or even diagonalizing the full 8 × 8 Hamiltonian
of the three bulk Γ8 valence bands and Γ6 conduction band in the present
effective mass approximation.
One of the purposes of this paper was to estimate the effect of classical
scattering for wire dimensions in the quantum confinement regime. Generally, a spatially varying EM field breaks the symmetry in the envelope part
of the transition matrix. We analyzed this by including the Mie scattering
terms found in Part I into the envelope integral. Although the qualitative
109
features indeed are different, since new peaks arize which are parity forbidden in the wire dipole limit, we conclude that the order of magnitude of
these kind of corrections is too small to overcome the parity selection rule
at kz = 0 significantly. The transitions that become weakly allowed are
weaker by a factor ∼ 10−5 for InP and InAs and wire radii up till 15 nm.
A second effect is that the strength of transitions which are already allowed
in the wire dipole limit changes, but also this effect is not large. For a InP
nanowire with a radius of 10 nm the Mie correction on these already allowed
states is about 5 %. Consequently, the effect of classical Mie scattering can
be neglected in the quantum regime.
Finally, in Chapter 5 we analyzed the full optical absorption process in
cylindrical nanowires by deriving an expression for the nanowire dielectric
function including the quantum confinement effects. Hereby we made a first
order approximation by assuming that the local field acting on a particular
electron at a lattice site is the same as the averaged field obtained from the
macroscopic Maxwell equations. Thus, we neglected the additional internal
field due to the induced polarization of the neighboring atoms. It is noted
that the inclusion of such a Lorentz correction has to be reconsidered within
a microscopic, semiclassical approach, when dealing with nanoscale systems
which are not ”microscopically large” any more: the resonant states are
extended over the whole volume by the envelope part of the wavefunctions
and consequently the induced polarization is position dependent.
As a first step, however, the dielectric function is approximated by a
constant and, based on the results in Chapter 4, the EM field can be considered in the dipole limit. In this framework we used the advantages of the
effective mass approach by deriving a simplified expression for the imaginary part of the nanowire dielectric function, in which the integral over kz is
replaced by an integral over the energy utilizing an explicit expression for
the 1D density of states.
In this paper we focussed on the band-to-band transitions close to the
band gap of III-V materials. In order to include also the dielectric response of all other transitions, a background dielectric function εbg is defined
which is basically the bulk dielectric function εbulk , in which the group of
band-to-band transitions is projected out. As a first approximation we took
εbg ' εbulk and neglected the quantum confinement contribution of the group
band-to-band transitions to the real part of the total dielectric function.
By making some rough simplifications, for instance neglecting the kz
dependence of the transition matrix and inserting line broadening of the
absorption peaks by hand, we obtained successfully the qualitative behavior
of the nanowire dielectric function. In agreement with recent literature
[4][5], we find that the dielectric response of the band-to-band transitions
is strongly polarization dependent, which completely relies on the results
of Chapter 4. Furthermore, comparing InP with InAs it is concluded that
110
the confinement has a significantly smaller effect on the dielectric function
of InP, which is explained by the larger bulk band gap of InP resulting in
smaller band-to-band transition probabilities.
Compared to the bulk polarization anisotropy due to the classical dielectric mismatch, which is almost a constant in the region of interest, it
is concluded that the polarization anisotropy due to quantum confinement
causes giant changes in the total polarization anisotropy of the wire. In
addition, by including the dielectric background the effect of quantum confinement disappears in a natural way if R is increased sufficiently.
After all, we have thus established that the effective mass approach provides a fast and flexible tool to analyze the diameter dependent properties
of nanowires for a wide range of semiconductor materials. Possible improvements of the current framework are achieved if the Γ8 spin-off band is
included, or eventually diagonalizing the full 8 × 8 Hamiltonian of the three
bulk valence bands and Γ6 conduction band in the present effective mass
approximation.
111
A Hole wavefunctions for
different kz
112
fz = 12 , kz R = 0. , H-L 1
fz = 12 , kz R = 0.125 , H-L 1
1.5
1
1
0.5
0.5
Χ jz
Χ jz
1.5
0
-0.5
0
-0.5
jz = 3 2
jz = 1 2
-1
0
0.2
0.4
0.6
jz = -1 2
jz = -3 2
0.8
jz = 3 2
jz = 1 2
-1
1
0
0.2
ѐR
fz = 12 , kz R = 0.25 , H-L 1
1
fz = 12 , kz R = 0.375 , H-L 1
1
1
0.5
0.5
Χ jz
Χ jz
0.8
1.5
0
-0.5
0
-0.5
jz = 3 2
jz = 1 2
-1
0
0.2
0.4
0.6
jz = -1 2
jz = -3 2
0.8
jz = 3 2
jz = 1 2
-1
1
0
0.2
ѐR
0.4
0.6
jz = -1 2
jz = -3 2
0.8
1
ѐR
fz = 12 , kz R = 0.5 , H-L 1
1.5
fz = 12 , kz R = 0.75 , H-L 1
1.5
1
1
0.5
0.5
Χ jz
Χ jz
0.6
ѐR
1.5
0
-0.5
0
-0.5
jz = 3 2
jz = 1 2
-1
0
0.2
0.4
0.6
jz = -1 2
jz = -3 2
0.8
jz = 3 2
jz = 1 2
-1
1
0
0.2
ѐR
0.4
0.6
jz = -1 2
jz = -3 2
0.8
1
ѐR
fz = 12 , kz R = 1. , H-L 1
1.5
fz = 12 , kz R = 1.125 , H-L 1
1.5
1
1
0.5
0.5
Χ jz
Χ jz
0.4
jz = -1 2
jz = -3 2
0
-0.5
0
-0.5
jz = 3 2
jz = 1 2
-1
0
0.2
0.4
0.6
ѐR
jz = -1 2
jz = -3 2
0.8
jz = 3 2
jz = 1 2
-1
1
0
0.2
0.4
0.6
jz = -1 2
jz = -3 2
0.8
ѐR
Figure 9: Radial part of the |fz | = 21 , − (1) hole envelope wavefunctions
for InAs. The value of kz R changes from 0 in the first picture to the
maximum value 1.125 (at the end of the band) in the last graph.
113
1
fz = 12 , kz R = 0. , H+L 1
fz = 12 , kz R = 0.166667 , H+L 1
1.5
1
1
0.5
0.5
Χ jz
Χ jz
1.5
0
-0.5
0
-0.5
jz = 3 2
jz = 1 2
-1
0
0.2
0.4
0.6
jz = -1 2
jz = -3 2
0.8
jz = 3 2
jz = 1 2
-1
1
0
0.2
ѐR
fz = 12 , kz R = 0.333333 , H+L 1
1
1
0.5
Χ jz
Χ jz
1
fz = 12 , kz R = 0.666667 , H+L 1
0.5
0
0
-0.5
jz = 3 2
jz = 1 2
-1
0
0.2
0.4
0.6
jz = -1 2
jz = -3 2
0.8
jz = 3 2
jz = 1 2
-1
1
0
0.2
ѐR
0.4
0.6
jz = -1 2
jz = -3 2
0.8
1
ѐR
fz = 12 , kz R = 1. , H+L 1
1.5
fz = 12 , kz R = 1.33333 , H+L 1
1.5
1
0.5
Χ jz
1
0.5
0
-0.5
0
-0.5
jz = 3 2
jz = 1 2
-1
0
0.2
0.4
0.6
jz = -1 2
jz = -3 2
0.8
jz = 3 2
jz = 1 2
-1
1
0
0.2
ѐR
0.4
0.6
jz = -1 2
jz = -3 2
0.8
1
ѐR
fz = 12 , kz R = 1.66667 , H+L 1
1.5
fz = 12 , kz R = 2.03333 , H+L 1
1.5
1
0.5
Χ jz
1
0.5
0
-0.5
0
-0.5
jz = 3 2
jz = 1 2
-1
0
0.2
0.4
0.6
ѐR
jz = -1 2
jz = -3 2
0.8
jz = 3 2
jz = 1 2
-1
1
0
0.2
0.4
0.6
ѐR
Figure 10: Radial part of the |fz | = 12 , + (1) hole envelope wavefunctions
for InP. The value of kz R changes from 0 in the first picture to the
maximum value 2.033 (at the end of the band) in the last graph.
114
0.8
1.5
-0.5
Χ jz
0.6
ѐR
1.5
Χ jz
0.4
jz = -1 2
jz = -3 2
jz = -1 2
jz = -3 2
0.8
1
fz = 12 , kz R = 0. , H-L 1
fz = 12 , kz R = 0.166667 , H-L 1
1.5
1
1
0.5
0.5
Χ jz
Χ jz
1.5
0
-0.5
0
-0.5
jz = 3 2
jz = 1 2
-1
0
0.2
0.4
0.6
jz = -1 2
jz = -3 2
0.8
jz = 3 2
jz = 1 2
-1
1
0
0.2
ѐR
fz = 12 , kz R = 0.333333 , H-L 1
1
fz = 12 , kz R = 0.666667 , H-L 1
1
1
0.5
0.5
Χ jz
Χ jz
0.8
1.5
0
-0.5
0
-0.5
jz = 3 2
jz = 1 2
-1
0
0.2
0.4
0.6
jz = -1 2
jz = -3 2
0.8
jz = 3 2
jz = 1 2
-1
1
0
0.2
ѐR
0.4
0.6
jz = -1 2
jz = -3 2
0.8
1
ѐR
fz = 12 , kz R = 1. , H-L 1
1.5
fz = 12 , kz R = 1.33333 , H-L 1
1.5
1
1
0.5
0.5
Χ jz
Χ jz
0.6
ѐR
1.5
0
-0.5
0
-0.5
jz = 3 2
jz = 1 2
-1
0
0.2
0.4
0.6
jz = -1 2
jz = -3 2
0.8
jz = 3 2
jz = 1 2
-1
1
0
0.2
ѐR
0.4
0.6
jz = -1 2
jz = -3 2
0.8
1
ѐR
fz = 12 , kz R = 1.66667 , H-L 1
1.5
fz = 12 , kz R = 2.76667 , H-L 1
1.5
1
1
0.5
0.5
Χ jz
Χ jz
0.4
jz = -1 2
jz = -3 2
0
-0.5
0
-0.5
jz = 3 2
jz = 1 2
-1
0
0.2
0.4
0.6
ѐR
jz = -1 2
jz = -3 2
0.8
jz = 3 2
jz = 1 2
-1
1
0
0.2
0.4
0.6
jz = -1 2
jz = -3 2
0.8
ѐR
Figure 11: Radial part of the |fz | = 12 , − (1) hole envelope wavefunctions
for InP. The value of kz R changes from 0 in the first picture to the
maximum value 2.767 (at the end of the band) in the last graph.
115
1
B Polarization selection rules
116
1
m0
hS ↑ | ε̂ · p | 32
3
2
i
ε̂x
ε̂y
ε̂z
propagation k to êx
impossible
Π
√
2
polF
propagation k to êy
−i √Π2
impossible
polF
propagation k to êz
−i √Π2
Π
√
2
impossible
Table 1: Selection rules on the atomic-like interband matrix elements
1
3 3
m0 hS ↑ | ε · p | 2 2 i. The propagation direction of the EM-wave is
denoted in the left column. The unit directions of E are denoted with εx ,
εy and εz ; Π is related to the Kane matrix element Ep by Ep = 2m0 Π2 .
The polarization forbidden transitions are denoted with polF .
1
m0
hS ↑ | ε̂ · p | 32
1
2
i
ε̂x
ε̂y
ε̂z
propagation k to êx
impossible
polF
2Π
i√
6
propagation k to êy
polF
impossible
2Π
i√
6
propagation k to êz
polF
polF
impossible
Table 2: Selection rules on the atomic-like interband matrix elements
3 1
1
m0 hS ↑ | ε̂ · p | 2 2 i.
1
m0
hS ↑ | ε̂ · p | 32 − 12 i
ε̂x
ε̂y
ε̂z
propagation k to êx
impossible
Π
√
6
polF
propagation k to êy
i √Π6
impossible
polF
propagation k to êz
i √Π6
Π
√
6
impossible
Table 3: Selection rules on the atomic-like interband matrix elements
1
1
3
m0 hS ↑ | ε̂ · p | 2 − 2 i.
117
1
m0
hS ↑ | ε̂ · p | 32 − 32 i
ε̂x
ε̂y
ε̂z
propagation k to êx
impossible
polF
polF
propagation k to êy
polF
impossible
polF
propagation k to êz
polF
polF
impossible
Table 4: Selection rules on the atomic-like interband matrix elements
1
3
3
m0 hS ↑ | ε̂ · p | 2 − 2 i.The propagation direction of the EM-wave is
denoted in the left column. The unit directions of E are denoted with εx ,
εy and εz ; Π is related to the Kane matrix element Ep by Ep = 2m0 Π2 .
The polarization forbidden transitions are denoted with polF .
118
C Interband matrix elements
119
ÈΡ T v c È
1.6
1.62 1.65
bL : kzR = 0.93
v7 -> c1
z - pol.
y - pol.
v7 -> c1
aL : kzR = 0.
1.55 1.58
v4 -> c1
v5 -> c1
v6 -> c1
1.62 1.65
v1 -> c1
v3 -> c1
v2 -> c1
1.6
v5 -> c1
v6 -> c1
v4 -> c1
1.53 1.55 1.58
0.14
0.12
0.10
0.08
0.06
0.04
0.02
v1 -> c1
v2 -> c1
v3 -> c1
ÈTv c È2 Harb. unitsL
ETrans HeVL, R = 4.83 nm
1.00
1.00
0.80
0.80
0.60
0.60
Ρ>0
Ρ<0
0.40
0.40
0.20
0
1.53 1.55 1.58
0.20
1.6
1.62 1.65
1.55 1.58
0
1.6
1.62 1.65
ETrans HeVL, R = 4.83 nm
Figure 12: Matrix elements |Tcv,k |2 and |Tcv,⊥ |2 and corresponding polarization anisotropy ρTcv of the first 7 transitions vi → c1 for InP, calculated including the scattering terms in the EM field at R = 4.85 nm.
The corresponding energy scale is given at the top and bottom of the
figure.
120
0.14
0.12
0.10
0.08
0.06
0.04
0.02
ÈΡ T v c È
1.46 1.46 1.47 1.47 1.48
v7 -> c1
v4 -> c1
v5 -> c1
v6 -> c1
bL : kzR = 0.93
v1 -> c1
v3 -> c1
v2 -> c1
v7 -> c1
v5 -> c1
v6 -> c1
v4 -> c1
1.45 1.46 1.46 1.47 1.47 1.48
0.14
aL : kzR = 0.
0.12
0.10
0.08
z - pol.
0.06
y - pol.
0.04
0.02
v1 -> c1
v2 -> c1
v3 -> c1
ÈTv c È2 Harb. unitsL
ETrans HeVL, R = 10. nm
0.14
0.12
0.10
0.08
0.06
0.04
0.02
1.00
1.00
0.80
0.80
0.60
0.60
0.40
Ρ>0
Ρ<0
0.40
0.20
0.20
0
1.45 1.46 1.46 1.47 1.47 1.48
1.46 1.46 1.47 1.47 1.48
0
ETrans HeVL, R = 10. nm
Figure 13: Matrix elements |Tcv,k |2 and |Tcv,⊥ |2 and corresponding polarization anisotropy ρTcv of the first 7 transitions vi → c1 for InP,
calculated including the scattering terms in the EM field at R = 10 nm.
121
D Reference articles
122
Figure 14: Califano and Zunger [4], pg. 6. In FIG. 3 the interband
matrix elements for a R = 4.8 nm InAs wire are shown based on an
atomistic, empirical pseudopotential plane-wave method. The C∞ v representations given in Table II differ from those derived in the present
123
paper.
Figure 15: Persson and Xu [5], pg. 3. In FIG. 3 ε00w is shown for a
R = 2.9 nm InAs wire based on an atomistic tight-binding approach.
124
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