Fine structure and size dependence of exciton and bi-exciton optical spectra in CdSe nanocrystals

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Fine structure and size dependence of exciton and bi-exciton optical spectra in CdSe
nanocrystals
Marek Korkusinski, Oleksandr Voznyy, and Pawel Hawrylak
Quantum Theory Group, Institute for Microstructural Sciences,
National Research Council, Ottawa, Canada, K1A0R6
Theory of electronic and optical properties of exciton and bi-exciton complexes confined in CdSe
spherical nanocrystals is presented. The electron and hole states are computed using atomistic
sp3d5s∗ tight binding Hamiltonian including an effective crystal field splitting, spin-orbit interac-
tions, and model surface passivation. The optically excited states are expanded in electron-hole
configurations and the many-body spectrum is computed in the configuration-interaction approach.
Results demonstrate that the low-energy electron spectrum is organized in shells (s, p, . . . ), whilst
the valence hole spectrum is composed of four low-lying, doubly degenerate states separated from the
rest by a gap. As a result, the bi-exciton and exciton spectrum is composed of a manifold of closely
lying states, resulting in a fine structure of exciton and bi-exciton spectra. The quasi-degenerate
nature of the hole spectrum results in a correlated bi-exciton state, which makes it slowly convergent
with basis size. We carry out a systematic study of the exciton and bi-exciton emission spectra as
a function of the nanocrystal diameter and find that the interplay of repulsion between constituent
excitons and correlation effects results in a change of the sign of bi-exciton binding energy from
negative to positive at a critical nanocrystal size.
PACS numbers: 78.67.Hc,78.67.Bf,73.21.La,71.35.-y
I. INTRODUCTION
Semiconductor nanocrystals (NCs) (Refs. 1–12) are
nano-sized crystalline particles with numbers of atoms
of the order of 102-105. NCs with controlled and tun-
able sizes as well as good optical properties are fabri-
cated in a colloidal growth process.13 This makes them
excellent candidates for use in low-cost optoelectronic
applications, including solar cells, biomarkers,14–16 light
emitting diodes,17–19 photodetectors,20,21 single-photon
sources in quantum cryptography,22 or lasers.23,24 In par-
ticular, it has been recently demonstrated that the opti-
cal gain in NCs can be blocked, created, and tuned by en-
gineering the NC confinement25 or the type of multiexci-
ton complex active in the stimulated emission process.26
The NCs are considered as a promising material for
the optically active media in solar cells. They of-
fer a potential way to increase the efficiency of so-
lar cells due to their tunable parameters amenable to
optimization,27–31 as well as by generation of multi-
exciton complexes (MEG) following absorption of a single
high-energy photon.7,9,32–37 During MEG a high-energy
photon with energy of at least twice the semiconduc-
tor bandgap, 2Eg, is absorbed creating an excited state,
which can be described as a superposition of config-
urations with one and more electron-hole pairs.32 Al-
ternatively, we can think of exciting a single exciton,
which is then converted via Coulomb interactions into
additional interacting electron-hole pairs. Energy relax-
ation of these multi-exciton complexes results in mul-
tiple carriers at the bottom of the conduction and the
top of the valence bands. These multi-exciton states de-
cay into exciton states by Auger processes, limiting the
number of additional charges generated in the MEG pro-
cess. The process of conversion of a single exciton into
multiple electron-hole pairs competes with the phonon-
assisted relaxation of exciton energy.37 Since the origi-
nal report by Schaller and Klimov,38 the MEG process
has been reported in PbSe, PbS, PbTe, CdSe, InAs, and
Si NCs,39 with efficiency reaching 700% (seven electron-
hole pairs out of one photon).40 However, proper as-
sessment of the MEG efficiency in these experiments is
nontrivial.11,36,41 The potential explanation of MEG has
been given by Shabaev, Efros, and Nozik42 and alterna-
tive interpretation proposed by Zunger and co-workers35
and others.43–45
The lowest-energy MEG process involves conversion of
an excited exciton into a low-energy bi-exciton following
absorption of a photon with energy of ∼ 2Eg. Therefore,
a detailed study of the electronic and optical properties
of the bi-exciton is needed. To date, theoretical attention
has been focused mainly on the properties of low-energy
exciton states, with the CdSe NCs being the most studied
system. The electronic and optical properties of an exci-
ton (X) confined in a NC have been explored utilizing the
multi-band k · p method,6,46, tight-binding47–50 and em-
pirical pseudopotential methods.51–53 These studies show
a fine structure in the low-energy states of X originating
from the electron-hole exchange, with the energy gap be-
tween the lowest - dark, and the higher - bright states of
the order of several meV.
Identification of bi-exciton (XX) signatures in emission
spectra is complicated by the presence of the inhomoge-
neous broadening in the ensemble measurements on NCs.
However, one can measure consistently the XX binding
energies both in ensemble measurements in CdSe54–57
and CdS NCs.25 First single-NC experiments have also
been reported.58 In particular, the exciton fine structure
has been studied in individual NCs as a function of the
magnetic field.59,60 Thus, the state of experimental tech-

2niques is approaching that in epitaxially grown quantum
dots, for which single-dot experiments, revealing details
of the fine structure of multiexciton complexes, are now a
standard.61 One of experimental tools utilized to obtain
spectroscopic information about bi-excitons confined in
NCs, the transient absorption, involves measuring with
a short probe pulse the change of absorption induced by
the pump laser pulse.1,54 By utilizing the transient ab-
sorption technique one can probe states of XX via both
emissive and absorptive experiments, which opens a pos-
sibility of probing the fine structure of XX directly.54,62,63
The experimental results of Ref. 54 obtained on CdSe
dots with the diameter of 5.6 nm have been compared
to the results of empirical pseudopotential calculations
carried out on dots with diameter of 3.8 and 4.6 nm.
However, to our knowledge, no systematic study of the
dependence of the bi-exciton spectra on the parameters of
the CdSe NCs have been carried out.64 The quantitative
analysis of these systems is computationally challenging
due to the NC size. With ∼ 105 electrons, NCs are too
large for ab initio methods. On the other hand, the k · p
methods are not accurate enough to capture important
atomistic details, such as the asymmetry of the crystal
lattice or the surface effects. This necessitates the use of
the semi-empirical atomistic methods in the theoretical
analysis.
Here we utilize the atomistic tight-binding approach
to perform a systematic study of the electronic and op-
tical properties of an X and XX confined in a single
CdSe NC as a function of NC size. We illustrate our
calculations on a spherical NC with the diameter of 3.8
nm for comparison with the empirical pseudopotential
of Ref. 54. To this end we utilize the QNANO compu-
tational platform.65 The atomistic single particle states
are used in computation of the Coulomb matrix elements,
describing the carrier-carrier interactions, and the opti-
cal dipole elements. The many-body multi-exciton states
are computed using exact diagonalization techniques.
The results show the s and p shells in the low-energy
electron spectrum as expected from a single-band effec-
tive mass theory. For holes we find a complex spectrum,
consisting of a band of four Kramers doublets forming
a quasi-degenerate hole shell separated from the remain-
ing hole levels by a gap. The energy separation of these
states is much smaller than the characteristic Coulomb
hole-hole interaction matrix elements. Therefore we pre-
dict the bi-exciton (XX) spectrum to be composed of a
manifold of closely lying correlated states of two electrons
residing mainly on the s-shell and a correlated complex
of two holes occupying almost degenerate hole states, re-
sulting in a fine structure of bi-exciton optical spectra.
The exciton (X) spectrum, on the other hand, reveals
the fine structure determined both by the hole shell de-
generacy and the electron-hole exchange interaction. We
find that the correlated character of both the X and XX
systems makes the computations challenging, with large
basis sizes necessary to obtain converged values of their
energies. In this work we discuss how this fine structure
influences the absorption and emission spectra of both X
and XX complexes. We find that for small NCs (with
diameter below 4 nm) the bi-exciton is unbound, whilst
for larger NCs it is bound. Also, the order of X and XX
emission peaks with model inhomogeneous broadening
depends on temperature. We show that due to details
of the electronic structure and assignment of oscillator
strengths, the thermal population of excited XX states
leads to a shift of the inhomogeneously broadened XX
peak to lower energies, whilst the analogous process leads
to the increase of the X emission energy. The shifts are
of the order of tens of meV and may lead to the reversal
of the order of emission peaks.
II. MODEL
We analyze the electronic and optical properties of
electrons and holes confined in a single, spherical CdSe
nanocrystal. The calculations are carried out utilizing
the QNANO computational platform and consist of the
following steps: (i) the definition of the geometry and
composition of the nanostructure on the atomistic level,
(ii) the calculation of single-particle quasi-electron and
quasi-hole states using the 20-band sp3d5s∗ tight-binding
(TB) model, (iii) the computation of many-body ener-
gies and states of N quasi-electron-quasi-hole pairs in
the configuration-interaction (CI) approach (in this case
N = 1 and 2), and (iv) calculation of emission and ab-
sorption spectra using Fermi’s Golden Rule. A detailed
review of the QNANO package and the computational
procedure is given in Ref. 65.
A. Atomistic tight-binding description of a
nanocrystal
The computational procedure starts with a definition
of the positions of all atoms present in the system. The
underlying crystal lattice of the CdSe nanocrystal is
taken to be in wurtzite modification, which is built out
of two hexagonal closely packed (hcp) sublattices, one
made up of cations and another of anions, shifted with
respect to one another. As a result, each atom is sur-
rounded by four nearest neighbors. The hcp structure
is described by two lattice parameters, a and c, which,
in principle, are independent. In this work we assume
however that the nearest neighbors of each atom form a
perfect tetrahedron. This relates the two lattice parame-
ters with one another such that we have a =
√
3
8c. With
c = 0.70109 nm (Ref. 66), this gives a = 0.42933 nm,
as compared to the experimental value of 0.42999 nm. If
one parametrizes all the distances with the lattice con-
stant c, the positions of the four atoms in the wurtzite
unit cell are as follows: we have two anions, in our case
Selenium, at (0, 0, 0) and (
√
6/8,
√
2/8, 1/2)c, and two
cations, in our case Cadmium, at (
√
6/8,
√
2/8, 1/8)c and

3(0, 0, 5/8)c.
Calculation of the single-particle states is carried out
in the linear combination of atomic orbitals (LCAO) ap-
proximation, in which the carrier wave function is written
as a linear combination
Ψi (~r) =
NAT
∑
R=1
20
∑
α=1
A(i)Rαϕα
(
~r − ~R
)
(1)
of atomistic orbitals of type α localized on the atom R,
with NAT being the total number of atoms in the sys-
tem. In our sp3d5s∗ model we deal with 10 doubly spin-
degenerate basis orbitals on each atom. The coefficients
A(i)Rα determining the ith single-particle state as well as
the corresponding single-particle energies are found by
diagonalizing the semi-empirical atomistic TB Hamilto-
nian
HTB =
NAT
∑
R=1
20
∑
α=1
εRαc+RαcRα +
NAT
∑
R=1
20
∑
α=1
20
∑
α′=1
λRαα′c+RαcRα′
+
NAT
∑
R=1
nn
∑
R′=1
20
∑
α=1
20
∑
α′=1
tRα,R′α′c+RαcR′α′ , (2)
in which the operator c+Rα (cRα) creates (annihilates) the
particle on the orbital α of atom R. The Hamiltonian
is parametrized by the on-site orbital energies εRα, spin-
orbit coupling constants λRαα′ , and hopping matrix el-
ements tRα,Rα′ connecting different orbitals located at
neighboring atoms. In our model we use the nearest
neighbor approximation, and therefore do not capture
directly the crystal field splitting, which is due to the
symmetry breaking on the level of third nearest neigh-
bors. Following Ref. 47 we include the crystal field split-
ting in an approximate manner by detuning the energy
of the orbital pz from that of the orbitals px, py which
remain degenerate. The TB parameters are obtained by
calculating the band structure of bulk CdSe and fitting
the band edges and effective masses at high symmetry
points of the Brillouin zone to the values obtained exper-
imentally or by ab initio calculations. The parametriza-
tion used in this work is given in Table I. Using this
parametrization we obtain the following parameters of
the bulk band structure. The bandgap Eg = 1.83 eV
(we fit to the low-temperature data), the crystal field
splitting ECFS = 0.0254 eV and the spin-orbit split-
ting ∆SO = 0.444 eV correspond closely to the exper-
imental values of Eg = 1.83 eV, ECFS = 0.026 eV
and ∆SO = 0.429 eV (Ref. 66). The electron effective
masses are m∗e(M) = 0.133m0 towards the M point, and
m∗e(A) = 0.134m0 towards the A point, while the mea-
sured value for both directions is m∗e = 0.13m0. In the
highest valence subband, the effective mass towards the
M point is m∗h1(M) = 0.455m0, while the measured value
is 0.45m0. In the same subband the mass towards the A
point is m∗h1(A) = 1.443m0 while the measured value
is 1.17m0. Finally, in the second valence subband we
compute the effective mass towards the M point to be
TABLE I: Tight-binding parameters for CdSe used in this
work. All values are in eV, and the notation follows that of
Slater and Koster
Parameter Value
Eas −10.9438
Eapx,py 1.3131
Eapz 1.2795
Ead 6.9721
Eas∗ 7.5610
λaSO 0.1307
Ecs 0.7855
Ecpx,py 4.7247
Ecpz 4.6844
Ecd 6.4424
Ecs∗ 6.4704
λcSO 0.1568
Vss −0.9470
Vsa,pc 2.6220
Vpa,sc 1.8608
Vppσ 3.1287
Vpppi −0.5674
Vsa,s∗c −0.0001
Vs∗a,sc −0.1685
Vpa,s∗c 0.4694
Vs∗a,pc 0.0004
Vs∗,s∗ −0.0937
Vsa,dc −0.0649
Vda,sc −0.0079
Vpa,dcσ −0.0137
Vda,pcσ −0.0005
Vpa,dcpi 0.0053
Vda,pcpi 0.0004
Vs∗a,dc −0.0748
Vda,s∗c −0.0121
Vddσ −0.0007
Vddpi 0.1479
Vddδ −0.1834
m∗h2(M) = 0.851m0 which is close to the experimental
value of 0.9m0.
Figure 1 shows the band structure of CdSe bulk com-
puted with the above TB parameters (a) compared to
the band structure obtained in DFT calculation (b), in
which the conduction band was rigidly shifted by 1.562
eV to reproduce the experimental value of the gap. Note
that in our parametrization the on-site energies of d or-
bitals on both the anion and the cation are above the
energies of orbitals s and p. This is in contrast to several
other parametrizations accounting for the d orbitals,67–69
where the cation d orbitals lie below the s and p orbitals.
In such parametrizations it is possible to reproduce the
flat d-band visible in Fig. 1(b) at the energy of about −8
eV. Since all our d orbitals lie high in energy, in our bulk
band structure in Fig. 1(a) the d-band is not present.
Our choice of the placement of d orbitals was dictated by
the fact that, according to the GW calculations, the ad-
mixture of the low-lying d orbitals in the wave functions
corresponding to the top of the valence band in the Γ
point is negligible.70 Since we set out to study the prop-

4erties of several lowest exciton and bi-exciton states, we
concentrate on an accurate reproduction of band edges
rather than deeper bands. Moreover, the small size of our
NCs necessitates an accurate description of the conduc-
tion band across the Brillouin zone, which in turn entails
the use of the high-energy s∗ orbitals. These orbitals are
taken to have higher on-site energies than the respective
high-energy d orbitals on both the cation and anion. We
thus have to account for all these high-lying orbitals and
neglect the low-energy cation d-band in order to treat
both types of atoms on equal footing.
Figure 2 shows the bulk density of states (DOS) com-
puted using three methods: the result of the density
functional (DFT) LCAO calculation using the SIESTA
package72 (top panel), the plane-wave approach used in
the package “Exciting”71 (middle panel), and the DOS
resulting from our TB approach (bottom panel). All
three panels shows the DOS within the energy range of
two gap energies into the valence and conduction bands,
i.e., the range of energies of interest for the multiexciton
generation process. Thus, our TB model gives results
consistent with the two other, ab initio approaches up to
3 eV into each of the conduction and valence bands.
The TB Hamiltonian in the above parametrization is
used to compute the single-particle states in a spherical
nanocrystal. The positions of all atoms in such a sys-
tem are determined by cutting a spherical sample out of
a bulk semiconductor, without any surface relaxation ef-
fects. The dangling bonds on the surface of the nanocrys-
tal are passivated by the procedure involving the follow-
ing steps: (i) rotation from the s − px − py − pz basis
to that of sp3 hybridized orbitals, (ii) identification of
the directions of resulting bonds and application of an
energy shift of 25 eV to those that are unsaturated, and
(iii) inverse rotation into the s − px − py − pz basis.73
B. Description of interacting electrons and holes
confined in the nanocrystal
The excited states of the NC are expanded in electron
and hole pair configurations. The electrons are defined
as occupied states in the conduction band and holes as
empty states in the valence band. With the operator
c+i (ci) creating (annihilating) an electron on the single-
particle state i, while the operator h+α (hα) creating (an-
nihilating) a hole on the single-particle state α, the ex-
cited states |ν〉 are written as
|ν〉 =
∑
i,α
Bνi,αc+i h+α |0〉+
∑
i,j,α,β
Cνi,j,α,βc+i c+j h+αh+β |0〉+. . . ,
(3)
where |0〉 is the ground state of the NC. The amount of
mixing among the configurations with different number
of excitations is defined by the amplitudes Bνi,α, Cνi,j,α,β
and depends on the energy of the state. The ground
exciton state, whose energy is of order of the semicon-
ductor gap Eg, will be built predominantly out of single
pair excitations, with a negligible contribution from the
two-pair (energy at least of order of 2Eg) or higher con-
figurations. On the other hand, the two-pair excitations
may be mixed with highly excited single-pair configura-
tions with similar energies. In this work we shall treat
the number of quasi-particles as a good quantum number
when labeling the states of the NC. A detailed analysis
of the mixing effects will be presented elsewhere.
The Hamiltonian of interacting Ne electrons and Nh
holes distributed on the single-particle states is
H =
∑
i
εic+i ci +
∑
α
εαh+αhα +
1
2
∑
ijkl
〈ij|Vee|kl〉c+i c+j ckcl
+ 12
∑
αβγδ
〈αβ|Vhh|γδ〉h+αh+β hγhδ
−
∑
il
∑
βγ
(〈iβ|Veh|γl〉 − 〈iβ|Veh|lγ〉) c+i h+β hγcl. (4)
In Eq. (4) the first two terms account for the single-
particle energies, the third and fourth terms describe the
electron-electron and hole-hole Coulomb interactions, re-
spectively, and the last term introduces the electron-hole
direct and exchange interactions. The Coulomb matrix
elements are computed using the single-particle TB wave
functions. In these computations we separate (i) the on-
site terms arising from the scattered particles residing on
the same atom, (ii) the nearest-neighbor (NN) terms in-
volving orbitals localized on adjacent atoms, and (iii) the
long-distance terms describing scattering between more
remote atoms. Using the general form of our LCAO wave
functions (1), each of these three elements can be written
as follows:
〈ij|Vee|kl〉 = Vons + VNN + Vlong, (5)
Vons =
NAT
∑
R=1
20
∑
αβγδ=1
A(i)∗Rα A
(j)∗
Rβ A
(k)
RγA
(l)
Rδ
× 〈Rα,Rβ| e
2
ǫons|~r1 − ~r2|
|Rγ,Rδ 〉, (6)
VNN =
NAT
∑
Ri=1
NN
∑
Rj
20
∑
αβγδ=1
A(i)∗RiαA
(j)∗
RjβA
(k)
RjγA
(l)
Riδ
× 〈Riα,Rjβ|
e2
ǫNN |~r1 − ~r2|
|Rjγ,Riδ 〉, (7)
Vlong =
NAT
∑
Ri=1
remote
∑
Rj
20
∑
αβ=1
A(i)∗RiαA
(j)∗
RjβA
(k)
RjβA
(l)
Riα
× e
2
ǫlong|~Ri − ~Rj |
. (8)
and analogously for the hole-hole and electron-hole in-
teractions. The necessary integrals in the on-site and
nearest-neighbor terms are computed by approximat-
ing the atomistic functions |R,α〉 by Slater orbitals.74
Note that in the above formulas we have assumed the

5two-center approximation. In an attempt to simulate
the distance-dependent dielectric function,52,75–78 each of
these terms is scaled by a different dielectric constant ǫ.
Typically we take ǫons = 1 and ǫlong = 5.8, the latter one
being the bulk CdSe dielectric constant, while ǫNN = 2.9,
i.e., half of the bulk CdSe value. In what follows we shall
present two computations, one with the nearest neighbor
term assuming the form identical to the remote term,
and another one, with the nearest-neighbor term as in
Eq. (7).
C. Calculation of optical spectra
Once the many-body states of the system of interact-
ing electron-hole pairs are established, we calculate the
emission spectra utilizing the Fermi’s Golden Rule
I (ω) =
∑
f,i
Pi(T ) |〈f |PX |i〉|2 δ (Ei − Ef − h¯ω) , (9)
where Ei is the energy of the initial state of N exci-
tons, Ef is that of the final state of N − 1 excitons,
and the sum is carried over all possible final states. The
temperature-dependent factor Pi describes thermal pop-
ulation of levels of the initial exciton complex. The tran-
sition intensity is determined by the interband polariza-
tion operator, which for the polarization x is defined as
PX =
∑
ij d
(x)
ij cihj . The single particle dipole elements
d(x)ij are defined as d
(x)
ij =
∫
d~rφ∗h,j (~r)xφe,i (~r). The po-
larization operators for polarizations y and z are defined
analogously. In our TB approach, the dipole matrix ele-
ments can be evaluated in the form:
d(x)ij =
NAT
∑
R=1
20
∑
α=1
A∗(j)Rα A
(i)
RαRx
+
NAT
∑
R
20
∑
αβ=1
A∗(j)Rα A
(i)
Rβ
∫
d~rφ∗α (~r)xφβ (~r). (10)
The integrals involving orbitals from the nearest and fur-
ther neighbors are neglected. Absorption spectra are ob-
tained using a formula analogous to Eq. (9), only the po-
larization operator PX is replaced by its Hermitian con-
jugate. In this case the initial state describes the system
of N excitons, with the appropriate thermal occupation
of levels, while the final state contains N+1 electron-hole
pairs.
III. SINGLE-PARTICLE STATES IN THE
SPHERICAL NANOCRYSTAL
Figure 3 illustrates the single-particle properties of a
spherical CdSe nanocrystal of diameter of 3.8 nm, whose
atomistic image is shown in the inset of Fig. 3(c). The
system consists of 1028 atoms, with the Cd (Se) atoms
rendered in blue (red). The energies of the single-particle
electron and hole states obtained in the tight-binding cal-
culation are shown in Figs. 3(a) and 3(b), respectively.
The structure of electron states is typical for a spherical
quantum confinement: the ground state of the s symme-
try is separated by a large gap (about 270 meV) from
three states of the p symmetry. For the valence holes,
however, we find four closely lying states, highlighted in
Fig. 3(b) by the blue rectangle, separated from the re-
mainder of the spectrum by a gap of about 120 meV.
This structure of the hole states is due to the interplay
of the spin-orbit interaction and the crystal field split-
ting. The characteristic gap is robust and appears also
for NCs with larger diameters, as illustrated in Fig. 3(c).
The existence of four closely-lying hole states appears to
be in agreement with results of earlier empirical pseu-
dopotential calculations.52,54
In the case of electron states, whose energies are shown
in Fig. 3(a), the p shell consists of three levels: almost
degenerate px and py states at a higher energy, and a
single non-degenerate pz level at a slightly lower energy.
This is a signature of the wurtzite symmetry of the NC,
which differentiates between the +z and −z directions,
leading to a corresponding asymmetry in the electron
wave function.
Insight into the symmetry of the four hole states em-
phasized in Fig. 3(b) can be gained by computing the
dipole matrix elements d(x)ij built out of the i-th elec-
tron and j-th hole states, with y and z matrix elements
constructed analogously. In Fig. 4(a) we plot the joint
optical density of states (JDOS), i.e., magnitude of dipole
elements |dij |2 versus the energy gap between the ground
electron (i = 1) and the four lowest hole states. For
polarizations x and y we obtain four nonzero elements,
while for polarization z the JDOS consists of only two
peaks. This structure of JDOS can be understood by ap-
proximating the atomistic wave functions as products of
the envelope and Bloch part, as is done in the k ·p model.
Since the envelope function changes slowly on interatomic
distances, one typically approximates the dipole element
by a product of the overlap of electron and hole envelope
functions and an integral involving the Bloch components
and the position operator appropriate for the x, y, or z
polarization. The electron ground state is built out of s-
type atomistic orbitals modulated by an s-type envelope,
while the hole states are built out of p-type atomistic or-
bitals. Due to the spin-orbit mixing the envelope func-
tions of the hole are mixtures of different symmetries.79
However, this projectional analysis will extract the part
of the envelope function of the hole which is of the same
symmetry as the electron envelope function (in this case,
symmetry s).
Using this approximation let us first analyze the lowest
(H1) and highest (H4) JDOS maxima. They are present
in the x and y polarizations, but absent in the z polariza-
tion. This means that the s-like term in the hole envelope
function is associated with the Bloch functions consist-
ing of px and py, but not pz atomic orbitals. The overlap
of the electron and hole envelope functions is large for