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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

6H1, but very small for H4, which indicates that the s-like
component dominates in the envelope function of the hole
ground state, while the state H4 is of a different symme-
try. The two middle JDOS peaks, H2 and H3, appear in
all polarizations, indicating that the Bloch components
of the corresponding hole functions are combinations of
all three atomistic p orbitals. Of those two, H2 is con-
sistently stronger than H3, which suggests that the hole
state H2 has a larger, and H3 - a smaller s-like compo-
nent.
Further confirmation of this assignment of symmetries
is obtained by computing the dipole matrix elements be-
tween the p electron states and the four hole states. This
procedure probes the p-like component in the hole enve-
lope functions. The elements are shown in Fig. 4(b) as
red, blue, and green bars, with the assignment of colors
explained in Fig. 4(c). We find that the ground state H1
gives a negligible dipole matrix element with either of the
three electron p states, which confirms that the state H1
is of the s type. On the other hand, the state H4 presents
large dipole elements, which indicates that it has a dom-
inating p-like component in its envelope function.
As previously with the s-type electron state, the two
middle peaks, H2 and H3, appear consistently in both
polarizations, but now H3 is larger. We conclude that
the hole states H2 and H3 are mixtures of s and p-type
envelopes. This assignment of symmetries is only ap-
proximate, as the details of the underlying crystal lattice
and surface roughness break the rotational symmetry of
the nanocrystal. However, a visual inspection of charge
densities of the hole states suggests a picture consistent
with the above analysis.
Note that all single-particle states are Kramers dou-
blets, whose degeneracy is due to the time-reversal sym-
metry of the single-particle Hamiltonian. In what follows
we shall distinguish the states forming the doublet by ar-
rows up and down, respectively. Due to the spin-orbit
interaction these labels cannot be identified with particle
spins, but rather with Bloch total angular momenta.
IV. EXCITONIC COMPLEXES IN THE
NANOCRYSTAL
A. Exciton
In order to find the energies and states of the exciton
(X) we generate all possible electron-hole configurations
in the single-particle basis, write the full Hamiltonian
(4) in a matrix form in the basis of these configurations,
and diagonalize this matrix numerically. Construction
of the Hamiltonian requires knowledge of the Coulomb
electron-hole scattering matrix elements. Typically one
distinguishes two types of Coulomb matrix elements: the
“direct” and the “exchange” ones, the latter originating
from the antisymmetric character of the many-body wave
function. This distinction is particularly clear in the case
of diagonal matrix elements, i.e., those arising when one
computes the expectation value of the Coulomb operator
against any configuration. In this case the direct terms
can involve pairs of particles with different spin, while the
exchange elements connect particles with the same spins.
Due to the spin-orbit interaction present in our TB model
the single-particle states cannot be characterized by a
definite spin. Moreover, as already mentioned, all single-
particle states are in reality Kramers doublets, and any
linear combination of the two constituent states is also
a good eigenstate of the TB Hamiltonian. In order to
be able to separate and analyze the Coulomb elements,
we perform a rotation of each pair of states forming the
Kramers doublet so as to optimize the expectation value
of the Pauli σz operator. With the states thus prepared
we compute the Coulomb elements using the formulas
(5).
Let us now comment on the magnitudes of various
Coulomb matrix elements for our NC with diameter of
3.8 nm. We will discuss these elements in two cases, de-
pending on the treatment of the nearest-neighbor contri-
butions: (i) the case when these contributions are com-
puted exactly using Slater orbitals, as shown in formula
(7), (ii) the case when they are expressed simply by the
formula (8) as for remote centers. In each case we scale
the nearest neighbor contribution by the dielectric con-
stant of 2.9, i.e., half of the CdSe bulk value.
If we denote the states composing the lowest-energy
electron doublet as |1e ↓〉 and |1e ↑〉, and the analo-
gous pair of hole states as |1h ↓〉 and |1h ↑〉, we find
the direct elements: 〈1e ↓ 1h ↓ |Veh|1h ↓ 1e ↓〉 = 〈1e ↓
1h ↑ |Veh|1h ↑ 1e ↓〉 = 212.76 meV in the case (i),
and 220.52 meV in the case (ii). These elements define
the interaction energy of an electron-hole configuration
c+1↓h+1↓|0〉 and c+1↓h+1↑|0〉, respectively, where |0〉 denotes
quasi-particle vacuum. The “spin-flip” electron scatter-
ing, described, e.g., by an element 〈1e ↓ 1h ↓ |Veh|1h ↓
1e ↑〉, is not possible, as the value of this element is neg-
ligibly small. However, due to the much stronger spin
mixing of the hole states resulting from the spin-orbit in-
teraction one might expect that the transitions involving
the hole spin flip should be possible. In fact, here the
only elements of note are 〈1e ↓ 1h ↓ |Veh|4h ↑ 1e ↓〉 =
〈1e ↓ 1h ↑ |Veh|4h ↓ 1e ↓〉 = 0.26 meV in the case (i), and
0.30 meV in the case (ii).
The scattering among the hole states, with the elec-
tron staying on the same level and without hole spin flip,
〈1e ↓ 1h ↓ |Veh|2h ↓ 1e ↓〉 and 〈1e ↓ 1h ↓ |Veh|3h ↓ 1e ↓〉
is very small. However, the scattering onto the fourth
hole Kramers doublet, 〈1e ↓ 1h ↓ |Veh|4h ↓ 1e ↓〉, is much
larger and its absolute value amounts to 5.2 meV in the
case (i) and 5.8 meV in the case (ii). Such an element
describes the Coulomb coupling between configurations
c+1↓h+1↓|0〉 and c+1↓h+4↓|0〉. Also, scattering with hole trans-
fer between the second and third Kramers doublet is size-
able and amounts to about 8.02 meV in the case (i) and
8.86 meV in the case (ii). The energy scales set by these
Coulomb elements are to be compared with the energy
separation of the hole states, which ranges from about 5

7to about 15 meV. Thus, the diagonal Coulomb electron-
hole terms, i.e., those that do not lead to a change of
the electron-hole configuration, are about 20 times larger
than the separation of the hole states. On the other hand,
the scattering elements, describing the change of configu-
ration, are approximately of the same order as this sepa-
ration. Therefore at this point it is not clear whether the
ground state of the X can be approximated by a single
configuration or it is rather a correlated system, with the
hole spread out among the four lowest Kramers doublets.
The second type of the Coulomb matrix elements in
play is the electron-hole exchange. Let us specify the ex-
change elements involving the lowest Kramers doublets -
one for the electron and one for the hole. As these ele-
ments are spin sensitive, let us first point out that the TB
model allows us to compute only the states of electrons in
the conduction and valence bands. In order to describe
the optics of our system in the usual language of quasi-
particles, we have to change the treatment of the valence
band by renaming the missing valence electron “spin up”
into the valence hole “spin down”. Having this in mind,
and working for the moment in the language of electrons
only, we find that the absolute value of the element 〈1e ↓
1h ↓ |Veh|1e ↓ 1h ↓〉 = 〈1e ↑ 1h ↑ |Veh|1e ↑ 1h ↑〉 = 6.36
meV, while 〈1e ↓ 1h ↑ |Veh|1e ↓ 1h ↑〉 = 〈1e ↑ 1h ↓
|Veh|1e ↑ 1h ↓〉 = 0.01 meV in the case (i). In the case
(ii) the absolute value of these elements is 7.2 meV and
0.01 meV, respectively. In the language of quasi-particles
these diagonal exchange terms describe the interaction
correction to the electron-hole pairs with opposite spin
(the former two elements) and parallel spin (the latter
two elements). Since the electron-hole exchange interac-
tion enters the total Hamiltonian with the positive sign,
the quasi-particle pairs with antiparallel spins (i.e., the
optically active ones) will have higher energy than those
with parallel spins. The remaining exchange terms within
this manifold of states are offdiagonal, and their abso-
lute values do not exceed 1 µeV. As a result, due to the
electron-hole exchange we expect a fine structure of the
X composed of two pairs of states, separated by a gap of
several meV, and each pair nearly degenerate.
To conclude the discussion of the electron-hole
Coulomb matrix elements, we comment on how these el-
ements are impacted by the difference in treatments of
nearest neighbors, i.e., the cases (i) and (ii). We find that
the treatment (ii) gives consistently larger magnitudes of
the elements, however the difference is only about 5% in
the direct terms and about 20% in the smallest exchange
terms. From the general formulas for Coulomb elements
given by Eqs. (7) and (8) we see that the nearest-neighbor
term scales as the number of atoms NAT , while the re-
mote term scales asN2AT . As a result, the change in treat-
ment of nearest neighbors will be more visible in smaller
nanocrystals. In what follows we will employ the simpli-
fied treatment (ii), as the resulting change of energies is
small compared to other energy scales in the system, and
the simplification of the treatment of nearest neighbors
leads to a considerable speedup in calculations.
Let us now move on to constructing the correlated
states of the interacting electron-hole pair. As was al-
ready mentioned, we accomplish this by diagonalizing the
Hamiltonian (4) set up in the basis of electron-hole con-
figurations. With 1028 atoms present in the system, and
the TB basis of 20 orbitals per atom, we can distribute
our particles on 20560 single-particle states, out of which
we have 3700 hole and 16860 electron states. As a result,
there exist 62.3 × 106 excitonic configurations. Since in
this work we focus on the low-energy excitonic config-
urations only, instead of dealing with the full basis we
shall build the electron-hole configurations out of single-
particle states closest to the bandgap. The computa-
tional effort grows rapidly with the increase of the num-
ber of single-particle basis involved in the calculation.
The most time- and resource-intensive part is the com-
putation of Coulomb matrix elements, as each element
involves ∼ N4AT operations, and for Me electron and Mh
hole states we require ∼ M4e electron-electron elements,
∼ M4h hole-hole elements, and ∼ M2eM2h electron-hole
elements.
The evolution of the X spectra as a function of the basis
size is visualized in Fig. 5(a). In the left-hand panel we
show the X energies resulting from the diagonalization
of the Hamiltonian built using Me = 2 electron states
(i.e., the lowest, s-shell Kramers doublet) and Mh = 8
hole states (i.e., the lowest four Kramers doublets sepa-
rated from the rest of the hole spectrum by a gap). This
results in 16 electron-hole configurations. In the mid-
dle panel we include more hole states by increasing Mh
to 28, whilst in the right-hand panel we compute with
Me = 8 (i.e., the s and p shells) and Mh = 28. This
increase of the single-particle basis gives respectively 56
and 224 configurations. We see, overall, that as the ba-
sis is increased, the energy of the lowest level decreases,
but not by a large amount compared to the bandwidth
of the excitonic states. Moreover, the excitonic states
are grouped into blocks separated by gaps. The lowest
energy block, consistent throughout the three spectra, is
built out of configurations from the lowest electronic dou-
blet and the four lowest hole doublets of single-particle
states. The second block, apparent in the middle panel,
involves the hole residing on higher single-particle states,
and the gap separating it from the lower section is con-
sistent with the gap in the single-particle hole spectrum.
Finally, in the right-hand panel we see two spectra from
the middle panel, stacked on top of one another. Fur-
ther, the top half of this ladder of states is denser than
the bottom half. Such a distinct structure of the spectra
is due to a large gap between the electron single-particle
s and p shells. As a result, the third block from the bot-
tom is composed of the hole residing on the four lowest
single-particle levels, but the electron occupying one of
the six p-shell levels. Similarly, the highest block con-
tains configurations with an electron on the p shell and
the hole on states deeper in the valence band.
In Fig. 5(b) we show the dependence of the X ground-
state energy on the hole basis size for three cases: elec-

8tron on the s shell only (black squares), on the s and p
shells (red circles), and on the s, p, and d shells (green
triangles). We see that as a function of 1/Mh the X
energy follows a power law and it is not completely con-
verged even for the largest hole basis used (Mh,max = 128
states). We notice also a marked decrease of the X en-
ergy as subsequent electronic shells enter the picture. As
the electron spreads to the p shell, the energy drops by
about 12 meV, while allowing the electron to spread to
the d shell results in a smaller decrease. Note that the
overall drop in energy is about 20 meV, most of it ac-
complished by using the three electronic shells, s, p, and
d (altogether 18 states) and increasing the hole basis to
Mh = 28 states. The energy drop is of the order of
Coulomb electron-hole direct scattering matrix elements
and is somewhat larger than the separation of the low-
est four hole single-particle states, but it is an order of
magnitude smaller than the diagonal direct Coulomb el-
ements describing the electron-hole attraction.
In view of the slow convergence of X energies it is nec-
essary to establish a scheme for extraction of their con-
verged values. To this end we first extrapolate each of
the curves from Fig. 5(b) to zero (i.e., infinite number
of hole basis states). Then we use the energies obtained
for each Me to extrapolate to infinite number of electron
states. In our case, the extrapolated ground-state energy
of the exciton is E∞X = 2.100 eV.
Because of the commensurability of the energy change
due to correlations and the single-particle energy scale
we now examine the spectral content of the several low-
lying X states in the case of Me = 8 and Mh = 28, i.e.,
in the regime when the energy is already relatively close
to convergence. Figure 6(a) shows the lowest section of
the energy spectrum of such an X, while panels (b) illus-
trate schematically the configurations dominant in the
respective wave functions. We find that the two lowest
X states are nearly degenerate (to within 0.1 µeV) and
are composed predominantly (in 95%) of the configura-
tions with the electron and hole occupying their respec-
tive lowest single-particle states, 1e and 1h, and assuming
the same spin character (i.e., approximately “spin paral-
lel”). Such configurations can be written as c+1↓h+1↓|0〉
and c+1↑h+1↑|0〉 and are shown schematically in the up-
per left-hand and upper right-hand panel of Fig. 6(b),
respectively The next pair of states is found 7.3 meV
higher in energy, this gap being due predominantly to
the electron-hole exchange. This pair is degenerate to
within 1 µeV. The constituent states are composed of
configurations, in which the electron and the hole have
“opposite spins”. One of such configurations, c+1↓h+1↑|0〉
is shown in the lower left-hand panel of Fig. 6(b). The
other one, c+1↑h+1↓|0〉, is shown in the lower right-hand
panel of that Figure.
B. Bi-exciton
We now proceed to calculating the energies and wave
functions of a system of two electron-hole pairs form-
ing a bi-exciton (XX). Since now we deal with pairs
of carriers of the same type, we need to establish the
electron-electron and hole-hole matrix elements. In com-
putations we also consider the two cases of treating
the nearest-neighbor contributions, as discussed previ-
ously for the electron-hole elements. Let us consider the
electron-electron elements first. The diagonal element
defining the interaction energy of the two-electron con-
figuration on the lowest single-particle levels c+1↑c+1↓|0〉 is
〈1e ↓, 1e ↑ |V |1e ↑, 1e ↓〉 = 197.79 meV in the case (i) and
203.71 meV in the case (ii). It is somewhat smaller (by
about 5%) than the fundamental electron-hole element
discussed in the previous Section. A similar element for
the holes, defining the interaction energy of the hole pair
h+1↑h+1↓|0〉 is 〈1h ↓, 1h ↑ |V |1h ↑, 1h ↓〉 = 271.77 meV
in the case (i) and 271.78 meV in the case (ii). It is
much larger than the fundamental electron-electron and
electron-hole elements. We find this to be the case for all
sizes of spherical CdSe nanocrystals studied (from 2 nm
to 7 nm). A possible reason for this disparity between
various types of matrix elements lies in the difference of
charge densities corresponding to the electron and hole
single-particle ground states. Figure 7(a) shows the ver-
tical cross-section of the ground-state charge density for
the electron computed by the QNANO package. As can
be seen, this density is distributed across the entire crys-
tal, it is largest in the center and tapers off towards the
surfaces. Small irregularities in this image are due to the
lack of the symmetry plane of the nanocrystal, which is
built out of 11 layers of atoms. This is why we find a
finite density on the lowest atomic layer, while the top
layer appears to carry no charge. An analogous profile
for the ground hole state is shown in Fig. 7(b). Here we
see a clear lack of symmetry, with the maximum charge
located in the lower half of the nanocrystal. Also, the
hole appears to occupy a much smaller volume than the
electron does. Note that neither of the wave functions is
centered in the NC. This is due to wurtzite structure of
NC, particularly due to polarity of the (0001) direction
which results in the development of internal dipole mo-
ment. We have confirmed the existence of such a dipole
in our DFT calculations (not shown here), however we
find that the direction and strength of this dipole is sen-
sitive to the type of ligands used to passivate the NC
surface. The decreased spatial extent of the hole state
leads to a large magnitude of the Coulomb repulsion of
two holes placed on the lowest Kramers doublet, since
the charge density of these two states is identical. In
the electron case the more uniform spread of the den-
sity across the crystal diminishes the electron-electron
element. The electron-hole element is also decreased, as
we deal here with a relatively localized hole interacting
with a distributed electron charge.
Let us now look at the scattering matrix elements,

9which describe the Coulomb coupling between different
configurations. For holes the largest element transfer-
ring the particle from the lowest Kramers doublet is
〈1h ↓, 1h ↑ |V |1h ↑, 4h ↓〉 = 2.73 meV in the case (i)
and 2.62 meV in the case (ii). To assess the strength
of this element in a meaningful way, let us first analyze
briefly a two-hole configuration h+4↑h+1↓|0〉. Compared to
the fundamental configuration h+1↑h+1↓|0〉, the excited con-
figuration is created by moving the hole from the first to
the fourth Kramers doublet and so its single-particle en-
ergy is higher by about 34 meV. On the other hand, the
interaction energy of the excited two-hole configuration
is given by the matrix element 〈1h ↓, 4h ↑ |V |4h ↑, 1h ↓〉,
which is 220.26 meV in the case (i) and 220.73 meV in
the case (ii). That is, in this excited configuration the
holes repel by about 50 meV weaker than in the funda-
mental one. So, altogether, the configuration h+4↑h+1↓|0〉
is lower in energy, even though it has a larger single-
particle energy part. The energy difference between the
two configurations is then only about 17 meV, suggesting
that the offdiagonal scattering matrix elements will lead
to the appearance of strongly correlated hole-hole states.
Let us now account for the presence of the two elec-
trons. The electron-hole attraction gives a negative con-
tribution to the total energy of the system. In any XX
configuration we have four constituent terms, as each
electron attracts two holes. For this discussion we need
also the electron-hole matrix element 〈1e ↓, 4h ↑ |V |4h ↑
, 1e ↓〉 = 195.38 meV in case (i) and 201.39 meV in
case (ii). Comparing this element with the fundamental
electron-hole direct element given in the previous section
we see that the electron attracts the hole on the level 4 by
about 18 meV weaker than it does the hole on the ground
single-particle level. Now we are in a position to com-
pare the energies of configurations c+1↑c+1↓h+1↑h+1↓|0〉 and
c+1↑c+1↓h+4↑h+1↓|0〉. Owing to the electron-hole direct terms
we find that the former, i.e., the fundamental configu-
ration of the two electron-hole pairs, is about 55 meV
lower in total energy than that with one excited hole. As
we see, the final alignment of levels results from cancella-
tions of large interaction terms of comparable magnitude,
and as such will be sensitive to the details of many-body
computation.
The smallest contribution to scattering comes from the
electron-electron interaction. Transfer of one electron
from the s-shell orbitals to the p-shell orbitals due to
this interaction is described, e.g., by an element 〈1e ↓
, 1e ↑ |V |1e ↑, 2e ↓〉 = 0.44 meV in the case (i) and 0.49
meV in the case (ii). This is to be compared with the
energy gap between the s and p electronic shells, which
amounts to 270 meV. So, the spread of electrons onto the
p shell due to the electron-electron interaction is expected
to be small.
We now proceed to diagonalizing the two-pair Hamil-
tonian as a function of the size of the single-particle basis.
Figure 8(a) shows the energy levels of the system with
Me = 2, Mh = 8 (left), Me = 2, Mh = 28 (middle), and
Me = 8, Mh = 28 (right). In the first case we populate
only the lowest electronic Kramers doublet and the four
lowest hole Kramers doublets. As a result we can cre-
ate 28 configurations. The left-hand panel of Fig. 8(a)
shows all resulting XX eigenenergies. As we increase the
hole basis, and later on also the electron basis, we allow
one or both particles of each type to populate higher-
energy single-particle states. This results in a buildup
of the density of XX states at higher energies, which
is clearly visible in the middle and right-hand panels of
Fig. 8(a). Also, the low-lying XX energy states appear
to shift down in energy by tens of meV. To analyze this
shift in greater detail, in Fig. 8(b) we plot the energies
of the XX states with dominant singlet-singlet configura-
tion c+1↑c+1↓h+1↑h+1↓|0〉 as a function of the size of the hole
basis in three cases: with both electrons on the s state
(black squares), with the electrons allowed to spread onto
the p shell (red circles) and with the electrons populat-
ing the s, p, and d shells (green triangles). Note that the
state under consideration is not always the ground state
of the system, which reflects the simple analysis outlined
above. This state stabilizes as the ground state for hole
basis of at least Mh = 28 states, while for smaller Mh
excitation of one of the holes is preferred.
As we can see, the energy decreases steadily if we in-
crease the hole basis size but keep a constant number
of electron states. On the other hand, for a constant
hole basis one large drop takes place as we increase the
electron basis from 2 to 8 states. Upon its further in-
crease to 18 states the energy change is much less signif-
icant. A systematic study of the convergence of ground
state energy is much more difficult here, as for the basis
Me = 18, Mh = 124 we already deal with 1.17 × 106
two-pair configurations. Using the procedure analogous
to that described in the previous Section, we extrapolate
the ground-state XX energy to the limit of infinite basis
and obtain E∞XX = 4.229 eV. Figure 8(a) shows that in
spite of the substantial energy change of the ground state,
the block of lowest 28 states appears to be separated from
the remaining spectra, suggesting that the configurations
with lowest single-particle energy contribute to their re-
spective eigenvectors the most. To demonstrate this, in
Fig. 9 we analyze the spectral content of several low-
est eigenstates of the system with Me = 8, Mh = 28.
Figure 9(a) shows the lowest 28 energy levels, consistent
with the number of possible configurations created out
of lowest electron and hole single particle blocks. As we
can see, the lowest 27 states are found just within a 65
meV window, i.e., a fraction of the value of the funda-
mental Coulomb matrix elements. This is the central
result of this work. We find that in the case of spheri-
cal CdSe nanocrystals the peculiar arrangement of hole
single-particle levels together with Coulomb interactions
lead to the appearance of a fine structure of bi-exciton
levels.
In Fig. 9(b) we show configurations dominant in the
lowest four XX states. The first two states, denoted re-
spectively as (I) and (II), correspond to the two configu-
rations analyzed before and behave as we predicted pre-

10
viously using the single-configuration arguments. The
configuration predominant in the ground state is the
“singlet-singlet” one, shown in panel (I), and created by
placing the pairs of carriers on lowest possible single-
particle levels. The first excited state, on the other hand,
is based on the configuration where the hole is excited
to the fourth Kramers doublet, shown in the panel (II).
Note that the gap between these two states, amounting to
about 9 meV, is of the order of the electron-hole exchange
energy, but it originates from correlations rather than ex-
change. The first excited state also contains significant
admixtures of configurations, in which the holes occupy
the two middle Kramers doublets in the low-energy sec-
tion (not shown). At even higher energy we find a pair of
nearly degenerate states, in which the holes have aligned
“spins”. The dominant configurations in these states are
shown in panels (III) and (IV), respectively.
C. Optical spectra of exciton and bi-exciton
Having described the electronic properties of the X and
XX confined in the NC we now proceed to discussing their
optical spectra. We will conduct our analysis for the sys-
tems built upon Me = 8 electron and Mh = 28 hole
single-particle states. We start with the absorption spec-
trum of the exciton, shown in Fig. 10 and focus on the
lowest 16 spectral lines of that spectrum, i.e., those cor-
responding to the electron on the s shell and the hole on
one of the four lowest Kramers doublets. We find that the
two lowest, nearly degenerate states of the exciton are op-
tically dark irrespective of polarization, while the second
pair of states is bright, but only in polarizations x and
y. The two pairs of states form the fine structure of the
exciton, in which the energy gaps are due to the electron-
hole exchange. In these two polarizations we also find a
third, even larger maximum at a slightly higher energy.
Analysis of the spectral content of the corresponding X
eigenstate reveals that it consists of a mixture of con-
figurations in which the hole resides on the second and
third Kramers doublet, with a smaller admixture of the
fundamental configurations shown in Fig. 6(b)(II). Each
of these configurations contributes constructively to the
large final absorption amplitude of this state. The spec-
trum in polarization z is different: it consists only of one
large maximum, larger than any of the maxima in po-
larizations x and y. This large amplitude comes from
the predominant distribution of the hole on the second
single-particle Kramers doublet, which introduces a large
oscillator strength in the z polarization, as evident from
Fig. 4(a). The general properties of the exciton spec-
trum, the multiplicities of the states and their oscillator
strength obtained in our calculation agree well with pre-
vious empirical pseudopotential,52 tight-binding47 and
qualitative k · p calculations.46
The absorption spectrum discussed above is equivalent
to the emission spectrum of the X at high temperature,
i.e., when the occupations of all X energy levels are sim-
ilar. However, at low temperature only the lowest states
will be occupied, and the emission spectrum will consist
predominantly of the lowest line of the exciton. As this
exciton state is dark, its radiative lifetime is expected
to be very long. This suggests that owing to the char-
acteristic alignment of the hole single-particle levels the
X emission spectrum will sensitively depend on the tem-
perature, with the dominant maxima appearing at higher
energy as the temperature is increased.
Let us move on to computing the optical spectra of
the XX. Figure 11 shows its absorption (two top panels)
and emission spectra (third panel from the top). Calcu-
lation of the absorption spectrum involves preparing the
single exciton system in the bright, second excited state
(top panel) or dark, ground state (second panel from the
top) and adding the second electron-hole pair to form
the ground and excited XX states. Due to the optical
selection rules, the carriers composing the photo-created
electron-hole pair must have antiparallel spins.
Let us first discuss the absorption spectra involving an
addition of the second electron-hole pair to the bright
exciton. We find that this spectrum is composed of sev-
eral peaks, one at energy 2.115 eV, and denoted as XX0,
the second one at energy about 2.124 eV, and the third
one at energy about 2.136 eV. The low-energy peak cor-
responds to addition of an optically active electron-hole
pair to the bright exciton configuration as in Fig. 6(b)(I)
and formation of the ground state bi-exciton XX0 as in
Fig. 9(b)(I). The two higher-energy peaks correspond to
the formation of an excited bi-exciton, in which the holes
are redistributed among the four lowest Kramers dou-
blets, however retaining their “spin unpolarized” charac-
ter.
If the exciton is prepared in the ground, dark state, the
XX absorption spectrum (Fig. 11, second panel from the
top) is dominated by two groups of peaks, one around
the energy of about 2.13 eV, denoted as XX∗, and the
second at energy 2.144 eV, indicating the formation of
excited XX states. This is because at any lower energy
the optical selection rules prevent us from adding a pho-
toexcited, spin-unpolarized electron-hole pair to the spin-
parallel ground X state. The resulting XX states contain
spin-polarized holes.
In the calculation of the emission spectrum (third
panel of Fig. 11) we prepare the XX in its ground, singlet-
singlet state. We find that the emission spectrum is dom-
inated by one maximum, corresponding to the bright ex-
citon final state [(II) in Fig. 6(b)]. It is accompanied by
a small maximum at lower energy. Note that the main
maximum is found at the same energy as the low-energy
absorption peak of the bright exciton (top panel). How-
ever, there is an energy gap between the emission and
dark exciton absorption peaks, due to the fine structure
of both the exciton and bi-exciton low-energy spectra.
Finally, the bottom panel of Fig. 11 shows the emission
spectra of the exciton. The low-energy peak corresponds
to the dark, ground state XD, whilst the high-energy
peak denotes the radiative transition from the bright,

11
excited state XB of the exciton.
Since in our calculations we have not accounted for the
dynamics of carriers, and in particular the relaxation pro-
cesses, the above emission and absorption spectra should
be understood as presentation of oscillator strengths for
various transitions rather than candidates for direct com-
parisons with experimental data. To make this point
clear, in Fig. 12 we present two possible absorptive and
emissive scenarios in our system. In Figure 12(a) and (b)
we assume that the relaxation from the excited, bright
state XB to the ground, dark state XD is faster than the
radiative recombination from either of these two states.
Panel (a) shows the absorption from the vacuum to the
XX state. We start with the optical creation of an ex-
citon in its bright state (the red arrow), followed by its
relaxation to the dark state (the black arrow). Accord-
ing to Fig. 10, the absorption process should take place
at the energy of about 2.122 eV. We can now create the
XX, but only in one of its excited states, such as the one
denoted as XX∗ in Fig. 11 (second panel from the top).
This absorption event should be observable at the energy
of about 2.128 eV, i.e., we predict the XX binding energy
in this absorptive process to be negative.
The emissive cascade under the assumption of fast re-
laxation is shown in Fig. 12 (b). We start with XX in its
ground state XX0, recombining radiatively and leaving
the system with the X in its excited, bright state XB. Ac-
cording to Fig. 11 (third panel from the top), this process
is seen in the emission as a maximum at 2.115 eV. The
exciton further relaxes to the dark state XD and may
recombine with a very long lifetime. The recombination
energy appears to be also 2.115 eV, but its degeneracy
with the XX emission energy is here accidental and can-
not be treated as the universal property of the NCs. To
summarize, assuming the fast X relaxation we find the
absorptive XX binding energy to be −6 meV, while the
emissive binding energy is zero.
Let us now consider the situation in which the X relax-
ation from its bright to dark state is the slowest process.
The absorption processes under this condition are visu-
alized in Fig. 12 (c). Here we create the X in its bright
state XB at the energy 2.122 eV, and then immediately
afterwards the XX in its ground state XX0. According
to Fig. 11 (top panel), the latter transition should be seen
as an absorption maximum at 2.115 eV, producing the
absorptive XX binding energy of +7 meV. The emissive
cascade under the same condition is shown schematically
in Fig. 12 (d). As the cascade utilizes the same X and
XX states as the absorptive process, we expect the XX
emissive binding energy in our system to be also equal to
+7 meV.
Let us now present a detailed comparison of the
XX and X emission spectra. As already mentioned,
the exciton recombines to vacuum, and therefore
the position of the respective emission peak equals
to the energy of the X. In the simplest treatment
we approximate the X state by one configuration
|X0〉 = c+1↓h+1↓|0〉. The emission energy of such a
state, neglecting for now the electron-hole exchange,
is EX0 = ε(e)1 + ε
(h)
1 − 〈1e ↓, 1h ↓ |Veh|1h ↓, 1e ↓〉.
The bi-exciton, on the other hand, recombines to
the final-state exciton. The energy of the funda-
mental XX configuration |XX0〉 = c+1↑c+1↓h+1↑h+1↓|0〉 is
EXX0 = 2ε(e)1 + 2ε
(h)
1 + 〈1e ↓, 1e ↑ |Vee|1e ↑, 1e ↓〉+ 〈1h ↓
, 2h ↑ |Vhh|1h ↑, 1h ↓〉 − 4〈1e ↓, 1h ↓ |Veh|1h ↓, 1e ↓〉
and accounts for the repulsion of the like carri-
ers and attraction of each pair of opposite carri-
ers. The position of the XX fundamental emission
peak can then be evaluated as ΩXX = EXX0 −
EX0 =
(
ε(e)1 + ε
(h)
1 − 〈1e ↓, 1h ↓ |Veh|1h ↓, 1e ↓〉
)
+
(〈1e ↓, 1e ↑ |Vee|1e ↑, 1e ↓〉 − 〈1e ↓, 1h ↓ |Veh|1h ↓, 1e ↓〉)+
(〈1h ↓, 1h ↑ |Vhh|1h ↑, 1h ↓〉 − 〈1e ↓, 1h ↓ |Veh|1h ↓, 1e ↓〉).
It consists of the exciton energy EX0 and self-energy
corrections. If the electron-electron and hole-hole
elements are equal to the electron-hole terms, the self-
energy corrections cancel out and the XX emission peak
matches that of the exciton. However, as demonstrated
above, the hole-hole repulsive interaction is significantly
larger than the electron-hole element, and so from that
simple analysis we expect the XX peak to occur at the
energy higher than that of X, i.e., we expect the XX to
be unbound, as is found in some epitaxial dots (see e.g.
Ref. 81).
Let us now compare the emission peak positions of X
and XX.25,55 Figure 13 shows these spectra on the same
energy scale. The bars in the two top panels show the
complete X emission spectrum assuming equal occupa-
tion of all levels (infinite temperature), while the black
and red curves account for finite temperature effects. At
the temperature of 4 K (top left panel) only the ground
state of the exciton will be occupied, as it is separated
from the excited states by the electron-hole exchange
gap. As a result, the low-temperature X recombination
is forbidden optically, resulting in a long excitonic life-
time. Assuming long integration time, and accounting
for a model inhomogeneous broadening, such a recombi-
nation maximum is schematically shown in Fig. 13(top
left panel) with a black line. Note that in this graph only
the position of this peak is meaningful, since, as we have
already mentioned, we do not model dynamical phenom-
ena taking place in the system. On the other hand, as
the temperature is increased to 300K, excited X states
are populated, resulting in optically allowed excitonic re-
combination.
The high-temperature emission maximum computed
assuming the Maxwell-Boltzmann thermal distribution of
carriers and including model inhomogeneous broadening,
is shown with the red continuous line in the top right
panel of Fig. 13. As already discussed, this maximum is
shifted towards higher energies by about 20 meV.
Let us now move on to the XX spectra (the two bot-
tom panels of Fig. 13). Since the XX ground state is op-
tically active, we deal with relatively short XX lifetimes
even at very low temperatures. Such a low-temperature
spectrum is plotted in Fig. 13(lower left panel) with black

12
bars, while the black continuous line is the emission enve-
lope accounting for a model inhomogeneous broadening.
As the temperature is increased, excited XX states will
become populated. This, however, does not have to mean
that the emission peak will shift towards higher energies,
as was the case for the exciton. Indeed, the XX states re-
combine to excited X states, while the X states can only
recombine to the vacuum. Depending on the oscillator
strengths appropriate for each of the possible XX-X tran-
sitions we may deal with additional higher-energy peaks
(when excited XX states recombine to low-lying X states)
or low-energy maxima (when the final X states lie high
in energy). Figure 13(lower right panel) demonstrates
that in the studied system we deal with the latter case.
If we also account for inhomogeneous broadening (red
continuous line), we find that the overall emission maxi-
mum moves towards lower energies, i.e., exhibits a trend
opposite to that of the exciton.
Let us now compare the two spectra. In the 4 K case
(the two left panels), due to the interplay of Coulomb
elements, we predict the XX emission peak at a slightly
higher energy than the X maximum, i.e., we find that the
XX becomes unbound. However, as the temperature is
raised to 300 K (the two right panels), the opposite shifts
in the X and XX spectra lead to a rearrangement of the
order of the peaks, so that the XX maximum is found
below the X.
In the last element of our analysis we study the relative
positions of the X and XX emission maxima as a function
of the NC size. In Fig. 14 we show several characteristic
quantities measuring the position of the XX peak rela-
tive to the X peak, i.e., the XX binding energy. Black
squares show the difference EXX0−2EX0 computed ear-
lier in the single-configuration approach neglecting the
electron-hole exchange. As can be seen, this difference is
positive (i.e., the XX is unbound80) for all NC sizes stud-
ied. Next we include the electron-hole exchange effects
and correlations by computing the XX binding energy
with the basis of Me = 8 and Mh = 28 single-particle
states. In this case we present two sets of results, shown
with filled symbols. The red (blue) circles show the differ-
ence between the XX lowest-energy peak and that of the
dark (bright) exciton; the splitting between these two sets
of data is due to the electron-hole exchange. We find that
if the position of the XX peak is measured relative to the
dark exciton, the XX becomes unbound for NC diameters
smaller than 4 nm. On the other hand, if the bright exci-
ton is considered, the XX is always bound, in contrast to
the single-configuration calculation. However, as demon-
strated earlier, the basis set taken in this calculation is
not sufficient to achieve convergence of the X and XX en-
ergies. To eliminate this systematic error, we extrapolate
the X and XX energies presented in Fig. 5 and 8, respec-
tively, to the infinite hole basis. The XX binding energy
computed in this limit relative to the bright X maximum
is shown in Fig. 14 with empty symbols. We find that the
XX is unbound for NC diameters smaller than 4 nm, and
bound for larger NC diameters. We compare our results
to those obtained by Sewall et al. (Ref. 54) in a finite elec-
tron and hole basis set. The XX binding energy obtained
in this empirical pseudopotential calculation is denoted
by the blue triangle for the NC diameter of 3.8 nm. In
agreement with our finite-basis calculation, Sewall et al.
predict a bound XX. Note, however, that after extrapo-
lation to the infinite basis set the XX becomes unbound,
which demonstrates the need for systematic convergence
study.
V. CONCLUSIONS
In conclusion, we have analyzed the electronic and op-
tical properties of an exciton and a bi-exciton confined in
a single, spherical CdSe nanocrystal. Using the atomistic
tight-binding approach we have calculated the single-
particle spectra and found that the lowest energy hole
states form a shell consisting of four states separated
from the rest of levels by a gap. The bi-exciton state
was computed using configuration interaction techniques
and found to be a strongly correlated state consisting
of a two electron singlet in the s-shell of the conduction
band and a strongly correlated state of two holes dis-
tributed on the degenerate hole shell, resulting in a fine
structure of bi-exciton energy levels. The fine structure
is also present in the exciton spectrum, however it is due
to the electron-hole exchange interaction. The bi-exciton
fine structure becomes apparent in the absorption of the
second exciton into the nanocrystal. We find that if the
initial state exciton is prepared in the bright configura-
tion, the maximum indicating the absorption of the sec-
ond pair is found at the same energy as the emission peak
from XX. However, if we prepare the exciton in the dark
state, the absorption takes place to higher XX states in
the quasi-degenerate manifold. As for the emission spec-
tra, we found that at a low temperature the bi-exciton
emission peak corresponds to an energy slightly higher
than the energy of the excitonic ground, dark state, i.e.,
the XX is unbound. However, at higher temperatures the
exciton emits from the excited bright state, so that the
inhomogeneously broadened X emission peak moves to
higher energies. On the other hand, thermal population
of higher XX levels leads to emission to excited final X
states, moving the broadened XX emission peak to lower
energies, even below the high-temperature emission max-
imum. A similar transition in the character of XX can
be achieved by changing the diameter of the nanocrystal:
for diameters of up to 4 nm the XX is unbound, while for
larger NCs the XX becomes bound. Due to the compli-
cated nature of the spectrum of the valence hole we find
that in all elements of our analysis the correlations play
a crucial role and that any qualitative conclusions as to
the electronic and optical properties of X and XX can be
drawn only after a careful convergence analysis.
Future work will focus on improving several aspects of
QNANO which at present received only a model treat-
ment. We plan to improve the description of screening

13
and carry out a computation of the distance-dependent
dielectric function.52,75–78 We are going to develop a more
realistic model of the surface passivation accounting for
the presence of ligands. The more microscopic descrip-
tion of the surface is also needed for a proper description
of the NC shape. We plan to develop a hybrid tb-DFT
QNANO package in order to account for a realistic sur-
face reconstruction and faceting.
Acknowledgment
The authors acknowledge discussions with G. Scholes,
Kui Yu, A. Stolow, P. Kambhampati, A. Efros, M. Zielin-
ski, and thank E. Kadantsev for providing the results
of DFT calculations with the EXCITING/elk software
(Figs. 1, 2). Funding from the NRC-NSERC-BDC Nan-
otechnology Project and CIFAR is gratefully acknowl-
edged.
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15
FIG. 1: Band structure of CdSe computed using the 20-band
tight-binding model of this work (a) and the DFT approach
with a rigid shift applied to the conduction band (b)

16
FIG. 2: Bulk density of states computed using the DFT proce-
dure of SIESTA (top panel),plane-wave approach of the pack-
age ”Exciting” (middle panel), and our tight-binding model
(bottom panel).

17
FIG. 3: (Color online) Energies of single-particle states of an
electron (a) and a hole (b) in a CdSe nanocrystal of 3.8 nm
diameter. (c) Energies of the hole states as a function of the
diameter of the nanocrystal; the characteristic gap separat-
ing the four lowest hole states from the rest of the spectrum
is visible for all nanocrystal sizes. Inset shows an atomistic
picture of the 3.8 nm nanocrystal.

18
































































  !"#$!%#&'
 !"#$!%#&$
































 !"#$!%#&'
 !"#$!%#&$













(

(

(

(

(


(










FIG. 4: (Color online) Joint optical density of states charac-
terizing the oscillator strength between each of the four lowest
hole states H1-H4 with the electron s-shell (a) and p-shell (b)
states. The color coding is explained in part (c).

19









































   
   































 !"
 !"!#


FIG. 5: (Color online) (a) Ground and excited energy levels
of an exciton with increasing electron and hole single-particle
basis size for a nanocrystal with diameter of 3.8 nm. (b)
Ground-state energy of the exciton plotted as a function of
inverted hole basis size for Me = 2 (black squares), Me = 8
(red circles), and Me = 18 (green triangles).

20
FIG. 6: (Color online) (a) Ground and excited exciton energy
levels computed with a basis of Me = 8 electron and Mh = 28
single-particle hole states for a nanocrystal with diameter of
3.8 nm. (b) Spectral content of the four lowest exciton states
(see text for analysis).

21
FIG. 7: (Color online) Vertical cross-section of the electron
(a) and hole (b) ground state charge density in a nanocrystal
of 3.8 nm diameter computed by the QNANO package.

22
















































  
 
  




































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FIG. 8: (Color online) (a) Ground and excited energy lev-
els of a bi-exciton with increasing electron and hole single-
particle basis size for a nanocrystal with diameter of 3.8 nm.
(b) Energy of the bi-exciton singlet-singlet state plotted as a
function of inverted hole basis size for Me = 2 (black squares),
Me = 8 (red circles), and Me = 18 (green triangles).

23
FIG. 9: (Color online) (a) Ground and excited bi-exciton
energy levels computed with a basis of Me = 8 electron and
Mh = 28 single-particle hole states for a nanocrystal with
diameter of 3.8 nm. (b) Spectral content of the four lowest
bi-exciton states (see text for analysis).

24






























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FIG. 10: (Color online) Absorption spectrum of the exciton in
polarization x and y (red bars), and z (blue bars). Black bars
denote positions of absorption maxima, whilst the height of
the color bars denotes the corresponding oscillator strength.

25
FIG. 11: (Color online) Optical spectra of the exciton and
bi-exciton for a nanocrystal with diameter of 3.8 nm. Upper
panel shows the absorption spectrum of the second exciton as-
suming that the first exciton is prepared in the bright excited
state. Second panel from the top shows the same assuming
that the first exciton is prepared in the dark ground state.
Third panel shows the emission spectra from the ground bi-
exciton to the ground and excited exciton states, whilst the
bottom panel shows the exciton emission spectrum. In all
panels, black bars denote positions of absorption or emission
maxima, whilst the height of the red bars denotes the corre-
sponding oscillator strength.

26
FIG. 12: (Color online) Schematic view of the exciton and
biexciton absorptive and emissive processes assuming fast (a
and b) and slow (c and d) relaxation from the excited, bright
to the ground dark state of the exciton. Panels (a) and (c)
show absorption processes, whilst panels (b) and (d) show
emission processes.
FIG. 13: (Color online) Exciton (top panels) and bi-exciton
(bottom panels) emission spectra at low (left panels) and high
temperatures (right panels). Black (red) curves show emis-
sion spectra at low (high) temperature accounting for model
inhomogeneous broadening of 50 meV. Void bars denote posi-
tions of the respective maxima, whilst the height of the solid
bars represents the respective oscillator strength multiplied
by thermal population of levels.

27
FIG. 14: (Color online) Relative position of exciton and bi-
exciton emission maxima at zero temperature as a function
of the nanocrystal size. Black symbols show the bi-exciton
binding energy calculated in a single-configuration approach
neglecting the electron-hole exchange and correlations. Full
circles denote the bi-exciton binding energy computed in the
basis of Me = 8 and Mh = 28 states; the blue (red) symbols
are computed in reference to the exciton bright (dark) emis-
sion peak. Empty symbols show the bi-exciton emission peak
relative to the exciton bright transition obtained by extrapo-
lation to infinite electron and hole basis. The triangle shows
the XX binding energy obtained in empirical pseudopotential
calculation of Sewall et al. (Ref. 54).