1.07 Quantum Dots: Theory · quantum dots, sometimes termed as electrostatic quantum dots, can be...

1.07 Quantum Dots: TheoryN Vukmirovic and L-W Wang, Lawrence Berkeley National Laboratory, Berkeley, CA, USA

ª 2011 Elsevier B.V. All rights reserved.

1.07.1 Introduction
189
1.07.2 Single-Particle Methods
190
1.07.2.1 Density Functional Theory
191
1.07.2.2 Empirical Pseudopotential Method
193
1.07.2.3 Tight-Binding Methods
194
1.07.2.4 k ? p Method
195
1.07.2.5 The Effect of Strain
198
1.07.3 Many-Body Approaches
201
1.07.3.1 Time-Dependent DFT
201
1.07.3.2 Configuration Interaction Method
202
1.07.3.3 GW and BSE Approach
203
1.07.3.4 Quantum Monte Carlo Methods
204
1.07.4 Application to Different Physical Effects: Some Examples
205
1.07.4.1 Electron and Hole Wave Functions
205
1.07.4.2 Intraband Optical Processes in Embedded Quantum Dots
206
1.07.4.3 Size Dependence of the Band Gap in Colloidal Quantum Dots
208
1.07.4.4 Excitons
209
1.07.4.5 Auger Effects
210
1.07.4.6 Electron–Phonon Interaction
212
1.07.5 Conclusions
213
References
213
1.07.1 Introduction

Since the early 1980s, remarkable progress in technology

has been made, enabling the production of nanometer-

sized semiconductor structures. This is the length scale

where the laws of quantum mechanics rule and a range

of new physical effects are manifested. Fundamental

laws of physics can be tested on the one hand, while onthe other hand many possible applications are rapidly

emerging for nanometer-sized semiconductors..The ultimate nanostructure where carriers are

confined in all the three spatial dimensions is called

a quantum dot. In the last 15 years, quantum dots

have been produced in several different ways in a

broad range of semiconductor material systems. Theproperties of quantum dots and their possible appli-

cations are largely dependent on the method they

have been obtained with, which can, therefore, be

used as a criterion for classification of different

types of quantum dots:Electrostatic quantum dots. One can fabricate quan-

tum dots by restricting the two-dimensional (2D)electron gas in a semiconductor heterostructure

laterally by electrostatic gates or vertically by etch-ing techniques [1,2]. The properties of this type ofquantum dots, sometimes termed as electrostaticquantum dots, can be controlled by changing theapplied potential at gates, the choice of the geometryof gates, or external magnetic field. The typical sizeof these dots is of the order of 100 nm.

Self-assembled quantum dots. Self-assembled quantumdots are obtained in heteroepitaxial systems with dif-ferent lattice constants. During the growth of a layer ofone material on top of another, the formation ofnanoscale islands takes place [3], if the width of thelayer (the so-called wetting layer) is larger than acertain critical thickness. This growth mode is calledStranski–Krastanov mode. The most common experi-mental techniques of the epitaxial nanostructuregrowth are molecular beam epitaxy (MBE) and meta-lorganic chemical vapor deposition (MOCVD) [4,5].Since the quantum dot material is embedded inanother material, we refer to these dots also asembedded quantum dots. Self-assembled quantumdots typically have lateral dimensions of the order of15–30 nm and height of the order 3–7 nm.

189

190 Quantum Dots: Theory

Colloidal quantum dots. A very different approach toobtain quantum dots is to synthesize single crystals

of the size of a few nanometers through chemical

methods. The dots obtained in this way are called

nanocrystals or colloidal quantum dots [6]. Their

size and shape can be controlled by the duration,

temperature, and ligand molecules used in the synth-

esis [7]. Colloidal quantum dots are typically of

spherical shape. They are often smaller than

embedded quantum dots with the diameter some-

times as low as 2–4 nm.Quantum dots have enabled the study of many

fundamental physical effects. Electrostatic quantum

dots can be controllably charged with a desired

number of electrons and therefore the whole periodic

system [8] of artificial atoms created, providing a

wealth of data from which an additional insight into

the many-body physics of fermion systems could be

obtained [1]. Single-electron transport and Coulomb

blockade effects on the one hand, and the regime of

Kondo physics on the other hand, have been investi-

gated [9,10].One of the most exciting aspects of quantum dot

research is certainly the prospect of using the state of

the dot (spin state, exciton, or charged exciton) as a

qubit in quantum information processing. Coherent

control of an exciton state in a single dot selected

from an ensemble of self-assembled quantum dots as

well as the manipulation of the spin state in electro-

static quantum dots [12,13] have been achieved [11].

The theoretical and experimental progress in the

field of spin-related phenomena in quantum dots

has been reviewed in Refs. [1 and 14]. These results

appear promising, although the control of a larger

number of quantum dot qubits is not feasible yet,

mainly due to difficulty in controlling qubit–qubit

interactions.The practical applications of quantum dots

certainly do not lag behind these exciting areas of

fundamental science with quantum dots. For exam-

ple, colloidal quantum dots have found several

cutting-edge applications such as fluorescent biolo-

gical labels [15], highly efficient photovoltaic solar

cells [16], and nanocrystal-based light-emitting

diodes [1]. Self-assembled quantum dots find the

main application as optoelectronic devices – lasers

[17], optical amplifiers [18], single-photon sources

[19,20], and photodetectors [21,22,23].This chapter focuses on theoretical methods used

for calculation of physical properties of self-

assembled and colloidal quantum dots.

1.07.2 Single-Particle Methods

While quantum dots seem to be small and simple

objects, a look at their structure from the atomistic

side reveals their high complexity. Bearing in mind

that the lattice constants of the underlying semicon-

ductor materials are typically of the order of 0.5 nm,

one can estimate that a single self-assembled quan-

tum dot contains �106 nuclei and even a larger

number of electrons interacting among each other

with long-range Coulomb forces. Even the smallest

colloidal quantum dots contain thousands of atoms.This clearly indicates that direct solution of the

many-body quantum dot Hamiltonian is not a feasi-

ble approach and that smart and efficient methods

need to be developed. Here, methods that reduce the

problem to an effective single-particle equation are

reviewed.More than two decades ago, Brus introduced

[24–25] a simple effective mass method to calculate

ionization energies, electron affinities, and optical

transition energies in semiconductor nanocrystals.

Within Brus’s model, the single-particle (electron or

hole) energies E and wave functions (r) satisfy the

Schrodinger’s equation given as

–1

2m�r2 þ PðrÞ

� � ðrÞ ¼ E ðrÞ ð1Þ

where m� is the electron or hole effective mass. Thesystem of atomic units where the reduced Planck’sconstant h�, the electron mass m0, and the electroncharge c are all equal to 1 was used in equation 1 andwill be used in what follows. For simplicity, equation1 assumes that the particle must be confined withinthe dot, that is, that the potential outside the dot isinfinite. This simplifying assumption can be easilyrelaxed by adding a more realistic confining potentialVconf (r).

P(r) in equation 1 is the additional potential causedby the presence of the surface of the quantum dot. It

has a certain analogy with electrostatic image poten-

tials in the case when a charge is near the surface of the

metal or the interface between two dielectrics. It can

be obtained by calculating the interaction energy

between a bare electron and its induced screening

potential. The extra interaction energy of an electron

at r inside the quantum dot compared to the corre-

sponding value in bulk is then P(r).To model the two-particle excitations (such

as electronþ hole¼ exciton), Brus introduced an

Quantum Dots: Theory 191

electrostatic interaction energy term among theseparticles as

V ðr1;r2Þ ¼ �e2

" r1–r2j j � PM r1;r2ð ÞþP r1ð ÞþP r2ð Þ ð2Þ

where " is the dielectric constant, PM corresponds tothe interaction of the charge of one particle withsurface-induced polarization potential of the otherparticle, while the P-terms describe the interactionof the charge of one particle with its own surface-induced polarization potential, as previouslydescribed. The plus (minus) sign is for the two par-ticles of the same (opposite) charge. The effectiveexciton Hamiltonian is then given as

Hexciton ¼ –1

2mer2

e –1

2mhr2

h –e2

� re – rhj j

– PMðre; rhÞ þ PðreÞ þ PðrhÞ ð3Þ

The solution of the eigenvalue problem of thisHamiltonian can be written down analytically as

E�. Eg þ�2

2R2

1

meþ 1

mh

� �–

ace2

�Rþ small term ð4Þ

where ac ¼ 2 –Sið2�Þ�þ Sið4�Þ

2�� 1:8 and Si(x) is the

sine integral function, SiðxÞ ¼R x

0

sint

tdt : The last

term in equation 4 originates from the last three termsin equation 3. One should note that P(r)¼ PM(r, r)/2;therefore, PM(re, rh) and P(re)þ P(rh) cancel out exac-tly when re¼ rh and lead to a small term when re and rh

are not equal. This small term can often be ignored inpractice for spherical quantum dots.

The cancellation of the polarization terms gives us aguide for a general approach for calculating the excitonsin nanocrystals. As a first step, one calculates the single-particle energies from equation 1 without the polariza-tion term. As a second step, the screened electron–holeinteraction is added perturbatively. One should, how-ever, have in mind that such an approach is anapproximation based on classical electrostatic considera-tion. It ignores the effects such as dynamic screening andthe local-field effects of the dielectric function. Thesingle-particle states obtained in this way are not thequasiparticles from the usual GW formalism. (Theeigenenergies of equation 1 with the P-term are thequasiparticle energies that correspond to the electronaffinity and ionization potential.) However, such single-particle states are the natural extension of single-particlestates considered in other nanostructures, such as quan-tum wells and superlattices. These are also fully in line

with eigenstates defined in the density functional theory(DFT) discussed below. This section is, therefore, com-pletely devoted to the theoretical frameworks andmethodologies for calculating these states.

1.07.2.1 Density Functional Theory

Within the DFT [26], the many-body Hamiltonianproblem reduces to a set of single-particle Kohn–Sham equations [27] that read as

–1

2r2 þ Vion þ VH þ VXC

� � iðrÞ ¼ "i iðrÞ ð5Þ

In equation 5, i(r) and "i are the wave functions andenergies of Kohn–Sham orbitals, Vion(r) is the poten-tial of all nuclei in the system, and VH(r) is theHartree potential of electrons given as

VHðrÞ ¼Z

dr9�ðr9Þr� r9j j ð6Þ

where

�ðrÞ ¼X

iðrÞj j2 ð7Þ

is the electronic charge density of the system. Thesummation in equation 7 goes over all occupiedKohn–Sham orbitals. The exchange-correlationpotential VXC in equation 5 is supposed to take intoaccount all the other effects of electron–electroninteractions beyond the simple Coulomb repulsion(described in VH). The exact form of this potential isnot known, and it needs to be approximated. Themost widely used approximation is the local densityapproximation (LDA) where it is assumed that VXC

depends only on the local electronic charge densityand takes the same value as in the free electron gas ofthat density [27]. Equations 5 and 7 need to be solvedself-consistently until the convergence is achieved.

DFT calculations are still computationallydemanding, partly due to the necessity of self-consistent calculations. One also needs to calculateall the orbitals i in each iteration, while in semicon-ducting systems one is often interested in only afew states in the region around the gap that determinethe optical and transport properties of the system.

An alternative approach that avoids the full self-consistent calculation without loss in accuracy is thecharge patching method (CPM) [28,29,30–32,33].The basic assumption of the CPM is that the chargedensity around a given atom depends only on thelocal atomic environment around the atom. This istrue if there is no long-range external electric field


that causes long-range charge transfer. This is oftensatisfied if there is a band gap in the material. Basedon this assumption, the idea is to calculate (e.g., usingDFT in LDA) the charge density of some smallprototype system �LDA(r), decompose it into contri-butions from individual atoms (charge densitymotifs), and then use these motifs to patch the chargedensity of a large system.

In particular, charge density motifs are calculatedfrom the charge density of the prototype system as

mI�ðr –R�Þ ¼ �LDAðrÞw� r – R�j jð ÞP

R�9w�9 r –R�9j jð Þ ð8Þ

where R� is the position of atom type � andmI� r –R�ð Þ is the charge density motif of thisatom type, w�(r) is an exponentially decayingfunction that defines the partition functionw� r –R�j jð Þ=R�9

w�9 r –R�9j jð Þ that divides thespace into (mutually overlapping) regions assignedto each atom. mI�ðr –R�Þ is, therefore, a localizedfunction that can be stored in a fixed-size numer-ical array. I� denotes the atomic bondingenvironment of the atom �. After the chargedensity motifs are obtained from small prototypesystems, the total charge density of the largenanosystem is obtained simply as the sum ofmotifs assigned to each of the atoms:

�patchðrÞ ¼XR�

mI� r –R�ð Þ ð9Þ

Once the charge density is obtained using thecharge patching procedure, the single-particleHamiltonian can be generated by solving thePoisson equation for the Hartree potential andusing the LDA formula for the exchange-correlationpotential. The energies and wave functions of a fewstates around the gap can then be found using themethods developed to find a few eigenvalues of theHamiltonian only, such as the folded spectrummethod (FSM) [34] (that is described in Section1.07.2.2).

The CPM was used to generate the charge den-sities of carbon fullerenes [33], semiconductor alloys[28], semiconductor impurities [29], organic mole-cules and polymers [35], and semiconductorquantum dots [32]. The resulting patched chargedensity is typically within 1% of the self-consistentlycalculated LDA charge density, and the correspond-ing energies are within 30 meV. Typical numericaluncertainty (due to basis function truncations anddifferent nonlocal pseudopotential treatments) of anLDA calculation is about the same order of

magnitude. Therefore, the CPM can be consideredto be as accurate as the direct ab initio calculations.

There are, however, cases where CPM cannot beused. One example is the total dipole moment of anasymmetric quantum dot [36]. Such a dipole momentcan also induce an internal electric field and causethe long-range charge transfer in the system. It is,therefore, necessary to solve the charge density self-consistently, which can be done using the DFT/LDAmethod. However, a much more efficient linearscaling method to do such calculations has beendeveloped recently: the linear scaling three-dimensional fragment (LS3DF) method [37].Within the LS3DF method, the system is dividedinto many small fragments. The wave functions andcharge densities of each fragment are calculatedseparately, each within the standard DFT/LDAmethod, using a group of a small number of computerprocessors. After the fragment charge densities areobtained, they are patched together to get the chargedensity of the whole system using a novel schemethat ensures that the artificial surface effects due tothe system subdivision will be cancelled out amongthe fragments. The patched charge density is thenused to solve a global Poisson equation for the globalpotential. An outside loop is iterated, which yieldsthe self-consistency between the global charge den-sity and the input potential. Due to the use of thisnovel patching scheme, the LS3DF is very accurate,with its results (including the dipole moments) essen-tially the same as the original direct DFT calculationresults [37], but with potentially 1000 times speed-ups, for systems with more than 10 000 atoms. As thesystem grows larger, there are more fragments (whilethe fragment size is fixed); thus, more processorgroups can be used to solve them. This provides aperfect parallelization to the number of processors.Meanwhile, the total computational cost is propor-tional to the number of fragments, and consequentlythe total number of atoms.

A well-known problem of the LDA-basedcalculations is that the band gap is severely under-estimated [38,39]. DFT is rigorously valid only forground-state properties, and there is no physicalmeaning for the Kohn–Sham eigen energies [27].This conceptual difficulty can be circumvented byusing time-dependent DFT, which is discussed later.In practice, however, one often restricts to the simpleempirical ways to correct the band-gap error. Onesuch way is to slightly modify the LDA Hamiltonianto fit the crystal bulk bandstructure, which can bedone, for example, by changing the s, p, and d


nonlocal pseudopotentials [29] to move the posi-tion of the conduction band while keeping theposition of the valence band unchanged. This approachis based on the assumption that the valence bandalignment predicted by the LDA is reliable.

For the treatment of colloidal quantum dots, onealso has to take care of the quantum dot surface. Thesurface of an unpassivated nanocrystal consists ofdangling bonds that introduce band-gap states. Oneway to remove these states is to pair the dangling-bond electron with other electrons. If a surface atomhas m valence electrons, this atom provides m/4electrons to each of its four bonds in a tetrahedralcrystal. To pair m/4 electrons in a dangling bond, apassivating agent should provide 2�m/4 additionalelectrons. To keep the system locally neutral, theremust be a positive 2�m/4 nuclear charge nearby.The simplest passivation agent can, therefore, be ahydrogen-like atom with 2�m/4 electrons and anuclear charge Z¼ 2�m/4. For IV–IV group mate-rials like Si, this is a Z¼ 1 hydrogen atom. For III–Vand II–VI systems, the resulting atoms have anoninteger Z; consequently, these are pseudohydro-gen atoms. These artificial pseudohydrogen atomsdo describe the essential features of good passivationagents and serve as simplified models for the realpassivation situations, where organic molecules withcomplicated and often unknown structure areinvolved.

1.07.2.2 Empirical Pseudopotential Method

The empirical pseudopotential method (EPM) wasintroduced in the 1960s by Cohen et al. [40,41] to fitthe bandstructure of bulk semiconductors. Withinthe EPM, the Schrodinger equation is given as

–1

2r2 þ V ðrÞ

� � iðrÞ ¼ Ei iðrÞ ð10Þ

with

V ðrÞ ¼Xatom

vatom r – Ratomj jð Þ ð11Þ

where Ratom are the positions of the atoms andvatom(r) are spherical atomic potentials that in aneffective manner take into account the effects ofnuclei, core, and valence electrons. The great successof the EPM was that it was actually possible to fit thebandstructure of the semiconductors using thissingle-particle approach.

In the EPM calculations, the plane-wave repre-sentation is typically used, that is, the wave function

is expanded as a linear combination of plane waves,where the summation is restricted only to reciprocallattice q vectors with kinetic energy smaller thancertain predefined value Ecut. To evaluate the result-ing Hamiltonian matrix in plane-waverepresentation, Fourier transforms of atomic poten-

tials vatom(jqj) are needed. Only a few of these arenonzero. These are used as adjustable parameters tofit the semiconductor bandstructure.

To apply the EPM to nanostructures, one needs tohave a continuous vatom(q) curve, since the supercell

is very large, and consequently q points are verydense. The continuous vatom(q) can be representedby a function of four parameters a1�a4

vðqÞ ¼ a1 q2 – a2ð Þa3ea4q2 – 1

or a sum of Gaussians

vðqÞ ¼X

ai e– ci q – bið Þ2

For a full description of the colloidal quantum dots,the pseudopotentials of surface passivating hydrogenor pseudohydrogen atoms need to be fitted as well.The pseudopotentials are fitted to experimental dataand first-principles calculations of bulk bandstruc-tures, clean surface work function, and the densityof states of chemisorbed surfaces.

Another approach to fit the pseudopotentials is tofit them directly to the LDA-calculated potential

[42] and then modify them slightly to correct theband-gap error. The vatom(q) obtained in such a man-ner are able to fit the band structure within 0.1 eVand have in the same time a 99% overlap with theoriginal LDA wave function. This approach, calledthe semiempirical pseudopotential method (SEPM),has been applied to CdSe [42], InP [43], and Si [42]nanostructures, representing II–VI, III–V, and IV–IVsemiconductor systems, respectively.

With the empirical or semiempirical pseudopo-tentials at hand, one is able to construct the single-particle Hamiltonian. The diagonalization of thisHamiltonian is a routine task in the case of bulksemiconductors due to a small number of atoms in

a supercell. However, this is no longer the case inquantum dots that contain a large number of atoms.Even the conventional conjugate gradient method[44] that is often used in ab initio calculationscannot be used since it scales as O (N3) due to anorthogonalization step, which is a necessary part ofthe algorithm. Fortunately, for the analysis of mostelectronic, transport, and optical properties of


semiconductor nanosystems, only the states in thespectral region close to the band gap are relevant,and there is no need to find all the eigenstates ofthe Hamiltonian. The FSM, specialized to find theeigenstates in a certain spectral region only, hastherefore been developed by Wang and Zunger[34]. The method is based on the fact that theHamiltonians H and (H� Eref)

2 have the sameeigenvectors and that the few lowest eigenvectorsof (H� Eref)

2 are the eigenvectors of H closest tothe energy Eref. The lowest eigenstates of(H� Eref)

2 are then solved using the conjugate gra-dient method. It turns out that the use of(H� Eref)

2 compared to H slows down the conver-gence but with the use of preconditioners and alarge number of iterations, convergence can still beachieved. The FSM within the plane-wave repre-sentation has been implemented in the parallelcode Parallel Energy SCAN (PESCAN) [45]. Itcan be routinely used to calculate systems with afew thousand atoms, or even near-million atomsystems [46]. Since only a few wave functions arecalculated, the computational effort scales linearlyto the size of the system. Linear scaling method forthe calculation of the total and local electronicdensity of states and the optical absorption spec-trum has been developed by Wang [47]. Thereader is referred to Ref. [47] for the descriptionof this method, called the generalized momentsmethod (GMM).

Another method for solving of the EPMHamiltonian is the linear combination of bulk Blochbands (LCBBs) method [48]. The disadvantage of theplane-wave expansion is that it does not lend itself tosystematic approximations. A basis set in which suchapproximations can be naturally made is the basis offull-zone bulk Bloch states. In this basis, the wavefunction expansion reads

ðrÞ ¼XNB

n

XNk

k

Ck;n �0knðrÞ ð12Þ

where

�0knðrÞ ¼

1ffiffiffiffiNp uknðrÞeik ? r ð13Þ

is the bulk Bloch function of the constituent bulksolid, where n is the band index, k is the supercellreciprocal-lattice vector, N is the number of primarycells in the supercell. The LCBB expansion allowsone to select the physically important bands n andk-points. As a result, the number of basis functions

can be significantly reduced compared to theplane-wave basis. It turns out that it is possible touse a fixed number of basis functions to achieve thesame degree of accuracy for different system sizes, incontrast to the plane-wave basis where the number ofbasis functions scales linearly with system size. Theorigin of this effect is the fact that when the size of thesystem increases, the envelope function of the elec-tronic state becomes smoother, and therefore themaximum value of the k-vector needed to representit becomes smaller. This makes the LCBB methodideal for studying very large systems such asembedded quantum dots.

1.07.2.3 Tight-Binding Methods

The tight-binding (TB) method [49] is the simplestmethod that still includes the atomic structure of aquantum dot in the calculation [50,51,52,53]. In theTB method, one selects the most relevant atomic-like orbitals ji�i localized on atom i, which areassumed to be orthonormal. The single-particlewave function is expanded on the basis of theselocalized orbitals as

j i ¼X

i�

ci�ji�i ð14Þ

and therefore the TB single-particle Hamiltonian isof the form

H ¼X

i�

"i�ji�ihi�j þ1

2

Xi�; j�

ti�; j�ji�i j�jh ð15Þ

where "i� are the energies of the orbitals (the on-siteenergies), while ti�, j� are the hopping integralsbetween different orbitals, which can be restrictedto include only nearest neighbors or next-nearestneighbors. For the sake of notational simplicity, theform that does not include the spin–orbit interactionand therefore does not mix the states of different spinwas presented. The extension to include spin orbitinteraction is straightforward. The most popular fla-vor of TB is the empirical TB where the parametersof the Hamiltonian are treated as phenomenologicaland fitted to reproduce the bulk band structureobtained from experiment or higher level calcula-tions. In such an approach, the atomic orbitals arenot treated explicitly, since the whole spectrum ofthe single-particle Hamiltonian is determined by theonsite energies and hopping integrals. The wavefunction is represented by the coefficients ci� thatslowly vary from site to site.


In the TB method, one restricts the atomic orbitalsto include only a few for each atom. Since one is

typically interested in states around the energy

gap, one has to select the orbitals that define these

states. In III–V, IV–IV, and II–VI semiconductors,

these are typically the s, px, py, pz orbitals and some-

times d orbitals. Quite often, an additional s-like

orbital called s� is added to provide an additional

degree of freedom in fitting of the TB parameters,

which leads to models such as sp3s� [54] and sp3d5s�

[55]. In these models, the size of the resulting

Hamiltonian matrix is nN� nN, where N is the num-

ber of atoms and n is the number of orbitals per atom

(n¼ 10 for sp3s� with spin and n¼ 20 for sp3d5s� with

spin). Due to nearest-neighbor approximation, the

matrix is sparse, and efficient methods for the diag-

onalization of sparse matrices can therefore be

exploited.One problem of the TB method is the lack of

explicit basis functions. Although these can be

added after the TB eigenstates have been calcu-

lated, these basis functions are not an intrinsic part

of the TB Hamiltonian and its fitting process; thus,

their compatibility is an issue. This causes problems

to calculate physical properties such as dipole tran-

sitions and Coulomb and exchange interactions.

Another issue in treating quantum dot heterostruc-

tures is how to choose the parameters at the

interface of two materials, since only the parameters

for bulk materials are available. An approximation

needs to be introduced, usually by assuming the

parameters at the interface as a certain average of

the TB parameters of the two materials. In colloidal

quantum dots, the surface has to be passivated.

Here, we give an example of how this is done in

the case of Si nanocrystals. The surface is passivated

by H atoms, where the TB nearest-neighbor matrix

elements VH–Si between H and Si are scaled from

the Si–Si matrix elements VSi–Si according to the

Harrison’s rule [56]: VH–Si¼VSi–Si(dSi–Si/dH–Si)2,

where dSi–Si and dH–Si are the bond distances [57].

Another way to treat the surface passivation is simply

to remove the dangling-bond states from the calcu-

lated results or even from the Hamiltonian before the

matrix is diagonalized. This is done by removing the

hybrid sp3 dangling-bond orbital from the TB

Hamiltonian basis set (e.g., by removing the

Hamiltonian matrix columns and rows expanded by

these sp3 bases) [58]. This is a unique way of artificial

passivation only applicable to TB calculations. The

ability to describe the surface atomistically is a big

advantage of the TB model compared to the k ? pmodel, which is described next.

1.07.2.4 k ? p Method

The previously described methods treat explicitlythe atomistic details of the nanostructure, whichtherefore leads to their high accuracy and reliabilitybut also to a significant computational cost. In thek ? p method, central quantities are the slowly vary-ing envelope functions that modulate the rapidlyvarying atomistic wave function. Historically, thek ? p method was introduced to describe the bulkband structure around a certain special point in theBrillouin zone, and later on it was extended todescribe heterostructures.

Let the Hamiltonian of an electron in a semicon-ductor be

H ¼ p2

2þ V0ðrÞ þ Hso ð16Þ

where p is the momentum operator, V0(r) the crystalpotential (including nuclei, core electrons, and self-consistent potential of valence electrons), and Hso thespin–orbit interaction Hamiltonian arising from rela-tivistic corrections to Schrodinger equation given by

Hso ¼�2

4rV0ðrÞ � p½ �?s ð17Þ

where � is the fine structure constant, and s is avector of Pauli matrices:

�x ¼0 1

1 0

" #; �y ¼

0 – i

i 0

" #; �z ¼

1 0

0 – 1

" #ð18Þ

The envelope representation of the wave function ofan electron is given by

�ðrÞ ¼X

i

iðrÞuiðrÞ ð19Þ

where ui(r) form the complete orthonormal set offunctions with periodicity of the Bravais lattice, and i(r) are slowly varying envelope functions. Themost common choice of the functions ui are bulkBloch functions at the � point. After the replacementof equation 19 in the Schrodinger equation and mak-ing an approximation that eliminates the nonlocalterms that appear in the derivation, one arrives at[59,60]

–1

2r2 mðrÞ þ

Xn

ð – iÞpmn?r nðrÞ

þX

n

HmnðrÞ nðrÞ ¼ E mðrÞð20Þ


Since the second term in equation 20 is crucial indetermining the Hamiltonian matrix (and (�i)rbecomes the k-vector if, e.g., the envelope functionis expanded in plane waves), the method beingdescribed is called the k ? p method. The terms inthe previous equation are given by

pmn ¼1

�

ZumðrÞ�punðrÞd3r ð21Þ

where the integration goes over the volume of thecrystal unit cell �, and Hmn(r) is the term that, awayfrom the interfaces, reduces to the bulk matrixelements of the Hamiltonian

Hmn ¼1

�

ZumðrÞ�HunðrÞd3r ¼ Emmn ð22Þ

where Em is the band edge of band m. In practice, onehas to restrict to a finite number of bands.Historically, the k ? p method was first applied tovalence band (six-band Hamiltonian) [61–62], andlater on the conduction band was added (eight-bandHamiltonian) [63].

The explicit form of the eight-band Hamiltonianfor the crystals with zinc-blende structure (such as,InAs, GaAs, AlSb, CdTe, GaP, GaSb, InP, InSb,ZnS, ZnSe, and ZnTe) is given below. ThisHamiltonian also perturbatively includes the effect

p

– P

p

p

of remote bands. Since the point Td symmetry groupof the zinc-blende crystal is a subgroup of the dia-mond group of Ge and Si, the same k ? pHamiltonian can be applied to these semiconduc-tors, as well. On the basis jJJzi that diagonalizes thebulk Hamiltonian at k¼ 0

j1i¼ 1

2; –

1

2

��¼ jS #i

j2i¼ 1

2;1

2

��¼ jS "i

j3i¼ 3

2;1

2

��¼ –

i ffiffiffi6p jðX þ iY Þ #i þ i

ffiffiffi2

3

rjZ "i

j4i¼ 3

2;3

2

��¼ i ffiffiffi

2p jðX þ iY Þ "i

j5i¼ 3

2; –

3

2

��¼ –

i ffiffiffi2p jðX – iY Þ #i

j6i¼ 3

2; –

1

2

��¼ i ffiffiffi

6p jðX – iY Þ "i þ i

ffiffiffi2

3

rjZ #i

j7i¼ 1

2; –

1

2

��¼ –

i ffiffiffi3p jðX – iY Þ "i þ i ffiffiffi

3p jZ #i

j8i¼ 1

2;1

2

��¼ –

i ffiffiffi3p jðX þ iY Þ #i – i ffiffiffi

3p jZ "i

ð23Þ

the eight-band k ? p Hamiltonian reads (where thedefinition k¼�ir was introduced)

ffiffiffi3V –

ffiffiffi2p

U –Uffiffiffi2p

Vþ

0 –Vffiffiffi2p

V U

R 0

ffiffiffi3

2

rS –

ffiffiffi2p

Q

0 R –ffiffiffi2p

R1ffiffiffi2p S

–Q Sþ1ffiffiffi2p Sþ

ffiffiffi2p

Rþ

S –P þ Qffiffiffi2p

Q

ffiffiffi3

2

rSþ

1ffiffiffi2

Sffiffiffi2p

Q – P –� 0

ffiffiffi2R

ffiffiffi3

2

rS 0 –P –�

377777777777777777777777777777777777775

ð24Þ
Hk ¼
A 0 Vþ 0

0 A –ffiffiffi2p

U –ffiffiffi3p

Vþ

V –ffiffiffi2p

U –P þ Q – Sþ

0 –ffiffiffi3p

V – S – P –Q

ffiffiffi3p

Vþ 0 Rþ 0

–ffiffiffi2p

U –Vþ 0 Rþ

–Uffiffiffi2p

Vþffiffiffi3

2

rSþ –

ffiffiffi2p

Rþ

ffiffiffi2p

V U –ffiffiffi2p

Q1ffiffiffi2p Sþ

266666666666666666666666666666666666664


where

A ¼ EC þ A9k2 þ k2

2

U ¼ 1ffiffiffi3p P0kz

V ¼ 1ffiffiffi6p P0 kx – iky

P ¼ – EV þ 1k2

2

Q ¼ 21

2k2

x þ k2y – 2k2

z

� �

R ¼ –

ffiffiffi3p

22 k2

x – k2y

� �– 2i3kxky

h iS ¼

ffiffiffi3p3kz kx – iky

In previous equations jSi, jXi, jYi, and jZi are thebulk Bloch functions that transform as s, x, y, and z

under the action of the symmetry group

P0 ¼ – i S px

�� X �¼ – ihSjpy jY i ¼ – i S pz

�� Z �ð25Þ

is the interband matrix element of the velocity opera-tor [64] usually reported in energy units as EP ¼ 2P2

0 ,the parameter A9 is related to the conduction bandeffective mass as

A9 ¼ 1

2m�–

P20

Eg þ 13 �

–1

2ð26Þ

� is the spin–orbit splitting, Eg is the energy gap (theactual energy gap after the effect of � was taken intoaccount) equal to Eg¼ EC� EV, while 1, 2, and 3

are the Luttinger parameters [61] of the eight-bandmodel that can be expressed in terms of the para-meters of the six-band model L

1 ; L2 ; and L

3

1 ¼ L1 –

EP

3Eg þ�

2 ¼ L2 –

1

2

EP

3Eg þ�

3 ¼ L3 –

1

2

EP

3Eg þ�

Since material parameters in the Hamiltonian of aquantum dot are position dependent and the k opera-tors do not commute with coordinate operators, anambiguity arises about the proper choice of operatorordering. It is necessary to choose the ordering insuch a way that the Hamiltonian remains hermitian;however, this condition still does not give a uniquechoice. The most widely used [65,66,67,68,69] opera-tor ordering in k ? p-based quantum dot electronic

structure calculations is heuristic, symmetricalarrangement of operators

f ðrÞki kj !1

2ki f ðrÞkj þ kj f ðrÞki

f ðrÞki ! 1

2ki f ðrÞ þ f ðrÞki

ð27Þ

It has been pointed out that such ordering of opera-tors can lead to unphysical solutions in somecircumstances [70]. One can derive the appropriateform of the envelope function Hamiltonian withproper operator ordering starting from the empiri-cal pseudopotential [59] or the LDA Hamiltonian[71]; however, such Hamiltonians are still notwidely used.

A variety of numerical methods can be used tosolve the k ? p Hamiltonian; these include the finite-difference methods [66,65,72,73] and the wave func-tion expansion methods, where the basis functionscan be plane waves [74,75,76,77,69,78], the eigen-functions of the particle in a cylinder with infinitewalls [79,80,81], or eigenfunctions of a harmonicoscillator [82].

While the k ? p model can be quite reliable for largeembedded quantum dots, the colloidal quantum dotsare often only a few nanometers in size. In reciprocalspace, this could correspond to the k point at 1/3toward the Brillouin zone boundary, where the k ? pmight no longer be adequate. Indeed, it was found thatthe k ? p result compared to the result of a moreaccurate calculation might differ by 50% in theconfinement energy [83], and sometimes it couldchange the ordering of the states [84]. Without care,spurious states in the energy gap might appear in k ? pcalculations [85]. These states appear as the conse-quence of the fact that k ? p Hamiltonian does notcorrectly represent the bandstructure for k-vectorsfar away from � point and can give states in the gapfor these k-vectors. The finite-difference method is, inparticular, susceptible to the appearance of thesestates. The wave function expansion methods are lesssusceptible to this [86] since by the expansion in afinite basis set, the high k components of the envelopefunction are effectively filtered out. Another issue isthat the k ? p Hamiltonian with a limited number ofbands has a larger symmetry group than the truesymmetry group of the system. This weakness fromthe fundamental point of view can be turned into astrength from the computational point of view, as itallows for block diagonalization of the Hamiltonianand therefore a more efficient solution of the problem[87,77,81].


1.07.2.5 The Effect of Strain

In previous sections, it was assumed that positions of

atoms in a quantum dot are known a priori and that

local arrangement is the same as in the bulk crystal.

However, in real structures this is certainly not the

case. Self-assembled quantum dots are grown by

depositing layers of material with a different lattice

constant than the substrate, and therefore the quan-

tum dot is strained. In colloidal quantum dots, there

is also some relaxation of atoms close to the surface.

It is well understood that strain has a strong effect on

the electronic structure of semiconductors.

Therefore, in this section, we describe how the effect

of strain can be included in each of the methods

described previously.Within the framework of DFT, the effect of strain

appears naturally in the formalism itself. One starts

with a reasonable initial guess for the positions of

atoms in the structure, then self-consistently solves

the Kohn–Sham equations and obtains the forces on

all atoms. One then moves the atoms in the direction

of forces and obtains the new atomic configuration

and solves the Kohn–Sham equations again, and the

whole procedure is repeated until forces become

close to zero. In such a way one obtains a new,

relaxed configuration for the positions of atoms in

the structure. Unfortunately, this procedure is prac-

tical only for small systems and is not feasible for

larger systems, such as quantum dots.A widely used methodology to overcome these

difficulties is to model the total energy of the system

through a classical force field, that is, to express it as a

function of atomic coordinates only. The valence

force field (VFF) model of Keating [88] and Martin

[89] is mostly used in inorganic semiconductors for

that purpose. Within the VFF model, the energy of

the system is modeled as [90]

E¼ 1

2

Xi

Xnn

j

3�ij

8ðd ð0Þij Þ2ðRi–RjÞ2–ðd ð0Þij Þ

2h i2

þ1

2

Xi

Xnn

j ;k>j

3�i;jk

8d 0ij d 0

ik

ðRj–RiÞ?ðRk–RiÞ–cos�0ðjikÞd 0ij d 0

ik

h i2

ð28Þ

where d 0ij are the equilibrium bond lengths between

atoms i and j, and �0(jik) is the equilibrium anglebetween bonds ij and ik, which is a constant inzinc-blende materials (�0� 109.47). In the case ofzinc-blende material, the constants � and � are relatedto elastic constants of the material through [90]

C11 þ 2C12 ¼ffiffiffi3p

4d 03�þ �ð Þ

C11 –C12 ¼ffiffiffi3p

d 0�

C44 ¼ffiffiffi3p

4d 0

4��

�þ �

ð29Þ

Although there are three elastic constants and onlytwo parameters � and �, it is possible to choose � and� to fit the C ’s of the most zinc-blende materialswithin a few percent. To obtain the relaxed atomicstructure, one again starts with a reasonable guess forinitial atomic structure and then minimizes E inequation 28 using some of the standard methods forfinding the local minimum of a function, such as theconjugate gradient method. The atomic structureobtained can be used as an input to any of theatomistic approaches previously described: chargepatching, empirical pseudopotentials, and TB.

It has been pointed out in Section 1.07.2.1 that thecharge density motifs used in the CPM depend on the

local environment of the atom. In strained structures,

bond lengths and angles change compared to the idealones, which therefore represents the change in the envir-

onment that affects the motifs. To include this effect, one

introduces the so-called derivative motifs, defined as the

change in the motif due to a particular bond length or

angle change. These motifs can also be extracted from

small-system calculations on prototype structures with

slightly changed bond lengths or angles. Once the motifs

and derivative motifs are obtained, the total charge

density is constructed and the calculation of the electro-

nic structure can be performed as previously described.It might seem at first sight that it is not necessary

to introduce any modifications to the empirical pseu-

dopotential Hamiltonian to include the effects of

strain. However, it turns out that within such an

approach it would be difficult to correctly describe

the dependence of the valence band maximum state

on the hydrostatic strain [91]. Therefore, a strain-

dependent term is introduced for the local part of the

pseudopotential of the atom of type � in the form

vloc� ðr; eÞ ¼ vloc

� ðr; 0Þ 1þ �TrðeÞ½ � ð30Þ

where � is a fitting parameter and Tr(e)¼ exxþ eyyþezz is the trace of the strain tensor. The SEPM strain-dependent Hamiltonian obtained this way can besolved by representing it on the basis of plane wavesor bulk Bloch bands. The extension of the unstrainedcases to the strained cases for the basis of plane waves isstraightforward. On the other hand, this is not true if


bulk Bloch bands are used since the Bloch functions ofthe unrelaxed system form a poor basis set for therelaxed system. Therefore, one needs to use thestrained linear combination of Bloch bands, and themethod is then referred to as the SLCBB method. Thereader is referred to Ref. [48] for technical details ofthe implementation of the SLCBB method.

The natural way to introduce strain in TB modelsis through the dependence of hopping integrals onbond lengths and bond angles. The dependence onbond lengths is modeled by scaling the Slater–Kostertwo-center integrals [92] from which the hoppingintegrals are constructed as

V ¼ V0d0

d

� ��ð31Þ

which is a generalization [52,93] of Harrison’s d �2

rule [56,50]. In the above equation, V0 is theintegral for equilibrium bond length d0 and V

the integral when the bond length is d. The

–

–

p

p

change in bond angles in the system leads todifferent relative orientation of orbitals of neigh-boring atoms and consequently to a differenthopping integral. This effect is naturallyincluded through the Slater–Koster [92] tablesof matrix elements in terms of the two-centerintegrals and direction cosines. Furthermore,there is a question whether the influence ofstrain on onsite energies should also be included.This is indeed done in many recent works[94,93,95], although different methods are used.Currently, there does not seem to exist a uniqueand simple model for the inclusion of this depen-dence as for the hopping integrals.

In k ? p models, the effect of strain is includedthrough the bulk deformation potential parameters

that can be either measured or determined from ab

initio calculations. In the case of eight-band

Hamiltonian for zinc-blende crystals, the strain con-

tribution to the Hamiltonian reads

ffiffiffi3p

vffiffiffi2p

u uffiffiffi2p

vþ

0 v –ffiffiffi2p

v – u

r 0

ffiffiffi3

2

rs –

ffiffiffi2p

q

0 r –ffiffiffi2p

r1ffiffiffi2p s

p – q sþ1ffiffiffi2p sþ

ffiffiffi2p

rþ

s – p þ qffiffiffi2p

q

ffiffiffi3

2

rsþ

1ffiffiffi2

sffiffiffi2p

q – p 0

ffiffiffi2r

ffiffiffi3

2

rs 0 – p

377777777777777777777777777777777777775

ð32Þ
HS ¼
ace 0 – vþ 0

0 aceffiffiffi2p

uffiffiffi3p

vþ

– vffiffiffi2p

u – p þ q – sþ

0ffiffiffi3p

v – s – p – q

–ffiffiffi3p

vþ 0 rþ 0

ffiffiffi2p

u vþ 0 rþ

u –ffiffiffi2p

vþffiffiffi3

2

rsþ –

ffiffiffi2p

rþ

–ffiffiffi2p

v – u –ffiffiffi2p

q1ffiffiffi2p sþ

266666666666666666666666666666666666664

32

where

e ¼ e11 þ e22 þ e33

p ¼ ave

q ¼ b e33 –1

2e11 þ e22ð Þ

� �

r ¼ffiffiffi3p

2b e11 – e22ð Þ – ide12

s ¼ – d e13 – ie23ð Þ

u ¼ 1ffiffiffi3p P0

X3

j¼1

e3j kj

v ¼ 1ffiffiffi6p P0

X3

j¼1

e1j – ie2j

kj

where ac and av are the conduction and valence bandhydrostatic deformation potentials, respectively, and


b and d are the shear deformation potentials. Thestrain tensor that enters the Hamiltonian (32) canbe obtained either from the VFF model (previouslydescribed) or from the continuum mechanical (CM)model.

In the CM model, the quantum dot structure ismodeled by an elastic classical continuum medium

whose elastic energy is given by

W ¼ 1

2

Xijkl

Zd3r ijkl eij ekl ð33Þ

where ijkl is the elastic tensor relating the stress andstrain tensor by Hooke’s law

�ij ¼X

kl

ijkl ekl ð34Þ

In the crystals with zinc-blende lattice, the elastictensor is of the form

0.03

Subtrate

(a) εxx

(b) εzz

(c) Tr(ε)

Dot Cap

AECE

0.00

–0.03

–0.06

0.00

Str

ain

0.00

–0.10

–0.06

–0.12–20 0

Position along [001]20 40

Figure 1 Strain profiles of InAs/GaAs square-based pyram

direction through the pyramidal tip (left-hand side). The "xx,shown respectively in parts (a), (b) and (c). The difference b

elasticity (CE) and atomistic elasticity (AE) is given in parts (

around the interfaces, while the strains in the barrier (GaAssignificant difference is also found inside the quantum dot w

about 7% due to the lattice mismatch. Reproduced with pe

and Zunger A (1998) Comparison of two methods for describ

Physics 83: 2548–2554.

ijkl ¼ C12ij kl þ C44 ikjl þ iljk

þ Can

X3

p¼1

ipjpkplp

ð35Þ

where C12, C44, and Can¼C11�C12� 2C44 are theelastic constants. The finite element discretizationand minimization of the functional (33) leads to asystem of linear equations that can be efficientlysolved.

There have been several comparisons in theliterature between the VFF and CM models

[66,90,96]. While certain differences have been

obtained, the results of the two models give

overall agreement, as can be seen from a

comparison between strain distribution in a pyr-

amidal InAs/GaAs quantum dot from Ref. [90]

that is given in Figure 1. From the fundamental

point of view, the advantage of the VFF model is

that it captures the atomistic symmetry of the

system, while the CM models have a higher

(d)

(e)

(f)

Dot

AE–CE

CapSubstrate

Position along [001]–20 0 20 40

–0.02

0.00

–0.01

0.00

0.01

0.02

–0.02

0.00

idal quantum dots and the differences along the z-

"zz and Tr(")¼ "xx þ "yy þ "zz components of strain areetween these components calculated using continuum

d), (e) and (f) respectively. The discrepancy is the largest

substrate and capping layer) agree well within 0.5%. Ahere the InAs experience large compressive strains at

rmission from Pryor C, Kim J, Wang LW, Williamson AJ,

ing the strain profiles in quantum dots. Journal of Applied


symmetry group. From the computational pointof view, the VFF model is more demanding as

the displacement of each atom is considered, incontrast to the CM models where a grid of thesize of lattice constant or even larger may be

used, leading to a smaller number of variablesto be handled. In several important cases, thereare analytical or nearly analytical solutions of the

CM model [97,98]. However, these advantages ofthe CM models are becoming less important as

modern computers can handle the VFF calcula-tions quite easily.

The nonself-consistent methods describedabove do not allow for long-ranged charge redis-tributions and therefore neglect the effects such aspiezoelectricity where charge is moved due to

strain. The piezoelectric potential then has to becalculated independently and added as an addi-tional potential. The components of piezoelectric

polarization in a crystal of arbitrary symmetry aregiven as

Pi ¼X3

k;l¼1

"ikl ekl ð36Þ

where "ikl are the piezoelectric constants of the mate-rial. In a crystal with zinc-blende symmetry, the onlynonzero components of "ikl are

"123 ¼ "132 ¼ "213 ¼ "231 ¼ "312 ¼ "321 ð37Þ

The charges induced by piezoelectric polarizationcan then be calculated, and the additionalpiezoelectric potential is obtained from the solu-tion of Poisson equation. It has been recentlyrealized that in addition to the first-orderpiezoelectric effect given by equation 36,second-order piezoelectric effects might beimportant as well [99].

1.07.3 Many-Body Approaches

The methods presented in Section 1.07.2 give a strat-egy for calculating the single-particle states. Thesecan be very useful for calculating the optical proper-

ties, as demonstrated, for example, in Section 1.07.4.3.Nevertheless, there are cases when the many-body

nature of electron–hole excitations should be directlyconsidered. The approaches along this line aredescribed in Section 1.07.3.

1.07.3.1 Time-Dependent DFT

Within the time-dependent DFT (TDDFT)[100,101], one solves the time-dependent Kohn–Sham equations

iqqt iðr; tÞ ¼ –

1

2r2 þ V ðr; tÞ

� � iðr; tÞ ð38Þ

where

�ðr; tÞ ¼XMi¼1

iðr; tÞj j2 ð39Þ

The potential V(r, t) should depend, in principle, oncharge density in all times before t. A widely usedapproximation is the adiabatic LDA in which it isassumed that V(r, t) depends only on �(r, t), and thatthe functional form of this dependence is the same asin LDA in time independent DFT. We refer to thisapproximation as time-dependent LDA (TDLDA).

The TDLDA can be used to calculate the opticalabsorption spectrum by adding the external electro-magnetic field perturbation potential to equation 38and solving the equations by explicit integration intime [102]. Another approach is to assume that theperturbation is small and use the linear responsetheory. The exciton energy can then be found fromthe eigenvalue problemX

jl

"i – "kð Þij kl þ ðfi – fkÞKik;jl ð!Þ� �

Cjl ¼ !Cik ð40Þ

where "i and "k are the LDA ground-state Kohn–Sham eigen energies, and fi and fk are the occupationnumber of Kohn–Sham eigen states i and k. WithinTDLDA, Kik, jl becomes independent of ! and isgiven as

Kik;jl ¼Z Z

d r1dr2 iðr1Þ kðr1Þ½ 1

r1 – r2j j

þ r1 – r2ð Þ qmLDAxc � r1ð Þð Þq� r1ð Þ

i j r2ð Þ l r2ð Þ

ð41Þ

where �LDAxc (p) is the LDA exchange-correlation

potential. The first term in equation 41 is theexchange interaction, while the second can be calledthe screened Coulomb interaction. The justificationof this assignment would require a comparison withequations from other approaches, such as the config-uration interaction and GW+Bethe–Salpeterequation (BSE). It might also seem surprising thatthe screened Coulomb interaction is not a nonlocalintegral between r1 and r2. This is because in theLDA, the exchange-correlation term is a localfunctional of charge density.


TDLDA appears to work quite well for optical spec-tra of small clusters and molecules. The results ofTDLDA can then agree quite well with experimentalmeasurements, as shown, for example, for the case ofSiH4 in Ref. [103]. These results are significantlyimproved compared to bare LDA results, which is dueto exchange interaction in equation 40, which can bequite strong in such small systems. The screenedCoulomb interaction in equation 40, however, does notplay a significant role then, as also shown in Ref. [103].

On the other hand, the TDLDA is not as accuratefor larger systems. For a bulk system, it is known thatthe TDLDA band gap will be the same as the LDAband gap [104,49]. The TDLDA does not provide abetter bulk optical absorption spectrum than theLDA, as shown in Ref. [103]. The origin of theseproblems is the screened Coulomb interaction inequation 40, which then gives a significant contribu-tion. However, such diagonal form of screenedCoulomb interaction is not able to correctly describeits long-range behavior.

Density functionals other than the LDA can alsobe used in conjuction with the TDDFT. A verypopular approach is to use the hybrid B3LYP func-tional [105]. Within the B3LYP approach, the totalenergy is modeled as a linear combination of theexact Hartree–Fock exchange with local and gradi-ent- corrected exchange and correlation terms. Thecoefficients in the linear combination were chosen tofit the properties of many small molecules. TheB3LYP method gives accurate bandgaps for variousbulk crystals [106]. Since it contains the explicitexchange integral, it introduces the long-rangeCoulomb interaction in equation 40. Therefore, theTDDFT–B3LYP can overcome the two difficultiesof TDLDA previously discussed.

1.07.3.2 Configuration Interaction Method

When the single-particle states are obtained, one canform many-body excitations by creating Slater deter-minants out of these single-particle states. One canthen diagonalize the many-body Hamiltonian in theHilbert space formed from a restricted set of suchdeterminants. This approach is called the configura-tion interaction (CI) method. When one is interestedin excitons, the wave function is assumed as

� ¼XNv

v¼1

XNc

c¼1

Cv;c �v;c ð42Þ

where �v,c is the Slater determinant when theelectron from the valence band state v is excited to

the conduction band state c. The eigenvalue problemof the Hamiltonian then readsX

v9c9

Hvc;v9c9Cv9c9

¼Xv9c9

½ Ev – Ecð Þv;v9c;c9þKvc;v9c9 – Jvc;v9c9�Cv9c9

¼ ECvc ð43Þ

where Ev and Ec are the single-particle eigenenergies,E is the exciton energy, and Kvc;v9c9 and Jvc;v9c9 are theexchange and Coulomb interactions, respectively

Kvc;v9c9 ¼Z Z

c9 r1ð Þ �v9 r1ð Þ v r2ð Þ �c r2ð Þ�" r1; r2ð Þ r1 – r2j j dr1dr2 ð44Þ

Jvc;v9c9 ¼Z Z

v r1ð Þ �v9 r1ð Þ c9 r2ð Þ �c9 r2ð Þ�" r1; r2ð Þ r1 – r2j j dr1dr2 ð45Þ

The effective dielectric screening used in equations44 and 45 is of the form

1

�" r1; r2ð Þ r1 – r2j j ¼Z" – 1

bulk r1; rð Þ 1

r – r2j j dr ð46Þ

where " – 1bulk r1; rð Þ is the bulk inverse dielectric function

that differs from the one of the quantum dot nanos-tructure, which also contains the surface polarizationpotential P discussed in Section 1.07.2. The use of bulkinverse dielectric function is in line with the fact thatthe single-particle energies Ec and Ev are obtained fromthe Schrodinger equation that does not contain thesurface polarization term. If the surface polarizationterm is used in the single-particle equation, then thefull inverse dielectric function should be used and thesurface polarization terms will roughly cancel out.

There are arguments that the exchange interac-tion Kvc;v9c9 should not be screened, which come from

the two-particle Green’s function construction,

where screening of the exchange term would cause

double counting [107]. Nevertheless, in practice, it is

found that the exchange consists of a long-range term

that should be unscreened and a short-range term

[108] that should be screened by the bulk dielectric

function [109,110]. The effective dielectric function�" r1; r2ð Þ used in equation 44 incorporates this

because �" r1 : r2ð Þ ! 1 for r1 – r2j j ! 0. The seeming

contradiction to the Green’s function argument can

be resolved by realizing that if only a limited config-

uration space is used in equation 43, the effect of

other unused configurations can be included in the

exchange screening term [107].The CI equation 43 has the same form as the

corresponding equation in the TDLDA (equation 40),

with the difference in the expressions for the exchange


and the Coulomb integrals. On top of a single-particlecalculation, the CI method was used to calculate verylarge systems, such as pyramidal quantum dots withnear by one million atoms [111]. It was also used tocalculate many-body excitations, such as multiexci-tons, and few electron excitations, and to study Augereffects [112]. All these calculations are made possibleby selecting a limited window of single-particle statesused in these configurations. It is difficult or impos-sible to study such systems using TDLDA orGWþBSE. One should nevertheless be cautiousabout the models used for screening in these multi-particle excitations.

1.07.3.3 GW and BSE Approach

Within this approach, one first calculates the quasi-particle eigenenergies, which is somewhat analogous tosingle-particle calculations in Section 1.07.2. These arethen used to solve the BSE for excitons, which is in somesense similar to CI equations of the previous section.

A quasiparticle is defined as the pole in frequencyspace in the single-particle Green’s function

G rt ; r9t 9ð Þ ¼ – i MjT rtð Þ y r9t 9ð ÞjMD E

ð47Þ

where rtð Þ is the particle creation operator, jMi theM particle ground state, and T the time-orderingoperator. Quasiparticle energies correspond to ener-gies for adding or removing one electron from thesystem [113]. Within the GW approximation [114],the appropriate single-particle equation reads

–1

2r2þ

Xatom

�vbareðr –RatomÞþZ

�ðr9Þjr – r9jd

3r9

" # iðrÞ

þZ X

ðr;r9;�iÞ iðr9Þdr9¼ �i ðrÞ ð48Þ

where

� r;r9;!ð Þ¼ –X

k

k rð Þ �k r9ð Þ

� fkW r;r9;"k–!ð Þþ 1

�

ZImW r;r9;!9ð Þ!–"k–!9þ i

d!9

� �ð49Þ

is the self-energy potential that replaces the LDAexchange-correlation potential of LDA single-particle equations.

W r; r9; !ð Þ ¼Z" – 1 r; r1; !ð Þ 1

r1 – r9j j dr1 ð50Þ

is the dynamically screened interaction, where"�1(r, r1, !) is the inverse dielectric function.

While equations 48 and 49 should, in principle, besolved self-consistently, one usually replaces the self-energy term with its expectation value with respect tothe LDA Kohn–Sham wave functions h kj�j ki,which constitutes the zeroth-order approximation ofthe GW procedure. It has been shown that the self-consistent calculations [115,116,117,118,119] makethe spectral properties worse. Such calculations areperformed with the use of pseudopotentials. It is pos-sible that self-consistency will not make the resultsworse if all-electron calculation is performed.

The two-particle Green’s function defined as

G r1t1;r2t2;r91t 91;r92t 92ð Þ¼ – M T r1t1ð Þ r2t2ð Þ y r92; t 92ð Þ y r91; t 91ð Þ

�� M �ð51Þ

contains the information about the excitonenergies. These can be retrieved by takingt1 ¼ t 91 þ 0 – and t2 ¼ t 92 þ 0 – and transforming to fre-quency space to obtain (in the condensed notation)G2(!), whose poles are the exciton energies. TheDyson equation for G2(!) reads [120,113]

G2 !ð Þ ¼ Gð0Þ2 !ð Þ þ G

ð0Þ2 !ð ÞK 9 !ð ÞG2 !ð Þ ð52Þ

where Gð0Þ2 !ð Þ is the noninteracting two-particle

Green’s function, and K9(!) is an electron–hole inter-action kernel. Equation 52 for the electron–hole pairis called the BSE [120]. It can be solved by expandingthe exciton wave function as

jM; Si ¼X

v

Xc

Cvc ayvbyc jMi ð53Þ

where avþ and bc

þ are the hole and electron creationoperators. The equations for Cvc coefficients are thengiven as

"c – "vð ÞCvc þXv9c9

Kvc;v9c9 – Jvc;v9c9

Cv9c9 ¼ �S Cvc ð54Þ

where "c , "v are the quasiparticle eigenenergiesobtained from equation 48, and �S is the excitonenergy. The Kvc;v9c9 is the same as in equation 44without the screening �" r1; r2ð Þ. The Jvc;v9c9, on theother hand, reads

Jvc;v9c9¼Z

drdr9 �c rð Þ c9 rð Þ v r9ð Þ �v9 r9ð Þ

� i

2�

Zd!e–i!0þW r;r9; !ð Þ

�h

�S –!– "c9–"vð Þþi0þð Þ–1

þ �Sþ!– "c–"v9ð Þþi0þð Þ–1i

ð55Þ

where W(r,r9,!) is the screened Coulomb interactiongiven by equation 50.


The GWþBSE approach is thought to be one ofthe most reliable methods for the calculation of theoptical absorption spectra and excited-state electro-nic structures. It has been used to calculate smallmolecules and bulk crystals. Unfortunately, its usefor the larger systems is hindered by the significantcomputational cost.

Excellent agreement with experimental resultswas obtained for the optical absorption spectrumof bulk Si calculated using the GWþBSE approach[121]. Within GWþBSE, the lower energy peakoriginating from the excitonic binding effect wasobtained for the first time. In contrast, previousLDA and TDLDA results were unable to predictthis peak due to the inadequacy of the local approx-imation for the Coulomb interaction.

The BSE (equation 54) is formally the same asthe linear response TDLDA (equation 40) and CI(equation 43), except for the meanings of single-particle energies and the exchange and screenedCoulomb interactions. In the GWþBSE approach,the quasiparticle electron (hole) energies are equal tothe total energy difference for adding (or removing)one electron from the system and include the surfacepolarization term P(r) from equation 1. On the otherhand, in the CI approach, the single-particle statesare obtained without including the P(r) term. Thesingle-particle states then do not correspond to thetotal energy differences for adding or removing oneelectron. A surface polarization term needs to beadded to relate them to ionization energies or elec-tron affinities [122]. In the linear response TDLDA(equation 40), the single- particle energy is theKohn–Sham LDA eigenenergy. It does not containthe surface polarization term, just like the single-particle energies in the CI approach. A differencebetween the TDLDA and CI approach is thatTDLDA contains LDA band-gap error, while in theCI approach the single-particle states can be found,for example, by EPM or SEPM that correct the LDAband-gap error.

The polarization term in the GW quasiparticleeigenenergy cancels out a term in J in equation 54because the same surface polarization term exists inW(r,r9,!) in equation 55. This cancellation corre-sponds to the cancellation of the PM(r1, r2) andP(r1), P(r2) terms in equation 2 of the classical phe-nomenological analysis. Delerue et al. [123] haveshown numerically that the Coulomb correctionterm cancels the polarization term in the self-energyof the quasiparticle eigenenergy. Consequently, theresults of the GWþBSE are expected to be similar

to the results of the CI where J is screened by thebulk dielectric function. In the TDLDA, wherethe Coulomb interaction is local, it is also screenedby the bulk dielectric function, in line with the factthe LDA single-particle states used in equation 40do not include the surface polarization term. Thedifferent cancellation schemes in TDLDA, CI, andGWþBSE (equations 40, 43, and 54) can be illu-strated by comparing the calculated absorptionspectra in these methods with the one obtainedfrom single-particle energies. It was shown in Ref.[124] that the BSE absorption spectrum of smallclusters of SinHm is red-shifted from the calculatedsingle-particle spectrum, which is mostly due tonegative surface polarization energies in theCoulomb interaction J. On the other hand, it wasshown in Ref. [103] that the TDLDA spectrum isblue-shifted from the single-particle LDA spectrum.There is no surface polarization in J or single-parti-cle energies then; thus, the exchange interactiondominates the spectrum shift. However, if totalLDA energy differences for adding or removing anelectron are used in equation 40, then the surfacepolarization must be considered [125] and Coulombinteraction cannot be calculated from equations 45and 46. The above-discussed cancellations are onlygood for spherical quantum dots. For quantum rods,wires, and the nanostructures of other shapes, theGWþBSE-like approach should be used. In the CIapproach, P(r) should be added to the single-particleenergy, while the full nanosystem inverse dielectricfunction (not the bulk one) should be used for�" r1; r2ð Þ in equations 44 and 45.

1.07.3.4 Quantum Monte Carlo Methods

Within the quantum Monte Carlo (QMC) method[126], the whole system is described by a many-bodywave function and the many-body Schrodinger equa-tion is solved using some of the Monte Carlotechniques such as variational Monte Carlo method(VMC) [127,128] or diffusion Monte Carlo method(DMC) [129,130].

In the VMC, the variational form of the many-body wave function �(X) is assumed as a Slaterdeterminant multiplied by a Jastrow term

� Xð Þ¼D" Rð ÞD# Rð ÞexpXM

i¼1

� rið Þ–XMi<j

u ri–rj

�� " #ð56Þ

where X¼ {ri , si} for i¼ 1, M. The Slater determi-nant D is usually constructed from single-particle


LDA or Hartree–Fock wave functions, while para-meterized forms are used to express � and u. Thetotal energy of the system is found by minimizing theexpectation value of the many-body Hamiltonian H,given as

E ¼ � Hj j�h i= �j�h i

The last expression can be rewritten as

E ¼

Z� Xð ÞH� Xð ÞdXZ

� Xð Þj j2dX

¼

ZH�ðXÞ� Xð Þ

� �� Xð Þj j2dXZ

� Xð Þj j2dX

ð57Þ

The last integral can be viewed as the average value ofthe quantity H� Xð Þ=� Xð Þ distributed in multi-dimensional space with a probability � Xð Þj j2. It canbe found using the Metropolis algorithm. In this algo-rithm, one simulates the path of the walker in themultidimensional space. The jump of the walker fromX to X9 is accepted if � ¼ � X9ð Þj j2= � Xð Þj j2 > 1 andaccepted with probability � if �< 1. The average valueof H� Xð Þ=� Xð Þ along the path of the walker gives E inequation 57.

In the DMC, one treats the many-body imaginarytime Schrodinger equation as the classical diffusionequation [129,130]. In this method, the many-bodywave function (not its square) corresponds to theequilibrium distribution of Monte Carlo walkers.However, for fermion system, antisymmetry isrequired for the many-body wave function. Thiscauses a sign problem, which is usually approxi-mately solved by a fixed nodal approximationwhere an auxiliary wave function is used to definethe fixed nodal hypersurface for the DMC wavefunction. Usually, the VMC wave function of equa-tion 56 is used as the auxiliary wave function.

With the use of pseudopotentials [131], bothVMC and DMC methods have been used for systemsup to a dozen atoms. Williamson et al. [132] showedthat QMC methods can be used for exciton energies.This is done by replacing one single-particle valenceband wave function with a conduction band wavefunction in the Slater determinant. The DMC witha nodal hypersurface defined by this new Slaterdeterminant is performed then, and it fully takesinto account the resulting correlation effects. Thisapproach gives the Si band structure that agrees

well with the experiments. QMC is one of the mostreliable methods for small-system calculations.

The development of a linear scaling QMCmethod [133] extended its applicability from adozen atoms to a few hundred atoms. Within the

linear scaling QMC method, Slater determinantsare represented on a basis of localized Wannier func-tions. This makes the Slater determinant sparse andtherefore the calculation time is proportional to thesize N, instead of N3 in the old scheme. Consequently,this allows the QMC calculation of a few hundredatoms and makes possible the use of the QMCmethod for small quantum dots [134].

1.07.4 Application to DifferentPhysical Effects: Some Examples

1.07.4.1 Electron and Hole Wave Functions

The shape of the single-particle wave functions andtheir energies determine many physical properties ofquantum dots. This section is, therefore, devotedto the analysis of electron and hole wave functions.The wave functions of the lowest four states in theconduction band and the top four states in thevalence band of a pyramidal [119]-facetedInAs/GaAs quantum dot are presented in Figure 2.The results presented in Figure 2 were obtainedusing the EPM, including the effect of spin–orbitinteraction.

The lowest conduction state is an s-like state,while the next two conduction states are p-like statesoriented in the directions of base diagonals, withnodal planes perpendicular to these directions.These are followed by d-like states. Due to lateraldimension larger than the quantum dot height, noneof the nodal planes is parallel to the pyramid base.There are, therefore, only two p-like states, in con-trast to spherical quantum dots where there are threep-like states.

The two p-states are relatively close in energyand their splitting is caused by several effects. Todiscuss each of these effects, we first assume that thestructure is unstrained.

1. In the simplest single-band effective mass model,these states are degenerate and can be split if thebase of the pyramid acquires a shape different thanthe square. The same is the case for the four-bandk ? p model (i.e., eight-band model without spin–orbit effects). Addition of spin–orbit effects tofour-band k ? p splits these levels by a small (lessthan 1 meV) amount [77,135]. Atomistic methodspredict the correct symmetry of the system andsplit the p-states, as well as k ? p models withlarger number of bands (14, for example).

Conduction and valence band states for b = 20a dot

Side view

CBM

VBM

VBM-1

VBM-2

VBM-3

CBM + 1

[110]

[110]CBM + 2

CBM + 3

Side view Top view

Figure 2 Isosurface plots of the charge densities of the conduction and valence band states for the square-based InAs/GaAs

pyramids with the base b¼ 20a, where a is the lattice constant of bulk zinc-blende GaAs. The charge density equals the wave-function square, including the spin-up and spin-down components. The level values of the green and blue isosurfaces equal 0.25

and 0.75 of the maximum charge density, respectively. Reproduced with permission from Wang LW, Kim J, and Zunger A (1999)

Electronic structures of [110]-faceted self-assembled pyramidal InAs/GaAs quantum dots. Physical Review B59: 5678–5687.


2. When strain (without piezoelectricity) is includedin the k ? p model within CM approach, it cannotcause the splitting, while the VFF model, due toits atomistic nature, splits the p-states.

3. Piezoelectricity added to any of the models alsocauses the splitting of the p-states.

The splitting of the p-states is therefore caused bythe shape anisotropy, spin–orbit effect, atomistic(a)symmetry, strain, and piezoelectricity. It is amaz-ing that a single quantity is determined by such alarge number of effects. Unfortunately, in a givenquantum dot, all these effects are present and cannotbe probed separately.

The conduction band states are formed essentiallyof a single envelope function and therefore these canbe classified as being s, p, and d-like. On the otherhand, the band mixing of the valence states is muchstronger and such a simple classification is not possi-ble. The valence band functions actually have nonodal planes. (This becomes obvious from Figure 2,when the isosurface values are additionally reduced

for valance band maximum (VBM-1) and VBM-2.)The approximation of using a single heavy holeband to describe the valence state, which is oftenused in quantum wells, is therefore not applicable toquantum dots due to stronger heavy and light holemixing.

As the dot size is reduced, the valence band ener-gies become lower and the conduction band energieshigher. The bound states are then less confined andthe effective energy gap increases. With the reduc-tion in quantum dot dimensions, some bound statesbecome mixed with wetting layer or continuumstates and the number of truly bound states decreases.

1.07.4.2 Intraband Optical Processes inEmbedded Quantum Dots

Most of the semiconductor optoelectronic devicesutilize transitions between the conduction-bandstates and the valence band states. The operatingwavelength of these devices is mainly determined

In-plane polarized

z-polarized

Figure 3 Scheme of energy levels and allowed optical

transitions in a parabolic quantum dot model with infinitepotential barriers. Only the levels with nxþny �2 and nz �1

are shown.


by the band gap of the materials employed and istherefore limited to the near-infrared and visible partof the spectrum. However, if one wishes to accesslonger wavelengths, a different approach is neces-sary, that is, the transitions within the same bandhave to be used. These transitions are called intra-band transitions. Intraband optical transitions in bulkare not allowed and therefore low-dimensionalnanostructures have to be used. Therefore, in thepast two decades, semiconductor nanostructures,such as quantum wells, wires, and dots, have beenrecognized as sources and detectors of electromag-netic radiation in the mid- and far-infrared region ofthe spectrum.

When the use of nanostructures as detectors isconcerned, several limitations of quantum well infraredphotodetectors (QWIPs) have motivated the develop-ment of quantum dot infrared photodetectors (QDIPs).The main origin of the undesirable dark current inQWIPs is thermal excitation (due to interaction withphonons) of carriers from the ground state to the con-tinuum states. The discrete electronic spectrum ofquantum dots as opposed to continuum spectrum ofquantum wells significantly reduces the phase space forsuch processes and therefore reduces the dark current.Higher operating temperatures of QDIPs are thereforeexpected. Due to optical selection rules, QWIPs basedon intersubband transitions in the conduction bandinteract only with radiation having the polarizationvector in the growth direction. This is not the case inquantum dots since these are 3D objects where thecorresponding selection rules are different.

For the QDIP applications, it is essential tounderstand the quantum dot absorption spectrum. Thesimplest model that is sufficient to qualitatively under-stand the quantum dot intraband absorption spectrum isthe parabolic dot model, where the potential is assumedin a separable form V rð Þ ¼ V1ðx; yÞ þ V2ðzÞ, where

V1ðx; yÞ ¼1

2m�!2 x2 þ y2

is the potential of a 2D

harmonic oscillator, and V2(z) is the potential of a quan-tum well confining the electrons in the z-direction. Thesolutions are of the form �ðrÞ ¼ hnx

ðxÞhnyðyÞ nz

ðzÞ,where hn(t) is the wave function of a 1D harmonicoscillator, and nz

ðzÞ are solutions of the 1DSchrodinger equation with potential V2(z) (correspond-ing to energies "nz

). The eigenenergies are then of theform E nx ; ny ; nz

¼ h�! nx þ ny þ 1

þ "nz

. The fac-tor h�! corresponds to the transition energy from theground to first excited state, and for modeling realisticquantum dots it should be set to h�! � 40 – 70 meV.Typical quantum dots are wide in the xy-plane

(diameters of the order of 20 nm and more) and have

very small height (of the order of 3–7 nm) in the z-

direction; therefore, the effective potential well repre-

senting the z-direction confinement is narrow (see

Figure 3). In a typical case, therefore, "1� "0 is of the

order of at least 100 meV.The optical absorption matrix elements on the

transitions between states are proportional to the

matrix elements of coordinate operators; therefore,

by calculating the latter, one obtains the following

selection rules on the transitions between states:

�nx¼� 1, �ny¼ 0, �nz¼ 0, for x-polarized

radiation; �ny¼� 1, �nx¼ 0, �nz¼ 0, for y-polarized radia-

tion; and �nx¼ 0, �ny¼ 0, for z-polarized radiation.

The transitions from the ground state are of primary

importance for QDIPs. From the selection rules

obtained, one concludes that only the transition to a

pair of degenerate first excited states is allowed for in-

plane polarized radiation, while in the case of

z-polarized radiation, only the transitions to higher

excited states are allowed, as demonstrated in Figure 3.Although the model presented considers the

quantum dot band structure in a very simplified

manner, it is excellent for understanding the results

of more realistic models. The strict selection rules

from this model are then relaxed, and strictly for-

bidden transitions become weakly allowed.

Nevertheless, qualitatively, the absorption spectrum

retains the same features as in this model.

0 0.1 0.2 0.3 0.4 0.5Energy (eV)

0

2

4

6

8

10

Abs

. cro

ss s

ectio

n (1

0–15 c

m2 )

Abs

. cro

ss s

ectio

n (1

0–15 c

m2 )

0 0.05 0.1 0.15Energy (eV)

0

20

40

60

80

1/2 3/2 5/2 7/2|mf|

0.6

0.65

0.7

0.75

0.8

0.85

Ene

rgy

(eV

)

x-polarized

z-polarized

(a) (b)

Figure 4 (a) The intraband optical absorption spectrum for a quantum dot of conical shape with the diameter D¼25 nm and

height h¼ 7 nm for the case of z-polarized radiation. The corresponding spectrum for in-plane polarized radiation is shown in

the inset. (b) The scheme of energy levels and the most relevant transitions. The quantum number of total quasi-angular

momentum mf, which is a good quantum number in the case of axially symmetric quantum dots within the axial approximationof eight-band k ? p model, is also indicated.


The absorption spectrum obtained by the eight-band k ? p model for an InAs/GaAs quantum dot of

conical shape with the diameter D¼ 25 nm and

height h¼ 7 nm is presented in Figure 4. The

dimensions were chosen to approximately match

those reported for quantum dots in a QDIP struc-

ture in Ref. [136]. The optical absorption spectrum

in the case of z-polarized radiation is shown in

Figure 4(a). The experimental intraband photocur-

rent spectrum exhibits the main peak at 175 meV

and a much smaller peak at 115 meV, in excellent

agreement with the results obtained for z-polarized

incident radiation where the corresponding peaks

occur at 179 and 114 meV, respectively. The corre-

sponding absorption spectrum for in-plane

polarized incident radiation is presented in the

inset of Figure 4(a). There is a single peak in the

spectrum, which is due to the transition from the

ground state to a pair of nearly degenerate first

excited states (see Figure 4(b)).The results presented and other similar calcula-

tions suggest that the in-plane polarized radiation

causes nonnegligible transitions only between the

ground and first excited states, these being located

in the region 40–80 meV in the far-infrared. On the

other hand, z-polarized radiation causes the transi-

tion in the �100–300 meV region in the mid-

infrared. The best way to understand the origin of

such behavior is through a simplified parabolic model

presented. Such behavior can be altered only if the

dot dimension in the z-direction becomes compar-

able to the in-plane dimensions.

1.07.4.3 Size Dependence of the Band Gapin Colloidal Quantum Dots

The size dependence of the band gap is the mostprominent effect of quantum confinement in semicon-ductor nanostructures. The band gap increases as thenanostructure size decreases. Many of quantum dotapplications rely on the size dependence of the opticalproperties. Therefore, studying the size dependence ofthe band gap is one of the most important topics insemiconductor nanocrystal research.

According to a simple effective-mass approxima-tion model, the band-gap increase of sphericalquantum dots from the bulk value is

�Eg ¼2h�2�2

m�d 2ð58Þ

where d is the quantum dot diameter and

1

m�¼ 1

m�eþ 1

m�hð59Þ

with m�e and m�h being the electron and hole effectivemasses.

The experiments usually measure the optical gapof a quantum dot. Therefore, in addition to the dif-ference in single-particle energies, one has to includethe interaction between created electron and hole, inorder to calculate the optical gap. One simpleapproach to do this is to calculate the exciton energyby including the electron–hole interaction on top ofthe single-particle gap. This procedure ignores theelectron–hole exchange interaction and possible cor-relation effects. However, in the strong confinement


regime, which is present in most colloidal nanocrys-tals and embedded quantum dots, these effects arevery small. Under this approximation, the excitonenergy can be expressed as

Eex ¼ "c – "v – ECcv ð60Þ

where "c and "v are the single-particle CBM andVBM energies, and EC

cv is the electron–holeCoulomb energy calculated as

ECcv ¼

Z Zdr1dr2

c r1ð Þj j2 v r2ð Þj j2

" r1 – r2ð Þ r1 – r2j j ð61Þ

where c(r) and v(r) are the electron and hole wavefunctions, and "(r1� r2) is a distance-dependentscreening dielectric function, which can be modeledas described in Ref. 32.

The dependence of calculated optical gap onCdSe nanocrystal size is presented in Figure 5. Afit of the theoretical results to the

Eg ¼ a?d –�

dependence yields values quite different from the simpled�2 law predicted from the effective mass approach. Inthe case of CdSe,�¼ 1.18. The� parameter is material-dependent and its values for III–V and II–VI semicon-ductors typically fall in the range of 1.1–1.7.

1.07.4.4 Excitons

In the previous section, we have presented the exci-ton calculations based on a simple, but usefulapproach. For the calculation of excitons, the meth-ods in Section 1.07.3 must be used in principle. The

10.0

0.4

0.8

1.2

1.6

ΔE

ΔEg = 2.12

CdSe QDs

ΔEg

(eV

)

Diame2 3

Figure 5 Comparison of the exciton energy shift from its bulk valpatchingmethod (withband-gapcorrectedpseudopotentials) (LDA

in this calculation. Experimental data is from Ref. [137]. Reproduced

corrected local density approximation study of semiconductor qua

results of these methods are shown in Figure 6 forH-passivated Si quantum dots [138]. The DMCmethod and GW–BSE method produce almost thesame band gap for the smallest quantum dots. TheDMC result is about 1 eV above all the other resultsfor somewhat larger quantum dots with the diameterup to 1.6 nm. It remains to be seen how accurate isthis DMC result, for example, when compared withwell-controlled experiments (perhaps for othermaterial quantum dots like CdSe). The TDLDAmethod gives almost the same results as the LDAKohn–Sham energy difference. This suggests thatboth the exchange and Coulomb interactions in theTDLDA results have a very small contribution.Besides TDLDA, TDDFT-B3LYP was used inRefs. [138,139]. The TDDFT-B3LYP band gap isbelow the DMC result, especially for relativelylarge quantum dots. However, in Ref. [139], it wasshown that for small molecules, the TDDFT-B3LYPresult agrees with the MR-MP2 quantum chemistrycalculations. The TB and EPM results in Figure 6can be considered as the lowest order results of theCI equation 43, where only the zero-order screenedCoulomb interactions between the VBM and CBMstates are taken into account. These agree well witheach other. However, they are between the TDLDAand TDDFT-B3LYP results.

To summarize these results, the DMC result isabove all the other methods for d¼ 1.5nm Si quan-tum dots. The LDA and TDLDA have the lowestband gap, followed by the TB and EPM-limited CIresults and the TDDFT-B3LYP results. For verysmall quantum dots, the DMC results agree wellwith the GW-BSE results.

g = 1.90/d1.18 (SEPM)

/d1.18 (LDA + C)

ExperimentSEPMLDA + C

ter (nm)4 5 6 7

ue of CdSe quantum dots (QDs) between experiment, chargeþC),andSEPMcalculations.Coulombenergiesareconsidered

with permission from Li J and Wang L-W (2005) Band-structure-

ntum dots and wires. Physical Review B 72: 125325.

0.00

2

4

6

8

101 2 5 10 17 2935 66 87 147

Number of Silicon atoms

293

TD-B3LYP

525

1.0 1.5

Diameter (nm)

Bulk gap

Opt

ical

gap

(eV

)

2.0 2.5 3.00.5

GW+BSE

DMC

Empirical pseudopotential

TDLDA

Tight binding

LDA HOMO-LUMO

Figure 6 Size dependence of optical gaps of silicon nanoclusters, calculated using diffusion Monte Carlo (DMC), GW-BSE,

LDA, and time-dependent LDA (TDLDA), time-dependent DFT with B3LYP functional (TDDFT-B3LYP), semiempirical tight-

binding, and semiempirical pseudopotential methods. Note the DMC and GW-BSE results are almost the same for the few

small clusters. Reproduced with permission from Williamson AJ, Grossman JC, Hood RQ, Puzder A, and Galli G (2002)Quantum Monte Carlo calculations of nanostructure optical gaps: Application to silicon quantum dots. Physical Review

Letters 89: 196803.


1.07.4.5 Auger Effects

Auger effects play a crucial role in carrier dynamicsin nanostructures when both types of carriers(electrons and holes) are present. They becomeimportant, in particular, in quantum dots that havediscrete electronic levels, which implies that thecompeting phonon-assisted relaxation processes arestrongly reduced. Different types of Auger processesare schematically illustrated in Figure 7.

According to Fermi’s golden rule, the formula forthe Auger rate is given as

Wi ¼2�

h�

Xn

ij i�Hj jfnij2 Efn – Ei

ð62Þ

where jii and jfni are the initial and final Auger states,Ei and Efn

their energies, and �H is the Coulombinteraction. At a temperature T 6¼ 0, the Boltzmannaverage over the initial states has to be taken. Itwould seem at first sight that the discreteness ofquantum dot energy levels and the requirement forenergy conservation in the process would not allowfor efficient Auger processes. However, other excita-tions, such as phonons, can be involved as well and

help satisfy the energy conservation. Their effect canthen be phenomenologically modeled by Lorentzianbroadening of the delta function in Fermi’s goldenrule expression as

Efn – Ei

! �

2�

1

Efn – Ei

2þ �=2ð Þ2ð63Þ

The most important step in the electron coolingprocess involves the transition of the electron from

the p-level (ep) to the ground s electronic state (es).

This process is mediated by a transition of the hole

from hs to hn. The calculated Auger lifetime for this

process is shown in Figure 8. Its value is of the order

of 0.1–0.5 ps, in agreement with experimental results

[140]. This result suggests that Auger processes are

sufficient to explain electron cooling in quantum

dots, although other mechanisms are not necessarily

ruled out.The same process can take place in the presence

of an electron and a hole that act only as spectators. It

is very interesting that the electron lifetime increases

by an order of magnitude in those cases, as demon-

strated in Figure 8.

Electron thermalization Thermalization with spectator

Tri-exciton recombination

Positive trion recombinationNegative trion recombination

Bi-exciton recombination

g.s. bi-exciton Excited mono-exciton g.s. Tri-exciton

Ground-state trion A– Ground-state trion A+ Hot electron Hot hole

(a) (b)

(d)

(f)(e)

(c)

Single-particle gap

Excited bi-exciton

τ2–1e–h τ3–2

e–h

τhτe

Figure 7 Illustration of different Auger processes: (a) electron thermalization; (b) thermalization with spectator; (c) bi-excitonrecombination; (d) tri-exciton recombination; (e) negative trion recombination; (f) positive trion recombination. In each of the figures,

the initial state in the process is shown in the left-hand side and the final one in the right-hand side. Reproduced with permission from

Wang L-W, Califano M, Zunger A, and Franceschetti A (2003) Pseudopotential theory of Auger processes in CdSe quantum dots.Physical Review Letters 91: 056404.

0.4

SP ClNo spectator

Clwith spectator

Cd232Se235

0.5εp–εs (eV)

0.30

2

4

6

Aug

er li

fetim

e (p

s)

Figure 8 Electron cooling. Auger lifetimes at T¼300 K calculated with EPM within the single-particle (SP) approximation(solid line), and with CI, both in the absence (dashed line) and in the presence (long-dashed line) of a spectator ground-state

exciton. The initial states include all three electron p-states and both hole s-states, and the final states "s and 30 hole states hn

with energy centered around En0 ¼ �hs – ��p þ ��s. Reproduced with permission from Wang L-W, Califano M, Zunger A, and

Franceschetti A (2003) Pseudopotential theory of Auger processes in CdSe quantum dots. Physical Review Letters 91: 056404.


10ΔE (meV)

0.66

0.68

0.7

0.72

0.74

0.76

Ene

rgy

(eV

)

20 30 40

Figure 9 Dependence of the polaron energy levels

obtained by direct diagonalization (circles) on the artificialshift �E. The corresponding single-particle levels

calculated using the eight-band k ? p model are shown by

full lines. Lens-shaped quantum dot with the diameter of20 nm and the height of 5 nm is considered.


1.07.4.6 Electron–Phonon Interaction

The theory and results presented so far covered onlythe stationary electronic structure of quantum dotswhen atoms are in their equilibrium positions.However, at finite temperatures the vibrations ofatoms around their equilibrium positions (phonons)create additional potential that perturbs otherwisestationary electronic states and causes transitionsamong them.

Phonons in quantum dots can be treated at variouslevels of approximations. The approximation that isoften used for large quantum dots is that the phononsare the same as in bulk material. The strongest cou-pling between electrons and phonons in polarcrystals is polar coupling to longitudinal optical(LO) phonons, while deformation potential couplingto longitudinal acoustic (LA) phonons might alsosometimes be important.

In order to calculate the transition rates amongdifferent electronic states due to the interactionwith LO phonons, it is tempting to apply Fermi’sGolden rule, which is a good approximation inquantum wells, for example, [141]. However, itsdirect application to quantum dots leads to theresult that transition rates are zero unless twolevels are separated by one LO phonon energyexactly [142]. Such an approach treats the electronand phonon systems separately with their interac-tion being only a perturbation. It is currentlyknown that electrons and phonons in quantumdots form coupled entities – polarons. Polarons inself-assembled quantum dots have so far been evi-denced experimentally by optical means in theintraband magneto-optical spectrum [143,144],magneto-photoluminescence spectrum [145], andRaman scattering [146], and it has been suggestedtheoretically that they could have transport signa-tures as well [147]. Polarons are usually evidencedby anticrossing of electron energy levels whenthese are gradually changed, such as, for example,by magnetic field. We illustrate this here by anumerical experiment where the energies of thepair of first excited states are shifted in oppositedirections by the same amount �E, which is var-ied. The polaron states were calculated by directdiagonalization of the electron–phonon interactionHamiltonian. The polaron energy levels that con-tain a contribution from at least one of theelectronic states larger than 10% are shown bycircles in Figure 9. Anticrossing features inpolaron spectrum are clearly visible.

There is therefore a widespread thought that car-rier relaxation in quantum dots should be treated byconsidering the carriers as polarons. The polaronlifetime is then determined by anharmonic decay ofan LO phonon into two low-energy phonons[148,143,149,150,151]. It is thought that the physicalprocess responsible for that decay process is thedecay either to two LA phonons [152,148] or to oneacoustic and one optical phonon [151]. Within suchassumptions, the polaron lifetime is in the case of atwo-level system given by [148]

W ¼ �

h�–

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 R –Xð Þ

ph�

ð64Þ

where R ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX 2 þ Y 2p

; X ¼ g2 þ �2 –�2

=4;Y ¼ ��=2;� ¼ Ei – Ef – h�!LO, g is the electron–

phonon coupling strength, �=h� the phonon decayrate, and Ei and Ef the energies of the single-particle states. Equation 64 has been used inseveral occasions to fit the experimental resultson intraband carrier dynamics in quantum dots[149,153].

The approximation of bulk phonon modescertainly fails in small quantum dots. In that case,one should use the atomistic description of phonons.To calculate the phonon frequencies and displace-ments, one needs a force field that describes thevibrations of atoms around their equilibriumpositions. VFF, for example, can be used for thatpurpose. To calculate the electron–phonon coupling,one needs to be able to calculate the change in single-particle Hamiltonian due to atomic displacements.


Any of the single-particle methods described inSection 1.07.2 can, in principle, be used for thatpurpose. However, if some of the empirical methodsare used, one should be sure that the fitted para-meters are appropriate for this purpose as well. Dueto large number of atoms and consequently phononmodes, such calculations could be quite expensivenevertheless they are sometimes practiced. Forexample, Delerue et al. [154] calculated the phononmodes in a Si nanocrystal using a VFF model, and thecoupling between the phonon modes and the transi-tion electronic states explicitly using the Harrison’srule [56] for changes of TB parameters following theatomic displacements. Most recently, Chelikowskyet al. [155] calculated the phonons of Si quantumdots using direct DFT calculations.

1.07.5 Conclusions

We have given an overview of theoretical methodsused for electronic structure calculations in quantumdots. We have emphasized the weaknesses andstrengths of each of the methods. An interestedreader can therefore choose the method of choicedepending on the desired application, the degree ofaccuracy required, and the available computationalresources.

For the treatment of single-particle states, thesimplest effective mass method is excellent forpedagogical purposes to illustrate the effect of quan-tum confinement. It is often even used in researchwhen one wishes to qualitatively take into accountthe effect of quantum confinement and the detailsof the electronic structure are not essential. Themultiband k ? p method gives a more quantitativedescription, especially for large quantum dots. It iswidely used in modeling of optical and transportprocesses in optoelectronic devices. Atomistic meth-ods give a very detailed description of quantum dotelectronic structure and are clearly the best choice inresearch for understanding the new physical effects.

For the treatment of excitations in quantum dots,Section 1.07.3 gives an overview of the methods thatcan be applied in principle. For application of thesemethods to quantum dots, linear scaling of themethod is an essential requirement. CI approachsatisfies this but it is based on classical model deriva-tions and physical intuitions. QMC also appears to bepromising. However, the method is relatively new,when the calculations of excited states and largesystems are concerned. A deeper understanding of

the accuracy, that is, the quality of the variationalform of the wave function or the nodal hypersurfaces,is required. Where the GW–BSE approach is con-cerned, it is a challenge to make it scalable to largersystems.

Acknowledgment

This work was supported by US Department ofEnergy BES/SC under contract number DE-AC02-05CH11231.

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1.07 Quantum Dots: Theory · quantum dots, sometimes termed as electrostatic quantum dots, can be...

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