1.07 Quantum Dots: Theory N 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 laterally by electrostatic gates or vertically by etch- ing techniques [1,2]. The properties of this type of Since the early 1980s, remarkable progress in technology quantum dots, sometimes termed as electrostatic has been made, enabling the production of nanometer- quantum dots, can be controlled by changing the sized semiconductor structures. This is the length scale applied potential at gates, the choice of the geometry where the laws of quantum mechanics rule and a range of gates, or external magnetic field. The typical size of new physical effects are manifested. Fundamental of these dots is of the order of 100 nm. laws of physics can be tested on the one hand, while on Self-assembled quantum dots. Self-assembled quantum the other hand many possible applications are rapidly dots are obtained in heteroepitaxial systems with dif- emerging for nanometer-sized semiconductors.. ferent lattice constants. During the growth of a layer of The ultimate nanostructure where carriers are one material on top of another, the formation of confined in all the three spatial dimensions is called nanoscale islands takes place [3], if the width of the a quantum dot. In the last 15 years, quantum dots layer (the so-called wetting layer) is larger than a have been produced in several different ways in a certain critical thickness. This growth mode is called broad range of semiconductor material systems. The Stranski–Krastanov mode. The most common experi- properties of quantum dots and their possible appli- mental techniques of the epitaxial nanostructure cations are largely dependent on the method they growth are molecular beam epitaxy (MBE) and meta- have been obtained with, which can, therefore, be lorganic chemical vapor deposition (MOCVD) [4,5]. used as a criterion for classification of different Since the quantum dot material is embedded in types of quantum dots: another material, we refer to these dots also as Electrostatic quantum dots. One can fabricate quan- embedded quantum dots. Self-assembled quantum tum dots by restricting the two-dimensional (2D) dots typically have lateral dimensions of the order of electron gas in a semiconductor heterostructure 15–30 nm and height of the order 3–7 nm.

189 190 Quantum Dots: Theory

Colloidal quantum dots. A very different approach to 1.07.2 Single-Particle Methods obtain quantum dots is to synthesize single crystals of the size of a few nanometers through chemical While quantum dots seem to be small and simple methods. The dots obtained in this way are called objects, a look at their structure from the atomistic nanocrystals or colloidal quantum dots [6]. Their side reveals their high complexity. Bearing in mind size and shape can be controlled by the duration, that the lattice constants of the underlying semicon- temperature, and ligand molecules used in the synth- ductor materials are typically of the order of 0.5 nm, esis [7]. Colloidal quantum dots are typically of one can estimate that a single self-assembled quan- spherical shape. They are often smaller than tum dot contains 106 nuclei and even a larger embedded quantum dots with the diameter some- number of electrons interacting among each other times as low as 2–4 nm. with long-range Coulomb forces. Even the smallest Quantum dots have enabled the study of many colloidal quantum dots contain thousands of atoms. fundamental physical effects. Electrostatic quantum This clearly indicates that direct solution of the dots can be controllably charged with a desired many-body quantum dot Hamiltonian is not a feasi- number of electrons and therefore the whole periodic ble approach and that smart and efficient methods system [8] of artificial atoms created, providing a need to be developed. Here, methods that reduce the wealth of data from which an additional insight into problem to an effective single-particle equation are the many-body physics of fermion systems could be reviewed. obtained [1]. Single-electron transport and Coulomb More than two decades ago, Brus introduced blockade effects on the one hand, and the regime of [24–25] a simple effective mass method to calculate Kondo physics on the other hand, have been investi- ionization energies, electron affinities, and optical gated [9,10]. transition energies in semiconductor nanocrystals. One of the most exciting aspects of quantum dot Within Brus’s model, the single-particle (electron or research is certainly the prospect of using the state of hole) energies E and wave functions (r) satisfy the the dot (spin state, exciton, or charged exciton) as a Schro¨dinger’s equation given as qubit in quantum information processing. Coherent control of an exciton state in a single dot selected 1 – r2 þ PðrÞ ðrÞ¼E ðrÞð1Þ from an ensemble of self-assembled quantum dots as 2m well as the manipulation of the spin state in electro- static quantum dots [12,13] have been achieved [11]. where m is the electron or hole effective mass. The The theoretical and experimental progress in the system of atomic units where the reduced Planck’s field of spin-related phenomena in quantum dots constant h, the electron mass m0, and the electron has been reviewed in Refs. [1 and 14]. These results charge c are all equal to 1 was used in equation 1 and appear promising, although the control of a larger will be used in what follows. For simplicity, equation number of quantum dot qubits is not feasible yet, 1 assumes that the particle must be confined within mainly due to difficulty in controlling qubit–qubit the dot, that is, that the potential outside the dot is interactions. infinite. This simplifying assumption can be easily The practical applications of quantum dots relaxed by adding a more realistic confining potential certainly do not lag behind these exciting areas of Vconf (r). fundamental science with quantum dots. For exam- P(r)inequation1 is the additional potential caused ple, colloidal quantum dots have found several by the presence of the surface of the quantum dot. It cutting-edge applications such as fluorescent biolo- has a certain analogy with electrostatic image poten- gical labels [15], highly efficient photovoltaic solar tials in the case when a charge is near the surface of the cells [16], and nanocrystal-based light-emitting metal or the interface between two dielectrics. It can diodes [1]. Self-assembled quantum dots find the be obtained by calculating the interaction energy main application as optoelectronic devices – lasers between a bare electron and its induced screening [17], optical amplifiers [18], single-photon sources potential. The extra interaction energy of an electron [19,20], and photodetectors [21,22,23]. at r inside the quantum dot compared to the corre- This chapter focuses on theoretical methods used sponding value in bulk is then P(r). for calculation of physical properties of self- To model the two-particle excitations (such assembled and colloidal quantum dots. as electron þ hole ¼ exciton), Brus introduced an Quantum Dots: Theory 191 electrostatic interaction energy term among these with eigenstates defined in the density functional theory particles as (DFT) discussed below. This section is, therefore, com- pletely devoted to the theoretical frameworks and e2 V ðr1;r2Þ¼ PMðÞþr1;r2 PðÞþr1 PðÞðr2 2Þ methodologies for calculating these states. "jjr1 –r2 where " is the dielectric constant, PM corresponds to the interaction of the charge of one particle with 1.07.2.1 Density Functional Theory surface-induced polarization potential of the other Within the DFT [26], the many-body Hamiltonian particle, while the P-terms describe the interaction problem reduces to a set of single-particle Kohn– of the charge of one particle with its own surface- Sham equations [27] that read as induced polarization potential, as previously described. The plus (minus) sign is for the two par- 1 – r2 þ V þ V þ V ðrÞ¼" ðrÞð5Þ ticles of the same (opposite) charge. The effective 2 ion H XC i i i exciton Hamiltonian is then given as In equation 5, i(r) and "i are the wave functions and 2 1 2 1 2 e energies of Kohn–Sham orbitals, Vion(r) is the poten- Hexciton ¼ – re – rh – 2me 2mh jjre – rh tial of all nuclei in the system, and VH(r) is the Hartree potential of electrons given as – PMðre; rhÞþPðreÞþPðrhÞð3Þ Z ðr9Þ The solution of the eigenvalue problem of this V ðrÞ¼ dr9 ð6Þ H 9 Hamiltonian can be written down analytically as jjr r where 2 1 1 a e2 . – c X E Eg þ 2 þ þ small term ð4Þ 2 2R me mh R ðrÞ¼ jj i ðrÞ ð7Þ

Sið2Þ Sið4Þ is the electronic charge density of the system. The where a ¼ 2 – þ 1:8 and Si(x)isthe c 2 summation in equation 7 goes over all occupied R x sint Kohn–Sham orbitals. The exchange-correlation sine integral function, SiðxÞ¼ 0 dt: The last t potential VXC in equation 5 is supposed to take into term in equation 4 originates from the last three terms account all the other effects of electron–electron in equation 3. One should note that P(r) ¼ PM(r, r)/2; interactions beyond the simple Coulomb repulsion therefore, PM(re, rh)andP(re) þ P(rh) cancel out exac- (described in VH). The exact form of this potential is tly when re ¼ rh and lead to a small term when re and rh not known, and it needs to be approximated. The are not equal. This small term can often be ignored in most widely used approximation is the local density practice for spherical quantum dots. approximation (LDA) where it is assumed that VXC The cancellation of the polarization terms gives us a depends only on the local electronic charge density guide for a general approach for calculating the excitons and takes the same value as in the free electron gas of in nanocrystals. As a first step, one calculates the single- that density [27]. Equations 5 and 7 need to be solved particle energies from equation 1 without the polariza- self-consistently until the convergence is achieved. tion term. As a second step, the screened electron–hole DFT calculations are still computationally interaction is added perturbatively. One should, how- demanding, partly due to the necessity of self- ever,haveinmindthatsuchanapproachisan consistent calculations. One also needs to calculate approximation based on classical electrostatic considera- all the orbitals i in each iteration, while in semicon- tion. It ignores the effects such as dynamic screening and ducting systems one is often interested in only a the local-field effects of the dielectric function. The few states in the region around the gap that determine single-particle states obtained in this way are not the the optical and transport properties of the system. quasiparticles from the usual GW formalism. (The An alternative approach that avoids the full self- eigenenergies of equation 1 with the P-term are the consistent calculation without loss in accuracy is the quasiparticle energies that correspond to the electron charge patching method (CPM) [28,29,30–32,33]. affinity and ionization potential.) However, such single- The basic assumption of the CPM is that the charge particle states are the natural extension of single-particle density around a given atom depends only on the states considered in other nanostructures, such as quan- local atomic environment around the atom. This is tum wells and superlattices. These are also fully in line true if there is no long-range external electric field 192 Quantum Dots: Theory that causes long-range charge transfer. This is often magnitude. Therefore, the CPM can be considered satisfied if there is a band gap in the material. Based to be as accurate as the direct ab initio calculations. on this assumption, the idea is to calculate (e.g., using There are, however, cases where CPM cannot be DFT in LDA) the charge density of some small used. One example is the total dipole moment of an prototype system LDA(r), decompose it into contri- asymmetric quantum dot [36]. Such a dipole moment butions from individual atoms (charge density can also induce an internal electric field and cause motifs), and then use these motifs to patch the charge the long-range charge transfer in the system. It is, density of a large system. therefore, necessary to solve the charge density self- In particular, charge density motifs are calculated consistently, which can be done using the DFT/LDA from the charge density of the prototype system as method. However, a much more efficient linear scaling method to do such calculations has been P wðÞjjr – R mI ðr – RÞ¼LDAðrÞ ð8Þ developed recently: the linear scaling three- w9ðÞjjr – R9 R9 dimensional fragment (LS3DF) method [37]. Within the LS3DF method, the system is divided where R is the position of atom type and into many small fragments. The wave functions and mI ðÞr – R is the charge density motif of this charge densities of each fragment are calculated atom type, w(r) is an exponentially decaying function that defines the partition function separately, each within the standard DFT/LDA method, using a group of a small number of computer wðÞjjr – R =R9 w9ðÞjjr – R9 that divides the space into (mutually overlapping) regions assigned processors. After the fragment charge densities are obtained, they are patched together to get the charge to each atom. mI ðr – RÞ is,therefore,alocalized function that can be stored in a fixed-size numer- density of the whole system using a novel scheme that ensures that the artificial surface effects due to ical array. I denotes the atomic bonding environment of the atom .Afterthecharge the system subdivision will be cancelled out among density motifs are obtained from small prototype the fragments. The patched charge density is then systems, the total charge density of the large used to solve a global Poisson equation for the global nanosystem is obtained simply as the sum of potential. An outside loop is iterated, which yields motifs assigned to each of the atoms: the self-consistency between the global charge den- X sity and the input potential. Due to the use of this patchðrÞ¼ mI ðÞr – R ð9Þ novel patching scheme, the LS3DF is very accurate, R with its results (including the dipole moments) essen- Once the charge density is obtained using the tially the same as the original direct DFT calculation charge patching procedure, the single-particle results [37], but with potentially 1000 times speed- Hamiltonian can be generated by solving the ups, for systems with more than 10 000 atoms. As the Poisson equation for the Hartree potential and system grows larger, there are more fragments (while using the LDA formula for the exchange-correlation the fragment size is fixed); thus, more processor potential. The energies and wave functions of a few groups can be used to solve them. This provides a states around the gap can then be found using the perfect parallelization to the number of processors. methods developed to find a few eigenvalues of the Meanwhile, the total computational cost is propor- Hamiltonian only, such as the folded spectrum tional to the number of fragments, and consequently method (FSM) [34] (that is described in Section the total number of atoms. 1.07.2.2). A well-known problem of the LDA-based The CPM was used to generate the charge den- calculations is that the band gap is severely under- sities of carbon fullerenes [33], semiconductor alloys estimated [38,39]. DFT is rigorously valid only for [28], semiconductor impurities [29], organic mole- ground-state properties, and there is no physical cules and polymers [35], and semiconductor meaning for the Kohn–Sham eigen energies [27]. quantum dots [32]. The resulting patched charge This conceptual difficulty can be circumvented by density is typically within 1% of the self-consistently using time-dependent DFT, which is discussed later. calculated LDA charge density, and the correspond- In practice, however, one often restricts to the simple ing energies are within 30 meV. Typical numerical empirical ways to correct the band-gap error. One uncertainty (due to basis function truncations and such way is to slightly modify the LDA Hamiltonian different nonlocal pseudopotential treatments) of an to fit the crystal bulk bandstructure, which can be LDA calculation is about the same order of done, for example, by changing the s, p, and d Quantum Dots: Theory 193 nonlocal pseudopotentials [29] to move the posi- is expanded as a linear combination of plane waves, tion of the conduction band while keeping the where the summation is restricted only to reciprocal position of the valence band unchanged. This approach lattice q vectors with kinetic energy smaller than is based on the assumption that the valence band certain predefined value Ecut. To evaluate the result- alignment predicted by the LDA is reliable. ing Hamiltonian matrix in plane-wave For the treatment of colloidal quantum dots, one representation, Fourier transforms of atomic poten- also has to take care of the quantum dot surface. The tials vatom(jqj) are needed. Only a few of these are surface of an unpassivated nanocrystal consists of nonzero. These are used as adjustable parameters to dangling bonds that introduce band-gap states. One fit the semiconductor bandstructure. way to remove these states is to pair the dangling- To apply the EPM to nanostructures, one needs to bond electron with other electrons. If a surface atom have a continuous vatom(q) curve, since the supercell has m valence electrons, this atom provides m/4 is very large, and consequently q points are very electrons to each of its four bonds in a tetrahedral dense. The continuous vatom(q) can be represented crystal. To pair m/4 electrons in a dangling bond, a by a function of four parameters a1a4 passivating agent should provide 2 m/4 additional a ðÞq2 – a electrons. To keep the system locally neutral, there vðqÞ¼ 1 2 a q2 must be a positive 2 m/4 nuclear charge nearby. a3e 4 – 1 The simplest passivation agent can, therefore, be a or a sum of Gaussians hydrogen-like atom with 2 m/4 electrons and a X 2 – ci ðÞq – bi nuclear charge Z ¼ 2 m/4. For IV–IV group mate- vðqÞ¼ ai e rials like Si, this is a Z ¼ 1 hydrogen atom. For III–V and II–VI systems, the resulting atoms have a For a full description of the colloidal quantum dots, noninteger Z; consequently, these are pseudohydro- the pseudopotentials of surface passivating hydrogen gen atoms. These artificial pseudohydrogen atoms or pseudohydrogen atoms need to be fitted as well. do describe the essential features of good passivation The pseudopotentials are fitted to experimental data agents and serve as simplified models for the real and first-principles calculations of bulk bandstruc- passivation situations, where organic molecules with tures, clean surface work function, and the density complicated and often unknown structure are of states of chemisorbed surfaces. involved. Another approach to fit the pseudopotentials is to fit them directly to the LDA-calculated potential [42] and then modify them slightly to correct the 1.07.2.2 Empirical Pseudopotential Method band-gap error. The vatom(q) obtained in such a man- The empirical pseudopotential method (EPM) was ner are able to fit the band structure within 0.1 eV introduced in the 1960s by Cohen et al. [40,41] to fit and have in the same time a 99% overlap with the the bandstructure of bulk semiconductors. Within original LDA wave function. This approach, called the EPM, the Schro¨dinger equation is given as the semiempirical pseudopotential method (SEPM), has been applied to CdSe [42], InP [43], and Si [42] 1 2 nanostructures, representing II–VI, III–V, and IV–IV – r þ V ðrÞ i ðrÞ¼Ei i ðrÞð10Þ 2 semiconductor systems, respectively. with With the empirical or semiempirical pseudopo- X tentials at hand, one is able to construct the single- V ðrÞ¼ vatomðÞðjjr – Ratom 11Þ particle Hamiltonian. The diagonalization of this atom Hamiltonian is a routine task in the case of bulk where Ratom are the positions of the atoms and semiconductors due to a small number of atoms in vatom(r) are spherical atomic potentials that in an a supercell. However, this is no longer the case in effective manner take into account the effects of quantum dots that contain a large number of atoms. nuclei, core, and valence electrons. The great success Even the conventional conjugate gradient method of the EPM was that it was actually possible to fit the [44] that is often used in ab initio calculations bandstructure of the semiconductors using this cannot be used since it scales as O (N3)duetoan single-particle approach. orthogonalization step, which is a necessary part of In the EPM calculations, the plane-wave repre- the algorithm. Fortunately, for the analysis of most sentation is typically used, that is, the wave function electronic, transport, and optical properties of 194 Quantum Dots: Theory semiconductor nanosystems, only the states in the can be significantly reduced compared to the spectral region close to the band gap are relevant, plane-wave basis. It turns out that it is possible to and there is no need to find all the eigenstates of use a fixed number of basis functions to achieve the the Hamiltonian. The FSM, specialized to find the same degree of accuracy for different system sizes, in eigenstatesinacertainspectralregiononly,has contrast to the plane-wave basis where the number of therefore been developed by Wang and Zunger basis functions scales linearly with system size. The [34]. The method is based on the fact that the origin of this effect is the fact that when the size of the 2 Hamiltonians H and (H Eref) have the same system increases, the envelope function of the elec- eigenvectors and that the few lowest eigenvectors tronic state becomes smoother, and therefore the 2 of (H Eref) are the eigenvectors of H closest to maximum value of the k-vector needed to represent the energy Eref. The lowest eigenstates of it becomes smaller. This makes the LCBB method 2 (H Eref) arethensolvedusingtheconjugategra- ideal for studying very large systems such as dient method. It turns out that the use of embedded quantum dots. 2 (H Eref) compared to H slows down the conver- gence but with the use of preconditioners and a large number of iterations, convergence can still be 1.07.2.3 Tight-Binding Methods achieved. The FSM within the plane-wave repre- sentationhasbeenimplementedintheparallel The tight-binding (TB) method [49] is the simplest code Parallel Energy SCAN (PESCAN) [45].It method that still includes the atomic structure of a can be routinely used to calculate systems with a quantum dot in the calculation [50,51,52,53].Inthe fewthousandatoms,orevennear-millionatom TB method, one selects the most relevant atomic- systems [46]. Since only a few wave functions are like orbitals jii localized on atom i, which are calculated, the computational effort scales linearly assumed to be orthonormal. The single-particle to the size of the system. Linear scaling method for wave function is expanded on the basis of these the calculation of the total and local electronic localized orbitals as X density of states and the optical absorption spec- j i¼ cijiið14Þ trum has been developed by Wang [47].The i reader is referred to Ref. [47] for the description and therefore the TB single-particle Hamiltonian is of this method, called the generalized moments of the form method (GMM). X X Another method for solving of the EPM 1 H ¼ "ijiihijþ ti; j jiihj j ð15Þ Hamiltonian is the linear combination of bulk Bloch i 2 i; j bands (LCBBs) method [48]. The disadvantage of the where " are the energies of the orbitals (the on-site plane-wave expansion is that it does not lend itself to i energies), while t are the hopping integrals systematic approximations. A basis set in which such i, j between different orbitals, which can be restricted approximations can be naturally made is the basis of to include only nearest neighbors or next-nearest full-zone bulk Bloch states. In this basis, the wave neighbors. For the sake of notational simplicity, the function expansion reads form that does not include the spin–orbit interaction

XNB XNk and therefore does not mix the states of different spin 0 ðrÞ¼ Ck;n knðrÞð12Þ was presented. The extension to include spin orbit n k interaction is straightforward. The most popular fla- where vor of TB is the empirical TB where the parameters of the Hamiltonian are treated as phenomenological 0 1ffiffiffiffi ik ? r knðrÞ¼p uknðrÞe ð13Þ and fitted to reproduce the bulk band structure N obtained from experiment or higher level calcula- is the bulk Bloch function of the constituent bulk tions. In such an approach, the atomic orbitals are solid, where n is the band index, k is the supercell not treated explicitly, since the whole spectrum of reciprocal-lattice vector, N is the number of primary the single-particle Hamiltonian is determined by the cells in the supercell. The LCBB expansion allows onsite energies and hopping integrals. The wave one to select the physically important bands n and function is represented by the coefficients ci that k-points. As a result, the number of basis functions slowly vary from site to site. Quantum Dots: Theory 195

In the TB method, one restricts the atomic orbitals advantage of the TB model compared to the k ? p to include only a few for each atom. Since one is model, which is described next. typically interested in states around the energy gap, one has to select the orbitals that define these 1.07.2.4 k ? p Method states. In III–V, IV–IV, and II–VI semiconductors, these are typically the s, px,py,pz orbitals and some- The previously described methods treat explicitly times d orbitals. Quite often, an additional s-like the atomistic details of the nanostructure, which orbital called s is added to provide an additional therefore leads to their high accuracy and reliability degree of freedom in fitting of the TB parameters, but also to a significant computational cost. In the which leads to models such as sp3s [54] and sp3d5s k ? p method, central quantities are the slowly vary- [55]. In these models, the size of the resulting ing envelope functions that modulate the rapidly Hamiltonian matrix is nN nN, where N is the num- varying atomistic wave function. Historically, the ber of atoms and n is the number of orbitals per atom k ? p method was introduced to describe the bulk (n ¼ 10 for sp3s with spin and n ¼ 20 for sp3d5s with band structure around a certain special point in the spin). Due to nearest-neighbor approximation, the Brillouin zone, and later on it was extended to matrix is sparse, and efficient methods for the diag- describe heterostructures. onalization of sparse matrices can therefore be Let the Hamiltonian of an electron in a semicon- exploited. ductor be 2 One problem of the TB method is the lack of pˆ Hˆ ¼ þ V0ðrÞþHˆso ð16Þ explicit basis functions. Although these can be 2 added after the TB eigenstates have been calcu- where pˆ is the momentum operator, V0(r) the crystal lated, these basis functions are not an intrinsic part potential (including nuclei, core electrons, and self- of the TB Hamiltonian and its fitting process; thus, consistent potential of valence electrons), and Hˆso the their compatibility is an issue. This causes problems spin–orbit interaction Hamiltonian arising from rela- to calculate physical properties such as dipole tran- tivistic corrections to Schro¨dinger equation given by sitions and Coulomb and exchange interactions. 2 Another issue in treating quantum dot heterostruc- Hˆ ¼ ½rV ðrÞpˆ ?s ð17Þ so 4 0 tures is how to choose the parameters at the interface of two materials, since only the parameters where is the fine structure constant, and s is a for bulk materials are available. An approximation vector of Pauli matrices: "# "# "# needs to be introduced, usually by assuming the 01 0 – i 10 parameters at the interface as a certain average of x ¼ ;y ¼ ;z ¼ ð18Þ the TB parameters of the two materials. In colloidal 10 i 0 0 – 1 quantum dots, the surface has to be passivated. The envelope representation of the wave function of Here,wegiveanexampleofhowthisisdonein an electron is given by the case of Si nanocrystals. The surface is passivated X by H atoms, where the TB nearest-neighbor matrix ðrÞ¼ i ðrÞui ðrÞð19Þ i elements VH–Si between H and Si are scaled from the Si–Si matrix elements VSi–Si according to the where ui(r) form the complete orthonormal set of 2 functions with periodicity of the Bravais lattice, and Harrison’s rule [56]: VH–Si ¼ VSi–Si(dSi–Si/dH–Si) , (r) are slowly varying envelope functions. The where dSi–Si and dH–Si are the bond distances [57]. i Another way to treat the surface passivation is simply most common choice of the functions ui are bulk to remove the dangling-bond states from the calcu- Bloch functions at the point. After the replacement lated results or even from the Hamiltonian before the of equation 19 in the Schro¨dinger equation and mak- matrix is diagonalized. This is done by removing the ing an approximation that eliminates the nonlocal hybrid sp3 dangling-bond orbital from the TB terms that appear in the derivation, one arrives at Hamiltonian basis set (e.g., by removing the [59,60] X Hamiltonian matrix columns and rows expanded by 1 2 – r mðrÞþ ð – iÞp ?r nðrÞ these sp3 bases) [58]. This is a unique way of artificial 2 mn X n ð20Þ passivation only applicable to TB calculations. The þ HmnðrÞ nðrÞ¼E mðrÞ ability to describe the surface atomistically is a big n 196 Quantum Dots: Theory

Since the second term in equation 20 is crucial in of remote bands. Since the point Td symmetry group determining the Hamiltonian matrix (and (i)r of the zinc-blende crystal is a subgroup of the dia- becomes the k-vector if, e.g., the envelope function mondgroupofGeandSi,thesamek ? p is expanded in plane waves), the method being Hamiltonian can be applied to these semiconduc- described is called the k ? p method. The terms in tors, as well. On the basis jJJzi that diagonalizes the the previous equation are given by bulk Hamiltonian at k ¼ 0 Z 

1 3 1 1 pmn ¼ umðrÞ pˆunðrÞd r ð21Þ j1i¼ ; – ¼jS #i 2 2  where the integration goes over the volume of the 1 1 j2i¼ ; ¼jS "i 2 2 crystal unit cell , and Hmn(r) is the term that, away  rffiffiffi from the interfaces, reduces to the bulk matrix 3 1 i 2 j3i¼ ; ¼ – pffiffiffi jðX þ iYÞ#iþi jZ "i elements of the Hamiltonian 2 2 6 3  Z 3 3 i 1 ˆ 3 j4i¼ ; ¼ pffiffiffi jðX þ iYÞ"i Hmn ¼ umðrÞ HunðrÞd r ¼ Emmn ð22Þ 2 2 2 

3 3 i where E is the band edge of band m. In practice, one j5i¼ ; – ¼ – pffiffiffi jðX – iYÞ#i m 2 2 2 has to restrict to a finite number of bands.  rffiffiffi

3 1 i 2 Historically, the k ? p method was first applied to j6i¼ ; – ¼ pffiffiffi jðX – iYÞ"iþi jZ #i 2 2 6 3 valence band (six-band Hamiltonian) [61–62], and  later on the conduction band was added (eight-band 1 1 i i j7i¼ ; – ¼ – pffiffiffi jðX – iYÞ"iþpffiffiffi jZ #i Hamiltonian) [63]. 2 2 3 3  The explicit form of the eight-band Hamiltonian 1 1 i i ð23Þ for the crystals with zinc-blende structure (such as, j8i¼ ; ¼ – pffiffiffi jðX þ iYÞ#i– pffiffiffi jZ "i 2 2 3 3 InAs, GaAs, AlSb, CdTe, GaP, GaSb, InP, InSb, ZnS, ZnSe, and ZnTe) is given below. This the eight-band k ? p Hamiltonian reads (where the Hamiltonian also perturbatively includes the effect definition k ¼ir was introduced)

2 pffiffiffi pffiffiffi pffiffiffi 3 A 0 V þ 0 3V – 2U – U 2V þ 6 7 6 ffiffiffi ffiffiffi ffiffiffi 7 6 p p p 7 6 0 A – 2U – 3V þ 0 – V 2VU7 6 7 6 rffiffiffi 7 6 7 6 pffiffiffi 3 pffiffiffi 7 6 V – 2U – P þ Q – Sþ R 0 S – 2Q 7 6 2 7 6 7 6 7 6 ffiffiffi ffiffiffi 7 6 p p 1 7 6 0 – 3V – S – P – Q 0 R – 2R pffiffiffi S 7 6 2 7 6 7 6 7 6 ffiffiffi ffiffiffi 7 ˆ 6 p 1 þ p 7 Hk ¼ 6 3V þ 0 Rþ 0 – P – QSþ pffiffiffi S 2Rþ 7 ð24Þ 6 2 7 6 7 6 7 6 rffiffiffi 7 6 pffiffiffi pffiffiffi 3 7 6 – – þ þ – þ 7 6 2U V 0 R S P þ Q 2Q S 7 6 2 7 6 7 6 rffiffiffi 7 6 pffiffiffi pffiffiffi pffiffiffi 7 6 þ 3 þ þ 1ffiffiffi 7 6 – U 2V S – 2R p S 2Q – P – 0 7 6 2 2 7 6 7 6 rffiffiffi 7 4 pffiffiffi pffiffiffi 1 pffiffiffi 3 5 2VU– 2Q pffiffiffi Sþ 2R S 0 – P – 2 2 Quantum Dots: Theory 197 where structure calculations is heuristic, symmetrical k2 A ¼ E þ A9k2 þ arrangement of operators C 2 ÀÁ 1 ˆ ˆ ˆ ˆ 1ffiffiffi f ðrÞki kj ! ki f ðrÞkj þ kj f ðrÞki U ¼ p P0kz 2 3 ÀÁð27Þ 1 ˆ ˆ ÀÁ f ðrÞki ! ki f ðrÞþf ðrÞki 1ffiffiffi 2 V ¼ p P0 kx – iky 6 It has been pointed out that such ordering of opera- k2 torscanleadtounphysicalsolutionsinsome P ¼ – E þ V 1 2 circumstances [70]. One can derive the appropriate  1 form of the envelope function Hamiltonian with Q ¼ k2 þ k2 – 2k2 2 2 x y z proper operator ordering starting from the empiri- pffiffiffi hi cal pseudopotential [59] or the LDA Hamiltonian 3 2 2 [71];however,suchHamiltoniansarestillnot R ¼ – 2 kx – ky – 2i3kxky 2 widely used. pffiffiffi ÀÁ S ¼ 33kz kx – iky A variety of numerical methods can be used to solve the k ? p Hamiltonian; these include the finite- In previous equations jSi, jXi, jYi, and jZi are the difference methods [66,65,72,73] and the wave func- bulk Bloch functions that transform as s, x, y, and z tion expansion methods, where the basis functions under the action of the symmetry group can be plane waves [74,75,76,77,69,78], the eigen-

functions of the particle in a cylinder with infinite P0 ¼ – iSpˆ X ¼ – ihSjpˆ jYi¼ – iSpˆ Z ð25Þ x y z walls [79,80,81], or eigenfunctions of a harmonic is the interband matrix element of the velocity opera- oscillator [82]. 2 tor [64] usually reported in energy units as EP ¼ 2P0 , While the k ? p model can be quite reliable for large the parameter A9 is related to the conduction band embedded quantum dots, the colloidal quantum dots effective mass as are often only a few nanometers in size. In reciprocal space, this could correspond to the k point at 1/3 1 P2 1 A9 ¼ – 0 – ð26Þ toward the Brillouin zone boundary, where the k ? p 2m E þ 1 2 g 3 might no longer be adequate. Indeed, it was found that is the spin–orbit splitting, Eg is the energy gap (the the k ? p result compared to the result of a more actual energy gap after the effect of was taken into accurate calculation might differ by 50% in the account) equal to Eg ¼ EC EV, while 1, 2, and 3 confinement energy [83], and sometimes it could are the Luttinger parameters [61] of the eight-band change the ordering of the states [84]. Without care, model that can be expressed in terms of the para- spurious states in the energy gap might appear in k ? p L L L meters of the six-band model 1 ;2 ; and 3 calculations [85]. These states appear as the conse- quence of the fact that k ? p Hamiltonian does not L EP 1 ¼ – correctly represent the bandstructure for k-vectors 1 3E þ g far away from point and can give states in the gap L 1 EP for these k-vectors. The finite-difference method is, in 2 ¼ 2 – 2 3Eg þ particular, susceptible to the appearance of these states. The wave function expansion methods are less L 1 EP 3 ¼ – susceptible to this [86] since by the expansion in a 3 2 3E þ g finite basis set, the high k components of the envelope Since material parameters in the Hamiltonian of a function are effectively filtered out. Another issue is quantum dot are position dependent and the k opera- that the k ? p Hamiltonian with a limited number of tors do not commute with coordinate operators, an bands has a larger symmetry group than the true ambiguity arises about the proper choice of operator symmetry group of the system. This weakness from ordering. It is necessary to choose the ordering in the fundamental point of view can be turned into a such a way that the Hamiltonian remains hermitian; strength from the computational point of view, as it however, this condition still does not give a unique allows for block diagonalization of the Hamiltonian choice. The most widely used [65,66,67,68,69] opera- and therefore a more efficient solution of the problem tor ordering in k ? p-based quantum dot electronic [87,77,81]. 198 Quantum Dots: Theory

pffiffiffi 1.07.2.5 The Effect of Strain 3 C þ 2C ¼ ðÞ3 þ 11 12 4d 0 In previous sections, it was assumed that positions of pffiffiffi 3 atoms in a quantum dot are known a priori and that C – C ¼ ð29Þ 11 12 d 0 local arrangement is the same as in the bulk crystal. pffiffiffi However, in real structures this is certainly not the 3 4 C ¼ case. Self-assembled quantum dots are grown by 44 4d 0 þ depositing layers of material with a different lattice Although there are three elastic constants and only constant than the substrate, and therefore the quan- two parameters and , it is possible to choose and tum dot is strained. In colloidal quantum dots, there to fit the C ’s of the most zinc-blende materials is also some relaxation of atoms close to the surface. within a few percent. To obtain the relaxed atomic It is well understood that strain has a strong effect on structure, one again starts with a reasonable guess for the electronic structure of semiconductors. initial atomic structure and then minimizes E in Therefore, in this section, we describe how the effect equation 28 using some of the standard methods for of strain can be included in each of the methods finding the local minimum of a function, such as the described previously. conjugate gradient method. The atomic structure Within the framework of DFT, the effect of strain obtained can be used as an input to any of the appears naturally in the formalism itself. One starts atomistic approaches previously described: charge with a reasonable initial guess for the positions of patching, empirical pseudopotentials, and TB. atoms in the structure, then self-consistently solves It has been pointed out in Section 1.07.2.1 that the the Kohn–Sham equations and obtains the forces on chargedensitymotifsusedintheCPMdependonthe all atoms. One then moves the atoms in the direction local environment of the atom. In strained structures, of forces and obtains the new atomic configuration bond lengths and angles change compared to the ideal and solves the Kohn–Sham equations again, and the ones, which therefore represents the change in the envir- whole procedure is repeated until forces become onment that affects the motifs. To include this effect, one close to zero. In such a way one obtains a new, introduces the so-called derivative motifs, defined as the relaxed configuration for the positions of atoms in change in the motif due to a particular bond length or the structure. Unfortunately, this procedure is prac- angle change. These motifs can also be extracted from tical only for small systems and is not feasible for small-system calculations on prototype structures with larger systems, such as quantum dots. slightly changed bond lengths or angles. Once the motifs A widely used methodology to overcome these and derivative motifs are obtained, the total charge difficulties is to model the total energy of the system density is constructed and the calculation of the electro- through a classical force field, that is, to express it as a nic structure can be performed as previously described. function of atomic coordinates only. The valence It might seem at first sight that it is not necessary force field (VFF) model of Keating [88] and Martin to introduce any modifications to the empirical pseu- [89] is mostly used in inorganic semiconductors for dopotential Hamiltonian to include the effects of that purpose. Within the VFF model, the energy of strain. However, it turns out that within such an the system is modeled as [90] approach it would be difficult to correctly describe the dependence of the valence band maximum state XXnn hi 1 3 2 E ¼ ij ðRi–RjÞ2 –ðd ð0ÞÞ2 on the hydrostatic strain [91]. Therefore, a strain- 2 ð0Þ 2 ij i j 8ðdij Þ dependent term is introduced for the local part of the hipseudopotential of the atom of type in the form X Xnn 2 1 3i;jk 0 0 þ 0 0 ðRj –RiÞ?ðRk –RiÞ–cos0ðjikÞdij dik 2 8dij dik loc loc i j ;k>j v ðr; eÞ¼v ðr; 0Þ½ð1 þ TrðeÞ 30Þ ð28Þ where is a fitting parameter and Tr(e) ¼ exx þ eyy þ 0 where dij are the equilibrium bond lengths between ezz is the trace of the strain tensor. The SEPM strain- atoms i and j,and0(jik) is the equilibrium angle dependent Hamiltonian obtained this way can be between bonds ij and ik, which is a constant in solved by representing it on the basis of plane waves zinc-blende materials (0 109.47 ). In the case of or bulk Bloch bands. The extension of the unstrained zinc-blende material, the constants and are related cases to the strained cases for the basis of plane waves is to elastic constants of the material through [90] straightforward. On the other hand, this is not true if Quantum Dots: Theory 199 bulk Bloch bands are used since the Bloch functions of change in bond angles in the system leads to the unrelaxed system form a poor basis set for the different relative orientation of orbitals of neigh- relaxed system. Therefore, one needs to use the boring atoms and consequently to a different strained linear combination of Bloch bands, and the hopping integral. This effect is naturally method is then referred to as the SLCBB method. The included through the Slater–Koster [92] tables reader is referred to Ref. [48] for technical details of of matrix elements in terms of the two-center the implementation of the SLCBB method. integrals and direction cosines. Furthermore, The natural way to introduce strain in TB models there is a question whether the influence of is through the dependence of hopping integrals on strain on onsite energies should also be included. bond lengths and bond angles. The dependence on This is indeed done in many recent works bond lengths is modeled by scaling the Slater–Koster [94,93,95], although different methods are used. two-center integrals [92] from which the hopping Currently, there does not seem to exist a unique integrals are constructed as and simple model for the inclusion of this depen- dence as for the hopping integrals. d0 V ¼ V0 ð31Þ In k ? p models, the effect of strain is included d through the bulk deformation potential parameters which is a generalization [52,93] of Harrison’s d 2 that can be either measured or determined from ab rule [56,50]. In the above equation, V0 is the initio calculations. In the case of eight-band integral for equilibrium bond length d0 and V Hamiltonian for zinc-blende crystals, the strain con- the integral when the bond length is d.The tribution to the Hamiltonian reads

2 pffiffiffi pffiffiffi pffiffiffi 3 a e 0 – vþ 0 – 3v 2uu 2vþ 6 c 7 6 pffiffiffi pffiffiffi pffiffiffi 7 6 þ 7 6 0 ace 2u 3v 0 v – 2v – u 7 6 7 6 rffiffiffi 7 6 7 6 pffiffiffi 3 pffiffiffi 7 6 – v 2u – p þ q – sþ r 0 s – 2q 7 6 2 7 6 7 6 7 6 ffiffiffi ffiffiffi 7 6 p p 1 7 6 0 3v – s – p – q 0 r – 2r pffiffiffi s 7 6 2 7 6 7 6 7 6 ffiffiffi ffiffiffi 7 ˆ 6 p 1 þ p 7 HS ¼ 6 – 3vþ 0 r þ 0 – p – qsþ pffiffiffi s 2r þ 7 ð32Þ 6 2 7 6 7 6 7 6 rffiffiffi 7 6 pffiffiffi pffiffiffi 3 7 6 þ þ – þ 7 6 2uv 0 r s p þ q 2q s 7 6 2 7 6 7 6 rffiffiffi 7 6 pffiffiffi pffiffiffi pffiffiffi 7 6 þ 3 þ þ 1ffiffiffi 7 6 u – 2v s – 2r p s 2q – p 0 7 6 2 2 7 6 7 6 rffiffiffi 7 4 pffiffiffi pffiffiffi 1 pffiffiffi 3 5 – 2v – u – 2q pffiffiffi sþ 2r s 0 – p 2 2 32 where s ¼ – deðÞ13 – ie23

X3 e ¼ e11 þ e22 þ e33 1ffiffiffi u ¼ p P0 e3j kj 3 j ¼1 p ¼ ave X3 ÀÁ 1 1ffiffiffi q ¼ be33 – ðÞe11 þ e22 v ¼ p P0 e1j – ie2j kj 2 6 j ¼1 pffiffiffi 3 where a and a are the conduction and valence band r ¼ beðÞ– e – ide c v 2 11 22 12 hydrostatic deformation potentials, respectively, and 200 Quantum Dots: Theory

ÀÁ b and d are the shear deformation potentials. The ijkl ¼ C12ij kl þ C44 ikjl þ il jk strain tensor that enters the Hamiltonian (32) can X3 ð35Þ be obtained either from the VFF model (previously þ Can ipjpkplp described) or from the continuum mechanical (CM) p¼1 model. where C12, C44, and Can ¼ C11 C12 2C44 are the In the CM model, the quantum dot structure is elastic constants. The finite element discretization modeled by an elastic classical continuum medium and minimization of the functional (33) leads to a whose elastic energy is given by system of linear equations that can be efficiently solved. X Z 1 3 There have been several comparisons in the W ¼ d rijkl eij ekl ð33Þ 2 ijkl literature between the VFF and CM models [66,90,96]. While certain differences have been where ijkl is the elastic tensor relating the stress and obtained, the results of the two models give strain tensor by Hooke’s law overall agreement, as can be seen from a comparison between strain distribution in a pyr- X amidal InAs/GaAs quantum dot from Ref. [90] ij ¼ ijkl ekl ð34Þ kl that is given in Figure 1. From the fundamental point of view, the advantage of the VFF model is In the crystals with zinc-blende lattice, the elastic that it captures the atomistic symmetry of the tensor is of the form system,whiletheCMmodelshaveahigher

Subtrate DotCap Substrate Dot Cap 0.03 ε (a) xx (d) 0.00 0.00

–0.03 AE AE–CE –0.02 CE –0.06

(b) ε (e) 0.02 0.00 zz 0.01 Strain –0.10 0.00 –0.01

(c) Tr(ε)(f) 0.00 0.00 –0.06 –0.02 –0.12 –20 0 20 40 –20 0 20 40 Position along [001] Position along [001] Figure 1 Strain profiles of InAs/GaAs square-based pyramidal quantum dots and the differences along the z- direction through the pyramidal tip (left-hand side). The "xx, "zz and Tr(") ¼ "xx þ "yy þ "zz components of strain are shown respectively in parts (a), (b) and (c). The difference between these components calculated using continuum elasticity (CE) and atomistic elasticity (AE) is given in parts (d), (e) and (f) respectively. The discrepancy is the largest around the interfaces, while the strains in the barrier (GaAs substrate and capping layer) agree well within 0.5%. A significant difference is also found inside the quantum dot where the InAs experience large compressive strains at about 7% due to the lattice mismatch. Reproduced with permission from Pryor C, Kim J, Wang LW, Williamson AJ, and Zunger A (1998) Comparison of two methods for describing the strain profiles in quantum dots. Journal of Applied Physics 83: 2548–2554. Quantum Dots: Theory 201 symmetry group. From the computational point 1.07.3.1 Time-Dependent DFT of view, the VFF model is more demanding as Within the time-dependent DFT (TDDFT) the displacement of each atom is considered, in [100,101], one solves the time-dependent Kohn– contrast to the CM models where a grid of the Sham equations size of lattice constant or even larger may be used, leading to a smaller number of variables q 1 i ðr; tÞ¼ – r2 þ V ðr; tÞ ðr; tÞð38Þ to be handled. In several important cases, there qt i 2 i are analytical or nearly analytical solutions of the where CM model [97,98]. However, these advantages of XM 2 the CM models are becoming less important as ðr; tÞ¼ jj i ðr; tÞ ð39Þ modern computers can handle the VFF calcula- i¼1 tions quite easily. The potential V(r, t) should depend, in principle, on The nonself-consistent methods described charge density in all times before t. A widely used above do not allow for long-ranged charge redis- approximation is the adiabatic LDA in which it is tributions and therefore neglect the effects such as assumed that V(r, t) depends only on (r, t), and that piezoelectricity where charge is moved due to the functional form of this dependence is the same as strain. The piezoelectric potential then has to be in LDA in time independent DFT. We refer to this calculated independently and added as an addi- approximation as time-dependent LDA (TDLDA). tional potential. The components of piezoelectric The TDLDA can be used to calculate the optical polarization in a crystal of arbitrary symmetry are absorption spectrum by adding the external electro- given as magnetic field perturbation potential to equation 38

X3 and solving the equations by explicit integration in Pi ¼ "ikl ekl ð36Þ time [102]. Another approach is to assume that the k;l¼1 perturbation is small and use the linear response theory. The exciton energy can then be found from where "ikl are the piezoelectric constants of the mate- rial. In a crystal with zinc-blende symmetry, the only the eigenvalue problem XÂÃ nonzero components of "ikl are ðÞ"i – "k ij kl þðfi – fkÞKik;jl ð!Þ Cjl ¼ !Cik ð40Þ jl "123 ¼ "132 ¼ "213 ¼ "231 ¼ "312 ¼ "321 ð37Þ where "i and "k are the LDA ground-state Kohn– The charges induced by piezoelectric polarization Sham eigen energies, and fi and fk are the occupation can then be calculated, and the additional number of Kohn–Sham eigen states i and k. Within piezoelectric potential is obtained from the solu- TDLDA, Kik, jl becomes independent of ! and is tion of Poisson equation. It has been recently given as ZZ realized that in addition to the first-order 1 piezoelectric effect given by equation 36, Kik;jl ¼ dr1dr2 i ðr1Þ kðr1Þ½ jjr1 – r2 second-order piezoelectric effects might be ð41Þ LDA i important as well [99]. qmxc ðÞðÞr1 þðÞr1 – r2 j ðÞr2 l ðÞr2 qðÞr1 LDA where xc (p) is the LDA exchange-correlation potential. The first term in equation 41 is the 1.07.3 Many-Body Approaches exchange interaction, while the second can be called the screened Coulomb interaction. The justification The methods presented in Section 1.07.2 give a strat- of this assignment would require a comparison with egy for calculating the single-particle states. These equations from other approaches, such as the config- can be very useful for calculating the optical proper- uration interaction and GW+Bethe–Salpeter ties, as demonstrated, for example, in Section 1.07.4.3. equation (BSE). It might also seem surprising that Nevertheless, there are cases when the many-body the screened Coulomb interaction is not a nonlocal nature of electron–hole excitations should be directly integral between r1 and r2. This is because in the considered. The approaches along this line are LDA, the exchange-correlation term is a local described in Section 1.07.3. functional of charge density. 202 Quantum Dots: Theory

TDLDA appears to work quite well for optical spec- the conduction band state c. The eigenvalue problem tra of small clusters and molecules. The results of of the Hamiltonian then reads TDLDA can then agree quite well with experimental X measurements, as shown, for example, for the case of Hvc;v9c9Cv9c9 SiH in Ref. [103]. These results are significantly v9c9 X 4 – – improved compared to bare LDA results, which is due ¼ ½ðÞEv Ec v;v9 c;c9 þKvc;v9c9 Jvc;v9c9Cv9c9 v9c9 to exchange interaction in equation 40, which can be ¼ ECvc ð43Þ quite strong in such small systems. The screened Coulomb interaction in equation 40,however,doesnot where Ev and Ec are the single-particle eigenenergies, play a significant role then, as also shown in Ref. [103]. E is the exciton energy, and Kvc;v9c9 and Jvc;v9c9 are the On the other hand, the TDLDA is not as accurate exchange and Coulomb interactions, respectively ZZ for larger systems. For a bulk system, it is known that c9ðÞr1 v9ðÞr1 vðÞr2 c ðÞr2 Kvc;v9c9 ¼ dr1dr2 ð44Þ the TDLDA band gap will be the same as the LDA "ðÞr1; r2 jjr1 – r2 band gap [104,49]. The TDLDA does not provide a ZZ better bulk optical absorption spectrum than the vðÞr1 v9ðÞr1 c9ðÞr2 c9ðÞr2 Jvc;v9c9 ¼ dr1dr2 ð45Þ LDA, as shown in Ref. [103]. The origin of these "ðÞr1; r2 jjr1 – r2 problems is the screened Coulomb interaction in The effective dielectric screening used in equations equation 40, which then gives a significant contribu- 44 and 45 is of the form tion. However, such diagonal form of screened Z 1 1 Coulomb interaction is not able to correctly describe ¼ " – 1 ðÞr ; r dr ð46Þ "ðÞr ; r jjr – r bulk 1 jjr – r its long-range behavior. 1 2 1 2 2 Density functionals other than the LDA can also " – 1 ; be used in conjuction with the TDDFT. A very where bulkðÞr1 r is the bulk inverse dielectric function popular approach is to use the hybrid B3LYP func- that differs from the one of the quantum dot nanos- tional [105]. Within the B3LYP approach, the total tructure, which also contains the surface polarization energy is modeled as a linear combination of the potential P discussed in Section 1.07.2. The use of bulk exact Hartree–Fock exchange with local and gradi- inverse dielectric function is in line with the fact that ent- corrected exchange and correlation terms. The the single-particle energies Ec and Ev are obtained from coefficients in the linear combination were chosen to the Schro¨dinger equation that does not contain the fit the properties of many small molecules. The surface polarization term. If the surface polarization B3LYP method gives accurate bandgaps for various term is used in the single-particle equation, then the bulk crystals [106]. Since it contains the explicit full inverse dielectric function should be used and the exchange integral, it introduces the long-range surface polarization terms will roughly cancel out. Coulomb interaction in equation 40. Therefore, the There are arguments that the exchange interac- TDDFT–B3LYP can overcome the two difficulties tion Kvc;v9c9 should not be screened, which come from of TDLDA previously discussed. the two-particle Green’s function construction, where screening of the exchange term would cause 1.07.3.2 Configuration Interaction Method double counting [107]. Nevertheless, in practice, it is found that the exchange consists of a long-range term When the single-particle states are obtained, one can that should be unscreened and a short-range term form many-body excitations by creating Slater deter- [108] that should be screened by the bulk dielectric minants out of these single-particle states. One can function [109,110]. The effective dielectric function then diagonalize the many-body Hamiltonian in the "ðÞr1; r2 used in equation 44 incorporates this Hilbert space formed from a restricted set of such because "ðÞ!r1 : r2 1 for jjr1 – r2 ! 0. The seeming determinants. This approach is called the configura- contradiction to the Green’s function argument can tion interaction (CI) method. When one is interested be resolved by realizing that if only a limited config- in excitons, the wave function is assumed as uration space is used in equation 43, the effect of XNv XNc other unused configurations can be included in the ¼ Cv;c v;c ð42Þ exchange screening term [107]. v¼1 c¼1 The CI equation 43 has the same form as the where v,c is the Slater determinant when the corresponding equation in the TDLDA (equation 40), electron from the valence band state v is excited to with the difference in the expressions for the exchange Quantum Dots: Theory 203 and the Coulomb integrals. On top of a single-particle While equations 48 and 49 should, in principle, be calculation, the CI method was used to calculate very solved self-consistently, one usually replaces the self- large systems, such as pyramidal quantum dots with energy term with its expectation value with respect to near by one million atoms [111]. It was also used to the LDA Kohn–Sham wave functions h kjj ki, calculate many-body excitations, such as multiexci- which constitutes the zeroth-order approximation of tons, and few electron excitations, and to study Auger the GW procedure. It has been shown that the self- effects [112]. All these calculations are made possible consistent calculations [115,116,117,118,119] make by selecting a limited window of single-particle states the spectral properties worse. Such calculations are used in these configurations. It is difficult or impos- performed with the use of pseudopotentials. It is pos- sible to study such systems using TDLDA or sible that self-consistency will not make the results GW þ BSE. One should nevertheless be cautious worse if all-electron calculation is performed. about the models used for screening in these multi- The two-particle Green’s function defined as particle excitations. GðÞr1t1;r 2t2; r91t91;r92t92 y y ¼ – MT ðÞr1t1 ðÞr2t2 ðÞr92;t92 ðÞr91;t91 M ð51Þ

1.07.3.3 GW and BSE Approach contains the information about the exciton energies. These can be retrieved by taking – – Within this approach, one first calculates the quasi- t1 ¼ t91 þ 0 and t2 ¼ t92 þ 0 and transforming to fre- particle eigenenergies, which is somewhat analogous to quency space to obtain (in the condensed notation) single-particle calculations in Section 1.07.2. These are G2(!), whose poles are the exciton energies. The then used to solve the BSE for excitons, which is in some DysonequationforG2(!)reads[120,113] sense similar to CI equations of the previous section. ð0Þ ð0Þ A quasiparticle is defined as the pole in frequency G2ðÞ¼! G2 ðÞþ! G2 ðÞ! K 9ðÞ! G2ðÞ! ð52Þ space in the single-particle Green’s function ð0Þ DEwhere G2 ðÞ! is the noninteracting two-particle y GðÞ¼rt; r9t9 – i MjT ˆðÞrt ˆ ðÞjr9t9 M ð47Þ Green’s function, and K9(!) is an electron–hole inter- action kernel. Equation 52 for the electron–hole pair where ˆðÞrt is the particle creation operator, jMi the is called the BSE [120]. It can be solved by expanding M particle ground state, and T the time-ordering the exciton wave function as operator. Quasiparticle energies correspond to ener- X X y ˆy gies for adding or removing one electron from the jM; Si¼ Cvc aˆvbc jMið53Þ system [113]. Within the GW approximation [114], v c þ þ the appropriate single-particle equation reads where av and bc are the hole and electron creation "#Z X operators. The equations for Cvc coefficients are then 1 ðr9Þ – r2 þ v ðr – R Þþ d3r9 ðrÞ given as 2 bare atom jr – r9j i atomZ XÀÁ X ðÞ"c – "v Cvc þ Kvc;v9c9 – Jvc;v9c9 Cv9c9 ¼ S Cvc ð54Þ þ ðr;r9;i Þ i ðr9Þdr9 ¼ i ðrÞð48Þ v9c9 where where "c , "v are the quasiparticle eigenenergies X obtained from equation 48, and S is the exciton ðÞ¼r;r9;! – ðÞr ðÞr9 k k energy. The Kvc;v9c9 is the same as in equation 44 k Z 1 ImW ðÞr;r9;!9 without the screening "ðÞr1; r2 . The Jvc;v9c9,onthe fkW ðÞþr;r9;"k –! d!9 other hand, reads !–"k –!9þi Z ð49Þ Jvc;v9c9 ¼ drdr9 c ðÞr c9ðÞr vðÞr9 v9ðÞr9 is the self-energy potential that replaces the LDA Z i – ! exchange-correlation potential of LDA single- d!e i 0þW ðÞr;r9;! h2 ð55Þ particle equations. þ –1 Z ðÞS –!–ðÞþ"c9 –"v i0 – 1 1 i W ðÞ¼r; r9;! " ðÞr; r1;! dr1 ð50Þ þ –1 jjr1 – r9 þðÞS þ!–ðÞþ"c –"v9 i0 is the dynamically screened interaction, where where W(r,r9,!) is the screened Coulomb interaction 1 " (r, r1, !) is the inverse dielectric function. given by equation 50. 204 Quantum Dots: Theory

The GW þ BSE approach is thought to be one of to the results of the CI where J is screened by the the most reliable methods for the calculation of the bulk dielectric function. In the TDLDA, where optical absorption spectra and excited-state electro- the Coulomb interaction is local, it is also screened nic structures. It has been used to calculate small by the bulk dielectric function, in line with the fact molecules and bulk crystals. Unfortunately, its use the LDA single-particle states used in equation 40 for the larger systems is hindered by the significant do not include the surface polarization term. The computational cost. different cancellation schemes in TDLDA, CI, and Excellent agreement with experimental results GW þ BSE (equations 40, 43, and 54) can be illu- was obtained for the optical absorption spectrum strated by comparing the calculated absorption of bulk Si calculated using the GW þ BSE approach spectra in these methods with the one obtained [121]. Within GW þ BSE, the lower energy peak from single-particle energies. It was shown in Ref. originating from the excitonic binding effect was [124] that the BSE absorption spectrum of small obtained for the first time. In contrast, previous clusters of SinHm is red-shifted from the calculated LDA and TDLDA results were unable to predict single-particle spectrum, which is mostly due to this peak due to the inadequacy of the local approx- negative surface polarization energies in the imation for the Coulomb interaction. Coulomb interaction J. On the other hand, it was The BSE (equation 54) is formally the same as shown in Ref. [103] that the TDLDA spectrum is the linear response TDLDA (equation 40) and CI blue-shifted from the single-particle LDA spectrum. (equation 43), except for the meanings of single- There is no surface polarization in J or single-parti- particle energies and the exchange and screened cle energies then; thus, the exchange interaction Coulomb interactions. In the GW þ BSE approach, dominates the spectrum shift. However, if total the quasiparticle electron (hole) energies are equal to LDA energy differences for adding or removing an the total energy difference for adding (or removing) electron are used in equation 40, then the surface one electron from the system and include the surface polarization must be considered [125] and Coulomb polarization term P(r) from equation 1. On the other interaction cannot be calculated from equations 45 hand, in the CI approach, the single-particle states and 46. The above-discussed cancellations are only are obtained without including the P(r) term. The good for spherical quantum dots. For quantum rods, single-particle states then do not correspond to the wires, and the nanostructures of other shapes, the total energy differences for adding or removing one GW þ BSE-like approach should be used. In the CI electron. A surface polarization term needs to be approach, P(r) should be added to the single-particle added to relate them to ionization energies or elec- energy, while the full nanosystem inverse dielectric tron affinities [122]. In the linear response TDLDA function (not the bulk one) should be used for (equation 40), the single- particle energy is the "ðÞr1; r2 in equations 44 and 45. Kohn–Sham LDA eigenenergy. It does not contain the surface polarization term, just like the single- particle energies in the CI approach. A difference 1.07.3.4 Quantum Monte Carlo Methods between the TDLDA and CI approach is that Within the quantum Monte Carlo (QMC) method TDLDA contains LDA band-gap error, while in the [126], the whole system is described by a many-body CI approach the single-particle states can be found, wave function and the many-body Schro¨dinger equa- for example, by EPM or SEPM that correct the LDA tion is solved using some of the Monte Carlo band-gap error. techniques such as variational Monte Carlo method The polarization term in the GW quasiparticle (VMC) [127,128] or diffusion Monte Carlo method eigenenergy cancels out a term in J in equation 54 (DMC) [129,130]. because the same surface polarization term exists in In the VMC, the variational form of the many- 9 ! W(r,r , ) in equation 55. This cancellation corre- body wave function (X) is assumed as a Slater sponds to the cancellation of the PM(r1, r2) and determinant multiplied by a Jastrow term P(r1), P(r2) terms in equation 2 of the classical phe- "# nomenological analysis. Delerue et al. [123] have XM XM ÀÁ " # shown numerically that the Coulomb correction ðÞ¼X D ðÞR D ðÞR exp ðÞri – u ri –rj ð56Þ < term cancels the polarization term in the self-energy i¼1 i j of the quasiparticle eigenenergy. Consequently, the where X ¼ {ri , si} for i ¼ 1, M. The Slater determi- results of the GW þ BSE are expected to be similar nant D is usually constructed from single-particle Quantum Dots: Theory 205

LDA or Hartree–Fock wave functions, while para- linear scaling QMC method, Slater determinants meterized forms are used to express and u. The are represented on a basis of localized Wannier func- total energy of the system is found by minimizing the tions. This makes the Slater determinant sparse and expectation value of the many-body Hamiltonian H, therefore the calculation time is proportional to the given as size N, instead of N3 in the old scheme. Consequently, E ¼ hijjH =hij this allows the QMC calculation of a few hundred atoms and makes possible the use of the QMC The last expression can be rewritten as method for small quantum dots [134]. Z Z HðXÞ ðÞX HðÞX dX jjðÞX 2dX ðÞX E ¼ Z ¼ Z ð57Þ 2 2 jjðÞX dX jjðÞX dX 1.07.4 Application to Different Physical Effects: Some Examples The last integral can be viewed as the average value of the quantity HðÞX =ðÞX distributed in multi- 1.07.4.1 Electron and Hole Wave Functions dimensional space with a probability jjðÞX 2. It can The shape of the single-particle wave functions and be found using the Metropolis algorithm. In this algo- their energies determine many physical properties of rithm, one simulates the path of the walker in the quantum dots. This section is, therefore, devoted multidimensional space. The jump of the walker from to the analysis of electron and hole wave functions. X to X9 is accepted if ¼ jjðÞX9 2=jjðÞX 2 > 1and The wave functions of the lowest four states in the accepted with probability if < 1. The average value conduction band and the top four states in the of HðÞX =ðÞX along the path of the walker gives E in valence band of a pyramidal [119]-faceted equation 57. InAs/GaAs quantum dot are presented in Figure 2. In the DMC, one treats the many-body imaginary The results presented in Figure 2 were obtained time Schro¨dinger equation as the classical diffusion using the EPM, including the effect of spin–orbit equation [129,130]. In this method, the many-body interaction. wave function (not its square) corresponds to the The lowest conduction state is an s-like state, equilibrium distribution of Monte Carlo walkers. while the next two conduction states are p-like states However, for fermion system, antisymmetry is oriented in the directions of base diagonals, with required for the many-body wave function. This nodal planes perpendicular to these directions. causes a sign problem, which is usually approxi- These are followed by d-like states. Due to lateral mately solved by a fixed nodal approximation dimension larger than the quantum dot height, none where an auxiliary wave function is used to define of the nodal planes is parallel to the pyramid base. the fixed nodal hypersurface for the DMC wave There are, therefore, only two p-like states, in con- function. Usually, the VMC wave function of equa- trast to spherical quantum dots where there are three tion 56 is used as the auxiliary wave function. p-like states. With the use of pseudopotentials [131], both The two p-states are relatively close in energy VMC and DMC methods have been used for systems and their splitting is caused by several effects. To up to a dozen atoms. Williamson et al. [132] showed discuss each of these effects, we first assume that the that QMC methods can be used for exciton energies. structure is unstrained. This is done by replacing one single-particle valence band wave function with a conduction band wave 1. In the simplest single-band effective mass model, function in the Slater determinant. The DMC with these states are degenerate and can be split if the a nodal hypersurface defined by this new Slater base of the pyramid acquires a shape different than determinant is performed then, and it fully takes the square. The same is the case for the four-band into account the resulting correlation effects. This k ? p model (i.e., eight-band model without spin– approach gives the Si band structure that agrees orbit effects). Addition of spin–orbit effects to well with the experiments. QMC is one of the most four-band k ? p splits these levels by a small (less reliable methods for small-system calculations. than 1 meV) amount [77,135]. Atomistic methods The development of a linear scaling QMC predict the correct symmetry of the system and method [133] extended its applicability from a split the p-states, as well as k ? p models with dozen atoms to a few hundred atoms. Within the larger number of bands (14, for example). 206 Quantum Dots: Theory

Conduction and valence band states for b = 20a dot

Side view Side view Top view

CBM VBM

[110]

CBM + 1 VBM-1

[110] CBM + 2 VBM-2

CBM + 3 VBM-3

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 b ¼ 20a,wherea 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 included for valance band maximum (VBM-1) and VBM-2.) in the k ? p model within CM approach, it cannot The approximation of using a single heavy hole cause the splitting, while the VFF model, due to band to describe the valence state, which is often its atomistic nature, splits the p-states. used in quantum wells, is therefore not applicable to 3. Piezoelectricity added to any of the models also quantum dots due to stronger heavy and light hole causes the splitting of the p-states. mixing. As the dot size is reduced, the valence band ener- The splitting of the p-states is therefore caused by gies become lower and the conduction band energies the shape anisotropy, spin–orbit effect, atomistic higher. The bound states are then less confined and (a)symmetry, strain, and piezoelectricity. It is amaz- the effective energy gap increases. With the reduc- ing that a single quantity is determined by such a tion in quantum dot dimensions, some bound states large number of effects. Unfortunately, in a given become mixed with wetting layer or continuum quantum dot, all these effects are present and cannot states and the number of truly bound states decreases. be probed separately. The conduction band states are formed essentially of a single envelope function and therefore these can 1.07.4.2 Intraband Optical Processes in be classified as being s, p, and d-like. On the other Embedded Quantum Dots hand, the band mixing of the valence states is much stronger and such a simple classification is not possi- Most of the semiconductor optoelectronic devices ble. The valence band functions actually have no utilize transitions between the conduction-band nodal planes. (This becomes obvious from Figure 2, states and the valence band states. The operating when the isosurface values are additionally reduced wavelength of these devices is mainly determined Quantum Dots: Theory 207 by the band gap of the materials employed and is (diameters of the order of 20 nm and more) and have therefore limited to the near-infrared and visible part very small height (of the order of 3–7 nm) in the z- of the spectrum. However, if one wishes to access direction; therefore, the effective potential well repre- longer wavelengths, a different approach is neces- senting the z-direction confinement is narrow (see sary, that is, the transitions within the same band Figure 3). In a typical case, therefore, "1 "0 is of the have to be used. These transitions are called intra- order of at least 100 meV. band transitions. Intraband optical transitions in bulk The optical absorption matrix elements on the are not allowed and therefore low-dimensional transitions between states are proportional to the nanostructures have to be used. Therefore, in the matrix elements of coordinate operators; therefore, past two decades, semiconductor nanostructures, by calculating the latter, one obtains the following such as quantum wells, wires, and dots, have been selection rules on the transitions between states: recognized as sources and detectors of electromag- netic radiation in the mid- and far-infrared region of nx ¼1, ny¼ 0, nz¼ 0, for x-polarized the spectrum. radiation; When the use of nanostructures as detectors is ny ¼1, nx¼ 0, nz¼ 0, for y-polarized radia- concerned, several limitations of quantum well infrared tion; and photodetectors (QWIPs) have motivated the develop- nx ¼ 0, ny ¼ 0, for z-polarized radiation. ment of quantum dot infrared photodetectors (QDIPs). The transitions from the ground state are of primary Themainoriginoftheundesirabledarkcurrentin importance for QDIPs. From the selection rules QWIPs is thermal excitation (due to interaction with obtained, one concludes that only the transition to a phonons) of carriers from the ground state to the con- pair of degenerate first excited states is allowed for in- tinuum states. The discrete electronic spectrum of plane polarized radiation, while in the case of quantum dots as opposed to continuum spectrum of z-polarized radiation, only the transitions to higher quantum wells significantly reduces the phase space for excited states are allowed, as demonstrated in Figure 3. such processes and therefore reduces the dark current. Although the model presented considers the Higher operating temperatures of QDIPs are therefore quantum dot band structure in a very simplified expected. Due to optical selection rules, QWIPs based manner, it is excellent for understanding the results on intersubband transitions in the conduction band of more realistic models. The strict selection rules interact only with radiation having the polarization from this model are then relaxed, and strictly for- vector in the growth direction. This is not the case in bidden transitions become weakly allowed. quantum dots since these are 3D objects where the Nevertheless, qualitatively, the absorption spectrum corresponding selection rules are different. retains the same features as in this model. For the QDIP applications, it is essential to understand the quantum dot absorption spectrum. The simplest model that is sufficient to qualitatively under- standthequantumdotintrabandabsorptionspectrumis the parabolic dot model, where the potential is assumed in a separable form V ðÞ¼r V1ðx; yÞþV2ðzÞ,where 1 ÀÁ V ðx; yÞ¼ m!2 x2 þ y2 is the potential of a 2D 1 2 z-polarized harmonic oscillator, and V2(z) is the potential of a quan- tum well confining the electrons in the z-direction. The solutions are of the form ðrÞ¼hnx ðxÞhny ðyÞ nz ðzÞ, In-plane polarized where hn(t) is the wave function of a 1D harmonic oscillator, and nz ðzÞ are solutions of the 1D Schro¨dinger equation with potential V2(z) (correspond- ing to energiesÀÁ"nz ). TheÀÁ eigenenergies are then of the form Enx; ny; nz ¼ h! nx þ ny þ 1 þ "nz . The fac- tor h! corresponds to the transition energy from the Figure 3 Scheme of energy levels and allowed optical ground to first excited state, and for modeling realistic transitions in a parabolic quantum dot model with infinite quantum dots it should be set to h! 40 – 70 meV. potential barriers. Only the levels with nx þ ny 2 and nz 1 Typical quantum dots are wide in the xy-plane are shown. 208 Quantum Dots: Theory

(a) 80 (b)

) 0.85 ) 10 2 2 cm

cm 60

–15 0.8

–15 8

40 0.75 z-polarized 6 0.7

20 Energy (eV) 4

Abs. cross section (10 Abs. 0.65 2 0 x-polarized Abs. cross section (10 Abs. 0 0.05 0.1 0.15 Energy (eV) 0.6 0 0 0.1 0.2 0.3 0.4 0.5 1/2 3/2 5/2 7/2 Energy (eV) |mf|

Figure 4 (a) The intraband optical absorption spectrum for a quantum dot of conical shape with the diameter D ¼ 25 nm and height h ¼ 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 approximation of eight-band k ? p model, is also indicated.

The absorption spectrum obtained by the eight- 1.07.4.3 Size Dependence of the Band Gap band k ? p model for an InAs/GaAs quantum dot of in Colloidal Quantum Dots conical shape with the diameter D ¼ 25 nm and The size dependence of the band gap is the most height h ¼ 7nm is presented in Figure 4.The prominent effect of quantum confinement in semicon- dimensions were chosen to approximately match ductor nanostructures. The band gap increases as the those reported for quantum dots in a QDIP struc- nanostructure size decreases. Many of quantum dot ture in Ref. [136]. The optical absorption spectrum applications rely on the size dependence of the optical in the case of z-polarized radiation is shown in properties. Therefore, studying the size dependence of Figure 4(a). The experimental intraband photocur- the band gap is one of the most important topics in rent spectrum exhibits the main peak at 175 meV semiconductor nanocrystal research. and a much smaller peak at 115 meV, in excellent According to a simple effective-mass approxima- agreement with the results obtained for z-polarized tion model, the band-gap increase of spherical incident radiation where the corresponding peaks quantum dots from the bulk value is occur at 179 and 114 meV, respectively. The corre- sponding absorption spectrum for in-plane 2h22 E ¼ ð58Þ polarized incident radiation is presented in the g md 2 inset of Figure 4(a). There is a single peak in the where d is the quantum dot diameter and spectrum, which is due to the transition from the 1 1 1 ground state to a pair of nearly degenerate first ¼ þ ð59Þ m m m excited states (see Figure 4(b)). e h The results presented and other similar calcula- with me and mh being the electron and hole effective tions suggest that the in-plane polarized radiation masses. causes nonnegligible transitions only between the The experiments usually measure the optical gap ground and first excited states, these being located of a quantum dot. Therefore, in addition to the dif- in the region 40–80 meV in the far-infrared. On the ference in single-particle energies, one has to include other hand, z-polarized radiation causes the transi- the interaction between created electron and hole, in tion in the 100–300 meV region in the mid- order to calculate the optical gap. One simple infrared. The best way to understand the origin of approach to do this is to calculate the exciton energy such behavior is through a simplified parabolic model by including the electron–hole interaction on top of presented. Such behavior can be altered only if the the single-particle gap. This procedure ignores the dot dimension in the z-direction becomes compar- electron–hole exchange interaction and possible cor- able to the in-plane dimensions. relation effects. However, in the strong confinement Quantum Dots: Theory 209 regime, which is present in most colloidal nanocrys- results of these methods are shown in Figure 6 for tals and embedded quantum dots, these effects are H-passivated Si quantum dots [138]. The DMC very small. Under this approximation, the exciton method and GW–BSE method produce almost the energy can be expressed as same band gap for the smallest quantum dots. The DMC result is about 1 eV above all the other results E ¼ " – " – EC ð60Þ ex c v cv for somewhat larger quantum dots with the diameter where "c and "v are the single-particle CBM and up to 1.6 nm. It remains to be seen how accurate is C VBM energies, and Ecv is the electron–hole this DMC result, for example, when compared with Coulomb energy calculated as well-controlled experiments (perhaps for other ZZ material quantum dots like CdSe). The TDLDA 2 2 C jj cðÞr1 jj vðÞr2 method gives almost the same results as the LDA Ecv ¼ dr1dr2 ð61Þ "ðÞr1 – r2 jjr1 – r2 Kohn–Sham energy difference. This suggests that both the exchange and Coulomb interactions in the where c(r) and v(r) are the electron and hole wave TDLDA results have a very small contribution. functions, and "(r1 r2) is a distance-dependent screening dielectric function, which can be modeled Besides TDLDA, TDDFT-B3LYP was used in as described in Ref. 32. Refs. [138,139]. The TDDFT-B3LYP band gap is The dependence of calculated optical gap on below the DMC result, especially for relatively CdSe nanocrystal size is presented in Figure 5.A large quantum dots. However, in Ref. [139], it was fit of the theoretical results to the shown that for small molecules, the TDDFT-B3LYP result agrees with the MR-MP2 quantum chemistry – Eg ¼ a?d calculations. The TB and EPM results in Figure 6 dependence yields values quite different from the simple can be considered as the lowest order results of the d 2 law predicted from the effective mass approach. In CI equation 43, where only the zero-order screened the case of CdSe, ¼ 1.18. The parameter is material- Coulomb interactions between the VBM and CBM dependent and its values for III–V and II–VI semicon- states are taken into account. These agree well with ductors typically fall in the range of 1.1–1.7. each other. However, they are between the TDLDA and TDDFT-B3LYP results. To summarize these results, the DMC result is above all the other methods for d ¼ 1.5nm Si quan- 1.07.4.4 Excitons tum dots. The LDA and TDLDA have the lowest In the previous section, we have presented the exci- band gap, followed by the TB and EPM-limited CI ton calculations based on a simple, but useful results and the TDDFT-B3LYP results. For very approach. For the calculation of excitons, the meth- small quantum dots, the DMC results agree well ods in Section 1.07.3 must be used in principle. The with the GW-BSE results.

1.6 CdSe QDs Experiment SEPM 1.2 LDA + C

Δ 1.18 Eg = 2.12/d (LDA + C)

(eV) 0.8 g

E 1.18 ΔE = 1.90/d (SEPM)

Δ g

0.4

0.0 1 234567 Diameter (nm) Figure 5 Comparison of the exciton energy shift from its bulk value of CdSe quantum dots (QDs) between experiment, charge patching method (with band-gap corrected pseudopotentials) (LDA þ C), and SEPM calculations. Coulomb energies are considered in this calculation. Experimental data is from Ref. [137]. Reproduced with permission from Li J and Wang L-W (2005) Band-structure- corrected local density approximation study of semiconductor quantum dots and wires. Physical Review B 72: 125325. 210 Quantum Dots: Theory

Number of Silicon atoms 125101729 35 66 87 147 293 525 10

DMC TDLDA Empirical pseudopotential 8 Tight binding LDA HOMO-LUMO GW+BSE TD-B3LYP 6

4

Optical gap (eV)

2 Bulk gap

0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Diameter (nm)

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 help satisfy the energy conservation. Their effect can then be phenomenologically modeled by Lorentzian Auger effects play a crucial role in carrier dynamics broadening of the delta function in Fermi’s golden in nanostructures when both types of carriers rule expression as (electrons and holes) are present. They become important, in particular, in quantum dots that have ÀÁ 1 E – E ! ÀÁ ð63Þ discrete electronic levels, which implies that the fn i 2 2 2 Ef – Ei þðÞ=2 competing phonon-assisted relaxation processes are n strongly reduced. Different types of Auger processes The most important step in the electron cooling are schematically illustrated in Figure 7. process involves the transition of the electron from According to Fermi’s golden rule, the formula for the p-level (ep) to the ground s electronic state (es). the Auger rate is given as This process is mediated by a transition of the hole X from hs to hn. The calculated Auger lifetime for this 2 ÀÁ W ¼ jjjii H f ij2 E – E ð62Þ process is shown in Figure 8. Its value is of the order i h n fn i n of 0.1–0.5 ps, in agreement with experimental results where jii and jfni are the initial and final Auger states, [140]. This result suggests that Auger processes are sufficient to explain electron cooling in quantum Ei and Efn their energies, and H is the Coulomb interaction. At a temperature T 6¼ 0, the Boltzmann dots, although other mechanisms are not necessarily average over the initial states has to be taken. It ruled out. would seem at first sight that the discreteness of The same process can take place in the presence quantum dot energy levels and the requirement for of an electron and a hole that act only as spectators. It energy conservation in the process would not allow is very interesting that the electron lifetime increases for efficient Auger processes. However, other excita- by an order of magnitude in those cases, as demon- tions, such as phonons, can be involved as well and strated in Figure 8. Quantum Dots: Theory 211

(a)Electron thermalization (b) Thermalization with spectator

Single-particle gap

(c) Bi-exciton recombination (d) Tri-exciton recombination

2–1 3–2 τe–h τe–h

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

(e) Negative trion recombination (f) Positive trion recombination

τ τ e h

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

Figure 7 Illustration of different Auger processes: (a) electron thermalization; (b) thermalization with spectator; (c) bi-exciton recombination; (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.

6

Cl Cd232Se235 with spectator 4

2 SP Cl No spectator Auger lifetime (ps) lifetime Auger 0 0.3 0.4 0.5 ε ε p– s (eV)

Figure 8 Electron cooling. Auger lifetimes at T ¼ 300 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. 212 Quantum Dots: Theory

1.07.4.6 Electron–Phonon Interaction 0.76 The theory and results presented so far covered only the stationary electronic structure of quantum dots 0.74 when atoms are in their equilibrium positions. 0.72 However, at finite temperatures the vibrations of atoms around their equilibrium positions (phonons) 0.7

create additional potential that perturbs otherwise Energy (eV) 0.68 stationary electronic states and causes transitions among them. 0.66 Phonons in quantum dots can be treated at various levels of approximations. The approximation that is 10 20 30 40 often used for large quantum dots is that the phonons ΔE (meV) are the same as in bulk material. The strongest cou- Figure 9 Dependence of the polaron energy levels pling between electrons and phonons in polar obtained by direct diagonalization (circles) on the artificial crystals is polar coupling to longitudinal optical shift E. The corresponding single-particle levels (LO) phonons, while deformation potential coupling calculated using the eight-band k ? p model are shown by to longitudinal acoustic (LA) phonons might also full lines. Lens-shaped quantum dot with the diameter of 20 nm and the height of 5 nm is considered. sometimes be important. In order to calculate the transition rates among different electronic states due to the interaction There is therefore a widespread thought that car- with LO phonons, it is tempting to apply Fermi’s rier relaxation in quantum dots should be treated by Golden rule, which is a good approximation in considering the carriers as polarons. The polaron quantum wells, for example, [141]. However, its lifetime is then determined by anharmonic decay of direct application to quantum dots leads to the an LO phonon into two low-energy phonons result that transition rates are zero unless two [148,143,149,150,151]. It is thought that the physical levels are separated by one LO phonon energy process responsible for that decay process is the exactly [142]. Such an approach treats the electron decay either to two LA phonons [152,148] or to one and phonon systems separately with their interac- acoustic and one optical phonon [151]. Within such tion being only a perturbation. It is currently assumptions, the polaron lifetime is in the case of a known that electrons and phonons in quantum two-level system given by [148] dots form coupled entities – polarons. Polarons in pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðÞR – X self-assembled quantum dots have so far been evi- W ¼ – ð64Þ denced experimentally by optical means in the h h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ÀÁ intraband magneto-optical spectrum [143,144], where R ¼ X 2 þ Y 2; X ¼ g2 þ 2 – 2 =4; magneto-photoluminescence spectrum [145], and Y ¼ =2; ¼ Ei – Ef – h!LO,gistheelectron– Raman scattering [146], and it has been suggested phonon coupling strength, =h the phonon decay theoretically that they could have transport signa- rate, and Ei and Ef the energies of the single- tures as well [147]. Polarons are usually evidenced particle states. Equation 64 hasbeenusedin by anticrossing of electron energy levels when several occasions to fit the experimental results these are gradually changed, such as, for example, on intraband carrier dynamics in quantum dots by magnetic field. We illustrate this here by a [149,153]. numerical experiment where the energies of the The approximation of bulk phonon modes pair of first excited states are shifted in opposite certainly fails in small quantum dots. In that case, directions by the same amount E, which is var- one should use the atomistic description of phonons. ied. The polaron states were calculated by direct To calculate the phonon frequencies and displace- diagonalization of the electron–phonon interaction ments, one needs a force field that describes the Hamiltonian. The polaron energy levels that con- vibrations of atoms around their equilibrium tain a contribution from at least one of the positions. VFF, for example, can be used for that electronic states larger than 10% are shown by purpose. To calculate the electron–phonon coupling, circles in Figure 9. Anticrossing features in one needs to be able to calculate the change in single- polaron spectrum are clearly visible. particle Hamiltonian due to atomic displacements. Quantum Dots: Theory 213

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