Electron states in semiconductor quantum dots.

Electron states in semiconductor quantum dots Suman S. Dhayal, Lavanya M. Ramaniah, Harry E. Ruda, and Selvakumar V. Nair Citation: The Journal of Chemical Physics 141, 204702 (2014); doi: 10.1063/1.4901923 View online: http://dx.doi.org/10.1063/1.4901923 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/20?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Moments-based tight-binding calculations of local electronic structure in InAs/GaAs quantum dots for comparison to experimental measurements Appl. Phys. Lett. 88, 053109 (2006); 10.1063/1.2171473 Size-dependent band gap of colloidal quantum dots J. Appl. Phys. 99, 013708 (2006); 10.1063/1.2158502 Biexcitonic blue Stark shift in semiconductor quantum dots J. Appl. Phys. 97, 123101 (2005); 10.1063/1.1927705 Magnetic-field and laser effects on the electronic and donor states in semiconducting quantum dots J. Appl. Phys. 92, 4209 (2002); 10.1063/1.1509110 Formation of quantum-dot quantum-well heteronanostructures with large lattice mismatch: ZnS/CdS/ZnS J. Chem. Phys. 114, 1813 (2001); 10.1063/1.1333758

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THE JOURNAL OF CHEMICAL PHYSICS 141, 204702 (2014)

Electron states in semiconductor quantum dots Suman S. Dhayal,1,a) Lavanya M. Ramaniah,2,b) Harry E. Ruda,3 and Selvakumar V. Nair3,c) 1

Department of Physics, University of North Texas, P.O. Box 311427, Denton, Texas 76203, USA High Pressure and Synchrotron Radiation Physics Division, Physics Group, Bhabha Atomic Research Centre, Trombay, Mumbai 400085, India 3 Centre for Nanotechnology, University of Toronto, 170 College Street, Toronto, Ontario M5S 3E3, Canada 2

(Received 22 May 2014; accepted 4 November 2014; published online 24 November 2014) In this work, the electronic structures of quantum dots (QDs) of nine direct band gap semiconductor materials belonging to the group II-VI and III-V families are investigated, within the empirical tight-binding framework, in the effective bond orbital model. This methodology is shown to accurately describe these systems, yielding, at the same time, qualitative insights into their electronic properties. Various features of the bulk band structure such as band-gaps, band curvature, and band widths around symmetry points affect the quantum confinement of electrons and holes. These effects are identified and quantified. A comparison with experimental data yields good agreement with the calculations. These theoretical results would help quantify the optical response of QDs of these materials and provide useful input for applications. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4901923] I. INTRODUCTION

Quantum dots (QDs) in the form of semiconductor-doped glasses have been used as coloured glass filters for several decades.1 An explosion of research activity in the later years driven by the interest in quantum confinement and the possibility of controlling electronic properties by size and shape of nanostructures promoted these materials into a class of their own. In particular, with developments in growth and characterization in the 1980s and 1990s,2 semiconductor nanocrystals in colloidal solutions and polymer or glass substrates have found potential applications in diverse fields including photovoltaics,3 optoelectronics,4, 5 and biological imaging.6 The analysis and understanding of the electronic properties that lead to these applications involve quantum confinement effects of the carriers, which crucially requires the accurate determination and characterization of the electron and hole energy levels. However, simplistic theories such as the single-band effective mass theory (EMA) continue to be used in the interpretation of experimental data often yielding misleading results. Thus, it is imperative to use more accurate theories, such as the pseudopotential or tight-binding (TB) theories, in order to understand and interpret the experiments. To the best of our knowledge, there has been no comprehensive study of the quantum size effects in all the elemental and compound semiconductors. Looking for trends across periods, or down groups, of the periodic table, should yield interesting insights into several aspects of the physics of quantum confinement. Some of the questions which arise as we look for trends in different materials, are, for instance, how do quantum confinement effects in narrow band gap materials compare with those in wide band gap materials? Would

the agreement between TB and EMA for these two classes of materials improve or worsen as a function of size? Such a wide-ranging study is, in our opinion, long overdue given the importance of this field, both from fundamental science and applications points of view. Hence, in this paper, the electronic structure of several semiconductor materials belonging to the III-V and II-VI semiconductor families is studied, at depth, using the empirical tight-binding approach. Only direct band gap materials, with bulk band structures which can be described reasonably well within a sp3 basis set, are considered. Section II contains a brief overview of the theoretical methods that are relevant to this work. The calculational details are given in Sec. III, followed by a discussion of the results in Sec. IV, and finally the conclusions in Sec. V. II. THEORETICAL METHODS

Quantum confinement effects in a semiconductor QD arise due to the confinement of the charge carriers (electrons and holes), within the finite volume of the QD. In order to understand these effects qualitatively and quantitatively, several theoretical approaches have been used which range, in sophistication, from the simple single-band effective mass theory7, 8 and multi-band effective mass theory,9–13 to empirical pseudopotential theory14 and empirical tight-binding theory.15–26 A comprehensive review of the electronic structure of QDs may be found in Ref. 27. A few first principles density functional theory calculations have been reported but these are restricted so far to small sized QDs, for reasons of computational complexity.28–34 A. Single band effective mass theory

a) Electronic mail: [email protected] b) Electronic mail: [email protected] c) Electronic mail: [email protected]

0021-9606/2014/141(20)/204702/13/$30.00

In the single band effective mass model, electron (conduction band) states and hole (valence band) states in a 141, 204702-1

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quantum dot are described by an effective Schrödinger equation −

spherical QDs Eexc

2

¯ ∇ 2 ψ(r) + V (r)ψ = Eψ(r), 2m0 m∗

(1)

where the effective mass m∗ and the confining potential V (r) depend on the material and the band of interest. m0 is the free electron mass and ¯ is the reduced Plank constant. ψ is called the envelope function and E is the energy of the state measured relative to the relevant band edge. In the simplest model, the potential V is given by the band-edge discontinuity between the QD and the surrounding material. In systems of interest to us, as the QDs are generally embedded in an insulating medium of large band gap, it is common to assume perfect confinement described by setting V = 0 in the above equation and demanding ψ(r) = 0 at the boundary of the QD. The resulting electron and hole states are that of a free particle-in-a-box where the shape and size of the box matches with that of the QD. In particular, for a spherical QD of radius R, the confined electron envelope functions ψ e and energies Ee are given by √ 2 j (ξ r/R)Ylm (θ, φ), (2) ψe(h) (r, θ, φ) = 3/2 R jl+1 (ξnl ) l nl ¯2 ξnl2 , l = 0, 1, 2, . . . n = 1, 2, 3, . . . , (3) Ee = 2m0 m∗e R 2 where r, θ , φ denote the spherical polar co-ordinates, ξ nl is the nth zero of the lth order Bessel function jl (r), and m∗e is the effective mass of the electron. The confinement leads to quantization of energy as evidenced by the discrete labels n and l describing the radial and angular quantum numbers. The corresponding expression for the hole energies Eh are obtained by replacing m∗e by the effective mass m∗h of the hole. For comparison with experiments involving optical absorption and emission, the single particle confinement picture should be extended to include the electron-hole Coulomb interaction. In QDs small compared to the bulk exciton Böhr radius, the Coulomb interaction may be treated perturbatively. The commonly used first-order perturbation theory gives an electron-hole binding energy of 1.786e2 /R where e is the elementary charge and is the dielectric constant responsible for screening of the Coulomb interaction. The above considerations lead to a simple expression for the ground state energy of a confined electron-hole pair in ⎛ H3−band =

1 1 + me mh

−

1.786e2 . R

(4)

B. Multiband effective mass theory

The effect of band-mixing in degenerate or closelyspaced bands can be taken into account within the multi-band EMA. Restricting ourselves to the most common case of p -type valence bands of III-V and II-VI semiconductors, two limiting cases of the multi-band valence band Hamiltonians are most useful: (i) the 3-band model described by the 3-band Luttinger Hamiltonian applicable to materials in which the spin-orbit interaction may be neglected and (ii) the 4-band model with the 4-band Luttinger Hamiltonian that describes the degenerate light-hole and heavy-hole bands, in materials where the spin-orbit interaction is large. In the 3-band model, the hole envelope function h is a column vector of three functions ψ x (r), ψ y (r), and ψ z (r), that describe the weights of the three valence band-edge Bloch functions uvx , uvy , and uvz in the hole wave function. h satisfies the matrix equation H3−band h = Eh ,

(5)

where E is the energy measured with respect to the bulk valence band-edge and H3-band is a matrix operator given by

C∇z ∇x

A∇y2 + B(∇x2 + ∇z2 ) C∇y ∇z C∇y ∇z

where A, B, and C are treated as empirical parameters given in terms of the experimental hole masses. The explicit relation between these parameters and masses is given later.

This simple theory very well captures the qualitative effects of confinement, viz., discrete nature of excitation energies, a blue shift of the absorption edge that increases as the size of the QD is reduced, and the enhancement of the Coulomb interaction with decreasing QD size. This model has also been generalized to two-electron states including the electrostatic potential of image charges.35 However, its quantitative validity is extremely limited24, 25 because of two major shortcomings. First, the effective mass picture is valid only close to the band edges where the bands are parabolic. However, particularly in small QDs, where the confinement energy is larger, the energy levels are pushed deep into the bulk bands where nonparabolicity effects are important.22 Second, in most semiconductors of interest, the valence band edge is degenerate leading to mixing of bands by the confinement potential.36 Proper treatment of band-mixing is important even for a qualitatively correct description of the electronic wave functions, as well as for several important properties such as oscillator strengths for optical absorption.

A∇x2 + B(∇y2 + ∇z2 ) C∇x ∇y

¯2 ⎜ ⎜ C∇x ∇y 2m0 ⎝ C∇z ∇x

¯2 π 2 = Eg + 2m0 R 2

⎞ ⎟ ⎟, ⎠

(6)

A∇z2 + B(∇x2 + ∇y2 )

The Bloch-functions describing the 3-fold degenerate band of cubic semiconductors have 5 symmetry. The inclusion of spin generates 6 bands out of these, which may be classified as four bands of 8 symmetry and two bands of 7


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symmetry. These two sets of bands are split by spin-orbit interaction. In all the materials considered in this paper, the 8 bands (the so-called heavy-hole light-hole bands) lie higher in energy relative to the 7 band (split-off band) and when the spin-orbit splitting is large, the 4-band model that describes hole levels derived from the 8 bands is more appropriate. The corresponding hole envelope function is a column vector of four functions and satisfies an equation analogous to Eq. (5) with the Hamiltonian matrix replaced by the 4-band Luttinger Hamiltonian given by ⎞ ⎛ P +Q −S R 0 ⎟ ⎜ −S † P −Q 0 R ⎟ ¯2 ⎜ ⎟, ⎜ H4−band = † 2m0 ⎜ 0 P −Q S ⎟ ⎠ ⎝ R 0 R† S† P +Q (7) where P = γ1 ∇ 2 , Q = γ2 (∇ 2 − 3∇z2 ), √ R = − 3[γ2 (∇x2 − ∇y2 ) − 2γ3 ∇x ∇y ], √ S = 2 3γ3 ∇z (∇x − i∇y ).

(8)

The matrix in Eq. (7) is in the Bloch function basis set 1 uv 3/2 = − √ (uvx + iuvy ) ↑, 2

√ 2 1 v v u 1/2 = − √ (ux + iuy ) ↓ + √ uvz ↑, 6 3 √ 2 1 uv −1/2 = √ (uvx − iuvy ) ↑ + √ uvz ↓, (9) 6 3 1 uv −3/2 = √ (uvx − iuvy ) ↓, 2 where ↑ and ↓ arrows denote spin = ±1/2. The Luttinger parameters γ 1 , γ 2 , and γ 3 can be related to the effective masses of heavy- and light-holes in the bulk by noting that the envelope functions in the bulk are simple plane waves and the resulting dispersions of the energy levels give the effective masses. The bulk heavy-hole and light-hole dispersions predicted by the 4-band model are given by

−E(k) = γ1 k 2 ± 4γ22 k 4 +12 γ32 −γ22 kx2 ky2 +ky2 kz2 +kz2 kx2 , (10) where the − sign applies to heavy-hole and the + sign to lighthole bands. Explicitly, the bulk heavy hole and light hole masses 100 along the [100] direction (m100 hh and mlh ) and those along the 111 111 [111] direction (mhh and mlh ) are given by v

(m∗hh )100 = 1/(γ1 − 2γ2 ),

(11)

(m∗lh )[100] = 1/(γ1 + 2γ2 ),

(12)

(m∗hh )[111] = 1/(γ1 − 2γ3 ),

(13)

(m∗lh )[111] = 1/(γ1 + 2γ3 ).

(14)

The parameters A, B, and C of the 3-band model are related to the Luttinger parameters as A = γ1 + 4γ2 , B = γ1 − 2γ2 ,

(15)

C = 6γ3 . C. Tight-binding approach

A more accurate method which incorporates both, nonparabolic and band-mixing effects, is semi-empirical TB theory. TB calculations have been found to give QD energy levels in good agreement with the measured size-dependence of the excitonic transition, the absorption edge, and the valence band edge.22, 24, 25 TB calculations of the optical spectra also yield absorption and photoluminescence excitation (PLE) spectra, which match excellently with experiments.23 The TB approach has several additional advantages: the detailed structure of the crystallite, including its size, shape, exact atomic positions, and local bonding structure can be incorporated in the model, and a wide range of sizes from molecular clusters to very large systems made up of hundreds of thousands of atoms can be studied with moderate computational power. Thus, the tight-binding approach is an excellent method for the interpretation of experimental data as well as the design of new structures. In this work, we employ a TB formalism based on bonding orbitals, viz., the empirical bond-orbital model (EBOM) which has been applied to QDs with great success by us22–26 as well as others18 in the past. Below we give a brief description of the method. In the EBOM approach, we consider only tetrahedrally bonded materials for which the conduction and valence band states can be described by sp3 -hybridized molecular orbitals. The wave functions of the electrons and holes in the QD are constructed as linear combinations of one s-like and three plike bonding orbitals on each anion-cation bond in the QD structure. Assuming the orbitals to be well-localized, only nearest neighbour interaction terms in the Hamiltonian are retained. The resulting Hamiltonian matrix can be described by the following matrix elements between the bonding orbitals: δR−R ,τ {Exy τα τα (1 − δαα ) Rα|H |R α = Ep δRR δαα + τ

+[Exx τα2

+ Ezz (1 − τα2 )]δαα } ,

Rs|H |R s = Es δRR + Rs|H |R α =

Ess δR−R ,τ ,

(16) (17)

τ

Esx τα δR−R ,τ ,

(18)

τ

where α = x, y, z, |Rα denotes a p-like orbital that transforms like α and is located at R, |Rs denotes an s-orbital at R, and τ stands for the nearest-neighbour displacement vector. The parameters Es , Ep , Exy , etc., that appear in the Hamiltonian are determined by applying the model to the corresponding bulk semiconductor and fitting the parameters to reproduce known symmetry point energies in the bulk band


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structure. Spin-orbit interaction is included as a local on-site term that reproduces the splitting between 7 and 8 symmetry bulk valence bands. Due to the absence of translational symmetry in the finite-sized QD, a straight-forward application of the above approach leads to a large Hamiltonian matrix of order 4N, where N is the number of atoms in the QD. For QDs of highly symmetrical shape, such as spherical QDs, the size of the Hamiltonian matrix can be reduced considerably by transforming to a symmetry-selected basis. In particular, for spherical QDs of zinc-blende structure considered by us, all states can be classified by the irreducible representations of the Oh double group, viz., 6 , 7 , and 8 . Increase in symmetry from Td to Oh is because of the bond orbital approximation in the tight-binding formalism. Details of the symmetry-selected basis may be found in Ref. 18. The calculation proceeds by constructing an atomistic model of the QD, and of the Hamiltonian matrix within each symmetry-selected sub-space of orbitals. The diagonalization of the Hamiltonian matrix gives eigenvalues (which are the confined electron and hole energies) and the corresponding eigenvectors. D. Other methods

Another atomistic method that has been successfully applied to QDs is the empirical pseudopotential model (EPM).14, 37–39 EPM starts from a microscopic model of atomic positions, as does the TB method, but uses empirical atomic potentials instead of the bonding parameters used in the latter. EPM has the advantage of providing an excellent description of bulk bands and thus could be an attractive alternative to TB. However, EPM is more intensive than TB computationally. More importantly, the application of the EPM to free-standing QDs of the kind considered in this paper is complicated by the need for special potentials to simulate the surface atoms. This issue has been addressed to an extent in the literature,40, 41 but the surface potentials depend on assumptions about surface structure and passivation. Nevertheless, good agreement between observed and calculated size-dependence of the excitonic states has been obtained by EPM calculations on a number of materials including Si, CdS, and CdSe.42 Further, strong deviations from the predictions of simplistic EMA models both in energies and symmetries of wavefunctions have been demonstrated. This work does not include the results of pseudopotential calculations. For details we refer the reader to Ref. 42. Apart from these empirical methods, a few ab initio calculations based on density functional theory (DFT) have been reported. Owing to computational complexity, most calculations have been so far restricted to small QDs of radius less than about 1–2 nm. Further, as DFT within the Kohn-Sham approach43 underestimates the band gap, the quasi-particle energies of the holes and electrons need to be calculated using more sophisticated methods such as GW or time-dependent DFT (TDDFT).44 GW calculations of hydrogen-terminated Si clusters containing up to 800 atoms were reported by Ogut et al.28 They showed a substantial size-dependence of the selfenergy correction of quasi-particle energies that precludes the

J. Chem. Phys. 141, 204702 (2014)

use of a simple scissor-shift correction to DFT band-gaps often applied to bulk materials. DFT calculations on very small clusters of ZnS (up to 18 atoms),29 ZnO (up to 36 atoms),30 and ZnSe (up to 26 atoms),31 and somewhat larger free-standing CdSe nanocrystals (up to 1.5 nm in diameter)32 have been reported. The latter work carried out structure optimization and identified mid-gap states arising from surface ligands. More recently, TDDFT has been used to calculate the confined electron and hole energies in Zn based II-VI QDs of diameter smaller than 1.5 nm.33 This work included optimization of the structure assuming hydrogen-termination of the surface. A DFT calculation of CdTe QDs of radius up to 2 nm by Haram et al.34 compares well with experimental quasi-particle and optical energy gaps. However, these authors have used DFT energy eigenvalues with a simple scissor-shift correction to estimate the bandgap, and excitonic effects were neglected. Although it is not yet computationally feasible to do ab initio calculations of excitonic states in QDs in the whole size range of experimental interest, the ability to optimize the geometrical structure and realistically model the surface atoms makes DFT-based approaches well-suited to study localized states and surface-dependent properties.

III. CALCULATIONAL DETAILS

For each material studied we have carried out TB and EMA calculations to understand the limitations and range of applicability of the latter. Here, we describe details of the models and parameters used.

A. Single-band EMA

As the conduction band in all materials studied is nondegenerate and isolated from the valence band, in the EMA calculations we use the single band model for the conduction band. Materials such as GaN and ZnS which have overlapping conduction bands are not considered. The effective mass values used for each material are given in Table I.45 For the dielectric function for the QD, we use the bulk value of the dielectric function. We use the same for screening the Coulomb interaction in both EMA and TB-based exciton calculations. It has been pointed out that the dielectric constant of QDs shows a significant sizedependence.46, 47 Although these calculations also underestimate the dielectric constant of the bulk material, the calculated is reduced compared to the bulk value by about 10% for a radius of 20 Å in Si QDs. Here, we neglect any sizedependence and treat as a phenomenological parameter. However, the question whether to use high-frequency or lowfrequency (ω) for screening needs some consideration. The most accurate way out would be to solve the exciton-phonon coupled problem, in which case (∞) would be the correct value to use for screening Coulomb interaction. But as this is a very complex problem, usually only a simplified effective problem that includes only the electron-hole Coulomb interaction perturbatively, as indicated earlier, is attempted. In this situation, the effect of phonons in screening is approximated by using an effective . For materials in which the screen-


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TABLE I. Material parameters used in the calculations45 (the values marked with ∗ are from Ref. 22 and 23, and those marked with ∗∗ are from Ref. 48). a is the lattice constant in (Å), so is the spin-orbit splitting energy in (eV), m∗e is the conduction band effective mass, and is the dielectric constant. m∗e GaAs GaSb InAs InSb InP CdS CdSe CdTe ZnSe

0.067 0.039 0.023 0.0118 0.0808 0.14∗ 0.11 0.095 0.170

a 5.65 6.096 6.058 6.479 5.87 5.82 6.05 6.46 5.667

so

0.34 0.75 0.38 0.810 0.108 0.060 0.424 0.950 0.450

12.8 15.7 15.15 16.2 12.56 6.0∗ 7.0∗ 8.8∗∗ 7.9∗∗

ing induced by polar phonons is complete (in other words, when the electron and hole polaron radii are much smaller than the exciton Bohr radius), one uses (0). At the other extreme, where exciton Bohr radius is much smaller than the polaron radii one would use (∞). This criterion could be alternatively specified in terms of the ratio of polaron and exciton binding energies leading to the same conclusion. In reality, all materials fall in between these extremes, and an intermediate value that best reproduces the bulk exciton binding energy is a better compromise. In this work, for III-V group materials such as GaAs, whose bulk exciton binding energy is much smaller than the LO phonon energy, we use (0). The values for these have been obtained from Ref. 45. For the II-VI group semiconductors, the values given in Table I have been obtained from Refs. 22 and 23 for the materials for which they are available. For the rest, the values quoted in Table I are the average value of (0) and (∞), with the latter values taken from Refs. 45 and 48.

B. Multiband EMA

We used four-band EMA to study valence bands of all the materials. The Luttinger parameters, γ 1 , γ 2 , and γ 3 used are given in Table II. These were taken from Ref. 49 for the semiconductors for which they were available (GaAs, GaSb, InAs, InSb, InP). For the other semiconductors, the γ ’s were calculated using Eqs. (11)–(14) using heavy-hole (m∗hh ) and light-hole (m∗lh ) effective masses determined from our tightbinding bulk band structures. For this purpose, the calculated band dispersion close to k = 0 was fitted to Eq. (10). We note that the energies, Elh (k) and Ehh (k), must be calculated for very small values of k, i.e., very close to the point, in order to obtain accurate results for the effective mass values.

C. Tight-binding theory

We work within the EBOM to set up the Hamiltonian for the QD. Diagonalization of the Hamiltonian matrix leads to the energy eigenvalues which correspond to the conductionand valence-bands. The EBOM parametrization scheme, that

TABLE II. Luttinger parameters (γ 1 , γ 2 , and γ 3 ), from Ref. 49, used for the multiband EMA calculations for the materials for which they are available. Also given for comparison (in brackets) are our values obtained from the effective masses of the tight-binding band structure; these were used for the multiband calculations for the materials for which no data were available from Ref. 49.

GaAs GaSb InAs InSb InP CdS CdSe CdTe ZnSe

γ1

γ2

γ3

6.98(7.09) 13.4(10.77) 20.0(14.76) 34.8(34.38) 5.08(6.013) (3.6) (4.53) (5.04) (3.35)

2.06(2.79) 4.7(4.64) 8.5(6.62) 15.5(16.39) 1.6(2.33) (1.22) (1.5) (1.925) (1.16)

2.93(3.13) 6.0(4.96) 9.2(6.93) 16.5(16.59) 2.1(2.64) (1.38) (1.80) (2.230) (1.31)

relates the parameters of the model to the bulk band structure described in detail earlier22, 23 are summarized below

Es + 12Ess = Ec 1 , X

Es − 4Ess = Ec 1 , L

Es = Ec 1 ,

Ep + 8Exx + 4Ezz = Ev 15 ,

(19)

X

Ep − 8Exx + 4Ezz = Ev 3 , X

Ep − 4Ezz = Ev 5 , L

Ep + 4Exy = Ev 3 . The experimental bulk band structure symmetry point energies45 used in the parametrization scheme are given in Table IV. Slight adjustments (to within ≈ 0.1 eV) to the valX L ues for Ec 1 and Ec 1 were required to be made for obtaining a self-consistent parametrization.22, 23 A comment about handling the surface atoms is in order. In a free standing quantum dot with no surface passivation, dangling bonds and defects at the surface lead to localized states within the energy gap as evidenced by broad low energy features in the optical emission spectrum and low TABLE III. Heavy- and light-hole effective masses (m∗hh and m∗lh ) in the [100] and [111] directions, as obtained from the tight-binding band structure. Also given (in brackets), for comparison, are the values from Ref. 45 wherever available.


(m∗hh )[100]

(m∗hh )[111]

(m∗lh )[100]

(m∗lh )[111]

0.660(0.34) 0.672(0.29) 0.662(0.35) 0.629(0.32) 0.740(0.45) 0.803 0.680 0.838 0.968(0.75)

1.192(0.75) 1.209(0.40) 1.152(0.85) 1.134(0.44) 1.383(0.45) 1.227 1.041 1.676 1.368(0.75)

0.079(0.094) 0.050(0.042) 0.036 0.015(0.016) 0.094(0.12) 0.1653 0.1278 0.1121 0.1761

0.075(0.082) 0.048(0.042) 0.035 0.015(0.016) 0.089(0.12) 0.1545 0.1200 0.1051 0.1672


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TABLE IV. Symmetry point energies (eV) in the bulk bandstructure45 (those marked with ∗ are from Refs. 22 and 23).

GaAs GaSb InAs InSb InP CdS∗ CdSe∗ CdTe ZnSe

X

L

X

X

L

Ec 1

Ec 1

Ec 1

Ev 15

Ev 3

Ev 5

Ev 3

1.52 0.822 0.418 0.235 1.423 2.55 1.9 1.475 2.82

1.984 1.5095 2.172 1.399 2.3764 4.693 3.826 3.3423 4.454

1.866 1.338 1.733 1.108 2.153 4.157 3.344 2.8754 4.046

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

− 6.88 − 6.711 − 6.907 − 6.416 − 6.174 − 4.5 − 5.1 − 5.149 − 4.96

− 2.89 − 2.441 − 2.512 − 2.311 − 2.392 − 2.24 − 2.45 − 1.742 − 1.96

− 1.2 − 1.017 − 1.081 − 0.961 − 0.9595 − 1.10 − 1.20 − 0.6534 − 1.04

quantum yield from excitons observed in such samples.50 Therefore, samples with high optical quality and strong excitonic emission require some form of surface passivation. While realistic modelling of the surface structure is important for understanding surface-sensitive phenomena such as blinking, photo-darkening, and persistent hole-burning51, 52 or fast non-radiative decay of biexcitons,4 the quantum confined states that are spread out over the QD are expected to be less sensitive to the microscopic surface structure owing to their nearly vanishing probability amplitude near the surface. EMA a priori assumes this to be the case through the slowly varying envelope approximation, but microscopic models such as EPM and TB need to take special care to isolate the localized surface states from the energy spectrum. We eliminate surface states by removing the dangling bonds at the surface from the Hamiltonian matrix. In EPM, the same result is achieved using surface atom pseudopotentials that are carefully chosen to avoid localized states in the band gap. This precludes applications of such models for a realistic study of surface states, although both TB and EPM are inherently capable of describing the surface through appropriate parametrization. However, this simplification is not a limitation for describing quantum confined excitonic states that are the central focus of this work.

IV. RESULTS AND DISCUSSION A. Band structure

Our parametrization scheme for the EBOM is given in Eq. (19), and the symmetry point energies, to which the bulk band structures are fitted, are given in Table IV. The values used for different material parameters22, 23, 45, 48 are given in Table I. Table III gives the values of the m∗hh and m∗lh along [100] and [111] symmetry directions, obtained from this band structure. Also given, for comparison, are the values from Ref. 49. We note that for most of the materials, our values compare well with those of the latter. However, we emphasize that while it is, in principle, important to have a parametrization scheme that gives accurate effective masses or Luttinger parameters, it is far more important from the point of view of obtaining accurate energy levels and wave functions in QDs that the overall features of the bulk band structure be

reproduced correctly, than just the effective masses or Luttinger parameters. The bulk band structures generated by the tight-binding parametrization for all the materials are shown in Figs. S1– S9 in the supplementary material.53 These band structures compare well with the nonlocal pseudopotential calculations by Chelikowsky and Cohen.54 B. Conduction band energy levels

Figs. 1(a)–1(e) and 2(a)–2(d) show the size-dependent near-band-edge conduction and valence band energy levels, as obtained from both the TB and the EMA theory. For the conduction band these are the 6+ , 7− , and 8− TB levels, and the corresponding l = 0, n = 1, and l = 1, n = 1 EMA energy levels. We discuss below the general characteristics of these levels and the factors that affect them. 1. Energy difference between and X, L points

Table V gives the difference between the tight-binding and EMA energy eigenvalues for the lowest two conduction band levels, CB1 and CB2 for all the materials, at a very small size (Rsmall ). Also indicated is the energy difference between the conduction band minima at -point and at X-point (E1 X = Ec 1 − Ec 1 ), as well as at -point and at L-point (E2 L1 1 = Ec − Ec ) for each material. The bulk band gap and the conduction band effective mass m∗e are also indicated in each case. We note that, for materials for which E1 and E2 are small, the deviation of the EMA results from the TB results (as seen from the values of CB1 and CB2 ) is more severe. For GaAs, this strange behaviour of the CB levels was highlighted and discussed in detail earlier.25, 37 Thus, for materials like CdS or for large sizes in direct gap semiconductors, the lowest few CB levels have a predominantly -character. However, as the size reduces and as the CB level gets pushed to higher energies, in materials for which E1 and E2 are small, new effects come into play: the levels and wave functions acquire an L- and X-character. In addition, as they encounter the maxima in the CBs (which occur at k values before the L and X points), the levels get pushed downwards (due to negative effective masses at the conduction band maxima) and the quantum confinement gets considerably suppressed. This results in the anomalous behaviour of the size dependence of the CB level for these materials, which was noted earlier in the case of GaAs. 2. Small effective mass

It is clear from Table V that for InAs, the difference between TB and EMA energies is almost double of that in GaAs, despite larger values of E1 and E2 compared to GaAs. The reason for this is the very small effective mass (less than half that in GaAs); this causes the quantum confinement to be stronger for a given (large) size, but that also implies that the bottom of the X- and L-bands are encountered at a larger size (or, to phrase it differently, the non-parabolicity of the


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2.4

J. Chem. Phys. 141, 204702 (2014)

(a) GaAs l=1, n=1 l=0, n=1

2.2

1.9

Γ 6+ Γ 7− − Γ8

1.5

0.0

≈

≈

−0.5 −1.0

F+3/2 F−5/2 F+5/2

−1.5 −2.0 0

10

1.1

0.8

≈

≈

50

−1.2

0

10

2.4

l=1, n=1 l=0, n=1 Γ6 + Γ7 − Γ8 −

0.6

≈

≈

−0.3 (n=1) F−3/2 F+3/2 F−5/2 F+5/2

−0.9 −1.2 −1.5 0

10

20 30 40 Radius (Å)

−1.0

50

60

0

10

1Γ8 2Γ8+ 1Γ8− 2Γ8− Γ6 − Γ6 +

20 30 40 Radius (Å)

1Γ8+ 2Γ8+ 1Γ8− 2Γ8− − Γ6 + Γ6 50

60

(e) InP l=1, n=1 l=0, n=1 Γ6+ Γ7− Γ8−

1.8 1.6 0.0

≈

≈

−0.3

+

−0.6

(n=1) F−3/2 F+3/2 − F 5/2 + F 5/2

−0.7

2.0

Energy (eV)

Energy (eV)

0.0

−0.4

2.2

0.4 0.2

≈

≈

−1.3

20 30 40 Radius (Å)

(d) InSb

0.8

1Γ8+ 2Γ8+ 1Γ8− − 2Γ8 Γ6− Γ6+

(n=1) F−3/2 F+3/2 F−5/2 F+5/2

−0.8

60

0.4

−0.1

−1.6

20 30 40 Radius (Å) 1.0

1.2

0.0

Γ 6+ − Γ7 − Γ8

l=0, n=1 l=1, n=1

2.0

1.3

0.9

(c) InAs

1.6

−0.4

1Γ8+ 2Γ8+ − 1Γ8 2Γ8− Γ6 − Γ6 +

(n=1) − F 3/2

Energy (eV)

Energy (eV )

1.8

2.4

+

Γ6 Γ7− Γ8−

l=1, n=1 l=0, n=1

1.7

2.0

1.6

(b) GaSb

Energy (eV)

204702-7

(n=1) F−3/2 F+3/2 F−5/2 + F 5/2

−0.6 −0.9

1Γ8+ 2Γ8+ 1Γ8− 2Γ8− − Γ6 Γ6+

−1.2 50

60

0

10

20 30 40 Radius (Å)

50

60

FIG. 1. The size dependence of the first few conduction and valence band levels in free-standing QDs of III-V materials calculated using TB theory and multiband EMA: (a) GaAs, (b) GaSb, (c) InAs, (d) InSb, and (e) InP.

bands starts to become important at a larger size), causing a deviation from a simplistic confinement picture. Thus, for materials with smaller effective masses, in addition to small E1 and E2 , the quantum confinement effect may be suppressed.

three-band or four-band EMA which is generally used for most materials. Further, for narrow-band gap materials, this discrepancy between EMA and TB energies would increase as the size is decreased, and at a much faster rate than for wide-band gap materials.

3. Narrow band gap materials

It is clear from Table V that there is an additional complication that arises in materials with a narrow band gap. In materials like InP, InAs, GaSb, at least some part of the deviation between TB and EMA is due to the small band gap, which causes mixing between the conduction and valence bands. Interestingly, this is taken into account in the TB theory (with the sp3 basis set) and in an eight-band EMA, but not in the

C. Valence band energy levels

For the VB, the energy levels plotted in Figs. 1(a)–1(e) and 2(a)–2(d) are 1 8+ , 2 8+ , 1 8− , 2 8− , 6− , and 1 6+ TB − + energy levels, and the corresponding F3/2 , n = 1, F3/2 , − + n = 1, F5/2 , n = 1, F3/2 , n = 1 EMA energy levels. We discuss below the general characteristics of these energy levels and the factors that affect them.


Dhayal et al.

J. Chem. Phys. 141, 204702 (2014)

(a) CdS

4.4

3.4 l=0, n=1 l=1, n=1

3.9

Γ 6+ − Γ7 − Γ8

3.4

Energy (eV )

≈

≈

−0.3 −0.6 −0.9

10

20 30 40 Radius (Å)

50

60

≈

≈ 1Γ8+ 2Γ8+ 1Γ8− 2Γ8− Γ6− Γ6+

(n=1) F−3/2 F+3/2 F−5/2 + F 5/2 0

4.4 l=0, n=1 l=1, n=1

10

20 30 40 Radius (Å)

60

l=0, n=1 l=1, n=1 Γ6+ Γ7− Γ8−

3.6 3.2

≈

≈ 1Γ8+ + 2Γ8 − 1Γ8 2Γ8− Γ6− Γ6+

−0.3 (n=1) F−3/2 F+3/2 F−5/2 + F 5/2

−0.6 −0.9

Energy (eV)

1.9

50

(d) ZnSe

4.0

Γ6 + Γ7 − Γ8 −

2.3

Energy (eV)

0.0

−1.2

(c) CdTe

2.7

0.0

1.8

−1.6 0

1.5

2.2

−0.8

−1.2

3.1

Γ6+ Γ7− Γ8−

−0.4

1Γ8+ 2Γ8+ 1Γ8− 2Γ8− Γ 6− Γ 6+

(n=1) F−3/2 F+3/2 F−5/2 F+5/2

l=0, n=1 l=1, n=1

2.6

2.9 2.4 0.0

(b) CdSe

3.0

Energy (eV)

204702-8

2.8 0.0

≈

≈

−0.3 (n=1) F−3/2 F+3/2 F−5/21 F+5/2

−0.6 −0.9

−1.2

1Γ8+ 2Γ8+ 1Γ8− 2Γ8− Γ6− Γ6+

−1.2 0

10

20 30 40 Radius (Å)

50

60

0

10

20 30 40 Radius (Å)

50

60

FIG. 2. The same as Fig. 1 but for II-VI materials: (a) CdS, (b) CdSe, (c) CdTe, and (e) ZnSe.

TABLE V. Difference between the EMA and tight-binding energies (Eema − Etb ), for the two lowest conduction-bands (CB1 and CB2 ) at some (an arbitrarily chosen) small size, (Rsmall (Å)). Eg (eV) refers to the bulk band L

X

gap, E1 (eV) = Ec 1 − Eg , E2 (eV) = Ec 1 − Eg , m∗e is the effective mass.


Rsmall

CB1

CB2

Eg

E1

E2

m∗e

11.4973 12.4004 12.3221 13.1793 11.9397 11.8380 12.3059 13.1398 11.5270

1.1781 5.8909 8.9225 1.7535 2.8529 1.3209 1.7070 1.7712 1.1667

2.5502 12.2038 18.4509 3.6536 5.9622 2.9010 3.6770 3.8128 2.5558

1.52 0.822 0.418 0.2352 1.423 2.55 1.90 1.475 2.82

0.346 0.516 1.315 0.8725 0.667 1.607 1.444 1.4004 1.226

0.464 0.6875 1.754 1.164 0.890 2.143 1.926 1.8673 1.634

0.067 0.039 0.0263 0.11 0.0808 0.14 0.11 0.095 0.170

Effects similar to those discussed in the case of the conduction band, viz., the effective mass values and the band gap, influence the valence band energy levels. From Table VI, it is clear that there is some difference (< 1 eV) between the TB and multiband EMA energies for CdS, CdSe, CdTe, and ZnSe, for the lowest two valence band states. In all these cases, the band gap is relatively large, and so are most of the effective masses (see Table III). In GaAs, the difference is larger even though the band gap is large, because of the smallness of the light-hole mass which pushes the EMA level deeper into the band to the region of non-parabolicity. Interestingly, in InP, despite a small band gap, the difference is less than that in GaAs by ≈0.5 eV because of the larger hole masses. In InAs, the masses are close to those in GaAs but the deviation is much larger (by up to ≈ 1 eV) because the band


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J. Chem. Phys. 141, 204702 (2014)

TABLE VI. Difference between the tight-binding and EMA energies (Etb − Eema ), for the two lowest valence-bands, V B1 and V B2 , (in eV), at some (arbitrarily chosen) small size Rsmall (Å) and the difference between the exciton energies (Eexc(ema) − Eexc(tb) ), from EMA and TB theories for the same small radii and for very large radii.


Rsmall

VB1

VB2

small Eexc

Rlarge

11.4973 12.4004 12.3221 13.1793 11.9397 11.8380 12.3059 13.1398 11.5270

1.0983 1.6858 1.4592 1.6752 0.6433 0.3679 0.3982 0.2885 0.3064

1.1009 1.7005 1.7555 2.3002 0.6150 0.4104 0.4535 0.3136 0.3428

3.5674 5.5539 8.5644 1.4555 2.5938 1.2792 1.5498 1.6290 1.0490

53.695 57.913 57.548 61.551 55.762 55.287 57.472 61.366 53.833

large

Eexc

0.1136 0.1928 0.3035 0.0530 0.0841 0.0708 0.0837 0.0639 0.0515

gap is much smaller. This causes mixing with the CB states to become important in InAs which is reflected as the deviation between EMA and TB results. In GaSb, the band gap is larger than that in InAs but the smallness of the mass causes a large deviation between EMA and TB energies. In InSb, the effect of the smallness of the masses is enhanced by the smallness of the band gap to cause the largest deviation between the EMA and TB energies. D. CB and VB energies in individual materials

In what follows, based on the observations made above, we discuss in detail the VB and CB energies in each of the semiconductor materials. 1. GaAs

This material has been discussed in detail in our earlier paper.25 The CB levels for the TB and EMA theories do not match even for the largest sizes studied, and they differ considerably at smaller sizes (CB1 and CB2 ≈ 1.2 and 2.6 eV). There is a clear flattening of CB energy levels at smaller sizes. The conduction band effective mass, though not the smallest of the ones studied, is still very small, so that the conduction band levels get pushed deep into the band. Here, however, they are bound by the bottom of the X and L valleys, (EX and EL are the closest to E from among all the materials, i.e., E1 and E2 are very small) and are pushed back to the point. Thus, the confinement effect for the CB energy levels is much smaller than that expected from EMA. It is important to note that the CB states have a large mixing between , X, and L energy levels. For the valence band levels, there is moderate disagreement with the EMA levels at the smaller sizes. The band gap is not too small, so the effect of CB levels on the VB levels is also relatively small. 2. GaSb

This is again an interesting material from the point of view of the CB levels. There is a large difference between the two theories at the smallest sizes studied. The band gap is small, implying a large mixing with the valence band states. The CB effective mass is also very small, so the confinement

energy would have been expected to be very large. But the proximity of the L- and X-points pushes the CB levels back towards the -point, reducing the confinement energy. There is mixing of L and X states as well, in the CB states. The valence band states also have a large discrepancy with EMA as seen from Table VI, but because of the larger effective masses, the size-dependence given by the two models is not as dramatically different unlike the CB levels.

3. InAs

This is another material that shows unusual sizedependence for the CB levels. Though there is no flattening of the energy levels at smaller sizes, the discrepancy with the EMA is largest at small sizes for InAs. The reasons are not difficult to understand. The band gap is small, so there is a large valence character in the CB levels. The effective mass is extremely small, so a large confinement shift leads to mixing with the L and X valleys. The valence band levels also show a large discrepancy mostly due to mixing with the CB across the small band gap.

4. InSb

The band gap is extremely small, so mixing with the VB levels is large leading to modification of both the VB and CB states and energies. The effective mass is moderately small. As the X and L energies are reasonably far away in energy from -point energy, there is no flattening of the CB levels at small sizes. The conduction band energy levels show similar ordering as that in InAs. In this case too, the TB and EMA theories are not comparable even for the largest radius. The 7− and 8− levels show a small splitting initially and then get closer to each other with the increasing crystallite sizes.

5. InP

In this material, the discrepancy with the EMA for the CB levels is moderately large, mainly due to the small effective mass which causes a large confinement effect which in turn leads to a larger mixing with the L- and X-point valleys which limits the energy shift at smaller sizes. The band gap is moderate, leading to smaller mixing with the valence band. This is also reflected in the fact that the valence band energy levels are not too far off from the multiband EMA results.

6. CdS, CdSe, CdTe, and ZnSe

Of these four materials, CdS and CdSe have been discussed by us as well as other authors in earlier papers. In all of them, the band gaps are relatively large, so the CB and VB do not mix much with each other and retain their character. Furthermore, the effective masses are comparatively larger and the X and L energies are relatively away from the -point. So no anomalous behaviour is expected in the confinement


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Dhayal et al.

effects. The CB and VB levels from TB theory agree reasonably well with those of multiband EMA.

E. Exciton energies

In this section, we present and discuss the excitonic energies for different materials obtained using TB theory and EMA, and compare these with the experimental results available in the literature. For the experimental data, we look only at absorption data to obtain the excitonic position and use only those data where the size is determined by independent measurements. The most common methods of measuring the size of QDs are transmission electron microscopy (TEM) and through a measurement of the widths of the peaks in an X-ray diffraction (XRD) study. Both methods may underestimate the size by ≈ 2.5 Å, as they are insensitive to the aperiodic surface layer in the QD.55 Often, there are also inaccuracies in the measurement process itself, reflected in the fact that the same group often gets rather different sizes when the measurement is made with different techniques. Finally, there is usually a size and sometimes a shape dispersion of QDs in the sample. The exciton binding energy was calculated as the screened Coulomb energy of the lowest electron-hole configuration that is optically allowed. In all the cases, this corresponds to the lowest energy electron state of 6+ symmetry and the highest energy hole state of 8− symmetry. The resulting eightfold degenerate configuration splits into three groups as 6+ ⊗ 8− = 3− ⊕ 4− ⊕ 5− , where only the 5− state is optically active. The sum of the single particle energy and the expectation value of the Coulomb potential in this threefold degenerate 5− configuration is quoted as the exciton energy. The Coulomb potential is statically screened by the dielectric constant given in Table I. Electron-hole exchange interaction is neglected. For all the III-VI materials considered, the exciton Bohr radius is larger than the QD radius so that the neglecting of mixing with excited configurations is justified. However, for the II-VI materials, the exciton Bohr radius is comparable to or larger than the QD radius at the largest end of the size-range. In those cases, the exciton energy reported here should be taken as an upper bound. 1. GaAs

Fig. 3(a) shows the exciton energies obtained using TB theory and EMA. As discussed in an earlier paper,25 the results obtained using the two do not match even for the largest crystallite sizes considered. The lone existing experimental data point56 clearly agrees well with our TB calculations. There do not seem to have been any other experiments in the literature on this material, perhaps because GaAs is considered an experimentally difficult material. 2. GaSb

The exciton energy levels are shown in Fig. 3(b). In case of GaSb too, the exciton energies obtained from the two theories do not match at all.

J. Chem. Phys. 141, 204702 (2014)

3. InAs

The exciton energy levels are shown in Fig. 3(c). The experimental data summarized as Expt1 and Expt2 are taken from Refs. 55 and 57, respectively. Clearly, they match better with the TB theory than with the EMA though discrepancies even with TB theory exist for intermediate sizes. For the Expt1 data, Banin et al.55, 57 carried out sizeselective PLE. The QDs, prepared by colloidal techniques, though faceted, were nearly spherical. The QD sizes, determined by high-resolution TEM, range from ≈ 10 Åto 35 Å, with a dispersion of ≈ 10%. The Expt2 data are from absorption measurements by Guzelian et al.57 on particles of radii 17 Å–30 Å, with the sizes determined from TEM and XRD, and the size dispersion ≈ 10%–15%. The QD shape ranged from spherical to slightly ellipsoidal, with aspect ratio ranging from 1.11 to 1.22.

4. InSb

Fig. 3(d) shows the exciton energies for InSb. The TB and EMA results differ greatly for smaller QD sizes but they are closer at the larger sizes considered by us.

5. InP

The exciton energy levels from the TB theory are shown in Fig. 3(e). The experimental data collated as Expt1, Expt2 obtained from Guzelien et al.58 are shown for comparison. Despite the slight scattering in the experimental results, clearly, they are in excellent agreement with the TB results. We note that the TB results differ slightly from the EMA values even for the largest sizes considered. In the Expt1 data, the QDs were found to be slightly ellipsoidal in shape with diameters ranging from 15 Å to 60 Å and ≈ 20% size dispersion as determined from TEM. Apart from the UV/visible absorption, Raman and PL measurements were also carried out. For the Expt2 data on the other hand, the sizes were measured by XRD. Other experimental results from the literature, e.g., Micic et al.59 who synthesized InP QDs of mean diameter 26.1 Å with a dispersion of 7.5 Å, as determined by TEM, also agree well with our TB theory.

6. CdS

The exciton energies obtained from our TB theory are shown in Fig. 3(f). CdS QDs have been extensively studied both theoretically and experimentally. Shown in the figure are the EMA values, as well as the experimental results from Katsikas et al.60 (Expt1) and Wang and Herron61 (Expt2). Wang and Herron61 have carried out absorption studies on slightly pyramidal QDs whose size, determined by XRD, ranged from 5 Åto 30 Å. The studies by Katsikas et al.60 included fluorescence and absorption on near-spherical QDS with sizes ranging from 25 Åto 35 Å as determined by TEM. The TB results, rather than the EMA, compare excellently with the experiments.


Dhayal et al.

2.8 Expt TB EMA

3.3 2.7 2.1 1.5

2.3 1.8 1.3

10 20 30 40 50 60 Radius (Å) 2.7

0.2

(e) InP Expt2

2.1

Expt3 TB EMA

1.8

3.9 Expt1 Expt2 Expt3 Expt4 TB

4.2 3.4 2.6

EMA

1.8 10 20 30 40 50 60 Radius (Å)

(f) CdS Expt1

3.7

Expt2

3.4

TB EMA

3.1 2.8

10 20 30 40 50 60 Radius (Å)

0

(h) CdTe

4.4 Expt1 Expt2 Expt3 TB EMA

3.3 2.7 2.1 1.5

0

10 20 30 40 50 60 Radius (Å)

2.5 0

Energy (eV )

Energy (eV )

2.4

10 20 30 40 50 60 Radius (Å)

(g) CdSe

5.0

1.0

4.0

1.5 0

TB

0

Energy (eV )

0.6

1.5

Expt1 Energy (eV )

Energy (eV )

1.0

2.0

Expt1 Expt2 EMA

10 20 30 40 50 60 Radius (Å)

TB EMA

(c) InAs

0.5 0

(d) InSb

1.4

5.8

2.5 TB EMA

0.8 0

1.8

(b) GaSb

Energy (eV )

(a) GaAs

Energy (eV )

Energy (eV )

3.9

J. Chem. Phys. 141, 204702 (2014)

Energy (eV )

204702-11

10 20 30 40 50 60 Radius (Å)

(i) ZnSe Expt1 Expt2 TB EMA

4.0 3.6 3.2 2.8

0

10 20 30 40 50 60 Radius (Å)

0

10 20 30 40 50 60 Radius (Å)

FIG. 3. The size dependence of the exciton energies in the QDs as obtained from tight-binding (TB) (dots) and multiband EMA (solid line) calculations for: (a) GaAs, (b) GaSb, (c) InAs, (d) InSb, (e) InP, (f) CdS, (g) CdSe, (h) CdTe, and (i) ZnSe. The experimental values, shown for comparison, are taken from Ref. 56 for GaAs, Ref. 55 (Expt1), Ref. 57 (Expt2) for InAs, Ref. 58 for InP, Ref. 60 (Expt1), Ref. 61 (Expt2) for CdS, Ref. 63 (Expt1), Ref. 64 (Expt2), Ref. 65 (Expt3), Ref. 66 (Expt4) for CdSe, Ref. 75 (Expt1), Ref. 76 (Expt2,Expt3) for CdTe, and Ref. 78 (Expt1), Ref. 79 (Expt2) for ZnSe.

Other experimental data from the literature include, e.g., Vossmeyer et al.,62 and are in equally good agreement with our TB theory. 7. CdSe

Fig. 3(g) shows our calculated CdSe exciton energy levels. CdSe too is a very well studied material, both theoretically,20–24 and experimentally, and for comparison, we include in the figure, the experimental data from various sources in the literature, viz., Expt1 from Nogami et al.,63 Expt2 from Alivisatos et al.,64 Expt3 from Woggon et al.,65 and Expt4 from Bawendi et al.66 Our TB results compare well with the experiments while the EMA values differ for all the sizes considered. We note that the experiments employ a variety of methods for studying the QDs. For instance, Woggon et al.65 carried out small-angle X-ray scattering, and linear and nonlinear (picosecond hole burning) optical measurements. Bawendi et al.66 carried out transient optical hole burning and photoluminescence studies of 32 Å CdSe QDs to investigate the electronic properties. They use TEM and XRD to estimate

the size and shape of the crystals, as do Alivisatos et al.64 and Nogami et al.63 There are several other experimental results available in the literature, viz., Park et al.,67 Murray et al.,68, 69 Kuno et al.,70 Shim and Guyot-Sionnest,71 Rogach et al.,72 Bowen Katari et al.,73 and Colvin et al.74 Most of these data match very well with our TB results. 8. CdTe

Our TB results for the exciton energies are given in Fig. 3(h). Also given for comparison are the EMA values and experiments from the literature. The experimental data at smaller sizes show a stronger deviation from our TB results than for any other material. The data from Rajh et al.,75 collated as Expt1, are obtained from PLE, with QD sizes determined from TEM, ranging from 29 Å to 40 Å mean diameter. The size distribution could be the reason for the discrepancy with our theory. The experimental data collated as Expt2 and Expt3 by Mastai and Hodes76 include QDs in thin films, with sizes ranging from 10 Åto 90 Å, as determined by XRD and TEM,


204702-12

Dhayal et al.

respectively. The experimental data show a large uncertainty in the measured size which should be the reason for the discrepancy with TB theory. Other experiments from the literature, e.g., Masumoto and Sonobe,77 also have a large size distribution, and show the same trend as the two earlier reports discussed above. 9. ZnSe

Fig. 3(i) gives the TB and EMA results for the excitonic energies for ZnSe QDs. Experimental values from the literature are summarized as Expt178 and Expt2.79 The TB calculations are in excellent agreement with the experiments. The Expt1 studies included optical absorption, photoluminescence, and photoluminescence emission studies for QDs of sizes 15 Å–45 Å in diameter estimated by XRD as well as TEM. The Expt2 studies included optical absorption and luminescence, elemental analysis, powder XRD, and HRTEM for characterization. Other experimental data from the literature such as Leppert et al.80, 81 and Chestnoy et al.82 compare rather well with our TB results. V. CONCLUSIONS

In this paper, empirical tight-binding calculations of the electronic structures of nine different materials belonging to the III-V and II-VI semiconductor families are reported. Trends in the quantum size effects on near-band-edge valence and conduction band levels as well as exciton levels have been identified, along with several interesting features arising from small effective masses, narrow band gaps, and a very narrow conduction band basin at -point. These features are counter-intuitive when explained on the basis of a simple effective mass approach that is often used in the literature. On the other hand, as we have shown, it is possible to obtain an elegant qualitative understanding, as well as quantitative explanation of experimental data, from tight-binding theory. The effects discussed here have important influence on the optical response of QDs of these materials, with consequences for devices made from them. This study has focused on direct bandgap materials with bulk bandstructures which can be described reasonably well within a sp3 basis set. Indirect-gap semiconductor materials and those with complicated bandstructures, including overlapping conduction bands, require a more complex orbital basis. This, along with the optical response of these materials, is left for future studies. ACKNOWLEDGMENTS

We gratefully acknowledge the BARC Computer Centre where these calculations were performed. S.D. was supported by a Junior Research Fellowship from the Department of Atomic Energy, India. 1 R.

K. Jain and R. C. Lind, J. Opt. Soc. Am. 73, 647 (1983). C. Rustagi and D. D. Bhawalkar, Ferroelectrics 102, 367 (1990); P. Horan and W. Blau, Phase Transitions 24–26, 605 (1990); V. S. Williams, G. R. Olbright, B. D. Fluegel, S. W. Koch, and N. Peyghambarian, J. Mod. Opt. 35, 1979 (1988).

2 K.

J. Chem. Phys. 141, 204702 (2014) 3 I. Gur, N. A. Fromer, C.-P. Chen, A. G. Kanaras, and A. P. Alivisatos, Nano

Lett. 7, 409 (2007). I. Klimov and M. G. Bawendi, MRS Bull. 26, 998 (2001). 5 T. Lundstrom, W. Schoenfeld, H. Lee, and P. M. Petroff, Science 286, 2312 (1999). 6 For a review see, X. Michalet et al., Science 307, 538 (2005). 7 Al. L. Efros and A. L. Efros, Fiz. Tekh. Poluprovdn. 16, 1209 (1982) [Sov. Phys. Semicond. 16, 772 (1982)]. 8 L. E. Brus, J. Chem. Phys. 80, 4403 (1984). 9 Semiconductor Quantum Dots, edited by Y. Masumoto and T. Takagahara (Springer, Berlin, 2002). 10 A. I. Ekimov, A. A. Onushchenko, A. G. Plyukhin, and Al. L. Efros, Zh. Eksp. Teor. Fiz. 88, 1490 (1985) [Soviet Phys. JETP 61, 891 (1985)]. 11 J. B. Xia, Phys. Rev. B 40, 8500 (1989). 12 P. C. Sercel and K. J. Vahala, Phys. Rev. Lett. 65, 239 (1990); Phys. Rev. B 42, 3690 (1990). 13 K. I. Kang et al., Phys. Rev. B 45, 3465 (1992). 14 L. W. Wang and A. Zunger, Phys. Rev. B 54, 11417 (1996). 15 J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 (1954). 16 D. J. Chadi, Phys. Rev. B 16, 790 (1977). 17 G. T. Einevoll and Y. C. Chang, Phys. Rev. B 40, 9683 (1989). 18 Y. C. Chang, Phys. Rev. B 37, 8215 (1988). 19 Y. R. Wang and C. B. Duke, Phys. Rev. B 37, 6417 (1988). 20 P. E. Lippens and M. Lannoo, Phys. Rev. B 39, 10935 (1989). 21 P. E. Lippens and M. Lannoo, Phys. Rev. B 41, 6079 (1990). 22 S. V. Nair, L. M. Ramaniah, and K. C. Rustagi, Phys. Rev. B 45, 5969 (1992). 23 L. M. Ramaniah and S. V. Nair, Phys. Rev. B 47, 7132 (1993). 24 S. V. Nair, L. M. Ramaniah, and K. C. Rustagi, Phys. Rev. Lett. 68, 893 (1993). 25 L. M. Ramaniah and S. V. Nair, Physica B212, 245 (1995). 26 P. D. Sawant, L. M. Ramaniah, and C. Manohar, J. Nanosci. Nanotechnol. 6, 241 (2006). 27 A. D. Yoffe, Adv. Phys. 50, 1 (2001). 28 S. Ogut, J. R. Chelikowsky, and S. G. Louie, Phys. Rev. Lett. 79, 1770 (1997). 29 J. M. Matxain, J. E. Fowler, and J. M. Ugalde, Phys. Rev. B 61, 053201 (2000). 30 B. Wang, S. Nagase, J. Zhao, and G. Wang, J. Phys. Chem. C 111, 4956 (2007). 31 S. P. Nanavati, V. Sundararajan, S. Mahamuni, V. Kumar, and S. V. Ghaisas, Phys. Rev. B 80, 245417 (2009). 32 A. Puzder, A. J. Williamson, F. Gygi, and G. Galli, Phys. Rev. Lett. 92, 217401 (2004). 33 J. M. Azpiroz, I. Infante, X. Lopez, J. M. Ugaldea, and F. De Angelis, J. Mater. Chem. 22, 21453 (2012). 34 S. K. Haram, A. Kshirsagar, Y. D. Gujarathi, P. P. Ingole, O. A. Nene, G. B. Markad, and S. P. Nanavati, J. Phys. Chem. 115, 6243 (2011). 35 D. Babi´ c, R. Tsu, and R. F. Greene, Phys. Rev. B 45, 14150 (1992). 36 J. M. Luttinger and W. Kohn, Phys. Rev. 97, 869 (1955). 37 M. V. Ramakrishna and R. A. Friesner, Phys. Rev. Lett. 67, 629 (1991). 38 M. V. Ramakrishna and R. A. Freisner, J. Chem. Phys. 95, 8309 (1991). 39 H. Fu and A. Zunger, Phys. Rev. B 56, 1496 (1997). 40 L. W. Wang and A. Zunger, J. Phys. Chem. 98, 2158 (1994). 41 L. W. Wang and A. Zunger, Phys. Rev. B 53, 9579 (1996). 42 A. Zunger, Phys. Stat. Sol. B 224, 727 (2001) and references therein. 43 W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). 44 R. M. Martin, Electronic Structure: Basic Theory and Practical Methods (Cambridge University Press, 2004). 45 W. Martienssen and H. Warlimont, Handbook of Condensed Matter and Materials Data (Springer, Berlin, 2001). 46 L. W. Wang and A. Zunger, Phys. Rev. Lett. 73, 1039 (1994). 47 F. Trani, G. Cantele, D. Ninno, and G. Iadonisi, Phys. Rev. B 72, 075423 (2005). 48 Numerical Data and Functional Relationships in Science and Technology, Physics of II-VI and I-VII Compounds, Semimagnetic Semiconductors, Landolt-Börnstein, New Series, Group III Vol. 41B, edited by O. Madelung, U. Rössler, and M. Schulz (Springer, Berlin, 1999). 49 I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, Appl. Phys. Rev. 11, 5815 (2001). 50 U. Woggon, Optical Properties of Semiconductor Quantum Dots (Springer, Berlin, 1997), Chap. 6. 51 Y. Masumoto, in Ref. 9, Chap. 5. 4 V.


204702-13 52 Al.

Dhayal et al.

L. Efros and M. Rosen, Phys. Rev. Lett. 78, 1110 (1997). supplementary material at http://dx.doi.org/10.1063/1.4901923 for bulk band structures generated by the EBOM parametrization. 54 J. R. Chelikowsky and M. L. Cohen, Phys. Rev. B 14, 556 (1976). 55 U. Banin, C. J. Lee, A. A. Guzelian, A. V. Kadavanich, A. P. Alivisatos, W. Jaskolski, G. W. Bryant, Al. L. Efros, and M. Rosen, J. Chem. Phys. 109, 2306 (1998). 56 M. A. Olshavsky, A. N. Goldstein, and A. P. Alivisatos, J. Am. Chem. Soc. 112, 9438 (1990). 57 A. A. Guzelian, U. Banin, A. V. Kadavanich, X. Peng, and A. P. Alivisatos, Appl. Phys. Lett. 69, 1432 (1996). 58 A. A. Guzelian, J. E. B. Katari, A. V. Kadavanich, U. Banin, K. Hamad, E. Juban, A. P. Alivisatos, R. H. Wolters, C. C. Arnold, and J. R. Heath, J. Phys. Chem. 100, 7212 (1996). 59 O. I. Micic, C. J. Curtis, K. M. Jones, J. R. Sprague, and A. J. Nozik, J. Phys. Chem. 98, 4966 (1994). 60 L. Katsikas, A. Eychmuller, M. Giersig, and H. Weller, Chem. Phys. Lett. 172, 201 (1990). 61 Y. Wang and N. Herron, Phys. Rev. B 42, 7253 (1990). 62 T. Vossmeyer, L. Katsikas, M. Giersig, I. G. Popovic, K. Diesner, A. Chemseddine, A. Eychmuller, and H. Weller, J. Phys. Chem. 98, 7665 (1994). 63 M. Nogami, S. Suzuki, and K. Nagasaki, J. Non-Cryst. Solids 135, 182 (1991). 64 A. P. Alivisatos, A. L. Harris, N. J. Levinos, M. L. Steigerwald, and L. E. Brus, J. Chem. Phys. 89, 4001 (1988). 65 U. Woggon, M. Müller, U. Lembke, I. Rückmann, and J. Cesnulevicius, Superlatt. Microstruct. 9, 245 (1991). 53 See

J. Chem. Phys. 141, 204702 (2014) 66 M.

G. Bawendi, W. L. Wilson, L. Rothberg, P. J. Carroll, T. M. Jedju, M. L. Steigerwald, and L. E. Brus, Phys. Rev. Lett. 65, 1623 (1990). 67 S. H. Park, R. A. Morgan, Y. Z. Hu, M. Lindberg, S. W. Koch, and N. Peyghambarian, J. Opt. Soc. Am. B7, 2097 (1990). 68 C. B. Murray, M. Nirmal, D. J. Norris, and M. G. Bawendi, Z. Phys. D26, 231 (1993). 69 C. B. Murray, D. J. Norris, and M. G. Bawendi, J. Am. Chem. Soc. 115, 8706 (1993). 70 M. Kuno, J. K. Lee, B. O. Dabbousi, F. V. Mikulec, and M. G. Bawendi, J. Chem. Phys. 106, 9869 (1997). 71 M. Shim and P. Guyot-Sionnest, J. Chem. Phys. 11, 6955 (1999). 72 A. L. Rogach, A. Kornowski, M. Gao, A. E. Echyller, and H. Weller, J. Phys. Chem. B103, 3065 (1999). 73 J. E. Bowen Katari, V. L. Colvin, and A. P. Alivisatos, J. Phys. Chem. 98, 4109 (1994). 74 V. L. Colvin, M. C. Schlamp, and A. P. Alivisatos, Nature (London) 370, 354 (1994). 75 T. Rajh, O. I. Micic, and A. J. Nozik, J. Phys. Chem. 97, 11999 (1993). 76 Y. Mastai and G. Hodes, J. Phys. Chem. B101, 2685 (1997). 77 Y. Masumoto and K. Sonobe, Phys. Rev. B 56, 9734 (1997). 78 V. V. Nikesh, A. D. Lad, S. Kimura, S. Nozaki, and S. Mahamuni, J. Appl. Phys. 100, 113520 (2006). 79 M. A. Hines and P. Guyot-Sionnest, J. Phys. Chem. B102, 3655 (1998). 80 V. Leppert, S. Mahamuni, N. Kumbhojkar, and S. H. Risbud, Mater. Sci. Eng. B52, 89 (1998). 81 V. Leppert, S. H. Risbud, and M. Fendrof, Philos. Mag. Lett. 75, 29 (1997). 82 N. Chestnoy, R. Hull, and L. E. Brus, J. Chem. Phys. 85, 2237 (1987).


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