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Tight-Binding model for semiconductor nanostructures

S. Schulz and G. CzychollInstitute for Theoretical Physics, University of Bremen, D-28334 Bremen, Germany

(Dated: November 9, 2018)

An empirical scp3a tight-binding (TB) model is applied to the investigation of electronic states in

semiconductor quantum dots. A basis set of three p-orbitals at the anions and one s-orbital at thecations is chosen. Matrix elements up to the second nearest neighbors and the spin-orbit couplingare included in our TB-model. The parametrization is chosen so that the effective masses, the spin-orbit-splitting and the gap energy of the bulk CdSe and ZnSe are reproduced. Within this reducedscp

3a TB-basis the valence (p-) bands are excellently reproduced and the conduction (s-) band is

well reproduced close to the Γ-point, i.e. near to the band gap. In terms of this model much largersystems can be described than within a (more realistic) sp3s∗-basis. The quantum dot is modelledby using the (bulk) TB-parameters for the particular material at those sites occupied by atoms ofthis material. Within this TB-model we study pyramidal-shaped CdSe quantum dots embedded ina ZnSe matrix and free spherical CdSe quantum dots (nanocrystals). Strain-effects are included byusing an appropriate model strain field. Within the TB-model, the strain-effects can be artificallyswitched off to investigate the infuence of strain on the bound electronic states and, in particular,their spatial orientation. The theoretical results for spherical nanocrystals are compared with datafrom tunneling spectroscopy and optical experiments. Furthermore the influence of the spin-orbitcoupling is investigated.

PACS numbers: 73.22.Dj, 68.65.Hb, 71.15.Ap

I. INTRODUCTION

Semiconductor quantum dots1,2 (QDs) are of particu-lar interest, both concerning basic research and possibleapplications. QDs are considered to be zero dimensionalobjects, i.e. systems confined in all three directions ofspace with a typical size of the magnitude of severalnanometers. Therefore, these systems are realizationsof “artificial atoms” whose form and size can be manip-ulated. Concerning basic research these nanostructures(QDs) are interesting, as the methods of quantum theorycan be applied to systems on new scales and with newsymmetries in between that of atoms or molecules andof macroscopic crystals. On the other side light emissionand absorption just from the localized states in such de-vices may be important for optoelectronic applications3,4,quantum cryptography5 and quantum computing6.

Semiconductor QDs can be realized by means ofmetallic gates providing external (electrostatic) confine-ment potentials7, by means of selforganized cluster-ing of certain atoms in the Stranski-Krastanow (SK)growth mode8,9,10 or chemically by stopping the crys-tallographic growth using suitable surfactant materi-als11,12,13,14. Here we deal only with the latter two typesof QDs. The QDs created in the SK growth mode emergeself-assembled or self-organized in the epitaxial growthprocess because of the preferential deposition of materialin regions of intrinsic strain or along certain crystallo-graphic directions. In epitaxial growth of a semiconduc-tor material A on top of a semiconductor material B onlyone or a few monolayers of A material may be depositedhomogeneously as a quasi two dimensional (2d) A-layeron top of the B-surface forming the so called wetting layer(WL). Under certain conditions and for certain materi-

als further deposited A-atoms will not form a furtherhomogeneous layer but they will cluster and form islandsof A-material because this may lower the elastic energydue to the lattice mismatch of the A- and B-material. Ifone then stops the growth process, one has free A-QDson top of an A-WL on the B-material. If one continuesthe epitaxial growth process with B-material, one obtainsembedded quantum dots (EQDs), i.e. QDs of A-materialon top of an A-WL embedded within B-material.The chemically realized QDs emerge by means of colloidalchemical synthesis11,12. Thereby the crystal growth ofsemiconductor material in the surrounding of soap-likefilms called surfactants is stopped when the surface iscovered by a monolayer of surfactant material. Thusone obtains tiny crystallites of the nanometer size in allthree directions of space, why these QDs are also called“nanocrystals” (NCs). The size and the shape of thegrown NCs can be controlled by external parameters likegrowth time, temperature, concentration and surfactantmaterial13,14. Certain physical properties like the bandgap (and thus the color) depend crucially on the size ofthe NCs. Typical diameters for both, EQDs and NCs,are between 3 and 30 nm, i.e. they contain between 103

up to 105 atoms. Therefore, EQDs and NCs can be con-sidered to be a new, artficial kind of condensed matterin between molecules and solids. For the in SK-modusgrown EQDs lens-shaped dots15, dome shaped and pyra-midal dots8,16,17, and also truncated cones18 have beenfound and considered.

Of course the fundamental task is the calculation ofthe electronic properties of EQDs and NCs. But hereone encounters the difficulty that these systems are muchlarger than conventional molecules and that the funda-mental symmetry of solid state physics, namely trans-lational invariance, is not fulfilled. Therefore, neither

2

the standard methods of theoretical chemistry nor theones of solid state theory can immediately be appliedto systems with up to 105 atoms. Conventional ab-initio methods of solid state theory based on densityfunctional theory (DFT) and local density approxima-tion (LDA) would require supercell calculations. But thesize of a supercell must be larger than the EQD or NC,and such large supercells are still beyond the possibil-ity of present day computational equipment. Therefore,only systems with up to a few hundred atoms can beinvestigated in the framework of the standard ab-initioDFT methods19,20,21. Simple model studies based on the

effective mass approximation15,22 or a multi-band ~k · ~p-model23,24,25 describe the QD by a confinement potentialcaused by the band offsets, for instance; they give qual-itative insights into the formation of bound (hole andelectron) states, but they are too crude for quantitative,material specific results or predictions. More suitablefor a microscopic description are empirical pseudopoten-tial methods26,27,28,29 as well as empirical tight-bindingmodels 30,31,32,33,34,35,36,37,38,39,40,41. The empirical pseu-dopotential methods allow for a detailed variation of thewave functions on the atomic scale. This is certainlythe most accurate description from a microscopic, atom-istic viewpoint, but it requires a large set of basis states.Within a TB-model some kind of coarse graining is madeand one studies spatial variations only on inter-atomicscales and no longer within one unit cell. The advantageis that usually a small basis set is sufficient, which allowsfor the possibility to study larger systems. Furthermorethe TB-model provides a simple physical picture in termsof the atomic orbitals and on-site and inter-site matrixelements between these orbitals. A cutoff after a fewneighbor shells is usually justified for orbitals localizedat the atomic sites.

Semiempirical TB-models have been used already todescribe ”nearly” spherical InAs and CdSe NCs for whichthe dangling bonds at the surfaces are saturated byhydrogen31,32,33,34 or organic ligands37,38,39. Also un-capped42 and capped35 pyramidal InAs QDs were ives-tigated by use of an empirical TB-model. In the latterwork an sp3s∗-basis was used leading to a 10N × 10NHamiltonian matrix, where N is the number of atoms,with 33 independent parameters. In the present paperwe apply a similar TB-model to II-VI nanostructures,namely CdSe EQDs embedded within ZnSe and spher-ical CdSe NCs. We show that a smaller TB-basis issufficient, namely an scp

3a-basis, i.e. 4 states per unit

cell and spin direction. This requires only 8 indepen-dent parameters and, in principle, allows for the inves-tigation of larger nanostructures than were accessible inRef. 35. Strictly speaking, the scp

3a-basis-set leads to a

smaller matrix-dimension and also to a smaller number ofnonzero matrix elements compared to sp3s∗ TB-model.So the scp

3a TB-model is numerically less demanding re-

garding both memory requirements and computationaltime. For the bulk system the valence p-bands are ex-cellently reproduced and the conduction s-band is well

CdSe ZnSe

Eg [eV] 1.7443 2.820143

∆so [eV] 0.4143 0.4343

me 0.1243 0.14744

γ1 3.3343 2.4544

γ2 1.1143 0.6144

γ3 1.4543 1.1144

C12 [GPa] 46.345 50.645

C11 [GPa] 66.745 85.945

TABLE I: Properties of the CdSe and ZnSe bandstructures.The lattice constants are given by 6.077 A and 5.668 A,respectively. Eg denotes the band gap, ∆so the spin-orbitcoupling and me the effective electron mass. The Kohn-Luttinger-Parameters are γ1,γ2 and γ3. The Cij are the ele-ments of the elastic stiffness tensor.

reproduced close to the Γ-point. Therefore, we expectthat also for the QDs all the hole states and at least thelowest lying electron states (close to the gap) are well re-produced. We investigate, in particular, the influence ofstrain effects on the electronic structure. To examine theaccuracy of our model we compare the results to othermicroscopic and macroscopic models. Furthermore TB-results obtained for CdSe-NCs are compared to experi-mental results, and very good agreement, for instance forthe dependence of the energy gap on the NC-diameter, isobtained. This demonstrates that our TB-model with areduced basis set is reliable and sufficient for the repro-duction of the most essential electronic properties of thenanostructures.This work is organized as follows. In Sec. II our TB-

model is presented. The formalism how to obtain theTB-parameters and how to apply them to the descriptionof EQDs and NCs is described. In Sec. III the inclusionof strain effects in our model is introduced. Results forthe pyramidal CdSe EQDs are presented. For the spher-ical CdSe NCs the results and the comparison with theexperimental data are presented in Sec. IV. Section Vcontains a summary and a conclusion.

II. THEORY

A. TB-Model for bulk materials

In this work we use a TB-Model with 8 basis statesper unit cell. Such a model has been succesfully used forthe investigation of optical properties in ZnSe-quantumwells46. For the description of the bulk semiconductorcompounds CdSe and ZnSe we choose an scp

3a basis set.

That implies that the set of basis states |ν, α, σ, ~R〉 isgiven by four orbitals α = s, px, py, pz with spin σ = ± 1

2.

One s-orbital at the cation (ν = c) and three p-orbitals

3

−6

−4

−2

0

2

4

6

8E

ner

gy

[eV

]

ΓL X

(a)

−8

−6

−4

−2

0

2

4

6

8

10

12

En

erg

y [e

V]

ΓL X

(b)

FIG. 1: Tight-binding band structures for CdSe (a) and ZnSe(b)

at the anion (ν = a) site in each unit cell ~R are chosen.The TB matrix elements are given by

Eα,α′(~R′ − ~R)ν,ν′ = 〈ν′, α′, σ′ ~R′|Hbulk|ν, α, σ, ~R〉 . (1)

The coupling of the basis orbitals is limited to nearestand next nearest neighbors. Following Ref. 47, the spin-orbit component of the bulk Hamiltonian Hbulk couplesonly p orbitals at the same atom. With the two centerapproximation of Slater and Koster48 we are left with

only 8 independent matrix elements Eα,α′(~R′ − ~R)ν,ν′ .

In ~k space, with the basis states |~k, ν, α, σ〉, the electronicproperties of the pure bulk material are modelled by an

8 × 8 matrix Hbulk(~k) (for each ~k from the first Bril-

louin zone). This matrix depends on the different TB-

parameters Eα,α′(~R′ − ~R)ν,ν′ . By analytical diagonal-

ization for special ~k directions, the electronic dispersion

En(~k) is obtained as a function of the TB-parameters;here n is the band index. Equations for the different

TB-parameters Eα,α′(~R′ − ~R)ν,ν′ can now be deduced asa function of the Kohn-Luttinger-parameters (γ1,γ2,γ3),the energy gap Egap, the effective electron mass me andthe spin-orbit-splitting ∆so. The zero level of the en-ergy scale is fixed to the valence-band maximum. Theused material parameters for CdSe and ZnSe are givenin Table I. The resulting numerical values for the differ-ent TB-parameters (obtained by optimizing them so thatthe resulting TB band-structure reproduces the param-eters given in Table I) are summarized in the Table II(with and without taking into account a site-diagonalparameter for the spin-orbit coupling). Within this ap-proach, the characteristic properties of the band struc-ture in the region of the Γ point are well reproduced,as can be seen from Fig. 1, which shows the TB-bandsof bulk CdSe and ZnSe (using the TB-parameters withspin-orbit coupling). When comparing with band struc-ture results from the literature49, one sees that the threevalence (p-) bands are excellently reproduced whereas thes-like conduction band is well reproduced only close tothe Γ-point. This is understandable, because higher (un-occupied) conduction bands are neglected, and can beimproved by taking into account more basis states perunit cell. But for a reproduction of the electronic prop-erties in the region near the Γ-point, which is important

Material Parameter TB TB-NO SO

ZnSe Exx(000)aa -1.7277 -2.0413

Ess(000)cc 7.0462 12.1223

Esx

(

12

12

12

)

ac1.1581 0.2990

Exx (110)aa 0.1044 0.2185

Exx (011)aa 0.1874 0.0732

Exy (110)aa 0.3143 0.4285

Ess (110)cc -0.3522 -0.7752

λ 0.1433 0

CdSe Exx(000)aa -1.2738 -1.7805

Ess(000)cc 3.6697 10.8053

Esx

(

12

12

12

)

ac1.1396 0.4260

Exx (110)aa 0.0552 0.2161

Exx (011)aa 0.1738 0.0129

Exy (110)aa 0.1512 0.3120

Ess (110)cc -0.1608 -0.7554

λ 0.1367 0

TABLE II: TB-parameters (in eV) with (TB) and without(TB-NO SO) spin-orbit coupling for ZnSe and CdSe, usingthe notation of Ref. 48.

for a proper description of the optical properties of thesemiconductor materials, the scp

3a-TB-model is certainly

sufficient and satisfactory.

B. TB-Model for Embedded Quantum Dots and

Nanocrystals

Having determined suitable TB-parameters for thebulk materials (here CdSe and ZnSe) a EQD or NC canbe modelled simply by using the TB-parameters of thebulk materials for those sites (or unit cells) occupied byatoms (or molecules) of this material. Concerning theon-site matrix elements this condition is unambiguous.Concerning the intersite matrix elements one also usesthe bulk matrix elements, if the two sites are occupiedby the same kind of material, but one has to use suitableaverages of the bulk intersite matrix elements for matrixelements over interfaces between different material, i.e.if the two sites (or unit cells) are occupied by differentatoms (or molecules). Concerning the surfaces or bound-aries of the nanostructure there are different possibilities.One can use fixed boundary conditions, i.e. effectivelyuse zero for the hopping matrix elements from a surfaceatom to its fictitious nearest neighbors, or (for the em-bedded QDs) one can use periodic boundary conditionsto avoid any surface effects, which artificially arise fromthe finite cell size used for the EQD-modelling. For theNCs the best thing to do is a realistic, atomistic mod-elling of the organic ligands covering the NC-surface, asdescribed in Refs.39,50,51. Within the restricted basis setthus selected the ansatz for an electronic eigenstate of

4

the EQD or NC is, of course, a linear combination of the

atomic orbitals |ν, α, σ, ~R〉:

|Φ〉 =∑

α,ν,σ, ~R

uν,α,σ, ~R

|ν, α, σ, ~R〉 . (2)

Here ~R denotes the unit cell, α the orbital type, σ thespin and ν an anion or cation. Then the Schrodingerequation leads to the following finite matrix eigenvalueproblem:∑

α,ν,σ, ~R

〈ν′, α′, σ′, ~R′|H |ν, α, σ, ~R〉uν,α,σ, ~R

−Euν′,α′,σ′, ~R′ = 0 ,

(3)where E is the energy eigenvalue. The shortcut nota-

tion 〈ν′, α′, σ′, ~R′|H |ν, α, σ, ~R〉 = Hl ~R′,m~R

is used in the

following for the matrix elements with l = ν′, α′, σ′ andm = ν, α, σ.The matrix elements for CdSe and ZnSe without strain

are denoted by H0

l ~R′,m~R. For these matrix elements the

TB-parameters Eα,α′(~R′ − ~R)ν,ν′ of the bulk materials,determined in Sec. II A, are used. For the off-diagonalmatrix elements over interfaces and the diagonal matrixelements of the selen atoms at the interface between dotand barrier, which can not unambiguously be referred tobelong to the ZnSe or CdSe, respectively, we choose themean value of the parameters for the two materials. Fur-thermore, a parameter for the valence-band offset ∆EV

has to be included in the model. This means that forCdSe in a heterostructure, i.e. surrounded by a barrierZnSe material, all diagonal matrix elements are shiftedjust by ∆EV compared to the bulk CdSe diagonal ma-trix elements. In the literature different values for ∆EV

can be found, they vary in the range of 10%-30% of theband gap difference between CdSe and ZnSe9,52,53. Wehave performed calculations with valence-band offsets of∆EV = 0.108 eV, ∆EV = 0.22 eV and ∆EV = 0.324eV, which corresponds to 10%, 20% and 30% of the dif-ference of the band gaps. We find that these differentchoices for ∆EV shift the EQD energy gap EQD

gap by lessthan 2%. This shows, that the results are not much af-fected by the specific choice of the valence-band offset∆EV . Therefore, in the following, an intermediate valueof ∆EV = 0.22 eV is chosen.Furthermore, in a heterostructure of two materials

with different lattice constants, strain effects have to beincluded for a realistic description of the electronic states,because the distance between two CdSe unit cells andthe bond angles are not the same as the correspondingequilibrium values in bulk CdSe. This means that theintersite TB matrix elements H

l ~R′,m~Rin the EQD differ

from the H0

l ~R′,m~Rmatrix elements in the bulk material.

In general, a relation

Hl ~R′,m~R

= H0

l ~R′,m~Rf(~d 0

~R′−~R, ~d~R′−~R

) (4)

has to be expected, where ~d 0~R′−~R

and ~d~R′−~Rare the

bond vectors between the atomic positions of the un-

strained and strained material, respectively. The func-

tion f(~d 0, ~d) describes, in general, the influence of thebond length and the bond angle on the intersite (hop-ping) matrix elements. For lack of a microscopic the-ory for the functional form we use as a simplified

model assumption f(~d0~R′−~R, ~d~R′−~R

) =(

d0~R′−~R/d~R′−~R

)2

.

With this d−2 ansatz, the interatomic matrix elements

Hl ~R′,m~R

, with ~R′ 6= ~R, are given by

Hl ~R′,m~R

= H0

l ~R′,m~R

(

d0~R′−~R

d~R′−~R

)2

. (5)

This corresponds to Harrison’s54 d−2 rule, the validityof which has been demonstrated for II-VI-materials bySapra et al.

55. More sophisticated ways to treat the scal-ing of the interatomic matrix elements, e.g. by calculat-ing the dependence of energy bands on volume effects anddifferent exponents for different orbitals, can be found inthe literature35,41,56. Furthermore the results of Berthoet al.

57 for the calculations of hydrostatic and uniaxialdeformation potentials in case of ZnSe show that the d−2

rule should be a reasonable approximation. Our model

assumption for the function f(~d0, ~d) means that we ne-glect the influence of bond angle distortion. Though en-ergy shifts due to bond angle distortions have been foundfor InAs EQDs35, here the negligence of bond angle dis-tortion can be justified when exclusively taking into ac-count the coupling between s- and p-orbitals at nearestneighbor sites. Piezoelectric fields, which are usually con-sidered to be less important for the zinc blende structuresrealized in CdSe and ZnSe24, are also not taken into ac-count in our model.The problem is now reduced to the diagonalization of a

finite but very large matrix. To calculate the eigenvaluesof this matrix, in particular the bound electronic statesin the QD, the folded spectrum method58 is applied tothe eigenvalue problem of Eq. 3.

III. RESULTS FOR A PYRAMIDAL CdSe

EMBEDDED QUANTUM DOT

A. Geometry and Strain

To model a CdSe QD embedded into a ZnSe barriermaterial we choose a finite (zinc blende) lattice within abox with fixed boundary conditions. Within this box weconsider a CdSe WL of thickness 1a (lattice constant ofthe conventional unit cell, i.e. about two anion and twocation layers), and on top of this wetting layer there isa pyramidal QD with base length b and height h = b/2.For the matrix elements corresponding to sites withinthe WL or the QD we choose the TB-values appropriatefor CdSe, for all other sites within the box the ones forZnSe. Figure 2 shows a schematic picture of this geome-try we use to model the EQD. We investigate EQDs with

5

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12.5 nm

CdSe QD

ZnSe

2.8 nm

CdSe wetting layer

[001]

12.5 nm

[100]

[010]

9.6 nm

5.7 nm

5.7 nm

FIG. 2: Schematic visualization of the pyramidal CdSe QDburied in a ZnSe matrix. The wetting layer has a thicknessof one lattice constant (1 a) of bulk ZnSe. The pyramidal QDhas a base length b of ten times the ZnSe lattice constant(b = 10 a).

a base lengths b of 6 a, 8 a and 10 a, where a = 5.668 Ais the lattice constant of the bulk ZnSe material. Cellswith the dimensions of 18 a× 18 a× 15 a (38 880 atoms),20 a×20 a×16 a (51 200 atoms) and 22 a×22 a×17 a (65824 atoms) are used for the calculations. Figure 2 showsthe EQD with a base length b of 10 a. Fixed boundaryconditions are applied to avoid a dot-dot coupling in con-trast to periodic boundary conditions35. The total size ofthe cells is chosen so that the boundary conditions affectthe energy gap of the EQD by less than 2%.To consider strain effects in our model the knowledge

of the strain tensor ǫ is necessary. The strain tensor ǫ isrelated to the strain dependend relative atomic positions~d~R′−~R

by

~d~R′−~R= (1+ ǫ)~d 0

~R′−~R. (6)

To appoint the strain tensor outside the EQD, the WLis treated as a quantum film. In the absence of a shearstrain (ǫi,j ∼ δi,j) for a coherently grown film, the straincomponents are given by59

ǫ|| = ǫxx = ǫyy =aS − aD

aD(7)

ǫ⊥ = ǫzz = −C12

C11

ǫ|| . (8)

Here aD is the lattice constant of the unstrained film ma-terial and aS denotes the parallel lattice constant of thesubstrate. In Table I the cubic elastic constants Cij ofthe bulk materials are given. The resulting strain profilefor a line scan in z-direction outside the dot is shown inFig. 3 (a). In Ref. 23 Stier et al. considered a similarstrain profile for an InAs/GaAs EQD. The lattice mis-match of approximately 7% in the InAs/GaAs system isnearly the same as for the CdSe/ZnSe system. So our

calculated strain profile shows the same behavior as theprofile in Ref. 23 for a line scan in z-direction outside theEQD.To obtain the strain profile inside the EQD we use a

model strain profile, which shows a similar behavior asthe strain profiles which are given in Refs. 23,60 for a linescan in z-direction through the tip of the pyramid. Thismodel strain profile is displayed in Fig. 3 (b). The shearcomponents, ǫxy, ǫxz and ǫyz, can be neglected, at leastaway from the boundaries of the dot22.

B. Bound single particle states

We have calculated the first five states for electronsand holes for three different EQD sizes. These calcula-tions are done with and without including strain effects.For the evaluations without strain we have chosen theexponent in Eq. (5) to be zero. The energy spectrumobtained from these calculations is shown in Fig. 4 (a)without strain and in Fig. 4 (b) including strain effects.The states are labeled by e1 and h1 for electron and holeground states, e2 and h2 for the first excited states, andso on. All energies are measured relative to the valence-band maximum of ZnSe. Figure 4 also shows the sizedependence of the electron and hole energy levels. Theenergies are compared to the ground state energies forelectrons and holes in the 1 a thick CdSe WL (WLe1 andWLh1

, respectively), which is calculated separately for acoherently strained quantum film (i.e. the WL withoutthe QD). As expected from a naive particle in a box pic-ture, the binding of electrons and holes becomes strongerin the EQD when the dot size is increased. The quan-tum confinement causes the number of bound states to

0 5 10 15 20−8

−6

−4

−2

0

2

4

6

z [nm]

Str

ain

(%

)

0 5 10 15 20−10

−8

−6

−4

−2

0

2

4

z [nm]

Str

ain

(%

)

(a) (b)

PSfrag replacements

ǫzz

ǫxx=ǫyy

FIG. 3: Strain distribution in and around the embedded pyra-midal CdSe QD with a base length of b = 10 a. The WL atthe base of the QD is 1 a thick. The whole structure is buriedin a ZnSe matrix. Line scans along the [001] direction throughthe WL outside the dot (a) and inside the dot through the tipof the pyramid (b) are displayed. The diagonal elements ofthe strain tensor ǫ are shown as solid (ǫzz) and dashed-dottedlines (ǫxx = ǫyy).

6

2.2

2.3

2.4

2.5

2.6

2.7

2.2

2.3

2.4

2.5

2.6

2.7

3 4 5 60

0.04

0.08

0.12

0.16

3 4 5 60

0.04

0.08

0.12

0.16

(b) (a)

e1

e2,e

3

e4

e5

WLe

1

e1

e2,e

3

e4

e5

WLe

1

h1h2

h3,h

4h

5

h1h2

h3,h

4h

5

WLh

1

WLh

1

PSfrag replacements

Pyramid base length b [nm]

Ele

ctr

on

Energ

y[e

V]

Ele

ctr

on

Energ

y[e

V]

Hole

Energ

y[e

V]

Hole

Energ

y[e

V]

FIG. 4: Electron and hole energies for embedded pyramidalCdSe/ZnSe QDs on a WL of thickness 1a (roughly 4 mono-layers) for different base length of the dot without strain (a)and with strain effects taken into account (b). The groundstate energies for electrons (WLe1) and holes (WLh1

) for theWL (of thickness 1a) alone are also displayed.

decrease when the dot size is reduced. For the EQDswith a base length b = 8 a and b = 10 a the calucatedhole states are well above the WL energy (WLh1

). Thisis valid for the strain-unaffected and strained EQD. Forthe system with b = 6 a we obtain at least four bound holestates in both models. The energy splitting between thedifferent states is only slightly influenced by the strain.Furthermore we see from Fig. 4 that the number of boundelectron states is influenced by the strain. For the sys-tem with a base length of b = 10 a we get at least threebound-electron states when we take strain effects intoaccount (Fig. 4 (b)). Without strain effects at least 5bound states are found. So the confinement potential forthe electrons is effectively reduced by the strain.The bound electron states e2 and e3 are energetically

not degenerate even without strain. This arises from theC2v symmetry of the system. Already from the geometryof the EQD-system it is clear that there is no (001) mir-ror plane. Furthermore, if one considers a (001)-planewith sites occupied by Se anions, the nearest neighbor(cation) planes in ±z-direction are not equivalent, as inthe zinc blende structure the nearest neighbors above theplane are found in [111]-direction and below the planein [111]-direction. So also for crystallographic reasons a(001)-plane is not a mirror plane. Finally, if one considersthe base plane of the EQD (or the WL) to be this anion(001)-plane, there are different cations, namely Cd aboveand Zn below this plane. Therefore, the QD-system hasreduced C2v-symmetry. In theories based on continuummodels, e.g. effective mass approximations15,22, the dis-cussed effects cannot be accounted for. These inter-facial effects also affect the one particle wave functions

in the system. In Fig. 5 the isosurfaces for the squaredelectron wavefunctions |Φi(~r)|

2 are displayed with andwithout strain, respectively. The light and dark isosur-face levels are selected as 0.1 and 0.5 of the maximumprobability density, respectively. For both calculations,the lowest electron state e1 is an s-like state accordingto its nodal structure. The next two states e2 and e3 arep-like states. These states are oriented along the [110]and the [110] direction, respectively. Due to the differentatomic structure along these directions we find a p-statesplitting ∆0

e2,e3= Ee3 − Ee2 for the unstrained EQD of

about 0.43meV. In conventional ~k · ~p models22,25 an un-strained, square-based pyramidal EQD is modelled witha C4v symmetry. In our microscopic model the resultingdegeneracy is lifted and a splitting occurs as a conse-quence of the reduction of C4v symmetry to a C2v zincblende symmetry.

The strain splits the states e2 and e3 further. Due tothe different atomic structure, the strain profile withineach plane (perpendicular to the growth z-direction)along the [110] and [110] direction is different60. Thiseffect contributes also to the anisotropy. Due to thefact, that the base is larger than the top, there is agradient in the strain tensor between the top and thebottom of the pyramid. In the EQD, the cation neigh-bors above each anion are found in [111] direction whilethe cation neighbors below are found in [111]-direction.Therefore, the cations along the [110] direction are sys-tematically more stressed than the cations along the [110]direction. In case of strain we find a p-state splitting of∆strain

e2,e3= 7.1 meV. Compared to the states e2 and e3

of the unstrained EQD, the two lumps of the light iso-surfaces are well separated. The states e2 and e3 revealnodal planes along the [110] and [110] direction, respec-tively.The state e4 for the strained dot is resonant with a WL-state, so the wave function is leaking into the WL. Alsothe wave function of the state e5 is localized at the baseof the pyramid but clearly shows already a finite proba-bility density inside the WL. The states e4 and e5 of theunstrained EQD are still mainly localized inside the dot.The classification of the state e4 by its nodal structure isdifficult. e4 is similar to a p-state which is oriented alongthe [001] direction. The electron state e5 is d-like.

Figure 6 shows the isosurface plots of the squared wave-function |Φi(~r)|

2 for the lowest five hole states h1-h5 withand without strain. The light and dark isosurface levelsare again selected as 0.1 and 0.5 of the maximum prob-ability density, respectively. Our atomistic calculationshows that the hole states cannot be classified by s-like(h1), p-like (h2 and h3) or d-like (h5) shape accordingto their nodal structures. With and without strain thehole states underly a strong band mixing. So the cal-culated hole states show no nodal structures. Thereforethe assumption of a single heavy-hole valence-band forthe description of the bound hole states in a EQD evenqualitatively yields incorrect results. In contrast to quan-tum well systems, the light-hole and heavy-hole bands are

7

Electron states for the b = 10 a base-length dot

With Strain

e1 e2 e3 e4 e5

Without Strain

e1 e2 e3 e4 e5

FIG. 5: Isosurfaces of the squared electron wavefunctions with and without strain for the embedded b = 10 a pyramidal QD.The light and dark surfaces correspond to 0.1 and 0.5 of the maximum probability density, respectively.

Hole states for the b = 10 a base-length dot

With Strain

h1 h2 h3 h4 h5

Without Strain

h1 h2 h3 h4 h5

FIG. 6: Isosurfaces plots of the squared hole wave functions with and without strain for the embedded b = 10 a pyramidal QD.See caption of Fig. 5 for more details.

strongly mixed in a EQD. This result is in good agree-ment with other multiband approaches23,24,25,26,27,28,29.

From Fig. 6 we can also estimate the influence of strainon the different hole states. Without strain the states h1

and h2 are only slightly elongated along the [110] and[110] direction, respectively. Due to strain these statesare clearly elongated along these directions. The statesh3-h5 are only slightly affected by strain.

Another interesting result is that strain effects shiftthe electron states to lower energies and the hole statesto higher energies as displayed in Fig. 4. Figure 4 alsoreveals that the WL ground-state for electrons and holesis shifted in a similar way due to strain. We observe thatstrain decreases the EQD gap EQD

gap = Ee1 −Eh1by about

1.4%, lowering it from the strain-unaffected value 2.12 eVto the value 2.09 eV. For a biaxial compressive strain ina zinc blende structure, the conduction-band minimumof a bulk material is shifted to higher energies while the

energy shift of the valence-band maximum depends onthe magnitude of the hydrostatic and shear deformationenergies59. So one would expect that the electron statesare shifted to higher energies due to the fact that CdSeis compressively strained in the ZnSe-Matrix. This is incontradiction to the behavior we observe here. To inves-tigate the influence of the WL states on the one-particlespectrum we use the same model geometry as shown inFig. 2 but with a considerably smaller WL thickness ofonly one monolayer (ML). A 1 ML thick WL was alsoused before by Santoprete et al.

35, Stier et al.23 and

Wang et al.26 for an InAs/GaAs EQD. Figure 7 shows

the comparison of the results for a strain-unaffected anda strained pyramidal CdSe EQD with a 1 ML thick WLand a base length of b = 10 a.

On the left-hand side of Fig. 7 the first five electronand hole-state energies for an unstrained EQD are dis-played while the right-hand side shows the energies for

8

2.4

2.6

2.8E

ner

gy

[eV

]

0

0.05

0.1

0.15

0=ZnSe VBM

2.8201=ZnSe CBM

∼

Nostrain Strain

h1

h2

h3,h

4h

5

h1

h2

h3,h

4

h5

e1

e2,e

3

e4,e

5

e4,e

5

e2,e

3

e1

∼

PSfrag replacements

∼

FIG. 7: First five electron and hole state energies for the em-bedded pyramidal CdSe QD with b = 10 a and a one mono-layer (1 ML) thick WL. On the left-hand side the results forthe unstrained EQD are shown while on the right-hand sidethe results for the strained EQD are displayed. The zeroof the energy scale is the bulk ZnSe valence-band maximum(VBM). The energies are compared with the conduction-bandminimum (CBM) of the bulk ZnSe.

the strained EQD. For a 1 ML thick WL the lowest elec-tron state is, by strain effects, shifted to higher ener-gies. This is what one would expect for biaxial com-pression of the bulk material. Furthermore the splittingof the p-like states e2 and e3 is larger compared to theresults for a 1 a thick WL. The splitting ∆0

e2,e3of the

unstrained EQD with a 1 a thick WL is ∆0e2,e3

= 0.43meV whereas for the system with a 1 ML thick WLone has ∆0

e2,e3= 0.5 meV. So the spltting ∆0

e2,e3is in-

creased by about 16%. With strain-effects, the splittingfor the system with 1 ML WL thickness ∆strain

e2,e3= 10.9

meV is about 54% larger than the splitting in the sys-tem with 1 a WL thickness ∆strain

e2,e3= 7.1 meV. Also the

energy splitting ∆e1,e2 between the ground state e1 andthe first excited state e2 is strongly influenced by the WLthickness, namely ∆e1,e2 = 162.8 meV for the unstrainedsytem with 1a WL but ∆e1,e2 = 204.1 meV for a 1 MLWL; with strain effects the splitting ∆e1,e2 is increasedby about 27% if the WL thickness is decreased from 1 ato 1 ML. The results are summerized in Table III. Thiseffect mainly arises from the fact, that the bound statesinside the dot are also coupled to the WL-states. Fora 1 a WL the wave functions of the bound states showalso a probability density inside the WL. For a thinnerWL the leaking of the states into the region of the WLis much less pronounced. In this case, the microscopicstructure inside the EQD and also the strain-affect aremuch more important. This explains the larger energysplittings in case of the 1 MLWL compared to the results

WL 1a 1 ML

No Strain Strain No Strain Strain

∆e1,e2 [meV] 162.8 161.5 204.1 221.2

∆e2,e3 [meV] 0.43 7.1 0.5 10.9

∆h1,h2[meV] 5.76 3.7 7.25 7.66

∆h2,h3[meV] 16.36 12.3 19.66 15.11

EQDgap [eV] 2.12 2.09 2.21 2.21

TABLE III: Energy splittings for electron and hole bound-states in case of different WL thicknesses. The influence ofstrain-effects on the splittings ∆e1,e2 = |Ee1 −Ee2 |, ∆e2,e3 =|Ee2 −Ee3 |, ∆h1,h2

= |Eh1− Eh2

| and ∆h2,h3= |Eh2

− Eh3|

is also displayed. The WL thickness is 1 a and one monolayer(1ML), respectively. The base length of the pyramid is b =10 a.

for a 1 a WL. The hole states are influenced in a similarmanner. In case of a 1 ML WL the energy spectrum ofthe hole states is shifted to higher energies due to thestrain-effects. This behavior is similar to the behaviorobtained from the calculations for a 1 a WL (Fig. 7). Inthe 1 ML WL system the energy splittings ∆h1,h2

and∆h2,h3

for the first three hole-states are larger than thevalues we obtain for the system with 1 a WL. These split-tings are also summarized in Table III. The WL thick-ness also influences the EQD energy gap EQD

gap . For a 1ML WL the electron-states are shifted to higher energiesin contrast to the behavior of the hole states (compareFigs. 4 and 7). In case of the 1 ML WL the gap energyEQD

gap is only slightly affected by the strain. We observehere that the strain has opposite effect for electrons andhole states: electron states become shallower, approach-ing the conduction-band edge, while the hole states be-come deeper, moving away from the valence-band edge.The knowledge of the single-particle wave functionsmakes the examination of many-particle effects in EQDspossible. The single particle wave functions can be usedfor the calculation of Coulomb- and dipole- matrix ele-ments as input parameters. For example the investiga-tion of multi-exciton emission spectra61, carrier captureand relaxation in semiconductor quantum dot lasers62 ora quantum kinetic description of carrier-phonon interac-tions63 is possible.

IV. RESULTS FOR CdSe NANOCRYSTALS

A. Geometry and Strain

In this section we investigate the single particle statesof CdSe nanocrystals within our TB-model. These nano-structures are chemically synthesized11,12. The nanocrys-tals are nearly spherical in shape12,64,65 and the surfaceis passivated by organic ligands. Due to the flexiblesurrounding matrix, these nanostructures are nearly un-

9

strained65. The size of these nanostructrues is in between10 and 40 A in radius11,64,66,67.We model such a chemically synthesized NC as an un-

strained, spherical crystallite with perfect surface passi-vation. The zincblende structure is assumed for the CdSenanocrystal. We neglect surface reconstructions21,39,51

and that the surface coverage with ligands is often notperfect68, though these effects can be important espe-cially for very small NCs. However, we concentrate onconsiderably larger NCs than in the before mentioned ref-erenceses. Therefore, unlike previous TB work we con-centrate here on size and the size dependence of the re-sults obtained for the electronic structure of the NCs.The TB-parameters, which describe the coupling be-tween the dot material and the ligand molecules, are cho-sen to be zero. This corresponds to an infinite potentialbarrier at the surface and is commonly used because ofthe larger band gap of the surrounding material69. Analternative approach to treat the ligand molecules is dis-cussed by Sapra et al. in Ref. 70. The influence of theorganic ligands on the electronic structure can also beinvestigated more realistically in the framework of mi-croscopic descriptions39,50,51.

B. Single particle states and comparsion with

experimental results

We have performed TB-calculations for finite, spher-ical, unstrained NCs of diameter between 1.82 nm and4.85 nm (corresponding to 3-8 a, when a ≈ 6.07 A isthe CdSe lattice constant of the conventional unit cell).The finite matrix diagonalizations yield both, the discreteeigenenergies and the eigenstates. For the largest NCs (ofdiameter 4.85 nm) results for the five lowest lying electronand hole eigenstates are shown in Fig. 8 again in the formof an isosurface plot. The lowest lying electronic state e1obviously has spherical symmetry and can be classifiedas a 1s-state. Correspondingly the second state e2 hasthe form of a 2s-state and the states e3,4 are p-states ande5 is a d-like state. Despite the spherical symmetry of thesystem this simple classification is no longer possible forthe hole states, however. Even the lowest lying hole stateh1 has no full rotational invariance, i.e. strictly speakingit cannot be classified as being an s-state. This is due tothe intermixing of different atomic TB-valence electronstates in the NC. Similarly the higher hole states h2−h4

cannot clearly be classified as an s- or p-like state. Thisis an effect, which simple effective mass models cannotaccount for, but which will have implications in the cal-culation of matrix elements between these states, whichenter selection rules for optical transitions etc.In the case of an ideal zinc blende structure as con-

sidered here we do not obtain any indications of quasi-metallic behavior, i.e. of a non-vanishing (quasi continu-ous) spectrum of states at the Fermi energy in contrastto previous work (assuming an ideal wurtzite structurefor CdSe nanocrystals)19,38,39. This is probably due to

the fact that this quasi-metallic behavior is due to sur-face states in the case when no passivation and surfacereconstruction is taken into account. These surface statesare formed by the dangling bonds of unsaturated Se atthe NC surface, which cause s-states in the band gapregion38. In our simplified and restricted TB scp

3a-basis

set these s-orbitals at the anions (Se) are not taken intoaccount. Therefore, these surface states, which in real-ity and in more realistic models are removed (i.e. ener-getically drawn down and filled) due to passivation andsurface reconstruction, do not occur.

The discrete electronic states of semiconductor NCsare experimentally accessible by scanning tunneling mi-croscopy (STM)64,66,67. The tunnel current I betweenthe metallic tip of the STM and the CdSe nanocrys-tal, which is e.g. epitaxially electrodeposited onto atemplate-stripped gold film, is measured as a functionof the bias voltage V . The conductance (dI/dV ) is re-lated to the local tunneling density of states. In thedI/dV versus V diagram, several discrete peaks can beobserved. These peaks correspond to the addition en-ergies (charging energies) of holes and electrons. Thespacing between the various peaks can be attributed tothe Coulomb charging (addition spectrum) and/or chargetransfer into higher energy levels (excitation spectrum).From these measurements the energy gap Enano

gap as wellas the splitting ∆e1,e2 between electron ground state e1and the first excited state e2 can be determined.

Alperson et al.67 investigated CdSe nanocrystals with

an STM. Here we compare our calculated energy gapEnano

gap , which is given by the difference between the elec-tron, e1, and hole, h1, groundstate, with measured datafrom Ref. 67. Figure 9 displays the results for CdSeNCs with diameters in between 1.82 nm and 4.85 nm.Alperson et al.

67 compare the STM results (dashed dot-ted line) with optical spectroscopy measurements (dot-ted line) from Ekimov et al.71. The overall agreementwith the TB results is very good, especially for the largerNCs. Deviations in the case of the small 2 nm NC arisefrom surface reconstructions19,21,39 which are neglectedhere. When the same calculation is done without spin-orbit coupling (TB-NO SO), the energy gap Enano

gap is al-ways strongly overestimated by the TB-model, in par-ticular for smaller nanocrystals. So the spin-orbit cou-pling is important for a satisfactory reproduction of theexperimental results. For the calculations without spin-orbit coupling, the TB-parameters are re-optimized toreproduce the characteristic properties (band gap, effec-tive masses) of the bulk material. The re-optimized pa-rameters are given in Table II. Taking into account theelectron spin, the lowest electron state e1 is twofold de-generated and s-like. This is consistent with the experi-mentally observed doublet67 in the dI/dV characteristic.The next excited level is (quasi) sixfold degenerated. Thespin-orbit coupling splits this into one twofold and onefourfold degenerate state34. In the STM measurementAlperson et al.

67 observed such a higher multiplicity ofthe second group of peaks. This behavior has also exper-

10

Electron and holes states for the nanocrystal

Electron states

e1 e2 e3 e4 e5

Hole states

h1 h2 h3 h4 h5

FIG. 8: Isosurfaces (at 30 % of the maximum probability density) of the squared electron and hole wavefunctions of sphericalCdSe nanocrystals of diameter d = 4.85 nm for the five lowest states

1.5 2 2.5 3 3.5 4 4.5 52

2.5

3

3.5

4

4.5STMTB−NO SOTBOptical

PSfrag replacements

Enano

gap

[eV

]

Diameter d [nm]

FIG. 9: Energy gap Enanogap as a function of the nanocrystal

diameter d. Compared are the results from our TB-modelwith (TB) and without (TB-NO SO) spin-orbit coupling, aSTM (STM)67 and an optical measurement (Optical)67.

imentally64 and theoretically34 been observed for InAsnanocrystals. The electron energy spectrum for NCs ofdifferent diameter is shown in Fig. 10 (a). Here the firstfive electron states e1−e5 are displayed. Note that everystate is twofold degenerated due to the spin.For the hole states the situation is more complicated.

Alperson et al.67 observed a high density of states at

negative bias. The distinction between addition and ex-citation peaks is difficult, due to the large number ofpossibilities and the close proximity between the charg-ing energy and the level spacing. For the holes we obtain

2

2.5

3

3.5

4

4.5

1 2 3 4 5−0.5

−0.4

−0.3

−0.2

−0.1

0

(a)

(b)

PSfrag replacements Ele

ctro

nenerg

y[e

V]

Hole

energ

y[e

V]

Diameter d [nm]

e5

e5

e4

e3

e2

e1

e1

e2, e3, e4

h1

h2

h3

h4

h5

h5

h1, h2

h3, h4

FIG. 10: Electron (a) and hole (b) energies as a function ofthe nanocrystal diameter d. For electrons (e1-e5) and holes(h1-h5) the first five eigenvalues are displayed. Each state istwofold degenerated.

that the first two states (h1, h2) and (h3, h4) are fourfolddegenerated. The energy splitting of these states is alsovery small. These results are consistent to the observa-tions of Alperson et al.

67. Figure 10 (b) shows the holeenergy versus diamater d for the spherical CdSe NCs.Obviously, for all diameters displayed the states h1 − h4

are almost degenerate, i.e. including spin there is almostan 8-fold degeneracy of these states.Furthermore the calculated splitting ∆e1,e2 = Ee2 −Ee1

11

1.5 2 2.5 3 3.5 4 4.5 5

0.5

1

1.5

2STMTB−NO SOTBOpticalIR

PSfrag replacements

∆e1,e

2[e

V]

Diameter d [nm]

FIG. 11: Splitting ∆e1,e2 = Ee2 − Ee1 between the lowesttwo electronic states as a function of the nanocrystal diame-ter d. The results from our TB-model, with (TB) and without(TB-NO SO) spin-orbit coupling, and from an STM measure-ment (STM, Ref. 67) are displayed. Besides this results frominfrared spectroscopy (IR, Ref. 72) and optical methods (Op-tical, Ref. 67) are shown.

between the first two electron states e1 and e2 is com-pared with experimentally observed results for this quan-tity. Figure 11 shows ∆e1,e2 as a function of the nanocrys-tal diameter d. The influence of the spin-orbit couplingon our results is also investigated. We have done thecalculations without (TB-NO SO) and with spin orbit-coupling (TB). The results of our TB-model for the split-ting ∆e1,e2 are compared with results obtained by STM67

and by optical methods (optical)71. This splitting ∆e1,e2

was independently determined experimentally by Guyot-Sionnest and Hines72 using infrared spectroscopy (IR).Without spin-orbit coupling the TB-model always over-estimates the splitting ∆e1,e2 . Especially for smallernanocrystals the spin-orbit coupling is very important todescribe the electronic structure. With spin-orbit cou-pling the results of the TB-model show good agreementwith the experimentally observed results.

V. CONCLUSION

We have applied an empirical scp3a TB-model to the

calculation of the electronic properties of II-VI semicon-ductor EQDs and NCs. Assuming a zinc blende lat-tice and (per spin direction) one s-like orbital at thecation sites and three p-orbitals at the anion sites, theTB-parameters for different materials (here ZnSe andCdSe) are determined so that the most essential proper-ties (band gap, effective masses etc.) of the known bandstructure of the (3 dimensional) bulk materials are wellreproduced by the TB band structure. Then a CdSe QD(on top of a two-dimensional, a few atomic layers thickWL) embedded within a ZnSe matrix is modelled by us-ing the TB-parameters of the dot material for those sites

occupied by CdSe and the ZnSe TB matrix elements forthe remaining sites; suitable averages have to be chosenfor intersite matrix elements over and for on-site matrixelements on anion (Se) sites at interfaces between QD andbarrier material. Spherical CdSe NCs can be modelledsimilarly by setting the intersite matrix elements betweensurface atoms and atoms in the monolayer of surfactantmaterial to zero. The effects of the spin-orbit interaction,the band offsets and for the EQDs strain effects are takeninto account.

For the EQD systems the numerical diagonalizationyields a discrete spectrum of bound electron and holestates localized in the region of the EQD. Energeticallythese discrete states are below the continuum of the WLstates. We have investigated the dependence on the EQDsize and find that the number of the bound states andtheir binding energy increases with increasing dot size,therefore the effective band gap decreases. We have alsoinvestigated the dependence of the bound eigenenergiesand their degeneracy on strain and on the thickness ofthe WL. Looking at the states themselves one sees thatconduction band (electron) states can be roughly clas-sified as s-like, p-like, etc. states but the valence band(hole) states cannot be classified according to such simple(s,p,d) symmetries because they are determined by a mix-ing between the different (anion) p-states. This cannotbe accounted for by simple effective mass models but itwill be important for instance for the calculation of dipolematrix elements between electron and hole states whichdetermine the selection rules for optical transitions. Forthe NCs the whole spectrum is discrete, but in spite ofthe spherical symmetry the hole states do not have thesimple s,p,d-symmetry but are intermixtures of atomicp-orbitals. Even the lowest hole state has no sphericals-symmetry but it is 4-fold (8-fold including spin) de-generate. The spin-orbit interaction is very important.Including the spin-orbit interaction we obtained nearlyperfect agreement with experimental results obtained bySTM for the dependence of the band gap and of the split-ting of the lowest electronic states on the diameter of theNC.

Compared to (two-band) effective mass15,22 and multi-

band ~k · ~p-models23,24,25 for EQDs our TB model clearlyhas the advantage of a microscopic, atomistic descrip-tion. Different atoms and constituents of the nanostruc-ture and their actual positions are considered, and thismay lead to a reduction of symmetries (for instance theC2v symmetry instead of a C4v-symmetry). This may au-tomatically lift certain degeneracies and lead automati-cally to a splitting, for instance between e2 and e3 states,

whereas an 8-Band-~k · ~p model still yields degenerate e2and e3 states25. The effects of inhomogeneous strain canbe easily incorporated into a TBmodel by considering thedeviations of the actual atomic positions from the idealposition in the bulk crystal. Only the (empirical) pseu-dopotential treatment26,27,28,29 may be still superior andmore accurate than the TB approach, but in a pseudopo-tential descripion a variation of the wave functions within

12

the individual atoms is accounted for and a large numberof basis states is required. Therefore, a TB descriptionis simpler and quicker and allows for the investigation oflarger nanostructures without loosing information on theessential, microscopic details of the structure. Comparedto other TB models of QD structures, we do not considerfree standing, isolated QDs (as in Ref. 42) but we can de-scribe realistic QDs (with a WL) embedded into anotherbarrier material. We show here that a reduced scp

3a-basis

is already sufficient for a satisfying reproduction of prop-erties like optical gaps, energy splittings, etc., and theirsize dependence. Much larger basis sets, namely a sp3s∗-basis35 or even a sp3d5s∗-basis56 were used in previousTB-models of EQDs. Our reduced, smaller basis set, ofcourse, leads to computational simplifications and allowsfor the treatment of larger QDs. Furthermore, we applyour TB-model to different materials than investigatedpreviously, namely II-VI CdSe nanostructures, and weinvestigate also NCs, for which excellent agreement withexperimental STM-results could be demonstrated.In the future further applications of our TB-model for

embedded semiconductor QDs and NCs are planned. Ofcourse applications to QDs of other materials, for in-stance nitride systems, and other (e.g. wurtzite) crys-

tal structures are possible. Furthermore, EQDs of othershape and size (dome-shaped, lens-shaped, truncatedcones, etc.) or two coupled QDs or freestanding (cappedand uncapped) QDs can be investigated. A combinationwith ab-initio calculations is also possible by determiningthe TB-parameters from a first-principles band structurecalculation of the bulk material. Furthermore the influ-ence of surface reconstructions and the surfactant mate-rial on the results for NCs should be investigated. Espe-cially for small NCs these effects are important. Finallymatrix elements of certain observables like dipole matrixelements between the calculated QD electron and holestates can be determined, which are important for selec-tion rules and the optical properties of these systems.

Acknowledgments

This work has been supported by a grant from theDeutsche Forschungsgemeinschaft No. Cz-31/14-1,2 andby a grant for CPU time from the John von Neumann In-stitute for Computing at the Forschungszentrum Julich.

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