Electronic Structure of Heterogeneous Materials Application to Optical properties

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Electronic Structure of Heterogeneous Materials

Application to Optical properties

Natália Leitão Marques Morais

Thesis to obtain the Master of Science Degree in

Engineering Physics

Supervisor: Prof. Dr. José Luís Rodrigues Júlio Martins

Examination Committee

Chairperson:

Prof. Dr. Pedro Domingos Santos do Sacramento

Supervisor:

Prof. Dr. José Luís Rodrigues Júlio Martins

Member of the Committee:

Prof. Dr. Eduardo Filipe Vieira de Castro

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Acknowledgments

I would like to thank the Instituto de Engenharia de Sistemas e Computadores, Microsistemas & Nan-otecnologias (INESC-MN) of Lisbon, that very kindly hosted me in as I predict to be the beginning of my career as a researcher in Physics, subject that I’ve always been passionate about since at a very young age.

I would like to thank my coordinator Jos é Lu´ıs Martins for guiding me on this project that concluded this Mestrado em Engenharia F´ısica Tecnol ógica (MEFT) 5 years Master Instituto Superior T écnico (IST, Lisbon) program and providing me an opportunity to perform a high-quality research in the domain of Condensed Matter Physics and also to his research fellow Carlos Reis who provided me some other help I needed.

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Resumo

O objectivo teste projecto é encontrar uma descriç ão para um s ólido cristalino de elementos de grupo IV Sil´ıcio, Carbono e Germ ânio, e calcular a estrutura electr ónica e propriedades ópticas. Para calcu-lar a estrutura de bandas v ão se usar pseudopotenciais. Vai ser usado o M étodo de Pseudopotencial Emp´ırico (EPM) para encontrar o melhor ajuste dos pseudopotenciais aos dados experimentais con-hecidos da estrutura de bandas de cada um destes elementos. Antes disso, começa-se por ajustar os pseudopotenciais a um gerador de pseudopotenciais ab initio, para que se encontre uma regi ão aceit ável de par âmetros dos pseudopotencias emp´ıricos. A partir da´ı, melhora-se o ajuste dos poten-ciais à experi ência. As propriedades ópticas dos cristais de Si, C e Ge s ão calculadas. Com estes pseudopotenciais, o objectivo é de gerar um pseudopotencial para cada um destes elementos que simule correctamente as propriedades, que depois seja transfer´ıvel para super-redes de Si-Ge-C. O objectivo final é prever o comportamento de nano-estruturas de Si-Ge-C.

Palavras-chave:

Nanotecnologias, Simulaç ão de Materiais, F´ısica da Mat éria Condensada, Aplicaç ões de F´ısica do Estado S ólido, Pseudopotenciais, Propriedades Ópticas

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Abstract

The objective of this work is to find a description of the group IV elements Silicon, Carbon and Germa-nium and calculate the electronic structure and optical properties of materials containing those elements. To calculate the band structure we will use pseudopotentials. We will use an Empirical Pseudopotential Method to find the better fitting to the pseudopotentials to the experimental known data about the band structure to each of this elements. We start by fitting the pseudopotentials to an ab initio pseudopotential generator, to find the first acceptable set of pseudopotentials’ parameters. From that we further adjust the potentials to the experiment, and find a better fit. After that the optical properties of the bulk Si, C and Ge are calculated. The purpose is to generate a pseudopotential to each of this elements that simulates correctly the properties and can be transferable to supercells of Si-Ge-C. We want to predict the behaviour of Si-Ge-C nanostructures.

Keywords:

Nanotechnologies, Simulation of Materials, Condensed Matter Physics, Solid State Applications, Pseudopotentials, Optical Properties

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4 Conclusions 63 Bibliography 67 A Clebsch-Gordon coefficients for mixing states ` and s = 1 2 71 B Spin-Orbit projectors for the pseudopotential 75 C Matrix elements of the momentum matrix operator between plane waves 77 D Orthogonality of the basis functions 79 E Pseudopotentials matrix elements 81 E.1 Local Pseudopotencial . . . 81

E.2 Nonlocal Potencial . . . 84

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List of Tables

1.1 Calculated values for the gaps (in eV) are shown . . . 2 3.1 The table shows previously obtained parameters for Silicon, that we first check in this

research. . . 38 3.2 Band structure calculated with the pseudopotential of Silicon, using the parameters from

Table 3.1 in eq. (3.9) scaled by a factor f with 0 ≤ f ≤ 1. The the opening of the gap is clearly shown . . . 39 3.3 Experimental and calculated in the current work transitions of Silicon in eV are calculated

with the parameters from Table 3.1 in equation (3.9) . . . 40 3.4 Results to the fit of equation (3.9) to the p pseudopotential of Silicon, generated with the

program in reference [SF], using the default weight function chosen by MATHEMATICA . . 43

3.5 The results to the fitting, using (3.22), to the p pseudopotential of Silicon generated with the program in reference [SF] are shown . . . 45 3.6 The table shows the obtained parameters of the fitting of (3.20) to the s “projector”

gener-ated with [SF] . . . 45 3.7 The initial parameters, used to calculate the important energetic transitions of Silicon are

shown . . . 46 3.8 Important energy transitions of Silicon where calculated with the parameters in Table 3.7 . 47 3.9 The final parameters for the pseudopotential of Silicon obtained after adjusting to the

experimental band structure . . . 47 3.10 Experimental and calculated in the current work transitions of Silicon in eV are calculated

with the parameters on Table 3.9 . . . 47 3.11 The table shows the results to the fit of equation (3.9) to the p pseudopotential of Carbon

generated with the program in reference [SF] . . . 48 3.12 The results to the fitting, using (3.22), to the p pseudopotential of Carbon, generated with

the program in reference [SF] are shown . . . 48 3.13 Results to the fit of equation (3.20) to the s “projector” of Carbon generated with the

program in reference [SF] are shown . . . 50 3.14 These are the obtained initial parameters, used to calculate the important energetic

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3.15 This important energy transitions of Carbon were calculated with the parameters in Table 3.14 . . . 52 3.16 The pseudo-parameters for Carbon, obtained after the adjustment to the experiment . . . 52 3.17 Experimental and calculated in the current work transitions of Diamond in eV , calculated

with the parameters on Table 3.16 . . . 53 3.18 The table shows the results to the fit of equation (3.9) to the p pseudopotential of

Germa-nium generated with the program in reference [SF] . . . 54 3.19 The table shows the results to the fitting to the p pseudopotential of Germanium generated

with the program in reference [SF] . . . 55 3.20 Results to the fit of equation (3.20) to the s “projector” of Germanium generated with the

program in reference [SF] are shown here . . . 55 3.21 This obtained parameters are used to calculate the some transitions of Germanium

with-out the spin-orbit part . . . 56 3.22 Important energetic transitions of Germanium, calculated without using the spin-orbit

split-ting, with the parameters on Table 3.21 . . . 57 3.23 Pseudo-parameters, after the better “adjustment” to the experiment, used to calculate the

band structure of Germanium, without the spin-orbit splitting. The “adjustment” is as close as possible because without the spin-orbit contribution we cannot fully describe Germanium. 57 3.24 Important transitions calculated for a bulk Germanium without the spin-orbit splitting, with

the parameters on Table 3.23 . . . 57 3.25 This set of parameters are used to calculate important transitions of Germanium, with the

spin-orbit splitting . . . 58 3.26 Important transitions of Germanium, including the spin-orbit splitting, are calculated from

the parameters in Table 3.25 . . . 58 3.27 The set of parameters used to calculate the band structure of Germanium, with the

spin-orbit splitting are shown . . . 58 3.28 Important optical transitions of Germanium where calculated using the parameters of

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List of Figures

1.1 The figure shows a) the diamond crystal structure and b) the Brillouin zone (BZ) of an fcc crystal lattice . . . 3 1.2 The energy-band structure of a) Si and b) Ge are calculated with a tight-binding model.

The top of the valence bands is set at zero energy. [TT93] . . . 3 1.3 Schematic representation of the band alignment between a substrate and a strained

het-erolayer. The three contributions shown are (i) the “material effect” for the unstrained case ∆Ea , (ii) the shift due to hydrostatic strain ∆Eh , and (iii) the splitting of a degenerated

band due to biaxial strain ∆Es. [Ost98] . . . 4

1.4 The unit cell of the (Ge)2/(Si)2superlattice and its band-structure are shown [Sch91] . . . 5

1.5 The figure shows a) electronic band structure of the (Ge)6/(Si)4 supperlattice on Si [001]

substrate and band structure of the free-standing b) (Ge)5/(Si)5c) (Ge)4/(Si)6superlattice

along lines of high symmetry [Sch91] . . . 5 1.6 Transition energies of various n + m = 10 superlattices s as a function of lateral strain in

the Si layer are represented [Sch91] . . . 6 1.7 Band structures E(~k) for the principal symmetry directions of the diamond lattice for (a)

Si0.2Ge0.8and (b) Si0.74Ge0.26where calculated [New84] . . . 6

1.8 The figures show the predicted single-impurity defect levels of a) T2and b) A1symmetry

as a function of the composition x. Shown are also the conduction and valence band edges [New84] . . . 7 1.9 Band structures of the α12magic sequence grown on Si0.4Ge0.6where calculated [d’A12] 8

1.10 The figure shows the comparison between Si6Ge4 superlattice and the magic sequence

of the direct absorption spectra [d’A12] . . . 8 1.11 Direct dipole-allowed band-gaps as calculated by [d’A12] . . . 8 1.12 Valence-band offsets for compressive strained Si1−xGex , and Si1−x−yGexCy (x =10%,

20%, and 30%) and tensile strained Si1yCyand Si1−x−yGexCy(y =1%, 2%, and 3%) are

plotted as a function of the effective lattice mismatch (expressed in “effective” Ge or C concentrations, respectively) [Ost98] . . . 9

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1.13 Conduction-band offsets for compressive strained Si1−xGex, and Si1−x−yGexCy(x =10%,

20%, and 30%) and tensile strained Si1yCyand Si1−x−yGexCy(y =1%, 2%, and 3%) are

plotted as a function of the effective lattice mismatch (expressed in “effective” Ge or C concentrations, respectively) [Ost98] . . . 10 1.14 The band-gap narrowing for ternary Si1−x−yGexCy alloys, strained on Si(001) is

repre-sented as a function of lattice mismatch as calculated by [Ost98] . . . 10 2.1 The figure shows the Venn’s diagram corresponding a potential in the space of all

po-tentials V (~r)to a ground state electron density in the space of the ground state electron densities ρGS(~r) . . . 16

2.2 The figure shows a schematic plot of a pseudopotential in reciprocal space with the G’s that correspond to G2_{= 3, 8, 11}_{with G}2_{in units of (2π/a)}2_{[CC92] . . . .} ₂₄

3.1 The figure shows a) the free electron bands in the fcc lattice and b) the Brillouin zone for the fcc lattice . . . 37 3.2 The graphic shows the functions that compose the local pseudopotential and the

pseu-dopotential itself, unscreened and screened . . . 38 3.3 The figure shows the LDA band structure of bulk Silicon calculated with the program of

reference [ea]. Experimental values are indicated by the double arrows. . . 39 3.4 The Figure shows the a) calculated density of states (blue line), the photo emission

inten-sity and inverse photo emission data obtained from reference [Che89] (yellow line) and b) the calculated joint density of states with the parameters from Table 3.1, for Silicon . . . . 40 3.5 Real part 1and imaginary part 2of the dielectric function of Silicon are calculated using

the parameters on Table 3.1 . . . 41 3.6 Comparison of the calculated dielectric function of Silicon, using the parameters on Table

3.1, with experimental results in reference [AS83] . . . 42 3.7 The figure shows the calculated reflectance for Silicon using Table 3.1 (blue line) and the

experimentally obtainced from reference [AS83] (yellow line) . . . 42 3.8 The figure shows the fit of the expression (3.9) (line) to the p pseudopotential of Silicon

generated with the program in reference [SF] (dots) . . . 44 3.9 The ballpark figure to fit the local part of the Silicon pseudopotential using function (3.21)

is shown. The green point is the result of the fit, in Table 3.4 . . . 44 3.10 The ballpark figure to fit the local part of the Silicon pseudopotential using function (3.21)

is shown. The green point is the result of the fit, in Table 3.4, the red points are the pairs of values (Ra,kT F) for which the eigenvalue of energy is the one of the Ep, the green

line is adjusted to these points, the orange points are the pairs of values (Ra,kT F) for

which the eigenvalue of energy is the one of the Ep+ 0.5eV and the yellow points are the

pairs of values (Ra,kT F) for which the eigenvalue of energy is the one of the Ep− 0.5eV .

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3.11 The figure has the s “projector” of Silicon, generated with [SF] (purple points) and the corresponding fit by MATHEMATICAof expression (3.20) . . . 46 3.12 The figure shows the contour plot of the function 3.23, used to fit the non-local part of

the pseudopotential of Silicon. The green point is the result of the fit (Table 3.6), the red points are the pairs of values (Rb,B) for which the eigenvalue of energy is the LDA value

of Es, the yellow line is a parabola adjusted to these points. The orange points are the

pairs of values (Rb,B) for which the eigenvalue of energy is the one of the Es+ 0.5eV and

the yellow points are the pairs of values (Rb,B) for which the eigenvalue of energy is the

one of the Es− 0.5eV . Es= −10.83eV is obtained from reference [SF] . . . 46

3.13 Band structure of Silicon was a) calculated using LDA, from [ea] and b) calculated using the parameters on Table 3.9 . . . 47 3.14 It is represented a) the calculated density if states (blue line),the photo emission

spec-troscopy and inverse photo emission data obtained from reference [Che89] (yellow line), b) the calculated joint density of states, c) calculated dielectric function, d) calculated (blue line) and experimental (yellow line, [AS83]) 1, e) calculated (blue line) and experimental

(yellow line, [AS83]) 2f),g) calculated (blue line) and experimental (yellow line, [AS83])

reflectance for Silicon with the local pseudopotential of equation (3.7) and non-local pro-jector for the pseudopotential of equation (3.19) with the parameters written in Table 3.9 . 49 3.15 The figure shows the fit of the expression (3.9) to the p pseudopotential of Carbon

gener-ated with the program in reference [SF] . . . 50 3.16 The figure to fit the local part of the pseudopotential using function (3.21) is shown. The

green point is the result of the fit using MATHEMATICA, in Table 3.11, the red points are the pairs of values (Ra,kT F) for which the eigenvalue of energy is the one of the Ep, the green

Ep= −5.41eV is obtained from reference [SF] for Carbon . . . 50

3.17 The fit of the expression (3.20) to the s “projector” of Carbon, generated with the program in reference [SF] is represented . . . 51 3.18 The figure to fit the non-local part of the pseudopotential of Carbon using function (3.23)

is shown. The green point is the result of the fit with MATHEMATICA, in Table 3.13, the red points are the pairs of values (Rb,B) for which the eigenvalue of energy is the one of the

Es, the red parabola is adjusted to these points, the orange points are the pairs of values

(Rb,B) for which the eigenvalue of energy is the one of the Es+ 0.5eV and the yellow

points are the pairs of values (Rb,B) for which the eigenvalue of energy is the one of the

Es− 0.5eV . Es= −13.63eV is obtained from reference [SF] . . . 51

3.19 Band structure of diamond a) form reference [ea] and b) calculated with the parameters in Table 3.16 . . . 52

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3.20 It is represented the a) DOS of Carbon, calculated here (blue line), the photo emission spectroscopy data from reference [ea74] (yellow line) divided by a factor of 20, b) the cal-culated joint density of states, the c) real part (blue line) and imaginary part (purple line) of the dielectric function, d) the comparison between the calculated (blue) and experi-mentally obtained (yellow, [RW67]) 1, e) comparison between the calculated (blue) and

experimental (yellow, [RW67] 2and f) the calculated (blue) and experimentally obtained

(yellow, [RW67]), divided by a factor of 100, reflectance. . . 53 3.21 The figure shows the fit of the expression (3.9) to the LDA p pseudopotential of

Germa-nium, generated with the program in reference [SF] . . . 55 3.22 The ballpark figure to fit the local part of the pseudopotential using function (3.21) is

shown. The green point is the result of the fit, in Table 3.18, the red points are the pairs of values (Ra,kT F) for which the eigenvalue of energy is the one of the Ep, the yellow

Ep= −4.05eV is obtained from reference [SF] for Germanium . . . 55

3.23 The fit of the expression (3.20) to the s “projector” of Germanium generated with the program in reference [SF] is shown . . . 56 3.24 The figure to fit the non-local part of the pseudopotential of Germanium using function

(3.23) is shown. The green point is the result of the fit, in Table 3.13, the red points are the pairs of values (Rb,B) for which the eigenvalue of energy is the one of the Es,

the yellow parabola is adjusted to these points, the orange points are the pairs of values (Rb,B) for which the eigenvalue of energy is the one of the Es+ 0.5eV and the yellow

points are the pairs of values (Rb,B) for which the eigenvalue of energy is the one of the

Es− 0.5eV . Es= −11.92eV is obtained from reference [SF] . . . 56

3.25 Calculated band structure of Germanium, without the spin-orbit splitting, with the param-eters on Table 3.23 . . . 57 3.26 The pseudopotentials a) Vlocal,screen(kx), b) ∆Vnonlocal(kx)and c) ∆Vspinorbit(kx)of

Ger-manium are represented graphically, using the parameters of Table 3.27 . . . 58 3.27 The figure shows the different pseudopotential contributions Vlocal,screen(kx), Vlocal,screen(kx)+

∆Vnonlocal(kx)and Vlocal,screen(kx) + ∆Vnonlocal(kx) + ∆Vspinorbit(kx)of Germanium . . . 59

3.28 Calculated band structure of Germanium, a) using reference [ea] (LDA) and with the ex-perimental values of some important transitions in eV, b) with the program, using the parameters of Table 3.27 . . . 59

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3.29 It is represented a) the calculated density if states (blue line),the photo emission spec-troscopy and inverse photo emission data obtained from reference [Che89] (yellow line) divided by a factor of 3, b) the calculated joint density of states, c) calculated dielectric function, real part (blue) and imaginary (purple), d) calculated (blue line) and experimen-tal (yellow line, [AS83]) 1, e) calculated (blue line) and experimental (yellow line, [AS83])

2 f),g) calculated (blue line) and experimental (yellow line, [AS83]) reflectance for

Ger-manium with the local pseudopotential of equation (3.7), the non-local projector for the pseudopotential of equation (3.19) with the parameters written in Table 3.9 and the spin-orbit projectors using equations (2.74) and (3.25-3.27) with lmax= 2 . . . 61

4.1 The figure shows the a) band structure and the b) density of states, calculated for Silicon using the pseudopotentials obtained here and using a DFT-MGGA calculation . . . 64 4.2 The figure shows the a) band structure and the b) density of states, calculated for Carbon

using the pseudopotentials obtained here and using a DFT-MGGA calculation . . . 64 4.3 The figure shows the a) band structure and the b) density of states, calculated for

Germa-nium using the pseudopotentials obtained here and using a DFT-MGGA calculation . . . . 64 4.4 The figure shows the unfolded band structure of Si29C . . . 65

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Chapter 1

Introduction

1.1 Motivation

In micro and nanotechnologies of today, there is an high interest in semiconductor materials that have a direct gap that can be grown in silicon, since it is the material that is widely used in integrated circuits. We are ultimately interested in the study of materials based in super-lattices of Si-Ge with C impurities, and to simulate its optical properties.

We are interested in simulating cells with many atoms. There are already quite a few methods to do so, each one with its positive and negative points. The “reference” method for electronic structure calculations are done uses the Kohn-Sham equations [KS65] with the local-density approximation (LDA) [PW92, KS65] for the exchange-correlation energy and potential (as we will describe later on). However this calculations can lead to results with very bad agreement with experiment for the band gap of semi-conductors and insulators. For example, in Si, with LDA, the band-gap is predicted to be one half of its value, while in Ge the band gap is very small or even disappears.

There are more recent exchange and correlations potentials, like the Tran-Blaha functions [TB09], which gives improved band gaps for a variety of insulators and semiconductors.

In a condensed matter system an excited electron and the hole it left behind, referred to collectively as an exciton, move through a sea of all the other electrons and a background of the much heavier ions. The Bethe-Salpeter equation approach [SB51] describes the time evolution of that electron-hole pair. The GW approximation (GWA) is used to calculate the self-energy of a many-body system of electrons [AG00]. The approximation to be made is that the expansion of the self-enerfy ε in therms of the single particle Green’s function G and the screened Coulomb interaction W can be approximated to the lowest order term. Both the Bethe-Salpeter equation approach and the GW method can yield very accurate band gaps, but require very heavy calculations.

The Empirical Pseudopotential Method (EPM) relies on the experimental results to fit a set of pa-rameters used to describe the potentials that act on the electrons. This has the advantage that, if the programming is efficient, the calculations can be made very quickly, and therefore a very large number of atoms can be included in the simulation cell. There are although “dangers” in this method, since it

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is very tempting to use a model with a big number of parameters that fit very well to the experience but don’t have any physical meaning, since with a large number of parameters we can fit anything.

The method we are going to use is the an EPM with only a few parameters with physical meaning, to be fit to experiment. It is possible for the band gap to be adjusted very precisely.

There is no black box in this work! This means that everything is rederived from the beginning, since this is a research project with pedagogical purposes. For this reason, the software used is written MATHEMATICA, since the programs are closer to the mathematical equations.

1.2 State-of-the-art

Silicon and germanium are indirect band-gap semiconductors. For this reason they cannot emit light effectively. It would be highly desirable to have efficient optoelectronic devices that could be integrated with the usual Si technology. Different ideas are under discussion to meet this goal. One of the options are the strained superlattices of Silicon and Germanium. By using strained layers it is possible to over-come indirect band behavior by the folding back mechanism, which might allow the use of Si/Ge SL as a light emitting structure [Zak01].

There is a limit on the number of strainedlayers that can be accommodated on a given substrate -it is called the cr-itical thickness. For the case of germanium grown ep-itaxially on silicon, the maximum number of germanium monolayers which can be deposited is six. Recently, it was shown that the addition of carbon into Silicon and Germanium layers can be helpful to eliminate this problems. Adding an element with a much smaller radius than that of silicon to a layer containing Ge atoms (= rSi= 1.17 ˚A,

rGe = 1.22 ˚A, but rC = 0.77 ˚A) also gives the possibility to manipulate the strain, helping as well with

thermal stability. [Pea87]

1.2.1 Si, Ge and C

Silicon, Germanium and Carbon are in the Group IV of the periodic table. They all crystalize in the diamond crystal structure (Figure 1.1) which has a face centered cubic Bravais lattice with a two-atom basis, with lattice constants a = 5.43 ˚A, a = 5.66 ˚Aand a = 3.57 ˚Afor Silicon, Germanium and Carbon, respectively. Carbon also appears as graphite, which is more stable at ambient conditions.

The band structure of Silicon and Germanium was calculated already for a lot of people. Here we show the one calculated in reference [TT93] (Figure 1.2) using a tight-binding model. For Silicon, the distance between the conduction-band minimum and the Γ point is equal to 0.89(2π/a), where a is the lattice constant. The values of the fundamental and direct gap for both Si and Ge are in Table 1.1 [TT93]. As we can see, we have for both Si and Ge indirect band gaps in their electronic structure. Silicon, being

Silicon Germanium

Direct 3.41 0.90

Fundamental 1.05 0.76

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Figure 1.1: The figure shows a) the diamond crystal structure and b) the Brillouin zone (BZ) of an fcc crystal lattice

Figure 1.2: The energy-band structure of a) Si and b) Ge are calculated with a tight-binding model. The top of the valence bands is set at zero energy. [TT93]

an indirect-gap semiconductor, is not used in photonics and optoelectronics.

1.2.2 Group IV Semiconductor Compounds and Alloys

When combining elements with different lattice constants, we have to take into account the effects of lattice mismatch and strain. Lattice mismatch happens when a compound is grown on a substrate with a different lattice constant. Both Si and Ge crystallize in the diamond structure, but their lattice constants differ by about 4.2%. As a result, the Si/Ge superlattices are under internal stress. This stress produces a distortion of the lattices and if the layers are too thick it generates dislocations to relieve that stress. For thin layers, the growth can be well behaved (pseudomorphic [TT93]), in which case the lateral lattice constant is the same in the Si and Ge layers and equal to that of the substrate.

When two semiconductors like Silicon and Germanium are put together, discontinuities can occur in the band structure. For a lattice matched interface (no strain), we need just to determine how the band structure of the two materials line up at the interface. When the materials are strained, we have two problems in the calculation of the band structure: Hydrostatic strain will produce additional shifts, and uniaxial or biaxial strain splits degenerate bands. Figure 1.3 shows the different contributions. Thus, the total change in a band can be expressed as [Ost98]

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∆E = ∆Ea+ ∆Eh+ ∆Es, (1.1)

where ∆Ea a stands for the material differences (for the unstrained case), ∆Eh is the shift due to

hydrostatic strain, and ∆Ess reflects the splitting due to biaxial strain. It should be noted that each one

of the contributions can have different signs, compensating one another.

Figure 1.3: Schematic representation of the band alignment between a substrate and a strained het-erolayer. The three contributions shown are (i) the “material effect” for the unstrained case ∆Ea , (ii)

the shift due to hydrostatic strain ∆Eh, and (iii) the splitting of a degenerated band due to biaxial strain

∆Es. [Ost98]

For epitaxial growth, the lattice constant along the growth axis is reasonably given by Poisson’s ratio ν = dεtrans

dεaxial, in which εtrans is the transverse strain and εaxial, the axial strain. The strain in each layer

will then be given by [TT93]

ε_||= a|| ai

− 1 ε⊥=

2ν

1 − νε||, (1.2)

in which ε||and ε⊥are the lateral and perpendicular strain, respectively, ai, a||, and a⊥are the equilibrium

(bulk) lattice constants of the strained material, of the substrate, and the lattice spacing perpendicular to the interface, respectively. The lattice constant parallel to the interface a||is the same along the structure

as long as the growth remains epitaxial.

Si and Ge heterostructures

Ultrathin (Ge)m/(Si)n [Sch91] strained-layer superlattices grown on Si substrates have attracted

con-siderable interest. Thus, the electronic and optical properties of these superlattices can be changed to specific needs. It is possible to obtain a direct or a quasidirect band gap based on two indirect semicon-ductors.

It is interesting the difference between tetragonal and orthorhombic symmetry. The orthorhombic symmetry occurs if the indices n and m are even. In Figure 1.4 [Sch91] we have the crystal structure of Si2Ge2 and its electronic structure on a Si substrate. The calculated lowest transition is indirect

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Figure 1.4: The unit cell of the (Ge)2/(Si)2superlattice and its band-structure are shown [Sch91]

Electroreflectance and photoluminescence experiments Ge/Si superlattices with n + m = 10 have stimulated interest in the possibility of the existance of quasidirect transitions in this material. The mini-mum of the conduction band occurs at 0.83X in Si. It is expected that it is folded back to k = 0 for a total periodicity of ten.

Besides the proper periodicity, another requirement has to be met in order to obtain a quasidirect transition; the proper strain distribution and thus a||. It is needed tensile strain in the Si layers to lower

the minimum of the twofold states (which are folded back to Γ) below those of the four other states. This is represented in Figure 1.5 [Sch91].

Figure 1.5: The figure shows a) electronic band structure of the (Ge)6/(Si)4 supperlattice on Si [001]

substrate and band structure of the free-standing b) (Ge)5/(Si)5c) (Ge)4/(Si)6superlattice along lines of

high symmetry [Sch91]

In Figure 1.6 the direct (EΓ−Γ) and the indirect transitions are plotted as a function of composition

and strain. The energy of the quasidirect EΓ−Γtransition remains constant for Si substrates. The biggest

variation of indirect gaps with composition is found for the EΓ−N gap. It shows an approximately linear

description with increasing Si content [≈ 0.14(m − n)eV ].

SixGe1−x alloys [New84]: Alloys of SixGe1−x have continuously variable lattice parameters and

band gap (although is indirect), and they have potential for practical applications. For example, they have been successfully used to create heterojunction bipolar transistors with cutoff frequencies bigger than 100 GHz, which is much higher than for the traditional Si ones. A problem facing Si-Ge technology

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Figure 1.6: Transition energies of various n + m = 10 superlattices s as a function of lateral strain in the Si layer are represented [Sch91]

is the mismatch that causes compressive strain in SixGe1−xlayers grown on Si. According to Vegard’s

law, in a SixGe1−xalloy, a||= xaSi+ (1 − x)aGe, in which aSiand aGeare the cubic lattice constants of

Si and Ge structures, respectively. The strain increases with increasing Ge concentration and the strain energy increases with Si-Ge film thickness as well.

The band structure for SixGe1−xalloys was calculated in reference [New84] (Figure 1.7)

Figure 1.7: Band structures E(~k) for the principal symmetry directions of the diamond lattice for (a) Si0.2Ge0.8and (b) Si0.74Ge0.26where calculated [New84]

The band gap, Eg, is indirect, with the valence-band maximum in the Γ point and the

conduction-band minimum changes from L [L = (2π/a)(1₂,1₂,1₂)]to near the point X [= (2π/a)(1, 0, 0)]. The change occurs at approximately x = 0.25 (for a temperature of 4K). It is this feature that makes the defect levels of this alloy interesting to study, since alloys with compositions near the x = 0.25 can possibly produce deep levels in the gap for impurities such as As and P [New84].

Near the band gap, every sp3 bonded impurity with a valence greater than that of tetrahedrally

bonded host by unity is expected to have an s-like level, a triply degenerate p-like deep level. In Figure 1.8 [New84] is shown the predicted single-impurity defect levels of p-like and s-like symmetries. Shown are also in these figures the conduction-band minima as functions of composition x where the zero of energy is taken to be the to of the valence band maxima for all x.

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Figure 1.8: The figures show the predicted single-impurity defect levels of a) T2and b) A1symmetry as

a function of the composition x. Shown are also the conduction and valence band edges [New84]

Magic sequence SiGe2Si2Ge2Si and the genetic algorithm [d’A12]: Using a genetic algorithm it

was identified the sequence of Si and Ge layers with strong transition across the electronic band-gap from amongst all possible superlattices [Sin0Gep0/Sin1Gep1/.../SinNGepN]∞including substrate

orienta-tion and strain.

It is used a efficient optimization method: A population of superlattices is “genetically” selected ac-cording to chance and their relative success, namely, their ability for light-emission at the band-edges. New superlattice candidates (offspring) are created from the previous population by interchanging ran-dom sets of layers in the superlattice between two parents (crossover), and by flipping ranran-dom Ge layers into Si layers and vice-versa in a single parent (mutation). At each generation, the worst individuals in the previous population are replaced by the best offspring, thus guiding the population as a whole to-wards the global optimum through survival of the fittest. To judge fitness, i.e., the strength of the optical transition, it is computed the dipole matrix element between the valence band minimum and conduction band maximum at Γ of each superlattice candidate, which is directly proportional to the strength of the optical transition.

The set of results are a variation of a magic sequence composed of α =SiGe2Si2Ge2Si followed

by a Germanium buffer layer of n = 12 − 32 monolayers. The magic sequence satisfies: wave vector directness and the dipole matrix element between the valence band maximum and the conduction band minimum is nonzero. The first condition is satisfied when the structure is grown on substrates Si1−xGex

with x ≥ 0.4 (Figure 1.9).

The second condition is shown by the spectrum of absorption in Figure 1.10 top, which also contains the spectrum of absorption for the superlattice Si6Ge4compared with the “magical” sequence.

They evaluated the effect of deviation of the best sequences in the “directness” of the band gap. By changing the substrate and the mutations in the magic sequence, it was constructed Figure 1.11. αn is

the magic sequence with a Ge buffer of n monolayers, while β is the sequence SiGe2Si2Ge2SiGe2SiGe9

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Figure 1.9: Band structures of the α12magic sequence grown on Si0.4Ge0.6 where calculated [d’A12]

Figure 1.10: The figure shows the comparison between Si6Ge4superlattice and the magic sequence of

the direct absorption spectra [d’A12]

Figure 1.11: Direct dipole-allowed band-gaps as calculated by [d’A12]

Si and Ge heterostructures with C impurities

A possible solution to the mismatch problem is the incorporation of C which has a lattice constant of 3.57 ˚Ain a Diamond crystal structure, which is much smaller than those of Si and Ge. Incorporation of C into SiGe material should reduce the lattice mismatch because of the smaller size of C, compensating for the larger size of Ge. The linear approximation for the lattice constant between Si, Ge, and diamond is [Ost98]

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a(x, y) = (1 − x − y)aSi+ xaGe+ yaC, (1.3)

resulting in a Ge:C ratio of 8.2 for strain compensation. The incorporation of a third component also adds additional flexibility in band-gap engineering. This could pose a challenge to the GaAs technologies. We define an “effective lattice-mismatch” mfeff as [Ost98]

mfeff =a(x, y) − a_a_Si Si, (1.4)

for ternary Si1−x−yGexCy on Si(001) substrates. A positive mfeff that the material is compressively,

mfef f < 0is tensile strain, and mfeff = 0 means strain-compensated Si1−x−yGexCyalloy. The

hydro-static contribution is [Ost98]

∆Eh= av,c(ε⊥+ 2ε||), (1.5)

where av,c is the appropriate hydrostatic deformation potential for the valence or conduction band,

re-spectively. For the material dependent term ∆Ea,

∆Ea(x, y) = ∆Ea(x) + ∆Ea(y), (1.6)

Figures 1.12, 1.13 and 1.14 summarize the results for the offsets of strained Si1−x−yGexCyon Si(001)

[Ost98]. The effective concentration corresponds to the concentration needed for identically strained binary layers.

Figure 1.12: Valence-band offsets for compressive strained Si1−xGex , and Si1−x−yGexCy (x =10%,

20%, and 30%) and tensile strained Si1yCy and Si1−x−yGexCy(y =1%, 2%, and 3%) are plotted as a

function of the effective lattice mismatch (expressed in “effective” Ge or C concentrations, respectively) [Ost98]

The band-gap narrowing is obtained by adding the valence and the conduction-band offsets. The band gap for the alloys is always smaller than that of silicon. The addition of C (Ge) into compressive strained Si1−xGex (tensile strained Si1−yCy) leads to a smaller change in band-gap narrowing than in

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Figure 1.13: Conduction-band offsets for compressive strained Si1−xGex, and Si1−x−yGexCy(x =10%,

20%, and 30%) and tensile strained Si1yCy and Si1−x−yGexCy(y =1%, 2%, and 3%) are plotted as a

function of the effective lattice mismatch (expressed in “effective” Ge or C concentrations, respectively) [Ost98]

Figure 1.14: The band-gap narrowing for ternary Si1−x−yGexCy alloys, strained on Si(001) is

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1.3 Thesis Outline

This thesis is organized as follows:

In chapter 2 we will make a theoretical introduction with the physics needed to this project. Starting by the by the many body problem, going through the Density Functional Theory (DFT) justify the use of an independent electron approximation for the problem. We use pseudopotentials, so we explain what is a pseudopotential, using not only local pseudopotentials but also the non-local (important in Carbon) and spin-orbit contributions (important in the heavier Germanium) that are needed to fully describe bulk group IV elements in question.

In chapter 3 we describe the search for the best pseudopotentials fitted to the experiment. We start from a previous description used for Silicon with only a local pseudopotential. After we improve the description by adding a non-local contribution to the pseudopotential. We search for the best fitted local and non-local parts of the pseudopotential to the experiment and calculate the optical properties. The same search is done for Carbon. For Germanium we add a spin-orbit contribution and find as well the best fitted potentials (local, non-local and spin-orbit) to the experimental results.

The software used is MATHEMATICA. It is good for the development and learning process, but has limitations such as the computation time increases highly with the precision requested for the calculation. In chapter 4 the pseudopotentials developed here are used in a FORTRANprogram, where numerically converged and obtained in a resonable amount of time, in particular for large simulation cells. An indication of future research was also made.

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Chapter 2

Theoretical Introduction

We want to calculate the properties of a solid - a condensed matter system. This task can be quite complicated, if not impossible to solve. Most of the time we have to use approximations to calculate the properties of the system, which means we have always to keep in mind those approximations and what they change in the final result. We start from the beginning of the problem: the many body problem, because that is what we have, a solid with a big number of electrons and nucleus. Afterwards we are going to describe the independent electron approximation with the use of pseudopotentials which is reasonable to describe solids. What is here written, from section 2.1 to 2.7, is based on the notes of reference [Mar].

2.1 The many-body problem

Consider a system of m electrons and n nuclei, each one with spatial coordinates ~ri and ~Rj and spin

ξi and Ξj. In the non-relativistic limit, the time-independent Schr ¨odinger equation that can be used to

calculate the properties of the system is

HΦ(~r1, ξ1, . . . , ~rm, ξm, ~R1, Ξ1, . . . , ~Rn, Ξn) = EΦ(~r1, ξ1, . . . , ~rm, ξm, ~R1, Ξ1, . . . , ~Rn, Ξn), (2.1)

where the many particle Hamiltonian is

The subscripts in the Laplacian ∇2 _{indicate on which coordinate they operate. M}

i and Zi are the

mass and the charge, respectively, of the ith nucleus. The terms in the Hamiltonian are, respectively, the kinetic energy of the nuclei and the electrons, the potential energy related to the potential caused by the nuclei and felt by the nuclei, the potential caused by the electrons and felt by them and the potential caused by the ions and felt by the electrons (or vice versa). Notice that it is written in an adimentional form, adequate to computational purposes. These units are called atomic units, a system in which the

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numerical values of following fundamental physical constants are all unity by definition: the electron mass, me, the elementary charge, e, the reduced Planck’s constant ~ = _2πh and the Coulomb’s constant

1

4π0, where 0is the dielectric permittivity of vacuum. In this work, all the equations will be written in the

atomic system of units unless it is said otherwise.

Since electrons are fermions, the wavefunction is anti-symmetric with respect to the electron coordi-nates, position and spin (Pauli’s exclusion principle),

Φ(. . . , ~ri, ξi, . . . , ~rj, ξj, . . .) = −Φ(. . . , ~rj, ξj, . . . , ~ri, ξi, . . .). (2.3)

An analytical solution for the equation (2.1) with the Hamiltonian (2.2) is only known for very simple systems such as the Hydrogen atom and the H2+ion, and numerical solutions can be obtained for atoms

and molecules with a small number of electrons such as the He atom or the H2molecule. These are

very simple systems. No such miracle is to be expected for larger systems. If we could solve the exact Hamiltonian for these more complicated systems we could in principle predict all of it’s properties. In a macroscopic solid, there are about 1023nuclei and a similar number of electrons [All10]. The equation (2.1) to be solved would have something in the order of 1023 _{variables, which is not possible with the}

current computational technology.

Therefore we are compelled to make approximations motivated by physical considerations, such as those described in the following sections.

2.2 The Adiabatic Approximation

The first approximation to be made takes into account that the mass of the nuclei is much larger (gen-erally 104_{− 10}5 _{times [All10]) than the mass of the electrons, M}

i me. Therefore we can say that

the nuclear motion is much slower (or say the nuclei are fixed) than the motion of the electrons. In this way we can neglect their Kinetic Energy, obtaining the Hamiltonian Hefor the electronic problem, which

depends on the nuclear coordinates,

and whose eigenvalues also depend on the position and spin of the nuclei and form a family identified by the quantum number (k). This is called the Born-Oppenheimer or adiabatic approximation [BH88]. The energy U(k) _{can be considered as the potential energy in the Hamiltonian used in the Schr ¨odinger}

equation that describes the motion of the nuclei

H_N(k)= n X i=1 − 1 2Mi ∇2_R~ i+ U (k) ( ~R1, . . . , ~Rn)

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H_N(k)χ(k,q)( ~R1, . . . , ~Rn) = E(k,q)χ(k,q)( ~R1, . . . , ~Rn), (2.5)

where the quantum number q concerns the vibrational, rotational and translational states. We say that the total wave function is the product of the solutions of the equations (2.4) and (2.5),

Θ(k,q)(~r1, . . . , ~rm; ~R1, . . . , ~Rn) ' Ψ(k)(~r1, . . . , ~rm; ~R1, . . . , ~Rn)χ(k,q)( ~R1, . . . , ~Rn), (2.6)

which is a complete set expansion of the wavefunction Φ = P

k,qck,qΨ

(k)_χ(k,q)_{. The decoupling of the}

electronic and nuclear motions can also be obtained using perturbation theory.

2.3 Separable Schr ¨

odinger equation

But the problems in the equation (2.2) are almost the same as the ones in equation (2.4) since we still have a number of electrons in the order of 1023 _{which interact with one another through this Coulomb}

interaction and thus we cannot separate the Schr ¨odinger equation in its different variables. We could neglect the Coulomb interaction but unfortunately we know this is a bad approximation. Another solution is to consider that a given electron is subjected by a potential depending on the average distribution of the electrons. This results in the Hamiltonian [All10]

Hs = m X j=1 −1 2∇ 2 ~ rj+ m X j=1 n X i=1 Vat(~rj− ~Ri) = m X j=1 " −1 2∇ 2 ~ rj + n X i=1 Vat(~rj− ~Ri) # = m X j=1 Hj, (2.7)

where Vat includes the average Coulomb electron-electron and the electron-nucleus interaction. The

nucleus-nucleus term is dropped, since it is a constant for the same position of the nuclei. Doing this, it is possible to separate the Hamiltonian into a sum of m independent terms, each acting on a different coordinate.

2.4 The Hartree-Fock Method

We can construct and anti-symmetric wave-function using a Slater determinant of orthonormal one electron spin orbitals, φi(~r), i = 1, ..., m,

D(~r1, . . . , ~rm) = 1 √ m! φ1(~r1) φ1(~r2) . . . φ1(~rm) φ2(~r1) φ2(~r2) . . . φ2(~rm) .. . ... . .. ... φm(~r1) φm(~r2) . . . φm(~rm) , (2.8)

in which hφi| φji = δij. In this case we can show that hD| Di = 1.

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wavefunction Ψ0of the many-body system, E0= min hΨ | H | Ψi hΨ | Ψi , E0= hΨ0| H | Ψ0i hΨ0| Ψ0i . (2.9)

In the Hartree-Fock method we search the wavefuntions in the form of Slater determinants,

EHF = minhD | H | Di, EHF = hDHF | H | DHFi, (2.10)

where we always have that EHF ≥ E0. To minimize the energy in equation (2.10) the Euler-Lagrange

method can be used,

δ δφ∗ j h hD | H | Di −X ij λij hφi| φji − δij i = 0, (2.11)

where λij are the Lagrange multipliers. The expectation value of the energy for the Slater determinant

that is a sum of five contributions: kinetic, external potential, ion-ion, Hartree and exchange. The ion-ion contribution is a constant. We can define the external ionic potential,

v(~r) =X

j

−Zj

|~r − ~Rj|

, (2.13)

and the total electronic charge density,

ρ(~r) = hD |X i δ(~ri− ~r) | Di = X j φ∗_j(~r)φj(~r). (2.14)

With this definitions we can rewrite the external potential and the Hartree contributions,

Z v(~r)ρ(~r)d3r, 1 2 Z Z _ρ(~_r)ρ(~_r0₎ |~r − ~r0_| d 3_rd3_r0_. (2.15)

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h −1 2∇ 2_{+ v(~}_{r) +}Z ρ(~r0) |~r − ~r0_|d 3_r0i_φ i(~r) − X j φj(~r) Z φ∗_j(~r0) 1 |~r − ~r0_|φi(~r 0_)d3_r0 ₌X j λijφj(~r). (2.16)

Knowing that the Slater determinants are invariant with respect to unitarian transformations Ψi →

Ψ0_j=P

iUijΨi, we can use this do diagonalize (2.16), obtaining

h −1 2∇ 2_{+ v(~}_{r) +}Z ρ(~r0) |~r − ~r0_|d 3_r0i_φ i(~r) − X j φj(~r) Z φ∗_j(~r0) 1 |~r − ~r0_|φi(~r 0_)d3_r0₌ iφi(~r). (2.17)

The correlation energy Ec = E0− EHF is the difference between the exact energy of a system

and the Hartree-Fock energy. This equation is a non-linear eigenvalue differential integral equation in 3 dimensional space. The Hartree-Fock method provides a connection between the many body wavefuntion to m one body wavefunctions.

2.5 Density Functional Theory

Density functional theory is a method to investigate the electronic structure in the ground state of atoms, molecules or condensed matter systems. It says that the properties of a many-electron system can be uniquely determined by the ground state electron density of the system, that depends on 3 spatial coordinates.

But before we have to make sure that the properties of the system can be indeed be uniquely asso-ciated to a determined electron density (ilustration in the Venn Diagram in Figure 2.1).

Figure 2.1: The figure shows the Venn’s diagram corresponding a potential in the space of all potentials V (~r)to a ground state electron density in the space of the ground state electron densities ρGS(~r)

If it is univocal, in principle, by knowing the electron density, we can obtain the potential that acts on this electrons and from that we can calculate the ground state eigenfunctions and all the other properties of the system.

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First let us assume that two different potentials can lead to the same ground state electron density, V1(~r) - Ψ1(~r) _XX XXXz V2(~r) - Ψ2(~r) : ρGS(~r), and consider the Hamiltonian

H = T + Vee+ Vext, (2.18) where, T =P i− 1 2∇ 2 ~ ri Vee =P_i<j_|~_r 1 i−~rj| Vext =Piv(~ri), (2.19)

are the Kinetic, Coulomb Potential and external Potential energies, respectively and v(~ri) is given by

equation (2.13). We calculate de ground state energy using V1,

EGS1 = hΨ1| T + Vee+ V1|Ψ1i

= hΨ1| T + Vee|Ψ1i +

Z

V1(~r)ρGS(~r)d3r. (2.20)

Using the variational principle,

But starting with the calculation of EGS2and following the same line of thinking we get

EGS2< EGS1+

Z

(V2− V1)(~r)ρGS(~r)d3r, (2.23)

which results with,

EGS1 > EGS2+ Z (V1− V2)(~r)ρGS(~r)d3r EGS1 < EGS2+ Z (V1− V2)(~r)ρGS(~r)d3r, (2.24)

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that is a contradiction. We reach then the conclusion that in the absence of degeneracies, two different potentials cannot lead to the same electron density in the ground state. This means that any property of the many-body system is a functional of ρ(~r). From that comes the name Density Functional Theory.

2.6 Hohenberg-Kohn Theorem

Considering a system of m electrons, the electronic Hamiltonian can be described by Equation (2.18), we define the set of all normalized anti-symmetric wavefunctions as

A = {Ψ | Ψ(. . . , ~ri, . . . , ~rj, . . .) = −Ψ(. . . , ~rj, . . . , ~ri, . . .) and hΨ | Ψi = 1}. (2.25)

We can define the ground state energy of the system as a functional of the external potential Vext(~r)

as

E[Vext] = min

Ψ∈AhΨ | H | Ψi. (2.26)

The subset Aρof A is the set of all wavefunctions that correspond to the charge density ρ(~r),

Aρ= {Ψ | Ψ ∈ A and m

Z

d3r2. . .

Z

d3rm|Ψ(~r, ~r2, . . . , ~rm)|2= ρ(~r)}. (2.27)

An universal functional of the charge density is [Lie83]

F [ρ] = min

Ψ∈Aρ

hΨ | T + Vee| Ψi, (2.28)

which is independent of Vext(~r). The ground state energy can be calculated as a functional of the

electron density ρ for each potential Vext,

E0= EGS[ρ] = min

Ψ∈AhΨ| T + Vee+ Vext|Ψi

= min ρ Ψ∈Aminρ hΨ| T + Vee+ Vext|Ψi = min ρ min Ψ∈Aρ

(hΨ| T + Vee|Ψi + hΨ| Vext|Ψi)

, (2.29)

in which hΨ| Vext|Ψi =R Vext(~r)ρ(~r)d3r, so

EGS[ρ] = min ρ Z Vext(~r)ρ(~r)d3r + min Ψ∈Aρ hΨ| T + Vee|Ψi = min ρ F [ρ] + Z Vext(~r)ρ(~r)d3r = min ρ EVext[ρ]. (2.30)

Therefore Hohenberg-Kohn theorem states that the minimum of the energy functional EVext is the

ground state energy E0 of the system. Note that contrary to the previous proof, we did not require the

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2.7 Kohn-Sham Equations

Consider the set of one electron wavefunctions φiand occupation numbers fithat have a charge density

equal to ρ , Bρ= {(f1, . . . , fk, φ1, . . . , φk) | hφi| φji = δij, k X i=1 fi|φi(~r)|2= ρ(~r) fi∈ [0, 1]}, (2.31)

where some authors define 0 ≤ fi ≤ 1, and we define an energy functional of the charge density ρ that

it is called kinetic energy functional but it is not the true kinetic energy of the system

T0[ρ] = min (f1,...,fk,φ1,...,φk)∈Bρ k X i=1 fihφi| − 1 2∇ 2_{| φ} ii. (2.32)

The exchange and correlation functional energy is defined as

Exc[ρ] = F [ρ] − 1 2 Z Z _ρ(~_r)ρ(~_r0₎ |~r − ~r0_| d 3_rd3_r0_{− T} 0[ρ], (2.33)

where F [ρ] was defined in equation (2.28). It contains the rest of the many-body contributions to the energy. Also since T0[ρ]is not the exact kinetic energy of the interacting energy but instead the kinetic

energy of the ground state of a system of non interacting electrons with density ρ(~r), Exc[ρ] contains

also kinetic energy terms. Rewriting it in a different way, we have

F [ρ] = T0[ρ] + 1 2 Z Z _ρ(~_r)ρ(~_r0₎ |~r − ~r0_| d 3_rd3_r0_{+ E} xc[ρ]. (2.34)

We can calculate the energy functional,

EVext[ρ] = F [ρ] + Z Vext(~r)ρ(~r)d3r = T0[ρ] + Z Vext(~r)ρ(~r)d3r + 1 2 Z Z _ρ(~_r)ρ(~_r0₎ |~r − ~r0_| d 3_rd3_r0_{+ E} xc[ρ], (2.35)

and the ground state energy,

EGS[ρ] = min ρ EVext[ρ] = min ρ T0[ρ] + Z ρ(~r)Vext(~r)d3r + 1 2 Z Z _ρ(~_r)ρ(~_r0₎ |~r − ~r0_| d 3_rd3_r0_{+ E} xc[ρ] = min ρ min (f1,...,fk,φ1,...,φk)∈Bρ k X i=1 fihφi| − 1 2∇ 2_{| φ} ii + Z ρ(~r)Vext(~r)d3r+ +1 2 Z Z _ρ(~_r)ρ(~_r0₎ |~r − ~r0_| d 3_rd3_r0_{+ E} xc[ρ] = min ρ (f1,...,fk,φmin1,...,φk)∈Bρ Xk i=1 fihφi| − 1 2∇ 2_{| φ} ii + Z ρ(~r)Vext(~r)d3r+ +1 2 Z Z _ρ(~_r)ρ(~_r0₎ |~r − ~r0_| d 3_rd3_r0_{+ E} xc[ρ] , (2.36)

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but this is the same as minimizing over all wavefunctions and therefore, EGS[ρ] = min (f1,...,fk,φ1,...,φk) _Xk i=1 fihφi| − 1 2∇ 2_{| φ} ii + Z ρ(~r)Vext(~r)d3r + 1 2 Z Z _ρ(~_r)ρ(~_r0₎ |~r − ~r0_| d 3_rd3_r0_{+ E} xc[ρ] . (2.37) The Euler equation that minimizes the expression above with respect to the one electron wavefunc-tions φiis δ δφ∗_i Xk i=1 fihφi| − 1 2∇ 2_{| φ} ii + Z ρ(~r)Vext(~r)d3r + 1 2 Z Z _ρ(~_r)ρ(~_r0₎ |~r − ~r0_| d 3_rd3_r0 +Exc[ρ] − X i,j λi,j(hφi| φji) − δij = 0, (2.38)

with Vextgiven by Equation (2.19), is called the Kohn-Sham equation. The result of the minimization is

−1 2∇ 2_{+ v(~}_{r) + v} H(~r; ρ] + vxc(~r; ρ] φi(~r) = iφi(~r), vH(~r; ρ] = Z _ρ(~_r0₎ |~r − ~r0_|d 3_r0_, vxc(~r; ρ] = δExc[ρ] δρ(~r) , ρ(~r) = k X j=1 fjφ∗j(~r)φj(~r), (2.39)

where vH is the Hartree potential and vxc is the exchange and correlation potential. The curve

paren-thesis in (~r; ρ]means that the function is dependent of a variable, and the square parenthesis indicate a functional dependence. Although the equation resembles the Schr ¨odinger equation for non-interacting particles, the dependence of vH and vxcin the charge density ρ makes it a non-linear system of

equa-tions. A common approximation is the Local Density Approximation (LDA) that says that the exchange and correlation energy depends only on the charge density in the point of interest vxc(~r) = vxc[ρ(~r)].

The way to solve this kind of equations is through iterative, self-consistent methods. The kind of logic is the following:

1. Guess initial ρin

2. Calculate vH[ρin]and vxc[ρin]

3. Solve the Kohn-Sham equation 4. Calculate ρout=P_i|φi(~r)|2

5. If ρin∼ ρoutstop. If not ρin= F (ρin, ρout)and start from 2.

Although it is a highly used method, the DFT with LDA gives the wrong value for the band gap, for example almost half of the value for the band gap of silicon.

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2.8 Pseudopotentials

The pseudopotential model describes a periodic solid as a sea of valence electrons moving in a back-ground of cores. The space can be divided into two regions: the region near the nuclei, the “pseudized core” composed primarily of tightly bound core electrons which are not very affected by the neighbour atoms; and the valence electron region which is involved in bonding the atoms together. The pseu-dopotential only acts on the valence electrons. This results that the atoms in the same group - such as Carbon, Silicon and Germanium (group IV, for ex.) - are treated in mostly the same way, apart from a few “details”. The focus of the calculation is only on the accuracy of the valence electron wavefunction away from the core. The potential in the ion core is strongly attractive for the valence electrons, but the requirement for the valence wavefuntions to be orthogonal to those of the core contributes to an effec-tive repulsive potential for valence states. This results in a net weakly attraceffec-tive potential that affects the valence electrons.

2.8.1 Ab initio pseudo-potentials

We first consider an atom of atomic number Z. An one-electron Hamiltonian can be written as [CC92]

H = 1 2∇

2_{+ V}

ion+ Vscr, (2.40)

where Vion = −Z_r is the ion core potential, that can be taken as a linear superposition of spherical

potentials, and Vscris the screening potential, a potential very important in many body physics. Usually

it is divided into two parts (as it was seen before), the Hartree potential, VH(~r, ρ], that comes from the

Poisson’s equation

∇2VH= −4πe2ρ(~r), (2.41)

where ρ(~r) is the valence electron charge density. The other part is the exchange and correlation potential Vxc, that was also mentioned before. If we use the Local Density Approximation (LDA), then

Vxc(~r) = Vxc[ρ(~r)]. The total potential is, thus

VT(~r) = Vion(~r) + VH+ Vxc(~r). (2.42)

If there is one state for which we know the the wavefuntion and the value of the energy we can invert the Schr ¨odinger equation to obtain the total potential [CC92]

VT =

1 2

∇2_Ψ

Ψ + E. (2.43)

This equation is well behaved if Ψ is nodeless, since it is highly preferable for the pseudopotential to be smooth and the wiggles associated with the nodes are undesirable. The quantity ∇2Ψ

Ψ is extremely

sensitive to numerical errors when Ψ → 0. If there are no numerical errors, what normally happens is that if Ψ → 0, then ∇2_{Ψ → 0}_{as well. If there is an error and ∇}2_Ψ_{doesn’t go to zero when Ψ → 0, this}

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quantity will diverge.

In an atom, we can extract the energy levels of interest by performing an atomic structure calculation starting from all electron atomic calculations. Within the density-functional theory this is done by assum-ing a spherical screenassum-ing approximation and self-consistently solvassum-ing the radial Kohn-Sham equation [TM91] −1 2 d2 dr2 + l(l + 1) 2r2 + VT(~r, ρ] rRnl(r) = εnlrRnl(n), (2.44)

that results in the “all electron” wavefuntions and energies. We have to take into account that we are going to perform an inversion of the Schr ¨odinger equation, which is only well behaved if the wavefuntions used have no nodes (see eq. (2.43)). This can be achieved by the construction of pseudo-wave functions with no nodes (for this reason, the quantum number n will be omitted in the further calculations) based on the wavefunctions of the equation (2.44) as it was done successfully in reference [TM91]. Other char-acteristics of this pseudo-wave funtion are [TM91]: the normalized atomic radial pseudo-wave-function with angular momentum l is equal to the normalized radial all-electron wave function after a cutoff radius rcl,

R_lP P(r) = RAE_l (r) for r > rcl; (2.45)

the charge enclosed for the two wavefuntions within rclmust be equal,

Z rcl 0 |RP P l (r)|2r2dr = Z rcl 0 |RAE l (r)|2r2dr, (2.46)

so that the norm of the wavefunction is conserved after normalization; and the valence all-electron and pseudopotential eigenvalues must be equal,

εP P_l = εAE_l . (2.47)

A pseudopotential under this conditions is called a “norm-conserving pseudopotential”. Once we have the pseudo-wave function we can calculate the pseudopotential by inversion of the Schr ¨odinger equation [TM91], V_lP P = εl− l(l + 1) 2r2 + 1 2rRP P l (r) d2 dr2[rR P P l (r)]. (2.48)

By inverting the Schr ¨odinger equation for each of the wavefunctions separately, we get with differ-ent potdiffer-entials for each quantum number l, Vl. This is called non-locality of the pseudopotential. The

pseudopotential is decomposed into a sum over angular momentum components [CC92],

VT = V0P0+ V1P1+ V2P2+ ..., (2.49)

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P`= |`mi h`m| , (2.50)

where h~r| `mi = Y_`m∗ (Ω) and Y`m∗ (Ω) is centered on the origin. Another complication is to take into

account the spin orbit effects in heavier elements (like Ge). Non-locality and spin-orbit considerations will be further developed in later sections.

2.8.2 Empirical Pseudopotential methods

The empirical pseudopotential method relies on experimental results for the construction of the pseu-dopotential and the predictions made with the pseupseu-dopotentials should converge as best as possible with experience.

Lets assume first that the pseudopotential is local, i.e., independent of `. The Schr ¨odinger equation for a periodic system is [Che96]

−1 2∇ 2_{+ V (~}_r) ψ~_k(~r) = E(~k)ψ~_k(~r). (2.51)

In a crystal, the potential V (~r)is periodic in the lattice. We can use a plane wave expansion that will only have plane waves with the periodicity of the lattice. With the local approximation, the general approximation for the pseudopotential is

V (~r) =X ~ G V ( ~G)S( ~G)ei ~G·~r=X ~ G U ( ~G)ei ~G·~r, (2.52)

where ~Gis a reciprocal lattice vector, V ( ~G)are the form factors and S( ~G)is the structure factor,

S( ~G) = 1 Na Na X i=1 ei ~G·~τi_. _(2.53)

Once the form factors are chosen, we can solve (2.51). We can assume that the wave functions ψ~k(~r)can be expanded in plane waves, with no loss of generality and solve the secular equation, which

is the Schr ¨odinger equation (2.51) in the reciprocal space [Che96, AM76],

det |H(~k, ~G − ~G0) − E(~k)I| = 0, (2.54)

where H(~k, ~G − ~G0) = 1 2(~k − ~G) 2_δ ~ G, ~G0 + V ( ~G − ~G 0_{)S( ~}_{G − ~}_G0_). (2.55) The form factors depend only on the magnitude of | ~G − ~G0| if the pseudopotential can be taken as spherically symmetric, which is generally the case for tetrahedral semiconductors [Che96], with

~

G discrete. In the Chelikowsky pseudopotential [Che96], for diamond or zinc-blende semiconduc-tors, generally only three form factors are enough to determine the pseudopotential, those for G2 ₌

3 2π_a 2

, 8 2π_a2

, 11 2π_a2

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en-ergy and S( ~G) = 0for G2 _{= 4} 2π a

2

, 12 2π_a2. So of the six smallest reciprocal lattice vectors lengths, only the form factors G2 _{= 3} 2π

a

2

, 8 2π_a 2, 11 2π_a2 are required to specify the crystalline potential [CC92][Che96] (Figure 2.2). These three values are fitted to optical transition energies and the whole band structure follows from them. The method is similar with the one discussed to DFT:

Figure 2.2: The figure shows a schematic plot of a pseudopotential in reciprocal space with the G’s that correspond to G2_{= 3, 8, 11}_{with G}2_{in units of (2π/a)}2_[CC92]

1. Estimate initial V ( ~G) 2. Solve secular equation

3. Calculate band structure and optical properties 4. Compare with experiment

5. If it agrees with experiment stop. If not change V ( ~G)and start from 2.

In reference [WZ95], a semi-empirical pseudopotential is used. First an ab initio method is used, in which spherical atomic potentials (with only the local part) vα_{(r), such that the solutions of}

( −1 2∇ 2_{+ V} nonlocal(~r) + X α X Rα v(α)(|~r − ~Rα|) ) ˜ ψi= ˜iψ˜i, (2.56)

where α is the chemical atomic type and ~Rαstands for all possible atomic positions of α, including those

related by lattice translations, will have large overlaps with the LDA solutions ψi and i, so that they

reproduce the LDA results for bulk systems with a good approximation. This means that it will suffer from poor reproduction of the observed optical energies. Those authors chose a potential described in the reciprocal space by

v(α)(q) =X

i

Cα(n)e−(q−an)2/b2n_, _(2.57)

where C(n), an and bn are free parameters (the author used 20 of each), to be adjusted, this time

empirically, to reproduce the experimentally observed excitation energies. Notice the big number of parameters used in the fit.

Electronic Structure of Heterogeneous Materials Application to Optical properties