Atomistic Modeling, reconstructions and surface facets of GaAs(112)B surface: structural and electronic properties by ab initio calculations

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1 Materials Physics and Technology Postgraduate Course Department of Physics - Aristotle University of Thessaloniki Atomistic Modeling, reconstructions and surface facets of GaAs(112)B surface: structural and electronic properties by ab initio calculations Post graduate student: Paul Mouratidis Supervisors: Assist. Prof. Joseph Kioseoglou Professor Philomela Komninou Aristotle University of Thessaloniki

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3 I would like to dedicate this work to my family, Mom Helen, Sister Helga and especially to my lovely Aunt Victoria, without their support this work would have never seen the light of day iii

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5 Acknowledgments I would like to thank all the group of Nanostructured Materials Microscopy Group (NMMG) and especially my supervisors Assist. Prof. Joseph Kioseoglou and Professor Philomela Komninou for providing direction and assistance through the course of my postgraduate studies. I should also thank Assoc. Prof. Thomas Kehagias and Assoc. Prof. George Dimitrakopoulos. Especially thanks to PhD student Theodoros Pavloudis for encouraging me when things went wrong, all my fellow students from the postgraduate course and especially to Thanasis Goudaras. v

6 Abstract The purpose of this study is the construction and analysis of GaAs (112)B surface as well as the growth mode of InAs QD s on this surface. The theoretical model predicts that the surface (112)B of GaAs is unstable and consists of the more stable {(111)B, (101), (011), (113)B or (123)B, (124)B, (214)B} surfaces with two possible shapes of pyramidal structure. The computational simulations of surfaces for the lattice constants, reconstructions, band structures and density of states, are calculated with the ab initio AIMPRO package using the LDA Hurtwigsen-Goedecker-Hutter pseudopotentials and Gaussian orbitals as its wavefunctions basis sets. The aim is the reconstruction and relaxation of the surfaces and not the bulk material, so we need to add a vacuum in the vertical direction of the surfaces. The structures of those surfaces emerge to be all orthorhombic with the surface on the top of the orthorhombic box. So if we want to run the slab mode of AIMPRO we need to discharge the bottom surface of the orthorhombic box and wipe out all the dangling bonds which will be the cause of interaction between the top and the bottom side of the box. An exception is the case where the top and bottom side of the box are completely symmetric. The well-established method that we followed is to use pseudohydrogen atoms with a fractional charge H 1.25 and H 0.75 in order to saturate both Ga and As dangling bonds respectively. It is known that the group-iii cation that in our case is Ga atom contributes 3/4e - to a covalent bond with the As atom, so each dangling bond needs a H 1.25 pseudohydrogen. On the contrary the group-v anion that in our case is As atom contributes 5/4e - to a covalent bond with the Ga atom and needs a H 0.75 pseudohydrogen. Through the conditions that reported previously, we performed the optimize run of the AIMPRO on slab mode for each 2x2 cleaved surfaces {(111)B or (100)B, (101), (011), (113)B or (123)B, (124)B, (214)B} of GaAs (112)B structure. The value of vacuum we used was 12 Å in the vertical direction of the surface and each box almost consists of 10 interlayers of atoms. We have set the k-point grid, as 6 6 1, whereas the direction z is perpendicular to the surface and the primitive vector along this direction was increased by the value of vacuum 12 Å. So in this direction we do not have periodic conditions and this is the reason that we set the grid mesh equal to 1 by the direction z. vi

7 Περίληψη Ο στόχος της παρούσας μελέτης είναι η κατασκευή και η ανάλυση της επιφάνειας (112)Β του GaAs καθώς και ο τρόπος ανάπτυξης των κβαντικών τελειών InAs πάνω σε αυτήν. Το θεωρητικό μοντέλο της μελέτης προβλέπει πως η (112)Β επιφάνεια του GaAs είναι ασταθής και αποτελείται από τις σταθερές επιφάνειες {(111)B, (101), (011), (113)B ή (123)B, (124)B, (214)B} με δύο πιθανές πυραμιδικές δομές. Οι υπολογιστικές προσομοιώσεις πρώτων αρχών των προαναφερθέντων επιφανειών για τον υπολογισμό των πλεγματικών σταθερών, την αναδόμηση, αφηρέμηση, τα ενεργειακά χάσματα καθώς και οι πυκνότητες καταστάσεων έγιναν με τον κώδικα του ab initio AIMPRO package χρησιμοποιώντας LDA Hurtwigsen-Goedecker-Hutter ψευδό-δυναμικά με Gaussian τροχιακά. Καταρχήν από την βελτιστοποίηση του πλέγματος υπολογίστηκαν οι βελτιστοποιημένες πλεγματικές σταθερές της συμπαγής δομής του GaAs και InAs οι οποίες είναι Α και Α αντίστοιχα. Με τις τιμές των βελτιστοποιημένων πλεγματικών σταθερών και ορίζοντας τέσσερα σημεία υψηλής συμμετρίας στην πρώτη ζώνη Brillouin του fcc πλέγματος L( ), Γ(0 0 0), X(1 0 0), K( ) έγινε η κατασκευή του ενεργειακού διαγράμματος των GaAs και InAs από τα οποία υπολογίστηκε το ενεργειακό χάσμα στο σημείο Γ μεταξύ της ζώνης σθένους και της ζώνης αγωγιμότητας, Eg=0.5 ev για το GaAs και Eg=0.0 ev για το InAs. Οι ιδανικές επιφάνειες από τις οποίες αποτελείται η (112)Β του GaAs και που αναφέραμε πριν δημιουργήθηκαν με το πρόγραμμα ATOMS, όλες 2x2 εκτός της (123)Β που κόπηκε 1x1. Ο στόχος της μελέτης είναι η αναδόμηση και αφηρέμηση των επιφανειών και όχι της συμπαγούς δομής του GaAs και αυτό επιτυγχάνεται με την προσθήκη κενού στην κάθετη διεύθυνση της επιφάνειας. Με την προσθήκη κενού οι δομές όλων των επιφανειών προέκυψαν να είναι ορθορομβικές με την επιφάνεια να βρίσκεται στην κορυφή της ορθορομβικής δομής, δημιουργείται ουσιαστικά ένα ορθορομβικό κουτί το οποίο περιέχει και κενό. Για να γίνει η προσομοίωση με τον κώδικα του AIMPRO πρέπει να γίνει πλήρωση όλων των σπασμένων δεσμών της επιφάνειας που βρίσκεται στο κάτω μέρος της δομής ώστε κατά την προσομοίωση η πάνω επιφάνεια της δομής να μην αλληλεπιδρά με την κάτω επιφάνεια. Εξαίρεση είναι η περίπτωση όπου η πάνω επιφάνεια της δομής είναι συμμετρική με την κάτω οπότε δεν υπάρχει αλληλεπίδραση μεταξύ τους κατά την διάρκεια των υπολογισμών του κώδικα. Η μέθοδος που ακολουθούμε είναι η πλήρωση των σπασμένων δεσμών της κάτω επιφάνειας με άτομα υδρογόνου αλλά με κλασματικό φορτίο δηλαδή vii

8 ψευδό-υδρογόνα Η 1.25 και Η 0.75 για κάθε σπασμένο δεσμό του Ga και του As αντίστοιχα. Είναι γνωστό ότι το κατιόν της 3 ης ομάδας του περιοδικού πίνακα που στην περίπτωση μας είναι το Ga συνεισφέρει 3/4e - στον ομοιοπολικό δεσμό με το As, οπότε κάθε σπασμένος δεσμός του πληρώνεται με ψευδό-υδρογόνο Η Αντίθετα το ανιόν της 5 ης ομάδας που στην περίπτωση μας είναι το As συνεισφέρει 5/4e - στον ομοιοπολικό δεσμό με το Ga, οπότε κάθε σπασμένος δεσμός του πληρώνεται με ψευδό-υδρογόνο Η Όλα όσα αναφέρθηκαν πριν ήταν η θεωρητική προετοιμασία της προσομοίωσης που έγινε με τον κώδικα του AIMPRO για τις επιφάνειες {(111)B, (101), (011), (113)B ή (123)B, (124)B, (214)B} από τις οποίες αποτελείται η (112)Β επιφάνεια του GaAs. Η τιμή του κενού που χρησιμοποιήθηκε είναι 12 Α κάθετα στην διεύθυνση της επιφάνειας και η κάθε ορθορομβική δομή αποτελούνταν περίπου από 10 στρώματα ατόμων. Το k-point grid στον κώδικα της προσομοίωσης ορίστηκε καθώς η διεύθυνση z είναι κάθετη στην επιφάνεια και το περιοδικό διάνυσμα της δομής κατά την διεύθυνση αυτή αυξήθηκε κατά την τιμή του κενού 12 Α με αποτέλεσμα να μην υπάρχουν περιοδικές συνθήκες κατά τη διεύθυνση αυτή και αυτός είναι ο λόγος που το k-point ορίστηκε να είναι 1 κατά την διεύθυνση z. Η ανάλυση του τρόπου ανάπτυξης των κβαντικών τελειών InAs πάνω στις επιφάνειες του GaAs έγινε με την αντικατάσταση του των ατόμων του Ga με τα άτομα του In, του επάνω στρώματος της δομής και του αμέσως επόμενου στρώματος από κάτω και εκτελέστηκε για κάθε μια περίπτωση ξανά η βελτιστοποίηση των θέσεων των ατόμων με τον κώδικα του AIMPRO. Τα αποτελέσματα έδωσαν για την επιφάνεια (111)Β του GaAs ενεργειακό χάσμα Eg=0.42 ev, με ένα στρώμα InAs ενεργειακό χάσμα Εg=0.29 ev και με δύο στρώματα InAs ενεργειακό χάσμα Εg=0.16 ev. Παρατηρούμε ότι το ενεργειακό χάσμα μειώνεται με την προσθήκη του In στην επιφάνεια του GaAs και το αποτέλεσμα αυτό ήταν αναμενόμενο από τη στιγμή που το ενεργειακό χάσμα του συμπαγούς GaAs υπολογιστικέ με τον κώδικα του AIMPRO με τιμή Eg=0.5 ev και του συμπαγούς InAs με τιμή Eg=0.0 ev οπότε όσο στη δομή αναπτύσσεται στρώμα με στρώμα το InAs το ενεργειακό χάσμα της επιφάνειας θα πρέπει να μειώνεται. Στη συνέχεια έγινε η ανάλυση των επιφανειών της οικογένειας {101} και συγκεκριμένα της (011) του GaAs της οποίας το ενεργειακό χάσμα βρέθηκε να είναι Eg=0.56 ev και με ένα και δύο στρώματα InAs Eg=0.51 ev και Eg=0.44 ev αντίστοιχα. Για την (113)Β του GaAs τα αποτελέσματα είναι Eg=0.49 ev, Eg=0.33 ev, Eg=0.28 ev. Για την (123)Β του GaAs Eg=0.56 ev, Eg=0.49 ev, Eg=0.46 ev, και για τις {124}Β και συγκεκριμένα την (124)Β του GaAs τα αποτελέσματα είναι Eg=0.65 ev, Eg=0.59 ev, Eg=0.57 ev. viii

9 Contents 1 Introduction Art of Modeling Ab Initio calculations Applications of DFT The Kohn Sham equation Local Density Approximation Generalized Gradient Approximation Hertree-Fock Theory The AIMPRO package Introduction Approximations Gaussian Orbitals Semiconductors Introduction Crystal structure Band structure GaAs (112)B surface Introduction Structure of GaAs (112)B surface Bulk GaAs and InAs band structures Analysis of GaAs (111)B surface Analysis of GaAs {011} surfaces Analysis of GaAs (113)B surface ,7 Analysis of GaAs (123)B surface Analysis of GaAs{124}B surfaces Conclusions 62 References ix

10 Chapter 1 Introduction 1.1 Art of modelling Today we are in a world where technology is constantly evolving. We see this daily with the use of computers which are a tool that has served to facilitate communication and the way how people work. Computers are a recent invention, during its early years computers used to be huge, occupying large rooms, but today there are such the smaller parts of how they used to be. Nowadays, supercomputers with a huge processing capacity speeds up the real time of calculations by a massive numbers of processors. So computer simula- tions have become a useful part of mathematical modeling of many natural systems in computational solid state physics. Simulation in solid state physics comprises ab initio first principles, molecular dynamics and Monte Carlo simulations which are represented as the processing of the system's theoretical model. It can be used to explore and gain new insights into new technology and estimate the performance of complex systems for analytical solutions. Although theory and experiment have recently been side to side in the long walk of understanding nature, it was the introduction of computers that allowed full quantum mechanical modelling and gave a more fundamental and qualitative edge to our understanding. Computing modelling can be thought as the intermediate step between theory and experiment. It can verify whether a theory matches an experiment or even predict various physical properties without performing the experiment. In the last few years the availability of the ever increasing computational resources meant that the modelling of 1

11 physical systems, and hence the research it was based on, could shift from simple models to complex ones. In the future these models which, in many cases, accurately predict a range of physical properties, will not need to be limited to a few hundreds of atoms, but will make modelling of thousands of atoms a task of the researcher's routine. Computer simulations have made possible to accurately compute a large number of electronic and structural properties of solids with first principles quantum mechanics. In materials science a typical system consists of many particles of the order of Avogadro s constant NA= mol -1. This evolution of computational physics has led to new dimensions in condensed matter studies. 1.2 Ab Initio calculations In the 1920's Ervin Schrodinger represented the wavefunction of a quantum system in terms of certain electronic parameters. This theory was used to roughly compute the electronic energy based on electron density distributions of DFT (density fanctional theoty), which can evaluate the system's properties. Quantum physics modelling techniques, are based on solving these equations within DFT. Even with today's standards this usually means very long and computationally intensive calculations. Importing experimental known data and using semi-empirical methods of approximations allows faster calculations, by simplifying Schrodinger's equation. Density functional theory DFT is an alternative theory to ab initio methods for solving the non relativistic, time-independent Schrodinger equation The ab initio approach is to assume that the wavefunction of the system may be decomposed into single electron wave functions. However, for a system with N electrons this means there are 3N continuous variables necessary to be fitted. Assuming that 3 p 10 parameters per variable are required to yield a fit of reasonable accuracy, this means that p 3N parameters must be optimized for an N electron system. Although this is a great oversimplification this exponential wall does exist and limits the traditional wave function methods to molecules with a small number of chemically active electrons. This problem shall be overcome by DFT, which is expressed in terms of the density n(r) in the Hohenberg-Kohn formulation and in terms of n(r) and the single-particle wave functions ψi(r) in the Kohn-Sham formulation. 2

12 The Hohenberg Kohn theorem: The groundstate density n(r) of a bound system of interacting electrons in some external potential u(r) determines this potential uniquely. A question that may arise immediately linked to the Hohenberg Kohn theorem is how one knows, after having minimized a given density, whether this density is arising from an antisymmetric N-body wavefunction and whether it actually corresponds to the ground state density of a potential u(r). 1.3 Applications of DFT Since u(r) depends on a set of parameters, lattice constants or nuclei positions, the energy may be minimized with respect to these quantities yielding molecular geometries and sizes, lattice constants and charge distributions. By looking at the change in energy with respect to these parameters one may calculate, lattice constant, bond geometry, compressibility, phonon spectra, bulk module and vibrational frequencies (molecules). Comparison of the energy of a composite system with its constituent systems gives dissociation energies. In principle it is sufficient to express the total energy of the system in terms of the density and minimize this functional to obtain the groundstate energy, denisty and wave function. The total energy of a typical molecule or solid is given by: 3 E( n) T( n) U( n) V ( n) T( n) U( n) n( r) u( r) d r (1.1) Here T(n) is the kinetic energy of the electrons, U(n) the potential energy due to the interaction between them and V(n) the potential energy from an external potential u(r), usually the electrostatic potential of the nuclei, which are taken to be fixed in space (Born-Oppenheimer approximation). Only this potential is easy to treat since it is a multiplicative operator. 3

13 1.4 The Kohn-Sham equation The idea of the Kohn-Sham approach is to reintroduce a special type of wave functions into the formalism, to treat the kinetic and interaction energy terms. In this approach the energy is taken to be composed of the following terms: E( n) T( n) U( n) V( n) T ( n) E ( n) V( n) here the kinetic energy is split into two contributions: T( n) T ( n) T ( n) s i xc (1.2) s c (1.3) Ts stands for the kinetic energy of non-interacting particles of density n with s denoting single particle and Ts being the remainder with c denoting correlation and is: T ( n) s 2 2m N i * i 2 i ( r) d 3 r (1.4) Hence Tc is the difference between Ts and T is a pure correlation effect. The interaction energy U is approximated by the classical electrostatic interaction or Hartree energy UH and in terms of density is given by 2 q 3 3 n( r) n( r ) U H d r d r 2 r r (1.5) The new energy term Exc stands for a correction of these approximation due to exchange (x) and correlation (c) effects, is given by Exc = (T Ts)+(U UH) = Tc+(U UH). It is often decom- posed as Exc = Ex + Ec, where Ex is the exchange energy due to the Pauli principle (antisymmetry) and Ec is due to correlations (Tc is then a part of Ec). 4

14 1.5 Local Density Approximation (LDA) The general idea of LDA is to take the known result for a homogeneous system and apply it locally to a non-homogeneous system. The exchange energy of a homogeneous system is known to be: E LDA xc 1/ 3 2 3q 3 4 / 3 3 ( n) n( r) d r (1.6) 4 Expressions for Ex LDA (n) are parametrizations of highly precise Quantum Monte Carlo calculations for the electron liquid. 1.6 Generalized Gradient Approximation (GGA) It was a major breakthrough, when it was realized that instead of power-series-like systematic gradient expansions one could experiment with more general functions of n(r) and n(r), which need not proceed order by order. Such functional have the general form: E GGA ( n) d 3 rf n( r), n( r) xc (1.7) The most popular GGAs are PBE (denoting the functional proposed in 1996 by Perdew, Burke and Ernzerhof) and BLYP (denoting the combination of the exchange functional by Becke and the correlation functional of Lee, Yang and Parr, both in 1988). GGAs give reliable results for chemical bonds, but mostly fail for van der Waals or dispersion interactions. 5

15 1.7 Hartree-Fock Theory Hartree-Fock theory HFT is an alternative theory to Kohn-Sham Density Functional Theory DFT, that was developed to solve the electronic Schrodinger equation which results from the time-independent Schrodinger equation after invoking the Born-Oppenheimer approximation. Many of these descriptions also apply to DFT, which bears a striking resemblance to Hartree-Fock theory but one difference, however, is that the role of the Hamiltonian different in DFT. The fundamentals of theory are about of electronic structure theory. It is the basis of molecular orbital theory, which posits that each electron's motion can be described by a single-particle function (orbital) which does not depend explicitly on the instantaneous motions of the other electrons. Hartree-Fock theory often provides a good starting point for more elaborate theoretical methods which are better approximations to the electronic Schrodinger equation. For symmetric energy expressions, we can employ the variational theorem, which states that the energy is always an upper bound to the true energy. Hence, we can obtain better approximate wave functions by varying their parameters until we minimize the energy within the given functional space. Hence, the correct molecular orbitals are those which minimize the electronic energy E. The molecular orbitals can be obtained numerically using integration over a grid, or as a linear combination of a set of given atomic orbital basis functions, usually atom-centered Gaussian type functions. The Hartree-Fock equations can be solved nume- rically or they can be solved in the space spanned by a set of basis functions. In either case, note that the solutions depend on the orbitals. Hence, we need to guess some initial orbitals and then refine our guesses iteratively. For this reason, Hartree-Fock is called a self-consistent field approach SCF. 6

16 Chapter 2 The AIMPRO package 2.1 Introduction AIMPRO package is an ab initio code which uses Gaussian orbitals as its wavefunctions basis sets. The package uses LDA (Local Density Approximation) pseudopotentials. The general idea of LDA is to take the known result for a homogeneous quantum system and apply it locally to a non-homogeneous system and so it changes the exchange-correlation energy Eex which is part of the pseudopotential s approximation in a manybody interacting systems. As we noticed in a previous chapter to determine the structural and electronic properties of a system that consists of many particles of the order of Avogadro s constant we need to solve the non relativistic time independent Schrodinger equation ΕΦ for both ions and electrons. Unfortunately it is impossible to give an exact analytical solution for anything more complicated than a hydrogen atom, so the progress can only be made if a set of computational approximations are made. 2.2 Approximations As well known the Hamiltonian operator in Shrodinger s equation for a system with a set of non interacting particles is: H i 2 2m i 2 i V ( r ) i (2.1) 7

17 however our problem is more complicated because of the interaction between the particles in the system, an so the Hamiltonian operator now consists of more terms which describe the interactions ann so: H K n V n n K e V e e V n e H i 2 2M 2 i i 1 2 i j Z Z R i i j e R 2 j k 2 2m i 2 k 1 2 k l e r k 2 r l 1 2 k i 2 Zie r R k i (2.2) where the positions R, r correspond to the positions of nuclei and electrons and Mi, m to their masses respectively. The first and third terms represent the kinetic energy of the nuclei and the electrons, while the remaining three the interaction between the nuclei, the interaction between the electrons and the interaction between the electrons and the nuclei. This coupling makes the solution of the system extremely difficult, and computationally very expensive, especially when it comes to large systems. The question arising is how one could simplify the Hamiltonian to give rise to a soluble problem. There are two main approximations that can be introduced. The first one takes advantage of the fact that only the valence electrons determine the properties of the solid. As the valence electrons are usually a small fraction of the total number of electrons, the Hamiltonian can be greatly simplified. The second approximation, which is usually referred to as the Born-Oppenheimer approximation, states that as the electrons react very quickly compared to the ion cores, the latter can be treated as being stationary, when considering electronic degrees of freedom. Born and Oppenheimer argued that, since the nuclear masses are so much larger than those of the electrons, then we can treat the nuclear motion classically and reduce the Schrödinger equation to one, which only involves the electrons movements in a potential of fixed nuclear sites. The Schrödinger equation of the electrons in a field of frozen nuclei still can't be solved because of the inter-electron electrostatic Coulomb interactions. Neglecting electronelectron interaction term is too extreme to make this theory useful and various attempts have been made to take into account this interaction. There are two principal schemes the Hartree-Fock theory HFT and density functional theory DFT. Both of these replace the electrostatic potential which acts on each electron by an average over the potentials experienced by all the electrons. 8

18 In these theories, the effective potential is composed of two terms. The Hartree potential is the electrostatic potential from all the electrons and the exchange-correlation potential a purely quantum mechanical contribution. In HF theory, the exchangecorrelation is a complicated function determined by all the orbitals. In DFT, it is rigorously determined only by the total electron density. The precise dependence of the exchangecorrelation potential on the density is known only for one particular problem: the homogeneous electron gas. For any other problem where the electron density varies throughout space, we can assume that the contribution from exchange-correlation potential at point r is given by the homogeneous electron gas value that involves the density, n(r), at the same point r. This is called the local density approximation LDA. This is the first successful approximate functional for DFT, and works well for many covalently solids. 2.3 Gaussian Orbitals AIMPRO uses Cartesian Gaussian orbitals to proceed and expand the wavefunction in terms of a basis φi(r), and so using Gaussian orbitals may significantly reduce the number of functions required. So the solution from the eigenvalue problem is: Φ ι c φ(r) ι ι (2.3) where φi(r) is: φ(r) ι x 2 1 n2 n3 ar n y z e (2.4) Deferent orbitals can be obtained by varying n1, n2 and n3. For example, if all of them were equal to 0, the orbitals are simply spherically symmetric Gaussian functions which called s-gaussian orbitals. If one of the ni is unity, and the others zero, then the functions are p-gaussian orbitals while if the sum of the ni is 2, the set of six orbitals generate five d-orbitals and one s orbital. The great advantage of Gaussian orbitals is that analytical expressions can be given for all the integrals. AIMPRO uses two separate basis sets based on Gaussians to expand the wavefunction and the charge density, which will be 9

19 denoted as n, m depending on the n number of exponents that the wavefunctions will be expanded to and the m number of exponents that the charge density will be expanded to. The exponents of the Gaussians used to expand the wavefunction are selected such as they minimize the energy of the atoms, while the Gaussians used to expand the charge density. So the exponents are selected in such a way to maximize the estimated Hartree energy. 10

20 Chapter 3 Semiconductors 3.1 Introduction A semiconductor is a material that has a conductivity value in between that of a conductor and an insulator in the range between 10 2 and 10-9 (Ωcm) -1. The conductivity of a semiconductor material can be varied under an external electric field. Devices made from semiconductor materials are the foundation of modern integrated electronics, computers, lasers, and may other devices. Semiconductor devices include the transistor, many kinds of diodes including the light emitting diode, the silicon controlled rectifier, and digital and analog integrated circuits. Solar photovoltaic panels are large semiconductor devices that directly convert light energy into electrical energy. The periodic system of elements which is a tabular arrangement of the chemical elements, organized on the basis of their atomic numbers, electron and structure configurations. It is one way of finding out, how many electrons an element can make available, either for conducting electrical current or for bonding to other elements. Electrons in incomplete outer electron shells are called valence electrons and the number of valence electrons defines the type of bond the element can undergo. In the middle of the periodic system between the metals on the left group III and the insulators on the right group V resides the group of elemental semiconductors group IV like silicon Si and germanium Ge the most known. The elements in this group have four valence electrons per atom, which means that four partly filled electron shells need to be filled from other atom. Therefore, elements of this group are reactive and can undergo bonds 11

21 with many other elements. Elements with four valence electrons tend to form covalent bonds when bonding to other elements of the same group (Si, Ge). Covalent bonds involve two electrons per bond and are fairly strong. The charge distribution in covalent crystals is not always homogeneous. Elemental semiconductors are perfectly covalent by symmetry electrons shared between two atoms are to be found with equal probability on each atom. Compound semiconductors always have some degree of iconicity cause the different electronegativity. In III-V compounds, like GaAs, the five-valent As atom retains slightly more charge than is necessary to compensate for the positive charge of the As 5+ ion core, while the charge on the Ga 3+ ion is not entirely compensated. The most important group IV semiconductors are silicon Si and germanium Ge. But semiconductors are not only from group IV, but can be binary combinations of group III and V elements, like gallium arsenide GaAs and indium phosphide InP, or binary combinations of group II and VI elements, like zinc selenide ZnS and cadmium telluride CdTe or even ternary alloys, having the same number of electrons in covalent bonds. As the same number of electrons is involved in all these materials they tend to have the same or very similar crystal structures in their solid state. As one would expect with carbon being in the same group of the periodic system, the crystal structure of Si and Ge is in fact the diamond structure. The crystal structure of the binary semiconductors, such as GaAs or InP, is the zincblende structure, which is the same structure for binary compounds with the two different elements occupying alternating positions. 3.2 Crystal structure Most semiconductor materials are single crystals. The element semiconductors, silicon Si and germanium Ge, have a diamond lattice structure as shown in Fig 3.1. This configuration belongs to the cubic-crystal family and can be envisaged as two interpenetrating fcc sublattices with one sublattice staggered from the other by one quarter of the distance along a diagonal of the cube. All atoms are identical in a diamond lattice, and each atom in the diamond lattice is surrounded by four equidistant nearest neighbors that lie at the corners of a tetrahedron. Most of the III-V semiconductors e.g. 12

22 GaAs have a zincblende lattice shown in Fig 3.2 that is identical to a diamond lattice except that one fcc sublattice has Column III atoms Ga and the other has Column V atoms As. The diamond structure, characteristic of Column-IV semiconductors, C diamond, Si, Ge shows in Fig 3.1 and it is not a Bravais lattice but has two atoms per unit cell. In Cartesian coordinates with axes along the side of a cube of side a the second atom is displaced by a vector a/4, a/4, a/4 relative to the first. The two-atom unit itself the basis is repeated perio- dically, forming a face-centered cubic fcc lattice. Fig 3.1 Diamond structure. The structure can be viewed as two interpenetrating fcc sublattices. The two fcc sublattices are fully interchanged by the inversion operation, so all atoms in the diamond structure are symmetry equivalent. This, however, does not make diamond a Bravais lattice, because there is no way to choose a set of three translation vectors that would generate the entire structure from a single atom. 13

23 The zincblende structure, of gallium arsenide GaAs is similar to diamond structure except that atoms in alternate planes are colored differently. Gallium atoms form a face centered cubic lattice. An identical sublattice but shifted along the body diagonal of the cube 1/4 th of its length is formed by atoms of As. One gallium Ga atom is surrounded by four atoms of arsenic As. In the tetrahedral arrangement of atoms in a diamond or zincblende structure the bonds are along the cube diagonals and the bond angle θ is given by cosθ = - 1/3. Four atoms in the corners define a regular tetrahedron. In a zincblende structure these four are different from the central atom, in the diamond structure they are all the same. Fig 3.2 The zincblende fcc structure of gallium arsenide GaAs. Red are the Ga atoms and blue As atoms 3.3 Band structure 14

24 When two atoms approach one another, the energy level will split into two by the interaction between the atoms. When N atoms come together to form a crystal, the energy will be split into N separate but closely spaced levels, thereby resulting in an essentially continuous band of energy. The detailed energy band structures of crystalline solids can be calculated using quantum mechanics. The energy band splits into two, the conduction band and the valence band, as the two atoms approach the equilibrium interatomic spacing. The region separating the conduction and valence bands is termed the forbidden gap or bandgap, Eg. Fig 3.3 exhibits the energy band diagrams of three classes of solids: insulators, semiconductors, and conductors. In insulators, the bandgap is relatively large and thermal energy or an applied electric field cannot raise the uppermost electron in the valence band to the conduction band. In metals or conductors, the conduction band is either partially filled or overlaps the valence band such that there is no bandgap and current can readily flow in these materials. In semiconductors, the bandgap is smaller than that of insulators, and thermal energy can excite electrons to the conduction band. The bandgap of a semiconductor decreases with higher temperature. For instance, for silicon, Eg is 1.12 ev at room temperature and 1.17 ev at zero Kelvin. Fig 3.3 Schematic energy band representations of a conductors, a semiconductor and dielectric or insulator At absolute zero temperature, in this energy diagram of semiconductor Fig 3.3, all states in the valence band are occupied and the conduction band is completely empty. The 15

25 semiconductor is insulating. At higher temperature or rather if energy in some form is absorbed by the electrons, lattice vibrations, photons, an electron electron bond can break and the electron becomes a free charge carrier capable of conducting electrical current, the hole left in its place, however, is immediately filled with other valence electrons from the surrounding bonds and thus is also considered to be a free charge carrier, but of positive charge. Holes, therefore, also contribute to the electric conductivity of the semiconductor. This process of free electron formation is called electron-hole pair generation. In pure semiconductors, free carriers are generated exclusively by the process of electron-hole pair generation described above. Therefore, the concentration of electrons in equilibrium equals the concentration of holes. Such semiconductors are called intrinsic semiconductors. But, whereas our simple band structure model only assigns an energy to the electrons, real band structures look somewhat different due to the fact that electrons also have momentum, which means that the direction of travel matters for the structure of the band. The real structure is based on the wave picture for the properties of the electrons, and this is derived from wave quantum mechanics. The wave provides a way of calculating the effects an electron can produce. In quantum mechanics the basics equations of energy, momentum and group velocity of an electron are given by: E p u k d dk de dp (3.1) where ħ is h/2π, ω is the radian frequency and k is the wave number the number of radians of phase change in unit distance. These equations are derived from a 3Ddimensional model of a solid, consisting of a row of N atoms with a distance a apart, each with two free electrons. When evaluating what values of k can be used to make allowed wave functions or states, by applying the boundary conditions for N basic cells in cube side L = Na occurs: 16

26 L na k 2 (3.2) k, k x y, k z 2 0, 4,,..., N 2 where n is integer and λ is a wavelength. The regular spacing of allowed values of ki occurs in three directions when a threedimensional lattice of atoms in ordinary space is analysed that is why ki-space or momentum space is useful to obtain a view of electron states. The 2N waves with ki running from 0 to N2π/L are all the waves one needs to make a band. Note, that they are just enough for the number of electrons in the crystal, and that the wavelength goes from the size of the crystal to the distance between the atoms. Band structures are periodic along the k axis because the real crystal is periodic in space. Usually only one interval of k is shown, conventionally the first Brillouin zone, which is called a reduced zone diagram. Energy diagrams against k for real semiconductors are specific for certain crystallographic directions and are also of a more complex structure than our simple band structure in Fig

27 Fig 3.4 The real electron Band Structure of silicon Si Fig 3.5 The real electron Band Structure of gallium arsenide GaAs 18

28 Figures 3.4 and 3.5 shows the real band structures of the most important semiconductors, Si and GaAs with their four valence bands on the bottom of the respective diagrams and some conduction bands on the top. The Γ point is the zone centre [000] with the L point denoting the [111] zone boundary and X denoting the [100] zone boundary and K denoting the [101] zone boundary, and shapes are not symmetric about the zone centre. It is now much more difficult to find the energy gap or band gap because of the irregularity of the structures. The band gap is given by the smallest distance between the conduction band and the valence band, the gap between the minimum in the conduction band and the maximum in the valence band. As shown in Fig 3.5 for GaAs the minimum in the conduction band and the maximum in the valence band occur at the same k value whereas for Si Fig 3.4 there is a k-shift between the minimum and the maximum. Semiconductors like GaAs are called direct band-gap semiconductors, those semiconductors with the minimum of the conduction band at a different value of k from the maximum of the valence band are called indirect band-gap. The curvature of the bands is also of significance. As the velocity is v = de/dp the acceleration a of an electron due to some external force can be expressed as de u (3.3) dp a du dp d dt de dp dp dt 2 d E 2 dp (3.4) But dp/dt is the rate of change of momentum, and hence equals the applied force F. Thus d 2 E/dp 2 replaces the mass in the equation of motion F = ma and we can describe the response of a carrier to a force by using (d 2 E/dp 2 ) -1 instead of the mass. This new term is known as the effective mass m * of a carrier and summarizes the way the interaction with the lattice affects the carrier motion. The effective mass of a free electron 19

29 is me * and the effective mass of an electron in the bottom of the conduction band is usually less than the free-electron mass, and may be much less. 20

30 Chapter 4 GaAs (112)B surface 4.1 Introduction It is well known that the atomic force field at the surface can substantially deviate from that in the bulk, as established by the observation of changes in the interlayer spacing that is relaxation, and even rearrangement in the two dimensional structure parallel to the surface that is reconstruction. The electrons from the dangling bonds have the tendency to spill over the surface in order to create a geometrically smooth surface and thus lowering their energy. This charge rearrangement leads to an increase of the charge density between the first and the second layer, which causes an attractive force between the atoms of these layers, a surface stress. This particular vertical surface stress is relieved by the relaxation of the first interlayer spacing. The charge redistribution leads however also to an increased charge density within the first layer, generating an in plane surface stress. This stress can only be relieved when the surface layer reconstructs to form more densely packed structures. The in plane stress of nonreconstructed surface can however be relieved by the charge rearrangement upon adsor- bing atoms or molecules from the environment or in the case of diffusion modes. This charge transfer creates a dipole layer with positively or negatively charged surface atoms. In the case of a positive dipole layer the Coulombic repulsion produces a weakening between the interlayer interatomic forces. The weakening of the bond 21

31 between the first and the second layer is usually accompanied by an increase in bond length and this is the case of outward relaxation of the clean surface. In the case of a negative dipole layer the situation is exactly inverse. The increased Coulombic attraction, due to the excess surface change, results in a stiffening of the interlayer bonds and so decrease the bond length and this is the case of inward relaxation of a clean surface. 4.2 Structure of GaAs (112)B surface The (112)B surfaces of GaAs are unstable and facet into more stable {(111)B, 2x(110), (113)B} surfaces pyramidal structure..the side walls of the GaAs (112)B prisms are build up of non-relaxed cleavage surfaces (111)B, (101), (011), (113)B or (123)B, (124)B, (214)B. In Figure 4.1 below shown the two possible shapes of GaAs (112)B surface within triangular prism method. Fig 4.1 Two possible shapes of GaAs (112)B surface construction 22

32 Fig 4.2 Possible shape (a) of GaAs (112)B view from ATOMS Fig 4.3 Possible shape (b) of GaAs (112)B view from ATOMS 23

33 Fig 4.4 non-relaxed cleavage surfaces (111)B, (101), (011), (113)B, (124)B, (214)B. 24

34 4.3 Bulk GaAs and InAs band structures From the lattice optimization run we obtained the values of lattice constants of bulk GaAs and InAs which are Å and Å respectively. With the values of the optimized lattice constants, using k-points grid and specify four paths through the k space which correspond the special points in the Brillouin zone of fcc lattice L ( ), Γ (0 0 0), X (1 0 0), K ( ), we constructed the band structures of bulk GaAs and InAs and obtained the band gaps of the point Γ, Eg=0.5 ev for GaAs and Eg=0.0 ev for InAs. Figures 4.5 and 4.6 are the results: Fig 4.5 GaAs band structure Eg=0.5 ev 25

35 Fig 4.6 InAs Band structure with Eg=0 ev 26

36 4.4 Analysis of GaAs (111)B surface The cleavage GaAs (111)B surface is polar and there is one As atom derived dangling bond per surface atom. This configuration of surface is unstable and the surface atoms will reconstruct to reduce the Coulomb forces between them. By forming a Ga atom vacancy, one As atom dangling bonds is created. Fig Side view of cleavage 2x2 GaAs (111)B with the surface on the top of the orthorhombic box. Ga atoms are blue, As atoms red The figure 4.7 shows the created structure of 2x2 GaAs(111)B surface with the program ATOMS. The structure of the box is orthorhombic and consists of 10 interlayers of atoms with the (111)B surface on the top of the box. As we see the bottom side of the box has 8 atoms of As with 8 dangling bonds of one Ga missing atom for each As atom respectively. We need to saturate those 8 dangling bonds with the use of 8 pseudohydrogen atoms with fractional charge H As we well know each Ga atom contributes 3/4e - to a covalent bond with As atom. We add also a vacuum along the perpendicular direction of the surface with value of 12 Å and increase the primitive vector by this direction. So the 27

37 created structure consists of 160 Ga and As atoms plus 8 pseudohydrogen atoms. The primitive vectors along the x, y and z directions are 7.94 Å, Å and Å respectively with the latter increased by the value of vacuum. We have set the k-point grid as and performed the optimization run of AIMPRO first only for the last bottom layer of the structure of 8 As atoms with the 8 pseudohydrogen atoms. Afterwards having the reconstructed configurations of those atoms we performed the optimization run for the rest structure. Fig Reconstructed GaAs (111)B surface In figure we have the result of the optimization run we performed. So as we see on the top of the structure this is the reconstructed (111)B surface of GaAs. For the analysis of the InAs growth mode made by the replacement of the top layer of the orthorhombic box and the next one by In atoms. So after the replacements now the new two structures consists of one and two interlayers of InAs. As we know the structures of GaAs and InAs are both fcc while the lattice constants of bulk GaAs and InAs are

38 Å and Å respectively. Due to the small difference between the lattice constants and small misfit the adjustment of those two materials is well, with satisfactory coherence and the reconstructed surfaces as we can see in the figures and below are similar to the structure of GaAs without the In ad atoms. Only the bond length of those two layers increased and this is due to bigger lattice constant of InAs. Fig Reconstructed GaAs (111)B surface with one InAs layer. In atoms are yellow 29

39 Fig Reconstructed GaAs (111)B surface with two InAs layers We also calculated the density of state by the use of AIMPRO to obtain the band gaps. In general it is useless to calculate the band structure because the structures under investigation consists of vacuum and pseudohydrogen atoms so the special points in the Brillouin zone differs from the fcc bulk structure of GaAs. We have set the k point grid as and the broadening 0.05 ev. The number of energies for which the density of states energy is calculated are 1000 and set all the bands to be used for the calculation of the density of states. 30

40 Fig Density of States of thegaas (111)B surface Calculating the density of state of (111)B GaAs surface we obtained the band gap equal to Eg=0.42 ev as we see in figure The first red line is the end point of the valence band and the second is the starting point of the conduction band respectively. The growth mode of InAs as mentioned is analyzed by the replacement of Ga atoms of the top layer and the next one by In atoms. In the first case with one InAs layer the calculation of the density of states leads to a band gap Eg=0.29 ev as we can see in figure below and in the second case with two layers InAs the result is a band gap of Eg=0.16 ev as we can see in figure below. We notice that the band gap decreases and this result was expected since the band gap of bulk GaAs is calculated equal to 0.5 ev and InAs equal to 0.0 ev so as the structure of InAs growths layer by layer, the band gap should be decreased. 31

41 Fig Density of States of the GaAs (111)B surface with one InAs layer Fig Density of States of thegaas (111)B surface with two InAs layers 32

42 4.5 Analysis of GaAs {011} surfaces The GaAs (101) and (011) are equivalent as we can see in Figure 4.4. So we will demonstrate the analysis of the (011) surface. Fig Side view of ideal cleaved 2x2 GaAs (011) with the surface on the top of the orthorhombic box. Ga atoms are blue, As atoms are red Fig Plan view of ideal cleaved 2x2 GaAs (011) 33

43 In figure we show the created structure of ideal cleaved 2x2 GaAs(011) surface from the program ATOMS. The structure of the box is orthorhombic and consists of 11 interlayers of atoms with the (011) surface on the top of the box. As we consider the top and the bottom side of the box are completely symmetric so we do not need to add a pseudohydrogen atoms to saturate the dangling bonds of the bottom side of the box. We add also a vacuum along the perpendicular direction of the surface with value of 12 Å and increase the primitive vector by this direction. So the created structure consists of 264 atoms, both Ga and As atoms. The primitive vectors along the x, y and z directions are Å, 9.95 Å and Å respectively with the latter increased by the value of vacuum. We have set the k-point grid as and performed the optimization run. In figure we show the result of the optimization run. Fig Reconstructed GaAs (011) surface Note that the GaAs(011) surface relaxes in the process of assuming a different electronic distribution, the As surface atoms move upward in relation to their Ga lateral neighbors. As we can see the top and bottom side reconstructed symmetric so this is a fact that the two surfaces do not interact due the optimization run of the AIMPRO and this is one proof of the state that we do not need to add a pseudohydrogen atoms on the bottom side of the orthorhombic box. 34

44 Fig Side view of reconstructed GaAs (011) surface with a rotation by 30 degree. The analysis of the InAs growth mode made by the replacement of the top layer of the orthorhombic box and the next one by In atoms in the same way we discussed previous in analysis of GaAs(111)B surface. So after the replacements now the new two structures consists of one and two interlayers of InAs. 35

45 Fig Reconstructed GaAs (011) surface with one InAs layer. In atoms are yellow Fig Reconstructed GaAs (011) surface with two InAs layer. In atoms are yellow 36

46 Fig Side view of reconstructed GaAs (011) surface with two InAs layer and rotation by 30 degree. We also calculated the density of state by the use of AIPMRO and obtain the band gaps for the (011) GaAs surface with one and two InAs layers in figures below. 37

47 Fig Density of states of the GaAs (011) surface Fig Density of states of the GaAs (111)B surface with one InAs layer 38

48 Fig Density of states of the GaAs (111)B surface with two InAs layers Calculating the density of state of (011) GaAs surface we obtained the band gap equal to Eg=0.56 ev as we can see in figure The growth mode of InAs as mentioned is analyzed by the replacement of Ga atoms of the top layer and the next one by In atoms. In the first case with one InAs layer the calculation of the density of states leads to a band gap Eg=0.51 ev as we can see in figure and in the second case with two layers InAs the result is a band gap of Eg=0.44 ev as we can see in figure

49 4.6 Analysis of GaAs (113)B surface We continuous to discuss and present the results of the 2x2 GaAs (113)B surface with the same exact way as before in the (111)B. The method is similar as in the (111)B surface and differs only in the construction of the box and the number of Ga atoms, As atoms and pseudohydrogen atoms. Fig Side view of ideal cleaved 2x2 GaAs (111)B with the surface on the top of the orthorhombic box The figure shows the created structure of ideal cleaved 2x2 GaAs(113)B surface with the program ATOMS. The structure of the box is orthorhombic and consists of 12 interlayers of atoms with the (113)B surface on the top of the box. We add a vacuum along the perpendicular direction of the surface with value of 12 Å and increase the primitive vector by this direction. So the created structure consists of 240 Ga and As atoms plus 24 pseudohydrogen atoms. The primitive vectors along the x, y and z directions are 7.94 Å, Å and Å respectively with the latter increased by the value of vacuum. 40

50 We have set the k-point grid as and performed the optimization run of AIMPRO first only for the last bottom layer of the structure of 32 As atoms with the 24 pseudohydrogen atoms and afterwards having with the configurations of those atoms we performed the optimization run for the rest structure. Fig Side view of reconstructed 2x2 GaAs (113)B surface 41

51 Fig Plan view of ideal cleaved 2x2 GaAs (113)B surface Fig Plan view of reconstructed 2x2 GaAs (113)B surface 42

52 For the analysis of the InAs growth mode made by the replacement of the top layer of the orthorhombic box and the next one by In atoms. So after the replacements now the new two structures consists of one and two interlayers of InAs. As we know the structures of GaAs and InAs are both fcc with the lattice constants of bulk GaAs and InAs calculated as Å and Å respectively Fig Reconstructed GaAs (113)B surface with one InAs layer. 43

53 Fig Reconstructed GaAs (113)B surface with two InAs layers. We also calculated the density of state for the reconstructed (113)B surface with one and two InAs layers respectively by the use of AIMPRO to obtain the band gaps. As mentioned before, in general it is useless to calculate the band structure because the structures under investigation consists of vacuum and pseudohydrogen atoms so the special points in the Brillouin zone differs from the fcc bulk structure of GaAs. We have set the k point grid as and the broadening 0.05 ev. The number of energies for which the density of states energy is calculated are 1000 and set all the bands to be used for the calculation of the density of states. 44

54 Fig Density of states of the GaAs (113)B surface Calculating the density of state of (113)B GaAs reconstructed surface we obtained the band gap equal to Eg=0.49 ev as we can see in figure The growth mode of InAs as mentioned is analyzed by the replacement of Ga atoms of the top layer and the next one by In atoms. In the first case with one InAs layer the calculation of the density of states leads to a band gap Eg=0.33 ev as we can see in figure below and in the second case with two layers InAs the result is a band gap of Eg=0.28 ev as we can see in figure below. 45

55 Fig Density of states of the GaAs (113)B surface with one InAs layer Fig Density of states of the GaAs (113)B surface with two InAs layers 46

56 4.7 Analysis of GaAs (123)B surface The (123)B surface is an alternative case in the place of (113)B GaAs surface and so we are going to discuss and present the results of the 2x2 GaAs (123)B surface with the same exact way as before in the (113)B surface. Fig Side view of cleavage 1x1 GaAs (123)B with the surface on the top of the orthorhombic box The figure shows the created structure of 1x1 GaAs(113)B surface with the program ATOMS. The structure of the box is orthorhombic and consists of 11 interlayers of atoms with the (123)B surface on the top of the box. We add a vacuum for this surface, along the perpendicular direction of the surface with value of 12 Å and increase the primitive vector by this direction. So the created structure consists of 191 Ga and As atoms plus 16 pseudohydrogen atoms. The primitive vectors along the x, y and z directions are 9.72 Å, Å and Å respectively with the latter increased by the value of vacuum. 47

57 We have set the k-point grid as and performed the optimization run of AIMPRO first only for the last bottom layer of the structure of 26 As atoms with the 16 pseudohydrogen atoms and afterwards having the configurations of those atoms we performed the optimization run for the rest structure. Fig Side view of reconstructed 1x1 GaAs (123)B surface 48

58 Fig Plan view of ideal cleaved 1x1 GaAs (123)B surface Fig Plan view of reconstructed 1x1 GaAs (123)B surface 49

59 The analysis of the InAs growth mode made by the replacement of the top layer of the orthorhombic box and the next one by In atoms. So after the replacements now the new two structures consists of one and two interlayers of InAs Fig Reconstructed GaAs (123)B surface with one InAs layer. 50

60 Fig Reconstructed GaAs (123)B surface with two InAs layers. We also calculated the density of state for the reconstructed (123)B surface with one and two InAs layers respectively by the use of AIMPRO to obtain the band gaps. As mentioned before, in general it is useless to calculate the band structure because the structures under investigation (123)B surface consists of vacuum and pseudohydrogen atoms so the special points in the Brillouin zone differs from the fcc bulk structure of GaAs. We have set the k point grid as and the broadening 0.05 ev. The number of energies for which the density of states energy is calculated are 1000 and set all the bands to be used for the calculation of the density of states. 51

61 Fig Density of statesn of the GaAs (123)B surface Fig Induced state in band gap from As-As atoms metallic of the GaAs (123)B surface 52

62 Calculating the density of state of (123)B GaAs reconstructed surface we obtained the band gap equal to Eg=0.56 ev as we can see in figure The growth mode of InAs as mentioned is analyzed by the replacement of Ga atoms of the top layer and the next one by In atoms. In the first case with one InAs layer the calculation of the density of states leads to a band gap Eg=0.49 ev as we can see in figure below and in the second case with two layers InAs the result is a band gap of Eg=0.46 ev as we can see in figure below. The induced energy state in band gap is from the As-As metallic bond as emerged from the Mulliken analysis of the AIMPRO Fig Density of states of the GaAs (113)B surface with one InAs layer 53

63 Fig Induced state in band gap from As-As atoms metallic of the GaAs (123)B surface with one InAs layer Fig Density of states of the GaAs (113)B surface with two InAs layers 54

64 Fig Induced state in band gap from As-As atoms metallic of the GaAs (123)B surface with two InAs layers 55

65 4.8 Analysis of GaAs {124}B surfaces The GaAs (124)B and (214)B are equivalent as we can see in Figure 4.4. So we will demonstrate the analysis of the (124)B surface. Fig Side view of ideal cleaved 2x2 GaAs (124)B with the surface on the top of the orthorhombic box. Ga atoms are blue, As atoms are red In figure we show the created structure of ideal cleaved 2x2 GaAs(124)B surface with the program ATOMS. The structure of the box is orthorhombic and consists of 16 interlayers of atoms with the (124)B surface on the top of the box. We add also a vacuum along the perpendicular direction of the surface with value of 12 Å and increase the primitive vector by this direction. So the created structure consists of 400 atoms, 188 Ga atoms, 170 As atoms and 42 pseudohydrogen atoms. The primitive vectors along the x, y and z directions are Å, Å and Å respectively with the latter increased by the value of vacuum. 56

66 We have set the k-point grid as and performed the optimization run of AIMPRO. In figure we show the result of the optimization run. Fig Reconstructed GaAs (124)B surface The analysis of the InAs growth mode made by the replacement of the top layer of the orthorhombic box and the next one by In atoms. So after the replacements now the new two structures consists of one and two interlayers of InAs. 57

67 Fig Reconstructed GaAs (124)B surface with one InAs layer. In atoms are yellow 58

68 Fig Reconstructed GaAs (124)B surface with two InAs layer. In atoms are yellow 59

69 We also performed the density of state run of AIMPRO to obtain the band gaps of (124)B surface with one and two InAs,layers respectively. Fig Density of states of the GaAs (124)B surface Fig Density of states of the GaAs (124)B surface with one InAs layer 60

70 Fig Density of states of the GaAs (124)B surface with two InAs layers Calculating the density of state of (124)B GaAs surface we obtained the band gap equal to Eg=0.65 ev as we can see in figure The growth mode of InAs as mentioned is analyzed by the replacement of Ga atoms of the top layer and the next one by In atoms. In the first case with one InAs layer the calculation of the density of states leads to a band gap Eg=0.59 ev as we can see in figure and in the second case with two layers InAs the result is a band gap of Eg=0.57 ev as we can see in figure