Point Defects in Semiconductors I
Theoretical Aspects
Published on 1. August 1981
Book
Hardback
XVII, 265 pages
978-3-540-10518-3 (ISBN)
Description
From its early beginning before the war, the field of semiconductors has developped as a classical example where the standard approximations of 'band theory' can be safely used to study its interesting electronic properties. Thus in these covalent crystals, the electronic structure is only weakly coupled with the atomic vibrations; one-electron Bloch functions can be used and their energy bands can be accurately computed in the neighborhood of the energy gap between the valence and conduction bands; nand p doping can be obtained by introducing substitutional impurities which only introduce shallow donors and acceptors and can be studied by an effective-mass weak-scattering description. Yet, even at the beginning, it was known from luminescence studies that these simple concepts failed to describe the various 'deep levels' introduced near the middle of the energy gap by strong localized imperfections. These imperfections not only include some interstitial and many substitutional atoms, but also 'broken bonds' associated with surfaces and interfaces, dis location cores and 'vacancies', i.e., vacant iattice sites in the crystal. In all these cases, the electronic structure can be strongly correlated with the details of the atomic structure and the atomic motion. Because these 'deep levels' are strongly localised, electron-electron correlations can also playa significant role, and any weak perturbation treatment from the perfect crystal structure obviously fails. Thus, approximate 'strong coupling' techniques must often be used, in line' with a more chemical de scription of bonding.
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Content
Content.- 1. Atomic Configuration of Point Defects.- 1.1 Definition of Point Defects.- 1.2 Geometrical Configuration of Point Defects.- 1.2.1 The Vacancy.- 1.2.2 The Divacancy.- 1.2.3 The Interstitial.- 1.2.4 Complex Defects.- 1.2.5 Aggregates.- 1.3 Lattice Distortion and Relaxation.- 1.4 Defect Symmetry and Group Theory.- 1.4.1 Factorization of the Hamiltonian.- 1.4.2 Irreducible Representations.- 1.4.3 The example of the Vacancy in the Diamond Lattice.- 1.4.4 Various Points of Interest.- 1.4.5 Basis Functions for Irreducible Representations.- 1.4.6 Simplification of Matrix Elements by Symmetry.- 1.5 Experimental Determination of Defect Symmetry.- 1.5.1 Splitting Under Unaxial Stress.- a) General Considerations.- b) Splitting of a Twofold Degenerate E State in the Group Td.- c) The Case of a T2 State.- d) Lowering of Symmetry and Orientational Degeneracy.- 1.5.2 Dipolar Transitions.- 1.5.3 Other Excitations.- 2. Effective Mass Theory.- 2.1 Simplified Presentation.- 2.2 Derivation in the One-Band Case.- 2.2.1 The Case of One Band with One Extremum.- 2.2.2 The Case of Equivalent Extrema.- 2.2.3 Validity of the Approximations.- 2.2.4 Generalization to the Case of a Degenerate Extremum.- 2.3 Pairing Effects.- 2.3.1 An Electron Bound to a Donor-Acceptor Pair.- 2.3.2 The Neutral Donor-Acceptor Pair.- 2.3.3 Density of States.- 2.4 Experimental Observation of Shallow Levels.- 2.4.1 Experimental Techniques.- 2.4.2 Results.- 2.4.3 Pairing Effects.- 3. Simpte Theory of Deep Levels in Semiaonductors.- 3.1 The Elementary Tight-Binding Theory of Defects.- 3.1.1 Basic Principles of the Tight-Binding Approximation.- 3.1.2 The Tight Binding Matrix Elements for Sp Covalently Bonded Solids.- 3.1.3 Formation of the Band Structure.- a) The Molecular Model.- b) Broadening Effects: A Simple Description.- c) Refi nements.- 3.1.4 The Vacancy in a Covalently Bonded Linear Chain.- 3.1.5 The Vacancy in a Covalent Crystal.- 3.1.6 The Interstitial.- 3.1.7 The Substitutional Impurity.- 3.2 Green's Function Theory of Defects: Tight Binding Application.- 3.2.1 Relation Between the Resolvant or Green's Operator and the Density of States.- 3.2.2 Local Densities of States and Green's Functions.- 3.2.3 Green's Function Treatment of Local Perturbations.- 3.2.4 Application to the Koster-Slater Problem.- 3.2.5 Green's Function and Method of Moments.- 3.2.6 Green's Function for a Semiinfinite Chain: Application to the Vacancy.- 3.2.7 Refined Tight-Binding Green's Functions Treatments.- 4. Many-Electron Effects and Sophisticated Theories of Deep Levels.- 4.1 One-Electron Self-Consistent Calculations.- 4.1.1 Charge-Dependent Tight-Binding Treatment.- 4.1.2 The Model of HALDANE and ANDERSON.- 4.1. 3 The CNDO Method.- 4.1.4 The Extended Hückel Theory.- 4.1.5 Self-Consistent Green's Function Calculations.- 4.1.6 A Thomas-Fermi Interpretation of the Self-Consistent Potential.- 4.2 Many-electron effects. The configuration interaction.- 4.2.1 The H2 Molecule.- a) Notations.- b) Exact Solution.- c) Restricted Hartree-Fock Solution and Configuration Interaction.- d) Unrestricted Hartree-Fock Solution.- e) Direct Use of Symmetry Properties.- 4.2.2 The Vacancy in Covalent Systems.- a) Description of the Model.- b) The Different Configurations.- c) Discussion of the Results.- d) Validity of the One-Electron Theories.- 5. Vibrational Properties and Entropy.- 5. 1 Vibrational Modes.- 5.1.1 The Dynamical Matrix.- 5.1.2 The Linear Chain.- 5.1.3 The Linear Chain with Two Atoms Per Unit Cell.- 5.1.4 The Covalent Solid.- 5.2 Localized Modes Due to Defects.- 5.2.1 The Mass Defect in the Monoatomic Linear Chain.- 5.2.2 The Mass Defect in the General Case.- 5.2.3 Expansion of the Greeen's Function.- 5.2.4 The Vacancy.- 5.3 Experimental Determination of Vibrational Modes.- 5.4 Vibrational Entropy.- 5.4.1 General Expression.- 5.4.2 Expansion in Moments.- 5.4.3 The Vacancy.- 5.4.4 The Vacancy in Covalent Materials.- a) Entropy of Formation.- b) Entropy of Migration.- 5.4.5 Exact Calculation of Entropy Changes.- 5.4.6 Experimental Determination of Entropies.- 6. Thermodynamics of Defects.- 6.1 Enthalpy of Formation.- 6.2 Defect Concentration at Thermal Equilibrium.- 6.2.1 The Case of One Kind of Defect at Low Concentration.- 6.2.2 Generalization to Several Kinds of Independent Defects and Internal Degrees of Freedom.- 6.2.3 Equilibriurn Between Charged States.- 6.2.4 Defects in Short-Range Interaction.- 6.2.5 The case of Long-Range Interaction.- 6.2.6 Defect Concentration in a Stoechiometric Compound.- 6.3 On the Nature of the Defects Present at Thermal Equilibrium.- 6.4 Experimental Determination of Enthalpies.- 6.5 The Statistical Distribution of Donor-Acceptor Pairs.- 7. Defect Migration and Diffusion.- 7.1 Jump Probability and Migration Energy.- 7.1.1 The Rate Theory.- 7.1.2 The Dynamical Theory.- 7.2 Experimental Determination of Migration Enthalpies.- 7.3 Charge-State Effects on Defect Migration.- 7.3.1 Weiser's Theory.- 7.3.2 Ionization Enhanced Migration.- a) Electrostriction Mechanism.- b) Normal Ionization Enhanced Mechanism.- c) Athermal Ionization Enhanced Mechanism.- d) Energy Released Mechanism.- 7.4 Diffusion.- 7.4.1 Fick's Law.- 7.4.2 Experimental Determination of a Diffusion Coefficient.- 7.4.3 Self-Diffusion.- 7.4.4 Substitutional Impurity Diffusion.- 7.4.5 Interstitial Impurity Diffusion.- References.