This document provides an overview of chapter 3 of the textbook "Introduction to the Quantum Theory of Solids". It discusses:
1) How the discrete energy levels of isolated atoms split into allowed and forbidden energy bands as atoms are brought closer together to form a solid. This is due to the wavefunctions of electrons overlapping and interacting between neighboring atoms.
2) How the width of energy bands increases as more atoms are added, resulting in a quasi-continuous distribution of energies.
3) Examples of how this splitting occurs for sodium and silicon, forming their distinctive band structure diagrams with allowed valence and conduction bands separated by a forbidden gap.
4) How the minimum energy configuration of electrons in solids is determined
This document discusses the density of states function for semiconductors. It begins by deriving the density of states function for a free electron gas confined to a 3D infinite potential well. It then extends this concept to semiconductors by approximating the conduction and valence band energy-momentum relationships as parabolic functions, with electron and hole effective masses. Finally, it provides an example calculation of the total number of states in silicon between the conduction band edge and the edge plus thermal energy (kT) at 300K.
Chapter3 introduction to the quantum theory of solidsK. M.
The document provides an introduction to the quantum theory of solids, including:
1. How allowed and forbidden energy bands form in solids due to the interaction of atomic electron wave functions when atoms are brought close together in a crystal lattice.
2. Electrical conduction in solids is explained using the concept of electron effective mass and holes, within the framework of the energy band model.
3. The Kronig-Penney model is used to quantitatively relate the energy, wave number, and periodic potential within a solid, resulting in allowed and forbidden energy bands.
The document discusses molecular orbital theory of bonding in hydrogen molecules. It explains how the bonding molecular orbital is formed from the linear combination of atomic orbitals of the two hydrogen atoms. As the hydrogen atoms approach each other, their atomic orbitals overlap to form the lower energy bonding molecular orbital which is filled by the two electrons, and the higher energy antibonding molecular orbital which remains empty. A diagram and energy level diagram are provided to illustrate this process.
This document discusses the history and properties of dielectric materials. It covers:
- The early history of dielectric materials from the 18th century experiments of Cunaeus and Musschenbroek to Maxwell's unified electromagnetic theory.
- The dynamic range over which dielectric spectroscopy can measure relaxation processes, from 10-5 to 10-12 seconds.
- The four main polarization mechanisms in dielectric materials: electronic, ionic, orientation, and space-charge polarization.
- Key dielectric concepts such as permittivity, dielectric constant, electric susceptibility, and the Clausius-Mossotti relation.
This document is a paper on inorganic chemistry that discusses line defects in solids, specifically edge and screw dislocations. It was written by Sakshi Mishra for their M.Sc. Part 2, Semester 3 course. The paper references common solid state chemistry textbooks and building construction resources.
The document provides an overview of the electronic band structure of solids from both the Sommerfeld and Bloch perspectives. It discusses key concepts such as:
1) Quantum numbers that label the eigenenergies and eigenfunctions of the Hamiltonian.
2) Bloch's theorem which describes the wavefunction of an electron as a plane wave modulated by a periodic function with the periodicity of the crystal lattice.
3) The band structure and energy levels that arise from Bloch's treatment, which has no simple explicit form unlike Sommerfeld's free electron model.
4) Key differences between the Sommerfeld and Bloch approaches regarding concepts like the density of states, Fermi surface, and wavefunctions
The document discusses the pn junction diode. It describes the ideal current-voltage relationship of a pn junction diode. When a forward bias is applied, it lowers the potential barrier and allows electrons from the n-region and holes from the p-region to be injected across the depletion region, becoming minority carriers. This creates an excess minority carrier concentration that diffuses away from the junction and recombines. The current is calculated from the minority carrier diffusion currents at the edges of the depletion region. The total current is expressed as a function of the applied voltage and follows an exponential relationship.
This document discusses the density of states function for semiconductors. It begins by deriving the density of states function for a free electron gas confined to a 3D infinite potential well. It then extends this concept to semiconductors by approximating the conduction and valence band energy-momentum relationships as parabolic functions, with electron and hole effective masses. Finally, it provides an example calculation of the total number of states in silicon between the conduction band edge and the edge plus thermal energy (kT) at 300K.
Chapter3 introduction to the quantum theory of solidsK. M.
The document provides an introduction to the quantum theory of solids, including:
1. How allowed and forbidden energy bands form in solids due to the interaction of atomic electron wave functions when atoms are brought close together in a crystal lattice.
2. Electrical conduction in solids is explained using the concept of electron effective mass and holes, within the framework of the energy band model.
3. The Kronig-Penney model is used to quantitatively relate the energy, wave number, and periodic potential within a solid, resulting in allowed and forbidden energy bands.
The document discusses molecular orbital theory of bonding in hydrogen molecules. It explains how the bonding molecular orbital is formed from the linear combination of atomic orbitals of the two hydrogen atoms. As the hydrogen atoms approach each other, their atomic orbitals overlap to form the lower energy bonding molecular orbital which is filled by the two electrons, and the higher energy antibonding molecular orbital which remains empty. A diagram and energy level diagram are provided to illustrate this process.
This document discusses the history and properties of dielectric materials. It covers:
- The early history of dielectric materials from the 18th century experiments of Cunaeus and Musschenbroek to Maxwell's unified electromagnetic theory.
- The dynamic range over which dielectric spectroscopy can measure relaxation processes, from 10-5 to 10-12 seconds.
- The four main polarization mechanisms in dielectric materials: electronic, ionic, orientation, and space-charge polarization.
- Key dielectric concepts such as permittivity, dielectric constant, electric susceptibility, and the Clausius-Mossotti relation.
This document is a paper on inorganic chemistry that discusses line defects in solids, specifically edge and screw dislocations. It was written by Sakshi Mishra for their M.Sc. Part 2, Semester 3 course. The paper references common solid state chemistry textbooks and building construction resources.
The document provides an overview of the electronic band structure of solids from both the Sommerfeld and Bloch perspectives. It discusses key concepts such as:
1) Quantum numbers that label the eigenenergies and eigenfunctions of the Hamiltonian.
2) Bloch's theorem which describes the wavefunction of an electron as a plane wave modulated by a periodic function with the periodicity of the crystal lattice.
3) The band structure and energy levels that arise from Bloch's treatment, which has no simple explicit form unlike Sommerfeld's free electron model.
4) Key differences between the Sommerfeld and Bloch approaches regarding concepts like the density of states, Fermi surface, and wavefunctions
The document discusses the pn junction diode. It describes the ideal current-voltage relationship of a pn junction diode. When a forward bias is applied, it lowers the potential barrier and allows electrons from the n-region and holes from the p-region to be injected across the depletion region, becoming minority carriers. This creates an excess minority carrier concentration that diffuses away from the junction and recombines. The current is calculated from the minority carrier diffusion currents at the edges of the depletion region. The total current is expressed as a function of the applied voltage and follows an exponential relationship.
Solid state physics (schottkey and frenkel)abi sivaraj
This document discusses different types of lattice defects in crystals. It describes Schottky and Frenkel defects. A Schottky defect occurs in ionic crystals when equal numbers of oppositely charged ions leave their lattice sites, creating vacancies while maintaining overall charge neutrality. A Frenkel defect occurs when an atom moves from its lattice site to an interstitial site, producing a vacancy and interstitial. Schottky defects lower the crystal's density while Frenkel defects do not change density.
1. Drude's classical theory of electrical conduction models a metal as composed of stationary ions and free-moving valence electrons. Electrons move randomly between collisions with ions or other electrons.
2. The drift velocity of electrons in an electric field is proportional to the field strength, resulting in a net current of electrons. This explains metals' conductivity.
3. However, the classical model fails to fully explain experimental observations such as the temperature dependence of resistivity and heat capacity. Quantum mechanics provides a more accurate description of electron behavior in metals.
The document outlines the syllabus for a semiconductor subject. It includes 15 chapters that cover topics like crystal structure of solids, quantum mechanics, equilibrium carrier transport, pn junctions, bipolar transistors, MOSFETs, and optical and power devices. It also provides details on some key concepts for semiconductors like electrical conductivity and resistivity. Examples are given for different crystal structures including simple cubic, body-centered cubic, and face-centered cubic lattices. Miller indices are introduced for representing crystal planes.
This theory, developed by Bardeen, Cooper and Schrieffer, states that electrons experience an attractive interaction through the lattice that overcomes their normal repulsive interaction, forming Cooper pairs. At low temperatures, these pairs move without resistance through the lattice, causing the material to become a superconductor. The electron-lattice-electron interaction must be stronger than the direct electron-electron interaction for superconductivity to occur.
This document summarizes key concepts about semiconductors including:
1) Semiconductors have electrical properties that can be controlled through doping with impurities, allowing them to be used in transistors and integrated circuits. Their purity must be very high, measured in parts per billion.
2) Intrinsic semiconductors like silicon have a band gap between the valence and conduction bands that requires energy like heat or light to excite electrons across.
3) Extrinsic semiconductors are doped with impurities that introduce electrons (n-type) or holes (p-type) that determine the semiconductor's conductivity type and shift the Fermi level. Charge neutrality must be maintained.
Electronic band structures in crystals can be understood using Bloch's theorem. Bloch's theorem states that the eigenstates of electrons moving in a periodic potential can be written as a plane wave multiplied by a periodic function. This leads to the formation of allowed energy bands separated by forbidden band gaps. The energy bands arise because the electron momentum is restricted to the first Brillouin zone of the crystal lattice. Bloch's theorem provides insights into the distinction between metals, semiconductors and insulators by explaining whether the Fermi energy lies in an allowed band or forbidden band gap.
1) The document discusses carrier transport in semiconductors, including drift and diffusion currents. Carrier drift occurs due to an electric field and is characterized by carrier mobility, while diffusion is due to concentration gradients and characterized by the diffusion coefficient.
2) Mobility is affected by phonon and ionized impurity scattering. The net mobility is the sum of these scattering components. Conductivity is directly proportional to carrier concentration and mobility.
3) The Hall effect can be used to determine the type of semiconductor (n-type or p-type), carrier concentration, and carrier mobility. Measurement of the Hall voltage polarity indicates type, and its magnitude relates to concentration and mobility.
Engineering physics 5(Quantum free electron theory)Nexus
This document discusses quantum free electron theory, which was developed by Sommerfeld in 1928 to explain physical properties that classical free electron theory could not. Quantum theory permits only a fraction of electrons to gain energy, as opposed to classical theory. It introduces the density of states and Fermi-Dirac distribution function to determine the number of electrons in a given energy range. The effect of temperature on the distribution function is also examined, showing that only states close to the Fermi level are affected at higher temperatures. An expression is provided for electrical conductivity based on quantum concepts involving the Fermi velocity and mean free path.
This document provides an overview of semiconductor theory and devices. It begins by introducing the three categories of solids based on electrical conductivity: conductors, semiconductors, and insulators. It then discusses band theory, which models the allowed energy states in solids as continuous bands separated by forbidden gaps. Semiconductors are defined as having energy gaps small enough for thermal excitation of electrons between bands. The document covers models like the Kronig-Penney model that explain energy gaps. It also discusses how temperature affects resistivity in semiconductors by increasing the number of electrons excited into the conduction band.
The document discusses the density of states in two-dimensional systems. It explains that the density of states function describes the number of available energy states in a system and is essential for determining carrier concentrations and distributions. In semiconductors, carrier motion is limited to two, one, or zero spatial dimensions, requiring the density of states to be known in quantum wells (2D), quantum wires (1D), and quantum dots (0D). The document then focuses on the density of states in 2D systems, noting that it is independent of energy and depends on the number of quantized levels in the confined dimension.
This document discusses the key principles of quantum physics including:
(1) The wave-particle duality of microparticles like electrons described by de Broglie's equation.
(2) Energy quantization described by Planck's equation.
(3) Heisenberg's uncertainty principle.
It describes how Schrodinger's equation is used to model the wave-like behavior of electrons in solids. The energy and behavior of electrons is quantized based on solutions to Schrodinger's equation under different boundary conditions, such as electrons confined in a potential well or interacting with a potential barrier. Quantum theory was needed to fully explain properties of electrons in solids and failures of classical free electron theory
This document provides an introduction to quantum mechanics concepts including:
1. It describes Schrodinger's wave equation and its applications, including quantized energy levels and tunneling effects.
2. Wave-particle duality is discussed through experiments demonstrating the wave-like and particle-like properties of electrons.
3. The uncertainty principle and solutions to Schrodinger's wave equation for simple potential wells are presented, showing energy levels are quantized.
This document provides an overview of quantum mechanics (QM) calculation methods. It discusses molecular mechanics, wavefunction methods, electron density methods, including correlation, Hartree-Fock theory, semi-empirical methods, density functional theory, and their relative speed and accuracy. Key aspects that can be calculated using these methods are also listed, such as molecular orbitals, electron density, geometry, energies, spectroscopic properties, and more. Basis sets and handling open-shell systems in calculations are also covered.
This document discusses the density of states (DoS) for bulk semiconductors. It begins by defining DoS as the number of available energy states per unit energy interval per unit dimension in real space. It then derives the DoS for bulk semiconductors using the Bloch theorem and shows that the DoS is proportional to the square root of energy. Finally, it defines the effective DoS, which accounts for occupancy based on the Fermi-Dirac distribution.
Derive the thermal-equilibrium concentrations of electrons and holes in a semiconductor as a function of the Fermi energy level.
Discuss the process by which the properties of a semiconductor material can be favorably altered by adding specific impurity atoms to the semiconductor.
Determine the thermal-equilibrium concentrations of electrons and holes in a semiconductor as a function of the concentration of dopant atoms added to the semiconductor.
Determine the position of the Fermi energy level as a function of the concentrations of dopant atoms added to the semiconductor.
This document provides an introduction to statistical mechanics and different types of statistics. It discusses classical statistics, which includes Maxwell-Boltzmann statistics, and quantum statistics, which includes Bose-Einstein (B-E) and Fermi-Dirac (F-D) statistics. Maxwell-Boltzmann statistics treats particles as distinguishable and applies to ideal gases, while B-E and F-D statistics treat particles as indistinguishable and apply to photons/bosons and electrons/fermions, respectively. The key differences between the statistics are whether particles can occupy the same state (B-E allows multiple occupancy, F-D allows only single occupancy) and the formulas that describe the most probable distribution of particles
Quantum tunnelling is a quantum mechanical phenomenon where a particle tunnels through a barrier that it classically could not surmount. Friedrich Hund first used quantum tunnelling to explain molecular spectra in 1927. George Gamow first applied it to calculate alpha decay in 1928. Max Born recognized it as a general result of quantum mechanics. Important applications include scanning tunneling microscopes, tunnel diodes, and the Josephson effect in superconductors. Recent research has explored tunnelling in other systems and potential uses in quantum computing.
Electrical transport and magnetic interactions in 3d and 5d transition metal ...ABDERRAHMANE REGGAD
The document discusses electrical transport and magnetic interactions in 3d and 5d transition metal oxides. It summarizes that for decades, transition metal oxides have been explored where exotic states like high-Tc superconductivity and colossal magnetoresistance emerge due to cooperative interactions between spin, charge, and orbital degrees of freedom. The document then examines various phenomena in transition metal oxides including Mott insulators, double exchange mechanism, and the Kitaev-Heisenberg model observed in iridate compounds like Na2IrO3 which may realize a spin liquid ground state.
This document discusses the magnetic properties of materials. It begins by explaining that magnetism in solids originates from the orbital and spin motions of electrons and spins of nuclei. It then defines key terms like magnetization, magnetic moment, magnetic susceptibility, permeability, and Curie temperature. The document classifies magnetic materials into five types - diamagnetism, paramagnetism, ferromagnetism, antiferromagnetism, and ferrimagnetism - and provides examples of each. It concludes by explaining Langevin's classical theory of diamagnetism using an electron orbit model.
Semiconductor theory describes how small amounts of impurities can be added to intrinsic semiconductors to create n-type and p-type materials. N-type semiconductors are created by adding elements with extra electrons, while p-type are created by adding elements with electron deficiencies. The junction between a p-type and n-type material allows current to flow in only one direction, forming the basis for important semiconductor devices such as diodes, transistors, and solar cells.
This document discusses band theory and semiconductor theory. It begins by introducing band theory, which models the allowed energy states in solids as continuous bands separated by forbidden gaps. Semiconductors are defined as having small band gaps (<1 eV), allowing thermal excitation of electrons across the gap. This explains their decreasing resistivity with increasing temperature. The Kronig-Penney model is presented to illustrate how periodic lattice potentials create energy bands and gaps. Semiconductors have filled valence bands separated from conduction bands by small gaps, whereas insulators have larger gaps preventing electrical conduction. Empirical models are discussed for describing the temperature dependence of resistivity in semiconductors.
Solid state physics (schottkey and frenkel)abi sivaraj
This document discusses different types of lattice defects in crystals. It describes Schottky and Frenkel defects. A Schottky defect occurs in ionic crystals when equal numbers of oppositely charged ions leave their lattice sites, creating vacancies while maintaining overall charge neutrality. A Frenkel defect occurs when an atom moves from its lattice site to an interstitial site, producing a vacancy and interstitial. Schottky defects lower the crystal's density while Frenkel defects do not change density.
1. Drude's classical theory of electrical conduction models a metal as composed of stationary ions and free-moving valence electrons. Electrons move randomly between collisions with ions or other electrons.
2. The drift velocity of electrons in an electric field is proportional to the field strength, resulting in a net current of electrons. This explains metals' conductivity.
3. However, the classical model fails to fully explain experimental observations such as the temperature dependence of resistivity and heat capacity. Quantum mechanics provides a more accurate description of electron behavior in metals.
The document outlines the syllabus for a semiconductor subject. It includes 15 chapters that cover topics like crystal structure of solids, quantum mechanics, equilibrium carrier transport, pn junctions, bipolar transistors, MOSFETs, and optical and power devices. It also provides details on some key concepts for semiconductors like electrical conductivity and resistivity. Examples are given for different crystal structures including simple cubic, body-centered cubic, and face-centered cubic lattices. Miller indices are introduced for representing crystal planes.
This theory, developed by Bardeen, Cooper and Schrieffer, states that electrons experience an attractive interaction through the lattice that overcomes their normal repulsive interaction, forming Cooper pairs. At low temperatures, these pairs move without resistance through the lattice, causing the material to become a superconductor. The electron-lattice-electron interaction must be stronger than the direct electron-electron interaction for superconductivity to occur.
This document summarizes key concepts about semiconductors including:
1) Semiconductors have electrical properties that can be controlled through doping with impurities, allowing them to be used in transistors and integrated circuits. Their purity must be very high, measured in parts per billion.
2) Intrinsic semiconductors like silicon have a band gap between the valence and conduction bands that requires energy like heat or light to excite electrons across.
3) Extrinsic semiconductors are doped with impurities that introduce electrons (n-type) or holes (p-type) that determine the semiconductor's conductivity type and shift the Fermi level. Charge neutrality must be maintained.
Electronic band structures in crystals can be understood using Bloch's theorem. Bloch's theorem states that the eigenstates of electrons moving in a periodic potential can be written as a plane wave multiplied by a periodic function. This leads to the formation of allowed energy bands separated by forbidden band gaps. The energy bands arise because the electron momentum is restricted to the first Brillouin zone of the crystal lattice. Bloch's theorem provides insights into the distinction between metals, semiconductors and insulators by explaining whether the Fermi energy lies in an allowed band or forbidden band gap.
1) The document discusses carrier transport in semiconductors, including drift and diffusion currents. Carrier drift occurs due to an electric field and is characterized by carrier mobility, while diffusion is due to concentration gradients and characterized by the diffusion coefficient.
2) Mobility is affected by phonon and ionized impurity scattering. The net mobility is the sum of these scattering components. Conductivity is directly proportional to carrier concentration and mobility.
3) The Hall effect can be used to determine the type of semiconductor (n-type or p-type), carrier concentration, and carrier mobility. Measurement of the Hall voltage polarity indicates type, and its magnitude relates to concentration and mobility.
Engineering physics 5(Quantum free electron theory)Nexus
This document discusses quantum free electron theory, which was developed by Sommerfeld in 1928 to explain physical properties that classical free electron theory could not. Quantum theory permits only a fraction of electrons to gain energy, as opposed to classical theory. It introduces the density of states and Fermi-Dirac distribution function to determine the number of electrons in a given energy range. The effect of temperature on the distribution function is also examined, showing that only states close to the Fermi level are affected at higher temperatures. An expression is provided for electrical conductivity based on quantum concepts involving the Fermi velocity and mean free path.
This document provides an overview of semiconductor theory and devices. It begins by introducing the three categories of solids based on electrical conductivity: conductors, semiconductors, and insulators. It then discusses band theory, which models the allowed energy states in solids as continuous bands separated by forbidden gaps. Semiconductors are defined as having energy gaps small enough for thermal excitation of electrons between bands. The document covers models like the Kronig-Penney model that explain energy gaps. It also discusses how temperature affects resistivity in semiconductors by increasing the number of electrons excited into the conduction band.
The document discusses the density of states in two-dimensional systems. It explains that the density of states function describes the number of available energy states in a system and is essential for determining carrier concentrations and distributions. In semiconductors, carrier motion is limited to two, one, or zero spatial dimensions, requiring the density of states to be known in quantum wells (2D), quantum wires (1D), and quantum dots (0D). The document then focuses on the density of states in 2D systems, noting that it is independent of energy and depends on the number of quantized levels in the confined dimension.
This document discusses the key principles of quantum physics including:
(1) The wave-particle duality of microparticles like electrons described by de Broglie's equation.
(2) Energy quantization described by Planck's equation.
(3) Heisenberg's uncertainty principle.
It describes how Schrodinger's equation is used to model the wave-like behavior of electrons in solids. The energy and behavior of electrons is quantized based on solutions to Schrodinger's equation under different boundary conditions, such as electrons confined in a potential well or interacting with a potential barrier. Quantum theory was needed to fully explain properties of electrons in solids and failures of classical free electron theory
This document provides an introduction to quantum mechanics concepts including:
1. It describes Schrodinger's wave equation and its applications, including quantized energy levels and tunneling effects.
2. Wave-particle duality is discussed through experiments demonstrating the wave-like and particle-like properties of electrons.
3. The uncertainty principle and solutions to Schrodinger's wave equation for simple potential wells are presented, showing energy levels are quantized.
This document provides an overview of quantum mechanics (QM) calculation methods. It discusses molecular mechanics, wavefunction methods, electron density methods, including correlation, Hartree-Fock theory, semi-empirical methods, density functional theory, and their relative speed and accuracy. Key aspects that can be calculated using these methods are also listed, such as molecular orbitals, electron density, geometry, energies, spectroscopic properties, and more. Basis sets and handling open-shell systems in calculations are also covered.
This document discusses the density of states (DoS) for bulk semiconductors. It begins by defining DoS as the number of available energy states per unit energy interval per unit dimension in real space. It then derives the DoS for bulk semiconductors using the Bloch theorem and shows that the DoS is proportional to the square root of energy. Finally, it defines the effective DoS, which accounts for occupancy based on the Fermi-Dirac distribution.
Derive the thermal-equilibrium concentrations of electrons and holes in a semiconductor as a function of the Fermi energy level.
Discuss the process by which the properties of a semiconductor material can be favorably altered by adding specific impurity atoms to the semiconductor.
Determine the thermal-equilibrium concentrations of electrons and holes in a semiconductor as a function of the concentration of dopant atoms added to the semiconductor.
Determine the position of the Fermi energy level as a function of the concentrations of dopant atoms added to the semiconductor.
This document provides an introduction to statistical mechanics and different types of statistics. It discusses classical statistics, which includes Maxwell-Boltzmann statistics, and quantum statistics, which includes Bose-Einstein (B-E) and Fermi-Dirac (F-D) statistics. Maxwell-Boltzmann statistics treats particles as distinguishable and applies to ideal gases, while B-E and F-D statistics treat particles as indistinguishable and apply to photons/bosons and electrons/fermions, respectively. The key differences between the statistics are whether particles can occupy the same state (B-E allows multiple occupancy, F-D allows only single occupancy) and the formulas that describe the most probable distribution of particles
Quantum tunnelling is a quantum mechanical phenomenon where a particle tunnels through a barrier that it classically could not surmount. Friedrich Hund first used quantum tunnelling to explain molecular spectra in 1927. George Gamow first applied it to calculate alpha decay in 1928. Max Born recognized it as a general result of quantum mechanics. Important applications include scanning tunneling microscopes, tunnel diodes, and the Josephson effect in superconductors. Recent research has explored tunnelling in other systems and potential uses in quantum computing.
Electrical transport and magnetic interactions in 3d and 5d transition metal ...ABDERRAHMANE REGGAD
The document discusses electrical transport and magnetic interactions in 3d and 5d transition metal oxides. It summarizes that for decades, transition metal oxides have been explored where exotic states like high-Tc superconductivity and colossal magnetoresistance emerge due to cooperative interactions between spin, charge, and orbital degrees of freedom. The document then examines various phenomena in transition metal oxides including Mott insulators, double exchange mechanism, and the Kitaev-Heisenberg model observed in iridate compounds like Na2IrO3 which may realize a spin liquid ground state.
This document discusses the magnetic properties of materials. It begins by explaining that magnetism in solids originates from the orbital and spin motions of electrons and spins of nuclei. It then defines key terms like magnetization, magnetic moment, magnetic susceptibility, permeability, and Curie temperature. The document classifies magnetic materials into five types - diamagnetism, paramagnetism, ferromagnetism, antiferromagnetism, and ferrimagnetism - and provides examples of each. It concludes by explaining Langevin's classical theory of diamagnetism using an electron orbit model.
Semiconductor theory describes how small amounts of impurities can be added to intrinsic semiconductors to create n-type and p-type materials. N-type semiconductors are created by adding elements with extra electrons, while p-type are created by adding elements with electron deficiencies. The junction between a p-type and n-type material allows current to flow in only one direction, forming the basis for important semiconductor devices such as diodes, transistors, and solar cells.
This document discusses band theory and semiconductor theory. It begins by introducing band theory, which models the allowed energy states in solids as continuous bands separated by forbidden gaps. Semiconductors are defined as having small band gaps (<1 eV), allowing thermal excitation of electrons across the gap. This explains their decreasing resistivity with increasing temperature. The Kronig-Penney model is presented to illustrate how periodic lattice potentials create energy bands and gaps. Semiconductors have filled valence bands separated from conduction bands by small gaps, whereas insulators have larger gaps preventing electrical conduction. Empirical models are discussed for describing the temperature dependence of resistivity in semiconductors.
The document discusses the transition from discrete atomic energy levels to energy bands in crystalline solids. As atoms come together to form a crystal lattice, the discrete energy levels of individual atoms broaden and overlap, forming continuous bands of allowed energies separated by forbidden gaps. Electrons near the top of the valence band can be excited into the conduction band, where they behave as free particles able to move through the solid and conduct electricity. Covalent bonding between atoms is also explained using the concept of energy bands.
The document discusses the band theory of solids, which explains how the discrete energy levels of individual atoms combine to form energy bands in solids. When many atoms come together to form a solid, their atomic orbitals overlap to form molecular orbitals with many near-continuous energy levels. This results in energy bands with small gaps between a very large number of allowed energy values. The band theory can be used to understand why some materials are conductors and others are insulators or semiconductors.
Introduction
Formation Of Bond.
Formation Of Band.
Role Of Pauli Exclusion Principle.
Fermi Dirac Distribution Equation
Classification Of Material In Term Of Energy Band Diagram.
Intrinsic Semiconductor.
a)Drive Density Of State
b)Drive Density Of Free Carrier.
c)Determination Of Fermi Level Position
Extrinsic Semiconductor.
a) N Type Extrinsic Semiconductor
b) P Type Extrinsic Semiconductor
Compensated semiconductor.
E Vs. Diagram.
Direct and Indirect Band Gap.
Degenerated and Non-degenerated.
PN Junction.
This document provides an overview of semiconductor theory and band theory of solids. It discusses how band theory can explain the differing electrical properties of conductors, semiconductors, and insulators based on their band structure and energy gaps. Semiconductors have a small energy gap between the valence and conduction bands that can be overcome by thermal excitation, allowing some electrons to reach the conduction band. Doping semiconductors with impurities can create n-type or p-type materials by introducing extra electrons or holes that increase conductivity. The document also covers thermoelectric effects in semiconductors.
The document discusses semiconductor theory and band theory of solids. It introduces band theory, which explains that in solids electron energy levels split into nearly continuous bands separated by forbidden gaps. Semiconductors have a small band gap (<1eV) allowing thermal excitation of electrons across the gap. This explains their decreasing resistivity with increasing temperature as electrons jump from the valence to conduction band. The Kronig-Penney model further illustrates band structure and forbidden gaps in semiconductors.
1) The document discusses energy band theory and how it relates to the electrical properties of semiconductors, insulators, and metals. It explains that semiconductors have a small forbidden band gap between the valence and conduction bands, allowing thermal or electromagnetic excitation of electrons.
2) The concept of effective mass is introduced, where electrons in a crystal lattice behave as if they have a different mass than free electrons due to the crystal potential. Effective mass depends on the curvature of electron energy-momentum diagrams.
3) Direct and indirect band gap materials are distinguished based on whether the minimum of the conduction band and maximum of the valence band occur at the same or different crystal momentum values.
Introduction to the structure of atoms from the view of a chemist - what are neutrons protons and electrons and how are they organized ? How are electrons organized - in 3 quantum numbers. Experimental evidence from the Bohr model.
This document provides an overview of atomic structure and bonding. It begins by defining key atomic properties like atomic number, mass, isotopes and electronic configuration. It then describes the Bohr and wave mechanical models of the atom. The document explains ionic, covalent and metallic bonding between atoms. It also discusses secondary bonding interactions and relates atomic properties to position on the periodic table.
Molecular orbital theory provides an approach to calculate molecular orbitals through a variational method. This involves taking linear combinations of atomic orbitals to form molecular orbitals. Electrons occupy these molecular orbitals according to certain rules. The molecular orbital theory can explain properties such as why some molecules are paramagnetic that valence bond theory cannot. Calculating molecular orbitals variationally involves using trial wave functions in the Schrodinger equation to find the lowest possible energy state.
This document provides an overview of atomic structure and quantum mechanics. It discusses early atomic models proposed by Rutherford and Bohr and limitations they faced. It then introduces the quantum mechanical model, which describes electrons as existing in distinct energy levels and orbitals. The document explains how electron configurations are written based on Aufbau principle, Hund's rule and Pauli exclusion principle. It also discusses atomic spectra and how light emitted during electron energy level transitions can be used to identify elements.
This document provides an overview of solid state physics concepts including the structure of solids, crystal structures, lattice structures, unit cells, Miller indices, and band theory. It begins by defining crystalline and amorphous solids based on the arrangement of atoms. Crystalline solids have long-range order while amorphous solids are randomly arranged. Key concepts covered include space lattices, Bravais lattices, unit cells, and Miller indices for describing plane orientations in crystal structures. The document also introduces band theory concepts such as conduction bands, valence bands, forbidden gaps, and the Kronig-Penney model for explaining electron behavior in periodic potentials. Intrinsic and extrinsic semiconductors are defined based on
1. The document discusses intrinsic and extrinsic semiconductors. Intrinsic semiconductors are pure semiconductors without dopants, while extrinsic semiconductors are doped by adding donor or acceptor atoms.
2. Extrinsic semiconductors are classified as n-type or p-type based on whether electrons or holes are the majority carrier. N-type uses donor dopants to increase electron concentration, while p-type uses acceptor dopants to increase hole concentration.
3. Semiconductor devices like diodes and transistors make use of the different properties of n-type and p-type extrinsic semiconductors. Diodes consist of a p-n junction
This Presentation "Energy band theory of solids" will help you to Clarify your doubts and Enrich your Knowledge. Kindly use this presentation as a Reference and utilize this presentation
This document provides information about atomic structure and bonding. It discusses:
- The basic components of atoms (protons, neutrons, electrons) and their properties.
- Electron orbitals and energy levels within atoms. Electrons occupy discrete shells and energy levels.
- Covalent bonding between atoms, where valence electrons are shared. This forms crystalline structures.
- Semiconductors like silicon and germanium have incomplete valence shells, so their atoms form covalent bonds to share electrons until each has a full outer shell. This tightly binds electrons.
- Doping semiconductors with impurities introduces extra electrons or holes, increasing conductivity and making the material n-type or p-
Atomic Structure and chemical BONDING.pptxSesayAlimamy
This document discusses fundamentals of atomic structure and interatomic bonding. It covers topics like atomic models, quantum numbers, electron configurations, and the periodic table. The key types of atomic bonding are also summarized, including ionic, covalent, metallic, hydrogen and van der Waals bonds. Interatomic forces are described as a function of separation distance, including both attractive and repulsive forces.
Spectroscopy is the study of the interaction of electromagnetic radiation with matter. There are different types of electromagnetic waves that make up the electromagnetic spectrum, including gamma rays, x-rays, ultraviolet light, visible light, infrared radiation, and radio waves. Spectroscopy techniques take advantage of the fact that molecules absorb specific wavelengths of light depending on their structure. Absorption spectra provide information about molecular structure through relationships between absorption wavelengths and transitions between molecular energy levels.
The Kronig-Penney model describes electron behavior in periodic solid structures by modeling the potential as a series of barriers and wells. Solving the time-independent Schrodinger equation under these conditions yields allowed energy bands separated by gaps. As the energy increases, the bands get wider and the gaps get narrower. The model also introduces the concept of effective mass to describe how electrons behave as if they have a different mass inside solid materials. Based on how the energy bands are filled, materials can be classified as metals, insulators, or semiconductors.
Similar to Semiconductor ch.3 part i, Introduction to the Quantum Theory of Solids (20)
Gender and Mental Health - Counselling and Family Therapy Applications and In...PsychoTech Services
A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!
This presentation was provided by Rebecca Benner, Ph.D., of the American Society of Anesthesiologists, for the second session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session Two: 'Expanding Pathways to Publishing Careers,' was held June 13, 2024.
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) CurriculumMJDuyan
(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 𝟏)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
𝐃𝐢𝐬𝐜𝐮𝐬𝐬 𝐭𝐡𝐞 𝐄𝐏𝐏 𝐂𝐮𝐫𝐫𝐢𝐜𝐮𝐥𝐮𝐦 𝐢𝐧 𝐭𝐡𝐞 𝐏𝐡𝐢𝐥𝐢𝐩𝐩𝐢𝐧𝐞𝐬:
- Understand the goals and objectives of the Edukasyong Pantahanan at Pangkabuhayan (EPP) curriculum, recognizing its importance in fostering practical life skills and values among students. Students will also be able to identify the key components and subjects covered, such as agriculture, home economics, industrial arts, and information and communication technology.
𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐍𝐚𝐭𝐮𝐫𝐞 𝐚𝐧𝐝 𝐒𝐜𝐨𝐩𝐞 𝐨𝐟 𝐚𝐧 𝐄𝐧𝐭𝐫𝐞𝐩𝐫𝐞𝐧𝐞𝐮𝐫:
-Define entrepreneurship, distinguishing it from general business activities by emphasizing its focus on innovation, risk-taking, and value creation. Students will describe the characteristics and traits of successful entrepreneurs, including their roles and responsibilities, and discuss the broader economic and social impacts of entrepreneurial activities on both local and global scales.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
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Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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The chapter Lifelines of National Economy in Class 10 Geography focuses on the various modes of transportation and communication that play a vital role in the economic development of a country. These lifelines are crucial for the movement of goods, services, and people, thereby connecting different regions and promoting economic activities.
A Visual Guide to 1 Samuel | A Tale of Two HeartsSteve Thomason
These slides walk through the story of 1 Samuel. Samuel is the last judge of Israel. The people reject God and want a king. Saul is anointed as the first king, but he is not a good king. David, the shepherd boy is anointed and Saul is envious of him. David shows honor while Saul continues to self destruct.
2. Introduction to the Quantum Theory of Solids
3.1 : ALLOWED AND FORBIDDEN ENERGY BANDS.
3.1.1 : Formation of Energy Bands.
3.1.2 The Kronig-Penney Model.
3.1.3 The k-Space Diagram.
3.2 : ELECTRICAL CONDUCTION IN SOLIDS.
3.2.1 The Energy Band and the Bond Model.
3.2.2 Drift Current.
3.2.3 Electron Effective Mass.
3.2.4 Concept of the Hole.
3.2.5 Metals, Insulators, and Semiconductors.
3.3 : EXTENSION TO THREE DIMENSIONS.
3.3.2 Additional Effective Mass Concepts.
3.4 : DENSITY OF STATES FUNCTION.
3.4.1 Mathematical Derivation.
3.5 : STATISTICAL MECHANICS.
3.5.1 Statistical Laws.
3.5.2 The Fermi-Dirac Probability Function.
3.5.3 The Distribution Function and the Fermi Energy.
REVIEW QUESTIONS and PROBLEMS
Chapter consist :
2 Ch. ...... 3
3. 3.1 : ALLOWED AND FORBIDDEN ENERGY BANDS.
3.1.1 : Formation of Energy Bands.
• The radial probability density function for the
lowest electron energy state of the single,
noninteracting hydrogen atom
𝒑(𝒓)
𝒂 𝒐
• The same probability curves for two atoms
that are in close proximity to each other.
𝒑(𝒓)
𝒂 𝒐
𝒑(𝒓)
𝒂 𝒐
• The wave functions of the two atoms
electrons overlap, which means that the
two electrons will interact. This interaction
or perturbation results a discretion in the
quantized energy level splitting into two
discrete energy levels
The splitting of the discrete state into two
states is consistent with the Pauli exclusion
principle.
ElectronEnergy
n= 1
n= 1
3 Ch. ...... 3
4. If we somehow start with a regular periodic arrangement of hydrogen type atoms that are
initially very far apart, and begin pushing the atoms together, the initial quantized energy
level will split into a band of discrete energy levels
ElectronEnergy
Interatomic Distancero
• where the parameter ro represents the
equilibrium interatomic distance in the
crystal
• At the equilibrium interatomic
distance, there is a band of allowed
energies, but within the allowed band,
the energies are at discrete levels.
• The Pauli exclusion principle states that the joining of atoms to form a system (crystal) does
not alter the total number of quantum states regardless of size. However, since no two
electrons can have the same quantum number , the discrete energy must split into a band
of energies in order that each electron can occupy a distinct quantum state.
• As an example,
suppose that we have a system with 1019 one-electron atoms and also suppose that, at the
equilibrium interatomic distance, the width of the allowed energy band is I eV. For simplicity,
we assume that each electron in the system occupies a different energy level and, if the
discrete energy states are equidistant, then the energy levels are separated by l0-19 eV. This
energy difference is extremely small, so that fur all practical purposes, we have a quasi-
continuous energy distribution through the allowed energy band. The fact that 10-19 eV very
small difference between two energy state can be seen from the following example.4 Ch. ...... 3
5. Calculate the change in kinetic energy of an electron when the velocity changes by a small
value. Consider an electron traveling at a velocity of 107 cm/s Assume the velocity increases by
a value of 1 cm/s .
Example :
Solution
The increase in kinetic energy is given by:
∆𝐸 =
1
2
𝑚 ( 𝑣2
2 − 𝑣1
2 )
𝑙𝑒𝑡 𝑣2
= 𝑣1
+∆𝑣
∆𝐸 =
1
2
𝑚 ( 𝑣1 +∆𝑣 )2
− 𝑣1
2
=
1
2
𝑚 𝑣1
2
+ 2 𝑣2
∆𝑣 + ∆𝑣2
− 𝑣1
2
𝐵𝑢𝑡 ∆𝑣 ≪ 𝑣1 , So we have that :
∆𝐸 =
1
2
𝑚 (2𝑣2
∆𝑣) = m 𝑣2
∆𝑣
∆𝐸 = 9.11 × 10−31
105
0.01 = 9.11 × 10−28
𝐽
∆𝐸 =
9.11×10−28
1.6×10−19 = 5.7 × 10−9 𝑒𝑉
A change in velocity of 1 cm/s compared with 107 cm/s results in a change in energy of
5.7x10-19 eV, which is orders of magnitude larger than the change in energy of 10-19 eV
between energy states in the allowed energy hand. This example serves to demonstrate that a
difference in adjacent energy states of 10-19 eV is indeed very small, so that the discrete
energies within an allowed band may be treated as a quasi-continuous distribution.
5 Ch. ...... 3
6. • Consider a regular periodic arrangement of atoms . Suppose the atom in this
imaginary crystal contains electrons up through the n = 3 energy level. If the atoms
are initially very far apart, the electrons in adjacent atoms will not interact and will
occupy the discrete energy levels.
• If the atoms continue to move
closer together, the electrons in the
n = 2 shell may begin to interact and
will also split into a band of allowed
energies.
• If these atoms are brought closer
together, the outermost electrons
in the n = 3 energy shell will begin
to interact initially, so that this
discrete energy level will split into a
band of allowed energies.
n=3
n= 2
• Finally, if the atoms become sufficiently close together, the innermost electrons in
the n = 1 level may interact. so that this energy level may also split into a band of
allowed energies
ElectronEnergy Interatomic Distancero
n= 1
6 Ch. ...... 3
7. The energy levels of the overlapping
electron shells are all slightly altered.
The energy differences are very small,
but enough so that a large number of
electrons can be in close proximity and
still satisfy the Pauli exclusion principle.
The result is the formation of
energy bands, consisting of many
states close together but slightly
split in energy.
7 Ch. ...... 3
8. When you bring two sodium atoms
together, the 3s energy level splits
into two separate energy levels.
Things to note: 4 quantum states
but only 2 electrons.
You could minimize electron energy by putting both 3s electrons in the lower energy
level, one spin up and the other spin down.
There is an internuclear separation which minimizes electron energy. If you bring the
nuclei closer together, energy increases.
8 Ch. ...... 3
9. When you bring five sodium atoms
together, the 3s energy level splits
into five separate energy levels.
The three new energy levels fall in
between the two for 2 sodiums.
There are now 5 electrons occupying these energy levels.
I’ve suggested one possible minimum-energy configuration. Notice how the sodium-
sodium internuclear distance must increase slightly.
9 Ch. ...... 3
10. When you bring N (some big
number) sodium atoms together, the
3s energy level splits into N separate
energy levels.
The result is an energy band,
containing N very closely-spaced
energy levels.
There are now N electrons occupying this 3s band. They go into the lowest energy
levels they can find.
The shaded area represents available states, not filled states. At the selected
separation, these are the available states.
10 Ch. ...... 3
11. Now let’s take a closer look at the energy
levels in solid sodium.
Remember, the 3s is the outermost
occupied level.
When sodium atoms are brought within
about 1 nm of each other, the 3s levels in
the individual atoms overlap enough to
begin the formation of the 3s band.
The 3s band broadens as the separation
further decreases.
3s band
begins to
form
11 Ch. ...... 3
12. Because only half the states in the 3s
band are occupied, the electron energy
decreases as the sodium-sodium
separation decreases below 1 nm.
At about 0.36 nm, two things happen:
the 3s energy levels start to go up
(remember particle in box?) and the 2p
levels start to form a band.
Further decrease in interatomic
separation results in a net increase of
energy.
3s electron
energy is
minimized
12 Ch. ...... 3
13. What about the 3p and 4s bands shown
in the figure?
Don’t worry about them—there are no
electrons available to occupy them!
Keep in mind, the bands do exist,
whether or not any electrons are in them.
What about the 1s and 2s energy levels,
which are not shown in the figure?
The sodium atoms do not get close
enough for them to form bands—they
remain as atomic states.
13 Ch. ...... 3
14. • Ten of the fourteen silicon
atom electrons occupy deep-
lying energy levels close to
the nucleus . The four
remaining valence electrons
are relatively weakly bound
and are the electrons
involved in chemical
reactions
• The band splitting of
silicon. We need only
consider the n = 3 level
for the valence
electrons, since the first
two energy shells are
completely full and are
tightly bound to the
nucleus
Bands of allowed energies that the electrons may occupy separated by bands of forbidden
energies. This energy-band splitting and the formation of allowed and forbidden bands is
the energy-band theory of single-crystal materials.
A schematic representation of an isolated silicon atom
14 Ch. ...... 3
15. Figure shows energy bands in carbon
(and silicon) as a function of
interatomic separation.
At large separation, there is a filled
2s band and a 1/3 filled 2p band.
But electron energy can be lowered if the carbon-carbon separation is reduced.
There is a range of carbon-carbon separations for which the 2s and 2p bands
overlap and form a hybrid band containing 8N states (Beiser calls them “levels”).
15 Ch. ...... 3
16. But the minimum total electron
energy occurs at this carbon carbon
separation.
At this separation there is a valence
band containing 4N quantum states
and occupied by 4N electrons.
The filled band is separated by a large gap from the empty conduction band. The
gap is 6 eV—remember, kT is about 0.025 eV at room temperature. The gap is too
large for ordinary electric fields to move an electron into the conduction band.
Carbon is an insulator.
16 Ch. ...... 3
17. Silicon has a similar band structure.
The forbidden gap is about 1 eV.
The probability of a single electron
being excited across the gap is small,
proportional to
exp(-Egap/kT).
However, there are enough 3s+3p electrons in silicon that some of them might make
it into the conduction band. We need to consider such a special case.
17 Ch. ...... 3
18. 3.1.2 The Kronig-Penney Model
• The potential function of a single, noninteracting
, one-electron atom
• The same type of potential function for
the case when several atoms are in
close proximity arranged in a one-
dimensional array
• The potential functions of adjacent
atoms overlap, and the net potential
function
• It is this potential function we would need
to use in Schrodinger's wave equation to
model a one-dimensional single-crystal
material.18 Ch. ...... 3
19. The solution to Schrodinger's wave equation, for this one-dimensional single crystal lattice,
is made more tractable by considering a simpler potential function.
The more interesting solution occurs
for the case when E < Vo, which
corresponds to a particle being bound
within the crystal. The electrons are
contained in the potential wells, but
we have the possibility of tunneling
between wells.
To obtain the solution to Schrodinger's wave
equation, we make use of a mathematical
theorem by Bloch. The theorem states that all
one-electron wave functions, for problems
involving periodically varying potential energy
functions, must be of the form:
𝜓 = 𝑢 𝑥 𝑒 𝑖𝑘𝑥 (Bloch funct. ) … … … … 1
The parameter k is called a constant of
motion and will be considered in more
detail as we develop the theory. The
function u(x) is a periodic function with
period (a t b).
𝜓 𝑥, 𝑡 = 𝑢 𝑥 𝜙 𝑡 = 𝑢 𝑥 𝑒 𝑖𝑘𝑥
. 𝑒
−𝑖
𝐸
ℏ
𝑡
the total solution to the wave equation is
the product of the time-independent
solution and the time-dependent solution
19 Ch. ...... 3
20. This traveling-wave solution represents the motion of an electron in a single-crystal
material. The amplitude of the traveling wave is a periodic function and the parameter k is
also referred to as a wave number.
𝜓 𝑥, 𝑡 = 𝑢 𝑥 𝜙 𝑡 = 𝑢 𝑥 𝑒
𝑖 𝑘𝑥−
𝐸
ℏ
𝑡
…………………….…(2)
If we consider region I (0 < x < a) in which
V(x) = 0 , take the second derivative of
Equation (1), and substitute this result into the
time-independent Schrodinger's wave
equation ,we obtain the relation
𝑑2 𝑢1 𝑥
𝑑𝑥2 + 2𝑖𝑘
𝑑𝑢1(𝑥)
𝑑𝑥
− (𝑘2 − 𝛼 2) 𝑢1 𝑥 = 0 ∶ 𝛼 2 =
2𝑚𝐸
ℏ2 …………..….(3)
Consider now a specific region II , -b < r < 0. in which V(x) = Vo. and apply Schrodinger's wave
equation. We obtain the relation
𝑑2 𝑢2 𝑥
𝑑𝑥2 + 2𝑖𝑘
𝑑𝑢2(𝑥)
𝑑𝑥
− (𝑘2 − 𝛽 2) 𝑢2 𝑥 = 0 ∶ 𝛽 2 = 𝛼 2 −
2𝑚𝑉𝑜
ℏ2 …….……….(4)
20 Ch. ...... 3
21. From Eq.s ( 3 & 4 ) we obtain :
𝑚𝑉 𝑜
𝑏𝑎
ℏ2
𝑆𝑖𝑛𝛼𝑎
𝛼𝑎
+ Cos𝛼𝑎 = Cos 𝑘𝑎 ∶ 𝑝′ =
𝑚𝑉 𝑜
𝑏𝑎
ℏ2 …….……….(5)
Equation (5) again gives the relation between the parameter k, total energy E , and the
potential barrier bVo . We may note that Equation (5 ) is not a .solution of Schrodinger’s
wave equation but gives the conditions for which Schrodinger's wave equation will have
a solution. If we assume the crystal is infinitely large, then k in Equation (5) can assume
a continuum of values and must be real
21 Ch. ...... 3
22. To begin to understand the nature of the solution, initially consider the special case for
which Vo= 0. In this case P' = 0. which corresponds to a free particle since there are no
potential barriers. From Equation (5). we have that
Cos𝛼𝑎 = Cos 𝑘𝑎 … … … … … … … … … .. 6
𝛼 = 𝑘 … … … … … … … … … .. 7
3.1.3 The k-Space Diagram
From Eq.(3) :
𝛼 =
2𝑚𝐸
ℏ2 =
2𝑚(
1
2
𝑚 𝑣2)
ℏ2 =
𝑝
ℏ
= 𝑘 … … … … … … … … … … . . (8)
where p is the particle momentum. The constant of the motion parameter k is related to
the particle momentum for the free electron. The parameter k is also referred to as
a wave number.
We can also relate the energy and momentum as
𝐸 =
𝑝2
2𝑚
=
𝑘2
ℏ2
2𝑚
… … … … … … … … … … . . (9)
22 Ch. ...... 3
: 𝑝′
=
𝑚𝑉 𝑜
𝑏𝑎
ℏ2
: 𝛼 2 =
2𝑚𝐸
ℏ2
23. If eq. (5) becomes as :
𝑝′ 𝑆𝑖𝑛𝛼𝑎
𝛼𝑎
+ Cos𝛼𝑎 = 𝑓(𝛼𝑎) …….……….(10)
When the parts of eq.(10) are plotted as following :
The shaded areas show the
allowed values of (𝛼a)
corresponding to real values
of k.
23
Ch. ...... 3
𝑚𝑉 𝑜
𝑏𝑎
ℏ2
𝑆𝑖𝑛𝛼𝑎
𝛼𝑎
+ Cos𝛼𝑎 = Cos 𝑘𝑎 ∶ 𝑝′ =
𝑚𝑉 𝑜
𝑏𝑎
ℏ2 …….……….(5)
24. Now from Equation (5).
Cos𝑘𝑎 = 𝑓(𝛼𝑎) …….……….(11)
we also have that
𝑚𝑉 𝑜
𝑏𝑎
ℏ2
𝑆𝑖𝑛𝛼𝑎
𝛼𝑎
+ Cos𝛼𝑎 = Cos 𝑘𝑎
24 Ch. ...... 3
25. Volume of k-space per quantum state :
If we consider , a small cubic
∆𝑘 𝑥
∆𝑘 𝑦
∆𝑘 𝑧
where V is the total volume of the crystal
We note the unit of Volume in k – space is (𝑚−3
)
∆𝑘 𝑥∆𝑘 𝑦 ∆𝑘 𝑧 =
2𝜋
𝐿 𝑥
.
2𝜋
𝐿 𝑦
.
2𝜋
𝐿 𝑧
=
8𝜋3
𝑉
… … … . . … . (12)
The final point we need to make is that we
need to take account of the internal spin of
electrons. This allows two electrons with
opposite spin to occupy each k- state. Thus,
the volume of k-space required for each
electron is given by:
∆𝑘 =
4𝜋3
𝑉
… … … … … … … (13)
25 Ch. ...... 3
26. 3.2 : ELECTRICAL CONDUCTION IN SOLIDS
3.2.1 The Energy Band and the Bond Model
SiSi
Si
Si
Si Si Si
Si
Si
Si
Covalent Bond
To the silicon atom be stabile the atoms combine So as to
have eight electrons in the valence orbit .For this each atom arranges
itself with four other atoms so that each neighbor shares an electron
with the central atom .
26 Ch. ...... 3
27. The semiconductor is neutrally charged. This means that, as the negatively charged electron
breaks away from its covalent bonding position, a positively charged "empty state" is
created in the original covalent bonding position in the valence band. As the temperature
further increases, more covalent bonds are broken, more electrons jump to the conduction
hand, and more positive "empty states" are created in the valence band.
The E versus, k diagram at the conduction and valence bands of a semiconductor
Valence band
Conduction band
Eg
e+
e
E
K
T = 0 K
E
K
T > 0 K
27 Ch. ...... 3
28. 3.2.2 : Drift Current
If we had a collection of positively charged ions with a volume density N (cm-3)
and an average drift velocity Vd (cm/s), then the drift current density would be :
Current is due to the net flow of charge
𝐽 = 𝑞𝑁𝑣 𝑑 (
𝐴
𝑐𝑚2
) … … … … 14 the drift current density
If, instead of considering the average drift velocity, we
considered the individual ion velocities, then we could
write the drift current density as :
𝑞 is the charge of ion , 𝑣 𝑑 is drift velocity of ion
𝐽 = 𝑞
𝑖=1
𝑁
𝑣𝑖 … … … … … … … . (15)
where 𝑣𝑖 is the velocity of the ith ion
net drift of electrons in the conduction band
will give rise to a current. The electron
distribution in the conduction band, as in the
beside figure. is an even function of k when no
external force is applied.
Recall that k for a free electron is related
to momentum so that, since there are as
many electrons with a (+k) value as there
are with a (- k) value, the net drift current
density due to these electrons is zero. This
result is certainly expected since there is no
externally applied force.
E
K
28 Ch. ...... 3
29. If a force is applied to a particle and the particle moves, it must gain energy. this effect is
expressed as:
𝑑𝐸 = 𝐹 𝑑𝑥 = 𝐹 𝑣 𝑑𝑡 … … … … 16
where F is the applied force, 𝑑𝑥 is the differential distance the particle moves, 𝑣 is the
velocity, and 𝑑𝐸 is the increase in energy
If an external force is applied to the electrons in the
conduction band, there are empty energy states into which
the electrons can move: therefore, because of the external
force, electrons can gain energy and a net momentum. The
electron distribution in the conduction band may look
like that shown in beside Figure, which implies that the
electrons have gained a net momentum.
We may write the drift current density due to the motion of
electrons as:
𝐽 = −𝑒
𝑖=1
𝑛
𝑣𝑖 ∶ 𝑞 = − 𝑒 (𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛 𝑐ℎ𝑎𝑟𝑔𝑒) … … (17)
where 𝑒 is the magnitude of the electronic charge and 𝑛 is the number of electrons per
unit volume in the conduction hand. Again, the summation is taken over a unit volume so
the current density is A/cm2
E
K
We may note from eq.(17) that the current is directly related to the electron
velocity; that is, the current is related to how well the electron can move in the crystal.
29 Ch. ...... 3
30. 3.2.3 Electron Effective Mass
In crystal the total force is caused from an externally applied force such as electric fields
, and the internal forces in the crystal due to positively charged ions or protons and
negatively charged electrons (ions and core electrons) , which will influence the
motion of electrons in the lattice.
• In a crystal lattice, the net force may be opposite the external force, however:
+ + + ++
Ep(x)
-
𝐹𝑒𝑥𝑡 = −𝑞𝐸𝑒𝑥𝑡
𝐹𝑖𝑛𝑡 = −
𝑑𝐸 𝑝
𝑑𝑥
𝐹𝑡𝑜𝑡𝑎𝑙 = 𝐹𝑥𝑒𝑡 + 𝐹𝑖𝑛𝑡 = 𝑚 𝑎 … … … … … … (18)
where 𝐹𝑡𝑜𝑡𝑎𝑙 , 𝐹𝑥𝑒𝑡, and 𝐹𝑖𝑛𝑡 , are the total force, the externally applied force, and the
internal forces, respectively, acting on a particle in a crystal. The parameter 𝒂 is the
acceleration and 𝑚 is the rest mass of the particle.
30 Ch. ...... 3
31. Since it is difficult to take into account all of the internal forces, we will write the equation
𝐹𝑡𝑜𝑡𝑎𝑙 ≈ 𝐹𝑒𝑥𝑡 = 𝑚∗ 𝑎 … … … … … … (19)
What is the expression for m*
where the acceleration 𝑎 is now directly related to the external force. The parameter m*.
called the effective mass, takes into account the panicle mass and also takes into account
the effect of the internal forces.
We can also relate the effective mass of an electron in a crystal
to the E versus k curves, such as was shown in figure
31 Ch. ...... 3
32. Mass of the free electron :
consider the case of a free electron whose 𝑬
versus 𝒌 curve was shown in Figure
𝐸 =
𝑝2
2𝑚
=
𝑘2ℏ2
2𝑚
… … … … … … … … … … . . (9)
Recalling Equation (9)
If we take the derivative of Equation (9) with respect to 𝒌, we obtain :
𝑑𝐸
𝑑𝑘
=
𝑑
𝑑𝑘
𝑝2
2𝑚
=
𝑑
𝑑𝑘
𝑘2
ℏ2
2𝑚
𝑑𝐸
𝑑𝑘
=
𝑘ℏ2
𝑚
=
ℏ 𝑝
𝑚
… … … … … … … … …(20)
1
ℏ
𝑑𝐸
𝑑𝑘
=
𝑝
𝑚
=
𝑚𝑣
𝑚
= 𝑣 … … … … … … … … … . (21)
If we now take the second derivative of E with respect
to k in eq. ( 9) , we have :
𝑑2 𝐸
𝑑𝑘2
=
ℏ2
𝑚
⇒
1
ℏ2
𝑑2 𝐸
𝑑𝑘2
=
1
𝑚
… … … … … … … … … . (22)
We may also note from
above figure that
𝑑2 𝐸
𝑑𝑘2 is a
positive quantity, which
implies that the mass of
the electron is also a
positive quantity .
32
Ch. ...... 3
33. Newton's classical equation of motion𝐹 = 𝑚 𝑎 … … … … … … (23)
𝐹 = −𝑞𝐸 = −𝑒 𝐸 … … … … … … (24) Faraday's classical equation of electric
force
𝑎 = −
𝑒 𝐸
𝑚
… … … … … … (25)
We note, the motion of the free electron is in the opposite direction to the applied electric
field because of the negative charge.
We may now apply the results to the electron in the
bottom of an allowed energy band. Consider the allowed
energy band in beside figure (a) . The energy near the
bottom of this energy band may be approximated by a
parabola, just as that of a free particle. We may write :
𝐸 − 𝐸𝑐 = 𝐶1(𝑘)2
… … … … … … (26)
The energy E, is the energy at the bottom of the band.
Since E > Ec the parameter C1 is a positive quantity.
𝐸 = 𝐶1(𝑘)2
− 𝐸𝑐 … … … … … … (27)
𝑑2 𝐸
𝑑𝑘2 = 2𝐶1 … … … … … … (28)
33
Ch. ...... 3
34. 1
ℏ2
𝑑2
𝐸
𝑑𝑘2
=
2𝐶1
ℏ2
… … … … … … (29) Multiply by
1
ℏ2
Comparing Equation (29) with Equation (22). We may equate
ℏ2
2𝐶1
, to the mass of the particle.
However, the curvature of the curve in Figure 3.16a will not, in general, be the same as the
curvature of the free-particle curve. We may write
1
ℏ2
𝑑2 𝐸
𝑑𝑘2 =
2𝐶1
ℏ2 =
1
𝑚∗ … … … … … … (30)
𝑚∗
=
ℏ2
2𝐶1
… … … … … … (31) Where m* is called the effective mass
If we apply an electric field to the electron in the bottom of the allowed energy
band, we may write the acceleration as :
𝑎 = −
𝑒 𝐸
𝑚∗
… … … … … … (32)
The effective mass m: of the electron near the bottom of the conduction band is a
constant.
34 Ch. ...... 3
35. 3.2.4 Concept of the Hole
Valence band
Conduction band
Eg
e+
e
In considering the two-dimensional
representation of the covalent bonding
shown in Figure , a positively charged
"empty state" was created when a
valence electron was elevated into the
conduction band.
For T > 0 K, all valence electrons may gain thermal energy; if a valence
electron gains a small amount of thermal energy, it may hop into the empty
state
What is happen ?
The crystal now has a second equally important charge carrier that can give rise
to a current
This charge carrier is called a hole and, as we will see, can also be thought of as a
classical particle whose motion can be modeled using Newtonian mechanics.
35 Ch. ...... 3
36. E
K
The drift current density due to electrons in the valence band, such as shown in below
figure , can be written as
𝐽 = −𝑒
𝑖(𝑓𝑖𝑙𝑙𝑒𝑑)
𝑣𝑖 … … … … … … … … (33)
where the summation extends over all filled states. This summation
is inconvenient since it extends over a nearly full valence band and
takes into account a very large number of states.
𝐽 = −𝑒
𝑖(𝑡𝑜𝑡𝑎𝑙)
𝑣𝑖 + 𝑒
𝑖(𝑒𝑚𝑝𝑡𝑦)
𝑣𝑖 … … … … … … … … (34)
If we consider a band that is totally full, all available states are occupied by electrons. The
individual electrons can be thought of as moving with a velocity as given by
𝑣 𝐸 =
1
ℏ
𝑑𝐸
𝑑𝑘
… … … … … … … … … … … . (36)
𝐽 = +𝑒
𝑖(𝑒𝑚𝑝𝑡𝑦)
𝑣𝑖 ∶ − 𝑒
𝑖(𝑡𝑜𝑡𝑎𝑙)
𝑣𝑖 = 0 … … … … … … … … (35)
The distribution of electrons will respect to k cannot he changed with an externally
applied force.
where the 𝑣, in the summation is the associated with the empty state.36 Ch. ...... 3
37. Equation ( 35) is entirely equivalent to placing a positively charged particle in the empty
states and assuming all other states in the band are empty, or neutrally charged.
Valence band with
conventional electron-filed
sates and empty states
Concept of positive charges
occupying the original empty states
Now consider an electron near the top of the allowed energy band shown in Figure
The energy near the top of the allowed energy band may again be
approximated by a parabola so that we may write
𝐸 − 𝐸 𝑣 = − 𝐶2 𝑘 2 … … … … … … … … … … (37)
The energy 𝐸 𝑣 is the energy at the top of the energy band
37 Ch. ...... 3
𝐽 = +𝑒
𝑖(𝑒𝑚𝑝𝑡𝑦)
𝑣𝑖
38. Since, 𝐸 < 𝐸 𝑣 , for electrons in this band, then the parameter C1 must be a positive
quantity.
𝑑2 𝐸
𝑑𝑘2 = −2 𝐶2 … … … … … … … … … … (38)
𝑏𝑢𝑡
𝑑2
𝐸
𝑑𝑘2
=
ℏ2
𝑚∗
𝑡ℎ𝑒𝑛 ,
1
ℏ2
𝑑2 𝐸
𝑑𝑘2 =
−2 𝐶2
ℏ2 =
1
𝑚∗ … … … (39)
where m* is again an effective mass. We
have argued that C2 is a positive quantity,
which now implies that m* is a negative
quantity. An electron moving near the top of
an allowed energy band behaves as if it has
negative mass.
𝑎 =
−𝑒 𝐸
− 𝑚∗
=
𝑒 𝐸
𝑚∗
… … . 40
An electron moving near the top of an allowed energy band moves in the same
direction as the applied electric field.
The net motion of electrons in a nearly full band can be described by considering just the
empty states, provided that a positive electronic charge is associated with each state and
that the negative of m* from Equation (39) is associated with each state. We now can
model this band as having particles with a positive electronic charge and a positive
effective mass. The density of these panicles in the valence band is the same as the
density of empty electronic energy states. This new panicle is the hole. The hole, then, has
a positive effective mass denoted by mp
* and a positive electronic charge, so it will move
in the same direction as an applied field.
38 Ch. ...... 3
40. Ch. ...... 340
3.3 : EXTENSION TO THREE DIMENSIONS
One problem encountered in extending the potential function to a three - dimensional
crystal is that the distance between atoms varies as the direction through the crystal
changes.
The (100) plane of a face-
centered cubic crystal showing
the [001] and [110] directions.
Electrons traveling in different directions
encounter different potential patterns and
therefore different k-space boundaries. The E
versus k diagrams are in general a function of
the k-space direction in a crystal.