A solid is a complex multiparticle system consisting of nuclei and electrons. It can be thought of as a collection of atoms brought together so that the electronic wave functions begin to overlap. In this case, the electrons of the outer shells cease to be localized near their atom.
Metals, dielectrics, semiconductors.
So, when atoms come closer together, energy levels split and zones are formed. It becomes clear where the concepts 2s - zone, 3p - zone, etc. arise; these are indications of the atomic terms from which this zone originated.
Different zones may overlap or remain separated by zones of forbidden energies. Let the zones not overlap. Then, from completely filled (completely empty, partially filled) atomic terms, completely filled (respectively, completely empty or partially filled) zones are formed. If the bands overlap (band hybridization), then an energy band partially filled with electrons can be formed from an atomic term occupied by an electron and a term with an unoccupied state. According to the Pauli principle, at T = 0 the band will be occupied by ZN/2 energy states with the lowest energy, where N is the number of atoms, Z is the number of electrons at the corresponding levels in the atom, 2 arose due to the spin. In total, one Brillien zone contains N states with different meanings j. Thus, by the charge of the Z ion one can judge the nature of the filling of the zone. For example, if Z is odd, partially filled zones will certainly appear. Indeed, such a situation occurs, for example, in alkali metals, where there is one electron in the upper filled level (Z = 1).
Table of contents
1 Basic methods and approximations for describing electronic states in a solid.
1.1 Adiabatic approximation
1.2 Self-consistent field approximation, Hartree-Fock method
1.3 Wave function of an electron in a periodic field
2 Spectrum of electrons in a solid, band structure
2.1 Spectrum of electrons in a solid
2.2 Model of almost free electrons
2.3 Approximation strong connection
3 Properties of Bloch electrons
3.1 Metals, dielectrics, semiconductors
3.2 Dynamics of a Bloch electron
3.3 Effective mass
3.4 Band structure of typical semiconductors
3.5 Density of states
4 Effective mass approximation.
4.1 Electrons and holes
4.2 Effective mass approximation equation
4.3 Impurity atoms
4.4 Wannier-Mott excitons
5 Statistics of charge carriers in metals and semiconductors.
5.1 Fermi-Dirac distribution
5.2 Degenerate electron gas. Metal
5.3 Non-degenerate electron gas
6 Dielectric constant of a solid. Lindhard's formula.
6.1 Spatial and temporal dispersion
6.2 Calculation of dielectric constant using perturbation theory
6.3 Shielding of static (w = 0) fields in conductors
6.4 Low frequency the dielectric constant dielectrics
6.5 Shielding at high frequencies. (q - 0, w - large)
6.6 Mott-Hubbard transition
7 The phenomenon of transfer to solids Oh. Kinetic equation
7.1 Boltzmann kinetic equation
7.2 Boltzmann kinetic equation
7.3 Pulse relaxation time
7.4 Form of the collision integral for scattering by phonons
7.5 Electron-electron collision integral
7.6 Time of pulse scattering by phonons
8 Kinetic phenomena. Solving the Boltzmann equation. Conductivity. Thermoelectric effects.
8.1 Solution kinetic equation in t - approximation. Response to a uniform field E
8.2 Stationary solution of the kinetic equation in the presence of electric and magnetic fields and a temperature gradient
8.3 Current in a non-uniform conductor and electrochemical potential gradient
8.4 Thermoelectric effects
9 Galvanomagnetic phenomena
9.1 Hall effect
9.2 Transverse magnetoresistance
10 Warming up the electron gas.
10.1 Energy dissipation time
10.2 Hot electrons, electron temperature
11 Contact potential difference
11.1 Work function
11.2 Metal-semiconductor contact
11.3 Two-dimensional electron gas
12 Superconductivity i
12.1 Effective electron-electron interaction in a system of electrons and phonons
12.2 Cooper pairs
12.3 Phase transition and spontaneous symmetry breaking
12.4 Self-consistent field method in the theory of superconductivity
12.5 Continuous current in a superconductor
Lecture course program on electronic properties of solids
Control questions.
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Electronic properties solids: basic experimental facts. Conductivity, Hall effect, thermoEMF, photoconductivity, optical absorption. The difficulties of explaining these facts based on classical theory Drude.
Basic approximations of band theory. Border conditions Born–Karman. Bloch's theorem. Bloch functions. Quasi-pulse. Brillouin zones. Energy zones.
Bragg reflection of electrons moving across a crystal. Banded energy spectrum.
Approximation of strongly bound electrons. Relationship between the width of the allowed band and the overlap of atomic wave functions. Law of dispersion. Inverse effective mass tensor.
Approximation of almost free electrons. Bragg reflections of electrons.
Filling energy bands with electrons. Fermi surface. Density of states. Metals, dielectrics and semiconductors. Semi-metals.
Magnetic properties of solids
Magnetization and susceptibility. Diamagnets, paramagnets and ferromagnets. Curie and Curie–Weiss laws. Paramagnetism and diamagnetism of conduction electrons.
The nature of ferromagnetism. Phase transition to a ferromagnetic state. The role of exchange interaction. Curie point and susceptibility of a ferromagnet.
Ferromagnetic domains . Reasons for the appearance of domains. Domain boundaries (Bloch, Neel).
Antiferromagnets. Magnetic structure. Neel's point. Susceptibility of antiferromagnets. Ferrimagnets. Magnetic structure of ferrimagnets.
Spin waves, magnons.
Movement magnetic moment in constant and alternating magnetic fields. Electronic paramagnetic resonance. Nuclear magnetic resonance.
Optical and magnetooptical properties of solids
Complex dielectric constant and optical constants. Absorption and reflection coefficients. Kramers-Kronig relations.
Light absorption in semiconductors (interband, impurity absorption, absorption by free carriers, lattice). Determination of the basic characteristics of a semiconductor from optical studies.
Magneto-optical effects (Faraday, Vocht and Kerr effects).
Penetration of high-frequency field into a conductor. Normal and abnormal skin effects. Skin layer thickness.
Superconductivity
Superconductivity. Critical temperature. High temperature superconductors. Meissner effect. Critical field and critical current.
Superconductors of the first and second kind. Their magnetic properties. Abrikosov's whirlwinds. The depth of penetration of the magnetic field into the sample.
Josephson effect.
Cooper mating. Coherence length. Energy gap.
Main literature
Kittel Ch. Introduction to solid state physics. M.: Nauka, 1978.
Ashcroft N., Mermin N. Physics of Solid State. T. I, II. M.: Mir, 1979.
Worth Ch., Thomson R. Physics of Solid State. M.: Mir, 1969.
Ziman J. Principles of solid state theory. M.: Mir, 1974.
Pavlov P.V., Khokhlov A.F. Solid state physics. M.: Higher. school, 2000.
Vonsovsky S.V. Magnetism. M.: Nauka, 1971.
Bonch-Bruevich V.L., Kalashnikov S.G. Physics of semiconductors. M.: Nauka, 1979.
Shmidt V.V. Introduction to the physics of superconductivity. MC NMO, M., 2000.
Note. When preparing for the exam in the technical science program Special attention must be referred to sections 7-10 of the program.
List of programs
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MINIMUM PROGRAM
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Plasma Physics"
in physics, mathematics, chemistry
And technical sciences
Introduction
This program is based on the following disciplines: statistics, elementary processes, physical kinetics, magnetohydrodynamics, electrodynamics continuum, physics of wave processes.
The program has been developed expert advice Higher certification commission Ministry of Education Russian Federation in physics with the participation of the Russian scientific center"Kurchatov Institute", Institute general physics RAS, Moscow Institute of Physics and Technology (state university), United Institute high temperatures RAS, Faculty of Physics Moscow State University named after. M.V. Lomonosov and Moscow State Engineering Physics Institute.
Plasma thermodynamics
The concept of plasma, quasineutrality, microfields, Debye radius, ideal and non-ideal plasma. Thermodynamic equilibrium condition, thermal ionization, Saha's formula, coronal equilibrium, decrease in ionization potential. Plasma degeneracy, Boltzmann and Fermi-Dirac statistics, Thomas-Fermi model.
Elementary processes
Collisions of charged particles, long-range action, collision frequencies, collisions of electrons with atoms (elastic and inelastic), collisions of heavy particles. Ionization, recombination, charge exchange and adhesion. Excitation and dissociation of molecules by electron impact.
Physical kinetics
Boltzmann and Vlasov equations, collision integral, Maxwellization time and temperature equalization rate various components plasma. The rate of ion formation and recombination of electrons and ions, the formation and destruction of excited atoms (ions). Transport phenomena in plasma, electrical conductivity, diffusion and thermal conductivity of particles in the presence and absence of a magnetic field. Kinetics of excited molecules in plasma.
4. Dynamics of charged particles
in electrical and magnetic fields
Movement in crossed electric and magnetic fields. Drift approach, types of drift motion. Charged particle in a high-frequency field. The concept of adiabatic invariant.
Electronic states in solids.
Let us first consider the change energy levels of an individual atom when an external or disturbing force is applied to it.
If a disturbing force affects the electrons of an atom, then the energy levels of the electrons shift, since this changes total energy electrons.
When a disturbing force is applied, electronic levels can split into levels with slightly different energies.
The reason for this splitting is that electrons that are in different quantum states but have the same energy can interact differently with the perturbing force.
When atoms come close to each other to form a solid, the interaction between them has a disturbing effect on the original atomic energy levels.
As a result, with a sufficiently strong approach, the symmetry of electronic states that existed in isolated atoms is broken, as a result of which the levels are split.
Then the only energy level of a solid with long distance between atoms in the lattice turns into big number levels of a solid body located close to each other with a small interatomic distance, forming a band (zone) of energy levels.
Some properties of energy level bands are quite obvious.
First, the binding energy of a solid must be determined by a shift in the energy levels of electrons, similar to what happens when a chemical bond is formed.
Therefore, during the formation of a solid, the energy levels should, on average, shift downward.
Secondly, the ones most distant from the nucleus, or valence electrons, are most susceptible to the disturbing action of neighboring atoms, since they are located closest to all other electrons to neighboring atoms.
Thirdly, the equilibrium distance between the atoms of the lattice must correspond to the minimum energy, since with further approach of the atoms the energy levels begin to shift upward.
Fourthly, the states of the original system must be continuously deformed as the atoms approach each other.
To make it clear physical origin energy structure of the crystal, at least three problems should be considered in detail:
1) the nature of the forces of attraction between atoms;
2) the nature of the repulsive forces acting when too close rapprochement atoms with each other;
3) the degree of splitting of energy levels due to interactions between atoms.
The answer to the first question is difficult to give, since it is different for different structures solid body.
Atoms repel when approaching each other mainly because each given electronic state corresponds to a well-defined region of space.
The Pauli exclusion principle states that identical wave functions different atoms cannot be localized in the same region of space, since in this case they would describe the same state.
If the atoms get closer together in such a way that the spatial region in which the wave functions are defined becomes smaller and smaller.
There is a spatial overlap of the wave functions and conditions arise in which the Pauli principle cannot be satisfied and, due to the action of the uncertainty principle, the energy of the system increases.
Thus, when atoms get too close to each other, their total energy increases.
This is equivalent to the action of a repulsive force.
The third question is the subject of the proposition that electrons in the zone of energy levels are mobile and not localized on individual atoms.
The mobility of electrons in solids can be explained by considering the changes in the wave function that occur when isolated atoms are brought closer together, when the wave functions overlap.
The overlap appears already at some finite distance between the atoms, but it becomes noticeable when the interatomic distance reaches a value of the order of 10 angstroms or less.
For an electron located at some point in time in the orbit of one of the atoms, there is a finite probability that it will be captured by a neighboring atom.
How more degree ceilings, so more likely electron migration from atom to atom.
At an interatomic distance corresponding to real crystal lattices, the overlap of wave functions is very large, so that an electron cannot remain in the orbit of a given atom for a long time and easily moves to a neighboring atom.
Since electron transitions from atom to atom occur quickly, the electrons in question should be considered to belong to the entire collective of atoms in the crystal, and not to individual atoms
The wave functions of electrons located below the valence shell are more strongly localized near the nucleus than the wave functions of valence electrons, so the degree of overlap of these functions is much less.
Consequently, internal electrons do not participate noticeably in the processes of transition from atom to atom.
Crystalline and amorphous state of matter.
Matter in the three-dimensional world around us can be in four states of aggregation: liquid, solid, gaseous and plasma (plus a fifth - nanostate).
According to classical definition in a solid state, a substance hardly changes volume and shape (it compresses and deforms a little), in a liquid it hardly changes volume, but easily changes shape (it compresses a little, but easily deforms), in a gas it easily changes volume,
and shape.
IN In these three states the chemical integrity and individuality of the atoms are preserved.
Recombination of nonequilibrium carriers in semiconductors.
Superconductivity.
Contact phenomena. Heterogeneous electronic systems.
Conditions for equilibrium of contacting conductors. Electron affinity, work function and contact potential difference. Distribution of electron concentration and electric field near metal-semiconductor and semiconductor-semiconductor contacts. Electric field shielding length. Current-voltage p-n characteristic transition and its physical interpretation.
Dimensional quantization and low-dimensional electronic systems.
Screening of electron-electron interaction by electrons and ions and effective attraction between electrons. Spectrum of elementary excitations in a superconductor. Continuous current.
Interband radiative recombination, impurity recombination (Hall-Shockley-Read recombination), interband Auger recombination. Dependence of the Hall-Shockley-Reed recombination rate on the concentration of recombination centers for a slight deviation of the semiconductor from the equilibrium state.
Literature
Main:
A.I. Anselm. Introduction to the theory of semiconductors. M., Nauka, 1978.
V.L. Bonch-Bruevich, S.G. Kalashnikov. Physics of semiconductors. M., Nauka, 1990.
N. Ashcroft, N. Mermin. Solid state physics. In 2 volumes. World, 1979
F. Blatt. Physics electronic conductivity in solids. M., Mir, 1971.
O. Modelung. Theory of solids. M., Nauka, 1980.
A.S. Davydov. Theory of solids. M., Nauka, 1976.
F. Seitz. Modern theory solid body. M.-L., State Publishing House of Technical and Theoretical
literature, 1949.
J. Zyman. Principles of solid state theory. M., Mir, 1966.
Adiabatic approximation and self-consistent field approximation:
,
J. Slater. Self-consistent field methods for molecules and solids. M., Mir, 1978.
A.S. Davydov. Quantum mechanics. M., Nauka, 1973.
R. McWeeney, B. Sutcliffe. Quantum mechanics of molecules. M., Mir, 1972.
V.A. Fok. The beginnings of quantum mechanics. M., Nauka, 1976.
A. Messiah. Quantum mechanics. volume 2, M., Nauka, 1979.
V. I. Smirnov. Well higher mathematics. Volume III, part 1., Ed. 8, M., Fizmatgiz, 1958
(about matrices and their diagonalization).
Bloch's theorem, quasimomentum, reciprocal lattice, Brillouin zone, General characteristics energy zones:
, , , ,
J. Callaway. Energy theory band structure. M., Mir, 1969.
Jones G. Brillouin zone theory and electronic states in crystals. M., Mir, 1968.
V. I. Smirnov. Course of higher mathematics. Volume II, Ed. 18, M., Fizmatgiz, 1961 (about the method
combinations of eigenfunctions to bring them to mutual orthogonality).
Avalanche reproduction of carriers:
Technique optical communications. Photodetectors. Ed. U. Tsanga. M.: Mir, 1988.
Grekhov I.V., Serezhkin Yu.N. Avalanche breakdown in semiconductors. L.: Energy, 1980.
V. A. Kholodnov. Carrier reproduction rates in p-n structures// FTP, vol. 30, no. 6, p. 1051-1063,
(June 1996).
Interzone tunneling:
Tunneling phenomena in solids. Ed. E. Burstein and S. Lundqvist. M., Mir, 1973.
Recombination of nonequilibrium carriers in semiconductors:
J. Bdeckmore. Statistics of electrons and holes in semiconductors. M., Mir, 1964.
R. Smith. Semiconductors. M., Mir, 1982.
V. A. Kholodnov. On the Hall-Shockley-Read recombination theory // FTP, vol. 30, no. 6, p. 1011-1025 (June 1996).
The state of motion of electrons in a solid would be precisely known if it were possible to solve the Schrödinger equation
and find the eigenwave functions and energy values of the Hamilton operator for the crystal in general case looks like
The first two terms in (2.2) are operators kinetic energy electrons with masses and nuclei with masses, the subsequent terms determine, respectively, the energies of the pairwise Coulomb interaction electrons, the interaction of all electrons with all nuclei and the interaction of nuclei with each other. The radius vectors of electrons and nuclei are designated by
Equation (2.1) contains the coordinates of particles, where is the number of atoms in the crystal; nuclear charge. Since the Schrödinger equation cannot be solved exactly even for individual atoms, with the exception of the hydrogen atom, it is natural that it is impossible to find exact solution(2.1). Therefore, the problem comes down to finding approximate solutions within the framework of physically justified simplifying assumptions.
The underlying band theory modern physics metals, dielectrics and semiconductors, is based on two approximations: adiabatic, or the Born-Oppenheimer approximation, and single-electron.
The adiabatic approximation takes into account the different nature of the motion of light particles - electrons and heavy particles - nuclei. Due to the sharp difference in their masses, the movement of electrons will be fast compared to the movement of nuclei. Therefore, when considering the movement of electrons at any moment in time, the nuclei can be considered motionless, and when considering the movement of nuclei, only the time-averaged field created by all electrons can be taken into account. In mathematical language, this means that the wave function in (2.1) can be represented as a product of two functions
one of which c describes the slow motion of the nuclei, and the second only parametrically depends on the coordinates of the nuclei. Then (2.1) breaks down into the equation for electrons
and the equation for nuclei
Typically, the motion of nuclei, i.e., thermal vibrations of the lattice, is considered as perturbations, and instead of the coordinates of the nuclei, the coordinates of fixed lattice nodes are substituted into equation (2.3). However, even after this, the Schrödinger equation can be solved
it is forbidden. The solution becomes possible only when the problem of the movement of many interacting particles is reduced to the problem of the movement of one electron in the field of all other particles. This is achieved by introducing the so-called self-consistent field
which is equal to the potential energy of all electrons, except at the point where the electron is located. Using the Hamiltonian of the system, the system is represented as a sum of Hamiltonians related to individual electrons
A wave function in (2.3) can be searched for as the product