scholarly journals The theory of electronic semi-conductors

1931 ◽  
Vol 133 (822) ◽  
pp. 458-491 ◽  
Keyword(s):  
Classical Theory ◽  
Energy Levels ◽  
Electron Gas ◽  
Mean Free Path ◽  
Pauli Principle ◽  
Critical Energy ◽  
Free Electrons ◽  
Conduction Electrons ◽  
Metallic Conduction ◽  
Direct Proportionality

The application of quantum mechanics to the problem of metallic conduction has cleared up many of the difficulties which were so apparent in the free electron theories of Drude and Lorentz. Sommerfeld* assumed that the valency electrons of the metallic atoms formed an electron gas which obeyed the FermiDirac statistics, instead of Maxwellian statistics, and, using in the main classical ideas, showed how the difficulty of the specific heat would be removed. He was, however, unable to determine the temperature dependence of the resistance, as his formulae contained a mean free path about which little could be said. F. Bloch took up the question of the mechanics of electrons in a metallic lattice, and showed that if the lattice is perfect an electron can travel quite freely through it. Therefore so long as the lattice is perfect the conductivity is infinite, and it is only when we take into account the thermal motion and the impurities that we obtain a finite value for the conductivity. On this view all the electrons in a metal are free, and we cannot assume, as we do in the classical theory, that only the valency electrons are free. This does not give rise to any difficulty in the theory of metallic conduction, as the direct proportionality between the conductivity and the number of free electrons no longer holds when the Pauli principle is taken into account. If there is no external electric field, the number of electrons moving in any direction is equal to the number moving in the opposite direction. The action of a field is to accelerate or retard the electrons, causing them to make transitions from one set of energy levels to another. This can only happen if the final energy levels are already unoccupied, and therefore only those electrons whose energies are near the critical energy of the Fermi distribution can make transitions and take part in conduction, as it is only in the neighbourhood of the critical energy that the energy levels are partly filled and partly empty. These electrons are few in number compared with the valency electrons, and are what should be called the conduction electrons. On the classical theory alone are the valency electrons, the free electrons and the conduction electrons the same.

scholarly journals On some electron properties of tellurium and Wilson's mechanism of semi-conductivity

1935 ◽  
Vol 148 (865) ◽  
pp. 648-664 ◽  
Keyword(s):  
Electrical Conductivity ◽  
Free Electron ◽  
Previous Investigation ◽  
Room Temperature ◽  
Specific Resistance ◽  
Mean Free Path ◽  
Free Electrons ◽  
Conduction Electrons ◽  
Classical Statistics ◽  
Limiting Case

In a previous investigation it was found that the unusually high value for the Wiedemann-Franz ratio of tellurium could be explained as being only a formal anomally. The amount of heat transferred by the bound atoms is the same in tellurium as in conducting metals; but, in tellurium, in contrast to good conductors, it is responsible for almost the entire heat conductivity because the heat transferred by the free electrons is especially small. This indicates that tellurium differs from true metals in that the density of free electrons is very small. Classical statistics is therefore applicable and the electrical conductivity is given by x = 4/3 e 2 ln (2 πmk T) -5/9 , (1) where n is the density of free (conduction) electrons and l is their mean free path. Taking the specific resistance of tellurium at room temperature as 0.3 ohm-cm and l as 5.2 X 10 -6 cm (Sommerfeld's value for silver, found by applying Fermi-Dirac statistics), n is 2.9 X 10 16 , or about one free electron per million tellurium atoms in contrast to good conductors in which there is approximately one free electron per atom. Even in the limiting case with l = 3.2 X 10 -3 cm (the distance between the tellurium atoms), n is 4.7 X 10 18 which is about one free electron for every 6000 tellurium atoms.

Thermal Properties of an Incompletely Degenerate Fermi Gas

1936 ◽  
Vol 32 (1) ◽  
pp. 108-111 ◽  
Author(s):  
N. F. Mott
Keyword(s):  
Thermal Properties ◽  
Specific Heat ◽  
Electron Gas ◽  
Electronic States ◽  
Fermi Gas ◽  
Paramagnetic Susceptibility ◽  
Free Electrons ◽  
Periodic Field ◽  
Degenerate Fermi Gas ◽  
Image Position

The purpose of this note is to calculate the specific heat and paramagnetic susceptibility of an electron gas obeying the Fermi-Dirac statistics for all temperatures, including those temperatures for which the gas is partially degenerate. The results are applicable to the electrons in a metal, whether free or moving in a periodic field, provided only that the number of electronic states per gram atom with energy between E and E + dE can be expressed in the formas for free electrons.

scholarly journals RESONANCES AND SPECTRAL SHIFT FUNCTION FOR THE SEMI-CLASSICAL DIRAC OPERATOR

2007 ◽  
Vol 19 (10) ◽  
pp. 1071-1115 ◽  
Author(s):  
ABDALLAH KHOCHMAN
Keyword(s):  
Dirac Operator ◽  
Upper Bound ◽  
Selfadjoint Operator ◽  
Energy Levels ◽  
Critical Energy ◽  
Spectral Shift ◽  
Approximation Formula ◽  
Shift Function ◽  
Spectral Shift Function ◽  
Electro Magnetic

We consider the selfadjoint operator H = H0+ V, where H0is the free semi-classical Dirac operator on ℝ3. We suppose that the smooth matrix-valued potential V = O(〈x〉-δ), δ > 0, has an analytic continuation in a complex sector outside a compact. We define the resonances as the eigenvalues of the non-selfadjoint operator obtained from the Dirac operator H by complex distortions of ℝ3. We establish an upper bound O(h-3) for the number of resonances in any compact domain. For δ > 3, a representation of the derivative of the spectral shift function ξ(λ,h) related to the semi-classical resonances of H and a local trace formula are obtained. In particular, if V is an electro-magnetic potential, we deduce a Weyl-type asymptotics of the spectral shift function. As a by-product, we obtain an upper bound O(h-2) for the number of resonances close to non-critical energy levels in domains of width h and a Breit–Wigner approximation formula for the derivative of the spectral shift function.

The Fermi Level

2020 ◽  
pp. 267-300
Author(s):  
Brian Cantor
Keyword(s):  
Fermi Level ◽  
Atomic Bomb ◽  
Energy Levels ◽  
Pauli Exclusion Principle ◽  
Mean Free Path ◽  
Maximum Energy ◽  
Exclusion Principle ◽  
Energy Bands ◽  
Anti Semitism ◽  
Dielectric Effect

The Fermi level is the maximum energy of the electrons in a material. Effectively there is a Fermi equation: EF = E max. This chapter examines the discrete electron energy levels in individual atoms as a consequence of the Pauli exclusion principle, the corresponding energy bands in a material composed of many atoms or molecules, and the way in which conductor, insulator and semiconductor materials depend on the position of the Fermi level relative to the energy bands. It explains: the concepts of electron mobility, mean free path and conductivity; the dielectric effect and capacitance; p-type, n-type, intrinsic and extrinsic semiconductors; and the behaviour of some simple microelectronic devices. Enrico Fermi was the son of a minor railway official in Rome. He had a meteoric scientific career in Italy, developing Fermi-Dirac statistics for the energies of fundamental fermion particles (such as electrons and protons), discovering the neutrino, and explaining the behaviour of different materials under bombardment from fast and slow neutrons. After initially joining Mussolini’s Fascist Party, he became unhappy at the level of anti-Semitism (his wife was Jewish) and left suddenly for America, immediately after receiving the Nobel Prize in Sweden. At Columbia and Chicago Universities and at Los Alamos National Labs, he played a key scientific role in developing controlled fission in an atomic pile, leading to the development of the atomic bomb towards the end of the Second World War, and the nuclear energy industry after the war.

scholarly journals The compton scattering and the new statistics

1929 ◽  
Vol 125 (796) ◽  
pp. 231-237 ◽  
Keyword(s):  
Compton Scattering ◽  
Electron Gas ◽  
Compton Effect ◽  
Maxwellian Distribution ◽  
Great Success ◽  
Free Electrons ◽  
Electron Theory ◽  
Conservation Of Momentum ◽  
New Statistics ◽  
Energy Principles

Great success has been achieved by Sommerfeld in the electron theory of metals by assuming that there are free electrons in them which obey the Fermi-Dirac statistics. It has been assumed in the case of univalent metals that on the average one electron per atom is free. In general, however, the valency electrons can be considered as free. These free electrons will take part in the Compton scattering. The analysis of such a Compton effect reduces to the analysis of the collisions between radiation quanta and an electron gas. The general features of such a scattering was first considered by Dirac. But he has assumed a Maxwellian distribution for the electrons which will not be applicable to the case under consideration, because the electrons in a conductor being degenerate do not obey the Maxwell's law, but the Fermian distribution. In considering such a process we take it that the conservation of momentum and energy principles are satisfied for each particular collision just as in Compton's theory—only we are here dealing with moving electrons instead of stationary electrons which Compton considers. Thus electrons of different momenta components will produce different Compton shifts, and the intensity of any particular shift will depend on the number of electrons in that state. Thus we have to average for the radiation falling on an assembly of electrons whose momenta are distributed according to the Fermi-Dirac law.

Electronic eigenvalues of copper

1953 ◽  
Vol 220 (1143) ◽  
pp. 513-529 ◽  
Keyword(s):  
Brillouin Zone ◽  
Metallic Copper ◽  
Copper Ion ◽  
Wave Functions ◽  
Free Electrons ◽  
Cubic Crystal ◽  
Conduction Electrons ◽  
Energy Gaps ◽  
The Face ◽  
Cellular Method

The cellular method as developed by Howarth & Jones (1952) is applied to the face-centred cubic crystal, and, in particular, the eigenvalues and wave functions of the 3 d and 4 s electrons in metallic copper are obtained for states whose wave vectors lie at the ends of the three- and four-fold axes in the Brillouin zone. The conduction electrons approximate closely to free electrons inside the first Brillouin zone, but the energy gaps at the zone faces are found to be sensitively dependent upon the potential used to represent the copper ion. The eigenvalues of the 3 d band agree closely with Fletcher’s results for nickel, and show a band width of 3·46 eV. An approximate solution of the Hartree equations for the metal shows the top of the d -band to lie 3·7 eV below the Fermi level in the conduction band.

Classical theory of the dielectric function for an inhomogeneous electron gas

1993 ◽  
Vol 286 (3) ◽  
pp. 346-354 ◽  
Author(s):  
Hideo Nitta ◽  
Shigeru Shindo ◽  
Mitsuo Kitagawa
Keyword(s):  
Dielectric Function ◽  
Classical Theory ◽  
Electron Gas

Nichtgleichgewichtscffekte im Argonkaskadenbogen im Druckbereich von 0,2 atm bis 5,0 atm / Non-LTE Effects in an Argon Cascade Arc in the Pressure Range from 0.2 atm to 5.0 atm

1973 ◽  
Vol 28 (9) ◽  
pp. 1459-1467 ◽  
Author(s):  
D. Garz
Keyword(s):  
Particle Temperature ◽  
Energy Levels ◽  
Heavy Particle ◽  
Electrical Current ◽  
Mean Free Path ◽  
Electrical Field Strength ◽  
Intensity Measurements ◽  
Plasma State ◽  
Significant Intensity

An analysis of the plasma state was performed by determination of the electron temperature and the distribution temperature from end -on intensity measurements in the arc axis and by determination of the heavy particle temperature from measurements of the electrical field strength. The electrical current of the arc was varied between 1.5 amp and 60 amp and the argon pressure was varied from 0.2 atm to 5.0 atm. It turned out that not the electron density but the mean free path of the electrons is the essential parameter for the adjustment of LTE. Moreover radial end-on intensity measurements of Ar-I lines with different upper energy levels revealed significant intensity anomalies spreading from the axis of the arc to the walls. These effects could be explained assuming that a diffusion of Ar atoms, which are excited to 4s-levels, takes place out of the arc axis. The diffusion equation and its solution led to a satisfying explanation of the observed radial non-LTE effects.

The band structure of aluminium III. A self-consistent calculation

1957 ◽  
Vol 240 (1222) ◽  
pp. 361-374 ◽  
Keyword(s):  
Band Structure ◽  
Electron Gas ◽  
Careful Consideration ◽  
Correlation Effects ◽  
Conduction Electrons ◽  
Hartree Fock ◽  
Exchange Effects ◽  
Self Consistent ◽  
Core Electrons ◽  
Self Consistency

The main feature of the present recalculation of the band structure is a careful consideration of the potential on which it is based. An effective potential acting on a conduction electron is defined intuitively so as to include the effects of correlation and exchange. This grafts the Bohm & Pines theory of a free-electron gas (Pines 1955) on to the Hartree—Fock treatment of the effect of the ion cores. Correlation and exchange effects among the conduction and ion-core electrons have been calculated, and the variation of the potential in the regions about half-way between the atoms has also been calculated and taken into account. Achieving self-consistency in the contribution to the potential due to the conduction electrons has not been difficult. Because of various approximations, including those in the potential, the calculated energy values are judged to be correct to about 0·03 Ry. The results are found to agree satisfactorily with the model of the band structure obtained in I by noting that the effective Fermi level for electrons and holes in the band structure may not be the same due to additional correlation effects.

Sign in / Sign up

Export Citation Format

Share Document