Frank J. Blatt

Theory of Mobility of Electrons in Solids

Publisher Summary This chapter discusses the theory of mobility of electrons in macroscopically homogeneous solids on mechanisms that inhibit the flow of electrons; that is, scattering by imperfections of various kinds. There is very little fundamental difference between the transport of electrons in metals and semiconductors. The most outstanding distinction between the two is that in the former, the electron distribution is highly degenerate at all temperatures of interest, whereas in the latter, Boltzmann statistics is frequency applicable. The chapter also presents a resume of the statistics obeyed by electrons and derivation of the equilibrium probability distribution of an assembly of electrons, and the formulation of the Boltzmann transport equation. The solution of the Boltzmann transport equation helps in obtaining formulas for various transport phenomena such as electrical and thermal conductivity, the thermoelectric effects, and effects which appear in the presence of a magnetic field. The chapter also discusses the diverse scattering mechanisms which permit a distorted distribution function to relax to equilibrium together with the “phonon-drag” effect, particularly its influence on thermoelectric properties and the “hot electron” problem in semiconductors.

The resistivity of fine platinum wires

We have measured the resistivity of pure platinum wires ranging in diameter from 16 mil to 0.3 mil in the temperature range 1.2 to 4.2°K, and observed size-dependent deviations from Matthiessen’s rule. The temperature dependent portion of the resistivity is dominated in this temperature range by a term of the formAT2, whereA increases from about 12×10−12 Ω cm/°K2 for the thickest wires to 18×10−12 Ω cm/°K2 for the thinnest ones. There is an additional resistivity contribution which appears to increase more rapidly thanT5, and which also evidences some increase with decreasing wire diameter. The observed deviations from Matthiessen’s rule display temperature and size variations consistent with the theory ofBlatt andSatz, and the magnitude of the deviations can be accounted for by this theory taking into account only that portion of the electrical resistivity produced by electron-phonon scattering. Thus the data are consistent with arguments suggesting that interband electron-electron scattering does not lead to size-dependent deviations from Matthiessen’s rule.ZusammenfassungWir haben im Temperaturbereich von 1,2°K bis 4,2°K den elektrischen Widerstand an dünnen Platindrähten mit Durchmessern von 0,32 mm bis 0,006 mm gemessen und „size-effect”-bedingte Abweichungen von der Matthiessen’schen Regel festgestellt.Der temperaturabhängige Teil des Widerstandes wird in diesem Temperaturbereich durch einen Term der FormAT2 beschrieben, wobeiA zwischen den Werten 12·10−12 Ω cm/°K2 für die dicksten Drähte, und 18·10−12 Ω cm/°K2 für die dünnsten Drähte variieren kann. Ein zusätzlicher Widerstandsanteil wächst etwa stärker alsT5 und zeigt auch einen leichten Anstieg bei abnehmender Drahtdicke. Die beobachteten Abweichungen von der Matthiessenschen Regel zeigen Temperatur- und „size-effect”-Abhängigkeiten, wie sie durch die Theorie vonBlatt undSatz vorausgesagt werden. Die Größenordnung der Effekte wird ebenfalls durch diese Theorie richtig wiedergegeben, wenn man nur den Anteil des Widerstandes berücksichtigt, der durch die Elektron-Phonon-Streuung verursacht wird. Die Ergebnisse zeigen, daß offenbar die Interband-Elektron-Elektron-Streuungen nicht zu den „size-effect”-bedingten Abweichungen von der Matthiessen’schen Regel beitragen.RésuméNous avons mesuré la résistance spécifique dans le domaine de température de 1,2°K à 4,2°K de fils de platine pur avec des diamètres variant de 0,4 mm à 0,01 mm et observé des déviations de la règle de Matthiessen en fonction de l’épaisseur des fils. La fonction décrivant la résistance spécifique possède une partie dépendante de la température qui est dominée, dans notre domaine de température, par un terme de la formeAT2, oùA passe d’une valeur de 12×10−12 Ω cm/°K2 pour les fils les plus épais à une valeur de 18×10−12 Ω cm/°K2 pour les fils les plus minces.De plus nous avons observé une contribution additionelle qui semble augmenter plus rapidement queT5 et qui en plus montre une certaine augmentation lors d’une diminution du diamètre des fils. Les déviations de la règle de Matthiessen que nous avons observées montrent des variations en fonction de la température et de l’épaisser des fils, qui sont en accord avec la théorie deBlatt etSatz et la grandeur de ces déviations peut être expliquée en ne considérant que la partie de la résistance provenant de la diffusion électron-phonon. Ainsi les résultats expérimentaux confirment l’hypothèse que la diffusion électron-électron entre les bandes de conduction ne produit pas de déviations de la règle de Matthiessen, fonctions de l’épaisseur des fils.

The Thermoelectric Power of Transition Metals

At any temperature the thermoelectric power of the 24 transition metals* is roughly an order of magnitude greater than that of “simple” metals. For this reason at least one limb of any thermocouple is generally a transition metal or alloy. It is also for this reason that the bulk of numerous measurements made on transition metals has been made with a view to determining their suitability and reliability as thermoelements, and not as a means of studying their electronic properties. Apart from the early work of Mott [36M1], who gave a measure of understanding as to why elements such as palladium and platinum had a large thermopower, the only scientific work of note has been carried out in the last ten years. Nevertheless, as we have said, for some transition metals there is no shortage of measurements.

Techniques in Thermoelectric Measurements

We shall digress here from the theoretical discussion of the TEP to consider in some detail the experimental techniques that have been employed in the past and that are currently used to measure thermoelectric effects. These measurements fall naturally into four categories: the first three are concerned with the determination of conventional thermoelectric parameters, the TEP, S, the Peltier heat (measurement of Π), and the Thomson coefficient μ ; the last is concerned with other parameters, specifically Πe, or G = l/Πe, introduced at the end of Chapter 1.

Dilute Magnetic Alloys

We shall devote a separate chapter to dilute “magnetic” alloys because both experimental results and theoretical approaches set such alloys apart from those discussed in Chapter 2. The adjective “magnetic” has been set in quotation marks because we do not refer here to magnetic systems in the conventional sense. On the contrary, we specifically exclude from discussion alloys which exhibit a phase transition to a magnetically ordered state, ferromagnetic or antiferromagnetic. Our interest is in alloy systems of nontransition-metal solvents with transition-metal or rare-earth solutes; moreover, the solute concentration is restricted to small values, typically less than a few tenths of an atomic percent and always less than a few atomic percent. The adjective “magnetic” arises from two sources: (a) First and foremost, there is the observation that the magnetic susceptibility of many such systems displays properties consistent with the presence of localized and free (or nearly free) magnetic moments, (b) There is a growing consensus that spin fluctuations are important for transport properties in many of these systems despite the apparent lack of significant temperature dependence in their magnetic susceptibilities. The concepts that have proved useful in understanding the properties of these alloys are also applicable to some alloys which do not require free magnetic moments or spin fluctuations being associated with solute atoms. Thus we shall include all these systems in our discussion.

Effects of Pressure and Magnetic Field on the Thermoelectric Power

The influence of pressure on the TEP of metals is of practical and fundamental interest. The first is obvious: In a study of the effect of pressure on the properties of materials at temperatures well above or below room temperature, direct determination of the specimen temperature is requisite for meaningful data.

Survey of the Theory of Electronic Conduction in Metals

In Chapter 1 we made no mention of the detailed mechanisms that lead to electronic conduction and associated thermoelectric effects. The general phenomenological equations (1.15) and (1.16) are equally valid for metals, semiconductors, or an ionized gas, and Onsager’s relations are correct for any of these. This book is concerned, however, only with metallic conductors, and our interest in thermoelectric phenomena is based on the belief that they can shed light on fundamental features of electronic energy levels and on the interaction of conduction electrons with their environment. To see this more clearly, we must now review briefly the elements of transport theory as it applies to these materials.