Electrical Conduction in Metals and Semiconductors
Abstract
Electrical transport through materials is a large and complex field, and in this chapter we cover only a few aspects that are relevant to practical applications. We start with a review of the semi-classical approach that leads to the concepts of drift velocity, mobility and conductivity, from which Matthiessen’s Rule is derived. A more general approach based on the Boltzmann transport equation is also discussed. We review the conductivity of metals and include a useful collection of experimental data. The conductivity of nonuniform materials such as alloys, polycrystalline materials, composites and thin films is discussed in the context of Nordheim’s rule for alloys, effective medium theories for inhomogeneous materials, and theories of scattering for thin films. We also discuss some interesting aspects of conduction in the presence of a magnetic field (the Hall effect). We present a simplified analysis of charge transport in semiconductors in a high electric field, including a modern avalanche theory (the theory of lucky drift). The properties of low-dimensional systems are briefly reviewed, including the quantum Hall effect.
A good understanding of charge carrier transport and electrical conduction is essential for selecting or developing electronic materials for device applications. Of particular importance are the drift mobility of charge carriers in semiconductors and the conductivity of conductors and insulators. Carrier transport is a broad field that encompasses both traditional bulk processes and, increasingly, transport in low dimensional or quantized structures. In other chapters of this handbook, Baranovskii describes hopping transport in low mobility solids such as insulators, Morigaki deals with the electrical properties of amorphous semiconductors and Gould discusses in detail conduction in thin films. In this chapter, we outline a semi-quantitative theory of charge transport suitable for a wide range of solids of use to materials researchers and engineers. We introduce theories of bulk transport followed by processes pertinent to ultra-fast transport and quantized transport in lower dimensional systems. The latter covers such phenomena as the Quantum Hall Effect, and Quantized Conductance and Ballistic Transport in Quantum Wires that has potential use in new kinds of devices. There are many more rigorous treatments of charge transport; those by Rossiter [2.1] and Dugdale [2.2] on metals, and Nag [2.3] and Blatt [2.4] on semiconductors are highly recommended.
2.1 Fundamentals: Drift Velocity, Mobility and Conductivity
Basic to the theory of the electronic structure of solids are the solutions to the quantum mechanical problem of an electron in a periodic potential known as Bloch waves. These wavefunctions are traveling waves and provide the physical basis for conduction. In the semi-classical approach to conduction in materials, an electron wavepacket made up of a superposition of Bloch waves can in principle travel unheaded in an ideal crystal. No crystal is ideal, however, and the imperfections cause scattering of the wavepacket. Since the interaction of the electron with the potential of the ions is incorporated in the Bloch waves, one can concentrate on the relatively rare scattering events which greatly simplifies the theory. The motion of the electrons between scattering events is essentially free (with certain provisos such as no interband transitions) subject only to external forces, usually applied electric or magnetic fields. A theory can then be developed that relates macroscopic and measurable quantities such as conductivity or mobility to the microscopic scattering processes. Principle in such a theory is the concept of mean free time τ which is the average time between scattering events. τ is also known as the conductivity relaxation time because it represents the time scale for the momentum gained from an external field to relax. Equivalently, 1 ∕ τ is the average probability per unit time that an electron is scattered.
The above semiquatitative description is sufficient to understand the basic principles of conduction. A more rigorous approach involves solving the Boltzmann charge transport equation and is addressed in Sect. 2.6.
2.2 Matthiessen’s Rule
Matthiessen’s rule is indispensable for predicting the resistivities of many types of conductors. In some cases like thin films, the rule is obeyed only approximately, but it is nonetheless still useful for an initial (often quite good) estimate.
2.3 Resistivity of Metals
2.3.1 General Characteristics
Typical resistivities at 273 K (0^{∘}C) ρ_{0} and thermal coefficients of resistivity α_{0} at 0^{∘}C for various metals above 0^{∘}C and below melting temperature. The resistivity index n in \(\rho=\rho_{0}(T/{T_{0}})^{n}\) is also shown. Note that n is fitted to resistivity data above 273 K but below the melting temperature of the metal. Data selectively combined from various sources, including [2.6, 2.7]
Metal | ρ_{0} (nΩ m) | n | \(\alpha_{0}\times{\mathrm{10^{-3}}}\,{\mathrm{(K^{-1})}}\) |
---|---|---|---|
Aluminium, Al | 24.2 | 1.20 | 4 |
Antimony, Sb | 302 | 1.27 | 4.7 |
Beryllium, Be | 30.2 | 1.73 | 8 |
Bismuth, Bi | 1070 | – | 4.5 |
Cadmium, Cd | 68 | – | 4.3 |
Calcium, Ca | 31.1 | 1.09 | 4.0 |
Cerium, Ce | 730 | – | 0.9 |
Cesium, Cs | 187 | 1.23 | 4.8 |
Cromium, Cr | 118 | 1.01 | 2.9 |
Cobalt, Co | 56 | – | 6.6 |
Copper, Cu | 15.4 | 1.16 | 4.3 |
Gold, Au | 20.5 | 1.13 | 4.0 |
Hafnium, Ha | 304 | 1.21 | 4.4 |
Indium, In | 80 | 1.31 | 5.0 |
Iridium, Ir | 47 | – | 4.5 |
Iron, Fe | 85.7 | 1.73 | 6.3 |
Lead, Pb | 192 | 1.14 | 4.2 |
Magnesium, Mg | 40.5 | 1.07 | 4.2 |
Molybdenum, Mo | 48.5 | 1.21 | 5.0 |
Nickel, Ni | 61.6 | 1.76 | 6.5 |
Niobium, Nb | 152 | – | 2.3 |
Palladium, Pd | 97.8 | 0.94 | 0.39 |
Platinum, Pt | 98.4 | 1.01 | 3.9 |
Rhodium, Rh | 43 | – | 4.4 |
Ruthenium, Ru | 71 | – | 4.1 |
Silver, Ag | 14.7 | 1.13 | 4.1 |
Strontium, Sr | 123 | 0.99 | 3.6 |
Tantalum, Ta | 122 | 0.93 | 3.6 |
Tin, Sn | 115 | 1.1 | 4.0 |
Titanium, Ti | 390 | 1.01 | 4.8 |
Tungsten, W | 48.2 | 1.24 | 4.8 |
Vanadium, V | 181 | 1.02 | 3.9 |
Zinc, Zn | 54.6 | 1.14 | 4.2 |
Zirconium, Zr | 388 | 1.00 | 4.2 |
It is instructive to mention that if ρ ≈ AT as we expect for an ideal pure metal, then \(\alpha_{0}=T_{0}^{-1}\). If we take the reference temperature T_{0} as 273 K (0^{∘}C), then α_{0} should ideally be 1/(273 K) or \({\mathrm{3.66\times 10^{-3}}}\,{\mathrm{K^{-1}}}\). Examination of a_{0} for various metals shows that ρ ∝ T is not a bad approximation for some of the familiar pure metals used as conductors, e. g., Cu, Al, Au, but fails badly for others, such as indium, antimony and, in particular, the magnetic metals, e. g., iron and nickel.
2.3.2 Fermi Electrons
2.4 Solid Solutions and Nordheim’s Rule
Nordheim’s rule assumes that the solid solution has the solute atoms randomly distributed in the lattice. For sufficiently small amounts of impurity, experiments show that the increase in the resistivity ρ_{I} is nearly always simply proportional to the impurity concentration X, that is, ρ_{I} ∝ X. For dilute solutions, Nordheim’s rule predicts the same linear behavior, that is, ρ_{I} = CX for X ≪ 1.
Solute in solvent (element in matrix) | Nordheim coefficient (nΩ m) | Maximum solubility at 25^{∘}C (at.%) |
---|---|---|
Au in Cu matrix | 5500 | 100 |
Mn in Cu matrix | 2900 | 24 |
Ni in Cu matrix | 1250 | 100 |
Sn in Cu matrix | 2900 | 0.6 |
Zn in Cu matrix | 300 | 30 |
Cu in Au matrix | 450 | 100 |
Mn in Au matrix | 2410 | 25 |
Ni in Au matrix | 790 | 100 |
Sn in Au matrix | 3360 | 5 |
Zn in Au matrix | 950 | 15 |
2.5 Carrier Scattering in Semiconductors
At low electric fields, ionized impurity scattering and phonon scattering predominate. Other types of scattering include carrier-carrier scattering, inter-valley scattering, and neutral impurity scattering; these may generally be ignored to a first approximation.
For phonon scattering, both polar and non-polar phonon scattering should be considered. In polar scattering, short wavelength oscillations of atoms on different sub-lattices vibrating out of phase produce an effective dipole moment proportional to the bond polarity. Since such vibrational modes are optically active (since this dipole moment can interact with an incident electromagnetic field), this type of lattice scattering is usually referred to as polar optical phonon scattering. Since a sub-lattice is necessary for optical modes, this scattering mechanism is not present in elemental semiconductors such as Si, Ge, or diamond.
Non-polar phonon scattering comes from long wavelength oscillations in the crystal, involving small displacements of tens to thousands of atoms. The wavelength depends on the material and its elastic properties. Such modes are very similar to sound vibrations and are thus referred to as acoustic modes. The associated atomic displacements correspond to an effective built-in strain, with local change in the lattice potential, causing carrier scattering known as deformation potential acoustic phonon scattering. Since the change in potential is relatively small, the scattering efficiency is relatively low as compared with polar optical phonon scattering.
2.6 The Boltzmann Transport Equation
- 1.
v ⋅ ∇_{ r }f represents diffusion through a volume element d^{3}r about the point r in phase space due to a gradient ∇_{ r }f
- 2.
\(\boldsymbol{\dot{k}}\cdot\nabla_{\boldsymbol{k}}f\) represents drift through a volume element d^{3}k about the point k in phase space due to a gradient ∇_{ k }f (for example, \(\hbar\,{\boldsymbol{\dot{k}}}=e\left({\boldsymbol{E}+\frac{1}{c}\boldsymbol{v}\times\boldsymbol{B}}\right)\) in the presence of electric and magnetic fields)
- 3.
\((\partial f/\partial t)_{\text{c}}\) is the collision term and accounts for the scattering of electrons from a point k (for example, this may be due to lattice or ionized impurity scattering).
2.7 Resistivity of Thin Polycrystalline Films
Resistivities of some thin Cu and Au films at room temperature. PC: Polycrystalline film; RT is room temperature; D = film thickness; d = average grain size. FS and MS refer to Fuchs–Sondheimer and Mayadas–Shatzkes descriptions of thin film resistivity. At RT for Cu, λ = 38–40 nm, and for Au λ = 36–38 nm. Data selectively combined from various sources [2.14, 2.19, 2.20, 2.21, 2.22, 2.24, 2.25, 2.26, 2.27]
Film | D (nm) | d (nm) | ρ (nΩ m) | Comment |
---|---|---|---|---|
Cu films | ||||
Cu encapsulated in SiO_{2}/Cu/SiO_{2} [2.20] | 45.3 31.7 | 101 41 | 28.0 35.5 | DC sputtering. Cu film sandwiched in SiO_{2}/Cu/SiO_{2}, annealed at 150^{∘}C. MS with R = 0.50 |
Cu encapsulated in SiO_{2}/Ta/Cu/Ta/SiO_{2} [2.21] | 34.2 | 39.4 | 37.3 | DC sputtering. Cu film sandwiched in SiO_{2}/Ta/Cu/Ta/SiO_{2},annealed at 600^{∘}C. MS with R = 0.47 |
Cu on Ta/SiO_{2}/Si(001) [2.21] | 35 | 40 | 31 | Sputtering and then annealing at 350^{∘}C |
Cu (single crystal) on TiN/MgO (001) [2.22] | 40 13 | ∞ ∞ | 21 29.7 | Cu single crystal epitaxial layer on TiN(100) on MgO surface (001). Ultra-high vacuum, DC sputtering. FS with p ≈ 0.6 in vacuum, p = 0 in air. |
Cu on TiN, W and TiW [2.14] | > 250 | 186 44 | 21 31 | CVD (chemical vapor deposition). Substrate temperature 200^{∘}C, ρ depends on d not D = 250–900 nm. MS. |
Cu on crystalline Si (100) surface [2.23] | 51.2 17.2 8.6 | D ∕ 2.3 | 32.2 70.5 126 | Ion beam deposition with negative substrate bias. Resistivity follows FS and MS equations combined; surface and grain boundary scattering. FS and MS, p = 0, R = 0.24, \(d=D/2.3\) |
Au films | ||||
---|---|---|---|---|
Au epitaxial film on mica | 30 | 25 | Single crystal on mica. p ≈ 0.8, highly specular scattering | |
Au PC film on mica | 30 | 54 | PC. Sputtered on mica. p is small | |
Au film on glass | 30 | 70 | PC. Evaporated onto glass. p is small, nonspecular scattering | |
Au on glass [2.24] | 40 40 | 8.5 3.8 | 92 189 | PC. Sputtered films. R = 0.27–0.33 |
It is generally very difficult to separate the effects of surface and grain boundary scattering in thin polycrystalline films; the contribution from grain boundary scattering is likely to exceed that from the surfaces. In any event, both contributions, by Matthiessen’s general rule, increase the overall resistivity. Figure 2.12 shows an example in which the resistivity ρ_{film} of thin Cu polycrystalline films is due to grain boundary scattering, and thickness has no effect (D was 250–900 nm and much greater than λ). The resistivity ρ_{film} is plotted against the reciprocal mean grain size 1 ∕ d, which then follows the expected linear behavior in (2.49). On the other hand, Fig. 2.13 shows the resistivity of Cu films as a function of film thickness D. In this case, the thin films are grown epitaxially (as a single crystal) on MgO single crystal surfaces and the grain boundary scattering is absent, which is a clear cut case for surface scattering; the transport is described by (2.51). Chapter 28 provides a more advanced treatment of conduction in thin films.
2.8 Inhomogeneous Media: Effective Media Approximation
The effective media approximation (EMA ) attempts to estimate the properties of inhomogeneous mixture of two or more components using the known physical properties of each component. The general idea of any EMA is to substitute for the original inhomogeneous mixture some imaginary homogeneous substance – the effective medium (EM) – whose response to an external excitation is the same as that of the original mixture. The EMA is widely used for investigations of non-uniform objects in a variety of applications such as composite materials [2.28, 2.29], microcrystalline and amorphous semiconductors [2.30, 2.31, 2.32, 2.33], light scattering [2.34], conductivity of dispersed ionic semiconductors [2.35] and many others.
Mixture rules for randomly oriented particles. (Compiled from Reynolds and Hough [2.36])
Particle shape | Mixture rule | Factors in (2.59) | References | |
---|---|---|---|---|
A | ε ^{∗} | |||
Spheres | \(\displaystyle\frac{\varepsilon_{\text{eff}}-\varepsilon_{\beta}}{\varepsilon_{\text{eff}}+2\varepsilon_{\beta}}=\chi_{\alpha}\frac{\varepsilon_{\alpha}-\varepsilon_{\beta}}{\varepsilon_{\alpha}+2\varepsilon_{\beta}}\) | \(\frac{1}{3}\) | ε _{ β } | |
Spheres | \(\displaystyle\frac{\varepsilon_{\text{eff}}-\varepsilon_{\beta}}{3\varepsilon_{\beta}}=\chi_{\alpha}\frac{\varepsilon_{\alpha}-\varepsilon_{\beta}}{\varepsilon_{\alpha}+2\varepsilon_{\beta}}\) | \(\frac{1}{3}\) | ε _{ β } | [2.42] |
Spheres | \(\displaystyle\chi_{\alpha}\frac{\varepsilon_{\alpha}-\varepsilon_{\text{eff}}}{\varepsilon_{\alpha}+2\varepsilon_{\text{eff}}}+\chi_{\beta}\frac{\varepsilon_{\beta}-\varepsilon_{\text{eff}}}{\varepsilon_{\beta}+2\varepsilon_{\text{eff}}}=0\) | \(\frac{1}{3}\) | ε _{eff} | [2.43] |
Spheres | \(\displaystyle\frac{\varepsilon_{\text{eff}}-\varepsilon_{\beta}}{3\varepsilon_{\text{eff}}}=\chi_{\alpha}\frac{\varepsilon_{\alpha}-\varepsilon_{\beta}}{\varepsilon_{\alpha}+2\varepsilon_{\beta}}\) | \(\frac{1}{3}\) | ε _{eff} | [2.44] |
Spheroids | \(\displaystyle\varepsilon_{\text{eff}}=\varepsilon_{\beta}+\frac{\chi_{\alpha}}{3\left({1-\chi_{\alpha}}\right)}\sum\limits_{i=1}^{3}{\frac{\varepsilon_{\alpha}-\varepsilon_{\text{eff}}}{1+A_{i}\left({\frac{\varepsilon_{\alpha}}{\varepsilon_{\beta}}-1}\right)}}\) | A | ε _{ β } | [2.45] |
Spheroids | \(\displaystyle\varepsilon_{\text{eff}}=\varepsilon_{\beta}+\frac{\chi_{\alpha}}{3}\sum\limits_{i=1}^{3}{\frac{\varepsilon_{\alpha}-\varepsilon_{\beta}}{1+A_{i}\left({\frac{\varepsilon_{\alpha}}{\varepsilon_{\text{eff}}}-1}\right)}}\) | A | ε _{eff} | [2.46] |
Lamellae | \(\displaystyle\varepsilon_{\text{eff}}^{2}=\frac{2\left({\varepsilon_{\alpha}\chi_{\alpha}+\varepsilon_{\beta}\chi_{\beta}}\right)-\varepsilon_{\text{eff}}}{\frac{\varepsilon_{\alpha}}{\chi_{\alpha}}+\frac{\varepsilon_{\beta}}{\chi_{\beta}}}\) | 0 | ε _{eff} | [2.43] |
Rods | \(5\varepsilon_{\text{eff}}^{3}+\left(5\varepsilon^{\prime}_{p}-4\varepsilon_{p}\right)\varepsilon_{\text{eff}}^{2}-\left({\chi_{\alpha}\varepsilon_{\alpha}^{2}+4\varepsilon_{\alpha}\varepsilon_{\beta}+\chi_{\beta}\varepsilon_{\beta}^{2}}\right)-\varepsilon_{\alpha}\varepsilon_{\beta}\varepsilon_{p}=0\) where \(\displaystyle\frac{1}{\varepsilon^{\prime}_{p}}=\frac{\chi_{\alpha}}{\varepsilon_{\beta}}+\frac{\chi_{\beta}}{\varepsilon_{\alpha}}\) and \(\displaystyle\frac{1}{\varepsilon_{p}}=\frac{\chi_{\alpha}}{\varepsilon_{\alpha}}+\frac{\chi_{\beta}}{\varepsilon_{\beta}}\) | \(\frac{1}{2}\) | [2.47] |
Mixture rules for partially oriented particles. (Compiled from Reynolds and Hough [2.36])
Particle shape | Formula | Factors in (2.59) | References | |||
---|---|---|---|---|---|---|
A | ε ^{∗} | cosα_{1} = cosα_{2} | cosα_{3} | |||
Parallel cylinders | \(\displaystyle\frac{\varepsilon_{\text{eff}}-\varepsilon_{\beta}}{\varepsilon_{\text{eff}}+\varepsilon_{\beta}}=\chi_{\alpha}\frac{\varepsilon_{\alpha}-\varepsilon_{\beta}}{\varepsilon_{\alpha}+\varepsilon_{\beta}}\) | \(\frac{1}{2}\) | ε _{ β } | \(\frac{1}{2}\) | 0 | |
Parallel cylinders | \(\displaystyle\chi_{\alpha}\frac{\varepsilon_{\alpha}-\varepsilon_{\text{eff}}}{\varepsilon_{\alpha}+\varepsilon_{\text{eff}}}+\chi_{\beta}\frac{\varepsilon_{\beta}-\varepsilon_{\text{eff}}}{\varepsilon_{\beta}+\varepsilon_{\text{eff}}}\) | \(\frac{1}{2}\) | ε _{eff} | \(\frac{1}{2}\) | 0 | [2.43] |
Parallel lamellae (with two axes randomly oriented) | \(\displaystyle\varepsilon_{\text{eff}}^{2}={\frac{\varepsilon_{\alpha}\chi_{\alpha}+\varepsilon_{\beta}\chi_{\beta}}{\frac{\varepsilon_{\alpha}}{\chi_{\alpha}}+\frac{\varepsilon_{\beta}}{\chi_{\beta}}}}\) | 0 | ε _{eff} | \(\frac{1}{2}\) | 0 | [2.43] |
Lamellae with all axes aligned (current lines are perpendicular to lamellae planes) | \(\displaystyle\frac{1}{\varepsilon_{\text{eff}}}=\frac{\chi_{\alpha}}{\varepsilon_{\alpha}}+\frac{\chi_{\beta}}{\varepsilon_{\beta}}\) | 0 | ε _{eff} | 0 | 1 | [2.49] |
Lamellae with all axes aligned (current lines are parallel to lamellae planes) | \(\displaystyle\varepsilon_{\text{eff}}=\varepsilon_{\alpha}\chi_{\alpha}+\varepsilon_{\beta}\chi_{\beta}\) | 0 | ε _{eff} | 1 | 0 | |
Spheroids with all axes aligned (current lines are parallel to one of the axes) | \(\displaystyle\varepsilon_{\text{eff}}=\varepsilon_{\beta}+\frac{\chi_{\alpha}\left({\varepsilon_{\alpha}-\varepsilon_{\beta}}\right)}{1+A\left({\frac{\varepsilon_{\alpha}}{\varepsilon_{\beta}}-1}\right)}\) | A | ε _{ β } | 0 | 1 | [2.51] |
Spheroids with all axes aligned (current lines are parallel to one of the axes) | \(\displaystyle\frac{\varepsilon_{\text{eff}}}{\varepsilon_{\beta}}=1+\frac{\chi_{\alpha}}{\left({\frac{\varepsilon_{\alpha}}{\varepsilon_{\beta}}-1}\right)^{-1}+A\chi_{\alpha}}\) | A | ε _{ β } | 0 | 1 | [2.52] |
Conductivity/resistivity mixture rules. (After [2.5])
Particle shape | Formula | Commentary |
---|---|---|
Lamellae with all axes aligned (current lines are perpendicular to lamellae planes) | \(\displaystyle\rho_{\text{eff}}=\chi_{\alpha}\rho_{\alpha}+\chi_{\beta}\rho_{\beta}\) | Resistivity mixture rule: ρ_{ α } and ρ_{ β } are the resistivities of two phases and ρ_{eff} is the effective resistivity of mixture |
Lamellae with all axes aligned (current lines are parallel to lamellae planes) | \(\displaystyle\sigma_{\text{eff}}=\chi_{\alpha}\sigma_{\alpha}+\chi_{\beta}\sigma_{\beta}\) | Conductivity mixture rule: σ_{ α } and σ_{ β } are the conductivities of two phases and σ_{eff} is the effective conductivity of mixture |
Small spheroids (α-phase) in medium (β-phase) | \(\displaystyle\rho_{\text{eff}}=\rho_{\beta}\frac{\left({1+\frac{1}{2}\chi_{\alpha}}\right)}{\left({1-\chi_{\alpha}}\right)}\) | ρ_{ α } > 10ρ_{ β } |
Small spheroids (α-phase) in medium (β-phase) | \(\displaystyle\rho_{\text{eff}}=\rho_{\beta}\frac{\left({1-\chi_{\alpha}}\right)}{\left({1+2\chi_{\alpha}}\right)}\) | ρ_{ α } < 0.1ρ_{ β } |
Mixture rules and corresponding spectral functions G ( L )
Mixture rule by Bruggemann [2.43]: \(\displaystyle\chi_{\alpha}\frac{\varepsilon_{\alpha}-\varepsilon_{\text{eff}}}{\varepsilon_{\alpha}+2\varepsilon_{\text{eff}}}+\chi_{\beta}\frac{\varepsilon_{\beta}-\varepsilon_{\text{eff}}}{\varepsilon_{\beta}+2\varepsilon_{\text{eff}}}=0\) \(\displaystyle G(L)=\frac{3\chi_{\alpha}-1}{2\chi_{\alpha}}\delta(L)\Theta(3\chi_{\alpha}-1)+\frac{3}{4\uppi\chi_{\alpha}L}\sqrt{\left({L-L^{-}}\right)\left({L^{+}-L}\right)}\Theta(L-L^{-})\Theta(L^{+}-L)\) where \(\quad\displaystyle L^{+/-}=\frac{1}{3}\left({1+\chi_{\alpha}\pm 2\sqrt{2\chi_{\alpha}-2\chi_{\alpha}^{2}}}\right)\) |
Mixture rule by Maxwell-Garnett [2.53]: \(\displaystyle\frac{\varepsilon_{\text{eff}}-\varepsilon_{\beta}}{\varepsilon_{\text{eff}}+2\varepsilon_{\beta}}=\chi_{\alpha}\frac{\varepsilon_{\alpha}-\varepsilon_{\beta}}{\varepsilon_{\alpha}+2\varepsilon_{\beta}}\) \(\displaystyle G(L)=\delta\left({L-\frac{1-\chi_{\alpha}}{3}}\right)\) |
Mixture rule by Looyenga [2.54]: \(\displaystyle\varepsilon_{\text{eff}}^{1/3}=\chi_{\alpha}\varepsilon_{\alpha}^{1/3}+\chi_{\beta}\varepsilon_{\beta}^{1/3}\) \(\displaystyle G(L)=\chi_{\alpha}^{2}\delta(L)+\frac{3\sqrt{3}}{2\uppi}\left({\chi_{\beta}^{2}\left|{\frac{L-1}{L}}\right|^{1/3}}+\chi_{\alpha}\chi_{\beta}\left|{\frac{L-1}{L}}\right|^{2/3}\right)\) |
Mixture rule by Monecke [2.55]: \(\displaystyle\varepsilon_{\text{eff}}=\frac{2\left({\chi_{\alpha}\varepsilon_{\alpha}+\chi_{\beta}\varepsilon_{\beta}}\right)^{2}+\varepsilon_{\alpha}\varepsilon_{\beta}}{\left({1+\chi_{\alpha}}\right)\varepsilon_{\alpha}+\left({2-\chi_{\alpha}}\right)\varepsilon_{\beta}}\) \(\displaystyle G(L)=\frac{2\chi_{\alpha}}{1+\chi_{\alpha}}\delta(L)+\frac{1-\chi_{\alpha}}{1+\chi_{\alpha}}\delta\left({L-\frac{1+\chi_{\alpha}}{3}}\right)\) |
Mixture rule for hollow sphere equivalent by Bohren and Huffman [2.56]: \(\displaystyle\varepsilon_{\text{eff}}=\varepsilon_{\alpha}\frac{\left({3-2f}\right)\varepsilon_{\beta}+2f\varepsilon_{\alpha}}{f\varepsilon_{\beta}+\left({3-f}\right)\varepsilon_{\alpha}}\) \(\displaystyle G(L)=\frac{2}{3-f}\delta(L)+\frac{1-f}{3-f}\delta\left({L-\frac{3-f}{3}}\right)\) where \(\displaystyle f=1-\frac{r_{\mathrm{i}}^{3}}{r_{\text{o}}^{3}}\) and r_{i∕o}is the inner/outer radius of the sphere |
2.9 The Hall Effect
2.10 High Electric Field Transport
2.11 Impact Ionization
At very high electric field (in the range \({\mathrm{2\times 10^{5}}}\,{\mathrm{V/cm}}\) or larger) a new possibility appears: a carrier may have kinetic energy in excess of the binding energy of a valence electron to its parent atom. In colliding with an atom, such a carrier can rupture the covalent bond and produce an electron-hole pair. This process is called impact ionization and is characterized by the impact ionization coefficient α (α_{e} for electrons and α_{h} for holes). The released electrons and holes may, in turn, impact ionize more atoms producing new electron-hole pairs. This process is called avalanche and is characterized by the multiplication factor, M, which is the ratio of number of collected carriers to the number of initially injected carriers.
Avalanche multiplication in crystalline semiconductors has been well studied over several decades and is now well developed and used in various commercial applications, such as avalanche photodiodes in optical communications, high sensitivity radiation detectors, and x-ray and nuclear detectors [2.69, 2.70]. In the case of amorphous semiconductors, the present work points to only a-Se exhibiting avalanche multiplication in the bulk for reasons explained in reference [2.71], with reports dating back to the 1980s [2.61, 2.62, 2.72]. Avalanche in a-Se has also been commercialized in ultrasensitive video tubes [2.73], with potential application in medical imaging [2.74].
Impact ionization theory in crystalline solids only reached an acceptable level of confidence and understanding in the 1980s and 1990s with the development of the lucky-drift model by Ridley [2.77] and its extension by Burt [2.78], and Mackenzie and Burt [2.79]. The latter major advancement in the theory appeared as the lucky drift (LD) model, and it was based on the realization that at high fields, hot electrons do not relax momentum and energy at the same rates. Momentum relaxation rate is much faster than the energy relaxation rate. An electron can drift, being scattered by phonons, and have its momentum relaxed, which controls the drift velocity, but it can still gain energy during this drift. Stated differently, the mean free path λ_{E} for energy relaxation is much longer than the mean free path λ for momentum relaxation.
It is worth noting that the model of lucky drift is successfully used not only for crystalline semiconductors but to amorphous semiconductors [2.83].
2.12 Two-Dimensional Electron Gas
Heterostructures offer the ability to spatially engineer the potential in which carriers move. In such structures having layers deposited in the z-direction, when the width of a region with confining potential t_{ z } < λ_{dB}, the de Broglie electron wavelength, electron states become stationary states in that direction, retaining Bloch wave character in the other two directions (i. e., x- and y-directions), and is hence termed a 2-D electron gas (2DEG ). These structures are notable for their extremely high carrier mobility.
2.13 One-Dimensional Conductance
The nature of carrier transport in quantum wires depends on the wire dimensions (i. e., length L_{Wire} and diameter d_{Wire}) as compared with the carrier mean free path, l_{Carrier}. When l_{Carrier} ≫ L_{Wire}, d_{Wire} the only potential seen by the carriers is that associated with the wire walls, and carriers exhibit wavelike behavior, being guided through the wire as if it were a waveguide without any internal scattering. Conversely, if d_{Wire} ≪ λ_{DeBroglie}, only a few energy states in the wire are active, and in the limit of an extremely small waveguide, only one state or channel is active, analogous to a single mode waveguide cavity – this case is termed quantum ballistic transport. In the limit, l_{Carrier} ≪ L_{Wire}, d_{Wire}, scattering dominates transport throughout the wire – with numerous scattering events occurring before a carrier can traverse the wire or move far along its length. In such a case the transport is said to be diffusive. As discussed previously, ionized impurity and lattice scattering contribute to l_{Carrier}, with l_{Carrier} decreasing with increasing temperature due to phonon scattering. For strong impurity scattering, this may not occur until relatively high temperatures. In the intermediate case of \(L_{\text{Wire}}\gg l_{\text{Carrier}}\gg d_{\text{Wire}}\) and where d_{Wire} ≪ λ_{DeBroglie} scattering is termed mixed mode and is often called quasi-ballistic.
2.14 The Quantum Hall Effect
Notes
Acknowledgements
The authors thank Dr. Robert Johanson for helpful discussions on the subject.
References
- 2.1P.L. Rossiter: The Electrical Resisitivity of Metals and Alloys (Cambridge Univ. Press, Cambridge 1987)CrossRefGoogle Scholar
- 2.2J.S. Dugdale: The Electrical Properties of Metals and Alloys (Arnold, London 1977)Google Scholar
- 2.3B.R. Nag: Theory of Electrical Transport in Semiconductors (Pergamon, Oxford 1972)Google Scholar
- 2.4F.J. Blatt: Physics of Electronic Conduction in Solids (McGraw-Hill, New York 1968)Google Scholar
- 2.5S.O. Kasap: Principles of Electronic Materials and Devices, 4th edn. (McGraw-Hill, Dubuque 2017)Google Scholar
- 2.6W.M. Haynes (Ed.): CRC Handbook of Chemistry and Physics, 97th edn. (CRC Press, Boca Raton 2016), Chap. 12Google Scholar
- 2.7Kaye & Laby Tables of Physical and Chemical Constants (Section 2.6) (National Physical Laboratory, Teddington 2016) provided by the website of the National Physical LaboratoryGoogle Scholar
- 2.8L. Nordheim: Ann. Phys. 9, 664 (1931)Google Scholar
- 2.9J.K. Stanley: Electrical and Magnetic Properties of Metals (American Society for Metals, Metals Park 1963)Google Scholar
- 2.10M. Hansen, K. Anderko: Constitution of Binary Alloys, 2nd edn. (McGraw-Hill, New York 1985)Google Scholar
- 2.11H.E. Ruda: J. Appl. Phys. 59, 1220 (1986)CrossRefGoogle Scholar
- 2.12M. Lundstrom: Fundamentals of Carrier Transport (Cambridge Univ. Press, Cambridge 2000)CrossRefGoogle Scholar
- 2.13R.H. Bube: Electronic Properties of Crystalline Solids (Academic, New York 1974) p. 211CrossRefGoogle Scholar
- 2.14S. Riedel, J. Röber, T. Geßner: Microelectron. Eng. 33, 165 (1997)CrossRefGoogle Scholar
- 2.15A.F. Mayadas, M. Shatzkes: Phys. Rev. B 1, 1382 (1970)CrossRefGoogle Scholar
- 2.16C.R. Tellier, C.R. Pichard, A.J. Tosser: J. Phys. F 9, 2377 (1979)CrossRefGoogle Scholar
- 2.17K. Fuchs: Proc. Camb. Philos. Soc. 34, 100 (1938)CrossRefGoogle Scholar
- 2.18E.H. Sondheimer: Adv. Phys. 1, 1 (1952)CrossRefGoogle Scholar
- 2.19J.S. Chawla, D. Gall: Appl. Phys. Letts. 94, 252101 (2009)CrossRefGoogle Scholar
- 2.20T. Sun, B. Yao, A.P. Warren, V. Kumar, S. Roberts, K. Barmak, K.R. Coffey: J. Vac. Sci. Technol. A 26, 605 (2008)CrossRefGoogle Scholar
- 2.21T. Sun, B. Yao, A.P. Warren, K. Barmak, M.F. Toney, R.E. Peale, K.R. Coffey: Phys. Rev. B 81, 155454 (2010)CrossRefGoogle Scholar
- 2.22J.S. Chawla, X.Y. Zhang, D. Gall: J. Appl. Phys. 110, 043714 (2011)CrossRefGoogle Scholar
- 2.23J.W. Lim, M. Isshiki: J. Appl. Phys. 99, 094909 (2006)CrossRefGoogle Scholar
- 2.24H.-D. Liu, Y.-P. Zhao, G. Ramanath, S.P. Murarka, G.-C. Wang: Thin Solid Films 384, 151 (2001)CrossRefGoogle Scholar
- 2.25J.-W. Lim, K. Mimura, M. Isshiki: Appl. Surf. Sci. 217, 95 (2003)CrossRefGoogle Scholar
- 2.26R. Suri, A.P. Thakoor, K.L. Chopra: J. Appl. Phys. 46, 2574 (1975)CrossRefGoogle Scholar
- 2.27R.H. Cornely, T.A. Ali: J. Appl. Phys. 49, 4094 (1978)CrossRefGoogle Scholar
- 2.28J.S. Ahn, K.H. Kim, T.W. Noh, D.H. Riu, K.H. Boo, H.E. Kim: Phys. Rev. B 52, 15244 (1995)CrossRefGoogle Scholar
- 2.29R.J. Gehr, G.L. Fisher, R.W. Boyd: J. Opt. Soc. Am. B 14, 2310 (1997)CrossRefGoogle Scholar
- 2.30D.E. Aspnes, J.B. Theeten, F. Hottier: Phys. Rev. B 20, 3292 (1979)CrossRefGoogle Scholar
- 2.31Z. Yin, F.W. Smith: Phys. Rev. B 42, 3666 (1990)CrossRefGoogle Scholar
- 2.32M.F. MacMillan, R.P. Devaty, W.J. Choyke, D.R. Goldstein, J.E. Spanier, A.D. Kurtz: J. Appl. Phys. 80, 2412 (1996)CrossRefGoogle Scholar
- 2.33C. Ganter, W. Schirmacher: Phys. Status Solidi B 218, 71 (2000)CrossRefGoogle Scholar
- 2.34R. Stognienko, Th. Henning, V. Ossenkopf: Astron. Astrophys. 296, 797 (1995)Google Scholar
- 2.35A.G. Rojo, H.E. Roman: Phys. Rev. B 37, 3696 (1988)CrossRefGoogle Scholar
- 2.36J.A. Reynolds, J.M. Hough: Proc. Phys. Soc. 70, 769 (1957)CrossRefGoogle Scholar
- 2.37R. Clausius: Die Mechanische Wärmetheorie, Vol. 2 (Vieweg, Braunschweig 1879)Google Scholar
- 2.38L. Lorenz: Ann. Phys. Lpz. 11, 70 (1880)CrossRefGoogle Scholar
- 2.39O.F. Mosotti: Mem. Math. Fisica Modena II 24, 49 (1850)Google Scholar
- 2.40V.I. Odelevskii: Zh. Tekh. Fiz. 6, 667 (1950)Google Scholar
- 2.41Lord Rayleigh: Philos. Mag. 34, 481 (1892)CrossRefGoogle Scholar
- 2.42K.W. Wagner: Arch. Electrochem. 2, 371 (1914)CrossRefGoogle Scholar
- 2.43D.A.G. Bruggeman: Ann. Phys. Lpz. 24, 636 (1935)CrossRefGoogle Scholar
- 2.44C.J.F. Bottcher: Rec. Trav. Chim. Pays-Bas 64, 47 (1945)CrossRefGoogle Scholar
- 2.45H. Fricke: Phys. Rev. 24, 575 (1924)CrossRefGoogle Scholar
- 2.46D. Polder, J.M. Van Santen: Physica 12, 257 (1946)CrossRefGoogle Scholar
- 2.47W. Niesel: Ann. Phys. Lpz. 10, 336 (1952)CrossRefGoogle Scholar
- 2.48J.A. Stratton: Electromagnetic Theory (McGraw-Hill, New York 1941)Google Scholar
- 2.49O. Wiener: Abh. Sachs. Ges. Akad. Wiss. Math. Phys. 32, 509 (1912)Google Scholar
- 2.50L. Silberstein: Ann. Phys. Lpz. 56, 661 (1895)CrossRefGoogle Scholar
- 2.51R.W. Sillars: J. Inst. Elect. Eng. 80, 378 (1937)Google Scholar
- 2.52F. Ollendorf: Arch. Electrochem. 25, 436 (1931)CrossRefGoogle Scholar
- 2.53J.C.M. Maxwell-Garnett: Phil. Trans. R. Soc. Lond. 203, 385 (1904)CrossRefGoogle Scholar
- 2.54H. Looyenga: Physica 31, 401 (1965)CrossRefGoogle Scholar
- 2.55J. Monecke: J. Phys. Condens. Mat. 6, 907 (1994)CrossRefGoogle Scholar
- 2.56C.F. Bohren, D.R. Huffman: Absorption and Scattering of Light by Small Particles (Wiley, New York 1983)Google Scholar
- 2.57D.J. Bergman: Phys. Rep. 43, 377 (1978)CrossRefGoogle Scholar
- 2.58W. Theiss: Adv. Solid State Phys. 33, 149 (2007)CrossRefGoogle Scholar
- 2.59P.Y. Yu, M. Cardona: Fundamentals of Semiconductors (Springer, Berlin, Heidelberg 1996)CrossRefGoogle Scholar
- 2.60K. Tsuji, Y. Takasaki, T. Hirai, K. Taketoshi: J. Non-Cryst. Solids 14, 94 (1989)CrossRefGoogle Scholar
- 2.61G. Juska, K. Arlauskas: Phys. Status Solidi 77, 387 (1983)CrossRefGoogle Scholar
- 2.62G. Juska, K. Arlauskas: Phys. Status Solidi 59, 389 (1980)CrossRefGoogle Scholar
- 2.63R.A. Logan, H.G. White: J. Appl. Phys. 36, 3945 (1965)CrossRefGoogle Scholar
- 2.64R. Ghin, J.P.R. David, S.A. Plimmer, M. Hopkinson, G.J. Rees, D.C. Herbert, D.R. Wight: IEEE Trans. Electron Dev. ED45, 2096 (1998)CrossRefGoogle Scholar
- 2.65S.A. Plimmer, J.P.R. David, R. Grey, G.J. Rees: IEEE Trans. Electron Dev. ED47, 21089 (2000)Google Scholar
- 2.66L.W. Cook, G.E. Bulman, G.E. Stillma: Appl. Phys. Lett. 40, 589 (1982)CrossRefGoogle Scholar
- 2.67C.A. Lee, R.A. Logan, R.L. Batdorf, J.J. Kleimack, W. Wiegmann: Phys. Rev. 134, B766 (1964)Google Scholar
- 2.68C. Bulutay: Semicond. Sci. Technol. 17, L59 (2002)CrossRefGoogle Scholar
- 2.69B. Nabet: Photodetectors: Materials, Devices and Applications (Elsevier, Amsterdam 2015)Google Scholar
- 2.70S.N. Ahmed: Physics and Engineering of Radiation Detection, 2nd edn. (Elsevier, Amsterdam 2014)Google Scholar
- 2.71A. Reznik, S.D. Baranovskii, O. Rubel, G. Juska, S.O. Kasap, Y. Ohkawa, K. Tanioka, J.A. Rowlands: J. Appl. Phys. 102, 53711 (2007)CrossRefGoogle Scholar
- 2.72M. Akiyama, M. Hanada, H. Takao, K. Sawada, M. Ishida: Jpn. J. Appl. Phys 41, 2552 (2002)CrossRefGoogle Scholar
- 2.73M. Kubota, T. Kato, S. Suzuki, H. Maruyama, K. Shidara, K. Tanioka, K. Sameshima, T. Makishima, K. Tsuji, T. Hirai, T. Yoshida: IEEE Trans. Broadcast. 42, 251 (1996)CrossRefGoogle Scholar
- 2.74O. Bubon, G. i DeCrescenzo, W. Zhao, Y. Ohkawa, K. Miyakawa, T. Matsubara, K. Kikuchi, K. Tanioka, M. Kubota, J.A. Rowlands, A. Reznik: Curr. Appl. Phys. 12, 983 (2012)CrossRefGoogle Scholar
- 2.75W. Shockley: Solid State Electron. 2, 35 (1961)CrossRefGoogle Scholar
- 2.76G.A. Baraff: Phys. Rev. 128, 2507 (1962)CrossRefGoogle Scholar
- 2.77B.K. Ridley: J. Phys. C 16, 4733 (1983)CrossRefGoogle Scholar
- 2.78M.G. Burt: J. Phys. C 18, L477 (1985)CrossRefGoogle Scholar
- 2.79S. MacKenzie, M.G. Burt: Semicond. Sci. Technol. 2, 275 (1987)CrossRefGoogle Scholar
- 2.80B.K. Ridley: Semicond. Sci. Technol. 2, 116 (1987)CrossRefGoogle Scholar
- 2.81J.S. Marsland: Solid State Electron. 30, 125 (1987)CrossRefGoogle Scholar
- 2.82J.S. Marsland: Semicond. Sci. Technol. 5, 177 (1990)CrossRefGoogle Scholar
- 2.83S.O. Kasap, J.A. Rowlands, S.D. Baranovskii, K. Tanioka: J. Appl. Phys. 96, 2037 (2004)CrossRefGoogle Scholar
- 2.84W. Walukiewicz, H.E. Ruda, J. Lagowski, H.C. Gatos: Phys. Rev. B 30, 4571 (1984)CrossRefGoogle Scholar
- 2.85K.V. Klitzing, G. Dorda, M. Pepper: Phys. Rev. Lett. 45, 494 (1980)CrossRefGoogle Scholar