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Table of Contents
                            Solid-State Physics for Electronics
	Table of Contents
	Foreword
	Introduction
	Chapter 1. Introduction: Representations of Electron-Lattice Bonds
		1.1. Introduction
		1.2. Quantum mechanics: some basics
			1.2.1. The wave equation in solids: from Maxwell’s to Schrödinger’s equation via the de Broglie hypothesis
			1.2.2. Form of progressive and stationary wave functions for an electron with known energy (E)
			1.2.3. Important properties of linear operators
		1.3. Bonds in solids: a free electron as the zero order approximation for a weak bond; and strong bonds
			1.3.1. The free electron: approximation to the zero order
			1.3.2. Weak bonds
			1.3.3. Strong bonds
			1.3.4. Choosing between approximations for weak and strong bonds
		1.4. Complementary material: basic evidence for the appearance of bands in solids
			1.4.1. Basic solutions for narrow potential wells
			1.4.2. Solutions for two neighboring narrow potential wells
	Chapter 2. The Free Electron and State Density Functions
		2.1. Overview of the free electron
			2.1.1. The model
			2.1.2. Parameters to be determined: state density functions in k or energy spaces
		2.2. Study of the stationary regime of small scale (enabling the establishment of nodes at extremities) symmetric wells (1D model)
			2.2.1. Preliminary remarks
			2.2.2. Form of stationary wave functions for thin symmetric wells with width (L) equal to several inter-atomic distances (L ≈ a), associated with fixed boundary conditions (FBC)
			2.2.3. Study of energy
			2.2.4. State density function (or “density of states”) in k space
		2.3. Study of the stationary regime for asymmetric wells (1D model) with L ≈ a favoring the establishment of a stationary regime with nodes at extremities
		2.4. Solutions that favor propagation: wide potential wells where L ≈ 1 mm, i.e. several orders greater than inter-atomic distances
			2.4.1. Wave function
			2.4.2. Study of energy
			2.4.3. Study of the state density function in k space
		2.5. State density function represented in energy space for free electrons in a 1D system
			2.5.1. Stationary solution for FBC
			2.5.2. Progressive solutions for progressive boundary conditions (PBC)
			2.5.3. Conclusion: comparing the number of calculated states for FBC and PBC
		2.6. From electrons in a 3D system (potential box)
			2.6.1. Form of the wave functions
			2.6.2. Expression for the state density functions in k space
			2.6.3. Expression for the state density functions in energy space
		2.7. Problems
			2.7.1. Problem 1: the function Z(E) in 1D
			2.7.2. Problem 2: diffusion length at the metal-vacuum interface
			2.7.3. Problem 3: 2D media: state density function and the behavior of the Fermi energy as a function of temperature for a metallic state
			2.7.4. Problem 4: Fermi energy of a 3D conductor
			2.7.5. Problem 5: establishing the state density function via reasoning in moment or k spaces
			2.7.6. Problem 6: general equations for the state density functions expressed in reciprocal (k) space or in energy space
	Chapter 3. The Origin of Band Structures within the Weak Band Approximation
		3.1. Bloch function
			3.1.1. Introduction: effect of a cosinusoidal lattice potential
			3.1.2. Properties of a Hamiltonian of a semi-free electron
			3.1.3. The form of proper functions
		3.2. Mathieu’s equation
			3.2.1. Form of Mathieu’s equation
			3.2.2. Wave function in accordance with Mathieu’s equation
			3.2.3. Energy calculation
			3.2.4. Direct calculation of energy when k = ± π/a
		3.3. The band structure
			3.3.1. Representing E = f (k) for a free electron: a reminder
			3.3.2. Effect of a cosinusoidal lattice potential on the form of wave function and energy
			3.3.3. Generalization: effect of a periodic non-ideally cosinusoidal potential
		3.4. Alternative presentation of the origin of band systems via the perturbation method
			3.4.1. Problem treated by the perturbation method
			3.4.2. Physical origin of forbidden bands
			3.4.3. Results given by the perturbation theory
			3.4.4. Conclusion
		3.5. Complementary material: the main equation
			3.5.1. Fourier series development for wave function and potential
			3.5.2. Schrödinger equation
			3.5.3. Solution
		3.6. Problems
			3.6.1. Problem 1: a brief justification of the Bloch theorem
			3.6.2. Problem 2: comparison of E(k) curves for free and semi-free electrons in a representation of reduced zones
	Chapter 4. Properties of Semi-Free Electrons, Insulators, Semiconductors, Metals and Superlattices
		4.1. Effective mass (m*)
			4.1.1. Equation for electron movement in a band: crystal momentum
			4.1.2. Expression for effective mass
			4.1.3. Sign and variation in the effective mass as a function of k
			4.1.4. Magnitude of effective mass close to a discontinuity
		4.2. The concept of holes
			4.2.1. Filling bands and electronic conduction
			4.2.2. Definition of a hole
		4.3. Expression for energy states close to the band extremum as a function of the effective mass
			4.3.1. Energy at a band limit via the Maclaurin development (in k = kn = n π/a)
		4.4. Distinguishing insulators, semiconductors, metals and semi-metals
			4.4.1. Required functions
			4.4.2. Dealing with overlapping energy bands
			4.4.3. Permitted band populations
		4.5. Semi-free electrons in the particular case of super lattices
		4.6. Problems
			4.6.1. Problem 1: horizontal tangent at the zone limit (k ≈ π/a) of the dispersion curve
			4.6.2. Problem 2: scale of m* in the neighborhood of energy discontinuities
			4.6.3. Problem 3: study of EF(T)
	Chapter 5. Crystalline Structure, Reciprocal Lattices and Brillouin Zones
		5.1. Periodic lattices
			5.1.1. Definitions: direct lattice
			5.1.2. Wigner-Seitz cell
		5.2. Locating reciprocal planes
			5.2.1. Reciprocal planes: definitions and properties
			5.2.2. Reciprocal planes: location using Miller indices
		5.3. Conditions for maximum diffusion by a crystal (Laue conditions)
			5.3.1. Problem parameters
			5.3.2. Wave diffused by a node located by ρm,n,p =m .a + n .b + p .c
		5.4. Reciprocal lattice
			5.4.1. Definition and properties of a reciprocal lattice
			5.4.2. Application: Ewald construction of a beam diffracted by a reciprocal lattice
		5.5. Brillouin zones
			5.5.1. Definition
			5.5.2. Physical significance of Brillouin zone limits
			5.5.3. Successive Brillouin zones
		5.6. Particular properties
			5.6.1. Properties of G h k l and relation to the direct lattice
			5.6.2. A crystallographic definition of reciprocal lattice
			5.6.3. Equivalence between the condition for maximum diffusion and Bragg’s law
		5.7. Example determinations of Brillouin zones and reduced zones
			5.7.1. Example 1: 3D lattice
			5.7.2. Example 2: 2D lattice
			5.7.3. Example 3: 1D lattice with lattice repeat unit (a) such that the base vector in the direct lattice is a
		5.8. Importance of the reciprocal lattice and electron filling of Brillouin zones by electrons in insulators, semiconductors and metals
			5.8.1. Benefits of considering electrons in reciprocal lattices
			5.8.2. Example of electron filling of Brillouin zones in simple structures: determination of behaviors of insulators, semiconductors and metals
		5.9. The Fermi surface: construction of surfaces and properties
			5.9.1. Definition
			5.9.2. Form of the free electron Fermi surface
			5.9.3. Evolution of semi-free electron Fermi surfaces
			5.9.4. Relation between Fermi surfaces and dispersion curves
		5.10. Conclusion. Filling Fermi surfaces and the distinctions between insulators, semiconductors and metals
			5.10.1. Distribution of semi-free electrons at absolute zero
			5.10.2. Consequences for metals, insulators/semiconductors and semi-metals
		5.11. Problems
			5.11.1. Problem 1: simple square lattice
			5.11.2. Problem 2: linear chain and a square lattice
			5.11.3. Problem 3: rectangular lattice
	Chapter 6. Electronic Properties of Copper and Silicon
		6.1. Introduction
		6.2. Direct and reciprocal lattices of the fcc structure
			6.2.1. Direct lattice
			6.2.2. Reciprocal lattice
		6.3. Brillouin zone for the fcc structure
			6.3.1. Geometrical form
			6.3.2. Calculation of the volume of the Brillouin zone
			6.3.3. Filling the Brillouin zone for a fcc structure
		6.4. Copper and alloy formation
			6.4.1. Electronic properties of copper
			6.4.2. Filling the Brillouin zone and solubility rules
			6.4.3. Copper alloys
		6.5. Silicon
			6.5.1. The silicon crystal
			6.5.2. Conduction in silicon
			6.5.3. The silicon band structure
			6.5.4. Conclusion
		6.6. Problems
			6.6.1. Problem 1: the cubic centered (cc) structure
			6.6.2. Problem 2: state density in the silicon conduction band
	Chapter 7. Strong Bonds in One Dimension
		7.1. Atomic and molecular orbitals
			7.1.1. s- and p-type orbitals
			7.1.2. Molecular orbitals
			7.1.3. σ- and π-bonds
			7.1.4. Conclusion
		7.2. Form of the wave function in strong bonds: Floquet’s theorem
			7.2.1. Form of the resulting potential
			7.2.2. Form of the wave function
			7.2.3. Effect of potential periodicity on the form of the wave function and Floquet’s theorem
		7.3. Energy of a 1D system
			7.3.1. Mathematical resolution in 1D where x r
			7.3.2. Calculation by integration of energy for a chain of N atoms
			7.3.3. Note 1: physical significance in terms of (E0 – α) and β
			7.3.4. Note 2: simplified calculation of the energy
			7.3.5. Note 3: conditions for the appearance of permitted and forbidden bands
		7.4. 1D and distorted AB crystals
			7.4.1. AB crystal
			7.4.2. Distorted chain
		7.5. State density function and applications: the Peierls metal-insulator transition
			7.5.1. Determination of the state density functions
			7.5.2. Zone filling and the Peierls metal–insulator transition
			7.5.3. Principle of the calculation of Erelax (for a distorted chain)
		7.6. Practical example of a periodic atomic chain: concrete calculations of wave functions, energy levels, state density functions and band filling
			7.6.1. Range of variation in k
			7.6.2. Representation of energy and state density function for N = 8
			7.6.3. The wave function for bonding and anti-bonding states
			7.6.4. Generalization to any type of state in an atomic chain
		7.7. Conclusion
		7.8. Problems
			7.8.1. Problem 1: complementary study of a chain of s-type atoms where N = 8
			7.8.2. Problem 2: general representation of the states of a chain of σ–s-orbitals (s-orbitals giving σ-overlap) and a chain of σ–p-orbitals
			7.8.3. Problem 3: chains containing both σ–s- and σ–p-orbitals
			7.8.4. Problem 4: atomic chain with π-type overlapping of p-type orbitals: π–p- and π*–p-orbitals
	Chapter 8. Strong Bonds in Three Dimensions: Band Structure of Diamond and Silicon
		8.1. Extending the permitted band from 1D to 3D for a lattice of atoms associated with single s-orbital nodes (basic cubic system, centered cubic, etc.)
			8.1.1. Permitted energy in 3D: dispersion and equi-energy curves
			8.1.2. Expression for the band width
			8.1.3. Expressions for the effective mass and mobility
		8.2. Structure of diamond: covalent bonds and their hybridization
			8.2.1. The structure of diamond
			8.2.2. Hybridization of atomic orbitals
			8.2.3. sp3 Hybridization
		8.3. Molecular model of a 3D covalent crystal (atoms in sp3-hybridization states at lattice nodes)
			8.3.1. Conditions
			8.3.2. Independent bonds: effect of single coupling between neighboring atoms and formation of molecular orbitals
			8.3.3. Coupling of molecular orbitals: band formation
		8.4. Complementary in-depth study: determination of the silicon band structure using the strong bond method
			8.4.1. Atomic wave functions and structures
			8.4.2. Wave functions in crystals and equations with proper values for a strong bond approximation
			8.4.3. Band structure
			8.4.4. Conclusion
		8.5. Problems
			8.5.1. Problem 1: strong bonds in a square 2D lattice
			8.5.2. Problem 2: strong bonds in a cubic centered or face centered lattices
	Chapter 9. Limits to Classical Band Theory: Amorphous Media
		9.1. Evolution of the band scheme due to structural defects (vacancies, dangling bonds and chain ends) and localized bands
		9.2. Hubbard bands and electronic repulsions. The Mott metal–insulator transition
			9.2.1. Introduction
			9.2.2. Model
			9.2.3. The Mott metal–insulator transition: estimation of transition criteria
			9.2.4. Additional material: examples of the existence and inexistence of Mott–Hubbard transitions
		9.3. Effect of geometric disorder and the Anderson localization
			9.3.1. Introduction
			9.3.2. Limits of band theory application and the Ioffe–Regel conditions
			9.3.3. Anderson localization
			9.3.4. Localized states and conductivity. The Anderson metal-insulator transition
		9.4. Conclusion
		9.5. Problems
			9.5.1. Additional information and Problem 1 on the Mott transition: insulator–metal transition in phosphorus doped silicon
			9.5.2. Problem 2: transport via states outside of permitted bands in low mobility media
	Chapter 10. The Principal Quasi-Particles in Material Physics
		10.1. Introduction
		10.2. Lattice vibrations: phonons
			10.2.1. Introduction
			10.2.2. Oscillations within a linear chain of atoms
			10.2.3. Oscillations within a diatomic and 1D chain
			10.2.4. Vibrations of a 3D crystal
			10.2.5. Energy of a vibrational mode
			10.2.6. Phonons
			10.2.7. Conclusion
		10.3. Polarons
			10.3.1. Introduction: definition and origin
			10.3.2. The various polarons
			10.3.3. Dielectric polarons
			10.3.4. Polarons in molecular crystals
			10.3.5. Energy spectrum of the small polaron in molecular solids
		10.4. Excitons
			10.4.1. Physical origin
			10.4.2. Wannier and charge transfer excitons
			10.4.3. Frenkel excitons
		10.5. Plasmons
			10.5.1. Basic definition
			10.5.2. Dielectric response of an electronic gas: optical plasma
			10.5.3. Plasmons
		10.6. Problems
			10.6.1. Problem 1: enumeration of vibration modes (phonon modes)
			10.6.2. Problem 2: polaritons
	Bibliography
	Index
                        
Document Text Contents
Page 2

Solid-State Physics
for Electronics













André Moliton

Series Editor
Pierre-Noël Favennec

Page 204

Electronic Properties of Cu and Si 185


in the direct space as only a single atom can occupy each node, be it copper or the
added metal). The number of electrons that fill this volume is now given by

3
4

2 ' 1.36N N N . In other words, for the sphere to be thought of as being

just-filled, the electronic concentration should be at 1.3 electrons per atom. This is
the sort of level of electron density that can be attained in copper alloys when zinc is
added and an alpha ( ) phase alloy is acquired. A higher level can be attained when
the alloy takes up a cubic centered beta ( ) phase with around 1.48 electrons per
atom (see the problem at the end of the section 6.6, question 5).

6.5. Silicon

6.5.1. The silicon crystal

Silicon is classed as the 14th element in the periodic table and it possesses four
peripheral electrons in the M layer. In a silicon crystal, each atom is engaged in four
bonds with four neighboring atoms. The silicon crystal is thus covalent and the four
bonds take up tetrahedral positions (with an angle of 109 28’) to give a structure not
unlike that of diamond.



Figure 6.6. Silicon’s: (a) electronic structure; and (b) crystal cell

Silicon atoms are placed at the nodes of two cubic face centered lattices that are
shifted with respect to one another by a quarter cube diagonal. This results in the cell
shown in Figure 6.6b, which can also be used to represent diamond carbon (hence its
common name, the “diamond lattice”) or germanium crystals. In the case of silicon,
the primitive cube has a lattice constant (d) equal to 5.43 Å, whereas for carbon,
d = 3.56 Å and for germanium d = 5.62 Å. Each primitive fcc cell with a volume of

(a)
(b)

K
L

M

d

Page 205

186 Solid-State Physics for Electronics


d3 contains on average eight atoms: there are four belonging to one fcc lattice and
four belonging to the other. We can assume that on average that there are four
atomic bases per fcc cell. With two atoms per base, there are in all eight atoms per
fcc cell.

6.5.2. Conduction in silicon

Given its structure, and that it crystallizes in an fcc system, with two silicon
atoms per node (base = two silicon atoms so as to account for the two imbricating
lattices), the N nodes are for 2N silicon atoms. Given that their valency is equal to
four, they liberate 8N electrons which can fill several successive Brillouin zones
(each with 2N electrons). The bands do not overlap and the width of the forbidden
band has been determined as being EG = 1.12 eV, indicating that silicon is
a semiconductor. The size of forbidden band decreases slightly as temperature
increases due to the dilatation of the crystalline structure, so that at 100 C, we find
that EG = 1.09 eV. Of note is that the forbidden band width of germanium is
0.66 eV.

6.5.3. The silicon band structure (see also section 8.4)

Determining the band structure of a material means finding the correspondence
between the wave vector and the energy at all points in different zones (in general
reduced to the first zone) or the different bands.


For a 1D material, the energy of the semi-free electrons is given by equation

[4.10] in section 4.3 that states
n

²
2m*

2
( ) (k ) E ( )k nE k k , where k0 n is the

value of k at the limit of the zone which has remained till now in the form

an
k n . For a 3D material (where the transport and therefore the effective mass

can differ with respect to the x, y and z directions), in the preceding expression m*
can take on various values, namely: mx*, my* and mz* for the respective directions x,
y and z. In addition, it is possible that the extreme values for the energy are not
obtained at the band limits (in the direction kx for kx = knx and similarly for ky and kz)
but at other points in the kx, ky or kz directions.

6.5.3.1. Properties of the conduction band

We thus find that silicon shows a conduction band in the [100] direction with a
minimum at 20.85

am
k , where 2

a
is the limit of the Brillouin zone in these

Page 407

388 Solid-State Physics for Electronics

dielectric polarons, 353, 354
diffraction, 129, 130, 134, 138, 140
diffusion, 128-141
direct gap, 190
direct lattice, 123, 134, 138, 139,

142, 144-149, 159, 161, 165, 168,
171-173

disorder, 302, 309, 312-316, 318,
320-326

dispersion curve, 220, 231, 250, 252,
294

dispersion, 339, 341, 342, 348, 358,
370, 371, 377, 381, 383, 384

distorted, 225, 227, 228, 231-234

E
effective mass, 1, 187, 195, 196, 255,

257, 289, 297, 300
electronic density, 17, 44
Esaki, 31
Ewald, 135
excitons, 336, 365, 367-369
face centered cubic, (fcc) 173, 174,

178, 181, 182, 184, 186, 192, 194,
254, 256-258, 276, 280, 295, 298,
300, 301

F
FBC, 19, 20, 29, 30-32
Fermi energy, 28, 44, 47
Fermi level, 18, 19, 38, 39, 42, 43,

47-49, 232
Fermi surface, 150, 151, 153, 154,

156, 163, 170
Fermi-Dirac, 15, 38, 40, 44
Floquet, 211-214, 217, 225, 246, 248
forbidden, 69, 72, 75, 76, 78
free electron, 1, 13, 17,20, 25, 29, 44,

49, 50
Frenkel (exciton), 365, 367, 368

G, H
GaAs, 24, 25, 30, 31
heterostructure, 22, 23, 31
holes, 8, 10, 13, 15-17, 25, 31, 32
Hubbard, 302-304, 306, 307, 309,

326
Hückel, 213, 214, 218, 220, 223
Hume-Rothery rules, 183
hybridization, 249, 258-262, 266,

267, 269, 270, 272, 288

I
index, 126, 138
indirect gap, 189, 190
insulator, 1, 11, 18, 21, 147, 156-158
Ioffe-Regel (conditions), 312, 313,

316, 318, 335
isolated, 306

L
lattice

1D, 146, 147 199, 210, 211, 215,
225, 234, 242

2D lattice, 125
3D lattice, 142, 249, 275

Laue, 128, 133, 134
longitudinal wave, 372, 384
Lyddane-Sachs-Teller, 376, 380, 383

M
Mathieu, 59, 61, 65, 70, 78
metal, 1, 18, 19, 21, 38, 147, 156-

158, 303-309, 320-323, 326, 331
Miller, 125-129, 138
mobility, 257, 314, 315, 321, 326,

333-335
mode density, 341, 375
Mott, 303, 305-307, 309-312, 315,

317, 318, 320, 325-327

Page 408

Index 389

N
nodes, 123-126, 128, 132, 134, 135,

137, 138, 164, 168

O
orbitals

molecular orbitals, 199, 203, 208-
210

p orbitals, 209
s orbitals, 200

oscillations, 337, 343
oscillator, 348, 357, 373
overlap, 12, 14, 15, 17, 18, 21

P
PBC, 25, 30-32, 50
Peierls, 229, 231, 232, 240
phonons, 335, 336, 350-352, 358,

377, 378, 381, 384
plasma, 369-374, 379
plasmons, 336, 373, 374
polarons, 335, 352-354, 357, 359,

362
polymers, 311, 312, 333-335
potential box, 32
proper function, 57-59, 76
proper value, 279-281

Q
quasi-particles, 335, 336, 352, 353

R
reciprocal lattices, 123, 173
reduced zones, 20, 22, 142, 145, 150,

151, 161, 174, 175
relaxation, 228, 229, 233
resonance integral, 223, 249
reticular plane, 125, 138, 144

S
semi-conductors, 17, 147, 156, 157,

163
semi-free electron, 56-59, 83, 137,

148, 151-153, 156, 159, 163-165
semi-Free, 1
semi-metal, 11, 12, 17, 21
silicon, 173, 186, 249
simple cubic, 124, 128, 142, 143,

148, 149, 154, 250-252, 255-257
state density, 12-14, 17, 19, 20, 38,

194, 195, 198, 229, 320, 329
stationary regime, 19, 23, 41
stationary wave, 73
superlattices, 1
symmetric well, 18, 19

T
transition, 189, 351, 366, 369
transverse modes, 347

V
vibration, 337, 338, 341, 347, 351,

357, 360, 364, 374, 384

W
Wannier (exciton), 365, 367, 368
wave function, 199, 200, 203-208,

211-214, 216, 226, 234, 237, 239,
241-249, 258, 262, 267, 275, 276,
279, 286, 287, 305, 313-319

weak bond, 55, 71, 72
Wigner-Seitz, 125, 135

Z
Z(E), 12, 13, 14, 17, 19, 18, 27, 28,

29, 30, 37, 38, 41, 42, 44, 46, 49,
50, 51, 53

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