Solid-State Physics: An Introduction to Principles of Materials Science: Fourth Extensively Updated and Enlarged Edition(Advanced Texts in Physics)

All the examples have been tested in our own lecture courses. A consideration of these two fundamental questions forms the content of the first two chapters.

The Periodic Table of the Elements

The width of the band (i.e. the broadening) depends on the overlap of the wave functions involved. What is important is simply the relative magnitude of the wavefunctions compared to the interatomic separation.

Covalent Bonding

A complete saturation of the covalent bond is possible for the group IV elements C, Si, Ge and a-Sn in the three-dimensional space-filling tetrahedral configuration. The total number of electrons per atom is thus the same as in the case of the carbon diamond structure.

Fig. 1.4. The tetrahedral configuration of nearest neighbors in the lattice of C, Si, Ge and a -Sn

Ionic Bonding

In most cases the binding is of an intermediate nature representing a mixture of the two extremes. The difference in the electronegativity of the two atoms is a measure of the ionicity of the bond.

Fig. 1.6. The two structures typical for ionic bonding in solids: ( a ) NaCl structure; ( b ) CsCl structure

Metallic Bonding

As previously mentioned, the d-electrons have a relatively small spatial extent (Fig. 1.9), and due to the correspondingly small overlap with the neighboring atoms, the d-band of the transition metals has a smaller energy width than the sp-. band. The large dispersion of the wave function of valence electrons in metals makes it particularly difficult to theoretically predict their binding energy.

Fig. 1.9. The amplitude of the 3d zz -wavefunction and the 4s-wavefunction of Ni [1.4]

The Hydrogen Bond

The van der Waals Bond

To do this, expand the approximate function in terms of exact eigenfunctionswi(exact eigenvalueEi). What is the origin of the bond system parallel to the hexagonal ring skeleton of six carbon atoms.

The Crystal Lattice

Instead of five possible base vector systems in the plane, one now has seven possibilities (Table 2.1). With the addition of centering, all possible grids of three-dimensional space can be constructed from these basic vector systems.

Table 2.1. The seven di€erent basis-vector systems or crystal systems. Most elements crys- crys-tallize in a cubic or hexagonal structure

Point Symmetry

Rotation with simultaneous inversion can be combined to give a new symmetry element ± the rotation-inversion axis. From this it appears that a 3-fold rotation-inversion axis corresponds to a 3-fold rotation together with inversion.

Figure 2.5 illustrates a 3-fold rotation-inversion axis. From this it is evident that a 3-fold rotation-inversion axis is equivalent to a 3-fold rotation together with inversion

The 32 Crystal Classes (Point Groups)

A cube has three 4-fold axes of rotation, four 3-fold axes of rotation, and planes of symmetry perpendicular to the 4-fold axes.

The Significance of Symmetry

If the Hamiltonian operator has a certain symmetry, for example mirror symmetry, then it makes no difference whether the reflection operation occurs before or after the Hamiltonian operator, i.e. both operators commute. Accordingly, the atoms can move symmetrically or antisymmetrically with respect to the two mirror planes of the molecule.

Fig. 2.7. The two symmetric and the antisymmetric vibrations of the water molecule. Together with the three rotations and three translations these give the nine normal modes corresponding to the nine degrees of freedom

Simple Crystal Structures

The structure of diamond is named after the structure of carbon atoms in diamond. The ZnS structure is found in the most important group III compound with group V elements.

Fig. 2.9. The close-packed layers of the fcc structure with the stacking sequence ABCABC

Phase Diagrams of Alloys

The free enthalpy of the melt is below that of the solid for all concentrations. Free enthalpy of the solid and liquid phases of a completely miscible alloy in the temperature range between the liquidus and solidus curves (schematic).

Fig. 2.14. Phase diagram for the continuously miscible alloy Ge/Si. In the range bounded by the liquidus and solidus curves a Ge-rich liquid phase coexists with a Si-rich solid phase

Defects in Solids

The modulus of the Burgers vector is equal to the distance of an atomic plane for the common screw or edge dislocations. The main diagonal of the rhombohedral lattice is parallel to the c-axis of the hexagonal lattice.

Fig. 2.20. Sectional drawing of a crys- crys-tal with an edge dislocation (sche-matic)

General Theory of Di€raction

Another important quantity, which differs significantly for the different probes, is the spatial extent of the scattering centers. We now assume that the scattering density is localized in the centers of the atoms.

Fig. 3.1. Schematic representation of scattering indicating the parameters used in deriving the scattering kinematics

Periodic Structures and the Reciprocal Lattice

It follows from the one-to-one correspondence of the lattice and its reciprocal lattice that any symmetry property of the lattice is also a symmetry property of the reciprocal lattice. The reciprocal lattice therefore belongs to the same point group as the real space lattice.

The Scattering Conditions for Periodic Structures

The vector Gis defined unambiguously by its coordinates h,k,l with respect to the basis vectors gi of the reciprocal lattice. The GK condition is satisfied whenever the surface of the sphere coincides with the reciprocal lattice points.

The Bragg Interpretation of the Laue Condition

The perpendicular distance of the grid plane hkl from the origin of the base a1,a2,a3 is. With the help of grid planes, an intuitively clear interpretation of the scattering conditions can be obtained.

Fig. 3.6. The Bragg interpretation of the scattering condition. Since the vector G hkl lies perpendicular to the lattice planes … hkl † in real space, the scattering appears to be a mir-ror reflection from these planes

Brillouin Zones

The production of two waves of equal intensity and a fixed phase relationship can also be used to construct an X-ray interferometer capable of imaging individual lattice defects (Panel II). For the case of electrons in a periodic solid, the production of reflected Bragg waves and their significance for the solid's electron bands will be discussed in more detail in chapter .

The Structure Factor

With this notation, the Fourier coefficients of the scattering density can be expressed as. Clearly it describes the interference of the spherical waves emanating from different points within the atom.

Fig. 3.10. Definition of the vec- vec-tors r n , r and r ' . The vector r n

Methods of Structure Analysis

One after the other, the vertices of the reciprocal lattice pass through the surface of the Ewald sphere. In other words, one observes all reflections that lie within a radius of 2k0 from the origin of the reciprocal grating.

Electrons

The strong absorption of electrons means that the third Laue condition for constructive interference between electrons scattered from atomic planes parallel to the surface is of little importance. As a result, dilation can be observed at all electron energies. it should be noted that the Ni(111) diractation pattern shows the true 3-fold body symmetry of the fcc crystal, as the scattering is not only from the surface layer but includes contributions from deeper layers. Figure I.2 b shows the diffraction pattern of the same surface after hydrogen adsorption. Additional diffraction spots show that hydrogen ± like many other adsorbates on surfaces ± creates an overlay with a new structure. In this case, the basic lattice of the hydrogen overlay is exactly twice that of the Ni(111) surface. .The additional spots therefore lie halfway between those on the nickel substrate.

Atomic Beams

Surfaces with regularly spaced monatomic steps can be produced by cutting the crystal at the appropriate angle and annealing in vacuum. The atomic beam used in the direction experiments is produced by a supersonic expansion of the gas. The Miller indices for this surface are (997). With respect to an optical echelon, maximum intensity is obtained in the diffraction orders corresponding to specular reflection from the contours of the interaction potential. In this case, it should be noted that the contours of these potentials are not exactly parallel to the terraces.

Fig. I.3. Diffraction of a He beam from a stepped platinum surface [I.3]. The Miller in- in-dices of this surface are (997).As for an optical echelon, one obtains maximum intensity in the diffraction orders that correspond to specular reflection from the c

3Neutrons

The Potential

As in the past, we number the unit cells by the triples n n1;n2;n3orm m1;m2;m3 and the atoms within each cell by,b. The ith component of the equilibrium position vector of an atom is then denoted byuni and the displacement from the equilibrium position byuni (Fig. 4.1). Equation (4.1) then represents an extension of the harmonic oscillator potential to the many-particle case.

The Equation of Motion

The neglect of the higher-order terms in (4.1) is therefore known as the ``harmonic'' approximation. Due to the translational invariance, the terms of the sum, as in (4.3), depend only on the difference m±n.

The Diatomic Linear Chain

Here the summation of the atoms must be referred to the smallest possible unit cell. At q= 0 in the optical branch, the two fcc substructures of the diamond structure vibrate against each other.

Fig. 4.2. The diatomic line- line-ar chain model

Scattering from Time-Varying Structures ± Phonon Spectroscopy

We separate each of the time-dependent vectors rn(t) into a grid vector rn and a displacement from the grid location un(t). In the sense of the conservation equations (4.28), one can thus consider these waves as particles.

Elastic Properties of Crystals

The sum of the elastic forces and the inertial force RdVu k must be zero. In many cases it is useful to work with the inverse of the elastic tensor number.

Fig. 4.6. Illustrations to elucidate the terminology in the theory of elasticity. ( a ) Strain along the x 1 -axis; ( b ) shear along the x 2 -axis, without separation of the rotational part of e 21 ; ( c ) the same shear after splitting o€ the rotational

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Here W is the angle between the observation direction ^ and the direction of the vibration of P. A condition for the observation of a Raman line is that the sensitivity v. III.5) has a non-vanishing derivative to the coordinate X of the elementary excitation.

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Thus, at low temperatures the intensity of the anti-Stokes lines is greatly reduced because the relevant elementary excitation is largely in its ground state. The intensity of the inelastically scattered radiation is typically a factor 106 weaker than the primary radiation.

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This modulated oscillation of the polarization leads to contributions in the scattered light from the so-called Raman sidebands with frequencies x0x(q). The phonon-induced modification of the susceptibility in the x-direction is coupled to a modification in the y-direction.

Fig. III.1. Schematic representation of the mechanisms of elastic ( a ) and inelastic ( b ) light scattering (Raman scattering): ( a ) if the electronic susceptibility is assumed to be constant in time, the polarization P oscillates with the frequency x 0

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The Density of States

The density of allowed values ​​of q in the reciprocal space is N divided by the volume of the unit cell of the reciprocal grid g1.(g2g3). Density of states is a concept of central importance in solid-state physics, including electronic properties (section 6.1).

The Thermal Energy of a Harmonic Oscillator

4.3, it is possible to consider the wave motion of atoms as non-interacting particles (phonons), whose state is determined by the wave vector q and the branch j. The number n then corresponds to the number of particles in state q ,j and hniT is the expected value of this number. It should be noted that the two different statistical distributions of Pn in (5.8) and hniTin (5.15), i.e. The Boltzmann and Bose distributions come from two different ways of studying the problem: The Boltzmann distribution gives us the probability that a single particle occupies a certain state;.

The Specific Heat Capacity

The specific heat is normalized to the Boltzmann constant, the density of unit cells N/Water, the number of atoms in the unit cell. Within the Debye approximation, the specific heat of a solid is completely determined at all temperatures by the characteristic temperature H.

Fig. 5.3. The specific heat capacity per unit volume according to the Debye model. The specific heat is normalized to the Boltzmann constant k , the density of unit cells N/V and the number of atoms in the unit cell r

E€ects Due to Anharmonicity

Since the specific heat in reality differs from that of the Debye model, it is not entirely clear how best to define H. An exact treatment as in the harmonic case is not possible, since one no longer has the neat decoupling of the equations of motion with the plane wave approach.

Thermal Expansion

For isotropic substances and cubic crystals, it is equal to one third of the volume expansion coefficient. For the simple calculation of the derivative (5.25) we consider the expanded free energy around the equilibrium position.

Heat Conduction by Phonons

An analogous relationship applies to the thermal conductivity of a gas and of the electron gas (Sect. 9.7). The full characteristic behavior of the thermal conductivity of a (non-conducting) single crystal is shown in Fig.

Fig. 5.5. Schematic representation of the thermal current through a cross-sectional area A

The Free-Electron Gas

We simply take the volume of a thin shell of the octant bounded by the energy. So for the density of states D(E) = dZ/dE of the free electron gas in the infinite potential well, we finally obtain.

Fig. 6.2. Spatial form of the first three wavefunctions of a free electron in a square well potential of length L in the x-direction

The size of the Fermi energy can therefore be estimated by using the number of valence electrons per atom to determine the electron concentration n. An interesting consequence of the Pauli exclusion principle is that the Fermi gas, in contrast to a classical gas, has a non-destructive internal energy at T =0 K.

Fig. 6.5a±c. Description of the quasi-free valence electrons of a metal at T = 0. ( a ) f … E † is a step function

Fermi Statistics

From this it immediately follows that the derivatives of the free energy with respect to the occupation numbers must be equal. The significance of the chemical potential l in the Fermi distribution is most easily seen in the limiting case of T= 0 K.

Fig. 6.6. The Fermi distribution function at various temperatures. The Fermi temperature T F =E 0 F / k has been taken as 5 10 4 K

The Specific Heat Capacity of Electrons in Metals

With TF=EF/ as the Fermi temperature, one obtains the following order of magnitude estimate for the specific heat of the electrons. Qualitative behavior of the density of states D E for the conduction band of a transition metal.

Fig. 6.7. Explanation of the specific heat capacity of quasi-free metal electrons. The e€ect of raising the temperature from 0 K to T is to allow elec-trons from 2 k T below the Fermi energy to be promoted to 2 k T above E F

Electrostatic Screening in a Fermi Gas ± The Mott TransitionThe Mott Transition

The screening process described here is responsible for the fact that the highest energy valence electrons of a metal are not localized. Below this critical electron concentration, the potential well of the screened field extends far enough for a bound state to be possible.

Thermionic Emission of Electrons from Metals

Sketch the wavefunctions of the free electrons in the p orbitals and match them with the LCAO orbitals. An explanation of such features is beyond the scope of the free electron gas model.

Fig. 6.12. ( a ) Schematic drawing of a diode circuit for observing thermionic emission of electrons from the heated cathode C (A = anode)

General Symmetry Properties

The original problem therefore splits into N problems (N= number of unit cells), each corresponding to a k-vector in the unit cell of the reciprocal grid. The strict periodicity of the lattice potential has further consequences that follow directly from the properties of the Bloch states.

Fig. 7.1. Example of the construction of a Bloch wave w k …r† ˆ u k …r† e i kr from a lattice-per- lattice-per-iodic function u k …r† with p-type bonding character and a plane wave

The Nearly Free-Electron Approximation

This increase and decrease in the energy of states at the zone boundary represents a deviation from the free electron energy parabola (Fig. 7.5). With E0k±G= (h2/2m) |k±G|2 as the free electron energy, the two solutions of this secular equation can be written.

Fig. 7.3. Bandstructure for a free electron gas in a primi- primi-tive cubic lattice (lattice constant a ), represented on a sec-tion along k x in the first Brillouin zone

The Tight-Binding Approximation

Using the fact that we know the solutions of (7.26) for the free atom, we write This is the origin of the insulating properties of diamond, as will be shown in Sects.

Fig. 7.7. Cross section of the potential used in the tight-binding approximation along the x-direction

Examples of Bandstructures

The Fermi "sphere" of Al (ÐÐ) lies beyond the edges of the first Brillouin zone. Filling the bands with available electrons continues up to the FermiEF energy (shown in Fig. 7.11).

Fig. 7.11. ( a ) Theoretical bandstructure E …k† for Al along directions of high symmetry ( C is the center of the Brillouin zone)

The Density of States

In the integration over k-space, the main contributions to the density of states are derived from the critical points. At the Fermi level, it is again the s-electrons that produce the density of states.

Density of States in Non-Crystalline Solids

A calculation of the electronic density of states of an amorphous solid requires the input of a particular distribution of bond angles and distances. Depending on the photon energy of the light sources, one distinguishes between Ultraviolet Photoemission Spectroscopy (UPS) and X-ray Photoemission Spectroscopy (XPS).

Fig. 7.15. Schematic density of states of an ideal amorphous material with saturated tetra- tetra-hedral bonds to the nearest-neighbors (full line)