Solid State Physics Bands & Bonds

PROBABILITY DENSITY The probability density P(x,t) is information that tells us something about the likelihood of obtaining one of a variety of possible results in a position measurement. As particles move through space, their wavefunction evolves with time and the probability density changes (along with the position information it contains). The quantum world is governed by statistics.

Postulates of Quantum Mechanics 1. Every physically-realizable state of a system is described by a state function  that contains all accessible physical information about the system in that state. 2. If a system is in a quantum state , then is the probability that in a position measured at time t the particle will be detected in an infinitesimal volume dv. 3. In quantum mechanics, every observable is expressed by an operator that is used to obtain physical information about the observable from state functions. For an observable from state functions.

ObservableSymbolOperation PositionMultiply by x momentum Hamiltonian Kinetic energy Potential energyMultiply by V(x,t)

Postulates of Quantum Mechanics 4. The time development of the state functions of an isolated quantum system is governed by the time- dependent Schrodinger equation (TDSE).

Classical versus Quantum The TDSE is the “equation of motion” in quantum mechanics just as Newton’s 2 nd is the equation of motion in classical mechanics.

Classical versus Quantum In classical physics a macroscopic particle moves through 2D space along a trajectory. In quantum physics a microscopic particle moves through 1D space and is represented by a complex state function, the real part of which is shown.

Translation Under translation, a macroscopic free particle must not alter its probability density, even though the state function will be altered.

Wave Packets To construct a wave packet with a single region of enahnced amplitude, we must superpose an infinite number of plane wave functions with infinitesimally differing wave numbers. Note: although this is developed for the free particle, it is applicable to many one-dimensional quantum systems.

The amplitude function A(k) is the Fourier transform of the state function at time t = 0. Time-independent Schrödinger In solid state physics, we’ll use the time-independent equation because in the study of most material properties solids are modeled assuming stationary nuclei at their equilibrium positions in the solid and other properties are statistically uniform.

Particle in a Box L0

Energies

Why do atoms bond to each other? When they are bound together, the energy of the system is lower than when the atoms are isolated. When they are bound together, the energy of the system is lower than when the atoms are isolated. In fact, most electrons occupy energy states that are lower than the ground state of an isolated atom. In fact, most electrons occupy energy states that are lower than the ground state of an isolated atom.

Consider the available energies for electrons in the materials. As two atoms are brought close together, electrons must occupy different energies due to Pauli Exclusion principle. Instead of having discrete energies as in the case of free atoms, the available energy states form bands. Conductors, Insulators, and Semiconductors

Wave Equation for an Electron in a Periodic Potential We’ve been considering electrons as moving freely in a box potential. In reality, the periodicity of the crystal leads to a periodic potential.

Bonding Mechanisms Covalent Covalent Electron concentration is greater in the neighborhood along the line “joining” two atoms together. Electron concentration is greater in the neighborhood along the line “joining” two atoms together. Metallic Metallic Electron concentration is uniform throughout the spaces in between atoms. Electron concentration is uniform throughout the spaces in between atoms. Ionic Ionic An electron can be transferred from the neighborhood surrounding one atom to that of another and the atoms can then be viewed as bonded through Coulomb interaction. An electron can be transferred from the neighborhood surrounding one atom to that of another and the atoms can then be viewed as bonded through Coulomb interaction. Can be qualitatively distinguished on the basis of electron distribution.

Quantum Mechanical View Calculation of the total energy of a solid begins with finding solutions to the Schrödinger equation for the electron energies and wavefunction. Electron wave functions overlap – forming a new wave function

Quantum Mechanical View The wave function can be written in terms of its spatial part and its temporal part. The wave function can be written in terms of its spatial part and its temporal part. The temporal part has the form… The temporal part has the form…

Quantum Mechanical View The real and imaginary parts of the wave function oscillate sinusoidally with time, but the probability density is independent of time. The real and imaginary parts of the wave function oscillate sinusoidally with time, but the probability density is independent of time. The angular frequency of the wave function is related to the energy of the electron. The angular frequency of the wave function is related to the energy of the electron.

Quantum Mechanical View The wave function and the probability density depend on the potential energy function for the electron. The wave function and the probability density depend on the potential energy function for the electron. The individual e  are replaced by a continuous distribution. The individual e  are replaced by a continuous distribution. Hartree Approach Hartree Approach

Hydrogen Bonding H 2 molecule R rrR Proton A Proton B electron

Antibonding States Not all molecular states result in a lowering of energy.

States When atoms are brought close together, an electron state is created for each core state of an atom – some bonding (lower energy) and some antibonding (higher energy). When atoms are brought close together, an electron state is created for each core state of an atom – some bonding (lower energy) and some antibonding (higher energy). These states are all occupied. Good thing we have the Pauli exclusion principle. These states are all occupied. Good thing we have the Pauli exclusion principle. Atoms are attracted by the outer electrons, occupying bonding states, and repelled mostly by ion core electrons. Atoms are attracted by the outer electrons, occupying bonding states, and repelled mostly by ion core electrons.

Covalent Bonding Let’s develop a bonding wave function through linear combination of localized atomic orbitals for each atom a:

Covalent Bonding For two atoms: For two atoms: Lots of solutions, but the lowest energy states have coefficients that make the overlap large. Lots of solutions, but the lowest energy states have coefficients that make the overlap large.

Covalent Bonding There are two types of covalent bonds : sigma and pi. When the covalent bonds are linear or aligned along the plane containing the atoms, the bond is known as sigma (  ) bond. Sigma bonds are strong and the electron sharing is maximum. Methane CH 4 is a good example for sigma bond and it has four of them.

Covalent Bonds When the electron orbitals overlap laterally, the bond is called the pi (  ) bond. In pi bonds, the resulting overlap is not maximum and these bonds are relatively weak. Covalent Bonds have well-defined directions in space. Attempting to alter those directions is resisted – making them both hard and brittle.

Ionic Bonding Consider two different atoms, such as Ga and As joined by a covalent bond. The  functions come from different orbitals. The C constants would be different. The wave function still spreads between the atoms, but the probability is that the shared electron will be nearer to one atom than another. Atoms in this situation act like oppositely charged ions that are attracted by the Coulomb force.

Ionic Bonding

11 +1+1

Bonding may be evaluated energetically via calorimetric measurements. 1. measure a-a bond energy in pure a material 2. measure b-b bond energy in pure b material 3. measure a-b material – This will be greater than the average of a-a and b-b bonds. The difference is the energy of the ionic bond.

van der Waals Bonding An atom has an electric dipole moment when the average position of its electrons does not coincide with that of the nucleus. An atom has an electric dipole moment when the average position of its electrons does not coincide with that of the nucleus. The dipole moment produces an electric field, even if the atom is neutral. The dipole moment produces an electric field, even if the atom is neutral. This field induces dipole moments in other, nearby atoms. This field induces dipole moments in other, nearby atoms.

p1p1 p2p2 E R van der Waals Bonding

The potential energy of a dipole in an electric field is...

Graphite 1.42 Å 3.4 Å A B A

Metallic Bonds The outer electrons of metallic atoms are loosely bound. When a solid formed, the potential energy barrier is substantially reduced. The outer electrons of metallic atoms are loosely bound. When a solid formed, the potential energy barrier is substantially reduced. Metals typically found in columns IA, IIA, IB, IIB of the periodic table have cohesive energies of around 1 to 5 ev/atom (for comparison, convalent bonds are in the 3 to 10 eV/atom range). Metals typically found in columns IA, IIA, IB, IIB of the periodic table have cohesive energies of around 1 to 5 ev/atom (for comparison, convalent bonds are in the 3 to 10 eV/atom range).