9.S: Condensed Matter Physics (Summary)
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Key Terms
acceptor impurity | atom substituted for another in a semiconductor that results in a free electron |
amplifier | electrical device that amplifies an electric signal |
base current | current drawn from the base n-type material in a transistor |
BCS theory | theory of superconductivity based on electron-lattice-electron interactions |
body-centered cubic (BCC) | crystal structure in which an ion is surrounded by eight nearest neighbors located at the corners of a unit cell |
breakdown voltage | in a diode, the reverse bias voltage needed to cause an avalanche of current |
collector current | current drawn from the collector p-type material |
conduction band | above the valence band, the next available band in the energy structure of a crystal |
Cooper pair | coupled electron pair in a superconductor |
covalent bond | bond formed by the sharing of one or more electrons between atoms |
critical magnetic field | maximum field required to produce superconductivity |
critical temperature | maximum temperature to produce superconductivity |
density of states | number of allowed quantum states per unit energy |
depletion layer | region near the p-n junction that produces an electric field |
dissociation energy | amount of energy needed to break apart a molecule into atoms; also, total energy per ion pair to separate the crystal into isolated ions |
donor impurity | atom substituted for another in a semiconductor that results in a free electron hole |
doping | alteration of a semiconductor by the substitution of one type of atom with another |
drift velocity | average velocity of a randomly moving particle |
electric dipole transition | transition between energy levels brought by the absorption or emission of radiation |
electron affinity | energy associated with an accepted (bound) electron |
electron number density | number of electrons per unit volume |
energy band | nearly continuous band of electronic energy levels in a solid |
energy gap | gap between energy bands in a solid |
equilibrium separation distance | distance between atoms in a molecule |
exchange symmetry | how a total wave function changes under the exchange of two electrons |
face-centered cubic (FCC) | crystal structure in which an ion is surrounded by six nearest neighbors located at the faces at the faces of a unit cell |
Fermi energy | largest energy filled by electrons in a metal at T=0K |
Fermi factor | number that expresses the probability that a state of given energy will be filled |
Fermi temperature | effective temperature of electrons with energies equal to the Fermi energy |
forward bias configuration | diode configuration that results in high current |
free electron model | model of a metal that views electrons as a gas |
hole | unoccupied states in an energy band |
hybridization | change in the energy structure of an atom in which energetically favorable mixed states participate in bonding |
impurity atom | acceptor or donor impurity atom |
impurity band | new energy band create by semiconductor doping |
ionic bond | bond formed by the Coulomb attraction of a positive and negative ions |
junction transistor | electrical valve based on a p-n-p junction |
lattice | regular array or arrangement of atoms into a crystal structure |
Madelung constant | constant that depends on the geometry of a crystal used to determine the total potential energy of an ion in a crystal |
majority carrier | free electrons (or holes) contributed by impurity atoms |
minority carrier | free electrons (or holes) produced by thermal excitations across the energy gap |
n-type semiconductor | doped semiconductor that conducts electrons |
p-n junction | junction formed by joining p- and n-type semiconductors |
p-type semiconductor | doped semiconductor that conducts holes |
polyatomic molecule | molecule formed of more than one atom |
repulsion constant | experimental parameter associated with a repulsive force between ions brought so close together that the exclusion principle is important |
reverse bias configuration | diode configuration that results in low current |
rotational energy level | energy level associated with the rotational energy of a molecule |
selection rule | rule that limits the possible transitions from one quantum state to another |
semiconductor | solid with a relatively small energy gap between the lowest completely filled band and the next available unfilled band |
simple cubic | basic crystal structure in which each ion is located at the nodes of a three-dimensional grid |
type I superconductor | superconducting element, such as aluminum or mercury |
type II superconductor | superconducting compound or alloy, such as a transition metal or an actinide series element |
valence band | highest energy band that is filled in the energy structure of a crystal |
van der Waals bond | bond formed by the attraction of two electrically polarized molecules |
vibrational energy level | energy level associated with the vibrational energy of a molecule |
Key Equations
Electrostatic energy for equilibrium separation distance between atoms | Ucoul=−ke2r0 |
Energy change associated with ionic bonding | Uform=Etransfer+Ucoul+Uex |
Critical magnetic field of a superconductor | Bc(T)=Bc(0)[1−(TTc)2] |
Rotational energy of a diatomic molecule | \displaystyle E_r=l(l+1)\frac{ℏ^2}{2I} |
Characteristic rotational energy of a molecule | \displaystyle E_{0r}=\frac{ℏ^2}{2I} |
Potential energy associated with the exclusion principle | \displaystyle U_{ex}=\frac{A}{r^n} |
Dissociation energy of a solid | \displaystyle U_{diss}=α\frac{ke^2}{r_0}(1−\frac{1}{n})\( |
oment of inertia of a diatomic molecule with reduced mass μ | \displaystyle I=μr^2_0 |
Electron energy in a metal | \displaystyle E=\frac{π^2ℏ^2}{2mL^2}(n^2_1+n^2_2+n^2_3) |
Electron density of states of a metal | \displaystyle g(E)=\frac{πV}{2}(\frac{8m_e}{h^2})^{3/2}E^{1/2} |
Fermi energy | \displaystyle E_F=\frac{h^2}{8m_e}(\frac{3N}{πV})^{2/3} |
Fermi temperature | \displaystyle T_F=\frac{E_F}{k_B} |
Hall effect | \displaystyle V_H=uBw |
Current versus bias voltage across p-n junction | \displaystyle I_{net}=I_0(e^{eV_b/k_BT}−1) |
Current gain | \displaystyle I_c=βI_B |
Selection rule for rotational energy transitions | \displaystyle Δl=±1 |
Selection rule for vibrational energy transitions | \displaystyle Δn=±1 |
Summary
9.1 Types of Molecular Bonds
- Molecules form by two main types of bonds: the ionic bond and the covalent bond. An ionic bond transfers an electron from one atom to another, and a covalent bond shares the electrons.
- The energy change associated with ionic bonding depends on three main processes: the ionization of an electron from one atom, the acceptance of the electron by the second atom, and the Coulomb attraction of the resulting ions.
- Covalent bonds involve space-symmetric wave functions.
- Atoms use a linear combination of wave functions in bonding with other molecules (hybridization).
9.2 Molecular Spectra
- Molecules possess vibrational and rotational energy.
- Energy differences between adjacent vibrational energy levels are larger than those between rotational energy levels.
- Separation between peaks in an absorption spectrum is inversely related to the moment of inertia.
- Transitions between vibrational and rotational energy levels follow selection rules.
9.3 Bonding in Crystalline Solids
- Packing structures of common ionic salts include FCC and BCC.
- The density of a crystal is inversely related to the equilibrium constant.
- The dissociation energy of a salt is large when the equilibrium separation distance is small.
- The densities and equilibrium radii for common salts (FCC) are nearly the same.
9.4 Free Electron Model of Metals
- Metals conduct electricity, and electricity is composed of large numbers of randomly colliding and approximately free electrons.
- The allowed energy states of an electron are quantized. This quantization appears in the form of very large electron energies, even at \displaystyle T=0K.
- The allowed energies of free electrons in a metal depend on electron mass and on the electron number density of the metal.
- The density of states of an electron in a metal increases with energy, because there are more ways for an electron to fill a high-energy state than a low-energy state.
- Pauli’s exclusion principle states that only two electrons (spin up and spin down) can occupy the same energy level. Therefore, in filling these energy levels (lowest to highest at \displaystyle T=0K), the last and largest energy level to be occupied is called the Fermi energy.
9.5 Band Theory of Solids
- The energy levels of an electron in a crystal can be determined by solving Schrödinger’s equation for a periodic potential and by studying changes to the electron energy structure as atoms are pushed together from a distance.
- The energy structure of a crystal is characterized by continuous energy bands and energy gaps.
- The ability of a solid to conduct electricity relies on the energy structure of the solid.
9.6 Semiconductors and Doping
- The energy structure of a semiconductor can be altered by substituting one type of atom with another (doping).
- Semiconductor n-type doping creates and fills new energy levels just below the conduction band.
- Semiconductor p-type doping creates new energy levels just above the valence band.
- The Hall effect can be used to determine charge, drift velocity, and charge carrier number density of a semiconductor.
9.7 Semiconductor Devices
- A diode is produced by an n-p junction. A diode allows current to move in just one direction. In forward biased configuration of a diode, the current increases exponentially with the voltage.
- A transistor is produced by an n-p-n junction. A transistor is an electric valve that controls the current in a circuit.
- A transistor is a critical component in audio amplifiers, computers, and many other devices.
9.8 Superconductivity
- A superconductor is characterized by two features: the conduction of electrons with zero electrical resistance and the repelling of magnetic field lines.
- A minimum temperature is required for superconductivity to occur.
- A strong magnetic field destroys superconductivity.
- Superconductivity can be explain in terms of Cooper pairs.