# 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 $$\displaystyle T=0K$$ 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 $$\displaystyle U_{coul}=−\frac{ke^2}{r_0}$$ Energy change associated with ionic bonding $$\displaystyle U_{form}=E_{transfer}+U_{coul}+U_{ex}$$ Critical magnetic field of a superconductor $$\displaystyle B_c(T)=B_c(0)[1−(\frac{T}{T_c})^2]$$ 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.

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