St. Lawrence University

Daniel W. Koon

Van der Waals bond, covalent, ionic, hydrogen, metallic. Madelung constant ideal crystal basis Bravais (space, plane) lattice unit cell |
point group plane group (real lattice) cubic lattice (cubic primitive) tetragonal lattice orthorhombic lattice BCC |
FCC HCP reciprocal lattice Brillouin zone vacancy interstitial |

You should know how many lattice groups and point groups there are in 2D, 3D.

You should know the names of the ten 2D point groups.

You should be able to determine the symmetries obeyed at points in the 2D Bravais lattice.

You should understand the various symmetries: Translation, Reflection, Rotation, Inversion, Glide, Screw

You should be able to sketch the planes corresponding to given Miller indices for cubic and HCP lattices.

You should recognize the following lattices on sight: BCC, FCC, HCP, zincblende (ZnS), diamond, rocksalt (NaCl), perovskite

You should understand which type of diffraction is most useful in which situation.

You should know Bragg's law.

You should be able to calculate the reciprocal lattice vectors.

You should know the reciprocal lattices of FCC, BCC, simple cubic.

phonon vibrational waves dispersion relationship group velocity phase velocity |
density of [phonon] states acoustic branch optical branch Einstein [characteristic] temperature Debye [characteristic] temperature |
lattice specific heat electronic specific heat phonon mean free path phonon scattering |

You should be able to convert between

You should know the general functional relationship for

You should be able to describe how longitudinal and transverse, and acoustic and optical vibrations differ from an atomic point-of-view.

You should be able to describe the Einstein model and the Debye model, and be able to recognize the origin of every term in either expression for

You should know a few of the sources of phonon scattering and be able to explain why each can scatter phonons.

You should know what materials have the largest ratio of electronic heat conduction to lattice heat conduction at low temperature.

1. Inverse space is also momentum space.

2. Each Brillouin Zone covers the same amount of volume in k-space.

3. The second BZ consists of two disconnected regions, the third consists of 3, etc.

4. The dispersion relation (E vs k) is discontinuous at the BZ boundaries.

5. If an electron experiences no scattering and no outside forces, it will remain where it

is in k-space.

6. In a 2- or 3- dimensional reduced zone scheme, one

moves the 2nd (or 3rd, or ...) zone into the first by

reflecting across the BZ boundary as shown to the left:

a® a'

b® b'

c® c'

d® d'

7. If a material is an insulator, then the Fermi energy lies at an energy which has a nonzero density of states.

8. If a material is an insulator, then the Fermi surface is equivalent to one of the Brillouin Zone boundaries.

9. Each Brillouin Zone is capable of holding two electrons for every atom that is in the lattice.

10. If, for a given value of k, the energy of an electron inside a solid (see the dispersion relation) is smaller than for a free electron, then it has a negative average potential energy and thus the effect of the ion cores on it is greater than the effect of other electrons.

Classical free electron theory: (CFE)f _{o} µ
exp(-b
E_{k})<E _{k}> = kTDrude Model:Scattering caused by the positive ionsSurvival probability: n=n _{o}e^{-t/t
}s
= ne^{2}t
/ mC _{eln} »
C_{lattice
}Lorentz Model: CFE using Boltzmann ea.:f = fR_{0} + _{H} = -1.18/neMagnetoresistance: r
= r
+_{o} M B
Quantized free electron theory: (QFE)^{2
} E = Density of states: g(k) dk = (k/p
)Fermion statistics: ^{2}dkg(E)dE = Significance of the Fermi Energy E _{f0} = Sommerfeld model: (QFE & Boltz. eq.)C
_{eln }= g
T »
(kT/E_{f}) C_{lattice
} |
g
= (p
^{2}k^{2}/3 ) g(E_{f})
Pauli Paramgnetism: c
µ
g(E,_{f})c
»
(kT/E l
>> than for Drude, Lorentz (Defects, phonons cause electron scattering.)_{f}) c
_{class}
s = [ne ^{2}t
(E_{f})]/mR _{H} = -1/neBand Theory:Bloch functions Kronig-Penney model Band gaps Fermi surface Brillouin zones (again!) k-space, inverse or momentum space Connection between Brillouin zone boundaries and energy gaps Effective mass, m _{ij}=Holes Superconductors Organic, heavy fermion, cuprate Zero resistance Persistant current Meissner effect Type I, Type II superconductors Flux quantiz.: 2.07 ´ 10 ^{-15}WbHeat capacity 'glitch' Tuyn's law. HBCS. electron-phonon coupling_{c} = H_{0}[1-(T/T_{c})^{2}]BCS. density of states in SC BCS: Cooper pairs: for Electromagnets, Josephson junctions, SQUIDS, etc. |

Be able to sketch density of states, g(e
) and Fermi filling factor, f(e
).
Impurities: e
_{d} = R_{y}(m_{c}/m_{e})(1/k^{2}), aMetal-Insulator Transition: _{d} = a_{0k
}(m_{e}/m_{c})nLaw of Mass Action: _{c}^{1/3}a_{d @
}1/4NDegenerate: e
_{a}^{-} + n = N_{d}^{+} + pClassical: e
_{F} = e
_{c} + ln(N_{F} = e
_{c} - kT_{c}/n_{0}) n Intrinsic: _{i} µ
T^{3} exp(-e
_{g}/2kT)n@
n _{i}Saturation: n@
N Freezeout: _{d} - N_{a
}n = [NConductivity: _{c}(N_{d} - N_{a})/N_{a}kT] exp (-e
_{d}/kT) s
= nem
Hall coeff.: _{e} + pem
_{h
} RBe able to sketch impurity and conduction bands as _{H} = (S
R_{is
}i^{2}/[S
s
_{i}]^{2})^{ ®
}_{
}(1/ne)(<t^{2}>/<t>^{2})tan q _{H} = r
_{xy}/r
_{xx} = w
_{c}t
= R_{Hs
}BN®
N._{c}Be able to explain why impurity bands broaden. |
P3D Variable-range hop:_{hop} µ
exp[-2a
R-b
D
E] s
µ
exp[-(TBe able to sketch g_{0}/T)^{1/4}](e
) for amorphous semis.Cyclotron resonance: w
r _{0} = s
_{0} / (s
_{0}^{2} + s
_{H}^{2}) r
_{H} = s _{H} / (s
_{0}^{2} + s
_{H}^{2}), etc.Quantum Hall Effect: r
_{c} = eB/m*Be able to sketch reciprocal lattice for a fcc space lattice._{H} = h/n
e^{2} = 25812W
/n
Be able to point out the gross features of the reciprocal space of Si, Ge, and describe location and number of equivalent valleys. Additional definitions:extrinsic semiconductorcompensation Seebeck effect Phonon-assisted hopping Fractional Quantum Hall Effect Indirect electronic transitions |

Miscellaneous E&M e _{0} = 8.85 ´
10^{-17}F/m m
_{0} = 4p
´
10^{-7}H/mDipole field: Kramers-Kronig: Dielectrics:Atomic polarizability: Clausius-Mosotti: S N_{j}a_{j}=3e
_{0
}Perm. dipoles:Langevin function: L(w) = coth(w)-(1/w) ®
(w/3) (t
=1/w
_{0})Ionic and Electronic polarization: Piezoelectricity Electrostriction Ferroelectricity |
Magnetism:Magnetic moment: Bohr magneton : m _{B} =9.27´
10^{-24}J/Tg = Lande splitting factor Atomic angular momentum: Diamagnetism: Paramagnetism: Curie Law: Quantum mech. ® Brillouin fctn: B _{J}(y) =Antiferromagnetism Neel temperature Ferrimagnetism Spin waves, magnons Exchange integral, UMagnetic domains_{ij} = -2J_{e
}Electron spin resonance n [GHz) = 14.00 g B[T];Spin-lattice & spin-spin relaxationg = 0.0715n /B Nuclear magnetic resonance: n [MHz] = 7.623 gProtons: m
_{n}B[T] = 2.793 m
Neutrons: m
_{n
} = -1.913 m
Negative temperature _{n
} |