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Solid State Physics: Key Concepts and Questions, Exams of Solid State Physics

Massachusetts Institute of Technology (MIT)Solid State Physics

A concise overview of key concepts in solid-state physics, presented in a question-and-answer format. It covers topics such as the born-oppenheimer approximation, crystal lattices, brillouin zones, phonon interactions, free electron models, and bonding in solids. The material is suitable for students seeking a quick review or a study aid for exams, offering explanations of complex phenomena like energy gaps, fermi surfaces, and semiconductor behavior. It also touches on the drude model and various types of scattering in solids, making it a valuable resource for understanding the fundamental principles of solid-state physics.

Typology: Exams

2025/2026

Available from 10/30/2025

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What is the Born-Oppenheimer approximation? And why is it a good
approximation.
-The motion of electrons and the nucleus can be treated independently. ie the
motion of the molecules is not needed when describing electrons
-this works as the mass of an electron is much less than the nuclear mass
Bond order
5 distinct 2D lattices
Distance between planes from miller indices
Bragg diffraction law
First Brillouin zone
-π/a ≤ k ≤ π/a
k- spring constant
a- interplanar distance
Transverse acoustic mode
-Both atoms/ions in basis oscillate in phase, entire basis is displaced
Transverse optical mode
Different classes of atoms/ions vibrate out of phase. Large oscillating dipole
moment is perpendicular to k (wave vector)
What happens when phonons interact with EM radiation
Oscillating dipole of LO phonon creates E-field that interacts strongly with EM
radiation causing photons to be emitted or absorbed.
-Phonons have many possible k values so allowed transitions are restricted by
conservation of momentum.
Crystal momentum
ħk
Assumptions for Free electron mode
-Valence electrons are free to move throughout solid.
-ignore interaction between valence electrons and atomic cores.
When does free electron model apply
Simple metals
Assumption for Nearly free electron model
-Assumes electrons are nearly free but there is a weak interaction between them
and the atomic cores.
Tight Binding Model assummptions
-Assumes valence electrons are quite tightly bound to individual cores and retain
some atomic character in the solid.
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What is the Born-Oppenheimer approximation? And why is it a good approximation.

  • The motion of electrons and the nucleus can be treated independently. ie the motion of the molecules is not needed when describing electrons
  • this works as the mass of an electron is much less than the nuclear mass Bond order 5 distinct 2D lattices Distance between planes from miller indices Bragg diffraction law First Brillouin zone
  • π/a ≤ k ≤ π/a k- spring constant a- interplanar distance Transverse acoustic mode
  • Both atoms/ions in basis oscillate in phase, entire basis is displaced Transverse optical mode Different classes of atoms/ions vibrate out of phase. Large oscillating dipole moment is perpendicular to k (wave vector) What happens when phonons interact with EM radiation Oscillating dipole of LO phonon creates E-field that interacts strongly with EM radiation causing photons to be emitted or absorbed.
  • Phonons have many possible k values so allowed transitions are restricted by conservation of momentum. Crystal momentum ħk Assumptions for Free electron mode
  • Valence electrons are free to move throughout solid.
  • ignore interaction between valence electrons and atomic cores. When does free electron model apply Simple metals Assumption for Nearly free electron model
  • Assumes electrons are nearly free but there is a weak interaction between them and the atomic cores. Tight Binding Model assummptions
  • Assumes valence electrons are quite tightly bound to individual cores and retain some atomic character in the solid.
  • Electrons can hop from one site to another to conduct electric current Assumptions for periodic boundary conditions
  • Assume crystal is infinite
  • Disregard effect of outer boundaries
  • Divide solid into boxes length L, and require wavefunctions are periodic with period L, i.e ψ(x+L, y, z) = ψ(x,y,z) Fermi wave vector Fermi surface the surface in reciprocal space which separates occupied from unoccupied electron states at zero temperature.
  • E = E_F Delocalisation energy for particle in a box Function which is periodic with lattice Why are there energy gaps in nearly free electron model
  • Diffraction of the electron wavefunction blocks the propagation
  • diffraction occurs because of strong interaction between electron wavefunction and lattice for wavelengths in registry with the lattice. Bloch function Band gaps in nearly free electron model Energy values where there are no allowed states.
  • Result of interference between electron wavefunction and the lattice. Crystal orbitals of solid in tight binding model Why is conventional unit cell sometimes used instead of primitive unit cell. Full symmetry of a crystal structure is not obvious from the primitive cell How many atoms are in a face centred cubic unit cell 4 How many atoms are in a edge centred cubic cell 4 Fermi Wavevector in terms of number density of electrons Fermi energy formula Fermi velocity Covalent solids- how does bonding work Definite bonds formed between pairs of atoms by electron sharing.
  • Bonding between pairs is directional Ionic solids- how does bonding work?
  • some atoms form stable ions by loss or gain of electrons.
  • Crystal is held together by long range electrostatic forces, giving a Madelung potential which must be summed to infinity over distant ions. Metallic solids- how does bonding work?
  • 2 electrons in the antibonding MO mostly cancel out the binding energy of the 2 electrons in the bonding MO First few MOs in homonuclear diatomic molecules Conductivity in drude model Mobility in Drude model Current density J = nev = σE E- Electric field n- electron density v- drift velocity σ - conductivity Key assumptions of Drude mode
  • Electrons collide with the ionic cores only and not with each other; these are instantaneous which change the electron velocity.
  • Between collisions electrons do not interact with ionic cores or each other.
  • Electrons achieve thermal equilibrium with their surroundings only through collisions. Heat capacity in terms of heat C = dQ/dT Occupation function for fermions Fermi-Dirac Distribution Volume of Fermi Sphere Wiedelann Franz Law Resistivity against Temperature graph for metal Thermal conductivity against temperature graph for metal Why do electrons scatter in solids
  • Scattering is associated with imperfections in periodicity of the lattice, caused by 2 main effects: 1: Electron-phonon scattering 2: Electron-impurity scattering describe electron-phonon scattering The thermal vibration of the lattice prevents atoms being on perfectly periodic sites at the same time. The corresponding collision time is denoted as τp which is temperature dependent. Describe electron impurity scattering The presence of impurity atoms and other defects upsets the lattice periodicity.
  • The collision time τi is independent of temperature. Drift velocity in a metal intrinsic semiconductor

A semiconductor where the conductivity is due to the intrinsic properties of the material itself (i.e it contains no impurities) Band gap in intrinsic semiconductors Qualitatively describe the main features of electric conductivity in intrinsic semiconductor

  • Small energy gap so at normal temperatures, a few electrons have enough energy to be excited into a conduction band and holes are left on valence band, so a small current can flow.
  • Conductivity is therefore due to electrons and hole and increases with temperature.
  • Therefore current carrying electron density, n, increases with T and because n=p so does concentration of holes Occupation function for electrons in an intrinsic semiconductor Fermi-Dirac function Number density of current carrying electrons in an intrinsic semiconductor Effective number density of accessible states at the bottom of the CB, Nc Effective number density of accessible states at top of Valence band Nv Number density of holes in an intrinsic semiconductor Drift velocity of electron in intrinsic semiconductor drift velocity of hole in intrinsic semiconductor Conductivity of an intrinsic semiconductor Electron mobility in terms of Hall coefficient The hole mobility in intrinsic semiconductors n in terms of p for intrinsic semiconductors n = p Extrinsic semiconductor Semiconductor which has been doped (controlled introduction of impurity atoms) n-type doping
  • impurities have more valence electrons than solid.
  • Above binding energy of electron, it is promoted into the conduction band where it can contribute to the conductivity. e.g Phosphorous dopant, silicon solid Donor ionisation energy of extrinsic semiconductor p-type doping
  • Doping with impurities with less valence electrons than the solid.
  • Dopant can accept electron from solid, leaving hole in valence band.
  • Acceptors accept electrons from Valence band and create holes in the valence band. e.g boron dopant in silicon

Impurity range for n type semiconductor

  • Conductivity rises with T as donors are ionised, dopant concentration is usually low. Number carrier density due to dopant atoms in extrinsic semiconductor Graph of molar heat capacity against temperature for noble metals (copper, gold, silver) Equipartition Principle The average energy in each accessible degree of freedom of a system in thermal equilibrium is kT/ Where k is boltzmann constant Summary of Einstein model for lattice molar heat capacity
  • Atoms are vibrating independently as a QM SHO, with a single excitation frequency for all.
  • we expect to see reduction in available quantim states when kbT < ħω, i.e T < ΘE
  • Cv - > 3R above ΘE.
  • as T-> 0 Cv ∝ exp(- Θ E/T) (Incorrect at very low T) Allowed energies of vibration Einstein temperature ΘE The limiting temperature below which the first excited vibrational state becomes inaccessible. Summary of classical model for lattice molar heat capacity Based on equipartition of energy, Cv = 3R independent on temperature Summary of Debye model for lattice molar heat capacity
  • Quantized lattice phonons distributed over whole lattice.
  • Maximum phonon frequency ωD with no minimum phonon frequency.
  • Predicts for T>> ΘD, Cv - > 3R and for T << ΘD, Cv ∝ T^3, close to experimental observations Electron (or hole) concentration in the impurity range of an extrinsic semiconductor Electron (or hole) concentration in the saturation range of an extrinsic semiconductor electron concentration in the intrinsic range of an extrinsic semiconductor why can n = ∫g(ε)f(ε)dε between infinity and 0 be written as n = ∫g(ε)dε between fermi energy and 0. Where f(ε) is the fermi dirac distribution Because occupation function f(ε,T) becomes negligible for ε>εF when T<TF, the fermi temperature which is ~ 10^4 K for most metals Dipole-Dipole interaction summary
  • Interactions between molecules due to uneven charge distribution within molecules.
  • Weak- dictate intermolecular forces which gives rise to molecular solids where weak induced dipole create Van der Waals forces Band structure of n-type doping at T=0K Band structure of p-type doping at T=0K Carrier density in an intrinsic semiconductor Conditions best for transistor
  • Only want a few charges in material- just below exhaustion range as there is a small impulse of energy to begin some ionsiations Conditions best for LED
  • lots of charges needed in conduction band for more energy
  • so intrinsic range is best Graph of molar heat capacity at constant volume for Cv vs absolute temperature for copper, highlighting low temp behaviour Describe einstein model
  • Assumes atoms vibrate independently as QHOs in x, y, z, so E = (nx + ny + nz + 3/2) ħω Why does Cv depends on T in einstein model of heat capacity
  • Lattice vibrations (phonons) described by bose-einstein distribution which gives rise to non linear T dependence in density of states in energy, which means Cv depends on T. Einstein temperature formula Number of degrees of freedom for free electrons 3 Contribution of free electrons to heat capacity according to classical model 3 dof (only translational) Describe how light is absorbed by a solid
  • The oscillating dipole of an optical vibration (phonon) creates an oscillating E- field that interacts strongly with EM radiation.
  • This only occurs when wavelength and frequencies of photon and phonon are matched at one defined point where the phonon and photon dispersions curves cross near the BZ zone centre at k~ Resistivity against absolute temperature for a metal Confinement potential graph for Einstein model with allowed energies Einstein model, Cv against T Density of vibrational states in Einstein Model Conductivity of a semiconductor Equation of motion of longitudinal vibrations of a monatomic 1D chain What happens to fermi sphere when current flows It shifts slightly in direction of direction of E-field Amount of shift ∝ drift velocity
  • Constructive interference of 2s orbitals of 2 atoms.
  • Hence increased electron density between nuclei, so bonding is stronger. Conductivity of intrinsic semiconductors Motion of particles in lower branch
  • longitudinal acoustic mode
  • Both atoms/ions in basis oscillate in phase, entire basis is displaced
  • long wavelength sound as k-> Motion of particles in upper branch
  • Atoms/ions vibrate out of phase but with fixed centre of mass
  • leads to large induced dipole moment parallel to k which interacts strongly with EM radiation.
  • Photons can be absorbed or emitted. Electronic excitation energy of diatomic molecule Vibrational excitation energy or diatomic molecule bond angle for hybridisation How can conductivity be explained in NFEM when valence band is full
  • 1st BZ is full.
  • 1st and 2nd BZ zones overlap How can conductivity be explained in tight bonding model when valence band is full
  • assume s and p bands overlap Wavefunction for hybridisation Model potential for pair wise interaction between ions of an ionic solid Tight bonding model ∫φnHφn dτ Tight bonding model ∫φmHφn dτ β** - interaction energy Energy spectrum, E against k for tight bonding model Electronic energy as a function of internuclear separation r in H2+ for bonding and antibonding state Diatomic molecular orbitals Binding energy and equilibrium configuration of a crystal sp2 hybridization summary
  • Combination of 2s and 2pxy to form 3-arm orbitals on a plane with 120° to each other- forming honeycomb lattice.
  • The remaining 2pz orbital forms π - orbital with its neighbours, describing the mobile electrons in graphene. sp3 hybridization summary
  • Combination of 2s and 2pxyz to form 4 arms in tetragonal directions with 109.47° to each other.
  • Each of these arms overlap with the arms of neighbour carbon to form to form strong σ bond, and extend in 3d to form structure of diamond. Density of states definition g(E)dE = the number of allowed energy levels per unit volume of solid in the energy range E-> E + dE Binding energy, Ui for ionic solids Total energy of ionic solid 2D lattice translation vector Trial solution for longitudinal vibrations of monatomic 1D chain Dispersion relation of 1D longitudinal vibration Trial solution for longitudinal vibrations of diatomoc 1D chain Dispersion relation for 1D diatomic crystal Optic and acoustic modes in 1st BZ Frequency in terms of interatomic force constant for optical mode phonons Energy levels of FEM Fermi velocity Energy band gaps in 1st BZ Dispersion relation for solid at different conductivities in NFEM Energy of a hybrid state 2s contribution to each hybrid Bonding of graphene sp^2 orbitals from each C atom form 3 σ - bonding orbitals and 3 σ* - antibonding orbitals with the sp^2 orbitals of the 3 neighbouring C atoms. Dirac point Point around boundary of BZ where filled and unfilled states touch at one point- in 2d they form dirac cones Energy spectrum of graphene Equipartition Theorem The average energy in each accessible degree of freedom of a system in thermal equilibrium is KT/ Total internal energy of one mole of substance Density of states for Debye model Energy-wave vector relation for Debye model Total heat capacity for fermion solid Failures of drude model Fails at low temp. Predicts Cv = 3/2 nk Group velocity of traveling wave quantum state of momentum k Momentum shift of QM electron Thermal conductivity against T for a metal What is a hole A hole is an empty state near top of valence band with positive hole effective mass and an effective positive charge e

Landau levels Refers to quantized energy levels an electron can occupy when a crystal is in presence of a plane perpendicular B field Fermi Surface the surface in reciprocal space which separates occupied from unoccupied electron states at zero temperature Angle Resolved Photo Emission (ARPEs) Technique for measuring fermi surface. Limitations: Only works for 2D and requires good surface Compton Scattering Compton scattering - Photon scattered by electron. Limitations: Need synchotron, slow measurements, low resolution Positron Annihilation Limitations: Positron may prevent measurements, slow measurements, better resolution than ARPEs but worse than quantum oscillations. Cooper Pairs Electrons move faster than lattice displacement, so the lattice is still moving after the first electron has stopped occupying so a second electron is attracted to the lattice. Elastic Scattering The magnitude of the new K vector equals that of the old K vector Metal Work Function The Meissner Effect The expulsion of a magnetic field by a superconductor. n Electron density of material Quantum Oscillations Changing B through a lattice causes oscillations in magnetisation (Haas de Van Effect) or Resistivity (Shubnikov-de Haas), oscillations in length (Magnetostriction). If can measure those oscillations as function of (1/B) can measure fermi surface area S using formula

Tight binding Model p Hole carrier density of Brillouin Zone Perpendicular Bisector of vectors to neighbouring lattice sites in k space. Weigner Seitz Unit Cell of reciprocal space. Boundary marks K values for Laue condition of diffraction Unit Cell Structure Factor Intrinsic Semiconductor Number of carriers is dominatedd bty electrons thermally excited from VB to CB Doped SemiConductor Acceptors or donors are added to the material, which may be ionized to donate electrons to the conduction band or ionzed to accept electrons from the valence band respectively. Photo (n-type-doped semiconductor) Dopant Acceptors and Donors. Donors have a small positive ionization energy gap to donate electrons to the conduction band. Acceptors have a small negative ionization energy gap to accept electrons from the valence band (donate holes) Metal-Semiconductor Junction Fermi energy of metal and chemical potential is set to equal. eX is energy between vacuum and conduction band. Rigid band bending of vacuum level with electron bands. Built in potential is height change from band bending. Equation relating K-vector and energy Equation relating Fermi Energy to number of electrons Electron Configuration 2nd Row Unlocks P6, 4th row Unlocks D10. D fills before P when possible. Diamagnetism a type of magnetism, associated with paired electrons, that causes a substance to be repelled from the inducing magnetic field Paramagnetism

Intermediate temp - > dopants ionized so T dependence of electron and hole mobility dominates High temp - > dominated by temp dependence of intrinsic carrier densities n,p. Band structure near Fermi Energy for metal, insulator, semiconductor Metals have partially filled bands. Insulators have fully filled bands. Semiconductors are insulators with a band gap of order kB T Langevin Paramagnetism Field induced magnetism wherein an external magnetic field has caused magnetic dipoles to align to magnetize the material. Langevin refers to paramagnetism in free atoms/ions Itinerant / Pauli paramagnetism Change in the number of occupied up/down spin states in a conductor due to an external magnetic field Cooper Pairs Electrons cause lattice deformation, lowering the potential energy leading new electrons to be attracted to the deformation, leading electrons to flow in cooper pairs. pn junction rules Electron mean free path Mean path before scattering, inverse to scattering rate