Unit 5: Solid State Physics

I. Orientation — Electronic Structure of Solids

Solid-state physics explains how the microscopic arrangement of atoms and electrons determines electrical behavior. In a crystal, closely spaced atoms interact, converting discrete atomic energy levels into allowed energy bands separated by forbidden gaps.

• Governing principle: Electrons obey quantum mechanics, the Pauli exclusion principle, and Fermi–Dirac statistics.
• Crystal assumption: Atoms occupy a periodic lattice, so electrons experience a periodic potential.
• Charge carriers:
  • Electrons carry charge (-e), where (e=1.602\times10^{-19}\ \text{C}).
  • Holes behave as mobile carriers of charge (+e).
• Electrical classification: The occupation and separation of the valence and conduction bands distinguish conductors, semiconductors, and insulators.
• Thermal effect: Temperature changes carrier occupation, scattering, conductivity, and the position of the Fermi level.

II. Free-Electron Transport — Classical and Quantum Foundations

A. Free electron theory (introduction)

Free electron theory treats conduction electrons in a metal as mobile particles moving through a background of fixed positive ions.

• Drude model: Classical electrons move randomly and undergo collisions characterized by a mean relaxation time (\tau).
• Response to an electric field: For an electron in field (\mathbf E),
  
  TEXT

    F = -eE,        a = -eE/m

  
  where (F) is force, (a) is acceleration, and (m) is electron mass.
• Drift velocity: Collisions limit the average directed velocity:
  
  TEXT

    v_d = -eEτ/m = -μ_eE

  
  Here (v_d) is electron drift velocity and (\mu_e=e\tau/m) is electron mobility.
• Conductivity:
  
  TEXT

    J = σE,        σ = ne²τ/m = neμ_e

  
  where (J) is current density, (\sigma) is conductivity, and (n) is the free-electron concentration.
• Sommerfeld improvement: Quantum free electron theory retains nearly free motion but applies the Pauli principle and Fermi–Dirac statistics.
• Limitations: The simple model does not explain positive Hall coefficients, semiconductor behavior, or the detailed influence of the crystal’s periodic potential.

B. Diffusion and drift current (qualitative)

Drift and diffusion are the two principal mechanisms by which mobile carriers produce current in a solid.

1. Drift current: An electric field produces directed carrier motion.

   • Electrons: Their velocity is opposite to (\mathbf E), but their conventional current is along (\mathbf E).
   • Holes: Their velocity and conventional current are both along (\mathbf E).
   • Total drift current density:
     
     TEXT

          J_drift = e(nμ_n + pμ_p)E

     
     Here (n,p) are electron and hole concentrations, while (\mu_n,\mu_p) are their mobilities.

2. Diffusion current: Carriers move from high concentration to low concentration because of random thermal motion.

   • Electron current:
     
     TEXT

          J_n,diff = eD_n(dn/dx)

   • Hole current:
     
     TEXT

          J_p,diff = -eD_p(dp/dx)

     
     Here (D_n,D_p) are diffusion coefficients and (x) is position.
   • Einstein relation: Under nondegenerate equilibrium conditions,
     
     TEXT

       D/μ = k_BT/e

     
     where (k_B) is Boltzmann’s constant and (T) is absolute temperature.
   • Equilibrium: In a nonuniform semiconductor, drift can exactly balance diffusion, giving zero net current.

III. Electron Energies and State Occupation

A. Fermi energy

Fermi energy is the highest occupied electron energy at absolute zero.

• Definition: At (T=0\ \text{K}), every available state below (E_F) is occupied and every state above it is empty.
• Three-dimensional free-electron result:
  
  TEXT

    E_F = (ℏ²/2m)(3π²n)^(2/3)

  
  where (E_F) is Fermi energy, (\hbar=h/2\pi), (m) is electron mass, and (n) is electron concentration.
• Fermi temperature:
  
  TEXT

    T_F = E_F/k_B

  
  In metals, (T_F) is commonly much greater than room temperature, so only electrons near (E_F) respond strongly to thermal excitation.
• Physical significance: (E_F) determines electron occupancy, carrier energy, and many thermal and transport properties.

B. Fermi-Dirac distribution function

The Fermi–Dirac function gives the probability that an available state of energy (E) is occupied by an electron.

• Distribution:
  
  TEXT

    f(E) = 1/{1 + exp[(E - E_F)/(k_BT)]}

  
  Here (f(E)) is occupation probability, (E_F) is the chemical potential at the stated temperature, and the other symbols have their usual meanings.
• At the Fermi level:
  
  TEXT

    f(E_F) = 1/2

  
  This result holds at every nonzero temperature.
• Zero-temperature limit: (f(E)=1) below (E_F) and (f(E)=0) above (E_F), producing a sharp step.
• Finite temperature: The step is broadened over an energy range of a few (k_BT); some electrons below (E_F) are thermally promoted to states above it.
• Classical limit: When (E-E_F\gg k_BT),
  
  TEXT

    f(E) ≈ exp[-(E-E_F)/(k_BT)]

  
  which is the Maxwell–Boltzmann approximation.

C. Density of states (qualitative)

The density of states specifies how many quantum states are available within a unit energy range and unit volume.

• Meaning: Carrier concentration depends on both available states (g(E)) and their occupation probability (f(E)).
• Three-dimensional behavior: Near a parabolic conduction-band edge,
  
  TEXT

    g_c(E) ∝ (m_n*)^(3/2)√(E-E_c),    E ≥ E_c

  
  where (m_n^*) is electron effective mass and (E_c) is the conduction-band minimum.
• Carrier counting:
  
  TEXT

    n = ∫[E_c to ∞] g_c(E)f(E)dE

  
  Thus, a high density of states does not guarantee many carriers unless those states are occupied.
• Forbidden gap: Ideally, (g(E)=0) between the valence-band maximum (E_v) and conduction-band minimum (E_c).

IV. Energy Bands and Material Classification

A. Band theory of solids

Band theory describes electron motion in a periodic crystal through allowed energy bands and forbidden energy gaps.

• Band formation: When (N) atoms form a crystal, each atomic level splits into approximately (N) closely spaced levels.
• Valence band: The highest band occupied at (0\ \text{K}), usually associated with bonding electrons.
• Conduction band: A higher band containing states in which electrons can move through the crystal.
• Band gap:
  
  TEXT

    E_g = E_c - E_v

  
  where (E_g) is forbidden-gap energy.
• Conduction condition: Current requires an incompletely filled band or accessible empty states into which electrons can move.
• Metals: A band is partially filled, or the valence and conduction bands overlap, providing carriers without thermal excitation.

B. Semiconductors and insulators

Semiconductors and insulators both possess band gaps, but differ mainly in gap size and resulting carrier concentration.

1. Semiconductors:

   • Gap: Typically of order (1\ \text{eV}); silicon has (E_g\approx1.12\ \text{eV}) near room temperature.
   • Excitation: Thermal or optical energy promotes electrons into the conduction band, leaving holes in the valence band.
   • Conductivity: Increases strongly with temperature and can be controlled by doping.

2. Insulators:

   • Gap: Usually several electron-volts, making ordinary thermal excitation unlikely.
   • Conductivity: Extremely low because both conduction-band electrons and valence-band holes are scarce.
   • Contrast with metals: Metal resistance generally rises with temperature because scattering increases, whereas semiconductor conductivity generally rises because carrier generation dominates.

C. Direct and indirect band-gap semiconductors

Direct and indirect semiconductors differ in the crystal momentum required for an electron to cross the band gap.

1. Direct band gap:

   • Alignment: The conduction-band minimum and valence-band maximum occur at the same wave vector (k).
   • Transition: A photon can create or annihilate an electron–hole pair while conserving momentum.
   • Examples: GaAs and GaN are efficient light-emitting materials.

2. Indirect band gap:

   • Misalignment: The band extrema occur at different (k)-values.
   • Transition: A phonon must participate to conserve crystal momentum.
   • Examples: Silicon and germanium emit light inefficiently.
   • Energy relation: Optical transitions satisfy approximately (h\nu\approx E_g), while indirect transitions additionally involve phonon energy and momentum.

V. Semiconductor Carrier Statistics and Dynamics

A. Fermi level for intrinsic and extrinsic semiconductors

The Fermi level controls equilibrium electron and hole populations and shifts when impurities are introduced.

• Carrier concentrations:
  
  TEXT

    n = N_c exp[-(E_c-E_F)/(k_BT)]
    p = N_v exp[-(E_F-E_v)/(k_BT)]

  
  Here (N_c,N_v) are effective conduction- and valence-band densities of states.
• Intrinsic semiconductor: With (n=p=n_i), the Fermi level lies near midgap:
  
  TEXT

    E_i = (E_c+E_v)/2 + (k_BT/2)ln(N_v/N_c)

  
  It is exactly at midgap when (N_c=N_v).
• n-type material: Donor atoms supply electrons, making (n\gg p) and moving (E_F) toward (E_c).
• p-type material: Acceptor atoms create holes, making (p\gg n) and moving (E_F) toward (E_v).
• Mass-action law:
  
  TEXT

    np = n_i²

  
  This applies to a nondegenerate semiconductor in thermal equilibrium at fixed temperature.

B. Concept of effective mass: electrons and holes

Effective mass represents a carrier’s dynamical response to force within the crystal’s periodic potential.

• Band-curvature definition:
  
  TEXT

    m* = ℏ²/(d²E/dk²)

  
  where (E(k)) is the energy–wave-vector relation and (m^*) is the effective mass for a one-dimensional parabolic band.
• Electrons: Near a conduction-band minimum, curvature is positive, so (m_n^*>0).
• Valence electrons: Near a valence-band maximum, curvature is negative, corresponding to negative electron effective mass.
• Holes: Missing valence electrons are described more conveniently as particles with positive charge (+e) and positive effective mass (m_p^*).
• Importance: Effective mass influences mobility, acceleration, density of states, cyclotron motion, and semiconductor conductivity.

VI. Magnetic Transport

A. Hall effect (with derivation)

The Hall effect is the development of a transverse electric field across a current-carrying material placed in a perpendicular magnetic field.

• Arrangement: Let current flow along (x), magnetic field (B_z) act along (z), and Hall field (E_y) develop along (y).
• Magnetic deflection: A carrier of charge (q) and drift velocity (v_x) experiences the Lorentz force:
  
  TEXT

    F = q(E + v × B)

• Transverse equilibrium: Charge accumulates until electric and magnetic forces balance:
  
  TEXT

    q(E_y - v_xB_z) = 0
    E_y = v_xB_z

• Current relation:
  
  TEXT

    J_x = nqv_x

  
  where (n) is the concentration of one carrier type.
• Hall coefficient derivation:
  
  TEXT

    R_H = E_y/(J_xB_z)
        = v_xB_z/(nqv_xB_z)
        = 1/(nq)

  
  Thus (R_H=-1/(ne)) for electrons and (R_H=+1/(pe)) for holes.
• Hall voltage: For specimen thickness (t), current (I), and magnetic field (B),
  
  TEXT

    V_H = R_HIB/t

  
  where (V_H) is the transverse potential difference.
• Applications: The sign of (R_H) identifies the dominant carrier type; its magnitude estimates carrier concentration, while (\mu=|R_H|\sigma) gives mobility in a single-carrier material.
• Two-carrier limitation: When both electrons and holes conduct,
  
  TEXT

    R_H = (pμ_p² - nμ_n²)/{e(pμ_p + nμ_n)²}

  
  so the simple relation (R_H=1/(nq)) no longer applies.