Unit 5: Solid State Physics - Subjective Questions

Basic assumptions:

• A metal contains a large number of free conduction electrons moving throughout its volume.
• Positive ion cores remain fixed at their lattice positions.
• Conduction electrons are treated as classical particles obeying Maxwell-Boltzmann statistics.
• The potential energy inside the metal is assumed to be constant.
• Electron-electron and electron-ion interactions are neglected except during instantaneous collisions.
• Collisions randomize electron velocities, and the average time between collisions is called the relaxation time, .

Successes:

• It explains electrical conduction and Ohm's law.
• It provides the conductivity relation
  
• It qualitatively explains thermal conductivity and the Wiedemann-Franz law.

Limitations:

• It predicts an electronic specific heat much larger than the observed value.
• It cannot correctly explain the temperature dependence of conductivity.
• It fails to explain positive Hall coefficients in some metals.
• It cannot distinguish conductors, semiconductors, and insulators.

These shortcomings are largely resolved by quantum free electron theory and band theory.

Drift current is produced by the directed motion of charge carriers under an applied electric field. Electrons drift opposite to the field, while holes drift along the field. The drift current density is

where and are carrier concentrations and and are mobilities.

Diffusion current is produced when carriers move from a region of high concentration to a region of low concentration because of a concentration gradient. In one dimension,

Main differences:

• Drift requires an electric field; diffusion requires a carrier concentration gradient.
• Drift is directional motion superimposed on random thermal motion; diffusion results from unequal random carrier flow.
• Drift depends on mobility, whereas diffusion depends on the diffusion coefficient.
• Both mechanisms may occur simultaneously in a nonuniform semiconductor.

Consider an electron of charge and mass in an electric field . The force and acceleration are

If is the mean relaxation time, the average velocity acquired between collisions is

The negative sign indicates that electrons drift opposite to the field.

For electron concentration , the current density is

Substituting the drift velocity,

Comparing this result with Ohm's law in microscopic form, , gives

The electron mobility is defined as the magnitude of drift velocity per unit electric field:

Therefore,

The resistivity is

Carriers in a semiconductor possess random thermal motion. If their concentration is nonuniform, more carriers cross from the high-concentration region to the low-concentration region than in the reverse direction. This net movement is called diffusion.

For variation along the -direction, the electron particle flux is

where is the electron diffusion coefficient. Since an electron has charge , the electron diffusion current density is

For holes,

and hence

The diffusion coefficient and mobility are connected through the Einstein relation:

Thus, mobility describes carrier response to an electric field, while the diffusion coefficient describes carrier response to a concentration gradient.

The Fermi energy, , is the energy of the highest occupied electron state in a solid at absolute zero.

For electrons confined in a three-dimensional volume , all available states are filled up to the Fermi wave vector at . Including the two possible spin states, the number of occupied states is

Therefore,

If is the electron concentration,

The free-electron energy is

At the Fermi surface,

Hence,

The corresponding Fermi temperature and Fermi velocity are

Because ordinary temperatures are generally much lower than in metals, only electrons close to participate significantly in thermal excitation.

The Fermi-Dirac distribution gives the probability that an available state of energy is occupied by an electron in thermal equilibrium:

Here, is the Fermi level, is Boltzmann's constant, and is absolute temperature.

At :

• For , ; all states are occupied.
• For , ; all states are empty.
• The distribution changes abruptly at .

At :

• Thermal energy excites some electrons from states below to states above it.
• The sharp step becomes a smooth transition over an energy range of a few .
• At ,
  
  at every nonzero temperature.

The distribution obeys the Pauli exclusion principle and is therefore appropriate for electrons, which are fermions.

The density of states, , is the number of allowed quantum states available per unit energy interval. Thus, gives the number of states between and .

For a three-dimensional free electron gas of volume ,

Per unit volume,

Qualitative features:

• The density of states is zero at .
• It increases as in three dimensions.
• It describes available states, not necessarily occupied states.
• The number of occupied states in an energy interval is
  
  where is the Fermi-Dirac distribution.

In a crystal, the density of states is arranged into allowed energy bands and is zero within forbidden energy gaps.

An isolated atom has discrete electronic energy levels. When a large number of atoms are brought together to form a crystal, their outer electron wave functions overlap and the electrons interact.

According to the Pauli exclusion principle, all electrons cannot occupy one identical quantum state. Each atomic energy level therefore splits into a very large number of closely spaced levels. For a crystal containing atoms, an atomic level may split into approximately levels. These closely spaced levels form an allowed energy band.

Energy intervals containing no permitted electron states are called forbidden energy gaps or band gaps. The two important bands are:

• Valence band: The highest band normally occupied by valence electrons at .
• Conduction band: The next higher band, in which electrons can move through the crystal and contribute to conduction.

The magnitude of the gap and the occupation of the bands determine whether a solid behaves as a conductor, semiconductor, or insulator.

Conductors:

• The highest occupied band is partially filled, or the valence and conduction bands overlap.
• Empty states are available very close to occupied states.
• Electrons can gain energy from a weak electric field and conduct readily.
• The effective band gap is approximately zero.

Semiconductors:

• The valence band is full and the conduction band is empty at .
• They have a small band gap, commonly of the order of .
• At finite temperature, some valence electrons cross the gap, producing conduction electrons and holes.
• Their conductivity generally increases with temperature.

Insulators:

• They also have a full valence band and an empty conduction band at .
• Their band gap is comparatively large, so ordinary thermal energy cannot excite many electrons into the conduction band.
• Their conductivity is consequently extremely low.

Thus, the decisive factors are band filling, band overlap, and the magnitude of .

In an intrinsic semiconductor, electrons and holes are generated in pairs, so

The equilibrium carrier concentrations are


where and are the effective densities of states of the conduction and valence bands.

Setting gives

Solving for the intrinsic Fermi level ,

Since

the expression can also be written as

If , the Fermi level lies exactly at the middle of the band gap. Otherwise, it is displaced slightly toward the band having the smaller effective density of states.

An -type semiconductor is produced by doping a pure semiconductor such as silicon with a pentavalent impurity such as phosphorus, arsenic, or antimony.

Four valence electrons of each donor atom form covalent bonds, while the fifth is weakly bound. The donor introduces an energy level slightly below the conduction-band edge . At ordinary temperatures, donor electrons are easily excited into the conduction band.

Important features:

• Electrons are the majority carriers.
• Holes are the minority carriers.
• The semiconductor remains electrically neutral because each donated electron leaves a positively charged immobile donor ion.
• Increasing donor concentration moves the Fermi level upward, closer to .

For a nondegenerate, fully ionized semiconductor with ,

Equivalently,

Thus, the Fermi level lies above the intrinsic level but normally remains inside the band gap.

A -type semiconductor is formed by doping a pure semiconductor with a trivalent impurity such as boron, aluminium, gallium, or indium.

The impurity has only three valence electrons and cannot complete four covalent bonds. It can accept an electron from a neighboring bond, thereby producing a hole. The impurity creates an acceptor level slightly above the valence-band edge .

Important features:

• Holes are the majority carriers.
• Electrons are the minority carriers.
• An acceptor becomes a negatively charged immobile ion after accepting an electron.
• Increasing acceptor concentration moves the Fermi level downward, closer to .

For a nondegenerate, fully ionized semiconductor with ,

It may also be expressed as

Therefore, the Fermi level lies below the intrinsic level and approaches the valence band as acceptor doping increases.

Intrinsic semiconductor:

• It is a chemically pure semiconductor.
• Electrons and holes are generated thermally in equal numbers:
  
• Its Fermi level lies approximately at the middle of the band gap.
• Its conductivity is
  

Extrinsic semiconductor:

• It is intentionally doped with donor or acceptor impurities.
• In -type material, electrons are majority carriers and .
• In -type material, holes are majority carriers and .
• Its Fermi level shifts toward the conduction band for -type doping and toward the valence band for -type doping.
• Its conductivity is
  

At thermal equilibrium, both types satisfy the mass-action law

Doping greatly increases the majority-carrier concentration and therefore usually makes extrinsic material more conductive than intrinsic material at the same temperature.

The distinction is based on the positions of the conduction-band minimum and valence-band maximum in crystal momentum or -space.

Direct band-gap semiconductor:

• The conduction-band minimum and valence-band maximum occur at the same value of .
• An electron can recombine with a hole while conserving momentum and emitting a photon directly.
• Radiative recombination is highly probable.
• Such materials are suitable for LEDs and semiconductor lasers.
• Examples include GaAs, InP, and GaN.

Indirect band-gap semiconductor:

• The conduction-band minimum and valence-band maximum occur at different values of .
• A phonon must participate to conserve crystal momentum during an optical transition.
• Radiative recombination is less probable, so light emission is inefficient.
• Examples include silicon and germanium.

Photon momentum is very small compared with electron crystal momentum. Therefore, direct transitions are nearly vertical on an - diagram, whereas indirect transitions require both a photon and a phonon.

An electron in a crystal experiences the periodic potential of the lattice. Its response to an external force is therefore generally different from that of a free electron. This response is described using the effective mass, .

The group velocity of an electron wave packet is

Differentiating with respect to time,

Under an external force , the semiclassical equation is

Therefore,

Comparing with gives

or

Thus, effective mass is determined by the curvature of the energy band:

• Large curvature corresponds to small effective mass and high carrier acceleration.
• Small curvature corresponds to large effective mass.
• Near a conduction-band minimum, curvature is positive and electron effective mass is positive.
• Near a valence-band maximum, electron effective mass is negative, motivating the hole description.

A completely filled valence band carries no net current because the contributions from states with opposite crystal momenta cancel. If one electron is removed from the band, an unoccupied state is created. This vacancy is called a hole.

When a neighboring valence electron moves into the vacant state, it leaves another vacancy behind. Repeated electron transitions make the vacancy appear to move through the crystal.

A missing electron of charge behaves electrically like a particle of charge . Near the valence-band maximum, the curvature is negative, so the electron effective mass is negative. Instead of describing many valence electrons with negative effective mass, the vacant state is represented by a hole with positive effective mass:

Under an electric field, holes drift in the direction of the field and contribute the current density

Thus, a hole is a quasiparticle representing the collective behavior of electrons in an almost full valence band.

Consider a rectangular semiconductor carrying current along the -direction. Let a magnetic field be applied along the -direction. A carrier of charge moving with drift velocity experiences the magnetic force

Carriers accumulate on one side, creating a transverse Hall field . At equilibrium, the electric and magnetic forces balance:

so

If the carrier concentration is , the longitudinal current density is

Hence,

The Hall coefficient is defined by

Therefore,

For electrons, , giving

For holes, , giving

If the specimen has width , thickness , and carries current , then . Since ,

The polarity of reveals the sign of the majority carriers.

In a Hall experiment, a known current is passed through a specimen of thickness placed in a perpendicular magnetic field . The transverse Hall voltage is measured.

The Hall coefficient is

Carrier type:

• A negative Hall coefficient indicates electrons as majority carriers, corresponding to -type conduction.
• A positive Hall coefficient indicates holes as majority carriers, corresponding to -type conduction.

Carrier concentration:

For conduction dominated by one carrier type,

for electrons, or

for holes.

Mobility:

If conductivity is measured independently, then for one carrier type

Using ,

where is resistivity.

Thus, a single Hall measurement combined with resistivity measurement provides major electrical properties of a semiconductor.

When both electrons and holes carry current, their Hall contributions oppose each other because they have opposite charges. In the weak-magnetic-field limit, the Hall coefficient is

Here, and are electron and hole concentrations, while and are their mobilities.

Consequences:

• The Hall coefficient is not generally equal to or when both carrier types are important.
• Its sign depends on the weighted terms and , not simply on whether or is larger.
• Because mobility is squared in the numerator, a less numerous but more mobile carrier can determine the Hall polarity.
• If , the Hall coefficient becomes zero even though the specimen conducts.

In strongly extrinsic material, minority-carrier effects are often negligible, and the expression reduces to the familiar one-carrier result.

The Hall effect has several scientific and engineering applications:

• Identification of carrier type: The sign of the Hall voltage distinguishes -type from -type material.
• Carrier concentration measurement: For one dominant carrier type, concentration is obtained from
  
• Mobility determination: Combining Hall coefficient and conductivity gives
  
• Magnetic-field measurement: Hall probes and gaussmeters determine magnetic flux density using
  
• Current sensing: The magnetic field around a current-carrying conductor can be measured without directly inserting a meter into the circuit.
• Position, proximity, and speed sensing: Hall sensors detect rotating magnets, gear teeth, and moving mechanical parts.
• Material characterization: Hall measurements help examine doping, carrier transport, and semiconductor uniformity.

Hall sensors are compact, reliable, and capable of contactless measurement, making them useful in automobiles, industrial control systems, and electronic devices.