Unit 1: Solid State Physics - Subjective Questions

Classical free electron theory, also called the Drude-Lorentz theory, treats the conduction electrons in a metal as a gas of free particles.

Main assumptions:

• Metals contain a large number of free electrons moving randomly throughout the crystal.
• Positive ion cores remain fixed at their lattice positions.
• Free electrons obey classical mechanics and Maxwell-Boltzmann statistics.
• Electrons move freely between two successive collisions.
• Collisions are instantaneous and are characterized by an average relaxation time .
• In the absence of an electric field, the average electron velocity is zero.

When an electric field is applied, the electron acquires an average drift velocity

The resulting current density is

where is the electron concentration and is the electrical conductivity.

Achievements:

• Explains electrical conduction and Ohm's law.
• Gives a qualitative explanation of thermal conductivity.
• Introduces useful concepts such as drift velocity, mobility, and relaxation time.

Limitations:

• Cannot correctly explain the electronic specific heat of metals.
• Predicts an incorrect temperature dependence of conductivity.
• Cannot explain why some solids are metals, semiconductors, or insulators.
• Fails to explain positive Hall coefficients observed in some materials.
• Ignores quantum mechanics and the periodic crystal potential.

Drift current is the electric current produced by the directed motion of charge carriers under an applied electric field.

For an electron of charge in an electric field , the force is

Therefore, its acceleration is

If is the mean relaxation time, the average drift velocity is

The magnitude of electron mobility is defined as

If is the free-electron concentration, the electron drift current density is

In a semiconductor containing electrons and holes, both carriers contribute to drift current. Thus,

where is the hole concentration and is the hole mobility.

Hence, the conductivity is

and the drift current density satisfies

Diffusion current is the current caused by the movement of charge carriers from a region of high carrier concentration to a region of low carrier concentration.

The movement occurs because charge carriers possess random thermal motion. A spatial concentration gradient produces a net carrier flow even when no external electric field is applied.

For one-dimensional transport, the electron and hole diffusion current densities are

and

where and are the electron and hole diffusion coefficients.

Drift current versus diffusion current:

• Cause: Drift current is caused by an electric field, whereas diffusion current is caused by a carrier concentration gradient.
• Requirement: Drift can occur in a uniformly doped sample; diffusion requires non-uniform carrier concentration.
• Dependence: Drift current depends on mobility and electric field, while diffusion current depends on diffusion coefficient and concentration gradient.
• Carrier motion: In drift, carriers acquire a directed average velocity; in diffusion, random thermal motion creates a net flow from high to low concentration.

The total current density is the sum of drift and diffusion components. For electrons and holes,

At absolute zero, electrons occupy all available quantum states from the lowest energy up to the Fermi energy .

For a cubical solid of volume , the number of allowed states inside a sphere of radius in wave-vector space, including the two possible spin states, is

On simplification,

If is the electron concentration, then

Therefore,

The energy of a free electron with wave vector is

At the Fermi surface, . Hence,

Substituting the value of gives

Using , the expression may also be written as

The corresponding Fermi temperature is

Thus, the Fermi energy depends on the electron concentration and is the highest occupied energy at .

The Fermi-Dirac distribution function gives the probability that an available electron state of energy is occupied at temperature .

It is given by

where is the Fermi energy or chemical potential, is Boltzmann's constant, and is the absolute temperature.

At :

• For , ; all states are occupied.
• For , ; all states are empty.
• The distribution changes abruptly at .
• Conventionally, is obtained as the limiting value.

At :

• Some electrons below gain thermal energy and move to states above .
• The sharp step near becomes smooth.
• States far below remain almost completely occupied.
• States far above remain almost completely empty.

At any nonzero temperature, setting gives

Therefore, the probability of occupation at the Fermi level is , independent of temperature. Only electrons within an energy range of a few around are significantly affected by temperature.

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

Formation of energy bands:

• Suppose identical atoms are initially far apart. Each atom has the same set of discrete energy levels.
• As the atoms approach one another, electrostatic interaction and wave-function overlap remove the degeneracy of each atomic energy level.
• Because of the Pauli exclusion principle, all electrons cannot occupy the same quantum state.
• Each atomic energy level therefore splits into approximately closely spaced levels.
• Since is extremely large, these levels appear nearly continuous and form an allowed energy band.

The energy ranges in which no electron states are permitted are called forbidden energy bands or band gaps.

The highest occupied or nearly occupied band is called the valence band, while the next higher allowed band is called the conduction band. The energy gap between them is

where is the conduction-band edge and is the valence-band edge.

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

The electrical behavior of a solid depends on the arrangement and occupancy of its valence and conduction bands.

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 contribute to conduction.
• The effective band gap is approximately zero.
• Examples include copper, silver, and aluminium.

Semiconductors:

• At , the valence band is full and the conduction band is empty.
• The two bands are separated by a small energy gap, typically of the order of .
• At ordinary temperatures, some electrons acquire sufficient thermal energy to enter the conduction band.
• Both conduction-band electrons and valence-band holes conduct current.
• Examples include silicon and germanium.

Insulators:

• The valence band is full and the conduction band is empty.
• The forbidden energy gap is large, commonly several electronvolts.
• Very few electrons can cross the gap under ordinary conditions.
• Their electrical conductivity is therefore extremely low.
• Examples include diamond, glass, and mica.

Thus, the principal distinction is based on band occupancy and the magnitude of

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

The group velocity of an electron wave packet is

Under an external force ,

Differentiating the group velocity with respect to time gives

Comparing this result with , the effective mass is

Physical significance:

• Effective mass represents the combined effect of the electron's actual mass and the periodic lattice potential.
• It allows crystal electrons to be treated as nearly free particles under external forces.
• A strongly curved relation gives a small effective mass and high carrier mobility.
• Near the bottom of a conduction band, , so the effective mass is positive.
• Near the top of a valence band, , so the electron effective mass is negative. Transport is then more conveniently described using positively charged holes with positive effective mass.

A hole is an unoccupied electron state in an otherwise nearly full valence band. It behaves as a mobile charge carrier having positive charge and a positive effective mass.

When a valence electron gains sufficient energy, it may move to the conduction band. This process creates:

• A free electron in the conduction band.
• A hole in the valence band.

A neighboring valence electron can move into the vacant state, leaving another vacancy behind. This sequence is equivalent to the hole moving in the direction opposite to the motion of the valence electrons.

In an applied electric field:

• Conduction-band electrons drift opposite to the field because their charge is negative.
• Holes drift in the direction of the field because they behave as positive charges.
• Despite their opposite directions of motion, both produce conventional current in the direction of the electric field.

The total drift current density is

where and are the electron and hole concentrations, and and are their mobilities.

A hole is not an independent fundamental particle. It is a useful quasiparticle representation of the collective behavior of electrons in a nearly filled valence band.

Consider a rectangular semiconductor carrying current along the -direction. Let a magnetic field be applied along the -direction. Charge carriers moving with drift velocity experience the Lorentz force

The carriers accumulate on one side of the specimen and establish a transverse Hall electric field . At equilibrium, the electric and magnetic forces balance:

Therefore,

For a material with one type of carrier of concentration , the current density is

Thus,

Substitution gives

The Hall coefficient is defined as

Hence,

For electrons, , so

For holes, , so

If the specimen has width and thickness , the Hall voltage across its width is

Since ,

Therefore,

The sign of the Hall voltage identifies the dominant carrier type, while its magnitude can be used to determine the carrier concentration.

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

The Hall coefficient is

and, for a material dominated by one carrier type,

Major applications:

• Identification of carrier type: A negative Hall coefficient indicates electron-dominated conduction, while a positive coefficient indicates hole-dominated conduction.
• Measurement of carrier concentration: For an electron-dominated material,

• Determination of mobility: If conductivity is known, the carrier mobility can be found from

• Magnetic-field measurement: Hall probes and gaussmeters use Hall voltage to measure magnetic flux density.
• Current sensing: The magnetic field generated by a current can be detected without direct electrical contact.
• Position and speed sensing: Hall sensors are used in motors, automotive systems, keyboards, and proximity detectors.

The Hall effect is particularly important in semiconductor physics because it provides direct information about the sign, concentration, and mobility of charge carriers.

An intrinsic semiconductor is a chemically pure semiconductor with no intentionally added impurity atoms. Silicon and germanium in pure form are common examples.

At , the valence band is completely filled and the conduction band is empty. At higher temperatures, thermal energy breaks some covalent bonds and excites electrons into the conduction band. Each excited electron leaves behind one hole.

Therefore, in an intrinsic semiconductor,

where is the intrinsic carrier concentration.

The electron drift current density is

and the hole drift current density is

The total current density is

Using ,

Since , the intrinsic conductivity is

The corresponding resistivity is

Intrinsic conductivity increases rapidly with temperature because the intrinsic carrier concentration increases approximately as

An extrinsic semiconductor is formed by adding a controlled amount of impurity to a pure semiconductor. This process is called doping.

-type semiconductor:

• It is produced by doping silicon or germanium with a pentavalent impurity such as phosphorus, arsenic, or antimony.
• Four impurity electrons form covalent bonds, while the fifth is weakly bound.
• The impurity introduces a donor level slightly below the conduction band.
• A small amount of energy moves the fifth electron into the conduction band.
• Electrons are the majority carriers and holes are the minority carriers.
• For complete donor ionization, when compensation is negligible.

-type semiconductor:

• It is produced by doping with a trivalent impurity such as boron, aluminium, gallium, or indium.
• One covalent bond remains incomplete, producing a hole.
• The impurity introduces an acceptor level slightly above the valence band.
• An electron from the valence band can occupy the acceptor level, leaving a mobile hole.
• Holes are the majority carriers and electrons are the minority carriers.
• For complete acceptor ionization, when compensation is negligible.

Doping does not give the semiconductor a net electric charge because the mobile carriers are balanced by ionized impurity atoms. At thermal equilibrium, both types obey the mass-action law

For a non-degenerate semiconductor in thermal equilibrium, the electron concentration in the conduction band is

and the hole concentration in the valence band is

For an intrinsic semiconductor,

and the Fermi level is denoted by . Therefore,

Taking natural logarithms and rearranging gives

Hence,

The effective densities of states are proportional to the carrier effective masses:

Thus,

If the electron and hole effective masses are equal, , and the intrinsic Fermi level lies exactly at the middle of the band gap:

The Fermi level represents the electron chemical potential and determines the equilibrium occupation of energy states.

For a non-degenerate semiconductor,

and

In an -type semiconductor:

For complete donor ionization and negligible compensation, . Therefore,

The Fermi level lies above the intrinsic level and moves toward the conduction band as donor concentration increases.

In a -type semiconductor:

For complete acceptor ionization and negligible compensation, . Therefore,

The Fermi level lies below the intrinsic level and moves toward the valence band as acceptor concentration increases.

Temperature dependence:

• At very low temperature, carriers remain bound to dopants; this is the freeze-out region.
• At moderate temperature, most dopants are ionized and the semiconductor shows extrinsic behavior.
• At high temperature, thermally generated electron-hole pairs dominate and the Fermi level approaches the intrinsic level .

Very heavy doping may move the Fermi level inside an allowed band, producing a degenerate semiconductor.

The distinction between direct and indirect band-gap semiconductors depends on the relative positions of the conduction-band minimum and valence-band maximum in wave-vector space.

Direct band-gap semiconductor:

• The conduction-band minimum and valence-band maximum occur at the same value of wave vector .
• An electron can recombine with a hole by emitting a photon without a significant change in crystal momentum.
• The transition is represented by an approximately vertical line in an diagram.
• Radiative recombination is efficient.
• These materials are suitable for LEDs and semiconductor lasers.
• Examples include gallium arsenide, gallium nitride, and indium phosphide.

The emitted photon energy is approximately

Indirect band-gap semiconductor:

• The conduction-band minimum and valence-band maximum occur at different values of .
• Electron-hole recombination requires both a photon and a phonon to conserve energy and momentum.
• Radiative recombination is less probable and less efficient.
• These materials are generally unsuitable for efficient light-emitting devices.
• Examples include silicon and germanium.

Both direct and indirect semiconductors can absorb light and be used in photodetectors or solar cells, but direct-gap materials usually absorb strongly over a much smaller thickness.

A solar cell is a semiconductor device that converts solar radiation directly into electrical energy through the photovoltaic effect.

Construction:

• A typical solar cell consists of a large-area junction, commonly made from silicon.
• The front surface has a thin semiconductor layer so that light can reach the depletion region.
• A metallic grid forms the front contact while allowing most of the light to enter.
• A continuous metal contact is provided on the rear surface.
• An antireflection coating reduces optical losses.
• A transparent protective layer protects the device from environmental damage.

Working principle:

• Photons with energy are absorbed and generate electron-hole pairs.
• Electron-hole pairs produced within or near the depletion region are separated by its built-in electric field.
• Electrons are driven toward the -side, while holes are driven toward the -side.
• This separation creates a photovoltage across the junction.
• When an external load is connected, electrons flow through the external circuit and deliver electrical power.

The illuminated current-voltage relation can be written as

where is the light-generated current and is the reverse saturation current.

Thus, a solar cell operates without external bias and converts incident photon energy into electrical output.

Under illumination, a solar cell supplies current to an external load. Its ideal current-voltage characteristic is

Important parameters:

• Short-circuit current : Current when . Ideally,

• Open-circuit voltage : Voltage when . It is

• Maximum power point: At voltage and current , the output power is maximum:

• Fill factor: It measures the squareness of the current-voltage curve:

• Conversion efficiency: If the incident optical power is , then

Efficiency is reduced by reflection, incomplete photon absorption, carrier recombination, series resistance, shunt leakage, and thermalization of carriers generated by high-energy photons.

Carrier transport in a semiconductor occurs through both drift and diffusion. At thermal equilibrium, these two processes can balance each other.

For electrons, the total current density is

At equilibrium, . Therefore,

The equilibrium carrier concentration varies with electrostatic potential according to the Boltzmann relation. Since , substitution and comparison give

Similarly, for holes,

and equilibrium gives

Hence, the general Einstein relation is

Physical meaning:

• Mobility describes carrier response to an electric field.
• The diffusion coefficient describes carrier spreading due to a concentration gradient.
• Both are related because they arise from the thermal motion and scattering of carriers.
• At equilibrium, an internal electric field can produce a drift current that exactly cancels diffusion current, resulting in zero net current.

The electrical behavior of a semiconductor changes significantly with temperature because temperature controls carrier generation and dopant ionization.

Intrinsic semiconductor:

The intrinsic carrier concentration is

Its conductivity is

As temperature increases, rises exponentially. Although carrier mobility generally decreases because of increased lattice scattering, the rapid increase in carrier concentration dominates, so conductivity increases.

Extrinsic semiconductor:

Its temperature behavior has three regions:

• Freeze-out region: At low temperature, dopant atoms are not fully ionized. Carrier concentration and conductivity are low.
• Extrinsic region: At moderate temperature, nearly all dopants are ionized. The majority-carrier concentration is approximately equal to the dopant concentration. Mobility decreases with temperature, so conductivity may decrease gradually.
• Intrinsic region: At high temperature, thermally generated electron-hole pairs exceed the dopant-generated carriers. The material behaves intrinsically and conductivity increases rapidly.

Unlike metals, semiconductors generally have a negative temperature coefficient of resistance because their conductivity usually increases as temperature rises.