Chapter 23: Band Theory, Semiconductors and Superconductivity (Set-1)
In the Kronig–Penney picture, why do energy bands form in a crystal
A Random atomic collisions
B Single isolated atoms
C Periodic lattice potential
D Vacuum boundary effects
In a crystal, electrons feel a repeating (periodic) potential from ions. Solving the wave equation with this periodicity gives ranges of allowed energies (bands) separated by forbidden gaps.
In a 1D periodic lattice, forbidden energy gaps mainly appear due to
A Bragg reflection
B Heat loss effects
C Electron charge decay
D Photon absorption only
At certain wavelengths, electron waves reflect strongly from lattice planes (Bragg condition). This causes standing-wave patterns and splits energies into two groups, creating a forbidden gap near zone boundaries.
In the nearly free electron idea, the lattice potential is treated as
A Very strong everywhere
B Small periodic perturbation
C Completely random noise
D Time-dependent field only
The model starts with free-electron energies and then adds a weak periodic potential. This small perturbation opens energy gaps at special k-values, especially at Brillouin zone boundaries.
Bloch theorem states that electron wavefunctions in a periodic potential are
A Purely constant waves
B Localized at one atom
C Plane wave × periodic
D Only real-valued waves
Bloch theorem says the wavefunction equals a plane-wave part times a periodic function with lattice periodicity. This explains how electrons can propagate while still “feeling” the crystal structure.
In band diagrams, the term “band edge” refers to
A Highest lattice temperature
B Center of energy gap
C Maximum impurity density
D Edge of allowed energy
A band edge is the boundary energy where an allowed band starts or ends. Near band edges, electron behavior is often approximated using curvature and effective mass ideas.
In the Kronig–Penney model, the main reason it is called “1D” is that it assumes
A Potential varies in one direction
B One electron only
C One lattice plane only
D Only one energy level exists
The model simplifies the lattice to a repeating potential along one axis. Real crystals are 3D, so this 1D assumption makes the math easier but limits full accuracy.
In k-space, a Brillouin zone is defined using
A Real-space distances
B Atomic mass values
C Reciprocal lattice vectors
D Magnetic field lines
Brillouin zones are regions in reciprocal (k) space built from reciprocal lattice geometry. Zone boundaries are where Bragg reflection conditions occur and energy gaps typically open.
Energy gaps tend to open most strongly at
A Random k-values
B Zone boundary points
C Very large temperatures
D Zero impurity limit
At zone boundaries, electron wavevectors satisfy Bragg conditions, causing strong mixing of states k and k±G. This mixing splits energies and produces a gap.
The reduced zone scheme is mainly used to
A Fold bands into first zone
B Remove lattice vibrations
C Increase electron charge
D Eliminate forbidden gaps
In the reduced zone scheme, all k-values are mapped into the first Brillouin zone. This makes it easier to compare bands and identify band gaps and band edges.
The extended zone scheme mainly helps to
A Convert metals to insulators
B Stop electron scattering
C Show bands without folding
D Remove Fermi level concept
The extended zone scheme keeps k increasing across multiple zones. It shows how energy varies continuously with k, while gaps appear at boundaries between zones.
The effective mass of an electron in a band is determined mainly by
A Electron spin only
B Crystal color
C Number of neutrons
D Band curvature d²E/dk²
Effective mass comes from how energy changes with k. A highly curved E–k relation means small effective mass, while flatter curvature gives larger effective mass.
A negative band curvature near the top of a valence band leads naturally to the concept of
A Neutrino emission
B Proton conduction
C Hole carriers
D Lattice melting
Near the valence band top, electrons behave as if they have negative effective mass. It is simpler to describe transport using “holes,” which act like positive charge carriers.
Mobility in a semiconductor depends most directly on
A Effective mass and scattering
B Crystal color
C Neutron number
D Nuclear binding energy
Mobility increases when effective mass is small and scattering is weak. Scattering comes from phonons and impurities, so mobility usually decreases when temperature or impurity concentration rises.
Cyclotron resonance is useful because it can help measure
A Thermal conductivity
B Effective mass values
C Lattice spacing only
D Photon wavelength only
In a magnetic field, carriers move in circular paths with a frequency related to charge and effective mass. Measuring this frequency helps estimate the effective mass in a crystal.
A metal typically conducts well because it has
A Completely filled valence band
B Very large band gap
C Partially filled band
D No available electrons
In metals, the highest occupied band is partially filled or overlaps with the next band. Electrons can easily change energy slightly and move under an electric field.
An insulator is best described as having
A Zero electrons present
B Huge carrier mobility
C Partially filled bands
D Large forbidden gap
Insulators have a large energy gap between valence and conduction bands. At ordinary temperatures, very few electrons gain enough energy to reach the conduction band.
In band theory terms, a typical semiconductor has
A Small band gap
B Infinite band gap
C No valence band
D Overlapping wide bands
Semiconductors have a moderate gap (smaller than insulators). Thermal energy can excite some electrons across the gap, producing measurable conductivity that increases with temperature.
The Fermi level indicates the energy where the probability of occupation is
A Exactly zero always
B Always 100%
C 50% at equilibrium
D Unrelated to temperature
At thermal equilibrium, the Fermi–Dirac function gives occupation probability 1/2 at the Fermi energy. Its position relative to band edges strongly affects carrier concentration.
In an intrinsic semiconductor, the Fermi level is located approximately
A Deep in conduction band
B Near mid-gap
C Far below valence band
D Outside band diagram
For a pure (intrinsic) semiconductor, electron and hole concentrations are equal. This usually places the Fermi level close to the middle of the band gap.
Adding a donor impurity to silicon most directly increases the number of
A Electrons only
B Holes only
C Neutrons only
D Photons only
Donor atoms contribute extra electrons close to the conduction band. These electrons are easily released at room temperature, making electrons the majority carriers in n-type material.
A p-type semiconductor is produced by adding impurities that create
A Donor levels near CB
B No levels at all
C Acceptor levels near VB
D Only deep trap levels
Acceptor impurities capture electrons from the valence band, leaving behind holes. These holes act as majority carriers, increasing conductivity by hole transport.
In n-type material, the majority carriers are
A Electrons
B Holes
C Ions only
D Neutrons
Donor doping supplies extra electrons, so electrons dominate conduction. Holes are still present but in much smaller numbers, so they are minority carriers.
In p-type material, the minority carriers are
A Holes
B Phonons
C Excitons only
D Electrons
In p-type semiconductors, holes dominate conduction as majority carriers. Electrons still exist due to thermal generation, but their concentration is lower, making them minority carriers.
The Hall coefficient sign helps identify whether a sample is mainly
A Hot or cold
B Amorphous or crystal
C p-type or n-type
D Magnetized or not
Hall measurements show the sign of dominant charge carriers. A negative Hall coefficient usually indicates electron-dominated conduction (n-type), while positive indicates hole-dominated conduction (p-type).
In simple terms, drift current is caused mainly by
A Random diffusion only
B Electric field force
C Nuclear reactions
D Crystal melting
Drift current happens when an electric field pushes carriers, giving a net motion. This is different from diffusion current, which comes from carriers moving from high to low concentration.
Diffusion current in a semiconductor mainly arises due to
A Concentration gradient
B Temperature independence
C Vacuum polarization
D Zero resistance state
When carriers are more concentrated in one region, they spread toward lower concentration. This movement creates diffusion current even without an external electric field.
The mass action law (intro) is commonly written as n·p equals
A Zero always
B Infinite at low T
C Constant at equilibrium
D Same as mobility
In thermal equilibrium, the product of electron concentration n and hole concentration p stays nearly constant (about ni² for a given temperature), linking majority and minority carrier levels.
A key limitation of the Kronig–Penney model is that it
A Ignores periodicity
B Uses oversimplified potential
C Predicts no band gaps
D Requires no math
Kronig–Penney often uses idealized square wells/barriers in 1D. Real crystals have complex 3D potentials, so the model gives qualitative understanding rather than exact band structures.
In band theory, an electron’s acceleration in a crystal is related to
A Gravitational force only
B Nuclear spin only
C Photon pressure only
D Applied field and effective mass
Under an electric field, the electron’s response depends on band curvature through effective mass. A smaller effective mass means larger acceleration for the same applied field.
A direct band gap semiconductor is especially useful for
A Mechanical springs
B Permanent magnets only
C Efficient light emission
D Nuclear shielding
In direct band gap materials, electrons can recombine with holes while emitting a photon without needing extra momentum change. This makes them good for LEDs and laser diodes.
A common method to estimate a semiconductor band gap is by studying
A Optical absorption edge
B Color of metal surface
C Mass of sample
D Earth’s magnetic field
When photon energy exceeds the band gap, absorption rises strongly. By measuring the absorption edge versus wavelength or energy, one can estimate the band gap value.
In many semiconductors, conductivity generally increases with temperature because
A Band gap grows sharply
B Lattice becomes empty
C Carrier generation increases
D Electrons disappear
Heating provides energy to excite more electrons from valence to conduction band. More electrons and holes means more charge carriers, so conductivity usually rises with temperature.
Superconductivity is most clearly identified by
A Very high resistance
B Zero electrical resistance
C Strong heating effects
D Permanent voltage drop
Below a critical temperature, a superconductor shows no DC resistance, allowing persistent current. This behavior is linked to Cooper pairing and a superconducting energy gap.
The critical temperature Tc is the temperature below which a material
A Becomes paramagnetic
B Turns into insulator always
C Loses all electrons
D Enters superconducting state
Tc is the transition temperature. Below Tc, the material becomes superconducting with zero resistance and expels magnetic flux (Meissner effect), provided the applied field is not too strong.
The critical magnetic field Hc is the field above which superconductivity
A Becomes stronger
B Becomes identical to metal
C Is destroyed
D Causes extra band gaps
If the applied magnetic field exceeds a critical value, superconductivity breaks down and the material returns to the normal state. The critical field usually decreases as temperature approaches Tc.
The Meissner effect means a superconductor
A Expels magnetic flux
B Traps all flux always
C Amplifies magnetic field
D Produces heat strongly
When cooled below Tc in a weak field, a superconductor pushes magnetic field lines out of its interior. This perfect diamagnetism distinguishes a true superconductor from a simple perfect conductor.
Perfect diamagnetism in the Meissner state corresponds to magnetic susceptibility approximately
A +1
B −1
C 0
D +10
In the Meissner state, the internal magnetic field is nearly zero, so the material strongly opposes the applied field. This corresponds to χ ≈ −1 in ideal conditions.
A key difference between a superconductor and a perfect conductor is that only a superconductor
A Has electrons
B Conducts electricity
C Shows Meissner expulsion
D Has lattice structure
A perfect conductor could keep existing magnetic flux trapped, but superconductors actively expel flux when entering the superconducting state. This is a fundamental property of the superconducting phase.
Type I superconductors are characterized mainly by
A Two critical fields Hc1/Hc2
B Always mixed vortex state
C No Meissner effect
D Single critical field Hc
Type I superconductors show complete Meissner effect up to one critical field. Above that field, superconductivity collapses suddenly, and the material becomes normal.
Type II superconductors are notable because they
A Have mixed vortex state
B Never allow magnetic flux
C Have no critical current
D Work only at 0 K
Type II materials allow partial flux penetration between lower and upper critical fields. This mixed state contains vortices, which makes type II superconductors useful in strong magnetic field applications.
The lower critical field in type II superconductors is denoted by
A Hc only
B Hc2
C Hc1
D H0
Below Hc1, the superconductor shows near-complete Meissner expulsion. When the field exceeds Hc1, vortices begin to enter and the mixed state starts.
The upper critical field in type II superconductors is denoted by
A Hc1
B Hc2
C Hc only
D Hmax always
As magnetic field increases toward Hc2, vortex density increases and superconductivity weakens. Above Hc2, superconductivity is destroyed and the material becomes normal.
Flux pinning is important in type II superconductors mainly because it
A Prevents vortex motion
B Increases band gap
C Removes Cooper pairs
D Eliminates Tc value
Moving vortices cause energy loss and resistance-like behavior. Pinning holds vortices in place, helping maintain zero-resistance current in strong fields, which is crucial for magnets and power uses.
London penetration depth λ describes how magnetic field
A Grows inside indefinitely
B Becomes uniform instantly
C Decays exponentially inside
D Turns into electric field
Magnetic fields do not penetrate deeply into a superconductor. Instead, the field falls off exponentially from the surface, with characteristic distance called the penetration depth λ.
The London equations primarily relate superconducting current to
A Temperature only
B Neutron diffusion
C Crystal fracture stress
D Electric and magnetic fields
London equations connect supercurrent with electromagnetic fields and explain key features like Meissner expulsion and finite penetration depth. They describe how superconductors respond differently from normal conductors.
The isotope effect shows that Tc is related to
A Electron charge value
B Ionic mass in lattice
C Sample color changes
D External pressure only
In many conventional superconductors, changing isotopic mass changes lattice vibration frequencies. Tc shifts with mass, supporting the idea that phonons (lattice vibrations) play a role in pairing.
The isotope effect strongly supports the importance of
A Nuclear fusion inside
B Purely magnetic binding
C Electron-phonon coupling
D Gravity-driven pairing
If Tc depends on isotopic mass, lattice vibrations must influence superconductivity. This points to electron–phonon interaction as a key mechanism, consistent with conventional BCS superconductors.
A Cooper pair consists of two electrons with
A Opposite spin and momentum
B Same spin, same k
C Different charges always
D No interaction at all
Cooper pairs form when electrons pair up with opposite spins and opposite momenta, creating a bound state through phonon-mediated attraction. Many such pairs act together to produce superconductivity.
In BCS theory, the energy gap refers to
A Gap between two metals
B Distance between atoms
C Minimum energy to break pairs
D Voltage across capacitor
The superconducting energy gap is the energy needed to break a Cooper pair and create excitations. It explains zero resistance, low-temperature behavior, and changes in heat capacity at Tc.
Flux quantization in superconductors occurs because
A Flux lines are continuous
B Superconducting wavefunction is coherent
C Electrons become neutral
D Lattice stops vibrating
The superconducting state has a single coherent quantum phase. This forces magnetic flux through a closed superconducting loop to take discrete values, a key quantum signature linked to Cooper pairs