Chapter 23: Band Theory, Semiconductors and Superconductivity (Set-3)
In Kronig–Penney, increasing barrier height mainly tends to
A Shrink band gaps
B Remove periodicity
C Fix Fermi level
D Widen band gaps
Higher potential barriers reduce electron tunneling between wells, making states more localized. This strengthens splitting near Bragg conditions and typically increases forbidden gap width between allowed energy bands.
In a 1D lattice, the first zone boundary occurs at k equal to
A 0
B ±2π/a
C ±π/a
D ±1/a
For a 1D lattice with spacing a, the first Brillouin zone extends from −π/a to +π/a. At these boundaries, Bragg reflection causes strong mixing and energy gaps.
The “nearly free electron” dispersion without lattice potential is
A E ∝ k²
B E ∝ k
C E constant
D E ∝ 1/k
A free electron has kinetic energy E = ħ²k²/2m, a parabola in k-space. The periodic lattice potential mainly modifies this parabola near zone boundaries by opening gaps.
The reciprocal lattice vector magnitude in 1D is
A a/π
B 2π/a
C π/2a
D 1/2πa
In a 1D lattice of spacing a, reciprocal lattice vectors are integer multiples of 2π/a. These vectors determine Brillouin zone boundaries and which k-states can mix to form gaps.
An energy gap opens when free-electron states become
A Non-degenerate
B Charge neutral
C Degenerate
D Temperature fixed
At zone boundaries, energies of states k and k−G match (degenerate). The periodic potential couples them and splits their energies into two distinct levels, creating a forbidden gap.
In reduced-zone plotting, a conduction band minimum may appear at
A A zone boundary
B Only k=0
C Random energies
D Only outside zones
After folding into the first zone, band extrema can occur at the zone center or boundary depending on crystal symmetry and band folding. Many gaps and extrema occur near boundaries.
The sign of effective mass becomes negative where E(k) is
A Strongly convex
B Perfectly flat
C Strongly concave
D Not defined
Effective mass is inversely related to curvature d²E/dk². If curvature is negative (concave downward), effective mass becomes negative, leading to the convenient hole description for transport.
In semiconductors, diffusion coefficient D relates to mobility μ via
A Einstein relation
B Drude law only
C London equation
D Bragg condition
For carriers in thermal equilibrium, D and μ are connected by the Einstein relation D/μ = kT/q. It links random thermal motion (diffusion) with field-driven motion (drift).
The “density-of-states effective mass” is used mainly in calculating
A Crystal hardness
B Carrier concentrations
C Work function only
D Magnetic levitation
Carrier concentrations depend on density of states near band edges. The density-of-states effective mass accounts for how the band structure supplies available states, improving n(T) and p(T) estimates.
A metal can be modeled as an insulator if its highest band is
A Half-filled
B Partially empty
C Completely filled
D Overlapping bands
If the highest occupied band is completely filled and separated by a gap from the next band, electrons cannot easily accelerate into nearby states, so conduction is suppressed, like an insulator.
Temperature dependence of metal resistivity is mainly due to
A Phonon scattering
B Band gap changes
C Cooper pairing
D Donor ionization
In metals, carrier density is nearly constant, so resistivity changes mostly because scattering increases with lattice vibrations (phonons). Higher temperature increases phonon scattering, increasing resistivity.
A key difference between Drude and band picture is that band theory uses
A Random potentials
B No Fermi level
C Classical gas only
D Quantum allowed states
Drude treats electrons like a classical gas, while band theory uses quantum states in periodic potentials and the Fermi level. This explains insulators, semiconductors, and band gaps naturally.
In an intrinsic semiconductor at equilibrium, the Fermi level usually lies
A Inside conduction band
B Outside energy gap
C Near middle of gap
D Inside valence band
In an intrinsic semiconductor, electrons and holes are generated in equal numbers. This balance places the Fermi level close to the middle of the band gap, with only a very small temperature-dependent shift.
In a doped semiconductor at room temperature, most donors are
A Un-ionized
B Ionized
C Turned to phonons
D Converted to holes
Donor levels lie close to the conduction band, so thermal energy at room temperature can free most donor electrons into the conduction band, leaving behind positively ionized donor ions.
In n-type material, conductivity rises mainly because
A Electron concentration increases
B Band gap increases
C Work function drops
D Holes dominate more
Donor doping supplies extra electrons that enter the conduction band easily. Higher electron concentration increases σ = q(nμn + pμp), so conductivity increases even if mobility slightly decreases.
A Hall measurement gives carrier type because Hall voltage depends on
A Heat capacity
B Lattice spacing
C Sign of charge
D Photon energy
Hall voltage direction reverses when dominant carrier charge changes sign. Electrons produce one Hall polarity, holes the opposite, so the Hall coefficient sign identifies majority carrier type.
The built-in potential of a p–n junction forms due to
A Charge separation
B Nuclear forces
C Meissner expulsion
D Lattice breakage
Diffusion of electrons and holes across the junction leaves behind fixed ionized donors and acceptors. This creates an electric field and built-in potential that opposes further diffusion.
Under reverse bias, the depletion width generally
A Decreases
B Stays zero
C Increases
D Becomes negative
Reverse bias adds to the built-in potential, pulling majority carriers away from the junction. This expands the depletion region, reducing current except for small minority-carrier leakage.
Zener breakdown is favored when the junction is
A Undoped only
B Heavily doped
C Lightly doped
D Superconducting
Heavy doping makes the depletion region very thin. A strong electric field develops at modest reverse voltage, enabling quantum tunneling across the junction and causing Zener breakdown.
Avalanche breakdown is mainly caused by
A Quantum tunneling
B Band folding
C Cooper pairing
D Impact ionization
In avalanche breakdown, carriers accelerated by a strong reverse electric field collide with atoms and create additional electron–hole pairs. This multiplication sharply increases reverse current.
In superconductors, zero resistance implies DC electric field inside is
A Large and constant
B Always increasing
C Essentially zero
D Randomly fluctuating
With zero DC resistance, a steady current flows without needing a sustaining electric field. In the superconducting state, current persists without energy loss, indicating no voltage drop in DC.
Critical field concept is essential because magnetic field can
A Destroy superconductivity
B Increase band gap
C Remove lattice ions
D Create donor levels
Above the critical field, superconducting order is broken and the material becomes normal. Magnetic fields disrupt pairing and create energy costs that exceed the superconducting condensation energy.
A type I superconductor shows complete Meissner effect until
A Tc only
B Hc only
C Hc2 only
D Hc1 only
Type I superconductors maintain perfect flux expulsion and zero resistance up to a single critical field Hc. When H exceeds Hc, the material abruptly returns to the normal state.
In type II materials, magnetic flux first starts penetrating at
A Hc1
B Hc
C Tc
D Hc2
Below Hc1, the superconductor is mostly in the Meissner state. At Hc1, vortices begin entering, marking the start of the mixed state with partial flux penetration.
In mixed state, vortices arrange often into
A Random gas
B Depletion layers
C Vortex lattice
D Crystal grains
Quantized flux vortices can form an ordered lattice (often triangular) because of mutual repulsion and energy minimization. This vortex structure is a hallmark of type II superconductors.
Flux pinning improves performance because it reduces
A Work function
B Band curvature
C Donor ionization
D Vortex motion losses
If vortices move under current, they dissipate energy and create resistance. Pinning centers hold vortices in place, allowing larger currents without significant energy loss in type II materials.
London penetration depth describes screening current flowing mainly within
A Surface layer
B Vacuum region
C Entire bulk
D Only in core
Supercurrents flow near the surface to cancel interior magnetic fields. The magnetic field decays exponentially over the penetration depth λ, so most screening happens within a thin surface region.
If λ increases with temperature, it indicates
A Larger band gap
B Stronger pairing
C Weaker superfluid density
D Higher donor density
Penetration depth is inversely related to superconducting carrier density. As temperature approaches Tc, fewer carriers participate in the superconducting state, screening weakens, and λ becomes larger.
In conventional superconductors, isotope effect suggests Tc depends on
A Crystal color
B Phonon frequencies
C Fermi surface area
D Electron charge
Isotope substitution changes ionic mass and therefore phonon frequencies. Since Tc shifts with mass, lattice vibrations must influence pairing, supporting phonon-mediated attraction in conventional superconductors.
In BCS, pairing is strongest for electrons mainly near
A Vacuum level
B Deep core levels
C Fermi surface
D Donor level only
Cooper pairing involves electrons close to the Fermi energy because these states are available for pairing and scattering. The BCS energy gap opens around the Fermi surface region.
The superconducting energy gap is observed experimentally using
A Tunneling spectroscopy
B Crystal diffraction
C Work function test
D Hall effect only
Electron tunneling between a normal metal and a superconductor reveals a reduced density of states inside the gap. Measuring the voltage dependence helps estimate the superconducting energy gap.
A Josephson DC effect refers to
A Band gap widening
B Supercurrent at zero V
C Resistance jump
D Hall voltage reversal
In a Josephson junction, Cooper pairs can tunnel through a thin barrier, creating a steady supercurrent even when no voltage is applied. This is a direct consequence of phase coherence.
SQUID sensitivity arises mainly from combining Josephson junctions with
A Band overlap
B Avalanche breakdown
C Flux quantization
D Donor ionization
SQUIDs use interference of supercurrents in a loop and the quantized nature of magnetic flux. Tiny changes in flux produce measurable changes in current or voltage, giving extreme sensitivity.
In semiconductors, band gap typically decreases with rising temperature because
A Zones disappear
B Lattice expands
C Electrons vanish
D Phonon effects grow
Increased lattice vibrations and thermal expansion modify the electronic band structure, usually reducing the band gap. This affects optical absorption and intrinsic carrier concentration strongly with temperature.
Intrinsic carrier concentration increases rapidly with temperature mainly due to
A Exponential factor Eg/kT
B Mobility increase only
C Bragg reflection
D Work function rise
Intrinsic carriers follow roughly ni ∝ exp(−Eg/2kT). As temperature rises, the exponential suppression weakens dramatically, producing many more thermally generated electron–hole pairs.
If mobility falls with temperature in doped semiconductors, a common reason is
A Fermi level fixed
B Band gap vanishes
C More phonon scattering
D Donors deactivate
At higher temperatures, phonon vibrations increase, causing more collisions with carriers. This reduces the mean free time and lowers mobility, even while carrier concentration may increase.
In k-space, a symmetry point like X or L is important because it often marks
A Band extrema points
B Depletion width
C Nuclear levels
D Meissner temperature
High-symmetry points in the Brillouin zone frequently host conduction band minima or valence band maxima. These points influence effective mass, optical transitions, and carrier transport properties.
If conduction band minimum and valence band maximum occur at different k, the gap is
A Direct
B Metallic
C Zero
D Indirect
In an indirect band gap semiconductor, the momentum (k) values differ for band extrema. Optical transitions then require a phonon to conserve momentum, reducing light emission efficiency.
The “effective mass” concept is most valid near
A Deep core states
B Band extrema regions
C Vacuum outside solid
D Highly irregular bands
Near band edges, E(k) can be approximated as parabolic, making effective mass meaningful and useful. Far from extrema, bands may be non-parabolic and anisotropic, reducing accuracy.
In metals, the Fermi surface is important because it determines
A Transport properties
B Crystal melting
C Penetration depth
D Donor activation
Only electrons near the Fermi surface respond significantly to small fields and contribute to conduction. The shape and curvature of the Fermi surface influence conductivity, effective mass, and magnetoresistance.
In superconductors, “perfect diamagnetism” means internal B is
A Same as outside
B Very large
C Nearly zero
D Randomly changing
In the Meissner state, the superconductor expels magnetic flux so the internal magnetic field is nearly zero (for fields below critical). This strong diamagnetism enables shielding and levitation effects.
The condensation energy refers to energy difference between
A Normal and SC states
B Two junction types
C Two band edges
D Two isotopes
Condensation energy is the energy saved when electrons form the superconducting state instead of remaining normal. It represents the stability of superconductivity and relates to critical fields.
Type II superconductors usually have penetration depth compared to coherence length that is
A Exactly equal
B Much larger
C Much smaller
D Always zero
Type II behavior occurs when λ is relatively large compared to coherence length ξ (large κ = λ/ξ). This favors vortex formation and allows mixed state between Hc1 and Hc2.
The isotope exponent α in Tc ∝ M^−α is about 0.5 for many
A All semiconductors
B High-Tc cuprates
C Conventional superconductors
D Insulators only
Many conventional superconductors show α near 0.5, matching expectations when phonon frequency scales as M^−1/2. Deviations suggest more complex interactions or reduced phonon dominance.
Cooper pairs behave collectively because they form a
A Coherent quantum state
B Thermal phonon cloud
C Classical gas
D Random lattice defect
Cooper pairs are boson-like and condense into a single coherent quantum state with a common phase. This coherence enables zero resistance, flux quantization, and Josephson effects.
In BCS, the energy gap generally decreases to zero when temperature
A Approaches Hc2
B Approaches 300 K always
C Goes below 0 K
D Approaches Tc
The superconducting gap is largest at low temperature and decreases as temperature rises. At Tc, pairing disappears and the gap closes, marking the transition to the normal state.
In semiconductors, “carrier lifetime” mainly affects
A Work function value
B Recombination rate
C Critical field
D Brillouin zone size
Carrier lifetime is the average time a carrier exists before recombining. Longer lifetime means carriers persist longer, influencing photoconductivity, minority-carrier diffusion length, and device performance.
Diffusion length depends mainly on lifetime and
A Diffusion coefficient
B Work function
C Zone scheme
D Reciprocal vector
Diffusion length L ≈ √(Dτ), where D is diffusion coefficient and τ is lifetime. It measures how far carriers travel before recombining, important for junctions and photodiodes.
A semiconductor with very small Eg tends to behave more like a
A Vacuum
B Perfect insulator
C Semimetal/metal
D Superconductor only
When the band gap becomes very small, thermal excitation produces many carriers even at modest temperatures. In the extreme case of overlap, the material behaves metallic due to abundant carriers.
A key reason superconductors are used in MRI magnets is their ability to
A Raise work function
B Increase band gap
C Reduce Hall voltage
D Carry large currents
Superconductors can carry very large currents with negligible resistive loss, creating strong stable magnetic fields. Type II superconductors with good flux pinning are especially suitable for high-field MRI systems