Chapter 23: Band Theory, Semiconductors and Superconductivity (Set-5)
In the nearly-free electron model, the first band gap size is mainly proportional to
A Lattice temperature only
B Electron charge squared
C Fourier component |U_G|
D Atomic mass number
The periodic potential can be expanded in Fourier components U_G. At the zone boundary, degenerate states mix and split by an amount proportional to |U_G|, giving the first energy gap size.
In Kronig–Penney dispersion, an energy is allowed only when the Bloch condition gives
A |cos(ka)| > 1
B cos(ka)=0 always
C k = 0 only
D |cos(ka)| ≤ 1
In the Kronig–Penney model, the dispersion can be written as cos(ka) = f(E). Only when |f(E)| ≤ 1 does a real k exist, giving propagating Bloch states (allowed bands).
In a periodic potential, group velocity becomes zero at a band extremum because
A dE/dk = 0
B dk/dt = 0 always
C d²E/dk² = 0
D E becomes negative
Group velocity is v = (1/ħ)(dE/dk). At band minima or maxima, the slope dE/dk becomes zero, so group velocity vanishes even though electrons still have energy.
A negative effective mass near the valence-band top implies electron acceleration is
A Opposite to E
B Same direction as E
C Independent of E
D Always zero
With negative effective mass, the acceleration a = qE/m* points opposite to the usual direction. This is why hole transport is introduced: holes behave like positive carriers with positive effective mass.
In 3D, an anisotropic effective mass is properly described by a
A Single scalar mass
B Simple constant μ
C Zone boundary only
D Mass tensor
When band curvature differs along directions, effective mass depends on direction. Mathematically it is represented by a tensor linking force and acceleration, explaining different mobilities along crystal axes.
The density-of-states effective mass differs from conductivity mass mainly because it weights
A Number of available states
B Scattering time only
C Magnetic field strength
D Junction depletion width
Density-of-states mass is defined so the density of energy states near band edges matches the real band structure. Conductivity mass instead relates to acceleration and current response under fields.
For nondegenerate semiconductors, the electron concentration n depends on (Ec−Ef) mainly through a
A Linear term
B Constant offset
C Sinusoidal term
D Exponential term
In the nondegenerate limit, n ≈ Nc exp[−(Ec−Ef)/kT]. Small shifts of Ef relative to Ec can change n dramatically, making Fermi level position crucial in doped materials.
In an n-type semiconductor under equilibrium, the minority hole concentration p is approximately
A p ≈ n/ni²
B p ≈ Nc exp(+)
C p ≈ ni²/n
D p ≈ constant always
Using the mass action relation np ≈ ni², if doping makes n large, then p must drop as ni²/n. This strongly affects reverse saturation current in p–n junctions.
The built-in potential of a p–n junction increases when doping levels
A Increase on both sides
B Decrease on both sides
C Become exactly equal
D Become intrinsic only
Higher doping pushes Fermi levels further into their bands, increasing the difference between p-side and n-side Fermi levels before contact. This increases the built-in potential and electric field in depletion region.
Reverse saturation current in a diode is mainly controlled by
A Majority carrier flow
B Lattice periodicity
C Supercurrent tunneling
D Minority carrier supply
Under reverse bias, majority carriers are blocked, so current comes from thermally generated minority carriers reaching the junction. Therefore temperature and carrier lifetimes strongly control reverse saturation current.
The strongest evidence that a superconductor is not just a perfect conductor is
A Low resistivity trend
B Meissner expulsion
C High work function
D Band overlap presence
A perfect conductor could keep trapped magnetic flux. A true superconductor expels magnetic field when cooled below Tc (Meissner effect), showing it is a distinct thermodynamic phase.
For type II superconductors, the parameter κ equals
A λ/ξ
B ξ/λ
C Tc/Hc
D Hc1/Hc2
The Ginzburg–Landau parameter κ is the ratio of penetration depth λ to coherence length ξ. Large κ typically corresponds to type II behavior with stable vortices and mixed state.
A material is more likely type II if κ is
A Much less than 1
B Exactly zero
C Always negative
D Greater than 1/√2
In Ginzburg–Landau theory, the boundary between type I and type II occurs near κ = 1/√2. If κ exceeds this value, vortex state becomes energetically favorable, giving type II behavior.
In the mixed state, each vortex core is locally
A Fully superconducting
B Vacuum region
C Normal-like region
D Perfect diamagnet
A vortex contains a tiny core where superconducting order parameter is suppressed, behaving like normal metal. Around it, supercurrents circulate and confine a quantized magnetic flux line.
Flux pinning increases critical current mainly because it prevents
A Vortex motion
B Cooper pair formation
C Gap opening
D Phonon creation
Moving vortices dissipate energy and produce resistance. Pinning holds vortices fixed, allowing higher currents without vortex flow, which raises practical critical current in type II superconductors.
London first equation implies a constant electric field in a superconductor would produce
A Constant resistive current
B Linearly growing current
C No current change
D Instant current stop
London’s first equation gives ∂Js/∂t ∝ E. So a constant E would keep accelerating the supercurrent, making current increase with time. In steady superconducting DC, E becomes essentially zero.
Penetration depth λ depends inversely on square root of
A Atomic mass M
B Band gap Eg
C Depletion width W
D Carrier density ns
In London theory, λ² ∝ m/(μ0 ns q²). So larger superconducting carrier density gives smaller λ and stronger screening. As ns decreases near Tc, λ increases.
In an n-type semiconductor, the Seebeck coefficient is usually
A Negative
B Positive
C Zero always
D Random sign
In n-type material, electrons are majority carriers. Under a temperature gradient, electron diffusion sets up an electric field with a typically negative thermopower sign. In p-type, the sign is usually positive.
In conventional superconductors, isotope exponent can deviate from 0.5 mainly due to
A Nuclear spin flips
B Crystal melting
C Vacuum polarization
D Coulomb and strong-coupling
The simple 0.5 value is a weak-coupling approximation. Coulomb repulsion (pseudopotential effects) and strong electron–phonon coupling can modify how Tc depends on mass, causing α to differ from 0.5.
In BCS, the energy gap at zero temperature is roughly related to Tc by
A Δ0 ≈ kTc
B Δ0 ≈ 1.76 kTc
C Δ0 ≈ 10 kTc
D Δ0 ≈ 0.1 kTc
In weak-coupling BCS theory, the zero-temperature gap satisfies 2Δ0 ≈ 3.52 kTc. This relation is supported by many tunneling and spectroscopy measurements in conventional superconductors.
The BCS gap causes resistivity to vanish mainly because scattering processes need
A Energy above gap
B No energy change
C Only phonon momentum
D Only impurity charge
Low-energy excitations are suppressed because states within the gap are absent. With no available low-energy states for scattering, current-carrying pairs flow without dissipation in the superconducting condensate.
A Josephson junction can carry supercurrent through an insulator because of
A Drift current
B Minority carriers
C Avalanche ionization
D Cooper-pair tunneling
Cooper pairs have a coherent wavefunction that can tunnel through a very thin insulating barrier. This produces a supercurrent at zero voltage, a key Josephson effect used in SQUIDs.
The AC Josephson frequency is given by f = (2e/h) times
A Current
B Temperature
C Voltage
D Field
Applying a constant voltage causes the superconducting phase difference to evolve in time, producing an alternating supercurrent. The frequency is directly proportional to voltage: f = (2e/h)V.
In a SQUID, maximum sensitivity comes from interference of
A Supercurrents
B Phonon waves
C Depletion charges
D Band electrons only
SQUIDs use two Josephson junctions in a loop. Supercurrents split and recombine, creating interference that depends on magnetic flux. Tiny flux changes shift the interference pattern measurably.
In a doped semiconductor, “degenerate” doping means the Fermi level lies
A Mid-gap region
B At vacuum level
C At Hc2
D Inside a band
When doping is extremely high, the Fermi level moves into the conduction band (n+) or valence band (p+). Then Fermi–Dirac statistics must be used and behavior becomes metal-like.
For degenerate semiconductors, the nondegenerate exponential formula fails mainly because
A Bands vanish
B Pauli blocking important
C No impurities exist
D Phonons disappear
At high carrier density, many states are already occupied near the Fermi level. Pauli exclusion and full Fermi–Dirac statistics control occupancy, so simple Maxwell–Boltzmann exponential approximations become inaccurate.
In band theory, the “effective mass” can differ for electrons and holes because
A Charges differ always
B Temperature fixed
C Band curvatures differ
D Lattice becomes random
Effective mass depends on curvature of E(k). Conduction and valence bands generally have different curvature shapes, so electrons near conduction minimum and holes near valence maximum have different effective masses.
The sign of Hall coefficient in a two-carrier (electrons+holes) material depends on
A Only band gap
B Only crystal type
C Only phonon mass
D Relative contributions
When both electrons and holes contribute, Hall coefficient reflects the weighted balance of their densities and mobilities. A high-mobility minority carrier can dominate the Hall sign even if less numerous.
In a p–n junction, diffusion length mainly controls
A Minority carrier reach
B Brillouin width
C Work function value
D Critical field Hc
Diffusion length L = √(Dτ) sets how far minority carriers travel before recombination. It determines how injected carriers spread, affecting diode current, transistor action, and photodiode collection efficiency.
In an indirect-gap semiconductor, radiative recombination rate is lower because
A Fermi level fixed
B Band overlap occurs
C Phonon required
D Mobility infinite
Momentum conservation requires a phonon to assist electron–hole recombination. This extra requirement reduces probability of photon emission, so indirect-gap semiconductors like Si are poor light emitters.
In nearly-free electrons, a gap appears at k = G/2 because states k and −k are
A Always unrelated
B Only thermally created
C Always empty
D Coupled by lattice
At zone boundaries, states with wavevectors differing by reciprocal vector G have the same energy and are coupled by the periodic potential. This coupling splits energies and produces the band gap.
In a metal, the electronic specific heat at low temperature is proportional to
A T
B T²
C 1/T
D Constant
Only electrons within about kT of the Fermi level can be thermally excited. This gives electronic specific heat Ce ∝ T, unlike lattice heat capacity which behaves differently at low temperature.
In superconductors, electronic specific heat below Tc behaves roughly as
A Linear in T
B Exponentially small
C Constant large
D Always zero
The energy gap suppresses low-energy excitations. At temperatures well below Tc, few quasiparticles exist, so electronic specific heat drops rapidly, often approximately exponentially with decreasing temperature.
The “critical current” limit in superconductors arises because large current creates large
A Band gap
B Work function
C Hall voltage
D Magnetic field
Current generates magnetic field around the conductor. If this self-field exceeds critical values locally, it can destroy superconductivity or depin vortices, limiting the maximum current a superconductor can carry.
In type II superconductors, resistive behavior under current often begins when
A Vortices start moving
B Tc increases suddenly
C Gap becomes larger
D λ becomes zero
In the mixed state, vortices experience Lorentz force from current and can move if not pinned. Vortex motion dissipates energy, producing voltage and resistance-like behavior.
A “band edge effective mass” approximation is most accurate when the band is
A Highly nonparabolic
B Completely flat
C Nearly parabolic
D Randomly broken
Effective mass is derived from second derivative of E(k). Near band edges many bands are close to parabolic, so constant effective mass works well. Strong nonparabolicity reduces accuracy.
In crystals, the relation ħ dk/dt = qE describes
A London screening
B Josephson tunneling
C Meissner expulsion
D Semiclassical dynamics
This equation gives how crystal momentum changes under electric field. Combined with band dispersion E(k), it predicts carrier velocity and acceleration, forming the basis of semiclassical transport theory.
In band theory, an electron’s “crystal momentum” is directly related to
A Band gap Eg
B ħk value
C Work function Φ
D Critical field Hc
In Bloch states, k labels the wave-like periodic state, and ħk acts like momentum in semiclassical equations. It is conserved modulo reciprocal lattice vectors in scattering processes.
In a periodic potential, k and k+G represent
A Equivalent states
B Different energies always
C Different charges
D Different temperatures
Reciprocal lattice vectors reflect periodicity. Adding G changes the phase of the Bloch factor but represents the same physical state in the crystal, which is why band structures repeat in k-space.
In the reduced-zone scheme, bands appear “folded” because one maps all k to
A Real lattice cell
B Vacuum region
C Depletion layer
D First Brillouin zone
Reduced-zone plotting forces all wavevectors into the first Brillouin zone by subtracting reciprocal vectors. This folding makes band gaps and extrema easier to analyze within a single zone.
The “forbidden gap” in Kronig–Penney corresponds to energies where the Bloch wavevector becomes
A Real number
B Exactly zero
C Complex number
D Infinite always
In forbidden gaps, solutions do not propagate as traveling Bloch waves. The wavevector acquires an imaginary part, so the wave decays exponentially, indicating no allowed extended electronic states.
In a type I superconductor, the Meissner state remains stable when applied field H is
A Between Hc1–Hc2
B Above Hc2
C At Tc
D Below Hc
Type I superconductors have a single critical field Hc. For H < Hc, they remain in the Meissner state with flux expulsion. When H exceeds Hc, superconductivity collapses into the normal state.
A type II superconductor in Meissner state occurs only when H is
A Greater than Hc2
B Less than Hc1
C Between Hc1 and Hc2
D Equal to Tc
Type II superconductors fully expel magnetic flux only below the lower critical field Hc1. Above Hc1 vortices enter and mixed state begins, so interior no longer remains field-free.
In tunneling experiments, a “coherence peak” near gap edge appears because density of states
A Diverges near edges
B Becomes zero everywhere
C Turns constant
D Becomes negative
BCS theory predicts a sharp rise in quasiparticle density of states just above the gap energy. This produces peaks in tunneling conductance, providing strong evidence for the superconducting gap structure.
In the Drude model, conductivity σ depends on scattering time τ as
A σ ∝ 1/τ
B σ ∝ τ²
C σ independent
D σ ∝ τ
Drude gives σ = ne²τ/m. Longer scattering time means carriers accelerate longer between collisions, increasing conductivity. Band theory modifies m to effective mass but keeps τ role similar.
For carriers with smaller effective mass, mobility tends to be higher if τ is same because
A μ = m*/qτ
B μ = q/m*τ²
C μ = qτ/m*
D μ fixed always
Mobility relates drift velocity to electric field. In the simple model, μ = qτ/m*. If scattering time is unchanged, smaller effective mass leads to larger mobility and faster response to electric fields.
In superconductors, magnetic flux quantization directly supports that charge carriers have charge
A 2e
B e
C 3e
D 0
Flux quantum is Φ0 = h/2e. The factor 2e indicates the superconducting condensate behaves as particles of charge 2e, consistent with Cooper pairing of two electrons.
The main reason many high-Tc materials can work at liquid nitrogen temperature is their Tc is above
A 4 K
B 273 K
C 1 K
D 77 K
Liquid nitrogen boils at about 77 K. If a superconductor has Tc above 77 K, it can operate using cheaper, easier cooling compared with liquid helium, enabling broader practical use.
In superconducting devices, “surface impedance” becomes very low mainly because
A No carriers exist
B Dissipation is suppressed
C Band gap closes
D Flux becomes continuous
In the superconducting state, resistive losses drop sharply because quasiparticle excitations are reduced by the energy gap. This leads to very low surface resistance and low microwave surface impedance.
In a strongly anisotropic band, the effective mass along one axis can be much larger because E(k) changes
A More steeply
B Not at all
C Randomly
D More slowly
Effective mass is inversely related to curvature of E(k). If energy changes only slowly with k along one direction, curvature is small and effective mass becomes large along that axis, reducing mobility