Chapter 23: Band Theory, Semiconductors and Superconductivity (Set-4)
In Kronig–Penney, band width mainly increases when adjacent wells have
A Higher tunneling
B Lower tunneling
C No periodicity
D Infinite barriers
When barriers are lower or thinner, electron wavefunctions overlap more between neighboring wells. Greater overlap increases coupling, spreads energy levels into wider bands, and generally reduces localization effects.
In a 1D lattice, the second Brillouin zone starts just beyond
A k = 0
B k = ±2π/a
C k = ±π/a
D k = ±a/π
The first zone spans −π/a to +π/a. Immediately beyond these boundaries begins the second zone in the extended zone scheme, where free-electron-like parabolas repeat and gaps appear at boundaries.
The dispersion relation E(k) in a periodic lattice is important because it determines
A Crystal color
B Atomic weight
C Nuclear size
D Carrier velocity
Group velocity of an electron wavepacket is v = (1/ħ) dE/dk. Therefore, the slope of the band E(k) directly controls how fast carriers move for a given crystal momentum.
Near a band edge, treating E(k) as parabolic is useful because it gives
A Work function directly
B Constant effective mass
C Fixed carrier density
D Zero scattering
Close to band minima or maxima, E(k) often looks like a parabola. This makes d²E/dk² nearly constant, so effective mass becomes a good simple parameter for transport calculations.
For a conduction electron (charge −e) in a parabolic band, acceleration under electric field E is
A a = +eE/m*
B a = m*E/e
C a = −eE/m*
D a = 0 always
In a parabolic band, the carrier behaves like a free particle with effective mass m*. Using a = qE/m*, for an electron q = −e, so acceleration is opposite to the electric field direction.
In metals, states contributing most to electrical conduction are mainly
A Deep valence states
B Core electron states
C Empty vacuum states
D Near Fermi level
Only electrons near the Fermi level can change energy slightly and move into nearby empty states under an electric field. Deep filled states cannot contribute because transitions are blocked.
A semiconductor becomes more insulating at low temperature mainly because
A Band gap increases
B Mobility becomes infinite
C Carriers freeze out
D Zones disappear
At low temperature, donors and acceptors may not ionize fully and intrinsic generation drops sharply. With fewer free carriers, conductivity falls and the material behaves closer to an insulator.
The mass action law in thermal equilibrium for nondegenerate case is roughly
A np = ni
B np = ni²
C n/p = ni²
D np = Eg
In equilibrium, the product of electron and hole concentrations is approximately the square of intrinsic concentration, np ≈ ni². This links majority and minority carrier concentrations in doped semiconductors.
In an n-type semiconductor, increasing donor concentration generally makes minority holes
A Increase strongly
B Stay unchanged
C Become majority
D Decrease strongly
With more donors, electron concentration n rises. Since np ≈ ni² in equilibrium, p must decrease. Thus minority hole concentration drops as donor doping increases.
In a p-type semiconductor, increasing acceptor concentration generally makes minority electrons
A Increase strongly
B Stay unchanged
C Decrease strongly
D Become majority
Higher acceptor doping increases hole concentration p. By the mass action relation, np ≈ ni², electron concentration n must decrease, so minority electrons become even fewer.
For a p–n junction at equilibrium, net current is zero because drift and diffusion currents
A Exactly balance
B Both vanish always
C Add together
D Reverse randomly
Diffusion current arises from concentration gradients, while drift current arises from the built-in electric field. At equilibrium, these two components cancel for electrons and holes, giving zero net current.
The built-in electric field in depletion region points from
A p-side to n-side
B Along junction plane
C Random directions
D n-side to p-side
Ionized donors on the n-side are positively charged and ionized acceptors on the p-side are negatively charged. The electric field points from positive to negative, i.e., from n-side toward p-side.
In forward bias, the injected minority carriers primarily cause current by
A Ion motion
B Photon emission only
C Recombination processes
D Vortex movement
Forward bias injects electrons into p-side and holes into n-side. These minority carriers diffuse and recombine with majority carriers, sustaining a large current through continuous injection and recombination.
The ideal diode equation current depends exponentially on
A Applied voltage
B Temperature only
C Band gap only
D Work function only
In the ideal model, diode current follows I ≈ Is (e^(qV/kT) − 1). Forward voltage lowers the barrier and greatly increases carrier injection, producing exponential rise in current.
Zener breakdown voltage is typically lower in junctions that are
A Lightly doped
B Perfectly intrinsic
C Heavily doped
D Superconducting
Heavy doping makes the depletion region thin, so the electric field becomes very strong at relatively small reverse voltage. This enables tunneling and results in low Zener breakdown voltage.
In Hall effect, the Hall voltage arises due to carriers experiencing
A Electric force only
B Lorentz force
C Gravitational force
D Nuclear attraction
A magnetic field exerts Lorentz force q(v × B) on moving carriers, pushing them sideways. Charge separation builds until an opposing electric field balances the magnetic force, producing Hall voltage.
Cyclotron resonance frequency depends mainly on
A Band gap size
B Crystal color
C Junction width
D Effective mass
Cyclotron frequency is ωc = qB/m*. Measuring resonance frequency in known magnetic field B allows determination of effective mass, revealing band curvature effects in semiconductors and metals.
A negative Hall coefficient typically indicates majority carriers are
A Holes
B Ions
C Electrons
D Cooper pairs
Hall coefficient sign follows the dominant carrier charge. A negative Hall coefficient implies negative charge carriers dominate conduction, which is typical of electron-majority (n-type) materials.
In band theory, a “semimetal” is commonly described as having
A Huge band gap
B Slight band overlap
C Fully filled bands
D No valence band
Semimetals have small overlap between valence and conduction bands, creating both electrons and holes but in low concentrations. This gives conductivity higher than semiconductors but not like good metals.
Work function is most directly related to energy difference between
A VB and CB
B Hc1 and Hc2
C Tc and 0 K
D Fermi and vacuum
Work function is the minimum energy needed to take an electron from the Fermi level inside the solid to the vacuum level just outside the surface, important for emission and contacts.
In superconductors, the Meissner effect implies magnetic field inside becomes
A Nearly zero
B Stronger than outside
C Uniformly increasing
D Randomly rotating
Below Tc and for fields below critical values, superconductors expel magnetic flux from their bulk. This makes internal magnetic field nearly zero, showing perfect diamagnetism in the Meissner state.
If a material shows zero resistance but does not expel flux, it behaves like
A True superconductor
B Strong ferromagnet
C Perfect conductor
D Band insulator
Zero resistance alone does not guarantee superconductivity. True superconductors also exhibit Meissner expulsion when entering the superconducting state; a perfect conductor can trap existing flux without expelling it.
The critical field Hc for type I superconductors is best interpreted as the field where
A Flux pinning starts
B Vortices form lattice
C Gap becomes maximum
D SC state ends
In type I superconductors, superconductivity exists fully up to a single critical field. When applied field exceeds Hc, superconductivity collapses and the sample becomes normal throughout.
In type II superconductors, increasing field from Hc1 to Hc2 mainly increases
A Band gap width
B Vortex density
C Work function
D Donor activation
Between Hc1 and Hc2, more magnetic flux penetrates as vortices. As the field increases, vortices become denser until superconductivity vanishes at Hc2.
Flux quantization value in a superconductor is approximately
A h/e
B 2h/e
C h/2e
D e/h
Because charge carriers are Cooper pairs with charge 2e, magnetic flux through a superconducting loop is quantized in units Φ0 = h/2e. This is observed in rings and SQUIDs.
London penetration depth appears because magnetic fields are screened by
A Supercurrents
B Normal resistors
C Donor ions
D Phonon waves
Superconducting electrons form currents near the surface that oppose and cancel the applied magnetic field in the interior. This screening causes an exponential decay of field with depth λ.
A larger penetration depth generally means magnetic shielding is
A More effective
B Unchanged
C Impossible always
D Less effective
If λ is larger, magnetic fields penetrate deeper into the material. That means the superconductor screens less strongly over a given thickness, so shielding is reduced compared with smaller λ.
The isotope effect is strong evidence for phonons because isotope substitution mainly changes
A Electron charge
B Crystal symmetry
C Lattice mass
D Band overlap
Isotopes change ion mass without significantly changing electronic structure. The resulting Tc shift points to a role of lattice vibration frequency, supporting electron–phonon interaction in conventional superconductors.
In BCS theory, the energy gap exists in the electronic density of states around
A Core levels
B Fermi energy
C Vacuum level
D Donor level
The BCS gap opens near the Fermi surface, removing states in a small energy range around EF. This gap suppresses scattering at low energy, producing zero resistance and altered heat capacity.
The BCS ground state is best viewed as a coherent state of
A Free electrons only
B Donor ions
C Phonon gas
D Cooper pairs
In BCS theory, many Cooper pairs overlap and share a common quantum phase. This collective coherence produces superfluid-like flow of charge without dissipation and explains key superconducting phenomena.
A key transport distinction: mobility mainly measures
A Carrier lifetime only
B Band gap per kelvin
C Drift speed per field
D Flux per area
Mobility μ relates drift velocity to electric field: vd = μE. It reflects how easily carriers accelerate between scattering events and depends on effective mass and scattering mechanisms.
In semiconductors, conductivity σ is best written as
A σ = q(nμn+pμp)
B σ = Eg/kT
C σ = h/2e
D σ = λ/ξ
Total conductivity comes from both electrons and holes. Each contributes q·(carrier density)·(mobility). This formula explains why doping and temperature change σ through n, p, and μ.
If donor ionization is incomplete at low temperature, the behavior is called
A Mixed state
B Meissner state
C Extended zone
D Freeze-out region
At low temperatures, donors may not have enough thermal energy to release electrons into the conduction band. Carrier concentration drops sharply, causing reduced conductivity called freeze-out.
In the intrinsic region at high temperature, conductivity is dominated by
A Donor carriers
B Flux vortices
C Thermally generated pairs
D Band folding
At sufficiently high temperature, intrinsic carrier generation overwhelms dopant contribution. Many electron–hole pairs form, and both carrier types contribute significantly to conductivity.
A direct indicator of an indirect band gap is that optical transitions typically need
A A phonon
B No momentum change
C A magnetic vortex
D Zero temperature
If band extrema occur at different k-values, momentum conservation requires assistance. A phonon supplies or removes momentum, making radiative recombination less efficient than in direct-gap materials.
In k-space, the condition for Bragg reflection can be written as
A k = 0 always
B k = 2G
C k = G/2
D k = 1/G
Bragg reflection occurs when an electron wavevector satisfies 2k ≈ G, where G is a reciprocal lattice vector. This is exactly where zone boundaries lie and gaps commonly open.
Effective mass becomes “infinite” in the ideal limit when band curvature is
A Very large
B Negative large
C Strongly oscillating
D Zero (flat band)
Since 1/m* ∝ d²E/dk², if curvature approaches zero, m* becomes very large. Then carriers respond very weakly to an electric field, implying extremely low acceleration.
The superconducting critical field is related to condensation energy because higher condensation energy implies
A Lower critical field
B Higher critical field
C No Meissner effect
D No energy gap
Condensation energy measures how much energy is saved by being superconducting. A larger energy difference means a stronger superconducting state, requiring a larger magnetic field to destroy it.
In type II superconductors, hysteresis in magnetization is mainly linked to
A Band overlap
B Work function
C Flux pinning
D Donor activation
Pinning causes vortices to enter and leave differently when field is increased or decreased. This produces magnetic hysteresis and energy loss in cycling fields, a key practical behavior of type II materials.
For many type I superconductors, the thermodynamic critical field varies with temperature approximately as
A Hc0(T/Tc)
B Hc0[1+(T/Tc)^2]
C Hc0 constant
D Hc0[1−(T/Tc)^2]
The thermodynamic critical field decreases as temperature rises and becomes zero at Tc. A common approximation is a parabolic dependence, showing superconductivity weakens as the transition is approached.
In BCS, the isotope effect exponent near 1/2 arises because phonon frequency scales as
A 1/√M
B √M
C M²
D M⁰
Lattice vibration frequency roughly follows ω ∝ 1/√M for ionic mass M. If Tc depends on phonon frequency in conventional superconductors, this leads to Tc ∝ M^−1/2 approximately.
For a Josephson junction, applying a constant voltage produces an AC current with frequency proportional to
A Temperature
B Voltage
C Band gap
D Zone width
The AC Josephson effect gives an oscillation frequency f = (2e/h) V. Thus a fixed voltage creates a precisely related oscillation, used for voltage standards and microwave applications.
In superconducting shielding, effectiveness is reduced if sample thickness is
A Much larger than λ
B Much larger than ξ
C Much smaller than a
D Comparable to λ
If thickness is only about one penetration depth, magnetic fields can leak through significantly. For strong shielding, thickness should be several times λ so field decays strongly inside.
In conventional superconductors, the coherence length is typically
A Much smaller than a
B Equal to a
C Much larger than a
D Zero always
In conventional superconductors, Cooper pairs overlap strongly, so the coherence length is usually tens to hundreds of nanometers, much larger than the atomic spacing a, indicating a highly collective paired state.
In semiconductor transport, “drift” refers to motion caused primarily by
A Concentration gradient
B Electric field
C Magnetic expulsion
D Lattice periodicity
Drift motion occurs when an applied electric field accelerates carriers, producing net velocity and current. It differs from diffusion, where carriers move from high to low concentration without an external field.
“Diffusion” current is strongest when there is a large
A Electric field
B Magnetic field
C Concentration gradient
D Work function
Diffusion results from random thermal motion, but a concentration gradient creates net flow from high to low density. This produces diffusion current, especially near junctions and under carrier injection.
In band theory, a “partially filled” band implies many nearby empty states and therefore
A Poor conduction
B No electrons
C Infinite band gap
D Good conduction
With a partially filled band, electrons can easily move into nearby empty states with small energy changes. This allows acceleration under an electric field and leads to metallic-like conductivity.
A main practical method to measure Tc in lab is by observing
A Resistance drop
B Work function shift
C Band folding change
D Hall sign change
As the sample is cooled, resistance decreases gradually in normal state and then drops sharply to nearly zero at Tc. This electrical measurement is a standard way to identify superconducting transition.
If a superconductor is cooled below Tc in zero field, then a field is applied slowly below critical, the interior remains
A Strongly magnetized
B Full of trapped flux
C Nearly field-free
D Always normal metal
With slow application of a field below critical values, the material stays in the Meissner state and expels the applied field from its interior, leaving nearly zero internal magnetic field.
In conventional superconductors, isotope effect plus tunneling gap measurements together strongly support
A Purely classical model
B Nuclear pairing only
C Band gap conduction
D Electron–phonon pairing
Isotope effect links Tc to phonons, while tunneling shows an energy gap consistent with BCS predictions. Together they strongly indicate pairing arises from electron–phonon mediated attraction in conventional superconductors