Quantum Mechanics and Statistical Foundations

Introduction

Nanoelectronic devices operate at dimensions comparable to electron wavelengths, mean free paths and electrostatic screening lengths. Under these conditions, classical particle trajectories and bulk material parameters become insufficient. Electrons exhibit wave–particle duality, allowed energies become quantized, barriers can be penetrated by tunnelling, and the availability and occupancy of states depend strongly on dimensionality.

These effects govern quantum wells, nanowires, quantum dots, resonant-tunnelling diodes, nanoscale field-effect transistors and low-dimensional sensors. Their analysis requires three connected foundations: quantum mechanics determines permitted states, band theory describes electronic states in solids, and statistical mechanics determines how those states are occupied. Quantum capacitance then links the microscopic state spectrum to measurable device electrostatics.

Learning Objectives

After studying this chapter, the learner should be able to:

• apply wave–particle duality to nanoscale carriers;
• formulate and solve one-dimensional Schrödinger equations;
• derive confinement energies for infinite and finite quantum wells;
• calculate transmission through a rectangular tunnel barrier;
• derive density-of-states relations in zero, one, two and three dimensions;
• interpret bands, band gaps, effective mass and subbands;
• apply Fermi–Dirac statistics to carrier populations;
• calculate quantum capacitance and its series combination with electrostatic capacitance; and
• evaluate the assumptions and limitations of nanoscale device models.

1. Wave–Particle Duality

1.1 Photon and electron duality

For a photon of frequency ν and wavelength λ,

E = hν = hc/λ,

p = h/λ,

where E is energy, p is momentum, h is Planck’s constant and c is the speed of light in vacuum.

De Broglie extended the momentum–wavelength relation to material particles:

λ = h/p.

For a nonrelativistic electron with kinetic energy E,

p = √(2mE),

and therefore

λ = h/√(2mE).

Here m is the relevant inertial mass. In a crystal, a band effective mass may be used near a parabolic band extremum; the free-electron mass should not automatically be substituted.

Worked Example 1: Electron wavelength

Find the de Broglie wavelength of a free electron with kinetic energy 0.10 eV.

Using m₀ = 9.109 × 10⁻³¹ kg and E = 0.10 × 1.602 × 10⁻¹⁹ J,

λ = h/√(2m₀E) ≈ 3.88 nm.

A structural feature only a few nanometres wide is therefore capable of producing substantial interference and confinement for such an electron.

1.2 Wave packets and uncertainty

A perfectly monochromatic wave is delocalized. A localized particle is instead represented by a wave packet constructed from a range of wavevectors. Position and momentum uncertainties obey

Δx Δpₓ ≥ ℏ/2,

where ℏ = h/(2π). This is not merely instrumental uncertainty; it follows from the noncommuting operators x̂ and p̂ₓ.

Confining an electron to a smaller region increases its momentum spread and generally raises its minimum kinetic energy. This is the physical origin of confinement energy.

2. Schrödinger Equation and Boundary Conditions

2.1 Time-dependent equation

For a nonrelativistic particle in a scalar potential V(r,t),

iℏ ∂Ψ(r,t)/∂t = [−ℏ²∇²/(2m) + V(r,t)]Ψ(r,t).

The wavefunction Ψ is a probability amplitude. Its normalization requires

∫|Ψ|² dτ = 1

for a bound single-particle state. The probability current density for a constant real mass is

J = (ℏ/2mi)(Ψ∇Ψ − Ψ∇Ψ).

2.2 Time-independent equation

For a time-independent potential, let Ψ(r,t) = ψ(r)e^(−iEt/ℏ). Substitution gives

[−ℏ²∇²/(2m) + V(r)]ψ(r) = Eψ(r).

The equation is an eigenvalue problem. Bound-state boundary conditions select discrete energies, whereas extended systems generally support continuous or quasi-continuous spectra.

2.3 Effective-mass equation

Near a band extremum, a slowly varying envelope function may satisfy

−(ℏ²/2)∇·[m*⁻¹(r)∇ψ] + V(r)ψ = Eψ.

For an abrupt one-dimensional heterojunction, this form gives the commonly used BenDaniel–Duke conditions:

• ψ is continuous;
• (1/m*)dψ/dx is continuous.

Ordinary derivative continuity is valid when the mass is the same on both sides. The effective-mass approximation may fail for atomically abrupt potentials, very high carrier energies or strong intervalley coupling.

3. Quantum Confinement

3.1 Infinite square well

Consider V(x) = 0 for 0 < x < L and V = ∞ elsewhere. Inside the well,

d²ψ/dx² + k²ψ = 0,

where k² = 2m*E/ℏ². The general solution is

ψ(x) = A sin(kx) + B cos(kx).

The boundary condition ψ(0) = 0 gives B = 0. The condition ψ(L) = 0 requires

sin(kL) = 0,

so kₙ = nπ/L, with n = 1, 2, 3, … . Thus

Eₙ = n²π²ℏ²/(2m*L²),

ψₙ(x) = √(2/L) sin(nπx/L).

The ground-state energy is nonzero, and the energy spacing increases as L⁻². This result assumes an infinite barrier, constant effective mass and a parabolic band.

Worked Example 2: GaAs quantum well

For an ideal 5.0 nm GaAs well with electron effective mass m* = 0.067m₀,

E₁ = π²ℏ²/(2m*L²) ≈ 0.224 eV,

E₂ = 4E₁ ≈ 0.898 eV.

Real barriers are finite, so actual confinement energies are lower and the wavefunctions penetrate adjacent layers.

3.2 Finite square well: derivation

Consider a symmetric well of width L = 2a:

V(x) = 0 for |x| < a,

V(x) = V₀ for |x| ≥ a,

with 0 < E < V₀ and equal mass m on both sides. Define

k = √(2mE)/ℏ,

κ = √[2m(V₀ − E)]/ℏ.

Because the potential is symmetric, eigenstates have definite parity.

For an even state,

ψ = A cos(kx), |x| < a,

ψ = Be^(−κ|x|), |x| ≥ a.

At x = a, continuity of ψ and dψ/dx gives

A cos(ka) = Be^(−κa),

−Ak sin(ka) = −κBe^(−κa).

Dividing the second equation by the first produces

k tan(ka) = κ.

For an odd state, the interior solution is A sin(kx), and matching gives

−k cot(ka) = κ.

Introduce z = ka and P = a√(2mV₀)/ℏ. Since κa = √(P² − z²), the eigenvalue equations become

z tan z = √(P² − z²) for even states,

−z cot z = √(P² − z²) for odd states.

These transcendental equations are solved numerically. They also show why a finite well supports only a finite number of bound states. In one dimension, any nonzero attractive finite well supports at least one bound state, but additional states appear only when the well is sufficiently deep or wide.

For L = 5.0 nm, V₀ = 0.30 eV and m = 0.067m₀ throughout, P ≈ 1.82. Numerical roots give approximately z ≈ 0.99 for the even ground state and z ≈ 1.78 for an odd excited state. The corresponding energies are about 0.090 eV and 0.287 eV, respectively. The second state lies close to the barrier and is therefore weakly confined. These values are model-specific; mass discontinuity would require modified matching conditions.

3.3 Confinement in multiple dimensions

For a rectangular three-dimensional infinite box with side lengths Lₓ, Lᵧ and L_z,

Eₙₓₙᵧₙ_z = (π²ℏ²/2m*)[nₓ²/Lₓ² + nᵧ²/Lᵧ² + n_z²/L_z²].

Confinement in one direction produces a two-dimensional electron gas; confinement in two directions produces a quasi-one-dimensional nanowire; confinement in all three directions produces quantum-dot-like discrete levels.

4. Quantum Tunnelling

4.1 Rectangular barrier

Let

V(x) = 0 for x < 0 and x > a,

V(x) = V₀ for 0 ≤ x ≤ a,

with 0 < E < V₀ and equal mass throughout. The wavefunctions are

ψI = Ae^(ikx) + Be^(−ikx),

ψII = Ce^(κx) + De^(−κx),

ψIII = Fe^(ikx),

where k = √(2mE)/ℏ and κ = √[2m(V₀ − E)]/ℏ.

Continuity of ψ and dψ/dx at x = 0 and x = a provides four linear equations. Eliminating B, C and D in favour of A and F gives the transmission probability

T = 1/{1 + [V₀² sinh²(κa)]/[4E(V₀ − E)]}.

Because the incident and transmitted regions have equal potentials and masses, their group velocities are equal, so T = |F/A|². In a more general structure, transmission must include the ratio of transmitted to incident probability-current velocities.

For κa ≫ 1, sinh²(κa) ≈ e^(2κa)/4, yielding

T ≈ [16E(V₀ − E)/V₀²]e^(−2κa).

The exponential is dominant, but the prefactor should not be discarded when quantitative accuracy is required.

Worked Example 3: Barrier transmission

For an electron with E = 0.10 eV incident on a barrier of V₀ = 0.50 eV, width a = 1.0 nm and mass m₀,

κ = √[(V₀ − E)/(ℏ²/2m₀)] ≈ 3.24 nm⁻¹.

The exact rectangular-barrier expression gives

T ≈ 3.9 × 10⁻³.

Increasing the width to 2 nm would reduce the dominant exponential factor by e^(−2κ × 1 nm) ≈ 1.5 × 10⁻³ relative to the original value. Tunnelling is consequently extremely sensitive to layer thickness.

4.2 Device relevance and model extensions

Tunnelling enables resonant-tunnelling diodes, flash-memory programming, tunnel FETs and scanning tunnelling microscopy. It also causes gate leakage and direct source-to-drain leakage.

Real barriers may require:

• transfer-matrix or scattering-matrix methods for multilayers;
• the WKB approximation for smoothly varying barriers;
• nonequilibrium Green’s functions for open, biased nanodevices;
• image-force and band-nonparabolicity corrections; and
• inelastic phonon-assisted tunnelling models.

The WKB transmission exponent through classically forbidden turning points x₁ and x₂ is approximately

T ∝ exp{−2∫ₓ₁ˣ² √[2m*(x)(V(x) − E)]/ℏ dx}.

This approximation is unreliable near abrupt turning points unless connection formulas are used, and it does not accurately describe sharp resonances without further treatment.

5. Band Theory and Effective Mass

5.1 Bloch states

In a periodic crystal potential V(r + R) = V(r), Bloch’s theorem states

ψₙk(r) = uₙk(r)e^(ik·r),

where uₙk has the periodicity of the lattice. The allowed eigenvalues form bands Eₙ(k), separated in some materials by forbidden energy gaps.

Band gaps arise from quantum interference and coupling of states near Brillouin-zone boundaries. They should not be interpreted simply as classical regions in which electrons lack sufficient energy.

5.2 Group velocity and effective mass

The semiclassical group velocity is

v = (1/ℏ)∇ₖE(k).

The inverse effective-mass tensor is

(m*⁻¹)ᵢⱼ = (1/ℏ²)∂²E/(∂kᵢ∂kⱼ.

Near an isotropic conduction-band minimum,

E(k) ≈ E_c + ℏ²|k − k₀|²/(2m*).

Near a valence-band maximum, band curvature is negative for electrons. Describing the missing electron as a positively charged hole produces a positive hole effective mass and simplifies transport analysis.

Effective mass is a local band-curvature property, not generally the physical rest mass. Strongly nonparabolic bands require energy-dependent or multiband models.

5.3 Confinement-induced subbands

If motion is confined along z but free in x and y,

Eₙ(kₓ,kᵧ) = Eₙ + ℏ²(kₓ² + kᵧ²)/(2m*).

Each quantized confinement level Eₙ becomes the edge of a two-dimensional subband. Device current and charge must therefore be computed from subband-resolved states rather than from an unmodified bulk density of states.

6. Density of States

The density of states (DOS) D(E) is the number of available one-electron states per energy interval. It is determined by the spectrum and boundary conditions. In a fixed noninteracting band model, DOS is independent of occupancy; the occupation function determines how many available states are filled. Interactions, disorder or self-consistent electrostatics can alter the spectrum and therefore the DOS.

Assume a parabolic band

E = E₀ + ℏ²k²/(2m*)

and a total degeneracy g, which may include spin and equivalent valleys.

6.1 State counting

With periodic boundary conditions, each k-state occupies volume (2π)ᵈ in d-dimensional k-space per real-space measure. Thus, the number of states is obtained by multiplying the k-space region by g and dividing by (2π)ᵈ. Differentiating the integrated number N(E) gives D(E) = dN/dE.

6.2 Three-dimensional DOS

The number of states per physical volume inside a k-space sphere is

N₃D/V = [g/(2π)³](4πk³/3) = gk³/(6π²).

Using k = √[2m*(E − E₀)]/ℏ and differentiating,

D₃D(E)/V = [g/(4π²)](2m*/ℏ²)^(3/2)√(E − E₀),

for E ≥ E₀, and zero otherwise.

The DOS rises as the square root of energy above the band edge.

6.3 Two-dimensional DOS

The occupied k-space region is a disk. Per unit area,

N₂D/A = [g/(2π)²]πk² = gk²/(4π).

Since k² = 2m*(E − E₀)/ℏ²,

D₂D(E)/A = gm*/(2πℏ²),

for E ≥ E₀. The DOS is constant within each parabolic subband. For several subbands,

D₂D(E)/A = Σₙ [gₙmₙ*/(2πℏ²)]Θ(E − Eₙ),

where Θ is the unit step function.

6.4 One-dimensional DOS

In one dimension, states occur at positive and negative k. Per unit length,

N₁D/L = [g/(2π)](2k) = gk/π.

Differentiation gives

D₁D(E)/L = (g/π)√[m*/(2ℏ²)](E − E₀)^(−1/2),

for E > E₀. The inverse-square-root divergence at a subband edge is a van Hove singularity. Real broadening prevents a truly infinite measured DOS.

6.5 Zero-dimensional DOS

An ideal isolated quantum dot has discrete energies Eₙ:

D₀D(E) = Σₙ gₙδ(E − Eₙ).

Coupling to contacts, finite lifetime, disorder and temperature broaden the delta-like peaks.

6.6 Dimensional comparison

The energy dependence near a parabolic band edge is therefore

• 3D: D(E) ∝ √(E − E₀);
• 2D: D(E) is step-like and constant within each subband;
• 1D: D(E) ∝ 1/√(E − E₀);
• 0D: discrete spectral peaks.

These differences strongly influence absorption, charge storage, capacitance and transport.

7. Fermi–Dirac Statistics

7.1 Occupation probability

For noninteracting fermions in thermal equilibrium,

f(E) = 1/{1 + exp[(E − μ)/(kBT)]},

where μ is the chemical potential, kB is Boltzmann’s constant and T is absolute temperature. At any nonzero temperature,

f(E = μ) = 1/2.

At T = 0, μ becomes the Fermi energy E_F: all states below E_F are occupied and all above are empty, subject to the degeneracy already included in the DOS.

7.2 Carrier concentration

The electron concentration is

n = ∫ D_c(E)f(E)dE,

and the hole concentration is

p = ∫ D_v(E)[1 − f(E)]dE.

For a nondegenerate three-dimensional semiconductor, where E_c − μ is several kBT,

n ≈ N_c exp[−(E_c − μ)/(kBT)],

with

N_c = g_v·2(m_d*kBT/2πℏ²)^(3/2)

when g_v is valley degeneracy and the factor 2 represents spin. Here m_d* is the appropriate density-of-states mass. This Maxwell–Boltzmann approximation becomes inaccurate in degenerate channels and heavily doped semiconductors.

For a single ideal two-dimensional subband with constant DOS D₂D/A = gm*/(2πℏ²), integration gives

n₂D = [gm*kBT/(2πℏ²)] ln[1 + exp((μ − E₀)/(kBT))].

This closed result is especially useful for quantum wells and inversion layers.

7.3 Equilibrium versus transport

A single Fermi–Dirac distribution describes equilibrium. A biased ballistic device is open and generally has separate source and drain occupation functions. The Landauer current is

I = (gq/h)∫ T(E)[f_S(E) − f_D(E)]dE,

where g is explicitly defined channel degeneracy and T(E) is transmission. Care is required not to count spin or valley degeneracy both in g and in the number of transport modes.

8. Quantum Capacitance

8.1 Definition

Adding charge to a low-dimensional conductor changes not only its electrostatic potential but also its chemical potential because the DOS is finite. If n is the carrier number per area, the small-signal quantum capacitance per area is

C_q/A = q² ∂n/∂μ.

Since

n = ∫ D(E)f(E − μ)dE,

then

C_q/A = q²∫ [D(E)/A][−∂f/∂E]dE.

At zero temperature, −∂f/∂E approaches δ(E − E_F), giving

C_q/A = q²D(E_F)/A.

This expression assumes a stable noninteracting or suitably quasiparticle-renormalized system. Exchange and correlation can modify electronic compressibility and require a more careful capacitance interpretation.

8.2 Series combination

A gate dielectric capacitance C_ox and channel quantum capacitance C_q commonly appear in series:

1/C_total = 1/C_ox + 1/C_q,

so

C_total = C_oxC_q/(C_ox + C_q).

For C_ox = 2.0 fF and C_q = 0.50 fF,

C_total = (2.0 × 0.50)/(2.0 + 0.50) = 0.40 fF.

Quantum capacitance therefore limits gate control when it is smaller than the electrostatic capacitance.

8.3 Two-dimensional parabolic channel

At low temperature with the Fermi level above one occupied 2D subband,

C_q/A = q²gm*/(2πℏ²).

A larger effective mass increases DOS and quantum capacitance, but it can reduce carrier velocity. Device design must balance electrostatic charge capacity against transport performance.

Graphene provides a contrasting example. Near an ideal Dirac point its dispersion is approximately E = ±ℏv_Fk, producing a DOS proportional to |E| rather than a constant. Its quantum capacitance consequently becomes small near charge neutrality, though disorder and thermal broadening prevent an ideal zero in practical samples.

9. Applied Case: Quantum-Capacitance-Limited Gate Stack

Background

This is an illustrative design case for a nanoscale transistor with a single occupied two-dimensional channel. The channel has m* = 0.05m₀ and total degeneracy g = 2. A 2.0 nm HfO₂ dielectric with relative permittivity 20 separates the gate from the channel.

Problem

A designer estimates gate capacitance using only the oxide. Determine whether this is defensible when the channel is degenerate and only one parabolic subband is occupied.

Analysis

The oxide capacitance per area is

C_ox/A = ε₀ε_r/t_ox

= (8.854 × 10⁻¹² × 20)/(2.0 × 10⁻⁹)

≈ 0.0885 F m⁻².

For the two-dimensional channel,

C_q/A = q²gm/(2πℏ²) = q²m/(πℏ²)

≈ 0.0335 F m⁻².

The series capacitance is

C_total/A = [(0.0885)(0.0335)]/(0.0885 + 0.0335)

≈ 0.0243 F m⁻².

For a gate area of 100 nm × 1 μm = 1.0 × 10⁻¹³ m²,

C_ox ≈ 8.85 fF,

C_q ≈ 3.35 fF,

C_total ≈ 2.43 fF.

Approach and Outcome

The oxide-only model predicts 8.85 fF, more than three times the series-limited result. The design team should therefore include subband-resolved charge and quantum capacitance in compact and numerical models. If additional valleys or subbands become occupied, C_q will increase in steps, subject to thermal and disorder broadening.

Learning Points

• A high-permittivity dielectric does not guarantee equally high total gate capacitance.
• DOS, degeneracy and effective mass directly influence electrostatics.
• Occupied-subband assumptions must be checked over the intended voltage range.
• Capacitance, carrier velocity, leakage and interface quality must be optimized jointly.

10. Modelling Challenges and Professional Practice

10.1 Important limitations

Simple analytical models commonly neglect:

• atomistic crystal and interface structure;
• nonparabolic and anisotropic bands;
• electron–electron and electron–phonon interactions;
• disorder, roughness and trap states;
• self-consistent potential redistribution;
• open-boundary and nonequilibrium effects; and
• quantum coherence loss and contact resistance.

A calculated confinement energy or tunnelling current is only as reliable as the potential profile, effective mass and boundary conditions used.

10.2 Best-practice modelling workflow

A defensible nanoscale analysis should:

1. identify confinement dimensions and relevant energy scales;

2. choose a band model appropriate to material and carrier energy;

3. specify all degeneracy conventions;

4. solve Schrödinger and electrostatic equations self-consistently when charge significantly modifies the potential;

5. distinguish available states from occupied states;

6. use probability current, not amplitude alone, to define transmission;

7. test convergence with mesh, domain size and energy resolution;

8. compare analytical limits with numerical results; and

9. validate against experimental data while reporting parameter uncertainty.

Machine-learning surrogate models may accelerate quantum transport or material screening, but they should preserve physical constraints and be validated outside their training set. An accurate fit does not establish that a model obeys charge conservation or correct asymptotic behaviour.

10.3 Governance, safety and sustainability

Quantum and nanoscale models affect material selection, fabrication decisions and claims of device performance. Reproducible parameter reporting, uncertainty disclosure and traceable simulation workflows are therefore professional obligations. Material toxicity, critical-mineral use, process energy and end-of-life disposal should be considered alongside electrical performance. Models used in safety-critical communication systems require verification beyond nominal operating conditions.

11. Contemporary Developments

Current research extends these foundations to two-dimensional semiconductors, van der Waals heterostructures, topological materials, spin–orbit devices, single-electron circuits and coherent quantum interconnects. Atomistic tight-binding and first-principles methods resolve band and interface effects beyond effective-mass theory, while nonequilibrium Green’s-function methods combine quantum transmission with open contacts and scattering.

Despite this sophistication, wave mechanics, DOS, Fermi statistics and quantum capacitance remain indispensable. Advanced simulations are interpretable only when their limiting behaviour is connected to these foundations.

Chapter Summary

Nanoelectronic behaviour arises from the wave character and fermionic statistics of charge carriers. The Schrödinger equation determines permitted states, while boundary conditions create confinement energies and tunnelling probabilities. Finite wells require transcendental eigenvalue equations and exhibit wavefunction penetration; tunnel transmission depends exponentially on barrier width and decay constant.

Periodic crystal potentials produce energy bands, whose curvature defines effective mass. State counting in k-space yields distinct density-of-states relations: square-root in 3D, constant per subband in 2D, inverse-square-root in 1D and discrete in 0D. Fermi–Dirac statistics determine occupancy but must be distinguished from the DOS itself. Finite DOS produces quantum capacitance, which can substantially reduce the total gate capacitance of a nanoscale device. Reliable design requires explicit assumptions, consistent degeneracy counting, appropriate boundary conditions and validation of simplified models.

Key Terms

Bloch state: A crystal eigenstate consisting of a plane-wave factor multiplied by a lattice-periodic function.

Chemical potential: The thermodynamic energy associated with adding a particle; it appears in the Fermi–Dirac distribution.

Confinement energy: Energy increase caused by restricting a carrier’s spatial motion.

Density of states: Number of available electronic states per energy interval and specified physical measure.

Effective mass: A parameter derived from band curvature that describes carrier response within a band.

Envelope function: A slowly varying function used to describe carriers near a crystal band extremum.

Fermi energy: The chemical potential of a fermion system at zero temperature.

Fermi–Dirac distribution: Equilibrium probability that a fermionic state of specified energy is occupied.

Finite quantum well: A confining potential with barriers of finite height and penetrable wavefunctions.

Group velocity: Carrier velocity given semiclassically by ℏ⁻¹∇ₖE.

Quantum capacitance: Differential charge-storage capacity arising from the finite electronic DOS.

Quantum dot: A structure in which carriers are confined in all three spatial dimensions.

Quantum well: A structure confined in one direction but extended in two directions.

Subband: A continuum of states originating from a quantized confinement level.

Transmission probability: Ratio of transmitted to incident probability current.

Tunnelling: Quantum penetration through a classically forbidden potential region.

Uncertainty principle: Lower bound on the product of uncertainties of conjugate observables.

Van Hove singularity: A DOS feature caused by a stationary point or reduced-dimensional band edge.

Wavefunction: A complex probability amplitude whose squared magnitude gives probability density.

Review Questions

Short Answer Questions

1. Why does reducing the width of a quantum well increase its confinement energy?

2. State the boundary conditions for an effective-mass heterojunction in the BenDaniel–Duke model.

3. Distinguish density of states from state occupancy.

4. Why is two-dimensional DOS constant within an ideal parabolic subband?

5. Define quantum capacitance and state its zero-temperature relation to DOS.

Long Answer Questions

1. Derive the normalized eigenfunctions and eigenenergies of a one-dimensional infinite square well.

2. Derive the even- and odd-parity eigenvalue equations for a symmetric finite square well.

3. Derive the density of states for parabolic bands in one, two and three dimensions.

4. Explain how Bloch’s theorem, band curvature and effective mass support nanoscale device modelling.

5. Discuss the derivation, assumptions and device implications of transmission through a rectangular tunnel barrier.

Analytical / Application Questions

1. A 7 nm quantum well has m* = 0.08m₀. Calculate its first two infinite-well energies and discuss how finite barriers would modify them.

2. For a 1.5 nm barrier with V₀ − E = 0.30 eV and m* = 0.20m₀, estimate the tunnelling exponential e^(−2κa) and assess sensitivity to a 0.1 nm thickness error.

3. A gate stack has C_ox/A = 0.06 F m⁻² and C_q/A = 0.02 F m⁻². Determine the total capacitance per area and explain the design error produced by neglecting quantum capacitance.

Multiple Choice Questions with Answers

Question 1

A nonrelativistic electron’s kinetic energy is increased by a factor of four. How does its de Broglie wavelength change?

A. It doubles

B. It is halved

C. It becomes four times larger

D. It remains unchanged

Correct Answer: B. It is halved

Question 2

For an electron in a one-dimensional infinite square well, what is the ratio E₂/E₁?

A. 2

B. 3

C. 4

D. 8

Correct Answer: C. 4

Question 3

Under the BenDaniel–Duke effective-mass model, which quantities are continuous across an abrupt heterojunction?

A. ψ and m*dψ/dx

B. ψ and (1/m*)dψ/dx

C. dψ/dx and m*ψ

D. ψ/m* and dψ/dx

Correct Answer: B. ψ and (1/m*)dψ/dx

Question 4

For a thick rectangular barrier with fixed energy and barrier height, increasing the width by Δa changes the dominant tunnelling factor by which multiplier?

A. e^(−2κΔa)

B. e^(−κΔa/2)

C. e^(2κΔa)

D. 1 + 2κΔa exactly

Correct Answer: A. e^(−2κΔa)

Question 5

What is the energy dependence of the DOS within one ideal two-dimensional parabolic subband?

A. It increases as √(E − E₀)

B. It decreases as 1/√(E − E₀)

C. It is constant above the subband edge

D. It consists only of delta functions

Correct Answer: C. It is constant above the subband edge

Question 6

At thermal equilibrium and nonzero temperature, what is f(E) at E = μ?

A. 0

B. 1/4

C. 1/2

D. 1

Correct Answer: C. 1/2

Question 7

Why are holes commonly assigned a positive effective mass near a valence-band maximum?

A. They are free protons inside the lattice

B. They represent missing electrons in a band with negative electron curvature

C. Their wavelength is always larger than an electron wavelength

D. The valence-band DOS is zero

Correct Answer: B. They represent missing electrons in a band with negative electron curvature

Question 8

An oxide capacitance of 2.0 fF is in series with a quantum capacitance of 0.50 fF. What is the total capacitance?

A. 0.40 fF

B. 0.50 fF

C. 1.25 fF

D. 2.50 fF

Correct Answer: A. 0.40 fF

Question 9

For a one-dimensional parabolic subband, how does the ideal DOS behave immediately above the subband edge E₀?

A. It is proportional to E − E₀

B. It is proportional to √(E − E₀)

C. It is constant

D. It is proportional to (E − E₀)^(−1/2)

Correct Answer: D. It is proportional to (E − E₀)^(−1/2)

Question 10

In the illustrative gate-stack case, C_ox/A = 0.0885 F m⁻² and C_q/A = 0.0335 F m⁻². What is the approximate series capacitance per area?

A. 0.122 F m⁻²

B. 0.0610 F m⁻²

C. 0.0243 F m⁻²

D. 0.00297 F m⁻²

Correct Answer: C. 0.0243 F m⁻²