Carrier Transport at the Nanoscale

Introduction and Context

Carrier transport describes how electrons and holes move through a solid, across an interface, or through a nanoscale device. In conventional semiconductor devices, transport is often modeled by drift and diffusion equations based on local carrier concentration, electric field, and mobility. As device dimensions approach the carrier mean free path, however, this local picture becomes incomplete. Carriers may cross a device without scattering, retain phase information, tunnel through classically forbidden regions, or become confined to discrete energy states.

Nanoelectronic transport therefore requires a hierarchy of models. The semiclassical drift–diffusion model is useful when many scattering events occur within the device. The ballistic and quasi-ballistic models apply when the device length is comparable with or smaller than the mean free path. Quantum-mechanical models become essential when confinement, tunneling, interference, or charging energy determines the current. Landauer transport provides a particularly useful framework because it relates conductance to the transmission probability of available quantum modes.

This chapter develops these models, identifies their assumptions, and explains how they are used in nanoscale transistors, nanowires, quantum dots, tunnel junctions, and interconnects.

Learning Objectives

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

1. Derive and interpret the drift–diffusion current equations.

2. Explain mobility, scattering, relaxation time, and mean free path.

3. Distinguish diffusive, ballistic, and quasi-ballistic transport.

4. Apply the Landauer formula to quantum conductors.

5. Explain tunneling through barriers and the dependence of transmission on barrier parameters.

6. Describe Coulomb blockade and derive the relevant energy and voltage conditions.

7. Select an appropriate transport model for a nanoscale device.

8. Evaluate practical limitations, variability, heating, and measurement issues in nanoscale transport.

1. The Transport Regime Is Set by Length Scales

The most important first step in transport analysis is comparison of characteristic lengths. Let \(L\) be the length of the active device region and let \(\lambda\) be the mean free path, the average distance traveled between momentum-randomizing scattering events.

• Diffusive transport: \(L \gg \lambda\). Many scattering events occur inside the device, and local equilibrium is approximately established.
• Ballistic transport: \(L \ll \lambda\). A carrier can cross the active region with negligible internal scattering.
• Quasi-ballistic transport: \(L\) is comparable to \(\lambda\). Some carriers cross without scattering, while others scatter one or more times.

Other relevant lengths include the phase-coherence length \(L_\phi\), screening length, de Broglie wavelength, and thermal wavelength. Quantum interference is important only when phase coherence persists over a significant part of the device. Electrostatic confinement is important when the device dimensions are comparable with the characteristic wavefunction extent. Charging effects become important when the electrostatic energy required to add one electron exceeds thermal broadening.

The transport regime can vary within one device. A long access region may be diffusive, a short channel may be ballistic, and a tunnel barrier may be dominated by evanescent-wave transmission. Consequently, a single global model is often less accurate than a coupled model using different descriptions in different regions.

2. Semiclassical Drift–Diffusion Transport

2.1 Drift current

In an electric field, carriers acquire an average drift velocity. For electrons,

\[

v_{n}= -\mu_n E,

\]

where \(\mu_n\) is electron mobility and \(E\) is the electric field. The negative sign reflects the electron charge. For holes, the conventional drift velocity is in the direction of the electric field:

\[

v_p=\mu_p E.

\]

The electron charge magnitude is \(q=1.602\times10^{-19}\,\mathrm{C}\). The electron current density due to drift is

\[

J_{n,\mathrm{drift}}=q n\mu_n E,

\]

and the hole current density is

\[

J_{p,\mathrm{drift}}=q p\mu_p E,

\]

where \(n\) and \(p\) are electron and hole concentrations in \(\mathrm{m^{-3}}\), and mobility is in \(\mathrm{m^2V^{-1}s^{-1}}\). Both conventional current densities have the same sign relative to the applied field.

The conductivity of a semiconductor is therefore

\[

\sigma=q(n\mu_n+p\mu_p),

\]

with resistivity \(\rho=1/\sigma\). This expression assumes local transport, well-defined mobility, and approximately linear response. At high electric fields, velocity saturation and carrier heating require more advanced models.

2.2 Diffusion current

A spatial concentration gradient produces diffusion. Electrons diffuse from regions of high electron concentration toward regions of low concentration. Because electrons carry negative charge, the conventional electron diffusion current is

\[

J_{n,\mathrm{diff}}=qD_n\frac{dn}{dx},

\]

where \(D_n\) is the electron diffusion coefficient. For holes,

\[

J_{p,\mathrm{diff}}=-qD_p\frac{dp}{dx}.

\]

Combining drift and diffusion gives the one-dimensional equations

\[

J_n=q n\mu_n E+qD_n\frac{dn}{dx},

\]

\[

J_p=q p\mu_p E-qD_p\frac{dp}{dx}.

\]

These equations assume nondegenerate statistics, local thermal equilibrium, and length scales sufficiently large that a local mobility and diffusion coefficient are meaningful.

2.3 Einstein relation

For nondegenerate carriers in thermal equilibrium, mobility and diffusivity are related by the Einstein relation:

\[

\frac{D_n}{\mu_n}=\frac{D_p}{\mu_p}=\frac{k_BT}{q},

\]

where \(k_B\) is Boltzmann’s constant and \(T\) is absolute temperature. At 300 K, \(k_BT/q\approx0.0259\,\mathrm{V}\). The relation does not generally apply without modification to degenerate electron gases, strongly hot carriers, or systems with nonparabolic bands.

2.4 Mobility and scattering

In a simple relaxation-time approximation,

\[

\mu=\frac{q\tau_m}{m^*},

\]

where \(\tau_m\) is the momentum relaxation time and \(m^*\) is the effective mass. Mobility is not a fundamental constant of a material under all conditions. It depends on temperature, impurity concentration, carrier density, surface roughness, strain, dimensionality, and electric field.

Important scattering mechanisms include:

• Ionized-impurity scattering, caused by charged dopants and defects.
• Lattice or phonon scattering, which generally increases with temperature.
• Neutral-defect scattering, caused by electrically neutral imperfections.
• Surface-roughness scattering, especially important in thin films and inversion layers.
• Interface-roughness and remote-phonon scattering, significant in heterostructures and high-k dielectric systems.
• Carrier–carrier scattering, which redistributes energy and momentum among carriers.

When independent mechanisms contribute, a commonly used approximation is Matthiessen’s rule:

\[

\frac{1}{\mu_{\mathrm{eff}}}=\sum_i\frac{1}{\mu_i}.

\]

This is an engineering approximation and may fail when scattering mechanisms are correlated or when transport is strongly nonlocal.

The mean free path can be estimated as

\[

\lambda=v\tau_m,

\]

where \(v\) is a representative carrier velocity. The appropriate velocity may be the thermal velocity, Fermi velocity, or another distribution-weighted velocity, depending on the carrier statistics.

3. From Diffusive to Ballistic Transport

3.1 Ballistic motion

In a ballistic channel, carriers are injected from contacts and traverse the channel with little or no internal scattering. Resistance does not disappear: contact injection and mode mismatch can still limit conductance. A ballistic device is therefore not necessarily a zero-resistance device.

The drift–diffusion equation becomes questionable when the carrier distribution at a point depends strongly on the contacts and cannot be described by a single local temperature and chemical potential. In this case, boundary conditions and transmission probabilities become central.

3.2 Quasi-ballistic transport

For a channel whose length is comparable with the mean free path, a useful engineering approximation is to express the effective mean free path as

\[

\lambda_{\mathrm{eff}}^{-1}=\sum_i\lambda_i^{-1},

\]

where each \(\lambda_i\) represents a distinct scattering process. A simple reduction factor often used in channel analysis is

\[

T_{\mathrm{sc}}\approx\frac{\lambda}{L+\lambda},

\]

where \(T_{\mathrm{sc}}\) is an approximate survival or transmission factor. This expression is model-dependent and should not be confused with the exact quantum transmission coefficient.

Quasi-ballistic transport produces several practical effects: apparent mobility becomes length-dependent, the source and drain influence the channel more directly, velocity overshoot can occur, and internal scattering may cause energy relaxation without complete momentum randomization.

3.3 Boltzmann transport equation

A more general semiclassical description is the Boltzmann transport equation for the distribution function \(f(\mathbf r,\mathbf k,t)\):

\[

\frac{\partial f}{\partial t}+\mathbf v\cdot\nabla_{\mathbf r}f+\frac{\mathbf F}{\hbar}\cdot\nabla_{\mathbf k}f=\left(\frac{\partial f}{\partial t}\right)_{\mathrm{coll}}.

\]

Here \(\mathbf v\) is group velocity, \(\mathbf F\) is force, and the collision term accounts for scattering. The drift–diffusion model can be derived as a low-field, near-equilibrium limit after taking suitable moments of this equation. In nanoscale devices, solving the full equation may be necessary when transport is nonlocal or strongly nonequilibrium.

4. Quantum States and Landauer Transport

4.1 Conductance as transmission

Landauer theory treats a conductor as a connection between reservoirs with electrochemical potentials \(\mu_L\) and \(\mu_R\). The current is

\[

I=\frac{2q}{h}\sum_n\int T_n(E)\,[f_L(E)-f_R(E)]\,dE,

\]

where \(h\) is Planck’s constant, \(T_n(E)\) is the transmission probability of mode \(n\), and \(f_L\) and \(f_R\) are Fermi functions in the two reservoirs. The factor 2 assumes spin degeneracy; it must be changed if spin degeneracy is lifted.

For small bias and approximately energy-independent transmission near the Fermi energy,

\[

G=\frac{2q^2}{h}\sum_n T_n(E_F).

\]

The conductance quantum per spin-degenerate mode is

\[

G_0=\frac{2q^2}{h}\approx77.5\,\mu\mathrm S,

\]

corresponding to a resistance of approximately \(12.9\,\mathrm{k\Omega}\) for one fully transmitting mode.

A perfectly transmitting single-mode channel has \(T_1=1\), giving \(G=G_0\). Partial transmission results from reflection, disorder, barriers, phonons, or imperfect contacts. Landauer resistance includes the finite resistance associated with reservoirs and interfaces; this is why a short ballistic conductor can exhibit measurable two-terminal resistance.

4.2 Modes in a nanowire

Confinement quantizes transverse motion. Each allowed transverse state acts as a transport mode. As gate voltage or Fermi energy changes, modes can open or close, producing conductance steps in sufficiently clean and cold nanostructures. At room temperature, disorder and thermal broadening may smear these steps.

For a multichannel conductor with weakly transmitting modes, the total conductance is the sum of the individual mode contributions. This differs conceptually from simply assigning a bulk conductivity to a very small cross-section.

4.3 Relation to semiclassical resistance

Landauer and drift–diffusion descriptions are not competing laws in all circumstances. In a long conductor with many modes and frequent scattering, the Landauer picture can reproduce an effective resistance that scales with length. Drift–diffusion is computationally simpler and often adequate. Landauer transport is more natural for short coherent or quasi-ballistic structures, especially when conductance quantization and contact resistance matter.

5. Quantum Tunneling

5.1 Physical basis

A classical particle with energy below a barrier cannot cross it. A quantum wavefunction, however, penetrates the forbidden region and produces a finite transmission probability. Tunneling is central to tunnel diodes, scanning tunneling microscopes, flash memory, resonant-tunneling devices, magnetic tunnel junctions, and leakage in aggressively scaled transistors.

For a rectangular barrier of height \(\Phi\) above the carrier energy and width \(a\), the approximate WKB transmission probability is

\[

T\approx\exp\left[-2\int_0^a\kappa(x)\,dx\right],

\]

where

\[

\kappa(x)=\frac{\sqrt{2m^*[V(x)-E]}}{\hbar}

\]

inside the classically forbidden region. For a uniform rectangular barrier this becomes

\[

T\approx\exp\left[-\frac{2a\sqrt{2m^*\Phi}}{\hbar}\right].

\]

The formula assumes a sufficiently smooth barrier and a semiclassical regime. It is an approximation, not an exact expression for all barrier shapes.

The exponential dependence means that a small change in width, height, effective mass, or electric field can produce orders-of-magnitude change in current. This sensitivity is useful for switching and sensing, but it also creates severe manufacturing variability.

5.2 Direct and Fowler–Nordheim tunneling

In direct tunneling, carriers traverse a relatively thin barrier without substantial barrier-shape deformation. Under a strong electric field, a triangular or trapezoidal barrier may result, leading to Fowler–Nordheim-type tunneling. The transition between regimes depends on barrier thickness, field, carrier energy, and material parameters.

A reliable analysis must use the actual band offsets, effective masses, image-force lowering, barrier shape, and electrode density of states. Treating every barrier as a rectangular barrier can lead to large errors.

6. Coulomb Blockade and Single-Electron Transport

6.1 Charging energy

A small conducting island connected through tunnel barriers has capacitance \(C_\Sigma\), the sum of capacitances to all nearby electrodes. Adding one electron changes the electrostatic energy by approximately

\[

E_C=\frac{q^2}{2C_\Sigma}.

\]

Coulomb blockade occurs when this charging energy is sufficiently larger than thermal energy:

\[

E_C\gg k_BT.

\]

A practical device also requires tunnel resistances large enough that charge remains localized and quantum fluctuations do not wash out the charge states. A commonly used condition is that each relevant tunnel resistance should be comparable to or greater than the resistance quantum scale \(h/q^2\), although the precise criterion depends on geometry and the desired blockade strength.

6.2 Single-electron transistor

A single-electron transistor consists of a small island connected to source and drain through tunnel junctions and capacitively coupled to a gate. The gate changes the island electrostatic potential. Current is suppressed over ranges of gate voltage, producing Coulomb oscillations or a stability diagram of conducting and blocked regions.

The electrostatic energy can be written in the constant-interaction approximation as

\[

U(N)=\frac{[-Nq+Q_0+C_gV_g+C_sV_s+C_dV_d]^2}{2C_\Sigma},

\]

where \(N\) is the number of excess electrons, \(Q_0\) is background charge, and \(C_g,C_s,C_d\) are gate, source, and drain capacitances. This model neglects detailed changes in wavefunction shape and interaction effects beyond a lumped charging energy.

At zero bias, conductance peaks occur when two neighboring charge states have equal energy. The gate-voltage period is approximately

\[

\Delta V_g=\frac{q}{C_g},

\]

provided the capacitances remain constant and the device is in the orthodox sequential-tunneling regime.

6.3 Limitations of Coulomb-blockade models

The orthodox model assumes well-defined charge states, weak tunnel coupling, rapid internal relaxation, and negligible coherent cotunneling. At very low temperatures or strong coupling, cotunneling, Kondo physics, level quantization, and nonequilibrium occupation may become important. Quantum dots may therefore require a combination of charging, wavefunction, and many-body models.

7. Worked Examples

Example 1: Diffusive conductivity and current density

A uniformly doped n-type semiconductor has electron concentration \(n=1.0\times10^{22}\,\mathrm{m^{-3}}\) and mobility \(\mu_n=0.10\,\mathrm{m^2V^{-1}s^{-1}}\). Neglect holes. Find the conductivity and current density for an electric field of \(1.0\times10^3\,\mathrm{V\,m^{-1}}\).

Using

\[

\sigma=qn\mu_n,

\]

\[

\sigma=(1.602\times10^{-19})(1.0\times10^{22})(0.10)=160.2\,\mathrm{S\,m^{-1}}.

\]

The current density is

\[

J=\sigma E=(160.2)(1.0\times10^3)=1.602\times10^5\,\mathrm{A\,m^{-2}}.

\]

The result is valid only in the low-field diffusive regime where mobility is approximately constant.

Example 2: Ballistic conductance

A nanowire supports two spin-degenerate modes, each with transmission \(T=0.80\). At low bias and low temperature,

\[

G=G_0\sum_nT_n=77.5\,\mu\mathrm S\times(0.80+0.80).

\]

Thus,

\[

G\approx124\,\mu\mathrm S,

\qquad R=1/G\approx8.06\,\mathrm{k\Omega}.

\]

This is a two-terminal Landauer result and includes contact-related resistance.

Example 3: Coulomb blockade temperature scale

An island has total capacitance \(C_\Sigma=1.0\,\mathrm{aF}=1.0\times10^{-18}\,\mathrm F\). Its charging energy is

\[

E_C=\frac{q^2}{2C_\Sigma}

=\frac{(1.602\times10^{-19})^2}{2(1.0\times10^{-18})}

\approx1.28\times10^{-20}\,\mathrm J.

\]

Dividing by \(k_B\), the equivalent temperature is

\[

\frac{E_C}{k_B}\approx928\,\mathrm K.

\]

The condition \(E_C\gg k_BT\) could therefore be satisfied at room temperature in an idealized device, although disorder, tunnel broadening, stray capacitance, and background charge may still prevent clear blockade.

8. Applied Case: Transport-Regime Selection in a Gate-All-Around Nanowire FET

Background

Consider an illustrative silicon gate-all-around nanowire transistor with a channel length of 20 nm, a nanowire diameter of 8 nm, and heavily doped source and drain contacts. The device is intended for low-power digital switching.

Problem

A conventional long-channel drift–diffusion simulation predicts current using bulk mobility and assumes local equilibrium. Experimental current, however, is higher than predicted at moderate gate bias, while the off-state current increases sharply when the gate dielectric is thinned.

Analysis

The channel length is comparable to typical room-temperature mean free paths in a high-quality nanoscale inversion layer. Transport is therefore quasi-ballistic rather than purely diffusive. Source injection and mode transmission must be included. The small cross-section also quantizes transverse states, although room-temperature broadening may obscure individual conductance steps.

The increased off-state current is not explained solely by mobility. It may result from direct source-to-drain tunneling, gate dielectric tunneling, reduced barrier height, and electrostatic drain-induced barrier lowering. A purely classical drift–diffusion model cannot accurately describe these processes.

Approach and solution

A practical multiscale workflow is:

1. Use a Poisson solver to determine the electrostatic potential and quantum-confined charge distribution.

2. Use a ballistic or quasi-ballistic Landauer model for the short channel.

3. Include source and drain contact transmission and an energy-dependent transmission function.

4. Add phonon and surface-roughness scattering through a nonequilibrium Green’s function or calibrated scattering model when necessary.

5. Calculate direct tunneling through the channel barrier and gate dielectric using a band-structure-based transmission model.

6. Calibrate uncertain interface and contact parameters against measured transfer and output characteristics.

Outcome

The combined model explains both observations: quasi-ballistic injection increases on-state current relative to a long-channel mobility model, while quantum tunneling increases off-state leakage. The design trade-off is not simply “higher mobility versus lower leakage.” A shorter channel may improve switching speed but reduce gate control and increase tunneling. Interface quality, dielectric thickness, contact resistance, and nanowire diameter must be optimized simultaneously.

Learning points

The case demonstrates that model selection must follow physical length and energy scales. It also shows why extracting a single “mobility” from a nanoscale transistor can be misleading: the measured current may contain contributions from injection, contact resistance, quantization, scattering, and tunneling.

9. Practical Measurement and Modeling Considerations

Two-terminal measurements combine channel, contact, and access resistance. Four-terminal measurements can separate some contributions, but nanoscale probe placement and invasive contacts remain difficult. Temperature-dependent measurements help distinguish phonon scattering, impurity scattering, thermionic emission, and tunneling. Magnetotransport can provide information about mobility, carrier density, and dimensionality.

Best practice includes:

• Identify all relevant length and energy scales before selecting a model.
• State whether resistance is channel, contact, or total two-terminal resistance.
• Use SI units consistently and distinguish mobility from transmission.
• Include temperature, degeneracy, and bias conditions in reported parameters.
• Use quantum corrections when confinement or tunneling is significant.
• Validate models against more than one observable, such as conductance, temperature dependence, and bias dependence.
• Report variability caused by surface roughness, defects, background charge, and process fluctuations.

10. Challenges, Reliability, and Emerging Directions

Nanoscale transport is highly sensitive to atomic-scale variation. A single defect can act as a scattering center, trap, resonant state, or charge fluctuator. Joule heating is also difficult to define locally because the lattice and electron temperatures may differ. In ballistic devices, energy may be dissipated mainly in reservoirs rather than uniformly in the channel.

Emerging directions include two-dimensional-material channels, topological and spin-dependent transport, molecular electronics, cryogenic CMOS, quantum-dot processors, and neuromorphic devices based on ionic or resistive switching. These systems require careful separation of coherent, incoherent, thermionic, and electrochemical mechanisms. Artificial intelligence can assist parameter extraction and compact-model fitting, but it cannot replace physical validation. A data-driven model trained outside its transport regime may produce plausible but incorrect predictions.

Ethically and professionally, transport models should not be used to claim device performance without specifying measurement configuration, uncertainty, temperature, and fabrication yield. For quantum and cryogenic technologies, reproducibility and calibration traceability are particularly important. Secure governance also requires protecting process data and clearly distinguishing experimentally demonstrated behavior from simulation projections.

Chapter Summary

Carrier transport changes fundamentally as device dimensions approach the mean free path, phase-coherence length, or quantum confinement length. Drift–diffusion theory describes local, near-equilibrium transport and leads to current equations containing drift and diffusion terms. Mobility is governed by scattering and depends on temperature, material quality, geometry, and carrier density.

When the channel becomes short, transport may be ballistic or quasi-ballistic. Contact injection, mode transmission, and nonlocal carrier distributions then become important. Landauer theory expresses conductance as the conductance quantum multiplied by the sum of transmission probabilities of available modes. Quantum tunneling produces exponentially sensitive transmission through barriers and is both a useful operating principle and a major leakage mechanism.

In small islands, charging energy can exceed thermal energy, producing Coulomb blockade and single-electron effects. No single model is universally correct: drift–diffusion, Boltzmann, Landauer, WKB, and Coulomb-blockade models apply in different physical regimes. Reliable nanoelectronic analysis therefore begins with scale identification, explicit assumptions, careful measurement separation, and validation across temperature, bias, and geometry.

Key Terms

Mean free path: Average distance traveled between momentum-randomizing scattering events.

Drift: Carrier motion caused by an applied electric field.

Diffusion: Carrier motion caused by a spatial concentration gradient.

Mobility: Proportionality between low-field carrier drift velocity and electric field.

Relaxation time: Characteristic time over which carrier momentum relaxes through scattering.

Quasi-ballistic transport: Transport in which device length and mean free path are comparable.

Ballistic transport: Transport with negligible internal scattering across the active region.

Transmission probability: Probability that a carrier or quantum mode crosses a structure.

Landauer formula: Quantum transport relation connecting current to reservoir distributions and mode transmission.

Conductance quantum: \(2q^2/h\) for a spin-degenerate mode.

Mode: An allowed quantum transport channel in a confined conductor.

Tunneling: Quantum transmission through a classically forbidden potential region.

WKB approximation: Semiclassical approximation used to estimate transmission through a varying barrier.

Charging energy: Electrostatic energy associated with adding an electron to a small island.

Coulomb blockade: Suppression of charge transfer because charging energy exceeds thermal and broadening effects.

Quantum dot: Nanoscale region in which carriers are confined in multiple dimensions.

Contact resistance: Resistance associated with injection, extraction, and interfaces between a device and its electrodes.

Degenerate semiconductor: Semiconductor in which carrier statistics require Fermi–Dirac rather than nondegenerate approximations.

Velocity saturation: High-field condition in which carrier velocity no longer increases linearly with electric field.

Nonequilibrium transport: Transport in which the carrier distribution cannot be described by a single local equilibrium distribution.

Review Questions

Short Answer Questions

1. What physical comparison distinguishes diffusive, ballistic, and quasi-ballistic transport?

2. State the electron and hole drift–diffusion current-density equations.

3. What is the physical meaning of the mean free path?

4. Why does a ballistic conductor still have finite two-terminal resistance?

5. State the condition for observing Coulomb blockade in terms of charging energy and thermal energy.

Long Answer Questions

1. Derive the drift–diffusion equations and explain the sign conventions for electron and hole currents.

2. Explain the role of scattering in determining mobility, mean free path, and the transition from diffusive to ballistic transport.

3. Derive the linear-response Landauer conductance and discuss the significance of conductance quantization.

4. Explain WKB tunneling through a rectangular barrier and discuss why tunneling is highly sensitive to barrier thickness.

5. Describe the operation of a single-electron transistor, including charging energy, gate control, and limitations of the orthodox model.

Analytical / Application Questions

1. A nanowire has three spin-degenerate modes with transmissions 0.90, 0.60, and 0.25. Calculate its low-bias conductance and two-terminal resistance using the Landauer model.

2. A device has channel length 30 nm and mean free path 10 nm. Use the approximate factor \(\lambda/(L+\lambda)\) to estimate the scattering transmission factor, and discuss why this is only an engineering approximation.

3. Compare two islands with capacitances 2 aF and 0.2 aF. Determine which is more suitable for room-temperature Coulomb blockade and explain the role of tunnel resistance and thermal broadening.

Multiple Choice Questions with Answers

Question 1

Which condition most strongly supports a ballistic transport model?

A. Device length much greater than mean free path

B. Device length much smaller than mean free path

C. Carrier concentration equal to zero

D. Very large contact resistance only

Correct Answer: B. Device length much smaller than mean free path

Question 2

The low-field electron drift current density is given by:

A. q n μn E

B. q E/(n μn)

C. q n Dn E

D. n μn/(qE)

Correct Answer: A. q n μn E

Question 3

At 300 K, the nondegenerate Einstein relation gives approximately:

A. D/μ = 0.0259 V

B. D/μ = 25.9 V

C. D/μ = 0.0259 A

D. D/μ = 300 V

Correct Answer: A. D/μ = 0.0259 V

Question 4

For one fully transmitting spin-degenerate Landauer mode, the conductance is approximately:

A. 12.9 μS

B. 77.5 μS

C. 155 μS

D. 1.60 × 10^-19 S

Correct Answer: B. 77.5 μS

Question 5

In the WKB approximation, increasing a rectangular barrier width generally:

A. Increases transmission linearly

B. Leaves transmission unchanged

C. Decreases transmission exponentially

D. Changes only the carrier charge

Correct Answer: C. Decreases transmission exponentially

Question 6

The charging energy of an island with total capacitance CΣ is:

A. qCΣ/2

B. q²/(2CΣ)

C. 2CΣ/q²

D. kBT/CΣ

Correct Answer: B. q²/(2CΣ)

Question 7

Which effect is most directly associated with a small island whose charging energy exceeds thermal energy?

A. Coulomb blockade

B. Velocity saturation

C. Ohmic breakdown

D. Classical equipartition only

Correct Answer: A. Coulomb blockade

Question 8

If two independent scattering mechanisms are represented by mobilities μ1 and μ2, a common engineering approximation is:

A. μeff = μ1 + μ2

B. 1/μeff = 1/μ1 + 1/μ2

C. μeff = μ1μ2

D. μeff = μ1/μ2

Correct Answer: B. 1/μeff = 1/μ1 + 1/μ2

Question 9

A major limitation of applying bulk mobility directly to a 20 nm transistor is that:

A. Mobility is always zero at nanoscale dimensions

B. Injection, contact effects, quantization, and nonlocal transport may dominate

C. Diffusion becomes independent of temperature

D. Electrons lose their charge

Correct Answer: B. Injection, contact effects, quantization, and nonlocal transport may dominate

Question 10

For two spin-degenerate modes with transmissions 0.5 and 0.25, the Landauer conductance is:

A. 0.25G0

B. 0.50G0

C. 0.75G0

D. 1.25G0

Correct Answer: C. 0.75G0