Sources and References

• Datta, S. Quantum Transport: Atom to Transistor. Cambridge University Press.
• Datta, S. Electronic Transport in Mesoscopic Systems. Cambridge University Press.
• Ferry, D. K.; Goodnick, S. M.; Bird, J. Transport in Nanostructures. Cambridge University Press.
• Lundstrom, M.; Guo, J. Nanoscale Transistors: Device Physics, Modeling and Simulation. Springer.
• Sze, S. M.; Ng, K. K. Physics of Semiconductor Devices. Wiley.
• Ashcroft, N. W.; Mermin, N. D. Solid State Physics. Saunders.
• Tsividis, Y.; McAndrew, C. Operation and Modeling of the MOS Transistor. Oxford University Press.
• Street, R. A. Hydrogenated Amorphous Silicon. Cambridge University Press.
• Ihn, T. Semiconductor Nanostructures: Quantum States and Electronic Transport. Oxford University Press.
• Kittel, C. Introduction to Solid State Physics. Wiley.
• Nathan, A.; Chalamala, B. R. Flexible Electronics: From Materials to Devices. World Scientific.
• Plummer, J. D.; Deal, M. D.; Griffin, P. B. Silicon VLSI Technology: Fundamentals, Practice, and Modeling. Prentice Hall.
• Taur, Y.; Ning, T. H. Fundamentals of Modern VLSI Devices. Cambridge University Press.

Chapter 1: From Bulk Semiconductor Electronics to the Nanoscale

1.1 Why Nano-Electronics Needs a Different Framework

Classical device physics, the kind that underlies long-channel MOSFET modelling, rests on three assumptions that slowly break as devices shrink. First, electrons are treated as semiclassical particles with a well-defined position and crystal momentum, moving under the influence of electric fields and bouncing off scattering centres. Second, the density of carriers is described by smooth Fermi-Dirac distributions inside each spatial region. Third, transport is diffusive: a carrier scatters many times on its way from source to drain, so averaged quantities like mobility \( \mu \) and drift velocity \( v_d = \mu E \) are meaningful. When the channel length becomes comparable to the mean free path, or when confinement wavelengths approach device dimensions, each of these assumptions needs to be re-examined.

Nano-electronics therefore borrows from two traditions. One is solid-state physics, which supplies band structure, effective-mass theory, and statistical mechanics of electrons in crystals. The other is mesoscopic physics, which provides a language for systems large enough to contain many atoms but small enough that quantum coherence and single-electron charging dominate. NE 471 sits at the intersection of these, treating nanoscale transistors (FinFETs, gate-all-around FETs, tunnel FETs, single-electron transistors, 2D-material FETs) and the fabrication techniques that make them possible.

1.2 Length Scales That Define the Physics

Several characteristic lengths mark transitions between regimes.

The de Broglie wavelength \( \lambda_{dB} = h/p \) at thermal energies \( p \sim \sqrt{2 m^{*} k_B T} \) determines whether confinement quantises the motion. In silicon at room temperature \( \lambda_{dB} \approx 8 \) nm; in GaAs 2DEGs at cryogenic temperatures it can exceed 40 nm.

The mean free path \( \ell = v_F \tau \) sets the scale over which a carrier’s momentum randomises. In high-mobility 2DEGs \( \ell \) can reach microns; in heavily doped silicon at room temperature it is closer to 5–10 nm.

The phase-coherence length \( L_\varphi \) is the distance over which the electron’s wavefunction retains a definite phase. Inelastic scattering (phonons, electron-electron) destroys phase memory. When \( L_\varphi \) exceeds device size, interference effects (weak localisation, Aharonov-Bohm oscillations, universal conductance fluctuations) appear.

The screening length \( \lambda_s \), or Debye/Thomas-Fermi length, governs how quickly a perturbing charge is screened. In metallic-density systems this is sub-nanometre; in lightly doped silicon it may be tens of nanometres.

The charging energy \( E_C = e^2/(2C) \) sets the scale at which adding a single electron to a small island matters. When \( E_C \gg k_B T \) and the tunnel resistance exceeds \( h/e^2 \approx 25.8 \) k\( \Omega \), Coulomb blockade governs transport.

1.3 Roadmap of the Course

We begin in Chapter 2 with quantum confinement and density of states, the starting point for 2DEGs, nanowires, and quantum dots. Chapter 3 develops band structure and effective-mass theory. Chapter 4 treats scattering and the classical Boltzmann transport equation. Chapter 5 introduces the Landauer-Büttiker formalism and the quantum of conductance, and Chapter 6 extends this to the non-equilibrium Green’s function (NEGF) framework. Chapter 7 surveys MOSFET scaling, short-channel effects, FinFETs, and gate-all-around geometries. Chapter 8 covers single-electron devices and resonant tunnelling. Chapter 9 looks at single-molecule transistors and 2D-material FETs. Chapter 10 treats large-area, flexible, organic, and amorphous-silicon electronics. Chapter 11 covers deposition, patterning, and metrology.

Chapter 2: Quantum Confinement and Density of States

2.1 The Particle in a Box

Consider an electron of effective mass \( m^{*} \) confined to a hard-wall well of width \( L \) in one dimension. Solving the time-independent Schrödinger equation gives stationary states with energies

\[ E_n = \frac{\hbar^2 \pi^2 n^2}{2 m^{*} L^2}, \quad n = 1, 2, 3, \dots \]

The spacing between the first two levels is \( 3 \hbar^2 \pi^2 / (2 m^{*} L^2) \). For an electron \( (m^{*} = m_0) \) confined to \( L = 5 \) nm, this spacing is roughly 0.09 eV, well above \( k_B T \) at room temperature (0.026 eV). For a GaAs conduction electron with \( m^{*} = 0.067 m_0 \) in a 10 nm well, the first excitation is about 0.17 eV. Confinement energies scale as \( 1/L^2 \), so shrinking from 10 nm to 3 nm raises the confinement energy by roughly an order of magnitude.

Real confining potentials are never hard-walled. A triangular well, appropriate for the channel near a MOSFET inversion layer, gives Airy-function solutions with eigenenergies

\[ E_n \approx \left( \frac{\hbar^2}{2 m^{*}} \right)^{1/3} \left( \frac{3 \pi e F}{2} \left( n - \frac{1}{4} \right) \right)^{2/3} \]

where \( F \) is the electric field perpendicular to the interface. A parabolic well gives the familiar harmonic-oscillator ladder \( E_n = \hbar \omega_0 (n + 1/2) \).

2.2 The 2DEG: Two-Dimensional Electron Gas

A heterostructure like GaAs/AlGaAs or a silicon MOS inversion layer confines electrons strongly in one direction (call it \( z \)) while leaving motion free in \( (x, y) \). The wavefunction factorises:

\[ \psi(x, y, z) = \frac{1}{\sqrt{A}} e^{i(k_x x + k_y y)} \chi_n(z) \]

where \( \chi_n(z) \) is the confined subband wavefunction and \( A \) is the lateral area. The total energy is

\[ E = E_n + \frac{\hbar^2 (k_x^2 + k_y^2)}{2 m^{*}} \]

The density of states per unit area per unit energy within a single subband is constant:

\[ g_{2D}(E) = \frac{m^{*}}{\pi \hbar^2} \]

(including spin). The total DOS is a staircase: each subband contributes a step of height \( m^{*}/(\pi \hbar^2) \) starting at \( E_n \). Most MOSFET channels at moderate inversion only populate the lowest subband, making the system effectively 2D.

2.3 Quantum Wires: 1DEG

Confining in two directions leaves transport in one. The DOS per unit length is

\[ g_{1D}(E) = \frac{1}{\pi \hbar} \sqrt{\frac{2 m^{*}}{E - E_{n,m}}} \]

for each transverse subband \( (n, m) \), and diverges as an inverse square root at each subband edge. These van-Hove singularities are the fingerprint of 1D transport.

2.4 Quantum Dots: 0DEG

Confining in all three directions produces a set of discrete energy levels, analogous to an artificial atom. The DOS is a sum of delta functions. Adding an electron to a small dot requires overcoming both the single-particle level spacing \( \Delta E \) and the charging energy \( E_C \). Dots fabricated from III-V semiconductors, silicon, or even colloidal nanocrystals exploit this discreteness for lasers, single-photon sources, and qubits.

2.5 A Unified View of the DOS

Collecting the spin-degenerate expressions, with \( E \) measured from the band (or subband) edge:

Dimensionality	Density of states	Scaling
3D	\( g_{3D}(E) = \frac{1}{2\pi^2} \left( \frac{2 m^{*}}{\hbar^2} \right)^{3/2} \sqrt{E} \)	\( \sqrt{E} \)
2D	\( g_{2D}(E) = \frac{m^{*}}{\pi \hbar^2} \) per subband	Step
1D	\( g_{1D}(E) \propto (E - E_{n,m})^{-1/2} \) per subband	Inverse-root singularity
0D	\( g_{0D}(E) = \sum_i 2 \delta(E - E_i) \)	Discrete

Chapter 3: Band Structure and Effective Mass

3.1 Why Band Structure Matters

In a crystal, the periodic potential forces electron states into Bloch form \( \psi_{n \mathbf{k}}(\mathbf{r}) = e^{i \mathbf{k} \cdot \mathbf{r}} u_{n \mathbf{k}}(\mathbf{r}) \) with energy bands \( E_n(\mathbf{k}) \). Near a band extremum the dispersion is approximately parabolic,

\[ E(\mathbf{k}) \approx E_0 + \frac{\hbar^2}{2} \sum_{i,j} (\mathbf{k} - \mathbf{k}_0)_i \left[ \frac{1}{m^{*}} \right]_{ij} (\mathbf{k} - \mathbf{k}_0)_j \]

and carriers respond to slowly varying fields as if they were free particles with the effective-mass tensor \( 1/m^{*}_{ij} = \hbar^{-2} \partial^2 E / \partial k_i \partial k_j \).

In silicon the conduction-band minimum lies along \( \Delta \) near the X-points, giving six equivalent valleys with longitudinal and transverse masses \( m_\ell^{*} = 0.92 m_0 \), \( m_t^{*} = 0.19 m_0 \). Valence bands are degenerate at \( \Gamma \), with heavy- and light-hole branches that complicate hole transport. In GaAs the conduction-band minimum is at \( \Gamma \) with \( m^{*} = 0.067 m_0 \), which explains the high electron mobility; the valence band has similar heavy/light/split-off structure.

3.2 Effective-Mass Equation

For envelopes \( F(\mathbf{r}) \) that vary slowly on the atomic scale, the Kohn-Luttinger effective-mass equation reads

\[ \left[ -\frac{\hbar^2}{2 m^{*}} \nabla^2 + U(\mathbf{r}) \right] F(\mathbf{r}) = E\, F(\mathbf{r}) \]

where \( U(\mathbf{r}) \) is the externally applied or heterostructure potential. This is what justifies treating carriers in a 2DEG or quantum well with the particle-in-a-box formulae of Chapter 2 simply by replacing \( m_0 \) with \( m^{*} \).

3.3 Limits of Effective-Mass Theory

The envelope approximation breaks down when potentials vary on the scale of the lattice constant (about 0.5 nm in silicon). Ultra-thin oxides, single-monolayer barriers, or dopant impurities require atomistic methods (tight binding, \( k \cdot p \), density functional theory). For NE 471 we mostly stay within effective-mass reasoning but note where it fails, for example in direct source-to-drain tunnelling in sub-5-nm channels or when modelling dangling-bond resonances.

Chapter 4: Scattering, Mean Free Path, and Mobility

4.1 Scattering Mechanisms

A real crystal is not perfectly periodic. Phonons, ionised impurities, neutral defects, interface roughness, and other carriers all scatter Bloch electrons and limit their mobility. Each mechanism has a characteristic momentum relaxation rate \( 1/\tau_i \), and by Matthiessen’s rule the total rate is (approximately) additive:

\[ \frac{1}{\tau} = \sum_i \frac{1}{\tau_i} \]

Acoustic-phonon scattering dominates at room temperature in pure silicon. Ionised-impurity scattering becomes important at low temperatures and high doping. Surface-roughness scattering at the Si/SiO\(_2\) interface limits inversion-layer mobility at high gate fields. Remote Coulomb scattering by charges in high-k dielectrics or in the gate stack is a design concern for modern MOSFETs.

4.2 Drude Mobility and the Semiclassical Picture

The Drude model gives mobility \( \mu = e \tau / m^{*} \) and conductivity \( \sigma = n e \mu \). Mean free path follows from \( \ell = v_F \tau \) (in a degenerate system) or \( \ell = v_{th} \tau \) (non-degenerate). At high fields the drift velocity saturates because carriers emit optical phonons as soon as they reach the phonon energy \( \hbar \omega_{LO} \approx 60 \) meV; saturation velocities in silicon are \( v_{sat} \approx 10^7 \) cm/s.

4.3 The Boltzmann Transport Equation

The semiclassical distribution \( f(\mathbf{r}, \mathbf{k}, t) \) satisfies

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

The collision integral is the hard part; in the relaxation-time approximation it reduces to \( -(f - f_0)/\tau \). Linearising around equilibrium and keeping first order in \( \mathbf{E} \) gives the Drude conductivity. Moments of the BTE yield drift-diffusion and hydrodynamic models still used in TCAD.

4.4 From Diffusive to Ballistic

When the channel length \( L \) is smaller than \( \ell \), the carrier crosses without scattering. The notion of mobility loses meaning because there is no momentum randomisation to define it. Drain current becomes controlled by the number of transverse modes and by how carriers are injected at the contacts, not by \( \mu E \). This is the Landauer regime we turn to next.

Chapter 5: Landauer-Büttiker Transport and the Quantum of Conductance

5.1 Conductance of a Ballistic Channel

Imagine a narrow channel (length \( L \ll \ell \)) connecting two large reservoirs held at electrochemical potentials \( \mu_L \) and \( \mu_R \) with \( \mu_L - \mu_R = eV \). Each transverse mode acts as a one-dimensional subband. Carriers in the energy window \( [\mu_R, \mu_L] \) are injected from the left and leave on the right. Counting states and velocities, the current carried by a single spin-degenerate mode is

\[ I_n = \frac{2 e}{h} (\mu_L - \mu_R) T_n \]

where \( T_n \) is the transmission probability of mode \( n \). Summing over modes gives the Landauer formula

\[ G = \frac{2 e^2}{h} \sum_n T_n \]

The quantity \( G_0 = 2 e^2/h \approx (12.9\ \text{k}\Omega)^{-1} \) is the quantum of conductance. A perfectly transmitting single-mode conductor has resistance 12.9 k\( \Omega \) regardless of its length, a deeply counter-intuitive result that reflects the unavoidable mismatch between a 1D conductor and bulk reservoirs.

5.2 Contact Resistance is Real

The 12.9 k\( \Omega \) is not a property of the channel but of the contacts. It arises because modes in a narrow channel must squeeze from a reservoir containing vastly more modes, producing an intrinsic contact resistance. This resolves the apparent paradox that a clean, scatterless channel still has finite resistance.

5.3 Multi-Terminal Büttiker Formula

With multiple reservoirs labelled \( p \) held at potentials \( V_p \), the current into lead \( p \) is

\[ I_p = \frac{2 e}{h} \sum_q \left[ T_{qp} \mu_p - T_{pq} \mu_q \right] \]

Current conservation and gauge invariance imply \( \sum_q T_{pq} = \sum_q T_{qp} = M_p \), the number of modes in lead \( p \). This framework handles Hall bars, four-probe measurements, and split-gate geometries on equal footing.

5.4 Quantised Conductance in Experiments

Point contacts formed by depleting a 2DEG with split gates show clear plateaus at integer multiples of \( 2 e^2/h \) as the gate voltage opens successive modes. Break junctions of gold wires show the same quantisation, now at \( 2 e^2/h \) per atomic-chain mode. These experiments are the cleanest demonstrations that electrons transport energy and charge through discrete quantum channels.

Chapter 6: Non-Equilibrium Green’s Functions for Nanoscale Devices

6.1 Why NEGF?

Landauer is elegant but assumes coherent transport and knows nothing about inelastic scattering inside the device. The NEGF formalism, developed by Keldysh, Kadanoff-Baym, and popularised for nanodevices by Datta, provides a self-consistent framework that handles open boundaries, coherent and incoherent scattering, and arbitrary bias.

6.2 The Basic Objects

Discretise the device Hamiltonian as a matrix \( [H] \). The coupling to each contact \( p \) is encoded in a self-energy matrix \( [\Sigma_p(E)] \); its anti-Hermitian part gives the broadening \( [\Gamma_p] = i([\Sigma_p] - [\Sigma_p]^\dagger) \). The retarded Green’s function is

\[ [G^r(E)] = \left[ E\, I - H - \sum_p \Sigma_p(E) \right]^{-1} \]

Transmission from lead \( p \) to lead \( q \) is

\[ T_{pq}(E) = \mathrm{Tr} \left[ \Gamma_p G^r \Gamma_q G^{r \dagger} \right] \]

and the current is obtained by a Landauer-style integral of \( T_{pq}(E) \) weighted by the difference of contact Fermi functions.

6.3 Scattering Inside the Channel

Inelastic scattering is incorporated through an additional self-energy \( [\Sigma_{scatt}] \) evaluated self-consistently (Born, self-consistent Born, or higher). For phonon scattering, \( \Sigma_{scatt} \) depends on \( G^{<} \) and on the phonon spectrum. Full inclusion is computationally demanding; practical TCAD tools truncate to Büttiker probes or phenomenological dephasing reservoirs.

6.4 Relation to Semiclassical Models

In the limit of strong dephasing and weak quantum effects, NEGF reproduces drift-diffusion and the BTE. For coherent transport it reduces to Landauer. NEGF thus provides a unifying framework that covers ballistic nanotransistors, resonant tunnelling diodes, molecular junctions, and 2D-material FETs within one language.

Chapter 7: MOSFET Scaling and Short-Channel Effects

7.1 Long-Channel MOSFET Refresher

The long-channel drain current in strong inversion (gradual channel approximation) is

\[ I_D = \mu C_{ox} \frac{W}{L} \left[ (V_{GS} - V_T) V_{DS} - \frac{V_{DS}^2}{2} \right] \]

in the linear region and saturates at

\[ I_{D,sat} = \mu C_{ox} \frac{W}{2 L} (V_{GS} - V_T)^2 \]

when \( V_{DS} \geq V_{GS} - V_T \). These expressions assume that the transverse field (set by the gate) is much larger than the longitudinal field (set by \( V_{DS} \)), and that scattering provides a well-defined mobility.

7.2 Dennard Scaling

Dennard’s 1974 recipe reduces every linear dimension by \( 1/\kappa \), every voltage by \( 1/\kappa \), and every doping by \( \kappa \), keeping electric fields constant. The resulting devices are faster (delay \( \propto L/v \)), denser (\( \kappa^2 \)), and dissipate less per operation. The recipe held for decades but broke when subthreshold leakage, gate tunnelling, and reliability concerns forced supply voltages to stagnate near 1 V.

7.3 Short-Channel Effects

As \( L \) shrinks without a corresponding reduction of depletion widths, several non-idealities appear. \( V_T \) roll-off: part of the channel depletion charge is supported by source/drain fields rather than by the gate, reducing the charge the gate must control and shifting \( V_T \) downward. Drain-induced barrier lowering (DIBL): the drain field lowers the source-channel barrier, producing a negative output conductance and shifting \( V_T \) with \( V_{DS} \). Subthreshold swing degradation: the swing \( S = \ln 10 \cdot (k_B T / e)(1 + C_d/C_{ox}) \) approaches the Boltzmann limit of 60 mV/decade at room temperature only for well-controlled electrostatics. Velocity saturation and ballistic injection: the current in short channels is limited not by mobility but by the injection velocity at the source-channel virtual source.

7.4 Gate Tunnelling and High-k Dielectrics

Scaling demands \( C_{ox} = \epsilon_{ox}/t_{ox} \) grow. SiO\(_2\) thickness below about 1.2 nm produces unacceptable direct-tunnelling leakage. High-k dielectrics like HfO\(_2\) (\( \epsilon_r \approx 20 \)) allow larger physical thickness for the same capacitance, trading tunnelling for reduced bandgap offsets and remote phonon scattering. The equivalent-oxide thickness (EOT) \( t_{EOT} = t_{high-k} \cdot \epsilon_{SiO_2}/\epsilon_{high-k} \) has been pushed below 1 nm in production nodes.

7.5 FinFETs and Gate-All-Around

Planar MOSFET electrostatics degrade when \( L \) drops below about 20 nm. FinFETs wrap a gate around a thin vertical silicon fin on three sides, reducing the drain’s electrostatic influence and restoring subthreshold control. Gate-all-around (GAA) structures, either silicon nanowires or stacked horizontal nanosheets, surround the channel entirely and give the best electrostatic integrity. Industry moved from planar to FinFET around the 22 nm node and from FinFET to GAA nanosheets around 3 nm.

The natural length \( \lambda \), measuring how far source/drain fields penetrate the channel, is a rough guide:

\[ \lambda \approx \sqrt{\frac{\epsilon_{Si} t_{Si} t_{ox}}{n \cdot \epsilon_{ox}}} \]

where \( n = 1, 2, 4 \) for planar, double-gate, and GAA respectively. Short-channel effects are controlled when \( L \) exceeds roughly \( 5 \lambda \to 10 \lambda \).

7.6 Quantum Corrections in the Channel

At the MOS interface the inversion layer is only a few nanometres thick and must be treated quantum mechanically. Carriers occupy discrete subbands, the inversion charge sits on average a few angstroms away from the interface (quantum-mechanical charge centroid), and the effective \( t_{EOT} \) is larger than the physical oxide thickness. This \( \Delta t_{EOT} \) is roughly 0.3–0.5 nm and no longer negligible at modern nodes.

Chapter 8: Tunnelling Devices, Single-Electron Transistors, and Resonant Tunnelling Diodes

8.1 Quantum Tunnelling Basics

The transmission through a rectangular barrier of height \( V_0 \) and width \( d \) for an electron of energy \( E < V_0 \) is approximately

\[ T(E) \approx \exp\left[ -2 d \sqrt{\frac{2 m^{*} (V_0 - E)}{\hbar^2}} \right] \]

For \( V_0 - E = 1 \) eV, \( m^{*} = m_0 \), the decay length is about 1 angstrom, so doubling the barrier thickness cuts current by roughly \( e^{-20} \). Tunnelling is thus exquisitely sensitive to geometry.

8.2 Resonant Tunnelling Diodes

A double-barrier heterostructure (for example AlGaAs/GaAs/AlGaAs) confines a quasibound state in the central well. Electrons incident from the emitter tunnel through the first barrier, reside briefly in the well, and tunnel out through the second barrier. When the emitter Fermi level aligns with the quasibound state, transmission exhibits a resonance approaching unity:

\[ T(E) \approx \frac{\Gamma_L \Gamma_R}{(E - E_0)^2 + (\Gamma/2)^2} \]

where \( \Gamma_L, \Gamma_R \) are the partial widths set by tunnelling rates and \( \Gamma = \Gamma_L + \Gamma_R \). Biasing the RTD sweeps the resonance through the emitter Fermi sea, producing a negative differential resistance (NDR) region in the I-V curve. Peak-to-valley ratios of 10:1 or more are achievable at room temperature in In(Ga)As-based designs. RTDs find use in THz oscillators and as building blocks for multi-valued logic, though they have not displaced mainstream CMOS.

8.3 Coulomb Blockade and the Single-Electron Transistor

Consider a small metallic island tunnel-coupled to source and drain and capacitively coupled to a gate. The electrostatic energy of the island with \( N \) extra electrons is

\[ E(N) = \frac{(N e - Q_G)^2}{2 C_\Sigma} + \text{const} \]

where \( Q_G = C_G V_G \) is the induced charge and \( C_\Sigma \) is the total island capacitance. Adding an electron costs \( \Delta E = e^2/C_\Sigma \cdot (N + 1/2 - Q_G/e) \). When \( E_C = e^2/(2 C_\Sigma) \gg k_B T \) and the tunnel resistance \( R_T \gg h/e^2 \), electrons move one at a time.

Two conditions must hold for Coulomb blockade to be observable:

\[ E_C \gg k_B T, \quad R_T \gg R_K = \frac{h}{e^2} \approx 25.8\ \text{k}\Omega \]

The first ensures thermal fluctuations cannot overcome the charging barrier. The second ensures the electron is well-localised on the island (weak tunnelling).

8.4 SET Operation: Coulomb Oscillations and Diamonds

Sweeping \( V_G \) at small \( V_{DS} \) reveals periodic conductance peaks (Coulomb oscillations) with period \( \Delta V_G = e/C_G \). Between peaks the island is “Coulomb-blockaded” and the current is exponentially suppressed. Plotting differential conductance in the \( (V_G, V_{DS}) \) plane produces Coulomb diamonds: rhombic regions of blockade bounded by lines where successive charge states become degenerate. The size of the diamonds gives \( E_C \) and the slopes give capacitance ratios.

SETs are extraordinarily charge-sensitive (below \( 10^{-4} e/\sqrt{\text{Hz}} \)) and are used as electrometers, qubit readout devices, and fundamental metrology standards. Their weakness is low drive current and exponential temperature sensitivity.

8.5 Single-Molecule Transistors

Replace the metallic island with a single molecule bridging two electrodes. The molecule’s discrete HOMO/LUMO levels play the role of island charge states. Conductance depends on alignment between metal Fermi levels and molecular levels, on vibronic coupling (inelastic electron tunnelling spectroscopy reveals vibrational sidebands), and on electrostatic control via a gate. Fabrication methods include mechanical break junctions, electromigrated nanogaps, and STM-based contacts. Single-molecule devices are mostly a research platform, probing electron-vibration coupling, spin states (Kondo physics in transition-metal complexes), and ultimate limits of device scaling.

Chapter 9: 2D Materials and Beyond-Silicon FETs

9.1 Graphene Field-Effect Transistors

Graphene is a one-atom-thick honeycomb of carbon with a linear (Dirac-cone) dispersion \( E(k) = \hbar v_F |k| \), \( v_F \approx 10^6 \) m/s. Its advantages: extremely high mobility (tens of thousands of cm\(^2\)/Vs at room temperature on hBN), compatibility with back-end processing, and ultra-thin channel with excellent electrostatic control. Its fatal weakness for digital logic: no bandgap. Graphene FETs do not turn off; the minimum conductivity is finite (the Dirac-point conductivity). Opening a gap via bilayer-graphene stacking, nanoribbons (quantisation-induced gap), or chemical functionalisation typically degrades mobility. Graphene has found roles in RF analog (high \( f_T \)), photodetectors, and as a passive interconnect or contact material.

9.2 Transition-Metal Dichalcogenide FETs

Monolayer MoS\(_2\), WS\(_2\), WSe\(_2\), and related compounds are direct-gap semiconductors (gap around 1.5–2.0 eV) with moderate mobilities (tens to a few hundred cm\(^2\)/Vs) and atomically thin channels that enable aggressive \( L \) scaling. Their bandgap makes them viable for digital logic in principle. Practical challenges include contact resistance (Schottky barriers are the norm), interface trap density, strain and dielectric integration. Industry roadmaps list TMDs as a post-silicon candidate channel material for sub-1-nm nodes, typically in GAA-like stacked configurations.

9.3 Tunnel FETs

In a tunnel FET the source injects carriers into the channel by band-to-band tunnelling rather than thermionic emission. Because the distribution is not the Maxwell-Boltzmann tail, the subthreshold swing can in principle be below 60 mV/decade, overcoming the thermodynamic limit that restricts conventional MOSFETs. TFETs thus promise ultra-low-voltage operation. Demonstrated TFETs often fall short of theoretical swing because of defect-assisted tunnelling and trap-assisted leakage, and drive currents are usually modest. III-V or 2D-material source/channel heterostructures are active research areas.

9.4 Carbon Nanotube FETs

Single-walled carbon nanotubes are quasi-1D conductors that can be metallic or semiconducting depending on chirality. Semiconducting nanotubes offer high carrier velocity, ballistic transport over hundreds of nanometres, and small diameter (1–2 nm) for excellent electrostatics. The manufacturing challenge is chirality-selective synthesis and placement at VLSI densities. Prototype CNT-FET microprocessors have been demonstrated but are not yet commercially competitive.

9.5 Spin-Based and Neuromorphic Devices

Beyond charge-based transistors, spintronic devices (magnetic-tunnel-junction transistors, spin-transfer-torque MRAM) and memristive devices (resistive RAM, phase-change memory) are increasingly integrated with CMOS for non-volatile memory and analog in-memory computing. These lie on the edge of NE 471’s scope but share the same fabrication toolkit.

Chapter 10: Large-Area, Flexible, Organic, and Amorphous-Silicon Electronics

10.1 Why Large-Area Electronics?

Displays, sensors, photovoltaics, X-ray imagers, and wearables demand electronics over square metres at modest performance — the opposite trade-off from VLSI, which values peak performance over small area. Large-area electronics forgoes crystalline silicon wafers for materials that can be deposited on glass or plastic: hydrogenated amorphous silicon (a-Si:H), low-temperature polycrystalline silicon (LTPS), amorphous metal-oxides (IGZO), organic small molecules, conjugated polymers, and solution-processed semiconductors.

10.2 Hydrogenated Amorphous Silicon

a-Si:H is produced by plasma-enhanced chemical vapour deposition (PECVD) of silane at 200–300 \(^\circ\)C. Hydrogen passivates dangling bonds, reducing the midgap defect density from \( 10^{19} \) cm\(^{-3}\) to \( 10^{16} \) cm\(^{-3}\). The mobility-gap density of states has exponential band tails and a residual midgap defect band. Conduction is by extended-state transport above the mobility edge for electrons, with trap-limited field-effect mobility typically 0.5–1 cm\(^2\)/Vs. The Meyer-Neldel rule and bias-stress-induced threshold shift (Staebler-Wronski and related effects) are defining behaviours. a-Si:H TFTs drove the first LCD panels and remain workhorses for low-cost backplanes.

10.3 Amorphous Metal-Oxide Semiconductors

InGaZnO (IGZO) and related amorphous oxides achieve electron mobilities of 10–30 cm\(^2\)/Vs because the conduction band is formed from spatially extended metal s-orbitals whose overlap is insensitive to bond-angle disorder. IGZO has displaced a-Si:H in high-performance backplanes for OLED displays, large-format TVs, and as selector transistors in advanced memory arrays.

10.4 Organic TFTs

Small-molecule semiconductors (pentacene, rubrene, DNTT) and conjugated polymers (P3HT, PBTTT, various donor-acceptor copolymers) offer solution processability and mechanical flexibility. Single-crystal rubrene has mobilities approaching 20 cm\(^2\)/Vs; solution-cast polymers sit at 0.1–10 cm\(^2\)/Vs. Carrier transport is a mix of band-like and hopping, depending on temperature, disorder, and trap density. Organic TFTs are used in flexible displays, electronic skin, and disposable RFID.

10.5 Mechanically Flexible Substrates

Replacing glass with a polymer (PEN, PI, PET) enables roll-to-roll processing and bendable/stretchable devices. Design constraints include thermal-budget limits (PEN tolerates about 180 \(^\circ\)C, PI tolerates 350 \(^\circ\)C), CTE mismatch, dimensional stability during patterning, and moisture/oxygen barrier performance for environmentally sensitive materials. Neutral-plane design minimises strain on active layers during bending. For stretchable electronics, rigid islands connected by serpentine interconnects decouple mechanical strain from device regions.

Chapter 11: Fabrication: Deposition, Patterning, and Metrology

11.1 Deposition Techniques

Physical vapour deposition (PVD) covers thermal evaporation, electron-beam evaporation, and sputtering. Sputtering uses energetic ions to eject target atoms, producing dense films with good step coverage for metal contacts and some oxides.

Chemical vapour deposition (CVD) exposes a heated substrate to reactive gases that decompose or react to deposit a solid film. Low-pressure CVD (LPCVD), plasma-enhanced CVD (PECVD), and metal-organic CVD (MOCVD) differ in pressure, activation, and precursor chemistry. PECVD is how a-Si:H and SiN passivation films are deposited at low temperatures; MOCVD grows III-V heterostructures and 2D transition-metal dichalcogenides.

Atomic layer deposition (ALD) alternates self-limiting surface reactions to deposit one atomic layer per cycle. The canonical example is Al\(_2\)O\(_3\) from trimethylaluminium and water, or HfO\(_2\) from HfCl\(_4\) or Hf-amide precursors and water. ALD produces pinhole-free conformal coatings on high-aspect-ratio structures, enabling the thin high-k dielectrics, metal-gate workfunction layers, and diffusion barriers that modern transistors demand.

Molecular-beam epitaxy (MBE) grows single-crystal heterostructures atom by atom under ultra-high vacuum. Primary use: research-grade III-V devices, quantum wells, and the high-mobility 2DEGs used in quantum-Hall metrology and qubit platforms.

Solution-based deposition: spin coating (for photoresists, polymers, perovskite precursors), dip coating, blade/bar coating, inkjet printing (for organics and functional materials on flexible substrates), and electroplating (for Cu interconnect damascene fill) round out the deposition toolkit. Their throughput advantage over vacuum methods is offset by uniformity and film-quality constraints.

11.2 Patterning: Photolithography

Photolithography transfers a mask pattern into a photoresist film by selective exposure. Resolution follows the Rayleigh criterion:

\[ R = k_1 \frac{\lambda}{NA} \]

where \( \lambda \) is the exposure wavelength, \( NA \) is the numerical aperture of the imaging system, and \( k_1 \) is a process-dependent factor. 193 nm ArF immersion lithography (\( NA \) up to 1.35) with multiple patterning pushed CMOS to the 7 nm node. Extreme-ultraviolet (EUV) lithography at \( \lambda = 13.5 \) nm, using all-reflective optics and pellicle-protected multilayer masks, now enables single-exposure printing at the 5 nm node and below. High-NA EUV (\( NA = 0.55 \)) is being deployed for sub-2 nm nodes. Throughput, mask defect control, and source power are the key industrial metrics.

Immersion lithography uses water between the lens and wafer to raise \( NA \) above 1. Multiple patterning (LELE, SADP, SAQP) splits a dense pattern across several masks or spacer-defined features to circumvent single-exposure resolution limits.

11.3 Patterning: Beyond Photolithography

Electron-beam lithography (EBL) writes patterns directly with a focused electron beam. Sub-10 nm resolution is routine in research. Throughput is limited by the serial nature of writing; e-beam is used for mask making, R&D, and photonic/quantum device prototyping, not volume production.

Focused-ion-beam (FIB) patterning mills material directly; used for circuit edit, sample preparation, and specialised devices.

Nanoimprint lithography (NIL) presses a rigid or soft mould into a thin resist layer, reproducing features below 10 nm mechanically. Hard-mould thermal NIL and UV-curable step-and-flash NIL trade throughput, defectivity, and mould wear differently. NIL is used in patterned media, nanophotonics, and niche nanoelectronics, and appears in pilot lines for some memory applications.

Self-assembled lithography uses directed self-assembly (DSA) of block copolymers (often PS-b-PMMA) to produce sub-20-nm periodic patterns aligned to lithographically defined guides. DSA complements EUV for certain pitch-doubling applications.

Etching transfers the resist pattern into the underlying film. Wet etching uses liquid chemistry; it is usually isotropic but offers high selectivity. Dry (plasma) etching — reactive-ion etching (RIE), inductively coupled plasma (ICP), deep-reactive-ion etching (Bosch process) — produces anisotropic, high-aspect-ratio features essential for fins, trenches, and vias. Chemical-mechanical planarisation (CMP) is an orthogonal workhorse that planarises wafers between process steps, indispensable for multilevel metallisation and for FinFET/GAA integration.

11.4 Doping

Ion implantation accelerates dopant ions (B, P, As) into the target wafer; a subsequent anneal activates the dopants and repairs lattice damage. Advantages: precise dose and depth control, low lateral straggle. For shallow junctions in ultra-scaled devices, alternative techniques include plasma doping, monolayer doping from surface precursors, and in-situ doping during epitaxial source/drain growth.

11.5 Metrology

Characterising nanoscale devices requires imaging, compositional, and electrical metrology.

Scanning electron microscopy (SEM) produces topographic images at resolutions from 10 nm down to 1 nm. Low-voltage SEM and environmental SEM extend applicability to insulating or delicate samples.

Transmission electron microscopy (TEM), with high-resolution and aberration-corrected variants, images atomic lattices directly at sub-angstrom resolution. Scanning TEM (STEM) with HAADF detectors provides Z-contrast. Electron energy-loss spectroscopy (EELS) and energy-dispersive X-ray spectroscopy (EDX/EDS) in a TEM give composition line profiles across heterointerfaces.

Atomic force microscopy (AFM) measures topography with sub-nanometre vertical resolution by scanning a sharp tip. Variants include conductive AFM (local I-V), Kelvin-probe force microscopy (surface potential), piezoresponse force microscopy (ferroelectric domain imaging), and scanning thermal microscopy.

X-ray methods: X-ray diffraction (XRD) and X-ray reflectivity (XRR) measure crystal structure and thin-film thickness/density; X-ray photoelectron spectroscopy (XPS) probes surface chemical states.

Secondary-ion mass spectrometry (SIMS) measures dopant concentration depth profiles with sub-ppm sensitivity but modest lateral resolution.

Electrical device metrology: wafer-probe I-V and C-V, noise spectroscopy, pulsed-IV for trap characterisation, and low-temperature transport (quantum Hall, Coulomb-blockade spectroscopy) for research devices.

Chapter 12: Synthesis and Outlook

12.1 Unifying Themes

Across the course three themes recur. First, geometry shapes physics: reducing dimensionality reshapes the density of states, opens or closes bandgaps, and moves the relevant length scales. Second, contacts matter as much as channels: Landauer’s resistance quantum, source-injection velocity in ballistic FETs, and Schottky barriers at TMD contacts all say the same thing. Third, electrostatics is king: short-channel effects, subthreshold slope, and Coulomb blockade all hinge on the capacitances that set how effectively a gate controls the channel charge.

12.2 Where the Field Is Going

The industry roadmap continues from FinFETs and nanosheets toward complementary FETs (CFET) that stack n-type and p-type devices vertically, and eventually toward 2D-channel FETs at the 1 nm node and below. Tunnel FETs, neuromorphic memristors, and cryogenic CMOS support for qubit control are on adjacent tracks. Flexible and large-area electronics continue to grow in display, medical, and IoT applications. Single-molecule electronics remains a research frontier with contributions to fundamental understanding rather than displacing CMOS.

12.3 Study Advice

The calculations that matter most for this course are: (1) particle-in-a-box confinement energies and density-of-states formulae for each dimensionality; (2) the Landauer formula, including how to count modes and how to interpret quantised conductance; (3) basic MOSFET scaling — DIBL, subthreshold swing, natural length \( \lambda \), EOT — enough to predict what happens when a dimension shrinks; (4) Coulomb-blockade energetics and the two conditions for single-electron behaviour; (5) the geometric resolution limit in photolithography and its workarounds. If you can reason quantitatively with these five, you can read the rest of the nanoelectronics literature.