Physics and Technology

This article collects the PC5101 Term Test 1 material from Lectures 1–4: quantum dots, nanowires, carbon nanotubes, graphene, and the functional devices discussed with them. It combines the lecture slides, supplementary derivations, handwritten class notes, exercises, and in-class assignments into one set of study notes.

The recurring question is how structure changes electronic states, and how those states produce a measurable optical, electrical, or mechanical response. Learn the assumptions behind each model as well as its equations: a formula for a spherical quantum dot, a parabolic electron band, or a metallic tunnelling barrier cannot be transferred unchanged to every nanomaterial.

Reading map and notation

Lecture	Main material	What to be able to do
1	Semiconductor and carbon quantum dots	Explain energy-level splitting and photoluminescence; derive the LCAO expressions; use the exciton and Brus models; identify materials and infer particle size or band gap.
2	Nanowires	Connect synthesis and surface chemistry to function; interpret optical spectra; derive and use Fowler–Nordheim plots; explain photodetection, lasing, and sensing.
3	Carbon nanotubes and low-dimensional states	Determine chirality and diameter; count electronic states; distinguish dimensional DOS curves; interpret STM/STS and nanotube devices.
4	Graphene and carbon devices	Obtain the tight-binding and Dirac dispersions; derive optical absorption; infer layer number; explain twisted layers, laser processing, thermal emission, and optomechanical devices.

The article uses SI units unless an equation explicitly specifies eV, nm, or micrometres. Useful conversions are

$$\hbar=\frac{h}{2\pi},\qquad 1\,\mathrm{eV}=1.602176634\times10^{-19}\,\mathrm{J},$$

$$E_\gamma=h\nu=\frac{hc}{\lambda},\qquad \frac{E_\gamma}{\mathrm{eV}}\simeq\frac{1240}{\lambda/\mathrm{nm}},$$

$$k_BT\simeq25.9\,\mathrm{meV}\quad(T=300\,\mathrm K).$$

Use the constants supplied in an examination question when reproducing its numerical answer. For example, one worksheet's rounded constants give $hc/e=1243.125\,\mathrm{nm\,V}$, equivalently a photon-energy conversion coefficient of $1243.125\,\mathrm{eV\,nm}$.

Symbols are defined where used. In particular, distinguish a quantum-dot radius from its diameter, a macroscopic field $E$ from a local field $F$, the eV value of a work function from its energy in joules, and a total DOS from a DOS per unit area. The overlap integral, field-enhancement factor, hopping integral, mobility, and fine-structure constant belong to different models even when lecture notation reuses a letter.

The dated worked examples below retain the classroom labels. Where a slide or supplied solution has an inconsistent sign, unit, or size convention, the correction is stated next to the calculation.

Lecture 1 — Quantum dots

Quantum dots connect three ideas: atomic orbitals combine into solid-state bands; spatial confinement changes the allowed electronic energies; and the resulting transitions determine optical behaviour. Size, composition and surface chemistry must be distinguished when interpreting an experiment.

1.1 From atomic levels to bands

An emitted photon's energy equals the energy released by the transition:

$$E_\gamma=h\nu=\frac{hc}{\lambda}=E_{\mathrm{high}}-E_{\mathrm{low}}, \qquad \frac{E_\gamma}{\mathrm{eV}}\simeq\frac{1240}{\lambda/\mathrm{nm}}.$$

Shorter wavelengths mean higher photon energies. At a fixed photon energy, greater intensity means a greater photon flux; intensity alone does not specify photon energy. The supplied approximations $h=6.63\times10^{-34}\,\mathrm{J\,s}$, $c=3\times10^8\,\mathrm{m\,s^{-1}}$ and $e=1.6\times10^{-19}\,\mathrm C$ instead give a photon-energy conversion coefficient of $1243.125\,\mathrm{eV\,nm}$. Use one consistent set.

Other useful constants are $\hbar=h/(2\pi)$, $m_0=9.109\times10^{-31}\,\mathrm{kg}$, $\epsilon_0\simeq8.85\times10^{-12}\,\mathrm{F\,m^{-1}}$ and $k_B=8.617\times10^{-5}\,\mathrm{eV\,K^{-1}}$.

A macroscopic solid contains enough atoms for closely spaced levels to resemble bands. For the lecture's copper piece,

$$V=\frac12(4)(2.5)(2.5)=12.5\,\mathrm{cm^3}.$$

$$N=\frac{\rho V}{M}N_A.$$

$$N=\frac{8.95(12.5)}{63.5}(6.02\times10^{23}) \simeq1.06\times10^{24}.$$

Here $\rho$ is mass density. Combining one equivalent orbital from each of $N$ atoms produces $N$ spatial states, each accommodating two opposite spins. Two atoms split an original level into two; many atoms produce a band. Greater overlap generally means greater splitting. Higher atomic orbitals extend farther and begin interacting at larger separations.

Electrons fill available states according to Pauli exclusion. A conductor has accessible empty states near occupied states, through partial filling or band overlap. A completely filled band alone carries no net current. An insulator has a filled valence band and a large gap; a semiconductor has a smaller gap, allowing excitation of conduction electrons and valence-band holes. A semimetal has a small valence/conduction-band overlap. There is no universal numerical gap separating every semiconductor from every insulator.

At zero temperature the highest occupied energy describes the Fermi energy of a simple metal. More generally, the Fermi level is the chemical potential; in a semiconductor it may lie in the gap rather than coincide with an occupied state.

A diffraction grating separates wavelengths: at normal incidence, $d_g\sin\theta=m\lambda$. The Aug14 paper-colour answer is approximately $620\,\mathrm{nm}$, a representative reflected wavelength, not proof of monochromatic reflection.

Sources: L01, pp. 3–16; Aug14 class notes, pp. 1–9; Aug14 yellow-paper Q1.

1.2 Why overlap produces bonding: the hydrogen molecular ion

For $\mathrm{H}_2^+$, fix the proton separation $r$ and denote electron–proton distances by $r_1,r_2$. The Hamiltonian includes electron kinetic energy, two attractions and nuclear repulsion:

$$\hat H=-\frac{\hbar^2}{2m_e}\nabla^2+ \frac{e^2}{4\pi\epsilon_0}\left(-\frac1{r_1}-\frac1{r_2}+\frac1r\right).$$

LCAO means linear combination of atomic orbitals. For real, individually normalized orbitals,

$$S=\int\psi_1\psi_2\,dV, \qquad \psi_\pm=\frac{\psi_1\pm\psi_2}{\sqrt{2(1\pm S)}}.$$

The positive combination increases electron density between the nuclei: a bonding $\sigma$ state. The negative combination has an internuclear node: an antibonding $\sigma^*$ state. A negative wavefunction is a phase, not a negative probability. A minimum in total energy versus separation gives stable bonding, with $dE/dr=0$ and $d^2E/dr^2>0$.

Define positive attraction-integral magnitudes and the nuclear repulsion:

$$A=\frac{e^2}{4\pi\epsilon_0}\int\frac{\psi_1^2}{r_2}\,dV, \qquad B=\frac{e^2}{4\pi\epsilon_0}\int\frac{\psi_1\psi_2}{r_1}\,dV, \qquad V_{pp}=\frac{e^2}{4\pi\epsilon_0r}.$$

For equivalent atoms of isolated energy $E_n$,

$$H_{11}=H_{22}=E_n+V_{pp}-A, \qquad H_{12}=H_{21}=(E_n+V_{pp})S-B.$$

The energy expectation of the unnormalized combination is

$$E_\pm= \frac{H_{11}+H_{22}\pm H_{12}\pm H_{21}}{2(1\pm S)} =E_n+V_{pp}-\frac{A\pm B}{1\pm S}.$$

The denominator follows from $\int(\psi_1\pm\psi_2)^2dV=2(1\pm S)$; omitting overlap there is incorrect. At large separation, $S,B\to0$ and the splitting vanishes. In this approximation, retaining only classical-looking terms does not produce the stable molecular minimum.

For hydrogen $1s$ orbitals, $\psi_i=(\pi a_H^3)^{-1/2}e^{-r_i/a_H}$. Now reuse $\rho$ locally for the dimensionless separation $r/a_H$, not mass density. With $C=e^2/(4\pi\epsilon_0a_H)$,

$$S=e^{-\rho}\left(1+\rho+\frac{\rho^2}{3}\right),$$

$$A=C\left[\frac1\rho-e^{-2\rho}\left(1+\frac1\rho\right)\right], \qquad B=Ce^{-\rho}(1+\rho).$$

The supplementary slides 13–14 contain two errors: the quadratic term in $S$ needs a plus sign, and the coefficients of $A,B$ need $1/a_H$. The corrected expressions follow from the integral definitions. For a reproducible integration route, use $\xi=(r_1+r_2)/r$, $\eta=(r_1-r_2)/r$, with $\xi\ge1$, $-1\le\eta\le1$ and

$$dV=\frac{r^3}{8}(\xi^2-\eta^2)\,d\xi\,d\eta\,d\phi.$$

Angular integration leaves elementary exponential integrals for $S$ and $B$; $A$ can also be evaluated using spherical coordinates about one nucleus. The separate derivation PDFs mentioned on slide 15 were not supplied.

Sources: L01, pp. 6–14; Energy Levels Splitting supplement, slides 1–15.

1.3 What a quantum dot is, and how it is made

A quantum dot confines carriers in all three directions: “zero-dimensional” means no freely propagating direction, not zero physical size. Semiconductor nanocrystals bridge molecular and bulk behaviour. Electrode-defined dots can instead contain a controlled number of electrons; their dimensions need not match colloidal nanocrystals. Confinement depends on relevant wavefunction length scales, not simply the label “nano”.

For the same semiconductor in the confinement regime, smaller dots generally emit bluer light. Across different materials, colour alone cannot rank sizes.

Colloidal synthesis creates nuclei from semiconductor precursors, then grows them by adding ions or monomers. Precursor supply, temperature, growth time and stopping the reaction control size. The lecture illustrates approximately $1.5$–$50\,\mathrm{nm}$ particles and batches with around $5\%$ size variation; not every particle in that range is strongly confined.

TEM measures size and shape through a scale bar; HRTEM reveals lattice structure. Absorption identifies optical thresholds and transitions; PL measures emitted energies, intensities and widths. Inferring size from PL requires a known material and emission mechanism.

Dangling bonds can trap carriers before radiative recombination. Organic ligands and semiconductor shells passivate these traps, suppress nonradiative loss, prevent aggregation and control solubility or chemical attachment. In typical type-I CdSe/ZnS dots, suitable offsets confine both carriers to the core. A wider shell gap alone does not guarantee this alignment. Brightness, quantum yield, lifetime and resistance to photobleaching are distinct properties.

Sources: L01, pp. 17–21, 32–35, 41–43, 53–68.

1.4 Carbon dots: structure, preparation and emission

Class	Structural and optical distinction
GQDs	Graphene-like sheets; lateral size exceeds thickness; conjugated domains and edges influence emission.
CQDs	Usually approximately spherical, with identifiable crystal lattices; intrinsic core states and confinement can contribute.
CNDs	Generally lack obvious crystalline or polymeric structure; defects, surfaces and internal subdomains dominate.
CPDs	Polymer/carbon hybrids with functional groups and polymer chains; cores range from carbonized structures to crosslinked polymer frameworks.

Carbon abundance, relatively low toxicity and possible biocompatibility motivate these materials, but actual properties depend on composition, residues and surface treatment.

Chemical exfoliation/oxidative cutting breaks larger carbon structures into smaller domains. Laser ablation ejects material from a target and forms nanoparticles in the surrounding medium; the class example uses graphite in water and a $248\,\mathrm{nm}$ KrF laser. Microwave synthesis rapidly heats precursors, promoting dehydration, condensation or carbonization, and can also assist cutting. Hydrothermal synthesis heats precursors in a sealed aqueous environment, where polymerization and carbonization occur. Microwave and hydrothermal describe processing conditions, not uniquely top-down or bottom-up mechanisms.

Carbon-dot emission can combine core, surface, defect, conjugated-domain and polymer-related states. Nitrogen configuration, oxygen groups, edge geometry and passivation can change energy levels without changing overall diameter. Therefore, a universal particle-size-only Brus interpretation is inappropriate, especially for CNDs.

Sources: L01, pp. 26–31, 36–40, 69–72; Aug17 class notes, pp. 1–2.

1.5 Photoluminescence, fluorescence and phosphorescence

PL follows absorption, relaxation and emission. Nonradiative relaxation commonly leaves the emitted photon less energetic than the absorbed photon: $\lambda_{\mathrm{em}}>\lambda_{\mathrm{abs}}$, a Stokes shift. Resonant emission can instead have essentially the same energy. Absorbing a $4\,\mathrm{eV}$ photon and emitting a $2\,\mathrm{eV}$ photon illustrates energy loss to other degrees of freedom; it does not imply automatic production of two photons.

In a Jablonski diagram, $S_0$ is the singlet ground state; $S_1,S_2$ and $T_1$ are excited singlet and triplet states. VR relaxes vibration within an electronic state; IC changes electronic states without changing multiplicity; ISC changes multiplicity. Fluorescence commonly follows $S_1\to S_0$ rapidly. Phosphorescence commonly follows ISC into $T_1$ and a slower $T_1\to S_0$ transition, allowing afterglow.

HOMO/LUMO are the highest occupied/lowest unoccupied molecular orbitals. They provide an occupied/unoccupied-state analogy to valence/conduction bands, but molecular orbitals remain discrete. Carbon-dot core and surface HOMO/LUMO descriptions permit multiple emission pathways; the lecture's example energies are not universal constants.

Quinine in tonic water, highlighter solutions and security markings demonstrate fluorescence. Persistent-emission tape demonstrates phosphorescence. Ordinary reflected colour and newly emitted PL must be distinguished.

Sources: L01, pp. 21–25, 69–74, 86–87; Aug14 notes, p. 9; Aug21 notes, pp. 1–2.

1.6 Excitons: binding energy and characteristic radius

An exciton is a Coulomb-bound electron–hole pair, whose energy lies below that of separated carriers. A weakly bound, spatially extended Mott–Wannier exciton permits a screened hydrogen-like model; a localized Frenkel exciton generally does not share that approximation. Biexcitons contain two excitons, negative trions contain $2e+h$, positive trions contain $e+2h$, and defects can bind excitons.

For relative motion,

$$\frac1\mu=\frac1{m_e^*}+\frac1{m_h^*}, \qquad U(r)=-\frac{e^2}{4\pi\epsilon_0\epsilon_r r}.$$

Replacing hydrogen's mass and dielectric response gives

$$E_B^*=13.6\,\mathrm{eV}\frac{\mu/m_0}{\epsilon_r^2}, \qquad E_n=E_g-\frac{E_B^*}{n^2},$$

$$a_B^*=\frac{4\pi\epsilon_0\epsilon_r\hbar^2}{\mu e^2} =0.0529\,\mathrm{nm}\frac{\epsilon_r}{\mu/m_0}.$$

Thus the lowest exciton absorption costs $E_g-E_B^*$, not $E_B^*$. Higher states approach the free-pair threshold $E_g$ from below. The Bohr construction combines $\mu v^2/r=e^2/(4\pi\epsilon_0\epsilon_r r^2)$ with $\mu vr=n\hbar$, giving $r_n=n^2a_B^*$. This characteristic radius need not equal a quantum-mechanical mean separation exactly. Screening and reduced mass make semiconductor excitons larger and less tightly bound than hydrogen.

Whole-exciton translation adds $\hbar^2K^2/[2(m_e^*+m_h^*)]$: use total mass there, reduced mass internally. For CdSe, $(m_e^*,m_h^*)=(0.13,0.45)m_0$ and $\epsilon_r=10.2$ give $\mu/m_0=0.1009$, $a_B^*\simeq5.35\,\mathrm{nm}$ and $E_B^*\simeq13.2\,\mathrm{meV}$. The lecture's approximate radii are CdSe $6$, PbS $20$, InAs $34$ and PbSe $46\,\mathrm{nm}$.

At $300\,\mathrm K$, $k_BT\simeq25.9\,\mathrm{meV}$ indicates thermal dissociation tendencies, not an absolute existence threshold. In indirect-gap materials, excitons near the higher direct gap may decay into lower-energy free carriers. Ultimately excitons recombine radiatively, emitting photons, or nonradiatively through other channels.

Sources: L01, pp. 45–52, 56; Aug17 class notes, pp. 3–7.

1.7 Confinement and the Brus model

Compare dot radius $R$ with $a_B^*$: $R\gg a_B^*$ is approximately bulk-like; comparable dimensions give appreciable confinement; $R\ll a_B^*$ is strong confinement. Tighter wavefunctions require larger kinetic energies. An infinite spherical well gives a lowest single-particle energy $\pi^2\hbar^2/(2m^*R^2)$.

For the $1S_e$–$1S_h$ optical transition,

$$E(R)=E_g+\frac{H}{R^2}-\frac{S_C}{R}.$$

$$H=\frac{\pi^2\hbar^2}{2}\left(\frac1{m_e^*}+\frac1{m_h^*}\right).$$

$$S_C=\frac{A_{\mathrm B}e^2}{4\pi\epsilon_0\epsilon_r}, \qquad A_{\mathrm B}=1.786.$$

Here $A_{\mathrm B}$ is dimensionless, unlike the earlier LCAO integral $A$. The positive confinement term scales as $R^{-2}$, faster than the attractive $R^{-1}$ correction. This supports separate electron/hole confinement followed by a Coulomb correction. Assumptions include spherical shape, effective masses, strong confinement and idealized barriers; surface effects, finite barriers and dielectric-polarization corrections are simplified.

Effective mass describes band response through $m^*=\hbar^2/(d^2E/dk^2)$. Use positive hole masses in the optical model. With $\alpha_e=m_e^*/m_0$, $\alpha_h=m_h^*/m_0$,

$$H\simeq0.376(\alpha_e^{-1}+\alpha_h^{-1})\,\mathrm{eV\,nm^2}, \qquad S_C\simeq\frac{1.440A_{\mathrm B}}{\epsilon_r}\,\mathrm{eV\,nm}.$$

Material	$E_g$ (eV)	$\alpha_e$	$\alpha_h$	$\epsilon_r$
InSb	0.24	0.015	0.39	15.6
InP	1.42	0.073	0.40	9.6
GaAs	1.52	0.067	0.20	12.8
CdS	2.58	0.190	0.80	5.7
CdSe	1.74	0.130	0.45	10.2
ZnO	3.44	0.240	0.45	3.7

Use the table supplied with a problem: lecture and exercise versions differ. In SI calculations, convert nanometres to metres and mass ratios to kilograms; do not add joules to electronvolts or substitute $h$ for $\hbar$.

The mathematical derivative is $(S_CR-2H)/R^3$. The truncated expression decreases only for $R<2H/S_C$, with a formal minimum $E_g-S_C^2/(4H)$. Consequently, monotonic decrease and $E\ge E_g$ are not universal statements for all radii; its large-radius extrapolation also leaves strong-confinement validity.

Sources: L01, pp. 53–64; Aug14 notes, p. 8; Aug17 notes, p. 8.

1.8 Worked optical problems and source inconsistencies

White-paper Q1: TEM to emission. For CdSe,

$$\frac{E}{\mathrm{eV}}\simeq1.74+ \frac{3.73}{(R/\mathrm{nm})^2}-\frac{0.252}{R/\mathrm{nm}}.$$

The reference answer treats the measured $2.5\,\mathrm{nm}$ size as $R$: $E\simeq1.74+0.597-0.101=2.236\,\mathrm{eV}$, hence $\lambda\simeq555\,\mathrm{nm}$, peak iv. The TEM arrow appears to span the particle width. If it is a diameter, $R=1.25\,\mathrm{nm}$ instead gives approximately $3.93\,\mathrm{eV}$ or $315\,\mathrm{nm}$, inconsistent with the visible candidate peaks. State the size convention explicitly.

Q2: materials and inverse size. CdS emits at higher energy primarily because its bulk gap exceeds CdSe's. More generally,

$$\Delta E=\Delta E_g+\frac{\Delta H}{R^2}-\frac{\Delta S_C}{R}.$$

Using the answer's $R=3.7\,\mathrm{nm}$ convention, net corrections are about $0.20\,\mathrm{eV}$ for CdSe and $0.059\,\mathrm{eV}$ for CdS, compared with their $0.84\,\mathrm{eV}$ gap difference. If the stated $3.7\,\mathrm{nm}$ is genuinely a diameter, corrections are larger; the schematic peaks are not precise fit data.

For the unknown CdSe peak at $575\,\mathrm{nm}$, $E\simeq2.157\,\mathrm{eV}$. Let $\Delta=E-E_g$; then

$$\Delta R^2+S_CR-H=0, \qquad R=\frac{-S_C+\sqrt{S_C^2+4H\Delta}}{2\Delta} \quad(\Delta>0).$$

This gives $R\simeq2.7\,\mathrm{nm}$, diameter $5.4\,\mathrm{nm}$. The answer text's “diameter $2.75\,\mathrm{nm}$” conflicts with both the radius convention and its embedded table's $2.7\,\mathrm{nm}$. Always substitute the result back.

Q3: identify materials using all three peaks. Treating the labelled sizes as radii, CdS predicts approximately $(296,403,449)\,\mathrm{nm}$ for $(1.1,1.8,2.6)\,\mathrm{nm}$: sample A. InP predicts approximately $(401,551,700)\,\mathrm{nm}$ for $(1.83,2.55,3.79)\,\mathrm{nm}$: sample B. Matching one colour is insufficient. The Q3 table lists CdSe's gap as $1.84\,\mathrm{eV}$ rather than the main table's $1.74\,\mathrm{eV}$; do not mix parameter rows between versions.

Aug17 yellow paper. Peak d above $750\,\mathrm{nm}$ is inconsistent with the specified CdSe strong-confinement band-edge transition. The bulk gap corresponds to roughly $713$–$715\,\mathrm{nm}$. Even the formal truncated-model minimum is approximately $1.736\,\mathrm{eV}$, far above that long-wavelength peak's energy. This does not prohibit other CdSe defect or surface emissions at longer wavelengths.

Sources: Aug14 Lecture 1 white-paper Q1–Q3, pp. 1–6; Aug17 yellow-paper Q1 and solution.

1.9 Three measurements determine an unknown bulk gap

White-paper Q4 gives corresponding radii $(2.5,3.5,6.0)\,\mathrm{nm}$ and wavelengths $(375,407,431)\,\mathrm{nm}$. Using its constants gives energies $(3.31500,3.05436,2.88428)\,\mathrm{eV}$. Retain precision before subtracting.

Write $x_i=1/R_i$ and $E_i=E_g+Hx_i^2-S_Cx_i$. Define

$$a=x_1-x_2,\quad b=x_1^2-x_2^2,\quad c=x_2-x_3,\quad d=x_2^2-x_3^2,$$

$$u=E_1-E_2,\qquad v=E_2-E_3.$$

Here $d$ is a difference coefficient, not particle diameter. Subtraction eliminates $E_g$: $u=bH-aS_C$, $v=dH-cS_C$. Multiplying by $c$ and $a$ eliminates $S_C$:

$$H=\frac{cu-av}{bc-ad},\qquad S_C=\frac{bH-u}{a},\qquad E_g=E_1-Hx_1^2+S_Cx_1.$$

The results are $H=3.65106\,\mathrm{eV\,nm^2}$, $S_C=0.222994\,\mathrm{eV\,nm}$ and $E_g=2.82003\,\mathrm{eV}\simeq2.82\,\mathrm{eV}$. Substitution at $R=2.5\,\mathrm{nm}$ recovers $3.315\,\mathrm{eV}$. Geometrically, $E$ is quadratic in $1/R$; its intercept is $E_g$. Two points cannot generally determine all three coefficients.

Source: Aug14 Lecture 1 white-paper Q4, pp. 7–9.

1.10 Connecting properties to applications

• Photovoltaics: absorption creates carriers, but separation, transport and collection are required for current. Size-tunable absorption and solution processing are useful. Multiple-exciton generation can reduce high-energy-photon losses; the lecture's PbSe example reaches seven excitons for a photon near $7.8E_g$, not a guaranteed sevenfold device-efficiency improvement. In the related sensitized TiO$_2$ architecture, excited electrons enter TiO$_2$, an electrolyte regenerates the sensitizer, and the external circuit completes charge flow.
• LEDs, displays and lasers: tunable emission supports colour control. A nearby quantum well can transfer energy to dots without separately wiring each particle; the lecture example reports about $55\%$ transfer. A laser additionally needs pumping, gain and optical feedback.
• Biological imaging: surface attachment targets biomolecules; multiple colours and photostability support extended imaging. Compatibility depends on the actual material and coating.
• Single-photon sources: controlled excitation of one dot produces one exciton whose radiative decay can emit one photon, supporting quantum-information applications.
• Carbon-dot functions: sensing, fluorescence labels, anticounterfeiting, photocatalysis, drug delivery, optoelectronics and lubricant additives exploit different optical or surface properties. The Aug21 example shows PL quenching after an NH$_3$-related surface interaction. Quenching establishes a sensing response but does not, alone, identify a unique microscopic charge-transfer mechanism.

Sources: L01, pp. 75–87; Aug21 class notes, p. 3.

Lecture 2 — Nanowires

Nanowires connect material structure to measurable optical and electrical behaviour. The central variables are geometry, composition, defects and surface chemistry. “L02 LV” below means the 75-page Nanowires: Synthesis, Properties and Applications lecture PDF; references to exercises and assignments identify the local course materials.

2.1 Structure, dimensionality and characterization

A nanowire has two small transverse dimensions and one long axis. Nanorods are usually shorter, nanoneedles have pointed tips, nanobelts are flattened, and nanotubes are hollow. A one-dimensional morphology is not automatically a strongly confined quantum wire: transverse dimensions must be sufficiently small for confinement energies to matter. Carriers can remain mobile along the axis while their transverse motion is restricted.

High aspect ratio supports electrical connections, optical waveguiding and tip-field enhancement. A large surface-to-volume ratio makes adsorption and functionalization influential. Core–shell wires introduce radial interfaces; segmented wires introduce axial composition changes. Arrays, networks and bundles add collective effects, including electrical connectivity and electrostatic screening. Applications span FETs, sensors, optical waveguides, field-emission displays, photodetectors, lasers, photocatalysis, batteries and mechanical-to-electrical energy conversion.

Characterization methods answer different questions. SEM measures morphology, alignment, dimensions and density. HRTEM reveals lattice fringes and local disorder; discrete SAED spots support single crystallinity, whereas rings commonly indicate many crystallite orientations. EDS identifies elements, XRD examines phases and lattice spacings, and PL probes radiative transitions. Fluorescence colour alone cannot establish a complete structure or composition.

Sources: L02 LV, pp. 1–5, 12–14; handwritten notes, 21 August, pp. 5–7.

2.2 Four synthesis approaches

Vapour–liquid–solid growth (VLS). A catalyst droplet collects vapour-phase precursor material and forms a liquid alloy. Supersaturation drives nucleation and crystallization at the liquid–solid interface. Continued supply extends the solid wire beneath the droplet, which often remains at its tip. Droplet diameter primarily controls wire diameter. Temperature controls decomposition, dissolution, diffusion and crystallization; pressure, flow and time affect supply and growth. Axial growth must dominate radial deposition. A tip particle supports a VLS interpretation but is not conclusive without composition and growth-condition evidence.

Template growth. Parallel pores in anodized aluminium oxide (AAO) constrain the cross-section. A deposited Au layer provides a working electrode for metal electrodeposition or pyrrole electropolymerization. Filling the pores and removing AAO releases the wires. To make a core–shell structure, deposit the core, widen the pore or shrink a polymer core by drying, and deposit a shell in the resulting annular space. Alternating Ni and Cu deposition instead produces axial Ni/Cu segments.

Hotplate oxidation. Heating copper in air promotes oxidation, transport through the developing oxide layers, nucleation and anisotropic growth:

$$\mathrm{Cu}\xrightarrow{\mathrm{O_2}}\mathrm{Cu_2O} \xrightarrow{\mathrm{O_2}}\mathrm{CuO}.$$

The classroom demonstration uses approximately $400\,^{\circ}\mathrm{C}$ and shows changes after minutes to an hour. The cited field-emission samples used $390$–$430\,^{\circ}\mathrm{C}$ for one to three days. These are different experiments. The handwritten phrase “molten state of Cu” should not be interpreted as bulk copper melting at $400\,^{\circ}\mathrm{C}$; this is an oxidation and growth process, not automatically VLS.

Chemical vapour deposition (CVD). A tube furnace combines precursor delivery, heating, a reaction chamber, vacuum/exhaust equipment and flow/pressure control. Carrier gases transport material; oxygen or hydrogen can alter the reaction environment. Temperature gradients determine evaporation and deposition locations. CVD describes delivery and deposition, whereas VLS describes a microscopic growth mechanism; both can apply simultaneously. The ZnO:C example uses ZnO/C source powder, a ZnO-coated Si substrate, approximately $850/900\,^{\circ}\mathrm{C}$, $2\,\mathrm{mbar}$ and $100\,\mathrm{sccm}$ of $99\%$ Ar/$1\%$ oxygen. These are sample-specific conditions.

Sources: L02 LV, pp. 6–22; handwritten notes, 21 August, p. 5; CuO reference article, PDF pp. 2–3.

2.3 ZnO defects and photoluminescence

ZnO is a wurtzite, direct-gap semiconductor. “Direct” means the conduction-band minimum and valence-band maximum occur at the same wavevector $k$, allowing near-vertical optical transitions without an additional phonon for momentum conservation. The horizontal axis of an $E(k)$ diagram is reciprocal-space wavevector, not position in the sample.

A vacancy removes an atom; an interstitial occupies an additional site; an antisite occupies another species’ lattice position. Oxygen and zinc vacancies, interstitials and carbon-related complexes can introduce gap states. Defects may create radiative channels or non-radiative losses, so more defects do not necessarily mean stronger total PL.

PL follows absorption, excitation, relaxation or trapping, and radiative recombination. An excitation source, filters/beamsplitter, collection optics, grating spectrometer and CCD separate and measure the emitted light. Fluorescence microscopy maps where emission occurs; spectroscopy measures its wavelength distribution.

$$E_\gamma=h\nu=\frac{hc}{\lambda},\qquad \frac{E_\gamma}{\mathrm{eV}}\simeq\frac{1240}{\lambda/\mathrm{nm}}.$$

ZnO commonly shows a near-band-edge UV peak around $380\,\mathrm{nm}$ and broad visible defect emission. A photon measures the energy difference between initial and final states, not an isolated defect level’s energy relative to an arbitrary zero.

White-paper Q1. Strong green, yellow and orange-red fluorescence under UV indicates sub-gap radiative channels and differing defect environments. The supplied diagrams contain several possible transitions, including approximately $504$, $544$, $554$ and $602\,\mathrm{nm}$, and carbon-related transitions around $1.81$, $1.87$ and $2.20\,\mathrm{eV}$. Colour supports a defect-related interpretation but does not uniquely identify a defect or its concentration.

21 August assignment. Matching visible spectral bands to the photographs gives $A\to2$, $B\to3$, $C\to4$, $D\to1$. Mark the approximately $380\,\mathrm{nm}$ peak as near-band-edge/excitonic and the visible broad bands as defect emission. Spectrum 2 lacks an obvious UV peak; the solution calls it highly defective, meaning near-band-edge emission is strongly suppressed under those conditions.

The $380\,\mathrm{nm}$ peak corresponds to $3.26\,\mathrm{eV}$. It estimates a gap of order $3.3\,\mathrm{eV}$; adding a known $0.06\,\mathrm{eV}$ free-exciton binding energy gives approximately $3.32\,\mathrm{eV}$. Neither is an exact derivation of ZnO’s quoted typical $3.37\,\mathrm{eV}$ gap: peak reading, exciton identity, temperature and Stokes shifts matter. In the separate slide-29 example, orange $\mathrm{ZnO}_1$ matches the $630\,\mathrm{nm}$ spectrum B, and green $\mathrm{ZnO}_2$ matches the $512\,\mathrm{nm}$ spectrum A.

Sources: L02 LV, pp. 16–17, 21–29; Lecture 02 white-paper exercise, Q1; 21 August assignment and solution; handwritten notes, 21 August, pp. 6–7.

2.4 Excitons and absorption spectra

An exciton is a Coulomb-bound electron–hole pair. Its formation costs less energy than creating a separated pair, so the lowest exciton absorption energy is $E_g-E_b$, not $E_b$. The screened hydrogenic model for a spatially extended Wannier exciton gives

$$U(r)=-\frac{e^2}{4\pi\epsilon_0\epsilon_r r},\qquad \frac1\mu=\frac1{m_e^*}+\frac1{m_h^*},$$

$$E_b=\frac{\mu}{m_0}\frac{13.6\,\mathrm{eV}}{\epsilon_r^2}, \qquad E_n=E_g-\frac{E_b}{n^2}.$$

Here $\mu$ is the reduced effective mass, $m_0$ the free-electron mass, $\epsilon_r$ the relative permittivity and $n=1,2,\ldots$. Greater screening weakens binding; greater reduced mass strengthens it. The effective-mass, continuum-dielectric approximation does not directly describe every localized defect.

On an energy axis, discrete exciton absorption precedes the continuum beginning at $E_g$; higher-$n$ lines approach the continuum. On a wavelength axis the order reverses: the exciton peak lies at longer wavelength than the continuum edge.

$$E_b=hc\left(\frac1{\lambda_g}-\frac1{\lambda_{\mathrm{ex}}}\right), \qquad \lambda_{\mathrm{ex}}>\lambda_g.$$

White-paper Q2. Estimate the continuum background separately from the exciton feature. Slide 33 labels $\lambda_g\simeq353\,\mathrm{nm}$ and $\lambda_{\mathrm{ex}}\simeq360\,\mathrm{nm}$, giving $E_g\simeq3.513\,\mathrm{eV}$, $E_{\mathrm{ex}}\simeq3.444\,\mathrm{eV}$ and $E_b\simeq68\,\mathrm{meV}$. This graph estimate differs from the same slide’s typical ZnO gap; report the graph-based estimate rather than forcing agreement. A wavelength difference is not itself an energy difference. ZnO’s typical $60\,\mathrm{meV}$ binding exceeds room-temperature $k_BT\simeq26\,\mathrm{meV}$, supporting room-temperature excitonic emission.

White-paper Q3. With $E_g=2.7\,\mathrm{eV}$ and $E_b=0.165\,\mathrm{eV}$,

$$E_{\mathrm{ex}}=2.535\,\mathrm{eV},\qquad \lambda_g\simeq459.3\,\mathrm{nm},\qquad \lambda_{\mathrm{ex}}\simeq489.2\,\mathrm{nm}.$$

Draw continuum absorption on the short-wavelength side of $459\,\mathrm{nm}$ and an exciton peak near $489\,\mathrm{nm}$, returning to a low background at longer wavelengths.

Sources: L02 LV, pp. 30–33; Lecture 02 white-paper exercise, Q2–Q3; handwritten notes, 24 August, pp. 1–3.

2.5 Carbon-incorporated ZnO: electrical colour control and polarization

Carbon incorporation changes occupied and unoccupied defect states. Oxygen-vacancy-related emission is commonly green; certain carbon-rich complexes produce orange-red emission. HRTEM shows that crystalline, amorphous and core–shell regions can coexist.

In electrically tailored metachrosis, an individual ZnO:C wire bridges Au electrodes while current and UV-excited fluorescence are measured together. Moderate bias extracts carriers before radiative recombination, quenching PL. Larger currents and heating can redistribute carbon-related defects. The cited device exhibits reversible changes around $\pm15$–$\pm25\,\mathrm{V}$ and irreversible changes beyond approximately $\pm30\,\mathrm{V}$, with carbon removal and more green-emitting regions proposed. These are device-specific thresholds. Negative differential resistance, $dI/dV<0$, accompanies changes in defect/surface-controlled resistance.

The non-LV slides also identify electroluminescence: after electrical modification, emission near $600\,\mathrm{nm}$ can occur without UV illumination. A proposed mechanism involves an internal junction between regions with different carbon contents and defect-mediated recombination. Distinguish optically excited PL, electrically excited EL and thermal radiation.

Nanowire geometry and dielectric mismatch often favour weak polarization along the axis, while many defect bands are almost unpolarized. ZnO:C’s approximately $2.2\,\mathrm{eV}$, $560\,\mathrm{nm}$ yellow band instead has stronger perpendicular polarization. For analyser angle $\theta$ measured from the wire axis,

$$I(\theta)=I_{\parallel}\cos^2\theta+I_{\perp}\sin^2\theta.$$

Maxima at $90^\circ$ and $270^\circ$ identify perpendicular emission; maxima at $0^\circ$ and $180^\circ$ identify parallel emission. The slide’s green and red bands show weaker parallel tendencies. Excitation polarization and analyser orientation are separate variables.

The proposed yellow-band mechanism connects the highest occupied defect level (HODL) and lowest unoccupied defect level (LUDL), separated by about $2.2\,\mathrm{eV}$. De-excitation from Zn $d$ to C $p$ orbitals produces a transition dipole predominantly perpendicular to the wire’s $c$ axis, with the shortest Zn–C bond contributing most strongly. This assignment concerns that emission component, not every ZnO defect band.

Sources: L02 LV, pp. 21–26, 34–43; non-LV p. 34; ACS Nano 2020 and Nano Research 2023 reference articles, PDF pp. 5–8.

2.6 Composition engineering in cadmium sulfoselenide

In $\mathrm{CdS}_x\mathrm{Se}_{1-x}$, $x$ is the sulfur fraction on anion sites. The endpoints are CdSe at $x=0$, with gap $1.74\,\mathrm{eV}$, and CdS at $x=1$, with gap $2.42\,\mathrm{eV}$. Composition changes orbital coupling and lattice structure, providing gap tuning without changing particle size.

$$E_g(x)=xE_{\mathrm{CdS}}+(1-x)E_{\mathrm{CdSe}}-bx(1-x).$$

With the lecture’s bowing parameter $b\simeq0.58\,\mathrm{eV}$,

$$\frac{E_g(x)}{\mathrm{eV}}=1.74+0.10x+0.58x^2,$$

$$x=\frac{-0.10+\sqrt{0.10^2+4(0.58)(E_g/\mathrm{eV}-1.74)}}{2(0.58)}.$$

Positive bowing places the gap below linear interpolation. More sulfur gives a larger gap and shorter-wavelength near-band-edge PL. Inverting a PL peak assumes its energy approximates the gap and neglects separately unspecified exciton, strain and Stokes corrections.

White-paper Q6. Read peaks, convert to energy, solve for $0\le x\le1$, and check the endpoints:

Sample	Approximate peak	Photon energy	Estimated sulfur fraction
A	$570\,\mathrm{nm}$	$2.175\,\mathrm{eV}$	$0.78$
B	$595\,\mathrm{nm}$	$2.084\,\mathrm{eV}$	$0.69$
C	$512\,\mathrm{nm}$	$2.422\,\mathrm{eV}$	$1.00$
D	$615\,\mathrm{nm}$	$2.016\,\mathrm{eV}$	$0.61$

These are graph/model estimates, appropriately reported as about $0.8$, $0.7$, $1$ and $0.6$. A slightly super-unity result for C reflects reading/endpoint approximations. The five compositions printed on slide 48 belong to a different dataset and must not be assigned to these four samples by colour alone.

Tube-furnace position controls the supplied S/Se ratio. Continuous XRD peak shifts, together with EDS and microscopy, support alloy formation; three elemental EDS peaks alone do not prove atomic-scale uniformity. Nanobelt-network FETs show larger current at more positive gate bias, light-induced transfer-curve shifts and repeated photocurrent switching.

Sources: L02 LV, pp. 44–49; Lecture 02 white-paper exercise, Q6; handwritten notes, 24 August, pp. 4–7.

2.7 Field emission: barriers, geometry and measurement

Field emission uses an electric field to narrow the surface barrier, allowing electrons near the Fermi level to tunnel into vacuum. Thermionic emission supplies thermal energy to cross the barrier; photoemission supplies photon energy. The work function and relevant fields are

$$\phi=E_{\mathrm{vac}}-E_F,\qquad E=\frac{V}{d},\qquad F=\beta E.$$

Here $E$ is the average field across electrode separation $d$, $F$ the local emitting field and $\beta$ the dimensionless enhancement factor. A small tip radius concentrates surface charge; the isolated-tip estimate $F\sim V/R$ explains the trend, not a complete device calculation. Neighbouring wires screen one another, particularly short wires between tall ones. More emitters therefore do not guarantee better emission; edge and exposed tips often dominate.

A departing electron induces an attractive image charge. Writing the work function in joules as $W$, the ideal planar image-corrected potential is

$$U(z)-E_F=W-eFz-\frac{e^2}{16\pi\epsilon_0z}.$$

Slide 53 uses an $8\pi$ denominator, corresponding to the direct electron–image pair energy. The induced-charge self-energy contains an additional factor of one-half, giving $16\pi$. The elementary FN derivation below neglects this correction entirely.

The nanowires form the emitting cathode; the collecting ITO/phosphor screen is the anode. Apply voltage, measure $I$, calculate $J=I/A_{\mathrm{sample}}$, and inspect phosphor emission for spatial uniformity. The example uses approximately $8\times10^{-7}\,\mathrm{Torr}$ and $260\,\mu\mathrm{m}$ spacing. Slide 58 reverses electrode labels, and slide 66 misidentifies the phosphor electrode; follow electron flow. The separation is micrometres, not millimetres. Transparent ITO/ZnO devices can transmit light while displaying an emission pattern.

Sources: L02 LV, pp. 51–58, 60–66; handwritten notes, 24 and 28 August; FN supporting notes, PDF pp. 1–2, 13–14.

2.8 Deriving the low-temperature Fowler–Nordheim equation

Assume a free-electron gas, Fermi–Dirac occupancy, a locally one-dimensional barrier, WKB tunnelling and dominant emission near $E_F$. Let $\phi$ be quoted in electronvolts and $W$ its joule value:

$$W=\frac{\phi}{\mathrm{eV}}\left(1.602176634\times10^{-19}\,\mathrm{J}\right).$$

All energies inside the following integrals use joules. Write transmission as $\mathcal T$ to distinguish it from temperature $T$.

Electron supply. Current is incident electron flux multiplied by transmission. Integrating over transverse momenta and both spins gives

$$J=\frac{4\pi emk_BT}{h^3}\int_0^\infty \mathcal T(E_x)\ln\left[1+\exp\left(\frac{E_F-E_x}{k_BT}\right)\right]\,\mathrm dE_x.$$

At $T\to0$, the thermal factor becomes $E_F-E_x$ below $E_F$ and zero above it. Thus

$$J=\frac{4\pi em}{h^3}\int_0^{E_F} (E_F-E_x)\mathcal T(E_x)\,\mathrm dE_x.$$

Here $E_x$ is kinetic energy normal to the emitting surface and $m$ is the electron mass. Equivalently the supply flux per energy is $N(E_x)=4\pi m(E_F-E_x)/h^3$ on this interval, with $J=e\int N\mathcal T\,\mathrm dE_x$.

Barrier transmission. Between classical turning points,

$$\mathcal T(E_x)\simeq\exp\left[-\frac2\hbar \int_{z_1}^{z_2}\sqrt{2m[U(z)-E_x]}\,\mathrm dz\right].$$

For the triangular approximation $U-E_F=W-eFz$, the barrier width is $W/(eF)$ and

$$\int_0^{W/(eF)}\sqrt{W-eFz}\,\mathrm dz =\frac{2W^{3/2}}{3eF},$$

$$\mathcal T(E_F)=\exp\left[-\frac{4\sqrt{2m}\,W^{3/2}}{3e\hbar F}\right].$$

Near-Fermi expansion. Set $u=E_F-E_x$ and expand the exponent about $E_F$:

$$d_F=\frac{e\hbar F}{2\sqrt{2mW}},\qquad \mathcal T(E_x)\simeq\mathcal T(E_F)e^{-u/d_F}.$$

Extending the $u$ integral to infinity is justified when the narrow near-Fermi window dominates, not because arbitrary negative kinetic energies are physical. Since

$$\int_0^\infty u e^{-u/d_F}\,\mathrm du=d_F^2,$$

$$J\simeq\frac{4\pi em}{h^3}\mathcal T(E_F)d_F^2,$$

we obtain

$$J=\frac{e^3F^2}{8\pi hW} \exp\left[-\frac{8\pi\sqrt{2m}\,W^{3/2}}{3heF}\right].$$

The practical form is

$$J=\frac{a_{\mathrm{FN}}}{\phi}F^2 \exp\left[-\frac{b_{\mathrm{FN}}\phi^{3/2}}F\right],$$

$$a_{\mathrm{FN}}=1.54\times10^{-6}\,\mathrm{A\,eV\,V^{-2}},$$

$$b_{\mathrm{FN}}=6.83\times10^9\,\mathrm{V\,m^{-1}\,eV^{-3/2}}.$$

Use $\phi$ in $\mathrm{eV}$, $F$ in $\mathrm{V/m}$ and obtain $J$ in $\mathrm{A/m^2}$. The lecture combines prefactors into $A$, writing $J=AF^2\exp[-B\phi^{3/2}/F]$, with positive $B$. Check definitions: some sources instead absorb $\phi^{3/2}$ into $B$.

The supporting derivation gives the finite-temperature multiplier $\pi c_T/\sin(\pi c_T)$, where $c_T=k_BT/d_F<1$; it tends to one as $T\to0$. Supply limitation, surface states, series resistance and changing geometry can invalidate an elementary FN fit. The supporting notes’ inconsistent sign for $B$ must not turn the tunnelling exponent positive.

Sources: L02 LV, pp. 54, 56; FN supporting notes, PDF pp. 3–12.

2.9 FN plots, UPS and worked emission problems

Substituting $F=\beta E$ and taking logarithms gives

$$\ln\left(\frac{J}{E^2}\right) =\ln(A\beta^2)-\frac{B\phi^{3/2}}\beta\frac1E, \qquad S=-\frac{B\phi^{3/2}}\beta.$$

Plot $\ln(J/E^2)$ against $1/E$, using consistent units. At fixed work function, larger $\beta$ gives a less negative, shallower slope. At fixed enhancement, larger $\phi$ gives a steeper slope:

$$\frac{S_B}{S_A}=\left(\frac{\phi_B}{\phi_A}\right)^{3/2} \frac{\beta_A}{\beta_B}.$$

A slope determines only $\phi^{3/2}/\beta$. Unknown emitting area also affects the intercept, so FN data alone do not generally determine both $\phi$ and $\beta$. Slope alone does not rank current at every field.

UPS independently probes the work function using photoelectron spectral edges. On a calibrated kinetic-energy axis,

$$\phi=h\nu-\left(E_F^{\mathrm{kin}}-E_{\mathrm{cut}}^{\mathrm{kin}}\right).$$

Use the separation between the Fermi edge and secondary-electron cutoff, correcting for bias and axis direction. The intensity maximum is not the work function. He I supplies $21.2\,\mathrm{eV}$ photons ($58.4\,\mathrm{nm}$); $40.8\,\mathrm{eV}$ is another helium line. UPS predominantly accesses valence and shallow states. Where the exercise directly labels a work-function axis, read the onset.

White-paper Q4 / 24 August assignment. Both lines drop by 10 vertical units, with horizontal runs 2.5 and 1.5:

$$S_A=-\frac{10}{2.5}=-4,\qquad S_B=-\frac{10}{1.5}\simeq-6.67,$$

$$\frac{\beta_A}{\beta_B}=\frac{S_B}{S_A}=\frac53\simeq1.67.$$

Positive “slopes” in the supplied solution denote magnitudes; the plotted slopes are negative.

White-paper Q5. Given $\beta_A/\beta_B=0.9$, $\phi_A=5.5\,\mathrm{eV}$ and $\phi_B=3.8\,\mathrm{eV}$,

$$\frac{S_B}{S_A}=0.9\left(\frac{3.8}{5.5}\right)^{3/2}\simeq0.5169.$$

Draw B through the specified common point with approximately half A’s downward slope; if $S_A\simeq-45$, then $S_B\simeq-23.3$. The stipulated vertical offset is a plotting instruction, not proof of equal physical intercepts.

28 August assignment. The onsets give $(\phi_1,\phi_2,\phi_3)=(3.0,3.5,4.0)\,\mathrm{eV}$. Applied $1000\,\mathrm{V}$ over $100\,\mu\mathrm{m}$ gives $E=10\,\mathrm{V}/\mu\mathrm{m}$. Local fields $(20000,38000,15400)\,\mathrm{V}/\mu\mathrm{m}$ give $(\beta_1,\beta_2,\beta_3)=(2000,3800,1540)$. Beware the diagram’s spatial order 2, 1, 3.

$$S_2=(-22.5)\left(\frac{3.5}{3.0}\right)^{3/2}\frac{2000}{3800} \simeq-14.92,$$

$$S_3=(-22.5)\left(\frac{4.0}{3.0}\right)^{3/2}\frac{2000}{1540} \simeq-44.99.$$

In the graph’s units, line 2 is shallowest, line 1 intermediate and line 3 steepest. Through $(x_0,y_0)$ use $y=y_0+S_i(x-x_0)$. The supplied solution omits minus signs in one formula but uses negative slopes numerically.

CuO example. Slopes $-17.6$ and $-44.3$ imply $\beta_I/\beta_{II}\simeq2.52$ for equal $\phi$. Finite-element values 1570 and 670 give 2.34. The cited model uses $B=6440$ with fields in $\mathrm{V}/\mu\mathrm{m}$, yielding work functions around $2.6$–$2.8\,\mathrm{eV}$ from $\phi=(\lvert S\rvert\beta/B)^{2/3}$; use that stated coefficient to reproduce its results, not the SI number directly. The ordinary FN coefficient converts to about 6830 in these field units. Example performance is a $3.5$–$4.5\,\mathrm{V}/\mu\mathrm{m}$ turn-on field and $0.45\,\mathrm{mA/cm^2}$ at $7\,\mathrm{V}/\mu\mathrm{m}$. Turn-on means reaching a specified current criterion, not zero tunnelling below it.

Sources: L02 LV, pp. 56, 60–65; Lecture 02 white-paper exercise, Q4–Q5; 24 and 28 August assignments and solutions; CuO reference article, PDF pp. 4–5.

2.10 Tungsten-oxide photodetection and surface oxygen

Bridge electrodes with tungsten-oxide wires, hold bias fixed and record $I(t)$ during light switching. Blue light increases current, which decays after switching off:

$$\sigma=e(n\mu_e+p\mu_h),\qquad I_{\mathrm{ph}}=I_{\mathrm{light}}-I_{\mathrm{dark}}.$$

Photogenerated carriers increase conductivity. In the $n$-type surface model, adsorbed oxygen traps electrons, creating depletion; photogenerated holes neutralize the adsorbed ions and promote desorption:

$$\mathrm{O_2(g)}+e^-\rightarrow\mathrm{O_2^-(ads)},$$

$$\mathrm{O_2^-(ads)}+h^+\rightarrow\mathrm{O_2(g)}.$$

Reduced depletion leaves more mobile electrons. At low pressure, stronger response and slow recovery indicate restricted oxygen readsorption. Returning to air restores the smaller response. Atmospheric-pressure nitrogen also produces a stronger response than air, while reducing nitrogen pressure to $0.35\,\mathrm{Torr}$ makes a smaller difference. This distinguishes oxygen chemistry from total-pressure effects. Red light produces little switching under comparable nitrogen conditions, supporting wavelength selectivity without precisely determining the gap.

Read dark current, photocurrent increment, rise time, decay time, persistence and repeatability separately. A large response may come with slow recovery because of surface or trap kinetics.

Sources: L02 LV, pp. 67–70; handwritten notes, 28 August, pp. 4–5.

2.11 Nanowire lasers and optical feedback

A laser requires optical gain and feedback, with stimulated emission overcoming losses. Pumping creates the carrier population required for gain. ZnO nanowires can simultaneously provide a gain medium, waveguide and Fabry–Pérot cavity: naturally faceted ends reflect light back through the excited wire.

Ignoring dispersion and additional reflection phases, longitudinal resonances satisfy

$$2n_{\mathrm{opt}}L=m\lambda,\qquad m=1,2,\ldots,$$

where $L$ is cavity length and $n_{\mathrm{opt}}$ refractive index. Above threshold, narrow UV emission emerges from a broad PL background and exits the end facets. Look for a threshold change in output versus pump, spectral narrowing and directional end emission together. Bright PL alone does not establish lasing. The cited threshold below approximately $50\,\mathrm{kW/cm^2}$ is optical pump intensity for that experiment, not a universal threshold or voltage. Such wires can supply nanoscale UV excitation for molecular detection.

Sources: L02 LV, pp. 5, 72–73; handwritten notes, 21 August, p. 4.

2.12 Functionalized sensors, logic and energy applications

A Si wire coated with silica can carry selective molecular probes. Surface silanols bind suitable silane coupling agents containing Si–Cl or Si–OR groups. Target binding changes local charge and potential, gating the narrow conducting channel. For pH sensing, amines can become $\mathrm{NH_3^+}$ and silanols can deprotonate. The sign of the conductance response depends on surface charge, channel doping and operating point; binding does not universally increase current.

Two-terminal nanowire diodes simplify connectivity; three-terminal FETs provide gate control and voltage gain. Their combinations implement AND, OR and NOR logic. AND requires both inputs high, OR requires at least one high, and NOR is high only when both are low. Crossed wires can define junctions and controlled channel regions.

Other examples include NiO nanowall electrodes, mechanical size–structure–property studies and piezoelectric generators, where stress changes polarization and electrical potential. The $\alpha$-$\mathrm{Fe_2O_3}$ nanoflake battery example reports approximately $680\pm20\,\mathrm{mAh/g}$ and discusses Fe/$\mathrm{Li_2O}$ conversion chemistry. Nanostructuring can shorten transport paths, increase active interfaces and change cycling behaviour. Across these devices, connect each geometry or chemical modification to its physical mechanism and measured signal.

Sources: L02 LV, pp. 4–5, 13–14, 71, 74–75.

Lecture 3 — Carbon nanotubes and density of states

Carbon materials show how bonding, geometry and quantum boundary conditions determine useful properties. The central chain is atomic structure → allowed electronic states → measurable response → device function.

Source notation: L3/L4 are the Lecture 3/4 carbon-material PDFs; “PPT” identifies additional PowerPoint slides. “Assignment” refers to the dated yellow-paper exercise, and “Notes” to the dated handwritten class notes. Page numbers refer to PDF pages.

3.1 Carbon bonding and allotropes

In $\mathrm{sp}^3$ bonding, four directed orbitals form a tetrahedral network. Diamond therefore has strong covalent bonds throughout three dimensions and, ordinarily, low electrical conductivity. In $\mathrm{sp}^2$ bonding, three in-plane $\sigma$ bonds leave a perpendicular $p_z$ orbital available for a delocalized $\pi$ system. Graphite consists of strongly bonded sheets with much weaker interlayer interactions: it conducts especially well within sheets and can be cleaved between them. Graphene is one such sheet. Graphite is thermodynamically more stable than diamond at ambient pressure, but a large transformation barrier makes diamond persist.

The fullerene $\mathrm C_{60}$ contains 12 pentagons and 20 hexagons. Pentagons introduce curvature into the otherwise hexagonal network, closing a molecular cage. Its discovery in 1985 led to the 1996 Chemistry Nobel Prize.

A carbon nanotube, or CNT, can be represented geometrically as rolled graphene. A single-walled CNT has one cylindrical wall; a multiwalled CNT has several concentric walls. The lecture gives representative diameters of $1$–$30\,\mathrm{nm}$, micrometre lengths, and wall spacings around $0.347\,\mathrm{nm}$, compared with approximately $0.337\,\mathrm{nm}$ in graphite. “Rolling” describes geometry, not necessarily a fabrication step. Electronic motion is effectively two-dimensional in graphene and one-dimensional along a CNT. Isolated fullerene molecules provide a zero-dimensional example with discrete molecular levels. Claims such as “strongest fibre” describe ideal-material potential; defects, orientation and contacts affect real samples.

Sources: L3 pp. 4–10, 33.

3.2 CNT chirality, radius and the August 28(b) exercise

Let $\mathbf a_1$ and $\mathbf a_2$ be equal-length graphene primitive vectors separated by $60^\circ$. Distinguish the carbon–carbon bond length from the lattice constant:

$$a_{CC}=0.142\,\mathrm{nm},\qquad a=\sqrt3a_{CC}\simeq0.246\,\mathrm{nm}.$$

The chiral vector connects equivalent points joined when the sheet closes. Its length is the circumference, and the tube axis is perpendicular to it:

$$\mathbf C_h=n\mathbf a_1+m\mathbf a_2,\qquad |\mathbf C_h|=a\sqrt{n^2+nm+m^2}.$$

$$d=\frac{|\mathbf C_h|}{\pi},\qquad R=\frac{d}{2},\qquad \tan\theta=\frac{\sqrt3m}{2n+m}.$$

Equivalently,

$$\cos\theta=\frac{2n+m}{2\sqrt{n^2+nm+m^2}}.$$

Under the usual convention $n\ge m\ge0$, zigzag tubes have $(n,0)$ and $\theta=0^\circ$; armchair tubes have $(n,n)$ and $\theta=30^\circ$; the remaining cases are chiral. Lattice symmetries reduce the classification to $0^\circ\le\theta\le30^\circ$, although opposite handedness can still matter experimentally.

To identify a model, unfold its lattice mentally, identify equivalent joined sites and count primitive-vector steps. Do not count each carbon–carbon bond as one lattice vector. The assignment answers are:

• A: $(13,9)$, chiral; $\tan\theta=9\sqrt3/35$, so $\theta\simeq24.0^\circ$. The calculated diameter is $1.500\,\mathrm{nm}$.
• B: $(14,14)$, armchair; $\theta=30^\circ$ and $d\simeq1.898\,\mathrm{nm}$.
• E: $(14,5)$, chiral; $\tan\theta=5\sqrt3/33$, so $\theta\simeq14.7^\circ$ and $d\simeq1.336\,\mathrm{nm}$.
• F: $(19,0)$, zigzag; $\theta=0^\circ$ and $d\simeq1.487\,\mathrm{nm}$.

The diameter calculations extend the official identification exercise. Remember that a radius requires division by $2\pi$, not $\pi$.

Sources: L3 pp. 11–16, 34; Aug28(b) Assignment and solution pp. 1–2.

3.3 Counting states: the three-dimensional electron gas

The density of states $D(E)$ counts available states per energy, not occupied electrons. The occupied number in an interval is $D(E)f(E)\,dE$, with

$$f(E)=\frac{1}{\exp[(E-\mu_{\mathrm{chem}})/(k_BT)]+1}.$$

At zero temperature, $\mu_{\mathrm{chem}}=E_F$: states below the Fermi energy are occupied and those above are empty. Here, ordinary electron-gas DOS means the total DOS of the sample, including spin degeneracy two. Divide by volume, area or length for an intensive DOS.

For constant potential, the Schrödinger equation has plane-wave solutions:

$$-\frac{\hbar^2}{2m}\nabla^2\psi=E\psi,\qquad \psi=C e^{i\mathbf k\cdot\mathbf r},\qquad E=\frac{\hbar^2k^2}{2m}.$$

Periodic boundaries on a cube of side $L$ require $k_j=2\pi n_j/L$, where $n_j$ may be zero, positive or negative. Each spatial state occupies $(2\pi/L)^3$ in $k$ space. Filling a sphere of radius $k_F$, including two spins, gives

$$N=2\frac{(4\pi/3)k_F^3}{(2\pi/L)^3} =\frac{Vk_F^3}{3\pi^2},\qquad k_F=(3\pi^2n_e)^{1/3},\quad n_e=\frac NV.$$

$$E_F=\frac{\hbar^2}{2m}(3\pi^2n_e)^{2/3}.$$

A spherical shell has volume $4\pi k^2dk$. Dividing by the single-state volume and including spin yields $dN_{\mathrm{states}}=Vk^2dk/\pi^2$. Convert from wavevector to energy using

$$D(E)=\frac{dN_{\mathrm{states}}/dk}{dE/dk},\qquad D_{3D}(E)=\frac{V}{2\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2}\sqrt E.$$

Integrating should recover $N$, providing a useful check on degeneracy factors. The mean energy follows from the ratio of energy and number integrals:

$$\langle E\rangle= \frac{\int_0^{E_F}ED_{3D}(E)\,dE}{\int_0^{E_F}D_{3D}(E)\,dE} =\frac35E_F,\qquad U=\frac35NE_F.$$

This result requires a zero-temperature, three-dimensional parabolic band. Energy is measured from the band minimum; use $E-E_c$ if the minimum is labelled $E_c$. The handwritten notes also emphasize that equal band gaps need not imply equal absorption: available initial/final states and transition matrix elements matter.

Sources: L3 pp. 17–23; Aug31 Notes pp. 1–3.

3.4 Quantum wells, wires and dots: subband thresholds

Free directions retain continuous wavevectors; confined directions have discrete quantum numbers. Each confined eigenstate defines a subband minimum $E_i$. Below that threshold the subband contributes no states. For hard-wall widths $a_j$,

$$E_i=\frac{\pi^2\hbar^2}{2m}\sum_{j\in\mathrm{confined}}\frac{n_j^2}{a_j^2}.$$

A quantum well confines one direction and leaves two free:

$$E=E_i+\frac{\hbar^2(k_x^2+k_y^2)}{2m},\qquad D_{2D}(E)=\frac{Am}{\pi\hbar^2}\sum_i d_i\Theta(E-E_i).$$

The circular annulus $2\pi kdk$ produces $dN_{\mathrm{states}}=Akdk/\pi$, including spin. Division by $dE/dk$ leaves a constant for each subband, producing a staircase. This constant DOS depends on parabolic dispersion, not dimensionality alone.

A quantum wire confines two directions:

$$E=E_i+\frac{\hbar^2k_z^2}{2m},\qquad D_{1D}(E)=\frac{L\sqrt{2m}}{\pi\hbar} \sum_i\frac{d_i\Theta(E-E_i)}{\sqrt{E-E_i}}.$$

Both $+k$ and $-k$, together with two spins, give $dN_{\mathrm{states}}=2Ldk/\pi$. Every term has its own denominator; evaluate its contribution only above its threshold. Near a band minimum, small $dE/dk$ packs many states into a small energy interval, producing a van Hove singularity. Temperature, disorder and finite lifetimes broaden actual peaks.

Here $d_i$ counts additional orbital degeneracy only. A square wire gives equal energies for $(n_x,n_y)=(1,2)$ and $(2,1)$, hence $d_i=2$ for that threshold. Do not multiply again if both combinations were already separately included. The lecture's one-dimensional prefactor is smaller by two; the expression above explicitly counts both propagation directions and spin. A single-spin convention gives half this DOS.

A quantum dot confines all directions:

$$E_{n_xn_yn_z}=\frac{\pi^2\hbar^2}{2m} \left(\frac{n_x^2}{a_x^2}+\frac{n_y^2}{a_y^2}+\frac{n_z^2}{a_z^2}\right), \qquad D_{0D}(E)=2\sum_i d_i\delta(E-E_i).$$

The delta-peak area, $2d_i$, counts states. An infinite ideal peak height does not mean infinitely many electrons. Sketch recognition: three-dimensional square-root increase; two-dimensional steps; one-dimensional threshold peaks; zero-dimensional discrete lines.

Sources: L3 pp. 24–32; Aug31 Notes pp. 3–4.

3.5 CNT electrical classification, STM and STS

The wavefunction must return to itself around a nanotube:

$$\mathbf k\cdot\mathbf C_h=2\pi q,\qquad q\in\mathbb Z.$$

This selects lines through graphene's two-dimensional band structure. If an allowed line passes through a Dirac point, conduction and valence bands meet; otherwise a gap appears. In ideal zone folding,

$$n-m=3\ell\quad\Longrightarrow\quad\text{metallic},\qquad \ell\in\mathbb Z.$$

Thus every armchair tube is metallic; zigzag tubes are metallic when $n$ is divisible by three. Examples: $(5,5)$ and $(9,0)$ are metallic, whereas $(10,0)$ is semiconducting. Roughly one-third of index combinations are metallic. Small non-armchair tubes can acquire curvature-induced gaps. Applying the rule to Aug28(b) gives A/F semiconducting and B/E metallic in the ideal model: chiral does not necessarily mean semiconducting.

In scanning tunnelling microscopy, tip and sample wavefunctions overlap across a vacuum barrier. Elastic tunnelling requires an occupied initial state and an empty final state at matching energy. At equilibrium and zero bias there is no net tunnelling current. Define

$$V_s=V_{\mathrm{sample}}-V_{\mathrm{tip}}.$$

For $V_s>0$, tip electrons probe empty sample states; for $V_s<0$, occupied sample states supply electrons to the tip. Equivalently, a negative tip raises its electronic energies relative to the sample. Always state the voltage convention.

STS holds tip position and height fixed while sweeping bias. Generally, current integrates the product of tip DOS, sample DOS, squared tunnelling matrix element and occupation difference. At low temperature, with approximately constant tip DOS and matrix element,

$$I(V_s)\propto\int_0^{eV_s}\rho_s(E_F+\epsilon)\,d\epsilon, \qquad \frac{dI}{dV_s}\propto\rho_s(E_F+eV_s).$$

Only with small bias and nearly constant sample DOS does this simplify further:

$$I\propto V_s\rho_s(E_F)e^{-2\kappa z},\qquad \kappa\simeq\frac{\sqrt{2m\phi}}{\hbar}.$$

Here $\phi$ is expressed in joules. The exponential distance dependence explains why constant height matters and why general $I(V)$ cannot simply be labelled DOS.

Aug31 exercise: all four $(8,8)$–$(11,11)$ tubes are metallic. Increasing diameter reduces circumferential wavevector spacing, approximately $2\pi/|\mathbf C_h|$, and corresponding subband energy separation. Compare the same-order peaks: figure spacing ranks $(b)>(c)>(a)>(d)$, corresponding to $(8,8),(9,9),(10,10),(11,11)$. Therefore $(9,9)$ is figure (c), the second-widest spacing. Peak height is not the criterion; a low-DOS interval between metallic peaks is not automatically a gap. The correctly matched solution is the Aug31 DOCX; its PDF filename contains the Sep04 handwritten solution instead.

Sources: L3 pp. 35–40, 49; PPT slides 38–47; Aug31 Assignment and solution DOCX.

3.6 Making CNTs and locating the catalyst

All synthesis routes supply active carbon and allow it to reorganize into tubular networks.

• Arc discharge: an arc between graphite electrodes vaporizes carbon in a controlled gas environment. Condensation produces nanotubes alongside material requiring purification.
• Laser ablation: a laser vaporizes a graphite target. The lecture shows an approximately $1200^\circ\mathrm C$ furnace, argon transport and a cooled copper collector. This creates feedstock vapour; it is different from trimming existing CNTs.
• Catalytic decomposition/CVD: hot catalyst particles decompose carbon-containing gases. The lecture's $\mathrm{Co}/\mathrm{La}_2\mathrm O_3$ example operates around $600$–$750^\circ\mathrm C$:

$$2\mathrm{CO}\rightarrow\mathrm C+\mathrm{CO}_2, \qquad \mathrm{CH}_4\rightarrow\mathrm C+2\mathrm H_2.$$

PECVD activates gases with plasma and can produce aligned arrays. The example uses Fe/Ni nanoparticles, Si or quartz, $\mathrm C_2\mathrm H_2$ with $\mathrm H_2$ or $\mathrm{NH}_3$, representative flows of 30 and 15 sccm, and $700$–$800^\circ\mathrm C$. Catalyst films may be deposited by sputtering or pulsed-laser deposition. The listed $10^{-6}\,\mathrm{Torr}$ is not clearly distinguished as base or working pressure, so it should not be generalized as a universal growth pressure.

In root growth, strong catalyst–substrate attachment keeps the particle at the base. In tip growth, weaker attachment allows the growing tube to lift the particle. Carbon adsorption/decomposition, migration and incorporation into the wall underlie both models.

Sources: L3 pp. 41–44; L4 pp. 42–43.

3.7 CNT devices, limitations and GNR junctions

A semiconducting CNT bridging source and drain above $\mathrm{SiO}_2$ can form a FET, with conducting silicon acting as back gate. Gate voltage changes carriers and injection, controlling current. The lecture example has approximately $1\,\mathrm{nm}$ diameter and $400\,\mathrm{nm}$ electrode spacing. Metallic tubes in a network provide paths that cannot be switched off effectively.

Ballistic transport occurs when device length is small compared with the relevant scattering length. Resistance then depends strongly on channels and contacts rather than increasing conventionally with length. The lecture's sweeping “no heating at any temperature” description should not be taken literally: scattering and contact dissipation remain possible. A small CNT island connected through resistive junctions can show Coulomb blockade when its charging energy dominates thermal fluctuations:

$$E_C\sim\frac{e^2}{2C},\qquad E_C\gg k_BT.$$

Low bias then cannot readily add another electron; a gate shifts addition energies.

Other property–application links include sharp tips and high aspect ratio for field emission; hollow interiors as nanoscale test tubes containing particles such as silver; strong bonding for mechanical reinforcement; and large surface area for membranes, sensors and proposed hydrogen storage. Dense emitter arrays can screen each other. Aligned CVD-grown CNTs favour directional transport, whereas solution-deposited networks offer scalable printing for flexible electronics. Uniform chirality, placement, purification and low-resistance contacts remain important limitations; sorting, doping and heterostructures provide tuning strategies. Moore's law is an empirical integration trend, not a physical guarantee of CNT replacement of silicon.

The PPT additionally shows a graphene-nanoribbon–CNT intramolecular junction. PMMA protects part of a nanotube on $\mathrm{SiO}_2/\mathrm{Si}$ during Zn sputtering; removing Zn and PMMA leaves an opened ribbon joined to the protected tube. Width and edges modify ribbon confinement. Light-switching measurements show photoconductivity; a scanned spot locates junction response; dark/light $I$–$V$ differences demonstrate optical functionality.

Sources: L3 pp. 45–55; PPT slide 62.

Lecture 4 — Graphene and carbon-based devices

4.1 Graphene properties, growth and graphene oxide

Graphene's honeycomb lattice has two equivalent sublattices, A and B, with two atoms per primitive cell. In-plane $\mathrm{sp}^2$ bonds provide stiffness, while $p_z$-derived $\pi$ states dominate low-energy electronics. Isolation in 2004 led to the 2010 Physics Nobel Prize for Geim and Novoselov. Representative lecture values include Young's modulus around $1$–$1.3\,\mathrm{TPa}$, resistivity near $10^{-6}\,\Omega\,\mathrm{cm}$ and mobility up to $200{,}000\,\mathrm{cm^2\,V^{-1}\,s^{-1}}$. These characterize particular high-quality samples, not every device.

Mechanical exfoliation exploits weak interlayer bonding to obtain excellent but relatively small flakes. CVD decomposes a carbon source on heated metals, producing larger films that can be transferred, including by electrochemical delamination. Carbon solubility differs between metals, so surface growth and cooling-induced segregation do not contribute identically on every catalyst. Large-area compatibility with lithography brings challenges from grain boundaries, wrinkles and transfer residues.

Graphene oxide, GO, contains oxygen functionalities such as hydroxyl, epoxy and carboxyl groups. They aid dispersion and functionalization but disrupt conjugation and reduce conductivity. Reduction restores part of the network; rGO still contains residual oxygen and defects. It is not equivalent to pristine graphene. The lecture also lists gating, strain, proximity, moiré patterns and light as tuning tools; this brief outlook does not require importing all of Lecture 5's magnetic-material theory.

Sources: L4 pp. 3–7, 9, 17–19, 31.

4.2 Bloch waves and the graphene tight-binding bands

For a periodic potential, Bloch's theorem gives

$$\psi_{\mathbf k}(\mathbf r)=e^{i\mathbf k\cdot\mathbf r}u_{\mathbf k}(\mathbf r), \qquad u_{\mathbf k}(\mathbf r+\mathbf R)=u_{\mathbf k}(\mathbf r).$$

The periodic modulation replaces the constant amplitude of a free-electron plane wave. Using one $p_z$ orbital per carbon, a linear combination of atomic orbitals constructs

$$\psi_{\mathbf k}=\sum_{\mathbf R}e^{i\mathbf k\cdot\mathbf R} [b_A\phi_A(\mathbf r-\mathbf R)+b_B\phi_B(\mathbf r-\mathbf R)].$$

Set the equal on-site energies to zero and retain nearest neighbours. Let $\alpha_{ov}$ denote orbital overlap and $\beta_{hop}$ hopping; the three neighbour phases combine as

$$\gamma(\mathbf k)=1+e^{-i\mathbf k\cdot\mathbf a_1}+e^{-i\mathbf k\cdot\mathbf a_2}.$$

Projection onto the two basis states gives

$$\begin{aligned} Eb_A+\gamma(\alpha_{ov}E-\beta_{hop})b_B&=0,\\ \gamma^*(\alpha_{ov}E-\beta_{hop})b_A+Eb_B&=0. \end{aligned}$$

Nonzero coefficients require

$$E^2-|\gamma|^2(\alpha_{ov}E-\beta_{hop})^2=0.$$

Writing $g=|\gamma|$, the roots are $E=\beta_{hop}g/(1+\alpha_{ov}g)$ and $E=-\beta_{hop}g/(1-\alpha_{ov}g)$. Neglecting small overlap yields

$$E_\pm(\mathbf k)=\pm t|\gamma(\mathbf k)|,\qquad t=|\beta_{hop}|.$$

With $\mathbf a_1=a(1/2,\sqrt3/2)$ and $\mathbf a_2=a(-1/2,\sqrt3/2)$,

$$E_\pm=\pm t\sqrt{1+4\cos\left(\frac{\sqrt3ak_y}{2}\right) \cos\left(\frac{ak_x}{2}\right)+4\cos^2\left(\frac{ak_x}{2}\right)}.$$

The two basis orbitals generate valence and conduction bands. They touch where $\gamma=0$, at Brillouin-zone corners. There are six corners but only two inequivalent valleys, $K$ and $K'$.

Sources: L4 pp. 8–14.

4.3 Dirac dispersion, effective mass and gate control

Near a valley, define $\mathbf q=\mathbf k-\mathbf K$. Linearization gives

$$E_\pm(\mathbf q)=\pm\hbar v_F|\mathbf q|, \qquad v_F=\frac{3ta_{CC}}{2\hbar}\sim10^6\,\mathrm{m\,s^{-1}}.$$

$$\mathbf v_g=\frac{1}{\hbar}\nabla_{\mathbf k}E, \qquad H_0=v_F\boldsymbol\sigma\cdot\mathbf p.$$

Here $\boldsymbol\sigma$ acts on sublattice pseudospin, not real electron spin. “Massless Dirac fermion” refers to this gapless effective Hamiltonian, with velocity scale $v_F$, not light speed. Do not infer zero mass by inserting linear dispersion into the parabolic-band curvature formula,

$$\frac1{m^*}=\frac1{\hbar^2}\frac{d^2E}{dk^2}.$$

A zero curvature is not a zero curvature-defined mass; the lecture conflates different mass concepts.

Count graphene states in a disk using spin two and valley two: $N/A=q^2/\pi$. Differentiating with $E=\hbar v_Fq$ gives the DOS per unit area,

$$D_{gr}(E)=\frac{2|E|}{\pi(\hbar v_F)^2}.$$

Unlike a parabolic two-dimensional gas, graphene has a linear DOS near neutrality.

A silicon back gate separated by oxide moves the Fermi level from hole-dominated transport through charge neutrality to electron-dominated transport. For oxide thickness $t_{ox}$,

$$C_g=\frac{\epsilon_0\epsilon_r}{t_{ox}},\qquad n_{2D}\simeq\frac{C_g(V_g-V_D)}{e},\qquad \mu\simeq\frac1{C_g}\left|\frac{d\sigma}{dV_g}\right|.$$

$C_g$ is capacitance per area; $V_D$ is the neutrality voltage. If measuring conductance $G$, use sheet conductivity $\sigma=GL/W$ and consider contact effects. Oxygen/water adsorption commonly causes p doping; slow traps and polar molecules cause shifts or hysteresis. High mobility supports rapid transport, but the zero gap complicates complete switching-off.

Sources: L4 pp. 14–16.

4.4 Deriving the approximately 2.3% optical absorption

Use SI units and a real incident field $E(t)=E_0\cos\omega t$, where $E_0$ is its peak amplitude. Assume normal incidence, in-plane polarization, weak light and unblocked interband transitions. A photon promotes an electron from $-\hbar\omega/2$ to $+\hbar\omega/2$; net absorption includes $f(-\hbar\omega/2)-f(+\hbar\omega/2)$, approximately one under these conditions.

The fine-structure constant is

$$\alpha=\frac{e^2}{4\pi\epsilon_0\hbar c}\simeq\frac1{137}.$$

From $\mathbf E=-\partial\mathbf A/\partial t$ and minimal coupling,

$$H_{int}=-ev_F\boldsymbol\sigma\cdot\mathbf A, \qquad |\mathbf A_0|=\frac{|\mathbf E_0|}{\omega}.$$

The sign convention for charge does not change the squared transition amplitude. Suitable valley pseudospinors are

$$\psi_\pm(\varphi)=\frac1{\sqrt2} \begin{pmatrix}e^{-i\varphi/2}\\\pm e^{i\varphi/2}\end{pmatrix}.$$

The opposite phases correct the repeated-sign typo in the slide. Their matrix element introduces an angular factor; averaging $\sin^2\varphi$ gives $1/2$. The real field's positive-frequency component has half the amplitude. Therefore

$$\overline{|M|^2}=\frac{e^2v_F^2E_0^2}{8\omega^2}.$$

Energy conservation is $\delta(2E-\hbar\omega)$, which contributes a Jacobian $1/2$. Using graphene's linear-dispersion DOS, the transition density per unit area is

$$D_{tr}(\hbar\omega)=\frac12D_{gr}\left(\frac{\hbar\omega}{2}\right) =\frac{\omega}{2\pi\hbar v_F^2}.$$

Fermi's golden rule gives transitions per area per second; multiplying by photon energy gives absorbed power density:

$$R=\frac{2\pi}{\hbar}\overline{|M|^2}D_{tr},\qquad P_{abs}=R\hbar\omega=\frac{e^2E_0^2}{8\hbar}.$$

$$I_{inc}=\frac{\epsilon_0cE_0^2}{2},\qquad \mathcal A=\frac{P_{abs}}{I_{inc}} =\frac{e^2}{4\epsilon_0\hbar c}=\pi\alpha\simeq0.023.$$

Field amplitude, frequency and $v_F$ cancel. Equivalently, the optical sheet conductivity is $e^2/(4\hbar)$. This consistent derivation avoids the slide's invalid substitution of $mv_F^2/2$ into the parabolic two-dimensional DOS. Doping-induced blocking, strong-field saturation, substrates, interference and multilayer coupling can alter the approximation. 2.3% is absorption; single-layer transmission is approximately 97.7%.

Sources: L4 pp. 20–30; corrected DOS, phase and unit conventions as explained above.

4.5 Transmission, layer count and graphene applications

Sep04(b): the AFM calibration gives $0.35\,\mathrm{nm}$ for one layer and $0.70\,\mathrm{nm}$ for two. The sketches are not to scale. For measured transmission $T=90.8\%$, use the exercise's first-order additive absorption:

$$T_N\simeq1-0.023N,\qquad N=\frac{1-0.908}{0.023}=4, \qquad h_X=4(0.35\,\mathrm{nm})=1.40\,\mathrm{nm}.$$

Thus X has four layers and thickness 1.4 nm. Multiplicative transmission, $0.977^N$, gives approximately 4.15 layers; this difference reflects approximation order rather than changing the intended integer answer. Real AFM heights also depend on substrate and adsorbates.

Transparency, conductivity and flexibility motivate electrodes for touch panels, LEDs, solar cells and smart windows; extra layers trade transmission against electrical performance. Graphene dispersions reinforce polymers through load transfer and interfacial bonding, supporting coatings, inks, packaging, thermal management, functional fluids, storage materials and tissue-engineering composites. Good dispersion matters as much as intrinsic strength.

Broad optical response supports photodetectors and saturable absorbers: intense illumination fills available final states and reduces absorption. Weak single-pass absorption limits collection but also preserves transparency. GO membranes use surface chemistry and interlayer channels for selective transport and separation. Claims of complete barriers or helium blocking describe particular membrane conditions, not every graphene-derived film.

Sources: L4 pp. 20–36; Sep04(b) Assignment and solution p. 1.

4.6 Twisted layers, flat bands and the September 4 exercise

Rotating adjacent sheets creates a larger moiré periodicity. Interlayer coupling reshapes electronic bands, giving twistronics: electronic control through relative orientation. Near the bilayer magic angle of approximately $1.1^\circ$, low-energy bands become narrow and nearly flat.

For the assignment, label flat bands near the Fermi energy in the twisted-bilayer plot. Label “fast” electrons on steep monolayer Dirac branches and “slow” electrons on nearly horizontal bilayer branches, because

$$v_g=\frac1\hbar\frac{dE}{dk}.$$

Velocity is determined by slope, not by the vertical energy coordinate. A wide, steep band gives a larger kinetic-energy scale; in a narrow band of width $W$, many states occupy a small energy range. High DOS and large interaction-to-bandwidth ratio promote correlated behaviour. With an effective attraction, Cooper pairing can become favourable. Flat bands do not guarantee superconductivity, and Coulomb repulsion is not automatically an attractive pairing mechanism.

The original bilayer example has $T_c\simeq1.7\,\mathrm K$, so the slide's “high-temperature” wording does not imply room-temperature superconductivity. The trilayer outlook gives an approximately $1.56^\circ$ angle and persistence under fields approaching $10\,\mathrm T$; triplet pairing is presented as an interpretation and MRI as a prospective application, rather than universally established conclusions. Layer number and stacking offer additional control variables.

Sources: L4 pp. 37–40; Sep04 Assignment solution pp. 1–3 and Notes pp. 1–2.

4.7 Wettability and focused-laser microfabrication

A vertical CNT forest is a micrometre-scale object built from nanoscale fibres. Its roughness and trapped air can produce water-repelling surfaces. Contact angles above $90^\circ$ indicate hydrophobicity; very large angles around $150^\circ$, together with low hysteresis, favour rolling droplets and self-cleaning. Large angle alone does not guarantee easy roll-off.

Oxygen reactive-ion etching changes chemistry and morphology, reducing contact angle in the lecture examples. Increasing treatment time or oxygen flow generally strengthens the change before a plateau. After treatment, laser-trimmed and untrimmed regions show approximately $126^\circ$ and $96^\circ$, enabling patterned wettability.

A focused beam, mirrors, beam splitters, microscope objectives, cameras and a moving stage form the trimming system. Local absorption heats and removes material above a threshold. Power, dwell time, focus, scan path and orientation set geometry. Top cuts define patterns and heights; side cuts release bridges and suspended elements; oblique cuts create slopes and pyramids.

The Stonehenge-style opening illustrates localized threshold removal: focus intensity is higher than along defocused portions of the beam. Side access, rotation and cutting order are also necessary; focusing cannot provide unrestricted transmission through an opaque forest. This is a mechanistic interpretation of the displayed structure, since the slides do not supply the full cutting trajectory. Video placeholders add no independently recoverable observations. Human-hair patterning and nano-art illustrate the wider fabrication concept.

Sources: L4 pp. 41–58, 87–90.

4.8 CNT–quantum-dot hybrids and the nanosieve

Depositing CdSe/ZnS QDs dispersed in toluene or chloroform onto patterned CNTs combines a fluorescent component with a conducting, mechanically stable scaffold. Wetting and solvent transport distribute particles; evaporation fixes them to the forest while retaining its pattern.

The nanosieve produces size-dependent sorting. Larger QDs are trapped earlier near the centre or upper surface; smaller QDs travel farther laterally and penetrate deeper. The lecture mixture differs in size by approximately $2\,\mathrm{nm}$. For comparable QD chemistry, larger particles emit at longer wavelengths: yellow emitters remain more central, while smaller blue emitters spread farther. Spatial PL spectra verify composition rather than relying only on image brightness.

Controlled removal of different depths then exposes different QD populations and colours. The reasoning chain is size-dependent mobility, spatial/depth sorting, evaporation, selective trimming and fluorescence verification.

Sources: L4 pp. 59–63.

4.9 Laser-induced incandescence

Laser-induced incandescence, LII, is broadband thermal radiation from heated CNTs, distinct from band-related QD photoluminescence. A notch filter suppresses scattered laser light before spectral collection. Temperature can be inferred by fitting an emissivity-weighted Planck spectrum:

$$B_\lambda(T)=\frac{2hc^2}{\lambda^5} \frac1{\exp[hc/(\lambda k_BT)]-1},\qquad I_\lambda\propto\epsilon(\lambda)B_\lambda(T).$$

The examples give approximately $2380$–$2500\,\mathrm K$. Greater laser power generally increases emission, but intensity also depends on emitting area, emissivity and structural change. An initial flash may decay toward a sustained signal. Comparing approximately $10^{-3}$–$10^{-1}\,\mathrm{mbar}$ shows environmental effects on cooling and material consumption; better vacuum supports emission lasting over two hours in the demonstrated case.

For interpretation, a Planck-like spectrum supports thermal emission, power dependence supports optical heating, and pressure-dependent persistence shows environmental losses. The application is a rapid localized heat source, without heating the entire substrate identically.

Sources: L4 pp. 64–68.

4.10 Rapid actuators, resonance and optical logic

A laser spot deflects a compliant CNT flap; removing illumination restores it. The lecture reports response faster than $1.5\,\mathrm{ms}$. Compare candidate mechanisms against predictions:

• Thermal expansion: heating occurs, but simple expansion struggles to explain both displacement magnitude and speed.
• Radiation pressure: absorbing and reflecting surfaces experience approximately $P/c$ and $2P/c$. Motion not consistently along the beam argues against a purely photon-momentum explanation.
• Gas expansion: continued actuation in vacuum excludes it as the sole mechanism.
• Electrostatic forces: optical charge separation, photovoltaic or thermoelectric effects can create forces. Similar motion under externally applied fields and Wimshurst-machine charging/discharging supports an electrostatic contribution without uniquely proving the microscopic charging mechanism.

An optical chopper makes forcing periodic. Resonance appears when drive frequency approaches a mechanical mode. For weak damping,

$$Q\simeq\frac{f_0}{\Delta f},\qquad Q=2\pi\frac{U_{\mathrm{stored}}}{\Delta U_{\mathrm{cycle}}},$$

where $\Delta f$ is the full width at half maximum of the power response. Vacuum reduces gas damping; the demonstrated quality factor increases by more than twenty times. The frequency examples include approximately 17.5 and $21.5\,\mathrm{kHz}$; the slide's “17.5 Hz” label is inconsistent with its caption. Ordinary 25-frame-per-second video cannot directly resolve kilohertz cycles, so frequency-response measurements matter.

A moving flap can close an electrical contact. Two independently controlled spots create an AND gate with contacts effectively in series, or an OR gate with alternative parallel paths. With light present as input 1 and conduction as output 1, AND conducts only for 11; OR conducts for 01, 10 or 11. Trace the conducting path before naming the gate. PECVD growth, laser-shaped compliance, localized driving and contact geometry together convert material properties into useful device behaviour.

Sources: L4 pp. 69–86.

Problem-solving checklist

Problem	Start here	Check before accepting the result
QD wavelength from TEM	Establish whether the measured width is a diameter; calculate the Brus transition energy; convert energy to wavelength.	Use the correct material parameters and a radius in the same length unit as the coefficients.
QD size from a PL peak	Convert wavelength to energy, subtract the bulk band gap, and solve the quadratic for the radius.	Keep a positive root within the model's confinement regime; report the diameter separately if requested.
Unknown bulk band gap	Convert three peaks to energies and solve three Brus equations, using differences to eliminate the bulk term.	Preserve the pairing of each size and spectrum.
Exciton absorption	Locate the continuum edge and the lower-energy exciton peak; subtract their photon energies.	The exciton wavelength is longer; subtract energies, not wavelengths.
Alloy composition	Convert near-band-edge PL to an approximate gap and solve the bowing equation.	The physical composition lies between zero and one.
Field emission	Identify the plotted field units, work functions, and enhancement factors; use the negative FN slope or its ratio.	One slope fixes a combination of work function and enhancement, not both independently.
CNT geometry and electronic type	Count lattice vectors, calculate the chiral vector and angle, then apply the ideal metallicity condition.	A C–C bond length is not the graphene lattice constant; curvature can modify the ideal prediction.
DOS or STS	Identify the dispersion and dimensionality, count degeneracies once, and establish the bias convention.	Distinguish available states, occupied states, and a measured tunnelling current.
Graphene transmission	Apply the stated thin-film absorption approximation and the AFM height calibration.	Use transmission as a fraction and avoid treating the approximation as exact for arbitrarily many layers.
Device explanation	Trace stimulus → microscopic change → measurable response.	State which observations support the mechanism and which simpler explanations they exclude.

Self-check questions

Try answering these without looking at the preceding sections, then check the short answers.

1. Why can two dots made from the same semiconductor emit different colours? Spatial confinement changes the allowed electron and hole energies. In the strong-confinement Brus model, the positive inverse-square size term grows faster than the attractive inverse-size Coulomb term as the radius decreases, usually producing a blue shift.
2. Is a carbon dot's emission necessarily set by the diameter visible in TEM? No. Conjugated domains, edges, surface states, defects, and molecular or polymer-related states can contribute. The relevant electronic length scale need not be the whole particle diameter.
3. What are the two ingredients of the elementary FN derivation? The supply of occupied electron states incident on the surface and their WKB tunnelling probability through the field-narrowed barrier. Integrating their product yields a field-squared prefactor and an exponential in the inverse field.
4. Does a flatter FN line prove that the work function decreased? No. A greater field-enhancement factor also makes the slope magnitude smaller. An independent measurement such as UPS is needed to separate these effects.
5. Why does a one-dimensional subband have a DOS peak at its onset? For a parabolic band, the group velocity tends to zero at the subband bottom, so many states occupy a small energy interval. The DOS is inversely proportional to the derivative of energy with respect to wave vector.
6. Why is graphene's DOS not the constant DOS of a usual two-dimensional electron gas? Dimensionality alone does not determine the DOS. Graphene has a linear Dirac dispersion, while the usual constant two-dimensional result assumes a parabolic dispersion.
7. Does a flat band automatically imply superconductivity? No. A narrow bandwidth increases the relative importance of interactions and often the DOS, but pairing and superconducting coherence require additional physical conditions.
8. How can CNT white-light emission be distinguished from QD PL? Examine the spectrum and mechanism. Laser-induced incandescence is broadband thermal radiation whose shape can be fitted by a temperature-dependent Planck spectrum; QD PL involves electronic excitation and radiative recombination.

Course-source guide

Source labels in the chapter notes refer to the supplied course files, rather than to independent online articles. Page references are PDF page numbers unless explicitly marked as PowerPoint slides.

Label	Material used
L01	Advanced Functional Materials — QDs, LV lecture PDF, pp. 1–87.
L01 additional notes	Energy levels splitting, supplementary PowerPoint, slides 1–15.
L02	NWs, LV lecture PDF, pp. 1–75; the non-LV PDF and PowerPoint versions supply additional electroluminescence material.
FN supporting notes	Supplementary field-emission derivation, supplied PDF pp. 1–14, corresponding to printed pp. 4–17.
L03	Carbon Part 1, LV lecture PDF, pp. 1–55, and PowerPoint, slides 1–63. The PowerPoint includes STM/STS and junction material missing from the PDF.
L04	Carbon Part 2, lecture PDF, pp. 1–90.
Exercises	Lecture 1 worksheet, Aug 14; Lecture 2 worksheet, Aug 21; and the dated in-class assignments and supplied solutions cited next to their worked examples.
Class notes	Handwritten notes dated Aug 14, Aug 17, Aug 21, Aug 24, Aug 28, Aug 31, and Sep 04.

For the Aug 31 nanotube-spectrum assignment, the matching DOCX question and solution are the relevant pair: the file labelled as its PDF solution duplicates the later magic-angle material. Some lecture pages contain video placeholders; the notes explain the mechanisms supported by the readable slides and accompanying materials rather than assuming observations from an unavailable video.