Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2020 / Paper 2 / Section B

A zero-dimensional example is buckminsterfullerene, ${\mathrm{C}}_{60}$: a closed cage with 60 vertices, 90 edges, 12 pentagons and 20 hexagons. Euler's relation is $60-90+32=2$. Each carbon has three neighbors in a curved, approximately $s{p}^2$-bonded network; bonds are not all equivalent. Its cage diameter is of order $0.7\mathrm{nm}$.

Confinement in all directions gives discrete molecular orbitals, unlike the extended dispersion of graphene. It is an electron acceptor with characteristic electronic and vibrational spectra. Its pentagons supply the positive curvature needed to close the carbon network. A molecular solid of cages is held together predominantly by intermolecular forces, so its mechanical and electrical properties differ from those of an individual strongly bonded cage. Undoped fullerene is not a high-conductivity graphene sheet; doping, packing and charge transfer can strongly change transport.

Three point-defect structures are a Stone-Wales defect, a reconstructed graphene monovacancy and a reconstructed graphene divacancy. The original sketch gives ring inventories rather than fitted atomic coordinates.

A Stone-Wales defect rotates one C–C bond, converting four hexagons into a 5–7–7–5 group without changing the number of atoms. A graphene monovacancy removes one carbon; a common reconstruction gives a 5–9 group with residual under-coordination. A graphene divacancy removes two adjacent carbons; a 5–8–5 reconstruction can eliminate dangling bonds. Other divacancy reconstructions exist, so atom loss does not uniquely select one topology.

Irradiation, energetic processing, etching and incomplete growth can create vacancies; thermally or beam-activated bond rearrangement can change their structures. Electron microscopy itself can create defects, so compare exposures before attributing a time sequence to unperturbed material.

Pentagons favor positive curvature and heptagons negative curvature. Such defects can bend tubes, help form caps or junctions, or change a local wrapping relationship. They scatter carriers and alter the local density of states; vacancies can add localized states and chemical reactivity. Whether a clean band gap appears depends on defect identity, concentration and arrangement, not merely on the presence of a defect.

Separated graphene sheets offer high accessible surface area, short lithium-transport distances and a conducting network. Edges and exposed sheet faces can supply additional storage sites; mechanical flexibility can help preserve contact during cycling. These advantages can improve rate capability when electrolyte access and electronic continuity are retained.

The graphite reference composition $\mathrm{Li}{\mathrm{C}}_6$ giveswhere the host-carbon mass is used. An idealized $\mathrm{Li}{\mathrm{C}}_3$ inventory doubles this figure, but two accessible faces do not prove a reversible $\mathrm{Li}{\mathrm{C}}_3$ electrode. Binding energies, aggregation, defects and electrolyte stability determine actual storage.

Large area increases electrolyte decomposition and solid electrolyte interphase formation, consuming lithium in early cycles. Restacking reduces accessible area, low packing density reduces volumetric capacity, and residual oxygen, defects or binder can lower conductivity and active mass fraction. A broad storage-voltage profile may lower full-cell energy efficiency. Compare electrodes at matched areal loading and practical current density, reporting first-cycle efficiency, retention, voltage profile and volumetric as well as gravimetric capacity.

I would choose liquid-phase exfoliation of graphite into few-layer graphene, followed by controlled electrode formation. It uses bulk feedstock and preserves much of the $s{p}^2$ framework; a battery electrode does not require a wafer-sized perfect monolayer.

Disperse graphite in a compatible liquid, apply shear or sonication, and classify the dispersion to separate thin flakes from unexfoliated material. Recover and wash the selected fraction, remove contaminants and dispersants, then coat a Cu current collector with a suitably bound, porous and electrically connected film. Drying and consolidation should establish contacts without collapsing every transport pore. Characterize thickness and size distributions, residual impurities and conductivity after processing.

The advantages are solution compatibility, accessible feedstock and less basal-plane chemical disruption than oxidation–reduction routes. Limitations include a distribution of layer numbers, limited single-pass yield, flake fragmentation, solvent recovery and drying-induced restacking. Solvent toxicity and retained surfactants matter for manufacturing. In the original exfoliation study, microscopy and spectroscopy distinguished thin flakes from graphite and checked structural preservation; those controls precede any performance claim.^[ref]

Oxidation–exfoliation–reduction is another scalable route, but irreversible lattice damage and residual oxygen make conductivity and first-cycle losses separate concerns.

Yes: carbon nanotubes can store lithium and form a porous, conducting anode network. Their external surfaces, defects and intertube spaces provide possible storage sites. Open ends and suitable defects may make internal surfaces accessible; a closed pristine tube should not be credited with its full internal area.

Axial conduction and an entangled mechanical scaffold can support either active storage or a composite electrode. Limitations include tube–tube contact resistance, low packing density, catalyst residues and large-area solid electrolyte interphase formation. Opening or functionalizing tubes may improve ion access while increasing defects and irreversible reactions. Compare reversible capacity, first-cycle efficiency, retention and full-cell lithium inventory under matched loading and voltage-window conditions. A high first-cycle capacity alone is insufficient evidence of a superior anode.

The infinite exterior potential enforces $\psi =0$ outside $0<x<L$. Within the well,The boundary conditions give $B=0$ and $kL=n\pi$, with $n\ge 1$. Since ${\int }_0^L{\sin }^2(n\pi x/L)dx=L/2$, normalization givesThese functions vanish outside the well. The ground state $n=1$ has energy $E_1$ and one density maximum at $L/2$. The first excited state $n=2$ has energy $4E_1$, a central node, and maxima at $L/4,3L/4$. The second excited state $n=3$ has energy $9E_1$, nodes at $L/3,2L/3$, and maxima at $L/6,L/2,5L/6$.

There is no normalized $n=0$ state: that choice gives the zero function. The original wells figure shows all three wavefunctions and probability densities. An arbitrary overall wavefunction sign does not affect the probability distribution.

An isolated nanometre-sized island has small total capacitance $C_{\Sigma }=C_S+C_D+C_G+\cdots$, so discrete charge carries appreciable charging energy. A simple constant interaction model isAway from charge degeneracy, adding or removing an electron costs more energy than a sufficiently small thermal/bias window supplies. Coulomb blockade suppresses sequential tunnelling between resolved charge states.

A single-electron transistor has source and drain reservoirs connected to an island through two tunnel junctions, plus an insulated capacitive gate. Each junction has resistance and capacitance; the gate shifts electrostatic energies without carrying the transport current. The original device schematic labels all components.

Require $k_{B}T\ll E_C$, a source–drain window below the relevant transition, and junction resistances well above $h/e^2\simeq 25.8\mathrm{k\Omega }$ to suppress strong quantum charge fluctuations. Electromagnetic noise and lifetime broadening must also be small. Gate-induced charge degeneracy permits conduction even at small bias; there is no gate-independent blockade threshold. Resolving orbital levels additionally requires temperature and broadening below their spacing between energy levels.

A gallium arsenide/aluminium gallium arsenide semiconductor heterostructure combines lower-gap GaAs with a higher-gap AlGaAs barrier. In a modulation-doped design, AlGaAs donors supply electrons to lower conduction-band states on the GaAs side. An undoped spacer separates the channel from ionized donors and reduces impurity scattering. The interface barrier and band bending quantize motion normal to the interface while allowing in-plane motion: a two-dimensional electron gas.

The normal confining potential is commonly triangular and finite rather than an infinite square well. At suitable density and low temperature, only the lowest normal subband is occupied. modulation doping reduces an important scattering source; it does not eliminate all disorder.

Use negatively biased surface gates to deplete the surrounding electron gas and leave a rectangular central region. Gate voltage controls lateral dimensions and barrier heights. Two constrictions can weakly couple the region to reservoirs, with a plunger gate controlling population. Etched boundaries are possible but may introduce edge disorder. A flat interior and steep edges approximate a box, givingFringing fields, finite barriers and penetration change the ideal result. A 2DEG confines only the normal direction; gates supply the extra lateral confinement. Gate-dependent spectroscopy can test level spacings, while charging measurements separate orbital confinement from Coulomb blockade.

A liquid crystal flows while retaining partial molecular order. A nematic liquid crystal has long-range orientational order but no crystal-like positional lattice. Its local mean axis is the nematic director $\mathbf{n}$, with $\mathbf{n}\equiv -\mathbf{n}$ for an ordinary apolar nematic. Order leads to anisotropic optical and dielectric response and an elastic cost for director distortions.

Nematic order selects an axis, not a common molecular head direction or density-layer stack. The original sketch uses short rods as a coarse-grained orientation map.

A cholesteric liquid crystal is locally nematic but its nematic director rotates helically, for exampleThe sign of $q_0$ specifies handedness and $p$ is the full-turn pitch. The sketch shows rods rotating through transverse planes. This is helical orientational order, not a smectic density-layer lattice. Although an apolar director repeats physically after a half-turn, the conventional pitch denotes a full $2\pi$ turn. A pitch comparable to visible wavelengths can give selective reflection.

The blue phases result when strong local chirality favors double twist that cannot fill three-dimensional space everywhere without frustration. Double-twist regions are connected through a network of nematic disclinations.

Blue phase I has body-centered cubic order and blue phase II simple-cubic order in their director/defect structures. The periodicity can cause selective optical reflection. Blue phase III is disordered or amorphous, not another ideal cubic lattice. These are structured fluid phases rather than ordinary crystals with molecules fixed to every lattice site.

A topological defect occurs where the local order cannot be continued smoothly while retaining its surrounding texture. At a nematic disclination core, the director model fails and the orientational order may decrease.

For a planar texture $\mathbf{n}=(\cos \varphi ,\sin \varphi )$,Because $\mathbf{n}\equiv -\mathbf{n}$, half-integer winding strengths are allowed. The original sketches use $\varphi =s\alpha +{\varphi }_0$ to show $s=+1/2,-1/2,+1$, corresponding to director turns $+\pi ,-\pi ,+2\pi$. A radial $+1$ pattern and an azimuthally rotated version have the same winding.

These are planar labels. In unrestricted three dimensions an integer line texture may escape by tilting out of the plane; half-integer lines have a different classification. A planar disclination strength must not be identified automatically with a three-dimensional hedgehog point charge.

homeotropic anchoring favors the nematic director normal to a surface. For a flat wall at $z=0$, uniform $\mathbf{n}=\hat{\mathbf{z}}$ is a simple compatible profile, shown as perpendicular rods. If a remote wall demands another orientation, elastic bending interpolates between the boundaries.

A common surface energy is $f_s=(W/2)[1-(\mathbf{n}\cdot \mathbf{\nu }{)}^2]$, with wall normal $\mathbf{\nu }$ and $W>0$. Its minima have $\mathbf{n}$ parallel or antiparallel to $\mathbf{\nu }$. The profile is controlled jointly by this preference and Frank elastic energy.

planar anchoring favors the nematic director tangent to a surface. A flat wall at $z=0$ permits uniform $\mathbf{n}=\hat{\mathbf{x}}$, as in the sketch. A rubbed surface can select one in-plane axis; degenerate planar anchoring selects only the tangent plane.

For the degenerate case, $f_s=(W/2)(\mathbf{n}\cdot \mathbf{\nu }{)}^2$ has minima at $\mathbf{n}\cdot \mathbf{\nu }=0$. A remote incompatible boundary produces elastic distortion. Tangency alone does not specify an azimuthal direction within the surface.

Strong homeotropic anchoring requires radial orientation just outside a sphere, while the remote director is uniform. A plausible dipolar texture contains a compensating hyperbolic hedgehog defect on one side. The particle's radial boundary has charge $+1$ under a consistent convention, balanced by an exterior point defect of charge $-1$ so the far field is uniform. The original cross-section depicts this configuration.

An alternative has an equatorial Saturn-ring defect. Its local disclination winding is half-integer; that local winding is not itself the net three-dimensional compensating charge. Particle size, elastic constants, confinement and anchoring strength select between configurations. The elastic theory explicitly compares dipolar, Saturn-ring and surface-ring states.^[ref]

A sphere with degenerate planar anchoring can have a bipolar texture with two surface boojums at opposite poles along the far-field axis. Surface director lines remain tangent, curve between the polar cores and relax toward uniform alignment far away. The original cross-section shows both cores.

A globally regular tangent orientation field cannot cover a sphere; the associated surface winding indices sum to its Euler characteristic, two. These are surface defects, not two arbitrarily placed bulk hedgehogs. Finite anchoring allows deviations near the cores, while an azimuth-selective coating could favor a different texture.

Choose an asymmetric diblock copolymer in a cylinder-forming regime: minority cylinders lie in a continuous majority matrix. Form and anneal a film to develop microphase separation, control substrate and top-surface preferences or apply a suitable field to orient cylinders normal to the film, then selectively remove the minority phase. The retained majority matrix becomes the porous support. Infiltrate the channels with liquid crystal. The majority phase forms the channel walls; removing that continuous matrix would not generally leave a continuous nanoporous membrane.

With homeotropic anchoring, the director is radial at a cylindrical side wall. A strictly transverse radial texture has a line core, but an escaped radial texture can avoid it:The director turns axially at the center and is normal to the wall at radius $a$. The original axial-section sketch shows this three-dimensional escape. End conditions, channel dimensions and anchoring strength determine whether escape or a defect-containing profile is favored.

With planar anchoring on the cylindrical side wall, uniform axial alignment $\mathbf{n}=\hat{\mathbf{z}}$ is a simple defect-free profile. The axial direction is tangent at every wall point, so it satisfies the lateral boundary everywhere without elastic distortion.

This assumes degenerate planar anchoring, or an azimuthal preference compatible with axial alignment. End constraints, selected circumferential anchoring or imposed twist can favor another state. planar anchoring alone does not require a circumferential vortex. The original axial-section sketch shows the uniform axial option.

Surface-treated colloids impose boundary conditions on a nematic liquid crystal, producing director distortions and defects. Bringing particles together can reduce total Frank elastic energy by sharing distortion or defect structures. The interactions are directional: particle–hedgehog dipoles can favor chains, while quadrupolar and confined configurations can favor other arrays. The defects provide orientation-selective bonds rather than an isotropic attraction alone.

Prepare a cell with controlled far-field alignment and particle anchoring. Image textures with a polarizing microscope; use optical tweezers to place two particles at selected separations and orientations, then release them and track their trajectories. Repeating over orientations tests anisotropy before increasing concentration to assemble larger arrays. Compare particles with different anchoring and compare with the isotropic phase to distinguish director-mediated binding from ordinary aggregation. Temperature-dependent viscosity and other surface forces must be accounted for in kinetic comparisons.

The reported two-dimensional nematic colloidal crystals used micrometre confinement and defect-mediated interactions to create ordered particle structures. This motivates the mechanism without implying that arbitrary particles will select the same lattice.^[ref]