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The important distinctions are changes of ring topology, missing or added atoms, and changes of local bonding:

Closed end caps require nonhexagonal rings, typically six pentagons per hemispherical cap; these are part of the cap topology rather than necessarily damage. Kinks, buckling, branches and incomplete closure are larger-scale structural imperfections.

A multi-walled carbon nanotube can contain any of these defects in individual shells, plus interrupted or terminated shells, extra incomplete graphitic layers, irregular interwall spacing, and covalent interwall bridges formed by displaced atoms. Such bridges change intershell coupling and sliding. Different shells having different wrapping indices is normal and is not, by itself, a defect.

Let ${\mathbf{a}}_1,{\mathbf{a}}_2$ be equal-length primitive translation vectors of the graphene lattice, separated by $60^{\circ }$. The carbon nanotube chiral vectoridentifies the two points joined when the sheet is rolled: $|{\mathbf{C}}_h|$ is the circumference, not the tube axis. The right-hand panel of the preceding original diagram shows ${\mathbf{C}}_h=6{\mathbf{a}}_1+6{\mathbf{a}}_2$ and ${\mathbf{C}}_h=7{\mathbf{a}}_1$.

The periodic boundary condition is $\mathbf{k}\cdot {\mathbf{C}}_h=2\pi \ell$, with $\ell$ an integer. Applying it at either Dirac valley gives the nanotube zone folding criterionfor a metallic tube. Thus $(6,6)$ is an armchair metallic example, whereas $(7,0)$ is a zigzag semiconducting example. Other choices with $n-m\not\equiv 0(\mathrm{mod}3)$ also give a semiconducting carbon nanotube.

This criterion is the ideal flat-sheet approximation. Curvature can open a small gap in nominally metallic non-armchair tubes; ideal armchair tubes remain gapless in that approximation. The property is determined by the wrapping vector, not simply by calling a tube “chiral” or by its diameter alone.

A single-walled carbon nanotube confines the electron around its circumference while allowing motion along its axis. Nanotube zone folding therefore turns the graphene bands into one-dimensional subbands. A semiconducting carbon nanotube has no allowed circumferential wave vector through a Dirac valley; a metallic tube has a band crossing at the Fermi energy in the ideal model. Each massive subband edge gives a one-dimensional van Hove singularity in the density of states, and corresponding peaks appear in the joint density of states.

For light polarized along the tube, the main electric-dipole selection rule conserves circumferential angular momentum: transitions connect matching valence and conduction subbands. These finite-energy transitions are conventionally $S_{11},S_{22},\dots$ in a semiconducting carbon nanotube and $M_{11},M_{22},\dots$ in a metallic tube. A metallic band crossing does not make its first strong interband peak have zero energy. Perpendicular polarization admits different angular-momentum channels, often much weaker because of depolarization. Excitons, dielectric screening and trigonal warping modify the simple band-edge transition energies.

Experimentally, measure absorption versus photon energy; Raman spectroscopy becomes resonantly enhanced when the excitation matches an optical transition. A photoluminescence excitation spectrum of dispersed semiconducting tubes correlates excitation near $S_{22}$ with emission near $S_{11}$ after relaxation. Metallic tubes generally lack comparable bright emission because nonradiative relaxation is efficient. Combining optical energies with radial-breathing-mode Raman spectroscopy or electron diffraction assigns tube diameter and roll-up indices; a Kataura plot organizes those assignments.^[ref]

Starting from the clean wafer, dehydrate it and use an appropriate adhesion treatment if required. Spin coat the positive resist to obtain a uniform film, then soft bake to remove solvent and stabilize handling. Align the mask and expose the clear pad windows. Develop away the exposed resist, rinse and dry, leaving openings at future metal locations.

Inspect the openings and clear residual resist gently if needed. Perform a compatible surface preparation immediately before metallization so that an unintended oxide or contaminant film does not dominate the contact. Deposit metal with a resist profile and film thickness that leave an accessible discontinuity at the resist edge; an undercut or suitable lift-off stack helps prevent continuous sidewall coating.

Dissolve the sacrificial resist and lift off metal above it, leaving the pads on the exposed semiconductor. Rinse, dry and inspect for residues or bridges. For source/drain test contacts, apply the intended contact anneal; for gate test contacts, preserve the intended Schottky interface. An aggressive final resist hard bake or a fully conformal bridging metal film can defeat lift-off, so these are not automatic extra steps.

Use equal-width rectangular contact windows in a row with several different edge-to-edge spacings. For positive-resist lift-off, the future metal pads correspond to clear windows in an otherwise chrome-covered mask: exposure removes resist there, and the unexposed background retains resist beneath unwanted metal.

For ordinary ohmic transmission line contact measurement, a conducting strip or isolated mesa of width $W$ connects adjacent pads. Measure several spacings $d$ so thatThe intercept separates two contact contributions from the sheet-resistance slope. Large enough pad length relative to the transfer length is needed for the simplest contact-resistivity extraction.

Prepare separate arrays on the appropriate surfaces: annealed source/drain metal on the n+ cap, and gate metal on cap-removed AlGaAs. A Schottky gate is nonlinear, so its interface cannot be assigned an ohmic zero-bias contact resistivity merely by fitting the same straight line. Use bias-dependent current–voltage or differential-contact analysis, accounting for the two rectifying contacts.

Higher donor concentration moves the semiconductor Fermi level toward the conduction band, reducing the semiconductor work function ${\Phi }_s=\chi +(E_c-E_F)$ when these quantities are expressed as energies. Equilibration with the metal aligns electrochemical potentials and creates band bending.

For an ideal n-type Schottky interface the barrier energy ${\Phi }_{Bn}={\Phi }_m-\chi$ is not automatically reduced merely by raising doping. The decisive change is the narrower depletion region:Large $N_D$ allows electrons to tunnel through the thin barrier, reducing the contact resistance and promoting an approximately Ohmic contact. Contact annealing can further modify the local interface and doping. This allows low-resistance source/drain injection while preserving a more rectifying gate contact; real interface-state pinning can modify the ideal work-function picture.

Choose process B, the citric-acid/peroxide wet etch. Its GaAs-to-AlGaAs etch selectivity is $200/1=200$, giving strong protection of the underlying barrier and a controllable removal rate for the thin cap.

For comparison, A has selectivity one and therefore no useful etch stop. It is also extremely fast: a nominal $20\mathrm{nm}$ GaAs layer would be removed in only $0.048\mathrm{s}$. C has selectivity $1000/20=50$, useful but inferior to B, and clears the same cap in about $1.2\mathrm{s}$. D has selectivity four, clears the cap in about $0.060\mathrm{s}$ and additionally risks plasma-induced surface damage or altered interface properties.

B clears the nominal cap in about $6\mathrm{s}$. Its slower underlying etch permits a more forgiving endpoint or modest overetch, although actual rates must be checked for the relevant composition and conditions. Wet-etch lateral undercut and contact-metal compatibility still matter; selectivity alone does not make every mask geometry acceptable.

A suitable sequence isFirst pattern the source/drain resist, then deposit and lift off the corresponding metal. Anneal these contacts to produce the low-resistance source/drain interfaces. Remove the exposed cap layer before gate lithography, leaving the cap beneath the contact metal; this uses the contact pads as an etch mask, with suitable protection where required. Finally define and deposit the gate onto exposed AlGaAs and lift off.

The key dependencies are that source/drain metal lands on the n+ cap, the gate lands on AlGaAs after cap removal, and the contact anneal precedes formation of the sensitive Schottky gate. A gate-recess variant can put gate lithography before the local cap etch if that resist is deliberately used as its mask, but it must additionally specify how unwanted conducting cap elsewhere is removed or isolated. The sequence above directly matches the cap remaining only beneath the source/drain contacts in the diagram.

Molecular beam epitaxy is a suitable research-scale choice. Independently controlled elemental beams and shutters allow abrupt composition changes, controlled silicon donor incorporation and thin spacer layers. Growth can proceed sequentially from the GaAs substrate through the undoped GaAs, undoped AlGaAs spacer, doped AlGaAs supply layer and highly doped GaAs cap.

A clean crystalline surface and controlled substrate temperature support epitaxial interfaces; separating the donor layer from the channel is modulation doping. Metalorganic vapor-phase epitaxy can also grow such structures, but MBE's controllable research-scale layer sequence is a strong match to the stated task. Merely depositing the right total elemental amounts would not guarantee the required crystallinity and interface abruptness.

For a face-centered cubic lattice, allowed reflections have indices all odd or all even. The first two are $\{111\}$ and $\{200\}$, withFor first-order Bragg diffraction, $2d\sin \theta =\lambda$. The detector intercepts the beam at the full scattering angle $2\theta$, givingThe inner detector radius supplies the minimum length for each ring:Thus both rings reach the annulus when $L\gtrsim 90.2\mathrm{mm}$. At this minimum the 200 ring is at roughly $3.12\mathrm{mm}$, well inside the $10\mathrm{mm}$ outer radius. Both remain within the detector for approximatelyEquality at the inner edge gives vanishing acceptance for an ideal infinitely narrow ring, so practical operation requires a small margin. The calculation concerns ring positions; individual nanoparticle orientation and the finite probe convergence also affect observed diffraction contrast. Using the Bragg angle instead of the full scattering angle would introduce an erroneous factor of about two.

An incident SEM probe undergoes elastic deflections by atomic potentials and inelastic interactions that excite or ionize the specimen. Repeated scattering generates an interaction volume wider and deeper than the incident probe. The measured resolution depends both on where a signal is generated and on whether it can escape.

Transmitted electrons from a sufficiently thin specimen can instead support scanning transmission electron microscopy. The values above are orders of magnitude, not intrinsic constants of the signal: beam energy, atomic number, specimen thickness, probe size and detector geometry all matter. The spatial origin explains why a fine topographic SEM image does not guarantee equally fine X-ray mapping.

The material figure of merit isLarge thermoelectric power factor supplies electrical output, while low thermal conductivity limits heat leaking without useful conversion. For a constant-property ideal generator, a common approximation isIncreasing the appropriately averaged figure of merit raises efficiency toward, but not beyond, the Carnot limit. Actual devices also incur contact, geometric and heat-transfer losses, and a single peak material ZT does not specify their efficiency.

The supplied trends show the trade-off with carrier density: conductivity rises, the magnitude of the Seebeck coefficient tends to fall, and electronic thermal conductivity rises approximately with conductivity through the Wiedemann-Franz law. Thus neither arbitrarily high doping nor maximization of conductivity alone gives the best ZT; the power factor and thermal terms produce an intermediate optimum.

Two complementary strategies are:

Both strategies must be checked for mobility penalties and electronic heat transport. Simply suppressing every kind of transport would also suppress electrical output.

The Shockley-Queisser limit is a radiative detailed-balance limit for a single-junction photovoltaic device under specified illumination and temperature. Its idealized absorber collects above-gap photons, thermalizes their excess energy and loses carriers only through the radiative processes required by absorptance and emission. Sub-gap photons are not harvested. A favorable gap balances current loss at large gap against voltage loss at small gap.^[ref]

For a step-like absorber,The dark radiative current decreases rapidly with gap, so the qualitative ordering for the supplied absorbers isThe curve rises beneath the diagonal $V=E_g/q$. The deficit $\Delta V=E_g/q-V_{\mathrm{OC}}^{\mathrm{rad}}$ reflects finite-temperature carrier statistics and the entropy associated with absorption and emission, including angular dilution of sunlight. Under ordinary unconcentrated illumination it is of order a few tenths of a volt, with a weaker gap dependence than $V_{\mathrm{OC}}$ itself. In a fixed blackbody-sun model its absolute value grows modestly with increasing gap, while its fraction of $E_g/q$ decreases. Above-gap thermalization is an efficiency loss but is not, by itself, the definition of this voltage deficit.

A measured perovskite device can have a larger deficit from nonradiative recombination and contact losses. The gaps alone do not determine those material-specific losses or a unique numerical voltage: solar spectrum, concentration, cell temperature and optical emission geometry must also be specified.

Absorption generates electrons and holes in the perovskite. After relaxation toward the band edges, their nonequilibrium populations are described by separate electron and hole quasi-Fermi levels. Selective contacts collect electrons through an electron-transport layer and holes through a hole-transport layer, while suppressing the opposite carriers.

For ideal contacts at open circuit,This quasi-Fermi-level splitting is not the band gap itself or simply the equilibrium built-in voltage. At equilibrium there is one common Fermi level and no extractable photovoltage. Under illumination at open circuit, generation is balanced by recombination while a nonzero population splitting persists.

Important parasitic paths include trap-assisted Shockley-Read-Hall recombination in the bulk, at grain boundaries and at interfaces, and recombination at poorly selective contacts. They lower the steady carrier populations and their quasi-Fermi splitting. Auger recombination can matter at high carrier densities but need not dominate an ordinary one-sun device. Radiative recombination is an unavoidable detailed-balance contribution, rather than an avoidable defect loss of the same kind.

An approximate illuminated diode relation isIncreased nonradiative recombination generally increases the dark recombination current and reduces $V_{\mathrm{OC}}$; it can also change ideality and fill factor. Shunts add a leakage path. Series resistance mainly reduces voltage under load and fill factor: its simple $I{R}_s$ drop vanishes at zero terminal current, so it should not be invoked as an automatic direct subtraction from open-circuit voltage.

Each formula unit of ${\mathrm{CF}}_{0.8}$ accepts $0.8$ electrons. Using the supplied Faraday constant and the periodic-table masses,The factor $3.6$ converts coulombs to mAh. This value is per gram of the active fluorinated carbon, not per gram of the entire electrode.

The composite value is underdetermined: the mass of active CF₀.₈ is not supplied. If that mass is $m$ grams, and the carbon black and binder are inactive in the stated conversion reaction, thenThis uses cathode composite mass and excludes the lithium counterelectrode and hardware. No unique numerical composite capacity follows from the two additive masses alone; assuming an unprovided active mass would change the requested quantity.

Graphite starts with approximately planar sp2-bonded carbon sheets and delocalized pi electrons. Increasing fluorination introduces carbon–fluorine bonding, disrupts the extended pi system and increasingly produces puckered, more sp3-like local carbon environments. Layer structure and spacing change rather than remaining those of pristine graphite.

The loss of connected graphitic electronic states generally lowers electronic conductivity and can open a large gap as highly fluorinated material is approached. Low-fluorine arrangements can show more complicated doping and defect effects, so a strictly monotonic conductivity curve for every composition is not implied.

Ratios above one fluorine per carbon require attention to edge, defect and multiply fluorinated environments such as CF2 or CF3; they should not be pictured as adding more than one fluorine to every intact carbon in an otherwise unchanged three-neighbor basal network. The variable empirical composition averages over the actual carbon structure.

This is a primary battery with a strongly reconstructive conversion electrode. Discharge produces stable LiF and a carbon phase while breaking the original carbon–fluorine framework. Charging would have to separate and reorganize those products, restore the fluorinated carbon structure and recover the initial lithium electrode.

LiF is highly stable and poorly conducting, the phase distribution and electrical contacts have changed, and the reverse pathway can require a large overpotential. Electrolyte oxidation or other side reactions can occur before efficient reconstruction is achieved. Thermodynamic reversibility of an overall reaction equation therefore does not guarantee a practical rechargeable electrode.

Make a composite positive electrode containing the active graphite fluoride, a connected conductive-carbon network and a binder, supported on a suitable current collector. Carbon black provides electronic pathways to distributed reaction sites; the binder maintains cohesion and contact.

Electrolyte-filled porosity provides ionic access, while particle dimensions and intimate mixing limit the distance electrons and lithium ions must traverse. Excess conductive additive or binder dilutes the capacity per gram of composite. The reaction can also alter the electronic network as carbon forms, but that does not remove the need for adequate initial connectivity or prevent insulating LiF from obstructing transport.