Advance the projected contact line by $dx$ per unit transverse length without wetting the gaps. A fraction $f$ converts solid–vapor interface into solid–liquid interface, contributing $f({\gamma }_{SL}-{\gamma }_{SV})dx$. The remaining fraction creates liquid–vapor interface above trapped gas, costing $(1-f){\gamma }_{LV}dx$. The outer droplet surface contributes ${\gamma }_{LV}\cos {\theta }^{\ast }dx$. ThusSubstituting Young's equation givesThis is the ideal Cassie–Baxter state relation. Its $f$ is projected solid-contact fraction, not the roughness factor of Wenzel wetting. A suspended interface over the gaps is essential; invasion of the texture changes the area bookkeeping.

At a straight contact line, take forces per unit transverse length and measure the contact angle $\theta$ through the liquid. Solid–vapor surface tension pulls outward along the solid, solid–liquid surface tension pulls inward, and the liquid–vapor force has horizontal component ${\gamma }_{LV}\cos \theta$. Tangential equilibrium givesThis is Young's equation. The substrate supplies the normal reaction balancing the vertical capillary component. The original force sketch labels directions at a right-hand contact line. The result assumes smooth, chemically uniform, rigid surfaces in partial-wetting equilibrium, with negligible pinning and line tension.

An exciton is a correlated electron–hole pair with a full wavefunction $\Psi ({\mathbf{r}}_e,{\mathbf{r}}_h)$. In strong confinement, a useful leading approximation is a product of lowest electron and hole envelope functions, followed by a Coulomb correction. A spherical hard-wall ground envelope is nodeless, proportional to $\sin (\pi r/R)/r$, and finite at the origin. Finite barriers produce exterior tails.

For a core-shell semiconductor nanocrystal with type-I band alignment, both conduction- and valence-band offsets confine the carriers mainly to the core. Their envelopes penetrate the shell but overlap less with external traps. Chemical surface passivation and spatial separation from defects can suppress nonradiative losses and increase photoluminescence quantum yield. Shell thickness, interfacial quality and strain must be controlled; a defective shell can introduce losses.

A larger shell band gap alone does not guarantee type-I alignment. A staggered type-II band alignment can move one carrier toward the shell, reduce electron–hole overlap and alter lifetime and emission energy. The sketch illustrates the conditional type-I case. In CdSe/ZnS experiments, optical spectra were combined with structural and compositional measurements to test shell-growth interpretations.^[ref]

The second infinite-well state hasIts maxima are at $L/4$ and $3L/4$, with nodes at the walls and center. The density is zero outside.

If the second bound state exists in a symmetric finite square well, odd parity preserves its central node and two equal lobes. However, the density is nonzero at the walls and has exterior tails proportional to $e^{-2\kappa |x-x_{\mathrm{wall}}|}$. Normalization includes those tails, and internal peak positions and heights change. Finite barriers change penetration and energy; symmetry preserves the central node. The original wells figure compares the two normalized second-state densities.

Choose zero potential for $0<x<L$. In an infinite square well,The first three levels have ratios $1:4:9$, with zero, one and two interior nodes; the wavefunctions vanish outside.

For a finite square well with exterior potential $V_0$, bound states have $0<E<V_0$, sinusoidal interiors and tails with $\kappa =\sqrt{2m(V_0-E)}/\hslash$. Matching wavefunction and derivative for equal masses giveswhen parity is taken about the well center. Corresponding energies fall below the infinite-barrier values, and the wavefunctions spread beyond the walls. There are finitely many bound states; a shallow well need not possess three. States above $V_0$ form a scattering continuum.

For standard deviations of position and momentum,Confinement to a region of size $R$ requires a momentum spread of order $\hslash /R$, hence kinetic energy of order ${\hslash }^2/(2m^{\ast }{R}^2)$. A sufficiently small semiconducting nanocrystal therefore has discrete energy levels and an optical edge shifted above the bulk band gap.

An illustrative strong-confinement model iswhere $\alpha$ is a geometry-dependent number of order unity. The positive confinement contribution grows as $R^{-2}$; electron-hole Coulomb interaction contributes an attractive term of order $-R^{-1}$. The uncertainty principle motivates the scale, not the exact coefficient or oscillator strength. Strong size dependence occurs near the exciton Bohr radius. Finite barriers, dielectric mismatch and surface states modify this model, while size distributions broaden ensemble spectra.

Spin-polarized injection produces different spin-resolved electrochemical potentials in Cu, ${\mu }_s={\mu }_{\uparrow }-{\mu }_{\downarrow }$. Their imbalance diffuses beyond the charge-current path. Oppositely directed spin-resolved currents can cancel in net charge while transporting spin.

The second Py contact is a spin-selective voltage probe. In a weakly invasive model,Its open-circuit potential adjusts until the total detector charge current vanishes. That potential changes when the detector magnetization reverses relative to the injected spins. Zero detector charge current does not imply zero spin imbalance or zero measured voltage.

Reverse injector current and each electrode's alignment to test the predicted signal symmetries. Remove spin-independent thermoelectric offsets and stray-field Hall contributions. A nonzero voltage alone would not prove spin accumulation.^[ref]

Measure nonlocal alignment-dependent resistance in otherwise comparable devices with several injector–detector spacings $L$. Steady spin accumulation obeys a diffusion–relaxation equation,In a weakly invasive detector limit, $\Delta R_{\mathrm{NL}}=(V_P-V_{AP})/I\propto e^{-L/{\lambda }_s}$; a semilog plot gives slope $-1/{\lambda }_s$. With transparent Py/Cu interfaces, fit a spin-resistance model including contact absorption and wire cross-section rather than assuming that every amplitude reduction is a bulk Cu loss.

Keep Cu processing, cross-section and temperature fixed, characterize contact resistances, reverse injector current and subtract alignment-independent offsets. Repeat at different temperatures to assess interface and bulk contributions. A Hanle spin-precession measurement provides an independent check of spin relaxation, with diffusion times and electrode rotation included in its model.

Sweep the in-plane magnetic field approximately along the wires' easy axes. At large negative field, both magnetizations are parallel. If the wires reverse at different positive coercive fields, an antiparallel interval separates this state from the final parallel positive-field state. In the usual positive giant magnetoresistance case, the antiparallel interval has higher resistance; the reverse sweep gives a mirrored interval and a hysteresis loop.

An ideal angular model isSpin-dependent injection and scattering favor transmission through similarly aligned electrodes. The two-current resistor model represents this using unequal majority- and minority-spin resistances. Both layers can reverse, hence pseudo spin valve, rather than an absolutely pinned reference layer.

The different thicknesses can produce different switching properties, but thickness alone does not identify which electrode reverses first. Measure their coercive fields. The local resistance also contains Cu, contact and Py anisotropic magnetoresistance contributions. Rotation scans and individual-wire controls distinguish them. Py/Cu experiments showed that local contact-related signals can exceed the spin-valve contribution, motivating nonlocal detection.^[ref]

Two applications are ultraviolet photodetection and ultraviolet/near-band-edge light emission. The bulk edge corresponds approximately toAbove-gap absorption creates carriers that can be collected in a photodetector. Measure dark current and wavelength-dependent response to separate photocarriers from heating or leakage. An optically or electrically excited emitter instead uses electron-hole recombination. quantum confinement in sufficiently small dots can shift absorption and emission above the bulk band gap. Surface states can produce broad defect luminescence or nonradiative loss, so a direct bulk gap does not guarantee an efficient, narrow-band emitter.

The upper island has lower contrast than the rod in HAADF imaging, consistent with a lighter Zn-containing region above a Ru-rich support. Thickness, orientation and detector conditions also affect contrast, so intensity alone is not a compositional assay. The island is a flattened supported cap, not an isolated regular bulk crystal.

Using the 2 nm scale bar as a ruler gives a projected lateral size of order 4–6 nm and height of order 1–2 nm, depending on placement of the diffuse edges. These are approximate image estimates, not a three-dimensional diameter or size distribution. The height/width ratio is roughly $0.2$–$0.5$, and only a few atomic rows span the height. Thus surface and interface atoms form a substantial fraction of the particle. The truncated, irregular cap and weak edge order depart from an infinite, fully coordinated, perfectly periodic wurtzite crystal.

Adhesion can favor spreading across the rod. epitaxy can orient ZnO against a compatible surface repeat of the rutile structure, despite the different bulk symmetries. Coherent growth initially introduces strain; free edges allow relaxation, while larger islands may develop misfit dislocations. Facet-dependent attachment, polar surface energies, ligand coverage and unequal precursor delivery can further alter the cap. A final image does not uniquely reveal which kinetic pathway operated.

To measure strain, first index the zone axis and projected lattices. For pixel spacing $s$ and scale-bar length $s_{\mathrm{bar}}$, $d_{\mathrm{image}}=2s/s_{\mathrm{bar}}\mathrm{nm}$. Compare an indexed spacing with its unstrained reference:The latter is the hexagonal wurtzite relation. An ideal tetrahedral geometry has $c/a=\sqrt{8/3}$ and internal coordinate $u=3/8$; real bulk ZnO itself need not attain those ideal values. Measure several periods and compare interface and free-edge regions to distinguish strain from tilt and calibration errors.

A unique numerical lattice strain, vacancy concentration or phase assignment is not established by this unindexed single projection. Oxygen columns are weak, so a missing bright spot does not prove an oxygen vacancy. Tilted images, electron diffraction, chemical mapping and thickness-aware simulations would test those claims. The full rutile structure is primitive tetragonal with a basis: its metal positions can appear body-centered in a conventional cell, but the oxygen basis prevents calling the entire structure a body-centered tetragonal Bravais lattice.

The competition between crystallization, exposed surfaces, adhesion and elastic strain can be summarized byHere $\Delta g>0$ measures the bulk driving force from supersaturation, and the $\gamma$ values are interfacial free energies. For a free spherical nucleus, $\Delta G=4\pi r^2\gamma -4\pi r^3\Delta g/3$ gives $r_{\ast }=2\gamma /\Delta g$. A support changes the heterogeneous nucleation barrier through adhesion and geometry.

Precursor activities and supply rate, temperature, time, solvent and ligands control nucleation, attachment, detachment and surface diffusion. Many nuclei divide a fixed solute inventory into smaller particles; continued growth or Ostwald ripening can enlarge them. Substrate defects determine preferred nucleation sites; lattice mismatch and interface bonding constrain orientation and elastic relaxation. Facet-selective adsorption can change both surface energies and growth rates.

Near equilibrium, a free nanocrystal follows a Wulff construction; a supported particle can follow a Winterbottom construction. Under kinetic control, slowly advancing facets remain visible while rapidly growing ones disappear. Equilibrium facet energy and kinetic growth rate are different controls, so a crystal structure does not uniquely determine particle shape.

Four advantages are strong electrostatic gate control, rapid charge transport, high current-carrying capability and mechanical flexibility.

A semiconducting single-walled carbon nanotube has a nanometre-scale channel cross-section, allowing efficient gate control in a field-effect transistor. A clean short tube can have a long mean free path and correspondingly little channel scattering. Strong covalent bonding supports large current densities for interconnect applications, while high axial thermal conductivity can help distribute heat. Small mass, large aspect ratio and mechanical compliance also permit conducting or semiconducting networks on flexible substrates. Metallic interconnects and low-leakage transistors require different electronic tube populations.

Manufacturing must control diameter, carbon nanotube chiral vector, purity, alignment and placement. Metallic contaminants short-circuit transistor networks; low network density reduces current, whereas excessive density makes isolation difficult. Tube–metal contact resistance, schottky barriers, tube–tube junctions, catalyst residues and substrate charge traps can dominate the intrinsic channel response. Thermal boundary resistance can limit cooling despite high axial thermal conductivity. Compare sorted and unsorted networks at matched density and contact geometry, measure contact and channel contributions separately, and assess uniformity across many devices. High intrinsic mobility alone is not evidence of a reproducible integrated circuit.

For an armchair single-walled carbon nanotube, $|{\mathbf{C}}_h|=3bn$. HenceThe integer bounds give $(n,m)=(8,8),(9,9),(10,10),(11,11),(12,12)$.

The neighboring values $d_7\simeq 0.9492\mathrm{nm}$ and $d_{13}\simeq 1.7628\mathrm{nm}$ lie outside the interval. “Exact” here means exact within the stipulated cylindrical model and input bond length; the decimals are rounded. No van der Waals wall thickness is added because the diameter is measured between atomic centers.

The carbon nanotube chiral vector identifies equivalent sites of graphene that become one point on rolling:The primitive vectors meet at $60^{\circ }$, $b$ is the C–C bond length, and one can choose $0\le m\le n$. The carbon nanotube chiral angle satisfies $\tan \theta =\sqrt{3}m/(2n+m)$. Armchair tubes have $(n,n)$ and $\theta =30^{\circ }$; zigzag tubes have $(n,0)$ and $\theta =0$. Other pairs are chiral, with mirror-related left- and right-handed forms. Armchair and zigzag tubes themselves have mirror symmetry.

The circumferential boundary condition requires $\mathbf{k}\cdot {\mathbf{C}}_h=2\pi \ell$. In nanotube zone folding, an allowed wavevector line intersects one of the Dirac valleys in graphene precisely when $n-m$ is divisible by three. Thus all armchair tubes are metallic in this approximation. Zigzag tubes are metallic for $n$ divisible by three and otherwise semiconducting; general chiral tubes follow the same modular rule. Curvature can open small gaps in nominally metallic non-armchair tubes. Strain, interactions and broken symmetries can modify the ideal result; metallic band structure does not imply zero device resistance.