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At lower salt concentration, the longer Debye–Hückel screening length allows the two diffuse layers to interact at greater separation. For sufficiently charged surfaces, this produces a positive free-energy barrier between separated plates and the short-range primary attractive region. Crossing a barrier costs free energy, so it can kinetically stabilize a dispersion even though a more attractive state exists at close approach. The relevant barrier in a finite object must be compared with thermal energy using the appropriate geometry and interaction area.

Over the experimentally relevant range of several screening lengths, the low-salt interaction can be predominantly repulsive, as drawn in the illustrative model. Strictly, at asymptotically infinite separation, the supplied nonretarded DLVO model still approaches zero from the attractive side, because $e^{-{\kappa }_{D}d}$ decays faster than $d^{-2}$ for any finite salt concentration. A shallow secondary minimum can therefore occur beyond the barrier. Calling the entire far field repulsive without specifying the distance range would conceal this distinction.

Lower salt alone does not guarantee a barrier: surface potential, charge regulation and attraction strength still matter. Extremely dilute or salt-free systems also require counterion and finite-cell constraints beyond blindly holding $B$ fixed while sending ${\kappa }_D$ to zero.

For like-charged plates, DLVO theory combines an attractive dispersion contribution and a screened electrostatic repulsion. A useful far-field schematic model isAt high salt concentration, ${\lambda }_D={\kappa }_D^{-1}$ is short. The repulsion then disappears over a short distance, leaving attraction across much of the accessible range. The usual high-salt sketch has a greatly reduced or absent repulsive energy barrier, so approach into a primary attractive minimum is easier. At very small separations, short-range repulsion cuts off the unphysical divergence of the simple $-d^{-2}$ continuum expression.

The large-distance tail of the stated model is attractive because an exponentially decaying repulsion eventually becomes smaller than the algebraic attraction. A barrier is not ruled out by the words “high salt” alone: its existence also depends on surface charge or potential, charge regulation, $A_H$ and the boundary conditions that determine $B$. The figure shows one declared barrier-free high-screening example rather than claiming a universal threshold concentration.

TODO — user-deferred biology. This part belongs to the explicitly biological molecular-system question and is preserved without a solution under the requested nonbiology scope.

TODO — user-deferred biology. This part belongs to the explicitly biological molecular-system question and is preserved without a solution under the requested nonbiology scope.

Scanning tunneling microscopy uses electron tunnelling between a conducting tip and a sample across a narrow vacuum gap. At fixed bias, the strong distance dependence of the current enables sensitive height or electronic-structure measurements; the result depends on local electronic states as well as topography.

A tunnel diode uses tunnelling between occupied and available states across a very thin, heavily doped semiconductor junction. The alignment of those states changes with bias and can produce a region of negative differential resistance, useful in fast switching or oscillation circuits. Both examples require a finite barrier with states available on the other side; an infinitely extended forbidden region is not an equivalent transport geometry.

Here the step is upward by $U=3\mathrm{eV}$, so the forbidden-region energy difference is $U-E=1\mathrm{eV}$. The bounded wavefunction isThe requested amplitude penetration depth is $\delta =1/\kappa$. Using the supplied mass and $h$,The probability density decays as $e^{-2\kappa x}$, so its $1/e$ depth is half this value. For an ideal lossless semi-infinite step below threshold, the evanescent tail does not imply net stationary particle current deep into the material; the reflected flux is unity.

For a finite step and equal masses, continuity of the wavefunction and its derivative givesHence the reflected and transmitted amplitude ratios areA travelling wave has probability current $j=(\hslash k/m)|A{|}^2$, so the flux coefficients areThey satisfy $R+T=1$. Using $|t{|}^2$ alone as the transmission probability would omit the different transmitted velocity.

For $E=2\mathrm{eV}$ and a downward step of $2\mathrm{eV}$, the transmitted kinetic energy is $4\mathrm{eV}$ and $k_2/k_1=\sqrt{2}$. ThereforeThe nonzero reflection is compatible with current conservation even though the potential drops.

For a stationary scattering state, the time-independent Schrödinger equation is $-{\hslash }^2{\psi }^{\prime \prime }/(2m)+V\psi =E\psi$. The left side contains the incident and reflected waves, while the solid has no incident wave arriving from positive infinity:Multiplying by $e^{-iEt/\hslash }$ gives the time-dependent wavefunctions. The kinetic energy rises inside the lower-potential solid, so $k_2>k_1$ and the transmitted wavelength is shorter. A downward step permits a propagating transmitted wave for every $E>0$, but still generally reflects some incident flux. There is no forbidden region merely because $E<V_0$; the relevant transmitted kinetic energy is $E+V_0$.

Classically every particle crosses a downward step and accelerates, so this partial reflection has no classical counterpart. If the step is instead upward to $U>0$, $E>U$ gives oscillatory transmission with possible reflection, while $E<U$ gives a decaying evanescent wave. A semi-infinite forbidden region carries no transmitted stationary flux; a finite barrier can permit quantum tunnelling to a propagating region beyond it.

Place the interface at $x=0$ and let the incident beam travel in the positive x direction. Set the vacuum potential to zero and the solid potential lower by $V_0>0$:Equivalently, $V(x)=-V_0\Theta (x)$. The incident kinetic energy is $E=m{v}^2/2$. This ideal potential step assumes a sharp, planar interface, constant particle mass and no absorption, inelastic processes or lateral structure. It models the prescribed decrease in potential without asserting that every real solid has the same electronic interface potential.

Under the same positive, real-permittivity assumptions,The particle is less polarizable than the surrounding medium and moves towards weaker-field regions. The sign describes the polarizability contrast, not a negative net electric charge. Conducting or lossy media require the real part of the full complex Clausius-Mossotti factor; a comparison of only the real dielectric constants is then insufficient.

For ordinary lossless dielectrics with positive real permittivities,The denominator is positive, soThe particle is more polarizable than the displaced medium and is driven towards the region where its polarization lowers the field-plus-source energy most effectively. If the two permittivities match, this ideal dipole contrast vanishes.

At fixed material response, frequency and field-strength gradient,Thus the force is proportional to particle volume. Doubling the radius multiplies the force by eight, not two. This scaling assumes that the particle is small enough for the dipole approximation and does not substantially distort the prescribed field distribution. The mechanical drift speed need not scale with volume, because the hydrodynamic drag also changes with particle size.

Positive dielectrophoresis acts up the gradient of the squared RMS field, towards stronger-field regions; negative dielectrophoresis acts down that gradient, towards weaker-field regions. The sign is the sign of $\Re \tilde{K}(\omega )$.

The force is not generally tangent to an electric field line. Field lines show the direction of the field itself, whereas $\nabla |{\mathbf{E}}_{\mathrm{RMS}}{|}^2$ describes how its strength changes in space. A spatially uniform field gives zero dipole-order dielectrophoretic force even when it is large. Reversing the field each half-cycle does not reverse this time-averaged force, because the relevant intensity-like quantity is quadratic.

Neutralizing the top boundary removes its preference for A, but the substrate still prefers A. The original integer-period constraint from two A-selective interfaces is therefore relaxed. The film can adjust its top termination without paying the same surface penalty, and thickness frustration can be reduced.

A possible morphology is an A-rich wetting layer near the substrate with lamellae turning towards a more perpendicular arrangement in the upper film, as schematically illustrated. A predominantly parallel stack is also possible if the substrate anchoring and thickness commensurability dominate. Neutrality of the top alone does not uniquely determine the orientation.

If both boundaries were neutral, perpendicular lamellae could expose comparable A and B areas without a selective-surface penalty and would avoid the same parallel-stack thickness constraint. That is a different boundary condition from the one given here. A correct sketch must retain the A preference at the bottom rather than silently replacing it with a second neutral interface.

With identical A-selective boundaries, stress-free parallel stacks favour approximately integer multiples of $L_0$. An arbitrary imposed thickness generally cannot preserve both the bulk period and those terminations everywhere.

A small mismatch can be accommodated by compressing or stretching the lamellae. A larger mismatch can favour terraces, islands or holes with locally commensurate heights, separated by steps of roughly $L_0$. For example, a mean height $\overline{h}=2.35L_0$ can be assembled from regions of heights $2L_0$ and $3L_0$, with area fraction $f=0.35$ at the larger height:The cartoon shows this volume-conserving alternative. Which response wins depends on elastic stretching energy, surface energies, lateral constraints, thickness mismatch and equilibration time. If the substrate and processing permit it, dewetting or a different domain orientation can also occur; an arbitrary thickness does not prove that a single specific terrace pattern must form.

Many independently oriented nuclei form on an amorphous support and then coalesce before they can become one large crystal. The SiO${}_2$ substrate provides no unique epitaxial orientation. At relatively low deposition temperature, short surface diffusion lengths and slow bond rearrangement restrict the distance over which deposited species can find a well-ordered configuration. Island collisions therefore leave grain boundaries; precursor residues and incomplete sulfurization can add disorder.

Raising temperature or annealing can improve crystallinity, but may conflict with the ALD reaction window, substrate compatibility or sulfur retention. Small grains are consequently a frequent process outcome, not a fundamental theorem that ALD can never form a well-ordered layer. A direct ALD experiment on SiO${}_2$-covered wafers found nanocrystalline MoS${}_2$ by diffraction and electron microscopy, while demonstrating self-limited precursor exposure behaviour.^[ref]

Potential advantages are reproducible thickness increments, large-area uniformity, conformal access to features and compatibility with moderate-temperature integration. Saturated exposure can make coverage less sensitive to small precursor-flux variations than an unsaturated deposition rate. The cycle count provides a useful process variable for matching an optical or electrical thickness target.

For a monolayer device, control of nucleation and coalescence is especially valuable, because excessive second-layer growth and uncovered patches both change the optical response. The advantages are conditional: a reliable ALD window and suitable precursor chemistry are required, and conformality does not imply a single-crystal film. Long exposure and purge steps reduce throughput, residual ligands or impurities can remain, and post-growth crystallization may require temperatures that the integrated device cannot tolerate.

In atomic layer deposition, expose the surface to a molybdenum precursor, purge unreacted precursor and volatile products, expose it to a sulfur reactant, and purge again. Each half-reaction should saturate the available reactive sites rather than continue indefinitely with dose. Repetition builds a molybdenum disulfide film, with nucleation, incubation and layer closure checked independently.

Self-limited surface chemistry is not the same as depositing exactly one complete crystal monolayer per cycle. A cycle may add only a fraction of a layer, and initially isolated islands can thicken before all bare substrate is covered. Reactions on hydroxylated silicon dioxide can also differ from those on the comparatively inert MoS${}_2$ basal surface. A precursor pair must actually show saturation under the chosen conditions; simply alternating gases is insufficient evidence of ALD.

In metalorganic chemical vapor deposition, metalorganic and sulfur-containing precursors are typically supplied simultaneously or with overlapping residence times, and film growth depends on decomposition, transport and surface reaction rates. Both methods involve chemistry, but ALD separates saturating half-reactions and purge steps. An ALD process need not use a metalorganic precursor, and pulsed CVD is not automatically ALD.

Deposit hydrogenated silicon nitride by plasma-enhanced chemical vapor deposition. A plasma activates a silicon-containing precursor and a nitrogen source, enabling a conformal thin film at a lower substrate temperature than a purely thermal process. Tune gas composition, power and time to control thickness, refractive index, stress and hydrogen content.

The film serves as an antireflection coating and can also provide chemical and field-effect surface passivation. As a simple optical check, a quarter-wave thickness is $d\simeq {\lambda }_{\mathrm{design}}/(4n)$; $d\simeq 75\mathrm{nm}$ and $n\simeq 2$ correspond to a design wavelength near $600\mathrm{nm}$. The textured surface and broadband illumination require a fuller optical optimization, so this check is an order-of-magnitude explanation rather than an exact reflectance calculation.

Diffuse a donor such as phosphorus into the textured p-type surface, then control the thermal drive-in to form the shallow n-type emitter. A phosphorus-containing vapour source can supply the donor over the exposed area. Diffusion and activation create an n-on-p p-n junction; remove the phosphorus-rich surface glass and isolate the rear and wafer edges as required to avoid shunts.

An alternative is ion implantation followed by an activation and damage-repair anneal. The diffusion route is particularly convenient for large-area coverage of a textured wafer. The indicated roughly $1\mu \mathrm{m}$ layer is a depth target for process control, not an instruction to grow an undoped film and assume that it becomes n-type.