Lateral oscillation or dithering is added to the vertical oscillation. It provides a tip-sidewall interaction signal that can drive lateral feedback while the vertical channel maintains height control. This is how a suitable flared tip can track steep or reentrant walls. Treating the additional signal as merely a larger vertical tapping amplitude would not supply the required sidewall sensing.^[ref]

A conventional tapping tip tapers toward its apex, so its shank contacts an overhanging feature before its apex can probe underneath it. Critical-dimension atomic force microscopy instead uses a calibrated flared or laterally extended tip, with a relatively narrow neck and side-sensitive control. The lateral extension can contact a reentrant sidewall while the neck clears the upper edge. The measured profile still includes the tip's finite shape and must be interpreted with a tip calibration; changing the tip does not automatically eliminate all convolution errors.^[ref]

In tapping-mode atomic force microscopy, the cantilever is driven near a resonance and its oscillation amplitude is monitored. Tip-surface interaction changes that amplitude. A controller compares it with a chosen setpoint and moves the vertical piezoelectric actuator to reduce the error. The recorded actuator displacement supplies the topography channel. A smaller amplitude during approach commonly triggers withdrawal and an increased amplitude during separation triggers approach, with the actual feedback sign set by the instrument's operating convention. Correct tracking requires enough response bandwidth without destabilizing the loop.

Excessive feedback gain can make the closed loop underdamped or unstable. Edge transitions then produce overshoot, ringing or oscillatory streaks in the recorded height. The apparent structure is a controller response rather than a real series of surface ridges. The exact amplitude and frequency depend on the actuator, cantilever, loop delay and controller settings; without those data one can sketch the qualitative ringing but cannot predict a unique quantitative trace.

With gains too low for the scan speed, the vertical response lags the surface. On entering the depression during the left-to-right trace, the tip descends too slowly and the recorded edge rounds or extends into the trench. On climbing out, the tip can approach or contact the rising feature before the controller lifts it sufficiently, again distorting the edge. Thus the error depends on scan direction and can differ between trace and retrace. Slower scanning can reduce this bandwidth limitation, but it does not correct the geometrical inability to probe an overhang.

With suitable atomic force microscopy feedback gains, the trace follows the accessible surface while maintaining the chosen oscillation-amplitude setpoint. The measured trench is narrower than the accessible opening because the finite tip encounters an edge before its apex reaches that edge's true lateral position. For an ideal tip flank angle $\alpha =15^{\circ }$ relative to the tip axis and height change $h=1.3\mu \mathrm{m}$, the geometrical setback is $h\tan \alpha \simeq 0.35\mu \mathrm{m}$ per side.

The conventional tapered tip cannot recover the underside of the overhang, even when feedback is perfect: the first contact with the upper edge or shank limits access. This geometrical envelope must be distinguished from errors caused by feedback dynamics. The stated tip height is sufficient for the vertical range; it does not remove the lateral-access problem.

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.