Basic electrostatics

The previous topic left a question hanging: what sets the width and shape of the depletion zone at a pn junction or bipolar membrane, that sliver where the $V^\circ_i$ ladder bends and neutrality locally breaks?

While the $V_i$ values want to flatten out for thermodynamic reasons, the same is not necessarily true of the $V^\circ_i$. To explain why $V^\circ_i$ might be flat (or not) at equilibrium, we finally have to open up the actual mechanics of electricity, kept offstage until now. The crucial point is that the $V^\circ_i$ ladder's spatial variations are (usually) equal to variations in the electrostatic potential $\phi$, and electrostatics ties variations in $\phi$ to charge. Bends in the ladder are therefore statements about charge.

$V^\circ_i$ as an agent of electrostatic screening

We have mentioned a few times that $\phi$ is generally ambiguous, but in the special case where we are inside an ideal material it does perfectly follow along with $V^\circ_i$. So there, at least, we can confidently discuss electrostatics. Specifically, within each material, $V^\circ_i$ variations are also variations in $\phi$, so their gradients are equal (and equal the negative of electric field):

$$\vec E = -\nabla\phi = -\nabla V^\circ_i.$$

Accordingly, inside each material, the $V^\circ_i$ ladder follows a Poisson equation, Gauss's law combined with the constitutive response of the background dielectric medium:^[1]

$$- \varepsilon \nabla^2 V^\circ_i = - \varepsilon \nabla^2 \phi = \rho_{\mathrm{free}},$$

where $\varepsilon$ is the material's background permittivity and $\rho_{\mathrm{free}}$ is the space charge density of the mobile carriers (plus any fixed charges). But as we saw in the charge neutrality topic, $\rho_{\mathrm{free}}$ is itself a function of the $V^\circ_i$ offset. This closes a feedback loop: any deviation of $\phi$ from its neutral point summons a charge that pushes it back, which is electrostatic screening.

One of the beautiful things about band diagrams is the perspective they give on screening: the $V_i$ are flat, and all the action sits in the ladder, where charge appears exactly as curvature of $\phi$ or $V^\circ_i$:

In fact, we already plotted the engine of this feedback loop back in charge neutrality; here is its central figure again, now carrying a second meaning:

The screening length $\lambda$ sets how far the screening tail reaches into the neutral bulk:

$$\lambda = \sqrt{\varepsilon/\chi},$$

where $\chi = -\mathrm{d}\rho_{\mathrm{free}}/\mathrm{d}\phi$ is a "screening capacitance"^[2] measuring how strongly the space charge pushes back as $\phi$ deviates from the neutral point (the slope at the neutral crossing just plotted). For ideal-dilute ions, electrons, or holes, $\chi$ is a sum of contributions from each species: $\chi = \sum_i z_i^2 F^2 c_i/(RT)$, which is exactly proportional to the ionic strength and the resulting $\lambda$ is known as the Debye length.

(The approach here of flat $V_i$ (flat-$\bar\mu_i$) is a very intuitive and direct yet underappreciated way of deriving the screening equation for ideal-dilute ions, which is known as the Poisson–Boltzmann equation. The ladder-moving feedback figure above is literally the "Boltzmann" part of it. In ideal Fermi gases this extends to Thomas–Fermi screening, where the Boltzmann charging curve gets swapped out for a new quantum-derived curve, and otherwise it works the same.^[3])

Diffuse layers are band bending

So far our screening charge has answered a defect buried in the bulk. The far more common provocation is an interface: nearly every real surface or contact nudges $\phi$ somehow (adsorbed charges, surface states, or simply contact with a different material), and the medium answers with the same screened response, a charge tail decaying according to the same screening length $\lambda$ defined above. Electrochemists know this tail as the diffuse (Gouy–Chapman) part of the electric double layer. Semiconductor physicists know the very same object as band bending: near a contact the band edges bend together while $V_{\mathrm{e}^-}$ runs flat. Two vocabularies, one picture:

Quasi-neutrality: a simplification of electrostatics

That is the heart of the electrostatics, and the good news is that we can mostly ignore it! This is due to the practical principle of quasi-neutrality:

Quasi-neutrality principle
"screening is very very strong"

In most conductors, the screening length $\lambda$ tends to be small (nanometres in metals and concentrated electrolytes, though reaching microns in lightly doped semiconductors or dilute solutions), so on sensible macroscopic length scales the system is usually almost exactly neutral everywhere.

Corollary: the concentration of mobile charge carriers needed to produce a gradually varying $\phi$ is a tiny and negligible fraction of the total carrier concentration.

There are exceptions to this: capacitors, field-effect transistors, and nano-devices all rely on electrostatics. But even for those devices, the bulk majority of the conducting system is neutral and only a tiny region deviates from neutrality.

Many interfaces in our devices are explicitly not of interest, yet each one carries its diffuse double layer all the same. One of the advantages of using electrochemical potentials like $V_i$ is that they 'ride flat' over this local craziness in $V^\circ_i$, so we can skip right over interfaces that are equilibrated.

The quasi-neutrality idea also gets applied out of equilibrium, in dynamic situations. For example, if you drive a current from a low-resistivity region to a high-resistivity region, you create a step in the electric field, but a step in electric field means some local charge accumulated at the interface. Yet we almost never talk about this charge, because (except at high frequencies or low conductivities, which show Maxwell relaxation) it is often negligible. Note however that the dynamic quasi-neutrality is distinct from the spatial quasi-neutrality described above: there, deviations from neutrality occupy negligible space; here, any accumulating charge relaxes away on the dielectric relaxation time $\varepsilon/\sigma$, too fast for slow dynamics to notice.

$V^\circ_i$ as a perfect neutralizer

As a convenient approximation, we can simply 'assert' local neutrality almost everywhere. Mathematically, we do this indirectly by sending $\varepsilon \rightarrow 0$, which converts the Poisson equation into the local electroneutrality approximation:^[4]

$$\begin{aligned} 0 & = \rho_{\mathrm{free}}, \\ \lambda & = 0, \\ \phi & = \text{anything}, \\ \end{aligned}$$

that is, there is no longer any constraint on the derivatives of $\phi$, and so $\phi$ at every point can independently 'float' to produce $0 = \rho_{\mathrm{free}}$.^[5]

So, our $V^\circ_i$ ladder floats to whatever value neutrality demands. In a homogeneous medium where the $V_i$ are flat (as they must be at equilibrium), $V^\circ_i$ must be flat too, except at its surfaces (within a few Debye lengths). This is the footing under the neutrality we assumed in the charge neutrality topic.

Interfaces: aligning two materials

Everything so far has concerned a single medium. The moment two different materials touch, a question appears that the rest of the book has quietly been assuming an answer to: how do their $V^\circ_i$ ladders line up? This is the essential interfacial question, and at the basic level the answer is electrostatic.

At equilibrium, charge rearranges into thin double layers on either side of the contact until any carrier that can cross has equalized its $V_i$. What is left is an offset between the two ladders: a built-in step in $\phi$, fixed by the interface. The bulk-to-bulk alignment is tied to the alignment right at the contact through the screening on each side: the double layers absorb the transition within a Debye length or so, and however much each side bends, the deep bulks settle at a fixed relative offset.

The useful part is what happens when the two sides screen very differently. A strongly screening material (a metal, $\lambda \to 0$) barely bends at all, so its level is effectively rigid right up to the interface and the weakly screening side does all the accommodating. The alignment then anchors to the strong screener's $V_i$. This is exactly why everything at a metal is referenced to its $V_{\mathrm{e}^-}$: the Schottky barrier height and the work function are both just "where the other phase's level sits relative to the metal's $V_{\mathrm{e}^-}$." The same rule, with gentler asymmetry, is behind the Donnan step of charge neutrality and the band offsets of semiconductor heterojunctions (below).

Predicting the size of that offset from scratch is a separate and much harder problem, which we will not attempt; the popular shortcut of lining materials up by a shared vacuum level is a rule of thumb that is notoriously unreliable, since every interface carries its own dipole. But the size is just one number. The structure of the alignment (electrostatic, screening-mediated, anchored to the stronger screener) is the essential part, and it is what lets the rest of this book draw interfaces at all.

There is one more thing to notice here: this is precisely where $\phi$ runs out of meaning. The figure draws the conduction edges, and their step at the contact is a definite, fixed number; the valence edges would take a different step at the same contact (the two offsets differ by the band-gap difference). A junction between two media has band offsets, plural, one per species, and none of them is "the" electrostatic step. The liquid version of this fact already appeared at the ITIES, where each ion's rung steps by its own transfer energy; it is why no liquid junction potential can be defined between two different solvents. Within one medium the distinction never bit, since following $\phi$ just meant following the ladder; between two different media only the per-species offsets remain, and $\phi$ names nothing. The full autopsy is deferred to $\phi$ under the microscope.

Beyond the simple case

It's worth noting what a blessing it is to have the quasi-neutrality principle, and how complicated things can get when we can't rely on it.

The full catalog of what can go wrong is banked in its own appendix: patchy and charged interfaces, graded media and their quasi-electric fields, nano-scale conductors that are all interface, constitutive relations beyond any effective $\varepsilon$, and the eventual breakdown of the local continuum picture itself.

All of those concerns are real and important when describing interfaces, especially interface kinetics and interface capacitance. And there is a huge engineering importance: as Herbert Kroemer said, "the interface is the device". But interfaces are always connected to bulky reservoirs, and the vast majority of those bulk regions in our devices and circuits are quasi-neutral, which means electrostatics can be largely ignored inside them and we only need to worry about the $V_i$ values (with $V^\circ_i$ acting as a mere neutralizer, as described above).

Takeaways

Electrostatics entered our picture mainly to justify leaving it out: screening is so strong that almost everywhere, the $V^\circ_i$ ladder simply floats to whatever offset keeps the bulk neutral, while the $V_i$ ride flat over the interfacial blips. Where two materials meet, the same screening decides how their ladders align, anchored to whichever side screens more stiffly.

The exceptions are narrow, and they are readable. Wherever the ladder does bend, the bend is paid for in charge — curvature of $V^\circ_i$ is local charge density, by the Poisson equation — and the payment always arrives in equal-and-opposite pairs a screening length apart: a defect and its cloud, a surface charge and its screened answer, the two lobes of a contact's double layer. That is what the charge panels under this topic's figures draw, and that read is worth keeping: the bends of a band diagram are a map of its charge. The screening length $\lambda = \sqrt{\varepsilon/\chi}$ says how far each bend reaches, and the response $\chi$ that sets it has a second life: read as charge banked against a voltage, it is a capacitance, which is where we turn next.

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