4
$\begingroup$

When discussing p n junctions, it is usually said that electrons diffuse from the n side to the p side, hence creating a separation of charges.

However, we know that electrons in a lattice are delocalized. Or more precisely, their state is given by the density operator $\rho = \frac{1}{Z} \sum_{k,n} e^{-\beta E_{k,n}}|\psi_{k,n}\rangle \langle \psi_{k,n}|$, with crystal momenta $k$ and additional quantum numbers $n$.

Even in an inhomogeneous system, the wavefunction should be spread out across the whole crystal.

These two pictures appear to be in conflict. Are electrons really localized at different sides of a p n junction?

Edit: Im looking for a quantum mechanical approach. From the microscopic point of view, we have to solve the stationary Schrödinger equation, and the only difference between the two sites are the different amount of impurity atoms. Im wondering how the density gradient in this system can arise from this microscopic point of view.

Improve this question
$\endgroup$
4
  • $\begingroup$ Atomic wave functions extend to infinity, yet we picture them close to the atom. And a function involving $E$ vs $k$ from band structure has nothing to say about $x$. Functionally they behave as localized entities in position. $\endgroup$
    – Jon Custer
    Commented Mar 2, 2021 at 16:23
  • $\begingroup$ I don't unterstand your comment. The Bloch theorem tells us that the position wavefunction of the electron is spread across the whole crystal. The only difference between the n and p side is presence of some impurity atoms. The Bloch waves should have no problem reaching from the n to the p side. I think one has to argue somehow that the amplitude on the p side is larger then on the n side. $\endgroup$
    – curio
    Commented Mar 2, 2021 at 18:29
  • 1
    $\begingroup$ Bloch functions are $E$ vs $k$. $\endgroup$
    – Jon Custer
    Commented Mar 2, 2021 at 19:02
  • $\begingroup$ Bloch functions are the probability amplitude as a function of space (k is just another quantum number characterizing the wavefunction). They are delocalized, meaning they are appreciably large across the whole crystal. Of course they are not really the eigenfunctions of a doped semiconductor because the impurity atoms destroy translational invariance. Still it seems to be a perfectly reasonable approximation when considering conduction in a homogeneous SC. The questions is what happens at a junction. $\endgroup$
    – curio
    Commented Mar 3, 2021 at 19:14

2 Answers 2

Reset to default
4
$\begingroup$

The magic words are "semiclassical model" (you can read about it for example in Ashcroft / Mermin, chapter 12), https://www.fzu.cz/~knizek/literatura/Ashcroft_Mermin.pdf).

You are right in that the solution of an electron with crystal momentum $\vec{k}$ for a lattice is itself periodic. You can however superimpose wave functions with differing $\delta k$, and get a localized wave-package. Under certain conditions, this wave package still has well defined enough $\vec{k}$ and $\vec{x}$ to be useful.

One example for the said condition: When you apply external fields (that is - every field which isn't generated by the electron you want to describe), the wavelength of the external field has to be big compared to the spatial extension of the wave package.

Improve this answer
$\endgroup$
3
  • $\begingroup$ I know that wavepackets are possible vectors in the Hilbert space and that they can have small fluctuations in k. Why are they appropriate to describe this problem here? The fundamental theory of statistical mechanics tells us that the state of the electrons is given by the gibbs ensemble $\rho \sim exp(-\beta H)$ and the Bloch theorem suggests that the corresponding eigenfunctions are delocalized. $\endgroup$
    – curio
    Commented Mar 29, 2021 at 16:13
  • $\begingroup$ Why look at the eigenfunctions? There can be solutions like the packets. They are not eigenfunctions, but that doesn't mean they can't be occupied. And for the purpose of the p-n-junction, they are way more appropriate than the eigenfunctions you mention, BECAUSE they are localized. In a p-n-junction, you have a position-dependent macroscopic potential, and a position-dependant doping. Strictly speaking, Bloch-eigenfunctions spanning the whole lattice are not a solution to this configuration (if you find one, everything would be fine), so working with wave-packets comes in handy. $\endgroup$
    – Quantumwhisp
    Commented Mar 30, 2021 at 8:49
  • $\begingroup$ We look at the eigenfunctions of H because this is what statistical mechanics tells us. The state of the electron is a density matrix (the Gibbs ensemble since we are in equilibrium) which is diagonal in the eigenstates of the Hamiltonian. Bloch functions should be a solution everywhere except at the position of the impurities, because the density of impurities is small. I know about Anderson localization but this is a completely different regime because there we have a random value of the potential at each lattice site. $\endgroup$
    – curio
    Commented Mar 30, 2021 at 11:33
3
+50
$\begingroup$

As @Quantumwhisp has correctly pointed out, the semiconductor junctions described in the classical textbooks are usually treated within the semiclassical approximation. However, nowadays we are able to fabricate devices where the wave function is coherent across the junction is everyday reality - these range from tunnel diodes (one tunnel junction) to superlattices, cascade lasers (hundreds of tunnel junctions with coherent tunneling along the length of the whole device).

Mean free path
The crucial notion here is the mean free path, $l$, which classically means the distance that electrons travel between collisions (with lattice imperfections, photons, other electrons, impurities). In the context of wave functions and tunneling mean free path really means the coherence length, i.e., the distance on which we can describe electron by a wave function. In classical p-n junctions the mean free path is much smaller than the size of the junction, $l\ll L$, whereas in tunneling devices the inverse is true: $l\gg L$.

Dephasing, decoherence, etc.
A finer point of view on the mean free path requires distinguishing the elastic and inelastic processes, which are sometimes referred to as dephasing and decoherence. As the OP correctly notes, even in an inhomogeneous system (but in absence of inelastic processes, such as electron-phonon or electron-electron collisions) the wave function in principle is coherent. While mathematically one is then tempted to average over all possible impurity configurations and imperfections, this is clearly not the correct approach when dealing with a specific sample with a specific disorder configuration. The subject has been an area of active research a couple of decades ago, and many interesting phenomena such as Anderson localization, weak localization, ballistic conductance were studied. (Note that use of the word *localization, echoing the title of the question.)

Diffusion
To finish with a short comment: diffusion is a process of random motion due to experiencing collisions with the environment particles (which in this case are impurities, photons,a nd other electrons). Diffusion is thus, by definition, an incoherent process, and its mentioning necessarily implies taking the limit of the short coherence length.

References

Improve this answer
$\endgroup$
14
  • $\begingroup$ You say that there is a finite distance on which "we can describe electron by a wave function". What do you mean by that? I know about Anderson localization, but this occurs in systems with very high disorder right? The texbooks treat homogeneous SC according to the band model and use conecpts like the crystem momentum of an electron. This can only be applicable if the system is very ordered, so that bloch functions are reasonable approximations for the eigenfunctions everywhere except at the positions of the impurities. $\endgroup$
    – curio
    Commented Mar 30, 2021 at 11:40
  • $\begingroup$ Apart from that, I know generally what dephasing, decoherence, scattering and mean free path are. I was hoping for a more specific answer which treats the problem quantum mechanically. You answer is basically "we can treat the system classically so there is no problem", right? $\endgroup$
    – curio
    Commented Mar 30, 2021 at 11:45
  • 1
    $\begingroup$ @curio This is what I am talking about: en.wikipedia.org/wiki/Boltzmann_equation Are you familiar with the Ashkroft and Mermin book? $\endgroup$
    – Roger V.
    Commented Mar 30, 2021 at 12:18
  • 1
    $\begingroup$ There are consistent derivations of the quantum kinetic equation, and systematic way to obtain approximations, as, e.g., in this review: journals.aps.org/rmp/abstract/10.1103/RevModPhys.58.323 But following it requires rather strong background in condensed matter physics and many-body physics. $\endgroup$
    – Roger V.
    Commented Mar 30, 2021 at 13:01
  • 1
    $\begingroup$ I realize that a quantum mechanical answer is too complicated. I accept this one because it points out that the semiclassical model is more complicated then saying electron wavefunctions are wavepackets. $\endgroup$
    – curio
    Commented Mar 30, 2021 at 19:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.