Pn junction to reach thermal equilibrium

[EmilyRuck]

Hello!
Some of the processes caused by a pn junction are not clear for me. Just after the contact between the p and the n region, a migration of charges happens in a semiconductor junction in order to reach an equilibrium condition. A valence band and a conduction band are present in both regions.
Initially, the p-region valence band is partially filled with holes and the conduction band is empty. The n-region conduction band is partially filled with electrons and the valence band is full.
After the contact:
1) Electrons migrate from the conduction band in n-region to the conduction band in the p-region and then recombine with holes by decreasing their energy?
2) Holes migrate from the valence band in the p-region to the valence band in the n-region and then the remaining electrons in the n-region conduction band recombine with them?

If 1 and 2 are true, how can electrons spontaneously go from a conduction band with a lower energy to a conduction band with a higher energy? (With reference to http://ecee.colorado.edu/~bart/book/book/chapter4/gif/fig4_2_3.gif figure).
And the recombination phenomena (electrons that drop from conduction band to valence band, emitting a photon/phonon depending on the material) present in both the p and n regions?
Thank you,

Emily

 

[Henryk]

1) Electrons migrate from the conduction band in n-region to the conduction band in the p-region and then recombine with holes by decreasing their energy?

TRUE

2) Holes migrate from the valence band in the p-region to the valence band in the n-region and then the remaining electrons in the n-region conduction band recombine with them?

TRUE
Note that electron has a negative charge and when they migrate to the p-side they charge it negatively. The same is for the holes migrating to the n-side and you have a double charged layer at the junction that results in electrostatic potential. The figure you quoted shows 'band bending' at the junction, the 'bending' is the result of adding electrostatic energy to the energy of a bulk semiconductor.
The force driving electrons up the potential barrier is thermal energy. In typical semiconductors, the electrons obey Boltzmann statistics, that is probability that a state is occupied is proportional to $p(E) = exp( - \frac E {k_BT})$ and there is a non-zero probability than an electron has enough energy to overcome the potential barrier at the junction and diffuse there.
In fact, the figure you quoted shows the junction under the equilibrium condition where the drift and diffusion processes cancel each other exactly.
Add an external voltage to the junction in either direction and you start having a net current flow.

 

[EmilyRuck]

TRUE
The force driving electrons up the potential barrier is thermal energy. In typical semiconductors, the electrons obey Boltzmann statistics, that is probability that a state is occupied is proportional to $p(E) = exp( - \frac E {k_BT})$ and there is a non-zero probability than an electron has enough energy to overcome the potential barrier at the junction and diffuse there.

Thank you for your clarifications. But in particular during the transient just after the contact between the n region and the p region, how can electrons increase their energy and move to the conduction band of the p side?
It is not for thermal energy. May the charge concentration gradient be the reason?

 

[Henryk]

But in particular during the transient just after the contact between the n region and the p region

Real junctions are made by controlled doping of acceptor and donor atoms. But let's assume you have n and p type pieces and bring them into contact. At the moment of contact, there is no potential barrier! The conduction and valence bands of both pieces have identical energies ! As the electrons migrate to the p side they leave unbalanced charge of donor atoms. The migrating holes leave the unbalance charge of the acceptor atoms. This double charge layer creates the potential barrier across the junction. The more carriers migrate the higher the barrier until an equilibrium is reached.
In any case, it is the thermal energy of electrons that let's them migrate against a potential barrier, regardless of its height.

I've discussed the p-n junction before, you might want to take a look
https://www.physicsforums.com/threa...rward-biased-pn-junction.854881/#post-5365418

 

[toutiao]

The confusing part for me is this: during this migration of electron from N to P region,
1. Is the electron a Bloch wave or a wave packet?
2. If it is the WAVE PACKET that moves from N to P region, where does the interference waves (that makes it a wave packet) comes from?
3. How to picture this thermal motion of electron at first place? According to Bloch wave function, the electron should really be "everywhere within the solid" and it should not be moving around like molecules in Brownian motions.

 

[Henryk]

Tautiao,
Interesting questions.
1. Bloch function describe quantum states extending everywhere within the solid. However, they are stationary states, they would correspond to actual electron states if electrons would stay in them for ever.
In real world, electrons change states due to things like interaction with other electrons, phonons, external fields, etc.
Bloch functions form a basis and 'real' electronic states can be constructed as a linear combination of Bloch functions.
For the description of electron dynamics in solids semiclassical transport theory is used.
In semiclassical transport theory, an electron is wave packet, that is a linear combination of Bloch states $\phi(k)$ of the form
$$ \psi = \sum_k A(k) \phi(k) $$
The function $A(k)$ is localized in the $k$ space, that is, it has a maximum near the average value $k$ and width $\Delta k$
Thus, it is also localized in real space with $\Delta r \approx. \frac 1 {\Delta k}$ (that answers your third question)
The velocity of an electron is the group velocity of the wave packet $v = \frac 1 \hbar \nabla _k \mathcal E(k)$
So, the answer to your second question, the interference comes from nothing, it is just decomposition of a wave packet into plane waves much the same as a radar pulse can be Fourier decomposed as a superposition of plane electromagnetic waves.

Additionally, when an external field is applied, the $k$ vector changes at the rate given by $\hbar \dot k = q E$
To complete the picture, one can also assign an effective mass as $\frac 1 {m_{eff} }= \frac 1{\hbar^2} \nabla ^2_k \mathcal E(k)$

I hope I answered your questions.

 

[toutiao]

Henryk,

After reading your reply, I checked out a solid state physics book and browse through the chapter regarding semiclassical transport theory. Based on my understanding, when external fields or phonons involved, the format of the Schrödinger's Equation changes correpondingly since we need to add the terms of electric field E and lattice defraction. In this case, the solution to the Schrödinger's Equation is no longer Block Function, but a wave packet instead, where the "basis" of the wave packet are Bloch functions with difinitive K. Adding those basis together (interference), we get a wave packet with momentum approximately K ("near the average value k and width Delta k) and position at approximate r (almost localized), without violating the uncertainty principle. Is this what you were explaining to me?

I have to confess that the derivation in the book is somewhat difficult to follow. Thanks for your prompt reply. I'll read more solid state physics books and hopefully things will clear up further. Looks like most semiconductor physics books don't go deep enough to unveil a clear picture of electron motion in crystal lattice.

You have a nice weekend.

 

[Henryk]

Pretty much yes