Do electrons have a net motion when a DC voltage is applied?

[fluidistic]

I am wondering whether electrons have a net motion against an applied constant electric field in a conductor. Intuition tells me that "of course they should", but so far the math has shown me otherwise.
Here are my current thoughts:
1) I cannot rely on the obsolute Drude's model. What's more, electrons aren't like classical particles in an ideal gas. There is no such thing as a drift velocity of the electrons, despite that it is useful to perform many calculations and obtain results that match experiments using that velocity. So it is just a useful concept but not what reality is.
2) I would have simply started with Bloch electrons, with a Bloch Hamiltonian to which I insert the action of an electric field. It turns out it's been somewhat done (of course, obviously) and this leads to Bloch oscillations. I.e. if we consider electrons to have a wavefunction of the form of the one given by Bloch, and if we assume that they have a well definite position and momentum (which is only valid if we leave the realm of QM and fall back to a semiclassical picture), then their motion is an oscillating motion, so that there's no net motion of electrons against the electric field. So this model, which looked the most promising to me to yield a definite answer, goes against intuition. What is this model missing if electrons do indeed posses a net motion against the E field?

How can I mathematically see, using QM theory only (please no semiclassical models), that electrons have a net motion against an applied electric field?

 

[mfb]

I am wondering whether electrons have a net motion against an applied constant electric field in a conductor.

Sure. You have a current flow. If the conductor is solid then ions don't contribute. You have electrons entering at one side and leaving at the other side, and a net motion of electrons inside.

In terms of the momentum distribution: As rough approximation you shift the whole distribution a bit.

 

[Cthugha]

2) I would have simply started with Bloch electrons, with a Bloch Hamiltonian to which I insert the action of an electric field. It turns out it's been somewhat done (of course, obviously) and this leads to Bloch oscillations. I.e. if we consider electrons to have a wavefunction of the form of the one given by Bloch, and if we assume that they have a well definite position and momentum (which is only valid if we leave the realm of QM and fall back to a semiclassical picture), then their motion is an oscillating motion, so that there's no net motion of electrons against the electric field. So this model, which looked the most promising to me to yield a definite answer, goes against intuition. What is this model missing if electrons do indeed posses a net motion against the E field?

How can I mathematically see, using QM theory only (please no semiclassical models), that electrons have a net motion against an applied electric field?

Well, for an ideal crystal they indeed do not have a net motion. Applying a dc electric field to a good crystalline solid will not give you any net current due to Bragg scattering at the end of the Brillouin zone. This has been first shown experimentally in 1992 by Karl Leo (Observation of Bloch oscillations in a semiconductor superlattice, Solid State Communications 84, 943 (1992)). Accordingly, an ideal crystal does not conduct electricity and this has absolutely nothing to do with the role of the ions. By the way, this is a pretty standard question for a PhD defense to see whether the student knows basic semiconductor physics.

Now of course this differs from our naive expectation and our everyday experience. This is a question of the timescales involved. Each full Bloch oscillation cycle takes some time. And there is no ideal crystal. All crystals contain some impurities or other imperfections which result in scattering of the electron state. If the time required for a full oscillation cycle is much shorter than the mean time between two scattering events, Bloch oscillations dominate and there will be no net current. If the time between two scattering events is much shorter than the time required for a full Bloch oscillation, the oscillation never takes place and the crystal conducts current. Even high purity crystals contain so many impurities and other imperfections that conducting crystals are the standard case. In order to observe Bloch oscillations, one usually has to employ special superlattice structures.

Sure. You have a current flow. If the conductor is solid then ions don't contribute. You have electrons entering at one side and leaving at the other side, and a net motion of electrons inside.

Just in "bad" crystals. Which fortunately is pretty much every crystal. ;)

 

[ZapperZ]

I am wondering whether electrons have a net motion against an applied constant electric field in a conductor. Intuition tells me that "of course they should", but so far the math has shown me otherwise.
Here are my current thoughts:
1) I cannot rely on the obsolute Drude's model. What's more, electrons aren't like classical particles in an ideal gas. There is no such thing as a drift velocity of the electrons, despite that it is useful to perform many calculations and obtain results that match experiments using that velocity. So it is just a useful concept but not what reality is.
2) I would have simply started with Bloch electrons, with a Bloch Hamiltonian to which I insert the action of an electric field. It turns out it's been somewhat done (of course, obviously) and this leads to Bloch oscillations. I.e. if we consider electrons to have a wavefunction of the form of the one given by Bloch, and if we assume that they have a well definite position and momentum (which is only valid if we leave the realm of QM and fall back to a semiclassical picture), then their motion is an oscillating motion, so that there's no net motion of electrons against the electric field. So this model, which looked the most promising to me to yield a definite answer, goes against intuition. What is this model missing if electrons do indeed posses a net motion against the E field?

How can I mathematically see, using QM theory only (please no semiclassical models), that electrons have a net motion against an applied electric field?

Be careful on what you ask for, because you might get it, and you might not like what you see.

To go beyond the Drude model, or more specifically, beyond the Boltzmann transport-type model, for electron transport in a metal, will be extremely messy. It is why we seldom use it, and it is why Landau went to all that trouble to come up with his Fermi Liquid theory, which allowed us to still use the semi-classical model on quasiparticles.

However, if you are a glutton for punishment and want to see the full QM treatment of it, then you need to dive into the Kubo formulation of electron transport.

http://felix.physics.sunysb.edu/~allen/Pubs/elec-trans-rev.pdf

When the article said that "... The derivations are quite tedious ..." in starting from the Kubo formula to get to the Drude/Boltzmann-type picture, it wasn't kidding!

Don't blame me if you go blind. You DID asked for it.

Zz.

 

[fluidistic]

Thank you people, that was extremely informative and interesting!

Well, for an ideal crystal they indeed do not have a net motion. Applying a dc electric field to a good crystalline solid will not give you any net current due to Bragg scattering at the end of the Brillouin zone.

I haven't checked the reference yet, but could you give a little bit more information about how Bragg scattering enters into play? As far as I know, this kind of scattering is the one responsible for X ray diffraction, I fail to see how a static electric field could cause such a scattering.

Now of course this differs from our naive expectation and our everyday experience. This is a question of the timescales involved. Each full Bloch oscillation cycle takes some time. And there is no ideal crystal. All crystals contain some impurities or other imperfections which result in scattering of the electron state. If the time required for a full oscillation cycle is much shorter than the mean time between two scattering events, Bloch oscillations dominate and there will be no net current. If the time between two scattering events is much shorter than the time required for a full Bloch oscillation, the oscillation never takes place and the crystal conducts current. Even high purity crystals contain so many impurities and other imperfections that conducting crystals are the standard case. In order to observe Bloch oscillations, one usually has to employ special superlattice structures.

I am not really understanding the part I put in bold. I would think that if the electrons are able to complete exactly an integer number of Bloch oscillations then there would be no DC. While if they aren't able to finish an integer number of Bloch oscillations (which would happen in most of the cases of course), there would be a DC. But apparently this isn't the case. Can you give more details to explain the bolded part?And thanks Zz, I'm going to try to understand as much as I can out of Kubo's treatment. It seems this is what I was looking for.

By the way guys, I think the approach I had in mind is not the one that would lead to Bloch oscillations. Instead, it would be the Pauli equation without magnetic vector potential (nor magnetic field). I would have to solve it for a free particle and see if I either get a DC or not and then try to think what would happen to Bloch electrons, having in mind that the free particle in an E field is just a special case of Bloch electrons in an E field.

Oh and last thing: perfect crystals are really strange then. They have an infinite electrical conductivity and yet they cannot sustain a DC. But they can sustain an AC.

 

[Cthugha]

I haven't checked the reference yet, but could you give a little bit more information about how Bragg scattering enters into play? As far as I know, this kind of scattering is the one responsible for X ray diffraction, I fail to see how a static electric field could cause such a scattering.

Do you know what the dispersion of an electron inside a crystal looks like? It looks similar to this in the repeated zone scheme:

The periodicity of the lattice also results in a periodic dispersion in momentum space. Now what is happening is that if you apply an external electric field, there will be a constant acceleration that results in a linear increase of the wave vector k with time. Now the group velocity of the electron is proportional to the derivative of the electron energy with respect to the wave vector (divided by Planck's constant). Now you can clearly see that for the lowest energy state the electron energy increases between k=0 and k=pi/a and it decreases between k=pi/a and k=2pi/a. Accordingly the group velocity changes sign. If you now plot the group velocity against the wavevector (which always increases due to the external field), you will find that it oscillates, which in turn also results in oscillations in real space. Now in periodic crystals, you will find that it is sufficient to consider the first Brilloin zone (as everything is periodic with respect to 2 pi/a. So one valid interpretation of what is happening is that Bragg scattering from one end of the Brilloin zone (pi/a) to the other end (-pi/a) as the electron wavelength reaches the lattice period. Anyway, independent of the interpretation, the physics stays the same. The group velocity oscillates and so does the electron position in real space.

I am not really understanding the part I put in bold. I would think that if the electrons are able to complete exactly an integer number of Bloch oscillations then there would be no DC. While if they aren't able to finish an integer number of Bloch oscillations (which would happen in most of the cases of course), there would be a DC. But apparently this isn't the case. Can you give more details to explain the bolded part?

Consider an electron that you place somewhere inside the crystal. As you introduce an electric field, it will start to move according to this field. This means that it will move into one direction initially, but it will perform an oscillatory motion around its starting point. Basically it will behave like a 1D harmonic oscillator and not move anywhere on average. However, it is important to notice that the deviations from the simple ballistic transport assumption occur at quite large values of k and thus at high energies. At these high energies, relaxation by scattering processes is quite likely, which "resets" the electron to the low wavevector region, where the electron motion will again directly follow the applied electric field.

As a simple intuitive picture consider a particle undergoing harmonic oscillation around some central point caused by the wave vector increasing linearly with time. Now every impurity or phonon scattering event "resets" this central point to the current position of the electron and the wave vector to some small value, from which the harmonic oscillation will start again. As the external field provides some preferred direction, you will end up with finite transport into that direction for frequent scattering events that will mostly occur when the electron is still at small wave vectors, while the harmonic oscillation will become dominant if scattering events are rare. You can simulate this quite easily.

And if you want to go the "hard" quantum theory way ZapperZ already provided a perfect starting point. But if you are not aware of Bloch oscillations at current, this might be a topic you want to save for later. This really is notoriously hard to grasp and involves non-trivial math.

 

[fluidistic]

Short update (I might reply to your nice post Cthugha in a few days).
I have been attempting my own approach, namely to solve the Schrödinger's equation by including the action of the electric field, for a free particle otherwise. In fact the equation is Pauli's one with $\vec A = \vec B=0$. It turns out that the solution involves the Airy function. I.e. in about half the space (in fact it depends on the strength of the E field) the particle oscillates with increasing frequency the farther it is, and in other other half of the space the wavefunction decays exponentially to 0. This is a surprise to me that the oscillations are not at a fixed frequency, but they depend on the position. And since the free particle is just a special case of a Bloch electron, then I can expect at least this behavior for an electron in a solid, though maybe there are some other surprising properties of psi in that case. Nevertheless, I find hard to believe Bloch oscillations with a well definite frequency would emerge from my approach, though I am maybe confusing psi with the dynamics of the particle. I'll think more about it.
I also tried to then solve the Pauli equation with the addition of a periodic potential (just like in a solid) with an explicit form of the potential. I picked a cosine but the equation was too hard to analytically solve for both Wolfram Alpha and Maxima, so I must resort to numerical solutions and I don't know how to perform them with say Scipy without invoking boundary conditions nor initial conditions, which is something I would have liked to keep undefined as long as I can since I seek the most general solution possible.

 

[ZapperZ]

You are approaching this the wrong way, and if you buy into Phil Anderson-Robert Laughlin's "More Is Different" approach, you will NEVER be able to derive all the emergent properties of conductivity, etc.

You need to look into the many-body "propagator" formalism which make use of the single-particle spectral function for a "quasiparticle". There's a reason why Landau's Fermi Liquid theory is still in use today, even with its limitations. So rather than reinventing the wheel, why not see if you can use the existing wheels and build from there?

Zz.