I Mean speed of electrons in a periodic potential / lattice

fluidistic

Gold Member
Hello people, I have 3 questions related to the mean speed of electrons in a period potential /lattice. I've read Ashcroft and Mermin's page 139 as well as the Apendix E.
From what I understood, if one applies the momentum operator on the wavefunction of a Bloch electron, one doesn't get a constant times that wavefunction, which means that the wavefunction of the Bloch electron is not a momentum eigenstate. Then it is stated that the crystal momentum is $\hbar \vec k$. My first question is:
1) Does this mean that the electrons have no well definite momentum, and thus, no well definite velocity?

Then the book arrives at the expression $$<\vec v_n(\vec k)>=\frac{1}{\hbar}\frac{\partial \varepsilon_n(\vec k)}{\partial \vec k}$$ (or something similar, I do not remember exactly).
I do not understand why it is associated to the speed of a Bloch electron in the nth band. I would think that the right hand side is related to the mean speed of a "quasiparticle" which would be the electron PLUS the interaction with the potential, whose momentum would be $\hbar \vec k$. Not that of the electron itself, only.
2) Can someone explain me why the right hand side corresponds to the mean velocity of the Bloch electron? I've read the Appendix E and I still have that question.

It is stated that because this mean speed is finite and non zero, it means that the electrons move forever without any effects on the mean velocity despite the interaction with the ions, which goes in contrast with the Drude model electrons which bump into them. I do not understand the logic (the implication). In Drude model, aren't electrons also having a constant through time mean velocity? After each collision, the electron is magically assigned a random velocity. So that in average I'd tend to think its mean velocity remains constant through time. Thus I do not see why the property of having a constant through time mean velocity implies a "striking difference" with the electrons in Drude model.
3) Can someone tell me what I'm missing here?

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DrDu

at 1) Yes, this is the speed of the quasiparticle. But on the long run, it is ugly to alway talk of quasi-electrons, so, as everybody knows what is meant, people call them simply electrons.
at 2) The Drude model describes only incoherent scattering from randomly oriented scattering centers while the scattering from a periodic potential is coherent.
Hence the Drude model will only describe scattering from perturbations of the lattice - either disturbances or phonons.
In the Drude model, the electrons only have a constant mean velocity in the presence of a driving electric field, while in QM, the electrons already have a mean speed without a driving field.

• fluidistic

fluidistic

Gold Member
at 1) Yes, this is the speed of the quasiparticle. But on the long run, it is ugly to alway talk of quasi-electrons, so, as everybody knows what is meant, people call them simply electrons.
I had to google quasi-electrons and it seems they are present in Landau theory of Fermi liquids which apparently is indeed useful to describe most metals at low temperatures. Hmm. If you could point me out a reference where it is stated that we deal with quasiparticles instead of electrons, I'd be glad.

DrDu said:
at 2) The Drude model describes only incoherent scattering from randomly oriented scattering centers while the scattering from a periodic potential is coherent.
Hence the Drude model will only describe scattering from perturbations of the lattice - either disturbances or phonons.
In the Drude model, the electrons only have a constant mean velocity in the presence of a driving electric field, while in QM, the electrons already have a mean speed without a driving field.
Oh wow, I had completely missed that. Indeed, that makes a lot of sense. Nice.

DrDu

To see the relation between the true momentum and the crystal-momentum of an electron (for simplicity you can neglect interaction between the electrons, so that the quasi-electron is equal to a real electron), it is useful to express the Bloch wavefunction in Fourier space in one dimension. So if the electron has crystal momentum q, its wavefunction is something like $\exp(iqx) u_q(x)$ where $u_q(x)$ has the periodicity of the lattice, $u_q(x+a)=u_q(a)$, where a is the lattice constant. Hence with $K_n=2\pi n/a$, $u_q(x)=\sum_n \exp(iK_nx) U_{q,K_n}$. Therefore, the total wavefunction is a superposition of momentum eigenstates with $p=K_n+q$.
Intuitively this means that the electron eigenfunction is a superposition of left and right moving waves which constantly interconvert due to Bragg scattering from the lattice planes.

PS: Note that I set $\hbar=1$.

"Mean speed of electrons in a periodic potential / lattice"

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