# Mean speed of electrons in a periodic potential / lattice

• I
• fluidistic
In summary, the conversation discusses the mean speed of electrons in a period potential/lattice. From the discussion, it is understood that the wavefunction of a Bloch electron is not a momentum eigenstate and the crystal momentum is \hbar \vec k. The expression <\vec v_n(\vec k)>=\frac{1}{\hbar}\frac{\partial \varepsilon_n(\vec k)}{\partial \vec k} is associated with the mean speed of a Bloch electron in the nth band. It is stated that this mean speed remains constant despite the interaction with the ions, which is in contrast with the Drude model. The Drude model only describes incoherent scattering while the scattering from a periodic potential
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?

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
DrDu said:
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.

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##.

## 1. What is the mean speed of electrons in a periodic potential/lattice?

The mean speed of electrons in a periodic potential or lattice is dependent on several factors such as the strength of the potential, temperature, and electron density. In general, it can range from a few meters per second to millions of meters per second.

## 2. How is the mean speed of electrons in a periodic potential/lattice calculated?

The mean speed of electrons in a periodic potential or lattice can be calculated using the Fermi-Dirac distribution function, which takes into account the energy levels and the probability of occupation for each energy level. It can also be estimated using experimental techniques such as electron diffraction or transport measurements.

## 3. What is the relationship between the mean speed of electrons and electrical conductivity in a periodic potential/lattice?

The mean speed of electrons in a periodic potential or lattice is directly related to the electrical conductivity. As the mean speed increases, so does the electrical conductivity. This is because higher mean speeds result in more frequent collisions between electrons and the lattice, leading to a higher probability of electron transport.

## 4. How does the mean speed of electrons in a periodic potential/lattice vary with temperature?

The mean speed of electrons in a periodic potential or lattice is strongly influenced by temperature. As the temperature increases, the mean speed also increases due to the increase in thermal energy. At low temperatures, the mean speed is limited by the lattice structure, but at high temperatures, the mean speed can approach the speed of sound.

## 5. Can the mean speed of electrons in a periodic potential/lattice be controlled?

Yes, the mean speed of electrons in a periodic potential or lattice can be controlled by varying the strength of the potential, temperature, and electron density. This can be achieved through various experimental techniques such as applying an external electric field or changing the composition of the material. Additionally, the use of different materials can also affect the mean speed of electrons in a periodic potential or lattice.

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