What does the wave vector k mean for an electron in a Bloch state?

[daudaudaudau]

Hi.

If we have a free electron with a certain wavenumber k, then this is equivalent to the momentum of the particle since $p=\hbar k$. For an electron in a Bloch state, this is not the case. Apparently k is not the momentum of the electron. But what is the physical meaning of k for a Bloch state then ? Can I think of k as telling me the direction in which the electron is moving?

 

[kanato]

k is the "pseudomomentum" or "crystal momentum" of the state. It is similar to the momentum in that it has a conservation law (up to modulo a reciprocal lattice vector, so this conservation law is violated for Umklapp processes). k does roughly tell you which direction the electron is moving, although the the true velocity is $$\nabla_k \varepsilon(k)$$ (I think there is a proof of this in an appendix of Ashcroft and Mermin). And there are some important cases where they are different. For instance, in the case of an electron in the valence band of a semiconductor near k = 0, you will find that there is a negative sign between velocity and k. This means that electron travel in the opposite direction of k, and is an indication that it may be easier to think in terms of holes rather than electrons.

 

[hess8]

I have a related question: the velocity of an electron in a bloch state is given by the gradient of $$E(k)$$, but the contribution to the current is given by the expectation value of the current density operator, which involves the bloch state wavefunction, and has no direct reference to $$E(k)$$.

I don't see the connection. Will this contribution to current density always be proportional to the velocity of the bloch state, and same direction?

 

[hess8]

OK, I see the connection now in Ashcroft and Mermin, appdx E. The two descriptions of velocity (current) are equivalent.

 

[daudaudaudau]

Now what about the physical meaning of the k-vector for a Bloch state? :-)

 

[hess8]

The physical meaning of k in the block state:

First, it describes the phase modulation of the Bloch state, i.e. its periodicity, wavelength, which is not the same as that of the lattice. It's a label for the quantum state (that's not really physical, but it's useful).

Second, it's something that's conserved in an infinite perfectly periodic crystal (except for arbitrary additions of a reciprocal lattice vector), and in processes that don't break that symmetry, so it's similar to true momentum.

Third, in regions where the band is approx. parabolic it is equal to the true momentum of the electron if we give the electron an effective mass that's related to the curvature of the parabola. This effective mass could be negative for negative curvature, so it accelerates to the left when a force is applied to the right. So we usually think of it as a "hole" (bubble) in a usually filled band, and instead assign a negative charge. Bubbles have "negative weight": they go up when gravity is down.