Questions regarding physical electronics [closed]

Question

I ran into some questions while studying physical electronics.

1. My textbook/lecture note says "An electric field applied to a semiconductor will produce a force on electrons and holes so that they will experience a net acceleration and net movement (if there are available energy states in the conduction/valence band)" In Fig. 1 shown below, does the bold part mean that the green electron can be accelerated (and gain momentum) only after the red electron accelerates and leave an available state? So, since energy states are discrete within bands (although the gaps are so small that the states are almost continuous), does the bold phrase mean that there's a tiny time difference between the acceleration of different electrons?

Fig. 1

2. As shown in Fig. 2 below, every k value is mapped to an energy level. However, since Kronig-Penney model deals with a single electron in a periodic potential, it seems like discrete k values should be allowed for such single electron. How is a continuous range of k values allowed?

Fig. 2

3. When calculating the Density of States, my textbook considers an electron confined in a 3D infinite potential well and uses its wave function to derive the k values, which turn out to be quantized as k = nπ/a. However, in a real crystal lattice, unlike the 1-electron assumption used in the calculation of DOS, multiple electrons interfere so energy level splitting occurs. Hence, shouldn't there be more k values (for example k = π/a + 0.0001) than k = nπ/a, which is the set of quantized k values assuming a single electron confined in 3D infinite potential well? Why is k = nπ/a used in the calculation of DOS?

4. Kronig-Penney model uses periodic potential model, which is more similar to the actual potential distribution in a lattice than a simple infinite potential well. Why is periodic potential not used in the calculation of DOS even though periodic potential better approximates a real crystal lattice than a 3D infinite potential well?

Answer 1

1. Not quite. Even in the discrete limit, you can solve the corresponding evolution using 2nd quantization, and you should get instep motion of Bloch oscillations (neglecting scattering).

2. I don't understand your expectation for discrete values of $k$. Perhaps this is due to the mix-up that for each value of $k$, you have a discrete energy spectrum? Remember that $e^{ika}$ is the translation operator's eigenvalue. While it is true that for a finite number of lattice sites $N$ with periodic boundary conditions, it can only have values in the $N$th roots of unity, you are typically interested in the infinite limit, where all the unimodular values are possible. Note that in any case, $k$ is defined up to a $2\pi/a$ multiple additive constant, hence all the talk of reduced phase space etc.

3. For DOS, you neglect electron interactions, so you are only interested in its interaction with the lattice. Actually, looking at a finite volume is merely a regularization trick. What you are more interested in analytical calculations is the continuum limit of these discrete energy level distribution obtained in the large volume limit, ie a free particle in space.

4. It's actually worse than that, you are not assuming an infinite well, but rather an entirely free particle. It becomes physically relevant when the relevant states in the physical phenomenon at stake is near the edge of a band. Indeed, what only matter in the DOS is the dispersion relation. Close to a minimum, you have an asymptotic quadratic dispersion, which is the same as in free space. The caveat is that the curvature of the band structure will give you what mass to use, and this has often little to do with the actual electron's mass. This is where the KP model comes in as it gives you a prediction of such an effective mass, which you plug into your effective free particle DOS.

In general, you want to reason in the continuum limit as it simplifies the formulas. Hope this helps, and tell me if something's not clear.