Optical transition rate for quantum wells

[Kcant]

So, let's say that you wanted to find the optical transition rate between subbands for electrons in a quantum well. To a good approximation, you can say that the wavefunctions are of the form

$\Psi = \psi(\vec r) u_c(\vec r)$

where u_c is the Bloch function and the lowercase psi is a slowly-varying envelope function that obeys an effective-mass Hamiltonian equation:

$\left( -\frac{\hbar^2}{2 m^*}\nabla^2+ V_{slow} \right)\psi = E\psi$

Obviously, finding the transition rates boils down to finding the momentum matrix elements between different states. Often, people simplify this by using the dipole moment instead of the momentum, using the following commutation relation:

$\vec p = \frac{i m}{\hbar}[H_0, \vec r]$

where H_0 is the crystal Hamiltonian. Then, to find the full momentum matrix element $\langle\Psi_f|\vec p|\Psi_i\rangle$, one can pull out a factor of the $i m/\hbar (E_f-E_i)$ from the element, becoming $i m/\hbar (E_f-E_i)\langle\Psi_f|\vec r|\Psi_i\rangle$. This can be further evaluated by exploiting the slow-varying nature of the envelope function and the orthonormality of the Bloch function, leaving $i m/\hbar (E_f-E_i)\langle\psi_f|\vec r|\psi_i\rangle$.

Ok, that's fine. But what if you don't use the commutation relation right away, and instead simplify the momentum element first? In that case,
$\langle\Psi_f|\vec p|\Psi_i\rangle = \int d^3\vec r \psi_f^* u_c^* \cdot -i \hbar (u_c \nabla \psi_i + \psi_i \nabla u_c)$.

Again using the slowness property of the envelope function, the second term is proportional to $\langle u_c|\vec p| u_c\rangle$, which vanishes for III-V materials. All you're left with is a factor of $\langle\psi_f|\vec p|\psi_i\rangle$. But wait! The commutation relation used before is now

$\vec p = \frac{i m^*}{\hbar}[H_{eff}, \vec r]$

where $H_{eff}$ is the effective-mass Hamiltonian from above. In that case, the matrix element reduces to $i m^*/\hbar (E_f-E_i)\langle\psi_f|\vec r|\psi_i\rangle$. So, in one case, the effective mass comes into the picture, while in the other, it is the free mass that does! What am I missing?

 

[eljose79]

First of all, it is great that you are thinking about the details of finding the optical transition rate between subbands for electrons in a quantum well. It shows that you are truly interested in understanding the physical principles behind this phenomenon.

Now, to address your question, let's first clarify the difference between the two commutation relations you mentioned. The first one, \vec p = \frac{i m}{\hbar}[H_0, \vec r], is the commutation relation for the momentum operator in the crystal Hamiltonian H_0. This is used to simplify the momentum matrix element in terms of the dipole moment, as you mentioned in your post.

The second commutation relation, \vec p = \frac{i m^*}{\hbar}[H_{eff}, \vec r], is the commutation relation for the momentum operator in the effective-mass Hamiltonian H_{eff}. This is used in the case where the electron is treated as a free particle with an effective mass, instead of considering the full crystal Hamiltonian.

Now, regarding your question about the effective mass coming into the picture in one case and the free mass in the other, this is because the two commutation relations are being used in different contexts. In the first case, we are considering the electron in the crystal Hamiltonian, where the effective mass is not explicitly present. Therefore, the free mass (m) is used in the commutation relation. In the second case, we are considering the electron as a free particle with an effective mass, hence the use of the effective mass (m^*) in the commutation relation.

In conclusion, both commutation relations are correct and are used in different contexts. It is important to understand the underlying assumptions and equations when applying them to a specific problem. I hope this clarifies your doubts. Keep exploring and asking questions!

 

[charng]

I would first commend your thorough understanding of the theoretical background of optical transition rates in quantum wells. It is clear that you have a strong grasp on the mathematical aspects of this topic.

In terms of your question, it seems that you are wondering about the difference between using the commutation relation for the momentum operator and simplifying the momentum matrix element first. Both approaches are valid and can lead to the same result, but they may involve different factors and considerations.

When using the commutation relation, the effective mass appears because it is related to the crystal Hamiltonian, which is used to calculate the momentum operator. On the other hand, when simplifying the momentum matrix element first, the effective mass does not appear because it is not directly involved in the calculation.

It is important to note that the effective mass is a parameter used to simplify the calculations in the effective mass Hamiltonian. It is not a physical property of the electron, but rather a way to mathematically describe its behavior in a quantum well. Therefore, its presence or absence in the calculations does not change the physical interpretation of the optical transition rate.

Overall, both approaches are valid and can lead to the same result. It ultimately depends on the specific problem and the convenience of using one method over the other. I hope this helps clarify your understanding of the topic.