It isn't.The effective mass of a hole is equal to that of the electron. A hole simply describes the absence of an electron.

Dear Prof. Nicolis,

Thank you so much for your response. Allow me to rephrase the reason for my question in a section you did not confirm.

It seems that in the band structure, the conduction band and valence band represent the energy states on a curvature. To fit this curvature and facilitate the use of formulas, we utilize the effective mass of electrons and holes. While a hole signifies the absence of an electron, considering this curvature, they have different effective masses. Even the effective masses of electrons in the longitudinal and transverse directions differ, and the effective masses of light and heavy holes also differ from each other.

The effective mass of electrons is generally about 10 times less than the free electron mass, and the effective mass of holes is slightly greater than the effective mass of electrons.

It is true that the curvature of the band defines the effective mass. This, however, is the effective mass of the electron, which is the same as the effective mass of the corresponding hole

Of course the effective mass of the electron is different, in general, from the mass of the free electron, since the effective mass expresses the effects of the interaction of the electron, with the medium. This doesn't affect the fact that a hole is nothing more or less than the absence of the corresponding electron, so its mass is, by definition of what's a hole, the same as the mass of the corresponding electron.

Dear Prof. Nicolis,

Thank you again. If we consider the effective mass of electrons and holes with these characteristics as equivalent, why is the intrinsic mobility of free electrons higher than that of free holes? In solar cells, we prefer our minority carriers in the active material to be electrons rather than holes because free electrons have higher mobility compared to free holes. That's why we prefer the light-absorbing material to be of the P-type.

Is this difference in mobility solely due to the fact that electrons are in a free state from the nucleus, while holes are confined to the nucleus?

A hole only makes sense in a medium, it doesn't make sense without it. A free electron makes sense, a free hole doesn't.

Once more, the physical objects are electrons, with a mass that takes into account the medium. Now it can be convenient to describe what happens in terms of holes, but this is just a choice of variables.

Dear Prof. Nicolis,

Thanks a lot. I got it with your help and nice descriptions. I appreciate your time and energy.

Milad

There aren't any holes, bound to nuclei, that's, simply, wrong. Electrons are bound to nuclei. The problem with legacy terms and ideas is that they get more confusing than they're useful.

In a medium it doesn't make sense to ask which electron is bound to which nucleus, because the nuclei are indistinguishable and the electrons are, also.

The only thing that matters is that electrons in a medium have a different mass than free electrons and that it's possible to describe the conduction properties of semiconductors in two, equivalent, ways: Eiither by the motion of real electrons, or by the motion of fictitious particles that describe the absence of electrons, called holes. Holes have the same mass, but opposite charge, as electrons. The effective mass of holes, that describe the result of p-doping, need not be the same as that of electrons, upon realizing n-doping; hence the confusing statement that holes can have different effective mass than electrons-the two statements refer to different systems.

Dear Prof. Nicolis,

Yes, as you mentioned, these definitions can be a bit confusing. Given that I work on doping and solar cells, the use of the abstract concept of holes is highly practical for me. So, I need to consider both the motion of electrons and the motion of holes in concepts related to diodes and electronic transitions. With the information you provided, I now have a clearer perspective. Thank you.

Glad to have helped!

Hi colleagues,

Just to contribute to your valuable conversation: when talking about electrons and holes in a semiconductor one generally means two different energy bands. This is always the case in the solar cell science.

The simplest tight-binding model of the electronic energy bands clearly demonstrates that lower bands are more narrow than the upper ones. It straightforwardly follows then that the electron effective mass (clearly determined near energy dispersion extrema) should decrease for higher bands.

That's why electrons in a completely filled valence band of an intrinsic semiconductor are expected to be "heavier" than electrons in a conduction band.

It's worth noting that right now we consider electrons in the states with minimum energy in the conduction band and therefore electron mass is positive (=the electron energy increases with changing the momentum value). Just oppositely, when turning to the valence band we deal with the electrons in the states with maximum energy there - near the top of the valence band - and therefore electron mass is negative (=the electron energy decreases with changing the momentum value).

The lack of few electrons in the top of the valence band (either due to their capture by acceptors or because of their transfer to the conduction band after photon absorption) can be described as the appearance of few new particles - holes. They dwell on the exactly same energy dispersion law as their 'parent' electrons - that's why their mass is the same as for electrons in the valence band, but has the opposite sign. Here is how the hole mass becomes positive, similarly to the electron in the conduction band.

There could be really complicated situations with the spin-orbit interaction, inversion of band orders etc., but the most intuitive picture is presented above.

Dear Prof. Solovyev,

Thank you very much for your precise and answer. This response has been very helpful for me. I appreciate your time and energy.