# Understanding Effective Mass in Solid State Physics

• tnho

#### tnho

Can anyone explains the concept of effective mass in solid state physics to me?
Thanks a lot~

Can anyone explains the concept of effective mass in solid state physics to me?
Thanks a lot~

You should not ask a rather open-ended and vague question such as this. This is because it is impossible to give you a "lesson" on the concept that you can find more effectively be done in a solid state class or textbook.

Please describe more explicitly what PART of the material that you do not understand, and parts that you do. Only then can the rest of us know where to start.

Zz.

Can anyone explains the concept of effective mass in solid state physics to me?
Thanks a lot~

Intuitively, effective mass is a concept that uncouples one many body problem into a set of one body problems that are more easu to solve (i mean, of which the Schrödinger equation, ie the SE, is easier to solve).

Suppose you have 100 electrons mutually interacting through a coulombic potential. The SE cannot be solved exactly because of the mutual coupling between all the electrons (electron 1 interacts with electron 2,3, etc). To uncouple this many body system, we convert it into a system where you have "new electrons" interacting in a background potential. So, you look at the problem as if the new electrons are no longer interacting with each other but with some background potential. Since both systems need to be equivalent (ie you need to describe the same physical reality) you need to compare both systems : the original system as mutually interacting electrons, the new system has non interacting electrons in a certain potential well. To make sure that both systems are equivalent, we change the mass of the electrons in the second system, in such a way that both SE are describing the same reality. This new mass is the effective mass, which can be defined as the electron mass + some interactions (ie energy of those interactions) to uncouple the many body problem into many one body problems.

That is the philosophy behind effective mass.

marlon

Intuitively, effective mass is a concept that uncouples one many body problem into a set of one body problems that are more easu to solve (i mean, of which the Schrödinger equation, ie the SE, is easier to solve).

Suppose you have 100 electrons mutually interacting through a coulombic potential. The SE cannot be solved exactly because of the mutual coupling between all the electrons (electron 1 interacts with electron 2,3, etc). To uncouple this many body system, we convert it into a system where you have "new electrons" interacting in a background potential. So, you look at the problem as if the new electrons are no longer interacting with each other but with some background potential. Since both systems need to be equivalent (ie you need to describe the same physical reality) you need to compare both systems : the original system as mutually interacting electrons, the new system has non interacting electrons in a certain potential well. To make sure that both systems are equivalent, we change the mass of the electrons in the second system, in such a way that both SE are describing the same reality. This new mass is the effective mass, which can be defined as the electron mass + some interactions (ie energy of those interactions) to uncouple the many body problem into many one body problems.

That is the philosophy behind effective mass.

marlon

Introducing the concept of effective mass, is it a kind of renormalization method?? =)

Introducing the concept of effective mass, is it a kind of renormalization method?? =)

What exactly do you mean here ? I explained what it means and if you refer to renormalization as the process of lumping in self interaction terms into the mass (like bare and dressed masses or particles) than YES there is an analogy with renormalization theory. But you really do not need to be thinking in those terms. Working with the simple formulation i used will do just fine to start.

marlon

To make sure that both systems are equivalent, we change the mass of the electrons in the second system, in such a way that both SE are describing the same reality. This new mass is the effective mass, which can be defined as the electron mass + some interactions (ie energy of those interactions) to uncouple the many body problem into many one body problems.

That is the philosophy behind effective mass.

marlon

Sorry to bump up an old thread, but I still don't quite follow.

so the effective mass is the mass that the electron/hole appears to posses in the field at the band gap.

what interactions are causing the apparent mass to be less than the real mass?

As the effective mass is related to the band curviture, are electrons near the tip of the gap curve 'lighter' than those at the lower part of the band curve?

so the effective mass is the mass that the electron/hole appears to posses in the field at the band gap.
To explain what i said i will show you the actual calculation of the effective mass : http://www.tf.uni-kiel.de/matwis/amat/semi_en/kap_2/backbone/r2_3_1.html

what interactions are causing the apparent mass to be less than the real mass?
What do you mean by real mass ? I suppose you are referring to the mass of a free particle. Electrons in the vacuum have mass m and electrons in a crystal have effective mass m'. m' exists exactly because the electrons have energylevels E determined by the symmetry (ie periodicity) of the crystal. This is also why some materials have band gaps and others do not.
As the effective mass is related to the band curviture, are electrons near the tip of the gap curve 'lighter' than those at the lower part of the band curve?

The effective mass is inversely proportional to the band curvature because of the second derivative (d²E/dk²) in the denominator. Large curvature gives small effective masses.

marlon

But what causes an electron in a crystal to have a larger acceleration under an applied force F than a free electron in vacuum under the same applied force? Why doesn't the effect from all the other electrons in a crystal just cancel out because the effect is the same in all directions?

There appears to be several different issues involved with this thread. I'm going to try and tackle all of them, and let's see if I don't make a bigger mess out of it.

First of all, for free electron gas, we basically have a quadratic dispersion curve, with no change in the curvature, or the 2nd derivative of the dispersion. So here, you get a constant effective mass everywhere.

Now, if we stick with that scenario again, but this time we impose a simple periodic boundary condition, we get the Bloch wavefunction and dispersion, which will result in band gap in the band structure. When this occurs, you have a change in curvature of the dispersion, and so you not only no longer get a constant effective mass everywhere, you also get a change in sign of the curvature. This results in a possibility of having an effective mass of a different sign.

What we need to keep in mind here is that, the "k" wavevector used the description, especially in the Bloch wavefunction is the crystal momentum, not the typical momentum that we are familiar with. So already we can't simply apply what we know to work in this case. Secondly, note that this change in curvature occurs at the Brillouin zone boundary and at the top of the band. This is why we deal with "holes" at the top of such band.

Now, as Marlon has mentioned, there really isn't a "mass renormalization" going on here, because we are dealing with still a "free, non-interacting electron gas", even when we impose a weak periodic boundary condition. We talk about mass renormalization when there is a deviation from the non-interacting scenario. When we include electron-phonon interaction, or electron-electron interaction, then the dispersion curve will actually change slightly (or a lot, depending on the strength of the interaction). See, for example, Fig. 1 and 2 in http://infrared.phy.bnl.gov/pdf/johnson/johnson_2001a.pdf" [Broken]. There's a slight "kink" in the band dispersion close to the Fermi level, which is a manifestation of all those many-body interaction (phonon, electron-electron, etc.), even in a "standard metal". In more exotic material, such as for high-Tc superconductors, this "kink" is even more pronounced (see Fig. 3), meaning that there's a lot of many-body effects that renormalize the effective mass.

Zz.

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Z---

The photo on p.157 looks like your 'image'/avatar---how closely are the two related?

Z---

The photo on p.157 looks like your 'image'/avatar---how closely are the two related?

Very close. I used to work in the same group when I was doing my postdoc at Brookhaven. That is why I am very aware of this work and results.

Zz.

Is the sloping curve of the intensity due to any aspect of the magnetic field/forces?

Is the sloping curve of the intensity due to any aspect of the magnetic field/forces?

Are are no external magnetic field a photoemission experiment. If there is, the momentum measurement will be completely ruined/useless.

Zz.

have you got a diagram of the set up?

I found this one, but it's a really simple diagram:

http://www.chem.qmul.ac.uk/surfaces/scc/images/scat5_3b.gif [Broken]

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consider a glass tube with water and glass tube with oil, with some marbles. the flow of marbles in water would be faster than the oil, depends on viscosity and of course gravity.
now becoz of slow movement in oil it seems to think that marbles with less mass. the same concept, if u consider a semiconductor crystal the free electrons attracted by the internal forces(ion's, free electrons interaction) and external forces(applied voltage) also... so total force is equal to
Ftotal=Fexternal(applied voltage)+ Finternal= ma
m is the mass of rest electron

now it is difficult to calculate the internal forces, so to compensate we are considering mass of electron as m*, now m* is the effective mass, force=m*.a
effective mass would be different for si, ge...