Talk:Effective mass (solid-state physics)

This article says that an electron hole in silicon has effective mass around 1.00me. But Electron hole says it's typically about 0.36me. Which is closer to the true value ?

0.36 -- Tim Starling 08:57, Jan 17, 2004 (UTC)

This article uses the density of states effective mass, while the other article you mention just uses the heavy hole effective mass. It might be good to add a few paragraphs to distinguish these. Also, to my eye, the effective mass quoted for electrons in GaAs in this article is just plain wrong... Am I missing something?

More theory needed[edit]

This article could use some discussion about how effective masses are calculated theoretically (e.g. k.p method), but I am not the person to write it. Alison Chaiken 20:03, 7 January 2006 (UTC)[reply]

I am adding something about this. I think that in general the problem is about non-parabolic band structures being mapped onto a single parabolic band model. That leads to different masses for different properties. The article already brings that point out but does not say much about the calculation. — Preceding unsigned comment added by 128.206.125.120 (talk) 13:23, 6 April 2018 (UTC)[reply]

I added a little "theoretical" methods section too, at least so that we can have a link to the k·p perturbation theory article etc. --Steve (talk) 14:45, 6 April 2018 (UTC)[reply]

Incorrect effective masses reported[edit]

I'm also confused by some of the reported effective mass values. According to the reference book: Madelung, Otfried (ed.), Semiconductors--Basic Data (rev. 2nd ed.), Springer, 1996 ...electron and hole effective masses in ZnO are 0.24-0.28 and 0.59, respectively. Should effective masses used for DOS calculations be different somehow? Willtiz 06:19, 2 March 2007 (UTC)[reply]

There is indeed a difference between conductivity effective mass and density of states effective mass. I don't really know enough at the moment to write it up, but http://ecee.colorado.edu/~bart/book/effmass.htm is a good reference on the topic. Essentially the difference is that for density of state calculations one takes a geometric mean of the effective masses, but for conductivity it is a harmonic mean. Mattski (talk) 04:12, 7 October 2010 (UTC)[reply]

Derivation[edit]

Why would anyone use the first derivation? And shouldn't it be noted, that the second derivation is only valid since we consider a parabolization of the (k,E)-curve. And what's with the epsilons? Why not use E for energy as the rest of Wikipedia? —Preceding unsigned comment added by 89.236.15.92 (talk) 12:40, 6 October 2009 (UTC)[reply]

Curvature-based effective mass is counterintuitive, e.g., infinite with linear dispersion[edit]

It's worth pointing out that the featured definition of effective mass, as based on the inverse of the second derivative of energy-momentum dispersion, can be counterintuitive. It becomes infinite when the second derivative drops to zero, as in a linear dispersion. In other words, photons, and the electrons in graphene have infinite effective mass. A simple way of thinking about it is: if I try to push a photon to make it go faster (giving it more momentum it in the direction it is already moving), then its velocity is unchanged, therefore (by mass = Force/acceleration) it must have infinite mass due to its lack of acceleration.

Of course, few people are comfortable with their photons to have infinite mass and so it might not be the best definition of effective mass. Perhaps it would be good to mention other distinct definitions of effective mass as well, such as relativistic mass (mass = pmomentum/velocity, a.k.a. cyclotron mass, though that still leaves some finite mass for photons), or rest mass which is finally zero for photons. It's also worth mentioning a good writeup distinguishing transverse and longitudinal mass. --Nanite (talk) 10:16, 4 August 2013 (UTC)[reply]

Good idea to discuss all the possible formulas and the contexts where they're useful. I know that the second-derivative formula is not the only one in use (but I don't remember the others).

When you do this (I hope you do) ... probably you already know this, but don't forget to write the formulas in such a way that the minimum energy is not necessarily at k=0. For example, the center of the graphene cone is not at k=0. Replace "k" by "k - k0" or whatever. :-D --Steve (talk) 14:19, 5 August 2013 (UTC)[reply]

Good point indeed about the K points. I'll mull the concepts over for a little while before getting to work on this. I'd like to make the page a little less complex and a little more detailed at the same time. I think there should be a first basic section that covers exclusively the most basic (

\hbar ^{2}k^{2}/2m^{*}

-type) dispersion relations where the effective mass is unambiguous---followed by an advanced section. So far I have heard about the following effective masses for the general anisotropic case:

Equation-of-motion effective masses from inverse mass tensor $[m^{-1}]_{ij}=\hbar ^{-2}(\partial ^{2}/\partial k_{i}\partial k_{j})E$ . Since it's symmetric I guess there are up to 6 independent effective mass coefficients in there, for 3D material.
Cyclotron effective mass $m(E,k_{z})=(2\pi )^{-1}\hbar ^{2}(\partial A/\partial E)_{k_{z}}$ for magentic field along z axis, where A is the k-space area in for a given valley at energy E and k_z.
Per-valley density of states effective mass (3D material), ${\rm {DOS}}=8\pi {\sqrt {2}}m^{3/2}{\sqrt {E-E_{0}}}/h^{3}$ .
Per-valley density of states effective mass (2D material), ${\rm {DOS}}=m/(\pi \hbar ^{2})$ . I think this is equivalent to the perpendicular-field cyclotron mass. It's been argued [1] that these are the "best" effective mass for 2D systems.
Per-valley density of states effective mass (1D material), not sure if anyone uses this.
[2] "Thermal velocity" effective mass, not sure when this is important.
[3] Conductivity effective mass, not quite sure how this is defined in the fully anisotropic case.

(I forgot to sign this comment from Aug 6 -- Nanite)

Okay, significant changes have been made. For the benefit of future readers of the talk page wondering what I was talking about, before and after. I left out the thermal velocity mass since it seems to be a minor concern (from Green, "This mass is required for calculating cross sections from capture rates."). --Nanite (talk) 08:56, 23 August 2013 (UTC)[reply]

Help with tensor indices needed[edit]

I need some help with the indices in the section Inertial effective mass tensor. When using tensor algebra, the balance between covariant and contravariant indices should be the same on both sides. But on the left hand side we have $a$ , which is a contravariant vector (so it should really be $a i$ , not $a i$ ), so there is one more contravariant index than covariant indices, while on the right hand side we have $F$ which is also a contravariant vector (so it should be $F j$ and not $F j$ ), but then we have $k$ two times in the denominator, which is a covariant vector but since it appears in the denominator it gets the reversed effect and the entire fraction effectively becomes a tensor with two contravariant indices. So we have one more contravariant index than covariant indices on the left hand side, but three more contravariant index than covariant indices on the right hand side!

I think the problem is that $E/\hbar ^{2}$ has the unit $m -2 kg -1$ and should actually provide minus two (the power over the length unit, by convention) to that balance, so the entire right hand side should also have just one more contravariant index than covariant indices, and thus be a contravariant vector. But I don't know how that should be written. Can $F j$ instead be written as $\delta _{jl}F^{l}$ ? Or should $E$ and $\hbar$ actually be tensors? Someone with some better understanding of tensor algebra, please help out! —Kri (talk) 20:46, 22 August 2014 (UTC)[reply]

You would never be using this equation in a relativistic context. So there's no point in distinguishing upper and lower indices. Some readers don't know special relativity, or don't know it very well. Just use exclusively lower indices, like they teach in freshman physics and mechanical engineering. :-D

I don't know what you meant by "a and F are the tensor equivalents of a and F". Can you explain? Are you using "tensor" when you mean "4-vector"? Are pedantically refusing to equate a vector v with its representation

(v_{x},v_{y},v_{z})

in a coordinate system? Well I guess it's a moot point since you deleted it. :-P

Also, F is the force on a particle. It's a vector, not a vector field. So you can always take it outside the gradient, without any caveats (you don't need to say that F is slowly varying in space). :-D --Steve (talk) 20:37, 24 August 2014 (UTC)[reply]

I wanted to get the indices right because I wanted to feel that I understood how to use them properly myself. But since it is not needed in this case, it is unnecessary to have that kind of notation in the article when equally well normal vector algebra can be used. I don't know what I meant with the tensor equivalents of a and F; I thought that one usually differed between vectors in vector algebra and vectors in tensor algebra (aren't these just tensors with only one index?), but maybe not...

And yeah, you're right, F is a constant. I came to the conclusion

d k /d t = F / ħ

by looking at the Schrödinger equation, and there F is a vector field since it is just the negative gradient of a potential energy. But it wouldn't have made any sense to take the derivative of F anyway, since

\nabla k

works in reciprocal space :P —Kri (talk) 15:40, 26 August 2014 (UTC)[reply]

Added sentence connecting to Bragg scattering[edit]

Simple_case:_parabolic.2C_isotropic_dispersion_relation I added a sentence referring to the fact that negative mass occurs just below the Brillouin zone. It seems to me like the applied force is pushing the Bloch wavefunction closer to the Bragg scattering situation where the reflected wave is sending the electron backwards. I chose not to say this in the WP article because none of the four references cited actually expressed that idea. --Guy vandegrift (talk) 17:10, 11 March 2016 (UTC)[reply]

I don't know what you mean by "just below the Brillouin zone". Brillouin zones don't have "above" and "below", they have inside and outside. I think you mean near the edge of the Brillouin zone. Anyway, it is not true that a negative curvature only occurs there, or even that it typically occurs there. The animation at the top of the article shows four semiconductors, and the valence band maximum is at k=0, right in the center of the Brillouin zone, for all four. Even in the simple 1D free electron model, negative curvature occurs at k=0 for half of the bands, and at the Brillouin zone edge for the other half. --Steve (talk) 00:04, 12 March 2016 (UTC)[reply]

I agree that my edit was faulty. I do believe I am correct for a 1-d crystal, where "just below" has unambiguous meaning. Also the word "typically" is not quite right. It would be better to include a brief section using a simple 1-d model, using the picture shown above. Why do this? Because it is important that Wikipedia bring topics to the lowest possible level. Before students start learning about 3D band structure, the simplest possible question should be answered first: How can negative effective mass possibly exist? The figure shown is similar to one found in Kittel.--Guy vandegrift (talk) 04:06, 12 March 2016 (UTC)[reply]

The different Brillouin zones are regions of the horizontal axis, i.e. certain values of k. That's why you can't be "above" or below" them.

If you extend that plot upwards just a little bit, you'll see a negative-curvature region right at k=0.

We do have an article about the nearly free electron model and I don't mind citing it. But my preference is to say that electrons are interacting with all the atoms and other electrons, and that the electron band structure is the solution to a complicated differential equation (given at Bloch wave) taking all these interactions into account. You have no reason to be surprised that the bands are not perfect parabolas, nor that they can curve in any which way. --Steve (talk) 19:56, 12 March 2016 (UTC)[reply]

But my understanding is that in the simplest models the electrons are treated as single non-interacting particles under the influence of a periodic potential. The fact that under this model the derivative of the group velocity ("group acceleration") goes backwards in the presence of an electric or magnetic field is surprising. And, I believe the intuitive ("handwaving") explanation is that if one applies a force to a particle just to the left of the first brillion zone, the electron goes in the opposite direction due to Bragg diffraction. This is not my article, but if you and/or another editor wants me to write a second One-dimensional nearly free electron model, I will. Otherwise I won't. You have a point in suggesting that nearly free electron model might be a better place for the image, but I have other things to do.--Guy vandegrift (talk) 04:20, 13 March 2016 (UTC)[reply]