Band structure of ferromagnetic metal

[Kit]

for a ferromagnetic metal, the there will be splitting of energy band(one for spin up e- and one for spin down e-) under the influence of external magnetic field. the it is known as exchange splitting.

here are my questions
1. what determine the degree of splitting? i guess it depends on the materials and how strong is the external H-field. any formula can calculate or approximate how strong is the splitting?

2. so if the energy band is splitted, would it affect the fermi level of the metal?

thanks

 

[Gokul43201]

Actually, for a paramagnetic or diamagnetic material, there will be spin split bands, in the presence of an external field, B. The spin splitting, or Zeeman, energy is given by $U=g\mu_B \mathbf{B \cdot S}$, where g is the Lande g-factor. [1]

In a ferromagnet, you don't need an external field to see spin splitting. The exchange interaction between charge carriers (with spin s at position r) and magnetic moments (with spin S_n at positions R_n) is given by $U_{ex} = -\sum_n J(|R_n-r|) \mathbf{s \cdot S_n}$ [2]. This can be calculated by your approximation of choice, and gives roughly, $\Delta E = SJ(M)$, where M is the temperature dependent magnetization within a domain, and can be approximated by the Brillouin function [3].

Whether your material is ferromagnetic or not, the effect of the spin splitting on the position of the fermi surface is negligible[4]. However, depending on the temperature (and in the case of a para/dia-magnet, also the applied field) the spin splitting can result in a significant polarization of the Fermi surface (whereas, in the absense of spin-splitting, the Fermi surface would be expected to have an equal number of spin up and spin down electrons, and hence, no net polarization) [5].[1] See any solid state textbook : Kittel, Ashcroft & Mermin, Marder, etc. talk about this.

[2] This is the basis of RKKY theory. You can look for references on RKKY or Indirect Exchange.

[3] I've seen this is Kittel and Ashcroft. Note the difference between the J used here - for the exchange integral - and the J you will come across in calculations of the Brillouin function - for the total angular momentum.

[4] There are some exceptions, notably in quantum Hall systems, at low filling factors.

[5] This is the basis of Spintronics. A review paper discussing the use of diluted magnetic semiconductors for spintronics will likely talk about this.

If you're having a hard time finding references, I could look some up for you.

 

[Kit]

thanks a lot ^^

let me look up the references first

 

[Kit]

i think i got a better idea but i need more infomation

1. i got the book of kittel but i cannot find the equation Delta E = SJ(M) in the chapter of ferronmagnetism and antiferromagnetism. where can i find more info about this, thanks.

2. i also want more details about the exceptional cases where the change in fermi level is not negligible.

thanks a lot

kit

 

[Gokul43201]

i think i got a better idea but i need more infomation

1. i got the book of kittel but i cannot find the equation Delta E = SJ(M) in the chapter of ferronmagnetism and antiferromagnetism. where can i find more info about this, thanks.

1. C. Haas, Phys Rev 168, 531 (1968)

2. C. Haas, "
Spin-disorder Scattering and Band Structure of the Ferromagnetic Chalcogenide Spinels", IBM Journal of R&D 14, 282 (1970)

3. Magnetism vol IIB, edited by Rado and Suhl (1966). See the chapter by Kasuya.

2. i also want more details about the exceptional cases where the change in fermi level is not negligible.

thanks a lot

kit

I'm not sure what kind of references to point to for this. The one example that comes to mind is in the high-field quantum Hall regime (ie: a 2D electron gas in a large, perpendicular B-field). You'll have to learn a good bit of 2DEG physics for this. Marder, I think, has a part of a chapter devoted to this. Also, you can read
John H. Davies, The Physics of Low-dimensional Semiconductors : An Introduction, Ch 6: Electric and Magnetic Fields

 

[Kit]

ok

thanks a lot^o^

kit