# Band structure of ferromagnetic metal

1. Apr 21, 2006

### 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

2. Apr 21, 2006

### Gokul43201

Staff Emeritus
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 Sn at positions Rn) 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].

[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.

3. Apr 23, 2006

### Kit

thanks a lot ^^

let me look up the references first

4. Apr 24, 2006

### 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

5. Apr 25, 2006

### Gokul43201

Staff Emeritus
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.

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

Last edited: Apr 25, 2006
6. Apr 25, 2006

### Kit

ok

thanks a lot^o^

kit