Understanding Landau Level Broadening in Magnetic Fields

[Muneer QAU]

what is meant by landau level broadening in a magnetic field?

 

[xiyangxixia]

The scope of energy level get large.

 

[genneth]

In a magnetic field, the transverse momentum of electrons are quantised into discrete Landau levels, separated in energy by the gyromagnetic frequency. Each level actually contains a continuum of states. In the presence of weak disorder, this continuum of states spread out (in energy) and gives rise to a non-degenerate density of states. Since these states are two dimensional, Anderson localisation occurs, i.e. the localisation length diverges (logarithmically). However, since in reality the samples are not infinite in extent, in each "blob" which comes from one Landau level, the middle region will still be spatially extended sufficiently to perform transport, but the edges will be insulating states.

 

[Muneer QAU]

thanks Genneth for this reply i am working on magnetotransport and have encounter with this term of Landau Level Broadening. i do not have idea of Anderson Localization please can you explain it simple term i read that landau level broadening occur due to impurity scattering how it occurs?

 

[Muneer QAU]

Thanks xiyangxixia can you elaborate the term the scope of energy level?

 

[genneth]

Roughly speaking, impurities can cause the electron to become trapped, bouncing between different impurities. Making this precise was Anderson's Nobel prize. If you are well versed in field theory, Ben Simons (google for him) has a couple of good sets of graduate lecture notes on his website.

 

[Muneer QAU]

Thanks genneth for such a nice favour :)

 

[Muneer QAU]

how to separate real and imaginary parts of a complex number??

i read that real and imaginary parts of a complex number can be separated by following equation
1/x+iη=P(1/x)-iδ(x)
where P is principal of x i dunt understand this equation can any 1 explain this?

 

[genneth]

It is the Cauchy principal value. See http://www.damtp.cam.ac.uk/user/stcs/courses/fcm/handouts/cauchy_principal_value.pdf for some notes.

Your equation is meant only in a distributional sense, i.e. they give the same thing if you integrate them with a test function.

 

[AlexKryuchkov]

i read that real and imaginary parts of a complex number can be separated by following equation
1/x+iη=P(1/x)-iδ(x)
where P is principal of x i dunt understand this equation can any 1 explain this?

the simplest example is if you have
\int \frac{f(x)}{\text{i$\epsilon $}+x} \, dx

so sometimes it is usefull to rewrite it in way
\int \frac{f(x)}{\text{i$\epsilon $}+x} \, dx=P \int \frac{f(x)}{x} \, dx-i \pi f(x)

where P means that you integrate in terms of principal value