Feeling the Fermi Surfaces
Table of Contents
- Methods to measure the Fermi surfaces
- de Haas-Van alphen effect
- Magneto acuostic effect
- Attenuation of ultrasound
- Skin anamoly effect
- Cyclotron Resonance Effect
- Size effect
1. de Haas-Van alphen effect ATTACH
- It is the widely used method to find the fermi surface. One measures the magnetic susceptibility \(\chi\) in these experiments. \(\chi\) is defined as the change in the magnetization of the material (\(M\)) due to applied magnetic field (\(H\)). Mainly two experimental ways are used to measure \(\chi\):
- Torque method
- Field modulation method
- The main idea is as follows:
- Under mangetic field the energy spectrum is represented by the Landau levels given as: \[\varepsilon = p_{z}^{2} + (\nu + \frac{1}{2})\hbar \omega_{c}\].
- For isotropic case, the Fermi sphere have a tube structure which are placed inside each other (known as landau tube). Each tube is named according to the number \(\nu\) named as landau levels. All the previously allowed states now reisde on these tubes.
- With increase in magnetic field the radius of the first tube, whose energy is less than fermi energy increases. At some given mangetic field the given tube crosses the fermi surface, the moment it crosses the Fermi surface one will have a large number of available states. When this happens it will affect the electronic properties of materials.
- Effect of spin on the properties is found by following formula:
Here \(g_{0}\) is the DOS without the spin degrees of freedom. - Explanation using Landau tube and Fermi surfaces
- One of the best explanation on internet: https://physics.stackexchange.com/questions/707695/landau-tubes-and-fermi-sphere
- Under magnetic field the distribution of allowed energy levels transforms to cocentric tubes.
- One then superimpose the concentric tubes with the fermi surface without magnetic field. It is done to find the number of electrons on the Fermi surfaces. One should remember that under magnetic field all the electrons live only on the surfaces of Landau tubes. However, as application of the magnetic field does not change the energy of the electrons, therefore the fermi energy remains same. Therefore the number of electron on the Fermi surface is found at the intersection of the Landau tube and intial fermi surfaces (initial fermi surface defines the fermi energy).
- With increase in the magnetic field the distance between the Landau tube increases, the moment when the initial fermi surface is tangential to the Landau tube one will have the large number of free electrons.
- When the spin of electron will taken into account the single circle will be divided into two cocentric circles with small gap in their energy.
2. Magneto acoustic Effects ATTACH
- In this case an oscillating magnetic field is applied to the system. The electron in this case, rotates perpendicular to the magnetic field with cyclotron frequency. If magnetic field is along $z$-axis then the applied sound wave travels on the $xy$-plane.
- As we are using long wavelengths hence their energy is very low, they can interact with electrons on the fermi surface only.
- When the diameter of the cyclotron orbit of the electrons is equal to the half wavelength then depending on the phase of the electronic orbits compared to the sound waves, the diameter of the electronic orbit decreases or increases, it means the electron absorbs energy from the sound waves. Hence, it is strongly coupled to the sound wave.
- When the diameter of the cyclotron orbit of the electron is equal to the full wavelength then no matter what the electronic orbit does not get affected. It means the electrons dont take energy from the sound wave. It is weakly coupled to the sound wave.
3. Resources ATTACH
- De haas-Van alphen effects
- Magneto-Acoustic effects
- The actual article: https://journals.aps.org/pr/pdf/10.1103/PhysRev.117.937