# Quantum oscillations (experimental technique)

Condensed matter
experiments
ARPES
ACAR
Neutron scattering
X-ray spectroscopy
Quantum oscillations
Scanning tunneling microscopy

In condensed matter physics, Quantum oscillations describes a series of related experimental techniques used to map the Fermi surface of a metal in the presence of a strong magnetic field.[1] These techniques are based on the principle of Landau quantization of Fermions moving in a magnetic field.[2] For a gas of free fermions in a strong magnetic field, the energy levels are quantized into bands, called the Landau levels, whose separation is inversely proportional to the strength of the magnetic field. In a quantum oscillation experiment, the external magnetic field is varied, which causes the Landau levels to pass over the Fermi surface, which in turn results in oscillations of the electronic density of states at the Fermi level; this produces oscillations in the many material properties which depend on this, including resistance (the Shubnikov-de Haas effect), Hall resistance,[2] and magnetic susceptibility (the de Haas-van Alphen effect). Observation of quantum oscillations in a material is considered a signature of Fermi liquid behaviour.[3]

Quantum oscillations have been used to study high temperature superconducting materials such as cuprates and pnictides.[1] Studies using these experiments have shown that the ground state of underdoped cuprates behave similar to a Fermi liquid, and display characteristics such as Landau quasiparticles.[4]

## Experiment

When a magnetic field is applied to a system of free charged fermions, their energy states are quantized into the so-called Landau levels, given by[5]

YBCO superconductor under high magnetic field. As field strength is increased, superconductivity is suppressed and Landau oscillations can be observed

${\displaystyle \varepsilon _{l}={\frac {eB}{m^{*}}}\left(\ell +{\frac {1}{2}}\right)}$

for integer-valued ${\displaystyle \ell }$, where ${\displaystyle B}$ is the external magnetic field and ${\displaystyle e,m^{*}}$ are the fermion charge and effective mass respectively.

When the external magnetic field ${\displaystyle B}$ is increased in an isolated system, the Landau levels expand, and eventually "fall off" the Fermi surface. This leads to oscillations in the observed energy of the highest occupied level, and hence in many physical properties (including Hall conductivity, resistivity, and susceptibility). The periodicity of these oscillations can be measured, and in turn can be used to determine the cross-sectional area of the Fermi surface.[6] If the axis of the magnetic field is varied at constant magnitude, similar oscillations are observed. The oscillations occur whenever the Landau orbits touch the Fermi surface. In this way, the complete geometry of the Fermi sphere can be mapped.[6]

## Underdoped cuprates

Studies of underdoped cuprate compounds such as YBa2Cu3O6+x through probes such as ARPES have indicated that these phases show characteristics of non-Fermi liquids,[7] and in particular, the absence of well-defined Landau quasiparticles.[8] However, quantum oscillations have been observed in these materials at low temperatures, if their superconductivity is suppressed by a sufficiently high magnetic field,[2] which is evidence for the presence of well-defined quasiparticles with fermionic statistics. These experimental results thus disagree with those from ARPES and other probes.[5]

## References

1. ^ a b Coldea, Amalia (2010). "Quantum oscillations probe the normal electronic states of novel superconductors". Philosophical Transactions of the Royal Society A. 368 (1924): 3503–3517. Bibcode:2010RSPTA.368.3503C. doi:10.1098/rsta.2010.0089. PMID 20603364. Retrieved 20 March 2012.
2. ^ a b c Doiron-Leyraud, Nicolas; et al. (2007). "Quantum oscillations and the Fermi surface in an underdoped high-Tc superconductor". Nature. 447 (7144): 565–8. arXiv:0801.1281. Bibcode:2007Natur.447..565D. doi:10.1038/nature05872. PMID 17538614. S2CID 4397560.
3. ^ Condensed-matter and materials physics: the science of the world around us. National Research Council. 2010. ISBN 978-0-309-13409-5.
4. ^ Broun, D. M. (2008). "What lies beneath the dome?". Nature Physics. 4 (3): 170–172. Bibcode:2008NatPh...4..170B. doi:10.1038/nphys909.
5. ^ a b Sebastian, Suchitra E.; Neil Harrison; Gilbert G. Lonzarich (2011). "Quantum oscillations in the high-Tc cuprates". Philosophical Transactions of the Royal Society A. 369 (1941): 1687–1711. Bibcode:2011RSPTA.369.1687S. doi:10.1098/rsta.2010.0243. PMID 21422021. Retrieved 23 March 2012.
6. ^ a b Ibach, Harald; Hans Lüth (1995). Solid-state physics: an introduction to principles of materials science. Berlin: Springer-Verlag. ISBN 978-3-540-58573-2.
7. ^ Alexandrov, A. S. (2008). "Theory of quantum magneto-oscillations in underdoped cuprate superconductors". Journal of Physics: Condensed Matter. 20 (19): 192202. arXiv:0711.0093. Bibcode:2008JPCM...20s2202A. doi:10.1088/0953-8984/20/19/192202. S2CID 117020227.
8. ^ Damascelli, Andrea; Hussain, Zahid; Zhi-Xun Shen (2003). "Angle-resolved photoemission studies of the cuprate superconductors". Reviews of Modern Physics. 75 (2): 473. arXiv:cond-mat/0208504. Bibcode:2003RvMP...75..473D. doi:10.1103/RevModPhys.75.473. S2CID 118433150.