Quantum Hall effect
The quantum Hall effect (or integer quantum Hall effect) is a
where VHall is the
The striking feature of the integer quantum Hall effect is the persistence of the quantization (i.e. the Hall plateau) as the electron density is varied. Since the electron density remains constant when the Fermi level is in a clean spectral gap, this situation corresponds to one where the Fermi level is an energy with a finite density of states, though these states are localized (see Anderson localization).[1]
The
Applications
The quantization of the Hall conductance () has the important property of being exceedingly precise. Actual measurements of the Hall conductance have been found to be integer or fractional multiples of e2/h to nearly one part in a billion. It has allowed for the definition of a new practical
In 1990, a fixed conventional value RK-90 = 25812.807 Ω was defined for use in resistance calibrations worldwide.[5] On 16 November 2018, the 26th meeting of the General Conference on Weights and Measures decided to fix exact values of h (the Planck constant) and e (the elementary charge),[6] superseding the 1990 value with an exact permanent value RK = h/e2 = 25812.80745... Ω.[7]
Research status
The integer quantum hall is considered part of exact quantization.
The
Currently it is considered an open research problem because no single, confirmed and agreed list of fractional quantum numbers exists, neither a single agreed model to explain all of them, although there are such claims in the scope ofHistory
The
The integer quantization of the Hall conductance was originally predicted by University of Tokyo researchers Tsuneya Ando, Yukio Matsumoto and Yasutada Uemura in 1975, on the basis of an approximate calculation which they themselves did not believe to be true.[15] In 1978, the Gakushuin University researchers Jun-ichi Wakabayashi and Shinji Kawaji subsequently observed the effect in experiments carried out on the inversion layer of MOSFETs.[16]
In 1980,
Integer quantum Hall effect
Landau levels
In two dimensions, when classical electrons are subjected to a magnetic field they follow circular cyclotron orbits. When the system is treated quantum mechanically, these orbits are quantized. To determine the values of the energy levels the Schrödinger equation must be solved.
Since the system is subjected to a magnetic field, it has to be introduced as an electromagnetic vector potential in the
where is the canonical momentum, which is replaced by the operator and is the total energy.
To solve this equation it is possible to separate it into two equations since the magnetic field just affects the movement along x and y axes. The total energy becomes then, the sum of two contributions . The corresponding equations in z axis is:
To simplify things, the solution is considered as an infinite well. Thus the solutions for the z direction are the energies , and the wavefunctions are sinusoidal. For the and directions, the solution of the Schrödinger equation can be chosen to be the product of a plane wave in -direction with some unknown function of , i.e., . This is because the vector potential does not depend on and the momentum operator therefore commutes with the Hamiltonian. By substituting this Ansatz into the Schrödinger equation one gets the one-dimensional harmonic oscillator equation centered at .
where is defined as the cyclotron frequency and the magnetic length. The energies are:
- ,
And the wavefunctions for the motion in the plane are given by the product of a plane wave in and Hermite polynomials attenuated by the gaussian function in , which are the wavefunctions of a harmonic oscillator.
From the expression for the Landau levels one notices that the energy depends only on , not on . States with the same but different are degenerate.
Density of states
At zero field, the density of states per unit surface for the two-dimensional electron gas taking into account degeneration due to spin is independent of the energy
- .
As the field is turned on, the density of states collapses from the constant to a Dirac comb, a series of Dirac functions, corresponding to the Landau levels separated . At finite temperature, however, the Landau levels acquire a width being the time between scattering events. Commonly it is assumed that the precise shape of Landau levels is a
Another feature is that the wave functions form parallel strips in the -direction spaced equally along the -axis, along the lines of . Since there is nothing special about any direction in the -plane if the vector potential was differently chosen one should find circular symmetry.
Given a sample of dimensions and applying the periodic boundary conditions in the -direction being an integer, one gets that each parabolic potential is placed at a value .
The number of states for each Landau Level and can be calculated from the ratio between the total magnetic flux that passes through the sample and the magnetic flux corresponding to a state.
Thus the density of states per unit surface is
- .
Note the dependency of the density of states with the magnetic field. The larger the magnetic field is, the more states are in each Landau level. As a consequence, there is more confinement in the system since less energy levels are occupied.
Rewriting the last expression as it is clear that each Landau level contains as many states as in a 2DEG in a .
Given the fact that electrons are
In order to get the number of occupied Landau levels, one defines the so-called filling factor as the ratio between the density of states in a 2DEG and the density of states in the Landau levels.
In general the filling factor is not an integer. It happens to be an integer when there is an exact number of filled Landau levels. Instead, it becomes a non-integer when the top level is not fully occupied. In actual experiments, one varies the magnetic field and fixes electron density (and not the Fermi energy!) or varies the electron density and fixes the magnetic field. Both cases correspond to a continuous variation of the filling factor and one cannot expect to be an integer. Since , by increasing the magnetic field, the Landau levels move up in energy and the number of states in each level grow, so fewer electrons occupy the top level until it becomes empty. If the magnetic field keeps increasing, eventually, all electrons will be in the lowest Landau level () and this is called the magnetic quantum limit.
Longitudinal resistivity
It is possible to relate the filling factor to the resistivity and hence, to the conductivity of the system. When is an integer, the Fermi energy lies in between Landau levels where there are no states available for carriers, so the conductivity becomes zero (it is considered that the magnetic field is big enough so that there is no overlap between Landau levels, otherwise there would be few electrons and the conductivity would be approximately ). Consequently, the resistivity becomes zero too (At very high magnetic fields it is proven that longitudinal conductivity and resistivity are proportional).[21]
With the conductivity one finds
If the longitudinal resistivity is zero and transversal is finite, then . Thus both, the longitudinal conductivity and resistivity become zero.
Instead, when is a half-integer, the Fermi energy is located at the peak of the density distribution of some Landau Level. This means that the conductivity will have a maximum .
This distribution of minimums and maximums corresponds to ¨quantum oscillations¨ called Shubnikov–de Haas oscillations which become more relevant as the magnetic field increases. Obviously, the height of the peaks are larger as the magnetic field increases since the density of states increases with the field, so there are more carriers which contribute to the resistivity. It is interesting to notice that if the magnetic field is very small, the longitudinal resistivity is a constant which means that the classical result is reached.
Transverse resistivity
From the classical relation of the transverse resistivity and substituting one finds out the quantization of the transverse resistivity and conductivity:
One concludes then, that the transverse resistivity is a multiple of the inverse of the so-called conductance quantum if the filling factor is an integer. In experiments, however, plateaus are observed for whole plateaus of filling values , which indicates that there are in fact electron states between the Landau levels. These states are localized in, for example, impurities of the material where they are trapped in orbits so they can not contribute to the conductivity. That is why the resistivity remains constant in between Landau levels. Again if the magnetic field decreases, one gets the classical result in which the resistivity is proportional to the magnetic field.
Photonic quantum Hall effect
The quantum Hall effect, in addition to being observed in
Topological classification
The integers that appear in the Hall effect are examples of
Concerning physical mechanisms, impurities and/or particular states (e.g., edge currents) are important for both the 'integer' and 'fractional' effects. In addition, Coulomb interaction is also essential in the
Bohr atom interpretation of the von Klitzing constant
The value of the von Klitzing constant may be obtained already on the level of a single atom within the Bohr model while looking at it as a single-electron Hall effect. While during the cyclotron motion on a circular orbit the centrifugal force is balanced by the Lorentz force responsible for the transverse induced voltage and the Hall effect, one may look at the Coulomb potential difference in the Bohr atom as the induced single atom Hall voltage and the periodic electron motion on a circle as a Hall current. Defining the single atom Hall current as a rate a single electron charge is making Kepler revolutions with angular frequency
and the induced Hall voltage as a difference between the hydrogen nucleus Coulomb potential at the electron orbital point and at infinity:
One obtains the quantization of the defined Bohr orbit Hall resistance in steps of the von Klitzing constant as
which for the Bohr atom is linear but not inverse in the integer n.
Relativistic analogs
Relativistic examples of the integer quantum Hall effect and quantum spin Hall effect arise in the context of lattice gauge theory.[27][28]
See also
- Quantum Hall transitions
- Fractional quantum Hall effect
- Quantum anomalous Hall effect
- Quantum cellular automata
- Composite fermions
- Conductance Quantum
- Hall effect
- Hall probe
- Graphene
- Quantum spin Hall effect
- Coulomb potential between two current loops embedded in a magnetic field
References
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F. D. M. Haldane (1988). "Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the 'Parity Anomaly'". Physical Review Letters. 61 (18): 2015–2018. PMID 10038961.
- ISBN 978-981-4360-75-3.
- ^ "2018 CODATA Value: conventional value of von Klitzing constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
- ^ "26th CGPM Resolutions" (PDF). BIPM. Archived from the original (PDF) on 2018-11-19. Retrieved 2018-11-19.
- ^ "2018 CODATA Value: von Klitzing constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
- S2CID 206528413.
- ^ "Haldane nobel prize Lecture" (PDF).
- ^ a b R. B. Laughlin (1981). "Quantized Hall conductivity in two dimensions". Phys. Rev. B. 23 (10): 5632–5633. .
- ISBN 978-1107404250.
- ^ Tong, David. "Quantum Hall Effect".
- ^ "1960 – Metal Oxide Semiconductor (MOS) Transistor Demonstrated". The Silicon Engine. Computer History Museum.
- ^ .
- ^ Tsuneya Ando; Yukio Matsumoto; Yasutada Uemura (1975). "Theory of Hall effect in a two-dimensional electron system". J. Phys. Soc. Jpn. 39 (2): 279–288. .
- ^ Jun-ichi Wakabayashi; Shinji Kawaji (1978). "Hall effect in silicon MOS inversion layers under strong magnetic fields". J. Phys. Soc. Jpn. 44 (6): 1839. .
- ^ K. v. Klitzing; G. Dorda; M. Pepper (1980). "New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance". Phys. Rev. Lett. 45 (6): 494–497. .
- ^ D. J. Thouless (1983). "Quantization of particle transport". Phys. Rev. B. 27 (10): 6083–6087. .
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K. S. Novoselov; Z. Jiang; Y. Zhang; S. V. Morozov; H. L. Stormer; U. Zeitler; J. C. Maan; G. S. Boebinger; P. Kim; A. K. Geim (2007). "Room-temperature quantum Hall effect in graphene". Science. 315 (5817): 1379. S2CID 46256393.
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Tsukazaki, A.; Ohtomo, A.; Kita, T.; Ohno, Y.; Ohno, H.; Kawasaki, M. (2007). "Quantum Hall effect in polar oxide heterostructures". Science. 315 (5817): 1388–91. S2CID 10674643.
- ISBN 9780511819070.)
{{cite book}}
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D. B. Kaplan (1992). "A Method for simulating chiral fermions on the lattice". Physics Letters. B288 (3–4): 342–347. S2CID 14161004.
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M. F. L. Golterman; K. Jansen; D. B. Kaplan (1993). "Chern–Simons currents and chiral fermions on the lattice". Physics Letters. B301 (2–3): 219–223. S2CID 9265777.
Further reading
- D. R. Yennie (1987). "Integral quantum Hall effect for nonspecialists". Rev. Mod. Phys. 59 (3): 781–824. .
- D. Hsieh; D. Qian; L. Wray; Y. Xia; Y. S. Hor; R. J. Cava; M. Z. Hasan (2008). "A topological Dirac insulator in a quantum spin Hall phase". Nature. 452 (7190): 970–974. S2CID 4402113.
- 25 years of Quantum Hall Effect, K. von Klitzing, Poincaré Seminar (Paris-2004). Postscript. Pdf.
- Magnet Lab Press Release Quantum Hall Effect Observed at Room Temperature
- Avron, Joseph E.; Osadchy, Daniel; Seiler, Ruedi (2003). "A Topological Look at the Quantum Hall Effect". Physics Today. 56 (8): 38. .
- Zyun F. Ezawa: Quantum Hall Effects - Field Theoretical Approach and Related Topics. World Scientific, Singapore 2008, ISBN 978-981-270-032-2
- Sankar D. Sarma, ISBN 978-0-471-11216-7
- A. Baumgartner; T. Ihn; K. Ensslin; K. Maranowski; A. Gossard (2007). "Quantum Hall effect transition in scanning gate experiments". Phys. Rev. B. 76 (8): 085316. .
- E. I. Rashba and V. B. Timofeev, Quantum Hall Effect, Sov. Phys. – Semiconductors v. 20, pp. 617–647 (1986).