Quantum Hall Effect

The quantum Hall effect ( QHE for short ) is expressed by the fact that at low temperatures and strong magnetic fields, the voltage that occurs perpendicular to a current does not increase linearly with the magnetic field, as in the classic Hall effect , but in steps. The effect occurs at interfaces where the electrons can be described as a two-dimensional electron gas .

The so-called Hall resistance , i.e. the ratio of the Hall voltage to the current intensity, only takes on whole-number fractions of the size as plateau values ( ), where is Planck's quantum and the elementary charge . Both are natural constants ; the plateau values ​​therefore depend neither on the material properties such as the charge carrier density, nor on the sample size, nor on the magnetic field strength. ${\displaystyle R_{\mathrm {H} }}$ ${\displaystyle R_{\mathrm {K} }=h/e^{2}}$ ${\displaystyle \approx 25{,}8\,\mathrm {k\Omega } }$ ${\displaystyle h}$ ${\displaystyle e}$

For this knowledge, Klaus von Klitzing received the Nobel Prize in Physics in 1985 .

The variable known as Von Klitzing's constant is now used to define the standard for electrical resistance . It is known exactly because and are used to define the units of measurement and an exact value has been assigned to them. ${\displaystyle R_{\mathrm {K} }}$ ${\displaystyle h}$ ${\displaystyle e}$

A distinction is made from the integral quantum Hall effect with only integer denominators of the fractional quantum Hall effect (also fractional QHE ), in which the denominators take the form of fractions (see below ). ${\displaystyle R_{\mathrm {K} }}$

Description of the phenomenon

Hall resistance ρ _xy and electrical resistance ρ _xx at low temperatures above the magnetic induction B in Tesla . For the highest shown plateau of ρ _xy , ν = 3 applies .

In the classic Hall effect , electrical current flows through a plate, which is penetrated by a magnetic field perpendicular to its surface . The charge carriers flowing in the magnetic field are laterally deflected by the Lorentz force , so that an electrical voltage can be measured at the edges of the plate across the direction of the current , which is referred to as the Hall voltage .

The ratio of the laterally applied Hall voltage to the current is referred to as the Hall resistance and is in two-dimensional Hall strips in the classic Hall effect

${\displaystyle R_{\mathrm {H} }={\frac {U_{\mathrm {H} }}{I}}={\frac {B}{ne}}}$, 

where the Hall voltage occurring across the total current is the total current (perpendicular to the direction in which the Hall voltage is measured), the magnetic field strength, the charge carrier density and the elementary charge . The classic Hall resistance is therefore particularly proportional to the applied magnetic field. You can see this in the picture for small field values. ${\displaystyle U_{\mathrm {H} }}$ ${\displaystyle I}$ ${\displaystyle B}$ ${\displaystyle n}$ ${\displaystyle e}$ ${\displaystyle B}$

However, at a sufficiently low temperature and a strong magnetic field, the Hall resistance takes one of the plateau values ​​regardless of the material

${\displaystyle R_{\mathrm {H} }={\frac {h}{\nu e^{2}}}={\frac {R_{\mathrm {K} }}{\nu }}}$(in the adjacent picture is marked with ) ${\displaystyle R_{\mathrm {H} }}$ ${\displaystyle \rho _{xy}}$

where are whole numbers, the Planck quantum and the " von Klitzing elementary resistance " is. ${\displaystyle \nu =1,2,\dots }$ ${\displaystyle h}$ ${\displaystyle R_{\mathrm {K} }}$

An increase in the strength of the magnetic field now leaves the Hall resistance constant until it changes to the next step value . The middle of the steps corresponds to the formula above, i.e. the classic Hall effect. Exactly in the middle of the step, the voltage applied to the sample in the direction of the current disappears , that is, the electrical resistance is zero there and the line becomes dissipation-free , apparently in the entire plateau area between the steps. At the steps themselves there are sharp maxima in the resistance. ${\displaystyle B}$ ${\displaystyle U_{x}}$

The plateau states of the quantum Hall effect are, similar to superconductivity , a macroscopic quantum state .

Test conditions

Attempts to observe the quantum Hall effect are usually in a simple helium - cryostat at 4.2  Kelvin performed. Lower temperatures, which are only possible with significantly more complex cooling technology, are usually not necessary, except for observing the fractional effect. However, nitrogen cooling is not sufficient, as the cooling temperature is around 70 Kelvin and as a result the mean free path of the electrons is still too short, i.e. the measurement is disturbed too much by interactions.

Depending on the sample, magnetic fields of a few Tesla are used and could be up to 40 Tesla with von Klitzing's apparatus, which corresponds to a multiple of the mean geomagnetic field strength in Germany of around 20 microtesla. For very strong magnetic fields, a pair of Helmholtz coils made of superconducting material is usually used, in which coil currents between 10 A and 100 A typically flow. In contrast, the current through the sample itself is only 0.1 to 10 µA.

The samples used in QHE experiments are MOSFETs ( metal oxide semiconductor field effect transistors ), in which the charge carrier density can be changed by a voltage applied to the transistor gate, or semiconductor-insulator heterostructures (e.g. Al _x Ga _1-x As / GaAs heterostructures), i.e. thin platelets that have a transition between an insulator and a semiconductor. At such a boundary layer the electrons lose a direction of movement: The direction in which the magnetic field is applied is fixed in the boundary potential by a quantum number , the probability of occupation of the next higher energy level is negligible. One speaks therefore of a two-dimensional electron gas . ${\displaystyle z}$

In the material graphene , which was first produced in 2004 , the quantum Hall effect was observed at room temperature , see also below in the section Unusual quantum Hall effect in graphene monolayers .

theory

Conductivity tensor

Due to a magnetic field or preferred conduction directions in a solid, Ohm's law can generally be written using a conductivity tensor: ${\displaystyle {\stackrel {\leftrightarrow }{\sigma }}}$

${\displaystyle {\vec {j}}={\stackrel {\leftrightarrow }{\sigma }}{\vec {E}}\,.}$

The conductivity and resistance tensor can be represented as 2x2 matrices in two dimensions : ${\displaystyle {\stackrel {\leftrightarrow }{\sigma }}}$ ${\displaystyle {\stackrel {\leftrightarrow }{\rho }}}$

${\displaystyle {\stackrel {\leftrightarrow }{\sigma }}=\left({\begin{array}{cc}\sigma _{xx}&-\sigma _{xy}\\\sigma _{xy}&\sigma _{xx}\end{array}}\right),\ \ {\stackrel {\leftrightarrow }{\rho }}={\stackrel {\leftrightarrow }{\sigma }}^{\,-1}={\frac {1}{\sigma _{xx}^{2}+\sigma _{xy}^{2}}}\left({\begin{array}{cc}\sigma _{xx}&\sigma _{xy}\\-\sigma _{xy}&\sigma _{xx}\end{array}}\right)=\left({\begin{array}{cc}\rho _{xx}&\rho _{xy}\\-\rho _{xy}&\rho _{xx}\end{array}}\right)}$.

If one chooses for the description of the QHE as the current direction, as the lateral direction in which the Hall voltage is applied, and as the magnetic field direction, then due to the arrangement . ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z}$ ${\displaystyle j_{y}=0}$

Orthogonal E and B fields

The classical movement of free electrons in mutually perpendicular (orthogonal) stationary electric and magnetic fields are is, one on spiral paths along the be construed -field and can as a superposition of the following components: ${\displaystyle B}$

• a circular motion with the cyclotron frequency around the field direction, ${\displaystyle \omega _{c}={\frac {eB}{m}}}$ ${\displaystyle B}$
• a drift movement with perpendicular to - and - field, ${\displaystyle v_{D}=-E/B}$ ${\displaystyle E}$ ${\displaystyle B}$
• an unaccelerated movement in the field direction. ${\displaystyle B}$

The cyclotron frequency also plays an important role in QHE, as we will see in a moment.

Quantum mechanical consideration

With the Coulomb calibration and the separation approach , the Schrödinger equation for the free electron, i.e. ${\displaystyle {\vec {p}}=i\hbar \nabla +e{\vec {A}}}$ ${\displaystyle {\vec {A}}=(0,xB,0)}$ ${\displaystyle \psi =\xi (x)\cdot e^{i(k_{y}y+k_{z}z)}}$

${\displaystyle {\frac {{\vec {p}}^{2}}{2m}}\ \psi \ =\ E\ \psi }$,

transformed into a differential equation for the -dependent function , which is the Schrödinger equation of a harmonic oscillator around the rest point . The only energy eigenvalues ​​obtained are the Landau levels : ${\displaystyle x}$ ${\displaystyle \xi }$ ${\displaystyle X={\frac {\hbar k_{y}}{m\omega _{c}}}}$

${\displaystyle E=E_{z}+\left(l+{\frac {1}{2}}\right)\ \hbar \omega _{c}\,}$, where . ${\displaystyle l=0,1,2,\dots }$

In the case of a sample dimension in the direction of the current or in the direction of the Hall voltage, the following applies: The wave number in the direction can assume the values with an integer , but it also appears in the rest position of the harmonic oscillator, for which applies. This results in the value range ${\displaystyle L_{x}}$ ${\displaystyle L_{y}}$ ${\displaystyle y}$ ${\displaystyle k_{y}={\frac {2\pi }{L_{y}}}\cdot \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle 0\leq X\leq L_{x}}$ ${\displaystyle \kappa }$

${\displaystyle 0\leq \kappa \leq L_{x}L_{y}{\frac {eB}{h}}}$.

Each Landau level in this component has a degree of degeneracy per unit area, g _L (" surface density "), for which the following relationship applies:

${\displaystyle g_{L}={\frac {eB}{h}}\,.}$  

At the edge of the sample and through potential for disorder in the sample, further effects occur that play a decisive role in understanding QHE and are explained below, because the QHE cannot be explained with the ideal Landau levels alone.

Simplified explanation of the QHE

Play media file

If the Fermi level is between two Landau levels, there is no scattering and plateaus occur.

By applying a magnetic field (perpendicular to the two-dimensional electron gas (2DEG)) the electrons are made to move on circular paths - the cyclotron paths . With the Coulomb calibration , the Hamiltonian of the system can be written as . This can be rewritten as a Hamiltonian of the harmonic oscillator in -direction with the cyclotron frequency . Its states are quantized and form the Landau levels . ${\displaystyle H}$ ${\displaystyle H={\tfrac {1}{2m}}(p_{x}^{2}+p_{y}^{2}+(p_{z}+eBx)^{2})}$ ${\displaystyle x}$ ${\displaystyle \omega _{c}}$

If you now apply an additional longitudinal electric field perpendicular to the magnetic field (for example by an external potential) parallel to the 2DEG, the electrons experience an additional deflection. In the ideal case (without scattering ) they are deflected in the direction perpendicular to the electric field and generate the Hall voltage U _H , i.e. That is, they describe a spiral path perpendicular to the electric and magnetic fields (movement is restricted by the 2DEG in these two dimensions). Since the scattering time τ approaches infinity without scattering, both the conductivity (in the direction of the external electric field / potential) and the associated resistance disappear, since the electrons move perpendicular to the potential. If you now include the scattering, the direction of an electron that was scattered at an impurity changes. As a result, the charge carriers experience a component in the direction of the electric field, which leads to a current.

In terms of quantum mechanics, the oscillations of resistance and conductivity can be explained in a simplified manner by the fact that depending on the position of the Fermi energy relative to the Landau levels, scattering may or may not take place. Due to the finite orbits of the electrons, the Landau levels are not delta-shaped , but broadened (half-width ). If the Fermi energy is within a level, scattering occurs because there are free states into which scattering can take place. However, if the Fermi energy lies between two Landau levels, the scattering is ideally completely suppressed due to the lack of free states and resistance-free transport takes place only via the edge channels (see below). ${\displaystyle \Gamma \propto 1/\tau }$

The position of the Landau levels to each other changes with the field. The Fermi edge , i.e. the energy value up to which free electrons are in the solid, lies between the levels and . As stated above, the component disappears in the middle of the plateaus; the Hall voltage , on the other hand, does not disappear. The current density can be determined from the charge carrier density , the respective charge and its drift speed : ${\displaystyle \hbar \omega _{c}}$ ${\displaystyle B}$ ${\displaystyle \nu }$ ${\displaystyle \nu +1}$ ${\displaystyle U_{x}}$ ${\displaystyle U_{\mathrm {H} }}$ ${\displaystyle n=\nu g_{L}}$ ${\displaystyle v_{\mathrm {D} }=E_{\mathrm {H} }/B}$ ${\displaystyle j_{x}}$

${\displaystyle j_{x}=-env_{\mathrm {D} }=\nu \cdot eg_{L}v_{\mathrm {D} }=\nu \cdot {\frac {e^{2}}{h}}E_{\mathrm {H} }\ {\stackrel {!}{=}}\ \sigma _{xy}E_{\mathrm {H} }}$.

The secondary diagonal component of the conductivity tensor is therefore an integral multiple ( ) of the von Klitzing's basic unit , from which it follows. If there is a change, the number remains constant until a new Landau level hits the Fermi edge and changes its value. ${\displaystyle \sigma _{xy}}$ ${\displaystyle \nu =1,2,\dots }$ ${\displaystyle e^{2}/h}$ ${\displaystyle R_{\mathrm {H} }={\frac {U_{\mathrm {H} }}{I}}={\frac {h}{\nu \cdot e^{2}}}}$ ${\displaystyle B}$ ${\displaystyle \nu }$ ${\displaystyle \nu }$

Strictly speaking, the Fermi level cannot lie between two Landau levels: If a Landau level is depopulated by a rising field, the Fermi energy jumps to the next lower level without remaining in between. However, this contradicts the assumption under which the occurrence of the oscillations is to be explained. The solution to this apparent problem are effects in real crystals . The above behavior only occurs in the case of completely pure crystals that do not have any lattice defects . The "smooth" Landau levels become "wavy" because of the imperfections that exist in reality. If the Fermi energy is now near such a level, there are no longer intersections ( “edge channels” ) only at the edge , but also inside the sample. Thus the Fermi level can also be between the Landau levels. ${\displaystyle B}$

Connection with magnetic flux quanta

If the degree of degeneracy is multiplied by the sample area, the following relationship is obtained between the number of electrons in the landing level and the number of flux quanta in the sample:

${\displaystyle N_{\text{Zustände pro Landauniveau}}=g_{L}\cdot L_{x}L_{y}={\frac {eB}{h}}\cdot L_{x}L_{y}={\frac {\Phi }{h/e}}=N_{\text{Flussquanten in der Probe}}}$.

In the plateau state, the same number of electrons rotates around each magnetic flux quantum. This relationship plays a role in the fractional quantum Hall effect, in which electrons and flux quanta form quasiparticles ( Robert B. Laughlin , Jainendra K. Jain ). ${\displaystyle \nu \,}$

Relationship with the fine structure constant

For elementary particle , atomic and molecular physicists or for chemists , the quantum Hall effect is u. a. Interesting because the reciprocal Von Klitzing resistance directly links the Sommerfeld fine structure constant, which is very important in these disciplines, with the electrical field constant : ${\displaystyle \alpha }$ ${\displaystyle \varepsilon _{0}}$

${\displaystyle \alpha \ =\ {\frac {1}{2c\,\varepsilon _{0}}}\;{\frac {e^{2}}{h}}=\ {\frac {1}{2c\,\varepsilon _{0}}}\;{\frac {1}{V_{\mathrm {K} }}}.}$

Necessity of the test conditions

On the one hand, the strong magnetic field is necessary to separate the Landau levels from one another. But it also brings the number of flux quanta in the same order of magnitude as the number of free charge carriers.

The transitions to higher Landau levels are only thermally probable at low temperatures. The restriction to two dimensions is also required in order to be able to view it as a fixed value. ${\displaystyle E_{z}}$

history

The QHE emerges continuously from the classic Hall effect when the temperature is lowered, samples with higher mobility of the electrons are examined and the magnetic field increases significantly. Depending on these parameters, the quantum Hall effect occurs at very high magnetic field strengths. The late discovery of the effect is based, among other things, on the fact that - in contrast to many other physical quantities - the apparatus-based generation of permanent magnetic fields is relatively severely limited (20–40  Tesla ). This is why the transition from the classic Hall effect, which has been known since 1879, to the quantum Hall effect took more than 100 years until enough highly mobile electron systems were available in semiconductor heterostructures .

Although the plateaus in the Hall resistance were observed earlier, the values ​​were only associated with natural constants by Klaus von Klitzing in 1980 at the high-field magnet laboratory in Grenoble (GHMFL) (at that time still German-French cooperation between MPI-FKF and CNRS ) .

Since the Von Klitzing constant is a universal reference value for the measurement of resistance that can be exactly reproduced anywhere in the world, it was established in 1990 by international agreement as the standard for the representation of the unit of measurement ohm . As mentioned above, it is related to the fine structure constant from quantum electrodynamics via two further quantities . ${\displaystyle R_{\mathrm {K} }}$ ${\displaystyle \alpha }$

Variants and related effects

Fractional Quantum Hall Effect (Fractional QHE)

A few years after the discovery of the quantum Hall effect , additional plateaus with non-integer numbers were found in GaAs , with many specific similarities to the integer quantum Hall effect occurring. Fractional quantum numbers for which or applies are easy to observe . ${\displaystyle \nu }$ ${\displaystyle \nu }$ ${\displaystyle \nu ={\frac {m}{2m+1}}}$ ${\displaystyle \nu =1-{\frac {m}{2m+1}}}$

The reason for the similarities is apparently the tendency of the electrons to form bound states ( composite fermions ) together with the magnetic field . The bound states each consist of one or more electrons and a suitable number of magnetic flux quanta .

For the discovery of the fractional quantum Hall effect, Horst Ludwig Störmer and Daniel Tsui, together with Robert B. Laughlin , who interpreted the effect as quantum fluid, received the Nobel Prize in Physics in 1998. Störmer and Tsui discovered the effect in 1981 at Bell Laboratories with Arthur Gossard .

Unusual quantum Hall effect in graphene monolayers

The quantum Hall effect was observed at room temperature in graphene , a material that was first produced in 2004 .

Because of the peculiarities of the dispersion , the staircase structure of the integer quantum Hall plateaus in this material (see graph ) , for all stages exactly “shifted by 1/2”, the “two-valley” structure of graphs and the spin -Degeneracy results in an additional factor of 4. The difference between the plateau centers is still an integer. ${\displaystyle \sigma _{xy}\propto \nu }$ ${\displaystyle \nu \to \nu +{\tfrac {1}{2}}\,,\,\forall \,\nu =\,1,\,2,\,\dots \,.}$

Quantum Spin Hall Effect

The quantum spin Hall effect was first proposed in 2005 by Charles L. Kane and Gene Mele, based on a paper by F. Duncan M. Haldane in graphs . and independent of Andrei Bernevig and Shoucheng Zhang . The underlying transport phenomena are topologically protected, for example topological isolators .

Researchers at Princeton University headed by Zahid Hasan and Robert Cava reported in the journal Nature on April 24, 2008 on quantum Hall-like effects in crystals made of bismuth - antimony , without the need to apply an external magnetic field. This bismuth-antimony alloy is an example of a topological metal . However, the spin currents could only be measured indirectly (using synchrotron photoelectron spectroscopy).

The direct measurement of spin currents in such Bi-Sb alloys was achieved in 2009 by an international team, including Charles L. Kane , Zahid Hasan, Robert Cava, and Gustav Bihlmayer from Forschungszentrum Jülich. The spin currents flow without any external stimulus due to the internal structure of the material. The flow of information is loss-free, even with slight contamination.

The group around Laurens Molenkamp achieved the first experimental evidence around 2007 in Würzburg in tellurium-cadmium quantum wells. In 2017 a proposal was made for a quantum spin Hall material at room temperature ( Werner Hanke et al.).

Schubnikow-de-Haas effect

The Schubnikow-de-Haas effect describes the oscillations in conductivity along the applied current path ( ), i.e. perpendicular to the direction of the quantum Hall effect. At first glance, paradoxically, both the conductivity and the resistance in a parallel direction (with high purity of the 2DEG ) drop to 0 exactly when the Hall voltage ( ) just reaches a plateau. The marginal canal model , which can be described by the Landauer-Büttiker formalism , provides a clear description . ${\displaystyle \sigma _{xx}}$ ${\displaystyle \sigma _{yy}}$

literature

• Zyun F. Ezawa: Quantum Hall Effects . Field Theoretical Approach and Related Topics. World Scientific, Singapore 2008, ISBN 978-981-270-032-2 (English).
• Benoît Douçot et al. (Ed.): The Quantum Hall Effect . Poincaré Seminar 2004. Birkhäuser, Basel 2005, ISBN 978-3-7643-7300-9 (English).
• Sankar D. Sarma, Aron Pinczuk (Eds.): Perspectives in Quantum Hall Effects . Novel Quantum Liquids in Low-Dimensional Semiconductor Structures. Wiley-VCH, Weinheim 2004, ISBN 978-0-471-11216-7 (English).
• Lucjan Jacak, Piotr Sitko, Konrad Wieczorek and Arkadiusz Wojs: Quantum Hall Systems . Braid Groups, Composite Fermions, and Fractional Charge. In: The International Series of Monographs on Physics . No. 119 . Oxford University Press, Oxford 2003, ISBN 0-19-852870-1 (English).
• JH Davies: The physics of low-dimensional semiconductors: An introduction . Cambridge University Press, Cambridge 1998, ISBN 978-0-521-48491-6 (English).
• " Preservation and representation of the unit of electrical resistance ohm ". Exhibit information sheet from the Physikalisch-Technische Bundesanstalt, Hanover Fair '82, April 21, 1982
• Klaus von Klitzing, Gerhard Dorda , Michael Pepper : New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance , Phys. Rev. Letters, Volume 45, 1980, pp. 494-497 (original work on the quantum Hall effect)
• Michael Lohse, Christian Schweizer, Hannah M. Price, Oded Zilberberg, Immanuel Bloch: Exploring 4D quantum Hall physics with a 2D topological charge pump, in: Nature , January 4, 2018, doi: 10.1038 / nature25000 , in addition :
  Leaving Flatland - Quantum Hall Physics in 4D , MPG press release from January 4, 2018

References and footnotes

1. ↑ Klaus von Klitzing: The Quantized Hall Effect, Nobel Lecture ( English ) Nobel Foundation. December 9, 1985. Retrieved December 11, 2009.
2. ^ Klaus von Klitzing: The quantized Hall effect . In: Rev. Mod. Phys. . 58, No. 3, 1986, pp. 519-531. doi : 10.1103 / RevModPhys.58.519 .
3. ↑ The SI system of units is used; in the Gaussian system would be against it by replace. ${\displaystyle B}$ ${\displaystyle B/c}$
4. ↑ Of course, in connection with the (two-dimensional) QHE, the charge carrier density is not a volume density, but an area density , total charge / (length times width of the Hall strip).
5. ↑ Regarding the experimental conditions: Imagine an area of ​​length and width . The "thickness" of the strip is only one atomic layer (monolayer) or a similarly small amount, while and are much larger and therefore an area view is possible. This experimental setup ensures that it is a two-dimensional electron gas . The electric field and the current are in the longitudinal direction ( direction), the Hall voltage acts in the transverse direction ( direction), across the width of the sample, and the direction of the magnetic field is the direction, i.e. the perpendicular direction the area formed from and . ${\displaystyle L_{1}}$ ${\displaystyle L_{2}}$ ${\displaystyle L_{1}}$ ${\displaystyle L_{2}}$ ${\displaystyle E}$ ${\displaystyle I}$ ${\displaystyle x}$ ${\displaystyle U_{H}}$ ${\displaystyle y}$ ${\displaystyle B}$ ${\displaystyle z}$ ${\displaystyle L_{1}}$ ${\displaystyle L_{2}}$
6. ↑ There is also another convention for ${\displaystyle \nu }$
7. ^ K. Kopitzki: Introduction to Solid State Physics , BG Teubner, ISBN 3-519-13083-1 .
8. ↑ A corresponding area is allocated to a given Landau state , whereby the size can also be referred to as “river quantum”. (In the theory of superconductivity is replaced by because the charge carriers there are Cooper pairs .) ${\displaystyle \Delta F=\Phi _{0}/B}$ ${\displaystyle \Phi _{0}=h/e}$ ${\displaystyle e}$ ${\displaystyle 2e}$
9. ↑ Wolfgang Nolting: Quantum Theory of Magnetism , Springer
10. ↑ J. Hajdu, B. Kramer: Der QHE , Phys. Leaves. 41 No. 12 (1985) 401.
11. ↑ ^{a b} K.v. Klitzing: The Fine-Structure Constant , A Contribution of Semiconductor Physics to the Determination of ${\displaystyle \alpha }$ ${\displaystyle \alpha }$ , Festkörperphysik, XXI (1981) 1.
12. ^ Resolution 6 of the 18th CGPM (1987). In: bipm.org. Bureau International des Poids et Mesures, accessed on January 21, 2020 . 
13. ^ Resolution 2 of the 19th CGPM (1991). In: bipm.org. Bureau International des Poids et Mesures, accessed on January 21, 2020 . 
14. ^ HL Störmer, M. Hill: Der fractional QHE , Phys. Leaves, No. 9 (1984).
15. ↑ This appropriate number is called, is even and does that in a many-body effect -fold increase in the magnetic field , through the "composite particle" -Näherung the value back to the valid in integer quantum Hall effect simple value reduced becomes; so ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle B\nearrow B^{*}=p\cdot B}$ ${\displaystyle B^{*}}$ ${\displaystyle B}$ ${\displaystyle B^{*}{\searrow }B\,.}$
16. ↑ KS Novoselov, Z. Jiang, Y. Zhang, SV Morozov, HL Stormer, U. Zeitler, JC Maan, GS Boebinger, P. Kim, AK Geim: Room-Temperature Quantum Hall Effect in Graphene . In: Science . tape 315 , no. 5817 , 2007, p. 1379 , doi : 10.1126 / science.1137201 ( sciencemag.org ).
17. ↑ Geim, AK, Novoselov, KS: The rise of graphene , Nature Materials 6 (2007) pp. 183-191
18. ^ Kane, Mele, Quantum Spin Hall Effect in Graphene, Physical Review Letters, Volume 95, 2005, p. 22608
19. ↑ Bernevig, Zhang, Quantum Spin Hall Effect, Physical Review Letters, Volume 96, 2006, p. 106802.
20. ↑ ^{a b} Werner Hanke, University of Würzburg , proposal for room temperature quantum spin hall
21. ↑ D. Hsieh, D. Qian, L. Wray, Y. Xia, YS Hor, RJ Cava, and MZ Hasan: A topological Dirac insulator in a quantum spin Hall phase, Nature, 452, pp. 970–974 (2008) . doi: 10.1038 / nature06843
22. ↑ Andreas Stiller, ct: Researchers discover quantum Hall effect without an external magnetic field. Retrieved April 23, 2009 . 
23. ↑ D. Hsieh, Y. Xia, L. Wray, A. Pal, JH Dil, F. Meier, J. Osterwalder, G. Bihlmayer, CL Kane, YS Hor, RJ Cava, MZ Hasan: Observation of unconventional quantum spin textures in topologically ordered materials. Science Volume 323, No. 5916, February 13, 2009, doi: 10.1126 / science.1167733 Press release FZ Jülich

Web links

• Physikalisch-Technische Bundesanstalt
• The quantum Hall effect. weltderphysik.de, 2000. 
• Quantum Hall Effect . In: Lexicon of Physics . Spectrum Academic Publishing House, 1998 ( Spektrum.de ).
• News from the quantum Hall effect. TU Berlin, July 23, 1996. 
• D. Tong: Lectures on the Quantum Hall Effect . 2016, p. 46ff , arxiv : 1606.06687 (English).
• Advanced internship: The quantum Hall effect. Giessen University, I. Institute of Physics, Dept. of Micro and Nanostructuring, Prof. Dr. Peter J. Sure. 
• Quantum Hall Effect, Poincaré Seminar 2004
• Yi-Xin Chen: Quasiparticle excitations and hierarchies of four-dimensional quantum Hall fluid states in the matrix models, Zhejiang U., Inst. Mod. Phys. (PrePrint), October 8, 2002, arXiv , ResearchGate , iNSPIRE HEP