Quantum Hall effect

The quantum Hall effect (short: QHE ) manifested by the fact that at low temperatures and strong magnetic fields perpendicular to the voltage occurring at a current does not increase linearly as the classical Hall effect with the magnetic field, but in stages. The effect occurs at an interface, in which the electrons can be described as a two-dimensional electron gas.

The so-called Hall resistance RH, ie the ratio of the Hall voltage to current, this takes only integer plateau values ​​as fractions of the very precisely known size ( ), where h is the Planck constant and e is the elementary charge. Both are natural constants; Thus, the plateau values ​​do not depend on the material properties such as the charge carrier density, nor on the sample size, or by the magnetic field strength.

For these findings Klaus von Klitzing was awarded the Nobel Prize in Physics in 1985.

Known as the von Klitzing constant RK size is now used for standard - definition of electrical resistance.

From the integral quantum Hall effect with only integer denominators of RK, a distinction is the fractional quantum Hall effect ( QHE also fractionated ), in which the denominator take the form of fractures ( see below).

• 5.1 the fractional quantum Hall effect ( QHE fractional )
• 5.2 Unusual quantum Hall effect in graphene monolayers
• 5.3 quantum spin Hall effect
• 5.4 Shubnikov - de Haas effect

Description of the phenomenon

The classical Hall effect, electric current flows through a plate, which is penetrated at right angles to the surface of a magnetic field. The current flowing in the magnetic field, charge carriers are deflected laterally by the Lorentz force, so that the edges of the plate transversely to the direction of current, an electric voltage can be measured, which is referred to as the Hall voltage.

The ratio of the side adjacent Hall voltage to the current is called the Hall resistance and is the classical Hall effect in ( zweidimensionalen! ) Hall strips

Wherein the transverse Hall voltage occurs to the total current, the total current (perpendicular to the direction in which the Hall voltage is measured ), the magnetic field intensity, the carrier density and the elementary charge. The classical Hall resistance in particular is thus proportional to the applied magnetic field. This can be seen in the picture for small B-field values ​​.

At sufficiently low temperatures and strong magnetic field, however, the Hall resistance increases regardless of the material of a plateau values

To, where ν = 1,2, ... are supposed to be integers here, h is the Planck constant and the RK mentioned should be " of Klitzing'sche elemental resistance."

An increase in the strength of the magnetic field B can now to the Hall resistance constant, until it changes to the next step value. The center of the stage corresponding to the formula above, which is the classic Hall effect. Exactly in the middle stages in the flow direction of the applied voltage of the sample disappears, that is, the electric resistance is zero and where the pipe is dissipationsfrei, apparently throughout the plateau region between the stages. At the stages themselves sharp maxima arise in the resistance.

In the plateau states of the quantum Hall effect is thus similar to the superconductivity to a macroscopic quantum state.

Experimental conditions

Attempts to observe the quantum Hall effect are usually carried out in a simple helium cryostat at 4.2 Kelvin. Lower temperatures, which are only possible through considerably more complex cooling technology are usually not necessary, except for the observation of the fractional effect. A nitrogen cooling, however, is not sufficient, since the mean free path of electrons is too low yet, the measurement is therefore to be strongly affected by interactions.

Depending on the sample magnetic fields of a few tesla are used, which corresponds roughly to the hundred thousand times the Erdmagnetfeldstärke (but v. Klitzing apparatus generated B- fields up to 40 Tesla). For this very strong magnetic fields usually a Helmholtz coil pair is used of superconducting material, typically flow into the coil currents between 10 A and 100 A. The current through the sample itself, however, is only 0.1 to 10 uA.

The samples used in experiments QHE are MOSFETs ( metal oxide semiconductor field effect transistor ), in which the charge carrier density can be modified by an applied gate voltage to the transistor, or semiconductor-insulator - heterostructure (eg AlxGa1-xAs/GaAs- heterostructures ), that thin plates, with a transition between an insulator and a semiconductor. At such a boundary layer, the electrons lose a direction of movement: the Z- direction in which the magnetic field is applied is fixed by a quantum number in the limit potential, the occupation probability of the next higher energy level is negligible. One speaks therefore of a two-dimensional electron gas.

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

Theory

Conductivity tensor

Due to a magnetic field or of preferred directions of conduction in a solid body Ohm's law is generally to write using a conductivity tensor:

In two dimensions, there are conductivity and the resistance of 2x2 matrices:

Is chosen for the description of the QHE as the current direction as the lateral direction in which the Hall voltage is applied, and when the magnetic field direction, as is true due to the arrangement.

Crossed E and B fields

The classical movement of free electrons, which are in perpendicular to each other stationary electric and magnetic fields, is an on spiral paths along the field and can be interpreted as a superposition of the following components:

• A circular motion with the cyclotron frequency to the field direction,
• A drift movement perpendicular to - and - field,
• An unaccelerated motion in field direction.

The cyclotron frequency also plays an important role in the QHE, as we shall soon see.

Quantum mechanical view

With, the calibration and the separation approach, the Schrödinger equation for the free electron, ie

Be transformed into a differential equation for the dependent function, which is the Schrödinger equation for a harmonic oscillator to the point of rest. Is obtained as the energy eigenvalues ​​only the Landau levels:

With a sample size of in the current direction or in the direction of the Hall voltage is given by: The wave number in direction may take the values ​​with an integer, but it also appears in the rest position of the harmonic oscillator, applies. The result for the range of values

Each Landau level has thus in this component as a degree of degeneracy per unit area size gL ( " state areal density " ), applies to the following relationship:

At the edge of the sample and by disorder potentials in the sample, there are further effects, which play a crucial role in understanding of the QHE and are explained in the following, because only with the ideal Landau levels can not explain the QHE.

Simplified explanation of the QHE

By applying a magnetic field (perpendicular to the two-dimensional electron gas (2DEG ) ) to bring electrons to be on circular orbits - the cyclotron orbits - To move. When an electron to the next scattering process creates more than a full revolution, it can interfere with itself. This splits the constant density of the electron gas into discrete energy levels. The Landau levels caused by the Bohr -Sommerfeld quantization.

By putting an additional magnetic field perpendicular to the longitudinal electric field (such as an external potential ) parallel to the 2DEG where the electrons experience an additional distraction. In the ideal case (no spread) they are distracted to the vertical direction to the electric field and generate the Hall voltage UH, i.e., they describe a spiral track perpendicular to the electric and magnetic field ( the movement by the 2DEG in these two dimensions limited). Because without scattering, the scattering time τ goes to infinity, disappear, both the conductivity since the electrons move ( in the direction of the external electric field / potential ) and the associated resistance perpendicular to the potential. If one includes now the scattering with a, then the direction of an electron that has been scattered at an impurity changes. Characterized the carriers experience a component in the direction of the electric field, which leads to a current.

Quantum- mechanical oscillations of the can and resistance conductivity simplified state that according to the position of the Fermi level can be held relative to the Landau levels scattering or not. The Landau levels are not delta-shaped by the finite orbits of the electrons, but broadened ( FWHM). The Fermi energy is within a level, so diffusion occurs because free states exist, in which can be scattered. If the Fermi energy, however, between two Landau levels, the scattering is lack of free states ideally completely suppressed and it is only on the edge channels resistant -free transport instead ( see below).

The position of the Landau levels to each other varies across the field. The Fermi energy, ie the energy value, up to the free electrons in the solid state are, lie between the levels and. As noted above, the Ux component disappears in the middle of the plateau; the Hall voltage UH however does not disappear. From the charge carrier density, the respective charge and their drift velocity, the current density jx can determine:

The off-diagonal component of the conductivity tensor is therefore an integral multiple of () of the base unit Klitzing'schen e2 / h, from which it follows. Is changed, the number remains constant until a new Landau levels abuts the Fermi level and changing its value.

Strictly speaking, the Fermi level does not lie between two Landau levels: If a Landau level depopulated by an increasing field, the Fermi energy jumps to the next lower level without staying in between. However, this contradicts the assumption under which the occurrence of the oscillations to be explained. The solution to this apparent problem, effects in real crystals. Only when completely pure crystals having no lattice defects, occurs above characteristics. By existing in reality impurities the "smooth" Landau levels are " wavy ". Is now the Fermi energy in the vicinity of such levels, there are no longer just on the edge intersections ( " edge channels"), but also in the interior of the sample. Thus, the Fermi level can also be between the Landau levels.

Related to magnetic flux quanta

If the degree of degeneracy multiplied by the sample surface, we obtain the following relationship between the number of electrons in Landau levels and the number of flux quanta in the sample:

At the plateau state rotated around each magnetic flux quantum that is, the same number of electrons. This relationship plays in particular the fractional quantum Hall effect a role in the form of electrons and flux quanta quasiparticles ( Robert B. Laughlin, Jainendra K. Jain ).

Relating to the fine structure constant

For elementary particle, atomic and molecular physicists or chemists of the quantum Hall effect is inter alia interesting because the reciprocal of the von Klitzing resistance mainly with the in these disciplines very important Sommerfeld's fine structure constant can be identified that very order is estimated. The precise relationship between the Sommerfeld constant and the reciprocal of the von Klitzing resistance, where c is the speed of light in vacuum and the known from Maxwell's equations exactly defined sizes.

Necessity of the experimental conditions

The strong magnetic field is on the one hand necessary that the Landau levels are separated from each other. But it also brings the number of flux quanta into the same order of magnitude as the number of free charge carriers.

Transitions to higher Landau levels are likely to be thermally at low temperatures. Similarly, the restriction is required in two dimensions in order to be regarded as a fixed value can.

History

The QHE goes continuously from the classical Hall effect produced when the temperature is lowered, samples with a higher mobility of the electrons (typical) are investigated and the magnetic field increases strongly. Depending on these parameters occurs on the quantum Hall effect at very high magnetic fields. The late discovery of the effect is based, among other things, that: - the apparatus producing permanent magnetic fields relatively very limited (20-40 Tesla) - in contrast to many other physical quantities. Why did the transition from the classical Hall effect, which is known since 1879, for the quantum Hall effect more than 100 years, were available until enough highly mobile electron systems in semiconductor heterostructures.

Although the plateaus have been observed in the Hall resistance earlier, the values ​​until 1980 at the High Magnetic Field Laboratory in Grenoble ( GHMFL ) (then German - frz. Cooperation of MPI - FKF and CNRS) by Klaus von Klitzing with natural constants have been associated.

Since the von Klitzing constant RK is a universal reference for the measurement of resistors that can be reproduced accurately anywhere in the world, the electrical unit ohm is defined by this size since 1990 by international agreement. It depends, as mentioned above, two other well-defined sizes with the fine structure constant of quantum electrodynamics together.

Variants and related effects

The fractional quantum Hall effect ( QHE fractional )

A few years after the discovery of the quantum Hall effect plateaux were additional non- integer found in GaAs, with many specific similarities to the integer quantum Hall effect occurs. Good observable are broken quantum numbers, or are expected to.

Cause of the similarities is apparently the tendency of the electron, together with the magnetic bound states ( fermions composite ) form. The bound states exist here in each of one or more electrons and a matching number of magnetic flux quanta.

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

Unusual quantum Hall effect in graphene monolayers

In the first produced in 2004 material graphene quantum Hall effect was observed at room temperature, which is quite unusual, because otherwise 300 times lower temperatures are needed.

Because of the peculiarities in the dispersion in this material (see graph ), the staircase structure of the integer quantum Hall plateau, " shifted by 1/2" for all levels accurately, the "two- valley" structure of graphs and the spin degeneracy provide an additional factor of 4, but the difference between the plateau centers is still an integer.

Quantum spin Hall effect

Researchers at Princeton University reported in the journal Nature on 24 April 2008 on the quantum Hall -like effects in crystals of bismuth - antimony without an external magnetic field had to be created. These bismuth -antimony alloy is an example of a topological metal. However, the spin currents could be measured only indirectly.

The direct measurement of spin currents in such a Bi - Sb alloys succeeded in 2009 an international team, including Dr. Gustav Bihlmayer from Forschungszentrum Jülich. The spinning currents flow without external stimulus, due to the internal structure of the material. The information flow is lossless, even for slight irregularities.

Shubnikov - de Haas effect

The Shubnikov - de Haas effect describes the oscillations of the conductivity along the applied current path ( ), ie perpendicular to the direction of the quantum Hall effect. At first sight paradoxical, both the conductivity and the resistivity in a direction parallel (at high purity of the 2DEG ) decreases if and to 0, if the Hall voltage () just reached a plateau. A clear description provides the edge channel model, which can be described by the Landauer - Büttiker formalism.