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Quantum Hall effect

From Wikipedia, the free encyclopedia
Electromagnetic effect in physics

Thequantum Hall effect (orinteger quantum Hall effect) is aquantized version of theHall effect which is observed intwo-dimensional electron systems subjected to lowtemperatures and strongmagnetic fields, in which the HallresistanceRxy exhibits steps that take on the quantized values

Rxy=VHallIchannel=he2ν,{\displaystyle R_{xy}={\frac {V_{\text{Hall}}}{I_{\text{channel}}}}={\frac {h}{e^{2}\nu }},}

whereVHall is theHall voltage,Ichannel is the channelcurrent,e is theelementary charge andh is thePlanck constant. The divisorν can take on either integer (ν = 1, 2, 3,...) or fractional (ν =⁠1/3⁠,⁠2/5⁠,⁠3/7⁠,⁠2/3⁠,⁠3/5⁠,⁠1/5⁠,⁠2/9⁠,⁠3/13⁠,⁠5/2⁠,⁠12/5⁠,...) values. Here,ν is roughly but not exactly equal to the filling factor ofLandau levels. The quantum Hall effect is referred to as the integer or fractional quantum Hall effect depending on whetherν is an integer or fraction, respectively.

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 theFermi 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 (seeAnderson localization).[1]

Thefractional quantum Hall effect is more complicated and still considered an open research problem.[2] Its existence relies fundamentally on electron–electron interactions. In 1988, it was proposed that there was a quantum Hall effect withoutLandau levels.[3] This quantum Hall effect is referred to as the quantum anomalous Hall (QAH) effect. There is also a new concept of thequantum spin Hall effect which is an analogue of the quantum Hall effect, where spin currents flow instead of charge currents.[4]

Applications

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Electrical resistance standards

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The quantization of the Hall conductance (Gxy=1/Rxy{\displaystyle G_{xy}=1/R_{xy}}) has the important property of being exceedingly precise.[5] Actual measurements of the Hall conductance have been found to be integer or fractional multiples of⁠e2/h⁠ to better than one part in a billion.[6] It has allowed for the definition of a new practicalstandard forelectrical resistance, based on the resistance quantum given by thevon Klitzing constantRK. This is named afterKlaus von Klitzing, the discoverer of exact quantization. The quantum Hall effect also provides an extremely precise independent determination of thefine-structure constant, a quantity of fundamental importance inquantum electrodynamics.

In 1990, a fixedconventional valueRK-90 =25812.807 Ω was defined for use in resistance calibrations worldwide.[7] Later, the2019 revision of the SI fixed exact values ofh ande, resulting in an exactRK =⁠h/e2⁠ =25812.80745... Ω.[8]

Research status

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The fractional quantum Hall effect is considered part ofexact quantization.[9] Exact quantization in full generality is not completely understood but it has been explained as a very subtle manifestation of the combination of the principle ofgauge invariance together with another symmetry (seeAnomalies). The integer quantum Hall effect instead is considered a solved research problem[10][11] and understood in the scope ofTKNN formula andChern–Simons Lagrangians.

Thefractional quantum Hall effect is still considered an open research problem.[2] The fractional quantum Hall effect can be also understood as an integer quantum Hall effect, although not of electrons but of charge–flux composites known ascomposite fermions.[12] Other models to explain the fractional quantum Hall effect also exists.[13]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 ofcomposite fermions and Non AbelianChern–Simons Lagrangians.

History

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TheMOSFET (metal–oxide–semiconductorfield-effect transistor), invented byMohamed Atalla andDawon Kahng atBell Labs in 1959,[14] enabled physicists to studyelectron behavior in a nearly ideal two-dimensional gas.[15]

In a MOSFET, conduction electrons travel in a thin surface layer, and a "gate" voltage controls the number of charge carriers in this layer. This allows researchers to explorequantum effects by operating high-purity MOSFETs atliquid helium temperatures.[15]

The integerquantization of the Hall conductance was originally predicted byUniversity 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.[16] In 1978, theGakushuin University researchers Jun-ichi Wakabayashi and Shinji Kawaji subsequently observed the effect in experiments carried out on the inversion layer of MOSFETs.[17]

In 1980,Klaus von Klitzing, working at the high magnetic field laboratory in Grenoble withsilicon-based MOSFET samples developed byMichael Pepper and Gerhard Dorda, made the unexpected discovery that the Hall resistance wasexactly quantized.[18][15] For this finding, von Klitzing was awarded the 1985Nobel Prize in Physics. A link between exact quantization and gauge invariance was subsequently proposed byRobert Laughlin, who connected the quantized conductivity to the quantized charge transport in a Thouless charge pump.[11][19] Most integer quantum Hall experiments are now performed ongallium arsenideheterostructures, although many other semiconductor materials can be used. In 2007, the integer quantum Hall effect was reported ingraphene at temperatures as high as room temperature,[20] and in themagnesiumzincoxide ZnO–MgxZn1−xO.[21]

Integer quantum Hall effect

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Animated graph showing filling of Landau levels asB changes and the corresponding position on a graph of hall coefficient and magnetic field|Illustrative only. The levels spread out with increasing field. Between the levels the quantum hall effect is seen. DOS is the density of states. Note however that if the electron density rather than the Fermi energy is hold constant, as in the actual experiments, this graph becomes a straight line. Existence of plateaus cannot be explained by this trivial model.

Landau levels

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Main article: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 theSchrödinger equation. The system considered is an electron gas that is free to move in the x and y directions, but is tightly confined in the z direction. Then, a magnetic field is applied in the z direction and according to theLandau gauge the electromagnetic vector potential isA=(0,Bx,0){\displaystyle \mathbf {A} =(0,Bx,0)} and thescalar potential isϕ=0{\displaystyle \phi =0}. Thus the Schrödinger equation for a particle of chargeq{\displaystyle q} and effective massm∗{\displaystyle m^{*}} in this system is:

{12m∗[p−qA]2+V(z)}ψ(x,y,z)=εψ(x,y,z){\displaystyle \left\{{\frac {1}{2m^{*}}}\left[\mathbf {p} -q\mathbf {A} \right]^{2}+V(z)\right\}\psi (x,y,z)=\varepsilon \psi (x,y,z)}

wherep{\displaystyle \mathbf {p} } is the canonical momentum, which is replaced by the operator−iℏ∇{\displaystyle -i\hbar \nabla } andε{\displaystyle \varepsilon } 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 contributionsε=εz+εxy{\displaystyle \varepsilon =\varepsilon _{z}+\varepsilon _{xy}}. The corresponding equations in z axis is:

[−ℏ22m∗∂2∂z2+V(z)]u(z)=εzu(z){\displaystyle \left[-{\frac {\hbar ^{2}}{2m^{*}}}{\partial ^{2} \over \partial z^{2}}+V(z)\right]u(z)=\varepsilon _{z}u(z)}

To simplify things, the solutionV(z){\displaystyle V(z)} is considered as an infinite well. Thus the solutions for the z direction are the energiesεz=nz2π2ℏ22m∗L2{\textstyle \varepsilon _{z}={\frac {n_{z}^{2}\pi ^{2}\hbar ^{2}}{2m^{*}L^{2}}}},nz=1,2,3...{\displaystyle n_{z}=1,2,3...} and the wavefunctions are sinusoidal. For thex{\displaystyle x} andy{\displaystyle y} directions, the solution of the Schrödinger equation can be chosen to be the product of a plane wave iny{\displaystyle y}-direction with some unknown function ofx{\displaystyle x}, i.e.,ψxy=u(x)eikyy{\displaystyle \psi _{xy}=u(x)e^{ik_{y}y}}. This is because the vector potential does not depend ony{\displaystyle y} and the momentum operatorp^y{\displaystyle {\hat {p}}_{y}} therefore commutes with the Hamiltonian. By substituting this Ansatz into the Schrödinger equation one gets the one-dimensionalharmonic oscillator equation centered atxky=ℏkyeB{\textstyle x_{k_{y}}={\frac {\hbar k_{y}}{eB}}}.

[−ℏ22m∗∂2∂x2+12m∗ωc2(x−lB2ky)2]u(x)=εxyu(x){\displaystyle \left[-{\frac {\hbar ^{2}}{2m^{*}}}{\partial ^{2} \over \partial x^{2}}+{\frac {1}{2}}m^{*}\omega _{\rm {c}}^{2}(x-l_{B}^{2}k_{y})^{2}\right]u(x)=\varepsilon _{xy}u(x)}

whereωc=eBm∗{\textstyle \omega _{\rm {c}}={\frac {eB}{m^{*}}}} is defined as the cyclotron frequency andlB2=ℏeB{\textstyle l_{B}^{2}={\frac {\hbar }{eB}}} the magnetic length. The energies are:

εxy≡εnx=ℏωc(nx+12){\displaystyle \varepsilon _{xy}\equiv \varepsilon _{n_{x}}=\hbar \omega _{\rm {c}}\left(n_{x}+{\frac {1}{2}}\right)},nx=1,2,3...{\displaystyle n_{x}=1,2,3...}

And the wavefunctions for the motion in thexy{\displaystyle xy} plane are given by the product of a plane wave iny{\displaystyle y} andHermite polynomials attenuated by the gaussian function inx{\displaystyle x}, which are the wavefunctions of a harmonic oscillator.

From the expression for the Landau levels one notices that the energy depends only onnx{\displaystyle n_{x}}, not onky{\displaystyle k_{y}}. States with the samenx{\displaystyle n_{x}} but differentky{\displaystyle k_{y}} are degenerate.

Density of states

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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

n2D=m∗πℏ2{\displaystyle n_{\rm {2D}}={\frac {m^{*}}{\pi \hbar ^{2}}}}.

As the field is turned on, the density of states collapses from the constant to aDirac comb, a series of Diracδ{\displaystyle \delta } functions, corresponding to the Landau levels separatedΔεxy=ℏωc{\displaystyle \Delta \varepsilon _{xy}=\hbar \omega _{\rm {c}}}. At finite temperature, however, the Landau levels acquire a widthΓ=ℏτi{\textstyle \Gamma ={\frac {\hbar }{\tau _{i}}}} beingτi{\displaystyle \tau _{i}} the time between scattering events. Commonly it is assumed that the precise shape of Landau levels is aGaussian orLorentzian profile.

Another feature is that the wave functions form parallel strips in they{\displaystyle y}-direction spaced equally along thex{\displaystyle x}-axis, along the lines ofA{\displaystyle \mathbf {A} }. Since there is nothing special about any direction in thexy{\displaystyle xy}-plane if the vector potential was differently chosen one should find circular symmetry.

Given a sample of dimensionsLx×Ly{\displaystyle L_{x}\times L_{y}} and applying the periodic boundary conditions in they{\displaystyle y}-directionk=2πLyj{\textstyle k={\frac {2\pi }{L_{y}}}j} beingj{\displaystyle j} an integer, one gets that each parabolic potential is placed at a valuexk=lB2k{\displaystyle x_{k}=l_{B}^{2}k}.

Parabolic potentials along thex{\displaystyle x}-axis centered atxk{\displaystyle x_{k}} with the 1st wave functions corresponding to an infinite well confinement in thez{\displaystyle z} direction. In they{\displaystyle y}-direction there are travelling plane waves.

The number of states for each Landau Level andk{\displaystyle k} can be calculated from the ratio between the total magnetic flux that passes through the sample and the magnetic flux corresponding to a state.

NB=ϕϕ0=BABLyΔxk=A2πlB2lB=AeB2πℏωc=m∗ωcA2πℏ{\displaystyle N_{B}={\frac {\phi }{\phi _{0}}}={\frac {BA}{BL_{y}\Delta x_{k}}}={\frac {A}{2\pi l_{B}^{2}}}{\begin{array}{lcr}&l_{B}&\\&=&\\&&\end{array}}{\frac {AeB}{2\pi \hbar }}{\begin{array}{lcr}&\omega _{\rm {c}}&\\&=&\\&&\end{array}}{\frac {m^{*}\omega _{\rm {c}}A}{2\pi \hbar }}}

Thus the density of states per unit surface is

nB=m∗ωc2πℏ{\displaystyle n_{B}={\frac {m^{*}\omega _{\rm {c}}}{2\pi \hbar }}}.

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 fewer energy levels are occupied.

Rewriting the last expression asnB=ℏωc2m∗πℏ2{\textstyle n_{B}={\frac {\hbar \omega _{\rm {c}}}{2}}{\frac {m^{*}}{\pi \hbar ^{2}}}} it is clear that each Landau level contains as many states as in a2DEG in aΔε=ℏωc{\displaystyle \Delta \varepsilon =\hbar \omega _{\rm {c}}}.

Given the fact that electrons arefermions, for each state available in the Landau levels it corresponds to two electrons, one electron with each value for thespins=±12{\textstyle s=\pm {\frac {1}{2}}}. However, if a large magnetic field is applied, the energies split into two levels due to the magnetic moment associated with the alignment of the spin with the magnetic field. The difference in the energies isΔE=±12gμBB{\textstyle \Delta E=\pm {\frac {1}{2}}g\mu _{\rm {B}}B} beingg{\displaystyle g} a factor which depends on the material (g=2{\displaystyle g=2} for free electrons) andμB{\displaystyle \mu _{\rm {B}}} theBohr magneton. The sign+{\displaystyle +} is taken when the spin is parallel to the field and−{\displaystyle -} when it is antiparallel. This fact called spin splitting implies that thedensity of states for each level is reduced by a half. Note thatΔE{\displaystyle \Delta E} is proportional to the magnetic field so, the larger the magnetic field is, the more relevant is the split.

Density of states in a magnetic field, neglecting spin splitting. (a)The states in each rangeℏωc{\displaystyle \hbar \omega _{\rm {c}}} are squeezed into aδ{\displaystyle \delta }-function Landau level. (b) Landau levels have a non-zero widthΓ{\displaystyle \Gamma } in a more realistic picture and overlap ifℏωc<Γ{\displaystyle \hbar \omega _{\rm {c}}<\Gamma }. (c) The levels become distinct whenℏωc>Γ{\displaystyle \hbar \omega _{\rm {c}}>\Gamma }.

In order to get the number of occupied Landau levels, one defines the so-called filling factorν{\displaystyle \nu } as the ratio between the density of states in a 2DEG and the density of states in the Landau levels.

ν=n2DnB=hn2DeB{\displaystyle \nu ={\frac {n_{\rm {2D}}}{n_{B}}}={\frac {hn_{\rm {2D}}}{eB}}}

In general the filling factorν{\displaystyle \nu } 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 factorν{\displaystyle \nu } and one cannot expectν{\displaystyle \nu } to be an integer. SincenB∝B{\displaystyle n_{B}\propto B}, 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 (ν<1{\displaystyle \nu <1}) and this is called the magnetic quantum limit.

Occupation of Landau levels in a magnetic field neglecting the spin splitting, showing how theFermi level moves to maintain a constant density of electrons. The fields are in the ratio2:3:4{\displaystyle 2:3:4} and giveν=4,83{\displaystyle \nu =4,{\frac {8}{3}}} and2{\displaystyle 2}.

Longitudinal resistivity

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It is possible to relate the filling factor to the resistivity and hence, to the conductivity of the system. Whenν{\displaystyle \nu } is an integer, theFermi 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 approximately0{\displaystyle 0}). Consequently, the resistivity becomes zero too (At very high magnetic fields it is proven that longitudinal conductivity and resistivity are proportional).[22]

With the conductivityσ=ρ−1{\displaystyle \sigma =\rho ^{-1}} one finds

σ=1detρ(ρyy−ρxy−ρyxρxx).{\displaystyle \sigma ={\frac {1}{\det \rho }}{\begin{pmatrix}\rho _{yy}&-\rho _{xy}\\-\rho _{yx}&\rho _{xx}\end{pmatrix}}\;.}

If the longitudinal resistivity is zero and transversal is finite, thendetρ≠0{\displaystyle \det \rho \neq 0}. Thus both the longitudinal conductivity and resistivity become zero.

Instead, whenν{\displaystyle \nu } is a half-integer, the Fermi energy is located at the peak of the density distribution of some Landau Level. This means that the resistivity will have a maximum due to increased scattering.

This distribution of minimums and maximums corresponds to ¨quantum oscillations¨ calledShubnikov–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.

Longitudinal and transverse (Hall) resistivity,ρxx{\displaystyle \rho _{xx}} andρxy{\displaystyle \rho _{xy}}, of a two-dimensional electron gas as a function of magnetic field. Both vertical axes were divided by the quantum unit of conductancee2/h{\displaystyle e^{2}/h} (units are misleading). The filling factorν{\displaystyle \nu } is displayed for the last 4 plateaus.

Transverse resistivity

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From the classical relation of the transverse resistivityρxy=Ben2D{\textstyle \rho _{xy}={\frac {B}{en_{\rm {2D}}}}} and substitutingn2D=νeBh{\textstyle n_{\rm {2D}}=\nu {\frac {eB}{h}}} one finds out the quantization of the transverse resistivity and conductivity:

ρxy=hνe2⇒σ=νe2h{\displaystyle \rho _{xy}={\frac {h}{\nu e^{2}}}\Rightarrow \sigma =\nu {\frac {e^{2}}{h}}}

One concludes then, that the transverse resistivity is a multiple of the inverse of the so-called conductance quantume2/h{\displaystyle e^{2}/h} if the filling factor is an integer. In experiments, however, plateaus are observed for whole plateaus of filling valuesν{\displaystyle \nu }, 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

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The quantum Hall effect, in addition to being observed intwo-dimensional electron systems, can be observed in photons.Photons do not possess inherentelectric charge, but through the manipulation of discreteoptical resonators and coupling phases or on-site phases, an artificialmagnetic field can be created.[23][24][25][26][27] This process can be expressed through a metaphor of photons bouncing between multiple mirrors. By shooting the light across multiple mirrors, the photons are routed and gain additional phase proportional to theirangular momentum. This creates an effect like they are in amagnetic field.

Topological classification

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Hofstadter's butterfly

The integers that appear in the Hall effect are examples oftopological quantum numbers. They are known in mathematics as the firstChern numbers and are closely related toBerry's phase. A striking model of much interest in this context is the Azbel–Harper–Hofstadter model whose quantum phase diagram is theHofstadter butterfly shown in the figure. The vertical axis is the strength of themagnetic field and the horizontal axis is thechemical potential, which fixes the electron density. The colors represent the integer Hall conductances. Warm colors represent positive integers and cold colors negative integers. Note, however, that the density of states in these regions of quantized Hall conductance is zero; hence, they cannot produce the plateaus observed in the experiments. The phase diagram is fractal and has structure on all scales. In the figure there is an obviousself-similarity. In the presence of disorder, which is the source of the plateaus seen in the experiments, this diagram is very different and the fractal structure is mostly washed away. Also, the experiments control the filling factor and not the Fermi energy. If this diagram is plotted as a function of filling factor, all the features are completely washed away, hence, it has very little to do with the actual Hall physics.

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 thefractional quantum Hall effect. The observed strong similarity between integer and fractional quantum Hall effects is explained by the tendency of electrons to form bound states with an even number of magnetic flux quanta, calledcomposite fermions.

Bohr atom interpretation of the von Klitzing constant

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The value of the von Klitzing constant may be obtained already on the level of a single atom within theBohr model while looking at it as a single-electron Hall effect. While during thecyclotron motion on a circular orbit the centrifugal force is balanced by theLorentz 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 chargee{\displaystyle e} is making Kepler revolutions with angular frequencyω{\displaystyle \omega }

I=ωe2π,{\displaystyle I={\frac {\omega e}{2\pi }},}

and the induced Hall voltage as a difference between the hydrogen nucleus Coulomb potential at the electron orbital point and at infinity:

U=VC(∞)−VC(r)=0−VC(r)=e4πϵ0r{\displaystyle U=V_{\text{C}}(\infty )-V_{\text{C}}(r)=0-V_{\text{C}}(r)={\frac {e}{4\pi \epsilon _{0}r}}}

One obtains the quantization of the defined Bohr orbit Hall resistance in steps of the von Klitzing constant as

RBohr(n)=UI=nhe2{\displaystyle R_{\text{Bohr}}(n)={\frac {U}{I}}=n{\frac {h}{e^{2}}}}

which for the Bohr atom is linear but not inverse in the integern.

Relativistic analogs

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Relativistic examples of the integer quantum Hall effect andquantum spin Hall effect arise in the context oflattice gauge theory.[28][29]

See also

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References

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