The Quantum Hall Effect

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The Quantum Hall Effect

Conrad Gr¨ alls June 26, 2020

Abstract

The quantum Hall effect occurs when a conductor carrying a current is placed in a perpendicular magnetic field. If certain conditions are met, such as strong magnetic field and low temperature, the resistivity becomes quantised, taking values of integer or fractional multiples of _e^h2. By analysing the movement of electrons in a magnetic field classically and quantum mechanically information about the integer quantum Hall effect and the fractional quantum Hall effect can be gathered, using the two different gauge potentials of Landau gauge and Symmetric gauge. Resistance Metrology is one field of study that the quantum Hall effect has greatly impacted by providing a way to universally maintain the ohm, with significantly less uncertainty than previously.

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1 Sammanfattning

Den kvantmekaniska hall-effekten uppst˚ar när en strömbärande ledare plac- eras i ett vinkelrätt magnetfält. Om vissa villkor är uppfyllda, s˚asom starkt magnetfält och l˚ag temperatur, blir resistiviteten kvantiserad. Given av heltal (integer ) eller fraktions-(fractional ) multiplar av _e^h2. Genom att anal- ysera elektroners rörelse i ett magnetfält klassiskt och kvantmekaniskt f˚as information om Hall-effekterna; integer quantum Hall effect och fractional quantum Hall effect, med hjälp av de tv˚a gauge potentialerna Landau gauge och Symmetrisk gauge. Resistansmetrologi är ett forskningsomr˚ade som kvant Hall-effekten har starkt p˚averkat genom att tillhandah˚alla ett sätt att universellt upprätth˚alla ohm-enheten med betydligt mindre osäkerhet än tidigare.

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2 Background

The classical Hall effect occurs when a 2-dimensional conductor carrying a current is placed in a magnetic field perpendicular to the current. The magnetic field exerts a force on the electrons in the current, called the Lorentz force. The Lorentz force is orthogonal to both the current and the magnetic field and in steady state conditions it will be balanced by an opposite force, arising from the electric field that is induced in the conductor. Hence there is a voltage difference across the conductor in the direction of the induced electric field, called Hall voltage. The Hall conductance is then the current divided by the Hall voltage. In the quantum case the Hall conductance becomes quantised taking on integer or fractional values that are multiples of

e²

h. Where e is the electron charge and h the Plank constant. These phenomena are called the integer quantum Hall effect (IQHE) and the fractional quantum Hall effect (FQHE).

Klaus von Klitzing discovered the IQHE in 1980 through his research in condensed matter, looking at semiconductors in a strong magnetic field at low temperatures. For this contribution he received the Nobel prize in Physics in 1985. The journey through von Klitzing’s career leading up to and after the discovery of the QHE is summarized in [9]. After the discovery of the integer quantum Hall effect (IQHE) several more related phenomena were uncovered such as the fractional quantum Hall effect (FQHE) and the quantum spin hall effect.

The IQHE and FQHE are important because of the many research areas they involve. The theoretical basis of the IQHE is well understood, however understanding the FQHE in generality is still a topic of active research. For example composite fermions, that underlie the physics of FQHE, are current research topics in quantum field theory [3]. Metrology is another field that has been impacted by the discovery of the QHE by its usage in yielding electrical standard units. Especially research of the QHE in graphene shows potential for new applications in metrology [5].

3 Problem formulation

What are some of the suggested explanations for the FQHE? And what are some applications of the QHE?

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4 Aim

The aim of the project is to explore FQHE. For conductances _mh^e² where m is an odd integer the FQHE can be explained by Laughlin’s states, which I will review. To achieve this aim I will first review explanations of the classical Hall effect and IQHE. After this I will study in detail motion of charged particles in a static magnetic fields in quantum mechanics. I will also discuss applications of the QHE to setting metrology standards and explain the practical importance of the QHE.

5 The classical Hall effect

The Hall effect occurs when moving charged particles are placed in a magnetic field, with direction orthogonal to the particles velocities. Say these charged particles are electrons in a current moving in a two dimensional slab of material, as in figure 1. The magnetic field causes a Lorentz force on the electrons, causing them to bend and accumulate at one edge of the material.

This accumulated charge give rise to an electric field, when this is strong enough to be equal and opposite to the applied magnetic field the electrons can move from the upper edge to the lower edge of the material. The voltage difference between the two edges is called Hall Voltage, after Edwin Hall who discovered it in 1879.[10]

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Figure 1: Current (electrons) flowing through a piece of material in the xy- plane with a magnetic field in the z-direction.

Charged particles in a magnetic field move in circular orbits with a fixed frequency, the cyclotron frequency ω_B. To get the expression for the cyclotron frequency solve the equation of motion for an electron in a magnetic field.

Given by plugging the expression for the Lorentz force into Newton’s second law.

md~v

dt = −e~v × ~B. (1)

With the magnetic field and the electron drift velocity B = (0, 0, B)~

~

v = ( ˙x, ˙y, 0),

the equations of motion become two coupled differential equations.

m¨x = −e ˙yB (2)

m¨y = e ˙xB (3)

They are solved by having the electron move in a circular motion described by

x(t) = X − Rsin(ω_Bt + φ) (4)

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y(t) = Y − Rcos(ω_Bt + φ) (5) ω_B = eB

m (6)

R, is the radius, (X, Y) the position of the circles centre and φ an arbitrary phase.[8, p.6-7]

To account for impurities in the sample include an electric field and a linear friction term.

md~v

dt = −e ~E − e~v × ~B − m~v

τ (7)

Eqn. 7 is the classical description of an electron, called the Drude model. τ represents the average time between collisions. ^d~_dt^v = 0 in equilibrium gives

0 = −e ~E − e~v × ~B − m~v

τ . (8)

Rearrange to get

−eτ

mE = −~ eτ

m~v × ~B + ~v. (9)

Replace the drift velocity with the current density, ~J . Also multiply the entire expression with −ne. Where n is the density of electrons.

~ v = −

J~

ne (10)

e²nτ m

E =~ eτ m

J × ~~ B + ~J (11)

To get to matrix notation, carryout the cross product with the current density in the xy-plane ~J = (Jx, Jy, 0).

e²nτ

m E_x = eBτ

m J_y+ J_x e²nτ

m E_y = −eBτ

m J_x+ J_y

Recognize the cyclotron frequency ωB = ^eB_m. The matrix notation becomes

1 ω_Bτ

−ω_Bτ 1

J =~ e²nτ

m E~ (12)

To get Ohm’s law ~J = σ ~E multiply with the invers matrix J =~ e²nτ

m

1 1 + ω_B²τ²

1 −ω_Bτ ω_Bτ 1

E.~ (13)

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The matrix expression for the conductivity is σ = σ_xx σ_xy

−σ_xy σ_xx

(14) the σ_xy terms are responsible for the Hall effect.[8, p.7-8]

In the sections about the QHE mostly the resistivity is used to describe the effect. The resistivity, ρ, is proportional to the resistance and takes geometry of the material into account. It is given by

ρ = σ⁻¹ = ρ_xx ρ_xy

−ρ_xy ρ_yy

= m

e²nτ

1 ω_Bτ

−ω_Bτ 1

(15) Plug in the cyclotron frequency to get the different resistivities

ρ_xx = m

ne²τ (16)

ρ_xy = B

ne (17)

The Hall coefficient is defined as R_H = − E_y

J_xB = ρ_xy

B . (18)

Using the Drude model it becomes [8, p.10]

R_H = ω_B

Bσ_DC = 1

ne. (19)

6 Movement of electrons in magnetic field

6.1 Classical movement

The effect of spin will be ignored in all the calculations below. The La- grangian for an electron moving in an external magnetic field, ~B = ∇ × ~A, is given by

L = 1

2m ˙~r²− e ˙~r · ~A (20) The particle is restricted to lie in a plane, ~r = (x, y), due to the 2-dimensional nature of the effect. Define two types of momentum, the normal mechanical momentum, π, and the canonical momentum, p.

~ p = ∂L

∂ ˙~r = m ˙~r–e ~A (21)

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Use the canonical momentum to convert the Lagrangian into the Hamiltonian H = ˙~r · ~p–L = 1

2m(~p + e ~A)². (22) The Hamiltonian describe an electron in an external field in classical mechanics. [8, p.15]

Written in terms of the mechanical momentum the Hamiltonian becomes π = m ˙~r = ~p + e ~A (23)

H = 1

2m~π² (24)

6.2 Quantum mechanical movement

The quantum Hamiltonian is the same as the classical one given by eqn.

22 except ~p and ~A are operators. Information of the electron’s movement is gained from the Hamiltonian by solving for the energy spectrum and the wavefunction.

In quantum mechanics the Hamiltonian in eqn. 24 is the Hamiltonian for a harmonic oscillator, which will be shown in the next section. It has the known solution for the energy levels

E_n = ¯hω_B

n + 1

2

(25) With n taking on positive integer values.

This says that there is a splitting between the energy levels depending on the magnetic field. These energy levels are called Landau levels. [8, p.17]

6.2.1 Landau Gauge

To get the energy levels and the wavefunction a gauge potential needs to be specified. Using Landau Gauge A = xB ˆy the Hamiltonian in eqn. 22 becomes

H = 1

2m p²_x+ (p_y + eBx)² . (26) The Hamiltonian does not depend on the y-coordinate making it translational invariant in the y-direction. Hence an ansatz of the following form is justified [8, p.18]

ψ_k(x, y) = e^ikyf_k(x). (27)

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Act on the wavefunction with the Hamiltonian operator, replacing the momentum in the y-direction with its eigenvalue.

Hψ_k(x, y) = 1

2m p²_x+ (¯hk + eBx)² ψ_x(x, y) = H_kψ_k(x, y) (28) Bring out a factor of eB from the second parenthesis, rearrange and use the cyclotron frequency to replace factors gives

H_k = 1

2mp²_x+mω_B²

2 x + kl²_B2

. (29)

l_B is the magnetic length useful for length scales affecting quantum phenomena.

l_B = r h¯

eB (30)

The Hamiltonian in eqn 29 is a harmonic oscillator with a displaced centre around x = −kl²_B. As seen before the energy levels for the harmonic oscillator are known to be

En = ¯hωB

n + 1

2

.

Using the fact that H_k is the harmonic oscillator the explicit wavefunction can be written down.

ψ_n,k ∼ e^ikyH_n x + kl²_B e(^−x+kl^B²)²^/2lB² (31) H_n are Hermite polynomial and ∼ indicates it is not normalised. The large degeneracy is caused by the wavefunction depending on two quantum number while the energy only depend on one quantum number, causing the wave- functions to be able to combine into any shape.[8, p.19]

6.2.2 Symmetric Gauge The symmetric gauge potential

A = −~ 1

2~r × ~B = −yB

2 x +ˆ xB

2 yˆ (32)

is helpful for explaining FQHE and provides a derivation of the degeneracy of the Landau levels. Symmetric gauge preserve rotational symmetry making angular momentum a good quantum number. The mechanical momentum

~

π = ~p + e ~A, can be used to build ladder operators to construct the Hamilto- nian similar to that for the harmonic oscillator.

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a = π_x–iπ_y

√2e¯hB (33)

a^†= π_x+ iπ_y

√2e¯hB (34)

These obey [a, a^†] = 1. Plugging the ladder operators into the Hamiltonian gives the harmonic oscillator Hamiltonian.[8, p.22]

H = 1

2m~π² = ¯hω_B

a^†a +1 2

(35) To show the degeneracy introduce another pair of ladder operators using another kind of momentum ~˜π = ~p–e ~A.

b = π˜x+ i ˜πy

√2e¯hB (36)

b^†= π˜_x− i ˜π_y

√2e¯hB (37)

Obeying [b, b^†] = 1. This new momentum and the mechanical momentum do not in general commute. However the commutators vanish in the symmetric gauge. Meaning ~˜π can be used to find out about other quantum numbers involved. [8, p.23]

The ground state |0, 0 > is annihilated by both the lowering operators a and b. Any general state can be constructed by acting on the groundstate with the raising operators.

|n, m >= a^†nb^†m

√n!m!|0, 0 > (38)

Compared to the expression for the energy these states are given by two quantum numbers, n and m, while the energy is only dependent on n causing the Landau levels to be degenerate. [8, p.23]

Now lets construct the wavefunction for the lowest Landau level |0, m >, given by n = 0. The lowest level is annihilated by the lowering operator, a. This can be expressed as two differential equation by plugging in the expressions for π_x and π_y using the symmetric gauge.

πx = px− yB

2 (39)

π_y = p_y +xB

2 (40)

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Put these into the lowering operator and replace p_x and p_y with their operators.

a = 1

√ 2s¯hB

−i¯h ∂

∂x − i ∂

∂y

− eB

2 (y + ix)

(41) Simplify by introducing complex coordinates z = x − iy, ¯z = x + iy and derivatives

∂ = 1 2

∂

∂x + i ∂

∂y

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∂ =¯ 1 2

∂

∂x − i ∂

∂y

(43) obeying ∂z = ¯∂ ¯z = 1 and ∂ ¯z = ¯∂z = 0. In terms of the new variables the ladder operators become

a = −i√ 2

lB∂ +¯ z 4l_B

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b = −i√ 2

l_B∂ + z¯ 4l_B

(45) l_B is the magnetic length. The lowest state |0, 0 > is annihilated by both a and b, is unique and given by [8, p.24-25]

ψ_LLL,m ∼ e^|z|²^/4l²^B (46)

The corresponding raising operators are a^† = −i√

2

l_B∂ − z¯ 4lB

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b^†= −i√ 2

l_B∂ −¯ z 4l_B

(48) However the lowest Landau level do not depend on m meaning m can take on any value. Acting on the wave function with b^† raise the m value and can be used to give the general expression for the wavefunction for the lowest Landau level.

ψ_LLL,m∼ z l_B

m

e^|z|²^/4l²^B (49)

Higher Landau levels can be constructed using a^†.[8, p.25]

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7 Integer quantum Hall effect

At low temperature and strong magnetic fields quantum effects cause the resistivity in the sample to take quantised values. The quantisation is a consequence of impurities (disorder) in the sample, causing the effect to become more prominent as disorder is increased, up to a certain amount. The quantisation cause the resistivity to take on a constant value as the magnetic field is increased. When the magnetic field is increased enough the resistivity will jump to another constant value. At the centre of the plateau the magnetic field is given by

B = 2π¯hn νe = n

νφ₀ (50)

and the resistivity is

ρ_xy = 2π¯h e²

1

ν. (51)

When ν is an integer the effect is called integer quantum Hall effect (IQHE).

φ₀ = ^2π¯_e^h is the flux quantum and n the electron density. [8, p.11-12]

However from the Drude model seen earlier the Hall resistivity is ρ_xy = B

ne.

For the two expressions of the resistivity to be equal the electron density would be

n = B

φ₀ν. (52)

This happens to be the density of electrons required to fill ν Landau levels.

[8, p.42-43]

Meaning the integer determining the energy level and the integer in the Hall resistivity are the same. The number of electron states, N , in one filled Landau level is then given by

N = AB

φ₀ (53)

A being the area.

In the jump between plateaus the longitudinal resistivity spikes. However when ρ_xy is sitting on a constant plateau value the longitudinal resistivity is zero ρ_xx = 0. This means that the conductivity also is zero σ_xx = 0, and that there is no dissipation of energy and no current flowing in the longitudinal direction. [8, p.12-13]

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7.1 Turning on electric field

Consider a free electron described by eqn. 26 using Landau gauge ~A = xB ˆy.

By adding an electric field, in the x direction, the new Hamiltonian becomes H = 1

2m p²_x+ (p_y+ eBx)² + eEx. (54) The wavefunction to this Hamiltonian is the same as for without the electric field, except for a shift in the x coordinate. [8, p.21]

ψ(x, y) = ψn,k x + mE/eB², y

(55) Now to derive the expression for the resistivity seen earlier in eqn. 51 consider the current ~I = −e ˙~r. With the velocity given by m ˙~r = ~p + e ~A, where p and x (inside the gauge potential A) are operators. The current in the quantum mechanical case is given by

I = −~ e m

X

states

< ψ| − i¯h∇ + eBx|ψ > . (56)

Plug in the wavefunction, eqn. 55, and examine the current component wise.

Ix= −e m

ν

X

n

X

k

< ψn,k| − i¯h ∂

∂x|ψn,k >= 0 (57) The expectation value of the momentum for a harmonic oscillator is zero casing I_x to vanish. Meaning the conductivity also vanish σ_xx = 0.

I_y = −e m

ν

X

n

X

k

< ψ_n,k| − i¯h ∂

∂y + eBx|ψ_n,k > (58) The momentum operators brings down ¯hk and the second term calculates the expectation value of x. The expectation value of x is the point it oscillates around which was displaced twice, most recently be the implementation of an electric field.

eB < ψ|x|ψ >= −¯hk–mE/B (59) The ¯hk term cancels the ¯hk brought down by the momentum operator, making the current [8, p.43-44]

I_y = e

ν

X

n

X

k

E

B. (60)

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The sum over n gives the number of filled states ν, and the sum over k gives the number of electrons given by eqn. 53. Divide by the area to get the current density

J_y = eνE

φ₀ . (61)

By Ohm’s law the conductivity and resistivity is σxy = eν

φ₀ ρ_xy = −φ₀

eν = −2π¯h e²ν.

Besides the minus sign this is the same expression for the resistivity stated earlier in eqn. 51. The minus sign is due to the direction of the electric field applied.

8 Fractional quantum Hall effect

The FQHE is more complicated than the IQHE and was like the IQHE first discovered through experiments before it was explained theoretically. FQHE is a consequence of the interactions between electrons while IQHE do not account for the interactions.

The ground state of a partially filled Landau level is macroscopically degenerate. This degeneracy is expected to be lifted by the coulomb interaction between the electrons. The inequality

¯

hωB ECoulomb Vdisorder (62)

must be satisfied. Having ¯hω_B V means the mixing between Landau levels can be neglected. [8, p.75-77]

8.1 Laughlin States

Laughlin explained the FQHE for filling factors of

v = 1/m, (63)

where m is an odd integer. To do this Laughlin simply wrote down the wavefunction using educated guesswork. [8, p.76]

It can be shown numerically, for a small number of particles, that Laugh- lin’s guess for the ground state of the wavefunction coincide 99% with what

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the actual ground state from the Coulomb potential would be. Doing this numerically is difficult and for more particles the overlapping becomes smaller.

[8, p.78]

To see what general form the wavefunction should have first consider two particles in an arbitrary central potential interacting in the lowest Landau level, then generalise to n number of particles.

To solve for the wavefunction of two particles in an arbitrary central potential in quantum mechanics work with angular momentum. This was done using symmetric gauge giving the single particle wavefunction as

ψm ∼ z^me^−|z|²^/4l²^B (64) where z = x–iy and m is for the angular momentum. For two particles in any potential the wavefunction is then

ψ ∼ (z₁+ z₂)^M(z₁–z₂)^me^−(|z¹^|²^+|z²^|²^)/4l²^B (65) M determins angular momentum of center of mass and m relative angular momentum, both are positive integers. [8, p.77]

Laughlin’s proposal for the ground state for some arbitrary number of particles was

ψ(z_i) =Y

i<j

(z_i–z_j)^me⁻^Pⁿⁱ⁼¹^|zⁱ^|²^/4l²^B. (66) This is just a product of the one particle state seen earlier and a generalisa- tion of the two particle state to more particles. When m is an odd integer the wavefunction is antisymmetric reflecting the fact that the particles are fermions. Even integers of m are for bosons. It can be shown that ν = 1/m is the filling factor for this wavefunction by considering what the wavefunction says for a single particle.[8, p.78-79]

8.2 Other filling factors

Other filling factors can be seen as charged excitations of the v = 1/m filling factor, called quasi-holes and quasi-particles. [8, p.85]

8.2.1 Quasi-Holes, Quasi-Particles & fractional charge

Quasi-holes are holes in the wavefunction carrying a fraction of the electric charge given by an electron with the opposite sign, e^∗ = e/m. With e^∗ being the charge of the quasi-hole. Introducing M quasi-holes at positions η_j with

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η_j ∈ C and η_j = 1, . . . , M the wavefunction becomes[8, p.85-86]

ψ(z; η) =

M

Y

j=1 N

Y

i=1

(zi− ηj)Y

k<l

(zk–zl)^me⁻^Pⁿⁱ⁼¹^|zⁱ^|²^/4l²^B. (67)

The fractional charge can be explained by having m quasi-holes at the same point η making the wavefunction

ψ(z; η) =

N

Y

i=1

(z_i− η_j)^mY

k<l

(z_k–z_l)^me⁻^Pⁿⁱ⁼¹^|zⁱ^|²^/4l²^B. (68)

This Laughlin wavefunction describes a deficit of one electron at the position η. However this deficit was created by m holes meaning each quasi-hole forms 1/m^th of an electron with charge +e/m.[8, p.86]

The fractionally charged quasi-objects act as independent particles. The fractional charge have been directly measured in so called shot noise experiments first described in [2].

The quasi-particles carry a charge of e^∗ = −e/m and is introduced into the wavefunction by differentiating. [8, p.87]

ψ(z, η) =

" _N Y

i<=j

2 ∂

∂z_i − ¯η

Y

k<l

(z_k–z_l)^m

#

e⁻^Pⁿⁱ⁼¹^|zⁱ^|²^/4l²^B (69)

8.2.2 Anyons

Anyon is a particle category similar to the boson and fermion categories. The boson and fermion categorisations come from the argument that the rotation of 360^◦ of a particle in three dimensions should return the particle to its original position. The expression for this is

ψ ( ~r₁, ~r₂) = e^2iπαψ ( ~r₁, ~r₂) (70) e^2iπα= 1

giving the two possibilities α = 1 and α = 0 for bosons and fermions respec- tively. [8, p.90]

However QHE works in two dimensions where the argument do not hold up. Depending on if the rotation occurs clockwise or anticlockwise the phase becomes different. One clockwise or anticlockwise rotation give a phase of

ψ ( ~r₁, ~r₂) = e^±iπαψ ( ~r₂, ~r₁) (71)

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+ for anticlockwise, - for clockwise exchange and α can take on any phase.

Particles are called anyons when α 6= 0, 1. [8, p.91]

For n quasi-holes the phase is α = n²/m and for quasi-particles α =

−n²/m. [8, p.96]

8.2.3 Excited Laughlin states

By changing the magnetic field uniformly the system moves away from a filling factor given by ν = 1/m. Moving away from the Laughlin filling factor causes quasi-holes to appear in the Laughlin wavefunction which affect the filling factor. The wavefunction of Laughlin states dropping the exponential term had the form

ψ ∼Y

I<j

(z_i–z_j)^m (72)

with m odd describing fermions and even describing bosons. If the quasi- holes by themselves form a quantum Hall state the Laughlin state would take the form

ψ ∼

N

Y

I<j

(η_i–η_j)^2p+α (73)

with η_i describing the position of the anyons given by α, p is a positive integer. With α = ±_m¹ positive for quasi-holes, negative for quasi-particles, the maximum angular momentum is

N

2p ± 1 m

. (74)

The contribution to the filling factor from the quasi-hole or quasi-particle, νq

can be calculated using the maximum angular momentum, the area and the number of electron states in a full Landau level. It is given by

ν_q = ∓ 1

2pm²± m (75)

Giving the total filling factor of

ν + νq = 1

m ± _2p¹ (76)

+ for quasi-holes and – for quasi-particles. To get the sequence of filling factors that have been observed by experiments consider a sequence p_i = p₁, p₂, . . . . Each quasi-object can by themselves form a quantum Hall state contributing to the filling factor. [8, p. 100 -101]

ν = 1

m ± 1

2p₁± _2p ¹

2±. . .

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9 QHE application to setting metrology stan- dards

9.1 The Ohm

Before the discovery of the QHE the electrical resistance unit, ohm, was maintained in each country by their respective national metrology institute (NMI). The ohm is defined as the resistance between two points of a conductor with a constant potential difference of 1 volt and a current of 1 ampere.

Factors such as temperature and pressure can affect the measurements. This has led to poor universal definition of the ohm between different NMI:s with an uncertainty around one in 10⁶.[7, p.207]

QHE causes a quantisation of the resistance in a two-dimensional electron system in terms of fundamental constants. Providing a better way to universally define the ohm with an uncertainty of some parts in 10⁹.[7, p.208]

Figure 2 illustrates the setup, with the Hall resistance R_H and the longitudinal resistance R_xx given by

R_H = V_H/I (78)

R_xx = V_xx/I. (79)

On the quantised plateau values the longitudinal resistance drops to zero and the Hall resistance takes on values given by

R_H = R_K/i. (80)

R_K = h/e² is the von Klitzing constant and i is the plateau integer index.

However resistors are often calibrated in terms of R_K−90 which is R_K with the uncertainty of some parts in 10⁷, in SI units, omitted. The value of RK−90

was assigned in 1990 to 25812.807Ω.[7, p.209]

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Figure 2: A two-dimensional slab of material carrying a current placed in a perpendicular magnetic field.

In real devices the measurements will slightly depend on things such as temperature and shape. The longitudinal resistance R_xx is no longer zero at a plateau affecting the value of R_H and can also depend on current and frequency. To get the value at Rxx = 0 an extrapolation is done. A voltage drop due to misalignment of the voltage terminals or a nonhomogeneous current density can affect the value of R_H as well. The effect of this can be measured and accounted for. [7, p223-224]

If the current goes beyond a certain critical value the QHE breakdown and the Hall resistance deviates from the expected. The cause being that the electronic temperature rises as the heat output is lower than the heat input.

Causing the temperature to abruptly rise and destroy the quantum Hall effects. The value of the critical current is suspected to depend on the width of the sample. Due to this large Hall bars in experiments are recommended for higher possible current. [7, p.224 -225]

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9.2 Multiple connection techinque

To find the Hall resistance in experiments have been limited by only being able to use the i = 4 and the i = 2 plateau in order to achieve a relative uncertainty of one part in 10⁹. With the other plateaus not being well quantised. [7, p.232] The Multiple connections technique (MCT) was proposed by Delahaye in [1].

The technique allows for building of a macroscopic Hall sample by con- necting several Hall bars. When two Hall bars are connected they get effected by an interconnection resistance effect. The technique accounts for this theoretically and allowed for a quantum Hall array resistance standard (QHARS) to be realized. [7, p. 232 - 233]

QHARS can be useful for improved stability of large resistance standards, for example of the order of 50R_k. QHARS can provide a more accurate way of international comparison while avoiding some of the setup required for other methods. [7, p.233]

9.3 Graphene

To observe QHE usually requires low temperature due to its nature of being a quantum phenomenon. However in graphene the QHE have been observed at room temperature (300K) due to the properties of graphene. [6]

Graphene is by design two-dimensional, constructed by a single layer of carbon atoms in a hexagonal crystal lattice structure. The energy gap between the first two Landau levels is larger in graphene than in other semicon- ductor materials due to the charge carriers in graphene behaving as massless.

This causes QHE to be observed at a lower magnetic field and/or higher temperature. [4, p.2]

Graphene is cheaper than other materials used to maintain the ohm unit.

[4, p.22] Not only is graphene a cheaper material the effect has also been observed at room temperature, driving the cost down additionally. Less money cost means smaller laboratories also can start experimenting with the effect, driving science and technology forward faster.

10 The importance of QHE

QHE has improved the field of resistance metrology by a great deal. The use of graphene to study QHE and the ohm unit continues to bring resistance metrology forward. Technical advances in this field can pave the way for a broader application of QHE and more applications of quantum mechanics overall.

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I think especially the discovery of QHE in graphene at room temperature can with further scientific exploration find applications useful for society.

Perhaps it can find some use in electrical devices.

The discovery of QHE is an example of experiments discovering something yet unexplained by theory. Showing the need for experiments to be performed in order to expand our knowledge of how the world works. New experiments continue to increase the knowledge of QHE and discover different kinds of QHE with their own areas of use.

As touched upon in the background section IQHE and FQHE are involved parts of many different research areas. With the theoretical basis for FQHE still being active research its potential applications may only be theorised on and may turn out to have a great impact on society, even if its applications are only to bring other scientific fields forward.

11 Conclusions

The fractional and integer quantum Hall effects are interesting quantum phenomena involving moving electrons in a magnetic field. Understanding the classical Hall effect and analysing the Hamiltonian of the moving electrons in Landau and symmetric gauges, helps in understanding the theory of the quantum effect. Laughlin states describes the fractional quantum Hall effect for filling factors of ν = 1/m. However they can also be used to describe other filling factors by introducing excitations known as quasi-holes and quasi-particles.

The quantum Hall effect have greatly impacted resistance metrology by universal property of the Hall resistance. The Hall resistance only being dependent on fundamental constants of physics provides a way to determine the ohm unit. Research into the quantum Hall effect in graphene continu- ous to push resistance metrology forward and may in the future have wider applications outside of metrology useful for society as a whole.

The impact of the quantum Hall effect can be seen by the amount of different research areas involving it. With the understanding of the fractional quantum Hall effect still being active research its usefulness can come to increase as understanding of it grows.

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The Quantum Hall Effect