The Quantum Hall Effect in Graphene
Daniel G. Flynn∗ Department of Physics, Drexel University, Philadelphia, PA (Dated: December 7, 2010) Graphene is a single layer of carbon atoms arranged in a hexagonal lattice. The two-dimensional nature of graphene leads to many interesting electronic, thermal, and elastic properties. One par- ticularly interesting property of graphene is that it exhibits the quantum Hall effect at room tem- perature.
I. INTRODUCTION sider the particle exchange operator, P, acting on n iden- tical particles constrained along a chain. Let xi be the Classically, when an external magnetic field is ap- position of the ith particle. Then the composite wave- plied perpendicularly to a current carrying conductor, function for the particles in the chain is the charges experience the Lorenz force and are deflected Ψ := Ψ(x , x , ..., x ) (1) to one side of the conductor. Then, equal but opposite 1 2 n charges accumulate on the opposite side. The result is For identical bosons, the particle exchange operator, P, an asymmetric distribution of charge carriers on the con- flips two of the coordinates and the resulting wavefunc- ductor’s surface. This separation of charges establishes tion has eigenvalue +1. an electric field that opposes further charge build-up. As long as charges flow, a steady electric potential exists PijΨ(x1, .., xi, .., xj, .., xn) = Ψ(x1, .., xj, .., xi, .., xn) called the Hall voltage and the resistivity of the conduc- (2) tor depends linearly on the magnetic field strength. This For identical fermions, the particle exchange operator is known as the classical Hall effect. flips two of the coordinates and the resulting wavefunc- In 1980, Klaus von Klitzing discovered that at low tem- tion has eigenvalue -1. peratures and high magnetic field strength, the plot of re- sistivity vs. applied magnetic field strength becomes an PijΨ(x1, .., xi, .., xj, .., xn) = −Ψ(x1, .., xj, .., xi, ..., xn) increasing series of plateaus. This implied that in quan- (3) h tum mechanics, resistance is quantized in units of e2 . For particles in two dimensions obeying parastatistics The plateaus corresponded to the cases where the resis- tivity was related to the magnetic field by integer and PijΨ(x1, .., xi, .., xj, .., xn) = exp(iα)Ψ(x1, .., xj, .., xi, .., xn) some fractional values of a quantity known as the filling (4) factor. These integer and fractional values led to the the- In this case, applying the particle exchange operator ory of the integer quantum Hall effect and the fractional swaps two of the coordinates and the resulting wavefunc- quantum Hall effect. Both of these effects have since been tion has an eigenvalue of arbitrary phase, α. Applying observed in graphene, a single layer of carbon atoms in a the exchange operator a second time will swap the two hexagonal lattice, at room temperature. [1][2][3][4] coordinates back to their original position returning the system to its initial wavefunction.
P P Ψ(x , .., x , .., x , .., x ) = exp(2iα)Ψ(x , .., x , .., x , .., x ) II. BACKGROUND ji ij 1 i j n 1 j i n = Ψ(x1, .., xi, .., xj, .., xn) (5)
A. Particle Exchange and Fractional Statistics This requires that the phase follows
The quantum Hall effect is only observed in two- exp(2iα) = 1 → exp(iα) = ±1 (6) dimensional systems. Thus the quantum Hall effect must be explained using the properties of two-dimensional For bosons the phase is given by α = 2nπ, and for systems. In three dimensions, particles with integer fermions the phase is given by α = π(2n + 1) where n spin (bosons) follow Bose-Einstein statistics while parti- is an integer. In two dimensions, however, the phase can cles with half-integer spin (fermions) follow Fermi-Dirac range between the fermionic and bosonic values, and the statistics. However, in two dimensions, particles can statistics corresponding to these intermediate phases are follow statistics that range continuously from fermionic called fractional statistics. to bosonic statistics. Such statistics are called para- statistics. To understand how these statistics work, con- B. Anyons
The quasi-particles that follow the fractional statistics ∗Electronic address: [email protected] described above were given the name “anyons” by Frank 2
Wilczek in 1982. One model for anyons is a spinless system where a particle of mass m/2 is orbiting a flux Φ. particle orbiting around a thin solenoid that produces The particle is spinless giving the boundary conditions a magnetic field that defines the z-axis. The charge of the particle is proportional to the applied flux Φ Ψ(r, θ + π) = Ψ(r, θ) (15)
q = CΦ (7) Making use of the gauge transformation where C is a constant. Before the magnetic flux is ap- A~0 → A~ − ∇Λ (16) plied to the system, the particle’s angular momentum is quantized, lz = n~. Then when a changing magnetic field where Λ = φθ , allows the Hamiltonian to be written as is applied, it creates a circular electric field according to 2π Maxwell’s equation P 2 p2 H = + (17) ∂B~ 4m m ∇~ × E~ = − (8) ∂t in the gauge where A~0 = 0. This Hamiltonian is equiva- This circular electric field exerts a torque on the particle lent to the Hamiltonian of two free particles. The bound- changing its angular momentum. This change in angular ary conditions in this choice of gauge are also trans- momentum is due to the change of magnetic field and formed, picking up a phase factor can be calculated usings Stoke’s theorem 0 iqΦθ Z Z Z Ψ (r, θ + π) = exp(−iqΛ)Ψ(r, θ) = exp(− )Ψ(r, θ) ~ ~ ~ ~ ~ ~ ~ ˙ 2π ∇ × E · dS = E · ds = E2πr = − B · S = −Φ (18) (9) Then using the original boundary conditions and calcu- where dS~ and ds~ are the infitesimal elements of the sur- lating Ψ(−~r) gives face and its enclosing curve, respectively. This leads to iqΦ Ψ0(r, θ + π) = exp(− )Ψ0(r, θ) (19) ˙ ˙ 2 ˙ qΦ CΦΦ lz = qEr = − = − (10) 2π 2π This proves that the wavefunction describing two anyons Thus, the change in the angular momentum associated is multiplied by a phase factor when the particles are with applying magnetic flux to the system is interchanged. The phase factor describing the resulting fractional statistics, also known as anyonic statistics is cΦ2 qΦ ∆lz = − = (11) qΦ 4π 4π α = (20) 2 If the particle’s angular momentum in the absence of flux is zero and the dimensions of the solenoid and the parti- Classically, the amount of magnetic flux produced by the cle’s orbit are limited toward zero, then the system may solenoid is an independent parameter in this system so be treated as a single object with spin determined by the the phase factor and the corresponding statistics are ar- magnetic flux passing through the solenoid. bitrary. [5] The statistical properties of this model for anyons can be described as an extension of the properties of a two- anyon system. Such a system follows the Hamiltonian C. Landau Levels
( ~p − qA~ )2 ( ~p − qA~ )2 H = 1 1 + 2 2 (12) Another important concept in the explanation of the 2m 2m quantum Hall effect is Landau levels. Consider an elec- tron confined to the x-y plane in the presence of a uni- where ~p are the momenta of the anyons and A~ are the i i form magnetic field in the z-direction. The Hamiltonian vector potentials at the position of each anyon given by is given by Φ ~r A~ = ± zˆ × (13) i 2 1 eA~ 2π r H = (~p + )2 (21) 2m c where ~r = ~r1 − ~r2. In terms of the center of mass coordi- ~ nate P = ~p1 + ~p2 and relative coordinate ~p = ( ~p1 − ~p2)/2, where A~ is the vector potential related to the applied the Hamiltonian is magnetic field by the Maxwell relation ∇ × A~ = B~ . The 2 2 electron will follow Schrodinger’s equation HΨ = EΨ P (~p − qA~rel) H = + (14) which is invariant under the Landau gauge transforma- 4m m tion: The first term in the Hamiltonian describes the motion of the center of mass while the second term describes a A~(x, y, z) = B(−y, 0, 0) (22) 3
In the Landau gauge, the Hamiltonian can be written in The degeneracy of the Landau states, G, is then terms of dimensionless quantities Nx 1 B 0 y G = = 2 = (30) y = − lk LxLy 2πl φ0 l x 0 lpy where is φ is the flux quantum φ = hc . Therefore, py = (23) 0 0 e ~ there is one state per flux quantum in each Landau level. The filling factor, ν, is the number of occupied Landau where kx is the x-component of the particle’s wavevector q levels for electrons in a given magnetic field. It is defined and l is the magnetic length l = ~c . The Hamiltonian eB as is then ρ 2 ρ 1 1 ν = = 2πl ρ = (31) H = ω [ y02 + (p0 )2] (24) G B/φ0 ~ c 2 2 y where ρ is the two dimensional density of electrons. eB where ωc is the cyclotron frequency ωc = mc . This is the Hence, the number of electrons that can exist in a given Hamiltonian for the familiar one-dimensional harmonic Landau level increases proportionally with the magnetic oscillator with energy eigenvalues field strength so that as the magnetic field strength 1 increases fewer and fewer Landau levels are occupied. E = (n + ) ω (25) Thus, the filling factor is a measure of both the applied n 2 ~ c magnetic field strength and the number of Landau levels where n=0,1,2,... The levels corresponding to each n are that are occupied in a system. called Landau levels, and the correspondeing eigenvectors η are given by III. THE HALL EFFECT 2n 2 1 1 y 2 y η (~r) = [π2 (n!) ] 4 exp[ik x− ( −lk ) ]H ( −lk ) n,kx x 2 l x n l x (26) A. Classical Hall Effect where Hn are Hermite polynomials. Some interesting properties of Lanau levels are that the energy of the level The Hall effect describes how current flowing through does not depend on kx while the y position does depend a sample will be effected by the application of a magnetic on kx. Because the energy is not dependent on kx, the field. Consider a two-dimensional conducting plate in an eigenstates with different values of kx in a given Landau applied electric field, E~ . According to Ohm’s law, the level are degenerate. current flowing through the plate, I, is proportional to the applied voltage, V, and is inversely proportional to the resistance of the plate, R. This is equivalent to D. Filling Factor J~ = σE~ (32) Another important feature in describing the quantum Hall effect is the filling factor. In the Landau gauge de- where σ is the conductivity of the plate, and J~ = qρ~v is scribed above, the electron orbital corresponding to a the current density for particles of charge q and density 2 given n value is localized at y = kxl according to [23]. ρ moving with a velocity ~v. Ohm’s law states that cur- If the electron is confined to a space of length Lx in the rent will flow in the same direction as the applied electric x-direction with the periodic boudary conditions field. In 1879, Edwin Hall discovered that in the presence ik (x+L ) ik x e x x = e x (27) of an applied magnetic field, the current in the plate wil actuallyl flow in a direction perpendicular to the applied then the allowed values of kx and nx are electric field. Consider the conducting plate in the pres-
nx Lxkx ence of an applied magnetic field perpendicular to the kx = 2π → nx = (28) x-y plane, B~ = Bzˆ. Then the charges flowing through Lx 2π the plate are subject to the Lorenz force: where nx is the corresponding Landau level quantum number. Counting the number of states in a region of 1 FLorenz = q(E~ + ~v × B~ ) (33) LxLy defined by y = 0 and y = Ly (For simplicity, the c state at y = 0 has nx = 0 and the state y = Ly corre- 2 ~ sponds to the wavevector kx = Ly/l .) gives the total If the applied electric field is in the y-direction, E = Eyˆ, number of states in this region, Nx. then the particle will move with velocity ~v given by L L E N = x y (29) qE~ = −q~v × B~ → ~v = c xˆ (34) x 2πl2 B 4
Then from Ohm’s law, the conductivity of the plate, or Integer values of the filling factor describes a system of Hall conductivity, is given by non-interacting electrons where the highest Landau level is completely filled. Once the Landau level is completely J~ qρ~v ρqc filled, a gap exists requiring a finite amount of energy σH = = = (35) E~ Ey B to reach the next degenerate Landau level. However, impurities in the sample create localized potentials that Similarly, the Hall resistance RH = 1/σ is defined as: can trap electrons in localized states. Therefore, if the B filling is changed slightly, the extra electrons fill the RH = (36) localized states and do not contribute to the current. ρqc Thus, in regions where the filling factor has an integer The Hall effect has been confirmed by countless ex- value, there is a plateau in the plot of resistivity vs. periments and is used routinely in solid-state physics magnetic field strength where the longitudinal resistance to determine the density of charge carriers ρ through of the sample disappears. measurements of a system’s Hall resistance.
Figure 1: Physical picture of classical quantum Hall effect
B. Quantum Hall Effect - Integer and Fractional
The integral quantum Hall effect (IQHE) was discov- ered by Klaus von Klitzing in 1980. Von Klitzing was studying the Hall effect of two-dimensional electrons in Figure 2: Plot of Hall resistivity and longitudinal resistivity vs. silicon MOSFET(metal oxide-semiconductor field-effect magnetic field strength exhibiting the integer quantum Hall effect transistor). He found that at low temperatures and high magnetic field, the Hall resistance of the system did not vary linearly with the strength of the magnetic field as predicted by the classical Hall effect. The plot of the re- D. Fractional Quantum Hall Effect sistivity as a function of magnetic field strength exhibited many plateaus which indicated that the Hall resistance In 1982, the fractional quantum Hall effect (FQHE) is quantized. was discovered by Horst Stormer and Dan Tsui. By re- In a two-dimensional system, the density of electrons,ρ, peating von Klitzing’s earlier experiments with cleaner can be written as: samples in higher magnetic fields, they found that the B plateaus in the plot of resistivity vs. magnetic field ρ = ν (37) strength also occured at some fractional values of the fill- φ0 ing factor. Because fractional values of the filling factor Then, the classical Hall resistance with q=e is refers to partially filled Landau states, the plateaus could only be explained in terms of interacting particles. The B B h R = = = (38) Hamiltonian for N interacting electrons can be written H ρec νBec νe2 φ0 as: ~ 2 2 h (~pi − eA(xi) e Therefore, the Hall resistance is quantized in units of 2 N N e H = Σi + Σi
Figure 4: Energy level diagram for graphene showing the Fermi level in the absence of any applied fields
The existence of the QHE at such high temperatures in graphene is due to the large energy gaps characteristic of Dirac fermions. These energies are given by:
p En = vf |2ne~B| (41) Figure 3: Plot of Hall resistivity vs. magnetic field strength 6 exhibiting fractional quantum Hall effect where vf is the Fermi velocity ( 10 m/s) and n is the Landau level quantum number. In a strong magnetic field (B=45T), the energy level spacing is 2800K. Graphene has a large concentration of charge carri- IV. THE QUANTUM HALL EFFECT IN ers which keeps the lowest Landau level completely GRAPHENE populated at high magnetic fields. Therefore, any carriers above the lowest Landau level will not be able Graphene is a single layer of carbon atoms in a to overcome the energy gap, and the quantum Hall two-dimensional hexagonal lattice. The carbon atoms effect is observed. The quantum Hall effect has been bond to one another via covalent bonds leaving one observed for the integer values of the filling factor 2p electron per carbon atom unbonded. The result is ν = 0, ±1, ±2, ±6, ±10, ... as well as the fractional filling 1 2 2 that the Fermi surface of graphene is characterized by factors ν = 3 , 3 , and 5 . Because graphene exhibits six double cones. In the absence of applied fields, the both the integer and fractional quantum Hall effects, Fermi level is situated at the connection points of these it is ideal for studying QHE, and may be proven use- cones. Since the density of electrons is zero at the Fermi ful in the development of quantum computers. [6][7][8][9] level, the electrical conductivity of graphene is very
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