Quantum Hall effect

The unconventionalquantum Hall effect (QHE) found in graphene [3, 4] was one of the driving forces in the early days of experimental graphene research and pushing investi-gations on the whole material system a lot. The QHE was one of the most significant

a)

b)

c)

e) d)

Figure 2.7: The sequence a)-c) describes the energy dependent density of states D(E) at a finite magnetic field perpendicular to the plane of the charge carriers a) for massless Dirac fermions in single layer graphene, b) for massive Dirac fermions in bilayer graphene and c) for Schrödinger electrons with two parabolic bands touching each other at zero energy [2]. d) and e) Resistivity (red) and Hall conductivity (blue) as a function of charge carrier density at finite magnetic field [10].

The measurement of the quantum Hall effect in d) corresponds to a monolayer [3] and shows the half-integer quantum Hall effect. The quantum Hall effect in e) reflects the quantum Hall effect in bilayer graphene, the missing plateau at zero carrier density is clearly visible [43].

discoveries [44] of the 1980s and is an effect that has so far only been observable in high quality semiconducting two dimensional charge carrier systems. Before going into the de-tails of the outstanding effects in graphene, QHE in conventional systems with a parabolic dispersion relation will shortly be revisited after a few words about the concept behind the classical Hall effect.

From classical Hall effect to (half-)integer quantum Hall effect

The classical Hall effect can be observed by measuring a conductor’s resistance perpen-dicular to the current, if a magnetic field is applied again orthogonal to current and voltage probes. The electric field induced by the voltage compensates the Lorentz force (produced by the magnetic field affecting the charge carrier. Thus the resulting voltage is proportional to the magnetic field: |Uxy|=I·B/(|dne|)withI the flowing current,Bthe absolute value of the magnetic field,d the width of the conductor,nthe charge carrier density andethe

2.2. TRANSPORT PROPERTIES 15

electron charge. By normalizing this voltage-drop by the currentIthe following expression for the Hall resistanceRxyis deduced.

R_xy= B

|dne| ^(2.5)

The Hall resistance is proportional to the magnetic fieldBand can be used to determine the charge carrier density n if the sample thickness d of the three dimensional sample is known. For two-dimensional charge carrier systems the Hall resistanceRxy is directly proportional toBand indirect toeandn.

But what happens if the same measurement is carried out with a high mobile two-dimensional charge carrier system at low temperatures and a high magnetic field? As demonstrated by K. v. Klitzinget al. in 1980 [44], under such conditions the Hall resistance gets quantized and exhibits constant values over a certain magnetic field range, what is referred to as quantum Hall plateaus. The resistance at the plateau takes the following values

R_xy= 1 ν

h e² = 1

ν·25812.8Ω (2.6)

withνbeing an integer. In the presence of a finite perpendicular magnetic field the density of states of a real two dimensional charge carrier system (only the lowest subband oc-cupied) condenses into equidistant highly degenerated delta shaped peaks, the so-called Landau levels (LL). These peaks are broadened by disorder. The integerνis the so called filling factor. For a non spin-degenerated system it is defined as:

ν= nh

eB ^(2.7)

The energy of such a LL in a conventional semiconductor is E_NLL =~ω_C(N_LL+¹₂) with ωC=eB/m^∗the cyclotron frequency and~the Planck constanthover2πand the effec-tive mass m*. The equidistant spacing of the LL level can be seen from this relationship. A schema of the LL of a conventional two dimensional system is depicted in Figure 2.7 c), a series of broadened peaks of the distance of~ωC. That is also the situation expected and already experimentally verified for FLG [45].

In the case of massless Dirac fermions as present in graphene monolayers the spectrum of the Landau levels takes a different form:

E_NLL=±v_F r

2eB~(N_LL+1 2±1

2) (2.8)

In this equationvF is the Fermi velocity that is approximately1/300of the speed of light (vF ≈c/300),~is again the reduced Planck constant andNLLrefers to the Landau level

index. This sequence of Landau levels is drawn in Figure 2.7 a). The ”½” term in equation (2.8) considers the spin and the ”±½” term the pseudospin, which is originated by the double valley degeneracy of the bandstructure as defined before. The ”±½” term gives rise to the unconventional so called half-integer QHE and a very big difference in the LL spectra compared to conventional QHE systems. For massless Dirac fermions there exists a zero energy state at the 0th LL, meaning that this lowest LL is occupied by both holes and electrons. For conventional parabolic systems with Schrödinger-like fermions this low-est energy level is shifted by ½~ωC. Additionally, the ±½ in equation (2.8) shifts the whole QHE spectrum by½ in comparison to conventional systems. Therefore, this QHE is denoted as half-integer QHE in contrast to the integer QHE in conventional parabolic system. However, the half-integer QHE must not be mixed up with the fractional quantum Hall effect, which can be described in the picture of composite fermions [46].

There is another noticeable property in equation (2.8). The distance between two Landau level peaks depends on the square-root of the energy. Together with high mobility and Fermi velocity this dependence (especially the big energetic distance between N_LL =0 andNLL=±1) allows to measure QHE at room temperature for the first time [47]. With respect to spin and pseudospin the degeneracy in the LL spectrum is f =4. This four-fold degeneracy of graphene was already mentioned in the discussion of the band structure. To sum up, the position of the quantum Hall plateaus are described in graphene monolayers by modifying equation (2.6):

In the case of massive Dirac fermions that are found in bilayer graphene another unusual QHE occurs. The Landau quantization is shown in Figure 2.7 b) and is described by

EN_LL =±~ωC

pNLL·(NLL−1) (2.10)

Evidently two possible solutions exist for a zero-energy state,N_LL=0andN_LL=1. Apart from a missing plateau for zero energy originated by this additional degeneracy, the Lan-dau level peaks are now equally spaced again and therefore a conventional sequence of quantum Hall plateaus is observed.

The experimental findings for both monolayer and bilayer confirm this results. The half-integer steps of4e²/hfor a graphene monolayer are shown in Figure 2.7 d) in blue and in the corresponding longitudinal resistance exhibiting SdH oscillations are clear fingerprints of massless Dirac fermions in graphene. In Figure 2.7 e) the same measurement for a bilayer graphene is given. There is really no plateau at zero energy and since the number

2.2. TRANSPORT PROPERTIES 17

of states in that situation is twice compared to higher levels, the step in the Hall resistance at the (missing) 0th LL must be twice, too. The amplitude of the SdH for this LL is also much higher and much more pronounced than those for higher LLs.