Mesoscopic Physics in Carbon Based Devices 

I/ Mesoscopic Physics and Quantum Transport in Graphene 

The fundamental properties of graphene (transport) 

Discovered in 2004 by Geim and Novoselov who received the Nobel Prize in 2010 

A perfect 2D metal: single sheet of atoms! 

Band Structure: a semi‐metal with relativistic particles. 

The valence band and the conduction band touch each other at two non‐equivalent points of 

the Brillouin’s zone (K & K’). Those points are called “Dirac points”. Very close to K and K’, the 

dispersion relation is linear:  ∆  (  stands for electrons or holes) 

This dispersion relation is typical of massless relativistic particles (for non‐relativistic 

particles, ∝  ). Indeed, the energy   in relativistic physics is given by   

where   is the momentum of the particle . Two different limits can be considered: 

a. The case of a massive particle ( ≫ ) 

In this case, the energy can be approximate by: 

 

112

2 

As   is constant, we recover the usual quadratic dispersion relation  ∝  of a massive 

particle. 

b. The case of a relativistic massless particle ( ≪ ) 

In this case, the energy can be approximate by: 

112

At the smallest order in  / , the dispersion relation is linear. This case is similar to the case 

of photons. Fundamental differences are nevertheless important to note: in the case of 

graphene, the particles are charged, they are fermions and the “speed of light” is about 106 

m/s instead of 3.108 m/s for photons.  

Very high mobility. 

mobilityμ:  drift velocity of the electron  μ  and the conductivity is directly related to the 

mobility :  μ ⇔ μ ⁄  

 

Metals  Semi‐conductors (2DEG)  Semi‐conductors (bulk) 

Typically, the mobility of graphene samples can be up to 200 000 cm²/Vs at liquid helium 

temperature. 

Band structure: ambipolar behavior 

changing the position of   thanks to a gate voltage allows us to measure either electrons or 

holes. 

 

II/ Magnetotransport in 2D‐metals: from the Hall effect to the 

quantum Hall effect 

We will have  interest  in this part  in the magnetotransport properties of massive charged particles  in 

conventional  metals  or  semiconductors.  We  start  with  the  classical  Hall  effect  and  introduce  the 

quantum mechanics to explain the Shubnikov‐de Haas oscillations. At higher magnetic fields, we show 

that  a  quantum  phase  emerges  due  to  the  quantum Hall  effect  (QHE).  This  regime  exhibits  purely 

quantum  transport  properties.  As  we  will  see  later  on, massless  particles  (graphene  or  topological 

insulators) behaves similarly at low magnetic field (in the classical regime of the Hall effect) but show a 

different  quantum  behavior  (SdH  oscillations  as  well  as  QHE)  so  that  such  magnetotransport 

experiments are a very convenient way to prove the massless nature of charge carriers. 

1. Classical Hall effect 

Dirty 2D‐metal embedded in a low magnetic field. Deviation of the charge carriers due to the Lorenz 

force:  ∧ ∧ . 

→ Charge non‐equilibrium that induces a transverse electric field   which counteracts the effect of 

the Lorenz Force once the steady state is reached:  ⇒ ∧ . 

We have then: 

∙ ∧ ∙  

∙  

 

We can define the Hall coefficient   

Remarks about   (or  ): 

=107 S/m  =3.10‐2 S/□  =10‐6 ‐104 S/m 

n=1029 m‐3  n=1‐2.1011 cm‐2  n=1015‐1020 cm‐3 

μ=6.10‐4 m2/Vs  μ up to  3,5.103 m2/Vs @4K  μ=10‐2 m2/Vs 

‐ It does not depend on the geometry of the sample 

‐ It is a direct way to measure the charge carrier density 

‐ It gives the sign of the charge carrier (test of the ambipolar behavior of graphene or Tis) 

2. Hall effect in Graphene 

Graphene is a semi‐metal that shows an ambipolar behavior when a sufficient gate voltage is applied. 

Indeed, the gate effect make possible the tuning of the Fermi energy that can move from the conduction 

band to the valence band or the opposite. It is then possible to study either the transport of electrons 

or holes and to switch on a single sample from the one to the other by choosing adequately the gate 

voltage. 

 

Fig. 1 Gate effect in a graphene sheet: applying a gate effect allow us to tune the position of the Fermi energy in the conduction band or the valence band 

Using standard fabrication technics, it is possible to pattern a Hall bar and to measure the classical Hall 

effect. 

Experimental method for producing a sample. 

For a given magnetic field, the Hall voltage is given by   and depends on the gate voltage 

through the charge carrier density  . It is judicious to plot the charge density   with 

respect to the gate voltage. A change in the sign of   (or of  ) is a clear indication that   changed 

its sign, namely that the type of charge carrier changed. 

 

Fig 2 Hall bar patterned in a graphene sheet and measurement of Hall effect that shows a ambipolar behavior (from Novoselov et al.) 

 

3. Case of an infinite 2D‐metal with no disorder (massive fermions) 

→ Classical deriva on

Cyclotron orbitals of the charge (no disorder).

∧ with

⇒ 0 where is the cyclotron frequency

The charge motion describes orbitals so that in the case of a non‐disordered material, the switching of

the magnetic field leads to a transition to an insulating state (I=0)!

The radius of the orbitals is given by the cyclotron radius:

cossin

⁄ sin ⁄ cos 1

We distinguish a clean metal from a disordered one by the ability for charge carriers under magnetic

field to form orbitals without to be scattered. A clean metal corresponds to the limit . A non‐

intuitive consequence of this model is that the introduction of disorder restores the conductance!

Two ways to reach this regime : to improve the material properties (longer ) or to use high magnetic

field (up to 80T!)

4. Landau Levels and Shubnikov‐de Haas oscillations 

To solve the problem of a clean metal under magnetic field, we need to introduce quantum mechanics.

First thing to do is to solve the Hamiltonian and to find the Eigen energies.

i) Naïve derivation of the Landau Levels in a semi‐classical approach In quantum mechanics, the charge carrier is represented by a complex wave function that has a

phase . Considering the latter case of charge carriers of a 2D metals with no disorder embedded in

a magnetic field, a semi‐classical approach would consist in keeping the classical trajectories and adding

a quantum phase that accounts for the circulation along the cyclotron orbital :

∮ ∙

In the presence of a magnetic field, the total phase acquired by a quasi‐particle will also be given by the

circulation of the potential vector and we end up with

/ ∙

According to Onsager and to the Bohr‐Sommerfeld quantification rules, the phase acquired on a closed

cyclotron orbital should satisfy: 2 with ∈ and ∈ . As the scalar products ∙

and ∙ are constant all along the orbital, we end up for an electron ( ) with:

/2 2 2

With ⁄ ⁄ ⁄

Finally, we have

/2 ⁄

2⁄

2⁄

It is not possible to determine the value of in a semi‐classical approach but the quantum mechanics

tell us that 1/2.

In the semi‐classical approach, we consider that the magnetic field only change the trajectories and not

the general expression of the Hamiltonian so that we have to consider the case of a Fermi sea with

massive free electrons. As a result, the energy of a massive charge coupled to a magnetic field will be

still given (only in our semi‐classical approach) by 2⁄ . We finally find the Landau solution

of the Hamiltonian for a free electron embedded in a magnetic field:

12

The consequence of switching on the magnetic field will be the quantification of the energy.

This rough approach reveals the underlying physics and gives the key ingredients to understand the

physics of the magnetotransport in a non‐disordered material: due to interferences along orbitals, we

open gaps in the density of states. In a similar way than for the classical case, a pure material is a material

for which the disorder is low enough (or the magnetic field high enough) to allow the formation of

cyclotron orbitals .

ii) Quantum mechanics treatment 

In a general way, we have: | | which can be simplified if 0. We have then

| | .

For a Fermi sea of massive Fermions without magnetic field, . If the magnetic field ∧

is turned on, one have to do the Peierl’s substitution: ↔ with . The

Hamiltonian that has to be solved in the case of electrons is:

2|

2| |

This is a classical problem of the quantum mechanics. This have been solved by Lev Landau in 1930 (he

was 21 years old!). As suggested by the semi‐classical model, gaps open in the energy spectrum and

only some discrete values of the energy will be allowed:

The major achievement of the quantum mechanics is to show the existence of a zero point energy

that is a direct consequence of the Heisenberg inequality∆ ∆ ~ . It is a fundamental property

and we will see that this zero energy points is not present for massless Dirac Fermions like graphene

quasi‐particles!

Fig. 1 Density of states at 2D  for a zero magnetic field, for a non‐disordered 2D metal and for a disordered 2D‐metal 

When the magnetic field is high enough to open some gaps into the DOS, the transport properties are

determined by the position of the Fermi energy (for an infinite sample, we have the relation

/ ):

‐ If lies between to Landau Levels (LL), the material will be insulating as predicted by the

classical model

‐ If is pined on a LL, the density of states is ≠ 0 at so that transport should occurs.

Nevertheless, even an infinitesimal disorder will localize those states such that 0. The

sample will also be in an insulating state!

In the case of an infinite 2D metal (with no edges) with an infinitesimal disorder, the switching of a

magnetic field induces a transition to an insulating state as predicted by the classical approach!

This simple picture does not take into account the effect of the edges (in the QHE regime, very

important!) as well as the effect of the disorder that induces broadening of the LL. In the regime where

the broadening induced by the disorder is larger than the energy spacing between two LLs ∆ ,

magnetic field is not strong enough to open gaps into the energy spectrum and the DoS does not vanish

for any energies : as in the classical case, the disorder restores the conduction into the bulk!

Nevertheless, the magnetic field induces a fluctuation of the DOS which exhibits maximum at energies

corresponding to a LL as soon as the magnetic field is strong enough to close the cyclotron orbitals

≲ . Such a fluctuating DOS has some strong implication for the transport properties of a the

metals: for a given sample, the position of the Fermi energy is fixed by the doping level in the sample.

Sweeping the magnetic field shifts the position of the LL with respect to since ∝ and the

fluctuates as well as the conductance of the sample. A minimum of the conductance (at a

magnetic field ) corresponds to the crossing of a LL.

Such oscillations, called Shubnikov‐de Haas (SdH) oscillations, are very useful to determine the charge

carrier density of a 2D metal (thanks to the value of the different ), the effective mass (the thermal

dependence of ∆ )and the mean free path (the dependence of with respect to the

magnetic field or the onset of the oscillations that corresponds to ≲ ⇔ ≲ ⇔ ≲ ).

iii) Temperature dependence of SdHO Not only the mobility is required to be good for the observation of SdH oscillations ( ≳ 1). The

temperature needs also to be low enough: the inequality: ≳ has to be satified. For usual

materials such as GaAs, this means that ≲ 10 . For graphene, the effective mass is much smaller and

→ 0 so that the cyclotron energy is much larger and the temperature requirement for the observation

of SdH oscillations becomes ≲ 1000 : SdHO can be observed at room temperature in graphene!

The temperature dependence of the SdHO gives a direct access to the cyclotron mass that is equivalent

to the effective mass in the case of massive fermions.

 

Fig. 2 Example of SdH oscillations measured in GaAs based heterostructures at low temperature (from Malcoff et al.) 

iv) SdHO for massless fermions In a very general case, it can be shown that the parameter that determines the temperature evolution

of the SdH oscillations will be the cyclotron mass that is expressed, in the case of masseless fermions

as:

For massive particles, the energy is dominated by the term which leads to an that is

independent on the charge carrier density.

For relativistic massless particles in graphene, we have:

√√

The cyclotron mass depends on the charge carrier density in the case of massless fermions and is

independent on then charge carrier density for massive charge carriers. The measurement of the

temperature dependence of the SdH oscillations and the study of the density dependence of the

cyclotron mass gives an unambiguous proof of the nature (classical or relativistic) of the charge carriers

that are involved in the transport properties of a sample.

Fig. 3 Example of the measurement of the cyclotron mass in a graphene sheet. The mass is found to depend on the charge carrier density n like n1/2 which an unambiguous proof that charge carriers in graphene are massless (from Novoselov et al.) 

5. The quantum Hall effect in non‐relativistic fermions (electrons) 

We have seen that  the classical model as well as  the quantum mechanics expect both an  insulating 

transition when the magnetic field is turned on for an infinite 2D metal with no disorder. As we will see, 

the introduction of edges drastically changes the conclusion of both models. 

i) Classical model of a finite (with edges) 2D‐metal with no disorder 

 

Fig. 1 Formation of chiral classical edge channels 

In the classical model, the “bulk” remains insulating but we have the formation of conducting stripes 

which width is about . Those channels should not be back‐scattered and should perfectly conduct the 

current. Very importantly, the transport is chiral (see the above figure 4). 

ii) Quantum mechanics 

The introduction of edges in the sample can be modeled in the Hamiltonian by the introduction of an 

infinite repulsive potential outside of the sample in order to retain the charge carriers into the sample. 

We assume here that this potential  is adiabatic:  its effect  is simply to shift  the position of the Eigen 

energies depending on the position considered in the sample. This is a good assumption if the potential 

changes slowly at the scale of  . 

2⇒

12

 

Into the bulk of the sample, we should have no influence of the edges: we have the formation of the 

previous LLs. Close to the edges, those LLs are shifted up such that even if the Fermi energy   lies in a 

gap into the bulk of the sample, it crosses a LL somewhere close to the edge. It can be shown that close 

to the edges, the LLs are not localized states anymore such that the introduction of edges induces a 

chiral  conduction  as  predicted  by  the  classical  model.  Those  states,  called  “edge  states”  or  “edge 

channels” have fascinating properties: 

‐ They are purely ballistic (ideal conductors). 

 

‐ They are chiral 

‐ They are purely 1D channels 

 

The number of edge channels  is given by the number of LLs filled into the bulk of the sample (filling 

factor ν). When the magnetic field increases, the number of edge channels decreases. When the Fermi 

energy lies between two LL into the bulk of the sample, the current will be carried by the edge channels 

only.  

6. Semi‐classical derivation of the conductance of a 1D channel 

The  determination  of  the  conductance  (in  a  two  point  configuration)  of  such  an  edge  channel  is 

equivalent  to  the determination of  a  1D  conductor which  transmission  1  (the probability  for  a charge carrier to from the left to the right or from the right to the left without to be backscattered). 

‐ Each contact absorbs all incoming charge carrier (black body model) 

‐ Energy relaxations occurs into the contact only (length of the channel  ) 

 

Fig. 2 Sketch of the principle of the conductance measurement of a 1D channel in a 2 probes configuration 

The current is given (at zero temperature) by: 

   

The factor ½ comes from the fact that half of the charge carriers have a positive velocity, the rest having 

a negative velocity. 

22

 

With   the spin degeneracy. Taking into account the fact that   we obtain finally: 

2 

Which results in   

This relation is very general and does not depend on the dispersion relation. 

Important remark: 

‐ the conductance is not infinite even if for a perfectly transmitted channel 

‐ the conductance is related to the fundamental constants e (the charge of an electron) and h 

(the Plank constant). It does not depend on the geometry of the sample nor on the charge density! 

7. Hall bar in the Quantum Hall effect regime 

In  this  section,  we  determine  the  typical  voltages  (Hall  voltage  and  longitudinal  voltage)  that  are 

measured on a Hall bar in the different field regimes (hall regime, Shubnikov‐de Haas regime and QHE 

regime). 

 

i) The classical Hall regime 

This regime corresponds to  ⁄ ≲ 1 

 

ii) The Shubnikov‐de Haas regime 

This regime corresponds to  ⁄ ≳ 1 and  ≪  where  is the magnetic 

length (see below). 

 

iii) The quantum Hall regime 

At  1,   exhibits plateaus which value is given by fundamental physics constant (e and h) and 

does not depend on the geometry nor on any other parameters of the metal (density, mean free path, 

effective mass,…). The corresponding Hall  resistance of  the plateau gives the Klitzing constant 

25812,…Ωwhich  defines  the  quantum  of  conductance/resistance.  Because  of  its 

independence to the various parameters, the quantum Hall effect is used in metrology to determine 

the value of . 

At , it is possible to show that the Hall voltage is given by . To do so, we need to take into 

account the fact that the emission rate of charge carrier of a contact is given by   per channel. 

Another important point is that in a four probe configuration, the longitudinal resistance of a ballistic 

conductor is founded to vanish. This results is in agreement with the common sense, on the contrary to 

the case of a two probe measurement. This indicates that the resistance of the 1D perfectly transmitted 

channel in a two probe configuration is an access resistance due to the strong reduction of the number 

of conducting channels between the macroscopic contact and the 1D conductor. 

iv) Experimental results 

8. Landau Levels and the quantum Hall effect in Graphene 

Considering again  the Hall  bar of graphene and  locking now at  the  longitudinal  resistance at higher 

magnetic  field  ( ≳ 1).  In  conventional  metal,  i.e.  with massive  charge  carriers,  the  temperature 

evolution of  the SdH oscillations  is directly  related to  the mass of  the charge carriers. This  is a very 

convenient and very conventional way to determine this mass. In the case of graphene, the situation is 

completely  different  since  the  charge  carriers  are  massless  fermions.  The  quantum  Hall  effect  in 

graphene 

Increasing further the magnetic field until   approaches few  , we could expect to enter the 

quantum Hall effect regime. Nevertheless this regime is expected to be very different than the one of 

massive fermions because of the vanishing of the effective mass of the charge carriers. Indeed, in the 

case of massive fermions, the cyclotron frequency is given by  → ∞ when  → 0. Hence, the 

energy of the LLs should diverge: 

12 →

∞ 

There is a breakdown of this model in the case of massless Dirac fermions! It can be shown than in the 

relativistic case, the cyclotron frequency can be given by  √2  and the cyclotron radius has to be 

written in a more general way as    that does not directly depend on the mass any more. Using 

similar argument than previously, one can try to infer what would be the energy spectrum of massless 

Dirac fermions under a magnetic field. Taking into account the fact that the  ‐vector is quantized along 

cyclotron  orbital  (⇔ /  with ∈ )  and  that  the  dispersion  relation  at  zero magnetic  field 

reads:  | |, we have: 

 (Wrong formulae!) 

Quantum Hall 

plateaus 

Spin‐ 

degenerescence 

lifted

Figure 3. First observation of the Quantum Hall effect by K. von Klitzing (Physical Review Letters 45, 494 (1980) )

As in the case of massive fermions, this formulae is wrong but gives the key ingredients of the quantum 

Hall effect with massless fermions and particularly the fact that  ∝ √  (for massive fermions,  ∝).  Solving  the  Hamiltonian  of  massless  Dirac  fermions  under  a  magnetic  field  gives  the  following 

eigenvalues : 

√  

Very important differences with the case of massive particles have to be mentioned: 

‐ The LLs are not equally placed in energy 

‐ There is a zero energy mode 

‐ There is positives (electrons) as well as negatives (holes) solutions 

‐ Even if the cyclotron energy in graphene does not→ ∞, it remains much larger than the one of 

usual massive particles. Typically,  ~1000  at 10 . This make possible the observation 

of the quantum Hall effect at room temperature! 

 

Figure 4 Graph LL massive vs Massless 

 

Example of the first experimental observation of such quantum Hall effect for massless Dirac Fermions 

(from Novoselo et al.): 

 

As we can see, on contrary to the case of the QHE in GaAs based heterostructures for which the LLs 

have just a spin degeneracy (the reason why we have plateaus at multiple of 2 ⁄  instead of ⁄ ), 

in the case of graphene, we have a spin degeneracy and a valley degeneracy (two valleys   and  ′ in the Brillouin zone) which leads to steps of 4 ⁄ . 

The LL of zero energy is equally composed of electrons and holes. In a similar picture than the one we 

used for electrons where the LLs are shifted at high energy when approching the edges of the sample, 

the LLs of holes are shifted at low energy (the holes need to be retained into the sample) and the zero 

energy LL will split into two LLs: one for electrons and the other one for holes. Each of those LLs has a 

two  times  lower  degeneracy  which  leads  to  plateaus  of2 ⁄   instead  of4 ⁄ .  The  symmetry 

electron‐hole is respected. 

 

Fig. 5 Shift of the LLs of graphene close to the edge of the ample and splitting of the zero energy mode 

Not only fundamental research: 

 

               

Ref : 

‐ Mesoscopic Physics : Datta 

‐ Graphene : Review of Das Sarma. The nature paper of Novoselov 

‐ Quantum Mechanics : Feynman / Landau / Cohen‐Tannoudjii