Transport Properties of Graphene

Contact Benjamin Huard (bhuard@), Kathryn Todd (kathrynt@), Nimrod Stander (nimrods@), or Joseph Sulpizio (jopizio@) for more information.

Over the past two decades Carbon has offered a testbed for measuring the electronic properties in reduced dimensions, with the advent of fullerenes (0D), nanotubes (1D), and graphene (2D), all of which are related to the ancient material graphite (3D). The recently-isolated graphene is not the first host which can confine electrons to 2 dimensions, but it has a unique combination of properties:

  • a good 2D conductor that can be contacted directly and patterned
  • semi-metal with a linear dispersion, symmetric for electrons and holes
  • new degree of freedom analogous to spin
  • finite conductivity at zero density
  • "half-integer" Quantum Hall effect observable even at room temperature

In our group, we are interested in understanding how charge carriers move in a non-uniform potential in graphene, where the potential is created either by charged impurities or by a tunable local gate. Our approach to this problem is two-fold: (i) applying a tunable local potential perturbation using a nanofabricated gate electrode, then performing standard transport measurements in the presence of this potential perturbation, and (ii) confining charge carriers to narrow regions and measuring transport properties as a function of the position of a mobile potential perturbation created by a "scanning gate".

Transport across a potential barrier in graphene

Tunneling effect in graphene
One of the most counter-intuitive phenomena explained by quantum mechanics is the possibility for a particle of a given energy to go through a small region of higher energy: the Tunneling Effect. It is well known that, in this situation, the probability that a particle crosses the potential barrier decreases exponentially when the barrier height increases. For relativistic particles though, this is not the case. A particle can transform into an anti-particle inside of the barrier, somewhat changing the barrier into a well and moving freely in this otherwise forbidden region. This phenomenon is called the Klein Tunneling.
Graphene offers a test-bed for observing the Klein Tunneling. The motion of carriers (electrons or holes) in graphene is governed by an Hamiltonian that is formally analog to the Dirac Hamiltonian, which describes a spin 1/2 relativistic particle. The speed of light is then replaced by the Fermi velocity in graphene (1.1 106 m/s), and the electron and hole bands are cone shaped and connect at a single point. Furthermore the electron-hole excitations in graphene evolve similarly as the electron-positron excitations for relativistic particles. Therefore, creating a high and long potential barrier in graphene should not prevent electrons from going through.
How to create a potential barrier on a graphene sheet?
In order to investigate the Klein tunneling, we created tunable potential barriers across a graphene sheet. We have developed a novel technique for modifying the potential profile in graphene using metallic gates on both sides of graphene sheets. A global gate (back gate) shifts the potential of the whole system, and a narrow gate across the sheet only shifts the potential underneath. The global gate is a heavily doped silicon substrate, capacitively coupled to the graphene sheet via a 300nm silica layer. On top of the sheet, a thin mono-layer of polymethylmethacrylate (PMMA) is spun and exposed to a high dose of 30keV electrons (21 mC/cm2) using a scanning electron microscope. This step crosslinks the PMMA molecules together and forms a network of larger molecules which is robust to solvents and forms a dielectric of about the same dielectric constant as silica. Finally, the top gate is patterned out of gold on top of this PMMA layer.


n-p-n junction in graphene: a) cross-sectional view of the device. b) electrostatic potential profile U(x) along the cross-section of a).
The combination of a positive voltage on the back gate and a negative voltage on the top gate produces a central p-doped region flanked by two n-doped regions. c) SEM picture of the device.

Experimental results
By applying a well chosen set of voltages on the back and top gates, we created potential barriers so high that electrons could cross them only if they got transformed as holes. A region where the main carriers are holes is called a p region, while a region where the main carriers are electrons is called an n region. With a single top gate device, we were able to create two regions with arbitrary polarities (n or p), below and outside the top gate region.
In the case of electrons entering in a p region inside the barrier (n-p-n), we observed that the higher the potential barrier got, the lower the electrical resistance was. This peculiar behavior was the first observation of the Klein Tunneling.


b) 2D plot of the resistance across the barrier as a function of the voltages Vb on the back gate and Vt on the top gate. Each color line corresponds to a cut at a given voltage Vb of the 2D plot and is shown on a).


If electrons actually transform into holes at the barrier boundaries, then the potential seen by the charges is reversed allowing them to go through. Therefore, the resistance in the cases n-p-n and n-n-n where the potentials inside of the barrier are exactly opposite should be about the same. In fact, we observed that an n-p-n configuration has always a higher resistance than its symmetric n-n-n (see how the resistance curves are asymmetric). The reason for this is that electrons do not transform into holes for free. Measuring the difference between these two resistances is a good way to measure this cost.
The transmission probability T of electrons into holes in clean graphene has been shown to depend mainly on three parameters:

  • the potential in both regions (T decreases with increasing potential)
  • the potential steepness at the interface (the steeper, the higher T)
  • the angle of incidence (the closer to normal incidence, the higher T). Actually, T=1 for exactly normal incidence

    • We were able to show that our measurements were consistent with these predictions. In further works, we will try to make cleaner samples of graphene in order to observe correlations in the tunneling events at both interfaces of the potential barrier. Optical effects such as the Veselago lens might then be observed.


    For further reading, B. Huard, J.A. Sulpizio, N. Stander, K. Todd, B. Yang, and D. Goldhaber-Gordon, "Transport measurements across a tunable potential barrier in graphene", Phys. Rev. Lett. 98, 236803 (2007).   Supplementary info.