# Graphene

## Shot noise in graphene

Electronic properties in graphene are being intensively studied since the discovery of the anomalous quantum Hall effect in this purely two-dimensional system [1]. Owing to its unique band structure, graphene conduction occurs via massless Dirac fermions. Graphene is a gapless semiconductor: the conduction and the valence band are touching in two inequivalent points (K and K', usually called Dirac points) where the density of state is vanished. However, the conductivity at the Dirac point remains finite. Indeed, at the Dirac point, the conduction occurs only via evanescent waves, i.e. via tunneling between the leads [2,3]. A first evidence of such mechanism has been recently given by studying the minimum conductivity in short and wide strips [4].

By biasing a conductor that has sizes smaller than the electron-phonon inelastic scattering length, it is possible to study the out of equilibrium current noise generated by the system: such noise is called shot noise [5]. These current fluctuations are due to the discreteness of charge. By probing shot noise, one can collect informations about disorder, interactions, contact quality, or carrier statistics for example. We have studied shot noise in short and large graphene strips (with different width over length ratio W/L). We used two ways to extract the Fano factor, using the formula defined in [6] and using a tunnel junction to calibrate the noise of our devices [7,8] (the noise generated by a tunnel junction is purely Poissonian, F = 1). We use a home made set-up especially design to work at high frequency (to avoid 1/f noise) [7]. In perfect short and wide graphene strips (W/L ≥ 3), for heavily doped graphene leads, at the Dirac point both minimum conductivity and Fano factor are expected to reach universal values of 4e^{2}/πh and 1/3 respectively [5]. Note that metallic leads does also work with the evanescent mode transport theory [8]. Astonishingly, the transmission coefficients at the Dirac point in perfect graphene show similar form as those found in diffusive systems. Disorder affects carrier transport. Recent theories taken into account disorder show that conductivity should increase [10,11,12] (the minimum conductivity is no longer 4e^{2}/πh), whereas things remain unclear for the Fano factor (F decreases in [10,11] or increases in [12] in the presence of disorder).

We have study graphene sheets exfoliated from natural graphite and deposited on top of Si/SiO_{2} wafer, where the substrate is used as a backgate. Our measurements show that for large W/L strips, both minimum conductivity and Fano factor reach universal values of 4e^{2}/πh and 1/3 respectively, as demonstrated in the evanescent mode theory [5] for perfect graphene. We see that the Fano factor is maximum at the Dirac point and diminishes at large carrier density. We also see that for smaller strips, the Fano factor is lowered as expected by the theory. While whether or not transport in graphene could be ballistic remains much debated, our findings tend to prove that transport at the Dirac point occurs via evanescent wave, i.e. that carriers can propagate without scattering. Note that during the completion of our manuscript, a shot noise study has been done in disordered graphene [13] showing density independent Fano factor that typically characterizes fully diffusive systems [14]. This work has been done in collaboration with Alberto Morpurgo's team from Delft University of Technology.

[1] For review see A.H. Castro Neto et al., condmat/07091163

[2] M.I. Katsnelson, Eur. Phys. J. B **51**, 157 (2006)

[3] J. Tworzydlo et al., Phys. Rev. Lett. **96**, 246802 (2006)

[4] F. Miao et al., Science **317**, 1530 (2007)

[5] For review see Ya.M. Blanter and M. Büttiker, Phys. Rep. **336**, 1 (2000)

[6] V.A. Khlus, Zh. Ekps. Teor. Fiz. **93**, 2179 (1987) [Sov. Phys. JETP **66**, 1243 (1987)]

[7] F. Wu et al., AIP Conf. Proc. **850**, 1482 (2006)

[8] F. Wu et al., Phys. Rev. Lett. **99**, 156803 (2007)

[9] H. Schomerus, Phys. Rev. B **76**, 045433 (2007)

[10] M. Titov, Europhys. Lett. **79**, 17004 (2007) and condmat/0611029v1

[11] P. San-Jose, E. Prada and D.S. Golubev, Phys. Rev. B **76**, 195445 (2007)

[12] C. H. Lewenkopf et al., Phys. Rev. B **77**, 081410(R) (2008)

[13] L. DiCarlo et al., Phys. Rev. Lett. 100, 156801 (2008)

[14] A.H. Steinbach et al., Phys. Rev. Lett. **76**, 3806 (1996)

**Summary of our mesurements**

Set-up, measurement principal and sample schematics (see also [1]):

Measurements in four different cases: Large W/L, small W/L, disordered and non parallel samples:

Fano factor for three large W/L samples:

**Related publications:**

- R. Danneau, F. Wu, M.F. Craciun, S. Russo, M.Y. Tomi, J. Salmilehto, A.F. Morpurgo, and P.J. Hakonen

*Evanescent wave transport and shot noise in graphene: ballistic regime and effect of disorder*

(unpublished), arXiv:0807.0157

- R. Danneau, F. Wu, M.F. Craciun, S. Russo, M.Y. Tomi, J. Salmilehto, A.F. Morpurgo, and P.J. Hakonen

*Shot noise in ballistic graphene*

Phys. Rev. Lett. **100**, 196802 (2008)