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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 4e2/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 4e2/πh), whereas things remain unclear for the Fano factor (F decreases in [10,11] or increases in [12] in the presence of disorder).


a) Minimum conductivity and (b) Fano factor at the Dirac point versus the width over length ratio W/L calculated using the evanescent wave theory [3], for three different boundary conditions: for metallic armchair edges in blue solid line (α = 0), for semiconducting armchair edges in green dashed line (α = 1/3) and smooth edges in red dotted line (α = 1/2). The black thin dotted line marks the universal values for the minimum conductivity and the Fano factor.


We have study graphene sheets exfoliated from natural graphite and deposited on top of Si/SiO2 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 4e2/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]):

(a) Experimental set-up for detecting shot noise at T = 4.2-30 K. (b) Schematic of the principle of our measurements in terms of the noise power reflection Γ. (c) Illustration of a typical graphene sample fabricated for our shot noise study.


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


Measurements on sample with W/L = 24: (a) Resistance R (left axis) and conductivity σ (right axis) as a function of charge carrier density. (b) Differential resistance dV/dI versus bias voltage Vbias at the Dirac point (red curve) and at high density (blue curve). (c) Current noise per unit bandwidth SI as a function of bias at the Dirac point, at T = 8.5 K, fitted (red curve) using the Klus formula [6] (F = 0.318). Note that the low bias data are perfectly fitted as well as the high bias (d) Mapping of the average Fano factor F as a function of gate voltage Vgate and bias voltage Vbias at T = 8.5 K.
(a) Measurements on sample with W/L = 2: (a) Resistance R (left axis) and conductivity σ (right axis) as a function of Vgate. (b) dV/dI versus Vbias at the Dirac point (red curve) and at high density (blue curve). (c) Current noise per unit bandwidth SI as a function of Vbias at the Dirac point, at T = 5 K, fitted (red curve) using the Khlus equation [6] (with F = 0.196). (c) Mapping of the average Fano factor F as a function of gate voltage Vgate and bias voltage Vbias at T = 5 K.
dc transport and shot noise measurements on a sample with W/L = 4.2 and L = 950 nm: (a) Resistance R (left axis) and conductivity σ (right axis) as a function of gate voltage Vgate. (b) Differential resistance dV/dI versus bias voltage Vbias at the Dirac point (red curve) and at high density (blue curve). (c) Noise spectral density SI as a function of bias voltage Vbias at the Dirac point, at T = 12 K, fitted (red curve) using Khlus formula (F = 0.256). (d) Mapping of the average Fano factor F as a function of gate voltage Vgate and bias voltage Vbias at T = 12 K.
dc transport and shot noise measurements on a sample with W/L = 1.8 and non-parallel contact (8 degres): (a) Resistance R (left axis) and conductivity σ (right axis) as a function of gate voltage Vgate. (b) Differential resistance dV/dI versus bias voltage Vbias at the Dirac point (red curve) and at high density (blue curve). (c) Noise spectral density SI as a function of bias voltage Vbias at the Dirac point, at T = 12 K, fitted (red curve) using Khlus formula (F = 0.087). (d) Mapping of the average Fano factor F as a function of gate voltage Vgate and bias voltage Vbias at T = 12 K.


Fano factor for three large W/L samples:


Average Fano factor extracted at large bias (Vbias = 40 mV) for three different samples, all having large aspect ratio (W/L ≥ 3). For the two unintentionally highly p-doped samples (orange and green dots), the Dirac point was estimated via extrapolation of the minimum conductivity at 4e2/πh.


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)

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