Session D12: Graphene Transport: Disorder Scattering

Theory of 2D Transport in Graphene for Correlated Disorder

We theoretically revisit graphene transport properties as a function of carrier density, taking into account possible correlations in the spatial distribution of the Coulomb impurity disorder in the environment. We find that the charged impurity correlations give rise to a density-dependent graphene conductivity, which agrees well qualitatively with the existing experimental data. We also find, quite unexpectedly, that the conductivity could increase with increasing impurity density if there is sufficient interimpurity correlation present in the system. In particular, the linearity (sublinearity) of graphene conductivity at lower (higher) gate voltage is naturally explained as arising solely from impurity correlation effects in the Coulomb disorder.   

Simultaneous Raman and electrical transport measurements of disordered graphene in situ in ultra-high vacuum

Resonant Raman scattering in graphene gives unique information about disorder, as the D peak is observed only in the presence of disorder which produces intervalley scattering of the electrons. The nature of disorder in graphene prepared by various techniques and on various substrates of the subject of significant research, with significant attention being paid to scattering by charged impurities; resonant scatterers due to vacancies, chemisorbed impurities, etc.; and non-resonant short-range impurities. In order to study the effect of these types of disorder on graphene's electronic properties and Raman spectra simultaneously, we have developed a facility combining thermal deposition, ion bombardment, electrical transport and micro-Raman measurements in an ultra high vacuum environment. We will discuss the capabilities of this facility and present the results of Raman and electrical transport measurements on controllably disordered graphene.   

Creating Vacancies in Supported and Suspended Graphene Field Effect Transistors

We have studied the creation of vacancy defects in both graphene samples supported on SiO$_2$ substrate and suspended graphene devices. The defects were created by irradiating graphene with a 30KeV Ga$^+$ ion beam in UHV; the effect of these defects on electrical transport in graphene was measured {\it{in situ}}. We find that the number of defects created in supported devices is dramatically higher in comparison with suspended devices. Using Monte-Carlo and SRIM simulations, we identify the formation of defects by secondary ions in the case of supported devices as a likely explanation of these results. We have also observed that the transport quality degrades drastically with respect to vacancy defect density and follows the same general trend for both supported and suspended graphene. These results may be important in designing devices for high radiation working environments, such as space electronics.   

Scattering mechanisms in graphene suspended in liquids. II. Flexural phonons (Theory)

Recent experiments reported strong scattering of charge carriers by flexural phonons in suspended graphene in vacuum.\footnote{Castro E. V., \textit{et al.}, ``Limits on Charge Carrier Mobility in Suspended Graphene due to Flexural Phonons,'' Phys. Rev. Lett. 105, 266601, 2010. } Our experimental data (previous talk) show that the carrier mobility observed for devices immersed in non-polar liquids, namely toluene and hexane, are significantly larger than the mobility limitation due to scattering by flexural phonons. We performed molecular dynamics simulations of graphene sheets suspended in hexane, toluene, and in vacuum at room temperature. We find that the interaction of molecules of the liquid with graphene suppresses the amplitude of the phonons by $\sim $50{\%}. We show computationally that this suppression is equivalent to an effective increase of the bending rigidity\footnote{Fasolino A., Los J. H., Katsnelson M. I., ``Intrinsic ripples in graphene.,'' Nature Mat. 6, 858, 2007.} of graphene from a free-space value $\sim $1.3 eV in vacuum to $\sim $3.6 eV in liquids. Therefore, we demonstrated that scattering by out-of-plane flexural phonons is reduced by the presence of liquids.   

Scattering mechanisms in graphene suspended in liquids. I. Coulomb scattering (Experiment)

Enhanced dielectric screening of charged impurities by high-$\kappa$ environment of graphene is predicted to improve the electronic quality of graphene devices by suppressing Coulomb scattering. However, experiments reported so far demonstrate that electronic transport in graphene is only modestly modified by a high-$\kappa$ environment. Here we fabricate large area multiterminal graphene devices suspended in liquids and study electronic transport in graphene as a function of liquid's dielectric constant. We observe a rapid increase of mobility $\mu$ with $\kappa$ due to dielectric screening in non-polar solvents ($\kappa\leq 5$). We also find that charged ions present in polar solvents ($\kappa \geq 18$) cause a drastic drop in mobility counteracting the gains by dielectric screening in polar high-$\kappa$ liquids. Furthermore, molecular dynamics simulations establish that scattering by out-of-plane flexural phonons is suppressed by the presence of liquids (next talk). We expect that our findings may provide avenues to control and reduce carrier scattering in future graphene-based electronic devices.   

Short range disorder in Graphene: A Green's function based approach

Electrons at the Fermi level in graphene monolayer behave like massless Dirac fermions. Using Green\'is function based technique we study the transport of such electrons in the presence of randomly located electrostatic impurities in different geometries and compare such transport in graphene with the transport in conventional semiconductors. This study would eventually be used to mimic short range disorder that would be superimposed on a regular structure. Comparison with the optical phenomena will be used to understand such transport. We further extend this technique to study the transport in the presence of magnetic scatterers.   

Universal conductance fluctuations in Dirac materials in the presence of long-range disorder

We study quantum transport in Dirac materials with a single fermionic Dirac cone (strong topological insulators and graphene in the absence of intervalley coupling) in the presence of long-range disorder [1]. We show, by calculating the conductance fluctuations, that in the limit of very large system size and disorder strength, quantum transport becomes universal. However, a systematic deviation away from universality is obtained for realistic system parameters. By comparing our results to existing experimental data on $1/f$ noise, we suggest that many of the graphene samples studied to date are in a non-universal crossover regime of conductance fluctuations, and provide an explanation for the origin of the 1/f noise in Dirac materials and in graphene in particular. \\[4pt] [1] E. Rossi, J. H. Bardarson, M. S. Fuhrer, S. Das Sarma, {\it Universal conductance fluctuations in Dirac materials in the presence of long-range disorder.} arXiv:1110.5652v1 (2011)   

Diamagnetism of Graphene with Long-Range Scatterers

We study weak-field orbital susceptibility of graphene containing scatterers with long-ranged potential. The effects of scattering from such impurities are taken up self-consistently by using Green's function technique within a self-consistent Born approximation. To see dependence on the potential range, we consider scatterers with a Gaussian potential and screened charged impurities. Because Green's function or self-energy for long-range impurities strongly depends on wave vector, we have to numerically calculate Green's function and vertex functions. In graphene, the susceptibility diverges at the Dirac point as a delta function of the Fermi energy. For the Gaussian potential, results show that the delta function in the ideal graphene is broadened by disorder, due to the mixing between the states at the Dirac point and other states. The susceptibility as a function of the Fermi energy rapidly decreases away from the Dirac point, with effective width determined by the potential range. As the scattering strength increases, the peak at the Dirac point is less prominent and the susceptibility has a long tail, corresponding to the strong mixing among states due to scattering. These behaviors are obtained only when the vertex corrections are properly taken into account. For charged impurities, the susceptibility shows a double-peak structure caused by the strong variation of the effective scattering strength as a function of the Fermi energy.   

Current decay rate due to electron--electron scattering in graphene

Electron--electron scattering in graphene does not conserve electrical current, because of the linear dispersion of the bands in graphene near the Dirac point. In graphene, when two electrons with initial momenta $\mathbf k_1$ and $\mathbf k_2$ undergo electron--electron scattering to final states $\mathbf k_1'$ and $\mathbf k_2'$, in general the total current $\mathbf v(\mathbf k_1) + \mathbf v(\mathbf k_2) \ne \mathbf v(\mathbf k_1') + \mathbf v(\mathbf k_2')$ [see {\em e.g.}, Li {\em et al.}, Appl.~Phys.~Lett. {\bf 97}, 082101 (2010)]. We calculate the electric current relaxation rate due to the electron--electron scattering of an electron that is injected into extrinsic graphene at low temperature. When the energy of injected electron relative to the Fermi energy is small compared to the Fermi energy, the current decay rate is small due to phase-space restrictions. The current decay rate grows monotonically as the energy of the injected electron increases.   

Electron energy relaxation in disordered graphene via e-phonon interaction

Motivated by recent experiments, we study theoretically the energy relaxation of hot electrons in disordered graphene via electron-phonon interactions. In contrast to previous treatments [1], we explicitly treat the effects of electronic disorder. Using the Keldysh diagram technique, and including vertex corrections, we identify various mechanisms through which disorder can significantly change the magnitude and temperature dependence of the electronic energy relaxation rate. [1]Felix Von Oppen, Francisco Guinea, Eros Mariani, Phys. Rev.B 80, 075420 (2009)   

Electron scattering at graphene edges

We theoretically study the reflection of electrons at edges in monolayer graphene in a low-energy limit. We consider both of zigzag and armchair edge cases. We find that in the case of zigzag edge, the reflection phase is determined by the rotation of electron pseudospin during the reflection, which is attributed to the chirality between the pseudospin and the electron momentum. On the other hand, in the case of armchair edge, the reflection phase does not contain the information of pseudospin rotation, because of intervalley mixing. The pseudospin rotation can be detected via the measurement of reflection phase in an interferometry setup in graphene with zigzag edge.   

Highly Defective Graphene: The thinnest insulating membrane

The electronic structure and transport properties of two-dimensional highly defective sp2 graphene are investigated theoretically. Using classical molecular dynamics, large planes of highly defective graphene-based sheets are first generated. An accurate empirical tight-binding Hamiltonian is then elaborated, allowing the prediction of elastic mean free paths, conductivities, and charge mobilities using a real-space order-N Kubo-Greenwood method. In sharp contrast to pristine graphene, the highly defective sp2 carbon sheet displays high density of states close to the Dirac energy. However, the dynamics of the corresponding electronic wavepackets reveals extremely short mean free paths (below 1 nanometer) and quantum interferences, both yielding to particularly strong localization phenomena. Consequently, these highly defective graphene-based sheets, although exhibiting a metallic character through the density of states, are from an electronic transport perspective among the most insulating two-dimensional one-atom-thick membrane ever made.   

Boundary scattering in quasi-ballistic graphene/hexagonal boron nitride mesoscopic wires

In a quasi-ballistic transport regime where the mean free path is larger than the width of conduction channel, diffusive boundary scattering results in an anomalous positive magnetoresistance due to a megnetic commensurability effect between cyclotron motion and sample width. In this work, we fabricate a high-mobility two terminal graphene mesoscopic wire on hexagonal boron nitride with a mean free path comparable to sample width $\sim $ 1 $\mu $m. Magnetoresistance of the graphene mesoscopic wire shows a peak structure at a magnetic field which scales with the ratio of the cyclotron radius $R_{c}$ to the wire width $w$. The peak field increases with back-gate voltage as a consequence that the ratio $w$/$R_{c}$ is modified due to the change in $R_{c}$. These results indicate the quasi-ballistic transport and diffusive boundary scattering in graphene on hexagonal boron nitride.   

Effects of disorder induced scattering in chemical vapor deposited Graphene.

The effect of the short-range scatters in chemical vapor deposited (CVD) graphene on the quantum interference effect of carrier scattering remains to be an interesting question. We study the magneto-resistance and low-frequencies noise of our CVD graphene by varying carrier density and temperature. Unlike previous studies of exfoliated clean graphene flakes, we have found in the vicinity of the Dirac-point (DP) WL signal cannot be fully described in terms of breaking the valley symmetry due to trigonal warping of the bands and atomically sharp disorder [1,2]. The discrepancy regime is coincident with the suppression of noise figures and the vanishing of Hall coefficient. Our data suggest that in low mobility CVD graphene an extra inter-valley elastic scattering process should be considered under the theoretical basis in Ref.1. More detailed experimental results and theoretical analysis will be presented and discussed. Ref[1]: E. McCann, et al., Phys. Rev. Lett. 97 146805(2006) Ref[2]: J.Phys. : Condens. Matter 22 205301 (2010)   

Efros-Shklovskii Variable Range Hopping in Reduced Graphene Oxide Sheets

Reduced graphene oxide (RGO) sheets consist of highly ordered graphene domain and structural defects including oxidized carbon atoms and topological defects. Charge transport properties of RGO sheets are strongly influenced by the degree of disorderliness which can be tuned by varying the amount of the reduction. We studied the hopping conduction of the RGO sheets with different reduction efficiency. We show that the low temperature charge transport properties of the RGO with various reduction efficiency can be well described by Efros-Shklovskii variable range hopping (ES VRH), $\rho $ $\sim $ exp[-(T/T)$^{1/2}$]. We will discuss how the localization length varies with the degree of reduction. The result indicates that the coulomb interactions between graphene domains play an important role in the charge transport of the RGO sheets.