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The nature of localization in graphene under quantum Hall conditions

Particle localization is an essential ingredient in quantum Hall physics [1,2]. In conventional high mobility two-dimensional electron systems Coulomb interactions were shown to compete with disorder and to play a central role in particle localization [3]. Here we address the nature of localization in graphene where the carrier mobility, quantifying the disorder, is two to four orders of magnitude smaller [4,5,6,7,8,9,10]. We image the electronic density of states and the localized state spectrum of a graphene flake in the quantum Hall regime with a scanning single electron transistor [11]. Our microscopic approach provides direct insight into the nature of localization. Surprisingly, despite strong disorder, our findings indicate that localization in graphene is not dominated by single particle physics, but rather by a competition between the underlying disorder potential and the repulsive Coulomb interaction responsible for screening.

high level of disorder and the behavior is similar to the one observed in GaAs two dimensional electron systems with their much larger carrier mobilities.
The preparation of graphene monolayers was performed in a similar manner to that reported in Ref. [ 23 , 24 ]. A detailed description is deferred to the method section. The sample was mounted in the preparation chamber of an ultra high vacuum scanning system and subsequently annealed by heating it above 100 °C at a background pressure . This conversion factor was extracted from two-terminal magneto-transport data.
We begin our investigations by measuring the energy level diagram as a function of the applied external magnetic field and density. Using a scanning probe microscope equipped with a single electron transistor (SET) on a glass fiber tip the local chemical potential µ and the inverse compressibility n ∂ µ ∂ are recorded. The SET measures the local electrostatic potential. However, under equilibrium conditions, any changes in the local chemical potential are exactly compensated for by changes in the local electrostatic potential. Hence, the SET enables to detect the local chemical potential as well as its local derivative n ∂ µ ∂ when weakly modulating the carrier density via the doped Si backgate. The latter quantity is inversely proportional to the local compressibility and directly related to the local density of states. The details of this technique have been described previously in Refs. [11,26 ]. Figure 1a depicts a schematic of the experimental arrangement with the SET hovering above the monolayer.
The energy spectrum of graphene can be determined by measuring the chemical potential as a function of the carrier density. In a fixed magnetic field and in the absence of disorder, the chemical potential undergoes sudden jumps equal to the energy between adjacent Landau levels. They arise when gradually increasing the density whenever the degeneracy of the highest filled Landau level is exhausted and the next Landau level starts to be filled. The chemical potential can either be measured directly with the SET or is obtained by integrating compressibility data. A color rendition of the inverse compressibility as a function of magnetic field B and carrier density n for a fixed position on the graphene flake is displayed in Fig. 1b. The gaps in the energy spectrum give rise to regions where the inverse compressibility is large. These regions appear as lines at filling factors ν= 4i-2 with i = -1, 0, 1, 2..... Fig. 1c depicts a measurement of the chemical potential (top curve) together with a cross section through the inverse compressibility data of Fig. 1b at a fixed magnetic field of 11.7 T. Prominent integer fillings have been marked. For the sake of comparison, the theoretical behavior of the chemical potential and the inverse compressibility at zero magnetic field have been included as well (dotted lines). Substantial disorder broadening [ 27 ] in graphene is responsible for the absence of abrupt jumps in the chemical potential. Only a smooth undulation is observed when an additional Landau level gets filled. From these data the Landau level energy spectrum can be reconstructed. A maximum in the compressibility (or a minimum in the depicted inverse compressibility) heralds the center of the individual Landau levels. The chemical potential values at the densities where such compressibility maxima appear yield the Landau level positions on the energy axis. A slight refinement of this simple procedure to extract the energy spectrum is required due to stray electric fields and is discussed in the method section. The resulting Landau level spectrum after the stray field correction is plotted in Fig. 1d. A similar spectrum has been recently obtained from scanning tunneling spectroscopy [ 28 ]. The energy of the center of each Landau level agrees well with the expected theoretical value as given by Here, i is the Landau level index ( which differs only by 10% from the value expected from band structure calculations [ 29 ]. This value is also consistent with our previous compressibility studies in the absence of a magnetic field from which a nearly identical value of the Fermi velocity was obtained [ 30 ]. Theoretical work addressing disorder and interactions confirm such weak renormalization of the Fermi velocity [13, 31 , 32 ].
The color rendition in Fig. 1b provides an overall view of the behavior of the local compressibility. We now zoom into the individual incompressible regions by recording data on a much finer grid of the density versus magnetic field plane. The outcome is depicted in Fig. 2a. Previous works on massive 2D electron systems made out of GaAs have shown that such local compressibility maps are dominated by sharp peaks which run parallel to the lines of constant integer filling when the system can still reach equilibrium on the time scale of the measurement [3]. Similar behavior is seen here in graphene at large magnetic fields. Many dark lines or spikes which run parallel to the line of constant filling factor 2 and -2 are visible. We will argue that these lines represent the charging spectrum of the localized states. Below we show that the number of charging lines and their parallel character constitutes a direct proof that despite the strong disorder, localization in graphene is not a single particle effect but rather involves an elaborate interplay between the disorder potential and local charge reorganization driven by the repulsive Coulomb interaction.
It is instructive to consider the expected behavior of the localized state spectrum for strong disorder where one can neglect the effects of the Coulomb repulsion and screening first. Under these conditions each localized state consists of a constant energy contour of the disorder landscape enclosing an integer number of flux quanta. When the magnetic field increases, these constant energy orbits must shrink in size in order to maintain a constant flux through their area and their energy changes accordingly. Fig. 3a displays a random disorder potential and two sets of localized states at fields  Finally, panel d highlights the evolution of the two selected localized states in the (n,B)plane. Localized states may cross and they do not run parallel to the line corresponding to complete filling of the Landau level. We conclude that the generic features of single particle localization such as the increase in the number of localized states with increasing field and their erratic, non-monotonic behavior in the (n,B)-plane are entirely absent in the data of Fig. 2a. Clearly, this picture does not hold for localization in graphene. We note that while the SET is a local probe it will nonetheless pick up the signature of all localized states within a spatial extent given by its size (of order 100 nm).
The behavior of the localized state spectrum described above should be contrasted with the expected behavior if Coulomb repulsion is strong enough to compete with disorder and to enforce local charge rearrangements to screen the bare disorder potential. is now seen to persist over a finite spatial extent which is set either by the localization length or by the resolution limit given by the size of the tip (~100nm). One can clearly see that electrons tend to localize at particular locations in space. We anticipate a strong correlation between these local density variations and the border of the incompressible band. It was shown previously that with a scanning SET it is also possible to extract the local density from a differential measurement of the surface potential in the compressible and incompressible regime [30]. The outcome of such a measurement has been included in Fig. 2b  The typical length scale of the density fluctuations in the line scan of fig. 2b is around 150 nm. It is resolution limited by the size of the SET sensor. The intrinsic disorder length scale of graphene was reported previously from the density interval spanned by the incompressible band around any of the well-resolved integer filling factors [30]. It was estimated to be approximately equal to 30 nm. A detailed study of spatial maps of the localized states spectra as in Fig. 2b offers however an alternative approach to estimate the characteristic length scale of the disorder. The level spacing between the localized states from a single dot or antidot is intimately connected with its size. Fig. 2c shows one example of a spectrum of equidistant compressibility spikes that apparently stems from a single dot. It exhibits large level spacing. In order to add one extra electron to this dot the average density in the sample needs to be changed by approximately 3 10 10 /cm 2 . Hence the electron is localized within an area with a diameter of approximately 60 nm. This size fits reasonably well with the previously reported estimated length scale from the width of the incompressible band.

Acknowledgements:
We would like to acknowledge useful discussions with Yigal Meir and Assa Auerbach.
This work is partly supported by the Harvard NSEC.

Sample preparation
An adhesive tape was used to peel a large graphite flake from an HOPG crystal. It was pressed onto a Si/SiO2 wafer. The conducting Si substrate served as a back-gate to vary the carrier density in the graphene sheet. Once a suitable monolayer was identified in an optical microscope, two electrical contacts were patterned with optical lithography. Typical dimensions of the monolayers were 10 × µm 2 . After lift-off of the ohmic contact metal (3 nm Cr, 30 nm Au), the graphene flakes exhibited their neutrality point at back gate voltages larger than 100V. Therefore, the sample surface was cleaned during an Ozone treatment and with an ammonia dip. Asprepared flakes had their neutrality point at back gate voltages near 30 V. The sample was annealed by heating it above 100 °C at a background pressure of 7 10 5 − × mbar in the ultra-high vacuum scanning system. This procedure apparently removed the doping adsorbents introduced during fabrication and exposure to humidity as the charge neutrality point shifted close to zero back-gate voltage.

Extraction of the Landau level spectrum
The procedure to extract the Landau level spectrum requires some refinement as some electrical field lines originating from the back-gate reach the SET. The parasitic charge associated with these fringing fields is also detected by the SET. The detected amount of charge depends both on the distance of the SET to the graphene edge as well as on the vertical distance between the tip and the flake. For the heights and distances to the sample edge used in these experiments, we derive a contribution to the compressibility from the fringing fields of approximately 0.15 meV 10 -10 cm 2 . This direct pickup will add a linear term to the chemical potential as we sweep the backgate voltage. The linear contribution from fringing fields to the chemical potential was determined from fitting the chemical potential data at zero magnetic field to the theoretically expected behavior of the chemical potential with an added linear term. The data in the presence of a magnetic field was then corrected with the linear term stemming from the fringing field.    a. The width of the incompressible region is determined from a best fit to the data. The fit function is composed of Lorentzians centered at the maxima of the inverse compressibility.
The variance of the Lorentzian serves as the fit parameter and is taken identical for all Lorentzians. A fit example is shown for the compressibility data recorded at 11.3 T. b. The dependence of the density width of the incompressible region on the applied magnetic field.
These widths are extracted as described under a. A sum of Lorentzians (blue squares) as well as Guassians (red circles) have been used. The solid line corresponds to 4eB/h and demarcates the transition from the regime of separated to overlapping levels.