Analysis of Scanned Probe Images for Magnetic Focusing in Graphene

We have used cooled scanning probe microscopy (SPM) to study electron motion in nanoscale devices. The charged tip of the microscope was raster-scanned at constant height above the surface as the conductance of the device was measured. The image charge scatters electrons away, changing the path of electrons through the sample. Using this technique, we imaged cyclotron orbits that flow between two narrow contacts in the magnetic focusing regime for ballistic hBN–graphene–hBN devices. We present herein an analysis of our magnetic focusing imaging results based on the effects of the tip-created charge density dip on the motion of ballistic electrons. The density dip locally reduces the Fermi energy, creating a force that pushes electrons away from the tip. When the tip is above the cyclotron orbit, electrons are deflected away from the receiving contact, creating an image by reducing the transmission between contacts. The data and our analysis suggest that the graphene edge is rather rough, and electrons scattering off the edge bounce in random directions. However, when the tip is close to the edge, it can enhance transmission by bouncing electrons away from the edge, toward the receiving contact. Our results demonstrate that cooled SPM is a promising tool to investigate the motion of electrons in ballistic graphene devices.


Introduction
Scanning probe microscopy (SPM) has been used in imaging electron motion in two dimensional electron gas (2DEG) inside a GaAs/AlGaAs heterostructures [1,2] and electronic states in a quantum dot [3,4,5]. Recently, we used SPM to image ballistic electron motion in graphene under a perpendicular magnetic field. The electrons follow cyclotron trajectories and regions in graphene corresponding to these cyclotron orbits were observed [6,7]. The sample is a hBNgraphene-hBN device etched into a hall bar geometry with two narrow (700 nm) contacts along each side, separated by 2.0 µm and large source and drain contacts at either end. The heavily doped Si substrate acts as a back-gate, covered by a 285 nm insulating layer of SiO 2 . The degree of magnetic focusing is measured by injecting current ! " into a narrow contact and measuring the voltage developed # $ at the second narrow contact on the same side of the hall bar device.
The trans-resistance % & = # $ /! " . We use a 20 nm wide conducting tip as a local probe which is raster scanned at a constant height above the sample surface as the change in conductance across the sample is measured. The image charge created by the tip on the sample surface scatters electrons away, thereby revealing the path of electrons through the sample.
In this paper, we present the theoretical model used for simulating the electron trajectories in graphene. The model is purely classical and the simulation results obtained agree quite well with the SPM data. Using these ray tracing simulations, we study the effects of the tip-created local charge density dip on the motion of electrons in graphene.
We also study the regions with dip (red) or increase in trans-resistance (blue) in the SPM images. We find that the scattering of electrons from the graphene edge is diffusive. However, when a tip is placed close to the edge, the scattering of electrons is specular due to the smoothly varying charge density profile created by the tip on the sample surface. As a result, there's an increase in the number of electrons reaching the receiving contact.

Method
We use classical ray tracing to simulate electron trajectories in graphene under an applied magnetic field. Each electron is modelled as a particle in the two-dimensional (2D) space of the sample with time-dependent position and velocity. External forces acting on the electrons are -(1) Lorentz force due to the perpendicular magnetic field B and (2) force from the tip-induced charge density profile. Depending on these forces, the electrons accelerate or decelerate.
The trajectories are injected from a source with cosine angular distribution that follows from the foreshortening of the apparent contact width. Fig. 1(a) shows the distribution of the injected electron trajectories from a narrow source 3 contact without a magnetic field. The color represents the intensity of the trajectories spatially with bright yellow as most densely populated region of such trajectories and black as the least populated.
For the simulations, the source is 700 nm wide with the injectors (point sources) uniformly spaced at an interval of 50 nm. The total number of electron trajectories N = 10,000. Fig 1(b) shows these trajectories after a perpendicular magnetic field B is introduced. Classically, when a magnetic field is applied perpendicular to the plane of two dimensional electron gas, the electrons travel in cyclotron orbits. The equation of motion is governed by the Lorentz force.
where, e is the elementary charge, v is the electron velocity and B the applied magnetic field. The radius of these cyclotron orbits is given by the following equation.
where 1 * is the dynamical mass of the electron. The conical band structure of graphene yields a linear dispersion relation 3 = ℏ, 5 6 at energies close to the Dirac point, where the speed , 5 ∼ 10 : 1 ; is fixed [8,9,10]. Unlike conventional semiconductors, the dynamical mass is density dependent 1 * = ℏ <= > ? , 5 , and the cyclotron radius graphene device, the first magnetic focusing peak in transmission between contacts occurs. The focusing field B F is given by where p is an integer.
The electrons feel a second force created by the tip, which is positioned at a fixed height ~70 nm above the graphene surface. The difference in work function between graphene and Si tip creates an image charge density profile in the graphene sheet.
where a is the radial distance from the tip location, h is the height of the tip from the graphene sample, and q is the charge on the tip. 6 the sample. Fig. 3(b) shows ray tracing simulations of the change in transmission ∆W caused by the tip vs. tip position at . = 0.14 W and = = 1.13 ×10 >? O1 P? . The simulations show features that are quite similar to the experimental results: an decrease in transmission along the cyclotron orbit that connects the two contacts, and enhanced transmission, when the tip bounces electrons away from the diffusely scattering edge of the sample.

Discussion
As shown in Fig. 4(a)  Maxwell's equations create an integrable divergence in the charge density of the graphene sheet at its edge. The graphene device is electrically gated by a conductive Si substrate below an insulating 280 nm thick oxide. The graphene sheet can be modelled as a semi-infinite parallel plate atop an infinite conducting plane. The electric field lines between these planes can be derived using conformal mapping technique [11]. At a distance dx near the edge of the graphene sheet at potential # e to ground, the electric field is: where a is the spacing between graphene and Si substrate. The electric field and the surface charge density are inversely proportional to the square root of the distance from the edge. Because the electron density n is diverging near the edge, the dynamic mass 1 * and the cyclotron radius r c increase correspondingly. In our simulations, we haven't taken this into account. It would be important to include these edge effects in future computations to get a more accurate view of the experimental data.

Conclusion
The ray-tracing method accurately describes images of cyclotron orbits in graphene taken by a cooled scanning probe microscope in the ballistic regime. Using classical equations of motion with two forces: 1) Lorentz force and 2) The force due the image charge density profile created by the tip, the simulations provide a good match to the SPM data. The force due to the image charge density profile created by the tip forces electrons away from the tip location.
The tip can image cyclotron orbits connecting two point contacts, by deflecting electrons away from the second contact. The tip can also enhance conductance between contacts by deflecting trajectories away from the rough edge into the second contact. The ray tracing technique could be further improved by adding the fringing electric field along the edges of the graphene device.