Imaging the Flow of Holes from a Collimating Contact in Graphene

A beam of holes formed in graphene by a collimating contact is imaged using a liquid-He cooled scanning probe microscope (SPM). The mean free path of holes is greater than the device dimensions. A zigzag shaped pattern on both sides of the collimating contact absorb holes that enter at large angles. The image charge beneath the SPM tip defects holes, and the pattern of flow is imaged by displaying the change in conductance between contacts on opposite sides, as the tip is raster scanned across the sample. Collimation is confirmed by bending hole trajectories away from the receiving contact with an applied magnetic field. The SPM images agree well with ray-tracing simulations.


Physics Subject Headings in the Discipline of Condensed Matter & Materials Physics: Physical-
systems / Two-dimensional-systems / Graphene; Transport phenomena / Ballistic transport; Techniques / Scanning-techniques / Scanning-probe-microscopy.
Imaging the flow of electrons, or holes, provides a direct way to understand ballistic flow in graphene. A cooled scanning probe microscope (SPM) can image the flow of electrons through a two-dimensional electron gas (2DEG) by displaying the change in conductance between two pointcontacts as the tip is raster scanned above the sample [13,14]. The tip creates an image charge in the 2DEG that deflects charge carriers and removes them from the pattern of flow. Displaying the conductance change vs. tip position provides an image of the carrier flow. Scanned probe imaging has proven to be a useful approach for understanding electron motion [15][16][17][18]. In previous work on graphene, we imaged the pattern of electron flow from a collimating contact [10] and the cyclotron orbits of electrons in the magnetic focusing regime [19,20].
In the present study, we use a cooled SPM to image the flow of holes in graphene from a collimating contact. Because the band structure of graphene is symmetric about the Dirac point for 3 small densities, the hole mobility is expected to be similar to the electron mobility [21,22]. The pattern of flow between two contacts on opposite sides of the sample is imaged by displaying the change in conductance as the SPM tip is raster scanned across the graphene. Confirmation that the collimating contact forms a hole beam is provided by applying a perpendicular magnetic field B that bends hole trajectories away from the receiving contact. Creating complimentary electron [10] and hole beams opens new approaches to ballistic graphene devices.
A scanning electron micrograph of the collimating contact device is shown in Fig. 1(a). An exfoliated monolayer graphene sheet was encapsulated between two layers of mechanically cleaved hexagonal boron nitride (hBN) to enhance the carrier mobility. The device sits on a heavily doped Si substrate topped by a 285 nm thick layer of SiO2, which acts as a back gate. Using a dry transfer technique, the bottom hBN, graphene, and top hBN flakes were stacked onto the substrate.
Using reactive-ion etching, the device was shaped into a Hall bar that has two wide end contacts 4 and two collimating contacts on either side. Each collimating contact consists of a narrow contact that emits electrons or holes into the graphene and two zigzag contacts, one on either side, that absorb carriers entering at large angles. Collimation can be turned off by floating the zigzag side contacts. For electrons, the half angle of the collimated beam is 9°, measured using our cooled SPM [10]. The rectangular graphene channel has dimensions 1.6 x 5.0 µm 2 and the collimating contacts on either side are separated by 1.6 µm. To make high quality contacts, chromium and gold layers were evaporated onto the freshly etched graphene edge immediately after etching [23].
The device was mounted inside our SPM and cooled to the temperature 4.2 K. For this geometry, the back-gate capacitance is CG = 30 fF. The hole density is p = CG(VD -VG)/e, where VG is the back-gate voltage, " is the back-gate voltage that puts the Fermi level at the Dirac point, and e is the fundamental charge. The transmission T of holes between collimating contact 1 and contact 3 was measured by passing a current between contact 1 and contact 2 and measuring the voltage difference Vs between contact 3 and contact 4. The accumulation of holes at contact 3 raises the potential of contact 3. The transmission Tm of holes from contact 1 to contact 3 is proportional to the measured transresistance Rm = (Vs/Ii).
A cooled scanning probe microscope (SPM) is used to image the motion of holes through the graphene sample. The technique is adapted from previous imaging experiments for twodimensional electron gases in graphene and GaAs/AlGaAs heterostructures [10,[13][14]16,[19][20].
Holes are emitted from the top collimating hole contact (contact 1) and travel along ballistic trajectories to the bottom contact 3, in Fig. 1(a). The zigzag sides on contact 3 are floated to turn off collimation, so holes can enter over a wide range of angles. To image the ballistic transmission of holes from contact 1 to contact 3, the silicon SPM tip is held above the encapsulated graphene creating an image charge below. As shown in Fig. 1(b), the local increase in hole density beneath 5 the tip acts as a lens that focuses and deflects ballistic hole trajectories. An image of ballistic hole flow can be obtained by displaying the change DRm as the SPM tip is raster scanned above the sample, as shown in the data below.
Working in the ballistic limit, we use ray tracing to simulate hole trajectories in graphene under the influence of a magnetic field B. The image charge forms a peak in hole density Dptip in the graphene sheet directly below the SPM tip: where the tip is modeled as a point charge q at height d = 70 nm above the graphene sheet, a is the radial distance in the sheet away from the tip position, e is the fundamental charge, and e is the dielectric constant of hBN.
We obtain the force that the image charge exerts on a ballistic hole traveling nearby by balancing the flow away from the tip caused by the peak in the Fermi energy EF, with the flow toward the tip caused by the dip in potential energy U that attracts holes to the tip. The total chemical potential EF(r) + U(r) of the hole gas is constant, where r is the position. The tip-modified Fermi energy is: where h is Planck's constant, and pi is the hole density without the tip present. Because the chemical potential is constant, the force on a hole from the tip is -∇U(r) = ∇EF(r). Using the dynamical mass of holes in graphene m * = h(p/p) 1/2 /2vF, where vF is the speed associated with the conical energy bands, we find the equation of motion: The SPM tip creates a force that pulls holes beneath the tip. Under a magnetic field, the Lorentz force F = ev × B acts on a hole with group velocity v. here DRm is the change in the measured transresistance, and DT is the simulated change in transmission between contacts 1 and 3. As B is increased, the hole paths curve away from contact 3, causing the signal to disappear. In blue regions, the tip focuses holes into contact 3, but in red regions, the tip deflects holes away from contact 3.
8 reducing the transmission. Simulations of the change in transmission DT displayed in Fig. 2(b), agree well with the experimental images.
We studied the degree of collimation by applying a perpendicular magnetic field B. The applied magnetic field bends the hole trajectories along cyclotron orbits that eventually curve away from the receiving contact and result in a reduction of flow. If a collimated hole beam with a small spread angle is emitted, the received signal DRm will fall away rapidly as B is increased, but if holes enter over a wide angle, the applied magnetic field will have less effect. These images are in good agreement with the simulations shown in Fig. 3(b). The blue regions in the center of the images in Figs. 2 and 3 show that the image charge beneath the tip acts to focus holes into the receiving contact, as illustrated in Fig. 1(b). The imaging signal is strongest at lower hole densities, where the image charge induced by the tip is a greater fraction of the original hole density.
As the magnetic field B is increased in Fig. 3(a), the hole paths bend away from the bottom contact, and the strength of the imaging signal decreases until it disappears. The curvature results from the Lorentz force, and the counterclockwise bend affirms that the carriers are positively charged, i.e. they are holes. The signal eventually disappears at B = 0.15 T, when the Lorentz force is strong enough to bend the hole beam entirely away from contact 3. The curvature of the hole paths in Fig. 3(a) is greatest at lower hole densities p, in agreement with the expression for the cyclotron orbit dc = h(p/p) 1/2 /eB.
To summarize, images of carrier flow taken by our cooled SPM show that a beam of holes is emitted into graphene by the collimating contact shown in Fig. 1(a). In addition, we find that the 9 image charge beneath the SPM tip can act as a focusing lens for holes. These results compliment our previous demonstration of a collimating contact for electrons in graphene [10], where the tip potential deflects electrons and defocuses the electron beam. The SPM images are in good agreement with ray-tracing simulations for experimentally relevant carrier densities and magnetic fields. The ability to make complimentary beams of electrons and holes paves the way for novel approaches to ballistic devices based on massless Dirac fermions in graphene.