Electrical Transport in High Quality Graphene pnp Junctions

We fabricate and investigate high quality graphene devices with contactless, suspended top gates, and demonstrate formation of graphene pnp junctions with tunable polarity and doping levels. The device resistance displays distinct oscillations in the npn regime, arising from the Fabry-Perot interference of holes between the two pn interfaces. At high magnetic fields, we observe well-defined quantum Hall plateaus, which can be satisfactorily fit to theoretical calculations based on the aspect ratio of the device.


I. Introduction
platform to observe Klein tunneling, in which the transmission coefficient of charges across high potential barriers strongly depends on the incident angle.
The most common failure mechanism of a suspended bridge is its collapse under sufficiently high voltages. Our previously fabricated device typically failed at critical voltages of ~40V-60V, due to the poor contacts between the vertical sidewalls and the horizontal bridge and electrodes. To overcome this deficiency, we perform an additional evaporation at 0º. Fig. 2a and  2b show two sets of suspended air bridges fabricated with two and three angle evaporations, respectively. Compared to Fig. 2a, the "joints" between different segments (outlined by dotted circles) in Fig. 2b are visibly strengthened.
To verify the mechanical robustness of these structures, we perform in situ SEM imaging while applying voltages to the top gate. As shown by the images in Fig. 2c-e, the air bridge remains suspended and undeformed under voltages of 70V and 100V, and ultimately fails at 110V. This surprisingly high critical voltage demonstrates significant improvement over our previous top gate structures.

Conductance of a pnp junction at B=0
Transport measurements on the graphene devices are performed at 260mK in a He 3 fridge using standard lock-in techniques. By varying voltages applied to the back gate (that controls the charge density and type in the entire device) and to the top gate (that controls charges directly under it), a graphene pnp junction can be created. A typical data set is shown in Figure 3a, plotting the 4-terminal device resistance R (color) as functions of V bg (vertical axis) and top gate voltage V tg (horizontal axis). The source-drain separation L of the device is 3.5 µm, and width W is 1 µm. The top gate is suspended d~100 nm above the center segment of graphene, with a length L tg ≈0.5 µm. In the non-top-gated or "bare" regions of the device, the charge density n 1 are modulated by V bg only, where e is the electron's charge, C bg is the capacitance per unit area between graphene and the back gate, and V D,bg is the Dirac point of the bare region of the device, which may be non-zero due to doping by contaminants. From Fig. 3a, V D,bg ~ 3.5V, at which device resistance reaches a maximum; since these regions account for 85% of the device area, their response dominate the device resistance, yielding the horizontal green-red band.
For the top-gated or "covered" region, the charge density n 2 are modulated by both V bg and V tg .  Fig. 3a, i.e. when the junction is in the npn regime. Compared with the neighboring unipolar (pp'p or nn'n) regions, the junction resistance is significantly higher, as expected at the boundary of a pn junction. More interestingly, we observe resistance oscillations as a function of both V bg and V tg , as indicated by the arrows in Fig. 3a. Notably, these oscillations are not found in the unipolar regions. Such oscillations were first reported by ref. 13, and arise from Fabry-Perot interference of the charges between the two p-n interfaces. Thus, the holes in the top-gated region are multiply reflected between the two interfaces, interfering to give rise to standing waves, similar to those observed in carbon nanotubes 24 or standard graphene devices 25 . Modulations in n 2 change the Fermi wavelength of the charge carriers, hence altering the interference patterns and giving rise to the resistance oscillations.
To analyze these oscillations in detail, we replot the data in Fig. 3a in terms of n 1 and n 2. Assuming a parallel plate geometry between the gate and the device, C bg /e ≈7.19 x 10 10 cm -2 . However, from quantum Hall measurements, we estimate the effective capacitance to be ~ C bg /e ≈6.51 x 10 10 cm -2 (see discussion in the next section). This small discrepancy may be attributed to a slightly thicker SiO 2 layer, slightly smaller ε bg , or additional screening by the electrodes. Using this value, we have (2b) The new plot is shown in Fig. 3b. The color scale is adjusted to accentuate the resistance oscillations, which appear as fringes fanning out from the Dirac point at n 1 = n 2 =0. Fig. 3c shows the device resistance vs. n 2 at n 1 =1.3 x 10 12 cm -2 , displaying clear oscillations.
Within the Fabry-Perot model, the resistance peaks correspond to minima in the overall transmission coefficient; the peak separation can be approximated by the condition k F (2L)=2π, i.e. a charge accumulates a phase shift of 2π after completing a roundtrip 2L c in the cavity. Here k F is the Fermi wave vector of the charges, and L c is the length of the Fabry Perot cavity. Under the top gate, ! k F 2 = "n 2 , so the spacing between successive peaks is estimated to be In Fig. 3d, we plot the measured peak spacing for the curve shown in Fig. 3c against ! n 2 . The data points fall approximately on a straight line. The best linear fit yields a line with a slope 0.95 x10 5 /cm, corresponding to L c =740 nm from Eq. (3). This agrees with the value estimated from electrostatics, L c =L tg +2d, as the electric field induced by the top gate on the device is expected to extend by a distance ~d away from either edge.
Finally, we note that the device in ref. 13 had extremely narrow gates L tg <~20 nm. In comparison, our top gate spans a much larger distance, L tg~5 00nm. Thus, the observation of clear Fabry-Perot interference patterns underscores the high quality of our pnp graphene devices.

Conductance of a pnp junction at B=8T
In high magnetic fields, the cyclotron orbits of charges coalesce to form Landau levels (LLs). For graphene with uniform charge densities, the energies of the LLs are given by For a graphene device with dual gates, the situations are complicated by the presence of regions with different filling factors, or the co-existence of n-and p-doped regions that result in counter-propagating edge states. The two-terminal conductance of the junction depends on the relative values of n 1 and n 2 , and can have fractional values of e 2 /h. A simple model is provided in ref. 26, assuming perfect edge state equilibration at the interfaces between different regions: for a unipolar junction (n 1 n 2 >0) with |n 1 | ≥|n 2 |, the non-top-gated regions act as reflectionless contacts to the center region, yielding a device conductance G=e 2 /h|ν 2 | (4) where ν 1 and ν 2 are the filling factor in the bare and top-gated regions, respectively, given by , where B is the applied magnetic field. If instead |n 2 |>| n 1 |, the conductance is For a bipolar junction (n 1 n 2 <0), the device behaves simply as three resistors in series, From Eqs. (4-6), the conductance of a graphene pnp junctions display plateaus at fractional multiples of e 2 /h. We emphasize that these fractional-valued plateaus are not related to the fractional quantum Hall effect; rather, they arise from the inhomogeneous filling factors within the device.
To observe these plateaus, we measure the two-terminal conductance G of the device as functions of V bg and V tg at B=8T. The data are shown in Fig. 4a, plotting G (color) vs. n 1 and n 2 , which are calculated from gate voltages using Eqs. (1) and (2). The conversion factor C bg /e is obtained by noting that, measured from the global Dirac point (n 1 =n 2 =0), center of the first finite density plateau should occur at ν=2, or n=3.9x10 11 cm -2 at 8T; the corresponding gate voltage difference is 6 V, yielding an effective C bg /e~6.51x10 10 cm -2 V -1 . This value is ~9% lower than that estimated from a simple parallel plate capacitor model, and is used for all plots in Fig. 3 and  4.
The data in Fig. 4a exhibit regular rectangular patterns, which arise from the filling of additional LLs as n 1 and n 2 are modulated. A line trace of G(n 2 ) at ν 1 =2 is shown in Fig. 4b, with equivalent values of ν 2 labeled on the top axis. As ν 2 varies from -2, 2 to 6, conductance plateaus with values of ~0.67, 2 and 1.2 e 2 /h are observed, in excellent agreement with those obtained using Eqs. (6)(7)(8). The solid line in Fig. 4c plots G(ν) for uniform charge densities over the entire graphene sheet, i.e., along the diagonal dotted line n 2 =n 1 in Fig. 4a. The ν =2 plateau is welldeveloped, indicating relatively small amount of disorder.
We now focus on the small conductance dips in Fig. 4c at ν~3 and ν~7, which are not expected to be present for a square device with L=W. Indeed, the two-terminal conductance of a conducting square includes both longitudinal and Hall conductivity signals, ! G = " xx 2 + " xy G(ν) appears as stepwise plateaus that increases monotonously for ν>0. However, for other device geometries the behaviors are more complicated. Depending on the aspect ratio of the device, the device conductance displays local conductance peaks or dips between the plateaus; if the device has significant Landau level broadening, the conductance will no longer be quantized at integer values of 2, 6, 10… e 2 /h. This was studied in detail in ref. 20, using an effective medium approach that yields a semicircle relation between σ xx and σ xy . To quantitatively examine the agreement between the data and the theory, we model the longitudinal conductivity as a Gaussian centered at a LL, are the incompressible densities corresponding to the Nth LL, and Γ describes the width of the Gaussian distribution. Following the procedures outlined in ref. 20, and using a fitting parameter Γ=0.67, we calculate G(ν) for our rectangular device with aspect ratio L/W=3.5.
The resultant curve is shown as the dotted line in Fig. 4c. The agreement with data is satisfactory at smaller values of ν, but deviates for ν>6. This is quite reasonable, since the energetic difference between Landau levels decrease for higher levels. Moreover, the value of Γ=0.67 obtained from the fitting is relatively small, again underscoring the high junction quality. Finally, we note that the model leading to Eq.s (4)- (6) is based on the single particle picture and assumes perfect edge state equilibration at the interfaces between different regions. The excellent agreement between our experimental results and model validates this assumption. On the other hand, if the electrical transport is fully coherent, one expects to observe universal conductance fluctuations (UCF) instead of well-defined plateaus. The exact origin of such modemixing mechanism and suppression of UCF is not clear, but is likely related to the presence of disorder, and/or dephasing due to coupling of the edge states to localized states in the bulk. Thus, an interesting future direction to explore is the mode-mixing mechanism (or the lack thereof) by studying ultra-clean pnp junctions.

Conclusion
Using suspended top gates, we are able to fabricate high quality npn junctions, which display Fabry-Perot resistance oscillations within a small cavity formed by the pn interfaces. In high magnetic fields, well-developed quantum Hall plateaus are observed, and the behavior can be quantitatively described by theoretical predictions for rectangular device geometry. In the long term, this technique may be important for study of transport of Cooper pairs [27][28][29] or spins 30, 31 through pn junctions, or the experimental realization of on-chip electronic lenses 19 , which require extremely clean graphene devices.