Introduction

Graphene, a two-dimensional (2D) material, is a useful material for electronic devices due to its excellent transport, mechanical, thermal, and optoelectronic properties1,2,3,4. The performance of devices, based on graphene field-effect transistors on Si/SiO2 was limited by scattering from impurities, doping effect, charged surface states, surface roughness, and surface phonons5,6,7,8. Recently, hexagonal boron nitride (BN) has been used as an ideal substrate for graphene devices because of its flat surface and crystal structure similar to graphene9,10,11. With the development of fabrication techniques for BN encapsulation and one-dimensional contacts, the carrier mobility of graphene was measured up to ~200,000 cm2 V−1 s−1 at room temperature12,13. This remarkable operation performance permits graphene to have a long carrier mean-free path (λ), which is required for a graphene-based ballistic transport device (GBTD).

GBTDs based on geometric asymmetry have recently attracted considerable attention. Unlike conventional diodes and transistors, GBTDs do not need a sizable band gap to operate, but its critical dimensions should be on the order of or less than λ14 and no scatterings are required except at graphene edges in order to achieve the ballistic behavior15,16. With promising features like high rectification efficiency, high responsivity, and low parasitic capacitance, GBTDs can be operated at GHz/THz frequencies17,18,19. These devices can be adopted for terahertz detection and imaging, optical rectennas, optical heterodyne detection, current rectifiers, and biomedical diagnostics20,21,22. Several GBTDs were fabricated and demonstrated the ballistic transport properties, including geometric diodes23,24,25,26,27, graphene ballistic rectifiers17,18,19,28, and geometric diodes for energy harvesting29,30. All reports show that the geometric features of graphene devices affect electron transport in the ballistic regime. However, the rectification efficiency and responsivity (0.22 − 0.24 A/W) are low24,25,26 due to unwanted doping and impurities on the graphene during fabrication processes. Many theories have been studied to predict the increased responsivity with a wide difference in carrier mobilities using extended Landauer-Büttiker formalism. The geometric dependence of graphene ballistic rectifiers has been studied in the nonlinear transport regime18. The formalism of nonlinear transport in mesoscopic conductors based on extended multiterminal Landauer-Büttiker formulas31,32 was presented to treat two-terminal devices at a finite applied voltage and temperature33. From Dirac-like equations, the geometrically induced rectification of graphene was found to be ambipolar29. The properties of the nanodevice are supposed to be very sensitive to the geometric shape of the nanostructure. So far, geometric designs for optimizing rectification of GBTDs are indistinct and still under debate. The geometry diodes need no potential barrier, implying that it operates with zero-threshold voltage for the energy harvesting application.

In this paper, we report the fabrication and electrical characterization of two-terminal GBTD fabricated from mechanically exfoliated graphene with different geometries operating at room temperature. These GBTDs can be considered as a geometric diode because of their asymmetric behavior depending on the direction of the current. Using a modified fabrication in nanoscale, the achieved results of GBTDs exhibit nonlinear electrical transports with high rectification efficiency. The DC current-voltage and a gate electrode were used to tune the I-V characteristics and charge carriers with the applied back-gate voltage (Vg). The ballistic transport characteristics of GBTDs are compared with numerical simulation based on the ballistic motion of particles. This study offers a pathway to optimize the other GBTDs and holds promise for potential application as graphene rectifiers and energy harvesting devices.

Results and discussions

Design and concept

GBTDs behave like a diode due to their asymmetric structure. Figure 1(a) shows a series of four designs of GBTDs, devices A, B, C, and D, with different geometries investigated in this study. All devices were designed to have the same tapered angle of around 30° in the forward direction and different angles in the backward. From the simulation result (Supplementary Fig. 8(d)), the device performance around this angle was estimated to be high enough, and the dimension of the device was suitable. These GBTDs were operated under an electric field given by the gate voltage (Vg), altering the carrier density in the channel. These devices were made from the monolayer graphene encapsulated by a few-layer hexagonal boron nitride (BN) on a silicon oxide/silicon substrate. The basic physical requirement for GBTD is that its minimum feature size must be of the order of or smaller than the mean free path (λ) of the charge carriers in graphene23,34,35, and graphene has a long λ (~1 µm) at room temperature1,19,36.

Fig. 1: Geometric designs and operation of ballistic transport devices.
figure 1

a Designs for ballistic devices with different geometries. b Configuration of device A shows dimensional parameters: length (L), channel width (dch), neck width (dneck), and tapered angle (30o). c The forward and backward directional motions of charge carriers in the ballistic regime. The ballistic region (blue region) is assumed to be inside a circle of radius λ, and the diffusive region (grey region) is outside the circle.

Full size image

Figure 1(b) shows the design of device A, where the critical dimension is the width of the neck (dneck), and the other parameters are length (L), channel width (dch), and tapered angle. In this model, the charge carrier moving toward the neck is considered ballistic, and the geometry and boundaries of the device have an impact on the transmission probability. The geometry of GBTDs is designed to allow charge carriers one-directional preference, acting as a funnel with a smooth flow and a restricted flow in opposite directions. The charges undergoing ballistic transport suffer collision and specular reflection at the edges of graphene as shown in Fig. 1(c). The charge carriers can move easily from the slowly tapered region (forward direction marked in green), while the carriers moving from the opposite direction may have difficulty in passing the abrupt constriction (backward direction marked in red). On the other hand, the effect of GBTDs depends on the majority carrier type (electrons or holes) of graphene controlled by Vg, as well as the geometric structure.

Performance ballistic diodes

Four GBTDs with graphene encapsulated by BN were fabricated on a Si chip as shown in Fig. 2(d). A detailed FESEM image of device A having dneck ≈ 200 nm exhibits the constricted region of the device in submicron scale as indicated in Fig. 2(e). The sharp edges of the BN-G-BN layer are clearly seen and are adequate for the device operating in the ballistic regime. Figure 2(f) and (g) show representative examples of transfer curves with resistance versus Vg to compare the devices prepared by modified and conventional methods (without edge defect treatment), respectively. The modified device exhibited minor hysteresis after 24 h in a vacuum (1 × 10−5 Torr). The Dirac point (DP) of the device is at around Vg = −1 V, which was slightly shifted toward n-type doping. On the other hand, the conventionally fabricated sample (Fig. 2(g)) showed a large hysteresis shifting DP into a heavily n-doped state due to impurity37,38. The carrier mobilities have been determined from the slope \(\frac{{dV_{sd}}}{{dV_g}}\) of the transfer characteristics curve of the graphene-based field-effect transistor at room temperature (Supplementary Fig. 4), giving values of 130,000 cm2 V−1 s−1 and 47,000 cm2 V−1 s−1 for electrons and holes, respectively. λ is estimated from the equation, \(\lambda = {\it{\hbar }}/2e\;\mu \sqrt {n_{th}/\pi }\)12,18, which gives λ ≈ 0.7 µm, where \({\it{\hbar }}\) is Planck’s constant, e is the electron charge, \(n_{th}\) is the threshold carriers density depends on Vg, \(\left| {n_{th}} \right| = \left| {\alpha \times \left( {V_{g,th} - V_{g,0}} \right)} \right|\)and α = 7.2 × 1010 cm−2 V−1 is the back gate lever arm39. These figures of merit of graphene quality ensure that the GBTDs can be operated in a ballistic transport regime. Of the 50 samples fabricated using conventional methods, only 5 samples showed ballistic behavior, whereas 12 samples were valid among 13 samples fabricated using the modified method.

Fig. 2: Fabrication process and characteristics of the sample.
figure 2

a Ballistic device with BN-Gr-BN layer was fabricated by RIE, the cross-section of which may include contaminations. b One-dimensional edge contact structure of the graphene encapsulated with BNs. c DC current-voltage measurement configuration is illustrated. The current was supplied with a series resistance of 1 MΩ to provide source-drain current (Isd). Vg was applied directly to the silicon substrate to control the carrier type and carrier concentration. d 100× optical microscopic image of a GBTD. Scale bar = 30 µm. e FESEM image of the device A shows a narrow neck width. Scale bar = 1 µm. f–g Transfer curves of resistance versus Vg of the GBTDs fabricated by the modified (f) and conventional method (g), respectively.

Full size image

Figure 3(a) and (b) show the I-V characteristic curves of device A with majority carrier at Vg = +2.5 V and −5 V, respectively. At Vg = +2.5 V with the n-type carrier, the I-V curve of the device is nonlinear, exhibiting that Vsd= +1.3 V was measured at Isd = +100 µA, while Vsd= −1.1 V was measured at Isd = −100 µA. At Vg = −5 V with p-type carriers, the nonlinear behavior is shown in the opposite direction where the higher absolute value of Vsd was measured at Isd = −100 µA. This asymmetric characteristic is evidence that the device was operated in the ballistic regime. However, the I-V curve was slightly bent when higher currents were applied for both negative and positive current biases, which can be explained by the heating effect in a nanodevice at room temperature40,41.

Fig. 3: Transport properties of device A at room temperature.
figure 3

a, b I-V characteristics at Vg = +2.5 V and −5 V, respectively. c The γ was plotted as a function of Vg at different Isd. d The γ versus Isd at Vg = +2.5 V, −5 V, and −2 V.

Full size image

To investigate the rectification ratio of the devices, Vg was tuned at the fixed Isd while carrier density was varied. As the direction of carrier motion in the devices depends on the charge carrier type, Vg was employed to change the majority charge carrier type. We define an asymmetric ratio (γ) as the ratio of the measured voltages (Vsd) between positive (+Isd) and negative currents (−Isd), which can be expressed as:

$$\gamma = \left| {\frac{{V\left( { + I_{sd}} \right)}}{{V\left( { - I_{sd}} \right)}}} \right|.$$
(1)

If γ > 1, the electron transport is dominant, and the hole transport is dominant in case of γ < 1 in Eq. (1). The γ was plotted as a function of Vg as shown in Fig. 3(c). Here, Vg at the DP (VD) is around −2 V, having zero conductivity as there is no carrier. γ is > 1 for Vg > VD and <1 for Vg < VD with the maximum value of γ = 1.22 at Vg = +2.5 V. γ has peaks around the Vg = +2.5 V and −5 V with charge carriers of electrons and holes, respectively, and vanishes as it moves away from DP. Notably, the electron transport has a higher γ than the hole transport due to higher carrier mobility with longer λ. On the other hand, resistance change was found to depend on the channel voltage (Supplementary Fig. 5), which is further evidence of the ballistic characteristic of the device as the incident angles of the carriers are varied by the strength of the electric force. To evaluate the change of γ with increasing current bias, the γ versus the source-drain current (Isd) with different Vg was plotted as shown in Fig. 3(d). The Isd was applied within the ±100 µA range for the device to be stably operated avoiding possible damage. At the Dirac point of Vg = −2 V, the γ ≈ 1 while the γ was deviated from 1 and almost monotonically increased with increasing current at Vg = +2.5 V and −5 V. This increment is not related to the self-heating effect because the heating effect cannot depend on the current direction. This can be explained by the fact that increasing current raises the directional uniformity of ballistic charges in graphene with a strengthened electric field. Furthermore, it can be interpreted that applying a high field renders reflections at the edges specular19. The localized edge states exist near disordered edges with dangling bond terminations, causing the Coulomb charging effect in the graphene channel42,43,44,45,46. When a low current is applied, the disordered edge causes diffuse reflection for ballistic charges. At a high current, the high electric field provides sufficient energy to the charge carriers to pass into the localized area. Initial carriers are trapped at the edges, and subsequent carriers will be reflected by the trapped charges in a specular way.

To investigate the geometric effect on the transport properties more evidently, device B with a modified design was prepared, and its I-V curves at Vg = +17 V and −2V are shown in Fig. 4(a). It has a long straight edge on one side, and the opposite side has an obtuse angle at the neck for the backward direction. With this special design at the neck region, the device exhibited non-linear I-V characteristic curves with a high γ. A common feature between the I-V curves for electron and hole carriers is noticeable, as it is linear in negative Vsd and strongly bent in positive Vsd. The increment of charge carriers renders specular reflection at the boundary due to the long-range Coulomb effect as mentioned in device A. The γ of the device was plotted as a function of Vg as shown in Fig. 4(b). It was found that near the Dirac point, γ has value ≈ 1, which increases with \(\pm \left| {V_g - V_D} \right|\). The highest value of γ = 1.58 was found at Isd = 60 µA, and \(V_g\) at peak γ was strongly shifted by Isd. This further clarifies the intensified specular reflection at the edges by high currents. The reason why γ of device B is higher than that of device A can be found from the wide structure in the backward input channel. As device B has a wider angle and width in backward input, the reflected carriers from the edges were spread, lowering the probability to pass the neck. It should be considered that the edge is not perfectly flat, causing partially diffusive reflections. Furthermore, the mean free path is not long enough to move far away from the neck once it is reflected.

Fig. 4: Transport properties of device B at room temperature.
figure 4

a I-V curves of device B at Vg= +17 V and −2 V. b The γ was measured as a function of the Vg at various currents Isd.

Full size image

To investigate the geometric dependence of the efficiency, we investigate other devices. Figure 5(a) displays the I-V characteristic curves of device C as a function of bias current with the fixed Vg = +2 V and −2 V. The I-V curves of device C are also non-linear depending on Vg, similarly to the devices A and B. As the current is increased, γ increases up to 1.32 as shown in Fig. 5(b).

Fig. 5: Transport properties of devices C and D at room temperature.
figure 5

a I-V curves of device C are plotted as a function of Isd at Vg = +2 V and −2 V. b γ of device C is shown as a function of Isd. c The I-V characteristic curves of device D at Vg = −2 V and +18 V. d γ of device D is shown as a function of Isd.

Full size image

Figure 5(c) shows the I-V characteristic curves of device D with majority charge carriers of electron and hole at Vg = +18 V and −2 V. Here, the higher γ than device C is achieved from the non-linear I-V characteristic curves. This is plausible considering that the backward resistance is increased due to the sharp angle at the neck in the right-hand region in the asymmetric structure. It is found that the forward direction of the device was markedly reversed by Vg. The device exhibits quite linear curves within Isd = ±20 µA range and became non-linear with Isd > ±20 µA for both electron and hole carriers. The current-dependent effect was analyzed from γ versus Isd curves measured at Vg = +18 V and −2 V as shown in Fig. 5(d). It is noted that the behavior of current bias has four distinct states (I, II, III, IV) for both electrons and holes. The γ is slowly increased for Isd < 20 µA (I), rapidly increased for 20 < Isd < 70 µA (II), saturated for 70 < Isd < 90 µA (III), and slowly decreased for 90 < Isd < 100 µA (IV). The highest γ = 1.66 is obtained at Isd = 80 µA, which is increased further by increasing the current bias, implying that γ is limited in the ballistic regime. As the power dissipation given by P = I.V in region IV is higher than those for other devices, a slight decrement of γ is attributed to the self-heating effect. In a high bias regime, the carriers are no longer ballistic, and hot electrons are induced and scattered by surface phonons in graphene47,48,49.

The highest values of γ along with geometry parameters for different types of ballistic devices are listed in Table 1. Due to the inaccuracy of the lithographic process, the dimension of each device varies slightly. Device B has higher γ than device A, and device D has higher γ than device C, though they have similar shapes. A common feature of B and D is a wider area in backward insertion, rendering the reflected charges spread easily. As device D has a typical shape for the ballistic rectifier, its efficiency can be compared with the previously published result. The highest value of γ = 1.66 of device D is much higher than that of the earlier report (γ = 1.31)24 and (γ = 1.4)26.

Table 1 The highest γ are listed with geometric parameters of different ballistic devices.
Full size table

Numerical simulation

Numerical simulation was performed, modifying the model suggested by Zhang and Hung to compare ballistic efficiencies of geometric devices qualitatively50,51. In Fig. 6(a), the ballistic region was considered as a semicircle with a radius λ, the center of which is the neck of the device. The charges in the green (yellow) region contribute to the forward (backward) transmission. The grey region is a dead region that may not contribute to the backward transmission. We assume that the charges can pass the neck either directly or after repeated specular reflections at the edges of the device as shown in Fig. 6(b). The allowed angles can be easily calculated using a mirroring process. The initial ballistic region (M0) is expanded into M1 and M2 by mirroring at the edge line to include the specular reflection cases. The allowed angle intervals, \(\theta _1^{min} \le \theta \; \le\) \(\theta _1^{max}\) and \(\theta _2^{min} \le \theta \le\) \(\theta _2^{max}\), from M0 can be obtained by considering straight lines (dotted) from the mirror points in M1 and M2. All possible charge carriers in the ABCD region are integrated, and a Gaussian distribution function with the standard deviation (σ) and the mean (\(\bar \alpha\)) is adopted to represent the ballistic angle distribution of the charges. The charges are assumed to be uniformly distributed in the area, and the integration intervals are depicted in Fig. 6(c–d).

Fig. 6: Simulation model for transport of charge carriers in the ballistic region of devices.
figure 6

a Ballistic regions are denoted as dotted circles with the radius λ. b A charge at point M0 can pass AB through processes: (1) direct passing with angles from \(\theta _0^{min}\) to \(\theta _0^{max}\) (blue arrow), (2) reflecting once from edge line (\(\overline {{{{\mathrm{CB}}}}}\)) with the angle (\(\theta _1^{min} \le \theta \le\) \(\theta _1^{max}\)) (orange arrow), or (3) reflecting twice from \(\overline {{{{\mathrm{AD}}}}}\) and \(\overline {{{{\mathrm{CB}}}}}\) with \(\theta _2^{min} \le \theta \le\) \(\theta _2^{max}\) (green arrow). c The considered areas for passing the neck in forward transmission: direct transport (no reflection: green region); 1 reflection (orange region); 2 reflection (blue region). d Schematic representation of possible transmission areas to be integrated.

Full size image

The transmission ratio (δ) was defined as the ratio of transmission probabilities for forward and reverse biases (see Supplementary Figs. 6 and7 for detail). Figure 7(a) displays δ for different values as a function of the dneck for the various devices depicted in Fig. 1(a) with σ = 0.55 rad, \(\bar \alpha\) = 0 (for device C and D), \(\bar \alpha\) = 7π/18 rad (for device A and B) and λ= 0.25 µm. The simulation results show that δ is increased monotonically with decreasing dneck. The model also indicates that δ depends strongly on λ, angular distribution, and geometry of the device. From Landauer-Büttiker formalism, δ can be approximated to γ, as it represents the asymmetry between the forward and backward directions51

Fig. 7: Simulation and experimental results of graphene ballistic transport devices.
figure 7

a δ of devices (A, B, C, D) were calculated as a function of dneck with σ= 0.55 rad, \(\bar \alpha\) = 7π/18 rad (for device A and B), \(\bar \alpha\) = 0 rad (for device C and D) and λ= 0.25 µm. b Comparing the δ of devices (A, B, C, D) with experimentally measured γ at the bias current of 40 µA.

Full size image

Figure 7(b) shows the experimental results of γ compared with the simulation results of δ. The experimental results were measured at 40 µA bias current as it was not affected by the heating effect. In principle, γ is expected to be equal to δ, and it can be seen that the overall trend of γ depending on the geometry fits reasonably with the analytical model. The experimentally measured γ is lower than the numerically estimated δ. Usually, theory predicts a larger rectification ratio as compared to the experiment. For example, El-Araby et al.30 predicted the asymmetry of ~10, and Berdiyorov et al. predicted ~5.27 Particularly, Stearns and Moddel predicted ~5 with geometry similar to device A,52 where the drift velocity was parallel to the direction of the applied voltage. However, the average orientation of the velocity in our simulation was set as the direction from the center of the allowed region to the neck because it was the overall drifting direction of the current. On the other hand, the experimental values for the current rectification do not exceed 2. Furthermore, the reflection from the graphene edge may be diffused partially due to non-uniformly etched edge lines with contaminations as mentioned in the fabrication section. These diffusion phenomena may reduce the rectification efficiency strongly as shown in the angular distribution effect (Supplementary Figs. 10, 11).

In this report, we have modified the fabrication process for encapsulated graphene to improve the quality of graphene, and corresponding GBTDs with different geometries were fabricated using electron-beam lithography. The devices exhibited ballistic behavior at room temperature. It was found that the GBTD devices with different geometries exhibit asymmetric I-V characteristics. It was shown that ballistic transport can be altered and enhanced by optimizing the geometries. The asymmetry ratio γ was found up to 1.58 (at Isd = 60 µA) depending on the back-gate voltage and bias current. A numerical simulation for the ballistic devices was performed based on the specular reflection of the carrier from graphene edges in the ballistic regime. The overall trend of measured γ from the I-V characteristics agreed with the numerical simulation results. This study provides a pathway for optimizing the ballistic graphene rectifiers to be utilized as potential applications for energy harvesting devices.

Methods

Fabrication method

GBTDs of different geometries were fabricated as shown in Fig. 1(a). The standard micromechanical exfoliation and dry-transfer method were used to prepare graphene53,54,55. To obtain graphene (Gr) layer, Kish graphite (Covalent Materials Corp.) was mechanically exfoliated on Si/SiO2 substrate with a 280 nm oxide layer using adhesive tape (Scotch Tape, 3 M Inc). Gr and BN flakes were identified by using an optical microscope. A thin top BN flake of thickness 10-15 nm was similarly exfoliated on a cover glass with spin-coated polyvinyl alcohol (PVA)/polymethyl methacrylate (PMMA), which was used to pick up Gr and the bottom BN layer. The resultant BN/Gr/BN layer was transferred onto a p-type silicon substrate with 300 nm thermal oxide used as a global back-gate (Vg) to tune the Fermi energy of Gr in the GBTD (Supplementary Fig. 1). The geometric designs were patterned by employing electron-beam lithography with a bilayer of PMMA 950 K (100 nm)/495 K (100 nm) and reactive ion etching (RIE) using a gas mixture of SF6 and Ar.

Some bubbles including residual gas or impurities appeared, probably degrading graphene quality as illustrated in Fig. 2(a). The encapsulated graphene was etched by RIE as designed, and the samples were kept under a high vacuum (1 × 10−5 Torr) for 24 h to remove the trapped molecules below the bubbles near the edges of the device. AFM images before and after the vacuum treatment show a clear change of morphologies of the graphene channel (Supplementary Fig. 3). The source and drain electrodes were patterned with the PMMA 495 K (100 nm)/950 K (100 nm) resist. For the ohmic contacts, Cr/Au (5 nm/60 nm) were deposited by an electron-beam evaporator on the devices12, and a lift-off process was performed in warm acetone. One-dimensional edge-contacts were formed on the active graphene layer with low contact resistances as illustrated in Fig. 2(b). The device morphology was measured using field-emission scanning electron microscopy (FESEM, SU8010 Hitachi, Japan) and atomic force microscope (AFM, Nano focus Inc.). Raman spectroscopy was carried out using a micro-Raman system (Renishaw, InVia systems) with a laser beam of 514 nm.

Measurements

The transport characteristics of two-terminal GBTDs were measured under a high vacuum (1 × \(10^{ - 5}\) Torr) condition to remove the absorbed molecules under ambient air. Figure 2(c) shows the measurement configuration for the electrical characteristics. The back gate voltage (Vg) was applied to control the carrier type and its concentration in the graphene channel by using a source-meter (Keithley, 2400-SMU). The bias current (Isd) was provided with a series resistor (1 MΩ) between the source and drain terminals to avoid accidental electrical shock. A Keithley 2000 multimeter was used for measuring voltages (Vsd) of devices. The graphene nanodevices suffered from hysteresis56,57 and self-heating effect at high bias58,59 at room temperature and under ambient conditions. To avoid a thermal electromotive force and the offset problem in I-V characteristic measurements, the source-drain currents were pulsed with a pulse width of 700 ms in the order of 0, +I1, −I1, +I2, −I2, … +Ii, −Ii, 0 by a computer control system.