DOI: 10.1039/C9CP04368C
(Paper)
Phys. Chem. Chem. Phys., 2019, Advance Article

F. Sánchez-Ochoa*^{a},
Jie Zhang^{b},
Yueyao Du^{b},
Zhiwei Huang^{b},
G. Canto^{c},
Michael Springborg^{d} and
Gregorio H. Cocoletzi^{e}
^{a}Universidad Nacional Autónoma de México, Instituto de Física, Apartado Postal 20-364, Cd. de México 01000, Mexico. E-mail: fsanchez@fisica.unam.mx; franciscosno88@gmail.com
^{b}Department of Environmental Science & Engineering, Huaqiao University, Xiamen 361021, People's Republic of China
^{c}Centro de Investigación en Corrosión, Universidad Autónoma de Campeche, Av. Héroe de Nacozari 480, 24079 Campeche, Campeche, Mexico
^{d}Physical and Theoretical Chemistry, University of Saarland, 66123 Saarbrücken, Germany
^{e}Benemérita Universidad Autónoma de Puebla, Instituto de Física, Apartado Postal J-48, Puebla 72570, Mexico

Received
6th August 2019
, Accepted 29th August 2019

First published on 13th September 2019

A systematic investigation is performed on the electronic transport properties of armchair-graphene nanoribbon (AGNR) heterojunctions using spin-polarized density functional theory calculations in combination with the non-equilibrium Green's function formalism. 9-AGNR and 5-AGNR structures are used to form a single-well configuration by sandwiching a 5-AGNR between two 9-AGNRs. At the same time, these 9-AGNRs are matched at the left and right to electrodes, 9 and 5 being the number of carbon dimers as width. This heterojunction mimics an electronic device with two potential barriers (9-AGNR) and one quantum well (5-AGNR) where quasi-bound states are confined. First, we study the ground state properties, and then we calculate the electron transport properties of this device as a function of the well width. We show the presence of electronic tunnelling resonances between the barriers by delocalized electron density inside the well's structure. This is corroborated by transmission curves, localized densities of states (LDOS), current-vs.-bias voltage results, and the trend of the resonances as a function of the well width. This work shows that carbon AGNRs may be used as resonant-tunnelling devices for applications in nanoelectronics.

Theory and experiments show that the electronic and magnetic properties of GNRs can be tuned by varying their geometry, width and chemical composition of the edge termination^{17} inducing 1D quantum confinement, resulting in semiconductors, metals or spin-polarized semiconductors (half-metals).^{18–21} On the one hand, GNRs with zigzag edge-termination (ZGNRs) have spin-polarized edge states with applications in spintronics.^{21} However, GNRs with armchair termination, hereafter AGNRs, have an electronic bandgap which changes as a function of the ribbon width.^{15,18,19} The AGNRs can be grouped into three different subfamilies, that is, N = 3p, N = 3p + 1 and N = 3p + 2, where p is an integer and N is the number of carbon atom pairs across the GNR.^{19} N = 3p and N = 3p + 1 families have wide bandgaps that scale inversely with the ribbon width. In contrast, simple models, such as nearest-neighbour tight binding, predict the family N = 3p + 2 to be metallic with a zero bandgap.^{19}

In the last decade, the design and study of more complex 1D nanostructures based on AGNRs, such as lateral heterojunctions, have attracted considerable attention.^{22–29} Computational and experimental efforts, mainly bottom-up chemical synthesis of GNRs from molecular precursors, have been made to explore quantum effects^{15,30–34} and demonstrate their usefulness.^{24–29} This is an interesting issue to address in order to develop new interconnection components with vanishing bandgaps and modern high-speed electronic devices as diodes (traditional p–n junctions)^{26} and transistors.^{35}

Here, we use the width of AGNRs as a critical factor for bandgap engineering to design 1D lateral-heterojunction devices and then to study quantum resonant-tunnelling. The ground states were obtained using total energy density functional theory (DFT) calculations; these results were combined with the non-equilibrium Green's function (NEGF) formalism to calculate the curves of current-vs.-bias voltage characteristics. Systems with two potential barriers and one quantum well as finite superlattices were studied. This work shows that quasi-bound/resonance states are present in these heterojunctions. These electronic resonances can be tuned as a function of the geometry of devices, in particular through the quantum well width. Through this systematic study, we demonstrate that AGNR heterojunctions may be used as resonant-tunnelling devices in nanoelectronics.

The content of the paper is organized as follows: first, we present the models used and computational details in the Methodology section. Then, we discuss the ground state properties, the transmission curves and density of states (DOS) for the scattering region, and current-vs.-bias curves for the first six devices as a function of the well's width in the Results and discussion section. Finally, the conclusions are drawn in the last section.

Once the ground state of the devices was determined with DFT, then the electronic transport properties of the optimized systems were calculated with the NEGF formalism as implemented in TranSIESTA.^{42} First, the calculation of the Hamiltonian for electrodes was done using a 1 × 1 × 1001 k-grid. Then employing the Hamiltonian of electrodes, the current (I) calculation was carried out through the scattering region sandwiched between two semi-infinite electrodes at a finite bias voltage (V_{b}) using the Landauer–Buttiker formula,^{43}

(1) |

T(e,V_{b}) = Tr[Γ_{L}(E,V_{b})G(E,V_{b})Γ_{R}(E,V_{b})G^{†}(E,V_{b})],
| (2) |

(3) |

System | C-atoms | H-atoms | E_{F} |
NR |
---|---|---|---|---|

9-AGNR | 22 | 3 | 0.0 | — |

HET-6 | 320 | 112 | 21.68 | 5 |

HET-5 | 310 | 108 | 21.65 | 4 |

HET-4 | 300 | 104 | 21.92 | 4 |

HET-3 | 290 | 100 | 21.65 | 3 |

HET-2 | 280 | 96 | 21.70 | 3 |

HET-1 | 270 | 92 | 21.35 | 2 |

5-AGNR | 10 | 4 | 22.02 | — |

The electronic properties of 5-AGNR and 9-AGNR unit cells reveal bandgaps in Fig. S1 of the ESI,† with a linear dispersion around the Fermi energy for the 5-AGNR system. Let us mention that the sizes of bandgaps in these materials are sensitive to the kind of theory employed.^{20} The 5-AGNR system can be considered as a metallic system due to the narrow bandgap, which is useful for leads in our device as well as a junction between two semiconducting structures forming two potential barriers. The calculated curve of transmission versus energy, only for spin-up states in Fig. S2 (ESI†), shows a small bandgap of 0.2 (0.87) eV for an ideal and periodic device constructed with 5-AGNR (9-AGNR) in both leads and the scattering region. Additionally, a transmission of 1 is observed from ∼±0.1 eV to ∼±1.6 eV in 5-AGNR and from ∼±0.4 eV to ∼±1.0 eV in 9-AGNR, with zero transmission around the Fermi level in Fig. S2 (ESI†).

Once we have determined the stability of 5- and 9-AGNR unit cells, and the band structure and transmission properties of ideal devices made of 5-AGNR and 9-AGNR units, we proceed with the calculation of the transmission versus energy for the first six heterojunctions in Table 1. The discussion will be focused on spin-up states, since the results for spin-down states are similar due to anti-ferromagnetic behaviour. In Fig. 2, we plot the square of transmission amplitude (on logarithmic scale) of six devices with two barriers, and one well, built with two 9-AGNR units. A sharp peak is manifested just at the zero eV (Fermi energy), and there are two broadened symmetric electronic resonances at ±0.65 eV corresponding to a single well system (one 5-AGNR unit), see Fig. 2(a). When the heterojunction has two units of 5-AGNR inside the well, the two resonances marked with black filled/empty circles in ±0.5 eV are closer to the Fermi energy. A pair of additional resonances located at ±1.4 eV, now red filled/empty circles, are generated as shown in Fig. 2(b). As the number of units inside the well goes from 3 to 6, the resonances shift towards the Fermi level where the sharp resonance at zero eV is always present. The heterostructure with six 5-AGNR units, see Fig. 2(f), shows five resonances in the valence and conduction bands by symmetry. This finding is clear evidence of quasi-bound states inside the quantum well formed by two potential barriers of 9-AGNR in Fig. 1. Thus, electronic resonances may be tuned by the well's width and, indeed, these resonances may split as a function of the number of potential wells and barriers in the heterojunctions.^{49} In Fig. S3 (ESI†), we show the calculated transmission curves for two heterojunctions with three barriers (2 wells) and four barriers (3 wells) with one unit of 5-AGNR inside the well. The energy splittings of electronic resonances from two and three peaks are clear at ∼±0.6 eV, which therefore means that the heterostructures under study are finite superlattices of AGNRs.^{49}

In order to get more insights into electronic transmission with the electronic properties such as the density of states (DOS) at zero eV (equilibrium conditions), we first calculate the DOS only for the scattering region in Fig. 1, using the Green's function formalism.^{42} Fig. 3 depicts the total DOS for the scattering region of six heterojunctions. The DOS calculated with Green's functions is more useful in this analysis, because quasi-bound/resonance states are differentiated from localized states by the size of peaks and by their position in energy in comparison with the total DOS calculation with the standard diagonalization method and Bloch states. With standard DFT calculations using the diagonalization method, the analysis of electronic states is quite complicated to determine the trend of quasi-bound/resonance and localized states. As shown in Fig. S4 (ESI†), peaks at −0.49 and 0.51 eV do not change in position and size for heterojunctions with four, five and six 5-AGNR units forming the well. Others peaks at −1.33 and 1.41 eV change in size but their energetic positions remain the same. These peaks can be ascribed to localized states. On the other hand in Fig. 3, resonances marked by coloured filled/empty circles can be grouped into four sets and identified by small peaks approaching the Fermi energy when the width of the well tends to increase. In some cases, resonances and localized states are mixed, such as those in filled/empty-black and red circles in Fig. 3(b), and filled/empty-blue circles in Fig. 3(e). The mixing of the first resonances with localized states around the Fermi energy is due to the same width of the potential well and barriers in terms of 5-AGNR and 9-AGNR units. The relationship between resonances in transmission curves, in Fig. 2, and those small peaks in the total DOS curves for every heterojunction is in excellent agreement, showing the validity of the use of AGNR heterojunctions as finite superlattices. Also, resonances and localized states in the valence regime are symmetric, in energy and size, like those in the conduction regime around the Fermi level. This implies that these semiconductor-lateral heterojunctions have a type-I or straddling band alignment.^{50}

In what follows the analysis will focus on the heterojunction of two barriers and one well with six 5-AGNR units. With the knowledge of localized and quasi-bound/resonances states from DOS and transmission plots in Fig. 2 and 3, we then calculate the local DOS (LDOS), using the standard DFT formalism, in order to show the distribution of unoccupied states around specific energies under equilibrium conditions (zero-bias voltage). For this purpose, we employ the energy values of resonances and localized states presented in Fig. 3 for the calculation of conduction states within an energy window of 0.04 eV. These are the first four resonances at 0, +0.23, +0.76 and +1.22 eV; and the localized states at +0.52 and +1.43 eV. At zero eV, the distribution of unoccupied states is through the electrodes and buffer atoms, see the upper panel in Fig. S5 (ESI†). In Fig. 4(a) and (c) the resonances at +0.23 and +0.76 eV show electronic states spreading inside the well with a symmetric distribution with quantum numbers n = 1 and n = 2 (n − 1 nodes), respectively. But the unoccupied states at +1.22 eV, see Fig. 4(d), are distributed inside the well with n = 3 (n − 1 nodes) and penetrate the barriers. This is because the heights of the potential barriers are finite, and the quantum confinement is weak owing to the narrow potentials. For each resonance, we can clearly note how the quantum confinement modifies the distribution of unoccupied states inside the 1D quantum well.

Red dashed rectangles show the quantized states with n = 1–3 of the sixth device. To demonstrate the recurrence of quantum confinement, we calculate the transmission and DOS of the scattering region for devices with seven to sixteen 5-AGNR units forming the quantum well. We plot the energy position of resonances as a function of the width of the quantum well, see Fig. S6 (ESI†). Thus for the system with fourteen 5-AGNR units, we investigated the spatial distribution of unoccupied states in LDOS of the first four resonances and plotted them in the bottom panel of Fig. S5 (ESI†). We only show the quantized states with n = 1–4, notwithstanding that this system has seven resonances within 0.1–1.6 eV. On the other hand, as depicted by Fig. 4(b) and (e), the localized states at +0.52 and +1.43 eV are highly confined on the 9-AGNR segments, respectively, and further are symmetric. In this context, both electronic quasi-bound and localized states in LDOS are due to C[2p_{z}] orbitals, and the particularity of resonances in transmission studies is due to the edge states in 5-AGNR.^{9,18,22} Note that the states at the center of C-dimers are absent along the transport direction inside the quantum well. Furthermore, we calculated the electrostatic potential of this system along the transport direction with DFT. Fig. S7 (ESI†) depicts the whole device and the planar (top panel) and averaged (bottom panel) electrostatic potentials. It should be noted that potential wells are formed inside 9-AGNRs sandwiched by 5-AGNR systems with an average of 0.96 eV and 0.27 eV, respectively, for the planar average and nanosmoothed one.^{51} The value of 0.27 eV is in agreement with the energy difference, of 0.33 eV, between the minima of conduction bands of 9-AGNR and 5-AGNR in Fig. S1 and S2 (ESI†). The electrostatic potential was calculated for occupied states. A symmetric potential is generated for unoccupied states.

Electronic transport property calculations of the first six heterojunctions are now described. In particular, we have calculated the current-vs.-bias voltage characteristics from 0 to 2.5 V using a symmetrical bias (±V_{b}/2) applied on the leads. Fig. 5 shows the current results with a similar trend to that found in Fig. 2 and 3 corresponding to resonances. The first small peaks at low bias, marked with empty-black circles, approach the zero-bias voltages as the width of the quantum well increases from one to six 5-AGNR units, in Fig. 2(a)–(f). The bias values where the first maximum current occurs are approximately two times the values of energy in Fig. 2, omitting the universal constant, e, of electric charge.^{49} To enhance the effect of electronic resonances in electronic transport measurements, we have calculated the differential conductance (dI/dV), displayed in Fig. S8 (ESI†), to point out the shift of resonances in terms of the quantum well width. Our findings are in excellent agreement with the behaviour of finite superlattices reported in the literature, with slabs of GaAs/AlGaAs heterostructures,^{49,52} or in Si/SiGe double-barrier diodes,^{53} to display a resonant tunnelling effect. Therefore, the heterojunctions made of AGNRs may be considered as finite superlattices with potential applications in nanoelectronics. Now we discuss an interesting non-linear effect in transport properties, i.e., the negative differential conductance (NDC), which is based on regions of negative current with positive voltage. 1D systems based on nanoribbons,^{54,55} metallic chains^{56,57} and nanotubes^{58,59} present the NDC effect due to the suppression or attenuation of conductive channels within a bias window. Comparing those 1D systems with the models considered here, we may mention that the resonant tunnelling is induced by the presence of quasi-bound states inside the quantum well. However, even when the well's width is three times larger than the barriers, for a heterojunction with six 5-AGNR units in Fig. 5(f), no decrease in electric current between the first (∼0.45 V) and second (∼1.5 V) resonant peaks is manifested. Note that Fig. S8(f) (ESI†) shows a dI/dV < 0, which is characteristic of NDC, and after that the dI/dV is almost constant with a vanishing value, up to the second resonance where the dI/dV is positive exhibiting a peak (red-point). The absence of NDC in these graphene heterojunctions is due to a saturation effect. To explain the saturation effect, in Fig. 6, we plot the transmission and DOS (only for scattering region) as functions of the bias voltage (applied on two terminals) and energy for the sixth device. Only positive energies are plotted by symmetry. Also, as a reference in Fig. 6, we indicate the bias window, +V_{b}/2, with a purple (orange) dashed line for the transmission (DOS) curve. In Fig. 6(a), the bias +V_{b}/2 marked by the purple dashed line intersects electronic resonances at ∼0.25 and ∼0.75 eV in good agreement with the results in Fig. 2, but additional resonances are absent between 0.25 and 0.75 eV, and from 0 to ∼1.5 V.

Note that the resonance at 0 eV splits into two peaks and follows the bias +V_{b}/2 in Fig. 6(a). On the other hand, looking at the calculated DOS for the scattering region in Fig. 6(b), the localized states at 0.5 and 1.4 eV also split into two branches as the bias voltage increases. In particular, the localized state at 0.5 eV mixes with the resonance at ∼0.25 eV starting from 0.5 V due to the perturbation of both resonance and localized states, and after that, the localized state follows the bias +V_{b}/2 window. Thus, the localized states saturate the electric current to a constant value, and dI/dV is zero around 1 V, see Fig. S8(f) (ESI†). When the applied bias on both electrodes is 1.5 V, the electrons can tunnel once again through the structure. This result can be appreciated by the small peak in the current-vs.-voltage curve of Fig. 5(f). After that the current saturates once again to a constant value. Note that the quantum well width is quite small and as a result the electronic resonances face a larger separation, see Fig. S6 and S8 (ESI†).

Finally, we study a graphene heterojunction with the quantum well formed by fourteen 5-AGNR units. With this structure, we have found the second and third resonances below the first localized state, see Fig. S6 (ESI†). The length of this device is 15.68 nm formed by buffer atoms, leads, and the scattering region. Transmission and DOS (only for the scattering region) at zero bias are plotted in Fig. 7(a) and (b), respectively. In addition to the electronic resonance at 0 eV in Fig. 7(a), seven resonances can be identified in the range of 0.1–1.6 eV. Note that these resonances and the two well defined localized states are at 0.5 and 1.4 eV, as previously. The sixth resonance is mixed with the second localized state; meanwhile, the seventh resonance is mixed with other conduction states. The width ratio between the well and barrier equals 7 is enough to obtain two consecutive resonances at 0.1 and ∼0.36 eV which are below the energy of the first localized state at 0.5 eV, see Fig. 7(b) and Fig. S6 (ESI†). The next step was to calculate the current-vs.-voltage properties. In Fig. 7(c), the electric current shows a linear (ohmic) behaviour for low bias until 0.22 V. The current has a non-linear feature when the bias is greater than 0.22 V. To be precise, the three peaks and valleys are manifestations due to NDC. The dI/dV shows three transitions from positive to negative values in Fig. S9 (ESI†). These are fingerprints of NDC in the device.^{49,52}

From Fig. 7(c), we can calculate the peak-to-valley current ratios (PVRs), which are crucial parameters for electronic applications of tunnel diodes. In this case, the PVR is calculated by the ratios of peak point – determined by peak voltage (PV) and peak current (PI), and valley point – determined by valley voltage (VV) and valley current (VI), leading to a dynamic resistance. In particular, the devices we are dealing with may be considered as resonant-tunnelling diodes because the quantum tunnelling is maximized when the applied bias is two times the energy of electrons in eV to tunnel. The conductance is negative for positive bias until a VV where the current increases again. As an example, in Fig. 7(c), the device with a quantum well formed by fourteen 5-AGNR units, between two potential barriers built with two 9-AGNRs, shows three PVRs. The PVR(1) is approximately 2, while the PVR(2) and PVR(3) approaches 1. It should be mentioned that because the PVR(1) ≃ 2, the heterojunctions studied here may not be suggested to form any logic device.^{35,60} However, changes in the geometrical configuration of potential barriers can induce higher PVRs such as wider barriers, or asymmetric barriers with different heights but with the same width.^{61} These topics will be addressed in future research. Furthermore, let us mention that similar 2D systems with different compositions could be used to achieve higher PVRs, such as TMDCs^{62} or allotropic forms of phosphorus.^{63,64}

To end this section, we describe the NDC. To do this, we have calculated the transmission and DOS for the scattering region as depicted in Fig. 7(d) and (e), respectively. The baseline of bias, +V_{b}/2, is marked with dashed lines as a reference. In Fig. 7(d), seven resonances are indicated by vertically shaded segments in yellow, orange and red from 0 → 1.6 eV. The baseline of bias, +V_{b}/2, is superimposed onto the resonance at the Fermi level as in the previous device with six 5-AGNR units, see Fig. 6(a). It crosses four resonances. This is corroborated by the results in Fig. 7(c), where the fourth resonance is located near +1.5 V. On the other hand, as can be seen in Fig. 7(e), the dashed line of +V_{b}/2 intersects the first resonance in ∼0.1 eV giving a maximum in current at ∼0.22 V, see Fig. 7(c). After that, the dashed line does not intersect resonances or localized states until 0.4 eV where the second resonance and the first localized state are mixed. At this energy the current increases by 0.23 μA at 0.75 V. Between the first [PV(1) = 0.225 V] and second [PV(2) = 0.75 V] maxima of current, the current undergoes an attenuation up to VI(1) = 0.06 μA. The current attenuation is due to the absence of states to be occupied within 0.10–0.35 eV, as displayed in Fig. 7(e). A similar behaviour happens with the second and third resonances. The localized states do not mix themselves for all the bias. However, they mix with their neighbouring resonances. The aforementioned nonmixing behavior is owing to the steric separation, between potential barriers in this device with fourteen 5-AGNR units, which is greater than that in six 5-AGNR units. Therefore, the geometry and size of these AGNR heterojunctions are important for applications in nanoelectronics and optoelectronics devices.^{65–67} Additionally, considering defects could be an attractive complementary route to characterize these nanodevices.

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c9cp04368c |

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