Charge transport through the multiple end zigzag edge states of armchair graphene nanoribbons and heterojunctions

Abstract

This comprehensive study investigates charge transport through the multiple end zigzag edge states of finite-size armchair graphene nanoribbons/boron nitride nanoribbons (n-AGNR/w-BNNR) junctions under a longitudinal electric field, where n and w denote the widths of the AGNRs and the BNNRs, respectively. In 13-atom wide AGNR segments, the edge states exhibit a blue Stark shift in response to the electric field, with only the long decay length zigzag edge states showing significant interaction with the red Stark shift subband states. Charge tunneling through such edge states assisted by the subband states is elucidated in the spectra of the transmission coefficient. In the 13-AGNR/6-BNNR heterojunction, notable influences on the energy levels of the end zigzag edge states of 13-AGNRs induced by BNNR segments are observed. We demonstrate the modulation of these energy levels in resonant tunneling situations, as depicted by bias-dependent transmission coefficient spectra. Intriguing nonthermal broadening of tunneling current shows a significant peak-to-valley ratio. Our findings highlight the promising potential of n-AGNR/w-BNNR heterojunctions with long decay length edge states in the realm of GNR-based single electron transistors at room temperature.

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1 Introduction

Graphene nanoribbons (GNRs) have been the subject of extensive study since the groundbreaking discovery of two-dimensional graphene in 2004 by Novoselov and Geim.¹ Known for their semiconducting phases resulting from quantum confinement, GNRs hold significant promise for next-generation electronics.^2–4 Among these, armchair GNRs (AGNRs) are particularly noteworthy due to their tunable band gaps, which are inversely proportional to their widths.^5–12 Recent research has focused on understanding the electronic properties of AGNRs under transverse electric fields, revealing transitions from semiconducting to semimetallic phases.^13–16 These transitions could be crucial for controlling plasmon propagation.

The terminal zigzag edge structures of AGNRs play a critical role in field-effect transistors (FETs), directly interfacing with the source and drain electrodes.^8,9 These structures harbor topological states (TSs), the response of which to electric fields remains poorly understood.^17,18 Wider AGNRs exhibit multiple terminal zigzag edge states in finite segments.^19,20 For instance, in 13-atom wide AGNR (13-AGNR) segments, each terminal may possess two distinct zigzag edge states, one with an exponential decay characteristic length and another with a longer characteristic length. Although the current spectra of 13-AGNR tunneling FETs have been experimentally reported, a systematic analysis of charge tunneling through the topological edge states and the subband states is lacking.⁸ Meanwhile, scientists want to know how the electronic and transport properties of GNRs are changed when GNRs are embedded into h-boron nitride sheets,¹⁶ which is a critical issue to realize GNR-based electronic circuits.^21–23

This study aims to elucidate the electronic and transport properties of multiple end zigzag edge states in finite-size AGNRs and AGNR/boron nitride nanoribbon (BNNR) heterojunctions under longitudinal electric fields. Fig. 1 illustrates an n-AGNR/w-BNNR heterojunction, where n and w denote the widths of AGNR and BNNR, respectively. Given the challenges associated with using density functional theory (DFT) to calculate the transmission coefficient of GNR segments under electric fields,^24,25 we employ a tight-binding model and Green's function technique to compute the bias-dependent transmission coefficient.^26–28 This approach enables us to elucidate the charge transport through multiple zigzag edge states in finite 13-AGNRs and 13-AGNR/6-BNNR heterojunctions. Spectra of transmission coefficients reveal interference between subband states and long characteristic length zigzag edge states in the finite 13-AGNR structure. Furthermore, the on-off switch behavior arising from resonant tunneling between the left and right zigzag edge states of the 13-AGNR/6-BNNR heterojunction is discerned in the transmission coefficient spectra. Intriguingly, nonthermal broadening tunneling current exhibits a large peak-to-valley ratio, which is highly useful for applications such as single-electron transistors at room temperature.

Fig. 1 (a) and (b) Schematic diagrams illustrating the armchair graphene nanoribbon (13-AGNR) and 13-AGNR/6-BNNR heterojunction under longitudinal electric fields, respectively. Schematic diagram (c) depicts the line-contacting of zigzag-edge atoms in a 13-AGNR structure to electrodes. Symbols Γ_L(Γ_R) represent the electron tunneling rate between the left (right) electrode and the leftmost (rightmost) atoms at the zigzag edges, and T_L(T_R) denotes the equilibrium temperature of the left (right) electrode. The probability densities of the zigzag edge states with short and long characteristic lengths in 13-AGNR segments without electric fields are plotted in (a) and (c), respectively. The radius of the circle represents the magnitude of the probability density.

2 Calculation methodology

To investigate the transport properties of 13-AGNR and 13-AGNR/6-BNNR heterojunctions connected to the electrodes, we employ a combination of the tight-binding model and the Green's function technique. The system Hamiltonian depicted in Fig. 1(c) is written as H = H₀ + H_GNR, where

The first two terms of eqn (1) describe the free electrons in the left and right metallic electrodes. a^†_k(b^†_k) creates an electron of with momentum k and energy ε_k in the left (right) electrode. describes the coupling between the left (right) lead with its adjacent atom in the row.

Here, represents the on-site energy of the orbital in the row and j-th column. The operators and create and annihilate an electron at the atom site denoted by . The parameter characterizes the electron hopping energy from site to site . For the tight-binding parameters of n-AGNR/w-BNNR, we assign E_B = 2.329 eV, E_N = −2.499 eV, and E_C = 0 eV to boron, nitride, and carbon atoms, respectively. We neglect variations in electron hopping strengths between different atoms for simplicity.^29,30 We set for the nearest-neighbor hopping strength. Additionally, the effect of electric field F_y is included by the electric potential U = eF_yy on , where F_y = V_y/L_a, with V_y as the applied bias and L_a as the length of the AGNR segment.

We calculate the bias-dependent transmission coefficient using the formula: ,²⁶ where Γ_L(ε) and Γ_R(ε) denote the tunneling rate (in energy units) at the left and right leads, respectively, and G^r(ε) and G^a(ε) are the retarded and advanced Green's functions of the GNRs, respectively. In terms of tight-binding orbitals, Γ_α(ε) and Green's functions are matrices. The expression for Γ_L(R)(ε) is derived from the imaginary part of the self-energies, denoted as , and is given by , where and denote the coupling strengths between the left and right metallic electrodes with their adjacent atoms (refer to Fig. 1). In the context of metals such as gold, where the density of states remains approximately constant near the Fermi energy, the wide-band limit serves as a suitable approximation. In this limit, the energy-dependent tunneling rates, Γ_L(R)(ε), are replaced by constant matrices denoted as Γ_L(R).²⁶ However, it's important to note that in some cases, only the diagonal entries of Γ_L(R) are non-zero. Additionally, if one adopts the alternative definition of , the factor “4′′ in the transmission coefficient is reset to one.^31–33 Notably, Γ_L(R) calculated within the wide-band limit can effectively replicate the transmission coefficient curves of graphene nanoribbon superlattices obtained using the surface Green's function method, as demonstrated in ref. 32 and 33.

3 Results and discussion

Before examining the effects of electric fields on the electronic and transport properties of end zigzag edge states of 13-AGNR and 13-AGNR/6-BNNR structures, we first present the electronic structures of these two configurations in Fig. 2(a) and (b), respectively. Both structures maintain semiconductor properties with direct band gaps. Specifically, the 13-AGNR and 13-AGNR/6-BNNR exhibit band gaps of 0.714 eV and 0.722 eV, respectively, consistent with values calculated by the DFT method.³⁰ It's noteworthy that the electronic structures of 13-AGNR/w-BNNR remain unchanged when w ≥ 6. This suggests that a 6-BNNR width is sufficient to model 13-AGNR embedded within an h-BN sheet. Although the impact of BNNR on the band gaps of 13-AGNRs is not significant, we will demonstrate later that BNNR structures have a notable effect on the end zigzag edge states of 13-AGNRs.

Fig. 2 Electronic structures of 13-AGNR and 13-AGNR/6-BNNR structures. (a) 13-AGNR structure and (b) 13-AGNR/6-BNNR structure. Energy levels of 13-AGNRs as functions of applied voltage V_y for two different N_a values. (c) N_a = 64 (L_a = 6.67 nm) and (d) N_a = 200 (L_a = 21.16 nm). Note that the energy levels are plotted with respect to the Fermi energy E_F, which is set as zero throughout this article.

3.1 13-AGNR segments under an electric field

To elucidate the end zigzag edge states of 13-AGNRs, we analyze the energy spectra of 13-AGNR segments under an electric field. Fig. 2(c) and (d) illustrate the energy levels of 13-AGNR segments with N_a = 64 (L_a = 6.67 nm) and N_a = 200 (L_a = 21.16 nm) as functions of V_y. In the absence of an electric field, four energy levels exist between E_C and E_V, representing the subband states as depicted in Fig. 2(c). These levels are denoted as Σ_C,S, Σ_C,L, Σ_V,S, and Σ_V,L. Here, C and V distinguish the energy levels above or below the Fermi energy, while S and L represent short and long decay characteristic lengths, respectively. These levels arise from the formation of bonding and antibonding states between the left and right TSs (Ψ_L,S(L) and Ψ_R,S(L)), as noted in ref. 19. For N_a = 64, Σ_C(V),L = ±7.13 meV and Σ_C(V),S = 0. For N_a = 200, Σ_C(V),L = ±7 μeV and Σ_C(V),S = 0. When considering N_a = 200, E_C and E_V are 0.3668 eV and −0.3668 eV, respectively, which closely correspond to the minimum conduction subband and the maximum valence subband of infinitely long 13-AGNRs shown in Fig. 2(a). Notably, in Fig. 2(c) and (d), we observe a blue Stark shift of TSs and a red Stark shift of the subband states. Consequently, the energy levels of end zigzag edge states cross the subband states. With increasing N_a, the applied bias inducing such a crossing is shifted toward lower bias, as observed in Fig. 2(d). Note that the Stark shift of energy levels for graphene nanoribbon (GNR) segments (or graphene quantum dots) responds sensitively to the applied electric fields, with directionality playing a crucial role.^34,35 Particularly, when transverse electric fields (F_x) are introduced to armchair graphene nanoribbons (AGNRs), we observe an intriguing semiconductor-to-semimetal transition in the electronic structures.^13,15

To further comprehend the charge transport through these topological states (Ψ_L(R),S and Ψ_L(R),L), we plot the transmission coefficient of a 13-AGNR segment with N_a = 64 for various V_y values in Fig. 3(a)–(c). As depicted in Fig. 3(a), the peak labeled by Σ₀ arises from Σ_C,L and Σ_V,L, rather than Σ_C,S and Σ_V,S, as their wave functions (Ψ_L,S and Ψ_R,S) are highly localized on the end zigzag edges (see Fig. 1(a)). With the application of bias voltage V_y in Fig. 3(b), the probability for charge tunneling through these topological states diminishes significantly due to the off-resonance energy levels of Ψ_L,L and Ψ_R,L. Upon reaching V_y = 0.918 V, two peaks labeled ε_C(V),B and ε_C(V),A emerge due to the interference between the end zigzag edge states Σ_C(V),L and the subband states E_C(V), as seen in Fig. 3(c). The peak at ε_C(V),B can be interpreted as charge tunneling through the long decay length edge state under the subband states (E_C(V)) assisted procedure. Fig. 3(d) illustrates the contact effect on this interference phenomenon. As Γ_t decreases, the resolution between the peaks (ε_C,B and ε_C,A) becomes more distinct. For Γ_t = 90 meV, the interference pattern resembles Fano interference. The probability distributions of the peaks labeled ε_C,B and ε_C,A are shown in Fig. 3(e) and (f), respectively. It is evident that ε_C(V),B and ε_C(V),A correspond to constructive and destructive interferences between the wave function of the conduction (valence) subband state and the wave function of the right (left) edge state with a long characteristic length (Ψ_R(L),L). Although many theoretical efforts predict the existence of topological states of GNRs,^10–12 revealing the quantum interference between the TSs and the subband states based on the tunneling spectra of GNR-based electronic devices remains a challenge.^8,9

Fig. 3 Transmission coefficients of 13-AGNRs with N_a = 64 for various V_y values at Γ_t = 0.09 eV. Panels (a), (b), and (c) correspond to V_y = 0, V_y = 0.45 V, and V_y = 0.918 V, respectively. Panel (d) shows the for various Γ_t values at V_y = 0.918 V. Panels (e) and (f) depict probability densities of ε_C,B = 0.337 eV and ε_C,A = 0.393 eV in a 13-AGNR segment with N_a = 64 and V_y = 0.918 V.

3.2 13-AGNR/6-BNNR segments under an electric field

Considering the influence of h-BN on the electronic and transport properties of 13-AGNRs, we present the energy levels of 13-AGNR/6-BNNR structures as functions of applied V_y for various N_a values in Fig. 4(a)–(c). In the gap region of the 13-AGNR/6-BNNR structure (refer to Fig. 2(b)), there are still four energy levels present (Σ_C,S, Σ_C,L, Σ_V,S, and Σ_V,L). Unlike the multiple end zigzag edge states of 13-AGNRs, the multiple zigzag edge states of finite 13-AGNR/6-BNNR heterojunctions are insensitive to variations in N_a, remaining nearly fixed at specific energy levels. For instance, at N_a = 44, we have Σ_C,S = 76.4 meV (Σ_V,S = −81.7 meV) and Σ_C,L = 0.253 eV (Σ_V,L = −0.268 eV). Similarly, at N_a = 64, we observe Σ_C,S = 76.4 meV (Σ_V,S = −81.7 meV) and Σ_C,L = 0.249 eV (Σ_V,L = −0.263 eV). This indicates that Σ_C,S, Σ_C,L, Σ_V,S, and Σ_V,L are not determined by the wave function overlaps between the left and right TSs. In finite 13-AGNR/6-BNNR heterojunctions, BNNRs have the left boron atom zigzag edge terminal and the right nitride atom zigzag edge terminal (see Fig. 1(b)). The energy levels of Σ_C,S, Σ_C,L, Σ_V,S, and Σ_V,L are significantly influenced by the energy levels of boron and nitride atoms.

Fig. 4 Energy levels of 13-AGNR/6-BNNR heterojunction as functions of V_y for various N_a. Panels (a), (b), and (c) correspond to N_a = 44 (L_a = 4.54 nm), N_a = 56 (L_a = 5.82 nm), and N_a = 64 (L_a = 6.67 nm), respectively. Probability densities of 13-AGNR/6-BNNR heterojunction with N_a = 44 (L_a = 4.54 nm) at V_y = 0. Panels (d) and (e) depict the probability densities of Σ_C,L and Σ_V,L, respectively.

Fig. 4(a)–(c) indicate that the multiple zigzag edge states of 13-AGNR/6-BNNR exhibit a linear function of the electric field. In the high bias region (V_y > 1 V), the interaction spectra resulting from the multiple end zigzag edge states and the subband states can be reproduced, akin to the interference scenario observed in 13-AGNRs (Fig. 2(c) and (d)). Before calculating the transmission coefficient of 13-AGNR/6-BNNR heterojunction, we plot the probability densities of Σ_C,L and Σ_V,L at V_y = 0 and N_a = 44 in Fig. 4(d) and (e), respectively. From the probability distributions of Σ_C,L and Σ_V,L, it is evident that charges are confined within the 13-AGNR segment, demonstrating that BNNRs act as potential barriers. Additionally, Σ_C,L(Σ_V,L) is determined by the unoccupied carbon–nitride antibonding state (occupied carbon–boron bonding state).^36,37

The maximum probability of Σ_C,L(Σ_V,L) occurs at sites labeled by and at j = 1 ( and at j = 44). The probability distribution of Σ_C,L(Σ_V,L) along the y direction exhibits a long decay length, spanning several benzene sizes, yet its probability density near the right (left) electrode sites is minimal. This indicates that the wave function of Σ_C,L(Σ_V,L) is weakly coupled to the right (left) electrode. The asymmetrical coupling to the left and right electrodes for the wave function of Σ_C,L(Σ_V,L) hinders charge transport through this energy level.

We now present the calculated transmission coefficient of the 13-AGNR/6-BNNR heterojunction with N_a = 44 for various applied bias V_y values at Γ_t = 90 meV in Fig. 5. In Fig. 5(a) and (b), the charge transport through Σ_C,L and Σ_V,L is hindered due to the very weak coupling strength between the electrodes and the zigzag edge states. However, a significant peak of Σ₀ occurs for ε near the Fermi energy at V_y = 0.702 V in Fig. 5(c). For V_y = 1.02 V and V_y = 1.458 V, the peaks for charge transport through Σ_C,L and Σ_V,L are shifted away from the Fermi energy, as shown in Fig. 5(d) and (e). A notable enhancement of charge transport through Σ_C,L and Σ_V,L is observed for V_y = 1.62 V, which is attributed to the assistance from subband states.

Fig. 5 Transmission coefficient of the 13-AGNR/6-BNNR heterojunction with N_a = 44 for various V_y values at Γ_t = 90 meV. Panels (a) through (f) correspond to V_y values of 0, 0.378 V, 0.702 V, 1.02 V, 1.458 V, and 1.62 V, respectively. Panel (g) displays the probability distribution of Σ₀ = 0 at V_y = 0.702 V. Panel (h) shows the transmission coefficient as functions of ε for various Γ_t values at V_y = 0.702 V.

The transmission coefficient of Σ₀ shown in Fig. 3(c) is attributed to the alignment between Σ_C,L and Σ_V,L. We plot the probability distribution of Σ₀ = 0 in Fig. 5(g). The symmetrical probability distribution explains the alignment of Σ_C,L and Σ_V,L. As these two levels approach alignment, their wave functions and generate constructive and destructive superpositions, forming the two splitting peaks. To resolve these two peaks, we present the transmission coefficient for various Γ_t values in Fig. 5(h). The results in Fig. 5(h) indicate that the transmission coefficient is affected not only by the applied voltage but also by the contact property between the electrodes and the 13-AGNR/6-BNNR heterojunctions. For weak Γ_t = 36 meV, we observe two separated peaks.

3.3 Tunneling current through the zigzag edge states of 13-AGNR/6-BNNR heterojunctions

In this subsection, we investigate the tunneling current through the end zigzag edge states with long decay length, utilizing the effective 2-site Hubbard model.³⁸ The transmission coefficient curves depicted in Fig. 3(d) and 5(h) can be faithfully replicated by the effective 2-site Hubbard model, as the zigzag edge states are well-separated from the subband states under low applied bias, and their wave functions are localized.²⁷ The tunneling current is determined by the expression:where f_L(R)(ε) denotes the Fermi distribution function of the left (right) electrode with the chemical potential μ_L(R) = E_F ∓ eV_y/2. The bias-dependent transmission coefficient, , includes one-particle, two-particle, and three-particle correlation functions in the Coulomb blockade region, computed self-consistently.³⁹ Furthermore, the bias-independent intra-site (U₀) and inter-site (U₁) Coulomb interactions are computed using with the dielectric constant ε₀ = 4, and U_cc = 4 eV at i = j. U_cc arises from the two-electron occupation in each p_z orbital. In Fig. 6, the non-interaction curve corresponds to the case depicted in Fig. 4(c). The inset of Fig. 6 illustrates charge transport in the Coulomb blockade region, where ε_C,L = Σ_C,L − ηeV_y + iΓ_e,L and ε_V,L = Σ_V,L + ηeV_y + iΓ_e,R. Here, ηeV_y = 0.336eV_y represents the orbital offset terms induced by the applied voltage (V_y), and Γ_e,L = Γ_e,R = Γ_e,t represents the effective tunneling rate of end zigzag states in the 2-site model, determined by Γ_t and the wave functions of the edge states Ψ_C(V),L.

Fig. 6 Tunneling current through the end zigzag edge state with a long decay length of the 13-AGNR/6-BNNR heterojunction with N_a = 64. Energy levels are defined as ε_C,L = Σ_C,L − 0.366eV_y + iΓ_e,L and ε_V,L = Σ_V,L + 0.366eV_y + iΓ_e,R. Other physical parameters include Σ_C,L = 0.249 eV, Σ_V,L = −0.263 eV, intra-site Coulomb interaction U₀ = 155 meV, inter-site Coulomb interaction U₁ = 42 meV, and inter-site electron hopping strength t_LR = 7.13 meV. We have Γ_e,L = Γ_e,R = Γ_e,t = 9 meV, E_F = 0, and J₀ = 0.773 nA.

The first peak of the red curve, calculated at temperature T = 300 K and labeled ε₁, corresponds to the charge transfer from ε_V,L + U₀ to ε_C,L when these two energy levels align. The second peak, ε₂, corresponds to electron transfer between ε_V,L and ε_C,L. However, the tunneling current for such a procedure is significantly suppressed due to electron Coulomb interactions. The blue curve with markers is calculated at T = 480 K. Distinguishing between the red and blue curves in the entire applied voltage region is challenging, illustrating the nonthermal broadening effect of the tunneling current. Additionally, the peak-to-valley ratio reaches 8.7. The results presented in Fig. 6 demonstrate that the edge states with long decay length act as effective single charge filters at room temperature, suggesting their potential application in single-electron transistors at room temperature.⁴⁰ Traditional single electron transistors formed by a single quantum dot exhibit temperature-dependent tunneling current spectra,^41–43 where the peak-to-valley ratio of tunneling current tends to decrease with increasing temperature. Our study demonstrates that the tunneling current through the zigzag edge states of 13-AGNR/6-BNNR heterojunctions maintains temperature stability, a significant advantage for practical applications.

4 Conclusion

In this study, we investigated the charge transport properties through the multiple end zigzag edge states of both 13-AGNR and 13-AGNR/6-BNNR heterojunctions under longitudinal electric fields, employing the tight-binding model and Green's function technique. For 13-AGNRs, we elucidated the distinct blue Stark shift behaviors exhibited by the zigzag edge states with short and long decay lengths in response to electric fields. We found that the latter can significantly interact with the subband states, thereby unveiling the mechanism of charge transport through the long decay length zigzag edge states assisted by the subband states, as evidenced by the spectra of the transmission coefficient.

Concerning 13-AGNR/6-BNNR heterojunctions, we observed that the energy levels of the end zigzag edge states of 13-AGNRs are notably influenced by the presence of BNNRs rather than their lengths. Even in long 13-AGNR/6-BNNR segments, Σ_C,L exhibits a significant orbital offset from Σ_V,L. We further revealed a remarkable resonant tunneling process for charge transfer between Σ_C,L and Σ_V,L through the bias-dependent transmission coefficient analysis.

Moreover, employing a 2-site Hubbard model, we analyzed the tunneling current through the end zigzag edge states of 13-AGNR/6-BNNR heterojunctions within the Coulomb blockade region. Our results demonstrate that these edge states function effectively as single charge filters at room temperature, exhibiting a high peak-to-valley ratio of tunneling current with a nonthermal broadening effect, as illustrated in Fig. 6. This unique characteristic suggests promising applications of 13-AGNR/6-BNNR heterojunctions in single electron transistors operating at room temperature.

Data availability

The data presented in this study are available upon reasonable request.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the National Science and Technology Council, Taiwan under Contract No. MOST 107-2112-M-008-023MY2.

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