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

David M T Kuo Department of Electrical Engineering and Department of Physics, National Central University, Chungli, 32001 Taiwan, China

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.

I 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 [Novoselovks, 1]. Known for their semiconducting phases resulting from quantum confinement, GNRs hold significant promise for next-generation electronics [JustinH, 2–WangHM, 4]. Among these, armchair GNRs (AGNRs) are particularly noteworthy due to their tunable band gaps, which are inversely proportional to their widths [Cai, 5–DJRizzo, 12]. Recent research has focused on understanding the electronic properties of AGNRs under transverse electric fields, revealing transitions from semiconducting to semimetallic phases [ZhaoF, 13–NikitaVT, 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 [LlinasJP, 8,JacobsePH, 9]. These structures harbor topological states (TSs), the response of which to electric fields remains poorly understood [TepliakovNV, 17,GolorM, 18]. Wider AGNRs exhibit multiple terminal zigzag edge states in finite segments [SanchoMP, 19,Zdetsis, 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 [LlinasJP, 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 [NikitaVT, 16], which is a critical issue to realize GNR-based electronic circuits [ChenLX, 21-GengDC, 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. Figure 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 [SonYW, 24,Matsuda, 25], we employ a tight-binding model and Green’s function technique to compute the bias-dependent transmission coefficient [Kuo1, 26–Kuo3, 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.

Refer to caption — Figure 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 $\Gamma_{L}$ ( $\Gamma_{R}$ ) 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 Figure 1(a) and 1(c), respectively. The radius of the circle represents the magnitude of the probability density.

II 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 $H=H_{0}+H_{GNR}$ , where

	$\displaystyle H_{0}$	$\displaystyle=$	$\displaystyle\sum_{k}\epsilon_{k}a^{\dagger}_{k}a_{k}+\sum_{k}\epsilon_{k}b^{\dagger}_{k}b_{k}$
		$\displaystyle+$	$\displaystyle\sum_{\ell}\sum_{k}V^{L}_{k,\ell,j}d^{\dagger}_{\ell,j}a_{k}+\sum_{\ell}\sum_{k}V^{R}_{k,\ell,j}d^{\dagger}_{\ell,j}b_{k}+h.c.$

The first two terms of Eq. (1) describe the free electrons in the left and right metallic electrodes. $a^{\dagger}_{k}$ ( $b^{\dagger}_{k}$ ) creates an electron of with momentum $k$ and energy $\epsilon_{k}$ in the left (right) electrode. $V^{L}_{k,\ell,j=1}$ ( $V^{R}_{k,\ell,j=N_{a}}$ ) describes the coupling between the left (right) lead with its adjacent atom in the $\ell$ -th row.

	$\displaystyle H_{GNR}$	$\displaystyle=$	$\displaystyle\sum_{\ell,j}E_{\ell,j}d^{\dagger}_{\ell,j}d_{\ell,j}$
		$\displaystyle-$	$\displaystyle\sum_{\ell,j}\sum_{\ell^{\prime},j^{\prime}}t_{(\ell,j),(\ell^{\prime},j^{\prime})}d^{\dagger}_{\ell,j}d_{\ell^{\prime},j^{\prime}}+h.c,$

Here, $E_{\ell,j}$ represents the on-site energy of the orbital in the $\ell$ -th row and $j$ -th column. The operators $d^{\dagger}_{\ell,j}$ and $d_{\ell,j}$ create and annihilate an electron at the atom site denoted by $(\ell,j)$ . The parameter $t_{(\ell,j),(\ell^{\prime},j^{\prime})}$ characterizes the electron hopping energy from site $(\ell^{\prime},j^{\prime})$ to site $(\ell,j)$ . 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 [GSSeal, 29,DingY, 30]. We set $t_{(\ell,j),(\ell^{\prime},j^{\prime})}=t_{pp\pi}=2.7$ eV for the nearest-neighbor hopping strength. Additionally, the effect of electric field $F_{y}$ is included by the electric potential $U=eF_{y}y$ on $E_{\ell,j}$ , 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 ${\cal T}_{LR}(\varepsilon)$ using the formula: ${\cal T}_{LR}(\varepsilon)=4Tr[\Gamma_{L}(\varepsilon)G^{r}(\varepsilon)\Gamma_{R}(\varepsilon)G^{a}(\varepsilon)]$ [Kuo1, 26], where $\Gamma_{L}(\varepsilon)$ and $\Gamma_{R}(\varepsilon)$ denote the tunneling rate (in energy units) at the left and right leads, respectively, and $G^{r}(\varepsilon)$ and $G^{a}(\varepsilon)$ are the retarded and advanced Green’s functions of the GNRs, respectively. In terms of tight-binding orbitals, $\Gamma_{\alpha}(\varepsilon)$ and Green’s functions are matrices. The expression for $\Gamma_{L(R)}(\varepsilon)$ is derived from the imaginary part of the self-energies, denoted as $\Sigma^{r}_{L(R)}(\varepsilon)$ , and is given by $\Gamma_{L(R)}(\varepsilon)=-\text{Im}(\Sigma^{r}_{L(R)}(\varepsilon))=\pi\sum_{k}|V^{L(R)}_{k,\ell,j}|^{2}\delta(\varepsilon-\epsilon_{k})$ , where $V^{L}_{k,\ell,j=1}$ and $V^{R}_{k,\ell,j=N_{a}}$ 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, $\Gamma_{L(R)}(\varepsilon)$ , are replaced by constant matrices denoted as $\Gamma_{L(R)}$ [Kuo1, 26]. However, it’s important to note that in some cases, only the diagonal entries of $\Gamma_{L(R)}$ are non-zero. Additionally, if one adopts the alternative definition of $\Gamma_{L(R)}(\varepsilon)=-i[\Sigma^{r}_{L(R)}(\varepsilon)-\Sigma^{a}_{L(R)}(\varepsilon)]$ , the factor ”4” in the transmission coefficient is reset to one [SunQF, 31–YuXX, 33]. Notably, $\Gamma_{L(R)}$ 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 references [NingXU, 32] and [YuXX, 33].

III 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 Fig. 2(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 [DingY, 30]. It’s noteworthy that the electronic structures of 13-AGNR/w-BNNR remain unchanged when $w\geq 6$ . 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.

III.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. Figures 2(c) and 2(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 $\Sigma_{C,S}$ , $\Sigma_{C,L}$ , $\Sigma_{V,S}$ , and $\Sigma_{V,L}$ . 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 ( $\Psi_{L,S(L)}$ and $\Psi_{R,S(L)}$ ), as noted in ref [SanchoMP, 19]. For $N_{a}=64$ , $\Sigma_{C(V),L}=\pm 7.13$ meV and $\Sigma_{C(V),S}=0$ . For $N_{a}=200$ , $\Sigma_{C(V),L}=\pm 7\mu$ eV and $\Sigma_{C(V),S}=0$ . 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 2(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 [ChenRB, 34,PedersenTG, 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 [ZhaoF, 13,Pizzochero, 15].

To further comprehend the charge transport through these topological states ( $\Psi_{L(R),S}$ and $\Psi_{L(R),L}$ ), we plot the transmission coefficient ${\cal T}_{LR}(\varepsilon)$ of a 13-AGNR segment with $N_{a}=64$ for various $V_{y}$ values in Fig. 3(a)-3(c). As depicted in Fig. 3(a), the peak labeled by $\Sigma_{0}$ arises from $\Sigma_{C,L}$ and $\Sigma_{V,L}$ , rather than $\Sigma_{C,S}$ and $\Sigma_{V,S}$ , as their wave functions ( $\Psi_{L,S}$ and $\Psi_{R,S}$ ) 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 $\Psi_{L,L}$ and $\Psi_{R,L}$ . Upon reaching $V_{y}=0.918$ V, two peaks labeled $\varepsilon_{C(V),B}$ and $\varepsilon_{C(V),A}$ emerge due to the interference between the end zigzag edge states $\Sigma_{C(V),L}$ and the subband states $E_{C(V)}$ , as seen in Fig. 3(c). The peak at $\varepsilon_{C(V),B}$ 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 $\Gamma_{t}$ decreases, the resolution between the peaks ( $\varepsilon_{C,B}$ and $\varepsilon_{C,A}$ ) becomes more distinct. For $\Gamma_{t}=90$ meV, the interference pattern resembles Fano interference. The probability distributions of the peaks labeled $\varepsilon_{C,B}$ and $\varepsilon_{C,A}$ are shown in Fig. 3(e) and 3(f), respectively. It is evident that $\varepsilon_{C(V),B}$ and $\varepsilon_{C(V),A}$ 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 ( $\Psi_{R(L),L}$ ). Although many theoretical efforts predict the existence of topological states of GNRs [DRizzo, 10-DJRizzo, 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 [LlinasJP, 8,JacobsePH, 9].

III.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 ( $\Sigma_{C,S}$ , $\Sigma_{C,L}$ , $\Sigma_{V,S}$ , and $\Sigma_{V,L}$ ). 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 $\Sigma_{C,S}=76.4$ meV ( $\Sigma_{V,S}=-81.7$ meV) and $\Sigma_{C,L}=0.253$ eV ( $\Sigma_{V,L}=-0.268$ eV). Similarly, at $N_{a}=64$ , we observe $\Sigma_{C,S}=76.4$ meV ( $\Sigma_{V,S}=-81.7$ meV) and $\Sigma_{C,L}=0.249$ eV ( $\Sigma_{V,L}=-0.263$ eV). This indicates that $\Sigma_{C,S}$ , $\Sigma_{C,L}$ , $\Sigma_{V,S}$ , and $\Sigma_{V,L}$ 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 $\Sigma_{C,S}$ , $\Sigma_{C,L}$ , $\Sigma_{V,S}$ , and $\Sigma_{V,L}$ are significantly influenced by the energy levels of boron and nitride atoms.

Figures 4(a)-4(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 Fig. 2(d)). Before calculating the transmission coefficient of 13-AGNR/6-BNNR heterojunction, we plot the probability densities of $\Sigma_{C,L}$ and $\Sigma_{V,L}$ at $V_{y}=0$ and $N_{a}=44$ in Fig. 4(d) and 4(e), respectively. From the probability distributions of $\Sigma_{C,L}$ and $\Sigma_{V,L}$ , it is evident that charges are confined within the 13-AGNR segment, demonstrating that BNNRs act as potential barriers. Additionally, $\Sigma_{C,L}$ ( $\Sigma_{V,L}$ ) is determined by the unoccupied carbon-nitride antibonding state (occupied carbon-boron bonding state)[PrunedaJM, 36,JungJ, 37].

The maximum probability of $\Sigma_{C,L}$ ( $\Sigma_{V,L}$ ) occurs at sites labeled by $\ell=10$ and $\ell=16$ at $j=1$ ( $\ell=10$ and $\ell=16$ at $j=44$ ). The probability distribution of $\Sigma_{C,L}$ ( $\Sigma_{V,L}$ ) 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 $\Sigma_{C,L}$ ( $\Sigma_{V,L}$ ) is weakly coupled to the right (left) electrode. The asymmetrical coupling to the left and right electrodes for the wave function of $\Sigma_{C,L}$ ( $\Sigma_{V,L}$ ) 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 $\Gamma_{t}=90$ meV in Fig. 5. In Fig. 5(a) and 5(b), the charge transport through $\Sigma_{C,L}$ and $\Sigma_{V,L}$ is hindered due to the very weak coupling strength between the electrodes and the zigzag edge states. However, a significant peak of $\Sigma_{0}$ occurs for $\varepsilon$ 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 $\Sigma_{C,L}$ and $\Sigma_{V,L}$ are shifted away from the Fermi energy, as shown in Fig. 5(d) and 5(e). A notable enhancement of charge transport through $\Sigma_{C,L}$ and $\Sigma_{V,L}$ is observed for $V_{y}=1.62$ V, which is attributed to the assistance from subband states.

The transmission coefficient of $\Sigma_{0}$ shown in Fig. 3(c) is attributed to the alignment between $\Sigma_{C,L}$ and $\Sigma_{V,L}$ . We plot the probability distribution of $\Sigma_{0}=0$ in Fig. 5(g). The symmetrical probability distribution explains the alignment of $\Sigma_{C,L}$ and $\Sigma_{V,L}$ . As these two levels approach alignment, their wave functions $\Psi_{C,L}(\ell,j)$ and $\Psi_{V,R}(\ell,j)$ generate constructive and destructive superpositions, forming the two splitting peaks. To resolve these two peaks, we present the transmission coefficient ${\cal T}_{LR}(\varepsilon)$ for various $\Gamma_{t}$ 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 $\Gamma_{t}=36$ meV, we observe two separated peaks.

III.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 [Mangnus, 38]. The transmission coefficient curves depicted in Figure 3(d) and Figure 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 [Kuo2, 27]. The tunneling current is determined by the expression:

J=\frac{-2e}{h}\int d\varepsilon~{}{\cal T}_{2-site}(\varepsilon)[f_{L}(\varepsilon)-f_{R}(\varepsilon)]

(3)

where $f_{L(R)}(\varepsilon)$ denotes the Fermi distribution function of the left (right) electrode with the chemical potential $\mu_{L(R)}=E_{F}\mp eV_{y}/2$ . The bias-dependent transmission coefficient, ${\cal T}_{2-site}(\varepsilon)$ , includes one-particle, two-particle, and three-particle correlation functions in the Coulomb blockade region, computed self-consistently [DavidK, 39]. Furthermore, the bias-independent intra-site ( $U_{0}$ ) and inter-site ( $U_{1}$ ) Coulomb interactions are computed using $U_{n}=\frac{1}{4\pi\epsilon_{0}}\sum_{i,j}|\Psi_{C(V),L}(\textbf{r}_{i})|^{2}|\Psi_{C(V),L}(\textbf{r}_{j})|^{2}\frac{1}{|\textbf{r}_{i}-\textbf{r}_{j}|}$ with the dielectric constant $\epsilon_{0}=4$ , 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 $\varepsilon_{C,L}=\Sigma_{C,L}-\eta~{}eV_{y}+i\Gamma_{e,L}$ and $\varepsilon_{V,L}=\Sigma_{V,L}+\eta~{}eV_{y}+i\Gamma_{e,R}$ . Here, $\eta~{}eV_{y}=0.336~{}eV_{y}$ represents the orbital offset terms induced by the applied voltage ( $V_{y}$ ), and $\Gamma_{e,L}=\Gamma_{e,R}=\Gamma_{e,t}$ represents the effective tunneling rate of end zigzag states in the 2-site model, determined by $\Gamma_{t}$ and the wave functions of the edge states $\Psi_{C(V),L}$ .

The first peak of the red curve, calculated at temperature $T=300K$ and labeled $\varepsilon_{1}$ , corresponds to the charge transfer from $\varepsilon_{V,L}+U_{0}$ to $\varepsilon_{C,L}$ when these two energy levels align. The second peak, $\varepsilon_{2}$ , corresponds to electron transfer between $\varepsilon_{V,L}$ and $\varepsilon_{C,L}$ . However, the tunneling current for such a procedure is significantly suppressed due to electron Coulomb interactions. The blue curve with markers is calculated at $T=480K$ . 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 [LeeYM, 40]. Traditional single electron transistors formed by a single quantum dot exhibit temperature-dependent tunneling current spectra [LeobandungE, 41–PostmaHWC, 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.

IV 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, $\Sigma_{C,L}$ exhibits a significant orbital offset from $\Sigma_{V,L}$ . We further revealed a remarkable resonant tunneling process for charge transfer between $\Sigma_{C,L}$ and $\Sigma_{V,L}$ 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.

Conflicts of interest
There are no conflicts to declare

Data availability

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

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

E-mail address: [email protected]

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