Thermal rectification through the topological states of asymmetrical length armchair graphene nanoribbons heterostructures with vacancies

We present a theoretical investigation of electron heat current in asymmetrical length armchair graphene nanoribbon (AGNR) heterostructures with vacancies, focusing on the topological states (TSs). In particular, we examine the 9-7-9 AGNR heterostructures where the TSs are well-isolated from the conduction and valence subbands. This isolation effectively mitigates thermal noise of subbands arising from temperature fluctuations during charge transport. Moreover, when the TSs exhibit an orbital off-set, intriguing electron heat rectification phenomena are observed, primarily attributed to inter-TS electron Coulomb interactions. To enhance the heat rectification ratio ($\eta_Q$), we manipulate the coupling strengths between the heat sources and the TSs by introducing asymmetrical lengths in the 9-AGNRs. This approach offers control over the rectification properties, enabling significant enhancements. Additionally, we introduce vacancies strategically positioned between the heat sources and the TSs to suppress phonon heat current. This arrangement effectively reduces the overall phonon heat current, while leaving the TSs unaffected. Our findings provide valuable insights into the behavior of electron heat current in AGNR heterostructures, highlighting the role of topological states, inter-TS electron Coulomb interactions, and the impact of structural modifications such as asymmetrical lengths and vacancy positioning. These results pave the way for the design and optimization of graphene-based devices with improved thermal management and efficient control of electron heat transport.

The interface states of 9-7-9 AGNR heterostructures exhibit topological states (TSs) with energy levels near the charge-neutral point, which are well-separated from the conduction and valence subbands [39,40].This unique characteristic allows for the manipulation of electron transport solely through TSs, avoiding the subband channels.The strong intra-TS and inter-TS electron Coulomb interactions will lead to remarkable current rectification and negative differential conductance in charge transport through these serially coupled TSs (SCTS) in the Pauli spin blockade (PSB) configuration, similar to the behavior of serially coupled quantum dots [49].
This article aims to theoretically investigate the electron heat rectification of SCTSs composed of asymmet-rical length 9-7-9 AGNRs with vacancies, as depicted in Figure 1(b) and 1(c).Asymmetrical length 9-7-9 AGNR heterostructures offer tunable tunneling rates for the left and right TSs, with the energy level of the right TS being modulated by the right gate electrode (shown in Figure 1(b)).The vacancies effectively suppress phonon heat current, while the TSs remain largely unaffected since their wave functions are located at the interfaces between 9-AGNR and 7-AGNR (Fig. 1(a)).Consequently, electron heat current dominates over phonon heat current in this scenario, leading to the observation of asymmetrical electron heat current with high rectification efficiency above liquid-helium temperature.

II. CALCULATION METHOD
We investigate the transport properties of heterostructures composed of 9-7-9 AGNRs connected to electrodes, as shown in Fig. 1.To analyze this system, we employ a combination of the tight-binding model and the Green's function technique [50][51][52].The Hamiltonian of the system consists of two parts: H = H 0 + H AGNR .Here, H 0 represents the Hamiltonian of the electrodes themselves, including the coupling between the electrodes and the 9-7-9 AGNR [51,52].H AGNR corresponds to the Hamiltonian for the 9-7-9 AGNR heterostructures, which can be expressed as follows: In this equation, E ℓ,j represents the on-site energy for the p z orbital in the ℓ-th row and j-th column.The study neglects the spin-orbit interaction.The creation (destruction) of an electron at the atom site labeled by (ℓ, j), where ℓ and j are the row and column indices, respectively, is denoted by d † ℓ,j (d ℓ,j ).The electron hopping energy from site (ℓ ′ , j ′ ) to site (ℓ, j) is described by t (ℓ,j),(ℓ ′ ,j ′ ) .For AGNRs, the tight-binding parameters used are E ℓ,j = 0 for the on-site energy and t (ℓ,j),(ℓ ′ ,j ′ ) = t ppπ = 2.7 eV for the nearest neighbor hopping strength.
To analyze the electron ballistic transport behavior of AGNR heterostructures, we need to calculate their transmission coefficient T LR (ε).This coefficient describes the probability for each electron trajectory between the left (L) and right (R) electrodes.Obtaining the closed-form expression of T LR (ε) is challenging, even though H AGNR does not include electron Coulomb interactions.These interactions will be accounted for by an effective two-site Hubbard model later to study charge transport through the SCTSs.To calculate T LR (ε), we use the numerical code, which is given by T Here, Γ L (ε) and Γ R (ε) represent the tunneling rates (in energy units) at the left and right leads, respectively.For simplicity, we consider the wide band limit for the electrodes and an energyindependent approximation for Γ L (ε) and Γ R (ε).The retarded and advanced Green functions of the AGNR, G r (ε) and G a (ε), respectively, can be calculated using the system Hamiltonian (H = H 0 + H AGNR ) [51,52].

III. RESULTS AND DISCUSSION
A. 9-7 AGNR Superlattices Topological states (TSs) hold significant promise for applications in electronics and optoelectronics due to their robust transport characteristics, which are resistant to defect scattering [53][54][55][56][57]. Recently, there has been extensive research focused on exploring the topological phases of various graphene nanoribbons (GNRs) [53][54][55][56][57].The first principle method, also known as density functional theory (DFT), has been employed to calculate the Zak phase of 9-7 AGNR heterojunctions.This calculation confirms the presence of a localized interface state with energy at the midgap [54].Notably, the distinctive feature of the topological phases in the 9-7 AGNR superlattice is the formation of minibands arising from the interface-localized states, a phenomenon well-captured by the Su-Schrieffer-Heeger (SSH) model [40,58].Using the Hamiltonian H AGN R (Equation (1)), we investigate the electronic structures of various 9 w − 7 x AGNR superlattices (SLs) with different AGNR segments, as illustrated in Fig. 2 (a)-(f).Here, the subscripts w, and x represent the length of each AGNR segment in terms of unit cells (u.c.).The presence of topological states at the interfaces between the 9 AGNR segment and 7 AGNR segment results in the formation of minibands within the energy gap between the conduction and valence subbands.These minibands can be effectively described using the Su-Schrieffer-Heeger (SSH) model [40,58], which provides a analytical expression of E SSH (k) = ± t 2 x + t 2 w − 2t x t w cos(kπ/L).In this expression, t x and t w represent the electron hopping strengths in the 7 AGNR segment and 9 AGNR segment, respectively, while L denotes the length of the super unit cell.The results depicted in Fig. 2 highlight the tunability of t x and t w in 9 w − 7 x AGNR SLs , making them an excellent platform for investigating the phases of the SSH model, which could potentially host Majorana Fermions and superconducting states [59].

B. Su-Schrieffer-Heeger Model
We have replicated the electronic structures of the minibands depicted in Fig. 2(a)-2(f) and represented them in Fig. 3(a)-3(f) using the E SSH (k) expression.Each diagram in Fig. 3 corresponds to its respective counterpart in Fig. 2. The curves in Fig. 3 are computed using E SSH (k) for specific parameter values: (a) t w = 0.084 eV, (b) t w = 0.0396 eV, and (c) t w = 0.0176 eV, with a fixed t x = 0.118 eV.In cases where w > x (a, b, c), we observe t x > t w , whereas in cases (d, e, f) with w < x, we find t w > t x .The minibands formed in (d), (e), and (f) have narrow bandwidths, and the corresponding t w and t x values in E SSH (k) are as follows: (d) t x = 13.77meV, (e) t x = 8 meV, and (f) t x = 4.69 meV, all with a fixed t w = 78 meV.The band gaps created by these minibands, as shown in Fig. 3(a)-3(f), are as follows: (a) ∆ T S = 68 meV, (b) ∆ T S = 157 meV, (c) ∆ T S = 202 meV, (d) ∆ T S = 129 meV, (e) ∆ T S = 140 meV, and (f) ∆ T S = 147 meV, respectively.Notably, the band gaps formed by the minibands in 9-7 AGNR superlattices are much smaller than those created by the topological states of other types of GNRs [36,41].It is worth noting that the values of t x and t w in 9-7 AGNR superlattices, as determined using the tight-binding method, might not be as accurate as those obtained through the DFT calculations [40].Nevertheless, the tight-binding method still captures the essential characteristics of 9-7 AGNR superlattices.The examples illustrated in Figures 2 and 3 serve to demonstrate these outcomes.The charge transport through a finite 1D SSH chain coupled to electrodes can offer valuable insights into the charge transport through the topological states of 9-7-9 AGNR heterostructures.To investigate this, we employ the finite-chain SSH model and present the calculated transmission coefficient T LR (ε) in Fig. 4 at Γ L = Γ R = Γ t = 10 meV.The T LR (ε) curves of Fig. 4(a)-4(c)correspond to the cases of Fig. 3(a)-3(c), respectively.These T LR (ε) spectra reflect their band widths and band gaps in Fig. 3(a)-3(c).Interestingly, the areas of these T LR (ε) curves exhibit arch-like shapes, rather than rectangular shapes, indicating that electronic states near the band edges of minibands have a low probability of transport between the electrodes.This behavior can be understood by the electron group velocities at the band edges.
In Fig. 4(a)-4(c), we investigate a finite SSH model with 64 unit cells.Subsequently, we analyze the impact of finite size on the SSH model's T LR (ε) due to the limited length of the 9-7 AGNR heterostructures synthesized using the bottom-up technique.The heterostructures 9 6 − 7 7 − 9 6 display transport channels (Fig. 4(d)) resulting from the serially coupled topological states (SCTS) corresponding to E SSH (t w = 0), which are solely determined by the energies of ε HO and ε LU .Moreover, Fig. 4(e) illustrates T LR (ε) for 9 6 − 7 7 − 9 6 − 7 7 − 9 6 AGNR heterostructures, presenting four transport channels.Among these, two channels originate from the outer topological states with an effective weak hopping strength t ef f,x , while the other two channels arise from the topological states of 7 − 9 − 7 with t w = 78 meV [40] and effective tunneling rates Γ e,t ≪ Γ t .Notably, the probability of transport through Σ outer−T Ss is significantly suppressed due to Γ t > t ef f,x = 2.1 meV.The effective coupling strengths (t ef f,x ) between the outer TSs of AGNR heterostructures have been observed in [48].If one tunes Γ t to 1 meV, Σ outer,T Ss will exhibit two peaks with a separated energy of |2t ef f,x |.This suggests that the electron transport between the electrodes is considerably influenced by the coupling strengths between the AGNR heterostructures and the electrodes, which, in turn, are determined by the contact types of electrodes [51,52].
In Figure 5(a), a symmetrical situation is depicted, showing two peaks labeled as ε HO and ε LU originating from the TSs with a coupling strength of t x (similar to Fig. 4(d)).The energy separation between ε HO and ε LU is determined by |2t x | = 0.2358 eV, with their maximum values reaching one.This indicates that the maximum electrical conductance is one quantum conductance (G 0 ), where G 0 = 2e 2 h .The energy levels of ε HO and ε LU are well separated from the bulk states within regions ε > E c and ε < E v , where E c and E v denote the minimum of the conduction subband and the maximum of the valence subband (or see Fig. 2(a)-2(c)).The localized charge density of ε LU (or ε HO ) with ±0.1179 eV is plotted in Figure 1(a).From the charge densities of ε HO and ε LU , SCTSs function as serially coupled quantum dots, each containing only one energy level.Therefore, it is expected that SCTSs can have promising applications in charge and spin quantum bits [40,41,[63][64][65][66][67][68].
Figure 5(b) shows that the peak positions of ε HO and ε LU remain robust against variations in the 9-AGNR segments, although the strength of these two peaks and the "bulk states" experience significant changes.The magnitudes of ε HO and ε LU are reduced to 2 3 .The variation in the 9-AGNR segments serves to adjust the "barrier width" between the TSs and the electrodes.This result indicates that the left TS (LTS) and the right TS (RTS) have different effective coupling strengths to the electrodes.When a triple vacancy (removing three carbon atoms shown in Fig. 1(b)) appears between the left electrode and the LTS, the peaks of ε HO and ε LU become narrower, as observed in Fig. 5(c).The role of the triple vacancy is to increase the "barrier height" between the left electrode and the LTS.Consequently, the widths of ε HO and ε LU become narrower, while their energy levels remain unchanged.In the case of two triple vacancies in 9 3 − 7 3 − 9 5 AGNR heterostructures, as illustrated in Fig. 1(c), the ε HO and ε LU peaks are further narrowed, as shown in Fig. 5(d).The results in Fig. 5 demonstrate the robustness of the TSs against vacancies when the vacancies occur in regions with low charge densities.These vacancies effectively suppress phonon transportation and reduce phonon heat currents [42][43][44][45].
In Figure 5, the spectra of the transmission coefficient demonstrate electron-hole symmetry.To enable heat rectification in practical applications, it is crucial to break the degeneracy between the LTS and the RTS.i.e., |ε HO | ̸ = ε LU .To achieve this, we introduce a right gate electrode that modulates the energy levels of the RTS associated with the p z orbital.The designed right gate electrode covers the carbon atoms within the region between j = (w+x−1)×4 and j = (w+x+1)×4.In Fig. 6(a) and 6(b), we present the calculated transmission coefficient T LR (ε) for 9 8 − 7 x − 9 6 and 9 6 − 7 x − 9 8 AGNR heterostructures, considering x = 8 with V gR = 45 mV and x = 10 with V gR = 27 mV.Upon applying the right gate electrode voltage, the peak near the ε = 0 corresponds to the LTS.
To gain insights into the behavior of the T LR (ε) spectra in Fig. 6(a) and 6(b), we introduce a two-site model with the following closed-form expression for T 2−site LR (ε): Here, Γ e,L and Γ e,R represent the effective tunneling rates of the LTS with energy level E e,L and the RTS with energy level E e,R , respectively.In Fig. 6(c

D. 2-site Hubbard model
Due to the localized wave functions of the TSs, the electron Coulomb interactions exhibit significant strength.To elucidate the electron heat current of 9 w − 7 x − 9 y AGNR heterostructures in the Coulomb blockade region, we employ a two-site Hubbard model.The Hamiltonian of this model is described as follows: where E j represents the spin-independent energy level of the TSs, U j = U L(R) = U 0 and U j,ℓ = U LR(RL) = U 1 denote the intra-TS and inter-TS Coulomb interactions, respectively, and n j,σ = c † j,σ c j,σ .The values of U 0 and U 1 are calculated using |ri−rj | with the dielectric constant ϵ 0 = 4, and U cc = 4 eV at i = j.U cc arises from the two-electron occupation in each p z orbital of Eq.
(1).Here, Ψ L(R) (r i ) represents the wave functions of TSs [69].In Figure 7, we present the calculated electron hopping strengths between the TSs, as well as the intra-TS and inter-TS electron Coulomb interactions as functions of x for the 9 3 − 7 x − 9 5 AGNR heterostructures.Figure 7(a) illustrates that the electron hopping strength (t x ) decreases rapidly with increasing x since it primarily depends on the overlap between the localized wave functions of the left and right TSs (Ψ L and Ψ R ).On the other hand, although U 0 decreases as x increases, this decrease can be attributed to the overall increase in the total area of the 9 − 7 − 9 AGNR.As a result, the sensitivity of U 0 to variations in x is small, particularly when x > 6 (Figure 7(b)).This observation suggests that the charge densities of the TSs are predominantly distributed at the interfaces (Fig. 1(a)).Furthermore, Figure 7(c) demonstrates a more rapid decrease in the inter-TS Coulomb interaction (U 1 ) compared to U 0 with increasing x.The trend of U 0 > U 1 > t x persists when x ≥ 3. Therefore, it is essential to consider the effects of electron Coulomb interactions on the charge transport through the TSs of 9 w − 7 x − 9 y AGNR heterostructures.

E. Electron heat current of SCTSs
The electron and heat currents leaving from the left (right) electrode are given by and where the Fermi distribution function of the α electrode is given by f α (ε) = . To discuss electron heat rectification, we consider the open-circuit condition (J L(R) = 0) under a temperature bias ∆T = T L −T R , where Here, T 0 denotes the average temperature of the junction system.Due to the Seebeck effect (for finite ∆T ), the thermal voltage V th induced by ∆T will balance the electrons diffusing from the hot electrode to the cold electrode, establishing the condition of open circuit (J L(R) = 0).Additionally, the chemical potentials also depend on the thermal voltage (µ L = µ+ eV th 2 and µ R = µ− eV th 2 , where µ is the equilibrium chemical potential of the electrodes).The transmission coefficient for charge transport through the SCTSs with U 0 and U 1 , denoted by T 2−site LR (ε), has a closed-form expression given by where we have ϵ L = ε − E e,L + iΓ e,L and ϵ R = ε − E e,R + iΓ e,R in Eq. ( 6).The intra-TS and inter-TS two-particle correlation functions are denoted by ⟨n ℓ,−σ n ℓ,σ ⟩ and ⟨n ℓ,σ n j,σ ⟩ (⟨n ℓ,−σ n j,σ ⟩), respectively.⟨n ℓ,−σ n ℓ,σ n j,σ ⟩ is the three-particle correlation function.These correlation functions can be solved self-consistently [48].While DFT serves as a potent tool for elucidating the electronic structures of materials, it remains a challenge to accurately portray electron transport within the Coulomb blockade region [53][54][55][56][57].The intricate interplay of many-body effects gives rise to two-particle and three-particle correlation functions, posing formulation difficulties within the mean-field theory framework [38].
The closed-form expression of Equation ( 6) presents a comprehensive explanation for the transport properties of serially coupled localized states in both the Coulomb blockade and PSB regions [70][71][72][73][74][75][76].When there are no intra-TS and inter-TS Coulomb interactions, equation (6) simplifies to the 2-site free electron model (equation (2)), which lacks temperature bias-dependent variables.As a result, the free electron model fails to exhibit electron heat rectification behavior, even when an orbital offset exists between the left and right TS, as depicted in the transmission coefficients of Figure 6.In contrast, the transmission coefficient described by equation ( 6) depends on ∆T or applied bias V bias , which arises from the electron occupation number and other correlation functions.Therefore, the presence of electron Coulomb interactions, along with the orbital offset, significantly contributes to elucidating the characteristics of electron and heat current rectification.The electron and heat current rectification in the PSB is illustrated in the appendix A.
To illustrate the aforementioned crucial statements, we present the calculated thermal voltage (a, c) and electron heat current (b, d) as functions of temperature bias (∆T ) for various average temperature values in Fig. 8, considering two different AGNR heterostructures (9 8 − 7 10 − 9 8 in (a, b) and 9 8 − 7 9 − 9 8 in (c, d)).The physical parameters t x , U 0 , and U 1 for x = 10 and x = 9 are obtained from the data in Fig. 7. Additionally, we set E e,L = E e,R = 0 meV and Γ e,L = Γ e,R = 1 meV.Moreover, throughout this article, we assume µ = 0.
The thermal voltage (V th ) is determined by the condition of zero electron current (J L(R) = 0).In the forward temperature bias (∆T > 0), the calculated V th is negative, while it is positive in the backward temperature bias (∆T < 0).The nonlinearity of V th at low average temperature (T 0 = 48 K) becomes nearly linear at T 0 = 72 K.The electron heat current is a complex function of V th and ∆T .The forward electron heat current (Q F ) and the backward heat current (Q B ) are calculated from Q L and Q R , respectively.Comparing the curves in Fig. 8(d) with those in Fig. 8(b), it can be observed that the electron heat current is enhanced with increasing t x , while the electron heat currents exhibit symmetrical behavior in Fig. 8 To observe asymmetrical electron heat current, it is necessary to have a temperature bias direction-dependent transmission coefficient.This implies that E e,L ̸ = E e,R (or Γ e,L ̸ = Γ e,R ) is required.Firstly, we consider the situation where E e,L ̸ = E e,R , which can be achieved by modulating the gate voltage as shown in Fig. 6.In Fig. 9, we present the calculated thermal voltage and electron heat current for various values of E e,R , assuming E e,L = 0 and T 0 = 48 K.As shown in Fig. 9(a), the thermal voltage V th is significantly enhanced by nearly an order of magnitude when E e,R = 10 meV, compared to the curve in Fig. 8(a) with T 0 = 48 K.However, the thermal voltage remains an odd function of ∆T , namely V th (∆T ) = −V th (−∆T ), even when E e,L ̸ = E e,R .On the other hand, the electron heat current Q e exhibits asymmetrical behavior, where Q F ̸ = |Q B |, as shown in Fig. 9(b).These results indicate that asymmetrical electron heat current behavior does not require asymmetrical thermal voltage behavior.
To highlight the importance of inter-TS Coulomb interaction U 1 in the asymmetrical behavior of Q e , we artificially set U 1 to zero and plot the calculated V th and Q e in Fig. 9(c) and 9(d).It should be noted that V th is less sensitive to changes in U 1 , but the asymmetrical behavior of Q e is suppressed when U 1 = 0.For instance, at k B ∆T = ±6 meV and E e,R = 10 meV with U 1 = 38 meV, we have Q F = 0.934Q 0 and Q B = −0.384Q0 , resulting in a ratio of Q F /|Q B | = 2.43.However, when we turn off U 1 (i.e., U 1 = 0), we obtain Q F = 0.936Q 0 and Q B = −0.738Q0 under the same conditions, resulting in a ratio of Q F /|Q B | = 1.27.Once E e,L ̸ = E e,R , asymmetrical electron heat current (Q F > |Q B |) can be observed.To investigate the effect of Γ e,L ̸ = Γ e,R , we analyze electron occupation number N j,σ , electron heat current Q e , and rectification efficiency η Q for different Γ e,L values at T 0 = 48 K and Γ e,R = 1 meV, as shown in Fig. 10.The variation in Γ e,L can be achieved by adjusting Γ t with different w values and introducing vacancies.As depicted in Fig. 10(a), N L,σ > N R,σ occurs when E e,L is closer to µ = 0. Additionally, N L,σ exhibits significant temperature bias direction-dependent behavior.These two factors contribute to the direction-dependent character of the transmission coefficient with temperature bias ∆T .The asymmetrical electron heat rectification behavior (Q F > |Q B |) displayed in Fig. 10(b) can be understood by considering the P 1 channel of Eq. ( 6).Notably, the probability weight of P 1 in T RL (ε) under reversed temperature bias can be obtained by exchanging the indices of L and R in T LR (ε).Since E e,R is far from µ = 0, the most influential factors in P 1 are the one-particle occupation numbers, while two-particle and three-particle correlation functions are negligible.Consequently, the proba-bility weight of P 1 is larger for ∆T > 0 compared to ∆T < 0, explaining the observed electron heat rectification behavior in Fig. 10(b).
In Fig. 10(c), the rectification efficiency η Q is enhanced when considering Γ e,L > Γ e,R .Here, 3h , which is 0.9464pW/K for T = 1 K) [45], and F s = 0.0287 is the reduction factor for Q ℓ due to the effects of vacancies and heterostructures [42][43][44][45].Notably, F s = 0.1198 provides an excellent fit for the calculated phonon thermal conductance of AGNR (κ ph ) at room temperature [77].The presence of vacancies and heterostructures in AGNRs not only reduces phonon thermal conductance but also creates asymmetrical phonon heat currents [5,6].It is worth noting that electron heat rectification exists even at very small temperature bias (∆T → 0), indicating that electron thermal conductances exhibit temperature bias direction-dependent behavior.This behavior has also been observed in phonon thermal conductances [6].In previous figures (Fig. 9 and Fig. 10) with t x = 2.7 meV, we have demonstrated electron heat current rectification when E e,L ̸ = E e,R .Now, to investigate the influence of electron hopping strength between TSs on heat rectification, we present the thermal voltage V th and electron heat current Q e as functions of temperature bias for various 7-AGNR segments at a low averaged temperature of T 0 = 24 K in Fig. 11(a) and 11(b), respectively.As shown in Fig. 11(a), for the case of x = 8 with t x = 8 meV, V th becomes very small, indicating that the electron-hole asymmetry lifted by E e,L ̸ = E e,R weakens for larger t x .Moreover, a larger t x results in a larger electron heat current Q e , as depicted in Fig. 11(b).
To illustrate the rectification efficiency arising from the enhancement of t x , we plotted η Q in Fig. 11(c) and 11(d) with and without the contribution of phonon heat current Q ph .For T 0 = 24 K, the rectification efficiency η Q in Fig. 11(c) can reach and exceed 0.5, even for very small temperature bias.This significant rectification is attributed to the dominance of electron heat current at low averaged temperature T 0 .Furthermore, by artificially turning off Q ph in Fig. 11(d), η Q can reach 1.6 at k B ∆T = 3 meV for the case of x = 10 with t x = 2.7 meV.This result indicates that Q F = 2.6|Q B | in this case, showcasing a remarkable rectification effect.Notably, the variation of t x in Fig. 11 is based on the change in 7 AGNR segment lengths.Additionally, the long-distance coherent tunneling mechanism [60][61][62]   Based on the findings presented in Fig. 11(c) and 11(d), it becomes evident that high-efficiency electron heat diodes (HDs) align with the condition Q e ≫ Q ph .Consequently, two strategies emerge: one involves the reduction of Q ph while preserving Q e , while the other entails enhancing Q e while maintaining Q ph at a constant level.In the former case, a viable approach involves increasing the vacancy density.In the latter case, a feasible strategy entails utilizing 9-7 AGNR superlattices in lieu of 9-7-9 AGNR heterostructures.Notably, earlier work [29] highlights that AGNRs containing one or two vacancies exhibit notably reduced phonon thermal conductance (κ ph ) when compared to vacancy-free AG-NRs.Particularly, a distinct decline in κ ph is observable in AGNRs harboring two vacancies over the temperature range of 180 K to 400 K.It is thus reasonable to expect that the rectification efficiency η Q will be superior for two triple vacancies as opposed to a solitary triple vacancy, as long as the electron heat current Q e remains unaffected by the vacancies.Fundamentally, the effective realization of proficient electron heat diodes and thermoelectric devices hinges on the effective management of phonon heat current.Recent strides in this direction showcase significant reductions in phonon heat flow within bulk junctions through the application of strains or magnetic fields [78,79].Considering the context of diminishing phonon heat flow in 9-7 AGNRs, strain manipulation proves more advantageous from a device perspective than the consideration of magnetic fields.Regarding the augmentation of electron density, it is foreseeable that such an undertaking might lead to the attenuation of η Q due to the diminished electron Coulomb interactions within the topological states.Specifically, the electron Coulomb interactions within the minibands generated by the topological states, as illustrated in Fig. 2, are comparatively weaker than those within the 9 − 7 − 9 heterostructures, a consequence attributed to the delocalized nature of their wave functions.

IV. CONCLUSION
In conclusion, our theoretical analysis focuses on achieving electron heat rectification in charge transport through the topological states (TSs) of asymmetrical length armchair graphene nanoribbon (AGNR) heterostructures with vacancies, specifically the 9 w − 7 x − 9 y configuration.Our results show that the topological states (TSs) of 9-7-9 heterostructures are effectively separated from the subbands, effectively reducing subband noise caused by temperature fluctuations during electron heat transfer.The localized charge densities of the TSs at the interfaces between the 9-AGNR and 7-AGNR segments allow for adjusting the TSs-electrode separation distance by varying the length of the 9-AGNR segments, offering control over the coupling strengths (or effective tunneling rates) between the electrodes and TSs.Notably, vacancies between the electrodes and TSs do not compromise the integrity of the TSs but instead aid in suppressing phonon heat current in the 9 w − 7 x − 9 y AGNR heterostructures.
Moreover, we find that the electron hopping strengths of the TSs (t LR ) can be tuned by varying the length of the 7-AGNR segment or using a long-distance coherent tunneling mechanism.The localized wave functions of the TSs exhibit significant electron Coulomb interactions, which decrease gradually as the length of the 7-AGNR segment increases, a critical factor for observing pronounced asymmetrical electron heat currents.This asymmetry arises from the temperature bias directiondependent transmission coefficient resulting from both intra-TS and inter-TS electron Coulomb in addition to the asymmetrical TS energy levels.Furthermore, the optimized orbital offset of the TSs can be modulated effectively using a designed gate electrode.Our proposed electron heat diodes achieve a remarkable heat rectification efficiency (η Q ) of 0.7 for very small temperature biases at an averaged temperature of T 0 = 24 K in the 9 5 − 7 8 − 9 8 (N a = 84, L a = 8.8 nm) configuration.This finding indicates a successful suppression of the short channel effect.
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: mtkuo@ee.ncu.edu.twIn this appendix, we investigate the electron and heat currents associated with charge tunneling through a serially coupled topological state (SCTS) composed of 9 8 − 7 10 − 9 6 AGNR heterostructures under Pauli spin blockade (PSB) configurations [49].Figure A.1 illustrates the calculated electron occupation number, two-particle correlation functions, and tunneling current as a function of applied bias at T 0 = 12 K and ∆T = 0 K.The parameters considered are E e,L = −U 1 + δ L eV bias , E e,R = −U 0 + δ R eV bias , Γ e,L = 1 meV, and Γ e,R = 3 meV.In diagrams (a), (b), and (c), δ L = δ R = 0 is used to ignore the applied bias-dependent orbital offset.In diagrams (d), (e), and (f), δ L = 1/6 and δ R = −1/4 are used, which are determined by the length of each 9-AGNR segment and the total length of the 9 8 − 7 10 − 9 6 AGNR heterostructure, for simplicity [80].Additionally, E e,L = −U 1 and E e,R = −U 0 are set by introducing two gate electrodes to modulate the energy levels of the left and right topological states (TSs).For instance, V g,L = −81 mV and V g,R = −360 mV can be set for the 9 8 − 7 10 − 9 6 AGNR heterostructure to obtain E e,L = −U 1 = −38 meV and E e,R = −U 0 = −134 meV.It should be noted that the regions covered by V gL and V gR are the first region from j = (w − 1) * 4 to j = (w+1) * 4 and the second region from j = (w+x−1) * 4 to j = (w + x + 1) * 4, respectively.
In the case of a forward applied bias (V bias > 0), the total electron occupation number N T = σ (N L,σ + N R,σ ) approaches two, with N L,σ = 0.5 and N R,σ = 0.5, indicating one electron occupation in the left and right TSs, respectively (see Fig . A.1(a)).Conversely, under a backward applied bias, the electron occupation number N R,σ of the right TS exceeds 0.5, while the electron occupation number N L,σ of the left TS is less than 0.5.correlation functions for inter-TSs with singlet state (2site-S) and triplet state (2-site-T).As V bias > 0, the SCTS is predominantly occupied by triplet states.Conversely, for V bias < 0, the curves of 2-site-T and 2site-S merge together, but they are lifted for V bias < −76 mV because electrons from the right electrode are tunneling through the resonant channels resulting from E e,L + 2U 1 = E e,R + U 0 + U 1 .In Fig. A.1(c), we observe significant tunneling current rectification under PSB configurations.The darkness of J F is attributed to the SCTS being predominantly occupied by the triplet state.Furthermore, in Fig. A.1(d), (e), and (f), we reveal the effect of biasdependent orbital offset on single-particle occupation numbers, two-particle correlation functions, and tunneling current, respectively.The bias-dependent orbital offset lifts the resonant channels arising from the condition of E e,L + U 1 = E e,R + U 0 = µ.Notably, when E e,L + U 1 + eV bias /6 ̸ = E e,R + U 0 − eV bias /4, a remarkable effect is observed not only on N j,σ and two-particle correlation functions but also on the tunneling current in the range of |V bias | > 12.5 mV.The current rectification spectra shown in Fig. A.1(f) have been experimentally measured in serially coupled GaAs quantum dots [49].A spin-current conversion device [49,81] requires a tunneling current that can highly resolve the spin singlet and triplet states.Consequently, it is necessary to suppress the effect of bias-dependent orbital offset of SCTSs.Based on the results shown in Fig. A.1, it can be concluded that the effect of applied bias-dependent orbital offset is significant only for large applied biases, specifi-cally |V bias | > 12.5 mV.Since the values of V th are small, we can safely neglect this effect when discussing the electron heat current under PSB configurations.To illustrate the electron heat current in 9 8 − 7 10 − 9 6 AGNR heterostructures with PSB configurations, we need to consider an open circuit condition (J L(R) = 0) to determine the thermal voltage (V th ) arising from the Seebeck effect and other self-consistently solved correlation functions.In To further analyze the effects of electron hole asymmetry, we present N j,σ , V th , and Q e for ε R = µ + 10 meV in diagrams (d), (e), and (f).As shown in Fig. A.2(e), the maximum V th is significantly enhanced by at least two orders of magnitude.Moreover, V th exhibits an Nshape behavior.The electron heat rectification behavior is observed when ε R ̸ = ε L = µ.Based on the results presented in Fig. A.2, it can be concluded that electron heat current rectification favors T2−site LR(RL) (ε) with a strong electron-hole asymmetry near µ.
FIG. 1: (a) Schematic diagram of a symmetrical heterostructure composed of 94 − 73 − 94 AGNR connected to electrodes.The tunneling rate of electrons between the left (right) electrode and the leftmost (rightmost) atoms of the AGNR is denoted by ΓL (ΓR), respectively.The temperature of the left (L) and right (R) electrodes is represented by TL and TR, respectively.The charge density of the topological state with energy ε = 0.1179 eV is plotted for a 94 − 73 − 94 AGNR heterostructure.(b) and (c) Schematic diagrams of asymmetrical length AGNR heterostructures, namely 93 −73 −95, with one triple-vacancy and two triple-vacancies, respectively.The subscripts w, x, and y in the notation 9w − 7x − 9y denote the segment lengths of the AGNR heterostructures in terms of the unit cell (u.c.).

FIG. 8 :
FIG. 8: (a) and (b) illustrate the thermal voltage (V th ) and electron heat current (Qe) as functions of temperature bias for various averaged temperature values in 98 − 710 − 98 AGNR heterostructures.(c) and (d) depict the thermal voltage (V th ) and electron heat current (Qe) as functions of temperature bias for different averaged temperature values in 98 − 79 − 98 AGNR heterostructures.Here, Γe,L = Γe,R = 1 meV and Ee,L = Ee,R = 0.The electron heat current is expressed in units of Q0 = 77.2pW.
Fig. A.2, we present the calculated electron occupation numbers (N j,σ ), thermal voltage (V th ), and electron heat current (Q e ) as functions of temperature bias at T 0 = 24 K.As observed in Fig. A.2(a), N R,σ > 0.5, and N L,σ < 0.5.The variation of N j,σ with respect to temperature bias ∆T is less sensitive.However, the thermal voltage V th shown in Fig. A.2(b) is extremely small compared to the results in Fig. 9.This negligible V th can be attributed to the condition ε L = ε R = µ (E e,L +U 1 = E e,R +U 0 = µ) (see the inset of Fig. A.2(a)).Consequently, there is a very weak electron-hole asymmetry near µ caused by Γ e,L ̸ = Γ e,R , resulting in an almost negligible electron heat current rectification shown in Fig. A.2(c).
provides an alternative means of modifying t x values.For instance, we can implement a central gate on the middle 9 AGNR segment of 9 − 7 − 9 − 7 − 9 AGNR heterostructures.By applying a central gate voltage, we can tune an effective hopping strength t ef f,x between the outer topological states of the 9 − 7 − 9 − 7 − 9 junctions.This approach offers further possibilities for controlling and optimizing the rectification efficiency.