Topological Interface States and Nonlinear Thermoelectric Performance in Armchair Graphene Nanoribbon Heterostructures

TL;DR

The paper addresses how topological interface states arise in armchair graphene nanoribbon heterostructures lacking translational symmetry and links these IFs to end states of the constituent segments. It develops a real-space, bulk-boundary perturbation framework and employs Stark-field spectroscopy to count ESs and map them onto IFs, deriving the rule $N_{IF,β} = |N_O,B(A) - N_C,A(B)|$ that governs IFs' number and chirality. It shows that IFs originating from the central segment ESs form a topological double quantum dot (TDQD), whose two in-gap transmission channels can be described by an extended Anderson model, enabling analysis of nonlinear thermoelectric power under Coulomb blockade. The results demonstrate enhanced nonlinear power output in TDQDs, with Coulomb interactions shaping transport and enabling rectification under structural or electrostatic asymmetries, suggesting TDQDs as viable high-temperature graphene-based thermoelectric devices.

Abstract

We investigate the emergence and topological nature of interface states (IFs) in N-AGNR/$(N-2)$-AGNR/N-AGNR heterostructure (AGNRH) segments lacking translational symmetry, focusing on their relation to the end states (ESs) of the constituent armchair graphene nanoribbon (AGNR) segments. For AGNRs with $R_1$-type unit cells, the ES numbers under a longitudinal electric field follow the relations $N = N_{A(B)} \times 6 + 1$ and $N = N_{A(B)} \times 6 + 3$, whereas $R_2$-type unit cells exhibit $(N_{A(B)} + 1)$ ESs. The subscripts $A$ and $B$ denote the chirality types of the ESs. The Stark effect lifts ES degeneracy and enables clear spectral separation between ESs and IFs. Using a real-space bulk boundary perturbation approach, we show that opposite-chirality states hybridize through junction-site perturbations and may shift out of the bulk gap. The number and chirality of IFs in symmetric AGNRHs are determined by the difference between the ESs of the outer and central segments, $N_O$ and $N_C$, according to $N_{IF,β} = |N_{O,B(A)} - N_{C,A(B)}|$, where $β$ labels the chirality. Depending on whether $N_O > N_C$ or $N_C > N_O$, the resulting IFs acquire B- or A-chirality, respectively. Calculated transmission spectra ${\cal T}_{GNR}(\varepsilon)$ reveal that AGNRHs host a topological double quantum dot (TDQD) when IFs originate from the ESs of the central AGNR segment. Using an Anderson model with effective intra-dot and inter-dot Coulomb interactions, we derive an analytical expression for the tunneling current through the TDQD via a closed-form transmission coefficient. Thermoelectric analysis shows that TDQDs yield enhanced nonlinear power output in the electron-dilute and hole-dilute charge states, with Coulomb blockade suppressing thermal current but not thermal voltage.

Figures (13)

• Figure 1: (a) Schematic illustration of a $9_w-7_x-9_y$ armchair graphene nanoribbon heterostructure (AGNRH) composed of three AGNR segments. The size of the AGNRH is characterized by ($N,M$), where $N$ and $M$ denote the row and column numbers. $R_1$ and $R_2$ represent the unit cells (u.c.) of the 9-AGNR and 7-AGNR segments, respectively. (b) A and B sublattice sites are indicated by white and blue colors. The inter-AGNR electron hopping strengths $t_{es}$ connect adjacent atoms between the 9-AGNR and 7-AGNR segments. (c) AGNRH segment with zigzag edge terminations coupled to left and right electrodes with equilibrium temperatures $T_L$ and $T_R$. $\Gamma_{L(R)}$ denote the tunneling rates for electrons tunneling from the left (right) electrode into the adjacent atoms at the zigzag edges.
• Figure 2: Energy levels of armchair graphene nanoribbon (AGNR) segments as functions of the applied voltage $V_y$. (a) 13-AGNR, (b) 15-AGNR, (c) 19-AGNR, (d) 21-AGNR, (e) 25-AGNR, and (f) 27-AGNR segments. All segments have a length specified by $M = 96$, and their widths are characterized by $N$.
• Figure 3: Energy levels of armchair graphene nanoribbon heterostructure (AGNRH) segments as functions of the applied voltage $V_y$ for different widths.(a) $9-7-9$ AGNRH, (b) $15-13-15$ AGNRH, (c) $21-19-21$ AGNRH, and (d) $27-25-27$ AGNRH. Here, we adopt $w = y = 4$ for the outer AGNR segments with $R_1$-type unit cells (u.c.) and $x = 16$ for the central AGNR segment with an $R_2$-type unit cell (u.c.).
• Figure 4: Energy levels of four AGNRH segments with different widths as functions of the inter-AGNR electron hopping strength $t_{es}$. (a) $9_{4}-7_{6}-9_{4}$ AGNRH segment, (b) $15_{4}-13_{6}-15_{4}$ AGNRH segment, (c) $21_{4}-19_{6}-21_{4}$ AGNRH segment and (d) $27_{4}-25_{6}-27_{4}$ AGNRH segment. The parameter $t_{es}$is illustrated in Fig. 1(b).
• Figure 5: (a-c) Probability densities corresponding to $\Sigma_{AB,c2} = 0.24355$ eV, $\Sigma_{AB,c1} = 0.20466$ eV and $\Sigma_{IF,c}= 0.12651$ eV in the $15_{4}-13_6-9_{4}$ AGNRH segment with $t_{es} = 0.1~t$. (d-f) Probability densities corresponding to $\Sigma_{AB,c2} = 0.47859$ eV, $\Sigma_{AB,c1} = 0.32919$ eV and $\Sigma_{IF,c}= 0.12399$ eV in the $15_{4}-13_6-15_{4}$ AGNRH segment with $t_{es} = 0.2~t$. (g-i) Probability densities of $\Sigma_{IF,c}$ in the $15_{4}-13_6-9_{4}$ AGNRH segment for various values of $t_{es}$:(g) $\Sigma_{IF,c}= 0.10343$ eV at $t_{es} = 0.5~t$, (h) $\Sigma_{IF,c}= 0.07926$ eV at $t_{es} = 0.8~t$, and (i) $\Sigma_{IF,c}= 0.06531$ eV at $t_{es} = 1~t$.