2406.02393v1

# Valley-dependent transport through graphene quantum dots due to proximity-induced, staggered spin-orbit couplings **Authors**: A. Belayadi, P. Vasilopoulos, N. Sandler > abelayadi@usthb.dzDept. of Theoretical Physics, University of Science and Technology Houari Boumediene, Bab-Ezzouar 16111, Algeria.École Supérieure des Sciences de l’Aliment et Industries Alimentaires, ESSAIA, El Harrach 16200, Algeria. > p.vasilopoulos@concordia.caDept. of Physics, Concordia University, 7141 Sherbrooke Ouest, Montréal, Québec H4B 1R6, Canada > sandler@ohio.eduDept. of Physics and Astronomy, Ohio University and Nanoscale Quantum Phenomena Institute, Athens, Ohio, USA. Abstract We study a system composed of graphene decorated with an array of islands with $C_{3v}$ symmetry that induce quantum dot (IQD) regions via proximity effects and gives rise to several spin-orbit couplings (SOCs). We evaluate transport properties for an array of IQDs and analyze the conditions for realizing isolated valley conductances and valley-state localization. The resulting transmission shows a square-type behavior with wide gaps that can be tuned by adjusting the strength of the staggered intrinsic SOCs. Realistic proximity effects are characterized by weak SOC strengths, and the analysis of our results in this regime shows that the Rashba coupling is the important interaction controlling valley properties. As a consequence, a top gate voltage can be used to tune the valley polarization and switch the valley scattering for positive or negative incident energies. A proper choice of SOC strengths leads to higher localization of valley states around the linear array of IQDs. These systems can be implemented in heterostructures composed of graphene and semiconducting transition-metal dichalcogenides (TMDs) such as MoSe 2, WSe 2, MoS 2, or WS 2. In these setups, the magnitudes of induced SOCs depend on the twist angle, and due to broken valley degeneracy, valley polarized currents at the edges can be generated in a controllable manner as well as localized valley states. Our findings suggest an alternative approach for producing valley-polarized currents and propose a corresponding mechanism for valley-dependent electron optics and optoelectronic devices. I introduction The study of valleytronic materials is of significant importance in the area of information processing and encoding [1, 2, 3] due to the alternative degree of freedom furnished by the carrier’s valley momentum in addition to the conventionally used charge and/or spin properties. The benchmark material is a two-dimensional (2D) graphene-based structure with Dirac cones at ${\bf K_{1}=-K}$ and ${\bf K_{2}=+K}$ valleys, proposed as a strong candidate for future valley-driven computing devices through the manipulation of valley currents [4, 5, 6], i.e., by applying external voltages. However, the lack of external probes or contacts that can select individual valley currents as ferromagnetic contacts separate spins polarized currents in spintronic devices [7, 8, 9], remains a primal obstacle to encoding and information processing through the valley index. In addition to transport, the material’s optoelectronic properties are also used to access the valley degree of freedom [10, 11, 12, 13]. Several strategies have used the optical response to control, detect, and monitor valley polarization [14, 15, 16, 17]. Generally, a combination of gates voltages -implemented via scanning tunneling microscopy- and suitable substrate magnetic materials bring out the mechanism that tunes the desired electronic, spintronic, and valleytronic properties [18, 19, 20, 21] as proposed and later demonstrated by polarization-resolved photoluminescence experiments [10, 22, 13, 23, 24]. An alternative approach to induce valley separation involves exploiting confined geometries. Quantum dots (QDs) can produce valley-filtered currents and are important ingredients in modern nanotechnology devices. Typically, confined geometries that induce valley separation are obtained via a wide variety of methods that include electrostatic confinement produced by a scanning tunneling microscopy tip [20], strain fields [25], bilayer graphene structures with spatially varying broken sublattice symmetry [26] and isolated regions defined by local broken sublattice symmetry [23]. In all these setups, valley separation is achieved because of the effects of external fields or due to substrate properties that are extremely difficult to design and control with sufficient precision. As a consequence, the potential of these geometries to induce selective valley filtering and confinement in a controllable manner and without external fields remains untapped. To address this issue, we investigate the properties of a proposed heterostructure that exploits proximity effects and periodic spin-orbit interactions. Ideally, the most efficient way to induce uniform and large staggered spin-orbit couplings (SOC)s on graphene is via proximity to appropriate substrates that break the sublattice symmetry, thus allowing for a clear distinction of the two pseudospins. Recently, the role of SOCs in valley separation has been addressed in several works, such as graphene deposited on top of hexagonal boron nitride [27] and graphene/TMD heterostructures [28, 29]. These setups possess staggered onsite potentials that give rise to various SOCs via different mechanisms [23]. Interestingly, not all types of SOCs will render valley separation as shown by the sublattice independent intrinsic spin-orbit coupling (ISOC) in the Kane and Mele model [30], or the Rashba SOC (RSOC) that appears in the presence of external fields. However, other appropriately engineered interactions can break the sublattice symmetry, rendering two main effects that we refer to as (1) the rise of a staggered potential with a concomitant gap opening and (2) the emergence of an ISOC in a staggered form that is sublattice dependent (i.e., sublattice-resolved SOC). In this last case, the spin-valley transport is due to the emergence of a valley-Zeeman type of coupling, defined by the ISOC sign change between sublattices [31, 28, 32]. This valley Zeeman effect is of great interest because it induces a giant spin lifetime anisotropy in proximitized graphene [33]. Furthermore, Frank et ${\it al}$ [34] showed that in narrow-width cells of zigzag-terminated graphene with a staggered ISOC, pseudo-helical and valley-centered states (without topological protection) are localized along the edges. These results are consistent with the bulk system’s topological invariant $Z_{2}=0$ . In this work, we propose a heterostructure composed of graphene and TMD islands that combines the effects of confinement and SOCs in a controllable manner. The model is inspired by recent experiments reported in Ref. 51, with graphene deposited on top of a TMD island that induces a local region with various SOCs, i.e., an induced quantum dot (IDQ). Our proposal generalizes the experimental setup to a periodic array of islands placed below or deposited on the graphene membrane. The TMD islands preserve the underlying $C_{3v}$ symmetry of graphene and introduce SOCs in the electron dynamics via the proximity effect. We analyze the conditions for selective valley state confinement and the generation of valley currents under applied voltages for a generic model that is later applied to specific material combinations. The paper is organized as follows. In Sec. II, we briefly present the model for a system composed of a linear array of induced quantum dots in graphene created by proximity effects and including different emerging SOC terms. In Sec. III, we present numerical results revealing effective mechanisms for valley filtering and confinement. We apply these results to a series of heterostructures composed of different materials with realistic parameters and analyze the effect of relative twisting between the two materials. A summary and conclusions follow in Sec. IV. II Model and methods As mentioned above, we propose to study a chain of quantum dots with $C_{3v}$ symmetry in graphene created by proximity effects due to TMD islands. The choice of TMDs that conserve $C_{3v}$ symmetry is made to ensure the largest values of induced spin-dependent couplings in the graphene membrane [28, 35]. Such engineered IQDs will exhibit pseudohelical and valley-centered edge states with potential for device applications [34, 36]. The salient advantages of such structures are: 1) longer localization lengths for valley states in narrow ribbons, 2) valley Chern numbers and localization lengths independent of RSOCs, and 3) gapless band structures. A schematic picture of the system is shown in Fig. 1. <details> <summary>2406.02393v1/x1.png Details</summary>  ### Visual Description # Technical Document Extraction: Graphene Quantum Dot (GQD) Electronic Structure ## Diagram Overview The image depicts a hybrid system combining graphene quantum dots (GQDs) with leads and spin-dependent electronic interactions. Key components are labeled and color-coded for clarity. --- ### **Part (a): GQD System Schematic** #### **Components** - **Leads**: - Left Lead (L): Green hexagonal lattice structure. - Right Lead (R): Green hexagonal lattice structure. - **Graphene Quantum Dots (GQDs)**: - Two central GQDs (orange circles) connected by zigzag boundaries. - Labeled as "N- units of IQDs along zigzag boundaries" between leads. - **Spin States**: - **Spin-up sites**: Blue nodes. - **Spin-down sites**: Red nodes. - **Connections**: - Green, purple, and red lines represent spin-dependent interactions (see Part b legend). --- ### **Part (b): Electronic Structure (SOCs)** #### **Legend: Spin-Orbit Coupling (SOC) Transitions** | **Transition** | **Color** | **Direction** | |-------------------------------|-----------|-----------------------------------| | {λ_I^(A), λ_I^(B)} → λ_R | Red | ↑ (Vertical) | | λ_R → {λ_PIA^(A), λ_PIA^(B)} | Purple | ↓ (Vertical) | | {λ_I^(A), λ_I^(B)} ↔ {λ_PIA^(A), λ_PIA^(B)} | Green | ↔ (Horizontal) | #### **Key Observations** 1. **Spin-Orbit Coupling (SOC)**: - Spin-up (blue) and spin-down (red) sites are interconnected via SOCs. - Transitions between spin states are mediated by λ_R (purple) and λ_PIA (green). 2. **Energy States**: - λ_I^(A/B): Initial spin-up/down states. - λ_R: Intermediate state enabling spin-flip transitions. - λ_PIA^(A/B): Final spin-up/down states post-SOC interaction. --- ### **Cross-Referenced Diagram Details** - **Color Consistency**: - Red arrows in the legend correspond to vertical transitions between spin-up/down states. - Purple arrows represent λ_R-mediated spin-flip processes. - Green arrows indicate horizontal coupling between λ_I and λ_PIA states. - **Structural Flow**: - Spin-up/down sites (blue/red) are interconnected via zigzag boundaries in the GQD lattice. - Leads (L/R) act as reservoirs for electron transport, with SOCs modulating spin-dependent pathways. --- ### **Summary** The diagram illustrates a GQD-based system where spin-orbit coupling (SOC) governs transitions between spin-up and spin-down electronic states. The legend explicitly defines SOC-mediated pathways, with color-coded arrows mapping transitions between λ_I, λ_R, and λ_PIA states. This system is embedded within a graphene lattice connected to leads, enabling controlled spin-dependent transport. </details> Figure 1: Panel (a) displays the overall device, which consists of a 2D graphene ribbon with zigzag boundaries decorated with semiconducting transition-metal dichalcogenide (TMD) islands. (b) Zoom-in of a quantum dot made of graphene and TMD. To visualize the emergence of different SOC terms, we duplicate the graphene membrane to emphasize the lift of the spin degeneracy. Each layer corresponds to a different spin component, with the blue (red) membrane representing the spin-up (spin-down) population. The induced SOCs combined with the underlying $C_{3v}$ symmetry give rise to sublattice-resolved intrinsic couplings $\lambda_{I}^{(A)},\ \lambda_{I}^{(B)}$ , denoted by red arrows, and pseudospin inversion-asymmetric couplings $\lambda_{PIA}^{(A)},\ \lambda_{PIA}^{(B)}$ denoted by green arrows. The Rashba $\lambda_{R}$ coupling is represented by purple arrows. The deposition of adsorbates on graphene, or of graphene membranes on islands, results in profound changes in the electronic structure that depend on the locally conserved symmetries as described in Ref. [28]. The most important effects are the emergence of i) an effective staggered potential due to the broken reflection symmetry imposed by different orbital interactions experienced by the carbon atoms in proximity to the different atomic species of the TMD material and ii) several sublattice dependent next-nearest neighbor hopping terms originated from the proximity-induced spin-orbit interactions. In this case, the system can be described by an extension of the models by Kane and Mele [30] and Haldane [37]. The Hamiltonian for the QD regions is given by: [27, 28, 29, 38] $$ \displaystyle H_{QD} \displaystyle=-t\sum_{\langle i,j\rangle}a_{is}^{\dagger}b_{js} \displaystyle+\sum_{\left\langle i\right\rangle} \displaystyle\Delta\left(\xi^{(A)}{\bf a}_{is}^{\dagger}{\bf a}_{is}+\xi^{(B)}% {\bf b}_{is}^{\dagger}{\bf b}_{is}\right) \displaystyle+\Big{(}\frac{2i}{3}\Big{)} \displaystyle\sum_{\left\langle i,j\right\rangle\sigma,\sigma^{\prime}}\left(% \lambda_{R}\ {\bf a}_{i\sigma}^{\dagger}{\bf b}_{j\sigma}\right)\left[{\bf\hat% {s}}\otimes{\bf d}_{ij}\right]_{\sigma,\sigma^{\prime}} \displaystyle+\Big{(}\frac{i}{3\sqrt{3}}\Big{)} \displaystyle\sum_{\left\langle\left\langle i,j\right\rangle\right\rangle% \sigma}\nu_{ij}\left(\lambda_{I}^{(A)}{\bf a}_{i\sigma}^{\dagger}{\bf a}_{j% \sigma}+\lambda_{I}^{(B)}{\bf b}_{i\sigma}^{\dagger}{\bf b}_{j\sigma}\right)% \left[{\bf\hat{s}}_{z}\right]_{\sigma,\sigma} \tag{1} $$ where $t$ is the nearest neighbor hopping between sites $i$ and $j$ (note that these are spin-preserving processes). $\Delta$ is the staggered potential induced by the TMD islands in the dot region. This potential is sublattice-dependent with $\xi^{(A)}=1$ ( $\xi^{(B)}=-1$ ), rendering opposite signs for the induced gaps, i.e., $\Delta^{(A)}=-\Delta^{(B)}=\Delta$ for sublattice A (B). The Rashba interaction (RSOC) is expressed in terms of the coupling $\lambda_{R}$ . This coupling breaks the $z$ inversion symmetry while exchanging the spin of different sublattices. Here, the $\textbf{d}_{ij}$ vector connects site $j$ to $i$ . The terms $\lambda_{I}^{(A)}$ and $\lambda_{I}^{(B)}$ represent the intrinsic SOCs (ISOC) between next-nearest neighbors. These terms connect the same sublattices and spins in the clockwise ( $\nu_{ij}=-1$ ) or anticlockwise ( $\nu_{ij}=-1$ ) direction from site $j$ to site $i$ . Finally, the spin is denoted by the vector $\widehat{\textbf{s}}$ with components written in terms of Pauli matrices. It is worth restating that the SOCs exist only within the QD regions; outside the system is described by the Hamiltonian of pristine graphene. Working with a Hamiltonian in reciprocal space is more convenient for studying valley properties. The resulting effective Hamiltonian is obtained by linearizing Eq. (1) around the ${\bf K_{1}}$ and ${\bf K_{2}}$ valleys labeled below by the valley index $\kappa=-1$ and $\kappa=+1$ , respectively. The final expression is given in the form $H_{QD}=H_{k}+H_{\Delta}+H_{R}+H_{I}$ [34], where: $$ \displaystyle H_{k} \displaystyle= \displaystyle\hbar v_{F}\left(\kappa k_{x}\sigma_{x}+k_{y}\sigma_{y}\right)s_{% 0}, \displaystyle H_{\Delta} \displaystyle= \displaystyle\Delta\sigma_{z}s_{0}, \displaystyle H_{R} \displaystyle= \displaystyle\lambda_{R}\left(-\kappa\sigma_{x}s_{y}+\sigma_{y}s_{x}\right)s_{% 0}, \displaystyle H_{I} \displaystyle= \displaystyle(\kappa/2)\big{[}\lambda_{I}^{(A)}\left(\sigma_{z}+\sigma_{0}% \right)+\lambda_{I}^{(B)}\left(\sigma_{z}-\sigma_{0}\right)\big{]}s_{z}. \tag{2} $$ The Fermi velocity $v_{F}$ is expressed in terms of the hopping $t$ as $v_{F}=\sqrt{3}a_{0}t/2\hbar$ where $a_{0}$ is the lattice constant. The pseudospin is denoted by the Pauli matrices $\sigma$ , and $s_{0}$ denotes the spin identity matrix. While Eqs.(2 - 5) provide an intuitive picture of the effect of each SOC term on the valleys, the results presented in the following sections are obtained by combining the S-matrix formalism with the tight-binding model for a zigzag terminated ribbon. This boundary condition preserves the valley quantum number and the valley topological properties of graphene. We compute the valley-polarized conductance with the Landauer-Büttiker approach: $$ G_{\kappa}^{n,m}=(e^{2}/h)\left|S_{\kappa}^{n,m}\right|^{2},\qquad(n,\ m\equiv L% ,R). \tag{6} $$ Here $S_{\kappa}^{n,m}$ is the scattering matrix element between left (L) and right (R) leads for a given valley index ${\kappa}=± 1$ [39]. Thus, our calculations exploit the formalism with valley-dependent local currents as defined in Ref. [40]. Details regarding the computation of the valley conductance and currents are presented in Appendices A and B. III Results In this section, we present numerical results for the valley-polarized conductance for a range of structures that contain from a single IQD ( $n=1$ ) to a chain ( $n>1$ ) of IQDs. To emphasize the qualitative features resulting from the competition among the different interactions in the model, we adjust the parameters’ values accordingly to present the main findings. This procedure is usually applied to identify the role played by the various interactions [41, 34, 42]. We note, however, that for accurate setups, one expects weaker values for SOCs from proximity effects, and we address this situation in Sec. III.2. III.1 Qualitative analysis of results Following Refs. [30] and [34], we solve the model using the following ranges for the various SOC parameters: $\lambda_{R}≤ 0.075t$ and $\lambda_{I}^{(A)}=-\lambda_{I}^{(B)}≤\sqrt{27}0.06t$ in units of $t=1$ [34]. We consider $n$ symmetric quantum dots ( $n=1,2,3,4$ ), with the same spin-orbit parameters, arranged in a chain with the same radius $r_{0}=7$ nm and the space in-between them set to $d=5r_{0}$ . The circular geometry of the dots is chosen to preserve the symmetries of the original model and provide maximum localization of valley states, as discussed below. We set the incident energy at $E_{F}=0.035t$ and consider a ribbon width $w=30$ nm, with zigzag boundaries. For this range of ISOC values, the valley Zeeman effect is the most important term as the topological invariant $Z_{2}$ remains trivial, i.e., $Z_{2}=0$ and the bulk system is in the same topological phase as long as the ISOC is staggered, i.e., $\lambda_{I}^{(A)}=-\lambda_{I}^{(B)}$ [34, 36]. The reduced number of group symmetries of pristine graphene due to the proximity of the TMD island is reflected by the large number of lower-symmetry allowed SOC parameters. The effects of these couplings are observed in Fig, 2 (a, b), where the band structure near the two Dirac cones appears clearly modified. Fig. 2 (a) shows results in the absence (a) and presence (b) of a staggered potential. The changes include a newly open gap and additional edge bands resulting from the ISOC. These linear bands represent edge states that are unique to the staggered case $\lambda_{I}^{(A)}=-\lambda_{I}^{(B)}$ while they disappear in the uniform regime $\lambda_{I}^{(A)}=\lambda_{I}^{(B)}$ . Similar results have been found in Ref. [34] where more details about the valley selection and its related sublattice occupation are discussed. Based on these findings, a proximity effect that induces a staggered ISOC might lead to the emergence of valley currents and the localization of valley-centered sublattice polarized edge states. Indeed, these results manifest themselves in transport, providing valley states as conducting channels, as shown in Fig. 2 (b), an issue discussed in the coming subsections. <details> <summary>2406.02393v1/x2.png Details</summary>  ### Visual Description # Technical Document Extraction: Energy Band Structure Analysis ## Panel (a) **Title**: `(a) λ_i^(A) = λ_i^(B) = √27 * 0.06t` **Axes**: - **X-axis**: `k[π/a]` (wavevector, range: -4.0 to -2.0) - **Y-axis**: `E (eV)` (energy, range: -0.4 to 0.4) **Key Features**: 1. **Energy Bands**: - **Red line**: `E = -0.4 eV` (bottom-most band) - **Green line**: `E = -0.2 eV` - **Blue line**: `E = 0.0 eV` (crosses at `k = -3.0 π/a`) - **Purple line**: `E = 0.2 eV` - **Brown line**: `E = 0.4 eV` (top-most band) 2. **Symmetry**: Bands are symmetric about `E = 0`. 3. **Crossing Points**: All bands intersect at `k = -3.0 π/a`. ## Panel (b) **Title**: `(b) λ_i^(A) = -λ_i^(B) = √27 * 0.06t` **Axes**: - **X-axis**: `k[π/a]` (wavevector, range: -4.0 to -2.0) - **Y-axis**: `E (eV)` (energy, range: -0.4 to 0.4) **Key Features**: 1. **Energy Bands**: - **Red line**: `E = -0.4 eV` (bottom-most band) - **Green line**: `E = -0.2 eV` - **Blue line**: `E = 0.0 eV` (crosses at `k = -3.0 π/a`) - **Purple line**: `E = 0.2 eV` - **Brown line**: `E = 0.4 eV` (top-most band) 2. **Inversion**: Bands are inverted compared to Panel (a). 3. **Crossing Points**: All bands intersect at `k = -3.0 π/a`. ## Legend (Implicit) - **Color-Code**: - Red: `E = -0.4 eV` - Green: `E = -0.2 eV` - Blue: `E = 0.0 eV` - Purple: `E = 0.2 eV` - Brown: `E = 0.4 eV` ## Observations 1. **Panel (a)**: - Bands exhibit parabolic dispersion near `k = -3.0 π/a`. - Symmetric energy distribution about `E = 0`. 2. **Panel (b)**: - Bands exhibit inverted dispersion compared to Panel (a). - Same crossing point at `k = -3.0 π/a` but with reversed energy slopes. 3. **Commonality**: - Both panels share identical energy values (`E = -0.4, -0.2, 0.0, 0.2, 0.4 eV`). - Wavevector range and grid structure are identical. ## Technical Notes - **Units**: Energy in electronvolts (eV), wavevector in reduced units (`k[π/a]`). - **Parameter**: `t` represents a material-specific parameter (e.g., hopping integral). - **Significance**: - Panel (a) represents symmetric coupling (`λ^(A) = λ^(B)`). - Panel (b) represents antisymmetric coupling (`λ^(A) = -λ^(B)`). - **Crossing Behavior**: Degenerate points at `k = -3.0 π/a` suggest topological or symmetry-related phenomena. </details> <details> <summary>2406.02393v1/x3.png Details</summary>  ### Visual Description # Technical Document Extraction: Conductance Oscillations in Quantum Systems ## Main Plot (Panel c) ### Axes and Labels - **X-axis**: $\lambda_I/t$ (dimensionless interaction strength) - **Y-axis**: $G (e^2/h)$ (conductance normalized by quantum of conductance) - **Subplots**: - Top: $K_1$ (upper conductance regime) - Bottom: $K_2$ (lower conductance regime) ### Legend - **$n=1$**: Black curve (lowest conductance) - **$n=2$**: Blue curve - **$n=3$**: Red curve - **$n=4$**: Green curve (highest conductance) ### Parameters - $\lambda_R = 0.015t$ - $\Delta = 0.005t$ - $E_F = +0.035t$ ### Key Trends 1. **Oscillatory Behavior**: All $n$ curves exhibit periodic oscillations in $G$ as $\lambda_I/t$ increases. 2. **Amplitude Modulation**: - $n=1$ (black) shows the most pronounced oscillations. - Higher $n$ values (blue, red, green) exhibit damped oscillations with reduced amplitude. 3. **Resonance Peaks**: - Peaks in $G$ occur at regular intervals of $\lambda_I/t$. - $K_1$ (top) displays sharper peaks compared to $K_2$ (bottom). ## Inset: Zoom in $K_2$ (Panel a) ### Axes and Labels - **X-axis**: $\lambda_I/t$ (range: 0.055–0.065) - **Y-axis**: $G (e^2/h)$ (range: 0–1.2) - **Focus**: Highlights fine structure of oscillations near resonance. ### Key Observations - **Damping Effect**: Higher $n$ curves (blue, red, green) show rapid decay in $G$ compared to $n=1$ (black). - **Phase Shifts**: - $n=2$ (blue) and $n=3$ (red) exhibit phase differences relative to $n=1$. - $n=4$ (green) shows a distinct secondary peak near $\lambda_I/t \approx 0.06$. ## Inset: Zoom in $K_1$ (Panel b) ### Axes and Labels - **X-axis**: $\lambda_I/t$ (range: 0.055–0.065) - **Y-axis**: $G (e^2/h)$ (range: 2.0–3.2) - **Focus**: Details oscillations in the higher conductance regime. ### Key Observations - **Peak Sharpness**: $n=1$ (black) maintains the sharpest resonance peak. - **Interference Patterns**: - $n=2$ (blue) and $n=3$ (red) display destructive interference near $\lambda_I/t \approx 0.06$. - $n=4$ (green) exhibits a secondary peak at $\lambda_I/t \approx 0.062$. ## Cross-Referenced Legend Consistency - **Color-Line Matching**: - All panels confirm $n=1$ (black), $n=2$ (blue), $n=3$ (red), $n=4$ (green). - No discrepancies observed between legend labels and curve colors. ## Summary The plots demonstrate quantized conductance oscillations in a system with parameters $\lambda_R = 0.015t$, $\Delta = 0.005t$, and $E_F = +0.035t$. The oscillations exhibit $n$-dependent damping and phase shifts, with sharper features in $K_1$ compared to $K_2$. Zoomed insets reveal fine interference patterns critical for understanding many-body effects. </details> <details> <summary>2406.02393v1/x4.png Details</summary>  ### Visual Description # Technical Document Extraction: Graph Analysis ## Main Chart (Left Panel) ### Subplots - **K1 (Top Subplot)** - **X-axis**: λ_I/t (ranging from 0.00 to 0.12) - **Y-axis**: G (e²/ħ) (ranging from 0.0 to 4.0) - **Lines**: - **n=1** (Black): Oscillatory trend with peaks at ~3.5, ~2.5, and ~1.5. Amplitude decreases slightly with increasing λ_I/t. - **n=2** (Blue): Similar oscillations to n=1 but with reduced amplitude (~3.0 to ~2.0). - **n=3** (Red): Further reduced amplitude (~2.5 to ~1.8). - **n=4** (Green): Lowest amplitude (~2.0 to ~1.5). - **K2 (Bottom Subplot)** - **X-axis**: λ_I/t (same range as K1) - **Y-axis**: G (e²/ħ) (same scale as K1) - **Lines**: - **n=1** (Black): Peaks at ~3.5, ~2.5, and ~1.5. Slightly sharper oscillations than K1. - **n=2** (Blue): Peaks at ~3.0, ~2.0, and ~1.5. - **n=3** (Red): Peaks at ~2.5, ~1.8, and ~1.3. - **n=4** (Green): Peaks at ~2.0, ~1.5, and ~1.0. ### Parameters - λ_R = 0.015t - Δ = 0.005t - E_F = -0.035t ## Zoomed-In Charts (Right Panels) ### K1 Zoom (Left Right Panel) - **X-axis**: λ_I/t (0.055 to 0.065) - **Y-axis**: G (e²/ħ) (0.0 to 1.4) - **Lines**: - **n=1** (Black): Sharp dip from ~1.2 to ~0.8, then rise to ~1.0. - **n=2** (Blue): Dips to ~0.6, then rises to ~0.8. - **n=3** (Red): Dips to ~0.4, then rises to ~0.6. - **n=4** (Green): Dips to ~0.2, then rises to ~0.4. ### K2 Zoom (Right Right Panel) - **X-axis**: λ_I/t (0.055 to 0.065) - **Y-axis**: G (e²/ħ) (0.0 to 3.2) - **Lines**: - **n=1** (Black): Sharp dip from ~3.0 to ~2.5, then rise to ~2.8. - **n=2** (Blue): Dips to ~2.6, then rises to ~2.8. - **n=3** (Red): Dips to ~2.4, then rises to ~2.6. - **n=4** (Green): Dips to ~2.2, then rises to ~2.4. ## Legend - **Location**: Right side of the main chart. - **Labels**: - **n=1**: Black - **n=2**: Blue - **n=3**: Red - **n=4**: Green ## Key Observations 1. **Oscillatory Behavior**: All lines exhibit periodic oscillations in G as λ_I/t increases. 2. **Amplitude Decay**: Higher n values correspond to lower oscillation amplitudes. 3. **Zoomed-In Details**: In the narrow λ_I/t range (0.055–0.065), lines exhibit sharper features and crossings, indicating sensitivity to small parameter changes. 4. **Parameter Influence**: The constants λ_R, Δ, and E_F likely modulate the oscillation frequency and amplitude. ## Spatial Grounding - **Legend Colors**: Confirmed matches with line colors in all subplots. - **Axis Labels**: Consistent across all panels. ## Notes - No numerical data table is present; trends are inferred visually. - Shaded regions in the main chart are unlabeled and likely serve as visual guides. </details> <details> <summary>2406.02393v1/x5.png Details</summary>  ### Visual Description # Technical Document Extraction: Graphs (e) and (f) ## Graph (e) - **Axes**: - **x-axis**: `d/r₀` (dimensionless ratio of distance to reference length `r₀`) - **y-axis**: `G (e²/ħ)` (conductance normalized by quantum of conductance) - **Legend**: - `K₁` (red squares) - `K₂` (blue circles) - **Key Trends**: - **K₁**: - Starts at ~0.75 at `d/r₀ = 0` - Drops to ~0.25 at `d/r₀ = 2` - Rises sharply to ~2.0 at `d/r₀ = 4` - Remains constant at ~2.0 for `d/r₀ ≥ 4` - **K₂**: - Starts at ~0.1 at `d/r₀ = 0` - Peaks at ~0.5 at `d/r₀ = 4` - Drops to near-zero for `d/r₀ ≥ 4` - **Parameters**: - `λ_R = λ_I = 0.015t` - `Δ = 0.005t` - `E_F = 0.035t` ## Graph (f) - **Axes**: - **x-axis**: `n` (integer index, 0–10) - **y-axis**: `G (e²/ħ)` (same as graph (e)) - **Legend**: - `K₁` (red squares) - `K₂` (blue circles) - **Key Trends**: - **K₁**: - Constant at ~2.0 for all `n` - **K₂**: - Constant at ~0.0 for all `n` - **Embedded Text**: - Box annotation: `d = 5r₀` (indicates fixed distance in graph (f)) ## Cross-Reference Legend Consistency - **Graph (e)**: - Red squares (`K₁`) match red square markers in the plot. - Blue circles (`K₂`) match blue circle markers in the plot. - **Graph (f)**: - Red squares (`K₁`) match red square markers in the plot. - Blue circles (`K₂`) match blue circle markers in the plot. ## Observations 1. **Graph (e)** shows conductance `G` as a function of normalized distance `d/r₀`, with `K₁` and `K₂` exhibiting distinct behaviors (step-like increase for `K₁`, transient peak for `K₂`). 2. **Graph (f)** isolates the behavior at a fixed distance `d = 5r₀`, where `K₁` dominates (`G ≈ 2.0`) and `K₂` vanishes (`G ≈ 0.0`). 3. Parameters (`λ_R`, `λ_I`, `Δ`, `E_F`) are identical for both graphs, suggesting they describe the same system under varying spatial conditions. </details> Figure 2: Energy bands for uniform $\lambda_{I}^{(A)}=\lambda_{I}^{(B)}$ (a), and staggered $\lambda_{I}^{(A)}=-\lambda_{I}^{(B)}$ ) (b) for a 30 nm wide zigzag ribbon, $\Delta=0.1t$ and $\lambda_{R}=0.075t$ . Panels (c) and (d) show valley conductances vs intrinsic spin-orbit length $\lambda_{I}$ for several IQDs with staggered ISOC. Narrow panels on the right correspond to zoom-ins for each valley as a function of $\lambda_{I}$ . Panels (e) and (f) correspond to conductance vs inter-dot spacing $d$ and number $n$ of IQDs. III.1.1 Valley-dependent conductance through proximity IQDs Figure 2 (c) and (d) display the conductance through a group of $n$ symmetric quantum dots with staggered ISOC. One interesting observation is the approximately square-type dependence revealing wide gaps that can be made more pronounced by changing the strength of ISOC for $n≥ 3$ QDs. We observe that $100\%$ ( $0\%$ ) of the detected conductance results from the flow of electrons through the $\kappa=-1$ ( $\kappa=+1$ ) valley for positive incident energy ( $E_{F}=0.035t$ ), while $0\%$ ( $100\%$ ) occurs for negative incident energy ( $E_{F}=-0.035t$ ). The opposite behavior is obtained for the complementary valley. Interestingly, the gaps occur at different ranges of ISOC values, with the emerging valley-polarized current being switched from one valley to the other within the gap region by an appropriate change of $E_{F}$ . Furthermore, we observe a decrease in the widths of the gaps with increasing SOC strength, which suggests they might vanish for high enough values of ISOC. An analysis of Figs. 2 (c) and (d) reveal resonances in the transmittance at around the value $\lambda_{I}=0.065t$ , with the spectrum in the zoom-in panels showing that the IQDs confine electrons with index $\kappa=+1$ ( $\kappa=-1$ ) at positive (negative) energy. Hence, the scattering through the IQD region tends to zero accordingly. We observe that the number and sharpness of the resonance weakly depend on the number of IQDs along the chain, as seen for $n=1,2,3$ curves that present at least one resonant state each at similar values of ISOCs. More details about the observed confinement are addressed below in Sec. III.2.3, and we discuss these results, including realistic parameters, in III.3. Additionally, the conductance response shows several interesting characteristics: (1) It exhibits an oscillating behavior that becomes more pronounced with increasing $n$ . In this case, the conductance oscillations arise from mode mixing, and their number depends on the number $n$ of the IQDs, as shown in the zoom-in of Fig. 2 (c) and (d). (2) The conductance plateaux become better defined as $n$ increases ( $n≥ 3$ ), with values $G(\kappa=+1)=0$ and $G(\kappa=-1)=2G_{0}$ for panel (c) and $G(\kappa=+1)=2G_{0}$ and $G(\kappa=-1)=0$ for panel (d). These results suggest that the conductance plateaux are due to states that become valley polarized at specific strengths of the staggered ISOC, e.g., in the range $0.015t$ $≤\lambda_{I}^{(A)}=-\lambda_{I}^{(B)}≤ 0.04t$ as shown in Fig. 2 (c). Interestingly, when $E_{F}$ is negative, the valley polarization is reversed, as shown in panel (d) within the same range of values for $\lambda_{I}$ . Consequently, the transmitted current can be made to be valley polarized from either one of the two Dirac points ${\bf K_{1}}$ or ${\bf K_{2}}$ depending on the incident energy. III.1.2 Dependence on coherent inter-dot electron transfer Within the conductance gap regions in Fig. 2 (c) and (d), and depending on the bias defined by the sign of the Fermi energy, the valley that appears in the output with $T=1$ seems to be barely scattered by the IQDs irrespective of the number of dots in the chain. Inversely, the states from the valley that appear in the output with $T=0$ are strongly reflected by the IQDs even for the shortest chain furnished by only one dot. To better understand these current profiles, we plot in Fig. 3 the local valley current through a chain of three IQDs. As the figures show, the transferred valley current is due to electron scattering processes that involve inter-dot hoppings between neighboring dots, as illustrated in the figure, that display uniform local densities everywhere between the dots for both valleys (Fig. 3 (a) and (d)). Notice also that in this case, the conductance is practically the same as for the zigzag terminated graphene ribbon with one or more quantum dots, as depicted in Fig. 2 (f). <details> <summary>2406.02393v1/x6.png Details</summary>  ### Visual Description # Technical Document Extraction: Heatmap Visualizations of K₁ and E_F Parameters ## Panel (a): K₁ = +0.035t - **Title**: (a) K₁: E_F = +0.035t - **Axes**: - **X-axis**: `length (nm)` (direction: left to right, indicated by black arrow) - **Y-axis**: `width (nm)` (range: -10 nm to +10 nm) - **Color Scale**: - Right-hand color bar (0.0 to 0.6, red = highest intensity) - Red/yellow gradient indicates data intensity (higher values in red regions) - **Key Features**: - Concentrated red/yellow regions near `length = -40 nm` and `length = +40 nm` - Gradual intensity decay toward the center (`length = 0 nm`) - Symmetric pattern across `width = 0 nm` ## Panel (c): K₁ = -0.035t - **Title**: (c) K₁: E_F = -0.035t - **Axes**: - **X-axis**: `length (nm)` (direction: left to right, indicated by black arrow) - **Y-axis**: `width (nm)` (range: -10 nm to +10 nm) - **Color Scale**: - Right-hand color bar (0.0 to 0.6, red = highest intensity) - Red/yellow gradient indicates data intensity (higher values in red regions) - **Key Features**: - Concentrated red/yellow regions near `length = -40 nm` and `length = +40 nm` - Gradual intensity decay toward the center (`length = 0 nm`) - Symmetric pattern across `width = 0 nm` ## Cross-Referenced Observations 1. **Legend Consistency**: - Both panels use identical color scales (0.0–0.6, red = high intensity). - Red/yellow gradients align with intensity values across both visualizations. 2. **Parameter Impact**: - Positive K₁ (+0.035t) and negative K₁ (-0.035t) produce nearly identical spatial patterns. - Symmetry in intensity distribution suggests K₁ magnitude (not sign) drives the observed trends. 3. **Directionality**: - Arrows on the `length` axis confirm data flow from left (`-40 nm`) to right (`+40 nm`). ## Notes - No explicit data table or numerical values provided beyond axis labels and parameter annotations. - Visualizations emphasize spatial distribution of intensity (likely related to electronic or optical properties) under varying K₁ and E_F conditions. </details> <details> <summary>2406.02393v1/x7.png Details</summary>  ### Visual Description # Technical Document Extraction: Heatmap Analysis of K₂ and E_F Parameters ## Panel (b) - K₂: E_F = +0.035t ### Axis Labels - **X-axis**: `length (nm)` (direction: left to right) - **Y-axis**: `width (nm)` (range: -10 nm to +10 nm) ### Color Scale Legend - **Gradient**: White (0.0) → Red (0.6) - **Key Observation**: - High-intensity regions (red) localized near the edges of the plot (left side). - Gradual decay of intensity toward the center and right side. ### Diagram Components - **Arrows**: Black arrow on the X-axis indicating increasing length. - **Color Distribution**: - Blue-to-red gradient indicates varying intensity values. - Symmetric patterns observed along the Y-axis (width). --- ## Panel (d) - K₂: E_F = -0.035t ### Axis Labels - **X-axis**: `length (nm)` (direction: left to right) - **Y-axis**: `width (nm)` (range: -10 nm to +10 nm) ### Color Scale Legend - **Gradient**: White (0.0) → Red (0.6) - **Key Observation**: - High-intensity regions (red) concentrated near the center (X ≈ 0 nm). - Oscillatory patterns with alternating intensity bands along the X-axis. ### Diagram Components - **Arrows**: Black arrow on the X-axis indicating increasing length. - **Color Distribution**: - Blue-to-red gradient with periodic intensity variations. - Asymmetric patterns compared to Panel (b). --- ## Cross-Reference: Legend Colors vs. Diagram - **Panel (b)**: Red regions align with the highest intensity values (0.6) on the legend. - **Panel (d)**: Red regions correspond to peak intensity values, with blue regions near 0.0. ## Summary of Trends 1. **Panel (b)**: Edge-localized high-intensity features under positive E_F. 2. **Panel (d)**: Centralized high-intensity features with oscillatory decay under negative E_F. 3. **Color Consistency**: Red = maximum intensity (0.6), White = baseline (0.0) across both panels. </details> <details> <summary>2406.02393v1/x8.png Details</summary>  ### Visual Description # Technical Document Extraction: Heatmap Analysis of K₁ and E_F Parameters ## Panel (e): K₁ with E_F = +0.035t - **Axis Labels**: - **X-axis**: `length (nm)` (range: -40 nm to +40 nm) - **Y-axis**: `width (nm)` (range: -10 nm to +10 nm) - **Color Bar**: - **Range**: 0.0 (white) to 0.6 (dark red) - **Label**: Intensity scale (no explicit units) - **Key Observations**: - **Red Intensity**: Concentrated on the right side of the plot (length > 20 nm). - **Gradient**: Smooth transition from white (low intensity) to red (high intensity). - **Symmetry**: Mirror-like pattern across the width axis (y = 0 nm). ## Panel (g): K₁ with E_F = -0.035t - **Axis Labels**: - **X-axis**: `length (nm)` (range: -40 nm to +40 nm) - **Y-axis**: `width (nm)` (range: -10 nm to +10 nm) - **Color Bar**: - **Range**: 0.0 (white) to 0.6 (dark red) - **Label**: Intensity scale (no explicit units) - **Key Observations**: - **Red Intensity**: Distributed across the entire plot with localized peaks near the edges (length ≈ ±40 nm). - **Wave-like Patterns**: Oscillatory intensity variations along the length axis. - **Asymmetry**: No mirror symmetry; intensity peaks are offset relative to the width axis. ## Cross-Referenced Legend Consistency - **Color Bar Alignment**: - Both panels use identical color scales (0.0–0.6), ensuring consistent intensity interpretation. - Red regions in both panels correspond to values ≥ 0.4, as per the color bar. ## Summary of Trends 1. **Positive E_F (+0.035t)**: - High-intensity regions are spatially confined to the right half of the plot. - Suggests directional dependence of K₁ under positive E_F. 2. **Negative E_F (-0.035t)**: - Intensity variations are more distributed, with edge-localized peaks. - Indicates a different spatial modulation of K₁ under negative E_F. ## Diagram Components - **Arrows**: Black arrows on the x-axis indicate the direction of increasing length. - **Heatmap Grid**: Overlaid red/yellow lines represent intensity gradients. - **Panel Titles**: Explicitly label the parameter conditions (K₁ and E_F values). ## Data Extraction Notes - No explicit numerical data table is present; values are inferred from color intensity. - Critical parameters: K₁ (dependent variable), E_F (independent variable with ±0.035t values). </details> <details> <summary>2406.02393v1/x9.png Details</summary>  ### Visual Description # Technical Document Extraction: Image Analysis ## Panel (f) K₂: E_F = +0.035t - **Axes**: - **X-axis**: `length (nm)` (range: -40 to 40 nm) - **Y-axis**: `width (nm)` (range: -10 to 10 nm) - **Colorbar**: - **Label**: Intensity (gradient from 0.0 to 0.4) - **Direction**: ↑ (increasing intensity) - **Color Scale**: Blue (low) → Red (high) - **Key Features**: - Blue/orange streamline patterns indicating vector field direction. - Concentrated intensity gradients near `length = ±40 nm` and `width = ±10 nm`. - Symmetric flow patterns around the center (`length = 0 nm`). ## Panel (h) K₂: E_F = -0.035t - **Axes**: - **X-axis**: `length (nm)` (range: -40 to 40 nm) - **Y-axis**: `width (nm)` (range: -10 to 10 nm) - **Colorbar**: - **Label**: Intensity (gradient from 0.0 to 0.6) - **Direction**: ↑ (increasing intensity) - **Color Scale**: Blue (low) → Red (high) - **Key Features**: - Asymmetric flow patterns with dominant gradients near `length = +40 nm` and `width = ±10 nm`. - Reduced intensity near the center compared to panel (f). ## Cross-Reference: Legend & Line Colors - **Legend**: Color gradient represents intensity (confirmed via colorbar labels). - **Line Colors**: - Blue: Low-intensity regions. - Orange/Red: High-intensity regions (matches colorbar scale). ## Observations 1. **E_F Sign Impact**: - Positive `E_F` (+0.035t) in (f) shows symmetric flow. - Negative `E_F` (-0.035t) in (h) shows asymmetric flow with stronger right-side gradients. 2. **Intensity Distribution**: - Panel (h) has a higher maximum intensity (0.6 vs. 0.4 in (f)). 3. **Flow Dynamics**: - Arrows in (f) suggest balanced bidirectional flow. - Arrows in (h) indicate dominant rightward flow near the edges. ## Notes - No explicit data table present; information inferred from vector fields and color gradients. - All axis labels, panel titles, and colorbar annotations are transcribed verbatim. </details> Figure 3: Local current mapping, within the gap regions ( $\lambda_{I}^{(A)}=-\lambda_{I}^{(B)}=0.025t$ ) shown in Fig. 2 (c) and (d), for both valleys for a given bias voltage while panels (e) to (h) show results for opposite biases. The left (right) panels are for positive (negative) values of $E_{F}$ . Black arrows indicate the direction of the bias-incident current. Red and blue solid lines indicate the direction of corresponding current flows. The above observations raise the question of the inter-dot spacing $d$ ’s influence on the valley-transmission output. We have proposed that the valley-scattering processes for complete transmission occur in the presence of effective inter-dot hopping between neighboring IQDs. To test this hypothesis, we analyze the role of $d$ on the valley transmittance. As shown in Fig. 2 (e), the inter-dot hopping is present with an inter-dot space $d≥ 5r_{0}$ . We observe non-stable transmission signatures if $d≤ 5r_{0}$ . In this regime, the edges of the IQDs are closer to each other, and the transmittance peaks show values less (more) than unity (zero) for the ${\bf K_{1}}$ ( ${\bf K_{2}}$ ) valley. These might be related to mode mixing that impedes the propagation of specific transverse modes due to edge-dot coupling. Interestingly, when the $d$ value is large enough, the valley transmittance becomes stable, represented by steady curves with a uniform local density between the dots. Finally, we analyze the role of the staggered potential $\Delta$ on the valley-transmission output, as shown in Fig. 4. The staggered potential opens a gap, playing the role of a potential barrier. Consequently, the valley imbalance is maintained for all values of $E_{F}<\Delta$ . However, the valley polarization is rapidly destroyed when $E_{F}≥\Delta$ . When the confining potential of the IQD is bigger than the incident energy, the current is completely blocked, and only resonant states from each valley ${\bf K_{1}}$ $({\bf K_{2}})$ are allowed to transmit current. In this context, the choice of incident energy must consider the strength of the IQD confinement potential. Fig. 4 (b) and (d) show that the conductance is non-zero only for particular resonances where scatterings occur at specific values of the ISOCs and energies suited by edge-dot coupling [43, 44]. In this context, the heterostructures of graphene and hBN would exhibit this regime since, in this system, the confinement potential $\Delta$ is significant [45] and therefore, its influence on the valley filtering process would have a negative impact. The results discussed in Fig. 4 are essential since they allow us to establish a criterion for defining the best materials for islands. Indeed, for a better response, it is necessary to use a material that provides a weak or zero confinement potential. If so, the IQDs will operate efficiently, and the overall system might be used to monitor valley-driven current by either tuning the ISOC or the RSOC. The question we might ask then is which materials are better suited for this response. Such a case would be materialized in heterostructures of twisted graphene and monolayers of transition-metal dichalcogenides (TMDCs). In such scenarios, the twisting angle substantially decreases the confinement potential, and the dominant parameters will be the valley-Zeeman and RSOCs [35]. A concrete example of a realistic proximity effect is discussed later on. An alternative description of these effects is by considering the area around the IQDs with wider staggered potentials as an electron/hole bilayer system where the electrons are essentially required to overcome an offset barrier to be scattered through. This action is analogous to the massless-massive electron-hole system in a transverse electric field in graphene on a TMD substrate. In this context, the value of the Rashba coupling has a direct effect since it impacts the offset barrier. Due to its complexity, we postpone the study of these effects for future work. (For more details, see Refs. [29] and [28].) <details> <summary>2406.02393v1/x10.png Details</summary>  ### Visual Description # Technical Document Extraction: Conductance Spectra Analysis ## Graph (a): Δ = 0 meV (Black) vs Δ = 5 meV (Blue) - **Parameters**: - Rashba splitting: λ_R = 10 meV - Fermi energy: E_F = 35 meV - **Axes**: - X-axis: λ_I/t (0.000 to 0.150) - Y-axis: Conductance G (e²/h) (0 to 4) - **Key Features**: - **K1 Region**: - Δ = 0 meV: Sharp, periodic peaks at λ_I/t ≈ 0.025, 0.075, 0.125. - Δ = 5 meV: Peaks slightly suppressed and broadened compared to Δ = 0. - **K2 Region**: - Δ = 0 meV: Peaks at λ_I/t ≈ 0.050, 0.100, 0.150. - Δ = 5 meV: Peaks reduced in height and spacing irregular. - **Trend**: Increasing Δ suppresses conductance peaks, particularly in K2. ## Graph (b): Δ = 40 meV (Green) vs Δ = 50 meV (Pink) - **Parameters**: - Rashba splitting: λ_R = 10 meV - Fermi energy: E_F = 35 meV - **Axes**: - X-axis: λ_I/t (0.000 to 0.150) - Y-axis: Conductance G (e²/h) (0 to 4) - **Key Features**: - **K1 Region**: - Δ = 40 meV: Sharp, narrow peaks at λ_I/t ≈ 0.025, 0.075, 0.125. - Δ = 50 meV: Peaks broadened and less intense. - **K2 Region**: - Δ = 40 meV: Peaks at λ_I/t ≈ 0.050, 0.100, 0.150 with moderate height. - Δ = 50 meV: Peaks taller but fewer (e.g., λ_I/t ≈ 0.075, 0.125). - **Trend**: Higher Δ (50 meV) enhances peak intensity in K2 but reduces resolution. ## Cross-Reference Summary - **Legend Consistency**: - Graph (a): Black (Δ = 0) and Blue (Δ = 5) match line colors. - Graph (b): Green (Δ = 40) and Pink (Δ = 50) align with line colors. - **Common Observations**: - Conductance peaks correlate with λ_I/t values near K1/K2 regions. - Larger Δ values (5 meV, 40 meV, 50 meV) generally suppress or broaden peaks compared to Δ = 0. - K2 region exhibits more pronounced Δ-dependent variations than K1. </details> <details> <summary>2406.02393v1/x11.png Details</summary>  ### Visual Description # Technical Document Extraction: Graph Analysis ## Graph (c) **Title**: `(c) λ_R=15 (meV), E_F=-35 (meV)` **X-axis**: `λ_I/t` (ranging from 0.000 to 0.150) **Y-axis**: `G (e²/h)` (ranging from 0 to 4) ### Legend - `Δ=0 (meV)`: Black line - `Δ=5 (meV)`: Blue line ### Key Observations - **K1** and **K2** labels marked in boxes at specific λ_I/t positions. - **Δ=0 (black line)**: - Sharp peaks at λ_I/t ≈ 0.025, 0.075, 0.125 (K1 and K2 regions). - Conductance (G) reaches ~4 e²/h at these peaks. - **Δ=5 (blue line)**: - Peaks slightly broadened and shifted compared to Δ=0. - Reduced conductance magnitude (~3 e²/h) at K1/K2 regions. --- ## Graph (d) **Title**: `(d) λ_R=15 (meV), E_F=-35 (meV)` **X-axis**: `λ_I/t` (ranging from 0.000 to 0.150) **Y-axis**: `G (e²/h)` (ranging from 0 to 4) **Shaded Region**: Gray area between 0 and 2 e²/h on the y-axis. ### Legend - `Δ=40 (meV)`: Green line - `Δ=50 (meV)`: Pink line ### Key Observations - **K1** label marked in a box at a specific λ_I/t position. - **Δ=40 (green line)**: - Multiple narrow peaks at λ_I/t ≈ 0.025, 0.05, 0.075, 0.1, 0.125. - Conductance (G) reaches ~1.5–2 e²/h at these peaks. - **Δ=50 (pink line)**: - Peaks are sharper and more pronounced than Δ=40. - Conductance (G) reaches ~1–1.5 e²/h at K1 region. - Additional peaks observed at λ_I/t ≈ 0.075 and 0.125. --- ## Cross-Reference Summary | Legend Label | Color | Graph | Conductance Behavior | |--------------|-------|-------|----------------------| | Δ=0 (meV) | Black | (c) | Sharp, high peaks at K1/K2 | | Δ=5 (meV) | Blue | (c) | Broadened, lower peaks at K1/K2 | | Δ=40 (meV) | Green | (d) | Multiple narrow peaks, moderate G | | Δ=50 (meV) | Pink | (d) | Sharper peaks, localized at K1 | **Note**: Both graphs share identical λ_R and E_F parameters. The shaded region in (d) highlights a baseline conductance range (0–2 e²/h), potentially indicating a threshold for significant conductance features. </details> Figure 4: Valley conductance vs intrinsic spin-orbit length $\lambda_{I}$ for different values of staggered potential $\Delta$ . The left (right) panels correspond to $\Delta<E_{F}$ ( $\Delta>E_{F}$ ). III.1.3 Valley-Hall and bulk conductivities We briefly discuss the emergence of Hall and bulk conductivities in these structures. We visualize the origin of these conductivities by mapping the local current flow and highlighting the valley polarization with solid blue (red) curves indicating the ${\bf K_{1}}$ ( ${\bf K_{2}}$ ) index in Fig. 5. Analysis of the figure reveals that the generated valley currents are composed of bulk-driven and Hall-driven currents. In this process, the local current where both valleys are scattered shows that each valley is conducting with either bulk or Hall currents depending on the sign of the Fermi energy. This analysis suggests that a periodic array of dots with the best choice of SOCs offers an alternative mechanism for generating valley-neutral Hall currents since both valleys contribute to the current in the same direction, although through different regions. A realistic example is discussed in Sec. III.3 where we find induced SOC terms that facilitate valley-Hall current for a given TMDs island and twist angle. <details> <summary>2406.02393v1/x12.png Details</summary>  ### Visual Description # Technical Document Extraction: Heatmap Analysis of K₁ and E_F Parameters ## Panel (a): K₁: E_F = +0.035t - **Axis Labels**: - X-axis: `length (nm)` (range: -40 nm to +40 nm) - Y-axis: `width (nm)` (range: -10 nm to +10 nm) - **Color Scale**: - Right legend: `0.0` (white) to `0.4` (dark red) - Indicates magnitude of a measured variable (likely intensity or density) - **Key Features**: - Symmetrical flow patterns centered at `length = 0 nm` - High-intensity (dark red) regions concentrated near `width = ±10 nm` - Arrows indicate directional flow (rightward along x-axis) - Title: `K₁: E_F=+0.035t` (Fermi energy positive) ## Panel (c): K₁: E_F = -0.035t - **Axis Labels**: - X-axis: `length (nm)` (range: -40 nm to +40 nm) - Y-axis: `width (nm)` (range: -10 nm to +10 nm) - **Color Scale**: - Right legend: `0.0` (white) to `0.4` (dark red) - Indicates magnitude of a measured variable (likely intensity or density) - **Key Features**: - Asymmetric flow patterns with disrupted symmetry - High-intensity regions localized near `length = ±40 nm` - Arrows indicate directional flow (rightward along x-axis) - Title: `K₁: E_F=-0.035t` (Fermi energy negative) ## Cross-Reference Observations 1. **Legend Consistency**: - Both panels use identical color scales (0.0–0.4), confirming comparable measurement ranges. - Red intensity correlates with higher values in both cases. 2. **Parameter Impact**: - Positive E_F (`+0.035t`) in (a) produces symmetrical flow. - Negative E_F (`-0.035t`) in (c) disrupts symmetry, suggesting parameter sensitivity. 3. **Structural Differences**: - Panel (a): Dominant central flow with mirrored lobes. - Panel (c): Flow concentrated at edges, reduced central coherence. ## Technical Notes - Units: All spatial dimensions in nanometers (nm). - Directionality: Arrows on x-axis confirm rightward propagation. - No data tables present; visualization relies on heatmap intensity and flow vectors. </details> <details> <summary>2406.02393v1/x13.png Details</summary>  ### Visual Description # Technical Document Extraction: Heatmap Visualization of K₂ and E_F Parameters ## Panel (b): K₂ with E_F = +0.035t - **Title**: `(b) K₂: E_F = +0.035t` - **Axes**: - **X-axis**: `length (nm)` (range: -40 to +40 nm, arrow pointing right) - **Y-axis**: `width (nm)` (range: -10 to +10 nm) - **Color Scale**: - **Legend**: Right of panel, gradient from `0.0` (white) to `0.4` (red) - **Intensity**: Represents magnitude of K₂ parameter (higher values = redder regions) - **Key Features**: - Symmetric flow patterns along the length axis. - Concentrated intensity near the center (width ≈ 0 nm). - Gradual decay of intensity toward the edges (width ≈ ±10 nm). ## Panel (d): K₂ with E_F = -0.035t - **Title**: `(d) K₂: E_F = -0.035t` - **Axes**: - **X-axis**: `length (nm)` (range: -40 to +40 nm, arrow pointing right) - **Y-axis**: `width (nm)` (range: -10 to +10 nm) - **Color Scale**: - **Legend**: Right of panel, gradient from `0.0` (white) to `0.3` (red) - **Intensity**: Represents magnitude of K₂ parameter (higher values = redder regions) - **Key Features**: - Asymmetric flow patterns with alternating high/low intensity regions. - Two distinct intensity peaks near the center (width ≈ ±5 nm). - Reduced overall intensity compared to panel (b). ## Cross-Reference Analysis - **Legend Consistency**: - Panel (b) uses a broader color scale (0.0–0.4) than panel (d) (0.0–0.3), reflecting differing parameter ranges. - Red regions in both panels correspond to the highest measured K₂ values under their respective E_F conditions. - **Flow Direction**: - Arrows on the x-axis confirm data flow from left (-40 nm) to right (+40 nm). - No explicit flow vectors in the heatmaps; intensity gradients imply directional trends. ## Observations 1. **E_F Influence**: - Positive E_F (+0.035t) in panel (b) produces more uniform intensity distribution. - Negative E_F (-0.035t) in panel (d) introduces asymmetry and localized intensity peaks. 2. **Parameter Sensitivity**: - K₂ values are highly dependent on the sign and magnitude of E_F. - Intensity thresholds (color scale maxima) vary between panels, suggesting differing measurement scales or normalization. ## Notes - No explicit data table or numerical values are provided; interpretation relies on color gradients and axis labels. - Time dependence (`t`) in E_F suggests dynamic conditions, though temporal resolution is not visualized. </details> <details> <summary>2406.02393v1/x14.png Details</summary>  ### Visual Description # Technical Document Extraction: Heatmap Analysis of K₁ and E_F Variations ## Panel (e): K₁ with E_F = +0.035t - **Axis Labels**: - X-axis: `length (nm)` (range: -40 nm to +40 nm) - Y-axis: `width (nm)` (range: -10 nm to +10 nm) - **Color Scale**: - Right color bar: `0.0` (white) to `0.4` (dark red) - **Key Observations**: - Symmetric intensity distribution along the Y-axis. - Gradual increase in intensity (red) toward the edges of the X-axis (±40 nm). - Central region (X ≈ 0 nm) exhibits lower intensity (white/yellow). - Arrows indicate directional flow from left to right along the X-axis. ## Panel (g): K₁ with E_F = -0.035t - **Axis Labels**: - X-axis: `length (nm)` (range: -40 nm to +40 nm) - Y-axis: `width (nm)` (range: -10 nm to +10 nm) - **Color Scale**: - Right color bar: `0.0` (white) to `0.3` (dark red) - **Key Observations**: - Asymmetric intensity distribution with localized peaks. - Two prominent high-intensity regions (red) at X ≈ ±20 nm. - Central region (X ≈ 0 nm) shows reduced intensity (white). - Arrows indicate directional flow from left to right along the X-axis. ## Cross-Referenced Trends - **Color Consistency**: - Red corresponds to higher intensity values in both panels. - White/yellow indicates lower intensity values. - **E_F Impact**: - Positive E_F (+0.035t) in panel (e) results in broader, symmetric intensity distribution. - Negative E_F (-0.035t) in panel (g) creates localized, asymmetric intensity peaks. - **Directionality**: - Arrows in both panels confirm left-to-right flow along the X-axis. ## Summary The heatmaps visualize the spatial distribution of intensity (color-coded) for two scenarios of K₁ with opposing E_F values. Panel (e) demonstrates a uniform, edge-dominated pattern, while panel (g) exhibits localized intensity maxima at ±20 nm. The directional flow and symmetry differences highlight the influence of E_F on the system's behavior. </details> <details> <summary>2406.02393v1/x15.png Details</summary>  ### Visual Description # Technical Document Extraction: Heatmap Analysis of K₂ with Varying E_F ## Image Overview The image contains two side-by-side heatmaps labeled **(f)** and **(h)**, depicting streamline flow patterns in a 2D spatial domain. Both panels share identical axis labels and color scales but differ in the sign of the Fermi energy parameter **E_F**. --- ### **Panel (f): K₂ with E_F = +0.035t** #### Labels and Axis Titles - **X-axis**: `length (nm)` ranging from -40 nm to 40 nm. - **Y-axis**: `width (nm)` ranging from -10 nm to 10 nm. - **Title**: `(f) K₂: E_F=+0.035t` - **Color Bar**: - **Range**: 0.0 (white) to 0.4 (red). - **Position**: Right of the panel. - **Spatial Grounding**: Legend located at `[x=45 nm, y=-10 nm]` (relative to the heatmap's coordinate system). #### Components and Flow - **Streamlines**: Blue arrows indicate vector direction and magnitude. - **Color Gradient**: - **High Intensity (Red)**: Central region (x ≈ 0 nm, y ≈ 0 nm). - **Low Intensity (White)**: Edges (x ≈ ±40 nm, y ≈ ±10 nm). - **Trend Verification**: - Arrows in the central region point outward radially, with density decreasing toward the edges. - Color intensity peaks at the center (red) and diminishes toward the edges (white). --- ### **Panel (h): K₂ with E_F = -0.035t** #### Labels and Axis Titles - **X-axis**: `length (nm)` ranging from -40 nm to 40 nm. - **Y-axis**: `width (nm)` ranging from -10 nm to 10 nm. - **Title**: `(h) K₂: E_F=-0.035t` - **Color Bar**: - **Range**: 0.0 (white) to 0.4 (red). - **Position**: Right of the panel. - **Spatial Grounding**: Legend located at `[x=45 nm, y=-10 nm]`. #### Components and Flow - **Streamlines**: Blue arrows indicate vector direction and magnitude. - **Color Gradient**: - **Low Intensity (White)**: Central region (x ≈ 0 nm, y ≈ 0 nm). - **High Intensity (Red)**: Edges (x ≈ ±40 nm, y ≈ ±10 nm). - **Trend Verification**: - Arrows in the central region point inward radially, with density increasing toward the edges. - Color intensity peaks at the edges (red) and diminishes toward the center (white). --- ### Key Observations 1. **Symmetry**: Both panels exhibit bilateral symmetry about the x=0 nm axis. 2. **E_F Dependency**: - Positive **E_F** (+0.035t) in (f) correlates with outward flow and central intensity. - Negative **E_F** (-0.035t) in (h) correlates with inward flow and edge intensity. 3. **Legend Consistency**: - Red (0.4) and white (0.0) values match the intensity peaks in both panels. - Arrows align with the gradient direction (e.g., outward flow in red regions of (f)). --- ### Notes - **Language**: All text is in English. - **Data Extraction**: No numerical tables or embedded text beyond axis labels and titles. - **Missing Information**: No explicit units for **t** (likely a material constant, e.g., thickness or temperature). </details> Figure 5: Local valley polarized currents at ${\bf K_{1}}$ and ${\bf K_{2}}$ valleys, in the presence of only staggered intrinsic SOC, $\lambda_{I}^{(A)}=-\lambda_{I}^{(B)}=0.075t$ . The left (right) panels are for $E_{F}>0$ ( $E_{F}<0$ ). III.2 Description of the model with weak couplings A quantitative description of the model with realistic parameters requires discussing the above results for the weaker strengths of the various SOCs. An important question is how the valley-conductance is affected by changes in the RSOC values. III.2.1 Valley-dependent conductance by tuning the Rashba coupling In this context, we consider weaker ISOCs as expected in realistic settings and an increased and controllable strength of RSOC. We notice that IQDs refer, in reality, to any heterostructure that exhibits $C_{3v}$ -symmetry. This section considers only ISO and RSO couplings, while the pseudospin inversion asymmetry (PIA) coupling in systems with broken inversion symmetry is discussed in the following section. A typical example of such a setup would be twisted graphene/transition-metal-dichalcogenides heterostructures since the relative rotation leads to a negligible value of PIA coupling. To this end, the IQDs with $C_{3v}$ -symmetry might lead to potential applications since the Rashba coupling can be tuned using a transverse electric field. As realized, we consider weak staggered couplings $\lambda_{I}^{(A)}=-\lambda_{I}^{(B)}=0.015t$ , we inject electrons with $E_{F}=± 0.035t$ into several-identical QDs, and by shifting the Rashba coupling with the aid of top gates, we compute the valley conductance through the system. In Fig. 6 (a) and (b), we show results for valley transmittance when the top gate is used to obtain a valley polarized conductance for coupling within the ranges $0.07t≤\lambda_{R}≤ 0.12t$ and $0.1t≤\lambda_{R}≤ 0.2t$ , for the two values of the staggered potential $\Delta=0$ anc $\Delta=0.02t$ correspondingly. The valley transmission for ${\bf K_{1}}$ , $({\bf K_{2}})$ jumps almost from 2 to 0 for positive (negative) incident energy, indicating that only electrons with either valley ${\bf K_{1}}$ or valley ${\bf K_{2}}$ go through the system. The most important conclusion from the results is that reasonable and better control of the valley transmission can be obtained by tuning a gate bias using IQDs with weak couplings. In this context, a quantitative understanding might easily be reached. By controlling the Rashba strength, the wave function may interfere destructively or constructively depending on the valley and the sign of the incident energy. It is seen that at $E_{F}>0$ and $\lambda_{R}≥ 0.05t$ , the wave function around ${\bf K_{1}}$ is transmitted through the IQDs region, where the wave function around ${\bf K_{2}}$ vanishes (is reflected), resulting on a polarized valley transmission, invertible by changes in the sign of the incident energy. These results can be understood from the analysis of the band structure in Fig. 6 (c) and (d). By analyzing the RSOC’s contributions to the edge state, we observe that for weak Rashba couplings, such as $\lambda_{R}=0.025t$ , the edge state is mixed with the bulk conduction bands. However, for $\lambda_{R}=0.075t$ , the system might sustain isolated and steady helical edge states, depending on the spin nature, as discussed in earlier works [29]. <details> <summary>2406.02393v1/x16.png Details</summary>  ### Visual Description # Technical Document Extraction: Graph Analysis ## Graph (a) - **Parameters**: - λ_I = 0.015t - Δ = 0 - E_F = 0.035t - **Axes**: - **y-axis**: G (e²/h) - **x-axis**: λ_R/t - **Key Features**: - **K₁ Region**: Peaks in G (e²/h) between λ_R/t ≈ 0.05–0.15. - **K₂ Region**: Lower G values (shaded area) between λ_R/t ≈ 0.15–0.25. - **Inset**: - **G_K1** (red line): Oscillatory behavior with peaks at λ_R/t ≈ 0.15 and 0.20. - **G_K2** (blue line): Oscillatory behavior with peaks at λ_R/t ≈ 0.10 and 0.25. ## Graph (b) - **Parameters**: - λ_I = 0.015t - Δ = 0.02t - E_F = 0.035t - **Axes**: - **y-axis**: G (e²/h) - **x-axis**: λ_R/t - **Key Features**: - **K₁ Region**: Reduced peak amplitude compared to graph (a), with a prominent peak at λ_R/t ≈ 0.10. - **K₂ Region**: Shaded area with minimal G values. - **Inset**: - **G_K1** (red line): Single peak at λ_R/t ≈ 0.15. - **G_K2** (blue line): Double-peaked structure at λ_R/t ≈ 0.10 and 0.20. ## Cross-Reference: Legend Colors vs. Line Placement - **Red (G_K1)**: - Peaks in both graphs align with K₁ regions. - **Blue (G_K2)**: - Peaks in both graphs align with K₂ regions. ## Observations 1. **Effect of Δ**: - Increasing Δ from 0 to 0.02t (graph a → b) reduces G_K1 amplitude and shifts K₁ peak positions. 2. **Shaded Area**: - Represents K₂ region with suppressed conductance (G < 1 e²/h). 3. **Inset Trends**: - G_K1 and G_K2 exhibit distinct oscillatory patterns, suggesting coupling between K₁ and K₂ states. </details> <details> <summary>2406.02393v1/x17.png Details</summary>  ### Visual Description # Technical Analysis of Energy Band Structure Plots ## Panel (c): λ_R = 0.025t - **Title**: `(c) λ_i^(A) = -λ_i^(B), λ_R = 0.025t` - **Axes**: - **x-axis**: `k[π/a]` (ranges from -4.0 to -2.0) - **y-axis**: `E (eV)` (ranges from -0.4 to 0.4) - **Key Features**: - **Energy Bands**: Multiple parabolic-like bands near the Fermi level (E=0). - **Band Separation**: Distinct separation between conduction and valence bands. - **Color-Coded Bands**: - Red: Highest energy valence band - Green: Intermediate valence band - Blue: Lowest energy valence band - Purple: Conduction band (symmetric to red) - Brown: Intermediate conduction band - Orange: Highest energy conduction band - Pink: Additional valence band (lower energy) - Cyan: Additional conduction band (higher energy) - **Notable**: Symmetric band structure with clear gaps between bands. ## Panel (d): λ_R = 0.075t - **Title**: `(d) λ_i^(A) = -λ_i^(B), λ_R = 0.075t` - **Axes**: - **x-axis**: `k[π/a]` (ranges from -4.0 to -2.0) - **y-axis**: `E (eV)` (ranges from -0.4 to 0.4) - **Key Features**: - **Energy Bands**: Bands near the Fermi level (E=0) exhibit linear dispersion. - **Band Proximity**: Reduced separation between conduction and valence bands compared to panel (c). - **Color-Coded Bands** (same as panel c): - Red: Highest energy valence band - Green: Intermediate valence band - Blue: Lowest energy valence band - Purple: Conduction band (symmetric to red) - Brown: Intermediate conduction band - Orange: Highest energy conduction band - Pink: Additional valence band (lower energy) - Cyan: Additional conduction band (higher energy) - **Notable**: Emergence of Dirac-like linear dispersion near E=0, indicating a topological transition. ## Cross-Referenced Legend - **Color-Label Mapping**: - Red ↔ Highest energy valence band - Green ↔ Intermediate valence band - Blue ↔ Lowest energy valence band - Purple ↔ Conduction band (symmetric to red) - Brown ↔ Intermediate conduction band - Orange ↔ Highest energy conduction band - Pink ↔ Additional valence band (lower energy) - Cyan ↔ Additional conduction band (higher energy) ## Observations 1. **λ_R Dependence**: - At λ_R = 0.025t (panel c), bands are well-separated with parabolic dispersion. - At λ_R = 0.075t (panel d), bands near E=0 exhibit linear dispersion, suggesting a transition toward a Dirac semimetal phase. 2. **Symmetry**: Both panels show inversion symmetry (λ_i^(A) = -λ_i^(B)), leading to symmetric band structures. 3. **Fermi Level**: All bands cross or approach E=0, indicating a metallic or semimetallic state. ## Structural Notes - **Dirac-like Dispersion**: In panel (d), the linear bands near E=0 resemble Dirac cones, characteristic of topological insulators or Weyl semimetals. - **Band Gaps**: Panel (c) shows larger gaps between bands, while panel (d) exhibits reduced gaps, consistent with increased λ_R. </details> <details> <summary>2406.02393v1/x18.png Details</summary>  ### Visual Description # Technical Document Extraction: Graph Analysis ## Panel (e) - **Parameters**: - λ_I = 0.015t - Δ = 0 - E_F = 0.035t ### Main Graph - **Axes**: - x-axis: λ_PIA/t (range: 0.00 to 0.15) - y-axis: G (e²/ħ) (range: 0 to 3) - **Regions**: - **K₁**: Shaded gray region (y ≈ 1 to 2) - Horizontal line at G ≈ 2 (black) - **K₂**: Shaded gray region (y ≈ 0 to 1) - Horizontal line at G ≈ 1 (black) ### Insets 1. **G_K₁** (Top-left inset): - x-axis: λ_PIA/t (0.00 to 0.4) - y-axis: G (e²/ħ) (0 to 2) - **Lines**: - Red (flat line at G ≈ 2) - Blue (slight upward slope from G ≈ 0 to 0.2) - **Legend**: - Red: G_K₁ - Blue: G_K₁ (alternate representation?) 2. **G_K₂** (Bottom-left inset): - x-axis: λ_PIA/t (0.00 to 0.4) - y-axis: G (e²/ħ) (0 to 1) - **Lines**: - Red (downward slope from G ≈ 1 to 0.5) - Blue (slight upward slope from G ≈ 0 to 0.2) - **Legend**: - Red: G_K₂ - Blue: G_K₂ (alternate representation?) ## Panel (f) - **Parameters**: - λ_I = 0.015t - Δ = 0.02t - E_F = 0.035t ### Main Graph - **Axes**: - x-axis: λ_PIA/t (range: 0.00 to 0.15) - y-axis: G (e²/ħ) (range: 0 to 3) - **Regions**: - **K₁**: Shaded gray region (y ≈ 1 to 2) - Horizontal line at G ≈ 2 (black, slightly higher than panel e) - **K₂**: Shaded gray region (y ≈ 0 to 1) - Horizontal line at G ≈ 1 (black, slightly lower than panel e) ### Insets 1. **G_K₁** (Top-left inset): - x-axis: λ_PIA/t (0.00 to 0.4) - y-axis: G (e²/ħ) (0 to 2) - **Lines**: - Red (flat line at G ≈ 2) - Blue (slight upward slope from G ≈ 0 to 0.2) - **Legend**: - Red: G_K₁ - Blue: G_K₁ (alternate representation?) 2. **G_K₂** (Bottom-left inset): - x-axis: λ_PIA/t (0.00 to 0.4) - y-axis: G (e²/ħ) (0 to 1) - **Lines**: - Red (steeper downward slope from G ≈ 1 to 0.5) - Blue (slight upward slope from G ≈ 0 to 0.2) - **Legend**: - Red: G_K₂ - Blue: G_K₂ (alternate representation?) ## Key Observations 1. **Panel Comparison**: - Panel (f) introduces Δ = 0.02t, altering the main graph's K₁/K₂ boundary lines. - Insets show similar trends but with modified slopes for G_K₂ (steeper decline in panel f). 2. **Legend Consistency**: - Red lines correspond to G_K₁/G_K₂ (flat or declining). - Blue lines correspond to G_K₁/G_K₂ (slight upward trend). 3. **Trends**: - G_K₁ remains relatively stable (flat/horizontal). - G_K₂ decreases with increasing λ_PIA/t, more pronounced in panel (f). </details> <details> <summary>2406.02393v1/x19.png Details</summary>  ### Visual Description # Technical Document Extraction: Energy Band Structure Analysis ## Image Description The image contains two side-by-side panels labeled **(g)** and **(h)**, depicting energy band structures as a function of wavevector **k**. Both panels share identical axis labels and ranges but differ in parameter values for **λ_p**. --- ### **Panel (g)** - **Title**: `(g) λ_i^(A) = -λ_i^(B), λ_p = 0` - **Axes**: - **x-axis**: **k** (wavevector) in units of **π/a**, ranging from **-4.0** to **-2.0**. - **y-axis**: **E** (energy) in electronvolts (eV), ranging from **-0.4** to **0.4**. - **Key Features**: - Symmetric energy bands due to **λ_p = 0**. - Multiple parabolic-like bands (color-coded lines) with **crossings at k = -3π/a**. - Red and green lines intersect at **E = 0**, indicating band degeneracy. - No legend present; color coding is unspecified but distinct. --- ### **Panel (h)** - **Title**: `(h) λ_i^(A) = -λ_i^(B), λ_p = 0.075t` - **Axes**: - **x-axis**: **k** (wavevector) in units of **π/a**, ranging from **-4.0** to **-2.0**. - **y-axis**: **E** (energy) in electronvolts (eV), ranging from **-0.4** to **0.4**. - **Key Features**: - Asymmetric energy bands due to **λ_p = 0.075t**. - **Avoided crossings** observed (e.g., red and green lines do not intersect). - Band structure retains symmetry about **k = -3π/a** but with energy shifts. - No legend present; color coding matches panel (g) but with modified band dispersion. --- ### **Common Elements** - **Gridlines**: Faint gridlines overlay both panels for reference. - **Symmetry**: Both panels exhibit **mirror symmetry** about **k = -3π/a**. - **Color Coding**: Lines are color-coded (e.g., red, green, blue, purple), but no legend is provided to define their significance. --- ### **Critical Observations** 1. **Parameter Impact**: - **λ_p = 0** (panel g): Results in symmetric, crossing bands. - **λ_p = 0.075t** (panel h): Introduces asymmetry and **avoided crossings**, suggesting hybridization effects. 2. **Band Structure**: - Bands are parabolic near **k = -3π/a**, indicating localized states or van Hove singularities. - Energy gaps or overlaps depend on **λ_p** magnitude. --- ### **Technical Notes** - **Units**: - Energy (**E**): electronvolts (eV). - Wavevector (**k**): normalized to **π/a**, where **a** is the lattice constant. - **Notation**: - **λ_i^(A/B)**: Interlayer coupling parameters. - **λ_p**: Intralayer perturbation parameter (proportional to **t**, likely hopping energy). --- ### **Conclusion** The panels illustrate how **λ_p** modulates the symmetry and hybridization of energy bands. Panel (g) represents a symmetric case with degeneracies, while panel (h) shows asymmetric, gapped bands due to finite **λ_p**. Further analysis would require a legend to interpret color-coded bands. </details> Figure 6: Panels (a) and (b): Valley conductance vs. Rashba spin-orbit coupling $\lambda_{R}$ for $\Delta=0$ and $\Delta=0.02t$ , respectively. Panel (c) and (d) show the energy bands in the staggered case ( $\lambda_{I}^{(A)}=-\lambda_{I}^{(B)}$ ) for $\lambda_{R}=0.025t$ and $\lambda_{R}=0.075t$ , accordingly. Panels (e) and (f) show the conductance vs the strength of the PIA coupling. Panel (g) and (h) show the energy bands for $\lambda_{P}=0$ and $\lambda_{P}=0.075t$ , accordingly. Insets in (a, b, e, f) show spin-resolved valley conductance per valley. In addition to the valley filtering, the interplay between the valley and spin degrees of freedom might realize new transport regimes. As shown in the inset of Fig. 6 (a) and (b), the wave functions around ${\bf K_{1}}$ or ${\bf K_{2}}$ (depending on the incident energy) are spin-dependent. Indeed, the Rashba coupling allows off-diagonal spin-flipping terms. Therefore, the spin conductance is also directly controlled by the Rashba strength, leading to valley-spin interplay dynamics. For a given incident energy, the spin-conductance $G_{\uparrow}({\bf K_{1}})$ oscillations have several phase shifts that correspond to the wavelengths of the oscillations determined by the Rashba coupling strength. Notice that the overall conductance of spin-polarized carriers, $G_{\uparrow}({\bf K_{1}})=G_{\uparrow\uparrow}({\bf K_{1}})+G_{\uparrow% \downarrow}({\bf K_{1}})$ remains constant. Our calculations suggest valley processes dominate the conductance as spin-flips between neighboring sites require wider IQDs. Details on spin-dependent conductance in these proximity effect structures have been fully addressed in Ref. [29]. III.2.2 IQDs with lower- or higher-order symmetries $C_{3v}$ symmetry is a property of hexagonal systems described by point group symmetry methods. To generalize the applicability of the model presented, it might be useful to lower (or raise) the symmetry of the IQDs. This strategy provides more options to describe novel regimes as it allows for removing (or adding) different spin-orbit terms. (i) In the case $\lambda_{R}=0$ , the point group is increased to $D_{3h}$ where the sublattice inversion asymmetry defines a point group of a planar system sustaining a triangular lattice with two staggered (non-equivalent) sublattices [27]. For such a case, the presence of the ISOC leads to the transmittance spectrum similar to Fig 2 (c) and (d). (ii) To lower the symmetry of the IQDs, one might include an additional staggered SOC term called PIA coupling (pseudospin inversion asymmetry for short). The spin-orbit Hamiltonian, in this case, will have an extra term: $$ \displaystyle H_{PIA} \displaystyle= \displaystyle(\sqrt{3}a_{0}/2)\big{[}\lambda_{PIA}^{(A)}(\sigma_{z}+\sigma_{0}) \displaystyle+ \displaystyle\lambda_{PIA}^{(B)}(\sigma_{z}-\sigma_{0})\big{]}(k_{x}\sigma_{y}% +k_{y}\sigma_{x}). \tag{7} $$ where $\lambda_{PIA}^{(A)}$ and $\lambda_{PIA}^{(B)}$ are the staggered PIA coupling and $a_{0}$ is the lattice constant. As shown in Fig. 6 (e) and (f), the PIA coupling does not affect the valley-conductance (even when it is a staggered spin-orbit term), neither for lower nor higher strength values. Therefore, we can confirm that only ISOC and RSOC control the valley process, as discussed in Fig. 2 and 6 (a, b). Hence, a QD with $\lambda_{PIA}≠ 0$ brings forth the same behavior as $\lambda_{PIA}=0$ , and a similar valley response will be observed. Indeed, the spectrum of the energy bands in Fig. 6 (g) and (h) supports these results, where we observe that for either neglected ( $\lambda_{PIA}=0$ ) or strong PIA terms ( $\lambda_{PIA}=0.075t$ ) the lower bulk and edge states are insensitive to the coupling strength. This result is exciting as it implies that one might use TMDs to create IQDs with $C_{3v}$ -symmetry but with a $\lambda_{PIA}≠ 0$ without affecting their valley transport properties. Indeed, we might easily tune the intrinsic and Rashba parameters in such a system using TMD quantum dots. For instance, based on density functional theory fittings, both parameters are strongly sensitive to the electric field, twisting, and/or vertical strain effects [28, 46, 47]. The possibility of controlling the decrease or increase in the coupling strengths is highly desirable to monitor the valley polarization. Additionally, the spin-conductance is weakly affected (see inset of Fig. 6 (c, d)) by the presence of PIA as compared to Rashba terms. For more details about spin dependence and the effect of PIA coupling on spin polarization, see [48]. III.2.3 Valley confinement with IQDs This subsection addresses the possibility of attaining confinement of valley-polarized electrons. Adopting the same sample parameters as in subsection III.1, we see that by zooming in on the transmission spectra from Fig. 2 (c, d), one can additionally recognize a resonance in the transmittance emerging in the IQDS, at $E_{F}=0.035t$ for higher values of ISOC ( $\lambda_{I}^{(A)}=-\lambda_{I}^{(B)}=0.065t$ ). The scattering through the induced region creates electron confinement for small-size IQDs ( $r_{0}=7$ nm). The resonance might be advantageously produced for weak ISOC at specific incident energy values. This case defines a real proximity effect where we tune $E_{F}$ to confine the valley states as shown in Sec. III.3. <details> <summary>2406.02393v1/x20.png Details</summary>  ### Visual Description # Technical Document Extraction: Heatmap Analysis of K₁ and K₂ Systems ## Panel (a): K₁, N=1, E_F=-35 meV ### Axis Labels - **X-axis**: `length (nm)` (range: -20 to 20 nm) - **Y-axis**: `width (nm)` (range: -15 to 15 nm) ### Color Scale - **Legend**: Right-aligned color bar with values from `0.0` to `3.0` (unitless intensity scale). - **Color Gradient**: - `White` (low intensity) → `Yellow` → `Orange` → `Red` (high intensity). ### Key Observations - **Central Peak**: A high-intensity region (red/orange) centered at `(length=0, width=0)`. - **Symmetry**: Radial symmetry with concentric intensity rings. - **Intensity Distribution**: - Maximum intensity: ~3.0 (red core). - Outer regions: Gradual decline to ~0.0 (white). --- ## Panel (b): K₂, N=1, E_F=-35 meV ### Axis Labels - **X-axis**: `length (nm)` (range: -20 to 20 nm) - **Y-axis**: `width (nm)` (range: -15 to 15 nm) ### Color Scale - **Legend**: Right-aligned color bar with values from `0` to `25` (unitless intensity scale). - **Color Gradient**: - `White` (low intensity) → `Yellow` → `Orange` → `Red` (high intensity). ### Key Observations - **Complex Structure**: - Central void (low intensity, white) surrounded by alternating high/low intensity regions. - Outer ring: High-intensity (red) with a secondary blue ring (lower intensity). - **Intensity Distribution**: - Maximum intensity: ~25 (red outer ring). - Blue ring: Intermediate intensity (~10–15). - Central void: Near-zero intensity. --- ## Cross-Reference: Color Legend Consistency | Panel | Color (Red) | Color (Blue) | Notes | |-------|-------------|--------------|-------| | (a) | 3.0 | N/A | Single-peak symmetry. | | (b) | 25 | ~10–15 | Multi-peak structure with void. | ## Notes - Both panels share identical axis ranges and energy parameters (`E_F=-35 meV`, `N=1`). - Panel (b) exhibits significantly higher intensity values (up to 25 vs. 3.0 in (a)), suggesting differing system properties or measurement scales. - The blue ring in (b) may indicate a secondary feature or interference effect. </details> <details> <summary>2406.02393v1/x21.png Details</summary>  ### Visual Description # Technical Document Extraction: Heatmap Analysis ## Image Description The image contains two side-by-side heatmaps labeled **(c)** and **(d)**, representing spatial intensity distributions under identical parameters: - **N=2** (likely a quantum number or system parameter) - **E_F=-35 meV** (Fermi energy) ### Key Components 1. **Axes** - **X-axis**: `length (nm)` ranging from **-30 nm to 30 nm** - **Y-axis**: `width (nm)` ranging from **-15 nm to 15 nm** 2. **Color Scales** - **(c)**: Red-to-white gradient, intensity range **0.0 to 3.0** (unitless) - **(d)**: Blue-to-red gradient, intensity range **0.0 to 25.0** (unitless) 3. **Structural Features** - **Heatmap (c)**: - Symmetrical **dipole-like lobes** centered at (±10 nm, 0 nm) - Central void (low-intensity region) with radiating intensity peaks - Smooth gradient transition from red (high intensity) to white (low intensity) - **Heatmap (d)**: - **Annular ring structure** with concentric intensity variations - Central void surrounded by alternating high/low intensity rings - Blue (low intensity) transitions to red (high intensity) at outer edges 4. **Legend Cross-Reference** - **(c)** Colorbar: Red = 3.0, White = 0.0 - **(d)** Colorbar: Blue = 0.0, Red = 25.0 ## Observations - **Similarities**: - Both heatmaps exhibit **central voids** and **symmetrical intensity distributions** about the origin. - Identical system parameters (**N=2**, **E_F=-35 meV**) suggest comparable physical conditions. - **Differences**: - **(c)** shows **dipolar symmetry** with localized intensity maxima at ±10 nm. - **(d)** exhibits **rotational symmetry** with distributed intensity in annular patterns. - **(d)** has a **10× higher intensity range** (25 vs. 3.0), indicating stronger spatial variations. ## Technical Notes - The heatmaps likely represent **quantum mechanical probability densities** or **electronic band structures** for a system with two nodes (N=2). - The negative Fermi energy (E_F=-35 meV) implies a **valence-band-dominated** system. - Structural differences between (c) and (d) may reflect variations in **spin-orbit coupling**, **magnetic field orientation**, or **interaction strength**. </details> Figure 7: Local density of valley-localized states near ${\bf K_{1}}$ (left) and ${\bf K_{2}}$ (right) at $E_{F}=0.035t$ , obtained for values of $\lambda_{I}^{(A)}=-\lambda_{I}^{(B)}=0.065t$ . The red and blue arrows show each valley state’s circulation direction. Panels (a, b) and (c, d) are for single and 2-chain-IQDs. A detailed analysis of the local current profiles of valley states, as depicted in Fig. 7, shows a better representation of such a kind of valley-dependent electron confinement. Indeed, we observe that valley state confinement can be achieved by using either a single or a chain of IQDs. Importantly, the incident valley-states are trapped around the IQD region for appropriate SOC choices. This results in the confinement of the states, a process that can be regarded as the product of multiple internal reflections that trap the Dirac fermions by interference processes. The system with symmetric IQDs appears to provide confinement for states at both valleys simultaneously in the same spatial region (we do not observe splittings of the valley confinement states). However, to achieve control over the valleys and multiple valleytronic and optoelectronic functionalities, using these IQDs, it will be necessary to create splittings and confining with unique quasi-bound states (belonging to either ${\bf K_{1}}$ or ${\bf K_{2}}$ ) in the local regions. The valley-bound states might be split by considering asymmetric quantum dots with different shapes and point group symmetries. This topic will be addressed in future work. III.3 Realistic IQDs in heterostructures of twisted graphene and TMD monolayers As discussed in Sec. III.2.2 and III.2.3, a quantitative study of the model has been addressed in the case of weak SOCs. To provide a comprehensive picture of concrete setups, we perform a real case study and show how an IQD might be realized in realistic experimental conditions. To start, we consider IQDs produced with four different semiconducting TMDs islands: MoSe 2, WSe 2, MoS 2, and WS 2 deposited on graphene monolayer, as shown in Fig. 1 (b). Furthermore, we analyze the proximity-induced spin-orbit couplings in the case of twisted TMDs on graphene. Based on the work of T. Naimer et et al. [35], twisting leads to tuning the magnitudes of the valley-Zeeman and Rashba spin-orbit couplings. Interestingly, the amplitude of the staggered potential $\Delta$ in twisted G-TMDs is significantly weak and tends to zero at some specific twisting angles, thus favoring valley filtering, as discussed in Fig. 4. Based on the tight-binding parameters derived from first principles calculations [35], we compute the valley conductance through four $(n=4)$ IQDs at $E_{F}=0.035t\simeq 0.095$ meV and show results in Fig. 8. <details> <summary>2406.02393v1/x22.png Details</summary>  ### Visual Description # Technical Document: Conductance Analysis of MoSe₂ and WSe₂ under Twisting Angles ## Graph (a): MoSe₂, E_F = +0.095 meV - **Title**: (a) MoSe₂, E_F = +0.095 (meV) - **X-axis**: Twisting angle (θ in °), ranging from 0° to 30° in 5° increments. - **Y-axis**: Conductance (G) in units of e²/h, ranging from 0.00 to 2.00. - **Legend**: - **K₁**: Red squares (■) - **K₂**: Blue circles (●) - **Key Trends**: - **K₁**: - Peaks at 0°, 5°, 10°, 15°, 20°, 25°, and 30°. - Sharp drop at 10° and 15°, with conductance values decreasing to ~0.00 e²/h. - Stabilizes at ~2.00 e²/h after 15°. - **K₂**: - Single peak at 5° (~0.25 e²/h). - Remains near 0.00 e²/h for all other angles (0°, 10°–30°). ## Graph (c): WSe₂, E_F = +0.095 meV - **Title**: (c) WSe₂, E_F = +0.095 (meV) - **X-axis**: Twisting angle (θ in °), ranging from 0° to 30° in 5° increments. - **Y-axis**: Conductance (G) in units of e²/h, ranging from 0.00 to 2.00. - **Legend**: - **K₁**: Red squares (■) - **K₂**: Blue circles (●) - **Key Trends**: - **K₁**: - Sharp drop at 5° (~0.00 e²/h). - Stabilizes at ~2.00 e²/h for angles 10°–30°. - **K₂**: - Flat line at ~0.00 e²/h for all angles (0°–30°). ## Cross-Reference: Legend Consistency - **K₁** (Red squares) corresponds to higher conductance values in both materials, with distinct angular dependencies. - **K₂** (Blue circles) shows minimal conductance in both materials, with only a transient peak in MoSe₂ at 5°. ## Observations 1. **Material-Specific Behavior**: - MoSe₂ exhibits multi-peak conductance for K₁, suggesting complex electronic interactions under twisting. - WSe₂ shows a single sharp drop for K₁, indicating a threshold-like response to twisting. 2. **Fermi Energy Impact**: - Both materials are analyzed at E_F = +0.095 meV, suggesting a controlled doping or energy level alignment. 3. **Twisting Sensitivity**: - Conductance in K₁ is highly sensitive to twisting angles, with critical transitions at specific angles (e.g., 5°, 10°, 15°). 4. **K-Point Differentiation**: - K₁ and K₂ exhibit divergent behaviors, highlighting the importance of valleytronics in transition metal dichalcogenides (TMDs). ## Conclusion The graphs demonstrate that MoSe₂ and WSe₂ exhibit distinct conductance responses to twisting angles at the K₁ and K₂ points. These results underscore the role of structural modulation (twisting) in tuning electronic properties in TMDs, with potential applications in valleytronic devices. </details> <details> <summary>2406.02393v1/x23.png Details</summary>  ### Visual Description # Technical Document Extraction: Conductance vs. Twisting Angle for MoSe₂ and WSe₂ ## Graph (b): MoSe₂, E_F = -0.095 meV - **Title**: Conductance (G) vs. Twisting Angle (θ) for MoSe₂ at E_F = -0.095 meV - **Axes**: - **X-axis**: Twisting angle (θ) in degrees (°), ranging from 0 to 30°. - **Y-axis**: Conductance (G) in units of e²/h, ranging from 0.00 to 2.00. - **Legend**: - **K₁**: Blue circles (solid line). - **K₂**: Red squares (dashed line). - **Data Trends**: - **K₁ (Blue Circles)**: - Starts at 2.00 e²/h at θ = 0°. - Drops sharply to 1.50 e²/h at θ = 5°. - Further decreases to 0.00 e²/h at θ = 10°. - Remains at 0.00 e²/h for θ ≥ 10°. - **K₂ (Red Squares)**: - Starts at 0.00 e²/h at θ = 0°. - Peaks at 0.25 e²/h at θ = 5°. - Returns to 0.00 e²/h at θ = 10°. - Remains at 0.00 e²/h for θ ≥ 10°. ## Graph (d): WSe₂, E_F = -0.095 meV - **Title**: Conductance (G) vs. Twisting Angle (θ) for WSe₂ at E_F = -0.095 meV - **Axes**: - **X-axis**: Twisting angle (θ) in degrees (°), ranging from 0 to 30°. - **Y-axis**: Conductance (G) in units of e²/h, ranging from 0.00 to 2.00. - **Legend**: - **K₁**: Blue circles (solid line). - **K₂**: Red squares (dashed line). - **Data Trends**: - **K₁ (Blue Circles)**: - Starts at 2.00 e²/h at θ = 0°. - Drops sharply to 1.75 e²/h at θ = 5°. - Further decreases to 0.00 e²/h at θ = 10°. - Remains at 0.00 e²/h for θ ≥ 10°. - **K₂ (Red Squares)**: - Starts at 0.00 e²/h at θ = 0°. - Peaks at 0.05 e²/h at θ = 5°. - Returns to 0.00 e²/h at θ = 10°. - Remains at 0.00 e²/h for θ ≥ 10°. ## Key Observations 1. **Material-Specific Behavior**: - **MoSe₂ (Graph b)**: - K₁ exhibits a larger initial drop (2.00 → 1.50 e²/h) compared to WSe₂ (2.00 → 1.75 e²/h). - K₂ shows a more pronounced peak (0.25 e²/h) in MoSe₂ vs. WSe₂ (0.05 e²/h). - **WSe₂ (Graph d)**: - K₁ retains higher conductance (1.75 e²/h at θ = 5°) compared to MoSe₂. - K₂’s peak is significantly smaller (0.05 e²/h) in WSe₂. 2. **Critical Twisting Angles**: - Both materials show abrupt conductance drops at θ = 10°, indicating a structural or electronic transition. - K₂ peaks at θ = 5° in both materials but with differing magnitudes. 3. **Energy Alignment**: - Both graphs share the same Fermi energy (E_F = -0.095 meV), suggesting comparative analysis under identical electronic conditions. </details> <details> <summary>2406.02393v1/x24.png Details</summary>  ### Visual Description # Technical Analysis of MoSe2 Heatmaps ## Plot (e): MoSe2 - E_F=95 meV, θ=15° ### Axis Labels - **X-axis**: `length (nm)` (range: -40 to 40 nm) - **Y-axis**: `width (nm)` (range: -10 to 10 nm) ### Colorbar - **Scale**: 0.0 (white) to 0.3 (dark red) - **Gradient**: Smooth transition from white (low intensity) to dark red (high intensity) ### Key Observations 1. **Intensity Distribution**: - Two high-intensity regions centered at (-20, 0) and (20, 0) with values near 0.3. - Symmetric intensity decay toward the edges of the plot. 2. **Streamline Flow**: - Red streamlines indicate directional flow patterns. - Flow converges toward the high-intensity regions and diverges outward. ## Plot (f): MoSe2 - E_F=95 meV, θ=15° ### Axis Labels - **X-axis**: `length (nm)` (range: -40 to 40 nm) - **Y-axis**: `width (nm)` (range: -10 to 10 nm) ### Colorbar - **Scale**: 0.0 (white) to 0.4 (dark red) - **Gradient**: Smooth transition from white (low intensity) to dark red (high intensity) ### Key Observations 1. **Intensity Distribution**: - High-intensity region localized on the left side (x < 0 nm) with a gradient from dark red (0.4) to white (0.0). - No significant intensity in the right half (x > 0 nm). 2. **Streamline Flow**: - Blue streamlines indicate directional flow patterns. - Flow originates from the left edge, converges toward the high-intensity region, and dissipates toward the center. ### Cross-Reference: Colorbar vs. Streamline Colors - **Plot (e)**: Red streamlines align with the red-dominated colorbar scale. - **Plot (f)**: Blue streamlines align with the red-dominated colorbar scale (note: colorbar uses red gradient despite blue streamlines; possible visualization inconsistency). ### Summary Both plots depict MoSe2 under identical Fermi energy (E_F=95 meV) and angle (θ=15°) conditions. Plot (e) shows symmetric intensity peaks, while plot (f) exhibits asymmetric intensity localization. Streamline colors (red/blue) may indicate flow directionality or visualization conventions. </details> <details> <summary>2406.02393v1/x25.png Details</summary>  ### Visual Description # Technical Document Extraction: MoSe2 Electron Flow Visualization ## Panel (g): MoSe2 Electron Flow at E_F = -95 meV, θ = 15° ### Labels and Axis Markers - **X-axis**: `length (nm)` with markers: `-40, -20, 0, 20, 40` - **Y-axis**: `width (nm)` with markers: `-10, 0, 10` - **Color Scale**: `0.0` (white) to `0.4` (red), labeled on the right ### Key Observations - **Streamline Pattern**: - Red streamlines concentrated on the **left edge** (length ≈ -40 nm to -20 nm). - Streamlines exhibit a **radial outward flow** from the left boundary. - Intensity decreases exponentially toward the center (length > 0 nm), transitioning to white. - **Color Gradient**: - Highest values (red) localized near the left edge. - Smooth gradient to white in the central and right regions. --- ## Panel (h): MoSe2 Electron Flow at E_F = -95 meV, θ = 15° ### Labels and Axis Markers - **X-axis**: `length (nm)` with markers: `-40, -20, 0, 20, 40` - **Y-axis**: `width (nm)` with markers: `-10, 0, 10` - **Color Scale**: `0.0` (white) to `0.3` (blue), labeled on the right ### Key Observations - **Streamline Pattern**: - Blue streamlines form **closed loops** around two bright regions near the center (length ≈ 0 nm). - Loops suggest **circular or vortex-like flow** around these regions. - Additional flow lines extend radially outward from the loops toward the edges. - **Color Gradient**: - Bright white regions (high intensity) at the center, surrounded by blue streamlines. - Intensity diminishes toward the edges, transitioning to white. --- ## Cross-Reference: Color Scale and Streamline Correlation - **Panel (g)**: Red streamlines align with the highest intensity values (0.4) on the color scale. - **Panel (h)**: Blue streamlines correspond to moderate intensity values (0.3) on the color scale, with white regions indicating peak intensities. ## Summary Both panels visualize electron flow in MoSe2 under identical Fermi energy (E_F = -95 meV) and angle (θ = 15°). Panel (g) emphasizes **edge-driven flow**, while panel (h) highlights **central vortex dynamics**. Color scales quantify intensity, with streamlines mapping directional flow patterns. </details> Figure 8: Panels (a) and (b) show valley conductance vs twist angle $\theta$ for IQDs made of MoSe 2 on graphene at $E_{F}=0.095$ meV and $-0.095$ meV, respectively. Panels (c) and (d) correspond to IQDs of WSe 2 on graphene for the same values of $E_{F}$ . In panels (e)-(f) and (g)-(h) we show valley-polarized currents for MoSe 2 for positive and negative bias ( $E_{F}$ ), respectively . Valley transmittance with first propagating modes in twisted TMDs: It is important to state that in all previous sections, we considered the transmittance spectrum at $E_{F}=± 0.035t$ ; a value that ensures that the valley-dependent conductance is addressed independently with the simultaneous excitation of the first propagating modes in ${\bf K_{1}}$ and ${\bf K_{2}}$ since the Fermi energy $E_{F}$ is larger than the valley-mode spacing gap. For more details, refer to our previous work [49]. In Fig. 8, we inject electrons with $E_{F}=± 0.095$ meV into four identical IQDs produced with MoSe 2 and WSe 2 islands. The induced SOCs are tuned by twisting the TMDs and allow for monitoring of the valley response in the system. At positive incident energy, the valley conductance from valley ${\bf K_{1}},({\bf K_{2}})$ is 2 (0) in the units of $e^{2}/\hbar$ . Only electrons from either valley ${\bf K_{1}}$ or ${\bf K_{2}}$ go through the system (see panels (a) and (c)). As previously discussed, we also confirm that changing the Fermi energy sign reverses the valley polarity, as shown in Fig. 8 (b) and (d). Additionally, as shown in 8 (a) and (b), twisting in the case of MoSe 2 (that induces a weak staggered potential [35]) leads to control of valley transmittances similar to what we found when discussing the model with weak couplings in Fig. 6. The obtained results with MoSe 2 can be applied to obtain pure valley polarized conductances with the twist angle outside the limits of $9^{\circ}-14^{\circ}$ . Hence, the valley transmission from ${\bf K_{1}}$ ( ${\bf K_{2}}$ ) jumps almost from 2 to 0 for positive (negative) incident energy, depending on twisting angles and full valley polarized currents. To emphasize these findings, we plot the local current for $\theta=15^{\circ}$ at $E_{F}=0.095$ meV ( $-0.095$ meV) for MoSe 2 (WSe 2). Indeed, one observes that only one valley is scattered, which leads to conducting Hall currents depending on the sign of the incident Fermi-energy. These results are important compared to those shown in Fig. 5, where both valleys conduct through bulk and Hall currents. Hence, the IQDs based on semiconducting TMDs MoSe 2 and WSe 2 might be used as promising islands for generating valley Hall signals. <details> <summary>2406.02393v1/x26.png Details</summary>  ### Visual Description # Technical Document Extraction: Conductance vs. Twisting Angle ## General Observations - **Graph Layout**: Four subplots (a-d) arranged in a 2x2 grid. - **Common Parameters**: - **X-axis**: Twisting angle (θ) in degrees (0–30°). - **Y-axis**: Conductance (G) in units of $ e^2/h $. - **Fermi Energy**: $ E_F = +0.2 \, \text{meV} $ for all graphs. - **Legend**: Red squares (K₁), Blue circles (K₂). --- ### Subplot (a): MoSe₂, $ E_F = +0.2 \, \text{meV} $ - **K₁ (Red Squares)**: - Peaks at θ = 10°, 20°, and 30°. - Conductance values: ~2.0, ~3.5, ~3.0 $ e^2/h $. - **K₂ (Blue Circles)**: - Peaks at θ = 10° (~0.7 $ e^2/h $) and 30° (~1.0 $ e^2/h $). - Minimal conductance at θ = 0°, 20°. --- ### Subplot (b): WSe₂, $ E_F = +0.2 \, \text{meV} $ - **K₁ (Red Squares)**: - Peaks at θ = 10° (~2.2 $ e^2/h $) and 30° (~2.5 $ e^2/h $). - Flat baseline at θ = 0°, 20° (~2.0 $ e^2/h $). - **K₂ (Blue Circles)**: - Single peak at θ = 10° (~0.3 $ e^2/h $). - Near-zero conductance at θ = 0°, 20°, 30°. --- ### Subplot (c): MoS₂, $ E_F = +0.2 \, \text{meV} $ - **K₁ (Red Squares)**: - Peaks at θ = 10° (~4.0 $ e^2/h $), 20° (~3.0 $ e^2/h $), and 30° (~4.5 $ e^2/h $). - Sharp drop at θ = 0° (~3.5 $ e^2/h $). - **K₂ (Blue Circles)**: - Peaks at θ = 10° (~2.5 $ e^2/h $) and 20° (~2.0 $ e^2/h $). - Minimal conductance at θ = 0°, 30°. --- ### Subplot (d): WS₂, $ E_F = +0.2 \, \text{meV} $ - **K₁ (Red Squares)**: - Peaks at θ = 10° (~2.0 $ e^2/h $) and 30° (~2.2 $ e^2/h $). - Flat baseline at θ = 0°, 20° (~1.8 $ e^2/h $). - **K₂ (Blue Circles)**: - Peaks at θ = 10° (~0.5 $ e^2/h $) and 30° (~0.3 $ e^2/h $). - Near-zero conductance at θ = 0°, 20°. --- ### Key Trends 1. **Material-Specific Behavior**: - **MoSe₂/WSe₂**: K₁ exhibits periodic peaks; K₂ shows weaker or absent peaks. - **MoS₂/WS₂**: K₁ and K₂ both display peaks, with K₁ generally stronger. 2. **Twisting Angle Sensitivity**: - Conductance peaks consistently occur at θ = 10° and 30° across materials. - θ = 20° shows reduced conductance for K₂ in MoSe₂ and WS₂. 3. **Fermi Energy Impact**: - All graphs use $ E_F = +0.2 \, \text{meV} $, suggesting consistent doping levels. --- ### Legend Consistency Check - **Red Squares (K₁)**: Matches all subplots. - **Blue Circles (K₂)**: Matches all subplots. - No discrepancies observed between legend labels and line styles. --- ### Axis Labels and Titles - **X-axis**: "twisting (θ in °)" for all subplots. - **Y-axis**: "G (e²/h)" for all subplots. - **Subplot Titles**: - (a) MoSe₂, $ E_F = +0.2 \, \text{meV} $ - (b) WSe₂, $ E_F = +0.2 \, \text{meV} $ - (c) MoS₂, $ E_F = +0.2 \, \text{meV} $ - (d) WS₂, $ E_F = +0.2 \, \text{meV} $ --- ### Data Points Summary | Material | θ (°) | K₁ (e²/h) | K₂ (e²/h) | |----------|-------|-----------|-----------| | MoSe₂ | 0 | 2.0 | 0.0 | | | 10 | 2.5 | 0.7 | | | 20 | 2.0 | 0.0 | | | 30 | 3.0 | 1.0 | | WSe₂ | 0 | 2.0 | 0.0 | | | 10 | 2.2 | 0.3 | | | 20 | 2.0 | 0.0 | | | 30 | 2.5 | 0.0 | | MoS₂ | 0 | 3.5 | 0.0 | | | 10 | 4.0 | 2.5 | | | 20 | 3.0 | 2.0 | | | 30 | 4.5 | 0.0 | | WS₂ | 0 | 1.8 | 0.0 | | | 10 | 2.0 | 0.5 | | | 20 | 1.8 | 0.0 | | | 30 | 2.2 | 0.3 | --- ### Notes - Conductance values are approximate, derived from peak heights in the graphs. - All trends align with the legend definitions (K₁ vs. K₂). - No additional text or data tables present in the image. </details> <details> <summary>2406.02393v1/x27.png Details</summary>  ### Visual Description # Technical Document Extraction: Conductance vs. Twisting Angle Analysis ## General Observations - **Graphs (b)-(h)** depict conductance (G) as a function of twisting angle (θ) for different 2D materials. - **EF = +0.5 meV** is constant across all graphs. - **Legend**: - **K₁** (red squares) and **K₂** (blue circles) represent distinct conductance pathways or states. - Dashed lines connect data points for clarity. --- ### Graph (b): MoSe₂ - **X-axis**: Twisting angle (θ) in degrees (0–30°). - **Y-axis**: Conductance (G) in units of e²/h. - **Key Trends**: - **K₁** (red): - Sharp peak at θ = 5° (G ≈ 8 e²/h). - Secondary peaks at θ ≈ 15° and 25° (G ≈ 6–7 e²/h). - **K₂** (blue): - Peaks at θ = 5° (G ≈ 4 e²/h) and θ = 25° (G ≈ 6 e²/h). - Minima at θ ≈ 10° and 20° (G ≈ 2 e²/h). --- ### Graph (d): WSe₂ - **X-axis**: Twisting angle (θ) in degrees (0–30°). - **Y-axis**: Conductance (G) in units of e²/h. - **Key Trends**: - **K₁** (red): - Peaks at θ = 5° (G ≈ 5 e²/h) and θ = 30° (G ≈ 9 e²/h). - Relatively flat between 10°–20° (G ≈ 4 e²/h). - **K₂** (blue): - Peaks at θ = 5° (G ≈ 3 e²/h) and θ = 30° (G ≈ 7.5 e²/h). - Minima at θ ≈ 15° (G ≈ 2 e²/h). --- ### Graph (f): MoS₂ - **X-axis**: Twisting angle (θ) in degrees (0–30°). - **Y-axis**: Conductance (G) in units of e²/h. - **Key Trends**: - **K₁** (red): - Peaks at θ = 25° (G ≈ 14 e²/h) and θ = 30° (G ≈ 15 e²/h). - Moderate values at θ = 0°–10° (G ≈ 8–10 e²/h). - **K₂** (blue): - Peaks at θ = 25° (G ≈ 12.5 e²/h) and θ = 30° (G ≈ 13.5 e²/h). - Sharp drop at θ = 15° (G ≈ 2 e²/h). --- ### Graph (h): WS₂ - **X-axis**: Twisting angle (θ) in degrees (0–30°). - **Y-axis**: Conductance (G) in units of e²/h. - **Key Trends**: - **K₁** (red): - Peaks at θ = 5° (G ≈ 7 e²/h) and θ = 30° (G ≈ 9 e²/h). - Relatively flat between 10°–20° (G ≈ 5 e²/h). - **K₂** (blue): - Peaks at θ = 5° (G ≈ 5.5 e²/h) and θ = 30° (G ≈ 6.5 e²/h). - Minima at θ ≈ 15° (G ≈ 0.5 e²/h). --- ## Cross-Graph Comparisons 1. **Peak Conductance**: - **K₁** consistently exhibits higher conductance than **K₂** across all materials. - **MoS₂** (graph f) shows the highest peak for **K₁** (15 e²/h at θ = 30°). 2. **Twisting Sensitivity**: - **MoSe₂** and **WS₂** (graphs b, h) show pronounced conductance oscillations with twisting. - **WSe₂** (graph d) exhibits less variability, with **K₁** remaining relatively stable. 3. **Material-Specific Behavior**: - **MoS₂** and **WS₂** demonstrate sharper peaks at higher twisting angles (25°–30°). - **MoSe₂** and **WSe₂** show more distributed peaks across the twisting range. --- ## Critical Notes - **Legend Consistency**: Red squares (**K₁**) and blue circles (**K₂**) are uniformly applied across all graphs. - **Axis Labels**: All graphs share identical axis titles ("twisting (θ in °)" and "G (e²/h)"). - **EF Uniformity**: The fixed EF value (+0.5 meV) suggests a controlled experimental condition for comparative analysis. </details> Figure 9: Valley conductance vs twist angle $\theta$ for IQDs made of MoSe 2, WSe 2, MoS 2 and WS 2 on graphene, respectively. Panels in the top (bottom) show conductances at $E_{F}=0.2$ meV (0.5)meV. Valley transmittance beyond first propagating modes in twisted TMDs: To observe transport involving states beyond the first propagating modes, we set the incident energy of the injected electrons at $E_{F}=± 0.1,± 0.3$ meV. In this case, we will consider all TMDs: MoSe 2, WSe 2, MoS 2, and WS 2 where the valley-dependent conductance is shown in Fig. 9. Similarly to what was discussed, the valley process is sensitive to the magnitudes of the induced valley-Zeeman and Rashba spin-orbit couplings and the incident Fermi energy. The TMD islands, as well as the twisting, allow a variety of options (several values of ISOCs) to monitor valley-driven currents, either in bulk or along the edges, where the best choice is to set the incident energy at $E_{F}=0.035t$ (within the first propagating modes). We observe that for some of the twist angle values, the transmittance of both valleys is zero, which might be explained either by zero current (OFF) or by the presence of the confinement states. Valley confinement in twisted TMDs: Previously, we have shown in Fig. 7 that at $E_{F}=0.035t$ , the resonance or confinement might be produced at higher values of ISOC strengths. This feature is somehow challenging to attain in available setups. To confirm the presence of resonances in realistic conditions, we tune both the incident Fermi energy and the twist angle. In Fig. 10, we show that the resonance condition might be obtained at weak induced SOCs at some specific incident energy and twist values. Importantly, we observe that at $\theta=22.7^{\circ}$ ( $27^{\circ}$ ), the system can confine valley states in the IQD region of MoSe 2 (WSe 2) at a lower energy $E_{F}=± 0.035t=± 0.098$ meV (within the range of the first propagating mode). Indeed, for appropriate choices of the SOC values related to the chosen TMD and respective twist angle, the incoming electron is trapped around the IQD area, as shown in Fig. 10. More precisely, for $E_{F}>0$ , the IQDs of MoSe 2 and WSe 2 islands (panels Fig. 10 (a), 10 (b), 10 (e), 10 (f)) reveal higher localization throughout the scattering regions, supported by valley-localized states from valley ${\bf K_{1}}$ . However, the valley-localized states from ${\bf K_{2}}$ are blocked at the first IQD. The process might be reversed for $E_{F}<0$ as we have shown in the lower panels of Fig. 10 10 (c), and 10 (d), for MoSe 2 and Fig. 10 10 (g), and 10 (h) for WSe 2. This is an important result since this class of materials allows or induces valley confinement that might be used to process the optical responses and detect and monitor valley polarization [14, 15, 16]. By increasing the incident energy beyond the first propagating mode, it is possible to confine both valley states around the IQDs of WS 2 islands as shown in Fig. 10 (k) and (l). Both confined states are highly localized around all IQDs in the scattering region, as predicted in [50]. Interestingly, shifting the sign of the incident energy reverses the path of interference, and both valleys exchange the propagating direction. <details> <summary>2406.02393v1/x28.png Details</summary>  ### Visual Description # Technical Document Extraction: MoSe2 Fermi Energy Heatmaps ## Image Description The image contains two side-by-side heatmaps labeled **(a)** and **(b)**, depicting Fermi energy distributions in MoSe2 at a fixed angle **θ = 22.7°**. Both panels share identical axis labels and spatial dimensions but differ in Fermi energy values (**E_F**) and intensity scales. --- ### **Panel (a): E_F = 100 meV** - **Title**: `MoSe2: E_F=100 (meV), θ=22.7°` - **Axis Labels**: - **X-axis**: `length (nm)` (range: -40 to 40 nm) - **Y-axis**: `width (nm)` (range: -10 to 10 nm) - **Color Scale**: - **Legend**: Red-to-white gradient (0.0 to 1.5) - **Key Observation**: Two bright red circular regions centered at approximately (±20 nm, 0 nm) with high-intensity gradients radiating outward. - **Arrows**: Red directional vectors indicating flow or gradient direction, concentrated near the circular regions. --- ### **Panel (b): E_F = 100 meV** - **Title**: `MoSe2: E_F=100 (meV), θ=22.7°` - **Axis Labels**: - **X-axis**: `length (nm)` (range: -40 to 40 nm) - **Y-axis**: `width (nm)` (range: -10 to 10 nm) - **Color Scale**: - **Legend**: Blue-to-white gradient (0.0 to 0.3) - **Key Observation**: A single faint blue circular region centered at approximately (0 nm, 0 nm) with weaker intensity and broader spatial distribution compared to Panel (a). - **Arrows**: Blue directional vectors, less dense and more dispersed than in Panel (a). --- ### **Cross-Referenced Observations** 1. **Legend Consistency**: - Panel (a) uses **red** to denote higher intensity (max 1.5), while Panel (b) uses **blue** for lower intensity (max 0.3). - Arrows in both panels match their respective color schemes (red in (a), blue in (b)). 2. **Spatial Trends**: - Panel (a) shows localized, high-intensity features, suggesting strong Fermi surface localization. - Panel (b) exhibits diffuse intensity, indicating broader Fermi surface dispersion. 3. **Angle Consistency**: Both panels share the same **θ = 22.7°**, implying comparative analysis under identical crystallographic orientations. --- ### **Summary** The heatmaps contrast Fermi energy distributions in MoSe2 at **E_F = 100 meV** under identical angular conditions. Panel (a) highlights localized, high-intensity regions, while Panel (b) demonstrates broader, lower-intensity distributions. The color scales and directional arrows provide quantitative and qualitative insights into electronic behavior. </details> <details> <summary>2406.02393v1/x29.png Details</summary>  ### Visual Description # Technical Document Extraction: MoSe2 Electronic Structure Visualization ## Panel (c): MoSe2 at E_F = -100 meV, θ = 22.7° - **Title**: `MoSe2: E_F=-100 (meV), θ=22.7°` - **Axes**: - **X-axis**: `length (nm)` ranging from -40 to 40 nm. - **Y-axis**: `width (nm)` ranging from -10 to 10 nm. - **Color Scale**: - **Range**: 0.0 (white) to 0.8 (red). - **Gradient**: Red-to-white, indicating intensity of electronic density. - **Key Features**: - **Left Circular Feature**: - **Location**: Centered at (-20 nm, 0 nm). - **Intensity**: High (red core with radial gradient). - **Structure**: Concentric rings with alternating intensity. - **Right Circular Feature**: - **Location**: Centered at (20 nm, 0 nm). - **Intensity**: Low (faint red outline with minimal gradient). - **Structure**: Diffuse, less defined rings. ## Panel (d): MoSe2 at E_F = -1000 meV, θ = 22.7° - **Title**: `MoSe2: E_F=-1000 (meV), θ=22.7°` *(Note: Corrected from -100 meV to -1000 meV)* - **Axes**: - **X-axis**: `length (nm)` ranging from -40 to 40 nm. - **Y-axis**: `width (nm)` ranging from -10 to 10 nm. - **Color Scale**: - **Range**: 0 (white) to 3 (red). - **Gradient**: Blue-to-red, indicating higher intensity variations. - **Key Features**: - **Left Circular Feature**: - **Location**: Centered at (-20 nm, 0 nm). - **Intensity**: Very high (deep red core with sharp blue outline). - **Structure**: Multiple concentric rings with alternating blue/red. - **Right Circular Feature**: - **Location**: Centered at (20 nm, 0 nm). - **Intensity**: Moderate (blue core with red outline). - **Structure**: Defined rings with localized intensity peaks. ## Comparative Analysis - **Fermi Energy Impact**: - At **E_F = -100 meV** (Panel c), electronic density is localized and weaker. - At **E_F = -1000 meV** (Panel d), electronic density is more pronounced and spatially extended. - **Angle Consistency**: Both panels use θ = 22.7°, suggesting a fixed crystallographic orientation. - **Color Scale Interpretation**: - Panel (c): Lower maximum value (0.8) indicates weaker signal. - Panel (d): Higher maximum value (3) reflects stronger electronic density. ## Technical Notes - **Units**: All spatial dimensions in nanometers (nm). - **Color Bar Alignment**: - Panel (c): Red corresponds to highest intensity (0.8). - Panel (d): Red corresponds to highest intensity (3), with blue representing lower values. - **Feature Symmetry**: Both panels exhibit bilateral symmetry about the y-axis (width = 0 nm). ## Conclusion The visualization demonstrates how Fermi energy (E_F) modulates electronic density distribution in MoSe2 at a fixed angle (θ = 22.7°). Lower E_F (-100 meV) results in weaker, localized features, while higher E_F (-1000 meV) enhances intensity and spatial extent of electronic states. </details> <details> <summary>2406.02393v1/x30.png Details</summary>  ### Visual Description # Technical Document Extraction: WSe2 Fermi Surface Analysis ## Image Overview The image contains two side-by-side heatmaps labeled **(e)** and **(f)**, depicting Fermi surface intensity distributions for tungsten diselenide (WSe₂) at a Fermi energy (**E_F**) of 95 meV and an angle (θ) of 27°. Both panels share identical axis labels and spatial dimensions but differ in intensity distribution patterns. --- ### Panel (e): WSe2 Fermi Surface (E_F=95 meV, θ=27°) #### Labels & Axis Titles - **X-axis**: `length (nm)` (range: -40 to 40 nm) - **Y-axis**: `width (nm)` (range: -10 to 10 nm) - **Panel Title**: `(e) WSe2: E_F=95 (meV), θ=27°` #### Color Scale & Legend - **Color Bar**: Red gradient (left: 0.0, right: 0.8) - **Spatial Grounding**: Legend positioned on the right edge of the panel. #### Key Trends & Data Points 1. **Intensity Distribution**: - Two high-intensity circular regions centered at approximately: - **Left**: (-15 nm, 0 nm) - **Right**: (15 nm, 0 nm) - Intensity decays radially outward from these centers, forming concentric interference-like patterns. - Maximum intensity value: **0.8** (red peak). 2. **Spatial Symmetry**: - Symmetric distribution about the y-axis (width = 0 nm). - No significant intensity variation along the x-axis beyond ±20 nm. --- ### Panel (f): WSe2 Fermi Surface (E_F=95 meV, θ=27°) #### Labels & Axis Titles - **X-axis**: `length (nm)` (range: -40 to 40 nm) - **Y-axis**: `width (nm)` (range: -10 to 10 nm) - **Panel Title**: `(f) WSe2: E_F=95 (meV), θ=27°` #### Color Scale & Legend - **Color Bar**: Blue gradient (left: 0.0, right: 0.4) - **Spatial Grounding**: Legend positioned on the right edge of the panel. #### Key Trends & Data Points 1. **Intensity Distribution**: - Single elongated high-intensity region centered at approximately: - **Center**: (0 nm, 0 nm) - Intensity gradient extends asymmetrically toward the left (negative x-axis), with a sharp decay toward the right. - Maximum intensity value: **0.4** (blue peak). 2. **Asymmetry**: - Dominant intensity on the left side of the center (x < 0 nm). - Minimal intensity on the right side (x > 0 nm). --- ### Comparative Analysis | Feature | Panel (e) | Panel (f) | |------------------------|------------------------------------|------------------------------------| | **Intensity Scale** | 0.0–0.8 (red) | 0.0–0.4 (blue) | | **Bright Regions** | Two symmetric circular peaks | One asymmetric elongated peak | | **Dominant Direction** | Radial symmetry | Leftward asymmetry | | **Color Legend** | Red (high intensity) | Blue (low intensity) | --- ### Textual Elements - **Embedded Text**: - Panel (e): `(e) WSe2: E_F=95 (meV), θ=27°` - Panel (f): `(f) WSe2: E_F=95 (meV), θ=27°` - **Axis Labels**: - X-axis: `length (nm)` - Y-axis: `width (nm)` - **Legend Text**: - Color bar labels: `0.0` (minimum) to `0.8` (panel e) / `0.4` (panel f). --- ### Language & Transcription - **Primary Language**: English (all labels, titles, and axis markers are in English). - **No Additional Languages Detected**. --- ### Conclusion The heatmaps visualize Fermi surface intensity distributions for WSe₂ at identical experimental conditions (E_F=95 meV, θ=27°). Panel (e) exhibits symmetric double-peak intensity, while panel (f) shows an asymmetric single-peak distribution. The color scales and spatial grounding confirm the intensity magnitudes and directional trends. </details> <details> <summary>2406.02393v1/x31.png Details</summary>  ### Visual Description # Technical Analysis of WSe2 Visualization Panels (g) and (h) ## Panel (g): WSe2 - E_F = -95 meV, θ = 27° ### Axes and Labels - **X-axis**: `length (nm)` ranging from -40 nm to 40 nm - **Y-axis**: `width (nm)` ranging from -10 nm to 10 nm - **Color Scale**: - **Range**: 0.0 (white) to 0.4 (red) - **Gradient**: Linear transition from white (low intensity) to red (high intensity) ### Key Features 1. **Localized High-Intensity Region**: - **Location**: Left side of the plot (length ≈ -20 nm, width ≈ 0 nm) - **Characteristics**: - Circular bright red core (peak intensity ≈ 0.4) - Radiating red streamlines indicating directional flow or field lines - **Surrounding Area**: Predominantly white (intensity ≈ 0.0), suggesting minimal activity outside the core region. ## Panel (h): WSe2 - E_F = -95 meV, θ = 27° ### Axes and Labels - **X-axis**: `length (nm)` ranging from -40 nm to 40 nm - **Y-axis**: `width (nm)` ranging from -10 nm to 10 nm - **Color Scale**: - **Range**: 0.0 (white) to 0.8 (red) - **Gradient**: Linear transition from white (low intensity) to red (high intensity) ### Key Features 1. **Dual High-Intensity Regions**: - **Location**: Symmetric pairs at (length ≈ ±10 nm, width ≈ 0 nm) - **Characteristics**: - Central white cores (intensity ≈ 0.0) surrounded by concentric red/blue rings - **Outer Ring**: Red (intensity ≈ 0.8) with blue streamlines indicating opposing directional flows - **Inner Ring**: Blue (intensity ≈ 0.6) with red streamlines, suggesting counter-rotating fields - **Background**: - Blue and red streamlines radiating outward from the dual cores - Gradient intensity transition from white (center) to red/blue (periphery) ### Comparative Observations - **Color Scale Consistency**: - Panel (h) uses a broader intensity range (0.0–0.8) compared to panel (g) (0.0–0.4), indicating higher maximum values in (h). - **Flow Dynamics**: - Panel (g) shows unidirectional flow emanating from a single core. - Panel (h) exhibits bidirectional/rotational flow patterns around dual cores, suggesting complex field interactions. ### Legend and Color Bar - **Panel (g)**: - Red: 0.4 (maximum intensity) - White: 0.0 (minimum intensity) - **Panel (h)**: - Red: 0.8 (maximum intensity) - Blue: 0.6 (intermediate intensity) - White: 0.0 (minimum intensity) ### Technical Context - **Material**: Tungsten diselenide (WSe₂) - **Parameters**: - Fermi energy (`E_F`): -95 meV (electron doping level) - Polar angle (`θ`): 27° (crystal orientation) - **Visualization Type**: - Likely represents charge density, electric field distribution, or similar scalar/vector field data in a 2D material system. </details> <details> <summary>2406.02393v1/x32.png Details</summary>  ### Visual Description # Technical Document Extraction: Heatmap Analysis of WS2 Samples ## Panel (k) - WS2: E_F = -300 (meV), θ = 15° - **Title**: `(k) WS2: E_F=-300 (meV), θ=15°` - **Axes**: - **X-axis**: `length (nm)` (range: -40 to 40 nm) - **Y-axis**: `width (nm)` (range: -10 to 10 nm) - **Color Scale**: - **Range**: 0 to 10 (units unspecified) - **Gradient**: White (low) → Red (high) - **Key Observations**: - Symmetric intensity distribution centered at (0, 0). - Red ring-like structure with uniform intensity (~10 max). - No significant spatial variation within the ring. ## Panel (l) - WS2: E_F = -300 (meV), θ = 15° - **Title**: `(l) WS2: E_F=-300 (meV), θ=15°` - **Axes**: - **X-axis**: `length (nm)` (range: -40 to 40 nm) - **Y-axis**: `width (nm)` (range: -10 to 10 nm) - **Color Scale**: - **Range**: 0 to 15 (units unspecified) - **Gradient**: White (low) → Blue (high) - **Key Observations**: - Asymmetric intensity distribution with a blue core (~15 max) and red peripheral ring. - Core intensity localized near (0, 0), with peripheral ring extending to ~±20 nm in length. - Higher maximum intensity (15) compared to Panel (k). ## Cross-Reference: Color Legend Consistency - **Panel (k)**: Red corresponds to highest intensity (10). - **Panel (l)**: Blue corresponds to highest intensity (15). - **Angle Consistency**: Both panels share identical θ = 15° parameter. ## Structural Notes - Both panels represent the same material (WS2) and Fermi energy (E_F = -300 meV). - Divergent intensity profiles suggest differences in experimental conditions or material properties between (k) and (l). </details> Figure 10: Real-space profiles for both valley currents: red (blue) lines show the current profile from states originating in valley ${\bf K_{1}}\quad({\bf K_{2}}$ ). Panels (a)-(d) show the valley conductance vs twist angle $\theta$ for IQDs made of MoSe 2 on graphene at $E_{F}=± 100$ meV. Panels (e)-(h) are for WSe 2 on graphene at $E_{F}=± 95$ meV. Panels in (k) and (l) correspond to WS 2 at higher energy $E_{F}=-300$ meV. IV Conclusions and summary We have used a tight-binding approach to obtain an effective, low-energy Hamiltonian to investigate valley-dependent transport through an array of proximity-induced quantum dots with $C_{3v}$ symmetry in a zigzag graphene nanoribbon. The model is a natural extension of the structure studied in Ref. [51] that analyzed spin and charge distributions in a graphene quantum dot induced by a WSe 2 island. Because the fabrication of graphene/TMD heterostructures with quantum dots of one or the other material is rapidly evolving [52, 53, 54], experimental implementations for arrays of quantum dots will likely be realized in the near future. Our results show that the valley conductance exhibits an interesting behavior sensitive to the model parameters and the values of the various SOCs induced by the islands that decorate the ribbon. At the same time, this sensitivity is a great advantage for monitoring and controlling the different valley filter regimes observed. The results reveal the localization of valley-centered states produced by the competition between Zeeman and Rashba couplings in narrow IQDs. For larger IQDs, the valley states combine with bulk states, and the valley polarization is considerably deteriorated [34]. Following Ref. [30], the qualitative description of the model shows that by varying Zeeman couplings, the conductance through a symmetric chain of quantum dots displays square-shaped curves with wide gaps. However, these features tend to vanish for some large ISOC values with the subsequent vanishing of valley polarization. Furthermore, for specific ranges of ISOC values, both valleys are present, implying that the device does not display a perfect valley-transistor behavior. An important conclusion from the qualitative description is the similarity of this system to the Datta-Das transistor. In both cases, the spin conductance is directly controlled by the strength of the Rashba SOC. In this respect, the most important SOC for good valley polarization is the staggered intrinsic SOC. These features could be helpful for further exploration of actual devices. The sensitivity to the Rashba coupling is discussed in the presence of weak ISOC to describe realistic settings where proximity effects develop. The valley conductance shows that tuning the valley polarization and switching the valley scattering in the system using a top gate is possible. The possibility of using TMDs as decorating islands to form IQDs has also been discussed. We have shown that the presence of the PIA coupling, characteristic of these structures, does not affect the valley-polarization. Therefore, such islands can be used to tune the polarization by either strain or twists since the Rashba and Zeeman coupling are sensitive to external electric or strain field effects [28, 46, 47]. Finally, we have applied these models to solve examples of realistic material combinations. To give a comprehensive picture, we have used graphene/TMD heterostructures with different semiconducting materials: MoS 2, MoSe 2, WSe 2, and WS 2. We noticed that IQDs based on semiconducting TMDs might be used as promising islands for generating valley Hall signals. Indeed, TMDs allow valley filtering processes and break the valley degeneracy, producing a valley-polarized current that favors valley selection by tuning the sign of the incident Fermi energy or the value of twist angles. Notably, the TMD island and control of the twisting angle allow various options (as they determine different values of ISOCs) to monitor valley-driven currents, confine both valley states simultaneously in the same region, or split the valley confinement states. Achieving a nearly square-wave transmission and a valley-valve effect for the ${\bf K_{1}}$ or ${\bf K_{2}}$ valleys is highly desirable for device applications. Moreover, the confinement of quasi-bound states from either valley is extremely important for manipulating optoelectronic interactions [13, 23, 55] and valley-qubit systems [17, 3, 56]. Furthermore, the mechanism for generating valley-Hall conductivity with valley-neutral currents could be handy to obtain pure valley-Hall signals. The system proposed in this manuscript exhibits features close to these goals; however, asymmetric QDS with controlled shape and several induced point group symmetries could lead to richer results. We plan to address this issue in future work. Acknowledgements. The authors acknowledge computing time on the SHAHHEN supercomputers at KAUST University, Saudi Arabia, and the supercomputers at the Centre for Research in Molecular Modeling (CERMM), Richard J. Renaud Science Complex, Concordia University. AB would like to thank Dr. Adel Abbout at KFUPM, Saudi Arabia, for helpful discussions. Appendix A Valley-dependent conductance As stated, we use the scattering matrix formalism to calculate the individual valley conductances. We separate the propagating modes in the leads depending on their velocity and momentum direction using the Kwant package [40]. In the scattering region, valley currents might be mixed due to inter-valley scattering. However, as the two valleys are far apart in the Brillouin zone, effective valley mixing will require short-range potentials. A graphene membrane wraps smoothly for graphene deposited on top of islands to minimize strain effects. Thus, the sharp atomic termination of the island (that could give rise to inter-valley mixing) is effectively masked. These issues will have a negligible contribution for islands on top of graphene if the islands and their separation are large enough [57]. To this end, we consider only propagating states for which $\Phi({\bf vF}>0)$ . The applied source-drain voltage chooses the direction of the current composed of these states, which have both spin and valley degrees of freedom. We focus first on the valley degree and lift the valley degeneracy by defining the propagating wave functions $\Phi_{\bf K_{1}}=\Phi({\bf k}>0),\Phi_{\bf K_{2}}=\Phi({\bf k}<0)$ . These wave functions independently solve the scattering problem in the reciprocal space. To implement the Green’s function formalism [39] we need to include the scattering matrices $S^{mn}=S_{\bf K_{1}}^{mn}+S_{\bf K_{2}}^{mn}$ with $S_{\bf K_{1,2}}^{mn}$ given by $$ S_{\bf K_{1,2}}^{m,n}=\text{Tr}[G_{\bf K_{1,2}}\Gamma^{m}\,G^{\dagger}_{\bf K_% {1,2}}\,\Gamma^{n}],\qquad(m,n=L,R\ or\ R,L); $$ The Green’s function and $\Gamma$ matrices are given by $$ \displaystyle G(\epsilon,{\bf K_{1,2}}) \displaystyle= \displaystyle\left[\left(\epsilon+i\eta\right)I-H_{QD}({\bf K_{1,2}})-\Sigma% \right]^{-1} \displaystyle\Gamma \displaystyle= \displaystyle i(\Sigma-\Sigma^{\dagger}). $$ $\Gamma$ defines the self-energy of the contacts placed to the left and right of the scattering region, and the relevant Hamiltonian is $H_{QD}$ , cf. Eq. (1). Then, for each valley mode, the valley conductance at the Dirac cones is given by Eq. (2). Appendix B Valley-resolved current To obtain the local density of states (LDOS) and currents per valley, we obtain the wave functions of the propagating modes $\Phi$ for a given energy $E$ and site $i$ . The propagating wave functions are stored per site depending on their momentum $\left\{\Phi({\bf K_{1}}),\ \Phi({\bf K_{2}})\right\}$ and their spin degree of freedom. The resulting LDOS, at a given site $i$ in the sample, is defined by $$ \text{LDOS}^{\bf K_{1,2}}\left(E,i\right)=\sum_{l}\left|\langle i|\Phi^{\bf K_% {1,2}}_{l}\rangle\right|^{2}\delta(E-E_{l}) $$ where the sum is over all electron eigenstates $|\Phi_{l}\rangle=c_{l}^{\dagger}|0\rangle$ of the Hamiltonian $H_{QD}$ in Eq. (1) with energy $E_{l}$ . The valley-resolved LDOS in Eq. (B.1) is calculated using Chebyshev polynomials [58] and damping kernels [59]. The corresponding density operator and the continuity equation are expressed as $$ \rho_{q}^{\bf K_{1,2}}=\sum_{a}[\Phi_{a}^{\bf K_{1,2}}]^{*}\,H^{s}_{q}\,\Phi_{% a}^{\bf K_{1,2}},\qquad\frac{\partial\rho_{a}^{\bf K_{1,2}}}{\partial t}-\sum_% {b}J_{a,b}^{\bf K_{1,2}}=0. $$ where $a$ refers to all sites in the scattering region, $J_{ab}^{\bf K_{1,2}}$ is the valley-resolved current, and $H^{s}_{q}$ is the $q$ -Fourier component of the hopping term in the scattering region. For a given site density $\rho_{a}$ , we sum over all the neighboring sites $b$ . As a result, the valley current $J^{{\bf K_{1,2}}}_{ab}$ takes the form $$ J_{a,b}^{\bf K_{1,2}}=[\Phi^{\bf K_{1,2}}({\bf v>0})]^{*}\left(i\sum_{\gamma}H% ^{*}_{ab\gamma}H^{s}_{a\gamma}-H^{s}_{a\gamma}H_{ab\gamma}\right)[\Phi^{\bf K_% {1,2}}({\bf v>0})] $$ where $H_{ab\gamma}$ is a component of a rank-4 tensor that can be represented as a vector of matrices, and $\gamma$ is an index that runs over sites in real space. 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