Edge state reconstruction in proximitized graphene and its probe using RKKY interaction

The gist: The Ruderman-Kittel-KasuyaYosida (RKKY) magnetic exchange interaction can be used to probe shifted band dispersion in edge states, revealing that an on-site edge potential reconstructs these states into purely intra-valley, spin and valley polarized modes, which charge conductance cannot capture.

Edge State Reconstruction

The investigation focuses on the effect of a finite on-site potential restricted to the edges of zigzag graphene nanoribbons on their edge states The intervalley edge states undergo reconstruction and become purely intra-valley, exhibiting both spin and valley polarization In the uniform SOC case, after applying edge potential, the inter-valley edge state crossings of the QSHI shift towards the valley gap and we are left with purely intra-valley edge states In the staggered case, where the pseudohelical edge states are already spin-valley locked, applying edge potential similarly reconstructs them into intra-valley edge states.

Probing Mechanism

The charge conductance calculations fail to capture this reconstruction because the number of transmitted channels remains unchanged Therefore, a probe directly sensitive to the spin and valley character of the edge states is required The RKKY exchange interaction between two magnetic impurities situated on the ribbon edge can be used to probe the shifted band dispersion, for both the uniform and staggered spin-orbit coupling. In presence of strong intrinsic and Rashba SOC, the RKKY interaction becomes fully anisotropic Both sublattice staggering and edge potential-induced reconstruction lead to distinct features in the RKKY interaction.

RKKY Interaction Analysis

The RKKY interactions become anisotropic and electrically tunable when the mediating states carry spin-orbit coupling, acquiring a nontrivial tensor structure dictated by the residual spin symmetries of the system The application of an edge potential enhances the oscillation frequency of the RKKY interaction terms In presence of staggered SOC, the Dzyaloshinskii–Moriya (DM) interaction exhibits higher amplitude, accompanied by the emergence of nonzero in-plane components. Additionally, one of the symmetric exchange terms shows a beating pattern when both staggered SOC and edge potential are present.

Sublattice Dependence and Symmetry

The study considers two different situations when both the impurities are located at the same sublattices (AA) of the nanoribbon or they are situated at different sublattices (AB) The results show that the application of an edge potential enhances the oscillation frequency of the RKKY interaction terms. In absence of edge potential, Jxx = Jyy varyies smoothly with ∆m while Jzz is strongly suppressed in comparison to the other components This is expected for ordinary helical edge states whose spin is locked to momentum At V = 0 both the regimes have Jxx = Jyy ≫ Jzz suggesting that the effective spin-quantization axis along z remains approximately preserved irrespective of the SOC staggering pattern in absence of edge potential.

Conclusion and Significance

These results establish RKKY interactions as a spin and sublattice resolved probe of proximity induced effects in graphene nanoribbons, of potential relevance to future spintronic and valleytronic device applications The dominant components among various contributions are governed by the underlying symmetries of the system We also find that the RKKY interactions are stronger when the two impurities sit on the same sublattice sites, while a sign reversal between AA and AB configurations occurs for the zero edge potential case, in the Heisenberg and DM terms. The amplitude of DM terms is enhanced in the presence of staggered SOC and nonzero in-plane components emerge. The faster oscillation is the result of momentum-space reconstruction of the edge states’ Green function entering into the spin susceptibility in Eq. (11).

How it works

The RKKY interaction is established between magnetic impurities through their coupling with the conduction electrons of the host material By applying the second-order perturbation theory, Hint can be reduced to an effective RKKY interaction given by HαβRKKY = J2cXk,lI1kχαβkl(r, r′)I2l(10) This term can be decomposed into the isotropic Heisenberg interaction (Jαβxx, Jαβyy), Ising interaction (Jαβzz), anisotropic symmetric interactions (Jαβxy, Jαβyz, Jαβzx), and anisotropic antisymmetric Dzyaloshinskii–Moriya (DM) interactions. The explicit form of the spin susceptibility tensor used in our numerical calculations, is given in the Appendix. The RKKY Hamiltonian can be written as HαβRKKY = Xl,mJαβlmI1l2 + Iml1I2 + J⃗αβDM · ⃗I1 ×⃗I2 (13).

Key Findings Summary

**: The charge conductance calculations do not provide a sufficiently sensitive probe to capture the changes in the band dispersion induced by the onsite edge potential for both the uniform and staggered regimes. **

**: The RKKY interaction is highly sensitive to their spatial, spin and valley degree of freedoms and provides a more complete probe of the newly formed valley-polarized edge states. **

**: The application of a finite edge potential V = t shifts the edge states toward the valley gap [Figs. 2 (b) and (d)]. **

**: In presence of the edge potential, for the uniform SOC case, Jxx and Jyy are degenerate and Jzz is vanishingly small compared to the other two components. **

**: In the staggered case, by contrast, applying edge potential lifts the Jxx = Jyy degeneracy and Jzz becomes comparable to the in-plane components [Fig. 4 (f) and (h)] when both the intrinsic sublattice asymmetry and edge potential act together. **

**: The amplitude of DM terms is enhanced in the presence of staggered SOC and nonzero in-plane components emerge. **

**: In the staggered case at V = t, staggered SOC [Figs. 6 (f) and (h)] shows another distinct feature where instead of the smooth, monotonically decaying oscillations seen in the uniform case with edge potential, Jyz shows a beating pattern for both AA/AB cases. **

**: These results establish RKKY interactions as a spin and sublattice resolved probe of proximity induced effects in graphene nanoribbons. **

**: The RKKY interaction comprises several contributions, namely Heisenberg, Ising, symmetric anisotropic, and Dzyaloshinskii-Moriya (DM) terms. **

**: We also find that the RKKY interactions are stronger when the two impurities sit on the same sublattice sites. **

**: A sign reversal between AA and AB configurations occurs for the zero edge potential case, in the Heisenberg and DM terms. **

The paper is relevant to future spintronic and valleytronic device applications. The gist.

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Seerat Khan1,∗ and SK Firoz Islam1, † 1 Jamia Millia Islamia, New Delhi-110025, India Proximity induced spin-orbit coupling (SOC) in graphene emerges when it is placed on a substrate, breaking sublattice symmetry and giving rise to staggered intrinsic SOC together with Rashba spinorbit coupling In zigzag graphene nanoribbons, this staggered SOC produces pseudohelical edge states exhibiting sublattice-locked rather than spin-locked helicity We investigate the effect of a finite on-site potential, restricted to the edges of the ribbon, on these edge states We find the intervalley edge states undergo reconstruction and become purely intra-valley, exhibiting both spin and valley polarization The charge conductance calculations fail to capture this reconstruction, since the number of transmitted channels remains unchanged We show that the Ruderman-Kittel-KasuyaYosida (RKKY) magnetic exchange interaction, between two magnetic impurities situated on the ribbon edge can be used to probe the shifted band dispersion, for both the uniform and staggered spin-orbit coupling In the presence of strong intrinsic and Rashba SOC, the RKKY interaction becomes fully anisotropic Both sublattice staggering and edge potential-induced reconstruction lead to distinct features in the RKKY interaction We show that the RKKY interaction terms acquire faster oscillations as a function of the spatial separation between two magnetic impurities, in response to the edge state reconstruction, establishing it as an effective probe of the edge states Additionally, the significant mismatch in the amplitudes of the DM term corresponding to the uniform and staggered SOC, can also act as a probe of the type of SOC.

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2 such edge states.

1. No longer limited to charge conductance as a probe, but utilizing RKKY interactions to directly measure the spatial and spin-dependent properties of low-energy states. This allows AI systems to perform spatially resolved mapping of spin and valley degrees of freedom in proximity-induced graphene nanoribbons.

2. Ability to distinguish between uniform and staggered SOC regimes by analyzing the oscillation frequency of RKKY interaction terms, which are enhanced in the presence of edge potential. This enables AI to classify the type of SOC (uniform vs. staggered) based on measured magnetic impurity interactions alone.

3. Development of a predictive model for RKKY exchange tensors that incorporates the effects of both intrinsic sublattice asymmetry and edge potential, as shown by how Jxx and Jyy are degenerate and Jzz is vanishingly small compared to the other two components at V = t in the uniform case. This allows AI to predict how external perturbations (like an edge potential) will alter magnetic coupling signatures.

4. Creation of a diagnostic tool that uses the magnitude and relative strength of DM interaction terms, noting that the amplitude of DM terms is enhanced in the presence of staggered SOC and nonzero in-plane components emerge. This enables AI to identify systems exhibiting specific spin textures by analyzing the anisotropy between different RKKY interaction components.

5. Implementation of a method to detect phase reversal between AA and AB configurations in zero edge potential cases, which occurs in the Heisenberg and DM terms. This allows AI to differentiate topological phases or material configurations based on the sign of magnetic exchange interactions measured at specific impurity placements.