Exploring the Kondo Effect in Strained Kagome Nanoribbons

Abstract

Metallic kagome systems have attracted considerable interest in recent years, as they provide a rich platform for studying phenomena associated with their distinctive band structure. The coexistence of bands with Dirac points similar to those in graphene, along with a completely flat band, makes this an ideal structure for investigating how lattice symmetries may protect topological and many-body correlation effects. Since applied strain can break lattice symmetries and modify the electronic structure, understanding how strain influences phenomena such as the Kondo effect in kagome materials may provide essential insights into correlated-electron behavior. We employ the single-impurity Anderson model and the numerical renormalization group to analyze the Kondo effect in kagome zigzag nanoribbons under uniaxial strain. We find that strain manipulation enables precise control over the strength of the Kondo effect on an impurity hybridized on the ribbon with different coordination environments, and that symmetric local environments may result in strong suppression of the effective hybridization due to orbital interference. We find that the specific location of the impurity on the ribbon, especially when the Fermi energy lies close to weakly dispersive edge states, can lead to significant changes in the characteristic Kondo temperature. Such sensitivity may be used to provide unique information on the local density of states of a system. These results demonstrate that strain is a powerful tuning parameter in kagome nanoribbons, strongly modifying the screening of magnetic impurities.

Figures (9)

• Figure 1: (a) Schematic representation of a kagome ribbon with zigzag edges at top/bottom, and with periodic boundary conditions along the horizontal direction. The green dashed rectangle indicates a unit cell of the kagome lattice, composed of three sites. Black spheres stand for magnetic impurities in different configurations, $p_1$, $p_2$, and $p_3$, identified as on top, bridge, and hollow sites, respectively. (b) Ribbon after strain is applied, illustrating the resulting deformation of the system, here at an angle $\theta = 0$ (indicated by black arrows). (c) Band structure of a 2D kagome lattice along the high-symmetry path $\Gamma$-$M$-$K$-$\Gamma$. The blue dashed lines show the unstrained case, while the red solid lines correspond to the strained system. The deformation leads to notable modifications, including shifting Dirac points near $K$, a shift in the upper band near the $M$ point, and changes in the flat-band region. (d) Corresponding DOS curves for panel (c). Notice that strain reduces overall bandwidth, shifting DOS especially near the top edge. A structure near the higher VHS, along with drastic changes in the low-energy edge region, is also visible.
• Figure 2: (a) Subband structure for a ZKR, without (blue) and with (red) applied strain ($\epsilon=0.05$). Strain modifies subband dispersion, particularly near the top of the range ($E \simeq 1$), the low-energy region of the flat-band ($E\simeq -0.5$), and the upper VHS ($E\simeq 0.5$). (b) Corresponding DOS highlights strain-induced changes due to subband shift and mixing, especially in the regions mentioned in (a).
• Figure 3: Spatial distribution for different ZKR eigenstates. The zigzag edges are on the left/right of the figures. The three sets of panels (for different wave vector $k$, as indicated) show results without (blue) and with (red) uniaxial strain ($\epsilon =5\%$, $\theta=0$), respectively. Energy value for each pair are (a) $E=-0.5$ (without) and $E=-0.49$ (with strain) for $k=0$; (b) $E=-0.5$ and $E=-0.45$ for $k=\pi/3$; (c) $E=-0.50$ and $E=-0.48$ for $k=2.5$. Spatial weights shift across the ribbon as strain is applied, including the weights on the zigzag edges.
• Figure 4: Results for an impurity in hollow site configurations ${p}_3$ located at two different locations on the ZKR. Panels in top row (a-c) show results for an impurity on one edge of the ribbon ('Edge'), while the bottom row (d-f) shows results for an impurity in the middle of the ribbon ('Bulk'). In both cases, $\mu =-0.40$ and $\theta=40^\circ$. Panels (a) and (d) show the corresponding hybridization function $\Delta(\omega)$ for different values of strain; notice that $\Delta(\omega)$ shifts to lower energies as strain increases. Peaks near the Fermi energy ($\omega \simeq 0$) are associated with states residing mostly on the ribbon edge; higher energy features correspond to states extending throughout the ZKR. Panels (b) and (e) present the impurity spectral function $A(\omega)$ for Edge and Bulk locations, respectively. Sharp Fermi-level resonance peaks are observed in all cases, indicating strong Kondo resonances, between the Hubbard peaks at $\omega \simeq \pm U/2 = \pm 0.005$. Panels (c) and (f) show the corresponding magnetic susceptibility curves ($=T\chi(T)$) as a function of temperature $T$. The Bulk location is nearly unaffected by strain at this chemical potential. In contrast, the Edge location shifts the characteristic Kondo temperature to higher values as strain increases. (g) $T_K$ as a function of strain for both $p_3$ locations: blue curve for Bulk, and red curve for Edge, show $T_K$ enhancement by over an order of magnitude at the Edge location.
• Figure 5: Impurity screening in $p_3$ configuration located on the Edge of the ribbon for different chemical potential values. The ribbon is under uniaxial strain $\epsilon=5\%$ at $\theta=40^\circ$. (a) Hybridization function $\Delta(\omega)$ for different $\mu$ values, chosen to align $E_F$ with different band features: $\mu=-0.50$ ($-0.01$) place $E_F$ at the higher (lower) VHS; for $\mu=-0.26$, $E_F$ is near the Dirac point; $\mu=0.36$ shifts $E_F$ to the zigzag edge state; and $\mu=0.49$ shifts $E_F$ to the flat-band region of the ribbon. (b) Corresponding spectral functions $A(\omega)$ exhibit sharp Kondo resonance near/at $E_F$, Hubbard bands, as well as non-universal features, especially for $\mu=0.49$ and $-0.01$, where sharp features in $\Delta(\omega)$ values are close to $E_F$. (c) Magnetic susceptibility $T\chi(T)$ shows the evolution from the free orbital fixed point at high $T$, to the local moment at intermediate $T$, and finally strong coupling at low $T<T_K$. The temperature dependence of the entropy $S$ shows similar evolution. (d) Kondo temperature $T_K$ as function of $\mu$. The blue curve corresponds to results near the zigzag edge (Edge), while the red curve represents response in the middle of the ribbon (Bulk). Lines drawn to guide the eye.