Spectral properties and the Kondo effect of cobalt adatoms on silicene
In terms of the state-of-the-art first principle computational methods combined with the numerical renormalization group technique the spectroscopic properties of Co adatoms deposited on silicene are analyzed. By establishing an effective low-energy Hamiltonian based on first principle calculations, we study the behavior of the local density of states of Co adatom on external parameters, such as magnetic field and gating. It is shown that the Kondo resonance with the Kondo temperature of the order of a few Kelvins can emerge by fine-tuning the chemical potential. The evolution and splitting of the Kondo peak with external magnetic field is also analyzed. Furthermore, it is shown that the spin polarization of adatom’s spectral function in the presence of magnetic field can be relatively large, and it is possible to tune the polarization and its sign by electrical means.
The discovery of graphene, a two-dimensional (2D) honeycomb lattice of carbon atoms, in 2004 (Novoselov et al., 2004) spurred an interest in other 2D materials, especially those sharing graphene’s crystal structure. The search was motivated by the hope that such materials would also share the defining feature of graphene, i.e. the presence of Dirac cones in their electronic structures (Castro Neto et al., 2009). The suppression of backscattering characteristic of Dirac fermions should then lead to similarly high carrier mobilities (Novoselov et al., 2005; Zhang et al., 2005; Bolotin et al., 2008; Du et al., 2008) and possible use of graphene siblings in ultra-fast electronic devices (Lin et al., 2009; Liao et al., 2010; Lin et al., 2011; Schwierz, 2010; Kim et al., 2012). The continuously growing list of elements and compounds for which the existence of such graphene analogues was predicted theoretically and in some cases confirmed experimentally includes: Si (so called silicene) (Cahangirov et al., 2009; Lebègue and Eriksson, 2009; Vogt et al., 2012; Feng et al., 2012; Lin et al., 2012; Jamgotchian et al., 2012; Chiappe et al., 2012), Ge (germanene) (Cahangirov et al., 2009; Lebègue and Eriksson, 2009), Sn (stanene) (Garcia et al., 2011; Saxena et al., 2016), Al (aluminene) (Kamal et al., 2015; Yuan et al., 2017) and hexagonal BN (white graphene) (Nag et al., 2010).
Of these, silicene is particularly interesting thanks to its compatibility with the current Si-based electronic technology. The first proof of the concept field effect transistor made out of silicene has already been demonstrated (Tao et al., 2015). Many of the characteristic properties of graphene are predicted to be present also in its silicon counterpart. The band structure of the free-standing silicene exhibits the expected Dirac cones (Cahangirov et al., 2009), which can be preserved also on suitably selected substrates (Quhe et al., 2014). The zigzag edges of silicene nanoribbons are predicted to be spin-polarized (Cahangirov et al., 2009; Weymann et al., 2015; Weymann and Krompiewski, 2016) just like in the case of graphene (Castro Neto et al., 2009). However, some notable differences also exist between the two materials. While graphene is planar, in silicene the two sublattices are shifted vertically (buckling) because of larger in-plane lattice constant (weaker bonds) and the element’s preference for forming hybrids (no graphite analogue exists for Si). Consequently the and bands are hybridized. The spin-orbit (SO) interaction in silicene is three orders of magnitude stronger than in graphene and is responsible for a small gap in the electronic structure (Liu et al., 2011a, b). This may lead to the realization of the spin Hall effect in experimentally accessible temperatures. The symmetry breaking effect due to buckling opens an interesting possibility of fine tuning the gap using vertical electric field (Drummond et al., 2012; Ni et al., 2012).
Another way of affecting the properties of 2D materials is by deposition of magnetic adatoms on the surface. In fact, the presence of impurities has been invoked to explain the spin relaxation time in graphene Kochan et al. (2014). Besides modification of material properties, individual magnetic adatoms themselves can pose very interesting objects to study. This is because strong coupling between localized states of adatoms and the band of 2D material can result in various nontrivial effects. One of such effects, which has been widely studied in the context of quantum dots and molecules, is undoubtedly the Kondo effect Kondo (1964). In this effect the magnetic moment of confined electrons, either in an adatom or a quantum dot, becomes screened by surrounding mobile electrons. This results in the formation of a resonance in the local density of states at the Fermi level Hewson (1997). The Kondo effect due to the presence of magnetic adatoms has already been considered in the case of graphene Wehling et al. (2010); Fritz and Vojta (2013). Moreover, the spectroscopic properties of Co adatoms on graphene have also been examined Brar et al. (2011), and recently the presence of the Kondo effect has been reported Ren et al. (2014). Thus, while there are several considerations of spectroscopic properties of magnetic adatoms on graphene, not much is known about the spectral features and, in particular, the Kondo effect for other 2D materials, such as silicene. The aim of this paper is therefore to shed light on the physics of Co adatoms on silicene, with an emphasis on the Kondo regime.
The paper is organized as follows. In Sec. II we discuss the first principles methods used to determine the lowest energy geometry and the density of states (DOS) of silicene with Co adatom. Section III is devoted to numerical renormalization group calculations. First, we formulate the effective Hamiltonian, describe the method and then discuss the numerical results. The paper is summarized in Sec. IV.
Ii First principles calculations
The first principles calculations were performed using the generalized gradient approximation (GGA) of density functional theory (DFT). The specific method applied was full potential linearized augmented plane wave (FLAPW) (Singh, 2006) as implemented in the WIEN2K package (Blaha et al., 2001). The Perdew-Burke-Ernzerhof parametrization (Perdew et al., 1996) of the exchange potential was used in all the cases. The 2D Brilloiun zone (2D BZ) integration was performed using the mesh densities corresponding to several hundreds k-points (or more) in the single unit 2D BZ. The convergence criteria for energy, charge per atom and forces were set to , and , respectively. In all the calculations the silicene planes were separated by ensuring the lack of hopping between the neighboring planes.
In the first step the lattice constant and the sublattices’
displacement of bulk silicene were optimized. For the lowest energy
configuration we have found the in-plane lattice constant and the
vertical displacement to be equal to and
Next we compared three possible high symmetry locations of the Co adatom over the silicene. These are indicated in Fig. 1 and include the locations over two non-equivalent lattice sites (“1” and “3”) and also the position over the center of the hexagon (“2”). The calculations were performed using 3x3 supercells with the full relaxation of the atomic positions within the supercell. It has been found, in agreement with earlier works Lin and Ni (2012); Kaloni et al. (2014), that the order of the total energies is as follows
that is the central position “2” is the most favorable energetically, followed by the position over the “lower” sublattice A. The ground state is separated from the other two configurations by . These conclusions were additionally confirmed by 4x4 calculations for “1” and “2” cases with the same overall results.
In order to make sure that the size of the supercell used is sufficiently large so that the limit of the single impurity is reached we next studied the convergence of the magnetic moment and the local density of states (LDOS) of the Co adatom against the size of the supercell. Because of the huge computational costs involved, the structural relaxation was not performed during these calculations. Instead we adopted the geometry obtained in 4x4 calculations for Co and its surroundings (up to radius, corresponding to 4 coordination zones) embedding so defined “cluster” into the bulk silicene. As the deviations of Si atoms from bulk positions were found to be negligible at this distance from Co, the procedure is reasonable. We have found that both quantities stopped changing meaningfully when 7x7 (corresponding to the supercell lattice constant ) or larger supercells were used. The total magnetic moment, located predominantly on Co, equals . We conclude therefore that the impurity can be effectively treated as a spin one-half in the Anderson model.
In the final step we have calculated global and local DOS using 7x7 supercell and including the spin-orbit coupling. The 2D BZ integration was performed at this point using suitably dense mesh corresponding to k-points in the original 1x1 2D BZ. The results for the total DOS of the structure, together with the DOS of pure silicene, are presented in Fig. 2. In the vicinity of the adatom the hybridization effect comes into play, and it drastically rebuilds the LDOS. In consequence, the DOS of the system with an adatom follows the same general outline as the DOS of pure silicene but with additional modulations visible in Fig. 2. The most notable of these is perhaps the sharp peak located exactly at the Fermi energy in the minority spin channel. Analysis of the orbital contribution to the local density of states of cobalt shown in Fig. 3 indicates that the peak at the Fermi energy in the total DOS of the structure originates mainly from the orbitals of the adatom.
Iii Numerical renormalization group calculations
iii.1 Effective model
Based on the first-principle results we can now establish an effective Hamiltonian for the Co adatom on silicene. The calculated magnetic moment of adatom justifies usage of a spin one-half single impurity Anderson model Anderson (1961). In order to find the parameters of the effective single-orbital model we follow the procedure described in Ref. [Újsághy et al., 2000]. From the calculated occupancy of Co orbitals, (in agreement with Ref. [Kaloni et al., 2014]), we conclude that charge fluctuations can occur between the states with , and electrons. Starting with the Anderson model with full five-fold degeneracy of the d-shell electrons, we note that in the mean field approximation the energies of the respective charge states enumerated by can be expressed as, , where is an adatom on-site energy and denotes the Coulomb correlations. Following Ref. [Kaloni et al., 2014], the latter parameter will be set equal to eV. From the minimum energy condition with respect to we can estimate the on-site energy . Focusing on consecutive energies of the states with , and electrons (with respect to the lowest energy), we can find from and , the parameters for the effective impurity Hamiltonian, eV, , which has the following form
Here, describes the electrons in silicene with the corresponding density of states, see Fig. 2. The second term models adatom and is given by , where is the energy of an electron occupying the impurity and denotes the Coulomb correlation energy. The last term accounts for the Zeeman splitting, with being the external magnetic field and denoting the spin of the adatom. The operator creates a spin- electron on the adatom, and is the corresponding creation operator for spin- electrons in silicene. Finally, the coupling between the substrate and the adatom is modeled by the tunneling Hamiltonian , where denotes the tunnel matrix elements assumed to be equal to eV and is the density of states of bulk silicene, cf. Fig. 2. The parameter has been evaluated by Harrison’s scaling method Harrison (1980) for the average Si-Co bond length Å.
Since we are interested in nonperturbative effects resulting from the hybridization of Co adatom and silicene, to get the most accurate information about system’s spectral properties we employ the numerical renormalization group (NRG) method Wilson (1975); Legeza et al. (2008); R. Bulla and Pruschke (2008). In NRG, the band is first discretized in a logarithmic way with a discretization parameter . Then, such discretized Hamiltonian is tridiagonalized numerically and transformed to a tight-binding Hamiltonian of the following form
where is the creation operator of an electron with spin on the th site of the chain, denotes the hopping integral and is the on-site energy. In NRG calculations we assumed and kept at least states at each iteration. To obtain the most accurate results for the spectral functions, we also optimized the broadening parameter appropriately Žitko and Pruschke (2009).
iii.2 Discussion of numerical results
We now focus on the behavior of the spectral function of Co adatom, , where is the Fourier-transform of the corresponding retarded Green’s function, . The total spectral function, , corresponds to the local density of states of adatom, which can be experimentally examined with a weakly coupled probe, such as a tip of a scanning tunneling microscope.
Spectral properties and the Kondo effect
Because experimentally the position of the Fermi level can be adjusted by gating, in Fig. 4 we present the energy dependence of calculated for different values of the chemical potential , . Let us first focus on the case of no gating, . One can see that the total spectral function exhibits Hubbard resonances for and . For typical spin one-half quantum impurity models, at low energies, the Kondo physics plays an important role Kondo (1964). In the Kondo effect the conduction electrons screen the impurity’s spin resulting in an additional resonance at the Fermi energy in the local density of states, the halfwidth of which is related to the Kondo temperature Hewson (1997). In the case considered here, however, due to the depletion of states at the Fermi energy the screening of the adatom spin is not possible and, consequently, the Kondo peak is not present, see Fig. 4 for .
One can consider if it is possible to reinstate the Kondo resonance by changing the chemical potential via gating. First of all, it can be seen that, quite naturally, by tuning the position of the Hubbard resonances changes accordingly, see Fig. 4. Moreover, by adjusting the Fermi energy, one can also considerably affect the low-energy behavior of the system. In fact, for values of selected in Fig. 4, a pronounced Kondo resonance develops. This can be clearly seen in Fig. 4(b), which shows the spectral function plotted on logarithmic scale, as well as in the inset of Fig. 4(a), which presents a zoom into the low energy behavior of . For the considered values of gating, there is sufficient number of states at the Fermi energy to screen the adatom’s spin. One can then observe the Kondo effect with relatively large Kondo temperature, of the order of up to a few Kelvins, as estimated from the halfwidth at half maximum of the Kondo peak, see Fig. 4(b).
To gain a deeper understanding of the effect of gating on the local density of states, we also analyze the behavior of the spectral function by continuously tuning the chemical potential. This is presented in Fig. 5, which shows the energy and chemical potential dependence of the local density of states, with the bottom panel zooming into the low-energy behavior of . First, we note that for eV and eV, the spectral function exhibits maxima related with resonant tunneling, since then the empty and doubly occupied states of the adatom cross the Fermi energy (the adatom enters the mixed valence regime). In the range of chemical potential between these two values, the impurity effectively hosts a spin one-half, see Fig. 6(a) presenting the chemical potential dependence of the occupation of the orbital level, where . Consequently, one can expect that the Kondo effect will emerge if the number of states is sufficient to screen the impurity’s spin. The occurrence of the Kondo resonance is thus strongly dependent on the hybridization, which changes with . As a matter of fact, one can indeed clearly identify regions where the resonance at the Fermi energy develops. These regions are better visible in Fig. 5(b), which shows the zoom into the low-energy behavior of the spectral function. From the inspection of this figure one can see that for values of chemical potential ranging from eV to eV, the Kondo resonance develops with Kondo temperature strongly dependent on . The Kondo temperature becomes enhanced for eV, where K. Moreover, a relatively large Kondo temperature can be also found for eV, where K, cf. Fig. 4(b).
We would like to recall that the Kondo temperature depends in an exponential fashion on the ratio of hybridization between the orbital level and the host and the Coulomb correlations Hewson (1997). Thus, from theoretical point of view, if the host’s density of states at the Fermi energy is finite, the Kondo peak should emerge even if DOS is very low. However, the corresponding Kondo temperature would be then extremely small and, thus, the Kondo peak would be completely undetectable experimentally. In Fig. 6(b) we present the chemical potential dependence of the spectral function at for several, experimentally relevant, values of temperature. By looking at the orbital level occupation, Fig. 6(a), one can recognize enhanced spectral function due to resonant tunneling in the mixed valence regime and due to the Kondo effect in the local moment regime. Moreover, one can also inspect how quickly the Kondo correlations become smeared out by thermal fluctuations, which is especially visible in the range of from eV to eV.
Effect of external magnetic field
Let us now consider how external magnetic field affects the behavior of the local density of states of Co adatom. The energy dependence of the spin-resolved and total spectral function calculated in the presence of external magnetic field T is depicted in Fig. 7, where the insets show the zoom into the low-energy behavior of . First of all, one can see that magnetic field strongly affects the behavior of the spectral function. An important observation is a strong spin polarization of Hubbard resonances: () becomes suppressed for (). Moreover, magnetic field also has a strong effect on the low-energy behavior of : the Kondo resonance becomes split if the Zeeman energy becomes larger than , . The spin polarization of the spectral function can be clearly seen in Figs. 7(a) and (b), while the splitting of the Kondo peak is nicely visible in the inset of Fig. 7(c). We note that the largest suppression of the Kondo peak occurs for eV. For this value of gating we estimated K, which is smaller than for the case of eV and eV. Consequently, larger suppression of the Kondo resonance is observed for smaller , since the condition is then better satisfied. We also notice that the effect of splitting and suppression of the Kondo peak in the presence of magnetic field is in fact similar to the effect of an exchange-field splitting of orbital level caused by the presence of ferromagnetic correlations Martinek et al. (2003); Hauptmann et al. (2008); Gaass et al. (2011); Weymann (2011); Csonka et al. (2012). Thus, if due to proximity effect with magnetic substrate, the density of states of silicene becomes spin-polarized, the Kondo effect may be also split and suppressed even in the absence of magnetic field.
It is also interesting to analyze the energy and chemical potential dependence of the local density of states for several values of external magnetic field. This is presented in Fig. 8, which shows for T. In this figure we focus on the low-energy regime, where Kondo effect can emerge, and the interplay between the Zeeman splitting and Kondo correlations is most revealed. One can see that, depending on the value of external magnetic field and gating, the Kondo resonance can become split and suppressed. This is especially visible for eV, where with increasing , one shifts the position of the split Kondo peaks. Moreover, there are such values of , especially those close to the mixed valence regime, where the magnetic field is not strong enough to suppress the Kondo resonance, see Fig. 8.
The effect of external magnetic field can be better revealed when one considers the spin polarization of the spectral function, which is defined as, . The dependence of the spin polarization on energy and chemical potential for a few values of magnetic field is shown in Fig. 9. This figure is generated for the same values of as those considered in Fig. 8, again focusing on the low-energy behavior. When the impurity is either empty or doubly occupied, the spin polarization is suppressed and approaches zero. Its behavior, however, becomes completely changed in the local moment regime, see Fig. 9. The first observation is that can change sign around and such sign flip occurs in the regime where the impurity is occupied by a single electron. This effect is associated with the spin-splitting of the adatom orbital due to the Zeeman field. One can notice that for higher energies the spectral function becomes fully spin-polarized, for and for . Interestingly, it can be also seen that the interplay of finite Zeeman splitting and the density of states of silicene, can result in a sign change of the spin polarization at around eV, which is most visible for T, see Fig. 9(c). At low energies and for eV ( eV), becomes positive (negative).
Finally, in Fig. 10 we study the evolution of the splitting of the Kondo peak with external magnetic field. This figure is calculated for selected values of the chemical potential, the same as considered in Fig. 7. It can be seen that the Kondo peak becomes suppressed when magnetic field is so strong that the condition is fulfilled. Thus, the suppression occurs first for the case of eV, while larger field is needed to destroy the Kondo resonance for the other two cases, see Fig. 10. When magnetic field is large enough such that , becomes suppressed and the spectral function shows only side peaks, which occur exactly at the spin-flip excitation energy . The position of those side peaks depends thus linearly on magnetic field, which can be nicely seen in Fig. 10, where the dotted lines mark the Zeeman energy .
We have theoretically considered the spectroscopic properties and the Kondo effect of Co adatoms on silicene. By using the first principle calculations, we have determined the total density of states of Co-silicene system and estimated the orbital level occupancy together with the magnetic moment of Co. Our DFT results allowed us to formulate an effective low-energy Hamiltonian for spin one-half impurity, which was further used to analyze the spectral properties of Co adatoms. This analysis was performed by employing the numerical renormalization group method with a non-constant density of states. We focused on the behavior of the local density of states (spectral function), which can be probed experimentally by using the scanning tunneling spectroscopy. The analysis involved the effects of external magnetic field and gating of silicene. We showed that by appropriately tuning the parameters one can obtain clear signatures of the Kondo effect. We also analyzed the evolution and splitting of the Kondo resonance with an external magnetic field. Finally, we studied the spin polarization of the spectral function in the presence of magnetic field, whose magnitude and sign were found to greatly depend on the position of chemical potential.
Acknowledgements.I.W. acknowledges discussions with C. P. Moca. This work is supported by Polish National Science Centre from funds awarded through the decision No. DEC-2013/10/M/ST3/00488.
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