Tunable Dirac-point resonance induced by a STM-coupled Anderson impurity on a topological insulator surface

The interaction effect between the surface states of a topological insulator (TI) and a STM-coupled Anderson impurity is studied by using equations of motion of the Green’s functions. Remarkably, we show that when a coupling between the Anderson impurity and the STM tip is included, the tunneling resonance and the Kondo peak can be tuned to be exactly at the Dirac point, by adjusting the impurity level and Fermi energy, such that the local density of states at the Dirac point is significantly enhanced. This is in contrast to the case of a STM-decoupled Anderson impurity, where both resonances are always fully suppressed at the Dirac point. Our finding suggests a pathway to experimentally control the fundamental properties of the electrons on a TI surface.

The spin-momentum inter-locked TSSs are promising for unique technological applications, involving the manipulation of the Dirac electronic properties or engineering the Dirac cones [14][15][16]. Recently, engineering the TSSs through magnetically doping the TI materials has triggered strong interest. It was found that magnetically doping the bulk of TI materials can gap the TSSs because of broken TR symmetry [17,18]. The effect of impurity doping on the TI surface also has exhibited interesting phenomena [8,15,[19][20][21][22][23][24][25]. For example, recent theoretical investigations of magnetic doping on the TI surface, based on the classic impurity model, showed that low-energy resonances are produced, which vizibly modify the electronic spectrum near the Dirac point, and even split the Dirac point into two nodes [15,21,26,27]. For quantum impurity, the internal excitations of the impurities can profoundly modify the low-energy TSSs either by exchanging spin angular momentum, energy, and particles during the electron scattering processes. Highly localized impurity resonances associated with quantum impurities were experimentally revealed by scanning tunneling microscope (STM) performed on the strong TI Bi Se 2 3 [28]. Moreover, the internal excitations of the impurities can result in the interesting Kondo effect [29][30][31][32][33][34] and tunable low-energy resonances [35]. However, the low-energy resonances will be totally suppressed at the Dirac point because of the vanishing local density of states (LDOS) of the TSSs,

Model and Method
To demonstrate the method and better understand the DPR, we first consider the impurity to be decoupled from the STM, and start from the low-energy effective Hamiltonian for the TI surface, with s being the pauli-matrix for electron spin, u F the Fermi velocity and x y the in-plane wave vector. Here, we denote the annihilation operator for the TI surface electrons as , we obtain for the total Green's function TI is the Green's function without perturbation of the is the retarded Green's function of the impurity, which will be given in the next section. The Green's function in real space

The Green's function of the impurity
The retarded Green's function of the impurity is defined as , which is the Fourier In the same way, we can also derive the following equations . From these two equations, it follows By substituting equation (6) into equation (5) and replacing the summation by an integral 2 and Λ being the high-energy cutoff. Here, we have assumed s V k and ⟪ | ⟫ † s s d d to be isotropic in the k-space, so that the integral of the second term in equation (6) , and the relating higher-order Green's functions are To close the iterative equations, we truncate the higherorder Green's functions following the standard approximate method adopted in [39][40][41] by contracting the operator pairs. Generally, there are three kinds of contractions for each higher order Green's function in equation (9). For example, the first term of the second line in equation (9c) can be contracted as . For a graphene or normal metal host, since there is no spin-flip when the electron hops from the impurity to the host or propagates in the host, the last two terms vanish. However, on a TI surface, due to the strong spin-orbit interaction, the electron spin can be flipped, and thus With these techniques, the equation of motion for the Green's function of the impurity is closed, yielding . The occupation number á ñ s n is determined self-consistently by the fluctuationdissipation theorem

The self-energies
We notice that the expression for S s 1 resembles that for S s 0 . In fact, one can obtain the final analytical expression for S s 1 from equation (8)  . Unlike S s 1 , S s 2 includes the higher-order effects, and contains several averages of the operators of the TSSs. These averages can be calculated by using the retarded Green's function of the TSSs, given in equation (1), and employing the spectral theorem being the Fermi-Dirac distribution function. Since they are global averages, on which the effect of the impurity scattering is small, we can keep only the first term of equation (1). After some straightforward algebra, we obtain Substituting these relations to equation (11), we arrive at  for weak impurity-surface coupling, so that we will neglect S s 3 hereafter, which does not influence the qualitative results.

Results and discussion
We first analyze the DOS of the impurity, which is helpful for us to understand the effect of the Anderson impurity on the properties of the Dirac electrons. Then we turn to discuss the characteristics of the modified LDOS of the TSSs.

DOS of the Anderson impurity
Before analyzing the numerical results, we can infer some properties of the DOS of the impurity, defined as , from the expressions for the self-energies. In the deep Coulomb blockade regime, e.g.,  ¥ U , the impurity Green's function is reduced to Im . In the absence of a magnetic field, both S s 0 and S s 2 contain a factor w m + . Therefore, the tunneling resonance peak will change its location and width, when the chemical potential μ is varied. The resonance effect is fully suppressed at the Dirac point (w m + = 0).
The above inferences are confirmed by the numerical results shown in figure 1(a), where the DOS on the impurity is shown for three different values of chemical potential μ. The DOS vanishes exactly at the Dirac point, as can be easily seen from the dark solid curve in figure 1(a), and the tunneling resonance peak becomes broader and lower as μ increases from 0 to 0. , the tunneling resonance peak will be lower when it becomes broader, as shown in figure 1(a).
Apart from the tunneling resonance, another narrow and sharp peak is also found exactly at the Fermi level in figures 1(a) and ( figure 1(b). Then we estimate the Kondo temperature from the calculated impurity Green's function. The relevant energy scale can be determined from the leading terms in the denominator of the impurity Green's function, i.e., by the temperature at which the real parts of ( ) w ss G r d,

vanishes. When
, we find that the temperature scales as , the Kondo resonance emerges, as can be seen from figures 1(a) and (b), and T K increases exponentially with increasing μ for  e E d 0 . However, with further increasing μ, E d will be pushed away from the Fermi level, and when ẽ E d 0 , T K decreases with increasing μ. Therefore, there exists a critical value of m ẽ -0 , determined by the equation: e -= E 0 d 0 , above which T K becomes smaller than T and the Kondo peak vanishes again. As a result, the Kondo peak reduces its intensity after reaching a maximum value.
It is interesting to see that, for a finite LDOS of the ). Both the tunneling resonance and Kondo resonance can affect the TSSs. In figure 2, we plot the total LDOS r TI ( r r = +   TI, TI, ) versus ω for some different values of e 0 . Obviously, there exists tunable low-energy tunneling resonance in the LDOS of the TI surface. With e 0 increasing from −0.15 to 0.02, the resonance peak first becomes sharper and closer to the Dirac point, and then it further passes through the Dirac point with decreasing height and broadening width. For a fixed e 0 , the strength of the low-energy tunneling resonance strongly depends on the energy difference between the Fermi level and the Dirac point, which is helpful for understanding the experimental observations, e.g., the low-energy resonances observed by STM performed on the surface of strong TI Bi Se 2 3 [28]. The low-energy resonances induced by a classic impurity were also studied before [21,26,27], but the physical mechanism is different from that of a quantum impurity investigated here. For a classical impurity, the low-energy resonances are bound states corresponding to the poles of the T-matrix, which can never be tuned to cross the Dirac point. The low-energy tunneling resonance observed here originates from the electron exchange interaction between the impurity and the TSSs, which is highly tunable by varying the internal energy level of the impurity. It can even pass through the Dirac point with changing the impurity level alone, as has been observed in figure 2.
In addition to the tunneling resonance in figure 2, one can notice that there emerges a narrow and sharp peak in the LDOS exactly at the Fermi level for e = -0.025 0 . The new peak comes from the Kondo resonance, which vanishes if the Dirac point is tuned to be at the Fermi level. For a larger chemical potential, the Kondo resonance will be less noticeable because of the reduction of the impurity DOS at the Fermi level mentioned above.
Interestingly, although the Dirac dispersion of the TSSs is heavily destroyed by the resonances at low energies, especially near the Dirac point, the Dirac point itself is insensitive to the Anderson impurity, as the zero LDOS at the Dirac point remains unchanged. This is easily understood as follows. In the absence of the magnetic field, the modification for the spin-resolved LDOS from equation (4)

DPR
As analyzed above, nonvanishing S t s Im at the Dirac point is a necessary requirement for generating the DPR. However, the requirement is impossible to meet for classic impurities, since the T-matrix, 0 . In contrast, for quantum impurities, it is easy to induce an additional imaginary part in the self-energy. For example, during the experimental measurement by STM [28], the quantum impurity could couple to the STM, which will contribute to the imaginary part of the self-energy. for the STM tip. The same coupling has been extensively employed for impurities adsorbed on the surfaces of ordinary metals [36,37].
We derive the impurity Green's function, as an instance, in the deep Coulomb blockade regime, given by   We first consider the case, where the temperature is beyond the Kondo temperature, and the result is presented in figure 3(a). Obviously, the DPR is generated, and strongly depends on the STM tip-impurity tunneling resonance. For the impurity level far away from the Dirac point, e.g., for e = -0.10 0 and 0.05 in figure 3(a), the LDOS of the TI surface peaks at the tunneling resonance energy, just slightly away from the Dirac point. With the impurity level approaching to the Dirac point, e.g., for e = -0.037 0 and −0.05, the DPR peak shifts toward the Dirac point, and gradually reaches its maximum when the peak coincides with the Dirac point. The behavior of the DPR observed here is quite different from that for a high-spin nanomagnet adsorbed on the TI surface, where the DPR strongly relies on the strength of magnetic anisotropy [16]. Here, the DPR is due to the interplay of the STM tip-impurity and TI-impurity interactions, and is strongly dependent on the location of the impurity level. In [16], the magnetic anisotropy (D) in high-spin nanomagnet is crucial. If the magnetic anisotropy vanishes, i.e., D=0, the DPR in [16] will dizappear, reducing to the present Anderson impurity case. Here, we show that the DPR can be induced simply by coupling the impurity to a STM tip, without requiring the magnetic anisotropy of the impurity.
To . It is interesting to see that the LDOS r s TI, of one spin σ is modified by the opposite spin s, and shows an inverse-square r 1 2 attenuation law without Friedel oscillations, which is quite different from the Kondo resonance or the bound states [16,29,30]. More interestingly, when the temperature is below the Kondo temperature, the DPR can be significantly strengthened by the Kondo resonance. In figure 3(b), we plot the evolution of the DPR with the Kondo resonance. It is obvious that, with the Kondo peak approaching to the Dirac point, the DPR stands out rapidly. Moreover, by varying the Fermi energy of the STM, the Kondo resonance can be tuned to be exactly at the Dirac point, which strongly enhances the DPR.

Summary
We investigated the scattering of Dirac electrons with a STM-coupled Anderson impurity adsorbed on the TI surface, and proposed an effective method to deal with the mutual interactions between the TSSs and the Anderson impurity. It was found that the STM-coupled Anderson impurity can heavily destroy the Dirac spectrum by creation of the DPR, Kondo resonance, and tunneling resonance. The DPR can be significantly strengthened either by the low-energy tunneling resonance or Kondo resonance, even exactly at the Dirac point, in sharp contrast to the case of an impurity decoupled from the STM, where all the resonances are fully suppressed at the Dirac point. Our finding is helpful for understanding the experimental observations, e.g., the low-energy resonances observed by STM performed on the surface of strong TI Bi Se 2 3 . Furthermore, the lowenergy tunneling resonance induced by the Anderson impurity is tunable to cross the Dirac point, essentially different from the low-energy bound states induced by a classic impurity.