Abstract
Currently, Weyl semimetals (WSMs) are drawing great interest as a new topological nontrivial phase. When most of the studies concentrated on the clean host WSMs, it is expected that the dirty WSM system would present rich physics due to the interplay between the WSM states and the impurities embedded inside these materials. We investigate theoretically the change of local density of states in threedimensional Dirac and Weyl bulk states scattered off a quantum impurity. It is found that the quantum impurity scattering can create nodal resonance and Kondo peak/dip in the host bulk states, remarkably modifying the pristine spectrum structure. Moreover, the joint effect of the separation of Weyl nodes and the Friedel interference oscillation causes the unique battering feature. We in detail an alyze the different contribution from the intra and internode scattering processes and present various scenarios as a consequence of competition between them. Importantly, these behaviors are sensitive significantly to the displacement of Weyl nodes in energy or momentum, from which the distinctive fingerprints can be extracted to identify various semimetal materials experimentally by employing the scanning tunneling microscope.
Introduction
Progress in material preparation and experimental techniques has led to a surge of interest in twodimensional (2D) Dirac materials such as graphene and surface states of topological insulators. Very recently, this concept is extended to 3D systems, known as topological Dirac semimetals (DSMs), which are newlydiscovered bulk analog of graphene as a new topological states of matter. Recent experiments have identified a class of materials^{1,2,3} (Bi_{1−x}In_{x})_{2}Se_{3}, Na_{3}Bi, and Cd_{3}As_{2} to be the DSMs. In these new Dirac materials, 3D massless Dirac fermions are excited around the doubly degenerate Dirac cones, which are protected by timereversal symmetry (TRS) or inversion symmetry (IS).
Breaking either the TRS or the IS will drives the DSMs into a Weyl semimetal (WSM) phase, which is manifested as the splitting of a pair of degenerate Weyl nodes with opposite chirality in momentum or energy space. As a new topological nontrivial phase, these massless WSM fermions are drawing great interest for their scientific and technological importance. The WSM Fermion states have been predicted theoretically and observed experimentally in a family of the noncentrosymmetric transitionmetal monosphides^{4,5,6,7,8,9,10} with preserving the TRS, e.g., TaAs, NbAs, NbP, and TaP. The nontrivial topology along with the node separation leads to many exotic phenomena and unique physical properties, such as the chiral anomaly^{11,12,13}, the unique Fermi arc surface states^{6,7,8,9,10,14}, the chiral Hall effect^{13,15}, the chiral magnetic effects^{11,16}, and the negative^{17,18,19,20} and extremely large magnetoresistance^{21}.
When most of the previous studies concentrated on the clean host WSM bulk states, it is expected that the dirty WSM system would present rich physics due to the interplay between the WSM Fermion states and the impurities embedded inside these materials. On one hand, the unique 3D spinmomentum locking can mediate the interaction between magnetic impurities in both Dirac and Weyl semimetals, leading to anisotropic RudermanKittelKasuyaYosida (RKKY) coupling and rich spin textures^{22,23,24}. On the other hand, the feedback effect of impurities on host bulk states can change the property of Weyl nodes, at which the differences between a Weyl phase and a normal metal are most pronounced. The stability of the nodal density of states (DOS) was investigated in the presence of various types of local impurities^{25,26}, and nonzero DOS at the degeneracy point was predicted for the disorder strength beyond a certain critical value^{27,28}. We would like to mention that these discussions, however, are limited to the classic impurity model and only the single node scattering is taken into account. The nodal resonance induced by impurities have also been extensively studied in graphene and topological insulators^{29,30,31,32,33}. As a representational feature of quantum impurities, the Kondo effect has been intensely discussed in threedimensional Dirac and Weyl systems^{34,35,36,37} of dilute magnetic impurities. The results showed that the nature of the Kondo effect of impurity is only affected strongly by the linear dispersion of Dirac/Weyl host bulk states but it is in general blind to the momentum splitting of TRSbroken Weyl nodes. In ref. 35, they found that the spatial spinspin correlation between the magnetic impurity and the conduction electron is sensitive to the displacement in the momentum splitting of Weyl nodes, where rich features are shown due to an extra phase factor. Even so, it is still challenging how to identify the TRSbroken WSM materials from the transport fingerprints.
In this paper, we study how the local density of states (LDOSs) in host WSM/DSM bulk states are modulated by the embedded quantum impurity in the resonance regime and in the Kondo regime. We specially pay attention to the response of nodal behavior to impurity scattering processes. It is found that the quantum impurity scattering can create a LDOS resonance or Kondo peak/dip in the host bulk states exactly at the Dirac point and thus remarkably destroy the pristine spectrum structure, which are sensitive to the degree of the splitting of two WSMs nodes. Compared with the single node scattering, the internode scattering possesses more information about the unique properties. Interestingly, by taking the intranode scattering into account, we find the unique battering feature for the TRSbroken WSMs, which is longrange measurable in real space with current scanning tunneling microscope technologies.
The rest of the paper is organized as follows. In Sec. II we present a general interaction model of Weyl fermions with Anderson quantum impurity and treat it by employing the standard equations of motion for Green’s functions. The lowenergy resonance, Kondo signature, and Friedel oscillation in host materials are discussed in Sec. III, and a short summary is given in the last section.
Model and Theory
Consider a 3D WSM with a pair of chiralityopposite Weyl nodes, whose lowenergy Hamiltonian can be described as^{22,34} , with
and the annihilation operator of electrons acting on the spin and chirality spaces. Here, χ = ±1 represents the pair of weyl nodes with the opposite chirality, v_{f} is the Fermi velocity, is the effective wave vector measured from the Weyl nodes, and denotes the vector of Pauli matrices, For , reduces to the Hamiltonian of degenerate DSM, possessing both the TRS and the IS , where with complex conjugation operation K is the timereversal operator, P = τ_{x} ⊗ σ_{0} is the inversion operator, and is the Pauli matrix on the chirality space. Breaking either TRS or IS Q_{0} ≠ 0 transforms a DSM into a Weyl system, the former splitting the two degenerate weyl points separately at different momentum but with the same energy while the latter shifting two Weyl nodes at different energy ω_{node} = ±Q_{0} but with the same momentum. This can be seen from the dispersion spectrum of ,
We utilize the typical Anderson impurity model to study the quantum impurity effect and spin1/2 Kondo screening in 3D Dirac and Weyl semimetals. The full Hamiltonian can be written as . The impurity Hamiltonian
is characterized by a singleorbital energy ε_{0} and the onsite Coulomb repulsion U. is the creation (annihilation) operator for impurity electrons. represents the hybridization between the impurity and the host material with the hybridization matrix
where the spinor and the coupling strength V_{x} is assumed to be dependent on the Weyl node χ but regardless of and under the assumed wideband approximation and spinconserved hoping. Here, we also assume that the magnetic impurities are embedded inside the WSM such that the effect of Fermi arc surface states can be neglected safely.
Using the method of standard equations of motion, the retarded Green’s function of Weyl electrons with respect to the full Hamiltonian can be derived as , where ω^{+} = ω + i0^{+} and all qualities are 4 × 4 matrix in the spin ⊗ chirality space. When performing the Fourier transform to the real space, its block matrix in chirality space is
The Green’s function is still 2 × 2 matrix in subspace of the electron spin and is measured from the impurity as a scattering center, whose position is chosen to be the origin of coordinates. The expression in Eq. (5) recalls the extensively applied Tmatrix approach^{30,31}, but here is expressed in terms of the Green’s function of magnetic impurity, defined as , which is the Fourier transform of . Similar relation can be found in Anderson impurities interacting with topological insulator^{29,33,38,39} or graphene^{40}. In Eq. (5), is the bare Green’s function of Weyl fermions with . At the impurity position , the bare Green’s function is given by
with the cutoff energy D, unitstep function Θ(x), and ω_{χ} = ω^{+} − χQ_{0}. Note that even for STRbroken case due to finite , is diagonal in spin space and independent of , remarkably different from the case of 2D topological insulator or graphene^{33,41}. For we can calculate by expanding the in terms of spherical harmonics according to the Rayleigh equation^{24,42}, and finally arrive at a simple analytical expression for D ≫ ω as
where we define and
The next task is to calculate the impurity Green’s function . Carrying out the equations of motion, we find
with the retarded selfenergies . Further calculation gives
where three highorder Green’s functions emerge. Performing the same procedure with the equation of motion, we obtain all highorder Green’s functions and in the following take as an example,
with
and , where we denote opposite to χ(σ). To form a set of close iterative equations, we truncate them following the standard method^{43}, for example, , where the operator pair with the same spin indices can be pulled out of the Green’s function as an average and is calculated with the Fluctuation dissipation theory
where f(ω) is the fermi distribution function. After carrying out lengthy but straightforward calculations, we finally derive the expression for the impurity Green’s function in the deep Coulomb blockade regime, i.e., U → ∞, as
with
where the coefficient . By comparison with the normal metals^{44} or 2D Dirac materials^{33}, the most distinction is the specific expressions of selfenergies ∑_{0}(ω^{+}) and ∑_{1}(ω^{+}).
Results and Discussion
Resonance states in LDOS
Our purpose is to explore the unique local properties of the WSM when the conducting elections are scattered off a quantum impurity. As the Weyl nodes are separated in energy or momentum, a very interesting question is whether or Q_{0} leads to some especial spectrum structures locally around the quantum impurity. Next, we focus on the LDOS in WSMs, which is defined as
where is the unperturbed LDOS, and contributed by the second term in Eq. (5), reflects the substantial modification of the LDOS by the doping impurity. Beyond the usual singlenode treatments, we here emphasize the impurity scatter processes between two Weyl nodes. We find that the introduce of quantum impurity not only scatters the electrons within the same Weyl node but also between two nodes. Specifically, we can split LDOS as , where collects the contribution from the intranode scattering process, equivalent to singlenode situation, while collects the contribution from scattering process between two nodes. After proceeding the calculations, we obtain readily the following analytical expressions
and
Equations (16) and (17) are our central results. In order to understand them deep, we in the following limit our discussions to the symmetrical coupling Γ_{+} = Γ_{−}, and first discuss the impurity effect in the DSMs, i.e., setting whose LDOS ρ(ω) for a fixed is illustrated in Fig. 1. Without the internode scattering (i.e., singlenode case), seeing Fig. 1(a), there a pronounced resonance structure, whose position depends on the impurity level ε_{0}. This resonance is a consequence of the backaction of the resonance in the impurity DOS, which is defined as and depicted in the corresponding inset, indicating the singlelevel resonance tunneling between the impurity and the reservoirs. In Fig. 1(a), with the increase of ε_{0} from −0.2 to 0.2 in step of 0.1 the lowenergy resonance is first shifted close to the Dirac point, accompanied with increasing magnitude, and then passes over the Dirac point into its other side, on whole exhibiting a symmetry with respect to the Dirac point. Intriguingly, a sharp pronounced resonance for ε_{0} = 0 can be located exactly at the Dirac point, completely destroying the 3D typical ω^{2} Dirac spectrum. Similar Diracpoint resonance appears in doping surface of topological insulators with quantum impurities^{33} or quantum magnets^{44,45}. Our further calculations confirm that the scenario of Diracpoint resonance cannot emerge for classic impurity model, i.e., replacing in Eq. (5) with , where stands for a classic impurity potential. If the internode scattering is taken into account, the scenario is very different from the singlenode case. We plot the LDOS ρ(ω) including both intra and internode scattering in Fig. 1(b). By comparison with singlenode case, most interesting in doublenode case is that the resonance peak becomes weaker and weaker when close to the Dirac point and is completely smoothed away at the Dirac point, in which ρ(ω) ∝ ω^{2} recovers the typical square dependence on energy. To understand it, we plot the change of DOS δρ_{inter}(ω) and δρ_{intra}(ω) for ε_{0} = 0 in the inset of Fig. 1(b), from which we know that the negative δρ_{inter}(ω) tends to suppress the resonance in δρ_{intra}(ω) and, at Dirac point ω = 0 they have the same amplitude but opposite sign and thus cancel each other exactly. This point also can be seen from Eqs (16) and (17).
From the above discussions for DSMs, we are known that the competition between intra and internode scatterings is crucial for the development of the Diracpoint resonance. In Fig. 2(a) we depict the change of LDOS δρ(ω) for the TRSbroken WSMs, i.e., Q_{0} = 0 but . Here, we just choose along zaxis and so the degenerate Weyl nodes are shifted by ±Q_{z} in the direction of but Q_{x/y} = 0. Obviously, the Diracpoint resonance for finite Q_{z} recovers since δρ_{inter} only partly offsets δρ_{intra}, as shown in Fig. 2(a). From Eqs (16) and (17), one can notice that is independent of but is less than −1 for the chosen parameter. The variation of LDOS for different Q_{z} is plotted in Fig. 2(b), in which the Diracpoint resonance peak increases first for small Q_{z} ∈ (0, π/2r) and then exhibits a periodic function of Q_{z}, seeing the inset. For Q_{z} = (2n + 1)π/4r (n = 0, 1 …) or , the internode scattering is prohibited due to destructive interference and thus dominates. Therefore, to probe the feature of TRSbroken WSMs, it is necessary to consider the impurityinduced scattering between Weyl nodes since only enters δρ_{inter} but not δρ_{intra}.
For noncentrosymmetric WSMs, i.e., Q_{0} ≠ 0 and , we from Eqs (16) and (17) see that Q_{0} contributes to both and but with different ways, thus their zeroenergy resonances cannot be completely compensated. Another most interesting effect for noncentrosymmetric WSMs is the emergence of Kondo resonance, which is expected to occur because of the nonzero LDOS at ω = 0 when two Weyl nodes are split to ω_{node} = ±Q_{0}. If we choose the proper parameters in Kondo regime, the impurity DOS presents a remarkable sharp Kondo resonance at ω = 0 as shown in the inset. The Kondo resonance is mainly attributed to the selfenergy in Eq. (14), which depends on rather than linear Q_{0}, distinct from graphene^{40} and topological insulator^{41}. The results are in agreement with those obtained by number renormalization group^{34}. Suffering from the scattering off the impurity potential, the electronic LDOS in the host semimetal material also exhibits the feedback of Kondo resonance in both and . They have opposite sign but cannot compensate each other and so the total exhibits a dip structure as depicted in Fig. 3(a). We plot the evolution of the total LDOS with Q_{0} in Fig. 3(b), where the Kondo dip becomes more and more prominent with the increase of Q_{0}, companied by overall lift upwards due to the Weyl node pair shifting away from the zero energy. Interestingly, if we further consider a finite Q_{z}, it will significantly reverse the Kondo structure from a dip to a peak, as illustrated in Fig. 3(c), as a consequence of the competition between two types of scattering processes. Similar to the Diracpoint resonance in Fig. 2(b), the evolution of Kondo peak from a dip to a peak is a periodic function of Q_{z}, greatly different from the monotonouslyincreasing dependence on Q_{0}. Note that the Kondo resonance develops only in the inversebroken case with Q_{0} ≠ 0, which is a feature of the linear dispersion, similar scenarios appearing in TI or graphene^{40,41}.
Spatial Friedel oscillation of LDOS
In this section, we discuss the characteristics of Friedel oscillation, namely, the oscillation behavior of LDOS with the spatial distance measured from the impurity position. This is caused by the interference of incoming and outgoing waves when conducting electrons are scattered off a local impurity potention. Since the dependence of LDOS on stems completely from the impurity scattering correction , in following analysis we only focus on .
Figure 4(a) shows the variation of with for the DSM materials (Q_{0} = Q_{z} = 0). Obviously, a typical pattern of Friedel oscillations is presented for both and . By comparison, the oscillation of dominates in long distance while the oscillation of is in short distance. The reason is that the former decays as an inversesquare r^{−2} law and the latter as r^{−3} law, which can be seen from Fig. 4(b) where and exhibit the equal amplitude oscillation. Figure 4(c,d) correspond to the case of noncentrosymmetric WSMs (Q_{0} = 2 and Q_{z} = 0). When and display a damped oscillatory behavior similar to Q_{0} = 0, there appears an interesting beating pattern in . This beating feature is originated from the combination effect of the energy separation of Weyl nodes by ±Q_{0} and the Friedel oscillation, manifesting itself by the factors cos (2Q_{0}r/v_{f})exp (2iωr/v_{f}) and sin (2Q_{0}r/v_{f})Exp (2iωr/v_{f}), derived from Eq. (16). For , the beating feature vanishes and both and show 1/r^{3}law decaying oscillation. When two oscillating frequencies have distinct difference, the beating effect emerges, as illustrated in Fig. 4(c,d) where we choose Q_{0} ≫ ω and the length of beating is determined by 2ω. Inversely, the beating length is determined by 2Q_{0} for ω ≫ Q_{0}. In real materials, it is reported Q_{0} = 23 meV for TaAs in ref. 6 and 36 meV for NbAs in ref. 7, which is within the range of lowenergy spectrum due to usually ħv_{F} ≈ 0.37 eV and D ≈ 300 meV. Experimentally, the electron energy can be set to be larger or smaller than Q_{0} to observation both beating scenarios as discussed above. One, however, can notice that for r ≫ v_{f}/ω, the longrange quickly dominates and is larger than the shortrange by at least one order in magnitude, which easily overwhelms this beat frequency in measurement of total LDOS. Therefore, to measure the Q_{0}induced beating structure, the electron scattering off the impurities must be limited to the same Weyl node.
In contrast to the noncentrosymmetric WSMs, the nonzero in TRSbroking WSMs adds an extra phase factor in but has nothing to do with seeing Eqs (16) and (17). The displacement of the Weyl nodes in the momentum will further induce complexity to the Friedel oscillation behavior of . Similarly, there are two periods associated with and , exhibiting a batter pattern for large difference , where we choose along zaxis and denote r_{z} = rsin θ_{r} with respect to the zaxis. Obviously, the beating characteristics is dependent on the spatial direction θ_{r} but independent of the azimuthal angle φ_{r}, which is a consequence of the azimuthal symmetry around the correcting line of a pair of Weyl nodes (i.e., chosen zaxis). The spatial direction dependence of is plotted in Fig. 5(a–c) for θ_{r} = 0, π/4, and π/2, respectively. Figures (a) and (b) exhibit a prominent beating behavior, where the number and length of batter frequency are changed with θ_{r}. For θ_{r} = π/2 (i.e., ), the beat frequency of dies away and recovers the typical decaying oscillation, as shown in Fig. 5(c). Importantly, the decaying rate of all oscillations in , abiding by 1/r^{2} law regardless of θ_{r} as illustrated in the insets, always dominates over 1/r^{3}law decaying for sufficiently large r ≫ 1. Therefore, the battering feature in the TRSbroken WSMs is accessible in measurement of the scattering between nodes, moreover unaffected by the intranode scattering, which is important for identifying the TRSbroken WSM in the real space. Notice that this beating structure does not occur in the typical surface state of topological insulators. As far as we know, the WSM phase by breaking of timereversal symmetry has not been yet experimentally reported and the beating feature maybe provide an alternative route to identify this new type of materials, e.g., Y_{b}MnBi_{2}.
Finally, we want to remark the influence of asymmetric coupling of the impurity to two Weyl nodes. From Eqs (16) and (17), one can find that the asymmetric coupling only changes quantitatively the weight between intranode scattering and internode scattering. Thus, the above obtained results are qualitatively suitable as long as we properly reset other parameters.
Conclusions
On conclusions, we have investigated the influence of quantum impurity on the DSM and WSM materials by looking at the modification of LDOS around the impurity. It is found that the quantum impurity scattering can create the LDOS lowenergy resonance, the Kondo signature, and the Friedel oscillation, all of which are sensitive to the displacement of Weyl nodes in energy or momentum. We in detail analyze the different contribution from the intra and internode scattering processes and present different scenarios as a consequence of competition between them. We further study the spatial dependence of LDOS and find that the separation of Weyl nodes along with the Friedel interference oscillation leads to the unique battering feature, which arises in the intranode scattering for the ISbroken WSMs but in internode scattering for the TRSbroken WSMs. Especially, the beating feature for the TRSbroken WSMs is remarkably dependent on the spatial direction of the probing position, which is longrange measurable in real space by employing current scanning tunneling microscope technologies.
Additional Information
How to cite this article: Zheng, S.H. et al. Resonance states and beating pattern induced by quantum impurity scattering in Weyl/Dirac semimetals. Sci. Rep. 6, 36106; doi: 10.1038/srep36106 (2016).
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Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos 11174088 and 11474106) and by PCSIRT in China (Grant No. IRT1243).
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Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou 510006, China
 ShiHan Zheng
 , RuiQiang Wang
 , Min Zhong
 & HouJian Duan
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Contributions
R.Q.W. conceived the idea. S.H.Z. performed the calculation and provided all of the figures. R.Q.W. and S.H.Z. contributed to the interpretation of the results and wrote the manuscript. M.Z. and H.J.D. joined in the data analysis and contributed in the discussion. All authors reviewed the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to RuiQiang Wang.
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