Abstract
We demonstrate that chirality of the electron scattering in Weyl semimetals leads to the formation of magnetic chemical bonds for molecular states of a pair of impurities. The effect is associated with the presence of timereversal symmetry breaking terms in the Hamiltonian which drive a crossover from s to pwave scattering. The profiles of the corresponding molecular orbitals and their spin polarizations are defined by the relative orientation of the lines connecting two Weyl nodes and two impurities. The magnetic character of the molecular orbitals and their tunability open the way for using doped Weyl semimetals for spintronics and realization of qubits.
Introduction
Recent years witnessed unprecedented penetration of the ideas of high energy physics into the domain of condensed matter. In particular, lot of attention is now attracted to the condensed matter realizations of three dimensional (3D) massless quasirelativistic particles known as Dirac or Weyl fermions^{1}. The experimental observation of Dirac fermions in such materials as Na_{3}Bi^{2,3} and Cd_{3}As_{2}^{4,5} made possible the study of the 3D analogs of graphene physics in a robust topologically protected material possessing both inversion (\( {\mathcal I} \)) and time reversal (\({\mathscr{T}}\)) symmetries^{6}. In Weyl semimetals, where one of these symmetries is broken, a Dirac node, which is the point where conduction and valence bands touch each other, splits into a pair of Weyl nodes with opposite chiralities. Such nodes are predicted to give rise to a plethora of interesting phenomena, including formation of Fermi arcs, unusual Hall effects, and chiral anomaly, among others^{6,7,8,9,10,11,12}. The material platform for realization of Weyl fermions is presented by such compounds as tantalum arsenide (TaAs)^{13,14,15,16,17}, niobium arsenide (NbAs)^{18}, and tantalum phosphide (TaP)^{19}.
One of the aspects of Weyl semimetals which recently received particular attention is the peculiar impurity physics^{20,21,22,23,24,25}. For instance, in the case of a single Kondo impurity, Zheng, S.H. et al.^{24} have observed beating patterns in the local density of states strongly dependent upon the \( {\mathcal I} \) and \({\mathscr{T}}\) symmetries. Additionally, some of us have found an unusual ground state of antibondingtype for a diatomic molecule immersed into a Dirac host, which corresponds to Weyl semimetal with the symmetries above preserved^{25}. Thus by working off these regimes, we consider the \({\mathscr{T}}\)– breaking Weyl semimetal and propose the formation of molecules due to an unprecedented chemical bond mechanism, which we reveal being of chiralmagnetic nature. In the present work, we clarify the role played by chirality of Weyl quasiparticles in the processes of impurity scattering by investigation of the local density of states. The latter can be experimentally addressed by means of the scanning tunneling microscopy (STM). We show that longrange Friedellike oscillations^{26} contribute to the formation of molecular states in a pair of distant impurities embedded in a 3D relativistic semimetal. We demonstrate that, the scenario of the impurity scattering is radically different in Dirac and Weyl semimetals and show that in the latter case magnetic molecular states can be formed. Their particular type is defined by the relative orientation of the lines connecting two Weyl nodes and two impurities. We report a crossover from s to ptype atomic orbitals for individual impurities and related formation of spinpolarized σ– and π–type molecular orbitals for an impurity pair.
Model
We set \(\hslash =1\) throughout the calculations and represent the total Hamiltonian as the sum of the three terms:
The lowenergy Hamiltonian of the host may be represented as
where \(\psi ({\bf{k}})={({c}_{{\bf{k}}+\uparrow },{c}_{{\bf{k}}+\downarrow },{c}_{{\bf{k}}\uparrow },{c}_{{\bf{k}}\downarrow })}^{T}\) is fourspinor operator whose components \({c}_{{\bf{k}}\chi \sigma }^{\dagger }\) (\({c}_{{\bf{k}}\chi \sigma }\)) stand for the creation (annihilation) operators of an electron with wave number k and spin σ,
where k = (k_{x}, k_{y}, k_{z}) is the threedimensional wave vector, σ stands for the vector of Pauli matrices, the index \(\chi =\pm \,1\) corresponds to the chirality of the Weyl nodes and \({v}_{F}\) is the Fermi velocity. For Q = 0, \({\mathscr{T}}\) symmetry is conserved and a pair of Weyl nodes is degenerated, which corresponds to the case of a standard Dirac semimetal. If \({\mathscr{T}}\) symmetry is broken (Q ≠ 0), two Weyl nodes are displaced with respect to each other towards two different points in the Brillouin zone located at ±Q, but maintain energetic degeneracy as it is depicted in the Fig. 1.
The impurities are modeled by the Hamiltonian
with ε_{jσ} being singleparticle energy and U_{j} the onsite Coulomb repulsion, whereas \({n}_{j\sigma }={d}_{j\sigma }^{\dagger }{d}_{j\sigma }\) corresponds to the number of electrons with spin projection σ at the site j with \({d}_{j\sigma }^{\dagger }\) and d_{jσ} being respectively creation and annihilation operators.
The hybridization between the host and the impurities is described by the term:
wherein \({\hat{d}}_{j}^{\dagger }=({d}_{j\uparrow }^{\dagger },\,{d}_{j\downarrow }^{\dagger })\) and
where \({V}_{j{\bf{k}}}=\frac{{v}_{0}}{\sqrt{N}}{e}^{i{\bf{k}}\cdot {{\bf{R}}}_{j}}\), with ν_{0} being the hybridization amplitude between electrons of the host and localized states of the impurities positioned at R_{j} (j = 1, 2), N is the normalization factor yielding the total number of the conduction states. As performed by some of us^{27}, we assume for a sake of simplicity, that the hopping term V_{jk} neglects the exponential decay of the Bloch states right above the Weyl semimetal surface that overlap with those from the STMtip apex. By employing such an assumption, STMtip measurements of the local density of states would be just attenuated with respect to the values extracted from our simulations, in such a way that none generality is lost as ensured by Plihal, M. et al.^{28}, which have analyzed this density for the singleimpurity problem.
Local Density of States (LDOS)
The electronic properties of the considered system are determined by the LDOS of the host which can be experimentally accessed by means of an STMtip. It can be calculated using standard equationofmotion (EOM) procedure^{29,30} as:
where \({\tilde{{\mathscr{G}}}}_{\sigma }(\varepsilon ,{{\bf{r}}}_{m})\) is the timeFourier transform of the retarded Green’s function in the time domain, defined as:
where θ(t) denotes the Heaviside function, \({\psi }_{\sigma }(t,{{\bf{r}}}_{m})\) is the field operator of the host electrons written in terms of the continuous variable r_{m}, the brackets \({\langle \cdots \rangle }_{ {\mathcal H} }\) denote the ensemble average with respect to the full Hamiltonian, \(\{\cdots \}\) determines an anticommutator between operators in the Heisenberg picture, \({\rho }_{0}(\varepsilon )=\frac{6{\varepsilon }^{2}}{{D}^{3}}\) is the pristine host DOS with D being the energy cutoff corresponding to the halfbandwidth, and
encodes the Friedellike oscillations describing the scattering of the conduction electrons by the impurities, where the terms \(j^{\prime} =j\) and \(j^{\prime} \ne j\) give rise to intra and interimpurity scattering processes, respectively, which are ruled by the spatial dependent selfenergy
r_{mj} = r_{m} − R_{j} and ±signs correspond to the vector direction (positive for r_{mj}, negative for r_{jm}). The Eq. (10) is spatially anisotropic and the LDOS as a result, due to mutual orientation of Q and r_{mj} inside the plane wave \({e}^{i\chi Q\cdot {{\bf{r}}}_{mj}}\). Consequently, it leads to the main finding of this work, which is ruled by the mixing of chirality (χ) and spin (σ) quantum numbers, as one can perceive by the presence of the crossed terms \(\pm \chi \sigma \varepsilon \pm i\frac{\chi \sigma {v}_{F}}{{{\bf{r}}}_{mj}}\). Such a set of terms, thus gives rise to the possibility of spinpolarized molecular states, which are chiraldependent as we will see below. To show the emergence of such, we should regard the system anisotropy by means of the condition \({e}^{i\chi Q\cdot {{\bf{r}}}_{jj^{\prime} }}\ne 1\), with the relative distance between the impurities given by \({{\bf{r}}}_{jj^{\prime} }={{\bf{R}}}_{j}{{\bf{R}}}_{j^{\prime} }\). As aftermath, the spin degree of freedom in the quasiparticle energy correction of the impurity j due to j′ (and viceversa), namely \({{\rm{\Sigma }}}_{\sigma }({{\bf{r}}}_{jj^{\prime} })=\sum _{\chi }{{\rm{\Sigma }}}_{\sigma }^{\chi }({{\bf{r}}}_{jj^{\prime} })\), becomes lifted. In this way, \({{\rm{\Sigma }}}_{\uparrow }({{\bf{r}}}_{jj^{\prime} })\ne {{\rm{\Sigma }}}_{\downarrow }({{\bf{r}}}_{jj^{\prime} })\) and spinpolarized molecular states are allowed. In the absence of anisotropy \(({e}^{i\chi {\boldsymbol{Q}}\cdot {{\bf{r}}}_{jj\text{'}}}=1)\), the degeneracy \({{\rm{\Sigma }}}_{\uparrow }({{\bf{r}}}_{jj^{\prime} })={{\rm{\Sigma }}}_{\downarrow }({{\bf{r}}}_{jj^{\prime} })\) holds and paramagnetic molecular states prevail. We should stress that, such an effect is entirely distinct from the corresponding caused by an external Zeeman filed. In the latter, it breaks naturally the spindegeneracy whatever the orientation \({\boldsymbol{Q}}\cdot {{\bf{r}}}_{jj^{\prime} }\). Thus, the addressed chiral magnetic effect reveals that an applied magnetic field is not capable of reproducing this peculiar magnetism and prevents the feasibility of the proposed chiral magnetic chemical bond as well. We guide the reader to the supplementary material, where a concise derivation of Eq. (10) can be readily followed.
\({\tilde{{\mathscr{G}}}}_{j\sigma j^{\prime} \sigma }(\varepsilon )\) is the timeFourier transform of the Green’s function of the impurities
Application of the EOM method to \({\tilde{{\mathscr{G}}}}_{j\sigma j^{\prime} \sigma }(\varepsilon )\) together with HubbardI decoupling scheme^{31}, yields
where \(\bar{\sigma }=\,\sigma ,j\ne {j}^{{\rm{^{\prime} }}}\), \({\lambda }_{j}^{\bar{\sigma }}=1+\frac{{U}_{j}}{{g}_{j\sigma j\sigma }^{1}(\varepsilon ){U}_{j}}\langle {n}_{j\bar{\sigma }}\rangle \) is the spectral weight, \({g}_{j\sigma j\sigma }(\varepsilon )=\frac{1}{\varepsilon {\varepsilon }_{j\sigma }{{\rm{\Sigma }}}_{0}}\) as the single impurity noninteracting Green’s function,
is the occupation number of an impurity with n_{F}(ε) being the FermiDirac distribution,
is the local selfenergy and
Results and Discussion
In order to understand the formation of the molecular states of a pair of impurities inside a Weyl semimetal, we should start from analyzing the case of a single impurity. As model parameters, we adopt the energy of an impurity ε_{jσ} = −0.07D, hybridization amplitude ν_{0} = −0.14D, onsite Coulomb repulsion U_{j} = 0.14D, \(\hslash {v}_{F}\approx 3\,eV{\rm{\AA }}\), \(D\approx 0.2\,{\rm{eV}}\) and temperature T = 0K. Concerning this latter, we clarify that none generality is lost, once finite T just introduces thermal broadening into the FermiDirac distribution \({n}_{F}(\varepsilon )\) as well as in the LDOS via Eq. (13) for the impurity occupation number \(\langle {n}_{j\bar{\sigma }}\rangle \) and, according to some of us have^{27}, phonon modes rise as effective by increasing T and restore the molecular ground state to bondingtype. Thus in the current work, the molecules show antibonding ground state^{25}, due to the T = 0K condition.
As one can see in the Fig. 2, the 2D map of the LDOS which can be probed by an STMtip over the system surface presents a crossover from s to ptype atomic orbitals as Q is increased and one moves from Dirac (Q = 0) towards the Weyl regime (Q ≠ 0). This happens due to the presence of the terms depending on \({v}_{F}\chi Q\) in the original Hamiltonian. Note, that the porbital is elongated along the direction of Q.
Now we can analyze the molecular state corresponding to a pair of impurities inside a Weyl semimetal with broken \({\mathscr{T}}\)–symmetry. We will consider two cases of the mutual orientation of the vectors Q and \({r}_{12}={R}_{1}{R}_{2}\) connecting the two impurities: (i) perpendicular orientation, \(Q\cdot {r}_{12}=0\) and (ii) parallel orientation \(Q\cdot {r}_{12}=Q{r}_{12}\). As we will demonstrate, the former case corresponds to the formation of spin degenerate molecular orbitals, while the latter case gives rise to the chiral magnetic chemical bonds. In all plots of \(\rho (\varepsilon ,{{\bf{r}}}_{m})\) and \(\delta {\rho }_{jj^{\prime} }(\varepsilon ,{{\bf{r}}}_{m})\) versus ε/D we present, the STMtip is pinned at the site \({{\bf{r}}}_{m}=(1,1,1)\,{\rm{nm}}\) right above the Weyl semimetal surface and the impurities are buried into the bulk at \({{\bf{R}}}_{1,2}=(0,\mp 1,0)\,{\rm{nm}}\).
Let us start from the case \(Q\cdot {r}_{12}=0\). In the case of an individual impurity, a single energy resonance appears within the valence band in \(\rho (\varepsilon ,{{\bf{r}}}_{m})\). Naturally, in the twoimpurity system a pair of peaks corresponding to bonding and antibonding states appears, as it is shown in the Fig. 3b and d for the cases of Dirac (Q = 0) and \({\mathscr{T}}\)breaking Weyl (Q ≠ 0) hosts. Note, that the coupling between the impurities is fully mediated by Friedellike oscillations of the electronic density of the host mathematically described by the selfenergy \({\lambda }_{j}^{\bar{\sigma }}{{\rm{\Sigma }}}_{\sigma }({{\bf{r}}}_{j\bar{j}}){g}_{\bar{j}\sigma \bar{j}\sigma }(\varepsilon ){\lambda }_{\bar{j}}^{\bar{\sigma }}{{\rm{\Sigma }}}_{\sigma }({{\bf{r}}}_{\bar{j}j})\) entering into the denominator of \({\tilde{{\mathscr{G}}}}_{j\sigma j\sigma }(\varepsilon )\) given by the Eq. (12).
The 2D map of the molecular orbitals on the host surface is presented in the panels (e–h) of the Fig. 3. Panels (e and f) correspond to the case of a Dirac host which was previously considered by some of us in the ref.^{25}. One clearly sees the emergence of bonding and antibonding molecular orbitals with σtype symmetry resulting from the interference between two swave atomic orbitals of individual impurities, as it is illustrated in the panel (i). Panels (g and h) correspond to the case of a Weyl semimetal with \(Q\cdot {{\bf{r}}}_{j\bar{j}}=0\), for which individual impurities reveal ptype atomic orbitals stretched in the direction perpendicular to the line connecting the impurities. Note, that for the considered case bonding and antibonding molecular orbitals have clear πtype symmetry. These orbitals remain spindegenerate, as it follows from the Eq. (10), which leads to \({{\rm{\Sigma }}}_{\sigma }({{\bf{r}}}_{jj^{\prime} })=\sum _{\chi }{{\rm{\Sigma }}}_{\sigma }^{\chi }({{\bf{r}}}_{jj^{\prime} })\) independent of the spin degree of freedom, i.e., \({{\rm{\Sigma }}}_{\uparrow }({r}_{jj^{\prime} })={{\rm{\Sigma }}}_{\downarrow }({r}_{jj^{\prime} })\).
The case of the parallel orientation of the vectors Q and r_{12} is illustrated by the Fig. 4. Note that in this case, according to the Eq. (10) the presence of the terms \({e}^{i\chi Q\cdot {{\bf{r}}}_{j\bar{j}}}\) with \(\chi =\pm \,1\) in the expression for the selfenergy \({{\rm{\Sigma }}}_{\sigma }({{\bf{r}}}_{j\bar{j}})=\sum _{\chi }{{\rm{\Sigma }}}_{\sigma }^{\chi }({{\bf{r}}}_{j\bar{j}})\), leads to the lifting of spin degeneracy and gives rise to the formation of chiral magnetic chemical bonds. Interestingly enough, this spindependency can not be considered as being fully equivalent to one induced by effective external magnetic field, once the sequence of the peaks in the LDOS presented in the Fig. 4(b) does not correspond to the alternation of spinup and spindown states as usual, but consists of the two inner spindown states flanked by the two outer spinup states as can be clearly seen from the Fig. 4(c). The profiles of the spinresolved orbitals corresponding to the bonding and antibonding states are shown in the panels (d–g) of the Fig. 4. These orbitals exhibit σtype symmetry and are formed due to the interference between two frontal pwave orbitals as sketched in the Fig. 4(h).
To shed more light on the splitting between spinpolarized components in the LDOS, we investigate the impurity magnetization characterized by the polarization degree \(p=(\langle {n}_{j\uparrow }\rangle \langle {n}_{j\downarrow }\rangle )/(\langle {n}_{j\uparrow }\rangle +\langle {n}_{j\downarrow }\rangle )\), where the occupation numbers are defined by the Eq. (13). The dependence of the magnetization on the separation between the Weyl nodes in the direction parallel to the line connecting the two impurities Q_{y} is shown in the Fig. 5(a). One clearly sees pronounced periodic behavior, stemming from the oscillations of the factor \({e}^{i\chi Q\cdot {{\bf{r}}}_{j\bar{j}}}\) in the expression for spinresolved selfenergy in the Eq. (10). The LDOS corresponding to the spindegenerate case, maximal positive and negative magnetizations, is shown in the Fig. 5(d). Note that for the spindegenerate situation corresponding to Q_{y} = 0.157, the shape of the molecular orbitals presented in the Fig. 5(b,c) can be represented as linear combination of the orbitals presented in the Fig. 4(d–g).
Conclusions
We analyzed the structure of the molecular orbitals corresponding to the pair of impurities placed within a Weyl semimetal focusing on the role played by the \({\mathscr{T}}\)symmetry breaking. For this purpose the corresponding LDOS was evaluated. It was demonstrated that the terms in the selfenergy stemming from the chiral dependent minimal coupling drive a crossover from spin degenerate σtype molecular orbitals characteristic for the case of a Dirac host to spin degenerate πtype orbitals or spinpolarized σtype orbitals for the case of a Weyl host. The type of the chemical bonding in this latter case can be controlled by variation of the mutual position of the impurities with respect to the vector Q describing the shift of the Weyl nodes. The magnetic character of the molecular orbitals and their tunability open the way for using doped Weyl semimetals for spintronics and realization of qubits.
Methods
The findings of this research depicted in the figures of section Results and Discussion, which concern the Eq. (7) for the system LDOS, were obtained by performing selfconsistent evaluations of the occupation numbers defined by the Eq. (13) for the impurities. Such calculations and the aforementioned figures were performed by using numerical packages in Python version 3.7.
Data Availability
The authors declare that the data supporting the findings of this study are available within the paper (and its supplementary information files).
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Acknowledgements
We thank the funding Brazilian agencies CNPq (Grant No. 307573/20150), CAPES, and São Paulo Research Foundation (FAPESP) Grant No. 2018/094130. I.A.S. acknowledges support from Horizon2020 project CoExAN, megagrant 14.Y26.31.0015 and Goszadanie no. 3.8884.2017/8.9 of the Ministry of Education and Science of Russian Federation.
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Y.M., M.deS., M.S.F., I.A.S. and A.C.S. formulated the problem and wrote the manuscript. Y.M., W.N.M. and R.S.O. derived the expressions, performed their numerical computing and plotted the figures. All coauthors taken part in the discussions and reviewed the manuscript as well.
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Correspondence to A. C. Seridonio.
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Marques, Y., Mizobata, W.N., Oliveira, R.S. et al. Chiral magnetic chemical bonds in molecular states of impurities in Weyl semimetals. Sci Rep 9, 8452 (2019) doi:10.1038/s41598019448428
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