Large enhancement of Edelstein effect in Weyl semimetals from Fermi-arc surface states
Introduction
Weyl semimetals [1], [2], [3], [4], [5], [6], [7] have recently attracted a great attention because they provide a solid-state realization of Weyl fermions [8]. Weyl semimetals exhibit many unconventional properties that deviate from the standard theories of metals and semiconductors. For example, the chiral anomaly, a nonconservation of the chiral charges, leads to anomalous transport properties such as the chiral magnetic effect [9], [10] and the negative magnetoresistivity [11], [12], [13]. Weyl fermions with broken time-reversal symmetry show the anomalous Hall effect which has a universal form that depends on the distance between Weyl nodes [14], [15].
One of the manifestations of the topological properties in Weyl semimetals is the appearance of the nontrivial Fermi-arc surface states [1], [15]. In the semimetallic phase, the Fermi arc is an open curve that terminates at the two Weyl nodes of opposite chiralities. When the chemical potential moves away from the Weyl points, the Fermi arc still survives and connects the two disjointed bulk Fermi surfaces enclosing the two Weyl nodes. These Fermi arcs on the surface were observed experimentally in TaAs with angle-resolved photoemission spectroscopy (ARPES) [5], [6]. An effective two-band model that describes a Weyl-node pair and the generation of the Fermi arc was proposed by Ref. [16]. The Bloch Hamiltonian of the model is given by where are the Pauli matrices acting on pseudospins, and , , and are constant. In this model, the system is a Weyl semimetal if and a band insulator if . When a surface is introduced, Ref. [16] showed that there is a Fermi-arc state on the surface in the phase .
Given that surfaces of a material are always present in a device application, it is crucial to understand the interplay between bulk states and the Fermi-arc surface states and how they affect the properties of Weyl semimetals. There are some studies that focused on the effect of Fermi arcs on the electrical transport. Ref. [17] investigated the electrical conductivity of the surface states, , in the presence of a quenched disorder using Kubo formalism. Since the Fermi-arc states can be effectively described by a one-dimensional chiral fermion, one expects that their electrical transport should be dissipationless. However, according to the calculation by [17], is finite. is maximum at the surface and decreases further inside the material. This phenomenon stems from the fact that the gapless bulk states are not totally decoupled from the Fermi-arc states. The surface states can scatter into bulk states and vice versa. This leads to a dissipative transport of the surface states. A study by Ref. [18] tried to understand the contribution of Fermi arcs to the total electrical conductivity (i.e., the sum of bulk and surface conductivities) of a time-reversal-invariant Weyl semimetal in a finite-size geometry. Ref. [18] split the system into the sum of a subsystem with broken time-reversal symmetry plus its time-reversal conjugate. Using the Landauer-type approach, they found that surface states’ electrical conductivity could be as large as the bulk conductivity. These studies highlight the significant effects that the Fermi arcs have on the transport properties of Weyl semimetals.
The focus of this paper is the current-induced spin polarization or the Edelstein effect [19]. A system with a strong spin–orbit coupling for which the spin degeneracy is lifted, such as a Rashba system and a topological insulator, is expected to exhibit this effect [20]. In such a system, an electric field can be used to induce a perpendicular spin-polarization or magnetization inside a material. This effect could potentially be useful for applications in spintronics because it allows a manipulation of a magnetization with an electric field inside a nonmagnetic material. Using the semiclassical Boltzmann theory, Ref. [21] computed an electric-field-induced magnetic moment in a Weyl semimetal TaAs. They found the surface Edelstein effect in Weyl semimetals is much stronger than in Rashba systems and topological insulators. Furthermore, the magnetic moment of the surface states near the surface was found to be greater than that of the bulk states by about two orders of magnitude. This large surface Edelstein effect was attributed to long momentum relaxation times of the surface states. Additionally, an experiment performed on a Weyl semimetal WTe indicated that the material exhibits a large charge-to-spin conversion effect [22].
In this work, we further investigate the behavior in which the surface Edelstein response near an interface is much stronger than the bulk response with Kubo formalism. We calculate the magnetoelectric susceptibility of a Weyl semimetal with broken inversion symmetry. The perturbation theory techniques from [17] are used to compute the surface self-energy and susceptibility. We include short-range scalar impurities in the model and compute vertex corrections within the self-consistent ladder approximation. We also compute the bulk states’ susceptibility in order to make a comparison with the surface’s result. We show that, near an interface, the surface states have a much stronger Edelstein response than the bulk states when the chemical potentials are close to the Weyl points. As the chemical potential moves away from the Weyl points, the surface Edelstein response sharply decreases, whereas the bulk response is approximately constant. At work here is the decoupling between the bulk and surface states close to the energy of the Weyl nodes. This can be seen from our calculation that the rate in which the surface states scatter into the bulk states vanishes at . The Fermi-arc states, effectively a chiral Fermion in 1D, must be almost dissipationless at low chemical potentials. This results in large enhancements of the surface states’ vertex correction and the Edelstein effect
Section snippets
Model of a Weyl semimetal with broken inversion symmetry
In this paper, we consider the model that Ref. [21] used to describe a Weyl semimetal with broken inversion symmetry. Eq. (1) is modified by introducing two pseudospin sectors denoted with . Each sector contains a pair of Weyl nodes centered at . Unlike Eq. (1), the Pauli matrices act on spins, but not on pseudospins. The Bloch Hamiltonian of this model is given by where , , , and are positive constants. Here, is a momentum with respect
Weyl semimetal/vacuum interface
To study the effects of the surface, let us assume that the system is a Weyl semimetal in the region and a vacuum for . This setup can be achieved by allowing in Eq. (2) to depend on , Under this assumption, the surface of this system, which is a plane located at , breaks the translational invariance along the direction and, hence, the momentum along the direction is no longer conserved. in Eq. (2) is replaced by . As a result, the
Green Function
The Green function can be computed from eigenstate wave functions, , and eigenenergies, , by with the sum being over all eigenstates of the Hamiltonian.2 The summation here can be separated into the sum over bulk and surface eigenstates. This means the total Green function is
Surface states’ scattering rate from random impurity scatterings
As in [21], we study the Edelstein effect of Weyl semimetals in the presence of short-range random impurities. We use the same quenched disorder model as Ref. [17]. The impurities are assumed to be dilute, so that the perturbation theory we use to calculate self-energies, vertex corrections, and response functions are valid.3
Surface magnetoelectric susceptibility
In this section, we calculate the Edelstein response or the surface states’ magnetoelectric susceptibility, , using linear response theory. In general, is defined through where is an induced magnetization and is an applied electric field. In order to compute , one considers an action of a form where is an unperturbed action, is a vector potential, and is an external magnetic field. The magnetization can be written
Bulk magnetoelectric susceptibility
To understand how strong the Edelstein response of the surface states is, one needs to compare it with that of the bulk states. It turns out that the calculation involving the nontranslationally invariant part of the bulk Green function, , is somewhat complicated. Hence, we make an approximation by only considering the bulk states from deep inside the Weyl semimetal. This approximation is equivalent to neglecting the term and the bulk-to-surface scattering process as
Discussion and conclusion
Let us try to understand the effects that a surface has on the Edelstein response. First, we consider two types of the surface-to-bulk scattering processes, i.e., the surface-to-translationally-invariant-bulk (STTI) and surface-to-nontranslationally-invariant-bulk (STNI) scatterings. The STTI process is always present even in the absence of an interface, whereas the STNI process arises due to the broken translational symmetry caused by an interface. In the approximation in which the STNI
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
We acknowledge Thailand Research Fund and Office of the Higher Education Commission, Thailand (Grant No. MRG6280130) for funding of this project. Additionally, this research was financially supported by Faculty of Science, Mahasarakham University, Thailand (Grant Year 2019). We thank Chandan Setty for helpful comments on the manuscript.
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