Brief summary of the charge current, spin current, spin, and charge density induced by localized spin dynamics on the disordered surface of a TI. The charge current and spin density have both a local and nonlocal contribution. The spin current and charge density are described by the nonlocal contribution.
Abstract
We theoretically show spin and charge transport on the disordered surface of a three‐dimensional topological insulator with a magnetic insulator when localized spin of the magnetic insulator depends on time and space. To ascertain the transports, we use a low‐energy effective Hamiltonian on the surface of a topological insulator using the exchange interaction and calculate analytically using Green's function techniques within the linear response to the exchange interaction. As a result, the time‐dependent localized spin induces the charge and spin current. These currents are detected from change in the half‐width value of the ferromagnetic resonance of the localized spin when the magnetic resonance of the localized spin is realized in the attached magnetic insulator. We also show spin and charge current generation in a three‐dimensional Weyl–Dirac semimetal, which has massless Dirac fermions with helicity degrees of freedoms. The time‐dependent localized spin drives the charge and spin current in the system. The charge current as well as the spin current in the Weyl–Dirac system are slightly different from those on the surface of the topological insulator.
Keywords
- Spin pumping
- Spin–momentum locking
- Surface of topological insulator
- Weyl–Dirac semimetal
- Massless Dirac fermions
1. Introduction
A crucial issue in spintronics is the generation and manipulation of a charge and spin current by magnetism, since these mechanisms can be applicable to magnetic devices. One way to generate charge and spin flow is called “spin pumping,” which pumps from the angular momentum of a magnetization's localized spin into that of electrons through the dynamics of magnetization as well as spin-orbit interactions [1, 2]. No other way of doing this has so far been discussed in the field of metallic spintronics.
Ever since the discovery of a topological insulator (TI) [3–6], spintronics using topology has been studied. A TI has a gapless surface, its bulk is insulating, but its surface is metallic as a result of two‐dimensional massless Dirac fermions on the surface [4–6]. Because of spin–orbit interactions on the surface the spin and momentum of Dirac fermions are perfectly linked to each other. The relation between a TI's spin and momentum is dubbed “spin–momentum locking,” and the direction in which they travel is perpendicular to each other. Because of spin–momentum locking, unconventional spin‐related phenomena—such as magnetoresistance [7–15], the magnetoelectric effect [16–21], diffusive charge–spin transport [22–25], and the spin pumping effect [26–31]—have so far been the only phenomena theoretically and experimentally studied.
Of the unconventional phenomena on the surface of a TI, spin pumping is one of the most interesting when it comes to spintronics. Here spin pumping on the surface is different from that in metals. As a result of spin–momentum locking on the surface the localized spin plays the role of a vector potential, whereas time‐differential localized spin effectively plays the role of an electric field acting on electrons on the surface of the TI [16–19]. As a result, even in the absence of an applied electric field, the charge current is generated by time‐dependent localized spin as shown in Figure 1. The induced charge current flows along
Recently, it has been reported that localized spin on the surface of a TI subject to magnetism depends on space and its spin texture seems to be a magnetic domain wall [34]. It is predicted that in the presence of a spatial‐dependent spin structure the charge current, which reflects the spin structure, is induced [28]. Moreover, the spin current as well as the charge density are induced when an inhomogeneous spin structure exists on the surface. Detailed results are shown in Section 2. This study may help the study of spin pumping on the surface of a TI with inhomogeneous spin textures [35, 36].
Recently, the next generation of spintronics has been theoretically and experimentally studied in Weyl–Dirac semimetals. A Weyl–Dirac semimetal possessing three‐dimensional massless Dirac fermions has attracted much attention in condensed matter physics [37–39]. Such a semimetal has been experimentally demonstrated [40–45]. In addition, Weyl–Dirac semimetals have been theoretically predicted in a superlattice heterostructure based on the TI. Such a heterostructure has been realized in the GeTe/Sb2Te3 superlattice [46].
Spin‐momentum locking occurs in a Weyl–Dirac semimetal, but the locking is slightly different from that on the surface of the TI. As a result of spin–momentum locking, the spin polarization (charge density) and the charge current (spin current) are linked to each other. Moreover, Dirac fermions have helicity degrees of freedom, which are decomposed into left‐ and right‐handed fermions. Note that the total charge flow of Dirac fermions of left‐ and right‐handed Weyl fermions is preserved. In a Weyl–Dirac semimetal the anomaly‐related effect, which is discussed in the field of relativistic high‐energy physics, has also been discussed in condensed matter physics. Studies up to the moment have asserted that the charge current is generated by magnetic properties with helicity degrees of freedoms [47–54]. Our goal is to introduce the helicity‐dependent spin pumping effect, one of the characteristic properties of Dirac fermions (as shown in Section 3).
2. Spin and charge transport due to spin pumping on the surface of a topological insulator
2.1. Model
We will calculate the charge and spin current due to spin pumping on the surface of a TI with an attached magnetic insulator (MI) (as illustrated in Figure 1). To do so, we consider the following low‐energy effective Hamiltonian, which describes the surface of the TI with localized spin of the MI [4]:
where
where
where
Nonmagnetic impurity scattering is taken into account for a delta function type [19, 21–24] and is considered within the Born approximation. Because of impurity scattering the Fermi velocity of bare Dirac fermions in Eq. (2) is modified by
2.2. Charge and spin current due to localized spin dynamics
We will calculate the charge and spin current as well as the charge and spin density due to magnetization dynamics on the surface of the TI. They are given using the Keldysh–Green function and the lesser component of Green's function [48] as
As a result of spin–momentum locking the charge current <
The charge and spin current are represented by
where
where
where Γ
where
where
Eq. (10) shows that time‐dependent localized spin induces the charge and spin current and that they can be decomposed into local and nonlocal contributions. The first term in Eq. (10) is the charge current due to spin dynamics at that position on the surface; its direction is along the
Because of spin–momentum locking on the surface of the TI, spin polarization is given by <
Eq. (11) shows that the spin current is induced by the time‐ and spatial‐dependent
This result shows that the relation between the charge and spin current is different from that in the metallic spintronics system [2]. The spin current is proportional to the charge current and spin current flow is perpendicular to the charge flow and its spin polarization.
Note that no out‐of‐plane localized spin
2.3. Spin torque
Based on these results, we look at localized spin dynamics after generation of the charge current and spin polarization on the surface [24]. We assume there is an external static and AC magnetic field on the surface—as shown in Figure 2(a). The static magnetic field arranges the localized spin texture and the AC magnetic field triggers its spin dynamics. The propagation direction of the microwave is parallel to the static magnetic field, which is along the
After applying the microwave the dynamics of the localized spin of the MI is induced by the in‐plane AC magnetic field of the microwave. The dynamics of the spin induces the charge and spin current. Then, from <
where
Since
where
As a result of spin–momentum locking the torque is
We now consider magnetization when it is given by
where
where
The coefficients
Hence,
Figure 3(a) shows the dependence of Ω on the imaginary part of the longitudinal magnetic permeability for several
Any change in the half‐width value ΔΩ indicates an induced charge current on the surface of the TI because
Any contribution from the applied magnetic field is of course a concern. Note that an in‐plane static magnetic field contributes no finite charge current generation or spin polarization [49], whereas the contribution from the applied magnetic field is negligible.
3. Spin pumping in a Weyl–Dirac semimetal
3.1. Model
We now consider spin pumping in a Weyl–Dirac semimetal subjected to magnetism. To do this, a Weyl–Dirac semimetal subjected to spin and momentum locking, such as a supperlattice hetrod structure constructed from a TI/normal insulator/TI [39, 46 – 48], is considered. The low‐energy effective Hamiltonian describing the Weyl–Dirac semimetals that have a spin–exchange interaction is given by [50–52]
The first term takes the form
where
The second term of Eq. (20) indicates the spin–exchange interaction:
where
This Hamiltonian is similar to that on the surface of the TI—see Eq. (4). In the following calculations,
3.2. Response function within the linear response to the exchange interaction
To calculate the charge and spin current within the linear response to
where < > denotes the expectation value in Eq. (20). Such a charge current can be decomposed as <
The spin current in the Weyl‐Dirac semimetal can be defined from the Heisenberg equation for the spin operator:
where
The superscript and subscripts of
The relaxation term can also be decomposed by <
The charge and spin current can be obtained by calculating the response function (Figure 4):
where
where
where Γ
where the above matrix component (
As a result, the charge current and spin current can be obtained by
where
where <
Note that the above results are obtained when
3.3. Charge and spin current due to spin pumping effects
From the above results the total charge current can be given by
The charge current is triggered by the dynamics of localized spin. The first term indicates the local term of the dynamics of localized spin. Its direction is parallel to ∂t
Note that the property of the charge current at each helicity links to that of spin polarization because of spin–momentum locking. However, after summation of the indices of helicity the relation between total charge current and total spin polarization is changed. As a result, even in the absence of population imbalance, nonzero spin polarization can be given by Eq. (25) as
Local spin polarization is along
Total spin current can be represented from Eqs. (35) and (36) as
Total spin current can be generated by spatial divergence of localized spin dynamics, where localized spin is the convolution with diffusion. As a result, the spin current can be regarded as a nonlocal spin current. Note that a nonzero spin current is generated when localized spin depends on space. Such a spin current becomes nonzero even in the absence of population imbalance.
The nonlocal spin current in this case is obtained from the diffusive motion of spin density, which is driven by the dynamics of localized spin, where localized spin depends on space. The spin diffusive motion of each helicity can be given from Eqs. (25), (34), (37) as
As a result, the diffusive motion of total spin becomes:
Time‐dependent and spatial‐dependent localized spin,
4. Conclusion
Our results on the charge and spin current due to spin pumping on the surface of a three‐dimensional TI (Section 2) [28] and in the bulk of a three‐dimensional Weyl–Dirac semimetal (Section 3) [51] are summarized in this chapter.
Section 2 summarizes our results on spin pumping on the surface of a TI attached to an MI. The results are calculated using the standard Keldysh–Green function method within the linear response to the exchange interaction between the conduction spin and localized spin of an MI. The purpose of this work is to derive charge and spin current generation due to localized spin dynamics on the disordered surface of a TI; in particular, when the localized spin depends on space on the surface. The main results on the surface of a TI are summarized in Table 1. Time‐dependent localized spin on the surface is a prerequisite to obtaining nonzero charge and spin current generation. Moreover, Table 1 shows that when the spin texture is spatially inhomogeneous, not only the local charge but also the nonlocal charge and spin current are generated by time‐dependent localized spin. The flow and spin polarization of the spin current are perfectly perpendicular to each other because of spin–momentum locking. The magnitude of the spin current is proportional to the charge density, which is induced by divergence between time‐dependent localized spin and the diffusive propagator on the surface—see Eqs. (10) and (11). Such pumping effects are caused by time‐dependent localized spin, which plays a role in driving the charge current and can be regarded as an effective electric field
Recently, it has been reported that the localized spin texture at the junction of the TI/MI is spatially inhomogeneous [34]. We suppose that the spin current we have obtained at the junction is generated when the spin texture moves temporally.
On the basis of these results, in Section 2.3 we discussed a way of detecting the charge current and spin current induced on the surface of a TI attached to an MI by using ferromagnetic resonance. We assume that the dynamics of localized spin is triggered by the applied static and AC magnetic field of the microwave. The dynamics of the localized spin induced both the charge current and the spin current. Such induced currents are related to spin density, and spin polarization acts on the localized spin in much the same way as spin torque—see Eq. (14). Hence, the half‐width of ferromagnetic resonance changes as shown in Figure 3.
Spin pumping in a Weyl–Dirac semimetal hosting massless Weyl–Dirac fermions is summarized in section 3. The results are obtained within the same formalism as laid out in Sections 2.1 and 2.2. The charge and spin current as well as the charge and spin density are given in Table 2. Semimetals are subject to spin–momentum locking. The spin direction of Weyl–Dirac fermions brought about by spin–momentum locking is perfectly parallel/antiparallel to its momentum and its locking is determined by the helicity degrees of freedom of Weyl–Dirac fermions. As a result, the charge current and spin polarization induced depend on the helicity indices. Eqs. (38) and (39) show that localized spin dynamics induces the charge current and spin polarization, respectively; hence, localized spin plays the role of an effective electric field
〈 |
Ref. | ||
---|---|---|---|
Local | [51] | ||
Nonlocal | [51] | ||
Driving force |
This work [51] | ||
Total | 〈 〈 |
This work |
Acknowledgments
This work was supported by a Grant‐in‐Aid for JSPS Fellows (Grant No. 13J03141), by a Grant‐in‐Aid for Challenging Exploratory Research (Grant No. 15K13498), and by the Core Research for Evolutional Science and Technology (CREST) of Japan Science.
References
- 1.
Tserkovnyak Y, Brataas A, Bauer GEW. Enhanced gilbert damping in thin ferromagnetic films. Phys Rev Lett. 2002;88(11):117601. DOI: 10.1103/PhysRevLett.88.117601 - 2.
Saitoh E, Ueda M, Miyajima H, Tatara G. Conversion of spin current into charge current at room temperature: Inverse spin‐Hall effect. Appl Phys Lett. 2006;88(18):13–6.DOI: 10.1063/1.2199473 - 3.
Hasan MZ, Kane CL. Colloquium: Topological insulators. Rev Mod Phys. 2010;82(4):3045–67. DOI:10.1103/RevModPhys.82.3045 - 4.
Qi XL, Zhang SC. Topological insulators and superconductors. Rev Mod Phys. 2011;83(4). DOI: 10.1103/RevModPhys.83.1057 - 5.
Ando Y. Topological insulator materials. J Phys Soc Japan. 2013;82(10):1–32. DOI: 10.7566/JPSJ.82.102001 - 6.
Hsieh D, Xia Y, Qian D, Wray L, Dil JH, Meier F, et al. A tunable topological insulator in the spin helical Dirac transport regime. Nature. 2009;460(7259):1101–5. DOI: 10.1038/nature08234 - 7.
Yokoyama T, Tanaka Y, Nagaosa N. Anomalous magnetoresistance of a two‐dimensional ferromagnet/ferromagnet junction on the surface of a topological insulator. Phys Rev B. 2010;81(12):3–6. DOI: 10.1103/PhysRevB.81.121401 - 8.
Schwab P, Raimondi R, Gorini C. Spin‐charge locking and tunneling into a helical metal. Europhysics Lett. 2011;93(6):67004. DOI: 10.1209/0295‐5075/93/67004 - 9.
Mondal S, Sen D, Sengupta K, Shankar R. Magnetotransport of Dirac fermions on the surface of a topological insulator. Phys Rev B. 2010;82(4):1–11. DOI: 10.1103/PhysRevB.82.045120 - 10.
Ma MJ, Jalil MBA, Tan SG, Li Y, Siu ZB. Spin current generator based on topological insulator coupled to ferromagnetic insulators. AIP Adv. 2012;2(3). DOI: 10.1063/1.4751255 - 11.
Kong BD, Semenov YG, Krowne CM, Kim KW. Unusual magnetoresistance in a topological insulator with a single ferromagnetic barrier. Appl Phys Lett. 2011;98(24):96–9. DOI: 10.1103/PhysRevB.89.201405 - 12.
Kandala A, Richardella A, Rench DW, Zhang DM, Flanagan TC, Samarth N. Growth and characterization of hybrid insulating ferromagnet–topological insulator heterostructure devices. Appl Phys Lett. 2013;103(20):2011–5. DOI: 10.1063/1.4831987 - 13.
Taguchi K, Yokoyama T, Tanaka Y. Giant magnetoresistance in the junction of two ferromagnets on the surface of diffusive topological insulators. Phys Rev B. 2014;89(8):1–5. DOI: 10.1103/PhysRevB.89.085407 - 14.
Fischer MH, Vaezi A, Manchon A, Kim E‐A. Spin–Torque generation in topological–insulator‐based Heterostructures. Phys Rev B. 2016;93(12):1-4. DOI: 10.1103/PhysRevB.93.12530 - 15.
Jamali M, Lee JS, Lv Y, Zhao Z, Samarth N, Wang JP. Room Temperature Spin Pumping in Topological Insulator Bi2Se3. [Internet.] 2014. Available from: http://arxiv.org/abs/1407.7940 - 16.
Qi XL, Hughes TL, Zhang SC. Topological field theory of time‐reversal invariant insulators. Phys Rev B. 2008;78(19):1–43. DOI: 10.1103/PhysRevB.78.195424 - 17.
Garate I, Franz M. Inverse spin‐galvanic effect in the interface between a topological insulator and a ferromagnet. Phys Rev Lett. 2010;104(14):1–4. DOI: 10.1103/PhysRevLett.104.146802 - 18.
Nomura K, Nagaosa N. Electric charging of magnetic textures on the surface of a topological insulator. Phys Rev B. 2010;82(16):3–6. DOI: 10.1103/PhysRevB.82.161401 - 19.
Ueda HT, Takeuchi A, Tatara G, Yokoyama T. Topological charge pumping effect by the magnetization dynamics on the surface of three‐dimensional topological insulators. Phys Rev B. 2012;85(11):1–6. DOI: 10.1103/PhysRevB.85.115110 - 20.
Linder J. Improved domain‐wall dynamics and magnonic torques using topological insulators. Phys Rev B. 2014;90(4):1–5. DOI: 10.1103/PhysRevB.90.041412 - 21.
Taguchi K, Shintani K, Tanaka Y. Electromagnetic effect on disordered surface of topological insulators. J Magn Magn Mater. 2015;400:188–10. DOI: 10.1103/PhysRevB.92.035425 - 22.
Burkov AA, Hawthorn DG. Spin and charge transport on the surface of a topological insulator. Phys Rev Lett. 2010;105(6):6–9. DOI: 10.1103/PhysRevLett.105.066802 - 23.
Fujimoto J, Sakai A, Kohno H. Ultraviolet divergence and Ward–Takahashi identity in a two‐dimensional Dirac electron system with short‐range impurities. Phys Rev B. 2013;87(8):1–5. DOI: 10.1103/PhysRevB.87.085437 - 24.
Sakai A, Kohno H. Spin torques and charge transport on the surface of topological insulator. Phys Rev B. 2014;89(16). DOI: 10.1103/PhysRevB.89.165307 - 25.
Liu X, Sinova J. Reading charge transport from the spin dynamics on the surface of a topological insulator. Phys Rev Lett. 2013;111(16):1–5. DOI: 10.1103/PhysRevLett.111.166801 - 26.
Tserkovnyak Y, Loss D. Thin‐film magnetization dynamics on the surface of a topological insulator. Phys Rev Lett. 2012;108(18):1–5. DOI: 10.1103/PhysRevLett.108.187201 - 27.
Tserkovnyak Y, Pesin DA, Loss D. Spin and orbital magnetic response on the surface of a topological insulator. Phys Rev B. 2015;91(4):041121. DOI: 10.1103/PhysRevB.91.041121 - 28.
Taguchi K, Shintani K, Tanaka Y. Spin‐charge transport driven by magnetization dynamics on disordered surface of doped topological insulators. Phys Rev B. 2015;035425:1–34.DOI: 10.1103/PhysRevB.92.035425 [ Figure 4(b) in reference [28] is a misprint: The values 0.38, 0.34, and 0.30 on the vertical axis in the figure should be 0.038, 0.034, and 0.030, respectively.Figure 3(c) in this chapter is revised.] - 29.
Semenov YG, Duan X, Kim KW. Voltage‐driven magnetic bifurcations in nanomagnet–topological insulator heterostructures. Phys Rev B. 2014;89(20):1–5. DOI: 10.1103/PhysRevB.89.201405 - 30.
Deorani P, Son J, Banerjee K, Koirala N, Brahlek M, Oh S, et al. Observation of inverse spin Hall effect in bismuth selenide. Phys Rev B. 2014;094403:1–19.DOI: 10.1103/PhysRevB.90.094403 - 31.
Shiomi Y, Nomura K, Kajiwara Y, Eto K, Novak M, Segawa K, et al. Spin–electricity conversion induced by spin injection into topological insulators. Phys Rev Lett. 2014;113(19):7–11. DOI: 10.1103/PhysRevLett.113.196601 - 32.
Mizukami S, Ando Y, Miyazaki T. Effect of spin diffusion on Gilbert damping for a very thin permalloy layer in Cu/permalloy/Cu/Pt films. Phys Rev B. 2002;66(10):1–9. DOI: 10.1103/PhysRevB.66.104413 - 33.
Gilbert TL. Classics in magnetic: A phenomenological theory of damping in ferromagnetic materials. IEEE Trans Magn. 2004;40(6):3443–9. DOI: 10.1109/TMAG.2004.836740 - 34.
Wei P, Katmis F, Assaf BA, Steinberg H, Jarillo‐Herrero P, Heiman D, et al. Exchange‐coupling‐induced symmetry breaking in topological insulators. Phys Rev Lett. 2013;110(18):1–5. DOI: 10.1103/PhysRevLett.110.186807 - 35.
Coop G, Przeworski M, Wall JD, Frisse LA, Hudson RR, Di Rienzo A, et al. Observation of skyrmions in a multiferroic material. Science. 2012;336(April):198–201. DOI: 10.1126/science.1214143 - 36.
White JS, Prša K, Huang P, Omrani AA, Živković I, Bartkowiak M, et al. Electric‐field‐induced skyrmion distortion and giant lattice rotation in the magnetoelectric insulator Cu2OSeO3. Phys Rev Lett. 2014;113(10):1–5. DOI: 10.1103/PhysRevLett.113.107203 - 37.
Murakami S. Phase transition between the quantum spin Hall and insulator phases in 3D: Emergence of a topological gapless phase. New J Phys. 2007;9. DOI: 10.1088/1367‐2630/9/9/356 - 38.
Yang B, Nagaosa N. Classification of stable three‐dimensional Dirac semimetals with nontrivial topology. Nat Commun. 2014;5:4898. DOI: 10.1038/ncomms5898 - 39.
Burkov AA, Balents L. Weyl semimetal in a topological insulator multilayer. Phys Rev Lett. 2011;107(12):1–4. DOI: 10.1103/PhysRevLett.107.127205 - 40.
Brahlek M, Bansal N, Koirala N, Xu SY, Neupane M, Liu C, et al. Topological‐metal to band‐insulator transition in (Bi 1‐x In x) 2 Se 3 thin films. Phys Rev Lett. 2012;109(18):1–5. DOI: 10.1103/PhysRevLett.109.186403 - 41.
Wu L, Brahlek M, Valdés Aguilar R, Stier AV, Morris CM, Lubashevsky Y, et al. A sudden collapse in the transport lifetime across the topological phase transition in (Bi(1-x)In(x))2Se3. Nat Phys. 2013;9(7):410–4. DOI: 10.1038/nphys2647 - 42.
Liu ZK, Jiang J, Zhou B, Wang ZJ, Zhang Y, Weng HM, et al. A stable three‐dimensional topological Dirac semimetal Cd3As2. Nat Mater. 2014;13(7):677–81. DOI: 10.1038/nmat3990 - 43.
Neupane M, Xu S‐Y, Sankar R, Alidoust N, Bian G, Liu C, et al. Observation of a three‐dimensional topological Dirac semimetal phase in high‐mobility Cd3As2. Nat Commun. 2014;05:3786‐ May 7;5. DOI: 10.1038/ncomms4786 - 44.
Novak M, Sasaki S, Segawa K, Ando Y. Large linear magnetoresistance in the Dirac semimetal TlBiSSe. Phys Rev B. 2015;91(4):1–5.DOI: 10.1103/PhysRevB.91.041203 - 45.
Huang S‐M, Xu S‐Y, Belopolski I, Lee C‐C, Chang G, Wang B, et al. A Weyl Fermion semimetal with surface Fermi arcs in the transition metal monopnictide TaAs class. Nat Commun. 2015;6:7373.DOI: 10.1038/ncomms8373 - 46.
Tominaga J, Kolobov AV, Fons P, Nakano T, Murakami S. Ferroelectric order control of the Dirac–semimetal phase in GeTe‐Sb 2 Te 3 superlattices. Adv Mater Interfaces. 2013;1:1‐7.DOI: 10.1002/admi.201300027 - 47.
Zyuzin AA, Wu S, Burkov AA. Weyl semimetal with broken time reversal and inversion symmetries. Phys Rev B. 2012;85(16):1–9. DOI: 10.1103/PhysRevB.85.165110 - 48.
Zyuzin AA, Burkov AA. Topological response in Weyl semimetals and the chiral anomaly. Phys Rev B. 2012;86(11):1–8. DOI: 10.1103/PhysRevB.86.115133 - 49.
Vazifeh MM, Franz M. Electromagnetic response of Weyl semimetals. Phys Rev Lett. 2013;111(2):1–5. DOI: 10.1103/PhysRevLett.111.027201 - 50.
Taguchi K. Equilibrium axial current due to a static localized spin in Weyl semimetals. J Phys Conf Ser. 2014;568(5):052032. DOI: 10.1088/1742‐6596/568/5/052032 - 51.
Taguchi K, Tanaka Y. Axial current driven by magnetization dynamics in Dirac semimetals. Phys Rev B. 2015;91(5):1–5. DOI:10.1103/PhysRevB.91.054422 - 52.
Nomura K, Kurebayashi D. Charge‐induced spin torque in anomalous Hall ferromagnets. Phys Rev Lett. 2015;115(12):1–5. DOI: 10.1103/PhysRevLett.115.127201 - 53.
Chan C‐K, Lee P A, Burch KS, Han JH, Ran Y. When chiral photons meet chiral fermions: Photoinduced anomalous Hall effects in Weyl semimetals. Phys Rev Lett. 2015;1(7):1–5. DOI: 10.1103/PhysRevLett.116.026805 - 54.
Taguchi K, Imaeda T, Sato M, Tanaka Y. Photovoltaic chiral magnetic effect in a Weyl semimetal.Phys Rev B(R). 2016;93(20):1–5. DOI: 10.1103/PhysRevB.93.201202 - 55.
Haug H, Jauho A‐P. Quantum Kinetics in Transport and Optics of Semiconductors. [Internet.] 2008. 362 pp. Available from: http://www.springer.com/physics/complexity/book/978‐3‐540‐73561‐8 - 56.
Tokura Y, Kida, N. Dynamical magnetoelectric effects in multiferroic oxides. Phil. Trans. R. Soc. A. 2011:369;3679–3694. DOI: 10.1098/rsta.2011.0150