Abstract
Directional transport and propagation of quantum particle and current, such as electron, photon, spin, and phonon, are known to occur in the materials system with broken inversion symmetry, as exemplified by the diode in semiconductor p–n junction and the natural optical activity in chiral materials. Such a nonreciprocal response in the quantum materials of noncentrosymmetry occurs ubiquitously when the timereversal symmetry is further broken by applying a magnetic field or with spontaneous magnetization, such as the magnetochiral effect and the nonreciprocal magnon transport or spin current in chiral magnets. In the nonlinear regime responding to the square of current and electric field, even a more variety of nonreciprocal phenomena can show up, including the photocurrent of topological origin and the unidirectional magnetoresistance in polar/chiral semiconductors. Microscopically, these nonreciprocal responses in the quantum materials are frequently encoded by the quantum Berry phase, the toroidal moment, and the magnetoelectric monopole, thus cultivating the fertile ground of the functional topological materials. Here, we review the basic mechanisms and emergent phenomena and functions of the nonreciprocal responses in the noncentrosymmetric quantum materials.
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Introduction
Chirality, which characterizes right and lift, is an important and fundamental notion in whole sciences, including physics, chemistry, and biology^{1}. For example, a molecule is called chiral when its mirror image does not overlap with the original molecule. In the language of symmetry, it has no inversion symmetry \(\hat I\) nor mirror symmetry \(\hat M\). The inversion \(\hat I\)can be expressed in terms of the product of the mirror symmetry \(\hat M\) and the 180° rotation C_{2} around the axis perpendicular to the mirror plane. Therefore, \(\hat I\) and \(\hat M\) are identical once C_{2} is assumed in free space. (Hereafter, we use \(\hat I\) symmetry representing either \(\hat I\) or \(\hat M\) depending on the crystal symmetry.) It often determines the biological functions and is crucial for the pharmaceutical biotechnology. In addition to these issues on static structures, the dynamics of the systems is the main interest in physics. Namely, the directional responses may occur in physical systems with broken \(\hat I\). In solids, the crystal structure often breaks the \(\hat I\) symmetry, but they do not necessarily guarantee the directional response. To see this, one can look at the onedimensional scattering problem with the asymmetric potential barrier V(x)(≠V(−x)). The transmission and reflection probabilities are identical, irrespective of the leftgoing or rightgoing incident wave. Namely, the directional dependence is missing here. It comes from the unitary nature of the scattering (S)matrix. Furthermore, the timereversal symmetry \(\hat T\) of the Schroedinger equation imposes the condition S = S^{T} (S^{T} means the transpose of S). It is natural that the right mover and left mover are exchanged by both \(\hat I\) and \(\hat T\), and hence these two symmetries play essential roles in the nonreciprocal responses.
Accordingly, as shown in Box 1, there are four categories of nonreciprocal responses depending on (i) the linear and nonlinear response, and (ii) \(\hat T\)unbroken and \(\hat T\)broken. While the physical mechanisms of the phenomena and effects listed in this table will be discussed below in detail, we stress here that nonreciprocal responses touch the most fundamental issues of condensed matter physics such as symmetries, quantum geometrical nature of electrons, and electron correlation.
Linear nonreciprocal responses
Onsager reciprocal theorem
In manybody systems in thermodynamic limit, the dissipative process occurs commonly, which determines the direction of time due to its irreversible nature. However, this is distinct from the timereversal symmetry \(\left( {\hat T} \right)\) breaking due to the magnetic field B or the spontaneous magnetization M. Onsager recognized the role of \(\hat T\)symmetry in the microscopic dynamics of the system appearing in the macroscopic response functions linear to the external stimuli^{2,3}. It is formulated by Kubo formula for linear response function K_{AB}(q,ω, B) corresponding to the input B and output A with the wavevector q and frequency ω, which satisfies the relation^{4}
where ε_{A} = ±1, ε_{B} = ±1 specifies the even–odd nature of the quantity A (B) for the timereversal operation. This Onsager’s reciprocal theorem, when applied to the conductivity tensor, gives
The magnetochiral effect described in Box 1 corresponds to the formula that σ_{αα}(q, ω, B) = σ_{αα}(ω) + κBq, which is consistent with Eq. (2). Another application of Eq. (2) is the natural circular dichroism which is expressed by σ_{αβ} (q, ω, B = 0) = ζε_{αβγ}q_{γ} with ε_{αβγ} being the totally antisymmetric tensor, which describes the rotation of the light polarization in a chiral crystal. This natural circular dichroism should be distinguished from the Kerr and Faraday rotation which follows from σ_{αβ}(q = 0, ω, B) = ξε_{αβγ}B_{γ}, and does not depend on the direction of light, i.e., q. Below, we discuss the linear nonreciprocal responses in more detail describing their microscopic mechanism.
Directional dichroism and birefringence in magnetoelectrics
The light wavevector (q) dependent optical response, typically termed directional dichroism or directional birefringence, can be observed ubiquitously in materials with broken symmetries of both spaceinversion \(\left( {\hat I} \right)\) and timereversal \(\left( {\hat T} \right)\) and in a wide photonenergy region ranging from a microwave to hard Xray. One of the early observations is the directional light emission (luminescence) from a chiral molecule with luminescent Eu^{3+} ion under the magnetic field reported by Rikken et al.^{5}; depending on the relative direction of parallel and antiparallel lightemission q and the applied magnetic field B, i.e., q·B > 0 or <0, the lightemission intensity is modulated, typically up to a few % for B of the order of a few tesla. This case is called the magnetochiral dichroism. Magnetoelectric (ME) directional anisopropy has been first observed in Er_{1.5}Y_{1.5}Al_{5}O_{12} with the modulation of the refractive index δn = γq·(E × B)^{6}. When luminescent rareearth ions are incorporated into a polar medium, e.g., in ferroelectric BaTiO_{3} doped with Er^{7}, modulation of luminescence intensity occurs depending on the case, q·(P × B)>0 or <0; here P denotes the electric polarization. When the breaking of \(\hat T\) is brought about by spontaneous magnetization M instead of applied magnetic field B, even a larger directional effect can show up. The directional dichroism is phenomenologically described by the Maxwell equations plugged in with the ME susceptibility tensor α; P = αB or M = ^{t}αE. In case of multiferroics where the spontaneous P and M coexist, the diagonal and offdiagonal components of α are outcomes of the free energy terms proportional to E·B^{8} and E × B^{6}, respectively.
Figure 1a–d exemplifies the directional dichroism observed for the ME spin excitation, termed electromagnon, which shows both magnetic and electric transitiondipole activities^{9}. The helical or conical spin order as shown in Fig. 1a can produce the electronic chirality depending on the magnetic helicity. In this particular compound, Gadoped CuFeO_{2} with a triangular lattice of Fe^{3+} ions^{10}, the hybridization of the metal (Fe^{3+}) d state, and ligand (O^{2−}) p state under the influence of spin–orbit interaction can produce the electric polarization P along the screw wavevector (Q) direction, as described by the d–p orbital hybridization model^{11}. This multiferroic character can allow one to obtain the single spinhelicity domain, or equivalently the single chirality domain, via the ME cooling procedure under B//E, since the sign of P depends on the spin helicity in this mechanism. Thus, when the magnetic field is applied along the Q direction (or the Q is directed along the applied magnetic field direction), the magnetochiral dichroism for the q^{ω}//Q (here q^{ω} being the light q vector with the frequency of ω, set parallel to the z axis) is anticipated to stem from the offdiagonal ME coefficient (α_{xy}) since the orthogonal light electric and mangetic fields are along x or y. Figure 1b, c exemplifies the large directional dichroism spectra, i.e., the spectra of real part (n) and imaginary part (κ) of the refractive index depending on the q^{ω}//P or –P^{12}. The change of complex refractive index is given by the offdiagonal component of the ME susceptibility, ±α_{ij}(ω), in conjunction with the Maxwell wave equation. The α_{xy}(ω) = Δn(ω)+iΔκ(ω) shows the characteristic anomalous dispersion spectrum due to the resonance of the electromagnon.
The diagonal ME susceptibility arises from the E·Btype coupling term in the free energy and is found in the system endowed with the ME monopoles. Here the ME monopole is defined by the magnetic structure with the inward or outward configuration of spin moment (m), i.e., div m ≠ 0^{13}. A classic example of this is the antiferromagnetically ordered state in Cr_{2}O_{3}, in which Cr dimers formed along the c (z) direction show the up and down spin moment configuration, i.e., the ME monopole, and the linear ME effect was first predicted^{14} and detected^{15}. While Cr_{2}O_{3} shows both the diagonal and offdiagonal ME effect, the diagonal component leads to the specific nonreciprocal optical effect, termed gyrotropic birefringence (GB). The GB represents the rotation of the optical principle axes depending on the light incident direction, as schematized in Fig. 1d. When the light propagates along the y direction (q^{ω}//y) with the light E^{ω}//z and H^{ω}//x, the additional P^{ω}_{x} and M^{ω}_{z} components are induced inside the material by the diagonal ME susceptibility of α_{xx} and α_{zz}, respectively. Therefore, the rotation of the optical principle axes is described in terms of the dynamic ME susceptibility α_{GB}(ω) = α_{xx}(ω)–α_{zz}(ω). This was first confirmed for the Cr d–d excitation region in the visible range^{16}.
The large GB has been observed for the electromagnon in multiferroic (Fe, Zn)_{2}Mo_{3}O_{8} with the ferromagnetic moment M and the inherent electric polarization P both along the c axis (Fig. 1d)^{17,18}. The ferrimagnetic spin alignment along the c axis also produces the ME monopoles. The electromagnon locating at 1.4 THz (5.6 meV) shows the resonant dispersion of α_{GB}(ω), giving rise to the 0.02rad rotation and 0.04rad ellipticity of the light polarization for the 430μmthick crystal at 5 K. The electromagnon resonance is turned out to contribute to about 8% of the DC (ω = 0) linear ME susceptibility, pointing to the important role of the electromagon excitation as the ME fluctuation in multiferroics^{18}.
Another important example of multiferroics to potentially show the GB is the axion insulator state constituted from the topological insulator (TI). In the TI, the Lagrangian of the electromagnetic field can contain the axion coupling term,\(\left( {\frac{\alpha }{{4\pi ^2}}} \right)\theta {\mathbf{E}} \cdot {\mathbf{B}}\), in addition to the conventional Maxwell term^{19,20}. Here, α = e^{2}/ħc is the fine structure constant and θ is equal to π (0 or 2π) in the topological (nontopological) insulator. When the top and bottom surfacestate dispersions both show mass gaps due to the exchange coupling with the opposite magnetizations (i.e., ±1 Chern number, Fig. 1e) and the Fermi level is inside these mass gaps, the compound can be an axion insulator with the topological ME effect, as characterized by the diagonal ME susceptibility α_{zz}~α. This state can also be viewed as possessing the ME monopole like Cr_{2}O_{3}, and be anticipated to show the topological GB in a quantized manner^{21}.
As exemplified above, the nonreciprocal dichroism/birefringence can be ubiquitously observed in a broad family of materials system with both broken \(\hat I\) and \(\hat T\) in a wide photonenergy region ranging from μeV (microwave)^{22} to keV (hard Xray) region^{23}. Apart from the case of the ME excitation resonance like the electromagnon^{24,25}, their magnitudes usually remain at most a few % to the original absorption intensity or to the π/2 rotation for GB. A notable exception is the case of sharp resonant crystalfield d–d excitations in the canted antiferromagnetic phase of CuB_{2}O_{4} with noncentrosymmetric space group I42d^{26}. There, zerophonon line of the d–d transition is observed to show the huge qdirectional change of the absorption or the nearly perfect cloaking at high magnetic field, which is ascribed to the strong interference between E1 (electricdipole) and M1 (magneticdipole) transitions caused by the spin–orbit coupling^{26,27}.
Nonreciprocal dynamics of magnons
When the quasiparticle with the wave vector k propagates along the magnetic field B (or magnetization M) in the chiralstructure compound, the relevant response function adds the additional term A_{MCh}, which is in proportion with σ[sgn(k·H)], to the conventional term A_{0}; here σ = ±1 depends on the lattice chirality, i.e., right or left handedness of the crystal. This relation generally holds for photon, magnon, and other quasiparticles which can be sensitive to the B or M. The case of photon is the magnetochiral dichroism as described in the previous section. Here we review the studies done on the magnon (spin wave) propagation.
The most archetypal example is the spin wave propagation in the chirallattice ferromagnet; this was recently observed in chiral magnets such as LiFe_{5}O_{8}^{28}, Cu_{2}OSeO_{3}^{29}, and other intermetallic compounds^{30,31}. The effective Hamiltonian of the chiral magnet can be described as the sum of symmetric (J) and antisymmetric (D) exchange interactions, cubic anisotropy (K) term, and Zeeman term proportional to H. There the asymmetric exchange interaction, called Dzyaloshinskii–Moriya (DM) interaction, originates from the relativistic spin–orbit coupling in the noncentrosymmetric (here, chiral) lattice and works as a source of nonreciprocity of magnon transport. The chiral magnet generally shows the helical (screw) spin state whose magnetic periodicity is given as aD/J with a being lattice constant. The application of H turns the helical spin ground state to the spincollinear (ferromagnetic) state, where the spin wave dispersion is given by
Here, C_{sym} = JSV_{0} κ^{2} + 2KV_{0}S^{3}+γħμ_{0} H is an even function of k; γ, μ_{0},V_{0}, and S are gyromagnetic ratio, vacuum magnetic permeability, unit cell volume, and vector spin density, respectively. The first klinear term causes the shift of the parabolic spin wave dispersion (C_{sym}) and hence the different group velocity between +k and –k, as shown in Fig. 2a, leading to the nonreciprocal propagation of the spin wave. (In the case of the plateshaped sample, the dispersion is modified by the magnetic dipolar interaction in the k~0 region, as discerned by the sharp spike around k~0 in Fig. 2a.)
The spin wave spectroscopy can directly demonstrate the nonreciprocal magnon transport. As the typical experimental setup (Fig. 2b), a pair of coplanar wave guides (ports 1 and 2) placed beneath the chiral magnet play roles of generator and detector of magnon. The wave number (k) distribution of the generated spin wave is represented by the Fourier transform of the wave guide pattern with the spacing of λ, i.e., k~2π/λ. The spectra of the mutual inductance L_{12} (L_{21}) represent the propagation of the spin wave or magnon from terminal 1 to 2 (2 to 1). Figure 2c exemplifies the imaginary part of the inductance spectra L_{12} and L_{21} with λ ∼12 μm, taken for the specificchirality (Dbody or σ = +1) crystal of chiral (space group: P2_{1}3) magnet Cu_{2}OSeO_{3} with the spincollinear state induced by the magnetic field of ±740 Oe applied parallel/antiparallel to the magnon k direction^{29}. The magnetochiral nonreciprocal propagation can be clearly discerned as the frequency shift between L_{12} and L_{21} and also as the reversal of the shift upon the reversal of the magnetic field direction. The results for the Lbody (σ = −1) crystal show the reversed relation, confirming the magnetochiral nature. Figure 2d summarizes the result of the spin wave spectroscopy; in the relatively high magnetic field (hatched) region where the spincollinear ferromagnetic state is stable, the difference Δv_{p} of the peak frequency of L_{12} and L_{21} (representing the magnon energy difference propagating from terminal 1to2 and 2to1) is opposite in sign in the positive and negative magnetic fields. The magnon group velocity v_{g} deduced by the L_{12} and L_{21} data differs between the 1to2 and 2to1 propagation and the difference is also reversed upon the reversal of magnetization. All these features point to the magnetochiral nonreciprocity of magnon transport, which is well accounted for with Eq. (3). By contrast, the helical spin state in the low magnetic field shows the minimal nonreciprocity although the symmetry argument would allow the nonreciprocal effect. The magnon energy difference between +k and –k is directly related to the DM interaction D, and in turn its observation can give the quantitative estimate of D, as demonstrated in a series of the chirallattice magnets hosting the magnetic skyrmion near room temperature^{31}.
The klinear term in the magnon dispersion caused by the DM interaction can be probed by other means of spin wave or magnon spectroscopy such as inelastic neutron spectroscopy^{30} and light Brillouin scattering spectroscopy^{32,33}. For example, the inelastic neutronscattering spectroscopy on chirallattice magnet MnSi (P2_{1}3 space group) has directly demonstrated the asymmetric magnon dispersion as described by Eq. (3)^{30}. The DM interaction produced at the interface between the heavyelement (i.e., large spin–orbit interaction) metal and ferromagnet can play an important role in spintronics function and also generate the nonreciprocal magnon transport along the lateral direction under the magnetic field applied laterally and normal to the k direction. This was also detected as the difference of the spin wave resonance frequency between +k and –k in terms of the light Brillouin scattering spectroscopy^{32,33}.
Nonlinear/nonreciprocal transport
Theorem of fluctuation
Compared with the linear response theory, its nonlinear generalization of nonreciprocity is far more difficult and not well explored. However, the generalization of Onsager’s reciprocal theorem has been intensively discussed recently, and is called “fluctuation theorem”, which also originates from the timereversal symmetry of the microscopic dynamics of the system^{34,35,36,37}. Let p(R) be the probability that the entropy change is R. Then the fluctuation theorem claims that
where R > 0 means the entropy production, while R < 0 the entropy reduction. B represents the magnetic field, and the denominator and numerator of the lefthand side of Eq. (4) are related by the timereversal symmetry. In the thermodynamics limit, R becomes very large, and hence the righthand side of Eq. (4) is infinite, and the probability of the entropy reduction is zero, i.e., the second law of thermodynamics. This theorem imposes the constraint on the fluctuation around the average, and the linear response theory as well as the Onsager’s reciprocal theorem can be derived from this theorem. Especially, the fluctuation phenomena in the mesoscopic systems are an ideal laboratory to study this theorem. As an application of the fluctuation theorem, the analysis of the counting statistics gives the relation between the nonlinear transport coefficient and noise in quantum transport of a mesoscopic conductor^{38}. Let I be the current and V the voltage between the electrodes. The nonlinear I–V characteristics can be written as
and the currentnoise S = <(δI)^{2}>/Δf (δI: the deviation of the current from the average, Δf: the frequency range) is expressed as
The fluctuationdissipation theorem in the linear response is S_{0} = 4k_{B}TG_{1}, which determines the Johnson–Nyquist noise S_{0} in thermal equilibrium. The generalization to nonlinear response reads S_{1} = 2k_{B}TG_{2}. The analysis including the magnetic field B as well as its experimental test has been done, and the readers would refer to the literature^{38,39}. Fluctuation theorems can provide the basis of the nonlinear responses beyond the Onsager’s reciprocal theorem, but their applications to the bulk transport phenomena are not well explored as yet.
Unidirectional magnetoresistance
The directional nonlinear dc resistance has been discussed by Rikken^{40}, who gave a heuristic argument generalizing the Onsager’s reciprocal theorem into the nonlinear regime, and suggested Eq. (B1) in Box 1 for the current (I)dependent resistivity R, where the second term with coefficient β represents the usual magnetoresistance, while the third term with coefficient γ the directional resistance, i.e., magnetochiral anisotropy. Although Eq. (B1) does not describe the vector nature of the current and magnetic field, there are two types of magnetochiral anisotropy according to the crystal structures with broken inversion symmetry. One is the polar structure where the directions of the polarization P, magnetic field B, and the current I (electric field E) are orthogonal to each other; R = R_{0}[1+γ′(P×B)·I]. The upper part of Fig. 3 presents the list of examples, i.e., Si FET^{41}, magnetic bilayer^{42}, BiTeBr^{43}, and surface state of TI^{42}, in this class. One typical example is the rectification effect in BiTeX (X = I, Br) in which Bi, Te, and X layers are stacking alternately so that the mirror symmetry along the c axis is broken, i.e., P//c. This material shows the giant spin splitting of the band structure due to the Rashba spin–orbit interaction^{44}. However, the timereversal symmetry leads to the relation
with k being the crystal momentum and σ is the spin component. Therefore, when the spin components are summed up as in the case of charge transport, the directional dependence disappears. Once the external magnetic field is applied in plane, e.g., along the yaxis, the energy dispersion becomes asymmetric between k_{x} and –k_{x} as shown in Fig. 4a, and I–V characteristic along adirection becomes nonreciprocal. Analysis in terms of the Boltzmann transport theory concludes that the coefficient γ in Eq. (B1) is independent of the lifetime of the electrons in the relaxation time approximation, similar to the case of Hall coefficient, and is an intrinsic quantity to the band structure. Therefore, one can predict quantitatively the magnitude of γ as a function of the electron density and temperature. Figure 4b shows the dependence of the nonlinear resistivity on the direction of the magnetic field in BiTeBr, which is consistent with the relation that R = R_{0}[1+γ′(P × B)·I]^{43}. Figure 4c presents the electron density dependence of γ measured by experiments and calculated theoretically without any fitting parameters since the band structure of this material has been already determined^{43}. Excellent agreement between theory and experiment indicates that the microscopic origin of the magnetochiral anisotropy in this material is the asymmetric deformation of the energy dispersion.
Another example of the (P × B) · Itype nonreciprocity is the case of the TI, as shown in Fig. 4d–f ^{45}. In the TI thin film (Bi, Sb)_{2}Te_{3} (BST), the upper layer part (denoted CBST) is doped with magnetic Cr ions to enhance the magnetic response as well as to introduce the asymmetry between the top and bottom surface layers where the conelike dispersion of the conduction electron is formed with strong spinmomentum locking (Fig. 4d). When the magnetic field B is applied along inplane and perpendicular to the current I direction, the large nonlinear component ΔR_{xx} shows up (Fig. 4e); note that the spontaneous magnetization is originally (at zero field) perpendicular to the plane but inclined to the inplane by the external inplane field. The important feature for ΔR_{xx} is its sign reversal upon the reversal of B and also upon the reversal of the stacking sequence structure of BST (nonmagnetic) and CBST (magnetic) layer; the latter procedure corresponds to the reversal of P, thus satisfying the condition of the magnetochiral anisotropy. Under the inplane B field, the Weyl cone should show a parallel shift in the kspace, yet this would give no effect on the nonlinear conduction if the quadratic term of the band dispersion could be neglected, being contrary to the case of the above example of the Rashba system. The unidirectional nonlinear magnetoresistance ΔR_{xx} observed in this semimagnetic TI system shows the characteristic magnetic field and temperature dependence; as increasing B to severaltesla region and as elevating temperature across the ferromagnetic ordering temperature, ΔR_{xx} shows a steep reduction. These observed features are all explained in terms of the relevance of the magnon excitations, i.e., their emission and absorption, in the originally forbidden backwardscattering process of the spinmomentumcoupled Weyl electrons (Fig. 4a). The application of B opens the spin wave gap, resulting in the reduction of the magnon scattering events at low temperatures. Furthermore, the tuning of the Fermi level around the Dirac (band crossing) point is found to enhance ΔR_{xx} because the magnon’s q vector and energy to mediate the backward scattering of the conduction electron can be small and more effectively thermally populated.
The other class of the materials showing unidirectional magnetoreistance is the chiral structure shown in the lower part of Fig. 3; in this case, the nonlinear nonreciprocal conduction is sometimes called the electrical magnetochiral effect in analogy to the abovedescribed optical magnetochiral effect. Examples of this class include Bi helix^{40,46}, molecular solid^{47}, and MiSi^{48}. The helix is a representative example of this class, and an early experiment on Bihelix wire found a finite γ value^{40}. The suggested mechanism of this effect is the magnetoresistance due to the magnetic induction b by the current I circulating along the helix, i.e., the effective magnetic field felt by the electrons is B + b, the magnitude of which depends on the direction of I. In this case, the nonlinear conduction is maximized when magnetic field B and the current I (electric field E) are parallel to each other, namely taking the form of B·I. A similar effect was also observed in a molecular semiconductor^{47}. Ferromagnets with chiral crystal structures are also expected to show the magnetochiral anisotropy.
Beyond the phenomenological or symmetry argument, there is known the case where a specific scattering mechanism can cause the electrical magnetochiral effect. In MnSi with the cubic chirallattice structure (P2_{1}3), in which spinhelical or skyrmion phase forms below T_{c} = 35 K because of Dzyaloshinskii–Moriya interaction inherent to the lattice chirality, the B·Itype nonlinear conduction is observed with the opposite sign for the two enantiomers^{48}. Symmetry allows the electrical magnetochiral effect in every T–B region. In reality, however, the effect becomes appreciable not in the helical magnetic order region but immediately above T_{c}, where the spin fluctuation is observed to retain the chiral character. This means that the scattering process of the conduction electron by the chiral spin fluctuation is proven to be the major source of the electrical magnetochiral effect in this case.
Nonreciprocal response in Weyl semimetals
Weyl semimetal is a class of materials where the Weyl fermions are at the Fermi energy typically described by the Hamiltonian^{49}
where σ = (σ^{x},σ^{y},σ^{z}) are Pauli matrices, v is the velocity, k_{0} is the band crossing point, and η = ±1 specifies the chirality. It has been known that the Weyl fermion acts as the magnetic monopole or antimonopole of the emergent magnetic field in the momentum space for η = +1 and η = −1, respectively. The Nielsen–Ninomiya theorem^{50} dictates the equal number of the Weyl fermions with opposite chiralities. One of the peculiar phenomena associated with Weyl fermions is the chiral anomaly, which is shared also by the Dirac fermion with 4 × 4 Hamiltonian. This is the phenomenon where the fermions are transferred between opposite chiralities in proportion to E·B under the external electromagnetic field. This results in the large negative magnetoresistance when the electric and magnetic fields are parallel or antiparallel to each other^{49,51}.
Note that, as for the Weyl fermions with 2 × 2 Hamiltonian, either the timereversal \(\hat T\) or the spaceinversion \(\hat I\) must be broken to lift the Kramer’s degeneracy except at k_{0}. Accordingly, there are two types of Weyl semimetals. One is the \(\hat T\)broken, but \(\hat I\)symmetric Weyl semimetal. \(\hat I\)symmetry relates the Weyl fermions at k_{0} and −k_{0}, and the chiralities of these are opposite to each other. On the other hand, in the \(\hat I\)symmetry broken Weyl semimetals, the Weyl fermions at k_{0} and −k_{0} have the same chirality. Therefore, according to the Nielsen–Ninomiya theorem^{50}, there must be at least another pair of Weyl fermions at k_{1} and −k_{1}. Namely, there are at least four Weyl fermions at ±k_{0} and ±k_{1}. The energy dispersions and also the Fermi energy shift at ±k_{0} and ±k_{1} are different and contribute differently to the conductivity. Therefore, once the external magnetic and electric fields shift the electrons in proportion to E·B between positive and negative chirality Weyl fermions due to the chiral anomaly, there occurs the current proportional to (E·B)E, which indicates the magnetochiral anisotropy^{52}. Because the Weyl fermions in \(\hat I\)broken system are solely due to the spin–orbit interaction, and also the Fermi energy shift from the Weyl point is usually small, the smallness of the perturbation discussed in the previous section is avoided, leading to the large nonreciprocal response. Quantitatively, γ value in Eq. (B1) can be four orders of magnitude larger than those discussed in the literature^{40}.
Nonreciprocal transport in a noncentrosymmetric superconductor
The magnetochiral anisotropy is usually a small effect because it requires both the spin–orbit interaction λ, which reflects the inversion symmetry breaking and the magnetic field B breaking the timereversal symmetry. For each perturbation, the energy denominator is typically the Fermi energy ε_{F}, and γ is expected to be proportional to (μ_{B}B/ε_{F})(λ/ε_{F}) and is usually very small. In the case of BiTeBr discussed above, the giant Rashba splitting leads to λ/ε_{F} ∼ 1 and the reasonably large γ. However, even larger γ is realized in the noncentrosymmetric superconductors^{53} as shown below. The physical picture for this enhancement is that the Cooper pairs with the large coherence length ξ feel the noncentrosymmetric nature of the potential sensitively in the low energy region below the superconducting gap Δ. In the I–V characteristics, the voltage is zero in the low current region. However, in 2D superconductors with an external magnetic field, the unpinned vortices produce the finite resistance and γ can be defined^{54}. In this case, the energy denominator in the expression of γ becomes Δ replacing ε_{F}, leading to the enhancement of γ by some power of the factor (ε_{F}/Δ). An example is the noncentrosymmetric superconductor MoS_{2} composed of the stacked layers with weak van der Waals interaction^{55}. The monolayer system is a 2D network with D_{3h} symmetry, leading to the trigonalwarping of the Fermi surfaces and outofplane spin polarization with effective Zeeman fields at Kpoints. Measurement of γ as a function of temperature shows its rapid growth below the temperature slightly higher than the mean field transition temperature T_{0}, the mean field transition temperature, reaching the value γ~8000 T^{−1}A^{−1} well below T_{0}^{56}.
Theoretical analysis has been restricted to the paraconductivity above T_{0}^{57}, concluding the enhancement of γ as \(\frac{{\gamma _S}}{{\gamma _N}} \sim \left( {\frac{{\varepsilon _F}}{{\mathrm{\Delta }}}} \right)^3\), and predicted γ ~400 T^{−1}A^{−1} . Although the analysis well below T_{0} has not yet been done, the rapid increase of γ observed experimentally is consistent with the theoretical prediction. The dynamics of vortices should be relevant to the nonreciprocal resistivity well below T_{0}, which is left for future studies.
Nonlinear/nonreciprocal photonic responses
Nonlinear optical effects in multiferroics
The nonlinear and nonreciprocal optical effects are also expected generically for the materials system with broken symmetries of spaceinversion and timereversal. One important source of such nonlinear optics (NLO) in the magnetic system is the toroidal moment T^{58} as defined by \({\mathbf{T}} = \frac{1}{2}\mathop {\sum }\limits_i {\mathbf{r}}_i \times {\mathbf{S}}_i\); here S_{i} is the spin moment on the site r_{i}. Since the effective spin–orbit coupling term in Hamiltonian is described as λL·S = λ(r × p)·S = −λ(r × S)·p = eA_{eff} ·p where e and p are electron’s charge and momentum, the toroidal moment T can be viewed as the builtin vector potential A_{eff} under the presence of spin–orbit interaction. Then T can mix with the ac vector potential A_{ω} from the light (frequency ω, i.e., T + A_{ω}). The compound may not show the secondorder nonlinear optical (NLO) effect responding to \(A_\omega ^2\), but generically does the thirdorder term responding to \(A_\omega ^3\). When the T is mixed in, this thirdorder term can effectively generate the secondorder response, i.e., responding to \(TA_\omega ^2\). Namely, when the light electric field E_{ω} (or A_{ω}) is parallel to T of the material, the effective secondorder NLO, typically secondharmonic generation (SHG with frequency of 2ω) is observed. Conversely, such magnetizationinduced SHG can be used as a probe for the toroidal moment.
Figure 5 exemplifies the toroidal momentinduced SHG in multiferroics^{59}. FeGaO_{3} (Fig. 5a) shows the polar structure along the b axis (P//b), while the magnetic moments on two Fe sites (Fe1 and Fe2) is antiferromagnetically coupled with their moments parallel to the c axis. Then Fe1 and Fe2 jointly form the toroidal moment T //a^{60}. The respective sublattice magnetization on Fe1 and Fe2 shows the unbalance, allowing the ferromagnetic moment M, which enables to control the sign of T //(P × M) by application of an external field. The SHG is originally active to produce E_{2ω}//b light along the polar axis, while it is also activated by the toroidal moment for E_{ω}//a. In the case of oblique incidence of the light with E_{ω}//a (spolarized) on the ac plane sample, both the SHG components (E_{2ω} //b from the original crystal polarity and E_{2ω} //a from the toroidal moment) can mix to generate the rotation of the polarization of the SHG light, as shown in Fig. 5b. This is called nonlinear Kerr rotation^{61} and the effect is usually much larger than the conventional Kerr rotation angle in ferromagnets. The reversal of either P or M can cause the sign reversal of the nonlinear Kerr rotation. The parallel component (say, xcomponent) of the oblique incident light kvector is either parallel or antiparallel to the applied magnetic field or magnetization direction (along x), and hence the reversal of k_{x} shows also the sign reversal of the nonlinear Kerr rotation. The phase (π) change of the E_{2ω} of the SH light upon the sign change of the toroidal moment enables one to image and map the toroidal moment domains in a multidomain state, as exemplified in Fig. 5c. In some multiferroics, the ferrotoroidic/antiferrotoroidic domains can be distinguished from ferromagnetic/antiferromagnetic domains, as exemplified for the case of multiferroic LiCoPO_{4}^{62}. This method has been also applied to probe the magnetization at the heterointerface with the effective polarity along the normal^{63,64}. It is also noted that the spin toroidization in periodic crystals has been formulated recently from the viewpoint of the Berry phase of Bloch wavefunctions^{65}.
Shift current in polar insulators
Up to now, we assume the broken timereversal symmetry to obtain finite nonreciprocal response, e.g., the current proportional to BE^{2}. However, one can ask if there is a possible mechanism of the current proportional E^{2} even without the broken \(\hat T\)symmetry. From this viewpoint, it is known that the photoexcitation produces the current even without the potential gradient, i.e., the external electric field, in noncentrosymmetric materials. This photocurrent is called shift current. Figure a of Box 2 shows the experiment on a perovskite ferroelectrics [KNbO_{3}]_{1–x}[BaNi_{1/2}Nb_{1/2}O_{3–δ}]_{x} (KBNNO), where the direction of the shift current is switched by that of the electric polarization^{66}.
Then, an important question is how the nonreciprocal nonlinear responses are possible without the broken \(\hat T\)symmetry? To answer to this question, one notes that the previous discussion was based on the energy dispersion satisfying the relation Eq. (7). Therefore, when the current does depend not only on the energy dispersion but also on the wavefunction, it is possible to realize the nonreciprocal responses without \(\hat T\)breaking. This means that one must go beyond the Boltzmann transport theory where only the energy dispersion ε_{n}(k) of the nth band and consequent group velocity appear in the equation. Namely, the group velocity v_{n}(k) = ∂ε_{n}(k)/∂k corresponds to the intraband matrix element of the current operator. On the other hand, the interband matrix elements of the current operator are related to the quantum geometry of the Bloch wavefunction and Berry phase^{67}, playing essential roles in the various quantum transport phenomena including quantum Hall effect, quantum charge pumping, anomalous Hall effect, and spin Hall effect^{68,69}. The Berry connection a_{n}(k) is defined as a_{n}(k) = i<u_{n}(k)∂/∂k u_{n}(k)> with u_{n}(k)> being the periodic part of the Bloch wavefunction. a_{n}(k) is related to the overlap of the two Bloch wavefunctions neighboring in the momentum space as \(< u_n\left( {\mathbf{k}} \right)u_n\left( {{\mathbf{k}} + {\mathrm{\Delta }}{\mathbf{k}}} \right) > = e^{{\mathrm{i\Delta }}{\mathbf{k}} \cdot {\mathbf{a}}_n\left( {\mathbf{k}} \right)}\), and has the geometrical meaning of “connection” of manifold in Hilbert space. Therefore a_{n}(k) is called Berry connection acting as the vector potential in the momentum space. From the vector potential, one can define the “magnetic field”, i.e., Berry curvature b_{n} (k) as b_{n}(k) = ∇_{k} × a_{n}(k). It is known that the Berry curvature causes the anomalous velocity of the electrons \({\mathbf{v}}_n^{{\mathrm{an}}}\left( {\mathbf{k}} \right) = {\mathbf{F}} \times {\mathbf{b}}_n\left( {\mathbf{k}} \right)\) under the force F acting on the electrons. This anomalous velocity is regarded as the origin of the intrinsic anomalous Hall effect in ferromagnets^{69}. When the symmetries \(\hat I\) and \(\hat T\) are valid, there occurs the Kramer’s doublet at each kpoint, and therefore the Berry phase should be defined as the 2 × 2 matrix, i.e., SU(2), instead of the U(1) phase as discussed before. In other words, the U(1) Berry curvature is zero in this case. On the other hand, when \(\hat I\)symmetry is broken, the Kramer’s degeneracy is lifted at each kpoint in the presence of the spin–orbit interaction, and U(1) Berry phase can be nonzero. Therefore, the noncentrosymmetry is encoded in the U(1) Berry phase. (Note that \(\hat T\)breaking also produces the Berry phase in the presence of the spin–orbit interaction.)
Since the realspace position r of the wavepacket made from the Bloch wavefunctions is given by the gauge covariant derivative as r_{n}(k) = i∇_{k} + a_{n}(k), a_{n} (k) has the meaning of intracell coordinate^{70}. The interband transition from nband to mband induced by the incident light results in the change of this intracell coordinate from a_{n} (k) to a_{m} (k) and the corresponding shift of the electron by the amount of r_{nm}(k) = ∇_{k}φ_{nm}(k) + a_{n}(k)–a_{m}(k) (shift vector) where the first term makes the r_{nm} gauge invariant with φ_{nm}(k) being the phase of the interband matrix element of the current operator. Therefore, in the steady state under the light irradiation, the continuous shift of the electrons by r_{nm} gives the dc current called shift current^{66,71,72,73,74,75}. This shift current has been estimated as a function of incident light energy for BaTiO_{3} by the firstprinciples calculation, and a good agreement was obtained between theory and experiment^{73}.
Shift current is essentially different from the conventional transport current in the sense that the former comes from the interband matrix elements of the current J, while the latter from the intraband ones^{71,72,73,74,75}. Therefore, it is analogous to the polarization current in ferroelectrics, but the remarkable feature is that it can be the dc current. The experimental results on tetrathiafulvalenepchloranil (TTFCA) which is a quasionedimensional compound consisting of mixed stacks of alternating donor (TTF) and acceptor (CA) molecules are shown in Box 2 ^{76}. Shift current is a rather ubiquitous phenomenon in noncentrosymmetric systems. Actually, shift current has been studied or discussed for GaAs^{77}, SbSI^{78}, and warped surface state of the threedimensional TI^{79}. Furthermore, the shift spin current can be also considered, which can have an application to spintronics^{79}.
Giant secondharmonic generation in Weyl semimetals
The shift current is the dc current induced by the optical excitation, and is the secondorder process where the current is proportional to E(ω)E(−ω). Similarly, the ac current with frequency 2ω proportional to E(ω)E(ω) can be produced, i.e., secondharmonic generation (SHG). It has been discussed that SHG has the similar expression in terms of the shift vector r_{nm}(k) = ∇_{k}φ_{nm}(k)+a_{n}(k)–a_{m}(k) defined above, and again has the geometrical meaning. Especially, the Weyl point corresponds to the magnetic (anti)monopole of the Berry connection^{49}, and hence can be a source of large SHG. Actually as shown in Fig. 6, in noncentrosymmetric Weyl semimetal TaAs, the large SHG has been observed with the magnitude of d≈3600 pmV^{−1} (d represents a component of the thirdrank tensor representing SHG). This large value is to be compared with corresponding values in other materials such as d≈350 pmV^{−1} for GaAs, d≈250 pmV^{−1} for ZnTe, and d≈15 pmV^{−1} for BaTiO_{3}^{80}. SHG of a model with two Weyl fermions has been analyzed, and reasonably consistent values with the experiment were obtained.
Nonlinear photocurrent induced by Berry curvature
The photocurrent discussed above is driven by the Berry connection a_{n}(k) rather than the Berry curvature b_{n}(k). Then it is natural to ask if there is a photocurrent induced by b_{n}(k). This question has been studied by Moore and Orenstein, who predicted the helicitydependent photocurrent due to Berry curvature^{81}. The idea is that the deviation δf_{n}(k,t) of the electron distribution function occurs in the presence of the electric field E and is proportional to τ(∂f(ε)/∂ε)v_{n}(k)E with being the relaxation time and v_{n}(k) the group velocity. The integral of the anomalous velocity \({\mathbf{v}}_n^{{\mathrm{an}}}\left( {\mathbf{k}} \right) =  e{\mathbf{E}} \times {\mathbf{b}}_n\left( {\mathbf{k}} \right)\) over the modified distribution function results in the photocurrent proportional to E^{2}. The photocurrent is induced by this mechanism for the quantum wells structure with the quantum confinement of electrons^{81}. A similar idea has also been applied to the nonlinear Hall effect in timereversal symmetric systems induced by Berry curvature dipole^{82}.
Perspective
In this article, we have seen ample examples of nonreciprocal responses from quantum materials with broken spaceinversion symmetry \(\hat I\). The directional transport/propagation of the quanta, such as electron, photon, spin, and phonon, enables the oneway transmission of the information carrier. As shown in Box 1, the diagonal linear nonreciprocal response is only allowed for the system with broken timereversal symmetry, namely in a magnetic field or in a magnetically ordered state such as the multiferroics. As the linear response, the nonreciprocal transport of magnon can substantiate the diode effect of spin current, which may append an important spintronic function. The directional dichroism response is also an important nonreciprocal function in the multiferroics that can support the ME excitations, typically electromagnons. The directional dichroism can realize the cloaking function; contrary to the conventional Faraday rotation, it does not need the additional analyzer of the light polarization to achieve the oneway transmission of light, if the effect can be tuned so as to nearly equalize the electricdipole and magneticdipole transition moments.
Let us turn our eyes to the nonreciprocal transport of an electron. The p–n junction (diode) is the most successful and spectacular example for this. Although we did not touch on this classic example in this article, it originates from the change in the width of the depletion layers at the junction, and can be attributed to the Coulombic interaction of electrons, i.e., the electron correlation effect. Recently, this issue has been revisited from the viewpoint of the multiband effect^{83}. As for the dc electric field, only the inclusion of the electron–electron interaction can produce the nonreciprocal current proportional to E^{2}, while the noninteracting electron system cannot.
The shift current is another example that does not need breaking of timereversal symmetry, but even the electron correlation is not required in this case. Its topological nature may ensure the ballistic photocurrent upon the photoexcitation. In addition to its importance as the initial process of the photovoltaic action, the shift current may also work as potentially ultrafast information transfer. As exemplified by the p–n junction and the shift current, the nonlinear nonreciprocity is the key to realize the actodc conversion. On this basis, the dc spin current generation is also possible by the ac (photo) excitation on the surface state of TI^{79}. In the magnetic system with broken timereversal symmetry \(\hat T\), an even more variety of the nonlinear and nonreciprocal current controls are possible, as described in this article for the nonlinear conduction for the Rashba system under a magnetic field, the semimagnetic TI, and the chirallattice magnet. There, one mechanism of unidirectional magnetoresistance is the field (magnetization)induced electronic structural change even with the constant scattering time as in the Rashba system, and the other is due to the carrier scattering itself, for example, the backward scattering via magnon emission/absorption in a TI and the directional scattering due to chiral spin fluctuation in the chirallattice magnet (MnSi). Thus, the study on the nonlinear and nonreciprocal charge transport can bring about the important information about the underlying charge dynamics in the simultaneously broken symmetries of timereversal and spaceinversion.
One interesting question is whether the nonreciprocal responses are possible when the product of the symmetries \(\hat I\hat T\) is intact while both \(\hat I\) and \(\hat T\) are broken. One can easily see that this product symmetry does not exclude the linear nonreciprocal responses. For example, the toroidal moment, which produces the directional dichroism, breaks both \(\hat I\) and \(\hat T\) symmetries, while it does not break \(\hat I\hat T.\). Of particular interest is the \(\hat I\hat T\)symmetry in nonHermitian systems. (Usually it is called PTsymmetry^{84}.) It has been found that PTsymmetry plays a crucial role in enhancing the optical isolator function^{85,86}. In addition to optics, the nonreciprocal propagation of heat is also a hot topic^{87}. Also the relation to the quantum Ratchet is an interesting problem to be explored more in depth^{88}. In this case, there is no \(\hat T\) symmetry breaking and the solitonlike nonlinear excitations are regarded as the source of the thermal rectification effect.
From the viewpoint of the basic principles, the noncentrosymmetry is encoded by the Berry phase, and the Bloch wavefunctions acquire the quantum geometric properties as discussed in section “Nonlinear/nonreciprocal photonic responses.” Berry phase is related to the interband matrix elements of the current operator, and produces new types of current in sharp contrast to the conventional transport current due to the intraband matrix elements which allows the particle picture through the wavepacket formalism. This “interband current” manifests itself as the shift current appearing as the photovoltaic effect. Furthermore, it should be noted that the electron correlation combined with the interband current matrix elements plays an essential role in the nonreciprocal responses. Therefore, the physics of nonreciprocal responses touch the most important elements of modern condensed matter physics, i.e., the symmetry, quantum geometry or topology, electron correlation, and also irreversibility. Reducing the symmetries in space and/or time in quantum materials have explored a fertile ground for condensed matter physics and electronics. The recent research of multiferroics exemplifies this. Exploring the nonreciprocal responses in quantum and topological materials as described here is a promising direction of the research in search for emergent functions.
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Acknowledgements
This work was partly supported by the Japan Society for the Promotion of Science through the Funding Program for WorldLeading Innovative R&D on Science and Technology (FIRST Program) on “Quantum Science on Strong Correlation” initiated by the Council for Science and Technology Policy and CREST, JST (Grant No. JPMJCR16F1).
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Tokura, Y., Nagaosa, N. Nonreciprocal responses from noncentrosymmetric quantum materials. Nat Commun 9, 3740 (2018). https://doi.org/10.1038/s41467018057594
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DOI: https://doi.org/10.1038/s41467018057594
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