Berry phase theory of planar Hall effect in topological insulators

The appearance of negative longitudinal magnetoresistance (LMR) in topological semimetals such as Weyl and Dirac semimetals is understood as an effect of chiral anomaly, whereas such an anomaly is not well-defined in topological insulators. Nevertheless, it has been shown recently in both theory and experiments that nontrivial Berry phase effects can give rise to negative LMR in topological insulators even in the absence of chiral anomaly. In this paper, we present a quasi-classical theory of another intriguing phenomenon in topological insulators – also ascribed to chiral anomaly in Weyl and Dirac semimetals– the so-called planar Hall effect (PHE). PHE implies the appearance of a transverse voltage in the plane of applied non-parallel electric and magnetic fields, in a configuration in which the conventional Hall effect vanishes. Starting from Boltzmann transport equations we derive the expressions for PHE and LMR in topological insulators in the bulk conduction limit, and show the important role played by orbital magnetic moment. Our theoretical results for magnetoconductance with non-parallel electric and magnetic fields predict detailed experimental signatures in topological insulators – specifically of planar Hall effect – that can be observed in experiments.

conductivity (PHC) is defined as the transverse conductivity measured along ŷ, in a direction perpendicular to the applied electric field and current along x, in the presence of a magnetic field in the x − y plane making an angle θ with the x axis. This effect is known to occur in ferromagnetic systems [33][34][35][36][37] where its origin is non-trivial spin topology. Interestingly, it has also been observed recently in the surface states of a topological insulator where it has been linked to magnetic field induced anisotropic lifting of the protection of the surface states from backscattering 38 . The main objective of our work is to suggest the existence of planar Hall effect, from the bulk states of 3D topological insulators, in systems exhibiting negative longitudinal magnetoresistance [8][9][10][11][12] . We use a semi-classical Boltzmann transport theory incorporating topological Berry phase effects to show this. In a complete theory we also derive the associated expressions for longitudinal magnetoconductivity in topological insulators in the bulk conduction limit as discussed previously 30 . We predict specific magnitudes and direction dependence of PHC and LMC on the applied fields that can be tested in experiments.
In this paper we have chosen Bi 2 Se 3 as a reference 3D strong topological insulator and study the PHC and LMC expected from its bulk states. This material has been clearly identified as a 3D strong topological insulator with a bulk band gap of 0.3 eV, with a single spin-helical Dirac cone on each surface, which has been confirmed in angle-resolved photoemission spectroscopy measurement 39 . Our work on planar Hall effect in this system, together with positive longitudinal magnetoconductance 30 , completes the quasi-classical description of Berry curvature induced anomalous magneto-transport phenomena in three dimensional topological insulators in the bulk conduction limit.
The rest of the paper is organized as follows. First, we introduce the effective Hamiltonian for the bulk states of a 3D strong topological insulator Bi 2 Se 3 . Next we derive the analytical expressions of LMC and PHC using semiclassical Boltzmann transport equations. Then we show our numerical results on LMC and PHC establishing the anomalous features in the transport properties. We also make comparison of our results with existing experimental data. Finally we discuss the experimental aspects of the phenomena observed in our study and end with a brief conclusion.

Results
Model Hamiltonian. The low-energy long-wavelength properties of a 3D topological insulator in the presence of time reversal and space inversion symmetries can be described by the effective k ⋅ p Hamiltonian. In the basis of (ψ v↑ , ψ v↓ , ψ c↑ , ψ c↓ ), where v and c denotes the valence and conduction band, the effective Hamiltonian can be written as and V i as model parameters. σ, τ are the Pauli matrices in spin and orbital space, respectively. In the present work we have taken C 0 = 0.048 eV, = .
V 2 512 eV-Å to represent the Bi 2 Se 3 topological insulator as suggested by the ab-initio bandstructure calculations 40,41 . The Hamiltonian described by the Eq. (1) includes uniaxial anisotropy along the z-direction and k-dependent mass terms. implies that this model belongs to a 3D strong topological insulator 42 . The 3D dispersion of the four bulk bands near the Γ point, two degenerate conduction bands and valencebands each, is depicted in Fig. 1(a). To compute the longitudinal conductivity and planar Hall conductivity in the bulk conduction limit we have assumed that the Fermi level crosses the conduction bands.
In addition to the band energy, the Berry curvature Ω(k) of the Bloch bands is required for a complete description of the electron dynamics in topological systems. The general form of the Berry curvature can be obtained via symmetry analysis. Under time reversal symmetry, the Berry curvature satisfies Ω(−k) = −Ω(k). On the other hand, if the system has spatial inversion symmetry, then Ω(−k) = Ω(k). Therefore, for 3D topological insualtors like Bi 2 Se 3 with simultaneous time reversal and spatial inversion symmetries, the Berry curvature vanishes identically throughout the Brillouin zone for both bulk conduction and surface bands 43 . However, in the presence of a magnetic field, the Zeeman splitting breaks the time reversal symmetry and generates non-trivial Berry curvature for these bands. Therefore, for discussing magneto-transport phenomena, which by defination are in the presence of magnetic field, the effect due to non-zero Berry curvature of the bulk band should be important for the electron dynamics.
To study the Berry curvature-induced magneto-transport phenomena, in the presence of an in-plane magnetic field, we add the Zeeman magnetic term  z to Eq. (1), where g B Here μ B is Bohr magneton and g pv , g pc are the Landé g factors for valence and conduction bands in the x − y plane respectively. The 2D dispersions of the valence and conduction bands of the topological insulator along k x , k y , and k z in the presence of an in-plane magnetic field of strength 5 T applied along x axis are shown in Fig. 1

Boltzmann Equation Approach
For Planar Hall effect. In this section, we derive the semiclassical formulae for the planar Hall conductivity and longitudinal electrical conductivity (LEC) in the low field regime starting from the quasi-classical Boltzmann transport equation. For completeness we include the effects of the orbital magnetic moment m, which is the angular momentum of the semi-classical wave packet and also of geometrical origin, modifying the expressions for LEC and PHC significantly. The complete theory produces the magnetic field and direction dependence of longitudinal magnetoconductivity and planar Hall conductivity in topological insulators that can be verified in experiments.
In the presence of an electric field (E) and temperature gradiant (∇T), the charge current (J) and the thermal current (Q) flowing through the system can be described by the linear response equations, where σ , α, and l are different conductivity tensors. The tensors α and α are related to each other by Onsager's relation α α =T . In the linear response theory, we can write J and Q as The phenomenological Boltzmann transport equation in the presence of impurity scattering can be written as 44 where on the right side C{ f k,r,t } is the collision integral which incorporates electron correlations and impurity scattering effects and f k,r,t is the electron distribution function. Using relaxation time approximation, the collision integral takes the form is the relaxation time and f eq is the equilibrium Fermi-Dirac distribution function in the absence of any external fields. In this paper we have ignored momentum dependence of τ and assume the parameter to be a constant in the semiclassical limit for simplifying the calculation 45 . Dropping the r dependence of f k,r,t , valid for spatially uniform fields, and assuming steady state the Boltzmann equation described by Eq. (6) takes the following form In the presence of electric field and magnetic field, transport properties get substantially modified due to presence of non-trivial Berry curvature which acts as a fictitious magnetic field in the momentum space 43 . The Berry curvature of the n th band for a Bloch Hamiltonian H(k), defined as the Berry phase per unit area in the k space, is given by Ω = ∇ × 〈 | ∇ | 〉 u i u n n n k k k k k . The wave packet of a Bloch electron also carries an orbital magnetic moment in addition to its spin moment due to the self rotation around its center of mass. The orbital magnetic moment associated with n th Bloch band can be defined as 46 It is clear from the above equation that the orbital magnetic moment does not depend on the actual shape and size of the wave packet but only depends on the Bloch functions. The orbital moment has exactly the same symmetry properties as the Berry curvature, namely, m(−k) = −m(k) and m(−k) = m(k) under time reversal and inversion symmetries, respectively. Therefore m(k) vanishes in the simultaneous presence of both these symmetries. In the present case, the orbital moment is non-zero because of broken time reversal symmetry due to the in-plane magnetic field.
As the orbital moment couples to the magnetic field (B) through a Zeeman-like term −m(k) ⋅ B, the unperturbed band energy ε 0,k is modified as ε k = ε 0,k − m(k) ⋅ B. In the presence of m(k) the group velocity of Bloch electrons is also modified as . Incorporating the effects due to Berry curvature and orbital magnetic moment, the semi-classical equation of motion for an electron takes the following form 46 where the second term of the Eq. (9) implies the anomalous velocity originating from the non-trivial Berry curvature. The coupled equations for  r and  k described in Eq. (9) and Eq. (10) can be solved together to obtain 47 , modifying the invariant phase space volume according to dkdx → D(B, Ω k )dkdx, gives rise to a noncommutative mechanical model, because the Poisson brackets of co-ordinates is nonzero 48 . For ease of notation we will simply denote D(B, Ω k ) by D for rest of the paper.
The second term of the Eq. (11) gives rise to the anomalous transport induced by the Berry curvature. The third term in the same equation gives rise to chiral magnetic effect modified by the orbital magnetic moment. The chiral magnetic effect, an interesting signature of transport phenomena in Weyl semimetals, appears in equilibrium i.e. E = 0 49-52 . This term implies an electric current ∝B to flow along the direction of the magnetic field in Weyl semimetals without any electric field in the presence of finite chiral chemical potential (μ + − μ − ) where μ + and μ − are the chemical potentials of two Weyl nodes 53 . There has been some controversy regarding the existence of the equilibrium chiral magnetic effect in condensed matter systems because the effect described above violates the Maxwell's equations 51,[54][55][56] . It has been discussed that in the dc limit i.e. when frequency is set to zero first, the system is in equilibrium and the chiral magnetic effect vanishes 51 . The second term in Eq. (12) implies the usual Lorentz force modified by m(k) whereas the last term proportional to E ⋅ B in Eq. (12) is the semi-classical manifestation of the topological effect known as chiral anomaly in the context of topological semimetals. The chiral anomaly in topological Weyl semimetals implies the non-conservation of a chiral current i.e. violation of the separate number conservation law of Weyl Fermions of a given chirality in the presence of parallel electric and magnetic fields. It is important to note that chiral anomaly is a purely quantum mechanical effect, and while the third term in Eq. (12) has been interpreted in the literature as the semi-classical manifestation of chiral anomaly in topological semimetals, the term itself may be non-zero in the presence of non-trivial Berry curvature even in systems that do not support chiral anomaly in the quantum limit.
To calculate planar Hall conductivity, we apply an electric field (E) along the x− axis and a magnetic field (B) in the x − y plane at a finite angle θ from the x − axis, i.e.
Here, θ is the angle between E and B as shown in Fig. 2 where  f eq is equilibrium Fermi-Dirac distribution with the energy dispersion ε ε = − ⋅ m B k k 0, modified due to the orbital magnetic moment.
Keeping only linear order dependence on the applied field E and B, we now assume following ansatz for the distribution function,  f k , which is a solution to the steady-state Boltzmann equation Eq. (7), can be written as Now, in the absence of any thermal gradient, we write the charge density (ρ) and current density (J) as 47 , 3 3 and the factor D arises from a field-induced change in the volume of the phase space. The second term of Eq. (16) is a contribution of magnetization current. As we are working with spatially uniform fields, in the present work the expression for the current density takes the following form, Plugging  f k into the above equation and comparing it with Eq. (4), we now arrive at the semiclassical formula for the longitudinal electrical conductivity including the effects due to Berry curvature and orbital magnetic moment, In the above equation the anomalous velocity factor eB k cos appears due to the topological term (E ⋅ B) and the orbital magnetic moment. This term is the origin of finite B-dependent longitudinal electrical conductivity which is independent of B for a regular Fermi liquid. When θ = 0, we recover the formula for LEC for parallel E and B fields as derived in earlier works 53,[57][58][59] . Now we will derive the expression of PHC. Inserting  f k in Eq. (17) and comparing it with Eq. (4), we write the following expression for the planar Hall conductivity,  and (19). Negative longitudinal magnetoresistance has recently been observed in several topological insulators in the presence of bulk conduction [8][9][10][11][12] . Although the planar Hall conductivity has recently been observed from the surface states of a 3D topological insulator 38 , it is not observed from bulk states till date. In the present work we consider only bulk states and neglect the contribution to conductivity from the surface states.
In the inset of Fig. 3(a) we have plotted the LMC as a function of the applied magnetic field at T = 24 K in the presence and absence of m where we have defined LMC as The LMC increases monotonically with the magnetic field in both cases and follows the B 2 -dependence. The orbital moment, first-order correction to the classical equations of motion, increases the Zeeman splitting between two conduction bands and enhances the LMC significantly. Therefore it is essential to take into account the effect of m in computing magnetoconductivity for topological insulators. Our results indicates the remarkable fact that the non trivial Berry curvature and orbital magnetic moment can produce an anisotropy in the magnetoconductivity even without the chiral anomaly effect. The LMC also follows cos 2 θ dependence at B = 5 T in both cases (presence and absence of m) as depicted in Fig. 3(b), leading to the anisotropic magnetoresistance (AMR). In Fig. 3(a and c) we have shown the amplitude and angular dependence of planar Hall conductivity in for the bulk states conduction in Bi 2 Se 3 . The amplitude of the PHC is finite at all field directions except at θ = 0 and θ = π/2 and follows a quadratic dependence on B which is similar to what has been observed in experiments on PHC on the surface states of Bi 2 Se 3 . The amplitude is enhanced significantly due to the presence of m leading to the fact that orbital moment plays a very important role in PHC. The planar Hall conductivity σ xy ph does not satisfy the familiar anti-symmetry relation (σ xy = −σ yx ) in the spatial indices and this property can be used to identify PHC in experiments. Within the regime of applicability of quasi-classical formalism, we have found that the PHC follows cosθ sinθ dependence for B = 5 T as depicted in Fig. 3(c). This is also similar what has been observed in experiments on PHC due to surface states of Bi 2 Se 3 38 .
In the presence of orbital magnetic moment, the LMC and PHC as a function of Fermi energy (E F ) for B = 5 T and T = 24 K are shown in Fig. 4(a). In experiment, the Fermi level can be tuned by gate voltage. It is clear from the figure that the LMC is enhanced as the Fermi level approaches the band bottom whereas the amplitude of PHC decreases. The temperature dependence of the LMC and PHC at B = 5 T in the presence of orbital magnetic moment is depicted in Fig. 4(b). It is clear from the figure that both LMC and PHC decrease with increasing temperature.

Conclusions
In this work we present a quasiclassical theory of planar Hall conductivity due to bulk conduction in 3D strong topological insulators such as Bi 2 Se 3 using the phenomenological Boltzmann transport theory. In the presence of bulk conduction, negative longitudinal magnetoresistance has recently been found in these systems [8][9][10][11][12] . Negative longitudinal magnetoresistance in topological semimetals such as Dirac and Weyl semimetals is typically associated with chiral or Adler-Bell-Jackiw anomaly [13][14][15][16][17][18][19][20][21][22] . It has recently been shown that this effect can occur also in 3D topological insulators in the presence of bulk conduction even in the absence of chiral anomaly 30 . In this paper we predict that TI systems supporting negative LMR in the presence of bulk conduction will also exhibit planar Hall effect when the electric and magnetic fields are not perfectly aligned with each other. Note that, a similar effect has recently been predicted also in Weyl semimetals from effects associated with chiral anomaly 31,32 . In the present work we show that such an effect exists also in 3D topological insulators due to Berry curvature of the conduction band even in the absence of chiral anomaly.
In the presence of in-plane electric and magnetic fields not perfectly aligned with each other, we find the non-zero planar Hall response in 3D strong TIs which is very different in nature from the usual Lorentz force mediated Hall response and even the Berry phase mediated anomalous Hall response, both of which are antisymmetric in spatial indices. We find that both longitudinal magnetoconductivity and planar Hall conductivity follow a quadratic dependence on B. Moreover, for a specific value of the magnetic field, the LMC follows cos 2 θ dependence whereas the PHC goes as B 2 cos θ sin θ, where θ is the angle between the applied E and B fields. Since in PHC, σ yx ~ cos θ sin θ where B makes an angle θ with the electric field E, which is taken parallel to the x axis, it follows that PHC is symmetric in the spatial indices, σ yx = σ xy . We find that, in the definition for LMC, Eq. (20), the numerator increases with increasing Fermi Energy E F , but the rate of increase of the Drude conductivity in the denominator is larger. Consequently LMC has defined in Eq. (20), decreases with E F while PHC increases with E F (Fig. 4a). We have derived an analytical expression for planar Hall conductivity taking into account the orbital magnetic moment along with the non trivial Berry curvature of the conduction band. It is clear from our results that orbital magnetic moment enhances the magnitude of both LMC and PHC and is important for their theoretical description. Our numerical results predict experimental observations of PHC together with LMC from the bulk states of 3D strong topological insulators which can be tested in experiments. ) in the presence of m for B = 5 T for the bulk states of Bi 2 Se 3 (Inset shows the temperature dependence of LMC (θ = 0)). The amplitude of PHC has been normalized by its maximum value in both figures.