Berry curvature induced magnetotransport in 3D noncentrosymmetric metals
BBerry curvature induced magnetotransport in 3D noncentrosymmetric metals
Ojasvi Pal, Bashab Dey and Tarun Kanti GhoshDepartment of Physics, Indian Institute of Technology-Kanpur, Kanpur-208 016, India
We study the magnetoelectric and magnetothermal transport properties of noncentrosymmetricmetals using semiclassical Boltzmann transport formalism by incorporating the effects of Berrycurvature and orbital magnetic moment. These effects impart quadratic- B dependence to the mag-netoelectric and magnetothermal conductivities, leading to intriguing phenomena such as planar Halleffect, negative magnetoresistance, planar Nernst effect and negative Seebeck effect. The transportcoefficients associated with these effects show the usual oscillatory behavior with respect to the anglebetween the applied electric field and magnetic field. The bands of noncentrosymmetric metals aresplit by Rashba spin-orbit coupling except at a band touching point. The difference in Fermi surfacetopology above and below the band touching point is reflected in the nature of magnetoresistanceand planar Hall conductivity. For Fermi energy below (above) the band touching point, giant (di-minished) negative magnetoresistance is observed. The absolute magnetoresistance and planar Hallconductivity show a decreasing (increasing) trend with Rashba coupling parameter for Fermi energybelow (above) the band touching point. I. INTRODUCTION
The interaction of electron’s spin with its motion,known as Spin-Orbit Coupling (SOC), has an exten-sive role in condensed matter physics. Spintronics emerges as a multidisciplinary field dealing with the ac-tive manipulation of spin degree of freedom as spin isa non-conserved quantity in spin-orbit coupled systems.The semiconductor heterojunction undergoes the break-ing of inversion symmetry due to the interfacial elec-tric field giving rise to Rashba spin-orbit interaction(RSOI) . The recent developments that capture RSOIacross various fields of physics and material science in-cludes spin Hall effect , topological insulator , spin-orbit torque , and spin galvanic effects - all cov-ering under the umbrella of emerging spin-orbitronics, abranch of spintronics that centers around the control ofnon-equilibrium material properties utilizing SOC .The scarcity of semiconductors having a large RSOIhinders the growth of spintronic devices in actual prac-tice. Later, various theoretical and experimental inves-tigations suggested that the systems like Bi/Ag(111) surface alloy, Bi Se , and 3D polar semiconductorBiTeX (X=Cl, Br, I) show stronger spin-orbit couplingthan conventional semiconductor heterostructures, whichenhances the spintronics applications. The potential can-didates in this category are bismuth tellurohalides hav-ing the trilayer structure with X-Bi-Te stacking. Thegiant RSOI appears in BiTeX compounds due tostructural asymmetry. Apart from BiTeX compounds,B20 compounds and noncentrosymmetric metals likeLi (Pd − x Pt x ) B show large RSOI as a result of inver-sion symmetry (IS) breaking. The broken IS also resultsin nontrivial Berry curvature (BC) in the system. Theelectrical and thermoelectric transport properties of thesystem in the absence of magnetic field are not affectedby BC and has been studied recently . However, in thepresence of weak magnetic field, the BC and orbital mag-netic moment (OMM) produces significant modification in the transport properties which is the prime focus ofour paper.The BC acts as a magnetic field in the momentumspace and leads to various interesting phenomena thatare typically absent in conventional condensed mattersystems. Some examples are anomalous Hall effect and anomalous Nernst effect which are predicted toexist in systems with broken time reversal symmetry.The presence of BC may also result in Chiral magnetic ef-fect (CME) in which equilibrium current is generated bymagnetic field without any external bias in the presenceof finite chemical potential . The existence of externalparallel electric and magnetic field results in semiclassicalmanifestation of the effect known as chiral anomaly .It is a purely quantum mechanical effect describing theanomalous nonconservation of a chiral current.To address the negative magnetoresistance (MR) inWeyl metal, a concept of weak anti-localization quan-tum corrections in the collision term and BC throughthe semiclassical equations of motion was included inthe Boltzmann transport framework . In topologicalsemimetals, a negative MR is explained as an effectof chiral anomaly. But in some systems like topologi-cal insulators, e.g., Bi Se , a chiral anomaly is not well-defined. In such cases, the negative MR is explained byanomalous velocity induced by BC and OMM . Usingsemiclassical equations of motion, one can observe thatvelocity contains an anomalous term due to the presenceof BC which is proportional to B in the linear-responselimit and in turn, a correction to the conductivity be-comes proportional to B . In this way, the BC resultsin negative MR while the OMM enhances it by modify-ing the band velocity through a Zeeman-like term in theenergy dispersion.One of the striking consequences of BC is the planarHall effect (PHE) . The PHE appears in a config-uration when the applied electric field, magnetic fieldand the induced voltage lie in the same plane such thatthe induced voltage is perpendicular to the electric field.When the electric field is applied along ˆ x direction and a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b the magnetic field is applied in the x - y plane making anangle with the x -axis, then the transverse conductivitymeasured along ˆ y defines the planar Hall conductivity(PHC).The BC driven transport also produces the Nernst ef-fect which describes the transverse electric response to athermal gradient. In conventional Nernst effect, the in-duced voltage, temperature gradient and magnetic fieldare mutually perpendicular to each other. An anomalousNernst effect demands the need of non-zero Berry curva-ture component parallel to the ( ˆ E × ∇ T ) plane. The pla-nar Nernst effect (PNE) occurs in a configuration wherethe temperature gradient, magnetic field and voltage areco-planar such that the induced voltage is perpendicu-lar to the temperature gradient and it is equivalent tothe transverse thermopower . A similar kind of phe-nomenon is known to exist in ferromagnetic systems .Motivated by the unconventional phenomena produceddue to BC, we calculate various magnetotransport coeffi-cients such as electrical conductivity, PHC, MR, Seebeckcoefficient (SC) and planar Nernst coefficient (PNC) ofnoncentrosymmetric metals. We have explicitly includedOMM in our calculations which arises due to the self-rotation of wave packet of the Bloch electron. .This work is organized as follows: In Sec. II, we presenta discussion on the low-energy band structure of noncen-trosymmetric metals. In Sec. III, we provide a review ofthe semiclassical Boltzmann transport formalism incor-porating the BC and OMM effects. We provide a generalexpressions of the magnetoconductivities in Sec. IV(A)and discuss the result of PHE and MR in Sec. IV(B).It is followed by the formalism of thermoelectric trans-port and general expression of thermoelectric coefficientsin Sec. V(A). The results of PNE and SC are given inSec. V(B). Finally, we conclude and summarize our mainresults in Sec. VI. II. LOW-ENERGY BAND STRUCTURE
The noncentrosymmetric metals such asLi (Pd − x Pt x ) B and B20 compounds exhibit cu-bic crystal structure. In such lattice geometry, theeffective low-energy Hamiltonian of conduction electronsbased on symmetry analysis is given by: H = (cid:126) k m ∗ σ + α σ · k . (1)Here, m ∗ is the effective mass of an electron, σ =( σ x , σ y , σ z ) is the vector consisting of the three Pauli ma-trices, σ is 2 × k is the electron wavevector with magnitude, k = q k x + k y + k z . The term α denotes the strength of Rashba spin-orbit interaction(RSOI). The Hamiltonian in Eq. (1) preserves the time-reversal symmetry (TRS) and breaks the inversion sym-metry. The energy dispersion is given by E kλ = (cid:126) k m ∗ + λαk, (2) FIG. 1: The sketch of low-energy band structure of noncen-trosymmetric metals: The two bands touch at k =0 whichis known as band touching point (BTP). The Fermi surfaceintersect both the bands when E F > E F < where λ = ± representing the two spin-split bands andtheir corresponding eigen states as ψ λ k ( r ) = u λ k e ι k · r / √ V with | u + k i = (cid:18) cos ( θ/ e ιφ sin ( θ/ (cid:19) , (3) | u − k i = (cid:18) sin ( θ/ − e ιφ cos ( θ/ (cid:19) . (4)The energy dispersion is depicted in Fig. 1. At k = 0,the two bands touch each other which is known as bandtouching point (BTP). The wave vectors correspondingto E > k λ = − λk α + q k α + 2 m ∗ E/ (cid:126) ,where k α = m ∗ α/ (cid:126) is the wave vector associated withthe SOI and the corresponding energy scale is E α = (cid:126) k α / m ∗ and k ± refers to the radii of two concentricspherical constant energy surfaces. It consists of both λ = − and λ = + bands and therefore contributes to thefollowing density of states (DOS): D >λ ( E ) = 14 π (cid:18) m ∗ (cid:126) (cid:19) / (cid:20) E + 2 E α √ E + E α − λ p E α (cid:21) . (5)For E <
0, only the energy band corresponding to λ = − exists with minima located at k = k α having E = E min = − E α . When energy lies in the range − E α < E <
0, theenergy surface will dissect the E − band into two spheri-cal shells of radii, k η = k α − ( − η − q k α + 2 m ∗ E/ (cid:126) ,where η = 1 , ≤ k ≤ k α and k α ≤ k ≤ k α ) contains the given DOS: D <η ( E ) = 14 π (cid:18) m ∗ (cid:126) (cid:19) / (cid:20) E + 2 E α √ E + E α − ( − η − p E α (cid:21) . (6)The BC can be calculated using Ω λ k = i ∇ k × ( h u λ k | ∇ k | u λ k i ) as Ω λ k = − λ k k . (7)The orbital magnetic moment (OMM) arises from theself-rotation of a wave packet formed by superposing theBloch states of a band. The OMM is calculated using m λ k = − ie/ (cid:126) h ∇ k u λ k | × ( H ( k ) − E ( k )) | ∇ k u λ k i to be m λ k = λαek (cid:126) Ω λ k = − αe k (cid:126) k . (8)The system has time-reversal invariance and broken spa-tial inversion symmetry, thus the BC satisfies Ω λ ( − k ) = − Ω λ ( k ). The OMM transforms exactly like BC undersymmetry operations. III. SEMICLASSICAL TRANSPORTFORMALISM
We use Boltzmann transport formalism to understandthe effects of Berry curvature and OMM on the electricaland thermoelectric transport properties of the system.The semiclassical Boltzmann approach works effectivelyunder the regime of low magnetic field and therefore, theformation of Landau levels is irrelevant in our case.The modified semiclassical equations of motion of aBloch wave packet (including the effects of BC andOMM) are given by ˙r λ = 1 D λ k (cid:20) ˜ v λ k + e (cid:126) ( E × Ω λ k ) + e (cid:126) (˜ v λ k · Ω λ k ) B (cid:21) , (9) (cid:126) ˙k λ = 1 D λ k (cid:20) − e E − e (˜ v λ k × B ) − e (cid:126) ( E · B ) Ω λ k (cid:21) , (10)where we have defined D λ k = (1 + e B · Ω λ k (cid:126) ). The phase-space volume is changed by a factor of [ D ( k )] − . Den-sity of states no longer remains constant for Berry phasemodified dynamics. In order to balance this changedphase-space volume, density of states is multiplied by[ D ( k )] such that the number of states in the volume ele-ment remains constant in time.The semiclassical band velocity is (cid:126) ˜ v λ k = ∇ k ˜ E λ k ,where ˜ E λ k = E λ k − m λ k · B is themodified energy including Zeeman-like coupling ofOMM with external magnetic field. The OMM modi-fied velocity is expressed as ˜ v λ k = v λ k + v m ,λ k , where v λ k isproportional to q E λ + E α ) m and (cid:126) v m ,λ k = − ∇ k ( m λ k · B ),which is dependent on the orientation of external mag-netic field.Equation (9) holds two important effects: the term( E × Ω λ k ) is responsible for the anomalous Hall effect (AHE) and the third term (˜ v λ k · Ω λ k ) B give rise tothe chiral magnetic effect . In Eq. (10), the first twoterms describes the Lorentz force whereas the last term( E · B ) Ω λ k denotes the chiral anomaly effect.The steady-state Boltzmann transport equation (BTE)to obtain the non-equilibrium distribution function ˜ f λ r , k is given as ˙r λ · ∇ r ˜ f λ r , k + ˙k λ · ∇ k ˜ f λ r , k = I coll { ˜ f λ r , k } . (11)The collision integral under the relaxation time approxi-mation can be written as I coll { ˜ f λ r , k } = − ˜ f λ r , k − ˜ f λ eq τ λ k , (12)where ˜ f λ eq = [1 + e β ( ˜ E λ k − µ ) ] − is the Fermi-Dirac distri-bution Function and β − ≡ k B T , µ denotes the thermalenergy and chemical potential respectively. The quantity τ λ k is the relaxation time. We consider the relaxation timeto be constant in our case. Thus, the BTE becomes ˙r λ · ∇ r ˜ f λ r , k + ˙k λ · ∇ k ˜ f λ r , k = − ˜ f λ r , k − ˜ f λ eq τ . (13)Within the regime of linear response theory, the non-equilibrium distribution function (NDF) is given as˜ f λ r , k = ˜ f λ eq + (cid:20) τD λ k (cid:18) − e E − ( ˜ E λ k − µ ) T ∇ r T (cid:19) · (cid:18) ˜ v λ k + e (cid:126) B (˜ v λ k · Ω λ k ) (cid:19) + ˜ v λ k · Γ λ (cid:21)(cid:18) − ∂ ˜ f λ eq ∂ ˜ E λ k (cid:19) . (14)The second term represents the deviation due to the scat-tering process and it contributes to the non-equilibriumpart of current induced by the electric field and statisticalforces like temperature gradient. The term (˜ v λ k · Γ λ ) inthe obtained NDF shows the contribution of the Lorentzforce . The main motivation behind our paper is tostudy the BC induced magnetotransport phenomena andtherefore, we have not included the Lorentz force contri-bution.The electric current upto first order in electric field andgradient in temperature is defined as j ei = σ ij E j + α ij ( −∇ j T ) , (15)where σ ij and α ij represents the elements of electricalconductivity matrix and thermoelectric conductivity ma-trix respectively. The current density is defined as j = − e X λ = ± Z [ d k ] D λ k (˙ r λ ) ˜ f λ r , k , (16)where [ d k ] = d k/ (2 π ) . But quantum mechanically, thecarrier is represented by the wave packet which has finitespread in phase space and size of the wave packet of aBloch electron has non-zero lower bound due to which itrotates about its center of mass position, which give riseto OMM. The total local current in the presence of finiteintrinsic OMM is given by j local = − e X λ = ± Z [ d k ] D λ k (˙ r λ ) ˜ f λ r , k + ∇ r × X λ = ± Z [ d k ] D λ k ( m λ k ) ˜ f λ eq . (17)The results for the electrical transport and the thermo-electric transport of this system are presented in the sub-sequent sections for the cases E F > E F < E F >
0, both the bands are included in thesummation over λ whereas for E F <
0, only the band cor-responding to the λ = − η ). For numerical cal-culation, we consider the following parameters: effectivemass of an electron m ∗ = 0.1 m e , m e is the free electronmass, α = 10 − eV-m, E F = 18.6 meV (for E F >
0) and E F = − E F < IV. ELECTRICAL TRANSPORT
A. Formalism
In a spatially uniform system, only the first term of Eq.(17) survives which is same as Eq. (16). SubstitutingEqs. (9) and (14) in Eq. (16), we obtained the Berrycurvature and OMM dependent electric conductivity tothe first order of the electric field as: σ ij = − e (cid:126) X λ = ± Z [ d k ] (cid:15) ijl (Ω λ,l k ) ˜ f λ eq + e τ X λ = ± Z [ d k ]( D λ k ) − (cid:18) ˜ v λi + e (cid:126) B i (˜ v λ k · Ω λ k ) (cid:19)(cid:18) ˜ v λj + e (cid:126) B j (˜ v λ k · Ω λ k ) (cid:19)(cid:18) − ∂ ˜ f λ eq ∂ ˜ E λ k (cid:19) , (18) where (cid:15) ijl is the Levi-Civita tensor and Ω λ,l k denotes the l -component of Berry curvature. The first term refers tothe intrinsic anomalous Hall effect which is independentof any scattering process (relaxation time). It arises dueto presence of the Berry curvature without any magneticfield.We can express the magnetoconductivity in termsof power of magnetic field by separating various B -contributions. Note that the distribution function ( ˜ f λ eq )contains B -dependence through modified energy ( ˜ E λ k ),hence the Fermi function is expanded in terms of B as˜ f λ eq = f λ eq + ( m λ k · B )( ∂f λ eq /∂E λ k ) with f λ eq is a distribu-tion function when B = 0 . We keep only first term ofexpansion in our calculations. Expanding σ ij in Eq. (18)as: σ ij = σ AHE ij + σ (0) ij + σ (1) ij + σ (2) ij , (19)where superscript indicates the order of magnetic field.The anomalous Hall conductivity is zero in our system.The diagonal component of conductivity without mag-netic field (known as Drude conductivity) is given by σ (0) ij = e τ X λ = ± Z [ d k ] v λi v λj − ∂f λ eq ∂E λ k ! . (20)The magnetoconductivity linear in magnetic field is givenas: σ (1) ij = e τ X λ = ± Z [ d k ] (cid:20) ( v λi v m,λj + v λj v m,λi ) − e (cid:126) ( B · Ω λ k ) v λi v λj + e (cid:126) ( v λi B j + v λj B i )( v λ k · Ω λ k ) (cid:21) − ∂f λ eq ∂E λ k ! . (21)The quadratic contribution of magnetic field to the con-ductivity as: σ (2) ij = e τ X λ = ± Z [ d k ] " e (cid:126) ( B · Ω λ k ) v λi v λj − e (cid:126) ( B · Ω λ k ) (cid:26) ( v λi v m,λj + v λj v m,λi ) + e (cid:126) ( v λi B j + v λj B i )( v λ k · Ω λ k ) (cid:27) + (cid:26) e (cid:126) ( v λi B j + v λj B i )( v m,λ k · Ω λ k ) + e (cid:126) ( v m,λi B j + v m,λj B i )( v λ k · Ω λ k ) + v m,λi v m,λj + e (cid:126) ( v λ k · Ω λ k ) B i B j (cid:27) − ∂f λ eq ∂E λ k ! . (22)We are working in the zero temperature limit and there-fore, while performing the calculations, a derivative ofthe Fermi-Dirac distribution function is substituted bythe Dirac-delta function. We consider the two orientations of magnetic field i.e., B ⊥ ˆ z and B k ˆ z . The analytical expressions of differentelements of the conductivity matrix is calculated for boththe cases. B. ResultsCase 1: B ⊥ ˆ z We consider the externally applied electric field is along x -direction and the magnetic field is applied along thein-plane direction at an angle β from the x axis, i.e., B = ( B cos β, B sin β, β = π/
2. The β 𝐵 𝑉 𝑥𝑦 𝐸 𝑥 x y z FIG. 2: Schematic illustration of the planar Hall effect ge-ometry. The electric field E is applied along the x -axis andthe magnetic field B is applied in the x - y plane at an angle β from the E . The planar Hall effect is measured as an in-planeinduced voltage ( V xy ) perpendicular to the direction of theexternal electric field. conductivity matrix equation takes the following form as J x J y J z = σ (0) xx + σ (2) xx σ (2) xy σ (2) yx σ (0) yy + σ (2) yy
00 0 σ (0) zz + σ (2) zz E x E y E z . (23)The magnetic field independent conductivity (Drude con-ductivity) is obtained using Eq. (20) as˜ σ (0) xx = ˜ σ D = (cid:18) √ π (cid:19)(cid:20) ( ˜ E F + 2) q ˜ E F + 1 (cid:21) , (24)and ˜ σ (0) yy =˜ σ (0) zz =˜ σ (0) xx . The analytical expression ofquadratic in-plane diagonal component of the magneto-conductivity is explicitly given by using Eq. (22) as˜ σ (2) xx = ˜ σ (2) ⊥ + ∆˜ σ cos β. (25)Here, ∆˜ σ = ˜ σ k − ˜ σ ⊥ with ˜ σ k = ˜ σ xx ( β = 0) = ˜ σ D + ˜ σ (2) k and ˜ σ ⊥ = ˜ σ xx ( β = π/
2) = ˜ σ D + ˜ σ (2) ⊥ , where˜ σ (2) k = (cid:18) √ B π (cid:19)" (cid:0) E F + 27 ˜ E F + 26 (cid:1) ˜ E F p ˜ E F + 1 , (26)˜ σ (2) ⊥ = (cid:18) √ B π (cid:19)" (cid:0) ˜ E F − ˜ E F + 2 (cid:1) ˜ E F p ˜ E F + 1 . (27)Now, as expected,˜ σ (2) yy ( β ) = ˜ σ (2) xx ( π/ − β ) . (28) β (degree) β (degree) (a) (b)(c) (d) B / B 𝐸 𝐹 > 0 𝐸 𝐹 <
0B / B β = 45˚β = 60˚ β = 20˚ β = 45˚β = 60˚ β = 20˚ (b)(c) (d) B = 5 TB = 3 T
B = 1 T
B = 5 T
B = 3 T
B = 1 T B = 5 TB = 3 T
B = 1 T (a) (b)
FIG. 3: Variation of the planar Hall conductivity as a functionof angle between E and B in the planar geometry for B = 5T when (a) E F >
0, (b) E F <
0. The B -dependence of PHCat β = π/ E F > E F < The zz-component of conductivity quadratic in B is givenby ˜ σ (2) zz = (cid:18) √ B π (cid:19)" (cid:0) ˜ E F − ˜ E F + 2 (cid:1) ˜ E F p ˜ E F + 1 . (29)We have defined ˜ E F = E F /E α , ˜ B = B/B with B = (cid:0) m α / e (cid:126) (cid:1) and ˜ σ = σ/ (cid:16) τe E / α m / (cid:126) (cid:17) as scaled Fermienergy, magnetic field and conductivity respectively. Thelinear dependence of conductivity in B is zero. Planar Hall effect : We are interested in the planarHall effect (PHE) observed in our case of planar geome-try. The PHE appears in a configuration when theapplied electric field, magnetic field and the induced volt-age are co-planar such that the induced voltage is per-pendicular to the electric field. The expression for planarHall conductivity (PHC) is˜ σ yx = (cid:18) √ B π (cid:19)" (cid:0) E F + 28 ˜ E F + 24 (cid:1) ˜ E F p ˜ E F + 1 (sin β cos β ) . (30)The planar Hall conductivity shows the sin β cos β depen-dence, whereas longitudinal magnetoconductivity (LMC)follows the dependence of cos β as shown in Eq. (25).The PHC attains the maximum value at an odd multi-ple of π/
4. The angular dependence of PHC is depictedin Fig. 3(a) and 3(b). The amplitude of PHC shows aquadratic dependence on B , i.e., ∆ σ ∝ B for any valueof β except for β = 0 and β = π/ B dependenceexcept at β = π/
2. We can also write PHC as:˜ σ yx = ∆˜ σ sin β cos β. (31)The PHC does not follow the anti-symmetry relation ofregular Hall conductivity. Thus ˜ σ xy = ˜ σ yx , as its originis associated with the Berry curvature term and not tothe Lorentz force. All the other off-diagonal componentsare zero. The expressions of magnetoconductivity for E F > E F < Magnetoresistance : To study the effect of magneticfield, we calculate the magnetoresistance which is definedas MR ii = ρ ii ( B ) − ρ (0) ρ (0) , (32)where i = x, y, z . The Drude resistivity is defined as M R β(degree ) β(degree)E F > 0 E F < 0 (a)(a) (b) FIG. 4: The angular dependence of magnetoresistance of non-centrosymmetric metals at B = 5 T for the planar geometry:(a) E F >
0, (b) E F <
0. In both (a) and (b), the red dashedcurve shows the planar MR (MR xx ( β )) and the blue line de-picts the out-of-plane MR (MR zz ). ρ (0) = 1 /σ D . The expression of planar MR (MR xx ( β ))is given by Eq. (A1). Here, the planar resistivity com-ponent ρ xx ( β ) is obtained by inverting the conductivitymatrix. We observed that the magnetoconductivity in-creases monotonically with the magnetic field and followsthe B dependence due to the presence of Berry curva-ture and OMM leading to the decrease in magnetorestiv-ity or negative MR. The planar MR [MR xx ( β )] followsthe angular dependence of cos β . The decrease in mag-netoresistivity is maximum at β = 0 and π and minimumat β = π/
2. In Fig. 4, we have plotted the variation ofplanar MR with the angle between the electric field andmagnetic field (red dashed curve). For E F >
0, when E F = 18 . B = 5 T, the negative MR resultingfrom the Berry curvature is about −
1% as shown in Fig.4(a) at β = 0 and π . When E F = − E F < −
50% and thus the Berry curvatureeffects on MR are considerably large as shown in Fig.4(b) at the same magnetic field and angles .The out-of-plane MR is denoted by MR zz and its ex-plicit expression is given by Eq. (A2). In Fig. 4, theblue solid lines show the variation of MR zz with the an-gle which appears to be constant as it is independent ofangle ( β ).The plots of variation of planar MR with the magneticfield for different angles between E and B are shown inFig. 5. For E F >
0, when
B/B = 0 .
4, the change inMR [MR xx ( β = 0) − MR xx ( β = π/ − .
2% as shown in Fig. 5(a).When E F <
0, the change in MR reaches about −
9% asshown in Fig. 5(b) at the same magnetic field. β = 90˚ β = 60˚ β = 30˚β = 0˚ β = 90˚β = 60˚β = 30˚β = 0˚ M R B/B B/B E F > 0 E F < 0 (a) (b) FIG. 5: The variation of magnetoresistance for the planargeometry with the magnetic field for different angles between E and B : (a) E F >
0, (b) E F < Dependence on Rashba coupling parameter ( α ) :The variation of magnetoresistance with α is plotted inFig. 6(a) and 6(b). For E F >
0, the absolute value ofMR increases with α whereas for E F <
0, it decreaseswith α . The dependence of planar Hall conductivity onRashba strength is depicted in Fig. 6(c) and 6(d). When E F >
0, the PHC increases with α whereas for E F < α . 𝐸 𝐹 > 0 𝐸 𝐹 < 0 (a) (a)(c) (d) ( S / m ) ( S / m ) FIG. 6: The variation of magnetoresistance with α at B = 5T and α = 10 − eV-m for the cases: (a) E F >
0, (b) E F <
0. The α -dependence of PHC for (c) E F > E F < The above results are obtained by neglecting theeffect of Lorentz force. When E and B are parallel, theLorentz force trivially vanishes and only the BC effectsprevail. When there is a finite angle between E and B ,the Lorentz force leads to the additional corrections inthe magnetoconductivities. However, it is known thatLorentz force does not contribute to the MR of singleband systems with parabolic dispersion . Since thedispersion in this system is approximately parabolic forlarge E F , the Lorentz force contribution in MR can beneglected. Case 2: B k ˆ z For the magnetic field along z -direction, i.e., B = B ˆ z .The conductivity matrix equation takes the following di-agonal form: J x J y J z = σ (0) xx + σ (2) xx σ (0) yy + σ (2) yy
00 0 σ (0) zz + σ (2) zz E x E y E z . (33)The diagonal component of conductivity quadratic in B is calculated to be˜ σ (2) xx = ˜ σ (2) yy = (cid:18) √ B π (cid:19)" (cid:0) ˜ E F − ˜ E F + 2 (cid:1) ˜ E F p ˜ E F + 1 , (34)˜ σ (2) zz = (cid:18) √ B π (cid:19)" (cid:0) E F + 27 ˜ E F + 26 (cid:1) ˜ E F p ˜ E F + 1 . (35) Magnetoresistance : The longitudinal MR (MR zz ) is M R B/B (a) (b) E F > 0 E F < 0 B/B (a) (b) FIG. 7: Plot of MR of noncentrosymmetric metals as a func-tion of magnetic field for the B k ˆ z case: (a) E F >
0, (b) E F <
0. The red dashed curve represents the longitudinalMR (MR zz ) and the blue curve depicts the perpendicular MR(MR xx ). given by Eq. (A3). And the expression of perpendicularMR (MR xx ) is obtained as Eq. (A4). In Fig. 7(a), when E F >
0, the negative longitudinal MR is about −
1% andthe perpendicular MR reaches around − .
04% at B =5 T. When E F <
0, the longitudinal MR reaches about −
50% and the perpendicular MR reaches about − V. THERMOELECTRIC TRANSPORT
A. Formalism
The magnetization current in Eq. (17) is not observ-able in conventional transport experiments, as it is lo-calized current calculated from the self-rotation of thewave packet. Thus, it does not contribute to transport.Therefore, the transport current is defined as j trans = j local − ∇ r × M ( r ) , (36)where M ( r ) is the total orbital magnetization in realspace. The transport current is related to the global motion of center of mass of the wave packet and con-tributes to the boundary current and is analogous to thefree current in the classical electrodynamics .The equilibrium magnetization density upto first orderin magnetic field at finite temperatures is defined as F = − β X λ = ± Z [ d k ] D λ k ln[1 + e − β ( ˜ E λ k − µ ) ] . (37)The magnetization for the given chemical potential andtemperature is given by M = − ( ∂F/∂ B ) µ,T and M ( r ) = X λ = ± Z [ d k ] D λ k ( m λ k ) ˜ f λ eq + 1 β X λ = ± Z [ d k ] (cid:18) e Ω λ k (cid:126) (cid:19) ln[1 + e − β ( ˜ E λ k − µ ) ] . (38)This is the general expression for equilibrium orbitalmagnetization density valid at non-zero magnetic fieldand arbitrary temperatures. Using Eqs. (17) and (38) inEq. (36), the transport current is given by j trans = − e X λ = ± Z [ d k ] D λ k (˙ r λ ) ˜ f λ r , k − ∇ r × β X λ = ± Z [ d k ] (cid:18) e Ω λ k (cid:126) (cid:19) ln[1 + e − β ( ˜ E λ k − µ ) ] . (39)The first term represents the usual charge current in-cluding the non-equilibrium correction to the first orderin gradient of temperature which leads to the result de-pending on relaxation process (depending on τ ). Thesecond term is the Berry phase correction to the magne-tization. It is also defined upto first order in the statisti-cal forces but independent of relaxation time and hencean intrinsic property of the system.The Berry curvature and OMM dependent thermoelec-tric conductivity matrix defined linear in ∇ T is obtainedas α ij = ek B (cid:126) X λ = ± Z [ d k ] (cid:15) ijl Ω λ,l k ξ λ k − eτ X λ = ± Z [ d k ]( D λ k ) − ( ˜ E λ k − µ ) T (cid:18) ˜ v λi + e (cid:126) B i (˜ v λ k · Ω λ k ) (cid:19)(cid:18) ˜ v λj + e (cid:126) B j (˜ v λ k · Ω λ k ) (cid:19)(cid:18) − ∂ ˜ f λ eq ∂ ˜ E λ k (cid:19) , (40)with ξ λ k defined as ξ λ k = β ( ˜ E λ k − µ ) ˜ f λ eq +ln[1+ e − β ( ˜ E λ k − µ ) ].The first term in Eq. (40) describes the purely anomalousthermoelectric effect in the absence of magnetic field.In the low-temperature limit ( k B T (cid:28) E F ), thethermoelectric conductivity can be expressed in terms ofelectrical conductivity using Mott relation given as α ij = − π k B T e (cid:18) ∂σ ij ∂E (cid:19) E = E F . (41) 𝑉 𝑥𝑦 x y z β 𝐵 T 𝛻𝑇 T FIG. 8: Schematic illustration for the measurement of the pla-nar Nernst coefficent in noncentrosymmetric metals. A tem-perature gradient dT /dx produces a BC induced transverseelectric field due to the co-planar component of the magneticfield.
Thermopower : The thermopower is defined for an opencircuit and therefore, we will keep electric current to bezero in Eq. (15) and thus, the temperature gradient gen-erates the electric field as E i = ν ij ( ∇ j T ) . (42)The thermopower matrix can be evaluated using ν = σ − α . The diagonal components ( ν ii ) denotes the con-ventional thermopower or Seebeck coefficient which de-scribes the electric response along the direction of tem-perature gradient. The off-diagonal components ( ν ij ) de- notes the Nernst coefficients (NC). An anomalous (con-ventional) Nernst effect measures the thermoelectric volt-age induced transverse to the temperature gradient inthe presence of out-of-plane Berry curvature (magneticfield). The planar Nernst effect occurs in the con-figuration when the temperature gradient, magnetic fieldand the induced voltage are co-planar such that the volt-age induced is transverse to the temperature gradient.The planar component of B due to BC gives rise to PNE(also known as transverse thermopower). B. ResultsCase 1: B ⊥ ˆ z The thermoelectric conductivity matrix takes the fol-lowing form α = α D + α (2) xx α (2) xy α (2) yx α D + α (2) yy
00 0 α D + α (2) zz . (43)The different elements of thermoelectric conductivity ma-trix can be obtained from the corresponding elements ofconductivity matrix, i.e., Eq. (23) using Eq. (41) as˜ α D = − √ E F + 4 q ( ˜ E F + 1) , (44)˜ α (2) xx = − √ B " − ˜ E F + 3 ˜ E F − E F − E F ˜ E F ( ˜ E F + 1) / ! sin β + − E F −
81 ˜ E F −
184 ˜ E F −
104 ˜ E F ˜ E F ( ˜ E F + 1) / ! cos β , (45)˜ α (2) yy ( β ) = ˜ α (2) xx ( π/ − β ) , and ˜ α (2) zz = − √ B − ˜ E F + 3 ˜ E F − E F − E F ˜ E F ( ˜ E F + 1) / ! , (46)˜ α (2) xy = ˜ α (2) yx = − √ B − E F −
84 ˜ E F −
176 ˜ E F −
96 ˜ E F ˜ E F ( ˜ E F + 1) / ! cos β sin β. (47)The thermopower matrix is given as: ν = ν D + ν (2) xx ν (2) xy ν (2) yx ν D + ν (2) yy
00 0 ν D + ν (2) zz , (48)where the various elements of thermopower matrix can be obtained using Eqs. (23) and (43) in Eq. (42) as˜ ν D = ˜ α D ˜ σ D = − π E F + 4( ˜ E F + 1)( ˜ E F + 2) ! . (49)We have defined ˜ α = α/ (cid:16) eτK B T E / α m / (cid:126) (cid:17) and ˜ ν = β (degree) β (degree)B/B B/B ν 𝑥𝑥 ν 𝑧𝑧 (a) (b)(c) (d) E F < 0E F > 0 ν 𝑥𝑥 ν 𝑧𝑧 FIG. 9: The angular dependence of the planar SC for B =5 T in the planar geometry when (a) E F >
0, (b) E F < E F > E F < ν/ (cid:16) K B TeE α (cid:17) as scaled thermoelectric conductivity andthermopower respectively.The SC in the planar configuration (˜ ν xx ( β )) is givenby Eq. (B1) with ˜ ν yy ( β ) = ˜ ν xx ( π/ − β ). The angulardependence of the planar SC is given by Fig. 9(a) and9(b). The out-of-plane SC ( ν zz ) is given by Eq. (B2).The longitudinal SC (˜ ν xx ( β = 0)) and the out-of-planeSC dependence on the magnetic field is depicted in Fig.9(c) and 9(d). The magnetic field reduces the Seebeckcoefficient in presence of Berry curvature, which resultsin negative Seebeck effect.The planar Nernst coefficient obtained for the planarconfiguration is given by Eq. (B3). The PNC shows thesame angular dependence of cos β sin β as in the case ofplanar Hall conductivity, therefore, it is finite in all direc-tions except at β = 0 and π/ π/
4. Fig. 10(a) and 10(b) showsthe dependence of PNC on the planar angle between E and B and the B -dependence of PNC is depicted in Fig.10(c) and 10(d). Case 2: B k ˆ z The thermoelectric conductivity matrix takes the fol-lowing diagonal form: α = α D + α (2) xx α D + α (2) yy
00 0 α D + α (2) zz , (50)where˜ α (2) xx = ˜ α (2) yy = − √ B " − ˜ E F + 3 ˜ E F − E F − E F ˜ E F ( ˜ E F + 1) / . (51) β = 45˚β = 60˚β = 20˚ β (degree) B/B B/B β (degree) E F > 0 E F < 0 (a) (b)(c) (d) β = 45˚β = 60˚β = 20˚ β = 45˚β = 60˚β = 20˚ (b) (c) (d) B = 5 T
B = 3 T
B = 1 T B = 5 TB = 3 TB = 1 T (a) (b)
FIG. 10: The angular dependence of planar NC with the anglebetween the E and B for the planar geometry: (a) E F > E F <
0. The plots (c) and (d) shows the variation of theplanar NC with the magnetic field for E F > E F < B/B B/B ν 𝑥𝑥 ν 𝑧𝑧 (a) (b) E F > 0 E F < 0 ν 𝑥𝑥 ν 𝑧𝑧 FIG. 11: Dependence of the SC as a function of magnetic fieldfor the B k ˆ z case: (a) E F >
0, (b) E F <
0. The blue curverepresents the longitudinal SC ( ν zz ) and the orange curvedepicts the perpendicular SC ( ν xx ). ˜ α (2) zz = − √ B " − E F −
81 ˜ E F −
184 ˜ E F −
104 ˜ E F ˜ E F ( ˜ E F + 1) / . (52)The thermopower matrix is obtained using Eqs. (33)and (50) in Eq. (42) as: ν = ν D + ν (2) xx ν D + ν (2) yy
00 0 ν D + ν (2) zz , (53)The thermopower matrix has no off-diagonal compo-nents in this parallel configuration, therefore no Nernstcoefficients. The diagonal components, i.e., the longitu-dinal SC (˜ ν zz ) and the perpendicular SC (˜ ν xx ) obtainedare given by Eqs. (B4) and (B5) respectively. VI. CONCLUSION
In this work, we have studied the magnetoelectric andmagnetothermal transport phenomena in noncentrosym-0metric metals using semiclassical Boltzmann transportformalism by incorporating the effects of BC and OMM.The OMM enters the energy-dispersion relation in theform of Zeeman-like coupling term which modifies thevelocity of the Bloch electrons. We have worked out themagnetoconductivity matrix for two orientations of mag-netic field with respect to the z -axis (representing theunit normal to the current-voltage plane) – B ⊥ ˆ z and B k ˆ z .We find that the magnetoconductivity increases mono-tonically with the magnetic field and follows the B de-pendence due to the presence of Berry curvature andOMM resulting in negative MR for both the orientations.For the case of B ⊥ ˆ z , the planar Hall effect is observedin the system. It is distinct from the Berry curvatureinduced anomalous Hall effect and the Lorentz force me-diated Hall effect as the transverse conductivities corre-sponding to these two effects are antisymmetric in spatialindices.The PHC and MR show the usual angular dependenceof sin β cos β and cos β respectively as shown in previ-ous works . For E F close to − E α , we get giant neg- ative magnetoresistance with maximum value of about −
50% (for E k B ) and minimum value of −
33% (for E ⊥ B ) for typical values of Rashba strength and when E F <
0, the maximum MR is about − B k ˆ z , the conductivity matrix takes the diagonal formwith quadratic- B dependence components implying theabsence of any Hall response. The absolute magnetore-sistance and planar Hall conductivity show a decreasing(increasing) trend with Rashba coupling parameter forFermi energy lesser (greater) than zero.The thermopower matrix is obtained from the conduc-tivity matrix using Mott relation. For the case of B ⊥ ˆ z ,we note that the PNE and negative Seebeck effect havesame angular dependence as that of PHC and MR re-spectively. When B k ˆ z , the conductivity matrix takesthe diagonal form with no Nernst coefficient. ACKNOWLEDGEMENTS
We would like to thank Sonu Verma and Pooja Kesarwanifor useful discussions.
Appendix A: Electrical transport
Here, we present the final expressions of magnetoresistance for the different orientations of magnetic field.For the case of B ⊥ ˆ z (planar geometry), the planar MR is calculated asMR = − (cid:20)
40 ˜ E F (cid:0) E F + ˜ E F (cid:1) (cid:26)
160 ˜ E F (cid:0) E F + ˜ E F (cid:1) + ˜ B (cid:0)
28 + 26 ˜ E F + 9 ˜ E F (cid:1) − ˜ B (cid:0)
24 + 28 ˜ E F + 7 ˜ E F (cid:1) cos 2 β (cid:27)(cid:21)(cid:30)(cid:20) E F (cid:0) E F + ˜ E F (cid:1) + ˜ B (cid:0)
52 + 28 ˜ E F + 15 ˜ E F + 19 ˜ E F + 8 ˜ E F (cid:1) + 80 ˜ B ˜ E F (cid:0)
56 + 136 ˜ E F + 124 ˜ E F + 53 ˜ E F + 9 ˜ E F (cid:1) (cid:21) . (A1)The out-of-plane MR is obtained asMR zz = − E F (1 + ˜ E F )(2 + ˜ E F )˜ B (cid:0) − ˜ E F + ˜ E F (cid:1) + 80 ˜ E F (cid:0) E F + ˜ E F (cid:1) . (A2)For the other case of B k ˆ z , we calculate the longitudinal MR to beMR zz = − E F (cid:0) E F + ˜ E F (cid:1)
80 ˜ E F (cid:0) E F + ˜ E F (cid:1) + ˜ B (cid:0)
26 + 27 ˜ E F + 8 ˜ E F (cid:1) . (A3)The perpendicular MR is given byMR xx = − E F (cid:0) E F + ˜ E F (cid:1) ˜ B (cid:0) − ˜ E F + ˜ E F (cid:1) + 80 ˜ E F (cid:0) E F + ˜ E F (cid:1) . (A4) Appendix B: Thermoelectric transport
In this Appendix, we provide the results of thermoelectric transport coefficients for the different configurations ofthe magnetic field.1For the case of B ⊥ ˆ z (planar geometry), the SC in the planar configuration is given by˜ ν xx = " π ( − E F (1 + ˜ E F ) (cid:0) E F + 3 ˜ E F (cid:1) −
40 ˜ B ˜ E F (1 + ˜ E F ) (cid:0) − −
80 ˜ E F + 9 ˜ E F (cid:1) + ˜ B (cid:0)
104 + 172 ˜ E F + 71 ˜ E F + 32 ˜ E F + 19 ˜ E F + 4 ˜ E F (cid:1) ! + 5 ˜ B ˜ E F (1 + ˜ E F ) (cid:20) ˜ B (cid:0) −
16 + 4 ˜ E F + 7 ˜ E F (cid:1) + 16 ˜ E F (cid:0)
96 + 272 ˜ E F + 264 ˜ E F + 105 ˜ E F + 14 ˜ E F (cid:1) (cid:21) cos 2 β ) E F (1 + ˜ E F ) (cid:26) E F (cid:0) E F + ˜ E F (cid:1) + ˜ B (cid:0)
52 + 28 ˜ E F + 15 ˜ E F + 19 ˜ E F + 8 ˜ E F (cid:1) + 80 ˜ B ˜ E F (cid:0)
56 + 136 ˜ E F + 124 ˜ E F + 53 ˜ E F + 9 ˜ E F (cid:1) (cid:27)(cid:21) . (B1)The out-of-plane SC is calculated as˜ ν zz = π (cid:2) −
80 ˜ E F (cid:0) E F + 3 ˜ E F (cid:1) + ˜ B (cid:0) E F − E F + ˜ E F (cid:1)(cid:3) E F (1 + ˜ E F ) (cid:2) ˜ B (cid:0) − ˜ E F + ˜ E F (cid:1) + 80 ˜ E F (cid:0) E F + ˜ E F (cid:1)(cid:3) . (B2)The coefficient of the planar Nernst effect is given by˜ ν (2) xy = 5 ˜ B π " ˜ B (cid:0) −
16 + 4 ˜ E F + 7 ˜ E F (cid:1) + 16 ˜ E F (cid:0)
96 + 272 ˜ E F + 264 ˜ E F + 105 ˜ E F + 14 ˜ E F (cid:1) sin 2 β (cid:30) " E F (cid:0) E F + ˜ E F (cid:1) + ˜ B (cid:0)
52 + 28 ˜ E F + 15 ˜ E F + 19 ˜ E F + 8 ˜ E F (cid:1) + 80 ˜ B ˜ E F (cid:0)
56 + 136 ˜ E F + 124 ˜ E F + 53 ˜ E F + 9 ˜ E F (cid:1) . (B3)For the other case of B k ˆ z , the longitudinal SC is calculated as˜ ν zz = π (cid:2) −
80 ˜ E F (cid:0) E F + 3 ˜ E F (cid:1) + ˜ B (cid:0)
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