Kerr effect in tilted nodal loop semimetals
Johan Ekström, Eddwi H. Hasdeo, M. Belén Farias, Thomas L. Schmidt
KKerr e ff ect in tilted nodal loop semimetals Johan Ekstr¨om, ∗ Eddwi H. Hasdeo,
1, 2
M. Bel´en Farias, and Thomas L. Schmidt † Physics and Materials Science Research Unit, University of Luxembourg, L-1511 Luxembourg Research Center for Physics, Indonesian Institute of Sciences, South Tangerang, Indonesia (Dated: February 8, 2021)We investigate the optical activity of tilted nodal loop semimetals. We calculate the full conductivity matrixfor a band structure containing a nodal loop with possible tilt in the x − y plane, which allows us to study the Kerrrotation and ellipticity both for a thin film and a bulk material. We find signatures in the Kerr signal that givedirect information about the tilt velocity and direction, the radius of the nodal loop and the internal chemicalpotential of the system. These findings should serve as guide to understanding optical measurements of nodalloop semimetals and as an additional tool to characterize them. I. INTRODUCTION
Topological phases in materials have attracted substantialinterest the last two decades, with the discovery of topologicalinsulators [1] at the starting point. As the years have passed,this research field has advanced significantly and we now havea large variety of topological phases at our disposal. Apartfrom topological insulators, this family of materials now in-cludes for instance Weyl and Dirac semimetals, as well asnodal line, loop and chain semimetals [2–4]. Similarly toDirac and Weyl semimetals, a nodal loop semimetal displaysband crossings in the energy spectrum. However, in contrastto Dirac and Weyl semimetals where those crossing occurs ata discrete set of points, the crossing points in this case forma continuous loop in the energy-momentum spectrum. Such anodal loop is depicted in Fig. 1 (top).There is by now a large range of materials that havebeen suggested to host nodal loops. Experimentally, thereis evidence that ZrSiS [5–7], PbTaSe [8], NbAs , [9] andYbMnSb [10] host nodal loops in their spectra. On the theo-retical side, Cu PdN [11], CaAgAs [12] and CaAgP [12, 13],among others, have been predicted to be nodal loop semimet-als. However, further experimental studies are required fortheir validation. Furthermore, experimental and theoreticalwork has been performed on ZrSiSe samples and it has beenshown that nodal loop semimetals may serve as a platformfor investigating strongly correlated phases in Dirac materials[14].An important experimental tool for studying materials andtheir properties is the magneto-optical Kerr e ff ect (MOKE).In such experiments, a linearly polarized light beam is di-rected towards a material surface and one measures the re-flected light. Depending on the properties of the material, thereflected light may pick up an orientation-dependent phasedi ff erence which can then result in an elliptical polarizationof the reflected light. The quantity representing this change inpolarization is the Kerr rotation.In recent years, theoretical works have reported large Kerrrotations for a variety of topological materials and havedemonstrated that it is possible to obtain information about ∗ [email protected] † [email protected] their properties from the Kerr rotations. In an early study,Tse et al. reported a giant Kerr angle in thin film topologicalinsulators [15, 16]. This prediction has since been extendedto other topological phases. Kerr rotations have been studiedin Refs. [17, 18] for di ff erent parameter regimes and it wasfound that Weyl semimetals show signatures of large Kerr ro-tations as well. Furthermore, a very recent paper by Parent etal. shows how the Kerr e ff ect in Weyl semimetals is a ff ectedby magnetic fields and demonstrates how a valley polarizationand the chiral anomaly can be observed in the Kerr angle [19].Some aspects of the optical responses of nodal loopsemimetals have already been reported in Refs. [20–23].However, Kerr and Faraday e ff ects have not been reported fornodal loop semimetals primarily because of a vanishing Hallresponse when the nodal loop is untilted. In this paper weinvestigate in particular how semimetals with a tilted nodalloop can be characterized by the Kerr e ff ect. For this pur-pose, we investigate the Kerr signal both of a thin film of sucha material as well as of the bulk material. Tilted nodal loopsemimetals have been observed in ZrSiS and ZrSiSe [5–7, 14]whose nodal loop energy form sinusoidal shapes. We expectthat tilted nodal line semimetals can be observed by break-ing time reversal symmetry (external magnetic field or inter-nal magnetization).Once tilt is introduced a Kerr signal is obtained and weshow how the information from this can give informationabout the tilt of the nodal loop in relation to the radius of thenodal loop. We further show how the Kerr signal depends onthe chemical potential of the system.The structure of this article is as follows: in Sec. II, wepresent the theoretical model we use to describe a semimetalwith a tilted nodal loop. In Sec. III we then derive its opticalconductivity tensor and show how its di ff erent componentscan be interpreted from a physical perspective. This quan-tity is of direct importance for determining the Kerr rotation.Thereafter, we present the general theory for obtaining theKerr rotation in Sec. IV, and discuss our findings for both athin film and the bulk geometry. We present our conclusionsin Sec. V. a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b -2 0 2-0.500.5 k x k y uθ FIG. 1.
Top:
A cut through the spectrum of nodal loop semimetal forparameters k = u = k z = Λ = Bottom left : Spectrumof a tilted nodal loop semimetal. The tilt shifts the nodal loop awayfrom energy E = Bottom right:
Tilt vector for a tilted nodal loopsemimetal.
II. MODEL
A nodal line in a band structure naturally emerges at theintersection between two parabolic band with opposite ori-entation. At low energies, this model captures the physicalproperties of a nodal loop semimetal and, taking into accounta possible tilt of the nodal loop, the corresponding Hamilto-nian for a nodal loop in the k x − k y plane is given by [23]ˆ H ( k ) = u · k τ + Λ (cid:16) k − k ρ (cid:17) τ x + v z k z τ z . (1)Here, we have defined k ρ = k x + k y , k is the radius of thenodal loop, τ x , y , z denotes the vector of Pauli matrices repre-senting two orbital degrees of freedom, v z is the Fermi veloc-ity in z direction, and τ is the identity matrix. The tilt velocity u = ( u x , u y , u z ) causes a tilt of the nodal loop and Λ is a massscale which determines the band curvature and depends onthe particular lattice realization used to derive the low-energyHamiltonian (1). Its spectrum is given by E k , ± = u · k ± (cid:114) v z k z + Λ ( k − k ρ ) . (2)This spectrum is plotted in Fig. 1 (top) for u =
0. A nodalring is located in the E = k . Figure 1 (bottom left) shows the e ff ect ofthe tilt. A nonzero tilt shifts the point on the nodal loop awayfrom E = u = ( u cos( θ ) , u sin( θ ) , u z ) , (3) where u = (cid:113) u x + u y and θ = arctan( u y / u x ). The angle θ represents the tilt-direction in the k x - k y plane and is indicatedin Fig. 1 (bottom right). Moreover, it is straightforward toobtain the following eigenfunctions for the Hamiltonian (1), Ψ ± = √ D ± (cid:32) − Bv z k z ∓ A , (cid:33) (4)where A = (cid:114) v z k z + Λ (cid:16) k − k ρ (cid:17) , B = Λ (cid:16) k − k ρ (cid:17) , D ± = | B | + ( v z k z ∓ A ) . (5)In the rest of the article, we will consider a nodal loop thatis tilted in the k x - k y -plane, so we set u z =
0. Furthermore,we will assume the zero-temperature limit when calculatingresponse functions. In the next section, we will present theoptical conductivity tensor for the tilted nodal loop system.
III. OPTICAL CONDUCTIVITY TENSOR
The coupling between light and matter can be describedwithin the electric dipole approximation, so we consider aHamiltonian ˆ H = ˆ H + E · r where E is the electric field and r is the position operator. To obtain the optical conductivity weapply the Kubo formula, which results in σ i j ( ω ) = π ) (cid:90) ∞ dk ρ k ρ (cid:90) π d φ (cid:90) dk z σ i j k ( ω ) . (6)where i , j ∈ { x , y , z } and the integral is over all momenta k ,written in cylindrical coordinates. Since the photon momen-tum is negligible compared to the electron momenta, only in-terband transitions need to be considered. The correspondingintegrand is then, σ i j k ( ω ) = − i (cid:88) s (cid:44) s (cid:48) f ( E k , s ) − f ( E k , s (cid:48) ) E k , s − E k , s (cid:48) j ss (cid:48) k i j s (cid:48) s k j ω + E k , s − E k , s (cid:48) + i + , (7)where f ( E ) denotes the Fermi distribution and j ss (cid:48) k i = (cid:104) Ψ s | ˆ j k i | Ψ s (cid:48) (cid:105) are the optical matrix elements, where s , s (cid:48) = ± corresponds to the two bands of our model. The i th compo-nent of the current operator is defined as ˆ j k i = ∇ k i ˆ H . Weobtain the following matrix elements, j ss (cid:48) k x = k x v z k z Λ A , (8) j ss (cid:48) k y = k y v z k z Λ A , (9) j ss (cid:48) k z = − v z | B | A , (10)for s (cid:44) s (cid:48) . Note that j ss (cid:48) k x is related to j ss (cid:48) k y by letting k x → k y .Hence, a symmetry is expected in quantities involving the x and y -components. The real and imaginary parts of Eq. (7) areobtained by using the identity lim γ → + ( x + i γ ) − = P x − i πδ ( x ),where P denotes the Cauchy principle value and δ ( x ) is theDirac delta function. We use this identity to obtain Re σ i j ( ω ).From the latter, the imaginary part is calculated by using theKramers-Kronig relation,Im σ i j ( ω ) = − π P (cid:90) ∞−∞ d ω (cid:48) Re σ i j ( ω (cid:48) ) ω (cid:48) − ω . (11)The above equations yield the principal ingredients for calcu-lating the full conductivity tensor. In the k ρ integral, we makethe change of variables k ρ → k ξ . Moreover, we introduce thefollowing dimensionless quantities,˜ ω = ωω , ˜ µ = µω ˜ u = Λ uk , Γ = k v z Λ , (12)where µ is the chemical potential and ω = k / Λ . A. Real part
The terms contributing to the real part of the optical con-ductivity can be collected in the following two integralsRe σ i j ( ω ) = (cid:90) k k d ξ Θ (1 − ˜ ω/ F i j ( ξ ) + (cid:90) k d ξ Θ ( ˜ ω/ − F i j ( ξ ) , (13)where F i j ( ξ ) = g + i j ( ξ ) − g − i j ( ξ ), i , j ∈ { x , y , z } , and Θ ( x ) de-notes the Heaviside step function. The functions f ± i j ( ξ ) to beintegrated are given in the Appendix in Eqs. (A2)-(A5). Fur-thermore, we used k = √ − ˜ ω/ k = √ + ˜ ω/
2. For avanishing tilt velocity (˜ u = ω (cid:28) ω < ˜ u and ˜ µ = ω , the followingresults for the real parts of the longitudinal conductivities σ ii and the Hall conductivity σ xy ,Re σ xx ( ω ) ≈ Γ π ˜ u (1 − cos 2 θ ) ˜ ω, (14)Re σ yy ( ω ) ≈ Γ π ˜ u (1 + cos 2 θ ) ˜ ω, (15)Re σ zz ( ω ) ≈ π ˜ u Γ ˜ ω, (16)Re σ xy ( ω ) ≈ Γ sin 2 θ π ˜ u ˜ ω. (17)The Hall conductivities along the other transverse directions, σ xz and σ yz , vanish because of our assumption that the tilt is inthe k x - k y plane ( u z = ω . This correction becomes important only when θ ≈ n π/ n ∈ Z ) where the linear-order term in σ xx and σ yy k ρ E − k k ˜ ω IV ˜ ω II ˜ ω III ˜ ω I ( e ) k ρ E − k k ˜ ω II ˜ ω I ˜ ω III ˜ ω IV ( f ) FIG. 2. Real part of the optical conductivity for varying ˜ u and ˜ µ . (a)˜ u = . µ =
0, (b) ˜ u = . µ =
0, (c) ˜ u = . µ = . u = . µ = .
15. For all plots
Γ =
1. The vertical lines showthe di ff erent transition thresholds as specified in the text and depictedin (e) and (f). (e) Possible transitions along the nodal loop for ˜ µ < ˜ u and (f) possible transitions for ˜ µ > ˜ u . can vanish. For θ =
0, our results reproduce the known scalingof the longitudinal conductivities with energy, i.e., σ xx ∝ ω and σ yy ∝ ω [21]. For general parameters, we have solved theintegrals numerically and obtained the results shown in Fig. 2.The analytical approximations agree with the numerical cal-culations within the valid approximations.From the analytical expressions we see that the optical con-ductivity depends strongly on the tilt angle θ . This applies inparticular to σ xy which vanishes for θ = n π/ n ∈ Z .At these tilt angles, the electric field excites an equal flowof electrons in opposite directions which hence cancel eachother. This is also observed for the dc Hall conductivity [23].Furthermore, an untilted nodal loop (˜ u =
0) does not give riseto a Hall conductance, as was already shown for the dc Hallconductivity in Ref. [23]. For the ac Hall conductivity this fol-lows from the fact that the integral over the angular coordinatein Eq. (6) separates and vanishes as (cid:82) π d φ sin φ cos φ =
0, seealso the matrix elements defined in Eqs. (8) and (9).Secondly, we observe that σ xy reaches its maximum for θ = n π/
4, in which case we also obtain σ xx = σ yy . To bespecific, let us discuss the case θ = π/
4. The numerical resultsare plotted in Fig. 2. Note that in all cases we are considering | ˜ u | , | ˜ µ | <
1. For the ensuing discussion we assume ˜ u , ˜ µ ≥ µ < ˜ u , one can define the following thresholdenergies, ˜ ω I = − µ − ˜ u + ˜ u (cid:113) ˜ u + + ˜ µ ) , ˜ ω II = µ − ˜ u + ˜ u (cid:113) ˜ u + − ˜ µ ) , ˜ ω III = − µ + ˜ u + ˜ u (cid:113) ˜ u + − ˜ µ ) , ˜ ω IV = µ + ˜ u + ˜ u (cid:113) ˜ u + + ˜ µ ) . (18)The thresholds are ordered such that ˜ ω I < ˜ ω II < ˜ ω III < ˜ ω IV . Inthe case ˜ µ = ω I = ˜ ω II and ˜ ω III = ˜ ω IV . The thresholdenergies mark the onset of allowed vertical transitions aroundthe nodal loop. As long as ˜ ω < ˜ ω IV the system is partiallyPauli blocked. As the energy of the incoming photons in-creases, more vertical transitions become allowed around thenodal loop. Once the energy of an incoming photon is largerthan ˜ ω IV vertical transitions are allowed everywhere on thenodal loop and the Pauli blockade has been lifted. The absorp-tion processes defining the threshold energies are depicted inthe graph in Fig. 2(e) and are indicated in the numerical plotsin Fig. 2(a-c).In the case ˜ u < ˜ µ we find that the energy thresholds aregiven by ˜ ω (cid:48) I = µ + ˜ u − ˜ u (cid:113) ˜ u + + ˜ µ ) , ˜ ω (cid:48) II = µ − ˜ u − ˜ u (cid:113) ˜ u + − ˜ µ ) , ˜ ω (cid:48) III = µ − ˜ u + ˜ u (cid:113) ˜ u + − ˜ µ ) , ˜ ω (cid:48) IV = µ + ˜ u + ˜ u (cid:113) ˜ u + + ˜ µ ) . (19)The corresponding allowed transitions are depicted inFig. 2(f). Going from ˜ u > ˜ µ to ˜ u < ˜ µ changes the allowed ver-tical transitions around the nodal loop. As can be seen fromFig. 2(d), for ω < ˜ ω (cid:48) I the system is Pauli blocked and no ver-tical transitions are allowed. Once ω > ˜ ω (cid:48) I the Pauli blockadeis overcome and transitions are partially allowed around thenodal loop. We further note the impact of ˜ u < ˜ µ has on σ xy . Itnow takes both positive and negative values.Finally, we note that for ˜ ω > ˜ ω IV , ˜ ω (cid:48) IV , transitions fromthe valence to the conduction band are allowed all around thenodal loop and the response of the system then resembles thatof an untilted nodal loop. As has been previously observed,the conductivity in this region reaches a constant value [20,21]. B. Imaginary part
To obtain the imaginary part of σ i j ( ω ), we apply theKramers-Kronig relation (11) to the results obtained for thereal part of the conductivity tensor. The integration is donenumerically. A cuto ff ω c has to be introduced to regularize alogarithmic divergence in the unbounded integral. The neces-sity of a cuto ff is not surprising because the Kramers-Kronigrelation involves an infinite integration range whereas the ef-fective model we consider is only valid for small energies. We FIG. 3. Imaginary part of the optical conductivity for varying ˜ u and˜ µ . (a) ˜ u = . µ =
0, (b) ˜ u = . µ =
0, (c) ˜ u = . µ = .
05, (d) ˜ u = . µ = .
15. For all plots
Γ = have chosen ω c = k / Λ and have verified that a larger cut-o ff energy has no pronounced e ff ect on the low-energy conductiv-ity.The imaginary part of the conductivity tensor is plotted inFig. 3. For zero chemical potential the imaginary parts arenegative for small ˜ ω , and σ xx , yy remains so even for larger fre-quencies. On the contrary, the imaginary part of σ xy increasesand takes on positive values for increasing ˜ ω . Thereafter itdecreases towards zero.To summarize this section, we have calculated the fullfrequency-dependent conductivity tensor for a nodal loopsemimetal tilted in the k x - k y plane. In the following, we willuse this quantity for the calculation of the Kerr response. IV. MAGNETO OPTICAL KERR EFFECT
The magneto-optical Kerr e ff ect acts as an optical tool forcharacterizing and understanding di ff erent materials. Whenan incident electromagnetic wave is reflected from the surfaceof a material, the reflected wave may pick up a polarization-dependent phase, thus corresponding to a change in the polar-ization angle as well as the ellipticity of the reflected wave.This phenomenon is referred to as the Kerr rotation and is de-picted in Fig. 4.The Kerr rotation is governed by the properties of the mate-rial, which are represented in Maxwell’s equations describingthe propagation of light in the vacuum and inside the mate-rial. We will consider the Kerr e ff ect when a linearly polar-ized electromagnetic wave with normal incidence is reflectedon the surface of a nodal loop semimetal. Both a free standingthin-film and a semi-infinite bulk material will be considered.The general theory is outlined in the following paragraphs,whereas the specific details that have to be applied for the twodi ff erent geometries as well as the results will be presented inthe following sections. E θ K n n n FIG. 4. Schematic picture illustrating the magneto-optical Kerr ef-fect. A linearly polarized incident light beam is reflected with a el-liptic polarization. The latter is characterized by the Kerr angle θ K and the ellipticity angle (cid:15) K . A linearly polarized incident wave can be represented asan equal superposition of two circularly polarized waves. Wewill limit ourselves to the case of normal incidence, so weconsider a wave travelling along the z direction. In thiscase, we can write a circularly polarized wave as E R , L = E R , L ˆ e R , L e i ( k · r − ω t ) , where ˆ e R , L = ˆ x ± i ˆ y and k = k z ˆ z . The basisvectors ˆ x and ˆ y are in the plane perpendicular to direction ofthe wave propagation and the plus and minus sign correspondto right-handed ( R ) and left-handed ( L ) circularly polarizedlight, respectively.The Kerr and ellipticity angles are now obtained by consid-ering the quotient between the reflection amplitudes of a right-and a left-handed circularly polarized beam with the complexamplitudes E Rr and E Lr [15, 24]. This complex quotient hasa magnitude and a phase, E Rr / E Lr = | E Rr | / | E Lr | e i ( α R − α L ) . Thephase and the magnitude define, respectively, the Kerr and theellipticity angles, θ K =
12 ( α R − α L ) , (20) (cid:15) K = | E Rr || E Lr | . (21)We will now explain how to calculate the reflection ampli-tudes for a light beam incident on a thin film and a semi-infinite bulk material. A. Thin film
The boundary conditions applied to Maxwell’s equationson the two sides of a thin metallic film state that the elec-tric field components parallel to the surface are continuousacross the boundary, while there is a discontinuity in the mag-netic field equal to the generated current [25]. In mathematicalterms they are written as [26]
FIG. 5. (a,b):
Kerr angle θ K for a thin film. (c,d): Kerr ellipticity (cid:15) K for a thin film. For (a) and (c) we fix the chemical potential at ˜ µ = ω = u . For(b) and (d) we fix the tilt velocity at ˜ u = . E (cid:107) = E (cid:107) , (22)ˆ n × B (cid:107) µ − B (cid:107) µ = π c J , (23)Here, E (cid:107) i (for the two regions i = ,
2) are the components ofthe electric field vector inside the surface plane above and be-low the film, respectively. Moreover, ˆ n is the surface normalvector pointing from medium 1 to medium 2. The magneticfield is given by B = c ω k × E and µ i is the magnetic perme-ability of medium i . Since we consider vacuum on both sidesof the film, we set µ i =
1. Finally the current density is givenby J = σ S E , where the surface conductivity tensor can be ap-proximated by the bulk conductivity and the film thickness, d such that σ Si j = d σ i j [17].We consider an incoming wave E = E e R + ˆ e L ) e i ( k z z − ω t ) , (24)which describes an electromagnetic wave propagating alongthe k z direction and linearly polarized along the x -axis (heredecomposed into two circularly polarized waves with left- andright-handed polarization) and with an amplitude E . In thiscase, the reflected (subscript r ) and transmitted ( t ) waves, re- FIG. 6. Real (a) and imaginary (b) parts of the components of thedielectric function for θ = π/ spectively, are given by E r = (cid:16) E Rr ˆ e (cid:48) R + E Lr ˆ e (cid:48) L (cid:17) e − i ( k z z + ω t ) , (25) E t = (cid:16) E Rt ˆ e R + E Lt ˆ e L (cid:17) e i ( k z z − ω t ) , (26)where ˆ e (cid:48) R / L = ˆx ∓ i ˆy is the circularly polarized basis for thereflected light. Calculating the corresponding magnetic fields,inserting the fields into the boundary conditions (22) and (23)and using that σ yx = − σ xy we obtain a system of equationsfrom which we can solve for E Rr and E Lr . We obtain E R , Lr = E C (cid:104)(cid:0) κ − σ ± (cid:1) σ Sxx ± i (cid:0) κ − σ ± (cid:1) σ Sxy (cid:105) , (27)where C = (cid:0) κ − σ − (cid:1) (cid:0) κ − σ + (cid:1) + (cid:0) κ − σ + (cid:1) (cid:0) κ − σ − (cid:1) ,σ ± = σ Sxx ± i σ Sxy ,σ ± = σ Syy ± i σ Sxy , (28)and κ = / (4 πα ) with the fine structure constant α ≈ / θ = π/ ω = u .This is the point where the real part of the transverse con-ductivity vanishes and the imaginary part reaches its maxi-mum. Hence, a finite tilt of the nodal line has a strong e ff ecton the Kerr signal. By increasing the chemical potential, seeFigs. 5(b,d), we notice a second peak emerging in the Kerr sig-nals. For small frequencies a very large Kerr angle is observedfor a chemical potential corresponding to ˜ µ = ˜ u . B. Bulk material
Next, we consider the Kerr reflection on the surface of abulk material. Compared to the thin film we now have toconsider the propagation of the electromagnetic waves insidethe material. Hence, it is necessary to find the allowed wavevectors inside the bulk material and this is done by solving the electromagnetic wave equation obtained from Maxwell’sequations. The Maxwell-Faraday and Ampere-Maxwell equa-tions are respectively given by ∇ × E = − c ∂ B ∂ t , (29) ∇ × H = c ∂ D ∂ t + π c J . (30)We are assuming that the material is non-magnetic and hence B = H . The constitutive relations further tells us that D = (cid:15) b E ,where (cid:15) b is the relative permittivity of the nodal loop material.Ohm’s law states that J = σ E . We take the curl of Eq. (29)and insert Eq. (30) along with the constitutive relations andOhm’s law, into the obtained expression and are left with ∇ × ( ∇ × E ) = − c (cid:34) (cid:15) b ∂ E ∂ t + πσ ∂ E ∂ t (cid:35) . (31)Expanding the curl on the left hand side and performing aFourier transform, we obtain( k · k ) E ( k , ω ) − k [ k · E ( k , ω )] = (cid:15) ω c E ( k , ω ) . (32)Here we have introduced the permittivity tensor (cid:15) ( ω ) = (cid:15) xx (cid:15) xy (cid:15) yx (cid:15) yy
00 0 (cid:15) zz . (33)and it is fully determined by the following relation to the con-ductivity tensor: (cid:15) i j = δ i j (cid:15) b + π i ω σ i j The components of thepermittivity tensor are plotted in Fig. 6 for a nodal loop with θ = π/ k z -axis. The transmitted field can be written as E t ( z , t ) = E t ( k (cid:48) z , ω ) e ik (cid:48) z z − i ω t , where k (cid:48) z is the wave vector inside the bulkmaterial. Inserting this into Eq. (32) it follows that we to havefind solutions to the following determinant (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:16) k (cid:48) z (cid:17) + ω c (cid:15) xx ω c (cid:15) xy − ω c (cid:15) xy (cid:16) k (cid:48) z (cid:17) + ω c (cid:15) yy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = . (34)From here we find that the allowed wave vectors inside thematerial are given by (cid:16) k (cid:48) z (cid:17) = ω c ( (cid:15) xx + (cid:15) yy ) ± ω c (cid:114) (cid:16) (cid:15) xx + (cid:15) yy (cid:17) − (cid:15) xx (cid:15) yy − (cid:15) xy ≡ k ± . (35)We further find the basis vectors for the allowed electromag-netic fields inside the material. They are given by e ± t = c ± ˆx + c ± ˆy , (36)with the coe ffi cients c ± = ( (cid:15) ± + (cid:15) xy − (cid:15) xx ) − / (cid:15) xy , c ± = ( (cid:15) ± + (cid:15) xy − (cid:15) xx ) − / ( (cid:15) ± − (cid:15) xx ) and (cid:15) ± = c k ± /ω . The total transmitted wave FIG. 7. ( a,b ) θ K for bulk material and ( c,d ) (cid:15) K for bulk material.For (a) and (c) we fix the chemical potential, ˜ µ = ω = u . For (b) and (d) wefix the tilt velocity, ˜ u = . E t can be written as the linear combination E t = E + e + t + E − e − t .The boundary conditions are now given by E (cid:107) = E (cid:107) , (37)1 µ B (cid:107) = µ B (cid:107) , (38)where again the (cid:107) superscript denotes the field componentsparallel to the surface. We take the incident and reflectedwaves, Eqs. (24) and (25), and insert these along with theexpression for the transmitted wave into the boundary condi-tions, Eqs. (37), and (38). By solving the system of equationswe obtain the reflected fields, E R / Lr = (cid:18) − c ω k + (cid:19) (cid:0) c + ± ic + (cid:1) E + + (cid:18) − c ω k − (cid:19) (cid:0) c − ± ic − (cid:1) E − , (39)where E ± = E (cid:16) + c ω k ± (cid:17) (cid:18) c ± − c ± c ∓ c ∓ (cid:19) , (40)where k ± is defined in Eq. (35). Using Eq. (20) we plot theKerr rotation for a bulk material hosting a nodal loop, seeFig. 7. The results show similar patterns as those for the thin-film geometry. The main di ff erence lies in the amplitude of FIG. 8. (a) Kerr angle and (b) Kerr ellipticity for a thin film. (c)Kerr angle and (d) Kerr ellipticity for the bulk material. The chem-ical potential and the tilt velocity are fixed to ˜ µ = u = . ω is set to thevalues ˜ ω = .
17 (blue), ˜ ω = . ω = .
23 (yellow). the Kerr angle patterns. In comparison to the thin film, theamplitude is much smaller and this is further reduced whenthe relative permittivity (cid:15) b is increased. A Kerr angle shouldhowever still be possible to be experimentally detectable sincewith nowadays techniques Kerr angles of order of 10 − radi-ans have been measured [27]. C. Varying the tilt angle
So far, we have fixed the tilt angle of the nodal loop to θ = π/
4. In this section present briefly further numerical re-sults on the Kerr signal when varying θ . The results are plottedin Fig. 8. As seen from the figure the Kerr rotation and ellip-ticity depends highly on the tilt angle. The e ff ect is especiallynoticeable in the ellipticity angle. We observe that the elliptic-ity reaches large positive values in the first and third quadrantwhereas it almost disappears in the second and fourth quad-rant. This is because E Rr ( θ ) = E Lr ( θ + π/
2) as a consequenceof that σ xy ( θ ) = − σ xy ( θ + π/
2) and hence if (cid:15) K is large in onequadrant it has to be small in the next and vice versa. V. CONCLUSION
In summary, we have studied the Kerr e ff ect in a nodal loopsemimetal and have described how the Kerr rotation can be re-lated to di ff erent characteristics of the nodal loop, in particularthe tilt of the nodal loop.We have calculated the full optical conductivity tensor fora nodal-loop semimetal, which in turn has made it possibleto determine the Kerr rotation. We have found that the tiltdirection plays a dominating role in the determination of theconductivity. Depending on the tilt direction, the transverseconductivity oscillates between zero and a finite value. Incontrast, the longitudinal conductivity always retains a finitevalue unless the system is Pauli blocked, and depending on thetilt angle varies between a linear and cubic behavior at smallfrequencies.The Kerr rotations as a function of various system param-eters, which we calculated for both a thin film and the bulkmaterial, are the main results of this paper. We have foundthat the Kerr rotation is strongly dependent of the tilt veloc-ity and the radius of the nodal loop, features which originatefrom the specific behavior of the transverse conductivity. Sim- ilarly to other topological materials, the obtained Kerr angleis generally large and could serve as an important tool for ex-perimentally characterizing nodal-loop semimetals. ACKNOWLEDGMENTS
The authors acknowledge helpful discussions withChristoph Kastl and Alexander Holleitner at the ini-tial stages of the project. All authors acknowledgesupport by the National Research Fund, Luxembourgunder grants ATTRACT A14 / MS / / MoMeSys,CORE C16 / MS / / PARTI, and C20 / MS / / -OpenTop. Appendix A: Real part of optical conductivity
As stated in the main text the real part of the optical conductivity is obtained from the following integrals, ( i , j ∈ { x , y , z } )Re (cid:110) σ i j ( ω ) (cid:111) = (cid:90) k k d ξ Θ (1 − ˜ ω/ F i j ( ξ ) + (cid:90) k d ξ Θ ( ˜ ω/ − F i j ( ξ ) , (A1)where F i j ( ξ ) = g + i j ( ξ ) − g − i j ( ξ ) and g ± xx = Γ π ˜ u ˜ ω π − (cid:32) µ ± ˜ ω u ξ (cid:33) − θ (cid:32) µ ± ˜ ω u ξ (cid:33) (cid:115) − (cid:32) µ ± ˜ ω u ξ (cid:33) Θ (cid:34) − µ ± ˜ ω u ξ (cid:35) × ξ (cid:115)(cid:18) ˜ ω (cid:19) − (cid:0) − ξ (cid:1) , (A2) g ± yy = Γ π ˜ u ˜ ω π − (cid:32) µ ± ˜ ω u ξ (cid:33) + θ (cid:32) µ ± ˜ ω u ξ (cid:33) (cid:115) − (cid:32) µ ± ˜ ω u ξ (cid:33) Θ (cid:34) − µ ± ˜ ω u ξ (cid:35) × ξ (cid:115)(cid:18) ˜ ω (cid:19) − (cid:0) − ξ (cid:1) , (A3) g ± xy = Γ µ ± ˜ ωπ ˜ u ˜ ω sin 2 θξ (cid:115)(cid:18) ˜ ω (cid:19) − (cid:0) − ξ (cid:1) (cid:115) ξ − (cid:32) µ ± ˜ ω u (cid:33) Θ (cid:34) − µ ± ˜ ω u ξ (cid:35) (A4) g ± zz = Γ π ˜ ω π − (cid:32) µ ± ˜ ω u ξ (cid:33) Θ (cid:34) − µ ± ˜ ω u ξ (cid:35) ξ (cid:16) − ξ (cid:17) (cid:113)(cid:16) ˜ ω (cid:17) − (cid:0) − ξ (cid:1) . (A5)In the limit ˜ ω (cid:28)
1, ˜ ω < ˜ u and ˜ µ = { σ xx ( ω ) } ≈ Γ π ˜ u (cid:34)
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