Faraday rotation and transmittance as markers of topological phase transitions in 2D materials

SciPost Phys. 16, 077 (2024) Faraday rotation and transmittance as markers of topological phase transitions in 2D materials Manuel Calixto1,2⋆, Alberto Mayorgas1†, Nicolás A. Cordero2,3,4‡, Elvira Romera2,5◦and Octavio Castaños6§ 1Department of Applied Mathematics, University of Granada, Fuentenueva s/n, 18071 Granada, Spain 2Institute Carlos I for Theoretical and Computational Physics (iC1), Fuentenueva s/n, 18071 Granada, Spain 3Departamento de Física, Universidad de Burgos, Plaza Misael Bañuelos s/n, 09001 Burgos, Spain 4International Research Center in Critical Raw Materials for Advanced Industrial Technologies (ICCRAM), Plaza Misael Bañuelos s/n, 09001 Burgos, Spain 5Department of Atomic, Molecular and Nuclear Physics, University of Granada, Fuentenueva s/n, 18071 Granada, Spain 6Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico, Apdo. Postal 70-543, 04510, CDMX, Mexico ⋆[email protected] , † [email protected] , ‡ [email protected] , ◦[email protected] , § [email protected] Abstract We analyze the magneto-optical conductivity (and related magnitudes like transmittance and Faraday rotation of the irradiated polarized light) of some elemental twodimensional Dirac materials of group IV (graphene analogues, buckled honeycomb lattices, like silicene, germanene, stannane, etc.), group V (phosphorene), and zincblende heterostructures (like HgTe/CdTe quantum wells) near the Dirac and gamma points, under out-of-plane magnetic and electric fields, to characterize topological-band insulator phase transitions and their critical points. We provide plots of the Faraday angle and transmittance as a function of the polarized light frequency, for different external electric and magnetic fields, chemical potential, HgTe layer thickness and temperature, to tune the material magneto-optical properties. We have shown that absortance/transmittance acquires extremal values at the critical point, where the Faraday angle changes sign, thus providing fine markers of the topological phase transition. In the case of non-topological materials as phosphorene, a minimum of the transmittance is also observed due to the energy gap closing by an external electric field. Copyright M. Calixto et al. This work is licensed under the Creative Commons Attribution 4.0 International License. Published by the SciPost Foundation. Received 31-05-2023 Accepted 19-12-2023 Published 18-03-2024 Check for updates doi:10.21468/SciPostPhys.16.3.077 1 SciPost Phys. 16, 077 (2024) Contents 1 Introduction 2 2 Some two-band 2D-Dirac material models 3 2.1 Graphene analogues: Silicene, germanene, etc 4 2.2 HgTe/CdTe quantum wells 5 2.3 Phosphorene as an anisotropic material 8 3 Magneto-optical conductivity 11 3.1 Magneto-optical properties of graphene analogues 12 3.2 Magneto-optical properties of zincblende heterostructures 14 3.3 Magneto-optical properties of phosphorene and effect of anisotropies 17 4 Conclusions 21 A Landau levels plot versus external magnetic field 22 B Silicene conductivity in the circularly polarization basis 23 C HgTe quantum well conductivity with Zeeman effect 24 D Animations of the energy spectrum and conductivities 24 E Faraday angle contour plots 25 References 28 1 Introduction Two-dimensional (2D) materials have been extensively studied in recent years (and are expected to be one of the crucial research topics in future years) especially because of their remarkable electronic and magneto-optical properties which make them hopeful candidates for next generation optoelectronic devices. Graphene is the archetype of a 2D nanomaterial with exceptional high tensile strength, electrical conductivity, transparency, etc. In spite of being the thinnest one, it exhibits a giant Faraday rotation (ΘF∼6◦) on polarized light in singleand multilayer arrangements [1–6]with experimental confirmation [7]. Faraday rotation is a fundamental magneto-optical phenomenon used in various optical control, laser technology and magnetic field sensing techniques. Magneto-optical properties of other buckled honeycomb lattices, like silicene [8], have been studied in [9–11], together with other monolayer transition metal dichalcogenides [12] and anisotropic versions like phosphorene [13]. Magneto-optical measurements also provide signatures of the topological phase transition (TPT; see [14–16]for standard textbooks on the subject) in inverted HgTe/CdTe quantum wells (QW), distinguishing quantum Hall (QH) from quantum spin Hall (QSH) phases [17], where one can tune the band structure by fabricating QWs with different thicknesses λ. A universal value of the Faraday rotation angle, close to the fine structure constant, has been experimentally observed in thin HgTe QW with critical thickness [18]. To determine experimentally the Faraday rotation effect in Dirac materials it is convenient to consider: (1) A transverse-magnetic-polarized wave incident from the left onto a single 2 SciPost Phys. 16, 077 (2024) topological insulator sandwiched by dielectric layers, which yields an enhancement of the Faraday rotation with an angle larger than 700 mrad and with a transmission higher than 90% [19]. (2) A graphene sheet sandwiched by one-dimensional topological photonic crystals also an enhancement of the Faraday rotation can be achieved with high transmittance [20]. (3) In thin films of 3D topological insulators [21]or by considering thin films of Floquet topological insulators where giant Faraday and Ker rotations have been observed under the action of a perpendicular magnetic field or in a non-resonant optical field [22]. The inverse Faraday effect (IFE) has been studied in Dirac materials in 2D and 3D, and these studies have concluded that IFE is stronger than in conventional semiconductors. Then the Dirac materials can be potentially useful for the optical control of magnetization in optoelectronic devices [23]. Information theoretic measures also provide signatures of the TPT in silicene [24–28]and HgTe/CdTe QWs [29], as an alternative to the usual topological (Chern) numbers. They also account for semimetalic behavior of phosphorene [30,31]under perpendicular electric fields. In this paper we perform a comparative study of the magneto-optical properties of several 2D Dirac materials, looking for TPT signatures when the band structure is tuned by applying external fields or by changing the material characteristics. For this purpose, we focus on transmittance and Faraday rotation near the critical point of the topological phase transition for topological materials such as silicene and HgTe quantum wells. We found that, for these materials, transmittance attains an absolute minimum T0at the critical TPT point for a certain value Ω0of the normal incident polarized light frequency. This minimal behavior does not depend on the chosen values of magnetic field, chemical potential and temperature, although the location of Ω0varies with them. An inflection point of the Faraday angle is observed at each peak of the transmittance, coinciding in frequency. As a novel perspective, we study that non-topological materials, such as phosphorene, also exhibit an extremal value of the transmittance when the energy gap is closed by an external electric field. The organization of the article is as follows. In Sec. 2we discuss the structure of time independent Bloch Hamiltonians for general two-band 2D-Dirac material models, their Chern numbers and their minimal coupling to an external perpendicular magnetic field. We particularize to graphene analogues (silicene, germanene, etc.) in Sec. 2.1, zincblende heterostructures (HgTe/CdTe quantum wells) in Sec. 2.2 and anisotropic materials like phosphorene in 2.3, calculating their energy spectrum and Hamiltonian eigenstates (Landau levels) and describing their topological phases (when they exist). In Sec. 3we recall the Kubo-Greenwood formula for the magneto-optical conductivity tensor σof a 2D electron system in a perpendicular magnetic field Band an oscillating electric field of frequency Ω. In particular, we are interested in analyzing the transmittance and Faraday rotation of linearly polarized light of frequency Ωfor normal incidence on the 2D material. Magneto-optical properties of graphene analogues, zincblende heterostructures and phosphorene are analyzed in Sections 3.1,3.2 and 3.3, respectively. For topological insulator materials, we find that the critical point is generally characterized by a minimum transmittance T0at a given light frequency Ω0, where the Faraday angle changes sign. The effect of anisotropies is also discussed in phosphorene in Section 3.3. Finally, Sec. 4is devoted to conclusions. 2 Some two-band 2D-Dirac material models The time independent Bloch Hamiltonian of a two-band 2D insulator has the general form H(k) = ε0(k)τ0+d(k)·τ, (1) where τ= (τx,τy,τz)is the Pauli matrix vector, τ0denotes the 2 ×2 identity matrix and d(k)parameterizes an effective spin-orbit coupling near the center Γor the Dirac valleys K 3 SciPost Phys. 16, 077 (2024) and K′of the first Brillouin zone (FBZ), with k= (kx,ky)the two-dimensional wavevector. The energy of the two bands is ε±(k) = ε0(k)±|d(k)|. To distinguish between band insulator and topological insulator phases, one can use the TKNN (Thouless-Kohmoto-Nightingale-Nijs) formula [32]providing the Chern-Pontryagin number (related to the quantum spin Hall conductance and the Berry phase [33]) C=1 2πZ ZFBZ d2k∂ˆ d(k) ∂kx×∂ˆ d(k) ∂ky·ˆ d(k), (2) with ˆ d=d/|d|, which counts the number of times (winding number) the unit vector ˆ d(k) wraps around the unit sphere as kwraps around the entire FBZ. The Chern number Cusually depends on the sign of some material and (external) control parameters in the Hamiltonian H(see later for some examples), taking different values in different phases. We shall see that magneto-optical conductivity measures also capture the topological phase transition. We shall consider the interaction with a perpendicular magnetic field B= (0,0,B). Promoting the wavevector kto the momentum operator k→p/ħh=−i∇, this interaction is introduced through the usual minimal coupling, p→P=p+eAwith A= (Ax,Ay)=(−B y,0) the electromagnetic potential (in the Landau gauge) and ethe elementary charge (in absolute value). After Peierls’ substitution, which results in kx→Px/ħh=a†+a p2ℓB ,ky→Py/ħh=a†−a ip2ℓB , (3) the Hamiltonian (1) can be eventually written in terms of creation a†and annihilation a=ℓB p2ħh(Px−iPy) = −1 p2ℓB (y−y0+iℓ2 Bpy/ħh), (4) operators, where ℓB=pħh/(eB)is the magnetic length and y0=ℓ2 Bkxis the coordinate of the conserved center of the cyclotron orbit. Let us review some relevant physical examples. 2.1 Graphene analogues: Silicene, germanene, etc Silicene, germanene, and other transition metal dichalcogenides (of the Xene type) exhibit an intrinsic non-zero spin-orbit coupling Hso =−1 2sξ∆soτz(s=±1 is the spin of the electron and ξ=±1 refer to the Dirac valleys Kand K′) due to second neighbors hopping terms in the tight binding model [34]. Spin-orbit interaction Hso combined with and external perpendicular electric field coupling H∆z=1 2∆zτz, gives d(k)=(vħhξkx,vħhky,∆sξ), where ∆sξ= (∆z−sξ∆so)/2 results in a tunable (Dirac mass) gap (see e.g. [35–38]). In Table 1 we show a comparative of spin-orbit coupling and Fermi velocity values for several 2D materials. The Chern number (2) turns out to be Csξ=ξsign(∆sξ), (5) where we have integrated on the whole plane, as corresponds to the FBZ in the continuum limit (zero lattice constant). Therefore, the topological phase is determined by the sign of the Dirac mass at each valley ξ. More precisely, there is a TPT from a topological insulator (TI, |∆z|<∆so) to a band insulator (BI, |∆z|>∆so), at a charge neutrality point (CNP) ∆(0) z=sξ∆so, where there is a gap cancellation between the perpendicular electric field and the spin-orbit coupling. 4 SciPost Phys. 16, 077 (2024) Table 1: Approximate values of model parameters ∆so (spin-orbit coupling), l(interlattice distance) and v(Fermi velocity) for two dimensional Si, Ge, Sn and Pb sheets. These data have been obtained from first-principles computations in [38](∆so and l) and [39,40](v). ∆so (meV) l(Å) v(105m/s) Si 4.2 0.22 4.2 Ge 11.8 0.34 8.8 Sn 36.0 0.42 9.7 Pb 207.3 0.44 – Using the general prescription (3), the minimal coupling with a perpendicular magnetic field Bthen results in a different Hamiltonian Hξfor each valley ξ=±1 H1=∆s,1 ħhωa ħhωa†−∆s,1 ,H−1=∆s,−1−ħhωa† −ħhωa−∆s,−1, (6) where ω=p2v/ℓBdenotes the cyclotron frequency. The eigenvalues of both Hamiltonians are simply: Esξ n=¨sgn(n)Ç|n|ħh2ω2+∆2 sξ,n=0, −ξ∆sξ,n=0, (7) and the corresponding eigenstates are written in terms of Fock states ||n|〉, for Landau level (LL) index n=0,±1,±2,... [valence (−)and conduction (+) states], as spinors |n〉sξ=  Asξ n|n|− ξ+1 2¶ Bsξ n|n|+ξ−1 2¶ , (8) with coefficients (see [9,41–43]for similar results) Asξ n=¨sgn(n) p2q1+sgn(n)cosθsξ n,n=0, (1−ξ)/2, n=0, Bsξ n=¨ξ p2q1−sgn(n)cosθsξ n,n=0, (1+ξ)/2, n=0, (9) where θsξ n=arctanħhωp|n|/∆sξ, that is, cosθsξ n=∆sξ/|Esξ n|. Note that Asξ nand Bsξ ncan eventually be written as cos(θsξ n/2)or sin(θsξ n/2), depending on sgn(n). In Figure 1we plot the low energy spectra of silicene, given by (7), as a function of the external electric field ∆z, together with the charge neutrality (critical) points ∆(0) z=±|∆so| (marked by vertical dashed lines) at which the TPT takes place. 2.2 HgTe/CdTe quantum wells In [44–47]it was shown that quantum spin Hall effect can be realized in mercury telluridecadmium telluride semiconductor quantum wells. Similar effects were also predicted in Type-II semiconductor quantum wells made from InAs/GaSb/AlSb [48]. The surface states in these 3D topological insulators can be described by a 2D modified effective Dirac Hamiltonian H=H+0 0H−,Hs(k) = ε0(k)τ0+ds(k)·τ, (10) 5 SciPost Phys. 16, 077 (2024) s=1, ξ=1 s=-1, ξ=1 -4-2 0 2 4 -3 -2 -1 0 1 2 3 Δz/Δso En sξ/Δso E0-- E0++ E0-+ E0+- Figure 1: Low energy spectra of silicene as a function of the external electric potential ∆z(in ∆so units) for B=0.05 T. Landau levels n=±1,±2 and ±3[valence (−)and conduction (+)], at valley ξ=1, are represented by thin solid lines, blue for s=−1 and red for s=1 (for the other valley we simply have Es,−ξ n=E−s,ξ n). The edge states n=0 are represented by thick lines at both valleys: solid at ξ=1 and dashed at ξ=−1. Vertical dashed gray lines indicate the charge neutrality points separating band insulator (|∆z|>∆so) from topological insulator ( |∆z|<∆so) phases. where s=±1 is the spin and H−(k) = H∗ +(−k)(temporarily reversed). The expansion of Hs(k)about the center Γof the first Brillouin zone gives [45] ε0(k) = γ−δk2,ds(k) = (αskx,αky,µ−βk2), (11) where α,β,γ,δand µare expansion parameters that depend on the heterostructure (the HgTe layer thickness λ). The most important one is the mass or gap parameter µ, which changes sign at a critical HgTe layer thickness λcwhen going from the normal (λ < λcor µ/β < 0) to the inverted (λ > λcor µ/β > 0) regime [49]. Typical values of these parameters for different HgTe layer thickness (below and above λc) can be found in [49]and in Table 2(γcan be neglected). The energy of the two bands is ε±(k) = ε0(k)±Æα2k2+ (µ−βk2)2. (12) Table 2: Material parameters for HgTe/CdTe quantum wells with different HgTe layer thicknesses λ[49]. λ(nm) α(meV·nm) β(meV·nm2)δ(meV·nm2)µ(meV) 5.5 387 -480 -306 9 6.1 378 -553 -378 -0.15 7.0 365 -686 -512 -10 6 SciPost Phys. 16, 077 (2024) The TKNN formula (2) for ds(k)provides the Chern number Cs=s[sign(µ) + sign(β)], (13) where we have integrated on the whole plane, as corresponds to the continuum limit. According to Table 2,βdoes not change sing and, therefore, the topological phase transition occurs when µchanges sign, as already mentioned. In reference [49], the normal and inverted regimes are equivalently given by the sign of µ/β. Using again the general prescription (3), the minimal coupling with a perpendicular magnetic field Bnow results in H+=  γ+µ−(δ+β)(2N+1) ℓ2 B p2α ℓBa p2α ℓBa†γ−µ−(δ−β)(2N+1) ℓ2 B  , H−=  γ+µ−(δ+β)(2N+1) ℓ2 B−p2α ℓBa† −p2α ℓBaγ−µ−(δ−β)(2N+1) ℓ2 B  , (14) with N=a†a. A Zeeman term contribution HZ s=−s 2BµBge τ0+τz 2+gh τ0−τz 2, (15) can also be added to the Hamiltonian, with µB≃0.058 meV/T the Bohr magneton and ge,h the effective (out-of-plane) g-factors for electrons and holes (conduction and valence bands). Using (Fock state) eigenvectors ||n|〉 of the (Landau level) number operator N=a†a, one can analytically obtain the eigenspectrum Es n=γ−2δ|n|−sβ ℓ2 B−sge+gh 4BµB+sgn(n)v t2α2|n| ℓ2 B +µ−2β|n|−sδ ℓ2 B−sge−gh 4BµB2, (16) for LL index n=±1,±2,±3,... [valence (−)and conduction (+)], and Es 0=γ−sµ−δ−sβ ℓ2 B−BµBs+1 4gh+s−1 4ge, (17) for the edge states n=0, s=±1. These eigenvalues coincide with those in [17,50,51]for the identification s={−1,1}={↑,↓}. The corresponding eigenvectors are |n〉s=As n|n|− s+1 2 Bs n|n|+s−1 2, (18) with coefficients As n=sgn(n) p2Æ1+sgn(n)cosϑs n,n=0, (1−s)/2, n=0, Bs n=s p2Æ1−sgn(n)cosϑs n,n=0, (1+s)/2, n=0, (19) 7 SciPost Phys. 16, 077 (2024) where ϑs n=arctan p2|n|α/ℓB µ−2β|n|−sδ ℓ2 B−sge−gh 4BµB . (20) As for the graphene analogues in (9), the coefficients As nand Bs ncan eventually be written as sine and cosine of half angle, depending on sgn(n). According to (17), the band inversion for edge states occurs when E+ 0=E− 0⇒Binv =µ eβ/ħh−µB(ge+gh)/4, (21) which gives the critical magnetic field Bcwhich separates the QSH and QH regimes [51]. For example, for the material parameters in Table 2corresponding to a QW thickness λ=7.0 nm and g-factors ge=22.7, gh=−1.21, one obtains Binv ≃7.4 T. See also Figure 2for a graphical representation of this band inversion. From now on we shall discard Zeeman coupling for the sake of convenience since our main conclusions remain qualitatively equivalent. We address the interested reader to Appendix C where we reproduce some results of Reference [17]for non-zero Zeeman coupling and contrast with the zero Zeeman coupling case. We shall use a linear fit µ(λ) = 77.31 −12.53λ, α(λ) = 467.49 −14.65λ, β(λ) = 283.58 −138.16λ, δ(λ) = 458.46 −138.25λ, (22) of the material parameters in Table 2as a function of the HgTe layer thickness λ(dimensionless units and λin nm units). In all cases the coefficient of determination is R2>0.99. Looking at µ(λ)in (22), we can obtain an estimation of the critical HgTe thickness at which the topological phase transition occurs as µ=0⇒λc=6.17 nm. (23) In Figure 2we plot the low energy spectra given by (16) and (17) as a function of the HgTe layer thickness λ, where we have extrapolated the linear fit (22) to the interval [4nm, 8nm]. When neglecting Zeeman coupling, the band inversion for edge states (21) occurs for B=ħhµ/(eβ) which, using the linear fit (22), provides a relation λinv(B) = 368.31 −2.05B 59.7 −B, (24) between the applied magnetic field B(in Tesla) and the HgTe layer thickness λinv(B)(in nanometers) at which the band inversion E+ 0=E− 0takes place. Note that λinv(B)≃λc=6.17 nm for low B≪1 T, and that E+ 0=E− 0≃0 meV at this point as shows Figure 2. 2.3 Phosphorene as an anisotropic material The physics of phosphorene has been extensively studied [53–67]. There are several approaches to the low energy Hamiltonian of phosphorene in the literature. Rudenko et al. [68] and Ezawa [69]propose a four-band and five-neighbors tight-binding model later simplified to two-bands [69]. Several approximations of this two-band model have been used in [13,70–72]. We shall choose for our study the Hamiltonian H=Ec+αxk2 x+αyk2 yγkx γkxEv−βxk2 x−βyk2 y, (25) 8 SciPost Phys. 16, 077 (2024) s=1 s=-1 45678 -40 -20 0 20 40 HgTe layer thickness λ(nm) Energy En s(meV) λinv E0+ E0Figure 2: Low-energy spectra Es nof a HgTe/CdTe quantum well as a function of the HgTe layer thickness λfor B=0.5 T. Landau levels n=±1,±2,±3[valence (−)and conduction (+)]are represented by thin solid lines, blue for spin s=−1 and red for s=1. Edge states (n=0) are represented by thick lines. A vertical dashed black line indicates the HgTe thickness λinv(0.5) = 6.20 nm ≃λcwhere the band inversion for edge states occurs for B=0.5 T according to (24). proposed by Zhou and collaborators [13]. This corresponds to a Bloch Hamiltonian (1) with ε0(k) = Ec+Ev+ (αx−βx)k2 x+ (αy−βy)k2 y 2, (26) d(k) = γkx,0, Ec−Ev+ (αx+βx)k2 x+ (αy+βy)k2 y 2. The Hamiltonian (25) provides a trivial Chern number (2), even in the presence of a tunable perpendicular constant electric field (see below), which means that monolayer phosphorene does not have a topological phase per se. It has been shown that topological transitions can be induced in phosphorene when rapidly driven by in-plane time-periodic laser fields [73]; these are called in general “Floquet topological insulators” (see e.g. [74–76]), but we shall not consider this possibility here. Although phosphorene is not a topological material, we will see in Sec. 3.3 that the critical magneto-optical properties (e.g., minimum transmittance) observed for silicene and HgTe QWs are still valid in phosphorene when closing the energy gap through an external electric field. Another possibility to modify the energy gap could be by applying strain [60,70](see later in Sec. 3.3). The material parameters of phosphorene can be written in terms of conduction (c) and valence (v) effective masses as (see [13]for more information) αx,y=ħh2 2mcx,cy ,βx,y=ħh2 2mvx,vy , (27) with mcx=0.793me,mcy=0.848me, mvx=1.363me,mvy=1.142me,(28) 9 SciPost Phys. 16, 077 (2024) -10 -5 0 5 10 15 λ=5.50 nm -10 -5 0 5 10 λ=6.17 nm -10 -5 0 5 10 λ=7.00 nm -15 -10 -5 0 5λ=5.50 nm -15 -10 -5 0 5λ=6.17 nm 0 10 20 30 40 50 60 -15 -10 -5 0 5λ=7.00 nm Conductivity σij(Ω)/σ0 ℏΩ (meV) μ F=12.5 meV B =0.5 T T=1 K η=0.5 meV Re[σxx(Ω)] Re[σxy(Ω)] Im[σxx(Ω)] Im[σxy(Ω)] Figure 6: Real and imaginary parts of the longitudinal σx x and transverse Hall σx y magneto-optical conductivities in a bulk HgTe QW of thickness λ=5.50,6.17,7.00 nm, as a function of the polarized light frequency Ωand in σ0=e2/hunits. We set the conductivity parameters as µF=12.5 meV, B=0.5 T, T=1 K and η=0.5 meV. and some representative values of the chemical potential µF=12.5 meV, temperature T=1 K and scattering rate η=0.5 meV. For ħhΩ∈[0,60]meV, we achieve convergence with 100 LLs, that is, restricting the sum in (36) as P∞ n=−∞ →P100 n=−100. More explicitly, for the parameters mentioned above,  n=100 X n=−100 σi j − n=99 X n=−99 σi j /σ0≤         10−5, if σi j =Re(σx x ), 10−4, if σi j =Re(σx y ), 10−3, if σi j =Im(σx x ), 10−7, if σi j =Im(σx y ). (46) Similar to silicene, we can see in Figure 6that there are multiple peaks in the absorptive components Re(σx x )and Im(σx y ), corresponding to transitions between occupied and unoccupied LLs obeying the selection rules |n|=|m|±1. At lower frequencies ħhΩ∈[0,30]meV, inside each curve of Figure 6, we find the main peaks corresponding to the transitions 0 →1 for spin s=1 and s=−1. Both peaks merge approximately at λ≃λc=6.17 nm. This is because the energy differences E+ 1−E+ 0and E− 1−E− 0are similar when λ≃λcfor low magnetic 16 SciPost Phys. 16, 077 (2024) 0 1 2 3 4 5 6.12 6.13 6.14 6.15 6.16 6.17 B ( T ) λ*(nm) Estimate Standard Error t-Statistic a1218.447 0.01218 000 a217.30 0.15 110 a335.4004 0.0017 20 000 a42.770 0.025110 λfit*(B)= a1-a2B a3-a4B R2=0.9999999951 Numerical Non-linear fit Figure 7: Numerical solutions λ∗(in nm, blue dots) of the equation E+ 1−E+ 0=E− 1−E− 0 (energies (16,17) of HgTe QW) for 50 different values of the external magnetic field B. In orange, non-linear fit (47) of the numerical values. fields B≪1 T, according to equations (16,17). In order to extend this result to higher values of the magnetic field, we insert the parameter fits (22) into the equation E+ 1−E+ 0=E− 1−E− 0, and solve it numerically for λ∗=λ∗(B), obtaining the values represented by blue dots in Figure 7. These values fit the equation λ∗ fit(B) = 218.4 −17.3B 35.4 −2.8B, (47) which is represented as an orange curve in Figure 7. Consequently, only for small magnetic fields, we can infer the critical thickness λcwhere the TPT in HgTe QW occurs from the conductivity Re(σx x )plot, that is, λ∗≃λc=6.17 nm for B≪1 T. The behavior of the Faraday angle and the transmittance as a function of the polarized light frequency Ωaround the critical HgTe layer thickness λc=6.17 nm (at which the material parameter µchanges sign/Chern number) is shown in Figure 8. As for silicene, we focus on the lower frequencies ħhΩ∈[0,30]meV where the main peaks are located, and find again a minimum of the transmittance, this time T0=0.78, at the critical point λcand ħhΩ=15.0 meV. For this material, the “minimal” behavior does depend on the particular values of magnetic field, as we saw in equation (47). However, for small magnetic fields like B=0.5 T in Figure 8, the minimum of the transmittance still takes place at λ∗≃λc=6.17 nm. The Faraday angle at the critical point (black curve in Figure 8) changes sign at the minimum transmittance frequency ħhΩ=15.0 meV, a behavior shared with silicene. For completeness, in Appendix Ewe show several contour plots of the Faraday angle using different cross sections in the {ħhΩ,λ,B,T,µF}parameter space. 3.3 Magneto-optical properties of phosphorene and effect of anisotropies From the phosphorene Hamiltonian (25), the current operator (35) is js x=e ħh(γτx+kx(τ0(αx−βx) + τz(αx+βx))) , js y=e ħhkyτ0(αy−βy) + τz(αy+βy), (48) 17 SciPost Phys. 16, 077 (2024) λ(nm) 6.97 6.70 6.43 6.17=λc 5.90 5.63 5.37 0.80 0.85 0.90 0.95 1.00  0.78 0 10 20 30 -3 -2 -1 0 1 2 3 ΘF 15. ℏΩ (meV) μ F=12.5 meV B=0.5 T T=1 K η=0.5 meV Figure 8: Transmittance Tand Faraday angle ΘF(in degrees) in a bulk HgTe QW as a function of the polarized light frequency Ω, and for thickness λ < λc,λ=λc and λ>λc, with λc=6.17 nm (black line). We set the conductivity parameters as µF=12.5 meV, B=0.5 T, T=1 K and η=0.5 meV. which, after minimal coupling, according to prescription (29), results in js x=e ħhγτx+a†+a p2αy x ℓB (τ0(αx−βx) + τz(αx+βx)), js y=e ħh αy x (a†−a) ip2ℓBτ0(αy−βy) + τz(αy+βy). (49) Plugging these matrix elements into the general expression (36) we obtain the magnetooptical conductivity for phosphorene. Note that, unlike silicene and HgTe QW, there is now a large asymmetry between σx x and σy y (about one order of magnitude difference), as evidenced by Figure 9. This asymmetry was already highlighted by [71], where tunable optical properties of multilayer black phosphorus thin films were studied for B=0. In Figure 9we plot the real and imaginary parts of the conductivity tensor components σi j (in σ0=e2/h units) of phosphorene as a function of the polarized light frequency Ω, for some values of the electric potential around ∆(0) z=−Eg=−1.52 eV (closing the energy gap), a magnetic field of B=0.5 T, like in Figure 3, and some representative values of the chemical potential µF=−0.417 eV, temperature T=1 K and scattering rate η=0.2 meV. We are using the same threshold of N=1000 Fock states that we used to find convergence in the first 6 Hamiltonian eigenstates of the numerical diagonalization in Figure 3. This convergence is ensured for ħhΩ∈[0,20]meV. The anisotropic character of phosphorene also implies that the current 18 SciPost Phys. 16, 077 (2024) Re[σxx(Ω)] Im[σxx(Ω)] -20 0 20 40 Δz=-1.533 eV -20 0 20 Δz=-1.529 eV -20 0 20 Δz=-1.525 eV -0.10 -0.05 0.00 0.05 0.10 Δz=-1.533 eV Re[σyy(Ω)] Im[σyy(Ω)] -0.10 -0.05 0.00 0.05 0.10 Δz=-1.529 eV -0.10 -0.05 0.00 0.05 0.10 Δz=-1.525 eV -1.5 -1.0 -0.5 0.0 0.5 Δz=-1.533 eV Re[σxy(Ω)] Im[σxy(Ω)] -1.5 -1.0 -0.5 0.0 0.5 Δz=-1.529 eV 5 10 15 20 -1.5 -1.0 -0.5 0.0 0.5 Δz=-1.525 eV Conductivity σij(Ω)/σ0 ℏΩ (meV) μF=-0.417 eV B=0.5 T T=1 K η=0.2 meV Figure 9: Real and imaginary parts of the longitudinal σx x ,σy y and transverse Hall σx y magneto-optical conductivities in a phosphorene monolayer, as a function of the polarized light frequency Ωand in σ0=e2/hunits. Phosphorene is under a perpendicular electric field potential ∆(0) z=−Eg=−1.52 eV closing the energy gap in Figure 3. The y-axis ticks have different values in each subplot as the conductivities σx y and σy y attain smaller values than σx x (phosphorene anisotropy). We set the conductivity parameters as µF=−0.417 meV, B=0.5 T, T=1 K and η=0.2 meV. js yis significantly lower than js x[the Hamiltonian (25) is of second order in ky]. This makes transversal components of the conductivity significantly lower than longitudinal components. This is why we have disposed Figure 9in a slightly different manner from Figures 4for silicene and 6for HgTe QW, which display a more isotropic structure. Due to the parity symmetry of the Hamiltonian (30), only the electronic transitions between LLs of different parities are allowed [30]. The main peak (smaller frequency) of the conductivity Re(σx x )in Figure 9corresponds to the electronic transitions Eeven 0→Eodd 3and Eodd 1→Eeven 2, which have approximately the same energy difference for all ∆z<−1.53 eV with a tolerance ≤10−14 eV. That is, Eeven 0and Eodd 1, and Eeven 2and Eodd 3, are degenerate for all ∆z<−1.53 eV as the spectrum in Figure 3shows. When the degeneration is broken around the electric potential ∆z≃−1.53 eV, the main conductivity Re(σx x )peak splits into two as we can see in Figure 9. The anisotropic character of phosphorene also affects the Faraday angle, which attains much lower values (in absolute value) than for silicene or HgTe QWs. Indeed, in Figure 10 we plot Faraday angle and transmittance as a function of the polarized light frequency Ωfor 19 SciPost Phys. 16, 077 (2024) different electric field potentials −1.535 ≤∆z≤ −1.519 eV. Like for silicene and HgTe QWs, we find a minimal behavior in the transmittance of phosphorene T0=0.50 for a polarized light frequency ħhΩ=2.6 meV at electric field potential ∆(0) z=−1.523 eV, which is close to minus the energy gap −Eg=−1.52 eV. Note that this value of the minimal transmittance of phosphorene is much smaller than for silicene and HgTe QWs; actually, the assumption of low absortance in formula (39) is no longer valid here and we have used the exact expressions for Tand ΘFin (39). Moreover, unlike for graphene analogues and HgTe QWs, this minimum of the transmittance does not seem to be related to the union of two conductivity peaks into a bigger one; rather, it is simply related to the energy gap closure. Actually, the critical electric potential ∆(0) zwhere the transmittance of phosphorene reaches a minimum depends on the magnetic field Bchosen, as Figure 11 shows. We perform a non-linear fit of the numerical values of ∆(0) z(B)and obtain the equation (Bin dimensionless units) ∆(0) zfit (B) = −77.4 −3.5B 50.9 +2.2BeV, (50) which is represented as a orange curve in Figure 11. For small magnetic fields, we can deduce that the critical electric field potential is similar to minus the energy gap −Egof phosphorene, that is ∆(0) z(B)≃ −Eg=−1.52 eV for B≪1 T. We have also checked numerically that the critical electric potentials ∆(0) z(B)are independent of the parameters µFand ηfor a fixed magnetic field B. However, we set different values of µFfor small fields B≤2 T (see caption of Figure 11), in order to avoid blocking the electric transition Eodd 1→Eeven 2of the main peak of the transmittance. We also increment Nas Bdecreases in order to achieve convergence in the diagonalization. Additionally, Figure 10 shows how one peak of the transmittance splits into two around ∆z≃ −1.53 eV (blue lines), since the LL Eeven 0breaks its degeneration approximately for ∆z>−1.53 eV (see Figure 3). For ∆z=∆(0) z=−1.523 eV (thick black line), the big peak on the left in Figure 10 corresponds to the electronic transition Eodd 1→Eeven 2, and moves toward smaller values of ħhΩwhen increasing ∆z. The other small peak in the black line corresponds to the electronic transition Eeven 0→Eodd 3, which moves toward bigger values of ħhΩwhen increasing ∆z. The Faraday angle also presents inflection points at the frequencies where the peaks of the transmittance are located. Therefore, we see that anisotropies affect the values of the Faraday angle and transmittance. There are mechanical ways of introducing anisotropies in 2D materials by subjecting them to strain (like for strained [86]or rippled [87]graphene). This kind of anisotropies can be treated by replacing the scalar Fermi velocity vby a 2×2 symmetric tensor v(see e.g. [82]). Namely, for graphene, the Hamiltonian (1) vector dcomponents dj=ħhvkjare replaced by dj=ħhkivi j,i=1,2,d3=0. Actually, for uniformly strained graphene with strain tensor ϵ, the Fermi velocity tensor is (up to first order) v=v(τ0−βϵ)(see e.g. [82,88]), where β∼2. The relation between the isotropic σ0and the anisotropic σmagneto-optical conductivity tensors is simply σ(Ω,B) = vσ0(Ω,B)v/det(v), with B=Bdet(v)/v2an effective magnetic field. Interesting discussions on how measurements of dichroism and transparency for two different light polarization directions can be used to determine the magnitude and direction of strain can be found in [81]. Also, photoelastic effects in graphene [86], strain-modulated anisotropies in silicene [89,90], etc. The band gap Eg=Ev−Ecof phosphorene can be furthermore modulated by strain and by the number of layers in a stack [60,70]. 20 SciPost Phys. 16, 077 (2024) 0.5 0.6 0.7 0.8 0.9 1.0  0.5 1 2 3 4 5 6 -0.4 -0.2 0.0 0.2 0.4 0.6 ΘF ℏΩ (meV) μ F=-0.417 eV B=0.5 T T=1 K η=0.2 meV Δz(eV) -1.519 -1.521 -1.523=Δz(0) -1.525 -1.527 -1.529 -1.531 -1.533 -1.535 E0 even →E3 odd ↖ E1 odd →E2 even ↖ 2.6 Figure 10: Transmittance Tand Faraday angle ΘF(in degrees) in a phosphorene monolayer as a function of the polarized light frequency Ω, and for electric fields −1.535 <∆z<−1.519 eV around the minus energy gap −Eg=−1.52 eV. The black line corresponds to the electric potential ∆(0) z=−1.523 eV≃ −Egwhere the transmittance attains a minimum of T0=0.5 atħhΩ=2.5 meV. We set the conductivity parameters as µF=−0.417 meV, B=0.5 T, T=1 K and η=0.2 meV. 4 Conclusions We have studied magneto-optical properties of different 2D materials, focusing on transmittance and Faraday rotation near the critical point of the topological phase transition for topological insulators like silicene and HgTe quantum wells. We have seen that, in all topological 2D materials analyzed, transmittance attains an absolute minimum T0at the critical TPT point for a certain value Ω0of the normal incident polarized light frequency. This is a universal behavior for graphene analogues, that is, the minimal behavior of the transmittance does not depend on the chosen values of magnetic field, chemical potential and temperature, although the location of Ω0varies with them. In addition, we have found that each peak of the transmittance coincides in frequency with an inflection point of the Faraday angle, for a fixed selection of the electric field, magnetic field, chemical potential and temperature parameters. This extremal universal behavior is shared with other topological 2D materials like HgTe quantum wells as long as the applied magnetic field remains small enough B≪1 T. In HgTe quantum wells we have verified that there is a minimum of the transmittance T0at the critical HgTe layer thickness at a given frequency Ω′ 0(for this material this minimal behavior depends on the magnetic field) and the Faraday angle at the critical point changes sign at the minimum transmittance frequency Ω′ 0. 21 SciPost Phys. 16, 077 (2024) 0 2 4 6 8 10 12 14 -1.545 -1.540 -1.535 -1.530 -1.525 -1.520 B(T) Δz(0)(eV) Estimate Standard Error t-Statistic a1-77.4090.022-3500 a23.49 0.438.2 a350.8806 0.0095 5300 a42.21 0.288.0 (Δz(0))fit (B)= a1-a2B a3-a4B R2=0.9999999501 Numerical Non-linear fit Figure 11: Electric field potential at which phosphorene transmittance reaches a minimum, as a function of different magnetic fields. In orange, non-linear fit (50) of the numerical values. In general, we set the conductivity parameters µF=−0.41 eV, T=1 K, η=1 meV, and use N=300 Fock state in the numerical diagonalization, for all B≥3 T. For smaller magnetic fields B=0.1,0.5,1,2 T, we set µF=−0.419,−0.418,−0.416,−0.416 eV respectively. In the case of B=0.1 T we also set η=1 meV and N=500 Fock states to achieve energy diagonalization convergence. For other non-topological anisotropic materials like phosphorene, this minimal behavior of the transmittance still remains when the energy gap is closed, the Faraday angle being much smaller (in absolute value) than in silicene and HgTe QWs. In this case the critical electric potential where the transmittance reaches a minimum depends on the magnetic field. Therefore, these extremal properties of transmittance/absortance and chirality change of Faraday angle at the critical point turn out to provide sharp markers of either the topological phase transition or the energy gap closure. Acknowledgments Funding information We thank the support of Grant PID2022-138144NB-I00 funded by Spanish MICIU and Junta de Andalucía through the projects FEDER/UJA-1381026 and FQM381. AM thanks the Spanish MIU for the FPU19/06376 predoctoral fellowship. OC is on sabbatical leave at Granada University, Spain, since the 1st of September 2023. OC thanks support from the program PASPA from DGAPA-UNAM. A Landau levels plot versus external magnetic field We provide an additional plot of the Landau levels of the three different materials as a function of the external magnetic field B. Critical values of the electric field and layer thickness are selected, that is, in the case of the silicene ∆z=∆so =4.2 meV, for the HgTe QW λ=λc=6.17 nm, and for the phosphorene ∆z=−Eg=−1.52 eV. 22 SciPost Phys. 16, 077 (2024) s=1, ξ=1 s=-1, ξ=1 0 1 2 3 4 5 -15 -10 -5 0 5 10 15 B(T) Energy En sξ/Δso E0-- E0++ E0-+ E0+- Δz=Δso=4.2meV (a) Silicene. s=1 s=-1 012345 -50 0 50 100 B ( T ) Energy En s(meV) λ=λc=6.17nm E0+ E0- (b) HgTe QW. Even Odd 012345 -0.46 -0.44 -0.42 -0.40 -0.38 B(T) Energy En(eV) E0even Δz=-Eg=-1.52 eV (c) Phosphorene. Figure 12: Energies (Landau levels) of (a) Silicene, (b) HgTe QW, and (c) Phosphorene as a function of the external magnetic field B. Critical values of the electric field and layer thickness are selected, that is, in the case of the silicene ∆z=∆so =4.2 meV, for the HgTe QW λ=λc=6.17 nm, and for the phosphorene ∆z=−Eg=−1.52 eV. B Silicene conductivity in the circularly polarization basis We complete the analysis of magneto-optical properties of graphene analogues by discussing the case of circularly polarized light. In this case, the conductivity is σ±(Ω)=σx x (Ω)±iσx y (Ω) for right-handed (+) and left-handed (-) polarization [91]. Therefore, the absorptive part is Re(σ±) = Re(σx x )∓Im(σx y ). In Figure 13, we present both absorptive parts Re(σ±)for a silicene monolayer under an electric potential ∆z=0.5∆so as a function of the frequency of the incident light Ω. The conductivity parameters are specifically chosen to reproduce the results in [9], that is, µF=3.0∆so,B/∆2 so =657 G/meV2,T=0 K and η=0.05∆so. Note that we have defined the conductance quantum as σ0=e2/h=38.8µS, whereas the authors in reference [9]take σ0=e2/(4ħh). 23 SciPost Phys. 16, 077 (2024) Re[σ+(Ω)] Re[σ-(Ω)] 0 5 10 15 20 0 10 20 30 40 50 ℏΩ/Δso Conductivity σ±(Ω)/σ0 Δ z=2.1 meV μ F=12.6 meV B =1.15895T T =0 K η =0.21 meV Figure 13: Conductivity absorptive parts Re(σ±) = Re(σx x )∓Im(σx y )for righthanded (+) and left-handed (-) polarization in a silicene monolayer under an electric potential ∆z=0.5∆so, as a function of the polarized light frequency Ω(in σ0=e2/h units). We set the conductivity parameters µF=3.0∆so,B/∆2 so =657 G/meV2, T=0 K and η=0.05∆so as in Ref. [9]. C HgTe quantum well conductivity with Zeeman effect We recalculate the conductivity of the HgTe quantum well with and without Zeeman coupling to support the argument that the results are qualitatively equivalent, the quantitative differences being small. A layer thickness of λ=7.0 nm is selected, so the material parameters are α=365 meV·nm, β=−686 meV·nm2,δ=−512 meV·nm2, and µ=−10 meV, as taken from Ref. [49]. In Figure 14, we plot the real and imaginary parts of the longitudinal σx x and transverse σx y conductivities as a function of the polarized light frequency Ω. The conductivity parameters are chosen to reproduce the results in [17]with Zeeman coupling, that is, µF=8 meV, B=5 T, T=1 K and η=1 meV. The conductance quantum used here is again σ0=e2/h=38.8µS, whereas the authors in reference [17]take σ0=e2/ħh. D Animations of the energy spectrum and conductivities Attached in the Supplemental Material [52]is a series of animations called: -Silicene_Conductivity_and_Energy_VS_Omega.gif, -HgTe_Conductivity_and_Energy_VS_Omega.gif, -Phosphorene_Conductivity_and_Energy_VS_Omega.gif, where we plot the energy spectrum at right, and the real part Re[σx x (Ω)] and Re[σx y (Ω)] of the conductivity components at left, for three different materials studied in the main text: silicene, HgTe QW, and phosphorene. The external electric field ∆zin the case of the silicene and phosphorene, and the layer thickness λof the HgTe QW, are used as “time coordinate” on the animations, so each frame corresponds to one value of these control parameters. The conductivities are plotted as a function of the polarized light frequency Ω, and they change in each frame according to the values of ∆zor λ. Therefore, we can observe how the main peaks of the longitudinal conductivity Re(σx x )merge for the critical values ∆(0) z=∆so =4.12 meV (silicene) or λ=λc=6.17 nm (HgTe QW), where the topological phase transition occurs in these 2D materials. 24 SciPost Phys. 16, 077 (2024) Re[σxx(Ω)] Im[σxx(Ω)] -10 -5 0 5 10 15 Re[σxy(Ω)] Im[σxy(Ω)] 0 50 100 150 200 -15 -10 -5 0 5 10 Conductivity σij(Ω)/σ0 ℏΩ (meV) λ =7 nm μ F=8 meV B =5T T=1 K η=1 meV (a) Zeeman Re[σxx(Ω)] Im[σxx(Ω)] -5 0 5 10 15 Re[σxy(Ω)] Im[σxy(Ω)] 0 50 100 150 200 -15 -10 -5 0 5 10 Conductivity σij(Ω)/σ0 ℏΩ (meV) λ =7 nm μ F=8 meV B =5T T=1 K η=1 meV (b) No Zeeman Figure 14: Real and imaginary parts of the longitudinal σx x and transverse Hall σx y (magneto-)optical conductivities in a bulk HgTe QW of a thicknesses λ=7.0 nm, as a function of the polarized light frequency Ω(in σ0=e2/hunits) with and without Zeeman coupling. We set the conductivity parameters µF=8 meV, B=5 T, T=1 K and η=1 meV, as in Ref. [17]. In the case of the phosphorene, we only observe the degeneration of Landau levels n=0 and n=1 in the conductivity around the electric potential ∆z≃ −1.53 eV. That is, the electronic transitions Eodd 1→Eeven 2and Eeven 0→Eodd 3have a similar energy and share a longitudinal conductivity peak (main peak at left in the gif), until the degeneration breaks for electric fields approximately higher than −1.53 eV, when both electronic transitions will have different energies so the main peak will split into two. On the other hand, the energy spectrum is static on the animation, as it is plotted as a function of all the values that ∆zor λtake. However, we plot a moving vertical dashed line on it, representing the value of ∆zor λin the conductivity frame. On top of this vertical line, we also draw arrows representing the electronic transitions allowed between Landau levels (LLs) for the specific value ∆zor λ, where the Fermi energy µFis represented by an horizontal dashed line. The color of the arrows is the same as the color of the points plotted on the top of the longitudinal conductivity main peaks. The length of the arrows represents the energy difference |En−Em|between the corresponding Landau levels in this particular electronic transition n↔m, which also coincides with the frequency ħhΩof the longitudinal conductivity peak associated with this transition. Therefore, when two arrows have the same length, we can observe two longitudinal conductivity peaks merging at the critical point. We have only drawn the arrows of the main peaks or lower Landau level electronic transitions for the sake of simplicity. E Faraday angle contour plots For completeness, in Figure 15 we show the variability of the Faraday angle for silicene across the parameter space: polarized light frequency ħhΩ, electric field potential ∆z, magnetic field B, temperature Tand chemical potential µF}, using several contour plots corresponding to different cross sections. 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