This is the accepted version of the article: Vila M., Hung N.T., Roche S., Saito R.. Tunable circular dichroism and valley polarization in the modified Haldane model. Physical Review B, (2019). 99. 161404: - . 10.1103/PhysRevB.99.161404. Available at: https://dx.doi.org/10.1103/PhysRevB.99.161404 Tunable Circular Dichroism and Valley Polarization in the Modified Haldane Model Marc Vila,1, 2 Nguyen Tuan Hung,3 Stephan Roche,1, 4 and Riichiro Saito3 1 Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain 2 Department of Physics, Universitat Autònoma de Barcelona, Campus UAB, Bellaterra, 08193 Barcelona, Spain 3 Department of Physics, Tohoku University, Sendai 980-8578, Japan 4 ICREA–Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain We study the polarization dependence of optical absorption for the modified Haldane model, which exhibits antichiral edge modes in presence of sample boundaries and has been argued to be realizable in transition metal dichalcogenides or Weyl semimetals. A rich optical phase diagram is unveiled, in which the correlations between perfect circular dichroism, pseudospin and valley polarization can be tuned independently upon varying the Fermi energy. Importantly, perfect circular dichroism and valley polarization are achieved simultaneously, a feature not yet observed in known optical materials. This unprecedented combination of optical properties suggests some interesting novel photonic device functionality (e.g. light polarizer) which could be combined with valleytronics applications (e.g. generation of valley currents). Circular dichroism (CD) is the ability of a material LCP and RCP light are equally absorbed. Merging to exhibit a significant differential absorption of left- these two properties together would result in novel and right-handed circularly polarized light (LCP devices possessing both photonic and valleytronic and RCP). This optical property appears in chiral functionalities. Several attempts on inducing CD materials and molecules, and presents large interest in TMDs have been reported [19, 20], but the effect for biopharmaceutical applications, as well as molec- was produced by external chiral molecules. There- ular spectroscopy or even quantum information pro- fore, a two-dimensional system inherently present- cessing [1–4]. However, CD is generally too small for ing both valley polarization and circular dichroism practical applications [5, 6] so that there is a great is still missing. interest in exploring if novel materials such as carbon A paradigm for a 2D lattice model was introduced nanotubes [7] or metamaterials [2, 5] cound manifest by Haldane [21]. Together with the Kane and Mele stronger CD. On the other hand, two-dimensional model [22], these models have pioneered the devel- materials, ranging from insulating hexagonal boron opment of time-reversal invariant topological insu- nitride (hBN) and semiconducting transition metal lators [23]. Very recently, Colomés and Franz have dichalcogenides (TMDs) to semimetallic graphene, proposed a modified Haldane model exhibiting dis- display optical properties that differ from their bulk persive antichiral edge states, which are modes prop- parental materials [8, 9]. TMDs exhibit particularly agating in the same direction at both parallel rib- strong light absorption, and owing to the broken in- bon edges and compensated by bulk counterprop- version symmetry and time reversal invariance, light agating modes [24]. Such edge states, resilient to with opposite handedness leads to preferential pop- disorder, are believed to resemble the Fermi arcs in ulation of the K or K’ valleys of the Brillouin zone. 3D Dirac and Weyl semimetals and could be use- Such effect is known as valley polarization (VP) [10] ful for low-power electronics or spintronics. While or valley-selective circular dichroism [11], and it is the experimental realization of the Haldane model of paramount importance to realize valleytronic ap- has been achieved using ultracold atoms in an op- plications [12, 13]. Strong VP and large quantum tical lattice [25], a realization of the modified Hal- efficiency have been achieved with TMDs [11, 14– dane version might be possible with Weyl semimet- 17]. Noteworthy, this valley polarization is not re- als, TMD monolayers [8, 24, 26–28] or ferromag- stricted to TMDs, but to any hexagonal lattice with net/graphene/TMD heterostructures [29, 30]. Be- broken inversion symmetry [10], including hBN or sides, given that the Haldane model has recently novel graphene-based materials [18]. Nevertheless, been predicted to exhibit perfect CD when the sys- the presence of VP as an intrinsic property suggests tem is topological nontrivial [31], the exploration of that CD is not simultaneously achievable, since both the fundamental optical properties of the modified nihilation) operator for a spinless electron on site Ri (Rj ) of the honeycomb lattice. The second term is a next-nearest-neighbor (NNN) hopping with com- plex value t2 e−iνij φ that breaks time reversal sym- metry [21, 24]. In the Haldane model νij = ±1 with +1 for counterclockwise (ccw) and -1 for clockwise (cw) hoppings. In contrast, in the modified Haldane model νij = ±1 for sublattice A while νij = ∓1 for sublattice B, which besides breaking time reversal symmetry, also introduces a scalar potential with an opposite sign in each valley, which is crucial for ob- taining both antichiral edge states [24] and optical absorption tunability, as discussed herebelow. The K’ third term breaks inversion symmetry when the po- tential is different for the sublattices A (µ = +1) and K B (µ = −1) and opens a gap with value Egap = 2∆ at the K/K0 points. It is also useful to express Eq. (1) in reciprocal space and expand it near the vicin- ity of K/K0 up to first order in momentum q: FIG. 1. Band structure of the modified Haldane model for a given set of parameters that produces an (a) indi- H = ~vF (κσx qx + σy qy ) − ta2 σ0 − κtb2 σ0 + ∆σz , rect semiconductor and a (b) semimetallic phase. In all (2) cases, t1 = 1 eV. The value of the pseudospin polariza- tion tan p (Eq. (3)) is given in color scale. The horizontal with vF√ being the Fermi velocity, ta2 = 3t2 cos φ, dashed lines highlight selected Fermi energies at which tan Vσ (Eq. (6)) and tan θ (Eq. (5)) are calculated in tb2 = 3 3t2 sin φ and κ = +1(−1) for the K (K0 ) Figs. (2) and (3). (c) |Mσ (k)|2 in units of the atomic valley. Here, σ0 is the 2 × 2 identity matrix and dipole vector, for the parameters used in (b), over the σi (i = x, y, z) are the Pauli matrices acting on the full Brillouin zone (superimposed white, dashed line) for sublattice space. The p energy dispersion results in LCP light (left) and RCP light (right). ε(q) = −ta2 − κtb2 ± ∆2 + (~vF q)2 , where the ± sign stands for the conduction band/valence band. As mentioned previously, the NNN hoppings results Haldane model might help to realize circular dichro- in a potential opposite in each valley, which depends ism in presence of valley polarization. on the terms t2 and sin φ (third term in Eq. (2)). In this Letter, we show that the phase diagram However, although the valleys are relatively shifted, of the modified Haldane model presents the possi- the gap is the same and only depends on ∆. We plot bility for simultaneous realization of valley polar- the band structure in Fig. 1(a) and Fig. 1(b) for ization together with a perfect circular dichroism. two different set of parameters ∆, t2 and φ. In both More remarkably, the combination of perfect VP and cases, we identify the Dirac cone shift and gap open- CD can be monitored by varying the Fermi energy, ing, but whereas Fig. 1(a) shows the case of an indi- while the tuning of the modified Haldane parameters rect semiconductor, Fig. 1(b) reveals a semimetallic ({∆, t2 , φ}, see Eq. (1)) allows the description of a phase. This indirect semiconductor√ to semimetallic broad class of materials such as semimetals and indi- transition occurs when | t∆2 | ≤ 3 3 | sin φ|. rect semiconductors. The modified Haldane model The optical properties of the modified Hal- is defined on a honeycomb lattice with broken inver- dane model are calculated based on the perturbed sion and time reversal symmetries. Its tight-binding electron-photon Hamiltonian in the dipole approx- version reads [24]: imation [10, 31, 32] and using the Fourier trans- form of Eq. (1). The optical absorption tran- c†i cj + t2 e−iνij φ c†i cj + ∆ µc†i ci . X X X H = t1 sition probability from an initial state in the va- hi,ji hhi,jii i lence band v to a final state in the conduction band (1) c depends on the squared optical matrix elements The first term is the nearest-neighbor hopping |Mσ (k)|2 = |P · D(k)|2 . Here, P is the polarization with magnitude t1 and c†i (cj ) is the creation (an- vector (P = √12 (1 iσ)T ) and D(k) stands for the 2 dipole vector, that has the form hc|∇|vi. We con- the valence and conduction bands with the Fermi- sider only the case of circularly polarized light prop- Dirac distribution (fFD ). Since the valley-selective agating in the direction perpendicular to the lattice, |Mσ (k)|2 is maximum at the Dirac points, we restrict whose polarization is defined by σ = +1(−1) for ourselves to photons with energies Eph = Egap = LCP (RCP) light. 2∆. Thus, the intensity of optical absorption for a Using the parameters of Fig. 1(b), we numerically circularly polarized light is calculated from calculate the optical matrix elements in the whole Brillouin zone for LCP and RCP. The result is plot- Z ted in Fig. 1(c). One sees that |Mσ (k)|2 presents Iσ (EF ) ∝ |Mσ (k)|2 δ(Ec (k) − Ev (k) − Eph ) valley polarization as in other 2D hexagonal mate- BZ rials with broken inversion symmetry [8, 31]. While × fFD (Ev (k)) (1 − fFD (Ec (k))) dk. (4) LCP (RCP) is maximum at the K0 (K) valley, it is 0 at the K (K0 ). The VP was related to different Here, Ec and Ev are the energy of the conduction signs of the Berry curvature at different valleys due band and valence band, respectively. In this way, to the ∆ term [33, 34]. We reach the same conclusion the circular dichroism at a given EF is expressed by by noting that ∆ produces pseudospin polarization the angle θ (PP), defined as the difference of wavefunction lo- calization between atom A and atom B. This PP is I+ (EF ) − I− (EF ) encoded in the wavefunction and consequently also tan θ(EF ) = , (5) I+ (EF ) + I− (EF ) in D(k) and |Mσ (k)|2 , resulting in VP. Thus, it is insightful to calculate the pseudospin polarization as where I+ and I− are the intensity of absorption for it is related to the optical absorption at the K and LCP and RCP, respectively. Although the optical K0 valleys. We define PP in analogy to the circular matrix elements always present VP, this may not be dichroism (see Eq. (5) and Ref. [6, 31]) as the case for the optical absorption intensity. There- fore, we define the VP measure of the absorbed light d as the difference of intensities between K (IσK ) and |CA (k)|2 − |CBd (k)|2 0 tan p = d (k)|2 + |C d (k)|2 , (d = v, c). (3) K0 (IσK ) [35] |CA B d d 0 Here, CA (k) and CB (k) are the coefficients of the IσK (EF ) − IσK (EF ) Bloch wavefunction in band d of sublattice A and B, tan Vσ (EF ) = . (6) IσK (EF ) + IσK 0 (EF ) respectively, which we obtain by numerical diagonal- ization of the Hamiltonian. The PP is color-plotted In this case, when tan Vσ is +1(−1), light with cir- in the bands of Fig. 1(a) and Fig. 1(b). We observe cular polarization σ is absorbed at the K (K0 ) valley. that the PP is opposite for the valence and conduc- We solve Eqs. (5) and (6) for different values of ∆, tion bands and equal for both valleys, resulting in t2 and φ at three different Fermi levels, identified by VP [31]. Moreover, the PP is maximum at K and dashed lines in Fig. 1(a) and Fig. 1(b). When the K0 which explains why the absorption is also maxi- Fermi energy lies at EFb only electrons at the K val- mum there as seen in Fig. 1(c). Using the notation ley can be excited with a photon energy of the band (c, v) we obtain tan p = (+1, −1) for positive ∆, but gap since the valence band maximum at K0 is un- when ∆ is negative, tan p = (−1, +1) and the VP occupied. In this case, according to Fig. 1(c), only reverses (LCP (RCP) is absorbed at K (K0 )). Fi- RCP will be absorbed and therefore tan V+ (EFb ) = 0, nally, for ∆ = 0, there is no pseudospin polarization tan V− (EFb ) = +1 and tan θ(EFb ) = −1. This is in and both valleys absorb both LCP and RCP lights. sharp contrast with other hexagonal 2D materials or We note that the wavefunction does not depend on the Haldane model where VP and CD never occur t2 or φ and therefore |Mσ (k)|2 is not affected by the simultaneously [31]. Moving the Fermi energy to the scalar potential that shifts the Dirac cones. conduction band (EFt ) will restrict the light absorp- Next, we show that by changing the Fermi energy, tion to the K0 valley since the conduction band at the modified Haldane model also exhibits perfect CD the K point is occupied. Hence, since the optical ma- concurrently with VP. To that end, we now compute trix elements at K0 are opposite from those at the K the optical absorption with the Fermi’s golden rule valley, we obtain tan V+ (EFt ) = −1, tan V− (EFt ) = 0 considering the occupation probability of states in and tan θ(EFt ) = 1. Thus, changing the Fermi energy 3 EF a semiconductor to a semimetal, but the combina- tan V+ tan V− tan θ tion of VP and CD do vary when ∆ or sgn(t2 )φ EFt changes sign. Moreover, one can identify that the =+1 pattern in EFb is the same as in EFt but inverted =−1 with respect of sgn(t2 )φ. The situation is differ- =0 ent for the case of EFm . Firstly, now the semicon- EFm tan p = (+1, − 1) ducting and semimetallic phases do not have the Indirect semiconductor same optical properties. As mentioned above, no Semimetal absorption occurs in the semimetallic case, leading tan p = (−1, + 1) to tan V+ = tan V− = tan θ = 0. However, the semi- EFb Indirect semiconductor conducting band structure allows optical absorption, Semimetal but since each valley absorbs opposite light polar- ization, CD vanishes. The valley polarization swaps when changing the sign of ∆ due to PP inversion, as pointed out previously, but it remains the same FIG. 2. Phase diagram of the modified Haldane model when changing sgn(t2 )φ. showing the rich combination of VP, CD and PP (see text for details). It is also interesting to look at the transition points. These are the lines sgn(t2 )φ = 0 and ∆/|t2 | = 0. The former case is the situation of a 2D material with broken inversion symmetry, as reverses both the CD and the VP. These two situ- in TMDs. In this case, since the Dirac cones are ations are the same for the semiconductor and the not shifted, the valence band maximum is empty semimetallic phases, but differ in the case EF = EFm , at EFb and the conduction band minimum is oc- that is, in the midgap for the indirect semiconduc- cupied for EFb . Thus, no absorption will occur for tor and crossing an electron and hole pocket for the EFb and EFt even though there is both VP and CD semimetal regime. In this circumstance, the indi- for finite values of φ. For ∆/|t2 | = 0, the Dirac rect semiconductor behaves like TMDs, where both cones are shifted in energy but the system is always valleys absorb opposite light polarization with van- (semi)metallic with no energy gap. Nevertheless, we ishing CD. On the other hand, in the semimetallic can promote optical absorption if we change the pho- band structure, no light is absorbed because the con- ton energy. If we take as an example sgn(t2 )φ > 0, duction band (valence band) is occupied (empty) at the K point is lowered down in energy. If we now the K (K0 ) point. place the Fermi level there (which we label as the EFb By studying all possible combinations of the modi- case) such that no states near the K0 are occupied fied Haldane parameters, we obtain a complex phase and use small but nonzero photon energy, only elec- diagram showing a large variety of CD and VP pos- trons at K will absorb light. Since the PP is zero, sibilities, like a traffic signal, and different depen- I+ (EF ) = I− (EF ) which leads to the peculiar case dence with the Fermi level. This is depicted in Fig. tan V+ = tan V− = 1 and tan θ = 0. Finally, for the 2, where tan Vσ (EF ) and tan θ(EF ) are shown for special case of sgn(t2 )φ = ∆/|t2 | = 0, corresponding the three selected Fermi energies (three planes in to graphene, we find that no matter which EF and Fig. 2) and for all possible choices of ∆, t2 and φ. Eph we choose, there is not PP, VP or CD. To give a visual illustration of the results, we draw three colored circles for each region of the phase We summarize all possible combinations of pres- diagram. From left to right, the circles represent ence and absence of PP, VP, and CD in Table 1. As tan V+ , tan V− and tan θ with values +1, −1 and 0 novel phases, mHd gives access to an exclusive PP for red, blue and grey colors, respectively. The plot- ( ) in absence of VP and CD (×). Differently, mHh ted red line indicates the phase boundary from the describes the situation with nonzero ∆ and φ values semiconductor to semimetallic transition as is shown and for Fermi energies EFb and EFt , which permits by the background colors, which in addition also dis- all polarizations and CD to be active. We note that play the pseudospin polarization. is not possible to obtain CD in absence of PP, since We observe that VP and CD occurs simultane- whenever PP vanishes, I+ (EF ) = I− (EF ) for both ously at any point for EFb and EFt except at φ = 0. valleys, which implies that we cannot achieve CD. In these regions, there is no change when going from Additionally, the mHh presents a particularly 4 PP VP CD Model/Material × × × Graphene, H & mHa × × - K K' × × Hb , mHc × - × × mHd × He × hBN, TMD, Hf , mHg mHh K K' TABLE I. Combination of presence ( ) and absence (×) of pseudospin, valley polarization and circular dichro- ism and the materials/models where such combination occurs. The Haldane/modified Haldane model is ab- breviated as H/mH. Parameters are a : ∆√= φ = 0, EFm (semimetal, ∆ = 0); b : ∆/t2 = 3 3; c : EFb,t (∆ = 0); d : EFm (semimetal, ∆ 6= 0), EFb,t (φ = 0) ; e : topological insulator; f :trivial insulator; g : EFm (indirect semiconductor); h : EFb,t (∆ 6= 0, φ 6= 0). FIG. 3. VP and CD as a function of the Fermi energy for the case corresponding to the cross marked in Fig. 2. The insets illustrate the band structure for these param- unique feature which is the tunability of both VP eters in addition to the proposed mixed device function- and CD simultaneously. Fig. 3 illustrates such prop- ality: a light polarizer and valley selective excitation. erty for varying the Fermi energy for a selected case in the phase diagram (marked with a cross in Fig. 2 and connected vertically with a grey line). This dospin and valley polarization and circular dichro- variation of the Fermi energy could be performed in ism. The realization of both perfect valley polariza- an experiment by applying an external gate voltage. tion and circular dichroism enable the concept of in- The resulting absorption profile of perfect PP, VP tegrating photonics and valleytronics in a single ma- and CD suggests interesting technological applica- terial or device. Moreover, this suggests an optical tions. Indeed, by combining perfect VP and CD, one readout for discovering materials in which antichiral could envision a device that can be useful for valley- edge states could form [24]. tonics and simultaneously serve as a tunable light M. Vila acknowledges the Graphene Flagship polarizer (as illustrated in Fig. 3, insets). At low grant and the Department of Physics of Tohoku Uni- Fermi energies, and using linear polarized light, the versity for its hospitality. The research leading to material would absorb the LCP component at the these results has received funding from ”La Caixa” K valley and consequently transmitting RCP light. Foundation by supporting M. Vila. S. Roche was These excited electrons could be driven into a circuit supported by the European Union Horizon 2020 re- as a source of valley current and at the same time search and innovation programme under grant agree- the RCP light could be used for any application re- ment No. 696656 (Graphene Flagship). ICN2 is quiring circularly polarized light. By increasing the funded by the CERCA Programme / Generalitat de Fermi energy, we could reverse the device function- Catalunya, and is supported by the Severo Ochoa ality and thus create valley current at the K0 val- program from Spanish MINECO (Grant No. SEV- ley while transmitting LCP light. The possibilities 2017-0706). R. Saito acknowledges JSPS Kakenhi could go even further: by having regions of the de- (Grant No. JP18H01810). N.T. 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