## Abstract

We theoretically investigate quantum phase transitions from topologically non-trivial to trivial states in three-dimensional hybridized topological insulator (TI) ultra-thin films. The interplay between the hybridization of the top and bottom surface states (SSs) and Zeeman energy gives rise to topological and normal insulating phases. By tuning the Zeeman energy, we can drive phase transition between these two phases by closing and reopening the band gap. Furthermore, we impinge a Gaussian beam on the surface of the 3D TI to study the Faraday rotation (FR) and magneto-optical Kerr effect (MOKE) in the presence of an external magnetic field, while explicitly taking into account the hybridization between the top and bottom Dirac SSs of the TI film. The FR and MOKE change sign when the polarization is changed from *s* to *p*. We demonstrate that giant FR and MOKE can be achieved by tuning the external magnetic field. Furthermore, the MOKE signal takes a sharp anti-phase peak at the charge neutrality point as the system goes from the quantum spin Hall insulator (QSHI) to the semimetallic state. Lastly, the MOKE and FR in the bottom and top SSs can be independently tuned for intra-band and inter-band transitions via n-type and p-type doping respectively.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Three-dimensional topological insulators (TIs) are quantum materials that behave like bulk insulators (BIs) in their bulk but there exist topologically protected metallic surfaces with massless Dirac-type band structure that are responsible for most unique and exotic electronic and optical properties [1–3]. The Dirac-like band structure of the surface states are robust against perturbations of the system parameters, as they are protected by topological properties of the bulk quantum mechanical wave functions. Time-reversal symmetry (TRS) in these 3D TI materials guarantees the pairing of top and bottom Dirac surface states (SSs) [4,5]. These characteristic SSs have a topological origin and are potentially useful for the design of nano-scale devices in spintronics and for practical applications in quantum computations [6,7].

The discovery of topological phases and topological phase transitions (TQPTs) in a wide range of new quantum materials, such as graphene, staggered 2D semiconductors (i.e. silicene, germanene, stanene), 2D transition metal dichalcogenides (TMDCs), Dirac-Weyl semimetals, and their artificial analogs have provided a new frame of mind to the understanding of the origin of quantum states of matter. Besides finding new quantum materials, searching the ways to tune a normal insulating (NI) phase into a topological one is of both theoretical and experimental importance. In 3D topological insulators thin films, one way to drive such TQPTs is to tune the thickness $L$ of the film. The topologically protected SSs of a TI thin film have a finite decay length (also called the penetration depth of the SSs) over which the SSs decay into the bulk region. As the thickness $L$ of the TI thin film becomes comparable to, or smaller than, the penetration depth, quantum tunneling between SSs occurs due to which the top and bottom SSs hybridize to open up an energy gap $\Delta _{H}$ at the Dirac points [8,9]. This can happen for 1 to 5 quintuple layers with a thickness of the order of $L$ =5 nm [10,11]. In the small $L$ limit, the hybridization gap can be approximated by the relation $\Delta _{H} = 2B_{1} \pi ^{2}/L^2$, where $B_{1}$ is a system parameter. For the band structure in the $Bi_{2}Se_{3}$ the parameter $B_{1}$ is equal to $10 \cdot \mathrm{eV} \AA^{2}$, which is obtained by first principles calculations [9,12].

However, tuning the thickness of the TI in real space is not a suitable way to drive TQPTs. There are other efficient ways in which TQPTs can be extrinsically induced in TIs, such as through band distortions in momentum space by in-plane or out-of-plane magnetic fields [13,14], pressure and temperature [15,16], and an external electric field [17,18]. There are studies that the response of TI thin films in an external magnetic field is highly nontrivial [19,20]. Recently, successful TQPTs from NI to quantum spin Hall (QSH) state in TIs through the application of electric and magnetic field have been experimentally reported [21].

In addition to having interesting topological features, TI thin films exhibit Faraday and Kerr rotations in the THz regime. Faraday and Kerr rotations are non-reciprocal magneto-optic (MO) effects that describe the rotation of linearly polarized light when it is respectively transmitted or reflected from a magnetic medium. At the surface of a TI, both of these effects originate from the breaking of TRS either by an external applied magnetic field or by doping the system with magnetic impurities [22]. Some features of TIs thin film are analogous to graphene and other 2D quantum materials, however, the MO effects observed in a mono-layer graphene sheet exist only at low frequencies ($<$ 3 THz) and that too at the expense of large magnetic fields [23,24]. The FR and MOKE have been studied in other 2D quantum materials like staggered silicene [25]. In the THz range, W. K. Tse *et. al.* theoretically studied the MO Faraday and Kerr effects of TIs thin films and found that MOKE exhibits a giant $\pi /2$ rotation [26,27]. Experimentally, Colossal MOKE and FR in the presence of a strong magnetic field in ${Bi_{2}Se_{3}}$ thin films [28,29] have been investigated. It has been demonstrated that THz FR in pure or magnetic TI films is equal to the fine structure constant [30]. Recently, FR and cyclotron resonances magneto-transmission spectroscopy of $(Bi_{1-x}Sb_{x})_2 Te_{3}$ and THz Faraday and Kerr rotation spectroscopy measurements on $Bi_{1-x}Sb_{x}$ thin films have been experimentally reported [31,32]. Nedoliuk *et. al.* observed colossal FR on Landau levels in high-mobility encapsulated graphene in the mid-infrared and THz ranges [33].

In this paper, we employ a different mechanism to drive TQPTs between different phases, and study FR and MOKE in a TI thin film. We make use of the Zeeman energy as a control knob to close and reopen the band gap in order to drive phase transition between a band insulator state and a topological insulator state in a hybridized TI thin film. Hybridization provides mass to the Dirac fermions on the top and bottom SSs. In the absence of Zeeman energy ($\Delta _{z}=0$), the energy spectrum of the 3D TI is gapped (because of the hybridization energy) and electron-hole symmetry which leads to Shubnokov-de Haas oscillations. When $\Delta _{z}$ is less than $\Delta _{H}$, the system is in the quantum spin Hall insulator (QSHI) regime. However, by tuning $\Delta _{z}=\Delta _{H}$ we enforce a TQPT at the charge neutrality point (CNP) where the energy of one of the $n$ = 0 LLs becomes zero.

In this article, we also study the FR and MOKE in the 2D SSs of a TI thin film, and the effects of a magnetic field and chemical potential. We believe that the magnetic field dependent MO effects provide a convenient scheme that can be directly utilized for optical measurement of different quantum topological features, magnetic field sensing, Faraday rotators, isolators, current sensors, optical modulation and communication [34–36].

## 2. Quantum transport properties of TI thin films

The effective Hamiltonian for an ultra-thin film system of hybridized TIs such as $Bi_{2}Te_{3}$, $Bi_{2}Se_{3}$, $Sb_{2}Te_{3}$ and $(BiSb)Te$ in the presence of an out-of-plane magnetic field in the $z$ direction may be written as [37,38]

For $\tau _{z}=+1$, the Hamiltonian in Eq. (2) can be written as

Here, $t=\textrm {sgn(n)}$ denotes the electron/hole band, $E_{B}=v_{F}\sqrt {2e\hbar B}$, $\Delta _{\tau _{z}}= (\Delta _{z}+\tau _{z}\Delta _{H})$ and $n$ is an integer, representing Landau quantization. We note that the zeroth LL is quantum anomalous: its magnitude is independent of the magnetic field, and its sign depends on the specific Dirac SSs. When $\Delta _{z}=0$, the energy spectrum is degenerate for the two spin states. When $\Delta _{z}$ is turned on, the bands become spin-split representing two energy gaps determined by $2\Delta _{min}$ and $2\Delta _{max}$. In the regime $\Delta _{z}<\Delta _{H}$, the energy spectrum remains gapped and represents a topological phase, i.e., the QSHI phase. When $\Delta _{z}$ is increased to $\Delta _{z}=\Delta _{H}$, the lower bandgap closes and the system transforms to the CNP state. Furthermore, increasing $\Delta _{z}$ beyond $\Delta _{H}$, the band gap opens up again and the system transforms to the BI phase. The corresponding solutions of eigenstates of the top and bottom surface states are

In the presence of a magnetic field, there are two contributions to the MO conductivities, one is longitudinal and the other one is the transverse Hall conductivity. The next task is to calculate the MO conductivity in the presence of a perpendicular magnetic field for the TIs thin film systems taking into account the Zeeman and hybridization interactions. The standard Kubo formula is used in the local regime to derive the following general expressions for the MO conductivity of TI thin films [26,40],

In the limit $\Delta _{z}=\Delta _{H}=0$, the above conductivity expressions reduce to the result for graphene for a single valley [41].

## 3. Fresnel coefficients, magneto-optical rotations and ellipticities

In order to calculate the MO effects, we consider an ultrathin film of a TI of thickness $L\rightarrow 0$. In these conditions the TI thin film SSs can be treated as 2D surfaces. We consider an arbitrarily polarized, well-collimated, monochromatic, Gaussian beam of light propagating in vacuum impinging from one medium to the planar interface of the TI-substrate system at an incidence angle $\theta _{\psi }$. The interface of air and TI thin film is located at the $z$ = 0 plane. The beam of light of frequency $\Omega$ has polarization in an arbitrary direction, and is propagating through the incident and transmitted materials with *relative* permittivity and permeability $\varepsilon _n$ and $\mu _n$ respectively, where $n=\textrm {(1,2)}$. The beam makes an angle $\theta _{\chi }$ in the substrate. Fresnel reflection and transmission coefficients can be obtained by modeling TI thin film as a surface current density $K=\sigma \cdot E|_{z=0}$ and applying electromagnetic boundary conditions on either side of the thin film at $z$ = 0. The electric and magnetic fields on the two sides are connected by the Maxwell boundary conditions:

The Fresnel reflection and transmission coefficients of the TI thin film substrate system with an external imposed magnetic field are obtained as [42]:

Here, $k_{1z}=k_{1}\cos (\theta _{\psi })$ and $k_{2z}=k_{2}\cos (\theta _{\chi })$. The conductivities $\sigma _{L}(\sigma _{T})$ are the longitudinal (transverse) components. For homogeneous, isotropic media, $\sigma _{L}=\sigma _{xx}=\sigma _{yy}$ and $\sigma _{T}=\sigma _{xy}=\sigma _{H}$. The wave vectors are $k_1$ and $k_2$, $k_{n}=\Omega \sqrt {\mu _{n}\varepsilon _{n}}$, $Z_n=Z_0\sqrt {\mu _{n}/\varepsilon _{n}}$ and $Z_0=\sqrt {\mu _{0}/\varepsilon _{0}}$, where $\mu _{0}$ and $\varepsilon _{0}$ are the vacuum permeability and permittivity respectively. In our case, for the sake of simplicity we consider that the medium 1 is vacuum ($\varepsilon _{1}$= 1, $\mu _{1}$=1) and medium 2 (the substrate) is a non-magnetic material $\mu _{2}$=1. Clearly, the Fresnel coefficients which subsequently determine the magneto-optic rotations and ellipticity are proportional to the MO longitudenal and Hall conductivities. With these reflection and transmission coefficients in hand we are finally able to evaluate the Faraday, Kerr rotations and ellipticities. For example, for the incident $s$ and $p$ polarization, Faraday rotation and ellipticity are computed using the expressions

Similarly, Kerr rotations angles and ellipticities are

The subscripts identify the Faraday (F) or Kerr rotation (K) as well as the polarization state ($s$ or $p$), while the superscripts specify the top and bottom SSs. In the limit $\xi \ll 1$, Eqs. (28) and (29) reduce to $\Theta ^{\tau _{z}=\pm 1}_{\textrm {F},\textrm {s}/\textrm {p}}\approx \textrm {Re}( \xi ^{\tau _{z}=\pm 1}_{\textrm {F},\textrm {s}/\textrm {p}})$ and $\eta ^{\textrm {s}/\textrm {p}}_{\textrm {F}}\approx \textrm {Im}( \xi ^{\tau _{z}=\pm 1}_{\textrm {F},\textrm {s}/\textrm {p}})$ and vis-a-vis for MOKE.

## 4. Results and discussion

To fully understand the FR and Kerr rotation in a TI thin film-substrate, we plot the real part of the longitudinal and Hall conductivities as a function of the photonic energy for three different values of magnetic field strengths, i.e. for 1, 3 and 5 T as shown in Figs. 1(a) and (b). The parameters for simulation are mentioned in the caption of the figure. Notably the angle of incidence $\theta _{\psi }$ is taken to be $30^\circ$ for all simulations. The magnitudes of the MO effects depend on $\theta _{\psi }$. For example, an increase in the incidence angle diminishes the amount of rotation. A similar trend can be seen in the MO response of graphene [43]. In the QSHI phase, the absorption peaks can be seen for interband transitions. Each transition is represented by a Lorentzian peak with a half width at half maximum $\eta$. Following the terminology mentioned in [44], the peaks corresponding to different transitions $E_{m}^{\tau _{z}=\pm 1} \rightarrow E_{n}^{\tau _{z}=\pm 1}$ are labeled as $T_{mn}^{{\tau _{z}=\pm 1}}$. In this paper, while investigating the FR and Kerr rotation in TI thin film-substrate system, we restrict ourselves to the lowest magneto-excitation transition frequencies (originating only from the $T_{01}^{{+1}}$ and $T_{-10}^{{-1}}$ transitions), unless specifically mentioned otherwise. We can see resonant peaks when the incident photon hits the magneto-excitation energy gap. As we increase the strength of the applied magnetic field, the MO excitations shift towards higher frequencies. In Fig. 1(a), the first two absorption peaks correspond to $T_{01}^{{+1}}$ (top surface) and $T_{-10}^{{-1}}$ (bottom surface) transitions in which one of the participating level is zeroth LL. For higher magnetic field we can see that the spectral weight increases with the increase in the strength of the applied magnetic field. The spectral weight decreases for higher magneto-excitation frequencies. To shed more light on MO excitations, we plot in Fig. 1(c)-(f) the $s$ and $p$ polarized magneto-transmission and reflection coefficients as a function of the incident photonic energy for different strengths of the magnetic field. We can see the normalized magneto-transmission and reflection peaks of the magneto-excitaion energy for different transitions. Table 1 summarizes these results.

Next, we study the Kerr and Faraday rotation of TIs thin-film-substrate system by tuning the magnetic field for finite hybridization $\Delta _{H}=4$ meV. The parameters used are $\theta _{\psi }=30^{\circ }$, $\eta =0.15 ~\Delta _{H}$ (estimated from experimental findings [45]), refractive index $n_{2}=1.84$ and chemical potential $\mu _{F}=0$, so that the inter-band transitions bridge across the valance and conduction bands. Figure 2(a) shows the $p$ polarized Kerr spectra as a function of the incident photonic energy for different values of the magnetic field strength. Borrowing ideas from [25], the signal originating from a single transition is dispersive Lorentzian, with a positive peak followed by a negative peak (or vice versa). We call this an anti-phase peak. We observed the first anti-phase peak of the Kerr rotations, which occurs when the incoming photon frequency resonates with a Landau level transition $T_{01}^{{+1}}$. Similarly we can see a relatively small rotation anti-phase peak (with a negative followed by a positive) for second magneto-excitation transition $T_{-10}^{{-1}}$. Like other 2D materials including graphene [40,46] the TI energy levels are strongly dependent on the magnetic field $B$. As we increase the strength of the magnetic field, the MO excitations shift towards higher frequencies. The amount of the giant Kerr rotation concomitantly increases with the increase of the magnetic field as shown in Fig. 2(a). The Kerr rotation angle increases uniformly with the magnetic field and reaches $\pm 15^{\circ }$ at a magnetic field of 5 T.

Figure 2(b) shows the $s$ polarized Kerr rotation spectra for different strengths of the magnetic field indicating that the anti-phase peaks switch sign. Furthermore, the sign of the MO rotation can be controlled by the polarization of the incident light. Figure 2(c) shows that $s$ polarized FR as a function of the incident photonic energy in the QSHI phase. The $s$ polarized FR angles for the first two anti-phase peaks are $\sim \pm 6.5^{\circ }$. The main role of the modulated magnetic field is to shift the position of the MO excitation energies and also to change the magnitude of MO effects.

Figures 2(d)-(f) represent the MO Kerr and FR rotations for different Zeeman interaction term $\Delta _{z}$ in the THz regime. We display in Fig. 2(d) the $p$ polarized Kerr rotation angle versus incident photonic energy for several values of the Zeeman field $\Delta _{z}$. The first and second anti-phase peaks correspond to $T_{01}^{{+1}}$ and $T_{-10}^{{-1}}$ transitions in all three regimes. In the QSHI phase $(\Delta _{z}=0.5 \Delta _{H})$ with $B=1~$T, these magneto-excitation energies are calculated as 20.3 meV (4.9 THz) and 25.1 meV (6.1 THz). The magneto-excitations frequencies are presented in Table 2 for the first two transitions ($T_{01}^{{+1}}$ and $T_{-10}^{{-1}}$) in the three distinct topological regimes. At the CNP ($\Delta _{z}=\Delta _{H}$) the $E_{0}^{{-1}}$ has exactly zero energy (see Supplement 1). For QSHI phase $E_{0}^{{-1}}$ is electron like and for NI phase it is hole like (see Supplement 1). The amount of Kerr rotation angles for both QSHI and CNP is $\sim \pm 6.0^{\circ }$ for $\tau _{z}=+1$. For higher MO excitation energies, we have $T_{-10}^{{-1}}$ LLs transitions and the Kerr rotation angle $\geq \pm 2.0^{\circ }$. In the NI phase ($\Delta _{z}=2\Delta _{H})$) the separation between the anti-phase pair keeps on growing and the peaks gradually shift towards higher frequencies (blue shifted). Here, the magnitude of the Kerr rotation angle is small as compared to the QSHI and CNP phase.

In Fig. 2(e) we plot the $s$ polarized Kerr spectra as a function of photonic energy for three distinct topological regimes, but only for two transitions. For $s$ polarized incident light the polarity of the Kerr rotation angle is switched. The FR rotation spectra shown in Fig. 2(f) follow a similar trend. The magnitudes of the FR rotation angles range between $\sim \pm 2^{\circ }$ for all three topological regimes for top and bottom surfaces, which, in general, are smaller than the Kerr rotation angle.

It is worth investigating the control of Kerr and FR spectra by varying the chemical potential $\mu _{F}$. The LLs of the TIs top and bottom topological SSs can be tuned independently by employing top and back gate electrodes [47–49]. Fine control of $\mu _{F}$ of the paired surface states in a dual-gated system has been recently reported [50]. In the subsequent analysis, the chemical potential is tuned near the Dirac point. For this purpose, we consider two different values of chemical potentials $\mu _{F}$=10 and −10meV while keeping the magnetic field 1 T in the QSHI regime ($\Delta _{z}=0.5\Delta _{H}$), and an angle of incidence of $30^{\circ }$.

In the first case we assume n-type doping and $\mu _{F}$=10 meV. The chemical potential lies in between the $n$=0 and $n$=1 LL’s as shown in Fig. 3(a). Blue lines represent LL’s for top surfaces and red lines represent LL’s for bottom surfaces. The same color scheme applies for the LL transitions with green line indicating the chemical potential $\mu _{F}$. We identify these LL transitions as $T_{-01}^{{+1}}$, $T_{-10}^{{-1}}$, $T_{01}^{{-1}}$, $T_{-12}^{{+1}}$ and $T_{-21}^{{-1}}$ as shown in Fig. 3(a). The intra-band transition is shown by a dashed black upward pointing arrow. In the second case we consider p-type doping where $\mu _{F}$=−10meV and the chemical potential is in between the $n$=0 and $n$=−1LL’s as displayed in Fig. 3(b). For n-type doping, certain transitions become Pauli blocked. For example the transition $T_{-10}^{-1}=20.3$ meV becomes forbidden and in their place the intra-band transition $T_{01}^{-1}=16.3$ meV in the bottom surface appears. Similarly for p-type doping, when the chemical potential $\mu _{F}$ jumps between the $n$=0 and $n$=−1LL’s, owing to the Pauli blocking, transition $T_{01}^{+1}=25.1$ meV disappears and $T_{-10}^{+1}=13.1$ meV emerges.

The first and second magneto-excitation energies corresponding to $T_{-01}^{+1}$ and $T_{-01}^{+1}$ for top and bottom surfaces are calculated and are summarized in Table 3. Higher-order inter-band transitions are weak where the LL’s are closely spaced and the Kerr and FR originating from these transitions are very small hence we ignore them here. The longitudinal conductivities for n-type and p-type doped TIs are shown in Figs. 3(c) and (d). The intra-band and inter-band transitions are shown as pronounced dips at incident photonic energies in the QSHI regime for $B=1$ T. The spectral weights of these transitions are larger for p-type doping and smaller for n-type doping. The $p$ polarized Kerr rotations anti-phase peaks are shown in Fig. 3(e) and (f) for $\mu _{F}$=10 meV and −10 meV, respectively. For n-type doping, these anti-phase peaks originate from the two lowest transitions $T_{01}^{-1}=16.3$ meV and $T_{01}^{{+1}}=25.1$ meV as shown in Fig. 3(e). The maximum Kerr rotation is achieved at the bottom surface for n-type doping, while for the top surface the amount of rotation is small. On the other hand, for p-type doping we have giant anti-phase peaks for top surface and comparatively small peaks for bottom surface as shown in Fig. 3(f). For p-type doping, the Kerr rotation angles of $\sim \pm 10^{\circ }$ and $\sim \pm 9^{\circ }$ are observed (data not shown) for the two lowest transitions $T_{-10}^{{+1}}=26.0$ meV and $T_{-10}^{-1}=33.5$ meV for bottom and top surfaces respectively, for $B=3$ T. This exceeds the maximum values reported in literature for graphene in the THz region [23,51]. Please note that the the asymmetric response from the top and bottom surfaces arises due to the difference in their energies and not due to the permittivity of the medium.

In the semi-classical approximation the LL spacing due to applied magnetic field becomes negligible, which occurs when the chemical potential is very large $|\mu _{F}|\gg |E_{1}|$. This usage of the term ‘semi-classical’ is unique to these low dimensional quantum systems and different from other literature in atom-mater interactions, where the term describes quantized electronic systems interacting with light which is treated as a classical electromagnetic wave (though this is the assumption in the current work as well). In this case the intra-band transitions between closely spaced LL’s are allowed. To compare the semi-classical results with the quantum mechanical ones, we study the real part of the longitudinal and transverse magneto-optical conductivities of the TI thin films (see Supplement 1 for details). These conductivities are shaped as absorptive and dispersive Lorentzians which are directly used to compute the Fresnel coefficients Eqs. (19)–(24) and subsequently the rotations. We also investigate the dependence of Kerr and Faraday rotations on photonic energy in semiclassical regime for n-type TIs thin film (see Supplement 1 for details). These rotation angles exceed the maximum values as discussed above in the quantum regime. At a field of $1$ T, we report the $p$ polarized Kerr rotation and $s$ polarized Faraday rotations $\Theta _{K,p}^{\tau _{z}=+1}=14^{\circ }$, $\Theta _{K,p}^{\tau _{z}=-1}=10^{\circ }$, $\Theta _{F,s}^{\tau _{z}=+1}=6^{\circ }$ and $\Theta _{F,s}^{\tau _{z}=+1}=5^{\circ }$, respectively.

## 5. Conclusion

In this work we studied the transitional FR and MOKE due to the TQPTs in hybridized 3D topological insulators in the THz frequency range. We have theoretically investigated the magnetic field modulated top and bottom SSs polarized Faraday and Kerr rotations for three distinct topological regimes. We reported that the magnitude of the maximum giant Kerr rotation angles for the first two anti-phase pair exceeds $\sim 15^{\circ }$ at a magnetic field of 5 T, which is an exceptionally large rotation for hybridized 3D TI-substrate system as compared to 3D $Bi_{2}Se_{3}$ films with $\mu _{F}$=30 meV above the Dirac point in the presence of magnetic field of $7$ T [52]. The FR and MOKE switch sign if we change the polarization of the incident light or switched from one SS to another. We further investigated the MOKE for different chemical potentials and found that for n-type and p-type doping we can tune the intra-band transitions in bottom and top SSs respectively, and the amount of FR and MOKE rotation is also enhanced. Importantly, carriers of opposite type reside on the top and bottom SSs, which are separated by the band gap. Such type doped Dirac fermions in the TI open the way for the observation of fantastic quantum phenomena, for example, magneto-electric effect and exciton condensation in TI thin films. Moreover, we also studied the MO effect in the semi-classical regime and noted that in this limit the role of hybridization becomes inconsequential and hybridized 3D TI behaves as graphene.

## Funding

Higher Education Commission, Pakistan (NRPU 10375).

## Acknowledgements

The authors would like to acknowledge financial support from the National Research program for Universities (NRPU), scheme number 10375 funded by the Higher Education Commission of Pakistan.

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

## Supplemental document

See Supplement 1 for supporting content.

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