1. Introduction

Linear dichroism (LD) refers to the anisotropic absorption of polarized light along different directions in a sample [1,2]. It is a powerful tool to study the orientation and alignment of molecules and materials [1,2]. In turn, materials with prominent LD can be utilized for various applications, such as polarization-resolved photodetectors and sensors [3–9] and polarization optical imaging [6–8]. Significant LD is often observed in low-dimensional crystals due to their anisotropic nature [6,7,9–11]. Atomically thin two-dimensional (2D) materials with low symmetry such as ReS$_{2}$ [10–12], black phosphorus [13,14], PdSe$_{2}$ [6], PdPS [7], and SiP [9] are potential candidates to produce strong LD performance. Remarkably, there is a LD conversion in 2D PdSe$_{2}$ [6], PdPS [7], and SiP [9], which significantly expand the optoelectronic applications of those materials. However, the LD conversion in those materials always occurs at certain wavelengths, which can only distinguish two wavelength bands as wavelength-selective photodetectors [6,9–11,13,14]. Tunable LD corresponding to arbitrary wavelengths pose a significant challenge within those materials.

Monolayer 1T’-MoS$_{2}$ is a naturally anisotropic material with distorted octahedral geometry as shown in the inset of Fig. 1(b), giving rise to exotic electronic and optical properties [15–21]. It has been theoretically predicted to be a large gap quantum spin Hall insulator [15] and confirmed by subsequent scanning tunneling spectroscopy experiment [18]. Moreover, in addition to its nontrivial topological band structure, monolayer 1T’-MoS$_{2}$ also exhibits superconductivity [22,23] and ferroelectricity [24]. In monolayer 1T’-MoS$_{2}$, the Mo atoms are arranged in a trigonal prismatic coordination, resulting in a distorted octahedral geometry [15–17,20]. This structure is anisotropic compared with the commonly studied 2H phase, leading to highly anisotropic band structure [15–17,20]. In the pioneer work, Qian and Li develop a four-band $k {\cdot }p$ Hamiltonian for monolayer 1T’-MoS$_{2}$, which accurately reproduces the low energy band structure obtained from the first-principles GW calculation [15]. In the model, the inter-band coupling is specifically observed in the $k_{x}$ direction, while the inter-band spin-orbit coupling (SOC) exclusively manifests in the $k_{y}$ direction. The inter-band SOC opens a gap at the $\pm \Lambda$=(0, $\pm$0.146 ) $\mathring {A}^{-1}$ points, around which the four-band model can be simplified into a two-band massive tilted Dirac model [25]. The two-band model is widely used to study the optical and transport properties of monolayer 1T’-MoS$_{2}$ [25,26]. However, the validity of the two-band model is limited to the low energy regime. For problems associated with a comparatively high energy regime, it becomes necessary to revert back to the original four-band model.

 

Fig. 1. (a) Band structure, (b) density of states, and (c) joint density of states versus photon energy of monolayer 1T’-MoS$_{2}$. The inset in (b) is the crystal structure of monolayer 1T’-MoS$_{2}$ in the top view, where the blue-dashed rectangle denotes the unit cell of the structure.

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In this work, we study the optical properties of monolayer 1T’-MoS$_{2}$ based on the four-band effective $k{\cdot }p$ Hamiltonian by using the Kubo formula. We find the $k$-resolved optical transition matrix elements associated with the armchair and zigzag direction polarized light exhibit a staggered pattern due to the anisotropic band structure. The anisotropy of the optical absorption spectrum is shown to sensitively depend on the photon energy, the light polarization and the Fermi energy which can be tuned by a gate voltage. The gate voltage can continuously modulate the anisotropy of the optical absorption spectra, rendering it isotropic or even reversing the initial anisotropy. This modulation leads to LD conversion across multiple wavelengths, which is crucial to design polarized photodetectors and sensors [3–9]. Our results are also applicable to other monolayer transition metal dichalcogenides with 1T’ structure because they share similar crystal and band structure to monolayer 1T’-MoS$_{2}$.

2. Model and method

In the basis $\left ( c\uparrow,c\downarrow,v\uparrow,v\downarrow \right )$, the low-energy physics of monolayer 1T’-MoS$_{2}$ can be described by an effective four-band Hamiltonian given by [15]

Within the linear response theory, i.e., the Kubo formula, the dynamical conductivity is [27–31]

3. Results and discussions

Next, we will present some numerical examples and discussions for the inter-band optical absorption spectra. Hereafter, unless explicitly specified, the conductivities are all in units of $\sigma _{0}$=$e^{2}/h$, temperature T = 4K, level broadening $\gamma$=10 meV. Figure 1(a) presents the band structure of monolayer 1T’-MoS$_2$. As shown in Fig. 1(a), the energy dispersion along the $k_x$ direction is significantly different from that along the $k_y$ direction, exhibiting a highly anisotropic feature. The reason is that the structure distortion in monolayer 1T’-MoS$_2$ causes an intrinsic inversion of the conduction and valence band [15]. Then, the inter-band SOC reopens the gap at the $\pm \Lambda$=(0, $\pm$ 0.146) $\mathring {A}^{-1}$ points located in the $\Gamma$-Y path because the inter-band SOC only presents in the $k_y$ direction [15]. Therefore, there are two important gaps in the dispersion. One is the real gap $E_g$=0.08 eV at the $\Lambda$ point, another is the inverted gap $E_{ig}$=$\delta _c$-$\delta _v$=0.66 eV at the $\Gamma$ point. Those two gaps are confirmed by the scanning tunneling spectroscopy [18]. The DOS plotted in Fig. 1(b) well reflects the dispersion. In low-energy regime, there is a finite band gap, indicating the system is a semiconductor. Notably, there are two peaks also know as the van Hove singularities in the DOS spectrum at the energies $\delta _c$ and $\delta _v$, which is the embodiment of the band inversion. Figure 1(c) presents the inter-band JDOS for different Fermi energies. As shown in Fig. 1(c), for pristine sample ($E_F$=0 eV), the JDOS becomes finite when the photon energy exceeds the band gap $E_g$. There is a prominent JDOS peak when the photon energy equals the inverted band gap $E_{ig}$. For $E_F$=0.4 eV, JDOS is zero when the photon energy falls below the Fermi energy. The JDOS peak is still evident albeit with a reduced magnitude. The JDOS represents all possible interband transitions but doesn’t provide information about the type of polarized light that excits the transition. To address this question, we have to examine the optical transition matrix elements.

In Fig. 2, we present the $k$-resolved modulus squares of inter-band optical transition matrix elements $|M_{v,s_z}^{c,s_z^{\prime }}|^2$, which indicate the coupling strength with the optical fields of linear polarization. As described in Eq. (8), there are eight inter-band optical transition matrix elements in total, but only four of them are distinct, as depicted in Fig. 2. As observed in the figure, the modulus squares of all inter-band optical transition matrix elements, $|M_{v,s_z}^{c,s_z^{\prime }}|^2$, are even functions of $k$, which is also evident from Eq. (8). Despite the degeneracy of spin in the energy dispersion, the coupling strength $|M_{v,s_z}^{c,s_z}|^2$ and $|M_{v,s_z}^{c,-s_z}|^2$ exhibit different patterns in $k$-space, regardless of the light polarization. Notably, $|M_{v,s_z}^{c,s_z}|^2$ is greater than $|M_{v,s_z}^{c,-s_z}|^2$ due to the stronger inter-band coupling compared to the inter-band SOC. Interestingly, for different light polarizations, the squared magnitudes of $|X_{v,s_z}^{c,s_z^{\prime }}|^2$ and $|Y_{v,s_z}^{c,s_z^{\prime }}|^2$ are completely separated in $k$-space. Specifically, the squared magnitude of $|X_{v,s_z}^{c,s_z^{\prime }}|^2$ near the $\Gamma$ point is significantly high, indicating a large value and corresponding to a high resonance photon energy. Conversely, the squared magnitude of $|Y_{v,s_z}^{c,s_z^{\prime }}|^2$ near the $\Lambda$ point is also notably large, suggesting a high value but corresponding to a low resonance photon energy. Consequently, it can be predicted that the light absorption spectrum of monolayer 1T’-MoS$_2$ is highly anisotropic, and this anisotropy can be adjusted by tuning the Fermi energy. The Fermi energy, in turn, can be controlled through a gate voltage or doping.

 

Fig. 2. The squared modulus of the $k$-resolved inter-band optical transition matrix elements $|M_{v,s_z}^{c,s_z^{\prime }}|^2$, where the specific elements in each figure are (a) $|X_{v,\uparrow }^{c,\uparrow }|^2$, (b) $|X_{v,\uparrow }^{c,\downarrow }|^2$, (c) $|Y_{v,\downarrow }^{c, \uparrow }|^2$, and (d) $|Y_{v,\downarrow }^{c,\downarrow }|^2$.

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In Figs. 3(a) and 3(b), we present the optical conductivities as functions of photon energy for different Fermi energies. In panel (a), $E_F$ is set to 0.0 eV, while in panel (b), $E_F$ is set to 0.4 eV. The red solid (blue dash-dotted) line represents the optical conductivity for light polarized along the $x$- ($y$-)direction. Owing to the separated optical transition matrix elements in $k$-space (see Fig. 2), the absorption spectra exhibit high anisotropy. For the pristine sample ($E_F$=0 eV), Re$\sigma _{xx}$ gradually increases when the photon energy exceeds the band gap $E_g$, reaching its maximum at the inverted gap $E_{ig}$, and then decreases. In contrast, Re$\sigma _{yy}$ reaches its maximum immediately when the photon energy exceeds the band gap and then gradually decreases. Interestingly, Re$\sigma _{yy}$ equals Re$\sigma _{xx}$ when the photon energy coincides with the inverted gap $E_{ig}$, resulting in an isotropic behavior. Consequently, the LD for the pristine sample [see the olive solid line in Fig. 3(c)] decreases to zero and then increases after the turning point where $\hbar \omega$=$E_{ig}$. The LD reaches its maximum (−0.58) when the photon energy is around the inverted band gap, where the ratio of Re$\sigma _{yy}$/Re$\sigma _{xx}$ is 3.76, implying the sample tends to absorb $y$-direction polarized light. Importantly, it’s evident that the absorption peaks for optical conductivities excited by linearly polarized light with different polarizations occur at different photon energies, specifically, for Re$\sigma _{xx}$ (Re$\sigma _{yy}$), the photon energy is $E_{ ig}$ ($E_{g}$), as can also be inferred from Fig. 2. On the other hand, we can use this feature to tune the anisotropy of the optical absorption spectra by adjusting the Fermi energy. Let’s consider the absorption spectra for $E_F$=0.4 eV depicted in Fig. 3(b) as an illustrative example. In this case, both Re$\sigma _{xx}$ and Re$\sigma _{yy}$ are zero when the photon energy is below the Fermi level since all electron states are filled. Consequently, the absorption peak of the pristine sample in Re$\sigma _{yy}$ disappears, while the absorption peak in Re$\sigma _{xx}$ persists, resulting in a strong linear dichroism conversion. The reversed LD reaches its maximum (0.86) when the photon energy is around the inverted band gap [see the orange dash-dotted line in Fig. 3(c)], where the ratio of Re$\sigma _{yy}$/Re$\sigma _{xx}$ is 0.075, indicating the sample only absorbs $x$-direction polarized light, which is crucial for designing polarized photodetectors [7]. As the photon energy continues to increase, Re$\sigma _{yy}$ increases correspondingly, and Re$\sigma _{xx}$ decreases, leading to a reversal in the absorption anisotropy and a negative LD.

 

Fig. 3. The real part of optical conductivities as a function of photon energy under different Fermi energies with $E_F$=0.0 eV in (a) and $E_F$=0.4 eV in (b), where the red solid (blue dash-dotted) lines indicate the optical conductivity for light polarized along the $x$-($y$-) direction. (c) Linear dichroism (LD) as a function of photon energy under different energies, where the olive solid (orange dash-dotted) line corresponds to $E_F$=0.0 (0.4) eV. (d) Contour plot of the LD as functions of both the photon energy and the Fermi level, in which the LD on the black solid line is zero.

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To comprehensively study the impact of the Fermi level on the LD, we plot LD as functions of both photon energies and Fermi energies in Fig. 3(d). As depicted in Fig. 3(d), for a specific Fermi energy, the LD undergoes two reversals as the photon energy increases. This intriguing behavior implies that we can achieve multi-wavelengths LD conversion in monolayer 1T’-MoS$_2$, which is useful to design multi-wavelengths polarized photodetectors. Meanwhile, for a fixed photon energy (wavelength), the LD also exhibits two reversals as the Fermi energy increases. This suggests that the absorption spectrum’s anisotropy can be precisely regulated, ranging from isotropy to reverse anisotropy through gate voltage manipulation. By carefully adjusting the voltage, we can manipulate the sample to selectively absorb either the $x$-direction or $y$-direction polarized light. Meanwhile, we can also finely tune the voltage to achieve an equal absorption for both $x$-direction and $y$-direction polarized light. This feature opens up a wide range of potential applications in fields such as photonics, optoelectronics, and material science [6,9–11,13,14]. It’s important to note that the LD is zero on the black solid line. Positive LD consistently exists within the region enclosed by the solid black line. A strong LD occurs when the photon energy closely matches either the band gap $E_g$ or the inverted band gap $E_{ig}$. Notably, strong LD holds practical significance primarily when the photon energy exceeds the Fermi energy. In this scenario, one of the optical conductivities will be relatively large, leading to a high-performance device.

So far, we have discussed the anisotropy of optical absorption spectra and demonstrated their effective regulation through the adjustment of a gate voltage. In fact, the specific crystal orientation of anisotropic materials is unknown before determined experimentally. Polarized photo-spectroscopy which measures the linear polarization-dependent absorptance is an effective method for this purpose. Figure 4 plots the linear polarization-dependent absorptance $A(\theta )$ for various photon energies and Fermi energies, where $\theta$ represents the orientation of the linearly polarized direction with respect to the $x$-direction. In Fig. 4(a), $A(\theta )$ at $E_F$=0 eV and $\hbar \omega$=0.1 eV manifests as a standing dumbbell, indicating that Re$\sigma _{yy}$ is larger than Re$\sigma _{xx}$ and resulting in a LD of $-$0.55, as illustrated in Figs. 3(a) and 3(c). Conversely, in Fig. 4(b), $A(\theta )$ at $E_F$=0 eV and $\hbar \omega$=0.66 eV adopts a circular shape, signifying that Re$\sigma _{yy}$ equals Re$\sigma _{xx}$ and yields a zero LD, as depicted in Figs. 3(a) and 3(c). Notably, at $E_F$=0.4 eV and $\hbar \omega$=0.66 eV, $A(\theta )$ transforms into a lying dumbbell configuration, implying that Re$\sigma _{yy}$ is smaller than Re$\sigma _{xx}$ and resulting in a positive LD of 0.86, as shown in Figs. 3(b) and 3(c). Therefore, we observe that the anisotropy of the absorptance can be tailored to be isotropic (Fig. 4(b)) or even reversed (Fig. 4(c)) by choosing proper photon energy and gate voltage (Fermi energy). Intuitively, the gate-tunable anisotropy of the absorption spectra arises from the unique band inversion of the system, inducing a prominent JDOS peak at $\hbar \omega$=$E_{ig}$ as shown in Fig. 1(c). This implies that there will be a pronounced absorption at $\hbar \omega$=$E_{ig}$. For pristine sample, given that the band inversion occurs along the $\Gamma$-$Y$ direction, Re$\sigma _{yy}$ is larger when $\hbar {\omega }<E_{ig}$, and Re$\sigma _{xx}$ is larger when $\hbar {\omega }>E_{ig}$. This suggests that a LD conversion can be achieved by selecting proper Fermi energy $E_{F}$ and photon energy $\hbar \omega$. On the other hand, the anisotropic $A(\theta )$ for pristine sample (Fig. 4(a)) can be utilized to identify the crystal orientation. Let the incident light linearly polarized in a chosen orientation and incident near-normally into the sample, $A(\theta )$ should vary when the sample is rotated. The specific crystal orientations such as armchair ($x$-) or zigzag ($y$-) direction can be identified by monitoring the absorption signal. Once the crystal orientation is determined, it is far easier to construct high performance polarized photo-detectors utilizing the anisotropic absorption properties of monolayer 1T’-MoS$_2$.

 

Fig. 4. Light polarization dependent absorptance for various photon energies and Fermi energies. The parameters are (a) $E_F$=0 eV, $\hbar{ \omega}$=0.1 eV, (b) $E_F$=0 eV, $\hbar{ \omega}$=0.66 eV, (c) $E_F$=0.4 eV, $\hbar{ \omega}$=0.66 eV.

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4. Conclusion

In conclusion, we explored the optical absorption spectra of monolayer 1T’-MoS$_{2}$ based on the four-band effective $k{\cdot }p$ Hamiltonian within the framework of linear response theory. We found that the anisotropy of the optical absorption spectrum is shown to sensitively depend on the photon energy, the light polarization and the gate voltage arising from the anisotropic band structure. A gate voltage can continuously modulate the anisotropy of the optical absorption spectra, rendering it isotropic or even reversing the initial anisotropy. This modulation results in linear dichroism conversions across multiple wavelengths. Our findings are useful to determine the crystal orientation of monolayer 1T’-MoS$_{2}$ and design polarized photodetectors based on it. The results are also applicable to other monolayer transition metal dichalcogenides with 1T’ structure since they share similar crystal and band structures to monolayer 1T’-MoS$_{2}$.