An Improved Theoretical Model to Extract the Optical Conductivity of Two-Dimensional Material from Terahertz Transmission or Reflection Spectroscopy

Abstract

The technique of terahertz time-domain spectroscopy (THz-TDS) enables us to simultaneously determine the real and imaginary parts of optical parameters. However, it is still a challenge to extract the optical parameters of a two-dimensional (2D) material (or an ultra-thin film) on a substrate accurately and flexibly for an arbitrary incident angle and different polarization. By treating a 2D material as a conductive boundary without thickness, we propose an improved theoretical model to extract the optical conductivity of the 2D material on a substrate from THz transmission or reflection spectroscopy. Importantly, the effects of wave polarization, incident angle, and multiple reflections in the substrate are considered in our model and the analytical formulae associated with the optical conductivity of the 2D material are provided. Furthermore, we verify the validation of our model based on the THz transmission and reflection experiments for mono- and few-layer MoS₂ on sapphire substrates. These results not only are of practical significance for investigating the THz properties of 2D materials but can also be extended to the situations of ultra-thin films and/or incoherent detection such as Fourier transform infrared spectroscopy.

1. Introduction

The terahertz (THz) time-domain transmission spectroscopy (THz-TDTS) and THz time-domain reflection spectroscopy (THz-TDRS) are two geometries of the THz-TDS technique. THz-TDTS technology is often used to study insulators, undoped/thin semiconductors, low-loss solutions, and thin semimetals, whereas THz-TDRS technology is usually used to investigate metals, doped/thick semiconductors, and high-loss solutions. In contrast to incoherent detection methods such as Fourier transform infrared (FTIR) spectroscopy, through THz-TDS one can simultaneously measure the amplitude and phase of the THz field transmitted or reflected from a material. Accordingly, the complex refractive index (n), complex optical conductivity (σ), and complex dielectric constant (ε) of a material can be flexibly extracted. Experimental approaches based on THz-TDS have become powerful tools for investigating the electromagnetic responses of liquid [1,2,3,4], bulk [5,6], and film materials [7,8,9,10]. The THz responses and optical parameters obtained by THz-TDTS/THz-TDRS can help us analyze the interactions between particles and low-energy elementary excitation in materials, including free carriers [1,3,5,11], phonons [12,13], excitons [6,14,15], dipoles [16], and plasmons [17,18].

Recently, owing to the fascinating electronic and optical properties, two-dimensional (2D) materials are expected to be one of the ideal platforms for next-generation optoelectronic devices. For device applications, it is very important to know the complex optical parameters of 2D materials. The THz spectroscopy technique mentioned above has been widely used to obtain these parameters [9,10,11,13,14,15,19,20,21]. Since the typical thickness of a 2D material is only a few nanometers [20], this kind of ultra-thin film is usually attached to a substrate, forming a 2D material/substrate system akin to the well-known thin film/substrate system. In this situation, one should extract the optical parameters of the 2D material by considering the whole system, including the substrate, rather than only considering the single 2D material layer. Based on THz-TDTS, researchers have proposed several methods to resolve this problem [22,23,24,25,26]. For example, a famous relation to extracting the optical conductivity/conductance of a superconducting thin film of Pb on a substrate from an infrared transmission experiment was proposed by M. Tinkham [27,28]. This relation was known as the Tinkham formula, which establishes a relation between the normal incident transmission spectrum of a thin film and its optical conductivity/conductance. When the film thickness is smaller than the skin depth and the film area is larger than the wavelength, the relative power transmission of the film can be expressed by using the Tinkham formula $T={\left[1+Z_0\sigma /\left(n+1\right)\right]}^{-2}$, where n is the refractive index of the substrate, σ is the optical conductivity of the film. Therefore, the optical conductivity of a thin film can be experimentally obtained by measuring the relative transmission. Obviously, the effects of the incident angle and wave polarization are not included in the formula. Recently, this formula has been widely used to calculate the optical parameters of a 2D material on a substrate based on the THz-TDTS experiment with normal incidence [7,8,9,10,19]. Then, the formula was extended to include the effects of multiple reflections in substrates [29] and to extract optical parameters from the data measured by THz-TDRS [30]. Furthermore, Gao F. and co-workers investigated YBa₂Cu₃O_7−δ film in the far-infrared regime. A concise method, in which the surface current in the thin film was considered, was proposed to obtain the optical parameters of YBa₂Cu₃O_7−δ. Their method can also be applied to the situation of 2D materials in the THz regime [31]. For THz-TDRS experiments, the optical parameters of ultra-thin films/2D materials are usually extracted from the reflection coefficient of a film/substrate system with non-zero film thickness [32,33,34,35]. For instance, by measuring the THz reflection signals from an air/silicon interface and a water/silicon interface as reference and sample data, respectively, the optical parameters of liquid water were derived based on the reflection coefficient [33]. Meanwhile, the extraction methods proposed for infrared or visible reflection measurements can also be used for THz-TDRS measurement. Although there are many approaches to determining the optical parameters of ultra-thin films and 2D materials in the infrared and visible spectroscopic measurements, it is unsuitable to directly apply them to the THz-TDS experiments. As mentioned above, THz-TDS can detect the phase change in the reflected or transmitted THz field, whereas infrared and visible spectroscopic measurements cannot obtain the phase information accurately. In addition, most extraction methods based on THz-TDTS and THz-TDRS were established for normal incidence and the influences of wave polarization were not considered. Although the effects of multiple reflections in the substrate are included in some of the existing methods, it is complicated to select the required reflection orders in the extraction process. Therefore, it is still a challenge to extract the optical parameters of a 2D material on a substrate accurately and flexibly for an arbitrary incident angle and different polarization.

To resolve this problem, we derive an improved theoretical model to extract the optical conductivity of a 2D material from the THz-TDTS and THz-TDRS measurements. By treating 2D material as a conductive boundary, the transmission and reflection coefficients of a 2D material/substrate system are obtained. The effects of incident angle, wave polarization, and multiple reflections in the substrate are all included in our model. Furthermore, we verify the validation of our model based on the THz transmission and reflection experiments for mono- and few-layer MoS₂ on sapphire substrates. The results are consistent with those extracted by the Tinkham formula from the THz-TDTS measurement with normal incidence. Furthermore, we also discuss extending our model to the situations of ultra-thin films and incoherent detections, such as FTIR and visible spectroscopic measurements.

2. Theoretical Model

At first, we consider a 2D material located at the interface (z = 0) between two semi-infinite media, as shown in Figure 1a. The dielectric constants of the cladding (z > 0) and substrate (z < 0) materials are ε_i and ε_j, respectively. Since the 2D material meets the condition of $d\ll \delta \ll \lambda$, where δ is the skin or penetration depth and λ is the wavelength of incident THz radiation, respectively, we can assume the 2D material is a zero-thickness film (d→0). Then, the response of the 2D material can be treated as a surface current $J=\sigma E_{\parallel }$. The boundary conditions require that the tangential components of E should be continuous across the film, whereas the tangential components of H should be discontinuous by the surface current in the film [31]. Then, by matching the boundary conditions, the Fresnel coefficients at the interface with a 2D material for a p-/s-polarized wave can be derived as [36]:

$$r_{{}_{Jij}}^{\left(p\right)}=\frac{C_{ij}^{(p)}-k_{jz}{\epsilon }_i}{C_{ij}^{(p)}+k_{jz}{\epsilon }_i},t_{Jij}^{(p)}=\frac{2k_{iz}\sqrt{{\epsilon }_i{\epsilon }_j}}{C_{ij}^{(p)}+k_{jz}{\epsilon }_i},$$

(1a)

$$r_{Jij}^{(s)}=\frac{1-C_{ij}^{(s)}}{1+C_{ij}^{(s)}},t_{Jij}^{(s)}=\frac{2}{1+C_{ij}^{(s)}}.$$

(1b)

where $C_{ij}^{(p)}=k_{iz}({\epsilon }_j+\sigma k_{jz}{k}_0/Z_0)$ and $C_{ij}^{(s)}=(k_{jz}+k_0\sigma Z_0)/k_{iz}$ are parametric terms associated with the response of the 2D material for p- and s-polarization, respectively. The wave vector in medium i is written as $k_i=k_0\sqrt{{\epsilon }_i}$ with $k_0=\omega /c$, where c is the light speed in the vacuum. The x and z components of $k_i$ are $k_x=k_{i}sin{\theta }_{\mathrm{in}}$ and $k_{iz}=\sqrt{(k_i{}^2-k_x{}^2)}$, respectively. When there is 2D material at the interface, leading to $J=\sigma E_{\parallel }=0$, the Fresnel coefficients will reduce to their conventional forms:

$$r_{ij}^{(p)}=\frac{{\epsilon }_j{k}_{iz}-{\epsilon }_i{k}_{jz}}{{\epsilon }_j{k}_{iz}+{\epsilon }_i{k}_{jz}},t_{ij}^{(p)}=\frac{2k_{iz}\sqrt{{\epsilon }_i{\epsilon }_j}}{{\epsilon }_j{k}_{iz}+{\epsilon }_i{k}_{jz}},$$

(2a)

$$t_{ij}^{(s)}=\frac{k_{iz}-k_{jz}}{k_{iz}+k_{jz}},t_{ij}^{(s)}=\frac{2k_{iz}}{k_{iz}+k_{jz}}.$$

(2b)

Next, we consider the situation of a 2D material/substrate system with finite substrate thickness L as illustrated in Figure 1b. The electric fields of the transmitted and reflected THz waves with different orders are derived based on Figure 1b and given in

Table 1. The electric fields of the transmitted and reflected waves with different orders.

Order	Transmission (p-/s-Polarized)	Reflection (p-/s-Polarized)
1st	${\tilde{E}}_{t1}(\omega )=E_{\mathrm{THz}}{t}_{J12}{t}_{23}{e}^{-i\delta }{e}^{-i{\delta }_{\mathrm{air}}}$	${\tilde{E}}_{r0}(\omega )=E_{\mathrm{THz}}{r}_{J12}{e}^{-i{\delta }_{\mathrm{air}}}$
2nd	${\tilde{E}}_{t2}(\omega )=E_{\mathrm{THz}}{t}_{J12}{r}_{23}{r}_{J21}{t}_{23}{e}^{-i3\delta }{e}^{-i{\delta }_{\mathrm{air}}}$	${\tilde{E}}_{r1}(\omega )=E_{\mathrm{THz}}{t}_{J12}{r}_{23}{t}_{J21}{e}^{-i2\delta }{e}^{-i{\delta }_{\mathrm{air}}}$
3rd	${\tilde{E}}_{t3}(\omega )=E_{\mathrm{THz}}{t}_{J12}{r}_{23}^2{r}_{J21}^2{t}_{23}{e}^{-i5\delta }{e}^{-i{\delta }_{\mathrm{air}}}$	${\tilde{E}}_{r2}(\omega )=E_{\mathrm{THz}}{t}_{J12}{r}_{23}^2{r}_{J21}{t}_{J21}{e}^{-i4\delta }{e}^{-i{\delta }_{\mathrm{air}}}$
……	……	……
m-th	${\tilde{E}}_{tm}(\omega )=E_{\mathrm{THz}}{t}_{J12}{r}_{23}^{m-1}{r}_{J21}^{m-1}{t}_{23}{e}^{-i\left(2m-1\right)\delta }{e}^{-i{\delta }_{\mathrm{air}}}$	${\tilde{E}}_{rm}(\omega )=E_{\mathrm{THz}}{t}_{J12}{r}_{23}^m{r}_{J21}^{m-1}{t}_{J21}{e}^{-i2m\delta }{e}^{-i{\delta }_{\mathrm{air}}}$

, detailed derivation can be found in Supplementary Materials. Here, $e^{-i{\delta }_{\mathrm{air}}}=exp(-i{\tilde{n}}_{\mathrm{air}}{k}_0(z-L))$ and $e^{-i\delta }=exp(-i{k}_0{\tilde{n}}_{2}L)$ are the propagation coefficients corresponding to the distances of z−L in air and L in substrate, respectively. Meanwhile, ${\tilde{n}}_{\mathrm{air}}$ and ${\tilde{n}}_2$ are the complex refractive indices of air and substrate, respectively. We ignore the thickness of the 2D material due to $\lambda \gg d$.

For the THz-TDTS measurement, the substrate should be selected as highly transparent in the THz regime (${\tilde{n}}_2\approx n_2$), such as sapphire and high-resistivity silicon [37]. In data processing, because the substrate is thick enough, the main pulse (the 1st order) and the satellite pulses (the m-th orders) can be easily distinguished in the time domain, as shown in Figure 2.

When the 1st-order THz pulse is selected for further processing, the electric fields transmitted through the reference substrate and through the 2D material/substrate system can be expressed in the following forms:

$${\tilde{E}}_{\mathrm{t}1-\mathrm{ref}}(\omega )=E_{\mathrm{THz}}(\omega )t_{12}^{(p/s)}{t}_{23}^{(p/s)}{e}^{-i\delta }{e}^{-i{\delta }_{\mathrm{air}}},$$

(3a)

$${\tilde{E}}_{\mathrm{t}1-\mathrm{sam}}(\omega )=E_{\mathrm{THz}}(\omega )t_{J12}^{(p/s)}{t}_{23}^{(p/s)}{e}^{-i\delta }{e}^{-i{\delta }_{\mathrm{air}}}.$$

(3b)

The ratio of (3b) to (3a) is the 1st-order transmission coefficient of the 2D material, taking p-polarization as an example:

$${\tilde{t}}_1(\omega )=\frac{{\tilde{E}}_{\mathrm{t}1-\mathrm{sam}}}{{\tilde{E}}_{\mathrm{t}1-\mathrm{ref}}}=\frac{t_{J12}}{t_{12}}{e}^{-i\mathsf{\Delta }\Psi }=t_1(\omega )e^{-i\mathsf{\Delta }\Psi }=\frac{1+n_2}{1+n_2+Z_0\tilde{\sigma }\cos {\theta }_{\mathrm{in}}}{e}^{-i\mathsf{\Delta }\Psi },$$

(4)

where t₁ (ω) and $\mathsf{\Delta }\Psi$ represent the transmission amplitude ratio and phase difference, between the 2D material/substrate and the reference (bare) substrate, respectively. It should be noted that when there is a thickness mismatch (ΔL₀) between the reference (bare) substrate and the sample substrate, we have $\mathsf{\Delta }\Psi =(n_2-1)\mathsf{\Delta }{L}_0{k}_0$. For the 2D material on the dielectric substrate, the complex optical conductivity, $\tilde{\sigma }(\omega )={\sigma }_{\mathrm{re}}(\omega )+i{\sigma }_{\mathrm{im}}(\omega )$, can be determined by the equations shown in

Table 2. The optical conductivity derived from the 1st-order transmission or reflection.

1st	p-Polarized	s-Polarized
Transmission coefficient	${\tilde{t}}_1(\omega )=\frac{1+n_2}{1+n_2+Z_0\tilde{\sigma }\cos {\theta }_{\mathrm{in}}}{e}^{-i\mathsf{\Delta }\Psi }$	${\tilde{t}}_1(\omega )=\frac{1+n_2}{1+n_2+Z_0\tilde{\sigma }\sec {\theta }_{\mathrm{in}}}{e}^{-i\mathsf{\Delta }\Psi }$
Complex optical conductivity	${\tilde{\sigma }}_{\mathrm{t}}(\omega )=\frac{1+n_2}{{\tilde{t}}_1(\omega )Z_0\cos {\theta }_{\mathrm{in}}}-\frac{1+n_2}{Z_0\cos {\theta }_{\mathrm{in}}}$	${\tilde{\sigma }}_{\mathrm{t}}(\omega )=\frac{1+n_2}{{\tilde{t}}_1(\omega )Z_0\sec {\theta }_{\mathrm{in}}}-\frac{1+n_2}{Z_0\sec {\theta }_{\mathrm{in}}}$
Real part	${\sigma }_{\mathrm{re}}(\omega )=\frac{(t_1(\omega )+\cos \mathsf{\Delta }\Psi )(1+n_2)}{t_1(\omega )Z_0\cos {\theta }_{\mathrm{in}}}$	${\sigma }_{\mathrm{re}}(\omega )=\frac{(t_1(\omega )+\cos \mathsf{\Delta }\Psi )(1+n_2)}{t_1(\omega )Z_0\sec {\theta }_{\mathrm{in}}}$
Imaginary part	${\sigma }_{\mathrm{im}}(\omega )=\frac{\sin \mathsf{\Delta }\Psi (1+n_2)}{t_1(\omega )Z_0\cos {\theta }_{\mathrm{in}}}$	${\sigma }_{\mathrm{im}}(\omega )=\frac{\sin \mathsf{\Delta }\Psi (1+n_2)}{t_1(\omega )Z_0\sec {\theta }_{\mathrm{in}}}$
Reflection coefficient	$\tilde{r}(\omega )=\frac{1-(n_2+Z_0\tilde{\sigma }\cos {\theta }_{\mathrm{in}})}{1+(n_2+Z_0\tilde{\sigma }\cos {\theta }_{\mathrm{in}})}{e}^{-i\mathsf{\Delta }\phi }$	$\tilde{r}(\omega )=\frac{n_2-1+Z_0\tilde{\sigma }\sec {\theta }_{\mathrm{in}}}{n_2+1+Z_0\tilde{\sigma }\sec {\theta }_{\mathrm{in}}}{e}^{-i\mathsf{\Delta }\phi }$
Complex optical conductivity	${\tilde{\sigma }}_{\mathrm{r}}\left(\omega \right)=\frac{\tilde{r}\left(\omega \right)(1+n_2)+1-n_2}{Z_0\cos {\theta }_{\mathrm{in}}(1-\tilde{r}\left(\omega \right))}$	${\tilde{\sigma }}_{\mathrm{r}}=\frac{\tilde{r}\left(\omega \right)(1+n_2)+1-n_2}{Z_0\sec {\theta }_{\mathrm{in}}(1-\tilde{r}\left(\omega \right))}$
Real part	$\begin{array}{l}{\sigma }_{\mathrm{re}}\left(\omega \right)=\\ -\frac{r{\left(\omega \right)}^2-1+(r{\left(\omega \right)}^2+1-2r\left(\omega \right)\cos \mathsf{\Delta }\phi )n_2}{Z_0\cos {\theta }_{\mathrm{in}}(1+r{\left(\omega \right)}^2-2r\left(\omega \right)\cos \mathsf{\Delta }\phi )}\end{array}$	$\begin{array}{l}{\sigma }_{\mathrm{re}}\left(\omega \right)=\\ -\frac{r{\left(\omega \right)}^2-1+(r{\left(\omega \right)}^2+1-2r\left(\omega \right)\cos \mathsf{\Delta }\phi )n_2}{Z_0\sec {\theta }_{\mathrm{in}}(1+r{\left(\omega \right)}^2-2r\left(\omega \right)\cos \mathsf{\Delta }\phi )}\end{array}$
Imaginary part	$\begin{array}{l}{\sigma }_{\mathrm{im}}\left(\omega \right)=\\ -\frac{2r\left(\omega \right)\sin \mathsf{\Delta }\phi }{Z_0\cos {\theta }_{\mathrm{in}}(1+r{\left(\omega \right)}^2-2r\left(\omega \right)\cos \mathsf{\Delta }\phi )}\end{array}$	$\begin{array}{l}{\sigma }_{\mathrm{im}}\left(\omega \right)=\\ -\frac{2r\left(\omega \right)\sin \mathsf{\Delta }\phi }{Z_0\sec {\theta }_{\mathrm{in}}(1+r{\left(\omega \right)}^2-2r\left(\omega \right)\cos \mathsf{\Delta }\phi )}\end{array}$

. When ${\theta }_{\mathrm{in}}=0$ and the phase difference is neglected, the equations for the p-/s-polarized wave will reduce to the Tinkham relation [27,28]. Then, the m-th-order transmission coefficients of the 2D material for the p-polarized wave can be obtained:

$$t_m(\omega )=\frac{t_{J12}{r}_{J2}^{m-1}{}_1}{t_{12}{r}_{21}^{m-1}}=\frac{{\left(1+n_2\right)}^m{\left(Z_0\tilde{\sigma }\cos {\theta }_{\mathrm{in}}+1-n_2\right)}^{m-1}}{{\left(1-n_2\right)}^{m-1}{\left(1+n_2+Z_0\tilde{\sigma }\cos {\theta }_{\mathrm{in}}\right)}^m}.$$

(5)

The transmission of adjacent orders changes with the same ratio (${\mathrm{r}}_{J21}/r_{21}$). In THz-TDS measurements, the m-th-order (m > 1) transmission coefficient corresponds to the m-th pulse (i.e., the (m-1)-th satellite pulse) in the time domain. In principle, one can select any satellite pulse rather than the main pulse to extract the optical conductivity by using the above formula. However, since errors in measurement can accumulate in the multiple reflection process, the accuracy of the optical conductivity obtained from a satellite pulse (m > 1) is poorer than that from the main pulse (m = 1). In addition, the higher the order selected, the more complicated the formula will be. Furthermore, the main pulse is usually comprised of the effective response of the sample and the intrinsic errors of the measurement setup or method (e.g., systematic error, interference induced by multiple reflections). The intrinsic errors usually induce a series of weak resonances in the transmission spectra, although they hardly change the overall trend and obvious spectral features of the effective response of the sample. To guarantee accuracy and simplicity, one can use a window function to intercept the principal part of the main pulse in the transmitted THz waveform. The effective response of the sample without weak resonances can be obtained by the principal part of the main pulse. An example of the interception is provided in Figure 2. The interception range is indicated by short black dashed lines. By calculating the relative transmission spectrum based on the principal part of the main pulse, we can use the coefficients of the 1st-order transmission to extract the optical conductivity of the 2D material through the formulae provided in Table 2. When the substrate is thin and the refractive index is small, the satellite pulse may superimpose on the main pulse [30], and the higher-order transmissions induced by the multiple reflections in the substrate should be considered. Its expression is:

$$\begin{array}{l}{\tilde{t}}_{\mathrm{total}}=\frac{t_{J12}(1-r_{21}{r}_{23}{e}^{-i2\delta })}{t_{12}(1-r_{J21}{r}_{23}{e}^{-i2\delta })}\\ =\frac{2n_2\left(1+e^{i2\delta }\right)+\left(e^{i2\delta }-1\right)\left(1+n_2^2\right)}{2n_2\left(1+e^{i2\delta }\right)+\left(e^{i2\delta }-1\right)\left(1+n_2^2\right)+Z_0\sigma \cos {\theta }_{\mathrm{in}}\left(n_2-1+e^{i2\delta }\left(n_2+1\right)\right)},\end{array}$$

(6a)

$$B=n_2-1+e^{i2\delta }\left(n_2+1\right),A=2n_2\left(1+e^{i2\delta }\right)+\left(e^{i2\delta }-1\right)\left(1+n_2^2\right),$$

(6b)

$${\tilde{\sigma }}_{\mathrm{total}}\left(\omega \right)=\frac{A\left(1-{\tilde{t}}_{\mathrm{total}}\left(\omega \right)\right)}{{\tilde{t}}_{\mathrm{total}}\left(\omega \right)B{Z}_0}.$$

(6c)

The transmission coefficients of s-polarization and p-polarization are similar. It is only necessary to replace $\cos {\theta }_{\mathrm{in}}$ in Formulas (4), (5) and (6a) with $1/\cos {\theta }_{\mathrm{in}}$. For THz-TDRS, the 1st-order reflected wave from the interface with 2D material is used as the sample signal, and the 1st-order reflected wave from the reference substrate is used as the reference signal. Their electric fields are as follows:

$${\tilde{E}}_{\mathrm{r}-\mathrm{ref}}(\omega )=E_{\mathrm{THz}}(\omega )r_{12}^{(p/s)}{e}^{-i{\delta }_{\mathrm{air}}},$$

(7a)

$${\tilde{E}}_{\mathrm{r}-\mathrm{sam}}(\omega )=E_{\mathrm{THz}}(\omega )r_{J12}^{(p/s)}{e}^{-i{\delta }_{\mathrm{air}}}.$$

(7b)

Thus, the 1st-order reflection coefficient of the 2D material for p- polarization can be written as:

$$\tilde{r}(\omega )=\frac{{\tilde{E}}_{\mathrm{r}-\mathrm{sam}}}{{\tilde{E}}_{\mathrm{r}-\mathrm{ref}}}=\frac{r_{J12}}{r_{12}}{e}^{-i\mathsf{\Delta }\phi }=r\left(\omega \right)e^{-i\mathsf{\Delta }\phi }=\frac{n_2^2-1+Z_0\tilde{\sigma }\cos {\theta }_{\mathrm{in}}\left(n_2+1\right)}{n_2^2-1+Z_0\tilde{\sigma }\cos {\theta }_{\mathrm{in}}\left(n_2-1\right)}{e}^{-i\mathsf{\Delta }\phi },$$

(8)

in which $r(\omega )$ and $\mathsf{\Delta }\phi$ are the relative reflection amplitude and phase difference, between the 2D material/substrate and the reference (bare) substrate, respectively. When we use a flat dielectric or metallic surface as a reference ($r_{\mathrm{metal}}=-1$ [32]), the 1st-order reflection coefficient and optical conductivity can be extracted via the equations shown in Table 2. By considering ${\theta }_{\mathrm{in}}=0$ and ignoring the phase difference, $\tilde{r}(\omega )$ for the p-/s-polarized wave is consistent with previous works [26,30]. As mentioned earlier, the position mismatch between the reflection surfaces of the reference and sample can result in an obvious error in calculating the relative transmission. Compared to the situation of THz-TDTS, it is more important to correct the inaccurate phase difference between the reference signal reflected by the mirror or substrate and the signal reflected by the sample. If the position mismatch $\mathsf{\Delta }{l}_1$ is inevitable, the phase difference can be corrected by $\mathsf{\Delta }\phi \pm 2k_0\mathsf{\Delta }{l}_1$.

3. Results and Discussion

To verify the validation of the improved model, we take the extraction of the optical conductivity of ML- and eight-layer (8 L) MoS₂ placed on sapphire as an example. In the experiment, the THz-TDS measurements were carried out via the standard setup shown in Figure 3, including transmission and reflection modules. A Ti: sapphire mode-locked laser amplifier was used to generate pump and probe pulses with a wavelength of 800 nm, a repetition rate of 1 kHz, and a pulse duration of 30 fs. The pump beam was focused on a LiNbO₃ crystal to generate a THz wave based on the tilted-pulse-front technique. The time delay between the pump and probe pulses was controlled by dual retroreflectors driven by a servo. Then, the THz wave is transmitted through the sample or reference material and irradiates a ZnTe crystal together with the probe femtosecond beam. By using a standard electro-optic sampling technique, the transmitted THz wave was detected. The humidity was maintained at ~4% in dry nitrogen while the temperature was fixed at 300 K.

The ML MoS₂ nanosheets with an area of 1 cm × 1 cm were synthesized on Cu foils by using the standard chemical vapor deposition (CVD) technique [38]. In a furnace with a quartz tube with double heating zones, MoO₃ powders (99.999% purity) were placed at the second heating zone while other loaded sulfur powders (99.999% purity) were placed at the first heating zone. When the temperature of the second heating zone reached 650 °C, the first heating zone was rapidly heated up to 180 ℃ and then maintained during the reaction process. This CVD growth was performed at 4000 Pa for 10 min by using argon as the carrier gas. Then, the samples were cooled to room temperature naturally. Subsequently, the ML MoS₂ nanosheets were transferred from the Cu foils onto the sapphire substrates. By repeating the procedure, we also obtained samples of 8L MoS₂. The substrate thickness is $L=260\mathsf{\mu }\mathrm{m}$. In the THz regime, the refractive index of sapphire is taken as 3.07 [19]. We extract the optical conductivity of ML MoS₂ from the THz-TDRS at normal incidence through the formulae provided in Table 2. One can see from Figure 4 that the result agrees very well with the optical conductivity for the TDTS at normal incidence, which is also consistent with the data reported in Ref. [19]. In Figure 5, the optical conductivity of ML MoS₂ of THz- TDTS at a non-zero incident angle is extracted and compared to the result for normal incident TDTS measurement. Since the accuracy of the Tinkham formula in extracting the optical conductivity of 2D materials from transmission data for normal incidence has been validated by previous works, here we treat the result of optical conductivity obtained by the Tinkham formula as the reference, which is represented by solid curves in Figure 5 and Figure 6. In order to estimate the accuracy of our model, we introduce a parameter ζ to calculate the ratio between the optical conductivity extracted from the transmission data for non-zero incident angle by using our model and the reference mentioned above. In principle, ζ approaching 1 can indicate the optical conductivity obtained by our model is as accurate as that extracted by the Tinkham formula. As we know, the optical conductivity of 2D materials can be also extracted roughly from the transmission data for non-zero incident angle by using the Tinkham formula, which means that the effect of incident angle is approximatively neglected. We compare the result obtained by this approach with that obtained by our model in the inset of Figure 5b. As ζ₁ and ζ₄ are closer to 1 than ζ₂ and ζ₃, respectively, it is found that the optical conductivity extracted by our model is more accurate, especially for the situation of large incident angles. For smaller incident angles (e.g., ${\theta }_{\mathrm{in}}<{10}^{\circ }$), it is tolerable to neglect the effect of incident angle and to directly use the Tinkham formula. Furthermore, even for our model, the accuracy of the optical conductivity obtained in situations of large incident angles is relatively low in the high- and low-frequency range. Nevertheless, the analytical form, more considered experiment parameters, and better accuracy in most situations make our model accurate and flexible. Similarly, it is confirmed in Figure 6 that our model is effective and correct for different polarizations and for different thicknesses of 2D material. This process is of practical significance for the investigation of THz-nontransparent samples. Therefore, the extraction of the optical conductivity based on our model is accurate and flexible.

Furthermore, three situations in the practical application of the model should be noted: (1) if the refractive index of the substrate is too small, and the thickness is too thin to distinguish the satellite pulses in the time-domain waveform, the higher-order reflections or transmissions in the frequency domain is necessary for the model to eliminate the multiple resonance pattern in the spectrum of optical conductivity. (2) By squaring the modules of the total transmission and reflection coefficients in our model (i.e., calculating the relative transmittance $T={\left|{\tilde{t}}_{\mathrm{total}}\right|}^2$ and relative reflectance $R={\left|{\tilde{r}}_{\mathrm{total}}\right|}^2$(${\tilde{r}}_{\mathrm{total}}$ in Supplementary Materials)), the formula can also be extended to the data extraction in incoherent detection experiments such as FTIR, especially for the case of a non-zero incident angle and polarized measurements. In incoherent detection experiments, one can determine the relative energy/power transmittance or reflectance of a 2D material instead of the transmission and reflection coefficients including the relative amplitude and phase change. Therefore, only the real part of the optical conductivity can be directly extracted based on the formula of T or R. Then, one can deduce the phase information and thereby the imaginary part of the optical conductivity with the help of the Kramers–Kronig relation [27,39]. (3) The model is also suitable for parameter extraction of transmission and reflection spectra of thin film/substrate systems. Considering the influence of film thickness d ($d\ne 0$), the complex optical conductivity of the film is extracted by a formula such as ${\tilde{t}}_1(\omega )=\frac{1+n_2}{1+n_2+Z_0\tilde{\sigma }d\cos {\theta }_{\mathrm{in}}}{e}^{-i\mathsf{\Delta }\Psi }$.

4. Conclusions

In conclusion, we derive an improved theoretical model to extract the optical conductivity of 2D material from THz-TDS measurement. The effects of wave polarization, incident angle, and multiple reflections in a substrate are considered in the model. Moreover, the validation of our model is verified by taking various measurements of ML and 8 L MoS₂/sapphire systems based on THz-TDS. These results not only are significant for investigating the THz properties of 2D materials but can also be extended to the situations of ultra-thin films and/or incoherent detection such as Fourier transform infrared spectroscopy.