Introducing a novel approach to linear and nonlinear electrical conductivity of MoS<sub>2</sub>

Mohsen Balaei; Rouhollah Karimzadeh; Tayebeh Naseri

doi:10.1364/OME.430243

1. Introduction

$MoS_2$ is one of the most well-known transition metal dichalcogenide monolayers (TMDs) that has drawn researchers’ attention for its exceptional electronic, optical, mechanical, chemical, and thermal properties. There are a wide cluster of important nonlinear processes in TMDs such as harmonic generation, four-wave mixing, saturable absorption, and two-photon absorption and their potential applications. The optical properties of $MoS_2$ and thickness are inextricably intertwined. In 3D structures, the $MoS_2$ exhibits indirect band gap but the monolayer (or few layer) of $MoS_2$ has direct band gap. Furthermore, the fewer the number of $MoS_2$ layers, the higher energy band gap would be [1]. Layers of $MoS_2$ can be easily separated as the adjacent layers attached to each other by the weak Van-der-Waals force. On other hand, the atoms attracted each other strongly in each layer in a single layer [2]. There are a plethora of various approaches and patents to synthesis the monolayer and few layers of $MoS_2$ [3–6]. The movement from indirect band gap towards direct band gap are facilitated as the bulk materials are scaled down to the monolayer owing to the reduced dielectric screen effect [7].

The most significant feature of $MoS_2$ in nonlinear optics stemming from its nonlinear absorption and nonlinear refractive index at high intensities [8]. The emergence of this nonlinear distinctive feature in high intensity indicates that the $MoS_2$ monolayer (few layers) has a high saturation intensity($I_s$) [8,9]. At the high incident intensities (nonlinear regime), the nonlinear electrical conductivities ($\sigma ^{(2)}$ and $\sigma ^{(3)}$) are by ne means negligible since the optical absorption is commensurate with electrical conductivity $(\frac {\sigma _{(\omega )}}{2}|E_\shortparallel |)$ [9,10]. In contrast to the linear optical properties, the straight forward approach towards calculating nonlinearity in 2D materials is still challenging. Several methods have been suggested to achieve this objective for obtaining the nonlinear surface current treatment, bond-charge model, and the first-principle calculation [11,12].

Second order nonlinear phenomena such as optical rectification (OR), second harmonic generation (SHG) and linear electro-optics effect(Pockels effect) can be observed in the accumulation of odd number of $MoS_2$ layers [13,14]. That is to say, if the number of $MoS_2$ layers is even, the only third order nonlinear phenomena such quadratic electro-optics (Kerr effect) is attainable [13].

Nowadays, Kerr and Pockels effects have become the most important phenomena in nonlinear optics field. Nonlinear Kerr effect states that the refractive index of the medium changes via applying the external DC electric field($\omega =0$). The refractive index is interconnected to the square of the external electrical field ($n_{(E)} - n_0\varpropto |E_0|^2$ where $E_0$ is the amplitude of applied electric field). It should be noted that in interaction between light and matter, existence the external DC electrical field is not compulsory to observe the Kerr effect [15–17]. Here, we demonstrate that how Kerr (or Pockels) effect can adjust the electrical conductivity($\sigma _{(\omega )}$). The refractive index and electrical permeability ($\varepsilon$) are related to each other ($n=\sqrt {\varepsilon }$), then the Kerr (or Pockels) effect would be correlated $\varepsilon _{(\omega )}$.

In this study, we use the linear electrical permeability of $MoS_2$ $(\varepsilon _{(\omega )}^{(1)})$ that measured experimentally in Uv-Vis of electromagnetism wave by Mukherjee and et al. in previous work [18]. According to their experimental data, the electrical permeability of $MoS_2$ can be illustrated by the Drude-Lorentz model and the Gaussian model such as Eq. (3). It should be noted that the real part of the electrical permeability is of paramount importance, while in previous studies only the imaginary part of the Gaussian model has been studied. Here, we are able to obtain the real part of this parameter through precisely solving the Kramers-Kronig integral. By using real and imaginary part of linear electrical permeability, the $\sigma ^{(1)}$, $\sigma ^{(2)}$ and $\sigma ^{(3)}$ are calculated.

Furthermore, the electrical conductivity consisting of two parts, intraband and interband, which raised by intraband and interband transitions of electron (and hole) whilst the interaction. The proposed method in such obviates the need of sophisticated calculations since the impact of these two types of transitions on the absorption spectrum is already taken into consideration.

2. Theoretical model and methods

By applying a set of Eq. (1) simultaneously, the relation between electrical conductivity and permeability can be achieved [19]:

(1)$$\begin{cases} \nabla.\textbf{P}={-}\rho\\ \nabla.\textbf{J}={-}\frac{\partial}{\partial t}\rho\\ \textbf{D}=\epsilon_0\textbf{E}+\textbf{P}\\ \textbf{D}=\epsilon_0\varepsilon_{(\omega)}\textbf{E}\\ \textbf{J}=\sigma^{bulk}_{(\omega)}\textbf{E}\\ \end{cases}$$

where $\textbf {P}$ is polarization vector, $\rho$ is electrical charge density, $\textbf {J}$ is current density vector and $\sigma ^{bulk}_{(\omega )}$ is electrical conductivity for a bulk medium. For a thick layer of $MoS_2$ with a finite thickness of $\Delta$,the $\sigma ^{bulk}=\frac {\sigma }{\Delta }$ [20] is applicable to link the $\sigma ^{bulk}$(electrical conductivity for bulk) and $\sigma$ (electrical conductivity for surface). Also, the time dependency considered to be as $e^{-i\omega t}$. From Eq. (1), electrical permeability can be expressed as below:

(2)$$\varepsilon_{(\omega)}=1+\frac{i\eta_0}{k_0\Delta}\sigma_{(\omega)}$$

where $\eta _0$ is the vacuum resistivity($\sim 377\Omega$), $k_0$ is the wave number of the incident beam and $\Delta$ is the layer’s thickness($\sim 0.65nm$) [21,22].

So according to Eq. (2), it is expected that the electrical conductivity varies by changing the magnitude of electrical field of incident electromagnetic beam because of adjustability of refractive index during the interaction according to Pockels or Kerr effect. In the interaction between light and matter, similar to the Kerr effect, the external DC electric field is not required to observe the Pockels effect [14] and this will be shown in the present study.

This work is based on the relation between the (${\varepsilon ^{(1)}_{(\omega )}}$ ) and (${\chi ^{(1)}_{(\omega )}}$ ). Furthermore, by using the nonlinear concepts, ${\chi ^{(3)}_{(\omega )}}$ and ${\chi ^{(2)}_{(\omega )}}$ can be illustrated in terms of ${\varepsilon ^{(1)}_{(\omega )}}$ [23]. The value of ${\varepsilon ^{(1)}_{(\omega )}}$ is conspicuous and can be calculated via the hybrid Lorentz-Drude-Gaussian (LDG) model [18,24].

(3)$$\varepsilon_{(\omega)}^{(1)}=\varepsilon_{(\omega)}^{L.D}+\varepsilon_{(\omega)}^{G}$$

The contribution of the Lorentz-Drude ($\varepsilon _{(\omega )}^{L.D}$) in the electrical permeability is given by:

(4)$$\varepsilon_{(\omega)}^{L.D}=\varepsilon_{\infty}+\sum_{j=0}^{5}\frac{a_j\omega_p^2}{\omega_j^2-\omega^2-ib_j\omega}$$

where $\varepsilon _{\infty }$ is DC electrical permittivity ($\sim 4.44$), $\omega _p$ is the plasma frequency($\sim 7 \times 10^{12}\frac {rad}{s}$) [18,25], $\omega _j$ is the response frequency and $a_j$ and $b_j$ are oscillator strength and damping coefficients, respectively. Values of $\omega _j$, $a_j$ and $b_j$ are listed in Table (1).

The Gaussian part of the electrical permeability ($\varepsilon _{(\omega )}^{G}$) can be broken down to imaginary and real parts ($\varepsilon ^{G}=\varepsilon _{R}^{G}+i\varepsilon _{I}^{G}$) (Fig. 1). It’s imaginary part has been mentioned in previous studies as below [18]

(5)$$\varepsilon_{I}^{G}(\omega)=\alpha \exp(-\frac{(\hbar\omega-\mu)^2}{2\sigma^2})$$

the constant coefficient values are: $\alpha =23.224$, $\mu =2.7723(ev)$ and $\sigma =0.3089(ev)$.

By applying the Kramers-Kronig relation, the real part of Gaussian contribution($\varepsilon _{R}^{G}(\omega )$) can be obtained:

(6)$$\varepsilon_{R}^{G}(\omega)={-}\frac{1}{\pi}\int_{-\infty}^{+\infty}\frac{\varepsilon_{I}^{G}(\Omega)}{\Omega-\omega}d\Omega$$

Above integral is divergent, so we should calculate its Cauchy principal value:

(7)$$\begin{aligned} \varepsilon_{R}^{G}(\omega)&=-\frac{\alpha}{\pi}\lim_{\epsilon\rightarrow0}{\bigg [} \int_{-\infty}^{\omega-\epsilon}\frac{\exp(-\frac{(\hbar\Omega-\mu)^2}{2\sigma^2})}{\Omega-\omega}d\Omega \\ &+ \int_{\omega+\epsilon}^{+\infty}\frac{\exp(-\frac{(\hbar\Omega-\mu)^2}{2\sigma^2})}{\Omega-\omega}d\Omega {\bigg ]} \end{aligned}$$

After simplification, $\varepsilon _{R}^{G}(\omega )$ is expressed in terms of Dawson’s Function as Eq. (8). In Fig. 2, the real part of ${\varepsilon ^{(1)}_{(\omega )}}$ is illustrated.

(8)$$\varepsilon_{R}^{G}(\omega)={-}\frac{2\alpha}{\sqrt{\pi}}.DawsonF_{(\frac{\mu}{\sqrt{2}\sigma}-\frac{\hbar\omega}{\sqrt{2}\sigma})}$$

Fig. 1. Imaginary part of $\varepsilon _{(\omega )}$ according to experimental measurement [18].

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Fig. 2. Real part of $\varepsilon _{(\omega )}$ calculated by Kramers-Kronig integral .

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By considering the Eq. (2) and the linear and nonlinear electric susceptibilities, $\sigma ^{(1)}$, $\sigma ^{(2)}$ and $\sigma ^{(3)}$ can be stated in term of ${\varepsilon ^{(1)}_{(\omega )}}$, ${\varepsilon ^{(2)}_{(\omega )}}$ and ${\varepsilon ^{(3)}_{(\omega )}}$ respectively.

2.1 Pockels effect (in the monolayer of $MoS_2$)

In this case because of the asymmetric structure of $MoS_2$ monolayer, the second order nonlinear phenomena can occur. It is considered that the incident electromagnetic beam has an electrical field as below :

(9)$$E_{(t)}=\frac{1}{2} E_0(e^{{-}i\omega t}+e^{i\omega t})$$

The electrical polarization can be written as

(10)$$\begin{aligned} P_{(t)}&=\epsilon_0\chi^{(1)}E_{(t)}+\epsilon_0\chi^{(2)}E_{(t)}E_{(t)}\\ P_{(t)}&=\epsilon_0\chi^{(1)}E_{(t)}+\frac{1}{4} \epsilon_0\chi_{S.H.G}^{(2)}E_0^2(e^{{-}2i\omega t}+e^{2i\omega t})\\ &+ \frac{1}{2}\epsilon_0\chi_{O.R}^{(2)}E_0^2 \end{aligned}$$

In the above equation, the first, second and third terms are allocated to the linear polarization, the second harmonic generation (S.H.G) and the optical rectification (O.R) phenomena, respectively. Since our objective is not investigating the S.H.G, its relevant term is not considered here. We should emphasize that S.H.G can be observed only under the particular circumstances.

(11)$$P_{(t)}=\epsilon_0\chi^{(1)}E_{(t)}+\frac{1}{2}\epsilon_0\chi_{O.R}^{(2)}E_0^2$$

The second term in Eq. (11) is the time-independent polarization due to the optical rectification effect. This term shows that the electromagnetic beam has created a DC electric field in the matter and shifted the polarization by $\frac {1}{2}\epsilon _0\chi _{O.R}^{(2)}E_0^2$. This time-independent term is seemed the same as when a DC external electric field is applied and consequently vital conditions for Pockels effect would be provided. This has been showed in previous work [14], thereby the total electric field in matter is obtained as

(12)$$\begin{cases} E_{(t) _{Total}}=\frac{1}{2} E_0(e^{{-}i\omega_1 t}+e^{i\omega_1 t})+\frac{1}{2} E_0(e^{{-}i\omega_2 t}+e^{i\omega_2 t}) \\ (\omega_2\rightarrow 0 \hspace{.5cm} and \hspace{.5cm} \omega_1\rightarrow \omega) \\ \end{cases}$$

The displacement vector can be written as

(13)$$\begin{aligned} D_{(t) Total}&=\epsilon_0\varepsilon E_{(t)Total}=\epsilon_0E_{(t) Total}\\ &+ \epsilon_0\chi^{(1)}E_{(t) Total}+\epsilon_0 \chi_{L.E}^{(2)}E_{(t) Total}E_{(t) Total} \end{aligned}$$

$(\chi _{L.E}^{(2)}$ refers to contribution of Pockels effect in $\chi _{Total}^{(2)}$). From the mentioned equations, the total electrical permeability can be calculated as

(14)$$\varepsilon_{(E_{(t)})}=1+\chi^{(1)}+\chi^{(2)}_{L.E}E_{(t)Total}$$

By time averaging of the above equation, the electric permeability is restated as:

(15)$$\varepsilon_{(E_{0})}=1+\chi^{(1)}+\chi^{(2)}_{L.E}E_{0}$$

Furthermore, two pair equations that are used in calculating the Pockels effect are shown below [17]:

(16)$$\begin{cases} \varepsilon_{(E_0)}=\varepsilon^{(1)}+\chi_{L.E}^{(2)}E_0\\ \sigma_{(E_0)}=\sigma^{(1)}+\sigma^{(2)}E_0 \end{cases}$$

By applying Eq. (16) to Eq. (2), the linear and nonlinear parts of electrical conductivity can be achieved as below:

(17)$$\begin{cases} \sigma^{(1)}=\frac{\Delta k_0}{i\eta_0}(\varepsilon^{(1)}_{(\omega)}-1)\\ \sigma^{(2)}=\frac{\Delta k_0}{i\eta_0}\chi_{L.E}^{(2)} \end{cases}$$

Owing to the isotropic 2D $MoS_2$ structure, the $\varepsilon ^{(1)}_{(\omega )}$ is calculated and measured as a scalar in the previous studies [18]. According to the crystalline structure bulk of $MoS_2$, the electrical conductivity is represented as a tensor [26]

\sigma^{(1)}= \begin{pmatrix} \sigma_{xx} & 0 & 0 \\ 0 & \sigma_{yy} & 0 \\ 0 & 0 & \sigma_{zz} \end{pmatrix} ; \sigma_{xx}=\sigma_{yy}\neq\sigma_{zz}

By reducing the dimensions of the $MoS_2$ bulk from 3D to 2D, the electrical conductivity tensor will change to a 2D isotropic tensor. In addition, by considering Eq. (17), $\sigma ^{(1)}$ is expressed in terms of $\varepsilon ^{(1)}_{(\omega )}$:

\sigma^{(1)}=\frac{\Delta k_0}{i\eta_0}(\varepsilon^{(1)}_{(\omega)}-1) \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}

The identity matrix in above equation can be ignored.

(18)$$\sigma^{(1)}_{(\omega)}=\frac{\Delta k_0}{i\eta_0}(\varepsilon^{(1)}_{(\omega)}-1)$$

In Fig. (3), the real and imaginary parts of $\sigma ^{(1)}$ have been illustrated according to Eq. (17).

Fig. 3. Real and imaginary parts of $\sigma ^{(1)}$.

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With the aim of achieving the $\sigma ^{(2)}$, we should first calculate the second order electrical susceptibility contribution of linear electro-optic effect($\chi _{L.E}^{(2)}$). By applying the Lorentz-Drude(LD) model, the $\chi _{(\omega _1-\omega _2)}^{(2)}$ can be expressed in terms of $\chi ^{(1)}$ as below [23]:

(19)$$\chi^{(2)}_{(\omega_1-\omega_2)}=\frac{m_e a \epsilon^2_0}{N^2e^3}\chi^{(1)}_{(\omega_1-\omega_2)}\chi^{(1)}_{(\omega_1)}\chi^{(1)}_{(-\omega_2)}$$

where N is atomic number density($\sim 10^{28}m^{-3}$) and coefficient $'a'$ can be estimated as $a\sim \frac {\omega _0^2}{d}$; coefficient $'d'$ is of the order of the dimensions of an atom ($d\sim 3$Å) [23].

By Introducing $\omega _1=\omega$ and $\omega _2\rightarrow 0$ conditions in the linear electro-optic (Pockels effect); also, by using $\chi ^{(1)}_{(\omega )}=\varepsilon ^{(1)}_{(\omega )}-1$:

(20)$$\chi^{(2)}_{L.E(\omega)}=\frac{m_e \omega_0^2 \epsilon^2_0}{ N^2e^3d}(\varepsilon^{(1)}_{(\omega)}-1)^2(\varepsilon^{(1)}_{(0)}-1)$$

by substitution this result in Eq. (17), $\sigma ^{(2)}$ can expressed as below:

(21)$$\sigma^{(2)}_{(\omega)}=\frac{\Delta.k_0}{i\eta_0}\frac{m_e\epsilon^2_0\omega_0^2}{ d.N^2.e^3}[\varepsilon^{(1)}_{(\omega)}-1]^2[\varepsilon^{(1)}_{(0)}-1]$$

Whereas according to Table (1), $MoS_2$ has several resonance frequencies, Eq. (21) includes resonant frequency $\omega _0$. We consider two wavelength spectra from $300 nm$ to $560 nm$ and from $560 nm$ to $700 nm$ with $\lambda =447 nm$ centrality($\omega _0=4.21\times 10^{15}\frac {rad}{s}$) and $\lambda =640 nm$ centrality ($\omega _0=2.94\times 10^{15}\frac {rad}{s}$), respectively. The real and imaginary parts of $\sigma ^{(2)}$ are plotted in Fig. (4).

Fig. 4. Real and imaginary parts of $\sigma ^{(2)}$. a)$\omega _0^2=4.51\times 10^{15}(\frac {rad}{s})$ $b)\omega _0^2=2.94\times 10^{15}(\frac {rad}{s})$.

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Table 1. Coefficients of Eq. (4) according to experimental measurement [18].

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2.2 Kerr Effect (in bilayer of $MoS_2$)

In this section, there are two pairs of equations to calculate Kerr effect(quadratic electro-optics) are considered to study $\sigma ^{(3)}$ in the bilayer of $MoS_2$ [15–17].

(22)$$\begin{cases} \varepsilon_{(E_0)}=\varepsilon^{(1)}+\chi_{Q.E}^{(3)}|E_0|^2\\ \sigma_{(E_0)}=\sigma^{(1)}+\sigma^{(3)}|E_0|^2 \end{cases}$$

By considering Eq. (2) and Eq. (22), the third order electrical conductivity can be obtained as

(23)$$\sigma^{(3)}=\frac{\Delta^{'} k_0}{i\eta_0}\chi_{Q.E}^{(3)}$$

where $\Delta ^{'}\sim 2\times \Delta$ is the effective thickness of the bilayer of $MoS_2$. By using the nonlinear optics concepts, $\chi _{Q.E}^{(3)}$ can be expressed in terms of $\chi ^{(1)}$ for Kerr effect [23]:

(24)$$\chi^{(3)}_{Q.E}(\omega)=\frac{3m_e b\epsilon^3_0}{N^3 e^4}[\chi^{(1)}_{(\omega)}]^3[\chi^{(1)}_{(-\omega)}]$$

Coefficient $'b'$ can be estimated as $b\sim \frac {\omega _0^2}{d^2}$. By using $\chi ^{(1)}_{(\omega )}=\varepsilon ^{(1)}_{(\omega )}-1$, $\sigma ^{(3)}$ can, consequently be expressed in terms of $\varepsilon ^{(1)}$:

(25)$$\sigma^{(3)}_{(\omega)}=\frac{2\Delta k_0}{i\eta_0}\frac{3 m_e \omega_0^2\epsilon^3_0}{d^2N^3 e^4}[\varepsilon^{(1)}_{(\omega)}-1]^3[\varepsilon^{(1)}_{(-\omega)}-1]$$

Similar to Eq. (21) and because of dependency $\sigma ^{(3)}$ on $\omega _0$, we consider two spectra of wavelength. The real and imaginary parts of $\sigma ^{(3)}$ have been illustrated in Fig. (5).

Fig. 5. Real and imaginary parts of $\sigma ^{(3)}$. a)$\omega _0^2=4.51\times 10^{15}(\frac {rad}{s})$ $b)\omega _0^2=2.94\times 10^{15}(\frac {rad}{s})$

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3. Conclusion

In conclusion, the unique structure with fascinating mechanism and outstanding chemical and electrical properties of $MoS_2$ have ignited intense research interest in both fundamental investigations and potential applications. In this study, Pockels and Kerr effects of $MoS_2$ are theoretically investigated. It is widely accepted that the nonlinear susceptibility is sensitive and dependent on the asymmetric charge distribution and the inter atomic interactions. It is shown that Kerr (or Pockels) effect is able to modify the electrical and photo-electrical properties of $\sigma _{(\omega )}$. The refractive index and electrical permeability ($\varepsilon$) are inextricably linked ($n=\sqrt {\varepsilon }$) ; hence, Kerr (or Pockels) effect can be adapted accordingly. These illustrated results for nonlinear optical properties of $MoS_2$ can be promising candidates for the on-chip photonics and optoelectronic applications.

Disclosures

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

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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j	$a_{j}$	$b_{j} (\frac{r a d}{s})$	$ω_{j} (\frac{r a d}{s})$
0	$2.00 \times 10^{5}$	$1.63 \times 10^{13}$	$0$
1	$5.75 \times 10^{4}$	$8.92 \times 10^{13}$	$2.85 \times 10^{15}$
2	$8.14 \times 10^{4}$	$1.70 \times 10^{14}$	$3.06 \times 10^{15}$
3	$8.22 \times 10^{4}$	$1.80 \times 10^{14}$	$4.19 \times 10^{15}$
4	$3.31 \times 10^{5}$	$4.27 \times 10^{14}$	$4.39 \times 10^{15}$
5	$4.39 \times 10^{6}$	$1.18 \times 10^{15}$	$6.50 \times 10^{15}$

Introducing a novel approach to linear and nonlinear electrical conductivity of MoS₂

Abstract

1. Introduction

2. Theoretical model and methods

2.1 Pockels effect (in the monolayer of $MoS_2$)

2.2 Kerr Effect (in bilayer of $MoS_2$)

3. Conclusion

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Tables (1)

Equations (27)

Optical Materials Express