Low dimensional materials [1] - [3] can be used as the basic elements to create the controlling and nonlinear metamaterial media in terahertz (THz) range [4] - [6] 0.1 - 30 THz. Nowadays there exists a variety of two-dimensional (2D) materials like graphene, silicene, germanene etc. [1] - [3] . The layered metamaterial structures “dielectric – 2D material -dielectric…” and the periodic sets of graphene strips on dielectric substrates are perspective to realize nonlinear wave propagation and interaction in THz range, like frequency conversion and generation of higher harmonics [7] [8] , self-action, switching [9] - [12] , and solitons [13] [14] . In THz range under the temperatures T ≥ 20 K in graphene and silicene the nonlinearity is due to pseudo relativistic dispersion law [7] [10] [11] [14] [15] of electrons. It is important for applications that in these materials the dynamic nonlinearity in THz range possesses the reactive, or non-dissipative, character [7] , and the collision frequencies are small there ν = 10¹² − 5·10¹² s⁻¹. The following inequality is valid: ν = ω, where ω ≥ 10¹³ s⁻¹ is the circular frequency of the electromagnetic (EM) fields in THz range.

It is interesting to compare the properties of 2D electron gas in graphene and silicene in THz range. The band structure in these materials is similar near the Dirac cone, but the Fermi velocity is essentially smaller in the silicene [1] [2] , whereas usually the collision frequency is smaller in the graphene [2] .

The application of external magnetic fields to the structures with the solid-state plasmas changes qualitatively the wave propagation and yields additional possibilities to realize linear and nonlinear resonant phenomena. Due to the electron band structure, in graphene and silicone the cyclotron frequencies are in THz range ω_B = 10¹³ − 10¹⁴ s⁻¹ when the bias magnetic field is about 0.5 - 2 T [12] [15] . In 2D materials the cyclotron frequencies depend not only on the bias magnetic field but also on 2D electron concentrations [12] .

There are different methods to analyze the electron properties of 2D materials in THz range. They are the direct quantum kinetic approach, including the Kubo one [14] , the quasi-classical kinetics based on the Boltzmann equation [7] [8] [11] , and the electron hydrodynamics [8] . At the temperatures T ≥ 20 K the simulations on the base of the Boltzmann equation are adequate to investigate the conductivity in 2D materials [7] [8] .

In this paper, the resonant linear and nonlinear properties of 2D electron gas in graphene and silicene are investigated. Based on the Boltzmann kinetic approach, the linear and nonlinear expressions for the 2D tensor conductivity are derived. The quasi-linear method is used to analyze nonlinear resonant conductivity when the bias magnetic field is applied. For realistic values of 2D electron concentrations and collision frequencies, the values of the resonant conductivity coincide with ones obtained from the simpler hydrodynamic approach. In addition, the nonlinear dependences of the surface density of the electric current on THz electric field practically coincide when obtained both from the kinetics and from the hydrodynamics in the absence of a bias magnetic field. This makes possible to use the simpler hydrodynamic method to analyze the nonlinear propagation of THz EM waves through the layered structures “dielectric-graphene or silicene—dielectric”.

The consideration is based on the Boltzmann kinetic equation [7] [8] [11] [12] [15] for the electron distribution function f, SI units:

$\frac{\partial f}{\partial t}+v\cdot \frac{\partial f}{\partial r}+e\left(E+v\times B\right)\cdot \frac{\partial f}{\partial p}=St\left\{f\right\}.$ (1)

Here p is the electron quasi-momentum, St{f} is the collision integral that below is considered within the collision frequency ν approximation $St\left\{f\right\}=-\nu \left(f-f_{00}\right)$, f₀₀ is the equilibrium Fermi distribution function. The bias magnetic field B₀ is directed along OZ axis. This equation is valid in THz range, where the inequality takes place: ν < ω; ω ≥ 5⋅10¹² s⁻¹ is the frequency of EM field, ν = 10¹² − 5·10¹² s⁻¹ is the electron collision frequency. In THz range it is possible to neglect by interband electron transitions when the 2D electron concentration is n₂₀ > 10¹⁰ cm⁻². The effects due to the quantizing magnetic field are not considered here, so the bias magnetic field is not high: B₀ < 2 T.

Below the case of the two-component THz electric field is considered in the XOY plane: E = (E_x, E_y, 0). The kinetic Equation (1) is rewritten as:

$\frac{\partial f}{\partial t}+e\left(E_x\frac{\partial f}{\partial p_x}+E_y\frac{\partial f}{\partial p_y}\right)+e{B}_0\left(v_y\frac{\partial f}{\partial p_x}-v_x\frac{\partial f}{\partial p_x}\right)=-\nu \cdot \left(f-f_{00}\right).$ (2)

The stationary solution of Equation (1) is the Fermi distribution function:

$f_{00}\left(p\right)={\left(1+\exp \left(\frac{E\left(p\right)-E_F}{E_T}\right)\right)}^{-1}.$ (3)

Here $E_F$ is the Fermi energy in equilibrium, $E\left(p\right)$ is the electron energy, $E_T\equiv k_{B}T$, T is a temperature in Kelvins. Our goal is to derive the dependence of the surface density of the electric current on THz electric field.

The following notations are used:

$p_F=\frac{E_F}{v_F},p_T=\frac{E_T}{v_F},m^{*}\equiv \frac{p_F}{v_F}>m_g^{*},{\omega }_B\equiv \frac{e{B}_0}{m^{*}}.$ (4)

Here p_F is the Fermi momentum, $m^{*}$ is so-called “kinetic” effective mass, which is utilized below, ω_B is the electron cyclotron frequency.

The equation for p_F is:

$n_{20}=\frac{4}{4{\pi }^2{\hslash }^2}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{f}_{00}\left(p\right){\text{d}}^{2}p}}$. (5)

Here, n₂₀ = 10¹⁰ – 10¹³ cm⁻² is the steady 2D concentration of the electron gas.

The dispersion law for 2D electrons in the graphene and silicene is [1] [2] [7] :

$\begin{array}{l}E=v_F{\left(p^2+p_g^2\right)}^{1/2},p^2\equiv p_x^2+p_y^2;\\ p_g\equiv m_g^{*}{v}_F.\end{array}$ (6)

Here, v_F is so-called the Fermi velocity, it is v_F ≈ 10⁶ m/s for graphene and v_F ≈ 4.5⋅10⁵ m/s for silicene [2] . For the sake of generality, the small effective mass $m_g^{*}$ ≤ 0.001m_e is introduced in Equation (6). The nonzero effective mass $m_g^{*}$ can occur in 2D materials placed onto some substrates, like SiO₂.

The electron velocity for the quasi-classical electron motion is obtained as:

$\begin{array}{l}v\equiv \frac{\partial E}{\partial p};v_x\equiv \frac{\partial E}{\partial p_x}=\frac{v_F{p}_x}{{\left(p^2+p_g^2\right)}^{1/2}},\\ v_y=\frac{v_F{p}_y}{{\left(p^2+p_g^2\right)}^{1/2}},p^2\equiv p_x^2+p_y^2;\\ p_x=p\cdot \cos \phi ,p_y=p\cdot \sin \phi .\end{array}$ (7)

Below the quasi-linear approach is applied [16] to derive the nonlinear conductivity. The solution of Equation (2) is searched in the form:

$\begin{array}{l}f=f_0\left(t,p\right)+\frac{1}{2}\left(\tilde{f}\left(p,\phi \right)\cdot \exp \left(-i\omega t\right)+c.c.\right);\\ E_{x,y}=\frac{1}{2}\left({\tilde{E}}_{x,y}\exp \left(-i\omega t\right)+c.c.\right).\end{array}$ (8)

Here f₀ is the almost constant, or basic, part of the distribution function. Another part of it oscillates with the frequency ω. In the quasi-linear approach, the inverse action of the high-frequency oscillations to the basic part of the wave function f₀ is taken into account. The generation of higher harmonics is not considered because it is essential under satisfying matching conditions, which are realized in a specified geometry [8] .

From Equation (2) there is the following equation for $\tilde{f}$:

$-\frac{e{B}_0{v}_F}{{\left(p^2+p_g^2\right)}^{1/2}}\frac{\partial \tilde{f}}{\partial \phi }+\nu \tilde{f}=-\frac{e}{p}\left(E_x{p}_x+E_y{p}_y\right)\frac{\partial f_0\left(p\right)}{\partial p}.$ (9)

$\begin{array}{l}E\cdot p=p\cdot \left({\tilde{E}}_{+}\exp \left(-i\phi \right)+{\tilde{E}}_{-}\exp \left(i\phi \right)\right),\\ {\tilde{E}}_{+,-}=\frac{1}{2}\left({\tilde{E}}_x\pm i{\tilde{E}}_y\right).\end{array}$

The equation for the perturbation of the distribution function can be rewritten as:

$\begin{array}{l}\left(-i\omega +\nu -\frac{{\omega }_B\cdot {\left(p_F^2+p_g^2\right)}^{1/2}}{{\left(p^2+p_g^2\right)}^{1/2}}\frac{\partial }{\partial \phi }\right)\tilde{f}\\ =-\frac{e{p}_{\perp }}{p}\left({\tilde{E}}_{+}\cdot \exp \left(-i\phi \right)+{\tilde{E}}_{-}\cdot \exp \left(i\phi \right)\right)\frac{\partial f_0\left(p\right)}{\partial p}.\end{array}$ (10)

The expression for the surface density of the electric current is [7] :

$i_{sx}=\frac{4e}{4{\pi }^2{\hslash }^2}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{v}_{x}f\left(p\right){\text{d}}^{2}p}}.$ (11)

Then in Equation (11) the solution of Equation (10) in 2D case is used, and the formula for the surface density of current is:

$\begin{array}{c}{i}_{sx}=-\frac{i{e}^2{v}_F}{\pi {\hslash }^2}\cdot \{{\tilde{E}}_{+}{\displaystyle \underset{0}{\overset{+\infty }{\int }}\frac{p^2}{{\left(p^2+p_g^2\right)}^{1/2}\left(\omega +i\nu \right)-{\omega }_B{\left(p_F^2+p_g^2\right)}^{1/2}}\cdot \frac{\partial f_0}{\partial p}\text{d}p}\\ +{\tilde{E}}_{-}{\displaystyle \underset{0}{\overset{+\infty }{\int }}\frac{p^2}{{\left(p^2+p_g^2\right)}^{1/2}\left(\omega +i\nu \right)+{\omega }_B{\left(p_F^2+p_g^2\right)}^{1/2}}\cdot }\frac{\partial f_0}{\partial p}\text{d}p\}.\end{array}$ (12)

The analogous expression is for i_sy.

It is possible to release in the surface current density the resonant part i_s+ and the non-resonant one i_s- for EM fields of different circular polarizations:

$\begin{array}{l}{i}_{sx}\equiv {\sigma }_{1s}{E}_x+{\sigma }_{Bs}{E}_y\equiv i_{s+}+i_{s-};\\ i_{sy}\equiv -{\sigma }_{Bs}{E}_x+{\sigma }_{1s}{E}_y\equiv \frac{1}{i}\left(i_{s+}-i_{s-}\right).\end{array}$ (13)

The quasi-linear equation for f₀ is:

$\frac{\partial f_0}{\partial t}+\frac{e}{4}\langle \left({\tilde{E}}_x\frac{\partial {\tilde{f}}^{*}}{\partial p_x}+{\tilde{E}}_y\frac{\partial {\tilde{f}}^{*}}{\partial p_y}+c.c.\right)\rangle +\nu \cdot \left(f_0-f_{00}\right)=0.$ (14)

Then the following relation is used that results from relation for ${\tilde{E}}_{+}$, Equation (9), the last line:

$\langle \left({\tilde{E}}_x\frac{\partial {\tilde{f}}^{*}}{\partial p_x}+{\tilde{E}}_y\frac{\partial {\tilde{f}}^{*}}{\partial p_y}+c.c.\right)\rangle =\frac{1}{p}\left[{\tilde{E}}_{+}\cdot \frac{\partial }{\partial p}\left(p\cdot {\tilde{f}}^{*}\right)+c.c.\right]$. (15)

Here $\langle \dots \rangle$ is the symbol of averaging over the oscillation period 2π/ω.

As a result, the equation for f₀ is:

$\frac{\partial f_0}{\partial t}+\frac{e}{4p}\left[ie{\tilde{E}}_{+}\frac{\partial }{\partial p}\left(p\frac{\partial f_0}{\partial p}\frac{{\tilde{E}}_{+}{}^{*}}{\omega -i\nu -{\omega }_B\cdot Q^{1/2}}\right)+c.c.\right]+\nu \cdot \left(f_0-f_{00}\right)=0.$ (16)

The following notation is used:

$Q\equiv \frac{p_F^2+p_g^2}{p^2+p_g^2}.$ (17)

Thus, the equation for f₀ is:

$\frac{\partial f_0}{\partial t}-\frac{e{\left|{\tilde{E}}_{+}\right|}^2}{2p}\frac{\partial }{\partial p}\left(p\frac{\partial f_0}{\partial p}\frac{\nu }{{\left(\omega -{\omega }_B{Q}^{1/2}\right)}^2+{\nu }^2}\right)+\nu \cdot \left(f_0-f_{00}\right)=0.$ (18)

This equation possesses the diffusive character in the space of quasi-momenta [16] . The boundary conditions for f₀ are [16] :

$f_0\left(p\to +\infty \right)\to 0,\frac{\partial f_0}{\partial p}\left(p=0\right)=0.$ (19)

The expressions for the resonant and non-resonant surface conductivities are:

${\sigma }_{s+}\equiv {\sigma }_{1s}-i{\sigma }_{Bs};{{\displaystyle }}^{}{\sigma }_{s-}\equiv {\sigma }_{1s}+i{\sigma }_{Bs}.$ (20)

From Equation (12), the following expression for the resonant surface conductivity is written down:

${\sigma }_{s+}=-\frac{i{e}^2{v}_F}{\pi {\hslash }^2}{\displaystyle \underset{0}{\overset{+\infty }{\int }}\frac{p^2\frac{\partial f_0}{\partial p}}{\left(\omega +i\nu -{\omega }_B\cdot Q^{1/2}\right){\left(p^2+p_g^2\right)}^{1/2}}\text{d}p}.$ (21)

In the limiting case T → 0 there are simplified expressions from the kinetic theory for the linear 2D conductivity components [12] that coincide with ones obtained from the hydrodynamic approximation with the “kinetic” effective mass m^*:

$\begin{array}{l}{\sigma }_{1s}=\frac{i{e}^2{n}_{20}\cdot \left(\omega +i\nu \right)}{{\left(\omega +i\nu \right)}^2-{\omega }_B^2};{\sigma }_{Bs}=-\frac{e^2{n}_{20}{\omega }_B}{{\left(\omega +i\nu \right)}^2-{\omega }_B^2};\\ {\sigma }_{s+}=\frac{i{e}^2{n}_{20}}{\omega -{\omega }_B+i\nu }.\end{array}$ (22)

The nonzero “kinetic” effective mass m^* used in Equations (22) is taken from Equation (6).

In the hydrodynamic approach the surface density of current is expressed through the hydrodynamic velocity v [12] :

$\begin{array}{l}{i}_{sx,y}=e{n}_{20}{v}_{x,y};v_{+,-}\equiv \frac{1}{2}\left(v_x\pm i{v}_y\right);\\ i_{s+}=e{n}_{20}{v}_{+}\equiv {\sigma }_{s+}{\tilde{E}}_{+}.\end{array}$ (23)

In the kinetic approach the expression for σ_s+ can be written down by means of complex integrals, Equation (21). It possesses the resonant dependence on frequency ω. In the hydrodynamic approach the linear formula for σ_s+ is simple and is obtained from the equation for the electron velocity:

$\begin{array}{l}\left(-i\omega +\nu \right)v_{+}+i{\omega }_B{v}_{+}\approx \frac{e}{m^{*}}{\tilde{E}}_{+};\\ \left(-i\omega +\nu \right)v_{-}-i{\omega }_B{v}_{-}\approx \frac{e}{m^{*}}{\tilde{E}}_{-}.\end{array}$ (24)

It is seen that EM wave with the circular polarization ${\tilde{E}}_{+}$ is subject to the resonant interaction with 2D electron gas; another EM wave with the non-resonant circular polarization ${\tilde{E}}_{-}$ is not considered here.

Therefore, the hydrodynamic expression for the linear resonant conductivity is

${\sigma }_{s+}\approx \frac{i{e}^2{n}_{20}}{m^{*}}\frac{1}{\omega -{\omega }_B+i\nu }.$ (25)

Here ω_B ≡ eB₀/m^* is the cyclotron frequency for a particle with the “kinetic” effective mass m^*, also see Equation 4.

Our simulations have demonstrated that in the bias magnetic field both graphene and silicene possess a strong resonant dependence of the surface conductivity on frequency and essential nonlinearity at low magnitudes of THz electric fields of about 100 V/cm. As a nonlinear material, graphene is preferential there. The non-resonant nonlinearity in the absence of a bias magnetic field is more expressed in silicene, due to lower values of the Fermi velocity. Nonlinearity cannot be considered moderate in all cases.

Now photonic crystals, metamaterials, and other layered structures, which are designed for nonlinear applications in THz range, use 2D materials like graphene and silicene. It is of interest to combine the resonant nonlinearity due to a bias magnetic field with the geometrical resonances due to the thicknesses of dielectric layers. Various approaches can be used to analyze these complex structures. Due to the complexity, simple but adequate methods are needed to study essentially nonlinear phenomena there. The most suitable approach is the quasi-classical kinetic one, but also the application of simpler methods like electron hydrodynamics can be useful.

Expressions of resonant linear dependences of the surface conductivity in graphene and silicene at terahertz frequencies have been simulated from the kinetic approach, where the finite temperatures of 2D electron gas are considered. These dependences practically coincide with those obtained from a simpler hydrodynamic approach with the nonzero electron “kinetic” effective mass. This “kinetic” mass can be obtained from the direct quantum approach. At the resonant frequencies the essential nonlinearity occurs both in graphene and silicene at low magnitudes of terahertz electric fields of circular polarization, but the resonant nonlinearity in graphene is higher. In absence of a bias magnetic field nonlinear dependences of surface current densities on the terahertz electric field can also be obtained from the hydrodynamic approach, where the pseudo-relativistic dependence of the “kinetic” effective mass on the electron velocity is used. The Fermi velocity plays the role of the speed of light. The non-resonant nonlinearity is higher in silicene.

Y.R. acknowledges the National Science Centre, Poland, for funding initiative under project number UMO-2023/49/B/ST10/03465. V.G., J.E.-A., and A.K. are grateful to SEP-CONAHCyT (Mexico) for partial funding.