Terahertz optical bistability of graphene in thin layers of dielectrics

Kwang Jun Ahn; Fabian Rotermund

doi:10.1364/OE.25.008484

1. Introduction

Optical bistability (OB) is the fundamental operation of components constructing optical computers such as optical memory, switch, and amplifier [1]. OB is typically characterized by a hysteresis response to the input field [2, 3]. For a same input value, nonlinear optical systems can have different responses depending on roots of the systems. For instances, a system governed by third-order nonlinearities can have three solutions while it has only one in linear optical regime. OB accompanied by memory effect can be observed mostly in strongly correlated electronic systems [4] and phase transition materials [5]. Anyways, the memory effect shown by such material is originated from different physical backgrounds. In general, OB is induced by abrupt changes of transmitted/reflected light caused by nonlinear refractive index or absorption coefficient of materials and strong electric field intensities in cavities as feedback [6, 7], although OB without feedback is also suggested [8].

Since single layer of carbon atoms has successfully been isolated from bulk graphite and can now be chemically synthesized on wafer scales [9], the unique electric and optical properties of graphene have been intensively studied for the next generation of electric transport and optical materials [10]. In special, many theoretical [11–13] and experimental studies [14,15] demonstrate that graphene should have strong third-order nonlinearities in terahertz (THz) frequency regime, which motivate to suggest diverse geometries and physical mechanisms incorporated with graphene to produce energy efficient OB in ultra subwavelength scales. Examples are graphene between a half-infinite linear dielectric and a Kerr medium [16], and graphene sandwiched by two finite dielectrics [17, 18].

In this report, we theoretically investigate THz OB of graphene embedded in thin dielectric layers. By solving boundary conditions and determining the electric field at the graphene position self-consistently, we can derive an analytical solution for THz OB of graphene. In special, we present a critical condition for the occurrence of THz OB with graphene at the interface between two dielectrics. While previous studies have mainly focused on graphene plasmon resonance [17] or OB near the angles corresponding to total internal reflection [18], our consideration is fixed to the normal incidence, which is more favorable situation for experiment. In addition, we show that the threshold power for THz OB can be reduced by one order of magnitude, if highly doped monolayer graphene is replaced by randomly stacked multilayer graphene, where the nonlinear optical properties of monolayer graphene can be well preserved [19].

2. Mathematical model

The configuration of the system considered in this study is presented in Fig. 1(a) where graphene with an optical conductivity σ_g is located at the interface between two thin dielectric layers ∊₁ and ∊₂ [20]. The thicknesses of both layers are d₁ and d₂ along the z-axis. The electric fields in four divided regions are expressed as

E_{x} (z) = {\begin{array}{l} E_{0} e^{i k_{3 z} (z + d_{1})} + r e^{- i k_{3 z} (z + d_{1})} & (z < - d_{1}) \\ A_{1} e^{i k_{1 z} z} + B_{1} e^{- i k_{1 z} z} & (- d_{1} < z < 0) \\ A_{2} e^{i k_{2 z} (z - d_{2})} + B_{e} e^{- i k_{2 z} (z - d_{2})} & (0 < z < d_{2}) \\ t e^{i k_{4 z} (z - d_{2})} & (z > d_{2}), \end{array}

where

k_{j z} = 2 π \sqrt{∊_{j}} / λ

are the wave numbers in each region (j = 1, · · · , 4). With the optical conductivity σ_g = σ₁ + σ₃|E_x (0)|², the amplitude coefficients A₁, A₂, B₁, B₂, and r are determined by the Dirichlet and Neumann boundary conditions, a similar procedure presented in [21]. It should be noted that due to the nonlinear optical conductivity coupled to the electric field σ₃|E_x (0)|², the amplitude coefficients must be self-consistently determined. Since we are mainly interested in OB of graphene at around 1 THz, the linear (first order) optical conductivity σ₁ is mainly determined by the intraband contribution

σ_{1} (ω) = σ_{1 r} + i σ_{1 i} = \frac{2 e_{0}^{2} k_{B} T}{π ℏ^{2}} \frac{i}{ω + i Γ} log [2 cosh (\frac{E_{F}}{2 k_{B} T})]

and the nonlinear conductivity is expressed by the third order optical conductivity σ₃ derived from the Boltzmann’s transport equation within the relaxation time approximation [22]

σ_{3} (ω) = - \frac{3}{4} \frac{e_{0}^{2} {(e_{0} v_{F})}^{2}}{π ℏ^{2} E_{F}} {\frac{2}{(Γ^{2} + ω^{2}) (Γ - i ω)} + \frac{1}{{(Γ - i ω)}^{2} (Γ - 2 i ω)}}

\approx σ_{3 i} = - i \frac{9}{8} \frac{e_{0}^{2} {(e_{0} v_{F})}^{2}}{π ℏ^{2} E_{F} ω^{3}} for ω ≫ Γ

where e₀ is the elementary charge, k_B the Boltzmann constant, T temperature, ħ the Planck constant over 2π, Γ the decay rate of plasma, and E_F (v_F) the Fermi-energy (velocity). Finally, the incident field amplitude E₀ has a cubic dependence on the transmission coefficient t as

\begin{array}{l} E_{0} = & [\frac{n_{2}}{2 n_{1}} (a_{1} - b_{1}) (a_{2} - b_{2}) + \frac{1}{2} (a_{1} + b_{1}) (a_{2} + b_{2}) + \frac{Z_{0} σ_{1}}{2 n_{1}} (a_{1} - b_{1}) (a_{2} + b_{2})] t \\ - \frac{Z_{0} σ_{3}}{2 n_{1}} (a_{1} - b_{1}) {(a_{2} + b_{2})}^{3} t^{3} = α t - β t^{3} \end{array}

where Z₀ denotes the free space impedance,

n_{j} = \sqrt{∊_{j}}

are the refractive indices, and a_1/2 and b_1/2 are defined as

a_{1} = \frac{1}{2} (1 + \frac{k_{1 z}}{k_{3 z}}) e^{- i k_{1 z} d_{1}}, b_{1} = \frac{1}{2} (1 - \frac{k_{1 z}}{k_{3 z}}) e^{i k_{1 z} d_{1}},

a_{2} = \frac{1}{2} (1 + \frac{k_{4 z}}{k_{2 z}}) e^{- i k_{2 z} d_{2}}, b_{2} = \frac{1}{2} (1 - \frac{k_{4 z}}{k_{2 z}}) e^{i k_{2 z} d_{2}} .

By performing the absolute square on the both sides of Eq. (5), one obtains an ordinary cubic equation

\begin{array}{l} Y = {| E_{0} |}^{2} & = {| α |}^{2} t^{2} - (α β^{*} + α^{*} β) t^{4} + {| β |}^{2} t^{6} \\ = A X - C X^{2} + B X^{3} \end{array}

where by new definitions Y = |E₀|² and X = t² a complex-valued incident field but the real-valued transmitted field amplitude are implicitly assumed.

Fig. 1 (a) The sample geometry considered in this study. Graphene with optical conductivity σ_g is sandwiched by two finite dielectric layers ∊₁ and ∊₂ with thicknesses d₁ and d₂. (b) THz OB of graphene at the interface between ∊₁ = ∊₃ = 2.25 and ∊₂ = ∊₄ = 1 (E_F = 1.2 eV, ω = 2π THz). The hysteresis curve is bounded by two lines, P_t /P_i = S_max for σ_g = 0 and P_t /P_i = S_min for σ₃ = 0. The critical powers for the up- and down-transition are read as P_u = 36.7 MW/cm² and P_d = 27.8 MW/cm², respectively.

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Fig. 2 (a) The second criterion for THz OB of graphene in single interface system (n_a = 1, n_b = 1.5) as a function of frequency for different values of Fermi-energy. (b) THz OB of graphene for different values of E_F for the same conditions as (a).

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3. Analytical approach and numerical results

It is well known that cubic equations such as Eq. (8) have three different solutions only when two extrema exist:

\frac{d Y}{d X} = 3 B X^{2} - 2 C X + A = 0 \Leftrightarrow X_{d / u} = \frac{1}{3 B} (C \pm \sqrt{C^{2} - 3 A B}) .

Because both X_d and X_u must have positive real values [cp. Fig. 1(b)], it is additionally required that

C > \sqrt{C^{2} - 3 A B} and C^{2} > 3 A B .

While the former condition is always true (AB = |α|²|β|² > 0), the latter must be discussed in explicit situations.

We consider THz OB of graphene in single interface system to make it as simple as possible. By substituting $n_{b}^{2} = ∊_{1} = ∊_{3}$ , $n_{a}^{2} = ∊_{2} = ∊_{4}$ and d₁ = d₂ = 0, Eq. (5) is simplified to

E_{0} = [\frac{1}{3} (\frac{n_{a}}{n_{b}} + 1) + \frac{Z_{0} σ_{1 r}}{2 n_{b}} + i \frac{Z_{0} σ_{1 i}}{2 n_{b}} (1 - \frac{σ_{3 i}}{σ_{1 i}} t^{2})] t .

Then, the second criterion of Eq. (10) requires

| Z_{0} σ_{1 i} | > \sqrt{3} | (n_{a} + n_{b}) + Z_{0} σ_{1 r} | .

Furthermore, the OB of graphene embedded in a multilayer structure must be restricted within the area bordered by two lines, as displayed in Fig. 1(b). The upper limit (X/Y ∝ S_max) corresponding to the maximum transparency is the case without graphene (σ_g = 0), and the lower one (X/Y ∝ S_min) for the maximum opacity is given when the third-order conductivity of graphene is zero (σ₃ = 0), due to its negative imaginary value.

In Fig. 2(a) we plot the second condition, Eq. (12) for different values of the Fermi-energy where T = 300 K and the decay rate of plasma is decided by mobility of graphene μ_g = 10⁴ cm²/Vs and the Fermi-energy ( $Γ = e_{0} v_{F}^{2} / (μ_{g} E_{F})$ ) [23]. Together with Fig. 2(b), it is clearly seen that the THz OB of graphene in the single interface system can be generated when E_F > 1.0 eV. However, because the solutions of the cubic equation between two critical values X_d and X_u are known to be physically unstable [24], the system’s response to the increasing (decreasing) input is abruptly changed to another branch as depicted by red dashed arrows in Fig. 1(b). Due to the sufficiently small real part of the linear conductivity compared to the imaginary one for higher Fermi-energies(E_F > 1 eV), two critical values X_d and X_u are determined as

X_{d / u} = \frac{1}{3} \frac{σ_{1 i}}{σ_{3 i}} (1 \pm \sqrt{1 - {(\frac{n_{a} + n_{b}}{Z_{0} σ_{1 i}})}^{2}}) .

For increasing Fermi-energy σ_1i increases, but σ_3i decreases as found in Eq. (4). Because X_d/u and their counterparts of input Y_d/u ∝ P_d/u depend critically on the ratio σ_1i/σ_3i, larger modulation powers (ΔY = Y_u − Y_d) and modulation depths (ΔX = X_d − X_u) are obtained for the increasing Fermi-energy, as demonstrated in Fig. 2(b).

Although THz OB can be produced by graphene in single interface system at the normal incidence, the required Fermi-energy and the critical input powers for the up (down)-transition are exceptionally very high. In order to reduce those values, graphene is supposed to be asymmetrically located in thin dielectric layers as shown in Fig. 3(a). Here, graphene is embedded in a relatively higher refractive index material (n₁ = n₂ = 1.55) covered by a lower one (n₃ = n₄ = 1.5). With a distance ratio of over ten (d₂/d₁ = 11, d₁ = 200 nm) the critical power for the up- and down-transition are reduced by a factor of two. Especially, the Fermi-energy of graphene needed for THz OB is decreased to below E_F < 1 eV, which is not sufficient for graphene in single interface systems [cp. Fig. 2(a) and 2(b)].

Fig. 3 (a) THz OB of graphene in an asymmetric four-layered system. The physical parameters used are ∊₁ = ∊₂ = 1.55², ∊₃ = ∊₄ = 1.5², d₁ = 200 nm, d₂ = 2200 nm, and E_F = 0.9 eV. (b) THz OB of a bilayer graphene in an asymmetric four-layered system. The physical parameters used are ∊₁ = ∊₂ = 1.55², ∊₃ = ∊₄ = 1.5², d₁ = 600 nm, d₂ = 4800 nm, and E_F = 0.4 eV.

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Further decrease of the critical powers and the Fermi-energy level for THz OB can be accomplished by using randomly stacked multilayer graphene. Recently, it has been shown that optical nonlinearities of graphene in THz frequency range can be preserved in randomly stacked multilayer graphene films [19]. By using the same combination of materials and changing both thicknesses to d₁ = 600 nm and d₂ = 4800 nm, a complete hysteresis curve can be generated by one order of magnitude smaller incident powers, as represented in Fig. 3(b). Here, a bilayer graphene with Fermi-energy E_F = 0.4 eV is assumed. Additionally, the critical powers are reduced to several MW/cm². Such a tendency of lowered critical powers numerically observed in Fig. 1(b), and Fig. 3(a) and 3(b) results from narrowing the allowed power range for THz OB. As already discussed, OB arising in thin dielectric layers is bounded by two lines. While the upper limit determined by the transmission without graphene (σ_g = 0) is almost same for three cases, the lower boundary varies from S_min = 0.07 in the single interface system of Fig. 1(b), S_min = 0.17 for Fig. 3(a), and S_min = 0.19 for Fig. 3(b).

Finally, the influence of thicknesses d₁ and d₂ on THz OB is investigated in Fig. 4. In Fig. 4(a) d₁ is changed from d₁ = 100 nm to d₁ = 200 nm, while the thickness ratio is fixed to d₂/d₁ = 11. On the contrary, in Fig. 4(b) the thickness ratio is varied from d₂/d₁ = 10 to d₂/d₁ = 12 for a constant thickness of d₁ = 200 nm. Note that the upper and lower limits do not change in both figures.

Fig. 4 (a) THz OB of graphene in an asymmetric four-layered system for different values of thickness d₁ with a fixed ratio d₂/d₁ = 11. Otherwise, the same physical parameters as Fig. 3(a) are used. (b) THz OB of graphene in an asymmetric four-layered system for different ratios d₂/d₁ with a fixed value of d₁ = 200 nm. Otherwise, the same physical parameters as Fig. 3(a) are used.

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4. Conclusion

In summary, we theoretically studied THz OB of graphene embedded in thin dielectric layered systems. By self-consistently solving nonlinear wave equations including the third-order conductivity of graphene, we could derive hysteresis behavior of the transmitted power as a function of the incident power. We were able to derive a fully analytical solution for THz OB of graphene in single interface system. However, the critical powers and the Fermi-energy level are critically high. In order to decrease those values, graphene is suggested to be asymmetrically placed in four dielectric layers. Further lowering of the critical powers and the Fermi-energy were numerically demonstrated by using a randomly stacked bilayer graphene. Our studies can significantly contribute to constructing optical memory and switching devices based on graphene for optical computers.

Funding

National Research Foundation of Korea (NRF) Grants funded by the Korean Government (MSIP) (2014R1A2A1A11049467, 2016R1A2A1A05005381); Center for Advanced Meta-Materials (CAMM) funded by Korea Government (MSIP) as Global Frontier Project (2014M3A6B3063709).

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Terahertz optical bistability of graphene in thin layers of dielectrics

Abstract

1. Introduction

2. Mathematical model

3. Analytical approach and numerical results

4. Conclusion

Funding

References and links

Cited By

Figures (4)

Equations (13)

Optics Express