Plasmon modes in MLG-2DEG heterostructures: Temperature effects

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Plasmon modes in MLG-2DEG heterostructures: Temperature effects Nguyen Van Men a,b , Nguyen Quoc Khanh c,∗ , Dong Thi Kim Phuong d a

Atomic Molecular and Optical Physics Research Group, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, Viet Nam Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Viet Nam c University of Science - VNUHCM, 227-Nguyen Van Cu Street, 5th District, Ho Chi Minh City, Viet Nam d University of An Giang, 18-Ung Van Khiem Street, Long Xuyen, An Giang, Viet Nam b

a r t i c l e

i n f o

Article history: Received 25 September 2018 Received in revised form 13 December 2018 Accepted 20 January 2019 Available online xxxx Communicated by R. Wu Keywords: Graphene Plasmon Collective excitations Temperature effects

a b s t r a c t We calculate the plasmon frequency and damping rate in a double-layer system made of monolayer graphene and GaAs quantum well at finite temperature using the random-phase-approximation dielectric function and taking into account the inhomogeneity of the dielectric background of the system. We show that the temperature, interlayer correlation parameters and dielectric background inhomogeneity affect significantly the plasmon frequencies and damping rates of the system. At low temperatures, acoustic (optical) plasmon frequency increases (decreases) with the increase of temperature. We also find that damping rates of both plasmon modes increase remarkably compared to the zero-temperature case. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Recently, graphene has attracted lots of attentions due to its unique properties [1–4]. Quasi-particles in graphene are chiral, massless fermions and have a linear energy dispersion near the Dirac points [1,3–6]. These special characters may lead to excellent electronic properties and possible technological applications of graphene [4]. The versatility of graphene makes graphenebased hybrid systems suitable for manufacturing novel optical devices working in different frequency ranges with extremely high speed, low driving voltage, low power consumption and compact sizes such as photodetector with high detectivity, biosensor for near infrared, . . .. During the past years, plasmon excitations have been studied intensively and have been used to create plasmonic and photonic devices [7–16]. It is well known that temperature affects significantly plasmon dispersions in ordinary twodimensional electron gas (2DEG) [17] and mono-layer graphene (MLG) [18]. In addition, it is found that the plasmon modes of single-layer systems differ significantly from those of double-layer ones [19–26]. The authors of Ref. [19] have investigated plasmon excitations in chiral – nonchiral MLG-2DEG double-layer systems at zero temperatures. These types of double-layer enable to create

structures with significantly different carrier densities in two layers, providing various application abilities. To our knowledge, up to now no similar calculations at finite temperatures have been done for these systems although it is shown that the temperature affects considerably plasmon modes of double-layer systems [4,20, 21]. Therefore, in this paper, we calculate the plasmon dispersion and damping rate of a double-layer system, consisting of MLG and 2DEG isolated in AlGaAs/GaAs/AlGaAs quantum well, embedded in an inhomogeneous dielectric medium, using finite temperature random-phase-approximation (RPA) dielectric function. 2. Theory We consider the double-layer system consisting of a MLG flake placed onto modulation-doped GaAs/AlGaAs heterostucture hosting a 2DEG, with the effective mass m∗ , in the thin GaAs quantum well as shown in Fig. 1. Two layers are assumed to be electrically isolated via a spacer of thickness d with dielectric constant κ2 so that the tunneling of electrons between two layers can be neglected. The plasmon dispersion relation of the system can be determined from the zeroes of the dynamic dielectric function [4,19,27]



*

Corresponding author. E-mail addresses: [email protected] (N.V. Men), [email protected], [email protected] (N.Q. Khanh), [email protected] (D.T.K. Phuong). https://doi.org/10.1016/j.physleta.2019.01.043 0375-9601/© 2019 Elsevier B.V. All rights reserved.



ε q, ωp − i γ , T = 0

(1)

where q is the wave-vector, ωp is the plasmon frequency at wavevector q, and γ is the damping rate of plasma oscillations. In the

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and inter-layer bare Coulomb interactions in momentum space [30],

U 2DEG/g (q) = U g−2DEG (q) =

4π e 2 q 8π e 2 q

f 2DEG/g (qd) ,

(5)

f g−2DEG (qd)

(6)

with

f g−2DEG (x) =

κ2 e x

(κ1 − κ2 ) (κ2 − κ3 ) + e 2x (κ1 + κ2 ) (κ2 + κ3 ) (κ2 − κ1 ) + (κ1 + κ2 ) e 2x f g (x) = , (κ1 − κ2 ) (κ2 − κ3 ) + e 2x (κ1 + κ2 ) (κ2 + κ3 ) (κ2 − κ3 ) + (κ3 + κ2 ) e 2x f 2DEG (x) = . (κ1 − κ2 ) (κ2 − κ3 ) + e 2x (κ1 + κ2 ) (κ2 + κ3 )

, (7) (8) (9)

3. Results and discussion Fig. 1. A MLG-2DEG double-layer system embedded in an inhomogeneous dielectric environment with dielectric constants κ1 , κ2 and κ3 .

case of weak damping, the plasmon dispersion and decay rate are determined from the following equations [4,26,27]





Re ε q, ωp , T = 0

(2)

and



γ = Im ε q, ωp , T





− 1  ∂ Re ε (q, ω, T )   ∂ω ω=ωp

(3)

Within RPA, the dynamical dielectric function of MLG-2DEG double-layer system has the following form [4,21,28]

ε2DEG−g (q, ω, T ) = [1 + U 2DEG (q) 2DEG (q, ω, T )] ×   × 1 + U g (q) g (q, ω, T ) − [U g−2DEG (q)]2 2DEG (q, ω, T ) g (q, ω, T )

(4)

where 2DEG (q, ω, T ) (g (q, ω, T )) is the finite-temperature noninteracting density-density response function of the 2DEG (MLG) given in Ref. [18] ([29]). U 2DEG/g (q) and U g−2DEG (q) are the intra-

In this section, we present the numerical results for plasmon dispersion and damping rate for κ1 = κAlGaAs = 12.9, κ2 = κSiO2 = 3.8, κ3 = κair = 1.0 and m∗ = 0.067m0 where m0 is the vacuum mass of the electron. In the following we denote the Fermi energy, Fermi wave-vector, and Fermi temperature of MLG by E F , kF and T F , respectively. Numerical results for the plasmon frequency (a) and damping rate (b) as a function of q/k F are shown in Fig. 2 for d = 50 nm, κ1 = κAlGaAs = 12.9, κ2 = κSiO2 = 3.8, κ3 = κair = 1.0, n2DEG = nMLG = 1010 cm−2 and T = 0.2T F . As in the case of semiconductor double quantum well systems, we find that Eq. (2) admits two solutions: the higher (lower) frequency solution corresponds to the optical (acoustic) plasmon mode. We note that in the absence of the quantum well, only optical plasmon mode exists in MLG layer as shown in Fig. 1(a) of Ref. [4]. In the presence of the quantum well, due to Coulomb interaction between electrons of MLG and 2DEG in the quantum well, the optical plasmon frequency of MLG is modified (see, for example, Eq. 6 of Ref. [19]) and the acoustic plasmon branch appears [25]. The optical (acoustic) mode corresponds to in-phase (out-of-phase) oscillations of electron densities in the two layers. It is well known that at zero temperature, for given electron density of MLG and 2DEG, a free Landau damping acoustic plasmon exists only for d > dc (see, for example, Eq. 16 of Ref. [25] and Eq. 8 of Ref. [19]). At long-wavelength

Fig. 2. Plasmon frequency (a) and damping rate (b) as a function of q/kF for d = 50 nm, κ1 = κAlGaAs = 12.9, κ2 = κSiO2 = 3.8, κ3 = κair = 1.0, n2DEG = nMLG = 1010 cm−2 and T = 0.2T F . The insets show the plasmon frequencies of quantum well using parameters given in Ref. [17] (a) and of double-layer graphene (DLG) system using parameters given in Ref. [4] (b).

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Fig. 3. Plasmon frequency (a), (c) and damping rate (b), (d) for d = 50 nm, κ1 = κAlGaAs = 12.9, n2DEG = nMLG = 1010 cm−2 (a), (b) and n2DEG = 2nMLG = 2.1010 cm−2 (c), (d).

Fig. 4. Plasmon frequency (a) and damping rate (b) versus temperature for d = 50 nm, several wave-vectors.

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κ2 = κSiO2 = 3.8, κ3 = κair = 1.0, in two cases T = 0 and T = 0.2T F with

κ1 = κAlGaAs = 12.9, κ2 = κSiO2 = 3.8, κ3 = κair = 1.0, n2DEG = nMLG = 1010 cm−2 and

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Fig. 5. Plasmon frequency (a), (c) and damping rate (b), (d) for two cases T = 0 (a), (b) and T = 0.2T F (c), (d).

κ1 = κAlGaAs = 12.9, κ2 = κSiO2 = 3.8, κ3 = κair = 1.0, n2DEG = nMLG = 1010 cm−2 and several values of d in

limit (q → 0), the critical interlayer distance dc and the zerotemperature acoustic-plasmon group velocity can be determined by using the power expansion suggested in Ref. 23. For finite q, due to complexity of the dielectric function of the MLG-2DEG system, the acoustic plasmon frequency can be calculated only numerically as shown in Fig. 4 of Ref. [19]. At finite temperatures, the imaginary part of dielectric function of MLG-2DEG double layer is different from zero (see, for example Eq. 9 of Ref. [21]) and electron-hole pairs can be created at all wave vectors, hence the plasmon modes are always damped as shown in Ref. [4]. It is seen from the Fig. 2(a) that meanwhile the dispersion curve of the optical (OP) branch stays out of the single particle excitation (SPE) region of the system, that of the acoustic (AC) branch is always in the SPE region for all wave-vectors. For a comparison, we show in the inset of Fig. 2(a) the plasmon frequency of 2DEG at T = 0.2T F with parameters as given in Ref. [17]. Fig. 2(b) demonstrates the plasmon decay rate of the OP (solid curve) and AC (dashed curve) mode. We find that both plasmon modes stay relatively well-defined only in long wave-length region and decay strongly at large q values. The AC branch gets overdamped and disappears at q ≈ 1.4kF while the OP one is damped more weakly. To check the validity of our numerical calculations we have calculated the plasmon frequencies of DLG at zero and finite temperature using the same parameters as given in Ref. [4]. The results shown in

the inset of Fig. 2(b) indicate that our results are identical to those of Ref. [4]. In order to understand the effects of temperature on plasmon modes, we plot on Fig. 3 the plasmon frequency and damping rate using T = 0 and 0.2T F for two cases n2DEG = nMLG = 1010 cm−2 (a), (b) and n2DEG = 2nMLG = 2.1010 cm−2 (c), (d). We observe that a slight increase in temperature leads to a slight (strong) decrease (increase) of OP (AC) plasmon energy. The temperature increases considerably the damping rate of both AC and OP plasmon modes especially in large-q region because the electrons with larger kinetic energies can be easier excited to the electron–hole pairs. The AC branch is well-defined at T = 0 but is always damped at T = 0.2T F . At T = 0, the OP mode remains well-defined until q = 0.9kF while at T = 0.2T F it becomes damped even in long wave-length limit. This behavior of both plasmon modes stems from the fact that, in contrast to the case T = 0, at finite temperatures the electron–hole pairs can be created at all wave vectors. Fig. 4(c) shows that, the carrier density imbalance between the two layers affects weakly on the OP branch, but it causes considerable changes in the AC one. We also notice that the temperature effect on the AC branch is more pronounced in the case of carrier density imbalance. To understand the temperature effect in more details, we plot in Fig. 4 the plasmon frequency as a function of temperature for

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Fig. 6. Plasmon frequency (a), (c) and damping rate (b), (d) for κ1 = κAlGaAs = 12.9, κ2 = κSiO2 = 3.8, κ3 = κair = 1.0 and for a homogeneous dielectric background with average permittivity κ¯ = 6.95 in two cases T = 0 (a), (b) and T = 0.2T F (c), (d) for n2DEG = nMLG = 1010 cm−2 , d = 50 nm.

several values of wave-vector q. The Fig. 4(a) indicates that at the given q, the AC plasmon frequency increases as temperature increases. For large q and T , the AC plasmon branch no longer exists due to strong damping as shown in Fig. 4(b). The OP plasmon frequency decreases with increasing T , gets a minimum at T ≈ 0.4T F then increases with temperature. The damping rate of OP branch increases, with the increase of temperature, from zero to a peak at T ≈ 0.4T F and then decreases with further increase in T . In Fig. 5 we illustrate the combined effect of the spacer thickness and temperature on plasmon frequency and damping rate using T = 0 (a), (b) and 0.2T F (c), (d). It is seen that in both cases, the plasmon frequency of two branches increases with increasing d. In the region of intermediate values of q, the frequency of OP (AC) branch depends significantly (slightly) on d. The decay rates of both plasmon branches show a considerable dependence on the spacer thickness. In case of T = 0.2T F , the AC modes get overdamped at a critical wave-vector qc , which increases with increasing d, and disappear before reaching the SPE boundary. The effects of inhomogeneity of dielectric background on plasmon frequency and damping rate are depicted on Fig. 6 for T = 0 (a), (b) and T = 0.2T F (c), (d). It is seen that the inhomogeneity of the dielectric background causes the considerable (slight) increase (decrease) in OP (AC) plasmon frequency in both cases T = 0 and

0.2T F . The effect of inhomogeneity of the dielectric background on plasmon frequency shows weak dependence on temperature. We note that in the DLG systems the effect of the dielectric background inhomogeneity on the plasmon energy becomes stronger with T [28]. Finally, we investigate the effects of inhomogeneity of the dielectric background for T = 0.1T F and T = 0.5T F (Fig. 7). It is seen that, as the temperature increases, the damping rate of the AC branch increases strongly and gets overdamped at smaller wavevector. 4. Conclusion In summary, we investigate theoretically, for the first time, the plasmon energy and decay rate in a double-layer system consisting of MLG and 2DEG at finite temperature. The obtained results indicate that the temperature and spacer thickness affect considerably the frequency and decay rate of both plasmon branches. Meanwhile, the effect of the imbalance of electron density on plasmon increases significantly with increasing temperature, the effect of the inhomogeneity of the dielectric background depends slightly on T . We also find that at very low temperature, the OP plasmon frequency decreases as temperature increases, gets a minimum and then increases again in larger wave-vector region. Finally we

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Fig. 7. Plasmon frequency (a), (c) and damping rate (b), (d) for κ1 = κAlGaAs = 12.9, κ2 = κSiO2 = 3.8, κ3 = κair = 1.0 and for a homogeneous dielectric background (with κ¯ = 6.95) in two cases T = 0.1T F (a), (b) and T = 0.5T F (c), (d) for n2DEG = nMLG = 1010 cm−2 and d = 50 nm.

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