Phổ Plasmon trong hệ ba lớp Graphene với điện môi nền không đồng nhất

 Dong Thap University Journal of Science, Vol. 9, No. 5, 2020, 51-58
 PLASMON MODES IN THREE-LAYER GRAPHENE WITH 
 INHOMOGENEOUS BACKGROUND DIELECTRIC
 Nguyen Van Men1*, Dong Thi Kim Phuong1, and Truong Minh Rang2
 1An Giang University, Vietnam National University Ho Chi Minh City
 2Student, An Giang University, Vietnam National University Ho Chi Minh City
 *Corresponding author: nvmen@agu.edu.vn
 Article history
 Received: 10/09/2020; Received in revised form: 24/09/2020; Accepted: 30/09/2020
 Abstract
 The aim of this paper is to investigate collective excitations and the damping rate in a multilayer 
structure consisting of three monolayer graphene sheets with inhomogeneous background dielectric 
at zero temperature within random-phase approximation. Numerical results show that one optical 
branch and two acoustic ones exist in the system. The lowest frequency branch disappears as touching 
single-particle excitation area boundary while two higher frequency branches continue in this region. 
Calculations also illustrate that the frequency of optical (acoustic) mode(s) decreases (increase) as 
interlayer separation increases. The inhomogeneity of background dielectric and the imbalance in the 
carrier density in graphene sheets decline signifi cantly plasmon frequencies in the system. Therefore, it 
is meaningful to take into account the eff ects of inhomogeneous background dielectric when calculating 
collective excitations in three-layer graphene structures.
 Keywords: Collective excitations, inhomogeneous background dielectric, random–phase–
approximation, three-layer graphene systems.
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 PHỔ PLASMON TRONG HỆ BA LỚP GRAPHENE VỚI ĐIỆN MÔI NỀN 
 KHÔNG ĐỒNG NHẤT
 Nguyễn Văn Mện1*, Đổng Thị Kim Phượng1 và Trương Minh Rạng2
 1Trường Đại học An Giang, Đại học Quốc gia Thành phố Hồ Chí Minh
 2Sinh viên, Trường Đại học An Giang, Đại học Quốc gia Thành phố Hồ Chí Minh
 *Tác giả liên hệ: nvmen@agu.edu.vn
 Lịch sử bài báo
 Ngày nhận: 10/09/2020; Ngày nhận chỉnh sửa: 24/09/2020; Ngày duyệt đăng: 30/09/2020
 Tóm tắt
 Bài báo này nhằm khảo sát kích thích tập thể và hấp thụ trong một cấu trúc nhiều lớp gồm ba lớp 
graphene với điện môi nền không đồng nhất ở nhiệt độ không tuyệt đối trong gần đúng pha ngẫu nhiên. 
Kết quả tính toán bằng số cho thấy một nhánh quang học và hai nhánh âm học tồn tại bên trong hệ. 
Nhánh có tần số thấp nhất biến mất khi chạm vào đường biên vùng kích thích đơn hạt trong khi hai 
nhánh có tần số cao hơn vẫn tiếp tục tồn tại trong vùng này. Các tính toán cũng cho thấy, tần số nhánh 
quang giảm xuống còn tần số các nhánh âm lại tăng lên khi khoảng cách các lớp tăng. Sự không đồng 
nhất của hằng số điện môi nền và sự mất cân bằng về mật độ hạt tải giữa các lớp graphene làm giảm 
đáng kể các tần số plasmon trong hệ. Do đó, việc tính đến ảnh hưởng của hằng số điện môi nền không 
đồng nhất khi xác định kích thích tập thể trong hệ ba lớp graphene là việc làm có ý nghĩa.
 Từ khóa: Kích thích plasmon, điện môi nền không đồng nhất, gần đúng pha ngẫu nhiên, hệ ba 
lớp graphene.
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Natural Sciences issue
 1. Introduction illustrate that the inhomogeneity of background 
 Graphene, a perfect two dimensional system dielectric has pronounced eff ects on plasmon 
consisting of one layer of carbon atoms arranged modes (Badalyan and Peeters, 2012; Principi 
in honey-comb lattice, has attracted a lot of et al., 2012; Men and Khanh, 2017; Khanh and 
attention from material scientists in recent years Men, 2018). However, most of previous works 
because of its interesting features as well as about multilayer graphene have neglected the 
application abilities in technology. Theoretical contributions of this factor to plasmon characters 
and experimental researches on graphene show due to diff erent reasons (Yan et al., 2012; Zhu 
that the diff erent characters of quasi-particles in et al., 2013; Men et al., 2019; Men, 2020). This 
graphene, compared to normal two-dimensional paper presents results calculated for collective 
electron gas, are chirality, linear dispersion at excitations and the damping rate of respective 
low energy and massless fermions. Due to these plasma oscillations in a multilayer structure, 
unique properties, graphene is considered a good consisting of three parallel monolayer graphene 
candidate, replacing silicon materials being used sheets, separated by diff erent dielectric mediums 
in creating electronic devices (DasSarma et al., in order to improve the model.
2011; Geim and Novoselov, 2007; McCann, 2011). 2. Theory approach
 Collective excitation (or collective plasmon) We investigate a multilayer system consisting 
is one of the important properties of material of three parallel monolayer graphene, separated 
because it is relevant to many technological fi elds, by a different dielectric medium with equal 
includi ... erating in this range 2
of frequency (DasSarma et al., 2011; Geim Graphene 1
and Novoselov, 2007; Hwang and DasSarma, K
 SiO2 1
2007; Sensarma, et al., 2010; Shin et al., 2015). 
It is well known that the Coulomb interaction 
between charged particles in multilayer structures Figure 1. Three –layer graphene system with 
lead to the signifi cant increase in the frequency inhomogeneous background dielectric
of undamped and weak-damped plasmon modes 
existing in the systems (Yan et al., 2012; Zhu et al., It is well known that the plasmon dispersion 
2013; Men et al., 2019; Men, 2020). Moreover, relation of the system can be determined from the 
publications on multilayer structures also zeroes of dynamical dielectric function (Sarma 
 and Madhukar, 1981; Hwang and DasSarma, 
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 Dong Thap University Journal of Science, Vol. 9, No. 5, 2020, 51-58
2009; Vazifehshenas et al., 2010; Badalyan and 
Peeters, 2012; Zhu et al., 2013; Khanh and Men, 
 1
2018; Men and Khanh, 2017; Men et al., 2019; §·w ReHZq ,
 ¨¸
Men, 2020): JHZ Imq ,p . (3)
 ¨¸wZ
 ZZ 
 ©¹p 
 HZqi,0.p  J (1)
 Within random-phase approximation (RPA), 
 Where ω is plasmon frequency at given wave the dynamical dielectric function of three-layer 
 p 
vector q, and J is the damping rate of respective graphene structure is written by (Yan et al., 2012; 
plasma oscillations. In the case of weak damping, Zhu et al., 2013; Men et al., 2019; Men, 2020):
the solutions of equation (1) can be found from 
 HZqvqq,det1 3ˆ ˆ ,. Z(4)
the zeroes of the real part of dynamical dielectric 
functions as (Sarma and Madhukar, 1981; Hwang 
and DasSarma, 2009; Vazifehshenas et al., 2010; Here, vqˆ is the potential tensor, 
Badalyan and Peeters, 2012; Zhu et al., 2013; corresponding to Coulomb bare interactions 
Khanh and Men, 2018; Men and Khanh, 2017; between electrons in graphene sheets, formed 
Men et al., 2019; Men, 2020): from Poisson equation and read (Scharf and 
 Matos-Abiague, 2012; Phuong and Men, 2019; 
 ReHZq , 0. (2) Men, 2019):
 p 
 The damping rate can be calculated from the 
 2S e2
following equation: vqij fq ij . (5)
 q 
 Where:
 22ªºNNNN NNNee24qd  NNNN qd
 fq ¬¼2334 323 2334 , (6)
 11 Mqd
 8e2qd ªºªºNNNN cosh qdqdqdqd sinh  cosh  sinh 
 fq ¬¼¬¼1234 , (7)
 22 Mqd
 22ªºNNNN NNNee24qd  NNNN qd
 fq ¬¼2321 232 1223 , (8)
 33 Mqd
 8NNeqdqd2qd ªº cosh N sinh 
 fq fq  23¬¼ 4 , (9)
 12 21 Mqd
 8NNe2qd
 fq fq  23 , (10)
 13 31 Mqd
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Natural Sciences issue
 8NNeqdqd2qd ªº cosh N sinh 
 fq fq  32¬¼ 1 , (11)
 32 23 Mqd
 with 
 Mx NNNNNN     2 e2x  NNNNNN     
 122334 231324 (12)
 4x
 e NNNNNN122334  .
 characters in an inhomogeneous three-layer 
 3ˆ q,Z is the polarization tensor of the graphene system, compared to homogeneous 
system. When electron tunneling between ones. The numerical results calculated for this 
graphene layers can be neglected (large system are demonstrated in the following. 
separation), only diagonal elements of the 
 3. Results and discussions
polarization tensor diff er from zero, so
 This section presents numerical results 
 ˆ i
 3qq,,.ZG ij 30  Z (13) calculated for collective excitations in a three-
 layer graphene system with inhomogeneous 
 i background dielectric at zero temperature. In 
 In equation (13), 30 q,Z is Lindhard 
polarization function of layer graphene at zero an inhomogeneous system, dielectric constants 
temperature (i 13y) in the structure observed used are NN 3.8, NN 6.1, 
 1 SiO2 2 Al23 O
by Hwang and DasSarma (2007).
 NN3 BN 5.0; NN4 air 1.0. In all figures, 
 Equations (5)-(12) show the complicated 
 E and k are used to denote Femi energy and 
dependence of Coulomb bare interactions on the F F 
inhomogeneity of background dielectric. This Fermi wave vector of the fi rst graphene sheet.
dependence leads to the diff erences in plasmon 
Figure 2. Plasmon modes (a) and damping rate (b) in three-layer graphene structure, plotted for 
 d = 10 nm and n = n = n = 1012 cm-2. The grey-shaded area in Figure 2(a) shows single-particle 
 1 2 3 
 excitation area of the system
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 Dong Thap University Journal of Science, Vol. 9, No. 5, 2020, 51-58
 Figure 2 plots collective excitations (a) and q = 1,6kF. The damping rate, presented in Figure 
damping rate (b) in a three-layer graphene system 2(b), demonstrates that although Op mode (thick 
shown in Figure 1. Similar to other multilayer solid line) can continue in the SPE region, this 
systems (Yan et al., 2012; Zhu et al., 2013; Men mode loses its energy quickly as the plasmon 
et al., 2019; Men, 2020), three plasmon modes curve goes far away from SPE boundaries. As 
exist in a three-layer graphene structure. The also seen from Figure 2(b), the damping rate 
largest frequency branch is called optical mode of the Ac2 branch increases from zero as this 
(Op), corresponding to in-phase oscillations, and plasmon line crosses intra SPE region boundary, 
two smaller frequency ones are named as acoustic and then decreases as this line approaches inter 
modes (Ac) illustrating out-of-phase oscillations SPE area boundary. This behavior diff ers sharply 
of carriers in the system. The fi gure shows that from that of Op and Ac2 branches. It is necessary 
Op and Ac1 branches continue in single-particle to note that the energy loss in the Op branch is 
excitation (SPE) area while the Ac2 branch similar to that in monolayer graphene, obtained 
disappears as touching SPE boundaries at about by Hwang and DasSarma (2007). 
Figure 3. Collective excitations in three-layer graphene structure for several interlayer separations. 
 12 -2
Parameters used are n1 = n2 = n3 = 10 cm , d = 10 nm; 20 nm; 50 nm and d = 100 nm. Dashed-dotted 
 lines present SPE boundaries
 Collective excitations in a three-layer (up), especially outside SPE region. As a result, 
graphene system with several separations are plasmon branches become closer to each other, 
illustrated in Figure 3. The figure shows that similar to those in multilayer graphene systems 
Op frequency decreases signifi cantly while Ac with homogeneous background dielectric in 
ones increase noticeably as separation increases. which plasmon curves approach that of single-
The changes in frequency occur mainly nearby layer graphene in limit of d of. However, the 
the Dirac points, in a small wave vector region, diff erence between the two cases is that plasmon 
and outside SPE area. Nevertheless, in the case curves in the inhomogeneous case are still 
of Ac branches, plasmon frequencies increase separated from each other for large separations 
slightly in a large wave vector region. It is seen while they are identical in the homogeneous case 
from the fi gures that the increase in the interlayer as observed in previous papers (Yan et al., 2012; 
distance makes Op (Ac) branch shifts down Zhu et al., 2013; Men et al., 2019; Men, 2020).
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Natural Sciences issue
 Figure 4. Plasmon modes in three-layer graphene structure for several carrier densities, ploted for 
 d = 20 nm Dashed-dotted lines show SPE area boundaries
 According to recent publications, carrier all branches decreases noticeably, in comparison 
density has pronounced contributions to plasmon with that of n3 = n1, but at diff erent levels. The 
properties of layered structures (Hwang and Op branch is affected more strongly than Ac 
DasSarma, 2007; Hwang and DasSarma, 2009; ones are. The lowest plasmon branch approaches 
Badalyan and Peeters, 2012; Men and Khanh, SPE area boundary and disappears at a smaller 
2017; Khanh and Men, 2018; Men et al., 2019). wave vector, about q = 1.2kF compared to 1.6kF 
Figure 4 plots plasmon modes in a three- in the case of equal carrier density. Moreover, 
layer graphene system with the variation of as carrier density in the third layer decreases, 
carrier density in graphene sheets. Figure 4(a) the SPE region boundary shifts down (thin- and 
demonstrates that the increase in carrier density thick-dashed-dotted line), so plasmon modes 
in graphene layers declines remarkably frequency are damped at a smaller wave vector. Similar 
of plasmon branches, found mainly outside SPE behavior has been observed for multilayer 
region. Besides, the imbalance in carrier density graphene structures in previous publications 
between graphene layers causes significant (Hwang and DasSarma, 2009; Vazifehshenas et 
eff ects to plasmon modes as seen from Figure al., 2010; Badalyan and Peeters, 2012; Khanh 
4(b). In the case of n3 = 0.5n1, the frequency of and Men, 2018; Men et al., 2019; Men, 2020).
 Figure 5. Plasmon modes (a) and damping rate (b) in three-layer graphene structure in homoge-
 12 -2 
neous and inhomogeneous background dielectric, plotted for d = 20 nm and n1 = n2 = n3 = 10 cm .
 Dashed-dotted lines present SPE are boundaries
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 Dong Thap University Journal of Science, Vol. 9, No. 5, 2020, 51-58
 It is proven that plasmon modes in double The imbalance in the carrier density in graphene 
layer structures consisting of two graphene sheets and the inhomogeneity of the environment 
sheets grown on an inhomogeneous environment cause a noticeable decrease in the frequency of 
have been studied and published. The results plasmon modes. 
show that plasmon properties in these systems Acknowledgements: This work is supported 
are aff ected strongly by the inhomogeneity of by Vietnam National University Ho Chi Minh 
background dielectric (Badalyan and Peeters, City (VNU-HCM)./.
2012; Khanh and Men, 2018). In order to study 
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