Plasmon modes in three-layer graphene with inhomogeneous background dielectric

Nội dung text: Plasmon modes in three-layer graphene with inhomogeneous background dielectric

1. 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. 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. 51
2. 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 including optics, optoelectronics, membrane layer thickness d, as presented in Figure 1. Each technology, and storage technology (Maier, 2007; graphene layer is considered as homogeneously Ryzhii et al., 2013; Politano et al., 2016; Politano doped graphene, so the carrier density is a et al., 2017). Therefore, scientists have been constant n1 (i 13y) over its surface. As a result, interested in calculations on plasmon characters the Fermi wave vector and Fermi energy in each of materials for many years. Collective excitations graphene sheet have uniform distributions. in the ordinary two-dimensional electron gas, in monolayer and in bilayer graphene at zero temperature have been studied and published K Air 4 intensively in the early years of the 21st century. z = 2d Recent theoretical and experimental papers on Graphene 3 K graphene demonstrate that collective excitations Spacer 3 z = d in graphene spread from THz to visible light, Graphene 2 so graphene is considered as a good material to Spacer K create plasmonic devices operating 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, 52
3. 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 53
4. 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 54
5. 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). 55
6. 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 56
7. 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 References the inhomogeneity eff ects, we plot in Figure 5 plasmon frequencies and the damping rate as a Badalyan, S. M. and Peeters, F. M. (2012). Eff ect function of the wave vector in the homogeneous of nonhomogenous dielectric background on and inhomogeneous cases for a comparison. the plasmon modes in graphene double-layer Figure 5(a) demonstrates that plasmon branches structures at fi nite temperatures. Physical in an inhomogeneous system are much lower Review B, (85), 195444. than those in the homogeneous one (with average DasSarma, S., Adam, S., Hwang, E. H. and Rossi, E. (2011). Electronic transport in permittivity NNN / 2 2.4 ) for the same 14 two dimensional graphene. Review Modern separation and carrier density. As seen in Figure Physics, (83), 407. 5(b) that the inhomogeneity of background Geim, A. K. and Novoselov, K. S. (2007). The dielectric decreases signifi cantly the damping rise of graphene. Nature Mater, (6), 183. rate of plasma oscillations at a given wave vector in all branches. Finally, plasmon curves Hwang, E. H. and DasSarma, S. (2007). Dielectric in the homogeneous case can merge together at function, screening, and plasmons in 2D the edge of SPE region with suitable parameters graphene. Physical Review B, (75), 205418. while those in the case of inhomogeneous system Hwang, E. H. and DasSarma, S. (2009). Exotic are always separated from each other. Similar plasmon modes of double layer graphene. behaviors have been obtained in previous works Physical Review B, (80), 205405. for double layer graphene structures (Badalyan Khanh N. Q. and Men N. V. (2018). Plasmon and Peeters, 2012; Khanh and Men, 2018). Modes in Bilayer–Monolayer Graphene 4. Conclusion Heterostructures. Physica Status Solidi B, In summary, collective excitations and the (255), 1700656. damping rate of plasma oscillations in a three-layer Maier, S. A. (2007). Plasmonics – Fundamentals graphene structure on inhomogeneous background and Applications. New York: Springer. dielectric within random-phase approximation at McCann, E. (2011). Electronic Properties of zero temperature have been numerical calculated. Monolayer and Bilayer Graphene. In: The results show that three plasmon branches Raza H. (eds) Graphene Nanoelectronics. exist in the system including one optical and two NanoScience and Technology. Berlin, acoustic modes. Two higher frequency branches Heidelberg: Springer. can continue in a single-particle excitation region org/10.1007/978-3-642-22984-8_8. while the lowest branch merges to the boundary of this region and disappears. The investigations Men, N. V. and Khanh, N. Q. (2017). Plasmon also demonstrate that the increase in interlayer modes in graphene–GaAs heterostructures. distance reduces significantly the separation Physics Letters A, (381), 3779. between plasmon branches at a given wave vector. Men, N. V. (2019). Coulomb bare interaction in 57
8. Natural Sciences issue three-layer graphene. Dong Thap University Satou, A. and Otsuji, T. (2013). Injection Journal of Science, (39), 82-87. terahertz laser using the resonant inter-layer Men, N. V. (2020). Plasmon modes in N-layer radiative transitions in double-graphene- gapped graphene. Physica B, 578, 411876. layer structure. J. Appl. Phys., (113), 174506. Men, N. V., Khanh, N. Q. and Phuong, D. T. Scharf, B. and Matos-Abiague, A. (2012). K. (2019). Plasmon modes in N-layer Coulomb drag between massless and bilayer graphene structures. Solid State massive fermions. Physical Review B, (86), Communications, (298), 113647. 115425. Phuong, D. T. K. and Men, N. V. (2019). Plasmon Sensarma, R., Hwang, E. H. and Sarma, S. D. modes in 3-layer graphene structures: (2010). Dynamic screening and low energy Inhomogeneity eff ects. Physics Letters A, collective modes in bilayer graphene. (383), 125971. Physical Review B, (82), 195428. Politano, A., Cupolillo, A., Di Profio, G., Shin, J-S., Kim, J-S. and Kim, J. T. (2015). Arafat, H. A., Chiarello, G. and Curcio, E. Graphene-based hybrid plasmonic (2016). When plasmonics meets membrane modulator. J. Opt., (17), 125801. technology. J. Phys. Condens. Matter, (28), Vazifehshenas, T., Amlaki, T., Farmanbar, M. 363003. and Parhizgar, F. (2010). Temperature eff ect Politano, A., Pietro, A., Di Profi o, G., Sanna, V., on plasmon dispersions in double-layer Cupolillo, A., Chakraborty, S., Arafat, H. and graphene systems. Physics Letters A, (374), Curcio, E. (2017). Photothermal membrane 4899. distillation for seawater desalination. Yan, H., Li, X., Chandra, B., Tulevski, G., Wu, Advanced Materials, (29), 03504. Y., Freitag, M., Zhu, W., Avouris, P. and Principi, A., Carrega, M., Asgari, R., Pellegrini, Xia, F. (2012). Tunable infrared plasmonic V. and Polini, M. (2012). Plasmons and devices using graphene/insulator stacks. Coulomb drag in Dirac/Schrodinger hybrid Nature Nanotech., (7), 330. electron systems. Physical Review B, (86), Zhu, J.-J., Badalyan S. M. and Peeters F. M. 085421. (2013). Plasmonic excitations in Coulomb- Ryzhii, V., Ryzhii, M., Mitin, V., Shur, M. S., coupled N-layer graphene structures. Physical Review B, (87), 085401. 58