- Email: [email protected]
Plasmon modes in N-layer gapped graphene
Journal Pre-proof Plasmon modes in N-layer gapped graphene
Nguyen Van Men PII:
S0921-4526(19)30758-6
DOI:
https://doi.org/10.1016/j.physb.2019.411876
Reference:
PHYSB 411876
To appear in:
Physica B: Physics of Condensed Matter
Received Date:
23 October 2019
Accepted Date:
07 November 2019
Please cite this article as: Nguyen Van Men, Plasmon modes in N-layer gapped graphene, Physica B: Physics of Condensed Matter (2019), https://doi.org/10.1016/j.physb.2019.411876
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier.
Journal Pre-proof Plasmon modes in N-layer gapped graphene Nguyen Van Men An Giang University – VNU HCM, 18-Ung Van Khiem Street, Long Xuyen, An Giang, Viet Nam (Email: [email protected]).
Abstract We investigate plasmon dispersions and Landau damping in multi-layer gapped graphene structures consisting of N, up to 5, gapped graphene sheets grown on dielectric spacer within random-phase-approximation at zero temperature. First of all, the results show that one in-phase optical and N-1 out-of-phase acoustic plasmon modes are found in the system. In addition, the imbalance of carrier density and the separation between layers affect remarkably on plasmon characters, and plasmon frequencies increase pronouncedly with the increase in the number of layers in the structure. Especially, the increasing band gap leads to the significantly decreasing plasmon frequencies in large wave-vector region. Finally, the number of low carrier density graphene sheets is equal to the number of plasmon branches separated from the others whereas the order of these layers affects weakly on plasmon curves.
PACS: 73.22.Pr; 73.20.Mf; 73.21.Ac
Keywords: Gapped graphene; Collective excitations; Zero temperature; Multilayer structures. _____________________________________________________________________________
1
Journal Pre-proof
Plasmon modes in N-layer gapped graphene
Nguyen Van Men An Giang University – VNU HCM, 18-Ung Van Khiem Street, Long Xuyen, An Giang, Viet Nam (Email: [email protected]).
Abstract We investigate plasmon dispersions and Landau damping in multi-layer gapped graphene structures consisting of N, up to 5, gapped graphene sheets grown on dielectric spacer within random-phase-approximation at zero temperature. First of all, the results show that one in-phase optical and N-1 out-of-phase acoustic plasmon modes are found in the system. In addition, the imbalance of carrier density and the separation between layers affect remarkably on plasmon characters, and plasmon frequencies increase pronouncedly with the increase in the number of layers in the structure. Especially, the increasing band gap leads to the significantly decreasing plasmon frequencies in large wave-vector region. Finally, the number of low carrier density graphene sheets is equal to the number of plasmon branches separated from the others whereas the order of these layers affects weakly on plasmon curves.
1. Introduction In recent years, scientists have been paid a lot of attention for graphene, a wonderful two-dimensional material consisting of carbon atoms arranged in honey comb lattice with unique electrical and optical properties. Most of previous publications consider graphene as a zero gap material with linear low energy dispersion nearby Dirac’s points [1-4]. However, later theoretical and experimental researches demonstrate that a small band gap between conduction and valance band can be generated by various factors. According to these researches, such factors as the symmetry breaking of A and B sub-lattices, causing by the interaction between charged particles in graphene sheet and the substrate and the spin-orbit interaction as well as external electric and magnetic field can lead to the existence of small band gap in energy pattern of graphene, now named gapped graphene [5-10]. When taking into account the finite band gap in calculations, the energy dispersion, polarization function and collective excitations in graphene differ significantly from those in gapless case. It is well known that collective excitations in two dimensional electron gas have been studied extensively and applied to create plasmonic devices. Moreover, lots of technology areas covering electronics, optics, biotechnology and energy storage are also particular applications of plasmon characters in layer structures [2-4,11-18]. Plasmon properties in graphene and graphene-based structures have been investigated carefully with interesting features, compared to those in ordinary semi-conductors in order to find out new materials with better qualities and less energy consumson [22-27]. Calculations on plasmon in mono-layer graphene (MLG) [28], mono-layer gapped graphene (MLGG) [10,14,29], double-layer graphene (DLG) [30-33] have been carried out at zero as well as finite temperature. Besides, previous researches observe that the interaction between charged particles in layers of a multilayer system bring lots of different characters to this kind of structure. The authors of Ref.[34], [35] and Ref.[36] study plasmon in a system consisting of N-layers, up to five, of graphene sheets and demonstrate some interesting characters. However, the band gap separated conduction and valance band is neglected although its contributions to the results are remarkable. Therefore, the aim of this paper is to consider a similar system, taking into account the band gap when calculating plasmon dispersions in order to improve the model. 2. Theory We investigate plasmon frequencies and broadening function, corresponding to Landau damping of plasmon dispersions in a multi-layer gapped graphene consisting of N, up to 5, parallel gapped graphene sheets grown on dielectric substrates. It is known that the plasmon dispersion relations of the system can be found from the zeroes of the real part of the dynamical dielectric function [30-43] Re q, p 0
(1) 2
Journal Pre-proof
where p is the plasmon frequency at critical wave-vector q . In case of N-layer graphene system, the dynamical dielectric function has the following form [31,35,36]:
ˆ q, q, det 1 vˆ q
(2)
where vˆ q vij q is the matrix kernel, corresponding to the bare Coulomb interaction between electrons in
ˆ q, is the irreducible polarization function of the system. spatially separated MLGG sheets, and In case of homogenous environment, the bare Coulomb interaction is a symmetric tensor [35,36]:
vij q v q e
i j qd
with i and j ( i, j 1 N ) are the layer indices, d is interlayer spacing, and v q
(3)
2 e 2 ( is dielectric q
constant of media environment). When the electron tunneling between electrical isolated layers can be neglected, the non-diagonal elements of the polarizability can be set to zero so that it is given [35,36]
ˆ q, i q, ij 0
(4)
Within random-phase-approximation, i0 q, is the polarization function of i th MLGG, formed in Ref.[10]. The broadening function of the respective energy dispersions, describing effectively the Landau damping of the plasmon modes, has form [35,36]: 1 Re i0 q, i i Im 0 q, p
(5)
Numerical solving equation (1) with the form of dielectric function in equation (2), we can find out the plasmon dispersions of the system. And then replacing the solutions into equation (5), we can get the broadening function of respective plasmon modes. 3. Results and discussion In this section, we demonstrate the numerical results calculated for plasmon excitations of a N-layer, up to 5, gapped graphene (N-LGG) system at zero temperature. In the followings, we denote Fermi energy and Fermi wave vector of gapless graphene with the same carrier density in the first layer by EF and k F ( k F n1 ), respectively, and we also set 2.5 in the whole paper. It is noted that when calculating plasmon frequencies, the imaginary part of polarization function is set to zero, following the Ref.[35, 36].
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Figure 1. Plasmon modes in N-LGG with N 2 (a), N 3 (b), N 4 (c) and N 5 (d), plotted for d 5nm , ni 1012 cm 2 ( i 1 5 ) and 0.1 . Thin short-dashed-dotted lines demonstrate plasmon in MLGG. Dashed-dotted lines show SPE area boundaries (color online). Figure 1 illustrates plasmon frequencies in N-LGG system with N 2 (a), N 3 (b), N 4 (c) and N 5 (d), plotted for d 5nm , ni 1012 cm 2 ( i 1 5 ) and 0.1 . The figures show that plasmon dispersions in the system includes one in-phase optical (OP) mode and N 1 out-of-phase acoustic (AC) ones. The frequency of OP plasmon mode is much larger while those of AC ones are smaller than that in MLGG. At the band gap of 0.1 , OP and AC plasmon frequencies increase significantly with the increase in the number of layers in small wave-vector range, outside the single-particle-excitation (SPE) area. When entering SPE region, frequencies of plasmon modes depend weakly on the number of layers of the system. This character is similar to that in multi-layer gapless graphene structures at finite temperature, obtained in Ref.[35] and Ref. [36].
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Figure 2. Broadening function of plasmon dispersions in N-LGG with N 2 (a), N 3 (b), N 4 (c) and N 5 (d), plotted for d 5nm , ni 1012 cm 2 ( i 1 5 ) and 0.1 . Thin short-dashed-dotted lines demonstrate broadening function of plasmon in MLGG (color online). In order to study the Landau damping of plasmon modes, we plot in figure 2 broadening function of plasmon dispersions in 2-LGG (a), 3-LGG (b), 4-LGG (c) and 5-LGG (d) for the same parameters in Fig.1. As seen from the figures that broadening functions increase from zero as the plasmon branches enter SPE region, at about q 0.6k F k F . It is seen that the function gets larger values as respective plasmon branch goes far from SPE area boundaries, similar to those in N-layer gapless graphene system at finite temperature [35]. Therefore, the broadening function of OP plasmon branch takes the largest value, compared to those of the others at critical wave-vector. On the other hand, the broadening functions of all plasmon branches increase noticeably with the increase in the number of layers in the system.
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Figure 3. Plasmon frequencies and broadening function in 3-LGG ((a) & (b)) and 4-LGG ((c) &(d)) with 0.1 and 0 , plotted for d 5nm and ni 1012 cm 2 ( i 1 4 ). Figure 3 plots plasmon modes and respective broadening function in 3-LGG and 4-LGG system with d 5nm ,
ni 1012 cm 2 ( i 1 4 ), 0.1 and 0 for a comparison. It is seen from the figures that taking into account the band gap when calculating plasmon dispersions decreases slightly the frequency of all plasmon modes, compared to the case of gapless situation. In addition, the broadening function in gapped graphene system gets larger values than that in gapless graphene one with the same parameters. The significant changes mainly occur in large momentum range when the plasmon curves cross the SPE area boundaries.
Figure 4. Plasmon modes in N-LGG with N 2 (a), N 3 (b), N 4 (c) and N 5 (d), plotted for ni 1012 cm 2 ( i 1 5 ), d 5nm , 0.1 and 0.3 . Dashed-dotted lines show SPE area boundaries (color online). 6
Journal Pre-proof Figure 4 presents plasmon modes in N-LGG ( N 2 5 ) for ni 1012 cm 2 ( i 1 5 ), d 5nm , 0.1 and 0.3 . As can be seen from the figures that in long wavelength region ( q 0.3k F ), the band gap affects weakly
on plasmon frequencies, therefore, plasmon branches in two cases are seemly identical. However, in sufficiently large wave-vector region, plasmon frequencies decrease noticeably with the increase in the band gap. The OP plasmon branch is affected strongest by the gap, compared to the others. Moreover, in case of 0.1 , plasmon branches enter SPE area meanwhile in the other case ( 0.3 ) they go into the separation between inter- and intraband. This behaviors are sharply different from those in N-layer gapless graphene system, obtained in Ref.[35]. It can be stated that only the large band gap affects remarkably on plasmon characters in the system.
Figure 5. Plasmon frequencies in 3-LGG ((a) and (b)) and 4-LGG ((c) & (d)) as functions of band gap with several values of wave-vector, plotted for d 5nm and ni 1012 cm 2 . Thin solid lines show plasmon frequency in MLGG at the same parameters (color online). In figure 5, we plot plasmon frequencies in 3-LGG and 4-LGG as functions of band gap, increasing from zero to 0.4 with d 5nm , ni 1012 cm 2 and several values of momentum. Thin solid lines show plasmon frequency in MLGG at the same parameters. It is seen from the figures that plasmon frequencies decrease with the increase in band gap for all chosen values of wave-vector, in general. The effects of the band gap on plasmon in multi-layer systems are weakly than that does on MLGG at the same parameters. Again, the considerable changes in plasmon frequencies happen with large values of band gap and large momentum.
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Figure 6. Plasmon modes in N-LGG with N 2 (a), N 3 (b), N 4 (c) and N 5 (d), plotted for ni 1012 cm 2 ( i 1 5 ), 0.1 , d 5nm and d 10nm . Dashed-dotted lines show SPE area boundaries (color online). Now we turn to consider the effects of interlayer separation on plasmon characters of N-LGG. Figure 6 illustrates collective excitations of 2-LGG (a), 3-LGG (b), 4-LGG (c) and 5-LGG (d) for ni 1012 cm 2 ( i 1 5 ), 0.1 with two separated distances d 5nm and d 10nm (and d 20nm for 2-LGG). As seen from the figures that OP plasmon frequency decreases pronouncedly with the increase in interlayer separation while the AC ones increase with that. Consequently, the separation between OP and AC branches becomes smaller, and the branches go nearby each other. It can be predicted that plasmon curves would be identical in SPE area with sufficiently large interlayer distances, similar to those in case of gapless graphene system observed in Ref.[35] and Ref.[36].
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Figure 7. Plasmon frequencies in N-LGG with N 2 (a), N 3 (b), N 4 (c) and N 5 (d), plotted for ni 1012 cm 2 and ni 2n2 1012 cm 2 ( i 2 ), d 5nm and 0.1 . Dashed-dotted lines show SPE area boundaries (color online). In order to study the effects of the imbalance in carrier density in layers on plasmon properties of the N-LGG structure, Figure 7 plots collective excitations in N-LGG with d 5nm , 0.1 in two case: balanced and imbalanced carrier density ( n2 0.5ni , i 2 ). As seen from the figures that plasmon frequencies in case of n2 0.5ni are remarkably lower than those in the other case. The noticeable decrease in OP plasmon frequency occurs with the decrease in electron density in the second layer, even in very small wave-vector region ( q 0.1k F ). Besides, the AC plasmon frequencies just decrease slightly, and the changes occur when wave-vector is sufficiently large, about q 0.3k F . It can be said that the density of electrons in layers affects significantly on plasmon frequencies of the system.
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Figure 8. Plasmon frequencies in 4-MLGG different values of carrier density, plotted for d 5nm and n1 1012 cm 2 , 0.1 . Dashed-dotted lines show SPE area boundaries (color online). In case of 4-LGG, plasmon modes are demonstrated in Figure 8 for d 5nm , n1 1012 cm 2 and 0.1 . Fig.8(a) and (b) present that plasmon frequencies in case of n2 0.5ni ( i 2 ) and in case of n4 0.5ni ( i 4 ) take approximate values. Similarly, plasmon modes in two cases n2 n3 0.5n1 (Fig.8(c)) and n2 n4 0.5n1 (Fig.8(d)) are also difficult to distinguish from the others. This means that plasmon properties depend weakly on the order of low carrier density graphene sheets in the system. Making a comparison between Fig.8(a), (b) and Fig.8(c), (d), it can be stated that the number of low carrier density layers affects pronouncedly on plasmon pattern. The number of plasmon branches separated from the others is equal to the number of low carrier density layers in the system. 4. Conclusion In summary, collective excitations and broadening function in N-LGG, up to 5, at zero temperature are calculated theoretically for the first time. Numerical results present that the zero-point equation of dynamical dielectric function of the system admits N solutions corresponding to one optical and N 1 acoustic plasmon modes. The OP (AC) plasmon branch (branches) is higher (are lower) than that in MLGG at the same parameters. The broadening function of OP mode gets the largest values, compared to other branches. In addition, plasmon frequencies in the system decrease with the increase in the band gap, with lower level in comparison with MLGG. On the other hand, both interlayer distance and carrier density in layers affect significantly on plasmon characters. Finally, the number of plasmon branches separated from the others depends on the number (not the order) of low carrier density layers in the system. References [1] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, A. K. Geim, Rev. Mod. Phys. 81, 109 (2009). [2] A. K. Geim, K. S. Novoselov, Nature Mater 6, 183 (2007). [3] A. K. Geim, A. H. MacDonald, Phys. Today 60, 35 (2007). [4] J. Wei, Z. Zang, Y. Zhang, M. Wang, J. Du, and X. Tang, Optics Letters 42, 911 (2017). [5] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva and A.A. Firsov, Science 306, 666 (2004). [6] Y. Zhang, Y.-W. Tan, H.L. Stormer and P. Kim, Nature 438, 201 (2005). [7] G. Li, A. Luican and E.Y. Andrei, Phys. Rev. Lett 102, 176804 (2009). [8] Y. Yao, F. Ye, X.-L. Qi, S.-C. Zhang and Z. Fang, Phys. Rev. B 75, 041401 (2007). [9] E.V. Gorbar, V.P. Gusynin, V.A. Miransky and I.A. Shovkovy, Phys. Rev. B 78, 085437 (2008). [10] P. K. Pyatkovskiy, J. Phys. Condens. Matter 21, 025506 (2009). [11] F. J. Garcia de Abajo, ACS Photonics 1, 135 (2014). [12] H. L. Koppens Frank, D. E. Chang and F. J. Garcia de Abajo, Nano Lett. 11, 3370 (2011). 10
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[13] P. Avouris and M. Freitag, IEEE 20, 6000112 (2014). [14] D. K. Patel, Syed S. Z. Ashraf and Ami C. Sharma, Phys. Stat. Sol. 252(8), 1817 (2015). [15] V. Ryzhii, M. Ryzhii, V. Mitin, M. S. Shur, A. Satou and T. Otsuji, J. Appl. Phys. 113, 174506 (2013). [16] A. Politano, A. Pietro, G. Di Profio, V. Sanna, A. Cupolillo, S. Chakraborty, H. Arafa and E. Curcio, Advanced Materials 29 (2), 201603504 (2017). [17] M. Polini, R. Asgari, G. Borghi, Y. Barlas, T. Pereg-Barnea, and A. H. MacDonald, Phys. Rev. B 77, 081411(R) (2008). [18] A. Politano, A. Cupolillo, G. Di Profio, H. A. Arafat, G. Chiarello and E. Curcio, J. Phys. Condens. Matter 28, 363003 (2016). [19] N. V. Men and D. T. K. Phuong, Int. J. Mod. Phys. B 33(16), 1950174 (2019). [20] G. X. Ni A. S. McLeod, Z. Sun, L. Wang, L. Xiong, K. W. Post, S. S. Sunku, B.-Y. Jiang, J. Hone, C. R. Dean, M. M. Fogler and D. N. Basov, Nature 557, 530 (2018). [21] M. B. Lundeberg, Y. Gao, R. Asgari, Ch. Tan, B. V. Duppen, M. Autore, P. A. González, A. Woessner, K. Watanabe, T. Taniguchi, R. Hillenbrand, J. Hone, M. Polini and F. H. L. Koppens, Science 357, 187 (2017). [22] S. S. Sunku, G. X. Ni, B. Y. Jiang, H. Yoo, A. Sternbach, A. S. McLeod, T. Stauber, L. Xiong, T. Taniguchi, K. Watanabe, P. Kim, M. M. Fogler, D. N. Basov, Science 362, 1153 (2018). [23] B. Y. Jiang, G. X. Ni, Z. Addison, J. K Shi, X. Liu, Sh. Y. F. Zhao, P. Kim, E. J. Mele, D. N. Basov and M. M. Foglera, Nano Lett. 17, 7080 (2017). [24] M. B. Lundeberg, M. B. Lundeberg, Y. Gao, R. Asgari, Ch. Tan, B. V. Duppen, M. Autore, P. A. González, A. Woessner, K. Watanabe, T. Taniguchi, R. Hillenbrand, J. Hone, M. Polini and F. H. L. Koppens, Nat. Mater. 16, 204 (2017). [25] G. X. Ni, L. Wang, M. D. Goldflam, M. Wagner, Z. Fei, A. S. McLeod, M. K. Liu, F. Keilmann, B. Özyilmaz, A. H. Castro Neto, J. Hone, M. M. Fogler and D. N. Basovet, Nat. Photon. 10, 244 (2016). [26] A. Politano, H. K. Yu, D. Farías and G. Chiarello, Phys. Rev. B 97, 035414 (2018). [27] F. H. L. Koppens, T. Mueller, Ph. Avouris, A. C. Ferrari, M. S. Vitiello and M. Polini, Nat. Nanotechnol. 9, 780 (2014). [28] E. H. Hwang and S. D. Sarma, Phys. Rev. B 75, 205418 (2007). [29] D. K. Patel, A. C. Sharma, and S. S. Z. Ashraf, Phys. Stat. Sol. B 252(2), 282 (2015). [30] E. H. Hwang, S. D. Sarma, Phys. Rev. B 80, 205405 (2009). [31] S. M. Badalyan, F. M. Peeters, Phys. Rev. B 85, 195444 (2012). [32] D. V. Tuan and N. Q. Khanh, Physica E 54, 267 (2013). [33] T. Vazifehshenas, T. Amlaki, M. Farmanbar, F. Parhizgar, Phys. Lett. A 374, 4899 (2010). [34] H. Yan, X. Li, B. Chandra, G. Tulevski, Y.Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, Nature Nanotech. 7, 330 (2012). [35] J.-J. Zhu, S. M. Badalyan and F. M. Peeters , Physical Review B 87, 085401 (2013). [36] N. V. Men, N. Q. Khanh & D.T.K. Phuong, Solid State Communications 298, 113647 (2019). [37] N. V. Men and N. Q. Khanh, Physics Letters A 381, 3779 (2017). [38] A. Principi, M. Carrega, R. Asgari, V. Pellegrini, and M. Polini, Phys. Rev. B 86, 085421 (2012). [39] S. D. Sarma and A. Madhukar, Phys. Rev. B 23, 805 (1981). [40] B. Tanatar and B. Davoudi, Physical Review B 63, 165328 (2003). [41] G. Gonzalez de la Cruz, Solid State Comm. 262, 11 (2017). [42] N. V. Men and N. Q. Khanh, Canadian Journal of Physics 96, 615 (2018). [43] D. T. K. Phuong and N. V. Men, Physics Letters A 383, 125971 (2019).
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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
Nguyen Van Men PII:
S0921-4526(19)30758-6
DOI:
https://doi.org/10.1016/j.physb.2019.411876
Reference:
PHYSB 411876
To appear in:
Physica B: Physics of Condensed Matter
Received Date:
23 October 2019
Accepted Date:
07 November 2019
Please cite this article as: Nguyen Van Men, Plasmon modes in N-layer gapped graphene, Physica B: Physics of Condensed Matter (2019), https://doi.org/10.1016/j.physb.2019.411876
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier.
Journal Pre-proof Plasmon modes in N-layer gapped graphene Nguyen Van Men An Giang University – VNU HCM, 18-Ung Van Khiem Street, Long Xuyen, An Giang, Viet Nam (Email: [email protected]).
Abstract We investigate plasmon dispersions and Landau damping in multi-layer gapped graphene structures consisting of N, up to 5, gapped graphene sheets grown on dielectric spacer within random-phase-approximation at zero temperature. First of all, the results show that one in-phase optical and N-1 out-of-phase acoustic plasmon modes are found in the system. In addition, the imbalance of carrier density and the separation between layers affect remarkably on plasmon characters, and plasmon frequencies increase pronouncedly with the increase in the number of layers in the structure. Especially, the increasing band gap leads to the significantly decreasing plasmon frequencies in large wave-vector region. Finally, the number of low carrier density graphene sheets is equal to the number of plasmon branches separated from the others whereas the order of these layers affects weakly on plasmon curves.
PACS: 73.22.Pr; 73.20.Mf; 73.21.Ac
Keywords: Gapped graphene; Collective excitations; Zero temperature; Multilayer structures. _____________________________________________________________________________
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Journal Pre-proof
Plasmon modes in N-layer gapped graphene
Nguyen Van Men An Giang University – VNU HCM, 18-Ung Van Khiem Street, Long Xuyen, An Giang, Viet Nam (Email: [email protected]).
Abstract We investigate plasmon dispersions and Landau damping in multi-layer gapped graphene structures consisting of N, up to 5, gapped graphene sheets grown on dielectric spacer within random-phase-approximation at zero temperature. First of all, the results show that one in-phase optical and N-1 out-of-phase acoustic plasmon modes are found in the system. In addition, the imbalance of carrier density and the separation between layers affect remarkably on plasmon characters, and plasmon frequencies increase pronouncedly with the increase in the number of layers in the structure. Especially, the increasing band gap leads to the significantly decreasing plasmon frequencies in large wave-vector region. Finally, the number of low carrier density graphene sheets is equal to the number of plasmon branches separated from the others whereas the order of these layers affects weakly on plasmon curves.
1. Introduction In recent years, scientists have been paid a lot of attention for graphene, a wonderful two-dimensional material consisting of carbon atoms arranged in honey comb lattice with unique electrical and optical properties. Most of previous publications consider graphene as a zero gap material with linear low energy dispersion nearby Dirac’s points [1-4]. However, later theoretical and experimental researches demonstrate that a small band gap between conduction and valance band can be generated by various factors. According to these researches, such factors as the symmetry breaking of A and B sub-lattices, causing by the interaction between charged particles in graphene sheet and the substrate and the spin-orbit interaction as well as external electric and magnetic field can lead to the existence of small band gap in energy pattern of graphene, now named gapped graphene [5-10]. When taking into account the finite band gap in calculations, the energy dispersion, polarization function and collective excitations in graphene differ significantly from those in gapless case. It is well known that collective excitations in two dimensional electron gas have been studied extensively and applied to create plasmonic devices. Moreover, lots of technology areas covering electronics, optics, biotechnology and energy storage are also particular applications of plasmon characters in layer structures [2-4,11-18]. Plasmon properties in graphene and graphene-based structures have been investigated carefully with interesting features, compared to those in ordinary semi-conductors in order to find out new materials with better qualities and less energy consumson [22-27]. Calculations on plasmon in mono-layer graphene (MLG) [28], mono-layer gapped graphene (MLGG) [10,14,29], double-layer graphene (DLG) [30-33] have been carried out at zero as well as finite temperature. Besides, previous researches observe that the interaction between charged particles in layers of a multilayer system bring lots of different characters to this kind of structure. The authors of Ref.[34], [35] and Ref.[36] study plasmon in a system consisting of N-layers, up to five, of graphene sheets and demonstrate some interesting characters. However, the band gap separated conduction and valance band is neglected although its contributions to the results are remarkable. Therefore, the aim of this paper is to consider a similar system, taking into account the band gap when calculating plasmon dispersions in order to improve the model. 2. Theory We investigate plasmon frequencies and broadening function, corresponding to Landau damping of plasmon dispersions in a multi-layer gapped graphene consisting of N, up to 5, parallel gapped graphene sheets grown on dielectric substrates. It is known that the plasmon dispersion relations of the system can be found from the zeroes of the real part of the dynamical dielectric function [30-43] Re q, p 0
(1) 2
Journal Pre-proof
where p is the plasmon frequency at critical wave-vector q . In case of N-layer graphene system, the dynamical dielectric function has the following form [31,35,36]:
ˆ q, q, det 1 vˆ q
(2)
where vˆ q vij q is the matrix kernel, corresponding to the bare Coulomb interaction between electrons in
ˆ q, is the irreducible polarization function of the system. spatially separated MLGG sheets, and In case of homogenous environment, the bare Coulomb interaction is a symmetric tensor [35,36]:
vij q v q e
i j qd
with i and j ( i, j 1 N ) are the layer indices, d is interlayer spacing, and v q
(3)
2 e 2 ( is dielectric q
constant of media environment). When the electron tunneling between electrical isolated layers can be neglected, the non-diagonal elements of the polarizability can be set to zero so that it is given [35,36]
ˆ q, i q, ij 0
(4)
Within random-phase-approximation, i0 q, is the polarization function of i th MLGG, formed in Ref.[10]. The broadening function of the respective energy dispersions, describing effectively the Landau damping of the plasmon modes, has form [35,36]: 1 Re i0 q, i i Im 0 q, p
(5)
Numerical solving equation (1) with the form of dielectric function in equation (2), we can find out the plasmon dispersions of the system. And then replacing the solutions into equation (5), we can get the broadening function of respective plasmon modes. 3. Results and discussion In this section, we demonstrate the numerical results calculated for plasmon excitations of a N-layer, up to 5, gapped graphene (N-LGG) system at zero temperature. In the followings, we denote Fermi energy and Fermi wave vector of gapless graphene with the same carrier density in the first layer by EF and k F ( k F n1 ), respectively, and we also set 2.5 in the whole paper. It is noted that when calculating plasmon frequencies, the imaginary part of polarization function is set to zero, following the Ref.[35, 36].
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Figure 1. Plasmon modes in N-LGG with N 2 (a), N 3 (b), N 4 (c) and N 5 (d), plotted for d 5nm , ni 1012 cm 2 ( i 1 5 ) and 0.1 . Thin short-dashed-dotted lines demonstrate plasmon in MLGG. Dashed-dotted lines show SPE area boundaries (color online). Figure 1 illustrates plasmon frequencies in N-LGG system with N 2 (a), N 3 (b), N 4 (c) and N 5 (d), plotted for d 5nm , ni 1012 cm 2 ( i 1 5 ) and 0.1 . The figures show that plasmon dispersions in the system includes one in-phase optical (OP) mode and N 1 out-of-phase acoustic (AC) ones. The frequency of OP plasmon mode is much larger while those of AC ones are smaller than that in MLGG. At the band gap of 0.1 , OP and AC plasmon frequencies increase significantly with the increase in the number of layers in small wave-vector range, outside the single-particle-excitation (SPE) area. When entering SPE region, frequencies of plasmon modes depend weakly on the number of layers of the system. This character is similar to that in multi-layer gapless graphene structures at finite temperature, obtained in Ref.[35] and Ref. [36].
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Figure 2. Broadening function of plasmon dispersions in N-LGG with N 2 (a), N 3 (b), N 4 (c) and N 5 (d), plotted for d 5nm , ni 1012 cm 2 ( i 1 5 ) and 0.1 . Thin short-dashed-dotted lines demonstrate broadening function of plasmon in MLGG (color online). In order to study the Landau damping of plasmon modes, we plot in figure 2 broadening function of plasmon dispersions in 2-LGG (a), 3-LGG (b), 4-LGG (c) and 5-LGG (d) for the same parameters in Fig.1. As seen from the figures that broadening functions increase from zero as the plasmon branches enter SPE region, at about q 0.6k F k F . It is seen that the function gets larger values as respective plasmon branch goes far from SPE area boundaries, similar to those in N-layer gapless graphene system at finite temperature [35]. Therefore, the broadening function of OP plasmon branch takes the largest value, compared to those of the others at critical wave-vector. On the other hand, the broadening functions of all plasmon branches increase noticeably with the increase in the number of layers in the system.
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Figure 3. Plasmon frequencies and broadening function in 3-LGG ((a) & (b)) and 4-LGG ((c) &(d)) with 0.1 and 0 , plotted for d 5nm and ni 1012 cm 2 ( i 1 4 ). Figure 3 plots plasmon modes and respective broadening function in 3-LGG and 4-LGG system with d 5nm ,
ni 1012 cm 2 ( i 1 4 ), 0.1 and 0 for a comparison. It is seen from the figures that taking into account the band gap when calculating plasmon dispersions decreases slightly the frequency of all plasmon modes, compared to the case of gapless situation. In addition, the broadening function in gapped graphene system gets larger values than that in gapless graphene one with the same parameters. The significant changes mainly occur in large momentum range when the plasmon curves cross the SPE area boundaries.
Figure 4. Plasmon modes in N-LGG with N 2 (a), N 3 (b), N 4 (c) and N 5 (d), plotted for ni 1012 cm 2 ( i 1 5 ), d 5nm , 0.1 and 0.3 . Dashed-dotted lines show SPE area boundaries (color online). 6
Journal Pre-proof Figure 4 presents plasmon modes in N-LGG ( N 2 5 ) for ni 1012 cm 2 ( i 1 5 ), d 5nm , 0.1 and 0.3 . As can be seen from the figures that in long wavelength region ( q 0.3k F ), the band gap affects weakly
on plasmon frequencies, therefore, plasmon branches in two cases are seemly identical. However, in sufficiently large wave-vector region, plasmon frequencies decrease noticeably with the increase in the band gap. The OP plasmon branch is affected strongest by the gap, compared to the others. Moreover, in case of 0.1 , plasmon branches enter SPE area meanwhile in the other case ( 0.3 ) they go into the separation between inter- and intraband. This behaviors are sharply different from those in N-layer gapless graphene system, obtained in Ref.[35]. It can be stated that only the large band gap affects remarkably on plasmon characters in the system.
Figure 5. Plasmon frequencies in 3-LGG ((a) and (b)) and 4-LGG ((c) & (d)) as functions of band gap with several values of wave-vector, plotted for d 5nm and ni 1012 cm 2 . Thin solid lines show plasmon frequency in MLGG at the same parameters (color online). In figure 5, we plot plasmon frequencies in 3-LGG and 4-LGG as functions of band gap, increasing from zero to 0.4 with d 5nm , ni 1012 cm 2 and several values of momentum. Thin solid lines show plasmon frequency in MLGG at the same parameters. It is seen from the figures that plasmon frequencies decrease with the increase in band gap for all chosen values of wave-vector, in general. The effects of the band gap on plasmon in multi-layer systems are weakly than that does on MLGG at the same parameters. Again, the considerable changes in plasmon frequencies happen with large values of band gap and large momentum.
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Figure 6. Plasmon modes in N-LGG with N 2 (a), N 3 (b), N 4 (c) and N 5 (d), plotted for ni 1012 cm 2 ( i 1 5 ), 0.1 , d 5nm and d 10nm . Dashed-dotted lines show SPE area boundaries (color online). Now we turn to consider the effects of interlayer separation on plasmon characters of N-LGG. Figure 6 illustrates collective excitations of 2-LGG (a), 3-LGG (b), 4-LGG (c) and 5-LGG (d) for ni 1012 cm 2 ( i 1 5 ), 0.1 with two separated distances d 5nm and d 10nm (and d 20nm for 2-LGG). As seen from the figures that OP plasmon frequency decreases pronouncedly with the increase in interlayer separation while the AC ones increase with that. Consequently, the separation between OP and AC branches becomes smaller, and the branches go nearby each other. It can be predicted that plasmon curves would be identical in SPE area with sufficiently large interlayer distances, similar to those in case of gapless graphene system observed in Ref.[35] and Ref.[36].
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Figure 7. Plasmon frequencies in N-LGG with N 2 (a), N 3 (b), N 4 (c) and N 5 (d), plotted for ni 1012 cm 2 and ni 2n2 1012 cm 2 ( i 2 ), d 5nm and 0.1 . Dashed-dotted lines show SPE area boundaries (color online). In order to study the effects of the imbalance in carrier density in layers on plasmon properties of the N-LGG structure, Figure 7 plots collective excitations in N-LGG with d 5nm , 0.1 in two case: balanced and imbalanced carrier density ( n2 0.5ni , i 2 ). As seen from the figures that plasmon frequencies in case of n2 0.5ni are remarkably lower than those in the other case. The noticeable decrease in OP plasmon frequency occurs with the decrease in electron density in the second layer, even in very small wave-vector region ( q 0.1k F ). Besides, the AC plasmon frequencies just decrease slightly, and the changes occur when wave-vector is sufficiently large, about q 0.3k F . It can be said that the density of electrons in layers affects significantly on plasmon frequencies of the system.
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Figure 8. Plasmon frequencies in 4-MLGG different values of carrier density, plotted for d 5nm and n1 1012 cm 2 , 0.1 . Dashed-dotted lines show SPE area boundaries (color online). In case of 4-LGG, plasmon modes are demonstrated in Figure 8 for d 5nm , n1 1012 cm 2 and 0.1 . Fig.8(a) and (b) present that plasmon frequencies in case of n2 0.5ni ( i 2 ) and in case of n4 0.5ni ( i 4 ) take approximate values. Similarly, plasmon modes in two cases n2 n3 0.5n1 (Fig.8(c)) and n2 n4 0.5n1 (Fig.8(d)) are also difficult to distinguish from the others. This means that plasmon properties depend weakly on the order of low carrier density graphene sheets in the system. Making a comparison between Fig.8(a), (b) and Fig.8(c), (d), it can be stated that the number of low carrier density layers affects pronouncedly on plasmon pattern. The number of plasmon branches separated from the others is equal to the number of low carrier density layers in the system. 4. Conclusion In summary, collective excitations and broadening function in N-LGG, up to 5, at zero temperature are calculated theoretically for the first time. Numerical results present that the zero-point equation of dynamical dielectric function of the system admits N solutions corresponding to one optical and N 1 acoustic plasmon modes. The OP (AC) plasmon branch (branches) is higher (are lower) than that in MLGG at the same parameters. The broadening function of OP mode gets the largest values, compared to other branches. In addition, plasmon frequencies in the system decrease with the increase in the band gap, with lower level in comparison with MLGG. On the other hand, both interlayer distance and carrier density in layers affect significantly on plasmon characters. Finally, the number of plasmon branches separated from the others depends on the number (not the order) of low carrier density layers in the system. References [1] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, A. K. Geim, Rev. Mod. Phys. 81, 109 (2009). [2] A. K. Geim, K. S. Novoselov, Nature Mater 6, 183 (2007). [3] A. K. Geim, A. H. MacDonald, Phys. Today 60, 35 (2007). [4] J. Wei, Z. Zang, Y. Zhang, M. Wang, J. Du, and X. Tang, Optics Letters 42, 911 (2017). [5] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva and A.A. Firsov, Science 306, 666 (2004). [6] Y. Zhang, Y.-W. Tan, H.L. Stormer and P. Kim, Nature 438, 201 (2005). [7] G. Li, A. Luican and E.Y. Andrei, Phys. Rev. Lett 102, 176804 (2009). [8] Y. Yao, F. Ye, X.-L. Qi, S.-C. Zhang and Z. Fang, Phys. Rev. B 75, 041401 (2007). [9] E.V. Gorbar, V.P. Gusynin, V.A. Miransky and I.A. Shovkovy, Phys. Rev. B 78, 085437 (2008). [10] P. K. Pyatkovskiy, J. Phys. Condens. Matter 21, 025506 (2009). [11] F. J. Garcia de Abajo, ACS Photonics 1, 135 (2014). [12] H. L. Koppens Frank, D. E. Chang and F. J. Garcia de Abajo, Nano Lett. 11, 3370 (2011). 10
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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: