DSpace at VNU: Plasmon modes of double-layer graphene at finite temperature

Physica E 54 (2013) 267–272 Contents lists available at ScienceDirect Physica E journal homepage: www.elsevier.com/locate/physe Plasmon modes of double-layer graphene at finite temperature Dinh Van Tuan a,b,c, Nguyen Quoc Khanh a,n a b c Department of Theoretical Physics, National University in Ho Chi Minh City, 227-Nguyen Van Cu Str., 5th District, Ho Chi Minh City, Vietnam ICN2—Institut Catala de Nanociencia i Nanotecnologia, Campus UAB, 08193 Bellaterra (Barcelona), Spain Department of Physics, Universitat Autonoma de Barcelona, Campus UAB, 08193 Bellaterra, Spain H I G H L I G H T S     We investigate the temperature effects on the plasmon dispersion mode and loss function of doped double-layer graphene The temperature acoustic mode ωÀ in the case of n2 ¼ is degenerate with the ω ¼ υF q line only when d-0 Even though n2 ¼0, when d-0 the temperature optical mode ỵ does not become the single-layer plasmon with the same density The effect of temperature on the plasmon dispersion and damping is significant art ic l e i nf o a b s t r a c t Article history: Received May 2013 Received in revised form 15 July 2013 Accepted 16 July 2013 Available online 26 July 2013 We calculate the dynamical dielectric function of doped double-layer graphene (DLG), made of two parallel graphene monolayers with carrier densities n1 and n2, and an interlayer separation of d at finite temperature The results are used to find the dispersion of plasmon modes and loss functions of DLG for several interlayer separations and layer densities We show that in the case of n2 ¼ 0, the finitetemperature plasmon modes are dramatically different from the zero-temperature ones & 2013 Elsevier B.V All rights reserved Keywords: Graphene Plasmon Collective excitations Introduction Graphene is a two-dimensional electron system that has attracted a great deal of attention because of its unique electronic properties [1] and its potential as a new material for electronic technology [2,3] The main difference between 2D graphene and a conventional 2D semiconductor system is the electronic energy dispersion In 2D semiconductor systems, the electron energy depends quadratically on the momentum, but in graphene, the dispersion relation is linear near the corners of the Brillouin zone [4] Because of this difference in the electronic band structure, there are many properties of graphene that are significantly different from ordinary 2D systems [5] In this paper, we consider a double-layer graphene (DLG) system formed by two parallel single-layer graphene (SLG) sheets separated by a distance d DLG is fundamentally different from the well-studied bilayer graphene system [6] because there is no interlayer tunneling, only an inter-layer Coulomb interaction Spatially n Corresponding author Fax: +848 38350096 E-mail address: nqkhanh@phys.hcmuns.edu.vn (N.Q Khanh) 1386-9477/$ - see front matter & 2013 Elsevier B.V All rights reserved http://dx.doi.org/10.1016/j.physe.2013.07.010 separated two-component DLG can be fabricated by folding SLG over a high-insulating substrate [7] Recently, Hwang and Das Sarma [8] have investigated the plasmon dispersion and loss function in doped DLG at zero temperature, and found that the plasma modes of an interacting DLG system are completely different from the double-layer semiconductor quantum well plasmons In the long wavelength limit the density pffiffiffiffiffi p dependence of the plasma frequency is given by ỵ n2 ị n1 ỵ p p pffiffiffiffiffi Þ ∝ n n for optical plasmons and ðωÀ = n ỵ n for acoustic 2 plasmons, compared to ỵ ị ∝N and ðω0 Þ ∝n1 n2 =N in ordinary 2D systems, where N¼n1+n2 In this paper, we investigate the effect of temperature on the plasmon modes and loss function of DLG for several interlayer separations and layer densities Actually, the plasmon mode of DLG at finite temperature has been considered in several articles [9,10] The effect of spin–orbit coupling on plasmons in graphene has also been investigated at zero [11] as well as at finite temperature [12] In addition, the collective excitations in bilayer graphene (BLG), the closest material to DLG, have been calculated at both zero [13] and finite temperature [14] In this paper, we consider a larger range of temperature and imbalanced densities and obtain some interesting results 268 D Van Tuan, N.Q Khanh / Physica E 54 (2013) 267–272 Fig Plasmon dispersions of DLG for several layer separations at T ¼ (bold dotted lines), T ¼0.5TF (bold dashed lines) and T ¼ TF (bold solid lines) The thin lines indicate the plasmon dispersion of SLG with the same density and temperature Here we use the parameters n1 ¼ n2 ¼ n¼ 1012 cm À and (a) d ¼20 Å (kFd ¼ 0.35), (b) d ¼100 Å (kFd ¼ 1.8), (c) d ¼300 Å (kFd¼ 5.3), and (d) d ¼500 Å (kFd¼ 8.9) Theory In graphene, the low-energy Hamiltonian is well-approximated by a two-dimensional Dirac equation for massless particles, the so-called Dirac–Weyl equation [15], H ẳ F sx kx ỵ sy ky ị; 1ị where υF is the Fermi velocity of graphene, sx and sy are Pauli spinors and k is the momentum relative to the Dirac points, and ℏ ¼ throughout this paper The energy of graphene for 2D wave vector k is given by εk;s ¼ sυF jkj; Fig The plasmon modes of DLG for layer separation d ¼ 100 at several momenta 2ị where s ẳ indicates the conduction (+1) and valence ( À 1) bands, respectively The density of states is given by Dị ẳ gjj=22F ị, where g ẳ g s g v ẳ accounts for the spin (g s ¼ 2) and valley (g v ¼ 2) degeneracies The Fermi momentum p ðkFffiffiffiffiffiffiffiffiffiffiffiffiffi Þ andffi the Fermi energy ðEF Þ of 2D graphene are given by kF ¼ 4πn=g and EF ¼ υF kF , where n is the 2D carrier density D Van Tuan, N.Q Khanh / Physica E 54 (2013) 267–272 In the random-phase approximation (RPA), the dynamical dielectric function of SLG becomes q; ; Tị ẳ 1c qịq; ; Tị; 3ị where c qị ẳ 2e2 =q is the 2D Fourier transform of the Coulomb potential and Πðq; ω; TÞ, the 2D polarizability at finite temperature, is given by the bare bubble diagram [16] Z q; ; Tị ẳ g limỵ -0 s;s0 ẳ d k ỵ ss cos k;kỵq ị nF k;s ịnF kỵq;s0 ị ỵ k;s kỵq;s0 ỵ i 2ị2 4ị ẩ ẫ1 is the FermiDirac distribution here, nF ị ẳ expẵ0 ị ỵ function The non-interacting chemical potential, ¼ μ0 ðTÞ, is determined by the conservation of the total electron density as   TF ¼ F ðβμ0 ÞÀF ðÀβμ0 Þ; T ð5Þ where ẳ 1=kB T and F n xị is given by Z F n xị ẳ t n dt ỵ exptxị 6ị 269 The forms of the chemical potential in the low and high temperature limits are given by [17] "   # π2 T T μ0 ðTÞ≈EF 1À ⪡1 ð7Þ ; TF TF μ0 ðTÞ≈ EF T F ; 4ln T T ⪢1 TF ð8Þ Recently, Ramezanali et al [18] have obtained the following semianalytical expressions for the imaginary and the real parts of the dynamical polarizability n g ðαÞ ∑ F qịq2 f F q; ịẵGị mq; ; Tị ẳ ỵ q; ; TịG q; ; Tị ẳ h io ỵF qịq2 f ; F qị ; ỵ H ị ; 9ị ỵ q; ; Tị ( g 2kB Tlnẵ1 ỵ e0 =kB T ị eq; ; Tị ẳ ỵ F qịq2 f ; F qị ẳ 2F h i h io ị Gị q; ; TịGỵ q; ; Tị ỵ F qịq2 f F q; ị ; ỵ H ị q; ; Tị ð10Þ Fig (a)–(c) Plasmon dispersions of DLG for several temperatures and layer densities Here we use n1 ¼1012 cm À and d¼ 100 Å, and (d) plasmon mode of DLG for n2 ¼ and T ¼ at several layer separations 270 D Van Tuan, N.Q Khanh / Physica E 54 (2013) 267–272 where f ðx; yị ẳ p ; x2 y2 Gị q; ; Tị ẳ H ị q; ; Tị ẳ Z ð11Þ Z du À1 pffiffiffiffiffiffiffiffiffiffiffiffi u2 À1  ð12Þ pffiffiffiffiffiffiffiffiffiffiffiffi 1Àu2   : jυF qu j20 exp ỵ1 2kB T 13ị exp  ; du jF qu j20 2kB T ỵ1 Numerical results The DLG dielectric function is obtained from the determinant of the generalized dielectric tensor and has the form εdo q; ; Tị ẳ q; ; Tị2 q; ; TÞÀυ12 ðqÞυ21 ðqÞΠ ðq; ω; TÞΠ ðq; ω; TÞ expðÀqdÞ=ðκqÞ are the interlayer Coulomb interaction matrix elements, with κ the background lattice dielectric constant In our calculations, we set κ ¼ The spectrum of the collective excitations can be obtained from the zeros of the real part of the double-layer dielectric function, and the imaginary part describes the damping of collective modes ð14Þ here, ε1 ðq; ω; TÞ and ε2 ðq; ω; TÞ are the dynamical dielectric functions of individual layers given by Eq (3), and υ12 qị ẳ 21 qị ẳ 2e2 In this section, we calculate the plasmon dispersion and loss function of DLG at zero and finite temperatures for several layer separations and densities 3.1 The plasmon dispersion Fig from (a) to (d) show the plasmon dispersions of balanced DLG (bold lines) for several temperatures with increasing layer Fig Density plot of the DLG loss function in ðq; ωÞ space for fixed densities of n1 ¼ n2 ¼ 1012 cm À 2, layer separations d ¼ 20 Å, 500 Å and temperatures T ¼ 0.5TF, TF (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.) D Van Tuan, N.Q Khanh / Physica E 54 (2013) 267–272 separation d We also show the plasmons of SLG (thin lines) at the same temperatures for comparison In the long wavelength limit, pffiffiffi the plasmon dispersion of SLG at T ¼0 has q dependence Moreover, the optical branch ỵ of DLG shows the behavior of the SLG plasmon dispersion with an electron density 4n at small d This is the main difference between DLG and BLG, which shares the same plasmon dispersion as conventional two-dimensional electron gas (2DEG) systems [14] These results are in complete agreement with Refs [5,8,10] One can observe that the plasmon frequencies for both SLG and DLG decrease with increasing temperature (see T ¼ and T ¼ 0:5T F ), which is consistent with Ref [10] But interestingly, at high temperatures (see T ¼ T F ), the plasmon frequencies increase again and are larger than the zerotemperature frequencies when T T C , where T C ≈0:6T F (more detail in Fig 2) For low temperatures, the acoustic mode ωÀ approaches the boundary of the intraband single-particle excitation (SPEintra) (dot-dashed line) in the high-energy region As d increases (especially for the case T ¼ 0), the acoustic mode and the optical 271 mode ỵ in the low-frequency region move toward each other and approach the SLG plasmon at the same temperature due to the decreasing Coulomb interaction, whereas they split in the highfrequency region This behavior is completely different from 2DEG systems [19] For large momentum ðq 4kF Þ, we observe that the mode dispersion is almost unchanged when d4100 Å In order to understand more about the effect of temperature, we show the plasmon frequency versus temperature for layer separation d¼100 Å at several momenta in Fig When ToT0 (T4T0), where T ≈0:4T F , the acoustic mode ωÀ decreases (increases) with increasing temperature In the case of low momentum, the optical mode ỵ decreases and then increases whereas it only increases in the case of large momentum The decrease of both modes at low momentum with increasing temperature has already been observed [10], but its increase at high temperature has not been mentioned before Fig also shows that the high temperature plasmon modes are dramatically different from the zero-temperature ones, especially in the case of large momentum Fig (a) Calculated plasmon mode dispersions of DLG for several layer separations for T ¼0.5TF, n2/n1 ¼0 (i.e., n2 ¼0 and n1 ¼ 1012 cm À 2), and corresponding loss functions for (b) d-0 Å, (c) d ¼10 Å, and (d) d¼ 100 Å 272 D Van Tuan, N.Q Khanh / Physica E 54 (2013) 267–272 Fig 3(d), the acoustic mode ωÀ is degenerate with the ω ¼ υF q line only when d-0 Å More interestingly, even though there are no free carriers in the second layer (n2 ¼0) and d-0, the optical mode ỵ (the dotted line) does not become the SLG plasmon (the dotdashed line) with the same density, showing the strong effect of temperature in this case In Fig 5(b)–(d) we show the loss function of DLG corresponding to Fig 5(a) As the layer separation decreases, the acoustic mode ωÀ approaches the ω ¼ υF q line and loses spectral strength In Fig 6, we consider the effect of temperature on the plasmon mode dispersion at several momenta for the extreme case n2 ¼0 and d-0 Å The optical modes ỵ approach the SLG plasmon modes at low temperature and shift to higher energy at high temperature Furthermore, we also considered the cases with no free carriers in the second layer at finite layer separations [22] and showed that the temperature strongly affects the plasmon modes of DLG Conclusions Fig The plasmon modes of DLG at several momenta for n2 ¼ and d-0 Å In Fig 3(a)–(c) we show the calculated DLG plasmon dispersions for different layer densities We observe that changing the density in the second layer does not affect the plasmon mode at high temperature (T≈TF); this means that the number of electron-hole pairs in this case is mainly controlled by the temperature As n2/n1 decreases, the zero-temperature acoustic mode ωÀ approaches the boundary of SPEintra(i.e.ω ¼ υF q) However, it shifts to higher energy when the temperature increases The differences between the acoustic mode ωÀ at zero and at finite temperature in the case of n2 ¼0 in Fig 3(c) are due to the fact that in the zero-temperature limit the undoped layer admits only interband transitions, whereas at finite temperature it admits both interband and intraband transitions In Fig 3(d) we show the DLG plasmon modes at zero temperature for several layer separations in the case of n2/n1 ¼0, i.e., the second layer is undoped and the first layer has a finite density n1 ¼1012 cm À It can be seen from the figure that the acoustic mode ωÀ is degenerate with the boundary of SPEintra, while the optical mode ỵ is degenerate with the SLG mode below the interband single-particle excitation (SPEinter1,2) As d-0 the dispersion of ỵ becomes exactly that of SLG plasmon with the same density 3.2 The loss function Fig shows density plots of the balanced DLG loss function (i.e.,Àℑm⌊εdo ðq; ω; TÞÀ1 ⌋) in qÀω space for two separations and temperatures, where the color scale represents the mode spectral strength The loss function is related to the dynamical structure factor Sðq; ωÞ, which gives a direct measure of the spectral strength of the various elementary excitations Thus, our results can be measured in experiments such as inelastic electron and Ramanscattering spectroscopies [20,21] The acoustic mode ωÀ corresponds to a broadened peak near theω ¼ υF q line and the optical mode ỵ corresponds to a broadened peak with higher energy Unlike the zero-temperature case in which the undamped plasmons show up as a well-defined δ-function peaks in the lost function below SPEinter [8], the finite-temperature plasmon modes are overdamped even in this region As T or d increases, the spectral strengths of both modes ω increase We also calculate the loss function in the case of imbalanced densities We find that the spectral strengths of both modes ω slightly increase when the density imbalance decreases (i.e., n2/n1 increases), and that the high temperature spectral strength (T≥T F ) is almost independent of n2/n1 In Fig 5(a) we show the plasmon modes at T ¼0.5TF for n2 ¼ and n1 ¼1012 cm À Unlike the zero-temperature case shown in In this paper, we have investigated the effect of temperature on the plasmon dispersion mode and loss function of doped DLG, made of two parallel graphene monolayers with carrier densities n1 and n2, and an interlayer separation of d We have shown that unlike the zero-temperature plasmon modes, the temperature acoustic mode ωÀ in the case of n2 ¼ is degenerate with the ω ¼ υF q line only when d-0 Å More interestingly, even though there are no free carriers in the second layer, when d-0 the temperature optical mode ỵ does not become the SLG plasmon with the same density Our results indicate that the effect of temperature on the plasmon dispersion and damping is significant and cannot be ignored when investigating many properties of graphene Acknowledgment This work is supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number 103.02-2011.25 References [1] A.H Castro Neto, F Guinea, N.M.R Peres, K.S Novoselov, A.K Geim, Reviews of Modern Physics 81 (2009) 109 [2] A.K Geim, K.S Novoselov, Nature Materials (2007) 183 [3] A.K Geim, A.H MacDonald, Physics Today 60 (2007) 35 [4] P.R Wallace, Physical Review 71 (1947) 622 [5] E.H Hwang, S Das Sarma, Physical Review B 75 (2007) 205418 [6] E McCann, V.I Fal'ko, Physical Review Letters 96 (2006) 086805; M Koshino, T Ando, Physical Review B 73 (2006) 245403; E.H Hwang, S Das Sarma, Physical Review Letters 101 (2008) 156802 [7] H Schmidt, T Ludtke, P Barthold, E McCann, V.I Fal'ko, R.J Haug, Applied Physics Letters 93 (2008) 172108 [8] E.H Hwang, S Das Sarma, Physical Review B 80 (2009) 205405 [9] T Stauber, G Gomez-Santos, Physical Review B 85 (2012) 075410 [10] T Vazifehshenas, T Amlaki, M Farmanbar, F Parhizgar, Physics Letters A 374 (2010) 4899 [11] A Scholz, T Stauber, J Schliemann, Physical Review B 86 (2012) 195424 [12] X Wang, T Chakraborty, Physical Review B 75 (2007) 033408 [13] G Borghi, M Polini, R Asgari, A.H MacDonald, Physical Review B 80 (2009) 241402(R) [14] X Wang, T Chakraborty, Physical Review B 75 (2007) 041404(R) [15] J.C Slonczewski, P.R Weiss, Physical Review 109 (1958) 272; T Ando, T Nakanishi, R Saito, Journal of Physical Society of Japan 67 (1998) 2857 [16] B Wunsch, T Stauber, F Sols, F Guinea, New Journal of Physics (2006) 318 [17] E.H Hwang, S Das Sarma, Physical Review B 79 (2009) 165404 [18] M.R Ramezanali, M.M Vazifeh, R Asgari, M Polini, A.H MacDonald, Journal of Physics A: Mathematical and Theoretical 42 (2009) 214015 [19] K Flensberg, Yu-Kuang Hu Ben, Physical Review B 52 (14) (1995) 796 [20] Y Liu, R.F Willis, K.V Emtsev, Th Seyller, Physical Review B 78 (2008) 201403 [21] T Eberlein, U Bangert, R.R Nair, R Jones, M Gass, A.L Bleloch, K.S Novoselov, A Geim, P.R Briddon, Physical Review B 77 (2008) 233406 [22] Dinh Van Tuan, Nguyen Quoc Khanh, Communications in Physics 22 (2012) 45  ... shows that the high temperature plasmon modes are dramatically different from the zero -temperature ones, especially in the case of large momentum Fig (a) Calculated plasmon mode dispersions of DLG... second layer at finite layer separations [22] and showed that the temperature strongly affects the plasmon modes of DLG Conclusions Fig The plasmon modes of DLG at several momenta for n2 ¼ and... n2, and an interlayer separation of d We have shown that unlike the zero -temperature plasmon modes, the temperature acoustic mode ωÀ in the case of n2 ¼ is degenerate with the ω ¼ υF q line only