- Email: [email protected]
Temperature effect on plasmon dispersions in double-layer graphene systems
Physics Letters A 374 (2010) 4899–4903
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Temperature effect on plasmon dispersions in double-layer graphene systems T. Vazifehshenas ∗ , T. Amlaki, M. Farmanbar, F. Parhizgar Department of Physics, Shahid Beheshti University, G.C., Evin 1983963113, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 16 August 2010 Received in revised form 10 October 2010 Accepted 12 October 2010 Available online 14 October 2010 Communicated by R. Wu Keywords: Graphene Plasmon Temperature dependent Double-layer
a b s t r a c t We investigate the plasmon dispersion relation and damping rate of a double-layer graphene system consisting of two separated monolayer graphenes with no interlayer tunneling at finite temperature. We use the temperature dependent RPA dielectric function which is valid for graphene systems to obtain the plasmon frequencies and damping rates at different temperatures, interlayer correlation parameters and electron densities and then compare them with those obtained from the zero temperature calculations. Our results show that by increasing the temperature, the plasmon frequencies decrease and the decay rate increases. Furthermore, we find that the behavior of a double-layer graphene system at small and large correlation parameters is different from the conventional double-layer two-dimensional electron gas system. Finally, we obtain that in a density imbalanced double-layer graphene system, the acoustic plasmons are more affected by temperature than the equal electron densities one. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Graphene is a sheet of carbon atoms with a thickness of only one atom as a strictly two-dimensional (2D) system. Carbon atoms in graphene arranged on a honeycomb structure which is composed of two sublattices [1–5]. The monolayer graphene is a gapless semiconductor and has unusual massless and chiral carriers near the Dirac points, K and K (two equivalent corners of the Brillouin zone) where the band structure of graphene is linear. This is different from the parabolic relation of conventional two-dimensional electron gas (2DEG) systems [1]. Graphene can be doped chemically or electrically to be either n-type or p-type material [3]. The novel properties of graphene not only attract the attention of many theoretical and experimental researchers, but also make it a good candidate for technological applications [6,7]. Many-body properties of graphene have been studied extensively [8,9]. The dynamical dielectric function and plasmon dispersion relation are two important many-body quantities in such structures [10–12]. Plasmons are the quanta of collective excitations of the electronic systems which can affect the response of systems to the applied fields. The plasmon dispersion relation in double-layer system differs significantly from the single layer one. Two branches, acoustic and optical, are appeared due to the in phase and out of phase electron oscillations in two layers, respectively [13]. The double-layer graphene (DLG) is a system consisting of two separate parallel single layer graphenes (SLG) which are placed
close together to make an effective interaction between them but far enough to prevent the electron tunneling, similar to the conventional double-layer 2DEG systems. The DLG is different from bilayer graphene (BLG) system in which two coupled single graphene layers stacked as in graphite and interlayer tunneling should be considered. As in the conventional double-layer 2DEG systems, many-body phenomena like the Coulomb drag due to the interlayer electron–electron interactions can be occurred [14,15]. The Coulomb drag rate has been calculated for the DLG system and its behavior compared to the conventional double-layer 2DEG system [16]. In this Letter, we investigate the plasmon dispersion of a DLG system in which the two graphene layers separated by a nanometer distance d. The plasmon modes of a DLG at zero temperature have been obtained by Hwang and Das Sarma [17]. In this work, all calculations are done at finite temperature and the results compared with the zero temperature ones. Considering the correlation parameter, k F d, where k F is the Fermi wave vector, we discuss the interlayer correlation effects on the plasmon dispersion relation of a DLG at finite temperature. We put an upper limit on the temperature T = 0.2T F (T F is the Fermi temperature) to ensure that the effect of phonons is negligible [18]. We also study the damping rate of plasmons at finite temperature. In addition, the plasmon frequencies and decay rate of a DLG system with different electron densities of layers are calculated at both zero and finite temperature and compared with those obtained from the equal electron density system. 2. Theory
*
Corresponding author. Tel.: +98 21 29902784; fax: +98 21 22431666. E-mail address: [email protected] (T. Vazifehshenas).
0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.10.026
We consider two parallel doped graphene layers with the electron densities, n1 and n2 . Throughout of this Letter, we set h¯ = 1.
4900
T. Vazifehshenas et al. / Physics Letters A 374 (2010) 4899–4903
Fig. 1. (a) Plasmon dispersion and (b) damping rate of a SLG system as functions of dimensionless wave vector at different temperatures. The straight lines show SPE boundaries.
The plasmon dispersion relation of an electronic system can be obtained from the poles of density–density response function, Π(q, ω), or equivalently, from the zeroes of dynamical dielectric function, (q, ω) [17]:
(q, ω P − i γ ) = 0
(1)
where ω P is the plasmon frequency at a given wave vector q and γ is the damping rate of plasma oscillations. In complex plane of ω, if the poles of Π(q, ω) place on the real axis, the plasmons are long-lived and well-defined; but for the poles away from it, we have the landau damped plasmons due to the electron scattering. In case of weak damping (γ ω P ), the plasmon dispersion and decay rate are determined from the following equations:
Re (q, ω P ) = 0
(2)
and
−1 ∂ Re (ω, k) γ = Im (ω P , k) . ∂ω ω =ω P
(3)
To obtain the dielectric function of the system, we use the random phase approximation (RPA) approach which is valid for small dimensionless interaction parameter, r s 1 (r s is defined as the ratio of the interparticle potential energy to the single-particle kinetic energy). For our DLG system, the graphene layers are on the SiO2 substrate so that we have r s 0.87; hence the RPA is a good approximation. At finite temperature, the dielectric function of a DLG system is given by [18]:
(q, ω, T ) = 1 − v (q)Π1 (q, ω, T ) 1 − v (q)Π2 (q, ω, T ) − v 2 (q)Π1 (q, ω, T )Π2 (q, ω, T )e −2qd
(4)
where v (q) = 2π e /κ q is the Fourier transform of the Coulomb interaction and κ is substrate dielectric constant (for SiO2 : κ ≈ 2.5), d is the interlayer spacing and Πi (q, ω, T ) is the non-interacting density–density response function of the i-th layer [19]: 2
Πi (q, ω, T ) = g lim
η→0
×
and valley degeneracies), θ is the angle between k and k + q, and n F i = (exp[β( E k,λ − μi )] + 1)−1 is the Fermi–Dirac distribution function, with β = 1/k B T (k B is the Boltzmann constant) and μi being the chemical potential of i-th layer which is obtained from the normalization condition:
d2 q 1 + λλ cos(θ) λ,λ =±
2π
n F i ( E k,λ ) − n F i ( E k+q,λ )
ω + E k,λ − E k+q,λ + i η
2
.
(5)
Here E k,λ = λk (with λ = 1 and λ = −1 refer to the valence and conduction bands), g = g s g v (with g s = 2 and g v = 2 being spin
∞ ni =
dE D i ( E )n F i ( E ),
(6)
−∞
where D i ( E ) = 2E /π v 2F is the non-interacting graphene density of states (with v F = 106 m/s is the Fermi velocity). In the case of equal electron density layers, Eq. (4) can be rewritten as:
1 − v (q) Re Π(q, ω, T )
= − v (q) Re Π(q, ω, T )e −2qd ± e −2qd 1 + 2v (q) Im Π(q, ω, T ) sinh qd.
(7)
Using above equations, we can determine the optical (+) and acoustical (−) plasmon modes in a DLG system. For a SLG system, the right-hand side (RHS) Eq. (7) simply equals to zero. 3. Results In this Letter, we investigate the effect of temperature on the plasmon dispersion relation of a DLG system and compare it with the SLG and conventional double-layer 2DEG system. In Fig. 1, the plasmon frequencies and damping rate are plotted at two different temperatures: T = 0 and T = 0.2T F . According to these figures, in the region of small q, the plasmon frequencies decrease by increasing the temperature. It can be explained by this fact that at finite temperature, the electrons are excited easier than the zero temperature case; so the collective motions appear at low frequencies (or energies). As it is expected, the damping rate increases by increasing the temperature because the electrons with larger kinetic energies can be easier excited to the electron–hole pairs. In the region of electron–hole excitation, the plasmons are not well-defined anymore and the single particle excitations (SPE) are dominated. In this region where Im Π(q, ω, T ) = 0, two kinds of excitation can be occurred: intraband transition (λ = λ ) and interband transition (λ = −λ ). The boundaries of these two SPE regions at zero temperature are shown in all plasmon dispersion graphs. It is easily seen that at finite temperatures, the mode damping starts at smaller wave vectors, before entering the SPE region. The behavior of the acoustic and optical plasmon branches in a DLG system is shown in Fig. 2 for the layers with equal electron densities (n1 = n2 ) at
T. Vazifehshenas et al. / Physics Letters A 374 (2010) 4899–4903
4901
Fig. 2. (a) Plasmon dispersions and (b) damping rate of a DLG system as functions of dimensionless wave vector at different temperatures with a correlation parameter k F d = 1.68 (d = 30 nm) and n1 = n2 = 1011 cm−2 . The straight lines show SPE boundaries.
Fig. 3. Plasmon dispersion and damping rate of a DLG system as functions of dimensionless wave vector with n1 = n2 = 1012 cm−2 for different correlation parameters k F d = 0.35 (d = 2 nm), k F d = 1.8 (d = 10 nm) and k F d = 8.9 (d = 50 nm). (a) and (c) at T = 0 and (b) and (d) at T = 0.2T F . The straight lines show SPE boundaries.
zero and finite temperatures. Again, the plasmon frequency decreases and decay rate increases by increasing the temperature. From these figures, it is found that at finite temperature in contrast to the zero temperature case, the electron–hole pairs can be created at all wave vectors, this results in the plasmon damping. In Fig. 3, we compare the DLG plasmon dispersion and damping rate for three different correlation parameter, k F d = 8.9, k F d = 1.8 and k F d = 0.35 at T = 0, Figs. 3(a) and (c), and T = 0.2T F , Figs. 3(b) and (d). These figures show that out of the SPE region, the difference between the acoustic and optical branches is small for large correlation parameter (or large d) at both zero and finite temper-
atures. This behaviour can be described by the RHS of Eq. (7) which goes to zero when the separation between two layers is large enough:
1 − v (q) Re Π(q, ω, T ) 0.
(8)
This is just the condition for obtaining the SLG single plasmon branch. In the SPE region, Eq. (7) can be approximated for large d as:
1 − v (q) Re Π(q, ω, T ) ± v (q) Im Π(q, ω, T ).
(9)
4902
T. Vazifehshenas et al. / Physics Letters A 374 (2010) 4899–4903
Fig. 4. Plasmon dispersion and damping rate of a DLG system as functions of dimensionless wave vector with a correlation parameter k F d = 5.3 (d = 30 nm). (a) and (c) for n1 = n2 and (b) and (d) for n2 /n1 = 0.5. The straight line show SPE boundaries.
Here the two branches (acoustic and optical) of a DLG are decoupled even at T = 0 in contrast to the conventional double-layer 2DEG system in which these two branches merge at large d and never split [20]. Also, in the limit of large d, the behavior of DLG plasmon dispersion relation is independent of correlation parameters. In the other hand, Fig. 3 indicates that by decreasing the distance between two graphene layers in a DLG system, the acoustic branch reaches the ω = v F q line (the boundary of SPEintra ) and the damping rate decreases. In addition, at small d, the optical branch of a DLG shows the behavior of the SLG plasmon dispersion with an electron density 4n [17] which is different from the result for a double-layer 2DEG system that behaves like a single layer 2DEG with the electron density 2n. Moreover, for a DLG system, the effect of temperature is more significant at larger k F d and the damping of optical branch is greater than the acoustic one because of the higher energy of plasmon. √ If we set n2 /n1 = α , then we obtain T 2 / T 1 = α (where T i = k B T / E F i ) for a DLG in contrast to 2DEG in which we have T 2 / T 1 = α . Therefore, in graphene systems, the density imbalance has weaker temperature effect compare to the 2DEG. Up to here, we assumed the DLG system with equal electron densities. The effect of different electron densities of two graphene layers on plasmon energies is depicted in Fig. 4. In this case, there are two different SPEinter regions; SPE1,2 and SPE2 . The SPE1,2 refers to the region where the electron–hole pairs can be excited in both graphene layers but in SPE2 region the electron–hole excitation occur only in lower electron density layer. As this figure shows, the difference between acoustic and optical plasmon frequencies for n1 = n2 is larger and the plasmon damping is stronger and starts at smaller wave vectors compared to the n1 = n2 case. Also, the acoustic branch frequencies for a DLG system with different elec-
tron densities of layer at T = 0 are smaller than the n1 = n2 case and more affected by increasing the temperature. In general for a DLG system, the plasmon energies and damping rate are functions of the electron wave vector, frequency, temperature and density. For a given wave vector and fixed density, by increasing the temperature the number of electron–hole pairs increase and plasmon frequencies decrease. In case of different electron densities of layers, n1 > n2 , the effect of temperature for the layer with lower density is stronger and the electron–hole pairs in layer 2 can be excited at smaller q relative to the layer 1, thus the damping rate of density imbalanced DLG grows. This explains why the SPE2 region is bigger than SPE1 region. 4. Conclusion In summary, we investigate the effect of temperature on the plasmon dispersion relation and damping rate of the DLG systems at different correlation parameters and layer electron densities. We find that by increasing the temperature the plasmon frequencies decrease and the damping increases in all cases. Also, we obtain that the behavior of a DLG is different from the conventional double-layer 2DEG at small and large correlation parameters. Finally, our calculations indicate that for n1 = n2 , the acoustic branch of the DLG is more affected by temperature. References [1] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A.A. Firsov, Science 306 (2004) 666. [2] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, Rev. Mod. Phys. 81 (2009) 109. [3] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos, A.A. Firsov, Nature 438 (2005) 197.
T. Vazifehshenas et al. / Physics Letters A 374 (2010) 4899–4903
[4] [5] [6] [7] [8] [9] [10] [11] [12]
A.K. Geim, K.S. Novoselov, Nature Mater. 6 (2007) 183. S. Das Sarma, S. Adam, E.H. Hwang, E. Rossi, arXiv:1003.4731v1, 2010. M.C. Lemme, Solid State Phenomena 156–158 (2010) 499. M.C. Lemme, T. Echtermeyer, M. Baus, H. Kurz, IEEE Electron Device Lett. 28 (4) (2007) 282. S. Das Sarma, E.H. Hwang, Wang-Kong Tse, Phys. Rev. B 75 (2007) 121406. B. Wunsch, T. Stauber, F. Sols, F. Guinea, New Journal of Physics 8 (2006) 318. G. Borghi, M. Polini, R. Asgari, A.H. MacDonald, Phys. Rev. B 80 (2009) 241402(R). E.H. Hwang, S. Das Sarma, Phys. Rev. B 75 (2007) 205418. Zhao-Zhe Li, Commun. Theor. Phys. 52 (2009) 361.
4903
[13] H. Bruus, K. Flensberg, Introduction to Many-Body Theory in Condensed Matter Physics, Oxford University Press, 2002. [14] K. Flensberg, Y.-K.H. Ben, Phys. Rev. Lett. 73 (1994) 3572. [15] A. Kamenev, Y. Oreg, Phys. Rev. B 52 (1995) 7516. [16] Wang Kong Tse, Ben Yu Kuang Hu, S. Das Sarma, Phys. Rev. B 76 (2007) 081401(R). [17] E.H. Hwang, S. Das Sarma, Phys. Rev. B 80 (2009) 205405. [18] S. Das Sarma, A. Madhukar, Phys. Rev. B 23 (1981) 805. [19] M.R. Ramzanali, M.M. Vazifeh, R. Asgari, M. Polini, A.H. Macdonald, J. Phys. A: Math. Theor. 42 (2009) 214015. [20] K. Flensberg, Yu-Kuang Hu Ben, Phys. Rev. B 52 (14) (1995) 796.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Temperature effect on plasmon dispersions in double-layer graphene systems T. Vazifehshenas ∗ , T. Amlaki, M. Farmanbar, F. Parhizgar Department of Physics, Shahid Beheshti University, G.C., Evin 1983963113, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 16 August 2010 Received in revised form 10 October 2010 Accepted 12 October 2010 Available online 14 October 2010 Communicated by R. Wu Keywords: Graphene Plasmon Temperature dependent Double-layer
a b s t r a c t We investigate the plasmon dispersion relation and damping rate of a double-layer graphene system consisting of two separated monolayer graphenes with no interlayer tunneling at finite temperature. We use the temperature dependent RPA dielectric function which is valid for graphene systems to obtain the plasmon frequencies and damping rates at different temperatures, interlayer correlation parameters and electron densities and then compare them with those obtained from the zero temperature calculations. Our results show that by increasing the temperature, the plasmon frequencies decrease and the decay rate increases. Furthermore, we find that the behavior of a double-layer graphene system at small and large correlation parameters is different from the conventional double-layer two-dimensional electron gas system. Finally, we obtain that in a density imbalanced double-layer graphene system, the acoustic plasmons are more affected by temperature than the equal electron densities one. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Graphene is a sheet of carbon atoms with a thickness of only one atom as a strictly two-dimensional (2D) system. Carbon atoms in graphene arranged on a honeycomb structure which is composed of two sublattices [1–5]. The monolayer graphene is a gapless semiconductor and has unusual massless and chiral carriers near the Dirac points, K and K (two equivalent corners of the Brillouin zone) where the band structure of graphene is linear. This is different from the parabolic relation of conventional two-dimensional electron gas (2DEG) systems [1]. Graphene can be doped chemically or electrically to be either n-type or p-type material [3]. The novel properties of graphene not only attract the attention of many theoretical and experimental researchers, but also make it a good candidate for technological applications [6,7]. Many-body properties of graphene have been studied extensively [8,9]. The dynamical dielectric function and plasmon dispersion relation are two important many-body quantities in such structures [10–12]. Plasmons are the quanta of collective excitations of the electronic systems which can affect the response of systems to the applied fields. The plasmon dispersion relation in double-layer system differs significantly from the single layer one. Two branches, acoustic and optical, are appeared due to the in phase and out of phase electron oscillations in two layers, respectively [13]. The double-layer graphene (DLG) is a system consisting of two separate parallel single layer graphenes (SLG) which are placed
close together to make an effective interaction between them but far enough to prevent the electron tunneling, similar to the conventional double-layer 2DEG systems. The DLG is different from bilayer graphene (BLG) system in which two coupled single graphene layers stacked as in graphite and interlayer tunneling should be considered. As in the conventional double-layer 2DEG systems, many-body phenomena like the Coulomb drag due to the interlayer electron–electron interactions can be occurred [14,15]. The Coulomb drag rate has been calculated for the DLG system and its behavior compared to the conventional double-layer 2DEG system [16]. In this Letter, we investigate the plasmon dispersion of a DLG system in which the two graphene layers separated by a nanometer distance d. The plasmon modes of a DLG at zero temperature have been obtained by Hwang and Das Sarma [17]. In this work, all calculations are done at finite temperature and the results compared with the zero temperature ones. Considering the correlation parameter, k F d, where k F is the Fermi wave vector, we discuss the interlayer correlation effects on the plasmon dispersion relation of a DLG at finite temperature. We put an upper limit on the temperature T = 0.2T F (T F is the Fermi temperature) to ensure that the effect of phonons is negligible [18]. We also study the damping rate of plasmons at finite temperature. In addition, the plasmon frequencies and decay rate of a DLG system with different electron densities of layers are calculated at both zero and finite temperature and compared with those obtained from the equal electron density system. 2. Theory
*
Corresponding author. Tel.: +98 21 29902784; fax: +98 21 22431666. E-mail address: [email protected] (T. Vazifehshenas).
0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.10.026
We consider two parallel doped graphene layers with the electron densities, n1 and n2 . Throughout of this Letter, we set h¯ = 1.
4900
T. Vazifehshenas et al. / Physics Letters A 374 (2010) 4899–4903
Fig. 1. (a) Plasmon dispersion and (b) damping rate of a SLG system as functions of dimensionless wave vector at different temperatures. The straight lines show SPE boundaries.
The plasmon dispersion relation of an electronic system can be obtained from the poles of density–density response function, Π(q, ω), or equivalently, from the zeroes of dynamical dielectric function, (q, ω) [17]:
(q, ω P − i γ ) = 0
(1)
where ω P is the plasmon frequency at a given wave vector q and γ is the damping rate of plasma oscillations. In complex plane of ω, if the poles of Π(q, ω) place on the real axis, the plasmons are long-lived and well-defined; but for the poles away from it, we have the landau damped plasmons due to the electron scattering. In case of weak damping (γ ω P ), the plasmon dispersion and decay rate are determined from the following equations:
Re (q, ω P ) = 0
(2)
and
−1 ∂ Re (ω, k) γ = Im (ω P , k) . ∂ω ω =ω P
(3)
To obtain the dielectric function of the system, we use the random phase approximation (RPA) approach which is valid for small dimensionless interaction parameter, r s 1 (r s is defined as the ratio of the interparticle potential energy to the single-particle kinetic energy). For our DLG system, the graphene layers are on the SiO2 substrate so that we have r s 0.87; hence the RPA is a good approximation. At finite temperature, the dielectric function of a DLG system is given by [18]:
(q, ω, T ) = 1 − v (q)Π1 (q, ω, T ) 1 − v (q)Π2 (q, ω, T ) − v 2 (q)Π1 (q, ω, T )Π2 (q, ω, T )e −2qd
(4)
where v (q) = 2π e /κ q is the Fourier transform of the Coulomb interaction and κ is substrate dielectric constant (for SiO2 : κ ≈ 2.5), d is the interlayer spacing and Πi (q, ω, T ) is the non-interacting density–density response function of the i-th layer [19]: 2
Πi (q, ω, T ) = g lim
η→0
×
and valley degeneracies), θ is the angle between k and k + q, and n F i = (exp[β( E k,λ − μi )] + 1)−1 is the Fermi–Dirac distribution function, with β = 1/k B T (k B is the Boltzmann constant) and μi being the chemical potential of i-th layer which is obtained from the normalization condition:
d2 q 1 + λλ cos(θ) λ,λ =±
2π
n F i ( E k,λ ) − n F i ( E k+q,λ )
ω + E k,λ − E k+q,λ + i η
2
.
(5)
Here E k,λ = λk (with λ = 1 and λ = −1 refer to the valence and conduction bands), g = g s g v (with g s = 2 and g v = 2 being spin
∞ ni =
dE D i ( E )n F i ( E ),
(6)
−∞
where D i ( E ) = 2E /π v 2F is the non-interacting graphene density of states (with v F = 106 m/s is the Fermi velocity). In the case of equal electron density layers, Eq. (4) can be rewritten as:
1 − v (q) Re Π(q, ω, T )
= − v (q) Re Π(q, ω, T )e −2qd ± e −2qd 1 + 2v (q) Im Π(q, ω, T ) sinh qd.
(7)
Using above equations, we can determine the optical (+) and acoustical (−) plasmon modes in a DLG system. For a SLG system, the right-hand side (RHS) Eq. (7) simply equals to zero. 3. Results In this Letter, we investigate the effect of temperature on the plasmon dispersion relation of a DLG system and compare it with the SLG and conventional double-layer 2DEG system. In Fig. 1, the plasmon frequencies and damping rate are plotted at two different temperatures: T = 0 and T = 0.2T F . According to these figures, in the region of small q, the plasmon frequencies decrease by increasing the temperature. It can be explained by this fact that at finite temperature, the electrons are excited easier than the zero temperature case; so the collective motions appear at low frequencies (or energies). As it is expected, the damping rate increases by increasing the temperature because the electrons with larger kinetic energies can be easier excited to the electron–hole pairs. In the region of electron–hole excitation, the plasmons are not well-defined anymore and the single particle excitations (SPE) are dominated. In this region where Im Π(q, ω, T ) = 0, two kinds of excitation can be occurred: intraband transition (λ = λ ) and interband transition (λ = −λ ). The boundaries of these two SPE regions at zero temperature are shown in all plasmon dispersion graphs. It is easily seen that at finite temperatures, the mode damping starts at smaller wave vectors, before entering the SPE region. The behavior of the acoustic and optical plasmon branches in a DLG system is shown in Fig. 2 for the layers with equal electron densities (n1 = n2 ) at
T. Vazifehshenas et al. / Physics Letters A 374 (2010) 4899–4903
4901
Fig. 2. (a) Plasmon dispersions and (b) damping rate of a DLG system as functions of dimensionless wave vector at different temperatures with a correlation parameter k F d = 1.68 (d = 30 nm) and n1 = n2 = 1011 cm−2 . The straight lines show SPE boundaries.
Fig. 3. Plasmon dispersion and damping rate of a DLG system as functions of dimensionless wave vector with n1 = n2 = 1012 cm−2 for different correlation parameters k F d = 0.35 (d = 2 nm), k F d = 1.8 (d = 10 nm) and k F d = 8.9 (d = 50 nm). (a) and (c) at T = 0 and (b) and (d) at T = 0.2T F . The straight lines show SPE boundaries.
zero and finite temperatures. Again, the plasmon frequency decreases and decay rate increases by increasing the temperature. From these figures, it is found that at finite temperature in contrast to the zero temperature case, the electron–hole pairs can be created at all wave vectors, this results in the plasmon damping. In Fig. 3, we compare the DLG plasmon dispersion and damping rate for three different correlation parameter, k F d = 8.9, k F d = 1.8 and k F d = 0.35 at T = 0, Figs. 3(a) and (c), and T = 0.2T F , Figs. 3(b) and (d). These figures show that out of the SPE region, the difference between the acoustic and optical branches is small for large correlation parameter (or large d) at both zero and finite temper-
atures. This behaviour can be described by the RHS of Eq. (7) which goes to zero when the separation between two layers is large enough:
1 − v (q) Re Π(q, ω, T ) 0.
(8)
This is just the condition for obtaining the SLG single plasmon branch. In the SPE region, Eq. (7) can be approximated for large d as:
1 − v (q) Re Π(q, ω, T ) ± v (q) Im Π(q, ω, T ).
(9)
4902
T. Vazifehshenas et al. / Physics Letters A 374 (2010) 4899–4903
Fig. 4. Plasmon dispersion and damping rate of a DLG system as functions of dimensionless wave vector with a correlation parameter k F d = 5.3 (d = 30 nm). (a) and (c) for n1 = n2 and (b) and (d) for n2 /n1 = 0.5. The straight line show SPE boundaries.
Here the two branches (acoustic and optical) of a DLG are decoupled even at T = 0 in contrast to the conventional double-layer 2DEG system in which these two branches merge at large d and never split [20]. Also, in the limit of large d, the behavior of DLG plasmon dispersion relation is independent of correlation parameters. In the other hand, Fig. 3 indicates that by decreasing the distance between two graphene layers in a DLG system, the acoustic branch reaches the ω = v F q line (the boundary of SPEintra ) and the damping rate decreases. In addition, at small d, the optical branch of a DLG shows the behavior of the SLG plasmon dispersion with an electron density 4n [17] which is different from the result for a double-layer 2DEG system that behaves like a single layer 2DEG with the electron density 2n. Moreover, for a DLG system, the effect of temperature is more significant at larger k F d and the damping of optical branch is greater than the acoustic one because of the higher energy of plasmon. √ If we set n2 /n1 = α , then we obtain T 2 / T 1 = α (where T i = k B T / E F i ) for a DLG in contrast to 2DEG in which we have T 2 / T 1 = α . Therefore, in graphene systems, the density imbalance has weaker temperature effect compare to the 2DEG. Up to here, we assumed the DLG system with equal electron densities. The effect of different electron densities of two graphene layers on plasmon energies is depicted in Fig. 4. In this case, there are two different SPEinter regions; SPE1,2 and SPE2 . The SPE1,2 refers to the region where the electron–hole pairs can be excited in both graphene layers but in SPE2 region the electron–hole excitation occur only in lower electron density layer. As this figure shows, the difference between acoustic and optical plasmon frequencies for n1 = n2 is larger and the plasmon damping is stronger and starts at smaller wave vectors compared to the n1 = n2 case. Also, the acoustic branch frequencies for a DLG system with different elec-
tron densities of layer at T = 0 are smaller than the n1 = n2 case and more affected by increasing the temperature. In general for a DLG system, the plasmon energies and damping rate are functions of the electron wave vector, frequency, temperature and density. For a given wave vector and fixed density, by increasing the temperature the number of electron–hole pairs increase and plasmon frequencies decrease. In case of different electron densities of layers, n1 > n2 , the effect of temperature for the layer with lower density is stronger and the electron–hole pairs in layer 2 can be excited at smaller q relative to the layer 1, thus the damping rate of density imbalanced DLG grows. This explains why the SPE2 region is bigger than SPE1 region. 4. Conclusion In summary, we investigate the effect of temperature on the plasmon dispersion relation and damping rate of the DLG systems at different correlation parameters and layer electron densities. We find that by increasing the temperature the plasmon frequencies decrease and the damping increases in all cases. Also, we obtain that the behavior of a DLG is different from the conventional double-layer 2DEG at small and large correlation parameters. Finally, our calculations indicate that for n1 = n2 , the acoustic branch of the DLG is more affected by temperature. References [1] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A.A. Firsov, Science 306 (2004) 666. [2] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, Rev. Mod. Phys. 81 (2009) 109. [3] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos, A.A. Firsov, Nature 438 (2005) 197.
T. Vazifehshenas et al. / Physics Letters A 374 (2010) 4899–4903
[4] [5] [6] [7] [8] [9] [10] [11] [12]
A.K. Geim, K.S. Novoselov, Nature Mater. 6 (2007) 183. S. Das Sarma, S. Adam, E.H. Hwang, E. Rossi, arXiv:1003.4731v1, 2010. M.C. Lemme, Solid State Phenomena 156–158 (2010) 499. M.C. Lemme, T. Echtermeyer, M. Baus, H. Kurz, IEEE Electron Device Lett. 28 (4) (2007) 282. S. Das Sarma, E.H. Hwang, Wang-Kong Tse, Phys. Rev. B 75 (2007) 121406. B. Wunsch, T. Stauber, F. Sols, F. Guinea, New Journal of Physics 8 (2006) 318. G. Borghi, M. Polini, R. Asgari, A.H. MacDonald, Phys. Rev. B 80 (2009) 241402(R). E.H. Hwang, S. Das Sarma, Phys. Rev. B 75 (2007) 205418. Zhao-Zhe Li, Commun. Theor. Phys. 52 (2009) 361.
4903
[13] H. Bruus, K. Flensberg, Introduction to Many-Body Theory in Condensed Matter Physics, Oxford University Press, 2002. [14] K. Flensberg, Y.-K.H. Ben, Phys. Rev. Lett. 73 (1994) 3572. [15] A. Kamenev, Y. Oreg, Phys. Rev. B 52 (1995) 7516. [16] Wang Kong Tse, Ben Yu Kuang Hu, S. Das Sarma, Phys. Rev. B 76 (2007) 081401(R). [17] E.H. Hwang, S. Das Sarma, Phys. Rev. B 80 (2009) 205405. [18] S. Das Sarma, A. Madhukar, Phys. Rev. B 23 (1981) 805. [19] M.R. Ramzanali, M.M. Vazifeh, R. Asgari, M. Polini, A.H. Macdonald, J. Phys. A: Math. Theor. 42 (2009) 214015. [20] K. Flensberg, Yu-Kuang Hu Ben, Phys. Rev. B 52 (14) (1995) 796.