Plasmon modes in double-layer gapped graphene at zero temperature

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Plasmon modes in double-layer gapped graphene at zero temperature Nguyen Van Men a,b,∗ , Dong Thi Kim Phuong c a b c

Atomic Molecular and Optical Physics Research Group, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, Viet Nam Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Viet Nam An Giang University – VNU HCM, 18-Ung Van Khiem Street, Long Xuyen, An Giang, Viet Nam

a r t i c l e

i n f o

Article history: Received 15 October 2019 Received in revised form 30 November 2019 Accepted 20 December 2019 Available online xxxx Communicated by R. Wu Keywords: Gapped graphene Collective excitations Zero temperature

a b s t r a c t Plasmon dispersions and damping rate of plasma oscillations in a double-layer gapped graphene made of two parallel mono layer gapped graphene sheets grown on dielectric separation are calculated within random-phase-approximation at zero temperature. By using long wavelength limit expansion, analytical expressions for optical and acoustic plasmon frequencies have been formed, and the formulae demonstrate that the considerable difference in analytical form for plasmon frequencies comes from the factor depending on the band gap, compared to gapless situation. Numerical results show that only large band gap decreases remarkably plasmon frequencies of two modes in the range of large wave vector. Acoustic plasmon branch becomes shorter than that in case of zero gap while optical one seems independent with small band gap. In addition, interlayer separation and carrier density affect on collective excitations and damping rate when taking into account the band gap quite similarly to those in case of zero gap. © 2020 Elsevier B.V. All rights reserved.

1. Introduction Graphene, a two-dimensional system consisting of carbon atoms arranged in honey comb lattice, has attracted considerable attention from scientists since its experimental exploration because of its unique properties, compared to all previous usual two-dimensional electron gas. Application of Dirac’s model for graphene shows that quasi-particles in graphene behave as massless chiral fermions with linear low energy dispersion and zero band gap between conduction and valence band [1–4]. However, it is experimentally and theoretically proven that a small gap can be generated by internal factors such as the spin-orbit interaction and symmetry breaking of A and B sub-lattices, causing by the interaction between electrons in graphene sheet and the substrate, as well as by external ones such as electric and magnetic fields [5–14]. With the existence of the band gap, the form of energy dispersion in gapped graphene completely differs from that in gapless one. As a result, the response function of gapped graphene also has dissimilar characters, and the collective excitation modes also illustrate lots of different features [12].

*

Corresponding author at: Atomic Molecular and Optical Physics Research Group, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, Viet Nam. E-mail addresses: [email protected] (N.V. Men), [email protected] (D.T.K. Phuong). https://doi.org/10.1016/j.physleta.2019.126221 0375-9601/© 2020 Elsevier B.V. All rights reserved.

Plasmon, one of important characters of a material, has been studied and applied in lots of technology areas including electronics, optoelectronics, plasmonics, photonics, membrane technology, THz technology, biotechnology, spectroscopy and energy storage in recent years [2–4,15–29]. It is known that plasmon in graphene and graphene-based structures has been theoretically and experimentally investigated due to their application abilities as a new wonderful candidate, replacing silicon materials with better qualities and less energy consumption [30–38]. Therefore, plasmon in layer structures consisting of graphene has been calculated extensively with a large number of publications, demonstrating significantly different characters, compared to those in ordinary twodimensional electron gas structures. It is seen that most of previous work considers graphene as a gapless material [24,39–44] although a small band gap can be generated by both internal and external factors [5–14]. The authors of Refs. [12], [18] and [45] calculate plasmon in a monolayer gapped graphene at zero and finite temperature with several values of band gap, and the results illustrate significant difference from those in mono layer gapless graphene. In addition, it is known that the Coulomb interaction between carriers of layers in multilayer structures leads to new interesting features for collective excitations of the systems. Plasmon characters of a double layer graphene (DLG) system has been carefully studied at zero as well as finite temperature [40–43]. To our knowledge, up to now, no calculations have been done for plasmon dispersion in DLG, taking into account the band gap at

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2

 f s,k

f s ,k

is Fermi-Dirac function,



E s,k − μ

f s,k = 1 + exp



f s ,k = 1 + exp

−1 ,

kB T E s ,k − μ

(7)

−1

kB T



√

Here μ = (¯h v F k F )2 + 2 (k F = π n is Fermi wave vector of monolayer gapless graphene) and k B are chemical potential and Boltzmann constant, respectively. Intra (v ii (q)) and inter (v i j (q)) bare Coulomb interactions in momentum space is given by [40–43]: Fig. 1. A DLGG system with homogeneous dielectric background.

2π e 2

v ii (q) = zero temperature although these maybe bring some new interesting properties.

v i j (q) = v ii (q) exp (−qd)

⎡

In this paper, we investigate a double layer gapped graphene (DLGG) consisting of two parallel gapped graphene sheets grown on dielectric substrates as shown in Fig. 1. The plasmon excitation relation of the system can be determined from the zeroes of non-temperature dynamical dielectric function [39,40,42,46–49]:





ε q, ω p − i γ = 0

(1)

where ω p is the plasmon frequency at a critical wave-vector q, and γ is the damping rate of plasma oscillations. In case of weak damping (γ << ω p ), the plasmon dispersions and damping rate can be found from the following equations [40–43]



Re ε q, ω p = 0

(2)

and



γ = Im ε q, ω p





 ∂ Re ε (q, ω)   ∂ω

− 1

⎛

i (q, ω) = −

i (q, ω) = −



L2

s ,k



k,s,s



 2

h¯ v F k k

1−

2π h¯ 2

q

ω

− α2

2

αi =

q

ω

2

F ss

(6)



q2

μ2i

(11)

ω2

e2 g v F

 q2 + 1 − e −2qd α1 α2 4 = 0

√

κ h¯

π ni



(12)

ω



2 1 − 2 .

μi

ω± =

q 2



α1 + α2 ± e

−qd





4α1 α2 + (α1 − α2

)2 e 2qd

(13)

In long wavelength limit (q → 0), we can expand and rewrite Eq. (13) into approximate form: 2

ω+ =

1 2

(α1 + α2 ) q ∼

 √



n1 1 −

(5)



h¯ ω + E s,k − E s ,k + i η

  

=s s

f s,k − f s ,k

2

Eq. (12) admits two physical solutions, read: 2

g 



g μi

Put Eqs. (5), (8), (9) and (11) into Eq. (4), we drive to the following equation:

Here

εi (q, ω) = 1 + v ii (q) i (q, ω)

(10)

where φ is the angle between k and k . At zero temperature, the Fermi – Dirac contributions become step function, and the polarization functions have analytical form, obtained in Ref. [12]. In long wavelength limit, this function is read:

ω =ω p

Here, εi (q, ω) and i (q, ω) (i = 1, 2) are the non-temperature dynamical dielectric functions and the polarization functions of the i-th layer gapped graphene, respectively, formed in Refs. [12], [18], [39] and [45]:

(¯h v F k )2 + 2

⎞⎤  2 (¯h v F ) kq cos φ ⎟⎥ ⎜ × ⎝ (¯h v F k)2 + 2 +  ⎠⎦ (¯h v F k)2 + 2

(3)

(4)

⎣1 + 

2

1 − α1

ε (q, ω) = ε1 (q, ω) ε2 (q, ω) − v 212 (q) 1 (q, ω) 2 (q, ω)

ss

1⎢

F ss =

Within RPA, the dynamical dielectric function of a DLGG system is written [40–43]

where E  s,k

(9)

F ss is the overlap of states, given [12,18,50]:

2. Theory



(8)

κq

n1 n2 1 −

2α1 α2

α1 + α2

q2 d ∼

√



n1 1 −

2

μ21



μ21



√ 2 ω− =

2



2

+

1−

√



n2 1 −



μ21

+

√



2



μ22

n2 1 −

2

2

μ22



.q (14)

 .q2 d (15)

μ22

It is seen from Eqs. (14) and (15) that in long wavelength limit,

+ 2 is electron energy in  

 ) being momentum, and s s deDirac’s model, with k (k = k + q notes band index (1 for conduction band and −1 for valence band). 2 and v F are energy band gap Fermi velocity, respectively. g is the spin degeneracy factor, and L is the area of the system.

√

ω+ ∼ q (ω− ∼ q) corresponds to in-phase (out-of-phase) oscillations of carriers in two layers of the system. It is also seen that the plasmon in DLGG differ from those in DLG [40] by  frequencies  a factor

2 1 − 2 , created by band gap. When  = 0, Eqs. (14)

μi

and (15) reduce to formulae observed in Ref. [40].

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Fig. 2. Plasmon modes (a) and damping rate (b) in DLGG, plotted for d = 20 nm, n1 = n2 = 1012 cm−2 and  = 0.1μ. Thin lines demonstrate analytical results, expressed by Eq. (13). Dashed-dotted lines show SPE area boundaries (color online).

3. Results and discussion In this section, we demonstrate the numerical results calculated for plasmon excitation of a DLGG 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 E F and k F , respectively, and we also set κ = 2.5 in the whole paper. Fig. 2 illustrates plasmon frequencies (a) and damping rate of plasma oscillations (b) in DLGG, plotted for d = 20 nm and n1 = n2 = 1012 cm−2 . As seen from Fig. 2(a) that two collective excitation modes exist in the system, corresponding to in-phase and out-of-phase oscillations of carriers in two layers. The former gets higher values, named optical (OP) mode, and the later get lower values, called acoustic (AC) one similar to those in other double layer systems [40–43]. Fig. 2(b) presents the damping rate of two plasmon branches with the same parameters. It is seen that the plasmon branches begin losing their energy when they enter single-particle excitation (SPE) area, at about q = 0.8k F . Moreover, plasmon branches are damped more strongly as the plasmon curves go far from SPE area boundary. In Fig 2(a), thin lines show analytical results, given in Eq. (13) as a comparison. The figure shows that the analytical and numerical results are identical in sufficiently small wave vector. In order to understand the effects of band gap on plasmon modes of the system, we plot in Fig. 3 plasmon modes and damping rate with several band gaps  = 0.1μ and  = 0.3μ in comparison with gapless situation. Fig. 3(a) presents that OP plasmon branch is seemly independent on small band gap meanwhile AC one is affected more significantly. AC branch in case of DLG disappears as touching the SPE region boundary, at about q = 2k F while AC one in the other case exits before encountering this boundary, at about q = 1.5k F . The damping rate of plasma oscillations with the same parameters, showed in Fig. 3(b), demonstrates that the energy loss of plasmon branches in case of  = 0.1μ is quite similar to those in case of  = 0 (see Fig. 3(b)). With larger band gaps, as showed in Fig. 3(c) for  = 0.3μ, both OP and AC plasmon frequencies decrease pronouncedly in SPE area, compared to respective frequency in case of DLG. It can be observed that the increase in band gap decreases plasmon frequencies in both OP and AC modes, and the decrease occurs mainly in SPE region. As seen from Fig. 3(c) and (d) that when the band gap is sufficiently large, plasmon curves enter the gap opened between inter- and intra-band. Similar character of plasmon in MLGG has also been explored in Refs. [12], [17] and [45].

In order to know clearly about the effects of band gap, we illustrate in Fig. 4 plasmon frequencies as functions of band gap with several different values of wave vector, for d = 20 nm, n1 = n2 = 1011 cm−2 (a) and d = 10 nm, n1 = n2 = 1012 cm−2 . In general, the increasing band gap reduces slightly plasmon frequencies in the system. In case of large wave vector the effects of band gap are more noticeably when the band gap increases from zero to a half of chemical potential. Especially, in case of q = k F , as seen from the figures that plasmon frequencies decrease with the increase in band gap, reach the lowest peak at about  ≈ 0.2μ and  ≈ 0.25μ for AC and OP branch, respectively. In range of band gap 0.2μ ≤  ≤ 0.3μ, plasmon frequencies may increase weakly as the band gap increases. Fig. 5 demonstrates plasmon modes in DLGG ( = 0.1μ, (a)) and DLG ( = 0, (b)) for several interlayer separations to make a comparison. It can be seen from the figure that the AC frequency in DLGG increases while OP one decreases with the increase in separated distance. Moreover, the changes of plasmon frequencies mainly happen outside SPE region where plasmon modes are completely undamped. On the other hand, with sufficiently large spacer width, two plasmon branches become identical before entering SPE area. These behaviors are similar to those in DLG as illustrated in Fig. 5(b) and obtained in previous papers [40–43]. It is known that carrier densities in graphene layers affect significantly on plasmon properties of double layer structures. Fig. 6 plots plasmon modes and damping rate of plasma oscillations in DLGG and DLG with carrier densities of n1 = n2 = 1011 cm−2 and n1 = n2 = 1012 cm−2 for d = 20 nm. It can be seen from the figure that AC (OP) plasmon frequency in DLGG and in DLG increases (decreases) remarkably with the increase in carrier density. When the carrier density is sufficiently large, two plasmon branches become identical at the SPE region boundary. It can be stated that the increasing carrier density in graphene layer affects on plasmon modes similar to increasing interlayer separation showed in Fig. 5. In addition, calculations taking into account the band gap do not change significantly the effects of electron density on collective excitations. Fig. 6(b) and (d) present that the damping rates of plasma oscillations in DLGG are quite similar to those in DLG and noticeably depend on electron density. When carrier density increases, AC (OP) branch is damped more strongly (weakly) at critical wave vector. Finally, we turn to consider the effects of the imbalance of carrier density in two layers on plasmon modes of the system. Fig. 7 plots plasmon modes in DLGG ( = 0.1μ) and DLG ( = 0) for

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Fig. 3. Plasmon frequencies and damping rate in DLGG with several band gaps, plotted for, n1 = n2 = 1011 cm−2 , d = 20 nm ((a) and (b)) and n1 = n2 = 1012 cm−2 , d = 5 nm ((c) and (d)). Dashed-dotted lines show SPE area boundaries (color online).

Fig. 4. Plasmon frequencies as functions of band gap for several values of wave vector, plotted for d = 20 nm, n1 = n2 = 1011 cm−2 (a) and d = 5 nm, n1 = n2 = 1012 cm−2 (b).

d = 5 nm, n1 = 1012 cm−2 with three values of electron density in the second layer n2 = 0.5n1 , n2 = n1 and n2 = 2n1 . It is seen from Fig. 7(a) that both OP and AC plasmon frequencies in DLGG increase pronouncedly with the increase in carrier density n2 . These behaviors are similar to those in DLG as demonstrated in Fig. 7(b) and in previous papers [40–43]. It can be observed that with small band gap taken into account, the effects of the imbalance of carrier

density in two graphene layers on collective excitations depend weakly on the band gap. 4. Conclusion In summary, calculations for collective excitations and damping rate of plasma oscillations of DLGG consisting of two parallel

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Fig. 5. Plasmon modes for several values of separated distance, plotted for n1 = n2 = 1012 cm−2 ,  = 0.1μ (a) and  = 0 (b). Dashed-dotted lines shows SPE area boundaries (color online).

Fig. 6. Plasmon modes and damping rate in DLGG ((a) and (b)) and DLG ((c) and (d)), plotted for d = 20 nm, n1 = n2 = 1011 cm−2 and n1 = n2 = 1012 cm−2 . Dashed-dotted lines shows SPE area boundaries (color online).

gapped graphene sheets separated by dielectric spacers at zero temperature are carried out for the first time. Analytical expressions for plasmon frequencies at long wavelength limit demonstrate that the OP and AC plasmon frequencies differ from those in DLG by a factor depending on the band gap. The increase in band gap only decreases significantly plasmon frequencies with large values of band gap and large wave vector. However, tak-

ing into account the band gap in calculations for plasmon dispersions makes AC plasmon branch shorter, compared to that in gapless system with the same parameters because this branch disappears before touching SPE area boundary. The dependence on interlayer correlation parameters and carrier density of plasmon dispersions in DLGG is similar to those in DLG obtained in previous work.

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Fig. 7. Plasmon frequencies with several values of carrier density n1 , plotted for d = 5 nm and n1 = 1012 cm−2 ,  = 0.1μ (a) and  = 0 (b). Dashed-dotted lines shows SPE area boundaries (color online).

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