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

Plasmon modes of double-layer graphene at finite temperatureDinh Van Tuana,b,c, Nguyen Quoc Khanha,n a Department of Theoretical Physics, National University in Ho Chi Minh City, 227-Nguy

Plasmon modes of double-layer graphene at finite temperature

Dinh Van Tuana,b,c, Nguyen Quoc Khanha,n

a

Department of Theoretical Physics, National University in Ho Chi Minh City, 227-Nguyen Van Cu Str., 5th District, Ho Chi Minh City, Vietnam

b ICN2—Institut Catala de Nanociencia i Nanotecnologia, Campus UAB, 08193 Bellaterra (Barcelona), Spain

c 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 ωin the case of n2¼0 is degenerate with the ω ¼ υFq line only when d-0

Even though n2¼0, when d-0 the temperature optical mode ωþdoes not become the single-layer plasmon with the same density

The effect of temperature on the plasmon dispersion and damping is significant

a r t i c l e i n f o

Article history:

Received 8 May 2013

Received in revised form

15 July 2013

Accepted 16 July 2013

Available online 26 July 2013

Keywords:

Graphene

Plasmon

Collective excitations

a b s t r a c t

We calculate the dynamical dielectric function of doped double-layer graphene (DLG), made of two parallel graphene monolayers with carrier densities n1and n2, and an interlayer separation of d atfinite temperature The results are used tofind the dispersion of plasmon modes and loss functions of DLG for several interlayer separations and layer densities We show that in the case of n2¼0, the finite-temperature plasmon modes are dramatically different from the zero-finite-temperature ones

& 2013 Elsevier B.V All rights reserved

1 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

inter-layer tunneling, only an inter-inter-layer Coulomb interaction Spatially

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 tem-perature, and found that the plasma modes of an interacting DLG system are completely different from the double-layer semiconduc-tor quantum well plasmons In the long wavelength limit the density dependence of the plasma frequency is given byðωþ

0Þ2

∝pffiffiffiffiffin1þ ffiffiffiffiffin

2

p for optical plasmons and ðω

0Þ2

∝pffiffiffiffiffiffiffiffiffiffin1n2

= pffiffiffiffiffin1

þpffiffiffiffiffin2

for acoustic plasmons, compared toðωþ

0Þ2∝N and ðω

0Þ2∝n1n2=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

atfinite 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 atfinite 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

Contents lists available atScienceDirect

Physica E

1386-9477/$ - see front matter & 2013 Elsevier B.V All rights reserved.

n Corresponding author Fax: +848 38350096.

E-mail address: nqkhanh@phys.hcmuns.edu.vn (N.Q Khanh)

2 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],

whereυF 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 1 throughout this paper The energy of graphene for 2D wave vector k is given by

where s¼ 71 indicates the conduction (+1) and valence (1) bands, respectively The density of states is given by DðεÞ ¼

gjεj=ð2πυ2

FÞ, where g ¼ gsgv¼ 4 accounts for the spin (gs¼ 2) and valley (gv¼ 2) degeneracies The Fermi momentum ðkFÞ and the Fermi energyðEFÞ of 2D graphene are given by kF¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffi4πn=g and

E ¼ υ k , where n is the 2D carrier density

Fig 1 Plasmon dispersions of DLG for several layer separations at T¼0 (bold dotted lines), T¼0.5T F (bold dashed lines) and T¼T F (bold solid lines) The thin lines indicate the plasmon dispersion of SLG with the same density and temperature Here we use the parameters n 1 ¼n 2 ¼n¼10 12

cm2and (a) d¼20 Å (k F d¼0.35), (b) d¼100 Å (k F d ¼1.8), (c) d¼300 Å (k F d ¼5.3), and (d) d¼500 Å (k F d¼8.9).

Fig 2 The plasmon modes of DLG for layer separation d¼100 Å at several

momenta.

In the random-phase approximation (RPA), the dynamical

dielectric function of SLG becomes

whereυcðqÞ ¼ 2πe2=κq is the 2D Fourier transform of the Coulomb

potential andΠðq; ω; TÞ, the 2D polarizability at finite temperature,

is given by the bare bubble diagram[16]

Πðq; ω; TÞ ¼ g lim

η-0 þ ∑

s;s 0 ¼ 7

Z d2k ð2πÞ2

1þ ss′ cos ðθk;kþqÞ 2

nFðεk;sÞnFðεkþq;s 0Þ

ω þ εk;sεkþq;s0þ iη ð4Þ here, nFðεÞ ¼ exp½βðεμ 0Þ þ 11 is the Fermi–Dirac distribution

function The non-interacting chemical potential, μ0¼ μ0ðTÞ, is

determined by the conservation of the total electron density as

1

2

TF

T

whereβ ¼ 1=kBT and FnðxÞ is given by

FnðxÞ ¼Z 1

0

tndt

The forms of the chemical potential in the low and high temperature limits are given by[17]

μ0ðTÞ≈EF 1π62 TT

F

; TT

μ0ðTÞ≈ EF

4ln 2

TF

T ; TT

Recently, Ramezanali et al.[18]have obtained the following semi-analytical expressions for the imaginary and the real parts of the dynamical polarizability

ℑmΠðq; ω; TÞ ¼4gπ ∑

α ¼ 7 ΘðυFqωÞq2fðυFq; ωÞ½GðαÞ

þðq; ω; TÞGðαÞ

ðq; ω; TÞ n

þΘðωυFqÞq2fðω; υFqÞ π2δα;þ HðαÞ

þðq; ω; TÞ

ℜeΠðq; ω; TÞ ¼4gπ ∑

α ¼ 7

2kBTln½1 þ eαμ 0 = k ð B T Þ

υ2 F

þ ΘðωυFqÞq2fðω; υFqÞ (

 Gð Þ α

ðq; ω; TÞGðαÞ

þðq; ω; TÞ

þ ΘðυFqωÞq2fðυFq; ωÞ 2πδα;þ HðαÞ

ðq; ω; TÞ

ð10Þ

Fig 3 (a)–(c) Plasmon dispersions of DLG for several temperatures and layer densities Here we use n 1 ¼10 12

cm2and d¼100 Å, and (d) plasmon mode of DLG for n 2 ¼0 and T¼0 at several layer separations.

fðx; yÞ ¼ 1

2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2y2

GðαÞ7ðq; ω; TÞ ¼Z 1

1

du

ffiffiffiffiffiffiffiffiffiffiffiffi

u21 p exp jυF qu 7 ωj2αμ 0

2kBT

HðαÞ7ðq; ω; TÞ ¼Z 1

1du

ffiffiffiffiffiffiffiffiffiffiffiffi

1u2

p exp jυF qu7 ωj2αμ 0

2k B T

The DLG dielectric function is obtained from the determinant of

the generalized dielectric tensor and has the form

εdoðq; ω; TÞ ¼ ε1ðq; ω; TÞε2ðq; ω; TÞυ12ðqÞυ21ðqÞΠ1ðq; ω; TÞΠ2ðq; ω; TÞ ð14Þ

here,ε1ðq; ω; TÞ and ε2ðq; ω; TÞ are the dynamical dielectric functions

of individual layers given by Eq (3), and υ12ðqÞ ¼ υ21ðqÞ ¼ 2πe2

expðqdÞ=ðκqÞ are the interlayer Coulomb interaction matrix elements, withκ the background lattice dielectric constant In our calculations, we setκ ¼ 1

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

3 Numerical results

In this section, we calculate the plasmon dispersion and loss function of DLG at zero andfinite temperatures for several layer separations and densities

3.1 The plasmon dispersion

Fig 1from (a) to (d) show the plasmon dispersions of balanced DLG (bold lines) for several temperatures with increasing layer

Fig 4 Density plot of the DLG loss function in q; ω ð Þ space for fixed densities of n 1 ¼n 2 ¼10 12 cm 2 , layer separations d¼20 Å, 500 Å and temperatures T¼0.5T F , T F (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

separation d We also show the plasmons of SLG (thin lines) at the

same temperatures for comparison In the long wavelength limit,

the plasmon dispersion of SLG at T¼0 has pffiffiffiq dependence.

Moreover, the optical branchωþ 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 T¼ 0 and T ¼ 0:5TF), which is consistent with

Ref.[10] But interestingly, at high temperatures (see T¼ TF), the

plasmon frequencies increase again and are larger than the

zero-temperature frequencies when T4TC, where TC≈0:6TF (more

detail in Fig 2) For low temperatures, the acoustic mode ω

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 T¼ 0), the acoustic mode ωand the optical

modeωþ 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 high-frequency region This behavior is completely different from 2DEG systems[19] For large momentumðq4kFÞ, 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 inFig 2 When ToT0(T4T0), where

T0≈0:4TF, the acoustic modeωdecreases (increases) with increasing temperature In the case of low momentum, the optical mode ωþ

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 2

also shows that the high temperature plasmon modes are dramatically different from the zero-temperature ones, especially in the case of large momentum

Fig 5 (a) Calculated plasmon mode dispersions of DLG for several layer separations for T¼0.5T F , n 2 /n 1 ¼0 (i.e., n 2 ¼0 and n 1 ¼10 12 cm 2 ), and corresponding loss functions for (b) d-0 Å, (c) d¼10 Å, and (d) d¼100 Å.

InFig 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

tempera-ture (T≈TF); this means that the number of electron-hole pairs in this

case is mainly controlled by the temperature As n2/n1decreases, the

zero-temperature acoustic mode ω approaches the boundary of

SPEintra(i.e.ω ¼ υFq) However, it shifts to higher energy when the

temperature increases The differences between the acoustic modeω

at zero and atfinite temperature in the case of n2¼0 inFig 3(c) are

due to the fact that in the zero-temperature limit the undoped layer

admits only interband transitions, whereas atfinite temperature it

admits both interband and intraband transitions InFig 3(d) we show

the DLG plasmon modes at zero temperature for several layer

separations in the case of n2/n1¼0, i.e., the second layer is undoped

and thefirst layer has a finite density n1¼1012cm2 It can be seen

from the figure that the acoustic mode ω is degenerate with the

boundary of SPEintra, while the optical mode ωþ is degenerate

with the SLG mode below the interband single-particle excitation

(SPEinter1,2) As d-0 the dispersion of ωþbecomes exactly that of SLG

plasmon with the same density

3.2 The loss function

Fig 4shows density plots of the balanced DLG loss function

(i.e.,ℑm⌊εdoðq; ω; TÞ1⌋) in qω 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

Raman-scattering spectroscopies [20,21] The acoustic mode ω

corre-sponds to a broadened peak near theω ¼ υFq line and the optical

mode ωþ corresponds to a broadened peak with higher energy

Unlike the zero-temperature case in which the undamped

plas-mons show up as a well-defined δ-function peaks in the lost

function below SPEinter[8], thefinite-temperature plasmon modes

are overdamped even in this region As T or d increases, the

spectral strengths of both modesω7 increase

We also calculate the loss function in the case of imbalanced

densities Wefind that the spectral strengths of both modes ω7

slightly increase when the density imbalance decreases (i.e., n2/n1

increases), and that the high temperature spectral strength (T≥TF)

is almost independent of n2/n1

InFig 5(a) we show the plasmon modes at T¼0.5TFfor n2¼0

and n ¼1012cm2 Unlike the zero-temperature case shown in

Fig 3(d), the acoustic modeωis degenerate with theω ¼ υFq line only when d-0 Å More interestingly, even though there are no free carriers in the second layer (n2¼0) and d-0, the optical mode

ωþ(the dotted line) does not become the SLG plasmon (the dot-dashed line) with the same density, showing the strong effect of temperature in this case InFig 5(b)–(d) we show the loss function

of DLG corresponding to Fig 5(a) As the layer separation decreases, the acoustic modeωapproaches theω ¼ υFq line and loses spectral strength

InFig 6, we consider the effect of temperature on the plasmon mode dispersion at several momenta for the extreme case n2¼0 and

d-0 Å The optical modes ωþapproach 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 atfinite layer separations[22]and showed that the temperature strongly affects the plasmon modes of DLG

4 Conclusions

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 n1and n2, and an interlayer separation of d We have shown that unlike the zero-temperature plasmon modes, the temperature acoustic mode

ωin the case of n2¼0 is degenerate with the ω ¼ υFq 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 ωþdoes 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

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