Optical conductance and transmission in bilayer graphene
H. M. Dong, J. Zhang, F. M. Peeters, and W. Xu
Citation: J. Appl. Phys. 106, 043103 (2009); doi: 10.1063/1.3200959
View online: /10.1063/1.3200959
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Optical conductance and transmission in bilayer graphene
H.M.Dong,1J.Zhang,2F.M.Peeters,3and W.Xu1,2,a͒
1Key Laboratory of Materials Physics,Institute of Solid State Physics,Chinese Academy of Sciences,
Hefei230031,China
2Department of Physics,Yunnan University,Kunming610015,China
3Department of Physics,University of Antwerp,Groenenborgerlaan171,B-2020Antwerpen,Belgium
͑Received5May2009;accepted10July2009;published online20August2009͒
We present a theoretical study of the optoelectronic properties of bilayer graphene.The optical conductance and transmission coefficient are calculated using the energy-balance equation derived from a Boltzmann equation for an air/graphene/dielectric-wafer system.For short wavelengths͑Ͻ0.2m͒,we obtain the universal optical conductance=e2/͑2ប͒.Interestingly,there exists an optical absorption window in the wavelength range10–100m,which is induced by different transition energies required for inter-and intra-band optical absorptions in the presence of the Moss–Burstein effect.As a result,the position and width of this absorption window depend sensitively on temperature,carrier density,and sample mobility of the system.These results are relevant for applications of recently developed graphene devices in advanced optoelectronics such as the infrared photodetectors.©2009American Institute of Physics.͓DOI:10.1063/1.3200959͔
I.INTRODUCTION
Graphene is the basis of a new class of nanostructures
where conduction occurs in single or few layers of carbon
atoms arranged in a hexagonal lattice.1Owing to its unique
electronic band structure and the corresponding quasirelativ-
istic features,graphene has attracted a great attention in re-
cent years.Furthermore,graphene can have a high carrier
density and exhibits high electronic mobility even at room
temperature.One of the major advantages of a graphene de-
vice is that the carrier density in the graphene layer can be
controlled very effectively through a gate voltage.1Hence,
graphene has been proposed as a“building block”for ad-
vanced electronic devices2such as graphene p-n and p-n-p
junctions,3transistors,4etc.In recent years,the study of the
electronic transport properties of Dirac quasiparticles in
graphene has rapidly become an important research topic in
nanomaterial science,condensed matter physics,and
nanoelectronics,5which is partly motivated by the possible
applications of graphene in advanced electronic devices.6
Recently,the optical and optoelectronic properties of dif-
ferent graphene systems have been investigated.In particu-
lar,it was found experimentally that the optical conductance
per graphene layer is given by a universal value=e2/͑4ប͒in the visible frequency and UV range.7As a con-sequence,the light transmittance of monolayer and bilayer
graphene devices are about0.98and0.96,respectively,in the
visible bandwidth.8This important discovery has resulted in
the proposal that graphene can be used to replace conven-
tional indium tin oxide electrodes for making better and
cheaper optical displays.9Kuzmenko et al.7recently found
experimentally that for photon energy smaller than0.2eV,
there is an optical absorption window.The width and depth
of this window depend strongly on temperature.This inter-
estingfinding implies that graphene may also be applied for infrared detection in ambient condition.Very recent experi-
mental work showed that graphene can have strong intra-
and inter-band transitions which can be substantially modi-
fied through electrical gating,similar to resistance tuning in
graphenefield-effect transistors.10These experimental results
show clearly that graphene can be used not only as advanced
electronic devices but also as optical devices for various ap-
plications.
In conjunction with experimental investigations into op-
toelectronic properties of graphene systems,theoretical study
in this area has been quite active.The universal optical con-
ductance in the visible regime,0=e2/4បper graphene layer, has been obtained theoretically.7,11The features of graphene
systems under far-infrared͑FIR͒or terahertz radiation have
also been investigated theoretically using various
approaches.12,13The results obtained from these theoretical
investigations have indicated that͑i͒in the short wavelength
such as visible regime,inter-band transition is the principal
channel for optical absorption and conductance in
graphene;7,11͑ii͒in the FIR or terahertz bandwidth,both inter-and intra-band transitions play important roles to cause optical absorption and conductance in graphene;12,13and͑iii͒the optoelectronic properties for graphene in the FIR or tera-hertz regime depend strongly on temperature and carrier den-sity in the system.12
The optical and optoelectronic properties of monolayer
graphene has been well documented.7,8,10In contrast to a
nearly linear energy spectrum in monolayer graphene,bi-
layer graphene has a quadratic energy spectrum in low en-
ergy regime.14,15Thus,the density of states in a bilayer
graphene system differs significantly from that in monolayer
graphene.It has been realized that bilayer graphene is of
equal importance as monolayer graphene for both techno-
logical applications and fundamental science.4,16In this pa-
per we present a detailed theoretical study of the optoelec-
tronic properties of bilayer graphene.In Sec.II,the
theoretical approach is developed to calculate the optoelec-
a͒Electronic mail:wenxu_issp@.
JOURNAL OF APPLIED PHYSICS106,043103͑2009͒
0021-8979/2009/106͑4͒/043103/6/$25.00©2009American Institute of Physics
106,043103-1
tronic coefficients in bilayer graphene.The main results ob-tained from this study are presented and discussed in Sec.III. Our conclusions drawn from this work are summarized in Sec.IV.
II.THEORETICAL APPROACH
In this study,we consider a configuration where the bi-layer graphene sheet is in the xy-plane on top of a dielectric wafer such as SiO2substrate.Bilayer graphene is formed by the Bernal stacking of two graphene layers.Both sublattices are displaced from each other along an edge of the hexagons by a distance of a0=1.42Å.Under the usual effective-mass approximation,the effective Hamiltonian to describe a car-rier͑electron or hole͒in the-bands near the K-point is given by17,18
H0=
ប2
2mءͫ0k−2−k0k+
k+2−k0k−0ͬ,͑1͒
where kϮ=k xϮik y=keϮiwith k=͑k x,k y͒being the wavevector or wavevector operator for a carrier andis the angle between k and the x-axis,mء=2ប2␥1/͑3a0␥0͒2Ϸ0.033m e is the effective mass for a carrier in bilayer graphene with m e being the free-electron mass and␥0 =3.16eV and␥1=0.39eV being the direct intra-and inter-layer coupling constants,respectively,k0=3a0␥3mء/ប2Ϸ106/ͱ3cm−1with␥3=0.315eV being the indirect inter-layer coupling constant.19The warping of the band is ignored which is only important near zero energy.The corresponding Schrödinger equation can be solved analytically and the ei-genvalue and eigenfunction for a carrier in bilayer graphene are given,respectively,as
E͑k͒=ប2kK
2mء
,͑2͒
with K=ͱk2+k
2−2kk
cos͑3͒and=+1refers to an elec-
tron and=−1to a hole,and
k͑r͒=͉k,͘=2−1/2͓e i,͔e i k·r͑3͒in the form of a row matrix,with r=͑x,y͒and e i=͑k0e i−ke−2i͒/K.
Next we apply a lightfield perpendicular to the graphene sheet which is polarized linearly along the x-direction of the graphene system.Including the effect of the radiationfield within the usual Coulomb gauge,the carrier-photon interac-tion Hamiltonian in a bilayer graphene is
HЈ͑t͒=បeA͑t͒
2mͫ0k0−2k−
k0−2k+0ͬ,͑4͒
where A͑t͒=͑F0/͒sin͑t͒is the vector potential of the ra-diationfield with F0andbeing the electricfield strength and the frequency of the lightfield,respectively.We limit ourselves to the case of weak radiationfield and neglect the contribution from the F02term.
Using Fermi’s golden rule,thefirst-order steady-state electronic transition rate induced by carrier-photon interac-tion is obtained as
WЈ͑k,kЈ͒=
2
បͩបeF04mء
ͪ2͉UЈ͑k͉͒22K2␦kЈ,k
ϫ␦͓E͑k͒−EЈ͑kЈ͒+ប͔,͑5͒which measures the probability for scattering of a carrier from a state͉k,͘to a state͉kЈ,Ј͘,with
͉UЈ͑k͉͒2=͑k04+4k4͒͑1+Јcos͑2͒͒
−2kk0͑k02+2Јk2͓͒2cos+cos͑3͒
+Ј͑cos+2cos͑3͔͒͒+k2k02͓5+4͑cos͑4͒
+cos͑2͒͒+Ј͑8+5cos͑4͔͒͒.
In this work,we employ the Boltzmann equation as the gov-erning transport equation to study the response of the carriers in bilayer graphene to the applied radiationfield.In the case of a nondegenerate statistics,the semiclassical Boltzmann equation can be written as
ץf͑k͒
ץt=g s g v
͚
Ј,kЈ
͓FЈ͑k,kЈ͒−FЈ͑kЈ,k͔͒,͑6͒
where g s=2and g v=2count for spin and valley degeneracy, respectively,f͑k͒is the momentum-distribution function for a carrier at a state͉k,͘,and FЈ͑k,kЈ͒=f͑k͓͒1−fЈ͑kЈ͔͒WЈ͑k,kЈ͒.Because the radiationfield has been included within the electronic transition rate,the force term induced by thisfield does not appear in the drift term on the left-hand side of the Boltzmann equation to avoid double counting.There is no simple and analytical solution to Eq.͑6͒with WЈ͑k,kЈ͒given by Eq.͑5͒.In the present study, we employ the usual balance-equation approach to approxi-mately solve the problem.20For thefirst moment the energy-balance equation can be derived by multiplying g s g v͚k E͑k͒to both sides of the Boltzmann equation.From the energy-balance equation,we can obtain the energy transfer rate for a carrier:P=g s g v͚k E͑k͒ץf͑k͒/ץt and the total energy transfer rate of the system
P=P++P−=P+++P+−+P−++P−−,͑7͒
with
PЈ=16ប͚
kЈ,k
FЈ͑k,kЈ͒.
With the energy transfer rate P,we can calculate the optical coefficients of the sample such as the optical conductance, absorption coefficient,and transmission coefficient.Note that the optical conductance͑͒can be obtained from P =͑͒F02and we have
͑͒=͚
,Ј
PЈ/F02=͚
,Ј
Ј͑͒,͑8͒
which is independent on the radiation intensity F0when F0is sufficiently weak.Moreover,the transmission coefficient for an air/bilayer-graphene/dielectric-wafer͑SiO2͒system can be evaluated through21
T͑͒=ͱ⑀2⑀14͑⑀1⑀0͒2
͉͑ͱ⑀1⑀2+⑀1͒⑀0+ͱ⑀1͑͒/c͉2,͑9͒where⑀1and⑀2=⑀ϱare the dielectric constants of free space and the effective high-frequency dielectric constant of the SiO2substrate,respectively,and c is the speed of light in vacuum.
One of the advantages of the balance-equation approach is that we can circumvent the difficulties of solving the Bolt-zmann equation directly by using a specific form of the dis-tribution function.In this study,we assume that the momen-tum distribution of a carrier in bilayer graphene can be described by a statistical energy distribution such as the Fermi–Dirac function f͑k͒Ӎf͓E͑k͔͒,where f͑x͒=͓1 +e͑x−ء͒/k B T͔−1withءbeing the chemical potential͑or Fermi energy E Fat T→0͒for electrons or holes.We note that for a doped or gated graphene device subjected to a radiationfield,the chemical potentials for electrons and holes can be different.After considering the effect of the broadening of the scattering states due to energy relaxation, we have
͑͒=
0
2A2
͑͒2+1
͵
d͵0ϱdkk K2
ϫG+͑k,͒f͓E+͑k͔͕͒1−f͓E+͑k͔͖͒͑10͒
for the case of intraband transition,where=Ј=Ϯ,is the corresponding energy relaxation time,0=e2/͑2ប͒is the uni-versal conductance,A=mء/ប,and G+͑k,͒=2͑k04 +4k4͒cos2−6kk0͑k02+2k2͓͒cos+cos͑3͔͒+k2k02͓13
+9cos͑4͒+4cos͑2͔͒.For interband transition,we have +−͑͒Ӎ0
and
−+͑͒=0
2A2
Sͩប2
ͪ͵
d͵0ϱdkk K2
ϫG−
͑k,͒
2͑−បkK/mء͒2+1,͑11͒
with G−͑k,͒=͑k02−2k2−2kk0cos͒2sin2and S͑x͒=f−͑−x͓͒1−f+͑x͔͒.Using Eqs.͑10͒and͑11͒we can evaluate the contributions from different transition channels to the optical absorption and transmission in bilayer graphene.It should be noted that when the radiationfield is sufficiently weak,the optical conductance and absorption and transmission coefficients do not depend on the radiation intensity.
III.RESULTS AND DISCUSSIONS
For our numerical calculations we consider a typical bi-layer graphene device in which the conducting carriers are electrons.If n0ϳ1012cm−2is the electron density in the absence of the radiationfield͑or dark density͒,the electron density in the presence of the radiation is n e=n0+⌬n e,where ⌬n eϳ5ϫ1011cm−2is the density of photoexcited electrons. Under the condition of the charge number conservation ⌬n e=n h is the hole density in the presence of radiationfield.At afinite temperature,the chemical potentialءfor elec-trons and holes in bilayer graphene can be determined,re-spectively,through
n e=
2
2
͵
d͵0ϱdkkf
ͩប2kK
2mءͪ͑12͒and
n h=
2
2
͵
d͵0ϱdkk
ͫ1−fͩ−ប2kK
2mءͪ
ͬ.͑13͒
In the calculation,we take⑀1=1and⑀2=2.25for an air/bilayer-graphene/SiO2-wafer system,where the effect of the dielectric mismatch between the bilayer graphene and the substrate has been taken into account.22Furthermore,it has been obtained experimentally23that in a graphene device,the energy relaxation time is aboutϳ1ps for high-density samples.Thus,we takeϳ1ps in the calculation.
In Fig.1we show the contributions from different elec-tronic transition channels to the optical conductance͑or ab-sorption spectrum͒in bilayer graphene.We notice the fol-lowing features.͑i͒The interband transition contributes to the optical absorption in the short-wavelength regime, whereas the intraband transitions give rise to the long-wavelength optical absorption.͑ii͒Optical absorption varies very weakly with increasing radiation frequency in the short-wavelength regime͑Ͻ0.2m͒,whereas the absorption coefficient depends strongly on radiation wavelength in the long-wavelength regime͑Ͼ0.5m͒.͑iii͒The optical con-ductance in the short-wavelength regime is a universal value 0=e2/͑2ប͒for bilayer graphene,in contrast to0 =e2/͑4ប͒observed for monolayer grapheme.7,8͑iv͒In the intermediate radiation wavelength regime͑0.2ϽϽ5m͒, optical conductance increases with radiation wavelength.͑v͒More interestingly,there is an absorption window in between 10and100m wavelength regimes.This absorption win-dow is induced by the completing absorption channels due to inter-and intra-band scattering events.͑vi͒In the very long-wavelength regimeϾ100m,optical absorption in-creases sharply with radiation wavelength.These interesting features can be understood with the help of Fig.2.When the radiationfield is absent,there is a single Fermi level in the conduction band in a n-type bilayer graphene͑or in the
pres-FIG.1.Contributions from different transition channels͑Ј͒to optical conductance at thefixed temperature T=150K and carrier densities n e =1.5ϫ1012cm−2and n h=5ϫ1011cm−2.Here the solid curve is the total optical conductance and0=e2/͑2ប͒.
ence of a positive gate voltage ͒.In this case all states below E F
e
are occupied by electrons,as shown in Fig.2͑a ͒.When a light field is applied to the system ͓see Fig.2͑b ͔͒,the elec-trons in the valence band are excited into the conduction band via absorption of photons.Thus,the electron density in
the conduction band increases and the Fermi level E F
e
for electrons in this band is also higher.Meanwhile,the holes
are left in the valence band and a Fermi level E F
h
is estab-lished in the valence band for holes.As shown in Fig.2͑b ͒,in the presence of a radiation field the intraband electronic transition accompanied by the absorption of photons can be achieved not only in the conduction band via a channel ␣++but also in the valence band via a channel ␣−−.The intraband transitions are a direct consequence of the broadening of the scattering states in the conduction and valence bands.Be-cause bilayer graphene is a gapless semiconductor,the elec-trons in the valence band can be more easily excited into the conduction band in contrast to a conventional semiconductor.Thus,there is a strong interband optical transition channel ͓i.e.,␣−+in Fig.2͑b ͔͒in bilayer graphene.Since optical ab-sorption is achieved for transition from occupied states to empty states,together with the presence of the Moss–Burstein effect 24or the Pauli blockade effect 25͑shown in Fig.2͒,intraband transitions require less photon energy whereas a relatively larger photon energy is needed for interband tran-sition.Consequently,an optical absorption window can be induced through different energy requirements for inter-and intra-band transition channels.
In Fig.3,we show optical conductance ͑͒and trans-mission coefficient T ͑͒as a function of radiation wave-length for fixed carrier densities at different temperatures.As can be seen,optical conductance and transmission depend sensitively on temperature,in line with the experimental finding.7It should be noted that for the fixed electron and hole densities,the chemical potential for electrons/holes decreases/increases with increasing temperature.Thus,due to the Moss–Burstein effect,the optical absorption window shifts to higher energy ͑or shorter wavelength ͒regime,as shown in Fig.3.We note that the strength of the optical absorption is proportional to the optical conductance.There-fore,the height of the optical absorption window decreases
with increasing temperature.A sharper cutoff of the optical absorption at the window edge can be observed at lower temperature.
The optical conductance and transmission coefficient are shown in Fig.4as a function of radiation wavelength at a fixed temperature T =150K and a fixed hole density n h for different electron densities n e .Because the chemical poten-tial for electrons in the conduction band increases with elec-tron density,the optical absorption window shifts to higher energy ͑or shorter wavelength ͒regime with increasing elec-tron density,as shown in Fig.4.The height of the absorption window increases with electron density and a sharper cutoff of the optical absorption at the window edge can be observed for larger electron density.For a gate-controlled bilayer graphene placed on a dielectric SiO 2wafer,the positive ͑negative ͒voltage across the gate can pull the electrons ͑holes ͒out from the SiO 2wafer and inject them into the graphene layer.By doing so,the electron density in the graphene layer can be varied by the gate voltage and the corresponding optoelectronic properties of the device system depend on the gate voltage applied.This mechanism has been verified by the recent experiments.10
In the calculation we take the energy relaxation time as an input parameter to count the effect of the broadening
of
FIG.2.͑a ͒A bilayer graphene system in the absence of the radiation field ͑i.e.,F 0=0͒.Here we show the case where the conducting carriers are elec-trons with a Fermi energy E F e in the conducting band.The shaded area refers to the occupied states.͑b ͒Optical absorption channels in the presence of the
radiation field ͑i.e.,F 0 0͒.Here E F e and E F h
are the Fermi energies for electrons and holes,respectively,and there are three optical absorption chan-nels:␣−+,␣++and ␣−−
.
FIG.3.Optical conductance and transmission as a function of radiation wavelength at the fixed carrier densities n e =1.5ϫ1012cm −2and n h =5ϫ1011cm −2for different temperatures T =10K ͑solid curve ͒,77K ͑dashed curve ͒,150K ͑dotted curve ͒,and 300K ͑dotted-dashed curve ͒.The corre-sponding transmission coefficients are shown in the
inset.
FIG.4.Optical conductance and transmission coefficient,/0and T ͑͒,as a function of radiation wavelength at a fixed temperature T =150K and a fixed hole density n h =5ϫ1011cm −2for different electron densities n e =1ϫ1012cm −2͑solid curve ͒, 1.5ϫ1012cm −2͑dashed curve ͒,and 2.5ϫ1012cm −2͑dotted curve ͒.The corresponding transmission coefficients are shown in the inset.
the
absorp-
tion and time is shown and hole
to a to a sample and deeper observed for a
sample
The optical
very little on of the sample.e2/͑2ប͒
m͒is a universal that the universal e2/͑4ប͒. Thus,
should from
7In the in the bilayer
As long we canfind matter what the be pointed
re-gime
is often obtained from the of Eq.͑9͒.7,8
The ab-sorption can of-ten be26The strong bilayer graphene that bilayer͑MIR͒
con-ductance
in the Fig.3͒.This suggests that graphene devices can be applied for high-
speed MIR detection at ambient temperature for various
applications.27
IV.CONCLUSIONS
In this paper,we conducted a detailed theoretical study
of the optical and optoelectronic properties of bilayer
graphene.Our theoretical approach is based on the energy-
balance equation derived from the semiclassical Boltzmann
equation.By considering a coupled bilayer graphene system
and including the intra-and interband transition channels,we
studied the dependence of optical absorption/transmission on
temperature,electron density,and energy relaxed time.The
main conclusions drawn from this study are summarized as
follows.
In the short-wavelength regime͑Ͻ0.2m for bilayer graphene͒the optical conductance is induced mainly through
interband electronic transition and depends very little on
temperature,electron density,and sample mobility.There-
fore,the optical conductance in the short-wavelength regime
is a universal value e2/͑2ប͒.We have found that such phe-nomenon can be observed no matter whether the coupling
between two graphene layers is present or not in a bilayer
graphene system.Thisfinding confirms a recent theoretical
prediction7that the universal optical conductance per
graphene layer is e2/͑4ប͒.In such wavelength regime,the optical transmission coefficient is about0.96,in agreement with the experimental data.8
In the intermediate wavelength regime͑0.2ϽϽ5m for bilayer graphene͒the optical conductance increases slightly with radiation wavelength.There is an optical ab-sorption window in between10and100m wavelength regimes.This absorption window is induced by different en-ergies required for intra-and inter-band transition channels. Therefore,the width,height,and position of such window depend sensitively on temperature,electron density,and mo-bility of the sample.Wefind that the strong cutoff of the optical absorption can be observed at the edge of the absorp-tion window.Such effect can be utilized for MIR detection. The results obtained from this study confirm that the graphene systems can be used as advanced optical and opto-electronic devices working in the visible and infrared band-widths.
ACKNOWLEDGMENTS
This work was supported by the Chinese Academy of Sciences,the National Natural Science Foundation of China ͑Grant No.10664006͒,the Department of Science and Tech-nology of Yunnan Province via the Special Funds for Distin-guished Professorship and the Project for the Promotion of Science and Technology͑Grant No.2007A0017z͒,and by the Flemish Science Foundation͑FWO-Vl͒.
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