Terahertz absorption window and high transmission in graphene bilayer

Optics Communications 283 (2010) 3695–3697

Contents lists available at ScienceDirect

Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m

Terahertz absorption window and high transmission in graphene bilayer H.M. Dong a,⁎, W. Xu a,b, J. Zhang b, Yi-zhe Yuan c a b c

Key Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences, China Department of Physics, Yunnan University, Kunming, China School of Science, Tianjin University of Science & Technology, 13th Street Teda, Tianjin 300457, China

a r t i c l e

i n f o

Article history: Received 29 March 2010 Received in revised form 8 May 2010 Accepted 11 May 2010 Keywords: Terahertz Absorption window Transmission Graphene bilayer

a b s t r a c t A detailed theoretical study of terahertz (THz) optical absorption and transmission in graphene bilayer is presented. Considering an air/graphene/dielectric-wafer system, it is found that there is an absorption window in the range 3–30 THz and the optical transmission is up to 96%. Such an absorption window is induced by different transition energies required for inter- and intra-band optical absorption in the presence of the Pauli blockade effect. As a result, the position and width of this THz absorption window depend sensitively on temperature and carrier density of the system. These results are pertinent to the applications of recently developed graphene systems as novel optoelectronic devices such as THz photodetectors. Crown Copyright © 2010 Published by Elsevier B.V. All rights reserved.

1. Introduction Graphene has attracted incessant attention and has been investigated widely in recent years. Owing to its massless Dirac quasiparticles, unique gapless and linear electronic band structure at low energy, graphene has become one of the most important research topics in condensed matter physics, nano-material science and nanoelectronics [1,2]. Moreover, monolayer and bilayer graphene, whose carrier density can be controlled effectively through a gate voltage (back gate), exhibits high carrier mobility and ballistic transport over sub-micron scales even at room temperature [3]. Thus, graphene has prompted a large number of investigations into graphene based on high speed electronic devices, such as field-effect transistors [4], p–n junction [5], and high-frequency devices [6]. Recently the optical and optoelectronic properties of different graphene systems have been intensively investigated. Up to now, it was found experimentally that the optical conductance per graphene layer is a universal value in the visible range [7]. The light transmittances for monolayer and bilayer graphene are respectively about 98% and 96% in the visible bandwidth [8]. This important discovery has resulted in a proposal that graphene materials can be used to replace conventional indium tin oxide (ITO) electrodes for making better and cheaper optical displays [9]. It was also shown experimentally [7] that when the radiation photon energy is smaller than about 30 THz, there is an optical absorption window. The width and depth of this window depend strongly on temperature. A very recent experimental work demonstrated that graphene can have ⁎ Corresponding author. E-mail address: [email protected] (H.M. Dong).

strong intra- and inter-band transitions which can be substantially modified via electrical gating, similar to resistance tuning in a graphene field-effect transistor [10]. These experimental results suggest that graphene systems can be used not only as advanced electronic devices but also as optical devices for various applications. It has been realized that bilayer graphene is of equal importance as monolayer graphene for both technological applications and fundamental science [11]. Therefore, here we present a detailed theoretical study of the optical properties of graphene bilayer.

2. Theoretical approach Here we consider that a linearly polarized radiation field is applied perpendicular to the graphene sheet of a coupled graphene bilayer system. The corresponding Schrödinger equation can be solved analytically [12], so that our investigation of graphene bilayer can be carried out based on their work. The eigenvalue Eλ(k) = λℏ2kK/2m⁎

and eigenfunction ψλk(r) = |k, λ N = 2− 1/2[eiψ, λ]eik⋅ r for a carrier in bilayer graphene are given respectively. Here eiψ = (k0eiϕ − ke− 2iϕ) / K, r = (x, y), k = (kx, ky) being the wavevector for a carrier and ϕ is the angle between k and the x-axis. m⁎ ≈ 0.033m e is the effective mass ffi for qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a carrier in graphene bilayer, K = k2 + k20 −2kk0 cosð3ϕÞ with pffiffiffi k0 ≈106 = 3 cm− 1, and here λ = + 1 refers to an electron and λ = −1 to a hole. The electronic transition rate induced by electron and hole interactions with the radiation field is calculated using Fermi's golden rule. The semi-classical Boltzmann equation is employed as the governing transport equation to study the response of the carriers in bilayer graphene to the applied radiation field. We apply the usual

0030-4018/$ – see front matter. Crown Copyright © 2010 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.05.023

3696

H.M. Dong et al. / Optics Communications 283 (2010) 3695–3697

energy-balance equation approach to obtain the energy transfer rate of the system, from which we can calculate the optical conductance σλλ′(f) induced by different transition channels, which reads σλλ ðf Þ =

σ0 2πf τ π ∞ dkk ∫0 dϕ∫0 2 π2 A2f ð2πf τÞ2 + 1 K     × Gþ ðk; ϕÞFλ Eþ ðkÞ 1−Fλ Eþ ðkÞ ;

ð1Þ

for the case of intra-band transition, λ = λV= ± and λ = + refer to the conduction band and λ = − to the valence band and f is the radiation frequency, τ is the corresponding energy relaxation time, σ0 = e2/(2ℏ) is the universal conductance, Af = 2πm⁎f/ℏ, and G+(k, φ) = 2(k40 + 4k4) cos2ϕ − 6kk0(k20 + 2k2)[cosϕ + cos(3ϕ)] + k2k20[13 + 9cos(4ϕ) + 4cos (2ϕ)]. For inter-band transition, we have σ+−(f) ≃ 0, and σ−

+

ðf Þ =

  2σ0 f τ hf π ∞ dkk ∫0 dϕ∫0 2 S 2 πA2f K ×

G− ðk; ϕÞ ;   τ2 2πf −ℏkK =m4 2 + 1

ð2Þ

with G−(k,ϕ) = (k20 − 2k2 − 2kk0cosϕ)2sin2ϕ and S(x) = F−(−x)[1 − F+(x)]. Here Fλ(x) is Fermi distribution function for carriers. The warping of the band is ignored which is only important near zero energy. We consider that the conducting carriers are electrons with n0 ∼ 1012 cm− 2 being the electron density in the absence of the radiation field (or dark density). In the presence of the radiation field, the electron density is ne = n0 + Δne where Δne ∼ 5 × 1011 cm− 2 is the photo-excited electron density. Under the condition of the charge number conservation Δne = nh is the hole density in the presence of radiation field. Considering air/bilayer graphene/SiO2 wafer systems, we can calculate the transmission coefficient. The effect of the mismatch of the dielectric constants between graphene layers and substrate can be included [13]. We also include the effect of the broadening for the scattering states within the calculation. Such broadening is induced by the energy relaxation with relaxation time typically τ ∼ 1 ps [14]. The transmission coefficient for an air/bilayer graphene/dielectric-wafer (SiO2) system can be evaluated through [15] T ðf Þ =

rffiffiffiffiffi ε2 4ðε1 ε0 Þ2 pffiffiffiffiffiffiffiffiffiffi  ; pffiffiffiffiffi ε1 j ε1 ε2 + ε1 ε0 + ε1 σ ðf Þ =cj2

ð3Þ

where ε1 and ε2 = ε∞ are, respectively, the dielectric constants of free space and the effective high-frequency dielectric constant of the SiO2 substrate, and c is the speed of light in vacuum.

Fig. 1. Contributions from different transition channels (σλλ′) to optical conductance at a fixed temperature T = 150 K and carrier densities ne = 1.5 × 1012 cm− 2 and nh = 5 × 1011 cm− 2. Here the solid curve is the total optical conductance σ and σ0 = e2 / 2ℏ.

In Fig. 2, the optical conductance and transmission coefficient are shown as a function of radiation frequency at fixed carrier densities for different temperatures. Optical conductance and transmission depend sensitively on temperature. In the experimental work, the optical conductances measured are very different at different temperatures and the absorption windows observed experimentally are in agreement with our findings [7]. For fixed electron and hole densities, the chemical potential for electrons/holes decreases/increases with increasing temperature. Thus, due to the Pauli blockade effect, the THz absorption window shifts to higher-frequency regime as shown in Fig. 2. We also note that a sharper cut-off of the optical absorption at the window edge can be observed at a lower temperature. The transmission is very high (up to 96%) for f N 3 THz at different temperatures in graphene bilayer system. The optical conductance and transmission coefficient are shown in Fig. 3 as a function of radiation frequency at a fixed temperature and a fixed hole density nh for different electron densities ne. Because the chemical potential for electrons in the conduction band increases with electron density, the THz absorption window shifts to higherfrequency regime with increasing electron density as shown in Fig. 3. The height of the THz absorption window decreases with increasing electron density and a sharper cut-off of the optical absorption at the window edge can be observed for larger electron

3. Results and discussions In Fig. 1 we show the contributions from different transition channels to the optical conductance (or absorption spectrum) in bilayer graphene. We notice the following features: (i) The inter-band transition contributes to the optical absorption in the high-frequency regime, whereas the intra-band transitions give rise to the lowfrequency optical absorption. (ii) The optical absorption varies very weakly with radiation frequency in the high-frequency regime (f N 1000 THz), whereas the absorption depends strongly on radiation frequency in the low-frequency regime (f b 500 THz). (iii) The optical conductance in the high-frequency regime is a universal value σ0 = e2 / 2ℏ for graphene bilayer, in contrast to σ0 = e2 / 4ℏ observed for graphene monolayer [7,8]. We find that the optical conductance of graphite per graphene layer is very close to σ0 = e2 / 4ℏ, which has been demonstrated experimentally [7,8]. (iv) More interestingly, there is an absorption window in between 3–30 THz bandwidth. This absorption window is induced by the presence of the inter-band (σ−+) and intraband (σ++, σ−−) electronic transition events which require different transition energies.

Fig. 2. Optical conductance and transmission (insert) as a function of radiation frequency at the fixed carrier densities ne = 1.5 × 1012 cm− 2 and nh = 5 × 1011 cm− 2 for different temperatures T = 10 K (solid curve), 77 K (dashed curve), 150 K (dotted curve) and 300 K (dotted–dashed curve).

H.M. Dong et al. / Optics Communications 283 (2010) 3695–3697

3697

4. Conclusions

Fig. 3. Optical conductance and transmission coefficient (insert) as a function of radiation frequency at a fixed temperature T = 150 K and a fixed hole density nh = 5 × 1011 cm− 2 for different electron densities ne = 1 × 1012 cm− 2 (solid curve), 1.5 × 1012 cm− 2 (dashed curve), and 2.5 × 1012 cm− 2 (dotted curve).

We have presented a detailed theoretical study of terahertz (THz) optical properties of graphene bilayer. The optical conductance and transmission for an air/graphene/SiO2 system have been calculated in the presence of the radiation field. We have found and confirmed that: (i) the universal optical conductance σ0 = e2 / 2ℏ and high transmittance T0 ∼ 96% can be measured when the radiation frequency f N 1000 THz; (ii) the very high optical transmission of graphene bilayer demonstrates that graphene layers can be used as transparency electrode; and (iii) there is an terahertz optical absorption window in between 3 ∼ 30 THz. The depth and width of the infrared absorption window depend sensitively on temperature and dark carrier density in the sample. These results can be applied to understand and reproduce those measured experimentally. This confirms that graphene bilayer system can be used not only as advanced electronic devices but also as novel optical and optoelectronic devices such as Terahertz phototransistors. Acknowledgements This work was supported by the Chinese Academy of Sciences, Department of Science and Technology of Yunnan Province.

density. There are also very high transmissions for different carrier densities in Fig. 3. For a gate-controlled bilayer graphene placed on a dielectric SiO2 wafer, 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]. The results obtained from this study indicate that there exists a THz absorption window in the range 3–30 THz in bilayer graphene. The position and width of this absorption window depend sensitively on temperature and carrier density of the system. We note that recently Li et al. investigated experimentally the optical conductivity of such graphene system [16], and there exist obvious THz absorption windows at different gated voltages, which are in line with our results. The strong THz cut-off of the optical conductance at the edge of the absorption window can be observed for typical sample parameters at relatively high-temperatures. This finding implies that bilayer graphene systems can be applied as THz photo detectors working at relatively high-temperatures (up to room temperature) for various applications. The interesting features of the optoelectronic properties for bilayer graphene, such as THz absorption window and strong THz cut-off of the optical absorption, confirm that the graphene systems can be used as advanced optoelectronic devices such as THz photodetectors.

References [1] Y.B. Zhang, Y.W. Tan, H.L. Stormer, P. Kim, Nature (London) 438 (2005) 201. [2] Claire Berger, Zhimin Song, Xuebin Li, Xiaosong Wu, Nate Brown, Cecile Naud, et al., Science 312 (2006) 1191. [3] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson, I.V. Grigoreva, et al., Nature 438 (2005) 197. [4] Eduardo V. Castro, K.S. Novoselov, S.V. Morozov, N.M.R. Peres, J.M.B. Lopes dos Santos, Johan Nilsson, et al., Phys. Rev. Lett. 99 (2007) 216802. [5] J. Gonzάlez, E. Perfetto, J. Phys.: Condens. Matter 20 (2008) 145218. [6] Yu-Ming Lin, Keith A. Jenkins, Alberto Valdes-Garcia, Joshua P. Small, Damon B. Farmer, et al., Nano Lett. 9 (2009) 422. [7] A.B. Kuzmenko, E. van Heumen, F. Carbone, D. vander Marel, Phys. Rev. Lett. 100 (2008) 117401. [8] R.R. Nair, A.K. Geim, P. Blake, A.N. Grigorenko, K.S. Novoselov, T.J. Booth, et al., Science 320 (2008) 1308. [9] Hank Hogan, Photonics Spectra 42 (2008) 19. [10] Feng Wang, Y. Zhang, C. Tian, C. Giri, A. Zetti, M. Crommie, et al., Science 320 (2008) 206. [11] J.M.B. Lopes dos Santos, N.M.R. Peres, A.H. Castro Neto, Phys. Rev. Lett. 99 (2007) 256802. [12] Xue-Feng Wang, Tapash Chakraborty, Phys. Rev. B 75 (2007) 041404(R). [13] H.M. Dong, W. Xu, Z. Zeng, T.C. Lu, F.M. Peeters, Phys. Rev. B 77 (2008) 235402. [14] Dong Sun, Zong-Kwei Wu, Charles Divin, Xuebin Li, Claire Berger, Walt A. de Heer, et al., Phys. Rev. Lett. 101 (2008) 157402. [15] T. Stauber, N.M.R. Peres, A.K. Geim, Phys. Rev. B 78 (2008) 085402 textbf. [16] Z.Q. Li, E.A. Henriken, Z. Jiang, Z. Hao, M.C. Martin, P. Kim, H.L. Stormer, D.N. Basov, Nat. Phys. 4 (2008) 532.