Absorption of surface acoustic waves by graphene

We present a theoretical study on interactions of electrons in graphene with surface acoustic waves (SAWs). We find that owing to momentum and energy conservation laws, the electronic transition accompanied by the SAW absorption cannot be achieved via inter-band transition channels in graphene. For graphene, strong absorption of SAWs can be observed in a wide frequency range up to terahertz at room temperature. The intensity of SAW absorption by graphene depends strongly on temperature and can be adjusted by changing the carrier density. This study is relevant to the exploration of the acoustic properties of graphene and to the application of graphene as frequency-tunable SAW devices. Copyright 2011 Author(s). This article is distributed under a Creative Commons Attribution 3.0 Unported License. [doi:10.1063/1.3608045]


I. INTRODUCTION
Since the discovery of graphene, 1 the electronic, transport, optic and optoelectronic properties of this ideal two-dimensional electron gas (2DEG) system have been widely and intensively investigated worldwide. 2Now it has become well known that graphene shows a wealth of excellent performance in all respects of electronics, optics and optoelectronics, such as high electron mobility at room temperature and high optical transparency from UV to mid-infrared bandwidth.The accumulation of these non-conventional properties has allowed us to develop graphene-based electronic and optoelectronic devices.Over the past few years, tremendous progress has been made to realize high-speed and high-frequency electronic devices such as field-effect transistors 3 and single-electron transistor. 4Moreover, as an excellent transparent conductor, graphene has been utilized as transparent electrodes to be applied in optoelectronic devices such as LCD, LED, touch screen and solar cells, where it can replace the conventional indium-tin-oxide (ITO) electrodes to improve the device performance and to reduce the device cost. 5Very recently, graphene has been proposed as transparent electrodes to be applied in infrared detectors, 6 due to its even better light transmittance in mid-infrared bandwidth. 7t present, the acoustic properties of graphene has not yet been explored considerably.We know that the non-conventional electronic and optoelectronic properties of graphene are mainly resulted from a roughly linear energy spectrum, in sharp contrast to a parabolic energy dispersion of the conventional 2DEG systems.One would expect that the linear energy dispersion in graphene can also lead to some interesting features for carriers in graphene to couple with sound waves.Moreover, in a conventional semiconductor-based electron gas the band gap is quite large so that electrons interact with sound waves most likely within the conductance or valence band.Because graphene is a gapless and massless 2DEG, 8 the electrons in graphene can absorb sound waves and are excited into higher energy states via both intra-and inter-band transitions.It is therefore of significance to look into the features of electron interaction with sound waves through different transition channels.From a fundamental point of view, it is important and necessary to examine the response of carriers in graphene to surface acoustic waves (SAWs).It is well known that SAW can couple effectively with charge carries in thin films and, hence, is an ideal and powerful tool to study acoustic properties of the 2DEG systems. 9The coupling between SAW and charge carrier in an electron gas system is induced mainly due to the presence of the piezoelectric potential which is accompanying the SAW launched on the surface of piezoelectric medium.The form of piezoelectric potential was studied theoretically for ungated 10,11 and gated surface. 12It has been shown that the screening of the gate voltage applied can reduce the amplitude of the piezoelectric potential but has very little effect on its shape. 12As a result, we are able to describe the piezoelectric potential by using a model form for both ungated and gated surface of piezoelectric medium, and an effective model potential had been used successfully to study the quantized acoustic-electric current in a narrow quasi-one-dimensional electron channel. 13Very recently, the SAW propagation in graphene was investigated theoretically 14 where Landau oscillations in SAW were predicted and examined.Furthermore, it showed experimentally that graphene can be deposited on SAW-transducers and can be applied for low-frequency (about 0.1 GHz) SAW gas sensors. 15o utilize modern SAW technology to investigate graphene, it is necessary to be aware of the frequency region where carriers in graphene can respond sensitively to SAWs.Such information was not clearly indicated in previous theoretical study. 14The prime motivation of the present study is to examine how carriers in graphene interact with SAWs via intra-and inter-band transition channels and to explore the basic acoustic properties of graphene.

II. THEORETICAL APPROACH
In this work, we develop a simple and tractable theory to study the interactions between SAWs and carriers in graphene.We consider the device structure proposed previously: 14 i) a graphene sheet is fabricated on dielectric wafer (e.g., SiO 2 ); ii) the gate voltage V g can be applied to alter charge density in graphene layer; and iii) the SAW-transducer is placed on top of the graphene sheet.When a SAW travels (taken along x direction) in a piezoelectric medium such as graphene sheet, an electric potential is accompanied which can be described as 13,17 V q x (x, t) = V q x e i(q x x−ωt) , ( with Here, ω = ω q x = v s q x is the frequency of the SAW with a wave vector q x and a corresponding amplitude A 0 , v s is the longitudinal sound velocity, κ is the dielectric constant of graphene, and e a is the effective piezoelectric modulus of the host material (i.e., graphene).Furthermore, is a device-dependent factor, where d is a distance between the transducer surface and graphene layer (see Fig. 1 in Ref. 14), A 1,2 and s are dimensionless coefficients which can be determined by the elastic constants of the host material.For graphene, the two-component electron wave function obeys the 2D Dirac equation 8 Here, σ = (σ x , σ y ) is Pauli matrices, p = ( p x , p y ) is 2D momentum operator.Thus, the wave function for an electron in graphene takes a form in a form of the raw matrix, where λ = ± refers to the conduction (+) or valence (−) band with an energy spectrum ) is the electron wave vector, γ = v F with v F 10 8 cm/s being the Fermi-velocity, r = (x, y), and φ is the angle between k and the x-direction.
In the presence of piezoelectric potential induced by SAW, the steady-state electronic transition rate in graphene can be derived using Fermi's golden rule, which reads It measures the probability for scattering of an electron from a state |λ, k to a state |λ , k due to coupling between electron and piezoelectric potential induced by the SAW.Here, only the absorption scattering is taken into consideration for electron coupling with SAWs, and From now on, we consider a n-type (or positively gated) graphene in which the conducting carriers are electrons in the absence of the excitation.With the electronic transition rate, we can calculates the electron-energy-loss rate (EELR) defined as 18 where is the energy transfer rate for an electron at a state |λ, k in case of degenerate statistics, n e is the areal electron density of graphene, f + (x) = [1 + e (x−μ e )/K B T + 1] −1 is the electron energy-distribution function (EDF) with μ e being the chemical potential (or Fermi energy) for electrons in the conduction band, f − (x) = 1 is the EDF for electrons in the valence band, and g s = 2 and g v = 2 count for spin and valley degeneracy respectively.For electron interaction with SAWs in graphene, we obtain where is the electron energy-absorption rate induced by intra-[P ++ (ω) and P −− (ω)] and inter-band [P +− (ω) and P −+ (ω)] transition events.For a n-type graphene, P −− (ω) = 0 because the valence band is fully occupied.Besides, P +− (ω) = 0 because it is less possible to excite electrons from higher energy conduction band to lower energy valence band through absorption scattering.Furthermore, we can verify easily (see the Appendix) that there is no inter-band transition channel for electron coupling with SAW in graphene.This important and interesting effect is mainly induced by the unique energy spectrum of graphene and by the requirement of momentum and energy conservations during an inter-band scattering event.Hence, P −+ (ω) = 0 as well.For intra-band transition within the conduction band, we have where we have defined dimensionless constant α = v F /v s and y ) is a dimensionless and device-dependent parameter, and 2 is a dimensionless parameter which measures a ratio between the SAW potential amplitude and the hight of the electrostatically induced potential barrier in the electronic transition channel. 12Here m 0 is the rest-mass of an electron and ρ is the areal density of graphene.And C 0 = 10 11 is a constant which guarantees 0.05 β 0 0.95. 12IG.1. Dependence of the power absorption on frequency of SAW at a fixed electron density for different temperatures as indicated.

III. NUMERICAL RESULTS
For numerical calculations, we take material parameters for graphene as: i) the longitudinal sound velocity: 19 v s = 2.1 × 10 6 cm/s and ii) the areal density: 16 ρ = 6.5 × 10 −8 g/cm 2 .In our calculations, as in Ref. 17, we only calculate the function P(ω)/β which shows the net contribution induced by electronic transition events.It is proper and convenient to study the net interaction between electrons in graphene and SAW.
Fig. 1 shows the dependence of the power absorption on frequency ω of SAW at a fixed electron density for different temperatures.We see that the intensity of the SAW absorption increases with increasing temperature.Over a wide frequency regime up to terahertz (10 12 Hz or THz), strong absorption of SAWs by graphene can be observed.In the low-frequency regime, the intensity of the absorption depends weakly on frequency at high temperatures (e.g., T = 300 K) and there is an absorption peak at low temperatures (e.g., T = 4.2 K).There is a cut-off of SAW absorption at high-frequency end and the cut-off frequency observed at THz regime moves to higher frequency with increasing temperature.In addition, a sharper cut-off of SAW absorption can be seen at a lower temperature.These features can be understood easily on the basis that the SAW absorption depends on the number of transition channels.Firstly, the absorption peak seen at low temperature comes from two competitive mechanisms.With increasing ω, more electrons can be excited from lower k or E + (k) states to higher k or E + (k) states via SAW absorption scattering and more absorption events can happen in the low-frequency regime according to the laws of momentum and energy conservations.However, the momentum or wave vector for SAWs increases rapidly with increasing frequency ω.This can limit the number of transition channels required by the momentum and AIP Advances 1, 022146 (2011) FIG. 2. Power absorption as a function of SAW frequency at room temperature for different electron densities as indicated.energy conservation laws.Thus, at higher frequencies the absorption intensity decreases with ω.Secondly, when ω excesses a critical value there is no channel allowed by the momentum and energy conservation laws for SAW absorption, namely the cut-off of absorption is observed.Thirdly, with increasing temperature at a fixed electron density, the electrons disperse to wider frequency range due to thermal broadening.As a result, the number of transition channels increase in the wider frequency range with increasing temperature.Therefore, the amplitude and frequency range of absorption increase with temperature and the blue-shift of the cut-off can be observed.Besides, with increasing temperature, the absorption of low frequency SAW increases more rapidly because less momentum or energy exchange for the low-frequency absorption scattering is required.A direct consequence of this effect is that the intensity of SAW absorption depends weakly on frequency before the absorption starts to decrease at high-frequency regime where the reduction of transition channels is caused by the momentum and energy conversation laws.It should be noted that because of high longitudinal sound velocity v s in graphene, the frequency range of SAW absorption is very wide and the strong absorption can occur at relatively high frequencies, in comparison with conventional semiconductor-based 2DEG systems. 17This important nature of absorption can serve to fabricate graphene-based superhigh frequency or hypersonic SAW transducer. 20n Fig. 2, we show the dependence of the power absorption by graphene on frequency ω of SAW for different electron densities at room temperature.It is known that the chemical potential or Fermi energy in graphene increases with increasing electron density at a fixed temperature.This means the higher energy states are occupied by electrons with increasing electron density.Because the electronic transitions accompanied by SAW absorption are achieved from occupied states to unoccupied states, a larger electron density implies that a larger momentum or energy exchange is needed for electronic transition.As a result, the intensity of SAW absorption per electron decreases with increasing electron density and the blue-shift of the cut-off frequency for SAW absorption can be observed.It should be noted that the results shown in Fig. 2 are for power absorption per electron.The total absorption power (i.e., ∼ n e P(ω)) for a sample should increase with increasing electron density.Therefore, our results show that the intensity of SAW absorption by graphene depends sensitively on electron density.Because graphene on dielectric wafer is naturally a high carrier density 2DEG system, our results indicate that graphene can interact strongly with SAWs via piezoelectric coupling.

IV. CONCLUSIONS
In this work we have studied the acoustic properties of graphene through examining the features of electron interactions with SAWs in graphene.We have found that because of the unique electronic band structure for graphene, SAW absorption cannot be achieved via inter-band electronic transition channel due to the requirement of momentum and energy conservation laws.For graphene, strong absorption of SAWs can be observed up to THz frequencies at room temperature and the absorption intensity increases with temperature.We have demonstrated theoretically that the absorption spectrum for SAWs can be adjusted by changing the electron density in graphene.Because the carrier density in graphene can be controlled easily and efficiently through, e.g.varying the gate voltage, 21 our results suggest that graphene can be utilized to realize very high frequency (up to THz) SAW devices working at room temperature.We hope this work can shed some lights on the application of graphene as advanced acoustic devices.