Multilayer-graphene-based amplifier of surface acoustic waves Stanislav O Yurchenko, Kirill A Komarov, and Vladislav I Pustovoit Citation: AIP Advances 5, 057144 (2015); doi: 10.1063/1.4921565 View online: http://dx.doi.org/10.1063/1.4921565 View Table of Contents: http://scitation.aip.org/content/aip/journal/adva/5/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Surface acoustic wave micromotor with arbitrary axis rotational capability Appl Phys Lett 99, 214101 (2011); 10.1063/1.3662931 A novel compensation method of insertion losses for wavelet inverse-transform processors using surface acoustic wave devices Rev Sci Instrum 82, 115003 (2011); 10.1063/1.3663073 Damascene technique applied to surface acoustic wave devicesa) J Vac Sci Technol B 25, 271 (2007); 10.1116/1.2404684 High-resolution imaging of a single circular surface acoustic wave source: Effects of crystal anisotropy Appl Phys Lett 79, 1054 (2001); 10.1063/1.1394170 Visualization of 10 μm surface acoustic waves by stroboscopic x-ray topography Appl Phys Lett 73, 2278 (1998); 10.1063/1.121701 All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 131.170.6.51 On: Fri, 10 Jul 2015 21:07:18 AIP ADVANCES 5, 057144 (2015) Multilayer-graphene-based amplifier of surface acoustic waves Stanislav O Yurchenko,1,a Kirill A Komarov,1 and Vladislav I Pustovoit2 Bauman Moscow State Technical University, 2-nd Baumanskaya str 5, Moscow 105005, Russia Scientific and Technological Center of Unique Instrumentation, Russian Academy of Sciences, Moscow, Russia (Received March 2015; accepted 11 May 2015; published online 19 May 2015) The amplification of surface acoustic waves (SAWs) by a multilayer graphene (MLG)-based amplifier is studied The conductivity of massless carriers (electrons or holes) in graphene in an external drift electric field is calculated using Boltzmann’s equation At some carrier drift velocities, the real part of the variable conductivity becomes negative and MLG can be employed in SAW amplifiers Amplification of Blustein’s and Rayleigh’s SAWs in CdS, a piezoelectric hexagonal crystal of the symmetry group C6v , is considered The corresponding equations for SAW propagation in the device are derived and can be applied to other substrate crystals of the same symmetry The results of the paper indicate that MLG can be considered as a perspective material for SAW amplification and related applications C 2015 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License [http://dx.doi.org/10.1063/1.4921565] I INTRODUCTION The interaction between acoustic waves and carrier currents is one of the central problem in applied physics The reason for its appeal is the wide range of acousto-electronic devices and their applications Therefore, the study of solid-state excitations in the presence of a carrier current is an important physical and technical problem In the presence of a carrier current, acoustic waves in solids can be amplified.1,2 The nature of this phenomenon is related to Cherenkov’s emission of phonons by moving carriers.3 A surface acoustic wave (SAW) can be amplified if the carriers move near the crystal surface and the carrier drift velocity is more than the SAW velocity.4 The amplification arises at high currents; therefore, the carrier mobility in perspective materials for SAW amplifiers should be high Graphene, a one-layer modification of carbon, is one of such new material with high-mobility carriers.5,6 The carrier energy spectrum is linear, ϵ = pvF , where ϵ and p are the carrier energy and momentum, respectively, and vF = 108 cm/s is the characteristic Fermi velocity Owing to the form of the spectrum, there are many features defining the charge transport,7,8 in particular, the high-current transport.9–11 A series of related studies was performed recently to understand (i) the conductivity of graphene and its carrier dynamics,12–17 (ii) plasmonic effects,18–21 (iii) the own phonon generation by the carrier current,22–25 (iv) the interaction between graphene electrons and SAWs,26–29 and (v) sandwich-like “graphene-piezoelectric” structures, which allow one to create a new class of opto-acousto-electronic devices.30–34 Nevertheless, the problem of modeling SAW amplification in graphene-based SAW amplifiers has not been studied systematically to date In this paper, we present the results of modeling SAW amplification in a multilayer graphene (MLG)-based amplifier We discuss the variable conductivity of unipolar degenerated carrier plasma in graphene in the presence of a constant carrier drift motion At some carrier drift velocity vd , the real part of the conductivity of the graphene layer becomes negative, and it can be employed as an active element of a SAW amplifier In the hydrodynamic regime, this condition is vd > vs = ω/q a Electronic mail: st.yurchenko@mail.ru 2158-3226/2015/5(5)/057144/12 5, 057144-1 © Author(s) 2015 All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 131.170.6.51 On: Fri, 10 Jul 2015 21:07:18 057144-2 Yurchenko, Komarov, and Pustovoit AIP Advances 5, 057144 (2015) (where vs is the SAW phase velocity); in the collisionless regime, the condition is 2vd > vs because of the carrier spectrum To increase the device efficiency, we propose to use MLG in a sandwich-like structure We consider the general problem statement and characteristic examples of amplified Blustein’s and Rayleigh’s SAWs in CdS, a piezoelectric hexagonal crystal of the symmetry group C6v In conclusion, we discuss the advantages of MLG structures for the applications II CONDUCTIVITY OF GRAPHENE CHARGE CARRIER PLASMA IN EXTERNAL ELECTRIC FIELD Let us consider a two-dimensional massless graphene carrier plasma that can be described using Boltzmann’s equation:21,35 ∂f ∂f ∂f + vα + Fα = Stcc{ f } + Sti { f }, ∂t ∂ xα ∂pα (1) where f is the carrier distribution function; vα , pα , and Fα are the carrier velocity, momentum, and force components in the α direction (α = x, y, Cartesian coordinates), respectively; Stcc{ f } and Sti { f } are collision integrals corresponding to carrier–carrier and carrier–impurity (or carrier– phonon) interactions; and summation over repeated indices is performed We have used the problem statement for the collision integrals given in Ref 35 We consider an external longitudinal force as α are the constant and variable terms of the longitudinal α ), where Eα0 and E the sum Fα = e(Eα0 + E electric field, respectively; e = ±|e| is the carrier charge; and |e| is the absolute electron charge In the presence of the weak longitudinal electric field E0, carrier drift appears; thus, we find the distribution function as the following expansion: f = f + f + f, (2) where f is the Fermi–Dirac distribution function corresponding to the system ground state, and f respectively and fare the distribution function perturbations caused by the electric fields E0 and E, The expansion in Eq (2) is appropriate until the ratio of the carrier drift velocity vd to the Fermi velocity vF is small and the carrier plasma perturbations generated by fare small compared with those generated by f 1; thus, the collision integrals on the right-hand side of Eq (1) can be linearized The main purpose of the expansion in Eq (2) is to find the perturbation f asa function of the because the nonequilibrium current density electric field E, jγ is defined as jγ = vγ fdΓ, and we have the following relation between the current and the electric field: α , jγ = σαγ E (3) where σαγ is the conductivity of the moving carrier plasma in the external electric field The derivation of the conductivity σαγ is described in detail in Appendix A We obtained the following expression for the conductivity: qβVγ ∆ βγ = δ βγ − , σα β = aαγ ∆−1 βγ , ω iτp qβ v β0 vγ0 ∂ f vα = aαγ = −e2 dΓτp × vγ + (1 + Q) + (1 + Q) ∂ε Q Q Q iτp qβ v β0 vγ0 dϕ vα = σ0 vγ + (1 + Q) + (1 + Q) , π Q Q Q iτ q v v p β β ∂ f0 γ vγ + + dΓ = Vγ = − √ ∂ϵ Q Q n0 Q 0 iτ q v vγ dϕ p β β = vγ + + , Q Q 2π Q (4) where σ0 = τ µ is the constant conductivity in taken units, because σα β | v=0,ω=0 = σ0δα β ; ∆−1 βγ is the tensor inverse to ∆ βγ All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 131.170.6.51 On: Fri, 10 Jul 2015 21:07:18 057144-3 Yurchenko, Komarov, and Pustovoit AIP Advances 5, 057144 (2015) Consider the longitudinal SAW, q ∥ E0 Then we have the following asymptotic expressions from Eq (4): −1 (qℓ)2 qvd −i , qℓ ≪ : σ ∥ = σ0 − ω 2ωτ ( ) 2σ0 (ωτ)2 2qvd qℓ ≫ : σ∥ = − + iωτ , qℓ ω (qℓ)2 (5) where ℓ = τvF is the carrier free-path length, and the drift carrier velocity vd is related to the carrier current, vd = j0 τE = √ n0 n0 The first and second limits in Eq (5) are with respect to the hydrodynamic and collisionless regimes, respectively The real part of the variable conductivity Re( σ ∥) in Eq (5) becomes negative when vd > vs in the hydrodynamic regime (see Fig 1) and when 2vd > vs in the collisionless regime In contrast to the case for the usual massive electron gas, the critical drift velocities are not identical owing to the linear carrier energy spectrum Figure shows the typical dependence of the conductivity on the drift velocity in the hydrodynamic regime, qℓ ≪ in Eq (5) The minimum corresponds to the following condition: ( )2 ( ) qℓ 2 vs vd vd = v s + ⇐⇒ ωmax ≃ −1 , (6) 2τ τ vF vs where we introduced the velocity of an external perturbation wave vs = ω/q To estimate the frequency ωmax, we take typical values for the parameters used: τ ≃ 10−13 s and vs ≃ 2.5 × 105 cm/s For one graphene layer, n0 ≃ (1012 1013)cm−2, and the current density can reach the value10,21 JG ≃ 10 A/cm, =⇒ vd = JG ≃ 107 cm/s en0 We see that the drift velocity in graphene can be up to ∼ 20 times sound velocity Even at high currents, vd /vF ≪ 1, and the expansion in Eq (2), and thus the result (4), are correct Further, for FIG Real part of graphene carrier conductivity in the hydrodynamic limit, qℓ ≪ 1, at the frequency ω = GHz The real part of the conductivity becomes negative when the carrier drift velocity exceeds an external perturbation velocity All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 131.170.6.51 On: Fri, 10 Jul 2015 21:07:18 057144-4 Yurchenko, Komarov, and Pustovoit AIP Advances 5, 057144 (2015) the frequency maximum ωmax and the free-path length ℓ, we find ωmax ≃ 1010 Hz, ℓ ≃ 10−5 cm Therefore, we can use the hydrodynamic approximation for ω ≃ GHz because ql ≃ 10−1 in this case Below we consider an MLG-based SAW amplifier When MLG structures are used,36,37 the number of carriers amplifying the SAW is much greater than in MLG, and the expected efficiency of the device increases III SURFACE ACOUSTIC WAVE AMPLIFICATION A Problem statement As we have discussed in Sec II, when the real part of the variable conductivity becomes negative, perturbations due to an external electric field can be amplified by the graphene carrier current, for example, if the graphene sheets are placed on a piezoelectric crystal Figure illustrates the concept of the SAW amplifier discussed here: the MLG structure (MLG) is placed on a piezoelectric crystal (PE); the carrier drift velocity is controlled by the source (S) and drain (D) The input and output transducers are needed to generate and register the SAW To estimate the device efficiency, we should find the SAW gain increment Small elastic perturbations and fields induced in the crystal are described by the equations3 uăi = cik l m ∇k ul m − el,ik ∇k El , ε ik ∇i Ek + ei,k l ∇i uk l = 0, (7) where u j are the crystal displacement components, Ek = −∇k ϕ are the electric field components, ϕ is the electric potential, ρ is the crystal density, ciklm is the elasticity tensor, el, j k is the piezoelectric tensor, ∇ j is the derivative with respect to the coordinate x j , and u j k represents the small deformation tensor, u j k = 21 ∇ j uk + ∇k u j The electric field in the MLG structure is described by Poisson’s equation: ∆ϕ = − (n N − n0)δ(z − N d), ε0 N (8) FIG (a) MLG-based SAW amplifier and (b) calculation scheme PE is the piezoelectric substrate, and MLG is the MLG structure A SAW propagates along the surface in the x direction, and the carrier drift velocity lies in the same direction, from the source S to the drain D All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 131.170.6.51 On: Fri, 10 Jul 2015 21:07:18 057144-5 Yurchenko, Komarov, and Pustovoit AIP Advances 5, 057144 (2015) where ε is the vacuum dielectric constant; N and n N are the sheet number and the carrier density in this sheet, respectively; and d is the distance between neighboring MLG sheets We can rewrite the right-hand side of Eq (7) approximately when λ ≫ d, where λ is the characteristic spatial length of charge density perturbations (SAW wavelength), because N f N (x, y)δ(z − N d) ≃ f (x, y, z)/d, and thus ∆ϕ = − (n − n0) ε 0d (9) In addition, the continuous equation for MLG is ∂n + ∇j = 0, (10) ∂t where the conductivity of the MLG structure is defined by Eq (4) and is considered to be zero in the normal direction to the MLG sheet The latter is the main difference between MLG structures and typical semiconductors, where weak current perturbations appear in the normal direction The boundary conditions in the contact zone between the crystal and the MLG structure are (2) (1) D (1) = ϕ(2), σ (1) n = 0, j n j = D j n j, ϕ jk j (11) where the indices and correspond to the crystal and the MLG, respectively; n j is the normal vector The dielectric displacement D j and stress tensor σik are introduced as Di = ε ik Ek + ei,k l uk l , σik = cik l mul m − el,ik El (12) In addition, we suppose that the MLG has no rigid mechanical coupling with the PE substrate Using Eqs (7) and (9)–(12) and the usual procedure of perturbation theory, we can find the dispersion relation ω(q) for small periodic perturbations as a function of the external electric field To demonstrate SAW amplification in the proposed amplifier, we consider a piezoelectric hexagonal crystal of the symmetry group C6v (CdS, ZnO) as the substrate and the propagation of two characteristic examples of SAWs, Blustein’s and Rayleigh’s SAWs The detailed equations, taking into account the specifics of the tensors owing to the crystal symmetry, and the corresponding calculations are presented in Appendix B B Results and discussion As a substrate material, we studied CdS,38 which has the following elastic constants (measured in 1010 N/m2): c11 = 8.380, c13 = 4.500, c44 = 1.577, c33 = 9.653; the piezoelectric tensor constants are (C/m2): e31 = −0.262, e33 = 0.385, e15 = 0.183; the dielectric permittivity constants are ε 11 = 8.67ε 0, ε 33 = 9.53ε We considered SAWs in the gigahertz range because the SAW damping is negligible According to Eq (5), we used the hydrodynamic limit expression for the MLG conductivity The following constants were used for the MLG conductivity calculation:9,11,13 σ0 = mSm/m (for one layer), τ = 100 fs, and d = 0.35 nm We calculated the real and imaginary parts of the SAW wavevector q as a function of the frequency and carrier drift velocity Negative values of Im[q(ω, vd )] indicate SAW amplification by the MLG carrier current Therefore, to estimate the efficiency of the amplifier, we used the following dimensionless parameter: α = −Im[q(ω, vd )]/q0(ω), where q0(ω) is the SAW wavevector without the MLG structure Figure 3(a) and 3(b) present the numerical results of the α parameter calculation for various frequencies and carrier drift velocities for Blustein’s and Rayleigh’s SAWs, respectively The amplification appears first for Rayleigh’s SAW [Fig 3(b), low drift velocities] and then for Blustein’s SAW [Fig 3(a), high drift velocities] We see that the parameter α has the expressed maximum, which corresponds to the most effective work regime of the amplifier, for each type of SAW The existence of maxima of α indicates3 that at some carrier drift velocity, the amplifying structure becomes a generator of SAWs Overall, the observed solution structure is similar for Blustein’s and Rayleigh’s SAWs Comparing the results in Fig 3(a) and 3(b), we see that Rayleigh’s SAW All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 131.170.6.51 On: Fri, 10 Jul 2015 21:07:18 057144-6 Yurchenko, Komarov, and Pustovoit AIP Advances 5, 057144 (2015) FIG Amplification parameter α(ω, v d ) for SAWs in CdS: parametric plots of the amplification parameter α versus the SAW frequency ω and the carrier drift velocity v d for (a) Blustein’s and (b) Rayleigh’s SAW; numbers near the contour lines indicate the α value is amplified more The main reason for this is that Rayleigh’s SAW is strongly localized near the crystal surface and therefore has less kinetic energy and can be amplified more intensively (αmax ≃ 1.75 × 10−2 dB for Blustein’s SAW, and αmax ≃ 11.2 × 10−2 dB for Rayleigh’s SAW) At the same time, the velocity of Rayleigh’s SAW is less than that of Blustein’s SAW Therefore, the Rayleigh’s wave is amplified first if the carrier drift velocity vd grows monotonically IV CONCLUSION We studied the conductivity of a graphene charge carrier plasma in the presence of external longitudinal electric fields and the amplification of SAWs in a piezoelectric crystal using MLG structures We found that the variable conductivity of graphene becomes negative when the carrier drift velocity exceeds the SAW velocity We discussed the MLG-based SAW amplifier scheme, a general problem statement and two characteristic examples—Blustein’s and Rayleigh’s SAWs—in CdS, a piezoelectric hexagonal crystal of the symmetry group C6v For the hydrodynamic regime (low frequencies), the condition under which the variable conductivity of graphene becomes negative is the same as that for a massive carrier plasma in All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 131.170.6.51 On: Fri, 10 Jul 2015 21:07:18 057144-7 Yurchenko, Komarov, and Pustovoit AIP Advances 5, 057144 (2015) typical semiconductors, but in the collisionless regime (high frequencies), the condition is half as strong because of the linear carrier energy spectrum The results are valid for topological insulators,39 the carrier spectra of which have a linear form, similar to that of graphene electrons In typical amplifying semiconductors, the electric field near the surface substrate induces normal currents in the semiconductor On the contrary, the normal current in the amplifying layer is zero for the MLG carrier plasma Consequently, the screening of the electric field and the corresponding damping are smaller in MLG structures, and because of the high carrier mobility, the MLG structures are perspective for SAW amplifiers The carrier density can be constant (chemically doped MLG, as we considered in this paper), but it can be changed by using a gate (electric doping); however, this case is outside the scope of the paper and should be considered in other work Another advantage of using MLG structures as SAW amplifiers arises from the high own sound velocity in graphene, which should give MLG-based amplifiers a large signal-to-noise ratio By using MLG structures, more complicated amplifying devices can be constructed on the basis of the discussed scheme, for example, meander-like amplifiers with a controlled SAW gain increment in two perpendicular directions We considered the interaction between the MLG carriers and the SAW due to induced electric fields in a piezoelectric crystal The mechanism of interactions between electrons (holes) and the SAW can also be provided by the deformation potential, and in this case the models proposed for Rayleigh’s and Blustein’s SAWs in the considered crystals are applicable Our results showed that the use of MLG structures has some advantages over the use of typical semiconductors Thus, we believe that the results will be useful for a wide range of studies of new acousto-electronic devices based on novel materials and physical principles ACKNOWLEDGMENTS The work of S.O.Y and K.A.K was supported by the Russian Scientific Foundation (Project No 14-29-00277) APPENDIX A: CONDUCTIVITY OF MOVING CARRIER PLASMA In Appendix A, we derive the expression in Eq (4) for the conductivity of a moving carrier plasma To find the conductivity under an external electric field, we used the following approach.2,3 The ground state of the carrier gas without any external forces is described by the Fermi–Dirac distribution ( ϵ − µ ) −1 µ f = + exp , ≫ 1, T T where µ and T are the chemical potential and temperature, respectively, and Stcc{ f 0} = The chemical potential defines the equilibrium carrier density n0 as follows: d 2p √ n0 = dΓ f = µ, dΓ = × , (A1) (2π )2 where the factor of four arises because there √ are two possible spins and valleys for the carrier Here we used the normalization vF = 1, = 1/ π, and e = The collision integral on the right-hand side of Eq (1) can be linearized as35,40 Sti { f } = −τp−1( f − f 0s ), f 0s = f 0(µ), (A2) where the function f 0s is the symmetrical part of the carrier distribution function in the presence of an external electric field, and τp−1 is the momentum relaxation rate (see, for example, the calculations in Ref 40) Using the linearization in Eq (A2), we suppose that the characteristic frequency of −1 carrier–carrier collisions, τcc , is much greater than that of collisions with impurities or phonons, τci−1: −1 τcc ≫ τci−1 All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 131.170.6.51 On: Fri, 10 Jul 2015 21:07:18 057144-8 Yurchenko, Komarov, and Pustovoit AIP Advances 5, 057144 (2015) For the perturbed chemical potential µ = µ0 + µ, using Eq (A1) at µ ≪ µ0, we have the following expansion: f 0s = f + ∂ f0 n ∂ f0 µ = f0 − √ ∂µ n0 ∂ϵ (A3) Suppose that all the perturbations are periodic: n ∝ exp(iωt − iqα x α ), µ, f, E, where ω and qα are the frequency and wavevector in the graphene plane, respectively A substitution of Eqs (2) and (A2) into Eq (1) gives the following equation hierarchy: Eα0 ∂ f0 = −τp−1 f 1, ∂pα i(ω − qα vα ) f+ Eα0 ∂ f ∂ f + Eα = ∂pα ∂pα ) ( n ∂ f0 = −τp−1 f+ √ n0 ∂ϵ (A4) With an accuracy up to the second order of Eα0 , the solution of Eq (A4) is f = −τp Eα0 ∂ f0 , ∂pα ( ) α ∂ f iqβ v β0 τp E + vα + − vα − Q ∂ϵ Q Q2 ) α vα v β E ( τp2 E ∂2 f β + 1+ − Q Q ∂ϵ ( ) iqα vα0 τEα0 vα ∂ f n ∂ f0 1− − √ − , n0 Q ∂ϵ Q2 Q2 ∂ϵ f = (A5) where Q = + iωτp − iqα vα τp In Eq (A5) we have introduced the drift velocity field v β0 as v β0 = τp Eα0 mα−1β , (A6) where mα−1β is the carrier inverse mass tensor: mα−1β ( ) δ α β − vα v β pα pβ ∂ 2ϵ = δα β − = = ∂pα ∂pβ p p p We see that for graphene carrier plasma, the inverse mass tensor is not diagonal because of the energy spectrum linearity This feature leads to the different critical drift velocities in the hydrodynamic and collisionless regimes [see Eq (5)] The average current density follows from Eq (A5): jα0 = vα f 1dΓ = τ µEα0 , where τ = τp | p=µ (in taken units ϵ = p), and we accounted for the degenerate state ∂ f 0/∂ϵ ≃ −δ(ϵ − µ), where δ(z) is the Dirac delta function The distribution function perturbation f defines the nonequilibrium current density jγ by the simple relation jγ = vγ f dΓ The right-hand side of Eq (A5) includes terms resulting from the carrier density and electric field perturbations; therefore, it can be written as α + jγ = aαγ E nVγ , (A7) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 131.170.6.51 On: Fri, 10 Jul 2015 21:07:18 057144-9 Yurchenko, Komarov, and Pustovoit AIP Advances 5, 057144 (2015) where the tensor aαγ and the vector Vγ are derived from Eq (A5) Finally, using the continuous equation jα ∂ n ∂ + = 0, ∂t ∂ xα (A8) we found from Eq (A7) the variable conductivity σα β in Eq (4) APPENDIX B: DISPERSION RELATIONS FOR SAW IN CRYSTALS OF THE SYMMETRY GROUP C6v For crystals of the symmetry group C6v , some components of the tensor of the elastic modulus, the piezoelectric tensor, and the dielectric permittivity are zero: c11 c12 c13 0 c12 c11 c13 0 c13 c13 c33 0 0 c44 0 0 c44 0 0 el,ik = 0 e31 0 e31 0 e33 e15 e15 0 ε ik = ε 11 0 ε 11 0 , ε 33 cik l m = 0 , 0 c11 − c12 0 , (B1) where the indices 1,2, and denote the Cartesian coordinates x, y, and z, respectively Dispersion relation for Rayleigh’s SAW In Rayleigh’s SAW, we have the following conditions for the crystal displacements: u x (x, z,t) 0, u y (x, z,t) = 0, uz (x, z,t) Using Eqs (7)–(10) in this case, we obtain y < (PE) : ∂ 2u x ∂ 2uz ∂ 2u x ∂ ϕ1 + (c13 + c44) + c44 + (e15 + e31) , ∂ x∂z ∂ x∂z ∂x ∂z ∂ 2uz ∂ 2u x 2uz uă z = c33 + (c13 + c44) + c44 + e15 + e33 ∂ x∂z ∂z ∂x ∂x ∂z ( ) ∂ ϕ1 ∂ ϕ2 ∂ 2u x ∂ 2uz ∂ 2uz ε 11 + ε 33 − (e15 + e31) − e15 − e33 = 0, ∂ x∂z ∂x z x z uă x = c11 (B2) y > (MLG) : ∂n ∆ϕ2 = − (n − n0), − ∇(σ∇ϕ2) = 0, ε 0d ∂t ∂ϕ1,2 y → ±∞ : → 0, u x, z → 0, ∂y y = : ϕ1 = ϕ2, D1y = D2y , σ y z = 0, σ y y = 0, where we have written c = c44, e = e15, and ε = ε 11ε All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 131.170.6.51 On: Fri, 10 Jul 2015 21:07:18 057144-10 Yurchenko, Komarov, and Pustovoit AIP Advances 5, 057144 (2015) In the MLG area, we find a solution in the form 2 ϕ F1 = ei(q x−ωt)e−κq y , n F2 (B3) where F1 and F2 are constants Substitution of Eq (B3) into Eq (B2) gives κ2 = + i σ∥ , ε 0ωd F2 = − iq2 σ∥ F1 ω (B4) In the crystal, we find an elementary solution for the displacements and electric potential perturbation in the form ux A j q−1 uz = ei(q x−ωt) B j q−1 e χ jq y, −1 j 1 ϕ C j e15(ε 0q) (B5) where A j , B j , and C j are constants; χ i are the roots of the following equation written in a matrix determinant form: c11 ρω2 − + χ2 c44q ( c44 ) c13 iχ 1+ c44 ( ) e31 i 1+ χ e15 ( ) c13 iχ 1+ c44 ρω2 c33 −1+ χ c44 c44q e33 χ −1 e15 e15(e15 + e31) ε 0c44 e15(e33 χ2 − e15) = 0, ε 0c44 ε 11 − ε 33 χ2 ε0 iχ (B6) and we should take only the roots having a positive real part, Re[ χ i ] > The constants B j and C j in Eq (B5) are proportional to A j because of the correspondence to the eigenvectors for exponents e χ jq y: ∆ Bj Aj, ∆j Bj = Cj = ∆Cj Aj, ∆j (B7) where ( i χj ∆j = ∆ Bj = ∆Cj = c13 1+ c44 ) e15(e15 + e31) ε 0c44 , e15(e31 χ2j − e15) i χj ρω2 c33 χ −1+ c44 j ε 0c44 c44q2 e (e15 + e31) ρω2 c11 15 + − χ2j i χ j − c ε 0c44 c44q 44 , ( ) e15(e31 χ2j − e15) c13 −i χ j + c44 ε 0c44 ) ( c13 ρω2 c11 i χj + − + − χ2j c44 c44q2 c44 ( ) ρω2 c33 c13 − + χ −i χ + j c44 j c44 c44q2 (B8) Substituting Eqs (B5), (B3), and (B4) into the boundary condition in Eq (B2) at y = 0, we find the dispersion relation as the matrix M determinant: All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 131.170.6.51 On: Fri, 10 Jul 2015 21:07:18 057144-11 Yurchenko, Komarov, and Pustovoit AIP Advances 5, 057144 (2015) det Mi j = 0, j M1 j = i ∆j e15e33 ∆C c13 , + χ j Bj + χ j c33 ∆ c33ε ∆ j e2 ∆ ∆ Bj M2 j = χ j + i j + i 15 Cj , ∆ c44ε ∆ ( ) j e15∆Cj e33∆ B ε 33 M3 j = i + χ j − κ + χ j e31∆ j ε0 e31∆ j j (B9) Solving Eqs (B6) and (B9) numerically, we find the dispersion relation ω(q) Dispersion relation for Blustein’s SAW In Blustein’s SAW, we have the following conditions for crystal displacements along the surface: u x (x, y,t) = 0, u y (x, y,t) = 0, uz (x, y,t) The problem statement in Eqs (7) and (9)–(12) has the following form: y < (PE) : y > (MLG) : y → ±∞ : y=0: uă z = cuz + e1, euz ε∆ϕ1 = 0, ∆ϕ2 = − (n − n0), ε 0d ∂n + ∇j = 0, j = −σ∇ϕ2, ∂t ∂ϕ1,2 → 0, uz → 0, ∂z ϕ1 = ϕ2, D1y = D2y , σ y z = 0, (B10) ∂ ∂ where ∆ = ∂x + ∂ y ; 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Center of Unique Instrumentation, Russian Academy of Sciences, Moscow, Russia (Received March 2015; accepted 11 May 2015; published online 19 May 2015) The amplification of surface acoustic waves. .. of surface acoustic waves (SAWs) by a multilayer graphene (MLG) -based amplifier is studied The conductivity of massless carriers (electrons or holes) in graphene in an external drift electric