Modulation of terahertz radiation from graphene surface plasmon polaritons via surface acoustic wave

We present a theoretical study of terahertz (THz) radiation induced by surface plasmon polaritons (SPPs) on a graphene layer under modulation by a surface acoustic wave (SAW). In our gedanken experiment, SPPs are excited by an electron beam moving on a graphene layer situated on a piezoelectric MoS 2 flake. Under modulation by the SAW field, charge carriers are periodically distributed over the MoS 2 flake, and this causes periodically distributed permittivity. The periodic permittivity structure of the MoS 2 flake folds the SPP dispersion curve back into the center of the first Brillouin zone, in a manner analogous to a crystal, leading to THz radiation emission with conservation of the wavevectors between the SPPs and the electromagnetic waves. Both the frequency and the intensity of the THz radiation are tuned by adjusting the chemical potential of the graphene layer, the MoS 2 flake doping density, and the wavelength and period of the external SAW field. A maximum energy conversion efficiency as high as ninety percent was obtained from our model calculations. These results indicate an opportunity to develop highly tunable and integratable THz sources based on graphene devices. Disciplines Engineering | Science and Technology Studies Publication Details Jin, S., Wang, X., Han, P., Sun, W., Feng, S., Ye, J., Zhang, C. & Zhang, Y. (2019). Modulation of terahertz radiation from graphene surface plasmon polaritons via surface acoustic wave. Optics Express, 27 (8), 11137-11151. Authors Sichen Jin, Xinke Wang, Peng Han, Wenfeng Sun, Shengfei Feng, Jiasheng Ye, C Zhang, and Yan Zhang This journal article is available at Research Online: https://ro.uow.edu.au/eispapers1/2847 Modulation of terahertz radiation from graphene surface plasmon polaritons via surface acoustic wave SICHEN JIN,1 XINKE WANG,1 PENG HAN,1,3 WENFENG SUN,1 SHENGFEI FENG,1 JIASHENG YE,1 CHAO ZHANG,2 AND YAN ZHANG1,4 Department of Physics, Beijing Key Laboratory for Metamaterials and Devices, Key Laboratory of Terahertz Optoelectronics, Ministry of Education, Beijing Advanced Innovation Center for Imaging Theory and Technology, Capital Normal University, Beijing 100048, China School of Physics and Institute for Superconducting and Electronic Materials, University of Wollongong, New South Wales 2522, Australia hanpeng0523@163.com yzhang@mail.cnu.edu.cn Abstract: We present a theoretical study of terahertz (THz) radiation induced by surface plasmon polaritons (SPPs) on a graphene layer under modulation by a surface acoustic wave (SAW). In our gedanken experiment, SPPs are excited by an electron beam moving on a graphene layer situated on a piezoelectric MoS2 flake. Under modulation by the SAW field, charge carriers are periodically distributed over the MoS2 flake, and this causes periodically distributed permittivity. The periodic permittivity structure of the MoS2 flake folds the SPP dispersion curve back into the center of the first Brillouin zone, in a manner analogous to a crystal, leading to THz radiation emission with conservation of the wavevectors between the SPPs and the electromagnetic waves. Both the frequency and the intensity of the THz radiation are tuned by adjusting the chemical potential of the graphene layer, the MoS2 flake doping density, and the wavelength and period of the external SAW field. A maximum energy conversion efficiency as high as ninety percent was obtained from our model calculations. These results indicate an opportunity to develop highly tunable and integratable THz sources based on graphene devices. We present a theoretical study of terahertz (THz) radiation induced by surface plasmon polaritons (SPPs) on a graphene layer under modulation by a surface acoustic wave (SAW). In our gedanken experiment, SPPs are excited by an electron beam moving on a graphene layer situated on a piezoelectric MoS2 flake. Under modulation by the SAW field, charge carriers are periodically distributed over the MoS2 flake, and this causes periodically distributed permittivity. The periodic permittivity structure of the MoS2 flake folds the SPP dispersion curve back into the center of the first Brillouin zone, in a manner analogous to a crystal, leading to THz radiation emission with conservation of the wavevectors between the SPPs and the electromagnetic waves. Both the frequency and the intensity of the THz radiation are tuned by adjusting the chemical potential of the graphene layer, the MoS2 flake doping density, and the wavelength and period of the external SAW field. A maximum energy conversion efficiency as high as ninety percent was obtained from our model calculations. These results indicate an opportunity to develop highly tunable and integratable THz sources based on graphene devices. © 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement


Introduction
Terahertz (THz) radiation, which describes electromagnetic waves with frequencies in the 0.1-30 × 10 12 Hz range, is one of the most important types of radiation for light sources in the fields of sensing and imaging because of promising properties that include low photon energy, broad spectral information, and high penetration of nonpolar materials [1]. At present, THz technology is widely used in fields including semiconductor science [2], noninvasive flaw detection [3], substance identification [4], and security inspection [5]. A THz radiation source with broad bandwidth, high intensity and frequency tunability is highly desirable. Several approaches, including photoconductive antennas [6], optical rectification [7], air plasmons [8], quantum cascade lasers [9], and free-electron beam excitation have been used to produce THz wave emission.
Among these approaches, free-electron THz radiation sources are of particular interest because of their high light radiation powers and continuously tunable radiation frequencies [10,11]. In contrast to traditional free-electron THz sources, in which a beam of electrons is accelerated to almost the speed of light c using an electron accelerator with large associated facilities requirements, excitation of THz radiation by a relatively low energy electron beam moving on top of graphene layers was recently proposed as a new THz source [11][12][13][14][15][16]. In this approach, surface plasmon polaritons (SPPs) with resonance frequencies in the 1-30 THz range are excited by a beam of moving electrons with speeds of less than 0.1c on top of graphene layers. The en waves when t Fig [19,20], quan des [24][25][26][27] [28]. In these studies, the SAW fields were used to generate "dynamic" grating on the metal surfaces [29][30][31] or graphene layers [32][33][34] to interact with surface plasmon or light.
In this work, we present a theoretical study of SAW-modulated THz radiation from SPP resonance in a graphene layer that has been excited using a beam of moving electrons. The system is illustrated schematically in Fig. 1(a). In our gedanken experiment, the graphene layer is aligned on an n-doped molybdenum disulfide (MoS 2 ) flake with an odd number of layers that has strong piezoelectricity properties and forms a heterostructure with the graphene layers. The graphene layer and the MoS 2 flake are laid on a quartz substrate with a dielectric constant of 0 4.2ε (where 0 ε is the permittivity of a vacuum). Application of an external SAW field to the MoS 2 flake layer causes the charge carriers of this piezoelectric semiconductor to be periodically separated in space and results in the material having the dielectric response of free electrons with the same period. By summing the dielectric responses of the ions and the SAW-modulated free electrons [35], a periodic permittivity structure is realized dynamically on the MoS 2 flake. In our system, this MoS 2 flake with periodic permittivity acts as a periodic dielectric microstructure to fold the excited graphene SPP dispersion into the center of the BZ and this leads to matching of the momentums.
To give an accurate description on the SAW-modulated THz radiations, we calculate the charge carrier distributions of the MoS 2 flake under the SAW field by self-consistently solving a drift-diffusion model that was coupled with a time-dependent continuity equation and the Poisson equation. The periodic permittivity is then obtained using the Drude model with the calculated charge distributions. The SPP dispersion curves and the power intensity of the THz radiation are calculated thereafter by solving the Maxwell equations with the boundary conditions at the interfaces between regions I, II, and III, as illustrated in Fig. 1(b). The crossing points of the SPP dispersion curve with the electron beam are folded into the cone of the light line around the center of the BZ under the applied SAW field. This results in conservation of the momentum of the SPPs on graphene and the electromagnetic wave in a vacuum, and this leads to THz wave emission. We also show that both the frequency and the intensity of the THz radiation can be tuned by varying the chemical potential of the graphene layer, the doping density of the MoS 2 flake, and the period and wavelength of the external SAW field. Additionally, a maximum conversion efficiency of as much as 0.9 can be obtained for the energy transition from the SPP resonance to THz radiation in free space.

Periodically distributed charge and dielectric response under the SAW field
The spatiotemporal distributions of the electrons ( ) , n z t and the holes ( ) , p z t on the MoS 2 flake under the applied SAW field can be described using a 1D drift-diffusion model coupled with a time-dependent continuity equation [36,37] as follows: where B k is the Boltzmann constant, T is the temperature, q is the electron charge, and n μ and p μ denote the electron and hole mobilities, respectively. The recombination rate  The built-in field can be calculated by solving the Poisson equation with the dielectric permittivity ε and the donor impurity density D N , while the piezoelectric field caused by the SAW field is written as with the SAW field wavelength SAW λ and period SAW T . The intensity of the piezoelectricity field is described using the parameter SAW A . When the spatiotemporal charge distributions modulated using the external SAW field are calculated self-consistently by solving the coupled Eqs. (1)-(4) using the parameters given in Table 1, the dielectric response of the free charges ( ) , r z t ε can be calculated approximately using the Drude model [38], given by with the electron mass m and the SAW field frequency SAW ω . Because of the donor doping of the MoS 2 flake, we only consider the dielectric response of the free electrons in the following. In principle, the mobile charge carriers in the graphene layer lead to an additional dielectric screening on the MoS 2 flake. However, due to the much faster transport speed of the electron beam comparing to the SAW field, the effect of this additional screening can be viewed as a homogeneous reduction of the relative permittivity in MoS 2 flake without breaking the periodic dielectric structures. Screening induced by the electron beam in graphene layer is therefore not taken into account in the periodic dielectric structures [39].

SPP dispersion and THz radiation with periodic dielectric structure
When an electron beam moves on top of the graphene layer, the electromagnetic fields in the vacuum region, the periodic permittivity region and the substrate (regions I, II and III, respectively, as labeled in Fig. 1 (6) and (7) with the following boundary conditions: and The electromagnetic field induced by the moving electron beam is then written as [11,15,16] ( ) , where the speed of the electron beam is 0 v . The electron conductivity of the graphene layer is then calculated using the Drude model as [16,43] ( ) where the tunable chemical potential is c μ and the electron lifetime is . τ Using the calculated electromagnetic field amplitude in region III ( 4 A ) with the boundary conditions given in Eqs. (8) and (9), the power intensity of the THz radiation is then calculated as ( ) with where i κ (i = 1, 2 and 3) denotes the equivalent wavevector of i k when folded into the center of the BZ.

Periodic dielectric structures induced by the SAW field
In Figs. 2(a) and 2(b), we plot the spatial distribution of the electron concentration and the corresponding dielectric response of the MoS 2 flakes when doped with D N = 1.0 × 10 10 cm −2 (black solid lines), 1.5 × 10 10 cm −2 (red dash-and-dotted lines), and 2.0 × 10 10 cm −2 (blue dashed lines). The amplitude, wavelength, and period of the applied SAW field were set at 8 kV/cm, 2 μm and 2 ns, respectively, in these calculations. Because of the high in-plane carrier mobility of the MoS 2 flake, the electrons and holes arrive at their equilibrium positions quickly, within 0.1 ps after application of the SAW field. The charge carriers are subsequently transported "slowly" along the z direction with the propagation of the SAW. As shown in Fig. 2(a), the electrons are localized within SAW-induced periodic "valleys" of the conduction band minimum (CBM). The dielectric responses of these periodically distributed free electrons lead to periodic permittivity in these spaces, as indicated in Fig. 2(b). Comparison of Figs. 2(a) and 2(b) shows that the "peak" permittivity values correspond to the "valleys" of the electron concentrations and vice versa, as indicated by Eq. (5). Additionally, we find that the dielectric screening effect decreases rapidly as the donor density of the MoS 2 flake increases. A negative permittivity, which corresponds to the dielectric response of the metal, is obtained when the doping density is as high as 2.0 × 10 10 cm −2 .

Dynamic
When an elec MoS 2 flake a structure indu BZ. The dispe line in the qua highlighted in (17) using the τ = 0.1 ps. T = 2 ns, and A electron beam cone of the lig and momentu emissions. As shown in Fig. 4, the radiation frequency is determined by the crossing point of the SPP dispersion curve and the electron beam. To tune this radiation frequency, we adjust the chemical potential of the graphene layer c μ over the range from 0.35 to 0.55 eV, the MoS 2 flake doping density D N from 1.0 × 10 10 to 1.4 × 10 10 cm −2 , the SAW field period SAW T from 1.0 to 1.8 ns and the SAW field wavelength SAW λ from 1 to 5μm. The SPP dispersion curves and their crossing points with the electron beam lines that were calculated using these parameters are presented in Figs. 5(a)-5(d). Because the size of the BZ varies with the different wavelengths of the SAW field, the x-axis in Fig. 5(d) is labeled with units of 2π / μm rather than 2π / λ . Figure 5 shows that the slope of the SPP dispersion curves varies with changes in the chemical potential, the donor density, and the period and wavelength of the SAW field, and this forms a crossing region with the dispersion curve of the electron beam. We labeled this crossing region as the working region of the THz radiation and have highlighted it in green.
In Figs. 6(a) and 6(b), we plotted the modulated THz radiation frequencies that were extracted from the working region by varying the wavelength and period of the SAW field. In this figure, the THz radiation frequencies were calculated using parameter sets of c μ = 0. The period and wavelength of the SAW field were fixed at 2 ns and 2 μm, respectively, in Figs. 6(a) and 6(b) by varying the SAW field propagation velocity. Figure 6(a) shows that the radiation frequencies decrease from approximately 20 THz to a few THz when the SAW field wavelength increases from 0.5 to 5 μm. The red shift in the THz emission is the result of a reduction in the size of the BZ with increasing SAW field wavelength. In contrast to Fig. 6(a), we see a blue shift in THz emission with increasing SAW field period in Fig. 6(b). This blue shift can be understood from the curves in Fig. 5(c), where the slopes of the SPP dispersion curves increase with increasing SAW T and thus shift the working region to a higher frequency range. Additionally, the blue shift in the THz radiation frequency with increases in the chemical potential of the graphene layer and the donor density in the MoS 2 flake can be understood from the curves in Figs. 5(a) and 5(b), respectively. (a) and 7(b), w wavelength an ing the same p n intensity incr g the peak valu MoS 2 flake do AW field with ue in the case o THz radiation ing the relation 6(a). Having considered modulation of the THz radiation via the SAW field wavelength, we now turn to the effect of the period of the SAW field on the THz emission. As Fig. 7(b) shows, the THz radiation intensity increases slowly when the SAW field period is less than 1.5 ns, and arrives at a peak value when SAW T increases to approximately 1.8 ns. When the chemical potential c μ and the doping density D N are reduced, a SAW field with a long period is required to obtain the peak THz radiation value. In addition, the peak THz radiation values remain nearly constant for various doping densities and chemical potentials. As indicated by Eq. (5), the permittivity of the free electrons is proportional to the square of the SAW frequency 2 SAW ω and is inversely proportional to the square of the SAW period . To obtain the value of M II ε , a SAW field with a short period is required to balance the effects of the high doping density D N and the chemical potential c μ . In Fig. 7(d), we have plotted the THz radiation intensity that was presented in Fig. 7(b) as a function of the radiation frequency by using the relationship between the radiation frequency and the SAW field period given in Fig. 6(b). Interestingly, Fig.  7(d) shows that both the intensity and the frequency of the THz radiation remain nearly constant for the various chemical potentials and doping densities. This behavior can be understood as follows. In systems with fixed chemical potential, the SAW field period changes with the variation of the doping density D N to keep the value of II ε constant and this leads to the same radiation intensity and frequency indicated by Eqs. (12)- (14) and Eq. (17). For systems with different chemical potentials, a SAW field with a long period is required to balance the reduction of the chemical potential for the peak THz radiation intensity value.
To estimate the efficiency of the conversion of the SPPs into THz radiation, we calculate the conversion efficiency  Figs. 7 and 8, we see the same tuning of both the conversion efficiency and the radiation intensity produced by variation of the wavelength and the period of the SAW field. These results indicate that the large THz radiation intensity values originate from the high efficiency of the energy conversion from the SPP resonance to the THz light. Additionally, the maximum conversion efficiency of as much as 0.9 presented in Fig. 8 indicates the feasibility of THz radiation generation using SAW field-modulated SPP resonance in graphene-MoS 2 devices. Before co between the g potential and der Waals het chemical pote model calcula systems.

Conclusio
In summary, electron beam by an external SAW field. The spatial periodic permittivity of the MoS 2 flake is obtained using the Drude model with self-consistently calculated charge carrier distributions that are modulated using the SAW field. By folding the crossing point of the SPP dispersion curve with the electron beam line in the center of the BZ to converge the momentum of the SPPs and the electron beam within the cone of the light line, the transformation of the SPPs into THz radiation is achieved. The frequency and intensity of the THz radiation can be tuned by varying the MoS 2 flake doping density, the chemical potential of the graphene layer, and the period and wavelength of the applied SAW field. Based on our calculations, a maximum conversion efficiency of as much as 0.9 is obtained for the energy transformation from the SPP resonance to the THz emission. Our results suggest an exciting opportunity for development of dynamically tunable THz sources based on SPPs in a graphene layer.