Acoustoelectric current in graphene nanoribbon due to Landau damping

We perform self-consistent analysis of the Boltzmann transport equation for momentum and energy in the hypersound regime i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ql \gg 1$$\end{document}ql≫1 (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}q is the acoustic wavenumber and l is the mean free path). We investigate the Landau damping of acoustic phonons (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$LDOAP$$\end{document}LDOAP) in graphene nanoribbons, which leads to acoustoelectric current generation. Under a non-quantized field with drift velocity, we observed an acoustic phonon energy quantization that depends on the energy gap, the width, and the sub-index of the material. An effect similar to Cerenkov emission was observed, where the electron absorbed the confined acoustic phonon energy, causing the generation of acoustoelectric current in the graphene nanoribbon. A qualitative analysis of the dependence of the absorption coefficient and the acoustoelectric current on the phonon frequency is in agreement with experimental reports. We observed a shift in the peaks when the energy gap and the drift velocity were varied. Most importantly, a transparency window appears when the absorption coefficient is zero, making graphene nanoribbons a potential candidate for use as an acoustic wave filter with applications in tunable gate-controlled quantum information devices and phonon spectrometers.

www.nature.com/scientificreports/ by the deformation of the material caused by the intraband transitions of electrons under an applied field 31 . Theoretical and experimental studies of acoustic wave effects in graphene [32][33][34][35] , quantum wells (QWs) 36 , carbon nanotubes [37][38][39][40][41] , and rectangular quantum wires 42 have been investigated in the megahertz (MHz), the gigahertz (GHz) and the terahertz (THz) regions. Thalmeier et al. 43 observed Landau oscillations as a function of gate voltage in graphene. Zhang et al. 44 obtained a strong absorption when the carrier density and the field were increased as a result of electrons colliding with the acoustic phonons under a drift electric field. Such interaction can also generate sound waves. Considering acoustic phonons as quantized sound waves of frequency ( ω q ), the conducting electrons can absorb the sound energy. This leads to damping of the acoustic phonons and, subsequently, the production of acoustoelectric current. This form of damping is referred to as Landau damping of acoustic phonon due to Cerenkov emission (LDOCE). LDOCE occurs when the drift velocity is less than the speed of sound in a material. The absorption of the phonon energy is determined by the energy balance of the system. As the frequency of the acoustic phonon increases, the absorption also increases, until there is a resonance beyond which the absorption decreases. This phenomenon has been observed in several graphenebased AE experiments and has been used in the fabrication of sensing devices such as humidity sensors 46 , photodetectors 47 , and gas sensors 48 .
The quantum Hall effect is observed when sound waves in a material are subjected to magnetic fields 45,56 . For low fields, a large attenuation occurs when the frequency of the sound is an integral multiple of the cyclotron frequency. At high fields, oscillatory attenuation resulting from geometric resonance occurs when the wavelength of sound is an integral or half-integral multiple. According to Zhang et al. 44 , the absorption in graphene depends strongly on temperature and can be adjusted by changing the carrier density, suggesting the influence of doping on the absorption of acoustic phonons. That is, doping or patterning single-layer graphene (SLG) into GNRs creates a tunable multiband absorption effects, which opens an energy gap ( ) that varies with the width (N), the quantized wave vector B and the electron momentum. B varies with the width of the GNR, the lattice constant ( a c−c ) and the sub-band index (Pi). To the best of our knowledge, there has been no theoretical investigation of LDOCE in GNR even though some experimental evidence of the phenomenon exists 35 . Poole et al. 35 reported a nonlinear AE in GNR of width N < 500 nm when stimulated with a DC current. They observed a resonance between the measured current maxima, after which the current decreased. Liang et al. 48 , Zheng et al. 49 and Okuda et al. 50 reported similar behavior, where acoustic charge transportation was induced by SAW propagation in the graphene. Morgado et al. 51 also reported negative Landau damping in bilayer graphene, where they measured a DC electric current induced by a static voltage across the graphene sheet. In this work, we study the nonlinear AE in SLG and GNR-500 with degenerate energy dispersion in the hypersound regime. We adopted the Boltzmann kinetic equation for the electron system interaction with the sound waves and calculate the absorption coefficient ( Ŵ q ) for the acoustic phonon population (N q (t)) . The phonon dynamics are assumed to distort the electron distribution function (f k ), thus affecting those electrons whose velocities in the direction of sound propagation are close to the sound velocity. Herein, we theoretically examine the effect of DC fields on LDOCE in an SLG/GNR-500.

Methods, results, and discussion
To gain insight into LDOAP, we analyzed the effect of electron-phonon interactions in a gate-controlled single SLG and GNR-500. An in-plane current I was applied along the x-direction of the graphene sheet, which was biased by the presence of the source-to-drain voltage v sd . The electronic transition rate induced by electron interaction with acoustic phonons is given by the kinetic equation for the acoustic phonon population N q (t) 10,19,29,52 , expressed as: Here, the spin and the valley degeneracies are g s = 2 and g v = 2 , respectively. C q = |�| 2 q/2ρv s , where is the deformation potential, ρ is the density of the graphene sheet, and τ is the relaxation constant. The factor f k 1 − f k ′ is the probability that the initial state k is occupied and the final electron state k′ is empty. f k is the unperturbed Fermi-Dirac distribution function. The factor N q f k ′ 1 − f k is that of the boson and fermion statistics, and ε k,k ′ is the energy dispersion. With A being the area of the material, the summation in Eq. (1) spans over k, k′ and can be transformed into an integral as where v f is the Fermi velocity, v s is the velocity of sound, and is Planck's constant, φ is the angle between k and the z-direction, and θ is the angle between k and k'. The energy dispersion of SLG varies linearly with k and is given as ε(k) = ± v f |k| 53 . We first analyze the effect of temperature change on the mobility of electrons in the SLG by switching off the applied voltage. This enables the study of the effect of carrier concentration under www.nature.com/scientificreports/ various temperatures, a consequence of energy conservation in the electron-phonon scattering process, The absorption coefficient relates to the AE current via the Weinrich 25 relation, as follows: Thus, the acoustoelectric current J ac can be expressed as Integrating and simplifying Eq. (5) yields From Eq. (6), J ac varies with temperature as T 4 , which, according to Mariana and Von Oppen 58 indicates the contribution of an in-plane acoustic phonon. In Fig. 1a, we show the dependence of J ac on frequency ( ω q ) at various temperatures (T = 20, 30, 50, and 70 K) using the following parameters, v F ≈ 10 8 ms −1 , τ = 5 * 10 −10 s , = 9 eV , V s = 2.1 × 10 3 cm s −1 and q = 10 5 cm −1 . The plot shows a nonlinear AE current J ac which decreases with an increase in temperature. The AE current does not exhibit a simple linear dependence on ω q , and temperature. At T = 20 K, the current initially decreased to a minimum at 2 THz, and then increased at higher frequencies. A similar trend was observed when the temperature was increased to 30K. However, at 50K and 70K, the increase in J ac is gradual, with turning points at 5 THz and 8 THz, respectively. Thus, in general, increasing the temperature decreases the current. This indicates the transport of holes in the material, and as the temperature increases, the lattice vibration also increases, limiting the flow of the acoustoelectric current. From the relation I = ω q , the intensity of the acoustic phonons is directly proportional to the frequency (ω q ). Thus, Fig. 1a is qualitatively in agreement with the experimental work of Bandhu and Nash (see Fig. 4a 34 ), where they measured the acoustoelectric current for several temperatures at various frequencies in the MHz region. However, in this study, the frequencies are in the THz region.
To further illustrate this, the simultaneous dependence of the Ŵ q on frequency ( ω q ) and temperature (T) is shown as a 3D plot in Fig. 1b. For the dependence of Ŵ q on T, the graph decreased to a minimum and then increased to a point and remained constant at higher temperatures, while the dependence of Ŵ q on ω q conformed to that of Fig. 1a. By switching on the drift field where Ŵ 0 = J 0 . Then, Eq. (7) can be numerically analysed for a normalized Ŵ q dependence on v D /v s and ω q . Shown in Fig. 2a is the dependence of Ŵ q on ω q for v D = 0.9v s , 0.92v s and 0.94v s , which depicts a linear relationship. However, Γ q decreases when v D increases. www.nature.com/scientificreports/ The Weinreich relation J ac = − 2eτ v f Ŵ q relates the absorption to AE current. Thus, Fig. 2a is qualitatively in agreement with a previous experimental report (see Fig. 3 57 ), where the AE current varied linearly with the frequency. Figure 2b shows the dependence of Ŵ q on v D /v s for various values of ω q when a non-quantizing electric field is applied along the axis of the SLG . Absorption and amplification occur when v D /v s < 1 and v D /v s > 1 , respectively, which is consistent with the work of Nunes and Fonseca 29 . In Fig. 2c, we show a 3D graph of the dependence of Ŵ q on v D /v s and ω q . Setting v D = 1.1v s , the maximum amplification is obtained at Ŵ q = − 0.16 for ω q = 2 THz. It is interesting to note that, our results are in good agreement with the work of Bandhu et al. 57 , where acoustic-phonon frequencies above 10 THz were attained. The field E in the SLG can be calculated using E D = v D /µ , where µ = 2.0 × 10 4 cm 2 /v s is the electron mobility in graphene. Using v s = 2.1 × 10 5 cm/s gives E D = 11.5 V/cm . For the source-to-drain voltage, V sd = v D L/µ , (L is the length from the source to the drain electrode in the graphene), the in-plane current I = env D L (n is the electron density) can be calculated.
Patterning SLG into GNR opens a band gap (�) 54 with the energy dispersion given by 53 where Δ is the energy gap and B is the quantized wave vector. By considering that the acoustic phonon and the electric field are directed along the GNR axis, k ′ = (k + q) cos θ , where θ is the scattering angle. When a field , N is the width of the graphene, a c−c is the lattice constant, and Pi is the sub-band index. The absorption reveals the characteristic feature of the acoustic phonon spectrum in the materials that occurs in the Terahertz frequency range. In addition to the parameters used in Fig. 1, the following are used: N ≈ 500 nm , = 0.02, 0.04, 0.06, 0.08 eV , and v D < v s . The plot of Ŵ q versus ω q in Eq. (9) is shown in Fig. 3, which depicts a twin peak with a varying peak heights. The gap between them shifts to the right as the frequency increases. This is similar to the experimental report by Wu (see Fig. 5a 54 ). The twin peaks occur as a result of electron transport in the dual-band formed in the GNR. In the first band, the electrons are initially absorbed until they encounter a gap, where they lose their energy. They then gains energy in the second band by absorbing the energy of the confined phonons. This occurs at low drift velocities of v D = 0.1v s where the electron energy is comparable to the band gap energy. The peak difference is due to a change in the Fermi energy.
The first gap occurs at = 0.02 eV while the second, the third and the fourth occur at 0.04, 0.06, 0.09 eV, respectively. From the plot, gaps occur at points where the Ŵ q = 0 . At A, we obtained a partial gap, but B, C, and D showed a complete gap. When, Ŵ q = 0 , from Eq. (9), we obtain www.nature.com/scientificreports/ Therefore, knowing, v D , v s and ω q , the energy gap ( ) of the material can be determined as in Eq. (10). Using the Weinreich relation and Eq. (9) Fig. 4a shows the acoustoelectric versus ω q at drift velocities of v D = 0.4v s , 0.5v s , 0.6v s . The current increases to a maximum point (resonant) and then decreases. At these drift velocities, the energy of the electron is able to overcome the bandgap energy. The resonant point is referred to as the threshold frequency ω TH q , beyond which the current decreases. The resonance peak is dependent on v D . The plot shifts to the right when v D is increased, making AE in graphene tunable. Figure 4a is qualitatively consistent with the experimental report by Poole et al. (see Fig. 3 35 ). In Figs. 3 and 4, the conduction mechanism is via intraband transitions. Unlike Fig. 3, for a certain quantized phonon energy, the absorption Ŵ q = 0 spectrum in Fig. 4a is due to conduction electrons crossing the energy gap at higher drift velocities and subsequently absorbing the energy of the confined phonons. For further elucidation, a 3D plot of J ac versus ω q and q is shown in Fig. 4b. Similar results were obtained experimentally in the Megahertz (MHz) range by Liag et al. 48 , Okuda et al. 50 , and Morgado et al. 51 . In the Terahertz range, the simulated results of absorption in graphene obtained by Ullah et al. 55 are in qualitatively agreement with Fig. 4.
The LDOAP could be achieved by stimulating the GNR with THz radiation under a gated voltage to modulate the carrier concentration in the GNR. The unique band structure permits absorption via intraband electronic transitions, which can be used to adjust the electron density in the material. The carrier density can be controlled easily and efficiently by varying the gate voltage. The field E in the SLG can be calculated by using E = v D /µ , where µ = 2.0 × 10 4 cm 2 /v s is the electron mobility in graphene. Using v s = 2.1 × 10 5 cm/s gives E = 11.5 V/cm . For the source-to-drain voltage, V sd = v D L/µ , (L is the length from the source to the drain electrode in the graphene), the in-plane current I = env D L (n is the electron density) can be calculated.

Conclusion
We have theoretically demonstrated the acoustoelectric current generation in graphene nanoribbon resulting from Landau damping of acoustic phonons. The AE current in a single layer of graphene was calculated when the temperature and the electric fields were applied. Graphene nanoribbon exhibited a larger acoustoelectric current than a single layer of graphene when a non-quantizing field was applied. The acoustoelectric current shifts as the drift velocity is varied. This makes acoustoelectric current in graphene nanoribbon tunable and a good acoustic wave filter for phonon spectroscopy. www.nature.com/scientificreports/