Substrate carrier concentration dependent plasmon-phonon coupled modes at the interface between graphene and semiconductors

Lei Wang; Wei Cai; Linyu Niu; Weiwei Luo; Zenghong Ma; Chenglin Du; Shuqing Xue; Xinzheng Zhang; Jingjun Xu

doi:10.1364/OE.23.029533

1. Introduction

Recently, graphene plasmons (GPs) have been focused much attention for their extremely large field enhancement and relatively long propagation length [1, 2]. For these unique optical properties, graphene is believed as one of the best plasmon materials in the infrared and terahertz regimes. Due to the large momentum mismatch between free light and GPs, GPs cannot be excited by free light along translation invariant directions. Therefore, structured graphene such as graphene ribbons [3], periodic ribbon arrays [4], graphene disks [5] and graphene rings [6] have been put forward to break up translation invariance. Meanwhile, several research groups have used a scattering-type scanning near-field optical microscope (s-SNOM) to generate and mapping GPs in real space [7–9 ], which takes use of the increased momentum of the incident light. However, the generation of highly wavelength controllable and long-lived GPs with a relatively simple experimental method is still the bottleneck for applications of GPs. Therefore, the realization of highly tunable plasmons in graphene is particularly important. It is well-known that GPs can be actively controlled by bias voltages or chemical doping [10]. Specially, some outstanding works regarding plasmon induced doping of low-dimensional materials open a new route for controlling the properties of materials in the past few years [11–13 ]. Despite so many efforts on doping of graphene itself to controlling the GPs have been employed, how to realize a relatively simple and effective way to tune GPs is far from solved. As far as we know, little attention has been paid on how to tuning the properties of GPs by doping of the substrates. To demonstrate the possibility of doping substrates to tuning the GPs, we turn to see the dispersion relation of GPs. For long wavelengths k ≪ k _F, where k _F is the Fermi momentum. The dispersion relation of GPs reduces to [2]

q = \frac{ℏ ε_{eff} ω_{pl}^{2}}{2 α c E_{F}},

where q is the plasmon momentum, ω_pl is the frequency of GPs, ε _eff = (1 + ε _∞)/2 is the effective average over the dielectric constants of the substrate and air, α ≈1/137 is the fine structure constant, and E _F is the Fermi energy linked to the carrier density of graphene. This equation means that the properties of substrates can be an equally important factor as the Fermi energy of graphene to affect the GPs. For example, when a graphene sheet is placed on a polar substrate, the coupling between GPs and surface phonons of the substrate exists [14–19 ]. These plasmon-phonon coupled modes (PPCMs) can reduce intrinsic damping of GPs and process longer lifetime than GPs, which have been verified experimentally [16,20]. However, the energy of PPCMs at the interface between graphene and a polar substrate usually locates near the surface phonon energy of substrates, which is determined by the phonon of substrates. As a result, these PPCMs can hardly be controlled. In this paper, we propose using a semiconductor material as the substrate of graphene. One can change the surface phonon energy and lifetime of the substrate, thus realize active control of the properties of PPCMs, which can be explained by the actively adjusting of the coupling between the free electron oscillations and phonons in the substrate. On the other hand, the carrier concentration of semiconductor substrates can be easily modified by electrical fields or light, therefore these PPCMs can also be controlled by applying external visible or near-infrared optical excitations or electrical fields.

2. Models and calculation methods

Our proposed scheme is indicated in Fig. 1. A graphene sheet is placed on a doped semiconductor substrate. Gallium arsenide (GaAs), which is a typical semiconductor material, is taken for example. The carrier concentration of GaAs can be tuned by applying an external electrical field or an optical wave. The dielectric function contributed from phonons and free electrons in polar semiconductor materials is described as follows [21, 22]:

ε (ω) = ε_{\infty} + \frac{(ε_{0} - ε_{\infty}) ω_{TO}^{2}}{ω_{TO}^{2} - ω^{2} - i ω Γ} - \frac{ε_{\infty} ω_{p}^{2}}{ω^{2} + i ω δ_{p}},

where ω _TO denotes the frequency of transverse optical phonons, Γ is the damping rate related to phonons,

ω_{p}^{2} = 4 π n e^{2} / m^{*} ε_{\infty}

denotes the plasma frequency of free electrons, δ_p = ħ/τ_p is the damping rate related to the electron scattering in GaAs. ε ₀ and ε _∞ are the static and high frequency dielectric constants, respectively. For GaAs, these parameters [23, 24] are ω _TO = 268.7/cm, ε ₀ = 13, ε _∞ = 11, m ^* = 0.067m_e and μ _GaAs = 8500 cm²/Vs. Thus the derived parameters

ω_{LO} = \sqrt{ε_{\infty} / ε_{0}} ω_{TO} = 291.1 / cm

, τ_p = μ _GaAs m ^*/e = 0.324 ps. On the other hand, the optical response of graphene is described by the in-plane complex conductivity which is computed with local random phase approximation (RPA) [25, 26]. The Fermi energy of graphene is set as 0.2 eV (corresponding to carrier density 2.94 × 10¹² cm⁻²), which is a typical value for graphene grown by chemical vapor deposition. The intrinsic electron relaxation time of graphene is set as

τ_{e} = μ E_{F} / e v_{F}^{2}

, where v _F ≈ c/300 is the Fermi velocity, μ = 10000 cm²/Vs is the measured DC mobility [27], and the ambient temperature is set as 300 K.

Fig. 1 The sketch of graphene placed on doped GaAs. The carrier concentration of GaAs can be tuned by field effect or photon excitation. The dots in substrate indicate the free electrons injected into GaAs.

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The dispersion relation of eigenmodes in graphene sandwiched by two dielectric materials can be obtained from the pole of p-polarized Fresnel reflection coefficients [2]:

ε / \sqrt{ε k_{0}^{2} - q^{2}} + 1 / \sqrt{k_{0}^{2} - q^{2}} = - 4 π σ (ω) / ω,

where ε is the dielectric function of the substrate, the superstrate is vacuum, and σ(ω) is frequency dependent conductivity of graphene. In order to get the lifetime of PPCMs, the coupling mechanism between graphene and GaAs can be considered as follows. The coupling strength between GPs and surface polar phonons is given by

M (q) = ℏ g e^{- q z_{0}} \sqrt{π c α ω_{so} / q}

, where g = [1/(1 + ε _∞) − 1/(1 + ε ₀)]^1/2 is the coupling constant, ω_so = [(ε ₀ + 1)/(ε _∞ + 1)]^1/2 ω _TO is the original surface phonon frequency, and z ₀ is the distance between graphene and substrate, which is set to zero throughout this paper. The exchange potential due to the phonon coupling can be written as v_so = |M(k)|² G ⁽⁰⁾(ω, τ_so), where

G^{(0)} (ω, τ_{so}) = 2 ω_{so} / ℏ [{(ω + i / τ_{so})}^{2} - ω_{so}^{2}]

is the surface phonon propagator [28, 29]. The total effective carrier interaction results from the Coulomb interaction v_c = 2πe ²/qε _eff, the screened phonon exchange potential v_sc−so and electron-electron interaction potential from free carrier of GaAs v_p, it reads [30]:

V_{eff} = v_{c} / ε_{rpa} \approx \frac{\frac{v_{c}}{ε} + \frac{M {(k)}^{2}}{ε^{2}} D (q, ω)}{1 - \frac{1}{v_{c}} (\frac{v_{c}}{ε} + \frac{M {(k)}^{2}}{ε^{2}} D (q, ω)) v_{c}^{'} (q) Π_{s}^{0} (q, ω)}

where

ε = 1 - v_{c} Π_{g}^{0} (q, ω)

is the purely graphene electronic dielectric function,

Π_{g}^{0} (q, ω) \approx E_{F} q^{2} / π ℏ^{2} {(ω + i δ_{e})}^{2}

is the noninteracting polarizability of graphene, depending on δ_e = ħ/τ_e.

D (q, ω) = D^{(0)} (q, ω) / (1 - {| M (k) |}^{2} D^{(0)} Π_{g}^{0} (q, ω) / ε)

is the renormalized surface phonon propagator.

Π_{s}^{0} (q, ω) \approx n q^{2} / m^{*} {(ω + i δ_{p})}^{2}

is the polarizability for free carrier of GaAs, depending on δ_p= ħ/τ_p. v′_c(q) = 4πe ²/ε _∞ q ² is the 3D Coulomb potential of GaAs. It is worth mentioning that the coulomb drag effect between graphene and 3D electron gas is ignored because the graphene layer and semiconductor touch each other and the drag resistivity is usually far less than layer resistivity [31–34 ]. From Eq. (4), we can obtain:

\frac{ε_{rpa}}{ε_{eff}} = 1 - \frac{ω_{pl}^{2} (q)}{{(ω + i δ_{e})}^{2}} - \frac{2 ε_{eff} g^{2} ω_{so}^{2}}{{(ω + i δ_{so})}^{2} - (1 - 2 ε_{eff} g^{2}) ω_{so}^{2}} - \frac{\frac{ε_{\infty}}{1 + ε_{\infty}} ω_{p}^{2}}{{(ω + i δ_{p})}^{2}}

the dispersion relation of PPCMs can be obtained by using Eq. (5) with ε _rpa(q, ω − i/τ) = 0, and the carrier concentration dependent lifetime τ of PPCMs can be obtained from the imaginary part of the energy.

3. Carrier concentration dependent plasmon-phonon coupled modes

First of all, the dispersion of PPCMs is calculated by using Eq. (3) without considering the damping of the substrate (Γ and δ_p are set to 0) for simplicity. The carrier concentration dependent dispersion curves of PPCMs are shown in Fig. 2(a). It is worth noting that the intrinsic carrier concentration of GaAs without additional doping is n = 2 × 10⁶ cm⁻³ at room temperature. Compared with the dispersion of GPs when a graphene sheet is placed on a nonpolar substrate, one can find that the dispersion of PPCMs splits into two branches due to plasmonphonon coupling, and these two branches are labeled as A and B, respectively. This splitting phenomenon is the result of coupling between graphene plasmons and the optical phonon mode in the GaAs. Furthermore, one can find that there are two cut-off frequencies with two branches of the surface phonon mode which are result of the coupling between free charges and the optical phonon mode in GaAs, while there is only one branch and one cut-off frequency for normal polar material such as SiO₂ [35]. Moreover, several unique features for our proposed system can be illustrated compared with the case where a graphene sheet is laid on a normal polar material [29]. Firstly, for both branches A and B, the plasmon energy increases as the carrier concentration increases, which provides us an active method to tune the PPCMs. Secondly, there is a low energy cutoff frequency for the branch B, and the energy range of the branch B becomes narrower when the carrier concentration increases. Due to the fact that the upper limit ω _TO exists and the branch B only exists near ω _TO, near-zero group velocity PPCMs can be realized, which can be applied in slow light propagation and optical storage. Thirdly, the branch A becomes nearly linear rather than quadratic (in the no coupling limit) when the carrier concentration is over 5 × 10¹⁷ cm⁻³, which leads to group velocity dispersiveness PPCMs. Figure 2(b) shows the lifetime of carrier concentration dependent PPCMs by using Eq. (5). For simplicity, the free electron damping in GaAs is not considered (in order to keep consistent with the dispersion of PPCMs in Fig. 2(a)) and the surface-phonon damping τ_so is set as 1 ps as reported in SiO₂ [20]. We can find that the lifetime of PPCMs becomes longer when the carrier concentration becomes larger.

Fig. 2 (a) The dispersion of plasmon-phonon coupled modes (PPCMs) for graphene laid on GaAs substrate with various carrier concentrations. The labelled modes A and B describe the coupled higher and lower energy branches, respectively. The black arrows mean the increasing of the carrier concentration in GaAs. The carrier concentrations are 0, 2 × 10¹⁶, 1 × 10¹⁷, 2 × 10¹⁷, 5 × 10¹⁷, 1 × 10¹⁸, 2 × 10¹⁸ and 5 × 10¹⁸ cm⁻³, respectively. The shadow triangle area indicates the Landau intraband loss. The parameters of graphene are assumed to be E_F = 0.2 eV, and relaxation time τ_e = 0.2 ps (correspond to a DC mobility of 10000 cm²/Vs) in the whole paper. (b) Plasmon lifetime of the two branches A and B with different carrier concentrations. The free electron scattering of substrate is not included in the calculation.

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To further understand the dispersion and lifetime of PPCMs, the surface phonon of the GaAs substrate is analyzed, which can be obtained by solving [ε(ω̃_so − i/τ̃_so) + 1]/2 = 0 with Eq. (2). The calculated results are shown in Fig. 3(a). One can find that there are two branches for ω̃_so. The higher and lower energies ω̃_so are the approximate cutoff frequencies for branches A and B in Fig. 2(a), respectively. Moreover, one can find that the cutoff frequency of branch A monotonously increases from the originally value ω_so and does not have upper limit in our calculated carrier concentration range, while the cutoff frequency of branch B monotonously increases from 0 (no cutoff frequency) to ω _TO. On the other hand, the energy of branch A of PPCMs is larger than the high energy ω̃_so, and the energy of branch B locates in the middle of the low energy ω̃_so and ω _TO. Thus, the behavior of carrier concentration dependent PPCMs shown in Fig. 2(a) can be understood, which results from the change of the surface phonon of substrate. Then we turn to analyze the strength of surface phonons. From the surface phonon strength definition S_m = |〈m|ϕ_k|0〉|², where |m〉 is the one-phonon excited state in the mth level, $ϕ_{k} = b_{- k} + b_{k}^{†}$ and b_k is surface phonon creation operator. Surface phonon strength is given by [22]

S_{m} = \frac{\sqrt{ε_{\infty} / ε_{0}} (ε_{\infty} / ε_{0} - 1) x_{m}^{3}}{(ε_{\infty} / ε_{0}) x_{m}^{4} + y^{2} {(1 - x_{m}^{2})}^{2}},

where x = ω̃_so/ω _TO is the normalized coupled surface phonon energy, and y = ω_p/ω _TO is normalized bulk plasmon of GaAs. The value S_m describes the weight of the two surface phonons, which contributes to the properties of total surface phonons. The strength of surface phonons is calculated and shown in the Fig. 3(b), we know that the phonon strength of branch A (B) decreases (increases) dramatically as the carrier concentration increases, similar to the behavior of phonon lifetime shown in Fig. 3(a).

Fig. 3 (a) The energy and lifetime of surface phonons in doping GaAs, The thick (thin) line indicates the high (low) energy branch, and the dashed lines indicate their lifetime. (b) The surface phonon strength of these branches in Fig. 3(a).

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The calculated lifetime of PPCMs in Fig. 2(b) does not include the contribution from the scattering of free electrons in GaAs, in other words, the lifetime of surface phonon is set to a constant value 1 ps in our pervious calculations. However, from Fig. 3(a) we know that the lifetime of coupled surface phonon will decrease (increase) for branch A (B) with electron scattering. So the electron scattering effect must be considered in further calculations if one wants to compare the theoretical results with actual experiments. By substituting the δ_p into Eq. (5), the total lifetime of GPs is calculated and shown in Fig. 4. The largest lifetime for the branch A of PPCMs decreases apparently compared to the case without considering electron scattering in substrate (shown in Fig. 2(b)). In contrast, for the branch B of PPCMs, the largest lifetime is still long enough for proper carrier concentration. In other words, although the electron doping in substrate will provide a new pathway for plasmon damping and result in the lifetime decrease of the PPCMs, the lifetime of PPCMs can also be accepted if the doping concentration is not so large. Moreover, the Coulomb drag effect between electrons in graphene and substrate can also be a damping mechanism for PPCMs, however, this is not included in our present physical model because of the touching between graphene and substrate.

Fig. 4 The plasmon lifetime of the coupled plasmon-phonon modes with the free electron scattering of the substrate GaAs. It is notable that the curve n = 0 is the same as it in Fig. 2(b).

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For actual experiments about GPs, we usually have a given light source with certain wavelengths or a given momentum (∼ 1/a) determined by the structures, where a is the structure size for a tip or a grating, due to the fact that the efficient excitation of GPs always requires to fulfill momentum conservation conditions. So several discrete wavelengths and momenta are utilized to realize GPs excitation and propagation in our proposed graphene/GaAs system. First, we analyze the dispersion of PPCMs with a given wavelength. Without loss of generality, two energies from branch A (ħω = 0.06 eV) and branch B (ħω = 0.03 eV) are chosen respectively. The momentum and lifetime of the carrier concentration dependent PPCMs are displayed in Fig. 5(a). One can find that the momenta of branches A and B (the solid lines) decrease to zero as the concentration increases. This result indicates that for a given monochromatic wave, we can always find a proper concentration to match the momentum out of the light cone. This effect opens a door for that all kinds of confined modes or evanescent waves can be coupled to PPCMs. The corresponding lifetime of these branches is indicated by the dashed lines. And the lifetime increases dramatically as the momentum decreasing. This comes both from the effect of phonon coupling and lower mode confinement as the increase of carrier concentration. For the photon energy 0.06 eV, a critical point (8.12 × 10¹⁷cm⁻³, 0.203 ps) appears and shown in Fig. 5(a). In this point, the lifetime of PPCMs is the same as the case with zero concentration of the substrate. Because of the absence of GaAs electron scattering damping pathway, the lifetime of PPCMs without substrate carrier doping is longer than low concentration substrate doping. However, when the concentration is larger than 8.12 × 10¹⁷ cm⁻³, the lifetime of PPCMs can be longer than zero concentration lifetime τ = 0.203 ps. For the energy ħω = 0.03 eV the lifetime is always larger than zero concentration lifetime (τ = 0.215 ps) in our calculation carrier concentration range. On the other hand, for a given momentum q = 5 μm ⁻¹, the energy and lifetime of PPCMs are shown in Fig. 5(b). From the figure, we know that the branches A and B show behaviors similar with ħω_so(q → ∞) (namely Fig. 3 (a)). If a wide spectrum pulse is given, one can tune the output frequency of PPCMs by changing the carrier concentration in a fixed experimental scheme.

Fig. 5 (a) Carrier concentration dependent plasmon momentum and lifetime for given photon energies. The thick line (ħω = 0.06 eV) is chosen from branch A, and the thin line (ħω = 0.03 eV) from branch B. The dashed lines show the corresponding lifetime of the branches. (b) Concentration dependent energy for a given momentum 5 μm ⁻¹, the thick and thin lines indicate the branch A and B, respectively.

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Finally, although GaAs is taken for example in our above analysis, our results can be extended to various semiconductor materials. To verify this point, the coupled surface phonon energy changing with carrier concentration for several different semiconductor materials is investigated and shown in Fig. 6. The carrier concentration dependent surface phonon energies for GaAs, AlAs, InAs, GaP and InP are compared. And the changeable energy range is mainly determined by the effective carrier mass of semiconductor. For InAs, surface phonon energy can reach 0.2 eV for concentration near 10¹⁹ cm⁻³, which means that all the energy below intrinsic graphene phonon can lead to PPCMs. Beyond that, we reaffirm that the direct Coulomb drag effect has been ignored in this paper. When the effect is considered, extra blue shift will be pull-in [31] in the system.

Fig. 6 Carrier concentration dependent surface phonon energy in different semiconductors, GaAs, AlAs, InAs, GaP and InP.

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4. Conclusions

In summary, in this paper we investigated analytically the effect of substrate carrier concentration on the properties of PPCMs. By using the semiconductor substrate GaAs, we found the coupled modes can be effectively controlled by changing the carrier density of the substrate. Specifically, the dispersion and lifetime of PPCMs can be controlled by the carrier density of GaAs. In further, the effect can be understood by the changing of surface phonons of substrate. Our proposed physical mechanism to tune PPCMs can also be extended in other graphene/semiconductor systems. In addition, although the doping in the substrate will result in additional damping for GPs, the lifetime of PPCMs can still be longer than uncoupled plasmons in graphene under proper doping concentration. The controllable long-live and easily excited PPCMs can find applications using GPs. Moreover, the long-range interaction between electrons of substrate and graphene may lead to additional damping for plasmons, which still an open question and will be worked out in future works.

Acknowledgments

This work was financially supported by the National Basic Research Program of China ( 2013CB328702), Program for Changjiang Scholars and Innovative Research Team in University ( IRT0149), the National Natural Science Foundation of China (NSFC) ( 11374006) and the 111 Project ( B07013).

References and links

1. M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009). [CrossRef]

2. F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11, 3370–3377 (2011). [CrossRef] [PubMed]

3. A. Y. Nikitin, F. Guinea, F. J. García-Vidal, and L. Martín-Moreno, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B 84, 161407(R) (2011). [CrossRef]

4. L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6, 630–634 (2011). [CrossRef] [PubMed]

5. Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. García de Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7, 2388–2395 (2013). [CrossRef] [PubMed]

6. P. Liu, W. Cai, L. Wang, X. Zhang, and J. Xu, “Tunable terahertz optical antennas based on graphene ring structures,” Appl. Phys. Lett. 100, 153111 (2012). [CrossRef]

7. Z. Fei, G. O. Andreev, W. Bao, L. M. Zhang, A. S. McLeod, C. Wang, M. K. Stewart, Z. Zhao, G. Dominguez, M. Thiemens, M. M. Fogler, M. J. Tauber, A. H. Castro-Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Infrared nanoscopy of dirac plasmons at the graphene-SiO₂ interface,” Nano Lett. 11, 4701–4705 (2011). [CrossRef] [PubMed]

8. Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012). [PubMed]

9. J. Chen, M. Badioli, P. Alonso-Gonzalez, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovic, A. Centeno, A. Pesquera, P. Godignon, A. Zurutuza Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012). [PubMed]

10. A. Grigorenko, M. Polini, and K. Novoselov, “Graphene plasmonics,” Nat. Photonics 6, 749–758 (2012). [CrossRef]

11. Z. Fang, Y. Wang, Z. Liu, A. Schlather, P. M. Ajayan, F. H. Koppens, P. Nordlander, and N. J. Halas, “Plasmon-induced doping of graphene,” ACS Nano 6, 10222–10228 (2012). [CrossRef] [PubMed]

12. Y. Kang, S. Najmaei, Z. Liu, Y. Bao, Y. Wang, X. Zhu, N. J. Halas, P. Nordlander, P. M. Ajayan, and J. Lou, “Plasmonic hot electron induced structural phase transition in a MoS₂ monolayer,” Adv. Mater. 26, 6467–6471 (2014). [CrossRef] [PubMed]

13. Z. Li, R. Ye, R. Feng, Y. Kang, X. Zhu, J. M. Tour, and Z. Fang, “Graphene quantum dots doping of MoS₂ monolayers,” Adv. Mater. 27, 5235–5240 (2015). [CrossRef] [PubMed]

14. S. Fratini and F. Guinea, “Substrate-limited electron dynamics in graphene,” Phys. Rev. B 77, 195415 (2008). [CrossRef]

15. E. H. Hwang, R. Sensarma, and S. Das Sarma, “Plasmon-phonon coupling in graphene,” Phys. Rev. B 82, 195406 (2010). [CrossRef]

16. Y. Liu and R. F. Willis, “Plasmon-phonon strongly coupled mode in epitaxial graphene,” Phys. Rev. B 81, 081406 (2010). [CrossRef]

17. E. H. Hwang and S. Das Sarma, “Surface polar optical phonon interaction induced many-body effects and hot-electron relaxation in graphene,” Phys. Rev. B 87, 115432 (2013). [CrossRef]

18. S. H. Zhang, W. Xu, F. M. Peeters, and S. M. Badalyan, “Electron energy and temperature relaxation in graphene on a piezoelectric substrate,” Phys. Rev. B 89, 195409 (2014). [CrossRef]

19. X. Zhu, W. Wang, W. Yan, M. B. Larsen, P. Bøggild, T. G. Pedersen, S. Xiao, J. Zi, and N. A. Mortensen, “Plasmon-phonon coupling in large-area graphene dot and antidot arrays fabricated by nanosphere lithography,” Nano Lett. 14, 2907–2913 (2014). [CrossRef] [PubMed]

20. H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea, P. Avouris, and F. Xia, “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photonics 7, 394–399 (2013). [CrossRef]

21. B. B. Varga, “Coupling of plasmons to polar phonons in degenerate semiconductors,” Phys. Rev. 137, A1896–A1902 (1965). [CrossRef]

22. A. Mooradian and G. B. Wright, “Observation of the interaction of plasmons with longitudinal optical phonons in GaAs,” Phys. Rev. Lett. 16, 999–1001 (1966). [CrossRef]

23. D. Lockwood, G. Yu, and N. Rowell, “Optical phonon frequencies and damping in AlAs, GaP, GaAs, InP, InAs and InSb studied by oblique incidence infrared spectroscopy,” Solid State Commun. 136, 404–409 (2005). [CrossRef]

24. J. S. Blakemore, “Semiconducting and other major properties of gallium arsenide,” J. Appl. Phys. 53, R123–R181 (1982). [CrossRef]

25. B. Wunsch, T. Stauber, F. Sols, and F. Guinea, “Dynamical polarization of graphene at finite doping,” New J. Phys. 8, 318 (2006). [CrossRef]

26. E. H. Hwang and S. Das Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75, 205418 (2007). [CrossRef]

27. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306, 666–669 (2004). [CrossRef] [PubMed]

28. S. Q. Wang and G. D. Mahan, “Electron scattering from surface excitations,” Phys. Rev. B 6, 4517–4524 (1972). [CrossRef]

29. J. Schiefele, J. Pedrós, F. Sols, F. Calle, and F. Guinea, “Coupling light into graphene plasmons through surface acoustic waves,” Phys. Rev. Lett. 111, 237405 (2013). [CrossRef]

30. G. D. Mahan, Many-Particle Physics (Plenum, 1993).

31. S. Kim, I. Jo, J. Nah, Z. Yao, S. Banerjee, and E. Tutuc, “Coulomb drag of massless fermions in graphene,” Phys. Rev. B 83, 161401 (2011). [CrossRef]

32. P. M. Solomon, P. J. Price, D. J. Frank, and D. C. La Tulipe, “New phenomena in coupled transport between 2D and 3D electron-gas layers,” Phys. Rev. Lett. 63, 2508–2511 (1989). [CrossRef] [PubMed]

33. M. Carrega, T. Tudorovskiy, A. Principi, M. I. Katsnelson, and M. Polini, “Theory of Coulomb drag for massless Dirac fermions,” New J. Phys. 14, 063033 (2012). [CrossRef]

34. A. Gamucci, D. Spirito, M. Carrega, B. Karmakar, A. Lombardo, M. Bruna, L. Pfeiffer, K. West, A. Ferrari, and M. Polini, “Anomalous low-temperature Coulomb drag in graphene-GaAs heterostructures,” Nat. Commun. 5, 5824 (2014). [CrossRef] [PubMed]

35. V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, “Highly confined tunable mid-infrared plasmonics in graphene nanoresonators,” Nano Lett. 13, 2541–2547 (2013). [CrossRef] [PubMed]

Substrate carrier concentration dependent plasmon-phonon coupled modes at the interface between graphene and semiconductors

Abstract

1. Introduction

2. Models and calculation methods

3. Carrier concentration dependent plasmon-phonon coupled modes

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (6)

Equations (6)

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