A method for measuring the phonon-exciton coupling strength in a bilayer graphene film system

Abstract

A method for measuring the phonon-exciton coupling strength in a bilayer graphene film system, characterized by the following steps: 1) using a frequency of ω pu Strong pump light E pu and frequency ω pr Weak detection light E pr The combined effects on the bilayer graphene film create tunable parametric coupling between electrons and vibrational modes; 2) the four-wave mixing expression is numerically calculated; by adjusting the excitation wavelength, pump intensity, and pump-probe detuning, the dynamic evolution of the four-wave mixing signal with these parameters is obtained; in the obtained bimodal four-wave mixing spectrum, the spacing between the two peaks is equal to twice the phonon-exciton coupling strength. The method of this invention can conveniently determine the phonon-exciton coupling strength in the bilayer graphene film system, providing a basis for in-depth research on the internal physical mechanisms of the bilayer graphene system.

Description

Technical Field

[0001] This invention relates to a method for measuring the phonon-exciton coupling strength in a bilayer graphene film system. Background Technology

[0002] Single-layer two-dimensional graphene film is a semiconductor without band gap and with linear energy distribution. This characteristic gives graphene film good conductivity. Bilayer graphene is also a zero-gap semiconductor. It has parabolic energy distribution characteristics and stronger carrier mobility. Many scholars at home and abroad have carried out a series of studies on the properties of bilayer graphene film. In 2007, Professor LM Malard et al. of Brazil proposed to use Raman scattering to detect the electronic structure of bilayer graphene. In 2010, Professor Wang Feng et al. of the University of California, Berkeley reported the many-body Fano resonance behavior in bilayer graphene system[2]. Some scholars have observed that bilayer graphene system has long spin relaxation time at room temperature[3]. Professor Juca et al. pointed out that the normal electric field perpendicular to the plane of bilayer graphene film can break its symmetry, thereby changing its electronic and vibrational characteristics, leading to the opening of the electronic band gap. Therefore, discrete phonons and electron-hole excitons (excitons for short) can couple through the interaction between phonons and electronic states, thus generating a new hybrid phonon system [4]. However, the coupling strength between phonons and excitons in the system is difficult to calculate directly, which to some extent limits people's in-depth research on the internal physical mechanism of bilayer graphene system. Therefore, it is necessary to propose a method to measure the coupling strength between phonons and excitons.

[0003] [1] LM Malard, J. Nilsson, DC Elias, JC Brant, F. Plentz, ES Alves, AH Castro Neto, and MA Pimenta, Probing the electronic structure of bilayer graphene by Raman scattering, Physics Review B 76(20),201401(R)(1-4), 2007.

[0004] [2] Tsung-Ta Tang, Yuanbo Zhang, Cheol-Hwan Park, Baisong Geng,Caglar Girit, Zhao Hao, Michael C. Martin, Alex Zettl1, Michael F. Crommie1, Steven G. Louie, Y. Ron Shen and Feng Wang, A tunable phonon–exciton Fanosystem in bilayer grapheme, Nature Nanotechnology. 5, 32-36 (2010).

[0005] [3] T.- Y. Yang, J. Balakrishnan, F. Volmer, A. Avsar, M. Jaiswal, J.Samm, SR Ali, A. Pachoudeng, M. Popinciuc, G. Güntherodt, B. Beschoten, and B. Ö. Zyilmaz, Observation of long spin-relaxation times in bilayer grapheme at room temperature, Physics Review Letters 107(4), 047206(1-4)(2011).

[0006] [4] Wenhao Wu, Kadi Zhu, Nonlinear coherent optical responses oftunable optomechanical system based on a bilayer grapheme, OpticsCommunications 342 (2015) 199–203. Summary of the Invention

[0007] The technical problem to be solved by the present invention is to overcome the shortcomings of the prior art and provide a method for measuring the phonon-exciton coupling strength in a bilayer graphene film system.

[0008] To solve the above-mentioned technical problems, the technical solution proposed by the present invention is: a method for measuring the phonon-exciton coupling strength in a bilayer graphene film system, characterized by comprising the following steps;

[0009] 1) Using a frequency of ω pu Strong pump light E pu and frequency ω pr Weak detection light E pr The combined effect in bilayer graphene films

[0010] This allows for tunable parametric coupling between electrons and vibrational modes; the bilayer graphene film system is equivalent to a discrete two-level system modified by an electron cloud; at the pump field frequency ω pu Under the spin-wave approximation of position, the total Hamiltonian of the system can be expressed as:

[0011] (1)

[0012] The Hamiltonian, representing the energy of a phonon; The Hamiltonian represents the energy of the exciton in an electron-hole pair.

[0013] The Hamiltonian represents the interaction between phonon and exciton;

[0014] The Hamiltonian represents the interaction between the pump light and probe light and phonons;

[0015] Where ω n ∆ represents the vibrational frequency of graphene. n = ω n -ω pu This represents the frequency detuning of the discrete two energy levels and the pump field. It is the exciton frequency, d k + and d k G represents the exciton production and annihilation operators, respectively. k E represents the coupling strength between the phonon and electronic states in the G mode. pu and E pr Let denot be the slowly varying wave packet amplitudes of the pump field and the probe field, respectively, and μ be the electric dipole moment of the exciton. For the pump field Rabi frequency, , To detect pump mistuning; define , , , ;

[0016] Using Heisenberg's equations of motion and commutation relations , , , We can obtain the Langevin equations:

[0017] (2)

[0018] (3)

[0019] (4)

[0020] Where Γ1 represents the exciton relaxation rate, Γ2 represents the exciton dephase rate, γ represents the electron-hole pair exciton relaxation rate, and g represents the phonon-exciton relaxation rate at the center frequency. The coupling strength at the point; .

[0021] To solve the system of equations (2)-(4), an approximation is made: , , ;in , , Substituting the approximate equation into equation (2-4), we obtain the inverted particle number w0 and the parameter p. -1 The expression:

[0022] (5)

[0023] (6)

[0024] The four-wave mixing signal of this system can then be expressed as:

[0025] (7)

[0026] Given:

[0027]

[0028]

[0029]

[0030]

[0031]

[0032] 2) Numericalize the four-wave mixing expression; adjust the excitation wavelength, pump intensity, and pump-probe detuning to obtain the dynamic evolution diagram of the four-wave mixing signal with these parameters; in the obtained bimodal four-wave mixing spectrum, the spacing between the two peaks is equal to twice the phonon-exciton coupling strength.

[0033] The method described above for measuring the phonon-exciton coupling strength in a bilayer graphene film system, preferably, involves the strong pump light E... pu Pump intensity Ωpu 2 At 0 THz 2 up to 8.39 THz 2 between.

[0034] Compared with the prior art, the advantages of the present invention are: the method of the present invention can conveniently determine the phonon-exciton coupling strength in the bilayer graphene film system, providing a basis for people to conduct in-depth research on the internal physical mechanism of the bilayer graphene system. Attached Figure Description

[0035] Figure 1 A schematic diagram of the structure of a bilayer graphene film under the combined action of a strong pump field and a weak probe field.

[0036] Figure 2 A schematic diagram of the energy levels in a bilayer graphene film system.

[0037] Figure 3 When Ω pu 2 At 1 THz, changing the magnitude of the phonon-exciton coupling strength g, the four-wave mixing signal changes with the probe-pump detuning δ. pr The changing relationship.

[0038] Figure 4 When Ω pu 2 At 1 THz, the peak value and position of the L and R peaks are related to the phonon-exciton coupling strength g.

[0039] Figure 5 When g = 0.3 THz, the pump intensity Ω is changed. pu 2 The magnitude of the four-wave mixing signal varies with the probe-pump detuning δ pr The changing relationship.

[0040] Figure 6 When g = 1 THz, change the pump intensity Ω pu 2 Size, four-wave mixer signal detection - pump detuning δ pr The changing relationship.

[0041] Figure 7 When g = 5 THz, change the pump intensity Ω pu 2 The magnitude of the four-wave mixing signal varies with the probe-pump detuning δ pr The changing relationship.

[0042] Figure 8 When δ prAt 0 THz, the four-wave mixing signal under three different coupling conditions (weak coupling, intermediate coupling, and strong coupling) varies with the pump intensity Ω. pu 2 The relationship diagram of the changes. Detailed Implementation

[0043] To facilitate understanding of the present invention, the present invention will be described more fully and in detail below with reference to preferred embodiments, but the scope of protection of the present invention is not limited to the following specific embodiments.

[0044] It should be noted that when a component is described as being "fixed to, attached to, connected to or connected to" another component, it can be directly fixed to, attached to, connected to or connected to the other component, or it can be indirectly fixed to, attached to, connected to or connected to the other component through other intermediate connectors.

[0045] Unless otherwise defined, all technical terms used herein have the same meaning as commonly understood by those skilled in the art. The technical terms used herein are for the purpose of describing particular embodiments only and are not intended to limit the scope of the invention.

[0046] Example 1

[0047] A method for measuring the phonon-exciton coupling strength in a bilayer graphene film system includes the following steps;

[0048] 1) such as Figure 1 As shown, using a frequency of ω pu Strong pump light E pu and frequency ω pr Weak detection light E pr The combined effects on the bilayer graphene film enable tunable parametric coupling between electrons and vibrational modes; a schematic diagram of the energy levels of the bilayer graphene film system is shown below. Figure 2 As shown.

[0049] The bilayer graphene film system is equivalent to a discrete two-level system modified by electron clouds; at the pump field frequency ω pu Under the spin-wave approximation of position, the total Hamiltonian of the system can be expressed as:

[0050] (1)

[0051] The Hamiltonian, representing the energy of a phonon; The Hamiltonian represents the energy of the exciton in an electron-hole pair.

[0052] The Hamiltonian represents the interaction between phonon and exciton;

[0053] The Hamiltonian represents the interaction between the pump light and probe light and phonons;

[0054] Where ω n ∆ represents the vibrational frequency of graphene. n = ω n -ω pu This represents the frequency detuning of the discrete two energy levels and the pump field. It is the exciton frequency, d k + and d k G represents the exciton production and annihilation operators, respectively. k E represents the coupling strength between the phonon and electronic states in the G mode. pu and E pr Let denot be the slowly varying wave packet amplitudes of the pump field and the probe field, respectively, and μ be the electric dipole moment of the exciton. For the pump field Rabi frequency, , To detect pump mistuning; define , , , .

[0055] Using Heisenberg's equations of motion and commutation relations , , , We can obtain the Langevin equations:

[0056] (2)

[0057] (3)

[0058] (4)

[0059] Where Γ1 represents the exciton relaxation rate, Γ2 represents the exciton dephase rate, γ represents the electron-hole pair exciton relaxation rate, and g represents the phonon-exciton relaxation rate at the center frequency. The coupling strength at the point; .

[0060] To solve the system of equations (2)-(4), an approximation is made: , ,

[0061] ;in , , Substituting the approximate equation into equation (2-4), we obtain the inverted particle number w0 and the parameter p. -1The expression:

[0062] (5)

[0063] (6)

[0064] The four-wave mixing signal of this system can then be expressed as:

[0065] (7)

[0066] Given:

[0067]

[0068]

[0069]

[0070]

[0071]

[0072] 2) Adjust the excitation wavelength, pump intensity, and pump-probe detuning to obtain the dynamic evolution diagram of the four-wave mixing signal with these parameters; in the obtained bimodal four-wave mixing spectrum, the spacing between the two peaks is equal to twice the phonon-exciton coupling strength.

[0073] In this embodiment, the phonon attenuation rate Γ1 = 2 THz, and the phonon dephase rate Γ2 = Γ1 / 2 = 1 THz; exciton relaxation rate γ = 0.015 THz; exciton electric dipole moment μ = 40D; phonon vibration frequency = 298.7 THz, Δ n = 0 THz; Δ ex = 0 THz.

[0074] At weak pump intensity (Ω) pu 2 The relationship between the four-wave mixing signal and the phonon-exciton coupling strength at δ = 1 THz is investigated. When the phonon-exciton coupling is absent (g = 0 THz), the four-wave mixing spectrum exhibits a single-peak structure. When g increases from 0.1 THz to 0.5 THz, the four-wave mixing spectrum shows a bimodal shape. We label the left and right peaks as the L-peak and the R-peak. We note that the peak values ​​of the L-peak and R-peak gradually decrease with increasing g. To further verify whether the L-peak and R-peak change with respect to δ... pr =0 is symmetric about the axis. Figure 4 In the middle we further explored Figure 3The relationship between peak value and peak position as a function of g is investigated. The results show that the line shapes of the L and R peaks completely overlap, with both peak values ​​decreasing as g increases. In other words, for a given g, the peak values ​​of the L and R peaks are equal. Specifically, the peak positions of the L and R peaks are strictly symmetrical about the central axis. Clearly, the intensity of the four-wave mixing signal in the bilayer graphene film system can be tuned by changing the pump field intensity and the phonon-exciton coupling strength.

[0075] Next, we studied the pump intensity Ω. pu 2 The impact on four-wave mixing signals, such as Figure 5 , Figure 6 and Figure 7 As shown in the figure, in the case of weak phonon-exciton coupling (g < Γ2), the four-wave mixing signal exhibits different peak shapes with increasing pump intensity. Figure 5 In the middle, when 0 THz 2 ≤ Ω pu 2 ≤ 8.39 THz 2 When the four-wave mixed spectrum has a bimodal structure; when 8.39THz ≤Ω pu 2 ≤ 9.98 THz 2 When the four-wave mixed spectrum has a three-peak structure; when Ω pu 2 ≥ 9.98THz 2 At that time, the four-wave mixing spectrum has a five-peak structure. For comparison, we also discussed two cases: g = 1 THz = Γ2; g = 5 THz > Γ2. The results are as follows... Figure 6 and Figure 7 As shown, the peak position of the four-wave mixing signal is linearly correlated with the coupling strength between the G-mode phonons and electronic states. When the phonon-exciton coupling strength g = 1 THz and g = 5 THz, the coupling strength between the G-mode phonons and electronic states suppresses the appearance of three-peak and five-peak structures, and only a double-peak structure exists in the four-wave mixing spectrum. The four-wave mixing spectrum exhibits a double-peak structure, and the peak value increases with the pump strength Ω. pu 2 The amplitude increases with the amplitude. Surprisingly, the spacing between the two peaks in the four-wave mixing spectrum remains constant, and the spacing is equal to twice the phonon-exciton coupling strength (i.e., 2g). This means that we can detect the phonon-exciton coupling strength in the system based on the peak shape of the four-wave mixing spectrum in the bilayer graphene film.

[0076] We further explored the four-wave mixing signal in (i.e., δ) pr = 0) The variation of phonon-exciton coupling strength g at the position (e.g.) Figure 8As shown). When g = 0.3 THz, the four-wave mixing signal first increases with the pump intensity Ω. pu 2 The increase in Ω enhances the effect; pu 2 ≈ 10 THz 2 The signal increases sharply to reach its peak; then the four-wave mixing signal increases with Ω. pu 2 The signal weakens sharply with further increases in pump intensity Ω, eventually reaching a stable value. When g = 1 THz and g = 5 THz, the four-wave mixing signal changes with the pump intensity Ω. pu 2 The intensity increases with the increase of pump strength and shows a linear relationship with pump intensity. This result will provide scientific guidance for our control of four-wave mixing signals.

Claims

1. A method for measuring the phonon-exciton coupling strength in a bilayer graphene film system, characterized in that: Includes the following steps; 1) Using a frequency of ω pu Strong pump light E pu and frequency ω pr Weak detection light E pr The combined effect on the bilayer graphene film enables tunable parametric coupling between electrons and vibrational modes. The four-wave mixing signal of this system can then be expressed as: ； (1) ； ； ； ； ； The bilayer graphene film system is equivalent to a discrete two-level system modified by electron clouds; at the pump field frequency ω pu Under the spin-wave approximation of position, the total Hamiltonian of the system can be expressed as: ；(2) The Hamiltonian, representing the energy of a phonon; The Hamiltonian represents the energy of the exciton in an electron-hole pair. The Hamiltonian represents the interaction between phonon and exciton; The Hamiltonian represents the interaction between the pump light and probe light and phonons; Where ω n ∆ represents the vibrational frequency of graphene. n = ω n -ω pu This represents the frequency detuning of the discrete two energy levels and the pump field. It is the exciton frequency, d k + and d k G represents the exciton production and annihilation operators, respectively. k E represents the coupling strength between the phonon and electronic states in the G mode. pu and E pr Let denot be the slowly varying wave packet amplitudes of the pump field and the probe field, respectively, and μ be the electric dipole moment of the exciton. For the pump field Rabi frequency, , To detect pump mistuning; define , , , ; Using Heisenberg's equations of motion and commutation relations , , , We can obtain the Langevin equations: (3) ， (4) ； (5) Where Γ1 represents the exciton relaxation rate, Γ2 represents the exciton dephase rate, γ represents the electron-hole exciton relaxation rate, and g represents the phonon-exciton relaxation rate at the center frequency. The coupling strength; ; Approximation processing: , , ;in , , Substituting the approximate equation into equation (2-4), we obtain the inverted particle number w0 and the parameter p. -1 The expression: ； (6) ； (7) 2) Perform numerical calculations on the four-wave mixing expression; By adjusting the excitation wavelength, pump intensity, and pump-probe detuning, a dynamic evolution diagram of the four-wave mixing signal with these parameters is obtained; in the obtained bimodal four-wave mixing spectrum, the spacing between the two peaks is equal to twice the phonon-exciton coupling strength.

2. The method for measuring the phonon-exciton coupling strength in a bilayer graphene film system according to claim 1, characterized in that: The strong pump light E pu Pump intensity Ω pu 2 At 0 THz 2 up to 8.39 THz 2 between.

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