† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant Nos. 61076092 and 61290303).
The exchange effect and the magneto–plasmon mode dispersion are studied theoretically for an anisotropic two-dimensional electronic system in the presence of a uniform perpendicular magnetic field. Employing an effective low-energy model with anisotropic effective masses, which is relevant for a monolayer of phosphorus, the exchange effect due to the electron–electron interaction is treated within the self-consistent Hartree–Fock approximation. The magneto–plasmon mode dispersion is obtained by solving a Bethe–Salpeter equation for the electron density–density correlation function within the ladder diagram approximation. It is found that the exchange effect is reduced in the anisotropic system in comparison with the isotropic one. The magneto–plasmon mode dispersion shows a clear dependence on the direction of the wave vector.
In recent years, black phosphorus has attracted a great deal of attention because of its interesting physical properties and its great potential in the electronic and electro–optical device applications.[1–3]
Many experimental works are devoted to the material growth, the physical property characterization, and the device exploration.[1,2,4–7] There is also a considerable amount of theoretical investigation that concerning the electronic band structures,[8–10] the Landau levels and the anisotropic optical properties,[11–13] the plasmon at zero magnetic field,[14–17] the topological and edge states,[18,19] the anisotropic composite fermions,[20,21] the electron substrate phonon coupling,[22] and the tuning of the band gap by a bias electric field.[23] It is interesting to note that the monolayer black phosphorus may provide an alternative two-dimensional electronic system to study the influence of interplay between the anisotropy and the electron–electron interaction. This interplay has been studied in a GaAs quantum well recently.[24] In these theoretical investigations, the many-body effects arising from the electron–electron interaction are less extensively studied. More studies on the many-particle effect are desired.
This motivates the present study. In this paper, we study the exchange effect in an anisotropic two-dimensional electronic system, in the presence of a perpendicular magnetic field, due to the electron–electron interaction within the self-consistent Hartree–Fock approximation. This exchange effect is an important ingredient for the explanation of the quantum oscillation displayed in the magneto-transport property of an isotropic two-dimensional electronic system. For an anisotropic system, one expects that it is important as well. We also study the magneto–plasmon mode dispersion beyond the random-phase-approximation. The magneto–plasmon was observed experimentally for the isotropic two-dimensional electronic system.[25]
The present paper is organized as follows. In Section 2, the approach used is briefly presented. In Section 3, our theoretical results are shown and discussed. Finally, a brief summary is provided in the last section.
In this paper, the anisotropic two-dimensional (2D) electronic system in the xy plane is modeled by the following Hamiltonian:
By choosing a gauge as
![](cpb_25_11_117801/cpb161589ieqn1.gif)
![](cpb_25_11_117801/cpb161589ieqn2.gif)
![](cpb_25_11_117801/cpb161589ieqn3.gif)
It is interesting to note that the well-known Kohn’s theorem is also valid for this anisotropic 2D electronic system.[27] This leads to a constrain on the magneto–plasmon mode dispersion at small wave vectors.
The exchange effect due to the electron–electron interaction is treated within the self-consistent Hartree–Fock approximation.[28,29] The imaginary time Green’s function can be written as
The magneto–plasmon mode dispersion is obtained by solving a Bethe–Salpeter equation for the electron density–density correlation function within the ladder diagram approximation.[28] The electron density–density correlation function can be written as
There is a simple scaling relation when the electron density–density correlation function is calculated for a non-interaction 2D electronic system. By denoting the electron density–density correlation function of a non-interacting isotropic 2D system, in the presence of a perpendicular magnetic field, as
In this paper, we use ℏωc as the energy scale, lB as the length scale. The wave vector is scaled by 1/lB. The electron density enters our calculation via the filling factor ν = ne/nB for a single spin. As the Landau level energy is scaled by ℏωc, and the magneto–plasmon mode frequency is scaled by ωc, only the ratio of two effective masses m1 and m2 is needed in our calculation. However, the values of m1 and m2 used in the calculation for a particular figure are given individually without causing any confusion.
For clarity, we let the effective g-factor be a vanishingly small value. However, the occupation of the Landau levels is assumed to be spin resolved. The spin-down Landau level is assumed to be occupied first, and the spin-up Landau level to be occupied next. This approximation scheme has to be modified when one considers the case of a tilted and large magnetic field that the Zeeman spin splitting may become larger than the separation of two neighboring Landau levels. The system temperature is assumed to be zero.
Another parameter in our calculation is the strength of the electron–electron interaction. This parameter is introduced via the following dimensionless quantity
In Fig.
For an odd filling factor of a value 2n + 1, there are n + 1 spin-down Landau levels and n spin-up Landau levels occupied. The (n + 1)-th spin-up Landau is empty. This results in a larger energy shift, due to the exchange effect, for the spin-down Landau level. Consequently, a spin-splitting of the Landau level occurs.
When the filling factor takes a fractional value, an integer number of Landau levels for one spin are fully occupied, while for the other spin, one Landau level will be partially occupied. Thus, one observes that the spin-down (spin-up) Landau level energy remains constant, and the spin-up (spin-down) Landau level energy decreases linearly as the filling factor increases from one integer to the next higher integer. This picture is qualitatively the same as in the case of an isotropic electronic system.[30–33] In Fig.
In Fig.
One also observes that, the depths of two smile curves shown in Fig.
Next, let us examine the magneto–plasmon mode dispersion. In Fig.
![]() | Fig. 3. The frequency of magneto–plasmon modes versus the amplitude of the wave vector, for some integer filling factors. |
In the case of ν = 1, only the n = 0 spin-down Landau level is occupied, the electron density excitation around ωc should arise from a transition between the n = 0 spin-down Landau level and the n = 1 spin-down Landau level. As the spin-up Landau levels are empty, there is no spin-up Landau level transition. Therefore, one would expect only one magneto–plasmon mode around ω/ωc = 1. In the cases of ν = 2 and ν = 3, following a similar argument, one would expect two magneto–plasmon modes. Our calculation corroborates this hand waving picture.
Note that there is no spin-density excitation shown, as the 2D electronic system studied here has no spin–orbit interaction, and we are limited ourselves to the electron density excitations. The spin-density excitation can be evaluated by summing up a series of slightly different ladder diagrams.[28,29] When the filling factor is not an integer, the spin-density excitation will show up as a collective excitation frequency less than ωc, around the Zeeman spin splitting.
The magneto–plasmon modes can be classified into E1 modes and E2 modes for small wave vectors. They are indicated explicitly in Fig.
For an isotropic 2D electronic system, the magneto–plasmon mode was observed experimentally by depositing a grating coupler to a surface close to the 2D electronic system.[25] However, the wave vector achieved at that time was small. We hope that this theoretical work will inspire more experimental investigation taking advantage of currently available more advanced chip technology.
The dispersions shown in Fig.
In Fig.
Calculations are carried out for some q values. It is found that E1 modes have a larger φ oscillations amplitude, and the amplitude for the E2 modes is smaller. In Fig.
![]() | Fig. 4. The frequency of magneto–plasmon modes versus the direction of the wave vector, for some integer filling factors. |
It should be pointed out that, the Landau level energy will not change if one exchanges the values of two effective masses m1 and m2. However, the magneto–plasmon mode dispersions shown in Figs.
In Fig.
In Fig.
![]() | Fig. 5. The frequency of magneto–plasmon modes versus the amplitude of the wave vector, for a non-integer filling factor. |
![]() | Fig. 6. The frequency of magneto–plasmon modes versus the direction of the wave vector, for a non-integer filling factor. |
The band structure of a monolayer of phosphorus is more complicated than that of the simple anisotropic model used in the present study.[1,2,8,23] However, we believe that the picture shown here for the magneto–plasmon mode will qualitatively remain when the influence of a more realistic band structure is taken into account. On the other hand, the influence of disorder, of spin–orbit interaction, and of electron–phonon coupling should be studied in the future.
In summary, we have used an effective low-energy model with two effective masses to investigate the influence of anisotropy in an interacting two-dimensional electronic system, relevant for a monolayer of phosphorus. The electron–electron interaction induced exchange effect and the magneto–plasmon mode dispersion were studied theoretically. The correction due to the exchange interaction to the Landau levels was evaluated within the self-consistent Hartree–Fock approximation. The magneto–plasmon mode dispersion was calculated by solving a Bethe–Salpeter equation for the electron density–density correlation function within the ladder diagram approximation, beyond the random phase approximation. It was found that the exchange effect is reduced in the anisotropic system in comparison with the isotropic one. The magneto–plasmon mode dispersion showed a strong dependence on the direction of the wave vector.
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