Spectral function of the two-dimensional system of massless Dirac electrons

https://doi.org/10.1016/j.physe.2013.12.012Get rights and content

Highlights

  • We determine the spectral function for the two-dimensional system of massless Dirac electrons within the G0W0 approximation.

  • We taking into account only a partially filled band above the Dirac point and pay particular attention to the contribution coming from the plasmon mode.

  • The spectral function shows a wide structure with a pronounced weight just below the Dirac point and a renormalized quasi-particle δ-peak at the chemical potential.

Abstract

We determine the one-particle spectral function for the two-dimensional system of massless Dirac electrons interacting via the long-range Coulomb interaction. The electron self-energy is calculated within the G0W0 approximation, paying particular attention to the contribution coming from the collective plasmon mode. We find that, due to the acoustic plasmon dispersion, the spectral function has a renormalized quasi-particle δ-peak at the chemical potential along with a side band for the Fermi wave vector together with a wide structure with a pronounced spectral weight near the Dirac point. The present approach is appropriate for the wide range of energy scales given by the plasmon energy which may be probed in angle-resolved photoemission spectroscopy (ARPES) experiments.

Introduction

In condensed matter physics Dirac fermions have been applied to study a variety of novel materials, including graphene with band energy that contains two Dirac cones [1], [2], [3], [4], [5] and recently discovered topological insulators with the surface states that contain a single Dirac cone [6], [7], [8], [9], [10], [11]. Furthermore, both systems have been investigated in numerous measurements by angle-resolved photoemission spectroscopy (ARPES) which provides a direct probe of the electronic structure giving energy and momentum information [6], [7], [9], [10], [11], [12], [13], [14], and its data can be related to the one-electron spectral function that can be theoretically determined. However, in order to understand the spectral properties of real materials one has to take into account the electron–electron Coulomb interaction while calculating the spectral function. The consequence of electronic correlations is collective modes that influence the spectral function. In particular, in one dimension, eigenstates of a system of Dirac fermions have been calculated exactly within the Luttinger liquid model [15], [16]. They are the long-range charge and spin acoustic collective modes. Moreover, the Luttinger liquid approach enables the analysis of the spectral properties of quasi-one-dimensional metals that do not show the essential property of Fermi liquids, namely the low-energy quasi-particle peaks in their spectra [17], [18]. This approach, formulated at low energies, is complementary to the G0W0 approximation [19] for the electron self-energy for the square lattice of parallel chains coupled through the three-dimensional electron–electron Coulomb interaction that introduced a wide feature into the spectral function originating from the anisotropic plasmon dispersion [20], [21], [22]. Namely, the self-energy in the G0W0 approximation is the product of Green's function G0 of non-interacting electrons and the screened W0 interaction calculated within the random phase approximation (RPA) [19] which comprises collective modes. The early G0W0 approach to the three-dimensional “jellium” model[19], [23], [24] resulted in the spectral function showing low energy quasi-particle peaks and additional features due to the plasmon mode. Plasmons diminish the electron spectral weight. Electrons can interact with the plasmon mode to form the so-called plasmaron. Analog results were obtained for the two-dimensional graphene theoretically [25], [26], [27], [28], [29], [30] and in ARPES measurements [12], [13]. Also, plasmarons in this system could be seen in near field optics [31] as described in Ref. [32].

In this work we extend our earlier G0W0 approach [20], [21], [22] to the two-dimensional system of massless Dirac electrons interacting via the long-range Coulomb interaction. We take into account only a partially filled electron band above the Dirac point. Our aim is to determine the electron spectral function starting from the dynamically screened Coulomb interaction W0 in the long-wavelength limit calculated within the random phase approximation (RPA). The main approximation comes from the neglect of the electron–hole continuum of single-particle excitations as we concentrate on the effects induced by the collective mode. After taking into account the linear band dispersion of Dirac electrons, the dynamically screened Coulomb interaction W0 shows an acoustic plasmon with a square-root dispersion in the long-wavelength limit [33]. We note that much work has been done on the subject of screening in graphene showing the same behavior of the plasmon dispersion in this limit [27], [34], [35], [36]. We also refer to work [37], [38], [39], [40], [41] devoted to the dielectric response of graphene under various conditions which is characterized by low-energy plasmon mode. Proceeding as in Ref. [20] we determine the dressed Dirac electron Green's function and the corresponding spectral function with a two-dimensional Coulomb electron–electron interaction screened by the plasmon mode. The obtained results show a wide structure with a quasi-particle-like feature in the spectral function just below the Dirac point and a quasi-particle δ-peak with reduced weight at the chemical potential for Fermi wave vector together with a side band due to coupling to the plasmon mode. The long-wavelength limit plasmon dispersion renormalizes the spectral function, redistributing the spectral weight towards higher energies. It is important to note that the long-wavelength limit plasmon dispersion has a dominant effect on the dressed Dirac electron propagator near the chemical potential, diminishing the weight of quasi-particle peak.

The paper is organized in the following manner. In Section 2 we calculate the wave vector dependence of the plasmon collective mode for a two-dimensional system of massless Dirac electrons within the approach of Ref. [33]. Section 3 is devoted to the calculation of the Dirac electron Green's function within the G0W0 method of Ref. [20]. In Section 4 we calculate and discuss the corresponding spectral function. The concluding Section 5 summarizes the main results.

Section snippets

Dielectric function and plasmon excitation

We begin by considering two-dimensional massless Dirac plasma, a system of electrons whose energy band dispersion is linear, i.e. Es(k)=svFk, where k is a two-dimensional wave vector, s=± denotes the conduction (+) and the valence (−) band, and vF is the Fermi velocity. We are interested in the long-wavelength plasmon collective mode in a conduction band. Here, we use the random phase approximation (RPA) for the dynamical dielectric function which describes the screening properties of electron

Green's function

The obtained plasmon mode affects the Green's function of the Dirac electrons, dressed by the dynamically screened Coulomb interaction (4), which is calculated within the random phase approximation (RPA). In the calculation of the reciprocal dressed Green's function G1(k,ω), we follow the previously used G0W0 approximation [20] and after taking into account the dielectric function (8) with the plasmon dispersion (9) we obtainG1(k,ω)=ωE(k)+iη[12n(k)]Σ(k,ω),whereΣ(k,ω)=Eex(k)+12d2q(2π)2V(q)[

Spectral function

The single-particle spectral function is defined byA(k,ω)=1π|ImG(k,ω)|.It can be directly expressed in terms of ReG1(k,ω) and ImG1(k,ω):A(k,ω)=1πImG1(k,ω)[ReG1(k,ω)]2+[ImG1(k,ω)]2,except when ReG1(k,ω) has a zero ω0(k) in which ImG1(k,ω) is infinitesimally small. In the latter case the spectral function can be represented by the quasi-particle δ-peak:A(k,ω)=Z(k)δ[ωω0(k)],where Z(k)=ReG1(k,ω0(k))/ω1. The spectral function A(k,ω) that corresponds to the Green's function obtained

Conclusion

In conclusion, the present analysis indicates the importance of the electron–electron Coulomb interaction in the analysis of the spectral properties of the two-dimensional system of massless Dirac fermions. It has a dominant contribution to the spectral function at low energies causing the renormalization of a low-energy free-electron spectral weight. Namely, for the Fermi wave vector our analysis shows Fermi liquid behavior with the renormalized quasi-particle δ-peak at the chemical potential

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