Abstract
We investigate theoretically the infrared optical response characteristics of metallic armchair/zigzag-edge graphene nanoribbons (A/ZGNRs) to an external longitudinally polarized electromagnetic field at low temperatures. Within the framework of linear response theory at the perturbation regime, we examine the optical infrared absorption threshold energy, absorption power, dielectric function, and electron energy loss spectra near the neutrality points of the systems. It is demonstrated that, by some numerical examples, the photon-assisted direct interband absorptions for AGNR exist with different selection rules from those for ZGNR and single-walled carbon nanotube at infrared regime. This infrared optical property dependence of GNRs on field frequency may be used to design graphene-based nanoscale optoelectronic devices for the detection of infrared electromagnetic irradiations.
1. Introduction
Intriguing experimental [1] and theoretical [2–11] attentions have been paid to graphene-based electronics, especially the unique electronic [3–11] and transport properties [12–19] of GNRs due to its potential applications in nanodevices. The optical spectra [20] of semiconducting GNRs and the excitonic effects [21, 22] in the optical properties of AGNRs with many-electron effects included have been investigated by density functional theory and the first-principles calculations, respectively. Liu et al. [23] have demonstrated that magnetic field can enhance and tune the optical response of GNRs in the terahertz to far-infrared regime. The effects of intrinsic spin-orbit and Coulomb interactions on low-energy properties of AGNRs have been studied [24]. The optical absorption selection rule [25] for ZGNR has been proposed to be qualitatively different from that for armchair carbon nonotubes (CNTs) [26, 27]. The optical properties of AGNRs [28] irradiated under an longitudinal polarized electromagnetic field have also been analyzed. Zhang et al. [29] have investigated the magneto-optical properties of graphene quantum dots. Furthermore, the optical response of the ZGNRs with the spin interaction included has been found to be dominated by magnetic edge-state-derived excitons other than the affections of the edges and confinement on the optical transition energies. [30, 31] However, to the best of our acknowledge, the optical properties for driven GNRs [20–25, 28–31] in the important frequency band of infrared have seldom been involved.
A -AGNR is either semiconducting or metallic depending on the number of dimer lines along the edge. However, an -ZGNR is always metallic due to its edge states. In this paper, we present a theoretical investigation on the infrared properties in the vicinity of neutrality points for the two types of metallic GNRs under the irradiation of an external longitudinal polarized low-frequency field at low temperatures. The dependence of the optical absorption threshold values, absorption power, dielectric function, and electron energy loss spectrum (EELS) on the irradiation energy are demonstrated in the framework of linear response theory under the dipole approximation [25, 28, 29]. Some new interband absorptions are predicted with the exception of the quantitative description of threshold energy on the width of GNRs, and the results are discussed and compared with those in the previous works [20–31].
The rest of the paper is organized as follows. The analytical expressions of the band spectra close to the neutrality points for A/ZGNRs, the optical absorption power, dielectric function, and electron energy loss spectra are calculated in Section 2. Some numerical examples and discussions for the results are demonstrated in Section 3. Finally, Section 4 concludes the paper.
2. Model and Formulism
The tight-binding electron dispersion relation for an ideal infinite graphene sheet can be solved exactly as [3, 4, 10, 11] where the hopping integral = 2.75 eV, the minimal translation distance of honeycomb lattice is = 2.46 , and the variation range of and should be specified for particular GNRs.
Employing the hard-wall boundary conditions to longitudinal wave function for a -AGNR, that is, (0)=+1)=0, one gets the discretized wave vector =/ [(+1)], and further obtains the rationalized band spectrum in terms of continuous (longitudinal) and discrete (transversal) quantum numbers near the point of neutrality [10, 11] where represents, respectively, the conduction bands (CB) and valence bands (VB), =2(+1)/3 is the lowest/highest CB/VB mode with = 0, 1, 2,, , and the longitudinal wave vector is confined within the first Brillouin zone 0.
Distinct from AGNR, the discrete values of for an -ZGNR are -dependent, [10, 11] which is determined by where =. Therefore, one obtains = +1/2)] by substituting (3) into (1) and the real solutions = (+1/2) of the extrema for the conduction bands by equation +1)] = +1), where = 0,1,2,,. Therefore, the band spectrum for a -ZGNR can be induced as [10, 11] with the exception of the edge states and the continuous variable is restricted to the range of . However, when falls into the interval , the imaginary solution of (3) reads which gives the edge states (make ZGNRs always metallic) localized on the edge and decay exponentially into the center. The in (4) and (5) represents the CB and VB, respectively.
Using the perturbation theorem in the dipole-transition approximation, the optical absorption power for perfect GNRs irradiated under a longitudinally polarized weak electromagnetic field can be expressed as where and are the intensity and frequency of the irradiation field, respectively, is the free-electron mass, the Fermi-Dirac distribution function, and the subband indices for the VB/CB, while is the -component of the momentum operator. Furthermore, the imaginary part of the complex dielectric function can be obtained as and one gets the real part of the dielectric function by Kramers-Kronig transformation [26–28] where denotes the integral principal value. Therefore, at the long-wavelength limit [26–28], one can obtain EELS by while represents the imaginary part, the refractive index, and reflectivity from the complex dielectric function for the systems.
3. Results and Discussions
In the following, we present the numerical examples of the calculated , and EELS for two types of GNRs under the irradiation of a longitudinal polarized electromagnetic field. In the calculation, the Dirac delta function in (6) and (7) is simulated [26–28] as /() with the Gaussian broadening factor = 0.014 eV. The coupling between the states and the state is neglected since we are only interested in the optical response of the GNRs near the zero-energy points, that is, at the low energy regime [29]. Under the irradiation of a weak electromagnetic field, the dipole transition matrix elements within the tight-binding single-electron picture [25, 28] are chosen as 0.206. Since the wavelength (several hundreds of nanometer) of the weak infrared field is much larger than the transversal size (about 42.5 nanometer) of the 173-AGNR and 100-ZGNR, one can ignore the local-field correction in the present systems. [21, 22, 26, 27] Furthermore, the excitonic effects can also be ignored since the electron-electron interaction has not been included in this work [21, 22, 26, 27].
In Figure 1, we present the electronic structures versus the longitudinal wave vector near the neutrality points of 173-AGNR and 100-ZGNR in panel (a) and (b), respectively. As is seen the 173-AGNR case from Figure 1(a), the CB, and the VB are mirror symmetric with respect to the Fermi level = 0, and the CB subbands at = 0 correspond (from bottom to top in sequential order) to = 0,1,2,13. Moreover, Figure 1(b) the 100-ZGNR case presents an armchair-CNT-like [25] electron band structure other than the lowest/highest conduction/valence band, which converts from an almost linear decrease/increase for to an exponential-like (governed by (5)) curve up to the edge of the first Brillouin zone, while the sequence for the other subbands of the CB is = 1,2,13 from bottom to top. When the electromagnetic field is polarized longitudinally, the allowed optical absorptions for CNTs [26, 27] are restricted to vertical excitations (i.e., and remain unchanged) between the VB and the CB, while those for ZGNRs [25] are from () to () in the low energy range, where and denote the highest valence and the lowest conduction subband, respectively, while the odd and even conduction and valence subbands for GNRs are denoted as and . As follows, different optical absorptions will be demonstrated from the calculated and at vanishing for longitudinally irradiated A/ZGNRs.
(a)
(b)
The threshold energy (the optical transition energy between the highest valence subband and the second conduction subband or that from the second highest valence subband to the lowest conduction one for AGNR, while it is the optical transition from/to the edge states for ZGNR) as a function of the GNRs width is shown in Figure 2. As expected, the wider the ribbons, the lower the threshold energy (decreasing from visible to infrared). It is observed that the discretized points can be fitted by and 1.32 for AGNRs (see the solid line in Figure 2) and ZGNRs (the dashed line in Figure 2), respectively, therefore, the threshold energy for AGNRs is more sensitive to width than that for ZGNRs since the electronic properties of AGNRs are more sensitive to their geometries [20, 21].
The optical absorption power (in arbitrary units) near the neutrality points of 173-AGNR and 100-ZGNR as a function of the irradiation energy is demonstrated in Figure 3. In the presence of an external irradiation field, a peak in () indicates a direct absorption photon with energy followed by a transition from the VB to the CB. As illustrated in Figure 3(a) for the 173-AGNR case, the absorption peaks at 0.041, 0.129, 0.217, 0.305, 0.393, 0.484, and 0.575 eV can be identified to the transitions ()(), as denoted by with = 1,3,,13 in Figure 3(a), while those at 0.085, 0.173, 0.261, 0.349, 0.44, and 0.531 eV result from the excitations ()() as denoted by with = 2,4,,12 in Figure 3(a), respectively. It is worthwhile to note that some vertical transitions (with unchanged and subband indices) have been shown. For example, the peaks at 0.616, 0.792, 0.974, and 1.152 eV should rely on the excitations from , , and , while the absorptions at 0.704, 0.88, and 1.062 eV may be from , and , respectively. However, one can owe the peaks at 0.66, 0.748, 0.836, 0.924, 1.015, and 1.108 eV to the transitions from () to () similarly as [28]. Furthermore, Figure 3(b) exhibits the 100-ZGNR case. One can identify those absorption peaks at 0.074, 0.168, 0.25, 0.347, 0.429, 0.486, and 0.512 eV to the transitions between the decaying modes (edge states) and the odd subbands, as denoted by with = 1,3,,13 in Figure 3(b), respectively. The resonance structures at 0.072, 0.215, 0.303, 0.388, 0.476, 0.561, 0.646, 0.732, 0.817, 0.902, 0.99, and 1.075 eV may be attributed to the vertical transitions between the VB and the CB. It should be pointed out that the current results are different from the absorption coefficient of the monolayer graphene [32] in the optical range of frequencies due to the quantum confinement and different optical transitions. It is noted that the optical absorptions for AGNR are stronger than those for ZGNR at the regime of 0 0.66 eV and decrease slower in the higher energy range.
(a)
(b)
Furthermore, the imaginary part of the dielectric function (in arbitrary units) as a function of photon energy is illustrated in Figure 4. As shown in Figure 4(a) for the 173-ANGR case, one notes a series of resonance structures, corresponding to the absorption peaks in Figure 3(a), which are much higher in the low energy range since . In correspondence to the optical transitions in Figure 3(b), the 100-ZGNR case (see Figure 4(b)) presents a set of relative lower resonance peaks, especially at the low energy regime. It seems that AGNR is more sensitive to the low frequency infrared than ZGNR, which is consistent with [23, 28]. One notices that the imaginary dielectric function for both 173-AGNR and 100-ZGNR demonstrates several zero points (corresponding to plasma frequencies of the systems) at the higher energy regime.
(a)
(b)
The EELS of the two systems as a function of the irradiation field energy is demonstrated finally in Figure 5. Since the system EELS is combined with the complex dielectric function simply by (see [26–28]), Figure 5(a) (the 173-AGNR case) shows a set of singular peaks at 0.55, 0.688, 0.77, 0.825, 0.853, 0.88, 0.935, 0.963, 0.99, 1.018, 1.045, and 1.155 eV corresponding to the plasmon frequencies (zero points of , not shown here) while the 100-ZGNR case (see Figure 5(b)) presents some sharp peaks at 0.765, 0.798, 0.847, 0.963, 0.99, and 1.073 eV with much lower ones at 0.935, 1.018, and 1.155 eV than the 173-AGNR case. It seems that there are more plasmon modes for AGNR than for ZGNR in the infrared range, while the fine details [26, 27] of the system EELS may be smoothed out by a larger broadening parameter.
(a)
(b)
4. Conclusion
In summary, using the linear response theory, we have investigated theoretically the optical properties of semi-infinite clean A/ZGNR under the irradiation of an external longitudinal polarized low-frequency electromagnetic field at low temperatures. Under the dipole-transition approximation, it is shown that the optical absorption power, dielectric function and electron energy loss spectrum of the systems are sensitive to the infrared irradiation depending on the chirality and the width of GNRs. Some new photon-assisted direct interband transitions are proposed. The predicted optical properties are expected to be observed by scanning tunneling microscopy optical spectroscopy [33, 34] and reflection contrast spectroscopy [35] experiments and used to design the graphene-based nanoscale optoelectronic devices [36–38].
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant no. 10974052), Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant no. 20060542002).