Goos-Hänchen-like shifts for Dirac fermions in monolayer graphene barrier
We investigate the Goos-Hänchen-like shifts for Dirac fermions in transmission through a monolayer graphene barrier. The lateral shifts, as the functions of the barrier’s width and the incidence angle, can be negative and positive in Klein tunneling and classical motion, respectively. Due to their relations to the transmission gap, the lateral shifts can be enhanced by the transmission resonances when the incidence angle is less than the critical angle for total reflection, while their magnitudes become only the order of Fermi wavelength when the incidence angle is larger than the critical angle. These tunable beam shifts can also be modulated by the height of potential barrier and the induced gap, which gives rise to the applications in graphene-based devices.
pacs:73.23.Ad, 42.25.Gy, 72.90.+y, 73.50.-h
Monolayer graphene has attracted much attention (1); (2) since the graphitic sheet of one-atom thickness has been experimentally realized by A. K. Geim et al. in 2004 (3). The valence electron dynamics in such a truly two-dimensional (2D) material is governed by a massless Dirac equation. Thus graphene has many unique electronic and transport properties (1); (2), including Klein tunneling (4). Recently, further investigations show that a method to produce a finite bandgap in graphene sheets by epitaxially on proper substrate (5) has been proposed, and the Dirac fermions in gapped graphene are described by 2D massive Dirac equations. The induced gap at the Dirac point is significant to control the transport of the carriers and integrating graphene into the semiconductor technology.
Motivated by these progress, the transport of massive or massless Dirac fermions in graphene opens a way to design various graphene-based electron devices in term of the electron optics behaviors (6); (7); (8); (9); (10); (11), such as focusing (6), collimation (7), Bragg reflection (9), and Goos-Hänchen effect (GH) (10); (11). In particular, Zhao and Yelin (10) have shown that, based on the electronic counterpart to the trapped rainbow effect in optics (12), the interplay GH effect and negative refraction (6) in graphene leads to the coherent graphene devices, such as movable mirrors, buffers and memories. Beenakker et al. (11) have further found that the GH effect at a - interface in graphene doubles the degeneracy of the lowest propagating mode, which can be observed as a stepwise increase by of the conductance with increasing channel width.
In this paper, we will investigate the negative and positive lateral shifts for Dirac fermions in transmission through a 2D monolayer graphene barrier, based on the tunable transmission gap (13). Generally, the magnitude of GH shift for total reflection in graphene is in the order of Fermi wavelength. However, the lateral shifts discussed here are similar to but different from the conventional GH shift, because they do result from the beam width of Dirac fermions, and can be enhanced by the transmission resonances. As a matter of fact, the lateral shifts in transmission have nothing to do with the evanescent wave, thus we term them as Goos-Hänchen-like (GHL) shift, which can be considered as an electronic analog of the lateral shifts in the optics (14) and atom optics (15). More interestingly, not only large positive but also large negative GHL shifts can occur in transmitted beam through the graphene barrier. The negative shift behaves like the phenomena of negative refraction in graphene (6). In addition, we will also discuss the effect of the induced gap on the GHL shifts in the gapped graphene barrier. All these tunable beam shifts in the monolayer graphene barrier can be applied in the design of graphene-based devices, such as electron wave switch, wave vector or energy filters and splitter.
Ii Theoretical Model
Consider the massless Dirac fermions with Fermi energy at angle with respective to the axis incident from zero-gap graphene upon a 2D gapped graphene barrier of hight in which the electron acquires a finite mass of , as shown in Fig. 1, where the tunable potential barrier is formed by a bipolar junction (--) within a single-layer graphene sheet with top gate voltage (16), and are the height and width of potential barrier, respectively. Since graphene is a 2D zero-gap semiconductor with the linear dispersion relation, , the massless electrons are formally described by the Dirac-like hamiltonian (4), , where is the Fermi velocity, is the Fermi wave vector ( is Fermi wavelength), and are the Pauli matrices. The wave functions of the plane wave components for the incident and reflected regions are assumed to be
so the corresponding wave function in the transmitted region can be expressed by
where , and are the perpendicular and parallel wave vector components outside the barrier, is the incidence angle of the plane wave component under consideration. For the general case of the gapped graphene barrier, the Hamiltonian of massive Dirac fermions can be written down as , where is equal to the half of the induced gap in graphene spectrum and positive (negative) sign corresponds to the () point, thus the wave functions in the barrier region have the following form:
where , , , , and are defined by
Accordingly, the critical angle for total reflection can be defined by
so that when , the wave function in the propagating case becomes evanescent wave by replacing with , where . According to the boundary conditions, the transmission coefficient is determined by
where and the phase shift is obtained by
From the above expression, it is clear that in the limit , we get and thus one can obtain the same expression for electronic transmission probability corresponding to the massless Dirac fermions (13).
where the subscript in this paper denotes the values taken at . The most intriguing behavior in propagating case is found for Klein tunneling, , where the GHL shifts can be negative, and also be enhanced by the transmission resonances, whereas the lateral shifts for classical motion, , are always large and positive. Examples of their dependence on the barrier’s width at different induced gaps are plotted in Fig. 2, where , , is less than the critical angle defined by Eq. (4), solid and dotted lines correspond to Klein tunneling , and dashed and dot-dashed ones correspond to classical motion . On the contrary, when the incident angle is larger than the critical angle , the lateral shifts become in the order of Fermi wavelength due to the evanescent wave, which is similar to those in total reflection at a single graphene interface (11); (10). Instead of the enhancement by the transmission resonances shown in Fig. 2, Fig. 3 illustrates that the GHL shifts for Klein tunneling and classical motion saturate respectively to negative and positive constants with increasing the barrier’s width in the evanescent case, where (a) and (b) , is re-scaled to , the other parameters are the same as in Fig. 2.
In this section, we will shed light on the properties of GHL shifts in details. For simplicity, we will find the following analytical solutions in the limit and discuss the negative and positive lateral shifts in cases of Klein tunneling and classical motion, respectively.
Case 1: Klein tunneling (). In this case, the critical angle (4) becomes
when the condition is satisfied. When the incidence angle is less then the critical angle obtained above, , the lateral shift is given by
where , and transmission probability is given by Eq. (5),
The GHL shifts obtained above can be positive as well as negative, depending on the influence of . Since the inequality
the lateral shifts can be positive only when for a very thin barrier, namely . However, the lateral shifts turn negative with increasing . It is interesting that the negative lateral shifts can be enhanced by the transmission resonances. Since the transmission coefficient is an oscillating function of tunneling parameters and can be exhibit any value from to (4), there are resonance conditions , at which the barrier is transparent, . At resonances, the lateral shifts in this case reach , which correspond to the maximum absolute values. At anti-resonance, , the lateral shifts becomes . The exotic behaviors of negative and positive GHL shifts are analogous to those of lateral shifts for the transmitted light beam though left-handed metamaterial slab (14), based on the link between Klein paradox and negative refraction (18).
On the contrary, when the incidence angle is larger than the critical angle , , the lateral shift becomes
In the limit of opaque barrier, , the lateral shift trends to a constant as follows,
which is proportional to , and implies that the GHL shift in the evanescent case is in the same order of electron wavelength as the GH effect in a single graphene interface (10); (11). More interestingly, the saturated GHL shift is negative when the incidence angle satisfies , where the critical angle is defined by
But the GHL shift in this case will becomes positive when . The sign change of GHL shifts described by Fig. 3 (b) appears at the incidence angle , which is similar to the result of the quantum GH effect in graphene, taking the the pseudospin degree into account (11).
Case 2: classical motion (). In this case, the critical angle is
When the incidence angle is less than the critical angle for total reflection, , the lateral shift can be written as
where , and transmission probability is
Similarly, the lateral shifts for classical motion also depend periodically on the barrier’s width, thus can be enhanced by the transmission resonances. The lateral shifts at resonances reach , while at anti-resonance they become . However, these GHL shifts in classical motion are always positive as those in the two-dimensional semiconductor barrier (17). When , the GHL shift in the evanescent case becomes
Then the lateral shift in the limit, , is given by
which is always positive constant.
Based on the properties in two cases of Klein tunneling and classical motion, the GHL shifts (a) and corresponding transmission probabilities (b) as the function of incidence energy are shown in Fig. 4,where , , and . Solid, dashed, and dotted lines correspond to , , and , respectively. It is shown that the GHL shifts is closely related to the transmission gap (13). Fig. 4 indicates that the lateral shifts change the sign near the Dirac point , and can also be enhanced by the transmission resonances near the boundaries of energy gap. In addition, the incidence angle has also great impact on the GHL shifts. The absolute values of the lateral shifts increase with increasing the incidence angles, and the positions of the maximum (absolute) values for the positive (negative) beam shifts can also be tuned because of the resonance conditions, as shown in Fig. 4.
Finally, we will turn to discuss the modulations of GHL shifts by the height of potential barrier and the induced gap . It is shown in Fig. 5 that the GHL shifts can be controlled by changing the height of potential barrier, which can be easily implemented by applying a local top gate voltage to graphene (16). Since the lateral shifts are in the forward and backward directions in the cases of and , respectively, it is suggested that the incidence energy can be selected by these tunable beam shifts. Thus, this phenomenon does result in an alternative way to realize the graphene-based electronic devices, for example, energy splitter and energy filter. Moreover, Fig. 5 (b) further investigate how the GHL shifts are affected by a gap opening at the Dirac points. Comparisons of Figs. 2 and Figs. 3 further show that the energy gap will increase (decrease) the absolute values of the shifts in the propagating (evanescent) case. And the influence of gap in the propagating case is more pronounced than that in the evanescent one. The method to generate the energy gap in graphene is through an inversion symmetry breaking of the sublattice due to the fact the densities of the particles associated with the on-site energy for A and B sublattice are different (5). Therefore, the periodical dependence of GHL shifts on the gap provides an efficient way to modulate the lateral shifts in a fixed graphene barrier.
In summary, we have investigated the GHL shifts for Dirac fermions in transmission through a monolayer graphene barrier. The lateral shifts, as the functions of the barrier’s width and the incidence angle, can be negative and positive in Klein tunneling and classical motion, respectively. Since the lateral shifts have a close relation with the transmission probability, the lateral shifts can be enhanced by the transmission resonances when the incidence angle is less than the critical angle for total reflection, while their magnitudes are only the order of Fermi wavelength when the incidence angle is larger than the critical angle. Compared with the smallness of conventional GH shift in graphene, the large negative and positive GHL shifts, which can also be modulated by the height of potential barrier and the induced gap, will have potential applications in various graphene-based electronic devices. We further hope that these similar phenomena in magnetic graphene barrier may lead to the graphene-based spintronic devices on spin filter and spin beam splitter (17); (19).
This work is supported by the National Natural Science Foundation of China (Grants No. 60806041), the Shanghai Rising-Star Program (Grants No. 08QA14030), the Science and Technology Commission of Shanghai Municipal (Grants No. 08JC14097), the Shanghai Educational Development Foundation (Grants No. 2007CG52), and the Shanghai Leading Academic Discipline Program (Grants No. S30105). X. C. acknowledges Juan de la Cierva Programme of Spanish MICINN and FIS2009-12773-C02-01.
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