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In this paper, we theoretically investigate the effect of an external magnetic field on the properties of photonic band structures in two-dimensional n-doped semiconductor photonic crystals. We used the frequency-dependent plane wave expansion method. The numerical results reveal that the external magnetic field has a significant effect on the permittivity of the semiconductor materials. Therefore, the photonic band structures can be strongly tuned and controlled. The proposed structure is a good candidate for many applications, including filters, switches, and modulators in optoelectronics and microwave devices.
© 2015 Optical Society of America
1. Introduction
Photonic crystals (PC) are a new type of artificial material and contain a composite structure with a periodic arrangement of refractive indices in one, two, and three dimensions [1]. The key motivation behind the creation of PCs is the possibility that these structures offer the ability to control the properties of light by creating photonic band gaps (PBGs). The existence of PBGs that forbid the propagation of electromagnetic waves is crucial to the occurrence of most of the important electrodynamic effects reported in the literature. Many of the studies devoted to the understanding of band-gaps are based on non-dispersive materials. Tunable PBGs have been created over the past decade, but other dispersive materials such as metals [2–6], plasma [7, 8] and superconductors [9–12], which are strongly dependent on external parameters, have received limited attention. A tunable PBG may be obtained by externally controlling the response functions and aim at achieving practical applications, such as the possibility that these structures offer the ability to control the properties of light.
Based on the tuning agent, it is possible to design and fabricate PCs for various optoelectronic and microwave device applications, such as optical modulators, switches, tunable filters, and tunable resonators [13]. The tunability of PCs relies on the modification of the permittivity and/or the permeability of one of the constituent materials using an external parameter such as the external electric field [14, 15], temperature [16], hydrostatic pressure [17], or applied magnetic fields [18–20].
It is known that using external electric or magnetic fields to tune PCs is much easier and faster compared with using temperature or strain [21]. The external magnetic field has a significant effect on the dielectric constant of the dispersive materials, particularly metals and semiconductors, because of the magneto-optical effects that occur as a result of the Faraday and Vigot geometries [22, 23]. The eigenmodes are either left or right circularly polarized in a Faraday geometry wherein the direction of the external magnetic is parallel to the direction of propagation of the incident electromagnetic radiation. In Vigot geometry, the direction of the external magnetic is normal to the direction of propagation of the electromagnetic radiation, and the eigenmodes are linearly polarized.
To the best of our knowledge, few reports have discussed the effect of an external magnetic field on the band structures of 2D semiconductor PCs [13, 18, 19]. Additionally, those reports have demonstrated the effect of the external magnetic field based on only a Vigot geometry. In this work, we demonstrate the effect that an external magnetic field has on the properties of the photonic band structures in a 2D semiconductor PC containing - doped semiconductor material based on the Faraday effect. Our calculation method is based on the frequency-dependent plane-wave expansion (PWE) method. The numerical results reveal that an external magnetic field has a significant effect on the position and the width of the PBG. This paper is organized as follows. In Sec. 2, we discuss the basic equations used during our analysis. In Sec. 3, we present the numerical results and discuss the photonic band structures in the 2D n-doped semiconductor PCs. The conclusions are presented in Sec. 4.
2. Model and basic equations
Figure 1 shows cylindrical rods of the n-doped semiconductor material that have been arranged into a square lattice in a dielectric medium. The rods have a lattice spacing a and are assumed to be parallel and have infinite length in the z-direction. Based on the Faraday geometry [23], the relative permittivity of the n-doped Si may be written as
where ε∞ is the high-frequency limit of the relative permittivity, γe(h) is the damping frequency of the electrons (holes) and ωpe(h) is the electrons (holes) plasma frequency. The subscripts + and – denote the right and left circular polarization, respectively. The cyclotron frequency ωc is given bywhere me is the effective mass of the electron and B is the intensity of the external magnetic field. To calculate the photonic band gaps for the 2D PCs shown in Fig. 1, we use the frequency-dependent PWE method [11, 24]. We focus our attention on the transverse-magnetic (TM) field, where the electric field is parallel to the axis of the rods and the nonzero components of the electric field E(r,t) are defined aswhere r is the position vector in the (x-y) plane, ω is the Eigen angular frequency, and Ez(r) is the Eigenfunction of the wave equations. Therefore, Maxwell’s equations for the electric field reduce towhere ɛ(r) is the permittivity. We expand the electric field component Ez(r) and the permittivity ɛ(r) using a Fourier series as where G = (Gx, Gy) is a two dimensional reciprocal lattice vector, whereas k is the wave vector in the first Brillouin zone, and ε(G), Ez(G) are the Fourier coefficients. We substitute Eqs. (5-)a) and (5-b) into Eq. (4) to yield the following Eigenvalue equation:where the expressions for the Fourier coefficients ε(G) are found in [11]. We set the damping frequencies = = 0 [19, 28]. Equation (6) can be rewritten aswhere the matrices A,B,C and D are given by The nonlinear eigenvalue problem given in Eq. (7) can be transformed into a set of linear equations as defined in [24]:where μ = ω/c. Equation (9) yields the frequency modes for left circular polarization (LCP). The frequency modes for right circular polarization (RCP) can be obtained by replacing ωc with −ωc.
Fig. 1 Transverse cross section of the 2D n-doped semiconductor PCs composed of cylindrical rods of Si with a dielectric constant ε(ω) that have been arranged into a square lattice of dielectric material with a dielectric constant εb. The first Brillouin zone of the square lattice is shown in (b).
3. Numerical results
In this section, we present the numerical results for the photonic band structures that correspond to the TM modes that propagate in 2D PCs and contain n-doped semiconductor cylindrical rods that have been arranged into a dielectric background in the presence of an external magnetic field. The rods are arranged in a square lattice. The reciprocal lattice vectors (RLVs) are G = (2π/a) (n1, n2) for the square lattice, where n1 and n2 are integers, i.e., 0, ± 1, ± 2… We limit the maximum value of n1 and n2 to 6. Therefore, in all of the numerical calculations, a total number of 169 plane waves are used, which ensures sufficient convergence for the frequencies of interest. The Γ, X, and M points in the first Brillouin zone of the square lattice, as shown in Fig. 1(b), are included in the calculations. We choose the dielectric and the semiconductor material to be air and n-doped Si, respectively. We use ni = 1.5 × 1010 cm−3, Nd = 4 × 1014 cm−3, ε∞ = 11.7, me = 0.26mo, and mh = 0.34mo, where mo = 9.31 × 10−31 kg is the mass of a free electron [25]. The electron and hole plasma frequencies are 6.4651 × 1011 Hz and 2.1149 × 107 Hz, respectively. We assume a lattice constant a = 2.9156 mm.
We first examine the effect of the external magnetic field on the permittivity of the n-doped Si. Then, we investigate its effect on the gap to mid-gap ratio (Δω/ωg), where Δω is the frequency width of the gap that appears between the first and second frequency bands and ωg is the central frequency of this gap.
3.1. The effect of the external magnetic field on the permittivity of n-doped Si
Figure 2 shows the real part of the complex permittivity, given by Eq. (1), versus the frequency at various values of the cyclotron frequency ωc. We use four different values of ωc = 0.05ωpe, 0.1ωpe, 0.25ωpe, 0.5ωpe, which correspond to magnetic field intensities of 0.114 T, 0.228 T, 0.57 T, and 1.14 T, respectively. For LCP, we observe that the permittivity increases with the cyclotron frequency, as shown in Fig. 2(a). The permittivity changes its sign from negative to positive at frequencies near the electron plasma frequency, especially for ωc > 0.25ωpe. Over the frequency range ω < ωpe, the permittivity is strongly affected by changes in the cyclotron frequency. Figure 2(b) shows the permittivity for the RCP, and it can be observed that the permittivity decreases with increases in the cyclotron frequency. We observe a sharp decrease in the permittivity at ωc = 0.5ωpe, particularly in the frequency range from ω < ωpe. These negative values of permittivity indicate that there will be no propagating electromagnetic waves because the waves are evanescent. Furthermore, the ability to tune based on the capabilities of the n-doped semiconductor materials is expected.
Fig. 2 The real portion of the Si permittivity vs frequency for various values of the cyclotron frequency for a) LCP and b) RCP.
3.2. The effect of ωc on the photonic band structures
In this subsection, we address the effect of the external magnetic field on the photonic band structures. The gap to mid-gap ratio vs. the filling factor (f.f) is plotted for different values of the cyclotron frequency for LCP in Fig. 3(a). The Δω/ωg is strongly affected by any changes in the cyclotron frequency. In the absence of the external magnetic field, the gap reaches a maximum value of Δω/ωg of 16.382% at f.f = 0.3421. The PBG structure for this case is plotted in Fig. 3(b). The gap width is Δω = 0.1402 (2πc∕a) or 9.0659 × 1010 Hz. By applying the external magnetic field, the 1st PGB shifts to lower frequencies, and its width decreases. The gap to mid-gap ratio decreases because of the decrease in the width of the PBG. At B = 0.114 T (ωc = 0.05ωpe), a maximum value of Δω/ωg of 15.221% is obtained at f.f = 0.3217. Further increases in the magnetic field intensity to 0.228 T, 0.57 T, and 1.14 T show that the maximum values of Δω/ωg decrease to 14.021%, 10.177%, and 3.277%, respectively at f.f = 0.3217. Figures 3(c) and 3(d) show the photonic band structures with B = 0.57 T and 1.14 T at f.f = 0.3217. We observe that the width of these gaps decreases by increasing the intensity of the external magnetic field. The applied magnetic field modifies the PBG structure due to variations in the dielectric function with the cyclotronic frequency.
Fig. 3 (a) The variation in Δω/ωg vs. f.f at different values of ωc for LCP. (b, c and d) The photonic band structures of the circular rods of n-doped Si in a square lattice of air at ωc = 0, 0.25ωpe, and 0.5ωpe, respectively.
Next, we consider RCP and investigate the response of Δω/ωg vs. the filling factor at different values of ωc, as shown in Fig. 4(a). A comparison between Figs. 3(a) and 4(a) reveals that the values of Δω/ωg for RCP increase with the cyclotron frequency, unlike that observed for LCP. In addition, the frequency bands shift towards the high frequency regions. At ωc = 0, a maximum value for Δω/ωg of 16.382% is obtained at an f.f = 0.3421. The PBG is plotted in Fig. 4(b) with a gap width Δω = 0.1402 (2πc∕a) or 9.0659 × 1010 Hz.
Fig. 4 (a) The variation of Δω/ωg vs. f.f at different values of ωc for RCP. (b, c and d) The photonic band structures of the circular rods of n-doped Si in a square lattice of air at ωc = 0, 0.25 ωpe, and 0.5 ωpe, respectively.
As ωc increases to 0.05ωpe, 0.1ωpe, 0.25ωpe, and 0.5ωpe, Δω/ωg increases to reach its maximum values of 17.512%, 18.597%, 21.607%, and 25.808%, at f.f = 0.3421, 0.3421, 0.3632, and 0.4072, respectively. The PBGs for ωc = 0.25ωpe and 0.5ωpe are plotted in Figs. 4(c) and 4(d), respectively, and show that the gap width Δω increases to 0.1991 (2πc∕a) or 1.2875 × 1011 Hz and 0.2615 (2πc∕a) 1.6910 × 1011 Hz, respectively.
Figure 5 shows the variation in the lower and upper frequencies of the 1st PBG as a function of ωc. For LCP, Fig. 5(a) shows that the PBG decreases linearly with an increase of the cyclotron frequency until disappearing for ωc > 0.6ωpe, when a transition from a semiconductor PC to a transparent dielectric material occurs [26]. However, for RCP, Fig. 5(b) shows that the PBG width increases with the cyclotron frequency until ωc reaches 0.8ωpe. For ωc > 0.8ωpe the n-doped semiconductor PCs switches from a transparent state to a reflecting state [26]. Therefore, the PBG width begins to decrease as a result of the lower frequency increasing.
Fig. 5 The variation in the lower and upper frequencies of the 1st PBG with ωc: (a) LCP and (b) RCP.
Finally, it is worth noting that for RCP, we observe the appearance of the nearly dispersionless or flat bands. We investigate the effect of the magnetic field on these flat bands, as shown in Fig. 6, where we plot the PBG structure at f.f = 0.2827 for different cyclotron frequencies. We observe that these bands are a strong function of the cyclotron frequency. As ωc increases, the number bands increases, and the upper portion of the flat bands shift upwards in the higher frequency regions. Thus, the lower dispersion bands are no longer flat. Furthermore, new gaps begin to appear within these lower bands. These bands appear because the permittivity of the n-doped semiconductor material is smaller than zero [27].
Fig. 6 The response of the dispersionless bands with the cyclotron frequency for RCP: (a) ωc = 0.25 ωpe and (b) 0.5ωpe.
We can determine the ratio of the imaginary part to the real part of ε(ω) using [19, 28]
We observe that this ratio decreases for frequencies larger than ωp. To reduce the absorption further, a lower damping frequency is preferred. Therefore, the frequencies of the tuned PBGs should be as far as possible from the plasma frequency. In our analysis, we set the damping frequencies γe = γh = 0, which is an acceptable assumption because the frequencies of the tuned PBGs lie in the terahertz regime or below.4. Conclusion
In this work, we investigated the tunability of 2D n-doped semiconductor PCs using an external magnetic field as a result of the Faraday effect. The permittivity of the n-doped semiconductor material is strongly affected by the external magnetic field, particularly below the electron plasma frequency. Therefore, the external magnetic field has a significant effect on the properties of the photonic band structures. In the case of LCP, the gap shifts to a lower frequency, and its width decreases until it disappears completely. However, for RCP, the gap width and middle frequency increases with the external magnetic field. Additionally, flat bands are shown that can be controlled by the applied magnetic field. Therefore, the external magnetic field offers a good way to tune the flow of light in 2D n-doped semiconductor PCs. The proposed structure can play an important role in many applications, such as filters, resonators, switches, and modulators in optoelectronics and microwave devices.
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