Effect of Temperature on Photonic Band Gaps in Semiconductor-Based One-Dimensional Photonic Crystal

Abstract

The effect of the temperature and angle of incidence on the photonic band gap (PBG) for semiconductor-based photonic crystals has been investigated. The refractive index of semiconductor layers is taken as a function of temperature and wavelength. Three structures have been analyzed by choosing a semiconductor material for one of the two materials in a bilayer structure. The semiconductor material is taken to be ZnS, Si, and Ge with air in first, second, and third structures respectively. The shifting of band gaps with temperature is more pronounced in the third structure than in the first two structures because the change in the refractive index of Ge layers with temperature is more than the change of refractive index of both ZnS and Si layers with temperature. The propagation characteristics of the proposed structures are analyzed by transfer matrix method.

1. Introduction

Since the pioneering works of Yablonovitch [1] and John [2], studies on the electromagnetic properties of photonic crystals which are artificial structures with periodically modulated dielectric constants have been attracting a great deal of interest among the researchers. Photonic crystals that exhibit electromagnetic stop bands or photonic band gaps (PBGs) have received considerable attention over the last two decades for the study of their fundamental physical properties as well as for their potential applications in many optoelectronic devices [3–13]. It was observed that periodic modulation of the dielectric functions significantly modifies the spectral properties of the electromagnetic waves. The electromagnetic transmission and/or reflection spectra in such structures are/is characterized by the presence of allowed and forbidden photonic energy bands similar to the electronic band structure of periodic potentials. For this reason, such a new class of artificial optical material with periodic dielectric modulation is known as photonic band gap (PBG) material [14]. Fundamental optical properties like band structure, reflectance, group velocity, and rate of spontaneous emission, can be controlled effectively by changing the spatial distribution of the dielectric function [5, 6].

A 1D PC structure has many interesting applications such as dielectric reflecting mirrors, low-loss waveguides, optical switches, filters, optical limiters. It has also been demonstrated theoretically and experimentally that 1D PCs have absolute omnidirectional PBGs [15–19].

In addition to the existence of wide photonic band gaps in some properly designed PCs, the feature of a tunability of PBGs in PCs attracts the attention of investigators in recent years. PBGs can be tuned by means of some external agents. For instance, band gaps can be shifted by changing the operating temperature, and such method of tuning may be called temperature tuning or simply T-tuning [20]. A superconductor/dielectric PC belongs to this type of photonic crystals. This happens because of the temperature-dependent London perturbation length in the superconducting materials [21–24].

However, in earlier reports on one-dimensional PCs based on dielectric media studied by the researchers the dielectric property of the constituent layers was considered to be independent of the temperature and nondispersive. In this paper, we consider semiconductor media as one of the constituents of a one-dimensional photonic crystal in which the dielectric property of semiconductors depends not only on temperature but also on wavelength. Here we consider the three multilayer systems, namely, ZnS/air, Si/air, and Ge/air. The refractive index of air is independent of temperature and wavelength. But the refractive indices of semiconductor layers are taken as a function of both temperature and wavelength [25, 26]. Therefore, this study may be considered to be physically more realistic.

2. Theoretical Analysis

The schematic representation of one-dimensional photonic crystal is represented in Figure 1.

Figure 1 

Schematic diagram of 1D photonic crystal structure.

We consider air/(𝐴𝐵)^𝑁𝐴/air structure in which 𝐴 and 𝐵 represent the high and low refractive index materials. To compute the transmission spectrum, we employ the transfer matrix method (TMM) [27]. In this method, the transfer matrix for each layer can be written as 𝑀_𝑗=𝐷_𝑗𝑃_𝑗𝐷_𝑗⁻¹, (1) where 𝑗 stands for 𝐴 or 𝐵 layers and 𝐷_𝑗 and 𝑃_𝑗 are called the dynamical matrix and the propagation matrix, respectively. The dynamical matrix is given by the following equations: 𝐷_𝑗=𝑛11_𝑗cos𝜃_𝑗−𝑛_𝑗cos𝜃_𝑗𝐷forTEmodeofpolarization,_𝑗=cos𝜃_𝑗cos𝜃_𝑗𝑛_𝑗−𝑛_𝑗forTMmodeofpolarization. (2) Also, the propagation matrix 𝑃_𝑗 can be defined as 𝑃_𝑗=𝑒^𝑖𝛿𝑗00𝑒^{−𝑖𝛿𝑗}, (3) where the phase is written as 𝛿_𝑗=2𝜋𝑑_𝑗𝜆𝑛_𝑗cos𝜃_𝑗. (4)

The transfer matrix for the structure embedded in air, the transfer matrix for the air/(𝐴𝐵)^𝑁𝐴/air structure, can be written as 𝑀𝑀=¹¹𝑀₁₂𝑀₂₁𝑀₂₂=𝐷₀⁻¹𝑀_𝐴𝑀_𝐵^𝑁𝑀_𝐴𝐷₀, (5) where 𝐷₀ is called the dynamical matrix for air.

The reflection and transmission coefficients in terms of the matrix elements given in (5) can be written as 𝑀𝑟=¹¹+𝑞_𝑓𝑀₁₂𝑞_𝑖−𝑀₂₁+𝑞_𝑓𝑀₂₂𝑀₁₁+𝑞_𝑓𝑀₁₂𝑞_𝑖+𝑀₂₁+𝑞_𝑓𝑀₂₂,𝑡=2𝑞_𝑖𝑀₁₁+𝑞_𝑓𝑀₁₂𝑞_𝑖+𝑀₂₁+𝑞_𝑓𝑀₂₂, (6) where 𝑞_𝑖,𝑓=𝑛_𝑖,𝑓cos𝜃_𝑗,𝑓 for TE wave and 𝑞_𝑖,𝑓=(cos𝜃_𝑗,𝑓)/𝑛_𝑖,𝑓 for TM wave, where the subscripts 𝑖 and 𝑓 correspond to the quantities, respectively, in the medium of incidence and the medium of emergence. Whereas, the reflectance and transmittance of the structure is given by 𝑅=|𝑟|²,𝑇=|𝑡|². (7)

3. Proposed Structures and Structural Parameters

We choose ZnS, Si, and Ge for the material 𝐴, air for material 𝐵, and 𝑁=10 in Figure 1. So, there will be three proposed structures: [air/(ZnS/air)¹⁰ZnS/air], [air/(Si/air)¹⁰Si/air] and [air/(Ge/air)¹⁰Ge/air]. We take ZnS, Si, and Ge as the high refractive index materials and air as the low refractive index material. The geometrical parameters are so chosen that the thicknesses of high and low refractive index materials are the same at 300 K temperature; that is, 𝑑₁ = 𝑑₂ for all structures. We confine our study in the temperature range from 300 to 700 K only so that the three structures can be compared.

3.1. Structure I: [air/(ZnS/air)¹⁰ZnS/air]

The refractive index of ZnS layer is taken as a function of both wavelength and temperature. The thermal expansion coefficient and melting point for ZnS layers are taken to be 6.1 × 10⁻⁶/K and 1830 K, respectively [28]. The refractive index of ZnS in the ranges 1200–14000 nm and 293–700 K can be expressed as a function of both the wavelength and temperature as [25] 𝑛²(𝜆,𝑇)=𝜀₁(𝑇)+𝐴(𝑇)𝜆²−𝜆²₁+𝐵(𝑇)𝜆/𝜆₂²,−1 (8) where 𝜆₁=0.23979+4.841×10⁻⁵𝑇₁,𝜆₂=36.525+4.75×10⁻³𝑇₁,𝑇₁𝜀=𝑇−293,₁(𝑇)=8.34096+1.29107×10⁻³𝑇₁+4.68388×10⁻⁷𝑇₁²−1.31683×10⁻⁹𝑇₁³−6.64356×10⁻¹²𝑇₁⁴,𝐴(𝑇)=0.14540+1.13319×10⁻⁵𝑇₁+1.05932×10⁻⁸𝑇₁²+1.06004×10⁻¹⁰𝑇₁³+2.27671×10⁻¹³𝑇₁⁴,𝐵(𝑇)=3.23924+1.096×10⁻³𝑇₁+4.20092×10⁻⁷𝑇₁²+1.1135×10⁻⁹𝑇₁³+7.2992×10⁻¹²𝑇₁⁴. (9)

3.2. Structure II: [air/(Si/air)¹⁰Si/air]

The thermal expansion coefficient and melting point for Si layer are taken to be 2.6 × 10⁻⁶/K and 1685 K, respectively [28]. The refractive index of air is independent of temperature and wavelength. But the refractive index of silicon layer is taken as a function of both wavelength and temperature. The refractive index of silicon (Si) in the ranges 1200–14000 nm and 20–1600 K is represented as [26] 𝑛²()(𝑇)+𝑒𝜆,𝑇=𝜀^{−3Δ𝐿(𝑇)/𝐿293}𝜆²×0.8948+4.3977×10⁻⁴𝑇+7.3835×10⁻⁸𝑇², (10) where 𝜀(𝑇)=11.4445+2.7739×10⁻⁴𝑇+1.7050×10⁻⁶𝑇²−8.1347×10⁻¹⁰𝑇³,Δ𝐿(𝑇)𝐿₂₉₃=−0.00071+1.887×10⁻⁶𝑇+1.934×10⁻⁹𝑇²−4.554×10⁻¹³𝑇³ (11) for 20 K ≤ 𝑇 ≤ 293 K, Δ𝐿(𝑇)𝐿₂₉₃=−0.00021−4.149×10⁻⁷𝑇−4.620×10⁻¹⁰𝑇²+1.482×10⁻¹¹𝑇³ (12) for 293 K ≤ 𝑇 ≤ 1600 K.

3.3. Structure III: [air/(Ge/air)¹⁰Ge/air]

The thermal expansion coefficient and melting point for Ge layer are taken to be 6.9 × 10⁻⁶/K and 1210 K, respectively [28]. The refractive index of Ge layer is taken as a function of both wavelength and temperature. The refractive index of Ge in the ranges 1200–14000 nm and 293–1000 K can be expressed as a function of both the wavelength and temperature as [26] 𝑛²()(𝑇)+𝑒𝜆,𝑇=𝜀^{−3Δ𝐿(𝑇)/𝐿293}𝜆²×(2.5381+1.8260×10⁻³𝑇+2.8888×10⁻⁶𝑇², (13) where 𝜀(𝑇)=15.2892+1.4549×10⁻³𝑇+3.5078×10⁻⁶𝑇²−1.2071×10⁻⁹𝑇³,Δ𝐿(𝑇)𝐿₂₉₃=5.790×10⁻⁵(𝑇−293)+1.768×10⁻⁹(𝑇−293)²−4.562×10⁻¹³(𝑇−293)³ (14) for 293 K ≤ 𝑇 ≤ 1000 K.

4. Results and Discussion

4.1. Structure I: [air/(ZnS/air)¹⁰ZnS/air]

The plot of the refractive index of ZnS as the function of wavelength and temperature is shown in Figure 2(a). From this figure, it is clear that the refractive index of ZnS layers increases with temperature. Therefore refractive index contrast increases with temperature. So the photonic band gaps can be shifted as a function of temperature. The variation of reflectance with temperature at different temperatures is shown in Figure 3 and corresponding data tabulated in Table 1 in which the complete photonic band gaps (CPBGs) and corresponding central wavelengths for band gaps are given for two angles of incidence (i.e., 0° and 85°) and at two different temperatures (i.e., 300 K and 700 K).

Table 1 

PBG width and central wavelength of BAND-GAP at different angles of incidence and temperatures for ZnS/air multilayer system.

(a)

(b)

(c)

Figure 2 

Variation of refractive index with wavelength and temperature, (a) ZnS, (b) Si, and (c) Ge.

(a) 𝜃=0∘

(b) 𝜃=85∘

Figure 3 

Reflectance with temperature for ZnS/air.

Two complete band gaps appear in the reflection spectra. From these figures and table, it is clear that as we increase the temperature (at constant angle of incidence) the band gaps shift slightly towards the longer wavelength region. The shifting of band gaps with temperature is slight because the refractive index of ZnS layers changes slowly with temperature. On the other hand, as we increase the angle of incidence (at constant temperature) the band gaps shift towards the shorter wavelength region, but the PBG width increases. The regions with 𝑅≈1 are corresponding to the complete photonic band gaps. The variation of PBG and band width with temperature is shown in Figure 4. Figure 4 shows the variation of PBGs and central wavelength of these PBGs with angle of incidence. From this figure it is clear that if we increase the temperature, the band width and central wavelengths of these PBGs increase, but the central wavelength shifts towards the shorter wavelength region as we increase the angle of incidence.

(a)

(b)

Figure 4 

Variation of PBG band width and central wavelength of PBG with temperature for ZnS/air.

4.2. Structure II: [air/(Si/air)¹⁰Si/air]

The plot of the refractive index of Si as the function of wavelength and temperature is shown in Figure 2(b). From this figure, it is clear that the refractive index of ZnS layers increases with temperature. Therefore refractive index contrast increases with temperature. In this case the change in refractive contrast is more than that in the case of structure I. The variation of reflectance with temperature at different temperatures is shown in Figure 5, and corresponding data are tabulated in Table 2.

Table 2 

PBG width and central wavelength of BAND-GAP at different angles of incidence and temperatures for Si/air multilayer system.

(a) 𝜃=0∘

(b) 𝜃=85∘

Figure 5 

Reflectance with temperature for Si/air.

Only one complete band gap (CPBG) appears in the reflection spectra for this case. From these figures and table, it is clear that as we increase the temperature (at constant angle of incidence) the band gaps shift slightly towards the longer wavelength region. The shifting of band gaps with temperature is more than that of structure I because the change in the refractive index of Si layers with temperature is more than the change in the refractive index of ZnS with temperature. On the other hand, as we increase the angle of incidence (at constant temperature) the band gaps shift towards the shorter wavelength region, but the PBG width increases. The variation of PBG and band width with temperature is shown in Figure 6. This figure shows the variation of PBGs and central wavelength of these PBGs with angle of incidence. From this figure it is clear that if we increase the temperature the band width and center of these PBGs increase, but the central wavelength shifts towards the shorter wavelength region for large angle of incidence.

Figure 6 

Variation of PBG band width and central wavelength of PBG with temperature for Si/air.

4.3. Structure III: [air/(Ge/air)¹⁰Ge/air]

The plot of the refractive index of Ge as the function of wavelength and temperature is shown in Figure 2(c). From this figure, it is clear that the refractive index of Ge layers increases with temperature. Therefore refractive index contrast increases with temperature. The variation of reflectance with temperature at different temperatures is shown in Figure 7, and corresponding data are tabulated in Table 3.

Table 3 

PBG width and central wavelength of BAND-GAP at different angles of incidence and temperatures for Ge/air multilayer system.

(a) 𝜃=0∘

(b) 𝜃=85∘

Figure 7 

Reflectance with temperature for Ge/air.

Two complete band gaps appear in the reflection spectra. From these figures and table, it is clear that as we increase the temperature (at constant angle of incidence) the band gaps shift slightly towards the longer wavelength region. The shifting of band gaps with temperature is more pronounced than in the first two cases, because the rate of change in the refractive index of Ge layers with temperature is more than the rate of change of refractive index of both ZnS and Si layers with temperature. On the other hand, as we increase the angle of incidence (at constant temperature) the band gaps shift towards the shorter wavelength region, but the PBG width increases. The variation of complete PBGs and band widths with temperature is shown in Figure 8. This Figure shows the variation of PBGs and central wavelength of these PBGs with angle of incidence. From this figure it is clear that if we increase the temperature the band width and center of these PBG increase but the central wavelength shifts towards the shorter wavelength region for large angle of incidence.

(a)

(b)

Figure 8 

Variation of PBG band width and central wavelength of PBG with temperature for Ge/air.

The shifting behavior for all cases can be explained by using the phase equation (4) [29]. According to this phase equation, as 𝑛(𝜆,𝑇) increases with temperature, the wavelength must increase accordingly to keep the phase 𝛿 unchanged. The shifting with angle of incidence is due to the cosine function on the right-hand side. As we increase the angle of incidence, the value of the cosine function on the right-hand side decreases. So the wavelength must decrease accordingly to keep the phase 𝛿 unchanged. This behavior is helpful in tuning the PBG width and central wavelength of PBG.

5. Conclusion

The effect of the temperature and angle of incidence on the PBG for semiconductor-based photonic crystals has been investigated. The refractive indices of semiconductor layers are taken as a function of both temperature and wavelength. So, the simulation can be used for the wide range of temperature as well as wavelength. Therefore, this study may be considered to be physically more realistic. We can use the proposed structures as temperature sensing device, narrow band optical filter, wavelength division demultiplexer, and tunable omnidirectional reflector and in many optical systems.

Acknowledgment

One of the authors, Vipin Kumar, wishes to express sincere gratitude to Professor S. P. Ojha for his valuable suggestions and critical comments.