## Abstract

The propagation properties of terahertz (THz) waves passing through heavily doped semiconductor slit have been numerically investigated by using the transfer matrix method. The effects of geometrical parameters, carrier concentration, and dielectric materials filling in the slit have been considered. The contour for carrier concentration and slit width show that as slit width and carrier concentration decreases, the effective indices increase and the propagation lengths decrease. For the case of water filling in the slit, temperature has more effect on the imaginary part of propagation constant than the real part. Most of the energy stored in the slit is in the form of electric energy, which firstly decreases and then increases with the decreasing of slit width. It is expected that the semiconductor slit structure is very useful for the practical applications of THz waves in the fields of biological specimen analysis and medical diagnosis.

© 2009 OSA

## 1. Introduction

Terahertz (THz) waves show great potential in many scientific research and application fields, such as free space communication, environmental sensing, medical imaging, and biology detection [1–3]. For instance, THz waves can be used to detect skin cancer and living cell membranes because many biological molecules demonstrate the distinctive optical response in the THz region. The development of THz technology is largely boosted by the rapid development of radiation sources [4–7] and detectors [8], while THz waveguide propagation methods and the design of relevant waveguide device are also of very importance. Firstly proposed by Wang and Mittleman [9], bare metal wire can be used to confine and propagate THz waves, showing the merits of simple structure and low attenuation [10,11]. Parallel-plate metal waveguide is another important kind of THz waveguide methods and demonstrates the advantages of low group velocity dispersion, low loss and no cutoff frequency [12–14]. Besides those waveguide methods, many relevant waveguide components, such as filter, polarizer and attenuator, are also closely related to the applications of surface plasmon polaritons (SPPs) [15,16]. SPPs are quasi two dimensional electromagnetic excitations bounded to the metal-dielectric interface, exhibiting subwavelength confinement and offering the possibility of realizing subwavelength waveguide [17,18].

Gap (slit) surface polariton plasmons (GSPPs) waveguide is an important kind of subwavelength plasmonic devices and can be regarded as the metal-dielectric-metal (MDM) structure. Much research has been carried out to investigate the MDM structure in the visible [13,19,20], infrared, and THz region [21–24]. MDM plasmonic waveguide demonstrates the merits of strong subwavelength localization of plasmon, weak dissipation, the possibility of single mode operation [25], showing potential in many practical applications fields. For instance, MDM plasmonic structure can be used to conduct single-molecule analysis requiring pico- to nanomolar concentrations of fluorophore [26], fabricate optical tweezers for transporting micrometer or nanometer dielectric particles (water molecular or DNA molecules) [27]. The solute (alcohol, sugar) concentration of the aqueous solution can be determined by measuring the changes of the dielectric properties of liquids compressed between the metal plate [28]. The thickness and refractive index of the nanometer water-layer [29] can also be acquired by comparing the dielectric constant from the empty guide with that from the waveguide containing the dielectric layer. With biological analyte compressed between the plates, MDM or modified MDM structure can also be utilized to develop SPPs biosensors by probing the interaction between analyte and THz waves [30].

Due to the fact that metals always show large dielectric constant in the THz region, the decay length of SPPs mode for metal-based MDM structure is very long, which weakens the SPPs mode and limits its practical applications. One of the possible solution methods is adopting active dielectric core materials [31–34]. In the THz region, the imaginary part of many dielectric material, such as doped semiconductor (GaAs), biological analyte containing water or other polar molecules, are always large, which is different from the case in the visible and infrared spectral region. To Investigate the propagation properties of MDM structure with complex dielectric constant core materials is very interesting and important. For the plasma frequencies within far-infrared, heavily doped semiconductors also show metallic characters in the THz region, and their dielectric constant are similar to that of metal (Ag and Au) in visible/UV spectral range. Heavily doped semiconductors have many merits and show more flexibility in fabricating waveguide devices. Furthermore, the SPPs modes on heavily doped semiconductor is significantly more sensitive to the dielectric layer than surface modes supported on a metal substrate [3]. Indium Antimonide (InSb) is a kind of narrow gap semiconductor and has high intrinsic electron density at room temperature. Therefore, the propagation properties of GSPPs mode in the THz region based on heavily doped semiconductor slit have been shown and discussed. In addition, the effects of geometrical parameter, slit materials, and the core dielectric materials with complex refractive index on waveguide propagation properties have also been explored.

## 2. Theoretic Model and Research Method

The MDM plasmonic structure has been schematically shown in Fig. 1
, which can be regarded as an infinitely long slit under the illumination of THz waves. The slit is filled with dielectric gain material in region 2 and bounded by semi-infinite regions of heavily doped InSb in region 1 and 3. The slit width is *w*, and the thickness of the InSb film is enough large. For this slit like structure, the transfer matrix method (TMM) has been used to acquire the propagation constant of GSPPs mode [35,36]. Compared with the analytic equation method [13,16,37], TMM has the merits of high efficiency and accuracy. Under the framework of TMM, the electric or magnetic field can be written as the summation of the incident waves and the reflected waves in each layer of planar multilayered structure. The magnitude of magnetic field of incident light at the first layer and that of the reflected waves at the last layer could be related via the following transfer matrix [35]:

*β*is the

*z*component complex propagation constant of fundamental mode in the gap. Because the electromagnetic field vanish at the top and bottom layer, the coefficients ${H}_{i}{}^{+}$and ${H}_{i}{}^{-}$ should be zero, leading to ${M}_{11}$ = 0 .

The field component of transverse magnetic (TM) mode in the slit materials $\left(\left|x\right|>w/2\right)$ can be written as [38]:

*k*is the wave vector, Eqs. (3)-(8) should be normalized with a common ${E}_{0}$.

The electric and magnetic energy densities in the dielectric core are [38]:

where ${k}_{m}=\sqrt{{\beta}^{2}-{\epsilon}_{m}{k}_{0}^{2}}$, ${k}_{i}=\sqrt{{\beta}^{2}-{\epsilon}_{d}{k}_{0}^{2}}$, and the factor of A should be chosen to make the following Eq. (38):

${U}_{e}^{i}+{U}_{m}^{i}+{U}_{e}^{m}+{U}_{m}^{m}=1,$ Eq. (15) in which ${U}_{e}^{i}$ and ${U}_{m}^{i}$ are the total electric energy and magnetic energy in the dielectric core, ${U}_{e}^{m}$ and ${U}_{m}^{m}$ are the total electric energy and magnetic energy in the metal, which could be acquired by integrating energy density above Eqs. (11)-(14).

The permittivity of heavily doped InSb in the THz region could be expressed as [39]:

*τ*is the Drude relaxation times, ${m}^{\ast}$ is the effective mass,

*n*is the carrier concentration. The skin depth in the heavily doped semiconductor can be expressed as [39]:

*L*can be expressed as [16]:

## 3. Results and discussion

Figure 2(a)
shows the effective indices and propagation lengths of GSPPs mode with different core dielectric materials. The dielectric materials filling in the slit are air, low-density polyethylene, water, and GaAs, with the corresponding refractive index of 1.0, 1.51 [42], 2.375 + 0.502*i* [40], and 3.284 + 0.106*i* [41], respectively; the carrier concentration of InSb is 8.0 × 10^{15}cm^{−3}; the radiation frequency is 1.0 THz. It could be found that as the slit width decreases, the effective indices increase and the propagation lengths decrease, which may result from the fact that the fraction of total electromagnetic energy of GSPPs mode residing in the InSb increases when slit width become smaller. The effective indices of GSPPs mode increase with the increasing of the real part of permittivity for dielectric materials filling in the slit, which may result from the fact that the fraction of GSPPs mode pushed into InSb layer increases. It could also be found from Fig. 2(a) that the propagation length is closely relate to the imaginary part of permittivity for dielectric materials filling in the slit. For metal MDM structure, the large propagation length in the THz region is one of the drawbacks to limit its application. It can be learned from our earlier publication [13] that the propagation length of GSPPs mode of heavily doped InSb is much smaller than that of metal structure. The propagation length could also be largely reduced by filling the slit with different dielectric materials, which has been shown in Fig. 2(a). For example, the propagation length are 1.28 × 10⁴$\mu m$ and 2.06 × 10^{2}
$\mu m$ for air and GaAs filling in the slit (the slit width is 100 $\mu m$). The absorbing core dielectric materials lead to the propagation length of SPPs reducing significantly, which is according with the results in Ref. 3. The GSPPs mode can be better confined by filling dielectric materials (water or GaAs) in the slit, which is very interesting and important for the application of semiconductor made MDM structure. Figure 2(b) displays that the effects of imaginary part of permittivity of core dielectric materials on the dispersive properties, the dielectric material filling in the slit is GaAs. The dielectric constant of GaAs can be changed with carrier concentration, shown in Eq. (19). The real part of dielectric constant of GaAs keeps constant because it changes litter with the changing of carrier concentration, the imaginary part of dielectric constant of GaAs are adopted as 0.0, 0.02, 0.05, 0.10, 0.20, and 0.50, respectively. It also could be found from Fig. 2(b) that the real part of effective indices change litter, while the propagation lengths decrease noticeably with the increasing of imaginary part of dielectric materials filling in the slit. The reason is as follows. As the imaginary part of permittivity of dielectric materials filling in the slit increases, the imaginary part of effective index of GSPPs mode increases, leading to the propagation length increasing.

The dispersive properties of GSPPs mode are closely related to the dielectric constant of InSb, which changes with the carrier concentration. Figure 3
demonstrates that the effective indices and propagation lengths of GSPPs mode versus slit width for different carrier concentrations based on heavily doped InSb slit. The insets in Figs. 3(a) and 3(b) are the effective indices and propagation lengths contour for carrier concentration and slit width; the radiation frequency is 1.0 THz; air is filled in the slit. As carrier concentration decreases, the effective indices increase and the propagation lengths decrease. This phenomenon is related to skin depth, which describes the region where THz waves penetrate into InSb, resulting in the surface electromagnetic properties changing. It could be found from Eq. (20) that the skin depth depends on the dielectric properties of InSb, which changes with the carrier concentration of InSb. For instance, when the carrier concentration of InSb slit are 2.0 × 10^{1}⁶cm^{−3} and 8.0 × 10^{1}⁶cm^{−3} (the slit width is 100.0 $\mu m$), the dielectric constant are −254.64 + 70.30*i* and −1065.82 + 281.21*i*, with corresponding skin depth are 27.32 $\mu m$ and 9.18 $\mu m$, respectively. The larger skin depth at lower concentration means that there are more THz waves penetration into InSb slit, leading to the effective indices decreasing and the propagation constants increasing.

Water is the major ingredient in biological materials and shows a large imaginary part of dielectric constant in the THz region. THz waves can be used to diagnose cancer or tumors by measuring water content in tumors or cancer, which contain a larger amount of water. Because the changes of propagation constant caused by THz waves with analyte is proportional to the refractive index change [30], subwavelength dimensions biological analyte can be investigated by using SPPs biosensors based on MDM structure. It could also be found from Eq. (17) that the dielectric constant of water is closely relative to temperature and frequency, which is different from the case in the visible and near infrared spectral region [13]. The effective indices and propagation length for the case of water filling in the slit at different frequencies have been shown in Fig. 4(a)
; the carrier concentration of InSb is 8.0 × 10^{1}⁶cm^{−3}; the water temperature is 292.3 K. As frequency increases, the effective indices and the propagation lengths decrease. The reason may come from the fact that the permittivity of water decreases with the increasing of frequency. The dielectric constant of water are 15.56 + 8.84*i*, 7.57 + 6.16*i*, 6.12 + 4.14*i*, and 5.39 + 2.39*i* with the corresponding frequency of 0.1 THz, 0.3 THz, 0.5 THz, and 1.0 THz. As shown above, the effective indices are mainly depended on the real part of dielectric materials filling in the slit. Therefore, the larger dielectric constant of water at lower frequency leads to larger effective index. The effects of temperature on the dispersive properties have been shown in Fig. 4(b), which manifests that the propagation length decreases with the increasing of temperature; the radiation frequency is 1.0 THz; the temperature are 278.8 K, 292.3 K, 315.0 K, and 366.7 K, respectively; their dielectric constant are 5.40 + 1.86*i*, 5.39 + 2.39*i*, 5.30 + 3.15*i*, and 6.00 + 4.54*i*, respectively. As temperature increases, the real part of dielectric constant of water increases slowly, while the imaginary part of water increases seriously. This case is similar to the results given in Ref. 30, which displays that the solute (sucrose, alcohol) concentration has larger effect on the imaginary part of refractive index than that of real part. The possible reason maybe that as temperature increases, the motion of water molecular and the friction between them increases, the imaginary part of permittivity of water increasing, leading to the propagation length dropping. Therefore, temperature has more effects on the propagation length than the effective indices.

Figure 5(a)
shows the field distribution of the electric component along *x* direction, the length along *x* axis is normalized by *w*, the slit materials is heavily doped InSb. The carrier concentration is 8.0×10^{16} cm^{−3}; the slit width is 20 $\mu m$; the radiation frequency is 1.0 THz; the dielectric materials filling in the slit are air, polyethylene, water, and GaAs, respectively. It can be found that the GSPPs mode shows the maximum at the metal-dielectric interface for different slit widths. Furthermore, as the refractive indices of dielectric materials filling in the slit increase, the modes decrease more quickly, which means that mode can be better confined with the increasing refractive index of the core dielectric materials. The penetration depth can be defined by the distance where the absolute value of ${E}_{x}$ field decreased by a factor *e* with respect to the value at the interface between InSb and dielectric core. As the permittivity of core dielectric materials increases, the propagation constant increases, the penetration depth decreases, leading to the fact that THz waves penetrate InSb more quickly. For example, the penetration depth are 1.44 $\mu m$, 0.96 $\mu m$, 0.60 $\mu m$, and 0.44 $\mu m$ for the case of air, polyethylene, water and GaAs filling in the slit, respectively. The normalized mode distribution at different slit width have been shown in Fig. 5(b), the slit width are 1.0 $\mu m$, 10.0 $\mu m$, 20.0 $\mu m$, 50.0 $\mu m$, 100.0 $\mu m$, and 200.0 $\mu m$, air is filled in the slit. Additionally, Fig. 5(b) displays that the mode can be well confined in the wide slit. The reason may be as follows. As slit width decreases, there are more THz waves penetrating into InSb, leading to the effective indices increasing, shown in Fig. 3. This phenomenon can be well explained by the ratio of penetration depth to slit width, which are 1.447, 0.145, 0.072, 0.029, 0.014, and 0.007 with corresponding for slit width of 1.0 $\mu m$, 10.0 $\mu m$, 20.0 $\mu m$, 50.0 $\mu m$, 100.0 $\mu m$, and 200.0 $\mu m$, respectively.

The ratio of electric energy, magnetic energy stored in heavily doped InSb and slit versus frequency for different slit width are shown in Figs. 6 (a)-6(d), respectively. The carrier concentration of InSb is 8.0 × 10^{16} cm^{−3}; air is filled in the slit; the slit width are 1.0 $\mu m$, 2.0 $\mu m$, 10.0 $\mu m$, 20.0 $\mu m$, 50.0 $\mu m$, and 100.0 $\mu m$, respectively. As frequency increases, the electric energy in heavily doped InSb increases (${U}_{e}^{m}$), the electric energy in slit (${U}_{e}^{i}$), the magnetic energy in the slit (${U}_{m}^{i}$) and heavily doped InSb (${U}_{m}^{m}$) decreases. It can be found that most of the energy stored in the slit is in the form of electric energy, which is similar to the results in the visible spectral region shown in Ref. 38. But the ratio of ${U}_{e}^{i}$ in the THz region is larger than that in the visible spectral region, which may result from the former has larger dielectric constant. Furthermore, ${U}_{e}^{m}$ firstly increases and then decreases with the decreasing of slit width, while the energy stored in the slit vice versa. This agrees with the experimental results in Ref. 23 that the percent of field energy in the metal increases with the decreasing the thickness of dielectric layer. The possible reason maybe as follows. As slit width decreases, more mode have been squeezed from the slit into heavily doped InSb, the interaction region where THz waves with InSb become larger, the effective indices increase and the propagation lengths decrease, resulting in the fact that the fractional electric field energy stored in conductor increasing. At certain value of slit width (which may be related to skin depth), the electric field of the GSPPs mode quickly approaches that of the electrostatic (capacitor) mode. Therefore, the electric energy stored in the slit (i.e. the magnetic energy stored in InSb reducing) increases with the decreasing of slit width. It should also be noted that the ratio of ${U}_{e}^{i}$ is always larger than ${U}_{e}^{m}$ in the THz region, while in the visible spectral region, the ratio of ${U}_{e}^{m}$ increases and can even exceeds than that of ${U}_{e}^{i}$ with the decreasing of slit width. In addition, MDM structure is very similar to the structure of THz quantum cascade lasers (QCLs), which is one of the most important kinds of semiconductor radiation sources. The results and conclusion can also be used to explain the relevant phenomena of QCLs.

## 4. Conclusions

The waveguide properties of THz waves through heavily doped InSb slit have been investigated by using the TMM method. The effects of geometrical parameters, carrier concentration and dielectric properties of core materials on the dispersion properties of GSPPs mode have been given and discussed. The results show that as the permittivity of dielectric material filling in the slit increases, the effective indices of GSPPs mode increase, the propagation lengths decrease. The effective indices and propagation lengths of GSPPs mode are closely related to the real and imaginary part of the core dielectric materials, respectively. The contour for carrier concentration and slit width shows that as slit width and carrier concentration decreases, the effective indices increase and the propagation lengths decrease. For the case of water filling in the slit, the effective indices decrease and propagation lengths increase with the decreasing of frequency, and the imaginary part of propagation constant is significantly affected by the temperature than the real part. As the slit width and the refractive index of dielectric materials filling in the slit increases, the mode field can be better confined in the slit. The ratio of electric energy stored in the slit is larger than that for the case of visible spectral, which may result from heavily doped InSb showing larger dielectric constant. As the slit width decreases, the electric energy in the slit ${U}_{e}^{i}$ firstly decreases and then increases. It is expected the semiconductor MDM structure is very useful for the practical applications of THz waves, such as in the fields of semiconductor biosensors, medical diagnosis, and security detection.

## Acknowledgments

This work is supported by the Doctoral Funding of Henan University of Technology (2007BS044).

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