# Dependences of propagation constants of cylindrical n-Si Rod on the material specific resistivity/Cilindrinio n-Si strypo sklidimo konstantos priklausomybe nuo medziagos savitosios varzos.

IntroductionCircular rod waveguides for many years were investigated due to their excellent electrodynamical characteristics like a large broadbandwidth and the wide possibilities for their implementation in microwave devices and solid-state electronics.

Circular semiconductor and ferrite rod waveguides placed into the longitudinal magnetic fields were studied in [1]. The dispersion characteristic analyze of circular p-Ge, p-Si and n-InSb plasma rod waveguides at several free charge carrier concentrations was made in [2, 3]. The investigation of dielectric and metamaterial rod and hollow-core waveguides were presented in [4]. Rod waveguides are successfully used in microwave devices such as filters, isolators and others [5-8].

Here we present the dispersion characteristic analysis of circular semiconductor n-Si rod waveguides (Fig. 1) dependent on the waveguide radius and the material specific resistivity.

We have investigated the complex longitudinal propagation constants, the transversal propagation constants of the waveguide in the wide frequency range from 15 to 300GHz.

[FIGURE 1 OMITTED]

Calculation algorithm

For the solution of our electrodynamical problem we used Maxwell's equations in this form:

[nabla] x [E.bar] = i[omega][[mu].sup.s.sub.r][[mu].sub.0][H.bar]; [nabla] x [H.bar] = i[omega][[[epsilon].bar].sup.s.sub.r] [[epsilon].sub.0] [E.bar] (1)

where [H.bar]--the magnetic field strength; [bar.E]--the electric field strength. The magnitude [[[epsilon].bar].sup.s.sub.r] = [[epsilon].sup.s'.sub.r] -i[[epsilon].sup.s".sub.r] is the complex permittivity of the n-Si semiconductor, where the real part is [[epsilon].sup.s'.sub.r] = Re([[[epsilon].bar].sup.s.sub.r]) and the imaginary part [[epsilon].sup.s".sub.r] = Im([[[epsilon].bar].sup.s.sub.r]). It is known that the last magnitude dependent on the frequency:

[[epsilon].sup.s".sub.r] = 1/[omega][[epsilon].sub.0][rho], (2)

where the value [rho] is the semiconductor material specific resistivity. The magnitude [[mu].sup.s.sub.r] = 1 is the semiconductor permeability. The magnitudes [[epsilon].sup.a.sub.r] = 1 and [[mu].sup.a.sub.r] = 1 are the permittivity and permeability of air.

The dispersion equation of the semiconductor waveguide is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

where [[eta].bar] = [J.sub.m] ([[k.bar].sup.s.sub.[perpendicular to]]r)--the Bessel function of the m-th order; [[chi].bar] = [H.sup.(2).sub.m] ([k.sup.a.sub.[perpendicular to]][r.sup.s])--the Hankel function of the second kind; [[eta]'.bar], [[chi]'.bar]--derivatives of magnitudes [[eta].bar] and [[chi].bar],; [[k.bar].sup.s.sub.[perpendicular to]] = [square root of [k.sup.2] [[[epsilon].bar].sup.s.sub.r][[mu].sup.s.sub.r] - [[h.bar].sup.2]]--the transversal propagation constant for the semiconductor medium; [r.sup.s]--the waveguide radius; [[k.bar].sup.a.sub.[perpendicular to]] = [square root of [[h.bar].sup.2] - [k.sup.2][[[epsilon].bar].sup.a.sub.r][[mu].sup.a.sub.r]]--the transversal propagation constant in air; m is the azimuthal index. The value [[[DELTA].bar].sub.s] = [([[k.bar].sup.s.sub.[perpendicular to]]).sup.4], k = [omega]/c--the wave number in a vacuum, [omega] = 2[pi]f, where f--an operating frequency, [[s.bar].sub.1] = i[omega][[epsilon].sub.0] [[[epsilon].bar].sup.s.sub.r] [([[k.bar].sup.s.sub.[perpendicular to]]).sup.2], [[f.bar].sub.1] = i[omega][[mu].sub.0][[mu].sup.s.sub.r] [([[k.bar].sup.s.sub.[perpendicular to]]).sup.4], [[f.bar].sub.2] = i[h.bar] [([k.bar].sup.s.sub.[perpendicular to]]).sup.2].

The complex longitudinal propagation constant [h.bar] can be written [h.bar] = h' - ih", where h' = Re([h.bar]) is the real part of the complex longitudinal propagation constant, h" = Im([h.bar]) is the imaginary part of the complex longitudinal propagation constant. The expression of real part of the complex longitudinal propagation constant h' = 2[pi]/[[lambda].sub.w], where [[lambda].sub.w]--the wavelength of the waveguide modes. In our calculations an azimuthal index is m = 1.

We use the Muller method for searching of complex roots of the dispersion equation (3).

Dispersion characteristic analysis

We have investigated the n-Si waveguide dispersion characteristics for four sets of parameters of the waveguide. Our calculations have been executed for two material specific resistivity [rho] and two waveguide radii [r.sup.s]. Values [rho] were equal to 0.3 [ohm] x m and 300 [ohm] x m. We see that the value of material specific resistivity differ in 1000 times. Radii of investigated waveguides were 0.25 mm and 1mm. We accepted in our calculations that the value [[epsilon].sup.s".sub.r] = 3.9 at the frequency f = 15.4 GHz. Our calculations of the values h' and h" are presented in Figs 2-5. The real and imaginary parts of value [[k.bar].sup.a.sub.[perpendicular to]] are shown in Figs. 6 and 7. The values of h' and Re([[k.bar].sup.a.sub.[perpendicular to]]) are normalized to the value k in Figs 2, 4, 6.

Here we presented the results of our examination of two hybrid modes [HE.sub.11] and [EH.sub.11]. The mode [HE.sub.11] is the main mode and the mode [EH.sub.11] is the first higher mode of the open n-Si semiconductor rod waveguide. The curves noted by points correspond to the waveguide with the radius [r.sup.s] = 0.25 mm. The curves designated by circles correspond to the waveguide with the radius [r.sup.s] = 1 mm.

In Fig. 2 we presented h' when [rho] = 0.3 [ohm] x m, radii [r.sup.s] = 0.25 mm (circles) and 1mm (points).

We see that dispersion characteristics of waveguides with different radii are considerably various. The behavior of dispersion characteristics of n-Si rods with various radii especially differ in the area of the cutoff frequency. The dispersion characteristic of mode [HE.sub.11] at the waveguide radius 0.25 mm is almost parallel f-axis in the frequency range from 15 to 100 GHz. It means that the wavelength of the n-Si rod with the radius 0.25 mm does not depend on the frequency in the wide frequency range.

We see that losses sharply increase in the area of the cutoff frequency of [HE.sub.11] mode when waveguide radius is 1 mm. The inclination of the curve h' rapidly increased in the area of [f.sub.cut] (see Fig. 2) too. The both cutoff frequencies of the [EH.sub.11] modes are located in the area where the parameter h'/k < 1. Though the working part of dispersion curves located in the area where the parameter h'/k > 1.

[FIGURE 2 OMITTED]

The broadbandwidths of waveguides with [r.sup.s] = 0.25 mm and 1 mm approximately equal to 61% and 62% respectively.

In Fig. 3 is shown losses of modes [HE.sub.11] and [EH.sub.11] at two radii 0.25 mm (points) and 1 mm (circles).

[FIGURE 3 OMITTED]

The losses of the mode [HE.sub.11] when the waveguide radius is 0.25 mm are small in the frequency range from 15 to 100 GHz. It is possible to explain by the fact that the magnitude h'/k [approximately equal to] 1 in this frequency range. It leads that the largest part of the electromagnetic energy extends outside of n-Si rod. The losses' minimum closed to [f.sub.cut] of [EH.sub.11] modes when radii 0.25 mm and 1 mm is possible to explain by a conception that these modes have h'/k < 1. The most portion of the [EH.sub.11] mode' electromagnetic energy concentrates outside of the n-Si rod where losses are small.

In Fig. 4 is presented h' when [rho] = 300 [ohm] x m, radii [r.sup.s] = 0.25 mm (circles) and 1 mm (points). The comparison of the dispersion characteristics in Fig. 2 and Fig. 4 shows that the influence of material specific resistivity on the real part of the complex longitudinal propagation is weak.

[FIGURE 4 OMITTED]

The losses of modes [HE.sub.11] and [EH.sub.11] at two radii 0.25 mm (points) and 1 mm (circles) is demonstrated in Fig. 5.

[FIGURE 5 OMITTED]

We see that when the material specific resistivity is large ([rho] = 300 [ohm] x m) the losses become extremely small. The comparison of Figs 3 and 5 shows when the specific resistivity changes from 0.3 [ohm] x m till 300 [ohm] x m the losses can decrease more than in 1000 times.

In Fig. 6 and Fig. 7 are presented the real and imaginary parts of the complex transversal propagation constant in air of the [HE.sub.11] and [EH.sub.11] modes at two radii 0.25 mm (points) and 1 mm (circles) when [rho] = 0.3 [ohm] x m.

The value Re([[k.bar].sup.a.sub.[perpendicular to]]) describes speed of the electromagnetic energy attenuation of a mode outside of the waveguide. The larger is Re([[k.bar].sup.a.sub.[perpendicular to]]) the larger part of electromagnetic mode energy propagates inside of the n-Si rod. We can see that the position of Re([[k.bar].sup.a.sub.[perpendicular to]]) curves' maximums coincide with the position of the loss curves' maximums on a scale of frequencies Fig. 3 and Fig. 5.

[FIGURE 6 OMITTED]

The behavior of the similar characteristics Re([[k.bar].sup.a.sub.[perpendicular to]]) when [rho] = 300 [ohm] x m is analogical to Fig.6. Magnitudes of Re([[k.bar].sup.a.sub.[perpendicular to]]) when [rho] = 300 [ohm] x m compare to Re([[k.bar].sup.a.sub.[perpendicular to]]) when [rho] = 0.3 [ohm] x m are differ in 1000 times but the curves' behavior is the same.

[FIGURE 7 OMITTED]

The magnitude Im([[k.bar].sup.a.sub.[perpendicular to]]) influences most considerably on the behavior and value of h' (Fig. 2). Here we do not placed the [[k.bar].sup.a.sub.[perpendicular to]] characteristics when [rho] = 300 [ohm] x m. Because the values and behaviors of Im([[k.bar].sup.a.sub.[perpendicular to]]) as well as the behaviors of Im([[k.bar].sup.a.sub.[perpendicular to]]) are the same for n-Si rods with [rho] = 0.3 [ohm] x m and [rho] = 300 [rho] x m.

Conclusions:

1. We investigated the dependences of n-Si rod waveguides' dispersion characteristics at two material specific resistivity (0.3 [ohm] x m, 300 [ohm] x m) and two radii (0.25mm, 1mm) in the wide range 15-300 GHz of frequencies.

2. The behavior of dispersion characteristics of n-Si rods with various radiuses especially differ in the area of the cutoff frequency.

3. The dispersion characteristic of mode [HE.sub.11] at the waveguide radius 0.25 mm is independent on the frequency in the range from 15 to 100 GHz.

4. The waveguide losses sharply decrease when the real part of the complex longitudinal propagation constant is less than the wave number in a vacuum.

Received 2009 02 15

References

[1.] Nickelson L., Asmontas S., Malisauskas V., Shugurov V. The open cylindrical gyrotropic waveguides.--Vilnius: Technika, 2007.--248 p. (in Lithuanian).

[2.] Asmontas, S., Nickelson, L., Malisauskas, V. Investigation of Magnetized Semiconductor and Ferrite Waveguides // Elektronika ir elektrotechnika. Kaunas: Technologija, 2006.--No. 2(66).--P. 56-61. (in Lithuanian)

[3.] Nickelson L., Asmontas S., Malisauskas V., Martavicius R. The dependence of open cylindrical magnetoactive p-Ge and p-Si plasma waveguide mode cutoff frequencies on hole on concentrations. //J. Plasma Physics, 2009.--No 1 (75).--P. 35-51.

[4.] Nickelson L., Gric T., Asmontas S., Martavicius R. Electrodynamical analyses of dielectric and metamaterial hollow-core cylindrical waveguides // Electronics and Electrical Engineering, 2008.--No. 2(82).--P. 3-8.

[5.] Safavi-Naeini S., Chaudhuri S. K., Goss A. Design and Analysis of Novel Multimode Optical Filters in Dielectric Waveguide // J. of lightwave technology, 1993.--No. 12(11).--P. 1970-1977.

[6.] Kanda M., May W. G. New millimetre-wave isolator containing a semiconductor rod in a circular waveguide // Electronics Letters, 1975.--No. 12(11).--P. 261-262.

[7.] Jankauskas Z., Kvedaras V. Electrical Field and Current Distribution in Semiconductor Plasma in the Strong Magnetic Field // Elektronika ir elektrotechnika. Kaunas: Technologija, 2007.--No. 2(74).--P. 41-44.

[8.] Skudutis J., Daskevic V. Investigation of Meander Delay System Properties using the "MicroWave Studio" Software Package // Elektronika ir elektrotechnika. Kaunas: Technologija, 2006.--No. 8(72).--P. 11-15.

S. Asmontas, L. Nickelson, D. Plonis

Semiconductor Physics Institute, Teraherzt's Electronics Laboratory, A. Gostauto g. 11, LT-01108 Vilnius, Lithuania, phone: +370 5 2627124; e-mail: asmontas@pfi.lt

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Title Annotation: | HIGH FREQUENCY TECHNOLOGY, MICROWAVES/AUKSTUJU DAZNIU TECHNOLOGIJA, MIKROBANGOS |
---|---|

Author: | Asmontas, S.; Nickelson, L.; Plonis, D. |

Publication: | Elektronika ir Elektrotechnika |

Article Type: | Report |

Geographic Code: | 4EXLT |

Date: | Aug 1, 2009 |

Words: | 2092 |

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