The Magnetoactive p-Ge Rod Waveguide Loss Analysis on the Concen- tration of Two Component Hole Charge Carriers

L. Nickelson, A. Bubnelis, A. Baskys Center for Physical Sciences and Technology, Gostauto 11, LT01108 Vilnius, Lithuania, e-mails: lucynickelson@gmail.com, arturas.bubnelis@gmail.com Faculty of Electronics, Vilnius Gediminas Technical University, Naugarduko str. 41, LT-03227, Vilnius, Lithuania, e-mail: algirdas.baskys@vgtu.lt R. Navickas Faculty of Electronics, Vilnius Gediminas Technical University, Naugarduko str. 41, LT-03227, Vilnius, Lithuania, e-mail: romualdas.navickas@vgtu.lt


Introduction
A semiconductor material placed in an external constant magnetic field can be called magnetoactive semiconductor plasma or gyrotropic material.The propagation of electromagnetic (EM) waves in the unbounded magnetoactive semiconductor plasma is analyzed in many works i.e. [1,2].The EM wave propagation in the metal waveguide filled with magnetoactive plasma is also studied sufficiently fully [3,4].The limited number of works is devoted to analyses of the open (without a metal screen) magnetoactive plasma waveguides.Our present article is dedicated to a study of such waveguides.
On the base of magnetoactive semiconductor plasma waveguides are worked out controllable microwave devices, i.e. phase shifters, modulators, converters, switches, filters [5][6][7].The microwave device phase and attenuation can be controlled by an external constant magnetic field as well as optically.Magnetoactive semiconductor waveguides are used also in the development of various optoelectronic, plasmonic devices and lasers [8,9].
The phase constant dependencies on the frequency of open circular magnetoactive semiconductor waveguides are given in [7,10,11].In the last works the magnetoactive semiconductor waveguide losses are not presented.
As the magnetoactive semiconductor plasma usually possesses high losses, for this reason the study of loss' dependencies on the carrier concentration and frequency are the topic of special interest.
In this article we present the dispersion characteristics of dissipative magnetoactive germanium (Ge) circular cylindrical waveguide when the material contains the different percentages of light and heavy holes.The solution of this boundary problem in the rigorous electrodynamical formulation of the problem was fulfilled by the partial area method with using of the Müller's method for the searching of complex propagation constants [7,12].
Here for the first time is presented dependencies of the complex longitudinal propagation constant i , h h h c cc where hc is the phase constant and hcc is the waveguide attenuation constant (losses) on the percentage of light hole l N and heavy hole h N concentrations and the operating frequency f.The investigation was accomplished for the EM wave with the left-handed circular polarization (looking from EM source) when the azimuthal dependence is expressed by i e M in the wide frequency range from 5 GHz till 200 GHz.These kinds of EM waves are also called the helicon or the extraordinary waves.We have used our created computer software in the MATLAB language.Here is shown a number of important properties of two component hole p-Ge waveguides that can be useful to design controllable microwave devices.

Permittivity tensor of magnetoactive p-Ge with two component hole carriers
Electrodynamical properties of p-Ge semiconductor placed in a constant longitudinal magnetic field are characterized by the relative permittivity tensor   In the present article we give the results of our calculations for the p-Ge waveguide when the semiconductor material has two component charge carriers, i.e. heavy and light holes.The tensor components depend on the percentages of certain kind of holes in the comparison with the total concentration N. The total concentration N is the sum of the light hole l N and heavy hole h N concentrations.The altering of rate h N N allow us to change the electrical semiconductor parameters.This makes it possible to select the required waveguide electrodynamical characteristics as the broadbandwidth, the losses, the wavelength and the kind of operating mode.6 is given the complex propagation constant dependencies of the main helicon and eight higher helicon modes on frequencies.It is known that the main mode HE 11 of cylindrical semiconductor waveguide without the external magnetic field B 0 is a hybrid mode with the first and second subscripts equal to 1.As it known the first subscript denotes the number of the EM field varia-tions in the azimuthal direction and the second subscript shows the variations along the waveguide radius.Here are only analyzed modes with the first subscript equal to 1, i.e. the same azimuthal symmetry by M as the main mode [7].

Dispersion characteristics of p-Ge waveguide are calculated when the ratio of heavy holes' concentration
We do not classify here the investigated hybrid modes of the dissipative gyrotropic waveguide because the mode kind can transform with the changing of frequency.
In Figs 1, 3, 5 we present the normalized real part 0 h k c of complex propagation constant h , where 0 k is the wavenumber in a vacuum.
The cutoff frequency f cut of the main mode is 9.4 GHz, 9.1 GHz and 8.2 GHz when the heavy holes' concentration is 10%, 30%, 90% of N, respectively (Figs 1, 3, 5).The cutoff frequency of the main mode with increasing of rate h N N slowly moves to the lower frequencies.The broadbandwidth of the p-Ge waveguide is equal 93% (Fig. 1), 107% (Fig. 3), 109% (Fig. 5), respectively.We have examined the dispersion characteristics at the heavy holes' concentration from 0 till 100% with the step equal to 5%.On this risen we can note the general properties of the dispersion characteristics dependent on concentration h N .We have observed the transformation of waves with a change of values h N and f.As an example in Fig. 1 we can see the intersection of the second and third higher modes' curves at f = 100 GHz.This means that the degeneration of second and third modes is observed at f = 100 GHz.The transformation of these modes occurs at frequencies higher than 100 GHz because the second subscript values of these higher modes change by the places.The second and third higher modes' curves detach and degeneration is removed when the heavy holes' concentration is grown till 30% and higher.The second higher mode' cutoff frequency changes noticeably (Fig. 3) in the comparison with the same mode f cut of the previous case (Fig. 1).We see that the first and second higher mode curves intersect at f = 102.2GHz when the heavy holes' concentration is 90% (Fig. 5).The transformation of these higher modes is happened at the frequencies higher than 102.2GHz.In Figs.2b, 4b and 6b are shown other five higher modes' losses.The fourth higher mode is denoted by a number 4 and so on.In Figs 2a, 4a, 6a we see that the loss of the main mode are much lower than losses of all higher modes in the entire frequency range.It is important to note that the first higher mode loss is higher than the main mode loss for all value h N .Initially the first mode loss grows till the maximum and then this loss slowly diminish with the growth of frequency.The maximum of the first higher mode loss is max hcc = 190 m -1 (f = 77.0GHz) at the h N equal to 10%, 379 m -1 (f = 100.2GHz) at the h N equal to 30%, 1907 m -1 (f = 102.8GHz) at h N equal to 90% of N.

2
the free carriers average collision frequency, Z is the angular operating frequency[7].The index n indicates that the frequencies are calculated for the heavy holes (n = 1) or for the light holes (n = 2).The tensor components (2) (4) depend on the germanium parameters such as the material lattice permittivity -Ge p k H , the operating frequency f , the magnetic induction B 0 of external constant magnetic field strength and other[1,2,7].

hN
is equal to 10%, 30% and 90% of the total free

Fig. 1 .Fig. 2 .
Fig. 1.Dependences of waveguide normalized phase constant on the frequency when the heavy holes' concentration is 10% of the total carrier concentration

Fig. 3 .Fig. 4 .
Fig. 3. Dependences of waveguide normalized phase constant on the frequency when the heavy holes' concentration is 30% of the total carrier concentration

Fig. 5 .Fig. 6 .
Fig. 5. Dependences of waveguide normalized phase constant on the frequency when the heavy holes' concentration is 90% of the total carrier concentration