Introduction

Electron cyclotron resonance is the motion of an electron in a helical path which occurs due to exposure of both magnetic field and electromagnetic radiation. It is named after the cyclotron, a particle accelerator that utilizes an oscillating electric field tuned to the resonance frequency to add kinetic energy to charged particles which was invented & patented by Ernest O. Lawrence in 1932^[1]. Resonance phenomena in solids have been studied extensively since the 1940s. Cyclotron resonance was first discovered by ionospheric physicists and it has been studied in ionized gases as well as with free electrons^[2]. Cyclotron resonance has shown its tremendous potential as a fundamental research tool. Many researchers considered the idea that it would be possible to perform cyclotron resonance experiments on solids. In 1951, Dorfmann^[3]published the suggestion of the possible application of cyclotron resonance to solids. Cyclotron resonance experiments in solids provide very different types of information. Cyclotron resonance experiments done in semiconductors have demonstrated the basic predictions of quantum mechanics applied to solids. The results have provided experimental evidence for the existence of energy bands and the effective mass of the electron in the solid body.

The cyclotron resonance is effectively used to determine the effective mass of charge carriers and to study the band structures in semiconductors. In 1953, the first cyclotron resonance studies were carried out in germanium and silicon crystals by G. Dresselhaus et al., 1953, 1955^[3], Lax et al., 1953^[2]which, in conjunction with the effective mass theory by Luttinger and Kohn, 1955. They were able to successfully determine the band-edge parameters for these materials with exceptional accuracy. Since then, this phenomena has been investigated in a large number of elementary and compound materials and their alloys and heterostructures^[4].

Principle

If a uniform magnetic field is applied to a material, the electrons in the material will move in a circular path or an orbit due to Lorentz force. This circular motion which occurs due to the magnetic field, is known as cyclotron. When an electromagnetic signal is applied into the same material, the electrons which were travelling in the circular path will be accelerated by absorbing the energy and the electrons will move in a helical path^[2][4].

Figure 1 shows a charged particle with an effective mass m*, charge q, rotating in a helical path due to magnetic force- B and electromagnetic radiation^[2].

Figure 1. Charged particle in a helical path (Figure: Wathsala Jayarathne).

Cyclotron resonance is the phenomenon when the frequency of the electromagnetic radiation (oscillating electric field) is the same as the rotation frequency of the charged particle.

The cyclotron resonance frequency can be written as^[4];

\( \begin{equation} \omega_{c} = qB/m^{*} \ (1) \end{equation} \)

In cyclotron resonance condition, the maximum energy will be absorbed by the charged particle. 

Cyclotron Resonance in Semiconductors (Silicon & Germanium)

When a magnetic field is applied into a semiconductor material, both the electrons and holes execute a cyclotron motion which is opposite in direction due to the opposite charges. So there are two distinguish cyclotron frequencies for electrons and holes respectively^[3].

\( \begin{equation} \omega_{ce} = qB/m_{e}^{*} \ (2) \end{equation} \)

\( \begin{equation} \omega_{ch} = qB/m_{h}^{*} \ (3) \end{equation} \)

When an electromagnetic radiation is applied to the semiconductor material externally, the charge carriers in the semiconductor will absorb the energy. Maximum absorption or the cyclotron resonance will occur when the frequency of the electromagnetic radiation is equal to either ω_ce or ω_ch depending on the majority charge carriers in the semiconductor. The experiments are done by applying a magnetic field and a microwave radiation to a particular semiconductor. The magnetic field is varied until resonance is achieved. The absorbed energy spectrum is obtained to characterize the semiconductor. The various peaks in the absorption spectra indicate the presence of holes and electrons. The effective mass of the carriers can be found by the above equations (2) & (3).

Experimental Procedure

The semiconductors used for the experiment are pure samples. A typical sample is a disk about 3 mm in diameter and about 0.5 mm thick. Samples have usually been prepared by rough-cutting from a single crystal, grinding to size with abrasives, and etching the surface for several minutes in an etch made up of 1 cc HF, 1 cc H₂O₂ (30%), and 4 cc H₂O^[3].

The experiments are carried out mostly by using microwave radiation and large magnetic fields. Low temperature is also used during the experiment to reduce scattering and thermal vibrations of the lattice. Low temperatures are provided by liquid helium. The magnetic field is varied and the absorption spectrum as a function of the magnetic field is obtained.

Cyclotron Absorption Spectrum

Figure 2 shows the cyclotron resonance results in Silicon at ω = 24 GHz and T = 4K. Magnetic field is applied in (110) plane at 30⁰ from [100] axis.^[3].

Figure 2. Cyclotron resonance absorption spectrum in Silicon (Figure: Wathsala Jayarathne).

According to the absorption spectrum, there are two resonance peaks for electrons which corresponds to longitudinal mass (m_l) and transverse mass (m_t) respectively. Other two resonance peaks belongs to holes which corresponds to light hole and heavy hole bands. In anisotropic materials, the cyclotron mass is given by Shockley equation^[5].

\( \begin{equation} m_{c}^{*} = [\cos^2 \theta / m_{t}^{2} + \sin^2 \theta / m_{l}m_{t}]^{-1/2} \ (4) \end{equation} \)

m_c* denotes the effective mass when the magnetic field makes an angle θ with the longitudinal axis of the energy surface.

Magnetic field make different angles with the longitudinal axis of the energy surface as it change from [001] direction to [110] direction, leading to different resonance peaks in the absorption spectrum. Each resonance peak leads to different effective mass. By fitting the absorbance data into the Shockley equation, it was obtained that m_l = 0.97m₀ and m_t = 0.19m₀ , where m₀  is the free electron mass.

Figure 3 shows the cyclotron resonance results in Germanium at ω = 24 GHz and T = 4K. Magnetic field is applied in (110) p1ane at 60⁰ from [100] axis.^[3].

Figure 3. Cyclotron resonance absorption spectrum in Germanium (Figure: Wathsala Jayarathne).

For Germanium it was obtained that m_l =1.58m₀ and m_t = 0.082m₀

Pure samples are taken to reduce the scattering events due to impurities and imperfections. The charged particles in the semiconductors should execute several circular orbitals, allowing to form a cyclotron before subjected to scattering. If there are impurities it’s difficult to identify whether the absorption is from charge carriers or impurities^[2].

The importance of anisotropy in the effective masses of silicon and germanium requires that data be obtained as a function of crystal orientation. A sample cut with its surface in the (110) plane, oriented so that the applied field can be directed along the [001], [110], and [111] directions by rotation of the sample^[3].

In order to verify the sign of the charge carriers involved, this experiment can be done by using circularly polarized microwaves. Absorption can be observed only when the direction of circular polarization corresponded to the direction of rotation of the charge carriers.

Benefits of Electron Cyclotron Resonance (ECR)

Apart from studying semiconductor band structures, Electron cyclotron resonance phenomena is used in several other fields.

Ion Cyclotron Resonance (ICR)

Electron cyclotron concept is converted to Ion cyclotron resonance to be used in various research experiments. Ion cyclotron resonance occurs when ions rotating in a magnetic field are resonated by external electric field. Fourier-transform ion cyclotron resonance mass spectrometry (FTICR) is the commonly used characterization method based on ICR concept^[6].

Fourier-transform ion cyclotron resonance mass spectrometry is a type of mass analyzer used to determine the mass to charge ratio of ions based on the cyclotron resonance phenomena of ions. This is used to study systems with large mass-to-charge ratios or systems that have very complicated isotope patterns and chemical characterization of natural complex mixtures.

Excitation Method in Plasma Engineering

Another main benefit is the usage of cyclotron resonance as an advanced excitation method in plasma engineering^[7].  Here ions are produced in a magnetically confined plasma, which is heated by microwaves using the resonance effect. The performance of the ion source plays a crucial role in lot of technical fields. Few examples are mentioned below.

• Advanced semiconductor manufacturing
• Advanced cancer treatment
• Electric propulsion devices for spacecraft propulsion
• For particle accelerators, on-line mass separation and radioactive ion charge breeding

References