CYCLOTRON RESONANCE

V. N. Lazukin

I. INTRODUCTION

In 1913 V. K. Arkad’ev[^1] observed selective absorption of short radio waves by ferromagnetic wires. The classical explanation of the effect that he developed was based on the idea of the resonant character of the oscillations of elementary magnetic dipoles under the action of the magnetic field H. Later Ya. G. Dorfman[^2] gave a quantum interpretation of the phenomenon.

In the time that has elapsed since Arkad’ev’s discovery, several kinds of resonant absorption of the energy of a high-frequency electromagnetic field by magnetic substances have been observed and investigated. All these varieties of “magnetic resonance” have a common nature: the absorption of the energy of electromagnetic waves by matter is the result of magnetic-dipole transitions between energy sublevels into which the system of levels of an atom is split by the magnetic field acting on it. These sublevels correspond to different spatial orientations of the magnetic moment of the atom with respect to the field H; their energy depends linearly on the field:

\[ \varepsilon = m_H g \beta H, \tag{1} \]

where \(\beta = \dfrac{e\hbar}{2m_0c}\) is the Bohr magneton, \(m_H\) is the magnetic quantum number, for which the selection rules have the form: \(\Delta m_H = 0;\ \pm 1\). Consequently, resonant absorption of electromagnetic energy must occur at the frequency:

\[ \omega = g\,\frac{eH}{2m_0c}, \tag{2} \]

coinciding with the precession frequency of the magnetic moment about the direction of the field H. The factor \(g\) of the spectroscopic splitting is equal to unity if the magnetic moment of the atom is produced by the orbital motion of electrons; then the frequency (2) is equal to the Larmor frequency.

In the general case, however, \(g \ne 1\), since the magnetic moment is the sum of the magnetic moments of the electronic orbits, the spins, and the nucleus.

In a weak field the magnetic moment of the atom precesses about \(\mathbf H\) as a whole; in a sufficiently strong field the spin-orbit and nuclear magnetic moments rotate about the field independently of one another, with different frequencies determined by the masses of the carriers of the moments.

Magnetic dipole transitions leading to absorption of energy are induced by the magnetic component \(\mathbf H_\nu\) of the electromagnetic field. The probability of these transitions is maximal if \(\mathbf H_\nu \perp \mathbf H\); under this condition an absorption maximum should also be observed.

In a strong field \(\mathbf H\), owing to the independence of the precession of the nuclear and spin-orbit moments, two kinds of magnetic resonance should be distinguished:

A. Nuclear (proton) resonance, observed when the frequency of the field \(\mathbf H_\nu\) coincides with the frequency of precession of the magnetic moment of the nucleus in the constant field \(\mathbf H\) \({}^{3,4}\).

B. Spin-orbit magnetic resonance—when the frequency \(\mathbf H_\nu\) coincides with the precession frequency of the spin-orbit moment of the atom.

When the total orbital moment is equal to zero, a purely spin resonance may occur, which in turn is divided into two varieties:

a) Paramagnetic spin resonance, observed in substances whose paramagnetic ions interact only weakly with one another \({}^{5}\), and

b) Ferromagnetic (or antiferromagnetic) resonance—in substances in which a strong magnetic coupling is established between the ions \({}^{6,7}\).

The experimental study of these effects constitutes one of the main tasks of radiospectroscopy \({}^{8,9}\) and has yielded many important results for physics and chemistry.

Spin paramagnetic resonance should also be observed in systems of free and quasi-bound charges—on electrons and holes in metals and semiconductors, on volume electronic and ionic charges. Indeed, resonant paramagnetic absorption has been observed on conduction electrons of alkali-metal solutions in liquid ammonia \({}^{10}\), in semiconductors \({}^{11}\), in gas-discharge plasma \({}^{12}\), and on electrons in a metal \({}^{13}\).

Dorfman \({}^{14}\) was the first*) to point out that in a system of free or quasi-bound charges there should occur still another resonance effect, fundamentally different from the paramagnetic resonance of electrons, although coinciding with it in frequency. The nature of this effect, called by the author diamagnetic resonan—

*) Somewhat earlier, Shokley \({}^{26}\), considering the band scheme of a crystal, came close to a similar result, but did not formulate it.

is due to the same causes as the diamagnetism of the electron gas, discovered by L. D. Landau^15. Dorfman at the same time pointed out the importance of studying the effect he had predicted for the theory of solids and especially of semiconductors. The subsequent experiments, still not numerous, show that the study of the new phenomenon is important for resolving certain questions of the band theory of semiconductor crystals.

Although the existence of diamagnetic resonance is due to causes related to those that determine the existence of diamagnetism of the electron gas, in essence this effect has very little in common with the magnetic-resonance phenomena listed above. Transitions between different energy states of a charge carrier in a constant magnetic field are induced by the electric vector \(\mathbf{E}\) of the electromagnetic field acting on the charge, when the frequency of this field coincides with the frequency of rotation of the charge around \(\mathbf{H}\). In this sense the mechanism of the phenomenon is akin to the mechanism of absorption of high-frequency energy by a charged particle accelerated in a cyclotron. Therefore the new effect came to be called cyclotron resonance; initially, however, this name did not take hold.

The basic laws of cyclotron resonance can be understood classically. Discussion of the effect from the quantum point of view as applied to a gas of free electrons was developed by Dingle^20. The classical and quantum theory of resonance in semiconductor crystals was developed by Dresselhaus, Kip, and Kittel^21. Somewhat earlier, the quantum theory of the effect in hole semiconductors was discussed by Kohn and Luttinger^22.

II. CYCLOTRON RESONANCE IN AN ELECTRON GAS

1. Electron gas in a magnetic field

According to the classical theory^15 an electron gas has no magnetic susceptibility. This follows from the following considerations. In a magnetic field \(\mathbf{H}\), electrons move along helical lines, with the radii of the circles in the plane perpendicular to \(\mathbf{H}\) connected with the velocity component \(v_\perp\) by the well-known relation:

\[ \frac{eH}{c}=\frac{mv_\perp}{\rho}. \tag{3} \]

In the case of a spatially unbounded system the total magnetic moment of the charges must be different from zero. For a bounded system this is not so: the magnetic moment of the electrons induced by the field \(\mathbf{H}\) is compensated by the oppositely directed moment of the electrons that have undergone collisions with the walls bounding the system. However, a quantum treatment^15 leads

to a different result*). In the plane perpendicular to the magnetizing field, the motion of the electrons is periodic, and the energy of such motion must be quantized; the distribution of electrons over energies in a magnetic field must change with a change of the field and, consequently, the total magnetic moment of the entire system will be nonzero. An electron gas occupying a bounded volume will thus possess diamagnetic properties. The transverse size \(R\) of the system is an important parameter of the theory: if \(R \gg \rho_{\max}\), then practically all the electrons of the system are capable of making at least one complete revolution without collisions with the boundaries of the volume. If, however, \(R \lesssim \rho_{\max}\), then only some of the electrons will be able to move without such collisions. The calculations are considerably simpler for the first case than for the second or for cases close to it.

A number of works\(^{17}\) were devoted to an analysis of the possible cases; in particular, it was shown\(^{13}\) that for \(R \sim \rho_{\max}\) the susceptibility depends on the dimensions of the system. An attempt to generalize the theory was made by Dingle\(^{19}\), who in this connection also considered\(^{20}\) the absorption of high-frequency energy by an ensemble of electrons situated in a field \(\mathbf H\) with \(\mathbf E_\nu \perp \mathbf H\).

Let a constant magnetic field \(\mathbf H\), parallel to the axis of the cylinder, be imposed on a system of free electrons bounded by a cylinder of radius \(R\). Then the Schrödinger equation \(\mathscr H\psi=\mathscr E\psi\) in cylindrical coordinates is written as follows:

\[ -\frac{\hbar^{2}}{2m} \left\{ \frac{1}{\rho}\frac{\partial}{\partial \rho} \left(\rho\frac{\partial \psi}{\partial \rho}\right) +\frac{1}{\rho^{2}}\frac{\partial^{2}\psi}{\partial \varphi^{2}} +\frac{\partial^{2}\psi}{\partial z^{2}} \right\} - \frac{i e H\hbar}{2mc}\frac{\partial \psi}{\partial \varphi} + \left( \frac{e^{2}H^{2}\rho^{2}}{8mc^{2}}-\mathscr E \right)\psi =0. \tag{4} \]

The term depending on \(z\) describes the behavior of the electron wave function when the latter moves along the axis of the cylinder and may be discarded if the problem is restricted to consideration of the energies of orbits in the plane perpendicular to \(\mathbf H\).

By a change of variables, equation (4) is brought to the form:

\[ y\frac{d^{2}M}{dy^{2}}+(l-y+1)\frac{dM}{dy}+nM=0, \tag{5} \]

where

\[ y=\frac{eH\rho^{2}}{2\hbar c}, \tag{6} \]

) L. D. Landau’s theory is set forth in detail in the book by F. Bloch, Molecular Theory of Magnetism, ONTI (1933); see also F. Seitz, The Modern Theory of Solids, p. 662, Gostekhizdat (1949), and S. V. Vonsovskii, The Modern Theory of Magnetism*, p. 10, Gostekhizdat (1952).

\(a\), \(l\), and \(n\) are positive integers (including zero). The solution of (5) that has physical meaning is the function \(^{23}\):

\[ M(n,\ l-1,\ y)=\frac{(-1)^n n!l!}{[(n+l)!]^2}\,L^l_{n+l}(y), \tag{7} \]

where

\[ L^l_{n+l}(y)=\left(\frac{d}{dy}\right)^l \left\{e^y\left(\frac{d}{dy}\right)^{l+n}\left(y^{l+n}e^{-y}\right)\right\} \tag{8} \]

is the Laguerre polynomial in powers of \(n+l\).

Using this solution, we obtain for the wave function:

\[ \psi=e^{il\varphi}\cdot y^{1/2}e^{1/2y}\cdot L^l_{n+l}(y). \tag{9} \]

For positive integer \(n\), the wave functions (9) tend to zero as \(y\to\infty\); in the cross section of a system much larger than \(\rho\), the quantum numbers \(n\) and \(l\) satisfy the inequality

\[ n\ll l \tag{10} \]

and, practically for all occupied states, the corresponding wave functions are localized in the region determined by the values \(y\sim l\). The electron energy is a function of the quantum number \(n\) and coincides with the energy of a linear oscillator:

\[ \varepsilon=\frac{e\hbar H}{mc}\left(n+\frac{1}{2}\right). \tag{11} \]

Thus, if in the absence of a field the electron possesses a continuous energy spectrum, then upon application of a magnetic field its energy splits into a series of discrete levels, the width of which is

\[ \Delta\varepsilon=\frac{e\hbar H}{mc}, \tag{12} \]

and each level is \(\frac{e\hbar H}{mc}\)-fold degenerate. In addition to the states whose energy is expressed by formula (11), there are states in which the electron energy also depends on \(l\):

\[ \varepsilon=\frac{e\hbar H}{mc}\left(n+l+\frac{1}{2}\right). \tag{13} \]

However, the number of such states is very small in comparison with the states (11), and their contribution to the total energy of the ensemble of charges is insignificant \(^{20}\). From the semiclassical point of view, quantization of the electron energy in a magnetic field is equivalent to quantization of the radius of its orbit. In this case both the semiclassical and the quantum theory lead to identical conclusions concerning the influence of \(l\) on the character of the orbits; in particular, it turns out that one and the same value of \(l\) corresponds to a certain family of electron trajectories.

2. Absorption of High-Frequency Energy as a Result of Dipole Interaction

The wave functions (9) with different values of \(n\) determine different energy states of the electron and are mutually orthogonal. These same functions with different values of \(l\) determine identical energy states.

Transitions between different states \((n,l)\) and \((n',l')\) can occur spontaneously, but with vanishingly small probability. If, however, an external electromagnetic field acts on the system of electrons, the probability of transitions increases sharply. To each transition \((n,l)\to(n',l-1)^{*)}\) there may be associated an electric dipole moment proportional to the quantity:

\[ D'(n,l;n',l-1)=\int_{0}^{\infty} y^{l} e^{-y} L_{n+l}^{l}(y)\,L_{n'+l-1}^{l-1}(y)\,dy, \tag{14} \]

which, after evaluating the integral with the aid of recurrence formulas \(^{24}\) and substituting the values of \(L(y)\) and \(y\) according to formulas (6) and (8), gives two values for \(D'\):

\[ D(n,l;n+1,l-1)=\left\{\frac{2e\hbar c(n+1)}{H}\right\}^{1/2}, \tag{15} \]

\[ D(n,l;n,l-1)=\left\{\frac{2e\hbar c(n+l)}{H}\right\}^{1/2}. \tag{16} \]

All other possible values of the moment (14) prove to be identically equal to zero. For most occupied states \(l\gg n\), and therefore the moment (16) is much larger than (15); but the transitions determined by (16) occur at identical energy levels (\(n\)—one and the same) and therefore cannot be accompanied by absorption of energy. The first electric dipole moment (15) is associated with transitions between different energy states, accompanied by resonant absorption of energy at the frequency

\[ \nu=\frac{eH}{2\pi mc}, \tag{17} \]

coinciding with the classical frequency of a free electron moving in a magnetic field \(H\). Transitions between two states \((n,l)\) and \((n',l')\) are induced by the electric component \(\mathbf{E}_{\nu}\) of the wave incident on the electron system; the probability of these transitions per unit time depends, in the general case, on the angle between \(\mathbf{E}_{\nu}\) and the moment \(\mathbf{D}\), and reaches a maximum for \(\mathbf{E}_{\nu}\perp\mathbf{H}\). For the moment (15) this maximum probability, propor-

\({}^{*)}\) Here \(l'=l-1\) has been taken in accordance with the selection rule for the number \(l\): \(\Delta l=0;\ \pm 1\).

tional to the square of the dipole, will be:

\[ P(n, n+1)=\frac{2\pi}{\hbar^{2}}D^{2}(n,n+1)I =\frac{4\pi e c I}{\hbar H}(n+1), \tag{18} \]

where \(I\) is the density of the radiation incident on the system per unit frequency interval. The probability of transitions in the reverse direction is also expressed by formula (18), but since at thermal equilibrium the number of occupied states with quantum number \(n\) is much greater than with \(n+1\), pure absorption will be observed.

In accordance with the value of the probability (18), the energy absorbed in \(1\ \mathrm{cm}^{3}\) per unit time, proportional to the number of electrons \(N\) in this volume, will be:

\[ \varepsilon_{1}=4\pi e^{2}\frac{N}{m}\cdot I \tag{19} \]

and for the relative absorption we have:

\[ w=\frac{\varepsilon}{u}=4\pi e^{2}\frac{N}{m\cdot \Delta \nu}, \tag{20} \]

where \(u=I\cdot \Delta \nu\) is the total energy of the electromagnetic field acting on \(1\ \mathrm{cm}^{3}\) of the system’s volume, and \(\Delta \nu\) is the frequency interval of this field.

3. Absorption as a Result of Quadrupole Interaction

The mixed electric quadrupole moments associated with transitions between the states \((n,l)\) and \((n',l')\) can be calculated using the general relation for a quadrupole:

\[ Q'(n,l;n',l')=e\int \psi_{n,l}\psi^{*}_{n',l'}\{x^{2},y^{2},z^{2},xy,xz,yz\}\,dV. \tag{21} \]

Owing to the independence of the wave functions from \(z\), the components \(Q'\) along the field \(H\) differ from zero only for \(n'=n\) and \(l'=l\). Thus, for \(E_y \parallel H\), quadrupole transitions to different energy levels are not induced, and, consequently, as in the case of dipole transitions with this orientation of the fields, absorption of high-frequency energy will not occur.

Taking account of the angular dependence of the wave functions makes it possible to obtain the selection rules for \(l\) in the case of quadrupole interaction: \(\Delta l=0\) or \(2\). Taking into account the values of the wave functions (9) and the selection rule for \(l\), formula (21) gives, for the quantity proportional to the quadrupole moment:

\[ Q'(n,l;n',l-2)=\int_{0}^{\infty} y^{l}e^{-y}L^{l}_{n+l}(y)L^{\,l-2}_{n'+l-2}(y)\,dy. \tag{22} \]

Application of the same recurrence formulas and substitution of the values \(L(y)\) and \(y\) lead to the following nonzero values of the quadrupole moments:

\[ \begin{aligned} Q(n,l;n+2,l-2)&=\frac{2\hbar c}{H}\{(n+1)(n+2)\}^{1/2},\\ Q(n,l;n+1,l-2)&=\frac{2\hbar c}{H}\{(n+1)(n+l)\}^{1/2},\\ Q(n,l;n+1,l)&=\frac{2\hbar c}{H}\{(n+1)(n+l+1)\}^{1/2},\\ Q(n,l;n,l)&=\frac{2\hbar c}{H}(2n+l+1),\\ Q(n,l;n,l-2)&=\frac{2\hbar c}{H}\{(n+l)(n+l-1)\}^{1/2}, \end{aligned} \tag{23} \]

of which the first three are associated with transitions between different energy states of the electron. These transitions cause absorption of electromagnetic energy at the frequency (17), or at a multiple of it. The other two quadrupoles are associated with transitions at the same energy levels and do not lead to absorption.

The intensity of the quadrupole interaction is much less than the intensity of the dipole interaction, since the probability of quadrupole transitions is much smaller than that of dipole transitions. In reality, the role of quadrupole transitions in resonant absorption will apparently be even smaller, because the quadrupole interaction is connected, by its origin, mainly with wave functions having large values of the quantum number \(l\). These functions must be strongly perturbed by electron collisions. This leads to an even greater decrease in the transition probability and to a considerable decrease in the energy absorbed as a result of the interaction.

In addition to the electric dipole and quadrupole interactions, one might also expect magnetic dipole transitions to participate in the absorption of high-frequency power. However, an estimate\(^{20}\) of the role of such transitions shows that they are not accompanied by absorption, since they occur at identical energy levels. In exactly the same way, the role of states with energy (13) proves to be negligibly small.

Consequently, as a result of the electric dipole and quadrupole interactions, the electron gas in a magnetic field must absorb the energy of electromagnetic waves at a frequency equal to the cyclotron frequency, i.e., the frequency of rotation of an electron in a magnetic field. The absorption will be maximal when the mutually perpendi-

cular orientation of the electric vector \(\mathbf{E}\) of the wave and of the constant magnetic field \(\mathbf{H}\) acting on the electron.

In the case of electrons, the quantum theory of cyclotron resonance leads to the same results as the classical theory\(^{24}\). This is connected with the fact that the approximation of the classical theory gives the correct values of the energy levels for a harmonic oscillator. In the case of resonance of holes in semiconductors the results turn out to be somewhat different.

4. Line Width of Absorption

The lines of cyclotron resonance absorption must have a considerable width, composed mainly of two terms: the natural width \(\Delta \nu_{\mathrm{nat}}\), due to dipole transitions, and the width \(\Delta \nu_{\mathrm{st}}\), arising from collisions.

The first term is expressed through the Einstein coefficient of spontaneous emission:

\[ \Delta \nu_{\mathrm{nat}} = \frac{64 \pi^4}{3 h c^3} \sum \nu D^2, \tag{24} \]

where the sum is taken over all possible transitions. Substitution of the values of \(D\) and \(\nu\) gives:

\[ \Delta \nu_{\mathrm{nat}} = \frac{32 \pi^2}{3}\frac{e^2}{m c^2}(n+1). \tag{25} \]

Thus, the normal line width increases strongly with increasing absorption energy.

The maximum value of \(n\) can be obtained from (11). Estimation of \(\varepsilon_{\max}\) by means of (19) and substitution in (25) makes it possible to estimate the approximate value of \(\Delta \nu_{\mathrm{nat}}\):

\[ \Delta \nu_{\mathrm{nat}} \sim 10^{-3}/H. \]

The term due to collisions is proportional to the collision time:

\[ \Delta \nu_{\mathrm{st}} \sim 2\tau \tag{26} \]

and the total width is

\[ \Delta \nu = \frac{32 \pi^2 e \varepsilon_{\max}}{3 c h H} + 2\tau. \tag{27} \]

For fields of the order of several kilooersteds, usually \(\Delta \nu_{\mathrm{nat}} \ll \Delta \nu_{\mathrm{st}}\), and the width is determined mainly by the magnitude of the collision time.

5. Cyclotron Resonance in Metals

The detection and study of cyclotron resonance absorption in metals apparently present great difficulties. For the effect to exist in any system of charge carriers, the collision time \(\tau\) must be such that the carrier has time to complete in the magnetic field at least one full revolution,

i.e., this time must be no less than the period of revolution of the charge. This leads to the condition

\[ \omega \tau > 1. \tag{28} \]

For most metals at room temperatures \(\tau \sim 10^{-14}\) sec, which requires frequencies of order \(10^{14}\) cycles/sec and, consequently, unattainably high fields of order \(10^7\) oersted. At extremely low temperatures the time \(\tau\) increases to \(10^{11}\) sec, and the corresponding value of \(\nu\) becomes \(10^{11}\) cycles/sec \((\lambda = 0.3\ \text{cm})\). In this case the resonance could be observed in fields of about 40 kilooersted.

Here, however, there is still another difficulty. For cyclotron resonance to exist, it is also necessary that the depth of penetration of the electromagnetic field into the metal be no less than the diameter of the orbit of revolution of the electron in the resonant field \(H_{\text{res}}\). This cannot always be fulfilled. Thus, for copper at hydrogen temperatures and frequency \(\nu = 10^{11}\) cycles/sec the skin depth will be about \(2 \cdot 10^{-6}\ \text{cm}\), while the radius of the orbit of revolution in fields corresponding to this frequency will be of order \(4 \cdot 10^{-7}\ \text{cm}\), i.e. not much smaller. At helium temperatures this ratio will be still less favorable.

The losses due to the skin effect are equal\({}^{20}\) to the losses due to cyclotron-resonance absorption, which leads to a reduction of the quality factor of the radio-frequency circuit by a factor of two and thus to the need to increase the sensitivity of the method by at least a factor of two. In reality, however, the sensitivity of the method for observing the effect in metals will have to be increased much more, because, owing to the low power of sources of radiation in the millimeter range, the intensity of the resonance lines will be very small. In addition, the polarization effect will apparently make the observations extremely difficult and in a number of cases even render them impossible.

All these circumstances make attempts\({}^{25}\) to observe cyclotron resonance in metals unpromising, at any rate at the present time. Although experiments on paramagnetic resonance at wavelengths of the order of millimeters and even fractions of a millimeter are already known, nevertheless the present state of the technique for these waves and, in particular, the lack of sufficiently powerful radiation sources make experiments in this range difficult. These difficulties increase considerably in the case of observations of the effect in metals, under conditions in which it is necessary to apply extremely low temperatures and much higher magnetic-field strengths. The conditions will be somewhat more favorable in experiments on alloys whose electrical conductivity at low temperatures is not too large.

Nevertheless, it would be mistaken to abandon attempts in this direction, in view of the fact that cyclotron resonance will apparently make it possible to obtain some information about the behavior of electrons in the surface layer of metals (in particular, about their distribution over energies), and hence about the properties of this layer.

III. CYCLOTRON RESONANCE IN SEMICONDUCTORS

1. Effective Mass and Energy Surfaces of Semiconductor Crystals

The conditions for observing cyclotron resonance in semiconductors are much more favorable than in metals, and they proved to be the first and, so far, the only representatives of substances in which this effect has been investigated.

The importance of studying cyclotron resonance in semiconductors for their theory was first pointed out by Dorfman in the work cited above. Later Shockley \(^{26}\) showed that germanium single crystals are the most suitable objects for carrying out the corresponding experiments, and noted that such experiments may help in solving the question of the form of the energy surfaces in the Brillouin zone—a question that is one of the fundamental ones for the band theory of the solid state.

A free charge under the action of a magnetic field moves along a helical line, whose parameters are determined by the magnitude of the field, the velocity of the carrier, and the ratio of its charge to its mass. In the case of a quasifree charge in a semiconductor crystal, its motion is complicated by the action of the crystalline field.

From consideration \(^{27}\) of the motion of a charge in the periodic field of the crystal lattice it follows that an external field accelerates an electron in a crystal as a free classical particle with mass \(m^*\), determined by the equation:

\[ \frac{1}{m^*}=\frac{1}{\hbar^2} \left| \begin{array}{ccc} \dfrac{\partial^2 \varepsilon}{\partial k_x^2} & \dfrac{\partial^2 \varepsilon}{\partial k_x \partial k_y} & \dfrac{\partial^2 \varepsilon}{\partial k_x \partial k_z} \\[6pt] \dfrac{\partial^2 \varepsilon}{\partial k_x \partial k_y} & \dfrac{\partial^2 \varepsilon}{\partial k_y^2} & \dfrac{\partial^2 \varepsilon}{\partial k_y \partial k_z} \\[6pt] \dfrac{\partial^2 \varepsilon}{\partial k_x \partial k_z} & \dfrac{\partial^2 \varepsilon}{\partial k_z \partial k_z} & \dfrac{\partial^2 \varepsilon}{\partial k_y^2} \end{array} \right|, \tag{29} \]

where

\[ \varepsilon=\frac{h}{2} \left( \frac{k_1^2}{m_1^*} + \frac{k_2^2}{m_2^*} + \frac{k_3^2}{m_3^*} \right) \tag{30} \]

is the electron energy as a function of the wave vector \(|\mathbf{k}|=\dfrac{mv}{h}\) of the wave function describing the behavior of the electron in the crystal; \(m_1^*, m_2^*, m_3^*\) are the principal components of the effective-mass tensor. The values of the components of the mass tensor depend on the composition and structure of the lattice; in crystals of any symmetry there exist, at a minimum, three axes along which the force acting on the electron and the acceleration of its motion coincide in direction. The principal components

mass tensor:

\[ m_1^*=\frac{\hbar^2}{\dfrac{\partial^2 \varepsilon}{\partial^2 k_1}}; \qquad m_2^*=\frac{\hbar^2}{\dfrac{\partial^2 \varepsilon}{\partial k_2^2}}; \qquad m_3^*=\frac{\hbar^2}{\dfrac{\partial^2 \varepsilon}{\partial k_3^2}}. \tag{31} \]

In crystals of arbitrary symmetry they are different, and only for crystals of cubic symmetry can they coincide with one another.

The effective mass takes into account the interaction of a charge moving in crystals (an electron or a hole) with the lattice. The tensor character of this quantity and its dependence on direction are due to the fact that the magnitude of the potential barriers traversed by the charge is different for different crystallographic directions.

Knowledge of the components of the effective-mass tensor for various directions in a crystal makes it possible to determine the form of the energy surfaces and to verify the basic conclusions of band theory.

Theoretical calculations28, numerous experimental data on magnetoresistance29, 30, piezoresistance31, and infrared absorption32 of germanium crystals lead to the conclusion that surfaces of constant energy in the Brillouin zone for these crystals do not have a spherical form, but are more complex. Thus, for n-type germanium these surfaces consist of a family of ellipsoids of revolution with the major axis along the [111] axis of the crystal; for n-type silicon crystals these surfaces are represented by a family of analogous ellipsoids with the axis of revolution along the [100] axis. In the case of p-type crystals these surfaces are still more complex21, 33.

2. Classical theory for the case of an isotropic carrier mass

For the case in which the mass of the charge carrier is isotropic, the conditions of motion will not depend on the orientation of the crystal relative to the external fields. The equation of this motion, neglecting the high-frequency magnetic field \(\mathbf H_1\), is written as follows:

\[ m^*\left(\frac{d\mathbf v}{dt}+\frac{1}{\tau}\mathbf v\right) = e\left(\mathbf E+\frac{1}{c}[\mathbf v\mathbf H]\right). \tag{32} \]

Assuming that \(\mathbf H \parallel Oz\) and that the electromagnetic field is plane-polarized, with \(\mathbf E_1 \parallel Ox\), (32) may be written:

\[ \left. \begin{aligned} m^*\left(i\omega+\frac{1}{\tau}\right)v_x &= eE_x+\frac{e}{c}v_yH,\\ m^*\left(i\omega+\frac{1}{\tau}\right)v_y &= \frac{e}{c}v_xH. \end{aligned} \right\} \tag{33} \]

Finding \(v_x\) from these equations and substituting it into the expression for

high-frequency conductivity, we find:

\[ \sigma=\frac{j_x}{E_x}=\frac{Nev_x}{E_x}=\sigma_0\left[\frac{1+i\omega\tau}{1+(\omega_c^2-\omega^2)\tau+2i\omega\tau}\right], \tag{34} \]

where

\[ \sigma_0=\frac{Ne^2\tau}{m^*} \]

is the static conductivity of the crystal, \(N\) is the number of charges in \(1\ \mathrm{cm}^3\), \(\omega_c\) is the cyclotron frequency, and \(\omega\) is the frequency of the high-frequency field acting on the crystal.

The losses of high-frequency power are proportional to the real part of the conductivity; separating in (34) the real part from the imaginary part, we obtain:

\[ \frac{\sigma_R}{\sigma_0}= \frac{1+\nu'^2-\nu_c'^2} {(1+\nu_c'^2-\nu'^2)^2+4\nu'^2}, \tag{35} \]

where \(\nu'=\omega\tau\) and \(\nu_c'=\omega_c\tau\).

In Fig. 1 this function is presented for four values of \(\omega\tau\); resonant absorption becomes distinctly noticeable at \(\omega\tau=3\).

The absorption of high-frequency power, expressed by formula (35), depends differently on the field \(H\) for different ratios between the frequency of the wave incident on the crystal and the cyclotron frequency:

Fig. 1. Resonant absorption as a function of magnetic field for different values of \(\omega\tau\).

a) \(\nu_c' \gg \nu'\); this case occurs in strong magnetic fields. \(\sigma_R\) and, consequently, the absorption are proportional to \(\frac{1}{H^2}\). The radii of the orbits, determined by formula (3), are very small, and the carriers have a weak drift along the electric field.

b) \(1 \gg \nu'_c \gg \nu'\)—the case occurring in weak fields. The absorption is proportional to \(H^2\).

c) \(\nu' \gg 1 \gg \nu'_c\)—the case realized in the infrared frequency region, in fields \(H\) considerably smaller than the resonant one. The absorption is proportional to \(\dfrac{1}{H^2}\).

d) \(\nu'=\nu'_c \gg 1\)—the case corresponding to resonant cyclotron absorption; \(\sigma_R=\dfrac{1}{2}\sigma_0\). Thus, upon the occurrence of cyclotron resonance, the high-frequency conductivity is equal to one half of the static conductivity. Hence one obtains

\[ \frac{\sigma_{R\,\mathrm{res}}}{\sigma_{R\,(H=0)}}=\frac{1}{2}\,\nu_c'^2, \tag{36} \]

i.e. the ratio of the high-frequency conductivity at resonance to this same conductivity in the absence of a magnetic field is equal to one half the square of the product of the cyclotron frequency and the carrier collision time.

The condition for observability of cyclotron resonance, \(\omega_c\tau>1\), may be written in terms of the mobility \(\mu\) of the carriers, using the relation between this quantity and the collision time: \(\mu=\dfrac{e}{m^*}\tau\). This gives

\[ \mu>\frac{e}{m^*\omega_c} \]

or:

\[ \frac{m^*}{m_0}\,\mu>\frac{e}{m_0\omega_c}. \tag{37} \]

For a frequency of \(24000\) Mc/s,

\[ \frac{m^*}{m_0}\cdot\mu>11000\ \mathrm{cm}^2/\mathrm{volt}\cdot\mathrm{sec}. \]

From the experimental point of view it is of interest to estimate the intensity of cyclotron resonance in comparison with spin resonance. While the energy densities of the high-frequency electric and magnetic fields in the resonance cavity are quantities of the same order, the probabilities of electric and magnetic dipole transitions, which respectively determine the absorption intensities in cyclotron and spin resonances, differ greatly from one another. Indeed, for a free electron:

\[ \frac{\mathcal{P}_{\mathrm{cycl}}}{\mathcal{P}_{\mathrm{spin}}}\simeq \frac{D^2}{\beta^2} = \frac{\dfrac{2e\hbar c}{H}(n+1)} {\left(\dfrac{e\hbar}{2m_0c}\right)^2} = \frac{8m_0^2c^3}{e\hbar H}(n+1), \tag{38} \]

i.e.

\[ \frac{\mathcal{P}_{\mathrm{cycl}}}{\mathcal{P}_{\mathrm{spin}}}\approx 10^{11}\div 10^{12}. \tag{39} \]

A similar result can also be arrived at by comparing the quality factor \(Q\) of the sample in cyclotron and spin resonances and understanding by \(Q\) the ratio of the energy of the high-frequency electric field released in the resonator in one period to the energy absorbed in the sample during the same time. The first quantity, calculated for \(1\ \mathrm{cm}^{3}\) of the resonant cavity, is equal to \(\varepsilon E^{2}/8\pi\), and the second, referring to the same volume of the sample\({}^{21}\), is equal to \(\sigma_R E^{2}/\nu\).

Consequently,

\[ Q_{\text{cycl}}=\frac{\varepsilon \nu}{8\pi\sigma_R} =\frac{m^{*}\varepsilon \nu}{4\pi N e^{2}\tau}. \tag{40} \]

which gives, for the case of resonance of free electrons at centimeter-range frequencies:

\[ Q_{\text{cycl}}=\frac{10^{12}}{N}. \]

The corresponding quantity for spin resonance turns out to be\({}^{41}\)

\[ Q_{\text{spin}}=\frac{kT}{N\beta^{2}\omega\tau}\simeq\frac{10^{24}}{N}. \]

Thus:

\[ \frac{Q_{\text{cycl}}}{Q_{\text{spin}}}\simeq 10^{-12}. \]

\(1/Q\) is a measure of the absorptive capacity of the sample, and from the estimate given it is clear that, at the same carrier density in the sample, the intensity of cyclotron-resonance absorption exceeds the corresponding magnitude of spin-resonance absorption by a factor of \(10^{12}\). In other words, cyclotron resonance can be detected at a carrier concentration \(10^{12}\) times smaller than that required for the detection of spin resonance. Taking into account the sensitivity of modern radiospectroscopic methods\({}^{9}\), one may expect the detection of cyclotron resonance in a sample containing \(10^{4}\div 10^{5}\) carriers per \(1\ \mathrm{cm}^{3}\).

3. Influence of the Polarization of the Sample

The circumstance that cyclotron resonance can be observed at a very low carrier concentration is favorable, especially in view of the effects associated with the polarization of the sample.

Under conditions of a high carrier concentration, an additional force acts on them, due in its origin to the dielectric polarization of the sample in the microwave electric field,

as a result of this, the field strength acting on the charges will be:

\[ \mathbf{E}_i=\mathbf{E}_\nu-D\mathbf{P}, \tag{41} \]

where the polarization \(\mathbf{P}\) is determined by the equation

\[ P=\chi_0 E_i-\frac{iNe v}{\omega} \tag{42} \]

and the depolarization factor \(D\) depends on the shape of the specimen; for a sphere, for example, \(D=\frac{4\pi}{3}\). Taking account of the change in the field, the equation of motion will be:

\[ m^*\left[i\left(\omega-\frac{D_iNe^2}{m^*\omega}\right)+\frac{1}{\tau}\right]\mathbf{v} = \frac{e}{1+D\chi_0}\mathbf{E} + \left[\frac{e}{c}\mathbf{vH}\right]. \tag{43} \]

Here:

\[ D_i=\frac{D}{(1+D\chi_0)} \quad \text{and} \quad \chi_0=\frac{(\varepsilon-1)}{4\pi}. \tag{44} \]

Thus, in the equation of motion of the carriers there appear additional terms, which leads to a change in the resonance equation (34): instead of \(\omega\), it will contain the quantity

\[ \omega_D=\omega\left(1-\frac{D_iNe^2}{m^*\omega^2}\right), \tag{45} \]

which depends on the carrier concentration and on the shape of the specimen. The shift of the resonance will depend on these same parameters. This shift will be the more noticeable, the larger the quantity

\[ \omega_p=\frac{D_iNe^2}{m^*}. \tag{46} \]

The critical concentration \(N_{\mathrm{cr}}\) is determined from the condition \(\omega_p=\omega\):

\[ N_{\mathrm{cr}}=\frac{m^*\omega^2}{D_i e^2}. \tag{47} \]

Substitution of values for the most typical specimens and frequencies gives:

\[ N_{\mathrm{cr}}\simeq 10^{12}\div 10^{14}\ \mathrm{cm}^{-3}. \]

Consequently, in germanium and silicon at \(4^\circ\) K, when the carrier concentration is considerably less than the critical one, the influence of this effect will be negligible, and specimens of any shape may be used.

However, resonance effects at temperatures above \(4^\circ\) K, depending on the shape of the specimen and on the temperature, have been observed experimentally\(^{21,34}\); this is precisely what distinguishes a resonance of polarization nature from a true cyclotron resonance.

The influence of this effect can be considerably reduced (even if \(N\) is sufficiently large) by choosing a more advantageous shape of the specimen and its orientation with respect to the high-frequency electric field in the resonator.

4. Theory of Cyclotron Resonance in the Conduction Band of Ge and Si (n-Type Resonance)

In accordance with the existence in semiconductors of two types of conductivity—electron and hole conductivity—one must also distinguish two types of cyclotron resonance. In the first case, quasifree electrons located in the conduction band of the crystal are responsible for the resonance absorption, while in the second case it is holes formed in the valence-bond band as a result of one cause or another. Experimentally, both of these varieties of cyclotron resonance are comparatively easy to distinguish.

The present state of the band theory of semiconductors is such that it does not make it possible theoretically to derive the principal features of the energy bands and, in particular, to derive analytically the equations of the energy surfaces for various cases. Even in the comparatively simple case of germanium, attempts made up to the present time[^28] have not yielded reliable results. Everything that is known about the form of the energy surfaces in the conduction band and in the valence-bond band is the result of interpretation of experimental data on magnetoresistance, infrared absorption, and, in part, cyclotron resonance[^21].

From these data it follows that in the conduction band of Ge and Si the energy surfaces constitute a system of spheroids, which can be described by an equation of the form

\[ \varepsilon(k)=\hbar^{2}\left(\frac{k_x^{2}+k_y^{2}}{2m_t}+\frac{k_z^{2}}{2m_l}\right). \tag{48} \]

Here \(m_t\) and \(m_l\) are the components of the effective-mass tensor along the minor and major axes of the spheroid, respectively.

The effective mass \(m^{*}\), which determines the cyclotron frequency in any direction, can be obtained from the solution of the equation of motion of the carrier in the fields \(\mathbf{E}\) and \(\mathbf{H}\):

\[ \frac{d\mathbf{P}}{dt}=e\left\{\mathbf{E}+\frac{1}{c}[\mathbf{vH}]\right\}, \tag{49} \]

where

\[ \mathbf{P}=\mathbf{p}-\frac{e}{c}\mathbf{A}; \]

\(\mathbf{p}\) is the momentum and \(\mathbf{A}\) is the vector potential, which determine the velocity \(\mathbf{v}\) by means of the equation

\[ \mathbf{v}=\nabla_p \mathscr{H}(\mathbf{P}). \tag{50} \]

Here

\[ \mathscr{H}(\mathscr{P})=\frac{\mathscr{P}_x^{2}}{2m_t}+\frac{\mathscr{P}_y^{2}}{2m_t}+\frac{\mathscr{P}_z^{2}}{2m_e}. \tag{51} \]

If the field \(\mathbf H\) makes an angle \(\theta\) with the \(Oz\) axis and lies in the plane \(xOz\), then its components will be:

\[ H_x=H\sin\theta;\quad H_y=0;\quad H_z=H\cos\theta . \]

Then, taking into account the difference in cyclotron frequency along the major axis of the spheroid and perpendicular to it, the equation of motion gives:

\[ \left. \begin{aligned} i\omega \mathscr P_x-\omega_t y\cos\theta&=0,\\ i\omega \mathscr P_y-\omega_l \mathscr P_z\sin\theta+\omega_t \mathscr P_x\cos\theta&=0,\\ i\omega \mathscr P_z+\omega_t \mathscr P_y\sin\theta&=0. \end{aligned} \right\} \tag{52} \]

The corresponding secular equation has the solution

\[ \omega^2=\omega_t^2\cos^2\theta+\omega_t\omega_l\sin^2\theta, \tag{53} \]

i.e., the effective mass determining the cyclotron-resonance frequency as a function of \(\theta\) will be:

\[ \left(\frac{1}{m^*}\right)^2=\frac{\cos^2\theta}{m_t^2}+\frac{\sin^2\theta}{m_t m_l}. \tag{54} \]

As has already been noted, all the experimental data suggest that, both in Ge and in Si, the energy surfaces in the conduction band are elongated along the crystallographic axes \([111]\) for the former and \([100]\) for the latter. The number of such surfaces, as follows from the multiplicity of the representation, must be equal to four. And, generally speaking, in each given direction inside the crystal four resonance peaks should be observed experimentally. However, not all four peaks will always be observed, because the spin-orbit interaction will in some directions reduce the multiplicity of the representation\(^{33,34}\), which will lead to the merging of the energy surfaces in these directions. Therefore, when resonance is observed for the corresponding orientations of the crystal, a merging of the resonance maxima will be observed.

5. Cyclotron resonance in the valence band of Ge and Si

(\(p\)-type resonance)

The edge of the valence band of Ge and Si crystals is located at \(\mathbf k=0\); the energy surfaces in this region have the form of a family of distorted spheroids, whose equations, obtained by the perturbation method\(^{21}\), may in first approximation be written in the form

\[ \varepsilon(k)=Ak^2\pm\{B^2k^4+C^2(k_x^2k_y^2+k_y^2k_z^2+k_z^2k_x^2)\}, \tag{55} \]

where \(A\), \(B\), \(C\) are constants determined by the values of the perturbation matrix—

CYCLOTRON RESONANCE

perturbations. The second term in the braces of equation (55) describes the distortions of the spheroids. These distortions of the energy surfaces near the edge of the valence band have a complex effect on the cyclotron-resonance frequency. If, in the case of \(n\)-resonance, each ellipsoid of energy corresponds to one value of the cyclotron frequency for a given field direction, then in the case of a \(p\)-type resonance, under the same conditions, a certain family of frequencies will be observed. The distribution of these frequencies in the general case depends on the values of the quantum numbers of the system \(^{22}\). Owing to the complications arising from the nature of the degeneracy at the top of the valence band, the quantum theory leads to different energy levels and selection rules for the case of small quantum numbers in hole resonance, whereas for large numbers the results coincide with the classical ones. In experiments carried out up to the present time \(^{35,36,37,38}\), the values of the quantum numbers were sufficiently large \((n \sim 5)\), and additional resonance peaks were in most cases not observed; by lowering the temperature to \(1 \div 1.5^\circ\mathrm{K}\), one may expect the appearance of additional resonance maxima. Weak traces of such resonance maxima have been observed experimentally \(^{34}\).

A general expression for the effective mass as a function of the orientation of the crystal relative to the fields can be obtained with the aid of the concept, introduced by Shockley \(^{40}\), of mass tubes, or cyclotron tubes.

In the classical approximation the magnetic field does not change the energy of motion of a carrier over the energy surface and does not change the projection \(\mathscr P_H\) of the vector \(\mathbf P=\hbar\mathbf k\) on the direction of the magnetic field. The motion of the carrier under the action of the field will take place in the region bounded by the values \(\mathscr P_H\) and \(\mathscr P_H+d\mathscr P_H\) on one side, and \(\varepsilon\) and \(\varepsilon+d\varepsilon\) on the other. This region is called a mass tube or a cyclotron tube. For spherical energy surfaces Shockley’s tube is a toroid, whose cross section has the shape of a parallelogram; for more complicated surfaces the tubes will have a more complicated form.

From equation (49) we have:

\[ c\,d\mathscr P = e v_\perp H\,dt, \]

where \(v_\perp\) is the projection of the carrier velocity on the plane perpendicular to \(H\). Hence \(^{40}\):

\[ \oint \frac{c\,d\mathscr P}{eH v_\perp}=\oint dt=\frac{2\pi}{\omega}. \tag{56} \]

When cyclotron resonance occurs,

\[ \omega=\omega_c=\frac{eH}{m^*c}, \]

and (56) gives:

\[ m^*=\oint \frac{d\mathscr P}{2\pi v_\perp}. \tag{57} \]

Introducing in phase \(k\)-space, in whose coordinates the equation of the energy surface is written, a cylindrical system instead of a Cartesian one and expressing \(\mathcal P\) in terms of the energy, we obtain in such a system:

\[ m^*=\frac{\hbar}{2\pi}\oint \frac{\rho\,d\varphi}{\dfrac{\partial \varepsilon}{\partial \rho}} . \tag{58} \]

Application of this result to concrete cases of energy surfaces in the valence band of germanium and silicon requires complicated calculations. Such a calculation was carried out in \(^{21}\) for the case of \(\mathbf H\) parallel to the plane \([110]\) of the crystal, with zero projection of the wave vector \(\mathbf k\) on the direction \(\mathbf H\), and gave the following relation for \(m^*\):

\[ m^*=\frac{\hbar^2}{2}\, \frac{1}{A\pm\left[B^2+\left(\frac{C}{2}\right)^2\right]^{1/2}} \times \]

\[ \times \left\{ 1+ \frac{C^2(1-3\cos^2\theta)^2} {64\left[B^2+\left(\frac{C}{2}\right)^2\right]^{1/2} \left\{A\pm\left[B^2+\left(\frac{C}{2}\right)^2\right]^{1/2}+\ldots\right\}} \right\}, \tag{59} \]

where \(\theta\) is the angle between \(\mathbf H\) and the \([100]\) axis of the crystal. This may be written in the form

\[ m^*=m_0\{a+b(1-3\cos^2\theta)^2\}, \tag{60} \]

where \(a\) and \(b\) are new constants depending on the constants \(A\), \(B\), and \(C\), which determine the shape of the energy surface.

By measuring the effective mass \(m^*\) for two values of \(\theta\), we obtain a system of two equations from which it is easy to calculate \(a\) and \(b\); then, using the connection between these constants and the quantities \(A\), \(B\), and \(C\), it is easy to calculate the values of the latter and thereby determine the approximate form of the energy surface.

IV. EXPERIMENTAL RESULTS

1. Experimental Method

The methods of experimental investigation of cyclotron resonance in semiconductors do not differ in any essential way from the well-known \(^{9}\) methods used in the study of paramagnetic resonance in the low-temperature region. The slight difference is that the specimen under investigation must be placed in the antinode of the electric field of a standing wave in the resonator, and the constant magnetic field \(\mathbf H\) must be perpendicular to the vector \(\mathbf E\) of the high-frequency field. A diagram of a typical experimental setup \(^{36}\) is presented in Fig. 2. A stabilized klystron gene-

rator feeds the resonant cavity, at the center of which the sample under study is placed. The magnetic field, perpendicular to the vector \(\mathbf{E}\) and to the chosen direction in the crystal, is produced by an electromagnet and can be varied from zero to some maximum value. During the experiment the sample, which is in liquid helium \((T = 4.2^\circ \mathrm{K})\), can be rotated, changing the angle

Fig. 2. Schematic of the experimental setup for observing cyclotron resonance at low temperatures. At the top the resonant cavity with the sample under study is shown.

between the magnetic field and a preselected crystallographic direction. Amplitude modulation of the microwave power, or modulation of the magnetic field, is necessary in order to record the course of the absorption as a function of \(H\). The detecting system and the recorder provide automatic recording of the absorption curve.

For the experiment to be successful, the following two conditions must be satisfied:

1. The presence in the sample of a sufficient number of charge carriers to produce resonance absorption of noticeable intensity. The estimate given above shows that, in order to detect a cyclotron-absorption signal, \(10^4\) carriers per \(1 \ \mathrm{cm}^3\) of the sample are sufficient. According to the estimate in \({}^{38}\), the number of carriers in a germanium crystal, depending on the frequency used and on the sensitivity of the recording circuit, lies within the range \(10^5 \div 10^7\) per \(1 \ \mathrm{cm}^3\); according to the estimate of the authors of work \({}^{36}\), this number was approximately \(10^9 \ \mathrm{cm}^{-3}\).

2. The mean free time \(\tau\) of the carriers must be sufficient for the carrier to complete at least one revolution between two successive collisions. An approximate criterion for this is relation (31); for values \(\omega \tau \lesssim 1\), the resonance becomes unobservable.

The requirement on the free path time, in addition to the need to carry out experiments at very low temperatures, restricts the range of possible objects of investigation to materials of high chemical purity, so that the possibilities of collisions of charge carriers with impurity atoms are reduced to a minimum.

The second of the indicated requirements contradicts the first: at helium temperatures the equilibrium number of carriers in the conduction band is much smaller than necessary. Therefore, for the success of the experiment some mechanism is needed for exciting carriers from nonconducting states into the conduction band. In germanium crystals at frequencies below \(2.4 \cdot 10^{10}\) c/s, the intensity of the high-frequency field \(\mathbf{E}\) in the resonant cavity is sufficient to ionize impurity atoms by impacts of the carriers present in the crystal and thereby to increase the number of the latter.

In the first experimental work of its time \(^{35}\) the process of carrier excitation was precisely of this kind. The cyclotron-resonance absorption signal appeared abruptly upon reaching a certain threshold value of the high-frequency power, which is connected with the existence of a threshold of the electric field of the wave, beyond which an avalanche process of ionization of donors or acceptors arises. Such an excitation mechanism is not possible in all semiconductors, since for its realization it is necessary that the energy interval between free and bound carriers be small. In germanium this condition is fulfilled (\(\Delta \varepsilon \approx 0.01\) eV); in silicon, however, \(\Delta \varepsilon \approx 0.05\) eV, and radio-frequency ionization is impossible. In these cases it is necessary to use another method of exciting carriers into the conduction band. An attempt \(^{38}\) to use heating of the sample for this purpose was unsuccessful because of the large decrease in the time \(\tau\) at temperatures ensuring a sufficient carrier concentration.

More successful were attempts to create a sufficient number of carriers by illuminating the sample with visible light \(^{34}\) and infrared radiation \(^{21, 34, 36}\). The subsequently applied modulation of the light illumination \(^{38}\) proved, apparently, still more encouraging, since it made it possible to obtain a higher signal-to-noise ratio. By using illumination close to monochromatic, it will probably be possible to determine the carrier excitation levels for each of the observed resonance peaks.

Confirmation of the generally accepted view concerning the signs of carriers in \(n\)- and \(p\)-semiconductors was obtained \(^{21, 38}\) by a small change in the experimental procedure. Instead of a rectangular resonant cavity, an axially symmetric one was used, in which, with the aid of the equivalent of a quarter-wave plate, high-frequency oscillations polarized circularly with the \(\mathbf{E}\)-vector perpendicular to the axis were excited (Fig. 3). When \(\mathbf{H}\) is applied parallel to the axis of the cavity, absorption should be observed only when the direction of rotation of the carriers in their orbits coincides with the direction

of rotation of the vector \(\mathbf{E}\) of the electromagnetic wave. Apparently, the use of such a method is advantageous only for determining the sign of the charge and is of little use for measuring effective masses, since, in the presence of a small asymmetry in the position of the sample with respect to the resonator axis, additional resonance peaks are observed, caused by the excitation of higher wave modes appearing in this case.

2. Resonance in \(n\)-type germanium and silicon

The first experimental observation\({}^{35}\) in single crystals of germanium of electron (\(n\)-type) and hole (\(p\)-type) conductivity was carried out at a frequency of \(9.05 \cdot 10^{9}\) cycles/sec at \(T = 42^\circ\) K with radio-frequency excitation of carriers.

In \(n\)-type crystals, resonance absorption was observed in a field of \(370 \pm 5\) oersteds, with a line width of about 100 oersteds. The calculated effective mass of the conduction electrons proved to be equal to \(0.11m_{0}\), where \(m_{0}\) is the rest mass of a free electron; the mean free time, calculated from the width of the curve, proved to be \(7 \cdot 10^{-11}\) sec.

In \(p\)-type crystals, two resonance peaks were observed at \(H_{1} = 125 \pm 5\) and \(H_{2} = 970 \pm 5\) oersteds, with half-widths of the lines of 50 and 100 oersteds, respectively. The effective masses of the holes proved to be \(0.04m_{0}\) and \(0.3m_{0}\), and the mean free times \(5 \cdot 10^{-11}\) and \(2 \cdot 10^{-11}\) sec. In crystals of both types the positions of the resonance peaks and, consequently, the values of the effective masses did not depend on the orientation of the crystal with respect to the fields.

Fig. 3. Resonance cavity for observing resonance in a circularly polarized \(E_y\)-field.

Subsequent experiments\({}^{34,36,37,38}\) with excitation of carriers by light or infrared irradiation made it possible to refine these results considerably.

Experiments by Lax and co-workers\({}^{36,37}\) confirmed the theoretical considerations concerning the anisotropy of cyclotron resonance absorption in single crystals of Ge and Si and the dependence of the electron effective mass on the orientation of the crystal. Figure 4 presents absorption curves in an \(n\)-type germanium crystal for four different directions of \(\mathbf{H}\) relative to the crystallographic axes. The change in the position of the maxima upon rotation of the crystal and the change in their number from one to three are clearly visible. Four resonance peaks were observed by the same authors on a specially etched crystal.

Figure 5 presents curves, calculated theoretically from equation (54), of the dependence of the effective mass of electrons in Ge

Fig. 4. Cyclotron absorption in \(n\)-type germanium for four different directions of H in the plane \(\{110\}\); the indices at \(P\) denote the direction of the field. Temperature \(4^\circ\) K, frequency \(8.9 \cdot 10^9\) cycles.

Fig. 5. Effective mass of electrons in \(n\)-type germanium as a function of the angle between H, lying in the \((1\bar{1}0)\) plane, and the \([001]\) axis.

of the \(n\)-type, as a function of the angle between \(\mathbf{H}\) and the \([001]\) axis for three energy surfaces, and the experimental points obtained in the same experiments. From these observations the values of the longitudinal and transverse masses for germanium were obtained: \(m_l = 1.4m_0\) and \(m_t = 0.083m_0\). Measurements by Dresselhaus \(^{21,34,35}\) and co-workers gave, for this same case, close results: \(m_l = 1.58m_0\) and \(m_t = 0.082m_0\).

The anisotropy of the effective mass of electrons in silicon, from the same measurements, proved to be in good agreement with theory. Figure 6 shows curves of the dependence of \(m^*\) for a silicon crystal on the angle between \(\mathbf{H}\) and the \([001]\) axis of the crystal, for a frequency \(2.4 \cdot 10^{10}\) cycles and a temperature of \(4^\circ\)K. The curves were calculated from formula (54), using the experimentally obtained values \(m_l = 0.98m_0\) and \(m_t = 0.19m_0\).

For the collision time, experiments \(^{36,37}\) gave a value of the order \(6 \cdot 10^{-11} \div 7 \cdot 10^{-11}\) sec.

Thus, measurements of effective masses by means of cyclotron resonance satisfactorily agree with the interpretation of magnetoresistance measurements \(^{42,43}\) and make more justified the considerations on the shape of the energy surfaces in the conduction band of Ge and Si that were put forward on the basis of those works.

Fig. 6. Effective mass of electrons in \(n\)-silicon as a function of the angle between \(\mathbf{H}\), lying in the \((110)\) plane, and the \([001]\) axis.

3. Resonance in \(p\)-type germanium and silicon

The use of infrared radiation for excitation of carriers in \(p\)-type germanium made it possible to observe the anisotropy of the resonant absorption of holes \(^{37}\). Measurements gave two values of the effective mass for holes; moreover, the resonance associated with holes of larger mass proved to be strongly anisotropic, whereas for holes of smaller masses the anisotropy was revealed to a much lesser degree. Figure 7 shows the resonance curves for Ge at a frequency \(8.9 \cdot 10^9\) cycles for three directions of the field \(\mathbf{H}\) in the crystal: along the axes \([001]\) \((a)\), \([111]\) \((b)\), \([110]\) \((c)\). Figures 8 and 9 show the dependences of the effective masses of holes in germanium crystals \(^{37}\)

and silicon$^{21}$ with theoretical curves obtained from equation (59) using the values of the constants \(A\), \(B\), and \(C\) calculated from the same measurements at two arbitrarily chosen angles \(\theta\) with the \([001]\) axis of the crystal.

Fig. 7. Cyclotron absorption in \(p\)-type germanium as a function of the direction of \(\mathbf{H}\) along the axes: \([001]\) (a); \([111]\) (b) and \([110]\) (c). Temperature \(4^\circ\) K, frequency \(8.9 \cdot 10^9\) cps.

Calculations of the constants carried out by different investigators and on different crystals gave, in general, coincident values.

	Measurements$^{21}$	Measurements$^{26}$
a) For germanium:
\(A\)	\(A = -(13.0 \pm 0.2)\dfrac{\hbar^2}{2m_0}\)	\(A = -13.6\,\dfrac{\hbar^2}{2m_0}\)
\(B\)	\(B = (8.9 \pm 0.1)\dfrac{\hbar^2}{2m_0}\)	\(B = 9.1\,\dfrac{\hbar^2}{2m_0}\)
\(C\)	\(C = (10.3 \pm 0.2)\dfrac{\hbar^2}{2m_0}\)	\(C = 11.2\,\dfrac{\hbar^2}{2m_0}\)
b) For silicon:
\(A\)	\(A = -(4.1 \pm 0.2)\dfrac{\hbar^2}{2m_0}\)	\(A = -4\,\dfrac{\hbar^2}{2m_0}\)
\(B\)	\(B = (1.6 \pm 0.2)\dfrac{\hbar^2}{2m_0}\)	\(B = 1.3\,\dfrac{\hbar^2}{2m_0}\)
\(C\)	\(C = (3.3 \pm 0.5)\dfrac{\hbar^2}{2m_0}\)	\(C = 3.6\,\dfrac{\hbar^2}{2m_0}\)

Fig. 8. Effective mass of holes in p-type germanium as a function of the angle between \(\mathbf{H}\), lying in the \((110)\) plane, and the \([001]\) axis. Temperature \(4^\circ\mathrm{K}\); frequencies: \(\nu_1 = 8.9 \cdot 10^9\ \mathrm{Hz}\) (points) and \(\nu_2 = 23 \cdot 10^9\ \mathrm{Hz}\) (circles).

Fig. 9. Effective mass of holes in p-type silicon as a function of the angle between \(\mathbf{H}\) and the \([001]\) axis.

These constants give equations for the energy surfaces in the valence band that are in good agreement with theoretical calculations[^28].

Analysis of the widths of the curves[^21] made it possible to determine the collision times, which proved to be:

\[ \mathrm{Ge}\left\{ \begin{array}{l} \tau \simeq 5\cdot 10^{-11}\ \text{sec.}\\ \tau \simeq 7\cdot 10^{-11}\ \text{sec.} \end{array} \right. \qquad \mathrm{Si}\left\{ \begin{array}{ll} \tau \geqslant 6\cdot 10^{-11}\ \text{sec.} & \text{(for the larger masses),}\\ \tau \simeq 7\cdot 10^{-11}\ \text{sec.} & \text{(for the smaller masses).} \end{array} \right. \]

The form of the absorption curves proved to be close to Gaussian, although from elementary theory one should expect Lorentzian curves.

The resonance curves look different for different types of excitation used. Thus, under infrared excitation of carriers in \(p\)—Ge, the resonance peaks corresponding to the large masses \((m_2^* = 0.3m_0)\) are approximately 8 times more intense than the peaks corresponding to the smaller masses \((m_1^* = 0.04m_0)\). Meanwhile, under radio-frequency excitation the peaks corresponding to \(m_1^*\) are more intense. This circumstance is connected with the fact that absorption is inversely proportional to the effective mass, the consequence of which is enhancement of ionization in high electromagnetic fields. The indicated intensity ratio agrees satisfactorily with that calculated theoretically[^21].

Fig. 10. Absorption curve obtained under excitation of carriers by modulated light irradiation of a \(p\)—Ge crystal. Peaks are visible that are not observed with other methods of excitation. Temperature \(4^\circ\) K, frequency \(23\cdot 10^9\) Hz.

Excitation of carriers by modulated illumination gives additional resonance peaks that are not detected by other methods. In Fig. 10 an absorption curve is presented[^38] obtained with the use of such illumination; on it one can see peaks that do not arise with other methods of carrier excitation. An explanation of these results is still lacking.

CONCLUSION

As is clear from the results presented, the study of cyclotron resonance in semiconductors proves useful for the theory of the latter; further work in this direction with other representatives of semiconductor substances will undoubtedly make it possible to obtain a number of new data.

In principle, it is possible to observe cyclotron absorption in the plasma of a gas discharge and in ion beams. Investigations in this direction may provide a number of useful data on the composition and properties of plasma.

Undoubtedly, experiments of this kind will encounter a number of methodological difficulties associated both with the dielectric and with the oscillatory properties of plasma and with the production of intense low-energy beams. If these difficulties prove surmountable, then it will apparently be possible to use the effect under discussion in gas analysis and in mass spectrometry of atomic nuclei.

CITED LITERATURE

1. V. K. Arkad’ev, Zh. R. F.-Kh. O. 45, 103 (1913).
2. Ya. G. Dorfman, Zeits. f. Phys. 17, 98 (1923).
3. F. Bloch, W. W. Hansen and M. Packard, Phys. Rev. 69, 37 (1946).
4. E. M. Purcell, H. C. Torrey and R. V. Pound, Phys. Rev. 69, 37 (1946).
5. E. K. Zavoiskii, Doctoral dissertation, FIAN, 1944; J. Phys. USSR 9, 211 (1945).
6. E. K. Zavoiskii, ZhETF 17, 883 (1947).
7. J. Griffiths, Nature 158, 670 (1946).
8. M. E. El’yashevich, UFN 54, 513 (1954).
9. V. Gordi, V. Smith, R. Trambarulo, Radiospektroskopiya, GTTI, 1955; B. Bleaney and K. Stevens, Rep. Progr. Phys. 16, 108 (1953).
10. C. Hutchison and Pastor, Phys. Rev. 81, 282 (1951); M. Harstens and Ryan, Phys. Rev. 81, 888 (1951).
11. A. M. Portis, A. F. Kip and C. Kittel, Phys. Rev. 90, 988 (1953).
12. Goldstein, Phys. Rev. 82, 956 (1951); Phys. Rev. 83, 1255 (1951).
13. Griswold, A. F. Kip and C. Kittel, Phys. Rev. 88, 951 (1952).
14. Ya. G. Dorfman, DAN 81, 765 (1951).
15. L. D. Landau, Zeits. f. Phys. 64, 629 (1930).
16. J. H. Van-Vleck, Theory of electric and magnetic susceptibility, Oxford, 1932; Ya. P. Terletskii, ZhETF 9, 796 (1939).
17. C. G. Darwin, Proc. Cambr. Phil. Soc. 27, 86 (1930); E. Teller, Zeits. f. Phys. 67, 311 (1931); M. Bleckman, Proc. Roy. Soc., A 166, 1 (1938); H. Welker, S. B. Bayer, Akad. Wiss. 14, 115 (1938); F. Hund, Ann. Phys. Leipz. 32, 102 (1938); M. F. M. Osborn, Phys. Rev. 81, 147 (1951).
18. A. Papapetrou, Zeits. f. Phys. 112, 587 (1939).
19. R. B. Dingle, Proc. Roy. Soc. A 211, 500 (1952).
20. R. B. Dingle, Proc. Roy. Soc. A 212, 38 (1952).
21. G. Dresselhaus, A. F. Kip and C. Kittel, Phys. Rev. 98, 368 (1955).
22. W. Kohn and J. M. Luttinger, Phys. Rev. 96, 529 (1954).

1. V. S. Smirnov, A Course of Higher Mathematics, III, 2, 422, GTTI, 1951.
2. E. Jahnke and F. Emde, Tables of Functions, 373, GTTI, 1949.
3. S. A. Altshuler, B. M. Kozyrev, S. G. Salikhov, ZhETF 24, 503 (1953).
4. W. Shockley, Phys. Rev. 79, 191 (1950); Phys. Rev. 90, 491 (1953).
5. E. I. Adirovich, Some Questions in the Theory of Luminescent Crystals, Ch. 3, GTTI, 1951; V. Shockley, Theory of Electronic Semiconductors, Ch. 7, GTTI, 1953.
6. F. Herman and J. Calloway, Phys. Rev. 89, 518 (1953); F. Herman, Phys. Rev. 95, 847 (1954).
7. F. Esterman and A. Foner, Phys. Rev. 79, 365 (1950).
8. G. L. Pearson and H. Suhl, Phys. Rev. 83, 768 (1951).
9. C. S. Smith, Phys. Rev. 94, 42 (1954).
10. H. B. Briggs and R. C. Fletcher, Phys. Rev. 91, 1342 (1953).
11. R. J. Elliot, Phys. Rev. 96, 266 (1954); Phys. Rev. 96, 280 (1954).
12. G. Dresselhaus, A. F. Kip and C. Kittel, Phys. Rev. 95, 568 (1954).
13. G. Dresselhaus, A. F. Kip and C. Kittel, Phys. Rev. 92, 827 (1953).
14. B. Lax, H. J. Zeiger and R. H. Dexter, Physica 20, 818 (1954).
15. H. J. Zeiger, R. H. Dexter and B. Lax, Phys. Rev. 95, 557 (1954).
16. A. F. Kip, Physica 20, 813 (1954).
17. C. Kittel, Physica 20, 810 (1954).
18. W. Shockley, Phys. Rev. 79, 191 (1950).
19. C. Kittel and J. M. Luttinger, Phys. Rev. 73, 162 (1948).
20. S. Meiboom and B. Abeles, Phys. Rev. 95, 31 (1954).
21. M. Shibuya, Phys. Rev. 95, 1385 (1954).