Spin-Phonon Magnetic Resonance of Conduction Electrons in Indium Antimonide Crystals

Resonance absorption of radio waves (with a frequency of 10 MHz) by c-band electrons in indium antimonide crystals doped with hydrogen-like donors (tellurium atoms) at room temperature in an external magnetic field is theoretically studied. Known experimental data obtained for samples with electron concentrations in the range from 6∙1015 to 5∙1018 cm–3 are analyzed and interpreted. Resonant absorption of radio waves by n-InSb:Te crystals in a magnetic field is calculated to be due to spin-phonon resonance based on the law of conservation of energy and the quasi-wave vector for electrons and optical phonons. The resonance arises as a result of a spin-flip interaction of a c-band electron with an optical phonon, which is assisted by resonant absorption of radio waves in a magnetic field. A physical picture of the phenomenon is given. Analytical relations are presented. Calculations are carried out and are consistent with experimental data that could not previously be interpreted at all.

the limiting weakly doped samples was ~0.7 meV. The degree of compensation (concentration ratio of acceptors, all in the -1 charge state, to Te atoms) was K ≈ 0.1 in all samples. The Te atoms at room temperature were fully ionized (i.e., had the +1 charge state). Therefore, samples Nos. 3-6 were located on the metallic side of the insulator-metal concentration phase transition [13].
At liquid He temperature (4.2 K), the high-fi eld tail of the resonance of sample No. 4 had Shubnikov-de Haas oscillations [11] related to quantization of c-band electron density of states in strong magnetic fi elds. They were observed at low temperatures and resulted from overlap of Landau and Fermi levels [14]. The oscillations disappeared and the resonance weakened as the temperature was increased to 300 K. However, the resonant magnetic strength B r practically did not change [11]. The width δB r of the resonance line increased (from δB r ≈ 0.13 T for n = 6·10 15 cm -3 to δB r ≈ 1.34 T for n = 5·10 18 cm -3 ) as the concentration of c-band electrons increased. However, the tail of the magnetic resonance line became more symmetric. It was also found that the centers of resonance lines determined from the condition Y′(B r ) = 0 (Fig. 1b) obeyed the equation B r = αf rw n 1/3 [1 -β(3 cos 2 φ -1)], where α = 0.97·10 -13 T·cm·s; f rw = ω rw /2π = 10 MHz is the radiospectrometer cavity resonance frequency; β = 0.1; and φ, the angle between the direction B of the constant magnetic fi eld and the crystallographic [211] direction in the sample [12]. The center B r , width δB r , and integrated intensity of lines Y (i.e., the integral of |Y ′(B)| over magnetic fi eld strength from 0 to 3B r ) had approximately the same dependence on the electron concentration n with δB r ≈ B r . The position of the center of resonance line B r depended on the angle between the direction of the magnetic fi eld and the [211] direction in the crystals because of anisotropy of the c-band electron g-factor [15,16].
Hypotheses. Interpretation of the measured resonant absorption of radio waves of frequency f rw = 10 MHz by n-InSb:Te crystals in a magnetic fi eld led to proposed and refuted hypotheses that the depth of the skin-layer increased in a magnetic fi eld; ultrasound generated by the measuring radiospectrometer coil was absorbed by conductivity electrons and/or 115 In and 121 Sb nuclei of the InSb crystal matrix; and magnetostriction appeared in the samples [17].
The goal of the present work was to propose a model that described quantitatively the magnetic resonance measurements in n-InSb:Te crystals at room temperature [11,12].
Let us also discuss hypotheses that were not considered before [11,12] regarding various types of magnetic resonances [18] in solids: 1) ultrasound absorption by 125 Te nuclei [the hypothesis was refuted because the position of the center of the magnetic resonance varied in the range 0.1-1 mT while the line width (from peak to peak of the fi rst derivative of the resonance line vs. the magnetic fi eld) varied by ~1 mT [19], which did not agree with the values given in Table 1]; 2) a Knight shift ΔB K of the center of the B nmr line for resonant absorption of radio waves by atomic nuclei magnetic moments because of the additional magnetic fi eld imposed on them by c-band electrons (the hypothesis was refuted because of the small shift of the NMR lines ΔB K /B nmr ≈ 2·10 -4 [20] depending on the concentration of c-band electrons as compared to the data in   (5), and 5·10 18 cm -3 (6); constant frequency f rw = ω rw /2π = 10 MHz of electromagnetic fi eld of radiospectrometer coil; fi gures adapted from the literature [12] (b).
the angular frequency of the electromagnetic fi eld created by the cavity coil in the studied sample; ω c = eB/m, the angular cyclotron frequency; e, the elementary charge; B, the external magnetic fi eld strength; and m, the c-band electron effective mass (the hypothesis was refuted because the condition for observation of cyclotron resonance was not fulfi lled because ω c >> ω rw [21]); 4) spin resonance in conductivity electrons: rw ω = |g n |μ B B r , where |g n | ≈ 51 is the g-factor of a c-band electron; μ B = 0 /2 e m , the Bohr magnetron (the hypothesis was refuted because the condition for resonant absorption was fulfi lled for magnetic fi eld strength B r ≈ 10 μT); 5) spin-phonon resonance: transition of a c-band electron between Zeeman sublevels of Landau levels in a magnetic fi eld upon absorption by the electron of an optical phonon. This transition led to a change of direction of the electron spin relative to the direction of the scanning magnetic fi eld induction and absorption of radio waves ( f rw = 10 MHz). The dependence of the c-band electron g-factor on its quasi-wave vector was responsible for the coupling of the optical phonon and electron spin [22,23].
General Relationships. Let us discuss hypothesis 5. The effects of optical phonons responsible for magnetic resonance signals, i.e., cyclotron-phonon [24], spin-cyclotron-phonon [25, and spin-magnetophonon resonances [26][27][28], were considered before. The sound absorption coeffi cient associated with transfer of a c-band electron between Landau levels and simultaneous inversion of its spin were calculated [25]. A model for spin inversion because of coupling of a c -band electron with the lattice but without a change in the number ζ of the Landau parabola (levels) was given [28] (Fig. 2). Also, magnetophonon resonance, i.e., transfer of a c-band electron between Landau levels upon absorption of an optical phonon without spin inversion is known [29,30]. It is noteworthy that localization of c-band electrons on donor ions in an external magnetic fi eld [31,32] was not considered in that work because the magnetic resonance in InSb crystals was measured at room temperature, i.e., with full thermal ionization of the Te atoms.
The motion of the electrons becomes unidimensional in a rather strong external magnetic fi eld [ω c τ > 1, where τ is the average relaxation time of the c-band electron (quasi)wave vector k]. They can move with kinetic energy ( k) 2 /2m only along or against the direction of the external magnetic fi eld vector B, i.e., k || B. The magnetic fi eld does not act on phonons so that they can move in any direction. Let us examine phonon absorption by a c-band electron with wave vector projection k 1z on the z axis parallel to the external fi eld vector with subsequent spin inversion and a transition of the electron into a Landau level corresponding to a higher-energy electron. In this instance, the law of conservation of energy for spin-phonon magnetic resonances is written: where k 2z is the z-component of the c-band electron wave vector [after absorption of a phonon with energy E ph (q) and (quasi)wave vector value q]; |g n |μ B B r , Zeeman splitting of each Landau level; ω c , angular cyclotron frequency at resonance (B = B r ); ζ i and ζ f , Landau level numbers before and after phonon absorption; and ω rw , a quantum of radio-wave signal energy (detecting the resonance) with angular frequency ω rw . It is noteworthy that electron transitions occur primarily from the most populated Landau levels (ζ i = 0 and ζ i = 1) into the fi nal Landau level ζ f > ζ i . Figure 2 shows a diagram of electron where q is the phonon wave vector modulus; θ, the angle between the directions of the phonon wave vector q in the crystal and the z axis. By substituting Eq. (2) into Eq. (1), we obtain: The condition for spin-phonon resonance with external constant magnetic fi eld strength B r considering the equal probability of all possible angles 0 < θ < π from Eq. (3) with averaging over angles θ is written: where g n and ω c = eB r /m are the g-factor and cyclotron frequency depending on the c-band electron concentration; ζ f -ζ i = 1, 2, 3, …, differences of Landau level numbers (fi tting parameter); and E ph (q), energy of a phonon with wave vector q. Let us examine separately absorption of acoustic and optical phonons.
Absorption of acoustic phonons. The energy of an acoustic phonon E ph (q) = E ph (q a ) = v a q a , where v a is the velocity of sound in the crystal (v a = 3.41·10 5 cm/s for InSb). According to Eq. (4), the magnetic fi eld B r reaches a maximum for q a = 2mv a / . The maximum max r B = 0.38 mT is reached for ζ f -ζ i = 0 for n-InSb with electron concentration n = 6·10 15 cm -3 . The quantity max r B = 3 mT for n-InSb with n = 5·10 18 cm -3 . The B r values calculated using Eq. (4) are much less than those observed ( Fig. 1b and Table 1). Thus, absorption of acoustic phonons was not the reason for the magnetic resonance observed before [12]. room temperature, i.e., the sample temperature during the magnetic resonance measurements. Then, only transverse optical phonons with wave vector q o = 0 (center of the Brillouin zone) were considered for simplicity in calculating B r using Eq. (4). Hence, optical phonons with wave vector q o ≠ 0 but still satisfying the resonance condition of Eq. (4) for various B r could contribute to the resonance line shape. The greater q o was, the smaller B r was. As a result, the resonance lines [transfer of Y′(B) line intensity from the low-fi eld tail B < B r into the high-fi eld part B > B r ] were slightly asymmetric in samples Nos. 1 and 2 and partially 3. Table 1 presents experimental (e) r B and (t) r B calculated using Eq. (4) upon absorption of optical phonons for discrete values of ζ f -ζ i in the range from 16 (for sample No. 1) to 6 (No. 6). Equation (4) also shows that the greater frequency ω rw was, the greater B r would be (under otherwise equal conditions) when the center of the electron spinphonon magnetic resonance line was recorded. This was experimentally confi rmed [11].
The dependence of the effective mass m on the concentration of c-band electrons had to be taken into account to calculate the properties of the n-InSb crystals. The experimental c-band electron effective mass [4,33] was approximated as ( where m 0 is the electron mass in a vacuum; n, electron concentration; and n m = 6·10 17 cm -3 . The Fermi energy E F in the c-band of degenerate InSb crystals (samples Nos. 4-6) without a magnetic fi eld can be estimated using the standard formula [34]: where the effective electron mass m = m(n) is determined by Eq. (5). The Fermi energy for sample No. 4 (n = 1·10 17 cm -3 ) was E F ≈ 53 meV, which was approximately twice the thermal energy (~26 meV) at room temperature.
where E p = 23.1 eV; E g3 = E g2 ; Δ 3 = 0.803 eV; E av = 3k B T/2, the average electron energy in a nondegenerate (Maxwell-Boltzmann) electron gas; k B , Boltzmann′s constant; E av = 3E F /5, the average energy of an electron with effective mass m = m(n) in a degenerate (Fermi-Dirac) electron gas. Figure 4 shows an approximation of the experimental data for a c-band electron g-factor from Eqs. (7)- (9). Magnetic Resonance Integrated Intensity and Line Width. The magnetic component B 1 cos (ω rw t) of an electromagnetic wave under the experimental conditions [12] penetrated the sample from the ends (area 1.5 × 2 mm 2 ; Fig. 1a) to a depth equal to the skin-layer thickness. Therefore, only c-band electrons located within the skin-layer near the sample surface will contribute to the integrated intensity of magnetic resonance signal Y [40,41].
The dependence of the specifi c electrical conductivity at constant current on the external magnetic fi eld strength can be approximated as [10]: where σ dc = enμ(n) is the electrical conductivity at constant current without a magnetic fi eld (Table 1); μ = μ(n), the mobility of c-band electrons [ Fig. 3; Eq. (6)]; S(μB) = 1/[1 + (μB) p ], a dimensionless function considering the decrease of electrical conductivity because of increased electron scattering rate in the bulk and on the surface of the samples, the Hall effect, and amplifi cation of the electrical component of the RF fi eld in the magnetized electron plasma with immobile positively charged donors and negatively charged acceptors, etc. [42][43][44][45]; and p, a fi tting parameter ( p = 1 was assumed in further calculations and was fully supported in experiments [12]).
The electrical conductivity of all examined n-InSb:Te samples according to the Drude model [46] without an external magnetic fi eld was independent of angular frequency ω rw of the measured signal, i.e., σ(ω rw ) = σ dc /(1 + ω rw τ) ≈ σ dc , where τ = μ(n)m σ (n)/e is the average relaxation time of the electron wave vector; m σ (n) = m(n), effective mass of the electrical conductivity (Fig. 3). In fact, ω rw τ ≈ 10 -5 for c-band electron concentrations in the range from n = 6·10 15 cm -3 to n = 5·10 18 cm -3 so that σ(ω rw ) ≈ σ dc . Then, ω c τ = B r μ(n) ≈ 2 for strength B = B r (Table 1) considering Eq. (10). The characteristic penetration depth of a variable magnetic fi eld B 1 cos (ω rw t) for a normal skin-effect into the studied sample (Fig. 1a) was defi ned as [40,47]: where ω rw = 2πf rw = 2π·10 7 rad/s is the angular frequency of the electromagnetic fi eld of the measuring cavity coil; μ 0 = 1.257 μH/m, the magnetic constant; σ dc (B), specifi c sample electrical conductivity depending on constant magnetic fi eld B [Eq. (10)]; σ dc = enμ(n), electrical conductivity for B = 0. Mobility μ = μ(n) was calculated using Eq. (6). Let us average skin-layer depth δ s (B) over the scanning range of constant magnetic fi eld strength B in which the magnetic resonance line is recorded. The average skin-layer depth for the sample with number j using Eqs. (11) and (10) where B rj is the resonant strength (i.e., the center of the resonance line; Table 1); δ sj = rw 0 dc 2 j ω μ σ , skin-layer depth without an external magnetic fi eld (for B = 0); μ j = μ j (n), electron mobility in the jth sample. The integrated intensity of the magnetic resonance line of the jth sample (original resonance line of radio wave absorption by the sample in a magnetic fi eld) is where ( ) j Y B ′ = dQ j /dB is the derivative of the cavity-coil gain Q j over the scanning magnetic fi eld strength B during recording of the sample magnetic resonance (Fig. 1b). According to Eq. (12), the number of electrons contributing to the magnetic resonance signal of the jth sample with electron concentration n j and average skin-layer depth 〈δ sj 〉 is n j 〈δ sj 〉A, where A ≈ 1.5 × 2 mm 2 is the area of the end surface of each of six samples (Fig. 1a). Then, the ratio of integrated intensities of the magnetic resonance signal of the jth sample according to Eq. (13) to the signal intensity of sample No. 1 is equal to the ratio of the numbers of electrons in regions of their resonant absorption of the radio-wave magnetic component: where j = 1, 2, …, 6 are the numbers of the n-InSb:Te samples. Next, let us consider that the spin-phonon magnetic resonance signal width is mainly due to fl uctuations of c-band electron potential energy because of their Coulombic interaction with the closest dopant ions and other c-band electrons. According to the literature [48,49], the mean-square fl uctuation of the electron electrostatic potential energy is where ε r = 17.2 is the relative dielectric permittivity of the InSb crystal matrix at room temperature [5]; ε 0 = 8.854 pF/m, an electrical constant; n = (1 -K)/N, the c-band electron concentration; N, the concentration of Te hydrogen-like donor atoms, which exist totally in the +1 charge state; KN, the acceptor concentration (charge state -1); (1 + K)N, the dopant ion concentration; K ≈ 0.1, the degree of compensation of samples Nos. 1-6. Parameter ζ f -ζ i = 1, 2, 3, … in Eq. (4) is a discrete value in the absence of electron potential energy fl uctuations. The Landau levels in the crystalline sample are inequivalent [50] because of electrostatic fl uctuations of the electron potential energy. Therefore, parameter ζ f -ζ i ≥ 1 obtained by harmonizing the experimental B r and those calculated using Eq. (4) can be approximated by a continuous variable depending on the electron concentration n at 300 K: The fl uctuations of c-band electron potential energy are due to the coexistence of vertical and nonvertical electron transitions on the single-electron-energy-spatial-coordinate diagram [51]. This condition determines the spin-phonon magnetic resonance linewidth during electron transitions between Zeeman sublevels of Landau levels. Then, the ratio of the linewidth (from peak to peak) in the jth sample δB rj to linewidth δB r1 in sample No. 1 calculated from Eq. (15) is Figure 5 shows ratios of experimental intensities Y j /Y 1 and widths (from peak to peak) δB rj /δB r1 of magnetic resonance lines for sample j to sample 1 ( Table 1 gives the sample parameters). Table 1 and Fig. 5 show that the ratio of magnetic resonance signal widths for samples with j = 1, 2, …, 6 to the width of the signal for the fi rst ( j = 1) sample (e) (e) r r1 j B B δ δ is practically the same as the ratio of the mean-square fl uctuations of the c-band electron potential energy for the jth and fi rst samples W nj /W n1 . Conclusions. A model for spin-phonon magnetic resonance in n-InSb:Te crystals at room temperature (resonance absorption of radio waves of frequency 10 MHz at constant magnetic fi eld) was proposed for the fi rst time. Spin-phonon resonance in n-InSb samples with c-band electron concentrations from 6·10 15 to 5·10 18 cm -3 was calculated. The position of the magnetic resonance signal line center was related to the magnetic fi eld strength required to produce that splitting of the c-band electron energy levels at which an electron transition was energetically possible. The transition occurred from a Zeeman sublevel of a lower Landau level (parabola) into one of the Zeeman sublevels of an upper Landau level with electron spin inversion via absorption by it of a transverse optical phonon of energy ~22.9 meV. Radio waves of frequency 10 MHz that detected magnetic resonance were also absorbed. The resonance line width (from peak to peak) was due to c-band electron potential energy fl uctuations because of Coulombic interaction of them with dopant ions and among themselves. The signal integrated intensity was related to the dependence of the skin-layer thickness at 10 MHz to the sample electrical conductivity at constant current in a magnetic fi eld. The angular dependence of the radio-wave absorption resonance line in a magnetic fi eld was determined by the anisotropy of the c-band electron g-factor. A quantitative description of known experiments that were not previously interpreted was given.