Wavelength Acoustic Modes Research Articles

The elastic behaviour of the ternary zincblende structure semiconductor CuGe 2P 3

A single crystal has been grown of CuGe 2P 3, a ternary semiconductor with a zincblende structure in which the copper and germanium atoms are randomly distributed on the cation sites. The second order elastic constants measured by the ultrasonic pulse echo overlap technique (C 11 = 13.66, C 12 = 6.17, C 44 = 6.66 and bulk modulus B = 8.67 in units of 10 10Nm −2 at 291 K) are closely similar to those of GaP and conform well to Keyes' correlation for zincblende structure crystals. The hydrostatic pressure derivatives of the second order elastic constants ( ∂C 11 ∂P = 4.40 , ∂C 12 ∂P = 3.9 , ∂C 44 ∂P = 0.93 and ∂B ∂P = 4.07 ) and the third order elastic constants ( C 111 = − 8.45, C 112 = − 3.49, C 123 = − 1.13, C 144 = − 2.82, C 155 = − 1.44 and C 456 = − 0.69 in units of 10 11Nm −2) also resemble those of GaP: even the anharmonicity of the long wavelength acoustic modes of this ternary semiconductor resembles that of its binary “parent” compound. The positive signs of the hydrostatic pressure derivatives and the negative signs of the third order elastic constants show that the acoustic modes at the long wavelength limit stiffen under pressure, an effect which is quantified here by computation of the mode Grüneisen parameters, which are all positive. The harmonic and anharmonic force constants, obtained in the valence force field model, fit well into the pattern shown by related zincblende structure compounds: the ratio ( β α ) for bond bending (β) to stretching (α) force constants is greater than 4:1; the dominating anharmonic force constant is γ: most of the anharmonicity is associated with nearest neighbour atoms. Analysis of the temperature dependence of the elastic constants on the basis of a simple anharmonic model again emphasises the similarity between the elastic behaviour of CuGe 2P 3 and GaP. The thermal expansion of CuGe 2P 3 varies almost linearly with temperature between 291 K and 1000 K, the linear coefficient of thermal expansion α being 8.21 ± 0.02 × 10 −6 °C −1. The similar lattice parameters and elastic behaviour of CuGe 2P 3 and GaP indicate that semiconducting devices made of epitaxial deposits of CuGe 2P 3 on a GaP substrate should prove to be almost strain-free.

Vibrational anharmonicity and the elastic phase transition of indium-thallium alloys

The anharmonic behaviour of the long wavelength acoustic modes in In-Tl alloys in the vicinity of the elastic, f. c. c. to f. c. t. phase transition has been determined from the composition, temperature, hydrostatic and uniaxial pressure dependences of ultrasound wave velocities in single crystals containing between 10 at. % and 30 at. %TI. The hydrostatic pressure derivatives show that the soft acoustic mode of f. c. c. alloys softens while that of the f. c. t. alloys stiffens with pressure. An empirical equation of state has been constructed for the f. c. c. alloys. A large negative mode Grüneisen parameter found for the soft transverse acoustic phonon (wave vector <110>, polarization <11̄0>) for f. c. c. alloys near the transition reflects the marked vibrational anharmonicity of the soft mode. The pressure effects on ultrasound wave velocity have been used to determine sets of independent, room temperature third order elastic constants for several alloy compositions. The third order invariants in the elastic strain energy have been evaluated to assess quantitatively the applicability of Landau theory, in conjunction with the soft mode concept, to an elastic phase transition. The relative magnitudes of the third order invariants conform to the nearly second order character of the phase transition. However, the first order character has been established by a marked hysteresis in ultrasonic wave velocities as crystals are cycled by temperature or pressure through the transition. There is a third order invariant in the order parameter: Landau theory predicts correctly the first order nature of the transition.

Spin–Lattice Relaxation and Elastic Properties for Rare-Earth Substituted LaCl3

We develop a new theory of spin–lattice relaxation in an effort to improve the agreement between detailed predictions and experiment. Our work is based on (a) a uniquely general and convenient formulation of the crystal field problem utilizing the superposition model developed by Newman and co-workers and (b) a knowledge of the detailed lattice motions, in particular the eigenvectors of long wavelength acoustic modes, in LaCl3. In conjunction with the latter, we give elastic constants, tables for the stress-induced crystal field, and sound velocities. While the velocities agree with experiment, they are sufficiently low to reveal the inadequacy of earlier spin–lattice relaxation calculations on these systems. Relaxation data due to Stapleton and co-workers permit a sensitive test of our model. It is possible to improve agreement with experiment and to achieve a consistent picture of direct and double direct (Orbach) relation with the dopants Pr3+, Nd3+, Tb3+, and to suggest an improved phenomenological estimated scheme. However, Raman rates are consistently overpredicted; also we find (as did earlier workers) special problems in connection with LaCl3:Er3+.

Magnetoelastic Coupling Constants of Terbium and Europium Iron Garnets

The purpose of this paper is twofold: (1) to present data on the magnetoelastic constants of rare-earth iron garnets, a subject which for various reasons has been little studied thus far, and (2) to describe a new method for obtaining these constants which is applicable to many ferromagnetic insulators and does not require the use of strain gauges. The new method was developed from a study of the coupling introduced between long wavelength acoustic modes and long wavelength spin-wave modes by ions in which the orbital moment is not quenched, such as rare-earth ions, Co2+, and Fe2+. The effect of the coupling caused by such ions is observed as a magnetic field dependence of the resonant frequency of an acoustic resonator made from the given material, even above dc saturation. The acoustic Q above dc saturation is also field-dependent. A theory has been obtained which explains the observed field dependences of both v(H) and Q(H) as well as other important features of the magnetoacoustic interaction. It will appear elsewhere.1 One of the results of this theory is a relation between the cubic magnetoelastic constants, B1 and B2, and the acoustic resonant frequency v of a thin disk vibrating in a thickness shear mode. For a (110) disk with the external field H0 in the plane along [100] and the rf driving field perpendicular to the disk, we obtain ν=ν∞(1−B222c44MHeff),where Heff=H0+Hanis+4πM, and c44 is the usual shear elastic modulus, which may be obtained directly from v∞ and the thickness of the disk. Equation (1) applies to the mode with displacement along [100]. B1 is obtained similarly, only now H0 is along [110] in the plane, and the shear displacement is along [110]. Here we have ν=ν∞(1−B122c44MHeff). All data thus far have given excellent fits to these equations as Heff is varied, making it easy to obtain the unknown constants. It should be noted that Eqs. (1) and (2) are high-temperature approximations and at low temperatures will not in general be as simple. For TbIG and EuIG at room temperature we find the following (all in ergs cm−3): For TbIG, |B1|≤3.5×106, |B2|=30×106; and for EuIG, |B1|=45×106 and |B2|≤4.2×106. The signs must still be determined from strain gauge measurements. The observed reversal of the magnetoelastic anisotropy (ratio of B1 to B2) is being investigated, as well as the temperature dependences of B1 and B2.

Broadening of Impurity Levels in Silicon

Sharp absorption lines have been observed in $p$- and $n$-type silicon whose absolute and relative positions lead to their interpretation as optical transitions between bound states of trapped holes or electrons that are approximately hydrogenic in character. These lines have a finite breadth of the order of 0.001 electron-volt at liquid helium temperatures. This breadth is determined by the zero-point vibrations of the lattice. At higher temperatures, the squared breadth increases in proportion to the mean squared amplitude of oscillation of those lattice modes that contribute significantly to the broadening. The theory indicates that the modes of importance have wavelengths of the order of the Bohr radius of the trapped carrier state. These are rather long wavelength acoustic modes whose energy $h\ensuremath{\omega}$ corresponds to 80\ifmmode^\circ\else\textdegree\fi{}K. Thus the squared broadening is expected to increase by a factor of two from helium to nitrogen temperatures---and this increase is confirmed experimentally. This confirms the hypothesis that the broadening is due to the interaction of the trapped electron with the acoustic lattice vibrations. The form of the electron-lattice interaction is taken to be that of the Bardeen-Shockley deformation potential, and its strength is determined from the experimental mobility. Thus the theory contains no adjustable parameters. In absolute magnitude, the theoretical line breadth turns out to be several times too large. Possible reasons for the discrepancy are discussed.