HYPERSONIC WAVES IN DIELECTRIC CRYSTALS

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HYPERSONIC WAVES IN DIELECTRIC CRYSTALS

K. Dransfeld

To cite this version:

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JOURNAL DE PHYSIQUE Colloque C 1, Supplkment au no 2, Tome 28, Fivrier 1967 page C 1-157

HYPERSONIC

WAVES

IN DIELECTRIC CRYSTALS

by K. DRANSFELD

Physik Department der Technischen Hochschule, MUNCHEN

Resume. - On passe d'abord en revue certaines methodes modernes de production et de detection d'ondes acoustiques de frkquences atteignant 7

x

1010 Hz. On dkrit ensuite brikvement l'absorption de ces ondes dans differents cristaux. Enfin, on discute quelques mkthodes qui pourraient per- mettre de detecter des ondes acoustiques non coherentes de frequences superieures B 1011 Hz. I1 n'est pas question ici des cristaux paramagnetiques, etudib par Joffrin dans ce m6me fascicule.

Abstract. - Here we will first review some of the modern methods which have been used for the generation and detection of acoustic waves at frequencies up to 7 x 1010 cps. Secondly, we will briefly describe the absorption of these waves in various crystals. Finally a few methods are discussed, which might be useful to generate and detect incoherent acoustic waves at frequencies above 101 1 cps. Not included here are paramagnetic crystal, already covered by Joffrin elsewhere

in this issue.

I. Generation and Detection of Hypersonic Wawes. - One of the simplest means to excite ultrasonic longitudinal or transverse waves is a piezoelectric plate half an acoustic wavelength thick, or an odd multiple of it. Since at a frequency of 10'' cps the acoustic wavelength is only a fraction of a micron in the hypersonic range relatively thick quartz plates are used in a rather high overtone. This piezoelectric plate is cemented to a flat face of the crystal into which one wants to transmit the soundwaves, and placed into the r. f. electric field of a microwave cavity [I]. The transfer of acoustic energy from the quartz plate through the bonding material to another crystalline sample is difficult, particularly at high microwave frequencies and for shear waves.

Recently it has become possible to evaporate films of piezoelectric Cds with a preferred orientation onto various substrates [2], thereby avoiding altogether the necessity of using bonding materials. Sandwiching the CdS-film between two metal film allows, in addi- tion, a strong concentration of the electric microwave field in the CdS.

In piezoelectric semiconductors, such as GaAs or CdS where the piezoelectric behaviour is often masked by conduction electrons, thin layers of depleted carrier density

-

and thus of piezoelectrical activity - can be established at a p-n junction or in regions of trapping centers diffused in from the surface [3,4].

If one exposes at least the surface of a large piezoelectric crystal to a microwave electric field, ultrasonic waves of the same frequency are excited at the surface

and travel subsequently into the interior of the sample. This surface excitation [5,6] has two advantages :

Firstly, there is no bonding problem and, secondly, there are no mechanical resonances and therefore no frequency limits arising from the geometry. Up to date, the highest frequency at which ultrasonic waves have been generated and detected by this method is 1,l x 10'' cps [7]. The efficiency of conversion from electrical into acoustic energy is, however, rapidly declining at increasing frequency so that it appears very difficult with this method to reach still higher frequencies.

Hypersonic waves can also be excited by a ferromagnetic film undergoing ferromagnetic resonance or spinwave resonances : The magnetostrictive deformation of the film accompanying the spin precession causes the emission of phonons into the substrate carrying the film 181. For example a film of nickel or permalloy is plated onto a substrate into which phonons should be emitted. The film is then placed into a dc-magnetic field (often perpendicular to the film) and a r. f. magnetic field, in order to excite ferromagnetic resonance and thereby phonons. In this way circularly polarized shear waves as well as longitudinal waves can be emitted into any substrate, without the necessity of special bonding materials.

The availability of high power lasers in the optical frequency range with wavelengths of sowewhat less than a micron makes also possible the excitation of hypersonic waves by the stimulated Brillouin scattering as reviewed by Kastler [9] in this same issue.

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C 1

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158 K. DRANSFELD

We like to mention here only that it is. not possible, by the stimulated Brillouin scattering, to excite phonons whose wavelength is considerably shorter than the wavelength of the exciting light.

11. The Absorption of Hypersonic Wawes in Crys- tals.

-

The absorption of hypersonic waves in dielectric crystals has now been measured by several groups [lo] in the microwave frequency range by using one of the methods described above. Figure 1

TEMPERATURE 1°K)

FIG. 1 . - Absorption of longitudinal hypersonic

waves in quartz along the X-axis.

shows the temperature dependance of the absorption in quartz which is representative for probably most dielectric crystals : Its main features are two temperature ranges with different behaviour of the absorption. At low temperatures T the absorption increases stron- gly and monotonically with T, while at high tempera- tures the absorption reaches a flat plateau. We will discuss these two cases separately.

At low temperatures the thermal phonons are few and

do not interact strongly with each other. Ultrasonic collisions with thermal phonons as indicated in figure 2. The probability of such a collision is pro-

FIG. 2. - Ultrasonic phonon of energy fiw collides with

thermal phonon of energy 81 E KT. Energy and momentum is

being conserved in this 3-phonon process.

portional to the ultrasonic phonon energy go, the thermal phonon energy kT and the number of thermal phonons N. Since N increases as T 3 , the absorption

coefficient varies as

a cc w . ~ ~

a relationship which was first derived by Landau and Rumer [ll]. As can be seen From the examples in figures 3 and 4, the wT4 law is apparently well obeyed

by both transverse and longitudinal waves [12].

O . O ! I

10 40 100 400 1000

T e m p e r a t u r e ( O K )

FIG. 3. - Absorption of transverse waves in MgO,

propagating along

<

100

>.

20 30 40 50 60 70 80 90100

Temperature ( O K )

FIG. 4. - Absorption of longitudinal waves in Ruby,

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HYPERSONIC WAVES IN DIELECTRIC CRYSTALS C 1

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159

At high temperatures the themal phonons rapidly

interact with each other, so that their lifetime is becoming shorter than the ultrasonic period. In the microwave frequency range this is generally the case for temperatures above about one tenth of the Debye temperature. Tn this range of temperatures the simple 3-phonon collision is not a good description because of the large uncertainties in energy and momentum, and it is more appropriate to consider the interaction of the hypersonic waves with the whole ensemble of thermal phonons [13, 141.

The simplest absorption process of this type arises from the finite heat conduction K of the medium :

A longitudinal acoustic wave causes temperature differencies between the compressed and rarefied regions of the wave to an extent which is determined by (( Gruneisen's constant n y. The absorption arising

from the heat conduction between these regions of different temperature is [14] proportional to

where v is the velocity and (012 n) the frequency of the hypersonic wave. Since at high temperatures K varies as T - l , a is temperature independent in agreement with the observations (see figure 1). Also the frequency dependence and the order of magnitude of this expression agree with the experiments fairly well.

But there is one discomforting fact against this explanation : The absorption of transverse waves,

which propagate along an axis of two-fold rotational symmetry (for example the 100-axis in cubic crystals) should disappear, if only heat conduction were res- ponsible. Because in this case crystal regions half a wavelength apart (along the direction of propagation) are in the identical state of deformation and can therefore not be at different temperatures. Thus the heat transport between such regions should be zero and there should be no absorption of transverse waves in these directions. However, as can be seen from table I, the transverse absorption is by no means negligible compared to longitudinal waves in the first six cases where the waves are propagating along highly symmetrical axes, even though the effective Gruneisen constant (last column) is clearly smaller for transverse waves in symmetrical directions. We must therefore conclude that heat conduction is a contributing factor but not the only important contri- bution to the absorption. The absorption of transverse waves travelling along symmetrical directions must be caused by a different process, perhaps a process first described by Akhiezer [15, 161. But since the frequency and temperature dependance of this Akhiezer process is the same as for the process of teat conduction we will not here differentiate between both processes in more detail.

It is perhaps of more interest to compare the high temperature absorption of longitudinal waves at one frequency for different crystals : Figure 5 shows such a comparison between crystals of widely different Debye temperatures, the absorption at room temperature and at one gigacycle is plotted vs. The Debye TABLE 1. - The hypersonic absorption of transverse waves relative to longitudinal

waves for different crystals at room temperature. Crystal and propagation direction propagation along 2-fold symmetry axis KBr < 100 > Yes 1.33 0.07 KC1

<

100

>

Yes 0.77 0.05 NaCl

<

100 > Yes 0.47 0.05 Ge < 100 > Yes 0.30 0.09 Si

<

100 > Yes 0.26 0.08 MgO

<

100 > Yes 0.31 0.09

TiO, C-axis Yes 0.11 0.01

A1,0, C-xis no 1.54 0.25

SiO, AC no 1.72 0.47

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160 K. DRA LONGITUDINAL WAVES F r e q u e n c y : 1 k M c T = 300 O K \ O NoCl

t

P r o p a g a t i o n direction:

\

C u b i c c r y s t a l s (100) \@ M g O S i 0 2 X-axis T i 0 2 C - a x i s

\

_\ Debye temperature, O K

FIG. 5. - Hypersonic absorption in different crystals

as a function of their Debye temperature.

temperature of the material. The alkali halide points were extrapolated form data by Merkulov [17], the germanium and silicon data were taken from Lamb 1181 and Dobbs [19]. There are no data yet available for diamond, but in view of its high debye temperature (8

-

2000 OK) it should have excellent acoustic transperency at room temperature at very high frequencies.

111. Further possible developments.

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In this section we will discuss the possibility to extend ultrasonic research into the frequency range above 10'' cps. First we will show that the coherent methods described in part I are unlikeley to be very useful above 10'' cps and therefore we propose, finally, incoherent methods for generation and detection of phonons at these high frequencies.

Before talking about methods for phonon generation in this frequency range it is important to discuss the small angle scattering which these phonons suffer in crystals. Von Gutfeld and Nethercot [20] working with a pulsed beam of thermal phonons of a frequency of about 10'' cps showed that most at these phonons suffer a scattering by a little more than 5O after travelling about 1 cm in crystalline quartz at He-temperatures. Although such small angle scattering is hardly of any importance for the heat transport it has a very damaging effect on the propagation of plane waves. In the presence of small angle scattering plane wavefronts will not remain plane as the wave travels along,

but will become corrugated, thereby making it very difficult to detect them with coherent methods discussed in section I, for example a piezoelectric quartz plate. This small angle scattering and its damaging effect on the exponential decay of the acoustuc pulse pattern is much more serious at 10'' cps than at lo9 cps, and we must therefore expect that it might be altogether impossible to work with really plane waves above frequencies of 10'' cps.

The methods mentioned in section I generate a coherent train of plane waves, and - assuming that they remain plane waves

-

detect them at a different part of the crystal. But even if they remained plane waves, it would still be very difficult to keep the detecting surface parallel to the wavefronts within half an acoustic wavelength. Remembering that the acoustic wavelength at 10'' cps is only about 500

a

it is immediately clear how very difficult it is to employ the coherent methods of section I for the detection of acoustic waves at frequencies above 10'' cps.

An incoherent detector would not suffer from this difficulty nor from the degradation of plane waves due to small angle scattering. It seem to be the only useful detector above 10'' cps.

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HYPERSONIC WAVES IN DIELECTRIC CRYSTALS C 1

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161

irregularities of the atomic mass, electronic bonding or ionic charge, direct excitation of acoustic phonons by incident infrared photons of the same frequency becomes possible at any temperature [23], It leads to a temperature independent far infrared absorption in impurity doped crystals which has been observed, for example, by Renk [24] for NO; centers in KI to be quite sizeable.

It may well be that the far infrared absorption in fused quartz and in soft alkali glasses arives from a similar cause. Figure 8 shows the measurements of

FIG. 6. - The spectral distribution of the fluorescence arising from bound excitons in ZnTe at various temperatures as mea- sured by Dietz et a1 [22].

the thermal Antistockes background. As soon as ultrasonic waves are switched on a satellite peak would appear in the fluorescence as shown in figure 7.

10 20 30 40 50 60 70 80 100

Frequency v (cm-')

2 4 v

(.lo1'

sec-') Ultrasonic frequency

FIG. 7. - The expected spectral distribution of the fluores- cence of ZnTe at 2 O K with an hypersonic field (of frequency 1.5 x 1011 cps) switched (( on n and (( off )).

The separation of this extra peak from the zero phonon line could give the phonon frequency, and thus bound excitons with narrow zero phonon lines may be very useful hypersonic detectors.

Incoherent acoustic phonons at frequencies above

10" cps may also be generated - with good conversion efficencies

-

by far infrared radiation in imperfect crystals. As is well known, in perfect crystals acoustic phonons are optically inactive. However, if the crystal contains imperfections, for example,

FIG. 8. - The far infrared absorption in fused quartz and in a soft alkali glass between 20 O K and 300°. Also included are measurements on crystalline quartz at T = 300 OK.

the absorption in fused quartz by Stolen [25] and of a soft glass by Bagdade [26] : The absorption varies roughly as m2 and is temperature independent. It seems to us that for example, an irregular charge distribution of about lo1'

-

lo2' charges per cm3

might well account for the size of the absorption and for its temperature

-

and frequency dependence. Thus it seems not unlikely that phonons of a frequency of about 1012 cps may be excited strongly if one

directs the far infrared radiation of one of the new molecular gas lasers, operating between 100 p and 700

to an imperfect region of a crystal. The phonons thus generated may be

-

after traversing the crystal

-

received at the other side by some incoherent detector as mentioned above.

I greatly appreciate several fruitful discussions with Mr. Bagdade and Mr. Stolen. Their measurements on glass have not previously been published.

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C 1

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162 K. DRANSFELD

[I] TEHOU (S. W.) et al. Proc. IEEE, Oct. 1964, 11 13. [2] FOSTER (N.), Proc. ZEEE, 1965,53,1400.

[3] WHITE, I R E Transact., 1962, UE-9,21.

[4] FOSTER (N.), IEEE- Transact., 1963, UE-10, 39. DE KLERK (J.) et al., Appl. Phys. Lett., 1964, 5, No. 1.

[5] BOMMEL (H. E.) et al. Phys. Rev. Lett., 1958, 1, 234. 1959, 2, 298.

[6] JACOBSEN (E. H.), Phys. Rev. Lett., 1959, 2, 249. Journ. Acoust., Soc. Am., 1960,32,949.

[7] JACOBSEN (E. H.), et al., Science, 1966, 153, 1 13. [8] SEAVY (M.), Proc. IEEE, a965,53,1387.

[9] KASTLER, communication orale.

[lo] see for example : Pomerantz, Proc. ZEEE, 1965, 53

1438.

[ll] LANDAU (L. D.) et al., Z. Physik der Sowjetunion, 1941,11, 18.

[I21 CICCARELLO (I.) et al., Phys. Rev., 1964, 134, 1517. [I31 WOODRUFF (T. 0 . ) et al., Phys. Rev., 1961, 123, 1553, [14] ORBACH (R.), Thesis, Berkeley, 1960.

[I51 AKHIEZER (A.), J. Phys. (USSR), 1939,1,277.

[16] BOMMEL (H. E.) et al., Phys. Rev., 1960, 117, 1245. [I71 MERKULOV (L. G.), SOV. Phys. ACOUS~., 1960, 5, 444. [18] LAMB (J.) et al., Phys. Rev. Lett., 1959, 3, 28.

[I91 DOBBS (R.) et at., Phys. Rev. Lett., 1959,3, 332. [20] VON GUTFELD (R. J.), et al., Phys. Rev. Lett., 1964,

12, 641.

[21] SHIREN (N.), Phys. Rev. Lett., 1961, 6, 168.

[22] DIETZ (R. E.) et al., Phys. Rev. Lett., 1962, 8, 391. [23] GENZEL (L.) et al.,. Phys. Stat. Sol., 1965, 12, 639. [24] RENK (K. F.), Phys. Lett., 1965,14,281.