Acoustic phonon spectroscopy

SURFACE

SCIENCE 37 (1973) 914921

ACOUSTIC

0 North-Hollanb

PHONON

C. H. ANDERSON RCA David Sarnoff Research

Publishing Co.

SPECTROSCOPY

and E. S. SABISKY

Center, Princeton,

New Jersey 08540,

U.S.A.

An introduction is given to an acoustic phonon spectrometer, which is based on optical modulation techniques to detect magnetic circular dichroism. The spectrometer is capable of detecting deviations from a thermal background at a few degrees Kelvin of 1 part in lo4 within a 4 mK (80 MHz) bandwidth over the phonon energy tange of 0.3 to 16K. Because the strongest phonon interactions in dielectric crystals at a few degrees Kelvin occur on the surfaces, many of the applications have been concerned with surface problems.

This paper presents a brief review of a new acoustic phonon spectrometer, which is based in part on optical modulation spectroscopy 1). While the spectrometer at first glance may appear to be out of the main theme of this conference, it has been used to measure the Van der Waals potential between liquid helium and alkaline earth fluoride substrates over the range of separations of 10 8, to over 200 A 2). These very precise measurements, which are one of the cleanest surface adsorption experiments ever carried out, are in excellent agreement with the Lifshitz theory of the Van der Waals force. The spectrometer has also been used to measure the acoustic properties of adsorbed films of liquid helium and the crystalline solid-liquid helium interfaces); problems that can be considered as surface physics, but traditionally are put into low temperature physics. It should be emphasized that there is great potential for using this technique as a probe of the acoustic properties of many surfaces and adsorbed layers in the 10 to over 300 GHz frequency range, which in most solids covers the acoustic wavelength range of 100 to 1000 A. The spectrometer is based on the fact that the spin temperature of certain trace paramagnetic impurities can be measured using the magnetic circular dichroism of the impurity absorption bands. At temperatures below about 10 K most spin 4 paramagnetic ions dynamically interact with the host lattice through the emission and absorption of resonant acoustic phonons, hence the spin temperature is determined by the effective temperature of the resonant modes within the paramagnetic resonance linewidth. In effect the spin system acts as a bolometric detector of acoustic phonons over a narrow bandwidth centered at a frequency determined by an external magnetic field, which can be readout in a localized region of the crystal by a light beam. 914

ACOUSTIC

PHONON

915

SPECTROSCOPY

The importance of this spectrometer can be appreciated when it is realized that at temperatures below about 10 K, the acoustic modes within the volume of a dielectric crystal become decoupled. Therefore the spectral and spatial distribution of acoustic radiation within the crystal depends primarily on the conditions on the surface, in exactly the same manner that the spectral and spatial distribution of electromagnetic radiation in a room depends on the temperature of the walls as well as the location and nature of the light sources present. The acoustic spectrum is therefore basically a blackbody Planck distribution, which may be perturbed by phonon sources, such as a heater, located on and within the crystal. Using CaF, :Tmzf it is possible to scan the phonon energy range of 0.3 to over 16 K with a 4 mK resolution and a sensitivity to charges from the thermal background of 1 part in 104. Details of how the spectrometer operates and examples of its uses are covered in a chapter in Physical Acoustics, Vol. 8, so only the briefest description of the basic apparatus will be given here, fig. 1. Unpolarized light from a 25 W tungsten lamp is filtered to those wavelengths where the impurity bands show large circular dichroism properties, passed through the sample and analysed for the fraction of induced circular polarization. The signals are generally large (up to 20% polarization) and so some may be sacrificed by passing the light through at an angle with respect to the magnetic field; this makes room for as large an experimental dewar as is possible for a given separation of the pole caps of the electromagnet. Also, no care

DRIVER

MAGNET POLE FACE

DEWARS SAMPLE

PHASE

8.

4 OUTPUT

Fig. 1.

Sketch of basic apparatus.

FOR PLATE

916

C.H.ANDERSON

AND E.s.SAEISKY

has been taken to obtain strain free optical windows. Standard round glass dewars are used, which are unsilvered for about a 10 cm length in the experimental region, and a special technique has been developed that completely eliminates liquid nitrogen in the outer dewar in the region of the experimental area. Over the past seven years we have built or bought almost every device conceived to do the phase modulation and at present use a photoelastic strain modulator4). A standard Kepco 2500 V power supply was converted to provide the necessary feedback to control the output of the photomultiplier, before they introduced the 2000 V operational amplifier, and is still in use. The overall system achieves a stability of I part in lo3 to IO4 with 1 set of integration, the limiting factor being changes in the depolarization in the dewar walls. This corresponds to a sensitivity to changes in the spin temperature of about a millidegree, which is close to the limit of the temperature stability of the bath. Greater sensitivity is achieved through modulation of the phonon source. The Van der Waals potential measurements are the most successful results achieved with the spectrometer to date2). These experiments also illustrate the sensitivity of the phonons within the volume of a crystal to surface conditions. In these experiments the spins are driven by resonant microwave radiation to generate phonons throughout the volume of a small, freshly cleaved chip of SrF, :TmZC covered with an adsorbed film of liquid helium. The phonons are lost primarily by leaking into the liquid helium, but because the acoustic mismatch between the liquid and solid is very large, the transmission is greatly enhanced when the film is an odd multiple of a quarter wavelength in thickness. The same spins which generate the phonons are used to detect their steady state level in the crystal and the resonant impedance matching condition is readily detected through a decrease in the spin temperature. This is summarized in fig. 2. An example of the experimental results is shown in fig. 3, where the film thickness was scanned by slowly punlping the helium gas out of the system. (The vertical scale in this figure is the direct output from the modulation system with an arbitrary zero offset.) In effect this is an acoustic Fabry-Perot interferometer that can be operated at 20 to 60 GHz, corresponding to wavelengths of 40 to 120 A. The wavelengths are so short that the resonances can be set up only on those parts of the cleaved surfaces that are atomically flat, hence the data are only obtained from the clean parts of the surface. The thickness of the helium film on resonance can be calculated through the formula d = Nc,[l + S(v)]/v + do(v), (1) where N is the order of the resonance

(4, +, . . . . etc.), co is the velocity

of

ACOUSTIC

PHONON

SPECTROSCOPY

917

vo Fig. 2.

Diagram of the helium film resonance experiment.

sound in liquid helium (237 m/set), 6( v) is a small correction of 1 to 3% due to the dispersion in the velocity of sound, and v is the driving microwave frequency. The additional length d,,(v) is a small frequency-dependent term, I to 3 A in magnitude, to take into account a phase-shift at the solid-liquid interface. The dispersion and phase-shift corrections were measured in a related series of experiments, where the helium film thickness was periodically modulated a few Angstriims3) thus constituting the shortest wavelength modulated Fabry-Perot interferometer ever operated. The system could detect a change of 0.1 A in a film thickness of IO0 A. While the dispersion and phase-shift results are of current interest in the theory of liquid helium, they make only a small correction to a determination of the film thickness, and thus are of lesser importance to the Van der Waals potential measurements. The chemical potential of a helium atom in the free surface of the liquid helium film is primarily determined by Van der Waals forces, and in equilibrium the potential is the same as that in the gas in the vicinity of the crystal. A plot of the potential due to the Van der Waals forces as a function of the film thickness can therefore be constructed by measuring the chemical potential at the resonance peaks and calculating the film thickness with eq. (1). The chemical potential of the gas is given by V = kT log [P,

(T)/P] ,

(2)

where k is the Boltzmann constant, T is the temperature, P,(T) is the vapor pressure of liquid helium at the temperature T, and P the pressure in the vicinity of the crystal. The gas pressure could be measured in the apparatus over a wide range using a commercial, capacitive type, differential gauge.

918

C. H. ANDERSON

AND

E. S. SABISKY

2

TIME

Fig. 3.

-

Helium film resonances, the vertical scale is the spin temperature in arbitrary units, and the horizontal scale is time as the helium film thins in a monotonic, nonuniform fashion.

To go to very low pressures, hence to small potentials and very thick films, the chemical potential was measured through the height of the crystal above a bulk liquid surface through the relationship I/ = mgh,

(3)

where m is the mass of a helium atom, g is the acceleration of gravity, and h is the height. The pressure gauge was sensitive enough to measure a pressure drop equivalent to going to a height of 8 cm above the liquid level, providing overlap between the two methods. The final results are plotted in fig. 4. Lifshitz has developed a macroscopic theory of the Van der Waals force that can be readily applied to the helium film case5). In order to apply the theory it is necessary to know the complex dielectric response function for liquid helium and the substrate material along the imaginary frequency axis.

ACOUSTIC

PHONON

SPECTROSCOPY

919

-I(

14

-II

33

-I(

-II

-z I2 2 + I !2 w I

3

I j

d

Fig. 4.

FILM

THICKNESS

(%I

Measured and calculated Van der Waals potential as a function of film thickness.

This can be approximated

by the expression

where each term c,,& is proportional to the oscillator strength of a band transition at the frequency o,, and the sum is taken over an appropriate set of band transitions. It is readily seen that this is a real, monotonic decreasing

920

C.

H.ANDERSON ANDE.S.SABISKY

function of the imaginary frequency, which makes the calculation very insensitive to the model parameters. Richmond and Ninhama) performed this calculation and demonstrated that our original results are in good agreement with the Lifshitz theory. More recently we have used their program and made detailed comparisons of our own2). It turns out that the contribution from the lattice vibrations can be ignored and the sum over the optical bands can be reduced to one transition. The most important input parameter is the optical index of refraction, while the energy for the transition has to be known in highly ionic solids only to within f 1 eV. The solid line drawn in fig. 4 is the result of a calculation using a 4 band model of the CaF, index of refraction and a two band model for liquid helium. The agreement between the theory and experiment is excellent not only in magnitude, but in that both also exhibit the same subtle deviation from a simple inverse cube law due to retardation. In summary we have tried to show how optical modulation detection of circular dichroism can be used to probe a wide range of interesting phenomena in the fields of physical acoustics, low temperature physics, paramagnetic resonance as well as surface physics. The example of the Van der Waals potential measurements contains a long string of concepts, based at the very beginning on the optical technique. And it should be emphasized that there are many branches along this string that lead to fascinating problems yet to be investigated. References 1) C. H. Anderson and E. S. Sabisky, Phys. Rev. Letters 18 (1967) 236; J. Acoust. Sot. Am. 49 (1971) 1052; and in: Physical Acoustics, Vol. 8, Eds. Mason and Thurston (Academic Press, New York, 1971). 2) C. H. Anderson and E. S. Sabisky, Rev. Letters 24 (1970) 1049; E. S. Sabisky and C. H. Anderson, Phys. Rev. A 7 (1973) 740. 3) C. H. Anderson and E. S. Sabisky, Phys. Rev. Letters 28 (1972) 80. 4) S. N. Jasperson and S. E. Schnatterly, Rev. Sci. Instr. 40 (1969) 761. 5) E. M. Lifshitz, Zh. Experim. Teor. Fiz. 29 (1955) 94 [Soviet Phys.-JETP 2 (1956) 731; I. I. Dzyaloshinskii, E. M. Lifshitz and L. P. Pitaevskii, Advan. Phys. 10 (1961) 165. 6) P. Richmond and B. W. Ninham, J. Low Temperature Phys. 5 (1971) 177; P. Richmond and B. W. Ninham, Solid State Commun. 9 (1971) 1045.

Discussion Question (by H. FRITZSCHE): The transverse and longitudinal acoustic phonons generated in the CaFa crystal must hit the superfluid Ge film at all angles. Could you please explain how one can observe such sharp interference patterns in such a case. Speaker’s reply (by C. H. ANDERSON) : The transverse phonons will couple to the longitudinal modes in the liquid He at all incidence angles other than the normal. The velocity of sound in liquid He is an order of

ACOUSTIC PHONON SPECTROSCOPY magnitude parallel

smaller

than

to the interface

few degrees Question

of the normal

it is in the crystals. in the crystal

Thus,

921

even those

are refracted

going

phonons

into

travelling

the liquid

almost

He to within

a

direction.

(by H. MORAWITZ):

Can

your

experimental

Van der Waals Casimir-Polder

technique

go

beyond

verifying

the

cancellation

potential to an effective l/R7 dependence due to retardation calculation considers only the lowest order nonvanishing

of the

l/R6

because diagram

the due

to fluctuations? Speaker’s reply (by C. H. ANDERSON): There is no evidence of a systematic Question (by M. CARDOEIA): How realistic do you think it would

deviation

between

be to propose

this technique

the ultimate quality of optical polishes? Materials other coupled (glued) to a CaFe piece and thus also studied. Speaker’s Some

have to go to lower for various reasons source

Comment I would theory simple

than

as a means

CaFz

could

of studying

be acoustically

reply (by C. H. ANDERSON): preliminary

phonon

the data and theory.

experiments

along

these

frequencies (i,e., longer we feel a conventional

for these

lines have been tried. wavelengths) piezoelectric

It is apparent

one will

to have lower resolution. Also transducer would be a better

experiments.

(by M. CARDONA): like to point

out the intimate

connection

we spectroscopists are engaged in. We usually models for it. Analytical continuation yields

related to the Van der Waals Van der Waals forces.

forces.

between

this work

and the kind

of

calculate E(W) for w real and develop e(iW) (w real), which is the quantity

Any of our standard

models

is useful

for calculating