Z. Phys. B - Condensed Matter 73, 33-41 (1988) Condensed ze,.c,r, Matter fLir Physik B

�9 Springer-Verlag 1988

Phonon focusing in [001] germanium W. Metzger and R.P. Huebener

Physikalisches Institut, Lehrstuhl Experimentalphysik II, Universitfit Tiibingen Federal Republic of Germany

Received May 17, 1988

Phonon focusing in [001J-oriented Ge crystals has been studied at temperatures near 2.0 K using electron-beam scanning for phonon generation and thin-film bolometers with an effective area of 10 gm x 10 gm or smaller for phonon detection. The thickness of the Ge crystals was 2 or 3 mm. In our experiments the angular resolution appeared to be dominated by the effective diameter of the local region heated by the beam and acting as phonon source. Whereas metallic overlay films did not appreciably effect the angular resolution, a distinct increase in resolution due to high-frequency beam modula- tion has been observed which can be understood in terms of the thermal skin effect. From a comparison between theoretical calculations and our experimental results the effective diameter of the phonon source has been found to be 20-30 gin. Our focusing images show appreciable deviations from theoretical calculations in the low-frequency limit. However, excellent agreement is obtained, if dispersion is included and if the domi- nant phonon frequencies are placed in the range 300-400 GHz.

I. Introduction

The propagation of ballistic phonons in single crystals at low temperatures is characterized by a distinct an- isotropy resulting from the anisotropic elastic crystal properties. Usually, this effect is referred to as phonon focusing. In recent years a powerful technique has been developed for the two-dimensional imaging of this phonon focusing effect. This method simply con- sists of scanning a planar crystal surface at low tem- peratures with a laser beam or an electron beam. The ballistic phonons emitted from the surface region lo- cally heated by the beam are detected with a small- area phonon detector attached to the opposite crystal surface. So far, superconducting thin-film bolometers or tunnel junctions have been used for phonon detec- tion. By detecting the ballistic phonon signal as a function of the coordinate point of the beam focus on the crystal surface, a two-dimensional image of the variation of the phonon intensity with the angle of propagation through the crystal is obtained. A more detailed description of these developments can be found in recent summaries [-1-5].

Whereas the principles of phonon focusing appear well understood, some questions remain which are

mostly connected with the phonon source experimen- tally generated during the scanning process. The an- gular resolution obtained in these experiments is de- termined, in principle, by the lateral extension of both the phonon source and the phonon detector. How- ever, since phonon detectors with only a few gm 2 effective area can be fabricated today using standard thin-film microfabrication techniques, it is the exten- sion of the phonon source which usually limits the angular resolution, In this paper we address ourselves mainly to the experimental aspects of the phonon source. As in our earlier work, [,,1-3, 6] our experi- ments utilize electron-beam scanning performed in a scanning electron microscope equipped with a liquid- He stage. All measurements are carried out with [001J-oriented single-crystalline Ge samples having a thickness of 2 or 3 mm. In particular we study the influence of various overlays (Chap. IV) and of the beam parameters (Chap. V) upon the angular resolu- tion. In Chap. VI the lateral extension of the phonon source is determined from a comparison of experi- mental line scans with theoretical calculations per- formed in the acoustic longwavelength limit. In addi- ton to these studies dealing with the source of the ballistic phonons, in Chap. VII we compare the exper-

34

imental and theoretical two-dimensional phonon fo- cusing images and point out the possibilities for ex- plaining slight discrepancies.

Preliminary results of this investigation have been reported at previous conferences [7-9]. A more de- tailed report can be found elsewhere [10].

II. Experimental procedures

The principle of our experiments is similar to that described in our previous paper [6]. The top surface of the crystal to be studied is scanned with the elec- tron beam of a scanning electron microscope (SEM) whereas the bottom surface is in direct contact with liquid He. A description of the low-temperature stage of the SEM is given elsewhere [11]. The phonons generated at the top crystal surface during the scan- ning process propagate through the crystal ballisti- cally and are detected on the backside using a Super- conducting thin-film bolometer attached directly to the crystal surface. As a bolometer we have used gran- ular A1 films prepared by evaporation of aluminum (MRC; 99.9995%) in the presence of oxygen. The crit- ical temperature of the films ranged between 1.7 and 2.0 K. The geometry of the bolometer film is shown schematically in Fig. 1. Usually the effective bol- ometer area between the much wider contact banks w a s 10 x 10 g m 2. (Some experiments have been per- formed also using bolometers with 2 x 2 gm 2 effective area. However, the latter bolometers showed a rela- tively high risk for burnout). For microfabrication of the bolometers standard photolithographic tech- niques were used. The thickness of the bolometer films was 40 nm. The bolometer current was taken in the range 20-40 gA. The bolometer current and the tem- perature of the liquid-He bath were always adjusted for obtaining the maximum spatial resolution of the focusing image.

The arrangement of the measuring electronics is shown schematically in Fig. 2. We have used three different operational modes.

In mode A the electron beam is chopped at 20 kHz frequency and 50% duty cycle. The bolometer voltage

I

W

uj

I

o-- ____2

Fig. 1. Geometry of the bolometer film. The hatched part in the center represents the sensitive area

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Fig. 2. Schematics of the measuring electronics

signal is taken using the standard four-point-probe configuration and fed to the differential amplifier in- put of a lock-in detector.

In mode B the electron beam is chopped at a fre- quency in the range 100 kHz-16 MHz and 50% duty cycle. The bolometer signal is fed to a preamplifier via 50 f2 coax cable and then to a fast lock-in detector. Phase-independence of the output signal is achieved by means of a vector adder which processes the two output signals UI and UQ of the lock-in detector shifted in phase by n/2. Finally, the signal passes a low-pass filter.

In mode C electron-beam pulses of 30-50 ns width and a repetition rate of about 400 kHz were generat- ed. Following preamplification the bolometer signal is fed to a boxcar-integrator via 50 ~2 coax cable. The boxcar-integrator can be operated in two different ways. First, for a locally fixed beam the temporally resolved bolometer signal is plotted using an x-y re- corder. Second, during the beam scanning process the bolometer signal can be selected for a specific time window.

In general, the bolometer signal obtained in all three modes A - C can be used for brightness modula-

35

tion of the cathode ray tube screen of the SEM or for y-modulation of individual line scans (where the signal causes a vertical deflection for a line scan in horizontal direction).

III. Samples

The experiments were performed using [001J-oriented single-crystalline Ge samples with 2 or 3 mm thick- ness. All samples except one had a circular shape with a diameter of 20 mm. The crystals were ultra- pure Ge with an impurity concentration INo-NAI

1.10 ~~ cm -3 and a density of edge pit dislocations EPD~IOOOcm -2. One sample was ion-implanted with B on one side (implantation energy=350 keV, dose= 5-10 l~ cm -2, maximum of the Gauss distribu- tion at about 0.8 gm depth with 0.2 ~tm halfwidth), the doping concentration reaching 10 t5 cm -3 at its maximum.

One sample was shaped as a square with 12 mm side length. For this specimen a special sample holder had to be designed. Here it is important that the sam- ple crystal itself constitutes the separation between the vacuum in the sample chamber of the SEM and the liquid-He bath [-6]. A vacuum-tight indium seal between the sample crystal and the top plate of the liquid-He stage [-11] could be accomplished for al- most arbitrary shape of the boundary of the planar sample crystal using the arrangement shown schemat- ically in Fig. 3. The mechanical pressure is applied to the sample and the indium seal in a ring-like area of only relatively small diameterl thereby reducing the risk of breaking the specimen.

During many experiments the top surface of the sample crystal carried a metallic thin-film overlay pre- pared by standard vacuum evaporation.

IV. Angular resolution: influence of metallic overlay films and ion implantation

In imaging the phonon focusing effect the angular resolution is limited by the lateral extension of the

~(1) ' 1 cm i ( 2 )

Fig. 3. Mounting arrangement for a planar sample crystal for arbi- trary shape of its outer boundary. (1) clamping screw, (2) press ring, (3) sample crystal, (4) indium seal, (5) threaded parts

phonon source and the phonon detector. Since the effective area of the latter can be made extremely small by means of standard microfabrication tech- niques (here 10 x 10 gm 2 or smaller), the extension of the phonon source is expected to become crucial for the angular resolution. Therefore, our first experi- ments were concerned with the question whether the lateral size of the phonon source is effected by metallic overlay films. These films were evaporated on the sur- face of the Ge crystal to be irradiated with the electron beam.

We have investigated the influence of granular A1 films of different thickness evaporated in the presence of oxygen (O2-partial pressure 1.10 -4 torr). A Ge crystal (20 mm diameter, 3 mm thickness) has been prepared where three of the four quadrants of the circular crystal surface were covered with an A1 film of different thickness (0.3, 0.8, and 3 gin) and one quadrant was left uncovered. Taking the backscat- tered electrons and the secondary electrons for imag- ing the surface of this specimen, one notes a distinct reduction of the signal magnitude with increasing thickness of the A1 overlay film, the largest signal originating at the uncovered Ge surface (beam volta- ge = 26 kV). This is to be expected since the backscat- tering coefficient varies proportional to Z 2, where Z is the electron number per atom [12] (AI: Z=13; Ge: Z=32). The penetration depth of 26 keV beam electrons in A1 amounts to about 3.5 gm [12].

Along with the decrease in intensity of the back- scattered electrons with increasing thickness of the A1 thin-film overlay we have observed a correspond- ing increase of the detected phonon signal, as ex- pected. Clearly, with decreasing intensity of the backs- cattered electrons, the beam power dissipated in the sample and converted into ballistic phonons becomes larger. Typical results on the ballistic phonon image are shown in Fig. 4. On the left the two-dimensional phonon-focusing pattern can be seen. Here and in the following bright regions always correspond to large values of the bolometer signal. The scanned are shown is 1.2 mm x 1.3 mm. The placement of the A1 overlay films with different thickness within the scanned area is indicated by the sketch on top.

On the right in Fig. 4 we show a series of line scans (y-modulation) performed along the white lines marked on the image on the left. The scan direction is located within the (011)-plane. The angles between the [-001J-direction and the propagation direction of the ballistic phonons projected on the (011)-plane is marked on the horizontal axis. The dashed arrow near the top trace indicates the location of the bound- ary between 0 gm and 3 gm overlay thickness. The solid arrow marks the boundary between the 0.3 gm and 0.8 gm overlay.

36

- lS' ~10" -S" O" 5" 10" 15'

Fig. 4. Left: Two-dimensional image of the time-integrated bol- ometer signal of a [001]-oriented Ge sample. The placement of the AI overlay films with different thickness within the scanned area is indicated by the sketch on top. Right: Bolometer signal for a series of line scans (y-modulation) performed along the white lines marked on the image on the left. Crystal thickness 3 mm; beam parameters: 26 kV, 10 hA; bath tem- perature 1.7 K. Further details are given in the text

For a quantitative evaluation of the spatial resolu- tion limit we have determined the slope of the steeply rising outer portion of the line scans. This slope would be infinite for zero lateral extension of both phonon source and detector [13]. First we have graphically determined the distance uM between the position of the sharp signal peak and the position where the sig- nal starts to rise (by linearly extrapolating signal and background). Then we have normalized the distance uu by the width bF of the square of the focusing pat- tern, yielding the ratio

rs=uM/bv. (1)

In this way data obtained for different thickness of the specimen crystals can be compared directly. The width bv was obtained from linescans determined the- oretically, be representing the distance between the outer two maxima. The method of these calculations is outlined in Chap. VI. For the overlay films of gran- ular A1 we have found the value r~5 .3 -10 _2 nearly independent of the overlay films.

A similar value of r, has been obtained for a 2 ~tm overlay film of Pb-10% In alloy and for a sample ionimplanted with B at the irradiated surface as de- scribed in Chap. III. Both samples yielded the value r~ = 5.9-10 -2. In summary we note that the metallic overlay films investigated and ion implantation with B did not appreciably effect the angular resolution

obtained in imaging the phonon focusing pattern of Ge by means of electron beam scanning at low tem- peratures.

V. Angular resolution: influence of the beam paramaters

If the beam voltage was reduced from 26 kV to 2 kV, the signal amplitude due to the backscattered elec- trons and the secondary electrons was found to be nearly the same for three granular A1 film overlays with thickness 0.4, 1, and 5 gin. Similarly, the magni- tude of the bolometer signal was observed to depend only weakly on the overlay thickness. This is to be expected, since the penetration depth of 2 keV elec- trons in A1 is only about 0.1 pm [12].

By performing line scans similar to those shown in Fig. 4 and by determining the ratio r~ of (1), we have investigated the influence of the beam voltage and the beam current upon the angular resolution. In the ranges studied, 5-26 kV and 0.5-500 hA, the angular resolution was nearly independent of both parameters.

A significant increase in angular resolution was observed if the chopping frequency of the electron beam was increased from 20 kHz up to 16 MHz (lock- in detection of the bolometer signal according to mode A or mode B). Typical results are shown in Fig. 5. On the left we see the phonon focusing pattern for the same specimen and the same overlay film con- figuration as in Fig. 4. On the right the bolometer

fch =16 MHz

;. ~. ~'~.

Fig. 5. Left: Two-dimensional image of the time-integrated bol- ometer signal. Beam parameters: 26 kV, 10 nA. Right: Bolometer signal for a series of line scans (y-modulation) performed along the white line marked on the image on the left for different beam chopping frequencies. Beam current = 1 gA. Same specimen and A1 overlay film configuration as in Fig. 4; bath tem- perature = 1.7 K. Further details are given in the text

signal is plotted for a series of line scans with different beam chopping frequencies f~h. The position of the line scan is marked by the white line on the left. The arrow on the top indicates the location of the bound- ary between the 0.3 gm and the 0.8 gm overlay for the line scans. The values of the ratio rs of (1) obtained for the different chopping frequencies are listed in Ta- ble 1.

Table 1. Ratio rs of (1) for the different chopping frequencies

2.104 4.2" 10-2 2' 10 s 3.0" 10-2 2.106 3.5.10 -2 1.6-107 1.9' 10 -2

The smallest observed halfwidth of a bolometer signal peak in a line scan for fch = 16 MHz was about 25 txm. This value corresponds to an angular resolu- tion limit of 0.5 ~ for the 3 mm crystal thickness. If a point-like phonon source is assumed and taking the 2 gm x 2 gm effective area of the bolometer used in these experiments and the 3 mm transit distance of the ballistic phonons, an angular resolution of 0.04 ~ is calculated. From this small value we see that the influence of the bolometer upon the experimentally obtained angular resolution limit appears to be negli- gible.

The increase in angular resolution for increasing beam chopping frequency demonstrated by the results in Fig. 5 and Table 1 is due to the thermal skin effect. At sufficiently high chopping frequencies f~h the later- al spreading of the modulated thermal perturbation due to the beam irradiation varies proportional to f ~ 1/2. Of course, for the corresponding increase in resolution to become effective the beam-induced modulated signal must be detected. A more detailed discussion of this point can be found elsewhere [1 33.

VL Theoretical line scan calculation and lateral size of the phonon source

The experiments we have discussed so far indicate that the angular resolution obtained by low-tempera- ture electron beam scanning for imaging the phonon focusing pattern is dominated by the lateral size of the phonon source. An accurate way for determining this size consists of comparing the results of theoreti- cal line scan calculations with experimental measure- ments. Here the line scan calculations must be per- formed for different assumed values of the effective phonon source diameter. We have performed such

37

calculations based on the computer program by Northrop [14] and a Monte-Carlo technique [15]. The computer program has been extended by Held [16] allowing arbitrary crystal orientations and easy variation of the range of the phonon propagation an- gle. Our theoretical results were obtained with this extended program.

In the Monte-Carlo technique a distinct number of phonons with an isotropic distribution in k-space is taken using a random number generator, and the corresponding propagation directions in space are calculated. Then the points where the phonons im- pinge on the chosen crystal surface are determined. The density of these points represents the two-dimen- sional phonon focusing pattern, locations with high point density corresponding to a large intensity of the phonon flux. For determining the local point den- sity the selected crystal surface is divided into square cells of typical size 1 gm x 1 gm and the number of phonons impinging upon the individual cells is counted for a large total phonon number (> 1. 107).

Next a circle with variable radius r is moved over this surface in a straight line typically in 1 gm steps. For each step the phonon numbers of all cells within the circle are added following the application of a proper weighting factor. For the weighting function we take a Gauss distribution such that the weighting factor drops from 1 in the center of the circle to 10 -2 at the radius r. The radius at which the weighting factor drops to e-a is denoted by ref f. The circular

2 area :g-ref f can be interpreted as the effective detector area with a Gaussian distribution of its sensitivity. The phonon source is then taken as pointlike. Inverse-

2 ly, the area ~.ref f can also be interpreted as the effec- tive area of the phonon source with a Gaussian inten- sity distribution. The detector is then taken as point- like. In the following we take Deff=2roff as the effec- tive diameter of the phonon source. Of course, the calculation outlined above can be performed for the combination of all three phonon modes or for each mode separately.

The theoretical results presented in the following are based on the following parameter values for Ge:

elastic constants [14, 17]:

cll :- 1.311.1011 N / m 2

c12 =4.92 �9 101~ N / m 2

c1r 6.82 �9 10 TM N / m 2

density = 5.34 g/cm 3 sound velocities [18] (used for weighting the individual modes)

average value of the transverse modes: 3.1 km/s longitudinal mode: 5.2 km/s

38

l I I I - 1 0 ~ - S ~ 0 o 5 �9

L

1 * - 1 0 ~ - 5 ~ 0 ~ 5 ~ 1 ~

Fig. 6. Left: Calculated phonon intensity (sum of all three modes; acoustic long-wavelength limit) for a series of line scans with differ- ent values of the effective diameter D~ff of the phonon source. The location of the line scan within the focusing pattern is indicated by the dashed line on the sketch at the upper right. Right: Corresponding bolometer signal experimentally obtained for the same line scan by means of a boxcar technique. Crystal thick- ness: 2 mm; beam parameters: 26 kV, 0.1 gA; bath temperature: 1.96 K

_J

J/

I I I I / I I I I /

-I0 ~ - 5 �9 0 ~ 5 ~ I0 o _10 o _ 5 ~ 0 o 5 ~ I0 o

Fig. 8. Left: Calculated phonon intensities (sum of all three modes, acoustic long-wavelength limit) for a series of line scans with differ- ent values of the effective diameter D0ff of the phonon source. The location of the line scan within the focusing pattern is indicated by the dashed line in the sketch at the upper right. Right: Time-integrated bolometer signal experimentally obtained for the same line scan. Crystal thickness: 2 mm; beam parameters: 26 kV, 4 nA; bath temperature: 1.95 K

=5

-72

o

.g 0

i [

I

[ m

2 t [ps]

Fig. 7. Time-resolved bolometer signal for a fixed electron beam position which is marked with A in the sketch on the upper right. The arrow indicates the boxcar time-window position at which the experimental trace in Fig. 6 was obtained. Crystal thickness: 2 mm; beam parameters: 26 kV, 0.1 gA; bath temperature: 1.96 K

Typical results are shown in Fig. 6. On the left we see the calculated phonon intensity (sum of all three modes; acoustic long-wavelength limit) for a series of line scans with different values of D~f e. The location of the line scan is indicated by the dashed line on the sketch at the upper right. The experimental trace can be seen on the right. It was obtained by means of the boxcar technique (mode C) using a 10 ns time window placed in the middle between the sharp onset and maximum of the time-resolved bolometer signal and the position of this time window is shown in

Fig. 7. In Fig. 6 the angle between the [001J-direction and the propagation direction of the ballistic phonons is marked on both horizontal axes. The nonsymmetry of the experimental line scan apparently resulted from nonuniform beam irradiation along the scanning line. For the results shown in Fig. 6 the best agreement between experiment and theory is obtained for a phonon source diameter of about Deff = 30 gin.

The arrows below the traces on the left and on the right in Fig. 6 indicate the position of the corre- sponding peaks in the curve. We note a slight discrep- ancy between the experimental and theoretical value of this position, a point which we take up again in Chap. VII.

Further results obtained for a different location of the line scan are shown in Fig. 8. Here the experi- mental scan was performed using the time-integrated bolometer signal (lock-in method, fob = 20 kHz). Tak- ing into account only the fine structure of the two main peaks, the comparison between experiment and theory suggests the value Deft= 10-20 gm. In addition to the results shown in Fig. 6 and 8 the comparison between experimental and theoretical line scans has been performed also for other line scan locations. From all these results the effective diameter of the phonon source for the chopping frequency fc~ = 20 kHz is found to be D~ee = 20-30 gm.

39

I . I

200 pm Fig. 9. a Two-dimensional image of the time-integrated bolometer signal. The points A - F and the angle ~ serve for quantifying the geometric pattern of the focusing image. Crystal thickness: 2 ram; beam parameters: 26 kV, 3 hA; bath temperature: 1.96 K. b Calculated phonon intensity (sum of all three modes, acoustic long-wavelength limit) of the two-dimensional focusing pattern for [001J-oriented Ge. The field of view corresponds to the experimentally obtained image of a. Locations with high point density correspond to a large intensity of the phonon flux. The ratios of the elastic constants are a = 1.92 and b=0.72, e Same calculation as in b, but with the modified ratios of the elastic constants a,, = 2.11 and bm= 0.91

VII. Two-dimensional phonon focusing image: comparison between experiment and theory

In the following we perform a detailed comparison between the experimental and theoretical two-dimen- sional phonon focusing image of [001J-oriented Ge. In Fig. 9 part a we show the experimental image of the time-integrated bolometer signal for a Ge crystal with 2 mm thickness. The points marked A - F serve for quantifying the geometric pattern of the focusing image of the slow transverse (ST) phonons. The angle c~ specifies the focusing image of the fast transverse (FT) phonons. Part b in Fig. 9 shows the calculated results obtained using the Monte-Carlo point density method outlined in Chap. VI and the parameter values for Ge listed in that chapter. These calculations were performed in the acoustic long-wavelength limit without taking into account dispersion.

The different size of the inner square of the focus- ing image represents the most prominent difference between experiment and theory. Defining the distance ratio u = CD/AB, the theoretical calculations and the experimental measurements yield the values uth = 0.77 and uex p = 0.67, respectively. The experimental values Uexp were determined from line scans and two-dimen- sional images of different samples, and the indicated value represents the average of these measurements. The difference between uth and U~xp corresponds to the different positions of the maxima in the theoretical and experimental line scans of Fig. 6 marked by the

arrows. (The theoretical and experimental value of the corresponding propagation angle is 5.54 ~ and 4.6 ~ respectively).

In the experimental focusing image the corners of the square appear more rounded than in the theo- retical image. In Fig. 9 b the two ST lines which are shown enhanced at the upper right intersect close to the edge of the focusing square, whereas this intersec- tion is located further outside in the experimental im- age. This difference is also reflected in the distance ratio v = EF/AB, attaining the theoretical and experi- mental values Vth= 1.29 and vex p = 1.22, respectively. Finally, the experiments yield for the angle c~ dis- played by the FT mode the average value e= 9.0 ~ whereas the theoretical value is e = 7 ~

For explaining these discrepancies between exper- iment and theory, we first concentrate on the elastic constants. For calculating the phonon focusing pat- tern without dispersion one only needs the ratios a = ca ~/c44 and b = c la/c44 of the elastic constants, in addition to the density. From the elastic constants listed in Chap. VI we have the values a = 1.92 and b = 0.72. We have calculated a series of phonon focus- ing patterns by systematically varying the ratios a and b. From the results we have determined the geo- metric ratios u and v and the angle ~ defined above. The best agreement between theory and experiment was found using the values a,,=2.11 and bin=0.91, and the corresponding theoretical image is shown in part c of Fig. 9. Based on these values one finds Uth

40

Fig. 10a and b. Lines with diverging phonon intensities (caustics), projected on the (001)- plane of Ge. The solid lines represent the ST- mode, the dashed lines the FT-mode. a shows the result for non-dispersive phonons (0 THz); b for dispersive phonons with a frequency of 0.7 THz (from Ref. 2t). The marked distance d is explained in the text

d I I

I I

',: ~ .~

[hq Oe ' ~ c ~ e ' ~ ~ l o b ) \~ ]

. _ % . -

= 0.68 and % = 1.21 in good agreement with the ex- perimental values indicated above. In Fig. 9c the corners of the focusing square are more rounded, and the intersection of the ST lines is located further out- side the square than in Fig. 9 b, similar to the experi- mental result. However, the angle c~ of the FT mode and the resulting discrepancy with experiment re- mains unchanged. Whereas the modified ratios a,, and bm clearly yield better agreement with the experimen- tal observation, their deviation from the literature values (a = 1.92; b =0.72) is relatively large and ap- pears difficult to understand.

A more likely explanation of the discrepancy may be possible from a consideration of phonon dispersion which has been neglected so far. The bolometer repre- sents a wideband phonon detector recording also the high-frequency phonons. The dispersion curves of Ge show particularly flat transverse-acoustic.\branches. Calculations by Tamura [19-21] indicate that in Ge the phonon focusing pattern changes appreciably al- ready above 0.3 THz due to dispersion effects.

In Fig. 10 we present theoretical results obtained by Tamura [-21] for Ge. The lines with diverging phonon intensities (caustics) on the (001)-plane are shown for the low-frequency limit (part a) and for 0.7 THz (part b). At the high phonon frequency we note the following changes: (1) the size of the inner square of the focusing pattern is reduced, (2) the angle enclosed by the FT mode becomes larger (above we have denoted this angle by c~), (3) the corners of the square are more rounded and, as a consequence, the ST lines intersect further outside these corners. All three points can be noted in our experimental images.

Tamura [21] also calculated the dependence upon phonon frequency of the size of the inner and outer square of the focusing pattern and of the polar angle

0 indicating the direction of the FT caustic as mea- sured from the [110]-direction. Therefore, the corre- sponding geometrical quantities we have found exper- imentally, in combination with Tamura's results, can be used for determining the dominant phonon fre- quency contributing to our observed focusing image. In this way, our results uexv = CD/AB = 0.67 suggests a dominant phonon frequency of about 400 GHz.

In Fig. 10 we have marked the distance d at the top. From this distance the polar angle 0, mentioned above, can be calculated using the relation d

= 2. ]/2. tan 0. We have measured the distance d in different samples by recording correspondingly large images. From these measurements we have found the average value 0exp = 2.5 ~ of the polar angle. According to Tamura's calculation [21] this corresponds to a phonon frequency of 350 GHz.

From a theoretical focusing pattern presented by Tamura [20] in polar coordinates for a phonon fre- quency of 300 GHz, the angle c~ of the FT mode and the geometric quantity v = EF/AB, defined above, can be found yielding the theoretical values ~th (300 GHz)=9.2 ~ and % (300 GHz)=I.21. These values compare well with our experimental results ~exv = 9"0~ and Vexp = 1.22.

In conclusion we note that the details of the phon- on focusing images in [001] Ge we have observed are in excellent agreement with theoretical results for phonon frequencies in the range 300-400 GHz. This fact appears consistent with an estimate of the range of the dominating phonon frequencies generated at the specimen surface by the electron beam irradiation. Such an estimate can be performed assuming that the local region acting as phonon source can be de- scribed by an effective excess temperature T* above the temperature of the remainder of the crystal (which

41

can be taken as the temperature of the liquid-He bath). The effective temperature T* can be found from simply balancing the power of the electron beam dissi- pated in the phonon source with the power emitted by the ballistic phonons, assuming a Planck spectrum at temperature T* and using the Stefan-Boltzmann law [-3, 10]. In this power balance the radiating sur- face area of the phonon source is determined from the effective source diameter Defer25 ~tm we have found from the results described in Chap. VI. Using this procedure we have obtained for the effective source temperature values in the range T* =4-6 K, yielding for the frequency Vm, x = 2.82 k B T*/h of the maximum in the Planck spectrum values in the range 250-350 GHz.

Northrop and Wolfe [133 also studied phonon focusing in [001J-oriented Ge using a crystal of 2.7 cm thickness and a laser-beam scanning technique. For the two inner intensity peaks marked by the arrows in Fig. 6 they found a propagation direction of 5.5 ~ in good agreement with the theoretical value of 5.54 ~ calculated in the low-frequency limit. On the other hand, our measurements yielded a propagation direc- tion of 4.6 ~ . It appears that this discrepancy between our results and those of Northrop and Wolfe can be explained in terms of the different thickness of the Ge crystals used. Whereas our crystal thickness of only 2-3 mm is likely to be traversed also by ballistic phonons in the frequency regime of 300-400 GHz where dispersion effects take place, these high-fre- quency phonons are more likely to be filtered out by isotope scattering in the crystal used by Northrop and Wolfe where the propagation distance is about ten times larger. Therefore, the low-frequency limit is expected td be more applicable to the experiment of these authors.

VIII. Summary and conclusions

We have investigated phonon focusing in [001]-ori- ented Ge crystals of 2-3 mm thickness using electron- beam scanning at low temperatures. Superconducting bolometers with an effective area of 10 gm x 10 gm or smaller, fabricated from granular A1, served for phonon detection. Because of the small phonon detec- tor area, the angular resolution of the focusing image is expected to be dominated by the lateral size of the phonon source generated by the beam irradiation. The influence of various metallic thin-film overlays of different thickness (granular A1, PbIn-alloy) on the Ge surface to be irradiated upon the angular resolu- tion has been studied, and no appreciable effect has been found. Similarly, a variation of the beam param-

eters in the range 5-26 kV and 0.5-500 nA showed very little influence upon the angular resolution. However, an appreciaNe increase in angular resolu- tion was observed if the beam was chopped at high frequencies (up to 16 MHz) and if the modulated bol- ometer signal was recorded. This increase can be well understood in terms of the thermal skin effect. The lateral size of the phonon source has been determined by comparing experimental line scans with theoretical calculations. In these calculations the effective diame- ter Def t of the phonon source has been folded into a Monte-Carlo technique. For a beam chopping fre- quency of 20 kHz an effective phonon source diameter D e f t = 20.30 gm has been found.

Our observed two-dimensional phonon focusing images show appreciable deviations from theoretical calculations for the low-frequency limit. It appears that these deviations are due to phonon dispersion. Comparing our experimental results with the recent calculations of Tamura for [001] oriented Ge includ- ing dispersion, we obtain excellent agreement between experiment and theory for phonon frequencies in the range 300400 GHz.

Financial support of this work by a grant of the Deutsche For- schungsgemeinschaft is gratefully acknowledged. E. Held has pro- vided essential help for performing the theoretical calculations of the phonon focusing pattern in the low-frequency limit. The Ge samples used in our experiments have been kindly supplied by P. Glasow, Siemens AG, Erlangen.

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W. Metzger, R.P. Huebener Physikalisches Institut Lehrstuhl Experimentalphysik II Universit/it Tiibingen D-7400 Tfibingen Federal Republic of Germany