Ballistic phonon signal for imaging crystal properties

Materials Science and Engineering, B5 (1990) 157-165

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Ballistic Phonon Signal for Imaging Crystal Properties R. P. HUEBENER, E. HELD and W. KLEIN

Physikalisches lnstitut, Lehrstuhl Experimentalphysik 11, Universitiit Tiibingen, D- 7400 Tiibingen (F.R.G.) (Received May 31, 1989)

Abstract

We present a summarizing overview of our recent experiments using the ballistic phonon signal generated by low temperature scanning electron microscopy for imaging defect structures and the anisotropic elastic properties (phononfocusing effect) in single-crystal materials. Extensive studies have been performed with dielectric crystals (sapphire and a-quartz) and semiconductors (GaAs, silicon and germanium). With a single phonon detector of small area (superconducting bolometers of size between 2 tzm x 2 / z m and 10 ~ m x 10 fzm), we have obtained two-dimensional images with a lateral resolution reaching about 5 /zm. Using superconducting tunnel junctions as frequency-selective phonon detectors, dispersive effects in the ballistic phonon propagation have been observed. The application of more than one phonon detector placed at different locations allows stereoscopic observations and yields information on the defect structure in all three dimensions. Using computer-controlled electron beam scanning and digital data processing, techniques for contrast enhancement in the three-dimensional acoustic imaging of defect structures have been developed. A three-dimensional acoustic tomography principle based on the ballistic phonon signal in combination with a detector array is outlined. Our present experiments indicate that a spatial resolution of about 20 Izm in all three dimensions can be achieved with this acoustic imaging principle.

1. Introduction

Since the pioneering experiments of von Gutfeld and Nethercot [1] 25 years ago, heat pulse propagation in solids has been intensively investigated. Here the transport of energy by means of ballistic phonons is a particularly important process. Ballistic phonon propagation at the velocity 0921-5107/90/$3.50

of sound takes place if the phonons can traverse the characteristic distance of the sample configuration without scattering. Therefore, ballistic phonon transport is restricted to low temperatures (where the scattering of phonons by other phonons becomes unimportant) and to crystals of sufficiently high purity. Studies of heat pulse propagation in solids at low temperatures have contributed significantly to our understanding of ballistic phonon propagation and phonon scattering [2, 3]. Recently, ballistic phonons have taken on a new role in imaging spatially structured properties in crystals. The phonons are generated by scanning the crystal surface with an electron or laser beam, while the crystal is cooled to liquidhelium temperatures. On the opposite crystal surface the phonons are detected with one or more spatially fixed phonon detectors of small area. During the scanning process the ballistic phonons propagate between source and detector through different parts and in different crystallographic directions of the crystal. In this way the anisotropy of the ballistic phonon propagation (phonon-focusing effect) [4, 5] and the scattering of phonons by defect structures can be imaged. This principle is usually referred to as phonon imaging. In this paper we summarize the results of our recent phonon-imaging experiments performed by electron beam scanning at liquid-helium temperatures. These experiments have been carried out in a scanning electron microscope equipped with a low temperature stage. The principal features of this technique and various other applications of low temperature scanning electron microscopy (LTSEM) have been discussed in recent reviews [5, 6]. A review of and further references on the recent phonon-imaging experiments performed by laser beam scanning can be found elsewhere [4, 7, 8]. © Elsevier Sequoia/Printed in The Netherlands

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2. Ballistic phonon signal in low temperature scanning electron microscopy The experimental arrangement based on LTSEM for phonon imaging is shown schematically in Fig. 1. The sample crystal is mounted on the low temperature stage of the scanning electron microscope [9] in such a way that its top surface can be scanned directly with the electron beam, while its bottom surface is in direct contact with liquid helium. At the coordinate point of the beam focus the crystal is heated locally by means of the dissipated beam power. The beam parameters are typically as follows: voltage, 26 kV: current, 1-1000 nA (depending on the electronics for signal detection); beam diameter, 1/~m or less. The local region irradiated by the beam during the scanning process is heated to an effective temperature T* and acts as the source of ballistic phonons. The phonon emission can be described in good approximation by Planck's radiation law. For a typical effective source temperature T* = 10 K the maximum of the Planck distribution of the emitted phonons corresponds to a phonon frequency of about 600 GHz. The equivalent acoustic wavelength is in the nanometre range for the crystalline materials that we have studied. Following their emission from the source region, the phonons traverse the crystal. At low temperatures and for sufficient purity of the crystal, the phonons propagate ballistically (i.e. without scattering) at the velocity of sound over distances in the millimetre range up to about 1 cm. Since phonon imaging is based on ballistic phonon propagation through the crystal, the

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thickness of the sample must not exceed these values of the phonon mean free path, At the lower crystal surface the phonons are detected with one or more spatially fixed phonon detectors. For phonon detection, thin film devices such as superconducting bolometers or tunnel junctions deposited directly on the lower crystal surface are used. During the scanning process, ballistic phonons originating from the different points on the upper crystal surface are recorded by the detector. In this way the anisotropy of the ballistic phonon propagation through the crystal (phonon-focusing effect) [4, 51 and the scattering or absorption of phonons by defect structures located along the acoustic path between phonon source and detector can be imaged. In the latter imaging process of defect structures the "acoustic shadow" of the defect seen by the detector is utilized. Since the acoustic wavelength of the relevant phonons is in the nanometre range, geometric optics concepts can be applied. The angular resolution of this imaging principle is determined by the effective size of the phonon source and of the detector. Therefore, both sizes should be made as small as possible. Using standard microfabrication techniques, the characteristic dimensions of the thin film devices used for phonon detection can be made as small as only a few micrometres. In contrast, experiments have shown that the effective diameter of the phonon source is typically 20-30 /zm '~1{), 111. From this it appears that the angular resolution is limited by the spatial extension of the phonon source. If we take 25 um as the source diameter and a crystal thickness of 3 mm and assume that the size of the phonon detector can be neglected, the angular resolution is found to be 0.5 °. In order to increase the sensitivity of the phonon detector, the electron beam power is modulated, and the detector signal is recorded using a lock-in amplifier. The beam modulation frequency is typically in the 10 kHz range. An additional advantage of high frequency beam modulation results from the fact that the region thermally modulated by the irradiation shrinks with increasing frequency (thermal skin effect), improving the spatial resolution of LTSEM [10-12]. In this operational mode the time-integrated phonon signal is detected, and the beam current can be kept relatively small (1-10 nA range). To study the temporal behaviour of the propa-

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gating phonons, a pulsed electron beam and boxcar averaging for signal detection are required. We have used beam pulses of width 30-50 ns at a repetition rate of about 400 kHz [11, 13]. For such time-resolved experiments a relatively high beam current is needed ( 100-1000 nA range). Computer-controlled manipulation of the electron beam and digital signal processing was used for contrast enhancement and for extracting special features in the imaging process. Further, these techniques are essential for the threedimensional reconstruction of the defect structures from the signals recorded with more than one phonon detector [14]. We shall return to these aspects in Section 5. It is important to note that the ballistic propagation of phonons without scattering over distances which are interesting in terms of phonon imaging (in the millimetre range up to about 1 cm) is restricted to crystals with negligible phonon scattering at free charge carriers. Therefore, in general, phonon imaging can be performed only in dielectric or semiconducting crystals. In the latter case, nearly all free charge carriers are frozen out and their interaction with the propagating phonons can be neglected at low temperatures. An interesting exception to this restriction to dielectric or semiconducting crystals is single-crystal superconductors at temperatures sufficiently below their superconducting transition temperature. In this case the electronphonon interaction is strongly reduced owing to the superconducting energy gap, and correspondingly large values of the phonon mean free path are possible. Electron beam irradiation of dielectric or semiconducting crystals at low temperatures can result in charging effects, causing severe perturbations during the scanning process. In order to avoid these charging effects, the crystal surface to be irradiated is usually coated with a thin metallic overlay film. For this purpose electrically conducting overlay films of granular aluminium of 50-100 nm thickness prepared in the presence of oxygen have been used with good success. Since the penetration depth of the beam electrons in our sample materials was typically in the micrometre range, the beam power was always predominantly dissipated in the underlying crystal. Therefore, beyond draining the charge carried by the beam currents, the granular aluminium overlay films had little influence on the phonon generation in the source region.

A more detailed discussion of the ballistic phonon signal and the technical aspects of phonon imaging based on LTSEM including the measuring electronics can be found in previous publications [6, 10, 11, 13-15]. As we have mentioned above, the spectrum of phonons emitted by the source region can be approximated by a Ptanck distribution with the frequencies extending into the terahertz range. At phonon frequencies of such magnitude we expect already appreciable dispersion, i.e. deviations of the propagation properties from the long-wavelength acoustic limit. To study such dispersive effects, the use of frequency-selective phonon detectors such as superconducting tunnel junctions becomes imperative. A detailed discussion of this application of superconducting tunnel junctions for phon0n imaging and of the magnitude of the detector signal has been given elsewhere [ 16]. The phonon detectors utilized so far in phonon-imaging experiments have all been thin film devices attached directly to the appropriate crystal surface by means of thin film deposition. In this way, each study of a new sample crystal also requires fabrication of a new phonon detector. An important simplification in phonon imaging would be achieved if a phonon detector becomes available which can be detached from the sample crystal. In addition to its repeated use for different specimens, such a detector could also be shifted to different locations along the surface of the sample crystal. This latter possibility would enhance considerably the lateral span of the crystal which can be analysed by phonon imaging. First experiments on the detection of ballistic phonons using a detachable phonon detector have been reported recently [17]. Comparing electron beam scanning and laser beam scanning [4, 7, 8] for phonon imaging, we note that at large working distances a smaller beam diameter can be achieved with an electron beam than with a laser beam. Therefore, electron beam scanning is expected to yield higher angular or spatial resolution for phonon imaging than laser beam scanning does.

3. Phonon-focusing experiments Because of the anisotropy of the elastic crystal properties the surface of constant phonon energy in wavevector space is, in general, not spherical

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and displays pronounced anisotropy. Therefore, the energy flux or group velocity of an acoustic plane wave is generally not parallel to the wavevector. This results in the preferential propagation of the phonon energy along distinct crystallographic directions, including directions in which the phonon energy flux diverges mathematically. This phenomenon is referred to as "phonon focusing", and its first experimental demonstration was reported by Taylor et al. [18,

191. In the past the different techniques of phonon imaging (laser beam scanning and LTSEM) have been applied extensively to the investigation of phonon focusing in a series of materials. In addition to the imaging of the anisotropy of the total phonon energy flux, the angular patterns generated by the individual phonon modes (longitudinal, fast and slow transverse modes) have been studied. In the latter studies the separation of the individual modes could be achieved by means of time-resolved experiments. Further, dispersive effects in phonon focusing have been investigated. The experimental observations have been compared with theoretical calculations based on specific models of the crystal lattices. It appears that all essential features of phonon focusing are now well understood. In the following, we briefly summarize our phonon-focusing experiments performed by means of LTSEM.

GaAs. Again, superconducting granular aluminium bolometers were used for phonon detection, and these experiments were carried out at a temperature below the 2 point of liquid helium. Additional experiments for studying dispersive effects have been carried out with GaAs [16], where superconducting PbIn tunnel junctions were used for phonon detection. The temperatures of the latter experiments were taken both above and below the ~ point of liquid helium. As a typical example, we present in Fig. 2 the phonon-focusing image of single-crystal [1 l 1 Ioriented silicon. Bright regions indicate high intensity of the phonon flux. The bottom part of

3.1. Dielectric crystals

We have studied phonon focusing in singlecrystal a-quartz and sapphire [13]. The thickness of the sample crystals was 2 ram. The experiments were performed at a temperature below the 2 point of liquid helium where the superconducting bolometers fabricated from granular aluminium films and used for phonon detection showed maximum sensitivity. In addition to the anisotropy of the total phonon energy flux, in aquartz we investigated the patterns generated by the individual phonon modes. For the latter measurements, sufficient time resolution was obtained using beam pulses of 35 ns width and box-car averaging of the detector signal.

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Phonon-focusing experiments have been carried out with germanium [10, 11], silicon [20] and GaAs [15, 16]. The single-crystal samples had the following thicknesses: 2-3 m m for germanium, 2 m m for silicon and 1.35 m m for

Fig. 2. (a) Phonon-focusing image of single-crystal [ I l l ] oriented silicon (crystal thickness, 2 mm). Bright regions indicate high intensities of the phonon flux. (b) Line scan where the bolometer signal is plotted vertically v s . the coordinate of the beam focus (y modulation). The scale marks refer to the surface area scanned by the electron beam. (From ref. 20.)

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Fig. 2 shows a linear scan in which the bolometer signal is plotted v s . the coordinate of the beam focus (y modulation). The location of the scanning line within the two-dimensional image at the top is indicated in the inset. Figure 2 refers to the time-integrated phonon energy flux containing the contributions of all three phonon modes. The phonon-focusing pattern of single-crystal [001]-oriented GaAs is shown in Fig. 3 (crystal thickness, 1.35 mm; scanned area, 0.8 mm × 0.8 ram). Again, the sum of the contributions of all three phonon modes is shown, and bright regions indicate high phonon energy flux. In Fig. 3(a) we present the experimental results, whereas Fig.

Fig. 3. Phonon-focusing pattern of single-crystal [001]oriented GaAs (crystal thickness, 1.35 mm; scanned area, 0.8 mm × 0.8 mm): (a) experimental results; (b) theoretical results calculated in the long-wavelength limit. Bright regions indicate high intensities of the phonon flux. (From ref. 15.)

3(b) contains the calculated phonon intensity distribution for the same geometric crystal configuration. The calculations shown in Fig. 3(b) were performed in the long-wavelength limit without considering dispersion. By comparing the experimental with the theoretical results in Fig. 3, we note qualitative agreement. However, a detailed comparison indicates deviations in the experimental phonon-focusing pattern due to dispersion [15]. Calculations of phonon focusing in GaAs at high phonon frequencies including dispersive effects have been reported by Tamura and Harada [21]. From a comparison of our experimental results with the calculations by Tamura and Harada, we conclude that phonons in the frequency range around 500 GHz have dominantly contributed to the bolometer signal. Dispersive effects in the phonon-focusing pattern of GaAs have been observed more clearly, using superconducting PbIn tunnel junctions for phonon detection [16]. In this case the detector signal is caused by the breaking of Cooper pairs due to the phonons. This results in an energy threshold of the detector given by twice the energy gap of the superconducting electrodes of the tunnel junction. For our PbIn junctions in the low temperature limit this energy threshold corresponds to a phonon frequency of about 0.65 THz in GaAs. From a detailed comparison between our experimental results obtained with this threshold detector below the temperature of the 2 point and the calculations of Tamura and Harada [21], we conclude that phonons in the frequency range around 1 THz contributed predominantly to the detector signal. For further discussions including a theoretical treatment of the magnitude of the detector signal we refer to ref. 16. Our phonon-focusing studies in germanium were carried out using [001]-oriented crystals [10, 11]. Again we have found that calculations performed in the long-wavelength limit are inadequate to account for all details of the experimental observations and that dispersive effects cannot be neglected. By comparing our experimental results with calculations performed in the dispersive frequency regime by Tamura and Harada [21] and Tamura [22, 23], we conclude that the dominant phonon frequencies in our experiments were placed in the range 300-400 GHz. Systematic studies relating to the angular resolution limit of phonon imaging by means of

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LTSEM have been performed with single-crystal germanium [10, 11] and GaAs [15]. For this purpose we have taken line scans across regions where particularly sharp structures appear in the phonon-focusing pattern. These experimental recordings were compared with theoretical calculations incorporating different assumed values of the effective diameter of the phonon source. In this way an effective source diameter in the range 20-30 /~m has been found. As has been discussed in Section 2, it is the effective diameter of the phonon source which limits the angular resolution of phonon imaging by means of LTSEM.

3.3. Influence of phonon scattering So far we have assumed in our discussion that the phonons emitted from the source region travel ballistically at the velocity of sound to the detector, without any phonon scattering taking place along their path. However, it is interesting to extend the phonon-imaging experiments into the regime where some elastic phonon scattering events occur along the ballistic propagation path and where a diffusive component is superimposed on the ballistic component of the phonon signal. In the limit where phonon scattering becomes dominant, the beam power injected into the crystal propagates through the crystal purely diffusively. In this case, distinct spatial structures cannot be observed any more and phonon imaging breaks down. Prom experiments in the regime where both ballistic and diffusive phonon transport are significant the value of the phonon mean free path can be obtained as an important piece of information. We have performed such measurements on GaAs crystals cut from wafers 2 in in diameter taken from differently prepared rods [15]. In addition, for each wafer we have taken crystals cut from different parts along the wafer radius. It is well known that the defect concentration in the singlecrystal semiconductor rods, from which the wafers are cut subsequently, can vary considerably along the radial and axial coordinates of the rod. Here phonon imaging can provide important information on the crystal quality. The starting point for determining the value of the phonon mean free path for a particular sample crystal is the calculation of the angular variation of the phonon intensities including the scattering processes. Such calculations can be performed straightforwardly using a Monte Carlo simulation [15]. By varying the value of the

phonon mean free path in these calculations and by comparing the theoretical results with the experimental observations the phonon mean free path can be found. A typical series of calculated phonon intensity patterns for i001I-oriemed GaAs and different values of the phonon mean free path is shown in Fig. 4. In this way we found values of the phonon mean free path in the range 0.35-0.80 mm for various GaAs crystals. Phonon focusing of elastically scattered phonons in GaAs has also been studied recently using laser beam scanning and following similar ideas [24]. In addition to the two-dimensional phononimaging experiments, important information on the ballistic and diffusive features of the phonon propagation through the crystal can also be obtained from time-resolved measurements and the temporal structure of the detector signal [1 1, 15]. Finally, we emphasize that phonon imaging looks promising as a novel technique for quality control of single-crystal semiconductor materials.

4. Imaging of defect structures

An important application of phonon imaging by means of LTSEM is the spatially resolved detection of crystal inhomogeneities which scatter or absorb the ballistic phonons. From our discussion in Sections 2 and 3 we note that the spatial resolution of this imaging technique is strongly influenced by the effective diameter of the phonon source of about 25 /~m. Therefore the spatial inhomogeneities to be detected must have characteristic dimensions not much smaller than this magnitude. This range of values is much larger than the nanometre range of the wavelength of the dominating phonons. Therefore the concepts of geometric optics can be applied to this imaging principle. If a single phonon detector is used during the imaging process, the three-dimensional configuration of the spatial inhomogeneity is projected onto the two-dimensional image plane by means of the acoustic shadow of this object [5, 6]. An unambiguous interpretation of the image recorded by the detector without overlap between the projections of different objects requires that the density of the objects in the three-dimensional sample volume be sufficiently small. If two detectors are used and placed at different locations, two two-dimensional images

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Fig. 4. Calculated phonon intensities for [001 J-oriented GaAs including elastic phonon scattering processes using a Monte Carlo simulation for the following values of the phonon mean free path 2 (image section, 3.2 m m × 3.2 mm; crystal thickness, 1.35 ram): (a) 2 = 1.0 ram; (b) 2 =0.7 ram; (c) ~, =0.5 ram; (d) 2 =0.3 mm. Bright regions indicate high intensities of the phonon flux. The intensity is given by the weighted sum of all three phonon modes. (From ref. 15.)

are obtained from which all three coordinates of the detected inhomogeneity can be reconstructed in principle (stereoscopic imaging). Recently, we have experimentally demonstrated this threedimensional acoustic imaging technique [25]. For more than two detectors these ideas can be extended to a general tomography principle. We shall return to this point in Section 5. As a typical example we show in Fig. 5 the phonon image of three laser-drilled holes having a diameter of 100 /~m or smaller in a sapphire single crystal of 2 mm thickness [25]. The holes were drilled sideways into the crystal near the middle between the upper and lower crystal surface. For comparison the optical image obtained by transmission light microscopy is also

shown. In the middle at the top, two holes crossing each other and placed at different values of the vertical coordinate in the crystal can be seen. Comparing both images we note that all details are well reproduced in the acoustic image. By performing line scans and plotting the detector signal in y modulation, the spatial resolution of the acoustic image has been found to be a few micrometres. The bright structural features in the acoustic image of Fig. 5 are caused by phonon focusing in the single-crystal sapphire. Compared with the optical image, the acoustic image is enlarged owing to the shadow projection of the imaging technique. The magnification by a factor of about 2 agrees well with the location of the holes at about half the thickness of the sample.

164 including stereoscopic imaging in three dimensions. A spatial resolution of a few micrometres has been achieved.

5. Digital image processing and acoustic tomography

Fig. 5. Section of a c-cut sapphire single crystal of 2 mm thickness with laser-drilled holes (dark structures): (a) optical image obtained by transmission light microscopy; (b) corresponding phonon image. The vanishing phonon intensity in the dark regions indicates the location of the holes. The bright pattern in the background is caused by phonon focusing. The scale mark refers to the surface area scanned by the electron beam. For further details see text. (From ref. 25.) Similar experiments have been performed for imaging laser-drilled holes with a diameter of 100 /~m or smaller in single-crystal a-quartz [25]. In this case, in addition to the holes, strain fields due to plastic deformations could be observed in the acoustic image which remained undetectable by conventional transmission light microscopy. Further, in single-crystal silicon, oxide precipitates have been detected by phonon imaging [20]. In concluding this section, we note that the imaging of defect structures by means of ballistic phonons has been demonstrated convincingly,

For three-dimensional reconstruction of crystal defects a quantitative analysis of the phonon signal and of the two-dimensional intensity distributions becomes necessary. For this we have used a real-time computer system to perform electron beam scanning and to digitize and store the output voltage of the lock-in amplifier as can be seen in Fig. 1. Then, using two bolometers located at different positions, stereoscopic imaging of the crystal defects becomes possible and the depth coordinate of the acoustic cross-sections of the defects can be calculated. The different detector positions result in a shift of the focusing pattern within the scanned image section whereas the shift of the acoustic shadow is caused by the different viewing angles. Where the bolometer positions are known within the phonon images, the three-dimensional configuration of the acoustic cross-section of the defect can be determined from the measured shifts [14]. Once the phonon intensity distributions are stored digitally, they can be analysed using techniques of digital image processing. We have applied algorithms for contrast enhancement to obtain improved representation of the defect shadow in the phonon image. So, even small variations in the phonon signal can be imaged with high contrast. In addition, convolution operations with special filter masks can be applied for edge detection [14]. The inhomogeneous phonon illumination due to the focusing effect results in a contrast variation of the imaged defects. This can be overcome by calculating the difference image of two intensity distributions with and without the acoustic shadows respectively. Then a binary representation of the acoustic cross-sections becomes possible ]14]. The focusing pattern can be used to calculate the bolometer position within the phonon image. When the crystal orientation is known, the detector coordinates are obtained from the symmetry of the phonon-focusing pattern or by comparing experimental images with theoretical images. Finally, cross-correlation calculations are useful for finding the corresponding parts of the defect shadow in the two images. Then depth profiles of

165 the acoustic cross-sections can be obtained directly f r o m the p h o n o n images [26]. For high densities of the defects in the crystal volume, no unique three-dimensional reconstruction can be achieved using only two bolometers. To obtain m o r e information, one- or two-dimensional detector arrays are necessary. T h e n our imaging technique can be extended by applying tomographic principles to obtain slices of the crystal with the e m b e d d e d defect structures. Following the concepts of conventional X-ray tomography, we have developed an algorithm to image different crystal planes parallel to the sample surface using multiple detectors [26]. Depending on the depth coordinate of the plane to be reconstructed and the bolometer positions, each p h o n o n image is shifted correspondingly. Then, after summing the intensity distributions, all features in the considered plane are imaged sharply with high contrast; the signals f r o m all other planes result in a diffusive background. Experimental results using this tomographic algorithm can be found elsewhere [26]. As we have d e m o n s t r a t e d by our experimental results obtained with the G a A s samples [15], the ballistic p h o n o n signal is a continuous function of the defect concentration. T h u s a p h o n o n absorption coefficient proportional to the defect concentration can be defined and used for defect reconstruction. We have developed algorithms (based on the principles of computer-aided tomography) to determine the three-dimensional distribution of the absorption coefficient that can be applied directly to the image geometry in LTSEM. To demonstrate the advantage of this technique we have reconstructed model structures f r o m their projections under different viewing angles [27].

Acknowledgments Financial support of the work described in this p a p e r by grants f r o m the Deutsche Forschungsgemeinschaft is gratefully acknowledged. T h e

single-crystal samples of g e r m a n i u m and G a A s have been kindly supplied by P. Glasow, Siemens AG, Erlangen.

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