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Scattering, Inelastic: Brillouin
Scattering, Inelastic: Brillouin 199
Relative intensity
4.0
E = 11107 eV
3.5
P = 600
3.0
H = 335
2.5 2.0 1.5 1.0 0.5 0 152.225
152.364 152.503 152.642 152.781 152.920 (degrees)
Figure 8 Azimuthal dependence of the 600 in germanium, near % Circles represent a peak due to three-beam diffraction (the 3 3% 5). experimental data. Solid line is theory. The agreement is excellent.
cell, and then one electron is placed in all other 47 equivalent positions, as dictated by the symmetry properties of the space group, the 600 structure factor for these electrons is always zero. Surprisingly enough, the 600 can be turned on again if resonant scattering is used. It is found that the 600 exhibits a beautiful asymmetry effect around a three-beam situation, and that interesting phase effects can be observed, which should provide a much more detailed description of the valence charge density in Ge. Figure 8 is an example of a three-beam experiment in Ge-600 at resonance. Again, the asymmetry effect is clearly visible, and phases can be reliably extracted from the azimuthal profiles. See also: Crystal Structure Determination; Mo¨ssbauer Spectroscopy; X-Ray Topography.
PACS: 61.10.Nz; 61.10.Dp; 61.44.Br with a triple invariant d ¼ 112.51, a clear indication of lack of centrosymmetry. While the deviations from centrosymmetry are probably very small, it is found that multibeam diffraction tends to overemphasize even a small amount of centrosymmetry. Another area in which three-beam diffraction is finding useful applications is that of resonant scattering. In germanium crystal, the 600 reflection is space-group forbidden. This means that the 600 cannot be turned on even for the most general charge density consistent with the symmetry properties of the Fd3m space group. This means that if an electron is put in a general position (x, y, z) within the unit
Further Reading Bonse U and Hart M (1965) An X-ray interferometer. Applied Physics Letters 6: 155. Colella R (1996) X-ray and neutron interferometry. In: Authier A, Lagomarsino S, and Tanner BK (eds.) X-Ray and Neutron Dynamical Diffraction. New York: Plenum. Deslattes RD and Henins A (1973) Physical Review Letters 31: 972. Rauch H and Werner SA (2000) Neutron Interferometry. Oxford: Clarendon. Shvyd’ ko YV (2004) X-Ray Optics. High Energy Resolutions Applications. Berlin: Springer.
Scattering, Inelastic: Brillouin T Blachowicz, Department of Microelectronics, Silesian University of Technology, Gliwice, Poland M Grimsditch, Argonne National Laboratory, Argonne, IL, USA Published by Elsevier Ltd.
Introduction Brillouin light scattering (BLS), in its historical sense, is the inelastic scattering of light by acoustic phonons. It thus provides information on the elastic properties of the scattering medium and it has been used to study gases, liquids, crystals, polymers, glasses, semiconductors, nontransparent thin layers, and superlattices. BLS yields information similar to that obtained using ultrasonic techniques but it has
certain advantages when dealing with small samples, reactive samples, or in cases where contact with a transducer poses experimental difficulties (e.g., high temperatures and pressures). The Fabry–Perot interferometer (FP) is, par excellence, the instrument of choice to achieve the 1–100 GHz resolution required in typical BLS experiments. Because of this, BLS has become synonymous for all experiments performed with an FP independent of whether the scattering is produced by acoustic phonons, by magnons, or by electronic states. Here BLS is used in this broader context. This article provides a brief historical review and a condensed description of the origin of the effect. Also, some key experimental aspects are briefly described. The emphasis of this article is to illustrate, via specific examples, the range of problems that can
200 Scattering, Inelastic: Brillouin
be investigated using BLS. This will include the determination of elastic constants with emphasis on its application under adverse experimental conditions (temperature, size, and pressure), as well as its application to magnetic systems and electronic levels.
Brief History of BLS The term Brillouin light scattering (BLS) was derived from the theoretical prediction made by L Brillouin in 1914 of a possible inelastic interaction between light and sound. The description of light-scattering experiments was published in his PhD dissertation. A similar prediction was made independently in 1926 by L I Mandelstam. The first experiment was performed by E Gross in 1930. BLS spectroscopy developed rapidly in the early 1960s as lasers became available. The first light-scattering experiments in pure liquids were done by G B Benedek et al. in 1964. The technique remained almost an academic curiosity until the multipassed and later the tandem FP, both introduced by J Sandercock, enabled the weak Brillouin signals to be observed in the presence of the usually overwhelming intensity of the elastically scattered light. At that point, BLS became a powerful and versatile tool for material characterization. This article describes how it has made significant contributions to a wide range of problems in condensed matter physics.
Origin of BLS Any excitation that produces a time-varying change in the polarizability of a medium (for a longitudinal sound wave this is related to the change in refractive index caused by the densification and rarefaction produced by the wave) interacts with electromagnetic radiation via the polarizability a. The polarization of the medium induced by the electromagnetic wave becomes Pðr; tÞ ¼ ½/aS þ daðr; tÞE0 eiðki ro0 tÞ and it can be shown that this leads to the re-radiation of electromagnetic waves with frequencies o0 7ophon . The origin of the polarizability changes (da) is different for different excitations: bulk phonons modulate the polarizability via elastooptic effects (strain-induced changes in the refractive index), surface phonons produce polarizability changes as the interface moves into and out of the vacuum (known as the ripple mechanism), and magnons interact via magnetooptic contributions. Interference between the radiation emanating from throughout the sample leads to the condition that
only excitations with wave vector q that satisfy q ¼ k0 k where k and k0 are the wave vectors of the incident and scattered radiation, respectively, will produce scattered radiation. Since k and k0 are determined by the directions of the incident and scattered radiation, it is clear from the above equation that in a given experiment only excitations with a fixed wave vector will be probed. This also makes it clear that the geometry chosen for the experiment is crucial for analyzing the resulting data.
Scattering Geometries and Frequency Analysis There are a number of scattering geometries that are often used in Brillouin experiments; these are shown schematically in Figure 1. Not only does each scattering geometry select different q values but it is clear that geometries (a), (b), and (c) are only suitable for transparent materials while geometries (d) and (e) can be used even if the sample is not transparent. The (e) geometry is a variant of the back-scattering geometry (d). The magnitude of the wave vector probed in each geometry, calculated from the q ¼ k0 k equation, is also indicated in Figure 1. In these expressions, jkj ¼ 2p=l (with l the wavelength of the incident radiation, and the refractive index of the medium). In opaque materials, only the component of q parallel to the surface is conserved so that the q ¼ k0 k equation yields only the value of q in the surface plane qs. Typically, the choice of scattering geometry depends on the nature of the sample and the constraints of the experiment being performed; for example, cryostat, magnet, and furnace. In the laboratory, the incident radiation depicted in Figure 1 is usually delivered by a laser. Since the frequency shifts are small, it normally requires a laser operating in a single longitudinal cavity mode. The scattered radiation is typically collected by a highquality lens (often a camera lens). Frequency analysis of the scattered light is accomplished using an FP interferometer. In all but a few cases, experiments rely on the high contrast provided by multipassed and/or tandem instruments. These very delicate instruments are technically complex. Excellent reviews on their operation and fabrication exist in the literature (see the ‘‘Further reading’’ section).
Examples To highlight BLS, a selection of examples are discussed. First, the classical application is described: the
Scattering, Inelastic: Brillouin 201
Incident light
Incident light 45°
Scattered light
Scattered light
k′
45°
k′ k
90° k
q
q
q = √2n k
(a)
q = √2 k
(b)
Incident light Scattered light
Incident light
k ≈ −k ′
Scattered light
qs
k′
Surface wave vector component qs conserved
q
(c)
k
qs = 2 k sin
q = 2n k sin(/2)
(d)
Scattered light ′
Incident light
k k′ Reflected light qs
Surface wave vector component qs conserved
(e)
q s = k (sin′−sin )
Figure 1 Scattering geometries: (a) a bulk-type analysis of excitations inside a transparent material using 901 geometry, (b) a thin transparent layer – this geometry is sensitive to excitations propagating in the plane of the sample, (c) a transparent layer with nearnormal incidence – this geometry is sensitive to wave vectors perpendicular to a sample surface, (d) a back-scattering geometry in an opaque material which is sensitive to in-plane, surface wave vectors, and (e) a variant of geometry (d), where k 0 is not along the direction of the incident light, is also used for studying surface excitations. For geometries (a) and (c), the material refractive index n is required for data analysis.
202 Scattering, Inelastic: Brillouin
determination of elastic constants in a bulk transparent material. In this subsection, references to examples are provided where the determination of cij has been done at high temperatures and high pressures. Then examples of BLS applied to liquids, glasses, opaque materials (surface wave BLS), magnetic materials (magnons), and electronic levels are given.
Determination of Elastic Constants of a Bulk Crystal
Figure 2 shows a Brillouin spectrum obtained from a diamond crystal in the backscattering geometry shown in Figure 1c. Since the diamond surface was (1 1 1) and since the phonons probed in the experiment propagate along the surface normal, the peaks observed in Figure 2 correspond to phonons along the [1 1 1] direction. Along this direction the longitudinal sound velocity is vL ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc11 þ 2c12 þ 4c44 Þ=3r and the velocity of the (degenerate) transverse phonons is vT ¼ ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc11 c12 þ c44 Þ=3r. The peaks in Figure 2 have therefore been labeled as L and T, respectively. Since diamond crystals are large enough to be cut and polished with any desired orientation, it is clear that from a combination of measurements in different scattering geometries and/or crystal orientations the whole set of elastic constants can be determined. In Table 1, the cij of diamond determined by both Brillouin scattering and by ultrasonic techniques are
listed. The results of both techniques are in very good agreement but it is interesting to note that while the BLS measurements were performed on crystals of around 1/3 carat, the ultrasonic measurements used a 22-carat diamond. In cases where large, transparent crystals are available, it is clear from Table 1 that BLS has no particular advantage over ultrasonic techniques. However, when only small samples are available the BLS technique becomes the technique of choice. Single crystals of cubic boron nitride (BN), for example, cannot be grown as single crystals larger than E500 mm. In this case, BLS becomes the technique of choice and yields the cij also included in Table 1. The preceding section outlines how BLS can be used to determine cij. At ambient conditions, cij can often be determined more accurately using ultrasonic techniques. In some cases, however, BLS becomes a much more versatile tool than ultrasonics. Such cases usually involve samples where transducer bonding is difficult, as was mentioned above, in small or thin samples, high-pressure and/or high-temperature experiments, or cases where chemical reactivity becomes severe. For example, the temperature dependence of the c11 and c44 elastic constants of Ba1–xLaxF2 þ x over the range 300–1600 K was determined by P E Ngoepe and J D Comins who found a linear decrease that they attributed to anharmonicity. A case involving high pressures is represented by measurements of liquid and solid argon in a diamond anvil cell up to 70 GPa.
400 L Diamond q // [111]
300 Intensity (a.u.)
Measurements of Spectral Features in Liquids
L
200
100 T 0 −7
−5
T −3
−1
1
3
5
7
Frequency shift (cm−1) Figure 2 Brillouin spectrum from a diamond crystal in the geometry from Figure 1c. The longitudinal (L) and transverse (T) phonons propagate along the [1 1 1] direction. Table 1 Elastic constants of diamond and boron nitride Material and method
c11 ðGPaÞ
c12 ðGPaÞ
c44 ðGPaÞ
Diamond, BLS Diamond, ultras Boron nitride, BLS
1076 1079 820
125 124 190
577 578 480
BLS can also be used to determine the elastic response of liquids. In contrast to solids, however, where the unshifted peak is dominated by the effects of impurities and surface scattering, in pure liquids the central peak is due to entropy fluctuations. The intensity of the unshifted component is related to that of the shifted BLS peaks via the Landau–Placzek ratio, Cp/ Cv–1, where Cp and Cv are the specific heats at constant pressure and volume, respectively. In pure liquids, therefore, BLS can be used to study the intensity and shape of the central peak. Frequency broadening of this peak can be caused by fluctuations in the orientation of anisotropically polarizable molecules or by diffusive processes governed by the equations of hydrodynamics. In liquids, it is also possible to investigate relaxation and damping effects via the phonon lifetime that translates into changes in the shape and line width of the phonon peaks. The same approach cannot usually be applied to most bulk crystals, because of a very low damping that results in line widths below the experimental resolution.
Scattering, Inelastic: Brillouin 203
Glasses
BLS has been extensively used to investigate the transition of liquid to solid in materials that do not crystallize: for example, silica, Rb1–x(NH4)x/H2PO4 (10–300 K), Rbx(NH4)y/H2PO4 (20–80 K), and Ca(NO3)/KNO3 (320–640 K). Across the transition region, the phonons in these materials become highly damped and the resulting line shapes can be interpreted in terms of the underlying mechanism of the glass transition. These types of investigations are also of practical importance, because these materials are often used in fiber optics technology and in optoelectronic devices. Gases
BLS in gases detects local fluctuations that lead to frequency shifts in the 0.1–1 GHz range. These shifts can also be viewed as a Doppler shift from the motion of individual atoms. In this context, BLS has also been used to investigate how the phonon spectrum evolves as a gas is pressurized to become a fluid. This evolution reflects the hydrodynamic transformation as the thermal mean free path for atomic motion becomes comparable and then shorter than the wavelength at which the medium is being probed by the scattering of visible light. These experiments can also be interpreted in terms of the hydrodynamic equations of motion. Investigations of CO2 up to 8 atm pressure, and Xe over the range 0.022– 0.66 atm are two instances where this phenomenon has been investigated. Surface Acoustic Modes in Thin Layers
Light scattering from acoustic excitations in opaque materials results from a purely surface-induced mechanism, viz. phonon-induced thermal ripples. In these cases, the scattering geometries (d) and (e) in Figure 1 must be employed. In a semi-infinite medium, the strongest coupling is to Rayleigh waves (reminiscent of waves in the ocean) with some weaker features induced by bulk waves reflecting from the free surface. In thin films and layered structures, however, it is also possible to couple to modes that are localized within the layers or at the interfaces; these modes are generally referred to as Sezawa
R
1400 R Scattering intensity (a.u.)
However, in temperature regions where structural phase transitions occur, damping processes influence Brillouin line widths enough so that the changes are detectable, for example, in molten and crystalline alkali halides, during order–disorder transitions in mixed crystals, or during ferroelastic phase transitions in some crystals.
1200 1000 800 600 S 400
S S
S
200 −40
−20
0
20
40
Frequency shift (GHz) Figure 3 BLS from surface acoustic phonons in the Co/Cu superlattice. Descriptions: R – the Rayleigh surface mode, S – Sezawa or higher-order Rayleigh modes. The measurement was carried out for (s–s) polarizations of the incident and scattered light.
modes. If the medium is not completely opaque, it is possible to couple to the phonons via both the ripple and elastooptic mechanisms. In these cases, unusual interference effects are possible. Figure 3 shows a BLS spectrum from a Co/Cu superlattice built from 60 bilayers of 1 nm Co/0.8 nm Cu in which the Rayleigh and Sezawa modes are observed. The experiment was carried out using a Sandercock-type tandem 3-pass interferometer. From these results, the effective elastic properties of the superlattice can be extracted. In cases where the surface layers become much thinner than the wavelength of light, the medium again becomes equivalent to a homogeneous solid. In these cases, Brillouin scattering has been a very efficient and reliable tool for determining the effective elastic constants of the heterogeneous solid. For example, R Danner et al. in the Ni/V superlattice, for different layer thicknesses from the range of 1.85– 63.3 nm, proved that this b.c.c./f.c.c. system reveals softening of elastic parameters. J A Bell and co-workers measured effective elastic constants in the Mo/Ta superlattice (c11, c13, c33, c44) with an elementary spatial bilayer thickness ranging from 0.7 to 20 nm. Observation of Bulk and Surface Spin Waves in Metallic Superlattices
Brillouin scattering can also be used to investigate the magnetic properties of materials via their magnetic excitations – magnons. In this sense, the information it yields is similar to that obtained in ferromagnetic resonance (FMR) experiments. However, contrary to
204 Scattering, Inelastic: Brillouin
H = 6.8 kG
400
300
BM
Scattered light intensity (a.u.)
Light scattering intensity (a.u.)
D-E
BM
200
100
H = 5.2 kG H = 4.1 kG H = 3.5 kG H = 2.8 kG H = 2.0 kG
H = 1.1 kG
0 −40
−20
0
20
−40
40
−20
Scatterings from Electronics Levels
More recently, BLS has been used to probe the lowlying excitations of B impurity atoms in a diamond host . In this case, since the excitations are local and
40
Figure 5 A set of Brillouin spectra measured for different magnetic field intensities in the Co/Cu superlattice. The peaks that appear only on the left side (Stokes signals) result from scattering from a unidirectional surface-like spin wave (D-E mode from Figure 4).
400
LT
Boron-doped diamond
300 Intensity (a.u.)
FMR that probes the infinite wavelength excitations, Brillouin scattering probes excitations with wavelengths comparable to the wavelength of light. As for phonons, magnons can be subdivided into surfacelike and bulk-like excitations. Expressions relating the magnon frequencies to the magnetic properties of the material – magnetization, anisotropies, applied field, and gyromagnetic ratio – can be quite complex and often cannot be rendered in analytical form. The Brillouin scattering technique has also been used to investigate the nature of magnetic excitations in nonconventional materials. In the case of the Co/Cu superlattice described in the previous paragraph, which is built from alternating magnetic and nonmagnetic materials, bulk-like collective excitations resulting from coupling between spin waves across the nonmagnetic spacer are detected (Figure 4). Additionally, a surface-like mode propagating on the surface of the sample was also found. The experiment was carried out with a tandem 3-pass spectrometer, with (s–p) polarization of the incident and scattered light, and a constant magnetic field lying in the sample plane. Figure 5 provides results for different magnetic field intensities, from which the gyromagnetic ratio and saturation magnetization can be determined.
20
Frequency shift (GHz)
Frequency shift (GHz) Figure 4 BLS from spin waves in a multilayered Co/Cu system. Descriptions: D-E, Damon–Eshbach surface mode, BM, (bulk magnon) collective excitations of a bulk-like nature resulting from coupling between spin waves through the nonmagnetic spacer layer. The measurement was carried out for (s–p) polarizations of the incident-scattered light.
0
T B
B
L
B
200
B
100
0
−17
−12
−7
−2
3
8
13
18
−1)
Frequency shift (cm
Figure 6 Brillouin scattering from B impurities in a diamond crystal as a host. Descriptions: (L) longitudinal phonons, (T) transverse phonons, and (B) electronic transitions between the hydrogen-like states surrounding the B atoms.
are not described by a wave vector, the results do not depend on the scattering geometry. Figure 6 shows a spectrum recorded at low temperature and with an applied field of 4 T; apart from the expected longitudinal (L) and transverse (T) phonon peaks, the additional peaks labeled B can be traced to electronic transitions between the hydrogen-like states surrounding the B atoms. Under the effect of an applied field these modes split and from the measured splittings, the degeneracy and nature of the transitions is extracted. This, in turn, provides information on how B impurities in diamond lead to its semiconducting properties.
Scattering, Inelastic: Electron 205
Acknowledgment Work at ANL was supported by the US DOE, Basic Energy Sciences, Materials Sciences under contract W-31-109-ENG-38. See also: Scattering, Inelastic: Raman; Thin Films, Mechanical Behavior of.
PACS: 43.35.G; 68.65. þ g; 63.20. e; 63.22. þ m; 72.10.Di; 75.30.Ds; 78.35. þ c; 81.07. b Further Reading Bell JA, Bennett WR, Zanoni R, Sttegeman GI, Falco CM, et al. (1987) Elastic constants of Mo/Ta superlattices measured by Brillouin scattering. Physical Review B 35: 4127–4130. Benedek GB, Lastovka JB, Fritsch K, and Greytak T (1964) Brillouin scattering in liquids and solids using low-power lasers. Journal of the Optical Society of America 54: 1284–1285. Blachowicz T (2003) Brillouin Spectroscopy in Crystal Lattices. Acoustic and Spin Waves. Gliwice: Silesian University of Technology Press. Brillouin L (1922) Diffusion de la Lumie`re et des Rayonnes X par un Corps Transparent Homoge´ne; Influence de l’Agitation The´rmique. Annales de Physique 17: 88. Broz A, Harrigan M, Kasten R, and Monkiewicz A (1971) Light scattered from thermal fluctuations in gases. The Journal of the Acoustical Society of America 49: 950–953.
Danner R, Huebener RP, Chun CL, Grimsditch M, and Schuller IK (1986) Surface acoustic waves in Ni/V superlattices. Physical Review B 33: 3696–3701. Grimsditch MH (1989) Brillouin scattering from metallic superlattices. In: Cardona M and Gu¨ntherodt G (eds.) Light Scattering in Solids, Superlattices and Other Microstructures, vol. 5, pp. 285–302. Berlin: Springer. Grimsditch MH and Ramdas AK (1975) Brillouin scattering in diamond. Physical Review B 11: 3139–3148. Grimsditch MH, Zouboulis ES, and Polian A (1994) Elastic constants of boron nitride. Journal of Applied Physics 76: 832–834. Gross EF (1930) Change of wave-length of light due to elastic heat waves at scattering in liquids. Nature 126: 201. Hyunjung K, Ramdas AK, Rodriguez S, Grimsditch MH, and Anthony TR (1999) Magnetospectroscopy of acceptors in ‘‘blue’’ diamonds. Physical Review Letters 83: 3254–3257. Mandelstam LI (1926) K Voprosu O Rassea˜nii Sveta Neodnorodnoj Sredoj. Zhurnal Russkovo Fiziko-Khimicheskovo Obshchestva 58: 381. Ngoepe PE and Comins JD (1986) Measurements of elastic constants in super-ionic B1 xLaxF2 þ x. Journal of Physics C: Solid State Physics 19: L267–L271. Sandercock JR (1982) Trends in Brillouin scattering: studies of opaque materials, supported films, and central modes. In: Cardona M and Gu¨ntherodt G (eds.) Light Scattering in Solids, Recent Results, vol. 3, pp. 173–206. Berlin: Springer. Shimizu H, Tashiro H, Kume T, and Sasaki S (2001) High-pressure elastic properties of solid Argon to 70 GPa. Physical Review Letters 86: 4568–4571.
Scattering, Inelastic: Electron M Erbudak, Laboratorium fu¨r Festko¨rperphysik, Zu¨rich, Switzerland D D Vvedensky, Imperial College, London, UK & 2005, Elsevier Ltd. All Rights Reserved.
Introduction Inelastic electron-scattering phenomena are used to obtain information on the electronic properties and atomic structure of materials. An interaction event between an electron and a target material is called inelastic if the electron loses some or all of its energy; otherwise, the event is called elastic. Any energy loss by the electron is equal to the energy gained by the material and is characteristic of the type of scattering event. The measurement of an inelastic event determines the energy transferred and the rate at which this transfer occurs, which provides physical information about the elementary excitations of the material. Inelastic interactions can be grouped according to different energy-transfer regimes that form the basis
of the principal experimental techniques. Historically, inelastic electron scattering was developed prior to the availability of continuous-energy photon sources from dedicated storage rings. In fact, under certain circumstances, electronic excitations are equivalent to photon excitations in that they can be described within a common conceptual and theoretical framework. An electron is an elementary particle with a mass m and, owing to its wave nature, has a wave number k ¼ p=_, where p ¼ mv is the momentum, v is its velocity, and _ is Planck’s constant. The kinetic energy is expressed classically by E ¼ ð1=2Þmv2, where v ¼ jvj. Electrons have a negative charge e and a nonclassical angular momentum (spin). These properties play a central role in the interactions between electrons and their environment and constitute the basis of a wide range of spectroscopic techniques. Owing to their unique properties, there are both advantages and disadvantages in using electrons as a probe in analytical applications. Spin-polarized or spin-averaged beams of electrons are readily created, electrons can be accelerated to any energy, and focused on a target at will. After they interact with
Relative intensity
4.0
E = 11107 eV
3.5
P = 600
3.0
H = 335
2.5 2.0 1.5 1.0 0.5 0 152.225
152.364 152.503 152.642 152.781 152.920 (degrees)
Figure 8 Azimuthal dependence of the 600 in germanium, near % Circles represent a peak due to three-beam diffraction (the 3 3% 5). experimental data. Solid line is theory. The agreement is excellent.
cell, and then one electron is placed in all other 47 equivalent positions, as dictated by the symmetry properties of the space group, the 600 structure factor for these electrons is always zero. Surprisingly enough, the 600 can be turned on again if resonant scattering is used. It is found that the 600 exhibits a beautiful asymmetry effect around a three-beam situation, and that interesting phase effects can be observed, which should provide a much more detailed description of the valence charge density in Ge. Figure 8 is an example of a three-beam experiment in Ge-600 at resonance. Again, the asymmetry effect is clearly visible, and phases can be reliably extracted from the azimuthal profiles. See also: Crystal Structure Determination; Mo¨ssbauer Spectroscopy; X-Ray Topography.
PACS: 61.10.Nz; 61.10.Dp; 61.44.Br with a triple invariant d ¼ 112.51, a clear indication of lack of centrosymmetry. While the deviations from centrosymmetry are probably very small, it is found that multibeam diffraction tends to overemphasize even a small amount of centrosymmetry. Another area in which three-beam diffraction is finding useful applications is that of resonant scattering. In germanium crystal, the 600 reflection is space-group forbidden. This means that the 600 cannot be turned on even for the most general charge density consistent with the symmetry properties of the Fd3m space group. This means that if an electron is put in a general position (x, y, z) within the unit
Further Reading Bonse U and Hart M (1965) An X-ray interferometer. Applied Physics Letters 6: 155. Colella R (1996) X-ray and neutron interferometry. In: Authier A, Lagomarsino S, and Tanner BK (eds.) X-Ray and Neutron Dynamical Diffraction. New York: Plenum. Deslattes RD and Henins A (1973) Physical Review Letters 31: 972. Rauch H and Werner SA (2000) Neutron Interferometry. Oxford: Clarendon. Shvyd’ ko YV (2004) X-Ray Optics. High Energy Resolutions Applications. Berlin: Springer.
Scattering, Inelastic: Brillouin T Blachowicz, Department of Microelectronics, Silesian University of Technology, Gliwice, Poland M Grimsditch, Argonne National Laboratory, Argonne, IL, USA Published by Elsevier Ltd.
Introduction Brillouin light scattering (BLS), in its historical sense, is the inelastic scattering of light by acoustic phonons. It thus provides information on the elastic properties of the scattering medium and it has been used to study gases, liquids, crystals, polymers, glasses, semiconductors, nontransparent thin layers, and superlattices. BLS yields information similar to that obtained using ultrasonic techniques but it has
certain advantages when dealing with small samples, reactive samples, or in cases where contact with a transducer poses experimental difficulties (e.g., high temperatures and pressures). The Fabry–Perot interferometer (FP) is, par excellence, the instrument of choice to achieve the 1–100 GHz resolution required in typical BLS experiments. Because of this, BLS has become synonymous for all experiments performed with an FP independent of whether the scattering is produced by acoustic phonons, by magnons, or by electronic states. Here BLS is used in this broader context. This article provides a brief historical review and a condensed description of the origin of the effect. Also, some key experimental aspects are briefly described. The emphasis of this article is to illustrate, via specific examples, the range of problems that can
200 Scattering, Inelastic: Brillouin
be investigated using BLS. This will include the determination of elastic constants with emphasis on its application under adverse experimental conditions (temperature, size, and pressure), as well as its application to magnetic systems and electronic levels.
Brief History of BLS The term Brillouin light scattering (BLS) was derived from the theoretical prediction made by L Brillouin in 1914 of a possible inelastic interaction between light and sound. The description of light-scattering experiments was published in his PhD dissertation. A similar prediction was made independently in 1926 by L I Mandelstam. The first experiment was performed by E Gross in 1930. BLS spectroscopy developed rapidly in the early 1960s as lasers became available. The first light-scattering experiments in pure liquids were done by G B Benedek et al. in 1964. The technique remained almost an academic curiosity until the multipassed and later the tandem FP, both introduced by J Sandercock, enabled the weak Brillouin signals to be observed in the presence of the usually overwhelming intensity of the elastically scattered light. At that point, BLS became a powerful and versatile tool for material characterization. This article describes how it has made significant contributions to a wide range of problems in condensed matter physics.
Origin of BLS Any excitation that produces a time-varying change in the polarizability of a medium (for a longitudinal sound wave this is related to the change in refractive index caused by the densification and rarefaction produced by the wave) interacts with electromagnetic radiation via the polarizability a. The polarization of the medium induced by the electromagnetic wave becomes Pðr; tÞ ¼ ½/aS þ daðr; tÞE0 eiðki ro0 tÞ and it can be shown that this leads to the re-radiation of electromagnetic waves with frequencies o0 7ophon . The origin of the polarizability changes (da) is different for different excitations: bulk phonons modulate the polarizability via elastooptic effects (strain-induced changes in the refractive index), surface phonons produce polarizability changes as the interface moves into and out of the vacuum (known as the ripple mechanism), and magnons interact via magnetooptic contributions. Interference between the radiation emanating from throughout the sample leads to the condition that
only excitations with wave vector q that satisfy q ¼ k0 k where k and k0 are the wave vectors of the incident and scattered radiation, respectively, will produce scattered radiation. Since k and k0 are determined by the directions of the incident and scattered radiation, it is clear from the above equation that in a given experiment only excitations with a fixed wave vector will be probed. This also makes it clear that the geometry chosen for the experiment is crucial for analyzing the resulting data.
Scattering Geometries and Frequency Analysis There are a number of scattering geometries that are often used in Brillouin experiments; these are shown schematically in Figure 1. Not only does each scattering geometry select different q values but it is clear that geometries (a), (b), and (c) are only suitable for transparent materials while geometries (d) and (e) can be used even if the sample is not transparent. The (e) geometry is a variant of the back-scattering geometry (d). The magnitude of the wave vector probed in each geometry, calculated from the q ¼ k0 k equation, is also indicated in Figure 1. In these expressions, jkj ¼ 2p=l (with l the wavelength of the incident radiation, and the refractive index of the medium). In opaque materials, only the component of q parallel to the surface is conserved so that the q ¼ k0 k equation yields only the value of q in the surface plane qs. Typically, the choice of scattering geometry depends on the nature of the sample and the constraints of the experiment being performed; for example, cryostat, magnet, and furnace. In the laboratory, the incident radiation depicted in Figure 1 is usually delivered by a laser. Since the frequency shifts are small, it normally requires a laser operating in a single longitudinal cavity mode. The scattered radiation is typically collected by a highquality lens (often a camera lens). Frequency analysis of the scattered light is accomplished using an FP interferometer. In all but a few cases, experiments rely on the high contrast provided by multipassed and/or tandem instruments. These very delicate instruments are technically complex. Excellent reviews on their operation and fabrication exist in the literature (see the ‘‘Further reading’’ section).
Examples To highlight BLS, a selection of examples are discussed. First, the classical application is described: the
Scattering, Inelastic: Brillouin 201
Incident light
Incident light 45°
Scattered light
Scattered light
k′
45°
k′ k
90° k
q
q
q = √2n k
(a)
q = √2 k
(b)
Incident light Scattered light
Incident light
k ≈ −k ′
Scattered light
qs
k′
Surface wave vector component qs conserved
q
(c)
k
qs = 2 k sin
q = 2n k sin(/2)
(d)
Scattered light ′
Incident light
k k′ Reflected light qs
Surface wave vector component qs conserved
(e)
q s = k (sin′−sin )
Figure 1 Scattering geometries: (a) a bulk-type analysis of excitations inside a transparent material using 901 geometry, (b) a thin transparent layer – this geometry is sensitive to excitations propagating in the plane of the sample, (c) a transparent layer with nearnormal incidence – this geometry is sensitive to wave vectors perpendicular to a sample surface, (d) a back-scattering geometry in an opaque material which is sensitive to in-plane, surface wave vectors, and (e) a variant of geometry (d), where k 0 is not along the direction of the incident light, is also used for studying surface excitations. For geometries (a) and (c), the material refractive index n is required for data analysis.
202 Scattering, Inelastic: Brillouin
determination of elastic constants in a bulk transparent material. In this subsection, references to examples are provided where the determination of cij has been done at high temperatures and high pressures. Then examples of BLS applied to liquids, glasses, opaque materials (surface wave BLS), magnetic materials (magnons), and electronic levels are given.
Determination of Elastic Constants of a Bulk Crystal
Figure 2 shows a Brillouin spectrum obtained from a diamond crystal in the backscattering geometry shown in Figure 1c. Since the diamond surface was (1 1 1) and since the phonons probed in the experiment propagate along the surface normal, the peaks observed in Figure 2 correspond to phonons along the [1 1 1] direction. Along this direction the longitudinal sound velocity is vL ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc11 þ 2c12 þ 4c44 Þ=3r and the velocity of the (degenerate) transverse phonons is vT ¼ ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc11 c12 þ c44 Þ=3r. The peaks in Figure 2 have therefore been labeled as L and T, respectively. Since diamond crystals are large enough to be cut and polished with any desired orientation, it is clear that from a combination of measurements in different scattering geometries and/or crystal orientations the whole set of elastic constants can be determined. In Table 1, the cij of diamond determined by both Brillouin scattering and by ultrasonic techniques are
listed. The results of both techniques are in very good agreement but it is interesting to note that while the BLS measurements were performed on crystals of around 1/3 carat, the ultrasonic measurements used a 22-carat diamond. In cases where large, transparent crystals are available, it is clear from Table 1 that BLS has no particular advantage over ultrasonic techniques. However, when only small samples are available the BLS technique becomes the technique of choice. Single crystals of cubic boron nitride (BN), for example, cannot be grown as single crystals larger than E500 mm. In this case, BLS becomes the technique of choice and yields the cij also included in Table 1. The preceding section outlines how BLS can be used to determine cij. At ambient conditions, cij can often be determined more accurately using ultrasonic techniques. In some cases, however, BLS becomes a much more versatile tool than ultrasonics. Such cases usually involve samples where transducer bonding is difficult, as was mentioned above, in small or thin samples, high-pressure and/or high-temperature experiments, or cases where chemical reactivity becomes severe. For example, the temperature dependence of the c11 and c44 elastic constants of Ba1–xLaxF2 þ x over the range 300–1600 K was determined by P E Ngoepe and J D Comins who found a linear decrease that they attributed to anharmonicity. A case involving high pressures is represented by measurements of liquid and solid argon in a diamond anvil cell up to 70 GPa.
400 L Diamond q // [111]
300 Intensity (a.u.)
Measurements of Spectral Features in Liquids
L
200
100 T 0 −7
−5
T −3
−1
1
3
5
7
Frequency shift (cm−1) Figure 2 Brillouin spectrum from a diamond crystal in the geometry from Figure 1c. The longitudinal (L) and transverse (T) phonons propagate along the [1 1 1] direction. Table 1 Elastic constants of diamond and boron nitride Material and method
c11 ðGPaÞ
c12 ðGPaÞ
c44 ðGPaÞ
Diamond, BLS Diamond, ultras Boron nitride, BLS
1076 1079 820
125 124 190
577 578 480
BLS can also be used to determine the elastic response of liquids. In contrast to solids, however, where the unshifted peak is dominated by the effects of impurities and surface scattering, in pure liquids the central peak is due to entropy fluctuations. The intensity of the unshifted component is related to that of the shifted BLS peaks via the Landau–Placzek ratio, Cp/ Cv–1, where Cp and Cv are the specific heats at constant pressure and volume, respectively. In pure liquids, therefore, BLS can be used to study the intensity and shape of the central peak. Frequency broadening of this peak can be caused by fluctuations in the orientation of anisotropically polarizable molecules or by diffusive processes governed by the equations of hydrodynamics. In liquids, it is also possible to investigate relaxation and damping effects via the phonon lifetime that translates into changes in the shape and line width of the phonon peaks. The same approach cannot usually be applied to most bulk crystals, because of a very low damping that results in line widths below the experimental resolution.
Scattering, Inelastic: Brillouin 203
Glasses
BLS has been extensively used to investigate the transition of liquid to solid in materials that do not crystallize: for example, silica, Rb1–x(NH4)x/H2PO4 (10–300 K), Rbx(NH4)y/H2PO4 (20–80 K), and Ca(NO3)/KNO3 (320–640 K). Across the transition region, the phonons in these materials become highly damped and the resulting line shapes can be interpreted in terms of the underlying mechanism of the glass transition. These types of investigations are also of practical importance, because these materials are often used in fiber optics technology and in optoelectronic devices. Gases
BLS in gases detects local fluctuations that lead to frequency shifts in the 0.1–1 GHz range. These shifts can also be viewed as a Doppler shift from the motion of individual atoms. In this context, BLS has also been used to investigate how the phonon spectrum evolves as a gas is pressurized to become a fluid. This evolution reflects the hydrodynamic transformation as the thermal mean free path for atomic motion becomes comparable and then shorter than the wavelength at which the medium is being probed by the scattering of visible light. These experiments can also be interpreted in terms of the hydrodynamic equations of motion. Investigations of CO2 up to 8 atm pressure, and Xe over the range 0.022– 0.66 atm are two instances where this phenomenon has been investigated. Surface Acoustic Modes in Thin Layers
Light scattering from acoustic excitations in opaque materials results from a purely surface-induced mechanism, viz. phonon-induced thermal ripples. In these cases, the scattering geometries (d) and (e) in Figure 1 must be employed. In a semi-infinite medium, the strongest coupling is to Rayleigh waves (reminiscent of waves in the ocean) with some weaker features induced by bulk waves reflecting from the free surface. In thin films and layered structures, however, it is also possible to couple to modes that are localized within the layers or at the interfaces; these modes are generally referred to as Sezawa
R
1400 R Scattering intensity (a.u.)
However, in temperature regions where structural phase transitions occur, damping processes influence Brillouin line widths enough so that the changes are detectable, for example, in molten and crystalline alkali halides, during order–disorder transitions in mixed crystals, or during ferroelastic phase transitions in some crystals.
1200 1000 800 600 S 400
S S
S
200 −40
−20
0
20
40
Frequency shift (GHz) Figure 3 BLS from surface acoustic phonons in the Co/Cu superlattice. Descriptions: R – the Rayleigh surface mode, S – Sezawa or higher-order Rayleigh modes. The measurement was carried out for (s–s) polarizations of the incident and scattered light.
modes. If the medium is not completely opaque, it is possible to couple to the phonons via both the ripple and elastooptic mechanisms. In these cases, unusual interference effects are possible. Figure 3 shows a BLS spectrum from a Co/Cu superlattice built from 60 bilayers of 1 nm Co/0.8 nm Cu in which the Rayleigh and Sezawa modes are observed. The experiment was carried out using a Sandercock-type tandem 3-pass interferometer. From these results, the effective elastic properties of the superlattice can be extracted. In cases where the surface layers become much thinner than the wavelength of light, the medium again becomes equivalent to a homogeneous solid. In these cases, Brillouin scattering has been a very efficient and reliable tool for determining the effective elastic constants of the heterogeneous solid. For example, R Danner et al. in the Ni/V superlattice, for different layer thicknesses from the range of 1.85– 63.3 nm, proved that this b.c.c./f.c.c. system reveals softening of elastic parameters. J A Bell and co-workers measured effective elastic constants in the Mo/Ta superlattice (c11, c13, c33, c44) with an elementary spatial bilayer thickness ranging from 0.7 to 20 nm. Observation of Bulk and Surface Spin Waves in Metallic Superlattices
Brillouin scattering can also be used to investigate the magnetic properties of materials via their magnetic excitations – magnons. In this sense, the information it yields is similar to that obtained in ferromagnetic resonance (FMR) experiments. However, contrary to
204 Scattering, Inelastic: Brillouin
H = 6.8 kG
400
300
BM
Scattered light intensity (a.u.)
Light scattering intensity (a.u.)
D-E
BM
200
100
H = 5.2 kG H = 4.1 kG H = 3.5 kG H = 2.8 kG H = 2.0 kG
H = 1.1 kG
0 −40
−20
0
20
−40
40
−20
Scatterings from Electronics Levels
More recently, BLS has been used to probe the lowlying excitations of B impurity atoms in a diamond host . In this case, since the excitations are local and
40
Figure 5 A set of Brillouin spectra measured for different magnetic field intensities in the Co/Cu superlattice. The peaks that appear only on the left side (Stokes signals) result from scattering from a unidirectional surface-like spin wave (D-E mode from Figure 4).
400
LT
Boron-doped diamond
300 Intensity (a.u.)
FMR that probes the infinite wavelength excitations, Brillouin scattering probes excitations with wavelengths comparable to the wavelength of light. As for phonons, magnons can be subdivided into surfacelike and bulk-like excitations. Expressions relating the magnon frequencies to the magnetic properties of the material – magnetization, anisotropies, applied field, and gyromagnetic ratio – can be quite complex and often cannot be rendered in analytical form. The Brillouin scattering technique has also been used to investigate the nature of magnetic excitations in nonconventional materials. In the case of the Co/Cu superlattice described in the previous paragraph, which is built from alternating magnetic and nonmagnetic materials, bulk-like collective excitations resulting from coupling between spin waves across the nonmagnetic spacer are detected (Figure 4). Additionally, a surface-like mode propagating on the surface of the sample was also found. The experiment was carried out with a tandem 3-pass spectrometer, with (s–p) polarization of the incident and scattered light, and a constant magnetic field lying in the sample plane. Figure 5 provides results for different magnetic field intensities, from which the gyromagnetic ratio and saturation magnetization can be determined.
20
Frequency shift (GHz)
Frequency shift (GHz) Figure 4 BLS from spin waves in a multilayered Co/Cu system. Descriptions: D-E, Damon–Eshbach surface mode, BM, (bulk magnon) collective excitations of a bulk-like nature resulting from coupling between spin waves through the nonmagnetic spacer layer. The measurement was carried out for (s–p) polarizations of the incident-scattered light.
0
T B
B
L
B
200
B
100
0
−17
−12
−7
−2
3
8
13
18
−1)
Frequency shift (cm
Figure 6 Brillouin scattering from B impurities in a diamond crystal as a host. Descriptions: (L) longitudinal phonons, (T) transverse phonons, and (B) electronic transitions between the hydrogen-like states surrounding the B atoms.
are not described by a wave vector, the results do not depend on the scattering geometry. Figure 6 shows a spectrum recorded at low temperature and with an applied field of 4 T; apart from the expected longitudinal (L) and transverse (T) phonon peaks, the additional peaks labeled B can be traced to electronic transitions between the hydrogen-like states surrounding the B atoms. Under the effect of an applied field these modes split and from the measured splittings, the degeneracy and nature of the transitions is extracted. This, in turn, provides information on how B impurities in diamond lead to its semiconducting properties.
Scattering, Inelastic: Electron 205
Acknowledgment Work at ANL was supported by the US DOE, Basic Energy Sciences, Materials Sciences under contract W-31-109-ENG-38. See also: Scattering, Inelastic: Raman; Thin Films, Mechanical Behavior of.
PACS: 43.35.G; 68.65. þ g; 63.20. e; 63.22. þ m; 72.10.Di; 75.30.Ds; 78.35. þ c; 81.07. b Further Reading Bell JA, Bennett WR, Zanoni R, Sttegeman GI, Falco CM, et al. (1987) Elastic constants of Mo/Ta superlattices measured by Brillouin scattering. Physical Review B 35: 4127–4130. Benedek GB, Lastovka JB, Fritsch K, and Greytak T (1964) Brillouin scattering in liquids and solids using low-power lasers. Journal of the Optical Society of America 54: 1284–1285. Blachowicz T (2003) Brillouin Spectroscopy in Crystal Lattices. Acoustic and Spin Waves. Gliwice: Silesian University of Technology Press. Brillouin L (1922) Diffusion de la Lumie`re et des Rayonnes X par un Corps Transparent Homoge´ne; Influence de l’Agitation The´rmique. Annales de Physique 17: 88. Broz A, Harrigan M, Kasten R, and Monkiewicz A (1971) Light scattered from thermal fluctuations in gases. The Journal of the Acoustical Society of America 49: 950–953.
Danner R, Huebener RP, Chun CL, Grimsditch M, and Schuller IK (1986) Surface acoustic waves in Ni/V superlattices. Physical Review B 33: 3696–3701. Grimsditch MH (1989) Brillouin scattering from metallic superlattices. In: Cardona M and Gu¨ntherodt G (eds.) Light Scattering in Solids, Superlattices and Other Microstructures, vol. 5, pp. 285–302. Berlin: Springer. Grimsditch MH and Ramdas AK (1975) Brillouin scattering in diamond. Physical Review B 11: 3139–3148. Grimsditch MH, Zouboulis ES, and Polian A (1994) Elastic constants of boron nitride. Journal of Applied Physics 76: 832–834. Gross EF (1930) Change of wave-length of light due to elastic heat waves at scattering in liquids. Nature 126: 201. Hyunjung K, Ramdas AK, Rodriguez S, Grimsditch MH, and Anthony TR (1999) Magnetospectroscopy of acceptors in ‘‘blue’’ diamonds. Physical Review Letters 83: 3254–3257. Mandelstam LI (1926) K Voprosu O Rassea˜nii Sveta Neodnorodnoj Sredoj. Zhurnal Russkovo Fiziko-Khimicheskovo Obshchestva 58: 381. Ngoepe PE and Comins JD (1986) Measurements of elastic constants in super-ionic B1 xLaxF2 þ x. Journal of Physics C: Solid State Physics 19: L267–L271. Sandercock JR (1982) Trends in Brillouin scattering: studies of opaque materials, supported films, and central modes. In: Cardona M and Gu¨ntherodt G (eds.) Light Scattering in Solids, Recent Results, vol. 3, pp. 173–206. Berlin: Springer. Shimizu H, Tashiro H, Kume T, and Sasaki S (2001) High-pressure elastic properties of solid Argon to 70 GPa. Physical Review Letters 86: 4568–4571.
Scattering, Inelastic: Electron M Erbudak, Laboratorium fu¨r Festko¨rperphysik, Zu¨rich, Switzerland D D Vvedensky, Imperial College, London, UK & 2005, Elsevier Ltd. All Rights Reserved.
Introduction Inelastic electron-scattering phenomena are used to obtain information on the electronic properties and atomic structure of materials. An interaction event between an electron and a target material is called inelastic if the electron loses some or all of its energy; otherwise, the event is called elastic. Any energy loss by the electron is equal to the energy gained by the material and is characteristic of the type of scattering event. The measurement of an inelastic event determines the energy transferred and the rate at which this transfer occurs, which provides physical information about the elementary excitations of the material. Inelastic interactions can be grouped according to different energy-transfer regimes that form the basis
of the principal experimental techniques. Historically, inelastic electron scattering was developed prior to the availability of continuous-energy photon sources from dedicated storage rings. In fact, under certain circumstances, electronic excitations are equivalent to photon excitations in that they can be described within a common conceptual and theoretical framework. An electron is an elementary particle with a mass m and, owing to its wave nature, has a wave number k ¼ p=_, where p ¼ mv is the momentum, v is its velocity, and _ is Planck’s constant. The kinetic energy is expressed classically by E ¼ ð1=2Þmv2, where v ¼ jvj. Electrons have a negative charge e and a nonclassical angular momentum (spin). These properties play a central role in the interactions between electrons and their environment and constitute the basis of a wide range of spectroscopic techniques. Owing to their unique properties, there are both advantages and disadvantages in using electrons as a probe in analytical applications. Spin-polarized or spin-averaged beams of electrons are readily created, electrons can be accelerated to any energy, and focused on a target at will. After they interact with