Definitions and History of the Brillouin-Mandelstam Spectroscopy

Brillouin-Mandelstam light scattering spectroscopy (BMS), also referred to as Brillouin light scattering spectroscopy (BLS), is the inelastic scattering of light by thermally generated or coherently excited elemental excitations such as phonons or magnons. Leon Brillouin and Leonid Mandelstam, the French and Russian scientists, independently predicted interactions between light and thermally excited phonons in solids in the early decades of the 1900s.1–3 Mandelstam, together with his associate Grigory Landsberg, proceeded to study the light scattering on thermal excitations experimentally in quartz crystals but succeeded in recording the scattering from optical phonons. The fascinating story of how this effort, independent from that of Chandrasekhara Raman who experimented by visual observation with fluorescence fluids under Sun illumination,4 is briefly narrated by Manuel Cardona and Roberto Merlin.5 A detailed account of the developments with the timeline for the theoretical prediction and experimental verification of light scattering on phonons, both acoustic and optical, is provided by Immanuil Fabelinskii6,7 It is commonly accepted that the first light scattering spectrum from thermal excitations, i.e. acoustic phonons, which we now refer to as BMS, was reported by Eugenii Gross in 1930.8–10 He succeeded in this endeavor by developing an optical setup of several cascaded interferometers capable of providing substantial frequency dispersion to analyze the fine structure of spectral lines. Gross, who worked in St. Petersburg, Russia, was aware of Mandelstam’s work in Moscow, his prediction of the light scattering on thermal excitations, and used his formula in the analysis of the experimental data.6,7 However, it was not until the 1960s that the experimental BMS – BLS research received an impetus with the invention of lasers, and, later, the introduction of the high-contrast multi-pass tandem Fabry–Pérot (FP) interferometers by John Sandercock.11

Basics of the Brillouin-Mandelstam Spectroscopy   

Brillouin-Mandelstam light scattering is complementary to Raman spectroscopy – another inelastic light scattering technique. In the realm of phonons – quanta of crystal lattice vibrations – Brillouin spectrometer measures energies of acoustic phonons while Raman spectrometer measures energies of optical phonons. In many cases, BMS is a more powerful technique in the sense that it measures not only the energy of phonons near the G point, like Raman spectroscopy but can provide data for determining the phonon dispersion in the vicinity of the Brillouin zone (BZ) center. Owing to the several orders of magnitude smaller energy shifts in the scattered light measured in Brillouin spectroscopy experiments than in Raman spectroscopy experiments, the BMS instrumentation utilizes FP interferometers rather than diffraction gratings. A single-plane parallel FP interferometer consists of two flat mirrors launched in a parallel configuration with respect to each other, at the spacing of . The low contrast of the single-plane parallel FP does not allow one to distinguish the low-intensity Brillouin scattering of light from that of the elastically scattered component. The multi-pass tandem FP interferometer enhances the spectral contrast by orders of magnitude making it possible to detect the low-intensity Brillouin scattering peaks even for opaque materials. The state-of-the-art triple-pass tandem FP systems provide a contrast of  [Ref. 12].

Fundamentals of the Brillouin – Mandelstam Light Scattering

It is well known that the light scattering processes depend strongly on the optical properties of materials and the penetration depth of light into the material.13 As a result, the light scattering must be treated separately for different classes of materials depending on their optical transparency at the excitation laser wavelength.11 In certain cases, the optical selection rules imposed by the size and opacity of material systems under investigation, create new opportunities for BMS technique. In BMS of crystal lattice vibrations, two different mechanisms contribute to the scattering of light: (i) bulk phonons via the volumetric opto-elastic mechanism, and (ii) surface phonons via the surface ripple mechanism.11,14 The dominance of one or the other in the BMS spectrum depends on the size, e.g. thickness, and optical properties of the material, e.g. transparency or opacity of the material at the given excitation wavelength (see Figure 1).

Figure 1: Fundamentals of Brillouin-Mandelstam light scattering. a) Schematic of the light scattering processes via bulk quasiparticles. b) Schematic showing typical spectra and accessible phonon frequency range using Raman, Low-Wave-Number-Raman, and BMS techniques. c) Phonon dispersion in silicon crystal along the [001] direction. d) Raman (up) and BMS (bottom) spectra of silicon showing TO and LO phonons at 15.6 THz and TA and LA phonons at 90.5 GHz and 135.2 GHz, respectively. e) Schematic showing light scattering by the surface ripple mechanism in semitransparent and opaque materials. f) Side view of the ripple scattering process with the indicated incident and scattering angles.

  Scattering of Light by Bulk Acoustic Phonons

In transparent or semi-transparent materials, where the penetration depth of light is sufficiently large compared to that of the incident laser light wavelength, scattering by the bulk phonons is the dominant mechanism. A propagating acoustic phonon with wavevector of  and angular frequency of  induces local variations in the dielectric constant of the medium which can scatter light through the opto-elastic coupling.14,15 A general schematic of the light scattering process by volumetric phonons is illustrated in Fig. 1a. In all light scattering processes, two equations of the conservation of momentum, ℏks – ℏki = ±ℏq, and the conservation of energy, ℏωs-ℏωi=±ℏωq, should be satisfied. Here ki and ks  are the wavevectors and ωi , and ωs are the angular frequencies of the incident and scattered light, respectively. The positive and negative signs on the right-hand side of these equations denote the “anti-Stokes” and “Stokes” processes. In the former, a phonon is annihilated (or absorbed) whereas in the latter a phonon is created (or emitted) during the scattering process. The angle between ki and ks, ϕ, is the scattering angle, which depends on the scattering geometry.11 Since the energy change in the light scattering processes by acoustic phonons is negligible compared to that of the energy of the incident light, one can assume that ki ≈ ks. Therefore, from the conservation of momentum, it is inferred that the magnitude of the phonon wavevector is q=(4πn ⁄ λ) sin⁡(ϕ ⁄ 2), where n and λ are the refractive index of the material and the wavelength of the excitation laser light, respectively. In the backscattering geometry where ϕ=180°, the maximum detectable phonon wavevector, qm=4πn/λ is achieved.11 

Apart from the substantial difference in the energies of the elemental excitations, the light scattering processes in both Raman and BMS techniques are similar. The Stokes and anti-Stokes optical peaks appear as doublets on both sides of the central line in the frequency spectrum (Fig. 1b). The spectral analyses of these peaks, including the frequency shift, intensity, and full width at half maximum (FWHM), provides information about the energy, population, and lifetime of the detected elemental excitations.16 Typical Raman systems can detect the frequency range between ~3 THz (~100 cm-1) to 135 THz (4500 cm-1). This is the range where the optical phonons reside. The modern low-wave-number Raman (LWNR) instruments utilize novel Notch filters and fine gratings, which allow for detecting phonons with minimum energies of ~360 GHz (~12 cm-1) or higher. The multi-pass tandem FP interferometer deployed in BMS can probe quasiparticles with much lower energies in the range of 300 MHz to 900 GHz. This spectrum interval is essential for probing acoustic phonons and magnons. Both Raman and Brillouin-Mandelstam light scattering techniques observe phonons with the wavevectors close to the BZ center limited by the wavevector of the excitation wavelength of the laser source. A combination of BMS, LWNR, and conventional Raman allows one to detect phonons and magnons in the frequency range from hundreds of MHz up to tens of THz at wavevectors close to the BZ center.

The dispersion of the fundamental acoustic phonon polarization branches is linear in the vicinity of the BZ center, where the phase velocity (υp=ω/k) and group velocity (υg=∂ω/∂k) are equal. In the BMS experiment, with calculating the probing phonon wavevector and measuring the peak frequency observed in the spectrum, one can determine υp and υg of the associated phonon mode. For example, in the backscattering geometry, which is widely used, υp=λf/(2n)  in which f is the spectral position of the associated peak observed in the BMS spectrum. Since optical phonons have a flat dispersion in the vicinity of the BZ center, determining q is not essential in most Raman experiments. Figure 1c shows a dispersion of the longitudinal (LA) and transverse (TA) acoustic and the longitudinal (LO) and transverse (TO) optical phonons in silicon along [001] direction. Figure 1d provides the actual accumulated Raman and BMS spectra for the same.

Scattering of Light by Surface Acoustic Phonons

In opaque and semitransparent materials, the BMS spectrum is dominated by the scattering of light by surface phonons via the ripple mechanism.11,41 The process is illustrated schematically in Fig. 1e,f. Owing to high optical extinction in these types of materials, the penetration depth of light is limited to the surface and only the in-plane component (parallel to the surface in the scattering plane) of the phonon momentum conservation law is satisfied. Therefore, only phonons with the in-plane wavevector component, q∥=ki sin⁡(θs)-ks;sin⁡(θs), where θi and θs are the incident and scattered light angle with respect to the normal to the surface, may contribute to the light scattering (Fig. 1f). In a complete backscattering geometry where θi = θs = θ, the in-plane phonon momentum is q∥=(4π⁄λ)sin⁡(θ). Note that, here, the phonon wavevector depends only on the incident angle and excitation laser wavelength. Under such conditions, one can probe phonons with different wavevectors by changing the incident angle of the laser light. The latter allows one to obtain the phonon dispersion that is energy (frequency) of the phonons as a function of wavevector. In the backscattering BMS configuration, considering a semi-infinite elastically isotropic material (Fig. 1e,f), the scattering cross-section is given as (d2σ) ⁄ (dΩdωs)=((ζωI4) ⁄ (16π2c4)) ξ2 ⟨|uz (0)|⟩2(q∥,ω) in which ζ and c are the sample area under illumination and the speed of light in vacuum, respectively. ξ is a coefficient which depends on θi, θs, and the relative permittivity of the scattering medium. The last term,⟨|uz (0)|2⟩(q∥,ω) is the mean square displacement of the surface perpendicular to q∥ in the scattering plane.17 In fact, it is the last term that determines which phonon modes contribute to the light scattering. As it is seen, only those surface phonons with vibrational displacement perpendicular to q∥ in the plane of scattering appear in BMS spectrum. Obtaining the energy dispersion of the elemental excitations is a distinctive advantage of the BMS technique over Raman spectroscopy. Modifications in the phonon dispersion, and, correspondingly, in the phonon density of states, contain a wealth of information on confinement and proximity effects in low-dimensional material systems.18,19,20

 

Balandin Group Brillouin-Mandelstam Experimental Capabilities

The group has developed a cryogenic integrated micro-BMS with advanced unique features enabling innovative research of low-dimensional spintronics, superconducting, and other materials. A unique instrument is located on the ground floor of the California NanoSystems Institute (CNSI) at UCLA. The advanced features of the BMS system include the specially designed rotating microscopy stage and imaging system for measuring the energy dispersion of phonons, magnons, and other elemental excitations in the temperature range from 4 K to 700 K. The instrument allows one to study samples of much smaller dimensions than those acceptable for conventional spectrometers – the atomic thickness and sub-micrometer lateral dimensions. The instrument is capable of recording phonon and magnon energies substantially below the current limit of conventional spectrometers. The advanced BMS instrument is connected to the micro-Raman system allowing one to investigate both acoustic and optical phonons. 

[Figure 2: Group members working in The Brillouin – Mandelstam Light Scattering Spectroscopy (BMS) Facility directed by Professor Alexander Balandin. The California NanoSystems Institute at UCLA.]

The unprecedented capabilities of the cryogenic sub-GHz to high-THz integrated micro-BMS instrument enable transformative science and engineering research for a better understanding of the properties of layered materials. The possibility of investigating atomically thin films with lateral dimensions in the sub-micrometer range allows the Balandin group to measure acoustic phonon energy dispersion, which is not possible with conventional micro-Brillouin spectrometers or other inelastic scattering techniques such as neutron or X-ray diffraction spectroscopies. The unique capabilities of the instrument provide fundamental knowledge of the phonon and magnon lifetime in layered materials; the strength of the magnon–phonon and spin-lattice interactions in the materials and heterostructures; phonon velocities in such materials; as well as characteristics of other quasiparticles in novel materials and heterostructures. The BMS instrument is used for measurements of the phase transition temperatures from the changes in the phonon and magnon peaks; phonon and magnon band structures, and their modification with the thickness, strain, and electric bias.

 

 

References

 

  1. Brillouin, L. Diffusion de la lumiere et des rayons X par un corps transparent homogene: Influence de l’agitation thermique. Ann. Phys.(Paris) 17, 88 (1922).
  2. Mandelstam, L. Light scattering by inhomogeneous media. Zh. Russ. Fiz-Khim. Ova. 58 381 (1926).
  3. Meng, Z., Traverso, A. J., Ballmann, C. W., Troyanova-Wood, M. A. & Yakovlev, V. V. Seeing cells in a new light: a renaissance of Brillouin spectroscopy. Adv. Opt. Photonics 8, 300 (2016).
  4. Raman, C. V. & Krishnan, K. S. A new type of secondary radiation. Nature 121, 501–502 (1928).
  5. Cardona, M. & Merlin, R. Light scattering in solids IX. Topics in Applied Physics vol. 108 (Springer, Berlin, Heidelberg, 2006).
  6. Fabelinskiǐ, I. L. The prediction and discovery of Rayleigh line fine structure. Uspekhi Fiz. Nauk 170, 107–108 (2000).
  7. Fabelinskiĭ, I. L. The discovery of combination scattering of light in Russia and India. Physics-Uspekhi 46, 1105–1112 (2003).
  8. Gross, E. Change of the wavelength of light due to elastic waves at scattering in liquids. Nature 126, 201 (1930).
  9. Gross, E. The splitting of spectral lines at the scattering of light by liquids. Nature 126, 400 (1930).
  10. Gross, E. über Änderung der WellenlÄnge bei Lichtzerstreuung in Kristallen. Zeitschrift für Phys. 63, 685–687 (1930).
  11. Sandercock, J. R. Trends in Brillouin scattering: Studies of opaque materials, supported films, and central modes. in Light Scattering in Solids III. Topics in Applied Physics (eds. Cardona, M. & Güntherodt, G.) vol. 51 173–206 (Springer, 1982).
  12. Scarponi, F. et al. High-performance versatile setup for simultaneous Brillouin-Raman microspectroscopy. Phys. Rev. X 7, 031015 (2017).
  13. Hayas, W. & Loudon, R. Scattering of light by crystals. (Wiley, 1978).
  14. Mutti, P. et al. Surface Brillouin scattering—Extending surface wave measurements to 20 GHz. in Advances in Acoustic Microscopy (ed. Briggs, A.) vol. 1 249–300 (Springer, 1995).
  15. Bottani, C. E. & Fioretto, D. Brillouin scattering of phonons in complex materials. Adv. Phys. X 3, 607–633 (2018).
  16. Olsson, K. S., An, K. & Li, X. Magnon and phonon thermometry with inelastic light scattering. J. Phys. D. Appl. Phys. 51, 133001 (2018).
  17. Loudon, R. Theory of surface-ripple Brillouin scattering by solids. Phys. Rev. Lett. 40, 581–583 (1978).
  18. Balandin, A. A. & Nika, D. L. Phononics in low-dimensional materials. Mater. Today 15, 266–275 (2012).
  19. Balandin, A. A. Phononics of graphene and related materials. ACS Nano 14, 5170–5178 (2020).
  20. Balandin, A. A. Thermal properties of graphene and nanostructured carbon materials. Nat. Mater. 10, 569–581 (2011).