Why Are the Raman Spectra of Crystalline and Amorphous Solids Different?

Reprinted from March 2017

Molecular Spectroscopy Workbench Why Are the Raman Spectra of Crystalline and Amorphous Solids Different?

The Raman spectra of crystalline and amorphous solids of the same chemical composition can be significantly different primarily because of the presence or absence of spatial order and long- range translational symmetry, respectively. The purpose or goal of this installment of “Molecular Spectroscopy Workbench” is to help readers understand the underlying physics that affect the Raman spectra of crystalline and amorphous solids. Wave vector, reciprocal space, and the Brillouin zone are explained with respect to Raman spectroscopy of solids.

David Tuschel

f you are a chemist, you most likely learned vibrational tional energy states as a result of these many and different spectroscopy as molecular spectroscopy. That is, you chemical interactions. I learned about the normal vibrational modes of discrete Interpreting the Raman spectra of compounds, or mate- molecules—as opposed to solid-state materials—and their rials in the solid state, requires the knowledge of concepts Raman or infrared activity based upon molecular spectro- and mathematical treatments other than those of molecu- scopic selection rules. Raman spectra of compounds in the lar spectroscopy. The Raman spectra of crystalline and liquid, or vapor, phase consist of narrow bands whose widths amorphous solids of the same chemical composition can be depend upon the degree of chemical interaction between the significantly different, primarily because of the presence or molecules. The weaker the chemical interaction, the nar- absence of spatial order and long-range translational sym- rower the band will be. In fact, the bands of a compound metry, respectively. Amorphous solids can be thought of as in the vapor phase will be narrower than those of the same a collection of formula units of the same chemical composi- compound in the liquid phase for that very reason (1). The tion, but with varying bond angles and lengths depending broad infrared absorption and Raman bands of water dem- upon chemical bond interactions with nearest neighbors. onstrate the effect of hydrogen bonding and the result of dif- There is no order to their arrangement in space. Conse- ferent degrees of chemical interaction between the molecules quently, one does not observe the narrow bands familiar in a neat liquid. The breadth of the O-H stretching modes to us in molecular spectroscopy from amorphous solids. in particular is a manifestation of the distribution of vibra- The distribution of formula units with varying bond angles

128948_wrmw.indd 26 4/27/17 4:43 PM and lengths produces a distribution of states of slightly varying vibrational energies. It may be helpful to think of this distribution of atomic and Microscope slide chemical bonding arrangements in the amorphous solid as analogous to the

distribution of vibrational energy states Fused quartz

in liquid water because of the presence Intensity of hydrogen bonding. Crystalline quartz

The spectroscopic selection rules 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 for solid-state and molecular spectros- Raman shift (cm-1) copy require making some important distinctions regarding symmetry. The Figure 1: Raman spectra of crystalline and fused quartz and a glass microscope slide. number of Raman and infrared active modes of a crystal depends upon the up to several hundred wavenumbers. Raman spectra of single crystal and space group symmetry of the crystal One might incorrectly conclude from ion-implanted Si obtained from differ- whereas the spectroscopically active the spectra that the chemical compo- ent locations on the same sample are vibrational modes of a molecule are sitions of these two solids were dif- shown in Figure 2. The Raman spec- determined by its point group. The ferent. However, they are not and it is trum of single-crystal Si consists of existence of a crystal lattice of repeat- only their solid-state structures that one triply degenerate optical phonon ing unit cells allows the propagation differ, thereby producing the dramatic at 520 cm-1 along with much weaker of lattice vibrational waves (also called differences in the spectra. The chemi- second-order transverse acoustic phonons) that originate in the repeti- cal composition of the microscope and optical modes that appear at ap- tive and systematic vibrational motions slide is different from that of quartz, proximately 305 cm-1 and 975 cm-1, of the crystal’s atoms whereas the vi- but like the fused quartz it has a glass respectively. The selection rules for brational modes of molecules are local solid-state structure with no long- first-order Raman scattering from and not coupled to those of neighbor- range translational symmetry. Conse- crystalline solids dictate that only ing molecules. It is the launching of a quently, its Raman spectrum is simi- those optical phonons at the Brillouin lattice vibrational wave or phonon by lar to that of the fused quartz insofar zone center are Raman active. (We the incident photon that leads to the as it consists of very broad bands. explain this rule later in this article.) scattering of a Raman photon by the Amorphicity can be induced in Consequently, the first-order Raman crystal. We define a phonon as a lattice crystalline solids through ion bom- band at 520 cm-1 is very narrow with vibrational wave propagating through bardment, which is also known as ion a full width at half maximum of only the crystal arising from repetitive implantation when used in semicon- 4 cm-1. This particular Si sample was atomic displacements. Furthermore, ductor fabrication. Dopants are placed implanted with a high dose of ar- the phonon has the characteristics of a at specific depths in semiconductors senic, a much heavier element than traveling wave insofar as it has a propa- through the process of ion implanta- the other common dopants boron or gation velocity, wavelength, wave vec- tion by carefully controlling the dop- phosphorus, and so the crystalline lat- tor, and frequency. ant flux and impact energy. Atoms are tice within the first several hundred Structurally, crystals differ from dislodged from their positions in the nanometers of the surface has been amorphous solids and glasses in that host crystal lattice as a result of this severely damaged and amorphized. the former possess long-range trans- ion bombardment. Consequently, the The Raman spectrum of the arsenic- lational symmetry whereas the latter semiconductor’s crystal lattice is bro- implanted Si consists of very broad lack any spatial order. The Raman ken and the long-range translational bands below 550 cm-1. The reason spectra obtained from quartz (crys- symmetry is disrupted. The extent of for the detection of Raman scatter- talline and fused) and a glass micro- damage to the crystal lattice is deter- ing over this broad region is that the scope slide (See Figure 1) very clearly mined by the mass, energy, and flux Raman selection rule for crystals re- demonstrate the importance of spatial of the dopant being implanted; for stricting Raman scattering to phonons order. The quartz samples are of the example, a heavier dopant mass and at the Brillouin zone center no longer same chemical composition but one is higher flux will induce greater dam- holds in amorphous Si and all of the crystalline whereas the fused quartz age to the crystal lattice. The semicon- phonons from Brillouin zone center is a glass with no long-range transla- ductor must be annealed following ion to edge are now sampled. The Raman tional symmetry. The Raman spec- implantation because of the damage spectrum of amorphous Si looks very trum of crystalline quartz consists done to the crystal lattice. The heat much like the calculated populations of sharp and mostly narrow bands treatment restores crystallinity and of phonon energy states within the whereas that of the fused quartz man- thereby activates the semiconductor first Brillouin zone of crystalline Si. ifests very broad peaks with widths of device. The purpose and goal of this in-

128948_wrmw.indd 27 4/27/17 4:43 PM goal. Therefore, we will explain each of these topics and finally bring them all together, leading to the conclusion that in a crystal only those phonons at the Brillouin zone center are Raman active. With the loss of long-range Ion-implanted amorphized Si translational symmetry in an amorphous solid, the Raman selection rules for crystalline solids no longer hold. Intensity Consequently, all of the phonons of the Brillouin zone become Raman Single-crystal Si active and the Raman spectrum of the amorphous solid resembles the

200 300 400 500 600 700 800 900 1000 phonon density of states of the corresponding crystal’s phonon dispersion Raman shift (cm-1) curve. If after reading this article you would like to view a detailed lecture Figure 2: Raman spectra of Si amorphized through ion implantation and single-crystal Si. on this topic, please see the companion video on YouTube (2).

Phonons and Reciprocal Space in a One-Dimensional Lattice a Linear crystal lattice Earlier, we defined a phonon as a lattice vibrational wave propagating through the crystal arising from a is the basis vector of the primitive crystal lattice repetitive and systematic atomic –4π –2π 2π 4π displacements. It should also be un- a a a a derstood that these displacements are O A k quantized vibrations of the atoms in Reciprocal lattice the lattice and are traveling waves. π π The phonon has the characteristics k = a k = a of a traveling wave insofar as it has a propagation velocity, wavelength, 2π A is the basis vector of the reciprocal lattice and is equal in length to wave vector, and frequency. It is im- a portant to note that all of the peaks Figure 3: Real and reciprocal lattices of a linear monatomic crystal. in a Raman spectrum of a crystalline solid are attributed to phonons and not only those at low energy or Raman shift. P.M.A. Sherwood makes that point abundantly clear in ω his fine book, Vibrational Spectros- First Brillouin zone copy of Solids: “All crystal vibrations involve the entire lattice and are thus lattice vibrations (a term sometimes unfortunately only applied to exter- nal vibrations) and such vibrations can be considered as a wave propa- −4 π −3π −2π −π o π 2π 3π 4π gating through the crystal lattice” (3). a a a a a a a a Unlike the normal vibrational mode –k +k of an individual molecule, a phonon is a lattice vibrational wave travel- Figure 4: Phonon dispersion curve for a linear monatomic lattice. ing through the crystal. This is why a mathematical treatment of the vibra- stallment of “Molecular Spectroscopy in crystalline and amorphous solids. tional motions of the crystal’s atoms Workbench” is to help readers un- We need to understand concepts such must take into account the medium derstand the underlying physics that as wave vector, reciprocal space, and through which the phonon is propa- produce such different Raman spectra the Brillouin zone to accomplish that gating.

128948_wrmw.indd 28 4/27/17 4:43 PM We describe the phonon as a traveling wave with one of its characteris- ω tics, the wave vector k, defined as Maximum ω 2π k = λ [1] Minimum ω Optical Gap where λ is the phonon wavelength. The wave vector is in units of wave- Acoustic numbers (cm-1) because the phonon wavelength is in the denominator. q Perhaps you can anticipate why a re- −π/2a 0 π/2a ciprocal lattice with units of distance in the denominator (for example, Figure 5: Phonon dispersion curve for a linear diatomic lattice. cm-1) would be compatible with our treatment of phonon wave vectors in the region between -π/a and +π/a in optical branch phonons are the origin units of cm-1. Depictions of a direct a linear monatomic lattice the first of the bands in the Raman spectrum linear monatomic lattice along with Brillouin zone and a k value of 0 is the of a solid. Of course, the number of its reciprocal lattice are shown in Fig- Brillouin zone center. An important Raman bands is dictated by the sym- ure 3. Each of the dots in the linear concept to grasp here is that because k metry of the crystal and their energies crystal lattice represents an atom and is inversely related to λ, the longer the by the masses of the atoms and force the distance between the atoms is phonon wavelength is the closer it will constants of the chemical bonds. It is given by the letter a. A reciprocal lat- be to the Brillouin zone center in the especially important to note in Figure tice can be constructed from the real phonon dispersion curve. 5 that there is a range of frequencies lattice wherein the basis vector A of Progressing in our understanding for a given optical phonon with the the reciprocal lattice is equal to 2π/a. of phonon propagation and dispersion maximum and minimum ω values at Our reason for choosing basis vector curves, we now consider vibrational the Brillouin zone center and edge, increments of 2π/a rather than 1/a waves in a linear diatomic lattice. respectively. The Raman bands would will soon become clear as we analyze a Again without providing the deriva- be very broad if all of those phonons vibrational wave propagating through tion or the mathematical expressions, from center to edge were sampled, and our one-dimensional lattice. we state that there are two solutions that is indeed the case for amorphous Let’s consider a phonon propagat- for waves traveling either to the right solids. The Raman intensity as a func- ing through our hypothetical linear or left in the linear diatomic lattice. tion of frequency for a given Raman monatomic lattice. Without provid- (Recall that there was only one solu- band is related to populations of the ing the derivation, we simply state tion for a traveling wave in a linear phonon state as a function of wave that there is one solution for a wave monatomic lattice. The derivations vector from the Brillouin zone center traveling either to the right or left in a and analytical expressions for vibra- to the edge, referred to as the phonon linear monatomic lattice. The angular tional waves in linear monatomic and density of states. In contrast with frequency (ω) of that traveling wave or diatomic lattices can be found in all amorphous solids, crystals restrict phonon is given by text books on solid-state physics.) A phonon sampling to the Brillouin phonon dispersion curve for the two zone center and therefore have narrow ½ ω ±2 f sin ka = m 2 [2] solutions of the wave equations of a Raman bands as we explain in the fol- linear diatomic lattice is shown in lowing section. where f is the vibrational force Figure 5. Note that for the acoustic constant, m is the atomic mass, k is branch solution to the linear diatomic The Brillouin Zone and the phonon wave vector, and a is the lattice a maximum value of ω is Conservation of Wave Vector distance between the atoms. A plot reached when k is equal to ± π/2a as Of course, the real world of crystals of ω as a function of wave vector k opposed to ±π/a for the linear mona- is not one-dimensional but is three- is known as the phonon dispersion tomic lattice. The lower curve is called dimensional. A detailed description curve and is shown in Figure 4. The the acoustic branch and is similar to of the construction of the Bril- dispersion curve is repeating and the phonon dispersion curve for the louin zone for a three-dimensional reaches a maximum value of ω when linear monatomic lattice. The upper crystal lattice is beyond the scope k is equal to ±π/a because of the sine optical branch is so named because of this article. That would involve function in equation 2. Furthermore, these lattice vibrational waves can be a mathematical description of the the dispersion curve for a linear excited with light of infrared wave- construction of the reciprocal lat- monatomic lattice is symmetric about lengths, energies much higher than tice from the real lattice (also called the center value of 0 and is repeat- those of the acoustic branch, espe- a direct lattice) and the subsequent ing beyond ±π/a. Therefore, we call cially at the Brillouin zone center. The construction of the Wigner-Seitz

128948_wrmw.indd 29 4/27/17 4:43 PM face-centered cubic Bravais F crystal lattice in real space. The first step Crystallographic unit cell First Brillouin zone is to construct the reciprocal lattice b 3 of the face-centered cubic F crystal. Following the mathematical proce- dure for doing so produces a body- Simple centered cubic I lattice. The next step cube b 1 is to construct a Wigner-Seitz cell from the body-centered I reciprocal lattice, thereby generating a truncated octahedron. The Wigner-Seitz b2 b 3 truncated octahedron is the shape of the first Brillouin zone of the real face-centered cubic F lattice. Depic- tions of the face-centered cubic F Body-centered b unit cell and its corresponding first cube 1 Brillouin zone are shown at the bot- tom of Figure 6. The first Brillouin zones of a simple cubic lattice and a b 2 body centered cubic I lattice are also b 3 shown in Figure 6. The significance of depicting the three-dimensional Brillouin zone is that we have just Face-centered made the leap from one-dimensional cube b 1 to three-dimensional reciprocal space, which prepares us for the typi- cal phonon dispersion curves found b 2 in the literature. Phonon dispersion curves generally plot the frequency of Figure 6: Real lattices of cubic crystals and their corresponding first Brillouin zones. the phonon from the Brillouin zone center to the various crystal faces and points within the first Brillouin zone. x You can find examples of the phonon dispersion curves and corresponding phonon density of states plots k = k + q ks i s Photon of various solids in the companion video lecture to this installment on YouTube (2).

θ Having developed our understand- ki z ing of the Brillouin zone and its basis Photon in reciprocal space, we relate it to the wave vector in Raman scattering from

1 2θ a crystalline solid. Energy, wave vec- q tor, and momentum are conserved Phonon in the Raman scattering process involving real crystals, which are y anharmonic. Therefore, the incident radiation transmits momentum to the Figure 7: Raman scattering and conservation of wave vector in a crystal. The wave vector of the crystal through the creation of a lattice vibrational wave or phonon. The phonon is given by q and those of the incident and Raman scattered photons are given by ki and ks, respectively. momentum of the incident photon is equal to ħki and that of the newly lattice from the reciprocal lattice. A excellent book by Richard Tilley created phonon is equal to ħq, where helpful description of the construc- (4). Nevertheless, we can describe ħ is Planck’s constant divided by 2π

tion of three-dimensional recipro- one example to give you a sense of and ki and q are the wave vectors of cal lattices, Wigner-Seitz lattices how the construction of a Brillouin the incident photon and phonon, re- and Brillouin zones is given in the zone is accomplished. Consider a spectively. A wave vector diagram of

128948_wrmw.indd 30 4/27/17 4:43 PM Raman scattering in a crystal is shown in Figure 7. The conservation of the wave vector is implicit in the diagram, Brillouin zone edge which shows the dependence of the direction at which the Raman scattered light is collected on the direction of phonon propagation. Of course, Raman scattering occurs in all direc- tions. However, the angle with respect to the incident light at which one col- lects the Raman signal determines the Brillouin zone center propagation direction of the phonon and Raman sampling that is being sampled. The angle of region phonon propagation is one-half that of the angle at which Raman scattering is collected. This relationship is important because in some uniaxial Figure 8: The Brillouin zone of a two-dimensional reciprocal lattice. Raman scattering excited with and biaxial crystals the energies of visible laser light will be generated only from the region very near the Brillouin zone center. certain phonons can vary depending upon the direction of phonon propa- ν, and are the wavelength, frequency 0 and the frequency ω of the phonon gation in the crystal lattice. Practi- and wavenumber of the incident laser does not vary much in that region. cally, that means that the Raman band light, respectively. Applying the laser It is this restriction of sampling only position can vary depending upon light wavenumber of approximately those phonons near the Brillouin the orientation of a single crystal with 20,000 cm-1 in equation 4, we obtain a zone center that results in the Raman respect to the direction and polariza- value for the phonon wave vector of bands from crystalline solids being tion of the incident light and the angle so narrow. q 2k 4π(20,000 cm–1)= 2.5 × 105 cm–1 at which the Raman scattered light is = i = [5] The diagram of the two dimen- collected. This phenomenon is known sional reciprocal lattice in Figure 8 as phonon directional dispersion, and Now here is where we make the should help you get a sense of the re- the variation of Raman band position connection between the wave vector stricted Raman sampling region near is quite significant in some ferroelec- of the phonon to the Brillouin zone of the Brillouin zone center. The region tric metal oxides. the crystal through which it is propa- from the circle around the Brillouin Creating a vector diagram for the gating. We calculate the k value at the zone center to the edge will have a wave vector equation shown in Fig- Brillouin zone edge (BZE), which is distribution of phonon energy states ure 7 and solving for q gives us the π/a (see Figure 3), where a is equal to as indicated in the phonon dispersion expression the real lattice spacing. Let us assume curve for any given solid. However, a lattice spacing of 3 Å = 3 × 10-8 cm those phonons will be inaccessible 2 2 2 q = ki + ks −2kiks cos θ [3] so that in Raman measurements of crystals π when using visible excitation because k = =1 × 108 cm–1 where ki and ks are the wave vectors BZE 3 × 10–8 cm [6] of the requirement to conserve wave of the incident photon and the Raman vector and momentum. Remember scattered photon, respectively. Raman That is a difference of three orders that k = 2π/λ so that k increases with scattering is generally performed with of magnitude between the k value of decreasing λ. We would need much visible laser light. Assuming incident the Brillouin zone edge and the wave shorter excitation wavelengths than blue-green laser light of approximately vector of the phonon that we generate those in the visible region of the 20,000 cm-1, a phonon of several because of the wavelength of the laser spectrum to probe phonons beyond hundred wavenumbers, and a Raman light in our measurement. Therefore, the Brillouin zone center out to the scattered photon also in the visible only the Raman scattering from pho- Brillouin zone edge at 108 cm-1. The region, we make the approximation nons of a very long wavelength, equal instructive publication by Kettle and

that ki ≈ ks. Consequently, equation 3 to several thousand atomic spacings, Norrby titled The Brillouin Zone – yields q ≈ 2ki for the 180° backscatter- and small k value near the Brillouin An Interface between Spectroscopy ing configuration. Substituting 2π/λ zone center out to 2.5 × 105 cm-1 will and Crystallography should be con-

for ki given by equation 1 yields be probed because of the requirement sulted for further information on this that wave vector and momentum be topic (5). 2(2π) 4πν ~ q =2k = = = 4πν [4] conserved. Furthermore, the slope i λ c of the phonon dispersion curve at Loss of Translational Symmetry Where c is the speed of light and λ, the Brillouin zone center is close to Our discussion of reciprocal space

128948_wrmw.indd 31 4/27/17 4:43 PM and the Brillouin zone led to our treatment of the con- Conclusion servation of wave vector and momentum in conjunction Energy and momentum are conserved in the Raman with the Raman scattering of crystals. The existence scattering process involving real crystals, which are an- of long-range translational symmetry in the medium harmonic. Therefore, the incident radiation transmits through which the phonons propagate was essential to momentum to the crystal through the creation of a lattice our treatment. However, what if there is no crystal lat- vibrational wave or phonon. Only phonons of a very long tice and the solid is amorphous or a glass? In an amor- wavelength, equal to several thousand atomic spacings, and phous solid the observed Raman bands are no longer small k value near the Brillouin zone center are probed. associated with traveling waves or wave vectors and are The slope of the phonon dispersion curve at the Brillouin technically therefore no longer phonons. “In an amor- zone center is close to 0 and the frequency ω of the phonon phous solid, the vibrational modes are no longer plane does not vary much in that region. It is this restriction of waves (and k has no meaning), but we will continue to sampling only those phonons near the Brillouin zone cen- use ‘phonons’ as a convenient abbreviation for the vibra- ter that results in the Raman bands from crystalline solids tional elementary excitations of the solid” (6). All of the being so narrow. The term “phonons,” as defined as lattice things discussed with regard to reciprocal lattice, Bril- vibrational waves, is still used for convenience in describ- louin zone, and conservation of wave vector in a lattice ing the vibrational modes of an amorphous solid because no longer apply. There is neither a real nor reciprocal the Raman spectrum so closely resembles the phonon den- lattice in an amorphous solid. One detects scattering sity of states. One might say that the entire Brillouin zone from local vibrational modes. The term “phonons,” is now sampled in an amorphous solid. as defined as lattice vibrational waves, is still used for convenience in describing the vibrational modes of References amorphous solids or glasses because the Raman spec- (1) D. Tuschel, Spectroscopy 29(9), 14–21 (2014). trum so closely resembles the phonon density of states. (2) https://www.youtube.com/watch?v=S3-M_RLXrlY. One might say that in an amorphous solid the entire (3) P.M.A. Sherwood, Vibrational Spectroscopy of Solids (Cam- Brillouin zone is now sampled. “In an amorphous solid, bridge University Press, London, 1972), p. 4. contributions from the entire phonon density of states (4) R. Tilley, Crystals and Crystal Structures (Wiley, Chichester, appear in the first-order infrared and Raman spectra” 2007), pp. 25–28. (6). The short-range order and distribution of chemical (5) S.F.A. Kettle and L.J. Norrby, J. Chem. Educ. 67, 1022– bond interactions are apparent in vibrational spectra. 1028(1990). All of the variations in bond angles and lengths in the (6) R. Zallen, The Physics of Amorphous Solids (Wiley, New York, formula units of the solid manifest themselves in the 1983), p. 270. Raman spectrum such that the bands then appear very (7) J.E. Smith, Jr., M.H. Brodsky, B.L. Crowder, M.I. Nathan and A. broad from the contributions of all of these modes. Pinczuk, Phys. Rev. Lett. 26, 642–646 (1971). The Raman spectrum of an amorphous solid resembles the phonon density of states for the crystalline form of the David Tuschel is a Raman applications material. It is for that reason that it is stated that the entire manager at Horiba Scientific, in Edison, New Jersey, where he works with Fran Adar. David is Brillouin zone and not just the center is being sampled in sharing authorship of this column with Fran. He the Raman spectrum of an amorphous or glassy solid. The can be reached at: [email protected] 1971 publication by Smith and coworkers (7) demonstrates the good agreement between the Raman spectrum of amorphous Si and the calculated phonon density of states of crystalline Si. Very broad bands appear in the first order Raman spectra of amorphous solids because of the absence of long-range translational symmetry and a corresponding For more information on this topic, please visit: www.spectroscopyonline.com/adar reciprocal lattice.

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