In the Laboratory
The Second-Order Raman Spectrum of 13C Diamond: An Introduction to Vibrational Spectroscopy of the Solid State Mikkel Nissum Department of Chemistry, Odense University, DK-5230 Odense, Denmark Elizabeth Shabanova CISMI, University of Copenhagen, DK-2100 Copenhagen, Denmark Ole Faurskov Nielsen* Department of Chemistry, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark;
[email protected] Raman spectroscopy of the solid state differs markedly from Raman spectroscopy of liquids and gases in that the evaluation of spectra of the solid state requires knowledge of solid-state physics. Therefore, university students are generally introduced to Raman spectroscopy in elementary spectroscopy courses by considering only the gas and liquid phases, and are not familiar with the interpretation of Raman spectra of the solid state. This area, however, is becoming increasingly important. New solid-state materials are continuously being produced for use in computer technology, in lasers, and as catalysts in oil refining, to mention only a few areas. Raman spectroscopy has been shown to be an indispensable tool in the characterization of these materials (1). As a consequence we feel inspired to show that a short introduction to vibrational spectroscopy of the solid state allows an understanding of the main features in Raman spectra of the solid state. We introduce the concept of dispersion curves and show how they can be used to explain the spectra. A focus will be placed on the second-order spectra because these contain a wealth of features that would be new to someone not familiar with solid-state vibrational spectroscopy. The second-order spectra of crystals correspond to overtone and combination bands in gas and liquid states. Reports on aspects of vibrational spectroscopy of the solid state have appeared earlier in this Journal. Excellent papers by Carter (2) and Kettle and Norrby (3) describe how symmetry can be used to approach the spectrum of the solid state and the concept of the Brillouin zone, to which we shall return. These papers are highly recommended for further insight into the area of vibrational spectroscopy of the solid state. For a full account on the present topic we refer to the books by Kittel (4 ) and Decius and Hexter (5). We have chosen 13C diamond as an illustrative example. Diamond has attracted a lot of attention owing to its extreme physical properties (6 ). It has strikingly high thermal conductivity (7 ) and is the hardest material known (8). Development of efficient ways to produce diamonds synthetically (9, 10) has made it possible to vary the isotopic concentration of 12C and 13C; this has led to the discovery that pure 13C diamond is even harder than natural diamond (8). These interesting properties have opened pathways to numerous applications of diamond in the areas of microelectronic devices, coatings, and optics (9). Raman spectra of diamond have recently been reported in this Journal (11, 12) but there was no attempt to interpret them in detail. Several reports of Raman spectra of diamond existing elsewhere describe spectral features thoroughly (13–
Figure 1. Photo and schematic drawing of the 13C diamond sample with crystal planes shown.
16 ). To understand these accounts, however, more than basic knowledge of solid state physics and spectroscopy is required. For those unfamiliar with the basics of Raman spectroscopy, we refer to accounts given in this Journal (17–21). Experimental Procedure The 30 mg diamond, kindly provided by J. P. F. Sellschop, University of Witwatersrand, South Africa, was pyramidal in shape with four {111} faces and a (100) base (Fig. 1). It had a greenish color due to the catalyst used in the synthesis (Ni, containing a few percent Ti). A low-pressure chemical vapor deposition method (9), followed by a highpressure, high-temperature gradient process (10) is employed to grow diamond with a high 13C content. Raman spectra were acquired using the 514.5-nm line of a Spectra-Physics argon ion laser model 2017. The scattered light was collected in a 180° configuration (backscattering) in a Dilor XY800 spectrometer with multichannel detection. The spectrometer consisted of a double monochromator in subtractive mode followed by a spectrograph and a 1152 × 298-pixel CCD camera. High quality, background-free Raman spectra were thus obtained with a low laser power of 50 mW at the sample and high resolution (1.5 cm1 spectral slit width). Acquisition times were only 2 s for first-order and 30 s for second-order spectra. Spectra were insensitive to the orientation of the sample. Results and Discussion
General Aspects of Vibrations in Crystals The vibrational modes in a crystal are described in terms of phonons. Phonons are the quanta of energy in elastic waves corresponding to the possible vibrational modes of a crystal,
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in analogy with the photon, which is the quantum of energy in electromagnetic waves. A phonon is characterized with its wave vector K. The length of the wave vector is equal to 2π/ λ, where λ is the wavelength. The direction of K can be either parallel or perpendicular to the particle motions, giving rise to longitudinal and transverse phonons, respectively (4, 22). _ The phonon energy is hc ω_ K, where h is Planck’s constant, c is the speed of light, and ω K is the frequency representation used in Raman spectroscopy (units: cm1), again just as for photons. Phonons are named either optical or acoustical. Long-wavelength optical phonons can be observed by optical spectroscopy, whereas long-wavelength acoustical phonons correspond to sound waves. Hence the names optical and acoustical phonons. The frequency (energy) of the phonon depends on the length and _direction of the wave vector. Dispersion curves that relate ω K to K can be established. The _ exact relation between ω K and K depends on the masses and the elastic constants of the crystals. Typical dispersion curves for optical and acoustical phonons of a linear lattice with two atoms in the unit cell and the cell length a are shown in the right part of Figure 2. The range of values of K physically significant for phonons is given by π/a ≤ K ≤ π/a. This range of K is referred to as the first Brillouin zone (3) and we need to look only at values of K in this zone, as values from outside merely reproduce motions of atoms already described by values of K within these limits. In fact, owing to symmetry around K = 0, only the range 0 ≤ K ≤ π/a needs to be considered, as shown in Figure 2. The wave vectors K = 0 and K = π/a are referred to as the zone center and zone boundary, respectively. At infinitely long wavelength K (= 2 π / λ) is zero. At the zone center, the acoustical phonon branch has a zero frequency value, whereas this is not the case for the optical branch. For diamond, dispersion relations similar to that in Figure 2 will be found for each direction in the crystal. However, they all show the same frequency value at K = 0 (5, 23). Now, the Raman scattering process is governed by conservation of energy: _
_
_
hc ω – hc ω ′ = ±hc ω K and conservation of linear momentum or wave vector:
Figure 2. In the right part is shown typical phonon dispersion curves for a lattice with two atoms in the unit cell. In the left part is shown density-of-states curves for both the optical and acoustical phonon branches.
a 180° scattering geometry k k' k = 2π/λ, λ = 5145 Å
k − k' ≈ 2×2π/(5145 Å) ≈ 0.0024 Å−1
b Two phonon process Kj
Ksum ≈ 0 Ki
Figure 3. (a) Illustration of the applied scattering geometry and evaluation of the incoming and scattered wave vectors. (b) The K ≈ 0 criterion for a scattering process involving two phonons.
k – k′ = ±K _ _ _ where ω, ω′ and ω K are the frequencies (cm1) for incoming
light, scattered light, and the created (+) or destroyed () phonon in the crystal, respectively, and k, k′, and K are the corresponding wave vectors. Considering our Stokes Raman experiment with backscattering geometry and a laser wavelength of 514.5 nm, the created phonons in the crystal will have magnitudes of ~0.0024 Å1, owing to the wave vector conservation law (Fig. 3a). When this value is compared to the much larger first Brillouin zone limit for 13C diamond of K = π/a ≈ 0.88 Å1, with a ≈ 3.57 Å (24 ) it becomes clear that only phonons in the zero range of the first Brillouin zone need to be considered in the Raman scattering process involving one phonon. With this K = 0 criterion in hand we expect only one frequency to be found in the Raman spectrum of diamond, _ namely the value ω 0 found for the optical branch at K = 0 (Fig. 2). This band corresponds to the carbon–carbon stretch vibration, and is found at 1331 cm1 in 12C diamond (11). In ordinary molecular Raman spectroscopy terminology this is the fundamental transition corresponding to the C–C 634
stretch. In the language used in spectroscopy of the solid state this is the first-order spectrum. The K = 0 restriction does not apply for inelastic_neutron scattering, and the full dispersion curves relating ω K to K can be established experimentally using inelastic neutron scattering (3, 4, 23).
Carbon-13 Content of the 13 C Diamond Sample Investigated First-order spectra of crystals are expected to show very sharp bands, because only the frequency value corresponding to the wave vector K = 0 is observed. This is not strictly correct, but the change in frequency with wave vector of the dispersion relation in Figure 2 is negligible for the very small wave vectors of importance for Raman spectroscopy. Thus firstorder spectra are expected to appear as infinitely sharp bands (lines) in a Raman spectrum. However, the Raman spectrum is affected by the isotopic composition of the diamond, since positions and widths of the K = 0 line depend on the con-
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Raman shift / cm−1
1270
Raman shift / cm−1
1320
a 1300
1941
2088
2237
2564
1295
1340
1282
Intensity (arb u)
FWHM 2.9 cm-1
2393
2364
Intensity (arb u)
In the Laboratory
1280
2200
1600
Raman shift /
1000
cm−1
Figure 4. Raman spectrum of 13C diamond acquired with 514.5-nm excitation wavelength and 2-s acquisition time in the [100] direction. The 1800–2800 cm1 region is also shown with 30-s acquisition time on the same scale. In the inset, the line width of the fundamental 13C diamond mode at 1282 cm1 is given in an expanded frequency scale.
centration of 12C and 13C (15). Accordingly, 13C content can be determined from the spectrum, and this was done using the position and width of the single line in the first-order spectrum (Fig. 4). Relations between these values and 13C content (Fig. 5) have already been established through an examination of the Raman spectra of diamonds with known isotopic composition (15). The first-order Raman spectrum obtained of the 13C diamond sample is shown in Figure 4. A frequency of 1282 cm1, compared to 1331 cm1 in natural diamond, and the full width at half maximum (FWHM) of 2.9 cm1 indicate 99–100% 13C enrichment of our diamond sample. The small FWHM is in accordance with the expectation for a first-order transition. The procedure is valid only for diamonds retaining their cubic structure because compressive distortion from the cubic structure will increase both the Raman frequency and the line width (25).
The Second-Order Spectrum We now focus on the second-order 13C diamond Raman spectrum shown in Figure 4 in the region 2600–1900 cm1. It has a much more complicated appearance than the single line of the first-order spectrum. However, by examining the concepts of vibrational spectroscopy of the solid state this can be explained by taking into account the vibrational density of states. This is illustrated in Figure 2, where the density of states is plotted to the left in the diagram. The density of states _ per unit interval of frequency g (ω K) is inversely proportional to the slope of the dispersion curve (5): g ωK ∝
d ωK dK
1
so whenever the dispersion curve is horizontal, the density of states is high. Inspection of Figure 2 shows that high densities of states are observed at both the zone center and the zone
FWHM / cm−1
2800
8
b
4
0 0
20
40 13C
60
80
100
concentration (%)
Figure 5. Dependence of (a) Raman shift and (b) line width of the firstorder spectrum of 13C diamond as a function of 13C concentration. • and 䊐 represent values from two independent experiments. Reproduced from ref 15 by permission of the American Physical Society.
boundary for the optical phonon branch, where the acoustical branch shows a maximum only at the zone boundary. Figure 6 shows the dispersion relations for 13C diamond in the [100] and [111] directions for longitudinal acoustical (LA), transverse acoustical (TA), longitudinal optical (LO), and transverse optical (TO) phonons. The 13C dispersion curves in Figure 6 were produced from the natural abundance curves (23) weighting the frequencies by a factor of (12/13) 1/2_. This procedure is justified by the (mass)1/2 dependence of ω K for a linear lattice with identical atoms (4, 5). Only a nearestneighbor interaction in a harmonic potential approximation is taken into account. The natural-abundance diamond is approximated as pure 12C and the diamond in the present study as pure 13C. A plotting of the densities of states for the diamond dispersion curves in Figure 6 would give a very complicated pattern, but to understand the second-order spectra it is sufficient to remember the results from Figure 2 concluding that the maximal densities of states are found at the zone boundaries and at the zone center (for optical phonons). The K ≈ 0 criterion still applies when two phonons are involved in the Raman scattering process. However, it may now be written as K sum = K i + K j ≈ 0 where indices i and j can represent either same or different
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phonon modes. This means that in the two-phonon process we are no longer restricted to the near-zero range of the first Brillouin zone. All K values are allowed as long as the sum of the wave vectors is close to zero, as illustrated in Figure 3b. This can occur even for the largest wave vectors at the zone boundary. As shown in Figure 3b, only the sum of two nearly equal and opposite-directed wave vectors is needed. This is possible for many randomly oriented sets of vectors in the Brillouin zone, and no dependence on crystal orientation is found for second-order spectra. Thus in principle all sums of the frequencies in the dispersion curves can be observed. The contribution to this sum will be a maximum for frequency regions with a maximal number of vibrational states. These regions are found at the zone center and the zone boundaries. Table 1 gives an assignment of the second-order spectrum in terms of the vibrational frequencies found for the zone center and zone boundaries of the dispersion curves in Figure 6. The discrepancies between the observed frequencies and the ones estimated from the dispersion curves are within the accuracy of our procedure, considering the approximations made in constructing the dispersion curves, the estimated errors in the frequencies from the neutron scattering experiments (23), and the accuracy with which frequencies could be determined in the Raman spectrum. In general, the FWHM values are larger for the second-order spectra. This has to do with the shapes of the density-of-states curves. They change gradually from higher to lower values. Consequently the second-order bands are much broader than bands in first-order spectra. In a molecular spectroscopist’s terminology, the second-order spectra are called overtones and combination bands. However, in this context it is important to note that the observed second-order spectra are not only obtained from the fundamental transitions, but all frequencies in the Brillouin zone can be involved. Conclusion We have introduced the concepts of vibrational spectroscopy of the solid state to readers with no background in this field and shown how a short introduction is sufficient to understand the most important features in Raman spectra of the solid state, where 13C diamond was chosen as example. Raman spectroscopy of the solid state differs from that of gases and liquids but is by no means more complicated. The most striking difference is found in the description of the second-order spectrum. For gases and liquids, overtones and combination bands can be assigned in terms of the fundamentals taking mechanical anharmonicity into account. In the crystalline state, the second-order spectrum includes combinations of vibrational modes inactive in the first-order spectrum. However, the observed bands are easily explained taking the densities of states in the Brillouin zone into account. Acknowledgments We wish to thank The Danish National Science Research Council for general funding, and Carlsberg, Denmark, for a postdoctoral fellowship (to MN, grant 970114/20-1306). We are grateful to J. P. F. Sellschop, University of Witwatersrand, South Africa, for the supply of the 13C diamond and to P. W. Jensen, Odense University, Denmark, for providing the Raman facilities. 636
Figure 6. Dispersion curves for 13C diamond in the [100] and [111] directions. A refers to acoustical and O to optical phonons. L means longitudinal and T means transversal phonons. The point indicated by 䊐 is the natural abundant Raman frequency determined from the first-order spectrum (24), weighting the value by (12⁄13)1/2. Regions with high density of states are marked with arrows. Modified reproduction from ref 24, by permission of the International Atomic Energy Agency, Vienna. Table 1. Frequencies from Dispersion Curves for Intensity Maxima in the Second-Order 1 3 C Diamond Raman Spectrum Raman EstiShift/ mated a/ 1 cm cm1 2564
2558
Assignment 2 × LO(K 艐 0); 2 × TO(K 艐 0); LO(K 艐 0 ) + TO(K 艐 0)
2393
2387
2 × LO(K 艐 π/a, [111])
2364
2356
TO(K 艐 π/a, [111]) + LO(K 艐 π/a, [111])
2237
2229
LO(K 艐 π/a, [111]) + TO(K 艐 π/a, [100])
2088
2071
2 × TO(K 艐 π/a, [100])
1941
1935
TO(K 艐 π/a, [111]) + TA(K 艐 π/a, [100])
a Calculated
from the frequencies corresponding to regions of high density of states (Fig. 6) listed in the Assignment column.
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