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A Quantitative Analysis of Raman Selection Rules in a Zincblende Crystal System D. A. Iozzo, A. N. Yaeger

Department of Physics Faculty Mentor: B. A. Weinstein Mentors: G. P. Lindberg, N. T. Gross

Introduction

Backscattering Experiment

Backscattering Results Right-Angle Scattering Experiment

References

Right-Angle Scattering Results

Discussion

Raman scattering occurs when incident photons are scattered inelastically on a material, resulting in the absorption or emission of a phonon, a vibration within a crystal lattice. In order to determine the direction of the cleavage plane of a crystal, one can use optical phonons as identifiers for crystal orientation. A polar οptical phonon is a standing wave vibration in the crystal. When the atoms of a lattice move in the direction of the vibration it is a longitudinal optical phonon (LO), if the atoms move perpendicular to the direction of the vibration it is a transverse optical phonon (TO), see figure above. The Raman intensity is dependent on the relationship,

(1)

where is the polarization of the scattered light, is the polarization of the incident light, and is the polarizability tensor [1]. For an optical phonon with amplitude , the polarizability tensor becomes [1]:

(2) Using these tensors and the polarization of the incident and scattered light, the selection rules of the phonons can be determined. Only certain phonon vibrational modes are allowed for a specific scattering geometry. From the intensity of the phonons, we can determine which scattering geometry is present, and thus the orientation of the cleavage plane.

A sample of Gallium Phosphide (GaP) was measured at STP with an Argon Ion 488nm laser at 10mW and a Krypton 530nm laser at 35mW. The sample was mounted on a rotating base with the laser light incident orthogonally on the crystal’s (1,0,0) face. A fixed vertical polarizer was used to ensure vertically polarized incident light, and a rotating polarizer was placed in front of a J.-Y. U1000 monochromator to select the polarization of the scattered light. In a backscattering measurement, the scattered light is collected after being reflected back from the crystal, .This can be described by the incident, scattered and normalized LO phonon wave vectors:

(3)

The polarizability tensor for the LO phonon is then:

(4) In going through the different possible polarizations for and for equation (1), the LO phonon should be allowed for an incident horizontal polarization and a scattered vertical polarization. TO phonons are forbidden for all polarizations, as all possible polarizations yield zero.

Direction of atom vibration

LO Phonon TO Phonon

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Average Intensity of LO Phonon (Backscattering)

Polarization Relative Intensity Orthogonal 3814.26 45o 3335.84 Parallel 1481.56

380 400 420

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Orthogonally Polarized Backscattering LO Phonon

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Orthogonally Polarized Backscattering LO Phonon

Using equation (1) with horizontally polarized , vertically polarized , and the polarizability of the LO phonon tensor yields:

(5)

The same calculation with vertically polarized and horizontally polarized also yields a value of one and is therefore also allowed. From this, we expected that we would get a strong LO peak whenever and are orthogonal to each other and almost no LO peak when they are parallel to each other. We measured the LO peak at orthogonal, parallel, and 45o scattered polarizations. At each polarization we rotated the sample 180o in steps of 30o. We averaged the intensities for each polarization across all sample orientations. We noticed that for each polarization, the intensity would change as the sample was rotated. We measured intensities for orthogonally polarized light while rotating the sample 110o in 10o increments.

We noticed that the LO phonon intensity varied as the sample was rotated about the (1,0,0) face. This does not appear in our calculation. This phenomenon occurs due to preferred orientation, a natural property of the crystal that favors one orientation over another. When plotting the area under the intensity peak as a function of the analyzer polarization angle, it distinctively demonstrates the shape of Malus’s Law cos2θ

The same sample of GaP was measured at STP with Krypton 568nm, 647nm, 676nm laser lines at 35mW. To achieve right angle scattering the beam must be incident on the edge of the crystal and the scattered light must be collected off the sides. It is important that backscattering off of other objects is not collected with the data, so we mounted the sample on a rotating dowel in such a way that there was no material behind the edges of the sample. In a right-angle scattering measurement, light is collected from the side of the crystal, . This scattering geometry can be described with the wave vectors:

From solving equation (1) for the different possible incident and scattered light polarizations, we found that both TO and LO are allowed for vertically polarized incident light and horizontally polarized scattered light, and vice-versa. Additionally, we found that LO is forbidden for horizontally polarized incident and scattered light. We performed the right-angle measurements in the same manner as the backscattering measurements. The area under the TO peak is plotted below along with the expected area curve. The data is shifted 10o from the curve which can be explained by the polarizers being 10o out of alignment initially. The table below presents the values for the TO and LO phonon intensities after correcting for the polarizer alignment.

!ki ⊥!ks

!ki = ( 1 0 0 )!ks = ( 0 1 0 )

!qLO = ( 1 1 0 )!qTO1 = ( 1 1 0 )

!qTO2 = ( 0 0 1 )

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—

[1] B.A. Weinstein, ECE509: Raman Scattering in Semiconductors and Their Heterostructures, Physics Department, University at Buffalo. (1995)

[2] K. Mizoguchi and S. Nakashima, J. Appl. Phys., 69, 8304 (1991)

[3] C.A. Arguello, D. Rousseau, S.P.S. Porto, Phys. Rev., 181, 1351 (1969)

[4] W.H.G. Lewin, 8.02 Physics II: Electricity and Magnetism, Physics Department, MIT. (2002)

In the backscattering experiment, the TO peak was forbidden and did not appear in the data, and the LO peak was present as expected throughout the dataset. Since the crystal orientation is not completely perfect, a preferred orientation exists at which the LO phonon has a higher intensity. In the right-angle experiment, the TO peak appeared for all polarizations and had a maximum intensity for parallel polarization as expected. The LO peak should have been forbidden for the parallel polarization but did not vanish in the measurements. The LO peak increased and dropped in intensity with the TO peak. One possible reason that accounts for the persistence of the LO peak could be a strong resonance affect that would allow the LO phonon to maintain a high intensity even for forbidden modes. We attempted to avoid this by using different laser lines further away from the resonance mode, but unfortunately encountered the same result. Another possible explanation could be that our laser may not have been perfectly perpendicular on the (1,0,0) crystal face. In this case, a different scattering geometry would occur and different selection rules must be calculated.

We expected the LO phonon to disappear for the parallel polarization, which did not agree with the data. The LO phonon remained throughout the measurement but did not affect the TO peak, since we expected the TO peak to show an increase in intensity for the parallel orientation and it is reflected in the data. Although the LO peak had a greater intensity than the TO peak, the ratio between the two was minimum for parallel polarization and maximum for orthogonal polarization as predicted.

Future measurements could involve taking measurements from different faces of the sample and testing various scattering geometries. We could also check the sample orientation through x-ray crystallography in order to address the problem of the present LO phonon in forbidden modes.

Future Work

Right-Angle Scattering Intensities Corrected

Polarization Average

Intensity of TO Phonon

Average Intensity of LO Phonon

Ratio of LO to TO

Intensity

Orthogonal 89.7 246.5 2.75

45o 124.8 305.1 2.44

Parallel 185.0 335.2 1.81