Measurement of Raman scattering cross-sections

In Raman scattering experiments, the measurable value is the scattering efficiency orrate S, which is defined as the power ratio of scattered light P_s to incident light P_l per length L in solid angle Ω (Fig. 2.10):

S = P_s

L·P_lΩ . (2.35)

2.6 Measurement of Raman scattering cross-sections 23

P

P

V

dW

j

V

Figure 2.10: Measurement of Raman scattering efficiency. Vef f is the effective scattering volume.

The differential scattering rate per unit cubic angle is measuredoutsidethe crystal.

Since Raman scattering efficiency relates to Raman tensor directly, by measuring Ra-man scattering efficiency one can obtain the RaRa-man polarizability. Notice that in RaRa-man experiments using photon counting electronics, one actually measures the scattered pho-ton rate outside the crystal, so that the incident (scattered) light transmission coefficients (Tl(s)) and absorption coefficients (αl(s)) should be taken into account. The measured scattering rate S⁰ in solid angle ∆Ω⁰ outside the crystal relates to the Raman tensor by [Men85, Soo87]

S⁰ = T_sT_lP_lω_s³[1−e^{−(αl+αs)L}](1 +n)∆Ω⁰

2c⁴(α_s+α_l)η_sη_lM ω₀v₀ |eˆs.R(q).eˆl|² , (2.36) where M is the reduced mass of PC, n the phonon occupation number; η_l(s), ω_l(s) the refractive index, frequency of the laser (scattered) light, respectively.

According to the expression of Eq. 2.36, the absolute scattering cross section can be calculated if all the components in the system are known. However, most experimental measurements of absolute cross sections in semiconductors employ the sample substitution method, in which the calibration is carried out by comparing the sample with a standard known scattering efficiency. The advantage of substitution method lies in the fact that the geometry measurement errors are cancelled. We applied the sample substitution method in our experiments. A pure bulk Si sample was used as a reference. We also referred to the known intrinsic RRS data of high-purity GaAs for our measurement calibration.

A rough estimate of the order of scattering cross-section (non-resonance) can be made:

it is 10⁻³⁰ m² for a single atom and about 10⁻⁴ −10⁻⁵ m² for a cubic meter of crystal [Hay78].

Chapter 3

Experimental

3.1 Set-up

The experimental set-up is sketched in Fig. 3.1.

Ar laser⁺

Prism-mono-chromator Dilor- triple

spectrometer

Immersion cryostat Computer

l/2plate

TSapphire laser

i-M 1

M 2 Color filter

M 3

M 4

M 5

O 1 O 2

L 1 LN

-CCD

2

S 1 S 2

M 0

M: Mirror O: Objective S: Stop L: Lens

Lyot

Sample

Figure 3.1: Experimental set-up.

25

The main components are explained in the following:

Laser: a self-made Ti³⁺-Sapphire laser ¹ was used as a tunable excitation source. A commercial two-quartz-plate Lyot-filter was inserted in the cavity for monochromatic mode tuning. In the wavelength range 720-890 nm, one of the cavity end-mirrors is high reflective while the other, i.e., the output coupling mirror has 2% transmission efficiency. The stable tuning wavelength range is 780-870 nm (1.588-1.424 eV). The pumping source is Ar⁺laser, operating in all-lines mode. Typically, 10-100 mW power of Ti³⁺-Sapphire laser was used in the experiment.

Spectrometer: in Raman scattering, the elastically scattered light is normally 4-6 orders of magnitude stronger than the Raman signal meanwhile the frequency separation between the laser light and scattered light is fairly small compared to the laser frequency (about 1%

of the laser frequency). Thus Raman scattering experiments require a spectrometer with an excellent stray light rejection capability. A 0.6 m triple grating spectrometer (Dilor) was employed in the experiments. The stray light rejection ratio is 10⁻⁴ −10⁻⁶. The subtract-mode of the spectrometer was utilized for optimal flux of the scattering light. For the resolution in such a mode, the full width of half maximum (FWHM) of a laser line in spectrum is about 2 cm⁻¹ with the slit being 150 µm wide.

Cryostat: a self-made immersion cryostat was installed to carry out measurements at low temperature. When the chamber is filled with liquid helium and pumped to a low pressure level, the superfluid helium state is formed, thus the temperature can be kept at about 2 K.

Detector: to record the spectra in a multi-channel manner, a commercial liquid-nitrogen-cooled charge-coupled-device (CCD) (Wright Company) was utilized as the pho-ton detector. The noise level of the CCD detector is less than 2 counts/second per pixel.

λ/2 plates: in order to measure RRS cross-sections in different configurations, a λ/2 plates were inserted in the optical arrangement to adjust polarization directions of the light.

For the consideration of light polarizations, notice that, to pump the Ti³⁺-Sapphire laser, a pumping light with polarization direction parallel to the plane of paper is demanded in this experiment. However, the out-put light of Ar⁺ laser has a polarization direction perpendicular to the plane of the paper. So a mirror set “M0” (see Fig. 3.1) was used to switch the polarization of the pumping light to the desired direction. Also notice that, the polarization direction of the scattered light detected by the spectrometer, however, is fixed, parallel to the slit-parallel direction, due to that fact that the gratings inside the spectrometer are much more sensitive to the light polarized in this direction than the light polarized in the slit-vertical direction.

Sample: commercial GaAs (Wacker-Chemitronic GmbH, named as 21412) was used in the experiments. The GaAs wafer has a (001) surface and the layer is 450±50µm thick. It is partially compensated GaAs with shallow donors and acceptors. It is p-type GaAs and the acceptors are mostly carbon. The impurity concentration: ND+NA∼10¹⁶ cm⁻³. The density of filled acceptors at 2 K is about 6×10¹⁵ cm⁻³, estimated based on the comparison

1It was designed by Dr. M. J. Gregor and P. G. Blome. Many thanks for the help in using this laser.

3.2 Procedure 27