Optical Study of Lattice Vibrations

Abstract

In this chapter we discuss the optical study of lattice vibrations.

Problems

22.1

How can the ratio of the Stokes to anti-Stokes intensities in the Raman scattering spectra of optical phonons be used to determine the lattice temperature of a 3D solid?

22.2

The ratio of the Stokes Raman intensity (phonon emission) to the anti-Stokes Raman intensity (phonon absorption) is given by the Maxwell-Boltzmann thermal factor:

$$\begin{aligned} \frac{I_{AS}}{I_S}=\exp (\frac{-E_{ph}}{k_BT}) \end{aligned}$$

(22.64)

• (a) Calculate the temperature of a carbon nanotube with a radial breathing mode frequency of 150 cm$^{-1}$ exhibiting an anti-Stokes/Stokes ratio of 0.25.

• (b) Calculate the temperature of the same nanotube exhibiting an anti-Stokes/Stokes ratio of 0.75.

22.3

The resonant Raman scattering process for a one-dimensional system , like a carbon nanotube , is given by the following equation:

$$\begin{aligned} I(E_{laser}) \propto \biggr | \frac{1}{(E_{laser}-E_{ii}-i\varGamma ) (E_{laser}\pm E_{ph}-E_{ii}-i\varGamma )} \biggr |^2 \end{aligned}$$

(22.65)

where $E_{laser}$ is the laser energy, $E_{ii}$ is the resonant electronic transition energy in the nanotube, $E_{ph}$ is the phonon energy, and $\varGamma$ is the linewidth of the resonance. The $+$ and − signs correspond to anti-Stokes and Stokes processes, respectively.

• (a) Plot both the Stokes and anti-Stokes Raman intensity profiles (I vs. $E_{laser}$) for the radial breathing phonon mode (180 cm$^{-1}$) and for the G-band phonon mode (1590 cm$^{-1}$) of a nanotube that has a resonant transition energy $E_{33}\,=\,$2.41eV. Assume a linewidth $\varGamma \,=\,$5meV.

• (b) Perform another set of calculations assuming $\varGamma \,=\,$17meV and compare the results in (a) and (b).

22.4

At room temperature the Stokes and anti-Stokes Raman intensities of a phonon mode at 150 cm$^{-1}$ are equal when taken with a 633 nm HeNe laser. This means that the nanotube is slightly off-resonance for this laser excitation energy. Using the equations from problems 24.1 and 24.2 calculate the true resonant transition energy ($E_{ii}$) of the nanotube assuming $\varGamma$ = 8meV.

22.5

Ferroelectrics (as opposed to dielectrics) are materials that have their atoms/mo-lecules all polarized in the same direction even when no external electric field is applied. That is, a ferroelectric material has a built-in non-zero fixed polarization vector $\mathbf {P}$ that is independent of any external field. Some important semiconductors like gallium nitride are ferroelectric. In this problem you will explore the consequences of such a built-in polarization. Consider a circular disc of a ferroelectric material of thickness d that is much smaller than the radius R, as shown in the figure. The built-in polarization vector is given by $\mathbf {P}\,=\,P_0$Z (Fig. 22.16).

Fig. 22.16

A circular disc of a ferroelectric material of thickness d that is much smaller than the radius R

• (a) Find the surface charge density due to the paired charges on the upper flat surface of the disc.

• (b) Find the surface charge density due to the paired charges on the lower flat surface of the disc.

• (c) Find the electric field (magnitude and direction) inside the ferroelectric disc. Hint: Use your answers from parts (a) and (b).

• (d) Find the D-field (magnitude and direction) inside the disc.

22.6

Derive the polariton dispersion relations for two-dimensional graphene , which has an optical phonon frequency of 1590 cm$^{-1}$. Sketch these dispersion relations for graphene with carrier densities of (a) $10^{12}$ cm$^{-2}$ and (b) $10^{13}$ cm$^{-2}$.

Fig. 22.17

Room temperature polarized reflectance spectra of single crystal La$_2$CuO$_4$ (Taken from Eklund et al., Journal of the Optical Society of America B, vol. 6, pp. 389 (1989).)

22.7

Based on the reflectance spectra of La$_2$CuO$_4$ , shown in Fig. 22.17, estimate the phonon energies which dominate the dielectric function in the Lyddane–Sachs–Teller relation .

22.8

Draw the Feynman diagrams for two Raman processes in which a photon is absorbed before a phonon is emitted, and write the corresponding matrix elements for the three vertices.

22.9

The optical phonon of graphene composed of $^{12}$C atoms is observed at 1590 cm$^{-1}$. Estimate the phonon frequency for graphene composed of $^{13}$C, based on the ratio of their atomic masses assuming that the force constants are the same in these two materials.

22.10

In addition to having a large Youngs modulus, carbon nanotubes can withstand a large amount of strain before breaking. The phonon modes of carbon nanotubes downshift with strain at a rate of 6.2 cm$^{-1}$/% strain, due to the weakening of the C-C bond. Under uniaxial strain, downshifts of up to 85 cm$^{-1}$ (from 1575 to 1490 cm$^{-1}$) have been observed. Estimate the amount of strain this corresponds to.

22.11

Suppose that you have a 2D superlattice sample of Si$_{1-x}$Ge$_x$/Si with a width of 10Å for both the Si$_{1-x}$Ge$_x$ quantum wells and the Si barriers on a Si$_{1-x}$Ge$_x$ substrate. Would you expect the Si–Si Raman frequency to be upshifted or down shifted relative to bulk Si? Why?

22.12

A crystal of a certain alkali halide has a static dielectric constant $\varepsilon (0)\,=\,5.9$. Its non-dispersive dielectric constant in the near infrared is $\varepsilon \,=\,2.25$. The reflectivity of the crystal becomes zero at a wavelength of 30.6$\,\upmu$m.

• (a) Calculate the longitudinal and transverse phonon frequencies at k $=$ 0. Express the results in eV, Kelvin, and $s^{-1}$ (angular frequency).

• (b) Using the results in (a) estimate the force constant $\kappa$ for the TO phonon mode assuming only nearest neighbor interactions .

• (c) From the splitting of the LO and TO phonon frequencies ($\omega _\ell - \omega _t$), find the magnitude of the lattice polarization contribution to the dielectric constant.

• (d) Plot the reflectivity as a function of wavelength.

• (e) Using standard tables in the literature (e.g., Kittel), identify the alkali halide.

22.13

• (a) If 2 $\sim$ eV light is incident on a semiconductor and is scattered by an angle of 60$^\circ$, what is the wave vector of the phonon that is generated?

• (b) What is the longest phonon wave vector that can be generated in this semiconductor by 2$\sim$ eV light in a Raman process?

• (c) Write an expression for the ratio between the Stokes (emission) and anti-Stokes (absorption) intensities for phonon ($\omega _q$) emission and absorption by the Raman process at room temperature $T\,=\,300$ K.

• (d) Why is the Stokes process in part (c) more intense at room temperature for a 50 meV phonon?

• (e) In an inelastic electron scattering process (electron energy loss spectroscopy), is the ratio of the Stokes to the anti-Stokes intensity the same or different for a 50 $\sim$ meV phonon with the same wave vector as in part (c)? Why?

• (f) Would electron scattering or light scattering be more sensitive to probing surface oxide formation (of a few monolayers) on a semiconductor surface?

Fig. 22.18

a Phonon dispersion diagram of graphite. b Band structure of graphite. c Double resonance Raman effect of graphite

22.14

The Raman spectra of Graphite exhibits three main features: (i) the G-band feature ($\omega _G\,=\,1580$ cm$^{-1}$) that comes from the $\varGamma$ point degenerate longitudinal optical (LO) and the in plane transverse optical (iTO) modes, (ii) the disorder-induced D-band ($\omega _D\,=\,1200-1400$ cm$^{-1}$) that comes from the LO phonon branch close to the K point, and (iii) its overtone, the $G^{\prime }$-band ($\omega _{G^{\prime }}\sim 2\omega _D=2400-2800$ cm$^{-1}$), which is a Raman process that involves two D-band phonons. The phonon diagram is shown in Fig. 22.18a.

• (a) Explain why the G-band and $G^{\prime }$-band appear in the Raman spectra of a perfect graphite crystal (HOPG - highly oriented pyrolytic graphite) while the D-band is observed only in defective graphitic materials.

• (b) The G-band scattering is a first-order Raman process, while the D-band and $G^{\prime }$-band scattering are second-order Raman processes. For the D-band, one of the scattering processes is elastic due to interaction of the electron/hole with a lattice defect. Draw one Feynman diagram for the G, D and $G^{\prime }$ scattering processes (3 diagrams in all).

• (c) (Optional) Graphite is a semi-metal since the valence and conduction band meet at the K point [see Fig. 22.18b]. Therefore, a resonance Raman effect is observed for electrons and phonons close to the K point. The reason why the second-order D and $G^{\prime }$ bands have enough Raman cross section to be visible in the Raman spectra with an intensity comparable to the first-order G-band is the resonance nature of their scattering processes, that involve two resonance processes, where not only the incident or scattered photons are associated with real electronic transitions , but also one of the intermediate scattering states, mediated by phonons (or by the defect in the case of the D-band), induces also an electronic transition between two real electronic states. One of these effects is illustrated in Fig. 22.18c. This process is called a double resonance process. Based on the various possible double resonance processes and on the phonon dispersion shown in Fig. 22.18a, explain why the D-band for defective graphite materials is composed of three peaks, where the intermediate frequency peak has twice the intensity of the lowest and highest frequency peaks (consider only Stokes scattering processes). Draw the Feynman diagrams for the possible Stokes processes of the D-band spectra.