Joaquin Rodriguez-Nieva, Millie S. Dresselhaus
Ph.D. student: 2011-2016

ABSTRACT

We studied the Double-Resonant (DR) Raman scattering process in disordered graphene, and showed the dependencies of the D and D’ band Raman intensities on laser energy, defect concentration and electronic lifetime. Several important features, which were contrasted with experiments, were discussed:

1. the laser energy dependencies of both the D and D’ band intensities are sensitive to the scattering potentials, thus providing detailed information about defects.

2. the D and D’ bands show a different laser energy dependence.

3. when the defect collision rate becomes faster than the electron-phonon collision, the ratio I_D/I_G saturates as a function of defect concentration.

MOTIVATION

Numerous theoretical and experimental works on the DR process are available, yet some of the most interesting and potentially useful questions remain to be answered:

1. distinguishing signatures of the different types of defects on the Raman spectra.

2. edges vs. point defects?

3. laser energy dependence?

4. D vs. D’ bands

5. Why I_D≫ I_D’???

INTEGRATED RAMAN INTENSITY

The Raman intensity is concentrated for backscattering of the photoexcited electron-hole pair:

Momentum transfer given by:

Raman Intensity can be integrated analytically using:

γ(~10meV) ≪ ω_q(~0.2eV) ≪ E_L(~eV)

Yielding:

$\frac{dI_{\mathrm{DR}}^{\mu}}{d \Omega_{\mathrm{f}}} = \frac{\alpha^2}{16 \pi} \frac{F^2_{\mu}}{\rho \nu_\mathbf{F} \omega_\mathbf{q}} \left(\frac{\nu_\mathrm{\mathbf{F}}}{c} \frac{E_L}{\omega_\mathbf{q}}\right)^2 \frac{n_{\mathrm{i}}|U_\mu(\textbf{q})|^2}{\nu^2_\mathrm{\mathbf{F}}}\mathrm{ln}(\frac{\omega_{\mathbf{q}}}{\gamma})$

ROLE OF THE SCATTERING POTENTIAL

Experiments typically show I_D≫ I_D’

What determines the ratio I_D/I_D’???

Two effects mainly determine I_D/I_D’:

1. Long wavevector scattering

2. Suppression of Backscattering

By (indirectly) measuring λ, we can identify the nature of the defects.

LASER ENERGY DEPENDENCE

Several laser energy dependencies of the integrated D and D’ band intensities are obtained in experiments... why?

a) Raman spectra is defect sensitive: potential probed at backscattering
b) γ ∝ E_L

Dispersive behavior of D and D’ bands explain their different laser energy.

I_D/I_D’ ∝ (ω_q≈κ/ω_q≈Γ)³

G-Band: $I_{\mathrm{G}} \propto E^4_{\mathrm{L}}$

Edge-Induced D Band [3]:

$$$I_D=\alpha^2 \lambda_K \frac{\nu^2_F}{c^2} \frac{E_{\mathrm{L}}}{\omega^2_{\mathbf{q}}} \frac{\nu_FL_e}{A} \mathrm{ln} (\frac{\omega_{\mathbf{q}}}{2\gamma})$

DEPENDENCE ON DEFECT CONCENTRATION

Saturation of the D band Intensity with defect concentration is controlled by the electronic lifetime due to electron-phonon (ep) and electron-defect (d) scattering:

$\gamma = \gamma^{\mathrm{d}} + \gamma^{\mathrm{ep}} \begin{cases} \gamma^{\mathrm{ep}}>\gamma^{\mathrm{d}} & \to I_{\mathrm{D}} \propto n_{\mathrm{i}}\\ \gamma^{\mathrm{ep}}<\gamma^{\mathrm{d}} & \to \frac{dI_{\mathrm{D}}}{dn_{\mathrm{i}}} = 0 \end{cases}$

Typical Values:

γ^ep ~ 15 meV

$\gamma^{\mathrm{d}} [\mathrm{meV}] \approx \frac{n_i|U_0|^2E_{\mathrm{L}}}{2(\hbar \nu_F)^2} \sim 10n_\mathrm{i} [10^{12} \mathrm{cm}^{-2}]$

CONCLUSIONS

Raman Spectroscopy can provide detailed information about the elastic scattering potential due to impurities, allowing to identify the nature of defects by using the laser energy dependence of the D and D’ bands, or the I_D/I_D’ ratio. Several experiments can be used to test our predictions, such as correlations with transport measurements or doping effects. Further computational work is required to model more accurately the scattering potential introduced by the different types of defects.

REFERENCES

JFRN, et al., PRB 90, 235410 (2014)