Disorder-Induced Double Resonant Raman Process in Graphene

Joaquin Rodriguez-Nieva (Ph.D. student 2011-2016) discusses some of his research.
by Joaquin Rodriguez-Nieva
Updated Apr 17, 2018 (1 Older Version)chevron-down
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Disorder-Induced Double Resonant Raman Process in Graphene
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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 ID/IG saturates as a function of defect concentration.


MOTIVATION

Figure from Eckmann, et al, Nano Lett 12, 3925 (2012)
Figure from Eckmann, et al, Nano Lett 12, 3925 (2012)

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 IDID’???


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) ≪ ωq(~0.2eV) ≪ EL(~eV)

Yielding:

dIDRμdΩf=α216πFμ2ρνFωq(νFcELωq)2niUμ(q)2νF2ln(ωqγ)\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 IDID’

What determines the ratio ID/ID’???

Two effects mainly determine ID/ID’:

  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 ∝ EL

Fig. 2. Laser energy dependence of the integrated Raman intensity ratio  I🇩/I🇬 between the D and G bands obtained from our model (solid line), and experimental points from Ref. [2]. The dashed line indicate the frequently used I🇩∝E🇱⁻⁴ fit.
Fig. 2. Laser energy dependence of the integrated Raman intensity ratio I🇩/I🇬 between the D and G bands obtained from our model (solid line), and experimental points from Ref. [2]. The dashed line indicate the frequently used I🇩∝E🇱⁻⁴ fit.

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

ID/ID’ ∝ (ωq≈κ/ωq≈Γ)3

G-Band:

IGEL4I_{\mathrm{G}} \propto E^4_{\mathrm{L}}

Edge-Induced D Band [3]:

ID=α2λKνF2c2ELωq2νFLeAln(ωq2γ)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

Fig. 3. Dependence of the integrated D band intensity on defect concentration nᵢ as obtained within our model (solid line), and experimental points of Ref. [4].
Fig. 3. Dependence of the integrated D band intensity on defect concentration nᵢ as obtained within our model (solid line), and experimental points of Ref. [4].

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:

γ=γd+γep{γep>γdIDniγep<γddIDdni=0\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:

γep ~ 15 meV

γd[meV]niU02EL2(νF)210ni[1012cm2]\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 ID/ID’ 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)

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