Application to graphite

The example of the last section – linear bands crossing at the Fermi energy – resembles very much the electronic band structure of graphene and graphite as introduced in Chapter 5.

We can thus expect qualitatively and even quantitatively a similar behavior as found in the textbook example. Before calculating the defect induced Raman spectra I take a closer look at the graphene Brillouin zone and the possible wave vectors for doubly resonant transitions.

Figure 6.4 shows the contour plot of the graphene π orbitals within the nearest neighbor approximation discussed in Section 5.1.2. Near the K point an electron (black circle) was resonantly excited to the conduction band by the incoming photon. A second resonant tran-Figure 6.4: Double resonant scattering in graphite:

The contour plot shows the electronic band structure of graphite in the nearest neighbor tight-binding ap-proximation of the π orbitals. Scattering from the K to a K⁰ point of the Brillouin zone (white ar-row) yields a phonon wave vector which is close to ΓK and thus has the D mode frequency. Scatter-ing between to K points gives a phonon wave vec-tor around the Γ point (but still large compared to k=0).

k

M

K

k

K'

K K

K'

K'

Dmode

sition is obtained by scattering the electron to one of the K or K⁰ points in the hexagonal Brillouin zone (white arrows and gray circles), because only there the electronic bands are close enough to the Fermi level. K and K⁰ are connected by a vector pointing from Γ to K, see Fig. 6.4. Hence a phonon which scatters the electron from the neighborhood of K to K⁰ has a phonon energy close to the K point frequency. In contrast scattering from K to the same or another K point in the Brillouin zone yields a phonon wave vector and energy close to theΓpoint. In the following I will only consider the first double resonant transition.

The second resonant transition results in the Raman peak slightly above the Γpoint vibra-tion which is seen in highly disordered graphite. This peak was observed, e.g., by Vidano et al.;¹⁸ in Fig. 6.1b) it is labeled D⁰. The excitation energy dependence is much weaker for this peak than for the D mode, because of the flat phonon dispersion and the overbending in the vicinity of theΓpoint.

To calculate the frequency and the excitation energy dependence of the graphite D mode I used the tight-binding nearest neighbor approximation with a carbon-carbon interaction energy γ0= 3.03 eV and s= 0.129 [see Eq. (5.3) and (5.5) Section 5.1.2.]. To evaluate Eq. (6.1) for graphite the calculation was restricted to the irreducible domain of the graphene Brillouin zone, i.e., the triangle formed by theΓM, MK, and KΓlines. Three simplifications were introduced to reduce the necessary computer power: Only the scattering from the K to the K⁰ point of the Brillouin zone was considered as explained in the previous paragraph (the K and K⁰ point coincide in the reduced zone scheme). As the phonon dispersion I used the LO optical branch of graphene, which I modeled by simple functions (linear and sin functions) such that they represent closely those obtained from force constants and ab initio calculations.106, 163, 164 TheΓ point frequency was 1580 cm⁻¹, the ones at the M and K point 1480 and 1270 cm⁻¹, respectively. The typical overbending of the optical branch to 1620 cm⁻¹ was included as well.⁹⁶ Lastly, instead of performing the full summation in Eq. (6.1) I searched the Brillouin zone for incoming resonances and, when I found one at kkk^o, summed

∑

1

{E₁−[Ec(kkk^o)−E_v(kkk^f)]−¯hωph(kkk^f−kkk^o)−i ¯hγ_}{^¯hphω(kkk^f−kkk^o)−i ¯hγ_}

2

(6.6) over kkk^f in the irreducible Brillouin zone. The subscripts v and c refer to the valence and conduction band in the tight-binding approximation. Note that the singularities of Eq. (6.6) which cancel by destructive interference in the full summation need to be excluded explic-itly from the calculation. The last approximation – fixed incoming resonances – must be treated with care when comparing the calculated results with experiments. First, it slightly shifts the calculated maximum of the Raman peak (≈10 cm⁻¹). Second, information on the relative intensities is lost completely, because “nearly resonant” electronic transitions are

Figure 6.5: Calculated Stokes Raman spectrum of the D mode in graphite for three different ex-citation energies. The frequency shift with laser energy is clearly seen in the calculations. Note that the relative intensities calculated at different E1cannot be compared to each other (see text for details). A reciprocal life time γ =0.1 eV was used for the calculations.

1300 1400 1500

Figure 3, Thom sen and Reich, LV7334 E1 = 4 eV

E1 = 3 eV E1 = 2 eV

|K 2f,10|2 (arb. units)

Ram an shift (cm^-1)

not considered. For the D mode in nanotubes, a nice study of the dependence of the calcu-lated Raman spectra on the mentioned approximations and the used parameters was done by Janina Maultzsch in her Diploma thesis.¹⁶⁵

In Fig. 6.5 I show |K_{2 f,10}|² as a function of the phonon frequencyωph for three different laser energies. Clearly the D mode Raman spectrum is obtained by considering double res-onances in graphite. The peak shifts with increasing excitation energy to higher frequencies as found experimentally; also the spectral shape is nicely reproduced. The calculated peak width (≈20 cm⁻¹) is smaller than the experimental one (≈40 cm⁻¹), which is partly due to having fixed the incoming resonance in the calculations and partly to a too small reciprocal life time. In Fig. 6.6 I plotted the maxima of the calculated peaks as a function of excita-tion energy. Also shown in the Figure are experimental results for the D mode frequencies obtained by various groups. The agreement is found to be excellent, in particular, since all parameters were fixed during the calculation and no fitting to experimental data was per-formed. The calculated slope of the D mode’s excitation energy dependence is 60 cm⁻¹/eV, slightly larger than the experimental values ranging from 44 to 56 cm⁻¹/eV.18, 156–158 The absolute frequency of the doubly resonant peak agrees excellently as well.

Figure 6.6: Measured and calculated frequency of the D band as a function of excitation energy.

The filled squares are the frequencies calculated from the double resonant condition; the full line is a fit to the calculated data with a slope of 66 cm⁻¹/eV. Open symbols represent measure-ments by P´ocsik et al.¹⁵⁶ (open circles), Wang et al.¹⁵⁷ (open diamonds), and Matthews et al.¹⁵⁸ (open triangles). The experimental slopes are

given in the figure ^{1.5 2.0 2.5 3.0 3.5}

1300 1350 1400

this work; 60 cm^-1 Pócsik et al.; 44 W ang et al.; 47 M atthews et al.; 51

F igure 4, T hom sen and R eich, LV 7334

Raman shift D-mode (cm-1 )

Excitation energy (eV)

When I discussed the experimental findings for the D band in graphite I mentioned three peculiarities of this mode: It appears only for disordered graphite and its intensity increases with the degree of disorder, the D mode frequency depends on excitation energy, and the Stokes and anti-Stokes frequencies are different. Obviously the first two characteristics are explained by defect-induced double-resonant Raman scattering. The difference in Stokes and anti-Stokes frequency also follows from the model. In the textbook example of double resonant scattering with two linear bands I depicted a Stokes process in Fig. 6.2. For anti-Stokes scattering – creation of a phonon – at the same incoming resonance a slightly larger phonon wave vector is required to meet the double resonance condition. Consequently, a larger phonon energy is expected for anti-Stokes scattering in graphite, since the phonon dispersion bends up when going away from the K point.^{106, 164} At a photon energy E1= 2 eV I obtained a difference of≈15 cm⁻¹in Stokes and anti-Stokes frequency compared to 7 cm⁻¹as found by Tan et al.¹⁵⁵

The double resonant process also occurs for two phonons, i.e., instead of scattering the elec-tron back by an impurity it is scattered back under the emission of a second phonon. This overtone of the D mode, the D^∗ band, is expected to have twice the frequency shift with varying excitation energy, which is indeed what is observed experimentally.¹⁸ In contrast to the D mode the second order D^∗mode is not defect induced and should also be found in the Raman spectra of graphite single crystals. Single crystal measurements are not available at present, but Nemanich and Solin⁹⁶ reported a strong D^∗ peak in highly-oriented pyrolytic graphite where the D mode is very weak. Moreover, the D^∗ band in their measurements was only half as large as the first order Γ point Raman signal. The large intensity of the second order signal as compared to first order Raman scattering independently confirms the interpretation of the D and D^∗Raman peaks as due to double resonances.