3. Functionalized Graphene: Adsorbates and Heterostructures 69
3.2. Optical Properties of Partially and Fully Fluorinated Graphene
3.2.4. Electronic and Optical Properties
-16 -8 0 8 16
(a)G0W0 and TB model DOS
-1 0 1
-2 -3
-4 2 3 4
(b) G0W0 LDOS
-1 0 1
-2 -3
-4 2
(c) TB model LDOS Figure 3.8.:
Total and local density of states for the graphene super cell (16 carbon atoms, 1 fluorine impurity). (a) Total density of states from the original G0W0 data (red) and the TB model (green). (b) and (c): Local density of states of the pz orbital of the fluorine atom (red), the fluorinated carbon atom (green) and a neighbouring carbon atom (blue) from (b) the original G0W0 calculation and (c)as resulting from the fitted TB model.
3.2.3. Simulation Details
In order to model realistic samples in the TB calculations, we perform simulations of systems on the scale of micrometer consisting of 2400ˆ2400 carbon atoms. Thereby we exclusively consider the chair configuration for neighbouring F atoms. The density of states and optical conductivity are calculated using the TB propagation method [189, 190] which is based on the numerical simulation of random wave propagation according to the time-dependent Schrödinger equation as discussed in section A.1.
ab-0 1 2 3 4 5 6 7 8 9 10 0.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
-4 -3 -2 -1 0 1 2 3 4 0.00
0.02 0.04 0.06 0.08 0.10
unpaired
/ 0
(eV)
Graphene
CF 0.1
CF 0.2
CF 0.3
CF 0.5
CF 0.7
CF 0.9
CF (a)
Graphene
CF 0.1
DOS(1/eV)
E (eV)
(a) unpaired fluorination
0 1 2 3 4 5 6 7 8 9 10
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
-4 -3 -2 -1 0 1 2 3 4 0.00
0.02 0.04 0.06 0.08 0.10 (b)
paired
/ 0
(eV)
Graphene
CF 0.1
CF 0.2
CF 0.3
CF 0.5
CF 0.7
CF 0.9
CF Graphene
CF0.1
DOS(1/eV)
E (eV)
(b) paired fluorination Figure 3.9.:
In-plane optical conductivity of partially and fully fluorinated graphene with different con-centration of randomly distributed(a)unpaired or(b)paired fluorine adatoms. The density of states of graphene and CF0.1 are plotted as insets of panels (a,b) and reveal mid-gap states in the unpaired case (see the sharp peak close to the neutrality point).
sorption of the unpaired case a clear enhancement around 2.8eV [see red curve in Fig. 3.9 (a)]. This enhancement can be explained by the broken sub-lattice symmetry in the unpaired scenario. Here, huge peaks near the neutrality point in the density of states arise [see inset of Fig. 3.9 (a)]. This allows for additional electron-hole exci-tations due to transitions between these mid-gap states and the π-band saddle-point singularities which manifest as enhancements of the optical conductivity at energies around 2.8eV for small fluorine concentrations of CF0.1, CF0.2 and CF0.3 in Fig. 3.9 (a). These enhancements of the optical conductivity around 2.8eV neither appear in the light absorption of partially fluorinated graphene measured by Nairet al.[see Fig.
3.5 (b)] nor in the simulated spectra of graphene with paired fluorine adsorbates. In-deed, the experimental absorption spectrum for partially fluorinated graphene is close to the one we obtain for CF0.3 in the paired fluorination case. This leads us to the conclusion, that the fluorine atoms tend to form pairs during the fluorination process.
Additionally, magnetic measurements for partially fluorinated graphene [191] show a small concentration of local spin one-half magnetic moments (roughly, one magnetic moment per thousand of fluorine atoms). Since these moments in graphene are as-sociated to mid-gap states [170] our conclusion that most of the fluorine atoms form pairs (which have no such states and are therefore obviously nonmagnetic) seems to be in agreement with this observation. However, this issue clearly requires further investigation.
As can be seen in Fig. 3.9 (a) and (b), the general trend of σ at energies below 10eV is to decrease with fluorination. Interestingly, there is a sharp resonance around 5eV which can not be found for concentrations below 30%, butintensifies for fluorine
concentrations up to 70% and vanishes afterwards. We will see in the following that this peak results from fluorine vacancies which are not well defined for small fluorine concentrations and which will nearly vanish for high concentrations. Thus, this peak arises not before a certain threshold and vanishes towards fully fluorination.
In general, for fluorine concentrations larger than F/Cą50%, the atomic structures in the paired and unpaired cases become comparable, leading to similar optical in-plane spectra. Thereby individual peaks below 8eV are the most prominent properties of the optical conductivity for these fluorine concentrations. As mentioned above, these peaks are fingerprints of certain atomic structures.
Fingerprints of Disordered Structures
To investigate these fingerprints in more detail, Fig. 3.10 displays the results of fully and highly fluorinated graphene with structural disorder, including carbon vacancies, fluorine vacancies and fluorine vacancy-clusters4.
The common effects due to the presence of structural disorder are defect states (partially) within the electronic band gap. The exact positions of these intra-gap states are defined by the type of disorder. For example, the defect resonances in the DOS aroundE “0.78eV are due to single carbon vacancies (top row of Fig. 3.10) while the resonances around E “ ´0.17{2.45eV are due to single/paired fluorine vacancies (second row of Fig. 3.10). The excitations between these intra-gap states and the states above or below the band gap lead to narrow or broad peaks in the optical spectrum below 6.3eV. For fluorine vacancies we find a pronounced peak at about 5eV (second row of Fig. 3.10) which has already been discussed above. For fluorine vacancy-clusters, there are many different intra-gap states due to different structures forming a continuous background noise within the optical gap as it can be seen in the last row of Fig. 3.10.
Altogether, our simulations of partially fluorinated graphene show optical excitations below6.3eV, but we do not find a clearly reduced optical gap of„3eV as it has been observed in the experiment in any of the considered disorder types. Thus, we conclude that the reduction of the optical gap is not due to structural disorder alone.
Out-of-plane Optical Conductivity
Optical experiments can work at normal as well as grazing incidence and measure polarization-dependent spectra. We therefore investigate the out-of-plane optical con-ductivity along thez-direction (σzz) and compare it to the in-plane optical conductiv-ity. The dipole operator associated with σzz contains two parts: one is the electron
4The latter can additionally be separated into clusters formed by broad areas and armchair or zigzag lines of missing fluorine atoms. Here, we solely show fluorine vacancy-clusters since the main conclusion is not altered by the results from the other structures. The interested reader is referred to the original article [189].
-4 -3 -2 -1 0 1 2 3 4 5 6 7 0.00
0.01 0.02 0.03 0.04 0.05
x v acancy
=0.5%
DOS(1/eV)
E (eV)
carbon vacancies CF
0 1 2 3 4 5 6 7 8 9 10
0.00 0.25 0.50 0.75 1.00 1.25
CF carbon vacancies
/ 0
(eV) x
v acancy
=0.5%
-4 -3 -2 -1 0 1 2 3 4 5 6 7
0.00 0.01 0.02 0.03 0.04 0.05
CF 0.9
fluorine vacancies fluorine vacancies
DOS(1/eV)
E (eV)
0 1 2 3 4 5 6 7 8 9 10
0.00 0.25 0.50 0.75 1.00 1.25
CF 0.95 fluorine vacancies
/ 0
(eV)
-4 -3 -2 -1 0 1 2 3 4 5 6 7
0.00 0.01 0.02 0.03 0.04 0.05
CF 0.95
fluorine vacancy-clusters fluorine vacancy-clusters
DOS(1/eV)
E (eV)
0 1 2 3 4 5 6 7 8 9 10
0.00 0.25 0.50 0.75 1.00 1.25
CF fluorine vacancy-clusters 0.95
/ 0
(eV)
Figure 3.10.:
Left column: Atomic structure with different types of structural disorder. The red dots indi-cate fluorine adatoms. Middle and right columns: density of states and optical conductivity of fully or highly fluorinated graphene with different types of structural disorder. From top to bottom: Fully or highly fluorinated graphene with randomly distributed carbon vacancies, fluorine vacancies, and fluorine vacancy-clusters.
0 1 2 3 4 5 6 7 8 9 10 0.00
0.02 0.04 0.06 0.08 0.10 0.12 0.14 c
unpaired
zz
/ 0
(eV) Graphene
CF 0.1
CF 0.2
CF 0.3
CF 0.5
CF 0.7
CF 0.9
CF
(a) unpaired fluorination
0 1 2 3 4 5 6 7 8 9 10
0.00 0.02 0.04 0.06 0.08 0.10
d
paired
zz
/ 0
(eV) Graphene
CF 0.1
CF 0.2
CF 0.3
CF 0.5
CF 0.7
CF 0.9
CF
(b) paired fluorination Figure 3.11.:
Out-of-plane optical conductivity of partially and fully fluorinated graphene with different concentration of randomly distributed(a) unpaired or (b)paired fluorine adatoms.
hopping between the carbon atoms which have different z-coordinates and the other is the hopping between carbon atoms and absorbed fluorine directly above or below.
This results in a zero optical conductivity along the z-direction in pristine graphene over the whole spectrum, since there are no differences in thez-positions of the carbon atoms. More generally, there are no inter-atomic contributions toσzz from anysp2-like carbon part of the sample. The evolution of σzz upon random and pair fluorination is shown in Fig. 3.11 (a) and (b), respectively. Unlike the in-plane optical conductiv-ity, the out-of-plane conductivities σzz are similar for both unpaired and paired cases in the energy range shown in Fig. 3.11, independently of the fluorine concentration.
There are in particular no features in σzz due to the chiral mid-gap states associated with local sub-lattice symmetry breaking in the randomly fluorinated graphene. Thus, polarization analysis of optical spectra yields clear fingerprints for spectral features associated with chiral mid-gap states.
Generally, the nonzero optical conductivity perpendicular to the sheets raises the possibility to rotate the polarization of passing polarized light. As nonzeroσzz requires the formation of sp3 orbitals, one is able to distinguish between impurity states origi-nating from adatoms and other in-plane disorder configurations (for example, carbon vacancies, in-plane carbon reconstructions like pentagon-heptagon rings, and coulomb impurities) by measuring the polarization angle.
Intra-Atomic Dipole Contributions
To study the influence of intra-atomic dipole contributions [arising from R2 from Eq.
(A.6)] we added them to the calculation of the optical conductivity and found that they are negligible. Fig. 3.12 shows the results for graphene and fluorographene with
0 5 10 15 20 25 30 35 0
2 4 6 8 10 12
Graphene
/ 0
(eV) <s|x|p
x
>=0
<s|x|p x
>=0.04699nm
0 5 10 15 20 25 30
0 1 2 3 4 5 6 7 8 9 10
Fluorographene
/ 0
(eV) <s|x|p
x
>=0
<s|x|p x
>=0.04699nm
Figure 3.12.:
Comparison of optical conductivity with and without intra-atomic dipole contribution in graphene (left) and fluorographene (right).
and without intra-atomic dipole contributions. The values of the overlap functions xs|x|pxy “ xs|x|pyy “0.04699nm are calculated from the overlap of carbon’s s and px
(py) wave functions using expressions given in Ref. [192]. In general, this intra-atomic dipole contribution slightly increases the value of the optical conductivity. However, a noticeable enhancement of the optical spectra of graphene or fluorographene appears not below energies of 17eV or 6.3eV, respectively. This dipole contribution does not change the optical spectrum qualitatively and has no effect on the value of the optical band gap in fluorographene. The same holds for the dipole terms in the case of the out-of-plane optical conductivity (data not shown).