# Band Gaps and Optical Spectra of Chlorographene, Fluorographene and Graphane from G0W0, GW0 and GW Calculations on Top of PBE and HSE06 Orbitals

**ABSTRACT** The band structures of three graphene derivatives (chlorographene, fluorographene, and graphane) were analyzed at three levels of many-body GW theory (G0W0, GW0, and GW) constructed over GGA (PBE) and screened hybrid HSE06 orbitals. DFT band gap values obtained with the HSE06 functional were notably larger than those from PBE calculations but were significantly lower than band gaps from all GW calculations. On the other hand, all GW-type calculations gave similar band gaps despite some differences in band structures. The band gap (4.9 eV at the highest GW-HSE06 level) was predicted to be smaller than that of fluorographene (8.3 eV) or graphane (6.2 eV). However, chlorographene can be considered a wide-band gap insulator analogous to fluorographene and graphane. Using the Bethe–Salpeter equation, optical absorptions of graphene derivatives were found to be at significantly lower energies due to large binding energies of excitons (1.3, 1.9, and 1.5 eV for chlorographene, fluorographene, and graphane, respectively). Point defects lowered band gaps and absorption energies. Taking into account the low concentration of defects in this type of material, their effect on the discussed electronic properties was rather small.

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**ABSTRACT:**Graphene derivatives containing covalently bound halogens (graphene halides) represent promising two-dimensional systems having interesting physical and chemical properties. The attachment of halogen atoms to sp2 carbons changes the hybridization state to sp3, which has a principal impact on electronic properties and local structure of the material. The fully fluorinated graphene derivative, fluorographene (graphene fluoride, C1F1), is the thinnest insulator and the only stable stoichiometric graphene halide (C1X1). In this review, we discuss structural properties, syntheses, chemistry, stabilities, and electronic properties of fluorographene and other partially fluorinated, chlorinated, and brominated graphenes. Remarkable optical, mechanical, vibrational, thermodynamic, and conductivity properties of graphene halides are also explored as well as the properties of rare structures including multilayered fluorinated graphenes, iodine-doped graphene, and mixed graphene halides. Finally, patterned halogenation is presented as an interesting approach for generating materials with applications in the field of graphene-based electronic devices.ACS Nano 06/2013; 7(8):6434-6464. · 12.03 Impact Factor

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Band Gaps and Optical Spectra of Chlorographene, Fluorographene

and Graphane from G0W0, GW0and GW Calculations on Top of PBE

and HSE06 Orbitals

František Karlický* and Michal Otyepka*

Regional Centre of Advanced Technologies and Materials, Department of Physical Chemistry, Faculty of Science, Palacký University,

Tř. 17. listopadu 12, Olomouc 771 46, Czech Republic

*

S Supporting Information

ABSTRACT: The band structures of three graphene

derivatives (chlorographene, fluorographene, and graphane)

were analyzed at three levels of many-body GW theory (G0W0,

GW0, and GW) constructed over GGA (PBE) and screened

hybrid HSE06 orbitals. DFT band gap values obtained with

the HSE06 functional were notably larger than those from

PBE calculations but were significantly lower than band gaps

from all GW calculations. On the other hand, all GW-type

calculations gave similar band gaps despite some differences in

band structures. The band gap (4.9 eV at the highest GW-

HSE06 level) was predicted to be smaller than that of

fluorographene (8.3 eV) or graphane (6.2 eV). However,

chlorographene can be considered a wide-band gap insulator

analogous to fluorographene and graphane. Using the Bethe−Salpeter equation, optical absorptions of graphene derivatives were

found to be at significantly lower energies due to large binding energies of excitons (1.3, 1.9, and 1.5 eV for chlorographene,

fluorographene, and graphane, respectively). Point defects lowered band gaps and absorption energies. Taking into account the

low concentration of defects in this type of material, their effect on the discussed electronic properties was rather small.

■INTRODUCTION

Covalently modified graphene derivatives prepared by attach-

ment of hydrogen and halogens have attracted considerable

interest over the past few years because of their potential

applications (e.g., in electronic devices).1,2The attachment of

atoms to sp2carbons changes its hybrid state to sp3, which

significantly alters the electronic properties and local structure

but preserves the 2D hexagonal symmetry. Such structural

changes induce opening of the zero band gap of graphene at the

K point and lead to loss of the π-conjugated electron cloud

present above and below graphene plane. Recently, fully

hydrogenated graphene (graphane, CH)3,4and fully fluorinated

graphene (fluorographene, also known as graphene fluoride,

CF) have been successfully prepared.5−8In contrast, the fully

chlorinated counterpart has not yet been prepared and partially

chlorinated graphene derivatives have only very recently been

reported.9,10Generally, wide band gap materials, such as CF,

CH, or BN, may be useful as 2D insulators for creating

semiconductor/insulator interfaces suitable for the develop-

ment of nanosized field-effect transistors (FETs).11Recently

proposed graphene-based ultracapacitors12also highlight the

importance of 2D insulator research.

Despite numerous theoretical and experimental studies,

much is still unknown about the electronic structure of these

types of materials. Therefore, we investigated the electronic

structure and band gaps of 2D halogenated graphene

compounds. Standard generalized gradient approximation

(GGA) to density functional theory (DFT) gives a band gap

value for CF only half that calculated using a high-level many-

body GW13approximation (GWA; Table 1),14−18which

includes electron−electron (e−e) interactions beyond DFT.

The CH band gap predicted by GWA is also much larger than

values obtained by using local density approximation (LDA) or

GGA of Perdew−Burke−Ernzerhof (PBE).14,15,17,19−21More-

Received:

Published: July 10, 2013

March 21, 2013

Table 1. Summary of Calculated and Experimental Band

Gaps, Eg(in eV), for Graphane (CH), and Fluorographene

(CF) based on a Literature Surveya

methodCH CF

DFT(PBE)

DFT(HSE06)

GW0, G0W0

BSE-G0W0(optical spectra)

exp. (optical spectra)

exp. (density of states)

exp. (transport measurement)

aFor details, see ref 22.

cFundamental band gap, ref 27.dFor C2.1F, ref 28.

3.5

4.5

5.4−6.1

3.8

3.1

5.1

7.3−7.5

5.4, 3.8

>3.8, >3.0b

>3.8c

∼3d

bOptical band gaps, refs 7 and 27.

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over, GWA predicts that the band gap of CF is greater than that

of CH, whereas GGA functionals give the opposite trend

(Table 1).22Using the Bethe−Salpeter equation (BSE)23,24on

top of GWA, which accounts for electron−electron and

electron−hole (e−h) correlations, the first exciton peaks in

the CF and CH absorption spectrum have been predicted to lie

between the PBE and GWA band gap values16,18,20(Table 1),

but without inclusion of e−h interactions (RPA over GW), are

close to GWA band gap values. BSE predicts the same trend of

band gaps as GWA: Eg(CF) > Eg(CH). In a previous paper,22

we showed that it is possible to obtain the same order of band

gaps with DFT as from GW and BSE if a screened hybrid

functional, such as those developed by Heyd, Scuseria, and

Ernzerhof (HSE03,25HSE0626), is used (Table 1).

Despite considerable research efforts, several questions

remain unanswered: (i) How do the results depend on the

level of GWA calculations (G0W0, GW0, and GW)? (ii) Are the

predicted values (GGA DFT band gaps are only half of band

gaps calculated using GWA; Table 1) different when HSE

orbitals are used in the GWA calculation? (iii) Is the

discrepancy between GGA DFT and GWA values observed

for different adsorbed halogens? (iv) Are the band gaps of the

studied systems significantly lowered in the presence of point

defects? In this work, we addressed all the above-mentioned

questions. We calculated the band gaps for CH, CF and CCl by

GWA over HSE06 orbitals and extended the previously

published GWA-PBE results for other graphene derivatives by

considering the fully chlorinated graphene (chlorographene,

CCl). We systematically investigated the effects of using G0W0,

GW0, and GW levels of many-body GWA in modeling the

aforementioned materials. Finally, we evaluated absorption

spectra from BSE on top of various levels of GWA and

discussed the role of excitons and point defects for comparison

of theoretical and experimental optical gaps.

■METHODS

GGA functionals systematically underestimate Kohn−Sham

band gaps (compared to experimentally determined values),

whereas the Hartree−Fock (HF) method systematically

overestimates them.29Therefore hybrid functionals, which

describe exchange using a mixture of the exact nonlocal HF

exchange (Ex

been developed. Hybrid functionals often give reasonably

accurate predictions of band gaps, lead to more accurate total

HF) and GGA exchange functional (Ex

GGA), have

energies and geometries. However, such functionals are

computationally demanding because of the slow decay of the

HF exchange, and therefore are intractable for extended

systems.30Thus, short-range functionals, such as the screened

hybrid functional HSE06,26seem to be an effective alternative

to standard hybrid functionals. In HSE06, the spatial decay of

the HF exchange interaction is accelerated by substitution of

the full 1/r Coulomb potential with a screened one, the

exchange energy term is split into short-range (SR) and long-

range (LR) components and the HF long-range component is

neglected but compensated by the PBE long-range contribu-

tion. The exchange-correlation energy is therefore calculated as

1

44

PBE

ωωω=++

+

EEEE

E

( )

3

( )( )

xc

HSE06

x

HF,SR

x

PBE,SR

x

PBE,LR

c

(1)

where Ec

is the screening parameter.26The HSE functional has been

shown to accurately predict the electronic properties of low-

dimensional carbon materials and optical transitions in both

metallic and semiconducting single-wall carbon nanotubes.31It

has also been studied and compared to GGA for its use as a

starting point in the calculation of quasiparticle (QP) excitation

energies of 3D materials. Single-shot G0W0 calculations

following calculations using the HSE03 functional have been

shown to give values very close to experiment.32

In the present work, the Vienna ab initio simulation package

(VASP)33implementing the projector augmented waves

(PAW) method34was used to perform single-point energy

GWA calculations on geometries from PBE/HSE06. In all

cases, the species were initially assumed to adopt a chairlike

conformation (Figure 1a and c for CCl) because it has been

shown to be the most stable conformation of both CH and

CF.15,19The optimized unit cell was obtained by minimizing

the total energy as a function of the lattice parameter. For each

value of the lattice constant, atomic positions (i.e., internal

degrees of freedom) were relaxed until the change in energy

was less than 1 × 10−5eV per cell (break condition for the

electronic step was an energy difference of 1 × 10−6eV). Band

structures were typically obtained by subsequent nonself-

consistent calculation of band energies along lines connecting

high-symmetry points K−Γ−M−K (for definition, see Figure

1b). Various tests were performed to determine the optimal

parameters in VASP. A k-point mesh of 16 × 16 × 1 points

PBEis the PBE correlation energy and ω = 0.11 bohr−1

Figure 1. (a) Geometrical structure (top view) of chlorographene (CCl) including the translation vectors (a1and a2) and elementary cell (blue).

The different colors represent chlorine above and below the graphene plane. (b) First Brillouin zone (1BZ) of CCl, basis vectors (k1and k2), and

high-symmetry points Γ, M, and K in reciprocal space. Black points correspond to original 16 × 16 k-point grid sampling of the 1BZ reduced by

VASP to 30 points using D3dsymmetry. (c) Tilted view of the 5 × 5 supercell of CCl and a (11̅0) plane (blue) used to construct the electron density

cuts shown in Figure 2.

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including the Γ point was used to sample the Brillouin zone

(BZ). This grid was reduced by VASP to 30 points with

different weights wk(defined such that they sum to 1) on the

basis of the D3dsymmetry of the considered structures (Figure

1b). A cutoff energy of 500 eV was applied for the plane-wave

basis set. As periodic boundary conditions were applied in all

three dimensions, a vacuum layer of 30 Å was included to

minimize any (spurious) interactions between adjacent layers.

In the GWA13calculations, usual Kohn−Sham equations

were first solved

ϕϕ+++=

−

TVVVE

()

kk

k

n e

nn

n

H xc

(2)

where T is the kinetic-energy operator, Vn−ethe potential of the

nuclei, VHthe Hartree potential, Vxcthe exchange-correlation

energy, and n and k the band and k-point indices, respectively.

Second, quasi-particle energies Enk

calculated as a first-order correction to the DFT single-particle

energies Enk

QPwithin the GWA were

ϕϕ=⟨|++ + Σ|⟩

−

E ReTVVG W E

( ,[;)]

k

k

k

k

n

n

n

n

QP,1

n eH xc

(3)

that is, both Green’s function G and the screened Coulomb

interaction W in the self-energy operator ∑ were calculated

using the DFT single-particle energies and wave functions.35

The corresponding single-shot total energy is henceforth

referred to as G0W0. The updated quasiparticle energy Enk

was obtained from the quasiparticle energy at the previous

iteration Enk

QP,i+1

QP,i

ϕ

ϕ

=+⟨|++

+ Σ| ⟩ −

+

−

EE Z Re

k

n

TVV

G W E

( ,

E

[

;)]

kk

k

k

k

k

n

i

n

i

n

n

i

n

n

i

QP,1 QP,

n eH

xc

QP, QP,

(4)

where Znk is the renormalization factor.35In the GW0

corrections, the screened Coulomb potential W was kept

fixed at the initial DFT value (W0) and Green’s function G was

iterated. In the GW case, W and dielectric matrix were

reevaluated in each iteration using the new quasi-particle

energies. The iterative procedure is generally carried out until

self-consistency is reached. In our case, we used four iterations

to obtain well converged results. In addition, a default cutoff

potential of 333.3 eV, 64 bands, and 192 frequency grid points

were employed in the GW calculations. The ratio of relative

computational costs is approximately 500:10:1 for our

calculations using G0W0/HSE06/GGA functionals, respec-

tively.

After the electronic ground states were determined, optical

properties of the considered materials were investigated. The

imaginary part of the macroscopic dielectric function ε(ω) was

calculated in the long-wavelength limit (q → 0) by summation

over the empty states36

∑

c v

, ,

εω

π

Ω

δ ε

(

εω=

| |

q

−−

× ⟨| ⟩⟨| ⟩*

αβ

→

++

αβ

Im

e

w

uuuu

( )

4

lim

q

1

2)

k

kkk

k e

c

kk e

c

k

cv

qvqv

2 2

0

2

(5)

where the band indices c and v are restricted to the conduction

and the valence band states, respectively (cmax+ vmax= nmax).

The vectors eαare the unit vectors for the three Cartesian

directions, Ω is the volume of the unit cell, and commonly used

notation ⟨un′k+q|eiGr|unk⟩ = ⟨ϕn′k+q|ei(q+G)r|ϕnk⟩ was adopted.

Note that the frequency ω has the dimension of an energy and

Imε(ω) was plotted for a light propagating along the CCl/CF/

CH plane. Local field effects, that is, changes in the cell periodic

part of the potential, were included in the random phase

approximation (RPA). To show the physical origin of different

features in the optical spectrum, we compared Imε calculated

(i) without taking into account both e−e and e−h correlation,

DFT+RPA,36(ii) with e−h interactions neglected, GW+RPA,35

and finally, (iii) from the full solution of the Bethe−Salpeter

equation23(BSE), which accounts for excitonic effects. The e−

h excited state was represented by the expansion

∑∑∑

cv

| ⟩ =

S

|⟩ k

cvA

k

k

cv

S

elec hole

(6)

where ASis the amplitude of a free e−h pair configuration

composed of the electron state |ck⟩ and the hole state |vk⟩. AS

was obtained by diagonalization of the excitonic equation37,38

implemented also in VASP,39according to eq 7, which

corresponds to the BSE

∑

′ ′ ′

k

c v

−+⟨| | ′ ′ ′⟩

c v

′ ′ ′= Ω

−

kk

EEAcv KAA

()

kkkkk

cv cv

S e h

c v

SS

cv

S

QPQP

(7)

Where ΩSis the exciton excitation energy and Ke−his the kernel

describing the screened interaction between excited electrons

and holes. Finally, BSE absorption spectra were obtained by

calculating the imaginary part of the dielectric function.39Note

that BSE was built on top of GWA. Therefore, corresponding

spectra were referred to as G0W0+BSE, etc.

■RESULTS AND DISCUSSION

DFT Calculations. The lattice constants (Table 2, d(X−X))

obtained from GGA(PBE) optimization were slightly larger

than those obtained using the HSE06 functional (the difference

was 0.03, 0.03, and 0.02 Å for CCl, CF, and CH, respectively).

The lattice constant and C−C distance in CCl (2.88 and 1.74

Å) were notably larger than the equivalent parameters for CF

(2.58 and 1.57 Å) and CH (2.52 and 1.53 Å) and deviated from

typical values for a single C(sp3)−C(sp3) bond (∼1.54 Å). This

is because chlorine atoms attached to the perturbed graphene

sp3lattice partially overlap with each other and the whole lattice

must balance C−C bonding with Cl−Cl repulsion. Notice also

decreasing electron density in the middle of the C-X and C−C

Table 2. Geometrical Parameters (Distances in Å and Bond Angles in deg) for Chlorographene (CCl), Fluorographene (CF),

and Graphane (CH) Obtained with PBE (Left Subcolumn) and HSE06 (Right Subcolumn) Functionals

d(C−C)d(C−X)d(X−X)a(C−C−C)

System PBEHSEPBEHSEPBE HSEPBEHSE

CCl

CF

CH

1.76

1.58

1.54

1.74

1.57

1.53

1.79

1.38

1.10

1.73

1.36

1.10

2.91

2.61

2.54

2.88

2.58

2.52

111.51

110.94

111.63

111.99

110.84

111.54

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bonds with increasing size of the halogen/hydrogen atom

(Figure 2). Consequently, CCl was predicted to be less stable

then CH and CF,22and from an experimental point of view,

CCl is likely to be unstable under ambient conditions. The

pristine parent material (graphite chloride) has been shown to

be unstable at temperatures >0 °C but stable at lower

temperatures.40However, nonstoichiometric graphene chloride

with low concentrations of approximately 8 or 30 at. % Cl has

recently been prepared9,10and its stability confirmed from

room temperature to 500 °C.

The CCl band gap of 1.41 eV obtained using the PBE

functional indicates it has semiconducting or insulating

properties. Surprisingly, fully chlorinated graphene, CCl, has

scarcely been mentioned in the literature and the first

theoretical remark14suggested it exhibits metallic behavior.

This probably stemmed from the formation of “nonbonded”

structures (composed of carbon sp2networks and very weakly

bound chlorine atoms) reported later,41as confirmed recently

in a study of the reaction mechanism between Cl atoms and

graphene,42or alternatively, Cl atom adsorption.43,44In

contrast, our PBE band gaps of CH and CF (3.53 and 3.09

eV) were in agreement with values reported earlier (see, e.g., ref

22 and references therein). CCl is much more polarizable than

CF due to the chlorine substituents, hence the band gap is

smaller. On the other hand, CF is rather ionic in character

(Bader analysis suggested charge transfer of ∼0.6 e from C to F

in CF versus ∼0.1 e from C to Cl in CCl), therefore the band

gap opens. The corresponding dielectric constants confirm this

(see Supporting Information). The HSE06 functional gave

notably larger band gaps than the PBE functional (Table 3; see

also refs 22 and 45). The difference was 1.40 eV, 1.84 and 0.85

eV for CCl, CF and CH, respectively. The inclusion of some

fraction of exact HF exchange not only led to a systematic

increase in calculated band gaps but also changed the trend in

band gaps; the HSE06 functional predicted Eg(CF) > Eg(CH)

> Eg(CCl) in agreement with GWA (see below), whereas for

PBE band gaps, Eg(CH) > Eg(CF) > Eg(CCl).

Calculations using the GW approximation. In our

calculations based on many-body GW theory, band gap values

were found to be sensitive to many parameters, such as k-point

sampling of BZ, cutoff energy, thickness of vacuum layer,

number of bands included and frequency grid. For example, for

CH, we obtained larger G0W0-PBE and GW0-PBE band gaps of

5.9 and 6.2 eV, respectively, for a sparse frequency grid (48

points) compared to band gaps of 5.7 and 5.9 eV, respectively,

for a frequency grid of 96 points (other parameters being

identical); similar values to the latter were obtained with a

denser grid (144 points). In a similar way, we tested other

parameters, e.g., a GW band gap error of about 0.6 and 0.1 eV

was introduced when the sampling of BZ was changed from 16

× 16 × 1 → 4 × 4 × 1 and 16 × 16 × 1 → 10 × 10 × 1,

respectively. Simultaneous tests of several parameters showed

that the band gaps predicted by the many-body GW theory

converged to within 0.02 eV. Finally, the predicted CH band

gap values (5.64 and 5.89 eV for G0W0-PBE and GW0-PBE,

respectively; Table 3) were in reasonable agreement with values

reported in the literature, i.e., 5.4 eV19(G0W0-LDA), 5.7 eV14

(G0W0-LDA), 5.4 eV20(G0W0-LDA), 5.7 eV21(G0W0-GGA),

6.1 eV15(G0W0-GGA), 6.1 eV46(G0W0-LDA) and 6.0 eV17

(GW0-LDA). However, the variation in literature values is

understandable considering the different conditions used, e.g.,

use of pseudopotentials or PAW method and different software

(VASP, Abinit+Yambo, Quantum Espresso), and sensitivity to

Figure 2. Total electron densities for chlorographene (CCl) (a,d), fluorographene (CF) (b,e) and graphane (CH) (c,f) calculated using the HSE06

functional. (a-c) Isosurfaces for isovalues of 0.1 au plotted for a 3 × 3 supercell. (d-f) Electron density cuts through the (11̅0) plane (see Figure 1c)

for a 2 × 2 supercell.

Table 3. Band gaps of chlorographene (CCl),

fluorographene (CF) and graphane (CH) in eV

SystemCCl CFa

CH

Method/orbitals PBEHSE06 PBEHSE06 PBE HSE06

DFT

G0W0

GW0

GW

1.41

4.07

4.46

4.89

2.81

4.54

4.73

4.93

3.09

6.98

7.48

8.12

4.93

7.69

7.95

8.28

3.53

5.64

5.89

6.28

4.38

5.83

5.95

6.17

aExperimental band gap >3 eV or >3.8 eV, see Table 1

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the aforementioned parameters. In addition, the level of

approximation used in calculations of band gaps based on the

many-body GW theory is sometimes not precisely documented

in the literature, e.g., G0W0and GW0calculations are often

generally labeled as GW. For CF, our G0W0-PBE and GW0-

PBE band gap values of 7.0 and 7.5 eV flank band gap values

reported in the literature: 7.4 eV14(G0W0-LDA), 7.4 eV15

(G0W0-GGA), 7.5 eV18(G0W0-GGA), 7.3 eV16(G0W0-LDA)

and 7.49 eV17(GW0-LDA). Again, small differences or

apparent similarities between the literature values can be

attributed to the different methods, parameters and software

used. Our GW0-PBE band gap value of 4.46 eV for CCl agrees

well with the only literature value of 4.33 eV47(GW0-LDA).

Whereas the HSE06 functional gave notably larger band gaps

than the PBE functional (as discussed above), differences

between bands gap from GWA constructed over HSE06 and

PBE orbitals were smaller (Table 3). G0W0-HSE06 band gaps

were larger by 0.47 eV, 0.71 and 0.19 eV (for CCl, CF and CH,

respectively) than G0W0-PBE band gaps, whereas GW0-HSE06

band gaps exhibited smaller differences of 0.27 eV, 0.47 and

0.06 eV, respectively, with respect to GW0-PBE band gaps. At

the highest level of theory, GW-HSE06 gave almost same

values as GW-PBE (i.e., differences 0.04 eV, 0.016 eV, −0.11 eV

for CCl, CF and CH, respectively). The predicted energy-band

structure was different for the DFT and GWA calculations.

However, further analysis suggested that the general shape of

individual bands was rather similar but distances between

individual bands were different. Surprisingly, similar trends

were observed when comparing band structures from

calculations with different DFT orbitals, i.e., HSE06 vs PBE,

G0W0-HSE06 vs G0W0-PBE, etc. A comparison of the band

structures for CCl, CF, and CH is shown in Figure 3.

Simplistically, band structures could be related by multiplying

the energies of occupied states by one coefficient and the

energies of unoccupied states by a different coefficient. All

compounds considered here were predicted to be direct band

gap materials. The bottom of the conduction band and top of

the valence band are located at the Γ point in the first Brillouin

zone and the maximum band gap is located at the K point. The

top of the valence band is doubly degenerate, belongs to the Eg

irreducible representation and has pxycharacter (from C and Cl

or F atom; from C only for CH). The bottom of the

(nondegenerate) conduction-band has pzcharacter and belongs

to the A2uirreducible representation.

Furthermore, the effects of different degrees of self-

consistency in GWA (G0W0, GW0and GW) were investigated.

For 3D materials, there is consensus that GWA systematically

improves the band gap of semiconductors and insulators;

calculated single shot G0W0band gaps are often within 10% of

experimental values (typically underestimated) and the best

agreement with experiment is achieved by GW0(see, e.g., ref

48; most calculations have considered LDA or GGA orbitals)

because GW band gaps are sometimes slightly overestimated.

Nevertheless, we observed some trends in the calculated values

notwithstanding the differences from experimental values.

Large differences in band gaps due to differing degrees of

self-consistency occurred in the case of the calculations with

PBE orbitals. In particular, for CCl and CF, there was a shift of

about 1 eV between the G0W0-PBE and GW-PBE band gap

(Table 3). Whereas this was apparent for PBE orbitals, the

calculations with more realistic HSE06 orbitals were more

consistent, i.e., there was only a slight difference between the

G0W0-HSE06 and GW-HSE06 values (Table 3). Application of

HSE06 orbitals as a zero order description for GWA

calculations has been recommended for 3D materials (also if

the PBE band gap is inverted or too small, HSE screening

properties are preferable).32Finally, the results suggested that

in predicting the band gaps of CCl, CF and CH, a similar

accuracy could be achieved by G0W0-HSE06 as by GW0-PBE

(Table 3). However, G0W0-HSE06 is computationally less

demanding since several iterations are required to converge G

in the GW0-PBE approach. Our findings agree with the general

trends observed in a previous study, in which GW0-PBE and

G0W0-HSE06 approaches were applied to model 3D

materials.32

Optical properties. In the case of insulating CF, GWA

band gaps (∼8 eV) were significantly different from the

experimentally observed absorption peaks (at 3 or 3.8 eV). A

large overestimation of band gap using GWA has also been

reported for a MoS2monolayer (PBE 1.6 eV → GW0-PBE 2.5

eV; exp. on bulk MoS2gives gap 1.7−1.8 eV)49and this casts

doubt on whether GWA is suitable for 2D structures. However,

calculated electronic band gaps and energies of electron

transitions derived from optical spectra do not match exactly

because the electron transitions observed in optical spectra

involve formation of an exciton. Therefore, DFT and GW band

gaps should not be directly compared with optical spectra. We

calculated optical spectra by using the BSE method including

Figure 3. The electronic band structure in the vicinity of the band gap for chlorographene (a), fluorographene (b), and graphane (c) along lines

connecting the high symmetry points K, Γ, and M in the Brillouin zone (for definitions, see Figure 1). Band structures were calculated using the PBE

functional (black line), HSE06 functional (blue line), GW over PBE (red circles) and GW over HSE06 (green dots). The Fermi level was set at zero

energy.

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excitonic effects and compared positions of the first absorption

peaks with available experimental data (cf. Table 1).

The theoretical absorption spectra directly correspond to the

imaginary part of the dielectric function, Imε(ω). From a

technical viewpoint, optical transition simulation requires

integration over the irreducible Brillouin zone using a

sufficiently dense k-point mesh. Naturally, the convergence of

k-point sampling is important. The optical absorption spectra

for light polarization parallel to the surface plane obtained with

different k-point meshes using G0W0(PBE)+BSE are illustrated

in Figure 4a for CF. The three absorption peaks with the lowest

energy (at about 5.1 eV, 6.1 and 6.6 eV; highlighted in Figure

4a), originating from bound excitons were found to converged

with an 18 × 18 × 1 k-point grid. As k was increased, an

increasing number of optically active bound excitons became

evident (from 6 to 12 excitons in Figure 4a) and their binding

energy converged within 0.02 eV (for convergence of the

binding energy of the first optically active exciton see inset of

Figure 4a). However, the dominant peak at 9−10 eV was still

not converged sufficiently with an 18 × 18 × 1 grid; previous

calculations have suggested that very dense grids are necessary

to obtain converged spectra for the 0−20 eV frequency window

(30 × 30 × 120or 40 × 40 × 116). Because the number of k

points directly relates to the rank of the BSE Hamiltonian,

causing extreme demands in terms of storage and CPU time for

the calculation of the spectra, we decided to use 18 × 18 × 1 k-

point grid and 192 frequency grid for evaluation of e−h kernel

in the BSE and the dielectric function (the important first

exciton peak was well converged). In addition, we included all

valence (11 for CCl and CF, 5 for CH) and 13 conduction

bands (maximal v and c indices in eqs 6 and 7) to obtain ε(ω)

at GW+ BSE level as we were only interested in the low energy

region of the spectrum.

We noted that the optical absorption spectrum for light

polarization perpendicular to the surface plane was almost

negligible up to 11 eV, as shown in Figure 4b. This indicates

that CF exhibits strong optical anisotropy because completely

different absorption spectra were obtained for light polarization

parallel to the surface plane. Similar is valid for CCl and CH

(data not shown). It is worth noting that the same behavior was

reported for CH and CF in previous works (ref 20 for CH and

ref 50 for CF). Optical anisotropy of CH, CF, and CCl is

induced by changes in the cell periodic part of the potential

(local-field effects) and, in addition, for CH there are low

energy excitons (see below) visible for the parallel polarization

but dipole forbidden for the perpendicular polarization.20The

first exciton peak of CF for light polarization parallel to the

surface plane was found to be located at 5.1 eV

(G0W0(PBE)+BSE), substantially lower than the corresponding

G0W0(PBE) band gap value at 7.0 eV (green arrow in Figure

4b). This difference corresponds to an exciton with a large

binding energy of 1.9 eV. To understand the origin of the

absorption peaks, we examined the oscillator strengths of the

excitonic transitions contributing to the absorption spectrum in

that region. We carried out a series of calculations that included

only selected valence and conduction bands. Only vertical

transitions from the two highest occupied bands to the lowest

empty band were found to be responsible for the exciton peaks

of bound excitons (indicated in Figure 4a by ovals). Major

transitions contributing to the dominant absorption peak at 9.2

eV originated from the two highest occupied bands and the

four lowest empty bands. The computation of oscillator

strengths (Figure 4b) included 11 valence and 13 conduction

bands.

Electron−hole excitations, which are not included in GW

itself, were clearly responsible for the optical absorption at

energies lower than GW band gaps. This is demonstrated in

Figures 5a, 5b, and 5c for CCl, CF, and CH, respectively, where

optical spectra calculated by (i) PBE+RPA (e−e and e−h

correlation neglected), (ii) G0W0(PBE)+RPA (e−e correlation

included and e−h correlation neglected), and (iii)

G0W0(PBE)+BSE (accounts both for e−e and e−h effects)

are presented. Comparison of the optical absorption spectra

obtained at the DFT+RPA and G0W0+RPA levels shows that

inclusion of e−e interaction led to a blue shift due to quasi

particle corrections. However, the shape of the spectra was

preserved. On the other hand, inclusion of e−h attraction

yielded a significant red shift of the absorption spectrum (insets

in Figure 5). Furthermore, redistribution of the spectral weight

to lower photon energies was observed. A number of

pronounced excitonic resonances also appeared. The most

prominent physical effect of the e−h interactions was

appearance of some bound excitons below the G0W0gap

(see insets in Figure 5), which were completely missing in the

G0W0+RPA. In the case of CCl, two low energy peaks were

evident at 2.82 and 3.33 eV and a shoulder at 3.86 eV

(corresponding to the resonant peak at 4.18 eV) due to bound

excitons. The binding energy of the exciton with the lowest

Figure 4. G0W0(PBE)+BSE absorption spectra of fluorographene

(CF) for light polarization parallel to the surface plane (a) obtained

using different k-point meshes. Inset: convergence of the binding

energy of the first optically active exciton (b) compared with

absorption spectra of CF for light polarization perpendicular to the

surface plane (red line). Oscillator strengths for all transitions are

shown as blue columns.

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energy was 1.25 eV. For CF, there were low energy peaks at

5.12 and 6.58 eV with a shoulder at 6.23 eV, and the lowest

energy exciton had a binding energy of 1.85 eV. The presented

values are shifted to lower energies with respect to values

reported in ref 18 (peaks at 5.4 and 7.2 eV, G0W0gap at 7.5

eV). The lowest energy of a bound exciton in CH was 4.11 eV

and had a binding energy of 1.53 eV, in agreement with ref 20

(binding energy of 1.6 eV), with the second peak located at

5.30 eV. The discussed exciton energies suggest that CF and

CH exhibit UV absorption, whereas CCl displays visible light

absorption.

For the sake of completeness, it should be noted that the

effect of using different degrees of self-consistency in GWA on

top of PBE and HSE06 orbitals on the first exciton peak

roughly corresponded to the effect on GWA band gap

discussed above. The same order of GW gaps and absorption

peaks was preserved, as shown in Figure 6. The first exciton

peak was on average shifted by ∼2 eV to lower energies with

respect to the corresponding band gap value from GWA. To

summarize this part, consideration of e−h interactions is crucial

for predicting absorption spectra of CCl, CF and CH that are

directly comparable with experimental optical spectra. There is

a large cancelation of e−e and e−h effects. This explains why

the experimental absorption value (>3.8 eV) is significantly red-

shifted with respect to GW band gaps (7−8 eV), whereas the

PBE band gap values fortuitously agree with this experimental

value.

Role of defects. The experimentally prepared materials

may contain some (usually small) amount of various defects,

which may influence the electronic structure and behavior of

the respective material. Therefore, we analyzed the role of point

defects on the electronic structure of the considered graphene

derivatives. It is worth noting that the presence of defects in CF

has been suggested as a possible explanation for the discrepancy

between experimental values and GWA band gaps.15,18,51

However, various conformations (chair, boat, zigzag and

armchair) for both CF and CH have been shown to have

similar band gaps on both DFT and GW levels.15,18,52,53Here,

we focused on selected point defects.

We investigated four types of point defects (Figure 7 for

CF): (i) Stone-Wales (SW) defect, (ii) Cl/F/H vacancy, (iii) C

vacancy, and (iv) interstitial Cl/F/H located at the site of a C

vacancy. SW defect is the simplest example of the graphene

lattice reconstruction by forming nonhexagonal rings:54four

hexagons are transformed into two pentagons and two

heptagons [SW(55−77) defect] by rotating one of the C−C

bonds by 90° (Figure 7a). We used a 3 × 3 supercell (with 36

or 35 atoms) and 5 × 5 × 1 k-point sampling for optimization

of atomic position in a fixed supercell at PBE level. The density

of states (DOS) and imaginary part of the dielectric function,

Imε(ω), were obtained for a 7 × 7 × 1 k-point grid. The

independent-particle (IP) picture was adopted within DFT

(DFT-IP) for Imε(ω) and no local field effect was included at

RPA level. We checked in the case of CF that the DFT-IP and

DFT+RPA spectra did not differ substantially, especially in the

low-energy region.

Figure 5. Absorption spectra of (a) CCl, (b) CF, and (c) CH for light

polarization parallel to the surface plane. Insets show amplified regions

of the absorption spectra in the vicinity of the first exciton peak and

G0W0(PBE) gap (dashed line).

Figure 6. The first exciton peak of the CF absorption spectra obtained

using different degrees of self-consistency in GWA on top of PBE and

HSE06 orbitals calculated using a 14 × 14 × 1 k-point grid.

Corresponding band gap values are shown as arrows.

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Stone−Wales defects (Figure 7a−c) did not affect the band

gap of CF, as seen from DOS plot, and the corresponding

absorption spectrum of defective CF was similar to pure CF.

The total magnetization equaled to zero, that is, the system was

nonspin-polarized (closed shell). The DOS and absorption

spectrum of the system with fluorine vacancies were very

similar to those of virgin CF (Figure 7d−f). However, peaks

corresponding to a dangling bond were evident in the DOS

plot. In the limit of a low density of defects (large supercell),

such peaks in DOS would be reduced to lines, and therefore,

the band gap would remain again rather unaffected. Spin

density (difference between α and β electron density, Figure

7d) was located in the vicinity of an unpaired carbon atom,

below and above the CF plane. On the other hand, C vacancies

led to a double fluorinated carbon atom, two carbon dangling

bonds and substantial shift of Fermi level. The original band

gap of ∼3 eV was significantly perturbed by these features,

which implies a broad peak in the visible light region of the

absorption spectra (Figure 7g−i). Spin density was located in

the vicinity of carbon atoms with unpaired electrons, here in

the CF plane. The final considered defect, interstitial F located

at the site of a C vacancy, led to substantial changes in the

DOS. This is a typical feature of n-type semiconductors, that is,

there are valence and dopant bands below the Fermi level. A

broad peak located at 0.4 eV appeared in the absorption

spectra. It is worth noting, that the last type of defect is not very

stable with respect to the preceding ones. Taking into account,

the rather low concentration of defects in 2D graphene

derivatives (here CF, CCl, and CH), we concluded that point

defects may lead to slight lowering of band gaps.

■CONCLUSIONS

The many-body GW approximation is often considered a

“benchmark” method against which other methods are

assessed. However, so far there has been no systematic study

of the effects of using various levels of GWA on the predicted

Figure 7. Geometrical structures, density of states (DOS) and optical absorption spectra for light polarization parallel to the surface plane of

defective fluorographenes. Stone-Wales defect (a-c), F vacancy (d-f), C vacancy (g-i), and interstitial F (j-l) were considered. DOS and spectra for

virgin CF are also shown for comparison (green dotted line). For DOS, the Fermi level was set to zero. Spin densities are added to plots containing

geometrical structures.

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band gap values of the recently developed 2D carbon-based

materials (e.g., chlorographene, fluorographene, and graphane).

Here, we presented values of many-body GWA band gaps

computed at G0W0, GW0, and GW levels constructed over

GGA (PBE) and hybrid (HSE06) orbitals. Whereas DFT

calculations at HSE06 level gave notably larger band gaps than

calculations at PBE level, the GWA calculations over HSE06

orbitals gave similar band gaps to GWA over PBE. The band

gap of CCl (4.93 eV at the highest GW-HSE06 level) was

predicted to be smaller than that of CH (6.17 eV) or CF (8.28

eV). However, CCl can still be considered a wide-band gap

insulator analogous to CF and CH. The optical properties of

the considered materials were investigated at BSE+GW level

and the inclusion of electron−hole interaction was found to be

crucial for predicting low energy absorption peaks of excitonic

nature. The formation of exciton peaks at 2.82, 5.12, and 4.11

eV with huge binding energy of 1.25, 1.85, and 1.53 eV for CCl,

CF, and CH, respectively, was observed. The large binding

energy of an exciton in CF can also explain the putative

discrepancy between the GW band gap (7−8 eV) and the

experimental observations (>3 or >3.8 eV). The role of point

defects was predicted to be synergistic but is likely to be small

considering the low concentration of defects in real materials.

■ASSOCIATED CONTENT

*

Outputs from Bader analysis, macroscopic static dielectric

tensors, and basic information about computational cost. This

information is available free of charge via the Internet at http://

pubs.acs.org/.

■AUTHOR INFORMATION

Corresponding Author

*E-mail: frantisek.karlicky@upol.cz (F.K.); michal.otyepka@

upol.cz (M.O.).

Notes

The authors declare no competing financial interest.

■ACKNOWLEDGMENTS

Financial support from the Czech Science Foundation (P208/

12/G016), the Operational Program Research and Develop-

ment for InnovationsEuropean Regional Development Fund

(project CZ.1.05/2.1.00/03.0058 of the Ministry of Education,

Youth and Sports of the Czech Republic), the Barrande project

(No. 7AMB12FR026) and the Operational Program Education

for CompetitivenessEuropean Social Fund (project CZ.1.07/

2.3.00/20.0017 of the Ministry of Education, Youth and Sports

of the Czech Republic) is gratefully acknowledged.

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