Graphane- and Fluorographene-Based Quantum DotsMozhgan N. Amini, Ortwin Leenaerts, Bart Partoens,* and Dirk Lamoen*

CMT-Group and EMAT, Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium

ABSTRACT: With the help of first-principles calculations, we investigate graphane/fluorographene heterostructures with special attention for graphane and fluorographene-based quantum dots. Graphane and fluorographene have large electronic band gaps, andwe show that their band structures exhibit a strong type-II alignment. In this way, it ispossible to obtain confined electron states in fluorographene nanostructures byembedding them in a graphane crystal. Bound hole states can be created in graphanedomains embedded in a fluorographene environment. For circular graphane/fluorographene quantum dots, localized states can be observed in the band gap if thesize of the radii is larger than approximately 4 to 5 Å.

Graphene and its chemical derivatives, graphane andfluorographene, are the subject of numerous investiga-

tions at the moment. The interesting physical phenomena thatare related to 2D electron gases as found in, for example,heterostructures, are readily obtainable in these naturally 2Dcrystals.Graphene is in its pristine form a zero-gap semiconductor,

but it is possible to create a substantial electronic band gap byconfinement or chemical functionalization. The gaps that canbe obtained by cutting graphene into nanoribbons range intheory from 0 to ∼2.5 eV,1 while experimental gaps are foundup to 0.5 eV.2 Chemical functionalization leads to band gapslarger than 3 eV.3−8 This has motivated some research ongraphene nanostructures (e.g., nanoribbons and quantum dots(QDs)) embedded in functionalized graphene materials such asgraphane9−11 (HG) and fluorographene12 (FG). Partialfunctionalization creates graphene islands or nanoroads ofwhich the boundaries are formed by the semiconductingfunctionalized graphene.12,13 In practice, such structures aresupposed to be formed by partial dehydrogenation (defluori-nation) of graphane (fluorographene) by exposure to, forexample, a laser beam,14 or by selective functionalization of thegraphene layer. By changing the size and the shape of thesenanostructures their electronic and magnetic properties can becontrolled.9,10,12,15 The realizability of such structures has beenexperimentally demonstrated by the creation of multiquantumdots in graphane.11

In this work, we examine the possibility of graphane-basedand fluorographene-based nanostructures, especially QDs.Graphane and fluorographene have similar band gaps butvery different ionization potentials.4 This can be expected tocause a type-II alignment of their band structures, which can beexploited in graphene-based heterostructures. Instead ofcreating graphene domains inside graphane or fluorographene,we consider domains of one functionalized material inside theother. We demonstrate that it is possible to build a graphane-based QD into a fluorographene crystal by substituting some

fluorine atoms with hydrogen atoms. Similarly, one can alsomake fluorine-based QDs in a graphane crystal by substitutingsome hydrogen atoms with fluorine atoms. We show thatgraphane dots contain localized hole states while fluorogra-phene dots have bound electron states.Some advantages of graphane and fluorographene QDs over

graphene dots can be expected. First, there is a smaller latticemismatch between graphane and fluorographene in comparisonwith graphene and graphane or graphene and fluorographene.Second, one can also expect these functionalized dots to bemore stable because all of the carbon atoms are saturated. Thisshould be contrasted with graphene dots embedded in HG orFG, where the boundary between both materials will becomemore chemically reactive and is often magnetic.15

This paper is organized as follows: We first give thecomputational details of our simulations, followed by a detailedcomparison of the properties of graphane and fluorographene.Because the band alignment is the most important factordetermining the properties of the graphane/fluorographeneheterostructures, we examine this property in the next section.This band alignment is used to construct graphane/fluorographene QDs, which are subsequently investigated.Finally, we give a summary of our work in the last section.

■ COMPUTATIONAL DETAILS

We perform first-principles density functional theory (DFT)calculations within the local density approximation (LDA), thegeneralized gradient approximation (GGA) of Perdew, Burke,and Ernzerhof,16 and the screened hybrid functional of Heyd,Scuseria, and Ernzerhof (HSE06),17 as implemented in theVienna ab initio simulation package.18 Electron−ion inter-actions are treated using projector-augmented wave poten-

Received: May 23, 2013Revised: July 15, 2013Published: July 15, 2013

Article

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tials.19−21 The C (2s22p2), H (1s1), and F (2s22p5) electronsare treated as valence electrons. For unit cell calculations ofpure HG and FG, the electron wave functions are describedusing a plane-wave basis set with a cutoff energy of 600 eV, anda 24 × 24 × 1 k-point grid is used to sample the Brillouin zone.Calculations for QD systems are performed with a lower energycutoff of 400 eV. Relaxations are done with a single k-point,while finer 4 × 4 × 1 k-point grids are used to calculate the(projected) density of states (P)DOS. A vacuum space of 15 Åis used to reduce the interaction between periodic images of thepure FG and HG system and a vacuum space of 10 Å for theQD structures. Convergence with respect to self-consistentiterations was assumed when the total energy differencebetween different cycles was less than 10−4 eV and thegeometry relaxation tolerance was better than 0.01 eV/Å.

■ RESULTSBefore discussing the formation of QDs in graphane/fluorographene heterostructures, we investigate the character-istics of these two materials separately. The properties thatconcern us here are both structural and electronic, and we makeuse of different exchange-correlation (xc) functionals (LDA,PBE-GGA, and HSE06) to examine these. The latter isimportant to understand the influence of the level ofcomputation on the obtained results. The electronic bandstructures of graphane and fluorographene are shown in Figure1. It is seen that both materials are large-gap semiconductors

with a valence band (VB) that is degenerate at the Γ point anda nondegenerate conduction band (CB). This gives rise tothree types of quasiparticles in the system, namely, electronsand heavy and light holes.A summary of the calculated structural (lattice parameters,

bond lengths, and angles) and electronic (band gap, effectivemasses, and ionization energy) properties of graphane andfluorographene is given in Table 1.Let us first compare the results of the various functionals.

Those obtained with the hybrid functional (HSE06) arebelieved to be the most accurate,22 especially for electronicproperties such as the band gap and the ionization potential.23

The structural parameters roughly vary with 1%, and theHSE06 functional gives values between those of LDA andGGA. Furthermore, the difference between graphane andfluorographene is consistent for all functionals. Therefore, wecan assume that, for our purpose, the structure is well-described, independent of the specific functional. Theelectronic properties show some substantial variation, althoughthe results from LDA and GGA are very similar. It can be seenfrom Table 1 that the electronic band gap and the ionizationenergy, defined as the difference between the valence bandmaximum (VBM) and the vacuum level, are significantly largerfor the hybrid functional.If we compare graphane to fluorographene, some important

differences can be observed. The lattice parameter of graphaneis ∼2% smaller than that of fluorographene. This is small

Figure 1. Electronic band structure of HG (a) and FG (b) calculated with the HSE06 xc-functional. The energy corresponding to the valence bandmaximum is put to zero.

Table 1. Structural and Electronic Properties of HG and FG for Different xc-Functionalsa

graphane fluorographene

LDA GGA HSE06 LDA GGA HSE06

a 2.508 2.541 2.522 2.557 2.609 2.582dCX 1.117 1.110 1.104 1.365 1.382 1.364dCC 1.516 1.537 1.526 1.555 1.583 1.568θCCX 107.3 107.4 107.4 108.3 107.9 108.1θCCC 111.6 111.5 111.5 110.7 111.0 110.8Egap 3.385 3.477 4.383 2.963 3.103 4.933me 0.761 0.768 0.750 0.363 0.366 0.352mlh 0.202 0.195 0.182 0.312 0.305 0.264mhh 0.463 0.473 0.438 0.863 0.893 0.731IE 4.962 4.740 5.383 8.066 7.911 8.952

aLattice constant, a, and the bond lengths, dCX and dCC (with X = H or F), are given in angstroms. The bond angles, θCCX and θCCC, are given indegrees (°) and the band gap, Egap, and ionization energy, IE, are give in electronvolts. The effective masses of electrons, me, and light and heavyholes, mlh and mhh, are given in units of the free electron mass.

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enough to match the two materials without introducing toomuch strain. However, this strain can change the size of theband gap and the effective masses of graphane andfluorographene in a heterostructure. Therefore, we performedsome test calculations on strained HG and FG. Thesecalculations show that the changes of the electronic propertiesare on the same order as the strain (∼2%) and can therefore beneglected.Another aspect that needs attention is the symmetry of the

wave functions at the top of the VB and the bottom of the CB,shown in Figure 2. Because the VB is degenerate for both HG

and FG, we show the sum of both wave functions to preservethe lattice symmetry in the Figure. It is seen that the VB ofgraphane has px,y character and is localized on the C atoms,while the CB wave functions are plane-wave-like states abovethe H atoms.24 The VB of FG also has px,y character, althoughin this case there is also a contribution of the F atoms. The CBhas a strong pz character and is located on the C and F atoms.So, in summary, the VBs of HG and FG have similar orbitalsymmetry, while their conductions bands do not.A simple approximation of the band alignment in

heterostructures can be obtained by the electron affinity rule,also known as Anderson’s rule.25 This rule states that thevacuum levels of two materials in a heterostructure should belined up. This can be easily done with the ionization energiesgiven in Table 1. The IE of fluorographene is more than 3 eVlarger than that of graphane, which means that the bands aresubstantially shifted with respect to each other. The resultingband alignment for the different functionals is given in Figure3a. There is a type-II alignment of the band structures in thecase of the hybrid functional and a type-III alignment in thecase of LDA and GGA. In the last case, the bottom of the CB ofFG is below the top of the VB of HG, which would imply acharge transfer from graphane to fluorographene. However, thedifference between the two levels is only ∼0.2 eV, which istypically smaller than the accuracy of Anderson’s rule.We can obtain another guess of the band alignment if we

place both materials a distance apart in the same supercell,which allows for a direct determination of the band offsets fromthe total band structure. This is possible because the lattice

mismatch between HG and FG is small enough to keep thechanges of the band gap and ionization energy limited. Inpractice, we place a graphane and fluorographene unit cellabove each other in a single supercell with 15 Å of vacuumbetween them and relax the supercell size to reduce the strain.A subsequent band structure calculation provides the requiredband alignment, which is shown in Figure 3b. All xc-functionalsresult in a type-II alignment and give consistent results for thedifference between the VBMs of HG and FG (≈ 2.7 eV). Thedifference between conduction band minima (CBMs),however, appears to vary significantly (2.5 to 3.4 eV).Furthermore, the band gap in graphane appears to differfrom the values in Table 1. Both discrepancies might beexplained by the plane-wave nature of the graphane CB, whichis largely localized in the vacuum (see Figure 2b) so that itsenergy level is more sensitive to perturbations, but this is only aminor issue because the CB of HG will not play an importantrole in the physics of the QDs, which are the main subject ofour investigation.Another difference of LDA and GGA as compared with

HSE06 is the size of the gap between the VBM of HG and theCBM of FG. This, too, is of minor importance because thecorresponding wave functions also have very differentsymmetry and barely interact.Because DFT is essentially a ground-state theory, we can

expect that the VBM is more accurately described than theCBM. The fact that the difference between VBM of the twomaterials is less consistent for Anderson’s rule than in a single-supercell calculation suggests that the latter is more accurate.From the discussions above, it is clear that the LDA xc-

functional leads to quantitatively similar geometries andqualitatively similar electronic properties as the HSE06functional. Therefore, we use the LDA xc-functional to relaxthe QD structures in the following and perform a subsequenthybrid functional calculation on these relaxed structures toobtain the electronic properties of the QDs. The advantage ofLDA above HSE06 is immense with respect to computation

Figure 2. Top and side views of the charge density of the states at thetop of the valence band (VB) and the bottom of the conduction band(CB) for graphane (a,b) and fluorographene (c,d).

Figure 3. Band alignment of graphane and fluorographene for differentfunctionals (FLTR: LDA, GGA, and HSE06) calculated withAnderson’s rule (a) and within the same supercell (b).

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time and resources. LDA allows us, therefore, to simulate muchlarger systems.The type-II band alignment of graphane and fluorographene

combined with the small lattice mismatch makes it possible toform two kinds of QDs from these materials: (i) an FG/HGQD with confined electron states, as shown in Figure 4a, and

(ii) an HG/FG QD with confined hole states, shown in Figure4b. We consider both types of dots and examine the possibilityof bound states in these dots in detail. Because of computa-tional restrictions, the size of the simulated QDs has to berather small. To have a rough estimate of the size of the dotthat is needed to observe these localized states, we can makeuse of a simple analytical QD model.If we assume the dots to be infinitely deep potential wells

with circular shape, we can write the Schrodinger equation for asingle particle with effective mass m* as follows:

− ℏ*∇ Ψ + Ψ = Ψ

mr V r r E r

2( ) ( ) ( ) ( )

22

(1)

where the confinement potential of the well is given by

= ≤∞

⎧⎨⎩V rr R

( )0 if

otherwise (2)

and r = (x2 + y2)1/2 and R is the radius of the dot. This equationis readily solved and the (radial) solutions are the well-knownBessel functions, Jl. The corresponding eigenenergies are givenby:

β=

ℏ*

Em R2n l

n l,

2,2

2 (3)

in which βn,l are the zeros of the Bessel functions, which can befound in the literature:26 for example, β0,0 = 2.4048. The loweststate is nondegenerate, but the higher states are all doublydegenerate (for l and −l).The electrons in the FG dot have an effective mass of

∼0.36me, and their confinement energy should be below 3.3 eVfor the bound state to fall inside the band gap. Therefore, thesize of the radius of the FG dot should be larger than 4.3 Å.For the graphane dot, we have two hole states, the heavy and

the light hole. Because the confinement energy of the heavyhole is smaller, we take the effective mass of the particle in thedot to be 0.46me. To have a confinement energy below 2.6 eV,the dot radius should again be larger than 4.3 Å.On the basis of the simple model above, we restrict our DFT

simulations to the two particular dots pictured in Figure 5a,b.

These dots contain 24 and 54 C atoms, respectively, and havehexagonal shapes that are close to circles with a radius ofapproximately 4.5 and 6.7 Å. Because of the periodic boundaryconditions in our simulations, there will be some interactionbetween the dot and its periodic images. A minimum distanceof ∼8 Å between periodic images is maintained to reduce thisinteraction.We first consider the case of a fluorographene QD embedded

in graphane. A schematic diagram of the band alignment isgiven in Figure 4a. The low-energy CB states of FG should giverise to confined electron states in the dot. These states areclearly visible in a projected density of states (PDOS) of the FGand HG part of the system as given in Figure 6. The different

symmetries of the states inside and outside the dot make surethat the electron states are well-confined inside the dot (seebelow). Several peaks, corresponding to confined states insidethe FG dot, can be observed. Some of these peaks are atenergies above the CBM of graphane. In other words, there aresome confined states of FG that fall inside the continuum of thegraphane CB. In fact, only one bound state is found below theCBM for the small dot and three bound states (of which twoare degenerate) are observed for the larger dot.The wave functions corresponding to the top of the VB and

the lowest bound electron states are shown in Figure 7. Thesewave functions were calculated with the hybrid functional ontop of a LDA relaxed structure. For the VBM and the secondbound state, the sum of two (almost) degenerate states is

Figure 4. Band alignment of (a) FG QD and (b) HG QD. Dottedlines denote possible confined states.

Figure 5. (a) Small QD containing 24 substituted atoms and (b) largerQD with 54 atom susbstitutions (red circles).

Figure 6. Partial and total PDOS of graphane and fluorographene partof the FG quantum dot system: (a) small QD and (b) larger QD. Theyellow band illustrates the band gap.

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shown to illustrate the symmetry. As expected from Figure 4a,the VBM wave function is mainly located in the graphane shell,but there is a substantial penetration into the FG dot. Thebound states are confined to the dot because they cannotpenetrate the graphane boundary due to the different symmetryof these states.Let us now consider a graphane dot in a fluorographene

environment. The graphane part in the middle of the supercellhas a higher VBM compared with the fluorographene part(Figure 4b), so we should have a stronger contribution ofgraphane to the VBM of the system and a larger contribution offluorographene to the CBM. The PDOS of the two graphanedots, shown in Figure 8, confirms this expectation.

There are no clear peaks visible above the VBM of FG,indicating that there are no bound states. A closer look,however, reveals the formation of a localized state of which theenergy is smeared out due to interaction with its periodicimages. This interaction is possible because the localized state isclose in energy to the VB states and has a similar symmetry,which makes hybridization possible. In other words, the boundstate is smeared out over the FG boundary and can interactwith neighboring dots. This is clearly illustrated by the shape ofthe wave function corresponding to the bound state (see Figure9a): The state is mainly located in the dot but decays ratherslowly away from the dot to the edge of the supercell. We canexpect that a larger boundary between the different dots willreduce the hybridization (smearing) of the hole states so thatdiscrete peaks could be observed in the DOS. Also, a larger dot

size will reduce the interaction because it confines the boundstate more inside the dot. This can already be observed for thelarger dot that we examined (Figure 5b). In the correspondingPDOS, shown in Figure 8, the formation of a small peak at theVBM of FG can be observed that indicates a bound HG state.

■ SUMMARYWe investigated the band alignment of graphane/fluorogra-phene heterostructures within the DFT formalism. Theobserved type-II alignment allows for the creation of graphaneQDs inside a fluorographene environment with confined holestates and fluorographene QDs inside graphane crystals withconfined electron states. The size of the QDs needs to be largerthan ∼4.5 Å for the confined states to fall inside the band gap.The electron states in the FG QD were found to be well-confined due to a difference in orbital symmetry between CBwave functions of FG and HG. This differs substantially fromthe bound hole states of the HG QDs, which spread out fromthe QD region into the FG boundary. This smearing is causedby the similar symmetry and energies of the confined hole stateand the VB orbitals that allow them to hybridize.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: Bart.Partoens@ua.ac.be; Dirk.Lamoen@ua.ac.be.NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSWe gratefully acknowledge financial support from the IWT-Vlaanderen through the ISIMADE project, the FWO-Vlaanderen, and a GOA fund from the University of Antwerp.This work was carried out using the HPC infrastructure of theUniversity of Antwerp (CalcUA), a division of the FlemishSupercomputer Center VSC, which is funded by the Herculesfoundation and the Flemish Government (EWI Department).

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