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Graphane and Fluorographene-Based Quantum Dots Mozhgan N Amini, Ortwin Leenaerts, Bart Partoens, and Dirk Lamoen J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/jp405079r • Publication Date (Web): 15 Jul 2013 Downloaded from http://pubs.acs.org on July 16, 2013

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The Journal of Physical Chemistry

Graphane and Fluorographene-Based Quantum Dots Mozhgan 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

E-mail: [email protected]; [email protected]

∗

To whom correspondence should be addressed

1

ACS Paragon Plus Environment


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Abstract

With the help of rst-principles calculations, we investigate graphane/uorographene heterostructures with special attention for graphane and uorographene-based quantum dots. Graphane and uorographene have large electronic band gaps and we show that their band structures exhibit a strong type-II alignment. In this way, it is possible to obtain conned electron states in uorographene nanostructures by embedding them in a graphane crystal. On the other hand, bound hole states can be created in graphane domains embedded in a uorographene environment. For circular graphane/uorographene quantum dots, localized states can be observed in the band gap if the size of the radii is larger than approximately 4 to 5 Å. Keywords:

Quantum dots, Graphene, Graphane, Fluorographene, Density functional

theory

Graphene and its chemical derivatives, graphane and uorographene, are the subject of numerous investigations at the moment. The interesting physical phenomena that are related with two-dimensional (2D) electron gases as found in e.g. heterostructures, are readily obtainable in these naturally 2D crystals. Graphene is in its pristine form a zero-gap semiconductor, but it is possible to create a substantial electronic band gap by connement or chemical functionalization. The gaps that can be obtained by cutting graphene into nanoribbons range in theory from 0 to about 2.5 eV, 1 while experimental gaps are found upto 0.5 eV. 2 Chemical functionalization, on the other hand, leads to band gaps larger than 3 eV. 38 This has motivated some research on graphene nanostructures (e.g. nanoribbons and quantum dots) embedded in functionalized graphene materials such as graphane 911 (HG) and uorographene 12 (FG). Partial functionalization creates graphene islands or nanoroads of which the boundaries are formed by the semiconducting functionalized graphene. 13 In practice, such structures are supposed to be formed by partial dehydrogenation (deuorination) of graphane (uorographene) by exposure to e.g. a laser beam, 14 or by selective functionalization of the graphene layer. By 2


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changing the size and the shape of these nanostructures their electronic and magnetic properties can be controlled. 9,10,12,15 The realizability of such structures has been experimentally demonstrated by the creation of multi-quantum dots in graphane. 11 In this work, we examine the possibility of graphane-based and uorographene-based nanostructures, especially quantum dots. Graphane and uorographene have similar band gaps, but very dierent ionization potentials. 4 This can be expected to cause a type-II alignment of their band structures which can be exploited in graphene-based heterostructures. Instead of creating graphene domains inside graphane or uorographene, we consider domains of one functionalized material inside the other. We demonstrate that it is possible to build a graphane-based quantum dot into a uorographene crystal by substituting some uorine atoms with hydrogen atoms. Similarly, one can also make uorine-based quantum dots in a graphane crystal by substituting some hydrogen atoms with uorine atoms. We show that graphane dots contain localized hole states while uorographene dots have bound electron states. Some advantages of graphane and uorographene quantum dots over graphene dots can be expected. First, there is a smaller lattice mismatch between graphane and uorographene in comparison to graphene and graphane or graphene and uorographene. Second, one can also expect these functionalized dots to be more stable because all the carbon atoms are saturated. This should be contrasted with graphene dots embedded in HG or FG, where the boundary between both materials will become more chemically reactive and is often magnetic. 15 This paper is organized as follows: We rst give the computational details of our simulations, followed by a detailed comparison of the properties of graphane and uorographene. Since the band alignment is the most important factor determining the properties of the graphane/uorographene heterostructures, we examine this property in the next section. This band alignment is used to construct graphane/uorographene quantum dots, which are subsequently investigated. Finally, we give a summary of our work in the last section. 3



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Computational details. We perform rst-principles density functional theory (DFT) calculations within the local density approximation (LDA), the generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof, 16 and the screened hybrid functional of Heyd, Scuseria, and Ernzerhof (HSE06), 17 as implemented in the Vienna ab initio simulation package. 18 Electron-ion interactions are treated using projector augmented wave potentials. 1921 The C (2s2 2p2 ), H (1s1 ), and F (2s2 2p5 ) electrons are treated as valence electrons. For unit cell calculations of pure HG and FG, the electron wavefunctions are described using a plane-wave basis set with a cuto energy of 600 eV and a 24 × 24 × 1 k-point grid is used to sample the Brillouin zone. Calculations for quantum dot systems are performed with a lower energy cuto of 400 eV. Relaxations are done with a single k-point while ner 4 × 4 × 1 k-point grids are used to calculate the (projected) density of states (P)DOS. A vacuum space of 15 Å is used to reduce the interaction between periodic images of the pure FG and HG system and a vacuum space of 10 Å for the quantum dot structures. Convergence with respect to self-consistent iterations was assumed when the total energy dierence between dierent cycles was less than 10 −4 eV and the geometry relaxation tolerance was better than 0.01 eV/Å.

Results. Before discussing the formation of quantum dots in graphane/uorographene heterostructures we investigate the characteristics of these two materials separately. The properties that concern us here are both structural and electronic and we make use of dierent exchange-correlation functionals (LDA, PBE-GGA, and HSE06) to examine these. The latter is important to understand the inuence of the level of computation on the obtained results. The electronic band structure of graphane and uorographene are shown in Fig. 1. It is seen that both materials are large-gap semiconductors with a valence band that is degenerate at the Γ point, and a non-degenerate conduction band. This gives rise to 3 types of quasiparticles in the system, namely electrons and heavy and light holes. A summary of the calculated structural (lattice parameters, bond lengths and angles) 4


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(a)

(b)

Figure 1: (Color online) The electronic band structure of HG (a) and FG (b) calculated with the HSE06 xc-functional. The energy corresponding to the valence band maximum is put to zero. and electronic (band gap, eective masses, and ionization energy) properties of graphane and uorographene is given in Table 1. Let us rst compare the results of the various functionals. Those obtained with the hybrid functional (HSE06) are believed to be the most accurate, 22 especially for electronic properties such as the band gap and the ionization potential. 23 The structural parameters vary roughly with 1% and the HSE06 functional gives values between those of LDA and GGA. Furthermore, the dierence between graphane and uorographene is consistent for all functionals. Therefore, we can assume that, for our purpose, the structure is well described, independent of the specic functional. The electronic properties, on the other hand, show some substantial variation although the results from LDA and GGA are very similar. It can be seen from Table 1 that the electronic band gap and the ionization energy, dened as the dierence between the valence band maximum and the vacuum level, are signicantly larger for the hybrid functional. If we compare graphane to uorographene, some important dierences can be observed. The lattice parameter of graphane is about 2% smaller than that of uorographene. This is small enough to get a matching of the two materials without introducing too much strain. However, this strain can change the size of the band gap and the eective masses of graphane 5



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Table 1: Structural and electronic properties of HG and FG for dierent xcfunctionals. The lattice constant, a, and the bond lengths, dCX and dCC (with X = H or F), are given in Å. The bond angles, θCCX and θCCC , are given in degrees (◦ ) and the band gap, Egap , and ionization energy, IE, in eV. The eective masses of electrons, me , and light and heavy holes, mlh and mhh , are given in units of the free electron mass.

a dCX dCC θCCX θCCC Egap me mlh mhh

IE

LDA 2.508 1.117 1.516 107.3 111.6 3.385 0.761 0.202 0.463 4.962

graphane GGA 2.541 1.110 1.537 107.4 111.5 3.477 0.768 0.195 0.473 4.740

HSE06 2.522 1.104 1.526 107.4 111.5 4.383 0.750 0.182 0.438 5.383

uorographene LDA GGA HSE06 2.557 2.609 2.582 1.365 1.382 1.364 1.555 1.583 1.568 108.3 107.9 108.1 110.7 111.0 110.8 2.963 3.103 4.933 0.363 0.366 0.352 0.312 0.305 0.264 0.863 0.893 0.731 8.066 7.911 8.952

and uorographene in a heterostructure. Therefore we performed some test calculations on strained HG and FG. These calculations show that the changes of the electronic properties are of the same order as the strain ( ≈2%) and can therefore be neglected. Another aspect that needs attention is the symmetry of the wavefunctions at the top of the valence band (VB) and the bottom of the conduction band (CB), shown in Fig. 2. Because the valence band is degenerate for both HG and FG, we show the sum of both wavefunctions to preserve the lattice symmetry in the gure. It is seen that the VB of graphane has px,y character and is localized on the C atoms, while the CB wave functions are plane wave-like states above the H atoms. 24 The VB of FG has also px,y character, although in this case there is also a contribution of the F atoms. The CB has a strong pz character and is located on the C and F atoms. So, in summary, the valence bands of HG and FG have similar orbital symmetry, while their conductions bands do not. A simple approximation of the band alignment in heterostructures can be obtained by the electron anity rule, also known as Anderson's rule. 25 This rule states that the vacuum levels of two materials in a heterostructure should be lined up. This can be easily done with 6


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Figure 2: (Color online) Top and side views of the charge density of the states at the top of the valence band (VB) and the bottom of the conduction band (CB) for graphane (a-b) and uorographene (c-d). the ionization energies given in Table 1. The IE of uorographene is more than 3 eV larger than that of graphane, which means that the bands are substantially shifted with respect to each other. The resulting band alignment for the dierent functionals is given if Fig. 3(a). There is a type-II alignment of the band structures in case of the hybrid functional and a type-III alignment in case of LDA and GGA. In the last case, the bottom of the conduction band of FG is below the top of the valence band of HG which would imply a charge transfer from graphane to uorographene. However, the dierence between the two levels is only about 0.2 eV, which is typically 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 osets from the total band structure. This is possible because the lattice mismatch between HG and FG is small enough to keep the changes of the band gap and ionization energy limited. In practice, we place a graphane and uorographene unit cell above each other in a single supercell with 15 Å of vacuum between them and relax the supercell size to reduce the strain. A subsequent band structure calculation provides the required band alignment which 7



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(a)

(b)

Figure 3: (Color online) The band alignment of graphane and uorographene for dierent functionals (FLTR: LDA, GGA, and HSE06) calculated with Anderson's rule (a) and within the same supercell (b). is shown in Fig. 3(b). All XC-functionals result in a type-II alignment and give consistent results for the dierence between the valence band maxima of HG and FG ( ≈ 2.7 eV). The dierence between conduction band minima, however, appears to vary signicantly (2.5-3.4 eV). Furthermore, the band gap in graphane appears to dier from the values in Table 1. Both discrepancies might be explained by the plane-wave nature of the graphane CB which is largely localized in the vacuum [see Fig. 2(b)] so that its energy level is more sensitive to perturbations. But this is only a minor issue because the CB of HG will not play an important role in the physics of the quantum dots which are the main subject of our investigation. Another dierence of LDA and GGA as compared to HSE06 is the size of the gap between the VBM of HG and the CBM of FG. This too is of minor importance because the corresponding wavefunctions have also very dierent symmetry and barely interact. Because DFT is essentially a ground-state theory, we can expect that the valence band maximum (VBM) is more accurately described than the conduction band minimum (CBM). The fact that the dierence between VBM of the two materials is less consistent for Ander8


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son'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 and qualitatively similar electronic properties than the HSE06 functional. Therefore we use the LDA xc-functional to relax the quantum dot structures in the following and perform a subsequent hybrid functional calculation on these relaxed structures to obtain the electronic properties of the quantum dots. The advantage of LDA above HSE06 is immense with respect to computation time and resources. LDA allows, therefore, to simulate much larger systems. The type-II band alignment of graphane and uorographene combined with the small lattice mismatch, make it possible to form two kinds of quantum dots (QD) from these materials: (i) a FG/HG QD with conned electron states as shown in Fig. 4(a) and (ii) a HG/FG QD with conned hole states, shown in Fig. 4(b). We consider both types of dots and examine the possibility of bound states in these dots in detail. Due to computational restrictions, the size of the simulated QD's has to be rather small. To have a rough estimate of the size of the dot that is needed to observe these localized states, we can make use of a simple analytical QD model.

(a)

(b)

Figure 4: (Color online) The band alignment of (a) a FG QD and (b) a HG QD. Dotted lines denote possible conned states. If we assume the dots to be innitely deep potential wells with circular shape, we can 9



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write the Schrödinger equation for a single particle with eective mass m∗ as follows: −

~2 2 ∇ Ψ(r) + V (r)Ψ(r) = EΨ(r), 2m∗

(1)

where the connement potential of the well is given by

V (r) =

   0

if r ≤ R

  ∞ otherwise,

and r =

(2)

p x2 + y 2 and R is the radius of the dot. This equation is readily solved and the

(radial) solutions are the well known Bessel functions, Jl . The corresponding eigenenergies are given by: En,l

2 ~2 βn,l , = 2m∗ R2

in which βn,l are the zeros of the Bessel functions which can be found in the literature

(3) 26

:

e.g. β0,0 = 2.4048. The lowest state is nondegenerate but the higher states are all doubly degenerate (for l and -l). The electrons in the FG dot have an eective mass of approximately 0.36 m e and their connement energy should be below 3.3 eV for the bound state to fall inside the band gap. Therefore, the size 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 connement energy of the heavy hole is smaller, we take the eective mass of the particle in the dot to be 0.46 me . To have a connement energy below 2.6 eV the dot radius should again be larger than 4.3 Å. Based on the simple model above, we restrict our DFT simulations to the two particular dots pictured in Fig. 5(a-b). These dots contain 24 and 54 C atoms, respectively, and have hexagonal shapes that are close to circles with a radius of approximately 4.5 and 6.7 Å. Due to the periodic boundary conditions in our simulations there will be some interaction 10


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(a)

(b)

Figure 5: (Color online) (a) Small QD containing 24 substituted atoms and (b) larger QD with 54 atom susbstitutions (red circles). between the dot and its periodic images. A minimum distance of about 8 Å between periodic images is maintained to reduce this interaction.

Figure 6: (Color online) Partial and total PDOS of graphane and uorographene part of the FG quantum dot system: (a) Small QD and (b) larger QD. The yellow band illustrates the band gap. We rst consider the case of a uorographene QD embedded in graphane. A schematic diagram of the band alignment is given in Fig. 4(a). The low-energy CB states of FG should give rise to conned electron states in the dot. These states are clearly visible in a projected density of states (PDOS) of the FG and HG part of the system as given in Fig. 6. The dierent symmetry of the states inside and outside the dot make sure that the electron states are well conned inside the dot (see below). Several peaks, corresponding to conned 11



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states inside the FG dot, can be observed. Some of these peaks are at energies above the CBM of graphane. In other words, there are some conned states of FG that fall inside the continuum of the graphane conduction band. In fact, only one bound state is found below the CBM for the small dot and 3 bound states (of which two are degenerate) are observed for the larger dot.

Figure 7: (Color online) Top and side views of the charge density of the states at the top of the valence band (a) and the lowest localized levels (b-c) for a uorographene QD. The wavefunctions corresponding with the top of the valence band and the lowest bound electron states are shown in Fig. 7. These wavefunctions were calculated with the hybrid functional on top of a LDA relaxed structure. For the VBM and the second bound state, the sum of two (almost) degenerate states is shown to illustrate the symmetry. As expected from Fig. 4(a), the VBM wavefunction is mainly located in the graphane shell but there is a substantial penetration into the FG dot. The bound states, on the other hand, are conned to the dot because they can not penetrate the graphane boundary due to the dierent symmetry of these states. Let us now consider a graphane dot in a uorographene environment. The graphane part in the middle of the supercell has a higher valence band maximum compared to the uorographene part [gure. 4(b)], so we should have a stronger contribution of graphane to the VBM of the system and a larger contribution of uorographene to the conduction band minimum. The PDOS of the two graphane dots, shown in Fig. 8, conrms this expectation. 12


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Figure 8: (Color online) Partial and total PDOS of graphane and uorographene part of the HG quantum dot system: (a) Small QD and (b) larger QD. 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 the energy is smeared out due to interaction with its periodic images. This interaction is possible because the localized state is close in energy to the VB states and has a similar symmetry, which makes hybridization possible. In other words, the bound state is smeared out over the FG boundary and can interact with neighboring dots. This is clearly illustrated by the shape of the wavefunction corresponding to the bound state [see Fig. 9(a)]: the state is mainly located in the dot but decays rather slowly away from the dot to the edge of the supercell. We can expect that a larger boundary between the dierent dots will reduce the hybridization (smearing) of the hole states so that discrete peaks could be observed in the DOS. Also a larger dot size will reduce the interaction because it connes the bound state more inside the dot. This can already be observed for the larger dot that we examined [Fig. 5(b)]. In the corresponding PDOS, shown in Fig. 8, the formation of a small peak at the VBM of FG can be observed which indicates a bound HG state.

Summary. We investigated the band alignment of graphane/uorographene heterostructures within the DFT formalism. The observed type-II alignment allows for the creation of graphane QDs inside a uorographene environment with conned hole states and uoro-

13



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Figure 9: (Color online) Top and side views of the highest localized level in the VB (a) and the state at the bottom of the conduction band (b) for a graphane QD. graphene QDs inside graphane crystals with conned electron states. The size of the QDs needs to be larger than ≈ 4.5Å for the conned states to fall inside the band gap. The electron states in the FG QD were found to be well conned due to a dierence in orbital symmetry between conduction band wavefunctions of FG and HG. This diers substantially from the bound hole states of the HG QDs which spread out from the QD region into the FG boundary. This smearing is caused by the similar symmetry and energies of the conned hole state and the valence band orbitals which allow them to hybridize.

Acknowledgments We gratefully acknowledge nancial 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 the University of Antwerp (CalcUA) a division of the Flemish Supercomputer Center VSC, which is funded by the Hercules foundation and the Flemish Government (EWI Department).

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