Graphene and Some of its Structural Analogues: Full-potential Density Functional Theory Calculations
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a r Graphene and Some of its Structural Analogues:Full-potential Density Functional TheoryCalculations
Gautam Mukhopadhyay ∗ and Harihar Behera † Department of Physics, Indian Institute of Technology, Powai, Mumbai-400076, India
Abstract
Using full-potential density functional calculations we have investigated the structuraland electronic properties of graphene and some of its structural analogues, viz., mono-layer (ML) of SiC, GeC, BN, AlN, GaN, ZnO, ZnS and ZnSe. While our calculationscorroborate some of the reported results based on different methods, our results on ZnSe,the two dimensional bulk modulus of ML-GeC, ML-AlN, ML-GaN, ML-ZnO and ML-ZnSand the effective masses of the charge carriers in these binary mono-layers are somethingnew. With the current progress in synthesis techniques, some of these new materials maybe synthesized in near future for applications in nano-devices.Keywords :
Graphene, Graphene-like materials, 2D crystals, Electronic structure,Firstprinciples calculations ∗ Corresponding author’s E-mail: [email protected]; [email protected] † E-mail: [email protected]; [email protected]
Introduction
Graphene is a crystal of carbon (C) atoms tightly bound in a two dimensional (2D) hexag-onal lattice. It is a monolayer of carbon atoms (ML-C), i.e., one atom thick. The exoticproperties of this 2D material were revealed only in 2004-2005 by a series of papers com-ing from the Manchester [1–3] and Columbia [4] groups. The unambiguous synthesis (bymechanical exfoliation of graphite), identification (by transmission electron microscopy(TEM)) and experimental determination of some of the exotic properties of graphene werereported first in 2004, by the Manchester group led by Novoselov and Geim [1]. In 2010,Konstantin S. Novoselov and Andre Geim were awarded the Nobel Prize in Physics for the“groundbreaking experiments regarding the two-dimensional material graphene”. How-ever, graphene research has a history which dates back to the 1859 work of Brodie [5–8].The term graphene was introduced by Boehm and his colleagues in 1986 [9]. Graphene =“graph” + “ene” and the term “graph” is derived from the word “graphite” and the suf-fix “ene” refers to polycyclic aromatic hydrocarbons. Now graphene is considered as thebasic building block of graphitic materials (i.e., graphite = stacked graphene, fullerenes= wrapped graphene, nanotube = rolled graphene, graphene nanoribbon = nano-scalefinite area sized rectangular graphene).The existence of free-standing 2D crystals were believed impossible for several years,because they would ultimately turn into a three-dimensional (3D) objects as predictedby Peierls [10], Landau [11] and Mermin [12]. The theoretical reason for this is thatat finite temperatures, the displacements due to thermal fluctuations could be of thesame magnitude as the inter-atomic distances, which make the crystal unstable. Fur-ther, experimentally one generally finds that thin films cannot be synthesized below acertain thickness, due to islands formations or even decomposition. Hence, the synthesisof graphene [1] was surprising which put a question mark on the predictions of Peierls,Landau and Mermin. However, this issue was (at least partially) solved when it was shownthat freestanding graphene sheets display spontaneous ripples owing to thermal fluctua-tion [13], and therefore real graphene is not perfectly flat. It is important to note thatsuch instabilities are the result of thermal fluctuations which disappear at temperature T= 0 ◦ K. This aspect will be used later in our discussions on the study of the stability ofgraphene and some of its structural analogues reported here on the basis of their groundstate (i.e., T = 0 ◦ K) total energies.Being a one-atom-thick planar crystal of C atoms, graphene is the thinnest nano-material ever known. It has exotic mechanical, thermal, electronic, optical and chemicalproperties, such as the high carrier mobility, a weak dependence of mobility on carrier con-centration and temperature, unusual quantum hall effect, hardness exceeding 100 timesthat of the strongest steel of same thickness and yet flexible (graphene can sustain elastictensile strain more than 20% without breaking, and is brittle at certain strain limit), high1hermal conductivity comparable to that of diamond and 10 times greater than that ofcopper, negative coefficient of thermal expansion over a wide range of temperature. Be-cause of these properties graphene has potentials for many novel applications [6, 7, 14–20].The rapid advancements of nanotechnology and the computing power have enabled theresearchers in the field to explore the unusual properties of graphene from many per-spectives of application and fundamental science. The study of graphene is possibly thelargest and fastest growing field of research in material science.The impressive growth in the research on graphene has inspired the study of othergraphene-like 2D materials [20, 21]. For instance, a number of 2D/quasi-2D nanocrystals(not based on carbon) have been synthesized or predicted theoretically in recent years.Representative samples of other 2D nanocrystals which have been synthesized includeBN, MoS , MoSe , Bi Te [20, 21], Si [22], ZnO [23]. Recently, the density functionaltheory (DFT) calculations of Freeman group [24] have shown that when the layer num-ber of (0001)-oriented wurtzite (WZ) materials (e.g., AlN, BeO, GaN, SiC, ZnO andZnS) is small, the WZ structures transform into a new form of stable hexagonal BN-likestructure. This prediction has recently been confirmed in respect of ZnO [23], whose sta-bility is attributed to the strong inplane sp hybridized bonds between Zn and O atoms.Graphene-like 2D/quasi-2D honeycomb structures of group- IV and III − V binary com-pounds have also been studied [25, 26] by using pseudo-potential DFT calculations. Here,we report our calculations on the structural and electronic properties of graphene andsome other graphene-like monolayer (ML) structures of the binary compounds, viz., SiC,GeC, BN, AlN, GaN, ZnO, ZnS, ZnSe using the DFT. We use the DFT based full-potential (linearized) augmented plane wave plus local orbital(FP-(L)APW+lo) method [27–29] as implemented in the elk-code (http:// elk.sourceforge.net/)and the Perdew-Zunger [30] variant of local density approximation (LDA) for our calcu-lations. The accuracy of this method and code has been successfully tested in our pre-vious studies [31–37]. For plane wave expansion in the interstitial region, we have used8 ≤ | G + k | max × R mt ≤
9, where R mt is the smallest muffin-tin radius, for deciding theplane wave cut-off. The Monkhorst-Pack [38] k -point grid size of 20 × × × × µ eV/atom. The 2D-hexagonalstructure was simulated by a 3D-hexagonal super cell with a large value of c -parameter(= | c | = 40 a.u.) as shown in Figure 1. For structure optimization, we have consideredtwo different structures, viz., (i) Planar structure (PL) and (ii) Buckled structure (BL)as shown in Figure 1 for a monolayer of BN (ML-BN), which is the prototype of all the2igure 1: Two possible structures of ML-BN: Planar (PL) and Buckled (BL) in ball-and-stick model. Sub-figures (a) and (b) show the top-down views of ML-BN in PLand BL structure respectively. In the buckled ML-BN, B atoms (large balls, red) and Natoms (small balls, green) are in two different parallel planes (sub-figure (d)), bucklingparameter ∆ is the perpendicular distance between those parallel planes and ∆ = 0 ˚A fora flat ML-BN (sub-figure (c)). Sub-figures (e) and (f) respectively depict the 3D supercell of hexagonal BN in PL and BL structure with a large value of c -parameter (the lengthalong the vertical axis).materials considered here (for graphene, B and N atoms are to be replaced by the Catoms). We have optimized the 2D hexagonal structures of graphene (ML-C) and the mono-layers of SiC (ML-SiC), GeC (ML-GeC), BN (ML-BN), AlN (ML-AlN), GaN (ML-GaN),ZnS (ML-ZnS) using the principle of minimum energy for the stable structure as perthe following procedure. Initially we assumed the planar structures for all the materialsconsidered here. With this assumption, we calculated the ground state in-plane latticeparameter ( a ) of each of these structures as listed in Table-1 along with the availablereported values by other authors. Then we investigated the possibility of buckling inthese structures at their assumed planar ground states (characterized by their respective a values in planar states) by introducing the concept of bucking (Figure 1) and the3rinciple of minimum energy for the most stable structure. Our calculated variation oftotal energy (E) with buckling parameter (∆) at fixed value of a was then plotted tolook for the value of ∆ which corresponds to the minimum energy. (LDA) E ne r g y : E - E r e f ( m R y / a t o m ) Buckling parameter: ( ¯ )ML-CE ref = -75.62 Ry/atoma = 2.445 ¯ Figure 2: Buckling probe of monolayer graphene (ML-C).In Figures 2-4, we present our calculated results selecting one example from eachgroup of materials considered here: graphene (ML-C) for Group-IV material, ML-BN forGroup-III-V material and MLZnS for Group-II-VI material. As seen in the Figures 2-4,these structures have minimum energy at ∆ = 0 .
00 ˚A , which means that these materialsadopt 2D planar structures in the ground state (T = 0 ◦ K) unlike the case with silicene(graphene analogue of Si) [22, 31] that adopts a buckled structure in its ground state. Itis to be noted that these results do not conflict with the theory of stability of 2D crys-tals [10–12]. Our calculated optimized structural parameters of the materials are listed inTable 1 along with available reported results based on other methods. The stability testsML-ZnO and ML-ZnSe are not yet complete.Considering the facts that LDA usually underestimates and GGA [39] usually over-estimates the lattice constant, the calculated value of a slightly depends on the c -parameter used for super cells in DFT based calculations as we have seen before [31]for graphene (0 .
12% lower value of a for c → ∞ ) and silicene (0 . a for c → ∞ ), and the different methods of study used by different authors, our results inTable 1 are acceptable.Using the definition of the 2D bulk modulus of planar a graphene-like 2D material as B D = A ( ∂ E/∂A ) | A min (where A is the area of the periodic cell of the 2D lattice and A min is the area with minimum energy), we have calculated the bulk modulus of each4able 1: Calculated LDA values of ground state in-plane lattice constant a (= | a | = | b | ), buckling parameter ∆ of monolayer graphene (MLC) and some structurally similarbinary compounds compared with available calculated data. PAW stands for the projectoraugmented wave.Material (2D) a (˚A) ∆ (˚A) Remark/Reference2.445 0.0 Our Work/ [31, 33]ML-C 2.46 0.0 PAW-potential/ [25]2.4431 0.0 Pseudo-potential/ [26]3.066 0.0 Our Work/ [37]ML-SiC 3.07 0.0 PAW-potential/ [25]3.0531 0.0 Pseudo-potential/ [26]ML-GeC 3.195 0.0 Our Work/ [37]3.22 0.0 PAW-potential/ [25]2.488 0.0 Our Work/ [33, 34]ML-BN 2.51 0.0 PAW-potential/ [25]2.4870 0.0 Pseudo-potential/ [26]ML-AlN 3.09 0.0 Our Work/ [34]3.09 0.0 PAW-potential/ [25]ML-GaN 3.156 0.0 Our Work/ [37]3.20 0.0 PAW-potential/ [25]ML-ZnO 3.20 0.0 (used) Our Work/ [35]3.283 0.0 PAW-potential/ [46]ML-ZnS 3.7995 0.0 Our Work/ [36]3.890 0.0 PAW-potential with GGA [39]/ [47]ML-ZnSe 3.996 0.0 (used) Our Work/ [37]5 .00 0.01 0.02 0.03-1.35-1.30-1.25-1.20-1.15-1.10-1.05 a = 2.488 ¯ E ne r g y : E - E r e f ( m R y / a t o m ) Buckling parameter: ( ¯ ) ML-BN E ref = -79.11 Ry/atom(LDA) Figure 3: Buckling probe of monolayer BN (ML-BN).Table 2: Calculated 2D bulk modulus ( B D ) value of monolayer graphene and someother mono-layers of binary compounds in graphene-like structure compared with availablecalculated data using other methods. Our cited values of B D from Wang [26] wereobtained from Wang’s calculated values of 2D elastic constants γ and γ and anotherdefinition of B D = ( γ + γ ) / B D (N/m) Remark/ReferenceML-C 223.85 This Work214.41 Pseudo-potential/ [26]220.00 Pseudo-potential/ [48]ML-SiC 125.66 This Work121.945 Pseudo-potential/ [26]ML-GeC 113.39 This WorkML-BN 188.03 This Work181.91 Pseudo-potential/ [26]212.0 Born’s Perturbation Method/ [49]ML-AlN 113.44 This WorkML-GaN 109.45 This WorkML-ZnO 94.78 This WorkML-ZnS 53.94 This WorkML-ZnSe 48.57 This Work6 .00 0.05 0.10 0.15-3.85-3.80-3.75-3.70-3.65-3.60 (LDA) E ne r g y : E - E r e f ( m R y / a t o m ) Buckling parameter: ( ¯ ) ML-ZnS E ref = - 2191.48 Ry/atoma = 3.7995 ¯ Figure 4: Buckling probe of monolayer ZnS (ML-ZnS).of the materials considered here. These calculated values of B D are listed in Table 2along with the available calculated data based on other methods. As seen in Table 2,the 2D bulk modulus of graphene has the highest value and that of ML-ZnSe has thelowest value among the materials considered here, implying the fact that graphene hasstronger in-plane bonds than the others and ML-ZnSe has the weakest bond among thesematerials. ML-GeC is seen to have same strength as that of ML-AlN. These data may beuseful in selecting the 2D materials having desired level of strength in nano-mechanicalapplications in specific situations.The LDA band structures of graphene (ML-C) and some other mono-layers of SiC,GeC, BN, AlN, GaN, ZnO, ZnS and ZnSe in graphene-like planar structure are depictedin Figure 5 and in Table 3, we have listed our calculated LDA values of the band gaps ofthese materials along with the reported values. As seen in the explicit case of ML-BN inTable 3, although LDA under-estimates the band gap (in our case the calculated value isabout 23% less than the experimental value of 5.971 eV) it correctly predicts the natureof the gap. However, the use of a more advanced approximation, such as the LDA + GW used in [25], improves the band gap problem (in the case of ML-BN although it overesti-mates band gap by 15% in this case), but the authors of [25] have presented their resultin terms of an indirect band gap for ML-BN as opposed to the experimental observationof direct band gap in ref. [40]. In all other cases, the LDA nature of band gap is the sameas that of the LDA + GW band gap. The important point is that the actual band gapis always more than the LDA value. 7able 3: Calculated LDA band gaps compared and listed with reported values.Material (2D) E g (eV) Remark/ReferenceML-C 0 ( K → K ) Our Work/ [31, 33]0 ( K → K ) PAW-potential/ [25]0 Pseudo-potential/ [26]ML-SiC 2.547( K → M ) Our Work/ [37]2.52 ( K → M ) PAW-potential/ [25]4.19 ( K → M ) PAW-potential (LDA + GW )/ [25]2.39 Pseudo-potential/ [26]ML-GeC 2.108( K → K ) Our Work/ [37]2.09 ( K → K ) PAW-potential/ [25]3.83 ( K → K ) PAW-potential (LDA + GW )/ [25]ML-BN 4.606( K → K ) Our Work/ [33, 34]4.61 ( K → K ) PAW-potential/ [25]6.86 ( K → Γ) PAW-potential (LDA + GW )/ [25]4.35 Pseudo-potential/ [26]5.971 (direct) Experiment/ [40]ML-AlN 3.037( K → Γ) Our Work/ [34]3.08 ( K → Γ) PAW-potential/ [25]5.57 ( K → Γ) PAW-potential (LDA + GW )/ [25]ML-GaN 2.462( K → Γ) Our Work/ [37]2.27( K → Γ) PAW-potential/ [25]ML-ZnO 1.680(Γ → Γ) Our Work/ [35]1.68 (Γ → Γ) PAW-potential/ [46]ML-ZnS 2.622(Γ → Γ) Our Work/ [36]2.07 (Γ → Γ) PAW-potential with GGA [39]/ [47]ML-ZnSe 1.866 (Γ → Γ) Our Work/ [37]8able 4: Calculated values of effective masses of the charge carriers in ML-C, ML-Si,ML-Ge, ML-BN, ML-AlN, ML-GaN, ZnO, ZnS, ZnSe. m is the rest mass of electron.Material (2D) Effective mass ( m ) Remark/ReferenceML-C m e ( K ) = 0 This Work m h ( K ) = 0 This WorkML-SiC m e ( M ) = 0 .
411 This Work m h ( K ) = − .
488 This WorkML-GeC m e ( K ) = 0 .
509 This Work m h ( K ) = − .
400 This WorkML-BN m e ( K ) = 0 .
920 This Work m h ( K ) = − .
617 This WorkML-AlN m e (Γ) = 0 .
523 This Work m h ( K ) = − .
470 This WorkML-GaN m e (Γ) = 0 .
266 This Work m h ( K ) = − .
157 This WorkML-ZnO m e (Γ) = 0 .
253 This Work m lh (Γ) = − .
374 This Work m hh (Γ) = − .
793 This WorkML-ZnS m e (Γ) = 0 .
173 This Work m lh (Γ) = − .
154 This Work m hh (Γ) = − .
665 This WorkML-ZnSe m e (Γ) = 0 .
107 This Work m lh (Γ) = − .
103 This Work m hh (Γ) = − .
652 This Work9igure 5: Band structures of graphene (ML-C) and some other mono-layers of SiC, GeC,BN, AlN, GaN, ZnO, ZnS and ZnSe in graphene-like planar structure within LDA.In Table 4, we present our calculated values of the effective masses of the charge car-riers in ML-C, ML-Si, ML-Ge, ML-BN, ML-AlN, ML-GaN, ZnO, ZnS, ZnSe, determinedat the band edges at the special points as appropriate for the material under study. Fromthe linear energy dispersion close to the K point of the hexagonal Brillouin Zone (BZ) asshown in Figure 6, mass-less carriers in graphene were inferred. As seen in Figure 6, nearthe K point of the BZ, the energy bad dispersion is linear in k : E ± = ± v F ~ k (1)where v F is the magnitude of Fermi velocity of the charge carriers in graphene and ( ~ k ) isthe magnitude of the momentum. Thus, charge carriers in graphene behave like mass-lessrelativistic particles. From the slope of the linear bands one obtains the value of v F , whichin our calculation corresponds to v F = 0 . × m/s. This value of v F is close to the10igure 6: Linear energy dispersion near the K point in monolayer graphenes band struc-ture.experimentally measured value of v F = 0 . × m/s in graphene monolayer deposited ongraphite substrate [41] and the reported calculated LAPW value of 0 . × m/s [42]and about 20% less than that measured in coupled multi-layers [43, 44] and the tight-binding value: v F = 1 × m/s [15]. The smallness in the value of v F in ref. [41] wasattributed to the electron-phonon interactions due to strong coupling with the graphitesubstrate. However, the closeness of our calculated value of v F for a freestanding graphenemonolayer with the results of ref. [41] suggests a very weak coupling of graphene to thegraphite substrate used in ref. [41]. Surprisingly, our calculated value of v F is close to themeasured value of v F = 0 . × m/s [45] in metallic single-walled carbon nanotube. Using full potential DFT calculations we have investigated the structural and electronicproperties of graphene and some other graphene-like materials. While our results cor-roborate the previous theoretical studies based on different methods, our calculations onML-ZnSe, the two-dimensional bulk modulus of ML-GeC, ML-AlN, ML-GaN, MLZnO,ML-ZnS, and the effective masses of the binary monolayer compounds considered hereare our new results. We hope, with the advancement of fabrication techniques, the hypo-thetical graphenelike materials discussed here will be synthesized in the near future forpotential applications in a variety of novel nano-devices.11 eferences [1] Novoselov K.S., Geim A.K., Morozov S.V., Jiang D., Zhang Y., Dubono S.V., Grig-orieva, I.V. and Firsov A.A., 2004. Electric field effect in atomically thin carbon films.
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