Influence of structure and thermodynamic stability on electronic

PHYSICAL REVIEW B 92, 075435 (2015)

Influence of structure and thermodynamic stability on electronic properties of two-dimensional SiC, SiGe, and GeC alloys I. Guilhon,* L. K. Teles,† and M. Marques‡ Grupo de Materiais Semicondutores e Nanotecnologia, Instituto Tecnol´ogico de Aeron´autica, DCTA, 12228-900 S˜ao Jos´e dos Campos, Brazil

R. R. Pela§ Grupo de Materiais Semicondutores e Nanotecnologia, Instituto Tecnol´ogico de Aeron´autica, DCTA, 12228-900 S˜ao Jos´e dos Campos, Brazil and Humboldt-Universit¨at zu Berlin, Institut f¨ur Physik and IRIS Adlershof, Theoretische Festk¨orperphysik Zum Groβen Windkanal 6, 12489 Berlin Germany

F. Bechstedt Institut f¨ur Festk¨orpertheorie und -optik, Friedrich-Schiller-Universit¨at, Max-Wien-Platz 1, D-07743 Jena, Germany (Received 2 June 2015; published 24 August 2015) The energetics and thermodynamic properties of two-dimensional binary graphene-like alloys made from graphene, silicene, or germanene are investigated by combining first-principles total energy calculations, and a statistical approach to account for disorder and composition effects. For the electronic properties the calculations are performed within the GGA-1/2 approach for an approximate quasiparticle bands. We derive lattice constants, first-neighbor distances, and buckling parameters as a function of composition x. The Si1−x Gex system is the only stable random alloy at usual growth temperatures. For Ge1−x Cx , we observe strong distortions of the lattice making the random configurations less favorable and leading to a pronounced tendency for phase separation. The situation for Si1−x Cx alloys is completely different. An ordered structure with composition x = 0.5 is stable up to T ≈ 1000 K, while intermediate compositions are mainly realized by silicongraphene and graphene or silicene. The ordering and decomposition effects have a strong influence on the average fundamental energy gap versus composition. Whereas large gaps appear for Si1−x Cx systems they almost vanish for Ge1−x Six and Ge1−x Cx . Moreover, the dependence of the Si1−x Cx energy gap on growth temperature is also obtained. The results can be very useful for chemical vapor deposition growth of these materials. DOI: 10.1103/PhysRevB.92.075435

PACS number(s): 73.22.Pr, 73.22.−f, 64.75.Jk

I. INTRODUCTION

The study of two-dimensional (2D) materials is an exciting field that has received an extraordinary amount of interest from both academia and industry after the synthesis of graphene. The world-wide attention has emerged from its exceptional mechanical, electronic, and thermal properties, which are closely related to the linear energy-momentum dispersion relations at the Dirac point. However, a key feature that makes the use of graphene unlikely in high-performance integrated logic circuits as a planar channel material is the absence of a band gap [1]. Another limitation of its application in electronics is the difficult integration into the current Si-based technology. These facts have triggered the search for other 2D atomically thin crystals, such as monolayers of hexagonal boron nitride (h-BN) [2], transition metal dichalcogenides (TMDs) [3], such as molybdenum disulfide (MoS2 ) [4], and many others [5]. Strong effort has been invested to study graphene-like 2D materials composed of other group-IV elements. It has been suggested that 2D sheets of Si and Ge, referred to as silicene and germanene, respectively, might exhibit similar

*

[email protected] [email protected] ‡ [email protected] § [email protected] †

1098-0121/2015/92(7)/075435(12)

properties as graphene, but with a low buckled structure instead of the planar one [6]. They have progressed from theoretical predictions to experimental observations in only a few years [7]. The formation of silicene-like adlayer was reported on Ag (1 1 1) [8–10], ZrB2 (0 0 0 1) [11], and Ir(1 1 1) [12] surfaces. Very recently, germanene was grown on the Au(1 1 1) surface [13], on Pt(1 1 1) substrate [14], and as a termination of Ge2 Pt crystals on Ge(1 1 0) [15]. Hydrogenated germanene, called germanane, has been recently synthesized by an exfoliation technique [16]. The angle-resolved photoemission spectroscopy revealed the presence of a linear dispersion in the band structure of silicene (so called Dirac cones) with a Fermi velocity of about 1.3 × 106 m/s [8], higher than expected from free-standing graphene (1.1 × 106 m/s). The corresponding measured value for 2D silicon adlayer [8] seems to be closer to velocities as theoretically predicted [17,18]. Alloys, so far, have not been studied experimentally. However, intensive research has been focused on fabricating SiC nanostructures [19], and recently the first photoluminescence results have been published for ultrathin SiC films, which may be considered as an ordered alloy of 2D silicon and carbon at 50% composition [20]. A common feature in devices is the use of three-dimensional (3D) alloys, which allows us to vary the band gap between the values of the end components. Considering the 2D group-IV materials, it is natural to try to tailor the electronic properties by alloying the elements with different compositions. This increases the potential to modulate the electronic structures

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PHYSICAL REVIEW B 92, 075435 (2015)

and to widen possible actual applications. Such a concept has been already realized for 2D TMDs. 2D-TMD alloys, such as Mo1−x Wx S2 [21], Mo1−x Wx Se2 [22], MoS2(x) Se2(1−x) [23], and CoMoS2 [24] have been synthesized recently for the study of continuous tunable optical properties. Recently, Wei et al. [25] discussed the phase stability of mixed single-layer TMDs, showing that the different alloyed TMDs have great distinct stability and electronic structures. More recently, the honeycomb structures of binary groupIV compounds Si1−x Cx , Si1−x Gex , and Ge1−x Cx have also been reported to exhibit interesting electronic properties. Theoretical predictions showed an energy gap for an ordered configuration of Ge0.5 C0.5 , Ge0.5 Si0.5 , and Si0.5 C0.5 of 3.16 eV, 0.285 eV, and 3.53 or 3.7 eV, respectively [26,27]. They raise the questions: (i) Is it possible to engineer their band gaps by varying the composition of the alloyed 2D binaries? (ii) Would these alloys be thermodynamically stable? (iii) How is the influence of alloying on the electronic properties? In this work, we present a rigorous and systematic theoretical study of 2D Si1−x Cx , Si1−x Gex , and Ge1−x Cx alloys, considering several different local atomic configurations and a statistical average to study the thermodynamic, structural, and electronic properties versus the composition of the compounds. The calculations performed here are based on an ab initio density functional theory (DFT) [28,29] and the generalized quasichemical approach (GQCA) combined with a cluster expansion of the thermodynamic potentials [30,31]. It has been demonstrated that this combination is able to successfully describe the physical properties of several 3D alloys [32–36]. We focus on the temperature-composition (T -x) phase diagram and discuss the miscibility versus decomposition. For random alloys we compute the lattice constants, the layer bucklings, the chemical bonds, and the band gap behavior as a function of the alloy composition x. The paper is organized as follows. Section II describes the theoretical methods adopted for the calculations, while the results and a detailed discussion of the alloy behavior and properties are given in Sec. III. Finally, Sec. IV is devoted to the conclusions.

II. METHODOLOGY

In order to analyze the disorder effect on the alloys, we use the GQCA, whose successful application to 3D alloys is described elsewhere [30,31,33]. Essentially, an alloy is described as an ensemble of individual clusters independent statistically and energetically of the surrounding atomic configuration. Each cluster j with a certain number of each atom is associated with a certain probability xj . In this work, the clusters are described by a 2 × 2 unit cell. The total energy of each cluster εj is calculated by adopting the state of art ab initio simulations, based on the DFT as implemented in the VASP code [37,38]. We use the generalized gradient approximation (GGA) for exchange and correlation (XC), as proposed by Perdew-Burke-Ernzerhof (GGA-PBE) [39,40]. The Kohn-Sham equations are solved by employing the projector augmented wave (PAW) scheme [41,42]. We use an energy cutoff parameter of Ecut = 400 eV and a 9 × 9 × 1 -centered k-point mesh.

FIG. 1. (Color online) Eight-atom calculations.

supercell

used

in

the

In order to simulate the quasiparticle shifts we apply the GGA-1/2 approach [43,44]. This method is a simple, low-time-consuming, parameter-free procedure to compute excitation energies. Therefore, it is adequate in our case for performing calculations for several different configurations. Considering 3D materials the GGA-1/2 method predicts energy band gaps in very good agreement with experimental data, including alloys [36]. For 2D materials it was recently applied to 2D allotropes of group-IV materials giving results in reasonable agreement when compared with gaps derived by hybrid functionals [45]. Within the GQCA the clusters with the energy εj used in the calculation of the internal energy have eight atoms, as shown in Fig. 1. Their properties are modeled by 2D crystals consisting of repeated unit cells corresponding to the clusters. These 2D sheets are arranged in 3D superlattices with about ˚ of vacuum between the sheets. Tests have shown a 20 A vanishing interaction between the neighboring planes, which allow us to consider them as isolated from each other. The atomic structure of each cluster is optimized with constant cell shape but varying 2D lattice constant via total energy minimization. Thereby, all the atomic coordinates in the supercell are relaxed until the Hellmann-Feynman forces are ˚ −1 . smaller than 0.01 eV A Here, we study A8−nj Bnj clusters with nj = 0,1, . . . ,8 and A,B = C, Si, and Ge, forming the 2D A1−x Bx alloys. We have a certain number J of configuration classes that cannot be a result of an application of a symmetry operation on a cluster of another class. Each one of these classes has different degeneracies gj and different physical properties, such as the cluster total energy εj , for example. Considering the sites shown in Fig. 1, we investigate the cluster symmetry and derive all 256 possible configurations, which can be arranged in J = 22 different classes with different degeneracies gj (j = 1, . . . ,22). The cluster classes determined for hexagonal 2D binary alloys with their respective degeneracies are described in Table I. The use of the next larger 3 × 3 cell with 18 atoms would, however, lead to the need for studying too many atomic configurations. Moreover, 16-atom supercells, which also leads to 22 nonequivalent configurations, are already sufficient in the 3D case [36,46]. Since the studied 2D systems are quantum systems with a finite extent in the direction of the layer normal, we generalize the GQCA method successfully to describe 2D alloys [30,31,33–36]. Within the GQCA, the Helmholtz free

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TABLE I. The 22 different cluster classes of hexagonal eightatom supercells to study 2D alloys. The “A atoms” column represents a possible configuration belonging to each j class. The site labeling in the cluster can be found in Fig. 1. nj

gj

A atoms

j

nj

gj

A atoms

1 2 3 4 5 6 7 8 9 10 11

0 1 2 2 2 3 3 3 4 4 4

1 8 4 12 12 8 24 24 6 8 2

– 1 1,8 1,3 1,2 1,3,5 1,2,7 1,2,3 1,2,3,4 1,2,3,5 1,3,5,7

12 13 14 15 16 17 18 19 20 21 22

4 4 4 5 5 5 6 6 6 7 8

6 24 24 24 8 24 4 12 12 8 1

1,2,7,8 1,2,3,7 1,2,3,6 1,2,3,4,5 1,2,3,5,7 1,2,3,6,7 1,2,3,6,7,8 1,2,3,4,5,7 1,2,3,4,5,6 1,2,3,4,5,6,7 1,2,3,4,5,6,7,8

energy of the system can be divided into two contributions:

J 

(3)

(4)

p(x,T ) =

J 

xj (x,T )pj ,

xj (x,T )εj − [xε1 + (1 − x)ε22 ]

+ kT [x log(x) + (1 − x) log(1 − x)]    xj (x,T ) xj (x,T ) log + kT , xj0 j

(10)

where pj is the property of each cluster class j . Calculating F (x,T ) it is possible to construct the temperature-composition (T -x) phase diagram of the considered A1−x Bx alloy. In the case that U (x,T ) > 0 at T = 0, there exists a finite critical temperature Tc , where the binary alloy is stable for all concentrations if the condition T > Tc is fulfilled. However, if T < Tc we have a miscibility gap for concentrations x in the interval x1 < x < x2 . The points x1 and x2 can be determined with the common tangent line that touches F (x,T ) from below. Since μ(x,T ) = ∂F∂x(x,T ) , then the alloys with average compositions x1 and x2 have the same chemical potential:

(5)

j =1

(9)

j =1

μ(x1 ,T ) = μ(x2 ,T ). xj (x,T )εj − (xε1 + (1 − x)ε22 ),

and S is called mixing entropy. Let xj (x,T ) be the probability of an individual cluster belonging to the j class, where j = 1,2, . . . ,J for an average alloy composition x at a temperature T . It is determined by minimizing the expression of mixing free energy of the alloy, which can be written as J 

(8)

where n = 8 is the number of atoms in each cluster and xj0 = gj x nj (1 − x)8−nj is the cluster probability of configuration j for a random alloy. Equation (7) is associated to the probability normalization condition and Eq. (8) to the concentration of B atoms in the alloy, which can be calculated from the probabilities xj . By using Lagrange multipliers formalism, we calculate the values for the probabilities xj that minimize the mixing free energy respecting these two constraints. The calculated xj is given by

(11)

With these concentrations it is possible to affirm that x2 − x x − x1 F (x1 ,T ) + F (x2 ,T ) < F (x,T ), (12) x2 − x1 x2 − x1

j =1

F (x,T ) =

nj xj (x,T ) = nx,

j =1

(2)

where U is called the mixing internal energy defined by U (x,T ) =

J 

where λ is a parameter determined from the condition Eq. (8). With the set of probabilities xj calculated, an arbitrary property p(x,T ) of interest of the alloy can be estimated as

This contribution describes how the two species mix, by comparing the alloy’s free energy with the free energy of the isolated 2D crystals. The knowledge of the mixing free energy is important to determine for which concentrations miscibility or phase separation may occur for a certain temperature. The mixing free energy can be written as F (x,T ) = U (x,T ) − T S(x,T ),

(7)

gj λnj exp(−εj /kB T ) xj (x,T ) = J , nj j =1 gj λ exp(−εj /kB T )

is the contribution associated with the interpolation of the free energies of the pure materials A and B, which may represent graphene, silicene, or germanene. The second contribution is called mixing free energy: F (x,T ) = F (x,T ) − F0 (x).

xj (x,T ) = 1

j =1

(1)

where F0 (x) = xFA + (1 − x)FB

J 

and

j

F (x,T ) = F0 (x) + F (x,T ),

with the conditions

so that phase separation is thermodynamically favored in the interval x1 < x < x2 into one phase with B concentration of x1 and the other one with B concentration of x2 . The amount of each phase is determined by the probabilities (x − x1 )/(x2 − x1 ) and (x2 − x)/(x2 − x1 ). The set of points (x1 ,T ) and (x2 ,T ) constitutes the binodal curve. There are certain compositions where there is no barrier to nucleation. One of these is the spinodal mode, which occurs for any alloy composition where the free energy curve has a negative curvature [47], i.e.,

(6)

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Thus, the corresponding composition region of spinodal decomposition is determined by x1 or x2 , which are given by the equation     ∂2 ∂2  F (x,T ) = F (x,T ) = 0. (14) 2 2 ∂x ∂x x 1 x 2 The set of points (x1 ,T ) and (x2 ,T ) represents the spinodal curve. An alloy with x1 < x < x2 , below Tc , is unstable because small fluctuations in composition that produces A-rich and B-rich regions will cause a decrease of total free energy. If the alloy lies outside the spinodal with composition x where x1 < x < x1 or x2 < x < x2 , small variations in the composition lead to an increase in the free energy and the alloy is therefore metastable. In other words, in these composition range there is an energy barrier that has to be overcomed. Besides the phase separation process, there is the possibility of chemical ordering, long-range- or short-range-ordered structures can be more stable than the decomposed alloy below some temperature Tc,ord . III. RESULTS A. Thermodynamic stability and energetic ordering

In an alloy where the mixing internal energy is not zero, the assumption that a random arrangement of atoms is the equilibrium geometry, or the most stable one, might not be true. The actual arrangement of atoms will be a compromise that gives the lowest internal energy consistent with sufficient entropy, or randomness, to achieve the minimum free energy

Eq. (4). However, also ordered structures with low excess energies may occur. In systems with U < 0 the internal energy of the system is reduced by increasing the number of A-B bonds, i.e., by ordering the atoms. If U > 0 the internal energy can be reduced by increasing the number of A-A and B-B bonds, i.e., by the clustering of the atoms in A-rich and B-rich domains. However, the degree of ordering or clustering will decrease as temperatures increase due to the increasing importance of entropy. First, we investigate the relative stability of ordered and disordered phases by observing the behavior of F at T = 0 K, which corresponds to analyze the values of U . In addition, for cluster class j we compute the excess energy:  nj  nj ε1 − ε22 . εj = εj − 1 − 8 8

(15)

They are listed in Table II. For the 22 configurations studied for Ge1−x Cx and Si1−x Gex the values are nonnegative, while for Si1−x Cx the configuration 11 exhibits an energy gain ε11 < 0, that indicates a tendency for ordering in Si1−x Cx . In Fig. 2, this fact is indicated by U < 0 for all x, despite the fact that, apart from configuration 11, any configuration has an energy greater or equal to the Si1−x Cx alloy consisting mainly of the configurations 1 (Si), 11 (SiC), and 22 (C). Indeed the U (x,T = 0 K) curve has one inflection point at x = 0.5, which is mainly due to the ordered structure corresponding to configuration 11 (displayed as an inset in Fig. 2). In this configuration the atomic sites represent an ideal honeycomb lattice and are occupied alternately. We conclude

˚ and energy gap Egj (eV) TABLE II. Excess energy εj (eV), statistical contribution xj (x = 0.5,T = 800 K), buckling parameter j (A), of the cluster configurations Ge8−nj Cnj , Si8−nj Genj , and Si8−nj Cnj . Metallic configurations are indicated by “metal.” Ge8−nj Cnj Class j 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

εj 0.00 2.57 3.76 4.16 2.51 4.91 5.19 3.13 3.26 5.49 5.49 5.31 5.70 4.78 5.28 6.64 6.22 4.40 6.90 7.63 5.34 0.00

Si8−nj Genj

xj

j

Egj

0.402 0.031 0.002 0.003 0.053 0.000 0.001 0.035 0.007 0.000 0.000 0.000 0.000 0.002 0.001 0.000 0.000 0.001 0.000 0.000 0.000 0.462

0.69 0.85 0.71 1.65 2.81 1.67 1.49 2.81 2.01 2.37 0.00 0.75 1.91 2.76 2.15 0.85 1.36 0.24 0.00 1.07 0.06 0.00

0.000 Metal Metal Metal Metal 1.442 0.446 1.095 Metal Metal 2.072 Metal 1.089 Metal 0.529 1.352 0.619 0.000 0.943 Metal 0.674 0.000

εj 0.00 0.10 0.17 0.18 0.16 0.26 0.21 0.20 0.21 0.23 0.32 0.23 0.24 0.22 0.20 0.26 0.22 0.17 0.19 0.16 0.11 0.00

Si8−nj Cnj

xj

j

Egj

εj

xj

j

Egj

0.006 0.038 0.017 0.048 0.050 0.028 0.091 0.093 0.023 0.029 0.006 0.022 0.086 0.091 0.093 0.028 0.091 0.016 0.048 0.051 0.038 0.006

0.43 0.48 0.51 0.51 0.51 0.55 0.55 0.55 0.59 0.58 0.58 0.58 0.58 0.58 0.62 0.61 0.62 0.64 0.65 0.65 0.67 0.68

0.000 0.003 0.000 0.005 0.003 0.007 0.005 0.003 0.008 0.004 0.007 0.008 0.006 0.000 0.005 0.005 0.005 0.000 0.002 0.006 0.001 0.000

0.00 1.41 1.26 2.10 1.44 0.82 2.76 3.67 2.56 2.82 −0.18 2.32 2.53 3.46 3.72 2.28 3.60 2.18 3.52 3.84 3.24 0.00

0.061 0.044 0.033 0.021 0.072 0.169 0.015 0.003 0.006 0.005 0.300 0.010 0.026 0.005 0.003 0.016 0.004 0.011 0.003 0.002 0.004 0.187

0.43 0.48 0.08 0.68 2.70 1.51 0.57 1.28 1.79 1.99 0.00 0.03 1.70 2.48 1.91 0.02 0.55 0.01 0.00 2.29 0.01 0.00

0.000 0.202 0.000 0.485 Metal 1.832 0.561 0.020 Metal Metal 2.534 0.003 1.374 Metal 0.563 1.429 0.579 0.000 1.295 0.079 0.766 0.000

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FIG. 2. (Color online) Plot of the excess energies Eq. (15) of each Si8−nj Cnj configuration. In addition, the internal energy U of Si1−x Cx , calculated through 5, is shown as a function of the concentration x. The inset figure corresponds to configuration 11, i.e., sp 2 -bonded SiC [27].

that the Si1−x Cx alloy consists of domains of ordered 2D SiC, the silicongraphene [27], graphene, and silicene depending on the alloy composition. For x = 0.5 at T = 0 K only silicongraphene is realized. This result is underlined by Fig. 3 for the composition x = 0.5. In Fig. 3(a) the values of xj are plotted as a function of the temperature. At very low temperatures the ordered configuration 11 is predominant. As the temperature increases other configurations begin to contribute. Besides SiC, mainly graphene and silicene contribute. Also configuration 6 with 5 Si and 3 C atoms per cluster may be visible. At temperatures about T = 1000 K, one can say that the alloy is almost randomly disordered. In order to quantify the degree of ordering, we calculate the short-range order (SRO) parameter, such that L = 1 for a fully ordered alloy and L = 0 for a completely random distribution. The SRO is defined as [47] L=

0 PSi-C − PSi-C , 0 max PSi-C − PSi-C

(16)

where PSi-C is the probability to find a Si-C bond in the considered alloy for a given x and T , which can be estimated by the means of Eq. (10), PSi-C (x,T ) =

J 

xj (x,T )PSi-C,j ,

(17)

j =1 0 = 2x(1 − x) describes this probability in a completely PSi-C max random alloy, and PSi-C = 2 min{x,1 − x} is its possible maximum value, given by the concentration x. The order parameter is plotted in Fig. 3(b) for x = 0.5 versus the temperature. It again indicates complete ordering for T = 0 K, but rapidly decreasing ordering for temperatures above T = 1000 K. For the 3D-SiC, it was observed an ordering, by forming self-organized 2D layers in a “natural” superlattice [48].

FIG. 3. (Color online) (a) The probabilities xj of all Si8−nj Cnj configurations versus temperature at x = 0.50. Configurations 1, 6, 11, and 22 are highlighted. (b) Degree of ordering in Si1−x Cx for x = 0.5, quantified as SRO parameter L, as a function of temperature.

In the case of Ge1−x Cx and Si1−x Gex the situation is completely different. It holds U (x,T = 0 K)  0 for all compositions and temperatures. This is obvious from the excess energies in Table II. Miscibility or spinodal decomposition should happen in dependence on composition and temperature. The application of the GQCA to these two binary alloys for different temperatures and compositions results in the (T -x) phase diagrams. The results are shown in Fig. 4. They represent the binodal and the spinodal curves that divide the (x,T ) points in zones of stable, metastable, and unstable phases. From Fig. 4(a) we observe a strong tendency of phase separation with a huge critical temperature of 22400 K for Ge1−x Cx . This temperature represents a minimum for which one has a completely random alloy of graphene mixed with germanene for any composition. Only small amounts of carbon can be mixed into the 2D germanium in order to form a random alloy also for temperatures that can be experimentally accessed. The immiscibility of these two 2D materials is analyzed in Sec. III B by results for the mean firstneighbor distances and the very deformed atomic geometries. This behavior follows the 3D-GeC alloys, for which it was observed that C has an extremely low solid solubility in Ge (1 × 108 cm−3 ) [49], and one reported low concentration of carbon (0–3%) in the GeC layer grown by molecular beam epitaxy [50].

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FIG. 5. (Color online) The lattice parameter as a function of the composition for Ge1−x Cx (black), Si1−x Gex (red), and Si1−x Cx (blue). The symbols represent the lattice parameters of an individual cluster configuration with the corresponding composition.

FIG. 4. (Color online) The (T -x) phase diagram of (a) Ge1−x Cx (black) and (b) Si1−x Gex (red). Solid lines, binodal curves; dashed lines, spinodal curves.

The situation is completely different if carbon is replaced by silicon as indicated by the phase diagram in Fig. 4(b). The Si1−x Gex alloy has a critical temperature of 550 K, indicating the possibility to synthesize a random alloy of germanene and silicene, in agreement with results obtained within special quasirandom structures (SQSs) [51]. The phase diagram is rather symmetric. For lower temperatures it indicates that in the equilibrium the random Si1−x Gex alloys tend to decompose into a low-concentration and a high-concentration alloy. This behavior follows the low critical temperatures obtained for its 3D counterpart of 170 K [52,53], which is lower than the growth temperatures, resulting in a random alloy. The probability of occurrence of each configuration xj depends on the average composition x in the alloy, the temperature and the excess energy of each configuration defined in Eq. (15). To illustrate the effect of εj values on xj calculations, we listed in Table II a column with xj in an alloy with composition x = 0.5 at T = 800 K. In Si1−x Gex , we observe smaller εj values explaining the achievable critical temperature of these alloys. On the other hand, most of the Ge1−x Cx configurations exhibit higher excess energies, contributing to small xj values, reflecting their immiscibility until very high temperatures. Consequently the details of the alloy statistics and thermodynamics can be related to the relative energies of the

individual cluster configurations. We observe from Table II that the excess energies of carbon-rich configurations are greater than germanium-rich ones. This means that adding germanium atoms to a graphene structure is more thermodynamically unfavorable comparing to incorporating carbon atoms in a germanene structure, since the first case demands bigger deformations of the atomic arrangements in the hexagonal unit cells. Both facts explain the asymmetry of the (T -x) phase diagram in Fig. 4(a). B. Structural properties

The configurationally averaged lattice parameter a is obtained by considering Eq. (10) as a=

J 

xj (x,T )aj ,

(18)

j =1

where aj is the equilibrium lattice constant of each cluster j . We perform explicit calculations at T = 800 K based on the growth temperatures of graphene on SiC found in literature [54]. However, the calculated lattice constant curves do not change significantly for 500 K, approximately the growth temperature of germanene and silicene on Ag(111) [55,56]. The resulting lattice constants projected onto a two-atom unit cell as in graphene, silicene, or germanene are shown in Fig. 5. One verifies that for 2D Ge1−x Cx , Si1−x Gex and Si1−x Cx alloys the lattice parameter varies approximately linearly with the composition. Only a small bowing appears for the carbides. We observe that some individual aj values in clusters forming Ge1−x Cx and Si1−x Cx , which have no significant statistical contribution for the calculated mean value, do not adhere to the obtained curve. This follows from the fact that these configurations are thermodynamically unfavored. In graphene, silicene, and germanene, all bond lengths are the same by symmetry. For the binary alloys, considering the ionic relaxation, the first-neighbor distances will change, depending very strongly on which elements are involved and

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FIG. 6. (Color online) The mean first-nearest-neighbor distance of Ge1−x Cx (black), Si1−x Gex (red), and Si1−x Cx (blue) as a function of the composition with respective mean values associated with the configuration in each cluster class. The A-A, A-B, and B-B bond lengths are shown with triangles, circles, and squares, respectively.

on the concentration. We study the different bonds in all clusters between atoms of the same kind (A-A or B-B bonds) and different kinds (A-B bonds), with A, B = Ge, Si, or C. Each cluster possesses 12 bonds between neighbor pairs. The mean neighbor distance dX-Y , using GQCA, can be written as J j =1 xj n(j,X-Y ) d(j,X-Y ) dX-Y = , (19) J j =1 n(j,X-Y ) xj where X and Y represents A or B, d(j,X-Y ) is the mean distance of X-Y bonds in cluster class j , and n(j,X-Y ) is the number of X-Y bonds in class j . The results are shown in the Fig. 6. First, we observe three distinct bond lengths, showing that Vegard rule fails for bond lengths as observed for 3D alloys [33]. We also observe a tendency to keep the bond length of pure materials, the mean bond lengths of A-B bonds mainly follow the covalent radii of the constituting atoms. We find that the difference between the bond lengths in Si1−x Gex is much smaller than in Ge1−x Cx and Si1−x Cx . This fact explains why Si1−x Gex is more likely to be a more stable alloy, with weak tendency to phase separation, in contrast to Ge1−x Cx and Si1−x Cx . The reason is clearly related to the different sizes of the atoms in carbides A1−x Cx (A = Si, Ge) and, hence, the strong internal strains. The obtained mean first-neighbor distances in Si1−x Gex alloys are compared in Table III, with results based on SQSs for particular concentrations [51]. Apart from generally slightly larger values obtained within the SQSs, the trends with the composition are quite similar.

Indeed, the mean bond lengths associated with different configurations in Fig. 6 are more diffuse in Ge1−x Cx and Si1−x Cx , while the calculated first-neighbor distance do not change appreciably with the composition x. These configurations present significant deformation on the honeycomb geometry, as we can see in Figs. 7(a) and 7(b). We believe that the main reason for such deformations are the big differences between the lattice parameters of graphene and the other 2D group-IV materials. For some specific configurations this effect is very pronounced, affecting not only the bond lengths but also the layer buckling (see explanation in Fig. 8). As an example, the geometries of a few cluster materials A8−nj Bnj are displayed in Fig. 7. In Figs. 7(a) and 7(b), despite the equal numbers of atoms, the covalent radii and the bonding are different, resulting in a strong deformation and change of the atomic geometries, which are related to neighboring constitutes of the same element. On the other hand, since germanium and silicon have similar covalent radii, the honeycomb geometry of these materials does not need to deform much to accommodate a different atom, as shown in Fig. 7(c). In Fig. 7(d) a symmetric and equally bonded

TABLE III. Comparison of mean first-neighbor distances obtained within GQCA and SQSs [51] (in parenthesis) for Si1−x Gex ˚ alloys of fixed compositions. All values are given in A. x 0 0.265 0.500 0.735 1

Ge-Ge

Si-Ge

Si-Si

– 2.40 (2.42) 2.41 (2.43) 2.42 (2.44) 2.44 (2.46)

– 2.34 (2.37) 2.36 (2.38) 2.37 (2.38) –

2.27 (2.31) 2.29 (2.32) 2.30 (2.32) 2.30 (2.33) –

FIG. 7. (Color online) Perspective views on (a) Ge4 C4 (configuration j = 14), (b) Si6 C2 (configuration j = 20), (c) Si4 Ge4 (configuration j = 11), and (d) Si4 C4 (configuration j = 11). C (Si, Ge) atoms are presented as brown (bright blue, dark blue) dots.

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FIG. 8. (Color online) Buckling amplitude of an arbitrary configuration.

TABLE IV. Layer bucklings () for the 2D Ge1−x Cx , Si1−x Gex , and Si1−x Cx alloys, considering the cluster statistics (x = 0.5) and for the configuration j = 11. Results are compared with literature data shown in parenthesis. ˚  (A)

Alloy

configuration (j = 11) is shown for Si1−x Cx , resulting in a completely hexagonal and planar honeycomb structure. Graphene has a planar honeycomb configuration, while silicene and germanene represent a slightly buckled structure, since the bonds between atoms are not pure sp2 but correspond to a mixed sp2 -sp 3 hybridization [57,58]. This hybridization is also expected for many bonds in the studied 2D alloys. As a consequence, some configurations exhibit large buckling ˚ (see Table II). They are associated with values above 1 A considerable deformations of the local hexagonal structure. Indeed, the bonding hexagons in Figs. 7(a) and 7(b) are strongly deformed. The buckling amplitude in each cluster is calculated as the maximum displacement between two atoms in the direction perpendicular to the sheet plane, as shown in Fig. 8. Besides the individual results in Table II, the mean alloy values averaged according to Eq. (10) are depicted in Fig. 9. The most stable configuration for graphene is the planar honeycomb structure, while silicene and germanene present more stable slightly ˚ buckled configurations with buckling parameters  of 0.46 A ˚ respectively, which are in agreement with other and 0.68 A, calculations [58,59]. As indicated in Table II, the buckling parameter varies ˚ Nevertheless, when consignificantly from zero to 2.81 A. sidering the cluster statistics Eq. (10), the buckling parameter of Ge1−x Cx exhibits an approximated linear behavior with composition x. This is due to the major statistical contribution of the cluster j = 1, which corresponds to pure graphene, and cluster j = 22, which corresponds to pure germanene. In Si1−x Gex and Si1−x Cx , we have contributions of different configurations. For Si1−x Gex , a more homogeneous distri-

Ge0.5 C0.5 (j = 11) Ge0.5 C0.5 Si0.5 Ge0.5 (j = 11) Si0.5 Ge0.5 Si0.5 C0.5 (j = 11) Si0.5 C0.5

0.00 (0.00)a 0.39 0.58(0.579,a 0.55,b 0.58c ) 0.58 (0.61d ) 0.00 (0.00a ) 0.19

a

Ref. [57]. Ref. [60]. c Ref. [61]. d Ref. [51]. b

bution of xj and j is observed (see, e.g., Table II), while for Si1−x Cx the most significant contributions come from the configuration j = 11 and pure configurations j = 1 and 22 as well as j = 6 (Si5 C3 ). The configuration 11 (Tables I and II) represents an ordered alloy of 50% Si and 50% C atoms [27], for which there is a maximum number of Si-C bonds. In particular, for Si1−x Gex , the only random alloy, the values previously obtained within SQSs [51] are depicted in Fig. 9 for comparison. The GQCA and the SQSs results agree very well. In Table IV we compare results considering the cluster statistics Eq. (11) and the configuration j = 11. We observe that the values for the layer buckling  differ when considering or not the statistics for Ge1−x Cx and Si1−x Cx alloys, being comparable only for Si1−x Gex . In the case of Si1−x Cx , the value depends on the growth temperature, which determines the contribution of the configurations and if the alloy consists of ordered domains. For Ge1−x Cx and Si1−x Gex , the mean buckling does not depend on the temperature. This is due to the fact that, in the first case, one has a phase separated alloy with a huge critical temperature, so for reasonable temperatures the only contributing configurations are the pure compounds, while in the second case, the dependence of the mean buckling in the random alloy is approximately linear, which will not change for a phase-separated phase at low temperatures. C. Electronic properties

FIG. 9. (Color online) Averaged buckling amplitude of Ge1−x Cx (black), Si1−x Gex (red), and Si1−x Cx (blue) as a function of the composition. The red diamonds represent previously reported results for Si1−x Gex alloys within SQSs methodology [51].

We focus our attention on the energy gap behavior by varying the alloy compositions. The obtained fundamental energy gaps Eg obtained within GGA and GGA-1/2 are shown in Fig. 10 for the random alloys Ge1−x Cx , Si1−x Gex and Si1−x Cx at growth temperature T = 800 K. In the case of GGA, when quasiparticle corrections are not taken into account, we expect to underestimate the gap values in comparison to the GGA-1/2 results [36]. In the case of 2D-SiGe and -GeC, in average, we observe that the energy gaps remain very small and no difference is noted between the GGA and GGA-1/2 curves. In the case of SiC, however, a difference in the magnitude is readily observed, even though the Kohn-Sham gaps give correct tendencies with the composition x, so that the qualitative analysis and inferences can still be done.

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PHYSICAL REVIEW B 92, 075435 (2015) TABLE V. Energy gaps (Eg ) for the 2D Ge1−x Cx , Si1−x Gex , and Si1−x Cx alloys, considering the cluster statistics (x = 0.5), and for the configuration j = 11. The values with the symbol † were calculated considering gap corrections. Eg (eV) Present work

FIG. 10. (Color online) The gap curves of random alloys of Ge1−x Cx (black), Si1−x Gex (red), and Si1−x Cx (blue) as a function of the composition for T = 800 K. Full line (dashed) curves correspond to results obtained within the GGA-1/2 (GGA) approach.

Alloy

GGA

GGA-1/2 (HSE)

Ge0.5 C0.5 Si0.5 Ge0.5 Si0.5 C0.5 Ge0.5 C0.5 (j = 11) Si0.5 Ge0.5 (j = 11) Si0.5 C0.5 (j = 11)

0.041 0.004 1.166 2.07

0.041 0.005 1.429 2.54

0.007

0.01

2.53

3.06 (3.4)

Other calc.

Exp.

0.010c 2.09,a 2.26h 3.16,†,b 3.37†,h 0.02,a 0.015i 0.275†,b 2.52,a 2.55d 2.5,e 2.56f 2.57h 3.53,†,b 3.63†,d 3.7,†,f 3.88†,h

3.3g

a

It is worth pointing out that the alloys may exhibit phase separation, or even ordered geometries. That means that the result for the 50% compositions can be different from the values predicted for the ordered cases [26]. This is the case for Ge0.5 C0.5 , for which the predicted values for the ordered configuration 11 is 2.54 (2.07) eV, within GGA-1/2 (GGA), while the predicted energy gap of the random alloy (Fig. 10) remains extremely small with 0.04 eV. This is due to the fact that, in spite of nonzero values for different concentrations, the only two configurations that have a significant statistical contribution are the two that correspond to pure graphene and pure germanene, which both have zero energy gaps. This is a good example of how mandatory is the use of statistics, showing that values previously obtained for a special arrangement of the C and Ge atoms [26,60] are only valid for an energetically unfavorable distribution of the atoms. For Si1−x Gex , which can be widely mixed, but especially Si1−x Cx we see from Fig. 10 that the values of Eg strongly deviate from the linear behavior but can be well described by a parabolic one. We apply the usual formula for the average energy gap Eg versus average composition x as well known from semiconducting alloys [30], where the deviations from the linear variation are described by a bowing parameter b as Eg (x,T ) = xEgA + (1 − x)EgB − bx(1 − x).

(20)

Germanene presents a metallic electronic band structure in the planar honeycomb geometry without buckling but near K in the BZ Dirac cones with zero gap when one considers the buckling effect, as we did. Silicene, on the other hand, present Dirac cones with zero gap independently if one considers the layer buckling or not [6]. Thus, here EgA = EgB = 0 holds because all group-IV-derived graphene-like systems possess a vanishing fundamental gap at K or K  in the Brillouin Zone, at least neglecting spin-orbit interaction [62,63]. In contrast to 3D semiconductors an antibowing b < 0 is found in Fig. 10. For Si1−x Gex (Ge1−x Cx ) we calculate b = −0.02 (−0.18) eV for the parabolic fit, whereas for Si1−x Cx , b = −5.37 eV, indicating the large gaps for SiC-like configurations. While

Ref. [60]. Ref. [26]. c Ref. [51]. d Ref. [65]. e Ref. [64]. f Ref. [27]. g Ref. [20]. h Ref. [61]. i Ref. [68]. b

Si1−x Gex and Ge1−x Cx alloys present energy gaps about a few meV, the Si1−x Cx alloys possess large direct energy gaps that can be very interesting for optoelectronic applications. Toward verifying deviations from the parabolic behavior, we consider the bowing depending on the composition as b(x) = Ax + B with constants A and B. In the case of GGA1/2 it gives bSiC (x) = −5.54 + 0.32x, bGeSi (x) = −0.03 + 0.01x, and bGeC (x) = −0.23 + 0.12x for SiC, GeSi, and GeC, respectively. In order to compare our values for Si1−x Cx and Ge1−x Cx with results previously obtained, we need to consider only the energy gap of configuration 11. The Eg values for this configuration, as well as considering the statistics, are shown in Table V. A nice agreement is observed among our results and the ones obtained considering LDA or GGA [27,60,64,65], and a reasonable agreement is verified among the GGA-1/2 results with other calculations [26,27,60,65] that use different (approximate) quasiparticle corrections (QPCs); these calculations obtain absolute gap values for Si4 C4 and Ge4 C4 (j = 11) about 0.5 eV higher than the values in Table V, as one can observe. We also made a simulation for a generalized Kohn-Sham scheme with a hybrid functional HSE06 [66,67], which has been successfully applied to band structures of silicene, germanene, and graphene [63] and gives rise to a gap value Eg = 3.4 eV close to the experimental finding of S. S. Lin, obtained by photoluminescence of ultrathin SiC nanosheets [20].

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with Tc,ord = (1022 ± 22) K. This temperature dependence is mainly a consequence of increasing probabilities for other atomic configurations aside the j = 11 one. The situation for the other carbide alloy is totally different. For Ge1−x Cx , there is a clear phase separation into pure graphene-like and pure germanene-like configurations, while all other configurations do not give rise to significant contributions. In contrast, for Si1−x Gex , from Fig. 10, we observe that all clusters appear and contribute to the statistics. Nevertheless, the resulting average gaps remain about few meV because of the small cluster gaps as listed in Table II.

IV. SUMMARY AND CONCLUSIONS

FIG. 11. (Color online) (a) The gap curves of Si1−x Cx for different temperatures within the GGA-1/2 approach. The black line corresponds to ordered phase 11. The black curve corresponds to T = 0 K, and the curves from the light gray to the red correspond to a variation of temperature from T = 300 to 900 K, in 100 K. (b) The maximum energy gap obtained for the Si1−x Cx alloy as a function of the growth temperature. The full line represents the best fit of the data using Eq. (21).

Another interesting feature of the Si1−x Cx alloys is the growth temperature dependence of the computed gap curves shown in Fig. 11(a). The same behavior is observed for GGA and GGA-1/2, so we only present the results within the GGA-1/2 approach. The gap curves vary substantially with the growth temperature. For low temperatures, the inclusion of the configuration 11 is favored, resulting in a wider energy gap, since this configuration has a gap value of 3.06 (2.53) eV in DFT-GGA-1/2 (GGA) (see also Ref. [27]). With increasing temperature other configurations begin to contribute resulting in the temperature dependence of Eg , and the maximum of the gap curves are shifted to lower average composition x as temperature increases. In Fig. 11(b) we show the maximum energy gap as a function of the growth temperature. It makes more obvious the temperature dependence of the average gap of SiC. The value for the ordered configuration 11 at T = 0 K is reduced to about 2 eV for T = 1000 K. The results can be well fitted by the expression Egmax (T ) = Egmax (0)e−T /Tc,ord ,

(21)

Using first-principles calculations combined with a generalized quasichemical approach we have studied the properties of Ge1−x Cx , Si1−x Gex , and Si1−x Cx alloys as a function of their average compositions. We found that the three alloy systems behave totally different versus composition. For Si1−x Cx the contribution of cluster configuration j = 11, which corresponds to a planar, stoichiometric, graphene-like SiC geometry, as well as pure silicene and graphene are energetically favored. This fact has direct consequences to the band gap engineering that is possible in this alloy. For the germanium-containing alloys, the obtained T -x diagrams show extremely different critical temperatures. They are related to the significantly different excess energies of the individual cluster configurations, which differ by about one order of magnitude between Si8−nj Genj and Ge8−nj Cnj clusters. For Ge1−x Cx , for which the majority of configurations have very high excess energies, the preparation of the respective random alloys is not likely to be achievable. A strong tendency for phase separation is expected to occur. Only small amounts of carbon can be mixed in the 2D germanium in order to form a random alloy. For Ge1−x Six the critical temperature approaches values of characteristic growth temperatures. Indeed, for higher temperatures random alloys should exist. For lower temperatures also the tendency to phase separation is visible. However, the two random alloys, which determine the decomposed Ge1−x Six system, are not so close to the pure materials as in the case of Ge1−x Cx . The thermodynamic properties together with the ionicity of the bonds between Si and C, Ge and Si, or Ge and C atoms drastically influence the average fundamental energy gap versus composition. Whereas large gaps appear for Si1−x Cx systems they almost vanish for Ge1−x Six and Ge1−x Cx .

ACKNOWLEDGMENTS

We thank the Brazilian funding agencies Fundac¸a˜ o de Amparo a` Pesquisa do Estado de S˜ao Paulo (FAPESP) from the grant 2012/50738-3 and Coordenac¸a˜ o de Aperfeic¸oamento de Pessoal de N´ıvel Superior (CAPES) from the PVE Grant No. 88881.068355/2014-01 and Alexander von Humboldt Stiftung from the grant BEX 1848/14-3 for financial support, and we thank Andr´e J. Chaves, Cleiton Ataide, and J. Furthm¨uller for fruitful discussions.

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