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Diamond & Related Materials

Atomic and electronic structure of crystalline–amorphous carbon interfaces

G. Kopidakis a,⁎, I.N. Remediakis b, M.G. Fyta b,1, P.C. Kelires b

a Department of Materials Science and Technology, University of Crete, P.O. Box 2208, 71003 Heraklion, Crete, Greeceb Physics Department, University of Crete, P.O. Box 2208, 71003 Heraklion, Crete, Greece

Available online 17 July 2007

Abstract

Interfaces of diamond with amorphous carbon (a-C) are investigated using tight-binding molecular dynamics simulations. Such interfaces arefound to be stable, with a-C atoms covalently bonded to the diamond surfaces. The atomic and electronic structure of the a-C region is consistentwith previous results on pure a-C and does not depend critically on the diamond face exposed. However, surface properties influence the relativestability of interfaces with high density a-C. In this case, the interfacial region is small and very dense homogeneous a-C grows on diamond. Atlower densities, carbon atoms nucleate on diamond surfaces and create an intermediate region between diamond and a-C. The shape of diamondcrystals embedded in a-C is predicted using the Wulff construction with appropriately defined energies for surfaces with overlayer material. Thesepredictions are verified by empirical potential simulations of nanodiamond inclusions in a-C matrix. The electronic density of states and thedielectric function, calculated for our samples containing a-C/diamond interfaces, show that optoelectronic properties of these composite materialsare dominated by a-C.© 2007 Elsevier B.V. All rights reserved.

Keywords: Diamond; Amorphous carbon; Interface; Nanocrystals

1. Introduction

Diamond–amorphous carbon heterostructures with promisingmechanical and optoelectronic properties have emerged as a resultof advances in carbon-based materials deposition and character-ization techniques [1–4]. These materials exhibit a variety ofstructural features and properties that depend on growthconditions. A large number of experimental studies revealrelations to several carbon allotropes and transformations betweenphases, particularly at the nanoscale [5]. Some, such asnanocrystalline diamond (with thin amorphous regions at thegrain boundaries), are fairly well described [1], but a more generalunderstanding of diamond–amorphous carbon nanocompositematerials, which are predicted to have superb mechanicalproperties [6], is highly desirable. In order to investigate thesemixed phases, it is crucial to understand the interaction betweendiamond surfaces and amorphous carbon (a-C).

⁎ Corresponding author.E-mail address: kopidaki@materials.uoc.gr (G. Kopidakis).

1 Present address: Physics Department, Harvard University, 17 Oxford Str.,Cambridge 02138, USA.

0925-9635/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.diamond.2007.07.013

We use Tight-Binding Molecular Dynamics (TBMD)simulations to study various a-C/diamond (a-C/D) interfaces.TBMD is a quantum mechanical method that allows foraccurate calculations for cell sizes and simulation times that aremuch larger than those feasible with first-principles methods[7]. Our previous TBMD studies of pure a-C networks, over thewhole range of possible densities, have resolved long-standingissues related to the structural, mechanical, electronic, andoptical properties of these materials, directly connecting ourtheoretical results with experiment [8–10]. The amorphous–crystal interface in silicon has been studied with simulationsbased on a combination of empirical and tight-binding models[11]. In the case of carbon, where several bond hybridizationsare available, these heterostructures are expected to have veryinteresting properties that depend both on the diamond surfaceand on the a-C overlayer. In the present work, TBMDsimulations allow us to obtain the detailed atomic-scale pictureof a-C/D interfaces and predict their structural and electronicproperties. We find that for high density a-C there is noextended intermediate phase at the interface. For lowerdensities, higher concentrations of carbon atoms close todiamond surfaces, compared with the rest of the a-C network,are observed. In all cases, a-C atoms are covalently bonded to

Fig. 1. Atomic structure of interfaces of high density a-C with low-index faces of diamond D(100) (left), D(110) (centre), D(111) (right). Blue atoms are 4-fold, red are3-fold coordinated. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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diamond surfaces. Systematic calculations of interface energiesbetween different faces of the crystal and a-C, and for several a-C densities and preparation conditions, determine the relativestability of various interfaces. We define a relevant energy perunit area for the a-C/D heterostructure and use it to predict theshape of faceted diamond crystals embedded in a-C. Our resultsare confirmed by empirical Monte Carlo simulations ofnanocrystals in a-C matrix. The electronic properties of thesesame element heterostructures and possibilities to tailor themare also briefly discussed.

2. Results and discussion

The a-C/D interfaces are obtained by melting part of ourinitial diamond simulation cells to liquid and quenching it downto zero temperature, while keeping the rest of the atoms frozen in

Fig. 2. Atomic structure of interfaces of low density a-C with low-index faces of diam3-fold coordinated. (For interpretation of the references to colour in this figure lege

their crystal positions. The as-quenched structures are thenrelaxed by allowing all atoms to move and reach the minimumenergy configuration. Periodic boundary conditions are used inall directions, so that the supercell is a layered structure withalternating amorphous and crystalline parts. In order to study theeffects of different preparation conditions, we used severalquenching rates, a-C densities, amorphous and crystal sizes, andvolume relaxation methods. Molecular dynamics simulationswere performed using the Environment-Dependent Tight-Binding model (EDTB) of Tang et al. [12]. Typical simulationcells used in our TBMD consist of hundreds of atoms and thetime step used is 1.05 fs. The results presented here are fromsamples obtained by quenching from temperatures of 12,000 Kdown to zero with quench time of 13.125 ps and 26.25 ps,corresponding to rates of about 914 K/ps (“fast”) and 457 K/ps(“slow”), respectively.

ond D(100) (left), D(110) (centre), D(111) (right). Blue atoms are 4-fold, red arend, the reader is referred to the web version of this article.)

Table 1a-C/diamond interface energies for low-index diamond faces (samples fastquenched, see text for definitions)

Interface a-C density(g/cm3)

Eint

(eV/Å2)Eint

(eV/atom)Eai

(eV/atom)Esa

(eV/Å2)

(100) 3.33 −0.06 −0.39 −6.881 0.444(100) 3.32 −0.06 −0.39 −6.887 0.423(110) 3.32 −0.04 −0.17 −6.858 0.465(110) 3.30 −0.05 −0.24 −6.884 0.380(111) 3.34 0.01 0.04 −6.814 0.618(111) 3.29 0.01 0.05 −6.808 0.683(100) 2.93 −0.10 −0.68 −6.885 0.474(110) 2.85 −0.09 −0.42 −6.863 0.510(111) 2.91 −0.08 −0.43 −6.840 0.643(100) 2.34 −0.09 −0.61 −6.814 0.554(110) 2.35 −0.08 −0.38 −6.806 0.567(111) 2.58 −0.10 −0.54 −6.820 0.691

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In Fig. 1, the atomic structure of interfaces of a-C with thelow-index faces of diamond D(100), D(110), D(111), ispresented. The initial configurations were diamond cells withthe appropriate orientation. The upper half of the samples wasmelt and “fast” quenched, thus becoming a-C. The density ofcrystal and liquid regions was the same during quenchingand equal to the density of diamond (3.47 g/cm3 within theEDTB model). In the specific examples, the number of atomsincluded in the simulation cells is 600, 576, and 600 for D(100), D(110), and D(111), respectively, and the thickness ofthe crystalline and liquid regions is approximately 11 to 13 Å.The final configurations were obtained by allowing all atoms(i.e., of the crystal and the amorphous regions) to move andrelax to the volume with the minimum energy. During this lastrelaxation procedure, a significant number of crystalline atomsconverts to amorphous and vice-versa. However, the interfa-cial region remains well defined and the crystalline andamorphous parts clearly separated. While the amorphous partrelaxes to lower densities, the crystal remains in the initialdiamond density. In the examples illustrated here, a-C isextremely dense, with densities 3.32, 3.32, and 3.29 g/cm3 forthe D(100), D(110), and D(111) interfaces, respectively, andwith a very high fraction of sp3 atoms (91%, 92%, and 89%,respectively).

Samples containing a-C of much lower density are shown inFig. 2. During the quenching process, diffusing atoms occupy alarger volume than the frozen diamond atoms. After the finalrelaxation, the a-C densities were found to be 2.34, 2.35, 2.58 g/cm3 for the D(100), D(110), D(111) interfaces, respectively,while the fraction of sp3 atoms is significantly reduced (68%,64%, and 69%, respectively). The crystalline region retains itsdiamond density. In such low density a-C samples, theinterfacial region is more dense than the rest of the amorphouspart. Evidently, sp3 atoms nucleate on the diamond surfacesserving as bridging material to the a-C region. On the otherhand, it is clear in Fig. 2 that voids are generated far from theinterface region and well into the amorphous part.

The interface energy is calculated using the expression

Eint ¼ Etot � NdEd � NaEað Þ= 2Að Þ ð1Þ

where Etot is the total cohesive energy of the system, Ed thecohesive energy per diamond atom (close to −7.365 eV), Ea thecohesive energy per atom of pure a-C, Nd the number ofcrystalline atoms, Na the number of amorphous atoms, and Athe area of the interface. The characterization of an atom asamorphous or crystalline is made by checking the value of itstetrahedral vector [11,13], which indicates deviations of bondlengths and angles from their nominal, tetrahedral values. Inorder to find Ea, several pure a-C samples are produced underthe same conditions used for the a-C/D cells. From the relationof Ea versus density (equivalently, average coordinationnumber), the appropriate Ea at the specific density is found.There are some uncertainties introduced in the estimation of Na

(equivalently Nd) and Ea. Another way to calculate Eint is byusing Eq. (1) for two samples that have been prepared under

almost the same conditions, so that their a-C regions havepractically the same density, but different number of amorphous(and crystalline) atoms. Assuming that Ea and Eint are the samefor both samples, the solution of the two equation system givesEa and Eint.

We have performed extensive TBMD simulations for severala-C densities and diamond faces. Some of our results aresummarized in Table 1. The first column describes the a-C/Dinterface with the indices of the exposed diamond surface andthe second column gives the density of the a-C of the sample.Interface energies Eint calculated from Eq. (1) are given in thethird column, while the same energies expressed per atom of theinitial diamond surface are in the next column. In the fifthcolumn, the energies per amorphous carbon atom Eai (contain-ing interface energies) are found from

Eai ¼ Etot � NdEdð Þ=Na: ð2Þ

All samples described in Table 1 were obtained by “fast”quenching, as described earlier. Most interface energies arefound to be negative, indicating stability, and rather small inabsolute value. It is evident that for very dense a-C samples,such as the ones depicted in Fig. 1, Eint100≃Eint110bEint111,where the indices correspond to the initial diamond surface ofthe interface. Calculation of Eint using the two samples for eachtype of interface produced under very similar conditions, asdescribed above, gives somewhat lower interface energies, butdifferences between them are practically the same. For smallera-C density (Fig. 2), interface energies are lower than dense a-C/D(100) and have approximately the same values for all samples,irrespective of the orientation of the exposed diamond surfaceand the a-C density. This is consistent with the observation(Fig. 2) that in low density samples, amorphous carbon atomsare not uniformly distributed but rather accumulate on the a-C/Dinterface region. These atoms have enough space to rearrangein an energetically favorable configuration, independent ofdiamond surface geometry. On the contrary, in high density a-Csamples, atoms are more restricted and the geometry of thediamond face seems to affect the interface energy. For the

Fig. 3. Left: Equilibrium crystal shape for free standing diamond from the Wulff construction using the unrelaxed, unreconstructed low-index surface energies, with(111) in green, (110) in yellow, and (100) in red. Right: Similarly, the Wulff construction for the diamond embedded in high density a-C using the energies Esa (see textfor details). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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“slowly” quenched samples we find that the differences amongthe several Eint become smaller, but the overall trends remainthe same.

Fig. 4. Top: Cubic diamond nanocrystal embedded in a-C matrix before relaxation (before relaxation (left) and after relaxation (right). On the left, blue atoms belong toatoms are 4-fold and 3-fold coordinated a-C atoms, respectively, while green are crysthe reader is referred to the web version of this article.)

In all cases, even at high a-C density (at least for the D(100)and D(111) faces), interface energies show that the formation ofa-C/D heterostructures is energetically favorable. Although Eint

left) and after relaxation (right). Bottom: Spherical diamond nanocrystal in a-Cthe a-C matrix and red atoms to the diamond crystal. On the right, blue and redtalline atoms. (For interpretation of the references to colour in this figure legend,

Fig. 5. Total (black), sp3 (blue), and sp2 (red) electronic density of states of a-C/D cells with density 3.29 g/cm3 (top) and 2.58 g/cm3 (bottom). (Forinterpretation of the references to colour in this figure legend, the reader isreferred to the web version of this article.)

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are rather small in absolute value, they are negative. It isinteresting to compare interface energies with the diamondsurface energies of D(100), D(110), and D(111), defined as

Es ¼ N Eslab � Edð Þ= 2Að Þ ð3Þ

where Es is the surface energy (per unit area), N is the numberof atoms in the simulation cell, Eslab is the energy per atom ofthe slab. The corresponding unrelaxed, unreconstructed surfaceenergies are E100=3.53, E110=2.03, E111=2.20 eV per surfaceatom, in reasonable agreement with the ab initio values of 3.89,2.09, 2.75, respectively [14]. These numbers correlate well withthe concentration of dangling (unsaturated) bonds. The D(111)surface we examine contains one dangling bond per surfaceatom. The D(110) contains also one, while D(100) contains twodangling bonds per surface atom. The lowest surface energyfound for D(111) is consistent with the fact that diamond'snatural cleavage occurs along (111) faces. In standard surfaceenergy units, Es100=0.55, Es110=0.45, and Es111=0.40 eV/Å2.

The stability of the a-C/D interface explains the fact thatdiamond microcrystals and nanocrystals have been found in a-Cmatrix, both experimentally and theoretically. Based on theenergetics of a-C/D interfaces, it is, in principle, possible topredict the shape of these embedded nanostructures. Using theunreconstructed, unrelaxed surface energies, the Wulff con-struction method [15] gives the equilibrium shape of a cleaveddiamond crystal shown in Fig. 3 (left). One might expect that theshape of diamond crystals embedded in a-C resembles the one inthis figure. After all, the relaxed, reconstructed surfaces ofdiamond should not be relevant when the surrounding a-Cmatrix is present. However, the construction of Fig. 3 (left) is notrelevant either, since it is based on the energetics of free surfaces.Our study of the a-C/D interface shows that amorphous carbonatoms are firmly attached in stable configurations and covalentlybonded to the diamond's exposed surfaces. In order to makepredictions about the shape of diamond crystals embedded in a-C using the Wulff construction, we define Esa, analogous to thefree surface energy of Eq. (3), as the quantity

Esa ¼ Na Eai � Edð Þ= 2Að Þ ð4Þwhere Eai is given by Eq. (2) and is the energy (per atom) of thea-C part (including the interface). Esa essentially gives the extraenergy per unit area (with respect to diamond) due to thepresence of the a-C, much like Es gives the energy due to thevacuum above the surface layer. From definitions of Eqs. (1),(2), and (4) it follows that the difference of interface energiesEint−Eint′ for two different samples with Eint and Eint′ is relatedwith the difference Esa−Esa′ by

Eint � E Vint ¼ Esa � E Vsa � Na

2AEa � Edð Þ þ N Va

2A VE Va � Edð Þ ð5Þ

where Ea and Ea′ are the a-C energies per atom, Na and Na′ thenumber of a-C atoms,A andA′ the surface area of the interface forthe two samples, respectively. When the samples have almost thesame a-C density it is Ea≃Ea′ and, in addition, if the geometricalcharacteristics of the cells are very similar, Na′/A≃Na′/A′. Thus,when comparing such samples, relative values of Esa equal the

relative values of Eint. The values of Esa for the “fast” quenchedsamples are recorded in the sixth column of Table 1. Using valuesfor high density a-C samples the Wulff construction of Fig. 3(right) is obtained. We expect diamond crystals embedded in a-Cmatrix to exhibit similar shape.

This was investigated by performing continuous-space MonteCarlo simulations using the Tersoff interatomic potential. Themethod is identical to the one employed in Ref. [13]. As shown inFig. 4, equilibrated nanostructures obtained under similar con-ditions as our dense a-C interfaces acquire a shape that resemblesthe one of Fig. 3 (right). Initially cubic (top) and spherical(bottom) crystalline structures embedded in a-C become facetednanocrystals according to our predictions (even though thesestructures including thousands of atoms were obtained by em-pirical Monte Carlo simulations). In Transmission ElectronMicroscopy (TEM) observations, this shape often appears to bespherical. However, it is practically impossible to make anyassessments about shape based on TEM pictures alone, due tolimited resolution. Simulations allow for a clear separation ofcrystalline and amorphous regions. Due to the rather smalldifferences in interface energies (i.e., stability), it is possible thatshapes change even with small perturbations of the system whichallow transitions from one local minimum configuration toanother. In actual experimental situations, these perturbationsmay occur at both preparation and characterization stages.

Although a-C/D interfaces are found to be stable and theirinvestigation provides a good starting point for understandingnanodiamond inclusions in a-C, the preparation conditionsshould be taken into account when dealing with specific experi-ments. Moreover, as nanostructured materials are very often farfrom equilibrium, further considerations should be madeconcerning their stability against thermal activation, irradiation,deformation, etc., that may lead to additional relaxation,recrystallization, and several other nanostructure propertiesmodifications. TBMD results related to these, and other

Fig. 6. Imaginary part of the dielectric function for a-C/D cells with density3.32 g/cm3 (black), 2.93 g/cm3 (red), 2.34 g/cm3 (green). (For interpretation ofthe references to colour in this figure legend, the reader is referred to the webversion of this article.)

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important issues, such as the size of nanodiamond inclusions,will be presented elsewhere.

Electronic and optical properties of a-C/D heterostructuresare of great interest since, besides connecting theory withexperiment, they may exhibit desired features of both crystallinediamond and amorphous carbon networks, combined. Theelectronic density of states (EDOS) of our samples with a-C/Dinterfaces has mainly the characteristics of pure a-C [9,10], withsome features associated with the crystal. In Fig. 5 the EDOSfor a very high density (top) and a low density (bottom) a-C onD(111) sample is plotted. Similar plots are obtained for the restof the interfaces, without any noticeable effect due to thedifferent diamond faces. As in pure a-C, the density of a-Cappears to determine electronic properties. A more detailedanalysis shows that at high a-C density the defect statesintroduced in the gap are very localized and associated both tosp3 and sp2 atoms. In the case of low a-C density, the σ–σ⁎ gapis filled with π and π⁎ states, but sp3 defects are still present.Diamond's presence is indicated by the highest energy peak.At the interface region diamond's electronic properties inter-mingle with those of the amorphous network.

Fig. 6 shows the imaginary part of the dielectric function ϵ2calculated from our TBMD simulations [9,10] for three sampleswith a-C/D(100) at different a-C densities. The main features ofϵ2 are similar to a-C. However, the lower density samples differfrom their pure a-C counterparts at low energies, where the π toπ⁎ transitions dominate, with their peaks becoming lesspronounced (since these are mixed phase samples). Anothersmall, but perhaps interesting, difference is observed in theposition of the main peak, which corresponds to the σ to σ⁎transitions at sp3 sites. In pure a-C, this peak shifts to higherenergies as the density of the samples decreases [9]. In Fig. 6, asmall shift is observed in the opposite direction. This issueneeds further investigation and a more extensive analysis of theoptical properties.

3. Conclusions

We have performed extensive TBMD simulations in order todetermine the atomic and electronic structure of diamond/amorphous carbon interfaces. We examined systematically a-Con the low-index diamond surfaces D(100), D(110), and D(110).In all cases, stable a-C/D heterostructures are formed, with carbonatoms covalently bonded on diamond surfaces. The structure of a-C in these heterostructures is consistent with our previous resultson pure a-C networks [8] and does not depend critically on thetype of diamond face exposed. Interface energiesEint are negativeand small in absolute value. For very high density a-C, the Eint ofa-C/D(100) and a-C/D(110) are similar and somewhat lower thana-C/D(111). Atoms in the a-C network are homogeneouslydistributed and an extended interfacial region is absent. At lowera-C densities, a-C atoms appear to nucleate on diamond surfacesin higher concentrations than in the rest of the amorphousnetwork. In this case, there is an intermediate region betweendiamond and low density a-C. Interface energies are found to havesimilar values for all diamond faces.

The properties of a-C/D interfaces should influencedecisively the shape of diamond crystals embedded in an a-Cmatrix. In order to make meaningful predictions, we define thediamond's surface energy with a-C overlayer Esa (Eq. (4)),much like free surface energies are defined. Esa is essentially ameasure of the load on diamond surfaces due to a-C. Whencomparing energies for a-C/D interfaces with the same a-Cdensities but different diamond faces, it follows from ourdefinitions that the relative Esa are almost the same as therelative Eint. Using these appropriately defined surface energiesin a Wulff construction, we determine the shape of theequilibrium diamond crystals embedded in a-C matrix. Ourpredictions agree with simulations of nanocrystals in a-C.

Our results point directly to mechanical and tribologicalapplications of both nanocomposite and multilayer carbonmaterials. Our preliminary results on the optoelectronic propertiesof a-C/D link simulations with experimental results and may beused in order to further explore the possibilities to optimizeelectronic structure of carbon-based materials for specific applica-tions through, for example, nanostructure modification or doping.

Acknowledgments

We are grateful to C.Z. Wang and K.M. Ho for providingtheir environment-dependent tight-binding code. This work wassupported by the programme “ΠYΘAΓOPAΣ”, action“EΠEAEK” of the Ministry of National Education andReligious Affairs of Greece.

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