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Surface Science 600 (2006) 298–304

A new structural model for the SiC(0001)(3 · 3) surface derivedfrom first principles studies

Yun Li *, Ling Ye, Xun Wang

Department of Physics, Surface Physics Laboratory, Fudan University, Shanghai 200433, China

Received 9 June 2005; accepted for publication 21 October 2005Available online 28 November 2005

Abstract

A new structural model with fluctuant Si-trimers and missing Si-adatom is proposed for Si-terminated 6H–SiC(0001)(3 · 3) recon-struction. The atomic and electronic structures of the model are studied using first principles pseudopotential density-functionalapproach. The calculated surface electronic density of states coincides quantitatively with the experimental results of photoemissionand electron energy loss spectroscopy. Based on the calculations, the Patterson map and scanning tunneling microscopic (STM) imagessimulated for the new model agree more satisfactorily with the experimental X-ray diffraction and STM observations than that for pre-viously proposed models. The calculations of formation energies suggest that the new structure would be formed under the environmentof dilute Si vapor around the surface in the preparation process.� 2005 Elsevier B.V. All rights reserved.

Keywords: Silicon carbide; (3 · 3) Surface reconstruction; First principles pseudopotential density functional; Surface electronic states; Formation energy

1. Introduction

Silicon carbide is a promising material for high-voltage,high-temperature and high-frequency electronic devices,because of its wide band-gap, high-saturation electronmobility, extreme hardness and thermal stability. In addi-tion, it has been utilized as the substrate for growingGaN light-emitting device structures since the lattice mis-match between SiC and GaN is fairly small . Therefore,the understanding of the structural and electronic proper-ties of SiC surfaces is scientifically meaningful. Among alarge number of SiC polytypes [1], the 6H–SiC(0001) sur-face has attracted a great deal of attention. Various recon-structions [2–4] on hexagonal SiC(0001) and cubicSiC(111) surfaces have been found. Among them, the Si-terminated (3 · 3) reconstruction has been extensively stud-

0039-6028/$ - see front matter � 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.susc.2005.10.029

* Corresponding author. Tel.: +86 2165643418.E-mail address: yun_lee@126.com (Y. Li).

ied theoretically and experimentally, but its atomic andelectronic structures have not been fully confirmed yet.

Several models [4–12] have been proposed to explain theatomic arrangement of (3 · 3) surface. Scanning tunnelingmicroscopic (STM) experiments [5–7] showed that the di-mer adatom stacking fault model [4] proposed by Kaplancannot be applied to SiC(3 · 3) surface, and the model pro-posed by Li and Tsong [5] does not agree with the holo-graphic low-energy electron diffraction [7–11]. Other twomodels are the Kulakov model [6] and the model proposedby Starke and co-workers [7–11] (refer to Starke model inthis work). The former model contains 11 Si atoms in theuppermost three layers of a (3 · 3) unit cell, and the lattercontains two more atoms than Kulakov model. Badziaghas calculated the surface energies of several models basedon a semi-empirical quantum chemical cluster calculationwithout considering the atomic relaxation which isunavoidable in the real reconstructed surface. His calcula-tions concluded that among the above proposed models theStarke model is more favorable than others in the Si-richlimit [12].

Y. Li et al. / Surface Science 600 (2006) 298–304 299

However, many experimental observations do not fullysupport the Starke model. The electronic structure of(3 · 3) surface derived from Starke model is contradictedto some experimental results. In the photoemission spec-troscopy (PES) [13–15] and high-resolution electron energyloss spectroscopy (HREELS) [16] experiments, three filledsurface states and one empty surface state in the bandgap of SiC were observed, while the local density approxi-mation (LDA) calculations on Starke model predicted onlya single half-filled dangling-bond surface state in the gap[17]. Even by using the LDA + U scenario [17], the inclu-sion of on-site Coulomb interaction only splits the singledangling-bond state into a filled-state and an empty staterather than four surface-state bands observed in exper-iments.

Recently, Aoyama et al. presented an X-ray diffractionstudy of the 6H–SiC(0001)(3 · 3) surface [18]. They com-pared the experimental Patterson map with the maps calcu-lated from all proposed models. Although the simulatedPatterson map of Starke model is the one fitting mostlyclose to the observation, actually it does not fully agreewith the experiment. In addition, previous STM experi-ment [19] revealed that the maximum height difference ofatoms in a (3 · 3) surface unit cell is about 1.5 A which isless than the value of 2.5 A in Starke model. All these dis-crepancies indicate that one must consider whether theStarke model really presents the structure of (3 · 3) surfaceor there exists another more reasonable structural modelfor the (3 · 3) reconstruction.

In this paper, we present a new model named fluctuanttrimer (FT) model based on the first principles studies. Inthis model, the topmost Si adatom in Starke model is re-moved and the remaining Si-trimer atoms sit no longeron the same height. The optimized atomic structure ofFT model obtained by relaxation calculations gives aheight fluctuation of the three Si-trimer atoms of about1 A. The simulated Patterson map of FT model is in bestfit with the experimental map. The calculated surface elec-tronic structure is in good agreement with the PES andHREELS experiments. The STM images of FT model aresimulated and agree basically with the previous STM pic-tures. Finally, the surface formation energies of Starkemodel and FT model are calculated and compared. Thedifferent formation conditions of these two structures aresuggested.

2. Calculation models and method

The first principles calculations were preformed withinthe framework of density-functional theory (DFT). Thegeneralized-gradient approximation (GGA) proposed byWang and Perdew [20] was employed for evaluating the ex-change-correlation energy. Vanderbilt-type ultrasoft pseud-opotentials [21] were used to describe the electron-ioninteractions and the wave functions were expanded by planewave basis set with the energy cutoff of about 21 Ry. Thedetails of the calculation methods were described in Ref.

[22]. The surfaces of 6H–SiC(0001)(3 · 3) were modeledin the supercell approach. The slabs, each contains six SiCbilayers, were separated by vacuum regions of about16 A, which was thick enough to isolate the interactions be-tween adjacent slabs. The carbon atoms in the carbon-ter-minal of the slab were saturated with hydrogen in orderto get rid of dangling bonds. The six atomic bilayers ofSiC in each slab were placed at their ideal bulk positionsfirst and then the atoms in the upper half slab were relaxeduntil the Hellman–Feynman forces vanish within 1 meV/A.Ten k points were used to sample the Brillouin-zone inrelaxation calculations and twenty k points were used to ob-tain more accurate electronic density of states.

3. Results and discussion

3.1. Surface atomic structures

The calculations lead to the optimized surface structuresof FT model and also that of Starke model as shown inFig. 1, where the upmost SiC bilayer of the substrate andthe topmost Si-adatom l, the Si-trimer layer atoms i, j, k,and Si adlayer atoms a � h and m in a (3 · 3) unit cellare illustrated. The heights, bond angles and bond lengthsof surface atoms in Stake model and FT model are listed inTable 1. The most remarkable feature of the Starke modelis that the three trimer atoms sit on the same height, andalso for the atoms f, g, h and atoms c, d, e, so it possessesa C3 symmetry. In Starke model the repulsions between thebonds l–i, l–j, l–k linking Si-adatom and Si-trimer and thebonds m–i, m–j, m–k linking atom m and Si-trimer cause adownward movement of atom m by 0.07 A. While in FTmodel the atom m moves up by 0.17 A due to the missingof adatom. Moreover, calculation showed that in order torelieve stress the atom m shifts by 0.2 A along ½2�1�10� andthe trimer rotation angle changes from 9.5� in Starke modelto 17.6� as shown in Fig. 1(a). The relief of stress alsocauses the downward shift of atom j and the upward shiftof atom k. Therefore, the trimer atoms are no longer sittingon the same height as those in Starke model, and the C3

symmetry is broken. The above surface geometry of FTmodel is more stable by reducing an energy of 0.48 eVper (3 · 3) cell than that without surface relaxation.

In the Starke model, the calculated bond angles betweenthe adatom and the trimer atoms are all around 90�. Theangles between the bonds linking the trimer atoms andthe atom m are a little larger than 90�. While these anglesin FT model increase to 105–106�, which are caused bythe upward shift of atom m. The bond lengths of i–m, j–m and k–m are the same in Starke model, but not in FTmodel. Comparing the bond angles between the three tri-mer atoms i, j, k and their nearest adlayer atoms, it canbe seen that the three angles around atom k become smallerin FT than in Starke model, the angles around atom j be-come much larger and the angles around atom i do notchange very much in FT. All these bond angle variationsimply the changes of bond characteristics and can be

Fig. 1. Schematic illustrations of 6H–SiC(0001)(3 · 3) surface structures: (a), (b) top and side views for FT model; (c), (d) top and side views for Starkemodel.

Table 1Heights relative to atom b, bond angles, and bond lengths of surface Siatoms for Starke model and FT model shown in Fig. 1

Height (A) Bond angle (degree) Bond length (A)

Atom Starke FT Atom Starke FT Atom Starke FT

a �0.18 �0.20 c i f 94.7 93.8 i c 2.37 2.34b 0.00 0.00 f im 103.0 114.1 i f 2.40 2.36c �0.01 �0.04 mic 98.8 96.3 im 2.34 2.33d �0.01 �0.08 d jg 94.7 109.6 j d 2.37 2.29e �0.01 �0.01 g jm 103.0 123.0 j g 2.40 2.28f 0.07 0.02 mjd 98.8 114.0 jm 2.34 2.29g 0.07 0.16 ekh 94.7 87.0 ke 2.37 2.40h 0.07 0.06 hkm 103.0 89.8 kh 2.40 2.42i 1.13 1.09 mke 98.8 88.7 km 2.34 2.38j 1.13 0.58 im j 96.0 105.0 l i 2.47k 1.13 1.49 jmk 96.0 106.1 l j 2.47m �0.07 0.17 kmi 96.0 106.0 l k 2.47l 2.57 i l j 89.3

j lk 89.3kl i 89.3

In FT model Si atom at site m shifts 0.2 A along ½2�1�10� relative to theunderneath Si atom in the uppermost substrate layer.

300 Y. Li et al. / Surface Science 600 (2006) 298–304

illustrated in the following electronic densities of states(DOSs).

As a result of nonequivalence of Si atoms at sites i, j andk in FT model, six configurations can be constructed byalternating the atomic height sequence at sites i, j and k.These six configurations can be marked as (i j k), (j k i),(k i j), (i k j), (k j i) and (j ik), in which the sequences of i, jand k in the brackets denote the heights of Si atoms at sitesi, j and k ranked from high to low. The configurationshown in Fig. 1(c) and Table 1 is (k i j). Other six mirrorconfigurations can be obtained by reflecting above config-urations on the planes in parallel with [0001] and½10�10�. The three configurations (i j k), (j k i) and (k i j) arethreefold states of point group C3 and can be transformedto each other by rotating the surface for 120� along the axis[0001]. This is also true for their mirror configurations.Thus, these three configurations and their mirror configu-rations (called group A) are degenerate. Obviously, above

arguments are also valid for another three configurations(i k j), (j ik), (k j i) and their mirror configurations (calledgroup B). Relaxation calculations show that the energyof group B is higher by 10 meV per (3 · 3) cell than thatof group A. It is reasonable to suppose that above twelveconfigurations can coexist as different domains in a widearea on the sample surface.

3.2. Patterson map

For comparison, the Patterson maps for FT, Starke andKulakov models were simulated according to their opti-mized structures by calculating the Patterson functions ofthe surface charge densities. The Patterson function is gi-ven by

Pðu; vÞ ¼Z Z

qsðx; yÞqsðxþ u; y þ vÞdxdy;

where qs(x,y) is the surface charge density. The calculatedresults as well as the experimental observation [18] areshown in Fig. 2.

It is quite obvious that the Patterson map of Kulakovmodel differs very much from the experimental one. So itcould not be a favorite candidate of (3 · 3) reconstruction.As indicated in Ref. [18], the major discrepancies betweenthe calculated Patterson map of Starke model and theexperimental map are: the spot E in Fig. 2(a) becomes alarge bright spot in Fig. 2(c), and the bright diffractionrod-like spot D in Fig. 2(a) splits into two separated spots.Our calculation shows that in the Patterson map of Starkemodel, the large bright spot is a threefold peak contributedby the three inter-atomic vectors ab

�!, bl!

and la!. While in

experimental map, E should be a onefold peak. For FTmodel, the missing of topmost atom l leads to the absenceof inter-atomic vectors bl

!and la

!, so the spot E becomes

weak. In addition, a little rotation and distortion of Si-tri-mer in FT model avoid the splitting of the rod-like spot D.It indicates that the calculated Patterson map of FT modelis in best agreement with the experimental Patterson map.

Fig. 2. (a)Experimental Patterson map reproduced from Ref. [18], and calculated Patterson maps from (b) FT model, (c) Starke model, (d) Kulakovmodel.

ΓΓ

Fig. 3. Electronic structure of 6H–SiC(0001)(3 · 3) surface for FT model.The projected bulk band structure are shown as shaded regions. Thevalence-band maximum (VBM) of the bulk is taken as energy zero. Forsimplicity, the detailed structures of valence and conduction bands areomitted.

Fig. 4. The partial DOSs of surface Si atoms at sites a, i, j and k in FTmodel. The VBM of the bulk is taken as energy zero.

Y. Li et al. / Surface Science 600 (2006) 298–304 301

3.3. Surface electronic structure

To understand the surface structures in more detail, wealso calculated the energy band structure and the DOSs forFT model as shown in Figs. 3 and 4.

In Fig. 3, four flatly dispersed surface states denoted byU1, S1, S2 and S3 exist in the bulk band gap at the energiesof 1.5, 1.0, 0.6 and 0.1 eV above the valence band maxi-mum (VBM) of the bulk. As shown in Fig. 4, the surfacestates U1, S1 and S2 are mainly contributed by the dan-gling-bond states located at atoms i, j and k, respectively,and the state S3 sitting a little above the VBM is derivedfrom the narrowing energy gap which are caused by insuf-ficient bonding of the atoms in the surface, especially theadlayer. As a representative the partial DOS of the atoma in the adlayer is also included in Fig. 4. These four sur-face states are in good agreement with the electronic struc-ture observed by direct photoemission (PES) and inversephotoemission (IPES) [13–15] and HREELS [16] experi-ments, where the U1, S1, S2 and S3 levels are located at2.6, 1.6, 0.6–0.8 and 0.2 eV above the VBM of SiC, respec-tively. Although these values are larger than those in ourFig. 3, it is known that the band gap energy is usuallyunderestimated in LDA calculation [23]. Our calculatedband gap of SiC is about 2.1 eV, i.e. 1.5 times smaller thanthe experimental value of 3.17 eV for 6H–SiC [24]. Takingthis systematic error into consideration, our calculated sur-face-state levels are in quantitative agreement with theexperiment.

Aforementioned electronic structure of FT model is dueto different relaxations of surface atoms. From the bondangles listed in Table 1 it can be deduced that in FT modelthe bonds i–e, i–f, i–m around atom i remain sp3 type, thebonds j–g, j–d, j–m around atom j become more sp2 type,

Fig. 5. The partial DOS of surface states in energy gap of SiC deduced forSi atoms at sites i, j and k in FT model. Solid and dashed curves represents- and p-like states, respectively. The VBM of the bulk is taken as energyzero.

302 Y. Li et al. / Surface Science 600 (2006) 298–304

and the bonds k–e, k–h, k–m around atom k become morep type. Correspondingly, the dangling bonds for the atomsat sites i, j and k are basically sp3, p and s types, respec-tively, which create three dangling-bond surface states inthe energy gap of SiC shown in Fig. 4. Since the energyof sp3 type bond is higher than that of s type bond but low-er than that of p type bond, the energies of these three dan-gling-bond states rank in the sequence of Ek < Ei < Ej.

Above arguments are also demonstrated by the partialDOSs of atoms i, j and k, as shown in Fig. 5. The chargetransferring from the p type dangling bond into the lowersp2 type back bonds for atom j and from the p type backbonds into the s type dangling bond for atom k would bein favor of lowering the energy. Thus, the remaining p typecharge in the dangling bond surface state of atom j willmove onto the three sp2 type back bonds, resulting in theevacuation of electrons from the surface dangling bonds.While for atom k, a part of charge in the three p type backbonds will move onto the remaining s type dangling bond,resulting in the dangling-bond surface state in the band gapalmost filled. So, the p type empty surface state U1 of atomj is located above the Fermi level, the s type filled surfacestate S2 of atom k is located below the Fermi level, andthe sp3 type surface state S1 of atom i is also filled if theFermi level lies above the middle of energy gap as that inthe experiment.

3.4. STM images

Based on the above calculated electronic DOSs, STMimages for the surfaces of FT model and Starke model weresimulated using Bardeen approach [25] and Tersoff–Hamann [26] model of the STM tip. In the simulatedSTM image of Starke model shown in Fig. 6(a), only a sin-gle protrusion appears in a (3 · 3) unit cell as that claimedby Starke et al. For Starke model, the tunneling currentthrough the topmost atom l is about ten times larger thanthat through atoms i, j and k because the value of DOS inthe band gap of atom l is about twice as that of atoms i, jand k and the atom l is located higher than that of atoms i,

j and k. Moreover, the simulated filled-state and empty-state STM images are the same in consistent with the state-ment by Starke et al. that their observed STM image is biasindependent. However, the experimental observationshown in Fig. 6(b) taken by Starke et al. [7] contains actu-ally not only a single protrusion, but also a rather dim spotnext to the big bright spot. On the other hand, the STMimages observed by Li and Tsong [5] are quite differentfrom that of Starke et al. They found that the STM imagemust be bias-dependent. So the overall coincidence be-tween the STM observation and Starke model is notsatisfactory.

The empty-state STM image simulated for FT model isshown in Fig. 6(c), where each bright spot connects to aless bright spots, so the image looks similar to the experi-mental one in Fig. 6(b). The simulation also illustratesthe experimental fact that the filled-state image shown inFig. 6(d) would be different from the empty-state imagein STM. The highest atom k on surface of FT model canbe observed in both filled-state and empty-state imagesand its height relative to Si-adlayer is about 1.5 A, whichis the same as the maximum height difference on (3 · 3) sur-face observed by previous STM experiment [19]. Thus, wesuggest that FT model would be the right structure of(3 · 3) surface observed in above STM experiments.

3.5. Formation mechanism

In order to evaluate which model is energetically morefavorable, surface formation energies of FT model andStarke model were calculated using the following expres-sion suggested by Qian et al. [27] and by Northrup andFroyen [28]:

X ¼ E �Xi

nili;

where X is the surface formation energy per unit cell, E isthe total energy, ni and li are the number and the chemicalpotential of atomic species i in the system, respectively. Thecalculated formation energies can be plotted as a functionof the chemical potential of surface Si atom. As mentionedin Ref. [27], the chemical potential of surface atoms cannotbe larger than that of the bulk phase lSi(bulk), and smallerthan lSi(bulk) � DHf if the surface is in equilibrium withthe bulk, i.e.

lSiðbulkÞ � DH f 6 lSi 6 lSiðbulkÞ;

where DHf is the heat of formation of SiC bulk. DHf is de-rived to be 0.56 eV per pair in our calculation and 0.72 eVper pair from experiment [29].

The calculation results are shown in Fig. 7. It can beseen that in the chemical potential range of lSi >lSi(bulk) � DHf, the Starke model is more stable than ourFT model. While in the range of lSi < lSi(bulk) � DHf, FTmodel is preferable. This later condition could be fulfilledif the surface were prepared under insufficient supplementof ambient Si-flux. The reason is that in the sample prepa-

Fig. 6. (a) Simulated STM image for Starke model under both positive and negative bias voltages, (b) experimental STM empty-state image reproducedfrom Ref. [7], (c) simulated empty-state image for FT model, and (d) simulated filled-state image for FT model. The (3 · 3) cell is labeled in whiteparallelogram.

∆

µ −∆ µ µ

∆

Fig. 7. Difference between the formation energies of Starke model and FTmodel.

Y. Li et al. / Surface Science 600 (2006) 298–304 303

ration process, the uppermost layer of the surface is inequilibrium with the gaseous phase instead of the bulk.Thus the chemical potential of uppermost Si atoms lSi isclose to lSi(gas) which depends on the partial pressure ofSi vapor [30]. The lower the partial pressure of Si vapor,the smaller the lSi(gas). In the work of Starke et al. [7],the (3 · 3) phase was prepared by annealing the sampleat 800 �C under a Si-flux for 30 min. Such long Si exposurewill form a very dense Si gas ambient on the surface. It im-

plies that lSi is close to the lSi(bulk) and Starke model ismore energetically favorable. However, in Aoyama�s exper-iment [18], Si was deposited on SiC at room temperaturefirst and then the sample was annealed at about 1075 �Cwithout the supplying of Si-flux. Annealing at such hightemperature without Si-flux will lead to the desorption ofsurface Si atoms [16] and then form a very dilute Si gasupon the surface. In such case, lSi would be much less thanlSi(bulk) and even less than lSi(bulk)�DHf, i.e. FT model ismore energetically favorable. Similar situation would occurin other experiments [5,13,14,16], where the (3 · 3) surfaceswere prepared by annealing either under a weak Si-flux orat higher temperature. Therefore, we believe there is thepossibility of forming two kinds of SiC(0001)(3 · 3) sur-faces. The Starke model can be formed under sufficientlystrong Si flux, while the FT model might be formed underan environment of very dilute Si vapor. These two cases aredetermined by the sample preparation conditions.

4. Conclusions

We have reported a first principles study on the atomicand electronic structures of 6H–SiC(0001)(3 · 3) recon-struction. It was found that besides the previously favor-able Starke model, 6H–SiC(0001)(3 · 3) can also beconstructed according to another model with fluctuantSi-trimers and missing Si-adatom. The calculated Patterson

304 Y. Li et al. / Surface Science 600 (2006) 298–304

maps verify that the new model could reach the best fit withthe experimental observation. In this new model, the threeatoms of Si-trimer in a unit cell fluctuate by a height ofabout 1 A and thus deduce s-like, sp3-like and p-like dan-gling-bond surface states in SiC energy gap, respectively.The characteristics of calculated surface electronic struc-ture agree fairly well with PES, IPES and HREELS results.The simulation of STM images further confirms the feasi-bility of the new model. The environment of Si vapor inpreparation process is suggested to be important for form-ing different (3 · 3) reconstructions. Above arguments for6H–SiC(0001) are also valid for the 3C–SiC(111) and4H–SiC(0001) surfaces, since the (3 · 3) reconstruction isindependent on the substrate polytype [1].

Acknowledgments

This work was support by the National Natural ScienceFoundation of China (Grant No. 60176005), the Founda-tion of National High Performance Computing Center,and Fudan High-end Computing Center, Shanghai, China.

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