First-principles calculations of the diamond (110) surface: A Mott insulator
Félix YndurainDepartamento de Física de la Materia Condensada, Universidad Autónoma de Madrid, 28049 Madrid, Spain
�Received 8 February 2007; published 30 May 2007�
The atomic and electronic structures of the free C�110� surface are studied by means of a first-principlescalculation based on the density-functional theory using the SIESTA code. In the calculations, the possibility ofdifferent spin populations at each atom is allowed. We find that the free C�110� surface is flat and does notreconstruct, but the two atoms at the surface unit cell have magnetic moment of the order of 0.5�B in anantiferromagnetic arrangement. The results are interpreted as being due to antiferromagnetic correlations at thenonbonding � p-orbitals at the quasi-one-dimensional zigzag chains of atoms at the surface, which, in turn, isa signature of a Mott insulator ground state. These correlations induce a gap in the electronic spectrum inagreement with experimental photoemission data. The similarities and differences between the C�110� andSi�110� surfaces are analyzed.
Carbon has been considered for many years as an elementwith important potential practical applications. This interesthas been fostered in the past years by the possibility of ob-taining carbon in different atomic configurations rangingfrom macromolecules like the fullerenes to two-dimensionalsheets like graphene and their wrapped configuration innanotubes. With the possible technological applications inmind, the possibility of developing magnetic moments andmagnetic ordering in carbon structures has been investigated1
in the various geometries, namely, nanotubes2,3 graph-ene,4 and other configurations.5,6 In all these works, carbonin its diamond structure has been overlooked. In this work,we consider the possibility of antiferromagnetic correlationbeing responsible for the opening of a gap in the electronicexcitations at the �110� surface of carbon in its diamondstructure. This has been partially motivated by the discrep-ancy between previous theoretical calculations7 that predict ametallic surface, whereas photoemission measurements8 donot register electronic states at the Fermi level.
Carbon �110� surface, unlike other surface orientations,does not present any reconstruction even when annealed athigh temperatures. Low-energy electron-diffraction �LEED�measurements indicate a surface close to an ideal diamondstructure surface9 �see Fig. 1�. There are, however, not manyexperimental detailed studies of this surface. Maier et al.8
performed a detailed study of this surface by LEED, angle-resolved photoelectron spectroscopy, and x-ray-inducedcore-level spectroscopy, concluding its nonmetallic charac-ter.
From the theoretical point of view, there have beenseveral calculations reported in the literature; Davidson andPickett,10 performed an electronic structure calculation usinga semiempirical tight-binding method. They found a Peierls-distortion-induced dimerized metallic surface. Alfonso etal.11 also found a buckled surface. Finally, Kern and Hafner7
performed a full first-principles calculation using the VASP
�Ref. 12� code. They found a symmetric nonbuckled metallicsurface.
II. METHODOLOGY
Here, in this work, we present the results of a differentfirst-principles calculation of the C�110� free surface. Thecalculations were performed within density-functional theory�DFT�,13 using the generalized gradient approximation14 toexchange and correlation. The calculations were obtainedwith the SIESTA �Ref. 15� method, which uses a basis ofnumerical atomic orbitals16 and separable17 norm-conservingpseudopotentials18 with partial core corrections.19 We havefound satisfactory the standard double-� basis with polariza-tion orbitals, which has been used throughout this work. Thebulk calculation yields a lattice constant of 3.585 Å, in goodagreement with the experimental value of 3.567 Å which isthe value used throughout this work.
To simulate the free C�110� surface, we have performedcalculations of different size slabs formed by the stacking of�110� planes. In particular, we have considered in this workslabs built by 7, 9, and 15 layers. In all cases, the “back”surface was properly saturated with hydrogen atoms. Theconvergence of the relevant precision parameters was care-fully checked. The real-space integration grid had a cutoff of250 Ry. Of the order of up to 900 k points were used in thetwo-dimensional Brillouin-zone sampling using the Monk-horst-Pack k-point sampling. To accelerate the self-consis-
FIG. 1. Atomic positions at the C�110� surface unit cell. Atomslabeled 1 are at the surface; atoms labeled 2 are at the layer under-neath. The lines joining the atoms indicate nearest-neighbor bonds.The arrows indicate possible antiferromagnetic arrangement consid-ered in this work.
tency convergence, a broadening of the energy levels wasperformed using the method of Methfessel and Paxton,20
which is very suitable for systems with a large variation ofthe density of states at the vicinity of the Fermi level, whichis the case in our system �see below�. It is necessary to men-tion that the energy differences between paramagnetic andmagnetic solutions are, in general, small, which requires avery high convergence in all precision parameters and toler-ances. To obtain the equilibrium geometry, the atomic posi-tions were relaxed using the standard conjugated gradientprocedure until the forces acting on them were smaller than0.02 eV/Å.
III. RESULTS OF THE CALCULATIONS
Before addressing the problem of the surface, we calcu-lated the bulk diamond. As indicated above, we obtained anequilibrium lattice constant of 3.585 Å. To study the �110�surface, we considered finite slabs of atoms such that the fivetopmost layers near the surface were allowed to relax. Re-sults of our calculation for the 15-layer case and its compari-son with previous calculations are shown in Table I. We firstobserved that, for a paramagnetic calculation, we have anexcellent agreement with the results of Kern and Hafner.7
This is a reassuring result since the SIESTA and VASP methodsof calculation start with very different basis sets and pseudo-potentials. We, like Kern and Hafner,7 obtained a flat surface.In some instances we started the calculation with an initialbuckled and/or dimerized surface obtaining in all cases, afterfull relaxation using a standard conjugated gradient proce-dure, a planar nonbuckled or dimerized surface as indicatedin Fig. 2.
After full atomic relaxation, we found that the surfacebonds are shorter than the bulk ones and the bond angles��122° � at the surface are larger than at the bulk �109.47°�,making the surface chains more one-dimensional than theideal structure ones.
We allowed in the calculation the possibility of differentspin populations in the atoms. We were not able to find astable solution with nonzero magnetic moment at the surfaceatoms in the ferromagnetic configuration. However, a stableantiferromagnetic solution was obtained. After atomic relax-ation, the antiferromagnetic �AF� solution for the 7-, 9-, and15-layer slabs was 0.065, 0.065, and 0.066 eV, respectively,lower than the paramagnetic ground state. This indicates thatthe atoms at the stable surface geometry have nonzero mag-netic moment arranged in an antiferromagnetic order. Thestable geometry in this case is shown in Table I and theMulliken population analysis results are given in Table II.We first observe the large magnetic moment at the surfaceatoms �of the order of 0.5�B�. This magnetism is essentiallyconfined at the surface layer. The atomic relaxations in theAF ground state are smaller than in the nonmagnetic case�see Table I�. We have checked these results using otherfunctionals, in particular, we have used “RPBE” �Ref. 21�and “LYP” �Ref. 22� functionals, obtaining an antiferromag-netic solution with 0.514 and 0.520�B per surface atom, re-spectively.
TABLE I. Atomic structure of the C�110� surface after atomicrelaxation. The results considering paramagnetic �Para� and antifer-romagnetic �AF� ground states are shown in columns two and three,respectively. The surface layer is labeled 1 and the adjacent layers2, 3, and so on. The primed and unprimed numbers distinguishbetween the two atoms in the surface unit cell. dij stands for thebond length between the two nearest-neighbor atoms in layers i andj. The values in parentheses are the change of the bond lengthrelative to the bulk value of 1.544 Å. The angle � refers to the bondangle at the surface layer �see Fig. 1�. �xi and �zi stand for therelaxations of the atoms in the ith layer.
FIG. 2. Self-consistent loop results of the calculation of aC�110� slab when an initial dimerized and buckled surface is con-sidered. In �a� the buckling �deviation from planar surface� of thesurface atoms, in Å, is indicated. �b� shows, also in Å, the dimer-ization �deviation from equal nearest-neighbor distances� along thesurface plane. �c� indicates the maximum force acting on the atomsand �d� shows, in eV, the total energy with respect to the convergedresult.
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In order to understand the origin of this AF ground state,we have calculated the electronic band structure. The resultsare given in Fig. 3, where the bands along the two-dimensional Brillouin zone for the C�110� 15-layer slab areshown. Similar results are obtained for the 7- and 9-layerslabs considered. In all cases the back surface was saturatedwith hydrogen atoms. We first observe in both Figs. 3�a� and3�b� the surface-state bands at the bulk band gap. In theparamagnetic case �Fig. 3�a��, we obtain similar results tothose of Ref. 7. A metallic surface is obtained with two bands
throughout the Brillouin zone, being degenerated along theJ-K direction due to the equivalence of the two atoms at thesurface unit cell. However, in the case of the AF solution, thedegeneracy is lifted, an absolute gap of 0.54 eV takes place,and the system is nonmetallic. The 1.65 eV splitting of sur-face states at the J point is compatible with the experimentalphotoemission result8 for the occupied surface level at this kpoint �1.1 eV below the Fermi level�. However, at the �point, our calculation gives a surface state slightly above thetop of the bulk valence band, in disagreement with the ex-perimental findings. This is probably due to the finite-sizecharacter of the slab calculation and the difficulty of distin-guishing between surface and bulk states at this particular kpoint. It should be stressed that a detailed comparison nearthe gap is not appropriate due to the gap narrowing of DFTcalculations. The energy gain associated with the opening ofthis gap is intimately related to the one-dimensional charac-ter of the surface chains. To analyze this in more detail, wehave calculated the surface atom local density of states pro-jected at the different atomic orbitals. Results of the calcula-tion are given in Fig. 4. Also, in this figure, the local densityfor the nonmagnetic solution is shown. We first notice theone-dimensional character of the problem. The density ofstates of the paramagnetic solution displays the characteristictwo-peak structure of the one-dimensional problems. In ad-dition, the magnetic moment only appears at the p� non-
TABLE II. Charge and magnetic moments at the C�110� surfaceatoms in the antiferromagnetic configuration. The charges are theMulliken populations. Charge �↑,↓� stand for the spin-up and spin-down electronic charges respectively.
Atom Charge �↑� Charge �↓�Magnetic moment
��B� Total charge
1� 2.27 1.78 0.49 4.05
1 1.78 2.27 −0.49 4.05
2 1.99 1.95 0.04 3.94
2� 1.95 1.99 −0.04 3.94
3� 2.02 1.99 0.03 4.01
3 1.99 2.02 −0.03 4.01
4 2.00 2.00 0.00 4.00
4� 2.00 2.00 0.00 4.00
-20
-15
-10
-5
0
5
10
Ener
gy
(eV
)
-25
-20
-15
-10
-5
0
5
10
KΓJ’KJΓ
Ener
gy
(eV
)
Wave vector
Paramagnetic
Anti erromagnetic
(a)
(b) f
FIG. 3. Band structure along the symmetric two-dimensionaldirections of a slab of 15 C�110� layers. One surface is kept free,whereas the other one is saturated with hydrogen. The energy originis at the Fermi level. �a� Assuming equal spin-up and spin-downpopulations and �b� allowing AF order within the unit cell �see Fig.1�. The surface-state band structure is easily identified at the energygap.
-2 2
C(110) AF Surface
FIG. 4. Partial densities of states at the up magnetic momentatom �say, atom labeled 1� in Fig. 1� of the C�110� surface. Ex-change of spin-up and spin-down curves provides the densities ofstates at the down magnetic moment atom �labeled 1 in Fig. 1� inthe unit cell. px, py, and pz stand for the p orbitals in the surfaceplane and normal to the chains, in the surface plane and along thechains, and normal to the surface, respectively. The inset indicatesthe total density of states of one surface atom in the paramagneticsolution. A small �0.2 eV� energy broadening has been included.
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bonding orbitals perpendicular to the chains. The s and p��py� orbitals do not contribute to the magnetic moment.
At this point is worth studying the difference betweencarbon and silicon. We have performed the same calculationas above for silicon, obtaining no magnetic solutions. More-over, we have found a buckled surface with an energy gain of0.53 eV. The atom displacement normal to the surface is0.81 Å, and a charge transfer between the atoms is of theorder of 0.3 electrons. In Fig. 5, the densities for the non-buckled and buckled Si�110� surfaces are shown. We firstobserve that, in this case, we obtain, for the nonbuckled sur-face, a typical two-dimensional density of states with a vanHove logarithmic singularity at the Fermi level, which dis-appears in the buckled surface. This energy-band splitting
inhibits the magnetic moment formation. These results can-not be compared with experiments since this surface under-goes a 2�16 reconstruction.23 This difference between Cand Si can be understood by the more localized character ofthe atomic orbitals in carbon that, in addition, inhibits theinteraction between the parallel chains at the �110� surface,becoming then more one-dimensional rather than two-dimensional like Si.
IV. CONCLUSIONS
In conclusion, theoretical evidence of opening a gap in theelectronic energy spectrum at the clean C�110� surface due tolocal magnetic correlations at the zigzag chains of the sur-face atoms has been presented. In our DFT calculation, thestatic magnetic moments at the atoms are arranged in anantiferromagnetic order, whereas there is no charge symme-try breaking. The existence of a gap is in agreement withexperiments8 and with many-body calculations of the Hub-bard model in one-dimensional chains24 �in our case, theHubbard parameters t and U can be estimated to be 0.5 and1.7 eV, respectively�. Due to the mean-field approximationused, we cannot assess, however, whether or not the AFlong-range order would prevail in the real system. The dif-ferent properties of the C�110� and Si�110� surfaces are in-terpreted as being due to the one-dimensional character ofthe former versus the two-dimensional one of the latter. TheC�110� surface has turned out to be a unique one-dimensional system, where magnetic correlations play a cru-cial role characteristic of a Mott insulator ground state neverdiscussed before.
ACKNOWLEDGMENTS
I am indebted to Nicolas Lorente for bringing my atten-tion to this problem. Very helpful discussions with him andwith G. Gomez-Santos, R. Miranda, and J. M. Soler are ap-preciated. I also would like to thank O. Paz for a criticalreading of the manuscript. Financial support of the SpanishMinistry of Education and Science through Grant No.BFM2003-03372 is acknowledged.
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