Curson, NJ

Radny, MW

Schofield, SR

Shah, GA

Smith, PV

English

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Radny et al. Reply: In response to the points raised in theabove Comment [1], we have carried out extensive calcu-lations of the Geð001Þc4   2 clean surface using a slabcontaining 8, 10, and 12 Ge layers, respectively. Thesecalculations have again been performed using the VASPcode. While the reconstruction of the surface was foundto be little different to that obtained in our original calcu-lations using 6 Ge layers, the electronic structure wasfound to vary signiﬁcantly with the number of layers, inagreement with the ﬁndings of Yan et al. The band struc-ture obtained using the primitive unit cell and 12 Ge layersis shown in Fig. 1. Comparison with the correspondingband structures for 6, 8, and 10 layers shows that the dis-persion of the bands at the top of the valence band (TVB)along the  -J,  -K1,  -J0, and  -K2 symmetry directions isstrongly layer dependent, with the dispersion along  -Jactually reversing in sign. The nature of some of the statesnear the TVB, and the size of the predicted energy gap, arealso layer dependent. All of these conclusions are consis-tent with the ﬁndings of Yan et al.The distribution of the different surface states and sur-face resonancesforthe12layerslabisalsoshowninFig.1.This shows, as stated by Yan et al., that the true back bondsurface states occur  0:5e Vbelow the Fermi Energy (EF)near the   point, and that these states become surfaceresonances as one approaches the J point.To further clarify the nature of the states, we havedetermined the spatial variation of their charge distributionnormal to the surface, by integrating the charge density foreach state over x and y. The highest occupied state at the Jpoint has an exponentially decaying envelope conﬁrmingit is a surface resonance. The corresponding state for ouroriginal 6 layer slab, which lies at the TVB, is also asurface resonance. Moreover, this state has the bulk of itscharge distributed along the bonds between the ﬁrst andsecond, and third and fourth, layer atoms, analogous to thestate shown in Fig. 4(c) of our original Letter [2]. Thus, thehighest occupied states for 6 layers are indeed back bondsurface resonances as we claimed.We agree with Yan et al., however, that this is no longerthe casewhen the number of layers is increased to 10 or 12.For these thicker slabs, the states at the TVB occur nearthe   point (see Fig. 1) and are really bulk states. Near thesurface, however, these states are characterized by acharge distribution arising from back bonds between theﬁrst layer (dimer) and second layer atoms. They thusreproduce charge density plots analogous to those shownin Figs. 4(a) and 4(b) of our original Letter, and low-biassimulated STM images dominated by double-lobed fea-tures on the up-Ge atoms, in agreement with experiment. Itis thus these states that give the low-bias Ge(001) imagestheir distinctly different appearance to low-bias Si(001)images, where the pi state dominates at the Fermi energy.The change in the electronic structure with increasingnumber of layers also affects the relationship between theback bond and dangling bond surface states. In our originalLetter, we concluded that the states at the TVB forGeð001Þc4   2 were back bond states, and that the dan-gling bond states were  0:35 eV below EF. While this iscorrect for the 6 layer slab, the 12 layer calculations yield avery different result, as shown in Fig. 1. We observe that inthis latter case, some of the dangling bond states occurquite close to the TVB, and are essentially degenerate withthe highest-lying back bond surface resonances. It thusfollows that the distinctions that we were drawing regard-ing the energy distributions of the back bond and danglingbond surface states of the Sið001Þc4   2 and Geð001Þc4  2 surfaces in the vicinity of EF are no longer so clear cut.M.W. Radny,1 G.A. Shah,1 S.R. Schoﬁeld,1,2 P.V. Smith,1and N.J. Curson21School of Mathematical and Physical SciencesThe University of NewcastleCallaghan 2308, Australia2London Centre for Nanotechnologyand Department of Electronic and Electrical EngineeringUniversity College LondonLondon, WC1H OAH, United KingdomReceived 10 September 2009; published 30 October 2009DOI: 10.1103/PhysRevLett.103.189702PACS numbers: 73.20.At, 68.37.Ef[1] Binghai Yan et al., preceding Comment, Phys. Rev. Lett.103, 189701 (2009).[2] M.W. Radny et al., Phys. Rev. Lett. 100, 246807 (2008).FIG. 1. Electronic structure of the Geð001Þc4   2 clean sur-face for the primitive surface unit cell and a periodic slabcontaining 12 layers, with the top of the valence band (TVB)indicated by the dashed horizontal line. The irreducible part ofthe Surface Brillouin zone is shown in the inset. The surfacestates and resonances are indicated by symbols, with the size ofthe symbol indicating the degree of surface localization: d backbond, j dangling bond.PRL 103, 189702 (2009) PHYSICAL REVIEW LETTERSweek ending30 OCTOBER 20090031-9007=09=103(18)=189702(1) 189702-1  2009 The American Physical Society

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