G Vasseur

Y Fagot-Revurat

B Kierren

M Sicot

D Malterre

Electronic surface potential from angle-resolved photoemission

Keywords: number(s): 73.20.At, 73.22.-f, 79.60.-i

We present a methodology to determine the periodic potential of a surface from angle-resolved photoemission spectroscopy. By mapping the band structure and measuring the energy gap widths at high symmetry points of the surface Brillouin zone, the first Fourier components of the potential can be estimated. This method has been applied in the case of the reconstructed Ag/Cu(111) surface where controlled potassium atom doping has been used to precisely tune the band gap energies. This potential determination allows one to calculate the local density of states which is found to be in very good agreement with scanning tunneling spectroscopy data.

Band structures of solids reflect the potential probed by the electrons. This electronic potential is usually only obtained from ab initio calculations in density functional theory approach and the corresponding band structures may be compared with experiments using essentially angle-resolved photoemission spectroscopy (ARPES). An approach was proposed to map the surface electronic potential from scanning tunneling microscopy (STM) by using the linear response theory [1]. We will show that it is possible to directly obtain the potential from ARPES experiments. This method is based on the measurement of band gaps at the high-symmetry points of the Brillouin zone (BZ) which allow the determination of Fourier components of the potential.

Let us recall some information about the relation between band structure and symmetry. The band degeneracy, especially at the high-symmetry points, is a direct consequence of the potential symmetry and more precisely of the symmetry of the wave vector group at the considered BZ point [START_REF] Tinkham | Group Theory and Quantum Mechanics[END_REF]. The wave vector group is the group of operations leaving the wave vector unchanged. A change of degeneracy or of the dispersion relation can result from a symmetry breaking (change of the wave vector group). For example, in graphene the Dirac points with their twofold degeneracy and the linear dispersion near the K points correspond to the two-dimensional (2D) irreducible representation of the C 3v wave vector group. A symmetry breaking which changes the wave vector group at the K point, achieved by putting a graphene layer on an appropriate substrate, leads to a gap opening and then to massive fermions [START_REF] Starodub | [END_REF]4].

Such a gap opening can be studied in reconstructed nanostructurated lattices at the surface of (111) noble metals. Surface Shockley states develop on these (111) surfaces with nearly-free-electron-like behavior [5]. Therefore parabolic dispersions are observed with deviations close to the surface BZ boundary. For example, confinement and/or gap openings have been observed for vicinal surfaces due to the onedimensional periodic potential corresponding to the step array [6]. In Au vicinal surfaces, additional superperiodicity parallel to the steps exists due to the herringbone reconstruction of the Au(111) surface on terraces leading to very small gaps [7]. Another famous example is the Ag/Cu(111) interface where an hexagonal array of triangular dislocations was observed [8,9]. The corresponding ∼ (9.5 × 9.5) superperiodicity induces the opening of gaps observed at the M point of the surface BZ of the reconstruction [10][11][12].

Due to the plane wave character of the Shockley state, the gap opening results from the Bragg mechanism at the BZ boundaries and their magnitudes are simply related to the corresponding Fourier components (V G ) of the potential in the first approximation of perturbation theory. At the M point of the hexagonal surface BZ, the gap magnitude is simply 2|V G |-like in a simple 1D nearly-free-electron gas discussed in solid state textbooks [START_REF] Ashcroft | Solid State Physics[END_REF]. Therefore the gap width only gives partial information (the modulus) about the potential. It has been shown by combining ARPES and scanning tunneling spectroscopy (STS) that both the magnitude and the phase of the first Fourier components of the Au reconstruction on a Au vicinal surface can be obtained [7]. In this Rapid Communication, we would like to show that a detailed description of the potential can be achieved by measuring the gaps at the different high-symmetry points ( , M, and K) and directions. However, only the first low energy gap corresponding to occupied states can be probed by ARPES in Ag/Cu(111). The technical key point of the present work is to use controlled potassium doping to precisely tune the energy position of the band gaps since K adsorption changes the work function and shifts down in energy the Shockley state [START_REF] Abd El-Fattah | [END_REF]15]. By using different K doping, we succeeded in measuring quantitatively several gaps in the different directions of the Brillouin zone and we obtained a complete determination of the Shockley band structure. The gap values at several high-symmetry points allow one to determine not only the norm but also the real and imaginary parts of the first Fourier components of the potential. It gives a satisfactory description of this potential since the calculated local density of states is found to be in very good agreement with local spectroscopic properties probed by STS. This demonstrates the interest and strength of this original methodology in estimating the surface potential.

The angle-resolved photoemission experiments were performed at 80 K with a Scienta 200 high-resolution hemispherical analyzer using a photon energy of hν = 21.2 eV. Data points in Fig. 2 are obtained from standard line fits of individual energy distribution curves (EDCs), using Lorentzian functions and Fermi edge functions. Features that lie within ±2k B T of E F are cut off by the Fermi edge. Peaks above E F can thus be visualized by normalizing the EDCSs with the Fermi-Dirac distribution function convoluted with the experimental resolution. STM and STS experiments have been carried out using an Omicron STM. The dI/dV spectra and maps were recorded at 5 K in the open feedback loop mode using the lockin technique with a bias modulation of 5 mV rms at 2300 Hz (stabilization parameters: set point 1 nA, bias 0.8 eV). Before transferring to the STM cryostat at 5 K, the Ag monolayer is evaporated at 150 K on the clean Cu(111) crystal, and then annealed to 350 K. This process leads to an ∼ (9.5 × 9.5) hexagonal lattice of triangular dislocations. Evaporation of K atoms in the submonolayer range was achieved by heating a standard K getter. Band structure dispersions are computed by a direct diagonalization of the Schrödinger equation with a potential defined by the two first Fourier components in a plane wave basis limited to the first 36 waves around . Note that additional plane waves have negligible effects. Local density of states (LDOS) maps and spectra are obtained resolving the Hamiltonian for more than 1500 wave vectors in the first BZ and by taking into account a 15-meV Gaussian energy broadening.

First of all, we will discuss angle-revolved photoemission spectroscopy measurements recorded along high-symmetry directions ( M and K) of the first BZ of the undoped and K-doped reconstructed Ag/Cu(111) surface. Figures 1(a) and 1(b) show the second derivative of the ARPES signal for a bare and a K-doped surface, respectively. These maps clearly show the parabolic dispersion of the Shockley surface state with energy gaps appearing at the boundary of the BZ. First, K deposition shifts the surface state energy at by 300 meV towards higher binding energy. This behavior is well known and reflects the lowering of the work function with alkali metals [16,17]. This shift of the surface state is proportional to the amount of K atoms evaporated on the surface. Secondly, the surface state remains well defined and only a small broadening is induced by the K deposition. For the bare interface, an 80-meV gap opens below the Fermi energy (E F ) [10][11][12] at the M point of the 2D BZ shown in Fig. 1(c) and although the gap at the K point which is very close to the Fermi energy is not clearly resolved, the analysis of the spectral shape reveals that a small gap (30 ± 15 meV) appears in agreement with previous measurements at room temperature [4]. In the K-doped surface, this gap is well below the Fermi energy and can be estimated more accurately, E K 2 -E K 1 = 32 ± 5 meV [inset of Fig. 1(b)]. In addition, a band is emerging at

E K 3 .
The characteristics of the energy spectrum at the highsymmetry points can be simply understood by considering the topology of the reciprocal space and the band energy calculated in the nearly-free-electron framework. Indeed, the gap widths depend on the Fourier components of the surface potential which can be decomposed:

V ( r) = G (p) n V G (p) n e i G (p) n • r . ( 1
)
Symmetry gives us some relations between the different Fourier components. In the hexagonal reciprocal space lattice, one has six first neighboring points ( G (1) n with n = 1,2, . . . ,6) around the central point [first ring in dashed line in Fig. 1(c)]. However, these points are not equivalent since there is no sixfold symmetry axis and only a threefold one (C 3 ). Then only G (1) 1 , G (1) 3 , and G (1) 5 or G (1) 2 , G (1) 4 , and G (1) 6 are equivalent. As a consequence, two different Fourier components are associated with these two sets of C 3 equivalent vectors. However, time-reversal symmetry which transforms a wave vector in its opposite connects these inequivalent points. As a consequence, if V G is the Fourier component for G, then

V -G = V * G .
In summary, the Fourier components in the first ring around are V G (1) n = V 1 for n = 1,3,5 and V G (1) n = V * 1 for n = 2,4,6.

Let us now investigate the effects of these Fourier components on the band energies at the M and K points in a very crude approximation. As the M point is located in between two neighboring reciprocal vectors, 0 ( ) and G (1) 1 , there are two states associated with the plane waves originating from these two reciprocal vectors. The diagonalization of the corresponding 2 × 2 Hamiltonian leads to a gap width

E M = E M 2 -E M 1 = 2|V 1 |.
The K point is located in the center of an equilateral triangle defined by three reciprocal nodes [Fig. 1(c)] and then three plane waves are involved and the resolution of the 3×3 Hamiltonian leads to a gap width

E = E K 2 -E K 1 = 2 √ 3 Im V 1 and an excited state at E K 3 (E K 3 -E K 2 = 3 Re V 1 - √ 3 Im V 1 )
. These simple calculations show that the gap at the K point is related to the imaginary part of V 1 , so that V 1 can be completely obtained from ARPES measurements.

We can go further than this first approximation described above by extending the plane wave basis. We then consider 36 plane waves associated with 36 reciprocal vectors around (additional vectors have nonsizable effects in the energy range corresponding to our spectroscopic measurements). We start from the V 1 value given by the analytical expressions above and we adjust it by using the extended basis in order to reproduce both the experimental energy gaps at M and K and the band dispersions in the high-symmetry directions. The energies have been determined by fitting the photoemission energy distribution curves for each k value by Lorentzian structures. The main intensity is found on the main parabola but small intensity is also observed on the folded bands. This method gives V 1 = (47 + i9.6) meV.

The gaps involving the Fourier components associated with reciprocal vectors ( G (2) n ) in the second ring are not accessible because they appear in unoccupied states. However, with increasing the amount of K adatoms, it is possible to shift the Shockley state in order to occupy states up to the second gap at the M point. This is illustrated in the inset of Fig. 2 where a highly doped surface exhibits a gap opening of about 50 meV. 1(c)] demonstrating that the second Fourier component (V 2 ) is purely real. The magnitude of this gap depends on V 2 at first order and on V 1 at second order. The fitting method leads to a value of V 2 ∼ 15 meV. It is worth mentioning that V 2 has a small effect on the dispersion and only affects the band structure close to this second gap at the M point. The band dispersions along the high-symmetry directions M and K calculated with V 1 = (47 + i9.6) meV and V 2 = 15 meV are shown in Fig. 2 (solid lines) and very good agreement with the experimental dispersions (symbols) is found. This surface potential directly deduced from experiments is one step beyond the potentials previously proposed to describe the Ag/Cu(111) surface [4,[START_REF] Abd El-Fattah | [END_REF]. In these papers, the shape of the surface potential was arbitrary postulated: It consists of areas with simple geometry (triangles and/or hexagons and are different in the two papers) corresponding to flat potentials whose values are free parameters adjusted in order to reproduce the experimental data.

The vertical mirror symmetry indicates that

V G (2) 2 = V G (2) 1 = V 2 [Fig.
Finally, in order to test the accuracy of the method, one can rebuild the 2D periodic potential and calculate local properties such as LDOS as a function of energy and position in the reconstructed supercell. By solving the Schrödinger equation with the determined surface potential, we obtain eigenfunctions k ( r) corresponding to a given energy E. The LDOS at position r and energy E is calculated by summing | k ( r)| 2 for all values of k in the first BZ. As the LDOS can be probed directly by STS, we will be able to give a quantitative comparison between calculations and spectroscopic measurements. The rebuilt 2D potential is plotted in Fig. 3(a) with a STM topographic image of the bare triangular reconstructed surface with the different regions characterized by their structural arrangement. We distinguish four areas: a triangular one corresponding to hexagonal compact packing (noted hcp), two face-centered-cubic packing (fcc1 and fcc2), and an intermediate region (side). From the determined potential, we have calculated local properties and have compared them with STS data. Figure 3(b) shows the differential conductance measured in the center of these four regions and Fig. 3(c) the corresponding LDOS spectra. The pronounced dip appearing close to the Fermi energy corresponds to the complete gap in the surface density of states. We would like to point out that the spectral intensity in the hcp region is significantly lower than in the three other regions for small energies (<0.1 eV) suggesting that the surface potential is high in the hcp triangles as obtained in our potential determination. Around 0.12 eV, the spectral intensity in the hcp region increases rapidly. This variation results from the third band at the M point (Fig. 2). Finally, a dip in the fcc and hcp regions and protrusion in the side region appear around 0.35 eV. This behavior is related to the opening of the second gap at the M point.

In Figs. 4(a) and 4(b), we compare differential conductance maps recorded at three characteristic energies and the corresponding calculated LDOS. The first energy (-70 meV) was chosen below the first gap at the maximum of the spectral peak, the second one (40 meV) is just above the gap, whereas the third one (360 meV) corresponds to the second gap at the M point. Very good agreement is obtained between the experimental and calculated maps. Below the first gap, the maxima of the local density of states are located in the fcc1 regions, whereas they are in the fcc2 regions above this gap. At the energy of opening of the second gap at the M point (350 meV), the electronic density is more structured and maxima appear in the side regions. The good agreement between measured and calculated conductance spectra (Fig. 3) and maps (Fig. 4) shows that the potential estimated from gap widths in ARPES describes remarkably the local properties.

In this Rapid Communication, we propose a methodological approach to obtain the surface potential from gaps associated with the nanostructurated surface of Ag/Cu(111). The determination of these energy band gaps by ARPES and symmetry arguments leads to an estimation of the first two Fourier components from which we can build an excellent approximation of the potential. Local properties such as the LDOS and surface band structure can be satisfactorily described by this potential. Such an original and simple approach could be used to model the electronic properties at the nanometric scale as soon as the nearly-free-electron model is a good approximation.

FIG. 1 .

 1 FIG. 1. (Color online) Experimental band dispersions (second derivative) of the Shockley state along the M and K directions for (a) the bare triangular reconstructed Ag/Cu(111) surface and (b) a K-doped surface (inset: energy cut very close to the K point exhibiting the three eigenenergies). (c) Reciprocal space showing the high-symmetry points of the 2D BZ and the first neighboring reciprocal vectors. We also indicate the symmetry mirrors (σ v ) and the Fourier components of the potential associated with the reciprocal vectors in the first two rings around the point (dashed circles).

FIG. 2 .

 2 FIG. 2. (Color online) Band dispersion in the M and K directions determined from the K-doped surface of Fig. 1(b). Symbols are experimental values and the solid lines represent the calculated band structure obtained with a potential built from the Fourier components [V 1 = (47 + i9.6) meV, V 2 = 15 meV] extracted from experimental gaps. Inset: experimental intensity of a highly K-doped surface illustrating the opening of a second gap at the M point (see text).

FIG. 3 .

 3 FIG. 3. (Color online) (a) Calculated potential and STM image corresponding to the same area showing the undoped reconstructed unit cell with the fcc1, fcc2, hcp, and side regions. Experimental dI/dV spectra (b) and calculated LDOS (c) corresponding to the center of the different regions of the reconstructed unit cell.

FIG. 4 .

 4 FIG. 4. (Color online) (a), (b) Experimental 2D dI/dV maps and calculated LDOS corresponding to the bare Ag/Cu(111) surface at different energies (-70, 40, and 350 meV).


Thomas Mazet

Daniel Malterre

Michel François

Lucas Eichenberger

M Grioni

C Dallera

G Monaco

M Franc ¸ois

Composition and temperature dependence of the

Keywords: number(s): 78.70.En, 71.28.+d, 75.30.Mb, 74.62.Bf

come

I. INTRODUCTION

Explaining the existence of atomic magnetic moments in metallic systems, including that of intermediate valent Yb, is an old and famous problem of condensed matter physics [1][2][START_REF] Maple | Demagnetization of Rare Earth Ions in Metals Due to Valence Fluctuations[END_REF][START_REF] Doniach | [END_REF][5]. Yb, Ce, or U based metallic solids may present a wealth of fascinating behaviors (Kondo effect, intermediate valence, heavy fermion, nonconventional superconductivity, quantum criticality, etc.) as a result of the interplay between very large intra-atomic Coulomb interactions in the f shell and a partial delocalization due to hybridization of f states with conduction electrons [START_REF] Coleman | Handbook of Magnetism and Avanced Magnetic Materials[END_REF]. These exotic phenomena may be tuned through a control parameter such as composition, pressure, or magnetic field. In the case of Yb, the energetically close 4f 14 (Yb 2+ ) and 4f 13 (Yb 3+ ) configurations are involved. The increase of the hybridization tends to favor the divalent state of larger atomic radius. While the divalent 4f 14 state is nonmagnetic (J = 0), the trivalent 4f 13 configuration is magnetic (J = 7/2). In intermediate valent Yb systems, the Kondo effect generally yields a screened moment with the formation of a nonmagnetic Fermi liquid ground state. However, in a rare few cases, intermediate valence is found to survive along with the stabilization of a magnetic order [START_REF] Eynatten | [END_REF][8][9][10].

The interplay of magnetism and intermediate valence in Yb based metallic materials remains an open question. Until recently, the rare known situations of magnetic ordering of intermediate valent Yb were restricted to low temperature and to Yb valence close to υ ∼ 3 [START_REF] Eynatten | [END_REF][8][9]. In materials where Yb is embedded in a nonmagnetic matrix, there is a competition between the Kondo effect and the RKKY-type magnetic interactions illustrated by the well-known Doniach's magnetic phase diagram [START_REF] Doniach | [END_REF]11]. For large hybridization between the f states and conduction electrons, the Kondo screening * thomas.mazet@univ-lorraine.fr dominates whereas for low hybridization strength the magnetic ordering prevails. Quantum criticality and the breakdown of the Fermi-liquid behavior are expected near the magnetic instability (i.e., for intermediate hybridization) [12][13][14]. Up to now, materials with coexisting intermediate valent Yb and magnetic 3d sublattice have been scarcely investigated [15,16].

In a former paper [17], we discussed the high temperature magnetic ordering (up to 90 K) of intermediate valent Yb (2.90 υ 2.95) in the YbMn 6 Ge 6-x Sn x alloys with x = 4.2 and 4.4. This unprecedented high magnetic ordering temperature of intermediate valent Yb is very likely related to the strong 3d Mn-5d Yb exchange interaction [18] and to the nonzero exchange field at the Yb site generated by the magnetic Mn sublattice.

The YbMn 6 Ge 6-x Sn x series (0 x 6) comprises three solid solutions (0 x 1.1, 3.2 x 4.6, and 5.3 x 6.0) and two miscibility gaps [19]. All these alloys crystallize in the hexagonal (P 6/mmm) HfFe 6 Ge 6 -type structure except the ternary stannide (x = 6.0) which adopts a slightly disordered variant [20,21]. The HfFe 6 Ge 6 type of structure comprises one Mn site in 6i position, one Yb site (1b position), and three metalloïd sites (2c, 2d, and 2e positions). The structure can be viewed as an alternative stacking of two kinds of slabs along the c axis: the Mn-Yb,2d-Mn slab and the Mn-2e-2c-2e-Mn one. In YbMn 6 Ge 6-x Sn x , the Mn sublattice magnetically orders at or above room temperature and, depending on composition and temperature, exhibits various kinds of magnetic arrangements (ferromagnetic, helimagnetic, or antiferromagnetic) with an atomic Mn moment m Mn ∼ 2.2μ B . Magnetic ordering of the Yb sublattice is observed only for x 4.6 (i.e., within the Ge-rich and intermediate solid solutions) at temperatures as high as 110 K. Besides the astonishingly high-magnetic ordering temperature of intermediate valent Yb in YbMn 6 Ge 1.8 Sn 4.2 and YbMn 6 Ge 1.6 Sn 4.4 , our former study further evidenced an unusual increase in the Yb valence upon cooling [17]. In the present work, we investigate the composition and temperature dependence of the Yb valence for the whole YbMn 6 Ge 6-x Sn x series by resonant inelastic x-ray scattering (RIXS).

The organization of the paper is as follows. Section II provides the experimental details. The results are presented and analyzed in Sec. III before a discussion in Sec. IV and a short conclusion in Sec. V.

II. EXPERIMENTAL DETAILS

The powder YbMn 6 Ge 6-x Sn x samples with x = 0.0, 3.8, 4.2, 4.4, and 5.5 were prepared from high-purity elements as described in Ref. [19]. The precise chemical composition of the alloys was checked by microprobe measurements using a CAMECA SX100. Long duration room-temperature x-ray diffraction patterns were collected using a Philips X'Pert Pro diffractometer (λ = 1.54056 Å) and refined with the Rietveld method using the FULLPROF software [22] to check the crystal structure of the compounds and the purity of the samples. All compounds were found to be isotypic with HfFe 6 Ge 6 (P 6/mmm), as expected. No attempt has been done to prepare and to study YbMn 6 Sn 6 (i.e., x = 6.0), which is known to crystallize in a slightly less ordered variant [20,21]. The refinements also indicated that the two richer Ge samples (x = 0.0 and 3.5) contain a few wt. % of the ferromagnetic Mn 5 Ge 3 impurity (T C ∼ 300 K), while the Yb 2 O 3 level was found below 2 wt. % in all cases. The diffractometer was also used to record patterns in the 300-15 K temperature range which were analyzed in pattern matching mode to determine the temperature dependence of the lattice parameters. dc magnetization data were recorded using a Physical Property Measurement System (PPMS) from Quantum Design. The temperature variation of the magnetization was recorded upon cooling between 350 and 5 K in a field of 500 Oe. Hysteresis loops were recorded at 5 K in field up to 90 kOe.

The partial fluorescence yield x-ray absorption spectroscopy (PFY XAS) and resonant inelastic x-ray scattering (RIXS) measurements were carried out at beamline ID16 of the European Synchrotron Radiation Facility (Grenoble, France). The beamline was equipped with a fixed-exit Si(111) monochromator and a toroidal Rh-coated focusing mirror providing a 130 × 50 μm 2 (H × V) beam spot at the sample position. The experiment was performed in reflection geometry. The emitted radiation was energy-analyzed using a Rowland circle spectrometer based on a 1-m radius, spherically bent Si(620) crystal, and a Si avalanche photodiode detector. The scattering plane was horizontal, i.e., parallel to the linear polarization vector of the incident beam. The spectra were collected at scattering angle 2θ = 90 • to minimize the nonresonant scattering contribution. The total energy resolution was ∼1.7 eV. Freshly scraped samples in the form of pressed pellets were mounted in a He cryostat and measured in high (10 -8 mbar) vacuum in the 300-10 K temperature range. The alloys with x = 4.2 and 4.4 were further investigated up to 450 K in an oven.

III. EXPERIMENTAL RESULTS

The RIXS and PFY XAS spectra recorded at 300 K for the five studied compounds are shown in Figs. 1(a respectively. The RIXS spectroscopy couples the initial state with 3d 9 4f 13 and 3d 9 4f 14 final states [23][24][25]. The weak Yb 2+ contribution can be enhanced by tuning the incident photon energy to the energy of the Yb 2+ edge in the L 3 PFY XAS. Due to the resonant nature of RIXS, the Yb valence cannot be obtained from a simple ratio of the Yb 2+ and Yb 3+ intensities. Then following a procedure previously proposed [23], we have normalized the RIXS ratio of these two structures with the ratio obtained from the low temperature L 3 PFY XAS data. Figure 1 evidences a strong valence change with composition. As the Sn content increases from x = 0.0 to x = 5.5, a marked spectral weight transfer from the "3+" component towards the "2+" one occurs.

The variation of the room-temperature Yb valence with Sn content is shown in Fig. 2(b) together with that of the cell volume [Fig. 2(a)]. While Yb is basically trivalent in YbMn 6 Ge 6 , there is an increasing departure from trivalence upon increasing the Sn content throughout the series, but even in the richest Sn compound studied (x = 5.5), Yb is found far from divalence (υ ∼ 2.72). The concomitant increase in the cell volume and reduction in the Yb valence upon the Sn for Ge substitution strongly suggests that the driving mechanism responsible for the Yb valence change throughout the YbMn 6 Ge 6-x Sn x series is a reduction of the chemical pressure, the increase in the interatomic distances favoring with that of LuMn 6 Sn 6 (data from Ref. [28]). The dotted line, which links the cell volume of YbMn 6 Ge 6 with that of YbMn 6 Sn 6 (data from Ref. [20]), corresponds to a simple linear variation of the cell parameters. (b) Variation of the room-temperature Yb valence with the Sn content. The shaded areas mark the miscibility gaps [19].

divalent Yb of larger atomic radius than trivalent Yb (see, e.g., Ref. [9]). It cannot however be excluded that electronic modifications arising from the nature of the Sn 5p compared to Ge 4p wave function also interfere. For instance, the rise in T C with Si content in the magnetocaloric Gd 5 (Si x Ge 1-x ) 4 alloys has been shown to be mainly due to an electronic effect, while the volume effect only plays a secondary role [26].

There is no linear relation between cell volume and Yb valence which only weakly evolves for 0 x < 3.8 and then varies rapidly for higher Sn content [Fig. 2(b)]. This shows that in YbMn 6 Ge 6-x Sn x the Yb valence is more sensitive to the Sn content for x around ∼4. Since the driving mechanism is expected to be the lattice expansion, this suggests that in this Sn composition range the Yb valence strongly depends on interatomic distances. We also observe that only for x = 5.5 the cell volume is larger than that expected for a fully trivalent Yb. Estimating the Yb valence in solids from lattice constants may thus be somewhat misleading. This also appears for instance in Table I of Ref. [27]. We speculate that Yb is not fully divalent even in the ternary YbMn 6 Sn 6 , unless the Yb valence hugely varies in the Sn-rich YbMn 6 Ge 6-x Sn x terminal solid solution.

The RIXS spectra obtained at 300 K and 10 K for x = 0 and 5.5 are shown in Fig. 3 together with the spectra recorded in the 8-450 K temperature range for x = 4.4. In both the Sn-rich ) in an applied field of 500 Oe. The weak ferromagnetic component seen on the curve of x = 3.8 below ∼300 K is due to the magnetic ordering of the Mn 5 Ge 3 impurity. The data for x = 0.0 are not shown: the Mn sublattice antiferromagnetically orders above 350 K (the highest temperature of the experiment) and the low-temperature behavior is overshadowed by the ferromagnetic Mn 5 Ge 3 impurity.

In YbMn 6 Ge 6-x Sn x , the Mn sublattice magnetically orders at or above room temperature and shows a complex dependence on temperature and composition [19]. To summarize: in the Ge-rich alloys, the Mn sublattice is antiferromagnetic while upon increasing the Sn content a high temperature ferromagnetic state develops at the expense of the low temperature spiral arrangements. In the Sn-rich solid solution (x 5.3), the Mn sublattice is purely ferromagnetic. From previous magnetization, neutron diffraction, and 170 Yb Mössbauer spectroscopy experiments, it has been shown that Yb does magnetically order at quite high temperature (up to 110 K) in YbMn 6 Ge 6-x Sn x with x 4.6 [19]. This manifests through the strong rise in the magnetization at low temperature on the thermomagnetic data we recorded on the present samples (Fig. 5). We identify the Yb ordering temperature (T Yb ) with the low-temperature minimum of ∂M ∂T , and find T Yb ∼ 40 K, ∼70 K, and ∼90 K for x = 3.8, 4.2, and 4.4, respectively. Hence we can conclude that in the YbMn 6 Ge 6-x Sn x alloys of the intermediate solid solution (3.2 x 4.6), intermediate valent Yb magnetically orders at quite high temperature, an unprecedented phenomenon that likely finds its origin in the interaction with the magnetic Mn sublattice. In YbMn 6 Ge 0.5 Sn 5.5 , intermediate valent Yb does not order and the magnetic behavior is only related to the ferromagnetic Mn sublattice. Thus the magnetic ordering of intermediate valent Yb disappears for 2.7 < υ < 2.9. The Yb valence also influences the Mn magnetic behavior. Numerous previous studies have shown that lowering the R valence in RMn 6 X 6 (X = Ge or Sn) or the average metalloid valence in pseudoternaries RMn 6 X 6-x X x (X = In or Ga) favors the stabilization of a ferromagnetic Mn sublattice: the RMn 6 X 6 compounds with trivalent R are antiferromagnets or helimagnets and only divalent R elements and/or partially substituted Ga or In alloys can bring the Mn sublattice to ferromagnetism (see Refs. [19,20,[29][30][31][32]). This dependence on the R valence is conspicuous in YbMn 6 Ge 6-x Sn x : the ferromagnetic region just below the Mn ordering temperature enlarges progressively as the Yb valence decreases throughout the series. In YbMn 6 Ge 0.5 Sn 5.5 , the Yb valence is low enough for the Mn sublattice to be ferromagnetic over the whole ordered temperature range.

The Yb valence may change with temperature. Increasing the temperature usually yields an increase in the Yb valence as described in the single impurity Anderson model [33]. The energy spectrum is characterized by a singlet intermediate valent ground state and a manifold of purely trivalent excited states (at an excitation energy δ = k B T K ). At low temperature (T T K ) the Yb valence is determined by the valence admixture in the ground state and increases with temperature due to the thermal population of the trivalent excited states. However, this picture does not describe the temperature dependence of the valence in the 3.8 x 4.4 composition range since a valence increase is observed upon cooling (Fig. 4). To explain this anomalous temperature dependence and the unusual Yb magnetic behavior in the YbMn 6 Ge 6-x Sn x alloys, we have proposed a toy model in Ref. [17] where the system is simply represented by one S = 1/2 impurity state with energy ε i < 0 and Coulomb interactions (U ) for Yb atoms hybridized (V ) with an infinitely narrow band at energy ε 0 = 0 corresponding to the itinerant electron gas. Moreover, magnetic interactions are modeled by a Zeeman interaction so that the Hamiltonian can be written

H = σ ε i n i,σ + ε 0 σ n 0,σ + V σ (a † i,σ a 0,σ + c.c.) + Un i,σ n i,-σ + g i μ B σ i,z B.
(

) 1
The last term is a Zeeman Hamiltonian on the localized impurity electron [34] (not on the itinerant one), whereas the other ones represent an Anderson-like Hamiltonian. In the infinite U limit, the double occupancy of the impurity orbital (corresponding to two holes in the 4f shell of Yb and then to the 4f 12 configuration) is forbidden and the basis states are |0 for the unoccupied impurity state (corresponding to the filled 4f shell and then to the 4f 14 configuration) and the |η,η with η,η = ↑ or ↓ for respectively the spin of the impurity and itinerant electrons (corresponding to the 4f 13 configuration). Without the Zeeman term, the ground state is a nonmagnetic Kondo-like singlet

| = c 0 |0 + c 1 |↑↓ -|↓↑ √ 2
at ε iδ, whereas the excited states correspond to the triplet (S = 1) states at ε i and a high energy singlet state corresponding mainly to the nonoccupied impurity state. An increase of the hybridization parameter V leads to an increase in |c 0 | (decrease of the valence) and δ. A finite magnetic field changes the ground state and leads to the appearance of a magnetic moment associated with the impurity state. In the high field limit, the impurity state is fully polarized and the ground state is the doublet |↑↑ ,|↑↓ . In Ref. [17], we showed the field dependence of the impurity valence (υ = 2 + |c 1 | 2 ) and of the localized magnetic moment. The increase in the magnetic field leads to a crossover from a nonmagnetic state to a completely polarized magnetic state and is in agreement with more elaborate calculations [35,36]. With such a very crude model, we were able to reproduce the anomalous temperature dependence of the Yb valence (in the x = 4.2 and x = 4.4 samples) with a temperature dependence of the Zeeman term [B(T )] resulting from the magnetization of the Mn lattice. A similar increase in the Yb valence upon increasing the external magnetic field has been observed in YbAgCu 4 [37].

In Fig. 6, we present the temperature dependence of the Yb valence obtained from our simple approach for different cases. The key parameter is the ratio r = μ B B(0)/δ comparing the energy stabilization of the nonmagnetic singlet state due to hybridization and the Zeeman energy. The qualitative evolution of r can be understood from the composition dependence of the Yb valence. Indeed, for x = 0, the valence is close to 3, corresponding to a very small Kondo energy and then a large r parameter. In this limit, the valence remains very close to 3 for the whole temperature range. The Kondo energy scale increases with increasing valence admixture leading to a decrease of the r parameter. The anomalous temperature dependence of the x = 4.4 alloy is reproduced with comparable characteristic energies (r = 1.25). For the x = 5.5 alloy, the large valence admixture associated with the nonmagnetic Yb sublattice suggests that the hybridization energy is larger than both the Zeeman energy and the thermal energy (k B T ). In this limit, the Yb valence is expected to be independent of the Mn magnetization and then independent of temperature. Our calculation reported in Fig. 6 confirms that the Yb valence in this limit is nearly temperature independent. Comparison with Fig. 4 shows that this crude model is able to qualitatively reproduce the Yb valence and its temperature dependence in this series.

The temperature dependence of the reduced cell volume of the five compounds studied is shown in Fig. 7. All the alloys present the usual lattice contraction upon cooling. The slight thermal variation of the Yb valence in the alloys with x = 3.8, 4.2, and 4.4 does not yield an observable anomaly. The enhancement of the thermal variation of the cell volume upon increasing x originates from a weakening of the chemical bonds. In such kind of intermetallics, the bonds with Sn are weaker than those with Ge [32,38]. In addition, the reduction of the Yb valence leads to less 5d valence electrons which further contributes to weaken the related interatomic bondings. Weaker chemical bonds yield a less steep Morse's potential and thus a wider thermal variation.

We further note (inset of Fig. 7) that the lattice contraction from x = 4.4 to x = 4.2 (δV /V ∼ -1.4% at 300 K) is similar to the thermal contraction ( V /V 0 ∼ -1.3% for x = 4.2 and 4.4, where V 0 is the cell volume at 300 K) and that the observed Yb valence increase due to composition change ( υ ∼ +0.05) is close to that measured upon cooling ( υ ∼ +0.02 and +0.03 for x = 4.2 and 4.4, respectively). Therefore, we propose that, owing to the apparent strong sensitivity of the Yb valence to interatomic distances in this composition range, the lattice contraction might play a role in the observed anomalous thermal variation of the Yb valence in addition to the model we proposed above.

IV. DISCUSSION

In YbMn 6 Ge 6-x Sn x , the Yb valence decreases upon increasing the Sn content, most likely because of negative chemical pressure effects associated with the replacement of Ge atoms by larger Sn atoms. The dependence of the Yb valence on the Sn content is more pronounced for x ∼ 4.0, which suggests that the Yb valence could be more sensitive to lattice variation in the vicinity of this composition range. Experiments under external pressure on intermediate valent Yb materials [16,27,39,40] often yield a nonlinear variation of the Yb valence, suggesting again that for a certain materialdependent hybridization range the Yb valence is more sensitive to (external or chemical) pressure variation (i.e., to lattice constant change). We propose that in the YbMn 6 Ge 6-x Sn x alloys with x ∼ 4.0, the electronic structure would be such that the Yb valence strongly depends on interatomic distances. Though Yb is already in the intermediate valence regime within the alloys with x = 4.2 and 4.4 (υ ∼ 2.9), their cell volume does not depart from that expected for a fully trivalent Yb and only the alloy with x = 5.5 (υ ∼ 2.7) has abnormally large lattice constants.

Except for the ternary germanide YbMn 6 Ge 6 within which Yb is basically trivalent, Yb was found in intermediate valent state over the whole investigated temperature range in all studied alloys. Accordingly, unlike the common belief, we showed that the high temperature magnetic order of the Yb sublattice in the alloys with x = 3.8, 4.2, and 4.4 (T Yb ∼ 40 K, 70 K, and 90 K, respectively) involves intermediate valent Yb. This likely results from the strong exchange field produced by the Mn sublattice at the Yb site, a situation that had never been observed before. The thermal variation of the Yb valence is another striking feature of YbMn 6 Ge 6-x Sn x in the intermediate solid solution. While the Yb valence of the alloys with x = 0.0 and 5.5 does not change with temperature, that of the magnetic intermediate valent Yb alloys (x = 3.8, 4.2, and 4.4) increases upon cooling, at odds to the usual behavior. We were able to explain both the occurrence of magnetic intermediate valent Yb and the anomalous temperature dependence of the Yb valence thanks to a simple model based on an Anderson Hamiltonian including a Zeeman term to account for the Mn-Yb exchange interactions. Because of the presumed sensitivity of the alloys near x ∼ 4 to changes of the interatomic distances, we also suggested that the lattice contraction might play a role in the unusual thermal variation of the Yb valence.

Figure 8 shows the composition dependence of the Yb ordering temperature (data from the present work and from Ref. [19]) and Yb magnetic moment (data from Ref. [19]). Weak changes of the Yb valence induced by subtle chemical composition variation in the intermediate solid solution are seen to yield drastic modifications in the Yb moment and Yb ordering temperature. The inspection of Fig. 2 and Fig. 8 leads to an a priori puzzling finding: the Sn for Ge substitution, resulting in an increase of the 4f -conduction electrons hybridization strength (i.e., a lowering of the Yb valence), causes an enlargement of the cell volume, a decrease in the Yb moment, but a sharp increase of the Yb magnetic ordering temperature. The reduction in the Yb moment is very likely a consequence of the enhancement of the Kondo screening. In this context, the concomitant increase in T Yb though the interatomic distances increase may appear more difficult to understand. Actually, the simultaneous reduction of the Yb moment and rise of the ordering temperature upon increasing the hybridization is predicted by the Doniach's [11,41], another hybridized 4f material with unusually high magnetic ordering temperature of the 4f sublattice. In YbMn 6 Ge 6-x Sn x , the main novelty compared with standard intermediate valent Yb materials is the interaction with the magnetic Mn sublattice which has two effects on the magnetic phase diagram. It yields Yb magnetic ordering temperatures that are one or two order(s) of magnitude higher than those of previously identified intermediate valent Yb materials [START_REF] Eynatten | [END_REF][8][9] and stabilizes the magnetic order for relatively low values of the Yb valence (υ ∼ 2.9).

The leads to an opening of the loop with nonzero coercivity (H c ∼ 300 Oe) and remanence, while the alloy with x = 5.5 behaves as a soft ferromagnet. This strongly suggests that a significant orbital contribution to the intermediate valent Yb magnetic moment still persists, as expected from localized magnetism. On the other hand, it is known from band structure calculations that intermediate valent Yb implicates partially delocalized f electrons [42,43]. Owing to the relatively strong hybridized character of the Yb 4f states in the YbMn 6 Ge 6-x Sn x alloys of the intermediate solid solution (3.2 x 4.6), it cannot be excluded that partially delocalized 4f electrons interfere in their astonishing magnetic behavior.

In YbMn 6 Ge 6-x Sn x , the Yb magnetic ordering is lost for 4.6 < x < 5.3. As recalled in the Introduction, quantum criticality and non-Fermi-liquid behaviors can be observed in the vicinity of the magnetic instability [12][13][14]. It would therefore be interesting to investigate the physical behaviors of YbMn 6 Ge 6-x Sn x close to the Yb magnetic instability. For that purpose, and also to clarify the role of the interatomic distances on the Yb valence in this series, various experiments under pressure are in progress. Pressure increase on initially nonmagnetic intermediate valent Yb solid can allow stabilizing Yb magnetic ordering and tuning the materials to a quantum phase transition [44,45].

V. CONCLUSION

We have investigated the composition and temperature dependence of the Yb valence in YbMn 6 Ge 6-x Sn x (x = 0.0, 3.8, 4.2, 4.4, and 5.5) using RIXS spectroscopy in the 10-450 K temperature range. The Yb valence decreases upon increasing the Sn content (from υ ∼ 3 for x = 0 to υ ∼ 2.7 for x = 5.5), most likely because of chemical pressure effects. The YbMn 6 Ge 6-x Sn x alloys with 3.2 x 4.6 present the unique situation of a strongly magnetized 3d sublattice exerting a significant exchange field at the intermediate valent Yb site. This yields astonishingly high magnetic ordering temperatures for intermediate valent Yb (up to ∼110 K) and shifts the Yb magnetic instability towards significantly lower valence (2.7 < υ < 2.9) than in standard intermediate valent materials where Yb is alloyed with nonmagnetic elements (magnetic instability close to υ ∼ 3). In addition, the alloys of the intermediate solid solution (3.2 x 4.6) show an increase of the Yb valence upon cooling, opposite to the usual behavior, which we have also tentatively ascribed to the presence of a magnetized Mn sublattice with the possible supplementary participation of the lattice contraction. Further experimental and theoretical works are needed for a better understanding of the unusual Yb behavior in this series.

FIG. 1 .

 1 FIG. 1. (Color online) Room-temperature RIXS (a) and PFY XAS (b) spectra for the alloys with x = 0.0, 3.8, 4.2, 4.4, and 5.5.

FIG. 2 .

 2 FIG. 2. (Color online) (a) Variation of the room-temperature cell volume with the Sn content. The continuous line marks volumes expected for a fully trivalent Yb: it joins the cell volume of YbMn 6 Ge 6with that of LuMn 6 Sn 6 (data from Ref.[28]). The dotted line, which links the cell volume of YbMn 6 Ge 6 with that of YbMn 6 Sn 6 (data from Ref.[20]), corresponds to a simple linear variation of the cell parameters. (b) Variation of the room-temperature Yb valence with the Sn content. The shaded areas mark the miscibility gaps[19].

  FIG. 3. (Color online) RIXS spectra for x = 0.0 and 5.5 at 300 K and 10 K. RIXS spectra for x = 4.4 between 450 and 8 K.

x 5 FIG. 4 .

 54 FIG. 4. (Color online) Thermal dependence of the Yb valence in YbMn 6 Ge 6-x Sn x (x = 0.0, 3.8, 4.2, 4.4, and 5.5).

8 FIG. 5 .

 85 FIG. 5. (Color online)Thermal dependence of the magnetization of YbMn 6 Ge 6-x Sn x (x = 3.8, 4.2, 4.4, and 5.5) in an applied field of 500 Oe. The weak ferromagnetic component seen on the curve of x = 3.8 below ∼300 K is due to the magnetic ordering of the Mn 5 Ge 3 impurity. The data for x = 0.0 are not shown: the Mn sublattice antiferromagnetically orders above 350 K (the highest temperature of the experiment) and the low-temperature behavior is overshadowed by the ferromagnetic Mn 5 Ge 3 impurity.

FIG. 6 .

 6 FIG. 6. (Color online) Calculated temperature dependence of the Yb valence obtained for several r = μ B B/δ parameters. The r 1 corresponds to the x = 0 sample (υ ∼ 3), whereas the Yb valence in the x = 4.4 and 5.5 alloys can be described with r = 1.25 and r = 0.15, respectively.

FIG. 7 .

 7 FIG. 7. (Color online) Thermal dependence of the reduced volume V /V 0 (V 0 is the room-temperature volume) of YbMn 6 Ge 6-x Sn x (x = 3.8, 4.2, 4.4, and 5.5) between 300 K and 10 K. The inset shows the thermal variation of the cell volume for x = 4.2 and 4.4.

FIG. 8 .

 8 FIG. 8. (Color online) Composition dependence of the Yb ordering temperature (green circles, left scale) and Yb magnetic moment (red squares, right scale). Data in open symbols are taken from Ref. [19]. The lines are guides to the eye.

unique YbMn 6 FIG. 9 .

 69 FIG. 9. (Color online) Hysteresis loops of x = 4.4 and x = 5.5 at 5 K in field up to 90 kOe.

  ) and 1(b),

	Yb Lα1 RIXS			Yb L3 PFY XAS
	7.40	7.41	7.42	8.93	8.94	8.95	8.96
	Scattered photon energy (keV)		Energy (keV)

ACKNOWLEDGMENTS

We are indebted to the ESRF (Grenoble, France) for the provision of research facilities. Work at Lausanne was supported by the Swiss NSF.


C Tournier-Colletta

B Kierren

Y Fagot-Revurat

D Malterre

Electron hopping at the Si(111):B-√ 3 surface: Insight from local impurity spectroscopy

Keywords: number(s): 73.20.At, 71.27.+a, 73.20.Hb, 71.55.Cn

Boron vacancies at the Si(111):B-√ 3 surface are model systems in the comprehension of strongly correlated semiconductor surfaces. By using scanning tunneling spectroscopy, the origin of the single-vacancy electronic structure is addressed. It is shown to originate from the localization of a well-identified dangling-bond surface state with significant B character. The bivacancy defect, which is characterized by energy-split bonding and antibonding states, is interpreted within the textbook diatomic molecule picture. From the hopping parameter, we determine the bandwidth of the surface state from which the impurity state derives and evaluate the strength of many-body effects. Our analysis supports the realization of the Mott-Hubbard insulator state in half-filled dangling-bond surface states on √ 3-reconstructed surfaces, as proposed recently for SiC(0001), Sn/Ge(111), and Sn/Si(111).

I. INTRODUCTION

Dangling bonds at semiconductor surfaces have been attracting considerable interest in the context of strongly correlated systems. Because of their sp orbital nature, dangling bonds are characterized by a Hubbard repulsion energy U ≈ 1 eV (Ref. 1) that seemingly makes them unsuitable to exhibit strong correlation physics, unlike d or f electron systems. However, surface-state bands deriving from such dangling bonds exhibit a small bandwidth W ≈ 0.1 eV because of the reduced dimensionality. That feature is strengthened by surface reconstructions that tend to seclude the dangling bonds from each other. A typical example is found on √ 3reconstructed surfaces where dangling bonds are separated by almost 7 Å. 2 As a consequence, dangling-bond surface states potentially verify the U W condition to trigger the Mott-Hubbard metal-to-insulator transition. 3 Prototypical examples are found at the pristine SiC(0001)-√ 3 surface 4 and in Sn/Ge(111)-√ 3 and Sn/Si(111)-√ 3 surface alloys, [5][6][7] where ab initio calculations fail to reproduce the insulating nature of these surfaces.

At the Si(111):B-√ 3 surface, the electronic properties of single boron vacancies have been studied recently by means of scanning tunneling microscopy/spectroscopy (STM/STS). [8][9][10] Their characteristic spectroscopic signature is an unoccupied resonance appearing in the intrinsic gap. 8 This resonance derives from sp z states and therefore is very similar to the dangling bonds that form upon creating a silicon surface. By further using very high tunneling currents, Nguyen et al. reached a permanent regime in which the impurity becomes singly occupied. 10 In that regime, they observed a new resonance at higher energy, attributed to the addition of a second electron and separated by U ≈ 1 eV, that is typical of silicon dangling bonds. 1 Independently of correlation effects, the transport process has been shown to be of polaronic nature as a result of strong electron-vibration coupling. 8 In this paper, we report a low-temperature STS study of the electronic hopping at the Si(111):B-√ 3 surface that completes the previously cited works on many-body aspects. To start with, we show that the in-gap resonance results from the localization at the impurity site of a well-identified danglingbond surface state with significant boron contribution. The second important result is the evidence of coupling between two neighboring boron vacancies. This so-called bivacancy defect shows a striking similarity to the textbook homonuclear, diatomic molecule case. Bonding and antibonding states are indeed observed, and their energy separation allows to determine the hopping parameter between impurity states (|t| = 80 meV). By combining these two independent results, we can estimate the bandwidth of the dangling-bond surface state (W = 0.64 eV). Finally, the strength of many-body interactions is addressed by comparing the surface-state bandwidth to available many-body parameters in the literature. The analysis supports the realization of a Mott insulator in the half-filled dangling-bond surface state on √ 3-reconstructed surfaces, as proposed recently for SiC(0001), 4 Sn/Ge(111), 5,7 and Sn/Si(111). 6,7

II. EXPERIMENT

The experiments were carried out in an ultrahigh vacuum setup, using a commercial LT-Omicron scanning tunneling microscope operated at either 5 or 77 K. Spectroscopic acquisition is achieved with the lock-in technique, in the open feedback-loop mode. Typical modulation bias and frequency were 30 meV and 700 Hz, respectively. The typical junction set point parameters were I = 0.5 nA and U = +2 V (unoccupied states). In order to produce the √ 3-reconstructed surface, we used highly boron-doped Si(111) wafers (degenerate p-type semiconductor, ρ ≈ 10 -3 cm). The sample is annealed by direct heating, starting with repeated flash annealings at 1400 K followed by an annealing of several hours at a lower temperature (1100 K). This procedure ensures bulk B impurities are segregated at the surface and substitute Si atoms at the S 5 site, leading to the √ 3 surface reconstruction with one B atom per reconstructed unit cell. 11 Those B atoms in the sublayer act as acceptors that accommodate the dangling bond electrons provided by Si adatoms, resulting in a passivated surface.

III. RESULTS

Figure 1 shows a constant-current STM image of an almostdefect-free surface (U = +2 V, I = 0.3 nA). Low-height protrusions (blue) correspond to passivated Si adatoms that are arranged in a √ 3 hexagonal array with 6.65-Å atomic spacing. In addition, two distinct bright features are observed. The first one (labeled single) is characterized by a single bright spot centered at an adatom position and surrounded by neighbors of higher apparent height than the passivated sites, giving this peculiar flower-like shape to the defect. This is attributed to a B vacancy in the sublayer. The strong contrast observed in the topographic image is a spectroscopic effect resulting from an unoccupied impurity state in the intrinsic gap that strongly contributes to the tunneling current. 8 The second bright feature (labeled double) consists of two adjacent defects of the first kind. This so-called bivacancy is statistically difficult to detect on well-prepared surfaces. Although not shown here, a drastic reduction of the segregation time yields more of these defects and even more complex multiple vacancies.

In the context of strongly correlated surfaces, the electronic structure of Si(111):B-√ 3 provides interesting insight as it is a model system involving dangling-bond surface states. In Fig. 2(a), we present dI/dV maps recorded on the passivated surface at 5 K. At low energy [(i), 1.30 eV], the density of states (DOS) is centered on Si adatoms (labeled ad). By increasing the energy, the DOS first becomes structureless [(ii), 1.55 V] before showing a honeycomb pattern with a minimum at the adatom sites and a maximum over second-layer Si 2 atoms [(iii), 1.80 eV]. This dramatic change, visualized as a contrast inversion, is the indication that different electronic states are probed at low and high energies. Alternatively, the point can be evidenced with energy-dependent dI/dV spectra [Fig. 2(b)]. On the adatom site (solid blue line), the local DOS shows an onset at 1.05 V and a pronounced maximum at 1.80 eV. The spectrum recorded on the second-layer site (dashed red line) exhibits a similar energy dependence but shows a clear transfer of spectral weight from high to low energies (see inset). Therefore, low-energy (high-energy) electronic states show an enhanced density on ad (Si 2 ) sites. As expected, at the crossing point of the two spectra, the corresponding dI/dV map (ii) shows no contrast.

The origin of these electronic states is addressed by comparing the STS measurements to the available DFT calculations [Ref. 11; Fig. 2(b), right]. Along the ¯ K surface direction, the electronic structure exhibits two surface states, labeled SS 1 and SS 2 , that overlap with the bulk conduction states (shaded gray area). The calculated band structure has been rigidly shifted in order to match the Fermi level with the top of the valence band, as expected in a degenerate p-type semiconductor. At the ¯ point, the onset and the pronounced maximum of the experimental DOS coincide remarkably with SS 1 and SS 2 , respectively. In the vicinity of ¯ , SS 2 has an almost flat dispersion that yields a peaked DOS as observed experimentally. In contrast, SS 1 disperses strongly, resulting in a spread DOS.

The analysis at ¯ only is not sufficient since STS is a k-integrating technique, whose finite range of integration depends on the tip properties. 12 However, in the present experiment, the loss of contrast allows one to estimate the momentum range. If the tip integrated over an infinite momentum range, the dI/dV maps would always exhibit atomic corrugations since SS 1 and SS 2 would provide a k state whatever the bias voltage. The loss of contrast rather suggests that the maximum momentum contributing to the tunneling current is smaller than k max = 0.6 ¯ K = 0.38 Å-1 , as defined in Fig. 2(b). The onset of the SS 1 DOS at 1.05 eV corroborates this estimation. Indeed, a larger k max would shift the onset to lower energy.

These surface states derive from the adatom sp z dangling bonds, which are the elemental bricks of strongly correlated surfaces. Indeed, theoretical studies predict that the sp z states weakly overlap on √ 3 surfaces, opening the way to strong correlation effects (large U/W ratio). [1][2][3] In the following, we propose a method to estimate the hopping parameter and, in turn, the strength of many-body interactions affecting dangling-bond surface states. We show that this information can be provided by local spectroscopy on impurities, namely, B (bi)vacancies. In Fig. 3(a), STS spectra recorded above a passivated adatom (dashed gray line) and a B vacancy (solid black line) are compared (T = 77 K). At the impurity site, an intense resonance appears in the intrinsic gap, as first pointed out by Berthe et al. 8 This in-gap state results mainly from the localization of the SS 2 surface state, as demonstrated by the dramatic DOS transfer. The point is further supported by DFT calculations 11 that show that the SS 2 Wannier function has a significant B contribution, while the latter is negligible in the case of SS 1 .

With STS, it is possible to extract the Hubbard repulsion energy U of the impurity state. Recently, Nguyen et al. employed very high tunneling currents in order to reach a permanent regime corresponding to a singly occupied impurity. 10 They observed a new spectral feature at higher energy, reflecting the charging of the impurity with a second electron. Their analysis yields U = 1.12 eV, in agreement with previous estimations. 2 Electrons occupying dangling bonds are affected not only by Coulomb repulsion but also by the coupling with local vibrations. The in-gap state, in principle decoupled from the other states, exhibits a surprisingly large linewidth (approximately 0.2 eV). Berthe et al. interpreted this feature within the Franck-Condon picture, the transport through the defect being dominated by a strong electronvibration coupling and the formation of a local polaron. 8 From a quantitative point of view, these authors extracted both the vibration energy hω 0 = 32.5 meV and the nondimensional coupling constant g 2 = 9.6.

In order to estimate the strength of many-body interactions and the possibility of a broken-symmetry ground state, the knowledge of the above parameters is not sufficient. The latter inform on the energy gained by electron localization. However, this effect competes with electron delocalization, whose energy scale is given by the hopping amplitudes. The analysis of the bivacancy defect allows us to extract this missing key parameter. In Fig. 3(b), we show STS spectra recorded at three different locations (A, B, and C; see inset) above the bivacancy defect. All spectra exhibit two resonances separated by 160 meV and shifted by ±80 meV with respect to the single vacancy energy. This behavior is interpreted by analogy to the homonuclear diatomic molecule case. In such molecules, electron hopping lifts the atomic states degeneracy, with the formation of bonding and antibonding states. Their energy separation is 2|t|, where t is the hopping integral; in the bivacancy case, we readily obtain |t| = 80 meV. Such a textbook problem was already realized, for example, in atomic chains built up by atomic manipulation. 13,14 While the spectra look very similar over the two adatom sites (A and C), the spectrum recorded in between (site B) shows a different shape, with a clear modification of the relative peak intensities. Indeed, the low-energy feature AS is weakened, while the high energy one S is strengthened. Due to inversion symmetry, the molecule eigenstates must be either symmetric or antisymmetric with respect to the center. As a general property, symmetric states show a maximum density between the atoms, whereas antisymmetric states exhibit a node at that same location. In the bivacancy, the analysis of the relative DOS at the center allows one to assign the symmetric state to the higher-energy state S and vice versa. This is at odds with the H 2 molecule case, where positive hopping between s orbitals ensures that the lowest-energy state is symmetric. Here, due to the large interadatom spacing (6.65 Å), the hopping between dangling bonds can only be indirect, hence the important role of the substrate in the hopping mechanism. 15 An odd number of Si atoms in the sublayer is involved, resulting in a negative hopping. The situation is depicted in Fig. 3(c), where the indirect path between two nearest-neighbor sp z orbitals involves bonds with p x -p y orbitals of three sublayer Si atoms. 2 Maximum bonding energy is achieved in the depicted configuration, with dangling bonds of opposite signs.

From the bivacancy hopping integral, we can, in turn, estimate the experimental SS 2 bandwidth. Indeed, both impurity and Bloch states derive from similar Wannier functions. On the hexagonal lattice, we get W = 8|t| = 0.64 eV, which compares fairly well with the bandwidth predicted by DFT 11 and measured by inverse photoemission. 16 We are now able to estimate the strength of many-body interactions. Regarding electron correlations, a U/W ratio of 1.75 is found. This value fulfills the U/W > 1 condition for the Mott-Hubbard transition. 1 Polaronic effects lead to a renormalized Hubbard term U eff = U -2g 2 hω 0 . 17 In our case, U eff = 0.50 eV. Therefore the interaction between polarons remains repulsive, so no bipolaronic ground state is expected to form. This finding supports the recent proposition that, contrary to a previous claim, 18,19 K/Si(111):B has no bipolaronic ground state. 20

IV. CONCLUSION

In summary, we investigated the local electronic structure of the Si(111):B-√ 3 surface, a model system in the context of strongly correlated surfaces. We determined strength of many-body effects affecting the dangling-bond surface states. Our finding strongly supports the realization of the Mott insulator state on prototypical √ 3-reconstructed surfaces. In order to tackle this problem, we developed a STS approach for bandwidth determination that relies on semiconductor impurities. This approach can be generalized to other semiconductor surfaces, where the usual standing-wave method 21 is not applicable because of strong atomic corrugations.

FIG. 1 .

 1 FIG. 1. (Color online) Atomically resolved STM image of the Si(111):B-√ 3 surface (U = +2 V, I = 0.3 nA). Besides passivated adatoms, we observe two kinds of defects associated with B vacancies in the sublayer: a single defect and a double defect, with the latter corresponding to two adjacent B vacancies.

FIG. 2 .

 2 FIG. 2. (Color online) (a) dI/dV maps of the defect-free Si(111):B-√ 3 surface at (i) 1.30 eV, (ii) 1.55 eV and (iii) 1.80 eV bias voltages. (b) STS spectra recorded above a Si adatom (ad, solid blue line) and a second-layer Si atom (Si 2 , dashed red line). The inset evidences the DOS difference in the low-energy sector. On the right, band dispersion of surface states SS 1 and SS 2 along ¯ K is shown; the shaded area represents the bulk continuum (from Ref. 11).

FIG. 3 .

 3 FIG. 3. (Color online) (a) Comparison of STS spectra recorded above a passivated adatom (dashed gray line) and a B vacancy (solid black line) (U = 1.8 V, I = 0.4 nA, T = 5 K). (b) Comparison of STS spectra recorded at different locations (A, B, and C) above a bivacancy and a single vacancy (U = 2.0 V, I = 0.3 nA, T = 77 K). (c) Schematics illustrating the indirect hopping path between nearestneighbor adatoms. The dark and light gray lobes depict the positive and negative parts of the wave functions, respectively.