Valence fluctuations and empty-state resonance for Fe adatom on a surface
Sergei N. Iskakov, Vladimir V. Mazurenko, Maria V. Valentyuk, Alexander I. Lichtenstein
VValence fluctuations and empty-state resonance for Fe adatom on a surface
S.N. Iskakov , V. V. Mazurenko , M.V. Valentyuk , and A. I. Lichtenstein , Theoretical Physics and Applied Mathematics Department,Ural Federal University, 620002 Ekaterinburg, Russia Hamburg University, Germany, Hamburg, Jungiusstraße 9 (Dated: September 4, 2018)We report on the formation of the high-energy empty-state resonance in the electronic spectrumof the iron adatom on the Pt(111) surface. By using the combination of the first-principles methodsand the finite-temperature exact diagonalization approach, we show that the resonance is the resultof the valence fluctuations between atomic configurations of the impurity. Our theoretical finding isfully confirmed by the results of the scanning tunneling microscopy measurements [M.F. Crommieet al., Phys. Rev. B 48, 2851 (1993)]. In contrast to the previous theoretical results obtained byusing local spin density approximation, the paramagnetic state of the impurity in the experiment isnaturally reproduced within our approach. This opens a new way for interpretation of STM datacollected earlier for metallic surface nanosystems with iron impurities.
I. INTRODUCTION
The modern scanning tunneling microscopy (STM)experiments are aimed to reveal, identify and operateexcitations of surface nanosystems, which is of crucialimportance for understanding the basic phenomena inquantum physics such as Kondo resonance, magneticanisotropy, exchange interactions between adatoms andfor constructing novel information devices on the atomiclevel. Typically, these excitations present the smallest en-ergy scale in the system, they are of a few millivolts andcan be reproduced by solving Heisenberg-type models. However, the picture of the STM experiment in thecase of an adatom on a metallic surface is not completewithout considering high-energy resonances of hundredsmeV. They carry information concerning inter-orbitalcharge or spin excitations in the impurity and can beused to study electronic structure of the adatom. In thissense there is one important example of such high-energyexcitations that is a pronounced peak at 0.5 eV above theFermi level in the tunneling spectra of the iron adatomdeposited on Pt(111), Pd(111) or W(110) surfaces.Importantly, it was proposed that the peak can be usedto identify iron species in surface nanosystems.The theoretical description of the high-energy excita-tions such as 0.5 eV peak for iron adatom on a metal-lic surface is a complex methodological and numericalproblem. Within the Lang’s model that establishes theconnection between experimental STM spectrum and thelocal density of states at the site of the adatom, this peakcan be attributed to the 4 s states of the Fe adatom. How-ever, the ab − initio calculations based on the densityfunctional theory do not confirm such a scenario. As wewill show below the non-spin-polarized density of statesexhibits a 3 d peak in the iron adatom spectral function atthe Fermi level (Fig.2). Above E F the density of states isa decreasing function without any features. The intensityof the 4 s states is very small.As it was demonstrated in Ref.9 the experimentallyobserved excitations can be reproduced in the frameworkof the spin-polarized local density approximation (LSDA) calculations. For that one should take into account theshift of the spin-down 3 d states to higher energies abovethe Fermi level due to the spin splitting. Then within theTersoff-Hamann approach the STS is associated withthe scattering of the s − p electrons at spin-down d z states. Thus to describe the experiment one should definesome ordered magnetic state of the system in the ab − initio calculations. As we will show below, this scenariois fulfilled in the case of Fe/Pt(111). At the same time,the experimental conditions (low temperatures and zeromagnetic field) correspond to the paramagnetic state ofFe/Pt(111). To describe the system in this regime, a real-istic five-orbital impurity Anderson model is constructedand solved by using the finite-temperature exact diag-onalization method. The correlated density of statesshows a peak at 0.5 eV above the Fermi level in accor-dance with the experiment. The composition analysis ofthe eigenvectors has shown that the 0.5 eV resonance isoriginated from the valence fluctuations, which are com-bined effect of the intra-atomic exchange coupling onthe impurity and strong hybridization with the surfacestates. We also show that the position of the resonanceis sensitive to the coupling with other adsorbates.
II. LDA AND LSDA RESULTS
The first step of our investigation was to define an equi-librium atomic structure of the Fe/Pt(111) nanosystem.For that we performed first-principles molecular dynam-ics simulations by using the Vienna ab initio simulationpackage (VASP) within local density approximation(LDA). In these calculations the energy cutoff of 300 eVin the plane-wave basis construction and the energy con-vergence criteria of 10 − eV were used. The atomic po-sitions were relaxed with residual forces less than 0.01eV/˚A. We used the PAW-PBE exchange-correlation po-tential as described in Ref.18.The simulations of the atomic structure of Fe adatomon the Pt(111) surface were carried out within a supercell a r X i v : . [ c ond - m a t . s t r- e l ] J u l Fe Pt FIG. 1. Schematic representation of the Fe/Pt(111) (left) andFeH/Pt(111) (right) nanosystems simulated in this work. approach. The supercell contains a three-layered (4 × Theunit cell parameters were fixed during relaxation proce-dure. The obtained vertical distance between Fe atomand Pt surface of 1.61 ˚A is in good agreement with thereported values. Fig.2 gives the partial densities of states obtained byusing the local density approximation. There is a peak atthe Fermi level. The width of the peak is about 1 eV thatis much larger than one would expect for a Kondo sys-tem. In contrast to the results obtained for Co/Pt(111)(Ref.7) we do not observe a strong orbital polarization ofthe LDA spectra. The integration of the density of statesgives the Fe-3d occupation of 6.5. One can see that LDAdensity of states does not reveal any feature at the en-ergy of the experimental resonance of 0.5 eV. 4 s statesof iron can be also excluded from the consideration sincethey give a negligible contribution to the spectral func-tion close to the Fermi level. As we will show below theobtained LDA spectra can be used to extract the hoppingintegrals between impurity and surface states. The latteris important to construct a realistic Anderson model forFe/Pt(111) system.The account of the spin-polarization within local spindensity approximation leads to a significant change of thephysical properties of Fe/Pt(111). In agreement with the FIG. 2. Partial densities of states obtained from LDA calcu-lations. Blue bold and red thin lines denote 3 d and 4 s statesof the iron adatom. The intensity of the Fe-4s is multipliedby ten. The dashed line corresponds to the Fermi level. Thedotted line denotes the energy of the experimental resonance. results of previous works the spin splitting produces thepeak at 0.5 eV in the spin-down channel (Fig.3). We ob-tain the magnetic solution with moments of the 3 d , 4 s and 4 p iron shells that are M d = 3.2 µ B , M s = 0.04 µ B and M p = 0.034 µ B , respectively. The value of the totalmoment, 3.3 µ B is in good agreement with the previousresults and smaller than 4 µ B one would expect forthe isolated iron being in the d atomic configuration. Astrong hybridization of the Fe and Pt states results in apartial magnetization of the surface, the induced mag-netic moment of the surface can be estimated as 1.6 µ B .We observe the peak for spin-down 3 d states of iron atthe energy of the experimental resonance. These elec-tronic and magnetic properties of Fe/Pt(111) agree withthat reported in the previous works. From Fig.3 one can see that the gravity center of theoccupied spin-up states shifts from the Fermi level to theenergies of about -3.5 eV. They are strongly hybridizedwith the surface state and becomes completely delocal-ized in comparison with the LDA picture. At the sametime, the width of the LSDA peak above the Fermi levelis smaller then that in LDA, which indicates the increaseof the localization of the spin-down 3 d orbitals. The spinsplitting that can be estimated from Fig.3 is in reasonableagreement with model estimate I d ( (cid:104) n ↑ d (cid:105) − (cid:104) n ↓ d (cid:105) ), where I d is the Stoner parameter of about 1 eV. Interestingly, thenumber of the 3 d electrons obtained in the LSDA solu-tion, (cid:104) n LSDAd (cid:105) = 6.13 is smaller than the LDA value of6.5. Since the spin-up states are almost occupied, thechange of the total number of the electrons is relatedto the change of the hybridization of the iron spin-downstates with the surface.Thus the 0.5 eV resonance observed in the STMexperiment can be reproduced and explained by usingthe results of the LSDA calculations, however, for thatone should assume the non-zero magnetization of the im-purity. In a real experiment the impurity is in the para-magnetic state and below we present the results of An-derson model simulations that describe the Fe/Pt(111)system without magnetic ordering. FIG. 3. Spin-resolved densities of states of the iron adatomobtained from LSDA calculations. The dashed line corre-sponds to the Fermi level. The dotted line denotes the energyof the experimental resonance.
III. ANDERSON MODEL
To take into account the paramagnetic state and dy-namical electron-electron correlations in Fe/Pt(111) wehave constructed and solved the following Andersonmodel H F e = (cid:80) i ( (cid:15) i − µ ) n iσ + (cid:80) pσ (cid:15) p c † pσ c pσ + (cid:80) ijklσσ (cid:48) U ijkl d † iσ d † jσ (cid:48) d lσ (cid:48) d kσ + (cid:80) ipσ (cid:16) V ip d † iσ c pσ + H.c. (cid:17) , (1)where (cid:15) i and (cid:15) p are energies of the impurity and surfacestates, d † iσ ( d iσ ) and c † pσ ( c pσ ) are the creation (annihila-tion) operators for impurity and surface electrons, V ip isthe hopping integral between impurity and surface states, µ is the chemical potential and U ijkl is the Coulomb ma-trix element. The impurity orbital index i ( j , k , l ) runsover the 3 d -states ( xy , yz , 3 z − r , xz , x − y ). A re-alistic simulation of the Fe/Pt(111) system requires anaccurate definition of these Hamiltonian parameters. Definition of the parameters.
The energies (cid:15) p and hop-pings V ip were calculated within the minimization of theLDA hybridization functions presented in Fig.4 by usingthe following expression,∆ i ( ω ) = N p (cid:88) p =1 | V ip | ω − (cid:15) p , (2)where N p is the number of the effective orbitals describingthe surface. The main limitation of the exact diagonal-ization approach is the number of the effective orbital inthe electronic Hamiltonian. In our study we have simu-lated five 3 d orbitals of the adatom. Depending on thesymmetry each impurity orbital is connected with a cer-tain number of the surface levels. For instance, we use N p = 2 for xz ( yz ) and x − y ( xy ) orbitals. Since thehybridization function of 3 z − r orbital demonstratesmore complicated structure than others then for this or-bital we used N p = 3 bath states. Thus the total numberof the effective orbitals in Eq.(1) is equal to 16, whichcorresponds to the maximal occupation of 32 electrons.The impurity orbital energies, (cid:15) i were varied within theexpression for the impurity bath Green’s function G i ( ω ) = [ ω − (cid:15) i − ∆ i ( ω )] − (3)to reproduce the LDA occupations for 3 d states of irondescribed in the previous section.In turn, the elements of the Coulomb interaction ma-trix, U ijkl were defined by using effective Slater integralsthat related to the averaged on-site Coulomb interac-tion, U d and intra-atomic exchange interaction J H asdescribed in Ref.22. Correlated spectral functions.
The constructed Andersonmodel was solved by using finite-temperature exact di-agonalization solver. To analyze the excitations near theFermi level we have calculated 26 lowest eigenvalues and -8 -6 -4 -2 0 2 4
Energy (eV) Δ ( ω ) x – y (xy) -8 -6 -4 -2 0 2 4 Energy (eV) Δ ( ω ) xz (yz) -8 -6 -4 -2 0 2 4 Energy (eV) Δ ( ω )
3z – r
FIG. 4. The imaginary part of the hybridization functions(dashed line) obtained from LDA calculations in comparisonwith fitting results by using Eq. 2 (blue solid lines). eigenvectors of the Anderson Hamiltonian. The main re-sult of our investigation that is the comparison of thecorrelated spectral function and experimental densitiesof states extracted from the STM spectra is presentedin Fig.5. One can see that the position and width of thetheoretical peak are in excellent agreement with the ex-perimental results. The analysis of the partial densitiesof states shows that all the 3 d orbitals equally contributeto the peak at 0.5 eV above the Fermi level.We found that the variation of U d from 2 eV to 4eV and J H from 0.6 eV to 0.8 eV does not significantlychange the correlated spectrum near the Fermi level. Oneshould mention that the choice of the chemical potential, µ is of crucial importance, since it controls the positionof the resonance above the Fermi level. If one choosesthe value of µ in such a way to reproduce the LDA totalnumber of 3 d electrons that is 6.5, then the resonanceis at 0.3 eV. The solution with correct position of theresonance at 0.5 eV above the Fermi level corresponds to6.13, which agrees with the LSDA solution. FIG. 5. The spectral function of the iron impurity atom calcu-lated by using the exact diagonalization approach (blue line).The dotted lines correspond to the densities of states deter-mined in the STM experiment Ref.1 with different tips.FIG. 6. The calculated spectrum of the Anderson model.Arrow denotes the excitation corresponding to the peak at 0.5eV above the Fermi level. N is the number of the electronsin the cluster, N = 27. To understand the microscopic origin of the peak weanalyzed the low-energy spectrum of the Anderson modelpresented in Fig.6. There are three excited states at 0.48eV, 0.49 eV and 0.56 eV. The excitations from the groundstate to first excited states correspond to the experimen-tal resonance above the Fermi level. These excitationslead to the change of the total number of the electronsin the system that is increased by one. The change ofthe impurity properties at this transition can be tracedby calculating the distribution of the atomic configura-
FIG. 7. Weights of the impurity atomic configurations com-prising the ground and first excited states. tions of the impurity for each eigenstate (Fig.7). For theground state we obtain 0 . d + 0 . d + 0 . d + 0 . d ,where the subscript index denotes the ground state.The iron impurity is mainly in d configuration, whichfollows from the Hund’s rules for isolated atom. Astrong coupling with the substrate leads to the valencefluctuations to d and d configurations within theground state. In turns, the first excited state at 0.48eV is characterized by the following composition of theatomic configurations 0 . d + 0 . d + 0 . d . Compar-ing the computed compositions for the ground and firstexcited states we observe the redistribution of the atomicconfiguration weights of the impurity, which correspondsto the d → d and d → d transitions.Experimentally, there is also a shallow rise in the den-sity of states at approximately -0.35 eV, which can not bereproduced by using the LSDA approach (Fig.3). Thisexperimental feature can be associated with the excita-tion at -0.7 eV in the theoretical spectrum of the Ander-son model. The latter has the symmetry of the in-plane xy and x − y states.Thus we obtain two different descriptions of the exper-imental STM resonance at 0.5 eV for Fe/Pt(111). Thefirst one is the LSDA solution where the peak is the re-sult of the splitting between spin-up and spin-down 3 d states of the adatom. On the other hand the solution ofthe five-orbital impurity Anderson model revealed a com-plex multiplet structure of the iron atom at low temper-atures, which is the result of the interplay between intra-atomic exchange coupling and strong impurity-surfacehybridization.Importantly, one can find a connection between theLSDA and Anderson model results. For an isolated atomwith a partially filled shell it was shown by Slater thatthe difference between one-electron energies with spin-up and spin-down is proportional to the terms that re-main even if there is no difference between spin-up andspin-down orbitals. Based on this result he found a con-nection between the spin-splitting energy computed inan one-particle approach and the difference between theaverage energy of all multiplets with S and those with S − S =2) calculated by using amany-particle method. In our case these results shouldbe revised by taking into account a strong hybridizationwith the surface, which leads to the valence fluctuationin the Fe/Pt(111) system. Since a further quantitativecomparison of the LSDA and Anderson model energiesrequires the calculation of high-energy ( ∼ Magnetic susceptibility.
To complete the picture of themagnetic properties of Fe/Pt(111 ), we calculated thespin-spin susceptibilities for 3 d states of the iron adatom, χ zzi ( ω ) = 1 Z (cid:88) nn (cid:48) | (cid:104) n (cid:48) | S zi | n (cid:105) | ω + ıδ + E n − E n (cid:48) (cid:0) e − βE n − e − βE n (cid:48) (cid:1) , (4)where Z is the partial function, E n is the eigenvalue ofthe Hamiltonian, Eq.(1). One can see that the lowestresonance in the spectral function of χ ( ω ) is at 1.7 eV(Fig.8). From Fig.6 it follows that the corresponding ex-citation does not change the total number of the electronsin the system. On the level of the correlated density ofstates (Fig.5) it refers to the excitation below the Fermilevel at the corresponding energy. IV. IRON-HYDROGEN DIMER
In the previous works it was proposed that theresonance above the Fermi level can be used for detec-tion of the iron atoms in surface nanostructures. Tocheck this proposition we consider the situation whenthe iron atom is hidden from the STM tip by anotheratom (Fig.1, right). Such a configuration can be real-ized experimentally. For that the hydrogen atom wasdeposited atop the iron impurity. By using the first-principles molecular dynamics simulations the equilib-rium distance between H and Fe atoms was obtained tobe 1.6 ˚A. Our LDA calculations show that the couplingwith hydrogen does not change the iron partial densitiesof states for the xy , x − y , xz , yz states. At the sametime, the 3 z − r atomic orbital of Fe and s orbital ofH form a molecular orbital. From Fig.9 one can see thatthe bonding and antibonding states are located at -2.4eV and +0.5 eV, respectively.The Anderson model describing the FeH/Pt(111) isgiven by H F eH = H F e + H H − F e + H H , (5)where H H = (cid:80) sσ (cid:15) s n sσ describes s -states of hydrogen. H H − F e corresponds to interactions between s -states ofhydrogen and d -states of iron and can be written as fol-lows: H H − F e = (cid:88) siσ (cid:16) V is d † iσ a sσ + H.c. (cid:17) . (6)Here (cid:15) s is energy of hydrogen state, V is is the hoppingbetween Fe and H states, a † s ( a s ) is the creation (annihi-lation) operator for H states.To define the parameters of the Anderson model de-scribing the FeH/Pt(111) system we assume that s -states FIG. 8. Spectral function of the total spin-spin susceptibilitycalculated by using Eq.4. -4 -2 0 2
Energy (eV) D o S ( s t a t e s / e V ) – r FIG. 9. Comparison of the 3 z − r densities of states calcu-lated for Fe/Pt(111) (blue dashed line) and FeH/Pt(111) (redsolid line) by using the LDA method.FIG. 10. Correlated densities of states for FeH/Pt(111). Blueline corresponds to the summary density of xy , yz , xz and x − y states. Red line denotes 3 z − r states. Dashedbrown line denotes the spectral function of the hydrogen atomdeposited atop the iron impurity. H is hybridized only with 3 z − r states of iron atom.Based on these assumption we minimize the hybridiza-tion functions of the 3 z − r orbital with two bath or-bitals for Pt and single effective orbital for hydrogen.The hopping integral between hydrogen and iron orbitalsequals to V is = 1.56 eV.The correlated spectral functions of the FeH/Pt(111)system is presented in Fig.10. One can see that there tworesonances above the Fermi level. Similar to Fe/Pt(111)the first one is at 0.5 eV which is due to the xy , x − y , yz and xz states. The second one at 1 eV is originatedfrom the iron-hydrogen hybridization. Thus, in contrastto Fe/Pt(111), the resonance of the 3 z − r states isshifted to +1 eV. We observe the excitation at the sameenergy in the hydrogen spectral function that will mainlycontribute to the STM spectrum in the simulated config-uration. V. CONCLUSIONS
In summary, we have studied the electronic structureand magnetic properties of the Fe/Pt(111) system bymeans of the first-principles calculations based on thedensity functional theory and model Anderson impurityHamiltonian that was solved by using exact diagonaliza-tion approach. It was found that both approaches suc-cessfully reproduce the main experimental feature thatis the peak at 0.5 eV above the Fermi level in the STMspectrum. While within LSDA the peak originated fromthe spin splitting of the 3 d shell, on the level of the An- derson model the excitation is associated with the valencefluctuations between atomic configurations of iron. Theconnection between these results on the basis of the semi-nal work by Slater is discussed. In addition, we predictthe shift of the STM resonance to 1 eV in FeH/Pt(111). VI. ACKNOWLEDGMENTS
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