Extension of the Anderson impurity model for finite systems: Band gap control of magnetic moments
aa r X i v : . [ c ond - m a t . s t r- e l ] J u l Extension of the Anderson impurity model for finite systems: Band gap control ofmagnetic moments
Konstantin Hirsch, ∗ Julian Tobias Lau, and Bernd von Issendorff Institut f¨ur Methoden und Instrumentierung der Forschung mit Synchrotronstrahlung,Helmholtz-Zentrum Berlin f¨ur Materialien und Energie GmbH,Albert-Einstein-Straße 15, 12489 Berlin, Germany Physikalisches Institut, Universit¨at Freiburg, Stefan-Meier-Straße 21, 79104 Freiburg, Germany (Dated: July 9, 2014)We study the spin magnetic moment of a single impurity embedded in a finite-size non-magnetichost exhibiting a band gap. The calculations were performed using a tight-binding model Hamil-tonian. The simple criterion for the magnetic to non-magnetic transition as given in the Andersonimpurity model breaks down in these cases. We show how the spin magnetic moment of the impuritythat normally would be quenched can be restored upon introducing a gap at the Fermi level in thehost density of states. The magnitude of the impurity spin magnetic moment scales monotonicallywith the size of the band gap. This observation even holds for a host material featuring a stronglydiscretized density of states. Thus, it should be possible to tune the magnetic moment of dopednano-particles by varying their size and thereby their band gap.
PACS numbers: 75.20.Hr, 75.75.-c, 73.22.-f, 71.10.-w
I. INTRODUCTION
Materials, especially metals, sparsely doped with mag-netic impurities have been widely investigated through-out the last 50 years primarily because some of themdisplay the fascinating Kondo effect . Such systems canserve as an ideal playground to explore the underlyingmany body physics. However, the prerequisite to observea Kondo effect at all is the survival of a local moment ofthe embedded impurity.The interaction of the localized impurity electronic stateswith the itinerant electrons of the host material is the-oretically described by the well established sd - or An-derson model . Within this model the size of the localmoment arises from an intricate interplay of the on-siteCoulomb repulsion U , the energy penalty for adding asecond electron to the localized state, and the width 2Γ ofthe localized state. The width of the localized state, alsoknown as virtual bound state, results from hybridizationof the electronic states of the impurity with the delo-calized states of the host material. In case of a sym-metric arrangement of the impurity spin levels E d, ± , i.e. E d, ± = E F ∓ U / n + − n − ) ( E F is the Fermi energyof the system and n ± the occupation of E d, ± ), a simplecriterion for the existence of a local magnetic momentcan be derived : U / Γ > π .In recent years, advances in experimental techniques andtheoretical methods allowed to push the research into thefinite size domain . In finite systems the itinerant elec-trons are confined and populate highly discretized energylevels. This in turn can be expected to have tremendousinfluence on the description within the Anderson impu-rity model, which accounts only for a continuous hostdensity of states. Indeed we found evidence that the sizeof the spin magnetic moment of a chromium impurity em-bedded in a small gold cluster is strongly affected by thediscretized density of states of the host particle . This becomes most evident in host particles that exhibit a shellclosure and therefore a wider highest-occupied–lowest-unoccupied molecular orbital (HOMO-LUMO) gap. Toget a more fundamental grasp on the influence of an en-ergy gap or a highly discretized host density of stateson the spin magnetic moment of an embedded impurity,we investigate such a systems using a modified Andersonimpurity model. II. MODEL HAMILTONIAN
We model the system in a tight-binding approach, us-ing the following model Hamiltonian: H T B = E d, ± a · · · aa E k, a E k,N (1)In H T B , a single localized orbital at energy E d, ± interactswith a finite number N of delocalized states at energies E k,i . Like in the Anderson model, the coupling strength a of the localized orbital to the continuum states is as-sumed to be the same for all states E k,i . Diagonalizationof the matrix H T B yields the N + 1 eigenstates φ i andeigenenergies ǫ i of the system. This is to be done sepa-rately for majority (+) and minority ( − ) impurity spinstates E d, ± = E F ∓ U / n + − n − ) to yield spin resolvedeigenfunctions φ ± i = ( c i, ± , c i, ± , . . . c i, ± N +1 ) and eigenener-gies ǫ ± i . The states E d, ± are separated by the on-siteCoulomb repulsion U , which is the energy neccessary toadd an electron to the localized orbital. Eigenfunctionsand eigenenergies obtained from diagonalization are usedto calculate the occupation numbers n ± of the majorityand minority spin states from the projected spin density FIG. 1. Comparison of the solutions for the occupation num-bers n ± of spin-up and -down states using U / Γ ≈ .
1, ob-tained analytically from the Anderson impurity model andthe tight-binding Hamiltonian equation (1), using a couplingstrength a = 0 .
02 eV, on-site Coulomb repulsion U = 2 eVand a host density of states of 180 eV − . Both models yield al-most identical results, the self-consistent solutions are markedby the orange circles. Inset: Impurity state projected densityof states for the spin polarized solution, obtained by solvingequations (1-3) self-consistently. The Lorentzian fit agreeswell with the density of states. of states ρ ± ( E ) as: ρ ± ( E ) = X i | c i, ± | δ ( E − ǫ i ) (2) n ± = Z E F −∞ ρ ± ( E ) dE (3)Here, δ ( E ) is the delta function and E F the Fermi en-ergy of the system.In order to find the spin polarization ( n + − n − )( n + + n − ) ofthe system the equations (1-3) have to be solved self-consistently, since the energetic position of the localizedorbital depends on the occupation n ± and vice versa.More specifically, the energetic position of E ∓ is deter-mined by n ± which in turn dictates the occupation of n ∓ . Like Anderson we solve this problem graphicallyby plotting the majority spin state occupation as a func-tion of the minority spin state occupation n + ( n − ) aswell as n − ( n + ). A self-consistent solution is found atthe intersections of both curves, as shown in Fig. 1. Wetest our model by comparing its results for dense butdiscrete levels, approximating a continuous band, to theanalytical solution of the Anderson impurity model. Inthis limit both models should yield identical results. Wechose a constant density of states of 180 eV − , which is comparable to the density of states at the Fermi level ofa free electron gas as can be found, for example, in a goldAu nano-particle if the level bunching due to electronshell effects is neglected. The analytical solution of theAnderson impurity model for the occupations of majorityand minority spin state is given by n ± = 1 π arctan (cid:18) U · ( n ∓ − . (cid:19) + 0 . . (4)A symmetrical arrangement of the impurity states E d, ± relative to the Fermi energy E F is assumed. Such asymmetrical arrangement of the levels is a reasonable as-sumption implying that the dopant remains charge neu-tral. Although in metallic systems the impurity can becharged to some extent, this will be well below one el-ementary electric charge, rendering its influence on theAnderson model negligible.Fig. 1 demonstrates that the numerical solution of H T B in the continuous band limit and the analytical solutionof the Anderson impurity model are nearly indistinguish-able. Furthermore, the inset of Fig. 1 shows the impuritystate projected density of states that results from the nu-merical calculation. The Lorentzian shape of the virtualbound states is also in very good agreement with whatone would expect from the Anderson impurity model andfurther confirms that our tight-binding model agrees withthe Anderson impurity model in the continuous bandlimit.The on-site Coulomb repulsion was chosen to be U =2 eV, since typical values of U for transition metals areranging from 1 eV to 6 eV . For a given density ofstates and on-site Coulomb repulsion the half-width ofthe virtual bound state is solely determined by the cou-pling strength a , which was set to 0 .
02 eV here. Thisset of parameters results in a width 2Γ of ≈ .
44 eVwhich is obtained from a lorentzian fit shown in the insetof Fig. 1 and compares well with the analytical value2Γ = πa ρ ( E F ) = 0 .
45 eV. Generally, a parameter rangeof the coupling strength a of 0 .
02 eV − .
08 eV yields linewidths which are consistent with line widths seen in UPSexperiments carried out on 3 d -transition metal impuritiesembedded in gold and silver , scanning tunneling ex-periments on adatoms as well as density functional the-ory calculations .It should be noted that the model introduced here is con-structed for a single impurity state only. An extension tomulti-orbitals as present in, e.g., 3 d -transition metals canbe done, but does not fundamentally alter the descrip-tion. The main impact is a further stabilization of theimpurity’s spin by the exchange interaction of the localorbital electrons.Having tested our model in the way described above, wecan now turn to studying the influence of a discretizedhost density of states on the spin polarization of the im-purity. We will proceed in two steps. First, we will keepthe host density of states quasi-continuous and introducean energy gap at the Fermi level. Second the host densityof states will additionally be discretized. FIG. 2. Upper panels: Impurity state projected density of states obtained using the tight-binding model Hamiltonian, equation(1), for an impurity interacting with a dense discrete host density of states (180 eV − ) without a gap (a) and with a gap of0 . a = 0 .
04 eV and U = 2 eV were kept constant. The insets show the uncoupled impurity and hostdensity of states. Lower panels (c) and (d) show the resulting solution for the occupation numbers for spin-up and -down states.The impurity magnetization is restored when introducing a small gap in the host density of states, panel (d). The values usedcorrespond to an Anderson criterion of U / Γ = 2 . < π . III. ENERGY GAP IN THE HOST DENSITY OFSTATES
The influence of an energy gap in the host density ofstates on the total occupation of the impurity states hasbeen studied in the seminal work of Haldane , which isan extension of the Anderson impurity model. Haldanewas able to explain the large variety of charge statesthat are observed in dilute magnetic semiconductors.However, the spin polarization was not addressed inHaldanes study. In this study we will concentrate onthe influence of a gap on the spin polarization of thesystem.Parameters of on-site Coulomb repulsion U = 2 eV,host density of states of 180 eV − and coupling strength a = 0 .
04 eV were chosen so that the spin polarizationof the system vanishes in the continuous band limit.The relative energies of the electronic states of theuncoupled host and impurity are depicted in the insetof Fig. 2 (a). Vanishing spin polarization of the systemis indicated by the degeneracy of the virtual boundmajority and minority spin states of the compositesystem, obtained from a self-consistent solution asdescribed in the previous section and shown in Fig. 2(a). Note that self-consistency is only reached for thetrivial non-magnetic solution n + = n − = 0 .
5, as can beseen in panel (c) of the same figure.However, upon introducing an energy gap in the hostdensity of states at the Fermi energy as small as 0 . cf. inset of Fig. 2 (b), the spin polarization is restored. FIG. 3. Comparison of systems incorporating an energy gap(0.1 eV and 0.5 eV) in the host density of states to a system ex-hibiting no energy gap. On-site Coulomb repulsion U = 2 eVand density of states 180 eV − were kept constant. Panel (a):Spin polarization as a function of coupling strength a . Onlyfor very small coupling parameters similar spin polarizationscan be found. The spin magnetic moment is quenched for a ≥ .
035 eV in the continuous band case, whereas it sur-vives in presence of a gap. Panel (b): Anderson criteriondrops below π for coupling strengths a ≥ .
035 eV markingthe magnetic-to-nonmagnetic transition.
Both, majority and minority spin states no longerfeature a lorentzian shape, but exhibit poles at theFermi level as shown in Fig. 2 (b). This in turn leads
FIG. 4. Spin polarization of the impurity as a function of thehost energy gap at constant coupling parameter a = 0 .
04 eV,on-site Coulomb repulsion U = 2 eV and host density ofstates of 180 eV − . to a transfer of density of states from the minorityto the majority spin state resulting in a finite spinpolarization. This can also be seen in Fig. 2 (d) whereadditionally to the non-magnetic solution n ± = 0 . n + = n − can be found. The depictedcurves are no longer in agreement with the analyticaldescription within the Anderson impurity model. Thedeviation from the arctan-function equation (4) is mostobvious in the regions exhibiting straight lines around n + = n − = 0 .
5, which suppress the quenching of thelocal spin magnetic moment.These observations indicate that the simple criterionof the Anderson impurity model for the magnetic tonon-magnetic transition, U / Γ = π , does not hold if anenergy gap is introduced in the host density of states.To be more specific, magnetic solutions can be foundalthough the criterion U / Γ yields values smaller than π in the continuous band limit. The stabilization of a mag-netic solution by introduction of an energy gap to thehost density of states seems to be a quite robust effect ascan be seen from Fig. 3 (a). Here, a comparison of thespin polarization as a function of the coupling strength a is shown between systems exhibiting an energy gapin the host density of states and a system lacking anenergy gap. For very small coupling parameters a theinfluence of the gap is negligible, as has already beenshown experimentally . For larger coupling strengths a , however, a severe deviation between the system withand without energy gap can be observed, most strikinglyat a ≥ .
035 eV. For that particular set of parameters( U = 2 eV, a ≥ .
035 eV and ρ = 180 eV − ) the spinpolarization of the system without a gap in the hostdensity of states vanishes while the spin polarizationsurvives in case of the systems exhibiting a gap. Thevanishing spin polarization can be associated to the dropof the Anderson criterion U / Γ below π , marking themagnetic to non-magnetic transition in the Andersonimpurity model as depicted in panel (b) of Fig. 3.Although the magnitude of the spin polarization willdepend on the particular choice of the parameters U and a as well as the host density of states, opening up anenergy gap reliably introduces a spin polarization in thesystem. This is even true for comparable large couplingparameters a corresponding to small values U / Γ < π inthe continuous band limit, cf. panels (a) and (b) of Fig.3. Again, this clearly shows the breakdown of the simplecriterion for the magnetic to non-magnetic transition asgiven in the Anderson impurity model.We can therefore state at this point that an energy gapin the host density of states has a profound influence onthe spin polarization. We can quantify this influence bycalculating the magnitude of the spin polarization as afunction of the size of the gap. The spin polarization asa function of the energy gap is plotted in Fig. 4. Allother parameters are kept constant as before U = 2 eV, a = 0 .
04 eV, and density of states 180 eV − . Again,opening up the gap immediately restores a spin polar-ization which then monotonically increases as a functionof the gap in the host density of states. The dependenceof the spin polarization on the coupling strength a willbe discussed in more detail in the next section. IV. DISCRETE HOST DENSITY OF STATES
The model Hamiltonian (1) enables us to tackle notonly bulk-like systems exhibiting a band gap but alsoto treat discrete energy levels of the host, which can befound, e.g. , in isolated systems consisting of only a fewatoms.It is reasonable to assume that the details of the spinpolarization in such a system will depend on the exactrelative arrangement of impurity and host electronicstates. However, we will show here that the overallscaling of the spin polarization with the energy gap,which in case of finite systems is the HOMO-LUMOgap, of the host will hold also for a highly discretizedhost density of states.To this end we calculated the spin polarization of asystem featuring 24 delocalized host states (12 occupied,12 unoccupied) interacting with a single impurity stateas a function of coupling strength a as well as of theenergy gap. The system introduced here could, e.g. , bea Na , Au or Cu host particle. As already pointedout the spin polarization will depend on the actuallevel arrangement. However, in order to get a detailedinsight into the influence of the energy gap, for everyset of parameters we generated thousand different hostsystems with randomly distributed host levels withina 20 eV energy range under the constraint to exhibit FIG. 5. Spin polarization of a system consisting of 24 ran-domly distributed host states within 20 eV as a function ofthe host energy gap for different coupling parameter a . Usingan on-site Coulomb repulsion of U = 1 eV. a certain energy gap at the Fermi level. The resultingdensity of states of 1 . − compares well with thedensity of states at the Fermi level of a free electrongas, again neglecting level bunching due to electronicshell effects, in a coinage metal particle of size 12. Theresults of these calculations are depicted in Fig. 5,where the mean value of the system’s spin polarizationis plotted versus the energy gap. The standard deviationis given by the error bars. This was done for differentcoupling strengths a = 0 . − . U = 1 eVconstant. The values for the coupling strength a usedin the calculation correspond to the parameters used inthe previous section, since a scales with the number ofhost states N as a = a / √ N .As can be seen from the figure, on average, the spinpolarization scales inversely with the coupling strength a . But more importantly the spin polarization scaleswith the size of the energy gap, comparable to thebehavior of a quasi-continuous band exhibiting a gapas presented in the previous section. The standard de-viation is larger for small energy gaps and intermediatecoupling strengths, cf. Fig. 5, since for parameters closeto the transition from a magnetic to a non-magneticstate the actual arrangement of the host and impuritystates becomes naturally more important, in contrastto systems exhibiting very small or large couplingstrengths a . In case of very weak interaction betweenhost and impurity, hybridization is small independentof the relative arrangement of the levels. In the largecoupling strength regime, hybridization is also mainlyindependent of the particular arrangement of the levels,since the host states are coupled to the impurity statedisregarding their energetic separation. V. CONCLUSION
The influence of an energy gap in the host density ofstates as well as the discrete nature of a host density ofstates on the spin polarization of an impurity was stud-ied using a tight-binding approach. The criterion for atransition from a magnetic to a non-magnetic system asstated within the Anderson impurity model is found notto be valid anymore. For cases where the magnetic mo-ment of the impurity is quenched in a system having acontinuous host density of states, we have shown thatthe opening of a gap can recover the spin magnetic mo-ment. The size of the spin polarization scales with thesize of the energy gap. This observation even holds for adiscrete host density of states. On average the spin po-larization follows the size of the energy (HOMO-LUMO)gap and the actual relative energetic position of impurityand host electronic states is of minor importance. This can severely influence the magnetic moment of impuritiesembedded in finite size matrices exhibiting a discretizeddensity of states. Although their bulk counterpart maylack a magnetization, finite systems can exhibit a spinpolarization, which can be tuned by the size of the hostenergy (HOMO-LUMO) gap. The described dependenceis expected to be observable by studying, e.g. cobaltdoped aluminum gas phase clusters combining x-ray mag-netic circular dichroism and ultraviolet photoelec-tron spectroscopy .Furthermore, the results even point at the possibility toswitch the impurity’s spin magnetic moment in a partic-ular class of bulk host materials, i.e., materials exhibitinga Peierls transition. By passing through the Peierls tran-sition temperature and thereby opening and closing anenergy gap at the Fermi level, respectively, the spin of anembedded impurity may be switched on and off. VI. ACKNOWLEDGMENTS
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