Determining surface magnetization and local magnetic moments with atomic scale resolution. W. A. Hofer and A. J. Fisher Department of Physics and Astronomy, University College, Gower Street, London WC1E 6BT, UK We propose a method to determine the direction of surface magnetization and local magnetic 2 momentsontheatomicscale. Themethodcompriseshighresolutionscanningtunnelingmicroscope 0 0 experimentsin conjunction with firstprinciples simulations of thetunnelingcurrent. Thepotential 2 ofthemethodisdemonstratedonamodelsystem,antiferromagneticMnoverlayersonW(110). We expect that it will ultimately allow to study the detailed changes of magnetic surface structures in n the vicinity of dopants or impurities. a J 72.25.-b,73.40.Jn,75.25.+z 5 1 ] i Until very recently magnetic properties of thin metal ordering on the atomic scale has already been observed c films or surfaces could be determined only by x-ray [6], it is justified to assume that magnetic surface prop- s - magnetic dichroism experiments [1–3]. In these exper- ertiesarenotdecisivelymodifiedbyanSTMtip. Andas l r iments x-rays of circular polarization, originating from the experiments are suitable to reveal electronic proper- t m synchrotron radiation, are used to probe the magnetic ties on an atomic scale, they also remove the resolution properties. With the help of second order perturbation problem one is confronted with in dichroism measure- . t theory the intensity of the adsorbed radiationcan be re- ments. a m lated to magnetic moments µ and the direction of spin Let us consider the situation in a tunnelling junction polarization M~ due to crystal anisotropies [4]. However, between a crystal surface and an STM tip in real space. - d the method suffers from two serious weaknesses, which Magnetic anisotropy in a crystal breaks the rotational n limit its applicability: (i) The local resolution obtained symmetry of electron spins. The spin states in this case o with synchrotron radiation is far too low for any atomic are projected onto the crystal’s magnetic axis. We as- c scale analysis and in the range of 50nm [5]. (ii) The in- sumeinthefollowingthatthissymmetrybreakingoccurs [ tensity of the x-raybeams is highenoughto leadto sub- in the two separate systems which form our tunnelling 1 stantialenergydissipation. Sincemagneticpropertiesare junction. Depending on the orientation of the magnetic v very sensitive to temperature changes, the measurement axes two limiting cases have to be distinguished. The 1 of groundstate properties is problematic. magneticaxisofsampleandtipareeither paralleloran- 4 Asolutiontothese problemscouldcomefromadiffer- tiparallel. Inthefirstcasewehavetosumupallelectrons 2 1 entmethod,onethatdoesnotaffecttheelectronicstates tunnellingfromspin-upstatesofthesample(n↑S)tospin- 0 of a magnetic layer in any substantial way. The spin po- up states of the tip (n↑). This ferromagnetic ordering is T 2 larized (SP) scanning tunneling microscope (STM) does described by the following transitions: 0 provide just such a method. In measurements with an at/ ilraoynercsoiatthedastubneegnstednemtiopnsotnrastiendglethaantttifheerrloomcaalgmneatgicneMtinc MS ↑ (cid:26)nn↑S↓ −−→→ nn↑T↓ (cid:27) ↑MT (1) m S T moment can be resolved on the atomic scale by STM - scans [6]. The theoretical simulation of these scans re- HereMS andMT describethemagneticaxesofsample d vealed that such a scan is very sensitive to the chemical and tip, respectively. We denote the tunnelling current n o natureofthe STMtipapex,whichallowsonetoidentify due to ferromagnetic ordering by IF. If the two vectors c the STM tip from the corrugation height of the surface areantiparalleltheelectronstunnelfromstates(n↑S)into v: [7]. However, these results were obtained under the con- states(n↓T)andvice versa. The antiferromagneticorder- i ditionofferromagneticorderingofsampleandtipstates, ing is therefore described by: X the direction of surface magnetization was therefore im- ar poIsnedthfrisomLetthteerowuteseptr.opose an extension of the method MS ↑ (cid:26)nn↑S↓ −−→→ nn↓T↑ (cid:27) ↓MT (2) S T to account for truly general orientations of the magnetic axis. Wewanttoshow,howtheorientationofthemagne- Thissetupyieldstheantiferromagnetictunnellingcur- tizationvectorM~ ofsampleandtipchangesthetunneling rent IA. Since the energy of the tunnelling electrons is very low, and since the overlap of the sample and tip current and the corrugation height measured on a sur- wavefunctions is computed far outside the core region face. This in turn allows one to determine the magnetic of surface atoms, spin-orbit coupling can generally be axis and the local moments from STM scans and first neglected in the theoretical treatment. Within density principles simulations. Since antiferrromagnetic surface functional theory (DFT) [8,9] the tunnelling current is 1 commonly described in terms of φM, the angle between modified the clean Fe-tip by two additional setups. In thetwomagneticaxes,andPS(T),thepolarizationofthe one case the apex atom was changed to Mn, in the sec- sample (tip) surface: ondalsothesurfacelayerconsistedofMnatoms(seeFig. 1). On the technical side we note that the free standing I(φM)=I0(1+PSPT cosφM) (3) film consisted of five Fe layers and two additional layers for the apex. 10 special k-points were used in the spin MS ·MT polarized DFT calculations. Given the c(2×2) unit cell cosφM = |MS||MT| (4) this amounts to 40 k-points for the elementary cell. The convergence in the final iterations was better than 0.01 For constant tunnelling matrix elements and within a e/au3. Since Mn and Fe are both 3d metals and relax- perturbation approach the current I and polarizations ations are therefore rather small, the STM simulations 0 PS(T) are given by: were based on the wavefunctions of bulk truncated crys- tals. The details of the electronic structure calculations I ∝ 1 n↑ +n↓ n↑ +n↓ (5) are published in a separate paper [12]. 0 2 S S T T In the Bardeen approach to tunnelling [10] the tun- (cid:16) (cid:17)(cid:16) (cid:17) nelling currentis computed by integratingthe overlapof sample andtipstates overthe separationsurface. Inour n↑ −n↓ S(T) S(T) program,which is described in detail elsewhere (bSCAN PS(T) = n↑ +n↓ (6) [11]), we integrate over a finite separation area and sum S(T) S(T) up all contributions from the Kohn-Sham states of sam- I0 and PSPT can be written in terms of the ferromag- ple and tip numerically. It has been shown previously netic and antiferromagnetic currents: that the STM current in the experiments on W(110)Mn is about one to two orders of magnitude higher than the 1 IF −IA currentobtainablewithinaperturbationapproachandat I0 = 2(IF +IA) PSPT = IF +IA (7) a reasonable distance [7]. Even though we could not pin down this discrepancy, it seems to be most likely due to If the tunnelling matrix element is not constant, current leakages of the STM circuit. In our implementa- the current has to be calculated numerically from the tion of perturbation theory it is implicitly assumed that Bardeen integral over the separation surface [10,11]. In all tunnelling current passes through a small area of the this case the current contributions have to account for separation surface, thus it cannot account for off center the spin orientation of a given eigenstate. In DFT the contributions in the circuit. Therefore we computed the energetic minimum for magnetic crystals is reached by constant current contours not for the actual values in optimizing the distribution of spin-up density n↑ and the measurements(whichwerein the rangeof30nA[6]), spin-down density n↓. The currents for ferromagnetic but chose contourscentered at about 4.5 to 4.6 ˚A above and antiferromagnetic coupling are computed by calcu- theW(110)Mnfilm. Thisdistanceisatthelowerlimitof lating the transition matrix elements for the spin polar- mechanicalstabilityonmetalsurfaces[13]. Thebiasvolt- ized Kohn-Sham states of sample and tip: age in the measurements was - 3 mV, we chose the same IF =I(n↑S −→n↑T)+I(n↓S −→n↓T) (8) vcoonltsatgaentfocruorruernstimcounltaotuiornssd.eIpnenthdeinfiggounretswwoesedpisapralatyepthae- rameters, important for every single STM experiment: IA =I(n↑S −→n↓T)+I(n↓S −→n↑T) (9) (i) The chemical nature of the tip surface. We report on simulationswiththreedifferentchemicalcompositionsof Calculating the tunnel current for different angles φM the tip. (ii) The angle of magnetization. We show simu- requires then only to compute the linear combination of lationsforselectedanglesφM betweenthe magneticaxis ferromagnetic and antiferromagnetic currents multiplied of the crystal surface and the STM tip. by the appropriatecoefficients. From these three dimen- The electronic structure of the W(110)Mn surface in sional current maps the constant current contours and antiferromagnetic ordering depends substantially on the the surface corrugations can be extracted in a straight- spin orientation of electron charge. For spin-up states forward manner. the density contours have a maximum at the position of We have used a full potential method to compute the Mn atoms with a negative magnetic moment and their electronic groundstate properties of model tips. The tip minimum at the position of atoms with the opposite po- in the experiments was a tungsten wire coated with sev- larization(seeFig. 2). Forspin-downstatesthesituation eral layers of iron [6]. We mimic this tip by an ideal is reversed: atoms with positive magnetic moments are Fe(100) surface with a single Fe atom in the apex (see now seen as protrusions. This indicates that the surface Fig. 1). Previous simulations of STM experiments re- has the highest corrugation for charge transport with a vealedthatthebestagreementbetweenexperimentsand high degree of polarization. In addition, it shows that simulations is often obtained with a tip model, covered the surface is comparatively flat if measured by a para- by impurities of the sample surface [11]. Therefore we magnetic tip (see Fig. 2, right frame). 2 Figs. 3 to 5 display the results of our STM simula- magneticfieldoftheSTMtipisstabilizedbyanexternal tions with clean and Mn contaminated Fe model tips. magneticfield,andthisexternalmagneticfieldisrotated. We show five angular settings, from φM = 0 (ferromag- In both cases images have to be recorded for a number netic charge transitions) to φM = 180◦ (antiferromag- of settings. From these images the direction in space of netic charge transitions). In general the obtained mag- the sample magnetization can be uniquely determined. neticcontrastdependsontheangleφM. Itisamaximum The method has a number of important implications. forthe limiting valuesof0◦ and180◦, anditvanishesfor First, it can lead to a detailed understanding, how mag- φM =90◦. This angle denotes the case where tunnelling netismdependsonthelayerstructureofamagneticmul- transitions from the sample into the tip are essentially tilayer. As the controlled growth of single layers is ex- unpolarized. We note two distinct features in our simu- perimentallyfeasible,magneticpropertiescanbedirectly lations: studied,whichchangewiththenumberoflayers,e.g. the orientation of the magnetization axis [2]. Second, it is • The obtained maximum of magnetic contrast de- possible to study the influence of impurities on the local pends on the chemical nature of the tip apex. magneticfield. Giventhatmagneticpropertieshavesuch a wide range of applications in current technology, this It is 68pm for a clean tip; 89pm for a tip contaminated possibilitytostudythesite-dependencyofmagnetismon with an Mn atom; and it nearly vanishes for a tip con- theatomicscalecouldleadtoamuchbetterunderstand- taminated by a surface layer of Mn (4pm, Fig. 5). We ing of the chemical determinants of magnetic systems. may therefore conclude that high resolution measure- We think that this potential alone makes the method, ments with a suitable degree of precision, measurements and the SP STM, the most promising tool in nanomag- which are already possible at present, allow to differen- netic research. tiate between different tips also in case of a magnetic In summary we have shown that the combination of tunnel junction. For non-magnetic tunnel junctions, we high resolution spin-polarized STM scans and first prin- have already shown the influence of the tip in previous ciplessimulationsofthe tunnel currentmakesit possible publications [11]. Comparing the simulations with the to determine all magnetic properties of a surface on the experiments [6], the most likely tip in the actual exper- atomic scale: the local magnetic moment as well as the iments seems to be tip model (c) (see Fig. 1): in this polarization and the direction of surface magnetization. case the corrugation is within the range of experiments Weexpectthatthismethodwillultimatelyallowtostudy over a wide range of angles (the range is about 2-4pm changesofthemagneticsurfacepropertiesinthevicinity and it varies, seemingly due to the lateral angle of sur- of dopants and impurities. face polarization [14]). This indicates that the STM tip The work was supported by the British Council and during the measurement is contaminated by at least a the National Research Council. Computing facilities at monolayerofsurfaceatoms;wewouldconcludefromthis the UCLHiPerSPACEcenterwerefunded bythe Higher feature that the tip is generally very close to the surface Education Funding Council for England. and that the experiments are indeed done at the lower limit of stability [13]. • Ingeneralthemeasuredmagneticcontrastdepends on the angle φM between tip and sample magneti- zation. Thereforethisanglecanbeuniquelydeterminedbycom- [1] J. St¨ohrand H. K¨onig, Phys.Rev.Lett. 75, 3748 (1995) paring experimental results with simulations. The ex- [2] D. Weller et al., Phys.Rev.Lett. 75, 3752 (1995) perimentally obtained magnetic contrast was 2 to 4pm [3] J.Goeringetal.,JournalofAlloysandCompounds328, [14]. In our simulations this value indicates either, that 14 (2001) the tip was covered by at least one monolayer of Mn, [4] P. Bruno, Phys. Rev.B 39, 865 (1989) or that the angle between the two magnetizationvectors [5] J. St¨ohret al., Phys.Rev.Lett. 83, 1862 (1999) was 87−89◦ (Fe tip apex or Mn tip apex). Given that [6] S. Heinze et al., Science 288, 1805 (2000) this seems highly improbable for two coplanar vectors, [7] W.A. Hofer and A.J. Fisher, Surf. Sci. Lett. (2002), in the most likely explanation of the experiments is that press [8] P.HohenbergandW.Kohn,Phys.Rev.B136,864(1964) they were performed with a tip close to our model (c) [9] W.KohnandL.J. Sham,Phys.Rev.A140, 1133 (1965) (see Fig. 1). However, this tip is the least suitable to [10] J. Bardeen, Phys. Rev.Lett. 6, 57 (1961) resolve the angle φM. With a clean tip, or a tip only [11] W.A. Hofer and J. Redinger, Surf. Sci. 447, 51 (2000) contaminated by single atoms of the sample, this angle [12] W.A. Hofer and A.J. Fisher, Phys. Rev. B (2002), sub- could be resolved accurately to about 1-2◦. mitted The last step in obtaining all magnetic properties on [13] W.A. Hofer, A.J. Fisher, R.A. Wolkow, and P. Gru¨tter, the atomic scale is the variation of the angle φM. This Phys. Rev.Lett. 87, 236104 (2001) canessentiallybedoneintwoways: (i)Eitherthesample [14] M. Bode, private communication (2001) ortheSTMtipcanberotated. (ii)Theorientationofthe 3 FIG. 3. Simulated STM scans for a Fe(100) tip with Fe a b c apex. Thesimulationsshowfourunitcells,theorientationof themagnetic axes for the individual scans is depicted by the Fe atom black arrows. The corrugation height ∆z is the difference in apparent height between atoms of positive (full circles) and Mn atom negative(emptycircle)magneticmoments. Forperpendicular moments thesurface corrugation vanishes. Fe(100) Fe(100)/Mn Fe(100)/Mn/Mn FIG.1. STMtipmodelsinoursimulation. Thetipismim- ickedbyac(2×2)Fe(100)freestandingfilmwithasingleapex atom (a). Contamination of the tip by atoms of the crystal surface is accounted for by a single Mn apex atom (b),or an Mn monolayer and a Mn apex atom (c). Spin-up Spin-down Added D z = -89 pm D z = -57 pm D z = +0 pm D z = +57 pm D z = +89 pm Mn atom FIG. 4. Simulated STM scans for a Fe(100) tip with Mn apex. The corrugation in this case is higher, the apparent W atom height is greater for surface atoms with a positive magnetic 10-2 10-10 moment. FIG. 2. Electronic structure of the W(110)Mn film. The plots show the local density of states in a vertical plane cut throughthecrystal(sketchatthebottom,left). Themagnetic momentinthesurfaceatomsiseitherpositive(emptycircles), or negative (grey circles). The LDOS above different atoms reveals these atoms either as protrusions, or as depressions, dependingontheelectronspin. Thecorrugationofthesurface layers vanishes for paramagnetic STM tips. D z = -4 pm D z = -3 pm D z = +0 pm D z = +3 pm D z = +4 pm D z = -68 pm D z = -46 pm FIG.5. SimulatedSTMscansforaFe(100)MntipwithMn apex. Themagneticcontrastinthiscasenearlyvanishes. The corrugation height remains unchanged over a wide range of angles, contrary to the simulations with clean or moderately D z = +0 pm D z = +46 pm D z = +68 pm contaminated STM tips. 4