A coherent description of the anomalous Hall effect (AHE) is presented that
is applicable to pure as well as disordered alloy systems by treating all
sources of the AHE on equal footing. This is achieved by an implementation of
the Kubo-St\v{r}eda equation using the fully relativistic
Korringa-Kohn-Rostoker (KKR) Green's function method in combination with the
Coherent Potential Approximation (CPA) alloy theory. Applications to the pure
elemental ferromagnets bcc-Fe and fcc-Ni led to results in full accordance with
previous work. For the alloy systems fcc-Fe$_x$Pd$_{1-x}$ and
fcc-Ni$_x$Pd$_{1-x}$ very satisfying agreement with experiment could be
achieved for the anomalous Hall conductivity (AHC) over the whole range of
concentration. To interpret these results an extension of the definition for
the intrinsic AHC is suggested. Plotting the corresponding extrinsic AHC versus
the longitudinal conductivity a linear relation is found in the dilute regimes,
that allows a detailed discussion of the role of the skew and side-jump
scattering processes.Comment: * shortened manuscript * slight rewordings * changed line style in
  Fig 1 * corrected misprinted S (skewness) factor * merged Fig. 3 with Fig. 1
  * new citation introduce

D. Ködderitzsch

H. Ebert

M. E. Rose

S. Lowitzer

V. A. Matveev

English

arXiv

A coherent description of the anomalous Hall effect is presented that is applicable to pure and disordered systems. This is achieved by an implementation of the Kubo-Středa equation using the fully relativistic Korringa-Kohn-Rostoker method in combination with the coherent potential approximation. Applications to the pure ferromagnets Fe and Ni led to results in full accordance with previous work. For the alloy systems FexPd1-x and NixPd1-x very satisfying agreement with experiment could be achieved for the anomalous Hall conductivity. To interpret these results a detailed discussion of the skew and side-jump scattering processes is given

Lowitzer, Stephan

Koedderitzsch, Diemo

Ebert, Hubert

University of Regensburg Publication Server

Coherent Description of the Intrinsic and Extrinsic AnomalousHall Effect in Disordered Alloys on an Ab Initio LevelS. Lowitzer, D. Ko¨dderitzsch, and H. EbertDepartment Chemie, Physikalische Chemie, Universita¨t Mu¨nchen, Butenandtstraße 5-13, 81377 Mu¨nchen, Germany(Received 13 August 2010; published 30 December 2010)A coherent description of the anomalous Hall effect is presented that is applicable to pure anddisordered systems. This is achieved by an implementation of the Kubo-Strˇeda equation using the fullyrelativistic Korringa-Kohn-Rostoker method in combination with the coherent potential approximation.Applications to the pure ferromagnets Fe and Ni led to results in full accordance with previous work.For the alloy systems FexPd1x and NixPd1x very satisfying agreement with experiment could beachieved for the anomalous Hall conductivity. To interpret these results a detailed discussion of the skewand side-jump scattering processes is given.DOI: 10.1103/PhysRevLett.105.266604 PACS numbers: 72.15.Gd, 72.15.Eb, 75.47.NpDuring recent years the anomalous Hall effect (AHE)has received great interest. This is partly caused by its closeconnection to the spin Hall effect, which possesses a largepotential for application in the rapidly growing field ofspintronics [1]. On the other hand, many theoretical inves-tigations are devoted to the development of a coherentdescription of these quite complex phenomena [2].As was already pointed out by Karplus and Luttinger [3],the ultimate origin for the AHE in ferromagnets is the spin-orbit coupling (SOC) that—together with the spontaneousmagnetization—leads to a symmetry breaking. As wasdemonstrated by experiment [4,5] and is obvious fromthe work of Karplus and Luttinger, the AHE is presenteven in pure systems. This so-called intrinsic AHE couldlater be connected to the Berry phase [6], and correspond-ing ab initio results could be obtained during recent yearsusing an expression for the anomalous Hall conductivity(AHC) xy in terms of the Berry curvature [7,8]. Fordiluted and concentrated alloys, on the other hand, theoccurrence of the AHE was primarily ascribed to thespin-dependent skew or Mott [9,10] and the so-calledside-jump [11] scattering mechanisms. The latter one iscaused by the anomalous velocity, a first-order relativisticcorrection to the nonrelativistic velocity operator con-nected to SOC. Interestingly, scaling laws connecting theAHC xy and the longitudinal conductivity xx (see be-low) could be derived for these two extrinsic mechanisms[2]. Their treatment in connection with a description ofelectronic transport in terms of wave packet dynamics wasdiscussed in detail recently by Sinitsyn [12]. When dealingwith the extrinsic AHE in disordered systems, however,disorder was treated thus far only by model potentials [13]or by a damping parameter [14,15]. Cre´pieux and Bruno[16] performed qualitative investigations on the AHE onthe basis of the Kubo-Strˇeda equation. This equation isderived from Kubo’s linear response formalism supplyinga suitable basis for investigations based on a realisticdescription of the underlying electronic structure (seeRef. [17] and below). An alternative description of theAHE with a wider regime of applicability is achieved byusing the nonequilibrium Green function formalism. Usinga suitable, but still tractable, model description for theelectronic structure, Onoda et al. [14,15] could dividethe range of xx covered typically by real materials intothree regimes with different scaling laws connecting xyand xx.In this Letter results for the AHC obtained using theKubo-Strˇeda equation are presented. Using a fully relativ-istic Green function formulation in combination with areliable alloy theory, a coherent description for pure aswell as diluted and concentrated alloys could be achievedthat treats intrinsic and extrinsic sources of the AHE onequal footing.The Kubo linear response formalism supplies an appro-priate basis to deal with electronic transport in magneticmetallic systems. Making use of a single-particle descrip-tion of the electronic structure and restricting to the caseT ¼ 0 K, one is led to the Kubo-Strˇeda equation for theelectrical conductivity tensor  [18]. For cubic systemswith the magnetization along the z direction, the AHE isdescribed by the corresponding off-diagonal tensor ele-ment or anomalous Hall conductivity xy given by [18,19]xy¼ @4NTrhj^xðGþGÞj^yG j^xGþj^yðGþGÞicþ e4iNTrhðGþGÞðr^xj^y r^yj^xÞic: (1)Here  is the volume of the unit cell, N is the number ofsites, while r^ and j^ are the position and current densityoperators, respectively. For the cubic systems consideredhere the last term is site diagonal for symmetry reasons. Asfurthermore all systems considered here are metallic, it hasbeen omitted [17]. The electronic structure of the system isrepresented in terms of the single-particle retarded (Gþ)PRL 105, 266604 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending31 DECEMBER 20100031-9007=10=105(26)=266604(4) 266604-1  2010 The American Physical Societyand advanced (G) Green functions at the Fermi energyEF. Within the present work these functions have beenevaluated by means of the multiple-scattering Korringa-Kohn-Rostoker formalism [20]. The chemical disorder inthe investigated random substitutional alloys has beenaccounted for by using the coherent potential approxima-tion [21]. This alloy theory supplies a reliable frameworkto perform the configurational average indicated by thebrackets h  ic in Eq. (1). It includes, in particular, a cleardefinition for differences of configurational averages likehj^xGþj^yGic  hj^xGþichj^yGic. These so-called vertexcorrections (VC) correspond to the scattering-in termswithin semiclassical Boltzmann transport theory [22].Dealing with the AHE requires us to account for theinfluence of spin-orbit coupling in an appropriate way. Thisis achieved by using the four-component Dirac formalism[23]. In combination with spin-density functional theory inits local approximation (LSDA) the corresponding DiracHamiltonian is given by [24]H D ¼ c  p^þ mc2 þ V þ zB: (2)Here p^ ¼ i@r is the canonical momentum operator, and  are the standard Dirac matrices [23], while V and Brepresent the spin-independent and spin-dependent, re-spectively, effective LSDA potentials for the magnetizationalong z. Within the fully relativistic framework adoptedhere, the current density operator j^ is given by [23]j^ ¼ cjej: (3)To allow for a more detailed discussion on the origin of theAHE it is useful to introduce the alternative current densityoperator [25],j^ p^ ¼ jejmþ E=c2p^þ Vcþ Bczðx; y; 0ÞT; (4)that is equivalent to j^ given by Eq. (3) and that can bederived from the anticommutator of j^ and the DiracHamiltonianH D given by Eq. (2).Recently, the intrinsic AHE of the pure ferromagnets Fe,Co, and Ni [7,8,26] as well as ordered FePt and FePd [27]has been investigated theoretically on an ab initio levelusing the formulation for xy in terms of the Berry curva-ture. Alternatively, the tensor element xy can be obtaineddirectly from the expression given in Eq. (1) that is eval-uated by Fourier transformation leading to a correspondingBrillouin zone integration [22]. As the integrand shows a-function-like behavior for pure systems, a small imagi-nary part  has to be added to the Fermi energy EF and anextrapolation to zero has to be made for . For the calcu-lations of xy performed for bcc Fe and fcc Ni,  has beenvaried between 103 and 106 Ry. To ensure convergenceof the Brillouin zone integration, about 109 k points havebeen used. The resulting AHC of bcc Fe and fcc Ni is givenin Table I together with experimental data as well as resultsof previous ab initio work [7,8,28]. Taking into accountthat SOC was treated by approximate schemes within thecorresponding calculations, the agreement is quitesatisfying.The expression for xy in terms of the Berry curvatureused within previous work is completely equivalent to theKubo-Strˇeda equation used here, as both approaches arebased on Kubo’s linear response formalism and adopt asingle-particle description for the electronic structure forT ¼ 0 K [17]. However, it should be stressed that theBerry curvature is usually formulated in terms of Blochstates implying translational symmetry this way.Calculations of xy for disordered alloys with broken trans-lational symmetry are therefore not possible on this basiswhile the Kubo-Strˇeda equation supplies an adequate frame-work for such investigations. As the k-dependent integrandconnected with Eq. (1) gets smeared out in this case due tothe chemical disorder, it loses its -function-like behavior.For that reason, broadening via a complex Fermi energywith  > 0 is not necessary for calculations on alloys.Within the present work, corresponding calculationshave been done for the alloy systems fcc FexPd1x andfcc NixPd1x. Both systems are formed by the nearlyferromagnetic transition metal Pd with an elemental 3dferromagnet leading to a very low critical concentrationxcrit for the onset of spontaneous ferromagnetic order(xFecrit  0 for FexPd1x and xNicrit  0:02 for NixPd1x). Ascan be seen in Fig. 1, calculation of the AHC via Eq. (1)leads to a very satisfying agreement with the experimentaldata [29], in particular, for FexPd1x, that are availableover a wide range of composition. In particular, the changein sign of xy with composition observed for both alloysystems is well reproduced by the calculations. As onemight speculate from the data for elemental bcc Fe andfcc Ni, in Table I the sign of xy on the Pd-poor side of fccFexPd1x is indeed positive while it is negative for fccNixPd1x. For the Pd-rich side, the situation is reversed,clearly showing that the elemental ferromagnet is theprimary source for the AHE in these two alloy systems(see below).To get a more detailed insight into the mechanismresponsible for the AHE in the investigated alloys, a de-composition of the AHC has been performed. A formalbasis for this is provided by the representation of the Kubo-Strˇeda equation in terms of Feynman diagrams [16]. Fromthis it can be seen that the skew and side-jump mechanismsTABLE I. The intrinsic AHC of bcc Fe and fcc Ni fromab initio theoretical as well as experimental (Exp.) investiga-tions.xy ðmcmÞ1 bcc Fe fcc NiThis work 0.638 1:635Yao et al. [7] 0.753Wang et al. [8] 0.751 2:203Yao [28] 2:073Exp. [4,5] 1.032 0:646PRL 105, 266604 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending31 DECEMBER 2010266604-2are exclusively connected to diagrams involving the vertexcorrections. The remaining diagrams are standing for prod-ucts of the type hj^xGþichj^yGic that correspond to theintrinsic AHE and correction terms due to chemical dis-order. It seems therefore sensible to extend the definition ofthe intrinsic AHC intrxy to the case of diluted and concen-trated alloys by combining all contributions not connectedto the vertex corrections. This obviously allows us tocalculate the total AHC (xy) and the intrinsic one (intrxy )by evaluating the Kubo-Strˇeda equation [Eq. (1)] with andwithout, respectively, including the VC, i.e., identifyingxy  VCxy and intrxy  no VCxy , respectively. As seen inFig. 1,intrxy gives a major contribution to the total AHCxyof fcc FexPd1x and shows, in particular, also a change insign with varying concentration. For fcc NixPd1x, on theother hand, intrxy varies weakly with composition and ex-trapolates rather well to the intrinsic AHC of pure Ni (seeTable I). For both alloy systems intrxy  1 ðmcmÞ1when xPd approaches 1, indicating that the intrinsic AHCis primarily determined by the properties of the Pd host inthe dilute regime. These findings obviously justify theextension of the definition for intrxy to represent all contri-butions not connected to the vertex corrections.The longitudinal conductivity xx of fcc FexPd1x andfcc NixPd1x lies nearly exclusively in the so-called super-clean regime with xx * ðcmÞ1 [14,15]. For thisregime the skew-scattering mechanism should dominatexy obeying the relation xy ¼ Sxx, with S being the so-called skewness factor [14,15]. Accounting for all threemechanisms one is therefore led to the decomposition [2],xy ¼ intrxy þ Sxx þ sjxy ¼ intrxy þ extrxy ; (5)that may be seen as a definition for the side-jump contri-bution sjxy [2]. In fact, a plot of xy versus xx with theconcentration as an implicit parameter was used in the pastto decompose the experimental AHC of alloy systemsaccordingly [30–32].In Fig. 2 the extrinsic AHC of fcc FexPd1x and fccNixPd1x defined as extrxy ¼ xy  intrxy is plotted versusthe longitudinal conductivity xx. Obviously, the relation0 0.2 0.4 0.6 0.8x-3-2-1012σxy (mΩ cm)-1Theo. (VC)Theo. (no VC)Exp.σxysj [∇] × 100fcc FexPd1-x0 0.2 0.4 0.6 0.8 1x-6-4-202σxy (mΩ cm)-1Theo. (VC)Theo. (no VC)Theo. pure NiExp.σxysj [∇] × 100fcc NixPd1-xFIG. 1 (color online). The AHC of fcc FexPd1x and fccNixPd1x. The total AHC xy (full squares) has been calculatedincluding the vertex corrections (xy  VCxy ), while the intrinsicAHC intrxy (open squares) has been obtained by omitting them(intrxy  no VCxy ). In addition, experimental data [29] for xy (fullcircles) determined at T ¼ 4:2 K are shown. Further, an estima-tion for the side-jump contribution sjxy to the extrinsic AHC offcc FexPd1x and fcc NixPd1x calculated as the differenceextrxy  extrxy ½r is shown (open diamonds; see text).0 0.1 0.2 0.3 0.4 0.5 0.6σxx (µΩ cm)-1-2-1.5-1-0.50σxyextr   (mΩ cm)-1Pd richfcc FexPd1-x0 1 2 3 4 5 6 7σxx (µΩ cm)-1-40-30-20-100σxyextr   (mΩ cm)-1Ni richPd richfcc Ni1-xPdxFIG. 2 (color online). The extrinsic AHC extrxy versus xxfor fcc FexPd1x and fcc NixPd1x. The straight lines representextrapolations of the data for xPd  0:9 (Pd rich) and xNi  0:9(Ni rich) to xx ¼ 0.PRL 105, 266604 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending31 DECEMBER 2010266604-3suggested by Eq. (5) is well fulfilled on the Pd-rich side ofboth systems as well as on the Ni-rich side of NixPd1x.Extrapolating for these regimes to xx ¼ 0 allows us todeduce the corresponding skewness parameters and side-jump term sjxy [FexPd1x for xPd  0:9, S ¼ 2:7 103and sjxy  0:1 ðmcmÞ1; NixPd1x for xPd  0:9, S ¼2:0 103 and sjxy  0:4 ðmcmÞ1; for xNi  0:9, S ¼6:6 103 and sjxy  4:8 ðmcmÞ1]. These resultsshow clearly that the skew-scattering mechanisms by fardominate extrxy in the dilute regimes. For the two alloysystems the corresponding skewness factor S is foundcomparable in magnitude but different in sign on thePd-rich side (see above). This once more demonstratesthat the skew-scattering mechanism has to be associatedprimarily with the solute component Fe or Ni, respectively.As emphasized above, Eq. (5) can be seen as a definitionfor various extrinsic contributions to extrxy according totheir scaling behavior. An alternative way to define theside-jump term sjxy is to make use of its connection withthe anomalous velocity, that is a correction to the non-relativistic current density operator j^nr ¼  jejm @i r. Withinthe relativistic approach used here, an estimate for sjxy canbe made using the alternative current density operator j^p^with the potential terms V and B suppressed [see Eq. (4)].The corresponding extrinsic AHC extrxy ½r ¼ xy½r intrxy ½r allows us to write sjxy  sjxy½r ¼ extrxy extrxy ½r. The results for sjxy obtained this way for fccFexPd1x and fcc NixPd1x are also shown in Fig. 1. Asone notes, there is obviously a non-negligible concentra-tion dependency for both alloy systems, in particular, onthe Pd-rich side. In both cases, however, the numericalresults are much smaller than for the intrinsic as well asthe skew-scattering contributions. While this once moresupports the conclusion that the AHE of the investigatedalloy systems is dominated by the latter mechanisms, italso shows that the quantitative results for the side-jumpterm sjxy may depend strongly on the definition used.In summary, a coherent description of the AHE for puremetals and diluted as well as concentrated alloys on anab initio level was presented based on a fully relativisticimplementation of the Kubo-Strˇeda equation using themultiple-scattering or Korringa-Kohn-Rostoker formalismin combination with the coherent potential approximationalloy theory. The intrinsic AHC obtained this way for bccFe and fcc Ni was found in satisfying agreement withprevious ab initio work using an equivalent expressionfor xy in terms of the Berry curvature. Correspondingcalculations for the alloy systems fcc FexPd1x and fccNixPd1x reproduced the available experimental data verywell. Identifying the contributions to xy that are notconnected to the vertex corrections with the intrinsicAHE of an alloy allowed us to decompose the remainingextrinsic AHE. Plotting xy versus xx it was found thatthe skew-scattering term by far dominates the side-jumpcontribution in the dilute alloy regime. This conclusioncould be supported by model calculations that supplied anestimate for the contribution to xy due to the anomalousvelocity.The authors would like to thank the DFG for financialsupport within the SFB 689 ‘‘Spinpha¨nomene in redu-zierten Dimensionen’’ for financial support.[1] D. D. Awschalom and M. Flatte´, Nature Phys. 3, 153(2007).[2] N. Nagaosa et al., Rev. Mod. Phys. 82, 1539 (2010).[3] R. Karplus and J.M. Luttinger, Phys. Rev. 95, 1154(1954).[4] P. Dheer, Phys. Rev. 156, 637 (1967).[5] J.M. Lavine, Phys. Rev. 123, 1273 (1961).[6] Z. Fang et al., Science 302, 92 (2003).[7] Y. Yao et al., Phys. Rev. Lett. 92, 037204 (2004).[8] X. Wang et al., Phys. Rev. B 76, 195109 (2007).[9] J. Smit, Physica (Amsterdam) 21, 877 (1955).[10] J. Smit, Physica (Amsterdam) 24, 39 (1958).[11] L. Berger, Phys. Rev. B 2, 4559 (1970).[12] N. A. Sinitsyn, J. Phys. Condens. Matter 20, 023201(2008).[13] A. A. Kovalev, J. Sinova, and Y. Tserkovnyak, Phys. Rev.Lett. 105, 036601 (2010).[14] S. Onoda, N. Sugimoto, and N. Nagaosa, Phys. Rev. B 77,165103 (2008).[15] S. Onoda, N. Sugimoto, and N. Nagaosa, Phys. Rev. Lett.97, 126602 (2006).[16] A. Cre´pieux and P. Bruno, Phys. Rev. B 64, 094434(2001).[17] T. Naito, D. S. Hirashima, and H. Kontani, Phys. Rev. B81, 195111 (2010).[18] P. Strˇeda, J. Phys. C 15, L717 (1982).[19] A. Cre´pieux and P. Bruno, Phys. Rev. B 64, 014416(2001).[20] H. Ebert, in Electronic Structure and Physical Propertiesof Solids, edited by H. Dreysse´, Lecture Notes in PhysicsVol. 535 (Springer, Berlin, 2000), p. 191.[21] P. Soven, Phys. Rev. 156, 809 (1967).[22] W.H. Butler, Phys. Rev. B 31, 3260 (1985).[23] M. E. Rose, Relativistic Electron Theory (Wiley, NewYork, 1961).[24] A. H. MacDonald and S.H. Vosko, J. Phys. C 12, 2977(1979).[25] G. Y. Guo and H. Ebert, Phys. Rev. B 51, 12 633 (1995).[26] E. Roman, Y. Mokrousov, and I. Souza, Phys. Rev. Lett.103, 097203 (2009).[27] K.M. Seemann et al., Phys. Rev. Lett. 104, 076402(2010).[28] This value has been mentioned in Ref. [8] as an unpub-lished result from Y. Yao.[29] V. A. Matveev and G.V. Fedorov, Fiz. Met. Metalloved.53, 34 (1982).[30] A. Majumdar and L. Berger, Phys. Rev. B 7, 4203 (1973).[31] Y. Shiomi, Y. Onose, and Y. Tokura, Phys. Rev. B 79,100404(R) (2009).[32] Y. Tian, L. Ye, and X. Jin, Phys. Rev. Lett. 103, 087206(2009).PRL 105, 266604 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending31 DECEMBER 2010266604-4

Coherent Description of the Intrinsic and Extrinsic Anomalous Hall Effect in Disordered Alloys on an Ab Initio Level

Crossref

Physical Review Letters

Coherent Description of the Intrinsic and Extrinsic Anomalous Hall Effect in Disordered Alloys on an<i>Ab Initio</i>Level

http://epub.uni-regensburg.de/19675/1/PhysRevLett.105.266604.pdf

Coherent description of the intrinsic and extrinsic anomalous Hall
  effect in disordered alloys on an $ab$ $initio$ level

Coherent description of the intrinsic and extrinsic anomalous Hall effect in disordered alloys on an $ab$ $initio$ level

Abstract

Similar works

Full text

Available Versions

University of Regensburg Publication Server

Crossref

Coherent description of the intrinsic and extrinsic anomalous Hall effect in disordered alloys on an ababab initioinitioinitio level

Abstract

Similar works

Full text

Available Versions

University of Regensburg Publication Server

Crossref

Coherent description of the intrinsic and extrinsic anomalous Hall effect in disordered alloys on an $ab$ $initio$ level