Thin Solid Films 519 (2011) 8419–8422

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Ab-initio study for the correlation effect on the magneto-optical properties ofCo-based full Heusler alloys

Miyoung Kim a,⁎, Hanjo Lim a, Jae Il Lee b

a Division of Energy System Research, Ajou University, Suwon 443-749, Republic of Koreab Department of Physics, Inha University, Incheon 402-751, Republic of Korea

⁎ Corresponding author.E-mail addresses: mykim.nu@gmail.com, mykim@aj

0040-6090/$ – see front matter © 2011 Elsevier B.V. Aldoi:10.1016/j.tsf.2011.03.096

a b s t r a c t

a r t i c l e i n f o

Available online 1 April 2011

Keywords:Heusler alloysMagneto-optical effectAb-initioFLAPW

Magneto-optical (MO) properties of ferromagnetic Co2YGe full-Heusler alloys (Y=Fe and Mn) areinvestigated by the first principles electronic structure calculations using the highly accurate FLAPW methodwithin GGA+U approach. The polar magneto-optical Kerr angles are calculated by solving Kubo's linearresponse formula using the FLAPW eigenstates and eigenfunctions to obtain the optical conductivity byinterband transitions. The typical features of polar Kerr rotation of transition metals with two major peakswere obtained while the low energy spectra are enhanced largely at 1–2 eV region compared to GGA results.Detailed electronic structures analysis revealed that this enhancement is due to the decreased interbandtransitions by reduced diagonal optical conductivity. Results indicate that the MO spectra can be used todetermine the appropriate correlation level for the present alloys.

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Searching for a material of large Magneto-optical (MO) Kerrspectrum at a specific energy range has been pursued for many yearssince it implies huge advantage in magneto-optical recording mediatechnology.Nowadays,MOspectra are also inwideuse as a convenienttool to characterize magnetic materials. One of the most investigatedclasses of MO materials for decades is so called full- or half-Heusleralloy with 3 d transition metals which drew attentions after the earlyreport of huge Kerr rotation up to 1.2° for the half-Heusler PtMnSballoy by Eugen, et al in 1983 [1]. Heusler alloys are also materials ofhighly interest in the Spintronics materials research since many ofthem are predicted to exhibit the half-metallicity where the spin has100% polarization at the Fermi level, which maximizes the spin-injection rate from ferromagnetic to semiconductingmaterials.[2] TheCo-based Heusler alloys in the form of Co2YX, where Y=Fe or Mn andX=group III or IV elements are the most promising half-metallicferromagnets for this usage since besides the half-metallicity they alsohas advantageous properties such as a structural compatibility withsemiconducting substrates, the high magnetization, as well as highCurie temperatures (Tc) up to 1100 K.[3] In spite of great efforts in bothexperiment and theory to realize the high spin-injection rate aspredicted, recent experiments have reported only a few percent ofspin-injection rate at room temperature for the Co-based Heusleralloy/semiconducting hetero-structures [4], indicating that there is alot to be desired in these materials research. The first-principleselectronic structure study of half-metallic materials has been im-

proved to address the reason for the low experimental spin injectionsby simulating the effects of crystal defects such as antisites, vacancies,or atomic swabs and interface with semiconductors. [5] Recently, theon-site Coulomb correlation effect through LDA+U approach also hasbeen pointed out to play an important role in providing betteragreementwith experiment for the electronic structure calculations ofthe Co2FeSi and Co2MnSi [6].

In this work, we performed the first-principles calculation formagneto-optical Kerr spectra of ferromagnetic Co2XGe alloys whereX=Fe and Mn within the GGA+U approach, aiming to reveal thecorrelation effect via on-site Coulomb interaction on the MOproperties. The precise all-electron full-potential linearized augment-ed plane-wave (FLAPW) [7]method is employedwhere the exchange-correlation potential is treatedwithin GGA+U [8,9] where the on-siteCoulomb interaction is characterized by a spherically averagedHubbard U and implemented by the second-variation procedureself-consistently in solving the Kohn-Sham equation. We adopt theCoulomb (U) and exchange (J) energies of 5 and 1 eV, respectively, inpresent calculation. Energy cut off of 220 eV for interstitial planewavebasis and the 12×12×12 special k-point meshes are used for 3DBrillouin zone integral. The spin-orbit coupling which is a crucialsource for MO Kerr effect is also treated in the second-variationtechnique. Experimental lattice constant (10.85 a.u.) is used forCo2MnGe [10] while a calculated value via the total energy calculation(10.84 a.u.) is adopted for Co2FeGe since no experimental value isavailable.

The magneto-optical polar Kerr spectra can be obtained from theoptical conductivity tensor. For present crystal with a four-foldrotational symmetry about the z-axis along which the wave vector of

Fig. 1. Calculated optical conductivity tensors for Co2YGe Heusler alloys for Y=Fe (leftpanels) and Mn(right panels) within GGA (dashed) and GGA+U (solid). Upper andlower panels are for the real part of diagonal and off-diagonal elements.

Fig. 2. CalculatedMO Kerr rotations for Co2YGe Heusler alloys for Y=Fe (left panel) andMn(right panel) within GGA (dashed) and GGA+U (solid). Angles are given in degrees.

8420 M. Kim et al. / Thin Solid Films 519 (2011) 8419–8422

incident light are assumed to be oriented, the complex conductivitytensor elements can be simplified by σxx=σyy and σxy=−σyx. Theoptical conductivity tensor can be determined from the microscopicformalism; by solving Kubo's linear order perturbation formula for thecurrent response to the time-dependent external field. In randomphase approximation, the diagonal and off-diagonal optical conduc-tivities are evaluated in the long wavelength limit using therelativistic Kohn-Sham eigenvalues and wave functions by [11,12]

σ xy ωð Þ = iv∫ dk

2πð Þ3 ∑ijf ωið Þ 1−f ωj

� �h i Πþij

��� ���2− Π−ij

��� ���24ωij

×1

ω−ωij + iδ− 1

ω + ωij + iδ

! ð1Þ

σ xx ωð Þ = iv∫ dk

2πð Þ3 ∑ijf ωið Þ 1−f ωj

� �h i Πþij

��� ���2 + Π−ij

��� ���24ωij

×1

ω−ωij + iδ+

1ω + ωij + iδ

! ð2Þ

where i and j go over the occupied and unoccupied states, re-spectively, ωij is the energy difference between i and j states andΠij

±=Πijx± iΠij

y correspond to be the relativistic momentum operatorfor left and right circularly polarized lights estimated with the wavefunctions of i and j states. The conductivity is integrated over thecomplex energy plane by introducing a finite phenomenologicalinverse relaxation time, δ=0.5 eV, which was adopted in previouscalculations for transition metal alloys to provide good agreementwith experiments. [12,13] These interband transitions are countedonly between the states which satisfy the selection rule for the electricdipole transition, that is, Δl=±1 and Δml=±1. Note that the off-diagonal optical conductivity is proportional to the difference ofoptical transition between the left- and right circular polarizationswhich indicates the magneto-optical phenomena. The diagonal termis, however, more represented by the average of optical transition ofboth polarizations. Therefore, diagonal conductivity is directlyproportional to the number of interband transitions between anoccupied initial state to an unoccupied final state satisfying theselection rule for the electric dipole transition which implies that onlytransition between s and p or between p and d levels are allowed forpresent alloys.

We show our calculated optical conductivities of Co2FeGe andCo2MnGe within GGA+U along with the GGA results for comparisonin Fig. 1. For Co2FeGe, there is obvious change in conductivity due tothe correlation effect; they shifted to the high energy region and thepeak heights are largely enhanced.While the diagonal element of GGA(cf. upper left panel in Fig. 1) conductivity develops its first peaksignificantly at the energy range of 1–2 eV andmaintains its height forthe higher photon energy, the GGA+U results exhibit a seriousreduction of the conductivity at this energy region (at 1–2 eV) and thepeak starts to occur at 2–4 eV showing a huge enhancement comparedto GGA results at higher energy regions. In this high energy region, theoff-diagonal conductivity also enhances a lot in GGA+U (compared toGGA result) while the change of off-diagonal conductivity at lowenergy region is relatively small. For Co2MnGe, the overall shift of theconductivity to higher energy region also clearly appears for thediagonal conductivity — with the clear reduction of diagonalconductivity element at 1–2 eV energy region. High energy diagonalconductivity and off-diagonal conductivity at most of the energyregions are less sensitive to the correlation effect for Co2MnGecompared to Co2FeGe case.

Once the optical conductivity tensors are determined, we cancalculate the polar Kerr angles, θK+ iεK, as a function of photon energy

(ω) simply by applying the macroscopic electromagnetic boundaryproblem as

θK ωð Þ + iεK ωð Þ = −σxy

σxx

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 + i

4πσxx

ω

� �s ð3Þ

where θK and εK are the Kerr rotation and ellipticity, respectively.While Eq. (3) accounts for the interband transitions, the contributionfrom intraband transitions also can be considered by adding a Drudeterm, σ0=ω2

p/[4π(1− iωτD)], into the diagonal element of theconductivity tensor adopting empirical values of the plasma frequen-cy(ωp) and relaxation time(τD).[14] We find that the Drude term onlyalters the MO spectra of low energy regime(b0.5 eV) slightly forpresent alloys, as reported in previous Kerr calculations for transitionmetals and some Heusler alloys [12,13].

Our calculated optical Kerr rotation angles of Co2FeGe andCo2MnGe are shown in Fig. 2. Overall, two alloys exhibit similar MOspectra. The GGA Kerr rotation can be described to have two majorpeaks, that is, a low energy peak at 0.5 eV with maximum magnitudeof 0.5° (for Co2MnGe) and a high energy peak located in broad energyregion of 2.0–4.0 eV. The height of the second peak is higher (lower)than the first peak for Co2FeGe (for Co2MnGe). While we don't have

Fig. 3. Spin-resolved projected density of states (DOS) of the Co (dotted) and Fe or Mn (solid) d-bands calculated within (a) GGA and (b) GGA+U for Co2FeGe, and (c) GGA and(d) GGA+U for Co2MnGe. The upper and lower panels are for the spin majority and minority channel, respectively.

8421M. Kim et al. / Thin Solid Films 519 (2011) 8419–8422

experimental results of MO spectra to directly compare with for thesealloys, the MO spectra with two major peaks resemble experimentalKerr rotation of other Heusler alloys with similar elements [12,13,15].Now, upon the +U correction, we find distinct changes in Kerrrotations; the significant enhancement in magnitude of MO spectraresulting in the maximum value of 1° for the low energy peak ofCo2MnGe and 1.3° for Co2FeGe. Also the major peak positions shift tohigher energy regime for Co2MnGe while the shift is not obviouslyobserved for Co2FeGe. It is interesting to note that the correlationeffect observed in present calculation is opposite to that of NiMnSbHeusler alloys where the correlation effect by the DMFT calculation[16] induced a red-shift and the reduction of the MO spectra. It isclaimed to be mainly induced from the change in unoccupied Mnd-bands since Ni d-bands are almost fully occupied. In contrast, thecorrelation effect comes from both Co and Mn (or Fe) d states inpresent alloys. To see the correlation effect more clearly, we showedour atom- and spin-projected density of states (DOS) in Fig. 3 for Co,Fe, and Mn d-bands. As can be seen from the DOS plots, the t2g–egsplitting of minority Co- and Mn (or Fe)- d bands around EF is largelyenhanced for GGA+U compared to GGA case. As a result, the Co2FeGeis predicted to be half-metallic in GGA+U with energy gap of ~1 eV.While the splitting is more dramatic for Co2MnGe, Fermi energy islocated at the top of the valence band resulting in the metallicproperty. The correlation effect on the MO spectra can be understoodby realizing that the optical interband transitions between theoccupied and unoccupied p and d states can start to occur at relativelylarge energy regime due to the splitting. We believe that this can be apartial reason that the diagonal element of optical conductivity inGGA+U is suppressed compared to GGA results especially at theenergy regions of 1–2 eV, which, according to Eq. (3), can result inprominent enhancement of Kerr rotation.

In summary, we performed the first-principles calculation results ofthe polar magneto-optical Kerr spectra for Co2FeGe and Co2MnGe full-Heusler alloys within the GGA+U approach. We found that thecorrelation effects via on-site Coulomb interactions of Co, Fe, and Mnd-bands induce a significant enhancement of the Kerr angles comparedto the GGA spectra in 1–2 eV energy region, which can be explained bythe reduced diagonal optical conductivity due to the increased t2g–egsplitting of d-bands. While the necessity of additional correction forcorrelation effect (by+U correction for example) to treat the half-metallicity is still in question, comparison of calculatedMO spectrawithexperiment may be used to decide the level of necessary correlationeffect in first-principles investigation of this material.

Acknowledgment

This work is supported by the National Research Foundation ofKorea (NRF-2009-0076970 and KRF-2008-313-c00218) and theKorean Research Foundation Grant by MOEHRD (2010-0029617).

References

[1] P.G. Van Eugen, K.H.J. Buschow, R. Jonegreur, M. Erman, Appl. Phys. Lett. 42 (1983)202.

[2] H. Ohno, Science 281 (1998) 951;G. Schmidt, D. Ferrand, L.W. Molenkamp, A.T. Filip, B.J. van Wees, Phys. Rev. B 62(2000) R4790.

[3] S. Wurmehl, G.H. Fecher, H.C. Kandpal, V. Ksenofontov, C. Felser, H.J. Lin, J. Morais,Phys. Rev. B 72 (2005) 184434.

[4] X.Y. Dong, C. Adelmann, J.Q. Xie, C.J. Palmstrom, X. Lou, J. Strand, P.A. Cowell, J.P.Barnes, A.K. Petford-Long, Appl. Phys. Lett. 86 (2005) 102107. Hickey, C.D.Damsgaard, S.N. Holmes, I. Farrer, G.A.C. Jones. D.A. Ritchie, C.S. Jacobsen, J.B.Hansen, and M. Pepper, ibid. 92, 232101 (2008).

[5] S. Picozzi, A.J. Freeman, J. Phys. Cond. Matt. 19 (2007) 315215;I. Galanakis, J. Phys. Cond. Matt. 14 (2002) 6329.

8422 M. Kim et al. / Thin Solid Films 519 (2011) 8419–8422

[6] Sh Khosravizadeh, S. Javad Hashemifar, Akbarzadeh Hadi, Phys. Rev. B 79 (2009)235203;Hem Chandra Kandpal, Gerhard H. Fecher, Felser Claudia, Phys. Rev. B 73 (2006)094422.

[7] E. Wimmer, H. Krakauer, M. Weinert, A.J. Freeman, Phys. Rev. B 24 (1981) 864.[8] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865.[9] V.I. Anisimov, F. Aryasetiawan, A.I. Lichtenstein, J. Phys. Cond. Matter 9 (1997) 767;

A.B. Shick, A.I. Liechtenstein, W.E. Pickett, Phys. Rev. B. 60 (1999) 10763.[10] P.J. Webster, J. Phys. Chem. Solids 32 (1971) 1221.[11] J. Callaway, Quantum Theory of the Solid State, Academic, New York, 1991;

C.S. Wang, J. Callaway, Phys. Rev. B 9 (1974) 4897.[12] M. Kim, A.J. Freeman, R. Wu, Phys. Rev. B 59 (1999) 9432.M. Kim, thesis,

Northwestern University press, (1999).[13] F. Ricci, S. Picozzi, A. Continenza, F. D'Orazio, F. Lucari, K. Westerholt, M. Kim, A.J.

Freeman, Phys. Rev. B 76 (2007) 014425.[14] The plasma frequency and relaxation time of 9.0×1015 sec -1 and 12.0×10-15 sec,

respectively, are used for present calculation refering to the bulk Co metal.[15] Y.V. Kudryavtsev, Y.P. Lee, J.Y. Rhee, Phys. Rev. B 69 (2004) 195104.[16] S. Chadov, J. Minar, H. Ebert, A. Perlov, L. Chioncel, M.I. Katsnelson, A.I.

Lichtenstein, Phys. Rev. B 74 (R) (2006) 140411.