Infrared magneto-optical properties of (III,Mn)V ferromagetic semiconductors Jairo Sinova,1 T. Jungwirth,1,2 J. Kuˇcera,2 and A.H. MacDonald1 1University of Texas at Austin, Physics Department, 1 University Station C1600, Austin TX 78712-0264 2 Institute of Physics ASCR, Cukrovarnick´a 10, 162 53 Praha 6, Czech Republic 3 (Dated: February 2, 2008) 0 We present a theoretical study of the infrared magneto-optical properties of ferromagnetic 0 (III,Mn)V semiconductors. Our analysis combines thekinetic exchange model for (III,Mn)V ferro- 2 magnetism with Kubo linear response theory and Born approximation estimates for the effect of n disorder on the valence band quasiparticles. We predict a prominent feature in the ac-Hall con- a ductivity at a frequency that varies over the range from 200 to 400 meV, depending on Mn and J carrier densities, and is associated with transitions between heavy-holeand light-hole bands. In its 2 zerofrequency limit, our Hallconductivityreduces tothe~k-spaceBerry’s phase valuepredictedby 2 a recent theory of the anomalous Hall effect that is able to account quantitatively for experiment. Wecomputetheoreticalestimatesformagneticcirculardichroism,Faradayrotation,andKerreffect ] i parameters as a function of Mn concentration and free carrier density. The mid-infrared response c feature is present in each of these magneto-optical effects. s - l PACSnumbers: 73.20.Dx,73.20.Mf r t m I. INTRODUCTION to the~k-space Berry’s phase expression that explains dc . t Hall effect observations.10,11,12 a m Rapid progress has been achieved over the past year In metallic ferromagnets, measurements of magneto- - in understanding how growth and annealing conditions optical coefficients on band energy scales provide very d influence the properties of (III,Mn)V diluted magnetic detailed information about the influence of broken time- n semiconductorferromagnets. Theseadvanceshaveledto reversal symmetry on itinerant electron quasiparticle o c therealizationofsampleswithhigherferromagnetictran- states. The appropriate band energy scale for the heav- [ sition temperatures and conductivities.1,2,3 (III,Mn)V ily p-doped (III,Mn)V ferromagnets, and for a number materials have normally been described using a phe- of other materials that have been studied recently13, is 1 nomenologicalmodel4,5,6 inwhichthevalencebandholes in the infrared. For this reason, we believe that exper- v 5 of the host (III,V) semiconductor are coupled by ex- imental infrared magneto-optical studies of (III,Mn)V 0 changeandCoulombinteractionstoMn2+ local-moment ferromagnets are highly desirable; we expect that they 4 ionswith spinS =5/2. The propertiespredictedbythis will be carried out in the near future and that compari- 1 modelaremostsimplyunderstoodinthestronglymetal- son with the predictions presented here will be very in- 0 lic regime for which disorder in the spatial distribution formative in clarifying the physics of these new ferro- 3 ofthe Mn2+ ions,andotherdefects ofthe materials,can magnets. They could, for example, reveal deficiencies of 0 be treated perturbatively. These approximations lead to the relatively simple theoreticalformulationthatwe em- / t a picture ofthe materialsin which spin-orbitcoupling of ploy. Thestudy ofthe magneto-opticalresponseofthese a m thevalence-bandholesubsystemplaysakeyrole4 inpro- ferromagnets is also potentially interesting for applica- viding detailed explanations for many qualitative effects tions, especially if room temperature ferromagnetism is - d discoveredinexperimentalstudiesofthermodynamicand achievedinthefuture. Magneto-opticalpropertiesofthe n transport phenomena. The model can account quanti- closely related (II,Mn)VI diluted magnetic semiconduc- o tatively for the critical temperature,5,7 strain sensitive tor paramagnets14 have already proveduseful from both c magnetic-crystalline anisotropy,5,8 anisotropic magneto- basic science and application points of view. : v resistance coefficients9, and the strong anomalous Hall Absorption and reflection measurements in the visible Xi effect.10,11 Golden rule estimates of quasiparticle scat- rangehavebeenusedtoestablishphenomenologicalesti- tering amplitudes evenprovidethe correctorderofmag- matesforthep-dands-dexchangecouplingconstantsin r a nitude for longitudinal dc conductivities.9 (II,Mn)VI materials, and in establishing the important Inthispaperwediscusscorrespondingtheoreticalpre- role of valence band holes in the (III,Mn)V’s.5,14,15,16,17 dictions for the infrared magneto-optical properties of Photoemission experiments, which explore the deeper these materials. We evaluate magnetic circular dichro- electronic structure have been used to explore the de- ism(MCD), Faradayrotation,andKerreffects inthein- gree of hybridization between the underlying host va- fraredregimeforseveraldifferentMnconcentrationsand lence band and Mn electronic levels, but suffer from be- carrier densities. From the microscopic point of view, ingsurfacesensitive.18,19,20 Intheinfraredregime,recent each of these effects reflects the non-zero value of the ac optical conductivity measurements have uncovered un- Hallconductivity,σ (ω). Ourlinearresponsetheoryfor usual non-drude behavior, including an optical absorp- xy the Hall conductivity reduces in the zero frequency limit tion peak21,22,23,24 connected to back-scattering local- 2 ization effects and to inter valence band transitions, in which is known to be less likely for the strongly metallic agreement with model calculations.25,26 (III,Mn)V ferromagnets on which we focus.28,29 We organize the paper as follows. In Sec. II we intro- The linear response theory Kubo formula expression duce the Kubo formula description of the ac anomalous for the real part of the ac Hall conductivity of disorder- Hall conductivity appropriate for the studied (III,Mn)V free non-interacting electrons is: ferromagnets. In Sec. III we detail the model Hamil- tonian and approximations used in our calculations. In Sec. IV we present an analytic evaluation of the anoma- e2~ d~k Re[σ (ω)] = (f f ) lous Hall conductivity for the case in which disorder xy −m2 (2π)3 n′,~k− n,~k Z n=n′ is neglected and the bands are approximated by the X6 four-band spherical model. (The six-band model that Im[ n′~k pˆx n~k n~k pˆy n′~k ] h | | ih | | i , (1) we use for numerical calculation reduces to the four- ×(ω E +E )(E E ) − n~k n′~k n~k− n′~k band model in the limit of infinite spin-orbit coupling strength.) The isotropic band dispersion of this model where n~k aretheBlochvalence-bandstatesandE the makes analytic calculations possible, although they are Bloche|igeinenergies within~k p~ theory (we use eithne~kr six still somewhat cumbersome. The details of this calcula- · or four band models here), m is the bare electron mass, tion, which builds intuition about qualitative properties f is the Fermi occupation number (0 or 1 at T = 0) of the σxy(ω) curves, are relegated to an appendix. In n,~k Sec. Vwepresentthe numericalresultsofthe fullmodel for the state n~k , and pˆ/m is the~k ~p velocity operator | i · Hamiltonian calculation for σxy(ω) and apply these re- obtained30 by differentiating the ~k ~p Hamiltonian with · sults to discuss all the common magneto-optical effects respect to wavevector. In the zero frequency limit Eq. 1 available for experiments in the present geometry. We reduces to the expression used by Jungwirth et al.10,11 summarize this work and presentour conclusions in Sec. to explain the dc anomalous Hall conductivity of these VI. materials. ThisrecentworksuggeststhatanomalousHall effects are more quantitatively useful in characterizing itinerantelectronferromagnetsthanhadpreviouslybeen II. THEORETICAL APPROACH thought, at least for the presentmaterials. In this paper we extend this advance to finite frequencies. Our theoretical model description starts by coupling Even though the Hall conductivity is finite in the ab- the host semiconductor valence band electrons, de- sence of disorder, we do anticipate that disorder will in- scribed within the ~k p~ or Kohn-Luttinger (KL) the- fluence the σAH(ω) curves, primarily by broadening out · xy ory, with S = 5/2 Mn local moments with a semi- features. Thesourcesofdisorderknowntoberelevantin phenomenological local exchange interaction treated at thesematerialsincludepositionalrandomnessofthesub- a mean-field level.5,8,9,25 At zero temperature this gives stitutional Mn ions with charge Q = e, random place- − rise to valence bands that are split by an effective ex- ment of interstitial Mn ions which act as double donors change field~h=N SJ zˆ, where N is the sub- and are believed to be non-participants31 in the ferro- Mn2+ pd Mn2+ stitutional Mn density and the strength of the exchange magneticorder,andAsanti-siteswhichalsoactashaving couplingistakentobeJ =55meVnm 3.27Weassume charge Q=+2e and are non-magnetic. We estimate the pd − in this paper that the magnetizationis alignedalongthe influence of disorder on the valence band quasiparticles growth (zˆ) direction by an applied small external mag- by calculating their lifetimes using Fermi’s golden rule netic field. We restrict ourselves to the T = 0 limit, al- including both screened Coulomb and exchange interac- lowingustoneglectscatteringoffthermalfluctuationsin tions of the valence electrons with the Mn ions and the the Mn moments orientation. We assume collinear mag- compensating defects.9 Including disorder broadening of netization in the ground state, ignoring the possibility the quasiparticle spectral functions, the Kubo formula of disorder induced non-collinearity in the ground state expression for the Hall conductivity becomes e2~ Re[σ (ω)]= Im[ n~k pˆ n~k n~k pˆ n~k ] xy −ωm2V h ′ | x| ih | y| ′ i ~kXn=n′ 6 dǫ f(ǫ)A (ǫ)Re[Gret(ǫ+~ω)+f(ǫ)A (ǫ)Re[Gadv(ǫ ~ω)], (2) × 2π n′,~k n,k n,~k n′,k − Z whereA (ǫ)=Γ /((ǫ E )2+Γ2 /4)isthedisorder broadened spectral function and Gret and Gadv are the n,~k n~k − n~k n~k 3 advanced and retarded quasiparticle Green’s functions Sinceweareinterestedinthefirstordereffectsofdisorder with finite lifetime Γ−n~k1/2,obtained fromthe goldenrule in σxy we approximate the above expression as, scattering rates from uncorrelated disorder (see Sec. II). Re[σ (ω)]= e2~ (f f )Im[hn′~k|pˆx|n~kihn~k|pˆy|n′~ki](Γ2n,n′ +ω(En~k−En,~k′)−(En~k−En,~k′)2), (3) xy −m2V n′,~k− n,~k ((ω E +E )2+Γ2 )((E E )2+Γ2 ) ~kXn=n′ − n~k n′~k n,n′ n~k − n′~k n,n′ 6 swchaetrteerΓinng,nr′a≡tes(Γavne+ragΓend′)o/v2eranbdanΓdnnaraestihneRgeof.ld2e5n.rWulee b = √m3~2γ3kz(kx−iky), use Eq. (3) to evaluate σxy(ω) below. √3~2 c = γ (k2 k2) 2iγ k k , 2m 2 x− y − 3 x y (cid:2) (cid:3) III. MODEL HAMILTONIAN √2~2 d = γ 2k2 (k2+k2) . (5) − 2m 2 z − x y In the virtual crystal approximation, the interac- (cid:2) (cid:3) tions are replaced by their spatial averages, so that the We focus here on GaAs for which γ = 6.98, γ = 2.06, 1 2 Coulomb interactionvanishes and hole quasiparticlesin- γ =2.93, and ∆ =341 meV .32 3 so teract with a spatially constant kinetic-exchange field. We treat the effects of disorder on the hole quasipar- The unperturbed Hamiltonian for the holes then reads ticles through a finite lifetime scattering rate Γ calcu- H = HL +~h ~s , where H is the host band Hamil- lated using the Fermi’s golden rule. For uncno~krrelated 0 h · tonian, and ~s is the envelope-function hole spin oper- disorder there are two contributions to the transport asctroirb.edTvhiae thhoestfoubranodr spixarbtaonfdtKheohHna-Lmuitlttoinngiaernmisoddeel-. weighted scattering rate Γn~k = ΓMn~kn2+ +ΓnA~ks−anti due to substitutional Mn impurities and As-antisites, given Choosing the angular momentum quantization direc- by tion to be along the z-axis, and ordering the j = 3(−/23/a2n,d1/j2,=−11//22,−b3a/s2is;1f/u2n,c−ti1o/n2s),atchceoLrduitntgintgoertHheamliislt- ΓMn,~kn2+ = 2~πNMn2+ (2dπ~k)′3|Mn~k,,n~k′′|2 tonian HˆL has the form8: Xn′ Z δ(E E )(1 cosθ ), × n,~k− n′~k′ − ~k,~k′ c b 0 b c√2 Hhh − − √2 c 0 b b∗√3 d and  − ∗ Hlh − √2 −  HˆL = −b0b∗∗ bb0∗√3 −Hdclh∗ H−ch√ch2 −c∗d√2 −√bb0√∗√223  ΓnA,s~k−anti = δ2~(πENAs−aEntiXn′)(Z1 (2cdoπ~ks)′3θ|M˜n~k),,n~k,′′|2  √2 −√2 ∗ − Hso  × n,~k− n′~k′ − ~k,~k′  c∗√2 −d∗ −b∗√√23 √b2 0 Hso  where the scattering matrix elements are approximated  (4) by the expressions (in S.I. units), Inthematrix(4)wehavehighlightedthej =3/2sector. The Kohn-Luttinger eigenenergies are measured down M~k,~k′ = J S z sˆ z from the top of the valence band, i.e. they are hole en- n,n′ pd h n~k| z| n′~k′i ergies. For completeness we list the expressions which e2 define the quantities that appear in HˆL: − ǫ ǫ (~k ~k 2+q2 )hzn~k|zn′~k′i, host 0 | − ′| TF ~2 and = (γ +γ )(k2+k2)+(γ 2γ )k2, Hhh 2m 1 2 x y 1− 2 z (cid:2) M˜~k,~k′ = e2 z z . n,n′ ǫ ǫ (~k ~k 2+q2 )h n~k| n′~k′i ~2 host 0 | − ′| TF = (γ γ )(k2+k2)+(γ +2γ )k2, Hlh 2m 1− 2 x y 1 2 z Here ǫhost is the host semiconductor dielectric constant, Hso = 2~m2 (cid:2)γ1(kx2+ky2+kz2)+∆so, |ozfnt~khieisHtahmeilstioxn-ciaonmHpˆohn,eanntdenthveeloTpheo-mfuansc-tFioernmeiigsecnresepnininogr 4 wavevector q = g(E )e2/(2ǫ ǫ ), where g(E ) TF F host 0 F is the density of states at the Fermi energy, E . The F p inter-band scattering broadening Γn,n′ in Eq. 3 is then 150 calculated by averaging Γn,n′(~k) (Γn(~k)+ Γn′(~k))/2 p=6 nm-3, x=6% over the allowed transitions betwe≡en bands n and n as p=2 nm-3, x=4% in Ref. 25. ′ -1m] 100 p=4 nm-3, x=6% c 1 -Ω IV. 4-BAND SPHERICAL MODEL (ω) [ 50 b 4 AH-xy Inthis sectionwe brieflysummarizeananalyticcalcu- σ lation of σxy(ω) for a disorder free 4-band model with 0 isotropic bands, the so-called spherical model. This model is realized by taking the spin-orbit coupling to 0 200 400 600 800 1000 infinity and taking γ = γ (equal to 2.5 for GaAs) in hω [eV] 2 3 the Kohn-Luttinger 6-band model of Eq. 4. This yields ~2 5 FIG.1: Anomalousac-Hallconductivitycalculatedwithinthe HˆL−4b = (γ1+ γ2)k2 2γ2(~k ~j)2 , (6) 4-bandsphericalmodelwithoutdisorderlife-timebroadening 2m 2 − · 0 (cid:20) (cid:21) for several itinerant hole and Mn concentrations. with the anti-ferromagnetic coupling between the local- ized moments and the holes given as before by, hsˆz = mlhEF/µ+(h/6)mlh/mhh+h/2<ω<mhhEF/µ h/6 (h/3)ˆjz. From the Hamiltonian in Eq. 6 one can imme- (h/2)mhh/mlh − − diatelysee oneofthe consequencesofastrongspin-orbit coupling: for a Bloch state labeled by wavevector~k, the e2 5 2µ~ω h A (ω)= . (10) spin quantizationaxis at h=0 is parallelto~k. It is pos- xy (2π~)24π ~2 ~ω r sible to evaluate σ4b(ω) fromEq. 1 in this modelto first xy order in h by completing a straightforward but lengthy For mhhEF/µ h/6 (h/2)mhh/mlh<ω< mhhEF/µ+ − − exerciseindegenerateperturbationtheory. This calcula- h/6+(h/2)mhh/mlh, tion is described in greater detail in appendix A.33 Here we simply state the final result: Axy(ω)=−e2(√2π2)µ2ω~/~ 83u2+ 4~hω[76u3−2u] 1−1∆˜+ (cid:20) − σx4by(ω)=Z−∞∞dω′Aωx−y(ωω′′), (7) +(cid:16)u8 (cid:16)q1− 34u2+ 23u(cid:17)(cid:0)− √123arcsin(√3u/2)(cid:17)(cid:1)(cid:12)(cid:12)(cid:12)11−−∆∆˜˜−+ 1 (cid:12)∆˜− where the spectral function Axy(ω) is given by dif- + h 7u3 12u + 4 4u2 126+351u2 −(cid:12) (11) fFeorrenmt eExpr/eµssion(sh/i6n)mthr/eemdifferhen/t2<eωne<rgmy inEter/vµal+s. 4~ω (cid:18) 1−2 q −4 (cid:16) 486 (cid:17)(cid:19)(cid:12)(cid:12)1−∆˜+# lh F − lh hh − lh F (cid:12) (h/6)m /m +h/2, with (cid:12) lh hh Axy(ω)=−e2(√2π2)µ2ω~/~ 83u2+ 4~hω[76u3−2u] 11−∆+ ∆˜± = h2(1+ ξ32)+ξ~ω˜∓h(3ξ1q+hξ22()1+ ξ32)−3ξ2~2ω˜2,(12) + u 1 3u2+ 3uh(cid:0) √3arcsin(√3u/2)(cid:1)(cid:12)(cid:12)1−∆+ 2 3 (cid:16)8 (cid:16)q − 4 2 (cid:17)− 12 (cid:17)(cid:12)1−∆− and ω˜ =ω−~kh2h/2µ, where khh is the heavy-hole band + h u+ 7u3 + 1 3u2 7 + 13u2 1(cid:12)(cid:12)−∆+ (8) Fermiwave-vectorinzeroexchangefield;andAxy(ω)=0 4~ω (cid:16)− 12 q − 4 (cid:16)27 18 (cid:17)(cid:17)(cid:12)1−∆−(cid:21) othWerewsihsoew. σ4b(ω)forseveralitinerantholeandMncon- (cid:12) xy with (cid:12) centrations in Fig. 1. From the above result (and from the details presentedin Appendix A) it is relativelysim- h(1+ ξ2)+~ω˜ ξ h2(1+ ξ2) 3~2ω˜2 ple to see the source of the feature observed in the mid- ∆ = 2 3 ∓ 3 3 − , (9) infrared regime. The spectral function Axy(ω), shown ± h(1q+ ξ2) in Fig. 2 for the parameters used in Fig. 1, has its 2 3 major contribution from transitions near the light-holes m m /(γ 2γ ), m m /(γ + 2γ ), bands Fermi wave-vectors (the lower frequency peak in hh 0 1 2 lh 0 1 2 ≡ − ≡ ξ m /m , µ m m /(m + m ), and A (ω)) and near the heavy-holes Fermi wave-vectors lh lh hh lh hh lh xy ω˜ ≡= ω ~k2 /2µ,≡ where k is the light-hole (thehigherfrequencypeakinA (ω)),visibleforx=4% band Fermi−wavleh-vector in zero lhexchange field. For andp=0.2nm 3. Thetransitioxnysthatcontributetofirst − 5 50 20 pp==62 nnmm--33,, xx==64%% 40 pp==00..44 nnmm--33 xx==66%% swpahrp -1m] 0 p=4 nm-3, x=6% -1m] 30 pp==00..22 nnmm--33 xx==66%% wspahrp -1Ω c −1Ωc 20 AH(ω) [Axy-20 AHσ(ω) [xy 100 -40 -10 -20 0 500 1000 0 500 1000 hω [eV] _hω [meV] FIG. 2: Spectral function Axy(ω) calculated within the 4- FIG. 3: Anomalous ac-Hall conductivity σxy(ω) for x = 6% band model theitinerant hole and Mn concentrations of Fig. Mnconcentrationandp=0.4and0.2nm−3,forsphericaland 1. non-spherical (band-warping) models. order in h are between heavy and light holes with oppo- now available in the infrared regime considered here) site polarization as shown in Appendix A. We also note α+ α Im[σ (ω)] that there is a considerable contribution to σx4by(ω) from MCD = − − = xy . (13) thehighfrequencypartofthespectralfunction(accounts α++α− Re[σxx(ω)] forrigidshiftsinthelowfrequencyrange)whichindicate Linearly polarized light propagatingthrough a magnetic thepossibleneedtoconsiderhigherbands,maybeinclud- mediumwillexperiencetheFaradayrotationofitspolar- ing the conductionbands,for morerealisticcalculations. ization angle and a transformationfrom linear to ellipti- cally polarized light due to MCD. The angle of rotation perunitlengthtraversed,againinthethinfilmgeometry, V. NUMERICAL RESULTS is (in cgs units) 4π θ (ω)= Re[σ ], (14) The qualitative physics behind the the 4-band model F (1+n)c xy calculation results still applies to the full model numer- ical calculations. However, the effects on σ (ω) due to where c is the speed of light and n is the index of refrac- xy the lifetime broadening of the quasiparticles, finite spin- tion of the substrate, in this case GaAs with n=√10.9. orbitcoupling,andthewarpingofthebands(γ =γ )at Perhapsthemoretechnologicallyrelevantmagneto-optic 3 2 6 higherconcentrationsareanimportantpartofthequan- phenomenaistheKerreffect,whichappearsinreflection titative numerical result. As an example, Fig. 3 shows from a magnetic medium. In this case, also within the the anomalousac-Hallconductivity ofdisorderedsystem Voigtgeometry,theKerrangleandellipticityaredefined for x = 6 % and p = 0.2 and 0.4 nm 3 calculated using as − the 6-band model with warping (γ = γ ) and without 3 2 r r 6 + warping (γ3 = γ2), which emphasizes the importance of θK +iηK ≡ r −+r−, (15) including the warping of the bands in obtaining reliable + − resultswhichcanbe compareddirectlywithexperiment. where r are the total complex reflection amplitudes The Hall conductivity must be non-zero in order to (with m±ultiple scattering taken into account) for right have non-zero magneto-optical effects, but most mea- and left circular polarized light. Note that the simple surable quantities are also influenced by other elements relations, θ Im[σ (ω)] and η Re[σ (ω)],14 ob- K xy K xy ∝ ∝ of the conductivity tensor. The most widely studied tainedinthethick-layerlimitdonotapplyforthetypical magneto-optical effects are the Faraday and Kerr ef- thin (III,Mn)V epilayers. In Fig. 4 we show the differ- fects. The Faraday effect reflects the relative difference ent magneto-optic effects for a concentration of x = 6% between the optical absorption of right and left circu- and p = 0.4 nm 3. The Faraday rotation in this case is − larly polarized light, referred to as magnetic circular larger than the giant Faraday rotation observed in the dichroism(MCD).IntheVoigtgeometry(magnetization paramagnetic(II,Mn)VI’s at opticalfrequencies14,34 and aligned with axis of light propagation) and assuming a shouldbe readilyobservableinthe currenthighly metal- thinfilmgeometry(applicableforall(III,Mn)Vepilayers lic samples. The Kerr angle and ellipticity we obtain for 6 0.2 3 2.5 2 0.15 2 m] c g/ m] CD 0.1 1.55 de eg/c 1 M0.05 01.5θ(ω) [10F 5θ(ω) [10 dF0 pp==00..24 nnmm--33 0 0 p=0.6 nm-3 -1 p=0.8 nm-3 1 1 0 250 500 750 1000 _hω [meV] 0.5 0.5 g] g] e e d d FIG.5: Faradayrotation anglefor x=6%Mn concentration ω) [ ω) [ with p=0.2,0.4,0.6, and 0.8 nm−3. (K (K θ 0 0 η howthesefeaturesappearindifferentmagneto-opticalef- fects(MCD, FaradayrotationandKerreffect)whichare -0.5 -0.5 relatively easily measured, finding strong signals. The 00 225500 550000 775500 11000000 _hω [meV] magnitude of the Faraday rotation is very large (one or- der of magnitude larger than that observed in param- agnetic (II,Mn)VI’s for example) and has a nontrivial FIG. 4: Faraday and Kerr effects for x= 6% Mn concentra- tion and p=0.4 nm−3. dependence on the free carrier concentration. The Kerr effect is also strong when compared to materials used in magneto-optic recording. The origin of the peaks is most easily understood within a simple 4-band spherical (Ga,Mn)As are comparable to the Kerr effects observed modelinwhichtransitionsbetweenheavyandlightholes in the optical regime in materials used for magneto- states with opposite spin-polarization give the strongest recording devices.35 The behavior as a function of free contributiontotheanomaloustransverseopticalconduc- carrier hole concentration can be seen in Fig. 5 where tivity. The four band model represents the infinite spin- the Faraday rotation angle is shown for several carrier orbitcouplingstrengthlimitofthesix-bandmodelweuse concentrations. The peaks and valleys in the different for numerical calculations. Our use of a six-band model quantities are present in all the concentrations, however can account only for transitions within the valence band the magnitude varies, even changing sign at severalcon- and not for transitions between conduction and valence centrations and frequencies. bands. Because of this limitation, we cannot address Rather than presenting many different graphs for all the crossover between intra-band and interband contri- the possible parameters (p,x, etc.), we direct the reader butions which are not completely separated in these ex- toadata-baselocatedathttp://unix12.fzu.cz/ms,where tremelyheavily-dopedsemiconductors,somethingthatis results for these quantities, together with other physi- clearly desirable and should be addressed in subsequent cal quantities, can be obtained and plotted vs. different theoretical work. nominal parameters.32 Ourpredictionsdependinintricatedetailonthemodel thatwehaveusedtodescribetheferromagnetismofthese materials. The model depends most essentially on the VI. CONCLUSIONS assumption that the Mn impurities act as reasonably shallow acceptors and introduce S = 5/2 local moment We have presented a theory of the ac Hall effect in degrees of freedom to the system. The specific calcu- the infrared regime by extending the Berry’s phase the- lations presented here assume that Mn impurities and ory of the dc-anomalous Hall effect to finite frequencies other scatterers in the system can be treated perturba- and treating the effects of disorder through a finite life- tively. This assumption enables quasiparticle scattering time of the valence-band quasiparticles. We observe fea- rates to be estimated in a simple way, but is a less es- tures (peaks and valleys) in the transverse conductivity sential part of the model. The magneto-optical proper- inthe rangebetween200and400meVatwhichthecon- ties studied here are directly dependent on valence-band ductivity changes by more than 100%. We have studied spin-orbit coupling, which we have argued elsewhere4,36 7 plays an essential role in understanding ferromagnetism contribution to the ac Hall conductivity calculated first in these materials. Confirmation by future experiment in the exchange field within the 4-band spherical model. of the detailed predictions made here for the magneto- ThehostvalencebandHamiltonianinthiscase,asshown optical properties of these materials would further vali- in Sec. IV, is given by date the approach we have taken to modeling these in- teresting new ferromagnets. We expect that the weak- ~2 5 quasiparticle-scattering approximations made here will HˆL−4b = (γ1+ γ2)k2 2γ2(k j)2 , (A1) 2m 2 − · bemorereliableinmoremetallicsamples,sincethescat- 0 (cid:20) (cid:21) tering rates are then smaller compared to other relevant The eigenspinors of HˆL 4b are given by energy scales, particularly the Fermi energy. We hope − that these calculations will help motivate magneto-optic experiments in the infrared regime for (Ga,Mn)As and |zn(0k)i=e−iˆjzφ/~e−iˆjyθ/~|ni, (A2) other (III,Mn)V ferromagnets. where n are the spinors with the axis of quantization alongt|heiz-directionandtotalangularmomentum3/2~. Acknowledgments The perturbation due to the antiferromagnetic coupling to the localized moments is Hˆ = hsˆ = (h/3)ˆj . The ′ z z The authors gratefully acknowledge stimulating con- eigenvalues to first order in h are then given by versations with D. Basov, B. Gallagher, T. Dietl, and Q. Niu. This work was supported by the Welch Foun- ~2k2 h dation, by the Office of Naval Research under grant Eh±h = 2m ± 2 cosθ (A3) N000140010951,and by the Grant Agency of the Czech hh Republic under grant 202/02/0912. and APPEND4I-XBAAN: DDESRPIHVEARTIICOANLOMFOσDxyE(ωL) IN THE El±h = 2~m2k2±h3 1− 43cos2θ = 2~m2k2±h6ccooss2θθ , (A4) lh r lh ′ We present in this appendix the details involved in where tan2θ = 2tanθ, hh labels heavy-holes and lh ′ derivingtheresultsshowninEqs. 7-12fortheanomalous labels light-holes. The dipole matrix elements in Eq. 1 are given by: m ∂H m(Enk En′k) ∂ hn′k|pˆα|nki= ~hzn′k|∂k |znki= ~− h∂k n′k|nki, (A5) α α so we can write m2 ∂ ∂ Im[hn′k|pˆx|nkihnk|pˆy|n′ki]= ~2 (Enk−En′k)2Im hzn′k|∂k znkihzn′k|∂k znki , (A6) (cid:20) x y (cid:21) where ∂ cosφcosθ ∂ sinφ ∂ ∂ z = z z +cosφsinθ z , (A7) nk nk nk nk ∂k i k ∂θ| i− ksinθ∂φ| i ∂k| i (cid:12) x (cid:12) and similarly (cid:12) (cid:12) ∂ sinφcosθ ∂ cosφ ∂ ∂ z = z z +sinφsinθ z . (A8) nk nk nk nk ∂k i k ∂θ| i− ksinθ∂φ| i ∂k| i (cid:12) y (cid:12) The perturbed spinor w(cid:12)ave function can be written as (cid:12) i |znki= Cnn′(θ,k)|zn(0′)ki= Cnn′(θ,k)e−i(ˆjz−jn(0))φ/~e−iˆjyθ/~|n′i≡|z˜nki− ~(cosφˆjy −sinφˆjx)|n′i, (A9) n′ n′ X X where jn(0) znk=kzˆˆjz znk=kzˆ . Inserting Eq. A9 into Eq. A7 and A8 gives ≡h | | i ∂ isinφ cosφcosθ ∂ z = (ˆj jn(0))z i [cosφˆj sinφˆj ]z +iz˜ +cosφsinθ z , |∂k nki ~ksinθ z − | nki− ~k y − x | nki | nki ∂k| nki x (cid:16) (cid:17) ∂ icosφ sinφcosθ ∂ z = − (ˆj jn(0))z i [cosφˆj sinφˆj ]z +iz˜ +sinφsinθ z , |∂k nki ~ksinθ z − | nki− ~k y − x | nki | nki ∂k| nki y (cid:16) (cid:17) 8 which can be inserted in Eq. A6 to yield m2 cosθ Im[hn′k|pˆx|nkihnk|pˆy|n′ki] = ~2 (Enk−En′k)2hzn′k|(ˆjz −jn(0))|znkiIm (~k)2sinθ(hzn′k|(cosφˆjy −sinφˆjx)|znki (cid:20) i ∂z +i~hzn′k|zn˜ki+ ~khzn′k| ∂knki , (A10) (cid:21) where hzn′k|(ˆjz −jn(0))|znki = Cnn1′(θ,k)Cnn2(θ,k)hn1|(ˆjz −jn(0))|n2i, nX1n2 Im hzn′k|(cosφˆjy −sinφˆjx)|znki = Cnn1′(θ,k)Cnn2(θ)hn1|ˆjy|n2i, h i nX1n2 hzn′k|zn˜ki= Cnn1′(θ,k)∂Cnn∂1(θθ,k) and hzn′k|∂∂zknki= Cnn1′(θ,k)∂Cnn∂1(kθ,k). Xn1 Xn1 Here we only need to consider six transitions since we only need the n = n terms and we will ignore transitions ′ 6 between bands with equal effective masses which can be shown to contribute to higher order in h. From degenerate perturbation theory we obtain the four eigenvectors to linear order in h: hµsinθ k,hh = k, 3/2 + k, 1/2 , (A11) | ±i | ± i √3(~k)2| ± i hµsinθ k,lh+ = cosθ k,+1/2 sinθ k, 1/2 [cosθ k,+3/2 sinθ k, 3/2 ], (A12) | i ′| i− ′| − i− √3(~k)2 ′| i− ′| − i hµsinθ |k,lh−i = sinθ′|k,+1/2i+cosθ′|k,−1/2i− √3(~k)2[sinθ′|k,+3/2i+cosθ′|k,−3/2i], (A13) where µ m m /(m m ). The Fermi wavevectorsto first order in h/E for each band are given by lh hh hh lh F ≡ − h h 3 kFhh±(θ)=kFhh(0)(cid:18)1± 4EF cosθ(cid:19) and kFlh±(θ)=kFlh(0) 1± 6EFr1− 4cos2θ!. (A14) After some lengthy algebra one obtains Im[ k,hh+ pˆ k,lh+ k,lh+ pˆ k,hh+ ] 3m2 hm2 1 h | x|(E+ iEh+ ) | y| i = 8µ cosθcos2θ′+ 2(~k)2(−4sin(2θ)sin(2θ′)+cos2θcos2θ′ lh− hh cos2θcos2θ 3 + ′ cos2θcos2θ ), (A15) ′ 4cos2θ − 4 ′ Im[ k,hh+ pˆ k,lh k,lh pˆ k,hh+ ] 3m2 hm2 1 h | x|(Eh−l−−iEhh+h) −| y| i = 8µ cosθsin2θ′+ 2(~k)2(+4sin(2θ)sin(2θ′)+cos2θsin2θ′ cos2θsin2θ 3 ′ cos2θsin2θ ), (A16) ′ − 4cos2θ − 4 ′ Im[ k,hh pˆ k,lh+ k,lh+ pˆ k,hh ] 3m2 hm2 1 h −| x|(Eh+l−iEhh−h) | y| −i = − 8µ cosθsin2θ′+ 2(~k)2(+4sin(2θ)sin(2θ′)+cos2θsin2θ′ cos2θsin2θ 3 ′ cos2θsin2θ ), (A17) ′ − 4cos2θ − 4 ′ Im[ k,hh pˆ k,lh k,lh pˆ k,hh ] 3m2 hm2 1 h −| x|(El−h−−iEhh−h) −| y| −i = − 8µ cosθcos2θ′+ 2(~k)2(−4sin(2θ)sin(2θ′)+cos2θcos2θ′) cos2θcos2θ 3 + ′ cos2θcos2θ ). (A18) ′ 4cos2θ − 4 ′ 9 Using Eqs. A15-A18 we can compute directly the dc conductivity (Eq. 1 for ω =0): 2e2~ (fn′,k fn,k)Im[ n′k pˆx nk nk pˆy n′k ] σ (0) = − h | | ih | | i xy m2V k,n>n′ (Enk−En′k)2 X e2 hkhh0 1 m 8 m = F 1 lh + lh , (A19) −(2π~)4πEF (cid:20) − 3rmhh 3mlh+√mlhmhh(cid:21) in agreement with the previously derived dc-anomalous Hall conductivity10 using the Berry’s phase contribution to the Bloch group velocity in the semi-classical equations of motion approach. To compute the ac-anomalous Hall conductivity given by Eq. 1 we rewrite it in terms of the spectral function A (ω) xy ∞ Axy(ω′) σ (ω)= dω , (A20) xy ′ ω ω Z−∞ − ′ with Axy(ω)≡−me22~V k,n=n′ (fn′,k−fn,k)I(mE[nhkn−′k|Epˆxn|′nk)kihnk|pˆy|n′ki]δ(~ω−(Enk−En′k)). (A21) X6 A (ω) is an odd function of ω and we need only to consider ω > 0. We need to consider three separate frequency xy ranges in what follows. First we look at the range m E m h h m E h m h lh F lh hh F hh + + <ω < , (A22) µ m 6 2 µ − 6 − m 2 hh lh and consider the different contributions to A (ω) from the four types of transitions, hh lh , separately. For xy ± ± → hh+ to lh transitions we have ± A (ω;hh+ lh ) xy → ± e2~ 1 µkIm[ k,hh+ pˆ k,lh k,lh pˆ k,hh+ ] x y = d(cosθ) h | | ±ih ±| | i −m2(2π)2 Z−1 ~2 (Eh±l−Eh+h) (cid:12)(cid:12)k=√2µ~ω(1∓12~hωccoossθ2θ′+4~hωcosθ) = e2 2µω/~ 1 d(cosθ) 3cosθ cos2θ′ + h [ 1sin(2θ)sin(2(cid:12)(cid:12)θ )+cos2θ cos2θ′ − p(2π)2~ Z−1 (cid:18)8 (cid:26) sin2θ′ (cid:27) 4~ω ∓4 ′ (cid:26) sin2θ′ (cid:27) 1 cos2θ cos2θ 3 cos2θ ′ cos2θ ′ ] . ±8cos2θ′ (cid:26) sin2θ′ (cid:27)− 8 (cid:26) sin2θ′ (cid:27)(cid:19) We can sum the two and obtain e2 2µω/~ 1 3 h 1 3 A (ω;hh+ lh+)+A (ω;hh+ lh ) = d(cosθ) cosθ+ [+cos2θ+ cos2θ cos2θ] xy → xy → − − (2π)2~ 8 4~ω 8 − 8 p Z−1 (cid:18) (cid:19) e2 5 2µ~ω h = (A23) (2π~)48π ~2 ~ω r For the hh to lh transition we obtain the same result, therefore within this range we have − ± e2 5 2µ~ω h A (ω)= . xy (2π~)24π ~2 ~ω r As one can see from its definition A (ω) changes most rapidly in the region where transitions near the Fermi xy surface are allowed. Let’s next consider transitions from hh+ to lh first in the lower range mlhE mlh h h < ± µ F − mhh 6 − 2 ωmlhE + mlh h + h: µ F mhh 6 2 1 A (ω;hh+ lh )= d(cosθ) ∞ dkf(θ,k)δ(~ω ∆E ), (A24) xy ± → ± Z−1 ZkFlh±(θ) − 10 with (~k)2 hcosθ h h 3 ∆E+± = 2µ ± 6cos2θ′ − 2 cosθ and kFlh±(θ)=kFlh(0) 1∓ 6EFr1− 4cos2θ!. The minimum of ∆E±(θ)+ at a fixed θ is then angular integration, θ˜, obtained by setting k = kFlh± so for hh+ to lh ∆E+±(θ) = 2~µ2kFlh(0)2±ξ6hccooss2θθ − h2 cosθ,(A25) ± ′ m h 3 h where we have defined ξ = m /µ+1=m /m , and ~ω˜ lh 1 cos2θ˜+ cosθ˜=0 lh lh hh − ∓m 3 − 4 2 the absolute minimum is given by hh r (A26) ∆E+±(θmin =0)= 2~µ2kFlh(0)2± mmlh h6 − h2. hh whose solution is For hh to lh we have instead − ± ∆E−±(θ) = 2~µ2kFlh(0)2± mmhlhh6hccooss2θθ′ + h2 cosθ, cosθ˜ = −~ω˜± 3ξ h2(1+ ξ32)−3~2ω˜2 1 ∆(A,27) ∆E−±(θmin =π) = 2~µ2kFlh(0)2± mmhlhhh6 − h2. ± qh2(1+ ξ3) ≡ − ± Let ~ω = ~2klh(0)2+ω˜ where ~ω˜ will be of the order of A similar procedure for the transitions from hh to lh h. For an ω2˜µthFat is too small there will be a limit on the yield cosθ˜ = 1+∆ . − ± ± − ∓ Combining the contributions for each transition we then obtain for m E /µ (h/6)m /m h/2 < ω < lh F lh hh − − m E /µ+(h/6)m /m +h/2 lh F lh hh 1 1 A (ω) = d(cosθ)A (ω,cosθ;hh+ lh+)+ d(cosθ)A (ω,cosθ;hh+ lh ) xy xy xy → → − Z1−∆+ Z1−∆− −1+∆− −1+∆+ + d(cosθ)A (ω,cosθ;hh lh+)+ d(cosθ)A (ω,cosθ;hh lh ) xy xy −→ −→ − Z−1 Z−1 e2 2µω/~ 1 3 h 1 = d(cosθ) cosθcos2θ + [ sin(2θ)sin(2θ )+2cos2θcos2θ − (2π)2~ 4 ′ 4~ω −2 ′ ′ p Z1−∆+ (cid:18) cos2θcos2θ 3 + ′ cos2θcos2θ ] ′ 4cos2θ − 4 ′ (cid:19) e2 2µω/~ 1 3 h 1 d(cosθ) cosθsin2θ + [+ sin(2θ)sin(2θ )+2cos2θsin2θ − (2π)2~ 4 ′ 4~ω 2 ′ ′ p Z1−∆− (cid:18) cos2θsin2θ 3 ′ cos2θsin2θ ] ′ − 4cos2θ − 4 ′ (cid:19) e2 2µω/~ 3 h 7 1 u 3 3 √3 1−∆+ = u2+ [ u3 2u] + 1 u2+ u arcsin(√3u/2) − p(2π)2~ "(cid:18)8 4~ω 6 − (cid:19)(cid:12)(cid:12)1−∆+ 8 r − 4 2 !− 12 !(cid:12)(cid:12)(cid:12)1−∆− h 7u3 3 7 1(cid:12)(cid:12)3u2 1−∆+ (cid:12)(cid:12) + u+ + 1 u2 + . 4~ω − 12 r − 4 (cid:18)27 18 (cid:19)!(cid:12)(cid:12)1 ∆− (cid:12) − (cid:12)  (cid:12) A similar procedure for the upper range h/6+(h/2)m /m yields A (ω) given in Eq. 11. For hh lh xy m E /mu h/6 (h/2)m /m <ω<m E /mu + hh F hh lh hh F − −