Symmetry analysis of second-harmonic generation at surfaces of antiferromagnets

Transcription

1 PHYSICAL REVIEW B VOLUME 60, NUMBER 2 1 JULY 1999-II Symmetry anaysis f secnd-harmnic generatin at surfaces f antiferrmagnets M. Trzeciecki Max-Panck-Institut für Mikrstrukturphysik, Weinberg 2, D Hae, Germany and Institute f Physics, Warsaw University f Techngy, Kszykwa 75, Warsaw, Pand A. Dähn Institute fr Theretica Physics, Freie Universität Berin, Arnimaee 14, D Berin, Germany W. Hübner Max-Panck-Institut für Mikrstrukturphysik, Weinberg 2, D Hae, Germany Received 22 Apri 1998; revised manuscript received 24 August 1998 Using grup thery we cassify the nninear magnet-ptica respnse at w-index surfaces f fcc antiferrmagnets, such as NiO. Structures cnsisting f ne atmic ayer are discussed in detai. We find that ptica secnd-harmnic generatin is sensitive t surface antiferrmagnetism in many cases. We discuss the infuence f a secnd type f magnetic atms, and as f a pssibe xygen subattice distrtin n the utput signa. Finay, ur symmetry anaysis yieds the pssibiity f antiferrmagnetic surface dmain imaging even in the presence f magnetic unit-ce dubing. S I. INTRODUCTION Optica secnd-harmnic generatin SHG has been prven t be a very usefu technique fr the investigatin f ferrmagnetism at surfaces. The bvius questin is if this technique can as yied sme new infrmatin in the case f mre genera spin cnfiguratins, such as antiferrmagnetic AF rdering. An experimenta answer t this questin has been prvided by Fiebig et a., 1 wh btained a prnunced ptica cntrast frm AF 180 dmains f rhmbhedra buk Cr 2 O 3. The authrs attributed this cntrast t the interference f magnetic and eectric dipe cntributins, the atter being present ny bew the Née temperature. Since it is knwn that, in cubic materias, within the eectric dipe apprximatin, ptica SHG riginates ny frm surfaces, interfaces, r thin fims, an imprtant questin is if SHG is as sensitive t antiferrmagnetism at surfaces f cubic antiferrmagnets. In this paper, we wi shw that the surface f a cubic materia can wer the symmetry f an AF fcc crysta tw-subattice antiferrmagnet in a way simiar t the trigna distrtin in a fur subattice antiferrmagnet Cr 2 O 3. Besides, even the imaging f AF dmains is pssibe as fr many cubic materias that exhibit unit-ce dubing. The first theretica expanatin f inear magnet-ptic effects in ferrmagnets has been given by Argyres 2 in the 1950s. He used inear-respnse thery fr current-current crreatin functins. His micrscpic expanatin was aready based n the cmbinatin f spin-rbit and exchange cuping. Experimenta techniques fr the detectin f AF dmain was using inear ptics in sme specia gemetries were eabrated a few years ater. 3 The interir f the dmains has been visuaized in piezeectric AF crystas using a inear magnet-ptica effect. 4 Hwever, inear ptica experiments suffer frm mixing the desired signa with a cntributin frm ther inear effects, such as birefringence r dichrism. A review f inear ptica experimenta methds fr the investigatin f AF dmains is given by Din. 5 The bservatin f dmain structure in antiferrmagnets is mre cmpicated than in ferrmagnetic materias since the reductin f the spatia symmetry is, unike fr ferrmagnets, nt inked t an imbaance in the ccupatin f majrity- and minrity-spin states. On the basis f gruptheretica cnsideratins, Brwn et a. 6 prpsed the use f inear ptica effects, namey gyrtrpic birefringence, fr the bservatin f AF dmains reated t each ther by the space-inversin peratin. A theretica review f effects fund by a grup-theretica apprach is presented by Eremenk and Kharchenk. 7 They perfrmed a cmprehensive study f inear ptica effects fr varius AF materias. Anther effect prpsed recenty by Dzyashinskii et a. 8 gives the pssibiity f detecting antiferrmagnetism taking advantage frm ptica path differences frm antiferrmagneticay cuped but intrinsicay ferrmagnetic panes. Nninear ptics exhibits an additina degree f freedm, since its eementary prcess invves three phtns instead f tw in inear ptics. Fr that reasn, sme authrs, e.g., Fröhich 9 suggested the appicatin f nninear ptics even fr k-seective spectrscpy, since mutiphtn phenmena aw fr the scanning f a sma part f the Briuin zne, at east fr semicnductrs. Recenty, nninear ptics has attracted mre and mre attentin fr the investigatin f magnetism due t its enhanced sensitivity t twdimensina ferrmagnetism. 10 The magnetic effects are usuay much strnger than in inear ptics rtatins up t 90, prnunced spin parized quantum we state sciatins, 11,12 magnetic cntrasts cse t 100%). 13,14 An exampe f ferrmagnetic effects measurabe ny by SHG deas with the existence f surface magnetism in very thin fims f Fe/Cu 001 and is given in Ref. 15. Nninear ptica effects were invked t expain the behavir f asers in magnetic fieds, 16 t investigate high-temperature supercnductrs, 17,18 and t study structures cmpsed frm aternatey ferr- and antiferrmagneticay rdered thin fims. 19 One theretica investigatin f the pssibiity t ap /99/60 2 / /$15.00 PRB The American Physica Sciety

2 PRB 60 SYMMETRY ANALYSIS OF SECOND-HARMONIC py nninear ptics t antiferrmagnetism was perfrmed by Kieich and Zawdny. 20 Hwever, the experiments cncerning the detectin f the AF dmains in materias such as Cr 2 O 3 were carried ut ny recenty. 21,22 In the 1970s, it was prpsed 23 that experimenta studies f dc magnetic and eectric-fied-induced SHG cud becme an effective methd f determining the crysta structure f sids, the symmetry f which cannt be investigated by ther methds. Extending this idea twards surface crystagraphy prvides us with a technique fr determining the spin cnfiguratin in a given surface structure. In turn, it permits us t use a knwn magnetic cnfiguratin fr the determinatin f the surface structure. A the mentined effects are mre difficut r even impssibe t btain in inear ptics, and mrever ther inear methds ike neutrn scattering have difficuties in prbing AF spin cnfiguratins. (2 ) The nninear magnet-ptica susceptibiity tensr e the surce fr SHG within the eectric dipe apprximatin has predminanty been investigated frm the symmetry pint f view. A cassificatin fwing this apprach, with tensrs f a rank up t six, has been perfrmed by Lyubchanskii et a ,13,27 In Ref. 13 the authrs incude the magnetizatin-gradient terms and appy the gruptheretica cassificatin t higher-rank susceptibiity tensrs. This apprach then aws them t study the thickness and the character Bch vs Née type f dmain was. An attempt by Muthukumar et a. 28 t cacuate the (2 ) e tensr eements fr the antiferrmagnetic Cr 2 O 3 bth frm grup thery as we as frm the micrscpic pint f view is rather unique. They impemented a (CrO 6 ) 2 custer, thus taking int accunt ny haf f the spins present in the eementary magnetic ce. In this apprximatin they expained the SHG frm Cr 2 O 3 as bserved by Fiebig et a. 1 and they were abe t give a quantitative estimate fr that. Tanabe et a., 29 hwever, pinted ut that the ccurrence f purey rea r imaginary vaues f the tensr eements pays a decisive re fr the existence f SHG frm this substance. They fund that fr a (CrO 6 ) 2 custer SHG can take pace ny in the case where the tensr eements are imaginary, and thus shud vanish in Muthukumar s apprximatin. They prpsed t take int accunt the fu unit ce with fur inequivaent Cr ins incuding their twisting interactin with the envirnment. Hwever, Tanabe et a. negected the dissipatin in the prcess f SHG, 30 which is a rather crude apprximatin. In (2 ) genera, taking int accunt the dissipatin makes the e tensr eements cmpex and invaidates their setin in purey rea and imaginary nes. 31 Lifting the inversin symmetry f a crysta is the surce fr SHG. Lyubchanskii et a. 24,26 suggested crysta attice defrmatins and dispacements as pssibe reasns fr SHG frm YIG fims. In the case f Cr 2 O 3 and YBa 2 Cu 3 O 6, described by Lyubchanskii et a., 25,26 AF rdering wers the symmetry f an therwise centrsymmetric crysta. In this paper, hwever, we rey n the idea that, rather than wering the crysta symmetry in the buk, SHG may as resut frm the breaking f inversin symmetry at the surface f a buk inversin-symmetric system. Magneticay active xide ayers are f imprtance fr the cnstructin f tunneing magnetresistance TMR devices, where a triayer structure is cmmny used. The centra ayer f TMR devices cnsists f an xide sandwiched between a sft and a hard magnetic ayer these tw ayers are ften cmpsed frm the same materia but f different thicknesses. Fr these techngica appicatins it is necessary t devep a technique t study buried xide interfaces. Such a technique can be SHG. One f the mst prmising materias fr the mentined devices is NiO. Hwever, t the best f ur knwedge, the understanding f its detaied spin structure is scarce even the spin rientatin n the ferrmagneticay rdered 111 surfaces is nt knwn. The technique presented here can shed sme ight n that issue. Our paper is rganized as fws: in Sec. II we present ur methds fr btaining sets f nnvanishing (2 ) e tensr eements. In Sec. III we present the resuts f ur anaysis, first fr the nndistrted surface f a simpe fcc structure Sec. III A, then fr the the distrted ne III B. Subsequenty, we discuss the infuence f a secnd kind f magnetic atms III C and f xygen subattice distrtin III D. The issue f dmain imaging is addressed in Sec. III E. Pssibe experimenta gemetries awing fr the detectin f the mentined structures and effects are discussed in Sec. IV. The cncusins are presented in Sec. V. II. THEORY Based n grup thery, Dähn et a. 32 prpsed a nninear magnet-ptic Kerr effect NOLIMOKE at the surface f cubic antiferrmagnets. They as gave an exampe f an antiferrmagnetic structure NiO and an ptica cnfiguratin, where this effect cud be bserved. Here, we perfrm a cmpete grup-thery based anaysis f cinear AF fcc w-index crysta surfaces. Surfaces f ther crysta structures are as we described by ur thery prvided they are simiar t fcc crysta surfaces, i.e., squares r hexagns. The resuts can be used t detect the magnetic rder f a specific surface under investigatin and aw fr the determinatin f the surface spin cnfiguratin in sme imprtant cases. Hwever, in rder t cacuate the SHG yied quantitativey, it is necessary t g beynd the present study and use eectrnic cacuatins f the nninear susceptibiity. Grup thery can give a unified picture f different experimenta bservatins and predict new effects, 33 whie the micrscpic rigins f the bserved phenmena may remain uncear. In rder t be cear with respect t the essentia ntin f time reversa we wud ike t emphasize the pint f view taken in this paper in the beginning. Here, we d nt divide (2 ) e int even and dd parts in the magnetic rder meter. Instead, the behavir f (2 ) e with respect t the magnetic rder meter which fr ferrmagnetic materias crrespnds t the dependence f (2 ) e n magnetizatin is fuy taken int accunt by the cnsideratins f the magnetic pint grup. At n stage f ur cnsideratin d we invke the ntin f time reversa, cnsequenty we d nt appy the characterizatin f the susceptibiity (2 ) as the c tensr changing its sign in the time-reversa peratin r i tensr invariant under the time-reversa peratin. 31 Befre we start ur grup-theretica cassificatin f the nninear ptica susceptibiities f AF surfaces we wud ike t emphasize the fwing fur imprtant pints: i We are nt interested in effects resuting frm the ptica path difference frm adjacent crysta panes which are ferrmagneticay rdered but ny antiferrmagneticay

3 1146 M. TRZECIECKI, A. DÄHN, AND W. HÜBNER PRB 60 cuped t each ther. We d nt cnsider this as an intrinsic AF effect. ii Cubic crystas that we are interested in revea a center f inversin in the -, ferr-, and a antiferrmagnetic phases. Thus, within the eectric dipe apprximatin, the SHG signa frm the buk vanishes. iii Whie, in principe, inear ptica methds can be sensitive t the presence f a spin structure, in practice they are nt usefu because, within the grup-theretica apprach, they cannt distinguish the AF phase frm either magnetic r ferrmagnetic, nr can they distinguish different AF cnfiguratins frm each ther. They have t resrt t methds ike ine-shape anaysis, where n strng statements characteristic fr symmetry anaysis can be made. iv Athugh the tensr eements fr a the magnetic pint grups are knwn and tabuated in the iterature e.g., Ref. 34, the cnnectin between the different spin cnfiguratins described by us and the mentined symmetry grups has nt been made, except fr sme easy cases. 32 Thus, fr SHG frm antiferrmagnetic surfaces there has been up t nw n cnnectin between the grup-theretica cassificatin and the rea situatins fund in experiments. The fwing part f the text shud expain the fundamentas f appying NOLIMOKE bservatins t investigate antiferrmagnetism f surfaces. Nw we turn t SHG, the surce f which is the nninear eectrica parizatin P (2 ) e given by P (2 ) e 0 (2 ) e :E ( ) E ( ). 1 Here, E ( ) is the eectric fied f the incident ight, whie (2 ) e dentes the nninear susceptibiity within the eectric dipe apprximatin, and 0 is the vacuum permittivity. The intensity f the utging SHG ight is 35 2 F,,2 xxx xyy xzz xyz xzx xxy I (2 ) I 0 yxx yyy yzz yyz yzx yxy, 2 zxx zyy zzz zyz zzx zxy f,, 2 where I 0 is the intensity f the incident ight, F() f() describe Fresne and gemetrica factrs fr the incident refected ight, and anges f incidence and refectin, respectivey ( ), and ( ) is utput input parizatin ange. Accrding t Neumann s principe, any type f symmetry which is exhibited by the crysta is pssessed by every physica prperty f the crysta. 34 T examine these physica prperties, we determine the magnetic pint grup f the crysta attice, thus determine its symmetries. The same symmetries must eave the investigated prperty tensr in ur case the nninear eectric susceptibiity (2 ) e ) invariant. This fact is mathematicay expressed by the fwing cnditin: (2 ) e,i j k i i j j k k (2 ) e,ijk, i,j,k,i,j,k x,y,z. 3 Here, n,n (n i, j,k,n i, j,k,) is a representatin f an eement f the magnetic pint grup describing the crysta. Fr symmetry peratins incuding the time reversa there shud be an additina sign in Eq. 3, but we d nt use it here since we excude the time reversa frm ur cnsideratin. In particuar, frm Eq. 3 it fws immediatey that par tensrs f dd rank such as (2 ) e ) vanish in inversin symmetric structures. This expains why SHG is pssibe ny at surfaces and interfaces, where this symmetry is brken. Fr a given spin cnfiguratin we appy Eq. 3 fr every symmetry peratin exhibited by the system. Thus, each f these symmetries gives rise t a set f 27 equatins with 27 unknwn eements f the tensr (2 ) e. This set can be reduced t 18 equatins, since (2 ) e,ijk (2 ) e,ikj, 4 which expresses the equivaence f the incident phtns f frequency, see as the reduced ntatin in Eq. 2. The anaytic sutin f even this reduced set f equatins seems cumbersme, but the set can be spit int severa decuped subsets. Fr exampe, an bvius subset in every case is the equatin zzz zzz, this tensr eement ccurs nwhere ese. The rank f ther subsets is, fr ur cases, never higher than six. In this manner, ne may btain a set f frbidden eements f the susceptibiity tensr as we as reatins between existing nes. III. RESULTS First, we wi define the ntins f phase, case, and cnfiguratin, used hencefrth t cassify ur resuts. Phase describes the magnetic phase f the materia, i.e., magnetic, ferrmagnetic, r AF. Secndy, the wrd cnfiguratin is reserved fr the descriptin f the magnetic rdering f the surface. It describes varius pssibiities f the spin rdering, which are different in the sense f tpgy. We describe up t 18 AF cnfiguratins, dented by itte etters a t r, as we as severa ferrmagnetic cnfiguratins, dented as, ferr2, etc. The number f pssibe cnfiguratins varies depending n surface rientatin. Thirdy, we describe different cases, i.e., additina structura features superimpsed n the symmetry anaysis. Case A des nt have such additina features. In case B we address distrtins f the attice. Case C deas with tw kinds f magnetic atms in an undistrted attice. In case D we take int accunt a distrted subattice f nnmagnetic atms, keeping the magnetic subattice undistrted. A the anaysis cncerns cinear antiferrmagnets, with ne easy axis. The tabes shw the SHG respnse types fr each cnfiguratin. The varius respnse types are encded by a key, which is then decded in Tabe I. This tabe presents

4 PRB 60 SYMMETRY ANALYSIS OF SECOND-HARMONIC TABLE I. Detais f SHG respnse types. We dente (2 ) ijk by ijk. Odd eements are in bd if a dmain peratin exists. Key Pint grup Symmetry peratins Dmain peratin Nnvanishing tensr eements a 4mm 1,2 z, 4 z,2 x,2 y,2 xy,2 xy xxz xzx yyz yzy, zxx zyy, zzz b m 1,2 x 2 z,2 y xzx xxz, xxy xyx, yxx, yyy, yzz, yyz yzy, zxx, zyy, zzz, zyz zzy 4 z,2 xy n infrmatin abut the parity c m 1,2 xy 2 z,2 xy xxx yyy, xyy yxx, xzz yzz, xyz yxz xzy yzx, xxz xzx yyz yzy, xxy yyx xyx yxy, zxx zyy, zzz, zxz zzx zyz zzy, zxy zyx 4 z,2 y xxx yyy, xyy yxx, xzz yzz, xyz xzy yxz yzx, xxz xzx yyz yzy, xxy yyx xyx yxy, zxx zyy, zzz, zxz zzx zyz zzy, zxy zyx d 4 1,2 z, 4 z 2 x,2 y,2 xy,2 xy xyz xzy yxz yzx, xzx xxz yzy yyz, zxx zyy, zzz e mm2 1,2 z,2 x,2 y 4 z,2 xy,2 xy xxz xzx, yyz yzy, zxx, zyy, zzz f 2 1,2 z 2 x,2 y xyz xzy, xxz xzx, yyz yzy, yzx yxz, zxx, zyy, zzz, zxy zyx 4 z,2 xy,2 xy xyz xzy, xxz xzx, yyz yzy, yzx yxz, zxx, zyy, zzz, zxy zyx g mm2 1,2 z,2 xy,2 xy 4 z,2 x,2 y xxz xzx yyz yzy, xzy xyz yzx yxz, zxx zyy, zzz, zxy zyx h m 1,2 y 2 z,2 x xxx, xyy, xzz, xxz xzx, yyz yzy, yyx yxy, zxx, zzz, zzx zxz 4 z,2 xy xxx, xyy, xzz, xxz xzx, yyz yzy, yyx yxy, zxx, zzz, zzx zxz i z A the eements are awed: xxx, xyy, xzz, xyz xzy, xzx xxz, xxy xyx, yxx, yyy, yzz, yyz yzy, yzx yxz, yxy yyx, zxx, zyy, zzz, zyz zzy, zzx zxz, zxy zyx 2 x xxx, xyy, xzz, xyz xzy, xzx xxz, xxy xyx, yxx, yyy, yzz, yyz yzy, yzx yxz, yxy yyx, zxx, zyy, zzz, zyz zzy, zzx zxz, zxy zyx 4 z,2 xy,2 xy n infrmatin abut the parity j m 1,2 xy 2 z,2 xy xxx yyy, xyy yxx, xzz yzz, xyz yxz xzy yzx, xxz xzx yyz yzy, xxy yyx xyx yxy, zxx zyy, zzz, zxz zzx zyz zzy, zxy zyx 4 z,2 y xxx yyy,xyy yxx,xzz yzz, xyz yxz xzy yzx, xxz xzx yyz yzy, xxy xyx yyx yxy, zxx zyy, zzz, zxz zzx zyz zzy,zxy zyx k mm2 1,2 z,2 x,2 y xxz xzx, yyz yzy, zxx, zyy, zzz m 1,2 x 2 z,2 y xzx xxz, xxy xyx, yxx, yyy, yzz, yyz yzy, zxx, zyy, zzz, zyz zzy m z A the eements are awed: xxx, xyy, xzz, xyz xzy, xzx xxz, xxy xyx, yxx, yyy, yzz, yyz yzy, yzx yxz, yxy yyx, zxx, zyy, zzz, zyz zzy, zzx zxz, zxy zyx 2 x xxx, xyy, xzz, xyz xzy, xzx xxz, xxy xyx, yxx, yyy, yzz, yyz yzy, yzx yxz, yxy yyx, zxx, zyy, zzz, zyz zzy, zzx zxz, zxy zyx

5 1148 M. TRZECIECKI, A. DÄHN, AND W. HÜBNER PRB 60 TABLE I. Cntinued. Key Pint grup Symmetry peratins Dmain peratin Nnvanishing tensr eements n 2 1,2 z 2 x,2 y xyz xzy, xxz xzx, yyz yzy, yzx yxz, zxx, zyy, zzz, zxy zyx m 1,2 y 2 z,2 x xxx, xyy, xzz, xxz xzx, yyz yzy, yyx yxy, zxx, zyy, zzz, zzx zxz p 6mm 1,2 z, 3 z, 6 z,6(2 ) xxz xzx yyz yzy, zxx zyy, zzz q 6 1,2 z, 3 z, 6 z 2 x,2 y xyz xzy yxz yzx, xxz xzx yyz yzy, zxx zyy, zzz r 3m 1, 3 z,2 y,2 S(xy),2 S( xy) zxx zyy, xxz xzx yyz yzy, zzz, xxx xyy yxy yyx s y A the eements are awed: xxx, xyy, xzz, xyz xzy, xzx xxz, xxy xyx, yxx, yyy, yzz, yyz yzy, yzx yxz, yxy yyx, zxx, zyy, zzz, zyz zzy, zzx zxz, zxy zyx t m 1,2 y xxx, xyy, xzz, xxz xzx, yyz yzy, yyx yxy, zxx, zyy, zzz, zzx zxz u 3 1, 3 z 2 y xxx xyy yxy yyx, xyz xzy yxz yzx, xzx xxz yyz yzy, xxy xyx yxx yyy, zxx zyy, zzz w 1 1 A the eements are awed the symmetries, dmain peratins, and nnvanishing tensr eements fr each respnse type. This is dne in rder t shrten the vera ength f tabes, because a given respnse type can appear in severa different cases. Severa spin structures depicted in Figs. 1 and 5 are distinct cnfiguratins ny in case B, and they are addressed in the tabes that cncern ny this case. Fr the rest f the cases they are dmains f ther, fuy described cnfiguratins, thus they are eft ut in these cases. The phisphy f the paper is that, t save sme space, we shw the spin structure in ne figure fr each surface Figs. 1, 4, and 5 fr a the fur cases A D, and depict the effects taken int accunt in cases B D ny fr the magnetic phase Figs Tabe I as cntains the infrmatin n the parity f the nnvanishing tensr eements: the dd nes are printed in bdface. In sme situatins an even tensr eement shwn in ightface is equa t an dd eement shwn in bdface, this means that this pair f tensr eements is equa in the dmain which is depicted n the crrespnding figure, but they are f ppsite sign in the ther dmain. This happens in the structures where tw pairs f dmains are pssibe tw distinct entries in Tabe I. The tensr eements that change their parity in the dmain peratin which is the inverse f the dispayed ne are shwn in itaic fnt. Fr exampe, entry j f Tabe I shws a tensr eement xxx, which is even under the peratin 4 z, this means that this tensr eement is dd under 4 z. This strange at the first sight behavir f tensr eements is caused by the fact that under these peratins, tensr eements are nt mapped n themseves. In ur exampe, after appying 4 z the tensr eement xxx becmes yyy, withut changing its sign. If we nw appy 4 z, yyy which is nw even becmes xxx, again withut changing the sign. The parity f the eements has been checked in the peratins 2 z, 4 z, and in the peratin cnnecting mirrrdmains t each ther fr the definitin f the mirrrdmain structure see Sec. III E. The dmain peratin s n which the parity depends is are, if appicabe, as dispayed in this tabe. If tw r mre dmain peratins have the same effect, we dispay a f them tgether. T make the Tabe I shrter and mre easiy readabe sme dmain peratins and the crrespnding parity infrmatin fr the tensr eements are nt dispayed, namey thse that can be created by a superpsitin f the dispayed dmain peratins. We as d nt address the parity f tensr eements in the 6 z nr 3 z peratins fr 111 surfaces nr any ther peratin that spits tensr eements, athugh these peratins as ead t a dmain structure. 36 As wi be discussed ater Sec. III E it is pssibe t define a parity f the tensr eements fr the 3 z and 6 z peratins, hwever the tensr eements then underg mre cmpicated changes. The situatins where the parity f the tensr eements is t cmpicated t be dispayed in the tabe are indicated by a hyphen in the cumn dmain peratin. Fr sme cnfiguratins, nne f the peratins eads t a dmain structure in thse cnfiguratins we dispay the infrmatin ne dmain. The reader is referred t the Appendix fr the particuarities f the parity check. As far as the first ayer is cncerned, we address a the spin cnfiguratins f the w index surfaces f fcc antiferrmagnets, with magnetic rder vectr ying in pane r perpendicuar t it and antiferrmagnetic cuping between nearest neighbrs. Fr the 001 surfaces we as discuss the cnfiguratins, where the antiferrmagnetic cuping exists

6 PRB 60 SYMMETRY ANALYSIS OF SECOND-HARMONIC FIG. 3. Spin structure f an antiferrmagnetic 001 biayer cnstructed frm a shift f the mnayer ang the psitive x(y) axis. Fied empty circes represent the tpmst secnd ayer. On the right-hand side the cnventina unit ces fr the resuting biayer structure are presented. Here, cnf a f the 001 mnayer serves as an exampe. FIG. 1. Spin cnfiguratins f an fcc 001 surface. Except fr cnfs ferr4 and r, the arrws aways indicate in-pane directins f the spins. In cnfs ferr4 and r ( ) dente spins pinting ang the psitive negative z directin, respectivey. between the secnd-nearest neighbrs cnfiguratins a, b, c, f, and, ang with d, g, and h fr case B. Wednt cnsider the cuping t the third and further neighbrs. This wud nt give rise t cnfiguratins f different symmetries in tw dimensins. It may at mst repace spins by grains bcks f spins in the cnfiguratins described by us. Thrughut this paper we take int accunt the spin structure ny f the first uppermst atmic ayer. This is sufficient t study a the symmetries f 001 and 110 surfaces bth in the magnetic and ferrmagnetic phases. Fr the 111 surface it is necessary t recgnize the atmic psitins but nt the spins in the secnd ayer fr the same purpse. Fr the sake f cmpeteness we as present a study f 111 surfaces withut this extensin. Hwever, in the antiferrmagnetic phase, the spin structure f the secnd and deeper ayers pays a re in determining the symmetry f the surface. This is presented in this paper using the 001 surface as an exampe. Fr the 110 and 111 surfaces it wi be pubished esewhere. 37 These structures can serve as simpe mdes fr deriving predictins fr mre cmpicated cases, whie the fu cnsideratin f the secnd ayer wud nt bring any new interesting resuts. Taking int accunt the spin structure f the secnd ayer deeper ayers d nt bring up anything new t the anaysis resuts in creating severa up t tw fr the 001 surface and three fr the 111 surface cnfiguratins ut f each ne addressed here by us. The symmetry f these cnfiguratins may remain the same r be wered smetimes even bew the symmetry f the ferrmagnetic phase with respect t the tw-dimensina cnfiguratins they are generated frm. Cnsequenty the distinctin f the cnfiguratins frm each ther may be imited, but the pssibiity f detecting the magnetic phase is nt severey affected. As ur remarks n dmain imaging remain vaid. Hwever the number f dmains is increased, thus the pssibiity t identify each f them might be hampered. Cnsequenty, ne can state that the symmetry f an AF surface depends n tw atmic ayers. They are as necessary and sufficient t define AF buk dmains. As wi be presented in ur resuts, SHG can prbe bth these ayers n AF surfaces. A. Equivaent atms The predicted nninear magnet-ptica effects resut frm the fact that the magnetic pint grups f antiferrmagnetic cnfiguratins are different frm thse describing - FIG. 2. Tp view f a spin structure n a 001 surface. The dashed ine depicts a cnventina unit ce, whie the sid ne utines the primitive unit ce. FIG. 4. Spin cnfiguratins f an fcc 110 surface. Except fr cnfs, g, h, and i, the arrws aways indicate in-pane directins f the spins. In cnfs, g, h, and i ( ) dente spins pinting ang the psitive negative z directin, respectivey.

7 1150 M. TRZECIECKI, A. DÄHN, AND W. HÜBNER PRB 60 FIG. 7. Surface structure f the nnequivaent magnetic atms case in the magnetic phase. Pictures present the 001, 110, and 111 surfaces, respectivey. Fied and empty circes represent the tw kinds f magnetic atms. Nte, the fragment representing the 111 surface des nt shw the cnventina unit ce but a bigger set f atms in rder t give a cear idea abut the surface structure. FIG. 5. Spin cnfiguratins f an fcc 111 surface. Except fr cnfs ferr5, k,, and m, the arrws aways indicate in-pane directins f the spins. In cnfs ferr5, k,, and m ( ) dente spins pinting ang the psitive negative z directin, respectivey. magnetic r ferrmagnetic phases f the same surface. Since, depending n the magnetic phase, different tensr eements vanish, it is pssibe t detect antiferrmagnetism pticay by varying the parizatin f the incming ight. The current subsectin discusses nnvanishing eements f the nninear susceptibiity tensr fr an fcc crysta cnsisting f ny ne kind f magnetic atms. The infuence f nnmagnetic atms in the materia wi be discussed ater. The cnfiguratins cnsidered here are, ferr2, ferr4, a, b, c, e, f, i, k, m,, p, and r fr the 001 surface see Fig. 1,,, ferr5, a, c, f, i, and k fr the 111 surface see Fig. 5, and a cnfiguratins depicted in Fig. 4 fr the 110 surface. Other depicted spin structures frm dmains f these cnfiguratins and are nt referred t in this subsectin nr in the tabes cncerning the current subsectin. 38 A pssibe cnfiguratins cnfs f a fcc 001 surface are shwn in Fig. 1, which dispays the cnventina rather than magnetic unit ces. Hwever, these are sufficient t fix the spin cnfiguratin f the whe surface impsing f the fwing cnventin : the fcc surface is cnstructed frm the depicted paquette in the way that neighbring spins ang the x and y directins pint the same way aternate if they are e antie n the paquette in these tw directins. The spins in rws and cumns where ny ne spin is presented are cntinued in the same way as the crner spins. Fr instance, in cnfiguratin a f the 001 surface, bth the right-hand side and eft-hand side neighbrs f the centra spin wi pint upwards, whie the spin directin wi be aternated ang the x axis. This cnventin wi be maintained hencefrth fr a 111 surface ne has t ater r keep the spins ang three axes, instead f tw. The smaest set that gives a cmpete idea abut the spin structure is presented in Fig. 2; 39 this magnetic primitive ce des nt give a cear picture f the crysta symmetries, hwever. The whe crysta attice can be reprduced by transatins f this ce, withut perfrming ther peratins such as refectins r rtatins. The SHG respnse types fr the 001 mnayer are given in Tabe II, fr the magnetic, ferrmagnetic, and a AF phases. We can bserve severa sets f awed tensr eements. r wi prduce the same signa as the magnetic phase. reveas a cmpetey different, distinguishabe set f tensr eements. In additin, cnf ferr2 prduces anther set f tensr eements, different frm any ther cnfiguratin. It is equivaent t the cnf rtated by 45. In cnfs a, b, e, and we find the same tensr eements as fr the magnetic phase. Hwever, due t the wer symmetry, their vaues are n nger reated t each ther. s c and f bring new tensr eements, thus awing fr the distinctin f these cnfs frm the previus nes. s i, k, m, p revea the same tensr eements as c and f but sme f these eements are reated. Thus ne may pssiby distinguish these tw sets f cnfiguratins. s ferr4 presents a cmpetey different, distinguishabe set f the nnvanishing tensr eements. Cnsequenty, in six cnfiguratins i.e., c, f, i, k, m, and p sme susceptibiity tensr eements appear ny in the AF phase, awing fr the detectin f this phase by varying the incident ight parizatin, as wi be utined in Sec. IV. In additin, a ther antiferrmagnetic cnfiguratins but r revea the breakdwn f sme f the reatins between the different tensr eements, cmpared t the - FIG. 6. Structure f the 001 and 111 surfaces f a fcc crysta with a rhmbhedra distrtin in the magnetic phase. Nte the changed rientatin f the crdinate system fr the 001 surface. FIG. 8. Surface structures f the case with a distrted xygen subattice white circes. Pictures present the magnetic phase f 001, 110, and 111 surfaces, respectivey. Nte, the fragment representing the 111 surface des nt shw the cnventina unit ce but a bigger set f atms in rder t give a cear idea abut the surface structure.

8 PRB 60 SYMMETRY ANALYSIS OF SECOND-HARMONIC TABLE II. SHG respnse fr a spin cnfiguratins f the 001 surface f a fcc attice Ref. 48. Fr the detaied descriptin f the respnse types see Tabe I. The cnfiguratins are depicted in Fig. 1. TABLE III. SHG respnse fr a spin cnfiguratins f the 001 surface f a fcc attice, with the spin structure f the secnd ayer taken int accunt. Fr the detaied descriptin f the respnse types see Tabe I. Fr the cnfiguratins see Fig. 1. a b ferr2 c ferr4 d a, b, e, e c, f f i, k, m, p g r a magnetic phase, and thus can be detected as we. Generay, a the phases can be distinguished frm each ther. There exists as we a pssibiity t distinguish different AF cnfiguratins prvided the crrespnding tensr eements can be singed ut by the prper chice f the experimenta gemetry. Fr the sake f cmpeteness, we nw present a shrt study f the 001 surface where the spin structure f the tw tpmst atmic ayers is taken int accunt. The magnetic phase and a the ferrmagnetic cnfiguratins remain unchanged with respect t the resuts f the previus graph fr the 001 mnayer. Hwever, mst f the AF cnfiguratins previusy addressed break up int tw different cnfiguratins smetimes even with a different symmetry. These cnfiguratins are cnstructed frm thse f the previus graph by assuming that the structure f the secnd atmic ayer is identica with that f the tpmst ne but shifted ang the psitive x axis indicated by x after the name f the rigina cnfiguratin r psitive y axis indicated by y after the name f the parent cnfiguratin in a prper way t frm a fcc structure; if ny ne cnfiguratin can be prduced in this way we use the name f the rigina ne. This cnstructin is depicted in Fig. 3, ang with the crrespnding cnventina unit ces fr the tw tpmst ayers f the AF fcc 001 surface. The resuting SHG respnse types are presented in Tabe III. In genera, seven types f respnse are pssibe. Firsty, the magnetic phase reveas a characteristic set f tensr eements. Thus it can be unambiguusy distinguished frm any ther magnetic phase. Secndy, cnfs, ax, x, bx, by, ex, and ey bring sme additina tensr eements int pay. The symmetry f cnfs ax and x is sighty different frm that f the rest f this grup, since the mirrr pane is rtated by 90 arund the z axis. A different set f tensr eements is brught up by cnfs ferr2, i, m, and p. The difference between the respnse yieded by cnf i and the ther cnfs in this grup, due t a sighty different symmetry, can be cmpensated by rtating the sampe by 90 arund the z axis. Anther, characteristic set f tensr eements is presented by cnf ferr4 ane. The fifth type f SHG respnse is given by cnfs ay, y, and r. Tensr eements, that d nt vanish in these cnfiguratins, are the same as fr the magnetic phase but sme reatins between them are brken due t a wer symmetry in the AF phase. s cx, fx, and a b ferr2 c ferr4 d ax, x h ay, y, r e bx, by, ex, ey b c, fx, fy i i j k f m, p c fy yied a tensr eements in an unreated way. The ast, characteristic type f respnse is presented by cnf k ane. Cnsequenty, the detectin pssibiities f an antiferrmagnetic biayer are sighty wrse than thse fr a mnayer. Especiay, a difficuty in distinguishing the ferrmagnetic phase frm the antiferrmagnetic ne may arise fr sme cnfiguratins where then the cmbinatin f SHG with inear magnet-ptics is definitey required. There exists a pssibiity t distinguish AF cnfiguratins frm each ther, simiary t the previus situatin. In mst cnfiguratins, the difference in terms f the SHG respnse between the biayer structure described here and the previusy addressed 001 mnayer can be detected. We nw turn t the 110 surface Fig. 4, which, in the magnetic phase, reveas a wer symmetry than the 001 surface. On the ther hand, the number f symmetry peratins in the AF cnfiguratins is cmbe t the 001 surface. In additin, as shwn in Tabe IV, the resuting SHG respnse types are nt very characteristic, s the detectin pssibiities fr this surface are very imited. In particuar, cnfs a, b, c, g, h, i, j, k, and give the same tensr eements as the magnetic phase. s d, e, f, and bring new tensr eements. Other ferrmagnetic cnfiguratins and ferr2 present different sets f new tensr eements, making these cnfiguratins distinguishabe frm the thers as we as frm each ther. ferr4 yieds a cmpetey different set f tensr eements, hwever this set is reated t the ne f cnf by 90 rtatin. The study f the 111 surface see Fig. 5 has t be seted in tw subcases, accrding t whether we take int accunt ny ne atmic mnayer r mre. In bth subcases, we cnsider the same cnfiguratins. The SHG respnse types fr the first subcase are isted in Tabe V, and fr the secnd subcase in Tabe VI. Fr the first subcase, cnfs a, i, and k revea the same tensr eements as the magnetic phase, hwever, due t the wer symmetry their vaues are nt reated t each ther. s c and f present new tensr eements. As fr the previus surfaces, the ferrmagnetic phase reveas cmpetey different sets f tensr eements, and the three ferrmagnetic cnfiguratins can be distinguished frm each ther since they bring differ-

9 1152 M. TRZECIECKI, A. DÄHN, AND W. HÜBNER PRB 60 TABLE IV. SHG respnse fr a spin cnfiguratins f the 110 surface f a fcc attice Ref. 48. Fr the detaied descriptin f the respnse types see Tabe I. The cnfiguratins are depicted in Fig. 4. TABLE VI. SHG respnse fr a spin cnfiguratins f the 111 surface f a fcc attice Ref. 48. Mre mnayers are taken int accunt. Fr the detaied descriptin f the respnse types see Tabe I. The cnfiguratins are depicted in Fig. 5. k ferr2 m n ferr4 a, b, c, g k d, e, f n ent tensr eements int pay. Unike fr the 110 surface, the axes x and y are nt tpgicay equivaent, and thus the fact that tensr eements f are reated t thse f by 90 rtatin des nt affect the pssibiity t distinguish these tw cnfiguratins. The ferrmagnetic cnf ferr5 brings up the same tensr eements as AF cnfs c and f, but the reatins between the eements are different. The secnd subcase mre ayers taken int accunt gives different sets f awed tensr eements cmpared t the first subcase fr each but the cnfiguratin. s a, i, k, and share the same set f awed tensr eements and can be easiy distinguished frm the magnetic phase. s c, f, and revea a tensr eements, with their vaues unreated. Simiary, cnf ferr5 presents anther, distinguishabe set f tensr eements. The pssibiity f distinguishing the magnetic phases is rather imited. The symmetry anaysis f nnvanishing tensr eements fr ferrmagnetic surfaces in the case A have been perfrmed by Pan et a. 10 Our anaysis yieds the same resuts, taking int accunt the crrectins made by Hübner and Bennemann. 40 B. Distrtins f mnatmic attice The rhmbhedra distrtin f the atmic attice, described here and shwn in Fig. 6, makes the x and y axes f the 001 surface inequivaent, even in the magnetic phase. On the 111 surface, the y axis is nt equivaent any nger t ther axes cnnecting the nearest neighbrs, TABLE V. SHG respnse fr a spin cnfiguratins f the 111 surface f a fcc attice Ref. 48. Ony ne mnayer is taken int accunt. Fr the detaied descriptin f the respnse types see Tabe I. The cnfiguratins are depicted in Fig. 5. p ferr5 q a, i, k k c, f n r s t ferr5 u a, i, k t c, f u namey S (xy) and S ( xy) fr the definitin f the S and H axes see Fig. 5, the magnetic cnf. These inequivaences f axes are the reasns fr the reductin f the number f symmetry peratins in the magnetic phase. Because f this reductin sme spin structures that previusy frmed different dmains f a singe cnfiguratin nw cannt be transfrmed int each ther and becme independent cnfiguratins. This happens fr amst a f the previusy addressed cnfiguratins f the 001 and 111 surfaces. Cnsequenty, a the depicted spin structures are in fact cnfiguratins, and are addressed in this subsectin. The resuting SHG respnse types fr the 001 surface are isted in Tabe VII. Fr this surface, ny tw f the ferrmagnetic cnfiguratins, namey and ferr2 can be easiy distinguished frm bth the magnetic as we as the antiferrmagnetic phases. These ferrmagnetic cnfiguratins can be as distinguished frm each ther. On the cntrary, a the AF cnfiguratins yied ny tw types f respnse, and in additin ne f them is equivaent t the respnse f the magnetic phase. Cnsequenty, it wi nt be pssibe t determine the surface spin structure, and the distinctin f the AF phase frm the magnetic ne can be successfuy perfrmed ny in cnfs a h and. Cmpared t the case A, there is an imprtant symmetry breaking fr mst cnfiguratins. Thus, the distinctin between the tw cases A and B is pssibe cmpare Tabes II and VII. A the 110 surfaces f an fcc crysta with a rhmbhedra distrtin are tpgraphicay equivaent t the 110 surface f the case A. The distrtin ny stretches the x r y axis, s the structure remains rectanguar. The anaysis f the 111 surface depicted in Fig. 6 in the subcase f ny ne mnayer reveas sets f symmetries very simiar t the 110 surface, as it fws frm Tabe VIII. In fact, the 111 surface f a fcc crysta with a rhmbhedra distrtin can be treated as tw rectanguar attices superimpsed n each ther. In turn, due t the distrtin, it is nt cnvenient any nger t describe the spin structures using S and H axes. The pssibiity f distinguishing AF cnfiguratins is very pr, and tw f the AF cnfiguratins a and k yied the same signa as the magnetic surface. In cnfs b j,, and m the AF phase can be distinguished frm the magnetic ne, but they give the same signa as cnf ferr5. ferr2 can be easiy distinguished since it reveas a characteristic set f

10 PRB 60 SYMMETRY ANALYSIS OF SECOND-HARMONIC TABLE VII. SHG respnse fr a spin cnfiguratins f the 001 surface f a fcc attice, distrted t a rhmbhedra structure. Fr a detaied descriptin f the respnse types see Tabe I. Fr the surface structure see Fig. 6, fr the spin cnfiguratins see Fig. 1. k m ferr2 ferr4 n a, b h, n i n, p r k a tensr eements. s and yied different sets f tensr eements, but they are reated t each ther by 90 rtatin. Mst f the cnfiguratins aw fr the distinctin f the cases A and B cmpare Tabes V and VIII. In the subcase f tw mnayers f the 111 surface, the symmetry is dramaticay reduced see Tabe IX. Even in the magnetic phase the grup f symmetries cnsists f ny ne nntrivia peratin, and this appears t ccur as in the AF cnfiguratins a, i, k, and. In a the ther cnfiguratins a tensr eements are awed due t the ack f any symmetry. Ony cnfs magnetic and ferr5 aw fr the unambiguus distinctin f the cases A and B cmpare Tabes VI and IX. Cnsequenty, this surface is nt very usefu t an anaysis f the magnetic structure, with the exceptin f stating the distrtin itsef. As the cncusin f the case f the distrted subattice f magnetic atms, the surfaces give extremey imited pssibiities t investigate the magnetic prperties. In ur further study, we wi imit urseves t attices f undistrted magnetic atms. TABLE VIII. SHG respnse fr a spin cnfiguratins f the 111 surface f a fcc attice, distrted t a rhmbhedra structure. Ony ne mnayer is taken int accunt. Fr the detaied descriptin f the respnse types see Tabe I. Fr the surface structure see Fig. 6, fr the spin cnfiguratins see Fig. 5. k, ferr4 ferr2 m ferr5 n a, k k b j,,m n TABLE IX. SHG respnse fr a spin cnfiguratins f the 111 surface f a fcc attice, distrted t a rhmbhedra structure. Mre mnayers are taken int accunt. Fr the detaied descriptin f the respnse types see Tabe I. Fr the surface structure see Fig. 6, fr the spin cnfiguratins see Fig. 5. t, ferr2, ferr4, ferr5 s t a, i, k s b h,j,,m t C. Structure with nnequivaent magnetic atms We assume nw that nt a the magnetic atms in the ce are equivaent. An exampe f such a structure is a materia cmpsed f tw magnetic eements, but as a situatin when the magnetic attice sites are inequivaent due t different bnds t a nnmagnetic subattice; distrtins f the subattice f nnmagnetic atms that preserve the center f tw-dimensina inversin prduce the same effect. Other distrtins f the subattice f nnmagnetic atms wi be discussed in Sec. III D. The magnetic mment at the distinguished psitins can be changed r nt this des nt affect the resuts btained by symmetry anaysis. The cnfiguratins cnsidered here are, ferr2, ferr4, a, b, c, e, f, i, k, m,, p, and r fr the 001 surface see Fig. 1,,, ferr5, a, c, f, i, and k fr the 111 surface see Fig. 5, and a cnfiguratins depicted in Fig. 4 fr the 110 surface. Other depicted spin structures frm dmains f these cnfiguratins and are nt referred t in this subsectin nr in the tabes cncerning the current subsectin. The structure is depicted in Fig. 7. Fr the sake f brevity, we shw the structure f the distinguished atms ny fr the magnetic phase. A the cnfiguratins are the same as in case A, fr a surface rientatins. The aready mentined cnventin f aternating r nt spin directins ang certain axes is appied regardess f the atm type. This aws us t btain the whe crysta surface frm the sma dispayed fragment. Our anaysis starts with the 001 surface f an fcc crysta. The SHG respnse types fr each cnfiguratin are isted in Tabe X. In genera, we can bserve seven types f respnse. The first f them is represented by the magnetic phase ane. The secnd type f respnse, exhibited by the ferrmagnetic and the AF a, b, e, cnfs, differs frm any ther type by sme tensr eements. s a and revea different tensr eements than the ther cnfiguratins frm the mentined grup. Hwever, the signa frm cnfs a and is the same as fr cnfs b, e, and if ne exchanges the axes x and y. Thus, if the directins f the spins cannt be determined by anther methd, cnfs a and cannt be distinguished frm b, e, and. The next type cnsists f cnf f and reveas a tensr eements, whie n reatins between them are enfrced by the symmetry anaysis. A cmpetey different type f respnse is presented by cnf c ane. Anther type, where cnfs i, m, and p beng, brings the same tensr eements as cnf c, but there exist mre reatins between the eements due t a higher symmetry in these cnfiguratins. The next type is given by cnfs ferr2 and k. As in cnf f a the tensr eements are present but this time there are sme reatins between them. In ad-

11 1154 M. TRZECIECKI, A. DÄHN, AND W. HÜBNER PRB 60 TABLE X. SHG respnse fr a spin cnfiguratins f the 001 surface f a fcc attice, with ne atm distinguished. Fr the detaied descriptin f the respnse types see Tabe I. Fr the surface arrangement see Fig. 7. Fr the cnfiguratins see Fig. 1. TABLE XI. SHG respnse fr a spin cnfiguratins f the 110 surface f a fcc attice, with ne atm distinguished. Fr detaied descriptin f respnse types see Tabe I. Fr the surface arrangement see Fig. 7. Fr the cnfiguratins see Fig. 4. a b ferr2 c ferr4 d a, h b, e b c f f i i, m, p e k j r d ditin, cnfs r and ferr4 yied a cmpetey new set f tensr eements due t the preserved furfd rtatina symmetry. Thus, assuming ne atm as distinguished may reduce the symmetry. New tensr eements appear in cnfs a, b, e, f, k,, and r cmpared t case A cmpare Tabes II and X. In these cnfiguratins it is therefre pssibe t distinguish the cases f equivaent and nnequivaent magnetic atms, prvided the tensr eements that make the cases different can be singed ut by the experimenta gemetry. There exists as a pssibiity t distinguish different AF cnfiguratins in case C. The antiferrmagnetic phase can be undubtedy detected in the surface cnfiguratins c, f, i, m, and p. Fr the 110 surface, there are mre pssibiities t distinguish the cnfiguratins with nnequivaent magnetic atms than in the case A. Hwever, the cnfiguratins sti prduce ambiguus signas see Tabe XI. Cnfiguratns b, c, h, i, k, and are equivaent t the magnetic phase. a is equivaent t the ferrmagnetic cnfiguratin, and cnfiguratin d t ferr2. In additin, cnfs e, f, and g are equivaent t the cnf and cnf j gives the same signa as cnf ferr4. Even the presence f nnequivaent atmic sites in the attice cannt be detected by SHG n this surface, since the symmetry f the 110 surface is usuay nt wered further by the existence f equivaent magnetic sites cmpare Tabes IV and XI. The ny exceptins are cnfs a, d, g, and j which give different tensr eements in the tw cases. As in the case f equivaent atms, the 110 surface is nt very usefu fr the anaysis. The study f the 111 surface must again be divided in the tw subcases f ne r mre mnayers, respectivey. Figure 7 depicts the situatin in the magnetic phase. The SHG respnse types are isted in Tabes XII and XIII fr the first and the secnd subcase, respectivey. In the first subcase ne mnayer the symmetry estabishes six different types f nninear respnse. The magnetic type fr the magnetic cnfiguratin ny is characteristic a the ther cnfiguratins have additina tensr eements. The next type f respnse the ferrmagnetic cnf and the antiferrmagnetic cnf a brings sme new tensr eements. Other tensr eements appear in k ferr2 m n ferr4 a b, c, h, i, k, k d 1m e, f, g n j cnf k. s and i shw anther set f nnvanishing tensr eements. s c and f revea a tensr eements in an unreated way. In additin, cnf ferr5 presents a characteristic set f tensr eements. In the secnd subcase, ny fur different SHG respnses are pssibe. First, the magnetic phase is characteristic a the ther cnfiguratins bring additina tensr eements int pay. The next type f respnse is presented by cnfs and i they yied sme additina tensr eements. s, a, c, f, and k revea a tensr eements and n reatins between them appear frm ur symmetry anaysis. Again, cnf ferr5 presents a unique set f nnvanishing tensr eements. Cnsequenty, fr the 111 surface, the symmetry breaking due t the presence f a secnd kind f magnetic atms has even mre imprtant cnsequences than fr the 001 surface. In the situatin f ny ne mnayer, the distinctin between the cases may be pssibe fr a the AF cnfiguratins cmpare Tabes V and XII. Cnsidering additina ayers eads t further symmetry breaking and renders the distinctin between the cnfiguratins impssibe. The distinctin between the cases A and C is pssibe in cnfs a and k cmpare Tabes VI and XIII. Besides, in mst cn- TABLE XII. SHG respnse fr a spin cnfiguratins f the 111 surface f a fcc attice, with ne atm distinguished. Ony ne mnayer is taken int accunt. Fr the detaied descriptin f the respnse types see Tabe I. Fr the surface arrangement see Fig. 7. Fr the cnfiguratins see Fig. 5. p ferr5 q a c, f m i k n

12 PRB 60 SYMMETRY ANALYSIS OF SECOND-HARMONIC TABLE XIII. SHG respnse fr a spin cnfiguratins f the 111 surface f a fcc attice, with ne atm distinguished. Mre mnayers are taken int accunt. Fr the detaied descriptin f the respnse types see Tabe I. Fr the surface arrangement see Fig. 7. Fr the cnfiguratins see Fig. 5. TABLE XIV. SHG respnse fr a spin cnfiguratins f the 001 surface f a fcc attice, with a distrtin f xygen subattice. Fr the detaied descriptin f the respnse types see Tabe I. Fr the surface arrangement see Fig. 8. Fr the cnfiguratins see Fig. 1. 3r s t ferr5 u a, c, f, k s i t figuratins it is pssibe t decide if these additina ayers pay any re cmpare Tabes XII and XIII. D. Distrted xygen subattice Due t the strng charge transfer between nicke and xygen in NiO the subattices may be distrted. This effect can wer the symmetry f the surface. A pint-charge mde cacuatin by Iguchi and Nakatsugawa 41 presented a shift f the xygen subattice rumping in the directin perpendicuar t the surface. Their methd did nt shw any inpane dispacement and thus n change f the surface symmetry. Hwever, if the rumping as has an in-pane cmpnent, i.e., if the xygen atms are dispaced as in the x and y directins, it wi as have a cnsiderabe effect n the symmetry f the crysta surface. Fr this paper, we have chsen a distrtin that can wer the symmetry f the surface and besides can be represented within ne cnventina unit ce. The cnfiguratins cnsidered here are, ferr2, ferr4, a, b, c, e, f, i, k, m,, p, and r fr the 001 surface see Fig. 1,,, ferr5, a, c, f, i, and k fr the 111 surface see Fig. 5, and a cnfiguratins depicted in Fig. 4 fr the 110 surface. Other depicted spin structures frm dmains f these cnfiguratins and are nt referred t in this subsectin nr in the tabes cncerning the current subsectin. As wi be shwn ater, the best cnditins fr the detectin f this kind f distrtin are presented by the 110 surface. The 111 surface cud shw equay gd pssibiities if ny a mnayer f magnetic atms is present. In the presence f an xygen subattice distrtin, the chemica unit ce is as dubed. This effectivey means that magnetic unit-ce-dubing describing the fact that the magnetic unit ce is twice as big as the chemica ne is ifted. In genera, taking int accunt distrted xygen atms in the magnetic phase des nt wer the symmetry f the prbem. The exceptin is the 111 surface, where the sixfd axis is repaced by the threefd ne. In the case f the distrted xygen subattice, the symmetry grup fr each cnfiguratin is a subgrup f the crrespnding nndistrted cnfiguratin, i.e., f the crrespnding spin cnfiguratin in the case where the xygen atms are nt cnsidered. As in case C we dispay ny the magnetic phase in Fig. 8 t depict the atm psitins. a b ferr2 c ferr4 d a, h b, e b c, f i i, k c m j p e r d A the spin cnfiguratins are the same as fr the crrespnding surfaces in case A, and the spins are assumed t be equivaent. As Tabe XIV shws, six different respnses can be expected frm the 001 surface. The magnetic surface wi give a characteristic respnse. The secnd grup is frmed by cnfs a, b, e,, and. Athugh cnfs a and have eements different frm the remaining cnfiguratins in this grup, this fact crrespnds simpy t rtating the sampe by 90 with respect t the z axis. s c and f revea a tensr eements withut reatins between them. s ferr2, i, k, and m revea a tensr eements with sme reatins. The ny difference between cnf m and thers frm this grup is ike fr the previus grup a 90 rtatin with respect t the z axis. Anther grup cnsists f cnf p ane. It reveas the same tensr eements as the magnetic phase, but certain reatins between tensr eements are brken due t a wer symmetry f cnf p. s r and frm the ast grup. A the cnfiguratins but k and can be distinguished frm thse f case A cmpare Tabes II and XIV. Hwever, ny cnfs c and g can be distinguished frm case C cmpare Tabes X and XIV. Thus, ny in these cnfiguratins wi it be pssibe t detect xygen subattice distrtins by SHG. The SHG respnse types fr the 110 surface are presented in Tabe XV. One can bserve that ny cnfiguratins c, f and i give rise t new cmpared t case A, Tabe IV tensr eements. Cmpared t case C Tabe XI, cnfs c, f, and i bring new tensr eements, and, surprisingy, cnfs a and g have ess tensr eements, due t higher symmetries in case D. Cnsequenty, cnfs a, c, f, g, and i aw fr an unambiguus determinatin f the xygen subattice distrtin frm the 110 surface. The pssibiity f distinguishing different cnfiguratins is rather imited. Oxygen subattice distrtin simiar t the ne presented in Fig. 8 fr a 111 surface was fund by Renaud et a. 42 and cacuated by Gian 43 in M 2 O 3 materias (M A, Fe). Since the nnmagnetic subattice symmetry grup has an infuence n SHG this distrtin can be detected as n surfaces f fcc crystas. In the previus cases A and C we di-