The RKKY coupling in diluted magnetic semiconductors. A. M. Werpachowska (1,2) and Z. Wilamowski (1)

Transcription

1 Mateials Science-Poland, Vol., No., 6 The RKKY coupling in diluted agnetic seiconductos A. M. Wepachowsa (, and Z. Wilaowsi ( ( Institute of Physics PAS, -668 Wasaw, Poland ( College of Science by the Polish Acadey of Sciences, Wasaw, Poland This pape is an attept to odify the classic Rudean-Kittel-Kasuya-Yosida (RKKY odel to allow the analysis of the agnetic esonance easueents. In ou calculations, we follow the teatent of the oiginal authos of the RKKY odel but include the finite band splitting,, as a phenoenological paaete. The RKKY exchange is not anyoe of Heisenbeg type and an anisotopy induced by the diection of caie agnetiation occus. Keywods: RKKY coupling, feoagnetic seiconductos, agnetic esonance I. INTRODUCTION The RKKY coupling [,], i.e. the exchange inteaction between localied coe spins ediated by etallic electon gas, has been nown fo 5 yeas as the basic inteaction in etallic feoagnets. The oscillatoy chaacte of the RKKY coupling causes a spin glass behaviou in diluted agnetic etals. It ules the intelaye coupling in agnetic layeed stuctues. As it has been shown fo the last few yeas, the RKKY inteaction is also the doinant spin inteaction in diluted feoagnetic seiconductos [,]. Howeve, in the case of seiconductos, stictly speaing of the sei-etallic phase of seiconductos, the ei enegy, E, is sall as copaed to classical etals and copaable to the exchange spin splitting of the conduction band,. The sall ei enegy causes a satuation of spin polaiation of caie spins and soe othe new effects [5]. In this pape we discuss consequences of the spin splitting on the RKKY coupling. The investigation of analytical expessions fo the distance dependence of the exchange coupling shows that in the pesence of spin splitting the RKKY exchange is not anyoe of Heisenbeg type but leads to agnetiation-induced anisotopy. With an incease of, which we teat as a phenoenological paaete, the RKKY coupling evolves fo the classical Heisenbeg coupling, via anisotopic inteaction to the Ising coupling. Moeove, the spin splitting leads to the occuence of vaious contibutions to the RKKY coupling which ae chaacteied by diffeent distance dependencies and vaious chaacteistic lengths. II. MAGNETIC RESONANCE IN (GA,MNAS The foation of agnetic ode in sei-etallic (Ga,MnAs is well descibed by the Mean ield Appoach (MA odels [6]. It is now coonly accepted that the p-d exchange is esponsible fo the hole ediated exchange. Within the MA, the Zene [7], RKKY [,] and Dietl [,] odels ae equivalent. All of the peit a good estiation of the citical tepeatue. The Dietl odel, which attibutes the p-d enegy to the caie spins and taes into consideation the details of the valence band stuctue, additionally allows fo the estiation of the agnetic anisotopy. Unfotunately, the afoeentioned odels calculate the enegy of the gound state of the syste but they do not analyse the eleentay

2 excitations. Theefoe, they cannot be diectly applied to the discussion of the agnetic esonance. Studies of the agnetic esonance show that ae two types of the agnetic esonance obseved in (Ga,MnAs [8,9]. None of the, howeve, can be attibuted to any agnetic esonance aleady obseved in siila systes [-]. Up till now, none of the is satisfactoily explained. In paticula, feoagnetic esonance in the sei-insulating (Ga,MnAs is chaacteied by the esonance fequency which atches pecisely the anganese spins esonance fequency. The absence of any influence of the hole spins eains unexplained. The esonance in the sei-etallic (Ga,MnAs is of a vey diffeent chaacte. It is chaacteied by a lage anisotopy, a coplex stuctue which can be attibuted to the spin wave esonance and g-facto consideably diffeent fo that coesponding to the g-facto of the Mn spins. These popeties allow to conclude that the esonance coesponds to the feoagnetic ode of the feiagnetic esonance [8,9]. It suggests that the caie spins ae odeed and fo a acoscopic agnetic oent. Till now that poble has not been consideed by any theoetical odel. The exchange coection within the ei liquid appoach is the only effect discussed by Dietl et al. [,]. All of the appoaches discussed postulate paaagnetic popeties of the caie spins. A quantitative desciption of the feiagnetic esonance equies the solution of the equation of otion of two inteacting spin subsystes. The pecise definition of the tenso coponents descibing spins coupling is of a cucial ipotance. In paticula, the two cases: the Heisenbeg exchange and the effective field lead to vey diffeent types of pecession. o exaple, accoding to the RKKY odel, which teats the caies as a paaagnetic ediu and postulates a Heisenbeg coupling between the localied spins, the esonance in the agnetically odeed sei-etallic (Ga,MnAs should coespond to the isotopic esonance with g =. On the othe hand, accoding to the Zene odel, which pedicts a huge ean p-d exchange field acting on each spin subsyste, the two esonances coesponding to Mn and hole spins ae expected at vey high fequencies. None of the odels coesponds to the expeiental obsevations. III. THE EECT O THE BAND SPLITTING ON THE RKKY INTERACTION The ai of this pape is a citical study of the RKKY inteaction and an attept to odify the classic RKKY odel to allow fo the analysis of the expeiental data of the agnetic esonance easueents. In ou calculations, we follow the teatent of the oiginal authos of the RKKY odel, the diffeence being that we assue a finite band splitting,. We do not specify whethe this splitting coes fo the spontaneous agnetiation of the local spins' subsyste, is the giant spin splitting typical fo the DMS's, N xmn S, o finally, whethe it is caused by the extenal agnetic field, B g B B. We obtain the analytical foula fo the diection dependent RKKY ange functions, which ae pesented in the Appendix. ig. shows an exaple of the dependence of the exchange integals on the distance fo =.5 E, hee E is the ei enegy in the absence of the spin splitting. All of the ae of oscillating chaacte but thei aplitudes and oscillation fequencies ae diffeent. As a consequence of one diection being pivileged by the splitting, a agnetiation-induced anisotopy aises. As opposed to the classic RKKY coupling, which has the fo of a stictly Heisenbeg-lie inteaction, the RKKY tenso (J RRKY coponents ae not all equal in ou calculations. The J coponent coesponds to the diection paallel to the conduction band agnetiation and the coponents J xx = J yy coespond to the pependicula diection.

3 RKKY unction RKKY unction RKKY Range (n The J ( coponent is a su of two coponents (see Eq. in the Appendix. Each of the is the classic RKKY function fo diffeent ei vectos and. (coesponding to the spin up and spin down subbands. Due to diffeing fequencies, the exchange inteaction ange is diffeent fo each of the two contibutions. The inset to ig. a shows the dependence of the distance, fo which the J coponent eaches eo value fo the fist tie. The distance fo the ajoity spin subband systeatically deceases, while the distance to the fist eo inceases and diveges fo a coplete polaiation Spin Splitting ( 5 5 J -5 (a (b -5 Distance (n J xx Distance (n ig. The coponents of the RKKY function as a function of the distance calculated fo a siple paabolic band with the electon concentation n c = 7 - : a The solid line epesents the J ( coponent, the dashed and the dotted line stand fo the contibutions fo the spin up and spin down subbands. The inset shows the dependence of the chaacteistic RKKY ange on the spin splitting,. The positions of the fist eo of J( functions, ae plotted. b The tansvese coponent J xx ( is plotted by the solid line. The dashed and the dotted line coespond to the two contibutions. The long ange contibution oscillates in the ange of lage distances beyond the figue fae. It is notable that the J coponent (ig. a does not vanish fo the half-etal case, when only one of the spin subands is occupied (fo lage than the ei enegy. In this egie, the exchange inteaction is caied by the total polaied chage density of the electon gas (iedel oscillations. Geneally, we can teat the classic RKKY inteaction as a su of the two iedel contibutions. In the liit the contibutions ae pecisely equal and the total chage density oscillations vanish, while the spin polaiation (RKKY oscillations eain. o the finite spin splitting the contibutions ae not equal anyoe. inally fo, when only the ajoity spin subband is occupied, the exchange is ediated by the iedel oscillations only. The J xx ( coponent (see Eq. in the Appendix ay be expessed as the su of two qualitatively diffeent contibutions (ig. b. One of the, shown in ig. b by the dashed line, is the odified RKKY function with the oscillations coesponding to the su of the ei vectos. The aplitude of this contibution gets salle as inceases and vanishes in the half-etal egie. The second contibution shown in ig. b by the dotted line oscillates with the fequency coesponding to the diffeence between the ei vectos. It is theefoe chaacteied by the long chaacteistic ange. The aplitude of this contibution gows with and in the half-etal egie deceases with the futhe incease of. The aplitudes of paticula contibutions ae bette seen in ig., whee the ean exchange field,, popotional to the volue integal of the J( function is plotted as a

4 RKKY Mean ield function of. o the whole ange of the longitudinal coponent of the exchange field,, is bigge than the tansvese one, xx. It eans that the agnetiation of the local spin has the tendency to be oiented paallel to the diection, as defined by the agnetiation of the caie spins. This tendency vanishes with vanishing. Apat fo the listed dependencies of the contibutions on the spin splitting, we see the onset of the agnetiation-induced anisotopy. The anisotopy, D = - xx, coesponds to the agnetiation-induced anisotopy. The coupling between the local spins evolves with, fo the pue Heisenbeg coupling fo, via anisotopic to the Ising coupling in the lage liit, whee the tansvese coponent vanishes. In this egie, the exchange field, as seen by a ando local spin, is paallel to the electon gas agnetiation. In this sense, the Ising fo of the exchange coupling between the localied spins coesponds to the Zene odel, whee the p-d coupling between both spin subsystes also has only the coponent paallel to the axis. The both odels, Zene and RKKY neglect the tansvese coponents. Howeve, the Zene appoach explains the lac of influence of the pependicula coponents by the postulated aveaging of the pependicula spin coponents (ando phases of spin pecession, while the lac of the tansvese coponent of RKKY exchange is the consequence of the half-etal chaacte and the lage spin splitting of the caie band Spin Splitting (/E ig. The dependence of the two RKKY ean field coponents on the spin splitting. The two contibutions to the tansvese coponent xx ae aed by the dotted and dashed lines. IV. CONCLUSIONS Contay to the intuition given by the Zene odel, which suggests an (antipaallel oientation of both agnetic oents, tilting of the diections of agnetiations is not that enegetically expensive. Only in the liit of lage, within the ange of the Ising exchange, the enegy of the agnetiation deviation as pedicted by the RKKY coupling and that estiated by the Zene odel, ae siila. o sall value of, howeve, the enegy gain fo the collinea agnetiations is not as significant as suggested by the Zene odel.

5 5 The anisotopic pat of the RKKY inteaction, which esults fo the band spin splitting, is not vey ipotant when discussing the gound state of the syste, i.e. when the diections of the Mn and the caie spins ae paallel. Its ipotance gows, e.g. when the caie agnetiation is tilted by an anisotopy field, o unde an extenal field when the pecession angles of both agnetiations ae diffeent. It ay be a possible explanation fo the quantitative desciption of the hysteesis loops. Magnetic anisotopy paaetes evaluated fo the agnetic esonance studies ae not able to descibe the hysteesis loops obseved in the tanspot, Ke and SQUID easueents. The descibed chaacte of the exchange integals should also lead to a peculia spin wave dispesion and doain wall stuctue. The appeaance of seveal contibutions to the ange functions coponents with the diffeent chaacteistic anges esults in the nonpaabolic spin wave dispesion. While the occuence of the long-ange coponent of the RKKY coupling ay lead to the big agnetic stiffness and consequently to the high enegy of the doain wall. Howeve, this ind of anisotopy has no diect influence on the obseved anisotopy of the spin wave dispesion. V. ACKNOWLEDGEMENT This wo was suppoted by PBZ-KBN-/P/. APPENDIX The function J is given by an analytical foula cos( (sin( cos( (sin( ( N J, ( whee is the effective ass, is the Planc constant ove and N is the exchange constant. The foula fo J xx = J yy function is H H N N J xx Si Si sin sin cos cos 8 ( ( whee dt t t sin ( Si and the Stuve function Hn( [, p. 96] fo intege n satisfies the diffeential equation!! ' " n y n y y n.

6 VI. REERENCES. M.A. Rudean and C. Kittel, Phys. Rev. 96, 99 (95. K. Yosida, Phys. Rev. 6, 89 (957. T. Dietl, H. Ohno,. Matsuua, J. Cibet, D. eand, Science 87, 9 (. T. Dietl, H. Ohno,. Matsuua, Phys. Rev. B 6, 955 ( 5. E.L. Nagaev, Phys. Rev. B 5, 668 ( J. König, J. Schlieann, T. Jungwith, A.H. MacDonald, eoagnetis in (III,MnV Seiconductos (, cond-at/ 7. C. Zene, Phys. Rev. 8, (95 8. O.M. edoych, Docto Thesis ( 9. O.M. edoych, Z. Wilaowsi, M. Potesi, M. Bysewsi, J. Sadowsi, J. of Supecond. 6, 5 (..J. Tean, M. Potesi, D.K. Maude, Z. Wilaowsi, A.K. Hassan, T.D. Plantie, T. Andeacy, J. Jaosińsi, T. Wojtowic, G. Kacewsi, Phys. Rev. Lett. 9, 77 (. T. Stoy, R.R. Gałąa, R.B. anel, P.A. Wolff, Phys. Rev. Lett. 56, 777 (986. S.E. Banes, Adv. Phys, 8 (98. M. Abaovit, I.A. Stegun, Handboo of Matheatical unctions (Dowe, New Yo, 97 6