Magnetoeectrc susceptbty tensor of mutferroc TbMnO 3 wth cycoda antferromagnetc structure n eterna fed Igor V. ychkov* Dmtry A. Kuzmn and Sergey J. Lakhov Cheyabnsk State Unversty 500 Cheyabnsk r. Kashrnyh Street 9 Russa Vadmr G. Shavrov The Insttute of Radoengneerng and Eectroncs of RAS 5009 Moscow Mokhova Street -7 Russa Magnetoeectrc deectrc and magnetc susceptbty tensors of mutferroc TbMnO 3 wth cycoda antferromagnetc structure n eterna eectrc and magnetc feds have been nvestgated wth takng nto account dynamcs of spn eectro-dpoe and acoustc subsystems. A components of tensors depend on vaues of eterna eectrc and magnetc feds. The possbty of contro of eectrodynamc propertes of mutferroc TbMnO 3 wth cycoda antferromagnetc structure by eterna eectrc and magnetc feds has been shown. The resonant nteracton of spn eectro-dpoe eectromagnetc and acoustc waves n such matera s observed. *E-ma address: bychkov@csu.ru
Nowads mutferrocs materas wth magnetc and eectrc orderng attracts researchers attenton for ther unusua physca propertes -3 : estng of eectromagnons and possbty to contro t by an eterna eectrc fed. Often mutferrocs have a moduated magnetc structure whch contrbute a number of features n the spectrum and dynamcs of spn acoustc and eectromagnetc ectatons n matera -6 : band structure s observed the nonrecprocty effect s manfested.e. dfference between the veocty of wave transmsson aong and aganst the moduaton as. Orthorhombc mutferroc TbMnO 3 (P bnm for eampe has a cycoda antferromagnetc structure at temperatures T < 8 K 7. The couped spn eectro-dpoe and eectromagnetc waves n TbMnO 3 wth cycoda antferromagnetc structure had been theoretcay nvestgated earer 89 however the nfuence of eterna eectrc and magnetc feds on dynamc propertes of such matera are not studed enough. The present work s devoted to studyng of eectrodynamca propertes of orthorhombc mutferrocs paced n an eterna eectrc and magnetc feds of dfferent drectons. Dervatons of dynamcs of the couped ectatons n the moduated magnetc structures are carred out n reg L a where L= π k s the perod of moduated structure a s the attce constant k s the wave number of moduated structure when the phenonoogca thod s appcabe. Method of Lagrange has been used for nvestgatng of dynamcs of TbMnO 3 wth cycoda antferromagnetc structure. The Lagrangan s epresson s L = E F where E - the knetc energy F - the Gnzburg-Landau functon. In case of orthorhombc mutferroc TbMnO 3 : ( a w u γ α A M P u r; { ( z y y V A A A A λ λ b Pz( Az yay Ay yaz E = d F A V μ μ λ ρ = β M M AM A M P PE ν c u u b AA u d PPu } dr. jm j m jm j m jm j m ( Where M = M M M3 M s the magnetzaton of the crysta; A= M M M3 M s the vector of antferromagnetsm; P s the vector of poarzaton; u ( u u = s the tensor of j j j deformatons; u s the dspacent vector; w s the constants of ansotropy; auβλ λ s the
constants of homogeneous echange; α γ s the constants of heterogeneous echange; b ν s the constants of eectrc and magnetoeectrc nteractons; cjk bjk d jk s the tensors of eastcty magnetostrcton and eectrostrcton; c λ= mv z where z and m s the charge and the reduced mass of the eentary ce wth the vou v c ; μ=χ 8g M 0 where χ s the statc transversa magnetc susceptbty g s the gyromagnetc rato M 0 s the magnetzaton of sub attces. The ground state of TbMnO 3 wth cycoda antferromagnetc structure s descrbed by vectors of antferromagnetsm and poarzaton wth foowng components: P 0 = P 0 = 0; P0 = P0 ; A 0 = 0; A0y = A sn ky ; A0z = A cos ky. The coordnate as s drecton defned by a y b z c. So y-as s the moduaton as z-as s the drecton of the spontaneous poarzaton n TbMnO 3. Epressons for determnng the paraters of the ground state can be obtaned from the mnmum of energy. In case of eterna eectrc fed E s conear to a spontaneous poarzaton P 0z and eterna magnetc fed have a non-zero components: y z λ λ λ ( β = ; ( β = ; γ ( ( 0 0; 3 ( a a ν = z; 3 ( p a my( y z 3 ( ( p a my y z M A A M A A A 0 0 0y z 0y z 3 α k A A k A A ν AA P = pp P p A p A b kaa E 3 0 0 0 a A A a P a A a M M M α k γ k a 8ν kp A = 0; 3 0 0 0 0 0 a A A a P a A a M M M α k γ k a w 8ν kp A = 0; 3 0 0 0 0 0 ( p a ere the foowng notaton has been ntroduced: det c j 3 = ( D p 3 =8 3 D3 ( ( a u ( a ( D = D3 p a = 3 D3 3 = 3 a = ( λ λ a a u c ( a u my ap =3 3 3 a = a = 8 3 3 D Ξ = ( Ξ Ξ Ξ p =3 3 3 = 3 3 3 a my = λ Ξ = ( Ξ Ξ Ξ Ξ= D = 3 j Ξ can be 3 obtaned from epresson for by changng of j coumn to vector The equbrum deformatons are heterogeneous: Ξ.
( 3 3 0 ( 3 3 0 ( 3 3 0 0 = D 0 yy = D 0 3 3 3 = D 0 0 0 y = z = 0; yz = ( sn cos. u A cos ky A sn ky P ; u A cos ky A sn ky P ; u A cos ky A sn ky P ; u u u c AA ky ky (3 For nvestgaton of dynamc characterstcs one have to take nto account the system of Lagrange equatons for A M P and u. Sovng the system of equatons by thod of ow oscatons nearzng and usng the form of harmonc seres for varabes n approach of frst harmoncs for the waves propagatng aong y-as the oscatng amptudes of poarzaton p and antferromagnetsm a em j j j j vectors can be epressed as p =α e κ h j j j j m =χ h κ e where α j χ j s the tensors of em deectrc and magnetc susceptbty consequenty * tensor (operaton * ans the compe conjugaton. The study shows that n case of the foowng components:.e. = ( κ = κ s the magnetoeectrc susceptbty j j 0 00 the tensors α j and χ j are dagonas wth pz m ( p m m pz m pz α =λ ; α =λ ; λ ( tz ( ( α yy = z zy z z y z z z py D tz D D3 D3 D χ =μ λαpzm ; χ yy =μ a my a y t ; χ =μ ( a zz ( ( a ; (5 y ere the foowng notaton has been ntroduced: a = t ( a ( t y z = ( ( ( ( 3 66 y = = ( μ y ( = tz ( ( ( tz z y tz tz 66 66 66 c A y z 3 = Ω y = Ω z = μ = ( μ c A 3 3 c A
z z py =D tz ( py ( tz pz D ( pz ( y = = ( μ 3 ( = m y z y z m c A {( ( ( ( } ( ( Az Ay ( Ay Az y z D3 y 3 z ( 3 = λ μ pz Ay Az z z y m y z D3 3 y z m pz = μ λ z y z z y z z = μ = ( μ c A 66 66 66 c A z = zy z y 3 3 0 D = λ c D P z D = λ c DP0 = ( μμ λ j yz M0 ja { u ( c AA Pk 0 } ± Ω =μ ± ν s = j = y z Ayz ( = μ ν A k t = stq tz = stzq = sq st = c66 ρ stz = c ρ = c ρ are the veoctes of transversa and y poarzed and ongtudna acoustc waves q s the 0p wave number of the propagatng wave ( D u p =λ b = y z 0p 3 u = ( D3 D3 D3 m =μ βλ ( A A my =μ βλ A A λa 0P aλ M 0 yz λ M k k ± 3 u γ α ayz ( =μ 0A 0A 0A 0A ( u{ A A} ( ( 3 u u ± ( u{ A A} ( ( 3 u u =μ βλ A A λa. The magnetoeectrc susceptbty tensor κ j has ony one non-zero component: pz z m m κ =α (6 In case of y.e. = ( 0 0 y0 the tensors α j and χ j aso are dagonas wth the components: py ( p yy tz py py α =λ ; α =λ ; ( my α = ; χ =aμ m a yy t ; λ ( pz my my pz my pz ; χ =λαyypyμ ( tz. χ =λα μ yy pz my 5 (7
Note that n (7: z zy y z py =py D tz = ( = pz pz ( 3 3 3 ( 3 ( 3 3 Ay Ay z z Az y z D D Az Az y z Ay z z D D Ay Ay z z 3 D3 my = yy yy yz yy yy yz ( my ( ( tz yy yz = = Az Az y z D3 Ay z z 3 D3 yy yy pz yz my = λ μ Az y z D3 yy yy yz { } = μ λ my Az Ay Ay Az pz yy yy yz yy yy yz ( py z z y z y = λ μ D tz yz yy yz yy my z z pz =D tz μ λ The other symbos have been ntroduced above. In ths case there are two non-zeros components of magnetoeectrc susceptbty tensor: pz py yz my my zy yy κ =α ; κ =α ; (8 In case of z.e. = ( 00 0z the tensors j α and χ j are: py my ( p yy tz my py my py α =λ ; α =λ ; ( my α = ; χ =μ a m a ; λ yy yy py ( pz pz pz ( tz ; χ =αpzμ my ( ; χ =λα μ (9 6
In (9 the foowng notaton has been used: ( ( my tz zy zy zy zy my = = zy zy ( ( py z z y z y my = Dtz λ μ zy zy my z z y z y py =D tz zy μ λ tz 3 3 ( 3 ( Ay z z pz D zy = λ μ Az y z D zy 3 3 ( 3 ( Ay z z D zy pz = μ λ( Az y z. D zy Aso there are two non-zeros components of tensor κ j : py pz yy yy my ; my κ = α κ = α. (0 Note that wthout an eterna magnetc fed the tensor of magnetoeectrc susceptbty has no nonzeros components. So an eterna magnetc fed actvates so components of the tensor dependng on the drecton of fed. FIG.. Frequency dependence of so components of the susceptbty tensor. (0 koe = T 00 CGSE 30 kv/cm 7
One can see that a components of a tensors have a resonant type. Ths effect s manfested n cause of resonant nteracton of subsystems of TbMnO 3. As we as k A A P 0 and M 0 are the functons from and E z defned from sovng of the system of equatons ( the resonant frequency vaues depend on vaues of both eectrc and magnetc feds and ts drecton. Fg. shows ths dependence. Let us now estmate so dependences. From ( ( A A u 0 β λ A A0 ( uβ ( λλ M 0 β ν A ( u = ( w L A ( u ( w L 0 z P k b AA b E 0 = 0 k γ α L = aγ α< 0. For vaues of constants 9- γ 0 cm α cm a 00 u 0. 0 8 w 0 β 00 b 0. ν 0 9 λ 0 we have 3 A0 0 Oe ( A ( u ( vary n range of 0 ( z z μμ β λ β λ λ of z 0 and take a mamum ( z 0 ma s - n fed 7 k 0 cm -. So for eampe wth changng ma 5 0 Oe. Decreasng of fed to 0 Oe w decrease ths vaue to 9 z 0 s -. The frequency ± Ω changes wth the eectrc fed on vaue 0 about Ω νkμ b E z 0 s - for Ez 00 CGSE (about 30 kv/cm. It shfts the resonant frequency correspondng to nteracton of eectromagnetc wave wth antferromagnetc subsystem to ower frequences on vaue about z and to hgher frequences on Ω. The deectrc and magnetc susceptbty tensors have dfferent epressons for ther components n magnetc fed of dfferent drectons. As we as refectance and transmttance of eectromagnetc waves depends on the deectrc and magnetc susceptbty these characterstcs aso depend on vaues of eectrc and magnetc feds and ts drecton. So t s possbe to contro the eectrodynamc propertes of TbMnO 3 by eterna eectrc and magnetc feds. 8
References P. Rovan M. Cous Y. Gaas M-A. Measson A. Sacuto. Sakata and M. Mochzuk Phys. Rev. Lett. 07 070 (0 Ymn Cu Yufeng Tan Aan Shan Chnpng Chen and Rongmng Wang App. Phys. Lett. 0 06 (0 3 5 6 7 N. Kda Y. Tokura J Magn. Magn. Mater. 3 35 (0 G. Lakshmnarana K.J. Pucnsk Materas Letters 9 35 (03 nbn Wu and Gerad J. Debod App. Phys. Lett. 00 60 (0 I.V. ychkov D.A. Kuzmn V.G. Shavrov J Magn. Magn. Mater. 39 (03 M. Kenzemann A.. arrs S. Jonas C. rohom J. Schefer S.. Km C.L. Zhang S.-W. Cheong O.P. Vajk and J.W. Lynn Phys. Rev. Lett. 95 08706 (005 8 9 0 I.E. Chups Low Temp. Phys. 35 858 (009 I.E. Chups e-prnt arxv:0.0 [cond-mat.mtr-sc] (00 G.A. Smoensk Usp. Fz. Nauk 6 (957 (rus G.A. Smoensk I.E. Chups Sov. Phys. Usp. 5 75 (98 I.E. Chups Low Temp. Phys. 38 75 (0 9