Triangular hysteresis loops in the spin-rotation region of orthoferrites

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1 Fizik Nizkikh Tempertur, 010, v 36, Nos 8/9, p Tringulr hysteresis loops in the spin-rottion region of orthoferrites YB Bzliy 1, nd LT Tsyml 3 1 Institute of Mgnetism of the Ntionl Acdemy of Sciences of Ukrine, 36 Verndskogo Blvd, Kyiv 03143, Ukrine University of South Crolin, Columi SC 908, USA 3 O Glkin Donetsk Physics & Technology Institute of the Ntionl Acdemy of Sciences of Ukrine 7 R Luxemurg Str, Donetsk 83114, Ukrine E-mil: lt_tsyml@yhoocom Received Jnury, 010 Suggested theory qulittively explins the shpes of the hysteresis loops in orthoferrites within the temperture intervl of the mgnetic reorienttion trnsition Tringulr loops result from the strong temperture dependence of oth the mgnetic moment nd the mgnetic domin wll structure PACS: 7560Ej Mgnetiztion curves, hysteresis, Brkhusen nd relted effects; 7560Jk Mgnetiztion reversl mechnisms; 7550Gg Ferrimgnetics; 7530Kz Mgnetic phse oundries (including clssicl nd quntum mgnetic trnsitions, metmgnetism, etc Keywords: orthoferrites, domin wll structure, hysteresis loops Dedicted to VG Brykhtr's nniversry Spienti supert mor * The orthoferrites hve chemicl composition RFeO 3, where R is the rre-erth element They re well studied mgnetic mterils It is known for long time [1] tht for R = Er, Tm, Sm, Nd, nd Y the mgnetic moment of n orthoferrite experiences 90 reorienttion trnsition in the temperture intervl ( T, T 1 Upon lowering the temperture the mgnetiztion continuously rottes from the crystlline c xis t T 1 to the xis t T The interest to the reorienttion trnsition ws renewed reltively recently y the pper [] which focused on the precise mesurements of the mgnetiztion M( T nd gve n explntion of the oserved temperture dependence in the frmework of the modified men field theory As side oservtion it ws pointed out in Ref tht the shpes of the hysteresis loops M( were mrkedly different inside the reorienttion region nd outside of it Outside of ( T, T 1 the hysteresis loops were rectngulr, whilst inside this intervl they developed tringulr tils or even hd two seprte tringulr su-loops (see Fig 4 in Ref The issue ws not investigted further t the time, ut lter it ws found [3] tht the sme types of hysteresis loops lso occur in ErFeO 3 t the tempertures T < 0 K, fr elow the reorienttion trnsition tht occurs t K The peculir loop shpes were dued the «tringulr til» nd «doule tringle» loops The explntion of the tringulr loops ws suggested in Ref 4 sed on the ssumption of the extremely simple two-domin mgnetic structure of the orthoferrite smple The two-domin stte of the reltively lrge ( mm smple ws supported y the theoreticl estimtes nd ws due to the very smll mgnetiztion of the orthoferrite The motion of the domin wll, nd the corresponding size chnges of the up nd down domins, ccounted for the mgnetiztion jumps on the hysteresis loop Since the loop shpes t low tempertures nd in the reorienttion intervl re similr, it is tempting to try to explin their shpe y the sme mechnism The tringulr loops re explined in Ref 4 s result of n interply etween the domin wll «expulsion field» exp nd the domin wll nucletion field n The former is the field t which the domin wll reches the oundry of the smple nd is expelled from it s the field is rised The ltter is * Wisdom overcomes difficulty YB Bzliy nd LT Tsyml, 010

2 YB Bzliy nd LT Tsyml the field t which the domin wll is nucleted in the smple s the field is lowered The existence of the nucletion field reflects the fct tht for domin wll to enter the smple the system hs to overcome certin wll nucletion rrier An pplied field cretes «mgnetic pressure» cting ginst this rrier nd cusing the wll to nuclete The nlysis of Ref 4 shows tht rectngulr hysteresis loops correspond to the cse n < exp, tringulr til loops correspond to exp < n < 0, nd doule-tringle loops to 0< n < exp In the model of Ref 4 the fields exp nd n re relted y n eqution n const = exp M ( T (1 This formul qulittively explined the evolution of the loop shpes using the experimentlly mesured temperture dependence of the mgnetiztion elow 0 K Consider now n experiment in the reorienttion region The field is pplied long the xis nd the temperture is rised from elow into the reorienttion intervl A forml ppliction of (1 gives result contrdicting the experiments Since M decreses ove T [1,5], this formul suggests tht n should ecome even more negtive This in turn would men tht the hysteresis loops hve to remin rectngulr, rther thn cquire the tringulr shpes To explin the discrepncy etween the theory nd the experiment we note tht ove T the nisotropy energy of the orthoferrite chnges nd thus the formuls otined for the unixil nisotropy elow T hve to e revised To properly consider the wll nucletion in the reorienttion region one hs to tke into ccount two circumstnces First, when mgnetiztion M points t n ngle to the xis the demgnetiztion energy chnges Second, in the reorienttion region the crystlline nisotropy is not unixil This chnges the structure nd energy of the domin wlls nd, s result, cn chnge the properties of the domin wll entrnce rrier The im of the present pper is to tke oth effects into ccount The mgnetiztion of n orthoferrite is sum of the iron mgnetiztion F ( T chrcterized y constnt solute vlue nd directed t n ngle θ F ( T with respect to the c xis, nd the temperture-dependent rre-erth mgnetiztion m ( T The iron moments re ordered due to the interctions etween them The interction etween the rreerth ions is negligile, ut they re mgnetized y the moleculr field of the iron moments This sitution cn e modeled [] y the free energy density 1 E = K ( cos ( cos (4 u T θ F + K θ F β β( Fχ m + Fcχ cmc + m, ( where K u, re the crystlline nisotropy constnts with Ku ( T chnging linerly with temperture in the reorienttion intervl, the coefficients χ, descrie the susceptiilities of the rre-erth moments to the moleculr field of iron ions, nd β chrcterizes the free energy of the prmgnetic system of rre-erth moments Minimiztion with respect to m gives the desired prmgnetic ehvior of the rre-erth system, mi = χ ifi ( i=, c, nd reduced expression for the free energy 1 E = K ( T cos ( θ + K cos (4 θ + const, (3 u F F u = u ( c / with K K βf ξ ξ For K >0 this energy form provides the reorienttion trnsition in the intervl 8 K < Ku ( T<8K [1], where the 8K end of the intervl corresponds to the high-temperture phse ( θ F = 0, F c nd the + 8K end corresponds to the lowtemperture phse ( θf = π /, F The energy profile E( θ F in the reorienttion region is shown in Fig 1 Its mxim re locted t θf = 0, π / nd its minimum point is found t the ngle θ * determined y the eqution Ku cos ( * = 8K θ (4 The four equilirium directions re given y θ F = = ±θ*, π± θ * First, we study the domin wll properties in the reorienttion region Two types of domin wlls re possile [6] In the «-wll» the ngle chnges etween θ * nd θ *, nd in the «c-wll» it chnges etween θ * nd π θ * The free energy density of the -wll strts from E( θ * in one domin, goes through the mximum vlue E (0 nd returns ck to E( θ * In the c-wll the mximum free energy density if E( π / For definiteness, consider the -wll which is nucleted in the experiment with ecuse it seprtes the domins tht differ in mgnetiztion projections M on the field direction We now wnt to estimte the width nd energy of this wll Assuming tht the (, c plne of mgnetiztion rottion is E * 0 / Fig 1 Left: ngulr dependence of the free energy E( θ F Right: equilirium directions of the iron mgnetiztion F F * F c 100 Fizik Nizkikh Tempertur, 010, v 36, Nos 8/9

3 Tringulr hysteresis loops in the spin-rottion region of orthoferrites lso the wll plne, ie, considering Bloch wll with zero demgnetiztion energy, we otin the totl free energy J dθ = tot = F E dx + E( θf dx, dx E (5 where J is the spin stiffness, nd x is coordinte long the xis perpendiculr to the (, c plne The exct shpe of the wll θ F ( x cn e determined from the eqution δetot / δθ F =0 s it hs een done in Ref 6, ut insted we will estimte the properties of the wll s follows Let the wll width e δ Then θ F θ* / δ nd the grdient contriution to its energy cn e estimted s Jθ* / δ The contriution of the crystlline energy inside the wll cn e estimted s F(0 δ The wll energy is the difference etween the sttes with nd without the wll Jθ* Δ Etot + [ E(0 E( θ* ] δ δ Minimizing it with respect to the wll width we find δ θ* J E(0 E( θ* The free energy vlues t the extremum points cn e clculted exctly s Ku π Ku E(0 = K +, E = K, Ku E( θ* = K, (6 3K which gives E(0 E( θ * = K(1 + Ku / 8 K The ngle θ * cn e pproximted from Eq (4 s π Ku θ* 1 + (7 4 8K Using these expressions we find J δ, (8 K Ku Δ E DW JK 1 + (9 8K Figure compres the exct result from Ref 6 with the pproximtion tht uses Eq (9 nd sets the numeric coefficient to π / π Ku ΔE DW JK 1 + (10 8K As one cn see, expression (10 turns out to e quite ccurte The -wll energy grdully decreses to zero s the temperture is rised from T to T 1 nd Ku ( T chnges from + 8K to 8K Physiclly this hppens ecuse the 8K Fig Energy of the -wll s function of K u The solid line shows the exct dependence (Eq (44 from Ref 6 The dshed line is given y the pproximtion (10 mgnetiztion directions in two domins get closer to ech other until the difference etween them disppers t T 1 The sme clcultion for the c-wll gives δc = δ nd c π Ku E DW JK 1 (11 8K Δ Next, we consider the demgnetiztion energy E d A rectngulr smple with n -wll in it is shown in Fig 3 The fces of the rectngle re ssumed to e cut perpendiculr to the crystl xis We will tret the dipole dipole energy s the interction energy of the surfce mgnetic chrges on the fces of the smple [7] For the domin wlls considered here the mgnetic chrges exist on the - nd c-fces (ie, fces perpendiculr to the nd c xis The demgnetiztion energy cn e divided into three prts: contriution from the interction etween the chrges on -fces, etween the chrges on the c-fces, nd DW c 8K Fig 3 Two-domin mgnetic structure of rectngulr smple Thick rrows show the mgnetiztion directions inside the domins tht re seprted y n -wll locted t X X K u Fizik Nizkikh Tempertur, 010, v 36, Nos 8/9 1003

4 YB Bzliy nd LT Tsyml cross-term from the interction etween the -fces nd c-fces Since the energy is proportionl to the product of the chrges, one cn write d = + cc c + c c, (1 E d M d M d M M where the coefficients d, d cc nd d c reflect the dimensions of the smple nd the position of the domin wll in it Note tht the dipole energy depends on the full mgnetic moment with M =(1 +χ Fsinθ F nd Mc =(1 +χc Fcosθ F The cross-term turns out to e zero y symmetry, d c =0 Indeed, it is ovious from Fig 3 tht the interction of mgnetic chrges on the upper -fce of the smple with chrges on ech c-fce is exctly compensted y the interction of the opposite chrges on the lower -fce with the sme c-fce Similr to Ref 4 we will pproximte the demgnetiztion energy dependence on the position of the domin wll y qudrtic function The position of the wll will e given y dimensionless coordinte X with X [ 1,1], (0 nd we will write Ed ( X= Ed + DX For n -wll the chrges on the c-fces do not depend on the position of the wll in the smple According to the rgument ove this mens D M, so tht (0 d + E ( X= E AM X, d where the coefficient A ccounts for the shpe nd size of (0 the smple By the sme rgument Edc ( X= E dc + + CMc X for the c-wll We cn now finlly proceed to the clcultion of the expulsion nd nucletion fields The totl energy of the smple with n -wll is [4] E ( X=const + AMx Mx+ UDW( X, (13 where UDW ( X is the position-dependent energy of the domin wll For X well inside the smple it is constnt UDW = ΔE DW owever, when the wll pproches the smple oundry, the energy UDW ( X ecomes essentilly position-dependent In the sence of n extr surfce pinning, UDW ( X drops from ΔE DW to zero on the distnce of the domin wll width δ A sketch of the free energy E ( X is shown in Fig 4 The expulsion field exp is otined from (13 using the condition X = ± 1 This gives exp = ± AM (14 The nucletion field is determined from the condition [4] de / dx x= ± 1=0, where the derivtive dudw / dx cn e estimted s ΔE DW / δ (Fig 4 This gives n = exp E(0 E( θ* (15 M ( θ* Using the expression for M( θ F nd Eq (4 for θ * one finds M =(1 +χ F (1 + K /8 K / This gives u 3/ K Ku exp n 1+ (16 (1 F 8K +χ This eqution is the min result of the pper After tking proper ccount of the chnge of nisotropy in the reorienttion region we hve found tht the difference exp n decreses s the temperture rises ove T The ehvior of oth fields is sketched in Fig 5 While the vlue of M in the denomintor of (15 does decrese s discussed in the introduction, the domin wll energy nd the corresponding nucletion rrier decrese fster, leding to theoreticl prediction tht is in qulittive greement with the experimentl findings: s n pproches, exp n Fig 5 Temperture dependence of the expulsion field nucletion field n in n experiment with DW Fig 4 Sketch of the domin wll energy position dependence The nucletion rrier is determined y the wll energy nd width exp T T 1 T x exp nd 1004 Fizik Nizkikh Tempertur, 010, v 36, Nos 8/9

5 Tringulr hysteresis loops in the spin-rottion region of orthoferrites the loop shpe chnges from rectngulr to tringulr tils nd then to doule-tringle loop For the c-wll the sme rguments give the reltionship 3/ K Ku nc (17 (1 c Fc 8K exp 1, +χ from which one concludes tht in mesurement with c nd temperture decresing from ove into the reorienttion region, the hysteresis loop shpe will chnge from rectngulr to tringulr due to the nucletion of c-wll Such prediction is gin in ccord with the experiments Our theory lso mkes predictions out the hysteresis loop shpe in the cse of mgnetic field tilted in the (, c plne ere either n -wll, or c-wll, or oth, cn e nucleted, depending on the vlues of n, nc nd the projections of the pplied field c, For the temperture vlues t which the nucletion field n lies in the intervl ( exp,, exp,, nd the nucletion field nc lies in the intervl ( exp, c, exp, c, one should oserve two jumps on the downwrd nd upwrd rnches of the hysteresis loop These jumps will correspond to the nucletion of - nd c-domin wlls In conclusion, we hve shown tht the temperture dependence of the domin wll structure in the mgnetic reorienttion intervl cn qulittively explin the evolution of the hysteresis loop shpes in orthoferrites We further suggested tht mesurement in tilted mgnetic field cn serve s n experimentl check of our theory YB ws prtilly supported y the NSF grnt DMR KP Belov, AK Zvezdin, AM Kdomtsev, nd RZ Levitin, Orienttion Phse Trnsitions in Rre Erth Mgnetic Mterils, Nuk, Moscow (1979 (in Russin YB Bzliy, LT Tsyml, GN Kkzei, AI Izotov, nd PE Wigen, Phys Rev B69, (004 3 LT Tsyml, YB Bzliy, GN Kkzei, FJ Plomres, nd PE Wigen, IEEE Trns Mgn 44, 933 (008 4 LT Tsyml, GN Kkzei, nd YB Bzliy, Phys Rev B79, (009 5 RL White, J Appl Phys 40, 1061 ( VG Br'ykhtr, BA Ivnov, nd AL Sukstnskii, ZhETF 78, 1509 (1980 [Sow Phys JETP 51, 757 (1980] 7 ME Sches nd A Ahroni, IEEE Trns Mgn 3, 388 (1987 Fizik Nizkikh Tempertur, 010, v 36, Nos 8/9 1005