Behera, S N

Parhi, N

Rout, G C

IACS Institutional Repository

W«/«./ Ph>’s. 77A(2), 153-158 (2003)%i  u p a  %Interplay of ferromagnetism and static |>and Jahn-Teller distortionin Lai_jcSr;cMn^ 3IN Parhi *^, G C Rout  ^ and S ^  Behera^‘P G Department of Physics, M P C College (AutonomousJ Baripada-757 001, Orissa, India ^Condensed Matter Physics Group, G M College (Automonoul, Sambalpur-768 004, Orissa, India ’Institute of Physics, Sachivalaya Marg, Bhubancsw|r-751 005, Orissa, India (>maii ' niIima(^jopb res.iilAbstract ; The manganitc system Lai„xSr;,Mn03 is a ferromagnetic metal in presence of static Jahn-I'cller distortion. The system is described by an electronic Hamiltonian containing the hopping terms as well as the ferromagnetic interaction term ari.sing out of the same iiiiterant Cf, electrons of Mn ion in presence of lattice distortion. The one particle Green’s functions of the electrons arc calculated by Zubarev technique using equations of motion method The magnetization and lattice strain are calculated and solved self-consistently taking into dt.u)unl the impurity and chemical potential ihe interplay between ferromagnetism and lattice distortions is discussed in this communication.Keywords Hlectron-electron interaction, magnetically ordered materials, exchange and super exchange.PA rs Nos. • 75. lOJm, 75.30 Kz1. IntroductionThe essential physics o f the “colossal magnetoresistance” materials RC) ,A,Mn03 (here Re is a rare earth such as La and A IS a divalent element such as Sr or Ca) is the interplay between a strong electron-phonon coupling and the “double exchange” effect o f spin alignment on electron kinetic energy [1], In the interesting doping range 0.2 < x ^ 0.5, Rci vAiMnOj is a ferromagnetic metal at low temperature and a poorly conducting paramagnet at high temperature; the paramagnetic-ferromagnetic transition occurs at an x dependent transition temperature T i^x) ~300 K and is accompanied by a large drop in resistivity [2,3]. The colossal magnetoresistance which has stimulated the recent activities m these materials is observed for temperature near T,{x)(4]. The electronically active orbitals are the Mn 3d  orbitals and the mean number o f d  electrons per Mn is 4 -  x". Ihe cubic anisotropy and Hund's rule coupling are so strong that three o f the electrons go into the crystal field /2k core states and make up an electrically inert core *>pin ,S’ of magnitude 3/2 and the remaining (1 - x )  electronsgo into conduction band of width ~2.5 eV made mostly of outer Cf, orbitals. The Cg electrons are aligned to the core states by a Hund's rule coupling Ju which is believed to be large [2,3].The large Jh modulates the hopping of an outer-shell electron between two Mn sites by the relative alignment of the core spins, being maximal when the core spins are parallel and minimal when they are antiparallel. Also, electron hopping promotes ferromagnetic order. This phenomenon, called double exchange [5-7], has been widely regarded [8-11] as the only significant physics in the regime0.2 5 X 5 0.5. However, it was previously shown [12] that double exchange alone cannot account for the very large resistivity of the T > phase or for the sharp drop in resistivity just below T^ , and was suggested that the necessary extra physics is a strong electron-phonon coupling due in part to a Jahn-Teller splitting o f the Mn eg states. The cubic- tetragonal phase transition observed for 0 ^ x i  0.2 is known to be arising due to a frozeri-in Jahn-Teller distortion wifli long-range order at the wave vector (;r. Ji, n).nrresponding Author ©  2003 lACS154 N Parhi, G C Rout and S N Behera2. FormalismThe prescnl model considers the itinerant electrons of Mn atoms disregarding the anli-ferroniagnctically ordered locali/cd tzf. electrons. The strong Hand's coupling aligns the localized spins in the slate. Because of the ionic character of the material, there exists strong electron correlation among the itinerant electrons. The ferromagnetism is induced in the conduction electrons because of their interaction with the localized spins. In the simplified model under consideration, we assume the existence of a direct Heisenberg type exchange interaction between the itinerant electrons to be responsible for their terromagnetic stale. Furthermore, calculations arc performed in the mean field approximation accounting for the ferromagnetic exchange interaction in the Ising limit. Since the t \  band is doubly degenerate a band Jahn-Tellcr distortion is invoked to remove the degeneracy. The aim of the calculation is to study the interplay between the transitions Irorn the paramagnetic to the ferromagnetic state and the structural transition induced by the band Jahn-Tcller distorlion. The different terms in the model Hamiltonian arc detailed below. I'hc band Hamiltonianrepresents the non-interacting electronic energy in a two-fold degenerate (c^ ,) band with single particle energy and chemical potential m. The operators c\,,^a{caia) being the creation (annihilation) operator of the electron in the band a of spin s  and energy „^(A:) (a  = 1, 2; is the orbital index).The Heisenberg exchange interaction between the spins of the c\, electrons at the /-th and y-th atomic sites isH m = - J . J u S i . S f . (2)The Ising limit was obtained by taking the s-component of the interaction Sj.S,  and then using the mean field approximation like 5; <5/> -< 5/ )<5j  ), theHamiltonian //«  can be written as(3)a,kwhere the ferromagnetic magnetization M = - J < S ‘> and J is the effective interaction energy. The electron hops in the Icrromagnetic metallic state. The exchange energy per atom is comparable to the kinetic energy and is taken to -  I cV.The coupling between the electron density in a degenerate electron band with static elastic strain mode is in general, of the formiJJ,m (4)where pim = ’Lk.a<^ ik(t<^ ik<T> the electron density of the unperturbed degenerate band, the static strain tensor (corresponding to the zone centre flr = 0 phonons) and O’ li the strength of the electron lattice interaction. In presence of tetragonal distortion, c,, is a constant and the cq. (4) simplifies to^e-L  ~ ^'2k,a^2k,a^ ■ (S)kTherefore, the interaction of the electrons in the degenerate conduction band with the lattice acts as external perturbation to the free electrons in the band which tries to create a population difference between the two bands. As the population difference increases, the strain builds up in the system resulting in a splitting of the single degenerate band into two with hand energies == 0)(k) ±Ge, providedthere is a net gain in the electronic energy due to the redistribution of electrons between tlio .split subbands m comparison to the cost in the elastic energy. Under tins situation there are two non-dcgcncralc bands separated h\ a gap of magnitude 2Ge.Hence the manganitc system exhibiting Jahn-Tcllcr static lattice distortion and ferromagnetism can he described by the total Hamiltonian/ /  = //( -h Hm + kif /^ . (()}3. Calculation of electron Green’s functionsThe double time single particle electron Green’s functions arc calculated hy equations ot‘ motion method of Zubarev 113]. The electron Green’s functions arc defined asB(kl , (o)  = . (7)The Green’s functions arc calculated and written asA {kf,(o) = 127t(0}- E \)A(ki,o>) 1• 2n:(o}-£2)B(kT,Q)) =B (k i,co ) =12k ((o - E ^ )r2;r(G »-£4)( 8 )Interplay of ferromagnetism and static hand Jahn-Teller distortion in La,_fSr^MnO^ 155The quasiparticic energies arc given by £, - e o ( k ) - M  + G e + ^ <  E 2 = e o ( k ) - p  + G e - ^ ^ .E, = e o ( k ) - p - G e +  -^^  .E^ = e o ( k ) - p - G e - ^ (9)where Ny Ji are Ihe number of atoms per unit volume and maiinctic moment of the manganese atom respectively.The number operators definedl>y ^'MT ~ »k" li  = X < " iU ‘ u l> -k«2TkkThe number operators n,s (/ = 1, 2) arc found to be ^ e \ p ( E t / k B ' r )  + l '" i i = X c x p ( £ 2/A:Br)+T"  ^ e x p ( £ , / * a D  + l "2i == X ’exp(£^/)kB7’) + lThe reduced magnetization m = Ml{NJi) is found to bekwhere b=: (kaT)-' and the Fermi function in general is given / (.v) “  • The sum over the momenta of the electronsin the conduction band is replaced by the integration over energy of the electron ^ (k )  with integration limit from -W 2 to + w n  asT ' f+W/2Here. N (^)  is the model density of states [14) and 2D is tlie total band width W of the conduction hand. To simulate the strong energy dcpendance of Ni^)) around the centre of the band we lakeN (€ o) ^ N { 0 )  1-- In i D ier^(14)4. Calculation of Magnetization (M) and Jahn-Teller di.stortionThe magnetization of the manganite system is defined as( 10)where A|[0) is unperturbed density of state of the free electron land ^  is the kinetic energy of the conduction band. !The cihcmical potential of the electron system changes with tenijerature (T) and the concentration of the dojxtd impurity|j :) in the system Lai_^Sr,MnOv Since the impurity produces^holes in the Mn03 plane, the chemical potential (jn) is dierm ined from the average number ol* electrons (n-.x) which is given by(n -  X) = u.fr > + (‘ 2* ,(T>]• (13)k ,CTUsing the expressions given in the cq. (12), this reduces toi n - . x ) ^ J ^ t f i P E ,  ) ,  (16)k /-Iwhere the Fermi functions for different quasiparlicle energies (£,) arc written earlier.The static band Jahn-Teller distortion energy is given byEjr  ( 1^ )k ,awhere G is the strength of the electron-lattice interaction and Cq is the static lattice distortion. The reduced lattice distortion (e)  is given by(IK)The physical quantities involved in the calculations arc the band energy (Q)(A:)), chemical potential (m), the strength ot the clectron-lallice interaction (G), the lattice distortion (e)  ^the effective interaction energy (7), the temperature (7), and the conduction band width (WO. Thc.se quantities arc sealed with respect to the band energy (WO* The dimensionless parameters arc given by>- vv ’ w ^,’ * wJNp e M '8 \ -  w ' e ^ — y 0^ NpW *(19)156 N Parhi, G C Rout and S N Behera5. Results and discussionThere exits ferromagnetism in manganitc system Lai ,Sr,MnOi accompanied by Jahn-Teller lattice distortion with impurity dependant ferromagnetic Curie temperature r,.(jc)« 250 K. There occurs a large drop in resistivity of the material below the Curie temperature signifying its ferromagnetic metallic character. A static Jahn-Teller distortion present in the lattice changes Curie temperature thereby changing a resistivity drop in the external doping concentration of range 0.2 2 x 5 0.5. In the present work we report a simple microscopic model to study the interplay of spontaneous magnetization (A/) due to ferromagnetism and lattice strain {e) due to static band Jahn-Teller distortion in the absence of external magnetic field (B). Different impurity concentrations (x) are given to the system by varying the chemical potential at varius temperatures.It is noticed that the spontaneous magnetization (Af) in eq. (10) the impurity concentration (x) in eq. (15) and the lattice strain (e) in eq. (18) are coupled to each other in the form of integral equations. To study the co-existance of magnetization and lattice .strain, we solve these equations numerically and self-consistently for different band fillings. The physical quantities involved in the calculations are made dimensionless with respect to the conduction band energy (fV). The dimensionless parameters are magnetization (m), lattice strain ( e ) ,  lattice coupling strength (g), magnetic coupling strength (gi). chemical potential (r) and reduced temperature (r). It is to note that Figure I gives the individual plot of OT ~ / and e ~ i whereas Figures 2-8 show the self- consistent plots.Figure 1. Individual plot of m vs r and e vs < for fixed values of g «  0, gl » 0.13995. s -  0 and g  -  0.074. gl -  0. z -  0.Figure 1 shows the individual plots of /n -  r and e  t in the absence of the magnetic field and chemical potential. Initially the magnetic coupling strength is kept constant at g l  = 0.13995 in order to give a transition temperature tc »0.024 corresponding to Curie temperature Tc »= 240 K.Similarly the lattice coupling strength g = 0.074 gives a lattice distortion temperature tj « 0.0285 corresponding to a distortion temperature Tj ~ 285 K. It is seen that < t,, initially. Then the magnetisation (m) and strain (e) are solved self-consistently under the half filling band situation. From Figure 2 it is observed that /<. > t j with » 0.029 and i j » 0.0225. It is also interesting to observe that the magnetization is also suppressed considerably from the initial value m » 0.84 to 0.47 due to the presence of lattice strainFigure 2. Sclf-consistcnl plot of m vb t and c vs t for fixed values ot g = 0.074, g\ -  0.13995 and r  -  0.Similarly the lattice strain is also suppressed considerably from its initial value e « 0.94 to 0.42. In co-existence phase both the magnetization and the strain are so suppressed that the strain curve lies below the magnetization curve in contrast to its original value in individual plots. The temperature depcndance of both magnetization and strain exhibit second order phase transition with respect to temperature. Experimental observations [15,16J show that the resistivity drastically reduces below Curie temperature indicating an insulating phase changing to ferromagnetic metallic state below /c. Here in the present case the presence of Jahn-Teller distortion in the system pushes the Curie temperature to higher temperatures where the ferromagnetic metallic state is thereby extended to the higher temperatures.Figure 3 and Figure 4 show the effect of lattice coupling strength (g) on magnetization (m) and lattice strain ( e )  respectively in their co-existance phase. As the lattice coupling increases from g  = 0.072 to 0.075 the spontaneous magnetization (m) increases accompanied by an enhancement of the Curie temperature from 0.0265 to 0.030 as shown in the Figure 3. It signifies that the ferromagnetic metallic state is enhanced due to the presence of lattice strain. Figure 4 shows the plot of  e ^ i for different values of lattice coupling strength (g). It is observed that the lattice stran e (0) at 0 K remains constant as the lattice coupling changes, but the lattice distortion temperature decreases with the increase of lattice coupling strength (g).Interplay of ferromagnetism and static band Jahn-Teller distortion in Loj-^r^MnOs 157o.«0.01 0.02t0.03 0.04h^urc 3. Plot o f m v s t  for different values of ^  0.075, 0,074, 0.072 Figure 5. P l4 o fw  v j/fo r different values of gl -0 .145 , 0.13995,0.130anti for fixed values o f g l «  0.13995, z = 0. and fixed varies of g = 0 074, z -  0.0.6 I .4 .. .. .■ . .T— ................ ................figure 4.aiitl fixedPlot of e vs t for different values of g  values of gl 0.13995, r  -  0.0.075, 0.074, 0.072 Figure 6. Plot of e vz / for different values of g l 0.130 and fixed values of g = 0.074, z «  0.0.040.145, 0.13995,Figure 5 shows the plot o l  m t for different values of Iciromagnetic coupling (g\). It is observed that the spontaneous magnetization w(0) at r = 0 as well as the C uric temperature ( 0  remain unaffected with the change of ferromagnetic coupling (gi).However, the spontaneous magnetization (m) decreases with the increase of ferromagnetic coupling (gi) in the intermediate temperature range lying between / = 0 and tc. Figure 6 shows the plot of e ^ t for different values of terromagnetic coupling (gi). It is seen that the lattice strain increases with the increase of ferromagnetic coupling (gi) i^nd the lattice distortion temperature (tj) is enhanced as well.Figure 7 shows the plot o f  m ^ t and e -  / for various (loping concentrations x = 0.00,0.10,0.15. The temperature dopendance of chemical potential is such that it allows only 0 maximum concentration of * = 0.15 for stability of the model system considered here in absence of the magnetic held and the Coulomb correlation energy. It is anticipated that higher doping concentration will definitely stabilize thesystem in presence of external magnetic field and tfie stronger Coulomb correlation. It needs further investigation.It is seen that the lattice strain ( e )  and lattice distortion temperature (td) decrease with increase of impurity concentration. It is also seen that there is a decrease of spontaneous magnetization (m) and a reduction in Curie temperature Oc) due to the weakening of exchange interaction (gi) with increase of impurity concentration at higher temperatures. The higher the temperature the greater the relative effect of the exchange weakening and towards tc doping actually causes a reduction in magnetization. However, the magnetization increases with increase of impurity concentration at lower temperatures. The opposing behaviour occurs at a critical temperature t* = 0.009.Figure 8 shows the plot of  m ~ t  with increase of impurity concentration (x) for three different sets of parameters. The critical temperature (/*) as discussed with respect to Figure 7 increases to a higher temperature as the elastic coupling strength (g) increases but t* decreases when magnetic coupling strength (gO increases.158 N Parhi, G C Rout and S N BeheraWe conclude that the spontaneous magnetization (m) and the ferromagnetic Curie temperature (4) can be variedFigure 7. Plot of m vs t and e vs 1 for different values of doped concentration x 0, 0.1, 0.15 and fixed values of g = 0,074, gl ~ 0.13995.by changing the lattice coupling, ferromagnetic coupling and impurity concentrations. The effects o f  die Coulomb correlation o f eg electrons and the extenuil magnetic field are studied and are to be reported elsewhere.AcknowledgmentTwo o f the authors (NP and OCR) gracefully acknowledge the research facilities offered by the Institute o f  Physics Bhubaneswar during their short stay.ReferencesmFigure 8. Plot o f  m vs t for different values of doped concentration jc“ 0 ,0.15 and fixed values of (a) g «* 0.074, gl = 0.13995, (b)g = 0.074, gl « 0.145 and (c) g = 0.075, gl = 0.13995.R von Heimolt, J Weeker, B Holzapfel, L Schultz and K Samwn Phys. Rev. Lett. 71 2331 (1993)E D Wollen and W C Koehler Phys. Rev. 100 545 (1955)O Matsumoto J. Phys. Soc. Jpn. 29 613 (1970)S Jin, T H Tiefel, M McCormack, R A Fastnack, R Ramesh and L H Chen Science 264 413 (1994)C Zener Phys. Rev. 82 403 (1951)P W Anderson and H Hasegawa P l^s. Rev. 100 675 (195.5)P G de Gennes Phys, Rev. 118 141 (1960)C W Searie and S T Wang Can. J. Phys. 48 2023 (1970)K Kubo and N Ohata J. Phys. Soc. Jpn. 33 21 (1972)R M Kusters, J Singleton, O A Keen, R McGieevy and W Hayes Physica (Amsterdam) 155B 362 (1989){11) N Furukawa J. Phys. Soc. Jpn. 63 3214 (1994); 64 2754 (1095). 3164 (1995)A J Minis, P B Littlewood and Boris I Shraiman Phys. Rev Leu 74 5144 (1995)D N Zubarev Soviet Phys. Uspekhi 3 320 (1960)B Dabrowski and Z Wang Phy. Rev. Lett. 76 1348 (1996)Y Tokura, A Urushibara, Y Moritomo, T Arima, A Asamitsu. G Kido and N Furukawa J. Phys. Soc. Jpn. 63 3931 (1994)A Urushibara, Y Moritomo, T Arima, A Asamitsu, G Kido andY Tokura Phys. Rev. B51 14103 (1995)12][3][4]t5][6117][8][9]flO][ 12][13][14][15]116]

Interplay of Ferromagnetism and Static Band Jahn-Teller Distortion in La1-xSrxMnO3

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Interplay of Ferromagnetism and Static Band Jahn-Teller Distortion in La1-xSrxMnO3

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