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Spin dynamics in the generalized 9 ferromagnetic Kondo model for manganites. 9 9 1 n N.B.Perkins and N.M.Plakida a J 4 Joint Institute for Nuclear Research, Dubna 141980, Russia 1 Dynamical spin susceptibility is calculated for the generalized ferromagnetic ] l e Kondo model which describes itinerant eg electrons interacting with localized t2g - electrons with antiferromagnetic coupling. The calculations done in the mean field r t s approximation show that the spin-wave spectrum of the system in ferromagnetic . t state has two branches, acoustic and optic ones. Self-energy corrections to the spec- a m trum are considered and the acoustic spin-wave damping is evaluated. - d n o c [ 1 v 1 4 1 1 0 9 9 / t a m - d n o c : v i X r a 1 Introduction Manganitesoftheperovskite structureoftheformR B MnO (whereRaretriva- 1−x x 3 lent rare-earthandB aredivalent alcalineions, respectively) andrelatedcompounds that present the phenomenon of ”colossal” magnetoresistance (CMR) have recently attractedmuchattentionbothfromthebasicpointofviewandduetotheirpotential application [1], [2]. The large magnetoresistance occurs close to the metal-insulator and the paramagnetic-ferromagnetic transitions where the interplay of transport, magnetic and structural properties is of the great importance (see [3]). The key elements of the manganese oxides are Mn ions. In the parent compound LaMnO 3 the electronic configuration of Mn+3 is (t3 e ). In this configuration due to a strong 2g g intra-orbital Hund’s coupling t3 electrons go into tightly bound d , d , d core 2g xy yz xz states and make up an electrically inert core Heisenberg spins S of magnitude 3/2. The t3 configuration is very stable and remains localized over the entire range of 2g doping. In the undoped case, with one eg electron per Mn ion, two eg orbitals, dx2−y2 and d3z2−r2 types, are splitted due to the Jahn-Teller effect. At low temperature eg electrons occupy d3x2−r2 and d3y2−r2 ordered alternately in ab plane with their spins aligned to the core spin by interorbital Hund’s coupling. Due to the Goodenough- Kanamori rules [4] it results in the A-type antiferromagnet (AFM) ground state (AFM vector Q=(0, 0, 0.5)) with spin S = 2 for LaMnO . Upon doping with holes 3 by substituting La with Sr or any other divalent ions system becomes ferromagnetic (FM) andconducting. The hopping between two Mn sites is maximal when the core spins are parallel and minimum when they are antiparallel.That results in effective ferromagnetic exchange between the nearest neighbor core spins and thus leads to the FM metallic ground state of doped compounds. This behavior is qualitatively well described within the framework of the double exchange (DE) mechanism (see [5],[6],[7]). At higher hole concentration, x ≥ 0.5, a charge ordering for holes is observed, and at x = 1 an insulating G-type AFM state (with Q = (0.5, 0.5, 0.5)) takes place for CaMnO compound. Therefore to describe the experimentally ob- 3 tained phase diagram ( see, for example, [8]) one should take into account both the Heisenberg type of AFM exchange between the core t electrons and the strong 2g Hund coupling between t and e electrons (see, e.g. [9, 10]). These competing 2g g interactions could be responsible for a coexistence of AFM and FM states observed recently in neutron scattering experiments in (La Pr ) Ca MnO [11] and 0.25 0.75 0.7 0.3 3 in the bilayer manganite La Sr Mn O [12]. Also a crossover from an ideal 1.2 1.8 2 7 isotropic FM spin-wave behaviour at low temperature to a diffusive spin propaga- tion observed in La Ca MnO [13] could be explained if one takes into account 0.7 0.3 3 boththe localized t spin (S = 3/2)and the itinerant e spin (σ = 1/2) subsystems. 2g g In the present paper we study the spin dynamics in manganites within the generalized ferromegnetic Kondo model (FKM) allowing for the AFM exchange interaction between t spins. Unlike to the DE model (see, [14]), where J /t → ∞ 2g H is considered and the system is treated as perfectly spin polarized with S = 2, in our work both the fluctuation of the localized and itinerant spins are taking into account. However, we ignore in the present calculations a possible orbital ordering 2 which is very important in explaining different types of AFMordering in the insulating phases [10, 15] but plays less essential role in the FM state considered here. To take into account strong Coulomb interaction between e electrons which excludes g the double occupancy of e electrons for a lattice site we employ the Hubbard op- g erator technique. The spectrum of spin waves in the FM state is calculated by employing equations of motion for the matrix Green function (GF) for the localized and itinerant spins. In the next Section the model and general formalism for the GF are presented. The spin-wave spectrum in a generalized mean field approximation (MFA) is calculated in Sec. 3 and self-energy corrections and spin-wave damping are evaluated in Sec. 4. 2 The model We consider an effective Hamiltonian of the generalized FM Kondo model which can be written in the following form [9]: J 1 H = − t Xσ0X0σ − H S σ + J S S . (1) ij i j 2S i i 2 ij i j i,j,σ i i,j X X X The first term of Eq. (1) describes an electron hopping between Mn-ions where Xσ0 i is the creation operator of an electron with spin σ in one of the e orbitals. Here we g neglect orbitaldegeneracy ofe electrons and introduce orbitalindependent hopping g parameter t with t = t for the nearest neighbors. The second term describes the ij ij ferromagnetic Hund coupling (J > 0) between e and t spins where S refers to H g 2g i the localized Mn core spin S = 3/2. The third term describes the antiferromagnetic coupling of localized spins between the nearest neighbor sites. In real materials the coupling of core spins is not the same in different directions and should be described in the matrix form, but for simplicity we are analyzing the isotropic case (J = J). ij We exclude the doubly occupied e state from the effective Hamiltonian by using g the Hubbard operator representation because the electron-electron interaction has the largest energy scale (intra-atomic Coulomb interaction in the e orbitals) and g can be estimated as 7 − 8 eV while J ∼ 1 eV. Due to large Hund energy we H neglect superexchange interaction between e electrons of the order of t2/U [10]. g The conduction bandwidth is smaller than the Hund coupling energy and from density-functional studies can be estimated as t ≃ 0.15 eV [16]. The HO’s in Eq. (1) are defined as Xαβ = |i,αihi,β| for three possible states at i the lattice site i: |i,αi = |i,0i, |i,σi for an empty site and for a singly occupied site with spin σ = (↑,↓) = (+,−) . The completeness relation for the HO’s reads as X00 + Xσσ = 1. (2) i i σ X For itinerant electrons the spin and density operators in Eq. (1) are expressed by HO’s as 1 σ+ = X↑↓, σ− = X↓↑, σz = (X↑↑ −X↓↓), n = X↑↑ +X↓↓. (3) i i i i i 2 i i i i i 3 The HO’s obey the following commutation relations Xαβ,Xγδ = δ δ Xαδ ±δ Xγβ . (4) i j ij βγ i δα i ± h i (cid:16) (cid:17) In Eq.(4) the upper sign stands for the case when both HO’s are Fermi-like ones (as, e. g., X0σ). The spin and density operators (3) are Bose-like and for them the i lower sign in Eq.(4) should be taken. It is assumed that the core spin operators Sα obey the standard commutation i relations, e.g., S+,S− = 2δ Sz . (5) i j i,j j h i To treat the fluctuations of localized and itinerant spins at the same level of approximation we introduce the dynamic spin susceptibility (DSS) of the system in the matrix form χ χ χ(q,ω) = 11 12 = hhA |A+ii , (6) χ χ q q ω 21 22 ! where σ+ A = q , A+ = σ− S− . q S+ q q q q ! (cid:16) (cid:17) Here ∞ 1 hhA |A+ii = −ı dte−ıωt e−ıq(l−m)h[A (t),A+]i (7) q q ω N l m Z0 q X denotestheFouriertransformedtwo-timeretardedcommutatorGreenfunction(GF) [18, 19]. The diagonal elements χ (q,ω) and χ (q,ω) stands for the itinerant and 11 22 core spin GF, respectively, while the nondiagonal elements χ (q,ω) and χ (q,ω) 12 21 define the crosscorrelations between the two spin subsystems. The GF (6) obeys the following equation of motion ωhhA |A+ii = h[A ,A+]i+hhıA˙ |A+ii , q q ω q q q q ω ωhhıA˙ |A+ii = h[ıA˙ ,A+]i+hhıA˙ |−ıA˙+ii . (8) q q ω q q q q ω These equations (8) could be easily combined in a more convenient form of the equation of motion [19]: ωhhA |A+ii = h[A ,A+]i q q ω q q 1 + h[ıA˙ ,A+]i+hhıA˙ |−ıA˙+iiirr · ·hhA |A+ii , (9) q q q q ω h[A ,A+]i q q ω (cid:16) (cid:17) q q where the current is defined as ıA˙ = ıdA/dt = [A,H] and in the matrix form can be given by the following expression: ıσ˙+ ıA˙ = q , (10) q ıS˙+ q ! and hhıA˙ |−ıA˙+iiirr = hhıA˙ |−ıA˙+ii q q ω q q ω 4 −hhıA˙ |A+ii hhA |A+ii−1hhA |−ıA˙+ii (11) q q ω q q ω q q ω is the irreducible part of the higher order GF. We can rewrite (9) in the Dyson form χ (ω) = [ωτˆ −Ω˜ −Π˜(q,ω)]−1·I, (12) q 0 q where τˆ is the unity matrix and 0 h[σ+,σ−]i h[σ+,S−]i 2hσzi 0 I = h[A ,A+]i = q q q q = (13) q q h[S+,σ−]i h[S+,S−]i 0 2hSzi q q q q ! ! where hσzi = hσzi and hSzi = hSzi . l l The matrix Ω˜ = Ω I−1 describes the mean field (MF) energy spectrum and q q Π˜(q,ω) = Π(q,ω) I−1 is the self-energy matrix. They are given by h[ıσ˙+,σ−]i h[ıσ˙+,S−]i Ω = h[ıA˙ ,A+]i = q q q q , (14) q q q h[ıS˙+,σ−]i h[ıS˙+,S−]i ! q q q q hhıσ˙+|ıσ˙−iiirr hhıσ˙+|ıS˙−iiirr Π(q,ω) = hhıA˙ |−ıA˙+iiirr = q q q q , (15) q q hhıS˙+|ıσ˙−iiirr hhıS˙+|ıS˙−iiirr ! q q q q with J ıσ˙+ = t (X↑0X0↓ −X↑0X0↓)− H(S+σz −Szσ+) , (16) l il i l l i 2S l l l l i X J ıS˙+ = − H(Szσ+ −S+σz)− J (SzS+ −SzS+) . (17) l 2S l l l l il i l l i i X 3 Mean field approximation Let us now examine the spectrum and DSS in mean field approximation (MFA). The spin-wave dispersion is determined by the following equation det ωτˆ −Ω˜ = 0 . (18) 0 q (cid:16) (cid:17) From (14) we obtain for the matrix elements of Ω˜ q [d+a(1−γ )]/2hσzi −d/2hSzi Ω˜ = q , (19) q −d/2hσzi [d−b(1−γ )]/2hSzi q ! where we are using the following notation: J d = H(2hσzSzi+hσ+S−i) , (20) 2S l l l l ↑ ↓ a = zt(n +n ) , b = zJN , (21) 1 1 1 5 with t = ztγ , γ = (2/z)(cosq +cosq +cosq ), where z = 6 for the simple three- q q q x y z dimensional cubic lattice with nearest-neighbor hopping t. In ( 21) the nearest neighbor particle-hole and spin correlation functions are defined as follows 1 nσ = γ nσ , nσ = hXσ0X0σi 1 N k k k k k k X 1 N = γ N , N = 2hSzSz i+hS+S−i . (22) 1 N k k k k −k k k k X Theequation(18)hastwosolutionsdescribing twobranchesofspinwaveexcitations: E1(2) = 1 Ω˜11 +Ω˜22 ∓ Ω˜11 −Ω˜22 2 +4Ω˜12Ω˜21 . (23) q 2 " q q r q q q q # (cid:16) (cid:17) For the model calculation we can expand this equation at q → 0 and for the finite value of d we obtain E(1) ≃ D q2, q 1 E(2) ≃ ∆+D q2 , (24) q 2 where E(1) corresponds to the gapless (acoustic) spin-wave excitation with the stiff- q ness D given by 1 a−b D = , (25) 1 12(hSzi+hσzi) and E(2) describes the optic mode of the spin fluctuations with the gap ∆ and the q effective stiffness D determined by the following expressions: 2 hSzi+hσzi ∆ = d , 2hSzihσzi ahSzi2 −bhσzi2 D = . (26) 2 12hSzihσzi(hSzi+hσzi) The ferromagnetic acoustic spin-wave becomes unstable when the stiffness D → 0 1 ora−b = 0in Eq.(25). It may happenfor smallconcentration ofitinerant electrons, n ≤ n ≃ 2SJ/t ≃ 0.3. The self-energy corrections considered below (see Eq. (47)) c even increase the critical value n . c The spectrum of spin fluctuations in MFA are given by the spectral functions 1 BMF(q,ω) = − ImχMF(q,ω +ıε) (27) αβ π αβ for the spin susceptibility Ω˜22 −E(1) E(2) −Ω˜22 BMF(q,ω) = 2hσzi q q δ(ω −E(1))+ q q δ(ω −E(2)) , (28) 11  (2) (1) q (2) (1) q  E −E E −E q q q q     6 Ω˜11 −E(1) E(2) −Ω˜11 BMF(q,ω) = 2hSzi q q δ(ω −E(1))+ q q δ(ω −E(2)) , (29) 22  (2) (1) q (2) (1) q  E −E E −E q q q q     d BMF(q,ω) = BMF(q,ω) = δ(ω −E(1))−δ(ω −E(2)) . (30) 12 21 (2) (1) q q E −E q q (cid:16) (cid:17) The spectral functions (28) - (30) obey the following sum rules: +∞ dωBMF(q,ω) = I , αβ αβ −∞ Z +∞ ωdωBMF(q,ω) = Ωαβ = h[ıA˙ ,A+]i , (31) αβ q q q αβ −∞ Z where the matrices I and Ωαβ are given by the Eqs.(13), (14). αβ q 4 Self-energy corrections The next step is to consider the self-energy corrections to the MF spectrum. Taking into account the self-energy corrections the equation for the spectrum transforms into the following form: det ωτˆ −Ω˜ −Π˜(q,ω) = 0. (32) 0 q First we compute the self-energ(cid:16)y matrix elements b(cid:17)y using mode-coupling approximation in terms of the dressed particle-hole and spin fluctuations (see, e. g., Götze et al., [17]). This scheme is essentially equivalent to the self-consistent Born approximation in which the vertex corrections are neglected. The proposed scheme is defined by the following decoupling of the time-dependent correlation functions: hX−0(t)X0+(t)X+0X0−i ≃ hX−0(t)X0−ihX0+(t)X+0i, (33) m i j l m l i j hσz(t)S−(t)σzS+i ≃ hσz(t)σzihS−(t)S+i , (34) i m j l i j m l hSz(t)S−(t)SzS+i ≃ hSz(t)SzihS−(t)S+i . (35) i m j l i j m l The self-energy matrix elements are obtained by using the above defined decoupling scheme (33)and(35)withthespectralrepresentation fortheGF.The diagonal elements involve two contributions: (1) (2) Π (q,ω) = Π (q,ω)+Π (q,ω) . (36) 11(22) 11(22) 11(22) The first one describes fluctuations of the internal degrees of freedom of the given spin subsystem. While the second one stems from the Hund’s term and describes the coupling between itinerant and core spins. For the itinerant spins the first term in Eq. (36) is due to decay of spin fluctuations into particle-hole pair excitations and reads as 1 +∞ +∞ n(ω −ω′)−n(ω ) Π(1)(q,ω) = dω′dω 1 1 11 N 1 ω −ω′+ıε −∞ −∞ Z Z 7 × t2 Aσ(k −q,ω −ω′)Aσ¯(k,ω ) (37) kq 1 1 k,σ X where t = zt(γ − γ ), n(ω) = (eβω + 1)−1, and Aσ(k,ω) is the single-electron kq k k−q spectral function. By using the MF approximation for that, Eq. (A.7), we can integrate over the frequencies in Eq. (37) and obtain the following estimation: 1 n(εσ )−n(εσ¯) Π(1)(q,ω) = t2 (1−nσ)(1−nσ¯) k−q k . (38) 11 N kq ω +εσ −εσ¯ +ıε k,σ k−q k X It has the standard form for a one-loop particle-hole contribution to the self-energy (see, e.g. [20]). The second terms in Eq.(36) are the same for both subsystem and coincides with the nondiagonal elements of the self-energy matrix due to the Hund coupling (2) Π (q,ω) = −Π (q,ω) = Π (q,ω) (39) 11(22) 12(21) H with J 2 1 +∞ +∞ 1+N(ω′ −ω )+N(ω ) Π (q,ω) = H dω′dω 1 1 H 2S Nπ2 1 ω −ω′ +ıε (cid:18) (cid:19) Z−∞ Z−∞ × [Imχz (k,ω )Imχ (k −q,ω′−ω )+Imχ (k,ω )Imχz (k −q,ω′−ω )] (40) 22 1 11 1 22 1 11 1 k X where N(ω) = (eβω−1)−1, χz and χz denotes the longitudinal susceptibility of the 11 22 itinerant and core spins, respectively. (1) Let us consider now the remaining Π term which describes the fluctuations 22 in the core spin subsystem. This contribution is due to the Heisenberg exchange between the localized spins and is given by 1 +∞ +∞ 1+N(ω′ −ω )+N(ω ) Π(1)(q,ω) = dω′dω 1 1 22 Nπ2 1 ω −ω′+ıε −∞ −∞ Z Z × J2 Imχz (k −q,ω )Imχ (k,ω′ −ω ) , (41) kq 22 1 22 1 k X where J = zJ(γ −γ ). kq k k−q In order to evaluate the longitudinal susceptibility in Eqs. (40), (41) for both subsystems we will use for them the simplest one-loop approximation (see, e.g. [20]). In this approximation the imaginary part of χz (q,ω) is given as the convolution of 11 the single-electron GFs 1 1 +∞ − Imχz (q,ω) = dω′[n(ω′ −ω)−n(ω′)] π 11 4N −∞ Z × Aσ(k,ω′)Aσ(k −q,ω′−ω). (42) k,σ X The imaginary part of the core spin susceptibility χz can be expressed in the linear 22 spin-wave approximation as 1 1 +∞ − Imχz (q,ω) = dω′(N(ω′ −ω)−N(ω′)) π 22 π24S2N −∞ Z 8 Imχ (k −q,ω′−ω)Imχ (k,ω′) , (43) 22 22 k X which follows directly from the Holstein-Primakoff representation. To study the spin wave spectrum including self-energy corrections let us consider the static limit for q → 0. For the self-energy matrix we can write −Aq2 −d d limΠ(q,0) = 1 1 (44) q→0 d1 −Bq2 −d1 ! where (2) (2) Π (q,0) Π (q,0) A = −lim 11 , B = −lim 22 , d = −ΠJH(0,0). (45) q→0 q2 q→0 q2 1 Here the coefficient A,B, and d are positive since Π(q,0) < 0 in the second 1 order of the perturbation theory. As it follows from Eqs. (44) and (45), the self- energy corrections coming from the Hund’s coupling does not renormalize the spin stiffness and gives input only into the gap. Hence the spin-wave spectrum in the longwavelegth limit can be written as ω(1) ≃ D˜ q2 q 1 ω(2) ≃ ∆˜ +D˜ q2 (46) q 2 where the renormalized spin stiffness and the gap are given by a−b−A−B D˜ = 1 12(hSzi+hσzi) (a−A)hSzi2 −(b+B)hσzi2 D˜ = (47) 2 12hSzihσzi(hSzi+hσzi) (d−d )(hSzi+hσzi) ∆˜ = 1 . 2hSzihσzi Now we consider the damping for the acoustic spin wave mode given by the imaginary parts of the self-energy matrix. For the damping induced by particle-hole excitations we get from Eq. (38) in the MF approximation for the single-electron GFs: (1) (1) Γ (q,ω) = −ImΠ (q,ω +ıε) 11 11 π = t2 (1−nσ)(1−nσ¯)[n(εσ )−n(εσ¯)]δ(ω +εσ −εσ¯) (48) N kq k−q k k−q k k,σ X The contribution, due to the finite k−independent gap in single-electron spectrum (1) in the ferromagnetic state disappears in the low frequency limit, Γ (q,ω) = 0 for 11 ω < h (see Eq. (A.14)). The damping due to the antiferomagnetic exchange interaction given by the (1) imaginary part of the self-energy Π (q,ω), Eq. (41), gives a small contribution 22 proportional to q2 in the longwavelength limit and can be disregarded due to small antiferromagnetic exchange interaction J. 9 The largest contribution to the damping of spin waves is given by the imaginary part of the self-energy due to the Hund coupling, Eq. (40): J 2 1 +∞ H Γ (q,ω) = −ImΠ (q,ω +ıε) = dω (1+N(ω −ω )+N(ω )) H H 1 1 1 2S πN (cid:18) (cid:19) Z−∞ × [Imχ (k −q,ω −ω )Imχz (k,ω )+Imχ (k −q,ω−ω )Imχz (k,ω )]. (49) 22 1 11 1 11 1 22 1 k X Here for the imaginary parts of the spin susceptibilities χ (k,ω) and χ (k,ω) we 11 22 will use their MF values in Eqs. (28), (29) taking into account only the acoustic E(1) q mode: 1 Ω˜22 −E(1) − ImχMF(q,ω) ≃ 2hσzi q q δ(ω −E(1)) = Λ11δ(ω −E ) , (50) π 11 E(2) −E(1) q q q q q 1 Ω˜11 −E(1) − ImχMF(q,ω) ≃ 2hSzi q q δ(ω −E(1)) = Λ22δ(ω −E ) . (51) π 22 E(2) −E(1) q q q q q For the longitudinal spin susceptibilities in (49) we will use Eqs. (42), (43). After integration over ω we get the following result 1 J 2 π Γ (q,ω) = (eω/T −1) H Λ22 (1−n )2 H 2S 4N2 k−q σ (cid:18) (cid:19) kX,k1,σ ×N(E )n(εσ )[1−n(εσ )]δ(ω −E +εσ −εσ ) k−q k1 k1−k k−q k1−k k1 J 2 π +(eω/T −1) H Λ11 Λ22Λ22 2S 4S2N2 k−q k1 k1−k (cid:18) (cid:19) kX,k1 (1) (1) ×N(E )N(E )[1+N(E )]δ(ω −E +E −E ) . (52) k−q k1 k1−k k−q k1−k k1 It describes a spin wave damping due to its decay into an electron-hole pair and another spin wave (the first term) and a three spin-wave scattering process (the second term). The latter has a standard form for three magnon scattering (see, (31.2.20) in [21]). At low energy regime (ω → 0) and at the longwavelenght limit the requirements for conservation of energy and momentum allow only small wave vectors and thus only small energies. Hence we can consider Λ11 and Λ22 as k- k k independent. In the limit k → 0 we obtain hσzi hSzi Λ11 ≃ Λ11 = 2hσzi , Λ22 ≃ Λ11 = 2hSzi . (53) k 0 hSzi+hσzi k 0 hSzi+hσzi In this approximation from the Eq. (52) follows that at low energies, (ω ≪ T), the damping has a linear ω-dependence, ΓJH(q,ω) ∼ ω, and does not depend explicitly on the wave vector q. In the low temperature region, (ω ≪ T ≪ ω ) where ω 0 0 is the maximal acoustic spin-wave frequency, estimations for Γ (q,ω)/ω show H that the first contribution in Eq. (52) is proportional to (T/ω )(J2/N(ǫ )v k ) 0 H F F 0 where N(ǫ ) is the density of state at the Fermi level and v is the Fermi velocity, F F k ≃ 2π/a and a is a lattice constant. The second term gives to the damping the 0 contribution proportional to (T/ω )(J /ω )2 . To give more accurate estimations 0 H 0 for the spin-wave damping numerical studies should be performed which will be considered elsewhere. 10