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Double-exchange is not the cause of ferromagnetism in doped manganites

The coexistence of ferromagnetism and metallic conduction in doped manganites has long been explained by a double-exchange model in which the ferromagnetic exchange arises from the carrier hopping. We evaluate the zero-temperature spin stiffness D(0) and the Curie temperature T_{C} on the basis of the double-exchange model using the measured values of the bare bandwidth W and the Hund's rule coupling J_{H}. The calculated D(0) and T_{C} values are too small compared with the observed ones even in the absence of interactions. A realistic onsite interorbital Coulomb repulsion can reduce D(0) substantially in the case of a 2-orbital model. Furthermore, experiment shows that D(0) is simply proportional to x in La_{1-x}Sr_{x}MnO_{3} system, independent of whether the ground state is a ferromagnetic insulator or metal. These results strongly suggest that the ferromagnetism in manganites does not originate from the double-exchange interaction. On the other hand, an alternative model based on the d-p exchange can semi-quantitatively explain the ferromagnetism of doped manganites at low temperatures.

The discovery of "colossal" magnetoresistance in thin films of the manganite perovskites Re 1−x D x MnO 3 (Re = a rare-earth element, and D = a divalent element) [1,2] has stimulated extensive studies of magnetic, structural and transport properties of these materials [3]. The coexistence of ferromagnetism and metallic conduction has long been explained by the double-exchange (DE) model [4,5], where the effective hopping for the manganese 3d conduction electrons varies with the angle between the manganese core electrons due to a strong Hund's coupling. However, Millis et al. [6] proposed that, in addition to the double-exchange, a strong electron-phonon interaction arising from a strong Jahn-Teller effect should be involved to explain the basic physics of manganites. In this modified model, the primary cause of the ferromagnetism of doped manganites is still the double-exchange interaction.
In the DE model, it is implicitly assumed that doped carriers are Mn e g electrons. This assumption is not justified by both electron-energy-loss [7] and photoemission spectroscopies [8], which have shown that the ferromagnetic manganites (x<0.4) are doped charge-transfer insulators with carriers mainly residing on the oxygen orbitals. Now a question arises: Does the ferromagnetism of doped manganites really originate from the DE interaction? If not, what causes the ferromagnetism in these compounds? One way to address this fundamental issue is to make a quantitative comparison between the predicted properties of the DE model and experiment.
There are two important parameters in the DE model, namely, the bare bandwidth W of the e g bands, and the Hund's rule coupling J H between e g and t 2g electrons. These parameters are related to an optical transition between the exchange splitted e g bands [9,10], and thus can be determined from optical data. With these unbi-ased parameters, one can calculate the zero-temperature spin stiffness D(0) and T C . Here D(0) is defined as ω q = D(0)q 2 with ω q being the magnon frequency. When one introduces interactions such as electron-phonon and electron-electron interactions, the magnitudes of both D(0) and T C are generally reduced.
Here we use the measured values of the bare bandwidth W and the Hund's rule coupling J H to calculate D(0) and T C on the basis of the double-exchange model. The calculated D(0) and T C values are too small compared with the observed ones. A realistic onsite interorbital Coulomb repulsion can reduce D(0) substantially in the case of a 2-orbital model. Moreover, experimental data show that D(0) is simply proportional to x in La 1−x Sr x MnO 3 system, independent of whether the ground state is a ferromagnetic insulator or metal. These results provide strong evidence that the ferromagnetism in manganites is not caused by the DE interaction. On the other hand, an alternative model based on the d − p exchange can well explain the ferromagnetism of doped manganites. Now we start with a Kondo-lattice type Hamiltonian [11], which leads to Zener's DE model when J H → ∞, Here d † iaα creates an electron in e g orbital a with spin α, t ab ij is the direction-dependent amplitude for an electron to hop from orbital a to orbital b on a neigboring site, and H IN T represents the other interactions. The calculated band structure is well fit by a t ab ij , which involves only nearest-neighbor hopping that is only nonzero for one particular linear combination of orbitals, i.e., t ab ij ∝ t, where t is a characteristic hopping amplitude that is related to the bare bandwidth W by W = 4t. Here we still call Eq. 1 as 2-orbital DE model rather than 2-orbital Kondo-lattice model for convienence. The quantum and thermal average of the hopping term in Eq. 1, defines a quantity K: The quantity K is related to the optical spectral weight by a familar sum rule, where σ 1 is the real-part optical conductivity contributed only from the e g electrons, and a • is the lattice constant. The quantity K generally consists of the Drude part K D , and incoherent part K I which, in general, involves interband and intraband optical transitions. The Drude part K D can be related to the plasma frequency Ω p as On the basis of Eq. 1, Quijada et al. [11] showed that, to the order of 1/J H , the spin stiffness D(0) is, where S = 3/2, S * = S + (1 − x)/2, and η = 1.04 when H IN T = 0. The presence of interactions may change the value of η. A similar result was obtained by Furukawa [9] for an 1-orbital DE model using the dynamical mean field method, but the value of η is doping dependent and less than 1. From the above equations, one can calculate K and D(0) using realistic values of the bare bandwidth W and the Hund's rule coupling J H . Both the local density approximation (LDA) [12] and "constrained" LDA [13] calculations show that J H ≃ 1.5 eV. The calculated J H value is very close to the atomic values for 3d atoms. This is reasonable because J H is not screened when the ion is put in a solid. The bare bandwidth W cannot be calculated reliably for 3d-metal based compounds due to a strong correlation effect. Fortunately, the values of both W and J H can be determined from an optical transition between the exchange splitted e g bands [9,10]. The peak position of this optical transition is about 2J H , and the width of the peak contains information about the bare bandwidth [9,10]. From the optical data of Ref. [11,14], one finds J H ≃ 1.6 eV and W = 1.6-1.8 eV by comparing the data with the theoretical predictions [9,10]. The value of J H obtained from the optical data is in excellent agreement with the calculated one. This implies that the feature appeared at about 3 eV in the optical data indeed arises from the optical transition between the exchange splitted e g bands.
The quantity K • for noninteracting 2-orbital model can be evaluated when the bare bandwidth W is known. Takahashi and Shiba [15] have calculated the optical conductivity using a tight binding (TB) approximation of the band structure. From their calculated result for the interband optical conductivity, we evaluate that K • I = 0.088t for x ∼ 0.3. One should also note that the magnitude of t defined in Ref. [15] is 1.5 times smaller than the t defined here. Since K • D = 1.2K • I [15], then K • D = 0.106t and K • = 0.194t. The LDA calculation for a cubic and undistorted structure shows that [16] W = 3 eV and hΩ • p = 1.9 eV. Using Eq. 4 andhΩ • p = 1.9 eV, one yields K • D = 78.6 meV. Since t = W /4 = 0.75 eV, one readily finds that K • D = 0.105t, in remarkably good agreement with that (K • D = 0.106t) estimated from the TB approximation. This justifies the relation K • = 0.194t obtained from the TB approximation. It is interesting to compare the present result with those reported in Ref. [10] and [11]. In Ref. [10], it is found that K • = 0.34t for x = 0.3 using the dynamic mean field method. In Ref. [11], Quijada et al., claimed K • = 0.46t for x = 0.3, which might be true if the two e g bands have the same dispersion. Therefore the quantity K • is significantly overestimated in Ref. [10] and [11].
When the Hund's coupling J H is turned on, the quantity K is reduced compared with K • . For J H = ∞ and x = 0.3, K = 0.77K • [10,17]. It was also shown that [10] the reduction factor (0.77) is basically the same for J H ≥ t. Therefore, we have K = 0.147t for x = 0.3. Using t = 0.4 eV, J H = 1.6 eV, and K = 0.147t, we yield D(0) = −25 mVÅ 2 from Eq. 5. The negative value of D(0) implies that the ferromagnetism is not sustainable with these unbiased parameters. It might be possible to have a small positive D(0) if one includes higher order terms in Eq. 5. Nevertheless, the theoretical D(0) value is too small compared with the measured ones (160-190 mVÅ 2 ) [18][19][20].
Now we turn to the calculation of T C for x = 0.3. For an 1-orbital DE model with J H = ∞, the dynamic mean field (DMF) calculation shows that T MF , or T C = 0.03t/k B [22]. This implies that the DMF calculation overestimates T C by a factor of about 2 due to the neglection of fluctuations. We would like to mention that the magnitude of t defined in Ref. [21] and [22] is 3 times smaller than the t defined here. For realistic parameters t = 0.4 eV, J H = 4t = 1.6 eV, the DMF calculation obtains T MF C = 0.038t/k B = 180 K [9]. Considering the fact that the DMF method can overestimate T C by a factor of 2, one has T C ∼ 100 K. For the 2-orbital DE model, the DMF calculation shows [10] T MF C = 0.07t/k B = 324 K. Since the quantity K calculated in Ref. [10] is overesti-mated by a factor of about 2 (as discussed above), and the DMF method itself can overestimate T C by about 2 times, the real T C should be about 100 K, which is comparable with the value for the 1-orbital model. This is reasonable because the bare values of K • for both 1-and 2-orbital models happen to be very similar in the case of x = 0.3. Therefore, both the 1-and 2-orbital DE models cannot explain the observed T C values with the unbaised parameters.
The above calculations have not taken into account any other interactions such as electron-phonon interaction and electron-electron correlation. The electron-phonon interaction can substantially reduce the K and thus D(0) if the coupling constant λ = E p /2t is greatly larger than 1 [23,24]. In reality, the polaron binding energy E p in manganites is estimated to be about 1 eV [25]. So λ ≃ 1, which suggests that the electron-phonon coupling does not lead to a sizable decrease in D(0). On the other hand, the electron-electron correlation in the 2-orbital model can lead to a large reduction in K, as demonstrated by Horsch and coworkers [26]. The parameter U = U ′ -J ab (where U ′ is the onsite interorbital Coulomb repulsion, and J ab is the interorbital Hund's coupling) has a strong influence on the value of K [26]. With realistic values of U = 3 eV [26,27] and t = 0.4 eV, K can be reduced by a factor of about 2 compared with the bare one [26]. Although their results are for 2-dimensional finite clusters, we would expect a similar result for a real 3-dimensional system.  [28], and the other data from Ref. [29]. It is evident that D(0) is simply proportional to x, independent of whether the ground state is a ferromagnetic insulator (e.g., x = 0.15), or a ferromagnetic metal (e.g., x = 0.175, 0.20, 0.30).
In addition, it is striking that the observed spin stiffness varies only with x, namely, D(0) ∝ x, as seen clearly from Fig. 1. The doping dependence of D(0) shown in Fig. 1 could be qualitatively explained by the 1-orbital DE model [9,30] where D(0) ∝ K ∝ x(1 − x) for J H = ∞ [30], as plotted in Fig. 2. Nevertheless, the doping dependence of D(0) is very different from that prediced from the 2-orbital model. By analogy to the 1-orbital model, one can easily show that D(0) ∝ K ∝ (1 − x)(1 + x) for the 2-orbital model, as demonstrated in Fig. 2. Here we have implicitly assumed that the two e g bands have the same dispersion. The solid circles in Fig. 2 are the results evaluated using the dynamical mean field method with J H = ∞ [10]. It is evident that the analytical expression is in excellent agreement with the numerical result. Comparing the results shown in Fig. 1 and Fig. 2, one clearly sees that the 2-orbital model cannot explain the observed doping dependence of D(0). The question is why the DE model cannot explain the ferromagnetism in doped manganites. As mentioned above, the DE model implicitly assumes that doped carriers are Mn e g electrons, which is not the case according to the electron-energy-loss and photoemission spectra [7,8]. Furthermore, the "constrained" LDA calculation [13] shows a large onsite Coulomb repulsion of about 8-10 eV, in agreement with the photoemission data [8]. The simple LDA calculation which ignores the strong correlation effect shows a bare plasma frequency of about 1.3 eV for x = 0.33 with a distorted structure determined by neutron scattering [31]. The bare plasma frequency calculated is much smaller than the one observed in Nd 0.7 Sr 0.3 MnO 3 (3.3 eV) [32]. The large bare plasma frequency observed in this material is consistent with the fact that doped holes reside mainly on the oxygen orbitals with a large bandwidth. The bare plasma frequency of about 3.3 eV for single conduction band (oxygen band) implies a bare K • of about 0.24 eV, which gives an upper limit for the K in the presence of interactions. The electron-phonon interaction with a coupling constant λ ∼ 1 will reduce the K slightly [23,24], but can significantly decrease the Drude weight. Optical data indeed show that the K for Nd 0.7 Sr 0.3 MnO 3 is about 0.2 eV [11], while the effective plasma frequency is about 0.57 eV [33]. The Drude weight is reduced by a factor of 33, implying small polaronic carriers in the low-temperature ferromagnetic state.  [13]. The energy scales are consistent with the optical data [11,33,37]. The density of the oxygen holes is equal to x per cell while the densities of the Mn 2+ and Mn 4+ ions are the same due to the charge disproportion (2Mn 3+ → Mn 2+ + Mn 4+ ) [36].
In Fig. 3, we plot a schematic band structure for doped manganites (x< 0.4), which is extracted from the LDA + U calculation. Here we have assumed that the local Jahn-Teller distortions still survive upon doping, but the average magnitude of the distortions decreases, in agreement with the experiments [34,35]. The doping with a divalent element shifts down the e 2 g band due to the decrease of the Jahn-Teller distortions. The density of the oxygen holes is equal to x per cell, while the electron carrier density in the majority e 2 g band (corresponding to the density of the Mn 2+ ions) is the same as the hole carrier density in the majority e 1 g band (corresponding to the density of the Mn 4+ ions). In other words, the doping does not change the average valence of the Mn ions, but lead to the charge disproportion (2Mn 3+ → Mn 2+ + Mn 4+ ). This is because the quench of the static Jahn-Teller distortions by doping makes the Mn 2+ -Mn 4+ pairs more stable than the Mn 3+ -Mn 3+ pairs [36]. The current band structure is consistent with the optical transitions at the photon energies of about 1.5 eV, 3.0 eV and 4.5 eV [11,33,37]. The optical spectral weight for the 4.5 eV transition should be much larger than for the 1.5 eV transition, as observed [37]. This is because the unoccupied state density for the former optical transition (i.e., 3 minority t 2g and 2 minority e g states per cell) is at least 5 times larger than that for the latter one (i.e., less than 1 majority e g states per cell). The optical transition at about 3 eV is related to the transition between the exchange splitted e g bands.
What is an alternative model for the ferromagnetism in doped manganites? If we consider an oxygen hole (spin 1/2) sitting in between two Mn ions, an exchange interaction between the oxygen and Mn spins (d−p exchange) will lead to a ferromagetic interaction between Mn spins [38]. In this case, the ferromagnetic exchange energy between two Mn spins is J = C(t 4 pd /∆ 3 ct ), where C is a numerical factor of order of 1, t pd is a hybridization matrix element between the d and p orbitals, and ∆ ct is a charge transfer gap. The long-wave spin stiffness for a cubic material is [39] where J ij is the exchange energy between pairs of spins at sites R i and R j , and the sum is over six neighboring sites. Clearly, J ij = J if there is an oxygen hole in between two Mn ions, while J ij = 0 if not. Since the oxygen hole density is equal to x per cell, then D(0) = (2/3)xJS * a 2 • , independent of whether the holes are localized or not. This mechanism can naturally explain why D(0) is simply proportional to x, as shown in Fig. 1. Using the Slater-Koster parameter V pdσ = 1.8 eV obtained from the analysis of photoemission data [8], or from the TB fit to the LDA band structure [40], we yield t pd = −( √ 3/2)V pdσ = −1.56 eV [41]. Taking ∆ ct = 4.5 eV [8,11,37], t pd =−1.56 eV, C = 1, and x = 0.3, we obtain D(0) = 0.5 eVÅ 2 , which is larger than the observed values by about three times. This is not unreasonable since the numerical factor C might be less than 1. A more quantitative calculation of J based on the d − p exchange is essential to address this issue.
It should be noted that the ferromagnetic "bonds" with the exchange energy J will be randomly distributed over the real space if the oxygen holes are localized, whereas the distribution of the bonds will become more homogeneous when the holes are more mobile (a motion narrowing effect). This can account for a conventional spin-wave dispersion in high T C materials [39] where the conductivity is high, and an unconventional magnon softening near the zone boundary for lower T C compounds [19] where the conductivity is lower. When the oxygen holes are ordered (charge-ordering), as observed in La 0.85 Sr 0.15 MnO 3 [28], the ferromagnetic bonds will form a "superlattice", leading to a splitting of spin-wave dispersion at a wave vector equal to the incommensurability of the superlattice as observed [28].
The above simple model would imply that the Curie temperature T C should be proportional to x. In reality, T C is strongly dependent on the cation radius r A of the Re 1−x D x site of the perovskite Re 1−x D x MnO 3 even if x is fixed [42]. Moreover, a giant oxygen isotope shift of T C has been observed in these compounds [43,44]. Therefore, the simple d − p exchange model cannot account for these unusual phenomena. Recently, Alexandrov and Bratkovsky [45] have proposed that, in addition to the d − p exchange interaction, there is a strong electronphonon interaction that may lead to the formation of small (bi)polarons. In the paramagnetic state, the singlet bipolarons (spin zero) are stable and the ferromagnetic interaction is produced by the thermally excited polarons (spin 1/2). Thus, the Curie temperature T C is selfconsistently determined by the polaron density at T C . Within this scenario, T C can strongly depend on the electronphonon coupling strength and the isotope mass [45], in agreement with experiment [43,44]. At zero temperature, all the carriers will be polarons if J pd S is much larger than the bipolaron binding energy ∆, where J pd is the exchange energy between Mn and oxygen-hole spins. In this case, D(0) ∝ x, as in La 1−x Sr x MnO 3 system. On the other hand, there is a mixture of polarons and bipolarons if J pd S is slightly larger than ∆. Then the relation D(0) ∝ x does not hold, and D(0) is proportional to the density of polarons which is less than x per cell. One should also expect that the mixture of mobile polarons and localized bipolarons would lead to a dynamic/static phase separation since bosonic and fermionic carriers tend to separate in real space.
Finally we would like to address whether the band structure shown in Fig. 3 can be consistent with an effective 1-orbital DE model with a large t ∼ 1.2 eV. We can show that in order to have a T C of 380 K with such a t, one needs an unphysically large value of J H , i.e., J H > 5 eV. Since the d − p exchange model can well explain the ferromagnetism, it would not be neccessary to have such an effective DE model for describing the physics of manganites. We believe that the d−p exchange + (bi)polaron model [45] is sufficient to account for the essential physics in manganites.