Effect of softening on colossal manganites

Physica B 324 (2002) 286–291

Effect of softening on colossal manganites Shih-Jye Suna,*, Wei-Chun Lub, Hsiung Choub a

Department of Electronic Engineering, Cheng Shiu Institute of Technology, Kaohsiung, Taiwan b Department of Physics, National Sun Yat-Sen University, Kaohsiung, Taiwan Received 30 May 2002; accepted 18 July 2002

Abstract The softening effects of ferromagnetic magnon on some ferromagnetic semiconductors and colossal magnetoresistance manganites have attracted much attention. Such effect can be calculated from the single-orbital ferromagnetic Kondo lattice model in proper conducting carrier numbers utilizing the equation of motion method with one magnon excitation and random phase approximations. However, if we take into account the Coulomb repulsion and use the Gutzwiller projection method to transfer this repulsion force to conducting bandwidth modulation, the softening effects disappear. This paper describes qualitively the effect of softening on properties of different colossal manganites. r 2002 Elsevier Science B.V. All rights reserved. PACS: 75.30.Vn; 75.40.Gb Keywords: Spin wave; Kondo lattice model; Double exchange

1. Introduction Many magnetic systems of current experimental interests, such as ferromagnetic semiconductors [1] and the colossal magnetoresistance (CMR) manganites [2], consist of itinerant electrons interacting with an array of localized magnetic moments with spin S: The magnetic and electronic properties of manganese oxides are believed to arise from the large Coulomb and Hund’s rule interaction of the manganese d shell electrons. Due to the almost octahedral coordination within the perovskite structure, the d levels split into two subbands, eg and t2g ; labeled according to their symmetry. In the case of zero doping ðx ¼ 0Þ; each Mn site *Corresponding author. Tel.: +886-728-26824. E-mail address: [email protected] (S.-J. Sun).

contains four electrons that fill up the three t2g levels and one eg level, forming a S ¼ 2 spin state. Doping of divalance elements removes the electrons from the eg level and forms a hole which becomes an itinerant bridging oxygen site. However, this hopping is constrained in a background of local spins S ¼ 3=2 formed by the t2g electrons, and its amplitude depends on the overlapping of the spin states in the neighboring sites. These systems can be described by the double exchange (DE) model or the ferromagnetic Kondo lattice model (FKLM) [3–5]. This model comprises a single tight binding band of electrons interacting with localized core spins by a ferromagnetic (Hund’s rule) exchange interaction JF bt X X þ tij ðCj;s Ci;s þ hcÞ  JF S i  si ð1Þ /i;jS;s

0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 1 4 1 4 - X

i

S.-J. Sun et al. / Physica B 324 (2002) 286–291

P where the /i; jS is restricted to the nearestneighboring sites, itinerant electrons spin density P si ¼ ab cþ ia tab cib with t being Pauli matrices, and Si is the on-site local magnetic moment. In this DE model, it is assumed that the hopping of eg electrons between neighboring sites is easier if the local spin on the sites is parallel. An effective ferromagnetic coupling between the local spins is induced by the conduction electrons that lower their kinetic energy. Therefore, in the classical point of view, the ground state of this model in multidimensional system must be ferromagnetic. This effect is called the double exchange. According to the conventional DE theory, the spin dynamics of the ferromagnetic state that evolves at temperatures below the Curie temperature TC can be described at a quasi-classical level by an effective nearest-neighbor Heisenberg P ~j with ferromagnetic ex~i  S model, Jeff /ijS S change integral Jeff ¼ t%=4S 2 ; where t% is the expectation value of the kinetic energy per bond in the lattice [5–10]. This picture seems to be reasonably accurate for manganites with high value of TC [11]. However, recent experimental results have shown a strong deviation of the spinwave dispersion (SWD) from the typical Heisenberg behavior. The unexpected softening of the SWD at the zone boundary has been observed in several manganites [12–15]. These observations are very important as they indicate that some aspects of spin dynamics in manganites remains unclear. Some theorists have suggested that this softening may be due to the influence of optical phonons [16]. Khaliullin and Kilian [17] proposed a theory of anomalous softening in ferromagnetic manganites due to the modulation of magnetic exchange bonds by the orbital degree of freedom of doubledegenerate eg electrons. They found out that charged and coupled orbital-lattice fluctuation can be considered as the main origin of the softening phenomena. Solovyev and Terakura [18] have argued that the softening of spin wave at the zone boundary and the increase in spin-wave stiffness constant with doping are purely of magnetic origin. It is well known that there are strong Coulomb repulsions among d orbitals of transition atoms [19]. This study examines how conduction

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bandwidth modulated by the strong Coulomb repulsion between d orbitals electrons influences the magnon softening effect. The intra-Coulomb repulsion X nim nik ð2Þ HU ¼ U i

exists between eg levels and affects itinerant electrons with higher energies. This is contrary to the local spin t2g electrons with much lower energies, which are less influenced by Coulomb repulsion for a restriction of double occupation of itinerant electrons according to the automatic double occupation restriction for core t2g electrons. We only consider a single eg energy level here. Although in d orbitals there are two degenerate eg levels, they will be split far away by some internal fields, e.g., Jahn–Teller effect [9], prominent in colossal manganites.

2. Theory Similar to how the Hubbard model treats large Coulomb repulsion problems [20] we utilize the Gutzwiller projection in the mean field approximation, and transfer Eq. (2) to modulate the hopping amplitude t to teff ¼ tð1  dÞ; where d is the carrier concentration. Naturally as d ¼ 1 (one orbital eg level of Mn element is half filling or manganite is undoped), the effective hopping amplitude will be zero. To study the magnon softening effect, we utilize the equation of motion method under one magnon excitation and random phase approximations (RPA) at zero temperature to formulate this modified FKLM Hamiltonian and to obtain the magnon excitation spectra of simple cubic structure along /1 0 0S: In this system, the magnon interacts with electrons throughout the whole Brillouin zone. In order to obtain these coupled magnon and electrons excitation energies using the equation of motion method, ðd=dtÞAðtÞ ¼ ½H; AðtÞ; under the Tyablikov decoupling scheme [21], we will derive the temperature Green functions //Siþ ; ðSj Þn ðSjþ Þn1 SS for the magnon in the general spin case and //ci;s ; cþ j;s SS for the

S.-J. Sun et al. / Physica B 324 (2002) 286–291

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electrons [22]. They are d //Siþ ; ðSj Þn ðSjþ Þn1 SS dt   d þ  n þ n1 S ; ðSj Þ ðSj Þ ¼ dðtÞdij jn þ ; ð3Þ dt i where jn ¼

To derive the equation of motion for the //Si ci;s ; cþ j;s SS Green function again, we obtain another relation of equation Mns ðion ; q; k  qÞ ¼

d //ci;s ; cþ j;s SS ¼ dðtÞdij þ dt



d ci;s ; cþ j;s dt

ð6Þ

Consequently, by combining Eqs. (5) and (6), we obtain electrons kinetic energies for different spins up and down:

 : ð4Þ

Through Fourier transformation, they are

ekm ¼ 2teff gk 

1 X inn t 1 X iqðRi Rj Þ e e gn ðinn ; qÞ b n N q

and

X

JF Z J2 1 /S S þ F /SZ S N 2 2 /cþ kqk ckqk S þ nq

ekm þ 2teff gkq þ JF /sZ S  ðJF =2Þ/S Z S

q

//Siþ ; ðSj Þn ðSjþ Þn1 SS

ð7Þ and

X

JF Z J2 1 /S S  F /SZ S N 2 2 /cþ kþqm ckþqm S  ðnq þ 1Þ

q

ekk þ 2teff gkþq  JF /sZ S þ ðJF =2Þ/S Z S

ekk ¼ 2teff gk þ

//ci;s ; cþ j;s SS X 1 1 X ikðRi Rj Þ eion t 2 e Gns ðion ; kÞ: ¼ b n N k The electrons in the midway of derivations for the //ci;s ; cþ j;s SS Green function will reduce automatically another higher order Green function //Si ci;s ; cþ j;s SS and its Fourier transformation form is //Si ci;s ; cþ j;s SS ¼

ion  JF /sZ S  2teff rkq  ðJF =2Þ/SZ S

Gns ðion ; kÞ:

/½Siþ ; ðSj Þn ðSjþ Þn1 S

and

¼

 þ JF /S Z S/cþ kq;s ckq;s S þ ðJF =2Þ/Sq Sq S

1 X ion t 1 X iqðRi Rj Þ e e b n N qk eikðRl Rj Þ Mns ðion ; q; kÞ:

After completing the derivation of the equation of motion of Eq. (4), we obtain an relation of equation   JF ion  /S Z S  2teff gk Gns ðion ; kÞ 2 JF 1 X ¼ Mns ðion ; q; k  qÞ  1: ð5Þ 2 N q



; ð8Þ

P respectively, where gk ¼ d eikd ; d is the nearestþ neighboring sites, nq ¼ /Sq Sq S=2/SZ S is the expectation value of the magnon number, and /sZ S and /SZ S are expectation values of magnetizations for itinerant electrons and local magnetic moment, respectively. For rationality and computer time considerations, we take only first-order parts for ekm ¼ 2teff gk  JF =2/SZ S and ekk ¼ 2teff gk þ JF =2/S Z S: Similarly, regarding the magnon excitation through derivations of //Siþ ; ðSj Þn ðSjþ Þn1 SS; we obtain another Green function and its Fourier transformation form  n þ n1 //cþ SS im cik ; ðSj Þ ðSj Þ 1 X inn t 1 X ik1 ðRi Rj Þ e e ¼ b n N2 k k 1 2

eik2 ðRi Rj Þ In ðinn ; k1 ; k2 Þ:

ð9Þ

S.-J. Sun et al. / Physica B 324 (2002) 286–291

Consequently, a relation of In ðinn ; k1 ; k2 Þ and gn ðinn ; qÞ can be obtained as  inn gn ðinn ; qÞ ¼ jn þ JF /SZ S

1 X In ðinn ; k þ q; kÞ N k

 JF /sZ Sgn ðinn ; qÞ:

ð10Þ

 n to derive ðd=dtÞ //cþ im clk ; ðSj Þ we will obtain another relation between In ðinn ; k1 ; k2 Þ and gn ðinn ; qÞ as

Further,

ðSjþ Þn1 SS;

In ðinn ; k þ q; kÞ ¼

þ JF =2ð/cþ kþqm ckþqm S  /ckk ckk SÞ

inn þ JF /S Z S þ 2teff ðgkþq  gk Þ

gn ðinn ; qÞ:

ð11Þ

After coupling Eqs. (10) and (11) with changing imaginary frequency inn to real nn ; we then obtain the magnon excitation at zero temperature: JF2 1 /S Z S N 2 X /cþ c S  /cþ kþq;m kþq;m k;k ck;k S

nq ¼ JF /sZ S þ

k

nq  2teff ðgkþq  gk Þ  JF /S Z S

:

ð12Þ

Because we consider only zero temperature, the magnitude of the magnetization of the core spin

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can be taken as the total spin sum of core electrons, /S Z S ¼ S; because these electrons’ energies are relatively low compared with their chemical potential.

3. Results and discussion To study the qualitative physics of CMR materials in our equations, we take the ferromagnetic coupling constant JF ¼ 0:3 eV; the total spin S ¼ 32 (for Mn element) and the bandwidth of conduction band W ¼ 1:0 eV when the band is empty. These equations are also suitable for some ferromagnetic semiconductor problems. Fig. 1 contains lines belonging to two groups. The upper group exhibits no softening effect for the bandwidth modulated by the strong Coulomb repulsion, which is contrary to the lower group which shows evident magnon softening effect [23]. This can explain the appearance of softening effect in some CMR materials with lower TC (with small bandwidth, i.e, La1x Cax MnO3 ) and not those with higher TC ’s (with large bandwith, i.e, La1x Srx MnO3 Þ [24] due to different degree of influence of Coulomb repulsion. The large bandwidth CMR comprises stronger Coulomb

Fig. 1. Upper group exhibits no softening effect for the bandwidth modulated by the strong Coulomb repulsion, and lower group shows evident magnon softening effect.

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Fig. 2. Magnon excitation energies in different occupied conducting carriers which exhibit that the transition temperatures are rising up quickly then falling down monotonically as carrier numbers increased.

repulsion in eg energy levels than the small bandwidth CMR. The different Coulomb repulsion may come from the different radius of doping di-valence atoms, in the cases of Ca and Sr doping, the larger atom with more electrons cloud have more influence on Coulomb force. Fig. 2 shows the magnon excitation energies which are very sensitive to the chemical potential in different conducting carriers occupied. It reveals that the bandwidth modulated by the Coulomb repulsion has intense effect on our results. According P to Nolting’s calculations, TC ¼ ðð1=NSÞ q 1=nq Þ1 for ferromagnetic transition temperature, TC [23], the spectrum curvatures enclosing larger areas will show higher TC : Our result shows that transition temperatures rise quickly and then fall monotonically with increasing number of carriers. This same qualitative property of TC is also found in ferromagnetic state of CMRs as di-valence atoms doping increases. Intuitively, it seems that the Coulomb repulsion destroys the magnon softening effect. Our result supports that CMR materials determine different magnon dispersion properties, and that the magnon softening effect in CMR materials can be

qualitatively formulated only by the simple FKLM along with one magnon excitation and RPA approximations [25]. The appearance of softening effect in CMRs is due to less Coulomb repulsion force in conduction bands.

Acknowledgements We thank Professor Ming-Fong Yang for his valuable comments and acknowledge the support of the National Science Council of the Republic of China under Grant No. NSC-89-2112-M-132-001.

References [1] H. Ohno, J. Magn. Magn. Mater. 200 (1999) 110. [2] A.P. Ramirez, J. Phys.: Condens. Matter 9 (1997) 8171; J.M.D. Coey, M. Viret, S. von Molnar, Adv. Phys. 48 (1999) 167. [3] C. Zener, Phys. Rev. 82 (1951) 403. [4] P.W. Anderson, H. Hasegawa, Phys. Rev. 100 (1955) 675. [5] P.-G. de Gennes, Phys. Rev. 118 (1960) 141. [6] K. Kubo, N. Ohata, J. Phys. Soc. Japan 33 (1972) 21.

S.-J. Sun et al. / Physica B 324 (2002) 286–291 [7] N. Furukawa, J. Phys. Soc. Japan 63 (1994) 3214. [8] N.B. Perkins, N.M. Plakida, Theor. Math. Phys. 120 (1999) 1182. [9] A.J. Millis, P.B. Littlewood, B.I. Schraimann, Phys. Rev. Lett. 74 (1995) 5144. [10] D. Golosov, Phys. Rev. Lett. 84 (2000) 3974; D. Golosov, J. Appl. Phys. 87 (2000) 5804. [11] T.G. Perring, et al., Phys. Rev. Lett. 77 (1996) 711. [12] M.C. Martin, G. Shirane, et al., Phys. Rev. B 53 (1996) 14285. [13] H.Y. Hwang, et al., Phys. Rev. Lett. 80 (1998) 1316. [14] P. Dai, et al., Phys. Rev. B 61 (2000) 9553. [15] J.A. Fernandez-Baca, et al., Phys. Rev. Lett. 80 (1998) 4012. [16] N. Furukawa, cond-mat/9905123.

291

[17] G. Khaliullin, R. Kilian, Phys. Rev. B 61 (2000) 3494. [18] I.V. Solovyev, K. Terakura, Phys. Rev. Lett. 82 (1999) 2959. [19] N.F. Mott, Metal–Insulator Transitions, 2nd Edition, 1990. [20] A.P. Balachandran, E. Ercolessi, G. Morandi, A.M. Srivastava, Int. J. Mod. Phys. B 4 (1990) 2057. [21] D.N. Zubarev, Usp. Fiz. Nauk 71 (1960) 71 [Sov. Phys. Usp. 3 (1960) 320]. [22] M.F. Yang, S.J. Sun, M.C. Chang, Phys. Rev. Lett. 86 (2001) 5637. [23] M. Vogt, C. Santos, W. Nolting, Phys. Stat. Sol. B 223 (2001) 679; cond-mat/0012086. [24] E.E. Dagotto, et al., Phys. Rep. 344 (2001) 1. [25] X. Wang, Phys. Rev. B 57 (1998) 7427.