## 行政院國家科學委員會專題研究計畫 期中進度報告

### (La,Sr)n+1 MnnO3n+1 低磁場巨磁阻之研究(1/3)

### 計畫類別： 個別型計畫

### 計畫編號： NSC90-2112-M-110-008-

### 執行期間： 90 年 08 月 01 日至 92 年 10 月 31 日

### 執行單位： 國立中山大學物理學系(所)

### 計畫主持人： 周雄

### 計畫參與人員： 孫士傑

### 報告類型： 精簡報告

### 處理方式： 本計畫可公開查詢

### 中 華 民 國 93 年 12 月 28 日

### Effect of softening on colossal manganites

### Shih-Jye Sun

a,### *, Wei-Chun Lu

b### , Hsiung Chou

ba_{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 magnetoresis-tance 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. r2002 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) man-ganites [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

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

elec-trons from the eg level and forms a hole which

becomes an itinerant bridging oxygen site. How-ever, 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 JFbt

X /i;jS;s tijðCj;sþCi;sþ hcÞ JF X i Si si ð1Þ

*Corresponding author. Tel.: +886-728-26824. E-mail address:sjs@cc.csit.edu.tw (S.-J. Sun).

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

where the P /i; jS is restricted to the nearest-neighboring sites, itinerant electrons spin density

si¼Pabcþiatabcibwith t being Pauli matrices, and

Siis 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

tempera-ture TC can be described at a quasi-classical level

by an effective nearest-neighbor Heisenberg

model, JeffP/ijSSS~i ~SSj with ferromagnetic

ex-change integral Jeff¼%t=4S2; 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 spin-wave dispersion (SWD) from the typical Heisen-berg 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 manga-nites due to the modulation of magnetic exchange bonds by the orbital degree of freedom of

double-degenerate 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

bandwidth modulated by the strong Coulomb repulsion between d orbitals electrons inﬂuences the magnon softening effect. The intra-Coulomb repulsion

HU ¼ U

X

i

n_{im}n_{ik} ð2Þ

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

elec-trons. 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 approx-imation, 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

//Sþ

i ; ðSj ÞnðSjþÞn1SS for the magnon in the

general spin case and //ci;s; cþj;sSS for the

electrons [22]. They are
d
dt//S
þ
i ; ðS
j Þ
n_{ðS}þ
j Þ
n1_{SS}
¼ dðtÞdijjnþ
d
dtS
þ
i ; ðS
j Þ
n_{ðS}þ
j Þ
n1
; ð3Þ
where
j_{n}¼ /½Sþ_{i} ; ðS_{j} ÞnðS_{j}þÞn1S
and
d
dt//ci;s; c
þ
j;sSS ¼ dðtÞdijþ
d
dtci;s; c
þ
j;s
: ð4Þ
Through Fourier transformation, they are

//Sþ
i ; ðSj Þ
n_{ðS}þ
j Þ
n1_{SS}
¼1
b
X
n
einnt 1
N
X
q
eiqðRiRjÞ_{g}
nðinn; qÞ
and
//ci;s; cþj;sSS
¼1
b
X
n
eiont 1
N2
X
k
eikðRiRjÞ_{G}
nsðion; kÞ:

The electrons in the midway of derivations for

the //ci;s; cþj;sSS Green function will reduce

automatically another higher order Green function

//S

i ci;s; cþj;sSS and its Fourier transformation

form is
//S
i ci;s; cþj;sSS ¼
1
b
X
n
eiont 1
N
X
qk
eiqðRiRjÞ
eikðRlRjÞ_{M}
nsðion; q; kÞ:

After completing the derivation of the equation
of motion of Eq. (4), we obtain an relation of
equation
ion
JF
2/S
Z_{S 2t}
effgk
Gnsðion; kÞ
¼JF
2
1
N
X
q
Mnsðion; q; k qÞ 1: ð5Þ

To derive the equation of motion for the

//S

i ci;s; cþj;sSS Green function again, we

ob-tain another relation of equation

Mnsðion; q; k qÞ
¼JF/S
Z_{S/c}þ
kq;sckq;sS þ ðJF=2Þ/S
qSqþS
ion JF/sZS 2teffrkq ðJF=2Þ/SZS
Gnsðion; kÞ: ð6Þ

Consequently, by combining Eqs. (5) and (6), we
obtain electrons kinetic energies for different spins
up and down:
ekm¼ 2teffgk
JF
2/S
Z_{S þ}JF2
2/S
Z_{S}1
N
X
q
/cþ
kqkckqkS þ nq
ekmþ 2teffgkqþ JF/sZS ðJF=2Þ/SZS
ð7Þ
and
ekk¼ 2teffgkþ
JF
2/S
Z_{S }JF2
2/S
Z_{S} 1
N
X
q
/cþ
kþqmckþqmS ðnqþ 1Þ
e_{kk}þ 2teffgkþq JF/sZS þ ðJF=2Þ/SZS
;
ð8Þ

respectively, where gk¼Pdeikd; d is the

nearest-neighboring sites, nq¼ /SqSþqS=2/SZS is the

expectation value of the magnon number, and

/sZ_{S and /S}Z_{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 e_{km}¼ 2teffgk JF=2/SZS

and e_{kk} ¼ 2teffgkþ JF=2/SZS:

Similarly, regarding the magnon excitation

through derivations of //Sþ_{i} ; ðS_{j} Þn_{ðS}þ

j Þn1SS;

we obtain another Green function and its Fourier
transformation form
//cþ
imcik; ðSjÞ
n_{ðS}þ
j Þ
n1_{SS}
¼1
b
X
n
einnt 1
N2
X
k1k2
eik1ðRiRjÞ
eik2ðRiRjÞ_{I}
nðinn; k1; k2Þ: ð9Þ

Consequently, a relation of Inðinn; k1; k2Þ and

gnðinn; qÞ can be obtained as

inngnðinn; qÞ
¼ j_{n}þ JF/SZS
1
N
X
k
Inðinn; k þ q; kÞ
JF/sZSgnðinn; qÞ: ð10Þ
Further, to derive ðd=dtÞ //cþ_{im}c_{lk}; ðS
j Þn

ðS_{j}þÞn1SS; we will obtain another relation

be-tween Inðinn; k1; k2Þ and gnðinn; qÞ as

Inðinn; k þ q; kÞ ¼JF=2ð/c þ kþqmckþqmS /cþkkckkSÞ innþ JF/SZS þ 2teffðgkþq gkÞ gnðinn; 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:

nq¼ JF/sZS þ
J2
F
2/S
Z_{S 1}
N
X
k
/cþ
kþq;mckþq;mS /cþk;kck;kS
nq 2teffðgkþq gkÞ JF/SZS
: ð12Þ

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

can be taken as the total spin sum of core

electrons, /SZ_{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

ferromag-netic coupling constant JF¼ 0:3 eV; the total spin

S ¼3_{2} (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 band-width modulated by the strong Coulomb repul-sion, 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, La1xCaxMnO3) and not those

with higher TC’s (with large bandwith, i.e,

La1xSrxMnO3Þ [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.

repulsion in eg energy levels than the small

bandwidth CMR. The different Coulomb repul-sion 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 to Nolting’s calculations, TC¼

ðð1=NSÞP_{q}1=nqÞ1 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 increas-ing 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 mag-non 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.

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