(La,Sr)n+1 MnnO3n+1 低磁場巨磁阻之研究(1/3)
執行期間： 90 年 08 月 01 日至 92 年 10 月 31 日
中 華 民 國 93 年 12 月 28 日
Effect of softening on colossal manganites
*, Wei-Chun Lub
, Hsiung Choub
aDepartment of Electronic Engineering, Cheng Shiu Institute of Technology, Kaohsiung, Taiwan bDepartment of Physics, National Sun Yat-Sen University, Kaohsiung, Taiwan
Received 30 May 2002; accepted 18 July 2002
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
Many magnetic systems of current experimental interests, such as ferromagnetic semiconductors  and the colossal magnetoresistance (CMR) man-ganites , 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:email@example.com (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 . 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 . Khaliullin and Kilian  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  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 . 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
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 , prominent in colossal manganites.
Similar to how the Hubbard model treats large Coulomb repulsion problems  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 , we will
derive the temperature Green functions
i ; ðSj ÞnðSjþÞn1SS for the magnon in the
general spin case and //ci;s; cþj;sSS for the
electrons . They are d dt//S þ i ; ðS j Þ nðSþ j Þ n1SS ¼ dðtÞdijjnþ d dtS þ i ; ðS j Þ nðSþ j Þ n1 ; ð3Þ where jn¼ /½Sþi ; ðSj ÞnðSjþÞn1S 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 nðSþ j Þ n1SS ¼1 b X n einnt 1 N X q eiqðRiRjÞg nðinn; qÞ and //ci;s; cþj;sSS ¼1 b X n eiont 1 N2 X k eikðRiRjÞG 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
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 eikðRlRjÞM 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 ZS 2t effgk Gnsðion; kÞ ¼JF 2 1 N X q Mnsðion; q; k qÞ 1: ð5Þ
To derive the equation of motion for the
i ci;s; cþj;sSS Green function again, we
ob-tain another relation of equation
Mnsðion; q; k qÞ ¼JF/S ZS/cþ 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 ZS þJF2 2/S ZS1 N X q /cþ kqkckqkS þ nq ekmþ 2teffgkqþ JF/sZS ðJF=2Þ/SZS ð7Þ and ekk¼ 2teffgkþ JF 2/S ZS JF2 2/S ZS 1 N X q /cþ kþqmckþqmS ðnqþ 1Þ ekkþ 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
/sZS and /SZS 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 ekm¼ 2teffgk JF=2/SZS
and ekk ¼ 2teffgkþ JF=2/SZS:
Similarly, regarding the magnon excitation
through derivations of //Sþi ; ðSj ÞnðSþ
we obtain another Green function and its Fourier transformation form //cþ imcik; ðSjÞ nðSþ j Þ n1SS ¼1 b X n einnt 1 N2 X k1k2 eik1ðRiRjÞ eik2ðRiRjÞI nðinn; k1; k2Þ: ð9Þ
Consequently, a relation of Inðinn; k1; k2Þ and
gnðinn; qÞ can be obtained as
inngnðinn; qÞ ¼ jnþ JF/SZS 1 N X k Inðinn; k þ q; kÞ JF/sZSgnðinn; qÞ: ð10Þ Further, to derive ðd=dtÞ //cþimclk; ðS j Þn
ðSjþÞn1SS; 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 ZS 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, /SZS 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 ¼32 (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 . 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  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=NSÞPq1=nqÞ1 for ferromagnetic transition
temperature, TC , 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 . The appearance of softening effect in CMRs is due to less Coulomb repulsion force in conduction bands.
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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