Sound velocity in La0.5Ca0.5MnO3

Physics Letters A 372 (2008) 5356–5360

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Physics Letters A www.elsevier.com/locate/pla

Sound velocity in La0.5 Ca0.5 MnO3 Xian-Sheng Cao, Chang-Le Chen ∗ Shaanxi Key Laboratory of Condensed Matter Structures and Properties, School of Science, Northwestern Polytechnical University, Xi’an 710072, China

a r t i c l e

i n f o

Article history: Received 30 May 2008 Accepted 13 June 2008 Available online 14 June 2008 Communicated by V.M. Agranovich PACS: 75.30.Mb 43.58.+z 72.10.Di

a b s t r a c t In this Letter the microscopic theory of the relative change in velocity of sound with temperature of La0.5 Ca0.5 MnO3 is reported. The phonon Green function is calculated using the Green function technique of Zubarev in the limit of zero wave vector and low temperature. The lattice model electronic Hamiltonian in the presence of the phonon interaction with hybridization between the conduction electrons and the l-electrons is used. The relative change in velocity of sound at various temperatures is studied for different model parameters namely the position of the l-level, the effective phonon coupling strength and hybridization strength. The phonon anomalies observed experimentally at different temperatures are explained theoretically. An abrupt change in velocity at Neel temperature (T N ) is observed clearly. It is observed that different parameters influence the velocity of sound. © 2008 Elsevier B.V. All rights reserved.

Keywords: Colossal magnetoresistance Sound velocity

1. Introduction Intense research on experimental and theoretical fronts have been focused on charge-ordered manganites due to the coexistence of charge, orbital, and spin orderings at various temperatures. Charge-ordered manganites in general show different types of ground states depending on the dominance of antiferromagnetic (AFM) and/or ferromagnetic (FM) interactions, and Jahn–Teller distortions. The half-doped Perovskite manganites R 0.5 A 0.5 MnO3 (R = trivalent rare earth ion and A = divalent ion Ca, Sr, Ba) exhibit a wide variety of magnetic structures and magnetotransport behaviors and have been extensively investigated [1–4]. However, the complex physics behind these have not been fully comprehended and, therefore, call for further studies. Qualitative explanation is given by the double exchange (DE) mechanism [5]. At higher doping (x > 0.5), the ground state becomes again an AFM insulator [6,7]. A phase boundary between the FM metallic and the AFM insulating ground states exists in a narrow range around x = 0.5 [8]. In addition, another intriguing phase, the charge ordered (CO) state has been found to exist in insulating La1/2 Ca1/2 MnO3 [9]. A direct evidence of the CO state is provided by the electron diffraction for La1/2 Ca1/2 MnO3 [10]. The CO state is characterized by the real space ordering of Mn3+ /Mn4+ in mixed valent R 1/2 A 1/2 MnO3 . The CO state is expected to become stable when the repulsive Coulomb interacts between carriers dominates over the kinetic energy of carriers [11]. In this respect, the electron lattice of the CO state may be viewed as the generalized Wigner crystal. Furthermore, the carriers formed the CO state are believed to be manifested in some types of polarons, which arise from the strong electron–phonon interaction, possibly, the Jahn–Teller effect [12]. In fact, ordering of such polarons is occasionally observed in 3d transition metal oxides. Ramirez et al. [8] have observed in La1−x Cax MnO3 (0.63 < x < 0.67) that the CO transition is accompanied by a dramatic increase (> 10%) in the sound velocity, implying a strong electron–phonon coupling. Another interesting aspect of the CO state is its relevance to the observed magnetic phases. In the half-doped LaCa manganites [9,10], the CO state has been realized with FM–AFM transition. The common feature is that the AFM structure of the specific CE-type [6] is observed in the CO state of manganites, suggesting a nontrivial effect of the CO state on the magnetic phase. The other noteworthy observation is the transport phenomena of the CO phase in the presence of the magnetic field. The high magnetic field induces the melting of the electron lattice of the CO phase to give rise to a huge negative magnetoresistance (MR) [13]. In this work, we report the microscopic theory of the velocity in La1/2 Ca1/2 MnO3 . For this purpose we describe the electronic Hamiltonian of Kondo lattice model and introduce an electron–phonon interaction in Section 2 and calculate phonon Green function in Section 3. The results and discussion are showed in Section 4.

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Corresponding author. Tel.: +86 029 88493979. E-mail address: [email protected] (C.-L. Chen).

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X.-S. Cao, C.-L. Chen / Physics Letters A 372 (2008) 5356–5360

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2. Model We start from the picture the characteristics of the systems with Jahn–Teller (JT) ions with an inter-play between electrons localized due to lattice distortions and those in the band states. This situation is typical of manganites and could be described in terms of the Kondo-lattice model in the double exchange limit with account taken for the Jahn–Teller distortions and the super-exchange interaction between the localized electrons [14]. The intra-atomic exchange coupling is assumed to be large enough to align the spins of e g electrons in a Mn ion parallel to spin S of core (t 2g ) electrons. The Jahn–Teller effect leads to the splitting of the double-degenerate e g level. Then, following [15], we can divide e g electrons into two groups: “localized” (l, t (l) → 0) producing the maximum splitting of the e g level and itinerant “band” (b) electrons with nonzero hopping integrals t, leading to smaller distortions of MnO6 octahedra. The corresponding effective Hamiltonian has the form [14] H eff = −t



cn+ cm − εJT



nm

nln + J



cos θnm + U



nm

n

nln nbn .

(1)

n

l and nbi = c + c . We Eq. (1) describes the intra-atomic Hubbard interaction between the l-electrons and l–b-electrons, where nli = l+ i i i i + + linearize the on-site l-electron interaction terms li li and c i c i in the Hamiltonian in the Hartree–Fock approximation as nli nbi → nli nbi + nli nbi , where nli  = l+ l , nbi  = c + c  are the expectation values of nli , nbi and we take nli  = nl , nbi  = nb for all i. The number of i i i i localized, nli , and band, nbi , electrons of per lattice site obeys an obvious relation nb + nl = 1 − x (x is doping concentration, value is 0.5). Then Hamiltonian H eff therefore reduces to H eff = −t



c+ c − εJT i j



i j 

nli + J



cos θi j + U



i j 

i

nli c + c + i i

i





, l+ ln i i bi

(2)

i

c ; nli = l+ l , εJT = − g 2 /2K (where K is the elastic energy and g is the electron-lattice coupling constant). nbi = c + i i i i The Fourier-transformed Hamiltonian of Eq. (2) is written as H eff = −



E k ck+ ck − E 0



k

nlk + J



cos θi j ,

(3)

i j 

k

where E 0 = εJT + U nb ; E k = εk + U nl ; εk = −2t (cos kx a + cos k y a + cos k z a). In the event of a large Jahn–Teller distortion resulting in the band splitting (∼ 1 eV) there is the possibility that the lower e g orbital hybridizes with the localized t 2g orbitals which will modify the magnetic ordering of the t 2g electrons. Thus the hybridization of the localized electrons with the conduction electrons of band one only is considered, and the Hamiltonian H ν which represents this hybridization effect is given by Hν = V





ck+ lk + lk+ ck .

(4)

k

The electron–phonon interaction term in the site representation can be written as H e− p =



   f (q) ck++q lk + lk++q ck b+ −q + bq .

(5)

k,q

Where bq (bq+ ) are annihilation (creation) operators for phonons with wave vector q and f (q) is the electron–phonon coupling constant. The free phonon Hamiltonian with phonon energy ωq is written as Hp =



ωq bq+ bq .

(6)

q

The total Hamiltonian of the system then becomes H = H eff + H ν + H e− p + H p .

(7)

3. Calculation of phonon Green function The phonon self-energy is evaluated by the double-time Green function technique of Zubarev [16] using the equation of motion method. The phonon Green function is defined as:









D qq (t − t  ) = A q (t ); A q (t  ) = −i Θ(t − t  ) A q (t ), A q (t  ) ,

(8)

A q = bq + b + −q

(9)

where

is the qth Fourier component of the displacement. To derive the phonon Green function D q,q (t − t  ). We use the equation of motions for phonon operators [17]. The equation of motion for the Fourier transformed phonon Green function D q,q (ω) is evaluated for the system using the total Hamiltonian H as given in Eq. (7). The Green function is expressed as D q,q = δq,q D q0 (ω) + 4 f 2 (−q) D q0 (ω)χq,q (ω) D q0 (ω). Where D q0 (ω) = 2ωq



ω2 − ωq2

−1

(10)

(11)

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are the free phonon propagators, and Green function can be written as D q,q (ω) = 2ωq



χq,q (ω) is the electron response function. Applying the Dyson approximation to Eq. (10), the phonon −1

ω2 − ωq2 − Σq (ω)

(12)

.

Where the phonon self-energy is

Σq (ω) = 4 f 2 (−q)ωq χq,q (ω). Where:

(13)





χq,q (ω) = ck+−q lk + lk+−q ck ; ck+ −q lk + lk+ −q ck .

(14)

The self-energy depends on the electron response function χq,q (ω). When evaluating these electron response functions, only the electronic Hamiltonian is considered, the electron–phonon interaction terms is neglected. These approximation amounts of neglecting changes in the electron hybridization as well as the l-electron density are due to interaction with phonons. In the limit of the long wavelength and finite temperature the phonon Green’s function is written as D q,q (ω, q = 0) =

ωq ω0 = , ω2 − ωq2 − Σ(ω, q = 0) ω2 − ω02 − Σ(ω, q = 0)

(15)

where the phonon self-energy is:

Σ(ω, q = 0) = −8g ω02 I ,

(16)

where: I =−

1 2g ω0

χq,q (ω, q = 0).

(17)

In evaluating the phonon self-energy, the summation over k is replaced by an integration over energy as density of states of conduction electrons at the Fermi level ε F . Then Eq. (17) reduces to W /2

I= − W /2

dεk

|D|

I1,



N (0) dεk . Here N (0) is the

(18)

 ( E k − E 0 )2  f (ω1 ) − f (ω2 ) , ω1 − ω2

εk + εJT + 0.5U ± [εk − εJT + U (2nl − 0.5)]2 + 4V 2 ω1,2 = , 2  2  | D | = ω − ( E k − E 0 )2 − 4V 2 . I1 = 2

The renormalized phonon propagator in Eq. (15) has poles at frequencies

ω2 − ω02 − Σ(ω, q = 0) = 0.

(19)

(20) (21)

ω as given by (22)

Here ω and ω0 are the renormalized longitudinal phonon frequency and longitudinal background phonon frequency, respectively. In the long wavelength limit q → 0, one has ω = ν q and ω0 = ν0 q where ν and ν0 are the normalized and bare longitudinal sound velocities. Hence Eq. (22) reduces to

ω2 Re Σ(ω, q = 0) −1− = 0. ω02 ω02

(23)

Finally, the reduced velocity ν˜ becomes



ν˜ = 1 − A 2 .

(24)

Where A 2 (ω) = 8s Re I ,  H (εk ) A 3 dεk  f (ω1 ) − f (ω2 ) . Re I =

(25)

ω1 − ω2

Where:



H (εk ) = 2 A3 =

2

εk − εJT + U (2nl − 0.5) ,

A4 A 24 + B 24

(26) (27)

,

A 4 = ω2 − ( E k − E 0 )2 − 4V 2 − η2 ;

B 4 = 2ωη.

(28)

To understand the influence of the physical quantities on the sound velocity, the parameters are scaled by frequency (ω D ). The dimensionless parameters are the phonon coupling to the hybridization between the l-electrons and conduction electrons are s = N (0) f 12 (0)/ω0 , the position of the bare l-level d = εJT /ω D , the hybridization between the l-electrons and the conduction ν = V /ω D , the reduced temperature t = k B T /ω D , the Coulomb interaction u = U /ω D and the energy of the conduction electrons x = εk /ω D . The velocity of sound of La0.5 Ca0.5 MnO3 at low temperature is evaluated numerically under the half-filling band situations in the long wavelength limit of the phonons. The Fermi level (ε F = 0) is taken at the middle of the conduction band.

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4. Results and discussion

The temperature variation of the relative velocity of sound of La0.5 Ca0.5 MnO3 exhibits certain anomalies at low temperatures for certain physical parameters like the position of the l-level (d) with respect to Fermi level and hybridization (ν ) between conduction electrons and l-level. The results are shown in Figs. 1–3. Fig. 1 shows velocity as a function of temperature for various electron–phonon (EP) coupling parameter(s). As the temperature decreases the velocity of sound decreases and attains a dip at tm = 0.23 and increases as temperature decreases further towards lower temperatures. The value of temperature tm remains unaltered with EP coupling. This dip arises due to the hybridization between the l-level and the conduction band intensity reduced. Fig. 2 shows the plot of (ν˜ ) vs. t for two different values of l-level positions d = −0.90, −0.96, for other fixed parameters. It is observed that as d changes from −0.90 to −0.96 (i.e. the l-level moves towards the Fermi level ε F = 0), the change in velocity is reduced. In other words, the hardening of velocity of sound is observed. We found that parameter d has no influence on the velocity of sound at higher temperatures where they attain the same values. Fig. 3 presents the plot of (ν˜ ) vs. t for different values of hybridization parameter ν . It influences the magnitude of velocity of sound as well as reduces Neel temperature in greater extent near Neel temperature. As hybridization parameter ν increases from 0.05 to 0.08 the softening of velocity of sound is enhanced. The charge ordering in hole-doped manganites is usually accompanied by a transition from FM metal to a AFM insulator, and the magnetic ground state for La0.5 Ca0.5 MnO3 is of the CE-type AFM order [18]. Zheng et al. [19] hold that the sound velocity anomalies near T N is due to the antiferromagnetic spin fluctuations alone. Therefore, it is suggested that the slight softening and dramatic stiffening of the sound velocity observed in La0.5 Ca0.5 MnO3 near charge-ordered temperature (T CO ) cannot be explained based on a conventional antiferromagnetic phase transition. It is worth mentioning that Radaelli et al. [20] observed a significant change of the lattice parameters near T CO in La0.5 Ca0.5 MnO3 , which is caused by a static Jahn–Teller distortion of the Mn3+ O6 octahedral, from synchrotron X-ray and neutron powder-diffraction experiments, although the average symmetry or the framework of the whole lattice does not change. It seemed undoubtedly that the dramatic ultrasonic sound velocity and attenuation anomalies near T CO result from the fine-structure change and relate to strong electron–phonon coupling via the Jahn–Teller effect. It also seemed that the electron–phonon coupling is very important

Fig. 1. Plot of reduced velocity ν˜ vs. reduced temperature (t) for different values of the phonon coupling to the hybridization s = 0.055, 0.058, with fixed values of d = −0.94, p = 1.3, u = 0.49 and ν = 0.05.

Fig. 2. Plot of reduced velocity ν˜ vs. reduced temperature (t) for different values of the position of the l-level d = −0.96, −0.90 with fixed values of s = 0.055, p = 1.3, u = 0.49 and ν = 0.05.

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Fig. 3. Plot of reduced velocity ν˜ vs. reduced temperature (t) for different values of the hybridization strength and d = −0.96.

ν = 0.05, 0.08 with fixed values of s = 0.055, p = 1.3, u = 0.49

in charge ordering, and the magnetic transition upon charge ordering is not the cause but the effect of the charge ordering, which agrees with the theoretical calculation of Ahn and Millis [21]. Note that, as the electron–phonon coupling is turned on, the AFM Neel temperature becomes reduced to T N (= T CO ). The behavior of sound reduced velocity ν˜ as a function of t (k B T /ω D ) can be obtained from Figs. 1 to 3. Due to a cusp maximum in the phonon selfenergy, there appears a cusp minimum in the sound velocity at T CO . The sound velocity is softened smoothly above T CO , but becomes substantially hardened below T CO . Quite interestingly, the behavior of the above sound velocity around T CO is qualitatively very similar to the observation of Ramirez et al. [8] for La1−x Cax MnO3 (0.63 < x < 0.67), and according to Lee et al. [22] theoretical calculation’s results. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant No. 60171034. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

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