A variational theory of Hall effect of Anderson lattice model: Application to colossal magnetoresistance manganites (Re1−x Ax MnO3)

Solid State Communications 266 (2017) 50–54

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Communication

A variational theory of Hall effect of Anderson lattice model: Application to colossal magnetoresistance manganites (Re1−x Ax MnO3)

MARK

⁎

Sunil Panwara, , Vijay Kumara, Ishwar Singhb,1 a b

Department of Applied Physics, Faculty of Engineering & Technology, Gurukula Kangri University, Haridwar 249404, Uttarakhand, India Physics Department, I.I.T. Roorkee, Roorkee 247667, Uttarakhand, India

A R T I C L E I N F O

A BS T RAC T

Communicated by E.L. Ivchenko

An anomalous Hall constant RH has been observed in various rare earth manganites doped with alkaline earths namely Re1−x Ax MnO3 (where Re = La, Pr, Nd etc., and A = Ca, Sr, Ba etc.) which exhibit colossal magnetoresistance (CMR), metal- insulator transition and many other poorly understood phenomena. We show that this phenomenon of anomalous Hall constant can be understood using two band (ℓ-b) Anderson lattice model Hamiltonian alongwith (ℓ-b) hybridization recently studied by us for manganites in the strong electronlattice Jahn-Teller (JT) coupling regime an approach similar to the two – fluid models. We use a variational method in this work to study the temperature variation of Hall constant RH (T) in these compounds. We have already used this variational method to study the zero field electrical resistivity ρ (T) and magnetic susceptibility of doped CMR manganites. In the present study, we find that the Hall constant RH (T) reduces with increasing magnetic field parameters h & m and the metal-insulator transition temperature (Tρ) shifts towards higher temperature region. We have also observed the role of the model parameters e.g. local Coulomb repulsion U, Hund’s rule coupling JH between eg spins and t2g spins, ferromagnetic nearest neighbor exchange coupling JF between t2g core spins and hybridization Vk between ℓ-polarons and d-electrons on Hall constant RH (T) of these materials at different magnetic fields. Here we find that RH (T) for a particular value of h and m shows a rapid initial increase, followed by a sharp peak at low temperature say 50 K in our case and a slow decrease at high temperatures, resembling with the key feature of many CMR compounds like La0.8Ba0.2 MnO3.The magnitude of RH (T) reduces and the anomaly (sharp peak) in RH becomes broader and shifts towards higher temperature region on increasing Vk or JH or doping x and even vanishes on further increasing these parameters. Our results of anomalous Hall constant (RH) have same qualitative behavior as the zero-field electrical resistivity. Moreover Hall Constant (RH) shows positive values indicating that the carriers in these manganites are holes.

Keywords: A. Colossal magnetoresistance D. Anomalous Hall constant E. Model parameters E. Variational method

1. Introduction Hole – doped Mixed-valence manganites with perovskite structure Re1 −x Ax MnO3 (where Re = rare- earth ions ; A = divalent ions such as Ca , Sr, Ba, Pb, etc.) have attracted for a long time a large proportion of researchers from varied fields because of their interesting physical properties such as phase coexistence, metal-insulator transitions, Colossal Magnetoresistance (CMR) and other multiferroic properties [1]. These properties make these systems promising for magnetic sensor and reading head device applications. Observation of CMR phenomenon in pervoskite manganites has been intensively studied, due to their distinctive structural, electrical, thermal and magnetic properties [2]. A number of works has been devoted to study the interplay between structure and transport properties of these manga-

⁎

1

Corresponding author. E-mail address: [email protected] (S. Panwar). Present address: H-202 , Sun Enclave Towers , Ropar , Punjab, India.

http://dx.doi.org/10.1016/j.ssc.2017.08.003 Received 12 June 2017; Received in revised form 31 July 2017; Accepted 2 August 2017 Available online 12 August 2017 0038-1098/ © 2017 Elsevier Ltd. All rights reserved.

nese oxides [3–5]. In these compounds, electrical resistivity decreases by orders of magnitudes upon influence of application of a magnetic field [1,6]. In doped manganites, metallic ferromagnetic (FM), insulating paramagnetic (PM), antiferromagnetic (AFM) and charge/orbital ordered states are among the competing ground states [7]. To explain the CMR phenomenon, Zener has proposed the double-exchange (DE) mechanism [8]. Recent studies suggest that the local Jahn-Teller (JT) distortion plays a key role in these manganites [9]. However, these materials became more complex due to various interactions among charge, spin and lattice and DE alone cannot explain the entire electrical transport behavior. Later on, various theoretical models have been proposed by considering electron-lattice and spin-lattice interaction and even today there is no comprehensive model to explain transport phenomena in manganites [2,9]. Recently,

Solid State Communications 266 (2017) 50–54

S. Panwar et al.

(a) 1.0

3.0

X=0.2 ,V=0.1 J =1.0 H h=0,m=0

0.8

(b) X=0.5,V=0.1,J =1.0 H h=0

h=0.01

2.5

h=0.01,m=0.1

h=0.03

2.0

0.6

h=0.02

1.5

(R H/ R0)

0.4

(RH / R0 )

h=0.03,m=0.3

1.0

h=0.05

0.5

0.2

0.0 0.0

0

100

200

300

400

500

0

T (K)

100

200

300

400

500

T (K)

Fig. 1. Variation of Hall constant (RH/R0) with temperature T(K) at U = 5, Ejt = 0.5 , V = 0.1, JH = 1.0 and JF = 0.1 for different values of h and m with a) x = 0.2 , b) x = 0.5.

of mobile carriers and/or the correlation of spin and charge systems. Chun et al. [13] have reported on low-field magnetization and the Hall effect in La2-2x Sr1+2x Mn2O7 (x = 0.40) , where they show the occurrence of spin-glass like behavior in the magnetization measurements and an enhancement of the anomalous Hall coefficient at low temperatures. The origin of these phenomena has been interpreted as a mixture of ferromagnetic and antiferromagnetic clusters. For CMR materials, such as La0.67 Ca0.33 MnO3 and Nd2/3 Sr1/3 MnO3 , anomalous Hall effect is usually observed [14,15]. The anomalous Hall coefficient depends on asymmetric electron scattering due to spinorbit coupling. A review on the detailed theoretical study of doped rare earth manganites Re1 −x Ax MnO3 has been reported earlier [16] where the authors presented a new model of coexisting localized JT polarons and broad band electrons for manganites and shown that it explains a wide variety of characteristic properties of manganites. The theory ignores the ℓ-b hybridization so it does not describe properly the low temperature behavior of manganites below T * ~ 100 K. We believe that a more general treatment of the model which includes ℓ-b hybridization can lead to a complete description of manganites as suggested by Ramakrishnan et al. (Ref. [16]) . Whereas Graziosi and co-workers [17] presented a polaron framework to account for transport properties in metallic epitaxial manganite films. They propose a model for the consistent interpretation of the transport behavior of manganese perovskites in both the metallic and insulating regimes. Some time ago, Panwar et al. [18] have developed a variational method to study the ground state and thermodynamic properties of

1.6

h=0.03, m=0.3,V=0.1, J H=1.0

1.4 1.2 1.0 0.8

x=0.1

0.6

(RH/R0)

x=0.2 0.4

x=0.3

0.2 0.0

x=0.4 0

100

200

300

400

500

T (K)

Fig. 2. Variation of Hall constant (RH/R0) with temperature T(K) at U = 5, Ejt = 0.5 , V = 0.1, JH = 1.0 and JF = 0.1 for different values of x with h = 0.03 and m = 0.3.

it has been found that manganese oxides display a rich phase diagram [10,11]. Percolation based on phase separation has been proposed to explain magneto- transport properties in these systems [12]. 1.1. Hall measurements on the manganites To reveal the origin of these anomalous transport properties, researchers have performed further detailed experiments, including Hall measurements, which give information about the sign and density

(a) X=0.1 , J H =1.0,h=0.03,m=0.3

0.8

(b) 1.0

X=0.2 , J H =1.0,h=0.03,m=0.3

0.8

0.6 0.6

V=0.1

V=0.1

0.4

R H /R 0

R H /R 0 0.2

0.2

V=0.2 0.0

0

100

200

300

400

0.4

V=0.2 0.0

500

0

T (K)

100

200

300

400

500

T (K)

Fig. 3. Variation of Hall constant (RH/R0) with temperature T (K) at U = 5, Ejt = 0.5, h = 0.03, m = 0.3, JH = 1.0 and JF = 0.1 for different values of V with a) x = 0.1, b) x = 0.2.

51

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S. Panwar et al.

(a)

(b)

1.0

X=0.2 , V=0.1,h=0.03,m=0.3

X=0.1 , V= 0.1,h=0.03,m=0.3

0.8

0.8

0.6

0.6

J H =1.0

J H =1.0

0.4

RH / R 0 0.4

J H =2.0

R H /R 0 0.2

0.2

J H =2.0

0.0

0.0 0

100

200

300

400

0

500

100

200

300

400

500

T (K)

T (K)

Fig. 4. Variation of Hall constant (RH/R0) with temperature T (K) at U = 5, E jt = 0.5 , V = 0.1, h = 0.03, m = 0.3, JH = 1.0 and JF = 0.1 for different values of JH with a) x = 0.1 , b) x = 0.2.

heavy fermion systems using Anderson lattice model. Recently, Panwar et al. had used this variational method in the study of the zero field electrical resistivity and Magnetic susceptibility of doped CMR manganites over a fairly wide temperature range [19,20]. More recently Panwar and co-workers have reported a theoretical study of magneto transport properties like electrical resistivity and thermoelectric power in the presence of magnetic field for the manganites systems [21]. In the present paper, we have carried out a systematic study of Hall constant RH (T) of hole doped CMR manganites using the variational method. The rest of the work is as follows. In Section 2, we give the basic formulation for RH (T). In Section 3, we discuss our results and finally we conclude our findings in Section 4.

∏ [1 + Akσ l +kσ bkσ ] Φd

ψlb =

(2)

kσ

where | Φd is the Fermi sea of broad d-states and Akσ is the variational parameter. With this wave function, the variational parameter Akσ is given by

⎡⎛ 1 ⎢⎜ U ∈k + nl σ _ ∈d σ 2Vk ⎢⎢⎜ 2 ⎣⎝

Akσ =

⎞ ⎟+ ⎟ ⎠

⎤ ⎞2 ⎛ U ⎥ ⎜ ∈k + nl σ _ ∈d σ ⎟ +4Vk 2 ⎥ ⎠ ⎝ 2 ⎥⎦

(3)

where

∈k = ∑ij tij e ik (Ri − Rj ), Ri & Rj are the position vectors of i & j sites (4)

2. Basic formulation

and ∈d σ = 2.1. Model Hamiltonian

m=

iσ

∙ Sj − ∑ VK

(l +

kσ bkσ + h .

kσ

iσ

c.)−μB ∑i Si.h

i

(5)

1 N

∑ σl+kσ lkσ

(6)

kσ

While the number of b-electrons and l-electrons are obtained by

nb σ =

∑ tij (b+iσ bjσ )− ∑ Ejt l +iσ liσ +U ∑ nl iσ nbiσ −JH ∑ si ∙ Si −JF ∑ Si ij σ

Ejt − JF σm –h

Here JF involves the number of nearest neighbors and m is the magnetization per site given by

We represent the doped CMR manganites by the two band (ℓ-b) Anderson lattice model Hamiltonian involving localized and itinerant states (i.e. ℓ, b states) as suggested earlier by both experimentalists and theorists [22–24]. In the model Hamiltonian, we consider the ℓ-b hybridization as an extra mechanism in order to address the low temperature properties of manganites (e.g. resistivity, Hall effect) [16,19]. The Hamiltonian in the presence of magnetic field H is given by

Hl b = −

U b J n σ + H nl σ − 2 2

1 N

∑

∑ nl kσ =

1 N

k

and nl σ =

ij

1 N

∑ nb k σ = 1 N

k

k

− f kσ

(1+Akσ 2 )

∑ k

(7)

− Akσ 2 f kσ

(1 + Akσ 2 )

(8)

f−

where kσ is the Fermi function for the lower branch of the quasi − particle spectra Ekσ given by

(1)

and h = g σ µB H. Here l+iσ creates the JT polaronic state of energy –E jt & spin σ localized at site i (liσ is the corresponding destruction operator) and b+iσ creates broad band electron having mean energy zero & nearest neighbor effective hopping amplitude tij . U is the local Coulomb repulsion between ℓ-polarons and b-electrons of the same spin at a particular site i. Vk is the ℓ-b hybridization between ℓ-polarons and belectrons of the same spin. JH is the strong Hund’s coupling between the eg spins si and the t2g spins Si. JF is the net effective ferromagnetic nearest neighbor exchange coupling between the t2g core spins (i Si, Sj) and the last term of Eq. (1) denotes the interaction of the t2g core spins with an external magnetic field H .

− − f kσ =1/exp[β (Ekσ −μ)+1]

(9)

Here µ is the chemical potential and β = 1/kB T. The expression for Ekσ− is given by

Ekσ − =

⎡ ⎤ ⎞2 ⎛ ⎞ 1 ⎢⎛ U U ⎜ (∈k + nl σ +∈d σ ⎟ − ⎜ (∈k + nl σ − ∈d σ ⎟ + 4Vk 2 ⎥ ⎥ ⎠ ⎝ ⎠ 2 ⎢⎣ ⎝ 2 2 ⎦

(10)

2.3. Electrical resistivity We use the electrical conductivity formula given by Mott considered by us in earlier works [19]

2.2. Wave-function

σ (T ) =

⎛

∂f − ⎞

∫ ⎜⎝− ∂Ekσ− ⎟⎠ σ j (Ekσ− ) dEkσ− kσ

In the finite interaction U case, the modified variational wavefunction [19] may be written as in k-space

where 52

(11)

Solid State Communications 266 (2017) 50–54

S. Panwar et al. − − 2 σ j (Ekσ ) = σ0 [N C (Ekσ )]

N C (E −

=

kσ )

and b) x = 0.5. The parameter h is related to the physical field through h = gµBScHphys/t. Using g = 2 t = 0.6 eV and Sc = 3/2, we find that h = 0.01 corresponds to Hphys = 15T [28] where as the parameter m stands for magnetization per site given by Eq. (6) in Section 2.2. In Fig. 1 ,Hall constant RH (T) is positive , increases rapidly in magnitude with temperature & reaches a maximum at Tp ~ 50K beyond which it falls off for a particular value of h & m resembling with the key feature of many CMR compounds like La0.8 Ba0.2 MnO3 [29]. Also our RH (T) curves have the similarity with the pervoskite cobaltite compounds with magnetic percolation behavior like La1−x Cax CoO3 (0.2 ≤ x ≤ 0.5) [30]. These materials involve additional physics, including high-spin/ low-spin transitions and inhomogeneous states, whose incorporation in our theory would be of interest. We also noticed from Fig. 1 that with the application of magnetic field i.e at higher values h and m, RH (T) of these materials decreases and the peak at Tρ shifts towards higher temperature side (see Fig. 1b). In Figs. 2–4, we have shown the temperature dependence of RH (T) with increasing the parameters like doping x, V and J H respectively keeping the value of h and m fixed at h = 0.03, m = 0.3 . In Fig. 2, we have plotted RH (T) against temperature for different values of x with V = 0.1, JH = 1.0 and JF = 0.1 . While in Fig. 3, we have plotted RH (T) against temperature for different values of V at JH = 1.0 and JF = 0.1 with a) x = 0.1, b) x = 0.2. Finally in Fig. 4, we have plotted RH (T) against temperature for different values of JH at V = 0.1 and JF = 0.1 with a) x = 0.1 , b) x = 0.2. From the Figs. 2–4, we find that the anomaly in RH (T) at low temperature becomes broader and shifts towards higher temperatures and even vanishes on increasing V or JH or doping concentration x. The reason is that our RH (T) curves have the same qualitative behavior as the zero-field electrical resistivity ρ(T) which depends on the quasiparticle (QP) density of states at Fermi energy εF which reduces on increasing x or JH or V hence RH value decreases and peak at Tρ becomes broader. This may be due to the delocalisation of charge carriers reported earlier which causes an increase of conductivity. We get a "hole – like" RH (T) for hole doped manganites, as observed at low temperature by Matl et al. [31] , can be obtained only when lattice distortions persist in the metallic state as T → 0. Our results for RH (T) have the same qualitative behavior as the zero field electrical resistivity [19]. It is due to the magnetic scattering processes which are responsible for both resistivity and extraordinary Hall effect through asymmetric spin-orbit coupling. We have also seen that the results of the simple model considered here are in qualitative agreement with the experimental results of a broad class of hole doped CMR manganites.

(12)

= Density of perturbed broad band eg – states (i.e. b – states)

N C σ (∈k )

⎛ dE − ⎞ (1+Akσ 2 ) ⎜ d ∈kσ ⎟ ⎝ k ⎠

with σ0 =

2πe 2ℏ3 m2

(13)

DE

2 av

(14)

The symbols in σ0 are given in Ref. [25]. σ0 has been taken as a constant in the calculations. Nc(εk) denote the density of unperturbed broad band eg-states (i.e. b – states). 2.4. Magnetic susceptibility The static magnetic susceptibility ' χ S' is given by

χ S = gσμB

∂ l [n σ –nl −σ ]B−−>0 ∂B

(15)

where σ and −σ represent the number ofℓ-electron of spin σ and - σ . We obtain χ S in the units of (g µB)2 as

nl

nl

⎛

χ S (U , JF ) =

⎞ ⎟ U U ⎟ ⎝ [1+( 2 ) I 3 − (( 2 ) + 2JF) χ 0(U )/2] ⎠

∑k σ ⎜⎜

χ 0(U )

whereχ 0 (U) = I1+I2

(∈k +

U l n _ 2 σ

∈d σ )2 +4Vk 2

(18)

4A2 k σ/(1+A2 k σ)2 (∈k +

I3 =

(17)

2Ak σVk I3

I1 =

I2 =

(16)

U l n _ 2 σ

A2 k σ (1+A2 k σ)

∈d σ )2 +4Vk 2

− − 2 β exp[β (Ekσ − μ)]( f kσ )

(19)

(20)

2.5. Hall constant The Hall constant due to intrinsic skew scattering is given by the expression [Fert formula (Ref. [18])]

RH (T) = γχ ″ (T)ρ(T)

(21)

where χ″ (T) is the normalized magnetic susceptibility given by χ″ (T) = χ s (U , JF )/C, (C being the Curie constant) and γ (Electronic specific heat coefficient) = 0.002 J/K2 mol for a band of width 2.0 eV (Ref. [16]) . whereas electrical resistivity ρ (T) = 1/σ(T) . Fert [26] have observed skew scattering phenomena in alloys with Cerium compounds. He applied these phenomena to explain the anomalous behavior of Hall effect in heavy fermion (HF) compounds [27]. This study will definitely help us in the present study of CMR manganites. We are not interested here in the absolute value of Hall constant RH but only in the variation of Hall constant (RH/R0) with temperature.

4. Conclusion We have analyzed the effect of magnetic field on Hall constant RH (T). In this study, we find a reduction in Hall constant with increasing magnetic field and the metal-insulator transition temperature (Tρ) shifts towards higher temperature region. This tendency of shifting the peak in RH (T) towards higher temperature for H > 0 is an expected one. We have also shown RH (T) for different values of the parameters like V, JH and x. The anomaly (sharp peak) in RH at low temperature becomes broader and shifts towards higher temperatures on increasing Vk or JH or doping x. Our results of anomalous Hall constant (RH) have same qualitative behavior as the zero – field electrical resistivity. Moreover Hall Constant (RH) shows positive values indicating that the carriers in these manganites are holes. In future, we are observing the effect of magnetic field on other finite temperature properties e.g. magnetic susceptibility and specific heat using Eq. (2) for variational wave -function.

3. Results and discussion In our calculations, we have taken the unperturbed band of three dimensional solid represented by simple semicircular density of states Ncσ (ϵk) = (2/π) √ (1 − ϵk2) (which is centered around zero energy) with band width W = 2.0 eV, U = 5.0 ,Ejt = 0.5, V = 0.1 and 0.2 , ∈F = -0.238 eV (for x = 0.3) and JH = 1.0 & 2.0 eV. Doping concentration x is varied from 0.1 to 0.5. In Fig. 1, we have shown the temperature dependence of Hall constant RH (T) for different values of parameter h, m with a) x = 0.2

Acknowledgment We would like to acknowledge our great appreciation to University 53

Solid State Communications 266 (2017) 50–54

S. Panwar et al.

Grants Commission (UGC), New Delhi (India) for the financial support (Grant No. F 42-765/2013 (SR) dated 30.03.2013).

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