Teori Formulasi Konduktivitas Optik dari La0.7Ca0.3MnO3 yang Menampakkan Transisi

Insulator Paramagnetik - Metal Feromagnetik

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Listiana Satiawati NIM 1206327916

Theoretical Formulation of Optical Conductivity of La0.7Ca0.3MnO3 Exhibiting Paramagnetic Insulator - Ferromagnetic

Metal Transition

Listiana Satiawati NIM 1206327916

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Black : Mn White : O Yellow : La, Ca

Crystalline structure of

La0.7Ca0.3MnO3 : Perovskite

LCMO has several fascinating properties : Magnetic properties vary with doping content

At Ca content of 𝒙≅𝟑𝟎% they reveal:

Parametic-Insulator to Ferromagnetic-Metal transition

CollosalMagnetoresistance (CMR) Strong T-dependent optical conductivity 0-22 eV

MOTIVATION

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LCMO has several facinating properties :

Magnetism property has excited since LMO

Parametic insulator to ferromagnetic metal transition / MIT ability

Magnetoresistance (MR) property, it will be very big after Ca doping, to be Collosal Magnetoresistance (CMR)

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MOTIVATION

To improve the formulation of the model previously proposed, by adding dynamic Jahn-Teller distortion, electron-phonon coupling, and crystal field effects in the Hamiltonian.

To get more accurate calculation results, especially in the region around the dc limit, where MIT should occur.

To address the influence of the improved model on the Collosal Magnetoresistance (CMR).

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Aim of the research

To obtain the formulation of the model, by adding Jahn Teller distortion, electronic and Jahn Teller distortion coupling and crystal field effect in the Hamiltonian have proposed before, is expected to be more precise, especially in the area around the dc limit, this is shown on the suitability of the the resulting graphs.

The progress of research in Collosal Magnetoresistant (CMR).

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Aim of the Research

A previous study by Majidi, et.al. [PRB 84, 075136 (2011)] has proposed a theoretical explanation to the experimental data of optical conductivity LCMO (with 30% Ca content) by A. Rusydi, et.al. [PRB 78, 125110 (2008)]. The theory has successfully captured the temperature-dependent trend in the photon energy range of 1-22 eV, but failed to explain the MIT at the dc limit.

Such a failure has been attributed mainly to the absence of Jahn-Teller phonons and the electron-phonon coupling in the Hamiltonian of the previous model.

In this research, we propose to improve the previous study by adding the relevant additional considerations into the model. To do this, we need to reformulate the algorithm and redo the computation.

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Problem

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Problem

Formulations of the model have proposed by M.Aziz Majidi,et.al. [PRB 84, 075136 (2011)] to prove the experimental data of the optical conductivity measurements by A. Rusydi,et.al. [PRB 78, 125110 (2008)] fails to capture the correct physics close to the dc limit in which metal-insulator transition occurs

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The formulation result of optical conductivity vs photon energy at several temperaturs around Currie

temperature : M.Aziz Majidi,et.al. [PRB 84, 075136 (2011)]

Optical Conductivity

σ (𝟏𝟎𝟑 𝝮−𝟏𝐜𝐦−𝟏 )

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Comparison of the experimental result and

the previous theoretical study

The presence of phonons and the electron-phonon coupling can influence the dynamics of the iterant electrons.

If the coupling is sufficiently strong, the phonons may trap the electrons and make them localized at the corresponding atomic sites (Mn in the case of LCMO), thus an insulating phase occurs at intermediate and high temperatures.

At low temperatures, magnetic correlations arise and compete with the phonons in such a way that eventually free the electrons from getting trapped by phonons. This process results in a transition from paramagnetic-insulator to ferromagnetic metal.

The process mentioned above also influence the overall profile of the optical conductivity in a wide photon energy range.

Crystal field effects and other additional terms in the Hamiltonian may insignificantly further modify the optical conductivity profile.

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Hypothesis

By proposing a Hamiltonian model with the addition of the influence of the crystal field, Jahn Teller distortion and electron-Jahn Teller distortion interaction is expected to produce charts that more fit with the experimental data.

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Hypothesis

the electronic interaction calculate with tight Binding approximation method . And Fourier transformation to transform the interaction to reciprocal lattice

Electronic-phonon interaction, Jahn Teller distortion, Crystal field effect, Coulomb repulsion force, and double exchange magnetic interaction is added to the system formulation to get the Hamiltonian of the system

Solution method using model and calculations based on tight-binding approach, and Green function method within the framework of Dynamical Mean-Field Theory (DMFT) algorithms

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Formulation

Electronic configuration

La Mn O3 or La3+Mn3+O32−

(LMO)

after doping with Ca (x=0.3) :

La0.7Ca0.3MnO3 or

La0.73+Ca0.3

2+Mn0.73+Mn0.3

4+O32−

(LCMO)

Double valence :

La0.73+Mn0.7

3+O32−

Ca0.32+Mn0.3

4+O32−

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Ca doping and double valence

Doping x = 0.3 causes the insulator turns into metal at Currie temperature of 260 K (chart 1) and the transition paramagnetic - ferromagnetic (chart 2). And a double valence (mixed phase) causes the double exchange magnetic interaction, electrons maintain a 'memory' spin when jumping from site to the next site (ferromagnetic property). The jump process is the state of metal. That is why this material featuring the behavior of metals and ferromagnetic at the same time.

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Ferromagnetic metal – paramagnetic insulator transiton (MIT) :

Chart 1 : P. Schiffer, A. P.

Ramirez, W. Bao, S.-W.

Cheong: Phys. Rev. Lett. 75,

3336 (1995)

Chart 2 : M. Euhara, B. Kim,

and S.W. Cheong, Phase

Diagram of 𝑳𝒂𝟏−𝒙𝑪𝒂𝒙𝑴𝒏𝑶𝟑,

2000

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1. The method is quite simple

2. Good enough to handle the problems of strong correlated system

3. Although the calculation is simple, a quantum effect is maintained

4. Small computing time

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Tight binding Method

Selecting the base, which is a quantum state that is relevant to represent the linear combination. There are 10 basis orbitals.

Fourier transformation.

Interaction between d shell in Mn.

Interaction between d shell in Mn and p shell in O.

Interaction between p shell in O.

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Tight binding method

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Manganite (MnO₃)

Crystal field and Jahn Teler distortion effect:

𝐻 =1

𝑁 η𝒌

†

𝒌

𝐻0 𝒌 η𝒌 + 𝑈

𝑖,𝜎,𝜎′

𝑛𝑎𝑖𝜎𝑛𝑏𝑖𝜎′

+ 𝑈1𝑛𝑎𝑖↑𝑛𝑎𝑖↓ +

𝑖

𝑈2𝑛𝑏𝑖↑𝑛𝑏𝑖↓ − 𝐽𝐻𝑺𝑖 . 𝒔𝑖

𝑖𝑖

Formulation: Hamiltonian 1 M.Aziz Majidi,et.al. [PRB 84, 075136 (2011)]

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Kinetic part of the Hamiltoian

1

𝑁 η𝒌

†

𝒌

𝐻0 𝒌 η𝒌

N is normalisation factor

𝐻0 𝒌 is 10 × 10matrix

η𝒌†is creation operator

η𝒌is annihilation operator

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Second part of the Hamiltonian

This form is the Coulomb repulsion forcebetween the electrons in the eg orbital of Mn ions up and down on one site (inter-orbital on-site Coulomb repulsions), include in the self energy matrix

𝑈𝑛𝑎𝑖𝜎

𝑖,𝜎,𝜎′

𝑛𝑏𝑖𝜎′

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The third and fourth part

Intra orbital on site Coulomb repulsions

𝑈1𝑛𝑎𝑖↑𝑛𝑎𝑖↓ + 𝑈2𝑛𝑏𝑖↑𝑛𝑏𝑖↓

𝑖𝑖

Approximation :

Elements with 1,2,6, and 7 index multiplied by half

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− 𝐽𝐻𝑺𝑖 . 𝒔𝑖

𝑖

This form is double exchange magnetic interaction between local spin of Mn (formed by a strong Hunt coupling among 3 electrons in 𝑡2𝑔skin and produce the amount 3/2 ) S, with spin contained in electrons in 𝑒𝑔subshell above and below s,include in the self energy matrix.

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The fifth part

− ∆𝜏𝑖𝑧

𝑖

Splitting the 5 orbitals in 3d shell into 2 energy level namely 𝑒𝑔 and 𝑡2𝑔 subshell, add to kinetic part of

Hamiltonian

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Crystal field efect

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Jahn Teller distortion effect Jahn Teller distortion stated that degenerate nonlinear octahedral molecules, when in a the ground state becomes distorted geometric, eliminates the degenerate due to lower overall energy. This usually occurs in molecules of the transition metal.

manganite : Q₂ and Q₃ model

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𝐻 =1

𝑁 η𝒌

†

𝒌

𝐻0 𝒌 η𝒌 + 𝑔 𝑄2𝑖𝜏𝑖𝑥 + 𝑄3𝑖𝜏𝑖

𝑧

𝑖

+ 𝐾𝑄2𝑖

2 + 𝑄3𝑖2

2− ε𝑧𝑄3𝑖

𝑖

− ∆𝜏𝑖𝑧

𝑖

+ 𝑈

𝑖,𝜎,𝜎′

𝑛𝑎𝑖𝜎𝑛𝑏𝑖𝜎′ + 𝑈1𝑛𝑎𝑖↑𝑛𝑎𝑖↓ +

𝑖

𝑈2𝑛𝑏𝑖↑𝑛𝑏𝑖↓ − 𝐽𝐻𝑺𝑖 . 𝒔𝑖

𝑖𝑖

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Hamiltonian 2

Green Function in terms of framework of Dynamical Mean Field Theory (DMFT) algorithm

Green function (lattice) :

𝐺(𝒌, 𝑧) = 𝐻 0(𝒌) + 𝛴(𝑧)−1

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Solution of the formulation

Algoritma DMFT

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Green function (lattice)

Green function (site)

Green function (effectif)

Green function (mean field)

Green function (local interaction)

Green function (average)

Value obtained from the self consistance

process

start

end

convergen ?

N

Y

guess value

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Green Function (site)

𝐺(𝑧) =1

𝑁 𝐺(𝒌, 𝑧)

𝒌

Green Function mean field

𝒢(𝑧) = 𝐺(𝑧) 𝑒𝑓𝑓−1

+ Σ(𝑧)−1

Green Function effective 𝐺(𝑧) 𝑒𝑓𝑓

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Local Self Energy Matrix

𝛴11 𝛴12 𝛴16𝛴61 𝛴66 𝛴67

= −

ħ

2𝐽𝐻𝑆 𝑐𝑜𝑠𝜃 +𝑈(𝑛𝑏↑ +𝑛𝑏 ↓) +𝑔𝑄3 𝑔𝑄2 −

ħ

2𝐽𝐻𝑆 𝑒𝑥𝑝(𝑖𝜙)

−ħ

2𝐽𝐻𝑆 𝑒𝑥𝑝(−𝑖𝜙)

ħ

2𝐽𝐻𝑆𝑐𝑜𝑠 𝜃+ 𝑈(𝑛𝑏↑ +𝑛𝑏 ↓)+ 𝑔𝑄3 𝑔𝑄2

𝛴21 𝛴22 𝛴27𝛴72 𝛴76 𝛴77

= 𝑔𝑄2 −

ħ

2𝐽𝐻𝑆 𝑐𝑜𝑠𝜃 +𝑈(𝑛𝑎↑ +𝑛𝑎↓)− 𝑔𝑄3 −

ħ

2𝐽𝐻𝑆 𝑒𝑥𝑝(𝑖𝜙)

−ħ

2𝐽𝐻𝑆 𝑒𝑥𝑝(−𝑖𝜙) 𝑔𝑄2

ħ

2𝐽𝐻𝑆 𝑐𝑜𝑠 𝜃+ 𝑈(𝑛𝑎↑ +𝑛𝑎↓) −𝑔𝑄3

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Local Interacting Green Function

𝐺𝑛𝑎𝜎,𝑛𝑏𝜎′𝑧, 𝜃, 𝜙, 𝑄2, 𝑄3

= 𝒢(𝑧) −1 − [Σ𝑛𝑎𝜎,𝑛𝑏𝜎′𝑧, 𝜃, 𝜙, 𝑄2, 𝑄3 ]

−1

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Green function average

average of local interacting green function over all possible θ and nl values

𝑃𝑛𝑎𝜎 ,𝑛𝑏𝜎′

𝜃, 𝜙, 𝑄2, 𝑄3, Probability of Mn spin S

having a direction with angle θ with respect to the direction of magnetization (which is defined as the z−axis)

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Green function average

[𝐺(𝑧)]𝑎𝑣𝑒 =

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Normalization factor

Ɲ = (1 − 𝑛𝑎 ↑ )(1 − 𝑛𝑎↓

)(1 − 𝑛𝑏 ↑ )(1 − 𝑛𝑏↓

) + 𝑛𝑎 ↑ (1 − 𝑛𝑎 ↓

)(1 − 𝑛𝑏↑ )(1 − 𝑛𝑏↓

) +(1 − 𝑛𝑎 ↑

) 𝑛𝑎↓ (1 − 𝑛𝑏↑

)(1 − 𝑛𝑏↓ )+(1 − 𝑛𝑎 ↑

)(1 − 𝑛𝑎 ↓ ) 𝑛𝑏↑

(1 − 𝑛𝑏↓ )

+(1 − 𝑛𝑎 ↑ )(1 − 𝑛𝑎↓

)(1 − 𝑛𝑏↑ ) 𝑛𝑏↓

+ 𝑛𝑎↑ (1 − 𝑛𝑎↓

) 𝑛𝑏↑ (1 − 𝑛𝑏 ↓

) + 𝑛𝑎 ↑

(1 − 𝑛𝑎 ↓ )(1 − 𝑛𝑏↑

) 𝑛𝑏 ↓ +(1 − 𝑛𝑎 ↑

) 𝑛𝑎↓ 𝑛𝑏↑

(1 − 𝑛𝑏↓ )

+(1 − 𝑛𝑎 ↑ ) 𝑛𝑎↓

(1 − 𝑛𝑏↑ ) 𝑛𝑏 ↓

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Normalized magnetization

𝑀

𝑀𝑆=

1

Ɲ (1 − 𝑛𝑎↑ )(1 − 𝑛𝑎↓ )(1 − 𝑛𝑏↑ )(1 − 𝑛𝑏↓ ) 𝑑𝛺 𝑑𝑄

2𝑑𝑄

3𝑃0,0 𝜃, 𝜙, 𝑄

2, 𝑄

3 𝑐𝑜𝑠𝜃

+ 𝑛𝑎↑ (1 − 𝑛𝑎↓

)(1 − 𝑛𝑏↑ )(1 − 𝑛𝑏↓

) 𝑑𝛺 𝑑𝑄2𝑑𝑄3𝑃1↑,0(𝜃 , 𝜙, 𝑄2 ,𝑄3

) 𝑐𝑜𝑠 𝜃 +(1 − 𝑛𝑎 ↑

) 𝑛𝑎 ↓ (1 − 𝑛𝑏↑

)(1 − 𝑛𝑏 ↓ ) 𝑑𝛺 𝑑𝑄2𝑑𝑄3𝑃1↓,0

(𝜃 , 𝜙, 𝑄2 ,𝑄3) 𝑐𝑜𝑠 𝜃

+(1 − 𝑛𝑎 ↑ )(1 − 𝑛𝑎↓

) 𝑛𝑏↑ (1 − 𝑛𝑏 ↓

) 𝑑𝛺 𝑑𝑄2𝑑𝑄3𝑃0,1↑(𝜃, 𝜙, 𝑄2 , 𝑄3

)𝑐𝑜𝑠 𝜃

+(1 − 𝑛𝑎 ↑ )(1 − 𝑛𝑎↓

)(1 − 𝑛𝑏↑ ) 𝑛𝑏↓

𝑑𝛺 𝑑𝑄2𝑑𝑄3𝑃0,1↓(𝜃, 𝜙, 𝑄2 , 𝑄3

)𝑐𝑜𝑠 𝜃

+ 𝑛𝑎↑ (1 − 𝑛𝑎↓

) 𝑛𝑏 ↑ (1 − 𝑛𝑏↓

) 𝑑𝛺𝑑𝑄2𝑑𝑄3𝑃1↑,1↑(𝜃, 𝜙, 𝑄2 ,𝑄3

) 𝑐𝑜𝑠 𝜃

+ 𝑛𝑎↑ (1 − 𝑛𝑎↓

)(1 − 𝑛𝑏↑ ) 𝑛𝑏↓

𝑑𝛺𝑑𝑄2𝑑𝑄3𝑃1↑,1↓(𝜃, 𝜙, 𝑄2 ,𝑄3

) 𝑐𝑜𝑠 𝜃

+(1 − 𝑛𝑎 ↑ ) 𝑛𝑎 ↓

𝑛𝑏↑ (1 − 𝑛𝑏↓

) 𝑑𝛺𝑑𝑄2𝑑𝑄3𝑃1↓,1↑(𝜃, 𝜙, 𝑄2 ,𝑄3

) 𝑐𝑜𝑠 𝜃

+(1 − 𝑛𝑎 ↑ ) 𝑛𝑎↓ (1− 𝑛𝑏↑ ) 𝑛𝑏↓ 𝑑𝛺 𝑑𝑄2𝑑𝑄3𝑃1↓,1↓(𝜃, 𝜙, 𝑄2 , 𝑄3) 𝑐𝑜𝑠 𝜃

𝑐𝑜𝑠𝜃𝑥𝑦

2= 𝑐𝑜𝑠

𝜃𝑧

2= 1 −

2

2

𝑀

𝑀𝑆

+ 2

2

𝐷𝑂𝑆(𝜔) = −1

𝜋 𝐼𝑚 𝑇𝑟[𝐺(𝜔 + 𝑖0+)]

Density of sate (DOS) :

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New self energy matrix : [Σ(𝑧)] = 𝒢(𝑧)−1 − [𝐺(𝑧)]𝑎𝑣𝑒

−1

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Optical Conductivity

𝜎𝛼𝛽 (𝜔) =𝜋𝑒2

ℎ𝑎𝑑 𝑑ʋ

𝑓(ʋ,𝑇)−𝑓(ʋ+𝜔 ,𝑇)

𝜔

1

𝑁 𝑇𝑟[ʋ𝛼(𝒌)][𝑨(𝒌, ʋ)]𝒌 ʋ𝛽 (𝒌) [𝑨(𝒌,ʋ + 𝜔)],

where : [ʋ𝜆(𝒌)] =

𝜕[𝐻0(𝒌)]

𝜕𝑘𝜆, and

[𝑨(𝒌, ʋ)] = ([𝐺(𝒌,𝜔 + 𝑖0+)] − [𝐺(𝒌, 𝜔− 𝑖0+)])/(2𝜋𝑖)

Outcome (in several temperatures values) :

Optical conductivity vs photon energy graph.

Density of states (DOS) vs energy graph.

Spectral Density vs photon energy graph.

Magnetic Susceptibility vs photon energy graph.

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Thank you for your attention

Chart 1 : P. Schiffer, A. P. Ramirez, W. Bao, S.-W. Cheong: Phys. Rev. Lett. 75, 3336 (1995)

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Chart 2 : M. Euhara, B. Kim, and S.W. Cheong, Phase Diagram of La1−xCaxMnO3, 2000

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LCMO phase diagram

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The study of CMR in rare-earth manganate perovskites brings forth novel features such as charge-ordering.

Charge ordering in the manganates is interesting because charge-ordering is resulting due to localization of charges

therefore it is associated with insulating and antiferromagnetic (or paramagnetic) behavior. But double-exchange gives rise to metallicity along with ferromagnetis m. Therefore a competition arises between ferromagnetic, metallic behavior and cooperative Jahn-Teller effect with

charge-ordering.

MIT ability of LCMO

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Studi tentang CMR di perovskites manganat langka bumi menumbuhkan fitur baru seperti charge-ordering. Charge ordering di manganates menarik karena Charge ordering

yang dihasilkan karena lokasi muatan karena hal ini terkait dengan isolasi dan antiferromagnetik (atau paramagnetik)

perilaku. Tapi double-exchange menimbulkan metallicity bersama dengan ferromagnetism. Oleh karena itu

kompetisi timbul antara feromagnetik, perilaku logam dan kooperatif efek Jahn-Teller dengan charge-ordering.

http://folk.uio.no/ravi/activity/ordering/colossal-magnet.html

MIT ability of LCMO

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Metal–insulator transitions are transitions from a metal (material with good electrical conductivity of electric charges) to an insulator (material where conductivity of charges is quickly suppressed). These transitions can be achieved by tuning various ambient parameters such as pressure or, in case of a semiconductor, doping. The basic distinction between metals and insulators was proposed by Bethe, Sommerfeld and Bloch in 1928/1929. It distinguished between conducting metals and nonconducting insulators. However, in 1937 de Boer and Evert Verwey reported that many transition-metal oxides (such as NiO) with a partially filled d-band were poor conductors, often insulating. In the same year, the importance of the electron-electron correlation was stated by Peierls. Since then, these materials as well as others exhibiting a transition between a metal and an insulator have been extensively studied, e.g. by Sir Nevill Mott, after whom the insulating state is named Mott insulator. The classical band structure of solid state physics predicts the Fermi level to lie in a band gap for insulators and in the conduction band for metals, which means metallic behavior is seen for compounds with partially filled bands. However, some compounds have been found which show insulating behavior even for partially filled bands. This is due to the electron-electron correlation, since electrons cannot be seen as noninteracting. Mott considers a lattice model with just one electron per site. Without taking the interaction into account, each site could be occupied by two electrons, one with spin up and one with spin down. Due to the interaction the electrons would then feel a strong Coulomb repulsion, which Mott argued splits the band in two: The lower band is then occupied by the first electron per site, the upper by the second. If each site is only occupied by a single electron the lower band is completely filled and the upper band completely empty, the system thus a so-called Mott insulator. https://en.wikipedia.org/wiki/Metal%E2%80%93insulator_transition 9/21/2021 9:11:07 AM 50

MR, GMR, and CMR

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MR (Magnetoresistance) is the change of material resistivity when subjected to a magnetic field

GMR (Giant Magnetoresistance) is a MR symptom that occurs in several thin layers of magnetic and

non-magnetic materials

CMR (Collosal Magnetoresistance) is a very big MR symptom

Magnetoresistance is the relative change in electrical resistance of a material on the application of magnetic field. The magnetoresistance

of conventional materials is quite small; but materials with large magnetoresistance have been synthesized now. Depending on the magnitude, it is called either as Giant magnetoresistance (GMR) or

Colossal magnetoresistance (CMR). Magnetoresistan adalah perubahan relatif dalam hambatan listrik dari

bahan dalam medan magnet. Magnetoresistance dari bahan konvensional cukup kecil, tetapi bahan dengan magnetoresistance besar telah disintesis sekarang. Tergantung pada besarnya, hal itu disebut baik sebagai giant magnetoresistance (GMR) atau collosal

magnetoresistance (CMR). http://folk.uio.no/ravi/activity/ordering/colossal-magnet.html

Magnetoresistance (MR)

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Giant magnetoresistance (GMR) is a quantum mechanical magnetoresistance effect observed in thin-film structures composed of alternating ferromagnetic

and non-magnetic conductive layers. The 2007 Nobel Prize in Physics was awarded to Albert Fert and Peter Grünberg for the discovery of GMR.

The effect is observed as a significant change in the electrical resistance depending on whether the magnetization of adjacent ferromagnetic layers are in a parallel or an antiparallel alignment. The overall resistance is relatively low

for parallel alignment and relatively high for antiparallel alignment. The magnetization direction can be controlled, for example, by applying an

external magnetic field. The effect is based on the dependence of electron scattering on the spin orientation.

The main application of GMR is magnetic field sensors, which are used to read data in hard disk drives, biosensors, microelectromechanical systems (MEMS) and other

devices. GMR multilayer structures are also used in magnetoresistive random-access memory (MRAM) as cells that store one bit of information.

Giant Magnetoresistance (GMR)

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Giant magnetoresistance (GMR) adalah efek magnetoresistance mekanika kuantum diamati dalam struktur film tipis terdiri dari beberapa lapisan

konduktif feromagnetik dan non-magnetik. Penghargaan Nobel dalam Fisika 2007 diberikan kepada Albert Fert dan Peter Grünberg untuk penemuan GMR.

Efeknya diamati sebagai perubahan signifikan dalam hambatan listrik tergantung pada apakah magnetisasi lapisan feromagnetik yang berdekatan

berada dalam paralel atau antiparalel alignment. Keseluruhan resistensi relatif rendah untuk penyelarasan paralel dan relatif tinggi untuk penyelarasan

antiparalel. Arah magnetisasi dapat dikontrol, misalnya, dengan menerapkan medan magnet eksternal. Efeknya didasarkan pada ketergantungan hamburan

elektron pada orientasi spin. Aplikasi utama dari GMR adalah sensor medan magnet, yang digunakan untuk membaca

data di hard disk drive, biosensor, sistem microelectromechanical (MEMS) dan perangkat lainnya. GMR struktur multilayer juga digunakan dalam magnetoresistive

random-access memory (MRAM) sebagai sel yang menyimpan satu bit informasi. http://en.wikipedia.org/wiki/Giant_magnetoresistance

Giant Magnetoresistance (GMR)

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Magnetoresistance has aroused great interest recently because of possible application in devices such as read/write heads in computer discs and in

sensors.

Colossal Magnetoresistance has been predominantly discovered in manganese-based perovskite oxides. This arises because of strong mutual

coupling of spin, charge and lattice degrees of freedom. Hence not only high temperature superconductivity, but also new magnetoelectronic properties

are increasingly discovered in materials with perovskite structures. Most of the perovskite oxides exhibit the following types of

ordering.They are, Charge ordering, Orbital ordering and Spin ordering

Collosal Magnetoresistance (CMR)

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Magnetoresistance telah menimbulkan minat yang besar baru-baru ini

karena aplikasi yang mungkin dalam perangkat seperti membaca / menulis head dalam disket komputer dan sensor.

Colossal Magnetoresistan banyak ditemukan pada oksida mangan berbasis perovskite. Hal ini timbul karena kopling spin mutual yang

kuat, derajat kebebasan charge dan kisi. Oleh karena itu tidak hanya superkonduktivitas suhu tinggi, tetapi juga sifat magnetoelectronic

baru yang semakin ditemukan dalam bahan dengan struktur perovskit. Sebagian besar oksida perovskite menunjukkan beberapa

macam jenis ordering.Yaitu, Charge ordering, Orbital ordering dan Spin ordering

http://folk.uio.no/ravi/activity/ordering/colossal-magnet.html

Collosal Magnetoresistance (CMR)

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Electronic and Jahn Teller disortion coupling

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Perovskite

The general formula for a perovskite is ABO3 where A and B are cations. The easiest way to visualize the structure is in terms of the BO6 octahedra which share corners infinitely in all 3 dimensions, making for a very nice and symmetric structure. The A cations occupy every hole

which is created by 8 BO6 octahedra, giving the A cation a 12-fold oxygen coordination, and the B-cation a 6-fold oxygen coordination. In

the example shown below, (SrTiO3, download cif) the Sr atoms sit in the 12 coordinate A site, while the Ti atoms occupy the 6 coordinate B

site. There are many ABO3 compounds for which the ideal cubic structure is distorted to a lower symmetry (e.g. tetragonal,

orthorhombic, etc.)

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Pictured is the undistorted cubic structure; the symmetry is lowered to orthorhombic, tetragonal or trigonal in many perovskites.

https://en.wikipedia.org/wiki/Perovskite_(structure)

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Perovskite https://www.princeton.edu/~cavalab/tutorials/public/structures/perovskites.html

http://glossary.periodni.com/dictionary.php?en=base-centered+orthorhombic+lattice

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orthorhombic

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https://en.wikipedia.org/wiki/Orthorhombic_crystal_system

Tetragonal

https://en.wikipedia.org/wiki/Tetragonal_crystal_system http://www.classicgems.net/Crystallography.htm

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Trigonal

http://chemistry.elmhurst.edu/vchembook/205trigpyramid.html

the trigonal system is a subsystem of the hexagonal. Most gem references will list

these as hexagonal.

http://gemologyonline.com/crystal_systems.html

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Electrical conductivity can be simply explained on the basis of drude model itself. Using band theory of solids, valence band and conduction band overlaps leading to have more free electrons in the conduction band. This conduction electrons are responsible for conduction, however due to so-called scattering mechanism, material possesses resistivity. Optical conductivity is rather the measure of electrical conductivity in the alternating field. Hence dielectric constant comes into play, which rules allowed propagation of light into matter. Here the term plasma frequency and relaxation time are used. Optical propagation is depend on this two important physical quantities. If the radiation is incident on matter, it drives the electron, and if there is no scattering, light gets reflected totally and hence metals are shiny. Above plasma frequency, reflection declines and transmission start to dominate. By understand this, we can simply differentiate electrical and optical conductivity. The color of the metals is also dependent on this factor.

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Optical Conductivity https://www.researchgate.net/post/Is_there_any_difference_between_optical_and_electrical_conductivity_If

_yes_how_can_one_explain_this_difference_physically2

Optical conductivity and AC electric conductivity experiments are indeed quite similar. However, they operate in different frequency regimes and measure slightly different quantities.

Optical conductivity refers to an experiment using light, such as a reflectivity measurement and then using a Kramers-Kronig transform to deduce the real part of the transverse conductivity, σT. The transverse conductivity is measured because the direction of propagation of the photon is perpendicular to the electric field. Another way to measure the transverse optical conductivity is through a light transmission experiment. In both cases, the conductivity is determined over the THz frequency range.

On the other hand, the AC electrical conductivity usually involves some sort of electrical circuit and usually measures the real part of the longitudinal conductivity, σL, in the MHz frequency range. In this case the probe and the electric field point in the same direction.

All in all, the measurements obtain slightly different experimental quantities and also over different frequency ranges. I say slightly different here because in the limit that the wavevector q→0

, the regime that we are in for the above experiments, it is expected that the longitudinal and transverse response become indistinguishable (the only known exception to this rule is in a superconductor).

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Optical Conductivity http://physics.stackexchange.com/questions/110169/what-does-optical-conductivity-mean

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unlcms.unl.edu/.../Section%2013_Optical_Properties_of_Solid.

Curie temperature (TC), or Curie point, is the temperature at which certain materials lose their permanent magnetic properties, to be replaced by induced magnetism.

The force of magnetism is determined by the magnetic moment, a dipole moment within an atom which originates from the angular momentum and spin of electrons. Materials have different structures of intrinsic magnetic moments that depend on temperature; the Curie temperature is the critical point at which a material's intrinsic magnetic moments change direction.

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Curie temperature

Below TC Above TC

Ferromagnetic ↔ Paramagnetic

Ferrimagnetic ↔ Paramagnetic

Antiferromagnetic ↔ Paramagnetic

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Permanent magnetism is caused by the alignment of magnetic moments and induced magnetism is created when disordered magnetic moments are forced to align in an applied magnetic

field. For example, the ordered magnetic moments (ferromagnetic) change and become disordered (paramagnetic) at the Curie temperature. Higher

temperatures make magnets weaker, as spontaneous magnetism only occurs below the Curie temperature.

https://en.wikipedia.org/wiki/Curie_temperature

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Curie and Neel points

Materials are only antiferromagetic below their corresponding Néel temperature. This is similar to the Curie temperature as above the Néel Temperature the material undergoes a phase transition and becomes paramagnetic.

The material has equal magnetic moments aligned in opposite directions resulting in a zero magnetic moment and a net magnetism of zero at all temperatures below the Néel temperature. Antiferromagnetic materials are weakly magnetic in the absence or presence of an applied magnetic field.

Similar to ferromagnetic materials the magnetic interactions are held together by exchange interactions preventing thermal disorder from overcoming the weak interactions of magnetic moments. When disorder occurs it is at the Néel temperatur.

https://en.wikipedia.org/wiki/Curie_temperature

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Antiferromagnetic and the Néel temperature

http://scienceworld.wolfram.com/physics/Magnetization.html

Definition of magnetization

: an instance of magnetizing or the state of being magnetized; also : the degree to which a body is magnetized

http://www.merriam-webster.com/dictionary/magnetization

The magnetization of a material is expressed in terms of density of net magnetic dipole moments m in the material. We define a vector quantity called the magnetization M by

M = μtotal/V

http://hyperphysics.phy-astr.gsu.edu/hbase/solids/magpr.html

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Magnetization

Along with the averaging process of the Green function, we also need to average the normalized magnetizatio M/Ms

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normalized magnetization

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http://farside.ph.utexas.edu/teaching/em/lectures/node73.html

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https://en.wikipedia.org/wiki/Magnetization

magnetic moment

Physical systems that we understand well correspond to ensembles of free particles. For example, semiconductors and most metals can be described as having non-interacting electrons. This simple approach is valid because the interaction (Coulomb) energy of electrons is much smaller than their kinetic energy. Another example is alkali atoms, that Bose condense at low temperatures. Alkali atoms can be treated as non-interacting bosons because their scattering length (i.e. the length at which they interact with each other) is much smaller than the average distance between the particles. Magnetic systems. Coulomb interaction between electrons may lead to a variety of spin ordering patterns, including ferromagnetism (spins of all the particles are alligned), antiferromagnetism (spins of the neighboring particles are antialigned). http://cmt.harvard.edu/demler/layman.html

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Strong correlated system

Unlike in semiconductors and novel metals, electrons strongly interact with electrons (or other quasiparticles) in strongly correlated electron systems. As a consequence, lots of novel phenomena, including high TC superconductivity, ferromagnetism, and ferroelectricity, have been observed in these materials. Interestingly, if we make ultrathin films or heterostructures of such systems, more exotic phases can emerge. For example, we can realize Weyl semimetal, which shows 3D Dirac dispersion and surface Fermi arc states, by breaking inversion symmetry. At the interface of two insulators, it has been shown that 2D electron gas emerges and it shows interesting properties including superconductivity and magnetism. Because of this, the interest in strongly correlated materials has grown as the field becomes more popular. https://www.ibs.re.kr/eng/sub02_03_01.do

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The physics of materials with strong electronic correlations is remarkably rich and complex and cannot be understood within the conventional theories of metals and insulators. In correlated materials, charge, spin, orbital and lattice degrees of freedom result in competing interactions. These lead to phase transitions and the emergence of exotic phases including the pseudogap state in cuprates and manganites, high-temperature superconductivity, charge stripes in cuprates, even phase separation in some manganites and cuprates.

Vanadium dioxide (VO2) is a canonical example of a transition metal oxide with correlated electrons. It undergoes a phase transition from the low temperature, monoclinic insulating phase to a high temperature, rutile metallic phase at Tc=340 K accompanied by a structural change. A controversial issue is whether the insulator-to-metal transition (IMT) is driven primarily by the structural change due to electron-phonon interactions leading to a Peierls insulator or by electron-electron interactions resulting in a Mott insulator. Our far-field infrared studies on VO2 and the related oxide V2O3 have revealed the importance of electronic correlations in these materials.

http://infrared.ucsd.edu/stronglycorrelatedsystems.html

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Typically, strongly correlated materials have incompletely filled d- or f-electron shells with narrow energy bands. One can no longer consider any electron in the material as being in a "sea" of the averaged motion of the others (also known as mean field theory). Each single electron has a complex influence on its neighbors.

The term strong correlation refers to behavior of electrons in solids that is not well-described (often not even in a qualitatively correct manner) by simple one-electron theories such as the local-density approximation (LDA) of density-functional theory or Hartree–Fock theory. For instance, the seemingly simple material NiO has a partially filled 3d-band (the Ni atom has 8 of 10 possible 3d-electrons) and therefore would be expected to be a good conductor. However, strong Coulomb repulsion (a correlation effect) between d-electrons makes NiO instead a wide-band gap insulator. Thus, strongly correlated materials have electronic structures that are neither simply free-electron-like nor completely ionic, but a mixture of both.

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https://en.wikipedia.org/wiki/Strongly_correlated_material

Fourier transformation

http://www.thefouriertransform.com/#introduction

https://en.wikipedia.org/wiki/Fourier_transform

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The Fourier transform of a function of time itself is a complex-valued function of frequency, whose absolute value represents the amount of that frequency present in the original function, and whose complex argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is called the frequency domain representation of the original signal. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. The Fourier transform is not limited to functions of time, but in order to have a unified language, the domain of the original function is commonly referred to as the time domain. For many functions of practical interest one can define an operation that reverses this: the inverse Fourier transformation, also called Fourier synthesis, of a frequency domain representation combines the contributions of all the different frequencies to recover the original function of time.

Coulomb’s law (https://en.wikipedia.org/wiki/Coulomb%27s_law)

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Coulomb repulsion force

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ajp.asj-oa.am/511/1/03-Rai.pdf

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Green function method http://mathworld.wolfram.com/GreensFunction.html

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Integral Kernel http://mathworld.wolfram.com/IntegralKernel.html

Crystal Lattice

•Periodic f(r+ T) = f(r) for any observable functions such as electronic density, electric potential….etc. which means they are all periodic functions because of the translational properties of the lattice vectors.

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Reciprocal lattice www.engr.sjsu.edu/rkwok/Phys175A/Chapter%202.pdf

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Direct and Reciprocal Lattice

•For every crystal, there is a set of space lattice (crystal lattice) –location of lattice points in real space where atoms and molecules are.

•There is also a set of reciprocal lattice in the momentum space (k-space) –something we see in diffraction measurement. However, there is no physical object present at the reciprocal lattice sites.

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www.engr.sjsu.edu/rkwok/Phys175A/Chapter%202.pdf brillouin zone

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s shell https://socratic.org/questions/what-is-an-atomic-orbital

The s sublevel appears as a single spherical shell. It can hold two electrons in a single orbital due to Pauli's Exclusion Principle.

The p sublevel are three dual-lobes projected along the x-, y-, and z-axes through the atom.Each set of dual-lobes is an orbital, so the p sublevel can contain up to six electrons.

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p shell https://socratic.org/questions/what-is-an-atomic-

orbital

The d sublevels are more complicated: four sets of quadruple-lobes and a pair of lobes through a toroid. There are five orbitals in the d sublevel.

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d shell https://socratic.org/questions/what-is-an-atomic-

orbital

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The five d orbitals in ψ(x, y, z)2 form, with a combination diagram showing how they fit together to fill space around an atomic

nucleus. http://en.wikipedia.org/wiki/Atomic_orbital

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F shell https://socratic.org/questions/what-is-an-atomic-orbital

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s, p, d, and f shell https://socratic.org/questions/what-is-an-atomic-orbital

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s, p, d, and f shell

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Ferromagnetism: The magnetic moments in a ferromagnetic material. The moments are ordered and of the same

magnitude in the absence of an applied magnetic field. https://en.wikipedia.org/wiki/Curie_temperature

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Paramagnetism: The magnetic moments in a paramagnetic material. The moments are disordered

in the absence of an applied magnetic field and ordered in the presence of an applied magnetic field.

https://en.wikipedia.org/wiki/Curie_temperature

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Ferrimagnetism: The magnetic moments in a ferrimagnetic material. The moments are aligned oppositely and have

different magnitudes due to being made up of two different ions. This is in the absence of an applied magnetic field.

https://en.wikipedia.org/wiki/Curie_temperature

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Antiferromagnetism: The magnetic moments in an antiferromagnetic material. The moments are aligned oppositely and have the same magnitudes. This is in

the absence of an applied magnetic field. https://en.wikipedia.org/wiki/Curie_temperature

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In a conductor, electric current can flow freely, in an insulator it cannot. Metals such as copper typify conductors, while most non-metallic solids are said to be good insulators, having extremely high resistance to the flow of charge through them. "Conductor" implies that the outer electrons of the atoms are loosely bound and free to move through the material. Most atoms hold on to their electrons tightly and are insulators. In copper, the valence electrons are essentially free and strongly repel each other. Any external influence which moves one of them will cause a repulsion of other electrons which propagates, "domino fashion" through the conductor.

Simply stated, most metals are good electrical conductors, most nonmetals are not. Metals are also generally good heat conductors while nonmetals are not.

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Metal http://hyperphysics.phy-

astr.gsu.edu/HBASe/hframe.html

Most solid materials are classified as insulators because they offer very

large resistance to the flow of electric current. Metals are classified as conductors because their outer electrons are not tightly bound, but in most materials even the outermost electrons are so tightly bound that there is essentially zero electron flow through them with ordinary voltages. Some materials are particularly good insulators and can be characterized by their high resistivities:

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Insulator http://hyperphysics.phy-

astr.gsu.edu/HBASe/hframe.html

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Energy band http://hyperphysics.phy-astr.gsu.edu/HBASe/solids/band.html#c4

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quantum effect https://www.quora.com/What-is-a-

quantum-effect#MoreAnswers

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Lanthanum manganite

Lanthanum manganite Properties Chemical formula LaMnO3 .

Lanthanum manganite is an inorganic compound with the formula LaMnO3, often

abbreviated as LMO. Lanthanum manganite is formed in the perovskite structure, consisting of oxygen octahedra with a central Mn atom. The cubic perovskite structure is distorted into an orthorhombic structure by a strong Jahn-Teller distortion of the oxygen octahedra.

Lanthanum manganite alloys Lanthanum manganite is an electrical insulator and an A-type antiferromagnet. It is the

parent compound of several important alloys, often termed rare-earth manganites or colossal magnetoresistance oxides. These families include lanthanum strontium manganite, lanthanum calcium manganite and others.

In lanthanum manganite, both the La and the Mn are in the +3 oxidation state. Substitution of some of the La atoms by divalent atoms such as Sr or Ca induces a similar amount of tetravalent Mn+4 atoms. Such substitution, or doping can induce various electronic effects, which form the basis of a rich and complex electron correlation phenomena that yield diverse electronic phase diagrams in these alloys.

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LMO

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LCMO

La Mn O3 or La3+Mn3+O32−

(LMO)

after doping with Ca (x=0.3) :

La0.7Ca0.3MnO3 or

La0.73+Ca0.3

2+Mn0.73+Mn0.3

4+O32−

(LCMO)

Double valence :

La0.73+Mn0.7

3+O32−

Ca0.32+Mn0.3

4+O32−

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Basic orbital http://www.introorganicchemistry.com/basi

c.html?i=1

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Quantum state https://en.wikipedia.org/wiki/Quantum_state

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In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).[1][2][3] The concept of linear combinations is central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.

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Linear combination https://en.wikipedia.org/wiki/Linear_combination

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http://www.mathbootcamps.com/linear-combinations-vectors/

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http://mathworld.wolfram.com/LinearCombination.html

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Normalisation factor http://farside.ph.utexas.edu/teaching/qmech/Quantum/node34.html

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Creation and annihilation operator

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Creation operator https://en.wikipedia.org/wiki/Ladder_operator

Ladder operator From Wikipedia, the free encyclopedia

In linear algebra (and its application to quantum mechanics), a

raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raising operator is sometimes called the creation operator, and the lowering operator the annihilation operator. Well-known applications of ladder operators in quantum mechanics are in the formalisms of the quantum harmonic oscillator and angular momentum.

Terminology Main article: Creation and annihilation operators

There is some confusion regarding the relationship between the raising and lowering ladder

operators and the creation and annihilation operators commonly used in quantum field theory. The creation operator ai† increments the number of particles in state i, while the corresponding annihilation operator ai decrements the number of particles in state i. This clearly satisfies the requirements of the above definition of a ladder operator: the incrementing or decrementing of the eigenvalue of another operator (in this case the particle number operator).

Confusion arises because the term ladder operator is typically used to describe an operator that acts to increment or decrement a quantum number describing the state of a system. To change the state of a particle with the creation/annihilation operators of QFT requires the use of both an annihilation operator to remove a particle from the initial state and a creation operator to add a particle to the final state.

The term "ladder operator" is also sometimes used in mathematics, in the context of the theory of Lie algebras and in particular the affine Lie algebras, to describe the su(2) subalgebras, from which the root system and the highest weight modules can be constructed by means of the ladder operators.[1] In particular, the highest weight is annihilated by the raising operators; the rest of the positive root space is obtained by repeatedly applying the lowering operators (one set of ladder operators per subalgebra).

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Annihilation operator www.cithep.caltech.edu/~fcp/physics/.../SecondQuantization.p...

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In a condensed matter context relevant to electrons moving in a material, the self-energy represents the potential felt by the electron due to the surrounding medium's interactions with it

Dalam konteks benda terkondensasi relevan dengan elektron yang bergerak dalam suatu material, self-energi merupakan potensi dirasakan oleh elektron akibat interaksi media sekitarnya dengan itu

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The self energy matrix https://en.wikipedia.org/wiki/Self-energy

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The intra orbital Coulomb on site repulsion

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www.icmm.csic.es/lenibascones/mottphysics3.pdf

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www.icmm.csic.es/lenibascones/mottphysics3.pdf

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Hunt coupling

F = k x

F gaya pegas Hooke

K konstanta pegas

x simpangan pegas

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Hooke law

Lattice = kisi

Crystal lattice, a repetitive arrangement of atoms https://en.wikipedia.org/wiki/Lattice

The Crystal Lattice. Most solids have periodic arrays of atoms which form what we call a crystal lattice. http://hyperphysics.phy-astr.gsu.edu/hbase/solids/lattice.html

with z the frequency variable

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Green function (lattice)

Site = situs = tempat

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Green function (site)

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Green function effective

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Σ 𝑧 is a self energy matrix

Fungsi Green mean field adalah fungsi Green effective ditambah harga tebakan suku self energy

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Green function mean field

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Probability 𝑃𝑛𝑎𝜎 ,𝑛𝑏𝜎′

𝜃, 𝜙, 𝑄2, 𝑄3,

average [Gnaσ,nbσ′(z,θ,φ,Q2,Q3)] over all possible θ and nl values

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Green function (average)

DOS https://www.physik.uni-augsburg.de/theo3/Publications/PT-Kotliar0304.pdf

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Spectral Density https://www.researchgate.net/post/What_is_power_spectr

al_density2

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https://en.wikipedia.org/wiki/Spectral_density

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Magnetic Susceptibility

Susceptibility is a measure of the extent to which a substance becomes magnetized when it is placed in an external magnetic field. A synonym for

susceptibility is "magnetizability".

When matter interacts with the magnetic field, an internal magnetization or polarization(J) is created that either opposes or augments the external

field.

If the polarization opposes the applied field, the effective field within the object is reduced, the lines are dispersed, and the effect is known as

diamagnetism. If the polarization is in the same direction as the external field, the magnetic lines are concentrated within the object, resulting in

paramagnetism, superparamagnetism, or ferromagnetism, depending on the degree of augmentation.

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Magnetic susceptibility, is denoted by the Greek letter chi (χ), is defined as the magnitude of the internal polarization (J) divided by the strength of the external field (B):

χ = J / Bo Since it is the ratio of two magnetic fields, susceptibility is a dimensionless number. Diamagnetic substances have negative susceptibilities (χ < 0); paramagnetic, superparamagnetic, and ferromagnetic substances have positive susceptibilities (χ > 0).

E = ℏ ω

E is photon energy in eV.

h is Planck constant.

ℏ is h/2π.

ℏ = ( 6,626 exp -34 )/ 2π J.s.

= ( 4,1357 exp -15)/ 2π eV.s.

= 6,582 exp -16 eV.s.

for E = 20 eV, ω is frequency

ω = E/ ℏ = 20 / ( 6,582 exp -16 ) 1/s

= 3,0385 exp 16 1/s (Hz)

Photon energy from DC limit (~ 0,1 eV) to 20 eV

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photon energy

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Double exchange magnetic interaction

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− 𝐽𝐻𝑺𝑖 . 𝒔𝑖

𝑖

https://en.wikipedia.org/wiki/Double-exchange_mechanism#/media/File:Double-exchange.PNG

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https://en.wikipedia.org/wiki/Superexchange#/media/File:MnO-superaustausch.GIF

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Super exchange

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Super exchange

A. CaMnO3 180° case (Kanamori8) In this system, the manganese occurs in the crystal field at the Mn4+, which means that there are three d-electrons: (3d)3. The crystal field at the Mn4+ sites is cubic. Under the effect of a cubic field, the five-fold degenerate orbital d state of a single d electron is split into an orbital triplet (d ε , t2g) and an orbital doublet (d γ , ε g). According to the Hund’s rule, the three d-electrons will go to one of the de orbitals with their spins up.

The superexchange involves the p-electrons of the O2-. The p-orbitals are described by p(x), p(y), and p(z), depending on the axis of rotation. These orbitals are classed into two types: (i) the p σ orbital (p-orbital whose axis points to one of the cations) and (ii) the p π orbital (p-orbital whose axis is perpendicular to the line connecting the anion and cation). The p σ orbital is orthogonal to the d(3z2 – r2), d(xy), d(yz), and d(zx), except for d(x2 – y2). A partial covalent bond between the pσ orbital and dγ state [d(x2 – y2)]. can be formed, Then the charge transfer occurs from the pσ orbital with the spin up-state( ↑ to the d γ

state [d(x2 – y2)] of the Mn4+, according to the Hund’s rule requiring that the total spin should be maximum. The remaining pσ orbital (spin-down state), which is orthogonal to the d ε ’ state, ferromagnetically couples to the d ε ’ orbitals of the other Mn4+. Thus the resultant superexchange interaction between Mn4+ is antiferromagnetic.

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www.binghamton.edu/physics/docs/super-exchange.pdf

The double-exchange mechanism is a type of a magnetic exchange that may arise between ions in different oxidation states. First proposed by Clarence Zener,[1] this theory predicts the relative ease with which an electron may be exchanged between two species and has important implications for whether materials are ferromagnetic, antiferromagnetic, or neither. For example, consider the 180 degree interaction of Mn-O-Mn in which the Mn "eg" orbitals are directly interacting with the O "2p" orbitals, and one of the Mn ions has more electrons than the other. In the ground state, electrons on each Mn ion are aligned according to the Hund's

rule: If O gives up its spin-up electron to Mn +4, its vacant orbital can then be filled by an electron from Mn +3. At the end of the process, an electron has moved between the neighboring metal ions, retaining its spin. The double-exchange predicts that this electron movement from one species to another will be facilitated more easily if the electrons do not have to change spin direction in order to conform with Hund's rules when on the accepting species. The ability to hop (to delocalize) reduces the kinetic energy. Hence the overall energy saving can lead to ferromagnetic alignment of neighboring ions. This model is superficially similar to superexchange. However, in superexchange, a ferromagnetic or antiferromagnetic alignment occurs between two atoms with the same valence (number of electrons); while in double-exchange, the interaction occurs only when one atom has an extra electron compared to the other.

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DMFT approximation is done to simplified the

complex to be a simple problem. Crystal problem is

approachable such that become one site problem

(Pendekatan DMFT dilakukan untuk menyederhanakan

problem yang kompleks menjadi problem yang sederhana.

Problem kisi didekati sedemikian rupa sehingga menjadi

problem satu site)

DMFT

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Derived by tight binding method for 10 basic orbital interaction, transferred to momentum space by Fourier transform, with the consideration of spin direction and

kind of interactions. Shaped 10 × 10 matrix.

(Diturunkan dengan metode tight binding untuk interaksi antar 10 orbital dasar, diubah keruang momentum dengan transformasi

Fourier, dengan pertimbangan arah spin, macam interaksi. Berbentuk matrik 10 × 10.)

Kinetic part of the Hamiltonian

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Jahn Teller distortion

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“Tight binding” has existed for many years as a convenient and transparent model for the description of electronic structure in molecules and solids. It often provides the basis for construction of many body theories such as the Hubbard model and the Anderson impurity model. titus.phy.qub.ac.uk/members/tony/.../WSMS-Paxton-Jan09.pdf

Tight binding is a method to calculate the electronic band structure of a crystal. It is similar to the method of Linear Combination of Atomic Orbitals (LCAO) used to construct molecular orbitals. Although this approximation neglects the electron-electron interactions, it often produces qualitatively correct results and is sometimes used as the starting point for more sophisticated approaches.

http://lampx.tugraz.at/~hadley/ss1/bands/tightbinding/tightbinding.php.

Tight Binding and Hubbard Model

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www.icmm.csic.es/lenibascones/mottphysics3.pdf

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In solid-state physics, the tight-binding model (or TB model) is an approach to the calculation of electronic band structure using an approximate set of wave

functions based upon superposition of wave functions for isolated atoms located at each atomic site. The method is closely related to the LCAO method

(linear combination of atomic orbitals method) used in chemistry. Tight-binding models are applied to a wide variety of solids. The model gives good

qualitative results in many cases and can be combined with other models that give better results where the tight-binding model fails. Though the tight-

binding model is a one-electron model, the model also provides a basis for more advanced calculations like the calculation of surface states and application to various kinds of many-body problem and quasiparticle

calculations. https://en.wikipedia.org/wiki/Tight_binding

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Modern explanations of electronic structure like t-J model and Hubbard model are based on tight binding model. Tight binding can be understood by working under a second quantization formalism. Using the atomic orbital as a basis state, the second quantization Hamiltonian operator in the tight binding framework can be written as: hc is hermitian conjugate

Tight Binding and Hubbard Model

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The one-band Hubbard model is the minimal lattice model for strongly correlated electron systems, i.e., for describing electronic properties of materials which cannot be treated without explicitly taking the Coulomb interaction between the (valence) electrons into account. The simplified valence-electron model still contains an infinite number of input parameters that cannot be reliably determined from first principles. This situation calls for a minimalistic approach where the number of parameters is chosen to be just large enough to capture the interesting effects, at least qualitatively.

Mott-Hubbard Metal-Insulator Transition and Optical Conductivity in High Dimensions (Nils Blümer: Phd thesis at the

University of Augsburg)

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Brillouin zone https://en.wikipedia.org/wiki/Brillouin_zone

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dao.mit.edu/8.231/BZandRL.pdf

dao.mit.edu/8.231/BZandRL.pdf

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The adjoint of an operator A may also be called the Hermitian adjoint, Hermitian conjugate or

Hermitian transpose (after Charles Hermite) of A and is denoted by A* or A† (the latter especially

when used in conjunction with the bra–ket notation).

h c : hermitian conjugate

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Hermitian conjugate http://quantummechanics.ucsd.edu/ph130a/130_notes/node133.html

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Hermitian matrix http://mathworld.wolfram.com/HermitianMatrix.html

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Today we understand that the static magnetic fields associated with lodestones and permanent magnets derive principally from the total

angular momentum of electrons within those materials. Lone electrons possess spin, a quantized fundamental property of nature denoted by the letter S. In addition to S, electrons orbiting a nucleus

also possess orbital angular momentum (L). Together S + L = J, or total angular momentum, is the property primarily responsible for bulk magnetism. Nuclei and other subatomic particles also possess spin

angular momentum, but this effect is too weak to affect gross magnetic properties of a material.

Total angular momentum

J = S+L

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𝐻0 𝒌 Matrix 10 × 10

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Electronic configuration

Crystal field effect

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DC limit

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Delta Dirac function https://www.deutsche-digitale-bibliothek.de/binary/.../full/1.pdf

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Heaviside function

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Fourier Transform

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Laplace Transform

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Error function