# The Lattice Compatibility Theory LCT: Physical and Chemical Arguments from the Growth Behavior of Doped Compounds in terms of Bandgap Distortion and Magnetic Effects

Authors: K. Boubaker
Journal: Advances in Physical Chemistry
 (2013)

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## Abstract

Physical and chemical arguments for the recently discussed materials-related Lattice Compatibility Theory are presented. The discussed arguments are based on some differences of Mn ions incorporation kinetics inside some compounds. These differences have been evaluated and quantified in terms of alteration of bandgap edges, magnetic patterns, and Faraday effect.


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## Body

## 1. Introduction

Bismuth oxides are nanocrystalline, fluorite-type materials which exhibit unexpected lattice expansion during doping stages. They are used in various domains, such as transparent ceramic glass, microelectronics, sensor technology, optical coatings, surface acoustic wave devices, and gas sensing [1–9].  Bismuth ternary oxides, such as Bi12SiO20, Bi4Ge3O12, and Bi4Ti3O12, usually exhibit high oxide ionic conductivity and hence can been used as high-efficiency electrolyte materials for several applications such as oxygen sensors and solid oxide fuel cells (SOFC) [7–10]. Bi4Ge3O12 (BGO, Bismuth germanate) is a high density scintillation inorganic oxide with cubic eulytite structure. It is used in detectors in particle physics, gamma pulse spectroscopy, aerospace physics, and nuclear medicine. Bi4Ti3O12 (BTO, Bismuth titanate) is a layered Aurivillius phase perovskite ferroelectric compound having a Curie temperature of about 675°C. In its monoclinic ferroelectric state, Bi4Ti3O12 has been pointed at as a good candidate for use in nonvolatile memories, thanks to its excellent fatigue resistance during repeated polarization reversals under electrical solicitation. In some recent studies [11, 12], manganese-doped bismuth oxides showed nearly 10 times the ionic conductivity of zirconia despite a low stability in reducing environments.In this study, a support to the Lattice Compatibility Theory LCT is presented in terms of alteration of bandgap edges, magnetic patterns, and Faraday effect. The paper is organized in the following way. In Section2, some relevant experimental details along with main manganese-doping features are presented. In Section 3, we present physical parameters alteration analysis along with LCT principles. Section 4 is the conclusion.


## 2. Samples Elaboration and Measurement Techniques

Bi4Ti3O12 (BTO), Bi12SiO20 (BSO), and Bi4Ge3O12 (BGO) compounds have been prepared using the polymeric precursor and Czochralski [11–14] methods using titanium tetraisopropoxide, Bismuth acetate, Bi2O3, GeO2 and SiO2 as precursors. Complexation and pH adjustment were achieved using wet ethylene glycol and ammonium hydroxide, respectively. Mn-doping has been achieved using manganese carbonate MnCO3 and manganese oxide MnO2 in various proportions.Static magnetization and field dependence of magnetization were measured at different applied fields in the temperature range 2–350 K with a SQUID magnetometer (Quantum Design for 0–5 T field range). Measurements have been carried out as guides to determine zero field cooling (ZFC) molar susceptibility.Verdet coefficientV measurement within the visible spectral domain has been obtained using a Faraday rotator which consists of a solenoid wrapped around a transparent dielectric material, along with four symmetric coils which produce controlled AC magnetic fields. The control unit was equipped with a “New Focus Model” 8702 PCB mountable single-axis driver.Finally, X-ray diffraction analysis of all prepared compounds was performed by a copper-source diffractometer (Analytical X Pert PROMPD), with the wavelengthλ=1.54056 Å while optical absorption spectra were measured on double-side polished parallel crystal plates using a SPM-2 monochromator within accuracy of ±2 nm.


## 3. Results and Discussion

### 3.1. Mn-Doping Patterns in terms of Bandgap, Magnetization, and Faraday Effects

In order to understand bandgap edges alteration following doping agent insertion in host structures, Urbach energyEu has been determined, for doped and undoped samples through the equations
(1)ln(α(hν))=ln(α0)+hνEu,Eu=α(hν)(d[α(hν)]d[hν])-1=h[ddν(lnα(ν))]-1,
where α(hν) represents, for each sample, the experimentally deduced optical absorption profile.Urbach energyEu is a measure of the inhomogenoeus disorder and atomic scale dispersion inside structures as it indicates the width of the band tails of the localized states in presence of defects (Figure 1). Its analytical formulation is deduced by taking into account three components: structural disorder, carrier-phonon interaction, and carrier-impurity:
(2)Eu=12kBUθD︷Structuraldisorder+F4π2Z2q4m*LD393ε2ℏ2︷Carrier-phononinteraction+F′coth(F′′2kBT)︷Carrier-impurity
with: Boltzmann constant, U: lattice strain related with the structural disorder, θD: Debye temperature, LD: Debye length, m*: Carrier effective mass, Z: impurity charge, q: electron charge, ε: static dielectric permittivity, ℏ: Planck’s constant, and  F, F′, F′′: constants.Figure 1
Urbach tailing and localized states in presence of defects.The width of the localized states (band tail energy or Urbach energyEu) has been estimated from the slopes of the plots of lnα(ν) versus energy hν (Figure 2).Figure 2
Plots of(lnα(ν)) versus energy hν (as guides for evaluating Eu).Figure3 reports the temperature-dependent zero field cooling (ZFC) molar susceptibility χmol for BTO-, BGO-, and BSO- doped samples. All the samples show clear transitions from a paramagnetic (P) to a ferromagnetic (F) state. For all the considered samples, saturation of the magnetisation was reached by applying magnetic field up to 5 T. The magnetic moment value per unit formula obtained from magnetization saturation is in good agreement with the expected ones from the stoichiometric formula.Figure 3
Zero field cooling (ZFC) molar susceptibility versus temperature for the doped samples.Faraday effect (or Faraday rotation effect) is a magneto-optical phenomenon which was revealed in the beginning of the last century by Michael Faraday [15–18] and which consists of an interaction between light and a magnetic field inside a given medium [18, 19]. It causes a rotation of the plane of polarization which is linearly proportional to the component of the magnetic field in the direction of propagation. The Faraday effect is based on the notion of circular birefringence, which causes a difference of propagation speed between left and right circularly polarized waves.Faraday effect has been evaluated, for the obtained samples, through the measurements of alterations of the Verdet coefficientV within the visible spectral domain. This coefficient [19] is deduced via the measurement of the polarisation rotating angle θ using the formula
(3)V=1Blθ
with B: applied magnetic field strength (in oersteds) and l: light path length through the medium.Verdet coefficientV changes for doped and undoped samples have been gathered in Table 1. XRD diagrams are also gathered in Figure 4.Table 1
Values of Verdet coefficient  for doped and undoped samples.
Sample
Verdet coefficient (a. u.)
BSO
Undoped
0.052
Mn-doped
0.039
BTO
Undoped
0.041
Mn-doped
0.038
BGO
Undoped
0.053
Mn-doped
0.047Figure 4
XRD diagrams of the prepared compounds.


### 3.2. Lattice Compatibility Theory LCT Fundaments and Analysis

Stability of Mn ions within host matrix does not occur in the same way inside the three studied lattice structures (BSO, BGO, and BTO). The Lattice Compatibility Theory [20–23] tries to give a plausible understanding of this disparity starting from intrinsic doping-element lattice properties in comparison to those of the host. In the studied materials, changes in the studied parameters have been associated to a Mn-doping-induced disorder in BSO matrices against a relative unaltered stability of both BGO and BTO. For explanation purposes main lattice constants of Mn intrinsic lattice have been compared to those of BSO, BGO, and BTO (Figure 5). Consecutively, a thorough study of BSO structures revealed a strong incompatibility with the Mn cubic lattice in terms of both bond length and incorporation kinetics (Figure 6).Figure 5
BTO and BGO lattices structure.Mn-element incorporation within host matrix: (a) interstitial position, (b) substitutional position.
(a)
(b)Finally, a possible explanation for the paradox of disparity of incorporation behaviors of doping agent is formulated as follows:“the stability of doping agents inside host structures is favorized trough geometrical compatibility, expressed in terms of matching patterns between doping agent intrinsic lattice and those of the host.”The many principles of this theory have been judged in good agreement with results published in the recent literature [17–23].


## 3.1. Mn-Doping Patterns in terms of Bandgap, Magnetization, and Faraday Effects

In order to understand bandgap edges alteration following doping agent insertion in host structures, Urbach energyEu has been determined, for doped and undoped samples through the equations
(1)ln(α(hν))=ln(α0)+hνEu,Eu=α(hν)(d[α(hν)]d[hν])-1=h[ddν(lnα(ν))]-1,
where α(hν) represents, for each sample, the experimentally deduced optical absorption profile.Urbach energyEu is a measure of the inhomogenoeus disorder and atomic scale dispersion inside structures as it indicates the width of the band tails of the localized states in presence of defects (Figure 1). Its analytical formulation is deduced by taking into account three components: structural disorder, carrier-phonon interaction, and carrier-impurity:
(2)Eu=12kBUθD︷Structuraldisorder+F4π2Z2q4m*LD393ε2ℏ2︷Carrier-phononinteraction+F′coth(F′′2kBT)︷Carrier-impurity
with: Boltzmann constant, U: lattice strain related with the structural disorder, θD: Debye temperature, LD: Debye length, m*: Carrier effective mass, Z: impurity charge, q: electron charge, ε: static dielectric permittivity, ℏ: Planck’s constant, and  F, F′, F′′: constants.Figure 1
Urbach tailing and localized states in presence of defects.The width of the localized states (band tail energy or Urbach energyEu) has been estimated from the slopes of the plots of lnα(ν) versus energy hν (Figure 2).Figure 2
Plots of(lnα(ν)) versus energy hν (as guides for evaluating Eu).Figure3 reports the temperature-dependent zero field cooling (ZFC) molar susceptibility χmol for BTO-, BGO-, and BSO- doped samples. All the samples show clear transitions from a paramagnetic (P) to a ferromagnetic (F) state. For all the considered samples, saturation of the magnetisation was reached by applying magnetic field up to 5 T. The magnetic moment value per unit formula obtained from magnetization saturation is in good agreement with the expected ones from the stoichiometric formula.Figure 3
Zero field cooling (ZFC) molar susceptibility versus temperature for the doped samples.Faraday effect (or Faraday rotation effect) is a magneto-optical phenomenon which was revealed in the beginning of the last century by Michael Faraday [15–18] and which consists of an interaction between light and a magnetic field inside a given medium [18, 19]. It causes a rotation of the plane of polarization which is linearly proportional to the component of the magnetic field in the direction of propagation. The Faraday effect is based on the notion of circular birefringence, which causes a difference of propagation speed between left and right circularly polarized waves.Faraday effect has been evaluated, for the obtained samples, through the measurements of alterations of the Verdet coefficientV within the visible spectral domain. This coefficient [19] is deduced via the measurement of the polarisation rotating angle θ using the formula
(3)V=1Blθ
with B: applied magnetic field strength (in oersteds) and l: light path length through the medium.Verdet coefficientV changes for doped and undoped samples have been gathered in Table 1. XRD diagrams are also gathered in Figure 4.Table 1
Values of Verdet coefficient  for doped and undoped samples.
Sample
Verdet coefficient (a. u.)
BSO
Undoped
0.052
Mn-doped
0.039
BTO
Undoped
0.041
Mn-doped
0.038
BGO
Undoped
0.053
Mn-doped
0.047Figure 4
XRD diagrams of the prepared compounds.


## 3.2. Lattice Compatibility Theory LCT Fundaments and Analysis

Stability of Mn ions within host matrix does not occur in the same way inside the three studied lattice structures (BSO, BGO, and BTO). The Lattice Compatibility Theory [20–23] tries to give a plausible understanding of this disparity starting from intrinsic doping-element lattice properties in comparison to those of the host. In the studied materials, changes in the studied parameters have been associated to a Mn-doping-induced disorder in BSO matrices against a relative unaltered stability of both BGO and BTO. For explanation purposes main lattice constants of Mn intrinsic lattice have been compared to those of BSO, BGO, and BTO (Figure 5). Consecutively, a thorough study of BSO structures revealed a strong incompatibility with the Mn cubic lattice in terms of both bond length and incorporation kinetics (Figure 6).Figure 5
BTO and BGO lattices structure.Mn-element incorporation within host matrix: (a) interstitial position, (b) substitutional position.
(a)
(b)Finally, a possible explanation for the paradox of disparity of incorporation behaviors of doping agent is formulated as follows:“the stability of doping agents inside host structures is favorized trough geometrical compatibility, expressed in terms of matching patterns between doping agent intrinsic lattice and those of the host.”The many principles of this theory have been judged in good agreement with results published in the recent literature [17–23].


## 4. Conclusion

The present study tries to give new arguments and fundaments to the Lattice Compatibility Theory (LCT). Main investigations have carried out comparative studies of the behavior of titanates (BTO), germanates (BGO), and sillenites (BSO) under transient-metal-doping. Urbach tailing and Faraday effect along with lattice constants alterations have been compared and discussed. Recoded results were in good agreement with this theory. A possible statement has been formulated:“the stability of doping agents inside host structures is favorized trough geometrical compatibility, expressed in terms of matching patterns between doping agent intrinsic lattice and those of the host.”Nevertheless, some additional investigation may be needed in order to confirm validity and universality.



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*Source: 578686-2013-07-09.xml*