A Thesis

Presented to

The Faculty of the Department of Chemistry

San José State University

In partial Fulfillment

of the Requirement for the Degree

Master of Science

By

Long Hoanghuan Nguyen

May 1999

© 1999

Long H Nguyen

ALL RIGHTS RESERVED

EXTENDED X-RAY ABSORPTION FINE STRUCTURE (XAFS) STUDIES OF PHASE TRANSITION IN HIGH T_c CUPRATE SUPERCONDUCTOR.

By Long H. Nguyen

XAFS measurements are used to investigate the change in the local structure about the Ba and Nd atoms in the orthorhombic Superconductor Nd_1.1Ba_1.9Cu₃O_7-d near the superconducting transition temperature T_c » 75K. The Cu and Nd local structures are stable while the Ba and some O sites are disordered as the material undergoes the transition to superconductivity. It is shown that the Ba and O position are sensitive to the superconducting phase transition while the Cu and Nd are not.

I would like to thank Professor Juana V. Acrivos, for her guidance, support throughout the program, and extensive proof reading of this thesis. I also want to thank Professor Bradley Stone and Professor Robert Richardson for being part on my research committee.

Thanks are also due to NSF/DMR Research Grants at SJSU to the Acrivos Group and DOE for support at SSRL.

I would like to thank Acrivos Group for all their helps.

A very special thanks to my mom and all my brothers and sisters for their love, support, and encouragement.

ACKNOWLEDGEMENT.......................................... v

TABLE OF CONTENTS........................................... vi

LIST OF TABLES.................................................. viii

LIST OF FIGURES................................................. ix,x

I. INTRODUCTION................................................. 1

1. Structure of Superconductor 123................................ 1

1.1 Tetragonal 123.................................................... 2

1.2 Orthorhombic 123................................................ 2

1.3 Variation of Cu Substitution..................................... 3

1.4 Oxygen in superconductor ....................................... 4

1.5 Substitution of Y and Ba ........................................ 4

1.6 Electronic Structure of 123....................................... 5

1.7 Structural change at Phase Transition........................... 7

1.8 Thesis Objective.................................................... 7

2. EXAFS Theory....................................................... 8

3. EXAFS Data Analysis.............................................. 10

3.1 Background Removal............................................. 10

3.2 Normalization and m ₀ Correction............................... 11

3.3 Conversion of E to k (extracting XAFS)..................... 12

3.4 Weighting Scheme................................................. 12

3.5 "Deglitching" and truncation..................................... 12

3.6 Fourier Transform................................................. 13

3.7 Curve Fitting....................................................... 14

3.8 Theory of Multiple Scattering.................................... 14

II. X-RAY ABSORPTION SPECTROSCOPY EXPERIMENT 17

1. Sample Preparation ................................................ 17

2. Data Collection at SSRL.......................................... 19

3. Computer Interface ................................................. 20

4. Analysis Procedure.................................................. 21

III. RESULTS AND DISCUSSION................................. 22

IV. CONCLUSION..................................................... 25

REFERENCES.......................................................... 26

Table 1: Distance between Y and Ba to other atoms in the unit cell.......... 55

Table 2: Standard matrices for Orthorhombic and Tetragonal systems. ...... 56

Table 3: X-ray Cross Sections and Atomic Weight of Atoms for sample

Calculation.................................................................. 57

Table 4: Model for Nd fitting...................................................... 58

Table 5: Model for Ba fitting....................................................... 59

Table 6: Fitting result for the lowest temperature run of Nd L_III edge........ 60

Table 7: Fitting result for the lowest temperature run of BaL_III edge......... 61

Figure 1: Tetragonal and Orthorhombic Structures of Yba₂Cu₃O_7-d (123).. 29

Figure 2: Sub-layers of 123 compound.......................................... 30

Figure 3: Energy bands of Yba₂Cu₃O₇ in the Brillouin Zone of .............. 31

Figure 4: XAFS of Sample KKD on Ba L_II as temperature increasing....... 32

Figure 5: Backscattering of photoelectron between absorber and scatterer.. 33

Figure 6: Unprocessed XAFS of Sample KKD on BaL_II...................... 34

Figure 7: Subtracted pre-edge and truncated of BaL_II spectra................ 35

Figure 8: Spline removal of BaL_II spectra....................................... 36

Figure 9: XAFS is converted into c (k) and graphed versus k vector......... 37

Figure 10: XAFS is converted into c (k)*k³ and graphed versus k vecto.... 38

Figure 11: Fourier transform of data and a sample of fitting.................. 39

Figure 12: Fitting curve compared with the data.............................. 40

Figure 13: Intensity of beam line and sample with thickness x............... 41

Figure 14: SPEAR storage ring at Stanford Linear Acceleration Center (SLAC). 41

Figure 15: Double Silicon-Crystal monochromator............................ 42

Figure 16: Transmission absorption setup for the experiment............... 42

Figure 17: Nd Fitting model...................................................... 43

Figure 18: Ba Fitting model....................................................... 43

Figure 19: A and Mu are plotted vs Temperature.............................. 44

Figure 20: X-ray Temperature Difference Absorption Fine Structure....... 45

Figure 21: X-ray Absorption Fine Structure of BaL_II versus Temperature. 46

Figure 22: X-ray Absorption Fine Structure of Nd L_III........................ 47

Figure 23: Distance from Nd to O(3A,B) as temperature decreasing......... 48

Figure 24: Distance from Nd to Cu(2) and Ba as temperature decreasing... 49

Figure 25: Distance from Ba to O(2) as temperature decreasing... .......... 50

Figure 26: Distance from Ba to O(1A, 3A) as temperature decreasing....... 51

Figure 27: Distance from Ba to O(3B) as temperature decreasing.............. 52

Figure 28: Distance from Ba to Cu(1, 2) as temperature decreasing........... 53

Figure 29: Motion of Ba along the b direction................................. 54

In 1986, J.G. Bednoz and K.A. Muller discovered the first high-T_c cuprate superconductor.¹ Soon afterwards, there was an explosion of theoretical and experimental work related to this field. Extended X-ray Absorption Fine Structure (XAFS) measurements have been used to study the system of High Temperature Superconductor Nd_1.1Ba_1.9Cu₃O_7-d (123), because it is a powerful tool for providing local structural information about an element.

The 123 are simple perovskite-like structures.² There are two Y₁Ba₂Cu₃O_7-d structures (d = 0 to 0.9): tetragonal and orthorhombic (Figure 1). The values for the orthorhombic 123 principal axes dimensions are (a, b, c) = (3.81, 3.88, 11.63 Å) with d =0 at 5K.³ These can change due to partial and/or full substitution of Y, Ba and/or Cu and also by the deficiency of oxygen (d > 0.3). The 123 structure consists of 7 sublayers.⁴ These planes are (Figure 2):

1/2 CuO (chain), BaO, CuO₂, Y, CuO₂, BaO, and 1/2 CuO (chain).

All these sublayers are normal to the c-axis direction. The CuO₂ layer is not flat as shown in Figure 2. Each Cu(2) is surrounded by four nearest O(3) (1.94Å ). In addition, Cu(2) also has another Cu(2) right below on the edge at a distance 2.45Å . The Cu(2)-O(3)-Cu(2) angle in 123 is about 167° . Oxygen atoms around Cu(1) do not form a perfect octahedral system, two Cu(1)-O(2) bond distances are 2.45Å and two to four Cu(1)-O(1) bond distances are at 1.8Å . In the Table 1, the bond distances between Y and Ba to other atoms in the unit cell are calculated base on dimensions and the standard matrix as in Table 2.⁵

The CuO₂ layers have mobile charge carriers that are believed to be the core of superconductivity. Due to localized carriers, 123 has weak contact between the planes. Hence, it has extreme anisotropic properties - it is a poor superconductor in the c direction in both the normal and the superconducting state. Many theories of superconductivity are based on the experimental and theoretical evidence that the cuprate oxide planes play a major role in superconductivity and chains act as electron reservoirs which can be filled or emptied by the change of percentage of oxygen or by other types of doping.^6,7

Tetragonal 123 transforms in the D⁷_4h Space Group.⁸ The symmetry is D_4h about the Y atom. Y is at the center of the shell with eight fold coordination: eight Y-O(3) at (2.4Å ) and eight Y-Cu(3) at (3.3Å ). Ba has two shells of four fold coordination: four Ba-O(2) at (2.7Å ) and four other Ba-O(3) at ( 2.8Å ). The total oxygen content in tetragonal 123 may exceed seven per unit cell.

Orthorhombic 123 transforms in the D¹_4h space group.⁹ Dimension b is less than a due to empty space at O(1B) sites. The coordination in the orthorhombic is different to the tetragonal 's. Y has two of four folds coordination shells: four of Y-O(3A), four of Y-O(3B), and eight fold coordinate to Cu(2).

The major difference between the orthorhombic and the tetragonal structures is the filling of oxygen at the position of O(1). In the ideal orthorhombic structure, O(1A) is fully occupied and the formula is YBa₂Cu₃O₇.

Cu has d orbital electrons. According to Fei et al,¹⁰ the controlling mechanism of superconductivity is associated with ionic radius primarily rather than with the charge of the doping element. There is a reverse relation between the oxygen content and cell's c dimension of the undoped material. This happens due to repulsion weakening between the Ba layers when the oxygen atoms fill up the chains. Increasing the coordination number for Cu(1) corresponds to expanding its ionic radius and lengthening the Cu(1)-O(1) bond. Increasing the charge of Cu(1) ions (smaller radius) causes decreasing of the cell dimensions and indirectly prevents the oxygen saturation of the lattice.

Oxygen stoichiometry significantly affects the physical properties of YBa₂O₃Cu_7-d . The fraction of the sample that was initially tetragonal (d =0.1) tends to remain tetragonal while the fraction that was initially orthorhombic (d =0.9) tends to remain orthorhombic.¹¹ The oxygen atoms in the Cu-O chains are most likely to move out of the structure to become vacancies in the oxygen deficient samples of 123. Superconductivity not only depends on the deficiency of oxygen but also on the arrangement of the oxygen ions.¹² Also, Fei et al.,¹⁰ have proposed an electronic charge model that has ignored the magnetic moment interacting terms for the explanation of the high T_c mechanism. They found that the oxygen deficiency in the copper oxygen chain serves more in enhancing the formation of the antiferromagnetic phase in the oxygen deficient compound. This deficiency impedes the existence of superconductivity. As oxygen decreases, the overlap between Cu d-orbital increases; thus increasing the spin-spin coupling. The correct amount of spin-spin interactions determines the superconductivity.

The substitution of other elements into the Y and Ba sites might either increase or decrease the charge balance of the material; in consequence, these substitutions will enhance or diminish the number of Cu-3d and O-2p holes.

The important structural features of the YBa₂Cu₃O_7-d compounds arise from the fact that (2 + d ) oxygen atoms are missing from the perfect structure of perovskite YCuO₃(BaCuO₃)₂. The O vacancies in Cu plane (between the BaO planes) give rise to the formation of Cu(1)O(1) chains. The double layers of Cu(2)-O(3A,B) planes in the superconductor yield a 2D structure.

The electronic structure is described by removing the linking oxygen atoms, without changing the interaction parameters. Starting from an empty chain, each additional oxygen atom introduces one anti-bonding level and two holes. If the level is above the Fermi energy, E_F, the antibonding level will absorb two holes and there is no doping in the cuprate layers. The whole count on the cuprate layer is modified only when the level lies below E_F.

The Cu configuration is 3d⁹ with one hole in the 3d shell and the O configuration is 2p⁶, a complete p shell. In the field of the crystal lattice the five Cu(3d) orbitals will lose their degeneracy, and the same will be true of the three O(2p) orbitals. In the independent electron tight-binding model, for instance, for a CuO₂ layer, there are 11 orbitals per unit cell and therefore, 11bands are constructed from them. Because this conducting layer has only one hole, the Fermi energy intersects only the highest of these bands. And these highest lying orbitals have lobes that lie within a-b plane.

The 123 superconductors contain one formula unit per primitive cell. Both planes CuO₂ and chains CuO cross the E_F line in the Brillouin Zone (Figure 3).¹³ These four bands consist of two Cu2(3d)-O2(p)-O3(p) orbitals and two of Cu1(d)-O1(p)-O2(p) orbitals. Two strongly dispersed band C:

Cu(d_x²_-y²)(2) - O(p_x)(2)-O(p_y)(3)

have 2D character. The anti-bonding band A:

Cu(d_z²_-y²)(1) - O(p_y)(1) - O(p_z)(2)

is 1D dispersion from the linear chain cuprate. The p bonding band B is Cu(d_zy)(1) - O(p_z)(1) - O(p_y)(2).¹⁴

From the charge density calculation, the two dimensional cuprate planes and one dimensional cuprate chains have dps bands. Y acts as an electron donor forming Y⁺³; it does not participate in the conduction electron density. The results of the calculation gives the average charge of Cu and O of 1.62 and -1.69 in the 1237 system. The main Cu(2P_3/2) peak and Cu(2P_1/2) are of d¹⁰ final state character where these electrons are transferred from ligands to Cu to fill the shell. There is also the appearance of the d⁹ final-state character which exhibits multiple splitting due to the interaction between a Cu(2p) core hole and an unpaired electron in the 3d shell.

The Y⁺³ and Ba⁺² ions contribute to the stability of the high superconducting critical temperature and explain why it changes little. The isolated Y atoms can be replaced by strongly magnetic elements.

The Acrivos group first discovered the increasing transparency of the superconductor only at near T_c (Figure 4).¹⁵ This indicates that the photoelectric effect is enhanced at the transition to superconductivity, suggesting that multiple scattering phenomena are affected by the fluctuation present at the transition to superconductivity. As the next step, the positions of atoms in this structure were speculated to change at the critical temperature. These changes in bond length between the Y-O(3A, B) can be caused by to two phenomena: the disorder of certain atoms and/or mixed valence states in an atom. For Pr doped in 123, there is distortion of Pr local structure, this distortion is due to interchanging between Pr³⁺ and Pr⁴⁺ but there is still a chance of the disorder of the other atoms. ¹⁶ There is evidence of the short distance of Cu(2)-O(4) that can be explained as the impedance charge transfer between planes and the chains. From the data of Y and Pr K edges studied by Boyce et. al.¹⁷, there is a large amount of disorder and distortion in the CuO₂ planes. Also in the Cu K edges, there is a lengthening of the Cu(2)-O(4) bond. This helps explain the distortion in the CuO₂ planes. As another approach, the distortion may be due simply to a hybridization of the O(2p) and Pr(4f) electrons. The Pr valence is not 4+ but ~3.33+.

X-ray Absorption Near Edge (XANES) and Extended X-ray Absorption Fine Structure (XAFS) Spectroscopies were used in this study to obtain the phase transition and the systematic disorder of atoms around Nd and Ba. Chapter I introduced systems of superconductor 123 and its related compounds, XAS techniques, and analysis processes. Experimental procedure including sample preparation, data collection... are in Chapter II. The result and its derivation in discussed in chapter III and concluded in chapter IV

XAFS measures the x-ray absorption coefficient m as a function of photon energy E above the threshold of an absorption edge E₀ for atoms either in a molecule or embedded in a condensed phase. An X-ray photon with energy E = hn above E₀ = hn ₀ release a photoelectron from the absorber atom with energy:

1/2 mv² = hn -hn ₀ = E_K = h ²k²/2m (1)

This photoelectron expands as a spherical wave with a kinetic energy E_K and propagates with a wavelength 2p /k, it has a probability for backscattering F_j2(k) from a neighboring atoms with phase shift f _j(k). The propagating electron coherently returns to original atom. This backscattered wave can interfere constructively or destructively with the original outgoing electron wave (Figure 5). As energy increases, the energy of the photoelectron increases and the wavelength decreases causing an alternating between destructive and constructing interference. This changing interference pattern modulates the transition matrix element of the photon absorption process. The transition of electrons into this continuum region of free electrons is modulated by an oscillatory structure, m _0c (k) which has been called XAFS. The theoretical relation is:

c (k) == (m - m ₀) / m =

where m ₀ is the absorbance extrapolated to k = 0, n_s is the number of backscattering shells, N_i is number of atoms in i^th backscattering shell at a distance R_i, l _i is effective photoelectron mean free path, s _i is thermal and static disorder parameter, and f_i(k,p ) is backscattering amplitude function for the i^th shell. c (k) depends on both the absorbing and scattering atoms and on the oxidation states of these atoms. The total phase difference between the leaving and returning wave is 2kR + a _j(k) in which R is the distance between the absorber and the backscatterer. The Sum of phase shifts due to the scattering process of nearby atoms and the potential well around the absorbing atoms is then:

a _j(k) = 2f(k) + f _j(k) - p , (3)

f (k,R_i) is the phase function (total phase shift experienced by photoelectron), it changes the origin with the frequency. It contains contributions from both the absorber and the backscatterer:

f _ijl (k,R_i) = f _il (k,R_i) + f _j(k,R_i) - l p (4)

where l = 1 for the K and L_I edges, and l = 2 or 0 for the L_II,_III edges. The electrons at the central atom experience a phase shift twice, once going out and then coming back. But electrons at the neighboring atoms experience a phase shift only once when arriving and returning back to the absorber. The sinusoidal XAFS oscillation is caused by interference, Sin (2kR_i + a _i(k)) with frequency 2R in k space. The larger R, the higher is the frequency of the oscillation. The XAFS amplitude is also reduced by 1/R². Short Range Order (SRO) where the scattering of electrons from close neighbors, producing XAFS a few lattices spacing before attenuating gives the closest fit to the data. As the energy increases above the edge, there is a gradual decline in the absorption cross section m _s. Simple theory predicts only this smooth decrease.

The interference function c (k) in (2) obtains:

c (E) = (m (E) - m ₀(E))/m ₀(E), (5)

the measurement obtains also the elemental absorption D m (E) from the experimental absorption (Figure 6). The pre-edge absorption curve can be fitted with a polynomial, a line, or a constant that is extrapolated beyond the edge (Figure 7). The elemental absorption still contains other background noises such as spectrometer base line, beam harmonics and elastic scattering. Spline fitting is a common method that uses least square procedure to remove the low frequency background components from m (E) without affecting the higher frequency XAFS oscillations. Spline fitting is the local fitting procedure where a polynomial function fits the changes. This method has problems when using higher order polynomials and many sections; it may remove parts of the XAFS oscillation and the signals. If low order or not enough sections are used, it will then distort the low distance peaks in Fourier transform.

Since the background m ₀(E) is affected by many other factors, it can not be used in the normalization procedure in real experiment. One commonly used method to normalize D m (E) is to divide it with theoretical value calculated m _victoreen or m ₀^th(E) (Figure 8):

m _victoreen = Cl ³ - Dl ⁴ (6)

with C and D are true absorption coefficient and tabulated by McMaster. The XAFS equation then is

c = (m _data - m _spline)/m _victoreen (7)

The next step is to convert the photon energy E to photon electron wave vector k using equation (1), k = 2p /l _e = {0.262(E - E₀)}^0.5 when E, k, and E₀ are in eV, Å ^-1, and the threshold energy is typically about 15eV above the calibration energy.

In order to compensate for the attenuation of the XAFS amplitude at high k value c (k) is multiplied by kⁿ to c (k) (Figure 9 and 10). This will balance the amplitude. The k³ multiplication gives more uniform EXAFS over the data range of k from 3 to 16Å^-1.

There are many discontinuities in data collecting such as glitches, spikes or steps in the data that will affect the background removal process. These discontinuities have to be removed before the background fit. Furthermore, data should be truncated at low and relatively high k. Shorter k range gives broader Fourier transform peaks and an effective resolution D R in bond lengths that decreases as D R = p /2k when the k range increases.

The Fourier transform is used to convert the data from k-space to R-space. Upon integrating, the Fourier transform gives a peak shift from the true value of R (Figure 11). The bond length can be extracted by correcting this shift versus the system with known distances. The Fourier transform of k^nc (k) in momentum (k) space gives a modified radial distribution function r (R) in distance R-space:

The Fourier transform equation including phase-correction is:

where f (k) and D are appropriate XAFS phase function and the shift from true distance. D approximately equals to half the average phase shift slope for a given interaction. The parameters in the Fourier transform of a closed system must comply with the conservation rule:

N = N_sA R²/(A_sR_s²), (10)

where the subscript S stands for standard, N, A, and R are the number of atoms, Fourier transform peak magnitude, and distance respectively.

The Fourier transform is accurate for a system with well separated peaks and provides a simpler physical picture of local structure around the absorber; however it does have many other problems such as side lobe background peak that must be taken into account.

This technique attempts to obtain the best fit to the function k^nc (k) versus k with standard XAFS models. Amplitude and phase functions can be parameterized from a simple sum of Lorentzian curves:

to a complicated ten-parameter form of Shulman:

The parameterization of the amplitude and phase functions has advantages of reducing these functions and parameters. These parameters can be obtained by fitting the known model to a model and then transfer to the unknown system for structural determination (Figure 12). Using a simple Gaussian model, the parameters R, N, and s ² are obtained with an interatomic distance accuracy of ± 0.02 Å, a coordination number to ± 1, and Debye-Waller factor (caused by thermal vibration and static disorder) in a typical range [0.0015, 0.008].

In addition to multidimensional parameter spaces, curve-fitting methods also include correlation among various parameters. The technique contains two sets of highly correlated variables: {F(k), s , l , N} and {f (k), E₀, R} in which N correlates with s ² and R correlates with E₀. The resolution by XAFS is limited by the Uncertainty Principle:

D R.D k ³ p /2, (13)

with D R is the spread of distance and D k is the data range.

The standard theory of x-ray absorption fine structure used for structural analysis is based upon a multiple-scattering formalism. Single scattering is mostly used to measure the nearest-neighbor distances, while higher-order multiple scattering contributions are important in quantitative calculation of XAFS. Multiple scattering terms are required for determining the second and third nearest neighbor distances and bond angles.

The standard theory of XAFS is based upon the assumption of a spherical muffin-tin scattering potential: where the potential seen by the photoelectron is composed of non-overlapping, spherically symmetric potentials around each atom and is flat between the different atomic-like potentials.

The multiple scattering (MS) problem is based on perturbation theory in the direction of the scattering potential. It is called a "path" approach.¹⁸ The perturbation expansion is expressed in terms of a hypothetical path in which an electron scatters and propagates atom to atom. Also in the high-energy region, the multiple scattering corrections are weaker and the simplifying plane-wave becomes more accurate. The path formalism depends on the lifetime effect that allows the possibility to truncate the multiple scattering expansion such that only path lengths less than a few mean free paths are retained.

The photoelectron wave is scattered from more than one scattering atom in multiple scattering. Photoelectron waves returning to the central atom from MS paths are thought to mostly cancel each other or undergo significant amplitude reductions because of inelastic scattering effects. However, in some compounds, forward scattering paths among linear or near linear atoms have been shown to increase amplitude relative to the single scattering predictions.

The Beer-Lambert Absorbance Law,

ln(I₀/I) = m x (14)

is written in terms of the mass absorption coefficient or cross section, a , in units of area/weight equal to, the density of the sample, d, in units of weight/volume, times the sample thickness, x (Figure 13). The absorbance is:

A = ln (I₀/I₁)/ln(10) = <a d x/2.303> (15)

The optimum value for relation (15) is 0.9. This gives the mass absorption coefficient a as function of the energy for the different atoms in units of cm²/g. The linear absorption coefficient, m , is equal to this constant, a , times the density, d, of the absorber. Relationship (15) holds for each individual component and is the sum of all individual atoms. Then for a given density d_i of component i in solution or in a solid of thickness x_i:

x_i m _i = a _i d_i x_i (16)

The average thickness times the linear absorption coefficient for a solution containing many components is then:

x m = S u_i x_i = S a _i d_i x_i (17)

Near the absorber edge energy of component 1, this contributes most the absorption of the system. The only changes that occur are in its linear absorption coefficient below the edge:

m _1- = a _1- d₁, (18)

and the value above the edge:

m ₁₊ = a ₁₊ d₁ (19)

Then x m changes as:

x D m = x₁ D m ₁ = D a ₁ d₁ x₁, (20)

where the change in the mass absorption coefficient:

D a ₁ = a ₁₊ - a _1- (21)

is obtained form the CRC Tables ¹⁹ and d₁ x₁ must be determined for maximum signal to noise ratio. From (15) it follows that:

A = x m /ln10 (22)

where A should change at the edge by 90% for maximum signal to noise ratio, i.e.,

D A/(x m ) = x₁ D m ₁/(m ln10) (23)

The calculation of the sample thickness is done to minimize the absorbance of the inert components at the interested edge. The inert component used is BN with a density of 2.25g/cm³. The calculation of the optimum sample concentration for the BaL_II edge is used as an example for a sample thickness of 0.2 cm with information on a _I of B, N, and Ba in Table 3:

A = x m = S u_i x_i = S a _i d_i x_i

= ((10.8*7.23+14.45*14)/(10.8+14))*2.25*0.2/2.3

= 2.21

At the BaL_II edge, Ba will contribute (90%) to the absorbance. The inert material is part of the background absorption. If the total absorbance is 1, relation (20) is used to calculate the weight of the sample that should be added to the sample, i. e.,

x D m = x₁ D m ₁ = D a ₁ d₁ x₁

or d₁x₁ = 1/(660-587) = 0.0137 g/cm²

For a sample area of 0.4 X 1.8 cm², the amount of 123 should be

W = 0.0137*1.8*0.4 = 0.00986 g

diluted with (0.2*0.4*1.8)*2.25 = 0.36 g of BN

The intense synchrotron x-ray source at SSRL is emitted when charged particles move near the speed of light in magnetized curved path of the SPEAR storage ring at the Stanford Linear Acceleration Center (SLAC) (Figure 14).²⁰ Synchrotron Radiation (SR) is used as a broad band, tunable x-ray source. The advantages of SR x-rays are high intensity, tunability over a wide range of energy, and precisely pulsed time structure. ( 10^-9s pulse at 10^-6s)

Continuous energy x-rays are produced using a double silicon-crystal diffraction monochromator (Figure 15). Two parallel crystals diffract the beam according to Bragg's law:

nl _hkl = 2d_hklsinq _B (24)

where l _hkl, n, and d_hkl are wavelength of the diffracted x-rays, harmonic order, and lattice spacing for that plane at a Bragg angle q _B from the beam direction. Under PDP-11 computer system control, the entire monochromator assembly is rotated so that q _B changes at a constant rate of motor step per unit time; thus, the x-ray wavelength is swept across the energy range of interest.

Solid state ion chambers are used at SSRL to measure the high intensity of the x-rays. Three ion chambers (Figure 16) are used to measure the intensity of the incident x-ray beams I₀ after absorption I₁ and to measure x-ray absorption reference edges for wavelength calibration. This consists of an ionization chamber mounted up-stream of the input beam collimator with associated current-sensing electronics to measure the incident beam flux. Usually the front and back chambers are 6" and 12" long. At the energy range less than 15keV, helium is used while with the range above 15keV nitrogen is used instead.

A MicroVAX II computer running the latest version of the CADS-4 software from Engraf-Nonius Corporation controls the CAD-4 system. The computer is tied into the SSRL ethernet and is equipped with a TK50 tape drive.

The energy of the x-ray beam is tuned by a 2-crystal monochromator. Detuning the monochromator crystals eliminates higher energy harmonics in the beam. For an energy range less than 17 keV, a Si crystal cut on 111 plane is used. There is a temperature control system available to cool the Si crystal. Liquid He is used to cool the sample down to as low as 5 K in an Oxford Cryostat. Program package XAS supports sample crystal alignment calibration of x-ray energy, crystallographic data collection, and processing of collected data.

FEFF 5¹⁸ is used for the data analysis. At first the model of fitting must be created from the XRF information. Distance and coordination of an atom from the center atom (absorber atom) are generated from this model. For Nd atom in Nd_1.1Ba_1.9Cu₃O_7-d , the model includes three single scattering shells and two multiple scattering shells (Figure 17 and Table 4). The first three single scattering shells are relatively closed to the center atom. Multiple scattering shells are distant from center absorber atom. The bonds to atoms in these distant shells must colinear with the absorber, first scatterer, and itself in Table 3. In the same manner, the model for Ba fitting includes 6 single scattering shells and three multiple scattering shells (Figure 18 and Table 5). Utilizing the FEFF 5 and EXAFSPAK²¹ to do the analysis, some constraints must be used to fit the model:

E₀ must be kept the same for all paths and is floated while distances and s ² are kept the same.

E₀ and s ² are then floated and linked in all paths, while distances are floated.

The results of the lowest temperature run is fitted first; then these results are used to fit the system at different temperatures.

The results are partially shown in table 6 and 7.

The XAFS analysis was done both on the X-Ray Abosrption Near Edge (XANES) and XAFS regions.

XANES provides information on the oxidation state of the absorber. The edge position increases linearly with the valence of a number of metals.²² The symmetry allowed transitions from Ba L core states P_3/2 and P_1/2 to s or d-symmetry state obtain the XANES white peak. The ratio of A/D A (Mu) measures the d character of the excited states for L_II and L_III edges.²³ The analysis was done on Nd_1.1Ba_1.9Cu₃O_7-d superconductors. A and m are plotted versus the temperature and energy (Figure 19). Also the threshold energy E₀ is measured but does not give quite reliable results.

X-Ray Temperature Difference Spectra (XTDAFS) were obtained:

XTDAFS º DC = kⁿ[C (T) - C (T₀)]=

The XTDAFS ( Figure 20) show that there is no change or very little change vs T in the XANES region. From the analysis of the data a number of results are observed:

The samples become transparent near T_c in the XAFS region of Ba L_II,III edges. The fluctuation of these two edges (Figure 21) maybe due to fluctuation of electron density.

XTDAFS of Nd L_III edge does not fluctuate (Figure 22).

Multiple scattering fits were first done on the structure of Nd-Substituted 123. The analysis was done on Ba L_III and NdL_III edges.

For Nd L_III analysis, distances from the absorbers were graphed against temperature. The distance between Nd and O(3A, 3B) and Cu(2) atoms do not change as the superconductor going through phase transition as in Figure 23 and 24. Assumption can be made that either Nd with Cu(2) and O (3A,B) in cuprate plane do not fluctuate or they all fluctuate at the same time. The second assumption seems to be more unreal.

In this analysis, the first two shells include 4 O(3A) and 4 O(3B), the next shell includes 8 Cu(2). These three shells are important in the fitting. The Fourier transform of the data obtains peaks at the 4 and 5Å. This shows that there are penetrations of the scattering electrons through these nearest neighbor shells. There must be backscattering atoms at these distance intervals. The next shell is Ba(2) that is in the same unit cell of the primary unit cell and is not important in the fitting. The rest are Nd(100, -100) and Nd(010, 0-10) that are 1 cell unit distance from the Nd of the primary unit cell, Ba(100, -100) and Ba(010, 0-10) that are also 1 cell unit distance from Nd of the primary unit cell, and Y(110, 1-10, -110, -1-10) that is diagonal in unit distance form Y of the primary unit cell.

In the analysis of Ba L_III, an increase in the Ba-O(1A) and Ba-O(3A) distances and an unchange in the Ba-O(3B) bond distance implies Ba moves along b direction parallel to cuprate chain (Figure 25, 26, 27 and 28).

The motion of the Ba atoms near T_c is mapped in Figure 29.

Nd does not move relatively to Ba.

Near T_c, the XTDAFS change the multiple scattering due to fluctuations near T_c that would give rise to slow changes in the first two terms in relation (24), changes in the potential sensed by the backscattered electron affect the phase shift which may then undergo the fluctuations of wave length of the order e .

The increased transparency near T_c in the XAFS region suggests a decrease in the electron density in path of x-rays. The correlation of the observed increased transparency with the increased bond distance should depend on the electron density.

No fluctuation of XTDAFS as a reason of Nd to be held tightly between two CuO layers (Figure 24).

Bond distances r are related to the overlap transfer integral t(b ) which gives rise to the conductivity:

t decreases with bond density giving rise to a greater bond distance r.

Change in the electron density allows the Ba atom to move in and out of the equilibrium.

The Ba motion along the b direction a distance 0.1Å may be caused by an increase of the electron density along chains near T_c.

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Figure 1: Tetragonal Structure and Orthorhombic Structure of YBa₂Cu₃O_7-d .²⁴ Orthorhombic structure has no Oxygen at O(1B) positions, causing a different between a and b dimensions.

Figure 2: The structure contains seven sub-layers that include 1 CuO chains, 2 BaO planes, 2 Cu₂O planes, and Y plane.

Figure 3: Energy bands or orthorhombic YBa₂Cu₃O₇ plotted along several directions in the Brillouin Zone.¹³ The Fermi energy is 0.0. Two strongly ispersed band C Cu(d_x²_-y²)(2) - O(p_x)(2)-O(p_y)(3) have 2D character. The anti-bonding band A Cu(d_z²_-y²)(1) - O(p_y)(1) - O(p_z)(2) is 1D dispersion from the linear chain cuprate. The p bonding band B is Cu(d_zy)(1) - O(p_z)(1) - O(p_y)(2).

Figure 4: Increasing transparency of Superconductor near T_c. Near critical temperature, the absorbance decreases in XAFS region.

Figure 5: Backscattering of photoelectron between absorber and backscatterer. Outgoing wave and backscattering wave can either constructively or destructively interfere causing a modulation of absorbance in XAFS region.

Figure 6: X-ray absorption spectrum (m x = abs) vs E for Ba L_II edge of Nd_1.1Ba_1.9Cu₃O₇. L_II edge of Ba corresponds to the ejection of 2p electron by absorption of a photon with E ³ E₀. E₀ is the threshold energy.

Figure 7: Subtracted pre-edge and truncated of the BaL_II spectra. The pre-edge region for BaL_II (5500-5480) is fitted with a polynomial that can be extrapolated beyond the edge. The pre-edge fit is then subtracted from the absorption data (raw data).

Figure 8: Spline removal of the BaL_II Speectra to convert the subtracted pre-edge data to XAFS data. Normalized D m is a difference between absorption data and Spline function (background function). XAFS c (E) is redefined as the result of division of normalized D m _e over m _Spline.

Figure 9: c (E) is converted to c (k) with k is photoelectron wave vector k = 2p /l _e.

Figure 10: c (k) is multiplied with kⁿ to compensate the attenuation of the data. c (k)*k³ is called weighted EXAFS spectra.

Figure 11: Fourier Transformation of weighted EXAFS of BaL_III edge in k space into radial distribution space is plotted thin curve. Thick curve is the Fourier transformation of the model.

Figure 12: c (k) Fitting curve from back transform. The thin spectrum is the weight EXAFS spectrum. The thick curve fits the data with the model by the curve fitting technique.

Figure 13: Intensity of beam line and sample with thickness x. The absorbance is defined as m x = ln (I₀/I₁)

Figure 14: SPEAR storage ring at Stanford Linear Acceleration Center (SLAC). X-ray is emitted by electrons moving at high speed in a magnetic curve path (storage ring).

Figure 15: Double Silicon-Crystal

Figure 16: Transmission absorption setup for the experiment

Figure 17: Nd fitting model includes 3 single scattering and 2 multiple scattering paths.

Figure 18: Fitting model for Ba atom includes 8 single scattering and 3 multiple scattering paths.

Figure 19: A and Mu are plotted versus Temperature. Mu measures the excited states for L_II edge of Ba. Two excited states are around 90K and 50K.

Figure 20: X-ray Temperature Difference Absorption Fine Structure of Ba L_II edge versus temperature. The graph shows significant fluctuation of absorption in XAFS region near critical temperature

Figure 21: X-ray Absorption Fine Structure of BaL_II edge versus temperature.

Figure 22: X-ray Absorption Fine Structure of NdL_III versus temperature. The graph shows no significant change of absorption in the XAFS region as temperature going through critical temperatures.

Figure 23: Distance from Nd to O(3A,3B) as temperature decreasing.

Figure 24: Distance from Nd to Cu(2) and Ba as temperature decreasing.

Figure 25: Distance from Ba to O(2) as temperature decreasing. Ba-O(2) bonds change at or near critical temperature. The change is significant and well above analysis error.

Figure 26: Distance from Ba to O(1A,3A) as temperature decreasing. Same observation as Figure 25.

Figure 27: Distance from Ba to O(3B) as temperature decreasing.

Figure 28: Distance from Ba to Cu(1, 2) as temperature decreasing.

Figure 29: Motion of Ba along the b direction.

Table 1: Distance between Y and Ba to other atoms in the unit cell

Tetragonal P4/mmm (D⁷_4h)

A = b @ 3.9 Å c @ 11.8 Å Z = 1

Atoms Site Symmetry x y z

Y 1d 4/mmm 1/2 1/2 1/2

Ba 2h 4mm 1/2 1/2 0.19

Cu1 1a 4/mmm 0 0 0

Cu2 2g 4mm 0 0 0.36

O1 2f mmm 0 1/2 0

O2 2g 4mm 0 0 0.15

O3 2i mm 0 1/2 0.38

Orthorhombic Pmmm (D¹_4h)

a = b @ 3.9 Å c @ 11.8 Å Z = 1

Atoms Site Symmetry x y z

Y 1h mmm 1/2 1/2 1/2

Ba 2t mm 1/2 1/2 0.19

Cu1 1a mmm 0 0 0

Cu2 2q mm 0 0 0.36

O1A 1b mmm 1/2 0 0

O1B 1e mmm 0 1/2 0

O2 2q mm 0 0 0.16

O3A 2r mm 0 1/2 0.38

O3B 2s mm 1/2 0 0.38

Table 2: Standard matrices for Orthorhombic and Tetragonal systems.⁵

Table 3: X-ray Cross Section and Atomic Weight of Atoms for sample Calculation

Table 4: Model for Nd fitting. (Ba* and Nd* are in cells 100 and -100, Ba** and Nd** are in cell 010 and 0-10)

Table 5: Model for Ba fitting (Ba* and Nd* are in cells 100, -100, and Ba** and Nd** are in cells 010, 0-10, Ba*** belongs to cells 110, -110, 1-10, -1-10)

Table 6: Fitting result for the lowest temperature run for Nd L_III edge . (Ba* and Nd* are in cells 100 and -100, Ba** and Nd** are in cell 010 and 0-10)

Table 7: Fitting result for the lowest temperature of BaL_III edge (Ba* and Nd* are in cells 100, -100, and Ba** and Nd** are in cells 010, 0-10, Ba*** belongs to cells 110, -110, 1-10, -1-10)