A Thesis

Presented to

The Faculty of the Department of Chemistry

San Jose State University

In Partial Fulfillment

Of the Requirement for the Degree

Master of Science

By

Thaddeus Jude Norman Jr.

May 1999

A X-RAY ABSORPTION STUDY OF THE METAL TO INSULATOR

TRANSION IN NiS_1.52Se_0.48

By Thaddeus Jude Norman Jr.

model for metal to insulator transitions proposed for these compounds.

ACKNOWLEDGMENTS

I would like to thank the NSF and DOE at SSRL for funding this work, and my advisor Dr. Juana V. Acrivos for all of her support, advice, guidance, and friendship. I also would like to thank the members of the Acrivos Group for their assistants.

A special thanks goes to Dr. J. M. Honig for providing the samples of NiS_1.52Se_0.48 measured in this study, and Dr. K. Kantardjieff and Dr. G. Crundwell for taking the x-ray diffraction data and walking me through the data analysis.

I would like to thank Dr. Scharberg and Dr Wagenknecht for being on my committee.

And thanks to my parents for all your support!

Acknowledgements.............................................................................. 2

Table of Contents................................................................................ 4

List of Tables .................................................................................... 6

List of Figures..................................................................................... 7

I. Introduction ........................................................................... 9

I.1. Overview of the NiS2-xSex system.................................................. 9

I.2. Thesis Statement............................................................. 13

I.3. X-ray Absorption Spectroscopy (XAS).......................................... 14

I.3.a. An Overview of XAS......................................................... 14

I.3.b. Theory of XAS.................................................... 15

II. Experimental.................................................................... 18

II.1. X-ray Diffraction ................................................................. 18

II.2. X-ray Absorption Measurements................................................... 18

II.2.a. XAS Sample Preparation......................................................... 18

II.2.b. Data Collection....................................................................... 19

III. Results............................................................................ 20

III.1. X-ray Diffraction....................................................... 20

III.2. XAS Data Analysis................................................................. 20

III.2.a. Ni Edge Data........................................................................ 23

III.2.b. Se Edge Data.................................................................... 24

III.3. X-ray Temperature Difference Absorption Spectroscopy (XTDAS)............. 25

IV. Discussion........................................................................... 28

V. Conclusion.............................................................. 30

References................................................................... 31

Table 1: The atomic positions and distances for NiS_2-xSe_x................................. 43

Figure 1: Phase diagram for the NiS_2-xSe_x system.......................................... 33

Figure 2: The unit cell of a crystal in the Pa3 space group.................................. 34

Figure 3: The molecular orbital diagram of an octahedral transition metal complex... 34

Figure 4: Qualitative band models for NiS_2-xSe_x.............................................. 35

Figure 5: The XAS taken at the Ni-K edge................................................... 36

Figure 6: The X-ray diffraction pattern for NiS_2-xSe_x....................................... 37

Figure 7: A block diagram of instrumentation at beam-line 2-3............................ 38

Figure 8: Vegard's Law plot for NiS_2-xSe_x................................................... 39

Figure 9: Figure 5 except the pre-edge region has subtracted............................... 40

Figure 10: The EXAFS spectrum as removed from Figure 8.............................. 41

Figure 11: The Fourier transform of Figure 9................................................... 42

Figure 12: The XAS spectrum of the Ni-K edge at different temperatures............... 44

Figure 13: Plot of the Ni-S bond distance versus temperature from the Ni-K edge...... 45

Figure 14: Plot of the Ni-Se bond distance versus temperature from the Ni-K edge..... 46

Figure 15: Plot of the Ni-Ni bond distance versus temperature from the Ni-K edge..... 47

Figure 16: The XAS spectrum of the Se-K edge at different temperatures................ 48

Figure 17: The Ni-Se bond distance versus temperature from the Se-K edge............ 49

Figure 18: A comparison of the Ni-Se bond distance plots................................. 50

Figure 19: Plot of the Se-S bond distance as measured from the Se-K edge............... 51

Figure 20: Plot of the Se-Se bond distances versus temperature from the Se-K edge.... 52 Figure 21: An overlay of the FT-EXAFS of the Ni-K edge data........................... 53

Figure 22: An overlay of the FT-EXAFS of the Se-K edge data........................... 54

Figure 23: A 3-D plot of the FT-EXAFS versus temperature for the Se-K edge data... 55

Figure 24: The XTDAS plot of the Se-K edge data.......................................... 56

Figure 25: A plot of The Debye-Waller factor for the Ni-Se bond......................... 57

Figure 26: A plot of The Debye-Waller factor for the Se-S bond.......................... 58

I. 1. Overview of the NiS_2-xSe_x system

There is only one exception to the simple band theory model that successfully predicts the electronic behavior of transition metal dichalcogenides possessing the pyrite structure. The theory predicts that NiS₂ should be metallic in nature, but experiments indicate that the compound is in fact a semiconductor¹. When NiS₂ is doped with Se forming the mixed crystal NiS_2-xSe_x a variety of electronic behavior is exhibited². Depending upon the temperature and Se composition (Figure 1), the material can be an anti-ferromagnetic metal, anti-ferromagnetic insulator, a paramagnetic metal, paramagnetic insulator, a ferromagnetic insulator, or a ferromagnetic metal. This behavior can not be easily explained using a simple band model, so more complicated theories have been developed³.

A good way of viewing the structure of pyrite is that of S₂ molecules dissolved in a face centered iron crystal lattice (Figure 2). Transition metal compounds with the pyrite structure posses a face centered cubic lattice and fall in the Pa3 space group. This necessitates that the transition metal occupies a site in the lattice with octahedral symmetry, and the chalcogenide atom occupies a site with tetrahedral symmetry. The metal orbitals interact with the orbitals on the chalcogenide atoms in a fashion similar to that of an octahedral transition metal complex. The octahedral symmetry of the metal atom causes a loss in degeneracy of the metal's d orbitals; two orbital sets of different energy and symmetry are created (Figure 3). The higher energy set contains two d orbitals of e_g symmetry while the lower energy set contains the remaining three orbitals and has t_2g symmetry in the Pa3 space group. The electrons fill the orbitals according to Hund's Rule and the Pauli Exclusion Principle. So, following these rules, the e_g^* orbitals are half filled and are the highest occupied molecular orbitals (HOMO), containing one electron in each e_g^* orbital. Since these compounds are not discrete molecules, the molecular orbital picture must be modified to account for the interactions of all the atoms in the lattice. These resulting orbitals become bands (Figure 4a), and, as should be expected, the band model for the electronic structure of the crystal resembles the molecular orbital model for a transition metal octahedral complex. As in the molecular orbital picture developed, there are bonding and anti-bonding band states. The symmetry is also preserved with the t_2g orbital forming non-bonding band state and the e_g orbitals forming bonding band states. Like in the molecular model the e_g band is the highest occupied state and is half filled with electrons.

The filling of the bands with electrons has been used to qualitatively predict the electronic properties of the transition metal dichalcogenides with the pyrite structure⁴. Bands that are completely filled with electrons cannot conduct electricity and are thus non-metallic while partially filled or empty bands can conduct electricity and are thus metallic. If a filled band lies close in energy (E_g) to an empty band electrons possessing the right energy can overcome the energy gap between the two bands allowing the material to conduct electricity with an activation energy, 1/2E_g. This phenomenon is typical in semiconductor materials. Following these rules one would predict that NiS₂ would be a conductor of electrons because the e_g band is half filled, but experiments reveal that NiS₂ is a semiconductor.

In NiS₂ electronic correlation effects, resulting from exchange interactions, alter the band model that has been qualitatively developed. Intra-atomic exchange occurs when electrons within an atom switch spin states. Inter-atomic exchange occurs when electrons in a molecule switch nuclei. Each phenomenon has an energy associated with it called the exchange energy. The exchange energies play a key role in the band model for NiS₂. The unpaired electrons in NiS₂ have a high intra-atomic exchange energy, which results in splitting the e_g band into two spin domains with the intra-atomic exchange energy being the energy required to move electrons between the two domains (Figure 4b)⁴. The spin domains require each electron be associated with another electron with the same spin; that is the electrons are correlated. This electronic effect has three consequences for the properties of NiS₂. The first is that NiS₂ will be a semiconductor due to the energy required to move electrons between the two bands. Secondly bonding in NiS₂ is more ionic in nature since intra-atomic interaction localizes the electrons on Ni. Thirdly NiS₂ will posses some type of magnetic order at low temperatures. Experiments show that NiS₂ posse anti-ferromagnetic ordering with T_Néel = 40 K⁵, where is T_Néel the onset temperature for the anti-ferromagnetic state.

In NiSe₂ the inter-atomic exchange is favored over the intra-atomic exchange, and the simple band model can be applied. As expected NiSe₂ is metallic⁴. When Se is added to NiS₂ forming NiS_2-xSe_x a competition between intra-atomic and inter-atomic exchange energies is created. With the increase in Se composition the compounds become more metallic in nature. This is indicated by the dependence of resistivity and magnetic susceptibility on the Se composition⁶. NiS_2-xSe_x 0 < x < 0.38 displays typical semiconductor behavior, an insulator that conducts electricity at high temperatures. In the Se composition range 0.55 < x < 2 the compound is metallic. However, in the Se composition range 0.38 < x < 0.55 the material undergoes a metal-insulator transition (MIT). At low temperatures the material is metallic but there is a sharp increase in resistivity near a specific temperature, which indicates the onset of an insulator phase². The temperature of the MIT increases with increasing Se composition. The NiS_2-xSe_x compounds in the MIT composition range are anti-ferromagnetic in both the metallic and insulator phases, and are slightly ferromagnetic at low temperatures.

The debate around the band model for NiS_2-xSe_x centers on the magnitude of the intra-atomic exchange energy. If the intra-atomic exchange energy is larger than the inter-atomic exchange energy then the MIT in these compounds is due to charge transfer between the nickel and chalcogenide atoms (Figure 4c). If the intra-atomic exchange energy is smaller than the inter-atomic exchange energy then conductivity occurs following the Mott-Hubbard model⁷ where the two e_g bands merge to form one band (Figure 4d). The truth is probably somewhere in between. Experiments show that at low temperature the Mott-Hubbard model can be applied to the conductivity behavior of the compounds at low temperatures while a charge transfer mechanism fits the data gathered at high temperatures. This would suggest that the two exchange energies be about the same magnitude².

An X-ray Absorption Spectroscopy (XAS) study was performed on NiS_1.52Se_0.48 in order to ascertain if the electronic behavior demonstrated in the phase diagram influences the absorption spectrum. The changes seen the in absorption spectrum at different temperatures correlate with the compound's metal insulator transition. The data also indicate that an order to disorder transition occurs in these compounds near the metal insulator transition suggesting that the anti-ferromagnetic metallic phase is a more ordered state than the anti-ferromagnetic insulating phase; a change in the band structure might be occurring in accordance with the Mott-Hubbard model.

The analytical tool used to study NiS_1.52Se_0.48 was Extended X-ray Absorption Fine Structure (EXAFS), an XAS technique. An XAS spectrum (Figure 5) can be broken down into four regions: the pre-edge, the edge, the x-ray absorption near edge structure (XANES) and EXAFS regions. The pre-edge region is in the lower energy side of the edge jump in the XAS spectrum. This region is essentially the background for XAS spectrum. The edge is the point on the spectrum where the x-ray energy matches the energy required to promote an electron from the absorbing atom to the continuum. In the spectrum the edge is the point where the quantum change in absorption occurs. Just after the edge is the XANES region, and the EXAFS region begins roughly 50ev beyond the edge. The changes in absorption seen in the XANES and EXAFS part of the spectrum are primarily due to scattering of the photoelectron wave off of the electron clouds of nearest neighbor atoms. However, the atoms immediately around the absorbing atom heavily influence the absorption in the XANES region. The symmetry of the absorbing atom and electronic environment around the absorbing atom play a more important role in influencing the XANES spectrum than the EXAFS. The EXAFS are generated from the electron wave scattering off of atoms as far as 10Å away from the absorbing atom.

The XAS phenomenon is not new for it is essentially an extension of the photoelectric effect. Photons of the right energy can promote a bound electron to the continuum, and X-rays posse the right energy to promote inner shell electrons to the continuum. In 1920 Fricke and Hertz were the first to detect the changes in absorption beyond the x-ray edge^{8, 9}. Further experimentation by others lead to the detection of more fluctuations in x-ray absorption hundreds of electron volts beyond the x-ray edge. In 1932 Kronig was the first to propose a theory that accurately explained the phenomenon¹⁰. Over time an equation was developed to relate the changes in absorption in the EXAFS region to inter-atomic distances between the absorbing atom and its' nearest neighbors^11-13. The derivation of this equation can be simplified, and the derivation presented here is based on that of Stern¹⁴. The EXAFS is the sum of two waves an outgoing wave and backscattered wave. It is the interference of these waves, which creates the EXAFS signal. The outgoing spherical electron wave can be described by the equation

r^-1e^(ikr), (1)

where r is the position of the wave and k is the electron wave vector 2p/l where l is the wavelength. The backscattered wave can be described by the equation

|r-r_i|^-1e^(ik|r-ri|), (2)

where r_i is the position of the backscattered wave. The simultaneous existance of the two waves at position r_i times an amplitude factor T_i(2k), which is dependent on the nature of the backscatterer, is the EXAFS relation

T_i(2k)[( r_i^-1e^{(ik ri)})( |r-r_i|^-1e^(ik|r-ri|))] . (3)

The equation must be corrected for other effects and simplified. Since r is the origin it can be assigned the value of 0, so equation 3 becomes

T_i(2k)[( r_i^-2e^{(i2k ri)})] (4).

A phase correction, [d_i(k)-(p/2)], must be added to the electron wave vector 2ikr to account for the varying electronic field experienced by the electron wave. Now the EXAFS equation becomes

T_i(2k)[( r_i^-2e^{(i2k ri +[di(k)-(p/2)]})] (5).

Only the real part of the equation is used, so (5) becomes

c_i(k) = K T_i(2k)[r_i^-2sin(2kr_i +d_i(k))] (6),

where c_i(k) is the EXAFS and K is a proportionality constant, which is a function of the electron wave vector.

The equation has to be further modified to account for the lifetime of the excited state. The hole created by the departure of the electron wave has a lifetime of on the order of 10^-15 seconds. The electron wave has a lifetime as a result of backscattering. The degree to which the two lifetimes are synchronous determines how well the backscattered wave interferes with the outgoing wave. This affects absorption. The lifetime of the EXAFS is defined as the time it takes the electron wave to go to the backscattering atom and return to the hole without scattering off of any other atoms. The probability of the electron wave returning to the hole before it is filled, e^{(-2ri /l(k))}, must be factored into the EXAFS equation. Thus equation 6 becomes

c_i(k) = K T_i(2k)*[e^{(-2ri /l(k))}]*[r_i^-2sin(2kr_i +d_i(k))] (7)

The K T_i(2k) term can be rewritten as m[2p(hk)²]^-1t_i(2k), where h = h/2p. Equation (7) represents the EXAFS from only one electron wave. In order to account for all the EXAFS the equations for the waves are summed. Atoms at similar distances can be treated as if they are at the same average distance R_i. In order to use this term to replace r_i a disorder term (the Debye-Waller factor), N_ie^{(-2(ksi)^2)}, must also be factored into the equation. In this term N_i represents the number of atoms of type i at the distance R_i and s_i is the standard deviation from R_i. Making these corrections to the EXAFS equation, it becomes

c_i(k) = m[2p(hk)²]^-1St_i{(2k)*[e^{(-2 Ri /l(k))}]*{[ R_i ^-2 * N_ie^{(-2(ksi)^2)}] sin(2kr_i +d_i(k))]} (8)

Over the years, many scientists have contributed to the development of XAS and EXAFS theory and techniques, but it wasn't until the availability of high energy x-ray sources at synchrotron radiation laboratories did the technique become popularized. Without the use of synchrotron radiation it is difficult to produce an x-ray spectrum with photon yields which would make EXAFS feasible.

The Honig Group of Purdue University¹⁵ supplied samples of NiS_1.52Se_0.48.

The x-ray diffraction data (Figure 6) was collected by Dr. Kantardjieff and Dr. Crundwell at the W. M. Keck Foundation Center for Molecular Structure.

XAS, like all other light absorption techniques, conforms to the Beer-Lambert Law

dI = -acIdx. (9)

Due to absorption the intensity of a beam of light (I) passing through a material decreases by an amount dI. The change in I (dI) can determined by the path length (dx) of the beam of light, the concentration (c) of the material within the path of the beam, and a cross section mass absorption coefficient (a). So upon integration the equation for the absorption phenomenon is

ln(I₀/I) = -act, (10)

where ln(I₀/I) is the molar absorption coefficient (A), and t is the thickness of the sample the beam of light passed through. In an XAS experiment each atomic species contributes to a, and the degree to which the atom contributes to a is determined by the atom's atomic number and the energy of x-rays used. As a general rule the higher atomic number the greater the contribution to a with the absorbing atom having the greatest contribution.

In this experiment to achieve efficient condition for x-ray absorption, ln(I_o/I)=10,

0.0057g of sample were diluted with 0.0988g of fine boron nitride powder then transferred to the sample holder whose area was 0.60955 cm². Boron nitride was chosen as a diluant because both boron and nitrogen have small a's for the energy of x-rays used in these experiments.

XAS measurements were taken at Stanford Synchrotron Radiation Laboratory (SSRL) at beam-line 2-3. Data was collected at the Ni K and Se K edges. While the XAS measurements were taken the temperature was lowered then raised between 4 K and 150 K at a rate of 1 K to 20 K an hour.

The block diagram in Figure 7 represents the layout of the beam station. The x-ray source was synchrotron radiation from Stanford Synchrotron Radiation Laboratory (SSRL). A monochromator is used to select x-rays of the desired energies. The resulting beam is passed through a series of chambers in the order: I₀, the sample chamber, I₁, the reference, then I₂. The current in the ionization chambers is used to measure the intensity of the x-rays. They are filled with He, N₂, and Ar gas respectively. A voltage is applied to the chambers so when x-rays pass through them, ionizing the gas, a current is created. This current is proportional the intensity of the x-rays passing through the chamber. An Oxford Instruments Cryostat was used to control the sample's temperature. The sample was cooled in liquid nitrogen before it was placed in the sample chamber. A Strawberry Tree Data Acquisition Card was used to collect the temperature data, and the XAS Collect software was used to collect the XAS data.

The diffraction pattern indicated that the compound was cubic with unit cell dimensions a=5.7539 Å. Earlier work established that NiS_2-xSe_x compounds obey Vegard's Law¹⁵. According to this principle the unit cell dimensions of a mixed crystal such as varies linearly with composition. From a Vegard's Law plot (Figure 8) the value of x was determined to be 0.48. The unit cell dimensions were used to construct the crystal system for NiS_1.52Se_0.48 (Table 1), and this model was used for the initial guess for the EXAFS fitting routine. Since both NiS₂ and NiSe₂ crystals have Pa3 symmetry the Pa3 space group was used to generate the x, y, z positions for the atoms in the lattice. X-ray diffraction data on NiS₂ and NiSe₂ reveal that Ni occupies the 4 a position in the lattice and the calcogenide atoms occupy the 8 c positions¹⁶. The distance between the atoms was then calculated using the equation

d = [(x_a-x_b)² + (y_a-y_b)² + (z_a-z_b)²]^1/2 (11)

where d is the distance between the atoms. In the unit cell of NiS_1.52Se_0.48 there are four Ni atoms. In the model compound used for the Ni edge each Ni atom is surrounded by three Ni atoms at 4.0634Å distance, and eight chalcogenide atoms two at 2.4196Å, two at 3.9098Å, two at 5.6390Å and two at 7.5988Å. The unit cell from the perspective of the chalcogenide atoms is more complicated. Each chalcogenide atoms does not have the same number of atoms at the same distance. So for simplicity the model position for the Se edge was based on the atom which had three Ni atoms at the closest distance. For the initial guess only one Se atoms was used at the 2.4196Å distance and none at the other distances.

The XAS data was analyzed using EXAFSPAK¹⁷. A series of mathematical operations are performed on raw XAS spectrum in order to extract the EXAFS data. The first operation subtracts the background from the data and the second involves the extraction of the EXAFS. The pre-edge data is fit to a polynomial function. This function is then subtracted from the data set (Figure 9). To extract the EXAFS data a spline function is used. The spline is a polynomial function that is fit to the EXAFS region of the XAS spectrum. This function is then used to create the EXAFS spectrum (Figure 10). The EXAFS is generated from three parts the spline (m_s) the raw pre-edge data (m_exp) and a correction function (m_vic) using the equation

c(k) = (m_exp - m_s)/m_vic (12).

The correction function is a victoreen function, and it is used to account for distortion in the data that might occur as a result of the subtraction of the pre-edge data. The equation for the victoreen function is

m_vic = C_vicl³ + D_vicl⁴ (13),

where the coefficients C_vic and D_vic are unique to each element and edge and l is x-ray wavelength.

After these manipulation the Fourier Transform of the EXAFS spectrum is obtained (Figure 11). Since the EXAFS data generated by the XAS experiments is in terms of the electron wave vector the Fourier Transform of the EXAFS changes the spectrum from the frequency dimension to the distance dimension. In theory each peak in the FT-EXAFS spectrum corresponds to one coordination shell and the amplitude of the peak is dependent upon the number of atoms in that shell. However, the resulting FT-EXAFS spectrum is the sum of many different EXAFS spectra, so for complicated samples each peak is a composite of different coordination shells. In these complicated samples the qualitative approach of examining the FT-EXAFS spectrum can be misleading since each peak may not represent a single atomic species or coordination shell. Also the Fourier transform contains artifacts, which increases the noise in the spectrum resulting in spurious peaks.

The next step in data analysis is getting the coordination shell and distance information from the FT-EXAFS. There are several approaches to this process. The EXAFSPAK software package uses a curve fitting program to determine distances and number of atoms in a coordination shell. The EXAFS equation the program used to fit the data was:

c(k) = S₀²S [N_iS_i(k,R_i)F_i(k,R_i)]/kR² * exp[-2R_i/l(k,R_i)] * exp(-2s_i²k_i²)

*sin[2kR_i + f_i(k,R_i) + f_c(k)] (14).

This equation is essentially the same as 8, but other terms are factored into the equation to account for certain phenomenon. The S₀² term takes electronic processes that compete with the creation of the electron wave into account. Thus when there are no competing processes S₀² is one. When these processes effect the creation of the electron wave S₀² becomes less than one. The term S_i is like S₀² but it represents electronic affects created by the absorbing atoms. The amplitude term m[2p(hk)²]^-1 and t_i(2k) are represented by the function F_i(k,R_i)/kR² in the equation used for the fit. Another phase term is added to distinguish the phase distortion created by the electronic fields and variations in atomic positions. The data was fit using a Marquardt algorithm. The model compound of NiS_1.52Se_0.48 (table 1) was used to generate the initial inter-atomic distances and coordination numbers guesses for the fit. The distance information is given in terms of R + d. To determine R, d was calculated by subtracting the Ni-Ni, Ni-S, Ni-Se, Se-S, and Se-Se distance calculated from the model from the R + d value from the highest temperature data for the Ni and Se K edges.

There were some limitations on the EXAFSPAK software, which affects the data. When the software asks for the nature of the backscatterer it will only accept one atom type as a backscatterer. For our sample each edge has two backscatterers. At the Ni edge the backscatterers are S and Se, and at the Se edge the backscatterers are Ni and Se. Not taking both backscatterers into consideration will alter the phase correction.

So for these analysis the atom with the largest atomic number was used as the backscatterer.

There were no changes in the pre-edge peak intensity or in its position with temperature at the Ni K-edge, indicating that there were no changes in the valence, the coordination number or the crystal field parameters (Figure 12). The Ni-X distances obtained from the EXAFS analysis (Figure 13, 14) showed changes only for the Ni-S and Ni-Se distances. As the temperature was lowered the Ni - Se distance shrank from 2.402 ±0.011 Å at approximately 30 K reaching a minimum distance of 2.366 ±0.010 Å at approximately 25 K. The distance returned to 2.402 ±0.011 Å as the temperature was lowered to 6 K. There also appeared to be a slight increase in Ni - Se distance in the 45 K to 60 K temperature range. Likewise the Ni - S distance decreased from 2.392 ±0.003 Å to 2.382 ±0.003 as the temperature approached 23 K then returned to 2.392 ±0.003 as the temperature was lowered. The same increase in Ni - S distance around 60 K was seen. The Ni - Ni distance remained constant at 4.052 Å ±0.011 Å over the measured temperature range (Figure 15). However, a data point exists at 24.5K, which would indicate a dramatic change in the Ni-Ni distance at this temperature. A Q test was performed on the data set to determine the validity of the point. Q_exp was calculated to be 0.75, which is far larger than Q_crit for the number of points in the data set. The results of the test suggest that the point is not real, ergo it is not likely an increase in distance is occurring at that temperature. Interestingly, this temperature is near the theoretical MIT temperature as extrapolated from Figure 1.

III. 2. b. Se K Edge Data

There were no changes in the pre-edge peak intensity or position with temperature at the Se K-edge indicating that there were no changes in the valence, the coordination number or the crystal field parameters (Figure 16). However, the Se K-post-edge absorbance changed in intensity with temperature. As the temperature was lowered from 50 to 24 K the Ni-Se distance decreased (Figure 17). There also appears to be an increase in Ni - Se distance in the 60 K to 45 K temperature range. There is a discrepancy in the Ni-Se distance measured from the Ni K edge and the Se K edge (Figure 18). This could result from errors in the analysis due to multiple scattering, which introduces factors that were not considered in the phase correction. In multiple scattering the electron wave scatters off of more than one atom, so the EXAFS equation developed here is not valid in this circumstance. However, both data sets display the same trend in bond compression as the temperature is lowered and expansion in the 45 K to 60 K temperature range. Yet again there remains a discrepancy in the minimum Ni-Se distance. The Ni edge data places this distance at 24K while the Se edge data suggests the minimum occurs in the 25K to 37K temperature range. The temperature discrepancy could be explained by an uncertainty in the temperature, which we did not take into account. The sparse Se edge data in that temperature range makes it difficult to pin point where the dip in distance occurs in this data set, which is necessary to ascertain the temperature uncertainty. The Se - S distance remained constant until approximately 39 K where it abruptly increased from 3.148 ±0.035 to 3.427 ±0.038 Å (Figure 19). The distance then slowly decreased as the temperature lowered, reaching 3.249 ±0.017 Å at 11 K. The Se - Se distance (Figure 20) remained constant at 3.276 ±0.025 Å over the temperature range.

An overlay of the FT-EXAFS plots (Figure 21-23) at different temperatures was used to qualitatively check to see if any structural changes occurred. Since the magnitude of peaks in the spectrum is dependent upon the number of atoms at that position, changes in peak position and intensity could indicate changes in the geometry and coordination number around the absorbing atoms. The overlay for the Ni edge data showed little change with temperature, indicating no structural changes with temperature. The overlay of the Se edge data shows more variability over the temperature range. The peaks that appear below 1.5 Å are probably anomalies; however the other variations could indicate changes in the coordination number around the Se atoms. Significant changes in the geometry around Se would be unlikely since no such changes were seen at the Ni edge, and the changes seen may be related to changes in the Se-S distance.

In order to compare changes in structure versus temperature we obtain the x-ray temperature difference absorption spectrum (XTDAS)¹⁸ relative to the phase transition temperature:

XTDAS = A(T) - A(T_transition)

where A(T) is the absorbance at the temperature T, and T_Curie = 7 K for the anti-ferromagnetic to ferromagnetic phase transition (Figure 1). The theory behind XTDAS is simple. The EXAFS equation represents the summation of the electron waves for a system. When the system is perturbed (in our case by changes in temperature) the characteristics of the electron wave may change, changing the EXAFS. So when the EXAFS taken from a standard state are subtracted from the EXAFS from the perturbed state only the changes in the electron wave remain. For example if the perturbed spectrum is identical to the standard state spectrum then the XTDAS will be a straight line with a zero slope. If the band structure changes as a result of the perturbation and or it cause a change in the relaxation time then the XTDAS will show changes in line shape and or slope. If the electron density changes around the absorbing atom and there are no changes in the bond angle or length then the XTDAS will show oscillations due only to those changes. Sinusoidal oscillations in the XTDAS can also result from changes in a bond's Debye-Waller factor and constructive canceling of EXAFS of different intensities. From the XTDAS plot (Figure 24) we can discern that the fine structure observed between 37 K is due to changes in the Debye-Waller factor s and a shift in the atomic distance around Se. Changes in the XTDAS slope reflect the increase in lifetime of the excited state, and band structure as the temperature decreases.

The Debye-Waller factor (s) for the Ni-Se and Se-S bond (Figure 25, 26) mirror the changes in the distance versus temperature plots and in the XTDAS. As the temperature is lowered, s for the Ni-Se bond remains constant until approximately 40 K. Then it begins to decrease, reaching a minimum at 7 K. This indicates that below 40 K the Ni-Se vibrational modes are different from those above 40 K. The value for s for the S-Se bond remained constant versus temperature, but showed discontinuities near 40 K.

IV. Discussion of Results

The decrease in the Se-Ni bond and increase in the Se-S bond occur near the MIT temperature in Figure 1. The changes in the Debye-Waller factor, the absorbance and the relaxation time begin above the same temperature. The compression of the Ni-Se bond produces more favorable conditions for charge transfer between the two atoms. All these changes appear to be related to the compound MIT. The correlation between changes in inter-atomic and MIT are corroborated by x-ray diffraction data on NiS_1.5Se_0.5, which show compression in the unit cell dimensions at the onset of the MIT¹⁹.

The key to EXAFS is that the electron density around the absorbing atom influences the change in absorption. Changes in the EXAFS are due to changes in the electron density as a result of changes in atomic position and changes in electronic structure. Therefore, the EXAFS act as a probe of the overlap necessary to form metallic band states in a semi-conducting material. Any change in the degree or nature of the scattering in the metallic bands should effect the post edge absorption and EXAFS spectrum. The changing slope of the XTDAS data (Figure 24) is indicative of changes in the band structure. In Figure 15 the dramatic change in the Ni-Ni bond distance was shown to be a statistical anomaly. If this is the case a shift of electron density away from the Ni atoms, which would indicate a change in the band structure, could produce this point. What has been developed is this: at 100K NiS_1.52Se_0.48 is an anti-ferromagnetic insulator; when the temperature is lowered past 40K the Ni-Se and Ni-S bond begins to shrink, and this shrinkage is accompanied by an a change in the bonds vibrational mode; at 24K the Ni-Se and Ni-S bonds reach a minimum distance, and the compound is an anti-ferromagnetic metal. The mechanism developed here for the MIT is that of a Mott-Hubbard transition. In the Mott-Hubbard model a compound starts off as an anti-ferromagnetic insulator. As the atomic distances shrink atomic orbitals overlap to form bands and the compound becomes an anti-ferromagnetic metal. As the distances become smaller the compound loses magnetic ordering but retains metallic. In terms of the band models presented in Figure 4a and Figure 4b the data would support model b or d. Since the changes in the absorption at the Se K edge are influenced by changes in the band structure in accordance with the Mott-Hubbard model the change sensed could be the closing of the band gap between the e_g and e_g* bands.

In the Mott-Hubbard model the magnetic ordering is lost as the inter-atomic distances decrease in the metallic region. This event can coincide with the distance need for metallic behavior, but it does not have to. The Se K edge data shows an increase in the lifetime of the states of high kinetic energy as the sample passes through the MIT. This suggests that the anti-ferromagnetic ordering in the insulator phase is not as stable as that of the metallic phase. This data and the emergence of ferromagnetic ordering at 7K present somewhat of a problem for characterizing MIT in these compounds transition as purely a Mott-Hubbard transition. In the Mott-Hubbard model one would expect the magnetic ordering to weaken at the onset of metallic behavior. The increased stability of the anti-ferromagnetic ordering could be related to the expansion of the Ni-Se and Ni-S distances below 24K. Further measurements are needed to satisfactorily answer this question.

The EXAFS versus temperature for NiS_1.52Se_0.48 indicate that the anti-ferromagnetic non-metal to anti-ferromagnetic metal transition follows the Mott-Hubbard model for MIT. The increase in the life time of the high kinetic energy excited state, at the Se K edge, as the temperature is decreased (near the transition temperatures T_Néel and T_Curie) indicates that as the sample goes from an anti-ferromagnetic insulator to an anti-ferromagnetic metal a more ordered state is obtained. At the metal to insulator transition for NiS_1.52Se_0.48 the anti-ferromagnetic domain order is not spoiled because another anti-ferromagnetic state is formed. The increased lifetime of the excited state indicates that the potential senses an increase in the anti-ferromagnetic domain order. The structural and absorption changes seen at the Ni and Se K edge support the Mott-Hubbard model for the MIT in these compounds.

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Figure 1: Phase diagram for the NiS_2-xSe_x system from reference 2. PI= Paramagnetic Insulator, PM= Paramagnetic Metal. WFI= Weak Ferromagnetic Insulator, AFI= Anti-ferromagnetic Insulator, AFM= Anti-ferromagnetic Metal.

Figure 2: The unit cell of a crystal in the Pa3 space group. The blue octahedrons represent the cations, and the yellow tetrahedrons represent the anions. Being cubic, the unit cell dimensions are equal (a=b=c; a=b=g).

Figure 3: The molecular orbital diagram of an octahedral transition metal complex. The crystal field created by the cations mandates the splitting of the d orbitals into e_g and t_2g sets. In terms of bonding orbitals of the same symmetry combine to form molecular orbitals. The metal s and p orbitals have a_1g and t_1u symmetry respectively, while the ligand orbitals have a_1g, e_g, and t_1u symmetry. Thus the metal s, p and e_g orbitals from bonding and anti-bonding orbitals with the ligand orbitals (anti-bonding orbitals are designated with a *), while the t_2g orbitals are not involved in bonding.

Figure 4: Qualitative band models for NiS_2-xSe_x. The yellow bands are an amalgamation of the bonding orbital states, while the red bands are the anti-bonding bands. The blue band evolves from the metal non-bonding t_2g orbitals, and the white bands are empty.

Figure 5: The XAS taken at the Ni-K edge at 9.5 K. The nickel K edge occurs at 8325 eV. The fluctuations in absorbance after the edge are referred to as the XANES and EXAFS.

Figure 7: X-Ray Diffraction Pattern for NiS_2-xSe_x.

Figure 8: Vegard's Law plot for NiS_2-xSe_x. The equation for the line is y=mx+b where m=0.1359, b=5.6890 and R²=0.999534. In the early 20th century Vegard demonstrated that for many compound unit cell dimensions are linearly dependent on composition.

Figure 9: The above is the same spectrum in Figure 5 except the pre-edge region has been fitted to a polynomial function and subtracted. This step is generally not necessary, but it does make the spectrum more aesthetically appealing, and assists in further analysis.