# Investigation of Cu-site disorder in undoped and doped BiCuSeO

Sheetal Jain  Christopher J. S. Heath  Dixshant Shree Shreemal  Blair W.
Lebert  University of Toronto, Toronto, Ontario, M5S 1A7, Canada  Ning Chen
Weifeng Chen  Canadian Light Source (CLS), Saskatoon, Saskatchewan, S7N 2V3,
Canada  Young-June Kim<EMAIL_ADDRESS>University of Toronto,
Toronto, Ontario, M5S 1A7, Canada

###### Abstract

We carried out X-ray diffraction and Extended X-ray Absorption Fine Structure
(EXAFS) studies to investigate the origin of the low lattice thermal
conductivity in BiCuSeO, and the role of silver (Ag) doping in doped samples.
BiCuSeO is a promising thermoelectric material with high thermoelectric
efficiency, which is significantly enhanced by doping either single Ag dopants
or dual dopants (Pb and Ag). We verified that the thermal displacement
parameters associated with copper (Cu) are significantly large in undoped
BiCuSeO. Ag dopant, which replaces Cu, was also found to have similarly large
thermal displacement parameters, that remain large down to low temperatures.
Our results point towards significant disorder on Cu-site in both undoped and
doped BiCuSeO, which is retained by Ag dopant on replacing Cu. The disorder is
observed to be localized on the Cu-site and seems to be independent of other
atoms in the crystal structure. Our observation of the disorder, which could
be either static or quasi-static, is consistent with a “rattling” mode
scenario.

##  I Introduction

Thermoelectric effects directly convert thermal energy into electrical energy,
and vice-versa. They have potential applications in energy generation from
waste heat, and refrigeration, and may play a major role in alleviating the
looming energy crisis and global climate change issues [1, 2, 3, 4]. Highly
effective thermoelectric materials must have high electrical conductivity and
low thermal conductivity. Since electronic thermal conductivity is linked to
electrical conductivity by the Wiedemann–Franz relation, the main goal of
designing thermoelectric materials is to optimize electrical transport while
limiting lattice thermal conductivity.

One of the promising examples of thermoelectric materials is BiCuSeO. Its
crystal structure consists of alternating layers of fluorite-like [Bi2O2]2+
oxide layer and anti-fluorite-like [Cu2Se2]2- chalcogenide layer, stacked
along the c-axis [5], as shown in Figure 1. This layered structure is similar
to the famous iron-based superconductor, LaFeAsO [6, 7]. The oxide layer is
insulating, with ionic bonds, while the conducting chalcogenide layer consists
of covalent bonds [8]. BiCuSeO has a moderate Seebeck coefficient and an
unusually low lattice thermal conductivity [8]. While undoped BiCuSeO exhibits
low electrical conductivity, it can be significantly enhanced by doping.
BiCuSeO has been doped with various alkaline earth metals [8, 9, 10, 11, 12,
13, 14], alkali metals [15, 16, 17], transition metals [18, 19, 20, 21], rare
earth metals [22, 23, 24, 25, 26] and other post-transition metals [27, 28,
29, 30, 31, 32, 33, 34, 35], with dopants replacing bismuth (Bi) to generate
holes in p-type BiCuSeO. In particular, lead (Pb) was observed to lead to the
highest enhancement of electrical conductivity [30], while calcium (Ca) doping
led to the lowest thermal conductivity [11]. The enhanced electrical
conductivity, coupled with the moderate Seebeck coefficient and intrinsic low
lattice thermal conductivity of BiCuSeO leads to high thermoelectric
efficiency in hole-doped BiCuSeO [36]. To further optimize thermoelectric
performance, substitution of copper (Cu) with transition metals was employed,
which was observed to reduce lattice thermal conductivity even further [37,
38, 39, 40, 41]. Furthermore, electrical and thermal transport can be
optimized synergistically by simultaneous introduction of two equimolar
dopants [42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55].

Despite promising thermoelectric properties of BiCuSeO, a fundamental
understanding of its physical properties is still lacking. In particular, the
unusually low lattice thermal conductivity has many proposed theoretical
explanations [56, 57, 58, 59, 60, 61, 62, 63, 64, 65], but direct experimental
test of these theories has been sparse and inconclusive [66, 67, 68].
Moreover, while the role of Ca doping in reducing the lattice thermal
conductivity in BiCuSeO is well understood, with Ca forming calcium oxide
nanoparticles that enhance phonon scattering [11, 53], the underlying
mechanism of silver (Ag) doping in reducing lattice thermal conductivity in
single and dual-doped BiCuSeO is unclear [37, 21, 39, 51].

This work presents a systematic study of undoped, Ag doped and (Pb,Ag)$-$dual
doped BiCuSeO, with the aim to examine the origin of the intrinsic low lattice
thermal conductivity, and mechanism of further reduction by introduction of
Ag. For this, the experimental techniques of powder X-ray diffraction (XRD)
and Extended X-ray Absorption Fine Structure (EXAFS) are employed to
understand the global and local crystal structure. Temperature dependent XRD
and EXAFS measurements are carried out, to investigate the dynamics through
thermal displacement parameters, in order to directly study the origins of the
low lattice thermal conductivity in BiCuSeO.

While XRD provides accurate results about the overall ‘global’ crystal
structure, it is not sensitive to dopants with small concentrations. In order
to study structure and dynamics of these dopants, EXAFS measurements are
conducted. EXAFS is an element specific spectroscopic technique, that can be
used to gain information about the doping site of trace elements in a crystal
structure, their local environment and motion along different atom-pair
directions. Thus, the combination of XRD and EXAFS provides a complete picture
of the global crystal structure and local sites of dopants in this structure,
along with information about thermal dynamics of the atoms.

Figure 1: Schematic representation of the layered crystal structure of
Bi1-xPbxCu1-xAgxSeO, with Pb dopants going to Bi-sites and Ag dopants going to
Cu-sites. The tetragonal unit cell, with the space group P4/nmm (No. 129), is
shown with the solid black line. Crystal structure visualization was done
using the VESTA software [69].

##  II Experimental methods

###  II.1 Sample Preparation

Polycrystalline (powder) samples of undoped BiCuSeO, Ag doped BiCuSeO
(BiCu1-xAgxSeO) and (Pb,Ag)$-$dual doped BiCuSeO (Bi1-xPbxCu1-xAgxSeO) for
doping levels of $x=\\{0.02,0.04,0.06,0.08\\}$ were synthesized via a solid
state reaction process. The raw materials used were Bi (Alfa Aesar, 99.5%),
Bi2O3 (Alfa Aesar, 99.9%), Cu (Alfa Aesar, 99.9%), PbO (Sigma Aldrich, 99.9%),
Ag (Sigma Aldrich, 99.9%) and Se (Alfa Aesar, 99.5%) powders. Stoichiometric
mixtures of these raw materials were mixed and ground in an agate mortar and
pestle. The mixtures were pressed into cylindrical pellets of 1.6 cm diameter
under 50 MPa. The pellets were sealed in quartz tubes under vacuum ($<$10-2
mbar). The sealed tubes were put in a box furnace, heated to 623 K and held
there for 12 h. This intermediate step ensures complete reaction of Se with
the elemental metals [37], and reduction in impurity phases, like Bi metal, in
the final synthesized sample [70]. Then, the temperature of the furnace was
raised to 973 K and held for 48 h. The tubes were then furnace-cooled to room
temperature. The obtained samples were ground, re-pelletized and reheated in
sealed quartz tubes at 973 K for another 48 h to improve purity and
crystallinity. All heating sequences were carried out at a ramp rate of 100
K/h. Finally, dark gray powder samples were obtained, which were found to have
greater than 99.5% phase purity, through XRD analysis. The primary impurity
phase was unreacted Bi2O3, as seen in Figure 2(b), which is a known limitation
of the solid state reaction process [71]. Details of phase purity analysis are
included in the Supplemental Material [[SeeSupplementalMaterialat]supp].

###  II.2 X-ray Diffraction

All XRD measurements were done using a Rigaku Smartlab diffractometer. The
powder samples were thoroughly ground before the measurement, and mounted in
Bragg-Brentano reflection geometry. Cu K$\alpha$ radiation was used, with Cu
K$\beta$ blocked using a nickel filter. Heating and cooling from room
temperature was done using Anton Paar DHS and DCS sample stages respectively,
under vacuum ($<$10-1 mbar). The DHS stage was used in the temperature range
of 298$-$1023 K, while the DCS stage was used in the 93$-$773 K range. In the
overlap temperature range of 303$-$773 K, all the XRD data and fitted
parameters were found to agree, within measurement uncertainty. Hence, for
visual clarity, data from the DCS stage is only shown for 93$-$298 K range,
with the DHS stage data displayed for the range of 298$-$1023 K. Note that the
measurements were only done up to 1023 K since BiCuSeO decomposes at higher
temperatures [73, 74, 75], which is intuitively expected given the synthesis
temperature of BiCuSeO.

Rietveld refinement of XRD data of all samples was done on the GSAS-II
software [76]. Reitveld refinement produced fits for the data and background,
and yielded the lattice constants (a, c) of the tetragonal unit cell and
isotropic mean-square displacements ($U_{iso}$) for each atom. Note that
$U_{iso}$ was computed to obtain values with reasonable error bars, since the
refinement of anisotropic displacement parameters with X-ray diffraction
increases uncertainty significantly. To overcome this difficulty, we used
EXAFS to estimate atom-pair mean-square displacement, as described in the next
section.

###  II.3 EXAFS

All EXAFS measurements were carried out at the Canadian Light Source (CLS) on
the Hard X-ray Micro-Analysis (HXMA) beamline 06ID-1. At room temperature (298
K), Ag K-edge (25514 eV) data was collected for all doped samples, while Cu
K-edge (8979 eV) data was collected for all doped and undoped samples. For
temperature dependence, the highest doping concentration ($x=0.08$) of single
doped and dual-doped BiCuSeO samples were chosen, along with the undoped
BiCuSeO sample. The energy of the incident X-ray beam from the wiggler was
selected using a double-crystal Si(111) monochromator, with higher harmonic
contributions suppressed by a combination of Rh-coated mirrors and a 50%
detuning of the wiggler. Relatively low concentration of Ag made transmission
geometry unsuitable. Ag K-edge EXAFS measurements were performed in a 90
degree, fluorescence geometry using a 13-element Ge solid-state detector, with
the sample 45 degree to the incident beam. Cu K-edge EXAFS data, on the other
hand, was collected in both transmission and fluorescence geometry
simultaneously, with the sample again 45 deg to the incident beam. Standard Ag
and Cu foils were utilized for energy calibration, respectively. As reference,
EXAFS spectra in transmission geometry were also collected on these standard
foils, along with each EXAFS measurement on the actual samples. Each sample
was diluted with pure boron nitride powder, finely ground, pressed into a
uniform pellet, and then affixed to polyimide tape. These procedures were done
in order to achieve suitable element concentration and uniformity. At least
three scans were taken at each edge for each sample in order to guarantee data
reproducibility and good signal-to-noise ratio.

EXAFS data processing and fitting was done on the ATHENA and ARTEMIS softwares
of the Demeter suite respectively [77]. In data processing, EXAFS data
obtained in energy-space was reduced using standard procedures, and the
resulting $k$-space data was Fourier transformed (F.T.) to $r$-space, yielding
peaks that correspond to different shells of neighboring atoms in the EXAFS
equation:

$\chi(k)=\sum\limits_{j}\frac{S_{0}^{2}N_{j}e^{-2k^{2}\sigma^{2}_{j}}e^{-2r_{j}/\lambda(k)}f_{j}(k)}{kr_{j}^{2}}\sin(2kr_{j}+\delta_{j}(k))$
(1)

where $j$ represents different coordination shells made up of $N_{j}$
identical atoms at approximately the same distance from the absorbing atom;
$S_{0}^{2}$ is the amplitude reduction factor which corrects for
multi$-$electron scattering and other experimental effects (generally between
0.7 and 1.2); $\sigma^{2}_{j}$ is the atom-pair mean-square displacement along
the atom-pair distance $r_{j}$; $f_{j}(k)$ and $\delta_{j}(k)$ are the
scattering factor and scattering phase shift, which are provided in the FEFF
program [78]; $\lambda(k)$ is the mean-free-path of the photoelectron;
$k=\sqrt{2m(E-E_{0})/\hbar^{2}}$, where $E_{0}$ is the theoretical Fermi level
position. In data fitting, the $r$-space data is fit to a sum of complex phase
and amplitude functions calculated using the FEFF program, to extract atom-
pair distances ($r_{j}$) and thermal factors ($\sigma^{2}_{j}$) for each atom-
pair. Another two fitting parameters, $S_{0}^{2}$ and $\Delta E_{0}$ (the
difference in edge energy between the value defined for the data and the
theoretical function), are also determined for each edge. In the preliminary
fits, the values of $S_{0}^{2}$ are found to be reasonable, very close to 1,
for all temperatures. Hence, in the final fits, $S_{0}^{2}$ is fixed to its
room temperature value for all edges. In this analysis, we fit both the real
and imaginary parts of the $r$-space data.

Table 1: Lattice constants a and c of the tetragonal unit cell, with the space group P4/nmm (No. 129), for BiCu1-xAgxSeO and Bi1-xPbxCu1-xAgxSeO samples, obtained from Rietveld refinement of XRD data taken at room temperature (298 K). Sample | BiCu1-xAgxSeO | Bi1-xPbxCu1-xAgxSeO
---|---|---
x |  $a$ [Å] |  $c$ [Å] |  $a$ [Å] |  $c$ [Å]
0.00 (undoped) | 3.928(4) | 8.931(9) | 3.928(4) | 8.931(9)
0.02 | 3.929(4) | 8.932(9) | 3.930(4) | 8.947(9)
0.04 | 3.929(4) | 8.932(9) | 3.932(4) | 8.956(9)
0.06 | 3.930(4) | 8.933(9) | 3.935(4) | 8.976(9)
0.08 | 3.930(4) | 8.934(9) | 3.935(4) | 8.984(9)

##  III Experimental results

###  III.1 X-ray Diffraction

####  III.1.1 Room Temperature

The XRD data obtained at room temperature for all samples was fit using
Rietveld refinement to extract the lattice constants of the tetragonal unit
cell. The lattice constants obtained from this fitting are tabulated in Table
1. It is worth noting that, in the case of Ag doping, the lattice constants
show almost no change as doping increases, within error bar. In contrast, in
the case of (Pb,Ag)$-$dual doping, the lattice constants show a more
pronounced increase with doping, in particular the c-lattice constant.

Isotropic mean-square displacements ($U_{iso}$) were also obtained for each
atom from Rietveld refinement of XRD data of undoped BiCuSeO, which will be
discussed in the next section. Due to the relatively low concentration of
dopants, $U_{iso}$ values for dopant atoms could not be estimated by Rietveld
refinement.

(a)

(b)

Figure 2: 2(a) XRD raw data for undoped BiCuSeO, as a function of temperature,
zoomed in near the most prominent XRD peak at 30 deg. Thermal expansion is
clearly observed, depicted by the decrease in 2$\mathrm{\theta}$ position of
peaks with increase in temperature. 2(b) Rietveld refinement profile of the
XRD data for undoped BiCuSeO at 573 K. The calculated profile, shown in black,
is seen to fit the observed data well, shown in red. The difference profile
and expected XRD peak positions are also shown in blue and green respectively.
The primary impurity XRD peak of unreacted Bi2O3 (less than 0.5%) is seen near
27 deg.

####  III.1.2 Temperature Dependence

XRD data obtained for undoped BiCuSeO as a function of temperature is shown in
Figure 2(a). The color-map shows intensity of XRD data as a function of
temperature. Only the limited 2$\theta$ range of 27.5$-$35 deg is shown to
illustrate the temperature dependence clearly. The colored lines of higher
intensity represent the XRD peak positions from the sample. Note that as
temperature increases, the peak position shifts to lower angles, resulting
from an increase in lattice constants due to thermal expansion. No
discontinuity is observed in the data over the entire temperature range,
signifying that no phase change occurs in this range.

The Rietveld refinement profile for XRD data of undoped BiCuSeO at 573 K is
shown in Figure 2(b), for the 2$\theta$ range of 20$-$60 deg. The observed
data is shown in red, while the calculated fit is the solid black line. The
fit is seen to match the data well, yielding R${}_{\mathrm{wp}}=5.33\%$ and
$\chi^{2}_{r}=2.19$. These goodness of fit parameters are found to be similar
in value for fits at all temperatures. The difference profile is shown in
blue, and is found to be within the measurement uncertainty throughout the
entire range. The expected XRD peak positions for BiCuSeO, based on the
crystal structure in Figure 1, are shown via the green vertical lines.

(a)

(b)

Figure 3: 3(a) Temperature dependence of lattice constant a and lattice
constant c of the tetragonal unit cell for undoped BiCuSeO. The dashed lines
represent quasi-harmonic fits up to 800 K, described by Eq. (2), based on the
Einstein model. 3(b) Isotropic mean-square displacements ($U_{iso}$) versus
temperature for Bi, Cu, O and Se in undoped BiCuSeO. The dashed lines
represent quasi-harmonic fits for the mean-square displacements up to 700 K.
Both lattice constants and isotropic mean-square displacements were obtained
from Rietveld refinement of XRD data of undoped BiCuSeO.

Rietveld refinement of the temperature dependent XRD data yields the
temperature dependence of the lattice constants of the tetragonal unit cell,
and isotropic mean-square displacements of each atom, for undoped BiCuSeO,
which is presented in Figure 3. The temperature dependence of the lattice
constant a is shown in the lower half of Figure 3(a). Uniform thermal
expansion is observed from base temperature (93 K) to room temperature (298
K). Above room temperature, there is a subtle increase in the rate of
expansion up to 700 K, following which the rate decreases up to 1023 K. The
relative thermal expansion behavior is almost identical for lattice constant
c, shown in the upper half of Figure 3(a). The error bars for both lattice
constants are smaller than the symbol sizes shown in the plot.

The Einstein model is often used as a quasi-harmonic fit for thermal expansion
of lattice constants [79, 80]:

$\ln\left(\frac{a}{a_{0}}\right)=\frac{A\;\Theta_{E}}{\exp(\Theta_{E}/T)-1}$
(2)

where $a$ is the length of the lattice constant at temperature $T$, $a_{0}$ is
the lattice constant at $T$ = 0 K, $A$ is a scaling constant and $\Theta_{E}$
is the Einstein temperature. A similar relation is used to fit lattice
constant $c$. The quasi-harmonic fits, based on Eq. (2), are shown via dashed
lines in Figure 3(a). These quasi-harmonic fits match the data for the
temperature range of 93$-$800 K, after which the data for both lattice
constants diverges subtly from the fits. The relative behaviour of lattice
constants is found to be similar to doped samples. The value of $\Theta_{E}$
is found to be 95 ($\pm$1) K, which matches the value quoted in previous
literature [66], within experimental uncertainty. $\Theta_{E}$ is also
consistent with the value obtained through harmonic fits of $U_{iso}$, as
discussed below.

Isotropic mean-square displacements ($U_{iso}$) of Bi, Cu, O and Se are shown
in Figure 3(b). Note that $U_{iso}$ for oxygen shows large uncertainty, due to
the small atomic form factor of oxygen. The dashed lines correspond to the
quasi-harmonic fits for the temperature dependence of $U_{iso}$ for each
element. The quasi-harmonic fits only work for the temperature range of
93$-$700 K, after which the data for all three elements is clearly seen to
diverge significantly from the harmonic fits. Both Einstein model and Debye
model were used as quasi-harmonic fits for the data [81, 79]:

$\displaystyle\mathrm{Einstein}:U_{iso}=\frac{\hbar^{2}}{2mk_{B}\Theta_{E}}\coth\left(\frac{\Theta_{E}}{2T}\right)+d_{E}^{2}$
(3)
$\displaystyle\mathrm{Debye}:U_{iso}=\frac{3\hbar^{2}T}{mk_{B}\Theta_{D}^{2}}\left[\frac{T}{\Theta_{D}}\int\limits_{0}^{\Theta_{D}/T}\frac{x}{e^{x}-1}dx+\frac{\Theta_{D}}{4T}\right]+d_{D}^{2}$
(4)

where $\hbar$ is the reduced Planck constant, $m$ is the mass of the atom,
$k_{B}$ is the Boltzmann constant, $\Theta_{E}$ is the Einstein temperature,
$\Theta_{D}$ is the Debye temperature, $T$ is the sample temperature and
$d_{E}$ & $d_{D}$ are empirical terms to describe temperature-independent
disorder. The two fits were found to overlap almost completely, with almost
same $d_{E}$ and $d_{D}$. Hence, only the Einstein fits are shown for visual
clarity.

The value of $\Theta_{E}$ and $\Theta_{D}$ from the fits are 95 ($\pm$2) K and
241 ($\pm$3) K respectively, which matches values reported previously by other
studies [66, 6]. It is interesting to note that these values are also very
similar to the values obtained for other thermoelectric materials like
clathrates [81] and skutterudites [79].

Note that the values of $U_{iso}$ are significantly higher for Cu, as compared
to Bi, O and Se, suggesting that the displacement amplitude for Cu is higher
than the other elements. In general, tightly bound atoms in a solid crystal
structure are expected to have $U_{iso}$ vales close to 0.007 Å2 at room
temperature [76, 82]. At room temperature, while Bi, Se and O values match
this expected value within error bar, the $U_{iso}$ value for Cu is 0.016 Å2.
Also note that $d_{E}^{2}$ for Cu is much higher than the other elements. This
indicates that disorder on Cu is significantly larger than that for other
elements in BiCuSeO. Our results is consistent with a previous neutron
diffraction study [66], confirming the large thermal parameters of Cu over a
wide temperature range, from low temperatures up to the decomposition
temperature of undoped BiCuSeO.

(a)

(b)

Figure 4: 4(a) Ag K-edge EXAFS data in $r$-space for BiCu0.92Ag0.08SeO as solid lines, and Bi0.92Pb0.08Cu0.92Ag0.08SeO as dashed lines, at different temperatures. The vertical brown dotted lines represent the first (Ag-Se), second (Ag-(Cu/Ag)) and third (Ag-(Bi/Pb)) shell locations in $r$-space, corresponding to Ag replacing Cu in the BiCuSeO crystal structure framework. The Fourier transform (F.T.) ranges are 3$-$11 Å-1. 4(b) The fit results for BiCu0.92Ag0.08SeO and Bi0.92Pb0.08Cu0.92Ag0.08SeO at 150 K. The fit ranges are shown via the window function in solid green lines. The oscillatory curve below the envelope is the real part of the Fourier transform ($FT_{R}$). The envelope is the magnitude of the Fourier transform, defined as $\sqrt{FT_{R}^{2}+FT_{I}^{2}}$, where $FT_{I}$ (not shown) is the imaginary part of the Fourier transform. Table 2: Mean-square displacement ($\sigma^{2}_{j}$) along atom-pairs of Ag-Se ($\sigma^{2}_{1}$), Ag-(Cu/Ag) ($\sigma^{2}_{2}$) and Ag-(Bi/Pb) ($\sigma^{2}_{3}$) for Ag doped BiCuSeO and (Pb,Ag)$-$dual doped BiCuSeO samples at different temperatures, obtained from fitting of Ag K-edge EXAFS data. Sample | BiCu0.92Ag0.08SeO | Bi0.92Pb0.08Cu0.92Ag0.08SeO
---|---|---
Temperature [K] |  $\sigma^{2}_{1}$ [Å2] |  $\sigma^{2}_{2}$ [Å2] |  $\sigma^{2}_{3}$ [Å2] |  $\sigma^{2}_{1}$ [Å2] |  $\sigma^{2}_{2}$ [Å2] |  $\sigma^{2}_{3}$ [Å2]
80 | 0.0005(1) | 0.0107(5) | 0.0003(1) | 0.0008(6) | 0.0107(8) | 0.0007(4)
150 | 0.0006(2) | 0.0103(7) | 0.0007(3) | 0.0006(5) | 0.0107(4) | 0.0004(1)
225 | 0.0007(6) | 0.0107(7) | 0.0007(6) | 0.0009(5) | 0.0106(6) | 0.0004(2)
298 | 0.0008(2) | 0.0106(5) | 0.0002(1) | 0.0006(2) | 0.0108(3) | 0.0005(1)

###  III.2 EXAFS

####  III.2.1 Ag K-edge

Having characterized the temperature dependence for undoped BiCuSeO, the doped
samples are now characterized using temperature dependent Ag K-edge EXAFS
measurements. The samples showed no significant doping dependence for room
temperature EXAFS measurements, which means that the local environment of the
dopant site remains the same as doping fraction increases. Further information
about doping dependence can be found in the Supplemental Material
[[SeeSupplementalMaterialat]supp]. Hence, the highest doping percentage sample
($x=0.08$) was chosen for both single and dual-doped BiCuSeO for temperature
dependent EXAFS, since higher concentration leads to better signal-to-noise
ratio.

Figure 4(a) shows the Ag K-edge EXAFS data (magnitude of Fourier transform) in
$r$-space as a function of temperature. The data for the single doped sample
is shown with solid lines in the upper half, while the dashed lines in the
lower half correspond to data for the dual-doped sample. The vertical dotted
lines are shown as visual aids to note the positions of the first three shells
expected in the EXAFS data for Ag replacing Cu in BiCuSeO, based on the
structure shown in Figure 1. These shells correspond to the Ag-Se ($r$ = 2.2
Å), Ag-(Cu/Ag) (2.7 Å) and Ag-(Bi/Pb) (3.5 Å) atom-pairs respectively. Note
that the $r$-space peak positions of shells are different from the exact atom-
pair distances, which are tabulated in the Supplemental Material
[[SeeSupplementalMaterialat]supp].

No significant temperature dependence is observed in this data across both
samples. It is generally expected that low temperature ($T$$\ll$$\Theta_{D}$)
EXAFS data has better signal-to-noise ratio, due to smaller Debye-Waller
factor. However, this temperature effect of the Debye-Waller factor is due to
a reduction in the mean-square displacements upon cooling. Thus, the lack of
temperature dependence indicates that the mean-square displacement of Ag
dopant does not change significantly in the temperature range studied, from
room temperature (298 K) down to base temperature (78 K). This mean-square
displacement behavior of Ag for this temperature range is very similar to that
of Cu, based on $U_{iso}$ from the XRD data shown in Figure 3(b).

To obtain quantitative information, the EXAFS data is fit to the first three
shells expected for Ag replacing Cu in BiCuSeO. This three-shell fit for the
data at 150 K is shown in Figure 4(b). The upper panel corresponds to the
single doped sample while the lower panel displays the fit for the dual-doped
sample. The fitting process involves fitting both the magnitude
($\sqrt{FT_{R}^{2}+FT_{I}^{2}}$) and real part ($FT_{R}$) of the Fourier
transform. In Figure 4(b), the oscillatory curve is the real part, while the
envelope is the magnitude of the Fourier transform. The fits match the data
well in the fitting window, yielding a goodness of fit parameter,
reduced-$\chi^{2}$, close to 1 for all temperatures.

The fits yield the atom-pair mean-square displacements ($\sigma^{2}_{j}$) for
the three fitted shells, tabulated in Table 2. The values of $\sigma^{2}_{j}$
for every shell are found to be almost temperature independent for both
samples. This was already expected, based on the lack of temperature
dependence seen in the raw data in Figure 4(a). The values of
$\sigma^{2}_{2}$, corresponding to Ag-(Cu/Ag) atom-pair, are an order of
magnitude higher than the values of $\sigma^{2}_{j}$ for the other atom-pairs.
These values are slightly smaller than the $U_{iso}$ of Cu at 150 K obtained
by XRD, and significantly larger than the $U_{iso}$ values of Bi and Se at the
same temperature, based on the $U_{iso}$ plot shown in Figure 3(b). This
suggests that the significant disorder on Cu in the BiCuSeO crystal structure
framework is retained by the Ag dopant, on replacing Cu at that site, for all
temperatures studied. This means that the disorder observed is actually linked
to the Cu-site, as opposed to just the Cu atom, since Ag dopant retains the
observed disorder.

Note that the comparison of $U_{iso}$ and $\sigma^{2}_{j}$ is possible since
both represent different measures of the same physical quantity, mean-square
displacement. Since both account for thermal vibrations of atoms about their
equilibrium positions, it is expected that all contributions from different
atom-pairs in the various $\sigma^{2}_{j}$ add up to give $U_{iso}$ [83].
Thus, the largest $\sigma^{2}_{j}$ is expected to influence $U_{iso}$ the
most. In the case of Ag K-edge EXAFS, it is found to be the second shell,
Ag-(Cu/Ag).

Table 3: Mean-square displacement ($\sigma^{2}_{j}$) along atom-pairs of Cu-Se ($\sigma^{2}_{1}$) and Cu-(Cu/Ag) ($\sigma^{2}_{2}$) for undoped BiCuSeO, Ag doped BiCuSeO (BiCu0.92Ag0.08SeO) and (Pb,Ag)$-$dual doped BiCuSeO (Bi0.92Pb0.08Cu0.92Ag0.08SeO) samples at different temperatures, obtained from fitting of Cu K-edge EXAFS data. Sample | BiCuSeO | BiCu0.92Ag0.08SeO | Bi0.92Pb0.08Cu0.92Ag0.08SeO
---|---|---|---
Temperature [K] |  $\sigma^{2}_{1}$ [Å2] |  $\sigma^{2}_{2}$ [Å2] |  $\sigma^{2}_{1}$ [Å2] |  $\sigma^{2}_{2}$ [Å2] |  $\sigma^{2}_{1}$ [Å2] |  $\sigma^{2}_{2}$ [Å2]
150 | 0.0012(1) | 0.013(2) | 0.0013(2) | 0.010(1) | 0.0013(3) | 0.013(2)
225 | 0.0011(3) | 0.011(5) | 0.0010(4) | 0.012(4) | 0.0012(2) | 0.012(3)
298 | 0.0010(5) | 0.013(3) | 0.0012(2) | 0.011(3) | 0.0013(1) | 0.011(2)

####  III.2.2 Cu K-edge

Since Ag is expected to replace Cu in BiCuSeO, Cu K-edge EXAFS measurements
were also performed in order to provide a reference point of the same site.
Also, having drawn the comparison of similarities in mean-square displacement
values of Ag (from EXAFS) and Cu (from XRD), these measurements provide a
direct comparison between the two elements based on the same technique of
temperature dependent EXAFS. These measurements were conducted on the undoped,
8% Ag doped and 8% (Pb,Ag)$-$dual doped samples.

The fit results for Cu K-edge EXAFS measurements are summarized in Table 3.
The atom-pair mean-square displacement values for the first two shells,
$\sigma^{2}_{1}$ (Cu-Se) and $\sigma^{2}_{2}$ (Cu-(Cu/Ag)) are shown as a
function of temperature. Note that higher noise levels of lower energy Cu
K-edge, coupled with significant disorder, limited the fitting range for the
EXAFS data, hence only two shell fits could be estimated with reasonable error
bars. No significant temperature dependence is observed, as expected for this
temperature range, based on the $U_{iso}$ plot for Cu, shown in Figure 3(b).

The values of $\sigma^{2}_{2}$, corresponding to Cu-(Cu/Ag) atom-pair, are an
order of magnitude higher than the values of $\sigma^{2}_{j}$ for the other
atom-pair. These values agree well with the $U_{iso}$ of Cu at 150 K obtained
by XRD, shown in Figure 3(b). Our combined XRD and EXAFS results indicate that
the displacement parameters for Cu do not change significantly below room
temperature. Note that this observation disagrees with an earlier neutron
diffraction study on undoped BiCuSeO, in which a fairly strong temperature
dependence of anisotropic atomic displacement parameters (ADPs) were reported
[67]. We also note that we decided not to estimate anisotropic parameters in
our studies due to the large uncertainty associated with estimating these
parameters from diffraction data.

The Cu K-edge EXAFS results confirm the observation of significant disorder at
the Cu-site in both undoped and doped BiCuSeO, which arises mainly along the
Cu-(Cu/Ag) atom-pair. Note that the introduction of Ag at Cu-sites in the
doped samples does not affect the disorder of Cu atoms, since the values of
$\sigma^{2}_{2}$ are almost same for all dopings, at all temperatures. The
values of overall mean-square displacement for Cu from EXAFS (sum of
$\sigma^{2}_{j}$) and XRD ($U_{iso}$) are found to match within error bar,
suggesting that contributions from further shells of neighbouring atoms are
not significant, compared to the second shell, Cu-(Cu/Ag), in particular.

##  IV Discussion

The main findings from our X-ray diffraction and EXAFS studies are our
observation of the unusually large $U_{iso}$ parameters for Cu in undoped
BiCuSeO and large $\sigma^{2}$ parameters for both Ag and Cu for doped BiCuSeO
samples. We also observe that these “thermal” displacement parameters are
independent of doping concentrations as well as temperature (at low
temperatures). We will first discuss the possible origin of the observed large
“thermal” parameters and the implication of our observations for
thermoelectric properties of doped and undoped BiCuSeO.

First, we note that $U_{iso}$ parameters as well as $\sigma_{j}^{2}$
parameters are usually called thermal parameters because the main source of
this displacement parameter is due to lattice dynamics (i.e. Debye-Waller
factor), but static disorder can also contribute to these parameters. Often
dynamic and static origins of the disorder can be distinguished from the
temperature dependence of thermal parameters, since the dynamic contribution
shows characteristic temperature dependence. However, temperature-independent
thermal parameter does not automatically indicate static disorder, since
quasi-static disorder, such as a low frequency fluctuation, will be
temperature independent for most of the temperature range probed.

While our observed large thermal parameters for Cu and Ag in Cu-site (this is
further discussed below) could be either due to static or quasi-static
disorder, we argue that static disorder is less likely to be the main source
of the disorder. What is peculiar about the $U_{iso}$ data shown in Figure
3(b) is that the lowest temperature $U_{iso}$ value of Cu is at least 50%
larger than other elements in BiCuSeO. This is surprising because we expect
similar level of static disorder for Se atoms, which is the other member of
the Cu-Se plane. In addition, we observed similarly large (albeit slightly
smaller) $\sigma^{2}$ for Ag regardless of doping concentration. Since it is
generally expected that doping could cause structural disorder, these
observations support the idea that the disorder is not due to static (i.e.
structural) disorder.

There exists a mechanism that could explain the quasi-static origin of the
observed thermal disorder of Cu/Ag. To explain the intrinsic low lattice
thermal conductivity of BiCuSeO, Vaqueiro et al. proposed a “rattling” mode
mechanism [66]. That is, a localized, independent vibrational “rattling” mode
at Cu-site could scatter phonons, lowering thermal conductivity. The
implication is that Cu/Ag is subject to a “flat” potential, making it easy for
it to be displaced from the equilibrium position. We could imagine the
displacement is incoherent and slow, showing up as quasi-static disorder in
our thermal parameter measurements. Of course, as pointed out by previous
studies for clathrates [84], careful examination of phonon dispersion is
required to confirm presence of a “rattling” mode; however, our result
indicates such a localized mode (or local disorder) as the origin of low
thermal conductivity as discussed next.

The origins of intrinsic low lattice thermal conductivity in BiCuSeO have been
debated to date. One of the proposed explanation suggested that the heavy Bi
atom is responsible [56, 62], with the higher atomic mass leading to lower
phonon frequencies and slower sound velocities, reducing heat transfer rate.
Another prediction was that flat phonon branches associated with Cu atoms
effectively block heat transfer [63, 66]. The flat phonon branches have
smaller group velocities and smaller phonon lifetimes, thus restricting
thermal transport [85]. Since the flat phonon branches are limited to Cu, it
suggests that a localized mode on Cu is responsible for these flat phonon
branches. This is the mechanism proposed by Vaqueiro et al., a low energy,
independent vibrational “rattling” mode on Cu atom, and our experimental
results seem to support this scenario.

Another important finding that backs up the above picture is our observation
of the lack of lattice expansion with Ag doping. The introduction of Ag dopant
in Cu-site does not significantly increase the lattice constants, as seen in
the case of BiCu1-xAgxSeO in Table 1. Since Ag1+ ion (129 pm) is much larger
in size than Cu1+ ion (91 pm) [86], this suggests that the void around Cu-site
is large, and the additional empty space accommodates Ag dopant without
requiring the lattice to expand. This is in contrast to Bi-site, where the
replacement of Bi3+ (117 pm) with Pb2+ (133 pm) [86] leads to a significant
increase in lattice constants, as seen for Bi1-xPbxCu1-xAgxSeO samples in
Table 1. Presence of a large void also supports the idea of a “rattling” mode
at Cu-site in BiCuSeO.

The introduction of Ag leads to a further decrease of lattice thermal
conductivity, in both single [37, 21, 39] and dual-doped samples [51]. Before
discussing the possible origin of additional suppression of lattice thermal
conductivity in Ag doped samples, we would like to comment on the dopant site
issue.

The dopant site for Pb is generally agreed upon, it is expected to replace Bi
in the BiCuSeO framework, as shown in Figure 1. This has been confirmed via
XANES measurements in a previous study [42]. The dopant site for Ag, however,
has been a matter of debate. Single doping studies of Ag in BiCuSeO have found
Ag in both Cu and Bi sites [37, 39, 40, 21]. A previous (Pb,Ag)$-$dual doping
study suggested that some of the Ag dopants did not enter the lattice and
produced inter-layer impurities [51]. Our Ag K-edge EXAFS results show no
evidence of any alternate dopant locations, except Cu-site. Hence, within
measurement uncertainty, the results point to Ag replacing Cu in BiCuSeO.

Given that Ag doping does not increase static disorder parameter in our doped
samples, we can conclude that the dopant is more or less uniformly (and
randomly) replacing Cu atoms. The mass defect therefore given by Ag replacing
Cu, coupled with the retained disorder of Ag dopant, could be the main reason
for the reduced thermal conductivity.

It is also worth noting that, while large $\sigma_{j}^{2}$ values are observed
along the Ag-(Cu/Ag) and Cu-(Cu/Ag) atom-pair in EXAFS results, the values of
$\sigma_{j}^{2}$ along Ag-Se and Cu-Se are found to be considerably smaller.
This result is somewhat unexpected. One possibility is that, within the Cu-Se
plane shown in Figure 1, the Se ‘sub-layer’ is relatively static, while the
Cu-site atoms vibrate in-plane with large amplitude. The Se-Cu-Se bond angle
changes in the CuSe4 tetrahedron during these vibrations, allowing for smaller
changes in Cu-Se bond length (smaller $\sigma_{j}^{2}$ along Cu-Se) and
smaller vibrations of Se (smaller $U_{iso}$ value of Se). Nevertheless, the
role of Se in the Cu-Se plane of BiCuSeO is unclear and should be clarified by
further study.

##  V Conclusion

This work presents X-ray diffraction (XRD) and EXAFS measurements on undoped,
Ag doped and (Pb,Ag)$-$dual doped BiCuSeO. The main observation is the
unusually large isotropic mean-square displacement of Cu in undoped BiCuSeO in
XRD results, and the large atom-pair mean-square displacement values along
Cu-(Cu/Ag) and Ag-(Cu/Ag) in all samples in EXAFS. These observations indicate
the presence of significant intrinsic disorder in Cu-site of BiCuSeO, which is
retained by Ag dopant on replacing Cu. No increase of lattice constants is
observed on Ag doping at room temperature, suggesting the presence of a large
void around Cu-site in BiCuSeO. The temperature independence and presence of
large void supports the “rattling” mode scenario, localized on Cu-site. This
may explain the intrinsic low lattice thermal conductivity of BiCuSeO. The
mass defect on introduction of Ag, coupled with the retention of “rattling”
behaviour, can also explain the further lowering of lattice thermal
conductivity of BiCuSeO on Ag doping. The combination of XRD and EXAFS, used
to comprehensively understand the structure and dynamics of undoped and doped
BiCuSeO, is shown to be a powerful technique which can be applied to the study
of other materials with interesting crystal structure and doping properties.

###### Acknowledgements.

The work at the University of Toronto was supported by the Natural Sciences
and Engineering Research Council (NSERC) of Canada [RGPIN-2019-06449 and
RTI-2019-00809], Canada Foundation for Innovation, and the Ontario Research
Fund. Research performed at the Canadian Light Source is supported by the
Canada Foundation for Innovation, NSERC, the University of Saskatchewan, the
Government of Saskatchewan, Western Economic Diversification Canada, the
National Research Council Canada, and the Canadian Institute of Health
Research. S.J. and D.S.S. acknowledges the receipt of support from the CLS
Graduate and Post-Doctoral Student Travel Support Program. B.W.L. acknowledges
support from the University of Toronto Faculty of Arts and Sciences
Postdoctoral Fellowship and NSERC [funding reference number PDF-546035-2020].

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