of 6
Determining conductivity and mobility values of individual components in multiphase
composite Cu1.97Ag0.03Se
Tristan W. Day, Wolfgang G. Zeier, David R. Brown, Brent C. Melot, and G. Jeffrey Snyder
Citation: Applied Physics Letters
105
, 172103 (2014); doi: 10.1063/1.4897435
View online: http://dx.doi.org/10.1063/1.4897435
View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/105/17?ver=pdfcov
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Determining conductivity and mobility values of individual
components in multiphase composite Cu
1.97
Ag
0.03
Se
Tristan W. Day,
1
Wolfgang G. Zeier,
1,2
David R. Brown,
1
Brent C. Melot,
2
and G. Jeffrey Snyder
1,
a)
1
Department of Materials Science, California Institute of Technology, MC 309-81, Pasadena,
California 91106, USA
2
Department of Chemistry, University of Southern California, Seeley G. Mudd Bldg.,
3620 McClintock Ave., Los Angeles, California 90089-1062, USA
(Received 12 August 2014; accepted 22 September 2014; published online 28 October 2014)
The intense interest in phase segregation in thermoelectrics as a means to reduce the lattice thermal
conductivity and to modify the electronic properties from nanoscale size effects has not been met
with a method for separately measuring the properties of each phase assuming a classical mixture.
Here, we apply effective medium theory for measurements of the in-line and Hall resistivity of a
multiphase composite, in this case Cu
1.97
Ag
0.03
Se. The behavior of these properties with magnetic
field as analyzed by effective medium theory allows us to separate the conductivity and charge car-
rier mobility of each phase. This powerful technique can be used to determine the matrix properties
in the presence of an unwanted impurity phase, to control each phase in an engineered composite,
and to determine the maximum carrier concentration change by a given dopant, making it the
first step toward a full optimization of a multiphase thermoelectric material and distinguishing
nanoscale effects from those of a classical mixture.
V
C
2014 AIP Publishing LLC
.
[
http://dx.doi.org/10.1063/1.4897435
]
Great strides have been made in improving thermoelectric
performance by combining solid phases. Microstructuring to
scatter phonons while maintaining high carrier mobilities
1
is a
proven method for reducing the lattice thermal conductivity.
2
The high potential for inclusions to improve the electronic
properties of thermoelectrics by doping,
3
,
4
electron filtering,
5
and composition modulation
6
has lead to values of the ther-
moelectric figure of merit
zT
greater than unity, approximately
the value needed for commercial modules.
The mechanisms described above use a second phase or
microstructure to affect the performance of the matrix phase
due to quantum mechanical or other nanometer size effects
on the transport properties. This is because the
zT
of a com-
posite material cannot be improved by the combination of
two phases in a parallel, series, or arbitrary mixture which
can be described by classical phenomena.
7
Secondary phases
are also of concern because phase-pure synthesis of some
materials is challenging, making measurement of the elec-
tronic properties of the target compound difficult.
8
Interpretation of transport measurements of multiphase
materials is critical for thermoelectric optimization and
application; however, no one has demonstrated a method for
separating the transport properties of each phase in a com-
posite from measurements of the bulk material.
In this work, we seek to provide a method for extracting
the conductivity and mobility of each phase in a composite
from measurements of the in-line and Hall resistivity of the
bulk, using effective medium theory. Effective medium
theory can be used to derive expressions for two-phase trans-
port properties of any kind, including the electrical conduc-
tivity,
7
,
9
the Seebeck coefficient,
7
and the thermal
conductivity,
10
,
11
meaning that effective medium theory can
be united with microstructure engineering to design electron-
ically optimized, low lattice thermal conductivity thermo-
electric materials. We take the first step toward such an
optimization by applying Stroud’s powerful coherent poten-
tial approximation
9
along with magnetic-field dependent re-
sistivity measurements to determine the conductivity and
mobility of each phase in a two-phase composite. Our
approach does not include nanoscale effects, so it can also be
used to distinguish between bulk and nanoscale contribu-
tions, or it can be used in conjunction with models of small-
size properties, as some researchers have already done.
10
This derivation yields quantitative correction terms that
can be applied to analyze transport data. In many cases, this
method would give quantitative justification for use of a sin-
gle phase approximation. In other cases, this method will
give a quantitative correction factor. The method also identi-
fies the volume-fraction weighted Hall conductivity of the
matrix phase and the magnetoresistance contribution of each
phase as the critical scaling factors which ultimately deter-
mine what level of correction or even appropriateness a sin-
gle phase transport model may have.
In general, the contribution of each phase in a composite
to the magnetoresistance and to the Hall effect depends on
its charge carrier mobility multiplied by the magnetic field
strength. Because the dependence is nonlinear, we can use a
magnetic field to distinguish the effect each phase has on
electrical transport in a composite. Cu
1.97
Ag
0.03
Se contains
an impurity phase with a high mobility that exerts a dispro-
portionate influence on the transport properties, providing us
with a model system to show the viability of this approach.
X-ray diffraction data show that this composition com-
prises two phases, the matrix with the crystal structure of
Cu
2
Se and an impurity phase with the crystal structure of
a)
Author to whom correspondence should be addressed. Electronic mail:
jsnyder@caltech.edu.
0003-6951/2014/105(17)/172103/5/$30.00
V
C
2014 AIP Publishing LLC
105
, 172103-1
APPLIED PHYSICS LETTERS
105
, 172103 (2014)
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131.215.70.231 On: Mon, 03 Nov 2014 15:39:38
CuAgSe, which dissolves into the matrix around 390 K.
12
,
13
The number of phases is confirmed by scanning electron mi-
croscopy (SEM) (inset of Figure
1
), which shows the gray
matrix phase and a light-colored impurity phase. The dark
spots are voids; according to geometric density measure-
ments, the sample has only 3% porosity. Cu
1.97
Ag
0.03
Se dis-
plays unusual electrical properties below its superionic phase
transition at 400 K (Figure
1
). It has a Seebeck coefficient
greater than zero indicating holes as the majority carriers,
but a Hall coefficient
R
H
less than zero, indicating electrons
have a strong influence on the electrical conductivity in this
composite. This opposite behavior is rare in materials but
occurs in some elemental metals like Li, Cu, Ag, and Au
14
and in some semiconductors such as AgSbTe
2
and in PbTe-
PbS alloys.
15
In these cases, band structure effects can
explain these properties.
16
However, in the case of
Cu
1.97
Ag
0.03
Se,
R
H
becomes positive as soon as the impurity
phase dissolves, suggesting that only the impurity phase is
n-type with an electron mobility much greater than that of
the matrix. Indeed, the CuAgSe impurity phase is n-type and
has high mobility, around 2000 cm
2
V

1
s

1
at room temper-
ature, two orders of magnitude greater than that of Cu
2
Se,
the apparent matrix phase.
17
The lack of a fully confirmed crystal structure for the
highly Cu-disordered Cu
2
Se
18
and these conflicting carrier
types make Cu
1.97
Ag
0.03
Se a challenging material on which
to develop an electrical transport model. The disparity of the
electron mobility values between its phases makes it a per-
fect model system for the combination of magnetic-field de-
pendent resistivity measurements and effective medium
theory. This technique allows us to extract the resistivity ten-
sor of each phase, the results of which we show below.
Cu
1.97
Ag
0.03
Se was synthesized by the method in Day
et al.
and was characterized by powder X-ray diffraction.
13
SEM micrographs were taken on a JEOL JXA-8200 electron
probe micro-analyzer.
The data in Figure
1
were taken on a custom-built Hall
effect system with van der Pauw geometry and a magnetic
field of
þ
/

2T.
19
The data in Figure
2
were taken on a
Quantum Design Physical Property Measurement System
(PPMS) with its stronger magnet and more sensitive meas-
urements. The in-line resistivity
q
xx
and the Hall resistivity
q
yx
were measured by the four-point method with alternating
current using the electrical transport option at a range of
magnetic field strengths between 14 T and

14 T. Electrical
leads were fixed to the sample using silver epoxy. A current
of 10 mA and a frequency of 3 Hz were used to measure
q
xx
.
A current of 4 mA and a frequency of 15 Hz were used to
measure
q
yx
. These parameters yielded the smallest phase
angles between the input current and output voltage for each
measurement.
The measurements of
q
yx
are offset by a term that
increases with magnetic field due to imperfect alignment of
the voltage leads. Therefore, the
q
yx
data shown in this work
were obtained by the following correction:
20
q
yx
;
data
B
ðÞ
¼
q
yx
;
measured
B
ðÞ

q
yx
;
measured
0
ðÞ
q
xx
;
measured
B
ðÞ
q
xx
;
measured
0
ðÞ
:
(1)
The effective medium model was fitted to the data with
a MATLAB script that minimized the differences between
the model values and the data values in both
q
xx
and
q
yx
and
between their derivatives as a function of magnetic field
strength. The matrix phase was modeled as a group of spher-
ical crystallites. The impurity phase was modeled as a group
of prolate spheroidal crystallites, with the ratio of the minor
axis to the major axis set at 0.1. This number was determined
from SEM micrographs (inset of Figure
1
) showing the im-
purity phase. The effect of the 3% porosity on the transport
measurements was estimated to be a 2% increase in
q
xx
and
a 3% increase in
q
yx
using the effective medium model and
treating the voids as spheres with zero conductivity. We
therefore neglect the effect of the porosity on the transport
measurements.
Stroud solved the electrostatic equations in a two-phase
composite to derive the coherent potential approximation,
which is given by
9
FIG. 1. Hall coefficient
R
H
(open symbols) and thermopower
S
(closed sym-
bols) data below the phase transition in Cu
1.97
Ag
0.03
Se. The positive
Seebeck coefficients indicate holes and the negative Hall coefficient indi-
cates electrons as the majority carriers, showing the conflicting properties in
this composite. Heating data are shown in red and cooling data in blue. The
inset is an SEM micrograph of the material at room temperature.
FIG. 2. Resistivity measurements as a function of magnetic field
B
at 300 K.
A composite of two phases guarantees non-linear behavior in the magnetore-
sistance
q
xx
(open symbols) and the Hall effect
q
yx
(open symbols) due to
the differences in the weighting of each phase by its electron mobility.
172103-2 Day
etal.
Appl. Phys. Lett.
105
, 172103 (2014)
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131.215.70.231 On: Mon, 03 Nov 2014 15:39:38
0
$
¼
X
2
i
¼
1
f
i
ð
r
$
i

r
$
e
Þð
I
$
r
$
i

r
$
e
Þ
C
$
i
Þ

1
:
(2)
f
i
is the volume fraction of phase
i
,
I
$
is the identity matrix,
and
C
$
i
is the depolarization tensor of phase
i
, given by Eq.
(S6).
9
C
$
i
relates the electric field inside a crystallite to that
outside it and therefore depends on the shape of the crystal-
lite and on the anisotropy of the effective conductivity ten-
sor.
r
$
e
is the effective conductivity tensor, given by Eq.
(S5).
r
$
i
is the conductivity tensor of phase
i
, given by
21
r
$
i
¼
r
0
;
i
1
1
þ
l
i
B
ðÞ
2
l
i
B
1
þ
l
i
B
ðÞ
2
0

l
i
B
1
þ
l
i
B
ðÞ
2
1
1
þ
l
i
B
ðÞ
2
0
001
6
6
6
6
6
6
6
4
7
7
7
7
7
7
7
5
:
(3)
r
0,
i
is the electrical conductivity at zero field,
B
is the mag-
netic field strength, and
l
i
is the electron mobility. The mag-
netic field always points in the
z
-direction, and both phases
are assumed to have zero longitudinal magnetoresistance,
i.e.,
r
i,zz
equals
r
0,
i
.
r
$
i
and
r
$
e
, in general, are not isotropic,
as evidenced by the different values of
q
xx
and
q
yx
in Figure
2
and in the supplementary material.
22
However, the behav-
ior of
r
$
i
with
B
is completely specified by just two parame-
ters,
r
0,
i
and
l
i
. Electronic bands in a phase are represented
by tensors with the form of Eq.
(3)
, and
r
$
i
for a multiband
material is the sum of these tensors. Since Eq.
(2)
is a tensor
equation, it contains two independent equations that can be
numerically solved for
r
e
;
xx
and
r
e
;
xy
, with
r
0,
i
and
l
i
as fit-
ting parameters.
To first order, the volume-fraction weighted Hall con-
ductivity of a single phase
r
i,xy
is
f
i
r
0,
i
l
i
, where
f
i
is the vol-
ume fraction of phase
i
. If the fractional contribution of the
matrix phase to the Hall conductivity
j
f
1
r
0,1
l
1
j
/
R
j
f
i
r
0,
i
l
i
j
is
less than unity, and at least one of the phases has a magneto-
resistance contribution
l
i
B
of order one, the impurity phase
will make a measurable contribution to both the in-line and
Hall resistivity, and both quantities will be nonlinear as a
function of magnetic field, providing data from which the
individual conductivity and mobility values can be extracted.
A representative measurement of the in-line resistivity
q
xx
and the Hall resistivity
q
yx
at 300 K, including the trans-
port model, is shown in Figure
2
. Data at higher temperatures
can be found in the supplementary material.
22
The in-line resistivity
q
xx
rises with positive and nega-
tive magnetic field, but its rate of change with
B
decreases
with increasing
B
. This is referred to as saturating behavior.
This is because each phase’s contribution to
q
xx
is weighted
by 1/(1
þ
(
l
i
B
)
2
), meaning that as
B
increases, the contribu-
tion of the high mobility phase decreases, causing the rate of
change of
q
xx
with
B
to decrease. The Hall resistivity
q
yx
exhibits linear behavior from 0 T to about
þ
/

4 T. Again,
this is because of the way
q
yx
is weighted with
B
. Each
phase’s contribution is weighted by
l
i
B
/(1
þ
(
l
i
B
)
2
). At low
values of
B
, the Hall resistivity will be dominated by the
high mobility phase. As
B
increases, the high mobility phase
will continue to dominate, but its contribution will decrease
as the denominator (1
þ
(
l
i
B
)
2
) increases, leading to decreas-
ing values of
q
yx
with
B
. Note, however, that if the Hall
resistivity
q
yx
is converted to a Hall coefficient
R
H
according
to
R
H
¼
q
yx
/
B
,
R
H
stays negative across the entire range of
B
values, meaning that the n-type impurity phase is the major-
ity contributor to the Hall effect in this material.
Each electronic band in each phase is characterized by a
zero-field conductivity
r
0,
i
and an electron mobility
l
i
.We
modeled the matrix phase as a single-band material and the
impurity phase as a two-band material, for a total of six free
parameters.
The estimates for
r
0,
i
are shown in Table
I
. Ishiwata
et al
. used two conduction bands to model the resistivity ten-
sor components of CuAgSe,
23
a strategy suggested by the
nonlinear behavior of
q
xx
and
q
yx
with magnetic field
observed in that work. For this reason, we have adopted the
same strategy. The total conductivity of the impurity phase
differs by up to 20% of that of CuAgSe. This suggests that
while the impurity phase has the crystal structure of
CuAgSe,
13
the impurity phase has a slightly different
composition.
The estimates for the carrier mobility of each phase are
shown in Table
II
. The units of 10

4
T

1
are numerically
equal to the units of cm
2
V

1
s

1
, but it is more instructive
to use inverse units of magnetic field strength to illustrate
that the carrier mobility is a weighting factor for the electri-
cal conductivity of each phase and each band.
The
l
i
B
/(1
þ
(
l
i
B
)
2
) term reaches a maximum when
l
i
B
equals one, which for the high mobility band of the impurity
phase is when
j
B
j
is between 6 and 9 T, depending on tem-
perature. At this value of
j
B
j
,
q
yx
reaches an extremum, and
q
xx
reaches an inflection point as the influence of the second
phase declines due to saturation (see Figure
2
).
Looking at Tables
I
and
II
reveals the cause of the non-
linear Hall effect in this material. At all temperatures stud-
ied, the material contains at least one band with a great
enough mobility such that
l
i
B
is of order one; the contribu-
tion of the matrix phase to the Hall effect
j
f
1
r
0,1
l
1
j
/
R
j
f
i
r
0,
i
l
i
j
is at most 85% and is as low as 50%. These two facts guaran-
tee that the impurity phase will have an enormous influence
on the Hall effect in the material, despite its making up less
than 3% of the sample volume.
The electron mobility of the high-mobility band in the
impurity phase is less than that reported for pure CuAgSe
(Table
II
). This is because the impurity phase has a greater
electron concentration that does pure CuAgSe (Figure
3
),
which leads to a lower electron mobility when electron scat-
tering is dominated by acoustic phonons.
24
Looking at the fit parameters together and as functions of
temperature indicates that me
tal atoms move between the ma-
trix and impurity phases. Computing the Hall carrier
TABLE I. Zero-field conductivity as a function of temperature.
r
0,
i
(S cm

1
)
300 K 333 K 350 K 366 K 380 K
Impurity phase, band 1
1169
1101
1006
985
846
Impurity phase, band 2
147
120
148
83
34
Impurity phase, total
1316
1221
1154
1067
880
CuAgSe
23
1305
1332
1208
1132
1052
Matrix
547
517
492
458
309
Cu
2
Se
12
853
768
731
650
497
172103-3 Day
etal.
Appl. Phys. Lett.
105
, 172103 (2014)
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concentration of the matrix with the relation
n
H
¼
r
/e
l
,where
e
is the elementary charge, and f
or the impurity phase with Eq.
(4)
(adapted from the equation for
R
H
25
in by the relation
R
H
¼
1/
n
H
e
), we see that the carrier concentration of the matrix
is reduced compared to that of Cu
2
Se, and that of the impurity
phase is increased relative to CuAgSe (Figure
3
)
n
H
¼
r
1
þ
r
2
ðÞ
2
e
r
1
l
1
þ
r
2
l
2
ðÞ
:
(4)
In the matrix, this must be due to Ag dissolving in the lattice.
This is supported by X-ray diffraction by Brown
et al.
12
in
which the reflections of Cu
1.97
Ag
0.03
Se are shifted to smaller
values of 2
H
, indicating a larger lattice, which is consistent
with the larger Ag cations located on vacant Cu sites.
26
Ordinarily these vacant sites create holes in Cu
2
Se; an Ag
atom on such a site donates an electron and reduces the num-
ber of positive charge carriers. Ag has been shown to reduce
the carrier concentration in Cu
2
Se,
13
further supporting the
idea that the matrix is Ag-doped Cu
2
Se. In general, because
the impurity phase dissolves into the matrix at higher tem-
peratures, slightly different compositions of the cation ratios
are not surprising.
The matrix phase and Cu
2
Se both show a gradual
decline of carrier concentration with temperature, but
between 366 K and 380 K the carrier concentration of the
matrix drops more sharply than does the carrier concentra-
tion of Cu
2
Se. In this temperature range, Ag becomes more
soluble in the matrix. The introduction of more Ag into the
matrix fills more holes in the valence band of the matrix,
reducing the carrier concentration. This is consistent with the
jump in the matrix phase mobility, as the mobility tends to
increase as the carrier concentration decreases.
The increase in the Hall carrier concentration of the im-
purity phase relative to CuAgSe could be due to an elevated
cation to anion ratio. The dependence of the Hall carrier con-
centration of the impurity phase on temperature is more com-
plicated because of the mobility-dependent contribution of
each band. It is difficult to distinguish from the model pa-
rameters in Tables
I
and
II
the effects of temperature, the
mass exchange between the phases, and the changing energy
difference between the two bands. However, Figure
3
shows
that the Hall carrier concentration of the impurity phase
gradually rises and then jumps between 366 K and 380 K.
Between these two temperatures, the conductivity of band 1
drops much less than does the conductivity of band 2 (Table
II
). This means that the carrier concentration will be more
influenced by band 1, the band containing more carriers, and
that the overall Hall carrier concentration will increase.
The classical influence of an impurity phase on the resis-
tivity tensor of Cu
1.97
Ag
0.03
Se and the interest in using impu-
rity phases to create beneficial quantum effects in
thermoelectrics drove us to introduce effective medium
theory as a tool for optimizing phase-segregated thermoelec-
trics. We have shown that useful information on each band
in each phase can be gathered by measuring the independent
components of the resistivity tensor at high magnetic fields.
In combination with X-ray diffraction and classic thermo-
electric characterization techniques, we present a powerful
tool to model and understand multiphase behavior in semi-
conductors, in order to optimize materials compositions for
high figure of merit thermoelectrics, such as composites of
Sb
2
Te
3
and PbTe,
27
Ag
2
Te and PbTe,
4
In
2
Te
3
and Bi
2
Te
3
,
28
or AgSbTe
2
with nanodot inclusions,
6
as well as to quantify
the negative effects of impurities.
T.W.D., D.R.B., and G.J.S. are grateful for the support
of the AFOSR MURI program under FA9550-12-1-0002.
D.R.B. acknowledges the support of the Resnick Institute.
B.C.M. gratefully acknowledges financial support through
start-up funding provided by the Dana and David Dornsife
College of Letters and Sciences at the University of
Southern California. W.G.Z. also acknowledges the support
of a fellowship within the Postdoc-Program of the German
Academic Exchange Service (DAAD).
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(11),
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TABLE II. Single band mobility as a function of temperature.
l
i
(10

4
T

1
)
300 K 333 K 350 K 366 K 380 K
Impurity phase, band 1

164

167

149

138

50
Impurity phase, band 2

1458

1504

1090

1235

1178
CuAgSe
23

2191
............
Matrix
24
22
24
25
50
Cu
2
Se
12
13
12
11
11
10
10
18
2
4
10
19
2
4
10
20
2
4
10
21
|
n
H
| [cm
-3
]
380
360
340
320
300
T
[K]
Cu
2
Se, Data
12
Matrix, Model
Impurity, Model
CuAgSe, Data
23
FIG. 3. Hall carrier concentrations measured on pure Cu
2
Se and CuAgSe
and extracted from the model for the matrix and impurity phases.
172103-4 Day
etal.
Appl. Phys. Lett.
105
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