LETTERS
PUBLISHED ONLINE: 21 APRIL 2013 | DOI: 10.1038/NMAT3621
Extremely high electron mobility in a
phonon-glass semimetal
S. Ishiwata1 *, Y. Shiomi1 , J. S. Lee1 , M. S. Bahramy2 , T. Suzuki2 , M. Uchida1 , R. Arita1 , Y. Taguchi2
and Y. Tokura1,2
The electron mobility is one of the key parameters that
characterize the charge-carrier transport properties of materials, as exemplified by the quantum Hall effect1 as well as
high-efficiency thermoelectric and solar energy conversions2,3 .
For thermoelectric applications, introduction of chemical disorder is an important strategy for reducing the phonon-mediated
thermal conduction, but is usually accompanied by mobility degradation. Here, we show a multilayered semimetal
β-CuAgSe overcoming such a trade-off between disorder and
mobility. The polycrystalline ingot shows a giant positive magnetoresistance and Shubnikov de Haas oscillations, indicative
of a high-mobility small electron pocket derived from the Ag
s-electron band. Ni doping, which introduces chemical and
lattice disorder, further enhances the electron mobility up
to 90,000 cm2 V−1 s−1 at 10 K, leading not only to a larger
magnetoresistance but also a better thermoelectric figure of
merit. This Ag-based layered semimetal with a glassy lattice
is a new type of promising thermoelectric material suitable for
chemical engineering.
High-carrier-mobility semiconductors and semimetals often
exhibit the giant magnetoresistance (GMR) effect, a pronounced
increase in resistivity ρxx by the application of a magnetic field B.
The classical magnetoresistance associated with the orbital motion
of carriers can be described in such a way that the magnetoresistance
ratio defined as 1ρxx (B)/ρxx (0) ≡ [ρxx (B) − ρxx (0)]/ρxx (0) is
proportional to (µB)2 (µ is electron mobility) at small B, followed
by saturation at large B. Although GMR showing a quadratic B
dependence has been known since the early twentieth century for
semimetallic Bi (ref. 4), particular interest has recently been directed
to positive GMR with a linear B dependence in a narrow-gap
semiconductor β-Ag2+δ X (X = Se and Te; refs 5,6). The origin of
the linear GMR without saturation in a wide temperature range was
discussed within the framework of the classical magnetoresistance
based on the electron trajectory due to chemical inhomogeneity
in addition to the high mobility5,7,8 . As an alternative mechanism,
quantum magnetoresistance has been proposed by assuming a zerogap semiconductor with a Dirac-cone-like linearly dispersing band
structure, which allows all conduction carriers to condensate at the
lowest Landau level9,10 . Although the band-structure calculations
suggest that β-Ag2 Se is a narrow-gap semiconductor with parabolic
electron and hole bands located at the 0 point11 , the linear
dispersion is expected to be realized through the inhomogeneity
caused by the slight off-stoichiometry of Ag (refs 9,12). In either
the quantum or classical case, chemical inhomogeneity plays an
essential role in the GMR in β-Ag2+δ Se. The quantum and classical
GMR have recently been discussed also for Dirac fermion systems
such as graphene13 , the topological insulator Bi2 Te3 (ref. 14) and
narrow-gap semiconductors such as InSb and HgTe (refs 15–17).
The multilayered semimetal β-Cu1−x Nix AgSe reported here
possesses both high-mobility electrons and lattice disorder as
evidenced by the GMR deviating from the quadratic B dependence
and the high thermoelectric performance. Whereas classical
semiconductors such as doped PbTe have been extensively studied
as high-efficiency thermoelectrics18 , semimetals have received less
attention, because the thermal broadening of the Fermi distribution
function almost equally activates two kinds of carrier with opposite
signs. However, the semimetal β-CuAgSe is found to possess a
significant electron–hole asymmetry in the band structure near the
Fermi level, as revealed by first-principles calculations. β-CuAgSe
lies in the middle of the solid solution of β-Ag2 Se and β-Cu2 Se but
has a unique layered structure consisting of alternate stacking of the
layers of Ag and CuSe (Fig. 1a)19 . Contrary to the naive expectation,
the Fermi surface has been found to be nearly isotropic, which
allows simple analyses of experimental data for the polycrystalline
samples. It should be noted here that β-Ag2 Se is a promising
thermoelectric material at room temperature20,21 . The other end
compound Cu2 Se also shows a high thermoelectric figure of merit
ZT ∼ 1.5 at 1,000 K, where the system forms a cubic lattice with
statistically distributed copper ions, and becomes superionic22 .
Likewise, CuAgSe forms a cubic lattice with superionic conductivity
in the α phase above 470 K (ref. 19).
As shown in Fig. 1b, β-Cu1−x Nix AgSe shows metallic behaviour
down to the lowest temperature. The residual resistivity ratio
ρxx (300 K)/ρxx (2 K) is 25 for the undoped sample (x = 0) and 17
for the Ni-doped sample (x = 0.1). The negative sign of the Seebeck
coefficient S indicates the dominance of electron-type conduction
(Fig. 1c). However, the sign of S for the undoped sample changes
from negative to positive at 400 K (see the inset), indicative of a
semimetallic band structure, as will be discussed later. Note that
the jump in ρxx as well as S at 470 K corresponds to the anomaly
of the phase transition to the superionic phase (α-CuAgSe). Lattice
thermal conductivity (κlattice = κ − κelectron ) as estimated from the
Wiedemann–Franz law is as low as 10 mW cm−1 K−1 for both
compounds (Fig. 1d). Such a small κlattice implies the presence of a
large lattice anharmonicity due to disorder, which is reminiscent of
the amorphous solids23 . Interestingly, the 10% Ni doping decreases
ρxx by a factor of 3–4, while reducing κlattice and maintaining S at
an almost comparable, or even larger value at low temperatures.
As a result, the Ni doping increases ZT from 0.15 to 0.25 at room
temperature, and from 0.02 to 0.1 at 100 K (Fig. 1e).
Figure 2a,b shows a positive GMR in β-Cu1−x Nix AgSe with x = 0
and 0.1. With decreasing temperature, 1ρxx (B)/ρxx (0) shows a
1 Department
of Applied Physics and Quantum-Phase Electronics Center (QPEC), University of Tokyo, Hongo, Tokyo 113-8656, Japan, 2 RIKEN Center for
Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan. *e-mail: ishiwata@ap.t.u-tokyo.ac.jp.
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2013 Macmillan Publishers Limited. All rights reserved.
1
NATURE MATERIALS DOI: 10.1038/NMAT3621
LETTERS
a
Se
Ag
Cu
10¬3
ρxx (Ω cm)
4
Cu1¬xNixAgSe
x=0
10¬5
0
100
S (μV K¬1)
¬50
6
4
0
2
¬100
8
6
10¬3
300
x = 0.1
x=0
400
500
Temperature (K)
¬100
κ (mW cm¬1 K¬1)
d
50
ρxx (Ω cm)
4
2
c
x = 0.1
2
10¬4
S (μV K¬1)
b
x=0
x = 0.1
40
30
20
x=0
10
0
e
0.2
ZT
x = 0.1
0.1
0.0
0
x=0
100
200
Temperature (K)
300
Figure 1 | Structure and transport properties. a, Crystal structure of
β-CuAgSe with a pseudo-tetragonal unit cell (shown by dotted lines). The
occupancy of the Cu site is 0.5. b–e, Resistivity ρxx (b), Seebeck coefficient
S (c), thermal conductivity κ (d) and dimensionless figure of merit ZT (e)
as a function of temperature for β-Cu1−x Nix AgSe. The blue shaded areas in
d correspond to the estimated lattice thermal conductivity on the
assumption of the Wiedemann–Franz law. The inset shows the Seebeck
coefficient and resistivity above room temperature for β-CuAgSe below
470 K and α-CuAgSe above 470 K.
significant increase especially below 100 K and reaches 24 at 2 K
under a transverse B of 14 T (B⊥ current) and that for the Ni-doped
sample exceeds 140 under the same conditions. The magnitudes
of magnetoresistance at low temperatures are 2–14 times larger
than that of β-Ag2+δ Se (ref. 5). As longitudinal magnetoresistances
(shown by dashed lines) are rather small, the spin-polarization
effect can be ruled out as a major origin of the transverse GMR. In
Fig. 2c,d, the negative slope of the Hall resistivity ρyx at low fields
reconfirms the predominance of the electron conduction below
300 K. However, nonlinear field dependence of ρyx can be found
for β-Cu1−x Nix AgSe with both x = 0 and 0.1 at low temperatures,
signalling two-carrier (electron and hole) conduction. For x = 0,
the sign of the slope ρyx /B changes from negative to positive
2
with increasing B at 50 and 2 K. In the high-B limit, ρyx /B is
approximately equal to (e(nh − ne ))−1 , where e and nh (ne ) denote
the elementary charge and the carrier concentration of holes
(electrons), respectively. Thus, the concentration of hole carrier is
expected to be larger than that of the electron for x = 0 and vice versa
for x = 0.1. The presence of the two kinds of carrier with different
τ can be confirmed for x = 0 by the deviation of 1ρxx (B)/ρxx (0)
from a scaling function f (B/ρxx (0)) (Fig. 2e). This scaling relation,
called Kohler’s rule, holds for the system with a constant scattering
time τ over the Fermi surfaces. Therefore, considering the negative
slope of ρyx at low B and its highly nonlinear B dependence, it
is reasonable to assume that β-CuAgSe has two kinds of carrier
with different τ , one is the high-mobility electron carrier and the
other is the hole carrier. As for the Ni-doped sample (Fig. 2f), the
magnetoresistance data at low temperatures tend to fall on a single
scaling curve with a power factor of 1.3, almost obeying Kohler’s
rule. This result implies the further predominance of the electron
conduction in the Ni-doped sample.
The conductivity tensors are analysed by the two-carrier model
as shown in Fig. 3a,b. All of the longitudinal and Hall conductivities
are fitted with the following formulae24 .
neµ
+ Cxx
(1)
σxx =
1 + (µB)2
1
σxy = nH eµ2H B
+ Cxy
(2)
1 + (µH B)2
Here, n(nH ) and µ(µH ) denote the carrier concentration
and mobility for the high-mobility carriers estimated from the
longitudinal conductivity, σxx (Hall conductivity σxy ), respectively.
Cxx (Cxy ) denotes the low-mobility component estimated from σxx
(σxy ). The fitting parameters are summarized in Supplementary
Tables SI and SII. We assume that the high-mobility electron
carriers dominate the positive magnetoresistance effect, and that
the fitting parameters, which are dependent on temperature, are
independent of B. Using the fitting results both for σxx (data not
shown) and σxy , we independently extract the longitudinal and Hall
mobilities, µ and µH (Fig. 3c), and carrier concentrations, n and nH
(see Supplementary Information), respectively. µ and µH coincide
perfectly with each other in a wide temperature range, indicating
the validity of our fitting procedure based on the two-carrier
model. The carrier concentrations at 10 K, n = 7.4 × 1018 cm−3 and
nH = 6.5 × 1018 cm−3 , are also comparable to each other, signalling
the semimetallic nature of the compound. On decreasing the
temperature to 20 K, the mobilities for both compounds increase
in proportion to T −1 (dashed lines in Fig. 3c). Thus, the carrier
scattering is dominated not by the defect scattering but by the
electron–phonon scattering in a wide temperature range down
to low temperatures. Notably, the mobilities at 10 K are about
20,000 cm2 V−1 s−1 and 90,000 cm2 V−1 s−1 for β-Cu1−x Nix AgSe
with x = 0 and 0.1, respectively, the latter of which is comparable
to the high-quality thin film of doped HgTe (ref. 15) suitable
for studying the quantum Hall effect. Although the deviation of
the magnetoresistance curves from the quadratic B dependence
indicates the effect of inhomogeneity, the main origin of the
positive GMR in this system can be ascribed to the extremely high
carrier mobility. As seen in Fig. 3d,e, the ratios of low-mobility
hole conductivity to high-mobility electron conductivity, Cxx /σxx
and Cxy , are fairly small and decrease either by doping Ni or
decreasing temperature, in accordance with the enhancement of
magnetoresistance.
Despite the random orientation of domains in the polycrystalline
β-CuAgSe, Shubnikov de Haas (SdH) oscillations are observed
for ρyx at low temperatures (Fig. 4a), reflecting the high carrier
mobility. The oscillatory behaviour cannot be discerned for ρxx
because of the GMR masking the SdH signals. The periodicity of the
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2013 Macmillan Publishers Limited. All rights reserved.
NATURE MATERIALS DOI: 10.1038/NMAT3621
a
LETTERS
b
CuAgSe
Cu0.9Ni0.1AgSe
B⊥I
8
100 K
2
50 K
B || I
2K
4
2
ρxx (Ω cm)
ρxx (Ω cm)
4
8
6
200 K
6
10¬4
10¬3
300 K
10¬3
B⊥I
300 K
200 K
100 K
10¬4
6
4
2K
2
6
4
50 K
2K
B || I
2K
10¬5
6
¬10
c
¬5
0
5
Magnetic field (T)
¬10
10
d
2
¬5
0
5
Magnetic field (T)
10
1.5
1.0
1
0
ρyx (mΩ cm)
ρyx (mΩ cm)
0.5
2K
50 K
0.0
2K
50 K
¬0.5
¬1
¬2
100 K
200 K
300 K
¬10
¬5
0
5
Magnetic field (T)
¬1.5
10
e
f
0.1
300 K
0.01
1014
1016
¬5
0
5
Magnetic field (T)
2K
20 K
50 K
100 K
200 K
300 K
1
0.1
0.001
1018
B2/ρxx(0)2 (Oe2 Ω¬2 cm¬2)
10
100
0.01
H2
H 1.3
0.001
300 K
¬10
10
2K
20 K
50 K
100 K
200 K
Δ ρ (B)/ρ xx(0)
Δ ρ (B)/ρxx(0)
10
1
100 K
200 K
¬1.0
H2
H 1.3
1014
1016
1018
B2/ρ xx(0)2 (Oe2 Ω¬2 cm¬2)
1020
Figure 2 | GMR and Hall resistance. a–f, Magnetic field dependence of resistivity ρxx (a,b) and Hall resistivity ρyx (c,d), and magnetoresistance 1ρ/ρ0
plotted as a function of B2 /ρ02 (e,f) for β-CuAgSe and β-Cu0.9 Ni0.1 AgSe, respectively (ρ0 ≡ ρxx (B = 0)). The resistivity under a transverse magnetic field
(B perpendicular to the current I) is shown by solid lines and the resistivity under a longitudinal magnetic field (BkI) is shown by the dashed lines in a,b.
SdH oscillation minima 1(1/B), 0.0076 T−1 , gives a Fermi surface
cross-section area of 0.013 Å−2 (Fig. 4a, inset). Assuming a spherical
Fermi surface, the carrier concentration at 2 K was estimated to
be 8.6 × 1018 cm−3 . As this value is close to that estimated by
the two-carrier analyses, one can ascribe the origin of the SdH
oscillation not to an impurity phase such as a silver metal but to
the intrinsic bulk nature of β-CuAgSe. The temperature dependence
of the SdH oscillation amplitude at 11 T is fitted to the standard
Lifshitz–Kosevich theory adopted for the Hall resistivity25 ,
1ρyx
2π2 kB T /h̄ωc
= exp(−2π2 kB TD /h̄ωc )
4ρyx
sinh(2π2 kB T /h̄ωc )
(3)
where TD and ωc denote the Dingle temperature and the cyclotron
frequency, respectively. The fitting gives the rather small cyclotron
mass mc (= eB/ωc ) of 0.13 m0 (m0 being the mass of free electron).
To further characterize the conduction carriers in β-CuAgSe, we
measured the reflectivity spectra at room temperature. Figure 4b
shows the plasmon loss function given by −Im[1/] = 2 /(12 + 22 ).
From the peak position at 0.06√eV, corresponding to the reduced
plasma frequency ωp∗ (= ωp / ∞ ), the effective mass m∗ is
estimated, using the values n = 7.0 × 1018 cm−3 and ∞ = 21,
to be 0.12 m0 , which is comparable to the cyclotron mass.
Reflecting the semimetallic nature with low carrier density, the
optical conductivity spectrum (Fig. 4b, inset) shows a small
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2013 Macmillan Publishers Limited. All rights reserved.
3
NATURE MATERIALS DOI: 10.1038/NMAT3621
LETTERS
a
c
6
20 K
μ (cm2 V¬1 s¬1)
10
30 K
5
50 K
0
0.0
0.5
40
2.0
Cu0.9Ni0.1AgSe
104
6
CuAgSe
100
200
Temperature (K)
2
CuAgSe
10¬3
6
Cu0.9Ni0.1AgSe
4
0
20 K
e
100
200
Temperature (K)
10¬4
Cxy
300 K
10¬5
Cu0.9Ni0.1AgSe
100 K
0.5
1.0
1.5
Magnetic field (T)
10¬6
2.0
300
CuAgSe
20
0
0.0
300
d
Cu0.9Ni0.1AgSe
50 K
μH
2
0
b
10 K
μ
μH
2
300 K
1.0
1.5
Magnetic field (T)
μ
4
4
100 K
σ xy (× 105 Ω¬1 cm¬1)
105
CuAgSe
Cxx /σ xx
σ xy (× 105 Ω¬1 cm¬1)
10 K
0
100
200
Temperature (K)
300
Figure 3 | Two-carrier model analyses for the conductivity tensors. a,b, Magnetic field dependence of Hall conductivity σxy for β-CuAgSe (a) and
β-Cu0.9 Ni0.1 AgSe (b). c–e, Temperature dependence of drift mobility µ and Hall mobility µH of high-mobility electron carriers (c), Cxx /σxx (d) and Cxy (e)
(see text), obtained from the fits for conductivity tensors.
Δ ρyx (μΩ cm)
a
0.5
Δ ρyx /ρyx (%)
0.4
2
4
c
2K
18 K
4
3
0
¬2
0.3
2
0.08 0.09 0.10 0.11
1/B (1/T)
0.2
1
0.1
¬Im [ε (ω )] (× 10¬3 m F¬1)
60
50
40
30
10
15
Temperature (K)
σ (ω) (Ω¬1 cm¬1)
b
5
20
25
1,200
¬1
¬2
800
400
0
0.0
E ¬ E F (eV)
0
0.0
0
¬3
0.5
1.0
Photon energy (eV)
¬4
20
¬5
10
0
0.0
0.1
0.2
Photon energy (eV)
0.3
¬6
Γ
X
M
Γ
Z
Figure 4 | Experimental and theoretical characterization of conduction electrons. a,b, Temperature dependence of Shubnikov de Haas (SdH) oscillation
amplitude 1ρyx /ρyx (%) at 11 T (a) and plasmon loss function as a function of the photon energy (b) for β-CuAgSe. The inset in a shows the oscillatory
part of ρyx , and the inset of b shows optical conductivity at room temperature. c, Calculated electronic band structure of β-CuAgSe (see text).
Drude weight and a pseudogap structure at around 0.1 eV, above
which interband transitions, for instance from Te 4p to Ag 5s
bands, are dominant.
4
Having the experimental evidence for the semimetallic band
structure with the light-mass electron carriers, we have performed
band-structure calculations for β-CuAgSe as shown in Fig. 4c.
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2013 Macmillan Publishers Limited. All rights reserved.
NATURE MATERIALS DOI: 10.1038/NMAT3621
Along the 0–M and 0–X lines, a highly dispersive Ag 5s band
is crossing the Fermi level, whereas a less dispersive band mainly
consisting of a Te 4p band and a Cu 3d band is located just below
the Fermi level. This feature is consistent with the two-carrier
analyses on the conductivity tensors and the sign change in S from
negative to positive on increasing temperature. Despite the layered
structure, the highly dispersive band is also discerned along the
0−Z line. The isotropic Fermi surface topology, that is, electron
pocket, is compatible with the observation of the SdH oscillation
in the polycrystalline sample. The averaged mass of the Ag s band
is about 0.4m0 , which is small but larger than the experimental
values. This discrepancy may arise from the inaccuracy of the
structure model used for the band calculation; for example, the
fivefold superlattice modulation in the c plane was confirmed by
transmission electron microscope observations26 . As the band-edge
structure is sensitive to external perturbations, the superlattice
modulation, the chemical disorder in β-CuAgSe and the chemical
pressure by Ni doping would cause non-trivial effects on the carrier
conduction in this semimetallic system.
Let us summarize the important features of β-CuAgSe. First,
it shows a positive GMR, neither linear nor quadratic to B, and
the magnitude of the magnetoresistance increases markedly as
the mobility increases at low temperatures. This is in contrast
to β-Ag2 Se showing positive and linear GMR without such a
remarkable temperature dependence5 . Second, it has a semimetallic
band structure with a small pocket of extremely high-mobility
Ag s electrons. The high mobility and relatively low carrier
concentration are the key factors for the large thermoelectric
power factor. Third, the high-mobility electron conduction is
rather robust against the chemical disorder and the randomly
oriented domains, as evidenced by the enhanced electron mobility
in the Ni-doped sample and by the observation of SdH oscillation
in the polycrystalline β-CuAgSe. Moreover, β-CuAgSe may have
substantial atomic-lattice disorder due to the proximity to the
superionic phase27,28 . For the origin of the compatibility of the high
electron mobility and the significant chemical disorder, we propose
two possibilities. One is the fairly large screening length due to the
large polarizability of Cu and Ag ions, which is advantageous also
for the superionic conduction. The other is the effective band-edge
engineering by the chemical inhomogeneity9,12,17 . Although our
band calculation suggests that the highly dispersive conduction
band and the less dispersive valence band are not very close to
each other, non-parabolic and gapless (Dirac-cone-like) energy
dispersion with a strong electron–hole asymmetry may be realized
by the inhomogeneity. In addition, the multilayered structure
is suitable for band tuning by chemical substitution. Thus, the
multilayered semimetal β-CuAgSe provides a fruitful playground
not only for new magnetoresistance phenomena but for a promising
thermoelectric performance near or below room temperature,
where the strong electron–hole imbalance near the Fermi level
remains intact. We regard this material as a candidate for a
new type of phonon-glass electron crystal29,30 , characterized by
high-mobility Ag s electrons in the presence of atomic and
macroscopic inhomogeneity, by which lattice thermal conductivity
is reduced effectively.
Methods
The polycrystalline ingots were prepared by a solid-state reaction in evacuated
quartz tubes (∼5 × 10−3 Pa). A mixture of Cu, Ni, Ag and Se powders at a molar
ratio 1−x: x: 1 : 1.01 (x = 0 and 0.1) was slowly heated to 673 K at a rate of 20 K h−1
and then heated to 1,073 K at a rate of 50 K h−1 and maintained at this temperature
for 24 h, followed by cooling to room temperature at a rate of 100 K h−1 . The
obtained sample was ground, pressed into pellets and subjected to the same heat
treatment process again.
The transport measurements were performed on bar-shaped rectangular
samples with dimensions of about 2×0.5×0.2 mm3 . Air-dried gold paste was used
as the electrodes. The Seebeck coefficient and thermal conductivity were measured
by a conventional steady-state method. The reflectivity spectrum was obtained in
LETTERS
the energy range between 0.01 and 6 eV with coated aluminium and silver mirrors
as references. After extrapolation of the reflectivity in unmeasured spectral regions
by adopting the Hagen–Rubens relation for the low-energy region, the optical
conductivity was derived through the Kramers–Kronig transformation.
The electronic structure calculations were carried out using the full-potential
augmented plane-wave plus local orbital method, as implemented in the WIEN2K
code31 . The exchange-correlation part of the potential was treated using the
modified Becke–Johnson functional32 . The muffin-tin radii RMT of Ag, Cu and Se
were set to 2.5, 2.04 and 1.81 Bohr, respectively, and the maximum modulus of
reciprocal vectors Kmax was chosen such that RMT Kmax = 7. The lattice parameters
and atomic positions were taken from experiment26 and the corresponding
Brillouin zone was sampled by a 11 × 11 × 7 k-mesh. In our calculations, the
unit cell contains two formula units of CuAgSe with two Cu atoms occupying
(3/4, 1/4, 0.895) and (1/4, 3/4, 0.105) sites, two Se atoms occupying (3/4, 1/4,
0.127) and (1/4, 3/4, 0.873) sites, and finally two Ag atoms located at (3/4, 3/4,
0.551) and (1/4, 1/4, 0.449).
Received 20 October 2012; accepted 5 March 2013;
published online 21 April 2013
References
1. Novoselov, K. S. et al. Two-dimensional gas of massless Dirac fermions in
graphene. Nature 438, 197–200 (2005).
2. Mahan, G., Sales, B. & Sharp, J. Thermoelectric materials: New approaches to
an old problem. Phys. Today 50, 42–47 (March, 1997).
3. Shah, A. V. et al. Thin-film silicon solar cell technology. Prog. Photovolt. 12,
113–142 (2004).
4. Kapitza, P. L. Bismuth crystals and its change in strong the study of the specific
resistance of magnetic fields and some allied problems. Proc. R. Soc. Lond. A
119, 358–443 (1928).
5. Xu, R. et al. Large magnetoresistance in non-magnetic silver chalcogenides.
Nature 390, 57–60 (1997).
6. Husmann, A. et al. Megagauss sensors. Nature 417, 421–424 (2002).
7. Parish, M. M. & Littlewood, P. B. Classical magnetotransport of
inhomogeneous conductors. Phys. Rev. B 72, 094417 (2005).
8. Herring, C. Effect of random inhomogeneities on electrical and
galvanomagnetic measurements. J. Appl. Phys. 31, 1939–1953 (1960).
9. Abrikosov, A. A. Quantum linear magnetoresistance. Phys. Rev. B 58,
2788–2794 (1998).
10. Lee, M., Rosenbaum, T. F., Saboungi, M-L. & Schnyders, H. S. Band-gap
tuning and linear magnetoresistance in the silver chalcogenides. Phys. Rev. Lett.
88, 066602 (2002).
11. Fang, C. M., de Groot, R. A. & Wiegers, G. A. Ab initio band structure
calculations of the low-temperature phases of Ag2 Se. J. Phys. Chem. Solids 63,
457–464 (2002).
12. Wang, X-L, Dou, S. X. & Zhang, C. Zero-gap materials for future spintronics,
electronics and optics. NPG Asia Mater. 2, 31–38 (2010).
13. Friedman, A. L. et al. Quantum linear magnetoresistance in multilayer epitaxial
graphene. Nano Lett. 10, 3962–3965 (2010).
14. Wang, X., Du, Y., Dou, S. & Zhang, C. Room temperature giant and linear
magnetoresistance in topological insulator Bi2 Te3 nanosheets. Phys. Rev. Lett.
108, 266806 (2012).
15. Thio, T. et al. Giant magnetoresistance in zero-band-gap Hg1−x Cdx Te.
Phys. Rev. B 57, 12239–12244 (1998).
16. Solin, S. A., Thio, T., Hines, D. R. & Heremans, J. J. Enhanced
room-temperature geometric magnetoresistance in inhomogeneous
narrow-gap semiconductors. Science 289, 1530–1532 (2000).
17. Hu, J. & Rosenbaum, T. F. Classical and quantum routes to linear
magnetoresistance. Nature Mater. 7, 697–700 (2008).
18. Heremans, J. P. et al. Enhancement of thermoelectric efficiency in PbTe by
distortion of the electronic density of states. Science 321, 554–557 (2008).
19. Miyatani, S. Electronic and ionic conduction in (Ag1−x Cux )2 Se. J. Phys. Soc. Jpn
34, 423–432 (1973).
20. Conn, J. B. & Taylor, R. C. Thermoelectric and crystallographic properties of
Ag2 Se. J. Electrochem. Soc. 107, 977–982 (1960).
21. Ferhat, M. & Nagao, J. Thermoelectric and transport properties of Ag2 Se
compounds. J. Appl. Phys. 88, 813–816 (2000).
22. Liu, H. et al. Copper ion liquid-like thermoelectrics. Nature Mater. 11,
422–425 (2012).
23. De Yoreo, J. J., Knaak, W., Meissner, M. & Pohl, R. O. Low-temperature
properties of crystalline (KBr)1−x (KCN)x : A model glass. Phys. Rev. B 34,
8828–8842 (1986).
24. Takahashi, H., Okazaki, R., Yasui, Y. & Terasaki, I. Low-temperature
magnetotransport of the narrow-gap semiconductor FeSb2 . Phys. Rev. B 84,
205215 (2011).
25. Shoenberg, D. Magnetic Oscillations in Metals (Cambridge Univ. Press, 1984).
26. Frueh, A. J., Czamanke, G. K. & Knight, C. H. The crystallography of eucairite,
CuAgSe. Z. Kristallogr. 108, 389–396 (1957).
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2013 Macmillan Publishers Limited. All rights reserved.
5
NATURE MATERIALS DOI: 10.1038/NMAT3621
LETTERS
27. Ramasesha, S. A phenomenological model for semiconductor–metal transition
in mixed conductors. J. Solid State Chem. 41, 333–337 (1982).
28. Fortner, J., Saboungi, M-L. & Enderby, J. E. Carrier density enhancement in
semiconducting NaSn and CsPb. Phys. Rev. Lett. 74, 1415–1418 (1995).
29. Slack, G. A. in CRC Handbook of Thermoelectrics (ed. Rowe, M.) 407–440
(CRC, 1995).
30. Snyder, G. J. & Toberer, E. S. Complex thermoelectric materials. Nature Mater.
7, 105–114 (2008).
31. Blaha, P., Schwarz, K., Madsen, G., Kvasnicka, D. & Luitz, J. WIEN2K package,
Version 10.1, http://www.wien2k.at.
32. Tran, F. & Blaha, P. Accurate band gaps of semiconductors and insulators
with a semilocal exchange-correlation potential. Phys. Rev. Lett. 102,
226401 (2009).
Acknowledgements
The authors thank D. Okuyama and T. Arima for experimental support and thank
J. G. Checkelsky, A. Tsukazaki, F. Kagawa and N. Kanazawa for useful comments.
This study was in part supported by a Grant-in-Aid for Scientific Research (Grant No.
6
23685014) from the MEXT, and by the Funding Program for World-Leading Innovative
R&D on Science and Technology (FIRST Program), Japan.
Author contributions
S.I. and Y. Tokura conceived the study and wrote the paper. S.I. prepared the
samples and performed the transport measurements. Y.S. and M.U. designed the
thermoelectric measurement systems. T.S. and Y. Taguchi performed thermoelectric
measurements at high temperatures. J.S.L. worked on the optical study. M.S.B. and R.A.
performed band calculations.
Additional information
Supplementary information is available in the online version of the paper. Reprints and
permissions information is available online at www.nature.com/reprints. Correspondence
and requests for materials should be addressed to I.S.
Competing financial interests
The authors declare no competing financial interests.
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2013 Macmillan Publishers Limited. All rights reserved.