The inhomogeneous distribution of excess or deficient silver atoms lies behind the large and linear transverse magnetoresistance displayed by Ag_(2±δ)Se and Ag_(2±δ)Te, introducing spatial conductivity fluctuations with length scales independent of the cyclotron radius. We report a negative, nonsaturating longitudinal magnetoresistance up to at least 60 T, which becomes most negative where the bands cross and the effect of conductivity fluctuations is most acute. Thinning samples down to 10   μm suppresses the negative response, revealing the essential length scale in the problem and paving the way for designer magnetoresistive devices

Betts, J. B.

Hu, Jingshi

Rosenbaum, T. F.

Caltech Authors - Main

PRL 95, 186603 (2005) P H Y S I C A L R E V I E W L E T T E R S
week ending
28 OCTOBER 2005Current Jets, Disorder, and Linear Magnetoresistance in the Silver Chalcogenides
Jingshi Hu and T. F. Rosenbaum
The James Franck Institute and Department of Physics, The University of Chicago, Chicago, Illinois 60637, USA
J. B. Betts
National High Magnetic Field Laboratory, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
(Received 26 May 2005; published 28 October 2005)0031-9007=The inhomogeneous distribution of excess or deficient silver atoms lies behind the large and linear
transverse magnetoresistance displayed by Ag2Se and Ag2Te, introducing spatial conductivity
fluctuations with length scales independent of the cyclotron radius. We report a negative, nonsaturating
longitudinal magnetoresistance up to at least 60 T, which becomes most negative where the bands cross
and the effect of conductivity fluctuations is most acute. Thinning samples down to 10 m suppresses the
negative response, revealing the essential length scale in the problem and paving the way for designer
magnetoresistive devices.
DOI: 10.1103/PhysRevLett.95.186603 PACS numbers: 72.20.My, 72.15.Gd, 72.80.JcThe silver chalcogenides provide a striking example of
the benefits of imperfection. Perfectly stoichiometric
Ag2Se and Ag2Te are nonmagnetic, narrow-gap semicon-
ductors whose electron and hole bands cross at liquid
nitrogen temperatures. They exhibit negligible magnetore-
sistance [1], as predicted from conventional theories [2].
By contrast, minute amounts of excess Ag or Se=Te—at
levels as small as 1 part in 10 000—lead to a huge and
linear magnetoresistance over a broad temperature range
[3–8]. The unusual linear dependence on magnetic field
down to 100 G indicates a particularly long length scale
associated with the underlying physics, while, at high field,
a nonsaturating response up to at least 0.5 MG exceeds by a
factor of 50 to 100 the expected cutoff where the product of
the cyclotron frequency and the scattering rate !  1 [9].
This remarkably robust linear magnetoresistive response
makes the silver chalcogenides promising candidates for
high field sensors. Missing at present, however, is experi-
mental evidence for the pertinent length scales of the
inhomogeneities that determine the unusual physics and
that are essential to the materials’ usefulness.
Abrikosov was the first to stress the importance of the
inhomogeneous distribution of the excess or deficient sil-
ver ions. In his effective medium theory of quantum linear
magnetoresistance [10], disorder and a linear dispersion
relation at band crossing [11] combine to produce a linear,
rather than a quadratic, magnetic field dependence for the
electrical conductivity. As pointed out by Parish and
Littlewood [12], fluctuations in the mobility are particu-
larly acute when the gap goes to zero and both positive and
negative values can be sampled. Their simulations of large
spatial conductivity fluctuations in strongly inhomogene-
ous semiconductors derive a linear magnetoresistance from
the Hall voltage picked up from macroscopically distorted
current paths caused by variations in the stoichiometry.
The spatial fluctuations in the conductivity are caused by
the random distribution of Ag ions, which may take the05=95(18)=186603(4)$23.00 18660form of highly conducting nanothreads or lamellae along
the grain boundaries of polycrystalline material [13]. The
distorted current paths seen in the simulations lead to the
emergence of a characteristic length scale that can be
associated directly with a magnetic field scale for the onset
of linear response.
The exact disposition of the excess or deficient Ag is
difficult to resolve via either scattering or imaging tech-
niques. In this Letter, we show that a systematic investiga-
tion of the longitudinal magnetoresistance of the silver
chalcogenides in magnetic fields up to 60 T can be used
to directly identify the physical length scale of interest.
Although the transverse magnetoresistance is always posi-
tive, a pronounced negative longitudinal magnetoresis-
tance, i.e., the reduction of the sample resistance in a
parallel magnetic field, emerges at high field. We observe
this effect over a wide temperature and composition range
for both n-type and p-type Ag2Se and Ag2Te. The appear-
ance of the negative magnetoresistance is due to an effect
analogous to current jetting [14,15], but is caused here by
conductivity fluctuations. Most significantly, its depen-
dence on sample thickness yields a characteristic length
scale set by the spatial inhomogeneity that is as large as
10 m.
High purity, stoichiometric Ag2Se was ground and
loaded into outgassed, fused silica ampoules inside a he-
lium glovebox, and appropriate weights of Ag or Se were
added to reach the desired compositions. The mixture was
baked at 50 K above the melting point (1170 K) and left to
cool for 24 h in a horizontal position. Polycrystalline
samples with typical dimensions 3:0 1:0 0:4 mm3
were cut perpendicular to the cylindrical axis to avoid
possible dopant variations due to temperature gradients
inside the furnace. Ag2Te polycrystals were prepared
in a similar manner, except that high purity elements of Ag
and Te (99.999%, Alfa Aesar) were directly used for melt
crystallization. We performed conventional four-probe re-3-1 © 2005 The American Physical Society
PRL 95, 186603 (2005) P H Y S I C A L R E V I E W L E T T E R S
week ending
28 OCTOBER 2005sistivity measurements T;H with electrical contacts
placed on the top, bottom, and sides of the samples to
measure the xx, xy, and xz components of the resistivity
tensor. It is important to note that the van der Pauw method,
with contacts placed on the corners of a square plate,
cannot be applied to the present situation because it re-
lies on a uniform current distribution to deconvolute the
tensor components of . We used a 14=16 T supercon-
ducting magnet for low field measurements and the 60 T
short-pulsed magnet at Los Alamos National Laboratory
for the high field response. The latter provides 16 ms long
field pulses from the discharge of a 1 MJ capacitor bank
and, when combined with the National High Magnetic
Field Laboratory-designed synchronous digital lock-in am-
plifier with response times much shorter than commercial
lock-ins, can provide low-noise data during each 16 ms
shot.
We contrast in the two panels of Fig. 1 the normalized
magnetoresistance H=0 of n-type Ag2Se 
0:0001 under transverse and longitudinal magnetic fields
0  H  55 T. The longitudinal and transverse magneto-
resistance differ in both magnitude and functional depen-
dence. The longitudinal response is an order of magnitude
smaller than its transverse counterpart and crosses through
zero to become negative at high field. Both quantitiesFIG. 1 (color online). Normalized magnetoresistance of n-type
Ag2Se under (a) transverse and (b) longitudinal magnetic field
at a series of temperatures. The transverse response is large,
positive, and essentially linear with magnetic field, while a
crossover to linear but negative magnetoresistance occurs in
the longitudinal configuration.
18660refuse to saturate at the highest fields and assume an
essentially linear dependence on field (except at the lowest
temperatures where quantum oscillations appear to domi-
nate). Similar negative longitudinal magnetoresistance was
observed on all Ag2Se samples investigated ( varying
from0:002 to 0.001). Although the negative longitudinal
magnetoresistance is consistently larger in p-type material,
there is no systematic dependence on .
The pronounced negative component in the longitudinal
magnetoresistance is consistent with the picture of inho-
mogeneous conductance. In such a model, the combination
of inhomogeneities and high magnetic fields leads to cur-
rent paths that behave in a counterintuitive manner, at
times flowing perpendicular to both the applied voltage
and the magnetic field [12]. This provides a linear contri-
bution to the transverse magnetoresistance due to Hall
resistances associated with the perpendicular flows.
In the present longitudinal geometry, where the mag-
netic field is parallel to the applied current, the perpen-
dicular current will flow across the thickness of the sample,
bending the equipotential lines. The voltage drop across
the length of the sample will be reduced and a negative
magnetoresistance can emerge. This is analogous to the
effect known as current jetting [14,15], differing only in the
fact that here it is a manifestation of inhomogeneous con-
ductivity rather than simple geometric effects. The maxi-
mum in the negative magnetoresistance at T  77 K
corresponds to the crossover from intrinsic to extrinsic
semiconducting behavior. At this point the distortions in
the current should be strongest as the fluctuations in the
conductivity can veer from positive to negative.
The inset of Fig. 2 shows a schematic visualization of
distorted current paths in the presence of a longitudinal
magnetic field. In addition to the negative magnetoresis-FIG. 2 (color online). A voltage drop across the thickness of
p-type Ag2Se develops under a longitudinal magnetic field,
revealing distorted current flows across the sample as the source
of the negative magnetoresistance (illustrated in the inset).
3-2
FIG. 3 (color online). Normalized longitudinal magnetoresis-
tance vs sample thickness for Ag2Se at T  30 K. The nega-
tive component of the magnetoresistance decreases with
thinning, vanishing by 15 m. This sets the inhomogeneity
length scale in the problem. Inset: The transverse magnetoresis-
tance is independent of sample thickness, shown here for 15 and
400 m.
PRL 95, 186603 (2005) P H Y S I C A L R E V I E W L E T T E R S
week ending
28 OCTOBER 2005tance, we expect a voltage drop across the sample thickness
for the depicted current pattern. In other words, in contrast
to what is observed in ordinary polycrystalline semicon-
ductors, a nonzero tensor component xzH, known as the
transverse-longitudinal coupling, should become apparent
due to the inhomogeneous nature of our system. We plot in
the main panel of Fig. 2 the voltage drop across a p-type
sample as a function of magnetic field for 0  H  14 T.
A conventional six-probe configuration was used to allow
for later correction [xzH  xz0
zzH
zz0
] of any pos-
sible lead misalignment. The observed voltage drop pro-
vides clear evidence of the predicted current distortion in
our system. The sign and functional form varies from
sample to sample depending on the direction of the bend
in the current and the location of the voltage leads. Again,
the maximum voltage drop found at T  77 K indicates
that the most crooked current paths occur near band
crossing.
The formation of spatial inhomogeneities in the silver
chalcogenides is closely related to the clustering of Ag ions
at lattice defects or grain boundaries when quenched from
the high temperature  phase to the semiconducting 
phase with lower Ag solubility [13]. The ultimate unit
size of these clustering networks sets a lower limit for
the effective length scale of the current distortion, in the
sense that the effect of spatial randomness would be com-
pletely suppressed if the size of the system becomes com-
parable to that length scale in one dimension. As a result,
the negative longitudinal magnetoresistance, a direct mani-
festation of inhomogeneity along the sample thickness,
should diminish in thinner crystals. We investigate the
thickness dependence of the negative longitudinal magne-
toresistance by mechanically shaving the same p-type
sample used for the xzH measurement of Fig. 2. We
plot in Fig. 3 the evolution of the longitudinal magnetore-
sistance at various sample thicknesses: 400, 50, and
15 m. The negative component decreases as the sample
grows thinner and eventually disappears at 15 m, indi-
cating a characteristic length scale of order 10 m. At the
same time, no change was found in the transverse magne-
toresistance at all thicknesses investigated, indicating that
the strain-induced magnetoresistance associated with
grinding is negligible, and in agreement with the claim
that the linear transverse effect results from the current
distortion within a 2D plane [12].
Experimental visualization of Ag clustering was pro-
vided recently by von Kreutzbruck et al., who report
micron scale silver lamellae in Ag2Se with extremely
high silver excess,  	 1:3 [16]. Such high doping levels
give rise to a relatively large diameter, interconnected
silver clusters, and an onset of percolation phenomena,
well away from the regime covered in this study, but the
silver threads observed with high resolution TEM are
consistent with the tens of microns length scales derived
from the longitudinal magnetoresistance.18660In clever experiments, Solin and co-workers have
crafted composite materials where current deflections
around highly conducting impurities combine a geometri-
cal magnetoresistance with the intrinsic physical magne-
toresistance of Hg1xCdxTe to greatly enhance the
response [17]. Such an approach points to the general
promise of disordered media, but differs from the silver
chalcogenides in underlying mechanism (dependence on
the intrinsic physical magnetoresistance of the semicon-
ductor Hg1xCdxTe), functional form (quadratic with
field), saturation at relatively low fields (
1 T), and mac-
roscopic homogeneity of the current flow.
To this point, we have concentrated on silver selenide as
the representative material for large, linear, and nonsatu-
rating magnetoresistance. We complement our study of
Ag2Se with a systematic investigation of the longitudi-
nal magnetoresistance in Ag2Te. As shown in Fig. 4,
even larger negative responses with smaller crossover
fields were found for n- and p-type Ag2Te ( 
0:0005). The measurement of the xzH components
also shows behavior similar to that depicted in Fig. 2.
The negative longitudinal magnetoresistance universally
observed in doped silver chalcogenides appears to be well
described by spatial fluctuations in the conductivity. The
associated length scale, which dominates the transport
mechanism and sets the effective dimensionality, was de-
termined to be approximately 10 m by investigating the
dependence of the magnetoresistive response on sample
thickness. The surprisingly large length scale opens the
gate to artificial fabrication of conducting networks with3-3
FIG. 4 (color online). Normalized longitudinal magnetoresis-
tance as a function of magnetic field for (a) electron- and
(b) hole-doped silver telluride at a series of temperatures.
Pronounced negative magnetoresistance emerges in both limits.
PRL 95, 186603 (2005) P H Y S I C A L R E V I E W L E T T E R S
week ending
28 OCTOBER 2005micron scale unit size for enhanced magnetoresistive ef-
fects. The narrow-gap semiconductor, InSb, may be a
promising candidate for such devices. A galvanomagnetic
study of n-type indium antimonide published almost 50
years ago [18] shows negative longitudinal magnetoresis-
tance of the type we observe in the silver chalcogenides
(but absent in modern epitaxial InSb wafers). Harnessing
the current jetting effects due to inhomogeneities could
linearize and extend the range of the material’s transverse
magnetoresistive response.
The authors are indebted to A. Husmann, M. Parish,
and P. Littlewood for illuminating discussions, to M.-L.
Saboungi for her insights into the silver chalcogenides,
and to Q. Guo for assistance with sample preparation.
The work at the University of Chicago was supported18660by U.S. Department of Energy Grant No. DE-FG02-
99ER45789. J. H. acknowledges support from the
Consortium for Nanoscience Research. The work at the
National High Magnetic Field Laboratory was carried out
under the auspices of the National Science Foundation, the
state of Florida, and the U.S. Department of Energy.3-4[1] P. Junod, Helv. Phys. Acta 32, 567 (1959).
[2] See, for example, C. Kittel, Quantum Theory of Solids
(Wiley, New York, 1963).
[3] R. Xu, A. Husmann, T. F. Rosenbaum, M.-L. Saboungi,
J. E. Enderby, and P. B. Littlewood, Nature (London) 390,
57 (1997).
[4] I. S. Chuprakov and K. H. Dahmen, Appl. Phys. Lett. 72,
2165 (1998).
[5] Z. Ogorelec, A. Hamzic, and M. Basletic, Europhys. Lett.
46, 56 (1999).
[6] H. S. Schnyders, M.-L. Saboungi, and T. F. Rosenbaum,
Appl. Phys. Lett. 76, 1710 (2000).
[7] S. S. Manoharan, S. J. Prasanna, D. Elefant-Kiwitz, and
C. M. Schneider, Phys. Rev. B 63, 212405 (2001).
[8] The addition of excess Ag or Se=Te does not alter the
nonmagnetic nature of the material. All samples reported
here have a dc magnetic susceptibility less than 5
109 cm3 at T  4:2 K.
[9] A. Husmann, J. B. Betts, G. S. Boebinger, A. Migliori,
T. F. Rosenbaum, and M.-L. Saboungi, Nature (London)
417, 421 (2002).
[10] A. A. Abrikosov, Phys. Rev. B 58, 2788 (1998); Europhys.
Lett. 49, 789 (2000).
[11] M. Lee, T. F. Rosenbaum, M.-L. Saboungi, and H. S.
Schnyders, Phys. Rev. Lett. 88, 066602 (2002).
[12] M. M. Parish and P. B. Littlewood, Nature (London) 426,
162 (2003).
[13] Y. Kumashiro, T. Ohachi, and I. Taniguchi, Solid State
Ionics 86–88, 761 (1996).
[14] A. B. Pippard, Magnetoresistance in Metals (Cambridge
University Press, Cambridge, 1989).
[15] Y. Ueda and T. Kino, J. Phys. Soc. Jpn. 48, 1601 (1980).
[16] M. von Kreutzbruck, B. Mogwitz, F. Gruhl, L. Kienle,
C. Korte, and J. Janek, Appl. Phys. Lett. 86, 072102
(2005).
[17] S. A. Solin and T. Thio, Appl. Phys. Lett. 72, 3497 (1998);
S. A. Solin, T. Thio, D. R. Hines, and J. J. Heremans,
Science 289, 1530 (2000).
[18] H. P. R. Frederikse and W. R. Hosler, Phys. Rev. 108, 1136
(1957).


English

Current jets, disorder, and linear magnetoresistance in the silver chalcogenides

PRL 95, 186603 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending28 OCTOBER 2005Current Jets, Disorder, and Linear Magnetoresistance in the Silver Chalcogenides
Jingshi Hu and T. F. Rosenbaum
The James Franck Institute and Department of Physics, The University of Chicago, Chicago, Illinois 60637, USA
J. B. Betts
National High Magnetic Field Laboratory, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
(Received 26 May 2005; published 28 October 2005)0031-9007=The inhomogeneous distribution of excess or deficient silver atoms lies behind the large and linear
transverse magnetoresistance displayed by Ag2Se and Ag2Te, introducing spatial conductivity
fluctuations with length scales independent of the cyclotron radius. We report a negative, nonsaturating
longitudinal magnetoresistance up to at least 60 T, which becomes most negative where the bands cross
and the effect of conductivity fluctuations is most acute. Thinning samples down to 10 m suppresses the
negative response, revealing the essential length scale in the problem and paving the way for designer
magnetoresistive devices.
DOI: 10.1103/PhysRevLett.95.186603 PACS numbers: 72.20.My, 72.15.Gd, 72.80.JcThe silver chalcogenides provide a striking example of
the benefits of imperfection. Perfectly stoichiometric
Ag2Se and Ag2Te are nonmagnetic, narrow-gap semicon-
ductors whose electron and hole bands cross at liquid
nitrogen temperatures. They exhibit negligible magnetore-
sistance [1], as predicted from conventional theories [2].
By contrast, minute amounts of excess Ag or Se=Te—at
levels as small as 1 part in 10 000—lead to a huge and
linear magnetoresistance over a broad temperature range
[3–8]. The unusual linear dependence on magnetic field
down to 100 G indicates a particularly long length scale
associated with the underlying physics, while, at high field,
a nonsaturating response up to at least 0.5 MG exceeds by a
factor of 50 to 100 the expected cutoff where the product of
the cyclotron frequency and the scattering rate !  1 [9].
This remarkably robust linear magnetoresistive response
makes the silver chalcogenides promising candidates for
high field sensors. Missing at present, however, is experi-
mental evidence for the pertinent length scales of the
inhomogeneities that determine the unusual physics and
that are essential to the materials’ usefulness.
Abrikosov was the first to stress the importance of the
inhomogeneous distribution of the excess or deficient sil-
ver ions. In his effective medium theory of quantum linear
magnetoresistance [10], disorder and a linear dispersion
relation at band crossing [11] combine to produce a linear,
rather than a quadratic, magnetic field dependence for the
electrical conductivity. As pointed out by Parish and
Littlewood [12], fluctuations in the mobility are particu-
larly acute when the gap goes to zero and both positive and
negative values can be sampled. Their simulations of large
spatial conductivity fluctuations in strongly inhomogene-
ous semiconductors derive a linear magnetoresistance from
the Hall voltage picked up from macroscopically distorted
current paths caused by variations in the stoichiometry.
The spatial fluctuations in the conductivity are caused by
the random distribution of Ag ions, which may take the05=95(18)=186603(4)$23.00 18660form of highly conducting nanothreads or lamellae along
the grain boundaries of polycrystalline material [13]. The
distorted current paths seen in the simulations lead to the
emergence of a characteristic length scale that can be
associated directly with a magnetic field scale for the onset
of linear response.
The exact disposition of the excess or deficient Ag is
difficult to resolve via either scattering or imaging tech-
niques. In this Letter, we show that a systematic investiga-
tion of the longitudinal magnetoresistance of the silver
chalcogenides in magnetic fields up to 60 T can be used
to directly identify the physical length scale of interest.
Although the transverse magnetoresistance is always posi-
tive, a pronounced negative longitudinal magnetoresis-
tance, i.e., the reduction of the sample resistance in a
parallel magnetic field, emerges at high field. We observe
this effect over a wide temperature and composition range
for both n-type and p-type Ag2Se and Ag2Te. The appear-
ance of the negative magnetoresistance is due to an effect
analogous to current jetting [14,15], but is caused here by
conductivity fluctuations. Most significantly, its depen-
dence on sample thickness yields a characteristic length
scale set by the spatial inhomogeneity that is as large as
10 m.
High purity, stoichiometric Ag2Se was ground and
loaded into outgassed, fused silica ampoules inside a he-
lium glovebox, and appropriate weights of Ag or Se were
added to reach the desired compositions. The mixture was
baked at 50 K above the melting point (1170 K) and left to
cool for 24 h in a horizontal position. Polycrystalline
samples with typical dimensions 3:0 1:0 0:4 mm3
were cut perpendicular to the cylindrical axis to avoid
possible dopant variations due to temperature gradients
inside the furnace. Ag2Te polycrystals were prepared
in a similar manner, except that high purity elements of Ag
and Te (99.999%, Alfa Aesar) were directly used for melt
crystallization. We performed conventional four-probe re-3-1 © 2005 The American Physical Society
PRL 95, 186603 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending28 OCTOBER 2005
sistivity measurements T;H with electrical contacts
placed on the top, bottom, and sides of the samples to
measure the xx, xy, and xz components of the resistivity
tensor. It is important to note that the van der Pauw method,
with contacts placed on the corners of a square plate,
cannot be applied to the present situation because it re-
lies on a uniform current distribution to deconvolute the
tensor components of . We used a 14=16 T supercon-
ducting magnet for low field measurements and the 60 T
short-pulsed magnet at Los Alamos National Laboratory
for the high field response. The latter provides 16 ms long
field pulses from the discharge of a 1 MJ capacitor bank
and, when combined with the National High Magnetic
Field Laboratory-designed synchronous digital lock-in am-
plifier with response times much shorter than commercial
lock-ins, can provide low-noise data during each 16 ms
shot.
We contrast in the two panels of Fig. 1 the normalized
magnetoresistance H=0 of n-type Ag2Se 
0:0001 under transverse and longitudinal magnetic fields
0  H  55 T. The longitudinal and transverse magneto-
resistance differ in both magnitude and functional depen-
dence. The longitudinal response is an order of magnitude
smaller than its transverse counterpart and crosses through
zero to become negative at high field. Both quantitiesFIG. 1 (color online). Normalized magnetoresistance of n-type
Ag2Se under (a) transverse and (b) longitudinal magnetic field
at a series of temperatures. The transverse response is large,
positive, and essentially linear with magnetic field, while a
crossover to linear but negative magnetoresistance occurs in
the longitudinal configuration.
18660refuse to saturate at the highest fields and assume an
essentially linear dependence on field (except at the lowest
temperatures where quantum oscillations appear to domi-
nate). Similar negative longitudinal magnetoresistance was
observed on all Ag2Se samples investigated ( varying
from0:002 to 0.001). Although the negative longitudinal
magnetoresistance is consistently larger in p-type material,
there is no systematic dependence on .
The pronounced negative component in the longitudinal
magnetoresistance is consistent with the picture of inho-
mogeneous conductance. In such a model, the combination
of inhomogeneities and high magnetic fields leads to cur-
rent paths that behave in a counterintuitive manner, at
times flowing perpendicular to both the applied voltage
and the magnetic field [12]. This provides a linear contri-
bution to the transverse magnetoresistance due to Hall
resistances associated with the perpendicular flows.
In the present longitudinal geometry, where the mag-
netic field is parallel to the applied current, the perpen-
dicular current will flow across the thickness of the sample,
bending the equipotential lines. The voltage drop across
the length of the sample will be reduced and a negative
magnetoresistance can emerge. This is analogous to the
effect known as current jetting [14,15], differing only in the
fact that here it is a manifestation of inhomogeneous con-
ductivity rather than simple geometric effects. The maxi-
mum in the negative magnetoresistance at T  77 K
corresponds to the crossover from intrinsic to extrinsic
semiconducting behavior. At this point the distortions in
the current should be strongest as the fluctuations in the
conductivity can veer from positive to negative.
The inset of Fig. 2 shows a schematic visualization of
distorted current paths in the presence of a longitudinal
magnetic field. In addition to the negative magnetoresis-FIG. 2 (color online). A voltage drop across the thickness of
p-type Ag2Se develops under a longitudinal magnetic field,
revealing distorted current flows across the sample as the source
of the negative magnetoresistance (illustrated in the inset).
3-2
FIG. 3 (color online). Normalized longitudinal magnetoresis-
tance vs sample thickness for Ag2Se at T  30 K. The nega-
tive component of the magnetoresistance decreases with
thinning, vanishing by 15 m. This sets the inhomogeneity
length scale in the problem. Inset: The transverse magnetoresis-
tance is independent of sample thickness, shown here for 15 and
400 m.
PRL 95, 186603 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending28 OCTOBER 2005
tance, we expect a voltage drop across the sample thickness
for the depicted current pattern. In other words, in contrast
to what is observed in ordinary polycrystalline semicon-
ductors, a nonzero tensor component xzH, known as the
transverse-longitudinal coupling, should become apparent
due to the inhomogeneous nature of our system. We plot in
the main panel of Fig. 2 the voltage drop across a p-type
sample as a function of magnetic field for 0  H  14 T.
A conventional six-probe configuration was used to allow
for later correction [xzH  xz0 zzHzz0 ] of any pos-
sible lead misalignment. The observed voltage drop pro-
vides clear evidence of the predicted current distortion in
our system. The sign and functional form varies from
sample to sample depending on the direction of the bend
in the current and the location of the voltage leads. Again,
the maximum voltage drop found at T  77 K indicates
that the most crooked current paths occur near band
crossing.
The formation of spatial inhomogeneities in the silver
chalcogenides is closely related to the clustering of Ag ions
at lattice defects or grain boundaries when quenched from
the high temperature  phase to the semiconducting 
phase with lower Ag solubility [13]. The ultimate unit
size of these clustering networks sets a lower limit for
the effective length scale of the current distortion, in the
sense that the effect of spatial randomness would be com-
pletely suppressed if the size of the system becomes com-
parable to that length scale in one dimension. As a result,
the negative longitudinal magnetoresistance, a direct mani-
festation of inhomogeneity along the sample thickness,
should diminish in thinner crystals. We investigate the
thickness dependence of the negative longitudinal magne-
toresistance by mechanically shaving the same p-type
sample used for the xzH measurement of Fig. 2. We
plot in Fig. 3 the evolution of the longitudinal magnetore-
sistance at various sample thicknesses: 400, 50, and
15 m. The negative component decreases as the sample
grows thinner and eventually disappears at 15 m, indi-
cating a characteristic length scale of order 10 m. At the
same time, no change was found in the transverse magne-
toresistance at all thicknesses investigated, indicating that
the strain-induced magnetoresistance associated with
grinding is negligible, and in agreement with the claim
that the linear transverse effect results from the current
distortion within a 2D plane [12].
Experimental visualization of Ag clustering was pro-
vided recently by von Kreutzbruck et al., who report
micron scale silver lamellae in Ag2Se with extremely
high silver excess,  	 1:3 [16]. Such high doping levels
give rise to a relatively large diameter, interconnected
silver clusters, and an onset of percolation phenomena,
well away from the regime covered in this study, but the
silver threads observed with high resolution TEM are
consistent with the tens of microns length scales derived
from the longitudinal magnetoresistance.18660In clever experiments, Solin and co-workers have
crafted composite materials where current deflections
around highly conducting impurities combine a geometri-
cal magnetoresistance with the intrinsic physical magne-
toresistance of Hg1xCdxTe to greatly enhance the
response [17]. Such an approach points to the general
promise of disordered media, but differs from the silver
chalcogenides in underlying mechanism (dependence on
the intrinsic physical magnetoresistance of the semicon-
ductor Hg1xCdxTe), functional form (quadratic with
field), saturation at relatively low fields (
1 T), and mac-
roscopic homogeneity of the current flow.
To this point, we have concentrated on silver selenide as
the representative material for large, linear, and nonsatu-
rating magnetoresistance. We complement our study of
Ag2Se with a systematic investigation of the longitudi-
nal magnetoresistance in Ag2Te. As shown in Fig. 4,
even larger negative responses with smaller crossover
fields were found for n- and p-type Ag2Te ( 
0:0005). The measurement of the xzH components
also shows behavior similar to that depicted in Fig. 2.
The negative longitudinal magnetoresistance universally
observed in doped silver chalcogenides appears to be well
described by spatial fluctuations in the conductivity. The
associated length scale, which dominates the transport
mechanism and sets the effective dimensionality, was de-
termined to be approximately 10 m by investigating the
dependence of the magnetoresistive response on sample
thickness. The surprisingly large length scale opens the
gate to artificial fabrication of conducting networks with3-3
FIG. 4 (color online). Normalized longitudinal magnetoresis-
tance as a function of magnetic field for (a) electron- and
(b) hole-doped silver telluride at a series of temperatures.
Pronounced negative magnetoresistance emerges in both limits.
PRL 95, 186603 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending28 OCTOBER 2005micron scale unit size for enhanced magnetoresistive ef-
fects. The narrow-gap semiconductor, InSb, may be a
promising candidate for such devices. A galvanomagnetic
study of n-type indium antimonide published almost 50
years ago [18] shows negative longitudinal magnetoresis-
tance of the type we observe in the silver chalcogenides
(but absent in modern epitaxial InSb wafers). Harnessing
the current jetting effects due to inhomogeneities could
linearize and extend the range of the material’s transverse
magnetoresistive response.
The authors are indebted to A. Husmann, M. Parish,
and P. Littlewood for illuminating discussions, to M.-L.
Saboungi for her insights into the silver chalcogenides,
and to Q. Guo for assistance with sample preparation.
The work at the University of Chicago was supported18660by U.S. Department of Energy Grant No. DE-FG02-
99ER45789. J. H. acknowledges support from the
Consortium for Nanoscience Research. The work at the
National High Magnetic Field Laboratory was carried out
under the auspices of the National Science Foundation, the
state of Florida, and the U.S. Department of Energy.3-4[1] P. Junod, Helv. Phys. Acta 32, 567 (1959).
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Current jets, disorder, and linear magnetoresistance in the silver chalcogenides

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