Electrical transport properties of semimetallic GdX single crystals (X=P, As, Sb, and Bi)

Article abstract of DOI:10.1103/PhysRevB.54.10483

The large single crystals of stoichiometric and nonstoichiometric Gd monopnictides GdX (X=P, As, Sb, and Bi) are grown by the mineralization method (for X=P and As) and Bridgman method (for X=Sb and Bi). A systematic investigation of the transport properties of GdX single crystals is presented. We report on measurements of the electric resistivity p(T), magnetoresistance p(H), and Hall effect performed on the stoichiometric and nonstoichiometric samples at temperatures between 1.6 and 300 K in magnetic fields up to 10 T. The stoichiometric samples behaved as the well-compensated semimetals that order antiferromagnetically at Neel temperatures TN=15.9 K for GdP, 18.7 K for GdAs, 23.4 K for GdSb, and 25.8 K for GdBi. The transverse magnetoresistance measured at low temperature follows a p(//)tx//2 law, and a larger positive ratio MRR =[p(H)- p(0)]/p(0) is observed at 10 T for the four stoichiometric samples. The temperature dependence of the resistivity can be explained by the d-f Coulomb exchange interaction at lower temperatures. The Halleffect measurements yield a carrier concentration ;i=2.1X1020 cm~3 for GdAs and ;i=4.2X1020 cm 3 for , GdSb, which are in a good agreement with the de Haas-van Alphen effect measurements. The nonstoichiometric samples showed some anomalies that could be explained qualitatively by the model of trapped magnetic polaren.

Full text of DOI:10.1103/PhysRevB.54.10483

PHYSICAL REVIEW B
VOLUME 54, NUMBER 15
15 OCTOBER 1996-I
Electrical transport properties of semimetallic GdX single crystals „X؍P, As, Sb, and Bi…
D. X. Li,* Y. Haga, H. Shida, and T. Suzuki
Department of Physics, Tohoku University, Sendai 980-77, Japan
Y. S. Kwon
Department of Physics, Suny Kyun Kwan University, Suwon, South Korea
͑Received 26 April 1996͒
The large single crystals of stoichiometric and nonstoichiometric Gd monopnictides GdX ͑XϭP, As, Sb, and
Bi͒ are grown by the mineralization method ͑for XϭP and As͒ and Bridgman method ͑for XϭSb and Bi͒. A
systematic investigation of the transport properties of GdX single crystals is presented. We report on measure-
ments of the electric resistivity ␳(T), magnetoresistance ␳(H), and Hall effect performed on the stoichiometric
and nonstoichiometric samples at temperatures between 1.6 and 300 K in magnetic fields up to 10 T. The
stoichiometric samples behaved as the well-compensated semimetals that order antiferromagnetically at N e´ el
temperatures T ϭ15.9 K for GdP, 18.7 K for GdAs, 23.4 K for GdSb, and 25.8 K for GdBi. The transverse
N
2
magnetoresistance measured at low temperature follows a ␳(H)ϰH law, and a larger positive ratio MRR
ϭ͓␳(H)Ϫ␳͑0͔͒/␳͑0͒ is observed at 10 T for the four stoichiometric samples. The temperature dependence of
the resistivity can be explained by the dϪf Coulomb exchange interaction at lower temperatures. The Hall-
2
0
Ϫ3
20
Ϫ3
effect measurements yield a carrier concentration nϭ2.1ϫ10 cm for GdAs and nϭ4.2ϫ10 cm for
GdSb, which are in a good agreement with the de Haas–van Alphen effect measurements. The nonstoichio-
metric samples showed some anomalies that could be explained qualitatively by the model of trapped magnetic
polaron. ͓S0163-1829͑96͒00239-1͔
I. INTRODUCTION
more, the 4f level in GdX is sufficient below the Fermi
energy, the usual type of d-f or p-f mixing effect is believed
to be much smaller, and thus the Ruderman-Kittel-Kasuya-
Yosida ͑RKKY͒ type d-f Coulomb exchange interaction is
the main part of the indirect exchange interaction between 4f
ions ͑where ‘‘d’’ represents the conduction electrons, i.e., 5d
or 6s electrons in rare-earth atoms, ‘‘p’’ represents the
Anomalous physical properties observed in the rare-earth
monopnictides RX ͑XϭN, P, As, Sb, and Bi͒ have attracted
much attention in recent years,1–3 because they are the typi-
cal low carrier strongly correlated systems with simple
rocksalt crystal structure. In the rare-earth monopnictides,
the conduction band is formed by the 5d orbitals of the cat-
ion R and has its minimum at the X point of the Brilloun
zone, while the valence band formed mainly by the anionic
np state of X ͑nϭ2, 3, 4, 5, and 6 for XϭN, P, As, Sb, and
Bi, respectively͒ has its maximum at the center of the Bril-
loun zone. A weak overlap between the bottom of the con-
duction band and the top of the valence band is sufficient to
make semimetals with a low carrier concentration out of
these materials. In most cases, the direct exchange interac-
tion between 4f ions in RX systems is weak due to the
majority of 4f electrons that lie well inside the 5s and 5p
closed shells and are well screened by them. Thus, the indi-
rect exchange interaction between 4f ions must be the main
mechanism, and charge carriers act as the intermediary of
this indirect exchange interaction. In the various RX com-
pounds, the indirect exchange interaction leads to different
magnetic and transport properties, depending on the free-
carrier concentration ͑the overlap between valence band and
conduction band͒, the position of 4f level as well as its split-
ting form in crystalline electric field ͑CEF͒. Among the RX
systems, Gd monopnictides are the most simple series, be-
cause Gd is located in the center of the series of the rare-
6
p-band holes of pnictogen͒. Therefore, GdX are the conve-
nient referent systems for other rare-earth monopnictides and
the convenient referent systems for the Eu monochalco-
7
–9
genides EuXP ͑X ϭO, S, Se, and Te͒. The studies
of
P
magnetic properties on powder compounds of GdP, GdAs,
GdSb, and GdBi suggest that they are type-II antiferromag-
nets below N e´ el temperatures T . However, to grow high-
N
quality single crystal of GdX is very difficult due to the high
weld point and high vapor pressure, and thus only a few
studies on the electronic-transport properties of Gd monop-
nictides have been done in the past in a nonsystematic way.
Moreover, their intrinsic electronic structure and mechanism
of conductivity are still a puzzle.
In order to study the magnetic and electronic behavior of
Gd monopnictides in a systematic way, recently, we have
succeeded in growing large stoichiometric single crystals of
GdX ͑XϭP, As, Sb, and Bi͒ and nonstoichiometric single
crystals of GdAs and GdP, and carefully measured the fun-
damental physical properties. Some magnetic properties have
been reported in our recent paper.10 In this paper, we will
present the magnetoresistance, electrical resistivity, and Hall-
effect measurements of these samples.
4
,5
3ϩ
earth metals in the Periodic Table of the elements, the Gd
II. SAMPLE PREPARATION AND EXPERIMENT
7
ion appearing in GdX has a 4f configuration and is an
7
S-state ion with spin 2 and no orbital momentum, the CEF
The GdX single crystals are grown by the mineralization
method ͑for XϭP and As͒ and Bridgman method ͑for XϭSb
effects in GdX are considered to be fairly weak. Further-
0
163-1829/96/54͑15͒/10483͑9͒/$10.00
54
10 483
© 1996 The American Physical Society

1
0 484
D. X. LI, Y. HAGA, H. SHIDA, T. SUZUKI, AND Y. S. KWON
54
TABLE I. Lattice constant a, N e´ el temperature T , residual resistivity ␳͑T→0͒, RRRϭ␳͑Tϭ300 K͒/␳͑T→0͒, MRRϭ͓␳͑Hϭ10 T͒
N
Ϫ␳͑Hϭ0͔͒/␳͑Hϭ0͒, carrier concentration n, size, and the preparing method of all the GdX single crystals.
␳
͑T→0͒
MRR
͑Tϭ1.7 K͒
n
Size
͑mm ͒
Preparing
methoda
Sample
a ͑Å͒
TN ͑K͒
͑␮⍀ cm͒
RRR
͑cmϪ3͒
3
GdP
GdAs
GdSb
GdBi
ns-GdP
ns-GdAs
5.709
5.864
6.219
6.295
5.717
5.895
15.9
18.7
23.4
25.8
14.4
17.2
8.5
5.2
0.7
0.6
46
10.0
16.7
85.0
53.3
2.4
0.8
8
3ϫ5ϫ5
5ϫ6ϫ8
␾14ϫ15
2ϫ2ϫ3
5ϫ5ϫ5
6ϫ8ϫ8
M
M
B
B
M
M
2.1ϫ102
4.2ϫ10
0
b
20
125
b
23
0.25
0.06
54
2.1
a
Mϭmineralization method, BϭBridgman method.
Tϭ4.2 K.
b
and Bi͒ in tungsten crucibles. The Gd metal of 99.9% purity
current I
ʈ
͓100͔ and the field H ͓010͔. The magnetic-field de-
ʈ
͑
turned into small flakes in a glovebox permeated with Ar
pendences of the transverse magnetoresistance measured at
fixed T are shown in Figs. 1͑a͒–1͑d͒ for the stoichiometric
samples GdP, GdAs, GdSb, and GdBi, respectively. The in-
sets in these figures show the plots of ͓␳(H)Ϫ␳͑0͔͒/␳͑0͒ vs
gas͒ and P, As, Sb, and Bi metals of 99.999% purity are
used. Because P and As are easy to evaporate at high tem-
perature, a prereaction of the constituent elements is first
carried out in a closed quartz ampoule at 900 °C for twelve
weeks ͑GdP͒ and 550 °C for six weeks ͑GdAs͒. The poly-
crystalline materials of GdP and GdAs obtained by the pre-
reaction are pressed into hard pellets at 720 °C and 1300 atm
using a glass capsule method. The hard pellets are then
sealed in cleared tungsten crucibles using an electron-beam
gun in vacuum. For GdSb and GdBi, the starting elements of
Gd and Sb or Bi are directly sealed in cleared tungsten cru-
cibles. The crucibles are slowly heated to above 2500 °C,
using a high-frequency induction furnace and kept at this
temperature for 72 h for GdP and GdAs, and 2 h for GdSb
and GdBi. Finally, the crucibles are drawn slowly at the rate
of 1.5 mm per hour. In this way, we obtained the stoichio-
metric and nonstoichiometric single crystals GdP, GdAs,
GdSb, and GdBi, with the dimensions shown in Table I. For
all the samples, the x-ray-diffraction patterns show a single
phase with NaCl structure. The room-temperature lattice
constants a of these samples are also listed in Table I. The
atomic ratios between Gd and pnictogen determined by
chemical analysis are 1:0.95Ϯ0.01 for the nonstoichiometric
GdAs ͑hereafter noted as ns-GdAs͒, 1:0.96Ϯ0.01 for the
nonstoichiometric GdP ͑hereafter noted as ns-GdP͒, and
2
H for the corresponding samples. At low temperature, the
transverse magnetoresistance increases with H in the inves-
tigated field range and follows approximately a ␳(H)ϰH2
law for all the four samples. The large positive rations of
͓
␳(H)Ϫ␳͑0͔͒/␳͑0͒, about 0.8, 8, 125, and 23 measured at 1.7
K ͑GdP, GdAs͒ and 4.2 K ͑GdSb, GdBi͒, are observed at 10
T for GdP, GdAs, GdSb, and GdBi, respectively ͑see Table
I͒. This indicates that the stoichiometric samples GdP, GdAs,
GdSb, and GdBi are the good compensated semimetallic
crystals of high-quality single, which are appropriate for the
de Haas–van Alphen ͑dHvA͒ effect measurements. In fact,
the clear dHvA oscillations have been observed for our sto-
ichiometric GdAs and GdSb samples at low temperature and
high applied field.11
␳(T,H) is also measured as a function of T at fixed H for
these samples. For instance, Figs. 2 and 3 illustrate the re-
sults of stoichiometric GdAs and GdSb. These data can be
used to calculate the carrier concentration and the mobilities
of electrons and holes as explained in Sec. IV.
2. Electrical resistivity
1
:1.00Ϯ0.01 for the four stoichiometric samples ͑hereafter
The electrical resistivity ␳(T) is measured between 1.6
and 300 K for the stoichiometric GdP, GdAs, GdSb, and
GdBi samples, as shown in Figs. 4 and 5. The four curves are
qualitatively similar. At low temperatures, the resistivity first
increases rapidly with increasing temperature, and shows a
noted as GdX with XϭP, As, Sb, and Bi͒.
The samples used for our experiments are cleaved from
the large single crystals. The electrical resistivity and mag-
netoresistance are measured by the standard four-probe
method. Instead, for Hall-coefficient measurements, a four-
contact geometry is used with the two voltage contacts per-
pendicular to the current. The magnetic field for Hall effect
and magnetoresistance measurements is provided by a super-
conducting magnet that enabled fields up to 10 T. The tem-
perature is obtained by an Au͑0.07%-Fe͒-Ag thermocouple
in the electrical resistivity, magnetoresistance, and Hall-
effect measurements.
kink at the Neel temperature T . At higher temperatures the
´
N
resistivity becomes linear in the temperature. The Neel tem-
peratures, as determined from the derivative ץ␳/ץT, are found
to be 15.9, 18.7, 23.4, and 25.8 K for GdP, GdAs, GdSb, and
GdBi, respectively, in good agreement with our specific-heat
measurements and magnetic-susceptibility data.12 The values
of residual resistivity ␳͑T→0͒ and ratio RRRϭ␳͑Tϭ300 K͒/
´
␳
͑T→0͒ are 8.5 ␮⍀ cm and 10.0 for GdP, 5.2 ␮⍀ cm and
1
6.7 for GdAs, 0.7 ␮⍀ cm and 85.0 for GdSb, and 0.6
III. EXPERIMENTAL RESULTS
A. Stoichiometric samples
␮⍀ cm and 53.3 for GdBi. For GdP and GdAs the metal-like
linear ␳(T) behavior curves appears above T Ϸ85 and 70 K,
L
respectively. In the temperature range between TN and TL ,
1
. Magnetoresistance
␳(T) deviates from the linear behavior. For GdSb and GdBi,
Temperature and field dependences of the magnetoresis-
tance were measured in the transverse configuration with the
however, the linear ␳(T) behavior remains even when T is
decreased near to T , i.e., T ϷT . Note that Kaldis, von
N
L
N

54
ELECTRICAL TRANSPORT PROPERTIES OF . . .
10 485
FIG. 3. Temperature dependence of the resistivity of stoichio-
metric GdSb measured at magnetic fields of 0, 5, and 10 T.
tive magnetoresistance is observed only in our nonstoichio-
metric ns-GdP and nonstoichiometric ns-GdAs samples in
the present study.14 This negative magnetoresistance effect
will be discussed in Sec. IV in detail.
3. Hall effect
The Hall coefficients of GdX are measured in two ways:
͑
1͒ measuring the Hall coefficient R as a function of H at a
H
fixed T, and ͑2͒ measuring the Hall coefficient R as a func-
H
tion of T at a fixed H. Both measurements are carried out
with I
ʈ
͓100͔ and H ͓010͔. Figure 6 shows the field depen-
ʈ
dence of the Hall coefficient for the stoichiometric GdAs up
to 5 T measured at Tϭ4.2 K. The Hall coefficient is a nega-
Ϫ8
3
Ϫ1
tive constant ͑R ϭϪ0.8ϫ10 m C ͒. At a fixed T, a con-
H
stant Hall coefficient is also observed for the stoichiometric
GdSb up to 8 T.
Figures 7͑a͒ and 7͑b͒ show the temperature dependences
of Hall coefficient of the stoichiometric GdAs and GdSb,
FIG. 1. Magnetic-field dependence of the transverse magnetore-
sistance of stoichiometric GdX. The insets ͕Y axis: ͓␳(H)Ϫ␳͑0͔͒/
respectively. Below TN ,
͉
RH
͉
of GdSb first decreases rapidly
with increasing T, after crossing zero at about 3 K, RH
changes sign from negative to positive and RH increases,
2
2
2
␳͑0͒; X axis: H (T )͖ show the plot of ͓␳(H)Ϫ␳͑0͔͒/␳͑0͒ vs H .
͉
͉
Schulthess, and Wachter13 observed a sharp peak of resistiv-
ity at TN and a negative magnetoresistance at low magnetic
field for their GdP sample prepared by a high-temperature
vapor growth method. In fact, a similar behavior of a nega-
above TN , then, RH goes through a broad maximum around
30 K and, finally, ͉RH͉ decreases slowly up to 100 K. At TN ,
no evident peak appears. This is very different from the be-
havior of RH in Ce monopnictides.15 The similar features are
FIG. 2. Temperature dependence of the resistivity of stoichio-
metric GdAs measured at magnetic fields of 0 and 5 T.
FIG. 4. Temperature dependence of the electrical resistivity of
stoichiometric GdX measured at zero field.

10 486
D. X. LI, Y. HAGA, H. SHIDA, T. SUZUKI, AND Y. S. KWON
54
FIG. 5. Temperature dependence of the electrical resistivity of
stoichiometric GdX measured at zero field and near the N e´ el tem-
perature.
also observed in the stoichiometric GdAs sample in which
R is negative up to 100 K, and a broad peak appears around
H
4
0 K. However, the RH values are different between them,
FIG. 7. Temperature dependence of the Hall coefficient for sto-
ichiometric GdAs and GdSb measured at Hϭ5 and 2 T, respec-
tively.
which indicates that the carrier concentrations in GdSb and
GdAs are different. The measurements shown in Figs. 7͑a͒
and 7͑b͒ will be used to calculate the mobilities of electron
and hole and the carrier concentrations in Sec. IV.
the ␳-H curves measured at 1.6 and 4.2 K can be divided into
four different regions, corresponding to four field ranges. In
the first region ͑0ϽHϽ0.5 T͒, ␳ decreases very rapidly with
increasing H. The decrease becomes even more pronounced
at lower temperature. Note that in the same temperature and
field region the magnetization of ns-GdAs strongly
increases.1 In the second region ͑0.5ϽHϽ1 T͒, although the
magnetoresistance remains negative, ͉ץ␳/ץH͉ is considerably
smaller than that in the first region. Correspondingly, the
magnetization in this field range also increases nonlinearly
B. Nonstoichiometric samples
The Gd monopnictides GdX are typical ‘‘exchange domi-
nating’’ systems similar to the Eu chalcogenides EuXP .
Their magnetic and transport properties are strongly depen-
dent on the deviation of stoichiometry of the sample. For the
nonstoichiometric EuX , there are many experimental and
theoretical reports on the anomalous physical properties.
Similar anomalies are considered to exist also in the nonsto-
ichiometric GdX. However, there is no systematical study on
them so far due to the difficulty of preparing GdX single
crystals. Recently, the growth of large nonstoichiometric
GdX single crystals has permitted the accurate measurements
of the magnetic and transport properties for them.
0
P
16–18
but ͉ץM/ץH͉ is smaller than that in the field range below
10
0
.5 T. The third region ͑1ϽHϽ7 T͒ corresponds to
Figure 8 shows the field dependence of the transverse
magnetoresistance for the nonstoichiometric ns-GdAs
͑Gd:Asϭ1:0.95͒ and ns-GdP ͑Gd:Pϭ1:0.96͒. For ns-GdAs,
FIG. 6. Magnetic-field dependence of the Hall coefficient for
FIG. 8. Magnetic-field dependences of the transverse magne-
stoichiometric GdAs measured at Tϭ4.2 K.
toresistance of the nonstoichiometric ns-GdAs and ns-GdP.

54
ELECTRICAL TRANSPORT PROPERTIES OF . . .
10 487
above T ϭ85 K, it is observed in the stoichiometric GdP.
L
Furthermore, for the nonstoichiometric ns-GdP, a shallow
minimum appears near 50 K, and a broad peak is observed
around TN instead of the kink that appeared in other GdX.
IV. DISCUSSION
A. Stoichiometric samples
1. Magnetoresistance
Transverse magnetoresistance measurements enable us to
check the sample quality of a semimetal. In general, the ex-
istence of lattice defects, originating from impurities or non-
stoichiometry, broadens the Landau levels. When the energy
interval between the Landau levels is smaller than the level
broadening, then ͓␳(H)Ϫ␳͑0͔͒/␳͑0͒Ͻ1 and the dHvA or
Shubnikov–de Haas ͑SdH͒ signals cannot be observed.19 For
a semimetallic single crystal of high quality, in general, one
observes the dHvA or SdH signals, a large value for MRR
ϭ͓␳(H)Ϫ␳͑0͔͒/␳͑0͒, a large value for resistivity ratio RRR
2
ϭ␳͑Tϭ300 K͒/␳͑T→0͒, and an approximate ␳(H)ϰH be-
havior. The stoichiometric GdP, GdAs, GdSb, and GdBi
samples can be considered to be the high-quality semimetal-
lic single crystals with a good compensation between elec-
trons and holes, because they show the large ratios of MRR2
and RRR ͑see table I͒, and follow an approximate ␳(H)ϰH
law at low temperature. Furthermore, clear dHvA signals
have been observed for GdAs and GdSb at low temperature
and high applied field.11 Thus the experimental results ob-
tained for these samples represent intrinsic features of pure
Gd monopnictides.
FIG. 9. Comparison between the resistivities of stoichiometric
GdP ͑GdAs͒ and nonstoichiometric ns-GdP ͑ns-GdAs͒.
the canted phase. Although the differential magnetoresis-
tance becomes positive above 1 T, ץ␳/ץH is small and almost
H independent. ␳ shows a shallow minimum around 1 T. In
the fourth region ͑HϾ7 T͒, although it is still in the range of
the canted phase, the increase of ␳ becomes more pro-
nounced at larger fields. Similar behavior of a negative mag-
netoresistance at low fields is also present in the nonstoichio-
metric ns-GdP ͓Fig. 8͑b͔͒.
Figures 9͑a͒ and 9͑b͒ are the electrical resistivity mea-
surements for ns-GdAs and ns-GdP, respectively. The data of
the stoichiometric samples are also shown in the correspond-
ing figures for the convenience of comparison between them.
For GdAs, the two curves ͑of GdAs and ns-GdAs͒ have simi-
lar ␳(T) behavior but are different in detail. As T increases,
2. Electrical resistivity
For the stoichiometric samples, the temperature depen-
dence of the electrical resistivity, shown in Fig. 4, has two
contributions. The first one is the normal resistivity ␳p ,
caused by the electron-phonon scattering, which is linear in
T at high temperatures. This is observed at the temperatures
␳
(T) of both GdAs and ns-GdAs first increases rapidly at
above T Ϸ85, 70, 24, and 26 K for GdP, GdAs, GdSb, and
GdBi, respectively. The second contribution is the magnetic
L
lower temperatures, then goes through a kink at the N e´ el
temperature T , and finally increases linearly at higher tem-
part ␳ of the resistivity. We are interested in the second part
N
m
peratures. The N e´ el temperatures TN determined from the
derivative ץ␳/ץT are 18.7 K for GdAs and 17.2 K for ns-
GdAs, The main differences between GdAs and ns-GdAs
consist in the residual resistivity, much larger in the nonsto-
ichiometric sample, and in the linear slope observed in the
paramagnetic phase. At Tϭ1.6 K, the lowest temperature of
this measurement, a residual resistivity of 5.7 ␮⍀ cm was
found for the stoichiometric GdAs. In the nonstoichiometric
ns-GdAs, however, the corresponding value is 55 ␮⍀ cm,
ten times larger than the value found for GdAs. Furthermore,
of the resistivity. The formal treatment of magnetic scatter-
ing mechanism has been carried out by Andersen and his
collaborators.20 For T→0, there are no magnon and phonon
excited in the crystal and both resistivities ␳ and ␳ vanish.
p
m
In the absence of any impurity, the conduction electrons
propagate through the perfectly periodic lattice without being
scattered incoherently and the resistivities ␳ and ␳ vanish.
p
m
With increasing temperature, the conduction electrons are
scattered off the thermally activated magnons and, in a fer-
romagnetic or antiferromagnetic metal, ␳ contributes to the
m
the metal-like linear ␳(T) behavior appears above T ϭ70 K
for the stoichiometric GdAs, while it appears at much higher
electrical resistivity together with the contributions of the
electron-phonon and electron-impurity scattering. Spin wave
͑magnons͒ are the collective excitations of the aligned spins,
just like phonons in the case of lattice vibrations. The
electron-magnon scattering resistivity increases nonlinearly
with increasing temperature. It may be the origin of the non-
linear increase of the resistivity observed in stoichiometric
GdX samples below TN ͑Fig. 4͒.
L
temperatures, above T ϭ200 K, for the nonstoichiometric
L
ns-GdAs. Similar anomalous behaviors are also observed in
ns-GdP ͓Fig. 9͑b͔͒. At Tϭ1.6 K, a residual resistivity of 47
␮⍀ cm was found for the nonstoichiometric ns-GdP, much
larger than the corresponding value of 8.8 ␮⍀ cm found for
the stoichiometric GdP. While the metal-like linear ␳(T) be-
havior appears above T ϭ200 K for the nonstoichiometric
ns-GdP, much higher than the corresponding temperature,
As the temperature is increased further, more and more
magnetic ions can have their spin orientations affected by
L

10 488
D. X. LI, Y. HAGA, H. SHIDA, T. SUZUKI, AND Y. S. KWON
54
thermal fluctuations. In this case, the spin-disorder scattering
is usually considered to be the origin of the scattering pro-
cess. In the Gd monopnictides, this spin-disorder scattering
may be caused by the d-f Coulomb exchange interaction,
i.e., the RKKY exchange interaction, in which the conduc-
tion 5d electrons interact with the f electrons, considered to
2
1
be well localized. Above T the spins of the magnetic ions
N
become randomly oriented and the magnetic part of the re-
sistivity, originated from the scattering between the conduc-
tion electrons and the spin disorder, saturates and becomes
independent of temperature. Kasuya22 calculated the contri-
bution to the magnetic part of the electrical resistivity origi-
nated from this d-f scattering process for magnetic metals
with magnetic ions in the S state ͑quenched orbital moment͒.
Above the N e´ el temperature, this calculation yields a mag-
netic resistivity inversely proportional to the carrier concen-
tration n. Our experimental results show that the value of
⌬
␳ϭ␳(TϭT )Ϫ␳(Tϭ0)Ϸ␳m decreases when going from
N
GdP to GdBi ͑Fig. 5͒. For GdAs, the value of ⌬␳ is about
twice the value found for GdSb. This is in accord with Ka-
suya’s theory, since our dHvA effect measurements show
that the carrier concentration in GdSb is about double, as in
GdAs. Thus, in stoichiometric GdX, the magnetic resistivity
is considered to originate mainly from the RKKY interaction
between the conduction 5d electrons and the 4f electrons.
On the other hand, the critical scattering due to the short-
range order is also important and affects the resistivity near
the N e´ el temperature.23 The fact that the warping of the
FIG. 10. Temperature dependencies of the mobilities ͑of elec-
tron and hole͒ and carrier concentration for stoichiometric GdAs
and GdSb calculated with a simple two-band model.
R ͑H,T͒ϭ͑␮ Ϫ␮ ͒/͓ne͑␮ ϩ␮ ͔͒,
͑3͒
͑4͒
H
h
e
h
e
␳(T)ϳT curve begins at a temperature slightly above the
2
͓
␳͑H,T͒Ϫ␳͑0,T͔͒/␳͑0,T͒ϭ␮ ␮ H .
h e
N e´ el point may be attributed to the critical scattering pro-
cess. When the orientations of 4f spins become completely
In Figs. 10͑a͒ and 10͑b͒, we show the resulting temperature
dependences of the parameters n, ␮ , and ␮ for stoichio-
metric GdAs and GdSb, respectively.
disordered at TӷT , the resistivity originated from this d-f
N
e
h
Coulomb exchange interaction eventually saturates and be-
comes independent of the temperature, thus the ␳(T)ϳT
curves show the metal-like linear behavior.
For GdAs, the mobility of the electrons is larger than that
of the holes in the investigated temperature range. This ex-
plains the measured negative Hall coefficient. Around TN ,
both mobilities ␮ and ␮ show an abrupt change in agree-
3
. Hall effect
e
h
In magnetic materials, the Hall resistivity comprehends
two contributions and is often expressed as
ment with the above given analysis of the magnetic scatter-
ing around T . Below T , the difference between ␮ and ␮
N
N
e
h
rapidly increases with decreasing T, and the carrier concen-
tration n slowly increases. According to Eq. ͑3͒, the increase
␳HϭR BϩR M,
͑1͒
0
S
of n will cause a decrease of
͒ have shown that RH increases still at low temperatures.
Thus, the rapid increase of RH below TN seems to be de-
͉RH͉, but the experiments ͑Fig.
7
͉
͉
where B is the applied magnetic field, and R and R are
0
S
͉
͉
referred to as the normal and the anomalous Hall coeffi-
cients, respectively. The stoichiometric GdX samples are
considered to be the well-compensated semimetals. There-
fore, three parameters are necessary to reproduce the electri-
cal transport measurements. These are the concentration of
conduction electrons and valence holes (nϭn ϭn ) and the
termined by an increase in the mobilities. Furthermore, the
slopes of all three curves of ␮ , ␮ , and n ͓Fig. 10͑a͔͒ vary
e
h
slightly near 45 K. This may be related to the broad maxi-
mum of RH that has appeared around the same temperature.
The origin of the relatively rapid increase of n and decrease
of ␮ and ␮ at TϾ45 K is not known yet.
e
h
e
h
mobilities (␮ ,␮ ) of the conduction electrons and the va-
e
h
For GdSb, the important feature is ␮ Ͼ␮ below 3 K and
e
h
lence holes. As a rough calculation, we can ignore the
anomalous Hall effect in Eq. ͑1͒ ͑Ref. 24͒ and the field de-
pendence of the mobilities and carrier concentration,25 then
the charge-carrier density and the mobilities of the electrons
and holes of the stoichiometric GdX samples are given by
measuring the zero-field resistivity ␳͑0,T͒, the transverse
␮eϽ␮ above 3 K, which leads to the change of sign of R
h
H
at this temperature. The faster variation of RH at low tem-
peratures seems to be also due almost entirely to the increase
in the mobilities for the same reason as that in GdAs. It is
interesting to note that the mobility difference between elec-
trons and holes in GdSb is very small at low temperature,
which is the same characteristic as in LaSb. In GdAs, how-
ever, this difference is relatively larger, this may be due to
the fact that the Fermi wave vector in GdAs is smaller, thus
the mixing of the p hole in conduction band is weaker, and
magnetoresistance ␳(H,T), and the Hall coefficient R in a
H
simple two-band model as follows:2
6–28
1
/␳͑0,T͒ϭne͑␮ ϩ␮ ͒,
͑2͒
h
e

5
4
ELECTRICAL TRANSPORT PROPERTIES OF . . .
10 489
magnetic scattering or phonon scattering causes the larger
mobility for the conduction-band electrons. It is also inter-
esting to notice that using Eqs. ͑2͒–͑4͒ a similar calculation
has been made for stoichiometric YbAs by Oyamada et al.29
The results confirm that the mobility of holes in YbAs does
not increase below 80 K, while the mobility of electrons
increases strongly. This indicates that at low temperatures, in
YbAs, the resistivity is mainly determined by the mobility of
the electrons. However, our results for stoichiometric GdAs
electron system. Wigner31 first pointed out that in low carrier
limit the strong correlation effect can localize the electrons to
form the electron lattice, which is called Wigner lattice.
Ϫ2
Since electron’s Fermi kinetic energy is in proportion to r s
,
,
Ϫ1
and the Coulomb correlated energy is in proportion to r s
evidently in a strongly correlated electron system, if the elec-
tron concentration is low enough, the formation of Wigner
lattice is more favorable energetically.
However, so far, no real Wigner crystal has been ob-
͓Fig. 10͑a͔͒ show that the mobility of the holes clearly in-
II
served except a two-dimensional one on the surface of He .
creases below T , though it is always lower than the mobil-
N
This is partly because the crystal has a large zero-point vi-
bration and thus the Wigner crystal is very easily melted into
the liquid phase and partly because even a very small amount
of impurity changes it to the impurity localized state.15 Fur-
thermore, in low carrier magnetic system with some defects,
the carriers are more easily localized in the defects, and due
to the existence of magnetic exchange interaction, so-called
trapped magnetic polarons around the localized carriers are
expected to be formed instead of the Wigner lattice. In fact,
the clear trapped magnetic polaron effects have been ob-
served in nonstoichiometric semiconductor EuTe ͑Refs. 16
ity of the electrons. Furthermore, for GdSb, we obtained
nearly equal mobilities of holes and electrons ͓Fig. 10͑b͔͒.
Thus, even at low temperatures, the contribution of the holes
to resistivity cannot be ignored in GdX. However, for differ-
ent RX system, one should normally expect different elec-
tron and hole mobilities, as a consequence of the difference
in the 4f level, the CEF splitting, and the electronic struc-
ture.
At 4.2 K, we find a free-carrier concentration of
20
Ϫ3
nϭ␮ ϭ␮ ϭ2.07ϫ10 cm ϭ0.010͑4͒ per Gd atom for
e
h
GdAs, this value is larger than the carrier number of 0.0024
per La atom in LaAs and is smaller than the carrier number
of 0.014 per Yb atom in YbAs. Recently, we could observe
clear dHvA signals for GdAs. The carrier number deter-
mined from the dHvA effect measurements is 0.011 per Gd
atom that is consistent with the Hall-effect measurements.
For GdSb, the carrier number determined from our Hall-
18
and 17͒ and other Eu chalcogenides with very low carrier
1
9
Ϫ3
concentrations ͑Ͻ10
cm ͒, and some theoretically
studies32 have been carried out for this series. Gd monopnic-
tides GdX are also the ‘‘exchange dominating’’ systems
similar to Eu chalcogenides EuX as described in the intro-
P
duction. However, our Hall effect and dHvA effect measure-
ments show that the carrier concentrations in GdX are much
20
Ϫ3
effect measurements is nϭ␮ ϭ␮ ϭ4.2ϫ10
cm
e
h
ϭ0.025͑2͒ per Gd atom at 4.2 K, this value is also larger
than nϭ0.014͑5͒ per La atom in LaSb and smaller than
nϭ0.026͑2͒ per Tm atom in TmSb ͑single crystal of YbSb
has not been grown so far͒ and is consistent with the dHvA
measurements, which gives nϭ0.024͑5͒ per Gd atom.30
larger than that in EuX . Then, the questions one may ask
P
are if the trapped magnetic polaron state could also be
formed in semimetallic GdX? If it exists, what are the dif-
ferences in the origin of formation of trapped magnetic po-
larons between GdX and EuX ? In our recent experiments,
P
the anomalous magnetic and transport properties have been
found for the Gd-rich nonstoichiometric GdX, which can be
considered to be the evidences of formation of trapped mag-
netic polaron states in nonstoichiometric Gd monopnictides.
On the basis of the trapped magnetic polaron model, the
anomalous magnetic behaviors of Gd-rich GdX have been
explained in our recent paper.10 In the following, we will
discuss the trapped magnetic polaron effects on the transport
properties of the nonstoichiometric GdX.
B. Nonstoichiometric samples:
The trapped magnetic polaron effects
It is well known that the average Coulomb potential ͑P.E.͒
and the average kinetic energy ͑K.E.͒ of one electron in an
2
electron system approximate to e /(r a )
and
s
H
Ϫ1
2
m
(ប/r a ) , respectively. Thus, as an estimate in order of
magnitude, there is
s H
The formation of the trapped magnetic polarons in non-
stoichiometric Gd monopnictides has already been described
in our recent short communication.15 The basic idea is the
following. In considering the state of a conduction electron
in an antiferromagnetic crystal, it is customary to assume
that it does not disturb the magnetic ordering of the crystal.
In some cases, however ͑for example, in low carrier sys-
tems͒, an energetically more favorable state is achieved
when the electrons become localized and interact with the
surrounding magnetic ions.33 In the nonstoichiometric GdX
samples, some electrons, originating from the X vacancies,
are trapped by the Coulomb potential of the X vacancies. The
strong d-f exchange interaction between the trapped elec-
trons and the neighboring 4f spins aligns the 4f spins. A
trapped electron generates a region with ferromagnetic or-
dered 4f spins around itself and a trapped magnetic polaron
is formed. When the X vacancies move through the crystal,
the trapped magnetic polarons automatically also move.
P.E.
K.E.
ϭrS ,
͑5͒
2
2
here, a ϭប /(me ) is the Bohr radius, r is a parameter
without a unit used to judge the electron concentration, and
H
s
4
3
3
␲
(r a ) represents the average volume of an electron
s H
sphere.
In the high electron concentration limit, r Ӷ1, the Fermi
s
kinetic energy of an electron is more important than the Cou-
lomb potential. Thus it is an extended state electron, and can
move through the lattice. Such an electron system is called
the electron gas. On the contrary, in low electron concentra-
tion limit, r ӷ1, according to Eq. ͑5͒, the Coulomb interac-
s
tion between electrons overcomes the kinetic energy, thus
the system does not exhibit the feature of Fermi electron gas.
Such a system ͑Coulomb-correlation energy is larger than
the electron’s kinetic energy͒ is called strongly correlated

1
0 490
D. X. LI, Y. HAGA, H. SHIDA, T. SUZUKI, AND Y. S. KWON
54
and GdP. Above 200 K, the values of resistivity difference
are smaller and approach to constants. Thus, the trapped
magnetic polaron states formed in ns-GdAs and ns-GdP are
considered to be disintegrated above 200 K, which is in ac-
cord with our susceptibility measurements.34 Further evi-
dence of the formation of trapped magnetic polaron states in
the nonstoichiometric Gd monopnictides are the nonlinear
magnetization in low fields, the lager susceptibility at low
temperatures, and the smaller entropy near N e´ el point,
10,14,34
etc.
It is well known that the trapped magnetic polaron effects
are evident only in some low carrier systems, i.e., in some
strongly correlated electron systems as described above. Al-
though the trapped magnetic polaron effects have been found
in nonstoichiometric semiconductors as EuTe ͑Refs. 16 and
FIG. 11. Temperature dependence of the resistivity difference
between nonstoichiometric ns-GdP ͑ns-GdAs͒ and stoichiometric
GdP ͑GdAs͒.
1
7͒ and other Eu chalcogenides18 with very low carrier con-
centrations, for semimetals, however, the effects of trapped
magnetic polarons are observed for the first time in our re-
cent experiments. In contrast to the Eu chalcogenides, the
usual type of d-f mixing can be ignored in GdX, because the
According to the trapped magnetic polaron model, the
anomalous magnetoresistance behavior observed in nonsto-
ichiometric Gd monopnictides, as an example in ns-GdAs,
can be explained as following: At zero field the trapped mag-
netic polarons have randomly oriented moments, yielding a
large contribution to the resistivity through the electron-
polaron scattering. With increasing temperature, the d-f ex-
change interaction within the polarons becomes gradually
weakened by thermal fluctuation. This explains why below
4
f level is deep and an additional RKKY interaction be-
comes important as discussed in Ref. 35. The carrier concen-
20
21
Ϫ3
trations in GdX ͑ϳ10 –10 cm ͒ are much larger than
19
Ϫ3
that in semiconductors Eu chalcogenides ͑ϳ10 cm ͒. Fur-
thermore, because of the different electron spin configura-
tions, the He-like model with singlet or triplet spin configu-
rations of impurity electrons used for EuTe ͑Ref. 32͒ is no
longer valid for Gd monopnictides. Thus, in order to acquire
a deep understanding of the formation of trapped magnetic
polaron states in nonstoichiometric Gd monopnictides, fur-
ther experimental and theoretical studies are necessary.
0
.5 T ͉d␳/dH͉ is smaller at 4.2 than at 1.6 K. Similarly, with
increasing field, the moments of polarons, canted in low
field, are rapidly aligned to the ferromagnetic configuration.
Thus, a pronounced drop appears in the resistivity with in-
creasing field in the low-field region. As the field is raised
further, the effect of magnetic polaron on the resistivity be-
comes smaller, and gradually the intrinsic positive magne-
toresistance of the semimetal becomes dominant. At high
enough fields ͑above 7 T͒, the scattering off the trapped mag-
netic polaron mechanism is not expected, and the magnetore-
V. CONCLUSION
The transport properties of Gd monopnictides GdX both
stoichiometric single crystals ͑XϭP, As, Sb, and Bi͒ and
nonstoichiometric single crystals ͑XϭP and As͒ have been
investigated by measuring their resistivity, magnetoresis-
tance, and Hall effect. Stoichiometric GdX behaves as ex-
pected for the well-compensated semimetals that order anti-
2
sistance of ns-GdAs approximately follows a ␳(H)ϰH law,
similar to the stoichiometric GdAs sample.
In the nonstoichiometric ns-GdAs and ns-GdP, a nonlin-
ear ␳(T) behavior was observed up to 200 K. Because the
difference of the N e´ el temperatures between stoichiometric
and nonstoichiometric samples are very small ͑about 1.5 K
for both GdAs and GdP͒, and the critical scattering is impor-
tant only near the N e´ el temperature, it is natural to believe
ferromagnetically at T ϭ15.9 K for GdP, 18.7 K for GdAs,
N
23.4 K for GdSb, and 25.8 K for GdBi. Their transverse
magnetoresistance measured at low temperature follows a
2
that the critical scattering vanishes near 70 K in ns-GdAs and
near 85 K in ns-GdP, as it does in GdAs and GdP, when the
resistivity observed for the stoichiometric samples becomes
linear in temperature. The origin of the nonlinear resistivity
between T and T in the nonstoichiometric sample is there-
␳(H)ϰH law, and the larger positive ratios ͓␳(H)Ϫ␳͑0͔͒/
␳͑0͒ are observed at Hϭ10 T. The temperature dependence
of the resistivity can be explained by the d-f Coulomb scat-
tering at lower temperatures. A calculation based on a simple
two-band model yields that the observed rapid change of the
Hall coefficient RH below TN is due to the increasing mo-
bilities of the conduction electrons and valence holes. At 4.2
K, the carrier concentration determined from the Hall-effect
measurements is 0.010͑4͒ per Gd atom for GdAs and
0.025͑2͒ per Gd atom for GdSb, which are in a good agree-
ment with the dHvA measurements. The scattering mecha-
nisms appearing in stoichiometric GdX are considered to ap-
pear also in nonstoichiometric GdX. But, comparing with the
stoichiometric samples, the nonstoichiometric samples ns-
GdAs and ns-GdP show some anomalies in the transport
properties such as a large negative magnetoresistance at low
N
L
fore thought to be not only due to the critical scattering, but
also due to another additional scattering process. We inter-
pret this anomaly as an indication of the formation of trapped
magnetic polarons. For the conduction electrons, the forma-
tion of trapped magnetic polarons in nonstoichiometric GdX
results in an additional scattering, superimposed on the in-
trinsic scattering of pure GdX. This leads to the large re-
sidual resistivity, the large change of d␳/dT around T , and
N
the nonlinear ␳ϪT behavior up to higher temperatures.
Figure 11 shows the resistivity difference between sto-
ichiometric and nonstoichiometric samples for both GdAs

54
ELECTRICAL TRANSPORT PROPERTIES OF . . .
10 491
temperature and low magnetic field, and a nonlinear resistiv-
ity at higher temperatures. These anomalies can be qualita-
tively understood within the framework of the trapped mag-
netic polaron model.
ACKNOWLEDGMENT
We would like to thank Dr. M. Umehara for helpful dis-
cussion.
*
20
Present address: The Oarai Branch, Institute for Materials Re-
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