Anomalous magnetotransport in (Y$_{1-x}$Gd$_{x}$)Co$_{2}$ alloys: interplay of disorder and itinerant metamagnetism

0.28 MB·English

by  A. T. Burkov

Anomalous magnetotransport in (Y1−xGdx)Co2 alloys: interplay of disorder and
itinerant metamagnetism.
A. T. Burkov, A. Yu. Zyuzin
A. F. Ioffe Physico-Technical Institute, Russian Academy of Sciences, Saint-Petersburg 194021, Russia.
T. Nakama, K. Yagasaki
4 Physics Department, University of the Ryukyus, Okinawa 903-0213, Japan
0 (Dated: February 2, 2008)
0
2 New mechanism of magnetoresistivity in itinerant metamagnets with a structural disorder is in-
troduced basing on analysis of experimental results on magnetoresistivity, susceptibility, and mag-
n
netization of structurally disordered alloys (Y1−xGdx)Co2. In this series, YCo2 is an enhanced
a
Pauliparamagnet,whereasGdCo isaferrimagnet (T =400K)withGdsublatticecoupledantifer-
J 2 c
romagnetically to the itinerant Co-3d electrons. The alloys are paramagnetic for x < 0.12. Large
4 positive magnetoresistivity has been observed in the alloys with magnetic ground state at temper-
1 atures T<T . We show that this unusual feature is linked to a combination of structural disorder
c

and metamagnetic instability of itinerant Co-3d electrons. This new mechanism of the magnetore-
]
l sistivity iscommon for abroad class ofmaterials featuring astatic magnetic disorderanditinerant
e metamagnetism.
-
r
t PACSnumbers: 71.27.+a,72.15.Gd,75.10.Lp,75.30.Kz
s
.
t
a
m I. INTRODUCTION of Y1−xGdxCo2 alloys has been published partly in our
previous article.6 Here we analyze these and new experi-
-
d Interplay of structural disorder and magnetic inter- mental results in order to revealthe physicalmechanism
n of anomalous megnetotransport properties observed in
actions opens a rich field of new physical phenom-
o the alloys. In this paper we will discuss the magneto-
ena. Among them are the actively discussed possi-
c transport properties of the FM alloys. The properties of
[ bility of disorder-induced Non-Fermi Liquid (NFL) be-
havior near a magnetic Quantum Critical Point (QCP) paramagnetic alloys will be published elsewhere.
2
as well as a broader scope of effects of disorder
v
on magnetotransport.1,2,3 Structurally disordered alloys
9 II. EXPERIMENTAL
3 Y1−xGdxCo2 are quasi-binary solid solutions of Laves
5 phasecompoundsYCo2andGdCo2. Thecompoundsbe-
2 long to a large family of isostructural composites RCo2. Samples of Y1−xGdxCo2 were prepared from pure
0 YCo is an enhanced Pauli paramagnet whose itinerant componentsbymeltinginanarcfurnaceunderaprotec-
2
3 Co-3delectronsystemiscloseto magneticinstability. In tive Ar atmosphere and were subsequently annealed in
0 externalmagnetic field of about 70 T this system under- vacuum at 1100 K for about one week. An X-ray analy-
/
t goes a metamagnetic transitioninto ferromagnetic(FM) sisshowednotracesofimpurityphases. Afour–probedc
a
ground state.4 GdCo is, on the other hand, a ferrimag- method was used for electrical resistivity measurements.
m 2
net with a Curie temperature of 400 K in which the Magnetoresistivity (MR) was measured with longitudi-
-
d spontaneousmagnetizationof4fmomentsisanti-parallel nal orientation of electrical current with respect to the
n to the induced magnetization of the Co-3d band. Com- magnetic field. The size of the samples was typically
o pounds of this family and their alloys provide a conve- about 1× 1×10 mm3. Magnetization was measured by
c nient ground for experimental studies of magnetotrans- a SQUID magnetometer for samples from the same in-
:
v port phenomena. The electronic structure in the impor- got as that used for the resistivity and AC susceptibility
i tantforthetransportvicinityoftheFermienergyiscom- measurements.
X
posed mainly of Co-3d states and is, to the first approx-
r
imation, the same for all compounds of RCo family. It
a 2
III. EXPERIMENTAL RESULTS
has been found that the main contribution to the resis-
tivity of RCo comes from the scattering of conduction
2
electrons on magnetic fluctuations due to strong s–d ex- The magnetic phase diagram of the Y1−xGdxCo2
change coupling,5 therefore the transport properties are system inferred from the transport and magnetic
expected to be especially sensitive to the magnetic state measurements6 is shown in Fig. 1. Curie temperature
of the sample. GdCo2 occupies a special place in RCo2 Tc decreases with increasing content of Y and eventu-
family since Gd 4f magnetic moment has no orbital con- ally drops to zero. A precise determination of the crit-
tribution and, therefore crystal-field effects are not im- ical concentration x which separates the magnetically
c
portant for this compound. orderedgroundstate andthe paramagneticregionis dif-
The experimental results on the transport properties ficult,sinceontheonsetofthelongrangeorderitssigna-
2
x 50
0.18
45
0 0.12 0.25 0.41 0.63 0.85
0.4 nits) 40
400 b.u 35
0.2 magneticnd state 300 bility, (ar 2350 0.1x4=0.15 0.2
∆ρρ/ 0.0 paragrou homogeneous 200T (K)c uscepti 1250 0.25 0.3
S 10 YCo
ferrimagnet C 2
100 A 5
-0.2 two-component
ferrimagnet 0
0 50 100 150 200 250
0
0.2 0.0 0.2 0.4 0.6 0.8 T, (K)
)0.6 FIG. 2: The ac susceptibility of the Y1−xGdxCo2 alloys.
K
0 Note, the experimental data for YCo2 and for the alloy with
0 x=0.14 are multiplied byfactor 20.
3
ρ(0.4
)/
K
2
ρ(0.2 The very surprising result is the positive MR in the
FM phase at low temperatures, see Figs. 1 and 3. The
well known theoretical result for MR of a localized mo-
0.0
ment ferromagnet was derived long ago by Kasuya and
-0.2 0.0 0.2 0.4 0.6 0.8
De Gennes.7 As it follows from their theory, MR of a
|x-x |0.75
metallic ferromagnetshould be negative, having a maxi-
c
mumabsolutevalueatCurietemperature,andapproach-
ingzeroasT→0,andinthe limitofhightemperatures.
FIG. 1: The upper panel shows the ordering temperature
T (cid:4) (right y-axis), and the MR (left axis) of the Qualitatively this behavior has been supported by ex-
c
Y1−xGdxCo2 system6. The MR was mLeasured at T = 2 K periment, as well as by later more detailed theoretical
in magnetic field of 15 T. The dotted vertical lines indicate calculations. The present experimental results are in a
phase boundaries at zero temperature. The lower panel dis- qualitativeagreementwiththistheoreticalbehavioronly
plays normalized resistivity ρ(2 K) . for alloys with large Gd content (x > 0.4) (Fig. 3). MR
ρ(300 K)
of the FM alloys with composition 0.3 > x > 0.14 fun-
damentally differs from this theoretical behavior. Let us
tures in the magnetic and transport properties are very note that this composition range falls into the region of
weak. The first firmevidence ofthe longrangeorderare the phase diagram between the paramagnetic phase and
found for alloy with x=0.14 in ac susceptibility at T=27 the additional phase boundary indicated by the kink in
K, Fig. 2. the T vs x dependency, see Fig. 1. MR of these alloys
c
Quantum critical scaling theory predicts that when is positive below Curie temperature and is very large.
CurietemperatureofaFMsystemcontinuouslydepends The known mechanisms of a positive MR can not ex-
onanexternalparameterx,thisdependenceisexpressed plainthe experimentaldata. AroughestimateofLorenz
as2: force-drivenMRonecangetfromacomparisonwiththe
Tc ∝|x−xc|d+zz−2 M(wRithofrepsuidreuaYlCreos2i.s8tiIvnitythoefmaboostutpu2reµΩsacmmp)letsheofLYorCeno2z
with critical index z = 3 for a FM system of spatial force-driven positive contribution to the total MR does
dimension d = 3. The experimental T vs. x depen- not exceed 5%. On the other hand the resistivity of the
c
dency does follow this relation, but with additional kink FM alloys at low temperatures falls into the region from
atx=x . Apossibleoriginofthiskinkwillbediscussed 30 to 100 µΩ cm, i.e. at least one order of magnitude
t
later. Linear extrapolation of the phase separation line larger than the resistivity of pure YCo2. Therefore, ac-
on the phase diagram Fig. 1 to T =0 gives as the crit- cording to Koehler’s rule, this mechanism can give MR
c
ical concentration x = 0.12. We do not claim however of only about 0.5%, this has to be compared with the
c
thatQCPexistsinthisalloysystem. Directexperimental experimental MR of almost 40%.
verification that T → 0 as x approaches x from mag- Weak localizationeffect is knownto givepositive MR.
c c
netically ordered state is difficult for a disordered alloy However our estimates show that this mechanism can
system. give a contribution which is at least two order of magni-
3
0.4 spatially fluctuating effective field induces an inhomoge-
x=0.4 neous magnetization of Co-3d electron system:
0.3 x=0.3
x=0.25
m(r)=χ(B )B (r).
x=0.2 eff eff
0.2 x=0.18
ρ x=0.16 Therefore even at zero temperature in the ferromagnetic
ρ/
∆ 0.1 x=0.15 ground state, there are two kind of static magnetic fluc-
tuations in the system: i. M (r); ii. m(r). These fluctu-
f
0.0 ations give an additional contribution to the resistivity.
Since at T=0 K in ferromagnetic phase the 4f magnetic
-0.1 moments are saturated the corresponding contribution
to the resistivity does not depend on external magnetic
0 1 2 3
T/T field. On the other hand, the 3d magnetic moment, as
c
we will see later, is not saturated even in the ferromag-
0.3 x= 0.4 netic groundstate. Thereforethe 3d magnetizationdoes
0.3 dependontheexternalfield. However,aslongas3dsus-
0.25
ceptibility is fieldindependent and uniform,the external
0.2
0.2 0.18 magneticfieldwillchangeonlythemean(non-fluctuating
ρ 0.16 part)valueofthe magnetization,whereasthemagnitude
∆ρ/ 0.1 of the fluctuations of m and corresponding contribution
to the resistivity remain unchanged. However actually,
the 3d system is close to the metamagnetic instability.
0.0 Therefore the 3d susceptibility χ is field dependent and
gives rise to static magnetic fluctuations which are de-
0 2 4 6 8 10 12 14 16 pendent on magnetic field resulting in non-zero magne-
Magnetic field (T)
toresistivity.
For a qualitative analysis of this new mechanism of
FIG. 3: The upperpanel shows the temperature dependence MR we make the following assumptions.
of the MR of the Y1−xGdxCo2 alloys, measured in field of 1.We consider the system only in its ground state.
15 T. Large positive MR of FM alloys (x ≤ 0.3) is observed 2.Correlations between potential and spin-dependent
atlowtemperatures. ThefielddependenciesofMR,measured
scattering are neglected.
at T=2 K, are presented on thelower panel.
3.Only the contributions which may be strongly depen-
dent on external magnetic field are retained, e.g. we do
not include the effects related to a change of d-density
tude smaller than the observed MR.
of states in magnetic field, and contributions due to
potential scattering and scattering on 4f moments. In
IV. DISCUSSION fact only the contribution related to the scattering on
fluctuations of Co-3d magnetization will be considered.
The key for understanding the mechanism of the pos-
Figure 4 shows schematically the dependency of the
itive MR is a combination of strong dependence of the
Co-3d magnetization of RCo compounds on effective
magneticsusceptibilityχofmetamagneticCo-3dsubsys- 2
magneticfield9 andthedistributionfunctionP{B (r)}.
tem on the effective magnetic field and of the structural eff
Themetamagnetictransitionisindicatedbytherapidin-
disorder in the R - sublattice of the alloys. In case of
creaseofthe magnetizationaround70T.Forthe further
GdCo as well as in the case of other magnetic RCo
2 2 discussionitisimportanttohaveanestimateofthemag-
compounds with heavy R-elements, the 4f–3d exchange
nitude of the fluctuations of the effective field, i.e. the
interaction is described by introducing an effective field
width of the distribution function P{B (r)}. We can
which acts on 3d electrons as: eff
get a hint considering experimental data on field (0 to
B =n M −B 7 T) dependency of the magnetization, Fig.5.
eff fd f
Two important points follow from these data: i. there
where B is external field, M is the uniform magneti- is a non-saturating para-processin these field dependen-
f
zation of R - sublattice, and n - is the f–d coupling cies above about 2 T; ii. the estimated susceptibility of
fd
constant (in case of GdCo n ≈ 50 T/f.u.µ 9). In this para-process (≈ 0.016 µ /T) is larger than the sus-
2 fd B B
Y1−xGdxCo2 alloys the Gd moments are randomly dis- ceptibility of 3d system below and above metamagnetic
tributedovertheR-sitesofthecrystallattice. Therefore, transition(≈0.002µ /T),howeveritissmallerthanthe
B
the effective field acting on 3d electrons depends on the susceptibility in the transition region (≈ 0.04 µ /T).9
B
local distribution of Gd moments and is therefore a ran- This indicates, thatthe scale ofthe fluctuations is larger
dom function of coordinate. This random field can be than the width of the metamagnetic transition. There-
characterizedbyadistributionfunctionP{B (r)}. The fore we can not treat the fluctuating part of the effec-
eff
4
1.0 GdCo becomes larger than B0, i.e. at this composition almost
) 2 wholevolumeisoccupiedbythehighmagnetizationcom-
o
C ponent, i.e. y ≈ 1. With a further increase of Gd con-
/Β 0.8
µ( tentthemeanmagnetizationofCoshouldincreasewitha
on 0.6 B)eff dsmenaclylerabraotvee,Bdet,ehrmowinevederbyy=th1esinlotpheisofretghieonm.(ABc)cdoerpdeinng-
etizati 0.4 P( mtoatxhimisusmcenvaarluio0eρamt xwiwllhiincchrecaosrerewspitohndxsattofiyrs=t,0re.5acahnda
n
g will decrease with further increase of x approaching to
a 0.2
M zero at x ≈ x . We believe that x corresponds to the
t t
kink on the phase diagram, Fig. 1. The expected vari-
0.0
0 50 100 150 200 250 300 ation of ρ with the alloy composition is schematically
m
B
0 Effective field (T) shown in Fig. 6. The total experimental resistivity in-
cludes additionally contributions coming from potential
scatteringandfromscatteringon4fmoments(ρ ),which
0
FIG. 4: Dependence of the magnetization of Co–subsystem
both are proportional to x(1−x) and also are depicted
in RCo compounds vs effective magnetic field (left axis)9
2 in Fig. 6. For comparison the experimental resistivity is
(shown schematically). The dotted line is a schematic repre-
shown in this picture too. We present here the resistiv-
sentationofthedistributionfunctionP(Beff)(rightaxis),the
shadedareaindicatesthepositionofP(B)inexternalfieldof ity measured at T=2 K in field of 15 T to exclude the
15 T. contributions related to spin-flip scattering.
0.6
x=0.18
0.2
u.) 0.4 0.6
µ/f.B ρm
M (
0.2
) y(x)=1
00 0.4 t
3
(
ρ
0.00 2 4 6 8 2)/ ρ
Magnetic field (T) ρ( 0
0.2
FIG.5: ThemagnetizationoftheY1−xGdxCo2alloysvsmag-
netic field.
tive field as a perturbation and have to resort to a phe- 0.0
nomenological analysis. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
The distribution function P(Beff) describing the fluc- x
tuating effective field depends on the alloy composition.
Fordilutedalloys(x≈0)themostprobablevalueB of
av
FIG. 6: Schematic of contributions to the low-temperature
B is close to zero. As x increases B shifts to higher
vaelffues and in a certain range of the coavncentrations, the resistivityofY1−xGdxCo2 alloys. •-experimentalnormalized
resistivity measured at T=2 K in field of 15 T. Solid line
function will have essentially non-zero weight for both
represents schematically ρm, broken line - ρ0.
Beff < B0 and Beff > B0, see Fig. 4. In this case there
shallberegionswithlowandwithhigh3dmagnetization
The experimental low-temperature resistivity shows
in the sample. The resistivity resulting from this static
the expected variation with x (one needs to keep in
disorder in 3d magnetization can be expressed as:
mind that the relation between x and y is in general
ρ =ρ ·y(1−y). non-linear, especially around magnetic phase boundary
m sd
x ). The resistivity attains the maximum value of about
∞ c
Where y = B0P (Beff)dB is the volume fraction of the 100µΩcmintheregion,whichcorrespondstomaximum
highmagnetRizationcomponent. Parametery dependson static magnetic disorder at y ≈ 0.5 (room temperature
the alloy composition x, and on external magnetic field resistivityofthealloysweaklydependsonxandisabout
B. In zero field, point y = 0 corresponds to x = 0. 150 µΩcm). About the same value was obtained for the
As the content of Gd increases B shifts to larger effec- high temperature limit (maximum magnetic disorder)of
av
tive fields, and finally, at some alloy composition x , it magneticpartoftheresistivity,arisingfromscatteringon
t
5
3dtemperature-inducedmagneticfluctuationsinYCo .5 random potential. Here χ (r−r′) is nonlocal suscepti-
2 m
This suggests that the main part of the experimental bility near metamagnetic transition. We estimate corre-
lowtemperature resistivity atx<x originatesfromthe lation function of fluctuating effective field as
t
scattering on the magnetic fluctuations of 3dmagnetiza-
hδB (r)δB (r′)i=(2S n )2δ(r−r′)x(1−x)a3.
tion, i.e. is identical to ρ . eff eff Gd fd
m
Let us show that ρm is of orderof the spin-fluctuation Here a3 is volume of the formula unit. In Born approxi-
contributiontoresistivityatlargetemperatures,i.e. isof
mation the corresponding contribution to the resistivity
order of 100 µΩcm. In case of YCo the spin-fluctuation
2 is given by the expression:
contributionisthemostimportantandisnearlyindepen-
dent of temperature above about 200 K. Hamiltonian of
1
s-d exchange interaction is given by
m 2k
ρ = G2N (2S n )2x(1−x)a3 dqq3χ2 q F
m ne2 s Gd fd Z m(cid:18) k∗ (cid:19)
F
0
Hsd =G drs(r)Sd(r) (3)
Z
Both expressions (1) and (3) give the contributions to
here s(r) and S (r) are spin density of s- and d- elec- theresistivityduetoscatteringonspinfluctuations,how-
d
trons, correspondingly.17 ever in the first case they are of thermal origin, whereas
Spin fluctuationcontributionto resistivityhasform10: inthesecondcasethefluctuationsareduetorandomness
of the effective field.
Assuming that nonlocality of the susceptibility is not
1 ∞
3m 1 dωω 2k important we obtain from (2) and (3):
ρ= G2N dqq3 Imχ q F,ω
4ne2 sT Z Z sinh2(ω/2T) (cid:18) k∗ (cid:19)
F
0 −∞ (1) ρm = (2SGdnfd)2x(1−x)a3χ2m
where N is density of states ofs-electrons, kF is ratio ρ 3πTχ
s kF∗
of Fermi momentum of s- and d- electrons. Dynamic
Using experimental results for χ and χ4,9,11 we find
m
susceptibility χ(q,ω) is given by the equation: ρm in the range from 0.5 to 3, i.e. the resistivity caused
ρ
by the static magnetic fluctuations is of the same order
χ−1(q,ω)=χ−1(q)(1−iω/Γ ) as the temperature–induced spin fluctuation resistivity.
q
The uncertainty is mainly due to determination of χ
m
here Γ is damping of the spin-fluctuations, whereas
static nonqlocal susceptibility χ−1(q) is given by nearthemetamagnetictransition. Takingχm asthesus-
ceptibility of YCo at B=70 T (at the field of metamag-
2
netictransition)4givestheupperboundfor ρm. Whereas
ρ
χ−1(q)=χ−1+Aq2/Nd χmforthedisorderedalloyofx=0.18,estimatedfromour
results on M(B), Fig. 5 gives the lower bound.
with χ = Nd/ζ(T) , where ζ(T) is inverse Stoner fac- In external magnetic field, the effective field Beff
tor, renormalized by spin fluctuations, Nd is density of decreases18, therefore y also decreases. Depending on
states of d-electrons, and A < 1 is a dimensionless con- the value of y – the volume fraction in zero field, ρ
0 m
stant. will either increase or decrease, resulting in positive or
At large temperatures T > Γq the expression (1) re- negative magnetoresistivity: for 0.5 < y0 < 1 it will
duces to: be positive, whereas for 0 < y < 0.5 we will have a
0
negative MR. In agreement with the model, the exper-
1 imental MR is positive at 0.15 < x < xt and quickly
ρ= 3πmG2N T dqq3χ q2kF ≃ 3πmG2N Tχ decreases at x > xt = 0.3 where y ≈ 1. Nearly lin-
ne2 s Z (cid:18) k∗ (cid:19) 4ne2 s ear field dependencies of MR, observed for x < 0.3, see
F
0 Fig. 3, implies that the width of P(B ) in this compo-
(2) eff
sition range is larger than our experimental field limit
∗
The last equality is valid when 2Ak /k < ζ(T) < 1
F F of 15 T. The region y < 0.5 almost coincides with the
which must be the case for YCo . Note that neglecting
2 paramagnetic region of the phase diagram. The model
momentum dependence of susceptibility means that s-
predicts negative MR for this region, and this predic-
electrons see the d-spin fluctuations as point scatterers.
tionagreeswiththeexperimentalresult. Themodelalso
Scattering of conducting s – electrons by the static
gives a satisfactory description of the residual resistivity
randomdistributionofspindensityofd-electronswecan
behaviourin this region,see Fig.6. This agreementsug-
estimate in the following way.
gests that at x < x the system is actually in spin-glass
Due to random distribution of magnetic moments of c
state. In this regionadditionalessentialcontributions to
Gd, s-electrons experience scattering by
MR are present. First, in the paramagnetic region there
G G is a negative MR due to suppression of magnetic disor-
hS (r)i= χ (r−r′)δB (r′)
2 d 2 Z m eff der in 4f magnetic moment system by external magnetic
6
field. This negative contribution may be the reason why sureeffectisinagreementwiththemodelprediction: the
the cross-over point from the positive to negative mag- resistivity decreases with pressure for x=0.1, whereas it
netoresistivity does not coincide with the maximum of increases with P for x=0.18 and x=0.3. Moreover,there
resistivity, see Fig. 1. Secondly, there can be additional is a good scaling of pressure and magnetic field depen-
contributions, both positive and negative, near to zero- dencies of the resistivity, Figs. 3 and 7 (P→ αB) with
temperature magnetic phase boundary due to closeness scaling parameter α which is close to literature data on
to QCP. the pressure dependence of B : dB0 ≈1.5 T/kBar.12,13
0 dP
An independent test of the model is based on the ob-
servationthat the criticalmagnetic field B of the meta-
0
magnetic transition in 3d subsystem increases under ex- V. CONCLUSION
ternalhydrostaticpressure(P).12,13 Therefore,basingon
the model, we expect that the resistivity of the alloys
It has been found that at low temperatures there is
withthecompositionleftoftheresistivitymaximum(see
a large contribution to the resistivity related to scat-
Fig.1)willdecreasewithpressure,whereasforthealloys
tering on magnetic fluctuations in metamagnetic itin-
right of the maximum it will increase with the increas-
erant 3d system, induced by fluctuating effective field
ing pressure. The experimental results for three alloy
of 4f moments. Large positive MR, found in the FM
compositions are shown in Fig. 7. The sign of the pres-
Y1−xGdxCo2 alloys and strong pressure dependence of
the resistivity are explained as arising from a combina-
0.8 tionofstaticmagneticdisorderandstrongmagneticfield
dependence of magnetic susceptibility.
x=0.18
Wewanttoemphasizethatthismechanismofresistiv-
0.6 ity (and of MR) is not material specific, rather it should
becommonforabroadclassofdisordereditinerantmeta-
magnets with strong coupling of conduction electrons to
ρ 0.4 the magnetic fluctuations. In Y1−xGdxCo2 the relevant
/ disorder originates from random distribution of d–f ex-
ρ
∆ change fields, however a similar effect should arise when
x=0.3
0.2 there is a random distribution of local susceptibilities (a
correspondingtreatmentforKondosystemswasrecently
developed in14). Positive MR observed in FM alloys
0.0 Y(Co1−xAlx)215 andinErxY1−xCo216 maybeexplained
x=0.1 by this mechanism.
-0.2
0 5 10 15 20 ACKNOWLEDGMENT
Pressure, kBar
This work is supported by grants 02-02-17671and 01-
FIG.7: TheresistivityoftheY1−xGdxCo2 alloysvsexternal 03-17794 of Russian Science Foundation, Russia. We
hydrostatic pressure at T = 2 K. thank Dr. P. Konstantinov for stimulating discussions.
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2
1159 (2001). pounds with light R-elements) the effective field increases
17 Hereafter we takeµB ≡1. with theexternal field.