Pressure effect on the Fermi surface and electronic structure of LuGa₃ and TmGa₃
The Fermi surfaces and cyclotron masses of LuGa₃ and TmGa₃ compounds are studied by means of the de Haas—van Alphen effect technique under pressure. The highly anisotropic pressure dependences of the de Haas—van Alphen frequencies and cyclotron masses have been observed in both compounds. Concurr...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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| Цитувати: | Pressure effect on the Fermi surface and electronic structure of LuGa₃ and TmGa₃ / V.B. Pluzhnikov, G.E. Grechnev, A. Czopnik, O. Eriksson // Физика низких температур. — 2005. — Т. 31, № 3-4. — С. 412-421. — Бібліогр.: 34 назв. — англ. |
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irk-123456789-1217612017-06-17T03:02:51Z Pressure effect on the Fermi surface and electronic structure of LuGa₃ and TmGa₃ Pluzhnikov, V.B. Grechnev, G.E. Czopnik, A. Eriksson, O. Электpонные свойства металлов и сплавов The Fermi surfaces and cyclotron masses of LuGa₃ and TmGa₃ compounds are studied by means of the de Haas—van Alphen effect technique under pressure. The highly anisotropic pressure dependences of the de Haas—van Alphen frequencies and cyclotron masses have been observed in both compounds. Concurrently, the ab initio calculations of the volume-dependent band structures have been carried out for these compounds, including ferromagnetic configuration phase of TmGa₃, by employing a relativistic version of the full-potential linear muffin-tin orbital method within the local spin-density approximation. The experimental data have been analysed on the basis of the calculated volume-dependent band structures and compared with the corresponding pressure effects in the isostructural compound ErGa₃. 2005 Article Pressure effect on the Fermi surface and electronic structure of LuGa₃ and TmGa₃ / V.B. Pluzhnikov, G.E. Grechnev, A. Czopnik, O. Eriksson // Физика низких температур. — 2005. — Т. 31, № 3-4. — С. 412-421. — Бібліогр.: 34 назв. — англ. 0132-6414 PACS: 71.18.+y, 71.20.Eh, 71.70.Gm http://dspace.nbuv.gov.ua/handle/123456789/121761 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| topic |
Электpонные свойства металлов и сплавов Электpонные свойства металлов и сплавов |
| spellingShingle |
Электpонные свойства металлов и сплавов Электpонные свойства металлов и сплавов Pluzhnikov, V.B. Grechnev, G.E. Czopnik, A. Eriksson, O. Pressure effect on the Fermi surface and electronic structure of LuGa₃ and TmGa₃ Физика низких температур |
| description |
The Fermi surfaces and cyclotron masses of LuGa₃ and TmGa₃ compounds are studied by means
of the de Haas—van Alphen effect technique under pressure. The highly anisotropic pressure
dependences of the de Haas—van Alphen frequencies and cyclotron masses have been observed in
both compounds. Concurrently, the ab initio calculations of the volume-dependent band structures
have been carried out for these compounds, including ferromagnetic configuration phase of
TmGa₃, by employing a relativistic version of the full-potential linear muffin-tin orbital method
within the local spin-density approximation. The experimental data have been analysed on the basis
of the calculated volume-dependent band structures and compared with the corresponding pressure
effects in the isostructural compound ErGa₃. |
| format |
Article |
| author |
Pluzhnikov, V.B. Grechnev, G.E. Czopnik, A. Eriksson, O. |
| author_facet |
Pluzhnikov, V.B. Grechnev, G.E. Czopnik, A. Eriksson, O. |
| author_sort |
Pluzhnikov, V.B. |
| title |
Pressure effect on the Fermi surface and electronic structure of LuGa₃ and TmGa₃ |
| title_short |
Pressure effect on the Fermi surface and electronic structure of LuGa₃ and TmGa₃ |
| title_full |
Pressure effect on the Fermi surface and electronic structure of LuGa₃ and TmGa₃ |
| title_fullStr |
Pressure effect on the Fermi surface and electronic structure of LuGa₃ and TmGa₃ |
| title_full_unstemmed |
Pressure effect on the Fermi surface and electronic structure of LuGa₃ and TmGa₃ |
| title_sort |
pressure effect on the fermi surface and electronic structure of luga₃ and tmga₃ |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2005 |
| topic_facet |
Электpонные свойства металлов и сплавов |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121761 |
| citation_txt |
Pressure effect on the Fermi surface and electronic structure of LuGa₃ and TmGa₃ / V.B. Pluzhnikov, G.E. Grechnev, A. Czopnik, O. Eriksson // Физика низких температур. — 2005. — Т. 31, № 3-4. — С. 412-421. — Бібліогр.: 34 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT pluzhnikovvb pressureeffectonthefermisurfaceandelectronicstructureofluga3andtmga3 AT grechnevge pressureeffectonthefermisurfaceandelectronicstructureofluga3andtmga3 AT czopnika pressureeffectonthefermisurfaceandelectronicstructureofluga3andtmga3 AT erikssono pressureeffectonthefermisurfaceandelectronicstructureofluga3andtmga3 |
| first_indexed |
2025-07-08T20:28:56Z |
| last_indexed |
2025-07-08T20:28:56Z |
| _version_ |
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| fulltext |
Fizika Nizkikh Temperatur, 2005, v. 31, Nos. 3/4, p. 412–421
Pressure effect on the Fermi surface and electronic
structure of LuGa3 and TmGa3
V.B. Pluzhnikov
International Laboratory of High Magnetic Fields and Low Temperatures
Gajowicka 95, 53-529 Wroc³aw, Poland
B. Verkin Institute for Low Temperature Physics and Engineering
of the National Academy of Sciences of Ukraine, 47 Lenin Ave., Kharkov 61103, Ukraine
G.E. Grechnev
B. Verkin Institute for Low Temperature Physics and Engineering
of the National Academy of Sciences of Ukraine, 47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: grechnev@ilt.kharkov.ua
A. Czopnik
W. Trzebiatowski Institute of Low Temperature and Structure Research
P.O. Box 1410, 50-950 Wroc³aw, Poland
O. Eriksson
Theoretical Magnetism Group, Department of Physics, University of Uppsala
Box 530, S-751 21 Uppsala, Sweden
Received August 31, 2004
The Fermi surfaces and cyclotron masses of LuGa3 and TmGa3 compounds are studied by means
of the de Haas—van Alphen effect technique under pressure. The highly anisotropic pressure
dependences of the de Haas—van Alphen frequencies and cyclotron masses have been observed in
both compounds. Concurrently, the ab initio calculations of the volume-dependent band struc-
tures have been carried out for these compounds, including ferromagnetic configuration phase of
TmGa3, by employing a relativistic version of the full-potential linear muffin-tin orbital method
within the local spin-density approximation. The experimental data have been analysed on the ba-
sis of the calculated volume-dependent band structures and compared with the corresponding pres-
sure effects in the isostructural compound ErGa3.
PACS: 71.18.+y, 71.20.Eh, 71.70.Gm
1. Introduction
In the recent years the de Haas–van Alphen effect
(dHvA) has been extensively studied in a number of
RM3 compounds (R is a rare-earth, M is a p element
from the group-III series), including RGa3 [1–4],
light RIn3 (R = La – Gd) [5], heavy RIn3 (R = Tb –
Lu) [6], TmAl3 [7], and CeIn3 [8]. The main objec-
tive of these studies was to determine the Fermi sur-
face (FS) geometry and effective cyclotron masses
in the representative series of RM3 compounds. The
role of magnetic ordering in reconstruction of the FS
has been also addressed in Refs. [5,6] (RIn3), [9]
(TmGa3), and [10] (ErGa3).
In the present work we study the effect of pressure
on the FS and cyclotron masses of the LuGa3 and
TmGa3 compounds by means of the dHvA effect. The
pressure derivatives of dHvA frequencies and cyclo-
tron masses are of particular interest due to their sen-
sitivity to details of the exchange interaction and
many-body effects in R systems. Therefore, the pres-
ent investigation can provide a critical test for re-
cently developed methods of ab initio calculations of
electronic and magnetic structures, and to stimulate
© V.B. Pluzhnikov, G.E. Grechnev, A. Czopnik, and O. Eriksson, 2005
the formulation of improved theories for electronic
structure of rare earths.
This work represents an extension of our recent
studies [1–3] of the FS and electronic structure in
RGa3 compounds at ambient pressure. Also, the pres-
sure effect on the FS of ErGa3 has been addressed in
Ref. 4. Information on physical properties of TmGa3
and LuGa3 is scarce. These compounds crystallize in
the AuCu3-type cubic structure. At TN = 4.26 K
TmGa3 orders antiferromagnetically [11] to the mul-
tiaxial 3k-type magnetic structure [12,13]. It has been
shown [14], that in the paramagnetic state an inter-
action between quadrupolar moments of the 4f shells
is strong, and it leads to their ordering just above TN ,
and the leading mechanism appears to be the pair
quadrupolar interaction via conduction electrons.
It can be expected that at low temperatures TmGa3
reveals large and field-dependent magnetization, in
the same manner as is the case of ErGa3 [3,4]. It
brings about, as in the case of ErGa3 [4], a number of
difficulties in the Fourier analysis of dHvA oscilla-
tions. As a result, one has to study the dHvA effect in
strong enough magnetic fields where magnetization
tends to saturate. Obviously, these fields are required
to be higher than the critical field destroying the
antiferromagnetic order. Therefore, the dHvA effect
can be studied in a paramagnetic phase of TmGa3, in
which sufficient magnetic fields lead to a quasi-ferro-
magnetic configuration of magnetic moments.
In this work, the experimental investigation of the
dHvA effect under pressure is supplemented by ab in-
itio calculations of the volume-dependent electronic
structures of TmGa3 and the reference compound
LuGa3. This provides the possibility of estimating
many-body enhancement of «bare» cyclotron masses,
and the mass enhancement factors � can be evaluated
with the observed (mc
*) and calculated (mc
b) cyclotron
masses.
Also, a comparison of the dHvA data under pres-
sure and the calculated volume-dependent band struc-
tures is expected to be very useful for development of
advanced theoretical models for electronic spectra of
rare-earth compounds. The evaluated parameters of
the electronic structure of TmGa3 and LuGa3 and
their pressure derivatives are compared with the corre-
sponding results obtained for the isostructural ErGa3
compound at high pressures [4]. This comparison pro-
vides the possibility of estimating the anisotropy and
volume dependences of the FS and the many-body en-
hancement of the cyclotron masses in heavy RGa3
rare-earth compounds.
A discussion is given on the role of different inter-
actions (exchange splitting, magnetic quadrupolar ex-
citations, spin waves, crystal field) in the revealed
pressure effects on the FS and cyclotron masses.
2. Experimental details
Single crystals of TmGa3 and LuGa3 were grown
by the flux method from a melt of the nominal compo-
sition 90 at.% Ga and 10 at.% Tm or Lu. The purity of
starting metals was 6N for Ga and 4N for Tm and Lu.
The feed placed in an alumina crucible and sealed in a
quartz tube in an argon atmosphere under a pressure of
150 Torr at room temperature, was heated in a resis-
tance furnace up to 920 �C, held at this temperature
for 48 h and then slowly cooled down at the rate
0.8 K/h. The synthesis was stopped at about 350 �C
and then sample was cooled fast down to room temper-
ature to avoid the formation of RGa6 in a peritectic
reaction [15]. The resulting crystals of TmGa3 and
LuGa3 were immersed in an excess of Ga which is easy
to remove. The crystals obtained had the form of cubes
with maximum dimensions 5�5�5 mm. According to an
x-ray examination the quality of the single crystals
was very good.
The magnetic phase diagram of TmGa3 in magnetic
field parallel to the � �100 axis is shown in Fig. 1 (cited
from Ref. 13). The antiferroquadrupolar phase exists
only in low fields up to 0.5 T and in the very narrow
range of temperatures: ( )T TQ N� � 0.1 K. The criti-
cal lines H Tc1( ) and H Tc2( ) are the lines of me-
tamagnetic transitions: at the field H Tc1( ) from the 3k
phase to an intermediate one and at H Tc2( ) from the
intermediate phase to a paramagnetic one. In a field
applied along the other two principal crystallographic
axes, � �110 and � �111 , the phase diagrams are similar to
that cited, but the critical field H Tc2( ) reaches much
lower values and does not exceed 2.2 T [13]. At tem-
peratures lower than TN and in magnetic field higher
than 7 T, magnetic moments reach an induced pa-
ramagnetic configuration, except for a small region
of angles at the � �100 axis. Above the second me-
tamagnetic transition the magnetization is large and
anisotropic. This fact has important consequences for
analysis of the dHvA effect in TmGa3.
First, in the Fourier analysis of the dHvA signal
Vosc one must take into account the magnetic induc-
tion B in the Lifshitz—Kosevich formula instead of
the applied magnetic field Happl ,
V A
F
Bosc �
�
�
� �
�
�
�
�
�
�sin ,
2�
� (1)
where the dHvA frequency F is proportional to the
extremal cross-sectional area of the FS, and the mag-
netic induction B H M N� � �appl 4 1( ) depends on
Pressure effect on the Fermi surface and electronic structure
Fizika Nizkikh Temperatur, 2005, v. 31, Nos. 3/4 413
the magnetization per unit volume M and on the de-
magnetizing factor N.
Secondly, the cyclotron effective mass, mc
*, is deter-
mined from the temperature dependence of the dHvA
amplitude A, namely, from the slope of the plot of
ln { [ exp ( )] }*A m T/B /Tc1 2� � � versus T, where � �
� 2 2� ck /eB �. Therefore, one has to take into account
the magnetic induction B instead of the applied mag-
netic field Happl .
Thirdly, in a strong magnetic field the magnetic
moments of Tm3+ ions reach spin-polarized paramag-
netic configuration. Then the k–f exchange interac-
tion leads to a splitting of the conduction band into
sub-bands, and the value of this band splitting is pro-
portional to the magnetic moment, density of states,
and the k–f exchange integral.
The dHvA effect measurements for the magnetic
TmGa3 were performed on a spherical sample (diame-
ter 2.5 mm) by using a standard field modulation
technique at temperatures down to 1.5 K and in mag-
netic fields up to 13 T applied along the principal
crystallographic axes. For a spherical sample, as we
have used, one has B H / M� �appl ( )8 3� . The magne-
tization in magnetic fields, used for the dHvA effect
study, depends rather weakly on the magnetic field
strength. Complementary magnetization measurements
were performed by a home-made vibrating sample
magnetometer.
A standard Cu–Be clamp was used for the pressure
effect study with an extracted benzine solvent as the
medium transmitting pressure to the sample. The max-
imum pressure employed was 6.4 kbar at 4.2 K. A
small Manganin coil with resistance about 60 � was
placed near the sample to measure the applied pres-
sure. Preliminarily this coil had been trained to cool-
ing-pressure and then calibrated by measuring the
superconducting transition temperature of Sn [16].
The deviation of the Manganin coil resistance due to
the residual magnetic field of the superconducting
magnet has been also taken into account. The sample,
the pick-up coil, and the Manganin coil, all were
placed in a Teflon cell, filled with the extracted ben-
zine solvent, and then the cell was put into the pres-
sure clamp. The deviation from hydrostatic pressure
and its effect on the measurements are estimated to be
negligible by observing that the superconducting tran-
sition width of Sn does not change noticeably, and the
amplitudes of the dHvA oscillations do not decrease
substantially under the pressures used in this work.
Since the pressure clamp is heated by the modulation
field, there is a difference in temperatures between the
helium bath and the sample in the pressure clamp. The
modulation amplitude and frequency used in the mea-
surements were 40 G and 38.5 Hz, respectively. These
amplitude and frequency were chosen to produce a
large enough dHvA signal, and, at the same time, to
reduce the heating power, which leads to a tempera-
ture difference not exceeding 0.02 K.
The applied pressure modifies the magnitude and
field dependence of the magnetization due to a pres-
sure effect on the crystal field (CF) splitting, as well
as on the exchange interaction [2,3]. It is known that
the CF of metallic rare-earth compounds contains con-
tributions from charges of surrounding ligands as well
as from the direct Coulomb and exchange interactions
of the R ion with conduction electrons. In order to
estimate the influence of pressure on the CF we have
restricted ourselves to the contribution from surround-
ing ligands within the point charge model. The ap-
plied pressure P brings about the volume dilatation
�V/V P/cB� � , were cB is the bulk modulus. Under
a pressure of 10 kbar �V/V is estimated to be –0.013,
provided the bulk modulus of TmGa3 is taken from
Ref. 14 (cB = 765 kbar). The change of CF due to this
dilatation causes a variation of the magnetic induction
not larger than 20 G at 1.7 K in an applied field of
13 T. One can also estimate the change of the magne-
tization in TmGa3 due to the variation of the ex-
change interaction parameter under pressure by using
the data obtained for the isostructural RIn3 com-
pounds [17,18]. The corresponding variation of the
magnetic induction at an applied pressure of 10 kbar is
about –10 G at 1.7 K in a field of 15 T. Therefore, the
total change of the magnetic induction reaches only
10 G, giving a relative variation of the dHvA fre-
quency �F/F � � �2 10 4 kbar �1, which can be ne-
glected in the Fourier analysis of the dHvA oscil-
lations.
414 Fizika Nizkikh Temperatur, 2005, v. 31, Nos. 3/4
V.B. Pluzhnikov, G.E. Grechnev, A. Czopnik, and O. Eriksson
100
50
0
1 2 3 4 50
TmGa
H 001�� � �
H
H
1 2 3 4 5 6 7
T, K
T, K
H
,k
O
e
20
15
10
5
0
Hc2
Hc1
3
Fig. 1. Low-field magnetic phase diagram of TmGa3 in
magnetic field applied along the � �001 axis (cited from
Ref. 13). The inset shows the whole diagram.
The dHvA effect measurements were carried out in
magnetic fields higher than 7 T, where the magnetiza-
tion does not change appreciably and the Fourier anal-
ysis of the dHvA oscillations can be performed. Other-
wise a dHvA frequency would change its value follow-
ing the strength of external magnetic field. Therefore,
the dHvA effect studies were carried out in the para-
magnetic phase well above the H Tc2( ) line of the
antiferromagnetic–paramagnetic transition (Fig. 1),
except for the � �100 axis, for which the critical field
reaches significant values. The magnetization in mag-
netic fields higher than 7 T tends to saturate, and the
magnetic moments settle into a quasi-ferromagnetic
configuration. Moreover, the magnetization along all
directions in a magnetic field higher than 7 T appeared
to be almost temperature independent in the range
1.7–4.2 K (Fig. 2, cited from Ref. 13).
The effects of the antiferromagnetic and antifer-
roquadrupolar order on the FS of TmGa3 have not
been examined in this study. For these phases the
large magnetization value and its strong dependence
on magnetic field have not allowed analysis of the
dHvA oscillations. Also, a very narrow temperature
range ( � 01. K) for the antiferroquadrupolar phase
has prevented the corresponding study of the dHvA ef-
fect.
3. Details of calculations
A treatment of localized strongly correlated 4f
electrons still presents a challenge to the band struc-
ture theory. The results of ab initio calculations (see,
e.g., Refs. [2,3,18–21]) together with a wealth of ex-
perimental data (including bulk and FS properties)
provide solid evidence that within the local spin-den-
sity approximation (LSDA) [22] a strict band treat-
ment of the 4f states is inadequate for heavy rare
earths. The f shell is not filled, and the 4f bands,
which act as a sink for electrons, would always cut the
Fermi level EF leading to absurd values of the specific
heat coefficients [19] and wrong 4f occupancies, close
to the divalent (i.e., atomic) configuration [23].
According to the photoemission data [23–25], the
4f spectral density for Er, Tm, and their compounds
were observed about 5 eV below EF . Therefore, in or-
der to describe the band structure of the ground state
of TmGa3 near EF , it is feasible to consider the 4f
states as semi-localized core states, in line with Refs.
[18,20,26]. The bulk and magnetic properties calcu-
lated within this approach, as well as the Fermi sur-
faces of Gd, Tb [20], and ErAs [21] appeared to be in
agreement with experimental data. Actually, the stan-
dard rare-earth model [19] is employed in this work in
the limit of large Hubbard repulsionU within the ab
initio LSDA scheme [22] for the exchange-correlation
effects. The localized f states of Tm were treated as
spin-polarized outer-core wave functions, contributing
to the total spin density, and the spin occupation num-
bers were fixed by applying the Russel–Saunders cou-
pling scheme to the 4f shell, which was not allowed to
hybridize with conduction electrons.
The ab initio band structure calculations were
carried out for the paramagnetic configuration phase
ofTmGa3 and non-spin-polarized LuGa3 by using
the full potential linear muffin-tin orbital method
(FP-LMTO) [27,28]. In the case of TmGa3, the spin
density of the 4f states polarizes the «spin-up» and
«spin-down» conduction electron states through the
local exchange interaction. The exchange split con-
duction electron states interact with the localized f
states at other sites, appearing as the medium for the
indirect f–f interaction [18,26]. In order to calculate
FS orbits for both TmGa3 and LuGa3, the charge den-
sities were obtained by including spin-orbit coupling
at each variational step, as suggested in Refs. [19,20].
The band structures and crystal potentials were calcu-
lated self-consistently on a uniform mesh of 455 k
points in the irreducible wedge of the cubic Brillouin
zone for a number of lattice parameters close to the ex-
perimental ones (a � 4196. Å and 4.180 Å for TmGa3
and LuGa3, respectively). The bulk moduli cB were
evaluated from the calculated total energies E V( ) as
functions of volumeV (i.e., from the theoretical equa-
tions of states, according to Ref. 27), and were esti-
mated to be about 800 kbar for TmGa3 and LuGa3,
which is close to the experimental value cB = 765 kbar
(TmGa3 [14]). This is a rather normal overestimation
Pressure effect on the Fermi surface and electronic structure
Fizika Nizkikh Temperatur, 2005, v. 31, Nos. 3/4 415
0 10 20 30 40 50 60 70
H, kOe
6
5
4
3
2
1
M
(
/T
)
�
m
B
T = 1.5 4 5 K TmGa 3
H 001�� � �
Fig. 2. Magnetization of TmGa3 in the magnetic field
applied along � �001 at different temperatures (cited from
Ref. 13).
of cB , presumably due to the overbonding tendency of
LSDA.
The calculated total and partial densities of states
(DOS) N E( ) for LuGa3 are presented in Fig. 3. There
are two fairly broad peaks (bonding and antibonding
states) arising due to hybridization of 5d states of Lu
and the p states of Ga. As can be seen in Fig. 3, these p
states give a substantial contribution to the conspicu-
ous peak in the total DOS at the Fermi energy EF .
The calculated total and partial DOS for TmGa3 ap-
peared to be in a qualitative agreement with the N E( )
of LuGa3, as well as with the previously calculated
TmGa3 DOS (see Fig. 7 in Ref. 2). The intersections
of the calculated FS of LuGa3 with faces of the cubic
Brillouin zone (Fig. 4) show the almost spherical elec-
tron FS centered at the R point and the complicated
multiply connected hole FS centered at the � and X
points, analogously to TmGa3 and also ErGa3 ([3]).
As a whole, the electron FS of RGa3 is nearly spheri-
cal, whereas the hole FS is a complicated multiply
connected surface.
In agreement with the results of Ref. 20 for Gd and
Tb, the incorporation of the spin–orbit coupling has a
small effect on the calculated dHvA frequencies and
cyclotron masses. It should be noted that for the
field-induced quasi-ferromagnetic configuration of
TmGa3 the exchange splitting is larger than the
spin–orbit splitting, and the dHvA spectrum of
TmGa3 can be compared with the results of band
structure calculations for the spin-polarized state.
4. Results and discussion
The Fourier spectra of dHvA oscillations in
TmGa3, observed along � �100 axis at different pres-
sures, are presented in Fig. 5, and the pressure effect
on the corresponding dHvA frequencies is exhibited in
Fig. 6. The branch a originates from the belly orbit in
the band 7 electron FS centered at the R point,
whereas the d orbit comes from the nearly spherical
416 Fizika Nizkikh Temperatur, 2005, v. 31, Nos. 3/4
V.B. Pluzhnikov, G.E. Grechnev, A. Czopnik, and O. Eriksson
–0.4 –0.2 0 0.2 0.4
E (Ry)
0
20
40
D
O
S
(s
ta
te
s/
R
y)
LuGa3
Fig. 3. Total (solid line) and partial densities of states
(DOS) N(E) relative to the Fermi energy EF = 0 for
LuGa3. The dashed line stands for the p states of Ga, and
the dashed-dotted line represents d states of Lu.
R
M
T Z
XS
T
MZ
�
!
"
M
Fig. 4. Intersection of the Fermi surface for LuGa3 with
the Brillouin zone faces.
0 40 80 120
0 kbar
h
b
d
a
4.39 kbar
6.4 kbar
h
b
d a
ad
b
h
Frequency, MG
d
H
vA
am
p
lit
u
d
e
,a
rb
. u
n
its
Fig. 5. Fourier spectra of the dHvA oscillations observed
in TmGa3 at 1.9 K for magnetic fields directed along � �001
axis at different pressure.
part of the hole FS at the � point (see Fig. 4). The or-
bit b is associated with the FS centered at the X point,
and the low frequency orbits h are related to «arms» in
the band 6 hole FS. At ambient pressure the experi-
mental angular dependent dHvA frequencies appeared
to be very close to those previously reported for
TmGa3 and LuGa3 (Figs. 2 and 3 in Ref. 2, respec-
tively), and also to the results of the present
FP-LMTO calculations.
In the range of high dHvA frequencies (branches a
and d) the spectra of LuGa3 and TmGa3 are very simi-
lar. In the low dHvA frequency range, instead of sin-
gle b and h branches for LuGa3, two b and several
h-type branches were observed for TmGa3. For all
principal crystallographic axes, � �100 , � �110 , and � �111 ,
the dHvA frequencies F at ambient pressure, their
pressure derivatives, d F/dPln , the corresponding cy-
clotron masses mc
* together with their pressure deri-
vatives, d m /dPcln * , are given in Tables 1 and 2 for
LuGa3 and TmGa3, respectively. The pressure coeffi-
cients d F/dPln are determined by fitting a straight
line to each set of data in ln F versus P plots (see,
e.g., Fig. 6). For comparison and further discussion,
the analogous results on F, d F/dPln , and mc
* in the
isovalent ErGa3 compound are taken from our paper
[4] and shown in Table 3. It is worth noting that the
«exchange-split» dHvA oscillations, which should
originate from the slightly spin-polarized sub-bands of
TmGa3, were not resolved in this work within the
limits of the large experimental errors. In contrast to
investigations of Refs. [2,3], in the present dHvA ex-
periments additional errors emerged due to the weak
signal from the pick-up coil, which had to be placed in
the pressure clamp, and also due to possible non-
hydrostatic pressure conditions at the sample. There-
fore, only solid and reliable high pressure results are
presented in the Tables, whereas questionable data are
omitted.
Pressure effect on the Fermi surface and electronic structure
Fizika Nizkikh Temperatur, 2005, v. 31, Nos. 3/4 417
0 2 4 6
–0,03
–0,02
–0,01
0
0,01
a
d
b
h
Pressure , kbar
"
F/
F(
0
)
Fig. 6. Fractional changes of the dHvA frequencies,
"F/F F P F /F( ) [ ( ) ( )] ( )0 0 0� � , in TmGa3 as a function of
pressure for the � �001 magnetic field direction at 1.9 K.
The frequencies are labeled according to Ref. 2 and Fig. 5.
The solid lines are guides for the eye.
Table 1. The dHvA frequencies F (in MG) and the corresponding cyclotron effective masses mc
* (in units of free electron
mass) inLuGa3 at ambient pressure, their logarithmic pressure derivatives (in 103 kbar�1), and the mass enhancement factor �.
Field direct-
ion, branch
F d F/dPln mc
* � d mc/dPln *
exper. exper. theory exper. theory exper. theory
� �100 , a 98.06 +1.2(0.2) 1.0 0.74 0.38 0.95 — —
d 41.3 +1.0(0.2) 1.3 0.63 0.48 0.31 —15(3) –5
b 11.87 —3.4(0.1) — 0.3 — — —6.5(0.3) —
h 4.92 — — 0.33 — — —38(9) —
� �110 , a 94.6 +1.2(0.2) 0.9 0.73 0.36 1.03 — —
#b 14.9 —2.6(0.2) — 0.36 — — +17(3) —
b 12.76 —3.0(0.2) — 0.47 — — +12(2) —
h 3.97 —4.2(0.4) — 0.23 — — +35(1) —
� �111 , a 88.6 +1.3(0.8) 1.0 0.57 0.36 0.58 –8.5(1) –3
d 35.6 +1.6(0.2) 1.4 0.53 0.39 0.36 –5.5(2) –4
h 4.8 –3.1(0.6) – 0.24 — — –20(1) –
As is seen from the Tables, the larger dHvA fre-
quencies a and d increase with pressure. On the other
hand, for the hole FSs b and h the derivatives
d F/dPln appeared to be negative. Also, the observed
pressure dependences of the dHvA frequencies are
quite different among ErGa3, TmGa3, and the non-f
reference compound LuGa3. The experimental pres-
sure derivatives d F/dPln , presented in Tables 1 and
2, are rather large in comparison with the free-electron
scaling prediction, which gives two-thirds of the vol-
ume compressibility, or 0.87� �10 3 kbar �1, provided
the available bulk modulus of TmGa3 [14] is ac-
cepted. This scaling estimation actually means that
with increasing pressure, the volumes of the Brillouin
418 Fizika Nizkikh Temperatur, 2005, v. 31, Nos. 3/4
V.B. Pluzhnikov, G.E. Grechnev, A. Czopnik, and O. Eriksson
Table 2. The dHvA frequencies F (in MG) and the corresponding cyclotron effective masses mc
* (in units of
free electron mass) in TmGa3 at ambient pressure, their logarithmic pressure derivatives (in 103 kbar �1), and the
mass enhancement factor �.
Field direct-
ion, branch
F d F dPln / mc* � d mc dPln * /
exper. exper. theory exper. theory exper. theory
� �100 , a 98.68 +1.8(0.3) 1.2 1.24 0.41 2.0 +1.6(1.6) —4
d 41.1 +2.0(0.4) 1.6 1.3 0.46 1.8 —14(3) —7
b 11.61 —3.8(0.2) — 0.83 — — +16(8) —
h 3.58 —4.5(1.7) — — — — — —
� �110 , a 94.41 +1.5(0.1) 1.1 1.04 0.38 1.7 —7(4) —4
b# 14.46 —2.9(0.3) — — — — — —
b 12.66 —2.8(0.1) — 0.96 — — —29(6) —
h# 4.8 —2.3(0.2) — — — — — —
h 3.57 —4.3(0.3) — 0.46 — — — —
� �111 , a 87.28 +1.8(0.3) 1.1 0.91 0.38 1.4 +22(10) —3
d 34.24 +2.1(0.5) 1.5 0.83 0.42 1.0 +3.6(0.7) —6
h 4.31 —6.0(0.3) — 0.46 — — —22(6) —
Table 3. The dHvA frequencies F (in MG) at ambient pressure, their logarithmic pressure derivatives (in
10 kbar3 �1), the corresponding cyclotron effective masses mc
* (in units of free electron mass), and the mass en-
hancement factor � in ErGa3 compound.
Field direct-ion,
branch
F d F/dPln mc
* �
exper. exper. theory exper. theory
� �100 , a 98.71 +2.3(0.3) 1.3 0.96 0.40 1.4
d 41.07 +1.7(0.2) 2.0 0.91 0.46 0.98
b 12.66 —2.7(0.1) — 2.8 0.44 — —
h 4.35 — — 0.55 — —
� �110 , a 95.17 +1.0(0.3) 1.1 0.89 0.37 1.4
b# 15.14 —1.1(0.1) — 0.57 — —
b 11.95 —2.4(0.2) —2.0 0.84 — —
h 3.37 —4.5(0.3) — 0.28 — —
� �111 , a 87.58 +1.7(0.1) 1.2 0.80 0.37 1.16
d 35.47 +2.3(0.2) 1.9 0.70 0.40 0.75
h 4.21 — — 0.51 — —
zone and the FS increase, and one can also expect that
both the hole and electron FS increase. However, this
effect cannot explain the negative pressure derivatives
for the hole FS.
In the framework of the band theory, the overlap of
the wave functions between the 4p bands of Ga and
the 5d bands of R increases with pressure, and the p –d
hybridization becomes stronger. Consequently, while
the volume of the big spherical FS sheets increases,
the volumes of small «arms» of the hole FS may
decreases due to strong hybridization and substantial
deviation from free-electron scaling, and this can pro-
vide the anisotropic dHvA frequency changes. Ba-
sically, there is qualitative agreement between the ex-
perimental and calculated derivatives d F/dPln for
the a and d orbits in LuGa3 (again, the bulk modulus
of TmGa3 was used to convert the calculated volume
derivatives to the pressure ones, listed in the Tables).
However, this band approach can not explain the dis-
crepancy between the experimental and calculated pres-
sure derivatives of the dHvA frequencies in TmGa3.
It was originally suggested in Ref. 29, that in ferro-
magnetic systems there are two contributions to the
pressure derivative of a dHvA frequency. The first
(«potential») contribution comes from an atomic vol-
ume effect on the crystal potential, and also from a
scaling effect due to the change of the Brillouin zone
size. It can be approximated by the corresponding de-
rivative for a nonmagnetic reference compound with a
close value of the compressibility (in our case it can be
LuGa3). The second («magnetic») contribution origi-
nates from the pressure induced redistribution of con-
duction electrons between exchange-split sub-bands
and the corresponding changes of the volume enclosed
by the FS sheets. According to the present calcula-
tions, the Fermi surfaces of TmGa3 do not change uni-
formly because of a strong k-dependent p–d mixing ef-
fect on the exchange-split conduction band. As a
result, the difference between pressure derivatives of
the spin-split FS cross-sectional areas is inconsistent
with simple estimations based on the Stoner–Wohl-
farth model.
Glancing at the experimental lattice parameters,
which are very close in TmGa3 and LuGa3, one may
assume that crystal potential in the two compounds
does not differ substantially, and that the differences
in the dHvA frequencies derivatives of TmGa3 and
LuGa3 compounds can be attributed to the unfilled 4f
shell in TmGa3. In the 4f spin-polarized state the dif-
ferences can arise, first, from the exchange splitting of
the conduction band of TmGa3, and, secondly, from
the magnetostriction effects due to the �5
1( ) CF ground
state of the 3H6 multiplet of Tm3+ ion. Unfortunately,
at the present stage the differences in d F/dPln can
not be ascribed confidently either to the conduction
band splitting or to the magnetostriction.
Cyclotron masses mc
* have been determined at am-
bient and high pressures for the most dHvA fre-
quencies in the field applied along the � �100 , � �110 ,
and � �111 axes, and are presented in Tables 1 and 2.
Band cyclotron masses mc
b were calculated for the a
and d branches, and were also given in the Tables.
Also, the cyclotron masses measured and calculated
for ErGa3 in Refs. [3,4] are listed in Table 3 for com-
parison.
The mass enhancement factor �, which is defined by
relation m mc c
b* ( )� �1 � , presents a measure of the in-
teraction strength of the conduction electrons with
low-energy excitations, and it can be determined by
comparison of the experimental cyclotron effective
masses with the corresponding calculated ones. The �
factors for electrons on the a and d orbits in LuGa3,
TmGa3, and ErGa3 are listed in the Tables. In the
nonmagnetic LuGa3 the � factor represents a measure
of the electron–phonon interaction, whereas in ErGa3
and TmGa3 this factor also contains contribution(s)
coming from magnetic excitations. As seen in Table 1,
in LuGa3 the � factor ranges from 0.3 (d branch) to
about 1 (a branch). Assuming that the values of � e- ph
in RGa3 are close to the corresponding ones in LuGa3,
we estimated the magnetic contributions �mag in
ErGa3 to be 0.4–0.6 and 0.4–0.7 for the a and d orbits,
respectively. In TmGa3 the corresponding values of
�mag are larger and more anisotropic, namely, 0.5–1
and 0.8–1.5.
The hybridization of conduction electrons with 4f
states could contribute to the larger cyclotron masses
observed in TmGa3 and ErGa3 as compared to LuGa3.
However, a strong hybridization with 4f bands in the
framework of the LSDA would lead to a substantial
reduction of the conduction band width in RGa3 and
thus to bulk properties remarkably different from that
in LuGa3. In fact, the lattice parameters decrease
slightly in a linear fashion in the series ErGa3,
TmGa3, and LuGa3 due to the lanthanide contrac-
tion, and it can be expected that conduction band
widths are close in RGA3, and therefore band cyclo-
tron masses should be also close. At the present stage
more elaborated analysis is necessary to estimate the
scale of the hybridization effects by employing modern
theoretical schemes (like the LSDA + U approach)
and additional experimental data (e.g., for related
RIn3 compounds).
One can also assume that the distinctions between
effective masses in RGa3 are probably due to the dif-
ferent ground state multiplets 3
6H and 4
15 2I / of
Tm3� and Er3+ ions in the CF of TmGa3 and ErGa3,
respectively. The triplet �5
1( ) with intrinsic magnetic
Pressure effect on the Fermi surface and electronic structure
Fizika Nizkikh Temperatur, 2005, v. 31, Nos. 3/4 419
and quadrupolar moments is the ground state in
TmGa3 [14], and most likely �7 is the ground state in
CF of ErGa3 [30]. Since the �5
1( ) state exhibits
quadrupolar moment and the �7 state does not, one
can expect large magnetostriction effects in TmGa3
and none in ErGa3.
In this connection one can expect that the presence
of strong quadrupolar interactions increases the cyclo-
tron effective masses. It is well known [31] that the
quadrupolar excitations does not occur in compounds
with the cubic symmetry. They may appear, however,
in TmGa3 as coupled magnetic-quadrupolar excita-
tions in applied magnetic field [14]. It was shown in
Ref. 31 that the corresponding excitations contribute
to the effective mass of the conduction electrons, and
the effect appears to be large and magnetic field de-
pendent. Also, in the quasi-ferromagnetic configura-
tion of magnetic moments, the exchange splitting of
the conduction bands can vary in ErGa3 and TmGa3
due to the difference in corresponding 4f-shell spin oc-
cupation numbers. At the moment, however, one can
not estimate the relative contributions of the conduc-
tion band splitting and the magnetostriction to the ob-
served differences in cyclotron masses and the pressure
dependences of the dHvA frequencies in TmGa3 and
ErGa3.
Also, one more mechanism can affect the cyclotron
masses in RGa3. It was shown in Refs. [31–33] that
virtual magnetic excitations can contribute substan-
tially to the effective mass of the conduction electrons
in rare-earth systems. These excitations are magnetic
excitons in a paramagnetic system (e.g., praseodym-
ium), and spin waves in magnetically ordered rare
earths. The corresponding mass enhancement is ex-
pected to be large, magnetic field dependent, and pro-
portional to the static susceptibility of the magnetic
system. According to estimations of the electronic spe-
cific-heat coefficients in Ref. 33, the corresponding ef-
fective masses increase in the series of heavy rare-earth
metals. This trend is consistent with the relation be-
tween observed cyclotron masses in ErGa3 and
TmGa3, and also in ErIn3 and TmIn3 compounds [6].
However, considerable work is needed to implement
findings of Refs. [32,33] for a quantitative description
of cyclotron masses in magnetic RM3 compounds.
In conclusion it should be noted that the calculated
LSDA band cyclotron masses mc
b did not reproduce the
highly anisotropic pressure derivatives of mc
* in
TmGa3 (see Table 2), which are presumably deter-
mined by various magnetic and many-body excita-
tions. On the other hand, the calculated d m /dPc
bln
in LuGa3 appeared to be in a qualitative agreement
with the corresponding experimental data in Table 1.
Summary
As a whole, the present results on the dHvA effect
and FS in TmGa3 and ErGa3 at ambient pressure are
in good agreement with the recent two-dimensional
angular correlation of the positron annihilation radia-
tion (2D-ACAR) studies (Refs. 9 and 10, respec-
tively). The calculated pressure derivatives of the
dHvA frequencies in LuGa3 and TmGa3 appeared to
be in qualitative agreement with the experimental
data, though the origin of some discrepancies found
for d F/dPln in TmGa3 is not clear. The estimated
magnetic contributions to the mass enhancement fac-
tor � in ErGa3 and TmGa3 appeared to be large and
magnetic field dependent. We admit that more work is
needed to elucidate the nature of the large cyclotron
masses observed in RGa3 and, in particular, to evalu-
ate the magnetic excitations effects on mc
*. Also, the
surprisingly large and highly anisotropic pressure ef-
fect on the cyclotron masses has been observed in
LuGa3 and especially in TmGa3, which can not be ex-
plained within the employed standard rare-earth mo-
del. It has to be emphasized that different interactions
(exchange splitting, CF and magnetic-quadrupolar
excitations, spin waves) have to be taken into account
in a further theoretical analysis of the revealed pres-
sure effects on the FS and cyclotron masses in RGa3.
Also, at the present stage a more elaborated study is
necessary to estimate the scale of the hybridization ef-
fects with 4f states in ErGa3 and TmGa3 within mod-
ern theoretical approaches (e.g. LSDA + U). These
tasks apparently go beyond the aim of the present
work, in which we tried to reach a true LSDA limit
within the standard rare-earth model in order to ex-
plain the experimental dHvA data. It can be also ben-
eficial to supplement such theoretical efforts with the
experimental study of the pressure effect on the dHvA
frequencies and cyclotron masses in the relative RIn3
compounds.
The authors dedicate this work to the 90th anniver-
sary of E.S. Borovik, who was one of pioneers of the
Fermi surfaces studies [34].
We are grateful to Professors J. Klamut, T. Pa-
lewski, V. Nizhankovskii, and I.V. Svechkarev for
their kind support and fruitful scientific discussions.
This work has been partly supported by The Swed-
ish Natural Science Research Council (VR) and The
Swedish Foundation for Strategic Research (SSF).
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Pressure effect on the Fermi surface and electronic structure
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