-
This content has been downloaded from IOPscience. Please scroll
down to see the full text.
Download details:
IP Address: 130.236.83.247
This content was downloaded on 08/05/2017 at 14:39
Please note that terms and conditions apply.
Topological transitions of the Fermi surface of osmium under
pressure: an LDA+DMFT study
View the table of contents for this issue, or go to the journal
homepage for more
2017 New J. Phys. 19 033020
(http://iopscience.iop.org/1367-2630/19/3/033020)
Home Search Collections Journals About Contact us My
IOPscience
You may also be interested in:
Electronic structure and de Haas–van alphen frequencies in
KFe2As2 within LDA+DMFT
Steffen Backes, Daniel Guterding, Harald O Jeschke et al.
A charge self-consistent LDA+DMFT study of the spectral
properties of hexagonal NiS
S K Panda, P Thunström, I Di Marco et al.
First-principles study of osmium under high pressure
Yongcheng Liang and Zhong Fang
Pressure-dependent semiconductor to semimetal and Lifshitz
transitions in 2H-MoTe2: Raman and
first-principles studies
Achintya Bera, Anjali Singh, D V S Muthu et al.
Generalized dynamical mean-field theory in the physics of
strongly correlated systems
Eduard Z Kuchinskii, Igor A Nekrasov and Mikhail V Sadovskii
Theory of the insulator ground state of the plutonium
monochalcogenides: An LDA+DMFT investigation
Michi-To Suzuki and Peter M Oppeneer
Implementation of LDA+DMFT with the pseudo-potential-plane-wave
method
Zhao Jian-Zhou, Zhuang Jia-Ning, Deng Xiao-Yu et al.
Dynamical screening in correlated electron systems—from lattice
models to realistic materials
Philipp Werner and Michele Casula
Ab initio calculations of the elastic and thermodynamic
properties of gold under pressure
N A Smirnov
http://iopscience.iop.org/page/termshttp://iopscience.iop.org/1367-2630/19/3http://iopscience.iop.org/1367-2630http://iopscience.iop.org/http://iopscience.iop.org/searchhttp://iopscience.iop.org/collectionshttp://iopscience.iop.org/journalshttp://iopscience.iop.org/page/aboutioppublishinghttp://iopscience.iop.org/contacthttp://iopscience.iop.org/myiopsciencehttp://iopscience.iop.org/article/10.1088/1367-2630/16/8/083025http://iopscience.iop.org/article/10.1088/1367-2630/16/9/093049http://iopscience.iop.org/article/10.1088/0953-8984/18/39/007http://iopscience.iop.org/article/10.1088/1361-648X/aa55a1http://iopscience.iop.org/article/10.1088/1361-648X/aa55a1http://iopscience.iop.org/article/10.3367/UFNe.0182.201204a.0345http://iopscience.iop.org/article/10.1088/1757-899X/9/1/012086http://iopscience.iop.org/article/10.1088/1674-1056/21/5/057106http://iopscience.iop.org/article/10.1088/0953-8984/28/38/383001http://iopscience.iop.org/article/10.1088/1361-648X/aa58ca
-
New J. Phys. 19 (2017) 033020
https://doi.org/10.1088/1367-2630/aa5f8e
PAPER
Topological transitions of the Fermi surface of osmium
underpressure: an LDA+DMFT study
QingguoFeng1,2,Marcus Ekholm1,2, FerencTasnádi2, H JohanM
Jönsson2 and IgorAAbrikosov2,3
1 Swedish e-Science ResearchCentre (SeRC), LinköpingUniversity,
SE-58183 Linköping, Sweden2 Department of Physics, Chemistry and
Biology (IFM), LinköpingUniversity, SE-58183 Linköping, Sweden3
MaterialsModeling andDevelopment Laboratory, NUST ‘MISIS’,
119049Moscow, Russia
E-mail: [email protected]
Keywords: electronic topological transitions, strong
correlations, Fermi surfaces
AbstractThe influence of pressure on the electronic structure
ofOs has attracted substantial attention recentlydue to reports on
isostructural electronic transitions in thismetal. Here, we
theoretically investigatethe Fermi surface ofOs from ambient to
high pressure, using density functional theory
combinedwithdynamicalmean field theory.We provide a detailed
discussion of the calculated Fermi surface and itsdependence on the
level of theory used for the treatment of the electron–electron
interactions.Althoughwe confirm thatOs can be classified asweakly
correlatedmetal, the inclusion of localquantumfluctuations between
5d electrons beyond the local density approximation explains
themostrecent experimental reports regarding the occurrence of
electronic topological transitions inOs.
1. Introduction
Osmiumhas the highest density of all the known elements.
High-pressure experiments and ab initio calculationsindicate that
the bulkmodulus of hexagonal close-packed (hcp)Os rivals that of
diamond carbon [1–6].Recently, a combined experimental and
theoretical studywas presented, identifying two types of
electronictransitions under pressure [1]. These transitions
expressed themselves as anomalies in the pressure evolution
ofthemeasured c/a-ratio of the hcp structure. At approximately 150
GPa, a so-called electronic topologicaltransition (ETT)was found,
which is a topological change of the Fermi surface, also known as a
Lifshitztransition [7–9]. Such transitions have previously been
demonstrated to cause anomalies in themeasured c/a-ratio [10] . It
is worth tomention that a second anomaly seen above 400 GPa could
not be identifiedwith anyETT, but was explained by the overlap of
core electron levels. Experiments also showed that the hcp
structure isstable up to the highest pressure achieved, above 770
GPa.
Earlier experimental and theoretical work has yielded
conflicting conclusions regarding the c/a-anomalies.Measurements
byOccelli et al [4]reported an anomaly in the c/a-ratio at 25
GPa.However, a similar study up to60GPa byTakemura [5] did not
report any suchfinding. On the theoretical side, Sahu andKleinman
[6]made afully relativistic ab initio calculation of the lattice
constants, and predicted a change of slope in the
pressuredependence of c/a-ratio at 9.5GPa, but they did not
consider the evolution of the Fermi surface under pressure.Ma et al
[11] calculated an equation of state (EOS) based on the generalized
gradient approximation (GGA) andfound an anomaly in the c/a ratio
at 25GPa, but without any sign of an ETTup to 80 GPa.
However, high precision calculations byKoduela et al [12],
demonstrated that although the formof theexperimental
pressure–volume curve is well reproducedwithGGA, it is generally
shifted towards largervolumes. Using the local density
approximation (LDA), an EOS in closer agreementwith experiment
wasobtained. Three ETTswere identified between 70 and 130GPa, at
different points in the Brillouin zone (BZ).Nevertheless, the
observed ETTswere concluded to be unrelated to the c/a anomaly.
Finally, a study by Liangand Fang [14]did notfind any peculiarity
in c/a-ratio or any sign of ETTs up to 150GPa.
In [1], the controversy was resolved. Using a highly accurate
experimental EOS,measured up to 770 GPa,and treating
electron–electron interactions with themore advanced scheme of
combining the LDAwith
OPEN ACCESS
RECEIVED
10August 2016
REVISED
12December 2016
ACCEPTED FOR PUBLICATION
9 February 2017
PUBLISHED
13March 2017
Original content from thisworkmay be used underthe terms of the
CreativeCommonsAttribution 3.0licence.
Any further distribution ofthis workmustmaintainattribution to
theauthor(s) and the title ofthework, journal citationandDOI.
© 2017 IOPPublishing Ltd andDeutsche
PhysikalischeGesellschaft
https://doi.org/10.1088/1367-2630/aa5f8emailto:[email protected]://crossmark.crossref.org/dialog/?doi=10.1088/1367-2630/aa5f8e&domain=pdf&date_stamp=2017-03-13http://crossmark.crossref.org/dialog/?doi=10.1088/1367-2630/aa5f8e&domain=pdf&date_stamp=2017-03-13http://creativecommons.org/licenses/by/3.0http://creativecommons.org/licenses/by/3.0http://creativecommons.org/licenses/by/3.0
-
dynamicalmeanfield theory (the LDA+DMFTmethod [15]), it was
demonstrated that the peculiarity of the c/aratio does exist at∼150
GPa and that it can be related to ETTs.However, no studies of the
Fermi surfaceevolutionwith pressure in hcpOswhich include
electron–electron interactions beyond the LDAor the semi-local
GGAhave been reported so far. The LDA+DMFTmethod does take
dynamical correlation effects intoaccount, and it is capable of
interpolating between themetallic regime and the strongly
localizedMott-insulatinglimit.
In this paper, we present further details of the electronic
structure ofOs in the pressure rangewhere theseETTswere detected,
and demonstrate the impact of electron correlations beyond the
LDA—local quantumfluctuations on the Fermi surface of
thismetal.
We calculate the electronic structure ofOswithin the
LDA+DMFTmethod for the range of pressures, fromzero to about 250
GPa, and compare the Fermi surface and charge distributions
obtainedwithin LDA+DMFTand LDA. Band structure, Fermi surface, and
charge density distribution plots give complementary informationon
this system fromdifferent aspects. In particular, earlier density
analysis ofOs has suggested the valence chargeto be highly
localized in the nearest-neighbor bonds [16], which could partly
explain its high bulkmodulus. It istherefore important to
investigate what happens to theOs electronic structure and charge
density when localquantumfluctuations are included, especially
under pressure.
2.Methods and details of calculation
2.1. Lattice constantsAlthoughDMFT calculations can nowadays be
carried out to determine the EOS for a transitionmetal [17],
theyare still quite time consuming. Therefore, in the present
workwe have used the experimental lattice constantspresented in [1]
in order to relate the unit cell volume to pressure. To avoid
additional complications incomparing LDA+DMFT andLDA results, the
latter were obtained at the experimental lattice parameters
aswell.Besides, the use of experimental lattice parameters allows
us to avoid complications related to inaccuracies of thecalculated
EOS forOs, clearly demonstrated in [1], as well as to skip the
necessity to compare results ofcalculations obtainedwithin local
(LDA) and semi-local (GGA) approximations of density functional
theory.Indeed, it is well documented, that the electronic structure
as calculatedwith either LDAorGGA at the samelattice constants, is
practically indistinguishable in nonmagnetic systems [13]. The
employed values aresummarized in table 1.
2.2. LDA calculationsIn our LDA calculations we have used the
all-electron full potential band structuremethod, provided in
theWien2k code [18].We used a ´ ´32 32 32 k-mesh, and kept the
product =·R K 10MT max , wherewe set themuffin-tin radius =R 2.5MT
a.u.
We have also performed calculations using the all-electron full
potential local orbital (FPLO) [19, 20]method.
2.3. LDA+DMFT calculationsFor LDA+DMFT calculations, we have
used the toolkit implemented in the TRIQS package. References
[21–23], which is fully self-consistent in both charge and local
self-energy in the LDA+DMFT cycles.We have treatedthe
partiallyfilled 5d shell as correlatedwithin theDMFT approximation.
The corresponding effectiveHamiltonian has the form:
= - + ( ), 1LDA DC U
å å=ss
ss s s= = ¢ ¢
¢ ¢ ¢ ¢ˆ ˆ ( )U n n1
2, 2
i i l l mmmm m m
U
,d d
Table 1. Lattice constants used in thecalculations. All values
are experimental andtaken from [1].
P (GPa) V (Å3) a (Å) c/a
0 27.960 2.734 1.580
86 23.948 2.592 1.588
134 22.572 2.540 1.590
247 20.304 2.449 1.596
2
New J. Phys. 19 (2017) 033020 QFeng et al
-
whereLDA is the energy of the system in LDA, fromwhich the 5d
contribution due to the on-site Coulombrepulsion is subtracted by
the double counting term,DC, whichwe approximate with the so-called
aroundmeanfield form [24]:
å=-
-
´ -s s
ds s
s s
¹ ¢ ¢
¢ ¢
¢ ¢
ss¢ ( ¯ )
( ¯ ) ( )
U Jn n
n n
2
, 3m m
mm mmm
m
DC
where
å=+
s s¯ ( )( )n
ln
1
2 1, 4
mm
andfinally,U is the Coulomb interaction term.Within eachDMFT
iteration, the impurity problemwas solvedwith the
hybridization-expansion
continuous-time quantumMonteCarlomethod [25]. In eachDMFT
iterationwe performed over 512millionMonte Carlo cycles at inverse
temperature =T1 40 eV−1. The impurity problemwas solved for the
self-energyon theMatsubara axis, after which an analytical
continuationwasmade to the real axis by a stochastic version
ofthemaximumentropymethod [26].
We have used theCoulomb interaction strength =U 2.80 eV
andHund’s coupling constant =J 0.55 eV,based on the estimations in
[27]. It should be noted that theU valuemay differ slightly,
depending on crystalstructure and different computationalmethods
[28–30]. However, for aweakly correlated compound asOs,small
variations in theU parameter should only slightly affect our
quantitative predictions, and not ourqualitative conclusions.
LDA+DMFT electronic structure calculations were performedwithin
the scalar-relativistic approximationand using a kmeshwith 32×32×32
points in the full BZ. Sincewe are interested in comparing the LDA
andLDA+DMFT approximations, we used the same parameters as for LDA.
As shown in [1], the inclusion of spin–orbit coupling in LDA
calculations will split some of the bands, however these features
are located far away fromthe Fermi level, even at high pressure.We
have therefore performed our calculations within the
scalar-relativistic approximation, neglecting spin–orbital
coupling.
2.4.Quasiparticle effectivemassWehave calculated the ratio of
the effectivemass of the correlated quasiparticles,m*, to that of
uncorrelatedelectrons,m, as:
* åå
r w w
r=
- ¶ S ¶¯ ( )[ ( ) ]( )
( )mm
E
E
1 Im i, 5l
l l
l l
F
F
where l is the orbital index, r ( )El F is the density of states
of the lth orbital inω space at the Fermi level andEF isthe Fermi
energy, wS ( )il is the local self-energy on theMatsubara axis.
Since there are several d orbitals and theoccupancy of each orbital
by the itinerant electronswill vary, we take the averaged value of
the effectivemass, *m̄ ,to represent the correlation strength for
all the 5d electrons.
2.5. Electron density distributionIn order to analyze details of
the density distributionwe have calculated the Fourier transformof
the chargedensity to k-space.We have used a very dense 200×200
k-point grid, which is considerably denser than in theFermi surface
calculations. The charge density at each k-point was calculated by
integrating the k-dependentdensity of states below the Fermi level,
i.e., with the following formula:
ò w r w w= w ( ) ( ) ( )n f k, d , 6kE
bottom
F
where the density of states, ρ, was obtained by summing over the
orbital channels:
år w r w=( ) ( ) ( )k k, , , 7l
l
labeled by l. In turn,
r wp
w= -( ) ( ) ( )k G k, 1 Im , , 8l l
where w( )G k,l is the k-dependent Green’s function of the lth
orbital on the real frequency axis. w = + bw( )f1
1 eis the Fermi–Dirac distribution functionwith b = k T1 B ,
where kB is the Boltzmann constant andT is thetemperature. The
charge density, nk, is thus defined on a 2D grid in k-space. In
fact, nk can bemeasuredexperimentally as well, e.g., by x-ray
scattering.
3
New J. Phys. 19 (2017) 033020 QFeng et al
-
Figure 1.Averaged quasiparticlemass for the 5d electrons inOs as
a function of pressure.
Figure 2. Illustration of relevant high symmetry points on the
hcp reciprocal lattice. In (a) and (b), theΓ point is at the center
of theBrillouin zone. In (c), the center of the cell is at the
A-point, andΓ is at the top and bottom faces. Figure (b) shows how
these are relatedto each other. This is amore suitable view of the
reciprocal lattice for the presentation of our results, and is used
infigures 3(b), 6 and 7.
4
New J. Phys. 19 (2017) 033020 QFeng et al
-
3. Results and discussion
3.1. Correlation strengthWebegin by addressing the strength of
electron correlations in hcp-Os. In 5d transitionmetals the effect
ofelectronic correlations is believed to be small. However,
previous studies onmoderately correlated isoelectronicsystems such
as hcp-Fe, have shown that a description of correlations beyond the
LDA level can lead to newfeatures in the electronic structure
[10].More specifically, using themore advanced description provided
byLDA+DMFT, an ETTwas found for hcp-Fe in precisely the pressure
rangewhere an anomaly in c/awas seenexperimentally.
The strength of electron correlations can be quantified by the
effectivemass of the quasiparticles, which arecompared to themass
of uncorrelated electronswith equation (5). As discussed in section
2.4, we express this asaweighted average over all the
5d-orbitals.
Infigure 1, we show the averaged effectivemass of the
quasiparticles plotted against pressure. From thefigure, one can
observe that the effectivemass *m̄ is very close to themass of
uncorrelated electrons. Therefore,
Figure 3.The Fermi surface of hcp-Os at 0 GPa calculatedwith
LDA, using (a) the conventional view of the reciprocal lattice
(seefigure 2(a)), and in (b), taking theA-point at the center of
the zone (seefigure 2(c)). The arrowpoints to the hole pocket at
theΓ point.
5
New J. Phys. 19 (2017) 033020 QFeng et al
-
Figure 4.Band structure obtained at the experimental lattice
constants (from [1]) corresponding to P=0 GPa, using (a)
theWien2kand (b) the FPlomethod.
Figure 5. So-called fat bands calculated at ambient pressure.
The size of the circles indicate the d-electron character of the
bands.
6
New J. Phys. 19 (2017) 033020 QFeng et al
-
this system is aweakly correlated system.Moreover, as expected,
we notice that the effectivemass decreases aspressure is increased,
and kinetic energy starts to be comparable to theCoulomb
energy.
3.2. Fermi surface3.2.1. Ambient pressureWehave calculated the
Fermi surface of hcpOs at different pressures using both the LDA
and LDA+DMFTapproximations. The relevant high symmetry points of
the hcp reciprocal lattice are shown infigure 2. Infigure 2(a), the
standard view of the reciprocal lattice is shown, with the BZ
clearlymarked. To showour results,it ismore illustrative to take
the A-point at the center of the cell, as shown infigure 2(c).
Infigure 3we show the Fermi surface ofOs at the pressure P=0
GPa, as obtainedwith LDA. Figure 3(a),shows the Fermi surfacewith
the convention offigure 2(a). Infigure 3(b), we have replotted the
Fermi surfacewith the convention offigure 2(c). One can see that
the Fermi surface consists of four sheets. Two electron
sheetssurround theΓ point (green and yellow infigure 3(b)), the
yellow one having awaist. In addition, one hole sheetis open (red),
and disconnected ellipsodials formhole pockets (blue). These are
all cut by the L–M–K–Hplane.In addition, we see a hole pocket
around theΓ point.
This topology of the Fermi surface at ambient pressure is in
agreement with the experimental work in [2] andthe theoretical work
in [11, 12]. However, as discussed in [1], details of the Fermi
surface at a given pressure issensitive to the specific choice of
the EOS,which can be obtained by calculations or experiment. For
example, attheΓ point, a hole pocket was seen in [14], in contrast
to [12]. Nevertheless, calculations at the experimentallattice
constants, whichwe have carried outwith the same
computationalmethod as in [12] (the FPLOmethod)clearly show a hole
pocket (seefigure 4).We therefore believe it is beneficial to
compare results atfixed volume,andwe therefore use the experimental
values presented in table 1.
We now compare results obtainedwith the LDA and LDA+DMFTmethods,
to see the impact of correlationeffects beyond the LDA. Figure 6
shows a comparative side-view of the Fermi surface obtainedwith
LDA+DMFT (left column) and LDA (right column). Comparing figures
6(a) and (b), we see howLDA+DMFT givesawider gap at the L point
compared to LDA. In addition, the hole pocket aroundΓ point seen in
the LDAcalculations, is not present at the Fermi surface
calculatedwithin LDA+DMFT. This part of the Fermi surface isnot
visible infigure 6, but is discussed in section 3.3.1.
Infigure 5we show so-called fat-bands, indicating the
d-character of the bands. It is seen that the band edgeat theΓ
point is of sp-character. At the L-point we alsofind a largeweight
of sp-electrons. Themain impact of theLDA+DMFT treatment is
concernedwith the d-states, forming our space of correlated states,
which are shifted
Figure 6.Comparison of the Fermi surface in side-view of hcp-Os
between LDA+DMFT and LDA calculations, where the left column(a),
(c), (e) corresponds to LDA+DMFT and the right (b), (d), (f) to LDA
calculations. In each column thefigures from top to
bottomcorrespond to 0GPa, 134 GPa, and 247 GPa, respectively.
Capital letters denote the high symmetry points (see figure
2(b)).
7
New J. Phys. 19 (2017) 033020 QFeng et al
-
alongwith the chemical potential. Comparingwith LDA, the bands
at theΓ and L-points appear shifted down inenergy.
3.2.2. 134 GPaUpon increase of pressure, the LDA+DMFT results
infigure 6(c) showhow the gap at the L point has only beenreduced.
This is in contrast to the LDA result shown infigure 6(d), where
one of the sheets (red), has becomeconnected,meaning that it has
undergone a so-called necking transition at the L–H line. Figure 7
shows the sameview of the reciprocal lattice as in 2(c), with theΓ
point clearly visible. In LDA (figure 7(b)), we see a
pronouncedpocket at theΓ point. There is no topological change of
the Fermi surface, as compared to ambient pressure (seefigure
3(b)), since the hole pocket is only seen to have become larger. In
LDA+DMFT, (figure 7(a)), the pocket issignificantly smaller. This
pocket is not present at ambient pressure, and the ETToccurs just
above 100 GPa [1],inmuch better agreementwith experiment as
compared to the LDA calculations.
3.2.3. 247 GPaIncreasing pressure further, to 247 GPa, LDA+DMFT
also reveals a necking transition of the Fermi surfacetopology at
the L point, as seen infigure 6(e). In this transition, holes
pockets appear at the L point, as seen inboth LDA and LDA+DMFT.
Thus, this ETT consists of a necking transition, followed by the
appearance of ahole pocket.We do not observe any of the hole
pockets along the L–Mline disappearing, whichwas suggestedin
[4].
Infigure 7we compare the Fermi surface evolution of hcpOs
between the pressure of 134 and 247 GPa.Wedo not see any additional
changes of the Fermi surface topology at theΓ point, neither with
LDA+DMFT(figure 7(a)) norwith LDA (figure 7(b)). Thus, the only ETT
thatwe see in this pressure range corresponds to the
Figure 7.The Fermi surface of hcp-Os at high pressure, where the
left column corresponds to LDA+DMFT (a), (c) and the right toLDA
(b), (d). In each column, the figures from top to bottom correspond
to P=134 GPa and P=247 GPa. Capital letters labelrelevant high
symmetry points. The arrow in (a) shows the hole pocket at theΓ
point. In all four cases, the hole pocket is presentaround theΓ
point.
8
New J. Phys. 19 (2017) 033020 QFeng et al
-
one at the L-point in the LDA+DMFT calculations. This result is
in agreement with the experimental variationof the c/a lattice
parameters ratio with pressure [1].
Concluding the discussion of the Fermi surface calculations for
hcpOs, we see that taking into account localquantumfluctuations
with LDA+DMFT,wefind the ETTs to occur at higher pressure than in
the LDAcalculations. Indeed, with LDA+DMFT the transition at the L
point is found to occur at around 180 GPa, ratherthan at 125 GPa,
as predicted by LDA [1]. The ETT at theΓ point cannot be seen at
all with LDA, since it occurs ata negative pressure. On the
contrary, with LDA+DMFT it shows up just above 100 GPa. In this
range ofpressure, anomalous values of the c/a ratio were observed
in [1].
3.3. Electron density distributionIn order to see the
pressure-induced features at theΓ and L pointsmore clearly, we have
calculated the Fouriertransformof the valence electron charge
density distribution on a plane in k-space containing theΓA, L,
andMpoints. In these 2D calculations onemay employ a denser k-mesh
to resolvefine changes due to the ETTs.Figure 8 shows the
results.
Figure 8.The Fourier transformof the charge density in k-space,
for the valence electrons inOs on theΓ–A–L–Mplane in k-space.The
left column (a), (c), (e) is fromLDA+DMFTand right one (b), (d),
(f) is fromLDA. In order from top to bottom, the pressure foreach
pair of figures is accordingly (a), (b): P=0 GPa, (c), (d):P=134
GPa, (e), (f):P=247 GPa. In (g), the region close to theΓpoint has
been replotted, showing a small depletion atΓ. The units for the
color bars are states.
9
New J. Phys. 19 (2017) 033020 QFeng et al
-
3.3.1. Ambient pressureInfigure 8we show the k-resolved charge
density at ambient pressure. TheΓ point is at the right-bottom
cornerand the L point is at the upper-left corner.With LDA+DMFT, at
0 GPa (figure 8(a)) the L point is at amediumstate density and theΓ
point is at a highest state density.With LDA (figure 8(b)), we
already see the pocket at theΓ point, with a clear depletion in
density. The ETThas thus taken place at a negative pressure.
3.3.2. 134 GPaAt 134 GPa, the L point stays at the same density
in our LDA+DMFT calculations (figure 8(c)), reflecting that
noETThas occured at the L point. However, with LDA,we see a clear
depletion in charge density at this point(figure 8(d)). At theΓ
point, we note a very small drop in the density with our LDA+DMFT
calculations. Thedrop is hardly visible at the scale offigure 8(c),
and the region around theΓ point has therefore been replottedwith
afiner scale infigure 8(g). The smallmagnitude of this charge
depletion is undoubtly connected to the smallsize of the pocket
seen infigure 7(a).Within LDA (figure 8(d)) the drop is
unmistakable, just as the pocket infigure 7(b).
3.3.3. 247 GPaAt 247 GPa, LDA+DMFT calculations (figure 8(e))
showhow the density at the L point has dropped, reflectingthe ETT
seen at the L point. The density at theΓ point shows a pronounced
drop.No qualitative changes are seenin the LDA results (figure
8(f)). These conclusions agrees well with accurate estimations of
the transitionpressure given in [1], 101.5 GPa at theΓ point and
183 GPa at L point. Evidently, these changes for
LDA+DMFTcalculations also agreewell withwhat is seen in the Fermi
surface plots.
3.4. Charge gradientInfigure 9, we examine the LDA+DMFT results
for theΓ point in yet greater detail, using a color
schemewhichallows us to see the gradient of the charge densitymore
easily.
As pressure increases fromP=0 GPa (figure 9(a)), the high
density region around theΓ point becomesnarrower in the kx
direction andwider in the kz direction (figure 9(b)). At 134 GPa
(figure 9(b)), there is a clearloss of electrons, which is
connected to the ETT. As pressure increases further to 247 GPa
(figure 9(d)), the dropat theΓ point is clear.
The Fourier transformof the charge density distribution gives
additional valuable information and has itsadvantages. It gives
information about the charge density at each k-point, instead of
the isoenergetic Fermisurface. Therefore, it is easier to notice
subtle changes. For example, from figure 9, one can already see an
earlysign of an ETT at 86 GPa from the charge depletion as
comparedwith 0 GPa. As pressure increases, we observethe ETT at
theΓ point between 86 and 134 GPa.
Figure 9.The gradient of the Fourier transformof the charge
density in k-space around theΓ-point (0, 0) obtained by
LDA+DMFTcalculations at: (a)P=0 GPa, (b)P=86 GPa, (c)P=134 GPa, and
(d)P=247 GPa. The color scheme here shows the gradient ofthe charge
density Fourier transform and should not be confusedwith that
offigure 8.
10
New J. Phys. 19 (2017) 033020 QFeng et al
-
4. Summary and conclusions
In this work a theoretical investigation of hcp-Os Fermi surface
under pressure was performed using the LDA+DMFT technique. Details
about the pressure-induced ETTs of its topology are presented.
Althoughwefind from the effectivemass of the quasiparticles
thatOs should be classified asweaklycorrelated systems, the
beyond-LDA treatment of electron correlations provided by the
LDA+DMFTmethodleads to noticeable changes in the predicted pressure
evolution of its electronic structure.
The previously suggested ETT at theΓ point is not observedwith
LDA calculations if the experimental latticeconstants are
used.However, including correlations beyond LDA, bymeans of the
LDA+DMFTmethod, weclearly see this ETT.Calculationswith the
LDA+DMFTmethod also see the other ETT at the L point, which
isobserved in LDA at a substantially lower pressure.Moreover, these
ETTs can also be clearly seen in the Fouriertransformof the charge
density distribution.We have demonstrated how early signs of an ETT
at theΓ point canbe seen already at 86 GPa. Thus,
LDA+DMFTcalculations put theoretical predictions inmuch better
agreementwith the experimentally detected anomalies connectedwith
ETTs.Our results thus confirm that local dynamicalquantum
correlation effectsmay be important even inweakly correlated
transitionmetals.
Acknowledgments
Wewish to thankDr LVPourovskii for helpwithDMFT
calculations.Wewould like to acknowledge thesupport from the
Swedish Foundation for Strategic Research SSF (SRL), grant 10-0026,
the Swedish ResearchCouncil (VR) grantNo 2015-04391, the Knut
andAliceWallenberg Foundation throughGrantNo. 2014-2019,the
SwedishGovernment Strategic ResearchArea SeRC and the
SwedishGovernment Strategic ResearchArea inMaterials Science on
FunctionalMaterials at LinköpingUniversity (Faculty Grant
SFOMatLiUNo 2009 00971).IAA acknowledges the support from theGrant
ofMinistry of Education and Science of the Russian Federation inthe
frameworkof Increase Competitiveness ProgramofNUSTMISIS
(K2-2016-013), implemented by agovernmental decreedated 16th
ofMarch 2013,N 211.We also acknowledge the travel support provided
byPHDDALÉNProject 26228RM. The calculations have been supported
through the SwedishNationalInfrastructure for Computing (SNIC), and
have been performed at the SwedishNational Supercomputer
Center(NSC) and the PDCcenter for high performance computing.
References
[1] Dubrovinsky L et al 2015Nature 525 226[2] KammGNandAnderson
J R 1970Phys. Rev.B 2 2944[3] CynnH,Klepeis J E, YooC S
andYoungDA2002Phys. Rev. Lett. 88 135701[4] Occelli F, FarberDL,
Badro J, Aracne CM, TeterDM,HanflandM,Canny B andCouzinet B
2004Phys. Rev. Lett. 93 095502[5] Kenichi T 2004Phys. Rev.B 70
012101[6] SahuBR andKleinman L 2006Phys. Rev.B 72 113106[7]
Lifshitz IM1960Zh. Eksp. Teor. Fiz. 38 1569
Lifshitz IM1960 Sov. Phys.—JETP 11 1130[8] Blanter YaM,KaganovM
I, Pantsulaya AV andVarlamovAA1994Phys. Rep. 245 197[9] Brunoa E,
GinatempoB,Giuliano E S, RubanAV andVekilov YK 1994Phys. Rep. 249
353[10] GlazyrinK et al 2013Phys. Rev. Lett. 110 117206[11]
MaYM,Cui T, Zhang L J, Xie Y, ZouGT, Tse J S, GaoX
andKlugDD2005Phys. Rev.B 72 174103[12] KoudelaD, RichterM, AMöbius,
Koepernik K and EschrigH 2006Phys. Rev.B 74 214103[13] RubanAV
andAbrikosov I A 2008Rep. Prog. Phys. 71 046501[14] Liang Y and
Fang Z 2006 J. Phys.: Condens.Matter 18 8749[15] Georges A, Kotliar
G, KrauthWandRozenbergM J 1996Rev.Mod. Phys. 68 13[16] Grossman
JC,Mizel A, CotéM,CohenML and Louie SG 1999Phys. Rev.B 60 6343[17]
Pourovskii L V,Mravlje J, FerreroM, Parcollet O andAbrikosov I A
2014Phys. Rev.B 90 155120[18] Blaha P,
SchwarzK,MadsenGKH,KvasnickaD and Luitz J 2001WIEN2k, AnAugmented
PlaneWave+Local Orbitals Program for
Calculating Crystal Properties (Vienna: Karlheinz Schwarz,
Techn.UniverstätWien)[19] KoepernikK and EschrigH 1999Phys. Rev.B
59 1743[20] Opahle I, Koepernik K and EschrigH 1999Phys. Rev.B 60
14035[21] Boehnke L,HafermannH, FerreroM, Lechermann F and
Parcollet O 2011Phys. Rev.B 84 075145[22] AichhornM, Pourovskii L,
VildosolaV, FerreroM, Parcollet O,Miyake T,Georges A andBiermann S
2009 Phys. Rev.B 80 085101[23] AichhornM, Pourovskii L andGeorges A
2011Phys. Rev.B 84 054529[24] CzyzykMTand SawatzkyGA1994Phys. Rev.B
49 14211[25] Gull E,Millis A J, Lichtenstein A I, RubtsovAN,
TroyerM andWerner P 2011Rev.Mod. Phys. 83 349[26] BeachK SD2004
arXiv: cond-mat/0403055[27] Solovyev I V,Dederichs PH andAnisimovV
I 1994Phys. Rev.B 50 16861[28] CococcioniM and deGironcoli S
2005Phys. Rev.B 71 035105[29] Shih B-C, Zhang Y, ZhangWandZhang P
2012Phys. Rev.B 85 045132[30] Vaugier L, JiangH andBiermann S
2012Phys. Rev.B 86 165105
11
New J. Phys. 19 (2017) 033020 QFeng et al
https://doi.org/10.1038/nature14681https://doi.org/10.1103/PhysRevB.2.2944https://doi.org/10.1103/PhysRevLett.88.135701https://doi.org/10.1103/PhysRevLett.93.095502https://doi.org/10.1103/PhysRevB.70.012101https://doi.org/10.1103/PhysRevB.72.113106https://doi.org/10.1016/0370-1573(94)90103-1https://doi.org/10.1016/0370-1573(94)90056-6https://doi.org/10.1103/PhysRevLett.110.117206https://doi.org/10.1103/PhysRevB.72.174103https://doi.org/10.1103/PhysRevB.74.214103https://doi.org/10.1088/0034-4885/71/4/046501https://doi.org/10.1088/0953-8984/18/39/007https://doi.org/10.1103/RevModPhys.68.13https://doi.org/10.1103/PhysRevB.60.6343https://doi.org/10.1103/PhysRevB.90.155120https://doi.org/10.1103/PhysRevB.59.1743https://doi.org/10.1103/PhysRevB.60.14035https://doi.org/10.1103/PhysRevB.84.075145https://doi.org/10.1103/PhysRevB.80.085101https://doi.org/10.1103/PhysRevB.84.054529https://doi.org/10.1103/PhysRevB.49.14211https://doi.org/10.1103/RevModPhys.83.349http://arxiv.org/abs/cond-mat/0403055https://doi.org/10.1103/PhysRevB.50.16861https://doi.org/10.1103/PhysRevB.71.035105https://doi.org/10.1103/PhysRevB.85.045132https://doi.org/10.1103/PhysRevB.86.165105
1. Introduction2. Methods and details of calculation2.1. Lattice
constants2.2. LDA calculations2.3. LDA+DMFT calculations2.4.
Quasiparticle effective mass2.5. Electron density distribution
3. Results and discussion3.1. Correlation strength3.2. Fermi
surface3.2.1. Ambient pressure3.2.2.134 GPa3.2.3.247 GPa
3.3. Electron density distribution3.3.1. Ambient
pressure3.3.2.134 GPa3.3.3.247 GPa
3.4. Charge gradient
4. Summary and conclusionsAcknowledgmentsReferences
