Electronic structure and phonon instabilities in the vicinity of the structural transition and superconductivity of (Sr,Ca)$_3$Ir$_4$Sn$_{13}$

1.2 MB·

by  David A. Tompsett

Electronic structure and phonon instabilities in the vicinity of the structural
transition and superconductivity of (Sr,Ca) Ir Sn
3 4 13
∗
D.A. Tompsett
Department of Chemistry, University of Bath, Bath BA2 7AY, UK.
(Dated: January 27, 2014)
Thenatureofthelattice instability connected tothestructuraltransition and superconductivity
of (Sr,Ca)3Ir4Sn13 is not yet fully understood. In this work density functional theory (DFT) cal-
4 culations of the phonon instabilities as a function of chemical and hydrostatic pressure show that
1 theprimarylatticeinstabilitiesinSr3Ir4Sn13 lieatphononmodesofwavevectorsq=(0.5,0,0)and
0 q = (0.5,0.5,0). Following these modes by calculating the energy of supercells incorporating the
2 mode distortion results in an energy advantage of -14.1 meV and -9.0 meV per formula unit re-
n spectively. However, the application of chemical pressure to form Ca3Ir4Sn13 reduces the energetic
advantage of these instabilities, which is completely removed by the application of a hydrostatic
a
J pressure of 35 kbar to Ca3Ir4Sn13. The evolution of these lattice instabilities is consistent with
experimentalphasediagram. Thestructuraldistortionassociated withthemodeatq=(0.5,0.5,0)
4
produces a distorted cell with the same space group symmetry as the experimentally refined low
2
temperature structure. Furthermore, calculation of the deformation potential due to these modes

quantitatively demonstrates a strong electron-phonon coupling. Therefore, these modes are likely
]
n tobe implicated in thestructural transition and superconductivity of this system.
o
c PACSnumbers: 71.20.Lp,71.15.Mb,74.25.Kc
-
r
p
u I. INTRODUCTION
s
.
t
a Understanding the mechanisms of superconductivity,
m despitethedocumentationofthisphenomenonforovera
- century, remains a pressing challenge to condensed mat-
d ter physics. Progress, particularly in enhancing the su-
n perconducting transitiontemperature (Tc) may drivefu-
o
ture technological innovation in energy storage, energy
c
[ transmission and information devices. Enhanced Tc is
oftenassociatedwithstructuralphasetransitions,asob-
1 served in copper oxides1,2, iron pnictides3,4 and CaC 5,
6
v
and recently the action of soft phonons has been impli-
2
1 cated in the superconductivity of BaNi2As26 (Tc = 3.3
4 K)andtheROxF1−xBiS2 (Tc =10K)compounds7. The
6 discovery of superconductivity in (Sr,Ca)3Ir4Sn13, with
1. Tc up to 8.9 K, presents a new opportunity to enhance
0 ourunderstandingoftheinterplaybetweensuperconduc-
4 tivity and structural instabilities in a system without an
1 associated magnetic effect8,9. Furthermore, a compre- FIG. 1. (Color online) Experimental phase diagram for
: hensive study of the phase diagram of (Sr,Ca) Ir Sn (CaxSr1−x)3Ir4Sn13 from Ref.10. Constructed by placeing
v 3 4 13 x = 0, 0.5 and 0.75 at -52, -26 and -13 kbar (c.f. top axis).
i has demonstrated the ability to tune its observed struc-
X The circled numbers correspond to the position of the three
tural distortion to a zero temperature quantum phase
calculations in Table I in thecomposition-pressure space.
r transition10. Consequently, there is a driving need to
a
understandthe underlying nature of the structuraltran-
sitionandsoftphononsin(Sr,Ca) Ir Sn ,whichwead-
3 4 13 by alloying calcium in (CaxSr1−x)3Ir4Sn13 results in the
dress here by a detailed first-principles study. suppression of the anomaly to T∗ =33 K in Ca Ir Sn
3 4 13
Recent experimental investigations9,10 into Sr Ir Sn and enhancement of the superconducting transition to
3 4 13
havedemonstratedthepresenceofsuperconductivitybe- Tc = 7 K. Proceeding beyond the chemical pressure
lowTc =5Kaswellasdistinctanomaliesintheelectrical regime by the application of hydrostatic pressure upon
resistivityandmagneticsusceptibilityatT∗ ∼147K.X- Ca Ir Sn results in suppression of the structural tran-
3 4 13
ray diffraction measurements indicate that the anomaly sitiontowardsazerotemperaturequantumphasetransi-
is associated with a structural transition to a superlat- tionatapproximately18kbar. A superconductingdome
tice structure. The transition has been traced with both extends beyond this pressure and peaks at 40 kbar with
chemicalandhydrostaticpressureto elucidate the phase a transition temperature of Tc =8.9 K.
diagram in Fig. 1. The application of chemical pressure (Sr,Ca) Ir Sn is only a small part of the R T X
3 4 13 3 4 13
2
FIG. 2. (Color online) Crystal structure of the I-phase of
(Sr,Ca)3Ir4Sn13. The large icosahedra are formed by Sn12
units with a further Sn atom at their center. The large blue
spheresindicatethe(Sr,Ca)positionandthesmallredspheres
are theIr.
FIG.3. (Coloronline)Densityofstatesof(a)Sr3Ir4Sn13 and
(b) Ca3Ir4Sn13 (f.u.≡formula unit.).
general stoichiometry, where R is an alkaline-earth or
rare-earth element, T is a transition metal and X is a
group-IV element. Many members of this family of ma-
2.25 a for Sn. A minimum of 4000 k-points was used
0
terialshavebeenobservedtoundergoastructuraldistor-
in the full Brillouin zone to obtain an accurate density
tion to a low temperature structure, the I′ phase, which
of states and potential for the calculcation of the band-
reduces the symmetry of the simple cubic parent struc-
structure.
ture (I phase, Pm¯3m). A series of diffraction studies
have not produced agreementin describing the low tem-
perature I′ phase11–14. The structure of (Sr,Ca) Ir Sn
3 4 13
inthehightemperatureI-phaseisshowninFig.2andis A. Electronic Structure
dominated by Sn icosahedra that are joined by trian-
12
gular prismal IrSn6 units. It is the transition from the I The intricate phase diagram of (Sr,Ca)3Ir4Sn13 is evi-
phase at elevated temperatures to the I′ that is thought dence of a delicate interplay between the electronic and
tobetheoriginoftheanomaliesatT∗in(Sr,Ca)3Ir4Sn13. lattice degrees of freedom in this system. We have
Given the presence of superconductivity and its intrigu- calculated the electronic structure for Sr Ir Sn and
3 4 13
inginterplaywiththestructuraltransitionitistimelyto Ca Ir Sn in the simple cubic I-phase using the ex-
3 4 13
apply DFT to study the evolution of the phase diagram perimental lattice parameters9,10 of 9.7968(3) ˚A and
in (Sr,Ca)3Ir4Sn13. In this workwe showthat the evolu- 9.7437 ˚A respectively. The smaller lattice parameter of
tion of the phase diagramin Fig. 1 may be explained by Ca Ir Sn correspondsto aneffective chemicalpressure
3 4 13
the pressuredependence ofthe phononinstabilities from and, as shownin the phase diagramof Fig. 1, this mate-
DFT. rial lies near to the critical pressure at which T∗ extrap-
olates to zero temperature. The density of states (DOS)
for both materials is shown in Fig. 3.
II. RESULTS AND DISCUSSION Both materials possess a large density of states, ap-
proximately12.5states/eVperformulaunit,attheFermi
The electronic structure has been calculatedusing the level which is indicative of good metallic behavior. The
Local Density Approximation (LDA). The VASP15 code overall appearance of the DOS for the two systems is
wasemployedforthecalculationofthephononspectrum similar. Projections of the DOS inside the muffin tin
inconjuntionwithPhonopy16. PAWpotentialswereem- spheresforthes,panddangularmomentaindicatethat
ployedwithacutofffortheplanewavebasissetof350eV. thestatesneartheFermilevelareprimarilyIr-dandSn-
A minimum of 4×4×4 k-point grid was used to achieve pcontributions. Thisreflectsthecrystalstructureshown
goodconvergenceoftheforceconstants. Theall-electron in Fig. 2 where IrSn triangularprisms connect the Sn
6 12
full-potenial code Wien2k17 was also employed to verify icosahedra. The majority of the Ir-d states are loosely
key results. Here RK was set to 7.0 and the radii of bound in a broad structure between -5 and -1 eV below
max
the muffin tins was 2.5 a for (Sr,Ca), 2.5 a for Ir and the Fermi level.
0 0
3
Attempts were made to stabilize spin polarized states
in both Sr Ir Sn and Ca Ir Sn in ferromagnetic
3 4 13 3 4 13
and various antiferromagnetic spin configurations. How-
ever, all calculations with the LDA exchange correlation
functional, as well as tests with Generalized Gradient
Approximations18, resulted in non-magnetic solutions.
The lack of magnetism in this system is consistent with
thelowexperimentalmagneticsusceptibiltiesandtheab-
senceofanisotropyinthesusceptibilitynearT∗10,aswell
as with recent µSR measurements19. Consequently, in
the remainder of this work all reported calculations are
non-magnetic.
We will return to discuss further features of the elec-
tronic structure, but given the importance of the struc-
tural transition at T∗ we first turn our attention to the
phonon driven instabilities.
FIG. 4. (Color online) Calculated phonon dispersion for
Sr3Ir4Sn13 and Ca3Ir4Sn13 in the I-phase. Only low energy
frequenciesareshownsincewearemostinterestedintheevo-
B. Structural Instabilities lution of soft modes.
The presence of a structural transition from the I-
energetic advantage of the phonon instabilities at X, M
phase has been observed in several members of the
andR the distortiondue to these modes has been incor-
R T X family. However investigations have met
3 4 13
with difficulty in indexing the low temperature I′ porated into supercells.
structure12,13. Themappingofthetemperature-pressure In Table I the energy of structurally minimized su-
phasediagramof(Sr,Ca) Ir Sn indicatesacomposition percells incorporating distortions due to the lowest en-
3 4 13
space in which the structural instability may be traced. ergy phonon modes at X, M and R are compared to
ThephonondispersionfortheI-phaseofbothSr Ir Sn the energy of the undistorted I-phase structure. For
3 4 13
and Ca3Ir4Sn13 has been calculated and the results are Sr3Ir4Sn13, denoted ➀ (c.f. Fig 1), the distortions due
shown in Fig. 4. There is a clear distinction between to the imaginary phonon modes at X and M deliver en-
the phonon dispersion in the two materials. Sr Ir Sn ergy advantages of 14.1 meV and 9.0 meV per formula
3 4 13
possessesimaginaryphononfrequenciesthatindicatethe unit respectively. These energies correspond to approx-
presenceoflatticeinstabilitiestowardsadisplacivephase imately 163 K and 105 K which are in reasonable cor-
transition. Strong instabilities lie at the X point of respondence with the experimental T∗ = 147 K for this
the Brillouin zone corresponding to q = (0.5,0,0) and material. Sr3Ir4Sn13 isalsounstabletowardsthephonon
M point corresponding to q = (0.5,0.5,0). A further, mode at R, but with significantly smaller energy advan-
but weaker, instability is also present at the R point tage of just 4.0 meV per formula unit.
(q=(0.5,0.5,0.5)). Table I also shows the energy of lattice distortions
In the case of Ca3Ir4Sn13, which lies much nearer the due to the phonon modes in Ca3Ir4Sn13, denoted ➁ (c.f.
zero temperature end point for the structural transition, Fig 1). The energetic advantage due to distortions at X
only a single imaginary mode remains at the M point. and M are now just 4.6 meV and 2.1 meV per formula
ThelowestmodeatXisnowreal,butwithasoft,lowfre- unit, which is consistent with the lower T∗ ≈ 33 K in
quency. The lowest mode at R has also clearly become Ca3Ir4Sn13. The distortion at R is no longer stable and
a real oscillatory frequency in Ca Ir Sn with signifi- relaxes back to become the undistorted I-phase.
3 4 13
cantly higher energy. The clearevolutionofthese modes In order to trace the evolution of the structural in-
as wemove fromconsideringSr3Ir4Sn13 to Ca3Ir4Sn13 is stabilities across the extrapolated zero temperature end
in concert with the suppression of T∗ in the phase di- point for T∗ the energies have also been determined for
agram of Fig. 1. In particular the near zero frequency a cell of Ca Ir Sn subject to a pressure of 35 kbar20,
3 4 13
of both the real mode at X and imaginary frequency at denoted ➂ (c.f. Fig 1). At this pressure,17 kbar beyond
M in Ca3Ir4Sn13 is consistent with the proximity of this theextrapolatedzerotemperatureendpoint,noneofthe
materialtothe zerotemperatureendpointforthe struc- mode distortions give an energetic advantage as shown
turaltransition. At the end point it is expected that the inthe finalcolumnofTableI.This givesstrongevidence
energy scale for the structural transition becomes very thatit is indeedthese modes that areresponsiblefor the
small and the associated phonon modes will soften. structural distortion associated with T∗. The presence
It is important to note that the phonon spectrum is of the superconducting dome in this region makes the
calculatedintheharmonicapproximation. Consequently phonon modes, particularly at X and M, strong candi-
anharmoniceffectsmayimpactthestabilityofstructural datestobeconsideredinthemechanismforthesupercon-
distortionsassociatedwiththesemodes. Toquantifythe ductivity in this system. The presence of these very soft
4
TABLEI.CalculatedstabilizationenergiesofstructuraldistortionsduetosoftmodesrelativetothehightemperatureI-phase.
The distorted structures for each wavevector are obtained by following the phonon modes of low frequencies in the phonon
dispersion of Fig. 4. Circle designations ➀ , ➁ and ➂ correspond to Fig 1.
Distortion ➀ Sr3Ir4Sn13 ➁ Ca3Ir4Sn13 ➂ Ca3Ir4Sn13 P =35 kbar
Undistorted I-phase 0 0 0
X q=(0.5,0,0) -14.1 -4.6 0
M q=(0.5,0.5,0) -9.0 -2.1 0
Rq=(0.5,0.5,0.5) -4.0 0 0
TABLE II. Predicted internal ion coordinates for the low
temperaturestructureofSr3Ir4Sn13modulatedbythephonon
mode at M. Calculated at the experimental lattice parame-
ters.
Species Wyckoff Position
Sr 48e (0.2491, 0.4997, 0.1259)
Ir1 16c (0.3747, 0.3747, 0.3747)
Ir2 48e (0.3750, 0.3750, 0.1249)
Sn1 16c (0.0004, 0.0004, 0.0004)
Sn2 48e (0.3267, 0.4012, 0.2500))
Sn3 48e (0.4235, 0.4997, 0.1506))
Sn4 48e (0.3494, 0.4237, 0.0002)
Sn5 48e (0.4997, 0.3494, 0.0764)
tural modulation that is the superposition of the three
symmetry equivalent M modes the symmetry is that of
the experimentally observed I¯43d space group. In this
structure there are eight inequivalent atomic sites. The
relaxed coordinates in this low temperature distorted
structure are shown in Table II for Sr Ir Sn . In the
3 4 13
high temperature I-phase there is only a single free Sn
coordinateatthe 24k site. Inthelowtemperaturestruc-
turethephonondistortionresultsinafreeSrcoordinate,
FIG. 5. (Color online) A 2×2×2 supercell is shown with the two free Ir coordinates and five free Sn coordinates Sn1-
lowest energy distortion due to the phonon mode at M. The Sn5. ItisthemovementoftheSncoordinatesinresponse
modemotionthatcompressesonesideoftheicosahedrawhile tothephononinstabilitiesthatdominatesthelowenergy
expandingtheother is evident.
mode. Fig.5depictsthestructurewhendistortedbythe
mode at M. The primary motion is a breathing of the
Sn cages that compresses one side of each Sn cage
12 12
modes also suggests that the structural phase transition while expanding the other side.
maybesecondorder,howeveradefinitiveconfirmationof Thepresenceofsubtle distortionsto the cubiccell, for
thiswillrequirehighqualityspecificheatmeasurements. instancetoatetragonalormonoclinicsymmetryasnoted
The distortions due to modes at X=(0.5,0,0) and in related structures21,22 can not be ruled out. How-
M=(0.5,0.5,0) give comparable energetic advantages ever, it is likely that high intensity synchrotron X-ray
(within the accuracy of DFT) compared to the I phase. diffraction on powdered samples will be required to dis-
Single crystal X-ray diffraction indicates a good refine- tinguishsuperlatticepeaksassociatedwithsuchadistor-
ment of the low temperature I’ structure to a body- tion, which are beyond the resolution capabilities of the
centred cubic (BCC) space group I¯43d10. This favors single crystal diffraction experiments performed on this
the mode at (0.5,0.5,0) as the driving structural insta- system so far. Nevertheless, in that scenario our conclu-
bility since, along with symmetry equivalent (0.5,0,0.5) sionregardingthe importanceofthe phononmode atM
and(0,0.5,0.5),theassociateddistortionproducesaface- will remain robust.
centred cubic (FCC) cell in reciprocal space. An FCC
reciprocal lattice corresponds to a BCC real space lat-
tice,whichisinagreementwithexperiment. Incontrast, C. Electronic Structure and Electron-Phonon
the distortion due to the low energy mode at X would Coupling
give rise to simple cubic cells in both reciprocaland real
space. The preceding results demonstrate that chemical and
Indeed, when we set into a 2×2×2 supercell a struc- hydrostatic pressure affect the phonon instabilities, but
5
Sr Ir Sn Ca Ir Sn
3 4 13 3 4 13 with recentspecific heat23,24 andthermalconductivity25
2
measurements, which indicate a nodeless gap structure.
1.5 A full determination of the electron-phonon coupling
1 for all modes in this 40 atom unit cell is beyond the
scope of this study and is a challenge to current meth-
0.5
ods. However, we may obtain a quantitative measure of
V)
e 0 the electron-phonon (EP) coupling due to a particular
gy ( modebyevaluatingthe Fermi-surfaceaverageddeforma-
ner -0.5 tion potential26,27 ∆ = [δǫ(k)−δµ]2 . Here, δǫ(k) is
E D E
-1
the change in the one-particle energy with momentum k
-1.5 due to the mode and δµ is the change in chemical po-
tential. hi denotes an average of k over the Fermi sur-
-2
face. We have calculated the deformation potential due
-2.5 to the lowest energy modes at X and M by setting the
R Γ X M Γ
distortions into a supercell. The unmodulated structure
FIG.6. (Coloronline)ElectronicbandstructureofSr3Ir4Sn13 of Ca3Ir4Sn13 at P = 35 kbar, which is beyond compli-
and Ca3Ir4Sn13. cations due to the structural transition, was employed
and the mode distortion set in with a Sn displacement
uSn ≈ 0.1 ˚A. We obtain a deformation potential of 0.19
the lattice degrees of freedom also couple to the elec- eV2 at X and of 0.13eV2 at M which indicates a strong
tronic structure. The force constants that deter- electron phonon coupling. This suggests that the low
mine the phononic instabilities are determined via the energy modes of (Sr,Ca) Ir Sn may be responsible for
3 4 13
Hellman-Feynman theorem and therefore depend di- not only the evolution of the structural transition, but
rectly on the electronic charge density and bonding. mayalsobethedriverofthecompetingsuperconducting
Theclearexperimentalevidenceforsuperconductivityin phase. However,whether other modes beyond these soft
(Sr,Ca)3Ir4Sn13, highlights the need to understand the modes are important to the superconductivity requires
evolution of the electronic structure. further theoretical and experimental investigation. The
We have calculated the bandstructure of analogywith the numerous examples of emergent super-
(Sr,Ca) Ir Sn along high symmetry directions as conductivity at magnetic quantum phase transitions is
3 4 13
shown in Fig. 6. The bandstructures possess dispersive tantalising, and this work invites detailed inelastic X-
states near the Fermi level along all directions which is ray diffractionstudies to experimentally identify the key
consistent with the cubic structure and metallic conduc- modes in this system of materials.
tivitymeasuredexperimentally10. Thebandstructuresof
the two materials are almost perfectly overlaid near the
Fermi leveland this is reflected in a near identicalFermi
III. CONCLUSIONS
surface, which is plotted in Fig. 7 for Ca Ir Sn . The
3 4 13
small surface 327 and the two large sufaces 328 and 328
exhibit flat sections, which may provide opportunities To conclude the key results include:
for enhanced superconducting pairing. The 330, 331 (1) Sr Ir Sn possesses imaginary phonons at wavevec-
3 4 13
and 332 surfaces are more free electron-like and likely tors at X (q = (0.5,0,0)), M (q = (0.5,0.5,0)) and R
to contribute to the good isotropic conductivity of the (q=(0.5,0.5,0.5)).
material in the normal state. (2) The application of chemical pressure by substitution
Away from the Fermi level, the bandstructure for the of Ca onto the Sr site results in the modes at X, M and
two materials, (Sr,Ca) Ir Sn , differ. For example, in- Revolvingtowardsrealfrequencies. Calculationoftotal
3 4 13
spectionofFig.6attheΓpointat-0.55eVshowsadense energies for supercells incorporating the three lowest
clusteringofstatesinSr Ir Sn thatpossessaratherflat lyingmodesatX,MandRrevealsthatthemodesatX
3 4 13
dispersion. However,thefeaturesinthisenergyrangefor and M give energy advantages of 4.6 meV/F.U. and 2.1
Ca Ir Sn differsignificantly. Theclusteringofstatesat meV/F.U. compared to the high temperature I phase.
3 4 13
-0.55 eV at Γ is now split into a less tightly bound set This is consistent with the transition temperature of 33
of states near -0.65 eV and a second set closer to -0.45 K for the structural transition in the Ca substituted
eV. Furthermore,these states areconsiderablymore dis- compound. The space group symmetry from the ex-
persive. The bandstructuresof the occupiedstates differ perimental X-ray refinement to a body-centred cubic
at other points in the Brillouin zone and this will deter- structure below the transition temperature corresponds
mine the force constants that drive the evolution of the to the instability at M (q=(0.5,0.5,0)).
structural instability as the composition changes. (3) Applying a pressure of 35 kbar to Ca Ir Sn in
3 4 13
The system’s superconductivity is driven by the cou- our simulations results in the removal of all instabilities
pling of the lattice instabilities to the electronic struc- towards these phonon modes. This is consistent with
ture. Phonon mediated superconductivity is consistent the end point of the structural phase transition at
6
FIG. 7. (Color online) Fermi surfaces of Ca3Ir4Sn13.
approximately18kbarinthe experimentaltemperature- mediated mechanism for its superconductivity.
pressure phase diagram. This implies a role for the
modes at X and M in the structural transition and
superconductivity of this system.
(4) The presence of low energy phonon modes that may
ACKNOWLEDGMENTS
be tuned to zero energy by the application of pressure
supportstheidentificationofaquantumphasetransition
in this system, which may be second order. I acknowledge discussions with S.K. Goh, F.M.
(5) Finally, the calculated deformation potentials due GroscheandP.J.SainesandfinancialsupportoftheEP-
to low energy modes in this system suggests a phonon SRC, UK.
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