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Title

Investigation into Structural Phase Transitionsin Layered Titanium‑Oxypnictides by aComputational Phonon Analysis

Author(s)Nakano, Kousuke; Hongo, Kenta; Maezono, Ryo

Citation

Inorganic Chemistry, 56(22): 13732‑13740

Issue Date

2017‑11‑02

Type

Journal Article

Text version

publisher

URL

http://hdl.handle.net/10119/15356

RightsThis is an open access article published under an
ACS AuthorChoice License(http://pubs.acs.org/page/policy/authorchoice̲termsofuse.html), which permits copying andredistribution of the article or any adaptations
for non‑commercial purposes. Kousuke Nakano,
Kenta Hongo, and Ryo Maezono, Inorganic
Chemistry, 56(22), 2017, 13732‑13740.
DOI:10.1021/acs.inorgchem.7b01709

Description

Japan Advanced Institute of Science and Technology

This is an open access article published under an ACS AuthorChoice License, which permits
copying and redistribution of the article or any adaptations for non-commercial purposes.

ArticleCite This: Inorg. Chem. 2017, 56, 13732-13740

pubs.acs.org/IC

Investigation into Structural Phase Transitions in Layered TitaniumOxypnictides by a Computational Phonon Analysis
Kousuke Nakano,*,⊥ Kenta Hongo,†,‡,§ and Ryo Maezono*,⊥†Research Center for Advanced Computing Infrastructure, JAIST, Asahidai 1-1, Nomi, Ishikawa 923-1292, Japan
PRESTO, Japan Science and Technology Agency (JST), Kawaguchi, Saitama 332-0012, Japan§
Center for Materials Research by Information Integration, Research and Services Division of Materials Data and Integrated System,
National Institute for Materials Science, Tsukuba 305-0047, Japan⊥
School of Information Science, JAIST, Asahidai 1-1, Nomi, Ishikawa 923-1292, Japan‡

ABSTRACT: We applied ab initio phonon analysis to layered
titanium-oxypnictides, Na2Ti2Pn2O (Pn = As and Sb), andfound a clear contrast between the cases with lighter/heavier
pnictogen in comparison with experiments. The result
completely explains the experimental structure at low
temperature, C2/m for Pn = As, within the conventionalcharge density wave, while there arise discrepancies when thepnictogen gets heavier. Our phonon calculation using the
GGA-PBE functional predicts that a Cmce polymorph is morestable than the experimentally observed one (Cmcm) for Pn = Sb. On the basis of further quantitative analysis, we suggest thepossibility that the GGA-PBE functional does not properly reproduce the electron correlation eﬀects for Pn = Sb, and this could
be the reason for the present discrepancy.

■

INTRODUCTION
Recently discovered layered titanium-oxypnictides, ATi2Pn2O[A = Na2, Ba, (SrF)2, and (SmO)2; Pn = As, Sb, and Bi],1−13
comprise a new superconducting family, and this superconducting mechanism has attracted intensive attentions
because the crystal and electronic structures are similar to
those of exotic superconductors such as cuprates14 or ironarsenides.15 The superconducting mechanism has been
expected to be electron−phonon driven,16−19 and theaccompanying singularities in resistivity and susceptibility at
low temperature are regarded as being due to conventionalcharge density wave (CDW).17,19−21 Though our latest phonon
evaluation using GGA-PBE without assuming any magnetic
orderings revealed that the singularities could be explained
within the conventional CDW for Pn = As,22 it is still unknown
if this is the case for Pn = Sb and Bi.16,22 Interestingly, recent
works are pointing out the possibility of exotic mechanism of
the structural transition for these compounds: For BaTi2Pn2O(Pn = As and Sb), Frandsen et al. reported the breaking of the
4-fold symmetry at low temperature without any superstructure
peaks by neutron and electron diﬀractions23 and suggested thepossibility of intra-unit-cell nematic CDW, which is observed incuprates and iron arsenides superconductors,24−26 to accountfor the breaking. A theoretical suggestion was made that suchan exotic CDW could be realized by the orbital ordering
mediated by spin ﬂuctuations.27 Thus, the mechanism ofstructural transition in layered titanium-oxypnictides is still
under debate.
Davies et al.28 have just synthesized large single crystals of
Na2Ti2Pn2O (Pn = As and Sb; Figure 1) in order to achieve© 2017 American Chemical Society

Figure 1. Crystal structure of Na2Ti2Pn2O (Pn = As and Sb). The
space group is I4/mmm.

reliable diﬀaction measurements and observed clear superstructure peaks of 2 × 2 × 2 and 2 × 2 × 1 for Pn = As and Sb,
respectively. They did not detect any indication of spinorderings, so they concluded that the structural transition could
be attributed to the conventional phonon-assisted CDW. From
the theoretical side, examining whether or not the observed
superstructures are explained by ab initio phonon calculations
can provide a critical clue to the mechanism of structuraltransition. In the present study, we performed phonon
Received: July 9, 2017
Published: November 2, 201713732

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Figure 2. Electronic band structures for undistorted Na2Ti2Pn2O structure ((a) Pn = As, (b) Pn = Sb). The Fermi energy is set to 0 eV.

calculations for Na2Ti2Pn2O (Pn = As and Sb) using the GGAPBE functional and compared the superstructures predicted by
the calculations with the experimentally observed ones. The
result shows that the phonon calculation can explain theobserved C2/m (monoclinic No.12) structure for Pn = As,
indicating that the present phonon calculation works well forthis system. For Pn = Sb, however, we obtained a Cmce(orthorhombic No. 64) polymorph, being inconsistent with theobserved one [Cmcm (orthorhombic No. 63)]. On the basis ofthese results combined with preceding studies16,22,29−32 andfurther quantitative analysis, we propose a new possibility that aheavier Pn causes stronger electron correlations and induces
the discrepancy.

■

RESULTS

Figure 2 shows band structures of undistorted Na2Ti2Pn2O (Pn= As and Sb). The results are consistent with the previous
calculations,32,33 meaning that there is no artifact due to the
choice of cutoﬀ, k-meshes, smearing parameters, and pseudopotentials. Figure 3a shows the phonon dispersions for Pn = As,giving imaginary frequencies appearing around N = (2π/a)·(1/2, 0, a/2c) and Σ′ = (2π/a)·(1/2, 0, 0), where a and c areconventional lattice constants for the undistorted structure. The
compatible modes with the experimental observation28 where
the twice larger periodicity along c-axis is realized are N = (2π/a)·(1/2, 0, a/2c) and N′ = (2π/a)·(0, 1/2, a/2c). We note thatthe previous phonon calculation34 could not reproduce thisdoubled periodicity. We therefore took these for further lattice
relaxations from the original I4/mmm structure along the mode
displacements to get the C2/m (monoclinic, No. 12)
superstructure (Figures 4a and 5a), being consistent with theexperiments.28 The results of geometry optimizations for theundistorted (I4/mmm) and distorted (C2/m) structures areshown in Tables 1 and 2. The resultant optimized geometry of
the C2/m superstructure gives fairly good agreement with theexperiments28 within deviations at most 2.5%.35 Although we
could not conﬁrm whether or not the negative modes disappear
in the C2/m superstructure even at high-symmetry points
because of intractable computational costs for the enlarged unit
cell, it should be conﬁrmed in the future.
For Pn = Sb, we obtained phonon dispersions as shown in
Figure 3b. Imaginary frequencies appear widely around X =(2π/a)·(1/2, 1/2, 0) and P = (2π/a)·(1/2, 1/2, a/2c). The
experiment28 reported that the twice larger super periodicity isrealized only in ab-plane, not along c-axis. Therefore, the X
mode is likely to realize the instability toward the superstructure. By optimizing the geometry along the mode

Figure 3. Phonon dispersions and phonon DOS for undistorted (I4/mmm) Na2Ti2Pn2O (Pn = (a) As, (b) Sb). The red cross marks in (b)represent the frequencies with spin−orbit coupling.

displacement, we obtained a Cmce (orthorhombic, No. 64,
Figures 4b and 5b), being inconsistent with the observed Cmcm
(orthorhombic, No. 63, Figure 4c) in the experiment.28 To
conﬁrm the present conclusion further, we examined thestability by seeing if the imaginary frequencies disappear at therelaxed structure, as shown in Figure 6. In addition to thestability check, we compared enthalpies between our Cmce andthe observed Cmcm. The optimized geometry with Cmce isfound to be more stable by 1.1 (0.7) mRy per formula unit thanthat with Cmcm when using Marzari−Vanderbilt (Gaussian)smearing. These results indicate that the experimentallyobserved Cmcm is unfavorable at 0 K at the GGA level of
DFT. It is, however, expected that GGA+U could reverse theorder as discussed later. A comparison of Helmholtz free13733

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Figure 4. In-plane superstructures obtained by our calculations for Pn = (a) As and (b) Sb, and (c) the observed in-plane superstructure for Pn =Sb.28 Pn is located above and below the Ti2O plane. Solid lines represent the original unit cells for I4/mmm. Thin dash lines represent the unit cells
for 2 × 2 superstructures. Solid dash lines represent redeﬁned unit cells for distorted superstructures. The conventional lattice vectors of the
⎯
→
→
⎯
⎯
→
superstructure for (a) Pn = As (C2/m) are redeﬁned as a′ = 2(a ⃗ − b ⃗), b′ = 2(a ⃗ + b ⃗), and c′ = 1/2( − a ⃗ + b ⃗ + c ),⃗ those for (b) Pn = Sb
⎯
⎯
→
→
→
⎯
⎯
→
⃗
⃗
⃗
⃗
(Cmce) are redeﬁned as a′ = c ⃗ , b′ = (a − b ), and c′ = (a + b ), and those for (c) Pn = Sb (Cmcm) are redeﬁned as a′ = 2(a ⃗ + b ⃗),⎯→→⎯b′ = 2(b ⃗ − a ⃗), and c′ = c ⃗ , where a ⃗ , b ⃗ , and c ⃗ are conventional lattice vectors of the undistorted structures.

Figure 5. 3D superstructures obtained from our calculations for Pn = (a) As (C2/m) and (b) Sb (Cmce).

Table 1. Results of Geometry Optimization for Undistorted
Na2Ti2Pn2O Structure (Pn = As and Sb)a
label

x

Na
Ti
As
O

0.50000
0.50000
0.00000
0.50000

Na
Ti
Sb
O

0.50000
0.50000
0.00000
0.50000y
Pn = As (I4/mmm)
0.50000
0.00000
0.00000
0.50000
Pn = Sb (I4/mmm)
0.50000
0.00000
0.00000
0.50000z

Table 2. Results of Geometry Optimization for Na2Ti2Pn2OSuperstructures (Pn = As and Sb) Obtained by Phonon
Calculationsa

wyckoﬀ

0.181460.00000
0.117520.00000

4e
4c
4e
2b

0.183690.00000
0.120150.00000

4e
4c
4e
2b

a
The relaxed conventional lattice parameters are a = 4.07 Å, c = 15.44
Å (a = 4.14 Å, c = 16.98 Å) for Pn = As (Sb), giving good agreementwith the experiments.2,54 The height of Sb toward Ti2O plane obtained
from the present calculation is 2.04 Å, which is consistent with theexperimental value at room temperature (2.01 Å).54 For Pn = As, theheight of As toward Ti2O plane has not been determined inexperiments, but the obtained value (1.81 Å) is quite consistent withthe experimental results of BaTi2As2O (1.77 Å)4 and (SrF)2Ti2As2O(1.81 Å).3

label

x

Na
Na
Na
Ti
Ti
As
As
As
O
O
O

0.909070.40943
0.159320.130310.880190.808770.30933
0.05905
0.00000
0.25000
0.00000

Na
Ti
Sb
O

0.184110.00000
0.378830.00000y
Pn = As (C2/m)
0.00000
0.00000
0.249810.11979
0.369690.00000
0.00000
0.249690.00000
0.25000
0.50000
Pn = Sb (Cmce)
0.00000
0.23266
0.00000
0.00000z

wyckoﬀ

0.63699
0.63694
0.637250.00000
0.00000
0.23625
0.23625
0.236230.00000
0.00000
0.00000

4i
4i
8j
8j
8j
4i
4i
8j
2a
4e
2b

0.00000
0.268120.00000
0.00000

8d
8f
8d
4a

a

The conventional lattice parameters after relaxation are a′=11.51 Å,
b′ = 11.51 Å, c′ = 8.23 Å, and β = 110.46° (a′ = 17.04 Å, b′ = 5.84 Å,and c′ = 5.83 Å) for Pn = As (Sb).

energies (E − TS) under quasiharmonic approximation36 isnecessary to discuss the stability more precisely, where theentropic term is calculated from phonon DOS.36 It was,13734

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We suggest the possibility that the GGA-PBE functional does
not properly reproduce the electron correlation eﬀects for Pn =Sb, and this could be the reason for the present discrepancy.
The conclusion was obtained by the following discussion: In
iron arsenide superconductors, the strength of electron
correlation is well-captured by a trend of h, a vertical distance
between Fe layer and Pn/Ch (Pn = P and As; Ch = Se andTe).30 Miyake et al. applied constrained random-phaseapproximation (cRPA) to iron arsenide superconductors andrevealed that the covalency of Fe-Pn/Ch bonding estimated by
DFT-LDA calculation and the resultant Wannier orbitals can beused for a measure of the magnitude of electron correlation in
the family.30 When h gets larger, the covalency gets weaker tomake Fe 3d more localized, and the spin/orbital polarizations
get enhanced as one of the electron correlation eﬀects. Thismeasure probably works well for layered titanium-oxypnictides
because they have crystal and electronic structures similar to
those of iron arsenides.8,15,23,30,31 In order to apply the measure
to the layered titanium-oxypnictides, we quantitatively analyzed
the covalencies of Ti−Pn bonding in Na2Ti2Pn2O (Pn = As andSb). The ratio of partial density of states (pDOS) of each atom
to total density of states (tDOS) can be used as a measure ofcovalency,37,38 which is one of the common methods toevaluate covalency within DFT framework.37−41 In thismethod, it is necessary to calculate a center of mass (CM) of
atomic orbital for each atom. The CM is calculated by thefollowing equation:38

Figure 6. Phonon dispersions for Na2Ti2Sb2O superstructure (Cmce).

however, intractable because the large unit cell and the low
symmetry of the Cmcm polymorph requires much morecomputational time to calculate its entropic term (TS). The
results of geometry optimizations for the undistorted (I4/mmm) and distorted (Cmce) structures, and those for theexperimentally observed superstructure (Cmcm) are shown in
Tables 1−3.
Table 3. Results of Geometry Optimization forExperimentally Observed Na2Ti2Sb2O Superstructurea
label
Na
Na
Na
Ti
Ti
Ti
Ti
Sb
Sb
Sb
O
O
O

x

y

z

wyckoﬀ

0.00000
0.00000
0.74999
0.374170.625840.875810.124180.00000
0.00000
0.75033
0.00000
0.00000
0.75000

Pn = Sb (Cmcm)
0.37498
0.125030.12500
0.000830.24918−0.00085
0.25082
0.37533
0.12532
0.124990.62500
0.12500
0.37500

0.932930.43294
0.932750.25000
0.25000
0.25000
0.25000
0.129090.62908
0.129070.25000
0.25000
0.25000

8f
8f
16h
8g
8g
8g
8g
8f
8f
16h
4c
4c
8g

ε

A

CM =

∫ε 1 εg A (ε) dε2ε∫ε 1 g A (ε) dε2(1)

where A is an element, ε is an energy, gA(ε) is pDOS of A at ε,and CMA is a center of mass of the element A. Note that theenergy window [ε2, ε1] in the equation has to be chosen in sucha way that it encompasses all relevant bands contributing to thebond.38 The diﬀerence in CM can be used as a measure ofcovalency of the bond.38 If the diﬀerence in the CM between X
and Y (|CMX − CMY|) is small, then the covalency of the bondis strong, and vice versa. Figure 7 shows the obtained pDOS
and center of masses of Ti 3d, As 4p, and Sb 5p orbitals for
Na2Ti2Pn2O (Pn = As and Sb). The pDOS indicates theexistence of two types of bonds: One is Ti−O bonding
contributing to [− 8.0 to −4.9 (4.5) eV], another is Ti−Pnbonding contributing to [− 4.9 (4.5) to 0.0 eV (Fermi energy)]
for Pn = As (Sb). Note that the result that Ti−Pn bonding isrelated to Fermi surface is consistent with the previous study bySingh.31 The lack of overlap between Pn and O states indicatesthat no Pn−O bonding is formed in the compound (Figure 7).
Figure 7 shows that Ti−O bonding does not change between
As and Sb. On the other hand, there is a signiﬁcant diﬀerence inpDOS between As and Sb, indicating that the nature of Ti−Pnbonding changes between them. To quantify the covalencies ofTi−Pn bonding, we calculated the center of masses according
to eq 1. The results are shown in Figure 7 as solid lines. We
adopted [ε2, ε1] = [− 4.9 eV, 0.0 eV] and [− 4.5 eV, 0.0 eV] for
As and Sb, respectively. As a result, we obtained |CMTi3d −CMSb5p| = 0.77 eV and |CMTi3d − CMAs4p| = 0.68 eV. The
smaller value of |CMTi3d − CMAs4p| indicates that Ti−As
bonding is more covalent than Ti−Sb bonding. According toelectronegativity, which is one of the simplest measures ofcovalency, Ti−Sb bonding seems more covalent than Ti−As
one. Our results, however, show that the simple measure isa
The conventional lattice parameters after relaxation are a′ = 11.74 Å,
b′ = 11.74 Å, and c′ = 16.83 Å.

■

DISCUSSION
The present results confront the sharp contrast between Pn =As and Sb in terms of whether or not the present phonon
calculations can reproduce the experimental structural phasetransitions. Since it is clearly reported28 that the measured XRD
results cannot be identiﬁed by the 2 × 2 × 1 superstructure(Figure 4b), the discrepancy would be intrinsic so that it shouldbe accounted for by the eﬀects that are not taken into accountin the present calculations. A spin−orbit eﬀect is likely to be theorigin of the discrepancy because Sb is heavier than As. To
conﬁrm, we again performed phonon calculations with fully
relativistic pseudopotentials at high symmetry positions (Figure3b). Although the frequencies slightly change because of thespin−orbit eﬀect, the X point that induces the Cmce
superstructure still shows the largest imaginary frequency.
This result indicates that the spin−orbit eﬀect is not the originof the discrepancy.
13735

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invalid for the layered titanium-oxypnicitdes. In the compounds, the distance of Ti−Pn (the height of Pn toward Ti2O
plane) governs the covalency as well as electronegativity. This is
the same trend as in iron arsenide superconductors. To conﬁrm
our conclusion, we also calculated electron density ofNa2Ti2Pn2O and visualize (0 2 0) plane including Ti−Pnbondings. Figure 8 shows that the electron density between
Ti−Sb (Ti−As) bonding is 0.03−0.04 electron/bohr3 (0.04−
0.05 electron/bohr3). The result indicates that Ti−As bonding
is more covalent than Ti−Sb one again, which is consistent
with the above pDOS analysis. Our analysis implies thecorrelation eﬀect gets more enhanced for Sb than As in layered
titanium-oxypnictides. The trend is again consistent with ourprevious study for BaTi2Pn2O (Pn = As, Sb, and Bi), where weshowed that the phonon calculation using GGA-PBE could
explain experiments only for Pn = As, but not for the otherheavier Pn.22 It might be a general tendency also applicable tothe layered titanium-oxypnictides that the larger h enhances theelectron correlation as the origin of the present discrepancy.
Calculating the absolute values of electron correlation (e.g.,
eﬀective Coulombic interaction, U) of the layered titaniumoxypnictides is an important future challenge.
Putting the discrepancy for Pn = Sb aside for a while, we canprovide a plausible explanation for the origin of the lattice
instability (Figure 3) from the above analysis: The lattice
instability mainly derives from in-plane Ti displacements(Figure 4). To account for the interatomic forces to restore
the lattice displacements, we can ignore (at least for the
discussion of the instability) the contribution from Na because
there are no bonds between Na and Ti (Figure 1). Starting with
X mode for Pn = As and Sb, the imaginary frequency appears
only for Pn = Sb. Figure 9a shows the change in potential
energy accompanying the displacement of Ti atoms according
to the phonon showing the imaginary (lowest) frequency at X
point in Pn = Sb (As). The result shows that the original
position is the most stable for Pn = As, but it is not for Pn = Sb.
A double-well potential is realized for Pn = Sb. These phonons
correspond to the Ti displacement toward Pn on the Ti2O
plane as shown in Figure 9b. One of possible restoring forces
would come from the Coulombic attraction from the valence
electrons between Ti and O atoms, but the force is almost

Figure 7. Partial density of states and center of masses of Na2Ti2Pn2O(a) Pn = As, (b) Pn = Sb.

Figure 8. Electron densities of Na2Ti2Pn2O (a) Pn = As, (b) Pn = Sb along Ti−Pn bonds. The electron densities are plotted in the same range(0.03−0.07 electron/bohr3), and the aspects are consistent with each crystal structure.
13736

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Figure 9. (a) Change in potential energy accompanying the displacement of Ti atoms according to the phonon showing the imaginary (lowest)
frequency at X point in Pn = Sb (As). The horizontal axis corresponds to the magnitude of the displacement from the original positon. (b) Twodimensional image of the displacement parttern of Ti atoms. (c) Three-dimensional image of the displacement parttern of the Ti atom enclosed in a
red broken line in (b).

orthogonal to the displacement and hence we can ignore it.42 In
contrast, Pn is located in the same direction of Ti displacement,
being responsible for the restoring forces. The displacement
heads for the intermediate point of Pn located above and below
the Ti2O plane (Figure 9c). Ti atom is surrounded by these Pn(Figure 9c), so forces in the z direction were canceled.
However, the forces in the y direction could be responsible forthe restoring forces (Figure 9c). To discuss the diﬀerence
between As and Sb, we focus on the high electron density
region surrounded by Ti and Pn atoms (Figure 8). When we
focus on the Ti atom shown in Figure 9c, it approaches(recedes from) the left (right) high electron density region.
The Ti atom feels the Coulombic repulsion from the left
region, because another Ti atom on the opposite side
simultaneously approaches the region (Figure 9b). They
compress the high electron density region, resulting in theincrease of the Coulomb energy. The magnitude of the highdensity region is 0.04 electron/bohr3 (0.03 electron/bohr3) for
Pn = As (Sb). This implies that the Coulombic repulsion eﬀect
in Pn = As is larger than that in Pn = Sb. In contrast, the Ti
atom feels Coulombic attraction from the right region. The
eﬀect of Coulombic attraction is again larger for Pn = As
because of the higher electron density of the region. These are
the reasons for the diﬀerence that the original position is themost stable for Pn = As, but it is not for Pn = Sb. For N mode,
the imaginary frequency appears commonly for Pn = As and Sb.
This is also accountable along the above Discussion section. In
this mode, the Ti−Pn interaction has little eﬀect on therestoring forces because Ti and Pn move in the same direction(Figure 4a). The magnitude of the Coulombic repulsion andattraction from the high electron density region are expected to
be much smaller because Pn moves in the same direction of Ti.
In this case, the possible restoring force would originate fromthe nearest oxygen; again, this is orthogonal to the displacement, giving too weak a contribution and leading to theinstability. This is the reason why the imaginary frequencies
appear at N mode for both Pn.
We return to the discussion of the discrepancy for Pn = Sb. If
the discrepancy is attributed to the electron correlation as wesuggest, then calculations using exchange-correlation functionals beyond GGA such as GGA+U and hybrid functionals

are necessary to reproduce the experimentally observed
superstructure of Pn = Sb within DFT framework. Though
we could not perform further phonon calculations with GGA+U or hybrid functional due to the limitation of theimplementation, we note that an interesting case has beenreported for BaTi2Sb2O that only structural optimization with
GGA+U could reproduce the observed transition from P4/
mmm to Pmmm, otherwise without +U it cannot be
explained.43 GGA+U has also been applied to Na2Ti2Sb2O32
to examine possible magnetic orderings, identifying a bicollinear antiferromagnetic ordering is the most stable. The
ordering is unfortunately not able to solely account for theobserved superstructure,28 but when combined with a phonon
evaluation under the supposed ordering, it might give further
instability toward the enlarged superstructure, which is possibly
consistent with the observed symmetry. A model Hamiltonian
can be used to explicitly take the electron correlation eﬀect into
consideration. In fact, plausible explanations for the observed
intra-unit-cell nematic CDW transition,23 which could not be
explained by the present phonon calculations, have been
proposed by using a model Hamiltonian.27,43 The ideas are
based on spin ﬂuctuations to form magnetic orderings at low
temperature, triggering another structural instability over theconventional prediction. Such a mechanism is not able to be
described within the present calculation,16,22 and could be apossible origin for the discrepancy. For the key issue (i.e. spin
ﬂuctuations), there are still a small number of experiments(NMR and μSR) giving the controversy about the existence ofthe ﬂuctuations.17−19,28,44 Inelastic magnetic neutron scattering
can be expected to provide conclusive information about this.

■CONCLUSIONS
We applied ab initio phonon analysis to layered titaniumoxypnictides, Na2Ti2Pn2O (Pn = As and Sb), and found a clear
contrast between the cases with lighter/heavier pnictogen incomparison with experiments. The result completely explains
the experimental structure at low temperature, C2/m for Pn =As, within the conventional charge density wave, while thecalculation predicts that a Cmce polymorph is more stable thanthe experimentally observed superstructure (Cmcm) for Pn =Sb. To reveal the origin of discrepancy, we focus on the height
13737

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Figure 10. Primitive Brillouin zones (a) for undistorted Na2Ti2Pn2O structure (I4/mmm) and (b) for Na2Ti2Sb2O superstructure (Cmce).
Fermi surfaces were depicted by using VESTA51 and XCrySDen,52
respectively.
To deal with all the compounds systematically, we checked theconvergence of plane-wave cutoﬀ energies (Ecut), k-meshes, q-meshes,and smearing parameters. The most strict condition among thecompounds was taken to achieve the convergence within ±1.0 mRy
per formula unit in the ground state energy, resulting in E(WF)
cut = 90 Ry
for wave function and E(ρ)
cut = 800 Ry for charge density. For
undistorted Na2Ti2As2O and Na2Ti2Sb2O, (6 × 6 × 6) k-meshes wereused for the Brillouin zone integration. Phonon dispersions were
calculated on (6 × 6 × 6) q-meshes. For Na2Ti2As2O (C2/m) andNa2Ti2Sb2O (Cmce), (4 × 4 × 4) k-meshes were used. For
Na2Ti2Sb2O, (4 × 4 × 4) q-meshes were used to calculate phonon
dispersions. We also calculated a ground state energy of theexperimentally observed superstructure for Na2Ti2Sb2O (Cmcm)28
with (6 × 6 × 3) k-meshes. The Marzari−Vanderbilt cold smearing
scheme53 with a broadening width of 0.01 Ry was applied to all thecompounds. All the calculations were performed with primitive
Brillouin zones.
The primitive Brillouin zone for undistorted Na2Ti2Pn2O (I4/mmm) is shown in Figure 10a. The special k and q points are Γ = (2π/a)·(0, 0, 0), N = (2π/a)·(1/2, 0, a/2c), P = (2π/a)·(1/2, 1/2, a/2c), X= (2π/a)·(1/2, 1/2, 0), Σ′ = (2π/a)·(1/2, 0, 0), and Z = (2π/a)·(0, 0,a/c) in Cartesian axis, where a and c are conventional lattice constantsfor the undistorted structures. The primitive Brillouin zone for
Na2Ti2Sb2O superstructure (Cmce) is also shown in Figure 10b. The
special k and q points are Γ = (2π/a′)·(0, 0, 0), X = (2π/a′)·(1, 0, 0),
S = (2π/a′)·(1/2, a′/2b′, 0), Z = (2π/a′)·(0, 0, a′/2c′), A = (2π/a′)·(1, 0, a′/2c′), R = (2π/a′)·(1/2, a′/2b′, a′/2c′) in Cartesian axis,
where a′, b′, and c′ are conventional lattice constants for thesuperstructure.

of Pn toward Ti2O plane in layered titanium-oxypnictides. Thisis because the strength of electron correlation is well-captured
by a trend of h, a vertical distance between Fe layer and Pn or
Ch in iron arsenide superconductors that have similar electronic
structures to our layered titanium-oxypnictides. On the basis ofquantitative analysis, we concluded that Ti−As bonding is morecovalent than Ti−Sb one and the distance of Ti−Pn (the heightof Pn toward Ti2O plane) governs the covalency. The result
indicates that the covalency of Ti−Pn bonding cannot be
explained only by electronegativity in layered titaniumoxpnictides, which is the same trend as in iron arsenide
superconductors. The result supports the similarity between
iron arsenide and layered titanium-oxypnictide superconductorsthat has been proposed so far and probably stimulates further
experimental and theoretical studies of layered titaniumoxypnictides. Our analysis could support the correlation eﬀect
gets more enhanced for Sb than for As in layered titaniumoxypnictides. Thus, we suggest the possibility that the GGAPBE functional does not properly reproduce the electron
correlation eﬀects for Pn = Sb. This could cause the present
discrepancy. If the discrepancy is attributed to the electron
correlation as we suggest, then calculations using exchangecorrelation functionals beyond GGA such as GGA+U andhybrid functionals are necessary to reproduce the experimentally observed superstructure of Pn = Sb. As shown in thepresent study, examining whether or not the observed
superstructures are explained by ab initio phonon calculations
can provide a clue to the mechanism of structural transition.
The combination of diﬀraction studies and phonon calculations
will be more important in the ﬁeld of inorganic chemistry.

■■

AUTHOR INFORMATION

Corresponding Authors

COMPUTATIONAL DETAILS*E-mail: kousuke_1123@icloud.com.
*E-mail: rmaezono@mac.com.

All the calculations were done within DFT using the GGA-PBE
exchange-correlation functional,45 implemented in Quantum Espresso
package.46 We chose the PBE functional because it has been used forthe layered titanium-oxypnictides so for and then no critical
discrepancy has been reported, for example, compared to the
ARPES measurement.21 We adopted PAW47 pseudopotentials inmost calculations. The present PAW implementation takes into
account the scalar relativistic eﬀects.48,49 Only for spin−orbit case, we
adopted Ultrasoft (US) pseudo potentials that take into account the
fully relativistic eﬀects.48,49 We restricted ourselves to spin unpolarizedcalculations. The valence electron conﬁgurations used for the US and
PAW pseudopotentials are 2s22p63s1 (Na), 3s23p63d24s2 (Ti), 4s24p3(As), 5s25p3 (Sb), and 2s22p4 (O). Lattice instabilities were detectedby the negative (imaginary) phonon dispersions evaluated for
undistorted and distorted structures. Taking each of the negative
phonon modes, the structural relaxations along the mode were
evaluated by the BFGS optimization scheme with the structuralsymmetries ﬁxed to C2/m for Na2Ti2As2O and Cmce for Na2Ti2Sb2O.
For phonon calculations, we used the linear response theory
implemented in Quantum Espresso package.50 Crystal structures and

ORCID

Kousuke Nakano: 0000-0001-7756-4355
Kenta Hongo: 0000-0002-2580-0907
Notes

The authors declare no competing ﬁnancial interest.

■

ACKNOWLEDGMENTS
The computation in this work has been performed using thefacilities of the Research Center for Advanced Computing
Infrastructure (RCACI) at JAIST. R.M. and K.H. are also
grateful to MEXT-FLAGSHIP2020 (hp170269, hp170220) for
their computational resources. K.H. is grateful for ﬁnancial
support from a KAKENHI grant (JP17K17762), a Grant-in-Aid
for Scientiﬁc Research on Innovative Areas “Mixed Anion”
project (JP16H06439) from MEXT, PRESTO (JPMJPR16NA)
and the Materials research by Information Integration Initiative
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Inorg. Chem. 2017, 56, 13732−13740

Article

Inorganic Chemistry(MI2I) project of the Support Program for Starting Up
Innovation Hub from Japan Science and Technology Agency(JST). R.M. is grateful to MEXT-KAKENHI (26287063,
17H05478), a Grant-in-Aid for Scientiﬁc Research on
Innovative Areas "Mixed Anion" project (JP16H06440), a
grant from the Asahi glass Foundation, and US-AFOSRAOARD for their ﬁnancial supports.

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