First-Principles Theory of Phase Transitions in IrTe2
We present a computational study of the electronic structure and lattice dynamics of IrTe2 that sheds light on the debated mechanism of the temperature-induced phase transitions of this material. At ambient temperature, IrTe2 adopts a hexagonal crystal structure typical of metal chalcogenides. Upon cooling, some Ir–Ir distances shorten, thus inducing lattice modulations. We demonstrate that this is due to the formation of multicenter bonds involving both Ir and Te atoms. We show how the formation of these bonds is energetically favorable but lowers the vibrational entropy; therefore, they are destabilized by temperature. The obtained model is exploited to rationalize the effect of Se doping and other experimental results from the literature.
<p>This paper published
ASAP on March 4, 2020 with incorrect Figures 2, 4, and 5. The figures
were corrected and the revised paper was republished on March 5, 2020.</p><p>The coupling among electronic,
orbital, and lattice degrees of freedom in materials, and the associated
solid–solid phase transitions, give rise to a rich variety
of intriguing phenomena such as superconductivity, metal–insulator
transitions, and multiferroicity.1 Equally
rich is the zoo of models developed to rationalize these phenomena,2 given their importance for both the fundamental
understanding of the matter and the development of new technologies.3 Among the materials involved, IrTe2 has recently attracted a great deal of interest because of the peculiarity
of its temperature-induced phase transitions. Upon cooling, new phases
form that are characterized by the shortening (by ∼25%) of
certain Ir–Ir bonds, disposed on a regular pattern throughout
the lattice,4 and by a drop in electrical
conductivity.5 The phase transitions are
of first-order type, with a significant hysteresis.6 They were originally thought to originate from charge density
waves (CDWs), but it was later recognized that IrTe2 lacks
the typical CDW signatures such as sinusoidal structure modulation7 and band gap opening.8 Moreover, the electronic structure rearrangement upon transition
extends well outside the Fermi level.8,9 Nonetheless,
similarly to CDWs, the suppression of bond length alternation (e.g.,
through doping) leads to the appearance of superconductivity.10 The driving force underlying the phase transitions
in IrTe2 is thus still a subject of debate. Numerous explanations
were put forward: in-plane intralayer Te–Te (p orbitals) bond
formation,11 interlayer Te–Te depolymerization,7 Jahn–Teller-like distortion,12 Ir charge ordering/disproportionation,5 and naturally also Ir–Ir bond formation.13−15 Understanding the electronic structure of IrTe2, and
what factors drive the phase transitions, represents a fundamental
step to rationalize these types of phenomena and to exploit them for
technological applications.</p><p>In this Letter, we present a thorough
computational investigation
of the electronic structure as well as lattice dynamics of IrTe2. We show that while the phases containing short Ir–Ir
contacts have a lower internal energy, they are penalized by temperature
as their formation lowers the vibrational entropy. The competition
between these two factors determines which phase is stable at a given
temperature. We study the nature of the Ir–Ir bonds appearing
at low temperature through a multitude of tools for chemical bonding
analysis, and we unveil their multicenter nature. As to the driving
force forming these bonds, we rule out some of the previously put
forward hypotheses while reconciling the remaining ones in a unified
picture. We also study the phase transition energy paths, whose shape
helps explain the observed hysteresis. Finally, we validate the gained
insights by applying them to rationalize X-ray absorption and photoemission
spectra from the literature and the change in transition temperature
upon doping IrTe2 with Se (IrTe2–xSex).</p><p>At ambient temperature
(T), IrTe2 adopts
the 1T structure typical of transition metal dichalcogenides (space
group P3m1; Figure 1a), where Ir is octahedrally coordinated
by 6 Te atoms. We shall refer to this phase as "HT".
Note that unlike in the lighter chalcogenide analogues, there is a
significant electron sharing among IrTe2 layers and the
compound is fully metallic.13 In the low-T phases (T < 280 K), the Ir–Ir
bond length alternation lowers the symmetry (Figure 1b–d); hence, the reciprocal lattice
cell shrinks. Each phase is characterized by the reciprocal space
modulation vector q = (1/n, 0, −1/n). We refer to these phases simply as qn and to short Ir–Ir bonds as "dimers" (in line
with literature nomenclature), even though we will demonstrate that
they are actually multicenter bonds. The n in the
modulation vector above was shown4 to vary
according to n = 3m + 2 (m = 1, 2, 3, ...). As temperature decreases, m increases and so does the density of dimers; that is, first the
q5 phase forms, then q8, q11, etc. Instead, when the temperature approaches
zero, n = 6, and the maximum dimer density is reached.
In this study, we consider the HT, q5, q8, and q6 phases,16 whose structures were taken from refs (4 and 13).</p><p>Crystal structures of the phases of IrTe2: HT (a), q5
(b), q8 (c), and q6 (d). Unit cells are represented by solid lines.
The inset in panel b shows the top-view of one IrTe2 layer.
In all figures, Ir (Te) atoms are colored in blue (orange), and short
Ir–Ir contacts are indicated by blue lines. Structures were
drawn with VESTA.17</p><p>IrTe2 is a challenging system for density functional
theory (DFT) because the energy differences among phases are in the
millielectronvolt per atom range. The situation is further complicated
by the presence of strong spin–orbit coupling (SOC). The electronic
energy (i.e., the internal energy without the vibrational contributions)
is expected to decrease in the order q6 < q8 < q5 < HT; that
is, at T = 0 K, the phases richer in dimers are more
stable (see section S1.2). This trend is
notoriously difficult to reproduce with DFT.4 In particular, the introduction of SOC is known to destabilize dimers,
thereby (over)stabilizing high-T phases, leading
to a wrong energy ordering when common DFT functionals like LDA, PBE,18 and PBEsol19 are
adopted.4,13 This is indeed what we observe (Figure S1). We found instead that the M06L functional,20,21 which goes beyond the generalized gradient approximation, produces
the correct energy ranking. These results also support the hypothesis
that q6 is the ground state of bulk IrTe2, which so far
has been inferred from surface-sensitive experiments15 and/or by analogy with the Se-doped compounds.4 When not otherwise stated, the results presented
below were obtained with the M06L functional, adopting a plane-wave
basis set (VASP code22). Section S1 reports a detailed description of the computational
strategies and settings.</p><p>We evaluated the (harmonic) vibrational
partition function of the
various phases in order to access their free energies, which govern
the thermodynamics of phase transitions (see ref (23) and section S1.4). While the vibrational enthalpy plays a minor
role, entropy dictates the stability ranges of the various phases
through the TS term (Figures 2a and S3). Our results show that
the more dimers a phase contains, the lower its vibrational entropy;
that is why temperature destabilizes dimer-containing qn phases. Generally, the formation of stronger (stiffer) bonds such
as the "dimers" of IrTe2 shifts the vibrational
modes toward higher frequencies, thereby decreasing the entropy of
the system.24 This effect is clearly visible
in the phonons density of states of IrTe2 (Figure 2), and it explains the entropy
lowering upon dimer formation. Note that while all our calculations
reproduce the formation of dimers as temperature decreases, we found
the value of transition temperatures to be heavily dependent on the
adopted computational method (Figure S2). This is because small energy changes are involved that are comparable
to DFT accuracy, e.g. a 5 meV/atom perturbation shifts the transition
temperature by about 250 K (Figure S2).
That being said, the qualitative insights discussed above hold true
for all the approaches tested (Figure S3). In summary, the stability of the various phases is determined
by the competition between the internal energy, favoring the formation
of dimers, and the vibrational entropy, destabilizing them. The rise
in temperature increases the weight of the latter term over the former.</p><p>Thermodynamics
of the q5-to-HT phase transition of IrTe2 at various temperatures.
Panel a shows the changes (HT minus q5)
in free energy (blue), vibrational entropy (gray), and vibrational
enthalpy (orange). ΔHel represents
the enthalpy difference without vibrational contributions ("electronic
energy"). Note that at ambient pressure (p), the pΔV term is negligible;
hence, ΔH ≈ ΔU (U, internal energy; V, volume).
Tc indicates the temperature of the phase transition. Tc estimations
are 280 K (experimental; refs (4) and (5)),
280 K (calculated, PBE, SOC neglected; this work), and 565 K (calculated,
M06L; this work). The energies on the left refer to M06L. The plots
including the q6 phase are shown in Figure S2. Panel b shows the q5 phonon density of states (PDOS) projected
onto Ir and Te atoms. Note how the atoms forming the "dimers"
(labeled "dim") display higher vibrational frequencies.
This effect is also visible by comparing the DOS of various phases
(Figure S2).</p><p>To clarify the chemical bonding pattern of IrTe2, we
first investigate the role played by interlayer (Te–Te) bonds.
We studied the energetics of isolated IrTe2 layers (interlayer
bonding "switched off"), in HT and qn geometries. The formation of dimers in isolated layers is even more
favorable than in bulk IrTe2 (e.g., ΔE for HT-to-q5 transition is −25.9 meV/atom in single layer
and −13.5 meV/atom in the bulk; Table S1). Therefore, interlayer Te–Te bonds can be ruled out as the
driving force for "dimer" formation. The electronic
stabilization of "dimers" is thus to be sought within
the layers. To that purpose, we explore the electronic density of
states (DOS), its decomposition into atomic orbital contributions
(p-DOS), and the partial charge density distributions of isolated
layers in the q5 phase (Figures 3 and 4). Five main DOS sections
can be identified (labeled with roman numerals in Figure 3), based on the contributing
orbitals. Of particular interest is the role of Ir(d) orbitals. The
t2g/eg splitting can be recognized from the
charge density distributions (Figure 4). For t2g states, most of the charge density
is localized around Ir, indicating a weak interaction with neighboring
atoms. Instead, the (formally) empty eg states display
a clear hybridization with Te(p) orbitals. Indeed, regions II and
V of the DOS correspond to Ir(d-eg)-Te(p) bonding and antibonding
states, respectively, as can be inferred from the accumulation (depletion)
of charge density in between atoms for states II (V) (see Figures 4c and S10). Note that the interaction of Ir(d-eg) with Te(p) orbitals results in bonding states that lie at
an energy lower than t2g. While the p-DOS of the atoms
lying far from the "dimers" is very similar to that
of the HT phase (Figure S7), the "dimers"
region displays three main differences: lower DOS around the Fermi
level (as already noticed in ref (13)), an additional set of empty bands (labeled
IV), and more occupied states at the bottom of region II (labeled
II′). The charge density of these two additional sets of states
is indeed localized in the "dimer" region (Figure 4a,b) and suggests
a bonding and antibonding nature for states II′ and IV, respectively.
Moreover, the two bridging Te atoms are clearly involved in the chemical
bonding. The "dimer" can thus be identified as a 4-center
Ir2Te2 bond formed by Ir(d) and Te(p) orbitals.
This multicenter nature is confirmed by the lack of a direct Ir–Ir
bond when the charge density topology is analyzed in the framework
of the quantum theory of atoms in molecules25 (Figure S12). Note that while we studied
isolated IrTe2 layers for their neater bonding features,
bulk DOS differs only by the peak broadening due to interlayer interactions
(Figure S6). It is noteworthy that the
DOS from PBE calculations neglecting SOC displays the same qualitative
features as in Figures 3 and 4 (Figures S8 and S9).</p><p>Density of states for an isolated layer of IrTe2 in
q5 phase. The total DOS is shown in gray in the top panel. The red
arrows represent the DOS sections discussed in the main text, identified
by roman numerals. by roman numerals. The middle and lowest panels
report atom and orbital projections (p-DOS), respectively. "(dim)"
indicates the p-DOS of atoms forming the "dimer". Note
how sections II′ and IV are missing in the lowest panel, which
displays the orbital projections of atoms far from the "dimers".</p><p>Charge density distribution of selected states of IrTe2 in q5 phase. Each panel shows the distribution relative to
the states
of a given energy window labeled in Figure 3, namely, (a) −6.75/–5.75 eV
(section II′ in Figure 3), (b) +0.4/+1.3 eV (section IV), (c) −6.5/–3.5
eV (section II; enlargement of the dotted box in the inset), and (d)
−3.5/0.4 eV (section III; enlargement of the dotted box in
the inset). In each panel, two isosurfaces are shown: red (higher
value) and yellow (lower value). Isovalues and additional plots, including
two-dimensional maps can be found in Figures S10 and S11. Note in panel c (d) the octahedral (cubical) shape
of the charge density around Ir, typical of the charge density distribution
of eg (t2g) d orbitals.26</p><p>The above chemical bonding scenario
can be put on energetic grounds
by the analysis of the crystal orbital Hamilton population (COHP).27,28 COHP partitions the energy of electronic states into bonding and
antibonding contributions from each atom pair (negative and positive
COHP, respectively), giving rise to DOS-like plots (Figure 5c). The COHP results fully
confirm the previous bonding–antibonding assignments and the
involvement of Te orbitals in the "dimer", manifested
as a significant Ir–Te COHP value in the "dimer"
regions of the DOS. ICOHP, i.e., the integral of COHP for all occupied
states, measures the contribution of a given bond to the electronic
energy of the crystal.29 In absolute ICOHP
value, Ir–Te is the dominating interaction in IrTe2 (Figure 5). However,
what matters for the phase transition energetics is the change in
ICOHP, which is shown Figure 5a. As expected, the main stabilization of the q5 structure
comes from the Ir–Ir and Ir–Te bonds forming the "dimer"
and from the Te–Te interaction just above it. This further
confirms that "dimers" are not simply Ir–Ir bonds.
Interestingly, the most destabilized bonds are the Ir–Te next
to the "dimers" (Figure 5a). This explains why even in the (low-T) ground state of IrTe2 there is an alternation of long
and short Ir–Ir bonds, which is reminiscent of Peierls distortions.
In order to achieve the overall electronic energy lowering, the formation
of "dimers" requires the lengthening and consequent
destabilization of the neighboring bonds. A hypothetical shortening
of adjacent Ir–Ir bonds would thus be energetically unfavorable.</p><p>Crystal
orbital Hamilton population of bulk IrTe2. (a)
Plot whose axes represent, for each bond in the unit cell, the change
in ICOHP30 (ΔICOHP) and bond lengths
(Δd) when passing from HT to q5 phase. Each
bond type is labeled with a different color, which is shown in the
bottom right legend. The average ICOHP values for each bond type in
q5 phase are −0.19 eV (Ir–Ir), −2.5 eV (Ir–Te),
and −0.15 eV (Te–Te). Bonds relevant for the discussion
are labeled in the plot, and their position is shown in panel b. The
top-left inset shows in a pictorial way that in the q5 phase, the
bonds within (next to) the "dimers" are stabilized (destabilized)
with respect to the HT phase: those bonds that in the plot are enclosed
in the green/violet shaded area are located in the crystal in the
region shaded by the same color. Panel c reports two representative
COHP plots: Ir–Ir and Ir–Te. The bonds within the "dimer"
are distinguished from those that are distant from it (in the case
of Ir–Te, Te is the atom bridging the "dimer",
i.e., Te1 and Te2 in panel b). Additional COHP plots are presented
in Figure S4.</p><p>From the kinetic point of view, nudged elastic band (NEB) calculations
indicate the presence of a barrier along the phase transitions (Figure S5); that is, a higher-energy transition
state is to be formed in order for "dimers" to build
or break. For first-order transitions, the NEB reaction path can be
viewed as simulating the growth of the forming phase domain. The height
of the barrier (66 meV/cell for HT-to-q5) is comparable to and greater
than the kBT thermal
energy at the transition temperature. This qualitatively explains
the observed hysteresis, although additional factors may hide in the
complex nucleation process. Along the NEB phase transition (minimum-energy)
path, all Ir atoms move in a concerted way (Figure S5e), thus supporting the mechanism sketched by Mauerer et
al.31 They showed that the perturbation
induced by one Ir–Ir "dimer" breaking extends
for several neighboring Ir–Ir pairs, a fact that was used to
explain the domain propagation anisotropy observed through atomic-resolution
microscopy experiments. We note that the height of our NEB barrier
is close to the one obtained in ref (32) with a different approach.</p><p>The insights
gained above for IrTe2 can be applied to
rationalize the behavior of the Se-doped compound, IrTe2–xSex (x = 0.10 for q5 and x = 0.04 for q6). Se-doping favors
dimer formation, thereby raising the transition temperatures.15 Our calculations capture this doping effect.
They indicate a preference for Se to lie in the "dimer"
position (Te1/Te2 of Figure 5b), although in q6 the possible substitution positions are
close in energy (ΔE = 1.5 meV/atom, Table S2). In fact, X-ray diffraction experiments
of Pascut et al.,4 carried out on a sample
with higher Se dopant concentration (IrTe1.6Se0.4), did not show any preferential Se distribution in the q6 phase.
Our calculations reveal that Se stabilizes qn phases
through the electronic energy term [ΔHel of Figure 2]. This effect can be explained by considering the orbital sizes:32,33 4p (Se) orbitals are smaller than 5p (Te) orbitals; hence, they
overlap better with the even smaller 5d (Ir) orbitals. A greater overlap
guarantees a larger stabilization of the Ir–Se bonding states
compared to Ir–Te. This is confirmed by Ir–Se ICOHP
values being more negative than Ir–Te and by the further shortening
of the Ir–Ir "dimer" distance upon Se doping
(Table S3). Finally, our NEB calculations
show a substantially barrierless phase transitions for IrTe1.90Se0.10, consistent with the observed hysteresis quenching
upon Se doping.4</p><p>Finally, we discuss
the hypotheses and experimental results from
the literature on IrTe2 in light of the insights gained
in the present study. Our chemical bonding analysis rules out Te–Te
bonds and Jahn–Teller distortion as driving forces for "dimer"
formation. Other authors proposed a Ir3+/Ir4+ charge ordering, the "dimer" Ir being more positively
charged. This model follows from the calculated Ir atomic charges
in qn phases12,13 and from the observed
photoemission spectra, showing a split of the Ir-4f signal upon dimer
formation.5 Both these observations, however,
can be simply explained by the electronic rearrangement taking place
on those Ir atoms that form "dimers" (Figures 3 and 4). In particular, atomic charges are evaluated by integrating the
charge density inside spheres of defined radius, generally nonoverlapping.
It is thus obvious that this population diminishes if more charge
is accumulated between atoms (Figure 4). The present study adds on to the idea of Ir–Ir
bond formation, demonstrating their multicenter nature and showing
why they break at high temperature. The Te(p)-Ir(d-eg)
bonding we unveiled is reminiscent of the "ligand hole"
concept used to rationalize the structure of AuTe234 and AgAgTe4.35 The involvement of Te(p) orbitals in the "dimers"
was also hypothesized by Takubo et al.9 Interestingly, their X-ray absorption experiments (Figure 2c of
ref (9)) show that
the "dimer" formation depletes the DOS just above the
Fermi level and creates additional empty states at slightly higher
energies, +1.3 eV. These results fully agree with our calculated DOS,
which also allow us to identify those additional states at 1.3 eV
as the antibonding orbitals relative to the "dimers"
(Figures 3 and 4).</p><p>In conclusion, we presented a model for
the debated mechanism of
IrTe2 phase transitions. We have shown that the temperature
ranges of stability for the various phases are determined by the competition
between electronic (internal) energy and vibrational entropy. We demonstrated
that the so-called "Ir–Ir dimers" forming at
low temperature are in fact multicenter bonds. Se-doping was simulated
and its effect explained. Our model can also rationalize X-ray absorption
and photoemission spectra from the literature. Besides elucidating
the hitherto poorly understood mechanism of IrTe2 phase
transitions, the present study may be considered as a fresh point
of view that can be helpful to rationalize those solid–solid
phase transitions that cannot be satisfactorily explained by commonly
adopted models, such as CDWs or Mott transitions.</p><p>Detailed description of
computational methods, results
obtained with other DFT functionals, NEB results, and details of the
electronic structure of pure and Se-doped IrTe2 (PDF)</p><p>jz0c00012_si_001.pdf</p><p>The authors declare no
competing financial interest.</p>