Lawrence Berkeley National Laboratory
Recent Work
Title
Double-hole-induced oxygen dimerization in transition metal oxides
Permalink
https://escholarship.org/uc/item/38b9d57v
Journal
Physical Review B - Condensed Matter and Materials Physics, 89(1)
ISSN
1098-0121
Authors
Chen, S
Wang, LW
Publication Date
2014-01-29
DOI
10.1103/PhysRevB.89.014109
Peer reviewed
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University of California
PHYSICAL REVIEW B 89, 014109 (2014)
Double-hole-induced oxygen dimerization in transition metal oxides
Shiyou Chen
1,2,*
and Lin-Wang Wang
1,†
1
Materials Sciences Division and Joint Center for Artificial Photosynthesis, Lawrence Berkeley National Laboratory,
Berkeley, California 94720, USA
2
Key Laboratory of Polar Materials and Devices (MOE), East China Normal University, Shanghai 200241, China
(Received 8 May 2013; revised manuscript received 21 October 2013; published 29 January 2014)
Rather than being free carriers or separated single-hole polarons, double holes in anatase TiO
2
prefer binding
with each other, to form an O-O dimer after large structural distortion. This pushes the hole states upward into
the conduction band and traps the holes. Similar double-hole-induced O-O dimerization (a bipolaron) exists
also in other transition metal oxides (TMOs) such as V
2
O
5
and MoO
3
, which have the highest valence bands
composed mainly of O 2p states, loose lattices, and short O-O distances. Since the dimerization can happen in
impurity-free TMO lattices, independent of any extrinsic dopant, it acts as an intrinsic and general limit to the
p-type conductivity in these TMOs.
DOI: 10.1103/PhysRevB.89.014109 PACS number(s): 71.38.Ht, 61.72.Bb, 71.38.Mx, 72.40.+w
I. INTRODUCTION
Transition metal oxides (TMOs) have been intensively
studied for their electronic, spintronic, photonic, and pho-
tocatalytic applications [1–5]. Quite often the synthesized
wide-gap TMOs have unintentional n-type conductivity and
poor p-type conductivity [6,7]. This has significantly limited
the applications of TMOs, especially in p-n junction devices
[8–10]. Traditionally the origin of the poor p-type conductivity
was attributed to the limited hole concentration: charge-
compensating donor defects, such as oxygen vacancies and
cation interstitials [11,12], will form spontaneously as the
Fermi energy shifts down to near the valence-band maximum
(VBM) level, which is low in the wide-gap TMOs [9,13]. This
mechanism follows the empirical doping-limit rule [14,15].
Besides the thermodynamic limit to the hole concentration,
the limit to the hole mobility can be another possible reason
for the poor p-type conductivity. A recent density functional
theory (DFT) study by Varley et al. using hybrid functionals
showed that, in the prototypical TMO, TiO
2
, the hole carrier
will form a self-trapped polaron, and thus has very low mobility
[16]. The formation of hole polarons results from the localized
nature of the O 2p orbitals, so it happens whether the holes
are created by p-type doping with extrinsic elements (like
Al, Ga, In, etc.) [17,18], or injected electrically or by light
absorption. The low mobility of hole polarons provides another
explanation for the poor p-type conductivity, an alternative to
the thermodynamic limit to the hole concentration. Because
the VBMs of many TMOs are derived from the O 2p orbitals,
polaron formation might occur widely in many TMOs. Indeed,
similar hole polarons have been found in HfO
2
, SnO
2
, and
Li
2
O
2
[19–22].
So far study has been focused on isolated hole polarons
[16–19,23], while the polaron-polaron interaction has been
ignored in TMOs. In general, the electronic structure of
an isolated single-hole polaron is characterized by a spin-
polarized localized state with an energy level in the band gap
(Fig. 1). When two or multiple polarons get close to each other,
*
shiyouchen@lbl.gov
†
lwwang@lbl.gov
how do the hole states interact? Intuitively, due to the Coulomb
repulsion, one might expect that two polarons will repel each
other and thus remain separated. However, in this paper we
will describe a phenomenon where two polarons bind to form
a bipolaron pair, even in an impurity-free TMO lattice. Such a
pair pushes the hole states all the way into the conduction band.
Thus, between the occupied and unoccupied states there is a
clean gap, as large as that of the original TMO, and the Fermi
energy can be pinned at the middle of the gap (as in an intrinsic
semiconductor) despite the existence of the positively charged
and localized holes. The rise of the Fermi energy suppresses the
spontaneous formation of charge-compensating donor defects,
so the conventional mechanism based on the limit to the hole
concentration becomes ineffective [14,15]. This phenomenon
highlights the complexity of TMO carrier dynamics, showing
that it is questionable to use our understanding of the main
group of semiconductors to explain the TMO behaviors.
Besides the abnormal electronic structure, there is signif-
icant change in the atomic structure near the bipolaron. Its
negative binding energy relative to two separated polarons
results from the hybridization of the two hole states plus
large O atomic relaxation, which can overcome the Coulomb
repulsion. The large displacements of two O atoms lead to the
formation of an O-O bond (dimer) which does not exist in the
original TMO lattice. The O-O bond (dimer) formed traps two
holes tightly, which causes a low hole mobility (even lower
than that of the single-hole polaron) and limits the p-type
conductivity. This effect is found to exist in TMOs with short
O-O distances and loose lattices, such as anatase TiO
2
,V
2
O
5
,
and MoO
3
, independent of any dopants or the formation of
defects, so it is an intrinsic and general limit.
II. STABILIZATION MECHANISM
We will first take the anatase TiO
2
as an example system
to show why the double-hole-induced O-O bipolaron is stable
vs free holes or separated single-hole polarons. When free
hole carriers are present in the lattice, there is no structural
distortion and the holes stay at the energy level of the VBM
state (Fig. 1). When a single-hole polaron is formed, one
Ti-O bond will be elongated by 0.2
˚
A, as shown in Fig. 2(b).
Meanwhile the hole wave function becomes localized on one
1098-0121/2014/89(1)/014109(6) 014109-1 ©2014 American Physical Society
SHIYOU CHEN AND LIN-WANG WANG PHYSICAL REVIEW B 89, 014109 (2014)
FIG. 1. (Color online) The band diagram of TMOs with a free
hole near the VBM (left), and two single-hole polarons binding to
form an O-O dimer bipolaron (right). The dashed lines show how
the coupling between two single-hole polaron states induces the
formation of an O-O dimer.
O atom with the energy level shifting up into the gap, as shown
schematically in Fig. 1 (right) and quantitatively in Fig. 3(b)
where the calculated partial density of states is projected on the
displaced O atom (the calculation details are given in Sec. V).
The hole polaron state is about 1.8 eV above the VBM, which
traps the holes to the localized O 2p orbitals, and on the
opposite-spin side the polaron also induces a localized state
−5.3 eV lower than the VBM.
When there are two holes in the system, intuitively we may
expect that they will form two separated hole polarons due to
FIG. 2. (Color online) Structural plots for (a) one Ti cation
coordinated by six O anions in the anatase TiO
2
lattice, (b) an anatase
supercell with two separated single-hole polarons, and (c),(d) an
anatase supercell with formation of two different O-O dimers. The red
and light blue balls show the O and Ti atoms without displacement,
respectively, and the green balls show the O atoms displaced from
their ideal positions. The dashed lines in (a) show the distances
between the O atoms. (c) is the top view along the c axis while
(b) and (d) show the side view along the b axis.
FIG. 3. (Color online) The calculated partial density of states
(DOS) projected on (a) a normal O atom in the anatase TiO
2
lattice,
(b) the polaron O atom, as shown by the green ball in Fig. 2(b),and
(c) the O-O dimer as shown in Fig. 2(c). The energy is relative to the
VBM eigenenergy. The top and bottom panels show the spin-up and
spin-down DOSs, respectively.
the Coulomb repulsion. However, if the coupling between the
two hole polaron states is considered, we find that there is a
possibility for two hole polarons to bind with each other and
induce an O-O dimerization. This can be understood according
to a simple band-coupling model, which is plotted in Fig. 1
(right). When two hole polarons get close to form an O-O
dimer, the two spin-up polaron states will couple, as will
the two spin-down states. As a result, the two bonding states
are pushed down into the valence band and two antibonding
states are pushed up into the conduction band. Despite the
large energy level separation between the initial spin-up and
spin-down states, the final spin-up and spin-down bonding
(and antibonding) states are degenerate, so the system becomes
non-spin-polarized. At the final stage, there is no state in the
band gap and the two electrons in the bonding states have
significantly lowered energies, thus gaining total energy for
the system. If the energy gain is larger than the strain energy
of the structural distortion plus the hole-hole Coulomb repul-
sion, the O-O dimerization is stabilized relative to the two
separated polarons.
To show whether this O-O dimerization is more stable than
two separated hole polarons, a direct calculation of their total
energy has been carried out (details are given in Sec. V). Here
we follow Varley et al.’s methods and define a self-trapping
energy for two holes as [16]
E
ST
= E(distorted, 2h
+
) − E(ideal, 2h
+
) + 2V , (1)
where E(ideal, 2h
+
) is the total energy of a 108-atom anatase
TiO
2
supercell with two delocalized free holes (no structural
distortion), and E(distorted, 2h
+
) is the total energy of the
014109-2
DOUBLE-HOLE-INDUCED OXYGEN DIMERIZATION IN . . . PHYSICAL REVIEW B 89, 014109 (2014)
same supercell with two holes trapped by structural distortion
[either two separated polarons as shown in Fig. 2(b),oran
O-O dimer, as shown in Fig. 2(c) or Fig. 2(d)]. Since the
supercell is charged with two holes, we follow the procedure
of Varley et al. and include a term 2V , where V is
the electrostatic potential difference between the free-hole
supercell and the trapped-hole supercell, calculated at a region
far from the structural distortion [16,24]. As the supercell size
becomes larger, V will approach zero, but for finite-size
supercells, the inclusion of 2V will make the calculated
E
ST
converge faster depending on the supercell size [24]. A
negative E
ST
means the self-trapped holes are more stable
than the free holes. If E
ST
of the O-O dimerization is lower
than that of two separated hole polarons, the O-O dimerization
is energetically more favorable, and their difference gives us
the polaron-polaron binding energy.
Three different configurations of O-O dimers in anatase
TiO
2
are considered. They are formed through moving the O
atoms together, starting from three O-O pairs which have orig-
inal separations 2.46, 2.78, and 3.03
˚
A, as shown in Fig. 2(a).
After local structural relaxation, the O-O dimers formed
from the 2.78 and 2.46
˚
A O-O pairs are stabilized at O-O
distances of 1.45 and 1.49
˚
A, respectively [shown in Figs. 2(c)
and (d)] with negative E
ST
=−1.50 and −1.18 eV per hole
pair (−0.75 and −0.59 eV/hole). However, the dimer formed
from the 3.03
˚
A O-O pair is unstable after relaxation. It should
be noted that it is the double holes that stabilize the O-O dimer,
and when there are no holes, all the O-O dimers are unstable.
We have also calculated E
ST
for two separated polarons,
as shown in Fig. 2(b). The corresponding E
ST
is −0.96 eV
per hole pair (−0.48 eV/hole). To test the convergence of
our calculation, we have also calculated E
ST
when there is
only one polaron in the 108-atom supercell; the corresponding
result is −0.53 eV/hole. The difference is only 0.05 eV/hole.
No matter which value is used, the energy of separated
polarons is higher than that of the O-O dimers. Relative to
two separated polarons in the supercell, the polaron-polaron
binding energies are −0.54 eV and −0.22 eV per hole pair
for the Fig. 2(c) and Fig. 2(d) O-O dimers, respectively [25].
The geometry of the O-O dimerization is rather like that of
the dopant-dopant binding (e.g., N-N, C-S, C-O in anatase
TiO
2
) as found interestingly by Yin et al. [2] In contrast with
the dopants (impurity atoms) which are locally charge neutral
when not ionized, the hole polarons are positively charged,
so it is unexpected that the strong Coulomb repulsion can be
overcome and the polaron-polaron binding is favored.
III. ELECTRONIC STRUCTURE
In Fig. 3(c), we plot the calculated partial density of states
(DOS) projected on the O-O dimer. It is obvious that the
O-O dimer produces localized electronic states in the lattice,
as demonstrated by the high peaks in the DOS, which can
be likened to the O
2
molecular orbitals. For an isolated O
2
molecule (or O-O bonds in peroxides [22]), the hybridization
of 2p atomic orbitals produces the bonding states σ
2p
z
, π
2p
x
,
and π
2p
y
, and the antibonding states π
∗
2p
x
, π
∗
2p
y
, and σ
∗
2p
z
.As
labeled in Fig. 3(c), the three highest peaks of the O-O dimer
2p DOS can be attributed to four O
2
molecular orbitals, whose
FIG. 4. (Color online) The wave functions of four electronic
states localized on the O-O dimer, which produce the three highest
peaks of the partial DOS as plotted in Fig. 3(c).Inthexyz coordinate
system shown here, these four states correspond to the σ
2p
z
, π
2p
x
,
π
2p
y
,andπ
∗
2p
x
orbitals of the O
2
molecule.
wave functions are also plotted in Fig. 4.(TwoO
2
molecular
orbitals, the occupied π
∗
2p
y
and unoccupied σ
∗
2p
z
, do not induce
high peaks in the partial DOS, since their wave functions
hybridize heavily with the Ti 3d states.) Although the O-O
dimer is in the TiO
2
lattice, these wave functions have obvious
characteristics of O
2
molecular orbitals, indicating that an O-O
bond is really formed. Therefore the O anions now have a
valence of −1, rather than the valence −2 of normal O anions.
This can be considered as a local superoxidation [26] induced
by two holes.
The calculated DOS (Fig. 3) of the O-O dimer is in good
agreement with the band coupling model as shown in Fig. 1:
(i) the σ
2p
z
energy level of the O-O dimer shifts down relative
to the 2p levels of the normal O atom or polaron O atom,
which lowers the energy of the O-O dimer (contributing to the
large and negative E
ST
); (ii) after the two polarons bind, the
system become non-spin-polarized with the same spin-up and
spin-down DOSs; (iii) the hole states are pushed up into the
conduction band, leaving a clean band gap as large as that of
the ideal anatase TiO
2
, so the holes are trapped at states high
in the conduction bands and their mobility should be seriously
suppressed. This sets an intrinsic limit to p-type conductivity
regardless of the hole concentration.
IV. OTHER TRANSITION METAL OXIDES
As well as in the anatase TiO
2
, double-hole-induced O-O
dimerization exists in other TMOs. One example is V
2
O
5
(shcherbinaite structure [27]) and another is MoO
3
[28].
Their trapping energies E
ST
for single-hole polarons and
double-hole bipolarons (O-O dimer) are shown in Table I.
In V
2
O
5
,anegativeE
ST
=−0.98 eV/hole is found for
the O-O dimer with a separation 1.37
˚
A (2.98
˚
Ainthe
nondistorted crystal), which is lower than the trapping energy
of the single-hole polaron (−0.85 eV/hole). In MoO
3
,alarge
014109-3
SHIYOU CHEN AND LIN-WANG WANG PHYSICAL REVIEW B 89, 014109 (2014)
TABLE I. Calculated self-trapping energy E
ST
(in eV/hole) for
the single-hole polaron and the O-O dimers in different TMOs.
V
2
O
5
MoO
3
Anatase TiO
2
Rutile TiO
2
Hole polaron −0.85 −0.54 −0.53 −0.08
O−Odimer −0.98 −1.31 −0.75 0.41
negative E
ST
=−1.31 eV/hole is found for the O-O dimer
with a separation 1.37
˚
A (2.82
˚
A in the nondistorted crystal),
which is much lower than the trapping energy of the single-hole
polaron (−0.54 eV/hole). The large negative E
ST
clearly
shows that two polarons in V
2
O
5
and MoO
3
prefer to bind
together to form O-O dimers. All these TMOs have the top
part of the valence bands composed of O 2p states, as in
TiO
2
, so the stabilization of the O-O dimers can be understood
according to the same band-coupling model as shown in Fig. 1.
However, not all TMOs will form O-O dimers when two
holes are present. In TMOs like zinc-blende ZnO or even rutile
TiO
2
, the O-O dimers are unstable. More specifically, in ZnO
the O-O dimer breaks up after structural relaxation, while in
rutile TiO
2
the O-O dimer is metastable with positive E
ST
=
0.41 eV/hole. (In contrast, single-hole polarons have negative
E
ST
, which is consistent with the experimental identification
of single-hole polarons in rutile TiO
2
[23].) The reason is
twofold: (i) In anatase TiO
2
,V
2
O
5
, and MoO
3
, there is a
large void space around the O-O pairs, which can tolerate the
structural distortion with low strain energy. But in ZnO or rutile
TiO
2
the crystal structure is relatively compact without such
a large void, which makes large structural distortion costly.
(ii) The shortest O-O distance in ZnO (3.21
˚
A) or rutile
TiO
2
(2.54
˚
A) is larger than that in anatase TiO
2
(2.46
˚
A),
so the formation of an O-O dimer needs larger structural
distortion and more strain energy. Although the present study
considered only a limited number of TMOs due to the
computational expense, we expect that double-hole-induced
O-O dimerization exists in other TMOs, as long as there are
large voids and loosely bonded O atoms around such sites and
the initial O-O distance is short. The band-coupling model of
O-O dimerization is general for all TMOs with the highest
valence bands consisting of localized O 2p states.
V. CALCULATION METHODS AND ANALYSIS
All the above calculations are performed within the
DFT formalism as implemented in the Vienna Ab-initio
Simulation Package (
VASP) code [30]. Projector augmented-
wave pseudopotentials [31] and an energy cutoff of 400 eV
were used in all cases. The hybrid exchange-correlation
functional Heyd-Scuseria-Ernzerhof (HSE) [29] is used with
one-quarter (α = 0.25) of exact electron exchange mixed
into the generalized gradient approximation (GGA) [Perdew-
Burke-Ernzerhof (PBE)] functional. The supercell size is 108
atoms for anatase TiO
2
, 216 atoms for rutile TiO
2
, 144 atoms
for MoO
3
,84atomsforV
2
O
5
, and 96 atoms for ZnO. To
ensure the proper occupation of the free-hole state, only the
point is included in the Brillouin-zone integration, following
the procedure of Varley et al. [16]. Test calculations with a
larger supercell (216 atoms) and more k points for anatase TiO
2
0 0.05 0.1 0.15 0.2 0.25
α (ratio of exact exchange)
-0.8
-0.6
-0.4
-0.2
0
0.2
ΔE
ST
(eV/hole)
hole polaron
O-O dimer (2.46 Å)
O-O dimer (2.78 Å)
FIG. 5. (Color online) The dependence of the calculated self-
trapping energy E
ST
on the α parameter of the hybrid functional
for holes trapped by polarons or two O-O dimer bipolarons in anatase
TiO
2
.
showed that the error in the calculated self-trapping energy is
less than 0.05 eV/hole.
Besides the hybrid functional HSE, we also tested our
calculations using different functionals. As we know, DFT
calculations with the local density approximation (LDA) or
GGA tend to delocalize electron wave functions of 2p states
due to an erroneous self-interaction energy [32], which makes
the formation of polarons unlikely in the first place. The
HSE functional largely corrects this problem, but nevertheless
the result might depend on the mixing parameter α of the
hybrid functional, which determines the percentage of the
exact exchange mixed into the GGA functional [29]. It is thus
necessary to test the dependence of our results on this mixing
parameter, and ensure that the phenomena we found exist for
the whole plausible parameter range.
The calculated trapping energies E
ST
as functions of α are
showninFig.5 for the single-hole polaron and two O-O dimers
in anatase TiO
2
. They decrease almost linearly with α. With
a larger α ratio, more exact exchange is mixed in the hybrid
functional, and the O 2p states becomes more localized. That
means the polaron and bipolarons (O-O dimers) become more
stable as the O 2p states become localized. As a result of the
linear dependence, the E
ST
differences between the single-
hole polaron and the O-O dimers are almost independent of α.
Within the whole plausible range of the α parameter from 0
(equivalent to a pure GGA functional in PBE form) to 0.25 (the
standard HSE functional), E
ST
of the most stable O-O dimer
[shown in Fig. 2(c)] is always negative and lower than that of
the single-hole polaron, Thus, it is rather safe to say that such
an O-O dimer is the stablest configuration regardless of what
α is used. This is also true for the lowest-energy O-O dimers
in V
2
O
5
and MoO
3
. Therefore our conclusion is independent
of the specific approximations to the functionals used.
On the other hand, as discussed above, the σ
2p
z
energy
level of the O-O dimer shifts down relative to the 2p levels
of the normal O atom or polaron O atom, which lowers
the energy of the O-O dimer (contributing to the large and
negative E
ST
). If the splitting between the bonding σ
2p
z
and antibonding σ
∗
2p
z
levels is exaggerated in DFT (PBE or
HSE) calculations, E
ST
will be overestimated (more negative
014109-4
