Polar Metals Taxonomy for Materials Classification and Discovery Daniel Hickox-YoungDanilo Puggioniand James M. Rondinelli Department of Materials Science and Engineering

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Polar Metals Taxonomy for Materials Classification and Discovery
Daniel Hickox-Young,
∗
Danilo Puggioni,
†
and James M. Rondinelli
‡
Department of Materials Science and Engineering,
Northwestern University, Evanston, Illinois 60208, USA
(Dated: October 18, 2022)
Over the past decade, materials that combine broken inversion symmetry with metallic conductivity
have gone from a thought experiment to one of the fastest growing research topics. In 2013, the
observation of the first uncontested polar transition in a metal, LiOsO
3
, inspired a surge of theoretical
and experimental work on the subject, uncovering a host of materials which combine properties
previously thought to be contraindicated [Nat. Mater.
12
, 1024 (2013)]. As is often the case in a
nascent field, the sudden rise in interest has been accompanied by diverse (and sometimes conflicting)
terminology. Although “ferroelectric-like” metals are well-defined in theory, i.e., materials that
undergo a symmetry-lowering transition to a polar phase while exhibiting metallic electron transport,
real materials find a myriad of ways to push the boundaries of this definition. Here, we review
and explore the burgeoning polar metal frontier from the perspectives of theory, simulation, and
experiment while introducing a unified taxonomy. The framework allows one to describe, identify,
and classify polar metals; we also use it to discuss some of the fundamental tensions between theory
and models of reality inherent in the terms “ferroelectric” and “metals.” In addition, we highlight
shortcomings of electrostatic doping simulations in modeling different subclasses of polar metals,
noting how the assumptions of this approach depart from experiment. We include a survey of
known materials that combine polar symmetry with metallic conductivity, classified according to
the mechanisms used to harmonize those two orders and their resulting properties. We conclude by
describing opportunities for the discovery of novel polar metals by utilizing our taxonomy.
I. INTRODUCTION
The concept of crystalline metals without inversion
symmetry, specifically those that lift parity symmetry to
support a polar crystal structure, has been frequently at-
tributed to a concise 1965 Letter by Blount and Anderson
titled “Symmetry Considerations on Martensitic Trans-
formations: ‘Ferroelectric’ Metals” [
1
]. What is less well
appreciated is that Blount and Anderson’s article focuses
on how a nominally first-order ferroelastic transition, ob-
served at the time in the metallic silicide V
3
Si, could
exhibit second-order character, presumably attributed
to a displacive component akin to the paraelectric-to-
ferroelectric transition found in insulating compounds,
which exhibit a well-defined polarization below the crit-
ical temperature. At the time, nearly all martensitic
(ferroelastic) transformations in metals exhibited strong
first-order character [
2
]; yet, V
3
Si exhibited second-order
behavior. The continuous response was rationalized by in-
ferring that the symmetry-break should be like that found
in second-order displacive ferroelectrics, hence the quotes
around “ferroelectric” in the title. Blount and Anderson
did not suppose that “ferroelectric” metals would possess
a switchable polarization. Gauss’s Law dictates that no
electric field may exist within a metal [
3
,
4
], so the atomic
structure of an ideal polar metal should be immune to
perturbation via external applied electric field.
Ironically, for years this same physical law seemed to
imply that polar metals should not exist at all. In proto-
typical ferroelectrics like BaTiO
3
it was shown that the
polar displacement was stabilized by long-range dipole-
dipole interactions [
5
,
6
]. In the presence of free charge
carriers, such interactions would be completely screened,
favoring the centrosymmetric structure. In the decades
following Anderson and Blount’s work, it appeared that
Gauss’s Law would prevail over the synthesis of a polar
metal. There were several candidates in the 2000s which
combined polar order and metallicity via compositional
ordering, but it was not until 2013 that Shi et al. showed
the metal LiOsO
3
exhibited a displacive transition [
7
].
Their work demonstrated that a “ferroelectric”-like tran-
sition need not rely on long-range interactions, but may
derive from local structural effects (geometrically-driven
Li displacements in the case of LiOsO
3
). Since the dis-
covery of LiOsO
3
and following the introduction of a
‘weak-coupling hypothesis’ by Puggioni and Rondinelli
outlining mechanisms by which to stabilize polar metals
[
8
], they have garnered significant interest (Fig. 1). It was
further accelerated by the discovery of nontrivial topolog-
ical metals and their ensuing phenomena and properties
[
9
–
11
]. During the rapid increase in scholarship, diverse
methods of combining these previously contraindicated
properties have been proposed and executed, ranging
from degenerately doped ferroelectrics [
12
–
15
] to metals
with hybrid improper polar distortions [
16
,
17
] to two-
dimensional thin films and interfaces [
18
–
20
]. Reviews of
the classification of different design strategies for polar
metals can be found in Refs. 21–23.
Accompanying these materials are a variety of terms.
Polar, ferroelectric, “ferroelectric,” ferroelectric-like, and
native ferroelectric are all qualifiers used to describe metal-
lic systems with broken inversion symmetry. Further-
more, many so-called “ferroelectric metals” either push
the boundaries of what may be called ‘metallic’ or do not
exhibit a switchable polarization. Given the advances in
dielectric, modern polarization, and soft-mode theories
[
24
–
28
], we find that the ferroelectric-like designation and
arXiv:2210.05110v2 [cond-mat.mtrl-sci] 15 Oct 2022
2
1968 1976 1984 1992 2000 2008 2016
year
0
10
20
30
40
citations
FIG. 1. Citations per year of Anderson and Blount’s 1965
paper on second-order martensitic phase transitions [
1
]. Data
collected from Google Scholar on October 10th, 2022.
its derivatives [
18
,
29
–
32
] are cumbersome and nonessen-
tial descriptions, obfuscating the physics displayed by
very different (and yet, equally interesting) classes of
materials. In addition, we note that both experimental
and computational approaches to studying these mate-
rials have at times been abused. Computationally, the
background-charge approach to electrostatic doping simu-
lations fails to model reality in multiple underappreciated
ways [
33
,
34
]. Meanwhile, in experiment, the application
of ferroelectric characterization techniques to materials
that are not formally ferroelectrics both erodes the key
differences separating metals from dielectrics and can lead
to misinterpretations of measured dielectric polarizations,
e.g., electric polarization hysteresis [29,35].
In this work, we propose using meaningful atomic and
electronic structure descriptions to distinguish materials
based on conductivity and symmetry considerations. We
begin by discussing the tensions between theory and ex-
periment, as exemplified by both the various methods for
combining polar order and metallicity and the shortcom-
ings of our current terminology. These issues are further
amplified in computational studies using electronic struc-
ture methods, e.g., with the popular background-charge
approach to electrostatic doping simulations, which can
lead to model results inconsistent with experiment. We
then present a survey of polar metals classified using
our taxonomy that combines conductivity with broken
inversion using terminology based on clearly defined class
descriptors.
II. TENSIONS IN TERMINOLOGY:
“FERROELECTRIC” “METALS”
Between the realms of theory and experiment, commu-
nication is critical in order to foster a productive relation-
ship. Since communication is built on having a common
vocabulary, this makes the terminology we use of utmost
importance in seeking to advance the current understand-
ing of our field and not just a pedantic dilemma. In
the field of polar metals, however, some classifiers that
seem obvious in theory are less well-defined in experi-
ment and the result has been confusing and at times
unintentionally misleading. We present a few examples
of the two most common sources of dissonance between
theoretical and experimental labels—namely, “metallic”
and “ferroelectric”—and suggest methods for relieving
the tension.
A. Ferroelectric “metals”
How to define metallicity?—The question of how one de-
fines a ‘metal’ carries significance for many disciplines, but
the distinction bears considerable weight when evaluating
doped ferroelectrics (FEs), especially as a large number of
reports describe doping known FEs as a route to achieve
polar or “ferroelectric” metals. At what point does a
doped ferroelectric become a polar metal? Is there an im-
portant fundamental difference between a polar structure
with intrinsic charge carriers or extrinsic charge carriers?
To answer these questions, let us consider fundamental
definitions of metallicity.
According to Kohn’s theory, while the wave function
of a metal in its ground state is delocalized, that of an
insulator is localized [
36
]. This distinction strictly defines
the difference between metals and insulators. Electron
transport considerations can, in principle, be used to
quite clearly separate metals from insulators (dielectrics).
According to Mott, “. . . a metal conducts, and a non-
metal doesn’t” [
37
,
38
]. This statement is strictly true
at
T
= 0 K and is often used as the discriminating factor
between a metal and an insulator (or semiconductor) such
that:
lim
T→0ρ(T) = (∞insulator
ρ0metal (1)
where
ρ
corresponds to the electric resistivity of the ma-
terial and
ρ0
is the residual resistivity due to electron
collisions with crystal impurities and imperfections at
T
= 0 K. Importantly, the value of
ρ
at room-temperature
does not matter; thus, although the definition is mathe-
matically well defined, experimental conditions (
T6
= 0 K)
can make the differentiation challenging, especially when
the carrier density of a semiconductor is sufficiently high
at room temperature. These may be intrinsic charge car-
riers or the result of degenerate doping, such that the
material is conductive through extrinsic doping [
15
,
39
–
41
]. These compounds are polar, and for reasonably large
temperature ranges there is a positive correlation between
resistivity and temperature, even though (strictly speak-
ing) the correlation between resistivity and temperature
becomes negative at low temperature. This fact may be
why the boundary between doped polar semiconductors
and polar metals is frequently blurred by experiment,
despite the clear theoretical distinction [36,42,43].
Band Theory—The distinction between a trivial insu-
lator and a metal can be understood using band theory.
3
For an insulator, the Fermi level is not well-defined, but
resides within an energy gap. Therefore, at 0 K, there are
no free charge carriers available and
ρ→ ∞
. With increas-
ing temperature, the electrons may thermally populate
the conduction band and become available for conduction
such that the resistivity decreases (
dρ/dT <
0). In metals,
the Fermi level is located within a band, giving rise to
free charge carriers. As temperature increases, so does
resistivity (
dρ/dT >
0). At high temperatures (
T
Θ
D
,
where Θ
D
is the Debye temperature),
ρ∼T
. At low
temperatures (
T
Θ
D
) and
ρ
has four contributions
[44]:
ρ(T) = ρ0+AT 2+BT 5+CT fexp −~ωmin
kBT.
The
T2
contribution originates from electron-electron scat-
tering. The
T5
contribution was evaluated by Bloch and
Gr¨uniessen [
45
,
46
] and is due to electron-phonon scat-
tering. The exponential term describes the contribution
of electron-phonon umklapp scattering, where
ωmin
is
the minimum phonon frequency below which umklapp
process are forbidden and
f
is an empirical parameter.
To be considered genuinely metallic, polar metals should
exhibit a positive correlation between resistivity and tem-
perature for all
T >
0 K. This is exactly what happens for
LiOsO
3
, which shows Fermi liquid-like behavior (
ρ∼T2
)
in its polar phase [
7
]. With this in mind we can answer
the question: At what point does a doped semiconduc-
tor become metallic? The threshold to achieve metallic
conductivity is determined by the carrier concentration
required to make dρ/dT > 0.
The ratio
dρ/dT
is useful for differentiating metals from
insulators. However, we note that metallic and insulating
phases are limiting situations. In transition metal oxides
and other strongly correlated systems
ρ
(
T
) exhibits a com-
plex behavior. For instance, the polar oxide Ca
3
Ru
2
O
7
exhibits metallic conductivity above 48 K, insulating be-
havior from 48 to 30 K, and shows again metallic transport
below 30 K [
47
]. Even doped ferroelectrics exhibit com-
plex transport properties. In La-doped BaTiO
3
,
ρ
(
T
)
shows insulating behavior from 350 to 260 K, metallic
transport between 260 and 70 K, and again insulating be-
havior with weak localization below 70 K [
40
]. Therefore,
the assessment of metal/non-metal status using
dρ/dT
alone is not an easy task and should be performed in
conjunction with other descriptors.
Drude Definition—Although the DC conductivity
would appear to be the natural way to separate met-
als from insulators, in practice assessing the frequency
dependent free-carrier response of a material allows one
to treat metals, doped-semiconductors, and insulators on
more equal footing [
49
]. For a perfect crystalline material
with non-interacting electrons, the optical conductivity
σrelates the electrical current density due to a spatially
uniform transverse electric field as
σ(ω) = ne2τ
m(1 −iωτ),
ω
σ(ω)
σ0
metal
insulator
doped semiconductor
FIG. 2. (top) Schematic illustration of the difference in opti-
cal conductivity response across materials classes. (bottom)
Experimental data from Ref.
48
illustrating the evolution of
the optical conductivity of BaTiO
3−δ
under oxygen vacancy
doping at T≈30 K.
where
m
is the mass of the carrier (free electron mass or
band renormalized mass) and
n
is the density of carri-
ers (electrons or holes). Fig. 2a shows that for an ideal
metal the dc conductivity
σ0
=
ne2τ/m
appears as a local
maximum of
σ
(
ω
) at zero frequency (Drude peak) and
then decays with a Lorentzian form due to finite relax-
ation time
τ
. In contrast,
σ
(
ω
) = 0 for 0
≤ω≤Eg
in
insulators with an optical gap
Eg
. This Drude definition
would categorize some doped FEs with very modest room
temperature conductivity as fundamentally closer to in-
sulators, providing some clarity. In addition, the nature
of the optical conductivity may vary with temperature
[
50
], allowing the metal-insulator classification to change
with temperature—thereby accounting for metal-insulator
transitions. However, sufficient doping of a ferroelectric
4
TABLE I. Plasma frequency and resistivity magnitudes as
measured at room temperature for band metals (top) and
doped semiconductors (bottom).
Material ωp(cm−1) Resistivity (10 −6Ω-m)
Al 1.19 ×1052.82×10−2
Cu 6.38 ×1041.70×10−2
Au 7.25 ×1042.44×10−2
Pb 6.20 ×1042.20×10−1
Ag 7.25 ×1041.59×10−2
n-GaAs 4.94 ×102∼103
n-Si 1.76 ×10310.7
p-Si 2.28 ×1036.4
n-InSb ∼2.1×102103–104
can still lead to optical conductivities that remain nonzero
as
ω→
0, but do not reach a local maximum at
ω
= 0
(Fig. 2b), which presents a somewhat ambiguous case.
Another low-frequency dynamical property, which is of
relevance to distinguishing between metals and insulators,
is the non-adiabetic Born effective charge (naBEC) [
51
].
Whereas Born effective charges are typically only well-
defined in insulators (as measurements of the changes in
dielectric polarization as a function of atomic displace-
ment), naBECs are measurable in metals by consider-
ing the current generated in response to atomic motion.
Atomic motion in this case is produced optically in a
regime such that
ω
is much greater than the inverse car-
rier lifetime (1
/τ
) while still being much smaller than
interband resonances. Effectively, i.e., for materials with
long carrier lifetimes, this amounts to
ω→
0 and the
naBECs stabilize as the “Drude weight” or the density
of free charge carriers available for conduction. As with
optical conductivity,
ω→
0 in insulators but reaches a
non-zero value in metals. The naBECs, as an analog to
BECs, are also advantageous as they allow for charac-
terization of polarizability in metals despite polarization
itself not being well-defined.
Order of Magnitude—Although for low-frequencies the
optical properties of (doped) semiconductors are qualita-
tively similar (
σ6
= 0), they are quantitatively different,
because of the difference in carrier masses and densities.
At high-frequencies, semiconductors and metals both ab-
sorb – as expected for an insulator with available conduc-
tion band states – owing to interband processes that give
rise to finite
σ
(
ω
). The frequency crossover at which the
behaviors change is given by the plasma frequency
ωp
,
corresponding to a zero in the real part of the dielectric
function. Neglecting any damping effects, the plasma fre-
quency can be expressed as
ω2
p
=
ne2/ε0m,
where
ε
=
∞
for a metal in that it includes only electronic contributions
from (high energy) interband transitions while an insula-
tor includes both electronic and ionic (static) polarization
contributions, i.e.,
ε
=
∞
+
ionic
and
ionic > ∞
. As the
carrier density increases, the plasma frequency increases.
In conventional metals
ωp
is in the UV region which gives
FIG. 3. Temperature-dependent resistivities of a variety of
metals, polar metals, and doped semiconductor. Although the
polar metals exhibit electron transport roughly 1-2 orders of
magnitude more resistive than typical elemental metals, they
are still easily identifiable as significantly more conductive
than the doped semiconductors (despite a positive slope in
resistivity for La:BaTiO
3
throughout the temperature range).
rise to the UV reflectivity edge (light of frequency
ω < ωp
is reflected). (In practice, this edge can be difficult to
assess experimentally due to interband transitions and in
some cases can be found just below visible frequencies.)
By contrast, the carrier density in doped semiconductors
places
ωp
in the 100s of meV (5-30
µ
m range) as in
n
-InSb
(Table I) and is highly tunable [52].
Table I also illustrates the order of magnitude gap in re-
sistivity between doped semiconductors and band metals
(as does Fig. 3). The order of magnitude is a meaningful
descriptor to aid classification as it similarly is a useful
materials selection when deploying materials under ap-
plication constraints. This distinction should be used
when comparing polar metals and doped FEs. However,
it is worth noting that some materials would be misla-
belled if one uses the order of magnitude of the plasma
frequency or resistivity alone. Doped SrTiO
3
exhibits
extremely low resistivity (including a superconducting
transition) but the plasma frequency, even at low tem-
perature, is comparable to other doped semiconductors
(
∼
1
.
5
×
10
3
cm
−1
)[
53
]. Meanwhile, many polar metals
exhibit relatively poor conductivity (Fig. 3), belonging to
the so-called class of “bad metals.” LiOsO
3
, for example,
has a room temperature resistivity of 15
×
10
−6
Ω-m and
a plasma frequency on the order of
∼
10
2
cm
−1
) [
7
,
54
].
Therefore, as with all other descriptors discussed thus far,
order of magnitude is best used in combination with other
5
criteria when assessing conductivity.
Doping Sensitivity Analysis—Although the descriptors
above are generally sufficient to clearly differentiate be-
tween polar metals and doped FEs, greater clarity may
be achieved by considering doping as a perturbation to
the initial state of a material and evaluating whether that
perturbation has been sufficient to change the material’s
classification. We consider the sensitivity of the electronic
and crystallographic structure with respect to the per-
turbation. The effect of doping on electronic structure is
direct and immediately distinguishes intrinsic conductiv-
ity from extrinsic conductivity. Doping shifts the Fermi
level, which in most metals (i.e., excluding semi-metals)
has little effect on the effective mass or concentration of
the free charge carriers. In FE insulators, on the other
hand, doping has an immediate impact, shifting the Fermi
level toward a band edge and often inducing defect states,
thereby altering the conduction mechanism. This distinc-
tion has practical considerations; the transport properties
of doped FEs will be more sensitive to changes in the
electron chemical potential than those of polar metals.
At sufficiently high concentrations, dopant atoms form
a partially occupied impurity band which may exhibit
metallic conductivity—so-called degenerately doped semi-
conductors. In this regime, the conductivity of the mate-
rial is less sensitive to small variations in the concentration
of impurity atoms than a traditional doped semiconductor.
However, the system should still be considered a perturba-
tion from the pristine state of the semiconductor and can
be distinguished from a band metal by both the relatively
smaller carrier concentration and the proximity of the
Fermi level to a band gap. These distinctions should be
clear from electron transport and optical measurements,
respectively.
Predicting the effect of doping on crystal structure is
less direct and requires an understanding of the structural
driving forces. In the case of polar metals or doped FEs,
the primary structural concern is the impact of doping
on the inversion-lifting mechanism. Once again, different
classes of materials will respond differently to doping as
a perturbation. In doped proper FEs, the asymmetric
structure is stabilized by a combination of dipole-dipole
interactions and covalent bonds which compete with short-
range repulsive forces (which favor a higher symmetry
structure) [
55
–
57
]. Although the addition of charge carri-
ers is not necessarily incompatible with the persistence
of broken symmetry, it cannot help but reduce and even-
tually eliminate the influence of long range dipole-dipole
interactions (due to the reduction of the screening length)
that cooperatively align the off-centering displacements
and may also interfere with bonding, depending on the
electronic structure of the material. By contrast, long-
range interactions in polar metals are always screened and
the atoms providing states at the Fermi level typically
display weak coupling with the atoms active in the soft
phonon(s) driving the symmetry-break [
8
]. Nonetheless,
for sufficient carrier densities local polar displacements
can persist. Therefore, beyond how one simulates dop-
FIG. 4. (a) Adding correlation to LiOsO
3
via increasing the
Hubbard
U
(applied to Os
d
states) enhances the amplitude of
the polar distortion. (inset) Schematic showing how the polar
distortion amplitude is defined by the relative long and short
distances between Li and Os along the polar axis. (b) Crystal
structure of polar (
R
3
c
) LiOsO
3
. (c) Increasing the degree
of correlation in cubic BaTiO
3
(by applying the Hubbard
U
to the Ti
d
states) reduces the critical doping concentration
to stabilize the soft Γ-point phonon mode of the cubic phase.
(inset) The crystal structure of cubic (P m3m) BaTiO3.
ing in these materials, as discussed in Appendix A, it is
also imperative to recognize that how we understand the
manner in which the atomic structure responds to dop-
ing relies intimately on whether the experimental probe
interrogates local or average structure [
58
,
59
]. In any
case, changes in the Fermi level of doped proper FEs will
almost always eventually affect the ground state crystal –
local and average – structure whereas similar changes in
the Fermi level of polar metals are more likely to leave
the crystal structure unaltered independent of the experi-
mental probe volume.
Electron Correlation and Magnetism—Correlation also
plays a significant role, both in the realization of po-
lar metals generally and in the potential to drive metal-
insulator transitions, often in concert with magnetic or-
dering. Evidence of the former is found in the number
of polar metals which exhibit “bad” metallic transport
from electron-electron interactions (Fig. 3). Although it
is now well-established that short-range interactions play
a dominant role in driving local off-centering in polar
metals, reduction of the screening length via correlation
may enable longer-range interactions to further enhance
the displacement magnitude, or at least allow for long-
range coordination of local displacements. It was shown
in Ref.
60
that the polar displacements in the predicted
polar metal SrEuMo
2
O
6
are enhanced by introduction of
additional correlation via a Hubbard
U
.interaction within
DFT. A similar effect is observed when plotting the ef-
fective polar amplitude in LiOsO
3
as a function of the
static
U
(Fig. 4a). However, just as correlation may help
to stabilize or enhance polar displacements, when cou-
pled with magnetic ordering it may also drive Mott-type
metal insulator transitions, as found in simulations of
LiOsO3/LiNbO3superlattices [61].
摘要：

PolarMetalsTaxonomyforMaterialsClassi cationandDiscoveryDanielHickox-Young,DaniloPuggioni,yandJamesM.RondinellizDepartmentofMaterialsScienceandEngineering,NorthwesternUniversity,Evanston,Illinois60208,USA(Dated:October18,2022)Overthepastdecade,materialsthatcombinebrokeninversionsymmetrywithmetallic...

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