Hybrid functionals applied to perovskites

Hybrid functionals applied to perovskites Cesare Franchini Faculty of Physics, University of Vienna and Center for Computational Materials Science, A-1090 Wien, Austria E-mail: [email protected] Abstract. After being used for years in the chemistry community for describing molecular properties, hybrid functionals has been increasingly and successfully employed for a wide range of solid state problems, which are not accurately accessible by standard density functional theory. In particular, the upsurge of interest in transition metal perovskites-based compounds motivated by their technological relevance and functional ductility has incentivated the use of hybrid functionals for realistic applications, as hybrid functionals appear to be capable to capture the complex correlated physics of this class of oxide materials characterized by a subtle coupling between several competing interactions (lattice, orbital, spin). Here we present a map of recent applications of hybrid functionals to perovskites, aiming to cover an ample spectra of cases,including the ’classical’ 3d compounds (manganites, titanates, nickelates, ferrites, etc.), less conventional examples from the the 4d (technetiates) and 5d (iridates) series, and the (non transition metal) sp perovskite BaBiO3 . We focus our attention to the technical aspects of the hybrid functional formalism such as the role of the mixing and (for range-separated hybrids) screening parameters, and on a extended array of physical phenomena: pressure- and doping-induced insulator-to-metal and structural phase transitions, multiferroism, surface and interface effects, charge ordering and localization effects, and spin-orbit coupling.

Hybrid functionals applied to perovskites

2

1. Introduction Over the past few years, great progress has been made in the development and understanding of novel transition metal oxide compounds, which are considered to be at the core of next generation electronics. In particular, the perovskite family is recognized for its stunningly rich physics (colossal magnetoresistance[1, 2], metal-insulator transitions[3, 4], superconductivity[5], two-dimensional electron gas[6], multiferroicity[7], bandgaps from the visible to the ultraviolet range[8], and catalytic activity[9]) and by an unprecedented array of functionalities, often being relevant to different fields of application, including optoelectronics, spintronics, (photo)catalysis, and piezoelectric devices. This multifunctional character of perosvkites has its ground in the chemical and structural flexibility as well as in the simultaneous activity of spin, orbital, and charge degrees of freedom. The necessity to handle this complex scenario poses challenges to solid state scientists at both technological and fundamental level. From the point of view of theory, the historically dominant method for describing the physical properties of solids has been density functional theory (DFT). Although some aspects of the ground state properties of transition metal perovskites can be sufficiently well described within the local or semilocal density functional approximations (DFA), in particular the volume and to a lesser extent the magnetic moment, important quantities are not correctly predicted. Specifically, DFT undervalues bandgaps, fails to reproduce structural distortions such as Jahn-Teller instabilities and breathing mode distortions of the octahedra, and underestimates the tendency toward magnetism[10]. This is due to the approximate treatment of electronic correlation effects, which are significant in systems, often referred to as strongly correlated materials, with partially filled d or f shells and narrow energy bands. Transition metal perovskites belong to this class of materials. A common remedy which enables the description of phenomena that are due to strong electron correlation is the inclusion of an Hubbard U in the DFT exchange-correlation formalism[11]. Alternative schemes may involve the approximate evaluation of the self-interaction energy (the self interaction correction method[12]), the employment of a quasiparticle picture within the GW approximation[13] or the application of model (Hubbard-like) Hamiltonians in the spirit of the dynamical mean field theory (DMFT)[14]. DFA+U has been widely exploited in a number of cases, and though the results depend to a certain extent on the choice of U it has been found that this computational method conveys a substantially improved account of the ground state properties of perovskites, in particular structural distortions, and bandgap[11, 15, 16, 17]. Equally reliable are the results obtained by SIC (or by its pseudopotenial version, pSIC[18]), which delivers an overall satisfactory accuracy for non-magnetic and magnetic correlated oxide. The quasiparticle GW equations are especially appropriate for the computation of bandgaps and optical spectra at a very high level of accuracy, but due to the considerable computational cost their application to perovskites is rather limited[19, 20]. The DMFT emerges as a powerful tool to investigate electronic and magnetic quantum


3

phase transition as well as optical spectra, but has to be combined with LDA to achieve quantitative predictions in realistic context. Numerous results are reported in literature for DMFT based applications to perovskites[21, 22, 23]. More recently, a novel pragmatic approach which combines the benefits of the DFT and the Hartree-Fock (HF) theory has been proposed in the quantum chemistry contest[24], which are being used more and more in solid state calculations. This class of methods are named hybrid functionals, as they are constructed by hybridizing the DFT exchange-correlation functional with a fractional component of the exact HF exchange. The performance of hybrid functionals in solid state problems have been assessed in several works initially focused on elemental solids and standard semiconductors (SC/40 semiconductors database). In the last few years, the increasing interest in compounds based on the perovskites building block has brought an upsurge of hybrid functional calculations for this class of materials[25]. The aim of this review article it to survey the most relevant aspects of the application of hybrid functionals to perovskites. We will start with a formal introduction to the hybrid functional methodology which will focus on the fundamental basis behind the construction of hybrid functionals and on the degree of applicability of the various hybrid functional recipes. The main body of the paper will illustrate several selected works covering a wide set of cases including 3d, 4d, 5d and sp perovskites, and many different physically important phenomena such as insulator-to-metal and structural phase transitions, multiferroism, the role of defects, surface and interface effects, charge ordering, polaron formation, and spin-orbit coupling. 2. Hybrid density functional theory First principles schemes are usually classified into two main approaches, the HF method and the DFT. In the HF theory the many-body problem is solved variationally with respect to a wave function expressed in terms of a Slater determinant. As such, the HF method treats the nonlocal exchange energy exactly but neglects completely correlation effects. In practical calculations this usually leads to a systematic overestimation of band gaps in insulators and semiconductors. DFT is a formally exact theory but its application requires the construction of suitable approximations for the exchangecorrelation (XC) kernel in the spirit of the Kohn-Sham (KS) scheme. The most common and widely used XC DFA are the local density and (semilocal) generalized gradient approximations (LDA and GGA, respectively).[26, 27] The accuracy of the results depends on the quality and complexity of the XC functionals, but in general the performance of standard local and semilocal functionals is poor for the electronic properties of strongly correlated materials, for which the band gap is significantly underestimated (or in some cases not captured at all). A possible way to improve the accuracy and predictivity of HF and DFT is the combination of these two theories by mixing a portion of the DFA exchange and correlation term with a fraction of the nonlocal exact HF exchange within a generalized


4

Kohn-Sham (GKS) theory[28]. The general form of the resulting hybrid functional reads: Hybrid HF DF A EXC = α1 E X + α2 E X + α3 ECDF A ,

(1)

HF where EX is the nonlocal exchange energy of the Slater determinant constructed with the Kohn-Sham orbitals:

HF EX (r, r′ )

1X =− 2 i.j

Z Z

φ∗i (r)φj (r)φ∗j (r′ )φi (r′ ) , d rd r |r − r′ | 3

3 ′

(2)

DF A whereas EX and ECDF A are DFA exchange and correlation functionals, respectively. The parameters α1 , α2 , and α3 ranges from 0 to 1, and control the portion HF DF A of EX , EX and ECDF A incorporated in the hybrid functional. The physical justification of the GKS formalism relies on the adiabatic connection formula[29, 24],

EXC =

Z

1

Exc,λ dλ, 0

Vee,λ =

X ij

λ |r − r′ |

(3)

where EXC is the exchange-correlation Kohn-Sham functional and λ is an intraelectronic coupling-strength parameter that tunes the electron-electron Coulomb potential Vee . This formula connects continuously the noninteracting Kohn-Sham system (λ = 0) with the fully-interacting real system (λ = 1) through intermediate partially interacting (0 < λ < 1) systems (see Figure 1). The exchange-correlation Kohn-Sham energy EXC corresponds to the shaded area in Figure 1 and is equivalent to the full (λ = 1) many-body exchange-correlation energy minus the kinetic contribution to the the XC term (TXC , white area in Figure 1). It can be decomposed over the exchange (EX ) and correlation (EC ) terms, as shown in Figure 1). Figure 1 suggests that the EXC can be obtained by a suitable mixing between EXC,λ=0 and EXC,λ=1 . The most simple approximation for EXC is given by the linear half-and-half expression: 1 1 EXC ≈ EXC,λ=0 + EXC,λ=1 , 2 2 that forms the basis for the hybrid functional proposed by Becke[24]:

(4)

1 HF 1 LDA Hybrid EXC = EX + EXC , (5) 2 2 where the λ = 1 limit is estimated by the LDA expression. A direct successor of the Becke hybrid formula is the 3-parameters B3LYP mixing scheme [30, 24, 31]:

B3LYP LDA HF LDA GGA LDA EXC = EXC + α1 (EX − EX ) + α2 (EX − EX ) + α3 (ECGGA − ECLDA )

(6)


5

0

Exc,λ

EX

EX =EXC, λ=0 EXC, λ=1 0

EC

TXC 0.2

0.4

λ

0.6

0.8

1

Figure 1. (Color online) Graphical interpretation of the adiabatic connection formula. Konh-Sham exchange-correlation energy EXC,λ (shadow area) as a function of the coupling constant λ. The extreme values at λ=0 and λ=1 (indicated by the circles) correspond to the non-interacting (exchange only, EX ) and truly interacting system (exchange plus full correlation, EXC,λ=1 ).

where the three mixing parameters α1 = 0.2, α2 = 0.72, and α3 = 0.81 are determined by fitting experimental atomization energies, electron and proton affinities and ionization potentials of the molecules in Poples G1 dataset. Despite its success the B3LYP is problematic for metals, since it does not attain the homogeneous electron gas (HEG) limit[32], and suffers for transferability issues when applied to surfaces and solids. These aspects are accounted for in the PBE0 functional proposed by Perdew, Burke, and Ernzerhof[33]‡, which reproduces the HEG limit and significantly outperforms B3LYP, especially in the case of systems with itinerant character (metals and small gap semiconductors)[32]. The PBE0 is also inspired by the adiabatic connection formula 3. By considering that Exc,λ is monotonic one can take advantage of the mean value theorem and approximate Exc with one parameter only: EXC ≈ αEXC,λ=0 + (1 − α)EXC,λ=1 ,

α ∈ [0, 1].

(7)

Based on this consideration, the PBE0 hybrid functional is constructed by using the GGA-type functional of Perdew, Burke, and Ernzerhof (PBE) and the optimum value of α=0.25, as rationalized by Perdew and Ernzerhof[35]. The final PBE0 expression is given by: 1 HF 3 P BE PBE0 + EX + ECP BE , (8) EXC = EX 4 4 Although the PBE0 method is applicable successfully to solids[36, 37], it is computationally costly to treat accurately the slow-decaying long-range (lr) part of ‡ This hybrid is also known under the acronyms of PBEh[34]-PBE1PBE[33].


6

the exchange interaction, especially for metals, where a dense k-points sampling is needed. To avoid this complication, Heyd et al.[38] proposed to replace the lr-exchange by the corresponding density functional counterpart. In the resulting Heyd, Scuseria and Ernzerhof (HSE) hybrid functional only the sort-range (sr) exchange is treated at HF level: HF,sr P BE,sr P BE,lr HSE EXC = αEX (µ)+(1−α)EX (µ)+αEX (µ)+ECP BE , α ∈ [0, 1], µ ∈ [0, ∞] (9)

The decomposition of the Coulomb kernel into a lr and sr part can be obtained by splitting the Coulomb operator by way of the error function erf: § 1 erfc(µr) erf(µr) + = r | {z r } | {z r } sr

(10)

lr

where r = |r − r |, erfc is the complementary error function, i.e. erfc(µr) = 1 − erf(µr), and µ is the screening parameter that controls the range separation. The inverse of the screening parameter, µ−1 , represents the critical distance (in ˚ A) at which the short-range Coulomb interactions can be assumed to be negligible. This is shown graphically in Figure 2. The HSE functional reduces to PBE0 for µ = 0, and to PBE for α=0 or µ → ∞. Moussa et al. have shown that the PBE limit is indeed achieved for much smaller value of µ of about 1[39]. ′

5 0.5 0.4

4

0.3

1/r

0.2

3

rs=2/µ

0.1

erfc(µr)/r

0

2

0

5

10

r

15

20

1

erf(µr)/r 0

0

1

2

r

3

4

Figure 2. (Color online) Plots of 1/r, erf(µr)/r, and erfc(µr)/r for the standard value of µ = 0.2. The inset show the three curves around the critical screening distance rs =2/µ=10 ˚ A, at which the short-range Coulomb interactions become negligible.

As the intent of HSE was to achieve accuracy equivalent to PBE0 at a reduced computational effort, the Fock exchange fraction was initially limited to its PBE0 value § The (Gauss) error function is an entire function defined by erf(x)

=

terms of the error function the the short-range Fock exchange is given by P R R 3 3 ′ erfc(µ)|r−r′ | × φ∗i (r)φj (r)φ∗j (r′ )φi (r′ ) − 21 i.j d rd r |r−r′ |

Rx

2 e−t dt. 0 HF,sr EX (r, r′ )

√2 π

In =


7

of α=0.25, but it has been shown that the optimum value α that lead to the best agreement with experimental data of bandgaps is material-specific and often deviates significantly from the 0.25 choice, for both PBE0 and HSE[40, 41, 39, 25]. The allowed range of variation of µ is 0 ≤ µ ≤ ∞, but in practical calculations its value is usually chosen in a relatively narrow (and physically relevant) range[39]. Based on molecular tests the value of µ was set to 0.2 ˚ A−1 (corresponding to a screening length rs = 10 ˚ A), which is routinely considered as the standard choice for HSE calculations[38]. Beside the computational convenience there is a formal justification for the incorporation of a certain amount of screening in the hybrid functional formalism, based on the consideration that in multielectron systems the unscreened HF exchange is effectively reduced by the presence of the other electrons in the system. This can be formally understood by considering the relation between hybrid functionals and the quasi particle GW approximation[13], as outlined for instance in Refs. [42, 39, 43, 44]. The main quantity in the GW approximation is the frequency dependent self energy Σ(ω) which is written in terms of the frequency dependent Green function G(ω) and the frequency-dependent screened Coulomb interaction W (ω): Σ(ω) = iG(ω)W (ω)

(11)

The screened Coulomb interaction is given by: W (ω) = ǫ(ω)−1 v

(12)

where v is the bare Coulomb interaction 1/|r − r′ | and ǫ is the frequency dependent dielectric function which depends on the polarizability χ(ω) in the following way: ǫ(ω)−1 = 1 + vχ(ω)

(13)

Within the so-called COHSEX approximation (the static limit of GW) the self energy Σ can be broken into statically screened-exchange (SX) term and ΣCH containing the static Coulomb hole (CH) and dynamical contributions[45]: Σ = ΣSX + ΣCH

(14)

The SX operator is similar to the Fock exchange operator (Eq. 2), except for the bare Coulomb interaction is replaced by the screened Coulomb interaction W . By replacing the screening in W by an effective static dielectric constant ǫ∞ = 1/α, we obtain the effective W of HSE: W HSE = α

erf c(ω|r − r′ |) |r − r′ |

(15)

Finally, by treating the CH at hybrid functional level the quasiparticle COHSEX equation has the same form as the hybrid functionals equation, in a similar fashion as the SX LDA method, where Σ is replaced by a Thomas-Fermi-like screening part of the


8

LDA XC functional[46]. This link between the GW and hybrid functionals theory gives the following relation for the optimal mixing coefficient[41, 47]: αopt ≈

1 ǫ∞

(16)

The ǫ∞ dependent α has proven to yield good band gaps[41, 47] but become unpractical when ǫ∞ is unknown, due to the complexity of the procedure to compute the dielectric constant, and is more justified for unscreened PBE0-like hybrid functionals than for screened hybrid functionals (like HSE). In screened hybrids, in fact, screening is already present to some extent in the range separation (see Eq.15). In Ref.[25] the effect of the HSE screening on α is quantified in a downward shift of about 0.07. Eq. 16 suggests that in the strongly screened metal limit αopt approaches zero, thereby indicating that for metal standard DFA represents a better methodological choice[48]. Indeed, it has been shown that hybrid functionals lead to a larger overestimation of the exchange splitting (originating from an overstabilization of the spin-polarized solution with respect to the non-magnetic out) and bandwidth in metals as compared to insulators[36]. Although it is tempting to try to render hybrid functionals parameter-free methods, as a matter of fact they rely on adjustable parameters (mixing factor(s) and screening length) and an universal recipe/parametrization valid for any materials does not exist. Several protocols have been proposed to obtain system-dependent parameters, either based on the similarity between hybrid functionals and quasiparticle theory[49, 41, 47, 25, 50, 51] (these schemes relies on the α ≈ 1/ǫ∞ assumption, Eq.16) or by means of a systematic analysis of the HSE parameters space[39] (for the SC/40 semiconductors set the best accuracy is achieved for α=0.313 and µ = 0.185˚ A−1 [39]). As a general rule, the bandgaps increase almost linearly with increasing α and for decreasing µ. As an illustrative example we show in Figure 3 the dependence of the bandgap of LaMnO3 at HSE level as a function of α and µ. The study of the effect of the hybrid functionals parameters on the properties of perovskites is very limited.[25, 52] Most of the results reviewed in the next section were obtained with the standard values of α and µ. We conclude this section with a final note on the fundamental limitations of hybrid functionals. Based on the construction scheme described above, hybrid functionals incorporate a portion of nonlocal exchange but do not treat electronic correlation at higher level of accuracy as compared to DFA. However, the role of nonlocal is sufficient to to adjust the DFA delocalization problem and to provide an overall improved description of strongly correlated oxides[66]. As a downside however, hybrid functionals have the tendency to overstabilize the magnetic solution over the non-magnetic one and, concomitantly, to overestimate the exchange splitting and bandwidth.[36] This can lead to the wrong description of non-magnetic system near a magnetic transition, like VO2 [53], and itinerant magnetic structures.[36]


9

2

0

0.3

µ

0.6

0.9

2.8 2.6

PBE [µ=0.2] 0

3

1 0

1 0

PBE

2

0.1

α

0.2

0.3

2.4

[α=0.25] 0

0.05

Gap (eV)

Gap (eV)

3

3.2

3 Gap (eV)

PBE0

0.1

µ

0.15

0.2

Figure 3. Effect of the HSE parameters α and µ on the band gap of LaMnO3 . (left) α dependence of the band gap with µ fixed to its standard value, 0.208. The PBE results for α=0 is also indicated. (right) µ dependence of the band gap with α fixed to its standard value, 0.25. The PBE0 (µ=0) and PBE (µ → ∞, inset) results are indicated by arrows.

3. 3d perovskites: Most of the applications of hybrid functionals to TM perovskites focus on the 3d subclass, in particular titanates[54, 55, 56, 57, 58, 40, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75], manganites[76, 77, 78, 79, 80, 81, 25, 52, 82, 83], nickelates[84, 25, 85, 86], ferrites[78, 87, 25, 88, 89, 90], aluminates[91, 83, 92, 93, 94], cobaltates[95, 25, 96], and cuprates[97, 25]. In this section we first review the general trends for the LaM O3 series discussed in Ref. [25], then we will focus on more specific cases including the pressure-induced insulator-metal transition (IMT) in LaMnO3 [52] (Section 3.2), ferroelectric Sr(Ti/Ba)O3 [58, 40, 60, 75, 68, 63, 54, 69] and (multi)ferroelectricity[57, 78, 87, 88, 80, 81, 74, 98, 99] (Section 3.3), defects[92, 62, 94, 73, 65, 74] (Section 3.4), and surface/interfaces [100, 101, 102, 103, 104, 105, 106, 107, 108, 109] (Section 3.5) 3.1. Structural electronic and magnetic properties of LaM O3 (M =Sc-Cu) Ref. [25] offers a systematic assessment of the HSE functionals for the 3d perovskite family LaM O3 , with M varying along the 3d row from Sc to Cu. The rich variety of electronic, magnetic and structural ground states encountered in this set of compounds represents a great theoretical challenge, and in fact this materials set is regularly use as a benchmark case for testing the predictivity and accuracy of computational schemes aiming to cure the lacking description of standard DFT: HF[110, 111], LDA+U[112], GW[20], pseudo self-interaction correction (pSIC)[18], and DMFT[21, 113]. The study of Ref. [25] derives a set of mixing factors which minimize the mean absolute relative errors for the structural (volume, lattice constants, and internal distortions) electronic (band gap) and magnetic (moment and ordering) properties on the basis of a detailed comparison of the α dependent HSE results with the available experimental data. It


10

was argued that the set of optimum α for LaM O3 (shown in the top panel of Figure 4(a)) correlates well with the inverse dielectric constant relation (Eq. 16) and with the progressive filling of the 3d manifold: α is large for the first member of the series, LaScO3 , a large band gap d0 insulator with ǫ∞ ≈ 3, it remains more or less constant from LaTiO3 to LaFeO3 , a group of compounds with a similar ǫ∞ of about 5, and finally decreases sharply towards the more strongly screened end-members (M =Co, Ni, and Cu) characterized by a completely filled t2g band.

b) 0.8 0.7 0.6 0.5 1 0.95 0.9

Ni RM

Cu

t

Ni

Cu

Co Intensity (arb. units)

0.2 0.1 0 Sc Ti V Cr Mn Fe Co 68 64 60 56 180 170 160 150 0.4 0.2 Q2 0 0 -0.1 Q 3 -0.2 6 CT/MH 4 BI MH CT/MH MH 2 CT MH 0 4

αopt

ε∞

m (µB) ∆ (eV)

JT (Å)

θ (deg) V (Å3)

a)

Fe Mn Cr V

2

Ti

0 10

Sc

5

-10

Sc

Ti

V

Cr

Mn

Fe

Co

Ni

Cu

-5

0

5

10

Energy (eV)

Figure 4. (Color online) (a) Trend of the optimum value of the α parameter (top panel: LaScO3 : α = 0.25; LaTiO3 : α = 0.15; LaVO3 : α = 0.10 − 0.15; LaCrO3 : α = 0.15; LaMnO3 : α = 0.15; LaFeO3 : α = 0.15; LaCoO3 : α = 0.05; LaNiO3 : α = 0; LaCuO3 : α = 0. ) along with a collection of selected structural (Volume V, tilting angle θ, and JT distortions Q2 and Q3 ), electronic (bandgap ∆), magnetic (magnetic moment m), and dielectric constant (ǫ∞ ) quantities along the LaRO3 series √ from M =Sc to M =Cu. We also show the trend of the tolerance factor t=(RA +R0 )/ 2(RM +RO ), where RA , RM and RO indicate the ionic radius for La, M =Sc-Cu and O, respectively, as well as RM . For LaTiO3 we used α=0.1. The character of the insulating gap is also indicated (BI = band insulator, CT = charge transfer, MH = Mott-Hubbard, CT/MH = mixed CT and MH character). (b) Comparison between experimental (blue squares) and calculated (red full lines) valence and conduction band spectra at the ’optimum’ value of the α parameter. The calculated and measured spectra have been aligned by overlapping the valence band maxima and conduction band minima. The experimental data are taken from the collection of spectra presented in Ref.[20], originally published in separated articles: (i) LaScO3 : Ref.[8]; (ii) LaTiO3 : Ref.[114]; (iii) LaVO3 : Ref.[115]; (iv) LaCrO3 : Ref.[116]; (v) LaMnO3 : Ref.[117]; (vi) LaFeO3 : Ref.[118]; (vii) LaCoO3 : Ref.[119]; (viii) LaNiO3 : Ref.[120]; (ix) LaCuO3 : Ref.[121]. Adapted from Ref.[25].

With this optimized set of α the overall description of the physical properties of LaM O3 is remarkably good, as demonstrated by the comparison between the HSE and experimental results given in Figure 4. The computed structural data match the measured values, which are however already well treated at PBE level with the


11

only exception of the Jahn-Teller (JT) instabilities which are loosely captured by standard DFA[25]. As for the electronic properties, HSE delivers (i) a excellent quantitative account of band gaps and dielectric constants; (ii) a correct qualitative description of the electronic ground state (M =Sc: band insulator, M =Ti, V, and Mn: Mott-Hubbard insulator, M =Cr and Fe: intermediate charge-transfer/Mott-Hubbard insulator, M =Co; charge-transfer and M =Cu and Ni: metal; see Figures 5(a) and 5); (iii) and a very satisfactory comparison between the HSE density of states and the experimental valence and conduction band spectra (see Figure 4(b)). As a downside, from the results of Ref. [25] it is clear that HSE is not capable to describe the weakly correlated metals LaNiO3 and LaCuO3 , which are ’better’ treated at PBE level (α=0). This is mostly due to the neglect of dynamical correlation effects in the hybrid functional framework, which ultimately weakens the performance of hybrid functionals for correlated metals, including materials with a potentially strong technological impact such as high-temperature superconductors and nickelates[97, 84, 85]. LaScO3

LaTiO3 T Z

Γ

S

LaCrO3

LaVO3 R

T Z

Γ

LaFeO3

LaMnO3

T Z

S R

Γ

LaCoO3

LaNiO3

LaCuO3

Γ Z H K Γ ML

S R

8

Energy (eV)

6 6

4

t2g

2

d

3

0

t2g

0 1

t2g

-2

Γ

S R

3

1

1

2

6

2

t2g eg

t2g eg

2

t2g

-4 T Z

3

t2g eg

6

t2g eg

Z Γ Y S X UR

T Z Γ

S R

Γ Z H K Γ ML

Z AM Γ Z R X

Figure 5. (Color online) Summary of the HSE electronic dispersion relations showing the complete trend from the band insulator LaScO3 to metallic LaCuO3 . The thick (red) lines demarcate the d bands responsible for the observed orbital-ordering in LaTiO3 (t2g ), LaVO3 (t2g ) and LaMnO3 (eg ). Taken from Ref.[25].

3.2. Insulator to metal transition: LaMnO3 under pressure Insulator-metal transitions are among the most outstanding effects in condensed matter physics and have been the subject of numerous studies[122]. Particularly challenging is the control and the understanding of the fundamental aspects that govern the phase transition, which may involve temperature, pressure, electron-electron correlation, chemical doping, spin-orbit coupling, confinement effects, etc. The most successful theories which are capable to achieve a theoretical understanding of IMT are the Hubbard model[123] and the related DMFT in which the full many-body problem


12

is mapped onto a quantum impurity model[124]. These methods relay on a model description based on adjustable parameters, which do not enable a quantitative microscopic comprehension of the IMT. In some cases DFA+U has proven to be a reliable method to trace and understand the transition[125, 126], but the uncertainty on the choice of U across the transition (U is expected to decrease going from the insulating to the metal phase) is a critical issue which can quantitatively affects the outcome of the calculations. As discussed in the methodology section PBE0 and HSE functionals, unlike B3LYP, attain the HEG limit and are in principle applicable to the metallic system (within the limits mentioned in the previous section about the overestimation of the exchange splitting and bandwidth, as well as the difficulties to apply HSE to correlated metals). Clearly, due to the dense k-point integration required to describe the metallic state and the associated huge computational cost, screened hybrids like HSE are highly preferable over PBE0 for practical applications, especially for perovskites. a)

HSE

B0

240

3

108 GPa

0.4

d)

1.2

200

0.2

0.8

180

0.1

FM

a b

5.6

0

1/2

c/2 Expt.

5.4

0

AFM 0

10

20

30

40

Pressure (GPa)

50

-0.1 0

10

20

30

40

Pressure (GPa)

50

5.2

f)

5

Pnma

4.8

cubic

2

c)

0.4

Q3

0.3

Q2 Expt.

0.2 0.1 0

0

eg t2g

1 0 -1 -2

-0.1 -0.2

Energy (eV)

JT distortions (Å)

Lattice Parameters (Å)

0.4

b)

e)

0.3

Gap (eV)

220

1.6

= 104 GPa

Expt B0 =

∆E (eV)

Volume (Å )

260

10

20

30

40

50

60

Pressure (GPa)

70

80

90

100

1 0.5 0 0.5 1 1 0.5 0 0.5 1 1 0.5 0 0.5 1 1 0.5 0 0.5 1 1 0.5 0 0.5 1 PDOS (P=0) PDOS (P=13.5) PDOS (P=43.2) PDOS (P=49.2) PDOS (P=55.7)

Figure 6. (Color online) Evolution of the structural and electronic properties of LaMnO3 as a function of pressure as predicted by HSE (triangles) and compared with the experimental data from Refs. [128, 129, 130]. Adapted from Ref.[52]. (a) Equation of state (the inset indicates the value of the bulk modulus);p(b) Structural parameters:pLaMnO3 undergoes an orthorhombic P nma (a 6= b 6= c/2) to cubic (a = b = c/2) transition. (c) Progressive quenching of the cooperative JT local √ √ modes Q2 = 2(l − s)/ 2 and Q3 = 2(2m − l − s)/ 6 with increasing pressure, where l, s and m indicate the long, short and medium Mn-O bondlength; the JT modes are almost completely quenched at the onset of metallicity. (d) Band gap as a function of pressure. (e) Energy difference ∆E between the AFM and FM spin arrangements: AFM/FM crossover. (f) Changes in the eg and t2g density of states around the Fermi level with pressure. The dashed (red) lines refer to the Oxygen p states.

There are very few examples of IMT studied by means of HSE. These include the pressure-induced IMT in LaMnO3 [52], the polaron-driven IMT in Ba1−x Kx BiO3 , and the relativistic Mott-Hubbard state in BaIrO3 [127]. In addition, Bruno and coworkers


13

have rationalized the strain-induced modulation of the metal-to-insulator transition temperature in rare-earth nickelates by inspecting the changes of the structural distortions and the bandwidth upon strain[85]. Here we summarize the main results for the IMT in LaMnO3 and in the following section we will also spend a few words on the IMT in Ba1−x Kx BiO3 (Sec. 5) and BaIrO3 (Sec. 4). Figure 6 depicts the collective structural, electronic and magnetic changes induced by pressure in the prototypical Mott-Hubbard Jahn-Teller insulator LaMnO3 discussed in Ref. [52]. At ambient conditions (pressure P=0, and temperature T=0) LaMnO3 crystalizes with the P nma orthorhombic structure characterized by the so-called stagger JT and GdFeO3 -type (GFO) distortions substantiated by the JT structural modes Q2 and Q3 , and exhibits an insulating antiferromagnetic (AFM) state. The progression of the structural properties upon pressure is shown in Figure 6(a-c). With increasing compression the system undergoes a structural transition towards a cubic phase, associated with a continuous quenching of the JT modes, the rectification of the GFO tilting distortions and the alignment of the lattice parameters to the same value. The structural modifications are concurrent to electronic and magnetic changes. At about 50 GPa, LaMnO3 is subjected to a IMT which HSE classifies as non-Mott because the closing of the gap is concomitant to a progressive reduction of the JT/GFO modes (a purely Mott transition would involve electronic interactions only), in agreement with experimental observations[131], LDA+U[125] and DMFT interpretations[132]. Finally, HSE predicts a strong competition between the distorted AFM and undistorted ferromagnetic (FM) configuration, in agreement with the X-ray absorption spectroscopy data of Ramos et al.[133], and gives indications for a magnetic moment collapse which drives the transition from an high-spin to low-spin transport half-metallic FM state[52]. This result opens up the possibility of realizing colossal magnetoresistance [134] behaviors in manganites without A-site hole-doping, in a genuine stoichiometric sample. 3.3. Ferroelectrics and multiferroics Multiferroics constitute an important class of compounds in which different ferroic orders such as ferromagnetism, ferroelectricity and/or ferroelasticity may coexist in a single compound[135]. To obtain a reliable description of multiferroic oxides it is necessary to go beyond standard DFT in order to treat structural, electronic, and magnetic properties on an equally accurate footing. This has been commonly done by applying the DFA+U method, semilocal functionals such as PBEsol[136] and Wu-Cohen (WC) [137]. and more recently hybrid functionals[58, 57, 78, 87, 88, 80, 81, 74, 98, 99, 72] The application of hybrid functionals to the classical model ferroelectric oxides SrTiO3 (STO) and BaTiO3 (BTO) has been discussed in details in the early works of Bilc et al.[57] and Wahl et al[58], as well as in the more recent study of the Scuseria group[59]. BTO and STO assume a paraelectric cubic phase at high-temperature and at lower


14

Table 1. Collection of computed and experimental band gaps of cubic SrTiO3 and BaTiO3 . The corresponding tetragonal values differ by about 0.1 eV[58].

SrTiO3 BaTiO3 a

LDA 1.81a , 2.04b 1.73a , 2.1c

PBE 1.80a , 1.99b 1.70a , 2.1c

PBEsol 1.82a 1.77a

Ref. [58], b Ref. [138], c Ref. [57], d Ref. [139],

e

HSE 3.07a 2.92a

B3LYP B3PW 3.63b 3.57b 3.7c

WC 2.1c

B1-WC 3.57c 3.39c

[140]

temperature exhibit a tetragonal structure with ferroelectric and antiferrodistortive distortions. In these studies it was reported that the B3LYP and HSE functionals cure the band gap problem, yielding value in excellent agreement with experiment (see Table 1), but overestimate the volume and the ferroelectric instability in the tetragonal phase, which gives rise to an incorrect c/a ratio[58]. This can be adjusted by adopting the so called B1-WC[57] hybrid functional, which mixes exact exchange with the WC GGA parameterization. B1-WC predicts exceptionally good ferroelectric displacements and polarizations. B1-WC and HSE work exceptionally well for the prediction of phonon frequencies, in particular the triply degenerate phonon mode Γ25 , which is not correctly accounted for at LDA and PBE level[58]. The adequate description of phonon frequencies and their coupling with the magnetic degree of freedom is crucial for the design of multiferroic behaviors driven by spin-phonon mechanisms as discussed by Hong et al.[88]. Also in this case, hybrid functionals (specifically HSE) turned out to yield accurate results and was used to construct a practical scheme to fit the U in order to perform less expensive DFA+U simulations on a large set of compounds, retaining an accuracy comparable to HSE for the calculation of spin-phonon couplings. A further valid illustration of the successful applicability of HSE to multiferroic materials is supplied by the work of Stroppa and Picozzi, in particular the detailed study of the structural, electronic, magnetic and ferroelectric properties of the two prototypical proper and improper multiferroic systems BiFeO3 and orthorhombic HoMnO3 , respectively[78]. These same authors have also contributed to disclose and explain, by means of HSE, the onset of a very large electric polarization in the magnetic ferroelectric PbNiO3 , an oxide that may become a prototype of a new class of Ni-based multiferroics[143, 86]. PbNiO3 , together with CdPbO3 , is a novel lithium niobate-type oxide (LNO) recently synthesized and characterized by Inaguma and coworkers[141, 142], whose ground state properties have been extensively investigated by PBE+U and HSE[143, 98, 86]. A compact representation of the structural, electronic and ferroelectric properties of PbNiO3 is given in Figure 7. In this compound the polarization is driven by the large Pb-O polar displacement along the [111] which is typical of the acentric LNO structure (like BiFeO3 ). The origin of the electric polarization in PbNiO3 is manifested by the

Expt. 3.25d 3.2e


15

comparison between the paraelectric and ferroelectric density of states (see Figure 7(b)) showing the 2 eV downshift and broadening of Pb 6s-O 2p spectral weight occurring along with the centrosymmetric-to-ferroelectric transformation. For PbNiO3 GGA not only gives inaccurate ferroelectric distortions (with errors exceeding 2 %) but also yields a wrong metallic ground state which prevents any further possibility to explore ferroelectric features[143]. In contrast both HSE and PBE+U lead to the correct picture, ultimately delivering an almost identical electric polarization of about 100 C/cm2 (see Figure 7(c)).

a)

b)

1

Total

0.5 0

Pbs

2

c)

Polarization (µC/cm ) Energy (eV/f.u.)

DOS (States/eV-atom)

0.2 0

Pbp

0.2 0

Op

0.3 0

0.6

Nid

1 0.3

0 100

GGA+U HSE

0

PTOT (GGA+U) PTOT (HSE)

2

Nid

50

0 PE

0 -10 0.2

0.4

λ

0.6

0.8

FE

-5

0

5

10

Energy (eV)

Figure 7. Ferroelectric properties of PbNiO3 , taken from Ref.[143] (Color online) (a) Structural model showing the ferroelectric displacements. (b) HSE calculated density of states in the acentric ferroelectric (full lines) and centric paraelectric (shadow) phases. (c) Total energy E and total polarization Ptot as a function of the polar distortion λ.

3.4. Oxide electronics: SrTiO3 & LaAlO3 Oxide electronics is a rapidly developing field, and STO and LaAlO3 (LAO) play a key role ever since the observation of a two-dimensional electron gas (2DEG) at the LAO/STO interface[144] and at the bare (001)[145, 146] and (110)[147] surfaces. STO and LAO have a wide band gap of 3.25 eV[139] and 5.6 eV[148], respectively. Their electronic and transport properties can be tuned by means of defects or strain effects in order to increase the performance of these oxides in oxide-based devices. Again, to successfully treat band gaps and structural distortions, which are crucial for the understanding of point defects[149] or in the IMT in oxides superlattices[150], it is necessary to adopt a theory with a high predictive quantitative power. Although DFA+U describes correctly the bang gap, structural instabilities such as rotation angles do not improve over LDA and GGA[93, 83]. The HSE functional has revealed to


16

be an excellent choice for this task. For instance, El-Mellouhi et al.[93] have shown that beside yielding good electronic properties, HSE identifies and interprets structural phase transitions in STO, LAO and LaTiO3 in excellent agreement with experiment and substantially more accurately than DFA+U. HSE was also used by Janotti et al. to study the effects of strain on the electronic structure of STO, an important issue in the context of epitaxial heterostructures that combines STO with other oxides[60]. It was found that tensile biaxial strain in the (001) and (110) planes induces a lowering of the effective mass of electrons, an effect that could be exploited for increasing the electron mobility in epitaxial films. The impact of defects in oxide semiconductors has been addressed in a few computational studies based on HSE[73, 94, 65, 74, 62]. Particularly interesting is the work of Choi et al.[73], dealing with the formation and role of different type of native point defects (O, La, and Al vacancies) in LAO. The HSE results indicate that cations vacancies are deep acceptors, whereas oxygen vacancies are deep donors and introduce two transition levels in the upper part of the gap (below the conduction band minima). By analyzing the positions of these defects levels with respect to the band edges of semiconductors used in metal-oxides-semiconductor devices, it was concluded that oxygen vacancies favor carrier traps, whereas cation vacancies are likely sources of negative fixed charges[73]. 3.5. Surfaces and interfaces We conclude this section with a brief overview of the few applications of hybrid functionals to perovskite surfaces[100, 101, 102, 103, 104, 105, 106, 108, 109]. The first hybrid functional study on surface properties of perovskites is probably the work of Heifets et al. focused on the calculation of the SrTiO3 (100) relaxation and rumpling using a variety of different schemes including B3LYP and B3PW (the latter is the Becke’s three parameter method combined with the non-local correlation functionals by Perdew and Wang[100]). The detailed structural and electronic properties of both the (001) and (110) surface terminations of STO were subsequentially investigated by Eglitis and Vanderbilt[103] using the B3PW functional. The results for the (001) surface are in overall good agreement with the LEED and RHEED data. The predictions for the (110) surface are more problematic to assess due to the intrinsic polarity of this surface plane. Different neutral slabs were proposed in Ref. [103] but none of them resembled the real reconstruction which was subsequently identified experimentally by Enterkin and coworkers[151]. Possible reconstructions of the (111) surface of SrTiO3 were proposed and investigated by an array of different methodologies (including GGA, meta-GGA and hybrids) in Ref. [104], which are to date not yet comforted by experimental results. A very representative application of HSE to surface science is the understanding of oxygen reduction reaction activity (OOR) on LaBO3 (100) (B=Mn,Fe,Cr) proposed by Wang et al., which is an important step forward for the design of fuel cell applications[105]. The comparison between the relative free energy diagrams based


17

on the computational hydrogen electrode model suggested by Nørskov[152] and the Poissons-Boltzmann implicit continuum model for solvation corrections[153] suggests that the HSE derived order of ORR activity is LaMnO3 > LaCrO3 > LaFeO3 , which is in better agreement with the most recent experimental results[154] than the corresponding GGA or GGA+U findings[105]. We will discuss one more example of the application of HSE to surface science in section 5 in the context of the physics of BaBiO3 . Hybrid functional results for perovskites-based interfaces are very scarce in literature, probably due to the associated large computational cost (especially for the structural relaxation)[107, 155, 106, 108, 109]. In Ref. [109] the equilibrium geometries and electronic ground states of LAO/STO interfaces were studied by HSE and compared with the GGA and GGA+U methods. The authors show that HSE is capable to predict the IMT between four and five unit cells of LAO, in agreement with experiment[156], and that the HSE density of states in the critical-thickness regime is qualitatively different to that delivered by GGA and GGA+U. The application of hybrid functionals (B1-WC in conjunction with the pSIC method) has proven to be effective also in the explanation of the nature of the 2DEG at the LAO/STO interface, as due to a spontaneous electron confinement at the Ti-dxy states, which appears to support the experimental attribution of 2DEG formation to a primarily electronic origin[155] . 4. 4d & 5d perovskites In recent years, there has been an upsurge of interest in 4d and 5d transition metal oxides in which exotic states may emerge from the subtle interplay between Hubbard’s U, Hunds J, the bandwidth, the spin orbit coupling (SOC), and the splitting of the crystal field. It is commonly expected that 4d and 5d oxides are more metallic and less magnetic than their 3d counterparts because of the extended nature of the 4d and 5d orbitals. In contrast with these expectations, many ruthenates, technetiates and iridates are found to be magnetic insulators and to display a large array of phenomena, seldom or never seen in other materials, including relativistic Mott-insulators[157, 158], Slater insulator[159, 160], Hunds correlated metals[22], molecular insulators[161], etc. Considering the complexity of the issues at stake hybrid functionals appear again to be particularly adequate, but so far their application to 4d and 5d perovskites has been very limited[160, 61, 162, 163, 164, 165, 127] 4.1. RTcO3 (R=Ca, Sr and Ba) In the last few years RTcO3 compounds (R=Ca, Sr and Ba) have generated a considerable interest due to the observed exceptionally high Néel temperature (TN ) [166, 167]. The origin of this unexpected behavior has been studied theoretically by means of DFT[166, 167], DMFT[168], DFA+U combined with model Hamiltonian[159] and also HSE[160]. These studied have shown that the proper treatment of the attenuated (but still important) electronic correlation and its coupling with the magnetic


18

exchange interactions is capable to explain the onset of the remarkable magnetic ordering temperature. In the DMFT picture the high TN is attributed to the fact that SrTcO3 lays on the verge between the itinerant-metallic to localized-insulating regime[168], in consistency with the beyond-DFT description. The microscopic origin of the enhancement of TN from CaTcO3 to BaTcO3 is explained by HSE (with α = 0.1) with the progressive rectification of the superexchange Tc-O-Tc path associated with the quenching of the residual JT distortion Q, which causes an increase of the t2g − Op hybridization and a steeply increase of TN (See Figure 8). It is worth to remark once more that these type of information are only obtainable by a method that treats at the same footing structural instabilities (Q and volume V), magnetic interactions (J, in this case deduced by mapping total energies of different magnetic orderings into a Heisenberg Hamiltonian) and electronic properties (the intra-atomic Coulomb repulsion included in HSE restores the correct insulating behavior which is not found by DFT)[160]. Ca

Sr

Ba

0.005

3

Vol (Å )

Q

0.01

0 260 240

J1 (meV)

220 36 32 28

TN (K)

1200 1000 800 150

155

160

165

170

175

180

Tc-O-Tc angle (°)

Figure 8. (Color online) Dependency of the relevant magnetic (TN and J1 ) and \ structural [Volume and Q=1/2(Q1 +Q2 )] quantities on the Tc − O − Tc angle (average between Tc −\ O1 − Tc and Tc −\ O2 − Tc) in the RTcO3 series. Taken from Ref. [160].

4.2. SrPdO3 and the band gap problem The predictive capability of HSE was recently helpful in characterizing the structural, electronic, and magnetic properties of a new member of the 4d oxide perovskites family, SrPdO3 , synthesized in 2010 by Galan and coworkers[169, 170]. By conducting a structural search based on lattice dynamics and group theoretical methods HSE has identified the orthorhombic Pnma space group as the most stable crystal structure, characterized by a large tilting of the BO6 octahedra. Moreover, HSE finds that SrPdO3


19

Table 2. Band gap of SrPdO3 calculated by using HSE with different α (α=0 refers to PBE calculations), GW0 and GW0 -TCTC. The value of ǫ∞ computed within GW0 TCTC is also shown. Data taken from Ref.[170].

α HSE(α) GW0 @HSE(α) GW0 -TCTC@HSE(α) ǫ∞

0.00 0.24 0.92 – 16.6

0.05 0.41 1.08 1.03 13.7

0.10 0.61 – – –

0.15 0.81 1.31 1.21 10.9

0.20 1.03 – – –

0.25 1.27 1.43 1.31 9.5

adopts a low spin state with a zero net magnetic moment.[170] Due to the lack of experimental information, the assessment of the HSE accuracy in predicting the band gap of SrPdO3 turns out to be particularly difficult. As already underlined a few times the HSE band gap unavoidably depends on the mixing factor α. This holds true for SrPdO3 as reported in the first row of table 2: by increasing α the band gap increases linearly from 0.24 eV (α=0) to 1.27 eV (α=0.25). Considering that the self-interaction error for 4d oxides is expected to be smaller than in 3d oxides, it would be reasonable to choose for SrPdO3 a mixing factor smaller than the average optimum α estimated for insulating 3d perovskites (≈ 0.15, see Figure 4 and Ref. [25]). This choice would yield a band gap of about 0.6 eV (see Table 2). Unfortunately, the inverse dielectric constant relation (Eq. 16) can not be used to support this conclusion, as ǫ∞ of SrPdO3 is unknown. The only way to overtake these uncertainties and to achieve a more reliable estimation of the band gap (which should serve as a benchmark for the HSE prediction) is to apply parameter free methods such as GW[13] (a possible alternative could also be post-HF quantum-chemistry methods, which are however not yet applicable to large systems like perovskites)[171]. Although GW is a substantial step forward with respect to GKS schemes for optical properties, including bandgap prediction, it is not free of complications, because the results critically depend on the treatment of the dynamical screen W (inclusion of excitonic effects), on the self-consistency in G, and on the starting orbitals used to solve the quasiparticle equations (DFT or HSE, usually)[170, 172]. The dependence on the starting orbitals could be in principle overtook by a fully self-consistent quasiparticle procedure both in W and G, but at a tremendous computational cost which hampers de facto any realistic application to distorted (large cell) and magnetic perovskites (to our knowledge, the only full self-consistent GW with vertex corrections calculation was reported in Ref.[173] for BaBiO3 ). Partially self-consistent results for SrPdO3 , also including excitonic effects at the test-charge/test-charge (TC-TC) level[174]) are reported in Table 2. One immediately notices that the α dependence is attenuated and that the accuracy is improved with respect to the HSE results (the band gap ranges from 1 to 1.3 eV going from α = 0.05 to α = 0.25). By applying the inverse dielectric constant relation by using the GW0 -TCTC dielectric constant and taking into account the 0.07 shift due to the HSE screening (see section 2 and Ref. [25]), an optimum α for


20

SrPdO3 in the range 0.13-0.18 is obtained, surprisingly similar (and not lower) to the best α acquired for the 3d perovskites series. 4.3. BaIrO3 : relativistic Mott-Hubbard insulator Iridium oxides are a primary class of materials which has recently attracted a great attention. Iridates lie at the intersection of strong SOC and electron correlation, in which the electrons entangle the orbital and spin degrees of freedom. One of the most stunning example is the Ruddlesden-Popper compound Srn+1 Irn O3n+1 (n = 1, 2, · · · ∞). The origin of the unusual insulating state in the parent n = 1 compound is long debated but a general consensus exists in attributing the opening of the gap to a mixed Slatertype and relativistic Mott-Hubbard type mechanism[157, 175, 176]. This and similar SOC-induced Mott-Hubbard states have been studied either within model Hamiltonian approaches or DFA+U, but not yet at HSE level. The inclusion of a modest U of about 1 eV is essential to open the gap, thus it is expected that HSE with a reduced mixing parameter should be able to capture the relevant physics. In Fig.9 we show unpublished HSE+SOC results on the onset of a SOC-Mott state in BaIrO3 [127], recently identified by Ju et al.[158]. Notice that BaIrO3 does not poss the perovskite-type structure, but rather a complicated assemblage of Ir3 O12 clusters forming almost one-dimensional chains along the c-axis[177]. Without the inclusion of SOC the system remains metallic for α smaller than 0.2, a much too large mixing parameter for 5d oxides (for α = 0.2 BaIrO3 would be assimilated to genuine MottHubbard insulator, which is not the response provided by the measurements). For a more realistic value of α = 0.1 and the incorporation of SOC BaIrO3 becomes a SOC-Mott insulator, with a band gap of about 100 meV, in good agreement with the experimental value, about 50 meV[178].

Figure 9. (Color online) Band structure and minimal phase diagram of BaIrO3 . The relativistic Mott-Hubbard insulating state emerges from the inclusion of a small fraction of exact-exchange at HSE level and spin-orbit coupling.[127]

5. sp perovskites: the case of multivalent BaBiO3 As a final example we focus now on the charge ordered mixed valence insulating perovskite BaBiO3 (BBO). Despite its apparently simple sp nature, BBO exhibits


21

a plethora of fascinating and absolutely unique behaviors[179, 180, 181, 182]. BBO exhibits a charge density wave insulating state, formed by alternating breathing-in and breathing-out distortions of oxygen O6 octahedra around inequivalent Bi5+ and Bi3+ sites[173]. Upon hole doping through Ba→K substitution Ba1 − xKx BiO3 undergoes an insulating to metal transition for x ≈0.33, eventually turning into a superconductor for higher doping[179]. The IMT was explained on the basis of HSE calculations as a progressive reduction of the Bi-O distortions modulated by the formation of holepolarons, i.e. the coupling between the excess holes induced by the K-doping trapped in Bi3+ sites and the surrounding phonon field[180]. This is shown in Figure 10: (i) At x = 0 a band gap is opened between the occupied Bi3+ s states and the unoccupied Bi5+ s band. The optical spectra is characterized by a main peak in agreement with experiment[183]. (ii) At x = 0.125 a very localized bi-polaronic mid gap states emerges, which is also recognizable in the optical spectra. (iii) Upon further hole doping additional bi-polaronic states are formed which start to interact among each other as reflected by the width of the mid-gap states, consistent with the experimental signal. As a consequence the band gap is progressively reduced and ultimately closes for x > 0.25, in good agreement with experiment[184]. d 3+

DOS (a.u.)

5+

5+

3+

Ba14K2(Bi )8(Bi )(Bi )7O48

c DOS (a.u.)

5+

5+

3+

Ba12K4(Bi )8(Bi )2(Bi )6O48

-2

-1

0

1

Energy (eV)

2

3

B

1 0

A

x=0.125

-1

f

x=0.25

h

-2

e

x=0.125

x=0.0

0

Energy (eV)

b

peak A

ε2 (a.u.)

5+

Ba2Bi Bi O6

g

2

Energy (eV)

DOS (a.u.)

x=0.0

1

Energy (eV)

a

B

A

x=0.25

0 -1 X

Γ

L

K

Γ

0

1

2

3

Energy (eV)

Figure 10. (Color online) Polaron driven insulator-metal transition in K-doped Bax−1 Kx BiO3 (a-c) Evolution of the DOS and (d-f) corresponding bandstructure. The gray shadows indicate the total density of states, and red curves indicate the bipolaronic states. (g) Comparison between the theoretical and measured imaginary part of the dielectric function for x=0, 0.125 and 0.25. Peak (A) corresponds to excitations from Bi3+ into Bi5+ states, peak (B) corresponds to excitations from Bi3+ into bipolaronic states [red curves in panel (b),(c),(e), and (f)]. The experimental curves (◦) are taken from Ref. [183] (x = 0) and Ref. [184] (x = 0.21). (h) Charge density corresponding to the bipolaronic band (red line) of panel (e) localized around the BiO6 octahedron at the converted Bi3+ →Bi5+ atom. Taken from Ref. [180]

The electron-phonon mechanism behind the formation of the superconducting state has been discussed by Yin et al. by combining HSE (along with other theoretical schemes) with a model approach to evaluate the electron-phonon coupling and the critical temperature (TC ) based on the McMillan equation[187]. It was shown that


22

HSE corrects the DFA overscreening (which yield to the underestimation of electronphonon coupling) and lead to a nice agreement with experiment in terms of TC for a Coulomb pseudopotential µ∗ =0.1. The surface properties of BBO are also a source of fascinating phenomena[181, 182]. Using time-of-flight and recoil spectroscopy Gozar and coworkers[185] identified the termination layer of the BBO(100) surface as BiO2 terminated. This results is reproduced well by HSE, which find the BiO2 termination more stable than the BaO one, as shown in Figure 11, in contrast to PBE which favors the BaO surface.

Figure 11. (Color online) Phase stability diagram of the BBO(100) surface in terms of the Ba and O chemical potential at PBE and HSE level. The (yellow) circles indicate the partial pressure conditions extracted from experiment[185].

Very recently Vildosola et al. based on GGA calculations with the modified BeckeJohnson potential (MBJ)[186] and HSE have disclosed the emergence of a 2DEG at the BiO2 terminated BBO(100) surface[181]. The 2DEG originates from strong reduction of the Bi-O breathing distortions near the surface, in intrinsic analogy with the K-doping induced structural distortions associated with the IMT in the bulk phase. Specifically, hole-doping and surface effects both drive the system towards a cubic-like and metallic state but in one case the IMT is modulated by the formation and coupling of holepolarons, whereas in the latter the attenuation of the structural distortion is due to surface effects, namely the Bi-O breathing distortions become progressively smaller going from the bulk to the surface. Like in the bulk, it could be speculated that the saturation of the breathing instabilities at the BBO(110) surface could be accompanied by the onset of a superconducting (2DEG) states. To enrich even further the physics of BBO Yan and coworkers have predicted that spin-orbit interaction induces a band-inversion in the conduction band BBO, a typical fingerprint of topological behaviors. Indeed, inspection of the SOC-induced changes of the surface band structure indicate that a topological insulator state associated with a well defined Dirac cone at the BaO-terminated BBO(100) surface can be obtained upon


23

electron doping (i.e. if the Fermi energy is shifted up into the band inversion energy window)[182]. This result was achieved at DFT level and benchmarked with HSE, especially for what concerns the band inversion feature. This extraordinary discovery leaves us with some open questions. First, considering the strong impact that hole and electron doping have on the structural and electronic properties of multivalent BBO (polaron formation) a more realistic treatment of doping effect beyond the simplistic rigid-band scheme would be needed to validate the effective formation of the topological insulator phase. Secondly, the experimental realization of this topological feature would need the stabilization of the BaO-terminated surface over the more stable metallic (2DEG) BiO2 . This could be possibly achieved by changing the partial pressure conditions (see Figure 11), by strain or by the interaction with adsorbates. Clearly, these mechanisms could in turn affect the topological/polaronic properties of the system or give rise to novel phenomena which will deserve further investigation. 6. Summary and Concluding remarks In summary, we have reviewed selected examples which illustrated the application of hybrid functionals to a wide class of perovskite compounds. The unique ability of treating at the same level of accuracy structural, orbital, and magnetic degree of freedom as well as energetics and activation energies, makes hybrid functionals a very powerful computational tool with a reach space of physical and chemical solutions. In particular, the PBE0 hybrid functional and its screened version (HSE) appears to be particularly suitable for this task, as they are capable to catch the metallic limit (but at the cost of an overestimation of the exchange splitting and bandwidth), and this allows the description of insulator metal transition, and have proven to be applicable with success to atom, molecules, solids and surface without loss of accuracy. This does give PBE0/HSE an edge over the renowned B3LYP which, despite its enormous success in quantum chemistry, has limited application in solids and surfaces. With respect to the widely used DFA+U method, hybrid functionals have the advantage that the self-interaction corrections are included at the same footing for every orbitals (which could be in principle taken into account into a multi-orbital DFA+U framework, conceptually similar to the SIC approach) and they can treat non-local effects through the non-local Hartree-Fock exact exchange (the short-range exchange energy in HSE is obtained through the exact exchange hole[188]). The critical point here is that the predictive power of hybrid functionals crucially depends on two parameters, the mixing factor α and the screening length µ (similarly, in DFA+U the results are affected by the value of U, which is not easily and univocally determinable k ). Moreover, hybrid functionals do not incorporate dynamical correlation k The U parameter can be computed either using a linear response approach[189] or within the constrained random phase approximation (cRPA)[190]. However, it remains disputable and formally not justified the use of this ab inito U, especially at the many-body cRPA level, within an approximated DFA+U scheme.


24

effects and tends to stabilize to much magnetic solutions over non-magnetic ones. For the screening parameter one can in principle bypass the problem by neglecting the screening (i.e by going from HSE to PBE0), but this would introduce all problematics, and the high computational costs, related to the treatment of the longrange HF exchange[38, 188]. As discussed in section 2, the standard value of µ = 0.2˚ A−1 arises from a fitting procedure on molecules data performed at a fixed value of the mixing parameter α = 0.25. However, when applied to LaMnO3 this standard (µ = 0.2˚ A−1 , α = 0.25) combination yields a band gap 0.8 eV smaller than the corresponding unscreened PBE0 estimate, clearly indicating that µ is a material-dependent parameter. Equally problematic is the treatment of α. It could be in principle computed at PBE0 level using the inverse dielectric formula relation but this would require the knowledge of ǫ∞ which is by far not trivial to compute accurately even within GW, as shown for SrPdO3 in Table 2. We have also noted that the optimum PBE0 α cannot be straightfully transferred to HSE, as HSE already incorporates a certain degree of screening through the range separation, controlled by µ. In practice, it appears that the only possible way to acquire a set of optimum parameters for HSE is through a systematic analysis of the HSE (µ, α) parameter space, similar to the fitting procedure that Moussa and coworkers carried out for the SC/40 semiconductors database[39]. On the basis of the available results, it can be concluded that for the perovskite family a mixing factor of about 0.1-0.15 seems to be more appropriate than the typical 0.25 setup. Despite the difficulties on the choice of the parameters and the overestimation of exchange splitting and bandwidth, the findings reviewed in the present paper show that hybrid functionals are among the best computational schemes tailored for accurate qualitative and quantitative description of a wide array of materials properties. This include: (i) Structural properties, especially GdFeO3 -type, Jahn-Teller, and polaron distortions, as well as phonon properties. (ii) Relative stability of different magnetic orderings (i.e. exchange interactions) and magnetic ordering temperatures; (iii) Energetics: stability of surface terminations (in BaBiO3 ) and oxygen reduction reaction activities (in LaM O3 ) in very good agreement with experiment; (iv) Correct description of the electronic groundstate in (relativistic)-Mott-Hubbard, charge-transfer, band, and charge-ordered insulators; (v) Ferroelectric instabilities and accurate estimation of the electric polarization in proper and improper ferroeletrics; (vi) Capability of delineate qualitatively and quantitatively insulator-metal-transitions (pressure induced IMT in LaMnO3 and polaron-induced metalization of BaBiO3 ). (vii) Improved calculations of formation energies and defect levels, and proper description of localization effects. (viii) Band gaps: this is the quantity that mostly dependents on the parameters α and µ. Within a reasonable range of variation of α for the perovskites family (0.1 < α < 0.2) the band gap changes by more then 0.5 eV (0.8 eV in LaMnO3 , and 0.4 eV in SrPdO3 ). This is a problem that is presently not easy to circumvent neither at hybrid functional level (for fundamental reasons), neither at many-body level (too large computing effort). Further methodological and technological developments are still necessary.


25

Acknowledgments I gratefully acknowledge valuable discussions and years long collaborations on the application of hybrid functionals to solids with Jiangang He, Xianfeng Hao, Sun Yan, G. Kresse, R. Podloucky, A. Filippetti, S. Sanvito and Xing-Qiu Chen.


26

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

[30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]

von Helmolt R, Wecker J Holzapfel B Schultz L and Samwer K 1993 Phys. Rev. Lett. 71 2331 Salamon M B and Jaime M 2001 Rev. Mod. Phys. 73 583 Mott N F 1990 Metal-Insulator Transitions (Taylor & Francis, London) Imada M Fujimori A and Tokura Y 1998 Rev. Mod. Phys. 70 1039 Bednorz J G and M¨ uller K A 1986 Z. Phys. B 64 189 Ohtomo A Hwang H Y 2004 Nature 427 423 Wang K F Liu J M and Ren Z F 2009 Adv. Phys. 58 321 Arima T Tokura Y and Torrance J B 1993 Phys. Rev. B 48 17006 Suntivich J Gasteiger H A Yabuuchi N Nakanishi H Goodenough J B and Shao-Horn Y 2011 Nature Chem. 3 546 Solovyev I Hamada N and Terakura K 1996 Phys. Rev. B 53 7158 Anisimov V I Zaanen J and Andersen O K 1991 Phys. Rev. B 44 943 Perdew J P and Zunger A 1981 Phys. Rev. B 23 5048 Hedin L 1965 Phys. Rev. 139 A796 Georges A Kotliar G Krauth W and Rozenberg M J 1996 Rev. Mod. Phys. 68 13 Hashimoto T Ishibashi S and Terakura K 2010 Phys. Rev. B 82 045124 Hsu H Umemoto K Cococcioni M and Wentzcovitch R 2009 Phys. Rev. B 79 125124 Guo G Grinberg I Rappe A M and Rondinelli J M 2011 Phys. Rev. B 84 144101 Filippetti A Pemmaraju C D Sanvito S Delugas P Puggioni D and Fiorentini V 2011 Phys. Rev. B 84 195127 Nohara Y Yamasaki A Kobayashi S and Fujiwara T 2006 Phys. Rev. B 74 064417 Nohara Y Yamamoto S and Fujiwara T 2009 Phys. Rev. B 79 195110 Pavarini E Yamasaki A Nuss J and Andersen O K 2005 New J. Phys. 7 188 Georges A De’ Medici L Marvlje J, 2013 Annual Review of Condensed Matter Physics 4 137 Held K Andersen O K Feldbacher M Yamasaki A and Yang Y F 2008 J. Phys.: Condens. Matter 20 064202 Becke A D 1993 J. Chem. Phys. 98 1372 He J and Franchini C 2012 Phys. Rev. B 86 235117 Kohn W and Sham L J 1965 Phys. Rev. 140 A1133 Perdew J P Ruzsinszky A Tao J Staroverov V N Scuseria G E and Csonka G I 2005 J. Chem. Phys. 123 062201 Seidl A Görling Vogl A P Majewski J A Levy M 1996 Phys. Rev. B 53 3764 Harris J Jones R O 1974 Phys. F: Met. Phys. 4 1170; Gunnarsson O and Lundqvist B I 1976 Phys. Rev. B 13 4274; Langreth D C and Perdew J 1977 ibid. 15 2884; Harris J 1984 Phys. Rev. A 29 1648 Stephen P J Devlin F J Chablowski C F Frisch M J 1994 J. Chem. Phys. 98 11623 Becke A D 1993 J. Chem. Phys. 98 5648 Paier J Marsman M and Kresse G 2007 J. Chem. Phys. 127 024103 Adamo C and Barone V 1999 J. Chem. Phys. 110 6158 Ernzerhof M and Perdew J P 1998 J. Chem. Phys. 109 Perdew J P Ernzerhof M and Burke K 1996 J. Chem. Phys. 105 9982 Paier J Marsman M Hummer K Kresse G Gerber I C and A´ ngya´ n J G J. Chem. Phys. 2006 124 154709 Franchini C Podloucky R Paier J Marsman M Kresse G 2007 Phys. Rev. B 75 195128 Heyd J Scuseria G E and Ernzerhof M 2003 J. Chem. Phys. 118 8207; erratum: 2006 J. Chem. Phys. 124 219906 Moussa Jonathan E Schultz Peter A and Chelikowsky James R 2012 J. Chem. Phys. 136 204117 Onishi T 2008 Int. J. Quant. Chem. 108 2856 Alkauskas A Broqvist P Pasquarello A 2011 Phys. Status solidi (b) 248 775

Hybrid functionals applied to perovskites [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82]

27

Henderson T M Paier J and Scuseria G E 2011 Phys. Status Solidi B 248 767 Del Sole R Reining L and Godby R W 1994 Phys. Rev. B 49 8024 Gygi F and Baldereschi A 1989 Phys. Rev. Lett. 62 2160. Hedin L and Lundqvist S 1970 in Advances in Research and Applications, edited by Seiz, F, Turnball, D, and Ehrenreich, H (Academic Press) Solid State Physics 23 1 Bylander D M and Kleinman L 1990 Phys. Rev. B 41 7868 Marques M A L Vidal J Oliveira M J T Reining L and Botti S 2011 Phys. Rev. B 83 035119 K¨ ummel S Kronik L 2008 Rev. Mod. Phys. 80 3 Shimazaki T and Asai Y 2008 Chem. Phys. Lett. 466 91 Koller D Balha P and Tran F 2013 J. Phys.: Condens. Matter 25 433503 Atalla V Yoon M Caruso F Rinke P Scheffler M 2013 Phys. Rev. B 88 165122 He J Chen M Chen X.-Q. Franchini C 2012 Phys. Rev. B 85 195135 Grau-Crespo R Wang H and Schwingenschlögl U 2012 Phys. Rev. B 86 081101(R). Ricci D Bano G and Pacchioni G Illas F 2003 Phys. Rev. B 68 224105 Wilson N C Russo S R Muscat J Harrison N M 2005 Phys. Rev. B 72 024110 Catti M 2007 Chem. Mater. 19 3963 Bilc D I Orlando R Shaltaf R Rignanese G M Í˜ niguez J and Ghosez P 2008 Phys. Rev. B 77 165107 Wahl R Vogtenhuber D and Kresse G 2008 Phys. Rev. B 78 104116 El-Mellouhi F Brothers E N Lucero M J Scuseria G 2011 Phys. Rev. B 84 115122 Janotti A Steiauf D and Van de Walle C G 2001 Phys. Rev. B 84 201304(R) Akamatsu H Kumagai Y Oba F Fujita K Murakami H Tanaka K and Tanaka I 2011 Phys. Rev. B 83 214421 Choi M Oba F and Tanaka I 2011 Appl. Phys. Lett. 98 172901 Evarestov R A Blokhin E Gryaznov D Kotomin E A and Maier J 2011 Phys. Rev. B 83 134108 Akamatsu H Fujita K Hayashi K Kawamoto T Kumagai Y Zong, Y Iwata K Oba F Tanaka I and Tanaka K 2012 Inorg. Chem. 51 4560. Mitra C Lin C Robertson J and Demkov A A 2012 Phys. Rev. B 86 155105 Iori F Gatti M Rubio A 2012 Phys. Rev. B 85 155105 Gruber C Bedolla-Velazquez P O Redinger J Mohn P and Marsman M 2012 EPL 977 67008 Florez J M Ong S P Onba¸sli M C Dionne G F Vargas P Ceder G and Ross C A 2012 Appl. Phys. Lett. 100 252904 Evarestov R Blokhin E Gryaznov D Kotomin E A Merkle R and Maier J 2012 Phys. Rev. B 85 174303 Evarestov R A and Bandura A V 2012 J. Comput. Chem. 33 1123 Evarestov R A Bandura A V Kuruch D D 2013 J. Comp. Chem. 34 175 Yu-Seong Seo Y S and Ahn J S 2013 Phys. Rev. B 88 014114 Choi M Oba F Kumagai Y and Tanaka I 2013 Adv. Mater. 25 86 Shimada T Ueda T Wang J and Kitamura T 2013 Phys. Rev. B 87 174111 Gryaznov D Blokhin E Sorokine A Kotomin E A Evarestov R A Bussmann-Holder A and Maier J 2013 J. Phys. Chem. C 117 13776 Mu˜ noz D Harrison N M and Illas F 2004 Phys. Rev. B 69 085115 Kotomin E A Evarestov R A Mastrikova Y A and Maier J 2005 Phys. Chem. Chem. Phys. 7 2346 Stroppa A and Picozzi S 2010 Phys. Chem. Chem. Phys. 12 5405 Ahmad E A Liborio L Kramer D Mallia G Kucernak A R and Harrison N M 2011 Phys. Rev. B 84 085137 Okuyama D Ishiwata S Takahashi Y Yamauchi K Picozzi S Sugimoto K Sakai H Takata M Shimano R Taguchi Y Arima T and Tokura Y 2011 Phys. Rev. B 84 054440 Prikockyt˙e A Bilc D Hermet P Dubourdieu C and Ghosez P 2011 Phys. Rev. B 84 214301 Franchini C Kov´ aˇcik R Marsman M Sathyanarayana Murthy S He J Ederer C Kresse G 2012 J. Phys.: Condens. Matter 24 235602


28

[83] Garc´ıa-Fernández P Ghosh S English N J Aramburu J A 2012 Phys. Rev. B 86 144107 [84] Gou G Grinberg I Rappe A M and Rondinelli J M 2011 Phys. Rev. B 84 144101 [85] Bruno F Y Rushchanskii K Z Valencia S Dumont Y Carréttéro C Jacquet E Abrudan R Bl¨ ugel S Leˇzaić M Bibes M and Barthélémy A 2013 Phys. Rev. B 88 195108 [86] Hao X F Stroppa A Barone P Filippetti A Franchini C Picozzi S 2014 New J. Phys. 15 015030 [87] Stroppa A Marsman M Kresse G Picozzi S 2010 New J. Phys. 12 093026 [88] Hong J Stroppa A Í˜ niguez J Picozzi S and Vanderbilt D 2012 Phys. Rev. B 85 054417 [89] Dong H Liu H and Shanying Wang S 2013 J. Phys. D: Appl. Phys. 46 135102 [90] McDonnell K A Wadnerkar N English N J Rahman M Dowling D 2013 Chem. Phys. Lett. 572 78 [91] Onishi T 2010 Int. J. Quant. Chem. 110 2912 [92] Choi M Janotti A Van de Walle C G 2013 Phys. Rev. B 88 214117 [93] El-Mellouhi F Brothers E N Lucero M J Bulik I W and Scuseria G E 2013 Phys. Rev. B 87 035107 [94] El-Mellouhi F Brothers E N Lucero M J and Scuseria, G E 2013 Phys. Rev. B 88 214102 [95] Gryaznov D Evarestov R A and Maier J 2013 Phys. Rev. B 82 224301 [96] Mukhopadhyay S Finnis M W and N. M. Harrison N M 2013 Phys. Rev. B 87 125132 [97] Rivero P Moreira I de P R Illas F 2010 Phys. Rev. B 81 205123 [98] Xu Y Hao X Franchini C and Faming Gao F 2013 Inorg. Chem. 52 1032 [99] Sim H Cheong S W and Kim B G 2013 Phys. Rev. B 88 014101 [100] Heifets E Eglitis R I Kotomin E A Maier J and Borstel G 2001 Phys. Rev. B 64 235417 [101] Piskunov S Kotomin E A Heifets E Maier J Eglitis R I Borstel G 2005 Surf. Sci. 575 75 [102] Evarestov R A Bandura A V and Alexandrov V E 2007 Surf. Sci. 601 1844 [103] Eglitis R I and Vanderbilt D 2008 Phys. Rev. B 77 195408 [104] Marks L D Chiaramonti A N Tran F Blaha P 2009 Surf. Sci. 603 2179 [105] Wang Y and Cheng H P 2013 J. Phys. Chem. C 117 2106 [106] Janotti A Bjaalie J Gordon L Van de Walle C G 2012 Phys. Rev. B 86 241108(R) [107] Franchini C Chen X Q Podloucky R 2011 J. Phys.: Condens. Matter 23 045004 [108] Sorokine A Bocharov D Piskunov S Kashcheyevs V 2012 Phys. Rev. B 86 155410 [109] Cossu F Schwingenschlögl U Eyert V 2013 Phys. Rev. B 88 045119 [110] Mizokawa T and Fujimori A 1996 Phys. Rev. B 54 5368 [111] Solovyev I V 2006 Phys. Rev. B 73 155117 [112] Solovyev I V Hamada N Terakura K 1996 Phys. Rev. B 53 7158 [113] Pavarini E and Koch E 2010 Phys. Rev. Lett. 104 086402 [114] Nakamura M Yoshida T Mamiya K Fujimori A Taguchi Y and Tokura Y 1999 Mater. Sci. Eng. B 68 123 [115] Maiti K and Sarma D D 2000 Phys. Rev. B 61 2525 [116] Sarma D D Shanthi N and Mahadevan P 1996 Phys. Rev. B 54 1622 [117] Saitoh T Bocquet A E Mizokawa T Namatame H Fujimori A Abbate M Takeda Y Takano M 1995 Phys. Rev. B 51 13942 [118] Wadati H Kobayashi D Kumigashira H Okazaki K Mizokawa T Fujimori A Horiba K, Oshima M Hamada N Lippmaa M Kawasaki M and Koinuma H 2005 Phys. Rev. B 71 035108 [119] Abbate M Fuggle J C Fujimori A Tjeng L H Chen C T Potze R Sawatzky G A Eisaki H and Uchida S 1993 Phys. Rev. B 47 16124 [120] Abbate M Zampieri G Prado F Caneiro A Gonzalez-Calbet J M and Vallet-Regi M 2002 Phys. Rev. B 65 155101 [121] Mizokawa T and Fujimori A Namatame H Takeda Y Takano M 1998 Phys. Rev. B 57 9550 [122] Imada Fujimori and Tokura 1998 Rev. Mod. Phys. 70 1039 [123] Hubbard J 1963 Proc. R. Soc. London, Ser. A 276 238; Hubbard J 1964 ibid. 277 237; Hubbard J 1964 ibid. 281 401 [124] Kotliar G Savrasov S Y Haule K Oudovenko V S Parcollet O Marianetti C A 2006 Rev. Mod. Phys. 78 865 [125] Trimarchi G and Binggeli N 2005 Phys. Rev. B 71 035101


29

[126] Franchini C Sanna A Massidda S and Gauzzi A 2004 Eurphys. Let. 71 952 [127] Yan S Chen X Q Franchini C 2014 Unpublished [128] Loa I Adler P Grzechnik A Syassen K Schwarz U Hanfland M Rozenberg G K Gorodetsky P and Pasternak M P 2001 Phys. Rev. Lett. 87 125501 [129] Pinsard-Gaudart L Rodri´ıguez-Carvajal J Daoud-Aladine A Goncharenko I Medarde M Smith R I and Revcolevschi A 2001 Phys. Rev. B 64 064426 [130] Elemans A van Laar A van der Veen K R and Loopstra B O 1971 J. Phys. Chem. Solids 3 238 [131] Baldini M Struzhkin V V Goncharov A F Postorino P and Mao W L 2011 Phys. Rev. Lett. 106 066402 [132] Yamasaki A Feldbacher M Yang Y F Andersen O K Held, K 2006 Phys. Rev. Lett. 96 166401 [133] Ramos A Y Souza-Neto N M Tolentino H C N Bunau O Joly Y Grenier S Itié J P Flank A M Lagarde P and Caneiro A 2011 Europhys. Lett. 96 36002. [134] Coey J M D Viret M and von Molnár S 1999 Adv. Phys. 48 167 [135] Khomskii D I 2009 Physics 2 20 [136] Perdew J P Ruzsinsky A Csonka G I Vydrov O A Scuseria G E Constantin L A Zhou X and Burke K 2008 Phys. Rev. Lett. 100 136406 [137] Wu Z and Cohen R E 2006 Phys. Rev. B 73 235116 [138] Piskunov S Heifets E Eglitis R I and Borstel 2004 Comp. Mater. Sci. 29 165 [139] van Benthem K Elsässer C and French R H 2001 J. Appl. Phys. 90 6156 [140] Wemple S H 1970 Phys. Rev. B 2 269 [141] Inaguma Y Yoshida M Tsuchiya T Aimi A Tanaka K Katsumata T Mori D 2010 J. Phys.: Conf. Ser. 215 012131 [142] Inaguma Y Tanaka K Tsuchiya T Mori D Katsumata T Ohba T Hiraki K Takahashi T and Saitoh H 2011 J. Am. Chem. Soc. 133 16920 [143] Hao X F Stroppa A Picozzi S Filippetti A Franchini C 2012 Phys. Rev. B 86 014116 [144] Ohtomo A Muller D A Grazule J L and Hwang H Y 2002 Nature 419 378 [145] Santander-Syro A F Copie O Kondo T Fortuna F Pailhés S Weht R G. Qiu X Bertran F Nicolaou A Taleb-Ibrahimi A Le Fiévre P Herranz G Bibes M Reyren N Apertet Y Lecoeur P Barthiéliémy A and Rozenberg M J 2011 Nature 469 189. [146] Meevasana W King P D C He R H Mo S K Hashimoto M Tamai A Songsiriritthigul P Baumberger F Shen Z X 2011 Nat. Mater. 10 114 [147] Wang Z Zhong Z Hao X Gerhold S Stöger B Schmid M Sánchez-Barriga J Varykhalov A Franchini C Held K Diebold U 2014 PNAS accepted [148] Lim S G Kriventsov S Jackson T N Haeni J H Schlom D G Balbashov A M Uecker R Reiche P Freeouf J L and Lucovsky G 2002 J. Appl. Phys. 91 4500 [149] Chen J Q Wang X Lu Y H Barman A R You G J Xing G C Sum T C Dhar S Feng Y P Ariando Xu Q H and Venkatesan T 2011 Appl. Phys. Lett. 98 041904 [150] Rondinelli J M and Spaldin N A 2010 Phys. Rev. B 82 113402 [151] Enterkin J A Subramanian A K Russell B C Castell M R Poeppelmeier K R Marks L D 2010 Nat. Mater. 9 245 [152] Nørskov J K, Rossmeisl J Logadottir A Lindqvist L, Kitchin J R Bligaard T Jónsson H. 2004 J. Phys. Chem. B 108 17886 [153] Sha Y Yu T H Liu Y Merinov B V Goddard W A 2010 J. Phys. Chem. Lett. 2010 1 856 [154] Suntivich J Gasteiger H A Yabuuchi N Nakanishi H Goodenough J B Shao-Horn Y 2011 Nature Chem. 3 546 [155] Delugas P Filippetti A Fiorentini V Bilc D I Fontaine D Ghosez P 2011 Phys. Rev. Lett. 106 166807 [156] Thiel S Hammerl G Schmehl A Schneider C W and Mannhart J 2006 Science 313 1942 [157] Kim B J 2008 Phys. Rev. Lett. 101 076402. [158] Ju W Liu G Q and Yang Z 2013 Phys. Rev. B 87 075112. [159] Middey S Nandy A K Pandey S K Mahadevan P Sarma D D 2012 Phys. Rev. B 86 104406

Hybrid functionals applied to perovskites [160] [161] [162] [163] [164] [165] [166] [167] [168] [169] [170] [171] [172] [173] [174] [175] [176] [177] [178] [179] [180] [181] [182] [183] [184] [185] [186] [187] [188] [189] [190]

30

Franchini C Archer T He J Chen X Q Filippetti A and Sanvito S 2011 Phys. Rev. B 83 220402(R) Mazin I I Manni S Foyevtsova K Jeschke H O Gegenwart P Valenti R 2013 Phy. Rev. B 88 035115 Piskunov S Gopeyenko A Kotomin E A Zhukovskii Y F Ellis D E 2007 Comp. Mat. Sci. 41 195 Bandura A V Evarestov R A and Kuruch D D 2011 Integrated Ferroelectrics 123 1 Evarestov R A Bandura A V and E. N. Blokhin E N 2009 Integrated Ferroelectrics 108 37 Evarestov R A Bandura A V 2011 Solid State Ionics 188 25 Rodriguez E E Poineau F Llobet A Kennedy B J Avdeev M Thorogood G J Carter M L Seshadri R Singh D J and Cheetham A K 2011 Phys. Rev. Lett. 106 067201 Avdeev M Thorogood G J Carter M L Kennedy B J Ting J Singh D J and Wallwork K S 2011 J. Am. Chem. Soc. 133 1654 De’ Medici L Marvlje J and Georges A 2011 Phys. Rev. Lett. 107 256401 Galal A Atta N F Darwish S A Fatah A A Ali S M 2010 J. Power Sources 195 3806 He J and Franchini C 2014 Phys. Rev. B 89 045104 Booth G H Gr¨ uneis A Kresse G and Alavi A 2012 Nature 493 365 Paier J Marsman M and Kresse G 2008 Phys. Rev. B 78 121201(R) Franchini C Sanna A Marsman M and Kresse G 2010 Phys. Rev. B 81 085213 Bruneval F Sottile F Olevano V Del Sole R and Reining L 2005 Phys. Rev. Lett. 94 186402 Arita R Kuneˇs J Kozhevnikov A V Eguiluz A G and Imada M 2012 Phys. Rev. Lett. 108 086403 Hsieh D Mahmood F Torchinsky D H Cao G Gedik N, 2012 Phys. Rev. B 86 035128 Siegrist T and Chamberland B L 1991 J. Less-Common Met. 170 93 Maiti K Singh R S Medicherla V R R Rayaprol S and Sampathkumaran E V 2005 Phys. Rev. Lett. 95 016404 Cava R J Batlogg B Krajewski J J Farrow R Rupp L W White A E Short K Peck W F and Kometani T 1988 Nature 332 814 Franchini C Kresse G Podloucky R 2009 Phys. Rev. Lett. 102 256402 Vildosola V G¨ uller F Llois A M 2013 Phys. Rev. Lett. 110 206805 Yan B Jansen M and Felser C 2013 Nature Phys. 9 709 Nishio Ahmad T J Uwe H 2005 Phys. Rev. Lett. 95 176403 (2005) Ahmad J Uwe H 2005 Phys. Rev. B 72 125103 (2005) Gozar A Logvenov G Butko V Y and Bozovic I 2009 Phys. Rev. B 75 201402(R) Tran D and Blaha P 2009 Phys. Rev. Lett. 102 226401 Yin Z P Kutepov A and Kotliar G 2013 Phys. Rev. X 3 021011 Janesko B Henderson T M and Scuseria G E 2008 Phys. Chem. Chem. Phys. 11 443 Cococcioni M and de Gironcoli S 2005 Phys. Rev. B 71 035105 Aryasetiawan F Karlsson K Jepsen O and Schönberger U 2006 Phys. Rev. B 74 125106