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a small distortion of the cubic. It is a semiconductor whose band gap lies in the Fe 3d

states of opposite spin (Figure 5.13).

There is no band gap in the LSDA method, even in the distorted structure. The

simplest way to create a band gap is to use the LDA þ U method, which does produce

a band gap, as in the calculation of Neaton et al. [85]. However, the U parameter must

be empirically chosen, which is unsatisfactory. We recently calculated the band

structure of BFO by the SX method for the experimental structure, with no adjustable

parameters [86, 87]. This gave a band gap of 2.7 eV, which was the ﬁrst available value

of the gap. It agrees with subsequent experimental evaluations [88, 89].

5.4

Summary

The SX method has been reviewed as a method to produce band structures of

insulators and semiconductors with more accurate band gaps. SX behaves as a hybrid

-12

-10

-8

-6

-4

-2

Energy (eV)

L F Γ Z L

-2

-1

Energy (eV)

L F Γ Z L Γ

Figure 5.13 Calculated SX band structure of BiFeO

in the R3c structure, and close up of bands

near the gap.

5.4 Summary

density functional, which can be used for energy minimization. Some examples

of the used of SX, particularly in oxides, are given.

Acknowledgements

The authors acknowledge valuable discussions with A. Zunger and S. Lany.

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5 Calculation of Semiconductor Band Structures and Defects by the Screened Exchange

Accurate Treatment of Solids with the HSE Screened Hybrid

Thomas M. Henderson, Joachim Paier, and Gustavo E. Scuseria

6.1

Introduction and Basics of Density Functional Theory

Theoretical predictions of electronic properties offer a clear complement to exper-

imental investigations. However, these theoretical predictions are only as useful as

they are accurate. Ideally, ﬁrst-principles calculations of electronic properties in

periodic systems would use some high level many-body technique such as coupled-

cluster theory; however these methods are horriﬁcally expensive and therefore have

restricted applicability in practice. While simpler many-body methods such as GW

theory are signiﬁcantly less expensive, the computational burden they impose

nevertheless restricts their scope. Therefore, we often must perforce resort to

single-particle descriptions, which are less computationally expensive but of course

also less reliable.

One important single-particle reference is the Hartree–Fock (HF) method. The HF

density matrix variationally minimizes the HF energy, given by

¼hWjTjWiþ

dr nðrÞu

ext

ðrÞþ

nðr

Þnðr

þE

½c; ð6:1Þ

where n is the density, c the density matrix, r

¼jr

r

j, and E

½c is the nonlocal

Fock exchange energy

½c¼

cðr

; r

Þcðr

; r

: ð6:2Þ

That is, HF takes the usual one-body terms and the Coulomb interaction between

electrons and adds a (known) functional of the density matrix to describe exchange.

We construct the wave function jWi, density and density matrix from HF orbitals,

obtained from a reference system of noninteracting electrons in which the external

potential is augmented by the Hartree potential

ðr

Þ¼

nðr

; ð6:3Þ

Advanced Calculations for Defects in Materials: Electronic Structure Methods, First Edition.

Edited by Audrius Alkauskas, Peter Deák, Jörg Neugebauer, Alfredo Pasquarello, and Chris G. Van de Walle.

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and by an additional nonlocal potential u

which accounts for the effects of the

exchange interaction on the orbitals. This nonlocal potential is the functional

derivative of the exchange energy E

with respect to the density matrix

dcðr

; r

¼hr

i¼

cðr

; r

; ð6:4Þ

so that matrix elements of u

between two single-particle functions are

hwju

jyi¼

hwjr

ihr

jyi¼



ðr

Þyðr

cðr

ð6:5Þ

HF has some advantages which should not be forgotten. It is exact for one-electron

systems and has no one-electron self-interaction. It also forms a variationally optimal

starting point for many-body methods. The orbital energies in HF have a clean

interpretation as ionization potentials and electron afﬁnities (i.e., Koopmans the-

orem holds), which justiﬁes the use of HF band energy differences to predict the

fundamental band gap (the difference between the ionization potential and electron

afﬁnity of the system). Numerically, however, the HF band energy difference is a poor

predictor of the fundamental gap.

Unfortunately, the advantages of HF are generally outweighed by disadvantages.

Because it neglects electron correlation effects entirely, HF is insufﬁciently accurate

for most applications. It has an innate tendency to overly favor electronically localized

states, a manifestation of its form of many-electron self-interaction [1–3]. Calculating

the nonlocal exchange interaction is computationally demanding in solids. The

nonlocal exchange interaction also prevents HF from describing metallic

behavior [4].

For all of these reasons, the single-particle method of choice is usually Kohn –Sham

(KS) density functional theory (DFT) [5–7], in which the energy is given by

¼hWjTjWiþ

drnðrÞu

ext

ðrÞþ

nðr

Þnðr

þE

½n: ð6:6Þ

In other words, KS-DFT includes the usual one-body terms and Coulomb

interaction between electrons and describes many-body exchange–correlation effects

with a functional E

½nof the density alone. The density is obtained from a reference

system of noninteracting electrons in which the external potential is augmented by

the Hartree potential and by an additional local potential u

which accounts for the

effects of the exchange and correlation interactions on the orbitals. This local

potential is the functional derivative of the exchange–correlation energy E

with

respect to the density.

In principle, the exchange component of E

can be taken directly from HF, with

the sole distinction being that the density matrix should be constructed from KS

orbitals. The functional derivative to build u

would then yield the optimized

effective potential (OEP) [8, 9] which is in some sense the best local variant of u

However, numerical computation of the OEP can be challenging, particularly in

6 Accurate Treatment of Solids with the HSE Screened Hybrid

a gaussian basis set [10]. Numerous approximations to OEP can solve this prob-

lem [11–16], but all remain computationally quite demanding in extended systems

because they must construct the nonlocal exchange operator. In practice, HF-type

exchange is almost never used in a genuine KS calculation.

The chief difﬁculty in KS-DFT is the need to approximate E

and u

practical calculations. Typical density functional approximations (DFAs) include

the local density approximation (LDA) and generalized gradient approximations

(GGAs) such as the functional of Perdew, Burke, and Ernzerhof (PBE) [17]. These

are examples of what we shall term semilocal functionals, in which the

exchange–correlation potential and energy density at a point depend only on the

density and possibly its derivatives at that point. In effect, they expand the

exchange–correlation energy around the homogeneous electron gas result, though

they include a rather general dependence on the density gradient which allows

them to satisfy known constraints on E

that the second-order gradient expansion

violates. Alternatively, one can view GGAs as curing the second-order gradient

expansions violations of known constraints on the exchange–correlation hole

[18, 19]. Meta-GGAs such as the TPSS functional of Tao, Perdew, Staroverov, and

Scuseria [20] add dependence on the local kinetic energy density, but are also

semilocal in character.

For simple solids, these semilocal functionals are reasonably accurate, presum-

ably because their electron densities tend to be slowly varying and the homoge-

neous electron gas is thus a reasonable starting point. This accuracy, however, does

not extend to the description of the band gap, which is severely underestimated.

For more complicated systems which may include localized electronic states,

magnetic effects, rapidly varying densities, or other complicating factors, semilocal

functionals are generally inadequate. Alternative semilocal functionals such as

PBEsol [21] and revTPSS [22] can be constructed, but while these yield better lattice

parameters and bulk moduli than their parents (respectively, PBE and TPSS), they

do not remedy the failures of semilocal DFT in systems with the foregoing

features, nor do they yield correct band gaps. Formally, this is largely because

semilocal functionals do not include the derivative discontinuity of exact KS (see

below) [23].

One possible solution which is very successful in molecular systems is to use a

global hybrid functional [24, 25], which mixes a fraction of nonlocal HF-type

exchange with conventional semilocal exchange:

hybrid

¼ E

DFA

þc

ðE

E

DFA

Þ: ð6:7Þ

Common hybrids include the three-parameter B3LYP hybrid [26] and the non-

empirical [27] PBEh global hybrid [28, 29]. It should be emphasized that hybrid

calculations are almost always done in what is called the generalized Kohn–Sham

(GKS) sense [30], in which the nonlocal exchange energy yields a nonlocal exchange

interaction.

Unfortunately, as already mentioned, nonlocal exchange is problematic in solids.

Therefore, Heyd, Scuseria, and Ernzerhof introduced the Heyd–Scuseria–Ernzerhof

6.1 Introduction and Basics of Density Functional Theory

(HSE) screened hybrid [31–35]. In HSE, the electron–electron interaction is split into

a short-range part and a long-range part, as

erfcðvr

|{z}

erf ðvr

r12

|{z}

; ð6:8Þ

with a screening parameter v which in the latest (HSE06) variant of the functional is

numerically v ¼ 0:11a

1

in terms of the Bohr radius a

. The short-range part is

treated as in the PBEh global hybrid – that is, it uses 25% short-range exact exchange

and 75% short-range PBE exchange – while the long-range part is treated purely by

PBE. As v goes to 0, the short-range part dominates and HSE reduces to PBEh; as v

goes to inﬁnity, the short-range part vanishes and HSE reduces to PBE. By screening

the electron–electron interaction in this way, one vastly reduces the expense of

calculating the exact exchange interaction. Note that while the exchange interaction is

short range in nature, the range over which exact exchange is included is approx-

imately 1/v

, allowing delocalization to nearest and next-nearest neighboring

atoms, which admits some form of nonlocality into the model without including all of

it. It is also worth noting that Savin introduced the concept of range-separation in

DFT [36–38], though he suggested the use of nonlocal exchange in the long range

where HSE includes it instead in the short range.

Remarkably, this simple adjustment to the PBEh global hybrid leads to a com-

putationally affordable functional which is accurate for a wide variety of properties of

solids. It is our purpose here to summarize what we believe are the key features of

HSE and explain roughly why the functional works; for a more detailed review of the

successes and failures of HSE, see Ref. [39]. We begin by discussing what is perhaps

the most notable success of HSE (its ability to predict band gaps) in Section 6.2 before

stepping back to more carefully consider the physics of screened exchange (SX) in

Section 6.3. Section 6.4 discusses several applications of HSE, and we offer a few

concluding remarks in Section 6.5.

6.2

Band Gaps

Probably the most important success of HSE is its ability to predict semiconductor

band gaps from its band energy differences. Here, we want to rationalize this success

and set up the discussion for what follows. Before we begin, however, we had best be

precise about what we mean when we say that HSE accurately predicts band gaps.

The fundamental band gap of the system deals with the energy required to add or

remove electrons. We deﬁne it as

ðNÞ¼EðN þ1ÞþEðN1Þ2EðNÞ; ð6:9Þ

where E(N) is the energy of the N-electron system. The optical gap instead measures

the energy required to excite an electron, and is just the lowest electronic excitation

100

6 Accurate Treatment of Solids with the HSE Screened Hybrid

energy of the system. The optical gap is lower than the fundamental gap by the exciton

binding energy, which represents the interaction between the excited electron and the

hole created upon excitation.

In HF, Koopmans theorem tells us that the HFfundamental gap is just equal to the

HF band energy difference. The HF optical gap can be computed from the time-

dependent (TD) linear reponse (generally just known as TD HF) and is lower than the

HF fundamental gap; in other words, HF predicts a nonzero exciton binding

energy [40, 41]. Due to self-interaction in the unoccupied energy levels, the HF

fundamental gap severely overestimates the experimental fundamental gap, at least

for semiconductors and typical insulators.

In exact KS, we do not have Koopmans theorem. However, we can use the

Janak–Slater theorem instead [42]. We obtain the well-known result that the exact KS

fundamental gap is

ðNÞ¼e

N þ1

ðNÞe

ðNÞþD

ðNÞ; ð6:10Þ

where D

ðNÞ arises from the discontinuity in u

as a function of N at integer N

[43, 44]. Standard semilocal functionals and hybrid functionals in the generalized KS

sense do not exhibit this discontinuity [23], so their prediction for the fundamental

gap is

DFA

ðNÞ¼e

DFA

N þ1

ðNÞe

DFA

ðNÞ; ð6:11Þ

where the superscript DFA means a conventional density functional

approximation.

The appropriate framework for the prediction of the optical gap in exact KS is its TD

linear response (i.e., TD KS) [45, 46]. Exact KS must predict the optical gap to be below

the fundamental gap, in agreement with experiment, simply by virtue of being a

formally exact theory. For semilocal functionals, the optical gap turns out to be [47–50]

ðNÞ¼e

N þ1

ðNÞe

ðNÞ: ð6:12Þ

In other words, semilocal functionals do not describe excitons, and predict the

optical gap and fundamental gap to be the same. One must therefore not be dogmatic

in claiming that the semilocal band energy difference constitutes a prediction for only

one of the optical gap or fundamental gap. Due to self-interaction error, the occupied

bands in semilocal DFTare too high in energy, and the semilocal optical gap severely

underestimates the experimental optical gap. Thanks to the their nonlocal exchange

component, global hybrids predict a nonzero exciton binding energy, and the quality

of their prediction for the experimental optical or fundamental gap depends on the

amount of nonlocal exchange they incorporate.

Because HSE is done in the generalized KS sense, the HSE band energy difference

is the HSE fundamental gap. As a practical matter, the HSE band energy differences

correspond better with the experimental optical gap than they do with the experi-

mental fundamental gap, especially as the gap gets larger. Therefore in practice we

claim that the HSE band energy difference should be understood as a prediction of

6.2 Band Gaps

101

the experimental optical gap; because the optical and fundamental gaps are so close in

semiconductors, for these systems the HSE band energy difference can also be used

as a predictor of the experimental fundamental gap. As the gap gets larger, HSE band

energy differences more dramatically underestimate the experimental fundamental

gap, and as they get larger still, the HSE band energy difference becomes a poor

predictor even for the experimental optical gap.

The success of HSE band energy differences can be rationalized quite straight-

forwardly, once one knows a few properties of the parent semilocal global hybrid

(PBEh) and its underlying GGA (PBE) [51]. As is typically the case for semilocal

functionals, PBE band energy differences underestimate the gap. For semiconduc-

tors, the PBEh band energy differences overestimate the gap. One can think of HSE

as an interpolation between PBE and PBEh; as the range-separation parameter v

varies between 0 and 1, HSE varies between PBEh and PBE. Thus, there is some

nonzero value of v for which the HSE band energy difference reproduces the

experimental gap. Schematically, this is illustrated in Figure 6.1. Note that because

HSE includes a fraction of screened exact exchange, it has only a small exciton

binding, so the TD HSE optical gap should not differ much from the band

energy difference.

While it is reasonably clear that there should exist a value of v that gives the correct

gap for a given system, it is not so clear that this v should be universal. In fact, it is not

(see Ref. [52]). Nonetheless, v ¼ 0:11a

1

as used in HSE06 seems to be a reasonable

system-averaged value across a wide variety of systems. The HSE band energy

difference accurately reproduces the optical gap in semiconductors, but severely

underestimates the gap in insulators, and the HSE band width in metallic systems is

Expt.

PBEh

Band Energy Difference

PBE

Figure 6.1 (online color at: www.pss-b.com)

Schematic illustration of the v-dependent band

energy difference in HSE. The x-axis indicates

that v ¼ 0 corresponds to PBEh and v !1

corresponds to PBE. The y-axis shows that the

experimental gap lies between the PBE and

PBEh band energy differences, so that

there is some v

whose band energy

difference reproduces the experimental gap.

102

6 Accurate Treatment of Solids with the HSE Screened Hybrid