Alkauskas A., Deak P., Neugebauer J., Pasquarello A., Van de Walle Ch.G. (Eds.) Advanced Calculations for Defects in Materials: Electronic Structure Methods

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electronic structure of a-SiO

has to be compared with the crystal phase that exhibits

the most similar density, i.e., cristobalite.

We have performed DFT calculations on two 108 atoms amorphous SiO

models,

WQ2 and FQ1 generated at two different quench rate 2.6 10

K/s and 1.1 10

s, respectively (where WQ and FQ stay for well-quenched and fast-quenched to

distinguish between the two quench rates, see Ref. [20] for further details), and on two

192 atoms cristobalite models (Fd3-m), one at T ¼0 K (C0) and one at T ¼300 K

(C300). Here, we use the temperature as a way to add to the system a small stochastic

disorder. Calculations have been performed at the C point, with a wave function cut-

off of 70 Ry and norm-conserving pseudopotentials. Hundred and eight atoms

supercell is big enough for having a system-size convergency within 0.1 eV at the

self-energy level. In Figure 12.2, we have plotted the density of states and the

localization of each state as a function of the energy for the two amorphous models

as well as for the perfectly ordered cristobalite, which we use as crystal reference. The

localization is described by means of the normalized SI, obtained by dividing the SI, i.

e., the Coulomb interaction between an electronic state and itself, as generally

deﬁned by the following equation:

SI ¼

ðð



ðrÞj



ðrÞj



ðr

Þj



ðr

jjrr

r d

; ð12:3Þ

by the SI of a plane wave normalized in the corresponding cell (which generalizes the

SI tool, extensively used in the quantum chemical community for ﬁnding maximally

localized basis sets, for the description of extended systems that are modeled through

periodic boundary conditions [21]): normalizing the SI allows for a quantitative,

Figure 12.2 (online color at: www.pss-b.com)

Upper panel: DFT–LDA energy density of states

for two a-SiO

models WQ2 and FQ1, obtained

by quenching from a melt with two different

quench rates, 2.6 10

and 1.1 10

K/s,

respectively, and a perfectly ordered crystobalite

(Cristobalite 0 K). Lower panel: states

localization by means of their |SI|. Each |SI|

point has been calculated by averaging within an

energy interval of 0.09 eV. The DOS have been

aligned maximizing the overlap between deep

valence levels.

206

12 SiO

in Density Functional Theory and Beyond

energy-independent estimate of localization for systems with different unit-cell

volumes (the boundaries being |SI| ¼1 for a fully delocalized state, i.e., a plane

wave, and divergent |SI| for a completely localized state, i.e., a Delta function)]. We

observe signatures of localization almost exclusively at valence edges, while the

bottom of the conduction band exhibits a perfectly delocalized character. As expected,

the increase of the quench speed increases the localization, and we see at the

neighborhood of the valence band top for the FQ1 model states highly localized, that

can be considered as localized band tail states. It is also easy to see that the cristobalite

gap is smaller than the mobility gap of the disordered systems. If we compare the

fundamental gap of the perfectly ordered cristobalite, 5.4 eV, with the system

perturbed just by thermal disorder, 5.3 eV [21], it is clear that a small degree of

disorder induces a small closure of the gap. Therefore, we recover completely the

Anderson, Mott, and Cohen picture [26–29], i.e., small disorder degree close the gap

while a strong disorder degree widen it.

The widening of the gap due to strong disorder is particularly enhanced when

moving to GW, as it includes non-diagonal screening, as explained above. Indeed, the

GW gap for cristobalite is 8.9 eV while the mobility gap for a-SiO

is 9.4–9.2 eV for

WQ2 and FQ1, respectively (for a more extended discussion see Ref. [21]).

12.4

Deep Defect States

Defect levels (donor or acceptor) and formation energies of charged defects are

difﬁcult to describe quantitatively within DFT [35]. Indeed, they are related to

ionization potentials (N 1 excitation)/electron afﬁnities (N þ 1 excitation) of the

defect state [36, 37]. Some attempts have been made to try to circumvent the need of

going beyond DFT [38, 39]. However, it is unlikely that one can ﬁnd semi-empirical

rules that work in a general case, and for defects in the already disordered structure of

an amorphous system this is even more critical. To illustrate the complexity of the

problem, we can compare results for the well-quenched model we showed above, to

the results of a model with connectivity defects.

We have performed DFT–LDA and G

(on top of LDA) on an amorphous SiO

model with two point defects: a non-bridging oxygen NBO (coordination 1) and a tri-

coordinated silicon (the simulation parameters are the same as in Ref. [21]). These

two atomic defects have produced ﬁve strongly localized defect states, a dangling

bond (Figure 12.3a), two occupied oxygen 2p non-bonding orbitals (Figure 12.3b and

c), and two unoccupied silicon non-bonding orbitals (Figure 12.3d and e).

The corresponding defect levels are located inside the gap (single lines in

Figure 12.4). Comparing the level alignment in the neighborhood of the gap, DFT,

and GW, it is evident that neither an ad hoc band stretching nor a rigid energy shift can

locate correctly the states from their DFT–LDA relative position, even if one tries to

treat separatelythe occupied and unoccupied states. In the case of the highest localized

states, such as the O dangling bond of the NBO, the many-body corrections even alter

their positionrelativeto the band tails,precludingscissor shift approximations. Evenif

12.4 Deep Defect States

207

Figure 12.3 (online color at: www.pss-b.com) Localized defect states due to a non-bridging oxygen [(a), (b), and (c)] and a tri-coordinated

silicon [(d) and (e)] in an a-SiO

model (in white oxygen atoms and in black silicon atoms).

208

12 SiO

in Density Functional Theory and Beyond

this kind of approximation has been successfully applied to ideal crystalline systems, it

is fundamental to include the exact many-body correction for quantitative studies

when treating states with markedly different characters. A qualitatively correct

ordering of localized states with a different nature is already provided at the

Hartree–Fock level, indicating that the inclusion of exact-exchange is mandatory for

a sound description of dangling bonds. However, the weight of the exchange part with

respectto the correlationpart is different from state to state and no semi-empirical rule

can be extrapolated from the results. It is also interesting to note the increase in the

energy difference between the two 2p non-bonding orbitals, which are seen almost

degenerate within DFT–LDA. The sensitivity to local ﬁeld effects in GW enhances the

energy difference, which is due to the different orientation of the orbitals. As for the

two unoccupied (and more delocalized) defect states, they maintain almost the same

relative position, and are just deeper in the gap, more detached from the mobility edge.

We also observe that the sensitivity of defect states to a proper treatment of many-body

effects is in agreement with a very recent paper [40], where the application of the GW

scheme to the analysis of a carbon vacancy in 4H-SiC turned out to be decisive to

correctly account for electron–electron correlations.

12.5

Conclusions

We have shown that the inclusion of local ﬁeld effects may be relevant even just for a

correct quantitative evaluation of the gap size. The example of the deep defects in

Figure 12.4 (online color at: www.pss-b.com)

Electronic structure of a 108 atoms a-SiO

system with defects introduced by one non-

bridging oxygen and one tri-coordinated silicon,

within DFT (upper panel) and G

(lower

panel). Shaded blue and red bands represent,

respectively, delocalized states and localized

tails due to disorder. Calculation for the defect

system was also performed through norm-

conserving pseudopotentials and C-point

sampling; an energy cutoff of 70 Ry was used for

the wave-functions and Fock operator. Single

gray lines stay for the strongly localized defect

states (occupied oxygen non-bonding and

unoccupied silicon non-bonding states). The

single red line shows the position of the oxygen

dangling bond level. Energies have been aligned

to the top of the valence mobility edge.

12.5 Conclusions

209

amorphous silica shows how it could be difﬁcult to try to extract general semi-

empirical models to circumvent the needs of going beyond DFT. Unfortunately, the

way to a bi-univocal modeling of the electronic properties in the gap neighborhood is

still under an active debate [36–38]. Indeed, the computational effort needed by a full

GW calculation, or GW including vertex-corrections, free from pseudo-potential

effects, precludes its application to realistic systems [41, 42], forcing the research

community to ﬁnd shortcuts, that need to be more extensively tested: pseudo-

potentials, approximations to the screening, one-shot GW starting from different

wave-functions and energies [42]. It is of fundamental relevance to systematically

explore all this shortcuts and give complete benchmarks.

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211

Overcoming Bipolar Doping Difficulty in Wide Gap

Semiconductors

Su-Huai Wei and Yanfa Yan

13.1

Introduction

Application of semiconductors as electric and optoelectronic devices depends

critically on their dopability. Failure to dope a material, i.e., to produce enough free

charge carriers beyond a certain limit at working temperature, is often the single most

important bottleneck for advancing semiconductor-based high technology. Wide

band gap (WBG) semiconductors, such as diamond, AlN, GaN, MgO, and ZnO, have

unique physical properties that are suitable for applications in short-wavelength and

transparent optoelectronic devices [1–9]. To realize these applications, the formation

of a high-quality homo p–n junction is essential. In other words, high-quality bipolar

(p- and n-type) doping are required for the same material. Unfortunately, most WBG

semiconductors experience a serious doping asymmetry problem, i.e., they can easily

be doped p- or n-type, but not both [10]. For example, diamond can be doped relatively

easily p-type, but not n-type [1–3]. On the other hand, ZnO can easily be made high-

quality n-type, but not p-type [7–9]. For materials with very large band gap such as AlN

and MgO, both p- and n-type doping are difﬁcult [11]. The doping difﬁculty also exists

in nanostructure semiconductors where the band gaps increase due to the quantum

conﬁnement [12, 13]. These doping problems have hindered the potential applica-

tions of many WBG and nanostructure materials.

Extensive research has been done to understand the origin of the bipolar doping

difﬁculties in WBG and nanostructure semiconductors and to ﬁnd possible solutions

to overcome the doping difﬁculty. In the past, we have proposed various approaches

to overcome the bipolar doping difﬁculty in WBG semiconductors. These approaches

have been tested by the systematical calculation of defect formation energies and

transition energy levels of intrinsic and extrinsic defects in various WBG and

nanostructure semiconductors using ﬁrst-principles density-functional theory

[14–23]. In this paper, we review the origins of the bipolar doping difﬁculty and

describe our approaches for overcoming the doping bottleneck in WBG semicon-

ductors.The paper is organized as follows. Section 13.2 discusses the salient features

of calculating defect properties, in which the issues related to the ﬁnite size of the

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.

Ó 2011 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2011 by Wiley-VCH Verlag GmbH & Co. KGaA.

213

supercell and the band gap errors are discussed. Section 13.3 analyzes symmetry

and occupation of defect levels. Section 13.4 describes what causes the bipolar

doping difﬁculty in WBG semiconductors and the origin of the doping limit rules.

Section 13.5 describes our proposed approaches to overcome the bipolar doping

difﬁculties, focusing mostly on ZnO and other WBG semiconductors. Section 13.6

brieﬂy summarizes the paper and provides an outlook for future research in this ﬁeld.

13.2

Method of Calculation

We performed the band structure and total energy calculations using the ﬁrst-

principles density-functional theory with local density approximation (LDA) or

general gradient approximation (GGA) [24, 25]. We used the supercell approach in

which the defects or defect complexes are put at the center of a supercell and the

periodic boundary condition is applied. For quantum dots (QDs) the surface are

passivated by hydrogen or pseudohydrogen [12, 13]. In all calculations, all the atoms

are allowed to relax until the Hellman–Feynman forces acting on them become

negligible. For charged defects, a uniform background charge is added to keep the

global charge neutrality of the supercells [19].

To determine the defect formation energy and defect transition energy levels, one

needs to calculate the total energy E(a, q) for a supercell containing defect a in charge

state q, the total energy E(host) of the same supercell without the defect, and the

total energies of the involved elemental solids or gases at their stable phases. It is

important to realize that the defect formation energy depends on the atomic chemical

potentials m

and the electron Fermi energy E

. From these quantities, the defect

formation energy, DH

(a, q), can be obtained by:

ða; qÞ¼DEða; qÞþ

þqE

; ð13:1Þ

where DE(a, q) ¼E(a, q) E(host) þ

E(i) þ qe

VBM

(host). E

is referenced to the

valence band maximum (VBM) of the host. m

is the chemical potential of constituent i

referenced to elemental solid/gas with energy E(i); n

is the number of elements and q

is the number of electrons transferred from the supercell to the reservoirs in forming

the defect cell. The transition energy e

(q/q

) is the Fermi energy at which the

formation energy of defect a at charge state q is equal to that at charge state q. Using

Eq. (13.1), the transition energy level with respect to the VBM can be obtained by

ðq=q

Þ¼

DEða; qÞDEða; q

 q

 e

VBM

ðhostÞ: ð13:2Þ

Typically, for ﬁnite supercell, the Brillouin zone integration for the charge density

and total energy calculations is performed using special k-points or equivalent

k-points in the superstructures. This approach gives better convergence on the

calculated charged density and total energy. However, it usually gives a poor

description on the symmetry and energy levels of the defect state, as well as the

214

13 Overcoming Bipolar Doping Difficulty in Wide Gap Semiconductors

VBM and conduction band minimum (CBM) states, and the results could be sensitive

to the k-points sampling because the band edges are determined by the k-point

sampling. To avoid this problem, a C-point-only calculation is often used to determine

the total energy and transition energy level because the symmetry of the defect and

the band edge states are well deﬁned at C, and both the shallow and deep levels are

correctly described (see Figure 13.1). However, for small supercell, C-point-only

approach may give poor total energy convergence. Here, we propose to use a hybrid

scheme to combine the advantages of both special k-points and C-point-only

approaches [14, 19]. In this scheme, we ﬁrst calculate the transition energy level

with respect to VBM, which is given by

eð0=qÞ¼e

ð0Þe

VBM

ðhostÞþ

Eða; qÞðEða; 0Þqe

ð0ÞÞ

q

: ð13:3Þ

For donor level (q > 0), it is usually more convenient to reference the ionization

energy level to the CBM, i.e., we can rewrite Eq. (13.3) as

ðhostÞeð0=qÞ¼e

CBM

ðhostÞe

ð0Þþ

Eða; qÞðEða; 0Þqe

ð0ÞÞ

;

ð13:4Þ

where a positive number calculated from Eq. (13.4) indicates the distance of the

transition energy level below the CBM. In Eqs. (13.3) and (13.4) e

ð0Þ and e

ð0Þ are

the defect levels at the special k-points (weight averaged) and at the C-point,

respectively; and e

VBM

(host) and e

CBM

(host) are the VBM and CBM energies,

respectively, of the host at the C-point. e

(host) is the band gap at the C -point. The

ﬁrst term on the right-hand side of Eqs. (13.3) or (13.4) give the single-electron defect

level at the C-point. The second term determines the relaxation energy U (including

both the Coulomb contribution and the atomic relaxation contribution) of the

charged defects calculated at the special k-points, which is the extra cost of energy

Figure 13.1 (online colour at: www.pss-b.com) Schematic plot of the defect level with respect to the

band edge. It is shown that in a supercell calculation, the energy level determined at the C-point is

correct for both shallow and deep levels.

13.2 Method of Calculation

215

after moving (q) charge to the neutral defect level with E ¼e

ð0Þ. Afterwards, the

formation energy of defect a at charge state q can be obtained by

ða; qÞ¼DH

ða; 0Þqeð0=qÞþqE

; ð13:5Þ

where DH

(a,0) is the formation energy of the charge-neutral defect and E

is the

Fermi level with respect to the VBM. This new approach has been used successfully

for studying defects in various semiconductors.

We want to point out that in writing down Eqs. (13.1)–(13.4), we assumed that the

common reference energy level is used in the calculation of defect a in different

charge states. Because in a periodic supercell calculation the zero potential energy is

not well deﬁned, therefore, we have to lineup the potential using a common

reference. This is usually done by lineup core level of an atom far away from the

defect center. In the case where core level is not available, average potential around

the atom can also be used.

In the formula above, we also assumed that the VBM energy position is given by its

eigenvalue e

VBM

(host). For small supercell, however, the Koopmans theorem may

not hold that is

dE ¼ e

VBM

ðhostÞ½Eðhost; NÞEðhost; N  1Þ ð13:6Þ

is not zero, especially for VBM with localized electron states. Similar situation may

also exist for the CBM energy position. In this case, the correction term dE should be

added to determine the VBM or CBM energy level.

It the supercell calculation, the periodic boundary conditions introduces a spu-

rious Coulomb interaction between the charged defects in different cells. To estimate

the magnitude of this interaction, point charges immersed in neutralizing jellium

are usually assumed. Attempts to including higher-order terms are difﬁcult, because

the higher-order multipoles of the defect charge are not uniquely deﬁned in the

supercell approach [26]. However, in reality, the charged defect does not have a delta-

function-like distribution, especially for shallow defects, which have a relatively

uniform charge distribution. Therefore, direct application of the Makov and

Payne [26] correction using only point charge often overestimates the effect [19].

Thus, for shallow low charge state defects we usually assume the Makov and Payne

correction are not important. However, this correction term could be large if the

defect level is localized or the defect is in a small QD.

Another issue that leads to uncertainty in defect calculation is caused by the fact

that LDA or GGA calculations underestimate the band gap of a semiconductor. One

way to correct this error is to project the defect level to the CBM and VBM states, and

shift the defect level accordingly when the band edges are shifted to correct the band

gap error [27]. Another way to correct this error is using high level DFT calculation

such as GWapproach [28]. However, currently, this type of full scale calculation is still

formidable. Recently, more empirical hybrid density-functional method [29, 30]

which mixes certain amount of Hartree–Fock potential with the GGA potential is

used for defect calculation. Although this kind of approach can correct the band gap

in some empirical way, the symmetry breaking caused by the Hartree–Fock potential

216

13 Overcoming Bipolar Doping Difficulty in Wide Gap Semiconductors