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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and the accuracy of the calculated defect levels, in our opinion, still needs experi-

mental veriﬁcation.

13.3

Symmetry and Occupation of Defect Levels

It is often very useful to know the symmetry and character of the single-particle defect

level before we start the calculation, because it can help us identify the defect level and

because defects with different symmetry and character will behave differently.

Moreover, to modify defect states or correct the band gap error, it is also important

to know the symmetry of the state. For example, although both anion vacancy and

cation interstitial have the same a

defect levels in II–VI semiconductors [23], anion

vacancy has the a

character derived from the valence band or cation dangling bond

states, whereas cation interstitial has instead the a

character derived from the

conduction band. Thus, the energy level of the cation interstitial is expected to follow

closely with the CBM, whereas the energy level of anion vacancy state will not.

For simple extrinsic impurities, one can predict in principle whether a dopant is

a donor with a single-particle energy level close to the CBM or an acceptor with

a single-particle energy level close to the VBM by simply counting the number of the

valence electrons of the dopants and the host elements. For example, in CdTe, one can

expect that group-I elements substituting on the Cd site, X

create acceptors,

whereas group-VII elements substituting on the Te site, Y

VII

creates donors.

Generally speaking, to produce a shallow acceptor, it is advantageous to use a more

electronegative dopant, whereas to produce a shallow donor, it is advantageous to use

a less electronegative dopant.

For intrinsic defects, the situation is more complicated. Figure 13.2 shows the

single-particle energy levels of tetrahedrally coordinated charge-neutral defects in

CdTe [23]. Generally speaking, when a high-valence atom is replaced by a low-valence

Figure 13.2 (online colour at: www.pss-b.com) Single-particle defect levels for the tetrahedrally

coordinated neutral intrinsic defects in CdTe. The solid (open) dots indicate the state is occupied

(unoccupied).

13.3 Symmetry and Occupation of Defect Levels

217

atom (e.g., Cd

) or by a vacancy V

and V

, defect states are created from the host

valence (v) band states that move upward in energy. The defect states consist of a low-

lying singlet a

state and a high-lying threefold-degenerate t

state. Depending on the

potential, both a

and t

can be above the VBM. These states are occupied by the

nominal valence electrons of the defect plus the valence electrons contributed

from the neighboring atoms (e.g., in CdTe, six electrons if the defect is surrounded

by four Te atoms or two electrons if it is surrounded by four Cd atoms). For example,

for charge-neutral V

, the defect center has a total of 0 þ 6 ¼6 electrons. Two of

them will occupy the a

state and the remaining four will occupy the t

states just

above the VBM, so V

is an acceptor. On the other hand, if a low-valence atom is

replaced by a high-valence atom (e.g., Te

), or if a dopant goes to an interstitial site

(e.g., Cd

and Te

), the a

and t

are pulled down and will remain inside the valence

band. Instead, the defect states a

and t

are created from the host conduction band

states that move down in energy. Depending on the potential, both the a

and t

states

can be in the gap. For example, for charge-neutral Te

,6þ 6 ¼12 electrons are

associated with this defect center. Eight of them will occupy the bonding a

and t

states, two will occupy the a

state, and the remaining two will occupy the t

state.

Since the partially occupied t

state is close to the CBM, Te

is also a donor. For the

interstitial defect, Cd

has two electrons that will fully occupy the a

state and is thus

expected to be a donor. The Te

defect center has six electrons. Two will occupy the a

state and the remaining four will occupy the t

states. Since the partially occupied t

states are closer to the VBM, Te

is expected to be a deep acceptor.

13.4

Origins of Doping Difficulty and the Doping Limit Rule

In general, there are three main factors that could cause the doping limit in

a semiconductor material [10, 14, 19, 22, 23]: (i) the desirable dopants have limited

solubility; (ii) the desirable dopants have sufﬁcient solubility, but they produce deep

levels, which are not ionized at working temperatures; and (iii) there is spontaneous

formation of compensating defects. The ﬁrst factor depends highly on the selected

dopants and growth conditions. The second factor only depends on the selected

dopants. Thus, these two factors can sometimes be suppressed by carefully selecting

appropriate dopants and controlling the growth conditions. The third factor is an

intrinsic problem for semiconductors; thus, it is the most difﬁcult problem to

overcome, especially for WBG semiconductors. This is because the formation energy

of charged compensating defects depends linearly on the position of the Fermi level,

[see Eq. (13.1)]. When a semiconductor is doped, the Fermi level shifts, which can

lead to spontaneous formation of the compensating charged defects. For example,

when a semiconductor is doped p-type, E

moves close to the VBM. In this case, the

formation energy of the charged donor defects decreases because they will donate

their electrons into the Fermi reservoir (Figure 13.3). In WBG semiconductors with

low VBM, the formation energy decrease of donor defects can be so large that at some

Fermi energy E

¼e

(p)

the formation energy of certain donor defect becomes zero,

218

13 Overcoming Bipolar Doping Difficulty in Wide Gap Semiconductors

i.e., it can form spontaneously, so further shift of the Fermi energy is not possible.

Moreover, low VBM also leads to high ionization energy. Therefore, a semiconductor

with low VBM is difﬁcult to be doped p-type. The trend for n-type doping is similar, i.e.,

a semiconductor with high CBM is difﬁcult to be doped n-type. This doping limit rule

explains why a semiconductor with large band gap usually cannot be doped one type

or even both types under equilibrium thermodynamic growth conditions. It also

provides a general guideline about whether a material can be doped p- or n-type if we

know the band alignment between different compounds. For example, Figure 13.4

Figure 13.3 (online colour at: www.pss-b

.com) Schematic plot of the dependence of the

formation energy of charged defects on

the Fermi energy position. The p-type

pinning energy e

(p)

is the Fermi

energy E

at which the formation energy of

donor A has zero formation energy.

Figure 13.4 (online colour at: www.pss-b.com) Band alignment and n- and p-type pinning energy

of II–VI and I–III–VI semiconductors (Ref. [10]).

13.4 Origins of Doping Difficulty and the Doping Limit Rule

219

shows the calculated band alignment for II–VI and I–III–VI semiconductors [10]. We

see that ZnO has very low CBM and VBM, so it can be easily doped n-type, but not

p-type. On the other hand, ZnTe with high VBM energy can be easily doped p-type,

but not n-type. For I–III–VI compounds, CuInSe

can be doped both p- and n-type,

but for CuGaSe

, n-type doping will be difﬁcult.

Based on the above understanding, we will search for corresponding solutions to

overcome the doping limit. We will focus on the following approaches: (i) increase

defect solubility by defeating bulk defect thermodynamics using non-equilibrium

growth methods such as extending the achievable chemical potential through

molecular doping or raising the host energy using surfactant; (ii) reduce defect

ionization energy level by designing shallow dopants or dopant complexes; and (iii)

reduce defect compensation and ionization level by modifying the host band

structure near the band edges. As examples, we will discuss doping in some

representative WBG semiconductors such as n-type doping in ZnTe and diamond

and p-type doping in ZnO. The principles discussed here are general and are

applicable to other WBG semiconductors.

13.5

Approaches to Overcome the Doping Limit

13.5.1

Optimization of Chemical Potentials

13.5.1.1 Chemical Potential of Host Elements

As Eq. (13.1) indicates, the formation energy of a defect, which determines the

solubility of dopants, depends sensitively on the atomic chemical potentials of both

the host elements and the dopants [14, 19]. Thus, optimization of the growth

conditions and dopant source is critical to enhance the doping ability. So far,

computational results and analysis have focused on the dependence of formation

energies on host-element chemical potentials [27, 31]. For example, N substituting O

) is expected to be a p-type dopant for ZnO. The formation energy of N

depends

on the chemical potentials of Zn, O, and N. Under thermal equilibrium growth

conditions, there are some thermodynamic limits on the achievable values of the

chemical potentials. First, to avoid precipitation of the elemental dopant and host

elements, the chemical potentials are limited by

 mðZn metalÞ¼0; ð13:7Þ

 mðO

gasÞ¼0; ð13:8Þ

 mðN

gasÞ¼0: ð13:9Þ

Second, m

and m

are limited to the value of maintaining ZnO. Therefore,

þm

¼ DH

ðZnOÞ: ð13:10Þ

220

13 Overcoming Bipolar Doping Difficulty in Wide Gap Semiconductors

Here, DH

(ZnO) is the formation energy of bulk ZnO. The calculated value is

about 3.5 eV. Finally, to avoid the formation of the Zn

secondary phase, m

is also

limited by

þ2m

 DH

ðZn

Þ: ð13:11Þ

The calculated formation energy of Zn

is about 1.2 eV. Using the equations

above, the achievable chemical potential region is shown in Figure 13.5.

Figure 13.6 shows the calculated formation energies of charge neutral defects as

a function of the O chemical potential. Here, m

is derived from N

gas. It is seen that

the formation energy of N

is lower at the O-poor condition, but higher at the O-rich

Figure 13.5 The achievable chemical potential region for N doped ZnO under equilibrium growth

condition.

Figure 13.6 (online colour at: www.pss-b.com) Calculated formation energies of charge-neutral

defects as a function of O chemical potential. The dashed line indicates the growth condition at

which (left region) Zn

will precipitate.

13.5 Approaches to Overcome the Doping Limit

221

condition. Thus, to enhance the solubility of N, ZnO ﬁlms should be synthesized at

O-poor conditions. It should be noted that the formation energies of other intrinsic

defects also depend on the growth conditions. At O-poor conditions, the formation

energies for acceptor-killer defects, such as Zn interstitials (Zn

) and O vacancies

), are decreased. Thus, there is an intrinsic problem for enhancing the p-type

doping using N as the dopant. In Section 13.5.4, we will discuss how this dilemma

may be eliminated by selecting appropriate dopants.

The left region of the dashed line indicates the growth condition at which Zn

will precipitate under equilibrium growth. This indicates that the O chemical

potential should not go into this region so as to avoid the precipitation of Zn

Thus, the precipitation of a secondary phase can limit the solubility of dopants. To

overcome this, it has been shown that the precipitation of a secondary phase can be

suppressed through epitaxial growth [32]. For example, the calculated thermody-

namic solubility of N in bulk GaAs is only [N] < 10

3

at T ¼650



C due to the

formation of a fully relaxed, secondary GaN phase. However, single-phase epitaxial

ﬁlms grown at T ¼400–650



C with [N] as high as 10% have been reported. Zhang

and Wei [32] found that if coherent surface strain is considered, the formation of

the secondary GaN phase could be suppressed during epitaxial growth. As a result,

the solubility of N can be enhanced signiﬁcantly. A similar approach could be used to

avoid forming Zn

in ZnO:N.

Avoiding the formation of secondary phase can also lead to some unexpected

consequences. For example, substituting Zn by Al for n-type doping in ZnO,

the formation energy of Al

depends on (m

m

), thus one may expect that the

formation energy of Al

should reach minimum under Zn-poor condition. How-

ever, to avoid the formation of Al

, we need to satisfy the following condition:

þ3m

 DH

ðAl

Þ: ð13:12Þ

Combine Eqs. (13.10) and (13.12), we have

ðm

 m

Þ

þ2DH

ðZnOÞDH

ðAl

: ð13:13Þ

That is, in the achievable chemical potential region, Al

has the lowest formation

under O-poor or Zn-rich condition.

13.5.1.2 Chemical Potential of Dopant Sources

Although the dependence of doping efﬁciency on the host elements chemical

potential has been studied extensively, the dependence of doping efﬁciency on

dopant chemical potential has not attracted much attention because normally there

are no signiﬁcant alternative dopant sources. However, for the case of N doping of

ZnO (or other oxides), there is a unique and unusual opportunity. There are at least

four different gases, namely N

, NO, NO

, and N

O that can be used as the dopant

source. If these molecules arrive intact at the growing surface, their chemical

potentials will determine the doping efﬁciency. We found that the N solubility can

be enhanced signiﬁcantly when metastable NO or NO

gases are used as the dopant

sources [15]. A key feature that underlies our idea of doping with NO or NO

is that

222

13 Overcoming Bipolar Doping Difficulty in Wide Gap Semiconductors

these molecules can supply single N atoms by breaking only the weak N–O bonds,

whereas one has to break the strong N–N bonds to obtain desirable single-N defects

when N

and N

O are used. For example, when the NO, N

O, or NO

molecules arrive

intact at the growing surface, the formation energy of a charge-neutral N

defect is

given by

ðN

; 0Þ¼EðN

; 0ÞEðhostÞþ2m

 m

ð13:14Þ

ðN

; 0Þ¼EðN

; 0ÞEðhostÞþ1:5m



; ð13:15Þ

ðN

; 0Þ¼EðN

; 0ÞEðhostÞþ3m

 m

: ð13:16Þ

Our calculated formation energy is about 0.9, 0.2, and 0.2 eV for NO, N

O, and NO

molecules, respectively. The NO and NO

molecules may simply be supplied as such

or they may be produced by a reaction in the gas phase. For example, NO molecules

can be created by N

O ,NO þ N. In this case, m

¼mN

O – m

. Figure 13.7 shows

the calculated formation energy of charge-neutral N

for four different gases. The

difference between N

O and NO/NO

is very clear, i.e., the use of NO/NO

leads

to signiﬁcantly reduced formation energies for N

because it does not entail any

energy to break the NN bonds. The negative formation energies of N

at Zn-rich

conditions indicate that NO or NO

molecules can be incorporated spontaneously to

form N

defects, if these molecules are intact before they are incorporated into ZnO.

However, in practical growth conditions, to avoid the precipitation of the secondary

phases such as Zn

, besides satisfying

þm

¼ m

for NO gas as dopant; ð13:17Þ

þm

¼ m

for NO

gas as dopant; ð13:18Þ

þ2m

¼ m

for N

O gas as dopant; ð13:19Þ

we also need to satisfy Eqs. (13.10) and (13.11), which set a low limit for the achievable

O chemical potential. In Figure 13.7, the left, middle, and right dashed vertical lines

indicate the low limits for O chemical potentials for N

O, NO, and NO

molecules,

respectively.

13.5.2

H-Assisted Doping

As we discussed above, the solubility of both the dopants and compensating defects

depend sensitively on the position of the Fermi level. If we can control the Fermi level

at a desirable position, then we may enhance the solubility of dopants and suppress

the formation of compensating defects. Electron or hole injection could be a method

to control the position of the Fermi level during ﬁlm growth. Another popular

13.5 Approaches to Overcome the Doping Limit

223

approach is passivating the dopants by H atoms. For example, in Mg-doped GaN, the

introduction of H can prevent such a shift. As a result, the concentration of Mg can

be enhanced [34]. After ﬁlm growth, H can be annealed out to achieve p-type

conductivity.

For N-doping in ZnO (also other oxides), the co-existence of H can, in addition to

preventing the Fermi level shift, directly passivate N dopants, forming a molecular NH

complex on O site [(NH)

]. The binding energy for (NH)

is 2.9 eV. The (NH)

complexes electronically mimic O atoms and cause smaller lattice distortion than N

Thus,theconcentration of (NH)

in ZnO can be much higherthanN

[14].Figure13.8

showsthe calculatedformationenergyfor (NH)

as a function of O chemical potential.

For comparison, the formation energies of N

and some hole-killer defects are also

shown. It is seen that the formation energy of (NH)

is lower than any other defects in

the O-poor condition. In addition, the existence of H also pins the Fermi energy level,

so the formation of compensating defects enhanced by a shifted Fermi level is also

suppressed. Therefore, p-type doping could be achieved after subsequently driving out

the hydrogen atoms from the sample by thermal annealing.

13.5.3

Surfactant Enhanced Doping

To lower the defect formation energy, which is the total energy difference between the

ﬁnal doped state and the initial state, we can either increase the initial dopant energy,

as discussed in the previous section, or increase the energy of the host. Recently, we

have shown that this can be done by introducing an appropriate surfactant during

Figure 13.7 (online colour at: www.pss-b.com) Calculated formation energy of charge-neutral

withfour different dopantsources:N

O, NO, and NO

. The left,middle, and rightdashedlines

indicate the low limits for achievable O chemical potentials for N

O, NO, and NO

molecule doping.

224

13 Overcoming Bipolar Doping Difficulty in Wide Gap Semiconductors

epitaxial growth [35]. The general concept for enhancing dopant solubility via

epitaxial surfactant growth is schematically described in Figure 13.9. It is known

experimentally that the surfactants in epitaxial growth ﬂoat on the top surface of the

growth front. The enhancement of dopant solubility initiates in the sublayers below

the surface. For p-type doping, dopants introduce acceptor levels with holes near the

VBM of the host system. On the other hand the surfactants on the growth surface will

introduce surfactant levels. If the surfactant levels are higher in energy than the

acceptor levels and have electrons available, the surfactants will donate the electrons

to the acceptors and consequently leads to a Coulomb binding between the surfactant

and the dopant. Such charge transfer reduces the energy of the system and

consequently leads to effective reduction on the formation energy of dopant incor-

poration in the host. The formation energy reduction is large if the energy level

Figure 13.8 (online colour at: www.pss-b.com) Calculated formation energy of (NH)

in ZnO.

Figure 13.9 (online colour at: www.pss-b.com) Schematic plot of the mechanism of surfactant

enhanced doping during epitaxial growth.

13.5 Approaches to Overcome the Doping Limit

225

difference between the surfactant and acceptor levels is large. The same principle

holds for n-type doping, except that the charge transfer is from dopant level to empty

surfactant level. In this case, the positions of dopant levels are close to the CBM and

the surfactant levels must be lower in energy than the dopant levels. Thus, the key for

this concept is how to ensure that the surfactant levels are higher (lower) in energy

than the dopant levels and there are indeed electrons (holes) in the surfactant levels in

p-type (n-type) doping. We have calculated the formation energy for substitutional

in the sublayer of ZnO ð000 1



Þ surface and found that the formation energy is

lowered by 2.3 eV with S as surfactant as compared to that without surfactant.

13.5.4

Appropriate Selection of Dopants

There are two general rules for choosing an appropriate dopant to produce shallow

defect levels. First, an appropriate dopant should favor the growth conditions that will

suppress the formation of compensating defects. As we discussed in Section 13.2, the

solubility of dopants and concentration of intrinsic defects depend sensitively on the

growth conditions. It is highly desirable to have a growth condition that enhances

the dopant solubility and suppresses the formation of intrinsic compensating

defects. This may be achieved by identifying the compensating defects and choosing

suitable dopants. For example, the major defects that compensate acceptors in ZnO

are Zn

and V

. Figure 13.6 shows that to suppress the formation of these hole-killer

defects, an O-rich growth condition is preferred. Of course, this condition is not

preferred for N incorporation. However, such a growth condition is preferred for

doping at cation sites.

Second, dopants at cation sites in compound semiconductors generally produce

shallower acceptor levels than dopants at anion sites. This is because for most

cation–anion compound semiconductors, the valance bands are derived mainly from

the anion atoms. Dopant substituting at cation site would, in general, cause smaller

perturbation than dopants at anion sites on the anion-derived valance band. Thus,

theoretical studies have found that Group-I elements such as Li and Na have low

acceptor levels, whereas Group-V elements such as N, P, As, and Sb have deep

acceptor levels in ZnO [33]. The large ionization energy for Group-Vacceptors in ZnO

can be understood as follows: The acceptor level, especially the shallow acceptor level,

has a wave-function character similar to that of the VBM state, which consists mostly

of anion p, and small amounts of cation p and cation d orbitals. Therefore, to have

a shallow acceptor level, the dopant should be as electronegative as possible, that is, it

should have low p orbital energy. For example, because the atomic p orbital energy

level of N is the lowest (Figure 13.10), i.e., most electronegative, among the Group-V

elements, N has been the preferred acceptor dopant for II–VI semiconductors

because it produces the lowest acceptor levels compared to the other Group-V

dopants. However, due to the low VBM of the oxides, the level of N

in ZnO is

still relatively deep [19, 33] at about 0.4 eVabove the VBM, making acceptor ionization

difﬁcult. The other Group-V elements are less electronegative than the N atom;

therefore, they have much larger ionization energy than N

[33]. This explains why it

226

13 Overcoming Bipolar Doping Difficulty in Wide Gap Semiconductors