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Contract No. DE-AC36-08GO28308

Carrier Density and 

Compensation in 

Semiconductors with Multi 

Dopants and Multi Transition 

Energy Levels: The Case of Cu 

Impurity in CdTe

Preprint

Su-Huai Wei, Jie Ma, and Tim A. Gessert

National Renewable Energy Laboratory

Ken K. Chin 

New Jersey Institute of Technology

Presented at the 37th IEEE Photovoltaic Specialists Conference 

(PVSC 37)

Seattle, Washington

June 19-24, 2011

Conference Paper

NREL/CP-5900-50683

July 2011
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1 

CARRIER DENSITY AND COMPENSATION IN SEMICONDUCTORS WITH 

MULTI DOPANTS AND MULTI TRANSITION ENERGY LEVELS: 

THE CASE OF Cu IMPURITY IN CdTe 

Su-Huai Wei1, Jie Ma1, T.A. Gessert1, and Ken K. Chin2 

1

National Renewable Energy Laboratory, Golden, CO 80401. 2 Department of Physics and Apollo CdTe Solar Energy Research Center, 

New Jersey Institute of Technology, Newark, NJ 07102

ABSTRACT

Doping is one of the most important issues in 

semiconductor physics. The charge carrier generated by 

doping can profoundly change the properties of 

semiconductors and their performance in optoelectronic 

device applications, such as solar cells. Using detailed 

balance theory and first-principles calculated defect 

formation energies and transition energy levels, we derive 

general formulae to calculate carrier density for 

semiconductors with multi dopants and multi transition 

energy levels. As an example, we studied CdTe doped 

with Cu, in which VCd, CuCd, and Cui are the dominant 

defects/impurities. We show that in this system, when Cu 

concentration increases, the doping properties of the 

system can change from a poor p-type, to a poorer p-type, 

to a better p-type, and then to a poor p-type again, in good 

agreement with experimental observation of CdTe-based 

solar cells.

INTRODUCTION

The CdTe solar cell is one of the most promising thin-film 

photovoltaic technologies. However, it has a relatively low 

efficiency (16.5%) [1] compared with its theoretical limit, 

~29%. [2] One of the main reasons is because of the low 

doping level or hole concentration of CdTe. Therefore, to 

improve the CdTe cell solar conversion efficiency, it is 

necessary to understand and control its doping properties. 

Traditionally, a dopant is distinguished as a donor or an 

acceptor, a single donor (acceptor) or a double donor 

(acceptor), and a shallow or a deep dopant. These 

classifications of dopants are well defined when a single 

dopant is concerned, as shown in most of the text books 

[3]; however, when a system contains multi dopants and 

each dopant has multi transition energy levels, then the 

situations are more complicated.

For example, Sb substitution on the Te site, SbTe is a 

single acceptor when it stays alone, but it can behave as a 

donor when it coexists with other shallower acceptors, 

such as Cd vacancy VCd, because in this case electrons at 

the higher SbTe level can donate to the lower VCd level. 

Moreover, a dopant can usually donate (accept) one or 

two or even more electrons, thus the dopant can have 

several charge states and the population of the dopants in 

each of the charge states depends on the transition 

energy level, temperature, the coexistence of other 

defects, and ultimately the resulting Fermi energy of the 

electrons. A comprehensive understanding of how to treat 

these complicated situations has not been presented in 

the common text books [3], which usually describe, for 

simplicity, only a single donor or acceptor.

In this work, we will first derive some general formulae to 

show how to calculate charge carrier concentrations as a 

function of temperature and dopant concentrations when a 

system contains multi dopants with multi transition energy 

levels [4,5]. In our approach, all the dopants are treated 

equal, i.e., we do not pre-classify the dopants. The charge 

states of the dopants are obtained self-consistently. 

To be more concrete, we will use the doping of Cu in 

CdTe as an example to describe how the formulae are 

derived and how dopants interact with each other. The 

concept can be easily extended to other semiconductor 

systems. We choose CdTe:Cu as an example because 

CdTe is one of the most important thin-film solar cell 

materials and it generally contains Cd vacancies as the 

most important intrinsic defect, which is a double acceptor, 

and Cu as an important impurity for improving the 

efficiency. Cu substitution on the Cd site, CuCd, is usually a 

single acceptor and Cu interstitial, Cui, is usually a single 

donor.

Experimentally, the observed effect of Cu on p-type 

doping in CdTe is very interesting [6]. After the CdTe 

polycrystalline thin film is deposited using a method such 

as close space sublimation (CSS), and a CdCl2 process is 

performed, the film is coated with ZnTe:Cu, followed by 

annealing at a certain temperature for back contact. The 

concentration of Cu in the sample is controlled by the 

annealing temperature. We noticed that a very low level of 

Cu inclusion may further reduce the already low 

conductivity (due to the low p-doping level) of the intrinsic 

CdTe thin film. Increasing the level of Cu involvement to a 

certain level, a reversal from deterioration to improvement 

of conductivity occurs, which leads to an optimized CdTe 

thin film confirmed by the high efficiency of the solar cell. 

Further incorporation of Cu into CdTe, however, will turn it 

from a p-type to an insulating type or even an n-type. Such 

a complicated and puzzling effect on the property and 

quality of the CdTe thin filmchanging from a poor p-type 

to a poorer p-type, then to a better and optimized p-type, 

and then to an insulating type, when Cu incorporation in 

CdTe increaseshas not been fully understood or 

explained.
2 

THEORETICAL DESCRIPTION

The concentrations of holes (positive charge) and 

electrons (negative charge) in the system are given by 

( )/

0

( )/

0

vf B

cf B

E E kT

v

E E kT

c

p Ne

n Ne

−

− −

=

=

 (1) 

where Ef is Fermi energy, Ev and Ec are valence band 

maximum (VBM) and conduction band minimum (CBM), 

respectively. It is usually convenient to set Ev = 0 and 

Ec = Eg, where Eg is the band gap. Nv and Nc are the 

effective density of states of the valence bands and 

conduction bands, respectively. After doping, the dopants 

in the system are partially ionized at a finite temperature. 

Suppose the concentrations of negative and positive 

charge induced by dopant ionizations are NA

- and ND+

. The 

charge neutrality condition is rewritten as 

0 0 D A pN nN + − + =+ (2) 

Solving this equation self-consistently, we can get the 

Fermi energy and hole/electron concentration at 

equilibrium as a function of temperature and dopant 

concentrations.

Figure 1 The hole concentration (black square) and 

Fermi energy (red dot) versus NCu. At very low Cu 

concentration, all Cu go to vacancy sites. The hole 

concentrations decrease as Cu density increases. In 

this calculation, initial Cd vacancy concentration is 

NVCd = 1015 cm−3

, μCd = −0.2 eV and T = 300 K. The 

result here does not depend on the chemical potential 

of Cd.

For a system with a dopant α that has multiple charge 

states, the number of the dopant α in charge state q is 

given by 

( , )/ ( ) H q kT f B

q n q ge αα

−∆ ∝ (3) 

So after we know the formation energy of dopant α at 

charge state q (e.g., using first-principles method [7]) and 

its degeneracy factor q g , we can calculate the population 

n q( ) α when the total number of defect α is known. For a 

single defect level that can hold one spin-up and one spindown electron, such as those derived from CBM, the 

number of possible electron configurations is given by 1, 

2, and 1, when this defect level has no electron, one 

electron, or two electrons.

For a doubly degenerate level that can hold up to two 

spin-up and two spin-down electrons, such as those 

derived from VBM of zinc-blende semiconductors with light 

hole and heavy hole states, the number of possible 

electron configurations is given by 1, 4, 6, 4, 1, when this 

defect level has 0, 1, 2, 3, or 4 electrons. In general,

exp( / ) q i i B g E kT = − ∑ with the summation over all the 

possible configurations, where Ei is the energy of the ith 

configuration relative to the lowest energy E1 = 0. For 

simplicity we assume the energies are all the same, i.e., 

Ei = 0 for all configurations, which is true for most cases. 

However, for some cases such as neutral VCd, the 

assumption does not hold exactly. Even in this case, our 

test calculations show that the reduction of the 

degeneracy factor does not affect our results below. A 

similar degeneracy factor can be derived for other 

situations.

For a defect with multi transition energy levels, such as 

VCd, the number of defects in charge state q is given by 

( , )/

( , )/ ( )

f B

f B

H q kT

q

H q kT

q

q

g e nq N

g e

α

α α α

−∆

−∆ =

∑ (4) 

where Nα is the total concentration of defect α and the 

summation is over all the possible charge states.

For a dopant atom that can form multi defects, such as Cu 

in CdTe, which can form either CuCd or Cui, the number of 

dopants in charge state q of ith type of defect is given by 

( , , )/

,

( , , )/

,

,

( ,)

f B

f B

H qi k T

q i

H qi k T

q i

q i

g e n qi N

g e

α

α α α

−∆

−∆ =

∑ (5) 

The meaning of every term in the above equation is 

obvious.

RESULTS AND DISCUSSIONS

Because there are Cd vacancies in the system and the 

formation energy of Cu occupying Cd vacancies is 

strongly negative, Cu occupies those vacancies 

spontaneously until no Cd vacancies exsit. The transition 

energy level of CuCd is higher than that of VCd, so as the 

concentration of Cu increases, the concentration of the 

hole decreases (see Fig. 1). This explains why initially the 

system turns from a poor p-type to a poorer p-type when 

the Cu is introduced at a very low concentration.
3 

Figure 2 The hole concentration (black square) and 

Fermi energy (red dot) versus NCu. In this calculation, 

μCd = −0.2 eV and T = 300 K. The hole concentration 

decreases monotonically and eventually the Fermi 

energy is pinned.

Next, we add more Cu into the system. Under the Cd-rich 

growth (μCd ~ 0) condition, the formation energy of Cui(+) 

in a p-type sample is smaller than that of CuCd, so Cu 

prefers to go to the interstitial sites that compensate the ptype dopants. Therefore, the hole concentration will keep 

decreasing as Cu concentration increases (see Fig. 2). 

However, under Cd-poor conditions (μCd ~ μCdTe), the 

formation energy of Cui is larger than that of CuCd and Cu 

prefers to be at the substitutional site as an acceptor. 

Therefore, the hole concentration increases (see Fig. 3). It 

is clear from the above analysis that the behavior of Cu 

doping depends sensitively on the chemical potential of 

Cd. With a low μCd, one may observe a turning from poor 

p-type to better p-type, which eventually saturates. 

However, the turning from a better p-type to an insulating 

type is not observed under these equilibrium doping 

conditions.

In realistic experiments, incorporation of Cu is usually 

done at a high temperature such as 600 K and the sample 

is then quenched to room temperature (300 K) for 

measuring the carrier density. In this case, the populations 

of dopants should be calculated at doping temperature 

and the hole concentration should be calculated at the 

measuring temperature. Because the formation energy of 

ionized defects also depends on the Fermi energy, we 

have to do self-consistent calculations twice: first, selfconsistently calculating the populations of CuCd and Cui at 

doping temperature and then, assuming the impurity has 

difficulty changing its sites preference at low temperature,

keeping the impurity populations fixed and calculating the 

hole concentration at measuring temperature selfconsistently. We have performed the calculations with μCd

= -0.6 eV. The results are shown in Fig. 4. It shows a clear 

peak of hole concentration, which is in agreement with the 

experimentschanging from a poor p-type to a better ptype and then to a poor p-type again. Similar results are 

obtained for μCd = -0.2 eV (see Fig. 5). 

Figure 3 The hole concentration (black square) and 

Fermi energy (red dot) versus NCu. In this calculation, 

μCd = −0. 6 eV and T = 300 K. The hole concentration 

increases monotonically.

Figure 4 The hole concentration (black square) and 

Fermi energies at growing temperature (blue triangle) 

and measuring temperature (red dot) versus NCu. In 

the calculation, the populations of CuCd and Cui are 

calculated self-consistently at 600 K and the hole 

concentration is calculated self-consistently at 300 K 

with μCd = −0.6 eV. A clear peak of hole concentration 

has been built up.
4 

Figure 5 The hole concentration (black square) and 

Fermi energies at growing temperature (blue triangle) 

and measuring temperature (red dot) versus NCu. In 

the calculation, the populations of CuCd and Cui are 

calculated self-consistently at 600 K and the hole 

concentration is calculated self-consistently at 300 K 

with μCd = −0.2 eV. A clear peak of hole concentration 

has been built up.

CONCLUSIONS

In conclusion, in this work general formulae are described 

to derive carrier concentration as a function of Fermi 

energy and dopant concentration in a multi dopant and 

multi transition energy level system. By analyzing the 

competition between the dominant defects VCd, CuCd, and 

Cui using the concept described in this work, we have 

successfully explained why the properties of the CdTe thin 

film show such puzzling changeschanging from a poor 

p-type, to a poorer p-type, to a better p-type, and to a poor 

p-type again when the Cu doping level is gradually 

increased. The model proposed here is general and can 

be applied to any other multi dopant and multi transition 

energy level system.

ACKNOWLEDGEMENT

This work was supported by the U.S. Department of 

Energy under Contract No. DE-AC36-08-GO28308 with 

the National Renewable Energy Laboratory.

REFERENCES

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solar cells”, Solar Energy 77, 803 (2004).

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multiple energy-gap terrestrial solar-cells”, J. Appl. Phys.

51, 4494 (1980).

[3] S. Sze, Physics of Semiconductor Devices, 2nd 

Edition, (Wiley, New York, 1981).

[4] K. K. Chin, S.-H. Wei, and T. A. Gessert, “The role of 

Cu impurity states in CdTe thin film solar cells”, 

Proceedings of 35th IEEE Photovoltaic Specialist 

Conference, Honolulu, June 20-25, 2010. 

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