Calculation of Fermi level location and point defect ensemble in CdTe single crystal and thin films
With the use of expressions obtained from the “first principles”, the ensemble of point defects was calculated, and the location of a Fermi level in undoped cadmium telluride single crystals and thin films depending on physico-technological conditions of their fabrication and annealing is determi...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
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Kosyak, V.V. Opanasyuk, A.S. 2017-05-28T18:06:33Z 2017-05-28T18:06:33Z 2007 Calculation of Fermi level location and point defect ensemble in CdTe single crystal and thin films / V.V. Kosyak, A.S. Opanasyuk // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2007. — Т. 10, № 3. — С. 95-102. — Бібліогр.: 21 назв. — англ. 1560-8034 PACS 68.55.Ln, 71.55.Gs https://nasplib.isofts.kiev.ua/handle/123456789/118133 With the use of expressions obtained from the “first principles”, the ensemble of point defects was calculated, and the location of a Fermi level in undoped cadmium telluride single crystals and thin films depending on physico-technological conditions of their fabrication and annealing is determined. The model in use accounts the most complete spectrum of defects in chalcogenide, including defects in the cadmium and tellurium sublattices, and the existence of an antistructural defect on the cadmium sublattice. Calculations of the concentration of neutral and charged defects are realized for two extreme cases – full equilibrium and quenching. The comparison of the obtained results with the data of modeling provided with the use of a quasichemical formalism for a number of models most used presently is carried out. It is shown that all models describe well the results of Hall measurements of the concentration of free carriers in single crystals in the range of high cadmium pressure, but give essentially different results in the range of high tellurium pressure. Dominant defects in single crystals at high cadmium pressure and annealing temperatures are twice charged tellurium vacancies or interstitial cadmium atoms, which is in agreement with experimental results, as just such defects can provide the dependence of the concentration of free carriers on cadmium pressure as n ~ P¹/³Cd . A type of defects which are dominant in a tellurium-enriched material is determined by the chosen model. This allows us to make conclusions about the validity of the considered models and to specify the thermodynamic parameters of the defect creation processes in a material. The offered model can be used for modeling the ensemble of point defects in any А₂В₆ compounds. Thus, the problem of the choice of models adequate to experimental data is reduced to the determination of the creation energy for uncharged defects and the depth of energy levels of charged defects. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Calculation of Fermi level location and point defect ensemble in CdTe single crystal and thin films Article published earlier |
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Calculation of Fermi level location and point defect ensemble in CdTe single crystal and thin films |
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Calculation of Fermi level location and point defect ensemble in CdTe single crystal and thin films Kosyak, V.V. Opanasyuk, A.S. |
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Calculation of Fermi level location and point defect ensemble in CdTe single crystal and thin films |
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Calculation of Fermi level location and point defect ensemble in CdTe single crystal and thin films |
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Calculation of Fermi level location and point defect ensemble in CdTe single crystal and thin films |
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Calculation of Fermi level location and point defect ensemble in CdTe single crystal and thin films |
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calculation of fermi level location and point defect ensemble in cdte single crystal and thin films |
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Kosyak, V.V. Opanasyuk, A.S. |
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Kosyak, V.V. Opanasyuk, A.S. |
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2007 |
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Semiconductor Physics Quantum Electronics & Optoelectronics |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Article |
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With the use of expressions obtained from the “first principles”, the ensemble
of point defects was calculated, and the location of a Fermi level in undoped cadmium
telluride single crystals and thin films depending on physico-technological conditions of
their fabrication and annealing is determined. The model in use accounts the most
complete spectrum of defects in chalcogenide, including defects in the cadmium and
tellurium sublattices, and the existence of an antistructural defect on the cadmium
sublattice. Calculations of the concentration of neutral and charged defects are realized
for two extreme cases – full equilibrium and quenching. The comparison of the obtained
results with the data of modeling provided with the use of a quasichemical formalism for
a number of models most used presently is carried out. It is shown that all models
describe well the results of Hall measurements of the concentration of free carriers in
single crystals in the range of high cadmium pressure, but give essentially different
results in the range of high tellurium pressure. Dominant defects in single crystals at high
cadmium pressure and annealing temperatures are twice charged tellurium vacancies or
interstitial cadmium atoms, which is in agreement with experimental results, as just such
defects can provide the dependence of the concentration of free carriers on cadmium
pressure as n ~ P¹/³Cd . A type of defects which are dominant in a tellurium-enriched
material is determined by the chosen model. This allows us to make conclusions about
the validity of the considered models and to specify the thermodynamic parameters of the
defect creation processes in a material.
The offered model can be used for modeling the ensemble of point defects in any А₂В₆
compounds. Thus, the problem of the choice of models adequate to experimental data is
reduced to the determination of the creation energy for uncharged defects and the depth
of energy levels of charged defects.
|
| issn |
1560-8034 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/118133 |
| citation_txt |
Calculation of Fermi level location and point defect ensemble in CdTe single crystal and thin films / V.V. Kosyak, A.S. Opanasyuk // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2007. — Т. 10, № 3. — С. 95-102. — Бібліогр.: 21 назв. — англ. |
| work_keys_str_mv |
AT kosyakvv calculationoffermilevellocationandpointdefectensembleincdtesinglecrystalandthinfilms AT opanasyukas calculationoffermilevellocationandpointdefectensembleincdtesinglecrystalandthinfilms |
| first_indexed |
2025-11-24T06:14:28Z |
| last_indexed |
2025-11-24T06:14:28Z |
| _version_ |
1850844157566779392 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 3. P. 95-102.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
95
PACS 68.55.Ln, 71.55.Gs
Calculation of Fermi level location and point defect ensemble
in CdTe single crystal and thin films
V.V. Kosyak, A.S. Opanasyuk
Sumy State University
2, Rimsky-Korsakov str., 40007 Sumy, Ukraine
Phone: 38(0542)392237, fax: 38 (0542) 334058, e-mail: opanasyuk@ua.fm, v_kosyak@ukr.net
Abstract. With the use of expressions obtained from the “first principles”, the ensemble
of point defects was calculated, and the location of a Fermi level in undoped cadmium
telluride single crystals and thin films depending on physico-technological conditions of
their fabrication and annealing is determined. The model in use accounts the most
complete spectrum of defects in chalcogenide, including defects in the cadmium and
tellurium sublattices, and the existence of an antistructural defect on the cadmium
sublattice. Calculations of the concentration of neutral and charged defects are realized
for two extreme cases – full equilibrium and quenching. The comparison of the obtained
results with the data of modeling provided with the use of a quasichemical formalism for
a number of models most used presently is carried out. It is shown that all models
describe well the results of Hall measurements of the concentration of free carriers in
single crystals in the range of high cadmium pressure, but give essentially different
results in the range of high tellurium pressure. Dominant defects in single crystals at high
cadmium pressure and annealing temperatures are twice charged tellurium vacancies or
interstitial cadmium atoms, which is in agreement with experimental results, as just such
defects can provide the dependence of the concentration of free carriers on cadmium
pressure as 31
Cd~ Pn . A type of defects which are dominant in a tellurium-enriched
material is determined by the chosen model. This allows us to make conclusions about
the validity of the considered models and to specify the thermodynamic parameters of the
defect creation processes in a material.
The offered model can be used for modeling the ensemble of point defects in any А2В6
compounds. Thus, the problem of the choice of models adequate to experimental data is
reduced to the determination of the creation energy for uncharged defects and the depth
of energy levels of charged defects.
Keywords: cadmium telluride, point defects, quasichemical formalism, “ab initio”
approach, single crystals, thin films.
Manuscript received 04.07.07; accepted for publication 27.09.07; published online 30.11.07.
1. Introduction
Due to unique physical properties, CdTe is the object of
careful studies during a long time, as a perspective
material to produce the detectors of x-ray and gamma
radiation, converters of the solar energy, and devices for
microelectronics [1]. At the same time, the expectations
of the researchers were not substantially justified in
connection with the complexity of obtaining the bulk
and film CdTe samples with predetermined
characteristics. For this reason, the main task of modern
scientific researches of CdTe is the development of
technologies of obtaining the single crystals and thin
layers with controllable point defects ensemble (PDE),
as just the PDE determines electrophysical parameters of
a material. The choice of optimum parameters of the
influence on chalcogenide is usually realized by
modelling the PDE in cadmium telluride in dependence
on technological conditions of its obtaining and annea-
ling with the use of the quasichemical formalism [2].
The procedure of modelling is reduced to the
solution of a system of equations, which describe the
introduction of defects in a solid from the gas phase,
jointly with the condition of electroneutrality of a
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 3. P. 95-102.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
96
material. In case of the formation of defects in the metal
sublattice, such quasichemical equations (QE) are as
follows:
−+↔ AgA VAA0 ,
)(
)(
0
A
AA
V
AN
PVN
K
A
−
= , (1)
+↔ ig AA ,
A
A
A
P
AN
K
i
)( +
= , (2)
)()( +− +=+ AA ANpVNn , (3)
where 0
AA − atom at the sublattice site; gA − atom in
the gas phase; −
AV − monovalent vacancy in the А sub-
lattice; +
i
A − monovalent interstitial atom; xK −
constant of QE; )(XN − the corresponding concen-
tration of defects; AP − pressure of the А component
during the annealing of samples; n and p − free electron
and hole concentrations in a material, respectively.
The type and number of the equations describing
the equilibrium of defects in a material [3-8] are
determined by a model of the creation of defects which
is accepted a priori, because the experimental data allow
one to determine only a possible type of dominant
defects. The constants of QE in system (1)-(3) are given
as:
∆
−=
kT
HKK exp01 , where 0K − constant that
does not depend on temperature; H∆ − enthalpy of the
creation of defects; k − Boltzmann constant; Т –
temperature, at which the reaction runs [2]. The
constants can be also determined by the comparison of
the results of modeling with those of experimental
measurements of the electrical characteristics of
chalcogenide at high temperatures (400-1000 ºС). As the
measurement of the concentration of neutral point
defects in a material by standard electrophysical
methods is impossible, QE describing the introduction of
defects in a solid from the gas phase are written for
charged defects. Further, the approximation like nA
i
≈+
is used. By determining the inclination angle for a
straight line on the plot Tn 310lg − and a segment that
is cut by a line on the axis of concentrations, it is easy to
find 0K and H∆ [2]. The reaction constants for the
creation of defects can be determined more reliably, if
the values of n calculated with the help of the given
constants are fit to experimental data by several
parameters: the inclination of isotherms and isobars, the
values of n on isotherms and isobars, and dependences at
maxCdP [6].
The first set of the QE constants for PDE in CdTe
was offered by Nobel [7]. By considering the balance of
defects in chalcogenide, he assumed that the creation of
defects in a material occurs according to Frenkel’s
mechanism over the cadmium sublattice. The alternative
point of view is the model which is offered by Chern [8].
By this model, a material can contain defects in both the
cadmium and tellurium sublattices. During the sub-
sequent researches, the model of defect creation became
more complicated, and simultaneously the meaning of
QE constants was specified. The basic models that were
more often used for the explanation of electrical
properties of CdTe were systematized by us in [9].
2. Models and results
The traditional approach to the study of PDE in CdTe
has a number of basic disadvantages. First, as a rule, the
majority of works deals with the calculation of only the
concentration of charged defects, though their
concentration depends really on both the concentration
of neutral centers and the Fermi level location in a
material which can depend, in its turn, on the
concentration of doped or uncontrollable impurities.
Second, the comparison of the results of modeling with
experimental data is a difficult task, because, as a rule,
the defect creation reactions are considered with
application a gas-phase reaction, while the significant
range of physical methods allow one to determine the
depth of the energy levels of point defects in the band
gap (BG) of a solid material.
Recently, the tendency to evaluate the
concentration of point defects in various materials by
“the first principles” approach [10] is observed. It
provides a more correct approach to the description of
PDE in a material: first, the concentration of neutral
interstitial defects loaded from the gas phase is
calculated, and then the processes of their ionization in a
solid state depending on the Fermi level location are
considered.
We now obtain the basic expressions which are
used in such an approach by an example of the formation
of a metal vacancy 0
AV . Under full equilibrium
conditions, the concentration of any point defect can be
expressed through thermodynamic potentials of the
defect creation process. According to the law of mass
action [11],
0)()( 00 =ξ−ξ+µ=µ∆ AAAAV AV , (4)
where AVµ∆ − change of the chemical potential;
AAA PkT ln0 +µ=µ ; Aµ − chemical potential of A
atoms in the gas phase; 0
Aµ − chemical potential of A
atoms in the gas phase under standard conditions;
)(ln)()( 0000
AAA VNkTVV +ξ=ξ – chemical potential of
a material with point defects; )( 00
AVξ − chemical
potential of point defects under standard conditions;
)(ln)()( 0000
AAA ANkTAA +ξ=ξ ; )( 0
AAξ – chemical
potential of an atom at the sublattice site (i.e., the
chemical potential of the perfect crystal); )( 00
AAξ −
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 3. P. 95-102.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
97
chemical potential of an atom at the sublattice site under
standard conditions
By taking the logarithm of expression (4), we get
µ+ξ−ξ
−=
kT
AV
P
AN
VN AAA
A
A
A
000000
0 )()(
exp
)(
)( . (5)
In modeling the defect creation process under
equilibrium conditions between the solid and the gas
phase, it is necessary to know the meaning of
thermodynamic functions (free energy, Gibbs energy,
and chemical potential) for each component. The free
energy of the gas phase can be found with the help of the
well-known expression
ZkTF ln−= . (6)
For a gas which consists of N identical molecules,
the thermodynamic potential Z can be expressed as the
molecular partition function
Nz
N
Z
!
1
= , (7)
where ∑
−=
i
i
kT
uz exp .
The total energy iu of a particle in the i-th state
can be written as
vibrrottrans0 uuuuui +++= , (8)
where 0u is the self-energy in the i-th state, and transu ,
vibru , and rotu are, respectively, the translation,
vibration, and rotational components of the total energy.
Then, in accordance with (7), we have
vibrrottrans0 zzzzz ⋅⋅⋅= . (9)
For a monatomic gas, 0vibrrot == uu , and
1vibrrot == zz . Then it is possible to write down the
relation trans0 zzz ⋅= . If N particles with mass m in the
gas phase occupy a volume V, then
2
3
2trans
21
π
=
h
Tkm
V
z , (10)
where h − Planck constant, and m − mass of a particle.
Let's find the free energy of a particle, by
substituting (7)-(10) in (6) and considering that
NnNa = and PnRTV = :
( )1ln0
trans +−= KnkTnuF , (11)
where ( )
3
2
5
2
3 )(
2
h
Tk
mK π= .
The Gibbs energy is connected with the free energy
by the expression nkTFG += . Then we have
( )KTknunG ln0
trans −= . (12)
By using the definition
TP
n
G
,
∂
∂
=µ , it is
possible to find the chemical potential of a particle in the
gas phase:
( )KTkuA ln0 −=µ . (13)
After simple transformations, we finally obtain
AAA PTk ln0 +µ=µ , (14)
where ( )KTkuA ln0
0 −=µ .
By considering the thermodynamic functions of
defects in a solid, we have no right to use the relation
1vibrrot == zz . On the contrary, the basic contribution
to iu will be brought by the vibration energy of a
particle: vibruui ≈ , because 0transrot == uu .
Therefore, it is necessary to research the complex
processes of interaction of an atom located in the crystal
lattice with adjacent atoms. In such a case, it is more
convenient to use the Gibbs energy instead of the
chemical potential. The corresponding chemical
potentials satisfy the relations
00000
000)(
AAA VVVA GTSHV =−=ξ , (15)
00000
000)(
AAA AAAA GTSHA =−=ξ , (16)
where 0S , 0H – entropy and enthalpy of structural
elements of the crystal lattice under standard conditions.
In the case of a solid, when the specific heats at
constant volume and pressure are approximately equal,
VP cc ≈ , the Gibbs energy can be replaced by the free
vibration energy vibF of the corresponding structural
elements in the crystal lattice:
0vibvibvib
0000
AAAA VVVV
GTSUF ≈−= , (17)
0vibvibvib
0000
AAAA AAAA
GTSUF ≈−= . (18)
Considering that 00
vib
0
AA VA
EUu =− is the defect
creation energy, the entropy vibvib
00
AA VA
SS << , and, in the
general case, 0
0 )( nAN A = , where 22
0 1048.1 ⋅=n cm−3
is the concentration of CdTe lattice sites, we finally
present (5) as
−+
−=
kT
TSUE
K
P
n
VN AAA VVV
A
A
vibvib
00
000
exp)( . (19)
By executing the simple transformation, we obtain
this expression in the standard form as
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 3. P. 95-102.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
98
+
−
=
kT
UE
k
S
K
P
n
VN AAA VVV
A
A
vibvib
00
000
expexp)( . (20)
By analogy with (20), it is possible to deduce the
formulas for the concentrations of other neutral point
defects in chalcogenide:
+
−
=
kT
UE
k
S
K
nP
AN iii
i
AAAA
vibvib
00
000
expexp)( , (21)
+
−
=
kT
UE
k
S
K
nP
VN BBB
B
VVVA
vibvib
00
000
expexp)( , (22)
+
−
=
kT
UE
k
S
K
P
n
BN iii BBB
A
i
vibvib
00
000
expexp)( , (23)
+
−
=
kT
UE
k
S
K
P
n
BN AAA BBB
A
A
vibvib
2
2
00
000
expexp)( , (24)
+
−
=
kT
UE
k
S
K
Pn
AN BBB AAAA
B
vibvib
2
2
00
000
expexp)( . (25)
The calculation of E , vibU , and vibS is reduced to
modeling the interaction of defects with the crystal
lattice by the ab initio method which is based on only
the quantum-mechanical and thermodynamic parameters
of Cd and Te atoms. By an example of the formation of
point defects in the cadmium sublattice in CdTe and
HgCdTe, the appropriate difficult mathematical com-
putations were carried out in [10, 12-15]. Some of them
were partially confirmed in a plenty of experiments
carried out on CdTe single crystals in [16, 17]. The
results of these researches are presented in Table 1. The
energy value designated by the asterisk corresponds to
two tetrahedral interstitial sites.
Table 1 shows the essential differences in the
values of the thermodynamic functions E, vibU , and
vibS determined by different authors, in particular in the
energies of formation of antistructural defects 0
CdTe and
tellurium vacancies 0
TeV . Therefore, in our opinion, the
most reliable results of calculations of the
thermodynamic functions of the defect creation process
are E , vibU , and vibS [16, 17] which have found a
sufficient confirmation by experiment data. We will use
them in the modeling in what follows.
If the concentrations of neutral defects are
determined, it is easy to find the concentrations of the
donors and acceptors of charged point defects by using
the Fermi-Dirac statistics and the condition of electro-
neutrality:
Table 1. Calculated parameters of the creation of neutral
point defects using the “ab initio” approach.
Type of
defects
vibUE + ,
H∆ , eV
vibS , bk References
1.65 −16.4 [3]
0.96 (1.16)*
1.29 (1.36)*
11.1 (16.4) [12-13]
[16]
1.97 (1.97) 38.46 (36.5) [18]
0Cdi
2.04 (2.26) − [14]
3.75 5.34 [3]
3.55 (3.58) −5.6 (−8.9) [16]
3.84 (2.11) −13.05 (33.8) [18]
0
CdV
2.67 − [14]
1.50 −19.7 [3]
1.97 (1.95) −38.54 (36.5) [18]
0Te i
3.41 – [14]
1.41 (4.06) 33.83 (55) [18]
1.57 19.7 [3]
0
TeV
3.24 − [14]
5.72 37.5 [3]
4.72 (6.3) −17 (−5.6) [16]
−0.81 (3.24) −23.71 (68.8) [18]
0
CdTe
3.7 − [14]
( )[ ] 1exp F +µ−
=
kTE
N
n
g
c , (27)
( ) 1exp F +µ
= υ
kT
N
p , (28)
( )[ ] 1exp
)(
)(
F
0
+µ−−−
=+
kTEEg
XN
XN
digd
diz
di , (29)
( )[ ] 1exp
)(
)(
F
0
+µ−
=−
kTEg
XN
XN
aid
aiz
ai , (30)
,)()()()()()(
)()()()(
222
22
++++++
−−−−
++++++=
=++++
AABBii
iiAA
BNBNVNVNANANp
BNBNVNVNn
(31)
where
2
3
2
)(
)(
2
2
π
=υ
h
kTm
N
pe
c − effective density of
states in the conduction band (valence band); )( pem −
effective mass of an electron (0.35 0m ) or a hole
(0.83 0m ); Eg − BG energy; Fµ − Fermi energy; z −
degree of defect ionization; g − degeneracy factor; Ea
and Ed − ionization energies of acceptors and donors,
respectively; )( 0XN − concentration of neutral defects;
)( +z
dXN − concentration of donors; and )( −z
aXN −
concentration of acceptors.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 3. P. 95-102.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
99
Table 2. Depths of the energy levels of native point defects in the CdTe band gap.
E∆ , eV Type of
defects
[8] [20] [12-13] [14] [16-17] [18] [15] [19]
+
iCd – cE −0.02 0 cE −0.33 0*
cE −0.21*
0
from [12]
cE −0.80 –
+2Cd i cE −0.15 cE −0.56 cE −0.20 cE −0.46*
cE −0.56*
cE −0.17*
cE −0.36*
cE −0.20
from [12]
− cE −0.54
cE −0.56
cE −0.61
cE −0.64
υE +0.88
+
TeV − − cE −0.40 − − cE −0.40
from [12]
cE −0.01
cE −1.1
cE −1.2
−
+2
TeV cE −0.035 cE −0.58 cE −0.50 cE −0.71 − cE −0.50
from [12]
cE −0.05
0
−
+
CdTe − − 0 cE −0.34 cE −1.44 0
from [12]
cE −0.60
0
−
+2
CdTe − − cE −0.40 cE −0.59 cE −1.6 cE −0.40
from [12]
cE −1.0 −
−
CdV υE +0.05 υE +0.05 υE +0.20 υE +0.13 υE +0.05
υE +0.20
υE +0.20
from [12]
υE +0.05 −
−2
CdV υE +0.90 υE +0.66 υE +0.80 υE +0.21 υE +0.47 υE +0.80
from [12]
υE +0.47
υE +0.60
υE +0.76
υE +0.18
υE +0.33
υE +0.38
υE +0.49
υE +0.51
υE +0.54
υE +0.60
υE +0.62
υE +0.63
υE +0.65
−
iTe υE +0.15 υE +0.15 – υE +0.67 – υE +0.10 υE –0.70 –
−2Tei – – – υE +0.74 – υE +0.60*
υE +0.70*
υE –1.0 –
The ionization energy of acceptors and the
location of the Fermi level are reckoned from the top of
the valence band, and the ionization energy of donors is
counted off the bottom of the conduction band. As the
factors of spin degeneracy of levels, we take g = 2
for single-charged donors of defects and g = 1/2
for acceptor levels, doubly charged centers, according
to [17].
At present, the spectrum of energy levels of point
defects in the BG of cadmium telluride is not enough
studied, and the identification of the majority of them is
disputable. The newest works on the determination of
the depth of local centers ∆E of a different nature were
systematized in [18]. Only the method of photoinduction
of currents gave more than one hundred fifty deep
energy levels. Most of them are caused just by native
point defects. Some hope to unequivocally determine the
energy E∆ of native defects is given by theoretical
works [12-14], where the depth of located centers was
found from the “first principles”. Unfortunately, these
methods are rather inexact. The results of some works,
where the energy levels of point defects in chalcogenide
are theoretically or experimentally determined, are
systematized in Table 2.
The calculation of the concentration of neutral
defects in CdTe single crystals and thin films was
carried by us with the help of Eqs. (21)-(26) with regard
for the values of the thermodynamic functions of the
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 3. P. 95-102.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
100
creation of defects given in Table 1. We have used the
model which takes into account the most complete
collection of defects in chalcogenide. It includes defects
in the cadmium ( 0
CdV − cadmium vacancy, 0Cd i −
cadmium interstitial atom) and tellurium sublattices
( 0
CdV − tellurium vacancy, 0Te i − tellurium interstitial
atom), as well as the existence of antistructural defects
0
CdTe . The antistructural defect 0
TeCd was not taken into
account, as its existence in a material is improbable [12,
13]. Then we calculated the Fermi level location and the
concentration of ionized defects according to Eqs. (27)-
(31). The type of defects and the positions of energy
levels which were used in the modeling are presented in
Fig. 1.
We compared the results of modeling with the
data obtained with the use of the traditional approach
for a number of models, which have found a wide
application now.
The results of modeling of PDE in single crystals
during annealing at different cadmium pressures are
given in Fig. 2, and Fig. 3 presents the result of
calculations of the Hall charge carrier concentration in
samples.
Figure 3 shows that the different models and sets of
QE constants offered in [2, 6, 9] at high cadmium vapor
pressures give a rather good correlation between the
results of modeling and the experimental value of n. It is
well clear, because the corresponding QE constants were
obtained as a result of optimization of the solution of the
multifactor task just in the range of high PCd. At a lower
cadmium pressure (in a material enriched by tellurium),
the large distinction is observed between the results of
modeling within different models, which requires
making a choice of the most reliable ones.
Fig. 1. Location of defect energy levels in the CdTe band
gap.
10-3 10-2 10-1 100 101 102 103 104 105107
109
1011
1013
1015
1017
1019
Cd2+
i
TS=700 0C
V 2-
Cd
V -
Cd
Cd+
i
Cd 0
i
V 0
Cd
p
n
a
N,cm-3
PCd,Pa
10-3 10-2 10-1 100 101 102 103 104 105
107
109
1011
1013
1015
1017
1019
PCd,Pa
Cd +
i
N,cm-3
TS=700 0C
V 0
Te
V 2+
Te
V +
Te
Cd0
i
Cd2+
i
Te0
i
V 0
Cd
Te2+
Cd
V 2-
Cd
Te-
iV -
Cd
p
n
c
10-3 10-2 10-1 100 101 102 103 104 105
107
109
1011
1013
1015
1017
1019
V+
Te
Cd+
i
Cd2+
i
V0
Te
Te-
i
Te2-
i
V2-
Cd
V-
Cd
Te0
i
Cd0
i
V0
Cd
Te2+
Cd
Te+
Cd Te0
Cd
PCd,Pa
N,cm-3
TS=700 0C
p n
d
10-1 100 101 102 103 104 105
1011
1012
1013
1014
1015
1016
1017
N,cm-3
PCd, Pa
n p
V2-
Cd
V-
Cd
Te-
i
b
TS= 700 0C
Cd2+
i
V2+
Te
Fig. 2. Concentration of native point defects and free charge carriers under different partial pressures of cadmium vapor PCd at
the annealing temperature of CdTe single crystals. Calculations within the traditional approach: а and b correspond to models in
[2] and [6], respectively; c − complete sets of defects [9]. Calculations within the “first principles” approach: d.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 3. P. 95-102.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
101
10-3 10-2 10-1 100 101 102 103 104 105
1013
1014
1015
1016
1017
1018
1019
PCd,Pa
N,cm-3
1
2
3
4
Fig. 3. Hall concentration of free charge carriers at different
partial pressures of cadmium vapor in CdTe single crystals and
the annealing temperature Ts =700 ºС. Calculations within the
traditional approach: 1 and 2 correspond to models in [2] and
[6], respectively; c − complete sets of defects [9]. Calculations
within the “first principles” approach: d. Experimental data are
taken from [8, 16, 21].
In Fig. 4, we present the results of modeling of
PDE in a material from the “first principles” during the
annealing of CdTe single crystals and in thin films on
their fabrication under different technological conditions
in the cases of full equilibrium and quenching [2, 9]. In
addition, we display the results of calculations of a
position of the Fermi level in a material.
It is obvious from Figs. 2 and 4 that the
dominant defects in single crystals at high cadmium
pressure and annealing are doubly charged tellurium
vacancies (Fig. 2, b) or interstitial cadmium atoms
(Fig. 2, c, d) in agreement with the experimental data
[8, 16, 17, 20]. Just such defects can ensure the
dependence of the concentration of free carriers on
the cadmium pressure as 31
Cd~Pn . A type of
predominant defects in the material enriched with
tellurium is determined by the chosen model. As a
rule, they are −
iTe or +
CdTe ( +2
CdTe ).
101 102 103 104 105 1061014
1015
1016
1017
1018
1019 E,eV
Te+
Cd
Te0
Cd
V0
Te
V2+
Te
Cd+
i
Te-
i
V2-
Cd
V-
Cd
V0
Cd
p n
N,cm-3
PCd,Pa
b
µF
10-5 10-4 10-3 10-21014
1015
1016
1017
1018
1019
E,eV
Te2+
Cd
Te+
Cd
Te0
Cd
Te2-
i
Te-
i
Te0
i
V2-
Cd
V-
Cd
V0
Cd
p
n
PCd,Pa
PTe,Pa
0.001541525.5 14.94 0.1542
N,cm-3
c
µF
10-5 10-4 10-3 10-21014
1015
1016
1017
1018
1019
N,cm-3 PTe,Pa
0.001541525.5 14.94 0.1542
Te+
Cd
Te0
Cd
Te0
i
Te-
i
V-
Cd
V2-
Cd
V0
Cd
p
PCd,Pa
E,eV
d
µF
101 102 103 104 105 106
1014
1015
1016
1017
1018
1019
V0
Te
V2+
Te
V+
Te
Cd2+
i
Cd+
i
Cd0
i
V2-
Cd
V-
Cd
V0
Cd
PCd, Pa
np
N,cm-3
µF
Te-
i
Te2-
i
Te0
i
Te+
Cd
Te2+
Cd
Te0
Cd a
E,eV
Fig. 4. Concentration of native point defects and the Fermi level location at different partial pressures of cadmium vapor: a, b –
CdTe single crystal annealing (Ts = 900 ºС), full equilibrium and quenching respectively; с, d – condensation of CdTe thin films –
full equilibrium (Ts = 550 ºС, Te = 600-1000 ºС) and quenching (Ts = 20 ºС, Tе = 600-1000 ºС), respectively.
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0 0.0
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
1.6
1.4
1.2
1.0
0.8
V
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 3. P. 95-102.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
102
3. Conclusions
The results of calculations presented in this work allow
choosing the technological conditions for the fabrication
of single crystals and CdTe thin films with the pre-
determined structure of point defects and electrophysical
properties.
The offered model is the most complete among
possible ones and, with regard for the corresponding
constants of materials, can be used for the modeling of
PDE in any А2В6 compound. Thus, the problem of the
choice of models adequate to the experimental data is
reduced to the determination of the defect creation
energy of uncharged defects and the position of the
energy levels of charged defects.
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