Theoretical approach to electrodiffusion of shallow donors in semiconductors: I. Stationary limit
Analysis is made for the possibility of redistribution of mobile point defects in a semiconductor after its exposure to the electric field till the stationary conditions in the crystal are reached. Two different ways of applying the voltage are considered: (i) directly to the sample, (ii) to a capac...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
1998
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| Zitieren: | Theoretical approach to electrodiffusion of shallow donors in semiconductors: I. Stationary limit / N.I. Kashirina, V.V. Kislyuk, M.K. Sheinkman // Semiconductor Physics Quantum Electronics & Optoelectronics. — 1998. — Т. 1, № 1. — С. 41-44. — Бібліогр.: 12 назв. — англ. |
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irk-123456789-1146672017-03-12T03:02:11Z Theoretical approach to electrodiffusion of shallow donors in semiconductors: I. Stationary limit Kashirina, N.I. Kislyuk, V.V. Sheinkman, M.K. Analysis is made for the possibility of redistribution of mobile point defects in a semiconductor after its exposure to the electric field till the stationary conditions in the crystal are reached. Two different ways of applying the voltage are considered: (i) directly to the sample, (ii) to a capacitor, with the sample located between the plates. This model is applicable also to the electric field of any nature, whether external or internal, e.g. that arisning at the metal-semiconductor interface. Дається аналіз можливості розподілу рухомих точкових дефектів в напівпровіднику після дії на нього електричного поля до встановлення стаціонарних умов. Розглядаються два способи прикладання напруги: а) безпосередньо до зразка, б) до обкладинок конденсатора, між якими поміщається зразок. Модель також можна застосовувати для електричних полів будь-якої природи . як зовнішніх так і внутрішніх, що виникають, наприклад, на контакті метал-напівпровідник. Приводится анализ возможности распределения точечных дефектов в полупроводнике после воздействия на него электрического поля до установления стационарных условий. Рассматривается два способа приложения напряжения: а) непосредственно к образцу, б) к обкладкам конденсатора, между которыми помещается образец. Модель также применима для электрических полей любой природы как внешних так и внутренних, возникающих, например, на контакте металл-полупроводник. 1998 Article Theoretical approach to electrodiffusion of shallow donors in semiconductors: I. Stationary limit / N.I. Kashirina, V.V. Kislyuk, M.K. Sheinkman // Semiconductor Physics Quantum Electronics & Optoelectronics. — 1998. — Т. 1, № 1. — С. 41-44. — Бібліогр.: 12 назв. — англ. 1560-8034 PACS 61.72; 71.55. http://dspace.nbuv.gov.ua/handle/123456789/114667 539.219.3 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Analysis is made for the possibility of redistribution of mobile point defects in a semiconductor after its exposure to the electric field till the stationary conditions in the crystal are reached. Two different ways of applying the voltage are considered: (i) directly to the sample, (ii) to a capacitor, with the sample located between the plates. This model is applicable also to the electric field of any nature, whether external or internal, e.g. that arisning at the metal-semiconductor interface. |
| format |
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| author |
Kashirina, N.I. Kislyuk, V.V. Sheinkman, M.K. |
| spellingShingle |
Kashirina, N.I. Kislyuk, V.V. Sheinkman, M.K. Theoretical approach to electrodiffusion of shallow donors in semiconductors: I. Stationary limit Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Kashirina, N.I. Kislyuk, V.V. Sheinkman, M.K. |
| author_sort |
Kashirina, N.I. |
| title |
Theoretical approach to electrodiffusion of shallow donors in semiconductors: I. Stationary limit |
| title_short |
Theoretical approach to electrodiffusion of shallow donors in semiconductors: I. Stationary limit |
| title_full |
Theoretical approach to electrodiffusion of shallow donors in semiconductors: I. Stationary limit |
| title_fullStr |
Theoretical approach to electrodiffusion of shallow donors in semiconductors: I. Stationary limit |
| title_full_unstemmed |
Theoretical approach to electrodiffusion of shallow donors in semiconductors: I. Stationary limit |
| title_sort |
theoretical approach to electrodiffusion of shallow donors in semiconductors: i. stationary limit |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
1998 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/114667 |
| citation_txt |
Theoretical approach to electrodiffusion of shallow donors in semiconductors: I. Stationary limit / N.I. Kashirina, V.V. Kislyuk, M.K. Sheinkman // Semiconductor Physics Quantum Electronics & Optoelectronics. — 1998. — Т. 1, № 1. — С. 41-44. — Бібліогр.: 12 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
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| first_indexed |
2025-07-08T07:46:56Z |
| last_indexed |
2025-07-08T07:46:56Z |
| _version_ |
1837064075556683776 |
| fulltext |
41© 1998 ²íñòèòóò ô³çèêè íàï³âïðîâ³äíèê³â ÍÀÍ Óêðà¿íè
Ô³çèêà íàï³âïðîâ³äíèê³â, êâàíòîâà òà îïòîåëåêòðîí³êà. 1998. Ò. 1, ¹ 1. Ñ. 41-44.
Semiconductor Physics, Quantum Electronics & Optoelectronics. 1998. V. 1, N 1. P. 41-44.
ÓÄÊ 539.219.3; PACS 61.72; 71.55.
Theoretical approach to electrodiffusion
of shallow donors in semiconductors:
I. Stationary limit
N. I. Kashirina, V. V. Kislyuk, M. K. Sheinkman
Institute of Semiconductor Physics, NAS Ukraine, 45 prospekt Nauki, Kyiv, 252028, Ukraine
Phone/Fax: +044 265 63 40; E-mail: moishe@photel.semicond.kiev.ua
Abstract. Analysis is made for the possibility of redistribution of mobile point defects in a semiconduc-
tor after its exposure to the electric field till the stationary conditions in the crystal are reached. Two
different ways of applying the voltage are considered: (i) directly to the sample, (ii) to a capacitor, with
the sample located between the plates. This model is applicable also to the electric field of any nature,
whether external or internal, e.g. that arisning at the metal-semiconductor interface.
Keywords: Electrodiffusion, CdS, mobile donors.
Paper received 14.07.98; revised manuscript received 25.08.98; accepted for publication 27.10.98.
1. Introduction
Mobile point defects in semiconductors result in degrada-
tion of semiconductor devices. Mobile point defects in
II�VI compounds are the principal reagent in photochemi-
cal reactions found to affect the photosensitivity [1] and lu-
minescence of laser screens [2], to give rise to conducting
channels [3] and to change the properties of metal-semicon-
ductor junctions [4, 5]. Since semiconductor devices are
based mainly on p-n- or hetero-junctions, there always is an
internal electric field to act on the mobile charged defects.
Manufacturers of very large scale Si solar cells have a prob-
lem with the mobility of drifting interstitials, which is sub-
stantially high even at moderate temperatures [6].
The drift of mobile defects can be utilized to redistrib-
ute them, decreasing thereby their content within some part
of a crystal [7, 8].
Cadmium sulfide is a non-stoichiometric compound with
inhomogeneity diagram shifted to the excess of cadmium at
room temperature [9]. The excess cadmium occupying
interstitials (Cdi) in CdS is a shallow donor (E
c
� E
d
=
= 0.03 eV) with a rather high mobility (activation energy
about 0.4�0.6 eV) at moderate temperatures (300�400 K).
This paper presents a model which describes the final
distribution of defects after exposure to the electric field.
The model assumes a constant charge state of the drifting
ions. This problem was considered for the case of cadmium
sulfide.
2. Theoretical background
One-dimensional diffusion-drift equation was used for the
electron subsystem in conjunction with Poisson�s equation
and Boltzmann distribution of charged Cd+ ions (stationary
conditions).
With D
n
being electron diffusivity, the electron current
is expressed as:
r r r
i en eD n
n n
= − ∇ + ∇µ ϕ , (1)
where ϕ is the electric field potential, and n is the electron
concentration.
The singly charged mobile donors are distributed in ac-
cordance with the Boltzmann law:
N x N
e x
kT
Cd Cd
L( ) exp
( )= −
ϕ
, (2)
where N
Cd
(x) is the concentration of Cd mobile interstitials.
With boundary conditions: N L N
Cd Cd
L( ) = ; ϕ( )L = 0
(cathode); ϕ ϕ( )0 0= (anode).
Equations (1) and (2) have to be complemented with
Poisson�s equation defining the relation between the poten-
tial ϕ( )x and charge density ρ( )x :
d
dx
x
2
2
4ϕ π
ε
ρ= − ( ) , (3)
N. I. Kashirina et al.: Theoretical approach to electrodiffusion...
42 ÔÊÎ, 1(1), 1998
SQO, 1(1), 1998
where ρ( ) ( ( )x e Z N Z N n x
d d a a
= + − + +
N x
Cd
( )) ,
(hereinafter Z
d
= 1; Z
a
= �1) (4)
Writing electron density as a function of the potential
from equation (1), with Boltzmann�s distribution (2) taken
into account for the concentration of the mobile Cd ions, we
obtain the integro-differential equation for the dimensionless
potential ψ ϕ( ) ( ) /x e x kT= :
∆−−+= −−∫ NeNendxe
L
e
eD
i
kT
e
dx
d L
Cd
L
x
Ln
ψψψψ
ε
πψ )()(
2
2
2 14
(5)
where i en D L
L
n0 = / , n n L
L = ( ) .
The diffusivity D
n
and mobility µn are connected by
Einstein�s equation.
Equation (5) can be easily rewritten in terms of the elec-
tric field:
∂
∂
∂
∂
π
ε
π
ε
ψ
2
2
24
2
4
0
E
x
e
kT
E
E
x
e
kT
N e N E
i
kT
e
kT
Cd
L
+ − ×
× + + =−( )∆
(6)
where ∆N N N
d a
= − .
At N
Cd
L = 0 eq. (6) transforms into the well-known equa-
tion for the field distribution in the space charge region [10].
The presence of mobile donors actually expands the space
charge region over Debye�s length, determined as:
L
kT
e n Z
D
k k
k
=
∑
ε
π4 2 2
1
2
,
where Zk is the charge of k
th
component, to the whole length
of the crystal.
Equation (6) can be solved by the successive approxi-
mation method. As a zero-order approximation, we can take
the N
Cd
(x) distribution calculated in the quasi-neutrality ap-
proach. Substituting this zero approximation N
Cd
(x) into
eq. (6), we can analyze the current-voltage dependence in
the first approximation. The convergence of this technique
is determined by L
D
/L ratio.
3. Results and discussion
The right side of eq. (5) is zero in the quasi-neutrality ap-
proach, and thus we get the potential Ψ as an implicit func-
tion in the expression:
x
L
F
F
= −1
0
( )
( )
ψ
ψ ,
where
F e
N
N
Cd
L
( ) ;ψ ψψ= − −− ∆
2
1 . (7)
Depending on the positive or negative charge of the
«background» (corresponding to the sign of ∆N), (7) tends
to behave in essentially different ways. While for ∆N ≥ 0
there is a continuous solution of eq. (7) (At ∆N = 0 it be-
comes identical to the expression used in the theory of bi-
nary electrolytes [11]), this does not occur if the «back-
ground» charge is negative.
In the latter case, the solution has a break, which indi-
cates that the electroneutrality approach is not valid. At any
rate, both cases have similar solutions for Ψ
0
3 (fig. 1).≤
Fig. 1. Redistribution of Cd+ concentration for a system which in-
cludes: (1) � three components (electrons + Cd
i
+ immobile accep-
tors; N
a
/NL
Cd+
= 0.1); (2) � two components; (3) � three components
(electrons + Cd
i
+ immobile donors; N
d
/NL
Cd+
= 0.1).
Fig. 2. Redistribution of Cd+ concentration and voltage
(in terms of dimensionless potential ψ) in the case of posi-
tive «background» for various values of current: 1 � i/i
0
=
= 3; 2 � i/i
0
= 5; 3 � i/i
0
= 10.
0,0 0,2 0,4 0,6 0,8 1,0
0
20
40
60
80
100
0
2
4
6
8
10
ψ
x/L
3
2
1
3
2
1
ψ
0,0 0,2 0,4 0,6 0,8 1,0
0,0
0,5
1,0
1,5
2,0
x/L
ψ
0
=3
1
2
3
N
N
Cd
Cd
=
i
0,00 0,01
0,08
0,12
2
1
3
x/L
cathode
N
N
Cd
Cd
=
N
N
Cd
Cd
=
anode anode catode
N. I. Kashirina et al.: Theoretical approach to electrodiffusion...
43ÔÊÎ, 1(1), 1998
SQO, 1(1), 1998
The result obtained for the system with immovable posi-
tively charged background is the most attractive, since these
conditions allow extending the low Cd+ content zone
(fig. 2, curves 1, 2, 3 correspond to i/i
0
= 3, 5, 10, respec-
tively).
When i i/ 0 value is rather low, the first term in the pa-
rentheses in eq. (5) becomes:
{ }
∂ ψ
∂
π
ε
ψ ψ
2
2
24
x
e
k T
n e N e N
L
C d
L
= ×
× − −− ∆
(8)
which describes the electric field between the plates of a
capacitor. Eq. (8) with ∆N = 0 and boundary conditions:
ψ ψ ψ ψ( ) / ; ( ) /0 2 20 0= = −L transforms into the
Poisson-Boltzman equation:
{ }∂ ψ
∂
π
ε
ψ ψ
2
2
2
04
x
e
kT
n e e= − −
(9)
which is solved analytically for x ≥ 0 at boundary condi-
tions taken as: x → ∝ ; ψ′ → 0, and the other condition
usually determines the value of the field at x = 0. The value
of the field in the sample at i/i
0
~ 0 is negligible at x far from
the edges of the crystal (fig. 3). The potential distribution
within the space charge region (SCR) is determined by the
following integral:
x
L
d
e eD
=
+ −−∫
ψ
ψ ψ
ψ
ψ
2
0 2/
. (10)
A surface charge confined by the external field within a
thin (~L
D
thickness) within surface layer is calculated by
Q eL n
e e d
e e
D= −
+ −
−
−∫0
2
2
1
0 ( )
/ ψ ψ
ψ ψ
ψ
ψ ψ
, (11)
where ψ
1
= ψ (L
D
); e, n
0
are the lectron charge and concen-
tration in the bulk where it equals the donor concentration
N
Cd
0 and can be obtained from the normality condition:
N L N e dxCd Cd
L
= = ∫0
0
ψ
(12)
The right � hand side of eq. (12) contains the total quan-
tity of Cd+ ions redistributed in the sample. The term from
the left side N
Cd
L determines the number of ions before the
electric field was applied, with the uniform initial distribu-
tion of the ions taken into account.
Fig. 4 shows that at the potential ψ
0
= 60�80 the con-
centration of mobile defects drops by 3�4 orders of magni-
tude (for L = 1 cm). Cd ions are confined within a thin layer
~10L
D
(L
D
� Debye�s length, L
D
= 10�5 cm for N
Cd
=
= 1015 cm�3, T = 300 K).
Obviously, Cd+ redistribution caused by the external elec-
tric field (fig. 4) can be formed by the electric field of a
different nature. The capacitance of the space charge layer
near the edge of the crystal is associated with the potential
Ψ
0
/2. This potential may originate from a phase interface.
Fig. 5 demonstrates the values of the bulk concentration of
mobile Cd which can be reached by the action of the electric
field on the metal � CdS interface for various metals. The
values of contact barrier heights Ψ
b
(in kT/e units) were taken
from [12]. Ψ
b
(in the terms used above) corresponds to Ψ
0
/2
in formula (10) and in fig. 4. Cd+ redistribution presented in
fig. 5 is calculated for crystals of length L = 0.2 cm under
the assumption that one face of a crystal is coated with a
metal deposited thereon.
4. Conclusions
The possibility is considered to clean out mobile charged
defects from the bulk of a crystal by means of the electric
field. Calculations are made only for the stationary limit,
which can be taken as a boundary condition for the transient
problem at t→ ∝. Hence, our analysis demonstrates the po-
tentialities (determined by the absolute minimum of the sys-
tem energy) that could be actualized by action of the electric
field. The conditions for «cleaning» are most appropriate at
∆N > 0, while at ∆N < 0 the situation is opposite. The transi-
tion from ∆N > 0 to ∆N < 0 could be reached using band-to-
band illumination of the crystal. Experimentally, this phe-
nomenon was observed in [8], where the crystal illuminated
with UV light was cleaned of mobile defects over the 1/4 of
the crystal length.
The Schottky barrier field was found to be effective for
reducing the concentration of mobile defects. Calculations
were made for various contacting metals.
Fig. 3. Distribution of ion concentration and potential of the elec-
tric field at i/i
0
~ 0 (capacitor).
3,0 2,5 2,0 1,5 1,0 0,5 0,0
10-3
10-2
10-1
100
10
1
10
2
10
-1
100
101
102
103
104
NCd/NCd
0
"CATHODE"
N
Cd
/N
Cd
0
|Ψ |
|Ψ |
(L-x)/LD
N. I. Kashirina et al.: Theoretical approach to electrodiffusion...
44 ÔÊÎ, 1(1), 1998
SQO, 1(1), 1998
109 1010 1011 1012 1013 1014 1015
1010
1012
1014
1016
1018
1020
N=
Cd (initial)
L=0.2 cm
Au (Ψb=40)
Sb (Ψb=28.1)
Ag (Ψb=24.2)
Co (Ψb=20.4)
Sn (Ψb=14.6)
109 1010 1011 1012 1013 1014 1015
1010
1012
1014
1016
1018
1020
N=
Cd (initial)
L=1 cm
Ψ0=50
Ψ0=40
Ψ0=20
Ψ0=60
Ψ0=80
Fig. 4 and 5. Diagrams to determine the anticipated value of the final (stationary) concentration of mobile Cd ions in the bulk (N0
Cd
) for
a certain value of the initial uniform concentration (N=
Cd
) at various magnitudes of the external potential y
0
applied (Fig. 4 for the case
of external electric field) or contact barrier potential yb (Fig. 5 for the case of Schottky barrier potential formed by the deposition of
metals Sn, Co, Ag, Sb, Au [12]). The potential values are presented in kT/e units.
5. Acknowledgments
The authors would like to thank Prof. N. E. Korsunskaya
and Prof. N. B. Lukyanchikova for fruitful discussions and
useful remarks.
References
1. M. K. Sheinkman, N. E. Korsunskaya, Photochemical reactions
in II-VI semiconductors. In: Physics of II-VI compounds, ed. by
A. N. Georgobiany and M. K. Sheinkman (in Russian), Moscow,
Nauka (1986).
2. I. V. Akimov, V. I. Korostelin, I. V. Akimova, V. I. Kozlovskiy,
Yu. V. Korostelin, Proc. of Lebedev Inst., 177, p. 142 (1987).
3. I. A. Drozdova, B. Embergenov, N. E. Korsunskaya, I. V. Markevich,
A. F. Singaevski, FTP (Soviet Phys. - Semicond., in Russian), 29,
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(Lithuanian Col. Pap.), 21, p. 37 (1981).
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FTP, 27, p. 630 (1993).
6. A. Zamouche, T. Heiser, A. Mesli, Appl.Phys.Lett., 66, p. 30 (1995).
7. N. E. Korsunskaya, I. V. Markevich, T. V. Torchinskaya,
M. K. Sheinkman, FTP, 13, p. 435 (1979).
8. V. V. Kislyuk, N. E. Korsunskaya, I. V. Markevich, G. S. Pekar,
M. K. Sheinkman, FTP, 30, p. 884 (1996).
9. V. P. Zlomanov, and A. V. Novoselova, P-T-x Phase Diagrams of Metal-
Chalcogen Systems (in Russian), Nauka Publ., Moscow (1987).
10. V. L. Bonch-Bruyevich, F. D. Kalashnikov, Physics of Semiconduc-
tors (in Russian), Nauka Publ., Moscow (1977).
11. J. S. Newman, Electro-Chemical Systems, Prentice-Hall Inc.,
Englewood Cliffs, NJ (1973).
12. N. M. Forsyth, I. M. Dharmadasa, Z. Sobiesierski, R. H. Williams,
Semicond. Sci. Technol., 4, p. 57 (1989).
N0
Cd
(final) N0
Cd
(final)
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I. ÑÒÀÖ²ÎÍÀÐÍÈÉ ÂÈÏÀÄÎÊ
Í. ². Êàø³ð³íà, Â. Â. ʳñëþê, Ì. Ê. Øåéíêìàí
²íñòèòóò ô³çèêè íàï³âïðîâ³äíèê³â ÍÀÍ Óêðà¿íè
Ðåçþìå. Äàºòüñÿ àíàë³ç ìîæëèâîñò³ ðîçïîä³ëó ðóõîìèõ òî÷êîâèõ äåôåêò³â â íàï³âïðîâ³äíèêó ï³ñëÿ 䳿 íà íüîãî åëåêòðè÷íîãî ïîëÿ äî
âñòàíîâëåííÿ ñòàö³îíàðíèõ óìîâ. Ðîçãëÿäàþòüñÿ äâà ñïîñîáè ïðèêëàäàííÿ íàïðóãè: à) áåçïîñåðåäíüî äî çðàçêà, á) äî îáêëàäèíîê êîíäåíñàòîðà,
ì³æ ÿêèìè ïîì³ùàºòüñÿ çðàçîê. Ìîäåëü òàêîæ ìîæíà çàñòîñîâóâàòè äëÿ åëåêòðè÷íèõ ïîë³â áóäü-ÿêî¿ ïðèðîäè � ÿê çîâí³øí³õ òàê ³ âíóòð³øí³õ,
ùî âèíèêàþòü, íàïðèêëàä, íà êîíòàêò³ ìåòàë-íàï³âïðîâ³äíèê.
ÒÅÎÐÅÒÈ×ÅÑÊÈÉ ÏÎÄÕÎÄ Ê ÅËÅÊÒÐÎÄÈÔÔÓÇÈÈ ÌÅËÊÈÕ ÄÎÍÎÐΠ ÏÎËÓÏÐÎÂÎÄÍÈÊÀÕ.
I. ÑÒÀÖÈÎÍÀÐÍÛÉ ÑËÓ×ÀÉ
Í. È. Êàøèðèíà, Â. Â. Êèñëþê, Ì. Ê. Øåéêìàí
Èíñòèòóò ôèçèêè ïîëóïðîâîäíèêîâ ÍÀÍ Óêðàèíû
Ïðèâîäèòñÿ àíàëèç âîçìîæíîñòè ðàñïðåäåëåíèÿ òî÷å÷íûõ äåôåêòîâ â ïîëóïðîâîäíèêå ïîñëå âîçäåéñòâèÿ íà íåãî ýëåêòðè÷åñêîãî ïîëÿ äî
óñòàíîâëåíèÿ ñòàöèîíàðíûõ óñëîâèé. Ðàññìàòðèâàåòñÿ äâà ñïîñîáà ïðèëîæåíèÿ íàïðÿæåíèÿ: à) íåïîñðåäñòâåííî ê îáðàçöó, á) ê îáêëàäêàì
êîíäåíñàòîðà, ìåæäó êîòîðûìè ïîìåùàåòñÿ îáðàçåö. Ìîäåëü òàêæå ïðèìåíèìà äëÿ ýëåêòðè÷åñêèõ ïîëåé ëþáîé ïðèðîäû êàê âíåøíèõ òàê è
âíóòðåííèõ, âîçíèêàþùèõ, íàïðèìåð, íà êîíòàêòå ìåòàëë-ïîëóïðîâîäíèê.
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