Concerning the depletion width of a radial - junction and its influence on electrical properties of the diode
Dependences of the depletion widths in a radial core-shell - diode on the radius of the metallurgical boundary of the - junction have been studied theoretically in detail. While the depletion width of the core increases with decreasing radius, the depletion width of the shell, on the contrary, decre...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2017
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| Cite this: | Concerning the depletion width of a radial - junction and its influence on electrical properties of the diode / V.L. Borblik // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2017. — Т. 20, № 2. — С. 168-172. — Бібліогр.: 11 назв. — англ. |
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| citation_txt | Concerning the depletion width of a radial - junction and its influence on electrical properties of the diode / V.L. Borblik // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2017. — Т. 20, № 2. — С. 168-172. — Бібліогр.: 11 назв. — англ. |
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| description | Dependences of the depletion widths in a radial core-shell - diode on the radius of the metallurgical boundary of the - junction have been studied theoretically in detail. While the depletion width of the core increases with decreasing radius, the depletion width of the shell, on the contrary, decreases. This is the consequence of the cylindrical symmetry of the structure. The total depletion width of the - junction can both increase and decrease depending on the doping levels of the core and shell. A number of cases are presented where the dependence of depletion width of the - junction on its curvature influences the diode current-voltage characteristics.
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Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 2. P. 168-172.
doi: https://doi.org/10.15407/spqeo20.02.168
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
168
PACS 73.63.-b, 85.30.Kk
Concerning the depletion width of a radial p-n junction
and its influence on electrical properties of the diode
V.L. Borblik
V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine,
41, prospect Nauky, 03680 Kyiv, Ukraine
E-mail: borblik@isp.kiev.ua
Abstract. Dependences of the depletion widths in a radial core-shell p-n diode on the
radius of metallurgical boundary of the p-n junction have been studied theoretically in
detail. While the depletion width of the core increases with decreasing the radius, the
depletion width of the shell, on the contrary, decreases. This is the consequence of
cylindrical symmetry of the structure. And the total depletion width of the p-n junction
can both increase and decrease depending on doping levels of the core and shell.
A number of cases is presented where the dependence of depletion width of the p-n
junction on its curvature influences on the diode current-voltage characteristics.
Keywords: nanostructures, core-shell nanowire, radial p-n junction, depletion width.
Manuscript received 26.01.17; revised version received 05.04.17; accepted for
publication 14.06.17; published online 18.07.17.
1. Introduction
It has been established that an interface curvature of
doped nano-dimensional semiconductor structures
enlarges their depletion length. For the samples of
cylindrical symmetry, it has been shown in paper [1],
and for the case of spherical symmetry – in the papers
[2, 3]. In those papers, it was talked of depleting toward
the interior of the sample.
A p-n junction is the interface where depletion
occurs on both its sides. This problem (with regard to a
radial core-shell p-n diode) has been considered in Ref. 4
where authors have developed a general electrostatic
theory for these structures and have marked out four
types of electric structure for radial p-n junction diode
depending on the set of parameters. However, any
explicit dependence of the depletion widths for both
sides of the p-n junction on its radius has been not
established. Partially this has made in the work [5] but
only for the depletion width of the core which (likely to
the case of semi-limited structures [1-3]) increases with
decreasing radius of metallurgical boundary of the p-n
junction. Behavior of the depletion width of the
nanowire shell as well as of the whole depletion width
has remained not studied.
Nothing of this has been also reported in the papers
[6-8] devoted to numerical calculations of radial core-
shell p-n junction solar cells. Meanwhile, the
dependence of depletion width for the nanowire shell on
radius of the p-n junction proves to be completely
different. And variation of the total depletion width
versus p-n junction radius proves to be non-trivial.
2. Calculation of the depletion widths
Let us consider the case of partially depleted p-core and
n-shell (Fig. 1). We will proceed from the known system
of 2 equations [4, 5], which allow us to determine the
depletion widths in the core pp rrw −= 0 and in the
shell 0rrw nn −= , where r0 is the core radius, rp is the
depletion region boundary in the core, rn is the depletion
region boundary in the shell, and rd is external radius of
the nanowire.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 2. P. 168-172.
doi: https://doi.org/10.15407/spqeo20.02.168
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
169
0 rn rd
p
n
rp r0
Fig. 1. Schematic view of the radial p-n structure.
From matching the electric fields in the point r0, we
have
( ) ( )2
0
222
0 rrNrrN nDpA −=− (1)
(NA and ND are the concentrations of acceptors and
donors, respectively) and matching the potentials in the
same point gives
bi
n
nD
p
pA
s
V
r
rrN
r
r
rNq
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+
ε 0
2
0
2 lnln
2
, (2)
where q is the electron charge, εs – dielectric constant of
the semiconductor, Vbi – built-in potential of the p-n
junction [9]
2ln
i
DA
bi
n
NN
q
kTV = , (3)
k is the Boltzmann constant, T – temperature, ni –concen-
tration of intrinsic carriers. Expressing rn in terms of rp
( ) ADpn NNrrrr 22
0
2
0 −+= , (4)
one can obtain the following transcendental equation
in rp:
.0
2
ln
11ln11
2
002
0
2
2
0
2
2
0
2
=
ε
−+
+
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
−+
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
−+
rqN
V
r
r
r
r
N
N
r
r
N
N
r
r
N
N
D
sbipp
D
A
p
D
Ap
D
A
(5)
For numerical solution of Eq. (5), the parameters of
silicon at room temperature were chosen as follows:
εs = 12ε0 (ε0 is the permittivity of free space), ni =
6.3·109 cm–3. The calculation results are presented in
Figs. 2a-c. Fig. 2a corresponds to the case when the core
is doped higher than the shell (NA =5·1018 cm–3,
ND = 5·1017 cm–3), Fig. 2b concerns the opposite case
(NA = 5·1017 cm–3, ND = 5·1018 cm–3), and Fig. 2c
represents the results for the case of equal doping levels
(NA = ND).
0
100
200
300
400
500
-10 0 10 20 30 40 50
wwn-wp
Depletion widths, nm
r 0
,
nm
NA=5x1018cm-3 , ND=5x1017cm-3
a
а
0
100
200
300
400
500
-60 -40 -20 0 20 40 60
NA=5x1017cm-3 , ND=5x1018cm-3
wwn
Depletion widths, nm
r 0
,
nm
-wp
b
0
100
200
300
400
500
-60 -40 -20 0 20 40 60 80
ww wnwn-wp
Depletion widths. nm
r 0
,
nm
NA= ND
5x1018 cm-3
5x1017 cm-3
-wp
c
Fig. 2. Dependence of the depletion widths in the core wp, in
the shell wn, and the total depletion width w on radius of the p-
n junction at NA >> ND (a), NA << ND (b) and NA = ND (c)
(abscissa equal to zero corresponds to metallurgical boundary
of the p-n junction).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 2. P. 168-172.
doi: https://doi.org/10.15407/spqeo20.02.168
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
170
As it follows from these figures, in all three cases
the depletion width of the core increases with decreasing
its radius, meanwhile, the depletion width of the shell,
on the contrary, decreases. As for the whole depletion
width of the p-n junction np www += , it can both
increase and decrease and even be nearly independent of
the p-n junction radius at equal and sufficiently high
doping levels of both its sides.
Opposite character of dependencies on the p-n
junction radius of the depletion widths for the core and
shell is a consequence of radial falling that is
characteristic feature for solutions of differential
equations in the cylindrical (as well as spherical)
coordinate system. In particular, this fact has shown
itself in the paper [10] where solution of the
discontinuity equation in the core-shell nanowire
demonstrates an accelerated decay of the non-
equilibrium carrier concentration in the shell and its
slowed down decay in the core.
Here, analogous situation is available. The built-in
electric field of the p-n junction, which is maximal at its
metallurgical boundary, decreases in direction of the
shell faster than in direction of the core. It is for this
reason that at equal doping levels in p- and n-sides, the
region of electric field in the shell proves to be always
shorter than in the core (as it has been found in [5]).
Note also that the “mysterious” increase in
depletion length in semi-limited semiconductor
nanostructures with cylindrical and spherical geometry
(as in [1-3]) has the same nature.
3. The current-voltage characteristics
Under biasing the diode by the voltage U, the total
depletion width changes its dimensions: it narrows under
the forward voltage and expands under the reverse
voltage. This means that the points where injection of
nonequilibrium carriers takes place, i.e. pr and nr , drift
to the region of different curvature of the p-n junction.
And this fact, as it has been shown in [10], varies the
current density at the expense of the curvature
additionally to the applied voltage.
In accordance with [10], the diode current density
from the core to the shell is
where ( )( )1exp(00 −Δτ= kTqUxpDqJ nppp ,
pnn Lrx = , pdd Lrx = , ppp DL τ= , Dp and τp are
the diffusion coefficient and the lifetime of non-
equilibrium holes in n-shell, I0 and K0 are modified
Bessel’s functions of the 1st and 2nd kinds, respectively,
)(0 nxpΔ is the equilibrium hole concentration at the
depletion region boundary in the n-shell (emitter of
holes), pp DSLS =* – dimensionless surface recom-
bination velocity at the external contact to the shell.
In the case of planar diode, the corresponding
current density takes the form
( ) ( ) ( )
( ) ( ) ( )ndnd
ndnd
xxxx
xxxx
p
pl
np eSeS
eSeSJJ
−−−
−−−
− −++
−−+
=+ **
**
0
11
11 . (7)
The ratio pl
npnp
JJ
−− ++ at the same parameters
xn, xd, and S* may be called as the enhancement
coefficient for the current density at the expense of the
diode curvature.
Under the forward bias, when the diode current
varies exponentially with the applied voltage, the
influence of this factor reduces to practically equal
current rise in all the range of applied voltages. But
under the reverse bias, variation of the enhancement
coefficient at the expense of the varying curvature can
result in qualitatively new effects.
Fig. 3 represents the reverse current-voltage
characteristics of the radial diode (at S*= 0 and S*= ∞) in
comparison with those of planar diode at the same
values of parameters. It is the variation of the
enhancement coefficients, connected with the curvature
variation (shown in the inserts), that results in decreasing
current density with the reverse bias.
Fig. 4 presents calculation results for another case,
when the boundary of depletion region in the shell
comes nearer and nearer to the external contact rd with
increasing the reverse voltage. In the radial diode, this
approach occurs at appreciably higher voltage values
than in the planar diode with the same parameter values
due to significant difference in the depletion region
widths.
( ) ( ) ( )[ ] ( ) ( ) ( )[ ]
( ) ( ) ( )[ ] ( ) ( ) ( )[ ]ddnddn
ddnddn
pnp
xISxIxKxKSxKxI
xKSxKxIxISxIxK
JJ
0
*
100
*
10
0
*
110
*
11
0
++−
−−+
=
−+
(6)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 2. P. 168-172.
doi: https://doi.org/10.15407/spqeo20.02.168
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
171
-1.0 -0.8 -0.6 -0.4 -0.2 0.0
-4
-3
-2
-1
0
-1.0 -0.8 -0.6 -0.4 -0.2 0.0
5.0
5.1
5.2
5.3
5.4
5.5
5.6
J n
/
J
p
l
n
U, V
planar diode
J n
/
J n
0
U, V
radial diode
S*=0
NA=1019 cm-3, ND=1018 cm-3, r0=30 nm, rd=1000 nm
Lp=1000 nm
a
-1.0 -0.8 -0.6 -0.4 -0.2 0.0
-7
-6
-5
-4
-3
-2
-1
0
-1.0 -0.8 -0.6 -0.4 -0.2 0.0
4.2
4.3
4.4
4.5
4.6
4.7
J n
/
J
p
l
n
U, V
S*=inf
Lp=1000 nm
radial diode
planar diode
NA=1019 cm-3, ND=1018 cm-3, r0=30 nm, rd=1000 nm
J n
/
J n
0
U, V
b
a b
Fig. 3. Reverse current-voltage characteristics of the radial diode at S* = 0 (a) and S* = ∞ (b) under conditions when the curvature
varies with applied voltage, in comparison with those of planar diode at the same values of the parameters; the inserts shows
dependences of the enhancement coefficients for the current density on the voltage.
-1.0 -0.8 -0.6 -0.4 -0.2 0.0
-0.04
-0.03
-0.02
-0.01
0.00
Lp=1000 nm
NA=1018 cm-3, ND=1017 cm-3, r0=120 nm, rd=250 nm
dn rw ≈
d
pl
n rw ≈
J n
/
J n
0
U, V
S*=0
a
radial diode
planar diode
-1.0 -0.8 -0.6 -0.4 -0.2 0.0
-500
-400
-300
-200
-100
0
planar diode
dn rw ≈
d
pl
n rw ≈
Lp=1000 nm
NA=1018 cm-3, ND=1017 cm-3, r0=120 nm, rd=250 nm
J n
/
J n
0
U, V
S*=inf
b
radial diode
a b
Fig. 4. Reverse current-voltage characteristics of the radial diode at S* = 0 (a) and S* = ∞ (b) under conditions when the depletion
region almost approaches the external boundary of the nanowire, in comparison with those of planar diode at the same values of
the parameters.
4. Conclusions
Thus, in radial diodes dependences of the depletion
widths in the core and shell on the p-n junction radius
have opposite character due to cylindrical symmetry of
the nanowire. And dependence of the total depletion
width on the p-n junction radius proves to be ambiguous
and depending on the doping levels of the core and shell.
Furthermore, the p-n junction curvature can result in not
only quantitative but also qualitative effects in the diode
current-voltage characteristics.
It is worth to note also that since under conditions
of high doping, the total depletion width of the radial
diode depends on the p-n junction curvature rather
weakly, inter-band tunneling probability remains nearly
the same as in planar diode. This means that the excess
tunnel current remains practically the same. For the
same reason, the p-n junction curvature does not
influence essentially on performance of tunnel core-shell
diodes. In particular, this fact is corroborated by
successful fabrication of a tunnel GaAs core-shell
diode [11].
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 2. P. 168-172.
doi: https://doi.org/10.15407/spqeo20.02.168
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
172
Acknowledgement
This work was supported by the National Academy of
Sciences of Ukraine [project 2.2.6.34].
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semiconductor nanosphere. Semicond. Sci. Technol.
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3. Borblik V.L. Depletion length in semiconductor
nanostructures with spherical symmetry. Solid-
State Electron. 2015. 114. P. 171–173.
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radial p-n junction nanowires. J. Appl. Phys. 2013.
114, No. 7. P. 074317.
5. Petrosyan S., Yesayan A., Nersesyan S. Theory of
nanowire radial p-n-junction. World Acad. Sci.
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Analytical modeling of the radial p-n junction
nanowire solar cells. J. Appl. Phys. 2014. 116, No.
2. P. 024308.
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Numerical simulation and a parametric study of
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9. Sze S.M., Ng K.K. Physics of Semiconductor
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10. Borblik V. Effect of circular p-n junction curvature
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|
| id | nasplib_isofts_kiev_ua-123456789-214938 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1560-8034 |
| language | English |
| last_indexed | 2026-03-18T12:56:45Z |
| publishDate | 2017 |
| publisher | Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| record_format | dspace |
| spelling | Borblik, V.L. 2026-03-04T12:54:47Z 2017 Concerning the depletion width of a radial - junction and its influence on electrical properties of the diode / V.L. Borblik // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2017. — Т. 20, № 2. — С. 168-172. — Бібліогр.: 11 назв. — англ. 1560-8034 PACS: 73.63.-b, 85.30.Kk https://nasplib.isofts.kiev.ua/handle/123456789/214938 https://doi.org/10.15407/spqeo20.02.168 Dependences of the depletion widths in a radial core-shell - diode on the radius of the metallurgical boundary of the - junction have been studied theoretically in detail. While the depletion width of the core increases with decreasing radius, the depletion width of the shell, on the contrary, decreases. This is the consequence of the cylindrical symmetry of the structure. The total depletion width of the - junction can both increase and decrease depending on the doping levels of the core and shell. A number of cases are presented where the dependence of depletion width of the - junction on its curvature influences the diode current-voltage characteristics. This work was supported by the National Academy of Sciences of Ukraine [project 2.2.6.34]. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Concerning the depletion width of a radial - junction and its influence on electrical properties of the diode Article published earlier |
| spellingShingle | Concerning the depletion width of a radial - junction and its influence on electrical properties of the diode Borblik, V.L. |
| title | Concerning the depletion width of a radial - junction and its influence on electrical properties of the diode |
| title_full | Concerning the depletion width of a radial - junction and its influence on electrical properties of the diode |
| title_fullStr | Concerning the depletion width of a radial - junction and its influence on electrical properties of the diode |
| title_full_unstemmed | Concerning the depletion width of a radial - junction and its influence on electrical properties of the diode |
| title_short | Concerning the depletion width of a radial - junction and its influence on electrical properties of the diode |
| title_sort | concerning the depletion width of a radial - junction and its influence on electrical properties of the diode |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/214938 |
| work_keys_str_mv | AT borblikvl concerningthedepletionwidthofaradialjunctionanditsinfluenceonelectricalpropertiesofthediode |