Impact of traps on current-voltage characteristic of ⁺--⁺ diode
A model of ⁺--⁺ diode is analyzed using analytical and numerical methods. First, a phase-plane analysis was conducted, which was aimed at further calculations for low and high injection approximations. A numerical method was used to calculate changes in the field, bias, and concentration throughout...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
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| citation_txt | Impact of traps on current-voltage characteristic of ⁺--⁺ diode / P.M. Kruglenko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2017. — Т. 20, № 2. — С. 210-216. — Бібліогр.: 16 назв. — англ. |
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| description | A model of ⁺--⁺ diode is analyzed using analytical and numerical methods. First, a phase-plane analysis was conducted, which was aimed at further calculations for low and high injection approximations. A numerical method was used to calculate changes in the field, bias, and concentration throughout the diode for different current values. Expected depletion of free-charge carriers near the anode and enrichment near the cathode was observed. Current-voltage characteristics were built for different concentrations of traps in the base. An increasing bias for the same value of current with increasing trap concentration was predicted.
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| first_indexed | 2026-03-21T12:40:38Z |
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| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 2. P. 210-216.
doi: https://doi.org/10.15407/spqeo20.02.210
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
210
PACS 85.30.Kk
Impact of traps on current-voltage characteristic of n+-n-n+ diode
P.M. Kruglenko
V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine,
41, prospect Nauky, 03680 Kyiv, Ukraine,
E-mail: p.kruglenko@gmail.com
Abstract. A model of n+-n-n+ diode is analyzed using analytical and numerical methods.
First, it was conducted a phase-plane analysis, which was aimed at further calculations
for low and high injection approximations. A numerical method was used to calculate
changes of the field, bias and concentration throughout the diode for different current
values. Expected depletion of free-charge carriers near the anode, and enrichment near
the cathode was observed. Current-voltage characteristics were built for different
concentrations of traps in base. Increasing bias for same value of current with increasing
traps concentration was predicted.
Keywords: semiconductor diode, phase-plane, trap influence.
Manuscript received 02.02.17; revised version received 13.04.17; accepted for
publication 14.06.17; published online 18.07.17.
1. Introduction
Semiconductors with defects were a subject of intense
researches as early as 1940s [1]. Later development of
semiconductor physics and technology allowed creating
the close to perfect defect-free materials and devices,
including diodes [2]. However, with increasing ability to
make nano-sized diodes, processes at surfaces of
interfaces became more influential. The presence of
traps in semiconductors contributes to different physical
phenomena. Unlike defect-free semiconductors, which
have quadratic current-bias characteristic [3], diodes
with defects exhibit slow rise of current until critical
voltage is reached, and power-law rise after critical
voltage [4]. The presence of dopants in regions with
traps also has a large effect on currents in semiconductor
devices [5]. Dopants together with the Frenkel effect
also control shape of the current-voltage characteristics
[6]. Another effect that is introduced by traps in
semiconductor devices are noise sources caused by
random trapping and detrapping of charge carriers (see
[7] for bulklike samples and [8] for nanowires/nano-
ribbons). The surface noise can be considerably sup-
pressed due to Coulomb correlations between trapped
and conducting electrons [9].
In this work, we calculate the current-bias
characteristics alongside distributions of the field,
potential and concentrations of free and trapped carriers
in the short diode with dopants and traps. We employ the
phase-plane method [10] that allows to make qualitative
conclusions on transport in the short diode. We include
in the model a finite-length base and infinite contacts
with higher doping concentration. Numerical
calculations were performed using the numerical
methods in general case, but analytical approximations
were also used for low and high injection modes.
2. Steady-state transport model and main equations
The Poisson equation describes the field change in the
base
bNNn
e
x
E
0
0=
d
d
(1)
and in contact
,=
d
d
0
0
cNn
e
x
E
(2)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 2. P. 210-216.
doi: https://doi.org/10.15407/spqeo20.02.210
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
211
where E is the field, n, N are electron density of free
charges and charges captured by traps, Nb, Nc – dopant
concentrations in the base and contacts, x is the
coordinate, e0 – positive electron charge, , 0 are the
relative and vacuum permeabilities.
For the density change, we consider continuity
equations for base with generation and recombination of
free particles
,=div RGi
t
n
(3)
.= RG
t
N
(4)
In the case of contact, generation and
recombination terms are absent:
0,=div i
t
n
(5)
where
0
=
e
j
i
is the charge flow equal to current
density divided by electron charge,
NNnRNG t =,= are the generation and re-
combination terms, Nt is the density of traps, +, – are
coefficients.
For the stationary problem, the time derivatives
disappear, and we obtain the Poisson equations (1), (2)
and constant current density that can be defined using
the drift-diffusion equation
,
d
d
=
0
nE
x
n
D
e
j
(6)
where D is the diffusion coefficient, μ – mobility.
Diffusion can be expressed through mobility
0
=
e
kT
D , where k is the Boltzmann constant, T –
temperature. The density of trapped charges we can get
from (4)
tN
n
n
N
= . (7)
Dimensionless equations look like
bt NN
n
n
n
x
E
=
d
d
(8)
for field in the base,
cNn
x
E
=
d
d
(9)
for field in the contacts and
nEj
x
n
=
d
d
(10)
for the drift-diffusion equation (6) throughout the diode.
The parameter
chN
= describes relation between
generation and recombination processes. Characteristic
parameters are Nch for concentration,
2
0
0
D =
eN
kT
l
ch
–
Debye length for the set concentration,
D
D =
el
kT
E –
Debye field,
D
=
l
NkT
j ch
ch
– for current density.
3. General case: the phase-plane analysis
We transform dimensionless equations (8), (10) into the
field-concentration differential equation
)()(
)(
=
d
d 2
nnEj
NNNnn
n
E bbt
(11)
which we can analyze using calculated separatrix
guidelines:
2
23
2
2
4)(4
=
crt
crt
btbcrt
crt
crt
N
N
kNNNN
jj
Nn
N
j
E
(12)
and such properties as: the null derivative on n = Ncrt
line, infinity derivative at
n
j
E = curve, derivative equal
to
j
Nb at n = 0. Ncrt here is the critical concentration for
Eq. (10) equal to
.
2
4
=
2 btbtb
crt
NNNNN
N (13)
Similar and simpler equations (no traps, hence
terms with Nt and disappear) can be written for
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 2. P. 210-216.
doi: https://doi.org/10.15407/spqeo20.02.210
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
212
equations describing movement of the charged particle
in the contact.
Examples of phase planes are shown in Figs 1, 2
and 3. All of them show relation of field (vertical axis)
and concentration (horizontal axis). Phase planes
shown in Fig. 1 describe all possible field and
concentration relations in the base and contact of
semiconductor, with all possible boundary conditions.
For instance, darker trajectories in the phase plane with
the critical point Ncrt correspond to any concentration-
field relations inside base (trajectories have lower
concentration inside the base, and higher concentration
at interfaces; and field is constantly decreased).
Likewise, darker trajectories in the phase plane with
the critical point Nc correspond to relations in the
contact (maximum of the concentration for particular
trajectory can be taken as starting and ending points,
with the concentration decreasing in cathode, and rising
in anode; and field growing in both contacts). It should
be noted that different intersections of darker
trajectories from those phase planes depict all solutions
to equations (8)-(10) in conditions of equal stronger
doped contacts and weaker doped base. When moving
along the particular line on the phase plane, one can
determine the distance between two points by
integrating the equation (10), for example. Movement
from or to critical point
crt
crt N
j
N , (along any of the
separatrix) gives infinite distance, which can be used to
model infinitely large contacts, as in our case.
Fig. 2 shows few integral paths for the base and
separatrix for the contact. A solution to the system of
equations (8)-(10) should start in the critical point for
contact
c
c N
j
N , , then moving along separatrix one
reaches the field maximum at the left interface, then
moving along the path integral for the base to the right
interface and then again one reaches the critical point in
the contact by using different separatrix.
j
N c r t
j
N c
Ncr t Nc
left
interface
right
interface
Fig. 2. Solutions for both the contact and base are shown on
the same phase plane. Solid line shows separatrices for the
contact, dashed – integral paths for different intercontact
distances; both critical points for the base and contact lie on the
dotted line
n
j
. The top intersection point of separatrix and
integral path represent the left interface of diode, bottom –
right interface.
0
Ncr t Nc
Fig. 3. Phase plane for diode without injection. Dotted line is
the separatrix for the base, dashed – for the contact, and solid
lines represent solutions for the base. Unlike previous Figs 1
and 2, these lines can be built analytically.
j
N c r t
Ncr t
j
N c
Nc
Fig. 1. Phase planes. Solid lines represent separatrix guidelines, dashed – characteristic lines. Four regions where the phase plane
is divided by the separatrix define different behaviour of Eq. (11).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 2. P. 210-216.
doi: https://doi.org/10.15407/spqeo20.02.210
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
213
4. Equilibrium distributions in the unbiased diode
In the absence of current, equations (8)-(10) can be
solved analytically. The field-concentration dependence
can be written as
,loglog2=
min
0
min
0
min00
N
n
N
N
n
NNnE bt
(14)
c
c N
n
NnE 0
00 log12= (15)
for the base and contact, respectively. Nmin defines the
minimum concentration in the base.
A dependence of the concentration on the coordi-
nate can be found from the integral expression
.
)(
d
=
000
0
n
nEn
n
x (16)
The equations give the expected result ( Fig. 3):
symmetrical distribution of charged particles concen-
tration and the potential in diode, as well as the asym-
metrical field distribution.
5. Regime of low injection
We consider a low current in the diode. In this case, we
can expect a solution in the form
*10=
N
j
EEE ,
n = n0 = n1, where E0, n0 denote unbiased solutions,
1*1 , n
N
j
E – corrections for low injection. N* is the
dopant concentration Nc in the case of contact and the
minimum concentration Nmin in the case of base.
Correction for the field is split by two terms for
convenience – it requires E1 term to be equal to zero in
the N* point.
Solving the equations (8)-(10) simplified to the first
order in respect to E1, n1 and current density j, we get
xx
Nn
E
EE
E
EjE
xx
dd
11
=
*
0
0
00
2
0
0
0
01 , (17)
x
Nn
E
E
n
j
E
E
Enn
x
d
11
=
*
0
0
00
0
0
1
001 , (18)
where
b
t N
n
Nn
nE
0
0
00 = , (19)
2
0
000 1=
n
N
EnE t (20)
for the base, and
cNnE 00 = , (21)
000 = EnE (22)
for the contact.
6. Regime of high injection under virtual cathode
approximation
Under the high current, we can neglect the diffusion
term
x
n
d
d
in (10), and equations become solvable in
respect to coordinate x
crt
b
crttcrtb
b
tb
crtbb
N
N
j
E
NNNN
N
N
j
E
NNNN
NNN
j
N
E
x
1ogl
1ogl
=
2
22
2
2
(23)
for the base and
c
cc
N
j
E
N
j
N
E
x 1log=
2
(24)
c
c
c
cc
c
c
N
j
E
EE
N
j
N
EE
xx 1log=
2
(25)
for the cathode and anode, respectively. Nc denotes
another critical point for the equation (11) with the
negative value, Ec, xc are the field and coordinate at the
interface between the base and anode.
Since Eq. (10) without diffusion gives the simple
field-concentration dependence
n
j
E = , all the integral
paths lay on the same line in the phase plane. Because of
it, we can’t use intersections of integral paths correspon-
ding to the base and contact to determine the values of
field and concentration at the interface. Hence, we
should use virtual cathode approximation that puts the
field maximum 0=
d
d
x
U
for x = 0 at the interface bet-
ween the cathode and base. This approximation requires
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 2. P. 210-216.
doi: https://doi.org/10.15407/spqeo20.02.210
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
214
the current density to be greater than not only diffusion
term, but also doping concentration of the contact Nc.
See Fig. 4 for comparison between numerical
results and approximations for high and low injection.
7. General case of arbitrary injection.
Numerical results
To solve equations (8)-(10) in general case, we will use
the Runge–Kutta method. Plotting the integral paths for
the base and contact, then we intersect them to determine
interface values of field and concentration along with
base width. By manipulating the minimal concentration
in the base Nmin, we can change the width of base, and
get it to the predefined value. After calculating basic
relations for the field, concentrations of free and trapped
charged particles, we can calculate and plot the potential
of the diode at various current densities.
As a result, we get the expected shift of the
concentration plot in direction to the anode with
increasing injection, as seen in the inlet of Fig. 5.
The above analysis was done in dimensionless
variables. To apply these results to particular diodes, we
present the normalization parameters in the table below.
Values for InAs at 77 K are not included, since the
calculated mean free path (Lfp) was comparable to
intercontact distance, which does not satisfy our model.
All the values are in SI units, except Lfp, which is
dimensionless. Values for mobility and other
characteristics of materials were taken from different
articles [11-14] and books [15, 16].
Fig. 6 depicts the dependence current-voltage on
the trap concentration in the base. A higher potential for
the same level of injection can be required with
increasing the concentration of traps. This shift to higher
voltages can be explained by trapped injected carriers
that generate push-back voltage until all the traps are
filled, at which point the current sharply rises [6]. Our
model doesn’t accommodate for breakdown field, but if
we take Si as example, its breakdown voltage would lie
near 240 V, and Fig. 6 goes only to 12 V.
T, K
3
1
,
m
N ch Lfp Ld, m
m
V
Ed ,
2
,
m
A
jch Uch, V
Si 300 1020 0.2 4.110–07 6104 105 0.03
Si 77 1020 1.7 2.110–07 3104 6105 0.007
Ge 300 1020 0.4 4.810–07 5104 3105 0.03
Ge 77 1020 2.8 2.410–07 3104 106 0.007
InAs 300 1022 4.9 4.710–08 6105 2109 0.03
GaAs 300 1021 1.1 1.410–07 2105 2107 0.03
GaAs 77 1021 2.0 6.910–08 1105 3107 0.007
GaN 300 1020 3.2 3.610–07 7104 5106 0.03
GaN 77 1020 0.8 1.810–07 4104 6105 0.007
0
0.005
0.01
0.015
0.02
0 50 100 150 200 250 300 350 400
j
U
20
40
60
80
100
120
140
160
180
200
600 800 1000 1200 1400 1600
j
U
Fig. 4. Comparison of approximation and numerical calculation. Solid line denotes the numerical result, dashed line – approxi-
mation for small injection and high injection, respectively.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 2. P. 210-216.
doi: https://doi.org/10.15407/spqeo20.02.210
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
215
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 50 100 150 200 250 300 350 400
j
U
Fig. 6. Current-voltage characteristics. Solid line calculated
according to Gurney–Mott law, dashed – numerical results for
different concentrations of trapped charges. Nt = 0, 0.5, 1, 1.5,
2, 2.5, 3 from shorter dashed lines to the longer ones,
respectively.
8. Conclusion
The introduced model is viable for both analytical and
numerical analysis. The phase-plane analysis gives
general prediction of numerical results for concentration
and field change inside the diode. From these results, we
can see the dependence of current-bias characteristic
from traps concentration in the base, with a linear
characteristic corresponding to the concentration of traps
being smaller than the dopant one, and a power like
characteristic for traps concentrations higher than that of
dopant, which shifts further to higher voltages with
increasing the traps concentration, due to push-back
voltage generated by trapped injected carriers.
References
1. Pekar S.I. Theory of the contact between metal and
dielectric or semiconductor. Zhurnal Eksperiment.
0
10
20
30
40
50
60
70
80
90
100
0 5 10 15 20
0
20
40
60
80
100
19 19.5 20 20.5 21
n
x
35
30
25
20
15
10
5
0
5
10
0 5 10 15 20
E
x
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
0 5 10 15 20
N
x
350
300
250
200
150
100
50
0
50
0 5 10 15 20
U
x
j
0.0001
0.01
0.08
0.26
Fig. 5. Concentrations of free (n) and trapped (N) particles, field (E) and potential (U) through diode, for different current
density. Legend for current density on potential graph applies to all others as well. Left and right interfaces located at the
coordinate values 0 and 20, respectfully. Inside free particles graph, the inset scales up the version of right interface. For
example, for particular case of Si, at room temperature, width of the base would be m108 6 , current density would be scaled
from 10 to
2
31026
m
A
, and maximum potential drop would be 9 V.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 2. P. 210-216.
doi: https://doi.org/10.15407/spqeo20.02.210
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
216
and Teoreticheskoi Fiziki. 1940. 10. P. 341–348 (in
Russian).
2. Weaver J.H. Formation of Defect-Free
Metal/Semiconductor Contacts. Minnesota
University, Minneapolis Department of Chemical
Engineering and Materials Science, 1992.
3. Mott N.F. and Gurney R.W. Electronic Processes
in Ionic Crystals. Clarendon Press, 1940.
4. Smith R.W. and R.A. Space-charge-limited
currents in single crystals of cadmium sulfide.
Phys. Rev. 1955. 97, No. 6. P. 1531–1537.
5. Zhang Yuan and Blom P.W.M. Field-assisted
ionization of molecular doping in conjugated
polymer. Organic Electronics. 2010. 11. P. 1261–
1267.
6. Zhang X.-G. and Pantelides S.T. Theory of space
charge limited currents. Phys. Rev. Lett. 2012. 108,
No. 26. P. 266602.
7. Kogan S. Electronic Noise and Fluctuations in
Solids. Cambridge University Press, 2008.
8. Sydoruk V.A., Vitusevich S.A., Hardtdegen H.
et al. Electric current and noise in long GaN
nanowires in the space-charge limited transport
regime. Fluctuation and Noise Lett. 2017. 16,
No. 1. P. 1750010 (12 p.).
9. Kochelap V.A., Sokolov V.N., Bulashenko O.M.,
and Rubi J.M. Coulomb suppression of surface
noise. Appl. Phys. Lett. 2001. 78, No. 14.
P. 2003–2005.
10. Sokolov V.N. et al. Phase-plane analysis and
classification of transient regimes for high-field
electron transport in nitride semiconductors.
J. Appl. Phys. 2004. 96. P. 6492–6503.
11. Li Sheng S. and Thurber W.R. The dopant density
and temperature dependence of electron mobility
and resistivity in n-type silicon. Solid-State
Electron. 1977. 20. P. 609–616.
12. Fistul V.I., Iglitsyn M.I. and Omelyanovskii E.M.
Mobility of electrons in germanium strongly doped
with arsenic. Fizika tverdogo tela. 1962. 4, No. 4.
P. 784–785 (in Russian).
13. Andrianov D.G. et al. Interaction of carriers with
localized magnetic moments in InSb-Mn and InAs-
Mn. Fizikai tekhnika poluprovodnikov. 1977. 11,
No. 7. P. 738–742 (in Russian).
14. Chin V.W.L., Tansley T.L. and Osotchan T.
Electron mobilities in gallium, indium, and
aluminum nitrides. J. Appl. Phys. 1994. 75. P.
7365–7372.
15. Sze S.M. and Ng Kwok K. Physics of
Semiconductor Devices. John Wiley & Sons, 2006.
16. Rode D.L. Ch. 1: Low-Field Electron Transport.
Semiconductors and Semimetals. 1975. 10. P. 1–89.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 2. P. 210-216.
doi: https://doi.org/10.15407/spqeo20.02.210
PACS 85.30.Kk
Impact of traps on current-voltage characteristic of n+-n-n+ diode
P.M. Kruglenko
V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine,
41, prospect Nauky, 03680 Kyiv, Ukraine,
E-mail: p.kruglenko@gmail.com
Abstract. A model of n+-n-n+ diode is analyzed using analytical and numerical methods. First, it was conducted a phase-plane analysis, which was aimed at further calculations for low and high injection approximations. A numerical method was used to calculate changes of the field, bias and concentration throughout the diode for different current values. Expected depletion of free-charge carriers near the anode, and enrichment near the cathode was observed. Current-voltage characteristics were built for different concentrations of traps in base. Increasing bias for same value of current with increasing traps concentration was predicted.
Keywords: semiconductor diode, phase-plane, trap influence.
Manuscript received 02.02.17; revised version received 13.04.17; accepted for publication 14.06.17; published online 18.07.17.
1. Introduction
Semiconductors with defects were a subject of intense researches as early as 1940s [1]. Later development of semiconductor physics and technology allowed creating the close to perfect defect-free materials and devices, including diodes [2]. However, with increasing ability to make nano-sized diodes, processes at surfaces of interfaces became more influential. The presence of traps in semiconductors contributes to different physical phenomena. Unlike defect-free semiconductors, which have quadratic current-bias characteristic [3], diodes with defects exhibit slow rise of current until critical voltage is reached, and power-law rise after critical voltage [4]. The presence of dopants in regions with traps also has a large effect on currents in semiconductor devices [5]. Dopants together with the Frenkel effect also control shape of the current-voltage characteristics [6]. Another effect that is introduced by traps in semiconductor devices are noise sources caused by random trapping and detrapping of charge carriers (see [7] for bulklike samples and [8] for nanowires/nanoribbons). The surface noise can be considerably suppressed due to Coulomb correlations between trapped and conducting electrons [9].
In this work, we calculate the current-bias characteristics alongside distributions of the field, potential and concentrations of free and trapped carriers in the short diode with dopants and traps. We employ the phase-plane method [10] that allows to make qualitative conclusions on transport in the short diode. We include in the model a finite-length base and infinite contacts with higher doping concentration. Numerical calculations were performed using the numerical methods in general case, but analytical approximations were also used for low and high injection modes.
2. Steady-state transport model and main equations
The Poisson equation describes the field change in the base
(
)
b
N
N
n
e
x
E
-
+
ee
-
0
0
=
d
d
MACROBUTTON GrindEQ.reference.UpdateGrindeqFields (1)
and in contact
(
)
,
=
d
d
0
0
c
N
n
e
x
E
-
ee
-
MACROBUTTON GrindEQ.reference.UpdateGrindeqFields (2)
where E is the field, n, N are electron density of free charges and charges captured by traps, Nb, Nc – dopant concentrations in the base and contacts, x is the coordinate, e0 – positive electron charge, (, (0 are the relative and vacuum permeabilities.
For the density change, we consider continuity equations for base with generation and recombination of free particles
,
=
div
R
G
i
t
n
-
+
¶
¶
r
MACROBUTTON GrindEQ.reference.UpdateGrindeqFields (3)
.
=
R
G
t
N
+
-
¶
¶
MACROBUTTON GrindEQ.reference.UpdateGrindeqFields (4)
In the case of contact, generation and recombination terms are absent:
0,
=
div
i
t
n
r
+
¶
¶
MACROBUTTON GrindEQ.reference.UpdateGrindeqFields (5)
where
0
=
e
j
i
r
r
-
is the charge flow equal to current density divided by electron charge,
(
)
N
N
n
R
N
G
t
-
g
g
-
+
=
,
=
are the generation and recombination terms, Nt is the density of traps, (+, (– are coefficients.
For the stationary problem, the time derivatives disappear, and we obtain the Poisson equations (1), (2) and constant current density that can be defined using the drift-diffusion equation
,
d
d
=
0
nE
x
n
D
e
j
m
+
MACROBUTTON GrindEQ.reference.UpdateGrindeqFields (6)
where D is the diffusion coefficient, μ – mobility.
Diffusion can be expressed through mobility
m
0
=
e
kT
D
, where k is the Boltzmann constant, T – temperature. The density of trapped charges we can get from (4)
t
N
n
n
N
+
g
g
-
+
=
.
MACROBUTTON GrindEQ.reference.UpdateGrindeqFields (7)
Dimensionless equations look like
b
t
N
N
n
n
n
x
E
+
+
k
-
-
=
d
d
MACROBUTTON GrindEQ.reference.UpdateGrindeqFields (8)
for field in the base,
c
N
n
x
E
+
-
=
d
d
MACROBUTTON GrindEQ.reference.UpdateGrindeqFields (9)
for field in the contacts and
nE
j
x
n
-
=
d
d
MACROBUTTON GrindEQ.reference.UpdateGrindeqFields (10)
for the drift-diffusion equation (6) throughout the diode. The parameter
ch
N
-
+
g
g
k
=
describes relation between generation and recombination processes. Characteristic parameters are Nch for concentration,
2
0
0
D
=
e
N
kT
l
ch
ee
– Debye length for the set concentration,
D
D
=
el
kT
E
– Debye field,
D
=
l
N
kT
j
ch
ch
m
– for current density.
3. General case: the phase-plane analysis
We transform dimensionless equations (8), (10) into the field-concentration differential equation
)
(
)
(
)
(
=
d
d
2
n
nE
j
N
N
N
n
n
n
E
b
b
t
+
k
-
k
+
-
k
+
-
-
MACROBUTTON GrindEQ.reference.UpdateGrindeqFields (11)
which we can analyze using calculated separatrix guidelines:
(
)
2
2
3
2
2
4
)
(
4
=
crt
crt
b
t
b
crt
crt
crt
N
N
k
N
N
N
N
j
j
N
n
N
j
E
+
k
+
k
-
-
+
±
-
-
+
+
MACROBUTTON GrindEQ.reference.UpdateGrindeqFields (12)
and such properties as: the null derivative on n = Ncrt line, infinity derivative at
n
j
E
=
curve, derivative equal to
j
N
b
at n = 0. Ncrt here is the critical concentration for Eq. (10) equal to
(
)
.
2
4
=
2
k
+
k
-
-
+
k
-
-
b
t
b
t
b
crt
N
N
N
N
N
N
MACROBUTTON GrindEQ.reference.UpdateGrindeqFields (13)
Similar and simpler equations (no traps, hence terms with Nt and ( disappear) can be written for equations describing movement of the charged particle in the contact.
j
N
c
r
t
N
c
r
t
j
N
c
N
c
jN
crt
N
crt
jNc N
c
Examples of phase planes are shown in Figs 1, 2 and 3. All of them show relation of field (vertical axis) and concentration (horizontal axis). Phase planes shown in Fig. 1 describe all possible field and concentration relations in the base and contact of semiconductor, with all possible boundary conditions. For instance, darker trajectories in the phase plane with the critical point Ncrt correspond to any concentration-field relations inside base (trajectories have lower concentration inside the base, and higher concentration at interfaces; and field is constantly decreased). Likewise, darker trajectories in the phase plane with the critical point Nc correspond to relations in the contact (maximum of the concentration for particular trajectory can be taken as starting and ending points, with the concentration decreasing in cathode, and rising in anode; and field growing in both contacts). It should be noted that different intersections of darker trajectories from those phase planes depict all solutions to equations (8)-(10) in conditions of equal stronger doped contacts and weaker doped base. When moving along the particular line on the phase plane, one can determine the distance between two points by integrating the equation (10), for example. Movement from or to critical point
crt
crt
N
j
N
,
(along any of the separatrix) gives infinite distance, which can be used to model infinitely large contacts, as in our case.
Fig. 2 shows few integral paths for the base and separatrix for the contact. A solution to the system of equations (8)-(10) should start in the critical point for contact
÷
÷
ø
ö
ç
ç
è
æ
c
c
N
j
N
,
, then moving along separatrix one reaches the field maximum at the left interface, then moving along the path integral for the base to the right interface and then again one reaches the critical point in the contact by using different separatrix.
j
N
c
r
t
j
N
c
N
c
r
t
N
c
l
e
f
t
i
n
t
e
r
f
a
c
e
r
i
g
h
t
i
n
t
e
r
f
a
c
e
j
N
crt
jNcN
crt
N
c
leftinterfaceright
interface
Fig. 2. Solutions for both the contact and base are shown on the same phase plane. Solid line shows separatrices for the contact, dashed – integral paths for different intercontact distances; both critical points for the base and contact lie on the dotted line
n
j
. The top intersection point of separatrix and integral path represent the left interface of diode, bottom – right interface.
0
N
c
r
t
N
c
0N
crt
N
c
Fig. 3. Phase plane for diode without injection. Dotted line is the separatrix for the base, dashed – for the contact, and solid lines represent solutions for the base. Unlike previous Figs 1 and 2, these lines can be built analytically.
4. Equilibrium distributions in the unbiased diode
In the absence of current, equations (8)-(10) can be solved analytically. The field-concentration dependence can be written as
,
log
log
2
=
min
0
min
0
min
0
0
÷
÷
ø
ö
ç
ç
è
æ
-
+
k
+
k
+
-
±
N
n
N
N
n
N
N
n
E
b
t
MACROBUTTON GrindEQ.reference.UpdateGrindeqFields (14)
÷
÷
ø
ö
ç
ç
è
æ
÷
÷
ø
ö
ç
ç
è
æ
+
-
±
c
c
N
n
N
n
E
0
0
0
log
1
2
=
MACROBUTTON GrindEQ.reference.UpdateGrindeqFields (15)
for the base and contact, respectively. Nmin defines the minimum concentration in the base.
A dependence of the concentration on the coordinate can be found from the integral expression
.
)
(
d
=
0
0
0
0
ò
-
n
n
E
n
n
x
MACROBUTTON GrindEQ.reference.UpdateGrindeqFields (16)
The equations give the expected result ( Fig. 3): symmetrical distribution of charged particles concentration and the potential in diode, as well as the asymmetrical field distribution.
5. Regime of low injection
We consider a low current in the diode. In this case, we can expect a solution in the form
*
1
0
=
N
j
E
E
E
+
+
, n = n0 = n1, where E0, n0 denote unbiased solutions,
1
*
1
,
n
N
j
E
+
– corrections for low injection. N* is the dopant concentration Nc in the case of contact and the minimum concentration Nmin in the case of base. Correction for the field is split by two terms for convenience – it requires E1 term to be equal to zero in the N* point.
Solving the equations (8)-(10) simplified to the first order in respect to E1, n1 and current density j, we get
x
x
N
n
E
E
E
E
E
j
E
x
x
¢
¢
¢
÷
÷
ø
ö
ç
ç
è
æ
-
¢
¢
¢
¢
¢
-
ò
ò
¢
d
d
1
1
=
*
0
0
0
0
2
0
0
0
0
1
,
MACROBUTTON GrindEQ.reference.UpdateGrindeqFields (17)
x
N
n
E
E
n
j
E
E
E
n
n
x
¢
÷
÷
ø
ö
ç
ç
è
æ
-
¢
¢
+
¢
-
ò
d
1
1
=
*
0
0
0
0
0
0
1
0
0
1
,
MACROBUTTON GrindEQ.reference.UpdateGrindeqFields (18)
where
b
t
N
n
N
n
n
E
+
+
k
-
-
¢
0
0
0
0
=
,
MACROBUTTON GrindEQ.reference.UpdateGrindeqFields (19)
(
)
÷
÷
ø
ö
ç
ç
è
æ
+
k
k
+
¢
¢
2
0
0
0
0
1
=
n
N
E
n
E
t
MACROBUTTON GrindEQ.reference.UpdateGrindeqFields (20)
for the base, and
c
N
n
E
+
-
¢
0
0
=
,
MACROBUTTON GrindEQ.reference.UpdateGrindeqFields (21)
0
0
0
=
E
n
E
¢
¢
MACROBUTTON GrindEQ.reference.UpdateGrindeqFields (22)
for the contact.
6. Regime of high injection under virtual cathode approximation
Under the high current, we can neglect the diffusion term
x
n
d
d
in (10), and equations become solvable in respect to coordinate x
(
)
(
)
(
)
÷
÷
ø
ö
ç
ç
è
æ
k
+
-
k
-
k
-
ç
ç
è
æ
-
÷
÷
ø
ö
ç
ç
è
æ
k
+
-
k
-
k
´
´
-
k
+
crt
b
crt
t
crt
b
b
t
b
crt
b
b
N
N
j
E
N
N
N
N
N
N
j
E
N
N
N
N
N
N
N
j
N
E
x
1
og
l
1
og
l
=
2
2
2
2
2
MACROBUTTON GrindEQ.reference.UpdateGrindeqFields (23)
for the base and
÷
÷
ø
ö
ç
ç
è
æ
-
+
c
c
c
N
j
E
N
j
N
E
x
1
log
=
2
MACROBUTTON GrindEQ.reference.UpdateGrindeqFields (24)
÷
÷
÷
÷
÷
ø
ö
ç
ç
ç
ç
ç
è
æ
-
-
+
+
-
+
c
c
c
c
c
c
c
N
j
E
E
E
N
j
N
E
E
x
x
1
log
=
2
MACROBUTTON GrindEQ.reference.UpdateGrindeqFields (25)
for the cathode and anode, respectively. Nc denotes another critical point for the equation (11) with the negative value, Ec, xc are the field and coordinate at the interface between the base and anode.
Since Eq. (10) without diffusion gives the simple field-concentration dependence
n
j
E
-
=
, all the integral paths lay on the same line in the phase plane. Because of it, we can’t use intersections of integral paths corresponding to the base and contact to determine the values of field and concentration at the interface. Hence, we should use virtual cathode approximation that puts the field maximum
0
=
d
d
x
U
for x = 0 at the interface between the cathode and base. This approximation requires the current density to be greater than not only diffusion term, but also doping concentration of the contact Nc.
0
0
.
0
0
5
0
.
0
1
0
.
0
1
5
0
.
0
2
0
5
0
1
0
0
1
5
0
2
0
0
2
5
0
3
0
0
3
5
0
4
0
0
j
U
2
0
4
0
6
0
8
0
1
0
0
1
2
0
1
4
0
1
6
0
1
8
0
2
0
0
6
0
0
8
0
0
1
0
0
0
1
2
0
0
1
4
0
0
1
6
0
0
j
U
0
0.005
0.010.0150.02050100150200250300350400
jU
20
40
60
80
1001201401601802006008001000120014001600
jU
See Fig. 4 for comparison between numerical results and approximations for high and low injection.
7. General case of arbitrary injection.
Numerical results
To solve equations (8)-(10) in general case, we will use the Runge–Kutta method. Plotting the integral paths for the base and contact, then we intersect them to determine interface values of field and concentration along with base width. By manipulating the minimal concentration in the base Nmin, we can change the width of base, and get it to the predefined value. After calculating basic relations for the field, concentrations of free and trapped charged particles, we can calculate and plot the potential of the diode at various current densities.
As a result, we get the expected shift of the concentration plot in direction to the anode with increasing injection, as seen in the inlet of Fig. 5.
The above analysis was done in dimensionless variables. To apply these results to particular diodes, we present the normalization parameters in the table below. Values for InAs at 77 K are not included, since the calculated mean free path (Lfp) was comparable to intercontact distance, which does not satisfy our model. All the values are in SI units, except Lfp, which is dimensionless. Values for mobility and other characteristics of materials were taken from different articles [11-14] and books [15, 16].
Fig. 6 depicts the dependence current-voltage on the trap concentration in the base. A higher potential for the same level of injection can be required with increasing the concentration of traps. This shift to higher voltages can be explained by trapped injected carriers that generate push-back voltage until all the traps are filled, at which point the current sharply rises [6]. Our model doesn’t accommodate for breakdown field, but if we take Si as example, its breakdown voltage would lie near 240 V, and Fig. 6 goes only to 12 V.
T, K
3
1
,
m
N
ch
Lfp
Ld, m
m
V
E
d
,
2
,
m
A
j
ch
Uch, V
Si
300
1020
0.2
4.1(10–07
6(104
105
0.03
Si
77
1020
1.7
2.1(10–07
3(104
6(105
0.007
Ge
300
1020
0.4
4.8(10–07
5(104
3(105
0.03
Ge
77
1020
2.8
2.4(10–07
3(104
106
0.007
InAs
300
1022
4.9
4.7(10–08
6(105
2(109
0.03
GaAs
300
1021
1.1
1.4(10–07
2(105
2(107
0.03
GaAs
77
1021
2.0
6.9(10–08
1(105
3(107
0.007
GaN
300
1020
3.2
3.6(10–07
7(104
5(106
0.03
GaN
77
1020
0.8
1.8(10–07
4(104
6(105
0.007
0
1
0
2
0
3
0
4
0
5
0
6
0
7
0
8
0
9
0
1
0
0
0
5
1
0
1
5
2
0
0
2
0
4
0
6
0
8
0
1
0
0
1
9
1
9
.
5
2
0
2
0
.
5
2
1
n
x
3
5
3
0
2
5
2
0
1
5
1
0
5
0
5
1
0
0
5
1
0
1
5
2
0
E
x
1
1
.
2
1
.
4
1
.
6
1
.
8
2
2
.
2
2
.
4
2
.
6
2
.
8
3
0
5
1
0
1
5
2
0
N
x
3
5
0
3
0
0
2
5
0
2
0
0
1
5
0
1
0
0
5
0
0
5
0
0
5
1
0
1
5
2
0
U
x
j
0
.
0
0
0
1
0
.
0
1
0
.
0
8
0
.
2
6
0102030405060708090100051015200204060801001919.52020.521nx 35 30 25 20 15 10 5051005101520Ex1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
05101520
N
x
350
300
250
200
150
100
50
0
50
05101520
U
x
j
0.0001
0.01
0.08
0.26
0
0
.
2
0
.
4
0
.
6
0
.
8
1
1
.
2
1
.
4
1
.
6
1
.
8
2
0
5
0
1
0
0
1
5
0
2
0
0
2
5
0
3
0
0
3
5
0
4
0
0
j
U
0
0.2
0.4
0.6
0.8
11.21.41.61.82050100150200250300350400
jU
Fig. 6. Current-voltage characteristics. Solid line calculated according to Gurney–Mott law, dashed – numerical results for different concentrations of trapped charges. Nt = 0, 0.5, 1, 1.5, 2, 2.5, 3 from shorter dashed lines to the longer ones, respectively.
8. Conclusion
The introduced model is viable for both analytical and numerical analysis. The phase-plane analysis gives general prediction of numerical results for concentration and field change inside the diode. From these results, we can see the dependence of current-bias characteristic from traps concentration in the base, with a linear characteristic corresponding to the concentration of traps being smaller than the dopant one, and a power like characteristic for traps concentrations higher than that of dopant, which shifts further to higher voltages with increasing the traps concentration, due to push-back voltage generated by trapped injected carriers.
References
1. Pekar S.I. Theory of the contact between metal and dielectric or semiconductor. Zhurnal Eksperiment. and Teoreticheskoi Fiziki. 1940. 10. P. 341–348 (in Russian).
2. Weaver J.H. Formation of Defect-Free Metal/Semiconductor Contacts. Minnesota University, Minneapolis Department of Chemical Engineering and Materials Science, 1992.
3. Mott N.F. and Gurney R.W. Electronic Processes in Ionic Crystals. Clarendon Press, 1940.
4. Smith R.W. and R.A. Space-charge-limited currents in single crystals of cadmium sulfide. Phys. Rev. 1955. 97, No. 6. P. 1531–1537.
5. Zhang Yuan and Blom P.W.M. Field-assisted ionization of molecular doping in conjugated polymer. Organic Electronics. 2010. 11. P. 1261–1267.
6. Zhang X.-G. and Pantelides S.T. Theory of space charge limited currents. Phys. Rev. Lett. 2012. 108, No. 26. P. 266602.
7. Kogan S. Electronic Noise and Fluctuations in Solids. Cambridge University Press, 2008.
8. Sydoruk V.A., Vitusevich S.A., Hardtdegen H. et al. Electric current and noise in long GaN nanowires in the space-charge limited transport regime. Fluctuation and Noise Lett. 2017. 16, No. 1. P. 1750010 (12 p.).
9. Kochelap V.A., Sokolov V.N., Bulashenko O.M., and Rubi J.M. Coulomb suppression of surface
noise. Appl. Phys. Lett. 2001. 78, No. 14.
P. 2003–2005.
10. Sokolov V.N. et al. Phase-plane analysis and classification of transient regimes for high-field electron transport in nitride semiconductors.
J. Appl. Phys. 2004. 96. P. 6492–6503.
11. Li Sheng S. and Thurber W.R. The dopant density and temperature dependence of electron mobility and resistivity in n-type silicon. Solid-State Electron. 1977. 20. P. 609–616.
12. Fistul V.I., Iglitsyn M.I. and Omelyanovskii E.M. Mobility of electrons in germanium strongly doped with arsenic. Fizika tverdogo tela. 1962. 4, No. 4. P. 784–785 (in Russian).
13. Andrianov D.G. et al. Interaction of carriers with localized magnetic moments in InSb-Mn and InAs-Mn. Fizikai tekhnika poluprovodnikov. 1977. 11, No. 7. P. 738–742 (in Russian).
14. Chin V.W.L., Tansley T.L. and Osotchan T. Electron mobilities in gallium, indium, and aluminum nitrides. J. Appl. Phys. 1994. 75. P. 7365–7372.
15. Sze S.M. and Ng Kwok K. Physics of Semiconductor Devices. John Wiley & Sons, 2006.
16. Rode D.L. Ch. 1: Low-Field Electron Transport. Semiconductors and Semimetals. 1975. 10. P. 1–89.
�
Fig. � SEQ Figure \* ARABIC �1�. Phase planes. Solid lines represent separatrix guidelines, dashed – characteristic lines. Four regions where the phase plane is divided by the separatrix define different behaviour of Eq. (11).
�
Fig. 4. Comparison of approximation and numerical calculation. Solid line denotes the numerical result, dashed line – approxi�mation for small injection and high injection, respectively.
�
Fig. 5. Concentrations of free (n) and trapped (N) particles, field (E) and potential (U) through diode, for different current density. Legend for current density on potential graph applies to all others as well. Left and right interfaces located at the coordinate values 0 and 20, respectfully. Inside free particles graph, the inset scales up the version of right interface. For example, for particular case of Si, at room temperature, width of the base would be �EMBED Equation.3���, current density would be scaled from 10 to �EMBED Equation.3���, and maximum potential drop would be 9 V.
PAGE
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
210
m
10
8
6
-
×
2
3
10
26
m
A
×
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| id | nasplib_isofts_kiev_ua-123456789-214930 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1560-8034 |
| language | English |
| last_indexed | 2026-03-21T12:40:38Z |
| publishDate | 2017 |
| publisher | Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| record_format | dspace |
| spelling | Kruglenko, P.M. 2026-03-04T12:52:08Z 2017 Impact of traps on current-voltage characteristic of ⁺--⁺ diode / P.M. Kruglenko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2017. — Т. 20, № 2. — С. 210-216. — Бібліогр.: 16 назв. — англ. 1560-8034 PACS: 85.30.Kk https://nasplib.isofts.kiev.ua/handle/123456789/214930 https://doi.org/10.15407/spqeo20.02.210 A model of ⁺--⁺ diode is analyzed using analytical and numerical methods. First, a phase-plane analysis was conducted, which was aimed at further calculations for low and high injection approximations. A numerical method was used to calculate changes in the field, bias, and concentration throughout the diode for different current values. Expected depletion of free-charge carriers near the anode and enrichment near the cathode was observed. Current-voltage characteristics were built for different concentrations of traps in the base. An increasing bias for the same value of current with increasing trap concentration was predicted. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Impact of traps on current-voltage characteristic of ⁺--⁺ diode Article published earlier |
| spellingShingle | Impact of traps on current-voltage characteristic of ⁺--⁺ diode Kruglenko, P.M. |
| title | Impact of traps on current-voltage characteristic of ⁺--⁺ diode |
| title_full | Impact of traps on current-voltage characteristic of ⁺--⁺ diode |
| title_fullStr | Impact of traps on current-voltage characteristic of ⁺--⁺ diode |
| title_full_unstemmed | Impact of traps on current-voltage characteristic of ⁺--⁺ diode |
| title_short | Impact of traps on current-voltage characteristic of ⁺--⁺ diode |
| title_sort | impact of traps on current-voltage characteristic of ⁺--⁺ diode |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/214930 |
| work_keys_str_mv | AT kruglenkopm impactoftrapsoncurrentvoltagecharacteristicofdiode |