1
Current-Voltage Characteristics for p-i-p Diodes
Angel Mancebo
Under the guidance of Selman Hershfield
Department of Physics, University of Florida, Gainesville, FL 32611
Abstract
We solve numerically the steady-state drift-diffusion equation for a p-i-p diode. The
electric field and carrier density profiles are obtained, as well as the voltage across the
insulating (i) base. We consider samples that are both long and short compared to
the characteristic charge relaxation length in the problem. In all cases the currentvoltage characteristic is linear at low voltages and follows a quadratic power law for
higher voltages. The quadratic dependence is the form expected by the Mott-Gurney
relation.
1
Introduction
Organic semiconductors are under active investigation because of applications such as organic
electronics and solar cells. They have the advantage over conventional semiconductors that
they are low cost to manufacture and can be made flexible. However, they also have some
disadvantages. They have a lower carrier density than conventional semiconductors, and
they typically have lower mobilities.
The lower carrier density means that when an organic semiconductor is connected to a
material like a conventional doped semiconductor or a metal, carriers from those materials
will go into the organic semiconductor and provide extra charge carriers. In many instances
the carriers from the leads become the dominant source of charge carriers. These extra
charge carriers mean that the sample becomes charged and the electric field is not constant
in the sample. This in turn leads to nonlinear current-voltage (I-V) characteristics. The
limit in which all the charge carriers are provided by the leads is called space charge limited
transport. This is the regime that we examine here.
2
Nonlinear current-voltage characteristics are common in organic semiconductors. The
data is often fit using the Mott-Gurney relation or one of its many extensions. The MottGurney law expresses the current, j, in terms of the mobility, µ, the length of the sample,
L, the voltage across the sample, V , and the permittivity, , in the sample (see Appendix A
for the full list of symbols):
j=
V2
9
µ 3 .
8
L
(1)
This law provides a closed-form expression for the space charged current density j in a
semiconductor, stating simply that j ∝ V 2 . Observation of j ∝ V 2 or something similar
is seen as an indication of space charge limited transport. The mobility, µ, is a measure of
how easily charge carriers flow through the sample. It is an important characteristic of each
sample. The Mott-Gurney law can also be used to extract the mobility.
The derivation of the Mott-Gurney law as expressed in Eq. (1) involves two crucial
assumptions.
1. The dominant source of charge carriers comes from the leads.
2. The current is dominated by the drift term - not the diffusion term.
The drift term is related to the motion of the charges in the material under the effect of an
applied electric field. The diffusion term accounts for the tendency of the space charge to
even out charge discrepancies over the sample region. In equilibrium, the two terms may
be non-zero but equal and opposite, hence resulting in zero net current. The conventional
derivation of the Mott-Gurney law above takes the diffusion term to be zero. However, the
diffusion term can not be neglected because the diffusion coefficient and the mobility are
related by the Einstein relation,
D = µkB T /q .
(2)
3
Since the Mott-Gurney law derivation neglects diffusion in the sample, it can be expected
that there is a regime – particularly when there is large charge buildup – where the diffusion term should become significant. One would expect there to be deviations from the
Mott-Gurney law in this regime. Deviations from the Mott-Gurney law are indeed seen experimentally [1]. They are often accounted for by introducing of charge traps into the model
and subsequently fitting is performed to better match the experimental measurements [1],[2].
In addition previous calculations have shown that for very low voltage j ∝ V [3]. The discrepancy in the power-laws for low voltages indicates that the Mott-Gurney is not exact, and
the diffusion term should be included. In this paper we include both the drift and diffusion
terms and solve numerically the differential equations for the charge and current density in a
sample with very low initial carrier density. By doing so we determine the effect of excluding
the diffusion term on the I-V characteristic.
2
Methods
The system studied is illustrated in Fig. (1). There are two hole doped leads labeled as the
p regions on the right and left. The center region is undoped and has no intrinsic carriers.
It is labeled by i for insulating. The system of equations solved is a set of two dimensionless,
non-linear differential equations in one independent variable:
dE
= p − θ(x)
dx
dp
j = pE −
,
dx
(3)
(4)
where p is the carrier density, E is the magnitude of the electric field, j is the current density
and is taken to be constant across the sample. The function θ(x) accounts for background
of the positive charges in the p-layers, valued at θ(x) = 1 in the p-doped leads and θ(x) = 0
in the insulating base. This dimensionless form of the differential equation was introduced
by Ref. [4].
4
Figure 1: Illustration of the p-i-p diode modeled by Eqs. (3) and (4). The dark regions represent
the p-doped semiconductor and the light regions the insulating base.
2.1
Solution within the p regions
After rearranging the derivatives on the left, the system of equations for the p-doped regions
becomes
dE
= p−1
dx
dp
= pE − j .
dx
(5)
(6)
Far away from the p-i junctions the system is uniform and hence
dE
≈ 0
dx
dp
≈ 0.
dx
(7)
(8)
Defining
δE = E − j
(9)
δp = p − 1
(10)
5
and combining with Eqs. (5) and (6) yields
dδE
= δp
dx
dδp
= j δp + δE + δp δE .
dx
(11)
(12)
Substituting Eq. (11) into Eq. (12) gives a second order differential equation,
d2 δE
1 d(δE)2
dδE
−
δE
=
.
−
j
dx2
dx
2 dx
(13)
Since δE is small very far away, Eq. (13) can be approximated to first-order as
d2 δE
dδE
−j
− δE ≈ 0 ,
2
dx
dx
which has solutions of the form eλx and λ2 − jλ − 1 = 0 =⇒ λ = j/2 ±
(14)
p
(j/2)2 + 1. A
general solution to Eq. (13) can be found of the form
δE(x) =
∞
X
cn enλx .
(15)
n=1
A good approximation comes from taking only the first term of the series in Eq. (15) so that
δE = c1 eλx . The coefficient c1 can be eliminated by taking a derivative of Eq. (15)
∞
δE(x) X
=
n λ cn enλx ≈ λc1 eλx
dx
n=1
(16)
dδE/dx
≈ λ.
δE
(17)
and then taking the ratio
6
Undoing the substitutions introduced in Eq. (9) and rearranging gives
E(x) = j +
1 dE
,
λ dx
(18)
which holds for both solutions of λ. Taking E to be continuous across the p-i junctions,
any solution of E(x) within the i region must also satisfy Eq. (18). This is the boundary
condition that we used for the two leads with λ > 0 in the left lead and λ < 0 in the right
lead.
2.2
Solution within the i region
Within the intrinsic region the system of equations is
dE
= p
dx
dp
= pE − j.
dx
(19)
(20)
The solution to Eqs. (19) and (20) can be handled as a boundary-value problem, which
requires two constants, e.g. the initial values of p and E. These two equations can be
combined to form a second order differential equation,
dp
dx
dE d2 E
= E
−
dx
dx2
2
1 dE
d2 E
=
−
,
2 dx
dx2
j = pE −
(21)
which can be solved for one dependent variable, E. Rearranging to have the highest order
term at the left and integrating once yields
dE(x)
1
= γ + E 2 (x) − jx,
dx
2
(22)
7
where γ is an integration constant [4]. It is found by evaluating Eq. (22) at the left side,
1
dE(−L+ /2)
γ = j (−L+ /2) − E 2 (−L+ /2) +
,
2
dx
(23)
where −L+ /2 is the left boundary of the sample region (accounting for discontinuity in p
across the junction). Putting all of the above together, the first order differential equation
we solve is
dE(x)
dE(−L+ /2) 1
2
2
+
=
+
E (x) − E (−L /2) − j(x + L+ /2).
dx
dx
2
(24)
While Eq. (22) could in principle be solved numerically as a boundary-value problem, it
is more useful to employ the shooting method which represents the equation as an initialvalue problem. In the shooting method, a guess is made at the left sample boundary. The
differential equation is then integrated or shot from the left side by providing E(−L+ /2),
eventually reaching a point at the right sample boundary. The positive value of λ calculated
from E(L− /2) using Eq. (18) is compared to the value of λ calculated from λ = j/2 ±
p
(j/2)2 + 1. When the ratio of these two calculations of λ agree to within 10−6 of unity,
the solution has been obtained.
The numerical computation was performed using the ODE solver lsode included in GNU
Octave. To solve Eq. (24), a guess is made for E(−L+ /2). The bisection method was used
to narrow in on numerically satisfactory initial conditions for each value of j and L.
3
Results
By solving numerically the above differential equations, the values of E, p, and V throughout
the sample were obtained. These quantities are useful in understanding the behavior of the
physical system and determining the validity of the results. Figs. (2), (3), (4), and (5) show
p, E, and V for sample lengths L = 1, 2, 4, and 8, respectively. Below we discuss the results
for each sample length before proceeding on to discuss the I-V characteristics.
8
The numerical results for the shortest sample considered, L = 1, are shown in Fig. (2).
In the j = 0 or equilibrium case (red curve) the solutions show a symmetry about the center
of the sample. At the middle of the E curve, the electric field is zero, E = 0, by Gauss’s law,
as the positive charge distribution gives an electric field that points away from the center of
the sample. This creates an electric field that is negative on the left and positive on the right
(Fig. 2). The electric field is nearly linear, especially for lower j, and hence corresponds to
a nearly quadratic potential across the sample at lower j. For a finite current, the carrier
density is still positive, but now the right side is less positive than the left side. The primary
effect of the increase in current is to shift the electric field upwards. The electric field remains
approximately linear, while the voltage deviates from being approximately quadratic.
The results for the L = 2 case are shown in Fig. (3). As in L = 1, the charge carriers
(holes) are injected at the leads, although in the L = 2 case the density of charge carriers is
smaller than in the L = 1 case. The electric field is approximately linear, crossing zero at
the middle, and the voltage is approximately quadratic at lower voltages. For finite current
in Fig. (3) the electric field remains approximately linear but is shifted up with increasing j.
The left side of the sample is now more positive than the right side. However, nonlinearities
are becoming more apparent, specifically the upward bending of the curve. The differences
in the solutions are easier to see in p, the hole density. For high current the charge density
shows the most rapid change near the edges. This is a form of screening – rapid changes
near the edges, slower near the center. This trend will continue at longer lengths.
As shown in Fig. (4) in the case of L = 4, the electric field is again zero at the center
in equilibrium, and the sample is positively charged. With increasing current, the left side
of p becomes more positive and the right side more negative. From the equilibrium nearly
parabolic hole density curve, p seems to shift to the right with increasing j. More rapid
changes are seen at the highest values of j, continuing the trend.
The numerical solutions for the longest samples that we considered, L = 8, are shown in
Fig. (5). Because of instabilities in the numerical solution, we were unable to solve for the
9
higher values of j that were calculated for the shorter lengths.
What does this mean for the current-voltage characteristics, which are what is observed
experimentally? The results for the current voltage characteristics are shown in Fig. (6).
Note that due to the numerical stability we were not able to get to the same current densities
in all cases. There is a clear linear regime for smaller voltages in all cases. As the voltage
increases the power law can be seen to increase from linear. To examine the quadratic
behavior more closely, the current density is plotted against the square of the voltage in
Fig. (7), which would show a straight line if j ∝ V 2 . In each sample length, above a
transition regime a clear quadratic regime extending into the higher voltages is shown.
Fig. (8) shows a comparison between the drift and diffusion terms across the sample. In
the case of L = 1, the drift and diffusion are fairly linear across the sample. In the case
of L = 2 similar considerations can be made. For L = 2 the drift term mostly dominates
the diffusion term, but nonlinearities can be seen in the behavior, especially near the edges
of the sample where each term can be seen to flare out. The nonlinearity becomes more
significant for longer sample lengths. In the case of L = 4 the flaring at the sample edges
of the diffusion term approaches the order of magnitude of the maximum (positive) and
minimum (negative) value of the drift term. Further flaring is noted for L = 8, where for
both drift and diffusion terms is much larger at the edges than in the middle of the sample.
At the edges the drift and diffusion terms are approximately the same order of magnitude
for all current densities.
10
L = 1
L = 2
0.85
j =
0.8
0.9
0
j =
0.8
j =0.25
j = 0.5
j =
0.7
1
j =0.9
j =1.25
0.6
j =1.2
0.5
j =1.5
0.65
0.4
0.6
2
-0.5
0.5
0.3
3
2.5
2
1.5
1
0.5
0
-0.5
1
0
0
1.5
E
E
1
0.5
0
j =0.6
0.7
j =0.75
p
p
0.75
0
j =0.3
-1
V
V
-0.5
-2
-1
-3
-1.5
-4
-2
-5
-0.4-0.2 0
x
0.2 0.4
-1
-0.5
0
0.5
1
x
Figure 2: p, E, and V for L = 1. Colors Figure 3: p, E, and V for L = 2. Colors
indicate distinct values of j.
indicate distinct values of j.
11
L = 4
j =
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1.5
0
j =0.1
j =0.2
j =0.3
j =0.4
p
p
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
2
L = 8
j =0.5
1.5
j =0.03
j =0.06
j =0.09
j =0.12
0.5
E
0.5
0
0
-0.5
-0.5
-1
1
-1
2
1
0
-1
-2
-3
-4
-5
0
-1
-2
V
V
0
1
1
E
j =
-3
-4
-5
-2-1.5-1-0.50 0.5 1 1.5 2
-4 -3 -2 -1 0 1 2 3 4
x
x
Figure 4: p, E, and V for L = 4. Colors Figure 5: p, E, and V for L = 8. Colors
indicate distinct values of j.
indicate distinct values of j.
12
L = 1
L = 2
1.4
2.5
1.2
2
1
1.5
j
j
0.8
0.6
1
0.4
0.5
0.2
0
0
0
0.5
1
V
L = 4
1.5
2
0
1.4
0.12
1.2
0.1
1
1
2
3
V
L = 8
4
5
6
2
V
3
4
5
0.08
j
j
0.8
0.06
0.6
0.04
0.4
0.02
0.2
0
0
-2
0
2
4
V
6
8
10
-1
0
1
Figure 6: j vs. V for L = 1,2,4, and 8. The relation is fairly linear for the lower voltages.
4
Conclusion
We solved numerically the current through a p-i-p diode in the space charge regime including
both the drift and diffusion terms. We found a regime at low voltage where current is linear
in voltage. The range of this region depends on the sample length, with larger linear regimes
for longer samples. The current density increases faster than linear at higher voltages and
approaches a quadratic dependence on the voltage. The drift and diffusion terms in the
current density have the same magnitude in shorter samples, and in the longest samples the
drift term is larger in the middle of the sample.
According to our results, small voltages, a j ∝ V dependence is expected. At higher
13
L = 1
L = 2
1.4
2.5
1.2
2
1
1.5
j
j
0.8
0.6
1
0.4
0.5
0.2
0
0
0
0.5
1
1.5
2
2.5
3
0
5
10
15
V2
L = 4
20
25
30
35
V2
L = 8
1.4
0.12
1.2
0.1
1
0.08
j
j
0.8
0.06
0.6
0.04
0.4
0.02
0.2
0
0
0
10
20
30
40
50
60
70
80
V2
0
5
10
15
20
25
V2
Figure 7: j vs. V 2 for L = 1,2,4, and 8. The linear regions confirm the V 2 dependence for large
V.
voltages, a j ∼ V 2 relation was observed, in agreement with the power law predicted by the
Mott-Gurney law which states that j ∝ V 2 .
Acknowledgments
I am thankful to Dr. Selman Hershfield for his contributions to the work and for overseeing
my portion, as well as for his helpful suggestions for the content of the paper. I am also
thankful to the National Science Foundation for supporting this work by the UF Materials
Physics REU Program through NSF grant DMR-1156737.
14
L = 1
L = 2
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-0.4
-0.2
0
0.2
0.4
-1
-0.5
0
x
0.5
1
x
L = 4
L = 8
0.6
0.4
0.3
0.4
0.2
0.1
0.2
0
0
-0.1
-0.2
-0.2
-0.3
-0.4
-0.4
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-4
-3
x
-2
-1
0
1
2
3
4
x
Figure 8: The calculated drift and diffusion terms across the sample. The sum of each pair of
same-colored solid and dashed lines is the current density. Colors indicate distinct values of j. The
color convention is the same as in Figs. (2), (3), (4), and (5) for each respective length.
A
D
E
j
kB
p
L
q
T
V
µ
θ(x)
List of Symbols
diffusion constant
dimensionless electric field
dimensionless current density
Boltzmann’s constant
dimensionless hole density
dimensionless insulating base length
carrier charge
temperature
dimensionless potential across the insulating base
dielectric permittivity
electric mobility
Heaviside function
15
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