Impedance model of two-carrier space-charge limited current in
organic light-emitting diodes.
Angeles Pitarch*, Germà Garcia-Belmonte, Juan Bisquert,
Departament de Ciències Experimentals, Universitat Jaume I, 12080 Castelló, Spain.
ABSTRACT
Impedance model of one-carrier space-charge limited current (SCLC) has often been applied to explain some
experimental features in organic light-emitting diodes (OLED). However double injection current occurs in working
devices and the impedance model has not been studied so far. We analyze the problem of double injection SCL current in
the limit of infinite recombination. In order to obtain the ac response of a biased OLED we solve the equations for time
dependent double-injection of space-charge-limited currents. We give an analytical expression for impedance as a
function of frequency. Calculations predict values for the static capacitance C(ω→0) similar to those encountered in case
of one-carrier SCLC, in which C/Cg=0.75 (being Cg the geometric capacitance), but shifted to higher frequencies. We
give the equivalent circuits representing the limits at low frequencies. This model will help to understand the behavior of
two carrier devices.
Keywords: Impedance; Capacitance; Modeling; Carrier distribution; Charge transport.
1. INTRODUCTION
Since the first report of electroluminiscence in polymers, the study and understanding of optical and electronic properties
of organic light-emitting diodes (OLEDs) have received considerable attention. Their potential for low cost
manufacturing and mechanical flexibility makes these materials promising for many display applications. In order to
optimize the device efficiency of OLEDs a balanced charge distribution is required. The calculation of carrier
distributions inside the sample results complex by the presence of both electrons and holes in working devices, and
consequently recombination and neutralization of carriers must be regarded.
The device operation is determined by three processes: charge injection, charge transport and recombination. The
dominant effect in holes and electron currents has been long discussed. Some authors proposed that hole current is
injection limited. However Blom demonstrated that hole and electron currents are space charge limited currents,2 SCLC,
in PPV. The same author discussed the crossover from SCLC to recombination-limited transport depending on the
electric field, being SCLC at low bias and recombination limited at high bias.3
Impedance spectroscopy is a powerful tool to investigate the charge transport and relaxation processes in solid state
devices. By means of this technique we can discern the different processes which occur on different time scales, and get
extended information about the carrier and field distributions inside the OLED. Several studies of impedance
spectroscopy where interpreted with one-carrier SCLC impedance models to explain some features in OLEDs.1,3-5
However to the best of our knowledge frequency-dependent model with double injection current has not been considered
previously.
Here we derive and discuss the impedance model of double injection SCLC. The assumption of large recombination
permits to find analytical solution for both steady-state and ac response. We give an equivalent circuit explaining the
impedance spectra at the low frequency limit for two-carrier devices, and compare it with the case of one-carrier device.
In the next section we revise the characteristics of both dc and ac currents in one-carrier devices. In Section 3 we solve
the new model for double injection in the frequency domain in order to determine an analytical expression for the
impedance.
Proceedings SPIE Int. Soc. Opt. Eng., 5519, 307-313 (2004)
2 ONE-CARRIER MODEL
The SCLC regime for a single carrier is well known.6-7 The basic equations describing transport and carrier distribution
are the current-flow equation for carrier drift and the Poisson equation:
J 0 = eµnE
ε dE
e dx
(1)
= −n
(2)
where µ is the drift mobility, which is assumed constant, n is the electron carrier density, e the elementary charge, J0 the
steady-state current density, and E the electric field intensity. J0 is constant across the sample. The boundary conditions
at the anode (x=0 ) and cathode (x=L) are:
E=0 at x=0 and at x=L
(3)
The solution of the above equations is given by the Mott-Gurney square-law equation
2
9 V
εµ 3
8
L
J0 =
(4)
being V0 the voltage applied across the sample.
In order to treat the time-dependent situation, we add the displacement current in the current-flow equation,
J = eµnE + ε
∂E
∂t
(5)
The impedance measurement corresponds to a small sinusoidal perturbation over a steady state. We separate the steadystate and modulated contributions by defining E(x,t)=E0(x)+E1(x,t), V(x,t)=V0(x)+V1(x,t), J(x,t)=J0(x)+J1(x,t), and
n(x,t)=n0(x)+n1(x,t). From the solution of the above equations the complex impedance is given as:
Z (ω ) =
1
⎡
2
−iωτ ⎤
⎥
⎢1 − iωτ + 2 (iωτ ) − e
g 0 (iωτ ) ⎣
⎦
6
3
(6)
Expanding the exponential terms of complex admittance, and solving the low frequency limit for the admittance, we
obtain the next complex admittance expression for low frequency limit:
Y = g 0 + iω
g 0τ
4
(7)
being
g0 =
τ=
9 V0
εµ 3
4
L
4 L2
3 µV0
(8)
(9)
This admittance represents a capacitance and resistance in parallel, as shown in Fig. 1:
C' =
3
Cg
4
(10)
R=
1
g0
(11)
where C g = ε / L is the geometric capacitance. Eq. (10) is a result of the modulation of the carrier distribution, at
frequencies lower than the reciprocal of the transit time.
Fig. 1. Equivalent circuit corresponding to low frequency limit for one-carrier device.
Note that, the conductance in the circuit shown in Fig. 1 agrees with the Mott-Gurney square-law (see Eq. 4).
dJ
9 V
(12)
g 0 = εµ 03 = 0
4
dV
L
0
The high frequency limit for the complex admittance gives
Y=
g τ
2g 0
+ iω 0
3
3
(13)
which represents the same equivalent circuit with different parameters.
C' = Cg
R=
(14)
3
2g0
(15)
3. TWO-CARRIER MODEL
In the model for two carriers, we consider the OLED as an insulator with double injection through ohmic injecting
contacts, free both of traps and thermal free carriers. Electrons are injected in the conduction band at the cathode, and
holes are injected in the valence band at the anode, and both of them drift to the plane of recombination at xp, where they
meet and recombine. We consider an infinite recombination rate, therefore on the anode side ( 0 ≤ x ≤ x p ) there are only
holes and current is a pure hole current. Similarly, on the cathode side ( x p ≤ x ≤ L ) we have only electrons, and a pure
electron current.
The equations characterizing double injection into a perfect insulator are the current-flow equation and the Poisson
equation, in the same way as in one-carrier problem. In the hole region, they are
J 0 = J 0 p = eµ p pE
ε dE
e dx
= p
(16)
(17)
Similarly, in the electron region:
J 0 = J 0 n = eµ n nE
ε dE
e dx
= −n 
(18)
(19)
where µn and µp are the electron and hole drift mobility respectively, n and p the free electron and hole densities
respectively, and J0p and J0n the hole and electron steady-state current density respectively and both of them are constant
across their corresponding region.
The solutions in both regions are tied together by continuity of the E field in the recombination plane:
E p ( x p ) = En ( x p )
(20)
where,
Ep( xp ) = −
3 V0 p
2 xp
(21)
En ( x p ) = −
3 V0 n
2 xn
(22)
From the last boundary condition, we derive the useful next relations:
V0 p
xp
=
V0 n
xn
(23)
µpL
µ p + µn
xp =
(24)
The steady state solution of the above Eq. 16-19 is given by the Mott-Gurney law in both regions:
J0p =
V02p
9
εµ p 3
8
xp
(25)
J 0n =
V2
9
εµ n 03n
8
xn
(26)
being V0p and V0n the applied voltage in the hole and electron region respectively, and xp and xn the width of each region.
By means of Eq. 23-24, and the fact that V0=V0p+V0n., we can give the potential drop in both regions in function of
mobilities and V0.
µp
V0 p = V 0
V0 n = V0
(27)
µn + µ p
µn
µn + µ p
(28)
By means of this relation, we derive the J-V characteristic in a simple form:7
J0 =
V2
9
ε µ n + µ p 03
8
L
(
)
(29)
It means that the total current density can be viewed as the sum of two SCL currents which traverse the entire sample,
independent of each other.
Fig. 2. Hole (dashed line) and electron (solid line) distributions in a double-injection device for different electron mobilities (hole
mobility is fixed on µp =1 10-10 m2V-1s-1).
We show in Fig. 2 carrier distributions for different µn, taking always µp =1 10-10 m2V-1s-1. In the case of low electron
mobility, holes reach the cathode before electrons can leave it. As electron mobility increases, recombination plane shifts
to the anode. When carrier mobilities are equal, electrons and holes recombine in the center of the film. Carrier
concentrations decrease as they leave the contacts, because of the increase of the electric field.
Next we solve the model equations in the frequency domain by adding the displacement current like in the case of onecarrier and proceeding in the same way. The steady-state and time dependent contributions are separated similarly as in
previous case. The two transport regions are linked by continuity of electrical field and potential at the recombination
plane. The solution provides us an analytical expression for the frequency dependent impedance:
Z(ω ) =
1
−iωτ p ⎤
⎡
2
⎢1 − iωτ p + 2 ( iωτ p ) − e
⎥
g p ( iωτ p ) ⎣
⎦
6
3
1
⎡
⎤
1 − iωτ n + ( iωτ n ) 2 − e −iωτ n ⎥
+
3 ⎢
2
g n ( iωτ n ) ⎣
⎦
6
(30)
where
gp =
V0 p
9
εµ p 3
4
xp
gn =
V0 n
9
εµ n
4
L − xp
(
(31)
)3
(32)
τp
2
4 xp
=
3 µ p V0 p
(33)
2
4 xn
3 µ nV 0 n
τn =
(34)
Eq. (30) corresponds to two impedances in series under SCLC regime, one corresponds to the hole region and the other
one to the electron region. This is a logical result because V0=V0p+V0n, and current is constant across the sample.
Similarly as in the one-carrier problem, we take the limit at low frequencies of the complex admittance and approximate
the exponential terms into a Taylor’s series expansion. Low frequency limit takes the form:
Y = g t + iω
gt 2
4
⎛τ p τn ⎞
⎟
⎜
+
⎜ g p gn ⎟
⎠
⎝
(35)
where gt-1=gp-1+gn-1. That is, gt corresponds to two resistances in series, each one describing one different region (hole
and electron region respectively).
At low frequencies the capacitance reaches a frequency independent value.
g t2
4
C' =
⎛τ p τn ⎞
⎟
⎜
+
⎜ g p gn ⎟
⎠
⎝
(36)
In order to understand the meaning of C’ we simplify this last expression. First we include the definitions shown in Eq.
31-34. Next, we use the relations expressed in Eq. 24, 27, 28. Finally, we obtain the simplified form of C’:
C' =
3ε
4L
(37)
It means that low frequency capacitance is identical to the one-carrier case and, consequently, independent of the
features of every region. This property can be viewed as two capacitance in series, each one corresponds to one different
region (holes or electron region).
1
1
1
=
+
C' C n C p
(35)
being,
Cp =
3 ε
4 xp
(36)
Cn =
3 ε
4 xn
(37)
As expected, the circuit corresponds to two equivalent circuits bounded together in series, each of them describing the
low frequency limit for holes and electron region respectively (See Fig. 3).
Fig. 3. Equivalent circuit at low frequency limit for two-carrier devices.
It is possible to understand the impedance at low frequency limit in another way. The total resistance in the circuit shown
in Fig. 3 (gt-1) is defined by gp and gn. Applying Eq. 24, 27, 28, again, gt can be expressed in terms of V0, and L:
gt =
(
)
V
9
ε µ n + µ p 03
4
L
(38)
This result agrees with the J-V characteristic expressed in Eq. 29.
gt =
V
dJ
9
ε ( µ p + µ n ) 03 = 0
4
dV0
L
(39)
Note that gt-1 represents two resistances in parallel as shown in Fig. 4, each one being the resistance by the one-carrier
model for holes and electrons respectively.
g t = g 0 p + g 0n
(40)
where,
g0 p =
V
9
εµ p 03
4
L
(41)
g 0n =
V
9
εµ n 03
4
L
(42)
Fig. 4. Equivalent circuit at low frequency limit for two-carrier devices. The circuit is equivalent to the circuit shown in figure 3.
By means of Eq.24,27,28 again, we can simplify the definitions of τp and τn. Both of them reduce to the same value:
τ n = τ p ≡ τ np =
4
L2
3 ( µ p + µ n )V0
(43)
Comparing τnp given in Eq. 43 with the transit time for one-carrier device shown in Eq. 9, we can say that the transit time
of two-carrier devices is always shorter as the transit time of the same device with a blocking electrode (one-carrier
device) under the same applied voltage, V0. Now the impedance expression can be given in a simplified form.
Z (ω ) =
1
− iωτ np ⎤
⎡
1 − iωτ np + (iωτ np ) 2 − e
⎢
⎥⎦
2
g t (iωτ np ) ⎣
6
3
(44)
It is obvious that the impedance of two-carrier devices is formally the same as the impedance of one-carrier device in Eq.
(6), but with a different apparent mobility ( µ p + µ n instead of µ ) and consequently shifted to higher frequencies. This
shift is due to a lower characteristic time in the case of two-carrier devices, which relates to the fact that carriers in twocarrier devices have to traverse a shorter way.
Fig. 5 illustrates the frequency dependence of the capacitance, C ′(ω ) = Re[1 / iωZ (ω )] , in the standard model of Eq. (30
or 44), showing the transition of the capacitance from geometrical value at high frequencies to Eq. (35) at low
frequencies. We show the capacitance spectra for different mobilities in order to compare the shift to higher frequencies.
Fig. 5. Capacitance spectra for different carrier mobilities: µ p + µ n =1 10-12 m2V-1s-1 (dotted line), 2 10-12 m2V-1s-1 (solid line), 4 10-12
m2V-1s-1 (dashed line).
5. CONCLUSION
The impedance model for double injection SCLC in OLEDs has been derived on the assumption of large recombination.
In this case the impedance is readily calculated as the series combination of two SCLC transport layers. We give an
equivalent circuit representing the impedance spectra at low frequencies. The circuit corresponds to two equivalent
circuits bounded together in series, each of them describing the low frequency limit for holes and electron region
respectively. The analytical expression for impedance of two-carrier devices reduces to the same analytical expression of
one-carrier device, but µ p + µ n instead of µ , and a shift to higher frequencies in the capacitance spectra. The model
provides a tool for understanding the features observed experimentally on the basis of reasonable and simple physical
considerations.
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