Drift-diffusion current in organic diodes

0.55 MB·

by  Gilles Horowitz

Drift-diffusion current in organic diodes
Gilles Horowitz∗
CNRS--LPICM, Ecole Polytechnique, 91128 Palaiseau, France
Because the conductivity of organic semiconductors is very low, a useful model for the organic
diode consists of treating the organic layer as an insulator, an approximation often referred to as
themetal-insulator-metal(MIM)model. Moreover,thedominantchargecarrierinjectionprocessis
diffusion, so that a modified Schottky’s theory can be used to derive a simple analytical equation
for the current voltage curve of the diode. Here, we carried out a full analysis of the MIM model
for the organic diode. We show that Schottky’s theory is only valid when charge injection is poor,
7 that is, for high injection barriers. When the injection barrier is lowered, the current given by
1 Schottky’stheoryisstillvalidintheweakinjectionregime,whentheappliedpotentialislowerthan
0 the diffusion potential. However, it becomes largely overestimated in the strong injection regime.
2 Wealsoshowthatinthestronginjectionregime,thecurrentgivenbytheMIMmodelmergeswith
Mott-Gurnery’s space-charge-limited regime.
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∗ [email protected]
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I. INTRODUCTION
Three mechanisms are usually invoked to rationalize charge carrier injection in semiconductor diodes: Thermionic
emission(TE),thedrift-diffusion(DD)modelandtunneling[1]. TEinvolvesballisticchargecarriertransportthrough
a depleted (aka space-charge) layer, and is the model of choice for silicon diodes. Because the mobility in organic
semiconductors is currently several orders of magnitude lower than that in single crystal silicon, the DD model is
generally recognized as more appropriate to describe the electrical current in organic diodes.
The DD model rests on two basic equations: Poisson’s equation (1) and the drift-diffusion equation (2).
d2V dF qp(x)
=− =− , (1)
dx2 dx ε
dp
j =qpµF −qD . (2)
dx
Here, the equations are written for positive charge carriers (holes). V is the electrical potential, F the electrical
field,andptheholedensity. εisthepermittivityofthesemiconductor,j theelectricalcurrentdensity,q theelemental
charge, µ the hole mobility and D the hole diffusion coefficient. We will assume the validity of Einstein’s relation,
which relates D and µ through D =µkT/q, where k is Boltzmann’s constant and T the absolute temperature.
In spite of they apparent simplicity, the exact resolution of these equations cannot be fully conducted by analysis;
numerical calculations become a necessity at some stage, which tends to hinder the physical meaning of the results.
Full calculations can be found in papers that date back to the early days of microelectronics [2–6], and in a more
recent work by K. Seki [7]. At variance with this analytical approach, the current trend is to perform numerical
resolutionsthroughthefiniteelementmethod(FEM)[8]. Variouscommercialpackagesareavailableforthatpurpose.
One prominent advantage of the FEM is that is allows for various refinement in the calculation, e.g., including
unconventional density of states (DOS) and non constant mobility. However, in spite of they usefulness for the
physical understanding of the process, these simulations are less appropriate in terms of compact modeling, which
requires the development of simple analytical equations.
The purpose of this paper is to delineates the various options to analytically resolve the drift-diffusion equation in
organic semiconductors, which are characterized by an extremely low density of thermal charge carriers.
II. THEORETICAL BACKGROUND
All the equations in this section are written for hole only devices. The extension to electrons would be straightfor-
ward.
A. Schottky’s diffusion theory
Thedevelopmentofthistheorycanbefoundintextbooks[1]. TheprincipleistoresolvePoisson’sandDDequations
in sequence. In the first step, (1) is used to determine the shape of the potential in the diode. The result is expressed
through the variation of the valence band edge E as a function of the distance x from the metal-semiconductor
v
junction:
q2N (cid:18) x2(cid:19)
E (x)=E (0)+ A W x− , (3)
v v ε sc 2
where :
(cid:115)
(cid:18) (cid:19)
2ε kT
W = V −V − , (4)
sc qN d a q
A
is the space charge layer width. N is the density of dopants (acceptors for a p-type semiconductor), V the diffusion
A d
(akabuilt-in)potential,definedasthedifferencebetweentheworkfunctionofthemetalandthatofthesemiconductor,
and V the applied potential.
a
The current is now established by rewriting (2) as :
(cid:18) (cid:19)
p dE dp
j =µkT v − , (5)
kT dx dx
3
which is next integrated using exp(E /kT) as an integrating factor:
v
(cid:90) Wsc (cid:18) E (cid:19) (cid:20) (cid:18) E (cid:19)(cid:21)Wsc
j exp − v dx=−µkT pexp − v . (6)
kT kT
0 0
Using the Fermi level of the metal as the reference energy, the boundary conditions are given by:
E (0)=−E (7)
v bp,
E (W )=−E +q(V −V ), (8)
v sc p d a
(cid:18) (cid:19)
E
p(0)=N exp − bp , (9)
v kT
(cid:18) (cid:19)
E
p(W )=N exp − p . (10)
sc v kT
E istheholebarrierheightatthemetal-semiconductorinterface,andE theenergydifferencebetweenthevalence
bp p
band edge and the Fermi level in the bulk of the semiconductor. N is the effective density of state at valence band
v
edge. An energy diagram of the junction and the relevant parameters are shown in Fig. 1
Figure 1. Energy diagram of a Schottky contact with a p-type semiconductor at equilibrium. Relevant parameters are: E :
bp
hole energy barrier; E : difference between the Fermi level and the valence band edge in the bulk of the semiconductor; V :
p d
diffusion (or built-in) potential; W : space-charge layer width.
sc
Combining all the above leads to:
(cid:18) (cid:19)
qV
j =j exp a −1 , (11)
d kT
(cid:18) (cid:19)
E
j =qµN F(0)exp − bp , (12)
d v kT
where F(0) is the electric field at the metal-semiconductor interface (x=0).
In an organic diode, the dopant density and semiconductor thickness are so small that it is generally accepted
that the space charge layer extends over the whole semiconductor layer, which is referred to as the full-depletion
or metal-insulator-metal (MIM) model. Under such circumstances, the potential at equilibrium varies linearly with
distance, and the electric field is constant. Attempts to adapt the Schottky model to such a geometry have been
recently made [9, 10]. The new energy diagram is shown in Fig. 2.
Now we have to consider both sides of the device; the hole injecting electrode is called anode, and the cathode is
hole blocking. Here, we restrict to a hole only diode, when the electron injection barriers are so high that only holes
can be injected at both electrodes.
The variation of the valence band edge is now given by:
x
E (x)=E (0)−q(V −V ) , (13)
v v d a d
where d is the thickness of the semiconductor . V is the diffusion potential, that is, the energy difference between
d
the work function of both electrodes, and V is the voltage difference applied between the anode and the cathode.
a
4
Figure 2. Energy diagram of a MIM diode at equilibrium (a) and low applied voltage (b).
Equation (6) must now be integrated over the whole thickness of the semiconductor, thus leading to:
V −V exp(qV /kT)−1
j =qµp d a a , (14)
0 d exp(qV /kT)−exp(qV /kT)
d a
where p = N exp(−Ean/kT) is the hole density at the anode (x = 0), Ean being the hole injection barrier at the
0 v b b
anode.
The main assumption that leads to (14) is that the shape of the potential profile (quadratic for a Schottky diode,
linear for a MIM diode) remains unchanged when a voltage is applied. This is basically true in a Schottky diode, as
shown in Equation (3). However, as will be shown in the following, the assumption only verifies in a MIM when the
injection barrier at the anode is high.
B. Full MIM model
We now focus on the following equation, which results of a combination of (1), (2) and Einstein’s relation:
(cid:18) dF kT d2F(cid:19)
j =εµ F − . (15)
dx q dx2
Following earlier works [5, 6], we will use the dimensionless reduced variables defined as follows:
x V d qd2 d3
x = , V = , F = F, p = p, j = j, (16)
r d r V r V r εV r εµV2
T T T T
where V =kT/q. (15) now writes:
T
d2F dF
r −F r +j =0. (17)
dx2 rdx r
r r
1. The diode without current
Integrating (17) with j =0 leads to:
r
dF F2
r − r +2g2 =0, (18)
dx 2
r
where g is an integration constant. The solution of this equation is given by:
F =−2gcoth(gx +argsinhgx ). (19)
r r r0
The electrical potential at a point x between the anode and the cathode is obtained by integrating (19) between
r
the anode (x =0) and x , thus leading to:
r r
(cid:18) (cid:114) p (cid:19)
V =2ln coshgx + 1+ r0 sinhgx . (20)
r r 2g2 r
Here, p = p (0) is the reduced density of holes at the anode. The integration constant g can be calculated by
r0 r
writing that the reduced potential at the cathode (x=d,x =1) is equal to to the reduced diffusion potential V .
r rd
5
2. Solution of the equation with current
The full DD Equation (17) has the following analytical solution:
c A(cid:48)(c +αx )+B(cid:48)(c +αx )
F (x )=−2α 2 i 1 r i 1 r , (21)
r r c A(c +αx )+B(c +αx )
2 i 1 r i 1 r
(cid:112)
where α = 3 j /2. A and B are Airy’s functions, and A(cid:48) and B(cid:48) their first derivative; c and c are integration
r i i i i 1 2
constants. Although (21) looks analytical, it does not allow for a direct computation of the current-voltage curve of
the diode, because the integration constants c and c must be estimated for each value of the reduced current j .
1 2 r
From the following relationship between Airy’s functions
A(cid:48)(cid:48)(z)=zA(z),
i i
B(cid:48)(cid:48)(z)=zB(z), (22)
i i
the reduced hole density p =dF /dx can be written as:
r r r
F2
p = r −j x −2α2c . (23)
r 2 r r 1
Equation (23) can now be used to estimate the integration constants. In a first step, we write the values of the
reduced hole density at the anode and cathode. Assuming a quasi-equilibrium, we postulate that these values are
those at thermodynamic equilibrium (no overall current). This leads at the anode (x =0):
r
(cid:40)(cid:20) c A(cid:48)(c )+B(cid:48)(c ) (cid:21)2 (cid:41)
p =2α2 2 i 1 i 1 −c , (24)
r0 c A(c )+B(c ) 1
2 i 1 i 1
or:
B(cid:48)(c )−P B(c )
c =− i 1 0 i 1 , (25)
2 A(cid:48)(c )−P A(c )
i 1 0 i 1
(cid:114)
p
P =± c + r0. (26)
0 1 2α2
A similar equation is obtained at the cathode (x =1):
r
B(cid:48)(c +α)−P B(c +α)
c =− i 1 1 i 1 , (27)
2 A(cid:48)(c +α)−P A(c +α)
i 1 1 i 1
(cid:114) (cid:114)
j +p p
P =± c + r r1 =± c +α+ r1. (28)
1 1 2α2 1 2α2
The sign in front of the square root in (26) and (28) depends on the orientation of the electric field at the anode
(x =0) and cathode (x =1). The constant c is now obtained by eliminating c between (25) and (27):
r r 1 2
[B(cid:48)(c )−P B(c )][A(cid:48)(c +α)−P A(c +α)]−
i 1 0 i 1 i 1 1 i 1
−[A(cid:48)(c )−P A(c )][B(cid:48)(c +α)−P B(c +α)]=0. (29)
i 1 0 i 1 i 1 1 i 1
The electrical potential is calculated by integrating (21) between 0 and x :
r
c A(c +αx )+B(c +αx )
V =2ln 2 i 1 r i 1 r . (30)
r c A(c )+B(c )
2 i 1 i 1
Replacing c by its value in (25) leads to:
2
V =2lnπ{[B(cid:48)(c )−P B(c )]A(c +αx )−
r i 1 0 i 1 i 1 r
−[A(cid:48)(c )−P A(c )]B(c +αx )}, (31)
i 1 0 i 1 i 1 r
where we used the identity A(z)B(cid:48)(z)−A(cid:48)(z)B(z)=1/π.
i i i i
The applied voltage V is connected to the reduced potential at x =1 through V(d)=V V (1)=V −V .
a r T r a d
6
C. Space-charge limited current
AusefulapproximationoftheDDmodelwasfirstintroducedbyMott[11],whichconsistsofneglectingthediffusion
component of the current. The SCLC regime becomes valid at high applied voltage, and also requires strong charge
carrier injection at the anode.
Neglecting the diffusion term leads to the following equation:
dF
j =εµF , (32)
dx
which can be integrated to:
2j
F2 = x+C. (33)
εµ
The integration constant C can be estimated by establishing the hole density at the anode to p(0) = p =
0
N exp(−Ean/kT):
v b
εdF j (cid:18)2j (cid:19)−1/2
p(x)= = x+C ,
q dx qµ εµ
j
p(0)= √ ,
qµ C
(cid:18) j (cid:19)2
C = . (34)
qµp
0
The potential a point at a distance x of the anode is obtained by integrating the electric field from the anode to
this point:
(cid:90) x εµ(cid:34)(cid:18)2j (cid:19)3/2 (cid:35)
V(x)= F(t)dt= x+C −C3/2 . (35)
3j εµ
0
Mott-Gurney’s model requires no limitation to charge carrier injection, so C → 0 (p → ∞) and the potential
0
profile becomes:
(cid:115)
8j
V(x)= x3/2. (36)
9εµ
Writing (36) at x=d leads to the well-known equation:
9 (V −V )2
j = εµ a d , (37)
8 d3
where the applied voltage is defined as V =V(d)+V .
a d
In the reverse case, when the hole density at the anode becomes low, we can develop the first term in the bracket
in the right side of Equation (35) to the first power of x:
εµ (cid:18)3 2jx(cid:19) √
V(x)= C3/2 = Cx, (38)
3j 2εµC
so the current now writes:
V −V
j =qp µ a d. (39)
0 d
A similar result is reported in Ref. [12].
Note that both (37) and (39) are only valid in the strong injection regime, when V >V .
a d
7
III. RESULTS
A. Potential profile
We conducted a numerical resolution of the full MIM model with the commercial package Mathcad. The first
step consisted of calculating the electric field profile F(x) for given values of the current, from which the potential
and charge carrier density profiles were obtained through numerical integration and derivation, respectively. The
parameters used for the calculations are gathered in Table I.
Table I. Parameters used for the calculation of the voltage profile of a MIM diode. ε is the permittivity of free space.
0
Temperature T =300K
Permittivity ε=4×ε
0
Mobility µ=1cm2/Vs
Density of states at valence band edge N =1020cm−3
v
Semiconductor thickness d=100nm
Hole injection barrier at the anode Ean =0.1 or 0.3 eV
b
Diffusion potential V =0.6V
d
The calculated potential profiles of the MIM diode for various values of the applied voltage are shown in Fig. 3. It
clearly appears that the MIM approximation, in which the potential linearly varies with distance, is only valid in the
caseofahighinjectionbarrier(Ean =0.3eV). Whentheinjectionbarrierislower(0.1eV),aslightcurvatureappears
b
at the anode (x=0), which is usually interpreted in terms of accumulation of holes at this electrode. Moreover, the
potential in the direct current regime (V > V ) is no longer a straight line; instead, it presents an upward
applied d
curvature. Comparing with the voltage profile described by Equation (36), this can be interpreted in terms of space
charge limited regime, as will be confirmed in the following.
Figure3. CalculatedpotentialprofileofaMIMdiodewiththeparameterslistedinTableIatvariousappliedvoltagesandfor
an injection barrier at the anode of 0.3 (a) and 0.1 eV (b).
B. Current-voltage curves
Calculatedcurrent-voltagecurvesofMIMdiodeswithaninjectionbarrierattheanodeof0.3and0.1eVaredrawn
in semi-log plot in Figures 4 and 5, respectively.
The exact MIM model is in good agreement with Schottky’s theory for a hole injection barrier of 0.3 eV, that is,
when injection efficiency is poor. When improving charge carrier injection by lowering the injection barrier down to
8
Figure 4. IV curve for a MIM diodes with the parameters in Table I and a hole injection barrier of 0.3 eV. Filled circles
correspond to data numerically calculated from the full MIM model, and the dashed line to the Schottky theory.
Figure 5. Same as Figure 4 for a hole injection barrier of 0.1 eV.
0.1 eV, the agreement remains good in the weak injection regime (applied voltage lower than the diffusion potential
V ). However, a discrepancy of nearly two order of magnitude is observed under strong injection, when the applied
d
voltage is in excess of the diffusion potential.
A log-log plot of the current-voltage curves is shown in Figure 6. Here, we have also calculated the space charge
limited current through Equations (34) and (37). Interestingly, the exact MIM curve now merges with the SCLC at
high voltages (strong injection regime). We also note that Schottky’s current is linear with the applied voltage in the
strong injection regime.
9
Figure6. Log-logplotofthecurrent-voltagecurveinFigure5inthestronginjectionregime(V >V ). Thespacecharge
applied d
limited current is shown by the dashed line.
IV. DISCUSSION AND CONCLUSION
Two distinct cases can be separated, depending on the injection barrier height: poor injection (high barrier) and
goodinjection(lowbarrier). Intheformercase, theorganicsemiconductorbehavesasaperfectinsulator; thevoltage
profile remains perfectly linear, including in the direct bias regime, when current is flowing through the diode. In
this particular case, Schottky’s theory leads to a current that is in perfect agreement with the full MIM calculation
in all regimes. It is worth pointing out that, in agreement with our calculation, poor injection also prevents SCLC to
occurathigh appliedvoltages. Accordingly, the currentathigh voltageisproportionalto thevoltageratherthan the
voltage to the square.
Foradiodewithgoodholeinjection, theagreementofSchottky’stheorywiththefullMIMmodelrestrictstoweak
injection, when the applied voltage is lower than the diffusion potential. At higher voltages, the current predicted
by Schottky’s theory in overestimated by a factor of nearly 100. This discrepancy is accompanied by two important
points. First, apart from a slight curvature near the anode due to hole accumulation, the voltage profile only remains
linear in the weak injection regime (V < V ). In the strong injection regime, the profile presents an upward
applied d
curvature. Thisobservationcanbeassociatedwiththefactthatunderstronginjection,theMIMcurrentmergeswith
the SCLC regime, with a current that is now proportional to the voltage to the square.
As a final remark, we note that the model developed here assumes that the organic semiconductor follows a non-
degenerate statistics. This assumption was used when estimating the density of holes at the electrodes. We have
recentlyshownthatthisassumptionisnotfulfilledinthecaseofaGaussiandensityofstates[13],whichbestdescribes
the vast majority of disordered organic solids. Further work will therefore be necessary to extend the model to a
degenerate statistics.
ACKNOWLEDGMENTS
I am profoundly grateful to Prof. Yvan Bonnassieux, Dr. Chang Hyun Kim and Sungyeop Jung for their constant
support during this work.
Appendix: Approximation at low current
The use of the exact equations (29) and (31) for the electrical potential and integration constant, respectively,
becomes problematic at low current because the value of the constant c becomes positive and large, so Airy’s
1
10
function B and its first derivative diverge. To work around this issue, we develop in this appendix an analytical
i
approximation of the equations at low current.
First, let us recall the asymptotic form of Airy’s functions:
exp(cid:0)−2z3/2(cid:1)
A(z)∼ √ 3√ , (A.1)
i 2 π 4z
√
(cid:18) (cid:19)
4z 2
A(cid:48)(z)∼− √ exp − z3/2 , (A.2)
i 2 π 3
exp(cid:0)2z3/2(cid:1)
B(z)∼ √ 3√ , (A.3)
i π 4z
√
(cid:18) (cid:19)
4z 2
B(cid:48)(z)∼ √ exp z3/2 . (A.4)
i π 3
The asymptotic form of the ratios A(cid:48)(z)/A(z) et B(cid:48)(z)/B(z) write:
i i i i
A(cid:48)(z) √
i ∼− z, (A.5)
A(z)
i
B(cid:48)(z) √
i ∼ z. (A.6)
B(z)
i
We can also write the asymptotic forms of the products of Airy functions as:
1
A(z)B(z)∼ √ , (A.7)
i i 2π z
1
A(cid:48)(z)B(z)∼− , (A.8)
i i 2π
1
A(z)B(cid:48)(z)∼ , (A.9)
i i 2π
√
z
A(cid:48)(z)B(cid:48)(z)∼− . (A.10)
i i 2π
Finally, developing (c +αx )3/2 in first of order of α/c leads to:
1 r 1
3√
(c +αx )3/2 (cid:39)c3/2+ c αx , (A.11)
1 r 1 2 1 r
so we can write the asymptotic form of Airy function at c +α as:
1
√
A(c +αx )∼A(c )exp(− c αx ), (A.12)
i 1 r i 1 1 r
√
A(cid:48)(c +αx )∼A(cid:48)(c )exp(− c αx ), (A.13)
i 1 r i 1 √ 1 r
B(c +αx )∼B(c )exp( c αx ), (A.14)
i 1 r i 1 1 r
√
B(cid:48)(c +αx )∼B(cid:48)(c )exp( c αx ). (A.15)
i 1 r i 1 1 r
Approximate electrical potential
Using the asymptotic forms of Airy’s functions, Equation (31) writes:
V (x )=2lnπ{[B(cid:48)(c )−P B(c )]A(c +αx )−[A(cid:48)(c )−P A(c )]B(c +αx )},
r r √i 1 0 i 1 i 1 r √i 1 0 i 1 i 1 r
=2lnπ[( c −P )B(c )A(c +αx )−(− c −P )A(c )B(c +αx )],
1 0 i 1 i 1 r 1 0 i 1 i 1 r
(cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
1 P √ P √
=2lnπ √0 +1 e c1αxr − √0 −1 e− c1αxr ,
2π c c
1 1
(cid:20) √ (cid:114) p √ (cid:21)
V (x )=2ln cosh c αx + 1+ r0 sinh c αx . (A.16)
r r 1 r 2c α2 1 r
1
√
Equation(A.16)issimilartothepotentialatzerocurrent(20)wheretheintegrationconstantg isreplaceby c α.
1
Hence we deduce that as the current tends to zero, c α2 →g2 and c tends to infinity.
1 1