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ResearchCite this article: Feicht SE, Schnitzer O,Khair AS. 2013 Asymptotic analysis ofdouble-carrier, space-charge-limitedtransport in organic light-emitting diodes.Proc R Soc A 469: 20130263.http://dx.doi.org/10.1098/rspa.2013.0263

Received: 25 April 2013Accepted: 2 July 2013

Subject Areas:mathematical modelling, power and energysystems, solid state physics

Keywords:asymptotic analysis, organic light-emittingdiode, drift–diffusion equations

Author for correspondence:Aditya S. Khaire-mail: akhair@andrew.cmu.edu

http://rsDownloaded from

Asymptotic analysis ofdouble-carrier,space-charge-limitedtransport in organiclight-emitting diodesSarah E. Feicht1, Ory Schnitzer2 and Aditya S. Khair1

1Department of Chemical Engineering, Carnegie Mellon University,Pittsburgh, PA 15213, USA2Department of Mathematics, Technion-Israel Institute ofTechnology, Technion City 32000, Israel

We analyse electron and hole transport in organiclight-emitting diodes (OLEDs) via the drift–diffusionequations. We focus on space-charge-limited transport,in which rapid variations in charge carrier densityoccur near the injecting electrodes, and in which theelectric field is highly non-uniform. This motivatesour application of singular asymptotic analysis to thedrift–diffusion equations. In the absence of electron–hole recombination, our analysis reveals three regionswithin the OLED: (i) ‘space-charge layers’ near eachelectrode whose width λs is much smaller thanthe device width L, wherein carrier densities decayrapidly and the electric field is intense; (ii) a ‘bulk’region whose width is on the scale of L, wherecarrier densities are small; and (iii) intermediateregions bridging (i) and (ii). Our analysis shows thatthe current J scales as J ∝ εμV2/L2λs, where V isthe applied voltage, ε is the permittivity and μ is theelectric mobility, in contrast to the familiar diffusion-free scaling J ∝ εμV2/L3. Thus, diffusion is seen tolead to a large O(L/λs) increase in current. Finally, wederive an asymptotic recombination–voltage relationfor a kinetically limited OLED, in which chargerecombination occurs on a much longer time scalethan diffusion and drift.

1. IntroductionOrganic light-emitting diodes (OLEDs) are solid-statelighting devices that convert electrical energy to lightthrough electroluminescence [1]. A standard device

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configuration consists of a transparent anode, typically indium tin oxide, an organicsemiconducting polymer film that is 10–1000 nm thick and a metallic cathode [2–5]. An appliedvoltage drives the injection of electrons through the cathode, and electron vacancies, or holes,through the anode into the organic thin film, where they diffuse and migrate, or drift, towardsthe oppositely charged electrode. When in close proximity, an electron and hole recombine toform an exciton, which radiatively decays to emit a photon.

The current in an organic device is limited by charge transport across the organic thin film [6]when the energy barrier to injection is lower than 0.3–0.4 eV [7], as opposed to being limited by theelectrode–film contacts that provide the majority of the resistance in inorganic devices [8]. At theelectrode–organic film interface, ‘space-charge layers’ exist where the charge carrier (electron orhole) density varies rapidly [9]. Quantifying the charge carrier distribution within these space-charge layers is crucial to understanding charge transport in OLEDs [9], as well as in othertypes of organic electronics such as light-emitting electrochemical cells [10] and organic solarcells [11].

The current and recombination rate are key parameters in analysing OLED performance. Thesimplest mathematical model that relates current, charge recombination and applied voltageis the drift–diffusion equations that account for diffusion, drift and recombination of electronsand holes [12–14]. The drift–diffusion equations are nonlinear coupled differential equations thatcannot be solved exactly. However, when simplifying assumptions are made, it is possible to makeanalytical progress. For example, Mott & Gurney [12] considered diffusion-free single-carrierinjection into a thin film with no intrinsic charge, for which the current is

J = 9με

8V2

L3, (1.1)

where V is the applied voltage; L is the width of the thin film; ε is the permittivity of the thinfilm; μ is the electric mobility, μ = De/kBT, where D is the diffusivity of the charge carrier, e is themagnitude of the charge of an electron, kB is Boltzmann’s constant and T is temperature. Notethat all hatted variables (e.g. V) are dimensional. Equation (1.1) is known as the Mott–Gurneylaw [12].

For single-carrier injection, the Mott–Gurney law has been extended to include the effectsof charge traps [15–19], uneven injection barriers [8,20] and intrinsic charge [17]. The latteris responsible for an ohmic regime (J ∝ V) that precedes the space-charge-limited regimedescribed by the Mott–Gurney law [12]. Mark & Lampert [17] add a term to the Mott–Gurney law that describes the linear increase in current with voltage owing to intrinsic charge.Murgatroyd [21] found that the current–voltage relation for single carrier, diffusion-free injectioncan be represented by a functional relationship between J/L and V/L2, which is consistent withthe Mott–Gurney law [12] and Mark & Lampert’s ohmic current at low voltage [17].

For double-carrier transport, Parmenter & Ruppel [22] obtained a solution to the drift–diffusion equations for diffusion-free, double-carrier injection and recombination into trap-freeinsulators, where recombination is modelled as a bimolecular reaction according to Langevinkinetics [23]. Parmenter & Ruppel [22] found that the Mott–Gurney law holds if the single-carrierelectric mobility is replaced with an effective mobility that is a function of the individual electronand hole mobilities, and a recombination mobility. In contrast to Parmenter & Ruppel [22], whoneglected diffusion, Baron’s [24] work included double-carrier diffusion, drift and recombination.In particular, Baron used Poisson’s equation to account for deviations from electroneutrality in thefilm, yet did not include the contribution of the gradient in charge density to the total carrier fluxacross the thin film. Baron found that the Mott–Gurney scaling, J ∝ V2/L3, holds.

The increase in current owing to diffusion and double-carrier injection is examined inTorpey’s [25] work. Torpey first addressed the diffusion-free case of double-carrier injectionwith recombination using a scaling analysis and confirmed that J ∝ V2/L3, in agreement withParmenter & Ruppel [22]. Torpey then solved the drift–diffusion equations numerically andfound that at small reservoir concentrations the current is limited by the inability of the contacts

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to provide sufficient electrons and holes. Torpey found a space-charge-limited current at largereservoir concentrations, where J ∝ V2. Notably, at large reservoir concentrations, the current ismuch greater than the J ∝ V2/L3 diffusion-free scaling (see fig. 11 in Torpey [25]).

Lastly, when diffusion and drift are included, but recombination is neglected, Neumannet al. [26] find J ∝ V3/L3 at high voltage, and J ∝ V/L3 at low voltage compared with the thermalvoltage, kBT/e (which is 26 mV at 300 K). They assert that there is only one length scale in thesystem, the device thickness L, to justify the scaling with L−3 in the current–voltage relation atall voltages. However, as we show below, when diffusion is included an additional length scalearises: the space-charge length.

The electrode–film boundary conditions imposed in the works discussed above feature ohmiccontacts [22,24,26,27], or reservoir conditions that fix the concentrations at the electrodes [4,9,12,17,25]. The ohmic condition assumes that the electric field is zero at the electrodes, whereasreservoir conditions assume a fixed voltage-independent concentration. In addition to theseboundary conditions, Walker et al. [28] and Malliaras & Scott [8] review boundary conditionsthat quantify the injection of charge due to thermionic emission, tunnelling and backflowinginterface recombination, where injection is a function of the local electric field at the electrode–organic interface.

In this work, we derive a current–voltage relation for space-charge-limited transport in thinorganic films owing to diffusion and drift of electrons and holes. Although this analysis does notaccount for carrier recombination, the results are subsequently used to derive a recombination–voltage relation for low recombination rate OLEDs.

As shown below, non-dimensionalization of the drift–diffusion equations reveals an importantparameter governing charge transport in the thin film: the width of the space-charge layer,

λs =√

εkBT2e2n0

, (1.2)

where n0 is the charge density at the electrode. This is mathematically equivalent to theDebye screening length in electrochemical systems [29]. In the present case, however, thespace-charge develops as a result of injection of charges into a film with no intrinsic charge,whereas the Debye length characterizes the extent of the space charge of mobile ions intrinsicto an electrolyte near a charged surface. Importantly, for space-charge-limited transport inOLEDs, the space-charge layer is often small in comparison with the width of the device,such that ε = λs/L � 1 [4,9,30]. For instance, in Pinner et al.’s [4] experiments on OLED chargetransport ε = 0.048 for an ITO/PPV/Au device. In Torpey’s [25] numerical simulation ofsmall-molecule organic semiconductors, based on the material parameters used in the paper,ε ranges from 0.022 to 0.049. Similarly, deMello’s [9] numerical work contains exampleswhere ε = 0.00098 for double-carrier injection into an OLED with ohmic contacts. In Penget al. [30], numerical analysis of charge transport in OLEDs, ε = 0.0012 for ohmic and injectionlimited cases.

When the space-charge layer is thin, ε � 1, numerical solutions of the drift–diffusion equationsare challenging, because the equations are singular as ε → 0 [9]. This is manifested in thedevelopment of sharp gradients in charge carrier concentration: the charge accumulates in thinregions (space-charge layers) at each electrode and rapidly decays to a low bulk concentration.Asymptotic analysis, specifically singular perturbation methods, are ideal for such problems [31];indeed, they exploit the fact that ε � 1 to obtain analytic, closed-form approximations that areasymptotic as ε → 0. In this paper, we use these techniques to derive a current–voltage relationfor OLEDs with double-carrier diffusion and drift.

This paper is structured as follows. In §2, we present the drift–diffusion equations thatdescribe charge transport and recombination in an OLED. In §3 through §5, we applyasymptotic analysis to the drift–diffusion equations to derive a current–voltage relation fordouble-carrier injection into OLEDs in the thin space-charge-layer regime, ε � 1. In §6, wecompare the current–voltage relation to the numerical solution of the drift–diffusion equations.

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In §7, we derive a recombination–voltage relation for kinetically limited OLEDs and comparewith the numerical solution of the drift–diffusion equations with recombination. We presentconclusions in §8.

2. Mathematical modelThe drift–diffusion equations describe the transport of electrons and holes in an organic film,caused by diffusion down concentration gradients and drift in an electric field. The thicknessL (10–1000 nm) of an OLED is typically much smaller than its width and depth; thus, the device istreated as one-dimensional, and charge transport occurs normal to the electrodes only, designatedas the x direction. The x-origin is located at the anode (figure 1). We assume that the OLED isoperating at steady state. The net fluxes of each species are

j± = −μ±kBT

edn±dx

∓ μ±n±dφ

dx, (2.1)

where the ± subscript refers to electrons (−) or holes (+), j± is the flux of the charge carrier, μ±is the carrier mobility and φ is the electric potential. The first term in (2.1) accounts for diffusion,the second drift. Poisson’s equation states that the variation in electric field across the OLED isproportional to the local charge density,

d2φ

dx2 = − eε

(n+ − n−). (2.2)

Charge carrier conservation equations equate the change in flux across the film to theconsumption of charge carriers due to recombination,

dj±dx

= −kn+n−, (2.3)

where k is the reaction rate constant for the recombination of electrons and holes, with unitsof length cubed per time, assuming Langevin kinetics [23]. In (2.3), we do not account for anyelectron or hole generation through exciton dissociation.

We assume that electrons are injected through the cathode and holes through the anode suchthat the number density of charge carriers at the electrode is finite and constant [25], and thatthe potential at the electrodes is equal to an applied potential. We assume that the concentrationof a charge carrier is zero at its counter electrode [9,26]. These assumptions form the boundaryconditions described in (2.12) and (2.13).

To non-dimensionalize the governing equations (2.1)–(2.3), the spatial coordinate, x, isnormalized by the device length, L; the electric potential φ by the thermal voltage; and the electronand hole densities by n0, the charge carrier density at the electrodes. The flux is normalized byDn0/L. For the sake of simplicity, we set the magnitude of the charge density at the anode andat the cathode to be equal, and set the mobility of electrons equal to the mobility of holes. Inaddition, we neglect the electric field dependence of the mobility. Henceforth, all equations arein terms of dimensionless variables (unless stated otherwise) and appear without a caret accent.From (2.1), the flux is

j± = −dn±dx

∓ n±dφ

dx. (2.4)

Poisson’s equation now readsd2φ

dx2 = − 12ε2 (n+ − n−), (2.5)

where ε is the dimensionless space-charge length, defined as

ε = λs

L= 1

L

√εkBT2e2n0

. (2.6)

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holes electrons

j+ j–

x = 0

anode

x = 1cathode

n+ = 0n– = 1f = –V/2

n+ = 1n– = 0f = V/2

x = 1/2

Figure 1. Holes are injected through theanodewith aflux j+,whereas electrons are injected through the cathodewith aflux j−;the fluxes will be determined in the proceeding analysis. The boundary conditions are shown at the anode (x = 0) and at thecathode (x = 1).

From (2.3), the charge conservation equations are

ddx

(−dn±

dx∓ n±

dφ

dx

)= −kn+n−, (2.7)

where the dimensionless reaction rate k = kn0L2/D is known as the Damkhöler number inengineering literature [32], and is equal to the ratio of the diffusive time scale (L2/D) to therecombination time scale (1/kn0). When the reaction is slow in comparison with diffusion (k � 1),the recombination is kinetically limited. If the inverse is true, then recombination is limited bycharge transport. Here, we assume that the system is kinetically limited, so the reaction term canbe neglected to a first approximation (k = 0) in (2.7) yielding

ddx

(−dn±

dx∓ n±

dφ

dx

)= 0. (2.8)

It is convenient to write the drift–diffusion equations (2.4)–(2.8) in terms of the mean chargecarrier concentration, c = 1

2 (n+ + n−), half the charge density, ρ = 12 (n+ − n−) and the current,

J = j+ − j−. Thus, Poisson’s equation becomes

− ε2 d2φ

dx2 = ρ. (2.9)

The net flux, j+ + j−, is zero across the film, because the flux of electrons and holes are equal inmagnitude but opposite in direction, which implies

ρdφ

dx+ dc

dx= 0. (2.10)

An integration of (2.8) shows that the current J satisfies

cdφ

dx+ dρ

dx= −1

2J. (2.11)

The aforementioned boundary conditions, in non-dimensional form, are

anode: x = 0 : n+ = 1, n− = 0 and φ = 12

V (2.12)

and

cathode: x = 1 : n+ = 0, n− = 1, and φ = −12

V, (2.13)

where V is the applied potential (normalized by the thermal voltage, V = Ve/kBT) minusthe normalized built-in potential drop owing to the difference in the work functions of the anodeand cathode [33]. The applied potential V is equal to the integral of the electric field E across

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the device,

V = −∫ 1

0E dx, (2.14)

where

E = −dφ

dx. (2.15)

A schematic of the device configuration is shown in figure 1. In the proceeding analysis, we solvefor the electric field, hole and electron density, and electric potential in the anodic half (0 ≤ x ≤ 1

2 )of the OLED. The electric field is symmetric around x = 1

2 , whereas the electric potential isantisymmetric. The electron density is symmetric to the hole density, so asymptotic expressionsfor the electron density in the cathodic region ( 1

2 ≤ x ≤ 1) can be derived from the expressions forthe hole density in the anodic region by replacing x with 1 − x, and vice versa.

In the absence of recombination, the problem presented here is mathematically similar toion transport across a permeable membrane at the interface between two reservoirs of fixedelectrolyte concentrations [34–36], relevant to ionic channels in biological systems. Note, however,that in [34–36] it is assumed that the concentrations of both species (cation and anion) are non-zeroin each reservoir. In the case of OLEDs, the appropriate boundary conditions set the concentrationof electrons (holes) to zero at the anode (cathode). Another related electrodiffusion problem is thatof current passage across a cation-selective surface, which is relevant to electrodialysis at overlimiting currents in thin-gap cells [37]. In that problem, the anions do not react at the electrodes;hence, their concentration within the cell is subject to an integral constraint. For an OLED, thereare no such constraints on the concentrations of electrons and holes in the semiconductor film.

3. Master equation for the electric fieldEquations (2.9)–(2.11) can be combined into a single master equation in terms of the electric fieldE. First, (2.9) is substituted into (2.10), yielding

ddx

(− ε2

2E2 + c

)= 0. (3.1)

Equation (3.1) is readily integrated, resulting in

− ε2

2E2 + c = A, (3.2)

where the constant of integration is

A = − ε2

2E2

� + c�, (3.3)

where E� and c� are the electric field and mean concentration, respectively, at the midpoint ofthe device. These two quantities are not known a priori, but must be determined as part of theasymptotic analysis. From (3.2) and (3.3), the carrier concentration can be written as

c = ε2

2(E2 − E2

�) + c�. (3.4)

Poisson’s equation (2.9) is inserted into (2.11) resulting in

ε2 d2Edx2 = cE − 1

2J. (3.5)

Equation (3.4) is combined with (3.5) to yield the master equation for the electric field across theOLED,

ε2 d2Edx2 −

[ε2

2(E2 − E2

�) + c�

]E = −1

2J. (3.6)

Similar master equations have been derived to analyse charge transport in liquid-stateelectrochemical systems [36,38] and in OLEDs without diffusion [25].

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4. Asymptotic analysisIn the operation of an OLED, a potential difference between the anode and the cathode drivesa current across the organic thin film. The applied voltage must be high enough to overcomethe energy barriers for the injection of charge carriers. Typically, the potential difference is muchgreater than the thermal voltage, V � 1 (or V � kBT/e � 26 mV) [2]. Therefore, we prescribe thevoltage

V = μ(ε)V�, (4.1)

where V� is an O(1) constant and μ(ε) � 1. The bulk field, EB, arises from the O(μ) drop inpotential across the O(1)-width bulk region. Owing to symmetry of the field about x = 1

2 , theintegral (2.14) can be written as

12μV� = −

∫ 1/2

0E(x) dx, (4.2)

where we have used φ( 12 ) = 0, and φ(0) = V/2 = μV�/2. Hence, the bulk field is

EB = μ(ε)V� + o(μ), (4.3)

which is spatially uniform to leading order. Therefore, the electric field at the midpointis E� = μ(ε)V� to leading order. The majority of the injected charge carriers in the deviceare concentrated at the space-charge layers near each electrode, so the concentration in theelectroneutral bulk is expected to be asymptotically small in ε,

cB = δ(ε)ca + o(δ), (4.4)

where δ(ε) � 1 is an a priori unknown function and ca is an O(1) coefficient. Thus, c� = δ(ε)ca toleading order. In the bulk, where x ∼ O(1), one can simply set ε = 0 in the master equation (3.6) toderive the leading-order current J as

J = 2cBEB = 2μδcaV� + o(μδ). (4.5)

The charge density is zero in the electroneutral bulk, to leading order. However, at the anode–thin-film interface (x = 0) the charge density ρ = 1

2 and ρ = − 12 at the cathode (x = 1), according

to (2.12) and (2.13). The bulk solution (4.3)–(4.5) does not satisfy these boundary conditions. Thisindicates the existence of an asymptotically small region near each electrode, a space-charge layer,where electroneutrality is violated. We derive an asymptotic approximation for the electric fieldin the space-charge layer next.

(a) Space-charge layerOwing to symmetry, we focus on the hole-rich space-charge layer near the anode (x = 0). In thisregion, the concentration of charge carriers varies rapidly; hence, we define a stretched ‘inner’coordinate x = x/ε, such that x ∼ O(1) as ε → 0. The electric field in the space-charge layer isexpected to be O(1/ε), resulting in an O(1) potential drop across the space-charge-layer (themajority of the O(μ) applied potential drops across the bulk of the OLED). We define E = Eε,where E ∼ O(1). Applying these scalings to the master equation (3.6) yields

1ε

d2Edx2 −

[12ε

E2 − ε

2E2

� + c�

ε

]E = − J

2. (4.6)

Recall that the midpoint concentration c� (4.4) is small, δ � 1, and the midpoint field E� (4.3) isO(μ). Thus, to ensure that the space-charge layer is in quasi-equilibrium to a first approximation,we choose μ � 1/ε, so that terms proportional to the midpoint field E� and current J (∼ μδ) in

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(4.6) do not enter the leading-order balance. That leading O(1/ε) balance is therefore

d2Edx2 = 1

2E3, (4.7)

with the boundary conditions

dEdx

= 12

andd2Edx2 = 1

2E at x = 0, (4.8)

derived from (2.9), (2.12) and (3.6). The solution to (4.7) and (4.8) is

E(x) = − 2x + 2

, (4.9)

which represents the leading-order solution for the electric field in the space-charge layer. Thisexpression was also found by Chu & Bazant [39] for the electric field in the Debye layer of anelectrochemical cell at the diffusion-limited current.

The decay to zero of the O(1/ε) space-charge field as x → ∞ is consistent with the prescribedO(μ) bulk field, where μ � 1/ε. We look to (4.6) for the scaling of the next term of the space-charge layer field. After the terms included in (4.7), the next largest term in (4.6) is c�E/ε ∼ δcaE/ε

from (4.4), because ca and E are O(1), and μ � 1/ε. Thus, the expansion of electric field in thespace-charge layer is

E(x) = 1ε

(E0(x) + δE1(x) + o(δ)), (4.10)

where E0 is the previously calculated leading-order term (4.9). An equation for the next, O(δ/ε),term of the electric field in the space-charge layer, E1(x), is derived by inserting (4.10) into (4.6).Because μ � 1/ε, the current J does not enter the O(δ/ε) balance. The term c�E/ε is to leadingorder δcaE0/ε from (4.4). The term containing the midpoint field, εE2

�E is O(εμ2) to leading orderin E. When μ is restricted to μ � δ1/2/ε, this term does not enter into the O(δ/ε) balance. With thisrestriction, the O(δ/ε) field is governed by

d2E1

dx2 − 32

E20E1 = caE0, (4.11)

with the boundary conditions

dE1

dx= 0 and

d2E1

dx2 = 12

E1 at x = 0. (4.12)

Substituting the result for E0 (4.9) into (4.11) and solving the resulting equation yields

E1(x) = ca12 + 12x + 6x2 + x3

3(2 + x)2 . (4.13)

The O(δ/ε) field (4.13) diverges as x → ∞ and hence clearly does not match the bulk, indicatingthat a transition, or intermediate, region exists between the electroneutral bulk and thespace-charge layer.

(b) Intermediate layerThe device length and the width of the space-charge layer are clear choices for the characteristiclength of the bulk, x ∼ O(1), and the space-charge layer, x ∼ O(ε), respectively; however, there isno obvious length scale for the intermediate layer. Through a scaling analysis, the intermediatelayer field and width are found to scale as

E = δ1/2

εE and x = ε

δ1/2 x, (4.14)

where E and x are O(1). The width of the intermediate layer is ε/δ1/2, and the magnitude of thefield within it is δ1/2/ε, which fit the criteria ε � ε/δ1/2 � 1 and μ � δ1/2/ε � 1/ε. Therefore, the

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leading-order equation that results in the intermediate layer from (3.6) is

d2Edx2 = 1

2E3 + caE. (4.15)

A solution to (4.15) that decays to zero as x → ∞ is [40]

E(x) = − 2√

ca

sinh (√

cax). (4.16)

The behaviour of the asymptotic expansion for the intermediate field must match that ofthe space-charge field as x → 0 and the bulk field as x → ∞. As x → 0, E → −2/x; as x → ∞,the leading-order space-charge field (4.9), rewritten in terms of x and E, approaches E = −2/x.Therefore, the behaviour of the intermediate field as x → 0 matches that of the space-charge field.Because we require μ � δ1/2/ε, the decay of (4.16) at large x is consistent with the O(μ) bulkfield (4.3). To determine a relation between μ, δ and ε, we need a second term for the field inthe intermediate layer that scales by μ to match to the leading-order bulk electric field EB = μV�.The expansion for the field in the intermediate region is thus written as

E(x) = δ1/2

εE0(x) + μE1(x) + O(μ), (4.17)

where the leading-order term E0 is given by (4.16). From (3.6), the equation for E1 is

d2E1

dx2 − 32

E20E1 − caE1 = cbE0 − caV�, (4.18)

where cb is the next order of the expansion of the midpoint concentration that is

c� = δca + μεδ1/2cb + o(μεδ1/2), (4.19)

which is added to account for a possible higher-order bulk concentration contribution to E1. Anexact solution to (4.18) cannot be obtained. However, only the behaviour of E1 as x → ∞ and asx → 0 is required for matching with the bulk and the space-charge layer, respectively. As x → ∞,it is clear that E1 → V�, because E0 decays exponentially, thereby matching the bulk field, but thebehaviour as x → 0 needs to be examined carefully.

A power series expansion around x = 0 reveals two homogeneous solutions to (4.18):

fh1 = a0

x2 + a0ca

6− 7a0c2

a120

x2 + O(x3), (4.20)

andfh2 = b0x3 + O(x4), (4.21)

where a0 and b0 are constants, as well as a particular solution,

fp = cb

3x + caV�

4x2 − c2

aV�

24x4 + o(x4). (4.22)

The series approximation of the O(μ) correction to the intermediate field as x → 0 is thus

E1 = a0

x2 + a0ca

6+ cb

3x +

(−7a0c2

a120

+ caV�

4

)x2 + b0x3 + O(x3). (4.23)

We now proceed to match the intermediate field, (4.16) and (4.23), to the electric field in the space-charge layer, (4.9) and (4.13), according to van Dyke’s matching procedure [31], in order to obtaina0 and δ. This is detailed in appendix A. The results of the matching procedure are

a0 = 4 and δ = με. (4.24)

However, we have not yet found an expression for ca, which can be determined by matching onlyto higher-order space-charge-field terms.

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(c) Higher-order space-charge-field termsTo determine an expression for ca, where cB ∼ δca is the leading-order bulk concentration, we needto determine the next two terms in the space-charge layer field, which include the contributionof current. The order of these terms in the space-charge layer expansion can be determined fromthe higher-order terms in the correction to the intermediate field (4.23), as x → 0. Equation (4.23)implies the expansion

E(x) = 1ε

E0(x) + δ

εE1(x) + δ

32

εE2(x) + δ2

εE3(x) + o

(δ2

ε

). (4.25)

Substituting (4.25) into (4.6), the differential equation for E2 is

d2E2

dx2 − 32

E20E2 = cbE0, (4.26)

subject to the boundary conditions

dE2

dx= 0 and

d2E2

dx2 = 12

E2 at x = 0. (4.27)

This differential equation has the same form as (4.11); the solution has the same form as E1 (4.13),except that ca is replaced by cb. Thus,

E2(x) = cb12 + 12x + 6x2 + x3

3(2 + x)2 . (4.28)

This term matches to the O(x) term in (4.23), however, this does not provide the value of ca.Another term of the expansion of the space-charge layer field is required here to determine ca.For the next term, E3(x), we first extend the expansion of the midpoint concentration c� by anadditional term

c� = δca + δ3/2cb + δ2cc + o(δ2), (4.29)

where we have applied μ = δ/ε from (4.24). Equation for the fourth term in the space-charge layerelectric field expansion is

d2E3

dx2 − 32

E20E3 = 3

2E0E2

1 − 12

V�E0 + caE1 + ccE0 − caV�, (4.30)

subject todE3

dx= 0 and

d2E3

dx2 = 12

E3 − caV� at x = 0. (4.31)

Equations (4.30) and (4.31) can be solved in closed form; however, we are interested only in thebehaviour as x → ∞. A power series expansion of the solution to (4.30) and (4.31) as x → ∞ yields

E3(x) = 4c2a − 21caV�

60x2 + c2

a − 9caV�

90x3 + O(x2). (4.32)

Next, we match the first four terms in the expansion for the space-charge field to the first twoterms in the intermediate solution. This matching procedure, in contrast to the last, matches theterms in each expansion that are larger than O(δ2/ε) in the space-charge field expansion andO(δ/ε) in the intermediate field expansion according to van Dyke’s rule [31]. This procedure,detailed in appendix A, yields

ca = 2V�. (4.33)

When (4.1) and (4.24) are inserted into (4.33), the bulk concentration is recovered to leadingorder as

cB ∼ 2εV, (4.34)

which is indeed small as V = μV�, where μ � 1/ε.

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5. Current–voltage relationFrom (4.3), (4.5) and (4.34), the leading-order current is

J ∼ 4εV2. (5.1)

The first correction to (5.1) can be found through a higher-order analysis of the integral constraint(2.14). The details of the analysis are given in appendix B, where it is shown that the constraintyields a logarithmic correction to the leading-order bulk electric field,

EB = V − 2 ln 2εV + o(ln δ). (5.2)

Substituting (4.34) and (5.2) into (4.5) yields the improved current–voltage relation,

J = 8εV(

V2

− ln 2εV)

+ o(δ ln δ). (5.3)

From (5.2) and (5.3), for the analysis to be valid the applied voltage must be sufficiently large suchthat μ � ln 1/δ. An upper bound on the applied voltage, μ � δ1/2/ε, is necessary to calculate thefirst correction to the space-charge field (4.23). The current–voltage relation (5.3) is therefore validover the range

ln1δ

� μ � δ1/2

ε, (5.4)

where μ is prescribed by the magnitude of the applied voltage V. The current (5.3), in dimensionalform, reads

J = 4εμ1

L2λs

(V2

2− VkBT

eln

2eλsV

LkBT

), (5.5)

where λs is given by (1.2). To leading order, the current exhibits a quadratic dependence on theapplied voltage V and the reciprocal of the film width L, J ∝ V2/L2λs, in contrast to the Mott &Gurney law for diffusion-free single carrier injection, where J ∝ V2/L3.

6. Comparison between asymptotic analysis and numerical solution of thedrift–diffusion equations

Our analysis has furnished asymptotic approximations for the field and carrier densities acrossthe OLED. To validate our analysis, we solved the drift–diffusion equations (2.4), (2.5) and (2.8)numerically using the Matlab BVP4C solver. The asymptotic results for the electric field arecompared against the numerical results for ε = 0.001 (figure 2), showing good agreement.

The density of holes n+ and electrons n− in the anodic region is easily calculated from theasymptotic expressions for the electric field. In figure 3, the hole and electron densities in theanodic region are plotted on a semilog axis. The solution for the hole density in the space-chargelayer includes the first correction to the density, calculated from (4.13). The asymptotic expansionfor the electron density in the intermediate region decays to zero as x → 0 (figure 3), as expected; infact, from (4.9), (4.13) and (4.28), it can be shown that the electron density is zero through O(δ3/2).The discrepancy between the asymptotic bulk electron and hole densities and the numerics canbe attributed to the fact that only the leading-order term for the bulk concentration (4.34) wascalculated. This error is seen in both the electron and hole density asymptotic expansions.

The steep gradient in hole density in figure 3 results in a diffusive current that must bebalanced by a negative electric field to conserve charge in (2.8). From the definition of the space-charge width (2.6), ε ∝ 1/

√n0. Hence, as the charge density in the electrode approaches infinity,

as in the case of ohmic contacts without diffusion, ε → 0. A plot of the potential at several valuesof ε is useful (figure 4) to elucidate the increase in current owing to diffusion. The leading-orderpotential φ is calculated from (2.15) and the electric field across the space-charge layer (4.9), theintermediate layer (4.16) and the bulk (5.2).

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–100010–4 10–3 10–2 10–1 0.5

–800

–600

–400

–200

0

200

position, x

elec

tric

fie

ld, E

space-charge layer

bulknumerics

intermediate

Figure 2. Asymptotic solutions for the electric field in the space-charge layer (dash), the intermediate layer (dash-dot) and inthe bulk (dot) are compared with the numerical solution (open circles) of the electric field in the anodic region (0≤ x ≤ 1

2 )for ε = 0.001 and V = 50. The asymptotic solutions plotted include two terms of the asymptotic series for the field in thespace-charge layer, (4.9) and (4.13), the leading-order intermediate field (4.16) and the bulk field (5.2).

position, x

space-charge layer intermediate layer

numericsbulk layer

space-charge layer intermediate layer

numericsbulk layer

10–4

hole

den

sity

, n+

elec

tron

den

sity

, n–

00.10.20.30.40.50.60.70.80.91.0

00.020.040.060.080.100.120.140.160.180.20(a) (b)

10–3 10–2 10–1 0.5position, x

10–4 10–3 10–2 10–1 0.5

Figure 3. Numerical solutions for the (a) hole density (circle) and (b) electron density (square) at ε = 0.001 and V = 50 arecompared with asymptotic expressions for the hole and electron densities in the space-charge layer (dash), intermediate layer(dash-dot) and bulk (dot). Note the vertical scales of (a) and (b) are different.

As ε → 0, a maximum in potential develops, resulting in ‘virtual ohmic contacts’ [25] near theelectrodes. The maximum results from the negative electric field near the anode that developsto balance diffusion in the charge conservation equation (2.8). The virtual contacts are locatedat the point where the electric field is zero. Between the virtual contacts near the anode and thecathodes, the voltage drops linearly. The potential at the virtual contact increases as ε decreases,and is larger than the applied potential at the electrode. Hence, the voltage drop across the bulk islarger than the naive estimate V/L, resulting in a higher current. This is reflected in the current–voltage relation (5.5), where J ∝ 1/λs. If diffusion is neglected this increase in current due to aninterior ‘virtual contact’ across the OLED is not accounted for.

The current–voltage relation (5.3) is compared with the numerical results in figure 5 at threevalues of ε. As ε decreases, the accuracy of the asymptotic expression increases. The current–voltage relation (5.3) is fundamentally different from the Mott–Gurney law for diffusion-free

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0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50position, x

pote

ntia

l, f

0

1

2

3

4

5

6

7

8

9

10

virtualcontact e = 0.001 : numerics

e = 0.01 : numerics

space-charge layerbulk

Figure 4. The asymptotic solutions (dash) for electric potential are compared to the numerical solution (symbols). The electricpotential at ε = 0.001 (circle) has a maximum (indicated by arrow), or virtual contact, closer to the anode than the potentialat ε = 0.01 (triangle). As ε → 0, the potential increases at the virtual contact, effectively increasing the voltage drop acrossthe bulk. In this figure, the applied voltage is V = 10.

0

10–1

1

curr

ent,

J

10

102

103

20 40 60 80 100voltage, V

120

asymptoticse = 0.01 : numericse = 0.05 : numericse = 0.001 : numerics

140 160 180 200

Figure 5. The asymptotic expression for current (5.3) (line) are comparedwith the numerical results at ε = 0.001 (circle), ε =0.005 (square) and ε = 0.01 (triangle). The asymptotics perform well as ε → 0.

injection into insulators. The major difference is the additional characteristic length scale, thespace-charge length λs, that results from including diffusion: instead of J ∝ V2/L3, we find thatJ ∝ V2/L2λs to leading order. Because the space-charge layer is much thinner than the film,ε = λs/L � 1, this predicts a large O(L/λs) increase in the current that passes through the film fordouble-carrier injection with diffusion (figure 5). This can be interpreted in terms of the virtualohmic contacts as discussed above.

The increase in current is consistent with experimental results for double-carrier injectioninto an ITO/MEH-PPV/Ca OLED performed by Parker [41]. The energy barrier to injectionacross the MEH-PPV/Ca interface is approximately 0.1 eV, which indicates a space-charge-limitedOLED [7]. Specifically, Parker reports current–voltage curves for OLEDs for varying widths of

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10–1

1

10–2

10–2 10–1 1 10 102

10–3

10–4

reaction rate constant, k

asymptoticsnumerics

reco

mbi

natio

n ra

te, R

Figure 6. The leading-order total recombination rate R= 4kε2V2 (line) is compared with the numerical solution of the drift–diffusion equations (circle) as a function of recombination rate constant k at V = 50 and ε = 0.001.

the polymer thin film, L. When the macroscopic electric field across the film, V/L, is plottedagainst the current, the curves for the various widths collapse onto a single current–voltage curve,indicating that the current–voltage relation should be a function of the ratio V/L, as seen in (5.5),where J ∝ V2/L2, as opposed to J ∝ V2/L3 for diffusion-free transport. In terms of the chargedensity at the electrode, J ∝

√n0, the current increases with the reservoir density, as expected.

When the charge density in the electrode is small, the current is small because it is limited by theelectrons and holes available in the electrode for transfer to the thin film.

7. The first effects of carrier recombinationThus far, we have set the recombination rate constant k equal to zero, thereby neglectingrecombination of carriers. The results reported in §6 for the electric field, hole and electrondensities, and electric potential, can be viewed as the leading-order terms in an asymptoticexpansion in terms of the recombination rate. That is, the leading-order local recombination rate is

r(x; ε, k) = kn+(x; ε)n−(x; ε), (7.1)

where n+(x; ε) and n−(x; ε) are the hole and electron profiles calculated for k = 0. In the space-charge layer adjacent to the anode, the electron density is zero to O(δ3/2); thus, the localrecombination rate in the space-charge layer is zero through kδ3/2. In the intermediate region,the leading-order recombination rate is equal to the leading-order bulk recombination rate.The total recombination rate across the OLED is the integral of the local recombination rate,R = ∫1

0 kn+(x)n−(x) dx. Because the space-charge layers are thin, the bulk recombination rater ∼ 4kε2V2, from (4.34), is the dominant contribution to the total recombination rate. Theleading-order asymptotic expression for the total recombination rate,

R ∼ 4kε2V2, (7.2)

is compared with numerical results in figure 6. In figure 6, the numerics solve the fulldrift–diffusion equations, including recombination, at ε = 0.001 and V = 50, for a variety ofrecombination rate constants, k. The asymptotic solution matches well to the numerics throughO(1) values of k, indicating that the simple expression R ∼ 4kε2V2 may be applied to OLEDs toapproximate the total recombination rate for low-to-moderate recombination rate constants.

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8. ConclusionsWe have quantified double-carrier drift, diffusion and recombination in OLEDs via asymptoticanalysis of the drift–diffusion equations. For space-charge-limited OLEDs, the carrier densitiesvary rapidly near the electrodes. We have shown this variation in carrier densities occurs onthe length-scale of the space-charge width λs, which is much smaller than the device thickness.Thus, the ratio ε = λs/L is a small parameter exploited in our analysis to yield asymptoticapproximations to the current–voltage relation and the electric field, carrier densities and electricpotential across the OLED. Chiefly, we found that the leading-order current across the OLEDis J ∝ V2/L2λs, in contrast to diffusion-free single-carrier injection, where J ∝ V2/L3 [12]. Thedifference is due to the characteristic length scale, the space-charge width λs, that emerges whenthe drift–diffusion equations are non-dimensionalized. This space-charge width is much thinnerthan the width of the device, so we predict a higher current than a diffusion-free analysis would,which can be interpreted as an increased potential drop between virtual ohmic contacts thatresult from diffusion. The scaling for the current derived here, J ∝ V2/L2λs, is consistent withexperimental data for ITO/MEH-PPV/Ca OLEDs by Parker [41], which shows that the currentscales as J ∝ V2/L2.

We assumed that the charge carrier densities at the electrode were equal, and the electrons andholes have the same mobility in the organic film. In addition, we assumed that the charge carriermobility does not depend upon the electric field. These assumptions can be relaxed; we plan todo so in future work. For example, the mobility can be adjusted to account for the electric field

through the functional form μ = μ0 exp√

E/E0, where μ0 and E0 are material parameters [7].Lastly, we analysed kinetically limited OLEDs with small recombination rates (k < 1). We

derived an approximate expression for the recombination rate across the OLED that compareswell with numerics for low-to-moderate reaction rate constants k. A solution extended to applyat larger k is desirable to accurately predict the recombination in an OLED beyond the kineticallylimited regime.

Acknowledgements. S.E.F. acknowledges support by the National Science Foundation Graduate ResearchFellowship under grant no. 0946825. O.S. thanks the United States–Israel Binational Science Foundation (BSF)for awarding him the Professor Rahamimoff travel grant, thereby facilitating this collaboration.

Appendix A. MatchingTo determine the value of a0 in (4.23), we match the field in the space-charge layer to that of theintermediate layer in the domain of overlap, x → ∞ and x → 0. As x → ∞, from (4.9) and (4.13)we have

E0(x) ∼ −2x

+ 4x2 − 8

x3 + O(

1x4

)and E1(x) ∼ ca

3x + 2ca

3+ 4ca

3x2 + O(

1x3

). (A 1)

The expansion of the leading-order intermediate field (4.16) as x → 0 is

E0(x) ∼ −2x

+ ca

3x − 7c2

a180

x3 + O(x4), (A 2)

whereas the next term E1 has the expansion (4.23). Next, we rewrite the space-charge layerexpansions (A 1) in terms of the intermediate coordinate x, collect terms, and then rewrite theintermediate expansion in terms of space-charge layer coordinate x. This matching procedureis discussed by van Dyke [31]. The space-charge layer expansions (A 1) in terms of x, wherex = xδ1/2, are

E(x) = δ1/2

ε

(−2

x+ ca

3x)

+ δ

ε

(4x2 + 2ca

3

)+ o

(δ

ε

). (A 3)

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Recall, δ � 1. The expansion of the intermediate field, (4.23) and (A 2), in terms of the innervariable is

E(x) = δ1/2

ε

(− 2

(δ1/2x)+ ca

3(δ1/2x)

)+ μ

(a0

(δ1/2x)2 + a0ca

6

)+ o

(δ

ε

)(A 4)

For (A 3) and (A 4) to be equala0 = 4 and δ = με. (A 5)

To determine ca we match the four terms in the space-charge layer expansion (4.9), (4.13), (4.28)and (4.32) to the two-term intermediate expansion (4.16) and (4.23). The series approximations forE0 and E1 as x → ∞ and E0 as x → 0 are (A 3) and (A 4), respectively. The O(δ3/2/ε) correction tothe space-charge field, E2, as x → ∞ is

E2(x) ∼ cb

3x + 2cb

3+ 4cb

3x2 + o(

1x3

), (A 6)

whereas the O(δ2/ε) correction, E3, is approximated as (4.32) as x → ∞. The first four terms inthe expansion for the space-charge layer field (E0, E1, E2 and E3) are rewritten as functions of theintermediate spatial variable, x, through the relation, x = x/δ1/2, which gives

E(x) = δ1/2

ε

(−2

x+ ca

3x + c2

a − 9caV�

90x3

)

+ δ

ε

(4x2 + 2ca

3+ cb

3x + 4c2

a − 21caV�

60x2

)+ o

(δ

ε

)(A 7)

The power series expansion of the first two terms representing the intermediate field as x → 0is given by (A 2) and (4.23). These terms are rewritten in terms of the space-charge layer spatialcoordinate x as

E(x) = δ1/2

ε

(− 2

(δ1/2x)+ ca

3(δ1/2x) − 7c2

a180

(δ1/2x)3

)

+ δ

ε

(4

(δ1/2x)2 + 2ca

3+ cb

3(δ1/2x) + 15caV� − 14c2

a60

(δ1/2x)2

)+ o

(δ

ε

). (A 8)

After rewriting (A 7) and (A 8), it is clear that the following matching conditions result:

c2a − 9caV�

90= − 7

180c2

a and4c2

a − 21caV�

60= 15caV� − 14c2

a60

. (A 9)

Both these conditions yield the same expression for ca, namely

ca = 2V�. (A 10)

Appendix B. Integral constraintAn integral constraint (4.2) relates the applied potential to the integral of the electric field acrossthe thin film. The integral (4.2) is broken up into three regions, corresponding to the electric fieldprofiles in the space charge layer, intermediate layer and in the bulk. Thus,

12μV� =

∫ η/ε

0E(x) dx +

∫ γ δ1/2/ε

ηδ1/2/ε

E(x) dx +∫ 1/2

γ

EB(x) dx. (B 1)

In the space charge layer, the lower bound is zero at the anode, whereas the upper bound is aregularization variable η normalized by the space-charge layer width ε, where ε � η � ε/δ1/2. Inthe intermediate layer, the bounds are the variables η and γ normalized by the intermediate layerwidth ε/δ1/2, where ε/δ1/2 � γ � 1. The purpose of η and γ is to avoid diverging contributionsfrom the space-charge layer and intermediate integrals as x → ∞, x → 0 and x → ∞, respectively.The final result cannot depend on η or γ .

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The integral of the space-charge field is∫ η/ε

0E(x) dx ∼

∫ η/ε

0E0(x) dx +

∫ η/ε

0δE1(x) dx + O

(δ3/2η2

ε2

), (B 2)

where the error originates from the integral of the next term, δ3/2E2(x), across the space chargelayer. The integral of the leading order field (4.9) is

∫ η/ε

0E0(x) dx =

∫ η/ε

0

−2x + 2

dx ∼ −2 lnη

ε+ 2 ln 2 + O

(ε

η

), (B 3)

whereas the integral of the first correction (4.13) is∫ η/ε

0δE1(x) dx =

∫ η/ε

0δca

12 + 12x + 6x2 + x3

3(2 + x)2 dx ∼ ca

6δη2

ε2 + O(

δη

ε

). (B 4)

In the intermediate layer, the integral of the leading-order field (4.16) is∫ γ δ1/2/ε

ηδ1/2/ε

E0(x) dx ∼∫∞

ηδ1/2/ε

−2√

ca

sinh(√

cax)dx ∼ ln δ + 2 ln

η

ε+ ln ca − 2 ln 2 − ca

6δη2

ε2 + O

(δ2η4

ε4

).

(B 5)Finally, the integral of the bulk field, whose lower limit γ is replaced by 0, because the bulk fieldis a constant to leading order, is

∫ 1/2

0EB(x) dx ∼ 1

2δ

εV� + O

(δ

ε

). (B 6)

The contributions (B 3)–(B 6) are summed, resulting in

12 μV� + ln caδ + · · · (B 7)

Thus, (B 1) is asymptotic to (B 7) to O(μ); however, there is an O(ln δ) mismatch. Therefore, thebulk field is adjusted to EB = μV� − 2 ln δca. Note that this adjustment of EB does not affect any ofthe preceding analysis.

The new expression for the bulk electric field, after substituting for μV�, δ and ca according to(4.1), (4.24), and (4.33), respectively, is thus

EB = V − 2 ln 2εV + o(ln δ). (B 8)

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