Influence of electronic properties on the threshold behaviourof organic laser diode structures

Christof Pflumma, Christian Gartnerb, Christian Karnutschb, Uli Lemmerb

aMerck KGaA, Liquid Crystals Division, Organic Lighting Technologies, P.O. Box,64271 Darmstadt, Germany

bUniversity of Karlsruhe (TH), Light Technology Institute (LTI), Kaiserstr. 12,76131 Karlsruhe, Germany

ABSTRACT

By employing a combined optical/electronic model, we investigate the effect of electronic properties on theperformance of three layer organic semiconductor structures, which are a potential candidate for future electricallypumped organic laser diodes. The drift-diffusion equations which describe particle transport are coupled to thespatially inhomogeneous laser rate equations to solve for the dynamics of the excited state and photon populationin the laser cavity. Due to the high current densities considered, high particle densities occur, which implies thatannihilation processes between the different particle species have to be considered. On the optical side, we takeinto account the absorption of the metal electrodes required for current injection to obtain the intensity profilesof the guided modes.

Our calculations show that the inclusion of annihilation processes leads to a strong dependence of the laserthreshold on the charge carrier mobilities, in contrast to the situation when exciton annihilation is neglected. Weobserve optimum values for the charge carrier mobilities in the emission layer regarding the threshold currentand power density. On the other hand, an increase of the mobilities in the transport layers leads to a reductionof these quantities. The threshold voltage decreases for increasing mobilities, regardless of the layer in whichthe mobility is increased. For optimised values, we obtain a threshold current density of jthr = 267 A/cm2 withannihilation processes taken into account.

The presented results can serve as guidelines in the search for material combinations and devices structuressuitable for electrically pumped organic semiconductor laser diodes.

Keywords: organic semiconductor laser, electrical pumping, laser threshold, simulation, modelling, drift diffu-sion, wave guiding, laser rate equations

1. INTRODUCTION

One goal of today’s research in the fields of optically pumped organic semiconductor lasers and organic lightemitting diodes (OLEDs) is the realisation of an electrically pumped organic laser diode. Such devices areexpected to provide advantages over their inorganic counterparts like emission over the whole visible spectrumand low cost, large area fabrication. Since organic semiconductors show a broad gain spectrum, they could alsobe used as broad-band optical amplifiers.

Optically pumped organic semiconductor lasers have been demonstrated in the mid 90’s [1–3] already. Re-cently, the threshold has been decreased sufficiently to allow excitation by inorganic laser diodes [4, 5]. Inaddition, highly efficient OLEDs have been manufactured [6–9] and successfully driven by very high currentdensities [10–12]. Despite this progress, no organic laser diode has been demonstrated yet. It seems that thereare still a lot of obstacle to overcome before a real breakthrough can be achieved, although Duarte et al. haveshown spatially coherent emission from a tandem OLED recently [13].

With the present article, we want to contribute to the scientific discussion by presenting simulation resultsobtained with an improved version of our organic laser diode model [14,15]. The focus is on the influence of thecharge carrier mobilities on the threshold of the devices. After a short explanation of the employed physical inthe following section 2, the results of the mobility variations are presented in section 3.

Further author information: Send correspondence to Christof Pflumm. E-mail: christof.pflumm@merck.de

Organic Light Emitting Materials and Devices X, edited by Zakya H. Kafafi, Franky So,Proc. of SPIE Vol. 6333, 63330W, (2006) · 0277-786X/06/$15 · doi: 10.1117/12.680733

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2. PHYSICAL MODEL

The physical model employed for the calculations presented in this article has been described in detail in variousother articles [14–18], so we will only give a short overview for readers not familiar with the material. Our modeldescribes the electronic and optical properties of three layer organic laser diode structures. Results of calculationsare, amongst others, the transient and spatially dependent particle densities, electron-hole recombination, excitonannihilation rates, the electric field as well as the photon density in the laser cavity. If carried out for differentexcitation current densities, the results of the calculations allow for the evaluation of macroscopic quantities likethe laser threshold.

2.1. Device structure

We consider an organic semiconductor device consisting of a hole transporting layer (HTL), an emission layer(EL) which acts as laser gain medium and an electron transporting layer (ETL). These layers are sandwichedbetween two metal electrodes which inject charges into the device (see figure 1). For a laser diode, it is notnecessary that one of the electrodes is transparent as light may be emitted along the direction of wave guiding,which is perpendicular to the current flow (the vertical direction in the figure).

Figure 1. Structure of the simulated three layer laserdiode with hole transport (HTL), emission (EL) andelectron transport layer (ETL).

We assume that the emission is perpendicular to the cur-rent flow and that the end facets of the laser are 100% re-flecting, which could be achieved to a good approximation byemploying DBR mirrors. The structure described is that of aclassical inorganic laser diode. We suggest a silicon wafer assubstrate due to its high thermal conductivity and flatness.For the anode, a high work function metal like platinum orgold could be employed for efficient hole injection.

2.2. Electronicproperties and annihilation processes

A commonly used drift-diffusion model is applied to accountfor the motion of electrons, holes as well as singlet and tripletexcitons. The excitons are formed by recombination of elec-trons and holes in the ratio 1/4 singlets and 3/4 triplets.The parameters necessary for the calculations are listed intable 1.

It is assumed that the HTL/EL interface completely blocks electrons while the EL/ETL interface completelyblocks holes. Furthermore, both interfaces are assumed to be blocking for excitons. These assumptions imply a100% exciton generation efficiency (triplet plus singlet). Very high internal quantum efficiencies can indeed beobserved in OLEDs made from small molecules [19, 20], which justifies these simplified assumptions.

We found that annihilation processes have a significant influence on the device properties due to the highparticle densities necessary to reach the laser threshold. As a consequence, we have supplemented our previousmodel [14, 15] by singlet-singlet, singlet-triplet, triplet-triplet, singlet-polaron and triplet-polaron interactions.For a detailed description of the annihilation model and the employed parameters, see the contribution in theseproceedings by Gartner et al. [18].

2.3. Modal profile and wave guide absorption

A commonly used slab approximation for dielectric wave guides (see [21] and references therein) was employedto determine the guided modes in the layer structure. The main results of the calculation are the normalisedmodal intensity profile I(x) of the guided modes and the corresponding absorption coefficients α. I(x) describesthe energy distribution in the guided mode, α determines the photon decay in the laser cavity [14, Eq. (53)].

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Electronic properties Optical properties

Parameter Value Parameter Value

µHTL = µETL 5·10-4 cm2/(Vs) λ 620 nm

µEL;h = µEL;e 5·10-6 cm2/(Vs) σstim 7·10−16 cm2

τS1 2 ns β 10−5

τT1 25 µs nSi 3.906-0.022i

DS1 = DT1 5·10-4 cm2/s nAu 0.13-3.16i

T 300 K nAl 1.29-7.48i

εHTL = εEL = εETL 4.0 nHTL = nETL 1.7

dETL = dHTL 400 nm nEL 1.75

dEL 500 nm nAir 1

Table 1. Overview of the employed electronic and optical material properties. Singlet and triplet excitons are denotedby S1 and T1, respectively. The parameter variations presented in this article concentrate on the variation of the chargecarrier mobilities in the emission and transport layers, all other parameters remain fixed.

2.4. Modelling of the laser dynamics

The two main equations which describe the interaction between the population of the upper laser level (thesinglet excitons, denoted by S1) and the photon density have been deduced in detail in [15, Eqs. (8) and (14)],so we will only summarise them shortly.

The change in the singlet exciton density is given by

∂nS1(x, t)∂t

= SS1(x, t)︸ ︷︷ ︸

Generation,Annihilation

− nS1(x, t)τ

︸ ︷︷ ︸

Decay

− ∂2 (DS1nS1(x, t))∂x2

︸ ︷︷ ︸

Diffusion

− κstimq(t)I(x)nS1(x, t)︸ ︷︷ ︸

Stimulated emission

(1)

where the source term SS1(x, t) summarises the various particle interaction processes that influence the excitondensity (electron-hole recombination, annihilation processes). Singlet exciton decay due to spontaneous emissionand non-radiative decay channels is represented by the second term, the third term represents exciton diffusion.The last term describes the exciton loss due to stimulated emission. It is proportional to the overlap between thesinglet exciton density nS1(x, t) and the normalised modal intensity profile I(x), multiplied by the areal photondensity q(t). κstim is essentially proportional to the stimulated emission cross section and thus related to thestrength of stimulated emission in the laser medium.

The photon density changes due to coupling of spontaneous emission into the wave guide and stimulatedemission from the singlet excitons (second and third term on the right side). Absorption in the wave guide isdescribed by the cavity lifetime τcav. The following equation is used to describe these processes:

∂q(t)∂t

= − q(t)τcav

︸ ︷︷ ︸

Wave guide losses

+β

τrad

∫ dx

0

nS1(x, t) dx

︸ ︷︷ ︸

Spontaneous emission

+ κstimq(t)∫ dx

0

I(x)nS1(x, t) dx

︸ ︷︷ ︸

Stimulated emission

(2)

β = 10−5 is a geometry factor which has virtually no influence on the results if chosen in the correct range.

3. RESULTS

3.1. Variation of the charge carrier mobilities in the emission layer

3.1.1. Balanced mobilities

In this section, we present results from calculations in which the mobilities of holes µEL;h and electrons µEL;e inthe emission layer were varied simultaneously. This means that µEL := µEL;h = µEL;e (“balanced mobilities”).All other parameters have been kept constant.

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Figure 2, left, shows the dependence of the laser threshold current density jthr on µELIf no annihilationprocesses are taken into account (long dashed line), the dependence is rather weak. A shallow minimum ofjthr = 177 A/cm2 is observed for µEL ≈ 5·10−7 cm2/Vs (not shown). The threshold increases to jthr = 227 A/cm2

for µEL = 5 · 10−4 cm2/Vs, which amounts to a factor of ≈ 1.3 for a change of mobility by three orders ofmagnitude. At first sight, even this weak dependence is unexpected because of the assumed blocking propertiesof the HTL/EL and EL/ETL interfaces: Every injected charge leads to the creation of an exciton, which meansthat the exciton creation rate is independent of the charge carrier mobilities. There is, however, a dependence ofthe rate of stimulated emission on the shape of the exciton density (see last term in equation (2)). As the chargecarrier mobilities influence the spatial distribution of the excitons, the change in threshold can be understoodas illustrated in figure 2, right. In the graph, the normalised modal intensity profile is shown together withthe singlet exciton density for three different mobilities at equal excitation current density (and thus excitongeneration rate) of j = 740 A/cm2. It can be seen that the exciton density in the middle of the device is highestfor µEL = 10−5 cm2/Vs (lowest threshold). This results in a high value of the overlap integral

∫ dx

0 I(x)nS1(x, t)dxand hence a high rate of stimulated emission. If µEL is increased, the value of the exciton density in the middleof the device drops, while it increases near the HTL/EL and EL/ETL interfaces, where I(x) is smaller. Thisleads to a smaller overlap integral and consequently to a higher threshold.

0

200

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1000

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1400

10−6 10−5 10−4 10−3 0

1

2

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4

5

6

7

j thr (

A/c

m2 )

Sing

let a

nnih

ilatio

n ra

te (

1030

/(m

3 s))

µEL;e = µEL;h (cm2/Vs)

jthr with annihilationjthr without annihilationSinglet annihilation rate

1021

1022

1023

1024

400 500 600 700 800 9000.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

n exc (

1/m

3 )

Nor

mal

ised

mod

al in

tens

ity

x (nm)

µEL = 10−5 cm2/Vs10−4 cm2/Vs10−3 cm2/Vs

Modal intensity

Figure 2. Left: Threshold current density jthr for various mobilities in the emission layer. Electron and hole mobilitiesare assumed to be equal. The solid line was calculated with, the long dashed line without annihilation processes. It canbe seen that annihilation processes lead to a strong influence of the threshold on the mobilities in the emission zone. Alsoshown is the total singlet annihilation rate at an excitation current density of j = 740 A/cm2. Right: Spatial distributionof singlet excitons for three different electron and hole mobilities in the emission layer. Electron and hole mobilities areequal in the three cases, µEL = µEL;h = µEL;e. The HTL/EL interface is at the left, the EL/ETL interface at the rightside of the figure. Also given is the normalised modal intensity profile I(x) to illustrate the overlap between the excitondensity and the guided mode. The calculations have been carried without annihilation processes.

The situation is completely changed if annihilation processes are included in the calculations (solid line infigure 2). A clear minimum of jthr = 267 A/cm2 now occurs at µEL = 10−4 cm2/Vs, which is very close to thevalue of µHTL = µETL = 5 · 10−4 cm2/Vs for the charge carrier mobilities in the transport layers. It seems thatsimilar values of the mobilities in all layers lead to low threshold values.

A decrease of the mobilities by two orders of magnitude results in a more than five fold increase in thresholdcurrent density, much more than in the case without annihilation processes. This strong dependence cannot beexplained by the rather weak change of the overlap integral with µEL. However, the jthr dependence can becorrelated to the rate of exciton annihilation Sann, shown as the short dashed line in figure 2. The shape ofjthr and Sann are similar, which shows that the change of the particle distributions (and thus annihilation rates)with µEL is responsible for the threshold behaviour. A quantitative analysis is not possible in the scope of thepresent article, as the laser onset is a highly nonlinear process. This implies that a simple connection betweenthe overlap integral in equation (2) and the laser threshold cannot be given.

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Besides the threshold current density, two other quantities are of interest in the design of an organic laserdiode: the voltage Uthr and the power density Pthr at threshold. The threshold voltage is of interest becauseit gives a guideline for the design of a suitable excitation source. Furthermore, it gives information about thefields occurring in the device, which is important to estimate whether dielectric breakdown might occur. Thethreshold power density is connected to the energy deposited in the device. From this, an estimated temperaturerise might be estimated, giving a feeling whether a device with certain parameters might be destroyed by heatingeffects.

0

2

4

6

8

10

12

14

10−6 10−5 10−4 10−3102

103

104

105

Uthr (

kV)

Pthr (

kW/c

m2 )

µEL;e = µEL;h (cm2/Vs)

UthrPthr

Figure 3. Voltage Uthr and power density Pthr at thresholdplotted against the electron and hole mobility in the emissionlayer. Both mobilities are equal. Though the threshold volt-age decreases monotonically, the threshold power density hasa minimum at ≈ 10−4 cm2/Vs, which is due to the increase ofthe threshold current density above mobilities of 10−4 cm2/Vs(see figure 2).

In figure 3, Uthr and Pthr are shown for differ-ent electron and hole mobilities in the emission layer.The mobilities are balanced, µEL;h = µEL;e, and an-nihilation processes were accounted for. It can beseen that the voltage required to reach thresholddrops monotonically in the considered mobility range.Above µEL ≈ 10−4 cm2/Vs, the threshold voltagedrops only slightly and reaches Uthr = 1.14 kV atµEL = 10−3 cm2/Vs. Though this value appearsquite high, pulsed current sources delivering compa-rable voltages are currently under test at the LTI.The minimum value for the threshold power densityis Pthr = 3440 kW/cm2, which is roughly 70 timeshigher than the value achieved by Tessler et al. in1998 in a single layer PPV OLED [22]. However, thework from the Adachi group shows that this limit canbe pushed a great deal further by employing thermallyoptimised device structures.

3.1.2. Variation of the electron mobility

For single layer polymer OLEDs, balanced mobilitiesin the emission layer are important to obtain goodefficiencies. The reason is that for unbalanced devices,exciton generation is shifted towards one of the electrodes, which favours fluorescence quenching at and excitondiffusion towards the corresponding electrode. Though both of these effects do not play a role in the devicespresented here, a similar effect can be observed (see figure 4, left): There is a clear minimum of the thresholdcurrent density with respect to the electron mobility µEL;e in the emission layer, if the hole mobility µEL;h =5·10−6 cm2/Vs is kept constant (“unbalanced mobilities”). Interestingly, the optimum does not occur at balancedmobilities, but at a slightly higher value of 10−5 cm2/Vs. The increase of the electron mobility from its balancedvalue first has a beneficial effect, as is the case when both electron and hole mobility are increased (see figure 2).A further increase however is detrimental, because the singlet exciton distribution is shifted towards the HTL/ELinterface. This results in a higher annihilation rate, which is also shown in the figure.

As can be seen from the graph with the annihilation processes neglected, the overlap between the excitondensity and the modal intensity profile only plays a minor role, as explained in the preceding section.

While the threshold current density exhibits a minimum, the voltage Uthr required to reach threshold decreaseswith increasing µEL;e (see figure 4, right). The relatively strong decrease of Uthr implies that the minimum thresh-old power density is reached at a slightly higher value of the electron mobility (around µEL;e = 2 · 10−5 cm2/Vs)than jthr.

3.2. Influence of the charge carrier mobilities in the transport layers

The preceding two sections addressed the electronic properties of the emission layer. It was shown that thereexist minima of the threshold current density and the threshold power density with respect to the charge carriermobilities in the emission layer. In the rest of this article, we discuss the influence of the charge carrier mobilitiesin the transport layers.

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0

200

400

600

800

1000

1200

1400

10−6 10−5 10−4 10−35.0

5.5

6.0

6.5

7.0

7.5

8.0

j thr (

A/c

m2 )

Sing

let a

nnih

ilatio

n ra

te (

1030

/(m

3 s))

µEL;e (cm2/Vs)

jthr with annihilationjthr without annihilationSinglet annihilation rate

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10−6 10−5 10−4 10−3 1000

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7000

8000

9000

10000

Uthr (

kV)

Pthr (

kW/c

m2 )

µEL;e (cm2/Vs)

UthrPthr

Figure 4. Left: Dependence of the threshold current density in the case of unbalanced charge carrier mobilities inthe emission layer. The solid line is for the case when annihilation processes are considered in the calculations, whileannihilation was neglected in the calculation for the long dashed curve. The total singlet exciton annihilation rate(calculated at j = 895 A/cm2) exhibits the same shape as jthr, indicating that changes of the particle distribution withµEL;e are responsible for the behaviour of the threshold current density. Right: Threshold voltage and threshold powerdensity in dependence of the electron mobility in the emission layer. Annihilation processes are taken into account andthe hole mobility was fixed at µEL;h = 5 · 10−6 cm2/Vs.

3.2.1. Equal hole and electron mobilities

Increasing the charge carrier mobilities of the transport layers always has a beneficial effect. Though for equalhole and electron mobilities in the respective transport layers (µTL = µETL = µHTL), there is a very slightincrease of jthr with increasing µTL (see solid line in figure 5, left), this is outweighed by the decrease in Uthr

and consequently Pthr (figure 5, right).

100

200

300

400

500

600

700

800

900

1000

10−5 10−4 10−3 10−24.8

5.0

5.2

5.4

5.6

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6.4

6.6

j thr (

A/c

m2 )

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let a

nnih

ilatio

n ra

te (

1030

/(m

3 s))

µETL = µHTL (cm2/Vs)

jthr with annihilationjthr without annihilationSinglet annihilation rate

2

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10−5 10−4 10−3 10−2103

104

105

Uthr (

kV)

Pthr (

kW/c

m2 )

µETL = µHTL (cm2/Vs)

UthrPthr

Figure 5. Left: Change of threshold current density with the electron and hole mobility in the transport layers. Bothmobilities were assumed to be equal (µETL = µHTL) and calculations were carried out with and without annihilationprocesses included. The total singlet exciton annihilation rate, calculated at j = 740 A/cm2, is shown to explain thesharp rise of jthr below a mobility value of ≈ 3 · 10−5 cm2/Vs. Right: Threshold voltage and threshold power density independence of the charge carrier mobilities in the transport layers, with annihilation processes included.

Decreasing the mobility in the transport layers below a value of ≈ 3 · 10−5 cm2/Vs however leads to a rapidincrease of the threshold, which is accompanied by an increase in the singlet exciton annihilation rate (shortdashed line in figure 5, left). This rise is intensified by the fact that the overlap between the exciton density

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and the modal intensity profile decreases rather strongly. This can be seen from the behaviour of jthr whenannihilation processes are neglected (long dashed curve in figure 5, left), in which case a change in threshold canbe attributed to the overlap integral (last term of equation (2)) alone.

3.2.2. Variation of the electron mobility in the electron transport layer

The case of varying the mobility in the ETL while keeping the hole mobility in the HTL constant at µHTL =5 · 10−4 cm2/Vs is illustrated in figure 6. As the explanation of the physical effects leading to the behaviourshown in the two graphs is the same as in the last section, it will not be discussed again at his point.

500

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850

10−5 10−4 10−3 10−25.0

5.2

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j thr (

A/c

m2 )

Sing

let a

nnih

ilatio

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te (

1030

/(m

3 s))

µETL (cm2/Vs)

jthr with annihilationSinglet annihilation rate

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10−5 10−4 10−3 10−2 1000

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7000

8000

9000

10000

Uthr (

kV)

Pthr (

kW/c

m2 )

µETL (cm2/Vs)

UthrPthr

Figure 6. Left: The threshold current density jthr only decreases very slightly with increasing electron mobility inthe ETL above a value of µETL ≈ 5 · 10−4 cm2/Vs. The hole mobility in the HTL was fixed at a constant value ofµHTL = 5 · 10−4 cm2/Vs. The sharp rise for lower µETL is due to the increasing singlet annihilation rate, shown as longdashed line, but the overlap between exciton density and modal intensity profile also contributes (not shown). Right:Voltage and power density required to reach threshold for different electron mobilities in the electron transport layer. Thecalculations considered annihilation processes and the hole mobility in the HTL was fixed at µHTL = 5 · 10−4 cm2/Vs.

4. SUMMARY

We have employed an improved version of our organic laser diode model to investigate the influence of the chargecarrier mobilities on the laser threshold of three layer organic semiconductor structures. In the presence ofexciton annihilation processes, the mobilities of the charge carriers have a pronounced influence on the thresholdcurrent density, in contrast to the case when annihilation is neglected.

It was found that the threshold current jthr and threshold power density Pthr exhibit minima with respectto the mobilities µEL;e and µEL;h in the emission layer. The reason is a change of the spatial distribution ofthe different particle species with µEL;e and µEL;h, resulting in different singlet exciton annihilation rates. Incontrast, jthr and Pthr decrease with increasing mobility of the charge carries in the transport layers. Thismeans that increasing the charge carrier mobilities in the transport layers always has a beneficial effect. Thevoltage required to reach threshold decreases with increasing mobility in all investigated cases. Our results showthat for device optimisation, annihilation processes need to be included in the calculations, because importantdependencies remain undiscovered otherwise.

We would like to mention that the effect of polaron absorption has not been included in our model yet. Asthis is considered to be another major problem for the realisation of organic laser diodes, work is underway toimprove our model to describe this effect.

5. ACKNOWLEDGEMENTS

The authors gratefully acknowledge financial support from the German Federal Ministry for Education andResearch (BMBF) within the OLAS consortium (FKZ 13N8168A).

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