Lasing in the Space Charge-Limited Current Regime
LLasing in the Space Charge-Limited Current Regime
Alex J. Grede ∗ and Noel C. Giebink Department of Electrical Engineering, Pennsylvania State University, University Park, Pennsylvania 16802 (Dated: September 29, 2020)We introduce an analytical model for ideal organic laser diodes based on the argument thattheir intrinsic active layers necessitate operation in the bipolar space charge-limited current regime.Expressions for the threshold current and voltage agree well with drift-diffusion modeling of complete p-i-n devices and an analytical bound is established for laser operation in the presence of annihilationand excited-state absorption losses. These results establish a foundation for the development oforganic laser diode technology.
Epitaxial inorganic semiconductors are the basis fornearly all laser diodes today. Organic semiconductorscould provide a new, wavelength-tunable laser diode plat-form, but are disadvantaged by lower charge carrier mo-bility, lower thermal conductivity, and more efficient ex-cited state quenching interactions. Beyond their materialproperties, however, organic and inorganic laser diodesalso fundamentally differ in their mode of electrical op-eration: gain in the inorganic case is driven by diffu-sion current in a p-n junction,[1], whereas in the organiccase, the need for an intrinsic active layer (since electri-cal doping severely quenches organic semiconductor ex-cited states[2, 3]) requires gain to be achieved by a spacecharge-limited (SCL) drift current. Although the impactof SCL current is well-appreciated for organic light emit-ting diodes (OLEDs)[2], its implications for organic laserdiode operation have not been explored in detail.Here, we derive expressions for the threshold voltageand current density of ideal organic laser diodes wheregain is provided by a bipolar SCL current in the intrinsicactive layer of a p-i-n device. The results are extendedto treat organic laser operation in the presence of tripletexciton and polaron losses, and are validated using thecommercial device simulator
Setfos [4]. As organic laserefforts accelerate following the initial demonstration bySandanayaka et al. [5], the framework established hereshould prove useful for guiding future development.Space charge effects become significant in a semicon-ductor when the transit time of electrons and holesdrifting across it ( τ tr = L /µ n,p ( V − V bi ) , where L is the layer thickness, V is the applied voltage, V bi isthe built in potential, and µ n,p is the electron or holemobility) is smaller than the dielectric relaxation time, τ rlx = (cid:15)/q ( µ n n + µ p p ) , set by its dielectric constant ( (cid:15) )and equilibrium free charge density ( n , p , with q rep-resenting the elementary charge). In essence, this situa-tion corresponds to injecting more charge into the semi-conductor than exists in equilibrium and it is frequentlythe case in undoped organic semiconductors[2, 3, 6]. Or-ganic laser diodes are generally expected to operate inthis regime because 1) they operate at high current den-sity where drift dominates diffusion and 2) although theirelectron and hole transport layers are typically dopedto achieve high conductivity[5], their active layer (i.e. gain region) must remain intrinsic to avoid strong exci-ton quenching with ionized dopant and polaron species.Thus, τ tr ,n , τ tr ,p (cid:28) τ rlx is well-satisfied and, notwith-standing the limited diffusion of charge carriers from thecontacts/transport layers into the active layer[7], opticalgain must be achieved in the bipolar SCL current regime.The two-carrier SCL current problem was originallysolved by Parmenter and Ruppel for the ideal case ofa trap-free insulator[8]. The result exhibits the same V /L dependence as the unipolar case, but predicts anenhancement of the overall current density that dependson the recombination rate between electrons and holes.In the following, we use this solution to establish analyt-ical bounds and scaling relationships for an ideal organiclaser diode with Ohmic electron and hole injection intothe active layer; any limitations on injection would con-stitute a source of non-ideality.In organic semiconductors with a large exciton bindingenergy and exchange splitting between bright (singlet)and dark (triplet) exciton states, the net modal gain canbe expressed as: g = Γ σ st ( N S − N tr ) − α , (1)where Γ is the modal confinement factor, σ st is the stim-ulated emission cross-section, N S is the singlet excitondensity, N tr is the transparency density, and α is the netoptical loss due to outcoupling, scattering, or parasiticabsorption from other materials or excited states. Atthreshold, g = 0 and thus the threshold singlet densityis: N th = N tr (cid:18) αN tr Γ σ st (cid:19) . (2)The singlet density is governed by the rate equation: d N S d t = χ S R − N S ( k S + k Q ) − gv g N ph , (3)where R is the exciton generation rate due to Langevinrecombination of electrons and holes (i.e. R = γnp ,where γ = q ( µ n + µ p ) /(cid:15) )[6], χ S = 1 / is the singletspin fraction[9], k S is the natural singlet decay rate, and k Q accounts for any additional quenching processes suchas exciton-exciton and exciton-polaron annihilation ex-plored below. The term on the right-hand side of Eq. (3) a r X i v : . [ c ond - m a t . m t r l - s c i ] S e p np E np . . . . . . − Position ( x/L ) N o r m a li z e d P r o fi l e s FIG. 1. Normalized carrier concentration, electric field, and np product as a function of position for bipolar space charge-limited current with a Langevin recombination rate, equalcharge carrier mobilities, and Ohmic electron and hole injec-tion. accounts for stimulated emission, with N ph and v g equalto the photon density and group velocity of the lasingmode, respectively.At threshold, g = 0 and thus Eq. (3) can be solved insteady state to yield the threshold recombination rate: R th = N tr k S χ S (cid:18) k Q k S (cid:19) (cid:18) α Γ σ st N tr (cid:19) , (4)which factors into an ideal rate multiplied by exciton andoptical loss terms in parentheses. Equation (4) is derivedassuming a spatially uniform exciton density, which isrigorously valid for the ideal case of µ n = µ p as shownbelow, and thus also justifies the neglect of exciton dif-fusion in Eq. (3).The Parmenter-Ruppel solution subsequently estab-lishes the link between recombination rate, voltage, andcurrent density in the device. The general result is pro-vided in the Supplementary Material; however, we focushere on the specific case of equal electron and hole mo-bilities ( µ n = µ p = µ ) and perfect charge balance (i.e.every injected electron recombines with a hole in the ac-tive layer) since it constitutes the limit of an ideal organiclaser diode.The current-voltage relationship is given by the Mott-Gurney law: J = 98 (cid:15)µ eff ( V − V bi ) L , (5)but with an effective mobility, µ eff = 256 µ/ (9 π ) ≈ . µ ,larger than that of the individual carriers due to the neu-tralization of space charge by recombination. For the assumed case of Ohmic electron and hole injection intothe active layer, the built-in potential is approximatedby the bandgap energy, V bi ≈ E g /q . The position, elec-tric field, and carrier densities are most conveniently ex-pressed parametrically in terms of the fractional electroncurrent j n = J n /J : x = Lj n E = (cid:112) j n (1 − j n ) (cid:115) LJ(cid:15)µ (6a) n = (cid:115) j n − j n (cid:115) (cid:15)J q µL p = (cid:115) − j n j n (cid:115) (cid:15)J q µL . (6b)Each quantity scales linearly with the applied voltage(square root of the current) and is shown normalized toits value at the midpoint of the active layer in Fig. 1. The np product is spatially uniform as evident from inspec-tion of Eq. (6b), justifying the neglect of spatial variationin deriving Eq. (4). As in the case of unipolar SCL cur-rent, these equations break down in the immediate vicin-ity of the contacts as the field drops to zero and transportbecomes diffusive.It is subsequently straightforward to evaluate the cur-rent density and voltage in terms of the recombinationrate at threshold: J th = qLR th and V th = πL (cid:115) qR th (cid:15)µ + V bi . (7)The expression for J th follows directly from particle con-servation due to the assumption of perfect charge balanceand shows that the simple dimensional analysis estimatecommonly used in the literature[10] is exact for a bipolarSCL current. We note that, although Eq. (7) is derivedunder the assumption µ n = µ p , it remains reasonably ac-curate even when the mobilities differ by up to an orderof magnitude provided that their average value is usedfor µ ; see the Supplementary Material for details.In evaluating R th from Eq. (4), we note that, while N tr can be obtained from thermodynamic considerations(see the Supplementary Material), it is immaterial inpractice because the cold cavity loss ( α cav ) typically sat-isfies α cav / Γ σ st N tr (cid:29) and thus R th ≈ N th , k S /χ S ,which is readily determined from the threshold singletdensity ( N th , ) measured experimentally under impul-sive optical pump conditions. Taking parameter val-ues typical of the 4,4 (cid:48) -bis[(N-carbazole)styryl]bi-phenyl(BSBCz) diode lasers reported by Sandanayaka et al. [5]: N th , = 2 × cm − , k S = 1 ns − , (cid:15) = 4 (cid:15) , µ ≈ × − cm V − s − , E g ≈ . eV, and L = 150 nm,we obtain J th = 190 A cm − , V th = 26 V and a thresh-old power density P th = J th V th = 5 . kW cm − aslower bounds for laser operation. For comparison, theexperimentally-recorded values are J th = 600 A cm − , V th = 34 V and P th = 20 kW cm − [5].Equation (7) highlights the importance of active layerthickness for organic laser diodes. Whereas J th depends threshold µ = 10 − cm /VsEGDMSCLCdrift-diffusion − Bias ( V − V bi ) [V] C u rr e n t D e n s i t y ( J ) (cid:2) A c m − (cid:3) FIG. 2. Drift-diffusion simulations of the current-voltage re-lationship for a BSBCz-like active layer with Ohmic majoritycarrier injection and equal electron and hole mobilities thatare either constant (blue line) or given by the extended Gaus-sian disorder model (EGDM) for disordered organic semicon-ductors (red). The black dashed line shows the analyticalspace charge-limited current (SCLC) prediction and solid cir-cles denote the lasing threshold in each case. weakly on L since Γ ∝ L to first order, the thresholdvoltage and power density both scale as L / . The is im-portant in the context of thermal management becauseorganic laser diodes not only generate more heat thantheir inorganic counterparts (due to their higher volt-age), but have more difficulty dissipating it (due to theirlower thermal conductivity) and are less able to with-stand high temperature without degrading. Electric fieldstrength, which scales as L / , is another concern sinceits maximum in the example above ( E max = 2 MV cm − )is comparable to the ∼ MV cm − breakdown field ofmany organic semiconductors[2, 3, 6]. To this point, di-electric breakdown was reported after roughly fifty pulsesnear threshold in Ref. [5].Given that the threshold relations above neglect theexistence of charge transport layers and also assume in-finite carrier densities at the active layer interfaces, it isimportant to test the accuracy of these results againstfull drift-diffusion modeling of a real device architec-ture. Retaining the BSBCz parameters from above, wetreat the case of an organic laser with a 150 nm-thick intrinsic active layer and fix the majority carrier con-centrations at each edge to . × cm − to simu-late Ohmic injection from heavily doped transport lay-ers. Drift-diffusion simulations are carried out usingthe commercial software Setfos [4] and the results arepresented in Fig. 2 for the case of a constant mobility( µ n = µ p = 1 × − cm V − s − ; blue) and for thecase in which it depends locally on electric field and car-rier density according to the extended Gaussian disordermodel (EGDM, red); details of the EGDM are providedin the Supplementary Material. The threshold currentdensity in both simulations is in good agreement withthe analytical prediction; however, the threshold voltagein the EGDM case is slightly lower. This is due to theincrease in mobility with field and carrier concentrationin the EGDM model, which highlights the importance ofusing mobility values that are congruent with the condi-tions at threshold.From Fig. 1, it is evident that to sustain the elec-tron and hole concentrations in the middle of the de-vice (and thus the np product everywhere), the majoritycarrier concentrations injected at the active layer edgesmust be roughly an order of magnitude higher. Insert-ing J th = 190 A cm − from Fig. 2 into Eq. (6b) suggeststhat edge carrier concentrations as low as ∼ × cm − are sufficient to maintain charge balance up to the las-ing threshold, in agreement with the full drift-diffusionmodel. This is important because it sets the minimumdoping concentration required for the transport layers ina laser diode (i.e. due to continuity of n and p ) which,notably, is within the range typical for p-i-n OLEDs [11];more detail on this point is provided in the Supplemen-tary Material.At this stage, it is also important to assess the im-pact of triplet exciton and polaron-related optical losses(due to respective absorption cross-sections σ TT [12] and σ PP at the lasing wavelength) and quenching interactions(with respective annihilation rate coefficients k STA and k SPA ) [13, 14]. Because the latter typically depend onthe former through Förster energy transfer[15], absorp-tion and annihilation losses must be treated on equalfooting. Assuming only one species of polaron (holes inthis case) is detrimental, the quenching rate in Eq. (3)becomes k Q = k SPA p mid + k STA N T and the loss in Eq. (1)becomes α = α cav +Γ( π p mid σ PP + N T σ TT ) , where p mid isthe hole density at the midpoint of the active layer. Us-ing the threshold singlet density in the absence of tripletand polaron losses, N th , (defined in Eq. (2) by the coldcavity loss), Eq. (4) can be rewritten in the form of animplicit quartic equation: R th = N th , k S χ S (cid:32) R th χ T t rise k STA k S + k SPA k S (cid:115) (cid:15)R th qµ (cid:33) (cid:32) R th χ T t rise σ TT N th , σ st + πσ PP N th , σ st (cid:115) (cid:15)R th qµ (cid:33) . (8)In this expression, the rise time ( t rise ) is important be-cause it determines the extent to which long-lived tripletexcitons accumulate before the full current density isachieved (i.e. N T ≈ Rχ T t rise by the time the full re-combination rate, R , is reached)[16]. Although t rise isnominally characteristic of the electrical pulse, it cannotbe significantly faster than the transit time of charge car-riers drifting across the active layer since this is the timeit takes to establish the SCL recombination profile to be-gin with ( τ tr ≈ ns for the BSBCz example above) [17].Note also that, in seeking to approximate the spatiallynonuniform polaron density with a single effective valuefor annihilation and absorption, it is more accurate touse the midpoint density for the former and the layer-averaged density ( π p mid ) for the latter as discussed inthe Supplementary Material.The form of Eq. (8) is useful because it allows the im-pact of each loss mechanism to be understood individu-ally. For example, singlet-triplet annihilation on its owndoubles the threshold when the middle term in the firstset of parentheses is equal to unity, which correspondsto the situation where t rise = χ S / (2 χ T N th , k STA ) . Moregenerally, for Eq. (8) to have any physically meaning-ful solution, the absorption and annihilation coefficientsmust satisfy the following inequalities: (cid:112) σ TT /σ st + (cid:113) N th , k STA /k S < (cid:112) χ S / ( χ T k S t rise ) (9a) and k SPA σ PP < χ S σ st qµ/ ( π(cid:15) ) . (9b)Figures 3(a) and 3(b) plot these bounds for triplet andpolaron losses, respectively, along with contours thatshow the relative increase in threshold caused by eachspecies. Figure 3(b) also includes the results from full Setfos numerical modeling (solid symbols) of the po-laron case, validating the midpoint and average carrierdensity approximations used to derive Eq. (8).These results highlight the difference between tripletlosses, where a small increase in absorption cross-sectionor annihilation coefficient can mean the difference be-tween a modest threshold increase and prevention of las-ing outright, and polaron losses, where the penalty ismore gradual and does not depend on external factors like α cav or t rise . A blunt way of characterizing this differenceis that triplet excitons are either insignificant or catas-trophic, whereas the region of parameter space betweenthese extremes is broader for polarons. This highlightsthe importance of designing new organic gain media likeBSBCz that have low overlap between their emission andtriplet-triplet absorption spectra [5], and of implementingdevice architectures that can deliver high-speed electri-cal pulses[19], since this is the difference between successand failure for the classic Alq :DCM[16] gain medium inFig. 3(a).Another intuitive, but important guideline is to seekorganic gain media with equal (and maximal) electronand hole mobilities. To the extent that perfect charge balance is maintained and the envelope of the lasing modevaries negligibly over the active layer, the general solutionfor µ n (cid:54) = µ p given in the Supplementary Material showsthat Eq. (7) is largely unaffected when the mobility ofone carrier dominates (in this case µ eff is just the mobil-ity of the more mobile charge carrier). The problem withimbalanced mobility, however, is that it concentrates re-combination toward the lower mobility carrier side of theactive layer. This not only exacerbates annihilation loss,but also makes it more challenging to maintain chargebalance since the less mobile carrier must be injected athigher density to sustain the same total recombinationrate in a narrower region of space; full details are pro-vided in the Supplementary Material.Finally, we note that the framework developed heremay also be relevant for metal halide perovskite (MHP)lasers. Though SCL current is less well-studied in thismaterial class (due in part to complications with ionmovement)[20, 21], the observation that MHP light emit-ting diodes use undoped active layers[22, 23] along withthe fact that electrical doping is a basic challenge forthese materials to begin with[20] makes it plausible thata future MHP laser diode will operate in the bipolarSCL current regime. In this case, however, recombi-nation tends to be sub-Langevin (i.e. the bimolecular J th = 2 J th , Alq :DCM t rise = 10 ns − − σ TT (cid:2) c m (cid:3) Setfos simulation J th = 2 J th , − − − − − − Annihilation Rate (cid:2) cm s − (cid:3) σ PP (cid:2) c m (cid:3) (a)(b) FIG. 3. Contours showing the relative increase in thresh-old current due to parasitic absorption and singlet excitonannihilation caused by (a) triplet excitons and (b) hole po-larons. Lasing is forbidden in the shaded region at any currentdensity. The solid lines are calculated using Eq. (8) and arecompared in (b) with the results of
Setfos numerical model-ing (solid circles). The Alq :DCM material system markedby the red dot in (a) lies in the forbidden region for the t rise = 10 ns case shown. All of the calculations are basedon the BSBCz parameters given in the main text along with σ st = 2 × − cm [18]. recombination coefficient is substantially smaller thanthe Langevin rate), which causes bipolar SCL currentto take place in the injected plasma regime where n ≈ p everywhere[24]. Unfortunately, the analytical solution inthis limit can greatly overestimate the threshold currentbecause diffusion strongly modifies the np product (andthus the recombination current) near the active layeredges (see the Supplementary Material for an example).A full drift-diffusion model is therefore required to accu-rately describe MHP lasers.In conclusion, we have put forth a model for organiclaser diodes that operate in the bipolar space charge-limited current regime. We have obtained analytical ex-pressions for the threshold voltage and current densityand have identified fundamental limits for laser opera-tion in the presence of parasitic annihilation and excitedstate absorption losses. These results, together with theexperiments of Sandanayaka et al. [5], emphasize a shiftin strategy from OLED-like architectures characterizedby many heterojunctions, blocking layers, and so forth,to a p-i-n structure based on a single, low threshold ma-terial that can be degenerately p- and n- doped, and thathas no triplet or polaron absorption overlapping with itsemission. With rational design of organic laser materialsnow emerging to meet these criteria [25] and the modelhere to predict device performance, the future of organiclaser diode technology is coming into clearer focus.This work was supported in part by AFOSR Award no.FA9550-18-1-0037 and DARPA Award no N66001-20-1-4052. ∗ [email protected][1] L. A. Coldren, S. W. Corzine, and M. L. Mashanovitch, Diode lasers and photonic integrated circuits , 2nd ed.,Wiley series in microwave and optical engineering No.218 (Wiley, Hoboken, N.J, 2012).[2] S. Forrest,
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