Charge carrier mobility in disordered organic materials is being actively
studied, motivated by several applications such as organic light emitting
diodes and organic field-effect transistors. It is known that the mobility in
disordered organic materials depends on the chemical potential which in turn
depends on the carrier concentration. However, the functional dependence of
chemical potential on the carrier concentration is not known. In this study, we
focus on the chemical potential in organic materials with Gaussian disorder. We
identify three cases of non-degenerate, degenerate and saturated regimes. In
each regime we calculate analytically the chemical potential as a function of
the carrier concentration and the energetic disorder from the first principles.Comment: 5 pages, 3 figure

Sharma, A.

Sheinman, M.

English

arXiv

Charge carrier mobility in disordered organic materials depends on the chemical potential. Chemical potential has an implicit functional dependence on the carrier concentration and the density of states. However, for efficient calculation of mobility in simulation programs it is highly useful to have an explicit and accurate approximation for the chemical potential. In this study, we focus on analytical approximation for the chemical potential in organic materials with Gaussian disorder and provide an accurate expression in both non-degenerate and degenerate regimes. © 2013 IOP Publishing Ltd

NARCIS 

Analytical approximation for chemical potential in organic materials with Gaussian density of states

VU Research Portal

A Sharma

M Sheinman

Crossref

Journal of Physics D Applied Physics

Sharma, Abhinav

OPUS Augsburg

                                                                                                                     Analytical approximation for chemicalpotential in organic materials withGaussian density of statesA Sharma and M SheinmanDepartment of Physics and Astronomy, VU University, Amsterdam, The NetherlandsE-mail: a2.sharma@vu.nl                                                                                                           AbstractCharge carrier mobility in disordered organic materials depends on the chemical potential.Chemical potential has an implicit functional dependence on the carrier concentration and thedensity of states. However, for efficient calculation of mobility in simulation programs it ishighly useful to have an explicit and accurate approximation for the chemical potential. In thisstudy, we focus on analytical approximation for the chemical potential in organic materialswith Gaussian disorder and provide an accurate expression in both non-degenerate anddegenerate regimes.                                                   Charge transport in disordered organic materials is exploitedin a wide range of devices, including organic light-emittingdiodes (OLEDs) [1], organic field-effect transistors (OFETs)[2] photoreceptors [3] and photovoltaic cells [4]. In disorderedorganic materials, the carrier mobility is due to thermallyassisted tunnelling ‘hopping’ between localized molecularstates [5, 6]. It is known that the carrier mobility depends onthe temperature, energetic disorder, and carrier concentration[7, 8]. For efficient device modelling it is useful to havea compact analytical expression for mobility. Coehoornet al [6] provided an analytical expression for mobility inorganic materials with Gaussian energetic disorder. However,the authors noted that the expression for mobility is notsuitable for practical numerical device modelling because noanalytical expression for the chemical potential as a functionof carrier concentration and energetic disorder is availablewhich is valid over a wide range of carrier concentrations.The main purpose of this paper is to show that an analyticalexpression for the chemical potential in organic materialswith Gaussian energetic disorder can be obtained from thefirst principles, with no free parameters. We show thatthe derived expression for the chemical potential is fairlyaccurate and the error involved is well below the thermalenergy.Within the Gaussian disorder model, it is assumed that thedensity of states (DOS) is given byg(E) = Nt√2πσexp(− E22σ 2), (1)where σ is the standard deviation of the DOS and is a measureof the energetic disorder whileNt is the total number of hoppingsites per unit volume. For a given charge carrier concentration,p, the chemical potential, µ, is related to the density in thefollowing way:∫ ∞−∞g(E)1 + exp(E−µkBT) dE = pNt, (2)where kB is Boltzmann’s constant and T is the temperature.In general, for given p and σ , equation (2) is solvediteratively to obtain the chemical potential. However, inthe limit of vanishing carrier concentration, an analyticalexpression for the chemical potential can be easily obtainedbecause in the limit of small p carriers can be consideredessentially independent of each other. In this case equation (2)can be solved for µ by replacing the Fermi–Dirac (FD)statistics with the Boltzmann statistics [6]. In this limit,referred to as the Boltzmann approximation (BA), the density                            1                                                                                               of occupied states (the product of DOS and the FD distributionfunction, DOOS) is to a very good approximation given by aGaussian, centred at the energy value E = −σ 2/(kBT ) [6].The corresponding expression for µ is given byµ = − σ22kBT+ kBT ln p. (3)The BA limit is applicable to OLEDs, where undertypical operating conditions, the concentration is 10−4–10−5 carriers per hopping site [9]. On the other hand inOFETs [9] application of high gate voltage can lead to aconcentration of 0.01–0.1 carriers per hopping site. At thesecarrier concentrations, interaction between carriers becomessignificant implying that Pauli’s exclusion principle must betaken into account. In this high concentration regime, the BAis no longer valid and FD distribution must be used. We showthat even in this regime, an analytical solution for µ can beobtained in a simple and intuitive manner. In the followingtext we identify three regimes referred to as the non-degenerate,degenerate and saturated regime. In each regime we calculateanalytically the chemical potential as a function of the carrierconcentration.Our objective is to solve equation (2) for the chemicalpotential, µ, for given values of carrier concentration, p,energetic disorder, σ , and temperature, T . We evaluate theintegral in equation (2) using a saddle point approximation. Weassume that the saddle point, E∗, of the integrand is known.Assuming this one gets two coupled algebraic equations forthe chemical potential, µ, and the saddle point of the integrandinstead of one integral equation for the chemical potential[10]. The first algebraic equation is obtained by imposingthe condition that E∗ is the saddle point of the integrand inequation (2). On doing so we get the following equation forthe chemical potentialµ = E∗ + kBT ln(− σ2E∗kBT− 1). (4)Saddle point approximation implies that the integrand(equivalently the DOOS) is a Gaussian centred at E∗with a standard deviation σ∗. Namely, equation (2) isapproximated by ∫ ∞−∞h(E) dE = pNt, (5)where the expression for the DOOS ish(E) = Nte− E2∗2σ2(E∗kBT + σ 2)√2πσ 3exp[− (E − E∗)22σ 2∗], (6)andσ∗ =√kBT σ 4σ 2(kBT − E∗) − E2∗kBT. (7)On evaluating equation (5) we obtain the second algebraicequation:e−E2∗2σ2(E∗kBT + σ 2)σ√σ 2 − E2∗ − E∗σ2kBT= p. (8)The two coupled equations equations (4) and (8) can besolved to obtain µ and E∗. We now solve equations (4) and(8) self-consistently in three limiting cases:(a) Non-degenerate regime. This regime corresponds tothe case of vanishing carrier concentration. equation (8)implies that in this regime the saddle point occurs atE∗  −σ 2/kBT . (9)The dependence of E∗ on the carrier concentration p isobtained asE∗ = − σ2kBT(1 − peσ 2/2(kBT )2). (10)Self-consistency in this regime requires that carrierconcentration is low enough such that the condition (9)holds. This implies that in this regime p  p1, wherep1 = e−12(σkBT)2. (11)The chemical potential in this regime is well known andis given byµ = − σ22kBT+ kBT ln p. (12)The functional form of µ in the non-degenerate regimeis well known in the literature [6, 11, 12]. The non-degenerate regime corresponds to the BA limit. In thisregime an increase in carrier concentration leads to anincrease in the maximum of DOOS with negligible shiftin E∗ [6].(b) Degenerate regime. For p  p1 the saddle point,E∗, shifts significantly from −σ 2/kBT and the BA limitis no longer valid. In this regime the carrier concentrationis such that saddle point satisfies −σ 2/kBT  E∗ −kBT , so equation (8) impliesE∗ = − σ√2√W[2(σ/kBT )2p4], (13)where W is the Lambert W function [13]. Self-consistencyin this regime requires p1  p  p2, wherep2 = e−12(kBTσ)2. (14)The chemical potential in this regime is given byµ = − σ√2√√√√W[2(σ/kBT )2(1p)4]+ kBT ln√2(σ/kBT )√W[2(σ/kBT )2(1p)4] − 1. (15)This analytical expression for the chemical potential in thedegenerate regime is the main result of this paper. Sinceour focus is on the systems with σ > kBT , the secondcrossover value, p2, is of the order of one and degenerateregime is valid until almost the full saturation of all theavailable states. For the sake of completeness we discussbelow this saturated regime.(c) Saturated regime. In this regime the carrierconcentration, p, approaches the limiting value of p2  1where almost all the states are occupied and the DOOS is2                                                    – – – – –– – Figure 1. Chemical potential µ versus the normalized carrierconcentration p for different values of σ/kBT (numbers on theplot). Full lines and dashed lines correspond to the non-degenerate,equation (12), and degenerate, equation (15), regime, respectively.Symbols correspond to the numerically calculated chemicalpotential.the same as the DOS. In this limit, the saddle point satisfies0 > E∗  −kBT , so equation (8) impliesE∗ = −σ√2 ln1p. (16)Self-consistency in this regime requires p > p2. Thechemical potential in this regime is given byµ = −σ√2 ln1p+ kBT ln σkBT√2 ln 1p− 1 . (17)This regime is irrelevant for organic devices because evenin OFETs, where carrier concentration can be tuned byapplying gate voltage, the concentration is typically muchsmaller than 1 [9].In figure 1, we plot the chemical potential as obtainedabove for different values of σ . As predicted by equation (11),with increasing σ , the crossover from the non-degenerate tothe degenerate regime occurs at lower concentrations. Thecrossover from the degenerate to the saturated regime isnot shown in the figure. The saturated regime, irrelevantfor organic devices, is a narrow regime with divergence ofchemical potential atp = 1. Our results for the non-degenerateregime are in complete agreement with those reported byOelrich et al [14] for σ = 4kBT . It is also clear fromthe figure that in the regime of low carrier concentration,the chemical potential is located in the tail of the DOS. Thechemical potential moves deeper into the tail of the DOS withincreasing width of the DOS. These findings are consistent withthose reported recently [15, 16] in which the authors showedthat in the non-degenerate regime, with increasing width ofthe DOS, the carriers become more localized. When chargecarriers are highly localized, the mobility can be well describedby thermally assisted tunnelling ‘hopping’ between localizedmolecular states [5, 6]. On the other hand when carrier densityFigure 2. Comparison of analytical result, equation (19) (lines), andthe numerical calculation (squares) for different values of σ/kBT(numbers on the plot). Full lines correspond to the degenerateregime. The horizontal dotted line corresponds to the limit of thenon-degenerate regime.is high (1%), the chemical potential is close to or abovethe mobility edge [15]. For such carrier concentrations,which can be obtained in single-crystal FETs, charge transportacquires a band-like nature [15, 16]. It is clear from thefigure that the analytically obtained chemical potential andthe crossover density, equation (11), are in good agreementwith the numerical calculations in both non-degenerate anddegenerate regimes.In addition to the mobility of carriers, to model chargetransport one needs to calculate the diffusion coefficient.Assuming equilibrium conditions, the diffusion coefficient isdetermined from the generalized Einstein relation [17] whichis given asγ = pq∂µ∂p, (18)where γ is ratio of diffusion coefficient to the mobility ofcarriers and q is the elementary charge. Roichman and Tessler[18] showed that in a Gaussian DOS, γ deviates significantlyfrom the low density limit value of kBT/q. They obtainedγ as an implicit function of the chemical potential, µ. Ourexpression for the chemical potential allows us to obtain ananalytical expression for the generalized Einstein relation ineach of the regimes as a function of p, σ and T :γ = kBTq×1 p  p1√2σkBT{2−W[2p4(σ/kBT )2]+√2σkBT√W[2p4(σ/kBT )2]}{√2σkBT−√W[2p4(σ/kBT )2]}{1+W[2p4(σ/kBT )2]} p2  p  p11 +√2σkBT√ln 1p− 2 ln 1p2 ln 1p− 2√2 kBTσln3/2 1pp > p2.(19)In figure 2 we compare our analytical expression for γ ,equation (19), with the numerical results (notice that the inverse3                                                    Figure 3. Relative error δ. For the range of σ considered the error does not exceed 20%.of γ is shown). As expected, fairly good agreement is obtainedfar from the crossover concentration, p1.The results above indicate that the analytical expressionsfor the chemical potential are fairly accurate. Nevertheless, it isuseful to estimate analytically the error in the derived formulae.The main source of the error is the saddle point approximation,equation (6), in evaluating the integral in equation (2). Underthe saddle point approximation, the integrand (or equivalentlythe DOOS) of the integral in equation (2) is a Gaussian centredat E∗. The integrand is well approximated by a Gaussian onlyin the neighbourhood of E∗ (within few standard deviations,σ∗). Away from E∗, the deviation of the integrand from theGaussian gives rise to the error in evaluation of equation (2).The relative error in saddle point approximation of the integralcan be written asδ =∣∣∣∣∣∣∣1 −∫∞−∞ h (E) dE∫∞−∞g(E)1+exp(E−µkBT)∣∣∣∣∣∣∣ . (20)For energies well below the saddle point, E  E∗ − 3σ∗, theFD distribution can be approximated as unity whereas wellabove the saddle point, E  E∗ + 3σ∗, it can be approximatedby the Boltzmann distribution. Therefore, the error can beestimated asδ ∣∣∣∣1 −[ ∫ ∞−∞h(E) dE][ ∫ E∗−3σ∗−∞g(E) dE+∫ E∗+3σ∗E∗−3σ∗h(E) dE +∫ ∞E∗+3σ∗g(E)e−E−µkBT dE]−1∣∣∣∣. (21)The integrals appearing in the above expression can besimplified using an error function. For given values ofcarrier concentration, p, and energetic disorder, σ , the otherparameters E∗, σ∗ and µ can be calculated as shown above.In figure 3 we show that for relevant range of σ and p values,the value of δ does not exceed 20%. This implies that theerror in the chemical potential is well below than the thermalenergy, kBT .Although we derived the expressions for the chemicalpotential only in the respective regimes, the algebraic equationsequations (4) and (8) can be easily solved numerically to anydesired degree of accuracy for any given values of p and σ .To conclude, in this paper we derive analytical expressionfor the chemical potential in organic materials with Gaussiandisorder. In the most relevant, degenerate regime our result,equation (15), is new and, as we demonstrate, is fairly accurate.We show that over the relevant range of carrier concentrationand energetic disorder, the error in calculation is well belowthe thermal energy. In addition, the existing iterative numericaltechniques to calculate chemical potential can use our derivedexpressions as fairly accurate starting point. This can lead tomuch more efficient algorithms to calculate chemical potentialand therefore the mobility in organic materials with Gaussiandisorder.AcknowledgmentThe authors thank M Depken and P A Bobbert for helpfuldiscussions.References[1] Friend R H et al 1999 Nature 397 121[2] Drury C J, Mutsaers C M J, Hart C M, Matters M andde Leeuw D M 1998 Appl. Phys. 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