Appl Phys A (2008) 93: 527–532DOI 10.1007/s00339-008-4729-2

Mobility–diffusivity relationship for heavily doped organicsemiconductors

Atanu Das · Arif Khan

Received: 5 December 2007 / Accepted: 4 March 2008 / Published online: 29 June 2008© Springer-Verlag 2008

Abstract The relationship between the diffusivity Dn andthe mobility μn of chemically doped organic n-type semi-conductors exhibiting a disordered band structure is pre-sented. These semiconductors have a Gaussian-type den-sity of states. So, calculations have been performed to eluci-date the dependence of Dn/μn on the various parameters ofthis Gaussian density of states. Y. Roichman and N. Tessler(Appl. Phys. Lett. 80:1948, 2002), and subsequently Penget al. (Appl. Phys. A 86:225, 2007), conducted numericalsimulations to study this diffusivity–mobility relationship inorganic semiconductors. However, almost all other previousstudies of the diffusivity–mobility relationship for inorganicsemiconductors are based on Fermi–Dirac integrals. An an-alytical formulation has therefore been developed for thediffusivity/mobility relationship for organic semiconductorsbased on Fermi–Dirac integrals. The Dn/μn relationship isgeneral enough to be applicable to both non-degenerate anddegenerate organic semiconductors. It may be an importanttool to study electrical transport in these semiconductors.

PACS 71.20.Rv · 72.90.+y · 73.50.h

A. DasDepartment of Physics and Techno Physics, VidyasagarUniversity, Midnapore 721 102, West Bengal, India

A. DasDept. of Electronic Engineering, Chang Gung University,Keei-Shan Tao-Yuan, Taiwan, 333, People’s Republic of China

A. Khan (�)Electrocom Corporation, P.O. Box 60317, Potomac,MD 20859-0317, USAe-mail: a.khan123@yahoo.com

A. Khane-mail: akhan@electrocom-corp.com

1 Introduction

Heavy doping of organic semiconductors (ORSs) [1–4], likethat of inorganic semiconductors (INSs) [5–9], is widelyused to enhance the electrical transport and luminescenceproperties. It is used to improve the constituent device per-formance, as well. In fact, it is the key to the realization ofmany important properties of semiconductors [1]. There aretwo types of ORS doping—physical doping and chemicaldoping. Physical doping enables guest molecules to modifythe luminescence properties of the host. Chemical doping,on the other hand, enables dopant molecules to increase theconductivity of the host. ORS is either amorphous or poly-crystalline with disordered crystal structure [10, 11]. Thisis because metal/ORS characteristics [12–16] are quite dif-ferent from the metal/INS characteristics [17–22], and anORS/INS interface acts almost as dielectric/INS interface[23–27]. Also, unlike the electronic transport orbital bandsof INS, those of ORS are split into a series of localizedstates. Unlike inorganic semiconductors [28, 29], the or-ganic semiconductors may not have a well defined energybandgap—a bandgap that may either be wide or narrow.

The diffusivity–mobility relationship (DMR) is very im-portant for electrical transport in semiconductors. Duringthe past years, although a series of investigations [30–59]have been carried out to study the DMR for INS, only a fewattempts have been made to study it for ORS [60]. A MonteCarlo simulation of charge carrier transport by Richert etal. [61] indicates that the Dn/μn ratio for disordered n-typesemiconductors deviates significantly from the Dn/μn ratiofor ordered semiconductors. On the other hand, the time-of-flight transient photocurrent measurements by Hirao etal. [62] show that the diffusion coefficient is proportional tothe logarithm of the mobility at a certain temperature. Thismeans that the Dn/μn ratio for the ORS system deviates

528 A. Das, A. Khan

from that for ordered semiconductors. It depends, however,on the nature of the Gaussian density of states.

Roichman and Tessler [55], and subsequently Penget al. [60], conducted numerical simulations to study themobility–diffusivity relationship for organic semiconduc-tors. However, almost all other previous studies on thediffusivity–mobility relationship of inorganic semiconduc-tors are based on Fermi–Dirac integrals. Our objective inthis investigation is, therefore, to provide an analytical for-mulation for the diffusivity–mobility relationship for or-ganic semiconductors based on Fermi–Dirac integrals. Ourobjective is also to analytically investigate the carrier con-centration, the density of states, and related properties forchemically doped organic semiconductors. Numerical sim-ulations [55, 60] carried out earlier to conduct such inves-tigations do not, unfortunately, appear to provide adequatetransparency of the functional dependence of the DMR onthe various parameters of the organic semiconductors.

2 Theoretical model

To a good approximation, the energy distribution of the den-sity of states (DOS) of the lowest unoccupied molecular or-bitals (LUMOs) and the highest occupied molecular orbitals(HOMOs) exhibit Gaussian characteristics [55, 63, 64]:

gL(E) = N0ξL(E)

= (1 − ςD)N0√2πσL

exp

[− (E − EL)2

2σ 2L

], (1)

gH(E) = N0ξH(E)

= (1 − ςD)N0√2πσH

exp

[− (E − EH)2

2σ 2H

], (2)

gDL(E) = N0ξDL(E)

= ςDN0√2πσDL

exp

[− (E − EDL)2

2σ 2DL

], (3)

gDH(E) = N0ξDH(E)

= ςDN0√2πσDH

exp

[− (E − EDH)2

2σ 2DH

], (4)

where gL and gH are the energy distributions in the DOS ofLUMO and HOMO, respectively, of the host; gLD and gHD

are the energy distributions of the dopant atoms inside or inthe vicinity of the LUMO and HOMO, respectively. E is theenergy of the HOMO/LUMO state, ζD is the fraction of thedoping concentration in this state, and N0 is the total densityof the HOMO or LUMO states, which equals the moleculardensity of the doped organic semiconductor. EL and EH arethe mean energy of the LUMO and HOMO states, respec-tively, and σL and σH are the variance of the LUMO and

Fig. 1 Schematic diagram of the Gaussian distributions of HOMO andLUMO levels of an organic semiconductor

HOMO states, respectively (see Fig. 1). EDL, EDH, σDL, andσDH are the corresponding parameters for the dopant atoms.There are statistical local fluctuations of the polarization en-ergy of charge carriers. We also have a van der Waals energyof adjacent molecules resulting from relative fluctuations ofthe intermolecular separation on a percent scale. All theselead to a width of the energy distributions in the DOS ofHOMOs and LUMOs typically between 0.05 eV and 0.2 eV.

The overall DOS is a superposition of the Gaussian en-ergy densities for the host and the dopant. We assume thatFermi–Dirac statistics is valid for the occupation of theLUMO and HOMO states, and the host and the dopant. Theelectron occupation function, for example, for the LUMOsof the host and the dopant is

f (E,Efn)−1 = [

1 + δ(E)] =

[1 + exp

(E − Efn

kBT

)], (5)

where Efn is the quasi-Fermi energy, kB is the Boltzmannconstant, and T is the absolute temperature. The total elec-

Mobility–diffusivity relationship for heavily doped organic semiconductors 529

tron concentration, for example, in the LUMO states exhibit-ing this quasi-Fermi energy Efn, is given by

n(Efn) =∫ ∞

−∞[gL(E) + gLD(E)

]f (E,Efn) dE. (6)

The basic relationship between the diffusivity Dn and themobility μn is given by

Dn

μn

= 1

q

(n

dn/dEfn

). (7)

Using (1) to (7) DMR may be obtained as

Dn

μn

=(

kBT

q

)

×∫ +∞−∞ [ξL(E) + ξDL(E)][1 + δ(E)]−1 dE∫ +∞

−∞ [ξL(E) + ξDL(E)]δ(E)[1 + δ(E)]−2 dE.

(8)

Equation (8) involves combinations of Gaussian functionsand can be solved only by numerical methods. However, thenumerical solution of (8) might not correctly uncover thereal influence of the various parameters on the DMR end re-sults. The fundamental physics underlying such a numericalsolution might also remain obscured. An analytical solutionfor DMR may overcome this problem. Almost always suchanalytical solutions for the DMR [30–59] are obtained interms of the Fermi–Dirac integral of order j , given, in gen-eral, by

�j (ηn) = 1

(j + 1)

∫ ∞

0

zj dz

1 + exp(z − ηn), (9)

where ηn = Efn/kBT . We intend to obtain our solutions alsoin terms of �j (ηn). Such an analytical solution may be bestaccomplished by making use of the following approxima-tion for a general Gaussian function exp(−x2):

exp(−x2) =

6∑ν=1

Bνe−νx, (10)

where x is a dimensionless parameter, B1 = 2.589 × 10−3,B2 = −2.183, B3 = 3.314 × 101, B4 = −7.625 × 101, B5 =6.825 × 101, and B6 = −2.196 × 101. As will become evi-dent later, (10) is an excellent approximation for a Gaussianfunction, and hence highly useful for a closed-form ana-lytical solution of the complex formula involving Gaussianfunctions. To solve our problem, we first simplify gL(E) andgDL(E). For this, we define the following parameters:

AL = (1 − ςD)N0√2πσL

, (11)

ADL = ςDN0√2πσDL

, (12)

αL = 1√2σL

, (13)

αDL = 1√2σDL

. (14)

Then, making use of (1), (10), (11), and (12), we get

gL(E) = AL

6∑ν=1

Cν

[1 − ναLE + ν2α2

LE2

2− ν3α3

LE3

6

+ ν4α4LE4

24− ν5α5

LE5

120+ · · ·

], (15)

where the terms within the brackets correspond to the Tay-lor’s series expansion of exp(−ναLE). For most practi-cal systems, σL < 1 and σDL < 1. Peng et al. [63, 64]used σL = 0.05 to 0.02 eV for their calculations. Also, no-tably E > 0 eV and 1≤ ν ≤ 6. Because of these values,exp(−ναLE) is highly convergent. In general, the larger thevalue of ν and E, the smaller is the value of exp(−ναLE).Also,

Cν = Bν exp (ναLEL) . (16)

Similarly, making use of (3), (10), (12), and (14), respec-tively, gDL(E) may be simplified to

gDL(E) = ADL

6∑ν=1

C′ν

[1 − ναDLE + ν2α2

DLE2

2

− ν3α3DLE3

6+ ν4α4

DLE4

24− ν5α5

DLE5

120+ · · ·

],

(17)

where the terms within the bracket correspond to the Tay-lor’s series expansion of exp(−ναDLE), and

C′ν = Bν exp(ναDLEDL). (18)

On the same grounds as for exp(−ναLE), the functionexp(−ναDLE) is highly convergent. The primary objectivefor the Taylor series expansion of (15) and (17) is again toobtain expressions that allow for the solution for DMR interms of the Fermi–Dirac integral �j (ηn).

If we consider a n-type semiconductor, the density ofstates below the conduction band edge, which is the originof our coordinate system, would hardly exist. Even if thereare fluctuating values below the origin, the average of themwould be negligible, if not zero. So, practically (6) may berewritten as

n(Efn) =∫ ∞

0

[gL(E) + gLD(E)

]f (E,Efn) dE. (19)

530 A. Das, A. Khan

Equations (15) to (19) yield

n(E) =∫ ∞

0

{6∑

ν=1

[B(0)

ν − B(1)ν E + B(2)

ν E2 − B(3)ν E3

+ B(4)ν E4 − B(5)

ν E5 + · · ·]}

f (E,Efn) dE, (20)

with

B(0)ν = ALCν + ADLC′

ν, (21)

B(1)ν = [

ALαLCν + ADLαDLC′ν

]ν, (22)

B(2)ν = [

ALα2LCν + ADLα2

DLC′ν

]ν2

2, (23)

B(3)ν = [

ALα3LCν + ADLα3

DLC′ν

]ν3

6, (24)

B(4)ν = [

ALα4LCν + ADLα4

DLC′ν

]ν4

24, (25)

B(5)ν = [

ALα5LCν + ADLα5

DLC′ν

] ν5

120. (26)

If we define ε = E/kBT , then (20) would simplify to theform

n(E) =6∑

ν=1

[(1)kBT B(0)

ν − (2)k2BT 2B(1)

ν �1(η)

+ (3)k3BT 3B(2)

ν �2(η) − (4)k4BT 4B(3)

ν �3(η)

+ (5)k5BT 5B(4)

ν �4(η) − (6)k6BT 6B(5)

ν �5(η)].

(27)

This is an analytical solution for the electron concentrationin HOMO. At room temperature kBT ≈ 0.024 eV, whichsuggests that the higher-order terms of (27) are very small.Equation (7) may be recast into the form

n(E) =(

qDn

μn

)dn(E)

dEfn. (28)

Comparing (27) and (28) we get

Dn

μn

=(

kBT

q

) ∑6ν=1

∑5τ=0(−1)τ (τ + 1)(kBT )τ+1B

(τ)ν �τ (η)∑6

ν=1∑5

τ=0(−1)τ (τ + 1)(kBT )τ+1B(τ)ν �τ−1(η)

.

(29)

This is the closed-form analytical relation for DMR for elec-trons in LUMOs. Following the same procedure, an analo-gous diffusivity–mobility relationship may be obtained forthe holes in the HOMOs.

3 Results and discussions

Closed-form analytical formulations have been obtained forcarrier concentrations and diffusivity–mobility relationshipsfor chemically doped organic semiconductors. These analyt-ical formulations may be very useful for analytical model-ing of organic light emitting diodes [65–67], organic field-effect transistors [68, 69], and organic solar cells [70]. All ofthese comprise single or multiple thin (10 to 100 nm) amor-phous/polycrystalline organic semiconductor layers.

It is interesting and remarkable that a simple and accu-rate approximation, (10), of the Gaussian function has ledto a closed-form solution of (6). It would otherwise be quiteintractable. Remarkably, the solution resembles the forms[30–39] derived almost always in terms of the Fermi–Diracintegral. The comparison of the exact value of exp(−x2)

with the approximation given by (10) is shown in Fig. 2,which indicates that the approximation is indeed quite excel-lent, and hence attests to the usefulness of the closed-formformulas obtained for carrier concentrations and diffusivity–mobility relationships. In the absence of heavy doping,E shifts away from Efn,�0(ηn) approaches �−1(ηn), and�1(ηn) approaches �0(ηn). Consequently (29) reduces to

Dn

μn

=(

kBT

q

). (30)

This is quite evident, because, as E moves away fromEfn, δ(E) � 1. So, 1 + δ(E) ≈ δ(E), and the right handside of (8) simplifies to (kBT/q). In the presence of heavydoping, 1+δ(E) = δ(E), and hence the DMR (Dn/μn) isquite dependent on the doping and on the Fermi–Dirac inte-grals.

Both (27) and (29) can be very useful for the study ofcarrier transport in ORS. We cite an example. Electrons are

Fig. 2 Comparison of the exact value of exp(−x2) with the approxi-mation of it given by (8). The solid line represents the exact result, andthe small solid circles represent the approximations given by (8)

Mobility–diffusivity relationship for heavily doped organic semiconductors 531

Fig. 3 The variation of the normalized value of Dn/μn with the rela-tive Fermi energy QFN = [(Efn − EL)/kBT ] for several different val-ues of ζD of a chemically doped organic semiconductor. The variousparameters used for the calculations are: EL = −2.4 eV, σL = 0.1 eV,EDL = −2.8 eV, σDL = 0.05 eV, EH = −5.0 eV, σH = 0.1 eV,N0 = 1020 cm−3, and T = 300 K. The solid curves represent the simu-lation results of Peng et al. [27]. The symbols (e.g., small solid circles,small open circles, small open squares, etc.) represent the present re-sults

injected from the cathode into the electron transport layer(ETL), and holes are injected from anode into the hole trans-port layer (HTL) of an organic light emitting diode (OLED).The current thus generated comprises the diffusion currentand the drift current. This current in the vicinity of the injec-tion contact is, however, dominated by the diffusion com-ponent if the energy barrier at the injection contact is quitelow (lower than 0.4 eV). Due to masking of the electric fieldby space charges, the drift component of the current at sucha low injection contact is negligibly small. This means thatthe injection current can be increased by increasing Dn anddecreasing μn, which is possible by judicious ETL dopingwith dopant atoms having an electron affinity lower than thatof the ETL host, or by HTL doping with dopant atoms hav-ing a work function lower than that of the HTL host.

Calculations were carried out to determine the depen-dence of the DMR Dn/μn on various parameters of theGaussian density of states. For example, the variation ofthe normalized values of Dn/μn (e.g., Dn/μn in units ofkBT/q) with the relative Fermi energy QFN = [(Efn −EL)/kBT ] for several different values of ζD is shown inFig. 3. While the solid curves are simulation results [60],the small symbols (e.g., small solid circles, small open cir-cles, small open squares, and small solid squares) are thepresent results. The correspondence between the two atteststo the accuracy of our approximations. One may note thatthe Dn/μn vs. QFN plot shows a peak at a certain value

Fig. 4 The variation of the normalized value of Dn/μn with the elec-tron concentration n for several different values of σDL of a chemi-cally doped organic semiconductor. The various parameters used forthe calculations are: EL = −2.4 eV, σL = 0.1 eV, EDL = −2.8 eV,ζD = 0.001 eV, EH = −5.0 eV, σH = 0.1 eV, N0 = 1020 cm−3, andT = 300 K. The solid curves represent the simulation results of Penget al. [27]. The symbols (e.g., small solid circles, small open circles,small solid squares, etc.) represent the present results

of QFN. Both the position and the height of this peak de-pend on ζD. As ζD increases, the peak becomes higher andshifts to higher values of QFN. For example, for ζD = 0.2,the peak occurs at QFN = −10 and has a value of 25. ForζD = 0.001, the peak occurs at QFN = −15 and has a valueof 1. There is essentially no peak in the Dn/μn vs. QFN

plot for ζD = 0. The existence of the peaks is attributed tothe two-Gaussian-peak density of states. It results from thecarrier filling of these double-Gaussian distributed energystates. As the Fermi energy increases, electrons occupy theenergy states of the dopant atoms, and the electron concen-tration increases with the quasi-Fermi energy.

The variation of the normalized Dn/μn with the electronconcentration n for several different values of σDL is shownin Fig. 4. While the solid curves are simulation results [60],the small symbols (e.g., small solid circles, small open cir-cles, small solid squares, and small open squares) are thepresent results. The correspondence between the two is al-most exact. The Dn/μn vs. n plot shows a peak at a cer-tain value of n. At lower electron concentration (n < 1012

cm−3), the normalized Dn/μn values change slowly withσD. However, for all values of σDL, the Dn/μn vs. n curveincreases exponentially with n until it reaches a maximumand then dies down rapidly. The position of the peak shiftsslowly to higher values of n. Also, the peak is higher forlarger values of σDL. At n > 1017 cm−3, they have essen-tially the same value for Dn/μn. The peaks in the Dn/μn vs.n plots arise from the consideration of a double-Gaussian-variable form for the density of states. Such a double-Gaussian density of states is essential for doped organic

532 A. Das, A. Khan

semiconductors, which are mixed organic molecule host-guest systems.

Acknowledgement The authors appreciate technical assistance ofAndrew V. Bemis.

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