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I-V characteristics of disordered organic layers, on the base of transport level concept. V. R. Nikitenkoa*, A. Y. Sauninaa. aNational Research Nuclear University ...
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ScienceDirect Physics Procedia 72 (2015) 438 – 443

Conference of Physics of Nonequilibrium Atomic Systems and Composites, PNASC 2015, 18-20 February 2015 and the Conference of Heterostructures for microwave, power and optoelectronics: physics, technology and devices (Heterostructures), 19 February 2015

I-V characteristics of disordered organic layers, on the base of transport level concept V. R. Nikitenkoa*, A. Y. Sauninaa a

National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoe highway, 31, Moscow, 115409, Russia

Abstract An analytic model of mobility dependence on charge-carrier concentration based on percolation theory was modified by the use of the transport level concept, including field dependence of transport level. This model was applied to calculations of I-V characteristics of a single organic layer under space-charge limited regime. ©©2015 Published by Elsevier B.V. B.V. This is an open access article under the CC BY-NC-ND license 2015The TheAuthors. Authors. Published by Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the N ational R esearch Nuclear U niversity M E P hI (M oscow E ngineering Peer-review under responsibility of the National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)

P hysics Institute) .

Keywords: disordered organic semiconductors; mobility; OLED; Gaussian Disorder Model.

1. Introduction Understanding of a charge-carrier transport mechanism in organic materials is a key factor for improving the organic polymer devices (organic light-emitting diodes - OLEDs, organic field-effect transistors - OFETs) perfomances. Today, the dominant viewpoint is that charge transport in such materials is due to a hopping of charge carriers between localized states, randomly distributed in energy. The main theory, describing the transport, is still the Gaussian Disorder Model (GDM) by Bässler (1993). One of the most important characteristics of charge transport is the mobility of charge carriers, which is usually theoretically studied by Monte-Carlo simulations.

* Corresponding author. Tel.: +8-495-788-5699. E-mail address: [email protected]

1875-3892 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the National Research Nuclear University MEPhI (Moscow Engineering Physics Institute) doi:10.1016/j.phpro.2015.09.089

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V.R. Nikitenko and A.Y. Saunin / Physics Procedia 72 (2015) 438 – 443

However, it is very useful to have an analytical form of mobility μ dependence on temperature T, energetic scale of disorder σ, charge carrier concentration n and electric field strength F. An analytic description of mobility was suggested and applied to I-V characteristics calculations by Pasveer et al. (2005), but its equations are rather difficult, not physically clear and even incorrect in a broad range of parameters, because they results from fitting of numerical simulations. Although a different simple analytic model of mobility based on percolation theory by Shklovskii and Efros (1984) has been known for a very long time, it has never been applied to calculations of I-V characteristics of organic layers. In present work the model of this type was modified on a base of transport level concept by Nikitenko and Strikhanov (2014) and applied to single-layer OLEDs I-V characteristics calculations. 2. Theoretical model Simplified theory of space-charge limited currents (diffusion is not considered) by Lampert and Mark (1970) was used in order to calculate the I-V characteristics of a single-layer OLED. The system of Poisson’s equation and continuity equation with Mott-Gurney boundary conditions at x=0 for this case:

eP ( x)n( x) F ( x)

j

eN0 P (c)cF (c) ,

wF wx

en

eN0

HH 0

HH 0

F (0)

0, V

(1)

c,

(2)

L

³ dxF ( x) ,

(3)

0

where j – current density, e – elementary charge, c – relative concentration of charge carriers, N0 – density of localized states, V – voltage, ɛ0 – the electric constant, ɛ – the relative dielectric constant of the polymer, L – thickness of the layer, sandwiched between two electrodes. The mobility dependence on charge carriers concentration and electric field is given by

P0

ª E ( F )  E F (c ) º *exp «  C », c kT ¬ ¼

P

(4)

where P0 | Z0 ah 2 6V exp 2J a0 , ah t a0 – typical hopping distance, calculated according to Nikitenko and Strikhanov (2014), a0 N 0 1/3 – average distance between the localized states, ω0 – transition frequency factor, γ – inverse locazization length, EC – ransport energy level, EF – quasi-Fermi level. One needs to point out that the concentration dependence of EC is neglected in this work, because it is not important up to the very high concentrations (c~10-2), which are not relevant for OLEDs according to Baranovskii et al. (2010). The dependence of quasi-Fermi level EF on concentration c, at quasi-equilibrium conditions could be expressed as follows: f

c

³ dH f

F

(H ) g (H ), g ( E )

f

§ E2 exp ¨  2 2SV 2 © 2V 1

· ¸, ¹

(5)

where f F ( E) 1 ª¬1  exp ( E  EF ) kT º¼ - the Fermi distribution function. In a low concentration limit the analytical expression for quasi-Fermi level is known, see works by Zvyagin (2008) and Coehoorn et al. (2005): EF0 (c) kT

2

1§ V · §1·  ¨ ¸  ln ¨ ¸ 2 © kT ¹ ©c¹

ln(2c*c), c

c* .

(6)

2 Critical value of concentration in Eq. (6), c* exp ª  V kT 2 º 2 , see Eq. (3) by Coehoorn et al. (2005) and ¬ ¼ Eq. (18) by Baranovskii et al. (2002), follows from the condition EF (c* )  V 2 kT . The analysis of Eq. (5) shows,

440

V.R. Nikitenko and A.Y. Saunin / Physics Procedia 72 (2015) 438 – 443

that even in general case dimensionless quasi-Fermi level, EF (c) kT , contains only one parameter – V kT . The analytical approximation of quasi-Fermi level was obtained using Eq. (6) to simlify the calculations:

EF ( c ) kT

1 ln(2c*c) , a (c )

(7)

where a c 1  v ln  h 1 c 1  c c* , v v1  v2 V kT , h v3  v4 V kT (in a range of 2.5< V kT