Consistent double Gaussian model with non-symmetric potential barriers at contacts for organic diodes

Abstract

A transport model with double Gaussian density of state (DOS) for organic semiconductors is proposed, with one Gaussian DOS for free carriers and one for trapped carriers. The variations of the density of trapped carriers with the density of free carriers, the effective mobility and the ratio of diffusion coefficient to effective mobility with the density of trapped carriers are analyzed. It is shown that the ratio of diffusion coefficient to effective mobility is the Einstein type for free carriers, but the ratio for total carriers is the non-Einstein type, and is an increasing function of the density of trapped carriers at low density, and a decreasing function at high density. The importance of non-symmetric barriers at contacts is emphasized to quantitatively describe the current–voltage relationships of typical organic layers sandwiched between two metallic electrodes. It is shown that slopes of current–voltage curves at low bias are very sensitive to the values of right barriers. The slopes in all bias are sensitive to the values of left barriers, and increase as the left barrier decreases. When applying the modified model to three organic diodes, an excellent agreement is obtained between theoretical results and experimental data.

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1. Introduction

The research of optoelectronic and electrical transport properties in conjugated semiconducting polymers has drawn intensive attention because of their potential applications in organic semiconductor devices such as light-emitting diodes (LEDs) and field-effect transistors (FETs).^1–18 Understanding their charge-carrier transport properties is very useful to improving the devices' performances, and to designing and synthesizing better materials. It was demonstrated that the current in single carrier diodes with amorphous polymer layers sandwiched by two electrodes is space-charge limited (SCL).^1–18

Davids et al.¹ proposed a unified device model considering two types of carriers for single layer organic LEDs, and Crone et al.² extended it into bilayer diodes. They also studied the impact of local electric fields on the injection barrier. Koster et al.³ proposed a model for organic solar cells considering two types of carriers. However, most organic diodes have a single type of carrier, and the SCL current in organic diodes is described by the solutions of drift and Poisson equations, with different expressions of mobility. The density dependent mobility of Vissenberg and Matters (VM)⁴ is popular, which originates from hopping in an exponential density of states (DOS). This description has been used to unify the charge transport in FETs and LEDs.^5–7 Pasveer et al.⁸ developed a description for the mobility incorporating both the density and field dependence, based on charge-carrier hopping within the Gaussian DOS. The mobility of Pasveer et al. has been widely applied to describe current–voltage (J–V) relationships of organic devices by many authors.^9–18

van Mensfoort et al.¹¹ and we^12,13 have numerically solved the drift–diffusion model (DDM) and drift model by using parameters optimized by Pasveer et al.⁸ for NRS-PPV and OC₁C₁₀-PPV. The obtained J–V curves are in good agreement with experimental data.^9–13 Zhou et al.¹⁴ studied the effect of high carrier density on mobility by using the Monte Carlo simulation. Coehoorn et al.,^15,16 and Bässler and Köhler¹⁷ reviewed recent advances in charge transport in disordered organic semiconductors. Myers and Xue¹⁸ reviewed organic semiconductors and their applications in photovoltaic devices. They discussed the fundamental electronic nature of organic semiconductors, processing techniques, and their application to two main classes of optoelectronic devices: light emitting diodes, and photovoltaics.

However, Zhang et al.¹⁹ demonstrated that electron transport is consistently described by the concept of free electrons in combination with deep traps. Recently Nicolai et al.^20,21 analyzed the reason that both the VM and Pasveer models provide a consistent description of the J–V relationships of organic devices. The reason is that the section of the Gaussian DOS that is being filled during a J–V scan may also be approximated by an exponential distribution, or vice versa.^20,21 Nicolai et al.^20,21 further emphasized that the trapped carriers are very important for description of J–V curves. If considering the trapped carriers with Gaussian DOS in the DDM and treating the mobility as a constant, the J–V curves of many organic diodes can be qualitatively described. However, the fitting procedure and values of parameters are not clearly presented and explained.^20,21

Lange et al.²² proposed a model for band bending in contacts of polymer layers with metallic electrodes. Cottaar et al.^23,24 presented a scaling theory for charge transport in disordered molecular semiconductors that extends percolation theory to the random resistor network representing charge hopping. A density dependent mobility for Miller–Abrahams and Marcus hopping on different lattices with Gaussian energy disorder was derived. Oelerich et al.²⁵ suggested a recipe for determining the density of states (DOS) in disordered organic semiconductors from the measured dependence of the charge carrier mobility on the concentration of carriers. The recipe is based on a theory for the concentration-dependent mobility. Fishchuk et al.²⁶ developed an analytical mobility model to describe hopping transport in organic semiconductors including both energetic disorder and polaronic contributions due to geometric relaxation.

Bruyn et al.²⁷ derived an analytic approximate JV formula for organic diodes. In the derivation of the analytic formula, mobility was treated as a function of temperature, being independent of density and electric field. However, the analytic formula cannot consider trapped carriers, and it merely describes JV data qualitatively. It is notable that Pasveer et al.,⁵ Bruyn et al.,²⁷ and Nicolai et al.^20,21 all belong to the group of Blom. They have proposed tens of models considering different physical effects, but the final model has not been found.^{3,5,8,19–21,27,42}

Except for the traps, the barriers at contacts between organic layers and electrodes are important for quantitative descriptions of JV data. Davids et al.¹ and Crone et al.² studied the impact of local electric fields on the injection barrier. Chen et al.²⁸ studied the electrostatic field and partial Fermi level pinning at the pentacene–SiO₂ interface. Recently, Yogev et al.²⁹ and Oehzelt et al.³⁰ independently proposed that Fermi level pinning is induced by gap states in organic semiconductors. Because the height of barriers is tightly related to Fermi levels, and since the Fermi level in organic semiconductors is pinned, it implies that the barriers are almost constant. Neumann et al.^31,32 developed a self-consistent theory of unipolar charge-carrier injection in metal/insulator/metal systems. They derived boundary conditions used for the drift–diffusion equations. The densities of electrons³¹ or holes³² were expressed as functions of barriers which were weakly dependent on electric field at interfaces. Thus, Bruyn et al.²⁷ directly adopted constant barriers in their derivation of an analytic JV formula.

In the DDM, the ratio between the diffusion coefficient D and the mobility μ is given by the Einstein relationship (ER), D/μ = k_BT/q. Some authors^33–36 have proposed that the ER should be replaced by the generalized Einstein relation (GER) with the ratio k_BT/q multiplied by a factor. However, Neumann et al.³⁷ have disproved the GER through theoretical analysis. Wetzelaer et al.³⁸ confirmed the validity of the ER in organic semiconductors by studying the diffusion-driven currents of single-carrier diodes.

In this paper, we present a detailed formulation of the DDM with Gaussian traps taken into account, give an explanation of the GER, and show that, except for Gaussian traps, the non-symmetric potential barriers at contacts are important factors for the quantitative description of J–V curves. Our work applies the DDM to three typical diodes.

2. Fundamental model

The fundamental energy-band model for p-type materials used in this paper is schematically plotted in Fig. 1. The model in Fig. 1 consists of two Gaussian DOS. One Gaussian DOS exists for free carriers and one exists for trapped carriers with widths σ and σ_t, and center energy levels E_v and E_t, respectively.N₀ is the total concentration of states at energy levels E_v, N_t is the total concentration of traps. The densities of free and trapped holes can be expressed as

f(E) = {1 + exp[(E_F − E + qφ(x))/k_BT]}⁻¹ ≡ {1 + exp[(E_F(x) − E)/k_BT]}⁻¹ (5)

Fig. 1 Schematic diagram of the energy-band model proposed in this paper.

Here f(E) is the Fermi–Dirac (FD) distribution function, E_F is the Fermi energy level without an external electric field, and E_F(x) = E_F + qφ(x) is the quasi-Fermi energy level. There is no constant E_F(x) in a device under SCL conditions, but the separated E_F should be a constant.

It should be pointed out that although the same FD distribution f(E) is used in both eqn (3) and (4), the literature^20,21 shows that the Einstein relationship is valid for organic semiconductors. This implies that the organic semiconductors should be non-degenerate, (E_F − E_v) ≫ kT. Therefore, the position of E_F is near E_t and deviated from E_v in Fig. 1. The boundary conditions in eqn (16) and (17) show that φ(x) > 0, so even under SCL conditions, the quasi-Fermi energy E_F(x) also satisfies the non-degenerate condition, (E_F(x) − E_v) ≫ kT. This means that the free holes can be seen as non-degenerate, and the trapped holes should be treated as degenerate. The Fermi–Dirac distribution in eqn (4) can and cannot be approximated by the Boltzmann distribution as applied to free holes and trapped holes, respectively. eqn (3) for free holes can be simplified as follows

Defining the effective DOS

the density of free holes can be expressed as:

p_f = N_f exp[(−qφ(x))/kT] (8)

Introducing the dimensionless variable, , we can evaluate the integral in eqn (7)

and obtain an expression for effective DOS as follows

N_f = N₀ exp[(E_v − E_F)/kT]exp[σ²/2(kT)²] (9)

where σ is the standard deviation of the Gaussian distribution. Although the Fermi–Dirac (FD) distribution in eqn (5) cannot be approximated by the Boltzmann distribution as applied to trapped holes, we can reformulate it by using eqn (8). Solving Fermi energy E_F from eqn (8) and substituting it into eqn (5), we obtain

f(E) ≈ {1 + (N_f/p_f)exp[(E_F − E)/k_BT]}⁻¹ (10)

One easy misunderstanding is that to derive eqn (8), the Fermi–Dirac distribution is approximated by a Boltzmann distribution and then this result is substituted in the Fermi–Dirac statistics in eqn (5) to arrive at eqn (10). This treatment is correct, and we explain the reason in more detail. The original Fermi distribution eqn (5) is applicable to both free and trapped holes, but the Boltzmann approximation in eqn (8) merely applies to free holes. However, as the transformed eqn (10) is applied to free holes, we may obtain the identity, p_f = p_f. Therefore there is no contradiction in eqn (10).

Nicolai et al.^20,21 pointed out that the trapped holes can be easily realized in the program based on the discretization method of Gummel and Scharfetter,^39,40 if the D_t(E) for trapped holes is taken as the single trap model. In contrast, the Fermi integral in eqn (4) with Gaussian DOS is not easy to realize.^20,21 It is necessary to use the accurate approximation of the Gauss–Fermi integral recently reported by Paasch and Scheinert.⁴¹ By using their approximation,⁴¹ the effect of Gaussianly-distributed traps on the transport can be accessed. However, from eqn (10), we can derive the following expression

p_f(∂f/∂p_f) = {1 + (N_v/p_f)exp[(E_F − E)/k_BT]}⁻²(N_f/p_f)exp[(E_F − E)/k_BT] (11)

Substitution of eqn (10) into eqn (11) yields

p_f(∂f/∂p_f) = f²(f⁻¹ − 1) = f(1 − f) (12)

Then the derivatives of trapped charges with respect to the free charges can be easily evaluated as

By combining eqn (13) with the discretization method of Gummel and Scharfetter,^39,40 the Gauss–Fermi integral can be easily and directly realized in the program, without the need for the approximation of Paasch and Scheinert.⁴¹

With trapped charges taken into account, the Poisson equation is as follows

Considering that the trapped charges do not contribute to current, and supposing a constant mobility, the drift–diffusion equation takes the following form

In order to solve eqn (14) and (15), we need to use eqn (4), (8) and (10).

Assuming the thickness of the organic layer is L, the left-side contact (x = 0) can be seen as Ohmic with a low potential barrier W_left. The right-side contact (x = L) can be seen as Schottky with a high potential barrier W_right. The difference between right and left potential barriers, V_bi = W_right − W_left, is the built-in potential in the devices. The boundary conditions for the Poisson equation is as follows²⁷

φ(0) = W_left + V, φ(L) = W_right (16)

V − V_bi = φ(0) − φ(L) (17)

For the boundary conditions for the drift–diffusion equations, Neumann et al.^31,32 have derived the following equations

where F(0) and F(L) are values of the electric field at the contacts and l is a characteristic length of the electrodes

The typical values of parameters in eqn (19) are as follows, n_∞ ≈ 4 eV, E_∞ ≈ 3 × 10²⁸ m³, and l ≈ 7.088 × 10⁻¹¹ m. In the devices studied in this paper, the values of F(0) and F(L) are about 10³ and 10⁷ (V m⁻¹), respectively. The values of ε_rql_leftF(0) and ε_rql_rightF(L) are about 10⁻⁶ and 10⁻² eV, respectively. The second terms in the exponential functions of eqn (18) can be neglected as compared with the first terms. Therefore eqn (18) can be simplified to the following form²⁷

p_f(0) = N_f exp(−W_left/kT), p_f(L) = N_f exp(−W_right/kT) (20)

The boundary conditions in eqn (16) and (20) have been used by Bruyn et al.²⁷

Ahead of concrete calculations, it is necessary to explain that our model goes beyond those proposed in previous publications. Firstly, Nicolai et al.^20,21 adopted the program of Koster et al.;³ they did not consider potential barriers at contacts, and adopted symmetric Ohmic contacts with W_left = W_right, as mentioned in last line of page 2 in their paper.²⁰ Our calculations show that asymmetric contacts with different barriers are very important to improve the fitting quality of experimental J–V data. Secondly, Nicolai et al.^20,21 did not mention boundary conditions used for the drift–diffusion equations. Thirdly, they did not use the non-degenerate condition for free holes, and did not extract effective DOS as a function of temperature. They also did not extract the temperature function of mobility. Fourthly, they did not derive relationships in eqn (11)–(13) ; they used an approximation of the Gauss–Fermi integral as proposed by Paasch and Scheinert.⁴¹

3. Analysis

Eqn (4), (8), (10), (14) and (15) show that the calculation of trap-limited currents requires a separation of the total carrier density into free (p_f) and trapped (p_t) carriers. We plot the (p_t–p_f) relationship with different values of σ in Fig. 2. The figure also shows that the influence of σ to (p_t–p_f) curves is very prominent. The p_t is always an increasing function of p_f and σ. The slopes of (p_t–p_f) curves are large when p_f is low and σ is small; the slopes decrease as p_f and σ increase. The p_t would reach saturation at high p_f. There exists a cross point of p_f for all curves, such that at this value of p_f, all curves have the same value of p_t.

Fig. 2 Schematic diagram for the (p_t–p_f) relationship calculated by using eqn (4) and (10) at 300 K, with N_f = 10²⁶ m⁻³, N_t = 10²³ m⁻³, E_tF = 0.3 eV, and with different values of σ_t; σ_t = 0, 0.05, 0.1, 0.15, 0.2 eV, respectively. The calculations in this figure do not involve the Gaussian width σ of free holes.

Now we give a qualitative explanation of the increase of mobility and GER based on trapped charges. For simplicity, we just consider the single trap case. In terms of eqn (4) and (10), if a single trap level E_t is assumed, the density of trapped charges can be expressed as

p_t = N_tf(E_t) = N_t{1 + (N_f/p)exp[(E_F − E_t)/k_BT]}⁻¹ (21)

In the situation considering GER, it is not necessary to divide the carriers into free and trapped types. We may introduce total density of carriers

p = p_f + p_t (22)

The Poisson and DD equations can then be reformulated by using p

The substitution of eqn (21) into eqn (22) results in the quadratic equation for p_f

p_f² − p_f(p − N_t − N_q) − pN_q = 0 (25)

The solution is as follows

p_f = (1/2)(p − N_t − N_q) + (1/2)[(p − N_t − N_q)² + 4N_qp]^1/2 (26)

N_q = N_f exp[(E_F − E_t)/k_BT] (27)

Its derivative can be evaluated as

By substituting eqn (26) and (28) into eqn (14) and (15), and comparing the resulting equation with eqn (23) and (24), we obtain the effective mobility and diffusion coefficient.

It should be pointed out that trap-modulated mobilities were introduced decades ago,^{1,2,4,23–26} although not within the Gaussian DOS leading to eqn (29), and there are many possible GERs, for example field-depend mobility.^33–37

Eqn (29) shows that although the mobility μ₀ in the original model in eqn (14) and (15) is a constant, where the carriers are divided into free and trapped types, the effective mobility μ and diffusion coefficient D in the transformed eqn (24) become functions of density of carriers. In Fig. 3, we plot the relationships of μ and D/μ with p. The figure shows that μ is an increasing function of p, and the increase becomes more dramatic as the depth of traps increases. This tendency is in agreement with that of VM theory.^4–7 The VM theory does not divide the carriers into free and trapped, and expresses μ as a function of density of carriers.

Fig. 3 The qualitative (μ–p) and (D/μ–p) relationships calculated by using eqn (26)–(30) at 300 K, with N_f = 10²⁶ m⁻³, N_t = 10²³ m⁻³, E_tF = 0.2 eV (solid lines) and 0.3 eV (dashed lines), and with σ_t = 0 eV. The calculations in this figure do not involve the Gaussian width σ of free holes.

Based on the analysis, one may deduce following assumption that the trapped carriers is equivalent to effective mobility as a function of carrier density; both factors cannot be considered at same time. The latest works of Zhang et al.¹⁹ and Nicolai et al.^20,21 also consider this situation. Since they consider trapped charges in their model,^19–21 they treat mobility as a constant, and do not use the mobility model of Pasveer et al.⁸ which treats mobility as a function of density of carriers and electric field. Both authors belong to same research group led by Blom.^8,19–21

Fig. 3 also shows that the ratio of D/μ deviates from the ER value kT/q, and the GER appears. However, the ratio also differs from the one in previous references,^33–35 in which the GER is due to degenerate effect of carriers. Since the viewpoint on GER is controversial in literature,^33–36 and the latest experiment³⁷ supports the ER for free carriers, the theoretical model dividing carriers into free and trapped types is consistent with the experiment.³⁷

The model of Nicolai et al.^20,21 does not consider the asymmetric barriers at contacts, and adopted symmetric Ohmic contacts with W_left = W_right.²⁰ In our calculations, it is shown that the potential barriers have a severe impact on the shape of J–V curves. In Fig. 4 we plot several J–V curves with different values of W_left and W_right. The figure shows that keeping W_left constant and increasing W_right causes the J–V curves to bend down dramatically at low voltage. Keeping W_right constant and increasing W_left causes the J–V curves to move down and the slope to slightly decrease. Therefore it is necessary to adjust values of W_left and W_right to fit the shapes of J–V curves.

Fig. 4 The J–V curves calculated at 300 K, with μ₀ = 2 × 10⁻¹⁰ m² V⁻¹ s⁻¹, N_t = 2 × 10²² m⁻³, σ = 0.1 eV, and E_tF = 0.22 eV. (a) W_left = 0.1 eV, and W_right = (0.1, 0.3, 0.5, 0.7, 0.9) eV. (b) W_right = 0.1 eV, and W_left = (0.1, 0.3, 0.5, 0.7, 0.9) eV.

4. Application to devices

Now we apply the above Gaussian model to polymer layers of poly[4′-(3,7-dimethyloctyloxy)-1,1′-biphenylene-2,5-vinylene] (NRS-PPV), poly(2-methoxy-5-(3′,7′-dimethyloctyloxy)-p-phenylene-vinylene) (OC₁C₁₀-PPV),^8,10 and poly(3-hexylthiophene) (P3HT),^5,42 with thicknesses of L = 560 nm, 275 nm and 95 nm respectively. In Fig. 5–7, we plot J–V curves for the three materials for comparison with experimental points. The three figures show that the agreement of theoretical curves with experimental J–V points is fairly good. For the three devices studied in this work, the agreement is quantitative for every isothermal line, with respect to the shapes and slopes from low to high voltage ranges, and also the values of current at every point within wide varying ranges, from 10⁻⁶ to 1, 10⁻⁵ to 1 and 10⁻¹⁹ to 1 A m⁻² for NRS-PPV, OC₁C₁₀-PPV and P3HT devices, respectively. Because the works of Nicolai et al.^20,21 do not consider non-symmetric barriers at contacts, they merely obtain qualitative agreement between theoretical results and experimental data. When compared with the works of Nicolai et al.,^20,21 it can be seen that the non-symmetric barriers are an important factor for reaching quantitative agreement between theoretical results and experimental data.

Fig. 5 Experimental (symbols)^8,10 and theoretical (lines) J–V curves of a NRS-PPV hole only diode, with thickness L = 560 nm and at different temperatures.

Fig. 6 Experimental (symbols)^8,10 and theoretical (lines) J–V curves of a OC₁C₁₀-PPV hole only diode, with thickness L = 275 nm and at different temperatures.

Fig. 7 Experimental (symbols)^5,42 and theoretical (lines) J–V curves of a P3HT hole only diode, with thickness L = 95 nm and at different temperatures.

Nicolai et al.^20,21 extracted parameters for the three materials as follows, σ ∼ 0.1 eV, N_t ∼ (1.3 × 10²³ to 4.0 × 10²³) m⁻³, (E_t − E_v) ∼ 0.6 eV. Although they do not analyze variation of parameters with temperature, they postulate that the electron traps have a common origin, which is most likely related to hydrated oxygen complexes. However, analysis of the variation of parameters with temperature is very important to test the internal consistency of a theoretical model. Therefore, we list all parameters optimized in Tables 1 and 2. Our modified model yields σ ∼ (0.055–0.15) eV, N_t ∼ (7.5 × 10²² to 7.6 × 10²⁴) m⁻³, (E_tF = E_t − E_F) ∼ 0.2 eV. The dispersion of N_t is not in line with the postulation of Nicolai et al.^20,21 The positive value of E_tF means that the position of traps is higher than the Fermi energy E_F. Table 1 also shows that the values of W_left and W_right are non-symmetric, with W_right > W_left and the built-in potential V_bi = W_right − W_left in the devices is positive.

Table 1 Temperature-independent parameters of the double Gaussian model with non-symmetric potential barriers at contacts optimized by fitting J–V data for NRS, OCC^8,10 and P3HT^5,42 organic diodes

	NRS	OCC	P3HT
σt (eV)	0.055	0.1285	0.15
Nt (m^−3)	7.55 × 10^22	1.3 × 10^24	1.7 × 10^26
EtF (eV)	0.2	0.11	0.215
Wleft (eV)	0.1	0.375	0.35
Wright (eV)	1.01	0.578	0.69

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Table 2 Temperature-dependent parameters N_f and μ₀ of the double Gaussian model with non-symmetric potential barriers at contacts optimized by fitting J–V data for NRS, OCC^8,10 and P3HT^5,42 organic diodes

NRS	T (K)	233	252	272	298
	Nf (m^−3)	2.12 × 10^25	2.2 × 10^25	2.3 × 10^25	2.4 × 10^25
	μ0 (m^2 V^−1 s^−1)	1.4 × 10^−11	2.75 × 10^−11	3.9 × 10^−11	8.5 × 10^−11
OCC	T (K)	235	255	275	293
	Nf (m^−3)	2.7 × 10^26	2.8 × 10^26	3.0 × 10^26	3.5 × 10^26
	μ0 (m^2 V^−1 s^−1)	1.1 × 10^−7	1.2 × 10^−7	1.4 × 10^−7	1.5 × 10−^7
P3HT	T (K)	215	235	255	275	294
	Nf (m^−3)	2.42 × 10^33	1.36 × 10^32	1.57 × 10^31	3.07 × 10^30	9.17 × 10^29
	μ0 (m^2 V^−1 s^−1)	8.35 × 10^−5	1.7 × 10^−4	3.23 × 10^−4	3.75 × 10^−4	4.25 × 10^−4

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We may further analyze variations of N_f and μ₀ with temperature. We can fit N_f by using eqn (8) and determine the characteristic energies of E_vF = E_v − E_F and σ. The μ₀ can be fitted by using the following non-Arrhenius expression.

μ₀ = μ₀₀ exp[Δ/kT + λ(Δ/kT)²] (31)

with three regression coefficients μ₀₀, Δ and λ. The fitted curves are compared with data points in Fig. 8. The figure shows that the agreement is satisfactory. The determined parameters for N_f and μ₀ are listed in Table 3. The values of E_vF = E_v − E_F are always negative, which means that the Fermi energy E_F is always higher than the valence level E_v. The free holes are indeed non-degenerate, and the formulae for free holes in eqn (8) and (9) are reasonable. We can calculate E_tv in terms of the relationship, E_tv = E_t − E_v = (E_t − E_F) − (E_v − E_F) = E_tF − E_vF. The results are about 0.22, 0.44 and 0.45 for NRS, OCC and P3HT, respectively. This is within the error range of Nicolai et al. – their results are about 0.6 eV. Therefore our results for E_tv are in line with those of Nicolai et al., and support the assumption that the electron traps have a common origin, which is most likely related to hydrated oxygen complexes.

Fig. 8 Variations of N_f(T) and μ₀(T) with temperature. Symbols plot data listed in Table 2, and lines are the smoothed curves obtained by using eqn (9) and (31), respectively.

Table 3 Temperature-independent parameters for functions N_f(T) and μ₀(T) determined by fitting data in Table 1

	NRS	OCC	P3HT
N0 (m^−3)	4.6756 × 10^25	8.5742 × 10^29	3.3709 × 10^28
EvF (eV)	−0.0214	−0.3350	−0.2524
σ (eV)	0.0149	0.0834	0.1305
μ00 (m^2 V^−1 s^−1)	2.2073 × 10^−6	2.6554 × 10^−6	5.755 × 10^−8
Δ (eV)	−0.3411	−0.1037	0.4744
λ	0.0175	0.0737	−0.0279

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5. Conclusion

In this work, we have analyzed variations of the density of trapped carriers with the density of free carriers, the effective mobility and the ratio of diffusion coefficient to effective mobility with the density of trapped carriers. It is shown that the density of trapped carriers is a dramatically increasing function of the density of free carriers at low density, and tends to saturation at high density. The density of trapped carriers is always an increasing function of the density of trapped carriers, and the slope dramatically increases at high density of trapped carriers. Although the ratio of diffusion coefficient to effective mobility is the Einstein type for free carriers, the ratio for total carriers is the non-Einstein type, and is an increasing function of the density of trapped carriers at low density, and a decreasing function at high density. This explains the recent argument about Einstein or non-Einstein types.

We emphasize the importance of non-symmetric barriers at contacts to quantitatively describe the current–voltage relationships of typical organic layers sandwiched in two metallic electrodes. It is shown that slopes of current–voltage curves at low bias are very sensitive to the values of right barriers. The slopes in all bias are sensitive to the values of left barriers, and increase as the left barrier decreases.

When applying the modified model to three typical devices with organic NRS-PPV, OCC-PPV and P3HT layers, we obtain excellent agreement between theoretical results and experimental data. Evident improvement can be seen with our model when compared with the qualitative agreement of Nicolai et al.^20,21 The extracted main parameters of materials are within the error range of Nicolai et al., and support the assumption that the electron traps have a common origin.

Acknowledgements

This work was supported by the Program for Excellent Talents of Sichuan Province of China under Grant no. 2011JQ0053, and the Science and Technology Foundation of State Key Laboratory for Shock Wave and Detonation Physics under Grant no. 9140C670103120C6702.

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