^{1,2,3}, A. Badinski

^{4}, R. A. J. Janssen

^{2}and R. Coehoorn

^{2,3,a)}

### Abstract

The development and application of predictive models for organic electronic devices with a complex layer structure, such as white organic light-emitting diodes, require the availability of an accurate and fast method for extracting the materials parameters, which determine the mobility in each of the layers from a set of experimental data. The absence of such a generally used method may be regarded as one of the reasons why so far relatively little consensus has been obtained concerning the most appropriate transport model, the shape of the density of states (DOS), and the underlying microscopic parameters, such as the width of the DOS and the density of hopping sites. In this paper, we present a time-efficient Gauss-Newton method for extracting these parameters from current-voltage curves for single-carrier devices, obtained for various layer thicknesses and temperatures. The method takes the experimental uncertainties into account and provides the correlated uncertainty margins of the parameters studied. We focus on materials with a Gaussian DOS with random and spatially correlated disorder. Making use of artificially generated as well as experimental data sets, we demonstrate the accuracy and limitations, and show that it is possible to deduce the type of disorder from the analysis. The presence of an exponential trap DOS, as is often observed for the case of electron transport, is found to significantly reduce the accuracy of the transport parameters obtained.

I. INTRODUCTION

II. METHOD

III. PARAMETER EXTRACTION ON ARTIFICIAL DATA—TWO CASE STUDIES

A. Ideal injecting contact and no trap states

B. Material with trap states

IV. APPLICATION TO HOLE AND ELECTRON TRANSPORT IN PF-TAA

A. Hole transport

B. Electron transport

V. SUMMARY AND CONCLUSIONS

### Key Topics

- Current density
- 16.0
- Carrier mobility
- 13.0
- Hopping transport
- 13.0
- Electric measurements
- 9.0
- Materials analysis
- 9.0

## Figures

Schematic energy level diagram of a double-carrier device indicating all materials and device parameters included in this study. For holes ( and electrons ( ), the widths of the DOS, , the hopping site distances , and the injection barriers Δ at the anode (*A*) and cathode (*C*) are indicated. For the electrons, the superimposed exponential DOS is indicated with the two additional parameters involved, *E* _{0} and . An applied voltage (*V*) leads to an energy difference between the Fermi levels ( ) of the anode and the cathode.

Schematic energy level diagram of a double-carrier device indicating all materials and device parameters included in this study. For holes ( and electrons ( ), the widths of the DOS, , the hopping site distances , and the injection barriers Δ at the anode (*A*) and cathode (*C*) are indicated. For the electrons, the superimposed exponential DOS is indicated with the two additional parameters involved, *E* _{0} and . An applied voltage (*V*) leads to an energy difference between the Fermi levels ( ) of the anode and the cathode.

*J*(*V*, *L*, *T*) characteristics, generated using the ECDM with 10% noise in *J* ( ), obtained from a fit using the ECDM (solid lines) and obtained from a fit using the EGDM (dashed lines), for an organic layer thickness equal to 67 nm (a) and 122 nm (b) and for a temperature equal to 272, 220, and 170 K. The parameter values used for generating the curves are given in the text.

*J*(*V*, *L*, *T*) characteristics, generated using the ECDM with 10% noise in *J* ( ), obtained from a fit using the ECDM (solid lines) and obtained from a fit using the EGDM (dashed lines), for an organic layer thickness equal to 67 nm (a) and 122 nm (b) and for a temperature equal to 272, 220, and 170 K. The parameter values used for generating the curves are given in the text.

Results of the ECDM and EGDM analyses presented in Sec. III A . (Un)converged results are indicated by open (filled) symbols. The plus symbols indicate the initial parameter values. (a) Sequences of -values as obtained using the ECDM ( ) and the EGDM ( and ), in the latter case resulting from two different starting points. (b) Trajectories in the -plane corresponding to the sequences shown in Figure 3(a) . (c) Temperature dependence of (symbols) and best fits based on Eq. (7) (solid lines). (d) Projection of the full 95% confidence ellipsoid on the -plane.

Results of the ECDM and EGDM analyses presented in Sec. III A . (Un)converged results are indicated by open (filled) symbols. The plus symbols indicate the initial parameter values. (a) Sequences of -values as obtained using the ECDM ( ) and the EGDM ( and ), in the latter case resulting from two different starting points. (b) Trajectories in the -plane corresponding to the sequences shown in Figure 3(a) . (c) Temperature dependence of (symbols) and best fits based on Eq. (7) (solid lines). (d) Projection of the full 95% confidence ellipsoid on the -plane.

Fit parameters (open symbols) obtained using the EGDM and the ECDM for *J*(*V*) curves generated using the other model and using the initial σ and parameter values indicated by the solid symbols. The calculations were carried out for 100 nm thick devices assuming *T* = 273 K. The other initial parameters are given in the text.

Fit parameters (open symbols) obtained using the EGDM and the ECDM for *J*(*V*) curves generated using the other model and using the initial σ and parameter values indicated by the solid symbols. The calculations were carried out for 100 nm thick devices assuming *T* = 273 K. The other initial parameters are given in the text.

Effect on the -value as obtained from the ECDM analysis presented in Sec. III B when constraining some of the parameter values. Solid square: end-point of the extraction procedure. Full (dashed) curves: (σ)-curve around the best fit σ-value as predicted using Eq. (2) without (with) constraints. Crosses: -values obtained from the actual model calculations using the predicted optimal values (Eq. (2) ) of all non-constrained parameters. Solid spheres: -values, obtained from the re-application of the parameter extraction method while constraining σ and all other parameter values specified. Open spheres: intermediate (non-converged) results. (a) All parameters free, except σ. The arrow indicates that the value of the upper datapoint (cross) at σ = 0.11 eV is actually far outside the frame of the figure, viz. at . (b) All parameters constrained, except for the mobility parameters . (c) All parameters free, except σ and . Arrow: see full text.

Effect on the -value as obtained from the ECDM analysis presented in Sec. III B when constraining some of the parameter values. Solid square: end-point of the extraction procedure. Full (dashed) curves: (σ)-curve around the best fit σ-value as predicted using Eq. (2) without (with) constraints. Crosses: -values obtained from the actual model calculations using the predicted optimal values (Eq. (2) ) of all non-constrained parameters. Solid spheres: -values, obtained from the re-application of the parameter extraction method while constraining σ and all other parameter values specified. Open spheres: intermediate (non-converged) results. (a) All parameters free, except σ. The arrow indicates that the value of the upper datapoint (cross) at σ = 0.11 eV is actually far outside the frame of the figure, viz. at . (b) All parameters constrained, except for the mobility parameters . (c) All parameters free, except σ and . Arrow: see full text.

Obtained best fit -values as a function of the width of the density of states σ, for the case studied in Sec. IV B . The uncertainty margins of every point and a guide-to-the eye (drawn curve) are shown.

Obtained best fit -values as a function of the width of the density of states σ, for the case studied in Sec. IV B . The uncertainty margins of every point and a guide-to-the eye (drawn curve) are shown.

Obtained best-fit parameters as a function of the width of the density of states, σ, for the case studied in Sec. IV B , for (a) the site density, (b) the trap site density, (c) the trap width, (d) the built-in voltage, and (e) the *C* parameter, defined by Eq. (7) .

Obtained best-fit temperature-dependent values of the mobility, , for three different values of σ (symbols), linear fits to data (lines), and the values of the slope parameters *C* (defined by Eq. (7) ).

Obtained best-fit temperature-dependent values of the mobility, , for three different values of σ (symbols), linear fits to data (lines), and the values of the slope parameters *C* (defined by Eq. (7) ).

Histogram of the measured current density at 273 K and at 5 V for a hole-only PF-TAA based device with a layer thickness of 122 nm, as discussed in Sec. IV A . The solid line shows the best-fit normal distribution.

Histogram of the measured current density at 273 K and at 5 V for a hole-only PF-TAA based device with a layer thickness of 122 nm, as discussed in Sec. IV A . The solid line shows the best-fit normal distribution.

Voltage dependence of the standard deviation of the measured current density for two temperatures (295 K and 273 K) in a PF-TAA based hole-only device with a layer thickness of 122 nm, as discussed in Sec. IV A .

Voltage dependence of the standard deviation of the measured current density for two temperatures (295 K and 273 K) in a PF-TAA based hole-only device with a layer thickness of 122 nm, as discussed in Sec. IV A .

## Tables

Overview of the cases studied in this paper. The description indicates which of the elements, shown in Figure 1 , are included. A = artificial data and E = experimental data.

Overview of the cases studied in this paper. The description indicates which of the elements, shown in Figure 1 , are included. A = artificial data and E = experimental data.

Initial and obtained parameter values for the ECDM data sets studied in Secs. III A and III B , analyzed using the ECDM model. A trap DOS, characterized by the parameters and *E* _{0} (see text), was only assumed in Sec. III B .

Initial and obtained parameter values for the ECDM data sets studied in Secs. III A and III B , analyzed using the ECDM model. A trap DOS, characterized by the parameters and *E* _{0} (see text), was only assumed in Sec. III B .

EGDM parameter values describing the current density in the PF-TAA based hole-only devices discussed in Sec. IV A , as obtained in Ref. ^{ 8 } and as obtained in this paper. As in Ref. ^{ 8 } , was taken. For the 67 and 122 nm devices, slightly different values of were found in both studies.

EGDM parameter values describing the current density in the PF-TAA based hole-only devices discussed in Sec. IV A , as obtained in Ref. ^{ 8 } and as obtained in this paper. As in Ref. ^{ 8 } , was taken. For the 67 and 122 nm devices, slightly different values of were found in both studies.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content