Modeling space-charge-limited currents in organic semiconductors: Extracting trap density and mobility

We have developed and have applied a mobility edge model that takes drift and diffusion currents to characterize the space-charge-limited current in organic semiconductors into account. The numerical solution of the drift-diffusion equation allows the utilization of asymmetric contacts to describe the built-in potential within the device. The model has been applied to extract information of the distribution of traps from experimental current-voltage measurements of a rubrene single crystal from Krellner et al. [Phys. Rev. B 75, 245115 (2007)] showing excellent agreement across several orders of magnitude in the current. Although the two contacts are made of the same metal, an energy offset of 580 meV between them, ascribed to differences in the deposition techniques (lamination vs evaporation) was essential to correctly interpret the shape of the current-voltage characteristics at low voltage. A band mobility of $0.13{\mathrm{cm}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$ for holes is estimated, which is consistent with transport along the long axis of the orthorhombic unit cell. The total density of traps deeper than 0.1 eV was $2.2\times {10}^{16}{\mathrm{cm}}^{-3}$. The sensitivity analysis and error estimation in the obtained parameters show that it is not possible to accurately resolve the shape of the trap distribution for energies deeper than 0.3 eV or shallower than 0.1 eV above the valence-band edge. The total number of traps deeper than 0.3 eV, however, can be estimated. Contact asymmetry and the diffusion component of the current play an important role in the description of the device at low bias and are required to obtain reliable information about the distribution of deep traps.

Figure 1

Energy-band diagram showing the HOMO and lowest unoccupied molecular orbital (LUMO) bands of the semiconductor. The horizontal line on both sides represents the position of the Fermi level at the contacts. (a) System in equilibrium, no current flow. (b) Applying a positive voltage on the right contact makes the current flow in the opposite direction of the built-in electric field.

Figure 2

Density of trap states calculated in Ref. 23 (red markers) and approximation by a piecewise exponential (blue line).

Figure 3

Measurement (circles) and model fit using the density of traps in Ref. 23 and symmetric contacts.

Figure 4

Measurement (circles) and model fit using an exponential density of traps and asymmetric contacts.

Figure 5

Comparison of the obtained exponential and Gaussian density of traps to that obtained in Ref. 23 using TD-SCLC spectroscopy.

Figure 6

Measurement (circles) and model fit using a Gaussian density of traps and asymmetric contacts.

Figure 7

Potential in the crystal at different applied voltages. The inset shows the position of the virtual cathode as a function of the applied voltage.

Figure 8

Position of the Fermi level, with respect to the band edge, near the extracting contact as a function of the applied voltage. For voltages beyond 2 V, where the diffusion term can be neglected, the Fermi level is always below 250 meV, thus, all states deeper than $\sim 300\mathrm{meV}$ essentially are occupied.

Figure 9

Comparison of the obtained Gaussian and piecewise exponential distribution of traps to that obtained in Ref. 23 using TD-SCLC spectroscopy. The three distributions agree in the 0.1–0.3-eV energy range. Beyond 0.3 eV, the distributions diverge, and the confidence intervals show that uncertainty in this range increases.

Figure 10

Measurement (circles) and model fit using a piecewise exponential density of traps and asymmetric contacts.

Figure 11

Discretization of the Gaussian DOS for the sensitivity analysis.

Figure 12

Sensitivity of the model to variations in the DOS. $D$ represents the sensitivity to variations in the total density of traps, while $S$ represents the sensitivity to the redistribution of traps (i.e., the shape of the trap distribution) keeping the total number constant.