Determination of charge-carrier transport in organic devices by admittance spectroscopy: Application to hole mobility in $\alpha$-NPD

Hole mobility in $N,N^{\prime }$-diphenyl-$N,N^{\prime }$-bis(1-naphtylphenyl)-$1,1^{\prime }$-biphenyl-$4,4^{\prime }$-diamine ($\alpha$-NPD) is evaluated by electrical characterization in the ac regime. The frequency-dependent complex admittance and impedance of the structure consisting of the organic layer, grown by thermal evaporation, sandwiched by indium tin oxide and aluminum electrodes, are measured as functions of the applied dc voltage. The capacitance response shows negative values for frequencies below a characteristic value depending on the bias and ranging from $0.1\mathrm{Hz}$ up to $20\mathrm{Hz}$. It increases with the modulation frequency and reaches a peak, the magnitude and position of which are functions of the applied voltage. For higher frequencies, a minimum can be observed before the capacitance increases again up to a constant value. A final decreasing occurs at frequency of $4\times {10}^6\mathrm{Hz}$. The analysis of the experimental data is performed by a detailed theoretical study of the steady-state and small-signal electrical characteristics of the device. Numerical calculations are based on the solution of the basic semiconductor equations for the system consisting of two electrodes connected by the semiconducting channel formed by the organic layer. The description explicitly includes a continuous distribution of trap density of states and a field-dependent carrier mobility. The spatially dependent charge carrier and occupied trap concentrations, as well as the various components to the total current density, are obtained for the dc and ac regimes and are analyzed for given bias and frequency. Based on a formalism used in the study of inorganic semiconductors, the results of the simulation show that the inductive contribution to the capacitance response originates from the modulation of the hole concentration in the organic material, leading to the corresponding carrier transit time. Moreover, the low-frequency behavior of the capacitance curves could be explained by the presence of a band of defect states which modifies the charge distribution within the organic layer and the injection of electrons from the cathode. We show that the latter contribution is also responsible for the negative values of the capacitance measured below $10\mathrm{Hz}$. Good agreement is observed between the experimental and theoretical electrical characteristics, in particular for the differential susceptance results and the subsequent hole mobility values. Our approach can be a useful contribution for the methodology of obtaining mobilities from admittance measurements as it allows one to clarify the physical origin of the measured frequency-dependent capacitance and to check for the experimental procedure. This work finally leads to the formulation of the conditions under which small-signal ac measurements can be used to determine carrier mobility in organic devices.

Figure 1

Experimental (a) capacitance $C$ and (b) conductance $G$ as functions of the modulation frequency $f$ for applied dc voltages $V_0$ between $3\mathrm{V}$ and $10\mathrm{V}$ at room temperature. The inset shows the capacitance $C$ as a function of frequency $f$ in the low-frequency range $(0.1–{10}^3\mathrm{Hz})$ for the same applied dc voltage values.

Figure 2

Negative differential susceptance $-\Delta B$ curve as a function of frequency $f$ for applied dc voltages between $3\mathrm{V}$ and $10\mathrm{V}$.

Figure 3

Hole mobility ${\mu }_h$ in $\alpha$-NPD as a function of the square root of the applied electric field $E$. The experimental points (solid squares) were obtained by relating the carrier transit time to the peak in $-\Delta B\left(f\right)$ for different applied dc voltages. The fit is shown as the solid line. Experimental (open squares) and fit (solid line) results of dark current injection measurements.

Figure 4

Experimental (solid squares) and theoretical (lines) current-voltage characteristics of the ITO/$\alpha -$NPD/Al system. The result including the effect of parallel conductance $G_p$ is shown as solid lines, with $G_p=4\times {10}^{-6}\mathrm{S}$. $R_d$ is the differential resistance. The inset shows the flat energy band diagram of the system.

Figure 5

Real parts of the small-signal ac hole concentration $\tilde{p}$, electron concentration $\tilde{n}$, and occupied trap concentration ${\tilde{n}}_t$ as functions of the $x$ coordinate for applied dc voltage $V_0=5\mathrm{V}$ and modulation frequency of $0.1\mathrm{Hz}$. The electron barrier height $q{\varphi }_n$ is taken as $0.4\mathrm{eV}$.

Figure 6

Capacitance $C$ as a function of frequency $f$ for applied voltage $V_0=3$, 6, and $9\mathrm{V}$. Set (a) is related to the case where the barrier height at the right contact (Al) $q{\varphi }_n$ is of $1.6\mathrm{eV}$ while set (b) corresponds to the situation where $q{\varphi }_n=0.4\mathrm{eV}$. Series resistance $R_s$ is set to zero.

Figure 7

Density of trap states $D_t$, steady-state occupied trap concentration $d_{t0}$, real part, and imaginary part of the small-signal occupied trap concentration, $\mathrm{Re}{\tilde{d}}_t$ and $\mathrm{Im}{\tilde{d}}_t$, as functions of the trap energy $E_t$, for $x=0$. The energy reference of $E_t$ is here the valence band edge. The small-signal quantities are shown for frequency $f=1\mathrm{Hz}$.

Figure 8

Capacitance $C$ as a function of frequency $f$ for applied voltage $V_0=3$, 6, and $9\mathrm{V}$. Set (a) is related to the case where the barrier height at the right contact (Al) $q{\varphi }_n$ is of $1.6\mathrm{eV}$ while set (b) corresponds to the situation where $q{\varphi }_n=0.4\mathrm{eV}$. Traps are present with a concentration of $8\times {10}^{13}{\mathrm{cm}}^{-3}$ and series resistance $R_s$ is set to zero.

Figure 9

Theoretical (a) capacitance $C$ and (b) conductance as functions of the frequency $f$ for dc bias $V_0$ between $3\mathrm{V}$ and $10\mathrm{V}$. Parameters of the simulation are of Table , with $R_s=45\Omega$ and $G_p=4\times {10}^{-6}\mathrm{S}$. The inset shows the capacitance $C$ as function of frequency $f$ in the low-frequency range $(0.1–{10}^3\mathrm{Hz})$ for the same applied dc voltage values.

Figure 10

Calculated negative differential susceptance $-\Delta B$ curve as a function of frequency $f$ for applied dc voltages between $3\mathrm{V}$ and $10\mathrm{V}$. Series resistance $R_s$ is equal to zero.

Figure 11

Hole mobility ${\mu }_h$ in $\alpha$-NPD as function of the applied electric field $E$, determined by numerical simulation. The straight solid line is the input data of the calculations, the full circles represent the values obtained by applying the admittance method. The inset shows the electric potential and field distributions in the simulated device for $V_0=5\mathrm{V}$.