Improved Determination of the Mobility and Built-In Voltage in Asymmetric Single-Carrier Devices

The steady-state charge-carrier mobility in semiconductors can be extracted from the space-chargelimited current in single-carrier devices. However, in many cases, a built-in voltage is present, which should be known accurately to obtain the correct mobility. Here, it is demonstrated that band bending at the injecting electrode has important consequences for the built-in voltage and the analytical description of the current-voltage characteristics. It is shown that the built-in voltage can be accurately determined from the slope of the current-voltage characteristics on a semilogarithmic scale. Knowing the effect of band bending on the injected current, a simple analytical equation for the drift-diffusion space-chargelimited current above the built-in voltage is derived, which allows for improved determination of the charge-carrier mobility from experimental data.

The steady-state charge-carrier mobility in semiconductors can be extracted from the space-chargelimited current in single-carrier devices. However, in many cases, a built-in voltage is present, which should be known accurately to obtain the correct mobility. Here, it is demonstrated that band bending at the injecting electrode has important consequences for the built-in voltage and the analytical description of the current-voltage characteristics. It is shown that the built-in voltage can be accurately determined from the slope of the current-voltage characteristics on a semilogarithmic scale. Knowing the effect of band bending on the injected current, a simple analytical equation for the drift-diffusion space-chargelimited current above the built-in voltage is derived, which allows for improved determination of the charge-carrier mobility from experimental data. DOI: 10.1103/PhysRevApplied. 13.034069 A widely used method to determine the charge-carrier mobility in semiconductors and organic semiconductors, in particular, is the measurement of space-charge-limited currents [1,2]. To do so, single-carrier devices are prepared, in which, depending on the work functions of the electrodes, either the electron or hole current through the semiconductor is measured as a function of the applied voltage. In the case of asymmetric electrode work functions, a built-in voltage exists [3,4], giving rise to a built-in electric field. For voltages below the built-in voltage, the current is dominated by diffusion of charge carriers, whereas above the built-in voltage the current is dominated by drift [5]. In the case of an Ohmic injecting electrode on an undoped semiconductor without traps, a space-charge-limited current is obtained, as described by the drift-only approximation given by the Mott-Gurney square law [6]: where J drift is the current density, V is the voltage, V BI is the built-in voltage, ε is the permittivity, µ is the charge-carrier mobility, and L is the layer thickness. Equation (1)  The built-in voltage has previously been obtained by electroabsorption measurements [3], photovoltaic measurements [4], and capacitance-voltage measurements [7]. However, these methods do not always give an accurate estimate of the built-in voltage [8]. Alternatively, builtin voltages can be estimated by fitting J -V characteristics with drift-diffusion simulations [5,9,10] or can be obtained by means of charge-extraction measurements [11]. Recently, a method was proposed to extract the builtin voltage directly from the ratio of the forward and reverse currents of a single-carrier device, based on drift-diffusion simulations [12]. While this method has potential, care should be taken to account for band bending at the injecting contact, which modifies the built-in voltage. Here, it is demonstrated that band bending has crucial implications for the built-in voltage and, as a result, characterization of the mobility with Eq. (1). By accounting for band bending, it is demonstrated that a near-perfect agreement between numerical simulations and analytical equations for the drift-diffusion current is obtained. Based on this result, a method is proposed to extract the built-in voltage directly from current-voltage characteristics. Having established the importance of band bending on the diffusion current and the built-in voltage, a simple analytical expression for the space-charge-limited current above the built-in voltage is derived, containing both drift and diffusion. This equation allows for direct determination of the mobility from experimental current density-voltage characteristics, leading to an improvement over the widely used Mott-Gurney law.
In Fig. 1, a schematic depiction of the band diagram of an electron-only device is given. The barrier at the injecting electrode (cathode) is zero, leading to sharp band bending in the vicinity of the cathode caused by the accumulation of diffused charge carriers. The amount of band bending is quantified by parameter b. The barrier ϕ b at the extracting electrode (anode) is nonzero and band bending near this electrode is therefore absent. As a result, a built-in voltage is present, which is given by Fig. 1. This demonstrates that band bending at the injecting contact has important implications for the internal field in the device. Neglecting band bending would lead to an overestimation of the internal electric field and V BI . In addition, disregarding the band-bending parameter results in severe overestimation of the diffusion current, as demonstrated in the following.
As derived previously [5], the diffusion-dominated current density in an organic metal-insulator-metal device is given by where q is the elementary charge, N C is the density of states, ϕ b is the barrier at the extracting contact, k is the Boltzmann constant, T is the temperature, and b is the band-bending parameter given by [5] b = kT q ln The inclusion of a band-bending parameter, b, is of critical importance not only for the correct description of the builtin voltage, but even more so for the correct magnitude of the current, especially in forward bias. This is displayed in Fig. 2(a), where the drift [Eq. (1)] and diffusion [Eq.
(2)] contributions to the current are compared with numerical drift-diffusion simulations [14], which is an approach that is also followed in Ref. [12]. The same parameters are used: N C = 10 25 m −3 , µ = 10 −8 m 2 /Vs, T = 295 K, and L = 100 nm. For boundary conditions in the numerical simulations, the barrier at the injecting electrode is set to zero, while varying the barrier ϕ b at the extracting electrode. These parameters lead to a band-bending Notably, for organic semiconductors, more realistic values of N C of approximately 10 27 m −3 (corresponding to a site spacing of 1 nm) lead to a parameter b of approximately 0.3 V, which has a larger effect on the built-in voltage. As demonstrated in Fig. 2(a), agreement between the analytical and numerical description is clearly superior to that obtained when neglecting band bending at the injecting contact (red lines). In this case, the built-in voltage is overestimated, leading to a disagreement between the Mott-Gurney law and the numerical simulations. In addition, the diffusion current is greatly overestimated due to the nonphysical boundary conditions at the injecting electrode. When including band bending, summing of the drift and diffusion currents, as described separately by Eqs. (1) and (2), results in a near-perfect agreement with the numerical simulations, as shown in Fig. 2(b). This agreement also holds when varying the input parameters, as shown in the Supplemental Material [15]. Also, the injection-limited reverse-bias current is now correctly reproduced by Eq.
(2), as displayed in Fig. 2(c). Notably, the Mott-Gurney law [Eq. (1)] is not applicable in reverse bias, since the injecting contact is non-Ohmic, preventing the buildup of space charge. The accurate description provided by Eqs.
(1) and (2) in forward and reverse bias is due to a combination of the correct definition of both the built-in voltage and the boundary conditions at the injecting electrode. It is clear that the analytical current equations only reproduce the numerical simulations when band bending is taken into account. Using the sum of the drift and diffusion current, a correct analytical description is obtained, which can be used to fit experimental data without using numerical modeling. This approach can even be applied to disordered semiconductors [13]. In the simplest case of using only Eq. (1) to fit experimental data, it is still important to use the correct built-in voltage, as shown below for the case of an organic-semiconductor device. When disregarding band bending, the built-in voltage will be overestimated, by typically 0.3 V, which leads to an overcorrection of the experimental J -V characteristics and, in turn, results in an overestimation of the charge-carrier mobility.
To obtain the built-in voltage and mobility from experimental data, the drift-diffusion current [sum of Eqs. (1) and (2)] can be fitted to the measured J -V characteristics. The built-in voltage will shift the calculated J -V characteristics horizontally along the voltage axis, while the mobility will shift the J -V characteristics vertically, so that both parameters can be obtained independently. Another, somewhat empirical, method to obtain the built-in voltage independent of the mobility has been described previously [16]. Here, the built-in voltage is obtained from the transition from the exponential diffusion regime to the quadratic drift regime, which can be observed on a semilogarithmic J -V plot, as shown in Fig. 3(a). The same simulation parameters are used as those in Fig. 2. To show the validity and to improve the precision of this method, the slopes of numerically simulated J -V characteristics, dln(J )/dV, are depicted in Fig. 3(b). At low voltages (V < V BI ) in the diffusion-dominated regime, the slope of the exponential J -V characteristics approaches q/ (kT). With increasing voltage, the slope reduces, marking the transition to the drift-dominated regime. The transition point is characterized by the voltage at which the slope reaches a value of q/ (2kT), which is half of the slope in the diffusiondominated exponential J -V region. Indeed, the derivative dln(J )/dV of the analytical diffusion current [Eq. (2)] in the limit of V → V BI equals q/ (2kT), which confirms that the transition in slope corresponds to the built-in voltage. As shown in Fig. 3(b), the extracted built-in voltages from the derivatives of the numerical simulations are also very close to V BI = ϕ b − b, demonstrating that the built-in voltage can be reliably extracted from the slope of the J -V characteristics. It should be noted that this method to obtain V BI is independent of the parameters used, such as the density of states, mobility, temperature, layer thickness, or the relative permittivity.
For voltages larger than the built-in voltage, a simplified expression for the drift-diffusion current can be derived. First, Eq. (3) is substituted into Eq. (2), followed by expressing Eq. (2) in terms of V BI . By summing Eqs. (1) and (2), the current density for V > V BI in an asymmetric single-carrier device (V BI kT/q), including both drift and diffusion, is then described by which is compared to numerical simulations in Fig. 3(a) and can be further reduced for V − V BI kT/q to the   relatively simple expression Notably, Eq. (5) contains only the mobility and built-in voltage as unknown parameters. With the built-in voltage determined from the transition from the exponential diffusion current to the quadratic drift current, Eqs. (4) or (5) can be used to extract the charge-carrier mobility from experimental J -V characteristics as an improved alternative to the Mott-Gurney law [Eq. (1)].
As an example of applying these methods to experimental data, a hole-only device of the conjugated polymer poly (9,9- Fig. 4(a)], as displayed in Fig. 4(b). The built-in voltage, corresponding to the voltage at which the slope has a value of q/2kT, amounts to 0.41 V.
With V BI known, the mobility can be obtained directly from experimental data with Eq. (4) or its simplified version, Eq. (5). With Eqs. (4) and (5), the mobility is obtained correctly, even at voltages very close to V BI , as displayed in Fig. 4(c). This can be seen from the negligible overestimation of the mobility at low voltage, which is due to the inclusion of diffusion in Eqs. (4) and (5). The obtained mobility is also in close agreement with transient electroluminescence measurements on similarly prepared films [17]. As expected, when estimating the mobility with the drift-only approximation [Eq. (1) (Mott-Gurney law)], the mobility can only be obtained at higher applied voltage, where diffusion is negligible. At low voltages, the apparent mobility is overestimated significantly, which is caused by neglecting the diffusion contribution to the current. The mismatch at low voltages becomes more severe when band bending is also neglected. With b calculated to be 0.26 V [Eq. (3)] and V BI = 0.41 V, ϕ b amounts to 0.67 V, which is in agreement with the expected barrier of 0.7 eV. Taking the built-in voltage to be equal to ϕ b leads to a severe overestimation of the mobility at low voltages. As a result, it is demonstrated that Eqs. (4) and (5), in combination with the proposed method to obtain V BI , constitute an improved method to obtain low-voltage mobility in single-carrier devices.
With the mobility known, the analytical calculations of the current density are compared with experimental data. These simple expressions for the drift-diffusion current, Eqs. (4) and (5), are in excellent agreement with experiment, using a mobility of 2 × 10 −9 m 2 /Vs. Equation (1) still works reasonably well when using the correct built-in voltage, if sufficient voltage is applied for the current to be fully drift dominated. When the effect of band bending on the built-in voltage is neglected (V BI = ϕ b ), the calculated current severely underestimates the experimental data.
In conclusion, it is shown that including band bending is critical for describing both the built-in voltage and the drift and diffusion currents in asymmetric single-carrier devices. It is demonstrated that the builtin voltage can be accurately obtained from the transition between the exponential diffusion-dominated regime and the quadratic drift regime. Based on these results, a simple