2.1.

Mean-Field Drift-Diffusion Approach

First consider a single metal/insulator contact, the insulator is semi-infinite in the positive half space in contact with a conductor reaching from 0 to –∞. It is assumed that the conductor can be described in the framework of the linearized Thomas–Fermi approximation. Far away from the conductor/insulator interface, the conductor is not supposed to be influenced by the contact and, thus, the derivative of the electric field vanishes for x → –∞, and the carrier density approaches the equilibrium density. As a consequence, the electric field distribution F_c for the contact material can be derived to be

where σ_c is the conductivity of the conductor, l_TF is the Thomas–Fermi screening length and j is the current density. The respective electrochemical potential κ_c reads: where E_b is the position of the conduction band bottom, κ_∞ the equilibrium value of the chemical potential in the electrode far away from the interface and e is the positive elementary charge.

The electric field distribution F_i in the insulator is described by the one dimensional drift-diffusion equation:

where T is the absolute temperature, k_B is the Boltzmann constant, μ_i is the carrier mobility, ɛ_i is the dielectric constant of the insulator, and ɛ₀ is the permittivity of vacuum. Far away from the conductor/insulator interface, the insulator is not supposed to be influenced by the contact and, thus, the derivative of the electric field and the carrier density (wide bandgap material) vanishes for x → +∞. The respective electrochemical potential κ_i considering Boltzmann statistics reads where Δ is the injection barrier at the insulator/conductor contact, p_i is the carrier density, N_i is the effective density of states, and ϕ_i is the electrostatic potential in the insulator.

The contact formation between insulator and conductor requires, in the first place, the following continuity conditions at the interface to be fulfilled:

Without loss of generality the restrictions to the lack of surface charges and dipole layers can be assumed. Applying these continuity conditions to Eqs. 1, 4, the following boundary condition can be found to solve Eq. 3:

Equation 5 connects the carrier density p_i(0) at the interface within the insulator to the prevailing electric field F_i(0). Apparently, an effective injection barrier Δ_eff = Δ – el_TF(ɛ_i/ɛ_c)F_i(0) can be defined that contains a field-induced change of the barrier Δ by the voltage drop within a surface near region of thickness l_TF in the conductor. Yet, keeping in mind that l_TF is on the order of angstroms, an essential contribution of the barrier modification requires rather high electric fields at the interface—respectively, external voltages—to match the expectations.¹⁸ Let us assume an injection-limited case where the electric field is constant across the insulator. Then, for an ITO/OS contact with an injection barrier of 0.4 eV, a l_TF of 8.6 Å, and ɛ_i/ɛ_c = 3/9.3, a reduction of Δ by half would require an electric field on the order of 10⁹ V/m, which would be 100 V for a 100-nm-thick insulator. Be aware that an effective injection barrier of 0.2 eV would yield already severe space-charge effects.^{14, 18, 19} A transition voltage of 100 V from injection to SCLC with an initial barrier height of only 0.4 eV is, however, an unrealistic high value.

2.2.

Single-Particle Influence

The discussed problematic stems from an inherent problem of the used one-dimensional mean-field approximation, implying averaging of all variables over the plane perpendicular to the injection current density. However, this procedure is restricted to relatively high carrier densities and does not account for single-particle phenomena, such as the well-known and considered Schottky effect,^{20, 21} describing the coulombic coupling of a single charge in the insulator to its image in the conductor.⁸ The superposition of the resulting image charge potential and the external potential yields a modification of the injection barrier, which is given by

where δϕ_sch is the barrier lowering induced by the Schottky effect, whereas x_m denotes the distance of the potential barrier maximum from the physical conductor/insulator contact. Apparently, the barrier lowering due to the Schottky effect is stronger than the aforementioned barrier modification by the field penetration in the contact. Considering again an injection barrier of 0.4 eV and a dielectric constant of the insulator of 3, a reduction of Δ by half would now require an electric field of only 8 × 10⁷ V/m, which would be 8 V for a 100-nm-thick insulator.

It is thus essential to estimate the validity region of the MF approximation and the SPD approach. This can be simply done by considering the force on a single carrier in the insulator¹⁷

being composed out of the force from an external field (first term), the force from the coulombic coupling to the individual image charge (second term), and the force from the coulombic coupling to all other charges (last term). It is feasible that the MF approximation is better suited when the interaction with other charges exceeds the one with its own image. On the other hand, the SPD approach should be applied once the coupling to the image charge dominates. Thus, the last term has to be compared to the second term of Eq. 7 at position x_m. A rough estimation given in Ref. 17 yields a transition criterion for the validity of the MF and SPD approach, expressed in terms of the two characteristic length scales of both approximations: the distance from the Schottky barrier maximum x_m to the physical contact and r_i = p_i(0)^−1/3 the average carrier distance in the insulator. The critical field at the contact can thus be extracted from the following transcendental equation:

Equation 9 can be solved graphically as depicted in Fig. 1, where the characteristic length x_m and 0.2 r_i are plotted, exemplarily, in dependence of the electric fields at the interface F_i(0) for typical parameters of ITO and organic semiconductors.^{22, 23, 24, 25, 26} Apparently, two points of intersection exist and, thus, there are an upper and lower limit for the use of the SPD regime. The dependence of the two corresponding critical fields on the barrier height Δ is depicted in Fig. 2. The SPD approximation is valid in the range of interface fields between F_min and F_max, where 0.2·r_i > x_m. The MF approximation is, on the contrary, below F_min and above F_max applicable. Yet, F_max lies for some fields in a region above the line F_lim, above which the applied Boltzmann statistics is violated and hence has no physical meaning. F_L marks the field that relates to a r_i equal to the insulator thickness L (100 nm) and thus, below F_L, the use of the continuous description is by all means inappropriate. This is the SP region. Be aware, that the transitions between regimes are not abrupt but rather indicate a transition region that is however much narrower than the indicated MF, SPD, and SP regimes, themselves.¹⁷

Using Eq. 3 in conjunction with the critical field values from Fig. 2 and the fact that the integral over the field along the MIM system yields the external voltage, a validity chart of the used SP, SPD, and MF approximations in the plane of external bias and the injection barrier can be established. The result is depicted in Fig. 2 as well. For the given example, the MF approximation holds over the entire voltage range as long as the injection barrier is <0.16 eV. With the growing Δ value, the bias window for the SPD region gains. Below V_min, the MF approximation always holds because, at low bias, the reservoir of the enrichment contact is not depleted and the diffusion field at the contact does not allow for a Schottky effect at all. Above the line V_max the barrier lowering due to the field penetration in the contact is so strong that, again, a substantial amount of carriers penetrate the insulator. The MF approximation holds again. In the striped SP region, the use of continuous particle densities is impossible because less than one carrier exists between the contact plates.

2.3.

Modified Mean-Field Approach

The deficit of the MF approximation to describe the carrier injection within the SPD regime but outside the striped parameter range results from the absence of the individual image force dragging on each single charge carrier. As long as x_m < 0.2r_i, such a strong but short-range deviation from the mean field near the interface can be modeled like a dipole layer, with a characteristic width x_m. Because each injected charge carrier traversing the dipole layer needs to perform the work Δ–eδϕ_sch due to its individual image charge force, the electrostatic potential at the interface will be shifted by

Equation 10, defined at the interface between insulator and conductor, implies that the dipole layer is supposed to be infinitesimally thin, thus leading rather to a dipole jump of the electrostatic potential right at the interface. The resulting modified boundary condition at the interface thus reads

where the effective barrier Δ_eff defined on basis of Eq. 5 is extended by the third term in the square brackets. Equation 11 holds in the SP region for max{V_min, V_L} ≤ V <min{V_max, V_lim}. The empirical interpolation factor 1 – x_m/(0.2 r_i) simply provides a smooth transition between regimes.

Because of the modification of the boundary condition, Fig. 2 must be reconsidered. Certainly, the critical voltages V_min, V_max do not change because, at these bias conditions, Eq. 5 is obtained. However, the unassailable SP regime framed by V_L is supposed to shrink because the density of injected charge carriers will increase within the insulator. In the remaining SPD regime, the modified MF (MMF) approximation is now applicable.

Despite the inviolable SP regime, the MMF approximation is suited to describe MIM systems across a large parameter space. The difference between the MMF-approximated drift-diffusion approach and the MF approximation is illustrated on basis of current voltage characteristics of a symmetric MIM system of ITO/intrinsic OS/ITO, assuming at both interfaces an injection barrier of 0.4 eV. The IV characteristics are depicted in Fig. 3. For both approximations, the IV characteristic is linear at low bias. The barrier lowering is small, and the contacts provide N_i exp[–Δ/k_BT] carriers. Because the conductivity is independent from the bias, an ohmic behavior is observed. For voltages of >0.1 V, the MMF approach starts to deviate from the MF approach. As the Schottky lowering of the injection barrier exceeds the potential drop in the injecting electrode, a steeper increase in the current density is observed for the MMF approximation approaching the space-charge limit faster. The transition to the SCLC indeed appears now at voltages of ∼10 V, whereas the MF approximation yields a transition voltage of 100 V. In the SCLC regime, both characteristics merge again because the OS bulk (drift term), but not the contacts (boundary conditions), limits the current density.

In Fig. 3, another example of a charge-transfer-doped OS within the very same MIM structure is shown. Be aware that depletion contacts develop in the considered case. The doping density of 10¹⁷ cm⁻³ increases, in the first place, the current density in the low-voltage regime if compared to the intrinsic case, but retains the ohmic behavior. This is due to the doping-induced background carriers providing a larger OS conductivity as the one embossed by the contacts. Yet, for the MF approach, the current density fades away above 10⁻² V, approaching the characteristic of the undoped MIM system. The doping-induced carriers are depleted from the OS because the supply of carriers from the weak injecting contact is hampered. At increased voltage, the barrier lowering due to the voltage drop in the contact gains, the current increase growth again, and the current density approaches the SCLC regime. For the MMF approximation, such a dependence is not observed. The space-charge region within the depletion width at the injecting contact provides a substantial electric field. In conjunction with the MMF approximation and the associated Schottky lowering, the deliverance of carriers across the depletion contact is sufficient to prevent the depletion of the OS bulk. As a consequence, the high current density level is maintained. At higher voltages, space-charge effects set in again.

The above-discussed topic is by far not restricted to OS but is also applicable to other insulators, such as inorganic dielectrics. The substantial material diversity opening wide parameter windows requires, however, the evaluation of the validity chart of each particular parameter set. An interesting parameter aspect is related to the prevailing dielectric constant of the insulator because it codetermines the strength of the barrier modifications given in Eq. 11. Because the dielectric constant of common organic materials is small, typically <10, the discrepancy between the MMF and MF approximations within the SPD regime is supposed to be stronger than for high-k materials. This is because the Schottky barrier lowering given by Eq. 6 shows a ɛ_i^−1/2 dependence and thus growth with decreasing ɛ_i. At higher ɛ_i, the Schottky effect becomes less important and the IV characteristics of the MMF and MF approximations approach each other. This fact underlines the importance of the modified mean-field approach for the calculation of the electrical response of MIM systems based on organic semiconductors.