Introducing the model

Our model is inspired by the earlier work of Vasquez et al.26,39 and, more recently, of Blakesley et al.27,28, as well as of Greiner et al.23 and Ley et al.25 Going beyond refs 23, 25, however, we take into account a continuous density of states (DOS) both for the occupied and the unoccupied levels in the organic semiconductor. For simplicity, we start with the common choice27,42,43 of one Gaussian peak, G(E−E H ,σ H ), arising from the energy distribution of the highest occupied molecular orbitals (HOMOs) and one, G(E−E L ,σ L ), from the distribution of the lowest unoccupied molecular orbitals (LUMOs) as indicated in Fig. 1b. To unambiguously define these peaks, both the energies of their maxima, E H and E L , and their standard deviations, σ H and σ L , need to be specified. The latter can be approximated, for example, by the width of the HOMO-derived feature observed in ultraviolet photoelectron spectroscopy (UPS). Here, we adapt the constant value of σ H =σ L =0.25 eV, found for the archetypical organic semiconductor pentacene44. From UPS and inverse photoelectron spectroscopy also the ionization energy (IE=4.8 eV)44 and the electron affinity (EA=2.6 eV)45 of pentacene are known. These quantities are typically determined from the energy difference between the vacuum level (VL) and the low-binding energy onset of the UPS spectrum and the high-binding energy onset of the inverse photoelectron spectroscopy spectrum, respectively. Choosing the onset of a Gaussian peak to be the energy, where the tangent through the inflection point crosses the base line, puts the peak centres 2σ away from the so obtained onsets, that is, at E H =5.3 eV and E L =2.1 eV, respectively. Also indicated in Fig. 1b is the work function, Wf film , given by the energy difference between the Fermi level and the VL outside the organic semiconductor film. Note that, in the specific situation shown, Wf film is actually higher than the IE of the organic semiconductor, where electrons reside at energies above E F before charge equilibration. It is, therefore, clear that in the course of establishing a common electron chemical potential, charge will flow across the interface. To quantify the amount of transferred charge in the general case, we discretize the organic semiconductor film into layers of thickness Δz and distance z from the electrode surface. We then calculate the charge density ρ(z) in each of these layers by integrating

with e the (positive) elementary charge, n the number of molecules per unit area and per discretization interval Δz. As the IE and EA values reported above pertain to pentacene molecules ‘standing’ on the electrode with their long molecular axes almost parallel to the surface normal44,45, we accordingly choose Δz=1.42 nm, the distance d 001 between the crystallographic (001) planes in the pentacene single crystal46; Δz then corresponds to the height of one molecular layer. For the sake of simplicity, we consider layer-by-layer growth of pentacene only, with n=4.1 × 1018 molecules/(m2·Δz). Note, however, that fractional coverages and different growth modes could easily be incorporated by accordingly choosing layer-specific values for n.

The first integral in the curly brackets of equation (1) yields the number of holes in the occupied DOS of the organic semiconductor and the second integral the number of electrons in the unoccupied DOS. While of noticeable consequence only for highly degenerate systems, we, nevertheless, follow refs 23, 25 by allowing, in contrast to refs 27, 28, only single charging of each molecule, which slightly modifies the Fermi function for each individual energy level under consideration to

Here, g H and g L are the HOMO and LUMO (spin-)degeneracies, both 2 in the case of pentacene, and β=1/k B T with k B the Boltzmann constant and T the temperature, set to 300 K throughout the paper. Finally, the energy distributions of HOMO- and LUMO-derived levels are D H [E+eV(z)]=G[E−E H +eV(z), σ H ] and D L [E+eV(z)]=G[E−E L +eV(z), σ L ] in our model case, unless otherwise noted. We stress, however, that (i) any peak shape can be used instead of a Gaussian, (ii) each discretization interval could have a different DOS to account, for example, for image-charge screening by the metal, (iii) more molecular states can easily be taken into account by adding the respective terms in the curly brackets and (iv) potential temperature effects can be included in the standard deviation σ=σ(T); for the values used here, 2σ=0.50 eV is about 20 times bigger than k B T at T=300 K and, therefore, the temperature dependence of f H/L (E) has only little impact on level alignment.

Finally, the electrostatic potential across the organic semiconductor film, V(z), is first set to zero for calculating an initial guess of the charge density via equation (1). Once known, this ρ(z) serves to find the next V(z), which, going beyond refs 23, 25, we obtain by numerically solving the generalized one-dimensional Poisson equation

with the discrete-difference method on a variable grid, using the second-order, central-difference scheme with mixed Dirichlet [V(0)=0] and von Neumann [V′(d)=0] boundary conditions, where d is the total thickness of the considered organic film and ε 0 the vacuum permittivity. For our first illustrative case of a homogenous pentacene film, we assume a static dielectric constant ε of ~3.6, as recently determined by impedance measurements47; note, however, that each discretization interval could, in principle, also be given a different dielectric constant. The DOS of each discretization interval is then shifted in energy by the respective −eV(z), as shown in Fig. 1c, and the next ρ(z) is again generated by occupying all states in the organic up to E F . This self-consistent process is iterated until the difference in charge density between cycles is less than one electron per layer and square meter and the final injection barriers are obtained as Δ e =E F −EA and Δ h =IE−E F .

Some illustrative results

To illustrate the results delivered by the method just described, we first consider 20 layers (~30 nm) of standing pentacene with E F set to lie right at the energy of the HOMO peak maximum. The self-consistent solution of the Poisson equation for this situation is depicted in Fig. 2a,b, showing the charge density and the electron potential energy, respectively. The charge density rapidly vanishes with increasing distance from the electrode surface and reaches almost zero at 5 nm. The corresponding potential energy, which locally shifts HOMO, LUMO and VL by the same amount (IE and EA remain constant), is characteristic for band bending—a term somewhat liberally borrowed from inorganic semiconductor physics despite the fundamental difference that, instead of actual bands there, a distribution of localized states is assumed here. Although this phenomenon has often been attributed to (impurity or intentional) doping29,48,49 in the past, it has recently been argued to arise also in intrinsic organic semiconductors, both of the conjugated-polymer27 and the small-molecule variety24,25. In trying to relate our results to experimental observations, it is important to recall that the work function measured by, for example, UPS or Kelvin probe only reveals the electron potential energy −eV(z⩾d) just outside the organic film (highlighted in Fig. 2b). As pointed out in refs 50, 51, 52, 53, however, the sequence of −eV(d) values obtained by increasing the total thickness d of an organic film in a step-wise manner and probing the surface potential at each step does not, in fact, recover the electron potential energy profile −eV(z) within the organic film for any thickness (Fig. 2c).

Figure 2: Exemplary results of the iterative electrostatic model. (a) Calculated charge density ρ(z), and (b) electron potential energy −eV(z) within a d=30 nm thick (≈20 layer) pentacene film. Surface sensitive experimental methods (for example, UPS or Kelvin probe) only determine the total electron potential energy drop −eV(d) across the entire organic-semiconductor film. (c) −eV(z) for a range of different film thicknesses d up to 30 nm, highlighting that the series of experimentally determined −eV(d), shown as line connecting open circles, does not reproduce the band-bending −eV(z) within the organic semiconductor for any thickness. Full size image

Fermi level pinning

With this in mind, we now plot in Fig. 3a the total potential energy change, −eV(d), across pentacene films of increasing d, for a series of different Fermi level energies. Clearly, when E F lies around mid-gap of the organic semiconductor, only very little charge is transferred from/to the metal electrode and, consequently, the potential energy profile remains close to flat. Upon E F approaching either the HOMO or the LUMO levels, however, pronounced charge transfer and, accordingly, ever higher and steeper potential energy changes across the organic film are obtained. Practically, the Fermi level is shifted by choosing electrodes of different work function, Wf elec , defined by the energy difference between E F and the VL just above the bare electrode surface with no organic material present. The energy barriers/losses for charge carriers, Δ e and Δ h , are then related to Wf elec and −eV(d) through

Figure 3: Impact of effective substrate work function on band bending and barriers. (a) Total electron potential energy change −eV(d) across pentacene films of varying thickness d for a series of Fermi level energies ranging from 0.15 eV below E L , the initial LUMO peak energy (top trace), to 0.05 eV above E H , the initial HOMO peak energy (bottom trace). (b) Data from a converted to injection barriers Δ e for electrons (left axis) and Δ h for holes (right axis). Dashed horizontal lines indicate respective values of zero and the vertical line highlights the cut visualized in c. (c) Injection barriers Δ e (left axis) and Δ h (right axis) as a function of effective substrate work function Φ eff for a d=10 nm thick pentacene film with (·) and without (○) polaronic relaxation energies λ+ (for holes) and λ− (for electrons) taken into account. The transitions between the regions with slope S=0 and S=1 mark the so-called pinning work functions, Φ− for LUMO pinning and Φ+ for HOMO pinning. The lowest achievable injection barriers are not related to λ+/−. Full size image

where ID is the interface dipole. Either because of contamination through atmospheric and/or solvent exposure or owing to deliberate passivation with inert buffer layers, the ID is often found to be (close to) zero for practically employed electrodes in organic electronics. However, microscopic charge rearrangements originating in a variety of physical and/or chemical phenomena occurring between the electrode surface and the first organic layer can, in general, lead to appreciable IDs22,37,38,54,55,56. If available, for example, through a fully atomistic and quantum-mechanical treatment on a particular system, then they can, of course, be readily incorporated into the present model through equations (4) and (5). To study the impact of Wf elec on Δ e/h on more general grounds, however, we refer to an effective electrode work function

in the following. Before charge equilibration, that is, with V(d)=0, the initial barriers are then simply given by Δ e =Φ eff −EA and Δ h =IE−Φ eff , respectively.

This now allows us to make the transition from relative potential energy changes for different values of E F (Fig. 3a) to absolute values of Δ e and Δ h for a series of effective electrode work functions, both of which are now quantities readily accessible through experiment. Figure 3b illustrates that changing Φ eff indeed permits tuning the interfacial energy barriers/losses, but only within a certain range. For our model case of pentacene, the achievable minimum separation between E F and EA/IE rapidly converges to Δ e =Δ h =0.4 eV with increasing film thickness, where decreasing Φ eff below ~3.0 eV does not further reduce Δ e and increasing Φ eff beyond ~4.4 eV does not further reduce Δ h . A more convenient way of visualizing these data is to directly plot the energy barriers as a function of Φ eff for a given film thickness, for example, d=10 nm as indicated by the vertical line in Fig. 3b.

The obtained functional dependence (open symbols in Fig. 3c) is characterized by a slope of zero (S=0) for low Φ eff , a region with S=1 for intermediate Φ eff , and again S=0 for high Φ eff . The transitions between the different regions are marked by the so-called pinning work functions, Φ−=3.0 eV for LUMO and Φ+=4.4 eV for HOMO pinning with the latter matching the experimental value for a 10-nm film of pentacene36. The term pinning refers to the observation that, for multilayer organic semiconductor films of device-relevant thickness, E F cannot cross the HOMO or LUMO level onsets (marked as horizontal lines) and, thus, energy barriers/losses cannot be completely eliminated; regardless of the choice for Φ eff , the Fermi level appears to be ‘stuck’ at specific energy separations (~0.4 eV for d=10 nm) well above/below the respective peak onsets.

In the past, this observation has led to the postulation of material-specific, discrete, quantum-mechanical states at these energies, the so-called ‘integer charge-transfer’ (ICT) states, with ICT+ shifted up in energy from the HOMO onset and ICT− shifted down in energy from the LUMO onset into the gap of the organic semiconductor29. As already pointed out in refs 25, 27, 28, however, the fact that the pinning energies of E F in the organic semiconductor gap emerge naturally (and quantitatively correct) from electrostatics alone: (i) lifts the necessity for invoking the existence of such states and (ii) invalidates the interpretation of their energy separation from the HOMO and LUMO onsets as the respective polaron-binding energies λ+ and λ−. We explicitly illustrate this latter point in Fig. 3c for our Gaussian DOS (cf. ref. 28). For our considered model case, pentacene, the polaron-binding energy, that is, the energy stabilization of an excess charge in the organic semiconductor by geometric relaxation of the charged molecule itself (internal relaxation energy) and of the surrounding lattice (external relaxation energy), was found to be only λ+/−=0.07 eV (refs 57, 58) with the latter being negligible compared with the former59. This value is taken into account in equation (1) by shifting the HOMO and LUMO levels symmetrically towards each other, thus effectively reducing the gap of the organic semiconductor. The magnitude of the resulting changes (closed symbols in Fig. 3c) is, not unexpectedly, comparable to the absolute value of λ+/− and, thus, its contribution to the observed minimum barriers is smaller than the typical experimental accuracy of ~0.1 eV; polaronic effects will, therefore, no longer be considered in the following.

Film thickness and Gaussian DOS

Briefly returning to Fig. 3b, it is evident that the relationship between Δ e/h and Φ eff changes with the total thickness of the organic film. In Fig. 4a, we show vertical cuts through the data presented in Fig. 3b not only for d=10 nm, as discussed in Fig. 3c, but for an entire series of thicknesses. As a direct consequence of band bending within the organic film, the substrate work functions where LUMO/HOMO pinning sets in successively decrease/increase with decreasing film thickness down to Φ−=~2.4 eV and Φ+=~4.7 eV, respectively, for a monolayer of pentacene, which, again, compares favourably to experiment35. Evidently, the thinner the organic semiconductor film, the lower Δ e and Δ h can be made by, respectively, decreasing or increasing Φ eff until, in the extreme case of a monolayer, they vanish for Φ eff =5.5 eV and Φ eff =1.8 eV, respectively (cf. refs 30, 31, 32, 33).

Figure 4: Impact of film thickness and width of Gaussian DOS. (a) Injection barriers Δ e for electrons (left axis) and Δ h for hole (right axis) as a function of effective substrate work function Φ eff for pentacene films of thicknesses from d=1.5 nm to 30 nm. (b) Injection barriers Δ e (left axis) and Δ h (right axis) as a function of Φ eff for a d=10 nm thick pentacene film with the standard deviations of the Gaussian HOMO and LUMO DOS varied from σ H/L =0.1 eV up to σ H/L =0.4 eV while retaining a constant IE and EA. (c) Electron potential energy profiles −eV(z) at the highest effective substrate work function Φ eff =6.5 eV for all σ H/L treated in b. Full size image

The same could, arguably, be achieved also for thicker layers if the DOS tailing into the gap of the organic could be eliminated; it is the presence and the occupation of these gap states, after all, that prevents E F from further approaching the HOMO/LUMO onsets. The origin of the gap states is commonly attributed to structural imperfection/defects28,42,60,61,62. Their density and energy distribution can experimentally be determined only on a case-by-case basis60,61,62, as they depend on specifics of sample preparation and environment, that is, exact film structure and composition60. To elucidate their impact on energy level alignment on more general grounds, we vary the width of the HOMO- and LUMO-related DOS while keeping it Gaussian-shaped, which has been shown to phenomenologically emulate also the impact of exponential tails27. Leaving, at the same time, the energies of the peak maxima, E H and E L , unchanged, trivially leads to the HOMO and LUMO onsets moving into the gap and, consequently, to E F getting pinned ever farther in the gap as well (not shown). A maybe less trivial result is presented in Fig. 4b. There, we keep the HOMO and LUMO onsets, which represent the physical material parameters IE and EA, respectively, at constant energy. Varying the width of the DOS then requires adjusting the energy separation between the peak maxima accordingly. Even under these conditions, however, reducing the Gaussian width of HOMO- and LUMO-related DOS leads to (i) the transition between the regimes of Fermi level pinning (S=0) and the Schottky–Mott limit (S=1) to become more abrupt, (ii) E F successively approaching the respective onsets in the pinning regimes and (iii) the spatial extent of the band-bending region to be reduced, as evidenced in Fig. 4c. The range of minimal energy barriers (~0.2 eV to ~0.6 eV) emerging from our model spans the entire range of values observed in experiment44,55, highlighting the decisive role of disorder in the organic semiconductor in determining interfacial energy level alignment60. Note also that, for T>0 K, finite barriers remain even in the limiting case σ H/L →0 of discrete molecular states, which is, in fact, the only case treated in refs 23, 25.

From band bending to interface dipole

Notably, the phenomenology described so far, the characteristic shape in Figs 3c and 4 in particular, is only ever observed for ‘weakly’ interacting electrode–organic interfaces. There, either through surface contamination or by design, the organic semiconductor is electronically decoupled from the metallic states by a thin, wide-gap buffer layer, which, in following refs 27, 28, we did not explicitly take into account so far. As shown in the Supplementary Discussion, even doing so does not alter the fact that the interfacial energy level alignment between organic semiconductor and metallic electrode always falls into one of three regimes, HOMO pinning, Schottky–Mott limit and LUMO pinning, characterized by slope parameters S=dΔ e /dΦ eff =−dΔ h /dΦ eff of (close to) zero, one and again zero, respectively (cf. Supplementary Figure 1). If, however, the buffer layer is eliminated completely in experiments, which can be achieved by, for example, depositing the organic semiconductor directly onto atomically clean metal surfaces in ultrahigh vacuum, also fractional slope parameters between 0 and 1 are frequently observed26,36,39,55,63. Despite having explored a good portion of the parameter space available in our model, we failed to generate non-integer values of S so far, whereas they are readily (and often quantitatively correct) predicted by the ‘induced density of interface states’ model26,39. This is surprising insofar, as this model is, in essence, quite similar to ours. Besides typically treating only molecular monolayers and assuming the organic DOS to be constant around E F for evaluating S, the main difference lies in the peak shape, which the authors take to be Lorentzian rather than Gaussian. This is entirely reasonable for the envisioned situation, where direct electronic coupling of the monolayer molecular levels to the metal leads to homogenous life-time broadening dominating over disorder-induced inhomogeneous (that is, Gaussian) broadening60. Having already elucidated the impact of the latter in Fig. 4b, we now choose a Voigt peak function with successively increasing Lorentzian weight to represent the DOS of the first organic layer, while retaining pure Gaussians for all subsequent layers. The Φ eff -dependence of the resulting injection barriers into a d=10 nm film of pentacene is shown in Fig. 5a, containing also the limiting, pure-Gaussian case already reported in Fig. 3c for comparison.

Figure 5: Fractional slope parameters and band bending versus interface dipole. (a) Injection barriers Δ e for electrons (left axis) and Δ h for hole (right axis) as a function of effective substrate work function Φ eff for a d=10 nm pentacene film. The peak shape of the HOMO- and LUMO-derived DOS in the first layer is successively varied from pure Gaussian over Voigt to pure Lorentzian. (b) Electron potential energy −eV(z) within a d=10 nm pentacene film and a Φ eff of 5.3 eV. Changing the peak shape of the HOMO- and LUMO-derived DOS in the first layer from pure Gaussian over Voigt to pure Lorentzian leads to a collapse of the spatially extended band-bending region to an interface dipole across the first molecular layer. (c) Injection barriers Δ e for electrons (left axis) and Δ h for holes (right axis) as a function of effective substrate work function Φ eff for a d=10 nm pentacene film. The DOS of the first layer is kept fixed to a Lorentzian peak shape of width 0.25 eV, whereas the standard deviations σ H/L of the Gaussian peak shapes in all subsequent layers increase from 0.1 eV up to 0.4 eV. Although the range of linearity decreases succeedingly, the value of the slope parameter S<1 remains unchanged. Full size image

Clearly, with increasing weight of the Lorentzian contribution, ever lower/higher substrate work functions are required to reach the same barriers for electrons/holes. More importantly, however, the Φ eff -region of non-vanishing slope parameter expands, where S itself decreases from unity to a value of 0.42 for a Lorentz dominated (75%), down to 0.35 for an exclusively Lorentzian DOS, which brackets the experimental observed values of 0.37 and 0.40 for pentacene55,63.