Introduction

Recent advances in polymer donors and non-fullerene acceptors have produced organic photoactive layers (PAL) that give more than 15% PCE in single-junction solar cells1,2,3. Yet, the understanding and design of their charge collection electrodes have remained relatively primitive. New PALs are often tested with one of several favored electrode systems, for example, sol‒gel ZnO as electron collection layer (ECL) and evaporated MoO3 as hole collection layer (HCL) in inverted cells. But these electrode materials may not necessarily be best or most suited for manufacturing. A better understanding of contacts would yield new insights in disordered semiconductors, and reveal design rules to optimize performance of any selected PAL. This would ultimately enable development of better solution-processable electrodes that may be more suited to manufacturing.

In organic photovoltaic cells, electrodes set up a built-in potential (Vbi) that creates the internal electric field to generate photocarriers4,5. The Vbi is determined by the difference of effective work functions (ϕeff) between the electron (e) and hole (h) contacts, that is, ϕeff,h − ϕeff,e, where ϕeff is the energy difference between Fermi level (FL) of the specified electrode, and vacuum level (VL) of the semiconductor away from the contact6. ϕeff can differ significantly from the vacuum work function ϕ of that electrode, if interfacial dipole or charge transfer occurs between the electrode and the semiconductor7,8,9,10. The latter case causes pinning of ϕeff due to counteracting electric field set up by the accumulated carriers. Generally, Vbi can be written as11,12: Vbi = Vo − ΔVel, where Vo is the so-called “bare potential” term given by the difference between FL and VL at each of the two contacts, and ΔVel is the total electrostatic band bending into the semiconductor due to carrier diffusion.

We have recently shown for P3HT:PCBM cells using HCLs with systematically “tuned” ϕ that both Vbi and Voc track the ϕeff of the hole contact13. This indicates the hole contact is nonselective. If it were selective, that is, allowing only the correct carrier sign to exit, the majority-carrier density at the contact will rise with illumination intensity, decoupling Voc from Vbi14,15. Furthermore, we found that charge collection is subjected to an interfacial contact resistance that varies strongly with ϕ13. A sharp Ohmic transition occurs a few tenths of an eV beyond FL pinning. The collection (or injection) resistance drops rapidly below the bulk resistance. Thus, while Voc rises strongly with ϕ and levels off at the onset of FL pinning, FF does so at the onset of the Ohmic regime. Together, they define an extended ϕ plateau where PCE broadly maximizes.

In this report, we have performed further detailed measurements in the Ohmic regime, and found that both the Voc − ϕ and FF − ϕ plateaus can in fact tilt slightly upwards. Consequently, PCE can continue to improve, albeit slowly, with overdriving of ϕ in the Ohmic regime. The increase in Voc with ϕ arises as a consequence of soft FL pinning. This occurs in face-stacked polymer semiconductors, where their π-conjugation plane is parallel to the film plane, as distinct from edge-stacked semiconductors, where they are perpendicular. The π orbitals of the frontier monolayer (ML) are exposed. This broadens the density of states (DOS), allowing coercion of FL up the DOS edge, and, consequently, gradual transition of the interface parameter7,8,9, given by S = \(\frac{{d\phi _{{\mathrm{eff}}}}}{{d\phi }}\), from unity to 0. Thus, ϕeff, Vbi, and Voc all continue to rise with ϕ well beyond the nominal FL-pinning threshold. On the other hand, the increase in FF with ϕ arises as a consequence of majority-carrier accumulation near the contact, as FL rises up the DOS edge. This causes carrier mobility to rise significantly, even in quality semiconductors with narrow widths, improving charge transport and collection. The phenomena together illuminate the importance of both interface and bulk DOS shapes for device physics, and opportunities for optimization beyond the Ohmic transition.

Results and discussion

Selection of model PAL system

We employ PBDTTPD (chemical structures, Fig. 1) as model of a face-stacked polymer donor, and PCBM as model of an isotropic molecular acceptor. PBDTTPD is donor‒acceptor (DA) polymer of the benzo[1,2-b:4,5-b′]dithiophene (BDT) family that has yielded record organic solar cell efficiencies16,17. It comprises an electron-rich BDT donor moiety conjugated to an electron-deficient thiophene acceptor moiety. DA polymers generally exhibit a much weaker electronic coupling than homopolymers to conformational disorder, which results in better electronic transport18,19. However, they also exhibit a less congruous π-stacking owing to the DA dissymmetry20,21,22,23. This induces face-stacking in thin films, which aids charge transport through the film thickness direction24. PCBM provides a highly reliable photoinduced electron acceptor for PBDTTPD25,26,27. It gives rise to a robust donor–acceptor morphology, and temperature-independent photogeneration efficiency, which greatly simplify device analysis.

Fig. 1: Chemical structure of photoactive layer and TAF materials.
figure 1

The value of x denotes doping level of the triarylamine–fluorene (TAF) polymers, in hole per repeat unit: x can vary between 0 (undoped) and 1 (fully doped). The value of x is 0.7–0.8 for all TAF polymers studied here. ϕ is vacuum work function given in eV (±0.05 eV), measured by ultraviolet photoemission spectroscopy.

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As polymer donor, PBDTTPD provides a contrast to P3HT in its polymer backbone orientation. Both their gas-phase HOMO energies are practically identical in the long-chain limit, ca. 6.15 eV, according to DFT/CAM-B3LYP/6-311G calculations (Supplementary Fig. 1). But their solid-state ionization energies (IE) are considerably different. Ultraviolet photoemission spectroscopy (UPS) gives 5.3 eV for the PBDTTPD film, which is 0.65 eV larger than P3HT (Fig. 2). These values are unaltered by blending with PCBM. Thus, the difference may be attributed to a stable orientation effect, edge-stacking for P3HT and face-stacking for PBDTTPD28. This is supported by the exclusion of PCBM from the surface layers of P3HT:PCBM films29,30, but not PBDTTPD:PCBM films (Supplementary Fig. 2). Surprisingly, despite its weaker energetic dispersion, both PBDTTPD and PDBTTPD:PCBM films show a broader valence band edge than the P3HT counterparts. Hemi–Gaussian fitting gives the standard deviation width for the former to be 0.25 eV, and the latter 0.12 eV. However, the interpretation of these widths is not straightforward, because of packing effects on ionization energies10,31.

Fig. 2: Ultraviolet photoemission spectroscopy of valence band.
figure 2

Legend: red, PBDTTPD:PCBM (1:1.5 w/w) film; blue, P3HT:PCBM (1:0.8 w/w) film. Ionization energy (IE) is marked by standard extrapolation of inflection point. Films were measured consecutively in identical configuration with same He-I radiation intensity.

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Estimation of transport DOS width

To estimate the width of the relevant transport DOS, we measured the current-density‒voltage (JV) characteristics of the films in the sandwiched diode configuration. If suitably strong hole injection layers (HIL) are used, the hole-only JV characteristics of PBDTTPD:PCBM are well-behaved, similar to P3HT:PCBM. These follow the ideal Mott‒Gurney law over one-and-a-half decades of J up to 2000 mA cm‒2, that is, J ~ μeff (V − V*)2, where V* is the apparent Vbi, and μeff is the effective carrier mobility (Supplementary Fig. 3). For PBDTTPD:PCBM, hole μeff is ca. 5 × 10–4 cm2 V–1 s–1, about three to five times as large as in P3HT:PCBM. The Gaussian disorder model thus suggests σ/kBT ≲ 3, i.e., σ ≲ 75 meV, where σ is the effective width of the transport DOS, that is, standard deviation of its forward edge (see discussions below). Good behavior suggests this edge extends at least 3.5σ below its effective center. Photothermal deflection spectroscopy indeed finds the optical DOS edge of “clean” polymer semiconductors is Gaussian out to 4σ, but charge transfer and polaron excitations can intervene in the presence of acceptors32,33,34,35. Such excitations can be readily seen in PBDTTPD:PCBM by index-matched optical spectroscopy (Supplementary Fig. 4).

Thus, both PBDTTPD and P3HT films and their PCBM composites exhibit rather similar (and narrow) transport σ, but apparently widely differing surface σ. This may be rationalized by DOS broadening of the frontier ML due to variations at its interface, including from Coulomb fluctuations of fixed charges and free carriers in the adjacent doped polymer layer36. The frontier DOS dominates energy-level alignment at the contact, while the bulk DOS determines quasi-FL and carrier mobility inside the film. However, their differentiation is often neglected in the literature.

Computational study of DOS effects: Heaviside model

To investigate the effects of these two DOS on energy-level alignment, we adopted the usual single-junction electrostatic model37,38,39, but added surface broadening, allowing surface σo to differ from bulk σ. We treat the polymer donor in the usual way as a series of MLs in thermal equilibrium with the electrode, taking into account image–charge polarization40,41 and electrostatic band bending11,42,43, but neglecting secondary influences from the opposite contact. The carrier density at thermal equilibrium is computed for each ML.

First, let us consider holes in a semiconductor with a hypothetical Heaviside DOS. This DOS rises abruptly from 0 to 5.5 × 1020 eV–1 cm–3 at the Heaviside energy EHS, set at 5.85 eV. It contacts an electrode with vacuum work function ϕ, which sets up a local work function ϕloc. This is the difference between the local FL and VL, which varies with distance (z) from the junction into the semiconductor. Under flatband condition for typical diode thicknesses, the electrostatic band bending vanishes at z ≈ 15 nm. The value of ϕloc at this z is thus taken to be the ϕeff that determines Vbi.

Figure 3a shows the plot of ϕeff vs ϕ for a temperature T of 298 K. Holes are stabilized by the usual image–charge interaction energy: \(\Delta E_{{\mathrm{pol}}} = - \frac{1}{2}\frac{{{\mathrm{e}}^2}}{{{\mathrm{4}}\pi \varepsilon _{\mathrm{o}}\varepsilon _{r}}}\frac{1}{{2d}}\), where the symbols have their usual meanings, which amounts to ca. 0.1 eV for the first ML ML0. Thus, most of the carriers in ML0 are trapped; only a small fraction is mobile, as noted previously11,12. For the Heaviside DOS, where σ = 0, the work function onset for pinning (ϕpin) occurs at 5.7 eV. This occurs at the intersection of the S = 1 segment (Schottky‒Mott limit) and the S = 0 segment (pinned-FL limit). The pinning depth given by: ΔEpin = EHS − ϕpin, is thus 0.15 eV at 298 K, which decreases to 0 as T → 0 K.

Fig. 3: Computational study of DOS effects in Heaviside model.
figure 3

a Fermi-level pinning plot for semiconductor with Heaviside DOS (5.85 eV; 5.5 × 1020 eV–1 cm–3) convoluted with variable Gaussian width σ: Effective work function against vacuum work function of electrode. Red cross marks Heaviside energy EHS. Other crosses mark valence-band-edge energy EV set up by corresponding σ. b Plot of hole density against ϕ offset by EHS, for the cases in (a), at z = 8 Å (i.e., middle of ML0), 24 Å (ML1), and 56 Å (ML3). c Fermi-level pinning plot for Heaviside DOS where only ML0 is convoluted with variable Gaussian width σo. d Plot of hole density against ϕ offset by EHS, for the cases in (c). ϕeff is evaluated at z = 15 nm. Other parameters: semiconductor dielectric constant, perpendicular to contact, εr = 2.1; charge-screening distance into electrode, do = 5 Å; ML thickness, dML = 16 Å; temperature, T = 298 K; numerical convergence quality, 1 mV.

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When the Heaviside DOS is convoluted with σ, its frontier edge rises less steeply. Extrapolation of the inflection point gives an onset, which defines the valence-band-edge energy EV, marked by crosses. The pinning depth now given by: ΔEpin = EV − ϕpin, becomes deeper, increasing from 0.15 to 0.27 eV as σ increases from 0.0 to 0.15 eV. The ΔEpin range is consistent with the energy-level alignment picture inferred from Vbi measurements44,45. Concomitantly, the transition from the Schottky‒Mott to the pinned-FL limit also becomes softer. Nevertheless, the value of S over the interval (ϕ − ϕpin) ∈ [0.2, 1.0 eV] remains smaller than 20 meV per eV. Thus, the FL pinning is still fairly stiff, which can be represented by a fixed level outside the HOMO or LUMO band edge, as widely assumed46. Figure 3b shows the computed plot of accumulated hole density np against (ϕ − EHS) at different depths into the semiconductor. These confirm that np builds up primarily in ML0, with diffuse tail extending into higher MLs. As ϕ increases beyond ϕpin, np builds up quickly to a few 1019 cm‒3, proportional to (ϕ − ϕpin). This contact-induced carrier density, called “δ-doping”, can be directly observed by sub-gap electroabsorption spectroscopy44,45.

Now, consider the same Heaviside semiconductor but with only ML0 subjected to Gaussian broadening of σo. Figure 3c shows the resultant FL pinning plot. Surface broadening is even more effective than bulk broadening to induce soft FL pinning. This is because of charge accumulation into ML0. Now ϕ has to be driven beyond (ϕpin + 2σo) to reach the final ϕeff. Thus, the use of an electrode with extreme work function can advantageously coerce FL up the DOS edge in these cases to produce a larger Vbi. This phenomenon is inherent to disordered semiconductors. It does not require the presence of exogenous gap states, different from inorganic semiconductor surfaces47. Figure 3d shows that the hole density is largely confined to ML0. The np inside the semiconductor diminishes (cf. Fig. 3b), because of screening by this hole density in ML0. This would eventually frustrate the formation of an Ohmic contact when σo becomes too large. In summary, this study suggests that soft FL pinning can occur with practical consequences for σo of a few tenths of an eV.

Computational study of DOS effects: realistic semiempirical models

To develop a simple yet realistic DOS model to parametrize σ and σo, we treated the polymer donor as a distribution of conjugated segments, in the usual way, of length ℓ (in repeat units), each with own HOMO energy EHOMO. For simplicity, we assume this distribution is uniform over the interval [\({\frac{2}{3}{{< }}\ell > },1{\frac{1}{3}}{{< }}\ell > \)], where <ℓ> is the mean conjugation length. This assumption imposes ℓmax/ℓmin = 2, ensuring each segment hosts only one transport site. We further assume that \(< \ell > \) is given where: \(\left. {\frac{{dE_{{\mathrm{HOMO}}}}}{{d\ell }}} \right|_{\ell {\mathrm{ = < }}\ell > } = k_{B}T\), and the hole-transport sites occupy 65 vol% of the donor matrix, which, in turn, occupies 50 vol% of the PAL. We then normalize this ℓ population with molecular volume to find states per unit volume, distribute in energy space, and convolute the native DOS with σ to get the transport DOS model. Figure 4a shows the results for P3HT:PCBM and PBDTTPD:PCBM. Their DOS shapes differ, because of different energy dispersion in ℓ. But their densities of hopping sites Nsite turn out to be fairly similar, ca. 2 × 1020 cm–3; and their DOS peak values also turn out to be similar, ca. 4 and 5 × 1020 eV–1 cm–3, for P3HT:PCBM and PBDTTPD:PCBM, respectively.

Fig. 4: Computational study of DOS effects with semiempirical model.
figure 4

a DOS models for P3HT:PCBM (red line) and PBDTTPD:PCBM (blue), both at 50 vol% donor: solid, transport DOS; dotted, first-monolayer DOS. Horizontal zero marks EV of the bulk semiconductor, EV,bulk. The optical DOS model is also shown for comparison, arbitrarily located: P3HT:PCBM (gray line) and PBDTTPD:PCBM (black), normalized to same intensity. b Fermi-level pinning plots for the two sets of DOS models. c Plot of hole density against ϕ offset by EV,bulk. σ = 75 meV, σo = 100 meV (P3HT:PCBM), 200 meV (PBDTTPD:PCBM). Other parameters are same as in Fig. 3, except for do = 12 Å; dML = 16 Å (P3HT, edge-stacked), 8 Å (PBDTTPD, face-stacked).

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To get the model for the surface ML, we convolute the native DOS with σo instead of σ. We take σo from an empirical equation: σo = A × exp(−z/zo), where z is distance from junction to center of ML, zo is characteristic distance (5.8 Å), and A is disorder amplitude (0.40 eV). This equation was parametrized by fitting the computed Coulomb fluctuation with distance from a polyelectrolyte layer (Fig. 4b in ref. 36). It gives σo to be 0.1 and 0.2 eV for P3HT:PCBM and PBDTTPD:PCBM, respectively, remarkably similar to the UPS results. In essence, the PBDTTPD surface layer is exposed to stronger multipolar Coulomb fluctuations because of its face-stacking. Our energy-level-alignment model then gives S ≈ 10 and 50 meV eV–1 for P3HT:PCBM and PBDTTPD:PCBM, respectively, as shown in Fig. 4b.

Thus, semiempirical DOS modeling predicts FL pinning to be softer for PBDTTPD:PCBM than P3HT:PCBM, due to energetic broadening of its frontier ML. Figure 4c shows the hole density in ML0 is consequently larger for PBDTTPD:PCBM than P3HT:PCBM, but of the same order of a few 1019 cm‒3 well beyond ϕpin. The hole density inside PBDTTPD:PCBM is smaller than inside P3HT:PCBM, but of the same order of 1018 cm‒3 at z = 2 nm, which is sufficient for an Ohmic contact44,45.

In summary, this study suggests that PBDTTPD:PCBM provides a contrast to P3HT:PCBM in its surface DOS due to polymer backbone orientation. Although soft pinning can be discerned in computed FL pinning plots for σ ≳ 0.2 eV7,8,9,10, its possible relevance is disregarded, because the disorder threshold is so much larger than the typical transport σ of 0.12 eV or smaller48,49. Here, we point out, however, that the typical energetic broadening of the frontier ML, particularly in face-stacked semiconductors—a common motif of “modern” DA polymers—potentially makes this phenomenon relevant.

Fabrication and characterization of PBDTTPD:PCBM solar cells

For experimental validation, we fabricated PBDTTPD:PCBM solar cells with a series of spin-on 20-nm-thick hole-doped triarylamine–fluorene (TAF) polymer50 layers with different ϕ as HCL. The chemical structures of this family are shown in Fig. 1. A spin-on 100-nm-thick PBDTTPD:PCBM (1:1.5 weight/weight ratio) is used as PAL. An evaporated 30-nm-thick Ca, capped with 120-nm-thick Al, is used as ECL (see “Methods”). The TAF polymers are used in the self-compensated, heavily hole-doped form51. Tethered CF3SIS or C2F5SIS counter-anions are used to confer improved ambient stability and control ϕ. This family spans ϕ ∈ [5.2, 5.9 eV] through a combination of semiconductor core and ion effects52,53. The lowest member, TFOMe-CF3SIS, gives 5.2 eV, same as the reference poly(3,4-ethylenedioxythiophene):poly(styrenesulfonic acid) (PEDT:PSSH), and the highest member, pTFF-C2F5SIS, 5.9 eV. Because of the amorphous polymer backbone and identical doping level of 0.7–0.8 hole per repeat unit, i.e., 6 × 1020 cm‒3, film resistance and Schottky depletion width do not significantly change across the series. We measured the solar cell characteristics under AM1.5 irradiance of 100 mW cm–2. Typical cell characteristics are shown in Fig. 5a, cell parameters in Fig. 5b, and values in Supplementary Table 1.

Fig. 5: Characteristics of PBDTTPD:PCBM solar cells and hole-only diodes.
figure 5

a JV curves for different hole collection layers in cell configuration: ITO/20-nm HCL/100-nm PBDTTPD:PCBM (1:1.5 w/w)/30-nm Ca/Al, measured under simulated AM1.5 G irradiance of 100 mW cm−2, spectral mismatch corrected, 298 K. b Cell parameters plotted against work function of hole collection layer. Data give population mean; standard error is smaller than symbol size. Blue lines are guides to the eye for TAF series. Yellow region corresponds to Ohmic regime. c Voc(ϕ, T) plot measured at 1.0 sun. Voc data are averaged for forward and reverse sweeps. Typical uncertainty: ±0.01 V at high ϕ; ±0.05 V at low ϕ. Colored lines are fits with constant slope of 80 mV eV‒1. The black line (\(\frac{{{d}V_{{\mathrm{oc}}}}}{{{d}\phi }}\) = 1) is anchored by low-temperature data set for ϕ = 5.2 eV. Below 50 K, Voc approaches Vo, which is the low-temperature Vbi. d Plot of effective hole mobility against work function of hole injection layer in hole-only diodes: ITO/20-nm HIL/100─120-nm PBDTTPD:PCBM (1:1.5 w/w)/Ag, for TAF series (circles) and PEDT:PSSH (square) as HIL. Error bar corresponds to standard error of mean. Green dashed line is from Gaussian disorder theory (see text).

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The results show cell performance improves systematically with work function of the HCL as it increases well into the Ohmic regime. For reference, PEDT:PSSH gives Voc of 0.92 V, FF of 0.55, Jsc of 9.25 mA cm–2, and thus PCE of 4.8%, similar to, or better than, reported literature results for similar PBDTTPD molecular weights and processing conditions16,17,54,55,56. The morphology of the PAL can be further improved by hot spinning and/or solvent additives to reach an even higher Jsc of 11 mA cm–2 57,58,59,60. However, this leads to a significant variability presumably due to donor–acceptor morphology drift that degrades our experiment. Thus, we use spin-cast films at room temperature without any solvent modifier. The open-circuit series resistance, given by Rs,oc = \(\frac{{{\it{dV}}}}{{{\it{dJ}}}}|_{V = V_{{\mathrm{oc}}}}\), of the cell with PEDT:PSSH is 22 Ω cm2. This gives the sum of bulk charge-transport and interfacial charge-extraction resistances13.

When PEDT:PSSH is substituted by TFOMe-CF3SIS having the same ϕ, FF increases to 0.64, and the JV hysteresis disappears. This improvement is associated with halving of Rs,oc to 11 Ω cm2, which we attribute to better hole collection by the TAF polymer without the tunneling layer on PEDT:PSSH61,62. As ϕ increases along the TAF series toward 5.9 eV, Rs,oc decreases and then levels off at 5.5–6 Ω cm2 for ϕ ≳ 5.35 eV. This marks the Ohmic transition of the PBDTTPD:PCBM contact13. However, Voc continues to rise steadily toward 0.96 V, and FF toward 0.72, while Jsc remains constant at 9 mA cm–2. Consequently, PCE increases from 5.1 to 6.1%. In contrast, the Voc of P3HT:PCBM cells declines slowly with ϕ beyond the Ohmic transition13.

Ruling out morphology and other artefacts

To check for absence of any confounding variation in the PAL morphology, we spin-cast PBDTTPD:PCBM films of various thicknesses on fused silica, and on selected TAF films with ϕ of 5.35 or 5.75 eV, and measured our optical spectra (Supplementary Fig. 5). We found the π → π* bandshape of PBDTTPD remains invariant, despite its strong dependence on film thickness. Spectral similarity is better than 1%, except for emergence of polaron band in the sub-gap of films on the HCLs due to interfacial hole doping. Thus, morphology artefacts can be ruled out.

To rule out device fabrication artefacts, we prepared “inverted” solar cells, where a self-compensated, electron-doped poly(fluorene–alt–benzothiadiazole) film (ϕ ≈ 3.3 eV)63 was first spin-cast as bottom ECL, then PBDTTPD:PCBM layer, then the TAF polymers top HCL was spin-cast from an orthogonal solvent, acetonitrile, over the PAL, followed by evaporation of Ag as top electrode (Supplementary Fig. 6). This eliminates any possibility that an HCL underlayer may have “primed” the PAL overlayer. Yet, we observed the same trends. As ϕ increases from 5.35 to 5.9 eV, Voc rises from 0.92 to 0.95 V, FF from 0.48 to 0.56, while Jsc remains constant at 10.8 mA cm‒2. Thus, device fabrication artefacts can also be ruled out for the basic trends.

For comparison, solar cells with sol–gel ZnO as bottom ECL, and evaporated MoO3 as top HCL, give Voc and FF of 0.93 and 0.60 V, respectively (Supplementary Fig. 7). Substituting MoO3 with hole-doped mTFF-C2F5SIS increases Voc to 0.97 V and FF to 0.66. Deep FL pinning is known to occur at the MoO3/PAL interface64. Thus, these spin-on ultrahigh-work function polymers appear to have an inherent advantage over the evaporated ultrahigh-work function oxides.

Evidence for soft Fermi-level pinning

The Vbi increases with ϕ of the HCL in the nominally pinned regime, as predicted by theory, confirming up-creep of FL at the hole contact. Vbi was estimated by the method of “photocurrent inversion”, which uses the saturated Voc at sufficiently low temperatures as proxy for Vbi11,65. This method applies for contacts that are nonselective and non-injecting, as manifested by symmetry of the illuminated JV characteristics about Voc, that is: J(Voc + δV) = −J(Voc − δV) for δV ≈ 0.1 V, which occurs here for T ≲ 50 K (Supplementary Fig. 8). The method fails if photogeneration efficiency falls strongly with decreasing temperature66. Figure 5c shows the Voc(ϕ, T) plot, measured at 1.0 sun. The Voc values for T ≲ 50 K converge to a limit. This limit corresponds to Vo, the low-temperature Vbi11. A break in the \(\frac{{dV_{{\mathrm{oc}}}}}{{d\phi }}\) slope for T ≲ 50 K occurs at ϕ = 5.25 eV, which gives the low-temperature ϕpin. This separates a lower segment with S = 1 from an upper segment that has a small but non-zero value of S. In P3HT:PCBM cells, the upper S value is 013. Here, it is 80 mV eV–1, as predicted by our DOS model. Clearly, the hole contact exhibits soft FL pinning for PBDTTPD:PCBM, but not P3HT:PCBM. At higher temperatures, Vbi is subjected to ΔVel loss, which scales logarithmically with carrier density at the contact, partially offsetting the FL upshift (Supplementary Fig. 9). This decreases ϕpin to ca. 5.1 eV at room temperature. In the absence of any FL upshift, the increasing ΔVel loss would cause Vbi to decline gradually with ϕ, as observed in P3HT:PCBM cells13. Thus, softening of FL pinning enables the Voc − ϕ plateau to tilt upwards.

Evidence for hole-mobility enhancement

However, drift–diffusion–generation modeling14,38,67,68 with the above Vo trend could not reproduce the FF results, unless we also allow the effective hole-mobility μeff to increase with ϕ. To check directly for this supposed enhancement, we fabricated hole-only diodes with different PBDTTPD:PCBM film thicknesses, employing the TAF series as HIL, and evaporated Ag as hole-exit layer. We found that the JV characteristics of PBDTTPD:PCBM films thicker than ca. 50 nm obey the Mott–Gurney equation with V* of 0.82 ± 0.01 V (Supplementary Fig. 10). Free fitting yields a Mott–Gurney index m of 2.0 ± 0.05, where m = \(\frac{{d{\mathrm{log}}J}}{{d{\mathrm{log}}\left( {V - V{\,}_ \ast } \right)}}\) (Supplementary Fig. 11). Thus, the condition is satisfied69 for reliable estimation of μeff from the Mott–Gurney equation: \(J = \frac{9}{8}\varepsilon _{r}\varepsilon _{o}\mu _{{\mathrm{eff}}}\frac{{(V - V{\,}_ \ast )^2}}{{d^3}}\) (Supplementary Fig. 12). The results are plotted in Fig. 5d. They show that as ϕ increases from 5.2 to 5.9 eV, μeff increases from 6 × 10–4 to 9 × 10–4 cm2 V‒1 s–1. This indicates mobility enhancement induced by the contact.

Origin of mobility enhancement

We show that this effect is a manifestation of carrier mobility enhancement due to filling up of the DOS in disordered semiconductors. We considered the simple Gaussian disorder model, wherein carriers exhibit a constant mobility at low carrier density—the Boltzmann transport regime—which increases as the density approaches and exceeds a certain threshold that depends on σ70,71,72. For a Gaussian DOS with σ/kBT of 3, this threshold occurs at a reduced carrier density (np/Nsite) of ca. 5.6 × 10‒3 73,74.

To model the expected size of effect, we first fitted the numerical results of Fig. 1 of ref. 73 to an empirical compact form: μp = μo,p (1 + α (np/Nsite)β), where μo,p is the limiting hole mobility at low density, α is the hole-density coefficient, and β is the exponent. This form describes the numerical results quite well. For σ = 75 meV, i.e., σ/kBT = 3, we obtain α = 20, β = 0.50, and μo,p = 4.3 × 10–4 cm2 V–1 s–1. The function is shown in Fig. 6a. Mobility begins to depart from the low-density limit at ca. 1016 cm–3. A larger σ would shift the onset to lower density.

Fig. 6: Simulation results.
figure 6

a Hole-mobility simulation, from Gaussian disorder theory: μp = μo,p (1 + α (np/Nsite)0.50), where np is hole density, μo,p is limiting hole mobility at low density, α is hole-density coefficient, and Nsite is transport site density. For parameter values, see caption of Table 1. b Drift–diffusion–generation simulation: red dots, simulation; blue line, experiment from Fig. 5b.

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With this density-dependent mobility, we computed the JV characteristics for the PBDTTPD:PCBM hole-only diodes. The results give a μeff − ϕ trend that matches the experimental one fairly well (dashed line, Fig. 5d). This is satisfying for a simple model without any tuning parameter. Thus, when ϕ increases from 5.2 to 5.9 eV, the charge density in ML0 of PBDTTPD:PCBM increases from 7 × 1018 to 3 × 1019 cm‒3 (Fig. 4c). The corresponding mobile hole density (Np), just beyond the contact, increases from 2 × 1017 to 1.2 × 1018 cm‒3, crossing the Boltzmann limit, enhancing carrier mobility over a substantial portion of the film.

To check for self-consistency with the solar cell characteristics, we also computed these using the same μp function. The other parameters, together with the apparent constant mobility needed to fit the characteristics μp,app, are shown in Table 1. The values of μp,app are close to those of μeff. The computed Voc − ϕ and FF − ϕ trends also match the experimental ones very well, again without any tuning parameter (Fig. 6b). Thus, contact-induced mobility enhancement tilts the FF − ϕ plateau upwards in the Ohmic regime. This is the dominant factor improving FF of the solar cells in this regime.

Table 1 Input parameters for drift–diffusion–generation model.
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The results here extend previous observations that both ultralow chemical doping75,76 and high-current density48,49 can enhance carrier mobility in diodes, as can high gate field in transistors48,49. Here, contacts with extreme work functions can accumulate carrier density sufficiently strongly to enhance carrier mobility, even in low-disorder semiconductors.

Generalization

To visualize carrier density profile within the PAL, we computed the depth-dependent np and nn for various device conditions, and ϕ at both limits of 5.2 and 5.9 eV (Supplementary Fig. 13). We find for the hole-collection half of the cell that np generally remains significantly larger for the larger ϕ, under all conditions, which leads to the higher hole mobility. This contact-induced mobility enhancement should be general, independent of soft FL pinning. To check this, we fabricated P3HT:PCBM cells with different HCLs—PEDT:PSSH, TFOMe-C2F5SIS, and mTFF-C2F5SIS (Supplementary Fig. 14). The FF indeed systematically increases from 0.57 ± 0.01 for PEDT:PSSH to 0.70 ± 0.01 for mTFF-C2F5SIS, as ϕ increases from 5.2 to 5.75 eV. The final FF attained is unprecedented for these cells. As expected, Jsc remains constant, while Voc declines marginally due to the ΔVel loss.

In an attempt to push the mobility enhancement in PBDTTPD:PCBM films further, we fabricated hole-only diodes with even thinner PALs, down to 40 nm. However, the polymer morphology changes in these very thin films, as suggested by changes in their π → π* spectra. This degrades μp severely, suppressing the expected enhancement (Supplementary Fig. 15).

Thus, simulation and experiment have firmly established two phenomena: (1) soft FL pinning, and (2) contact-induced mobility enhancement. Both these effects indicate that driving work function of the charge collection layer to extreme values well beyond the Ohmic transition can significantly improve Vbi and Voc of organic photovoltaic cells. The strategy is simple, and can be used to generate best performance, even for low-molecular-weight polymer semiconductors. However, there must ultimately exist a trade-off from the increased minority-carrier recombination in the near-contact region, though this does not yet pose a problem here, presumably because bimolecular recombination is very weak11,77.

Methods

Materials

PBDTTPD (1-Material), P3HT (Rieke Metals), and PCBM (1-Material) were obtained from commercial sources, and used as received. PBDTTPD was measured to have Mn = 29 kD by gel permeation chromatography in 1,2,4-trichlorobenzene at 160 °C, and also chloroform at 40 °C. 1:6 wt/wt PEDT:PSSH solution (Clevios P VP Al 4083, Heraeus Precious Metals GmbH) was purified by dialysis against 1-M semiconductor-grade HCl solution, followed by Millipore® water, through a 12-kDa molecular-weight-cutoff membrane to remove cation and acid impurities78. TAF polymers were synthesized following procedures reported in Tang et al.51. To hole dope the TAF polyelectrolytes, the dry TAF polymers were baked on a hotplate at 120 °C in a N2 glovebox for 1 h, then dissolved in anhydrous acetonitrile to give a 20 mM solution, mixed with 1.0 equivalent of nitrosonium hexafluoroantimonate (NOSbF6), also dissolved in anhydrous acetonitrile (40 mM), then purified by precipitation in diethyl ether and redissolution in acetonitrile, twice, to give the respective self-compensated, hole-doped TAFs. The final acetonitrile solutions were used directly for spin casting. The doping levels were evaluated by UV–Vis spectrophotometry51.

Solar cell fabrication and measurements

For PEDT:PSSH as HCL, 40-nm-thick films were spin-cast from the PEDT:PSSH solution onto oxygen-plasma-cleaned ITO substrates, and baked on hotplate at 140oC in air for 10 min; for TAF as HCL, 20-nm-thick films were spin-cast from acetonitrile solutions onto the ITO substrates in the N2 glovebox. Then a solution of PBDTTPD:PCBM in chlorobenzene (1:1.5 w/w; 30 mg mL–1) was spin-cast over the HCLs in the glovebox to give 100-nm-thick films. A 30-nm-thick Ca layer, followed by 120-nm-thick Al layer, was thermally evaporated through a shadow mask at a pressure of 10–6 mbar to define eight 4.29-mm2 pixels on each substrate. Ag paint was applied on the metal electrodes to eliminate external contact resistance. JV characteristics of the cells were collected using a semiconductor parameter analyzer (Keithley 4200) in a N2 chamber, in the dark, and under 100 mW cm–2 simulated AM1.5 solar irradiance (Oriel Sol2A), spectral mismatch corrected (factor, 1.08). Some of the cells were characterized from room temperature (298 K) to 30 K in a cryostat.

Hole-only diode fabrication and measurements

The ITO/HCL/PBDTTPD:PCBM (1:1.5 w/w) stacks were built as described above, but with PBDTTPD:PCBM thicknesses varied over 45–100-nm range. These thicknesses were measured as the step thickness of the scratched stack profile; repeatability was better than ±5 nm. Then a 120-nm-thick Ag layer was thermally evaporated through a shadow mask at a pressure of 10–6 mbar to define eight 4.29-mm2 pixels on each substrate. Ag paint was applied on the metal electrodes to eliminate external contact resistance. JV characteristics of the diodes were collected on a probe station in the N2 glovebox using a semiconductor parameter analyzer (Keithley 4200).

Ultraviolet photoemission spectroscopy (UPS)

UPS was performed on films in an ESCALAB UHV chamber equipped with an Omicron EA 125 U7 hemispherical electron energy analyzer at a base pressure of 10–9 mbar. 10-nm-thick PBDTTPD:PCBM and P3HT:PCBM films were spin-cast in N2 glovebox, following the standard protocol, onto O2-plasma-cleaned Au-coated Si substrates. The substrates were transported in N2 and loaded into the UPS chamber without exposure to ambient air. UPS was performed using He-I radiation (21.21 eV). The photoemission normal to the film surface was collected and analyzed at a pass energy of 5 eV, giving a resolution of 50 meV.

Low-temperature JV measurements

Low-temperature JV measurements were performed in a closed-cycle helium cryostat (Janis APD HC-2). The cells were loaded into the cryostat in the N2 glovebox, and evacuated to 10–6 mbar by a turbomolecular pump. JV characteristics were measured at 1 sun following the standard protocol.