Introduction

The power conversion efficiencies (PCE) of organic solar cells have recently breached the 10% benchmark in both small-molecule- and polymer-based organic solar cells with single-junction photoactive layers (PAL)1. This stimulates renewed interest in urban solar, indoor, and wearable applications and also in tandem organic–silicon hybrid cell structures. As the short-circuit current density (Jsc) of an organic solar cell reaches the photogeneration limit of its donor–acceptor PAL, further improvement in performance can only come from increasing its open-circuit voltage (Voc) and fill factor (FF). Therefore numerous studies have focused on the underlying physics of Voc2,3,4,5,6 and FF7,8,9. In particular, detailed analyses of the cell voltage deficit referenced to the semiconductor charge-transfer gap have been attempted, i.e., Δ = ECT/e − Voc, where e is the electronic charge, and ECT is the energy difference between the band edges of the donor highest-occupied molecular orbital and the acceptor lowest-unoccupied molecular orbital (LUMO)2,3,4,5,6.

In addition, contacts can further limit what is thermodynamically possible in the bulk of the cell. The role of contacts between the PAL and each of the two electrodes has not been well understood. Often cell properties are analyzed assuming that contacts do not limit behavior. However, contacts set the diode built-in potential Vbi, which does limit the attainable Voc10,11,12. Clearly Voc cannot significantly exceed the illuminated Vbi, as the internal field will reverse to oppose power generation. The notion that Vbi limits Voc is not surprising, for Vbi determines the injection current–voltage characteristic of the cell. Two constituent voltage deficits may thus be defined: Δ1 = Vbi − Voc, which turns out to be small, of the order of a few kT/e10; and Δ2 = ECT/e − Vbi, which turns out to be large, of the order of a few tenths of a volt, due to the combined effects of Fermi-level (FL) pinning, polarization band-bending, and electrostatic band-bending losses at the contacts10, suggesting much room for improvement through careful design of these contacts11.

Yet only rudimentary understanding is available for the desired ohmic contact. By definition, such a contact exhibits a contact resistivity that is small compared to the bulk resistivity of the device. Little is known however about contact resistivity of organic semiconductors, and their ohmic contact criteria beyond the requirement for shallow FL pinning to the relevant band edge of the semiconductor13,14. However, FL pinning relates to charge transfer equilibrium15,16,17, while ohmic contact refers to charge transfer kinetics; the two phenomena are not synonymous18. Detailed investigations of these questions have been hampered by the lack of suitable model electrodes whose work function ϕ can be varied in fine steps over a wide range, without accompanying changes in their morphology or doping density that would complicate analysis.

Here we have developed a family of hole-doped poly(3,4-ethylenedioxythiophene)-based polymers (PEDT:PSSCsxH1‒x) to provide the requisite model electrodes. We used these as the hole collection layers (HCL) to investigate both hole injection and collection in organic solar cells with poly(3-hexylthiophene):phenyl-C61-butyrate methyl ester (P3HT:PCBM) as the PAL. We show that besides setting Vbi, the work function of the charge collection layer can also limit FF through the charge-extraction resistance at the contact. We chose P3HT:PCBM because of its good batch-to-batch repeatability and high carrier mobilities that yield non-transport-limited photocurrents8. In fact, P3HT:PCBM is a high quantum-efficiency PAL, with internal photon-to-electron conversion efficiency approaching 0.919. By tuning the HCL ϕ in fine steps over the range 4.4 ≲ ϕ ≲ 5.2 eV, we first established the work function for the onset of FL pinning ϕpin and then showed that while both Vbi and Voc level off beyond ϕpin, FF continues to improve until a second threshold is reached, which we denote the optimal work function ϕop. The product of Voc and FF, and hence PCE, maximizes at this ϕop, which lies ca. 0.3 eV beyond ϕpin. We then developed a general methodology to extract contact resistivity ρc. We showed from current detailed balance theory that this ρc corresponds to the intrinsic charge-transfer resistance at the semiconductor/electrode interface. Finally, we established that ϕop marks the ohmic transition that arises from the strong exponential decrease of ρc with work function. While the deleterious effects of series resistance, whether contact or otherwise, are well known10,12,20, the existence of this intrinsic source of contact resistance has not been appreciated, for it cannot be elucidated from fitting to phenomenological expressions9,21,22.

Results

Tunable work function films

We have developed the hole-doped PEDT:PSSCsxH1−x system to give composition-tunable ϕ over a 0.8-eV-wide range by extending our previous work on spectator-ion effects23. Figure 1a shows the low-energy cutoff (LECO) region of selected PEDT:PSSCsxH1−x films measured by ultraviolet photoemission spectroscopy (UPS). ϕ was obtained in the usual way from the photoelectron kinetic energy difference between LECO and FL: ϕ = KELECO – KEFL + hν, where hν is the He I photon energy (21.21 eV). As x increases from zero to unity, ϕ decreases from 5.2 eV to 4.4 eV; linear regression gives ϕ = 5.2 − 0.8x (Fig. 1b). This ϕ shift arises from the local Madelung potential effect of the mixed ‒SO3H/‒SO3‒Cs+ local ion cluster on the electrochemical potential of the hole carriers23,24. The valence band edge of PEDT shifts rigidly with FL, which excludes any change in its electronic structure or doping level (Fig. 1c). This ability to vary work function without altering carrier density crucially avoids changing resistivity, Schottky barrier width and band bending in the HCL. The valence band features of PEDT, PSS, and Cs+ are also fixed in energy relative to the vacuum level, which excludes any change in the surface-dipole component of the work function (Supplementary Figures 1a and b). JV measurements on indium tin oxide (ITO)/45-nm PEDT:PSSCsxH1‒x/ Al sandwich structures gave resistivity of <0.3 Ω cm2. The PEDT:PSSCsxH1−x system thus provides model electrodes with well-defined work functions that can be manipulated freely without altering surface morphology, surface dipole, or doping density of the films.

Fig. 1
figure 1

UPS of PEDT:PSSCsxH1−x films. a Normalized ultraviolet photoemission spectra, showing the low-energy cutoff (LECO) region plotted against kinetic energy (KE) of the emitted photoelectrons (hν, 21.21 eV) for films with different x. b Plot of vacuum work function ϕ against composition x. Blue line gives the linear fit, ϕ = 5.2 − 0.8x. Error bar gives the standard error. c Valence band (VB) edge region, plotted against binding energy (BE) relative to vacuum level (VL). Fermi energy (EF) is marked for each film. Dots are data (color coding same as in a); magenta lines are global fits to a master shape function as guide to the eye

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Solar cells with different hole collector work functions

P3HT:PCBM diodes were then fabricated by spinning 85-nm-thick P3HT:PCBM films, with composition 1.0: 0.8 weight/weight (w/w), over 45-nm-thick PEDT:PSSCsxH1‒x films that were coated on glass/ITO substrates. This PAL composition was chosen based on the Jsc, FF, and PCE vs thickness‒composition landscapes established from lightly crosslinked P3HT networks19. Thirty-nm-thick Ca films, capped with 130-nm-thick Al films, were then thermally evaporated to give the electron collection contacts, which are expected to be ohmic10.

Figure 2a shows the JV characteristics measured at an illumination intensity of 110 mW cm−2, while Fig. 2b shows the cell performance parameters plotted against ϕ of the HCL. All measurements were performed in a single batch to avoid calibration uncertainties. Four representative diodes are displayed for each HCL to show repeatability. Cells with ϕ = 5.17 eV give FF = 0.63 ± 0.02, Jsc = 9.4 ± 0.3 mA cm–2, Voc = 0.595 ± 0.01 V, and PCE = 3.2 ± 0.2 %, similar to literature reports10,25,26,27,28. As ϕ decreases below 5.0 eV, the JV slope in the vicinity of Voc becomes noticeably gentler, indicating an increase in the open-circuit series resistance \(\left( {R_{{\mathrm{s,oc}}} = \left. {\frac{{dV}}{{dJ}}} \right|_{V_{{\mathrm{oc}}}}} \right)\), even though Voc is unchanged. This shifts the maximum power point down-voltage, degrading both FF and PCE. As ϕ crosses ca. 4.73 eV, Voc downshifts with unity slope, as may be expected of a depinned hole contact29. Thus the apparent ϕpin is 4.73 eV. This has been confirmed by Vbi measurements (vide infra). As soon as depinning occurs, the JV characteristic collapses to S shape30,31,32, forcing PCE to zero within a narrow ϕ range of 0.1 eV, even though the photocurrent at reverse bias voltage of −1.0 V remains unchanged. Particularly relevant to cell optimization, FF increases with ϕ beyond ϕpin until 5.02 eV, at which it levels off, while Voc rolls off gently at a rate of ‒60 mV eV‒1. Therefore, ϕop exceeds ϕpin here by ca. 0.3 eV, and the electrode work function has to be brought to ϕop for best performance. The FF improvement from ϕpin to ϕop is primarily due to Rs,oc reduction. While FF remains optimal beyond ϕop, cell performance eventually degrades due to the Voc roll-off.

Fig. 2
figure 2

Characteristics of organic solar cells with different hole collector work functions. a JV characteristics of ITO/PEDT:PSSCsxH1−x/P3HT:PCBM (1.0:0.8 w/w)/Ca/Al cells measured at 110-mW cm−2 illumination, spectral-mismatch corrected. Four diodes are shown in each panel. Vacuum work function ϕ and composition x of the PEDT:PSSCsxH1−x hole collection layers are given. b Cell performance parameters plotted against vacuum work function of the hole collection layer. Symbols are data; lines are guides to the eye. Error bar gives the standard error. The standard error in work function is 0.03 eV. The Fermi-level pinning work function ϕpin and optimal work function ϕop are indicated. “S-shape” and “Distorted” describe the shape of the illuminated JV characteristics in a

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Confirmation of ϕ pin transition

To confirm the nature of the ϕpin transition, we show here that Vbi indeed becomes pinned at ϕpin. We obtained Vbi from Voc measurements at 1 sun, and below 150 K where shutdown of carrier injection leaves photocurrent to freely track the direction of the internal field, which reverses at the flatband condition10,33. Figure 3 shows the Voc (ϕ) plots for selected temperatures. Vbi exhibits a temperature dependence of the form: Vbi = Vo − βT, where Vo is the zero-K term and β is of the order of a few k/e10. β receives contributions from the temperature dependences of bandgap and also electrostatic band bending due to the accumulated carrier tail34,35. This band bending has been confirmed by UPS on polymer organic semiconductors36, including monolayer to multilayer stacks of P3HT on PEDT:PSSH37. Lowering temperature contracts this carrier tail, increasing Vbi. Thus the Voc (ϕ) plots below 70 K collapse to a “universal” characteristic, indicating that Vo has been approached within experimental resolution. This characteristic exhibits a unity slope (i.e., \(\frac{{dV_o}}{{d\phi }}\) = 1) that turns over to zero slope (\(\frac{{dV_o}}{{d\phi }}\) = 0) at a well-defined threshold, a classic signature of the FL pinning transition15,16,17 (Supplementary Note 1). The threshold gives ϕpin for FL pinning to P3HT:PCBM to be 4.80 eV for temperatures <150 K. In contrast, FL pinning to edge-on, π-stacked P3HT lamellae occurs at 4.40 eV38, presumably due to order and orientation effects. Linear extrapolation to room temperature (295 K) gives Vbi = 0.66 ( ± 0.02) V. This intersects with the temperature-independent (unity slope) segment of the plot to give ϕpin = 4.75 (±0.02) eV at room temperature. This confirms that the kinks in both Voc(ϕ) and Jsc(ϕ) indeed occur at the FL pinning transition for room temperature.

Fig. 3
figure 3

Built-in potential of organic solar cells with different hole collector work functions. Cell open-circuit voltage plotted against vacuum work function of the hole collection layer. Conditions: 1.1-sun illumination; temperature, 295 K → 30 K. For T ≲ 150 K, Voc corresponds to Vbi; for T ≲ 30 K, Vbi approaches the zero-Kelvin value Vo (0.73 V). Typical uncertainties: for large ϕ: Voc, ±0.01 V; for small ϕ: Voc, ±0.05 V. Voc data are averaged for forward and reverse sweeps

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Extracting contact resistivity

To understand the dependence of Rs,oc on ϕ, we first develop a general methodology to extract contact resistivity characteristic ρc(J), and then show that this is a property of the contact, applicable to both dark and illuminated conditions. The methodology is based on drift‒diffusion simulation of JV characteristics with bimolecular bulk recombination in the PAL between its two surface boundaries next to the contacts20,39, with optical transfer-matrix simulation of light absorption within the PAL when illuminated9,10. We imposed globally self-consistent boundary conditions, and assigned the residual “excess” resistivity to the total contact resistivity of the semiconductor/electrode interfaces. For dark JV characteristics, we computed the contact resistivity as: \(\rho _c(J) = \frac{{V(J) - V_{{\mathrm{int}}}(J)}}{J}\), where V is the applied bias, and Vint is the internal voltage across the PAL given by drift‒diffusion simulation. Thus this approach is advantageously model-free40,41,42; it does not presume the diode equation nor any particular charge injection (or extraction) model. The self-consistent boundary conditions were found by systematic search for global best fit to the entire set of JV characteristics (all ϕ) in the diffusion-current regime, where ρc can be neglected compared to bulk resistivity, subjected to the constraints that Nn is constant, but Pp ∼ (ϕ ‒ ϕpin), where Nn is the electron density at the boundary to the electron contact and Pp is the hole density at the boundary to the hole contact43. All other input parameters are known from previous work and assumed constant10,25 (see Supplementary Table 1).

We first point out that the dark JV characteristics outside of the diffusion regime cannot be altogether fitted to any self-consistent {Nn, Pp} set with zero ρc(J) (Fig. 4a). The discrepancy is particularly severe for the cells with sub-optimal HCLs. This disparity can obviously be eliminated by including a free-fitting ρc(J) function to represent excess resistivity (Fig. 4b). The requisite dark ρc(J) characteristics are shown in Fig. 4c. These generally level off at low J to a high limit that increases strongly with ϕop – ϕ, e.g., 0.7 Ω cm2 at ϕ = 5.15 eV increasing to 10 Ω cm2 at ϕ = 4.9 eV; they also level off at high J to a low limit of ca. 0.5 Ω cm2. Probe and lateral electrode resistances together contribute <0.2 Ω cm2, with the balance likely dominated by ρc of the electron contact. Since this contact is unchanged, the strong ϕ dependence of ρc(J) can directly be attributed to the hole contact. For ϕ ≳ 5.15 eV, ρc(J) shows little dependence on J, a non-dispersive behavior found also for notionally ohmic contacts in organic field-effect transistors44,45. Near ϕpin however, ρc(J) exhibits an apparent inverse power law dependence, with exponent of –0.67, between the two limits.

Fig. 4
figure 4

Simulation and modeling of contact resistivity. a Experimental and drift–diffusion-simulated dark JV characteristics without contact resistance. Symbols are data; lines are simulations for the specified electron (Nn) and hole (Pp) densities at the electron- and hole-contact boundaries, respectively. Data color coding is given in b. Adjacent lines in each family (blue, magenta) differ by a factor of 3.16. b Same as a but with free-fitting contact resistivity function ρc(J). Simulation parameters are given in Supplementary Table 1. c Dark ρc(J) characteristics. Inset shows equivalent-circuit model. Symbols are data; lines are guide to the eye. Data color coding is same as b. d Experimental and drift–diffusion–generation-simulated illuminated JV characteristics without contact resistance. Symbols are data; lines are simulations for the self-consistent {Nn, Pp} found in b. Data color coding is same as b, with additional sky blue = 4.71 eV. e Illuminated ρc(J) characteristics. Inset shows equivalent-circuit model. Data color codingis same as d. f Schematic energy-level diagram of the cell under illumination with external load, showing boundary planes and (expanded) contact interfaces, and illustrating the electron and hole quasi-Fermi levels in the near contact regions for high and low ρc. The drop in FL across the contact interface is due to contact resistance. g Zero-current total contact resistivity plotted against vacuum work function of the hole collection layer. Error bar gives typical standard error

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Similarly, the illuminated JV characteristics in the power-generation quadrant, particularly for cells with sub-optimal HCLs, also cannot be fitted to any self-consistent {Nn, Pp} set with zero ρc(J) (Fig. 4d). In this case, Vint(J) is only slightly modified by the expected variation in Pp. So the marked dependence of Rs,oc on ϕ provides clear evidence for contact resistance. For illuminated JV characteristics, we computed this contact resistivity as: \(\rho _c(J) = \frac{{V_{{\mathrm{int}}}(J) - V(J)}}{J}\), where the order of terms is swapped due to the reverse direction of flow of the photocurrent. The illuminated ρc(J) characteristics thus obtained are shown in Fig. 4e. The close match between the dark and illuminated ρc at zero current ρc(0), together with their strong ϕ and J dependences, confirms that the excess resistivity is due to the contact resistivity at the semiconductor/electrode interface. This is not negligible until well beyond the threshold of FL pinning.

The FL drop at the contact caused by this ρc(J) creates a barrier to charge extraction by lowering the internal field (Fig. 4f), promoting non-geminate recombination in the bulk of the semiconductor and near-surface region of the contact. Figure 4g plots ρc(0) against ϕ. It reveals an inverse exponential-like dependence: ρc(0) ~ exp( − (ϕ − ϕpin)), decreasing from 100 Ω cm2 at ϕ = 4.7 eV to 1 Ω cm2 at ϕ = 5.2 eV, with \(\left( {\frac{{d\log \left( {\rho _c{\mathrm{/}}\Omega cm^2} \right)}}{{d\phi }}} \right)\) slope of ca. 5 eV–1 in the vicinity of ϕpin.

Current detailed balance analysis

To clarify the mechanistic origin of ρc(J), we first formulate a contact resistance theory in the current detailed balance formalism and then show that this leads to an intrinsic charge-transfer resistance that quantitatively agrees with ρc(J). Therefore ρc(J) is a fundamental feature of hopping charge injection/collection. The current detailed balance model of the contact is shown in Fig. 5a. Contact resistivity is modeled by site-to-site hopping between the two frontier monolayers (denoted i and j) across the contact. The net hole current density Jij from j to i under the influence of a small local bias ∂V, i.e., applied potential difference across the two monolayers of the contact (positive for forward bias), is given by detailed balance to be:

$$\begin{array}{*{20}{l}}J_{{\boldsymbol{ij}}} = e{\int}_{{\boldsymbol{i,j}}} \alpha \left[w_{{\boldsymbol{ij}}}N_{\boldsymbol{i}}(\varepsilon _{\boldsymbol{i}})f_{\boldsymbol{i}}(\varepsilon _{\boldsymbol{i}})N_{\boldsymbol{j}}(\varepsilon _{\boldsymbol{j}})({\mathrm{1}} - f_{\boldsymbol{j}}(\varepsilon _{\boldsymbol{j}}))\right.\\ -\,\left. w_{{\boldsymbol{ji}}}N_{\boldsymbol{j}}(\varepsilon _{\boldsymbol{j}})f_{\boldsymbol{j}}(\varepsilon _{\boldsymbol{j}})N_{\boldsymbol{i}}(\varepsilon _{\boldsymbol{i}})({\mathrm{1}} - f_{\boldsymbol{i}}(\varepsilon _{\boldsymbol{i}})) \right]\delta (\varepsilon _{\boldsymbol{i}}-\varepsilon _{\boldsymbol{j}} + e\partial V){\it{d\varepsilon }}\end{array}$$

where α is the effective charge-hopping cross-section area; wij = wji is the transfer integral given by Fermi golden rule: \(w_{{\boldsymbol{ij}}} = \frac{{4\pi ^2}}{h}V^2\chi ^2\), where V2 is the electronic Hamiltonian and χ2 is the Franck–Condon overlap; Nj(εj) and Ni(εi) are the density-of-state (DOS) per unit area per unit energy interval, generally different for j and i; εj and εi are the respective (electron) energies; fj(εj) and fi(εi) are the usual Fermi‒Dirac functions; and δ(…) is the Kronecker delta that enforces energy conservation during the hop. The corollary gives the electron current, if required. Assuming both wij and α are independent of εj and εi over the small energy range relevant to injection or collection (i.e., ±2kT), we can write

$$\begin{array}{*{20}{l}}J_{{\boldsymbol{ij}}} = e\alpha w_{{\boldsymbol{ij}}}N_{{\mathrm{o}}{\boldsymbol{,i}}}N_{{\mathrm{o}}{\boldsymbol{,j}}}{\int}_{{\boldsymbol{i,j}}} \left[\xi _i(\varepsilon -e\partial V){f}_{\boldsymbol{i}}(\varepsilon -e\partial V)\xi _{\boldsymbol{j}}(\varepsilon )\left( {{\mathrm{1}} - f_{\boldsymbol{j}}(\varepsilon )} \right)\right. \\ \qquad - \left. \xi _{\boldsymbol{j}}(\varepsilon )f_{\boldsymbol{j}}(\varepsilon )\xi _{\boldsymbol{i}}(\varepsilon -e\partial V) \left( {{\mathrm{1}} - f_{\boldsymbol{i}}(\varepsilon -e\partial V)} \right)\right]{\mathrm{\it{d\varepsilon }}} \\ = e\alpha w_{{\boldsymbol{ij}}}N_{{\mathrm{o}}{\boldsymbol{,i}}}N_{{\mathrm{o}}{\boldsymbol{,j}}}J_{{\boldsymbol{ij}}}^{\prime}, \end{array}$$

where the product No,iNo,j is the joint DOS per unit area of  i and per unit area of  j, per unit energy interval, such that ξ(ε) = N(ε)/No) and Jij’ is the reduced current density (in units of energy). The zero-current contact resistivity is thus obtained from: \(\rho _c\left( 0 \right) = \lim_{{J_{{\boldsymbol{ij}}} \to 0}}\frac{{\partial V}}{{J_{{\boldsymbol{ij}}}}}\) . It is more instructive, however, to display the reduced contact resistivity: ρc’(0)\(\,\,=\, \lim_{{J_{{\boldsymbol{ij}}}^{\prime} {\to} 0}}\frac{{\partial V}}{{J_{{\boldsymbol{ij}}}^{\prime}}}\), which is ρc(0) normalized by eαwijNo,iNo,j.

Fig. 5
figure 5

Current detailed balance analysis. a Schematic energy band model of hole contact under forward bias. b Plot of theoretical reduced contact resistivity ρc’ (in units of 1/e) against reduced current density Jij’, for different ϕ – ϕpin values, and Gaussian width σG = 0.15 eV. c Plot of ρc’(0) against ϕ – ϕpin. Symbols are computed results; yellow line is guide to the eye for 0.15 < σG < 0.2 eV

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Semiconductor with Heaviside DOS

We first treat a simplistic contact between a heavily doped electrode j, modeled by constant Nj(εj) over the relevant energy range in the vicinity of εF, and a semiconductor i, modeled by Ni(εi) that is a Heaviside step function with transition edge at εb. This contact presents a thermodynamic barrier for hole injection given by: Eb = εF – εb. For Eb > 2kT at zero ∂V, one can readily show that: ρc(0) = \((e^2{\mathrm{\alpha w}}_{{\boldsymbol{ij}}}N_{{\boldsymbol{o,i}}}N_{{\boldsymbol{o,j}}})^{-1}{\mathrm{exp}}\left( {\frac{{E_b}}{{{\mathrm{\it{kT}}}}}} \right)\). This equation reveals two key features. First, both injection and collection are subjected to the same ρc(0); second, ρc(0) is finite. The reason for the first feature is that carrier collection also requires FL offset as carrier injection. This leads to the following consequence: a good injection contact is also a good collection contact, and vice versa. An energy barrier that suppresses the electrode-to-semiconductor injection current also suppresses the semiconductor-to-electrode collection current in the same way because of depletion of carriers in the semiconductor DOS.

Semiconductor with Gaussian DOS

We then treat a more realistic contact of the same electrode but with a disordered semiconductor characterized by a hemi-Gaussian tail with disorder width σG in its frontier DOS. This requires numerical integration of the detailed balance equation. The results reveal that the two features above are preserved. Incidentally, this justifies the assumption of identical source and drain contact resistivities in organic field-effect transistors with symmetrical contacts44,45. To obtain the theoretical dependence of ρc on ϕ and σG, we stepped FL through the DOS tail, evaluating electrode ϕ at each step and taking into account the carrier density accumulated at the contact (Supplementary Note 2). We found that, for σG over the reasonable range of 0.15‒0.2 eV, the computed form of ρc’(J’) matches experiment well without any fitting parameters. In particular, for ϕ ≳ ϕpin, ρc’(J’) is substantially flat; for ϕ just smaller than ϕpin, ρc’(J’) levels off at low J’ but gives power law dependence with exponent –0.67 at higher J’ (Fig. 5b). The computed ρc’(0)(ϕ) characteristic also exhibits a form similar to experiment, with \(\left( {\frac{{d\log (\rho _c\prime )}}{{d\phi }}} \right)\) slope of 5 eV–1 near ϕpin (Fig. 5c). For reasonable parameters: No,iNo,j ≈ 1 × 1025 cm–4 eV–1, a ≈ 30 Å2, wij ≈ 1 × 1010 s–1, the Jij’ = 1 eV level corresponds to 100 A cm–2, and ρc’ = 1 e–1 level corresponds to 0.03 Ω cm2. This agrees with experiment to within half an order of magnitude. Therefore, the measured ρc(J,ϕ) characteristics are indeed manifestations of the hopping injection/extraction process at a disordered semiconductor contact.

Nevertheless, the steep dependence of ρc(0) on ϕ extends further beyond ϕpin than predicted by theory. This suggests a possible ϕ dependence in wij. Subgap electroabsorption spectroscopy found spectral changes that suggest polaron interactions within the accumulation layer at carrier densities above a few 1011 hole cm−2 33. The results here suggest that wij also becomes larger, which may arise, for example, if the doped polymer segments exhibit a higher wij than the undoped segments.

Ohmic transition

The inverse exponential-like dependence of ρc on ϕ produces a fairly sharp transition that corresponds to emergence of ohmic contact in the device when ϕ is tuned over a narrow range across ϕop. This can be characterized as the ohmic transition, akin to the glass transition that follows from the inverse exponential dependence of viscosity on temperature. Signatures of such behavior may be discerned in data on other systems, even though the limited resolution and range of earlier work have precluded analysis46,47,48,49. Whether a contact is ohmic or not, however, is determined by whether its resistivity is small compared to that of the bulk of the device. Therefore the precise location of the ohmic transition depends on bulk resistivity. One may thus specify a total ρc criterion for the ohmic transition at which device behavior is no longer dominated by contacts.

For solar cells, one may specify that ρc needs to be smaller than about half of the bulk resistivity at Voc, i.e., ρc(Voc) < \(\left. {\frac{1}{2}\frac{{dV_{{\mathrm{int}}}}}{{{{dJ}}}}} \right|_{V_{{\mathrm{oc}}}}\). Drift–diffusion modeling indicates for typical P3HT:PCBM cells under 1 sun, the bulk resistivity is ca. 5.5 Ω cm2 in the vicinity of Voc. Thus assuming the electron-contact ρc is negligible, the hole-contact ρc needs to be smaller than ca. 3 Ω cm2. Two phenomenological regimes can then be distinguished—a bulk-limited regime for ϕ ≳ ϕop where ρc ≲ 3 Ω cm2; and a contact-limited regime for ϕ < ϕop, where J(V) is at first lightly degraded (ϕpin ≲ ϕ ≲ ϕop; 3 ≲ ρc ≲ 50 Ω cm2) and then heavily degraded (ϕ ≲ ϕpin; ρc ≳ 50 Ω cm2). For injection diodes on the other hand, one may specify that ρc needs to be smaller than about half of the bulk resistivity at the desired J, i.e., ρc(J) < \(\left. {\frac{1}{2}\frac{{dV_{{\mathrm{int}}}}}{{{{dJ}}}}} \right|_J\). For P3HT:PCBM diodes with Nn, Pp ≳ 2 × 1017 cm‒3, the computed bulk resistivity is 7.5 Ω cm2 at J = 100 mA cm‒2. Therefore, assuming again the electron-contact ρc is negligible, the hole-contact ρc needs to be smaller than ca. 4 Ω cm2 at this current density.

Conclusion

In summary, we have successfully developed a model PEDT:PSS(CsxH1–x) electrode system whose inherent work function can be continuously tuned over a wide range without changing morphology, surface dipole or doping level. This enables us to show that the work function of the hole collection electrode that is needed to maximize the FF of organic solar cells is larger than that required to maximize their open-circuit voltage by few tenths of an eV. This arises as a consequence of the charge-transfer contact resistance at the semiconductor/electrode interface continuing to decline strongly with work function even in the Fermi-level pinned regime, due to the increasing accumulation of carrier density at the semiconductor side of the contact. Hence the ohmic transition occurs beyond the Fermi-level pinning transition. Although particularly important for solar cells, which are highly sensitive to the resulting series resistance, the clarification here of the ohmic transition closes a key gap in our understanding of disordered semiconductor contacts, pointing to a simple rule for the design of contacts relevant also to injection diodes and field-effect transistors. The development of heavily doped contacts, whether imposed by charge-counterbalancing polyelectrolyte monolayers45, or the use of self-compensated, heavily doped polymer injection layers50, is thus key to realizing ohmic contacts at will.

Methods

Preparation of PEDT:PSSCsxH1─x hole collection electrode films

PEDT:PSSCsxH1−x solutions were prepared by mixing PEDT:PSSH and PEDT:PSSCs solutions in the desired stoichiometric ratio, filtering through a 0.45-μm filter, and spin-casting to give the desired films. The PEDT:PSSCs solution was prepared from commercial PEDT:PSSH solution (Heraeus P VP AI4083, 1:6 w/w) by ion-exchange dialysis51. The PEDT:PSSH solution was first rigorously purified of ions by dialysis52. Rigorous ion removal is important to obtain reproducible results. Twenty-nm-thick PEDT:PSSCsxH1−x films were spin-cast on Au-coated Si substrates, annealed at 140 °C (hotplate, 10 min) in a N2 glovebox (pO2, H2O < 1 ppm) to remove water, and transferred in N2 to the UPS chamber equipped with an Omicron EA 125 energy analyzer and five channeltrons for ϕ measurements.

Diode fabrication and measurements

Standard P3HT:PCBM diodes were fabricated by spinning 85-nm thick P3HT:PCBM films (1.0:0.8 w/w) over PEDT:PSSCsxH1‒x films. The films were annealed at 140 °C (hotplate, 10 min) in a N2 glovebox before and after PAL deposition to remove water and solvent, respectively. Thirty-nm-thick Ca capped with 130-nm-thick Al were then evaporated to define 4.2-mm2 electron-collection electrodes.

Data availability

The data supporting the findings of this study are available within the article and its Supplementary Information files.