Efficiency limits for photoelectrochemical water-splitting

Abstract

Theoretical limiting efficiencies have a critical role in determining technological viability and expectations for device prototypes, as evidenced by the photovoltaics community's focus on detailed balance. However, due to their multicomponent nature, photoelectrochemical devices do not have an equivalent analogue to detailed balance, and reported theoretical efficiency limits vary depending on the assumptions made. Here we introduce a unified framework for photoelectrochemical device performance through which all previous limiting efficiencies can be understood and contextualized. Ideal and experimentally realistic limiting efficiencies are presented, and then generalized using five representative parameters--semiconductor absorption fraction, external radiative efficiency, series resistance, shunt resistance and catalytic exchange current density--to account for imperfect light absorption, charge transport and catalysis. Finally, we discuss the origin of deviations between the limits discussed herein and reported water-splitting efficiencies. This analysis provides insight into the primary factors that determine device performance and a powerful handle to improve device efficiency.

ARTICLE

Received 29 Feb 2016 | Accepted 21 Oct 2016 | Published 2 Dec 2016

Theoretical limiting efciencies have a critical role in determining technological viability and expectations for device prototypes, as evidenced by the photovoltaics communitys focus on detailed balance. However, due to their multicomponent nature, photoelectrochemical devices do not have an equivalent analogue to detailed balance, and reported theoretical efciency limits vary depending on the assumptions made. Here we introduce a unied framework for photoelectrochemical device performance through which all previous limiting efciencies can be understood and contextualized. Ideal and experimentally realistic limiting efciencies are presented, and then generalized using ve representative parameters semiconductor absorption fraction, external radiative efciency, series resistance, shunt resistance and catalytic exchange current densityto account for imperfect light absorption, charge transport and catalysis. Finally, we discuss the origin of deviations between the limits discussed herein and reported water-splitting efciencies. This analysis provides insight into the primary factors that determine device performance and a powerful handle to improve device efciency.

DOI: 10.1038/ncomms13706 OPEN

Efciency limits for photoelectrochemical water-splitting

Katherine T. Fountaine1,2,3,4, Hans Joachim Lewerenz3,4 & Harry A. Atwater3,4

1 NG Next, 1 Space Park Drive, Redondo Beach, California 90278, USA. 2 Deparment of Chemistry and Chemical Engineering, California Institute of Technology, 1200 East California Boulevard, Pasadena, California 91125, USA. 3 Division of Engineering and Applied Sciences, California Institute of Technology, 1200 East California Boulevard, Pasadena, California 91125, USA. 4 Joint Center for Articial Photosynthesis, California Institute of Technology, 1200 East California Boulevard, Pasadena, California 91125, USA. Correspondence and requests for materials should be addressed to K.T.F.(email: mailto:[email protected]

Web End [email protected] ).

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In the photovoltaics community, the detailed balance limit serves as a gold standard to which all device efciencies are compared1. The seminal paper written by Shockley and

Queisser in 1961 presents photovoltaic limiting efciencies as a function of a single parameter, the semiconductor bandgap, under the assumption that the only loss mechanism is radiative recombination in the semiconductor. While many corollaries to this limit exist29, the transcendence of this analysis is enabled by its elegance, analytic simplicity and basis in the ultimate limit of semiconductor device physics.

Due to their complex, multicomponent nature, an equivalent analogue to the detailed balance limit does not exist for photoelectrochemical (PEC) devices. A number of articles have been written on the limiting efciencies of photoelectrochemical devices3,1015, each with slightly different approaches and assumptions to arrive at different limiting efciency values. In this article, we aim to present a unied framework for photoelectrochemical device performance through which all previous limiting efciencies can be understood. To do so, we rst present the analytic equations and solutions for the limiting efciencies of photoelectrochemical water-splitting devices based on the ultimate limits of device physics; limiting efciencies reported here are consistent with those presented in references3,13,15. Subsequently, we examine the validity of these ideal limits and consider more realistic limits based on existing materials, similar to those presented in references1012,14; the realistic limits are presented and discussed as a function of ve parameters: semiconductor absorption fraction, semiconductor external radiative efciency (ERE), series resistance, shunt resistance and catalytic exchange current density. These ve parameters directly correlate with the three governing physical phenomena of photoelectrochemical device operationlight absorption (absorption fraction), charge carrier transport (ERE, series resistance and shunt resistance), catalysis (catalytic exchange current density); the parameter variation study demonstrates the varying impact of each phenomenon on overall device efciency and efciency limits. Finally, this analysis is contextualized via a comparison of the discussed limits with reported water-splitting efciencies.

ResultsOutline. The following analysis of water-splitting device efciencies is divided into ve parts. First, we derive the analytic equation that governs a variable-junction photoelectrochemical device and its efciency. This set of equations is generally applicable to any photoelectrochemical device. Second, we present the absolute limiting efciencies for a photoelectrochemical device for water-splitting for both single and dual junction photodiode units. Subsequently, we present two sets of realistic limiting efciencies based on currently available high performance (real1) and Earth abundant (real2) materials. Next, we consider the effects of ve representative parameters on the limiting efciency and the corresponding semiconductor bandgap(s). Finally, we contextualize this theoretical efciency analysis via a brief discussion of reported efciencies. This analysis provides insight into the primary factors currently limiting device efciency and guidance to researchers on (1) the most powerful handles to improve device efciency and (2) routes to maximize device efciency for a given set of material and device parameters.

Analytic equation for PEC device operation and efciency. In previous work, we derived analytic equations for a variable-junction photoelectrochemical device16,17. This derivation is briey summarized below. The characteristic current-voltage

relationship for a photoelectrochemical device lends itself to an inverse formulation, VPEC(j), where VPEC is the voltage generated by the device that is available for conversion into chemical energy, j is the device current density, VPV,i(j) is the inverse current-

voltage relationship for the ith photodiode component, Vcat,a(j)

and Vcat,c(j) are the current-dependent overpotentials of the anodic and cathodic catalysts, respectively, Vseries is the series

resistance due to electrolyte transport through solution and membrane, formulated as jRseries, and Erxn is the electrochemical

potential of the desired chemical reaction.

VPECj XiVPVij Vcat;aj Vcat;cj Vseriesj Erxn 1

The photodiode voltage is described by an inverse formulation of the diode equation, where j0 is the reverse saturation (dark)

current, nd is the ideality factor, kB is the Boltzmann constant and T is the device temperature.

VPVj

ndkBTq ln

jL jj0 1

2

Butler-Volmer kinetics are selected to describe the current density-dependent catalytic overpotentials, Vcat(j); this model

and the further simplied Tafel equation are commonly used to t electrocatalyst behaviour, however, it should be noted that Butler-Volmer kinetics are only accurate for outer sphere single electron transfer reactions18. Specically, we employ an inverse formulation of the Butler-Volmer equation, found by assuming that the charge transfer coefcients of a specic catalyst in the forward and reverse directions, af and ar, are equal, where R is the universal gas constant, ne is the number of electrons associated with the reaction, F is Faradays constant and j0,cat is the catalytic

exchange current density. A comparison of this formulation, the more standard Tafel equation and the complete Butler-Volmer equation is provided in Supplementary Note 1 and Supplementary Fig. 1, as well as in previous work17.

Vcatj

RTaneF sinh 1

3

The reaction proceeds when the photoelectrochemical device voltage is greater than or equal to the electrochemical potential required to drive the reaction, Erxn, as dened in equation (1).

The maximum efciency occurs when the voltage is precisely equal to the required electrochemical potential because this maximizes the device current density; this point is dened as the device operating point, Vop(jop) Erxn (see Supplementary

Note 2 and Supplementary Fig. 2 for visualization). The device operating current density is directly proportional to the device efciency, ZPEC, according to the following equation, where fFE is the Faradaic efciency and Pin is the incident solar power.

ZPEC

j 2j0;cat

jopErxnfFE

Pin 4

Absolute limiting efciencies. To determine the absolute limiting efciencies of single and dual junction photoelectrochemical devices for water-splitting, the following assumptions were made (and are also summarized in Table 1):

1. Illumination with the AM1.5G spectrum:

Pin q

Z

10 AM1:5GEdE 52. Complete absorption of all photons above the bandgap of the semiconductor.

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Table 1 | Selected parameter values for the three limiting efciency cases.

fabs ERE j0,cat(mA cm 2)* rsw rshw Legend

Ideal 1 1 N 0 N

Real1 0.9 0.03 {1,10 3} 0 N

Real2 0.9 10 6 {1,10 5} 0.1 10

*j includes both cathodic and anodic catalyst exchange current densities as {j ,j }. wr and r are the normalized series and shunt resistivities.

30

25

20

15

[afii9834] PEC(%)

10

5

0 1 1.5 2

Eg (eV)

3. A detailed balance model for the photodiode dark current, assuming only radiative recombination in the semiconductor:

j0 Eg

q 4p2 3c2

Z

1 Eg

E2 eE=kBT 1

2.5 3

Figure 1 | Single junction limiting efciencies. Limiting efciencies (ZPEC)

versus semiconductor bandgap (Eg) for ideal case (blue solid line, Zmax

30.6%, Eg 1.59 eV), high-performance realistic case (green dashed

line, Zmax 15.1%, Eg 2.05 eV) and Earth abundant realistic case (red

dotted line, Zmax 5.4%, Eg 2.53 eV); parameter values used for each

case are tabulated in Table 1.

dE 6

This equation assumes a perfect antireective coating and perfect back reector. For the dual junction calculations, the dark current of the upper junction is multiplied by a factor of 2 to account for emission from the upper and lower surfaces. A detailed treatment of angular emission probability, more formally known as the etendue, can be found in Markvart et al.194. Diode ideality factor, nd, of 1.5. Catalytic overpotentials are assumed to be negligible. Mathematically, this assumption corresponds to innite catalytic exchange current densities. This condition can be approached by (1) discovery of new catalysts with very high-catalytic exchange current densities and (2) high-surface area catalyst and high catalyst loading, which increases the effective catalytic exchange current density when normalized to device area.

6. Charge transfer coefcients of 0.5.7. No series resistance.8. The electrochemical potential for water-splitting at standard conditions, Erxn 1.23 V.

9. Unity Faradaic efciency.

The analysis is restricted to single and dual junctions because additional junctions do not result in any efciency gains. In fact, the maximum ideal triple junction efciency for water-splitting is28.3%, which is signicantly lower than the maximum dual junction efciency (40.0%). This drop in efciency with increasing junction number (beyond 2) is contrary to photo-voltaic efciencies and occurs because additional photovoltage beyond that required to kinetically split water does not increase efciency; furthermore, the increased number of current-matched junctions reduces device photocurrent, which directly lowers efciency, as shown in equation 4.

Figure 1 (blue solid line) and Fig. 2a display the limiting efciencies as a function of semiconductor bandgap for single and dual junction photoelectrochemical devices, respectively, that result from this ideal set of assumptions. The maximum single and dual junction efciencies are 30.6% at a bandgap of 1.59 eV, and 40.0% with bandgaps of 0.52 and 1.40 eV, respectively. The single junction efciency trend, Fig. 1(blue solid line), strongly resembles that of the single junction photovoltaic detailed balance limit, with a few notable differences. The efciency decreases for bandgaps smaller and larger than the maximum efciency point

due to the inverse correlation between light absorption and voltage generation. The efciency decreases with increasing bandgap due to decreased absorption of the incident solar spectrum, but the photoelectrochemical efciency decreases more rapidly than the photovoltaic efciency because the voltage converted to chemical energy remains constant (Erxn)

despite the larger photovoltages supplied by wide bandgap semiconductors. According to equation (4), the decrease in efciency is directly proportional to the decrease in photocurrent. For small bandgaps, the photoelectrochemical efciency drops off sharply to zero, in contrast with the gradual decay of photovoltaic efciency with decreasing bandgap; this sharp cutoff in efciency occurs due to insufcient photovoltage to drive the reaction. Another critical difference is that the maximum single junction photoelectrochemical device efciency is about 3% lower than that of a single junction photovoltaic device, despite the fact that this calculation has neglected any catalyst- or solution-related losses. This difference exists because the output voltage of a photovoltaic device is variable, whereas the required output voltage of a water-splitting device is xed at 1.23 V. The result of this requirement is that the maximum efciency occurs for the semiconductor with a sufcient bandgap to generate 1.23 V of photovoltage, which is a bandgap of 1.59 eV assuming detailed balance.

The ideal dual junction efciency contour plot (Fig. 2a) exhibits a similar trend, but in two dimensions; it has a sharp turn-on of efciency for bandgaps just large enough to supply the water-splitting efciency voltage and a gradual decline of efciency beyond the peak due to decreasing light absorption. The dual junction water-splitting efciency falls short of the dual junction photovoltaic efciency for the same reason: xed photovoltage.

Realistic limiting efciencies. The absolute limiting efciencies presented in the previous section represent theoretical limits for an ideally constructed device with ideal photodiodes and catalysts. This section presents realistic limiting efciencies based on material and device parameters reported in literature for(1) high performance (real1) and (2) Earth abundant photodiodes

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and catalysts (real2). Five material parameters capture non-ideal photodiode and catalyst performancesemiconductor absorption fraction, fabs, semiconductor ERE, series resistance, RS, shunt resistance, RSh and catalytic exchange current density, j0,cat. Absorption fraction and catalytic exchange current density represent the efciency of light absorption and catalysis. The efciency of charge carrier transport is divided into three parameters, where ERE represents intrinsic semiconductor material quality in accordance with the photovoltaic community20, and series and shunt resistance represent device fabrication quality. ERE serves as a straightforward, linear modication to the detailed balance limit (as shown in assumption #2, below). In the detailed balance limit, the only loss mechanism is radiative recombination, as required by black body emission of the semiconductor above its band edge. This limit implies that a photodiode at open circuit voltage will re-emit all photons that were absorbed. This ideal case corresponds to an ERE of 1. In real absorbers, non-radiative recombination mechanisms exist, so that the re-emitted photons at open circuit voltage are only some fraction of those absorbed; this fraction is the ERE. Additional details and tabulated values for photovoltaic materials can be found in ref. 20. For semiconductorliquid junctions, the ERE factor can also be treated as a simple dark current modication factor (see equation 8) to align the open circuit voltage of the model to the built-in voltage of the semiconductor-electrolyte interface18,21. The series and shunt resistance terms account for non-idealities in charge transport that are not captured by the ERE parameter and primarily affect the device ll factor. Series resistance reduces the ll factor and, at high values, also the short circuit current density; sources of series resistance include solution resistance to electrolyte transport, interfacial resistance at the semiconductor|catalyst interface, and resistance to majority carrier ow in the semiconductor. Shunt resistance pathways lower the ll factor and, at signicantly low values of shunt resistance, can also reduce open circuit voltage; shunt resistance arises from partial shorting of diode junctions, which can occur quite readily in semiconductorliquid junctions due to the ease with which liquid electrolyte can intercalate into pinholes in the semiconductor2224.

High-performance realistic efciencies. We rst consider realistic limiting efciencies for high-performance materials and devices. In these calculations, we include the effects of non-ideal light absorption, ERE and catalytic exchange current density, but neglect the effects of series and shunt resistances because ll factors of current high-performance photodiodes are approaching their ideal values25. Specically, the following modications to the ideal efciency calculations in the previous section were made (also summarized in Table 1):

1. Absorption of 90% of incident photons above the bandgap of the semiconductor. Reection and incomplete and parasitic absorption by catalyst materials or other device components contribute to this 10% loss. Analytically, we express this non-unity absorption fraction, fabs, as a modication to the ideal diode equation:

VPVj

kBTq ln

a

[afii9834]max=40.0%

1.5

Bandgap, E g, 2(eV)

1

0.5

1.5

Bandgap, E g, 2(eV)

1 1.5 2

Bandgap, Eg, 1 (eV)

Bandgap, Eg, 1 (eV)

b

[afii9834]max=28.3%

1

0.5

1.5

Bandgap, E g, 2(eV)

1 1.5 2

Bandgap, Eg, 1 (eV)

[afii9834]max=16.2%

c

1

0.5

1 1.5

0 10 20 30 40

2

[afii9834]PEC (%)

Figure 2 | Dual junction limiting efciencies. Limiting efciencies (ZPEC)

versus semiconductor bandgaps (Eg) for (a) ideal case (Zmax 40.0%,

Eg 1.40, 0.52 eV), (b) high-performance realistic case (Zmax 28.3%,

Eg 1.59, 0.92 eV) and (c) Earth abundant realistic case (Zmax 16.2%,

Eg 1.93, 1.38 eV), where contour lines mark every 5% and maximum

efciency points are indicated; a constant colour scale is used for ac; parameter values used for each case are tabulated in Table 1.

fabsjL jj0 1

7

2. An ERE of 3%, meaning that the detailed balance radiative recombination represents 3% of the total (radiative and nonradiative) recombination, and thus 3% of the total dark current. This value is characteristic of high-performance IIIV materials20. Analytically, ERE modies the ideal photodiode

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dark current:

[afii9834]PEC (%)

0

j0

j0;ideal

ERE 83. Catalytic exchange current densities of 1 mA cm 2 and 10 3 mA cm 2 for the hydrogen (cathodic) and oxygen (anodic) evolution reactions, respectively. These values are consistent with the best reported values in literature for Pt and IrO2 (refs 26,27).

This analysis is limited to single and dual junction devices because triple junction maximum efciencies are lower than that of dual junctions under these assumptions (25.4 versus28.3%); although, the gap between dual and triple junction device efciencies has narrowed in comparison to the ideal case.

Figure 1 (green dotted line) and Fig. 2b display the limiting efciencies as a function of semiconductor bandgap for single and dual junction photoelectrochemical devices, respectively, under these assumptions. An identical colour scale is used for each contour plot in Fig. 2 for facile visual comparison. The maximum efciency is 15.1% for a single junction device with a bandgap of2.05 eV, and is 28.3% for a dual junction device with bandgaps of0.92 and 1.59 eV. The high-performance realistic efciency plots exhibit similar trends to the ideal efciency plots, but the efciencies are signicantly lower and the optimum bandgaps are substantially higher. Due to the series addition of photo-voltages, the inclusion of realistic experimental values in this calculation does not affect the dual junction device efciency as dramatically as the single junction device efciency. All three of the modications stated above have a role in lowering the efciency and increasing the optimum bandgaps, but as discussed in greater detail in the parameter variation section, the introduction of nite catalytic exchange current densities has the most dramatic effect.

Earth abundant realistic efciencies. Next, we consider realistic limiting efciencies for Earth abundant materials and devices that currently have less than optimal performance characteristics. In these calculations, we include the effects of all ve parametersnon-ideal light absorption, ERE, series resistance, shunt resistance and catalytic exchange current density. Specically, the following modications to the ideal efciency calculations in the previous section were made (and are also summarized in Table 1):

1. Absorption of 90% of incident photons above semiconductor bandgap.

2. An ERE of 10 6, consistent with reported values for Earth abundant materials20.

3. Catalytic exchange current densities of 1 mA cm 2 and 10 5 mA cm 2 for the hydrogen (cathodic) and oxygen (anodic) evolution reactions, respectively. These values are consistent with reported values for Earth abundant catalysts in literature, such as NiMo for hydrogen evolution, and NiZn, CoFe and NiMoFe for oxygen evolution28.

4. Normalized series and shunt resistance values of 0.1 and 10, respectively, which each result in approximately a 10% reduction in ll factor. Series and shunt resistances are normalized to the characteristic resistance of an ideal photodiode, which is the ratio of the open circuit voltage to the short circuit current density. This normalization results in an approximately bandgap-independent reduction in ll factor. A more detailed explanation and discussion of these parameters can be found in Supplementary Note 3. Analytically, incorporation of the shunt and series resistance terms results in a transcendental equation for the

photodiode current-voltage equation, where RS and RSh are the absolute (non-normalized) series and shunt resistances, respectively:

VPVj

10 20 30 40

E g,3(eV)

1.1

1

0.9

0.8

0.71.8

1.4

Eg,2 (eV)

[afii9834]max=17.3%

Figure 3 | Triple junction limiting efciencies. Limiting efciencies (ZPEC)

versus upper and middle semiconductor bandgaps (Eg,1, Eg,2) for the Earth

abundant realistic case at three lower bandgap values (Eg,3 0.73, 0.93 and

1.13), where the middle plane displays the maximum triple junction efciency under these conditions (parameter values are tabulated in Table 1); contour lines mark every 5% and the maximum efciency point is indicated (Zmax 17.3%, Eg 1.91, 1.36, 0.93 eV); colour scale matches that

used in Fig. 2.

1

2

0.6 1.2

1.6

Eg,1 (eV)

kBT

q ln

"

fabsjL j VPV jRS

=RShj0 1

jRS 9

This analysis includes single, dual and triple junction devices because the additional assumed reductions in material and device parameters lead to a maximum triple junction maximum efciency that exceeds that of a dual junction (17.3 versus 16.2%).

Figure 1(red dotted line), Fig. 2c, and Fig. 3 display the Earth abundant limiting efciencies as a function of semiconductor bandgap for single, dual and triple junction photoelectrochemical devices that result from the above specied set of assumptions. Figure 3 displays the triple junction efciency as a function of its upper two bandgaps for three values of the lower bandgap(0.73, 0.93 and 1.13 eV), and uses a colour scale identical to that of Fig. 2 for facile value comparison. The maximum efciency is5.4% for a single junction device with a bandgap of 2.53 eV, 16.2% for a dual junction device with bandgaps of 1.38 and 1.93 eV, and17.3% for a triple junction device with bandgaps of 1.91, 1.36 and0.93 eV (contained in the middle contour of Fig. 3). As expected, the Earth abundant realistic (real2) efciency plots also exhibit similar trends to the ideal efciency and high-performance realistic (real1) efciency plots, but efciencies are signicantly lower and the optimum bandgaps are substantially higher; the dual and triple junction devices are also affected signicantly less than the single junction devices by non-ideal charge carrier transport. Additionally, the inclusion of series and shunt resistance terms that affect the ll factor result in a softening of the efciency turn-on with increasing bandgap, which is most evident in the single junction device trend (Fig. 1, red dotted line). It is also important to note that the maximum triple junction

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device efciency exceeds that of a dual junction device due to non-ideal charge carrier transport and large catalyst over-potentials, indicating that use of Earth abundant materials may (presently) necessitate the use of three or more junctions to maximize device efciency.

Effect of parameter variation on device efciency. A detailed analysis of the dependence of device efciency on key parameters provides insight into the most powerful handles to improve device performance. To this end, we present the dependence of the maximum achievable device efciency and the corresponding semiconductor bandgap(s) on the aforementioned parameters for single (Fig. 4) and dual (Fig. 5) junction devices; specically, parameter variations are performed around the three base cases presented in the previous sections (ideal, high-performance realisticreal1, and Earth abundant realisticreal2). The absorption fraction is varied between 0.7 and 1.0 (Figs 4a and 5a), the ERE between 10 6 and 1 (Figs 4b and 5b), the catalytic exchange current density between 10 4 and 102 mA cm 2 (Figs 4c and 5c); the normalized series resistance between 0 and0.2 (Figs 4d and 5d), and the normalized shunt resistance between 5 and 103 (Figs 4e and 5e). The maximum device efciency for a given set of parameters is plotted on the y axis and colour is used to display the corresponding bandgap(s); for the dual junction devices, the coloured line is a double band, where the upper component represents the wider bandgap, Eg,1, as indicated

in the gure legends. The colour variation representation of the optimum bandgaps is designed to illustrate trends; for easier extraction of the precise numeric bandgap values, plots of bandgap versus parameter corresponding to Figs 4 and 5 can be found in Supplementary Figs 3, 4 and 5. The maximum efciency

points corresponding to the ideal, high-performance realistic (real1) and Earth abundant realistic (real2) cases presented in previous sections are marked on Figs 4 and 5; each point on Figs 4 and 5 originates from a calculation analogous to those in the previous sections with one modied parameter. Note that the ideal, real1 and real2 points are omitted from the j0,cat variation plots because the anodic and cathodic exchange currents were lumped into a single parameter for simplicity. An animation is provided in Supplementary Movie 1 to visualize the connection between Fig. 3b and Fig. 1.

This sensitivity analysis reveals that the efciency of a single junction device is predominantly controlled by catalyst performance (catalytic exchange current densityj0,cat), whereas the

maximum efciency of a dual junction device is strongly affected by the optoelectronic performance of the semiconductor photodiodes in addition to catalyst performance (all ve parameters j0,cat, fabs, ERE, rs and rsh).

For single junction devices, variation of the catalytic exchange current density around all three cases (ideal, real1, real2) results in the largest modulation in device efciency and semiconductor bandgap (Fig. 4c). As the catalytic exchange current density decreases and, thus, kinetic overpotential increases, the semiconductor bandgap required to drive the water-splitting reaction increases, which leads to a precipitous drop in efciency due to reduced solar spectrum conversion, as illustrated in Fig. 1 and previously discussed. Series resistance (Fig. 4d) also has a distinct effect on single junction device efciency because, even at moderate values, series resistance shifts the maximum power point to lower voltages and thus signicantly lowers the photocurrent near the maximum power point, thereby lowering the efciency for a given bandgap and pushing the bandgap for maximum efciency to higher values. Variation of the ERE

Eg (eV)

1.7

a b

30

20

30

c

Ideal

30

20

10

0

Ideal

20

Ideal

[afii9834] PEC(%)

Real1

[afii9834] PEC(%)

[afii9834] PEC(%)

Real1

Real2

Real1

10

00.7 0.8 0.9 1 1 1 ERE j0,cat (mA cm2)

0 106 104 104 102

0 101 102 103

Real2

10 Real2

102

102

fabs

d e

30 Ideal

2.6 Key

[afii9834] PEC(%)

30

20

10

0 0 0.1rs rsh

0.2

Ideal

[afii9834] PEC(%)

[afii9834]max,ideal (Fig. 1)

[afii9834]max,real1 (Fig. 1)

[afii9834]max,real2 (Fig. 1)

2.3

Real1

Real1

10 Real2

2

Real2

Figure 4 | Single junction device efciency parameter dependence. Limiting efciencies (Zmax) for each single junction case (see Table 1) as a function of a single parameter variation(a) absorption fraction (fabs), (b) external radiative efciency (ERE), (c) catalytic exchange current density (j0,cat inmA cm 2), (d) normalized series resistance (rs), and (e) normalized shunt resistance (rsh); the colour variation indicates the semiconductor bandgap corresponding to the maximum device efciency; the blue squares, green circles and red triangles indicate the position of the maximum efciencies for the ideal, high-performance realistic and Earth abundant realistic cases shown in Fig. 1.

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a b

c

40

40

40

Efficiency (%)

20

Ideal

Real1

Real2

Efficiency (%)

Ideal

Real1

20 Real2

Efficiency (%)

Ideal

30 Real1

20 Real2

j0,cat (mA cm2)

30

30

0.7 0.8 0.9 1

106 104 102 1

104 102 1 102

fabs

ERE

d e

40

40

Ideal

Real1

Real2

Key

Efficiency (%)

20

Efficiency (%)

Ideal

Real1

20 Real2

30

30

[afii9834]max,ideal (Fig. 2a)

[afii9834]max,real1 (Fig. 2b)

[afii9834]max,real2 (Fig. 2c)

1.4 1.6 1.8

2

Eg,1 (eV)

Eg,2 (eV)

0.6 0.8 1 1.2 1.4

0 0.1 0.2

101 102

rs

rsh

Figure 5 | Dual junction device efciency parameter dependence. Limiting efciencies (Zmax) for each dual junction case (see Table 1) as a function of a single parameter variation(a) absorption fraction (fabs), (b) external radiative efciency (ERE), (c) catalytic exchange current density (j0,cat in mA cm 2), (d) normalized series resistance (rs) and (e) normalized shunt resistance (rsh); the dual colour band of the line indicates the two semiconductor bandgaps (upper bandgap material over lower bandgap material) corresponding to the maximum device efciency; the blue squares, green circles and red triangles indicate the position of the maximum efciencies for the ideal, high-performance realistic and Earth abundant realistic cases shown in Fig. 2.

(Fig. 4b), absorption fraction (Fig. 4a) and shunt resistance (Fig. 4e) have signicantly less inuence than catalytic exchange current density and series resistance on the single device efciency. Unlike series resistance, shunt resistance primarily lowers the photocurrent, but, at moderate values, does not signicantly shift the voltage at maximum power; therefore, shunt resistance lowers the efciency due to photocurrent, but the effect is more moderate because the optimum bandgap remains relatively constant. The dependence of device efciency on absorption fraction is almost linear, and the bandgap corresponding to maximum efciency is nearly unaffected due to the weak logarithmic dependence of photovoltage on photocurrent (equations (2) and (7)). The ERE exhibits a similar, nearly log-linear correlation with device efciency; however, the semiconductor bandgap and device efciency are more strongly correlated with ERE than with absorption fraction because a lower ERE translates to a lower photovoltage and, therefore, a large bandgap semiconductor is required to generate an equivalent photovoltage. The optimum bandgap trends provide guidance to experimentalists for (i) initial semiconductor selection in device design given known achievable material parameters, and (ii) when a re-design of their device would be benecial to its performance. For instance, a signicant improvement in catalytic exchange current density signicantly shifts the optimum bandgap for maximum device efciency and may suggest the use of a different semiconductor material with a lower bandgap whereas improvements in semiconductor absorption do not.

Unlike single junction devices, all ve parameters have a signicant effect on the overall dual junction device efciency (Fig. 5). The effect of poor catalyst performance on dual junction device efciency is mitigated due to the series addition of two

photovoltages. Similar to the single junction device, an increase in photovoltage is needed to compensate for the increase in kinetic overpotential, but in a dual junction device, this photovoltage increase is split between two semiconductors, which translates to a smaller increase in required bandgaps and, consequently, less overall effect on device efciency. Conversely, variation of the charge transport parameters (ERE, series resistance and shunt resistance) and absorption fraction affect the performance of each semiconductor individually and, therefore, the effect of these four parameters on overall efciency is largely the same across single and dual junction devices. The kinks in the efciency curves for ERE and catalyst exchange current density variations are a direct product of the AM1.5G spectrum; dips in atmospheric transparency translate to quick jumps in the ideal semiconductor bandgap, and thus, the efciency. These kinks are not observed in all curves because the ideal bandgaps do not cross these values.

Theoretical limits versus reported efciencies. To place the above theoretical analysis into context, this nal section contains a brief consideration of discrepancies between theoretical and experimental efciencies, based on three exemplary water-splitting devices(1) a minimally integrated, dual junction Ptblack|Si|AlGaAs|RuO2 device with 18.3% efciency from

Licht et al. in 2000 (ref. 29); (2) an integrated dual junction Rh|GaInP|GaInAs|RuO2 device with 14% efciency from May et al. in 2015 (ref. 30); and (3) a triple junction Co-Pi|BVO| a-Si|nc-Si|Pt device with 5.2% efciency from Han et al. in 2014 (ref. 31). A recent review paper from Ager et al.32 provides a more comprehensive summary of record water-splitting efciencies as a function of time and device subtype. These three devices were selected because they mimic our three cases (ideal, high performance and Earth abundant) and highlight the varying

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limiting factors in different device design strategies. In the device by Licht et al., a high efciency, dual junction photodiode component is wired to large surface area, high-performance catalyst components. The device by May et al. also uses a high-performance dual junction photodiode, but is monolithically integrated with a high-performance catalyst and fully immersed in solution. The device by Han et al. takes a very different tact by employing all Earth abundant materials and sacricing performance.

Despite the use of high-performance materials and large catalyst areas, the 18.3% efciency device by Licht et al., recognized as the current record water-splitting efciency by Ager et al. as of February 2015, falls far short of the ideal and high-performance realistic (real1) limiting efciencies (40.0% and 28.3%, respectively).

The primary reason for this efciency gap is non-ideal bandgap selection. The ideal (real1) efciency limits for the bandgap combinations in the Licht device, 1.1 and 1.6 eV, and the May device, 1.26 and 1.78 eV, are 27.2% (24.5%) and 22.8% (20.5%), respectively. The ideal (real2) efciency limit for the bandgap combinations in the Han device, 2.4, 1.7, and 1.1 eV, is 9.2% (4.2%). The integration of ideal bandgap materials for multijunction devices is a challenge that also faces the photovoltaics community. For high-performance materials, the principal issue is that high-quality material, which translates to high ERE, requires lattice-matched materials, and the lattice-matching requirement restricts the available bandgap combinations33. Lattice-mismatched multijunction photovoltaics are the subject of much ongoing research, including strategies such as inverted metamorphic and pseudomorphic designs, to name a few, and progress in this area will be instrumental for water-spitting device efciency improvement34,35. In the realm of Earth abundant materials, the challenge is identication and optimization of a material with the appropriate bandgap, and for liquid junctions in the photoelectrochemistry community, also the appropriate band alignment and stability. When their non-ideal bandgap combinations are taken into account, both the Licht and the May devices are within roughly 6% of the adjusted high-performance realistic (real1) efciency limits; and the Han device is within 4% of the ideal limit and actually exceeds the Earth abundant realistic (real2) limit, clearly illustrating the primary limitation in Earth abundant devicesphotoelectrode bandgap.

The factors accounting for the remaining 6% discrepancy between realized and theoretical efciency are different for the Licht and May devices. The Licht device uses large area electrodes consisting of high-performance catalysts that are positioned in such a way as to not interact with light incident on the photodiode unit. Therefore, the majority of the losses in this device are due to less than ideal photodiode performance. An examination of the JV curve of their photodiode reveals that both incomplete light absorption, non-ideal EREs and shunt and series resistances have a role in the reduced efciency; most notably, their ll factor is only 77% (B10% below the ideal). The high-performance real efciency limit trend with shunt resistance (Fig. 5e) shows that a 10% drop in ll factor (to rsh of 10) results in a B5% drop in maximum efciency; this value agrees well with the observation that the Licht device operating current was B5% below their short circuit current and well-aligned with their maximum power point. The May device faces different, and additional challenges due to the direct integration of catalyst on the photodiode surface. As illustrated in the supplementary information of May et al. and discussed theoretically in ref. 16, the loading of catalyst on the light incident side of the device has a tradeoff30; low catalyst loading results in slow catalyst turn-on and high-catalyst loading blocks light transmission. As a result, the May device loses signicant current density (B4 mA cm 2) due to parasitic catalyst light absorption.

May et al. also cite surface resistance and recombination at the

semiconductor catalyst interface as sources of loss, indicating non-ideal EREs and series resistance. Conversely, the primary factor limiting the Han device is the BVO performance; despite the use of gradient W-doping, the high resistivity of BVO drastically reduces the device ll factor, and thus, its efciency. Ultimately, these three exemplary devices illustrate that there is room for improvement both in individual component performance, particularly for Earth abundant materials, as well as their integration into water-splitting devices.

DiscussionIn summary, we presented limiting efciencies of water-splitting photoelectrochemical devices under ideal and realistic conditions, and arbitrary intermediate conditions through parameter variation studies. We rst dened the general analytic equation that governs the current-voltage characteristic of a variable-junction photoelectrochemical device and its efciency. Second, we presented the limiting efciencies (both ideal and experimentally realistic) of a water-splitting photoelectrochemical device for both single and dual junction photodiode units. Subsequently, we considered the effects of ve parameterssemiconductor absorption fraction, semiconductor ERE, series resistance, shunt resistance and catalytic exchange current densityon the limiting efciency and the corresponding semiconductor bandgap(s) and, thus, illustrated the varying impacts of the three main phenomena (light absorption, charge carrier transport and catalysis) on device performance. Finally, we contextualize this theoretical efciency analysis by examining reported experimental results for three exemplary water-splitting devices. This analysis provides a framework through which one can understand all previous reported limiting efciencies with various assumed values and also provides insight into the primary factors limiting device performance and the most powerful handles to improve device efciency.

Data availability. The AM1.5G spectrum data used for the efciency calculations was derived from the public domain resource, NREL-RREDC: http://rredc.nrel.gov/solar/spectra/am1.5/

Web End =http://rredc.nrel.gov/solar/spectra/am1.5/ . Additional data that support the ndings of this study, including source code, are available from the corresponding author upon request.

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Acknowledgements

This material is based upon work performed by the Joint Center for Articial Photo-synthesis, a DOE Energy Innovation Hub, supported through the Ofce of Science of the U.S. Department of Energy under Award No. DE-SC0004993. We are grateful toDr E. Warmann for useful discussions regarding the impact of external radiative efciency on photodiode efciency.

Author contributions

K.T.F. and H.J.L. designed the study and wrote the paper, K.T.F. executed the calculations and analysis, and H.J.L. and H.A.A. advised. All authors reviewed and commented on the manuscript.

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How to cite this article: Fountaine, K. T. et al. Efciency limits for photoelectrochemical water-splitting. Nat. Commun. 7, 13706 doi: 10.1038/ncomms13706 (2016).

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