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Limits of contact selectivity/recombination on the open-circuit voltage of a photovoltaic Ellis T Roe, Kira E Egelhofer, and Mark C Lonergan ACS Appl. Energy Mater., Just Accepted Manuscript • DOI: 10.1021/acsaem.7b00179 • Publication Date (Web): 21 Feb 2018 Downloaded from http://pubs.acs.org on February 25, 2018

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ACS Applied Energy Materials

Limits of Contact Selectivity/Recombination on the Open-Circuit Voltage of a Photovoltaic Ellis T. Roe,† Kira E. Egelhofer,‡ and Mark C. Lonergan∗,‡ Department of Physics, University of Oregon, Eugene, OR 97403, and Department of Chemistry and Biochemistry, The Materials Science Institute, University of Oregon, Eugene, Oregon 97403 E-mail: [email protected]

Abstract We analytically calculate fundamental limits on the open-circuit voltage (Voc ) of a solar cell imposed by contact selectivity and contact recombination. To do so, we consider a simple model consisting of only carrier generation in an absorber and charge transfer to its contacts enabling an algebraic solution for the relevant partial currents and the calculation of a contact-determined Voc . An expression for Voc is determined assuming the partial currents at the contacts linearly depend on the product of the appropriate equilibrium exchange current density and excess carrier concentration at the contact. Quantitatively defining contact selectivity and contact recombination, we illustrate the roles of the exchange current densities, recombination, and selectivity on Voc . Additionally, we use complete device physics simulations to show that our simplified model is valid in practically relevant situations. The framework we have developed ∗

To whom correspondence should be addressed Department of Physics, University of Oregon, Eugene, OR 97403 ‡ Department of Chemistry and Biochemistry, The Materials Science Institute, University of Oregon, Eugene, Oregon 97403 †

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elucidates the physics underlying more qualitative discussions of selectivity often invoked to describe the impact of contacts on Voc , thereby enabling better-motivated improvements in contact design.

Keywords solar cell, selectivity, recombination, contact, open-circuit voltage, quasi-Fermi level

1

Introduction

In the field of photovoltaics, the term “selectivity” is frequently used in the literature to describe a contact’s ability to selectively transport one charge carrier in favor of the other. Indeed, Green 1 discussed the selective collection of electrons or holes as a fundamental principle of photovoltaics that competes against recombination to determine the overall efficiency of a photovoltaic. 2 The importance of selectivity and recombination in photovoltaic performance has led to intense interest in the development of carrier-selective contacts that do not contribute significant recombination currents. 3–7 In the vast majority of studies, however, selectivity is invoked as a qualitative concept, with the precise connection between it and device parameters such as the open-circuit voltage (Voc ) unclear. The goal of this work is to analyze a relevant model system where the concepts of contact selectivity and contact recombination can be precisely defined and analytically related to Voc . By doing so, the work clarifies the role of contact selectivity and recombination in determining Voc . For instance, Voc is commonly related to the light-generated current density JL and a recombination current density J0 by: 8

Voc = VT ln

JL . J0

(1)

Here, VT = kT /q is the thermal voltage where k is the Boltzmann constant, q is the ele-

2


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mentary charge, and T is the temperature. Equation 1 relates Voc to the characteristic rate of a single rate (J0 ), whereas the concept of selectivity implies a comparison between rates. How does the selectivity factor into eq. 1? Does it enter entirely through the recombination term J0 or is another relation more appropriate? This work seeks to answer such questions by providing relations of similar form to eq. 1 that clearly show the limits on Voc imposed by contact selectivity and recombination, as defined herein. Perhaps because of the range of systems to which the concept is applied, there is no generally accepted definition of selectivity. U. Würfel et al. described selectivity in terms of the differences in electron/hole conductivities in variously doped regions found in common solar cell architectures. 9 This is similar to the definition used by Brendel and Peibst 10 in considering passivating selective contacts to silicon; they define selectivity in terms of the ratio of the apparent contact resistivities of majority to minority carriers to a doped silicon absorber. Mora-Seró and Bisquert defined an ideal selective contact by setting the partial current for one carrier (electron or hole) at the boundary with a contact equal to zero. They then relax the condition of ideal selectivity with a non-zero partial current related to the resistivity for the carrier or to a current through interfacial gap states. 11 This is similar to the definition from Tress et al. 12 where the surface recombination velocity for one carrier is set to zero. Although there is no universally accepted definition of selectivity as a continuously variable parameter, it follows from the examples given that there is an accepted definition for the concept of ideal selectivity. Namely, it corresponds to the situation where the partial current of one carrier in some part of the system becomes zero, either due to bulk or interfacial charge-transport limitations. As described below, we define contact selectivity in terms of a ratio of the electron and hole exchange current densities at an appropriate contact to a solar absorber. By appropriate, we mean a contact that is described adequately by the physics in the Theory section. Ideal (i.e. infinite) contact selectivity is achieved by setting one of either the electron or hole exchange current densities to zero. The fundamental relationship between selectivity and photovoltaic performance has been

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investigated theoretically 2,10,11,13 and through simulation. 12–15 Perhaps the simplest theoretical concept illustrating how selectivity limits the photovoltaic performance comes from considering the basic p-n homojunction, with identical electron and hole carrier mobilities. The Voc in such a structure cannot exceed the initial difference between the Fermi levels of the isolated n- and p-type semiconductors. These Fermi levels are in turn related through the carrier concentration to the electron and hole conductivities and hence to selectivity. The selectivity of the system, just like radiative recombination, 16 places limits on the maximum Voc obtainable. 2 Studies explicitly relating selectivity to photovoltaic performance have generally treated one non-ideal contact paired with an ideal selective contact. 10,11,13 These studies have illustrated the intimate relation between selectivity and recombination, as any deviation from ideal selectivity is seen as introducing a recombination current that decreases cell performance. The interplay between bulk charge transport processes and interfacial charge-transfer processes contributing to contact selectivity has also been an important consideration 12–15,17 and will be further investigated in the simulations we perform to complement our theory. As described in the Theory section, we explicitly consider the relation between Voc and the exchange current densities for electrons and holes at both the electron-collecting and holecollecting contacts to an intrinsic absorber material. Exchange current densities characterize the rate of charge carrier exchange between two materials in a state of dynamic equilibrium. They are readily measured from current-voltage characterization of a simple semiconductor junction based on an appropriately doped semiconductor, n-type for the electron and p-type for the hole exchange current densities. 18–20 It is also possible to measure the electron and hole exchange current densities at a particular semiconductor contact simultaneously using more complex device geometries 21,22 Our theoretical treatment provides a picture of the overall selectivity properties of a complete photovoltaic, rather than focusing on a single contact. We do so in a way that neglects limitations due to radiative and other forms of bulk recombination as well as bulk transport processes, focusing instead on the specific role of charge collection by the contacts.

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A complete device model for a photovoltaic would not neglect bulk recombination, but we seek an understanding of the upper limit due to contacts, not a universal device model. For comparison, consider the Shockley-Quessier limit. 16 There, the limiting performance of solar cells was calculated using the detailed balance of radiative generation and recombination while assuming the contacts are able to perfectly separate and extract electrons and holes. This is essentially the opposite of what we are considering here. In the end, both are theoretical upper limits of solar cell performance that allow for an understanding of how measurable quantities limit said performance. Our model resembles that of Nelson et al. for molecular photovoltaics. 23 They use Marcus rate expressions to treat electron transfer rates to and from both sides of a two-level molecular absorber, in addition to allowing for recombination in the absorber. Our model differs in two major ways: 1) our absorber is a macroscopic bulk semiconductor with (in general) spatially varying concentrations of electrons and holes requiring a solution to the continuity equation, and 2) our boundary conditions for the absorber are the familiar charge-transfer rates used in a variety of semiconductor systems (see eq. 2 below). Herein, we provide a general analytic expression for the Voc of our system over a wide range of conditions, in addition to explicitly solving for Voc in terms of well-defined contact selectivity and contact recombination parameters.

2 2.1

Theory Model and Assumptions

Figure 1 illustrates the model system treated herein. It consists of an intrinsic absorber with position variable x. The absorber is interfaced on one side with a contact termed the α contact, at position xα , and by a second contact termed the β contact on the other side, at position xβ . The direction of positive x runs from the α contact to the β contact. Throughout, we use the superscripts ‘α’ and ‘β’ to label quantities associated with the α and 5



contact α α

Jn(x α) = J0n

n(x α) -1 n0(x α)

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contact β

intrinsic absorber

β

Jn(x β) = -J0n

Ecb

n(x β) -1 n0(x β)

GL Jp(x α) = -J0p α

p(x α) -1 p0(x α)

Evb xα

β

Jp(x β) = J0p xβ

x

p(x β) -1 p0(x β)

Figure 1: Rate processes incorporated into our model system. Only bulk generation and charge transfer at the contacts are considered. In particular, the latter provides the only source of recombination in the system. Radiative and other forms of bulk recombination are not considered so the results place limits on the Voc due to contact recombination/selectivity. β contacts. Later on, it will be useful to consider one contact as the electron contact and one as the hole contact in order to understand certain limits. Note that positive current is defined to be in the positive x direction. Furthermore, we use the electrochemical potential, µ ¯ = µ + zqφ, where µ is the chemical potential, q is the elementary charge, φ is the electrostatic potential, and z is +1 for holes and -1 for electrons. We make the following assumptions: 1. Charge carriers in the absorber are thermalized within their bands so that the electrons in the conduction band can be described by an electron electrochemical potential µ ¯n and holes in the valence band by a hole electrochemical potential µ ¯p . These are equivalent to the electron and hole quasi-Fermi levels. 2. There are no transport limitations in the absorber such that d¯ µn /dx = d¯ µp /dx = 0. 3. The rate equations for the electron and hole partial currents Jn and Jp at the interfaces

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are given by: n(xα ) Jn (x ) = −1 n0 (xα ) p(xα ) α α −1 Jp (x ) = −J0p p0 (xα ) n(xβ ) β β Jn (x ) = −J0n −1 n0 (xβ ) p(xβ ) β β Jp (x ) = J0p −1 , p0 (xβ ) α

α J0n

(2a) (2b) (2c) (2d)

where n and p are the electron and hole carrier concentrations, n0 and p0 their equilibrium values as set by the contacts, and J0n and J0p the electron and hole equilibrium exchange current densities (i.e. saturation current densities). 4. All forms of bulk recombination are neglected. 5. The equilibrium carrier concentrations follow the law of mass action:

n0 (x)p0 (x) = n2i ,

(3)

where ni is the intrinsic carrier concentration. 6. Each of the chemical potentials in the contacts, µβ and µα , is a constant independent of voltage, and there is no electric field in the contacts. The condition for equilibrium across the device is that the electrochemical potentials of the contacts are equal, i.e. that µ ¯β = µ ¯α . Equation 2 is the central set of expressions that describe charge-transfer processes at the contacts. They are appropriate for a variety of treatments of semiconductor interfaces including the Schottky model describing thermionic emission at a semiconductor/metal 18 or related interfaces 19,24,25 or the Shockley model describing the diffusion and recombination of injected carriers in the quasi-neutral, field-free region of a doped semiconductor contact. 26,27

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β β α α Equation 2 defines the system in terms of a set of four J0 values: {J0n , J0p , J0n , J0p }. We

present general results in terms of these parameters. When the J0 ’s of a contact are specified, i.e. when we know which J0 for a given contact is the larger one, we use the symbols J0 and j0 to refer to the larger and smaller, respectively, of the two J0 values describing a contact. The charge-transfer process associated with J0 is referred to as the “majority” process, and that with j0 the “minority” process.

2.2

Definition of Contact Selectivity and Contact Recombination Parameters

We define a contact selectivity parameter S for a contact governed by the physics of eq. 2 as: S≡

J0 . j0

(4)

β β For a hypothetical ‘β’ contact where J0n > J0p (i.e. electron selective), the selectivity is β expressed in the J0 , j0 convention by S β = Jβ0n /j0p . The parameter S characterizes selectivity

because it describes the relative rates of the electron and hole processes at an interface; the greater S, the greater the asymmetry in the rates of these processes. On a log scale, S is the difference between the two J0 values. We define a contact recombination parameter Rm that depends on the nature of the recombination. We consider quasi-first-order recombination (m = 1) to effectively depend on the minority process at the contact, and define an associated first-order recombination parameter R1 as: R1 ≡ j0

(5)

Second-order recombination (m = 2) depends on both minority and majority processes, and we define a second-order recombination parameter R2 as:

R2 ≡

p J 0 j0 8


(6)

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The parameter R2 characterizes second-order recombination because it captures an average of sorts of the electron and hole rate processes at the interface. Specifically, R2 is the average of J0 and j0 on a log scale. Equation 2 specifies two values of J0 for each contact. These can equivalently be written in terms of S and Rm , with the natural choice of R1 or R2 depending on the relation of the exchange current densities to the current generated by photoexcitation as discussed further below. We present general results in terms of J0 values and then incorporate S and Rm where appropriate to demonstrate their important connections to Voc .

2.3

Derivation of Voc and ∆EF

Our aim is to calculate an expression for both Voc and the quasi-Fermi-level splitting (∆EF ) as a function the light current, JL , and the equilibrium exchange current densities from eq. 2 using our assumptions. The comparison between ∆EF and Voc helps facilitate understanding of the nature of Voc . The ∆EF is defined as the separation between the electron and hole electrochemical potentials: ∆EF ≡ µ ¯n − µ ¯p . We first relate the partial currents at one contact via their sum under open-circuit conditions. An additional constraint for the partial currents is obtained from the continuity equation. Each partial current is related to the corresponding surface carrier density, and a constraint for the carrier densities is obtained from assumption 2. Finally, the definitions of Voc and ∆EF are used, with our assumptions, to relate them to the carrier densities and hence the partial currents. The derivation is layed out below, starting with the continuity equation. As recombination in the bulk of the absorber is neglected, the steady-state continuity

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equations are: 1 dJn q dx 1 dJp Gp = , q dx

Gn = −

(7a) (7b)

where Gn and Gp are the generation rates for electrons and holes. Integration over the absorber layer yields two equivalent constraints:

where JL = q

R xβ xα

JL = Jn (xα ) − Jn (xβ )

(8a)

JL = Jp (xβ ) − Jp (xα ),

(8b)

GL (x)dx is the light-generated current density with GL = Gp = Gn , the

generation rate due to illumination. These equations express the balance of generation by light absorption in the absorber and recombination through electron and hole transfer at the contacts. The partial currents of each carrier at the contacts are easily related because the total current must be equal to zero across the device at open circuit:

Jn (xα ) + Jp (xα ) = 0

(9a)

Jn (xβ ) + Jp (xβ ) = 0.

(9b)

The combination of eqs. 8a, 9a, and 9b restrict the possibilities for the profiles of the partial currents. Figure 2 depicts an example of partial current profiles consistent with these equations. Because the partial currents must sum to zero at Voc , they mirror each other, i.e. Jn (x) = −Jp (x). Additionally, neither current’s magnitude can exceed JL at any point, as this would necessitate carriers entering the device from the contacts (this is not possible at Voc ; the only source of carriers can be generation in the absorber). The size of the partial currents at the contacts is determined by relating them to the concentrations of carriers at 10


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J(x) JL α Jn(x )

dJ

n

dx = - q G

L

0

Jp(xβ ) x Jn(xβ )

Jtot

dJ p = q dx

Jp(xα ) -JL α x

GL

xβ

Figure 2: Sample plot of the hole partial current (red), electron partial current (blue), and total current (purple) across the device model at Voc . The slope of both partial currents is uniform and equal in magnitude to the electron charge times the generation rate. the boundaries, as is discussed below. One more constraint is needed on the four partial currents (or equivalently, the four surface concentrations) in order to solve for them: ∆EF is a constant throughout the absorber (assumption 2). This and the definition of electrochemical potential (refer to assumption 6) yield: n(xβ ) n(xα ) − φ(xα ) = VT ln − φ(xβ ) ni ni p(xα ) p(xβ ) VT ln + φ(xα ) = VT ln + φ(xβ ). ni ni

VT ln

(10a) (10b)

Here, the reference electrochemical potential is taken as that for n = p = ni and φ = 0. Eliminating φ(xα ) − φ(xβ ) from these equations yields:

n(xα )p(xα ) = p(xβ )n(xβ ).

(11)

Equation 11 is simply a non-equilibrium law of mass action that applies everywhere in the absorber because we have assumed that the electrochemical potentials are flat.

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The Voc and quasi-Fermi level splitting (∆EF ≡ µ ¯n − µ ¯p ) are calculated from the carrier concentrations in the system. The ∆EF is given by:

∆EF ≡ µ ¯n − µ ¯p = qVT ln

np np = qVT ln . 2 ni n0 p0

(12)

The voltage V across the device is given by: −qV = µ ¯β − µ ¯α . Note that µ ¯α and µ ¯β are not equivalent to µ ¯n (xα ) and µ ¯n (xβ ), respectively. The first two refer to electrochemical potentials in the electrodes whereas the second two refer to electrochemical potentials for electrons in the semiconductor at the electrode boundaries. Using the definition of µ ¯ yields:

−qV = µβ − qφβ − (µα − qφα ) .

(13)

At equilibrium, the electrochemical potentials of the contacts are equal:

µβ0 − qφβ0 = µα0 − qφα0 .

(14)

Equations 13 and 14 combined with assumption 6 yield:

V = φ(xβ ) − φ(xα ) − φ0 (xβ ) − φ0 (xα ) .

(15)

From this equation and eq. 10 (also evaluated at equilibrium): V = VT ln

n(xα )n0 (xβ ) n0 (xα )n(xβ )

= VT ln

p(xβ )p0 (xα ) p0 (xβ )p(xα )

.

(16)

The relationships between the carrier densities at the contacts, their equilibrium values, and Voc and ∆EF are summarized in Fig. 3. Consistent with Fig. 2, the carrier concentrations are all above their equilibrium values so that both carriers are flowing out of the device at both contacts. The arrows indicate the ratios of the surface concentration of each carrier to its equilibrium concentration, thus in log space they are a measure of how far the carriers 12


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n,p p(xα ) p0(xα )

- =

n(xα ) ni

n0(xα )

+ =

Voc VT

ΔEf q VT

n(xβ )

= -

n0(xβ ) p(xβ )

p0(xβ )

= +

xα

xβ

Figure 3: Schematic of electron (blue) and hole (red) surface concentrations consistent with Fig. 2 via eq. 2 (note the logarithmic vertical axes). At the α contact, the electron concentration is higher than the equilibrium concentration set by the contact, thus electrons flow out of the device to the left, which is consistent with a positive Jnα using our convention. At the β contact, the electron concentration is also above equilibrium, leading to electrons flowing out of the device to the right, producing a negative value for Jnβ . Analogous logic applies for holes. The arrows represent the ratios (differences in log space) of the carrier concentrations to the equilibrium concentrations, set by the contacts, at the respective interfaces. These ratios determine the Voc and ∆EF as shown schematically in the center insets. are being driven from their surface equilibrium. Due to eq. 12, the sum of the electron and hole arrows at one contact as well as their sum at the other equals ∆EF /qVT . Due to eq. 16, however, the difference between the electron arrow at one contact and the electron arrow at the other as well as that difference for holes equals Voc /VT . This contrast highlights the fundamental difference between Voc and ∆EF . The ∆EF can be maximized when both arrows at each contact have equal length, however maximizing Voc requires asymmetry so that the size of one arrow at a contact goes to zero (i.e. the carrier associated with that arrow must not appreciably exceed its equilibrium value). In other words, one can easily maximize the ∆EF in a semiconductor under illumination by contacting it with a highly insulating material on both sides. This would not generate a Voc , of course, because there is no asymmetry to encourage the electrons to go one way and holes to go the other. Above, we have described the physics that relate the constraints implied by our model to

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each other. Mathematically, this simply amounts to solving a system of four variables (the four partial current values at the contacts, or equivalently the four carrier concentrations at each contact) with four constraint equations (eqs. 8a, 9a, 9b and 11). The first step of this is to re-write the modified law of mass action (eq. 11) in terms of the partial currents at the contacts using eq. 2. The other constraints, eqs. 8a, 9a, and 9b, are then used to eliminate all but one of the partial currents, which can then be solved for using the quadratic equation. Once one of the partial currents is obtained, the others are all easily calculated using eqs. 8a, 9a, and 9b again. Note that due to the non-linearity of the system, there are two solutions for each partial current/carrier concentration. One solution gives values of carrier concentrations at both ends of the device that are below their equilibrium values. This solution is non-physical, indeed it can result in Voc values larger than ∆EF . The physical solution is used exclusively in this work. To obtain expressions for Voc and ∆EF , one can write eqs. 12 and 16 in terms of the now-known partial currents, again using eq. 2. For a detailed algebraic derivation, refer to the Supporting Information.

3 3.1

Results and Discussion General Results

We calculate ∆EF and Voc both generally and in various limits. The ∆EF describes how far the carrier concentrations can be driven from their equilibrium values as determined by the balance of generation and contact recombination. We use ∆EF as a measure of how contact recombination limits Voc . The Voc cannot exceed ∆EF /q, and herein any deficit is necessarily due to insufficient asymmetry in the system. We associate the asymmetry in the system with selectivity, and hence consider a Voc lower than ∆EF /q as a measure of how selectivity limits Voc .

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The analytic expressions for ∆EF and Voc are: ∆EF = VT ln q

"

fβ



JL β J0n

+1

1−f

Voc = VT ln 

β

f

JL α J0n

β JL β J0p

+1

!# +1

 .

f β JJβL + 1

(17)

(18)

0n

Here, f β is the fraction of photogenerated current recombined at the β contact, specifically written for the electron partial current:

fβ ≡ −

Jn (xβ ) . JL

(19)

It is related to the exchange current densities by:

fβ =

i √ 1 h 2 β α 1 + A Γ + Γ 1 − 1 + Y , 1 − A2

(20)

where Y =

2A2 Γβ + Γα + (A2 Γβ + Γα )2

1 2

.

(21)

The Γ’s are the averages of the equilibrium exchange current densities at a particular contact reduced by JL : β β J0n + J0p Γ = 2JL α α J 0n + J0p α Γ = , 2JL β

(22a) (22b)

and A is a recombination ratio: s A=

α α J0n J0p β β J0n J0p

(23)

Equations 17 and 18 could equivalently be written in terms f α instead of f β ; we have omitted the former as they are equivalent. Note that eq. 20 has a removable singularity when A → 15



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1 with: 1 lim f = A→1 2 β

1 + 2Γα 1 + Γ β + Γα

.

(24)

Further, eq. 20 is one of two possible solutions and corresponds to the physical regime 0 ≤ f β ≤ 1 (f β < 0 corresponds to the flow of electrons and holes into the absorber instead of out, and f β > 1 corresponds to magnitudes of partial currents exceeding JL ). Note that in the 0 ≤ f β ≤ 1 regime, Jn (xβ ) ≤ 0, which is consistent with the flow of electrons out of the absorber. We now proceed to analyze sensible cases that allow for simplification of the general results to facilitate intuitive understanding. We will dub the β contact the electron contact, and the α contact the hole contact. Consistent with this naming of the contacts, we assume that:

α Jα0p j0n

(25a)

β j0p Jβ0n .

(25b)

Because the J0 ’s of the contacts are now specified, we employ the J0 , j0 notation to emphasize the difference between minority and majority J0 ’s. We emphasize that it is not necessary to assume eq. 25; the general result works for any values of the four J0 ’s. The purpose of making the assumption is to more easily understand how the Voc and ∆EF behave in sensible limits. In the discussion below, it is natural to define two regimes based on the order of the contact recombination process. First-order recombination depends on the concentration of a single carrier. This occurs when JL J0 because the concentration of carriers associated with J0 does not need to appreciably exceed its equilibrium value to support JL , and hence, can be considered constant. We define JL J0 as the low-injection regime. In contrast, second-order recombination depends on the concentrations of both carriers. This occurs when JL J0 , and both carrier concentrations need to significantly exceed their equilibrium values 16


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to support JL . We define JL J0 as the high-injection regime. Note that the terms highand low-injection are not used in the classical sense of relating the carrier concentration in the body of a semiconductor under illumination to the majority carrier concentration in the dark. However, they do in the same way separate regimes of quasi-first versus second-order recombination. We assume that JL j0 at each interface in both cases. Otherwise, the interfacial recombination currents result in a negligibly small Voc . The definitions for both regimes of injection are summarized in Table 1. Table 1: Definition of low and high injection injection low high

3.2

electron contact β Jβ0n JL j0p β JL Jβ0n j0p

hole contact α Jα0p JL j0n α JL Jα0p j0n

Two-J0 Case

It is illustrative to consider a simplified case that reduces the four J0 problem to only two J0 ’s. To do this, we assume the J0 values at the two contacts to be asymmetrically related β β α α so that J0n = J0p = J0 and J0p = J0n = j0 . By symmetry, f β = 1/2, and eqs. 17 and 18

become: ∆EF = VT ln q

JL +1 2J0

Voc = VT ln

JL 2j0 JL 2J0

+1 +1

JL +1 2j0 ! .

(26)

(27)

Figure 4 shows contour plot representations of eqs. 26 and 27; ∆EF /q and Voc are plotted as a function of log(J0 /JL ) and log(j0 /JL ). The contour plots are clipped at values of ∆EF /q and Voc greater than one so that the minimum and maximum values of the color scheme run consistently from 0 to 1. This can be seen as artificially imposing a maximum value of ∆EF /q and Voc due to radiative recombination (Vocmax ), and plotting ∆EF /qVocmax and Voc /Vocmax . The contours of both Figs. 4a and 4b clearly show two distinct regimes of behavior, separated by the grey horizontal dashed line in both figures. This dashed line represents the 17



�

(a) ΔEF / q

� 0.9

0.7

0.5

0.3

0.1

(b) |Voc |

0.9

-��

-��

0.7

0.5

0.3

0.1

�

��[� � /� � ]

�

��[� � /� � ]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 35

-�

-�

-��

-��

-��

-�� -��

-��

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-�

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��[�� /�� ]

-��

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Figure 4: Contour plots of eqs. 26 and 27 showing: (a) ∆EF /q and (b) |Voc | as a function of log(J0 /JL ) and log(j0 /JL ). The value of the contours run from 0 V (dark green) to 1 V (yellow) with every other contour labelled in Volts. The plots are clipped with the gray regions representing values greater than 1 V. The dashed gray line at log(J0 /JL ) = 0 divides the low (above line) and high (below line) injection regimes. In low injection, ∆EF /q = Voc , and the direction of the contours shows that they are limited by only the exchange current density for the minority process (j0 ). In high injection, ∆EF /q > Voc , and ∆EF /q is determined by J0 j0 while Voc is determined by J0 /j0 as evidenced by the fact that the contours between (a) and (b) are orthogonal. Note that the region above the dotted gray line marking S = 0 corresponds to J0 > j0 , as considered in the text. boundary between the high- and low-injection regimes, i.e. when J0 = JL . The gradient of ∆EF is different in the two regimes. In high-injection, it runs along the line log[J0 ] + log[j0 ] = constant or equivalently with the product J0 j0 , i.e. R2 . As ∆EF depends on contact recombination, this reflects the second-order nature of recombination in high-injection. In low injection, the gradient is parallel to the j0 axis; ∆EF only depends on j0 , i.e. R1 , reflecting the quasi-first-order nature of the recombination. The quantitative dependencies of ∆EF on Rm can be readily seen by evaluating eq. 26 in either of the two limits J0 JL and J0 JL yielding: ∆EF ≈ mVT ln q

JL 2Rm

(28)

with m = 1 for J0 JL , and m = 2 for J0 JL . The factor of two in the denominator of the log accounts for the two interfaces in the system. The behavior of Voc is also different in the two regimes. Comparison of Fig. 4a and 4b 18


Page 19 of 35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


show that Voc = ∆EF in low injection. Indeed, evaluating eq. 27 in the limit J0 JL yields: Voc ≈ VT ln

JL 2R1

= VT ln

JL 2j0

,

(29)

identical to the low injection expression for ∆EF . In low injection, the Voc is recombination limited depending on the balance of generation and recombination. In contrast, the highinjection Voc does not follow ∆EF . Rather, the gradient of Voc in high injection runs along log J0 − log j0 = constant lines. Thus, it depends on the ratio J0 /j0 , i.e. S. Indeed, evaluating eq. 27 in the limit J0 JL yields:

Voc ≈ VT ln

J0 = VT ln S. j0

(30)

The Voc is selectivity limited in high injection with Voc always remaining smaller than ∆EF ; this is because the selectivity in high injection is never sufficient to support the full quasifermi level splitting, which is always limited by recombination. To summarize, Voc , like ∆EF , is determined solely by JL /j0 as long as the majority process is faster than JL (J0 JL ). Selectivity does not limit the device, and the only way to improve a device in this regime is to decrease j0 . However, when J0 JL , the device selectivity is not large enough to support the recombination-determined ∆EF , and Voc becomes both selectivity limited and illumination independent.

3.3

Both Electrodes in High or Low Injection

The case of the previous section clearly illustrates how contact selectivity and recombination combine to determine Voc , but because of its symmetry, it conflates the asymmetry at a given contact with the asymmetry of the entire system. Further, it is unlikely that in a practical system, one could explore contact selectivity while preserving the symmetry of the two-J0 case. Hence, we return to treating the four J0 ’s considering both electrodes either in low or high injection. Recombination at the two contacts no longer has to be balanced; f β no 19



Page 20 of 35

longer has to be 1/2. The value of f β is largely determined by the relative recombination rates at the two contacts. To understand why, we simplify the general result, eq. 20, under the assumptions of low and high injection. In low injection, the term Y in eq. 20 is much less than one, and f β can be reasonably approximated with a first-order expansion of the square root term. √ In high injection, f β can be simplified with an additional constraint namely that Y 1, which amounts to:

q 2

α β j0p j0n

α + j0n

s

β j0p

JL JL 1. Jβ0n Jα0p

(31)

β α and j0p This inequality is satisfied when j0n are not too different from one another as com-

pared to the difference between JL and the J0 ’s. With these approximations, the limiting form for f β is: fβ ≈

1 α /Rβ 1 + Rm m

.

(32)

with m = 1 for J0 JL , and m = 2 for J0 JL . The partitioning of recombination between the two contacts is determined by their relative Rm values. The ∆EF is calculated by inserting eq. 32 into eq. 17 in the appropriate limit yielding: ∆Ef ≈ mVT ln q

JL α + Rβ Rm m

,

(33)

with the appropriate value of m. This is analogous to eq. 28 of the two-J0 case, but it allows for asymmetries in the Rm values. When there is a significant imbalance, the interface with the larger Rm dominates the recombination and ultimately limits ∆EF /q. The Voc in the low-injection case equals ∆EF /q: Voc ≈ VT ln

JL

= VT ln

R1α + R1β

JL β α j0n + j0p

! .

(34)

As in the two-J0 case, the Voc is limited by contact recombination, not selectivity, in low injection. The classic expression for Voc relating the photocurrent to the minority recombi20


Page 21 of 35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


β β α α nation current (j0p + j0n ) is obtained. If there is a significant imbalance between j0n and j0p ,

∆EF and Voc are determined by the interface with the larger j0 value. The high-injection result for Voc is: v u α β u J0p J0n Voc ≈ VT ln t β α . j0p j0n

(35)

The Voc in the high-injection case is less than ∆EF /q. It is limited by the selectivity as in the two-J0 case, but there is an important distinction. The Voc is determined by the system or device selectivity, SD , as defined by: v u α β u J0p J0n √ SD = t β α = S β S α . j0p j0n

(36)

With this definition and assumption 31, the Voc in eq. 35 is simply:

Voc ≈ VT ln SD .

(37)

The SD is the average in log space of S β and S α . We see that the selectivity-limited Voc cannot be described by the selectivity of only one of the contacts; the asymmetry in the rate processes for the entire system must be considered.

3.4

Implications for Contact Optimization

A practical photovoltaic with contacts governed by the type of physics described herein will be operating in or near the low-injection regime. We have treated Voc , but of course, device efficiency is the most important metric and depends on the ability of the cell to generate power away from open-circuit conditions. Here, the value of J0 becomes particularly important. If it is much smaller than JL (i.e., in the high-injection regime), a significant overpotential will be needed to drive the majority process at the contact, thereby potentially

21



Page 22 of 35

degrading the fill factor and efficiency of the cell. If one is working with contacts that already yield relatively high cell efficiencies, they are likely operating near or in the lowinjection regime. Optimizing such contacts then focuses on further reducing the exchange current density for the minority process as suggested by the low-injection result for Voc (see eq. 34). The development of such low-j0 contacts has long been recognized as a critical need in, for instance, silicon technology. 28 The contact or device selectivity, as defined herein, is not the limiting factor; contact recombination limits Voc . One can describe developing low-j0 contacts as improving contact selectivity because reducing j0 (without affecting J0 ) increases S, but the resulting improvement in Voc will be due solely to decreased recombination. Why then is it important to distinguish between selectivity and recombination? Is recombination really all that matters in practically relevant contacts? Perhaps it is, but during the development of contacts for emerging photovoltaics, a wide range of behaviors may be encountered, and a clear understanding of how contact kinetics affect Voc is required for rational progress. It is also possible that as one pushes the performance of a contact, in particular for concentrated light applications, it could shift from operating in the low- to high-injection regime (one such example involving the introduction of tunneling barriers is described below). The practical importance of understanding the relation between Voc and fundamental charge-transfer processes to the rational development of contacts for high efficiency photovoltaics can be illustrated by considering Schottky-type contacts. The partial currents for such contacts are given by eq. 2 with J0 values given by: φbn J0n = qνn Nc exp − VT φbp J0p = qνp Nv exp − , VT

(38a) (38b)

where ν and φb are the interfacial charge transfer velocity and barrier height with the subscript n or p to specify electron or hole processes, and Nc and Nv are the effective

22


Page 23 of 35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


conduction and valence band densities of states. Equation 38 can also be written in terms of a Richardson constants (A∗x ) and transmission coefficients (κx ): κx A∗x = qνx Nx /T 2 . The transmission coefficient for each interfacial charge-transfer process describes the extent to which the charge transfer velocity is reduced relative to the ideal metal limit, 29 which is determined by the material parameter A∗x . A wide range of contacts to semiconductors, including those based on metals, 18 conductive polymers, 24 and liquid redox couples, 19 have been described using rate eq. 2 with J0 given by eq. 38. This is particularly true when κ is reduced from the classic metal limit (κ = 1). Two ways a Schottky-type contact can be modified to improve the efficiency of the photovoltaic are: (1) introducing a tunneling barrier that reduces ν and (2) changing the work function of the contact. In the simplest case, methods to accomplish the former would reduce ν for both the electron and hole processes at a contact by the same factor. This is the same as an equivalent reduction in both J0 and j0 resulting in a change in Rm but not S. Changing only the recombination characteristics of the contact in this way will result in a Voc improvement if the contact is a significant recombination pathway and if the cell is operating in the low-injection regime. For example, metal-insulator-semiconductor structures use an insulator layer to passivate the contact to recombination. 30 There is a limit, however. Continued reduction of J0 and j0 (in relation to JL ) together will eventually result in the system shifting to the high-injection regime. At this point, the selectivity of the system could limit Voc , and further reduction of both J0 ’s together would no longer improve Voc . The reduction in J0 may also eventually introduce transport limitations that degrade the fill factor. Changing the work function of the contact will change S while having no affect on R2 , which can improve Voc in either the low- or high-injection regime. This assumes that φbn and φbp are sensitive to the work function of the contact, which is the case for an ideal interface with barrier heights not yet approaching the band gap. The R2 is not affected because Eg = φbn + φbp regardless of the work function; hence J0 j0 = q 2 νn νp Nc Nv exp (−Eg /VT ) remains unchanged. Assuming the work function change is appropriate to increase S, Voc

23



is expected to increase in both the high- and low-injection regimes. In low injection, it will improve because j0 is decreased. In high injection, it will improve because the selectivity of the system improves according to eq. 36. It is also possible that the system can move from high- to low-injection with a continued increase in S. The extent of control is limited by the requirement that φbn and φbp cannot be greater than the band gap. Ideally, one could control the barrier height and each of νn and νp independently at a Schottky-type contact. This would provide the greatest independent control over J0 and j0 . It also motivates the development of contacts with selective tunneling barriers that modify νn and νp differently.

3.5

Comparison to Device Physics Simulations

Bulk transport and recombination were neglected in our theoretical model to isolate the contact-determined Voc , but these processes can certainly be critical to the performance of a solar cell. Beyond the theoretical insight our treatment provides, we now demonstrate that there are practically relevant systems where it is quantitatively accurate. To do so, we compare our results to full device simulations using COMSOL (see Supporting Information for more details). The simulated device is based on an intrinsic silicon absorber because silicon-based systems are routinely modeled to a high level of accuracy and because the development of contacts to silicon remains a barrier to achieving ideal bulk recombination limits. 28,31,32 Standard material and carrier transport parameters are used, and direct, Auger, and Shockley-Reed-Hall (SRH) recombination are included with carrier lifetimes set to 5 ms. The silicon absorber was taken to be 200 µm thick and contacted on either side with ideal Schottky contacts. The work functions for the two contacts place their Fermi levels symmetrically about the center of the semiconductor gap, and their difference is set by the work function difference parameter ∆φ: φαbn = Eg /(2q) + ∆φ/2 = (Eg /q) − φαbp and φβbn = Eg /(2q) − ∆φ/2 = (Eg /q) − φβbp . The electron and hole transmission coefficients at the α and β contacts are all identical. As A∗n ≈ A∗p for Si, parameterization in terms of ∆φ and κ is approximately equivalent to the "two-J0 " case. The simulations were performed as 24


Page 24 of 35

0.8

(a) κ = 10-3

● ● ● ● ● ●

0.6

V oc / V

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


● ●

0.4 ● ●

0.2

● ●

● ●

● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0.0 0.0

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0.2

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0.6

● ●

●

● ●

● ● ●

●

● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0.4

●

● ●

●

● ●

● ●

●

● ●

● ● ●

● ● ●

● ● ●

0.8

0.8

2 1

(b) κ = 1

0 -1

0.6

-2 -3

V oc / V

Page 25 of 35

-4

● ●

0.4

-5

0.2

-6

-7

1.0

●

● ● ● ● ● ● ● ● ● ● ● ●

0.0 0.0

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0.2

Δϕ / V

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0.4

● ● ● ● ● ● ●

● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ●

0.6

● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ●

● ●

1 0

● ● ●

-1 -2 -3

● ● ● ●

● ●

0.8

2

-4 -5 -6 -7

1.0

Δϕ / V

Figure 5: Plots comparing simulated Voc data for a silicon-based devic (solid circles) to Voc ’s calculated using eq. 18 (solid lines) as a function of ∆φ. The black dashed line represents the boundary between high and low injection (above the dashed line is high injection). Part (a) shows results for κ = 10−3 , reducing the J0 values at both contacts from the ideal metal limit. The only situation where the theory does not match the data is when the contacts are highly selective (high ∆φ), and under impractically low light intensities, when bulk recombination starts to limit the device. Part (b) shows results for κ = 1, representing the ideal metal limit for transmission. In this case, it is possible for the mobility to limit the speed at which carriers reach the contacts and recombine, resulting in a Voc higher than the theoretical values. This effect does not occur in high injection, however, and disappears as ∆φ is increased. a function of ∆φ and κ (which combine to determine the four J0 ’s), and the illumination intensity (ΦL ). Figure 5 compares the simulation results to the theoretical Voc calculated using eqs. 18, 20, and 38. Figure 5a shows Voc as a function of ∆φ for illumination levels ranging from 10−7 to 102 Suns and with κ = 10−3 . This corresponds to an ideal metal Schottky contact modified by a thin tunneling barrier that reduces electron and hole transfer by a factor of one thousand (thereby reducing the contact recombination). The black dashed line is the dividing line between the high- and low-injection regimes (i.e. eqs. 18 and 20 evaluated with JL = Jβ0n ≈ Jα0p ). There is excellent agreement between the simulations (data points) and eq. 18 (solid lines) for most values of light intensity and 25



Page 26 of 35

∆φ in Fig. 5a. As is expected from the high-injection limit (eq. 36), above the dashed line, the simulated Voc is independent of light intensity and has a slope of one. In low injection, Voc only depends on JL and the sum of the j0 ’s, hence the slope of 1/2 and separation with light intensity seen in the data. There is some deviation from theory in Fig. 5a when ∆φ becomes sufficiently large so that the rate of interfacial recombination becomes slower than bulk recombination (i.e. precisely when our limit should not apply). This is indicated by the simulated Voc ’s saturating as ∆φ increases. Such plateaus are only observed at impractically low illumination levels for the case of κ = 10−3 . Figure 5b shows the effect of increasing κ to the metal limit of one. The faster interfacial recombination rate results in overall lower Voc values than in Fig. 5a. With metal contacts, interface recombination typically overpowers bulk recombination. Indeed, the plateau region corresponding to bulk recombination is barely noticeable in the simulation data of Fig. 5b. There is, however, more significant deviation between the theory and simulation. This is due to bulk transport limiting interfacial recombination, as has been discussed and modeled by Kirchartz et al. 15 Recombination at an interface can be viewed as a series process involving transport of carriers to the interface followed by recombination there. Sluggish bulk transport kinetics will suppress interfacial recombination processes resulting in slower effective interface recombination rates and higher possible values of Voc . Indeed, the simulated values of Voc in Fig. 5b are higher than predicted by our theory in the low-injection, interface-recombinationlimited regime. The selectivity-limited Voc remains a true upper limit in the high-injection regime. A crude check to determine when bulk carrier transport will restrict interface recombination can be made by comparing the expression for interface current (eq. 2) to the standard expression for bulk drift and diffusion current, using qVoc /l for dEF /dx, leading to:

Voc /VT < ν/(D/l),

26


(39)

Page 27 of 35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


where l is the length of the device, D is the diffusion constant and ν is the surface recombination velocity (whichever carrier has the smallest ν/(D/l) ratio here is expected to be the limiting carrier). If the inequality holds, the device’s Voc can exceed the limit set by the contact recombination. The Supporting Information contains plots of simulated values of Voc versus mobility for different ∆φ’s and κ’s, showing that the Voc ’s become mobility independent once inequality 39 no longer applies. At the larger values of ∆φ in Fig. 5b, the interface recombination decreases to a rate that can be supported by bulk transport, and the simulated results again approach the theory. The comparison between our theoretical model and more complete device simulations validates our model in practically relevant situations. This is perhaps obvious for the lowinjection regime because of the wide use of the classic eq. 1 with j0 characterizing an interfacial charge-transfer process in describing high efficiency photovoltaics. As can be seen from Fig. 5a, the high-injection regime becomes increasingly important in driving to higher Voc ’s, for instance using solar concentration. Our model clearly illustrates the complete contact-determined Voc including the transition from the low- to high-injection regime. As mentioned above, it is possible that the high-injection Voc ’s may disguise poor performance because the interfacial majority collection process could be too slow, leading to degradation of the fill factor. However, simulation results shown in Fig. S1 indicate that this may not always be the case. These simulations show the emergence of so-called S-shaped curves upon transitioning into high injection reflecting majority process limitations, but that the rollover can remain outside the power quadrant resulting in minimal degradation of the fill factor.

4

Conclusion

We provide a quantitative framework for understanding the effect of contact selectivity and contact recombination on Voc for contacts governed by eq. 2. Contact recombination effects are defined as those that reduce ∆EF . The ∆EF /q of eq. 17 is a contact-recombination-

27



determined Voc expressed in terms of the J0 values for the electron and hole processes at the contacts. In general, an experimentally measured ∆EF /q will have contributions from all recombination pathways, but it, along with Voc , cannot exceed the voltage calculated from eq. 17 under the assumptions of our model. It can be exceeded, however, when bulk transport processes restrict interface recombination. Contact selectivity effects are defined as those that reduce Voc below ∆EF /q and arise from the extent of kinetic asymmetry in the system. When Voc is less than ∆EF /q, eq. 18 provides the contact-selectivity-limited Voc . Although the relation of contact parameters to Voc can be completely described by the set of J0 values, it is conceptually valuable to define the contact selectivity and recombination parameters, S and Rm . When both electrodes are in high injection and the j0 values are not too different from one another, the Voc is selectivity limited; it is given by eq. 37, which is a function of the S for both contacts (i.e. the device selectivity). The asymmetry of the entire system, not just one of the contacts, determines the selectivity-limited Voc . The R2 value is important because it determines the contact contribution to ∆EF in high injection (see eq. 33). When both electrodes are in low injection, as is expected to be the case for many practical realizations of photovoltaics, the j0 values alone are the important contact parameters; Voc is contact recombination limited, and this limit is given by the classic eq. 1 β α . with J0 = j0p + j0n

The theory developed continuously spans two limits commonly discussed in the literature when considered in terms of Schottky contacts. The first stems from metal-insulator-metal models and states that the Voc is limited by the difference in the contact work function. 33–36 As parameterized in §3.5, ∆φ corresponds to a pure change in selectivity, i.e. moving diagonally up and to the left in Fig. 4. Such a path is along the gradient of Voc when in the highinjection regime. Further, when all the charge-transfer velocities are equal, ∆φ = VT ln S: the contact work function difference limits Voc according to eq. 30. Note this remains an upper limit regardless of bulk transport restrictions. It does not generally arise from any built-in potential but from the kinetic asymmetries introduced by ∆φ through the J0 val-

28


Page 28 of 35

Page 29 of 35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


ues. The second limit states that the Voc is recombination limited and cannot exceed the quasi-Fermi level splitting. 37,38 This is the low-injection limit herein; the exchange current densities for the minority processes limit Voc in our model rather than ∆φ. It is not always possible to assign a change in Voc due to contact modification to a change in the contact selectivity. For instance, in low injection, an improvement in Voc results from a decrease in the largest j0 . Typically, however, it is not known whether or not J0 has changed. Without such knowledge, it is impossible to say if the result was due to an increase in selectivity (at least as defined herein) or a decrease in both J0 and j0 , as would occur with the introduction of a tunneling barrier. One could choose to define any decrease in j0 as an increase in selectivity because it corresponds to slowing the kinetics for collection of the “wrong” carrier, but this is not the terminology we have adopted here. In many cases, it is perhaps preferable to simply state that the j0 at the contact is reduced, rather than state that the selectivity has improved. Regardless, the use of terms such as selectivity without definition obfuscates the underlying physics. As mentioned earlier, photovoltaics are typically operated in or near the low-injection regime. Consequently, there has been a natural emphasis on calculating j0 . The measurement 22 of how certain contact modifications affect both J0 and j0 coupled with the results herein, however, can provide useful insight that can help guide the rational development of solar cell contacts.

References (1) Green, M. A. Photovoltaic Principles. Physica E 2002, 14, 11–17. (2) Würfel, P. Physics of Solar Cells: From Basic Principles to Advanced Concepts, 2nd ed.; Physics textbook; Wiley-VCH: Weinheim, 2009. (3) Ratcliff, E. L.; Zacher, B.; Armstrong, N. R. Selective Interlayers and Contacts in Organic Photovoltaic Cells. J. Phys. Chem. Lett. 2011, 2, 1337–1350.

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Acknowledgement This work was funded by the Division of Chemical Sciences, Geosciences, and Biosciences, Office of Basic Energy Sciences of the U.S. Department of Energy through DE-SC0012363. 33



Supporting Information Available Algebraic derivations of the general results. Details on numerical simulations and additional simulation results. This material is available free of charge via the Internet at http://pubs.acs.org/.

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