Letter pubs.acs.org/JPCL
Thermodynamic Limit of Exciton Fission Solar Cell Efficiency Murad J. Y. Tayebjee,† Angus A. Gray-Weale,‡ and Timothy W. Schmidt*,† †
School of Chemistry, The University of Sydney, New South Wales 2006, Australia School of Chemistry, The University of Melbourne, Victoria, 3010, Australia
‡
ABSTRACT: A detailed balance chemical kinetic model is applied to exciton fission in photovoltaic devices. Energy conversion efficiency limits are found to be significantly higher than previously calculated. We lift unnecessary restrictions on the relative energies of the electronic states involved and show that the entropically favorable nature of the exciton fission process can be harnessed to bring about higher energy conversion efficiencies. By accounting for endothermic fission, the calculated limiting efficiency of a device is increased from 41.9 to 45.9% under standard conditions. By considering the exciton binding energy in organic cells efficiency limits of around 40% are calculated. SECTION: Energy Conversion and Storage; Energy and Charge Transport exciton fission. To explain what at first may appear to be an energy-conservation violating process, we rederive the SQ limit for a single threshold solar energy conversion device using a detailed balance model, then extend the model to include the exciton fission process. We analyze and interpret the results in terms of the entropy generation upon fission, which permits carrier energy to be drawn from a heat bath. Entropy is most easily accounted for from a statistical mechanical perspective, where an ensemble of absorbing chromophores can be assigned to “states”. To account for the entropy generation upon singlet fission, we first introduce a framework that reproduces the SQ limit for single threshold solar energy conversion devices. The single threshold solar energy conversion device is composed of an array of absorbers of area σ that may exist in the ground state |1⟩, or an excited state |2⟩. Positive work is done when an absorber makes an absorptive transition from |1⟩ to |2⟩, with rate constant k, and negative work is done when radiation is emitted, with rate constant k′ (see Figure 1). The excited absorber may also return to state |1⟩ nonradiatively, sacrificing an amount of energy Er − eV, corresponding to a device operating at a voltage, V, where e is the elementary charge. The resulting energy profile is depicted in Figure 1, which is analogous to a photon-driven ratchet.14 Work is performed following absorption of photons with rate constant k, followed by liberation of an amount of heat, Q. Each cycle corresponds to doing an amount of work, eV. The transitions of the system (motion of the pawl) across the Q = Er − eV gap are much faster than the radiative transitions. Therefore, if we define p1 and p2 to be the normalized occupancies of pawl states |1⟩ and |2⟩, then the rapid thermal equilibration ensures that
U
pon absorption of a photon by a single threshold photovoltaic device, the energy in excess of the band gap, Eg, is rapidly transferred to the lattice vibrations, creating a multitude of phonons. The entropy produced is such that this free-energy loss is irreversible and is one of the two major limitations to the energy conversion efficiency of single threshold solar cells (STSCs).1,2 This limit under the (unconcentrated) 6000 K blackbody spectrum is η ≈ 31% (the Shockley−Queisser (SQ) limit). One strategy to circumvent the SQ limit is to generate multiple excitons or carriers with each incoming photon. These may be generated within the material due to impact ionization processes in inorganic materials,3,4 singlet fission in organic materials,5−7 or multiple exciton generation in adjacent quantum dots.8 Previous models of these mechanisms predicting limiting efficiencies for such carrier multiplication (CM) devices have ignored the thermodynamic details of the process and limited thresholds for multiple exciton generation (MEG) to whole-number multiples or simple functions of the lower energy threshold (representing the band gap of the solar cell). However, since entropy may be generated upon the creation of particles (excitons), this thermodynamic driving force can be exploited to lower the threshold and bring about higher efficiencies. In essence, an endothermic exciton fission process may be rendered exergonic due to entropy generation. In previous reports, CM has been assumed to occur in an automated fashion once an absorption threshold is reached, corresponding to ≥nEg for n-tuple carrier generation.3,9−13 Under the standard air-mass 1.5 global spectrum (AM1.5G), Hanna and Nozik obtained an efficiency limit of 41.9% for n = 2.3 In their derivation, the ratio of the energies of the higher (Eb) and lower (Er) band gaps was limited to Eb/Er ≥ 2, which we will show is an unnecessary restriction. Indeed, allowing the upper threshold to drop below twice the lower threshold lifts the energy conversion efficiency limits for devices employing © XXXX American Chemical Society
Received: July 31, 2012 Accepted: August 31, 2012
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calculated under various spectra are all reproduced using the above equations, confirming the generality of this approach. STSC energy conversion efficiency limits are given in Table 1 for both the 6000 K blackbody spectrum and the AM1.5G solar spectrum. Table 1. Maximum Efficiency for a CM Device Operating Under 6000 K Black Body and AM1.5G Illumination17 a device
Figure 1. Left: The energy profile for work performed by solar absorbers analogous to the ratchet (right). Absorption of photons above threshold Er causes the ratchet to turn one cycle, moving the pawl from position |1⟩ to position |2⟩. The pawl then snaps back to position |1⟩, sacrificing Er − eV of energy. Between absorption events, the pawl is in constant and rapid thermal equilibrium between states |1⟩ and |2⟩, with photon emission completing the detailed balance.
p2 = p1 exp( −β(Er − eV ))
STSC CM
p2 = (xc/xr)/(1 + xc/xr)
(3)
STSC (0.3 eV) STSC (0.5 eV) CM (0.3 eV) CM (0.5 eV)
I(xc)/e = kp1 − k′p2 (4)
E2
, dE
k′ = kA(E1) =
σ 4π c ℏ
2 2 3
∫E
1
2 1 ⎛ σE1 ⎞ ⎜ 2 2 3⎟ 4 ⎝ βπ c ℏ ⎠
− (1.34) 1.72 (0.95)
27.6
− (1.38)
30.0
− (1.38)
22.4
− (1.53)
23.9
− (1.39)
39.9 35.4
2.13 (0.82) 2.14 (0.78)
42.8 37.8
2.11 (0.82) 2.17 (0.66)
(9)
(10)
where Ωi and Ωf are the initial and final number of microstates. From eq 1, we find
(7)
T ΔS(p) = Er − eV
where the factor of 1/4 accounts for Lambertian emission. By assuming β(E2 − E1) ≫ 1 and ignoring low-order terms and stimulated emission (since βE1 ≫ 1),2 the rate constant of spontaneous emission of the absorber is15 k′ = kA(E1) =
33.7 45.9 (41.9)
⎛p ⎞ = kB ln⎜⎜ 1 ⎟⎟ ⎝ p2 ⎠
2
E dE exp(βE) − 1
− (1.30) 1.73 (1.05)
⎛Ω ⎞ ΔS(p) = kB ln⎜ f ⎟ ⎝ Ωi ⎠
where , is the solar photon spectral density in s−1 m−2 J−1 as a function of photon energy, E, and E1 and E2 define the portion of the solar spectrum to be absorbed. The absorption crosssection, σ, is equal the cross section of the absorber over the spectral region of interest − equivalent to assuming an absorptivity of 1. Now, from detailed balance E2
31.1 42.6 (39.6)
where N is the (large) number of absorbers and p1 and p2 are the normalized occupancies of states |1⟩ and |2⟩, as above. The components, {pn}, of p are functions of the device voltage, V, and the lower energy bounds of the states, as given in eqs 2 and 3. Upon photon absorption, the entropy produced in the limit where N → ∞ is
(6)
1
Eb/Er (Er)
⎛ N ⎞ ⎛ N ⎞ ⎟=⎜ ⎟ Ω(p) = ⎜ ⎝ p1 N ⎠ ⎝ p2 N ⎠
It remains to define expressions for the rates of absorption and emission. The rate of photon absorption is
∫E
η (%)
The utility of the statistical mechanical approach becomes clear when we consider the entropy generation upon photon absorption, |1⟩→|2⟩. The number of microstates, Ω, is given by
Dividing by the area of absorbers in state |1⟩, p1σ, gives the current density of the device e J(xc) = (k − k′xc/xr) (5) σ
k = k S(E1 , E2) = σ
Eb/Er (Er)
a Efficiency limit values in parentheses denote the cases where the restriction Eb/Er = 2.0 has been applied. The corresponding value of Er is shown in parentheses next to the ratio Eb/Er.
where xc = exp (βeV) and xr = exp (βEr). The current, I, through the single threshold device is given by
= (k − k′xc/xr)/(1 + xc/xr)
η (%)
excitonic (−ΔECT)
where β = 1/kBT. Boltzmann’s constant and the device temperature are, respectively, kB and T. Unless otherwise stated, the device operates at 300 K. Ensuring that p1 + p2 = 1 (2)
AM1.5G
ideal
(1)
p1 = 1/(1 + xc/xr)
6000 K blackbody
ΔF = ΔE − T ΔS = eV
(11)
and thus the free energy change, ΔF, of one turn of the ratchet, corresponding to absorption of a photon of energy E > Er, is eV, as expected. Having shown the efficacy of the above approach, we now analyze a CM device using a detailed balance model that links three absorber states, |1⟩, |2⟩, and |3⟩, with respective energies 0, Er, and Eb. The device consists of an ensemble of N absorbers, wherein absorption of photons of energy Er < E < Eb brings about transitions from |1⟩ to |2⟩ and photons of energy
(8)
The maximum rate of work is calculated by maximizing the quantity VJ as a function of the band gap, Er = E1, and operating voltage, V. The limiting efficiency results that Shockley and Queisser;1 Hanna and Nozik;3 and Brown and Green16 2750
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E > Eb effect the transition |1⟩ → |3⟩. As such, |2⟩ and |3⟩ are radiatively coupled to |1⟩ but not to each other. Yet absorbers rapidly react as |1⟩|3⟩ ⇋ |2⟩|2⟩, mimicking the singlet fission or triplet−triplet annihilation processes of organic molecules. As such, absorption to |3⟩ can bring about two absorbers in |2⟩. Equilibration between levels |2⟩ and |1⟩ in a nonradiative manner is fast, as in the single threshold device analyzed above. This yields normalized occupancies of the three states of p1 = 1/Z, p2 = xc/(xrZ), and p3 = x2c /(xbZ), where the partition function of the system is Z = (1 + xc/xr + x2c /xb), and xb = exp (βEb). The absorption rates due to radiation are kr = kS(Er,Eb), for |1⟩ → |2⟩, and kB = kS(Eb,∞), for |1⟩ → |3⟩. The corresponding radiative recombination rate constants are k′r = kA(Er) and k′b = kA(Eb). The current density is e J(xc) = (k r − k r′xc/xr + 2k b − 2k b′xc2/xb) (12) σ
Figure 3. Limiting energy conversion efficiencies of the model device operating at 300 K under 6000 K blackbody illumination as a function of band gap and the ratio of the energies of the upper and lower thresholds. The white line shows the work that was previously conducted by Hanna and Nozik3 (although these authors used the AM1.5G spectrum).
where σ is the absorption cross section. The maximum energy conversion efficiency of the system is η = (VJ )max /
∫0
∞
E , dE
(13)
Table 1, along with calculations performed on the single threshold device. By lowering the ratio Eb/Er from a value of 2, the exciton multiplication process becomes endothermic (ΔEfission = 2Er − Eb > 0). However, the overall driving force is the free energy change, ΔFfission = ΔEfission − TΔSfission, which also accounts for entropy generated in the process, ΔSfission. Endothermic exciton fission has been experimentally observed in crystalline anthracene, where ΔEfission ≈ 0.5 eV.18 Nevertheless, at low temperatures the quantity TΔS is diminished, and the value of Eb/Er must be increased to exploit CM. This is illustrated in Figure 4, where the optimized ratio Eb/Er (solid
where (VJ)max is the maximum power per unit area of absorber. This equation for the current density as a function of the operating voltage V is identical to that of the equivalent circuit of this device. Here two photocells with band gaps of Eb (“blue”) and Er (“red”) are connected in parallel across a load. An efficient Buck convertor (DC transformer) obeying Kirchoff’s law is placed such that the operating voltage of the blue cell is double that of the red cell. Geometrically, the “blue” photocell is placed atop the “red” photocell so that it absorbs all photons with E > Eb, leaving that part of the spectrum with Er < E < Eb to irradiate the “red” photocell. The device is illustrated in Figure 2.
Figure 2. Equivalent circuit of an exciton fission solar cell. The element labeled ‘BC’ is a Buck convertor, a 100% efficient directcurrent step-down transformer. Figure 4. Optimized ratio Eb/Er (solid line, left axis) at the maximum efficiency (dashed line, right axis) of a CM device where Er = 1 eV as a function of operating temperature.
Figure 3 shows the limiting energy conversion efficiencies of the CM device. The ratio of the upper to lower energy threshold, Eb/Er, is varied from 1.2 to 2.8. Importantly, the maximum energy conversion efficiency is found when Eb < 2Er. Moreover, the limit is substantially higher than what was previously determined; under a 6000 K blackbody spectrum, we determine a limit of 42.6% at Eb/Er = 1.73. Under an AM1.5G spectrum, the upper limit derived by Hanna and Nozik, 41.9%, at Eb/Er = 2, is raised to 45.9% at Eb/Er = 1.72 (plot not shown). The results obtained with both the 6000 K blackbody spectrum and the AM1.5G solar spectrum17 are summarized in
line, left axis) is plotted as a function of operating temperature for a CM device with Er = 1 eV. As T → 0, the optimal value of Eb/Er → 2. At higher temperatures, however, Eb/Er < 2; as such, ΔEfission > 0, meaning that the fission process is endothermic: the excess energy required for the entropically favorable reaction is taken from the thermal bath. This thermal contribution is a diminishing proportion of the exciton energy 2751
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Figure 5. Current density (mA cm−2) generated by (a) the lower threshold absorption and (b) the upper absorption threshold as a function of Eb/Er and cell voltage. (c) Total current density as a function of Eb/Er and cell voltage. (d) Power of the cell as a function of Eb/Er and cell voltage.
as Er is increased. Thus, there is a slight trend toward higher optimum Eb/Er values as Er is increased (Figure 3). The advantage of the statistical model is that the entropy change for the CM process may be readily computed. Considering the system of N chromophores and the vector p, which contains the normalized state occupancies p1, p2, and p3, respectively, for |1⟩, |2⟩, and |3⟩, the number of microstates is given by ⎛ N ⎞⎛ N − p1 N ⎞ ⎟ ⎟⎜⎜ Ω(p) = ⎜ ⎟ ⎝ p1 N ⎠⎝ p2 N ⎠
⎛ 1 x 2 Z2x 2 ⎞ c r ⎟ ΔSfission(p) = kB ln⎜ 2 ⎝ Z xbZ xc ⎠ = (2Er − E b)/T
As expected then ΔFfission = ΔEfission − T ΔSfission = (2Er − E b) − T (2Er − E b)/T = 0
(17)
The question remains, what is to stop minimization of Eb if fission is always free-energy neutral given rapid equilibrium of the states through nonradiative pathways? What must be analyzed is the contribution from each energy threshold. The current density given by eq 12 can be rewritten.
(14)
The components of p are functions of the device voltage, V, and the lower energy bounds of the states, as given above. Upon exciton fission, the entropy produced in the limit where N→∞ is ⎛pp ⎞ ⎛ Ωf ⎞ ΔSfission(p) = kB ln⎜ ⎟ = kB ln⎜⎜ 1 23 ⎟⎟ ⎝ Ωi ⎠ ⎝ p2 ⎠
(16)
J(xc) = Jr + Jb
(18)
2k′bx2c /xb).
where Jr = e/σ (kr − k′r xc/xr) and Jb = e/σ(2kb − Figure 5a shows how Jr varies with the ratio Eb/Er and the cell voltage V for Er = 1 eV. Horizontal cross sections of this plot resemble I−V characteristics for the lower threshold. As Eb/Er is increased, the lower threshold harvests more of the solar spectrum, and J increases from
