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Fundamental Limitations to Plasmonic Hot-Carrier Solar Cells Yu Zhang, ChiYung Yam, and George C. Schatz J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.6b00879 • Publication Date (Web): 02 May 2016 Downloaded from http://pubs.acs.org on May 3, 2016

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The Journal of Physical Chemistry Letters

Fundamental Limitations to Plasmonic Hot-Carrier Solar Cells Yu Zhang,∗,† ChiYung Yam,‡ and George C. Schatz∗,† †Department of Chemistry, Northwestern University, Evanston, Illinois, 60208, USA ‡Beijing Computational Science Research Center, Haidian District, Beijing 100193, China E-mail: [email protected]; [email protected]

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Abstract Detailed balance between photon-absorption and energy loss constrains the efficiency of conventional solar cells to the Shockley-Queisser limit. However, if solar illumination can be absorbed over a wide spectrum by plasmonic structures, and the generated hot-carriers can be collected before relaxation, the efficiency of solar cells may be greatly improved. In this work, we explore the opportunities and limitations for making plasmonic solar cells, here considering a design for hot-carrier solar cells in which a conventional semiconductor heterojunction is attached to a plasmonic medium such as arrays of gold nanoparticles. The underlying mechanisms and fundamental limitations of this cell are studied using a non-equilibrium Green’s function method, and the numerical results indicate that this cell can significantly improve the absorption of solar radiation without reducing open-circuit voltage, as photons can be absorbed to produce mobile carriers in the semiconductor as long as they have energy larger than the Schottky barrier rather than above the bandgap. However, a significant fraction of the hot-carriers have energies below the Schottky barrier, which makes the cell suffer low internal quantum efficiency. Moreover, quantum efficiency is also limited by hotcarrier relaxation and metal-semiconductor coupling. The connection of these results to recent experiments is described, showing why plasmonic solar cells can have less than 1% efficiency.

Graphical TOC Entry


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Surface plasmon polaritons (SPPs) are collective excitations of electrons driven by light at the interface between metal and dielectric materials. The quantum-mechanical nature of SPPs is important in the scattering and decay processes, 1–3 with an important example being hot-carrier generation from the SPP (surface plasmon polariton) decay. SPPs can decay either radiatively via emission of photons or non-radiatively through electron-phonon interactions or by the generation of excited carriers, i.e., hot-carriers. Hot-carrier generation often dominates the energy dissipation of SPPs, 4 and this limits the application of plasmonics due to the short plasmon lifetime. However, the generated hot-carriers are found to have applications in several branches of applied physics, chemistry and energy science. 5–16 Applications to energy conversion

17,18

have been of particular interest, and indeed many

recent studies have tried to extract hot-carriers for photocurrent generation. As is well known, the energy conversion efficiency of semiconductor heterojunction based photovoltaics is limited by the Shockley-Queisser (SQ) limit under unconcentrated light illumination. 19 SQ concerns two major energy loss mechanisms: (1) the bandgap inherent in the semiconductor prevents photons with energy smaller than the gap from being absorbed; (2) photons with energy above the bandgap contribute to energy conversion but the generated electrons and holes quickly relax to the band edge by phonon scattering. These two intrinsic energy loss mechanisms result in an upper-bound to single-junction solar cell efficiencies of 31%. 20 In order to break the SQ limit, many different solar cell designs have been proposed, including plasmonic solar cells, 21 multi-junction solar cells, 22 multi-exciton generation, 23–26 etc. Despite the different designs, the basic idea in all cases is to improve light absorption and reduce energy dissipation. In recent years, an emerging solar-cell design utilizing hot-carriers has been proposed, i.e., the hot-carrier solar cell (HCSC), 27–30 which collects hot-carriers generated from electron-hole pair excitation in metals as a source of photocurrent. It should be noted that the concept of plasmonic hot-carrier solar cells considered in this paper is different from that of traditional (nonplasmonic) hot-carrier solar cells in which photogenerated hot carriers with energy greater than bandgap in a semiconductor are separated for energy



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conversion before cooling. In order to use these hot-carriers for photocurrent generation, detailed knowledge of carrier generation and extraction should be known. Even though previous quantum mechanical calculations have provided an understanding of the underlying mechanism of carrier generation from plasmon decay, 31–36 the transport of hot-carriers to collection surfaces,leading to photocurrent generation and energy conversion efficiency under operating conditions hasn’t been characterized. In this letter, a new HCSC design is proposed. The detailed mechanism of this cell is studied and fundamental limitations of HCSC performance are explored by using a non-equilibrium theory. From this, the opportunities and pitfalls associated with plasmonic solar cells are determined, including an analysis of recent experiments on related structures.

Figure 1: Schematic diagram of HCSC. Upper panel: geometry of HCSC. Arrays of spherical nanoparticles are deposited on the top of pn junction. Left bottom panel: Schematic of hotcarrier generation and injection to a semiconductor; Right bottom panel: A two-level system is used to model the conduction and valence band of semiconductor. Hot-carriers transfer to the semiconductor after they are generated from plasmon decay. The collected hot-carriers in the semiconductor are driven in different directions by the built-in potential, resulting in photocurrent generation. For simplicity, the semiconductor is modeled by a two-level system (TLS). As shown by Fig. 1, the lower and upper bounds of the TLS are assumed to couple more strongly to the left and right electrodes, respectively. In a conventional solar cell, light illumination excites electrons across the band gap, followed by charge transfer from the conduction band to right electrode and hole transfer from the valence band to the left electrode as driven by the built4 ACS Paragon Plus Environment

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in potential. Consequently, net photocurrent is generated from the right to left electrode against the chemical potential gradient. However, a different solar cell design, HCSC, is proposed in this letter. The HCSC is different from a conventional solar cell in that a metal structure (here considered to be arrays of spherical nanoparticles), attached to the depletion region of the pn junction, is used to absorb the photons. In principle, light illumination will also directly excite carriers within the pn junction, which provides a source of photocurrent. But in this work, carrier generation from semiconductor excitation is neglected since we only want to investigate the efficiency of photocurrent induced by plasmonic hot-carriers. The working principle of plasmon to electron conversion is illustrated in Fig. 1. The metal forms a Schottky contact with the semiconductor. Light illumination on the metal nanostructure excites the plasmon. The plasmon can lose energy via both radiative and non-radiative decay, and one non-radiative pathway involves non-Fermi distributed electron-hole pairs, i.e., hot-carriers. The hot electrons and holes can then transfer to the conduction and valence bands of the heterojunction, respectively, where the photocurrent is driven by the built-in voltage. Recent experiments have demonstrated a similar device in which plasmon to electron conversion and amplification is observed. 37 However, the experiment in Ref. 37 employed a pure semiconductor rather than a pn junction, which can only observe current amplification. In our design, the pn junction can separate electrons and holes and then convert plasmon excitation to photocurrent. Moreover, in our model, both hot electrons and hot holes can be injected into the semiconductor from the metal nanoparticle. Indeed, if only electrons (or holes) can transfer to the semiconductor, the metal nanoparticle will be continuously discharged (or charged) since the metal nanoparticle is not connected to external electrode, which is unphysical. Experiments on similar devices have not observed depletion of electrons with light illumination for a long times, 37 so there must be both hole and electron transport in order to maintain neutrality of the metal particle. The system we consider can be described by a Hamiltonian of the form: H = Hs + P

α=L,R

[Hα + Hsα ] + Hm + Hsm + Hp + Hm−p . Different from a conventional solar cell, a



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metallic plasmonic medium is introduced and the light-matter interaction in the system is P † replaced by the electron-plasmon interaction in the metal. Here Hs = µ µ dµ dµ is the Hamiltonian for the TLS. The energies of the two levels by 1 = −∆/2 and 2 = ∆/2 P † with ∆ being the gap. Hα = kα kα ckα ckα represents the Hamiltonian of electrode α. P Hsα = kα µ [Vkα µ c†kα dµ + H.c.] denotes the coupling between semiconductor and electrode α. P P Hm = km km c†km ckm and Hsm = km µ [Vkm µ c†km dµ + H.c.] are the Hamiltonian of metal and semiconductor-metal coupling, respectively. Due to the barrier at the metal-semiconductor interface, the metal-semiconductor coupling strength is assumed to be energy-dependent, Vkm i = Vms if |km | > |Φe/h | and Vms e−(|km |−|Φe/h |)/kb T otherwise, where Φe/h is the Schottky barrier for the electron/hole. The plasmon Hamiltonian is Hp = (~ω +1/2)b† b. The electronh i P † 0 0 0 = M c c b + H.c. with Mkm km plasmon interaction is taken to be Hm−p = km km 0 km km km km 0 (r)i being the electron-plasmon coupling strength, where V (r, ω) is the hφkm (r)|V (r, ω)|φkm

total potential. Upon illuminating the system with x-polarized light, Vext (r, ω) = −Ex, the external field results in the excitation of a plasmon with an induced electric potential (for a spherical particle in the quasistatic limit) via Vp (r, ω) =

(ω)−1 Ex. (ω)+2

Thus, the total potential

becomes V (r, ω) = Vext (r, ω) + Vp (r, ω). (ω) is the dielectric function of the metal, which is described by the improved Drude-like model in this work, 38 (ω) = b −

2 ωpl ω(ω+iγ)

+ i∆(ω)

with the background dielectric taken to be a constant b , the plasmon frequency ωpl and a plasmon damping factor γ. ∆(ω) =

A 1+exp[−(ω−ωc )/∆]

is used to account for the frequency-

dependent contribution of the interband transitions in gold. 38 In second quantization, the 1/2 electric field is E = η0 ~ω F(ω) , where F(ω) is incident photon flux. The photon flux is N P ~ω

for monochromatic light illumination or ζω 2 N (ω) for solar radiation (modeled as blackR body radiation) with ζ being the normalization factor making F(ω)ωdω = P, where P is the incident power which is fixed at 1 in this letter. η0 is introduced to control the strength 0 = hφk (r)|x|φk 0 (r)i = d of the electric field. And the dipole matrix is introduced as dkm ,km m m 0 are above and below the Fermi energy, respectively. Thus η0 d determines when km and km

the electron-plasmon coupling strength.


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The hot-carrier generation, collection and photocurrent of the hot-carrier solar cells are calculated within the non-equilibrium Green’s function (NEGF) formalism 39–41 (See Supporting Information for detailed derivation). Within the NEGF formalism, the three steps (hot-carrier generation, transport to the metal-semiconductor interface, and hot-carrier injection and separation by semiconductor) involved in HCSC are naturally considered on an equal footing. Conventional semiclassical models usually assume a probability function for hot-carrier collection. 42,43 Scattering of hot-carriers back into the metal is not considered in the conventional semiclassical model, but it is taken into account in the present model. Moreover, previous studies of hot-carrier collection have usually employed Fowler theory. However, Fowler theory does not consider the metal-semiconductor coupling. In contrast, all the detailed processes are taken into account in the present model on an equal footing. The processes of light absorption and excitation of SPPs are quantum-mechanical phenomena in this picture, as described by the NEGF formalism. Numerical simulation indicates that pinning the Fermi level of metal at the midpoint within the gap of semiconductor gives best performance. Hence, the Fermi energies of the semiconductor and metal are set as 0 in the following simulations. To mimic the effect of the built-in voltage, the upper (lower) band is assumed coupled to the left (right) electrode weakly at low forward bias. Consequently, the line-width function is set as Γ0L,11 = Γ0R,22 = 1.0 eV and Γ0R,11 = ζ1 Γ0L,11 ; Γ0L,22 = ζ2 Γ0R,22 , where ζ1/2 = min{1, exp[−|1/2 − µL/R |/kT ]} and µR = −µL = µ0 /2 with µ0 being the bias voltage. Note that as result of these choices, ζ1/2 varies from 0 to 1 as the voltage µL/R varies from 0 to 1/2 . Unless specified otherwise, the metal-semiconductor coupling strength is set as Vms = 1 eV and the phonon-induced decay rate is set as Γphonon = 10 meV. The electronic structure of the metal is modeled by 200 states which are uniformly distributed from −4 eV to 4 eV. Parameters of the Drude-like model for the dielectric function of gold metal are b = 9.84, ωpl = 9.01 eV, γ = 0.072 eV, A = 5.6 eV, ωc = 2.4 eV and ∆ = 0.17 eV, which leads to a plasmon resonant frequency for a spherical particle ωp = 2.49 eV. 38 The electron-plasmon coupling strength is η0 d = 0.02 eV.



300

with nanoparticle w/o nanoparticle

250 Current (uA)

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200 150 100 50 1.8

2.0

2.2 2.4 2.6 Photon energy (eV)

2.8

3.0

Figure 2: Amplification of current in a single semiconductor when it collects plasmoninduced hot-carriers from metal nanoparticle. As a preliminary application, we apply the theory to study the plasmon to electron conversion and amplification demonstrated in Ref. 37 to demonstrate the validity of our model. It order to observe the plasmon induced amplification, the conductivity of the semiconductor should be low. Hence, in this case we choose Γ0 = 0.1 eV and η0 d = 0.1 eV. ζ1/2 is set to be 1 since there is no built-in voltage in a single semiconductor. The bandgap is taken to be ∆ = 3.3 eV. Fig. 2 shows the photoresponse of the device at a bias voltage 4.0 eV. As shown by the figure, when the intrinsic conductivity of the semiconductor is relatively low, plasmon to electron conversion can greatly improve the conductivity. The amplification of current is attributed to the injection of plasmon-induced hot-carriers from the metal nanoparticle. This numerical result is consistent with a recent experimental measurement. 37 It should be noted that, in realistic devices, the magnitude of the amplification effect is determined by the difference between rate of plasmon-induced hot-carrier injection and intrinsic conductivity. The amplification is only notable when the intrinsic conductivity is relatively low and plasmon-induced hot-carrier injection rate is relatively high. In the next application, we employ the model to study the performance of HCSC. The bandgap is taken to be ∆ = 2 eV. The DOS of the system is shown in the left panel of Fig. 3(a). The photocurrent versus photon energy is shown Fig. 3(b). It is well known that the photocurrent of a conventional solar cell is only nonzero with a photon energy larger than the bandgap. In contrast, the HCSC can generate current even by absorbing photons with energy lower than the bandgap, which is shown by Fig. 3(b). In principle, a photon with arbitrary energy can excite electrons in the metal, but electrons excited by photons 8 ACS Paragon Plus Environment

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(a) Photocurrent (uA)

DOS

8 6 4 2 0 -2.0 -1.5 -1.0 -0.5 0.0

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(c) Dark current Light current Power

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Voltage (V)

Figure 3: (a) DOS of the system, the gap is exactly 2 eV. (b) photocurrent of the HCSC. A SPP induced resonant absorption is observed at ωp , at which point a pronounced photocurrent is generated; (c) Current-voltage characteristics of HCSC. (d) IQEs of HCSC: the IQE of HCSC is much smaller than that of a conventional solar cell. Parameters: 1 = −1 eV; 2 = 1 eV; with energies smaller than the Schottky barrier are unable to transfer to the semiconductor efficiently. Consequently, photocurrent is significant only when the incident photon energy is larger than the Schottky barrier, which is set as

∆ 2

in the present model. Although photons

with energy smaller than the Schottky barrier cannot contribute to the photocurrent, the HCSC is able to make better use of solar radiation compared to a conventional solar cell. This is also evident in Fig. 3(b), where we see that the photocurrent spectrum is much broader than that of a conventional solar cell. Current-voltage characteristics under solar radiation are shown in Fig. 3(c). The open-circuit voltage is found to be smaller than the bandgap because of thermal excitation at finite temperature and leakage current arising when the bias voltage approaches the bandgap. Fig. 3(c) also shows that the open-circuit (OC) voltage of the HCSC is close to bandgap, which indicates that the HCSC is able to make better use of solar radiation without reducing the open-circuit voltage. Moreover, the OC voltage is much larger than that of a metal-semiconductor junction in which the OC voltage is close to the Schottky barrier. 42 At first glance, the advantages of the HCSC seem to be overwhelming compared to the conventional counterpart. But is the quantum efficiency of HCSC is good enough?



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Here the internal quantum efficiency (IQE) of the HCSC is examined with comparison to a conventional solar cell. The IQE of the HCSC is shown in Fig. 3(d). As is well known, conventional organic or inorganic SCs can have an IQE as high as 80% ∼ 100%. However, the HCSC IQE is found to be less than 50%, much smaller than that of a conventional solar cell for photon energies just slightly larger than the Schottky barrier. This calculated IQE is consistent with a recent experimental measurement on Au-Nanorods/TiO2 electrode in the visible regime (Fig. 2(d) from Ref. 44). Here IQE values less than 15% were commonly found, and always less than 60% over the solar spectrum. The device in this experiment is different from our design (no pn junction), but the working principle of exciting hot-carriers to a semiconductor is similar. Thus the consistency in IQE values indicates the accuracy of our model in predicting the quantum efficiency. Based on the calculated IQE, the energy conversion efficiency (ECE) can also be estimated. Suppose the absorptivity of photons is ξ, then the ECE is determined by ECE = R

F (ω)IQE(ω)dω×ξV oc FF R , F (ω)ωdω

where Voc and FF are the open-circuit voltage and fill factor, respec-

tively. Substituting our calculated IQE(ω) into the above equation and considering the fact that for F F < 100%, the resulting ECE is less than 8.06 × ξVoc %. Using the experiment in Ref. 44 as an example, then the absorptivity is approximately 30% and Voc ' 0.3 V, then the corresponding ECE is less than 1%. Hence, the numerical results indicate that the bottleneck for a hot-carrier solar cell in energy conversion originates from the low IQE. This also shows why the efficiency of plasmonic hot-carrier solar cells is generally expected to be below a few percent In order to figure out why the IQE of HCSC is low, we have analyzed the initial distribution of hot-carriers generated from plasmon decay. Fig. 4 shows that when photons of energy ~ω are incident to the metal/semiconductor interface, hot-carriers with energy Ef < E < (Ef + ~ω) for electrons and (Ef − ~ω) < E < Ef for holes are produced in the metal. When ~ω is comparable with or larger than the Schottky barrier, hot electrons and holes can transfer across the interface and inject to the semiconductor.


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(a) ω=3.5 eV

Hot carriers

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(b) ω=ωp

Hot electron Hot hole

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-1

0

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Figure 4: Initial hot-carrier distribution as function of the incident photon energy. When the photon energy is small, the majority of hot-carriers are below the Schottky barrier and do not contribute to photocurrent generation. The metal-semiconductor coupling strength is set to be small (Vms = 1 meV) to avoid the influence of the semiconductor on the electronic states of the metal. When the photon energy is smaller than the Schottky barrier (ω

Fundamental Limitations to Plasmonic Hot-Carrier ... - ACS Publications

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