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ENERGY TRANSFER BETWEEN QUANTUM DOTS OF DIFFERENT SIZES FOR QUANTUM DOT SOLAR CELLS Timothy P. Holme", Cheng Chieh Chao", Fritz B. Prinz 1 ,2 1 Mechanical Engineering Department, Stanford University, 440 Escondido Mall, Stanford, CA 94305 USA 2Materials Science and Engineering Department, Stanford University, 496 Lomita Mall, Stanford, CA 94305 USA DETAILS OF CALCULATIONS excitons, this electric potential should be at least on the order of exciton binding energies, 50-200meV [7]. We present calculations showing a means to create built-in potentials on that order, as well as experiments designed to test the effect. Exciton splitting Exciton transfer Quantum simulations using VASP [10] were performed to solve the Schrodinqer equation in density functional theory using a plane-wave basis set expansion for charge density. Projector augmented wave pseudopotentials [11] were employed with exchange and correlation functional Ceperley-Alder as parameterized by Perdew and Zunger with relativistic corrections [12]. The calculation supercell contains one or two quantum dots with >10 A of vacuum on all sides to minimize fictitious interactions introduced by periodic boundary conditions. Only the r k-point was used due to the system aperiodicity. Fig. 1: Models of energy transfer in a graded bandgap solar cell: (a) exciton transfer (b) hot electron transfer and hole collection, creating a larger voltage. Step 1 is carrier excitation by light, steps 2 and 3 are carrier transport. Since quantum confinement increases the bandgap of the confined material, light absorbing active regions of the QDSC should be metals or low bandgap semiconductors in bulk so that the confined material has a bandgap in the range of 1-1.6 eV for optimal light absorption. For this study, we chose the model systems of Au and PbS as representative, well studied examples of a metal and low bandgap semiconductor respectively. That QDs of Au have a bandgap is well characterized experimentally [8] [9]. Due to short lifetimes of multiexcitons in QDs on the order of 10- 1°_10-11 s [6], the time scale of exciton separation must be on that order to collect a significant fraction of the available energy. To efficiently split ABSTRACT INTRODUCTION Quantum dot solar cells (QDSCs) offer several routes to increased solar cell efficiency, including the ability to tune light absorption, the prospect to collect hot carriers through efficient transport through "minibands" [1], and the possibility of multiple exciton generation [2]. Since the bandgap of a quantum dot (QD) changes with the size of the QD, one means to match the absorption of a solar cell to the solar spectrum is by including QDs of different size. Several recent reports observe that excitons in QDSCs with different size QDs transfer from the large bandgap (small QD) to the small bandgap (large QD), losing energy in the process.[3,4,5] We investigate, through quantum simulations and experiment, the possibility of using different sized QDs to split excitons rather than transfer excitons, as illustrated in Fig. 1. If possible, this would allow hot carrier collection, increasing the efficiency of the cell. Exciton recombination and slow charge carrier transport, major limitations of advanced photovoltaic cells, may be mitigated by designing cells with strong electric fields in the active regions. This may be done by combining quantum dots (QDs) of different Fermi levels in close proximity. While previous reports of quantum dot solar cells utilizing QDs of different sizes indicate that electrons and holes are transferred together from large bandgap QDs to small bandgap quantum dots, lowering the efficiency of the solar cell, we report a mechanism that may be able to use different bandgap QDs to split excitons and drive charge carrier transport, increasing the efficiency of solar cells. Quantum simulations of band structures of QDs show indications of this behavior, and experiments on solar cells with quantum dots of different sizes separated by thin insulating layers show improved photocurrent compared to solar cells with QDs of the same size. 978-1-4244-2950-9/09/$25.00 ©2009 IEEE 000085

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ENERGY TRANSFER BETWEEN QUANTUM DOTS OF DIFFERENT SIZES FORQUANTUM DOT SOLAR CELLS

Timothy P. Holme", Cheng Chieh Chao", Fritz B. Prinz1,2

1Mechanical Engineering Department, Stanford University, 440 Escondido Mall, Stanford, CA 94305 USA

2Materials Science and Engineering Department, Stanford University, 496 Lomita Mall, Stanford, CA 94305 USA

DETAILS OF CALCULATIONS

excitons, this electric potential should be at least on theorder of exciton binding energies, 50-200meV [7]. Wepresent calculations showing a means to create built-inpotentials on that order, as well as experiments designedto test the effect.

Exciton splittingExciton transfer

Quantum simulations using VASP [10] wereperformed to solve the Schrodinqer equation in densityfunctional theory using a plane-wave basis set expansionfor charge density. Projector augmented wavepseudopotentials [11] were employed with exchange andcorrelation functional Ceperley-Alder as parameterized byPerdew and Zunger with relativistic corrections [12]. Thecalculation supercell contains one or two quantum dotswith >10 A of vacuum on all sides to minimize fictitiousinteractions introduced by periodic boundary conditions.Only the r k-point was used due to the systemaperiodicity.

Fig. 1: Models of energy transfer in a graded bandgapsolar cell: (a) exciton transfer (b) hot electron transfer andhole collection, creating a larger voltage. Step 1 is carrierexcitation by light, steps 2 and 3 are carrier transport.

Since quantum confinement increases thebandgap of the confined material, light absorbing activeregions of the QDSC should be metals or low bandgapsemiconductors in bulk so that the confined material has abandgap in the range of 1-1.6 eV for optimal lightabsorption. For this study, we chose the model systems ofAu and PbS as representative, well studied examples of ametal and low bandgap semiconductor respectively. ThatQDs of Au have a bandgap is well characterizedexperimentally [8] [9].

Due to short lifetimes of multiexcitons in QDs on theorder of 10-1°_10-11 s [6], the time scale of excitonseparation must be on that order to collect a significantfraction of the available energy. To efficiently split

ABSTRACT

INTRODUCTION

Quantum dot solar cells (QDSCs) offer several routesto increased solar cell efficiency, including the ability totune light absorption, the prospect to collect hot carriersthrough efficient transport through "minibands" [1], and thepossibility of multiple exciton generation [2]. Since thebandgap of a quantum dot (QD) changes with the size ofthe QD, one means to match the absorption of a solar cellto the solar spectrum is by including QDs of different size.Several recent reports observe that excitons in QDSCswith different size QDs transfer from the large bandgap(small QD) to the small bandgap (large QD), losing energyin the process.[3,4,5] We investigate, through quantumsimulations and experiment, the possibility of usingdifferent sized QDs to split excitons rather than transferexcitons, as illustrated in Fig. 1. If possible, this wouldallow hot carrier collection, increasing the efficiency of thecell.

Exciton recombination and slow charge carrier transport,major limitations of advanced photovoltaic cells, may bemitigated by designing cells with strong electric fields inthe active regions. This may be done by combiningquantum dots (QDs) of different Fermi levels in closeproximity. While previous reports of quantum dot solarcells utilizing QDs of different sizes indicate that electronsand holes are transferred together from large bandgapQDs to small bandgap quantum dots, lowering theefficiency of the solar cell, we report a mechanism thatmay be able to use different bandgap QDs to split excitonsand drive charge carrier transport, increasing the efficiencyof solar cells. Quantum simulations of band structures ofQDs show indications of this behavior, and experiments onsolar cells with quantum dots of different sizes separatedby thin insulating layers show improved photocurrentcompared to solar cells with QDs of the same size.

978-1-4244-2950-9/09/$25.00 ©2009 IEEE 000085

the field between ODs may be on the order of excitonbinding energies, but the effect of the insulator betweenODs plays an important role.

Density functional theory (DFT) relies on thevariational principle to calculate ground state electrondensity, and DFT calculations of excited states such as thelowest unoccupied molecular orbital (LUMO) are suspectsince they are not necessarily orthogonal to the groundstate [13]. While high quantitative accuracy is not oftenavailable from DFT simulations of excited states, manymethods exist to calculate excited states with at leastsemi-quantitative accuracy [14]. Recognizing thelimitations of DFT for excited state calculations, we alsoperformed calculations of band structure using themethods: Hartree Fock, configuration interaction (singles),and a density functional theory hybrid 83LYP asimplemented in Gaussian '03 [15]. All calculationsqualitatively agreed in all major conclusions presentedbelow.

RESULTS

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A simple model for how equilibration adjusts thedensity of states of each OD shows that, depending on thesizes of the two ODs, the bands may bend enough toallow preferential tunneling of an excited electron towardsthe larger OD and the hole towards the smaller OD. Fig. 4shows the change in density of states of two PbS ODsafter charge transfer, demonstrating that in the equilibratedstructure, an exciton generated in the smaller OD will have

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(b) charge transfer [e)

Fig. 2: Charge polarization resulting from Fermi levelequilibration of two PbS ODs as a function of (a) fermilevel mismatch and (b) diameter difference.

PbS ODs show similar trends to Au, with a chargetransfer that results from Fermi level equilibration, thetrends are shown in Fig. 2. After equilibration, a built-involtage exists between the ODs, as shown by theelectrostatic potential (Fig. 3). The electron electrostaticpotential tilts down towards the smaller OD, as would beexpected from charge transfer from the smaller dot to thelarger dot.

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For the AU13sAu43 system with separation of 5.3A, thetransfer is 0.67e. For the system of AU177Au43 with 8.oAseparation, the transfer is 0.95e. From Coulomb effectswith vacuum between the ODs, the resulting potential inboth cases is -1.7V. Assuming a low-k dielectric insulatorwith &=15 is placed between the ODs without affecting thecharge polarization, the potential is >100mV. Apparently,

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where gc(v) is the density of states of the conduction(valence) band.

The difference in Fermi level between two ODs whenthey are brought into close proximity results in chargetransfer to equilibrate the quasi Fermi levels. The chargetransfer is defined to be positive when charge istransferred from the smaller OD to the larger OD. This isthe direction of charge transfer seen in most cases (Fig.2), and charge is transferred in the other direction onlywhen the size difference between ODs is small and theFermi level of the larger OD is below the Fermi level of thesmallerOD.

For Au ODs, the midgap level, approximately theFermi level, changes with size of the OD. From a bandstructure calculation, the effective Fermi level is calculatedby setting the intrinsic number of electrons equal to theintrinsic number of holes in the formulas:

978-1-4244-2950-9/09/$25.00 ©2009 IEEE 000086

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a driving force for separation; the electron will be driventowards the larger QD, while the hole will not feel a drivingforce for transfer.

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Charge transfer between QDs of different size may bestacked in sequence. A calculation performed on a seriesof three PbS QDs of different size shows that the chargetransfer results in a large voltage gradient across the threeQDs.

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Fig. 4: Density of states of (PbS)za and (PbS)ao beforecharge transfer (a) and after charge transfer (b)resulting in equilibration of the quasi Fermi levels .

The above results were all performed for QDs invacuum. To test the effect of the dielectric embeddingmedium that must surround QDs in a QDSC , PbS QDswere placed in an MgO matrix . MgO was chosen due toits having the same crystal structure as PbS and a widebandgap, so that it will not significantly absorb solarirradiation.

Given the promising results of calculations, anexperiment was designed to test charge transfer betweenadjacent QDs of different size.

000087

EXPERIMENTAL DETAILS TOpelectrode

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Fig. 6: (a) Schematic of a solar cell with QD sizegradients. (b) TEM cross section of a Zr02/PbS 5nmlZr021 PbS 2nm sample on Si. Measured thicknesses(in nm) are, from top to bottom, 1.9 ± 0.2, 4.8 ± 0.2,2.2 ± 0.2, 1.7 ± 0.3, and nominal thicknesses are 2,5, 2, and 2nm.

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An experimental setup was designed to test theproposed model of charge separation in QDSCs . Quantumdots of Au and PbS were purchased from NN Labs andEvident Technologies respectively. A ligand exchangewas performed with 3-aminopropyl-diethoxy-methlsilane(APDES) to reduce the tendency for the particles toagglomerate. After the ligand exchange, 5 and 10nm QDsof Au and 2, 3, 5, 7, and 11nm QDs of PbS (Fig. 5) aredeposited by Langmuir Blodgett (LB) deposition on quartzsubstrates with 2nm Zr02 barrier layers deposited byatomic layer deposition (ALD). Cross sectionaltransmission electron microscopy (TEM) of the structure isshown (Fig. 6).

Fig. 5: (a) High resolution TEM image of a single PbS11nm diameter particle . (b) SEM micrograph of Au 10nmQDs after LB deposition. [TEM by Hee Joon Jung].

Size dependent light excitations are performed, andelectrical measurements of light and dark current i-Vcurves taken in a probe station are presented in Fig. 7.Under weak illumination, more photocurrent is observed ina QDSC with different QD sizes that with either large orsmall QDs. This may be due to the electric field betweenQDs of different size which aids charge transport.

Fig. 7: Current-voltage curves of QDSCs with two layers ofPbS QDs of different sizes with Zr02 barrier layers.

Semi-transparent, symmetric, top and bottom Ptelectrodes of 5nm thickness are deposited by evaporation.Mott-Schottky analysis of band bending elucidates atwhich interfaces band bending occurs. Sample results areshown inFig. 8 for a cell with two layers of Au 5nm QDs. The i-Vand impedance spectra demonstrate that the electrodesare symmetric. The minimum in capacitance at zero

978-1-4244-2950-9/09/$25.00 ©2009 IEEE 000088

voltage shows the flat band interface and the ohmic natureof the contacts . The non-linearity of the Mott-Schottky plotof C-2 vs. V shows that the exponential behavior of the i-Vcharacteristic does not come from a Schottky barrierformed at the interface . We attribute the exponentialincrease in current with increase in voltage to a tunnelingprocess of electron transport through the Au QDs.

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Calculations of electronic structure of QDs show themidgap level changes with QD size, as expected. Theresult of placing two differently sized QDs next to eachother is an equilibration of the quasi Fermi level that isbrought about by electron transfer from the QD with higherFermi level to that with lower Fermi level. The chargetransfer results in a strong electric field between the QDsthat may be used to drive charge transport and splitexcitons . Experimental results of QDSCs utilizing twodifferent sizes of QDs give preliminary indications that thesize effect may be realized for QDs separated by thinbarrier layers. Specifically, enhanced photocurrents wereseen for samples with a gradient in QD size, indicating thatthere may be an enhanced driving force for chargetransport in size-graded QDSCs.

REFERENCES

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(a) V [V]1 R.D. Schaller, V.1. Klimov, Phys. Rev. Lett., 92 (2004)186601.2 R.T. Ross, A.J. Nozik, J. Appl. Phys., 53 (1982) 3813.3 EA Weiss, R.C. Chiechi, S.M. Geyer, V.J. Porter , D.C.Bell, M.G. Bawendi, G.M. Whitesides, J. Am. Chem. Soc.,130 (2008) 74.4 EA Weiss, V.J. Porter, R.C. Chiechi, S.M. Geyer, D.C.Bell, M.G. Bawendi, G.M. Whitesides, J. Am. Chern. Soc.,130 (2008) 83.5 T. Franzl, TA Klar, S. Schietinger, A. L. Rogach, J.Feldmann , Nano Lett. 4 (2004) 1599.6 V. I. Klimov, A. A. Mikhailovsky, D.W. McBranch , CALeatherdale, M.G. Bawendi. Science 287, (2000) 1011.7 A. Franceschetti, A. Zunger, Phys. Rev. Lett., 78 (1997)915-918 ; G. D. Scholes, G. Rumbles, Nature Mater., 5~2006) 683-696 .

M. Vaiden, X. Lai, D. W. Goodman , Science, 281, (1998)1647.9 K. Taylor et al., J. Chern. Phy., 96 (1992) 3319.10 G. Kresse, J. Hafner, Phys. Rev. B, 47(1993) 558; ibid.49 (1994) 14251; G. Kresse, J. FurthmOller, Comput. Mat.Sci. 6 (1996) 15; G. Kresse, J. FurthmOller, Phys. Rev. B,54 (1996) 11169.11 P.E. Blochl, Phys. Rev. B, 50 (1994) 17953; G. Kresse,J. Joubert, Phys. Rev. B, 59 (1999) 1758.12 D. Ceperley and B. Alder , Phys. Rev. Lett. 45 (1980)566; J. P. Perdew and A. Zunger, Phys. Rev. B, 23 (1981)5048.13 R. Parr, Ann. Rev. Phys. Chern., 34 (1983) 631, andreferences therein .14 Prasanjit Samal et ai, J. Phys. B: At. Mol. Opt. Phys. 39(2006) 4065; M. Harbola et al., Comput. Methods Sci.Eng., 1108 (2009) 54.15 Gaussian 03, Revision D.01, M. Frisch et ai, Gaussian ,Inc., Wallingford CT, 2004.

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CONCLUSIONS

978-1-4244-2950-9/09/$25.00 ©2009 IEEE 000089