MODELING THE FUNDAMENTAL LIMIT ON CONVERSION EFFICIENCY OF QD SOLAR CELLS Ա.Մ.Կեչիյանց Ara Kechiantz Institute of Radiophysics and Electronics (IRPhE), National Academy of Sciences (Yerevan, Armenia) Marseille Nov 19, 2009
My Business Card:Anomalous Photo-Voltaic Effect in Semiconductor Films Discovered by Starkiewicz J., Sosnowski L. and Simpson O., Nature 158, 26(1946) 0 +1000V CdTe, PbS, Si, GaAs etc n p n p n Studied by Tauc J., Adirovich E., Pankove J., and many others. although anomalous photovoltaic phenomenon is spectacular and very intriguing, a little progress has been made in explaining it in a detailed and definitive way (J. Pankove Optical Processes in Semiconductors, NY, Dover, 1976) Explained by A.M. Kechiantz and H.M. Kechiev, J. Phys. C, 13, 5715 (1980)
Direct conversion of solar energy into electricity is very attractive Reasons Feeding on Demand in Solar Energy - environmental - economic - special application
In principle the 2nd low of thermodynamics limits the efficiency for solar energy conversion to the Carnot value SUN 5780K (T sun -T cell )/T sun = 95% Earth 300K It is a very challenging task to design a solar cell operating under the Carnot cycle condition This device should exploit ideal electron gas
Principles of Photovoltaic Operation Photons transfer electrons from left electrode to right electrode.
Barrier (2): separates generated electrons from holes (built-in at the junction or at the interface) Contacts (3): transfer carriers into the external circuit (at the front and rear side of the cell) Key Elements of Solar Cells Sensitizer (1): converts light into electron-hole pairs. (bulk semiconductor, nanoparticles, dye, quantum dots)
Key Parametrs of Solar Cells Conversion efficiency η= V OC FF I V SC P P S incident power; I SC - short circuit current; Open circuit voltage FF fill factor n ideality factor I D dark current S nkt ln e I I SC D OC in fact both FF and n strongly depends on the dark current I D
Fill-Factor I V FF= mp I V mp SC OC I mp and V mp refer to the maximum output power
Dark Current I ε I exp - g nkt D 0 for the radiative recombination n=1 2 2πkT 2 2.96 10 A εg I 0 =I 0R =e ε 3 2 g = 2 hc cm ev 2 for the best processed materials n=1 I =I = 5 1.5 10 A 0 0I 2 cm Green M A 1998 Solar Cells in Modern Semiconductor Device Physics ed S M Sze (New York: Wiley)
Recombination Channels (Radiative and Nonradiative): 1 at the external surface and contacts; 2 in the bulk 3 in p-n-junction; Recombination channels determine the dark current 4, 5 flow of electronhole pairs towards the recombination centers; 6 generation of electron-hole pairs;
Actual PhotoCurrent ( ) ( ) V-I V R I( V ) =I -I exp S - PC D nkt V-I V R R Sh S I ε I exp - g nkt D 0 I 0 is strongly dependant on recombination centers
Waste of Energy in Conventional Cells Energy Waste occurs because: 1. High-Energy Photons Bring Excess Energy 2. Low-Energy Photons Bring No Enough Energy 3. Induced Voltage is Less than the Band gap
Limiting Efficiency of 1-junction Conventional Solar Cells Conventional solar cells are limited by 31% conversion efficiency under 1 sun irradiation. The efficiency roses to 43% under 1000 sun concentration of irradiation.
Under concentrated light illumination, solar cells operate only a little more efficiently Conversion efficiency η= FF (X I SC (X P S ) V ) OC The light is concentrated X times. FF fill factor; P S incident power; I SC - short circuit current; Open circuit voltage V OC nkt ln e X I n ideality factor; I D dark current I D PC
Concentrators Solar cells operate only a little more efficiently under concentrated light In practice, concentrators are mainly used to replace expensive semiconductor materials by inexpensive optics and inexpensive mechanics
New Concentrators Dye or QDs in Polymer or in Inorganic materials Solar Cell
Quantum Dot Intermediate Band Solar Cells InAs-GaAs QDs have shown a partial operation under the IBSC principles (Luque A et al 2004 J Appl Phys 96 903-909) The minority carrier diffusion current and the recombination current in the intrinsic region of Si cells are enormously increased as the number of Ge QD layers increases (Alguno A, Usami N et al 2006 Thin Solid Films 508 402 405)
Ge/Si Quantum Dots p-si R Ge ΔE 0 R0 R 3R 0 kt = 3 L LD ΔE V p-si Screening Layer at QD/Semiconductor Interface 2/3 Screening Layer at Metal/Semiconductor interface L L L D ΔE D 2 D = ε π kt kt 4 e p V 18-3 p=1. 10 cm L = 4nm
Quantum Dots Buried in p-n-junction p-si Ge Si ΔE p-si Ge R 0 n-si 0 ΔE In fact, the first Ge QD screens essential part of the built-in field (ΔE v ) p-si Ge R 0 0 n-si R 0 ΔE 0
Conventional QD Solar Cells J C Photons and the bias V split the Fermi level into three quasi-fermi levels: p-si Ge J t F C for the conduction band, F V for the valence band, F I for the confined QD state ε g V F C ε I J C Due to the tunneling holes generated by sub-band gap photons may escape QDs and produce an extra photocurrent J V J t F V F I ΔE 0 J V ε g n-si
Calculation of the Efficiency Limit of QD Solar Cells (The Main Assumptions I): the detailed balance principle (opposite electron transition is allowed if the direct transition is allowed) materials are ideal (only radiative transitions are allowed) radiative electron transitions from the confined state into the valence band are not allowed the extra photocurrent is generated in two consecutive electron transitions: p-si 1. from the confined QD state into the J t conduction band 2. tunneling from the valence band of p-si into confined state in Ge QDs. J V ε g V J C F V F C Ge ε I F I ΔE 0 J t J V ε g J C n-si
Escaping above band gap photons produce the conventional dark current that is the net photoelectron transition from the conduction band into the valence band. The Main Assumptions II: Photons are continuously absorbed and emitted while photoelectrons appear and disappear inside the material until the emitted photons escape the cell A perfect mirror is placed at the back of the cell so that the emitted photons escape the cell by the front area only Escaping photons produce the dark current that is the net photoelectron transition towards the lower energies: Escaping sub-band gap photons produce the extra dark current that is the net photoelectron transition from the conduction band into the confined QD state
Photon Modes Total radiation in the cell consists of: incident solar photons, recycled photons, thermal background photons ν(ζ) ζ Total radiation can be developed in a set of photon modes (plane waves of certain photon energy and wave vector) ν(ζ) the number of given mode photons; ζ the distance calculated along propagating ray of given photon mode inside the cell; Variation of photons in the mode may be written as: dv( ζ ) n = ( 1- )( ( ζ ) + 1) - ( 1- ) ( ζ ) hcivj fci fvj v hvjci fvj fci v d c ζ i, j i, j n + hiivj fii fvj v + hvjii fvi fij v c ( 1- )( ( ζ ) 1) - ( 1- ) ( ζ )
Development of Photon Modes v Photons in the mode may be written as α v { } ( ς ) = CV CV IV IV 1 exp[ ( α α ) ς] v α CV + + α α IV ( ) [ ( α α ) ς] + 0 exp + ; v CV + + IV CV IV v = 1 n ; α = h ( f f ); CV exp ( ) CV CiVj Vj Ci ε μcv kt 1 c i, j 1 μ ( ) CV = FC FV ; μiv = FI FV ; f εi = ; exp ( ε ) i F kt + 1 Luque A and Martí A 1997 Phys Rev Lett 78 5014 7
Equilibrium Distribution Though the recycling photons are not in thermodynamic equilibrium inside the cell, the recycling mechanisms trend to bring photons in the modes to the equilibrium distribution when photons leave the cell after some reflection v ( ) v ( ς ) = α v α + + α CV CV IV IV CV ς ε ε = v ; > 0 CV g α IV v v ( ς ) v exp( α ς ) + v( 0) exp( α ς ) = 1 ; 0 IV IV 0 IV 0 ε ε ε I < < g
I T p-si J V ε g V J C J t F V F C Tunneling Current (I) B T - Tunneling Transparency: B T =0 - opaque barrier ; B T =1 - transparent barrier Ge ε I F I J t ΔE 0 J V ε g J C n-si N T Density of QDs; v T Thermal Velocity ; E Built-in Field; d I the distance from QDs to p-si layer. Tunneling current reduced to (( V μ ) kt) eb 1 exp TNIv T IV = 1+ exp ( V + d ) 1 exp ( ) IE εi kt + εi die μiv kt
p-si J V ε g V J C J t F V F C Tunneling Current (II) Ge ε I F I J t ΔE 0 J V ε g J C n-si N SCV 2 2 π kt ε exp ε S g g 3 2 hc kts Random Photon Flux 2 2π ktεg μcv εg NCV exp 3 2 hc kt Balance of absorption and tunneling reduce the tunneling current to nh ς N c IV IV 0 I T = SIV IV0 exp 1 exp kt 1+ exp ( μiv εi + de I ) kt I e N N μ Solar Photon Flux
Total and Dark Currents I = e N ( ) SCV NCV 0 exp V kt + e N ( ) SIV NIV 0 exp V kt + N IV 0 NSIV εi EdI V εi + EdI 1+ + exp 1 exp BTvTN I NIV0 kt + kt In the lack of sunlight, T S =T, 2 the total current I reduces to 2 π kt ε 0 exp ε g g NCV 3 2 the dark current I D hc kt I ( ) D = encv0 1 exp V kt + en ( ) IV 0 1 exp V kt + NIV 0 ε I EdI V + EdI ε I 1+ 1 exp 1 exp BvN + + kt kt T T I
Detailed Balance Limit on the Currents (I) No Tunneling from QDs, B T =0 : I = e NSCV N V kt CV 0 exp ( ) 0 1 exp ( ) I = en D CV V kt If the barrier is tunneling-opaque, QDs do not introduce any change in the currents. Then, the same value limits conversion efficiency of otherwise same conventional and QD solar cells.
Detailed Balance Limit on the Currents (II) If the barrier is tunneling-transparent, QDs increase both total and dark currents. If there are a lot of QDs, N I v T >> N IV0, and intensive tunneling, B T =1, the currents reduce to : Then, the limits on the currents and, hence, conversion efficiency of QD solar cells reduce to that of QD-band gap conventional cells. I = e NSI N V kt I 0 exp ( ) 0 1 exp ( ) I = en D I V kt N N SI 2 2 πktsεi ε I exp 3 2 hc kts 2 2 π kt ε 0 exp ε I I I 3 2 hc kt
Thanks for your attention Yerevan, Armenia Ա.Մ.Կեչիյանց Ara Kechiantz Institute of Radiophysics and Electronics (IRPhE), National Academy of Sciences (Yerevan, Armenia)