How Solar Cells Work Basic theory of photovoltaic energy conversion Peter Würfel University of Karlsruhe, Germany
Three Messages Sun is a heat source, solar cells must be heat engines Important conversion step solar heat chemical energy of electron-hole pairs limited by thermodynamics Conversion of chemical energy electrical energy can be 100% efficient and needs more than a pnjunction
Carnot process P 1 T adiabatic 4 isothermal isothermal 2 adiabatic 3 T A T 0 isothermal 1 2 adiabatic adiabatic 4 isothermal 3 V S Important: generation of (entropy-) Free Energy by cooling F = E - TS
Solar cells are heat engines Principle I = T I E,in S S,in I E,out I E,in (1 - T 0/T S) I S,in I I S,out S,in TI 0 S,out
The solar cell as a heat engine? Questions: What is the working medium (the gas)? What kind of free energy is produced?
Conversion of solar heat into chemical energy of electrons and holes ε e ε C ε V g εf ε FC ε FV dn /dε e > m eh dn /dε h e e -14 10 s -12 10 s Energy per photon ħω chemical energy µ eh per e-h pair by thermalisation energy gap is necessary
Maximum chemical energy Reduce entropy generation during thermalisation by reducing the energy range, for which Fermi-distribution is established ideal: isoenergetic thermalisation in narrow energy ranges is isentropic e e De C e dn /de e e e FC ω e F μ eh De V h e FV dn /de h e -14 10 s -12 10 s tandem cells
Tandem cells 2 cells 4 Fermi-energies for 4 energy ranges 4-level system 3 Fermi-energies for connection in series e F4 e F3 e F2 e F1
Recombination recombination at the surface non-radiative in the material non-radiative radiative surface recombination velocity bulk lifetime effective lifetime
Recombination Direct optical transitions in 2-level system 2 d jγ ( ω) d rup ( ω) = M D12 f( ε1) [ 1 f( ε2) ] d( ω) d( ω) 2 d jγ ( ω) d rstim( ω) = M D12 f( ε2) [ 1 f( ε1) ] d( ω) d( ω) 2 c0 d rspont ( ω) = M Dγ ( ω) D12 f( ε2) [ 1 f( ε1) ] d( ω) n upwards downwards e 2 ε ε = ω 2 1 dr up dr spont dr stim e 1 density of states for photons 3 Ω n Dγ ( ω) = ω 4π c 3 3 3 0 ( ) 2
Absorption coefficient d r ( ω) = d r ( ω) d r ( ω) abs up stim 12 2 [ ] = M D f( ε ) f( ε ) d j ( ω) 12 1 2 = α ( ω)d j ( ω) γ absorption coefficient α ( ω) = M 2 D [ f( ε ) f( ε )] γ 12 12 1 2 α ( ω) < 0 for f( ε ) > f( ε ) 12 2 1 amplification ( ) ( ) j ( ω, x) = j ( ω,0) 1 r( ω) exp α ( ω) x γ γ 12 j x g( ) x
Spontaneous emission replace M α ( ω) 2 12 D12 = 1 2 [ f( ε ) f( ε )] in spontaneous emission rate [ f ε1 ] f ε2 [ ( ε ) ( ε )] c 1 ( ) ( ) ω α ω ω ω 0 d rspont ( ) = 12( ) Dγ ( ) d n f 1 f 2 with f 1 1 ( ε1) = and f( ε2) = ε1 εfv ε2 εfc exp + 1 exp + 1 kt kt and ε ε = ω 2 1 c0 d( ω) d rspont ( ω) = α( ω) Dγ ( ω) n ω ( εfc εfv ) exp 1 kt
Production of chemical energy dj eh = dg eh dr eh only radiative recombination, monochromatic dj g,abs dj eh = dj g,abs dj g,emit dj g,emit 2 Ω ( ω) generalized Planck law djγ ( ω) = a( ω) d ω 3 3 2 4π c ω ( εfc εfv ) exp 1 kt { } absorptance a( ω) = ( 1 R( ω) ) 1 exp ( α( ω) d) Sun: T = T S ε FC ε FV = 0 Semiconductor: T = T 0 ε FC ε FV 0 e FC e FV = m eh
Characteristic for production of chemical energy by monochromatic light dj eh (m eh )= dj g,abs dj g,emit (m eh ) dj g dj g, emit dj g, abs h = dj eh m eh,sc m eh,mp m eh,oc w m eh
Production of chemical energy from monochromatic radiation efficiency / % 100 90 maximum concentration 80 70 no concentration 60 50 40 30 20 10 0 0 1 2 3 4 5 photon energy / ev Infinite tandem: η = 86% max. concentration
Production of chemical energy in wide band semiconductor with total solar spectrum (Shockley-Queisser) AM0 spectrum η 0.4 0.3 max. concentration 0.2 no concentration 0.1 0 0 1 2 3 ε G / ev
Optimal materials full absorption above energy threshold (in a thin film) minimum recombination for given difference of Fermienergies: radiative recombination good materials for solar cells should be luminescing advantage for organic materials? difference of the Fermi-energies (chemical energy per eh-pair) is obtained from luminescence intensity
Chemical energy electrical energy Charge current e e j Q 0 x -ej e C e FC j Q = -e (G -R) G R µ eh e FV e V
Separation of electrons and holes with semi-permeable membranes H 2, O 2 e, h H 2 H 2 O 2 H 2 O 2 O 2 O 2 O2 e e 0 -ej n-type absorber p-type x H 2 H 2 O 2 O 2 O 2 O 2 H 2 H 2 H 2 O 2 H 2 e C e F,left e F,C µ eh e F,V e F,right Voltage: ev = e F,right - e F,left = μ eh e V
Transport properties, drift current acceleration a i = zee i m i drift current mobility b i = eτ m Ci, i
Diffusion current diffusion current n i μi = μi,0 + kt ln Ni chemical potential of particles i Fick s law of diffusion Einstein relation
total charge current electrochemical potential η = μ + zeϕ i i i for electrons (z i = -1) and holes (z i = +1) η = μ eϕ = ε e e FC η = μ + eϕ = ε h h FV field and diffusion currents do not exist
Dye Solar Cell e h Problem: e and h bound in exciton
Problems with excitons in organic semiconductors large exciton binding energy e C lumo exciton V exciton homo electron bound to free hole hole bound to free electron
exciton dissociation 0 1 2
Requirements for solar cell structures Conditions for optical and electrical properties of absorbers splitting of Fermi-energies selective transport Sufficient condition: L e, L h >> t a >> 1/a rules out low mobility absorbers Necessary condition: L e, L h >> distance between membranes n + p + absorber 1/α + - L e t a n + p + t a >> 1/a bulk heterojunction
Advantage of nano-structures in conventional solar cells distance between membranes on nm-scale absorbers can have poor transport properties Problem: large interface area may increase recombination
luminescence as a tool to prove energy conversion efficiency spectral intensity of luminescence 2 Ω ( ω) dr x = α ω d ω ( ) emission ( ) 4 3 3 2 π c ω εfc ( x) εfv ( x) exp 1 kt 0 spectral emission of photons through surface of homogeneous system 2 Ωemission ( ω) djγ, emit = a( ω) d ω 4 3 3 2 π c ω ( εfc εfv ) exp 1 kt0 a( eh ω = [ R ω ] ( α ω L ) ) 1 ( ) 1 exp ( ) e
Luminescence as a characterization tool Electroluminescence l < 1000 nm counts per pixel 16000 14000 12000 10000 8000 6000 4000 2000
Physics of Solar Cells