Lecture 19: Electronic Contributions to OCV in Batteries and Solar Cells. Notes by MIT Student (and MZB) March 18, 2009

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1 Lecture 19: lectrnic Cntributins t OCV in Batteries and Slar Cells Ntes by MIT Student (and MZB) March 18, In many situatins the µ e cnstant fr metal electrdes, this due t the abundance and freedm f electrns at the Fermi surface -Frm what we have previusly stated the OCV depends mainly n µ i fr reactants i. Hwever at the end f the reactin, electrns can becme lcalized at which pint there culd be a purely electrnic cntributin t OCV. An example f this exceptin is Li-in insertin electrdes, where Li can act as an electrn dnr and thse electrns may participate in Faradic reactins. 1) Metallic insertin electrde Apprximate cnductin band electrns as free and can mdel as a parablic functin 2 3π 2 N c Δ = s c =, dimensin (d)=3, where N c = # f cnducted electrns 2m L 3 L 3 = V = vlume, η c = N c L 3 density f electrns. Nw lets add lithium with filing fractin X 2 3 Nd = Li xl 3 f d V Li = xn s f d, f d = prbability that Li dnates an electrn, and N s = # f sites fr 2 3π 2 (N c + N d ) Δ = s c = 2m L

2 N = Δ d f 2 d 1+ N c 2 d xfd = Δ f 1+ V Li η c where V Li η c = the number f cnductin electrns per Li cell r site als, nte that the cnductin band is cnfined t d dimensins 1 d 3 2 xf Δ x 1+ d d ( ) = Δ f V Li η c f d # f dnated Li electrns # f dnated e where α = = V Li η c # f cnductin electrns per site = # f cnductin e fr a gd cnductr and large Li unit cell => α <<1 therefre, x 2 1+ αx +... d ( ) ~ Δ f 1+ αx Δµ ( x ) = Δ( x ) Δ ( 0) = Δ e f ( ) 2 α <<1 ~ Δ f µ Li ( ) + µ e ( ) ( ) = e x 2α = V x e de µ Li ( ) 2α x d x x V x V d 1 2

3 2) Semicnducting Insertin lectrdes Nte fr insulatrs: fr insulatrs where energy gap is very large and α = 0, because δd 0, Li cannt liberate electrns. f d 0 N L = 0 2 3π 2 N d Δ = 2m L 3 Δ( x = 0) = 0 Δµ e ( x ) = Δ( x ) = (cnst)( f d x ) 2 d 2 3 3

4 3) Slar Cells drps in energy ges int phnns (vibratins) and is lst, if yu can find ways t harvest all the energy and nt lse it yield large imprvements Light quanta (phtns) with energy larger than the band gap can be absrbed as excitns r electrn-hle pairs. The absrbed phtn energy can be lst by radiative recmbinatin (the reverse prcess f adsrptin, where the electrn and hle recmbine and release a phtn) r by nn-radiative prcesses, such as scattering interacting with the crystal atms, which transmits energy t quantized lattice vibratins (phnns). Since there are many available electrn states in the cnductin band and hle states in the valence band, nn-radiative prcesses tend t ccur very quickly and cause the excitn s energy t settle dwn t the band gap, with the electrn at the lwer cnductin band edge and the hle at the upper valence band edge. A phtvltaic cell is designed t extract the energy f the excitn as electrical energy in the external circuit, by cllecting electrns at the cathde and hles at the ande. The pen circuit vltage f the cell is the mean excitn energy, which is clse t the band gap f the semicnductr due t nn-radiative prcesses. With these cncepts, we are ready t calculate the ultimate efficiency f a slar cell (Shckley, Quiesser 1961), based n the principle f detailed balance: -very phtn striking the slar cell w/ p > g yields an excitn at the gap = g that is harvested as pen circuit vltage, withut any ther energy lsses. 4

5 g Pwer ut N( )d η = = g Pwer in N s ( )d N s ( )= density f phtn energy. The slar spectrum is well apprximated by a black bdy at temperature T sun = 6000K, where N() is given by Planck s frmula η = ~ x 2 g ~ dx g e x 1 x 3 dx e x 1 An amazing and frtunate fact: At rm temperature, kt =25mV is much smaller than typical energies f chemical bnds in materials, which are at the scale f Vlts. Hwever, the temperature f the sun at 6000K is such that kt=0.5v ~ ~ g g g = = kt s 0.5V and thus the maximum in the SQ frmula fr ultimate efficiency is cmparable t actual bandgaps in readily available materials, s that efficient slar cells can be cnstructed n max eff ~ earth. In particular, the SQ maximum efficiency f 44% ccurs at g 1.1 Vlts, which happens t be clse t band gap energy f silicn, an abundant element, making it an ideal slar cell material. 5

6 There are several ways t design slar cells t harvest pht-induced excitns. In traditinal silicn slar cells, the excitn is generated in a p-n junctin between p-type silicn (dped with acceptr impurities which attract valence-band hles) and n-type silicn (dped with dnr impurities which attract cnductin-band electrns). The junctin separates the electrns frm the hles, which are cllected at ppsite electrdes. In a dye-sensitized Graetzel cell, the cathde cnsists f a semicnducting ande (e.g. TiO2) cated with a pht-sensitive dye (e.g. rutenium based). The dye adsrbs slar phtns and transfers the electrns t the semicnductr electrde, while hles are remved by xidatin reactins with an electrlyte (e.g. idide slutin), which carries current t the cathde fr reductin and cmpleting the circuit. 6

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