Nanomaterials for Photovoltaics (v11) 6. Homojunctions

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1 Naomaterials for Photovoltaics (v11) 1 6. Homojuctios / juctio diode The most imortat device cocet for the coversio of light ito electrical curret is the / juctio diode. We first cosider isolated ad regios with o electric cotact. I this case, we caot assume the materials to be i electrical equilibrium with oe aother. I other words, the Fermi level is ideedet i each material: I equilibrium, the Fermi level becomes costat by a redistributio of mobile carriers i the viciity of the juctio to form a electric field, with a cocomitat bad bedig. Electros ted to lowest available eergy, so we ca ifer that the juctio has a built-i electric field oit from to We ca fid the built-i otetial (voltage) bi V without ay detailed kowledge of the carrier distributios ear the juctio. First fid the work fuctio o each side of the juctio. O the -side:

2 Naomaterials for Photovoltaics (v11) E E E E E E vac F vac C C F E E E E F V C V N V kt l E N A O the -side: E E E E E E vac F vac C C F NC kt l ND The voltage differeces is g qv E E E E I equilibrium: vac vac F F E E E F F F This gives NC NV qv Eg kt l NA ND Eg kt i e Eg ktl NA ND NA ND qv kt l i The equilibrium voltage dro across the juctio is called the built-i voltage: V bi kt NA ND l q i Deletio aroximatio: sace-charge regio We ofte eed to estimate E ear the juctio. A useful aalysis uses the deletio aroximatio. It is assume that adjacet regios o either side of the juctio are fully deleted of mobile carriers, with oly the immobile, ioized doats cotributig to the electric field. The roblem is the oe of electrostatics. This charged volume ear the juctio is called the deletio regio or the sace-charge regio (SCR).

3 Naomaterials for Photovoltaics (v11) 3 Deletio aroximatio: charge desity We are left with ioized accetors ad doors i the SCR. Withi this aroximatio, the charge desity is give by: 0, xw A, 0 N w x x ND, 0 x w 0, w x The ukow arameters here are the widths w ad w.

4 Naomaterials for Photovoltaics (v11) 4 Deletio aroximatio: electric field The Poisso equatio relates the charge desity to the electric field. d x dx Overall charge eutrality dictates that the field must vaish far from the juctio. The 0, xw qna x w, w x 0 q x Ε dx x qnd w x, 0 x w 0, w x Assumig o surface charges are reset at the juctio, the field must be cotiuous there: x0 x0, which requires that NAw ND w. The maximum field is the qna w qnd w max Deletio aroximatio: otetial The electric field is the gradiet of the otetial, or dv dx i 1-D. Itegratig 0, xw A qn x w, w x 0 Ε V x x dx x q N A w N D w w x, 0 x w q N A w N D w, w x We fid that the built-i otetial ca be writte as

5 Naomaterials for Photovoltaics (v11) 5 V q N w N w bi A D Deletio aroximatio: deletio width We kow that ND A w w, w N w NA ND We ca write the built-i otetial i terms of either w or 1 1 q 1 1 A D q V N w N w bi N A N D N A N D As we already kow V bi from eergy cosideratios, we ca the solve for the dimesios of the sacecharge regio: w 1 1 V bi N, A q 1 1 NA ND The total deletio width is the: w : w 1 1 V bi N D q 1 1 NA ND bi ww V 1 1 w q NA ND Deletio aroximatio: eergy diagram Fially, we ca lot the comlete bad diagram withi the deletio aroximatio for the / juctio i equilibrium.

6 Naomaterials for Photovoltaics (v11) 6 / juctio: o-equilibrium (I) Let s iclude the ifluece of a exteral voltage V, tyically, which may be due to a alied bias or to hotogeeratio. The voltage dro across the SCR is aaretly w q Εxdx NAw NDw Vbi V xw Usig the revious derivatio, the width of the SCR is: Vbi V 1 1 ww w q NA N D Notice that the SCR becomes arrower i forward bias ( V 0 ) ad wider i reverse bias ( V 0 ). I the eutral regios (outside of the SCR),we exect E F A e i E kt E F Ei kt N i, ND i e I the SCR, the quasi-fermi levels are aroximately determied by those of the majority carrier o either side. (We will look at this more carefully later). E F E F, E F E F Cosider the roduct F F i i NA ND i e i e Let s fid the quasi-fermi level slittig i the SCR: E E E E kt qvbi V kt NA ND kt l EF E F q V V bi qv i This shows that the quasi-fermi level slittig i the SCR is aroximately equal to the exteral voltage.

7 Naomaterials for Photovoltaics (v11) 7 / juctio: o-equilibrium (II) Based o the recedig discussio, we ca sketch the eergy diagram for the / juctio i forward bias ( V 0 ) as show below: / juctio: o-equilibrium (III) We ca sketch a slightly more comlete eergy diagram for a / juctio i forward bias ( V 0 ). If there are o hotogeerated carriers there is o quasi-fermi level slittig outside the SCR: It is fairly easy to see that the Fermi level far from the juctio is aroximately determied by the majority carrier cocetratio. / juctio: o-equilibrium (IV) First, aalyze without hotogeeratio. Far from the juctio: i 0, 0 N a I the SCR: N i d i e kt

8 Naomaterials for Photovoltaics (v11) 8 where qv. The deletio aroximatio gives: w Na w Nd, qv kt qv kt w 0 e w 0 e The excess miority-carrier cocetratios at the edges of the SCR are qv kt ww0 0 e 1 qv kt w w 0 0 e 1 These are critical for determiig the hotocurret. Photogeeratio rate Absortio d b E, x E, x b E, x dx where Ex, is the local absortio coefficiet ad be, x is the sectral hoto flux desity i the material. We ca treat each eergy searately. Let s assume uiform absortio [ Ex, E]. Searatig variables bx db x b l b x x b 0 This gives bb0 x0 be, x be,0 e Note that x dx x Ex be,0 1 RE b S E Carrier geeratio rate is d ge, x be, x dx EbE, x, 1 E x g E x R E E bs E e / juctio solar cell (I) Let s assume weak absortio, so that there is o atteuatio of the icidet illumiatio. The Ex e 1. The hotogeeratio rate is g E, x 1 REEb 1 s E g E This gives uiform geeratio, so that E 0 G x de g E G If this is bad-to-bad absortio, the G G G. I steady state, o the -side: g 0

9 Naomaterials for Photovoltaics (v11) 9 0 g G U G So, g G. O the -side 0 g 0 g G U G which gives g G. / juctio solar cell (II) With uiform hotogeeratio, there are excess miority carriers i the eutral regios, so the quasi- Fermi levels are slit: With moderate hotogeeratio, the majority carrier desities are ot largely affected, but the miority carrier desities are. / juctio solar cell (III) Cosider geeratio i the quasi-eutral regios (assumig uiform geeratio). O the () side, outside the SCR x x, x x At the edges of the SCR w w, w w The carrier cocetratios are still related to the quasi Fermi-level sittig. qv kt w, 0 e qv kt w 0 e So the excess miority-carrier cocetratio at the edges of the SCR are: w 0 e qv kt 0 0 e qv kt 1 w e qv kt e qv kt 1 g g 0 0 g 0 g

10 Naomaterials for Photovoltaics (v11) 10 / juctio solar cell (IV) The total curret desity is J J x J x where the sig covetio used for PV is adoted, such that J is ositive whe the device is deliverig ower. This curret must be costat at every oit i the circuit, so J J w J w J w J w The deletio aroximatio imlies that all the otetial differece occurs across the SCR. So, E=0 at the SCR edges ( x w, x w ). At these oits: d d J q qd qd dx dx d d J q qd qd dx dx / juctio solar cell (V) I the eutral regios, we have diffusio oly: x w L xw L x A e B e, x w xw L xw L x A e B e w x, Assume ifiitely thick eutral regios, 0 ad 0. The boudary coditios give xw L xw e, x w xw L x w e, w x The currets are d qd J x qd x dx L d q D J x qd x dx L So we kow that q D q D J w w ad J w w L L From revious results So qv kt w 0 e 1 g qv kt w 0 e 1 g q D qv kt J w 0 e 1 g L q D qv kt J w 0 e 1 g L / juctio solar cell (VI) J w ad J w. To fid We kow

11 Naomaterials for Photovoltaics (v11) 11 J J w J w J w J w we eed either or J w J w J w,scr J w J w J,SCR w where ad J w q UxGxdx,SCR J w,rec xw J,ge J w q UxGxdx,SCR J w,rec xw J,ge / juctio solar cell (VII) Let s igore recombiatio i the SCR, so J, J, 0. If there is uiform geeratio i the SCR,ge,ge w J J q G dx q G w So, we ca say that xw J,SCR w J,SCR w q G w We just eed oe or the other. Let s fid J J w J w. We kow q D qv kt J w 0 e 1 L g J w J w J, SCR w where rec rec ad q D qv kt J w 0 e 1 g L J,SCR w qg w Puttig these together q D qv kt J w J w J, SCR w qgw 0 e 1 gqg w L / juctio solar cell (VIII) We eed J J w J w We have q D qv kt J w 0 e 1 L ad g

12 Naomaterials for Photovoltaics (v11) 1 q D qv kt J w qgw 0 e 1 gqg w L We ca write qd L The qd qd qd g G qg L ad G qg L L L L g ad q D qv kt J w 0 e 1 qg L L q D qv kt J w 0 e 1 qgl w L / juctio solar cell (IX) The total curret desity is or q D q D qv kt J qg w L L 0 0 e 1 L L qv kt JJ hoto J0 e 1 where the hotocurret is J q G w L L The dark (diode) curret is hoto D D D D J0 q 0 0 q i L L L NA L N D Fially, we have the most commo form for the / juctio solar-cell curret: qv kt JV J hoto J0 e 1 SCR trasit time It is difficult to estimate the amout of recombiatio occurrig i the SCR. The carriers move quickly through the SCR, because of the large electric field there. Let s estimate the trasit time for carriers to cross the SCR. For this, we comute the drift velocity i a costat electric field give by the average field i the SCR, which is E Emax i the deletio aroximatio. Usig v E, we have a trasit time of w w v E max The maximum field is qna w qnd w E max Usig w w w

13 Naomaterials for Photovoltaics (v11) 13 1 NA w w 1 1 NA ND This gives q 1 E max w 1 1 NA ND We eed the ratio Emax q 1 w 1 1 NA N Fially D 1 1 q N A N D We ca ut i arameters for a umerical estimatio. Assumig 10 0, 100 cm V s, ad 16 3 N, we get s, which is short comared to tyical miority carrier NA D 10 cm lifetimes i high-quality material, so the aroximatio is reasoably valid, but should be used with aroriate discretio. Plots: deletio arox. (I) Case 1: equilibrium ( V 0 V, G 0 ) Plots: deletio arox. (II) Case : forward bias, o illumiatio ( V 0.5 V, G 0 )

14 Naomaterials for Photovoltaics (v11) 14 Plots: deletio arox. (III) 19 3 Case 3: short circuit, illumiated ( V 0 V, G (cm s) ) Plots: deletio arox. (IV) 19 3 Case 4: forward bias, illumiated ( V 0.5 V, G (cm s) )

15 Naomaterials for Photovoltaics (v11) 15 Plots: carrier cocetratios i deletio arox. The lots below show the mechaisms for curret i the cases of forward bias with o geeratio ad forward bias with geeratio. I the deletio aroximatio, the miority-carrier cocetratios at the edges of the SCR are ideedet of their cocetratios farther from the juctio. So, if o hotogeeratio occurs, miority carriers will diffusive away from the juctio, which costitutes the diode curret. If adequate hotogeeratio occurs, miority carriers will diffuse towards the juctio ad cotribute to the hotocurret.

16 Naomaterials for Photovoltaics (v11) 16 Aalysis without the deletio arox. I the followig simulatios, we have assumed the rimary recombiatio mechaism is radiative, by secifyig log SRH lifetimes. Dashed curves were comuted usig the deletio aroximatio. Solid curves were comuted by varyig the quasi-fermi levels to satisfy the uderlyig equatios: 1) Poisso equatio dv qx dx ) electro ad hole curret desities def de J, J dx dx 3) electro ad hole cotiuity 1 dj G U 1 dj, G U qdx qdx Plots: o deletio arox. (I) F Plots: o deletio arox. (I) We used the deletio aroximatio to estimate the dimesios of the SCR, from which the rest of the / juctio roerties were derived. We assumed o recombiatio i the SCR. These assumtios

17 Naomaterials for Photovoltaics (v11) 17 rovide a geeral isight ito the J-V characteristics of the device. It is ossible to fid more accurate descritios of these characteristics without so may assumtios. Case 1: forward bias, o geeratio ( V 0.7 V, G 0 ) Plots: o deletio arox. (II) 3 3 Case : oeratig, illumiated ( V 0.7 V, G (cm s) )