Excess carrier behavior in semiconductor devices

Transcription

1 Ecess carrier behavior i semicoductor devices Virtually all semicoductor devices i active mode ivolve the geeratio, decay, or movemet of carriers from oe regio to aother Carrier oulatio (, ) that is differet from the oulatio at rest (0, 0) is by defiitio ecess carriers (d=-0 d=-0) The ecess carrier behavior determies how a device work

2 Itroductio Devices i active states ivolve o-equilibrium, ad/or o-uiform, ad/or trasiet (o-steady state), ad/or o-ohmic carrier ijectio No-equilibrium carrier behavior Distributios (, t), (, t) Relaatio ad recombiatio Migratio: drift ad diffusio Some device eamles Photodetectors, solar cells, LEDs ad lasers Cathodolumiescece (Field-emissio dislays FEDs)

3 No-equilibrium carrier behavior Some Key Cocets Ecess carriers: quasi-fermi levels Carrier ijectio: otical ijectio (otical absortio), e-beam ijectio (electro umig) Carrier relaatio, recombiatio, lifetime: Ioizatio or avalache. Radiative or o-radiative recombiatio, direct recombiatio or tra-mediated recombiatio Carrier trasort: drift ad diffusio

4 No-equilibrium carrier behaviors

5 Desity of state (liear scale) ad Fermi distributio (log scale) Itrisic carrier desity Slightly -tye carrier desity Slightly -tye carrier desity Both ad, ot i thermal equilibrium

6 No-equilibrium carrier behaviors

7 Ecitatio ad Relaatio

8 Semicoductors i Noequilibrium Coditios Ijectio A rocess of itroducig ecess carriers i semicoductors. Geeratio ad recombiatio are two tyes: (i) Direct bad-to-bad geeratio (G) ad recombiatio (R) ad (ii) the recombiatio through allowed eergy states withi the badga, referred to as tras or recombiatio ceters.

9 Direct bad-to-bad geeratio ad recombiatio Thermal equilibrium: The radom geeratio-recombiatio of electros-holes occur cotiuously due to the thermal ecitatio. I direct bad-to-bad geeratio-recombiatio, the electros ad holes are created-aihilated i airs: G 0 = G 0 R 0 = R 0 At thermal equilibrium, the cocetratios of electros ad holes are ideedet of time; therefore, the geeratio ad recombiatio rates are equal, so we have, G = G = R = R

10 Idirect recombiatio: I real semicoductors, there are some crystal defects ad these defects create discrete electroic eergy states withi the forbidde eergy bad. Recombiatio through the defect (tra) states is called idirect recombiatio The carrier lifetime due to the recombiatio through the defect eergy state is determied by the Shockley-Read-Hall theory of recombiatio. Shockley-Read-Hall recombiatio: Shockley-Read-Hall theory of recombiatio assumes that a sigle tra ceter eists at a eergy E t withi the badga.

11 Four basic tra-assisted recombiatio rocesses The tra ceter here is accetor-like. It is egatively charged whe it cotais a electro ad is eutral whe it does ot cotai a electro.

12 Ecess carrier geeratio ad recombiatio Symbol 0, 0, δ = - 0 δ = - 0 G 0, G 0 Defiitio Thermal equilibrium electro ad hole cocetratios (ideedet of time) Total electro ad hole cocetratios (may be fuctios of time ad/or ositio). Ecess electro ad hole cocetratios (may be fuctios of time ad/or ositio). Thermal electro ad hole geeratio rates (cm -3 s -1 ) R 0, R 0 g, g R R, τ 0,τ 0 Thermal equilibrium electro ad hole recombiatio rates (cm -3 s -1 ) Ecess electro ad hole geeratio rates (cm -3 s -1 ) Ecess electro ad hole recombiatio rates (cm -3 s -1 ) Ecess miority carrier electro ad hole lifetimes (s)

13 Noequilibrium Ecess Carriers Whe ecess electros ad holes are created, () t = + δ(), ( t) + δ( t) 0 t = 0 ad i The ecess electros ad holes are created i airs, g ( ) ( ) ad δ t = δ t g = The ecess electros ad holes recombie i airs, () t R = R The ecess carrier will decay over time ad the decay rate deeds o the cocetratio of ecess carrier. d dt [ () ()] d() t t t i or, α r is the costat of roortioality for recombiatio. dt = α r [ () ()] t t i

14 d dt () t dδ dt = α () t Noequilibrium Ecess Carriers r [ () ()] t t i () t = + δ(), ( t) = 0 + δ( t) ad δ () t = δ( t) 0 t ()( t [ + ) + δ() t ] = α rδ 0 0 For low-level ijectio coditio: ( ) >> δ( t) For -tye material, 0 >> 0 Thus, dδ dt () t = α r δ δ () t α 0 ( ) ( 0) ( ) 0 t t 0 / τ t = δ e = δ e r The solutio for miority carrier is where τ 0 is the ecess miority carrier lifetime for electro ad τ 0 = 1/(α r 0 ) 0

15 Ecess carriers We cosider here absece of surface or bulk recombiatios Ecess carrier cocetratio i E VB ad E CB deeds o: - Carrier life time - Absortio rofile - Temerature

16 Ecess carriers Itrisic carrier cocetratio similar to Si i = i = cm -3 For -tye doig with majority carriers cocetratio = cm -3 Mass actio law: 10 ( 10 ) 4 3 = = cm 10 = i Otical ecitatio erturbs this relatio Miority carriers cocetratio =10 4 cm -3 Statioary ecess carrier cocetratio P- hoto flu cm - s -1 for hν=ev (red light) AM 1.5 at 84.4 mwcm - τ- carrier lifetime 1µs X α - absortio of hotos 10-3 cm 3 withi a volume of 1 cm µm deth = = = = 10 cm Pτ 3 α 6 1 s s [ cm 3 ]

17 i - itrisic carriers SC i =10 10 cm -3 - electros i doed SC i the dark =10 16 cm -3 - holes i doed SC i the dark =10 4 cm -3 - electros i doed SC created by illumiatio =10 14 cm -3 - holes i doed SC created by illumiatio =10 14 cm -3 - electros i doed SC uder illumiatio = + = = holes i doed SC uder illumiatio = + = = For majority carriers chage by illumiatio is oly 1% For miority carriers chage is illumiatio is drastical te orders of magitude For -tye semicoductor: - cocetratio of electros comig from doig ad thermal ecitatio is much higher tha cocetratio of electros comig from illumiatio - cocetratio of holes comig from illumiatio is much higher tha holes comig by thermal ecitatio

18 satially deedet carrier cocetratio rofiles i equilibrium (dark) ad uder illumiatio i comariso with the light absortio rofile. Light itesity decay Whereas the ecess majority carrier rofile chages little (the chage has bee magified i the figure), the ecess miority carrier cocetratio deviates strogly from the costat dark cocetratio ().

19 Quasi Fermi levels, defiitios For statioary illumiatio ad sufficietly log carrier life time, ecess miority ad majority carriers eist statioary at the resective bad edges. Their ecess carrier cocetratio relatio defies a ew quasi equilibrium ad attemts have bee made to describe this situatio i aalogy to the dark equilibrium termiology. Therefore oe describes the Fermi level for a illumiated semicoductor i the framework of the equatios derived for the o illumiated semicoductor. For -tye ad -tye semicoductors, E F was give by which ca be writte, based o the aroimatios derived as Carrier cocetratio for illumiatio: () = + () = + E F ( ) = E CB kt l N CB E F( ) = EVB + kt l N VB

20 For electro carriers NCB NVB EF ( ) = ECB - kt l[ ], E ( ) l[ ] F = EVB, kt. NCB EF ( ) = ECB - kt l[ ], ad kowig we ca write NCB NCB EF = ECB - kt l[ ] ad ECB = EF + kt l[ ], NCB NCB EF ( ) = EF + kt l( ) -kt l( ) NCB NCB EF ( ) = EF + kt[l( ) -kt l( )] N CB EF ( ) = EF + kt l[ ] N CB EF ( ) = EF + kt l[ ] Because = +D, we ca write,d D EF ( ) = ECB + kt l[ ] = ECB + kt l[1 + ] Quasi Fermi level for electro is eergetically located above the dark Fermi level.

21 For hole carriers, NVB EF ( ) = EVB + kt l[ ]. Kowig we ca write l[ N VB NVB EF = EVB + kt ] ad EVB = EF - kt l[ ], NVB NVB EF ( ) = EF - kt l( ) + kt l( ) NVB NVB EF ( ) = EF - kt[l( ) + kt l( )] N VB EF ( ) = EF -kt l[ ] N VB EF ( ) = EF -kt l[ ] Because = +D, we ca write,d D EF ( ) = EF - kt l[ ] = EF - kt l[1 + ] Quasi Fermi level for hole is eergetically located below the dark Fermi level.

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23 Cotiuity equatios: The cotiuity equatio describes the behavior of ecess carriers with time ad i sace i the resece of electric fields ad desity gradiets. Δ F () F ( + Δ) Area, A + Δ The et icrease i hole cocetratio er uit time, t +Δ = lim Δ 0 F + + ( ) F ( + Δ) Δ + g F =F + reresets hole article flu (# / cm s) R

24 R g F t + = + The hole flu, e J F / = + The uit of hole flu is holes/cm -s R g J e t + = 1 R g J e t + = 1 Similarly for electros, Cotiuity equatios These are the cotiuity equatios for holes ad electros resectively

25 Recall: I oe dimesio, the electro ad hole curret desities due to the drift ad diffusio are give by: ed E e J + = μ ed E e J = μ ( ) R g E D t + + = μ ( ) R g E D t + = μ R g E E D t = μ R g E E D t + + = μ Or, Substitute these i the cotiuity equatios:

26 The thermal-equilibrium cocetratios, 0 ad 0, are ot fuctios of time. For the secial case of homogeeous semicoductor, 0 ad 0 are also ideedet of the sace coordiates. So the cotiuity may the be writte i the form of: R g E E D t = δ μ δ δ R g E E D t + + = δ μ δ δ ad δ δ + = + = 0 0 Time deedet diffusio equatios

27 Semicoductor Equatios The oeratio of most semicoductor devices ca be described by the so-called semicoductor device equatios, icludig, from Poisso s equatio, where N is the et charge due to doats ad other traed charges; ad the hole ad electro cotiuity equatios where G is the otical geeratio rate of electro hole airs. Thermal geeratio is icluded i R ad R. The hole ad electro curret desities are give by. The two terms, φ ad φ, are the bad arameters that accout for degeeracy ad a satially varyig bad ga ad electro affiity. A comlete descritio of the oeratio of otoelectroic devices ca be obtaied by solvig the comlete set of couled differetial equatios Eq. (1), Eq. (), ad Eq. (3).

28 A case study of i solar cell (ecerted from the aer of E.A. Schiff, Low-mobility solar cells: a device hysics rimer with alicatio to amorhous silico, Solar Eergy Materials & Solar Cells 78 (003) ) Aalytical Model The fudametal structure of the solar cells cosists of three layers: a -tye electrode layer that collects holes, a itrisic layer i which hotocarriers are geerated, ad a -tye electrode layer to collect electros. The itrisic layer is assumed to be a isulator uder dark coditios. Ecess electros (holes) are doated from the - (-) tye layer to the - (-) tye layer, leavig these layers ositively (egatively) charged, ad creatig a built-i electric field.

29 The hotos are absorbed i the itrisic layer, where each absorbed hoto will geerate oe electro ad oe hole. The hotocarriers are swet away by the built-i electric field to the -tye ad -tye layers, resectively thus geeratig solar electricity. The electroic structure of the material is revealed with some critical arameters: the eergy badga, the effective desities of states ad of the coductio ad the valece badedges, ad effective mass ad mobility of carriers i the coductio ad valece bads. I the i-layer, hotos with eergies (ev) greater tha the badga ca ecite a electro out of the filled electroic levels i the valece bad ad ito the coductio bad; leaves oe

30 hole behid i the valece bad. The geeratio rate will be deoted by. The ecess carriers raidly shed eergy ad coalesce oto the levels close to the badedges. The average seed of these carriers i a electric field E is characterized by the electro (hole) mobility. The recombiatio rate is roortioal to the desity of electros, the desity of holes, ad the recombiatio coefficiet.

31 A ositive voltage V corresods to a eteral electric otetial that is larger at the -layer electrode tha at the -layer electrode. The value of V at which J=0 is called the oe-circuit voltagev OC. At short-circuit (V=0), the magitude of the curret desity is deoted the short-circuit curret desity J SC. No ower is delivered to the eteral circuit uder either short-circuit or oe-circuit coditios. The ower desity P (V)= J(V) V is delivered to the eteral circuit is maimized at V m (the maimum ower oit). Uder short-circuit coditios with a costat geeratio rate throughout the absorber layer, every hotocarrier that is geerated by hoto absortio will be swet across the thi absorber layer by the iteral electric field ad collected.

32 Therefore,. The short-circuit curret desity is roortioal to thickess for d<150 m. The oe-circuit voltage is ideedet of thickess, ad i terms of the fudametal arameters of the itrisic layer material

33 G evoc = Eg + kt l[ ] bnn. R c v The fill-factor FF, which is defied from the ower relatio P=(FF) J SC V OC, saturates at about the same thickess as the short-circuit curret desity. For itrisic layers eceedig some collectio widthd C (determied by the buildu of a sace-charge of driftig carriers), the outut ower desity P saturates at some maimum value. The ower desity saturates for smaller thickesses tha does the short-circuit curret desity or the fill-factor. This is due to that the electric otetial across the cell is larger uder short-circuit coditios tha it is at the maimum ower oit. Photocarriers geerated deeer i the cell thad C may thus be collected uder short-circuit coditios, but this additioal collectio does ot cotribute to ower geeratio.

34 If the mobile electro desity (i.e., the desity of electros occuyig levels above the coductio badedge ) is deoted by, the the electro quasi-fermi level is defied by For holes, a similar eressio ca be defied.

35 EF = Ev -kt l[ ] N. Thus, hotocarrier geeratio i a material creates two electroic reservoirs with differig chemical otetials. Electros i the coductio bad are oe such reservoir; holes i the valece bad are the secod. Electro ad hole currets may be eressed i terms of quasi-fermi eergy gradiet EF( ) J = E = ( - ). Uder oe-circuit coditios, the itrisic layer is i isolatio (i.e., without gradiets or electric fields that would trasort electros ad holes i sace). The -layer serves as a ideal electrode, ermittig us to measure E ; ad similarly, the -layer serves as a ideal electrode to measure E F. A voltmeter F coected to the - ad -layers the measures ( )/ V = E - E e. OC F F Assumig the semicoductor layer i a device is electrically eutral (=). Uder cotiuous illumiatio, v

36 i steady state the rate of geeratio G is equal to the rate of recombiatio ( G = R = b ), leadig to the carrier desities = = GbR. The quasi-fermi eergies of free carriers i the layer ca be deduced to be R E E F F kt = Ec + kt = Ev - G l[ ] bn R c G l[ ] bn. R v Thus the oe-circuit voltage ca be eressed as G evoc = Eg + kt l[ ] bnn. R c v Uder oe-circuit coditios, the rofile of the electric field (dashed curve) ehibits a quite large electric field ear the /i iterface. Thus, hole hotocarriers i this high field regio drift raidly towards the -layer. However, this drift simly cacels the diffusio of holes out of the -layer, resultig i a zero et curret (drift-diffusio).

37 Whe the cell is biased at its maimum ower oit, the electric field eetrates more deely ito the cell, ad et curret is flowig. The field declies liearly towards zero at a deth collectio width., idetified as the The geeratio rate rofile G() is essetially a costat. For dee i the device, all hotocarriers that are geerated recombie without beig collected (i.e., G=R). However, every hole geerated i ca

38 reach the -layer ad is collected. Thus, the curret that flows i the eteral bias circuit may be estimated by j = egd. C The electric field, which declies liearly to zero with deth, roduces a uiform charge desity r withi collectio zoe. From semicoductor equatios i the collectio regio ( 0 < < d ) C Jh = eg ( -R) (cotiuity equatio) Jh( ) = s( ) he ( ) (drift relatio). E ( ) s( ) = (Poisso's equatio) f At =0, R=0 ad E (0) =- rdc e. From the cotiuity equatio, J (0) =- egd. From the drift relatio, C (0) (0) C J = rm E =- r m d e. Thus, r = eg e m. We ote that d = D V (4 mf eg ) = D V [4 f r] is determied by r ad the c electric otetial C D V = Vd ( ) - V (0) = d r e across the collectio width. C The outut ower desity is give by PV ( ) = JV ( ) V, i which JV ( ) ca be aroimated with the hole

39 curret J (0) = egdc at =0. D V = V OC - V is the differece betwee the oe-circuit voltage ad the alied otetial V. Thus, PV ( ) µ V - V V, which reaches a maimum OC 3/ 3 3 1/4 m ( OC 3) ( me ) = at P V eg V m = V 3. Ad OC 1/ 1/4 C ( OC 3) ( me / ) d = V eg, imlyig that the maimum ower desity is a fuctio of the hole mobility. As the voltage across the cell is reduced fromv OC, hotocarriers are collected i the regio where the recombiatio rate R of electros ad holes falls below the geeratio rate G. The curret desity ca be aroimated with RV ( ) j( V ) = egd(1 - ) G, where d is the thickess of the itrisic layer. The alied otetial V(R) associated with a give G recombiatio rate R ca be determied from evoc = Eg + kt l[ ] bnn but relacig the geeratio rate G with the recombiatio rate, which yields R c v

40 R ev( R)» Eg + kt l[ ] bnn R c v for a arbitrary voltage V. Thus, we ca deduce e [ VOC - V( R)] = kt l[ ], ad J( V ) = egd [1 -e(-ed V kt)]. The ower desity at a arbitrary voltage V becomes PV ( ) V JV ( ) maimum ower desity ca be achieved with V m, which shall fulfill the coditio of ( e kt) Vm = ( e kt) VOC - l[1 + ( e kt) Vm ]. G R =. A For br = 10 cm s, we estimated VOC 1.09 = V. Solvig the eressio by iteratio, we obtaiv = 0.997V, which is fairly close to the value V m =0.97 V for a 50-m thick i-layer cell. m

41 Effects of Valece Badtail Width No-crystallie semicoductors do ted to have fairly low bad mobilities. The best documeted case is that of hydrogeated amorhous silico, for which the electro bad mobility is about cm /Vs, comared to the value i crystal silico (1800 cm /V s at 300 K). I ractice, the effective mobility is further reduced by traig of carriers ito localized states betwee the badedges. A satisfactory accout for drift-mobility measuremets i amorhous a-si:h could be give usig a eoetial distributio of these tra states. For the valece badtail, the tra distributio vs. level eergy is writte 0 3 V V V V g ( E) = g ( cm / ev ) e[ - ( E - E ) E E ]. EE V is the width of the valece badtail, which was foud to be aroud 50 mev i a-si:h. The coductio badtail width is smaller aroud mev, ad ear room-temerature (kt=5 mev) coductio badtail traig ca be eglected.

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