F'Je='-'l:"J,,== -e-n= :l--p. (1) 4n iP

Bull. KrTtshu lnet. Teeh.

csf. & ] . .g..) No,fi. Ig6o

Role of Trapping Processes in the interpretatinon of Lifetime of Carriers

Nobuo HAyANo

Department of Physics, Kyushu Institute of Technology,

I introduction

.," ,2.`.h,:.Mg;hg,d.O.f,le,".e,sgi,g,ct,i,",g,th,e",BLu,e?.e,,zn,g,e,u.ef,a:e.,re,e,om.bBagl?e, lig,g.t.'se.ieitr,?,mdi.g",et,i:.ke,,Se,?t.ai,e.},iegE:,2q,s,g?,`Ly.3o.S.etflm,g'e,e,gh,,e,sa,Evhe,I

iexepd,estshi:v,e`i.hy:,Oililo,rO:fl,egM2nffes,filt:;g,9s,oiah,ne,a[xaid,Xilei.gce..il13,dpgM.ME,liÅÄ,ta.asstrSm:g..:.e,i

ler than tbose deduced from PC data. Improving this point he took into ae- g:iUaEhtihfeetl•"mfleUedne:eucOefdt}raoPhlei;:idiSEinditha9forbiddenbandandcou]dexpiainthe

lifl ," ti//g,t,r,e,",t[,le,"i.t,'IZ,ltlX'tll:O,PqO.S,etdi,t.he.fq",g::,iki.e,P,{;;n,a,ndT,(electronandhole

F'Je='-'l:"J,,== -e-n= :l--p. (1) 4n iP

Soj.tti,"lingeIS./eh8zS2pii2ti2"2,:,efesMh,ZsO.•/LtS/i,M,:,l2e,`r:d,eisX/P.2atS:,XhEgCi,g,C:"m.isti/7fitm;Si;,:v:a:n:,/h,e:r'

IVoitphr:eteieVaailhttaYt8rfig?se. PeSitiOn of the trap ievels and to make them cerrespond

ll Dyllamics of Recomhination 1'rocess

d, :LZL.:1!!.L At first, we eonsider two types of trapping centers;

electron trap levels which is located at E. and have the density N,/cm'a measured from the bottom of the 'rLt Nt conduetion band, hole trap levels with energy E. and

the density P,/em3 measured from the top of the fi11ed level. The width of the energy gap is E. and the Pt Pt values of E. and E, can not exceed E.; E.ÅrE., E.ÅrEn•

`n' is the density of free electrons, n, is the density of trapped electrons and p, p, are respective quantities for holes. Accordingly, there remain (N, -n,)/emS

29

In Fig. 1 we as$ume that all the donor and acceptor levels are ionized beeause the doner levels lay elosely to the bottom pf the conduction band and aeeeptor levels are elesely Saying te the tep of the filled band.

Then, follcwing proce$ses are as$umed;

-

-- - - - --• " " xi . ---ny- -

- ---yE---"-" - T----pt---

rc

-e--.

-9(- ----

- pt - ---

pt --

-m- -- - --

-- - .----

---- - .- --. --

--. ew

-g--X

ny - - ny -b- -- -b --- ---

Fig. 2

1. recembination "f free eleetrens with trapped heles 2. thermal liberation of trapped hoies

3, ereatien ef free electrong by means of thermal er eptiea1 precess 1eaVing

-N --

-- --

ew

Role of Traping Procrres in the rnterpretation of Lifetime of Carriers 31

with trapped holes 4, trapping of free holes

5. thermal liberation of trapped electrons

6. recombination of free holes with trapped electrons 7. trapping of free electrns

8' tCrraepaBieOdn 81feefEreoenhsOleS bY Means of thermal or optical proeess leaving with

9. direct recombination of free electrons and free holes 10. direct pair creation by means of thermal process.

.Above proce.sses are illustrated in Fig. 2, In Fig.2 we ean immediately ob- tain the following complementary relations between each proeess; 9elO, le3, 6e8, 4Åq-2, 7e5. In other word, if we set our emphasis to the variation of n, n,, p, p, with time, we obtain the feur differential equation;

-d d7 =f(1, 3, 5,7, 9, 10) = - vsip,n + po3e-CEpirrpt + vose'CEn•'kTnt- vs7n(iVs - nt) - vsgnp+g

(2)

(lni

--

EiJt =:f(5, 6,7, 8) -- - vose-`EntkTn, -- vs6ntp+ vs7n(trNT, -n,) + :,ose-eEntlti'n, (3)

tLe- ==f(2,4,6,8,9, 10) = vo2e-eEprL'T( l', -p,)-vs,,p(P, -p,)-z's6n,p+ vose-CE"rkrni -- vsE}np+g

(4)

dp; --

-ai- -f(1,2,3, 4) : : -vstp,rt- vo2e-eEpta-T(P, -p,)+ vo3e'eEpikTp,+ vs,p(P, -p,) (5)

In the above formula, we assume that by the incident light, g pairs of elec- trons and holes are created in one second only through process 10. v is the thermal velocity of free carrier and s represents the trap cross section (cm2).

Expressions (2) to (5) represent the phenomena which are eommonly suitable to all types of the traps. These equations are mutually dependent because holes and eleetrons are create and annihilate simultaneously in pair form, and so {(2)+(3)} -- {(4)+(5)}=O. If we set vs,p,= -1- , we get the mean free rl

time of electron in process 1 and if we set -1 - ==vosc-eE,,"kr, we ebtain the 7s

Me an free capture time of electron or hole with energy E,. Here, vos is the OSeillating frequency of electron in process 3 and has numerieal value 10"/see.

Flirom the same procedure we ean difine T,, r2,---T, of all the processes• Using these Ti notations we can written down (2) to (5) as folloows;

lll't=--l n+ 1i,-p`+-iig nt-ii-, n- 1ii-n+q (6)

11 du

11

-ilÅí,S== -- ig-nt-- qn:+Tn+-:i;nt (7)

Ztil-:: lii-(p,-p,År--- -ll;-p-- il-",+ Iilll-u,- j:-u+g (s)

'"I'l3Pevt' =v l?n-" -lt- (pt "-pt)+ nv3;pt+ zll p (gÅr

In what fol!ows, we classify above general exprGs$ion to more practical enes.

J.; ptype material with deep trap levels.

In this material, enly prceesse$ 1, 2, 3, 4, 9 & le are available and preeess 3 is pessible te Reglect because of its small probability. !f we also set s2 :;: s,, trap leveis acts like reeombination centers Df two step$.

du 1 1

rEr, =" " -i-n- Li7n+g Åqle)

-{/I"`=O n,=eonstant=verysmall Åq11)

-[ltt "' uzi\ ÅqP` -p`År"{?p" l;-n+g (12) tll/l;` == --• +-}n--- ilt (a --pt)+tp a3)

ls; if siÅrs4, these centers act like electron traps and equations (6) to (9) reduae ue (14), (15), (16År, (Z7).

dn. 1 1

i, ta ke "Tn- -;;n+g (14)

-lll.lilft--o (ls)

-[S/ 'lts-(pt-p:)ny l;n+g a6År

":lli/l:f' --"-tn--jl,;- (p, -p,) a7)

lci if We Set siÅqs4, in this ease center$ become to hQle traps and expressiOnS (6) te (9) reduee te (18År te (21),

dn l

'zir, =e -Fg"n+g ÅqlsÅr

dR,

-S/=w'iig', (Pt-pt)- l:-pny -ik-g+g (2o)

Role of Traping Proesres in the rRterpretatie: gÅí Lifetime ef Carriers 33

dpt-e 1

uEr, -keF-'7, ÅqPtinpt)+-lp (2i)

ll. ; n-type imaterial with $haiow trap levels, IR this material enly preeesges 5, 6, 7, 8, 9, & le are avai}able and process (8) ean be eonsidered as negligible small probability. In ease gf s, ;:;g s7, we get the follewing expressions ;

dn 111

T, = "tk'rei-P T. n- i5Ln+g (22)

dns 1 1 1

ua,l- `="?g- nt- ;,- n`+ T.n (23) 2!Xt =: "- t,i•-,- ni-ta+g (24År

-cl--de-`th=O pt=ecnstant----verysmalL (25)

Ub; it we take s7Års6, in this case, eenters become electron trap levels and they are

dn 1 1

1

-dii`"'E,' iii- ., i2- -.-,-R+{7 (26)

nydalsP! =- l,lr n:+ "m;"-n (27)

' dp - 1

'-di'-- ., n+q (28År

II.; if we take sTÅqse, this ca$e corresponds to dominant hole trap proee$s and we obtain

11

clri

ZitT=-f, Rr- -.'6 R+9 (3e)

{IZ/` `d -.1, 'i," um.1, n; (31) g,ig --.,i-gn,- .i, re+e Åq32År

dP` ,,. e. (33)

(]e`t

Ill; In the small energy gap materials, proeess 9 has an important meaning and in seme material, direct recembikati"n akd recembimatien through traps beeome the eompeting processes•

M; In large energy gap material$, proces$ (3), Åq8) and (9) have aimost no

34 N.HAYANO

influence to the total probability and usually are omitted.

It is worthwhile to draw nomographs to survey the probability of individual processes with varying n, n. , p, P» Eo, En and E _{p ,} and then we can immediately estimate the range of lifetime of carriers in any type of traps and semi-con- ductors.

Acknowledgment

The author are greatly acknowledged to Prof. A. Okazaki of Kyushu University to point out the importance of the relation between lifetime of free carriers and PME effect.

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[6J KURNICK & ZITTER; J. App, Phys. 27 278 (1956) [7J DIXON; Phys, Rev. 107 374 (1957)

[8J Moss; Photoconductivity in the Elements, Academic Press (1952) [9J ROOSBROECK; Phys. Rev. 101 1713 (1956)

[10J ZITTER; Phys, Rev. 112 852 (1958)

[lIJ WALTON & Moss; Proc. Phys. Soc. 602 (1959)