Sharma, S K

Shekhar, Chandra

Yadav, K S

IACS Institutional Repository

Indidri J. Phya. 50, 986-990 (1976)Threshold energies of electrons and holes for impact ionization in siliconK . S !kfADAtl, S K STTAKi^ l/V, CllAls'DKA iSllKKIlAll* AND K  S B aLATKCentral JCleclionics /f^ ngnicti intj Uvsearvh Jnt,Uiu1r, l*ilani {Ilajasthan) [deceived ili March 1975. revised 19 March 1970)Impact jouizaiio]! iu .silicon lias bi^ cn cxpcM JiiU'utally studied by vavious workers (Mclcay 1 9 5 1 ,  Miller 1 9 5 7 ,  Cbyiiowetli 1 9 5 8 .  1 9 6 0 ,  Ckietzberfrer (d at 1 9 0 H ,  Moll & Ov'crstiueteii 1 9 0 i i ,  Let^  et al 1 9 0 4 . )  They have measured ioiiiza- t<iou coeiricieiit (number of ionizations pei unit path length from tlu^  measuie- meiits of breakdown volrages and iire-breakdo\ui mnlf-iplicalion factors. The expoiiiuontally moasiired ionization ecKiffieients are fitted lo the theoretical einves ol‘ VVolir ( 1 9 5 4 ) ,  SluH'kley ( 1 9 0 1 )  or Biiiaff ( 1 9 0 2 )  by adjusting three paia- meteis (1) tin* threshold energy of ionization, (2) the iiican tree, path for phonon emission n and (li) tlxe lainstant energy loss associated witli the si'afh'rmg Avhieh IS ecpial lo the optical phonon energy in silicon as optical inode ol scatteiing dominafes in this mateiial. Litt.le attempt seems to have boon made for eal- culatmg the tln'oshold energy tor impact ionization vitJi the help of band striic- turc of the material. Ahmed iK: Khoklc ( 1 9 0 7 )  havi‘, however, calculated the tliroshold eiitJ'gv of ionization for electrons nsirig only lieavy hole mass employing the gni]thical method ol f'rfintz ( 1 9 5 0 ) .We in this conimuiiu'ation ha\ (' attempted to ealculcite the threshold cneigy of elections and holes Lor imijact ionization by an analytical appioach described by Shekhai & »Sharnia (1974) foi gal limn arsenideSilicon has six valleys in it.s conduction band, each at a distance of 0'80x27r/a from the center of Brillonin zone in < 100 >  directions. Density of states mass of electrons is Tin* A^ aleucc baiul is degenerate having throe branches,trwo (>r which have coincideiu. tops at k (<K)0) and the third the split, off branch also has its top at k(OOO) but 0*04 ev bclov' the tops ol the other two branches. The lattice constant a ^ 5'4S A and the energy gap Eg — 1-107 eV at 300°K. The masises of hea^'y light tind sjilit ofl holes are 0-5 Wq, 0-16 and 0-23 vIq respectively.* Birla Inutiinto of Toohnology and Science, Pilani (Kajasbhan), India.980Threshold energies o f electrons 987The values ot tineshold ofioiiizaiioJi of olectroiis and hol(‘s lor vai lous ]>ossibJe impact pi'ocosses am given in tables 1 and 2 b(‘loNV Values of tlireshold oi ioniza­tion more than 4 ev are not reported as they are not ineaninglul,Table 1. Ionization thresholds for tdectrons m silicon Positions of Elorironsliolo massIniiizai'ion onorpy in ev forP r im a r y Jh’ im ar^ S c'ioondary n o i  in a l I h n k la j ip  1’ roeoHKt le d r m i e le e tru ii ( ' lo c i i o n jir o c e s s ------- ---------------a fte i in ip iie l ii f ie i im pac'i. f/ -  -  iTT/r/; G - 1(1 0 0 ) (1 0 0 ) (1 0 0 ) .oO 2 1C _ 3 18.IT) 2 1C 3.7C.2:1 2.159 - - 3 02(1 0 0 ) (1 0 0 ) (1 0 0 ) TjO 2 .1 5 3 . IS.1C 2 .4 5 3 .7 5 -2?. 2 3 02(IhO) (1 0 0 ) (1 0 0 ) CO _ 1 41.JC - ] 55---------- ---- - - ----------- - -----------1 55Table 2. ionization thresholds (or lioles in siliconI’osition of hojesPrimary Primaryhole before hole aftei‘ impact impart--------- —  Position ofSecondary elortron Normal hole ]iroeessIonization onerg)'^  in ev foiUmklaiip Piocess-47r/a G -Cojiduclion1 1 1 V a l l r v 1.81 — 2.641 1 1 *d o - 2 01 — 2 .9 81 2 2 tlo - 2.30 3 551 1 -l lo - 1 99 2 911 3 3 -d n - 2 20 3 301 2 ■5 -d n - 2 25 3 412 1 1 -ilo - 2 45 — 3.642 1 2 -d n - 2.81 —2 2 2 -do^ 3 47 —2 1 3 -d o - 2 74 —2 :l 3 -d o - 3 17 — —2 - :i -d o - 3.31 — _ _3 1 1 -d o - 2 oi 3 263 1 2 -d o - 2,50 — 3 HI3 2 2 -d o - 3 02 _3 1 3 -d o - 2.89 3.093 3 3 -d o - 2 79 —,3 2 3 -d o - 2 89 —9 8 8 K. S. Yadaii et alPoi ilui ca kill alio 11 ol thiivsliold ciiorgios ior impaot ionization lK>tli normal and IwocosHCH havt) bcfon oonsidorcd. 7’1k‘ momojiium conftor'vaiion equaljims for normal and Umldapp procissHOH rospoetivoly areki -- k\-\-k'»-\~kh k\ ~ 2'^ rkh-\~G... (la) -  (lb)whore, k\ in the wave veetoj- of jjiim ary el(‘eti'on befoie impact, fc/, k \  and kh ave WiiA i^ veetors of piiraaiy (di^ ctron Hi^ condary electj'on and secondary lioJe after the impact G in iee/q>j‘oc«l latlit-e vect.or. The enei'gv of eleeti’oii in tlie conduction hand is given by(2)wlieie fc„ IS till' Auu'-e v"ect,or of the eonespondmg conduction band ininimuni Since energy minima m conduction liadd lie in equivalent 100 directions expect the minimum energy for ionization avjII be Avhen the motion ol the particles iiivelvcf!, in tlie impacl process js confined in 1100] direction; hence ihe wave veetors can be considiMvd as scalarsAssuming the eneigy relerenci^ at the bottom ol the conduction band, tlie energy couservatiou equation for the first piocess listed in table 1 toi the impact. Kinization by electrons is2mr 2vi,. ' "  2 W f '  “ 2maAlter (‘limiiial.iag /.>, bom  ei]. (;i) with the help o f eij (la) ami then minimizing it with respect to and Z:',., cq (If) hermues, / \ I . ,_ 2 m,t:g \... (4)Equation (4) is qiiadrat iis ni and is solved numencally to give tAio A^ alues of Z;,,. Then energy is calculated by using eq. (2) for both values of Z-^ ,. The lower valuer of onci'gy is takmi as threshold energy provided that corresjxmding Z'„ is real, and within the first Brillouni zone; corresponding k'o and kh are all real and within first Brilloiim zone and the wave factors of electrons must correspond to their valley assumed in a particular process Such physical restrictions are apx l^ied in the calculations of threshold energy for all the processes listed, m tables 1 andThreshold energies o f  electrons 989From the iahle J it eau be seen that tlie thresholcl energy for impaei. ioniza­tion by electrons js lowest- when primary eleetron is in (100) A'alley and both the electrons are m (100) valley after impact. This is bccans(‘ the ITmklapj) process reduces the energy requirement for ionization in cases Avhero largo momentum transfers are involved in the ionization proc-ess. The values of threshold enorgj'- for this process range from 144 eV to 1-55 oV The loAVor A'-ahios of threshold energy for electrons foi normal process range from 245 eV to 245 cV This indicates that Avheii the energy of <*loctrons is increased by applied field, impact ionization by Umklapp process will start lielorc the onset ol impaid- ionization by normal process The impact ionization ivill increaso with the increase ol electron energy, consequently with the increase of applied electric field. Thus there will be a range threshold energy for electrons from 144 eV to 245 eV. These values are in excellent agreement with those deteimined from experimental data by Lee el al {E E 1-5 £  with most probable valiu? of l-b eV). Moll k  Overstraeten {E — 1-8 eV), Mckay &. Chynowetb {E — 2*5 eV)From table 2 we can see t-luit tluesliold eiu r^gy of holes is less lor normal process than that for IJmldapp process The lower \aJnes of threshold energy for holes range irom 1*84 oV to 3 47 eV Those values arc also m good agreement Avith the experimentally determined \’'alues of Moll & OA e^rstraeten {E — 24 eV). TiCe el al luivo fitted their oxporiraoiitally dat-a of ionization coefficient- (number of ionizations per cm) by assuming the same value of tlire-shold energy for electrons and holes {Ei~— 1-0 eY), but they baA'c taken different values ol mean free path n lor phonon soat-iering by electrons (50 A <  Ap <  70 A) and by holes (30 A <  A <  45 A) Tf the moan free path for holes is assumed the same as for electrons, then Lee et ctl data AAdll give the value of threshold energy of 2*6 eV.Similarly Chynoweth and Mekay’s data are also fitted to Wolff theory by assuming same threshold energy but different mean free path for optical phonon scattering for electrons and holes. T1 the scattering mean free- path is assumed the same, the thrc^shold cnei’gy of holes wT>uld then be 3-5 cW Our values of threshold energy for impact ionization are thus in excellent agreement Avith the c^xpofimental results. Tliese e-alculatic>ns further show that the impact ioniza­tion is not very sharp, boc^ ause all processes do not start at the same energy and consequently at the same field.Our values of threshold energy fer* electrons are higher than thosc^  calculated graphically by Ahmed & Khokle {E =  M 8 to 2-1 eV) the difference seem,s to be due to the different mass used for heavy hole by thorn (w =  0-78ReferencesAhmed S. & Khokle W. S. 1967 J. Phys. Chem. Solida 2B, 2499. Boraff G. A. 1662 Ph^a. Eev. 128, 2507,990 K . 8. Yadau et alCliyiKHveth A. Q. 1968 Phya Ecv. 109, 1637 Chynoweth A. G. 1960 J. A ppl Fhya. 31, 1101.Franz W. 1950 Handbook de? Phyaik 17, 185.Guotzborgor A., McDonald B , Hartz R. H. & Scarlett R M. 1903 J, Appl. Phya. 34, 1691. Loo C. A., Logan R. A., Batdorf R. L., Klomack J, J. & Woighmann W. 1964 Phya. Bev. 134,A661.McKay K. 0. 1954 Phya. Rev. 94, 873.Eillor S. L. 1967 Phya Rev 105, 1241Moll ,T. L. & Ovorstraotoii R. Van 1963 Solid State Electronica 6, 147 Moll ,1. L., Mover N T 1961 Solid State Electron, 3, 165Shokliur Chandra &. Sbarma S. K. 1974 Indian ,/. Pure tV Ap]d. Phya. 12, 179.Shekhar Chamhn & Shiunm S. K 1974 Phya Letter.^  50.Shockley W. 1901 Solid State Electron 2, 36.Wolff P A. 1054 Phya. Rev. 95, 1415(nd-ian J. Plvya- 50, 990-092 (1976)C om parison  o f  probe and sp ectroscop ic  e lectron  tem perature m easurem ents in a h-f. h ydrogen  plasm aD R  G uptaDepart?ycnl of Physic^ i^, Umv(o\sj{y of Jodhjmt, Jodhyur {Rmivml J4 January 1970. revit>eA 12 May 1970)MoGi'hgor (1900) Jias i!Xt )^orimontally loiuid tlial i.s au oqtuva]onc('- ol' t]u‘nlociron ininporatuj’oH (found Ly t]io pjoho mnihod) and tJxo excitation tonniera- turoH (found "by tlu^  ,si)i>ctroscopic im4hod) for an ai-gon d.c plasma. Kcrdiniind (1970) ]ias (!onij)a3'o(l llu' results for tlio (‘[(jctroii lomx)craturoS moa>sured by tlie sjiiMd'J'oscoxjic and xnobo inotJiods ior the jda'^ma of an H0-O2 flaino. The author has j)’‘'0‘^ <''i.tly comparc^ cL oloctJ’on Icnupcraturos in an h.f. plasma of ]iydTog(m gas by tlu'! ->x>uctroscoxnc. and tho double probo mtdhods and Iho results of Te hav(' born oondus c^ ixtlv (‘boehod by calculatm.g Te itsjug Stoejibock’s formulaFoj- tho xirobo moasuronnuits tlxo j)yi‘(vx glass dischai'go tube having a length of 51 5 cm and diameter 4 5 cin itsod The probe AA'as made of a tungsten wii'e 0 2 mm in diamot(‘r and proAudod Avith a cajAillary ciovor It Avas located at th(^  middle of th<A tube and projected radially inside the tube, having a oollect- ing length of 1 cm tei’minating at the axis of tho tube. Tho anti-probe Avas in tho form of a grid having a suiface area about 160 times as compared to the probo aroa. Ihe distance betAA^ oen tho probe and anti-probe was 3 cm Two external sloovo electrodes of 1 cm AAude thin copper foils Avore used and coimccled to tho Hartley oscillati u' for excit ing the discharge Hydrogen gas Avas introducedMectron iemjperaiure measurements in h, f . plasma 99linto the tube by ditfusion tlirough a palladium tube. The discharge was studied at five pressures viz. 0.03, 0.07, 0.13, 0 19 and 0.25 to]T. and tlxo pressures wore measured by tlie Pirani gaug(>. The excitation fVequoncy was 18 MHz, and voltage 300 volts. Tlu^  cleotroix temporat-iu'es Avorc> calculated from tlui slop>(V J \-1  j and r A>Jiert* j.s total iojxir current to theprobe assembly and i , the c l^ectroixic cum nt to 11m x)vobi‘^ and F, tjio p d between the j>robe asstunbly Tlxe J'e.siilts for electi’on i.oTnj)(vriitur(‘,s are repjaxluced bt^ loAV ixx the table.Tlxe elootJ'on temperatures have 1)0(mx also (;alculated by usiixg tlxe theoretical formula of Eixgel & SteeixluKh (1934)\/x1 i()\'io7r;^jf^2/?2Avlxere x — eVijlcTe, in which c is electronic charge, F, m onisation potential, h Boltzmamx’s constant Tg eleetjojx temporal,me and C is a constant whose value; lor ixydrogon is given to be 1 3 5 v l0 -- , Jl tlu' radius of dii'ehargo tube P  the pressui'o in mm HgThe si^ t, ux> used fox sx)octroscoi>ic tletermmatiojx of the ivlecti’on temperatures c.onsistod oi the dihiOiargo tiibi', and a vacuum systoixi IJie same as used for the probe study, llui discharge is stiidiotl undo.r the condilioxxs idojxtically the same a.s tho,so for 1]xe jxrobe oxperimejits Tint central £)aj’t of tlii' ghnv column is focusiswl oxx the slit of a Hilgoj’ (11) sjAectrogi-ajilx havnxg a dispersion of about 18 A.U./mra at 4(K)() A U. Avhich is about the mean wave li^ ngtlx rogioix of the SDxglei and tiipb^ts to be nioasurid Tlxe jxhotograplxs of the .specti'a aj’e takoix on Kodak plates foj‘ identificatioix oi tbo linos. The intensity ol lixxos is moasui'od by a Moll self recording inicrophotometer. The electron temperatures were computed from the- cur\ es given by Brasoliold (1930) The ro,suits arc- j‘oi>roduiHHlbelow" :Table 1S.No.Pressure of hydrogen gat? in inm. IJg.Te, measured by probe method‘‘KTecalculatedT .byspeotrOBOOpicmethod“Ki. 0.03 60.000 56,000 68.0002. 0.07 57,600 50,000 54.0003. 0.13 46,000 36,000 30,0004. 0.19 42,000 30,600 26,0006. 0.26 41,000 28,000 26,000&92 D. R. GuptaJl i.s soon that at highor pressures the spectroscopic values of the electron tem- pc^ ’aturcs do not agree with the p'’obe vtilues This may be due to non-roliability of the values beoaus(\ of increased number of moJcculai collisions at those pressures which may affect proci'-ss of excitation and tJie intensity of lines from which determination of Tc is madi  ^ by spectroscopic method However we could liavi^  a check for tlie prolu  ^ value of 7’,. by an alternative experimental set npMy llianlts ai’(i due to J)r. for this work.G 1\ MiOita for pioudiiig valuable suggOiStionSR eI’EKENOESBraaeaeld, 1930 Phys Eeo. 35, 92Engel V. & Stoonbeek, 1934 Electnschc (jUaerU Ladungen, 11, 8C. Fordmard J. 1970 Phya. Letters 31A, 370.McGregor W K. & Browei L E 1966 Phys, IHmda 9, 820.

Threshold Energies of Electrons and Holes for Impact Ionization in Silicon

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Threshold Energies of Electrons and Holes for Impact Ionization in Silicon

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