BSTJ 30: 4. October 1951: Hot Electrons in Germanium and Ohm's Law. (Shockley, W.)

21.4 MB·English

Hot Hlectrons in Germanium and Ohm's Law
by W. sHocELEY

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a iltteaucg neoaea ia ee
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ARTES rary

ae
‘AS. Ammosinate Equivalence tn Ea Se Bt
3 Rca cena ae

1, Iemanvcroy: Fenpnervtat: Destartoxe ston Bria's Law
NHI starting pola of may rsches of ysis ie Une relaton
Aung the Rost punsicot ofthese ane Hooke’ law, whl teltes

rom ane ara for ait ves, Newon'e axond hon af toto

ttn (hme le, I al of thew case, the linea ration fs onan 3

Powimatin tha mye regen the Sis term in Tay’ expansion

bf the fonction mintonip between oke Pan wriahle, lepine poy

‘principles ae ong to aitention when the ravEness rage = reseed.

‘Of the thee laws mentioned, Newton's i, of cour, the one ia whic the
‘nse of finearity ithe most signtrantrepresnting a it dots the ev
trance of relative ess nto the Ins of maton

‘The ahre of Honke’s lw may be of ether a primacy or seconde fom.
OF MaernOss a GLAMASGLAL As QHD LA oon

1 sold contains vids then ander a certain pressure it will crumble and
Fil he voids. This is seondry elect. Ifthe sample is homogerenn
however, high pressures wil produce fundamental deviations from Iooke'’s
la, these devictions asking em the nonleatity ofthe ‘ores bebmeen
stirs. Stes ofthese asin fete hy Hein bor amg, her
{ings puta Grr ai ewer of the fora Velen
in oni estas and che presooes of electzongase in metal

‘Deviations froin Ohm's law for electsnie luc in serconduciors
are almest the rule ater haa the exception, but the meet emia eases
fare secondary rather than primary, The primary Inear relation for the
‘uti proces sat eee the dst lacy af an electra, or hol,
‘ard the re tht drives, This relent

se wT) K an

hes ne bility (7) in function of de empertune of the specimen.
5By 2 fndomental destin fron Obs law we sal men a deviation is tis
laces rltinship essing from the largeness of B aticr then other causes

“Thomistor action ia typical of a secondary devinton from Ob’ law
A thermistor is umaly a tke-torsnal Geult clement in which th current
fons through an clectonis semiconductor. The semiconductor kas the
plopety that ia sestance deseans apis othe sensperatuze incest;
tt the pyc is for this eres sere ne nmber oc
Aeting wetrons for oles or both) ith incrasing emperaine. The
pusaye of current eats the themistor and ite resstanoe changes conse
‘quel the linear ation belveen careut and voltage ful and infact
there ray real x decree of vllage with increasing cueane go that a
Aieentat negative resistance i chnerved, The elctre feldeaze 20m,
hhonever, hat equation (1-1) is valid provided the depeadanceof uy an the
Temperature i cake ncn acon. Ab eaperinenta! poof that ne funda
rental deviation of Oha’s law eccureieSumnished by applying « small
2 test imal on tp of dias ht produces heating the fequeney
iro higher then te thermal ramaton rate, the we. resistance i fount
toe srl that expert for ve verve empenstane

‘The etal nonin of erst rectier, nr vation of ae
isons ane clay secondary sad are sana th changing sumer uf

Th this tcl ee ball discs some experimental evidence of fu
ert éeviation in Obm' ae for ceca n te rma sblaine
bp J Ryder af Bll Tepe Labora! We sal eseibe bis ex

ye and W, Sete, Pe 2,330 95H
992 wn wich everew Teeca, JoUNNAL, OoTORE 19ST

pevimeatal ecaigus and reels bel nthe nest section an sal the
ret ei spore ofthe qiantitative Coty tht explains Chm, leaving
the bale of te ratheritice ipultions for che nents,

‘Bere clacurirg Ryders reulls, we may indie why is procodere
succeded whens previous allem, of whieh there have sopare'y been
{ numsber,lngly snus, have failed, Ryser work takes adwanrae
thie factors (1) the avalaity of elect pales of rirosoeond dara
tian, (2) the high resvity af jenmanium, wrt (8) the igh mobi of
{ceeton in germ, Racaaae of (2), See posh uo deliver energy to
Elections at eatvely igh ates by elcai ks.To eect this heat”
the ebvtronsabuve the tempeatre of te erystal and lowers thei mobility,
The gnerand equation i then

wee 0, DE a2)

where the fact that depen aE epeeseats the fondamental nontinear
ity. We shall show that Ryle Lechuiquee rise the “temperature” ofthe
‘dewions by a ecto of about thirteen fold to shove UDOT. Sine the re
[Sst i high, sy 10 uhm ea, the power celivern to the specimen is
‘Sulsinty low thatthe healing ia one pale is nehle. The pul pet
‘Con tate i hen kept a0 lon hat serum Heat negligible al,

Tere cnltans are enormoasy ore faweable than coe mel with
in melas, Tx cna tae average lotro energy fs evel eset Wot
‘non t double this energy, enh eleiran would acquire 2 aie exergy
‘oughly walt the ces energy per atm ofthe erystal,Puethersote,
ina mela there fab one conduction electro per atom, compar with
10" pet ator io Ryde’s srmpls, Thus Lie stored exesy dee to “hat”
lec ina ical would be enoagh tn vapor, whereas in geranium,
tr» sinlar enisonductor, 9 temperature of JOGOITK forthe electrons
“would be enongh tn xine the eystal less chan O01°R. From tis reasoning
Fe appaare that i mill be eviren'y dificult, if nt impose, co proaoe
“khitcantfandsmental devitons fom Ohm's a ia ieals and ceainy
imposible to penlue fla of the magnitnde dsersed below.

Tt should be pointed out that the babevior of electzuna in erga in
fill an hig ta eqasio (1 file have Boon set to bth experimental
and theories) fvetpr ton fn coanazdon with, deste breakdown?
"The work does not apply to ase in which the spevimens bey Ohm's ley
at low fslis however and the experiments da rot peat accurate der
nt pees ih cA Si, Phe. Ro 8 (40 x Fite
aie ena ae ca gen Sete In ear Appecm, owe, we give mi™ nore
mination of a8 fanetion uf B. Feom the thence ede alo the emp
Sb ns bon ot lds so igh Ghat the near rae gested aU tbe
trunsiton trom linet to nonlinear is a see.

“he current theories of distestrie breakin are used the pre

reordary gavel” of "easton aaukisealns,” Thus fan easton
Sequins enough exanay fom the elected, i wll be capa "peo
‘dating secondaries hy calsian wih bound eects, ans Une raion of
(his proces il xd Uo an avalanche, Our theory inate Chat in per
rvnlam, even ats sighs 20,000 wll, fe elezons wll ave
nough energy to Produce sccondarien AL about chase Fels, humever,
brother phenomenon cus

T1838 C. Zane proposed that electric breakdown was duct a pi
trary elest the Bld isctced generation of aoleelectton pais. ls mathe
reatcal toon is ems to that for fl emia Sram cle met’ pints
tnd to tht for radioactive decey. Ie invelvee the “unaeling” af elatroae
CHivugh regions in whi tir wavefunctions are atenuste, rather than

‘Zenet’s theory doesnot seem to spply to breshdown; however, it does
apply tothe high electric Selds produced in roctiving pr junctions in ac
friar wen thes are biased i fe svers iscton, Unda thee on
lone fis oF the acer of ADOT wot are ced. Phe abies
Deletes, i thee falls "ave wot bean weasuted. Te has beet
own,‘ however, thee secondary production ig very etal At these Beds
‘sort af "breaksows eflec oscus and above a extcal value ofthe vat
sym wety rail ince in eurent fs eimerved, Tila curent appears ta
te of che nature picid hy Zener. Te iss at a given volte, has &
small tsperatice eueicent and wil probably be usfl ts semiconductor

taegue of vollage sepulstor ts" and procective device.

"hei shows im she trentvent given fg iatoe terms in orton 3 woe
in more detail n te Arpertics, te explain of te feared
tions ora O's lai ved on the Cer of weteon weve. The invest
tits desert inthis ay ths be eared a arn evidence

for the wave mate of emletion elena germane wl are sre
Inte tothe rsscches of C.J. Davisaon, to wham tis volume ig dedicated,
ot hiecolldnolor LH. Cerner. he Davin Germer xprize ere
fomcernel chiefly with ett weaves fT space a wh high energy
dectrons i cyetal, Both f thes exe ane npr than that deat within
this paper, Estoue i Cre colution bad in gennaniun appt to Bele
hgh they wete na ll pyafracting motara a which they my ave

SE PRAISED fae Wy ant MS, Pe Je 0 85
DM Fe WELL settoy MRONUEAT JDRNAL, OcTOMER 1951

several velocities of propagation for any specie ination of propagation
Ad frequency 1. to be hoped tat more detailed analy of the data ob-
tine by Ryder, together wits guantiestive interpretations af eresin ob-
feevations of ruyesovesstance, ay lead to a qe ealution of the
eitactve constant lan fr tne electron aves hie peg
‘nvlenaned billy in Seclian 8. Te ietutnn in the Zener careat age
Sond anetnor new npjartuily to ety eesron wewee fey. The
weaves involve iv bs st au thowe with energies in Ue enemy BS 3S
Theor the eomfuctios baad sa! the valence han wae inthis range Bove
‘ceived ile llantion from eter the experimental or Ueotetial ie.

2. By J, Rowe's Reseuts

(Oe of gemcanium’s must. soil atrbutes i ite ebiity to pioe ans
leasing Gf elestial sigras wes wade aco a tnsistr, ‘The bese phe
Inomesen fos many ype of tranetor ia thet of Mere nection.” As is

1g hey

i Be wl oJ. ie eae or ring cy 9 BR

sell kaowny gemaniuas may ratry current eer by ee elec mecha
tibm ir whr coe it is called oelybe germaniam, ve by the rechanim of
bole center in which cave ix cael peype. HF eit prepared
‘ettode is pleecd ona agp specimen at usent is eae to Hw in
he sense hat removes lovin tom fhe specimen, then tye proces may
ate "lee jection.” In hie ese addon to removing conduction
(eet oom the yormuniu, elrioas are removed fon the valence
Sums £0 hat foes are lajeted, Ths lads to noninesr effects hoo,
ca he cutentpasies sual Ye specngn, che nliee of caer in the
{Speci charge onl Goes it en

Thanet to void tye seondary dowations om Obes law due to exer
infec, Ryder bas dese species of se frm shown iv Fig. These
{pecs have large ess co eh the meal eletroses are alice. The
eshte ares chiefly from x thin zetion ofthe mace connecting the
abr exyertons bs cemeasTew ako ome taw os

lorgn ens, Snr the elds a he large ends aye small, eamice injection
leegely suppresed; furiterwr, the elie feds are applied for sch a
shox tine during the pulee tit, ever i bles were injected atone ai the
‘ns, they would na hve time vo rea the iow eetin of the bridge
land medulste its conductivity during the ptiod ofthe pale

‘Purther caus of porlinearity can eis from inhomngeniies in the
gerianiam materi ite, For emp grin Lounéns in golyeeystlne
ferinnrium are knon ta have nce etre resistance, Dien ee ce

7 ,
t 1 Trae

eL_| | Ra
Z| Z a Ef
i HHT
. 1
5 SI Torun [th
i i j
El = rhe
i l \ see

ste Rhee fe Soma 1 ebm user ibe Fie ecm
Tena dure = (29 210 cat tacts We mes ewe

te inkomagenccy save largely been clininated in these espeimenss by
the owe of bigs somigeis sigle eystal gemmaniam mae fur
ithe by 0, Kel and hixeisoratam

‘her expetnentl precanions ere necessary, ach wx marin eat
plished sutiace on te filament; if tix nt done, apres lee te
Ingsecad fou the suriace :regulrites of the tv section beer the
lenge erst who nes I rake carts fo welt ice
some of the resistance aes thin he linge eck thers

Some ofthe etn obtained by Ryder are shows in Fig 2. ‘The dil ve
lacey, soto au ordinate, ot mens Vet Ina dsr fra
the meaeured curentethrocgh tke sperimen by the fcr reno
906 rms men svt aemsTeAL.yovinAt, OorORRR 195

Ina specimen nt rm temperate the eft velcty of wets is given
by the equation

t= 300 emer ea

ivhen £ is expend in wlts per em. AL ICD volts per em, for expe
(eich is edo the nonnear rong) the selciy of electrons shoul he
306 10, Tos establsee the drift velocLy sal for room temperature
om olter meawuremer's of the germans spacizn of Fig 2, ti com
ced that the uber of electrons wee or conductin is eas
tilly independent of teperature don to Heid air tancoratutee, Cre
[Geely for this tempeatnes range the edt walt enamide wey
Troperteal tthe cient i the specien. The other ten sts of dala
Ene ncotdigly snp cele ia propor on fe thir crete

‘nthe figure me alsa entapolated res tS gree conesponding
to Olas Iw, Brom these we sce that decrees in roiity of tenfold ot
Inore have beea procuce Lies experiments fr high Lela conto

"Tac data for cer lenjeracue de pprovimaley thre Les: Te lw
fei or Ohm's lw reins tcanedate region ove> whieh ie pOpOE
tional to 22 and wf proposal to 2 rnd saturation sepa. The
telat lefevre a dit velocity of about 3 % 10% cm/sec for all
thee cavee ‘This nek, asouling to theory, emo cere when the eit
Yeleiy i several Lies the spon of sewed i germanita, the eps! af
ov! bing ssut $1 3108 emer ‘The limiting fe wleity at Hier
Felis i vu with the energy reine bn exit a particular type of
toric vibration, called an "epic mae.” Te comes nt approximately
the ve of crit velocity prec! hy then. The ekzore-x] uve,
‘compoia’ fx the appendices do a0! rk fi che sharp Line eesion supe
[Rated on Fg. 2 However, thoy show the distinct inuenos of arate
‘Shove ithe dat reasons ell a ne sell show below in ennection
with Fg 5,

8, maw oF Devuatione anon Om’s Law

a, Biro a 8 Te Germain

"The specimens we sll consider ore of sezype germanium ai tave
sitvites of seve ae, The combsetivity ares from she prewree

7,8, Basrotané Wen Hh 8 885 SD,

gph ETE SOE TE UT AB
Do? ELACIHONG BF GRIMES asm cms a oon

of donors: chemical impties such ag arsenic or antinony. These donors
fubstiuts iomzcle for genmeniom atone in the cxyta structure, Zora,
ceetton-pairbonés with thir four negara and release their Gh valance
cecttan to the conduction end, The deasty ef donors 'sebout 10%em
rene pet cube 10~ ca ~ JGUD A or ea edge, Tu canorsare x pasitve
epee not grove in elect ds, Tete charges reateliae tote
Uf the elton ‘Te eloctroetate energy of interaction betrern eects
tel donors lade ton de ton of the eacronszaaoo. For the tempers
irs of Fig. 2, however, his lot is rimetaat compared tothe eet
of thermal vraions of te ators.

‘Tae elstons in the conduction band tave fn anvonlance with a wave
equation. They may, hovever, be thought of 23 parle. The justineation
|S dat, under many expeinental condor, the nave factions wil actu-
ally be wave packets. These wave packet, it canbe showa, behave much
5 particles and can be dealt wil 45 parts, at last provided tha phe-
omens considered do not involve distaes emaler than the sie of the
save pocket,

‘ner con tions af (her! eqvirinm sxe may think of the 108 ele
trons in eae) cable cewtncer af an clcion gae with the ectons (3
‘wave packs} moving al aodooe vil: an aveage hineli energy of ttn
at G72).

Af the ets ofthe crystal wor ld vgly at seat © pect regu
cayetal stroctee an eleetson wave, and a wave pact too, would be ane
smite throug it with no srcterng, Ac SOO°K the vibrations ar such tat
the wave poset moves for ony 2500 A before eng seated. At low tea
pereture, the mean free path is longer and at cud aydrogen temper
tno it es long that therm eit wre les iapertant tn the
felis ofthe ont donors, As site ba, ever, we may neler this
scattering by ions over the temperatre range of Hig. 2

elon proteding lh the tase the interactions of leet
sd ier vation ve shall pitt at te problems ust ese
‘efor the dependence of mabilty upon elacrc Geld canbe explained

Tist, the mechanisms of the indivdusl proseses must be atalyd
“This the base physi preblen. In onder solve = we ust spply une
tum mestenie tothe model repreanting the elctesa anosng inthe eat
ame dotermine Ce pookabiitis of vasous type af trusilons and sae
appropriate averages

Sceond, the eaistical consequences must be worked out. On che basis
of tae individual process, the entice of the acenblage of exctons
rust be analyzed and sted state oltisn found.

Thea ery pte mare jin! problems adi gen the most
966 rue au seerme reeusteaL owns, ocTosER 1951

tention, Te acon robles more ell mathematica. I sven
nly an apprwrate traest which ie adequate, however, to idiate
that the tliion tothe fas problem contains che necesery fetes to
‘explain the experiiets
Ah, The Pronons

Wie must rext conser how to descbe the derma ibeations and tn
Covaluste icin interttis wil ty electors, We aha preseat one the
Principal teuteof tae matkematenlana= nem leaving the details for
the appends, ‘The cles treatment of therma! ivan in a ert
foe tht of Bistern, who rnd each atom to be axparate barman
Uelletor. This mode! wae improvel on by Debye who tested the crystal
ten clsticcontinuuss Chat eould export remalng waves. Debye's eta
ie regarded as evventhlly covet and wey therefore, revolve the storie
‘motions into set of rung waves, or mortal modes. There are thre:
times ee mony ilyendent ermal modes ox there are atoms in the eral
Crone per degre of Iason, al any posse atric motion of he erytal
Shay be nade up as «tort of Fontier sre in these bomal modes,

Track normal sne aust treted ne x Planch collar al Is
syste af erey level ith values

et 19 fo)

svheve wis ke Eequency of whaling, aes qoansum of exergy seer
thas plan ia nonmel me rakes a cagition wit fe = 1, Westy
tpl as beea ented and Hx = 1, we say oe hs been absstbed,

TT ripen of se crysal i Unie of phonons i in ose anslogy
sth the deetrpon ot kcLinmagaetic waves Tv ext ia texas off
Tams Tor the case of Tigh the clgtomagne'e stat ofthe cavity ie deer
Inined by finding he nora wae whieh ace cated ab quanti! ei
Jator, sal trains wid ~ 1 ermespons fo photon emvian aan
beonption.

"The normal modes for the etal are unlike those for light. Far ow
Frequerses ie waves ace czentilly Ue croscope rasyerse al hg
tucinal woves ol sll Ae the save legs becomes sorter, however,
the sound velky vars and there ie a Kiting minicom ve lg
Gehich aout ince ae spacing eter aoa. Te ander to wlersane
Ue eneray sce of electons in ‘igh Skis, we taut conider Ue role of
{is minuuy weve length Foe Cie puapoee wo tl deecibe the depen
fice of feceney pon sone Teg fora longitednal ree

‘ecology we consider the “ejcacy af the nrwal medescomespont
ing to's bugiudial wave propagating along a cube axis, Rather than
ur EAEeONS I OARS AN OHS La 00

sng the wave length as a variable, we se the wave member oF (1,
or omg mers the speenry i ily

+= a= altp fo)

“This comeeponds to the straghe lite porion for low frequencies in Fi 3
“Tai portion extends ca « wave lensth equal to trie the lattice constant 2
ofthe erst

Figuie 3 tons auether euve leh bas a ied foequency ever for
(A) ~ Gor inte wae Ingth. The presence ofthis branch ofthe “ie
‘racoal spectrum” ir dae to the Tat thee the diamond structure he 18

Fe, Sti 5 ein I sn a
Saale ath es a etter

torn per wit. (Vhe Hema seucuce is wale a te fae entered
fubie ery a? stm, jsp thal cack ata a ane sia Is een
taal stated is respact oa tewabedron of four ats of the other nay
Frith which fans four action jit bonds. The anit cel comtairs one
tore 9 ees wry.) Nn consequctos of thie its peste to have « b:o-
‘on in nich oni atom bates in the plus ection while the othe ator
‘Shnvosayiy and to have tis stave motions eects in ae in every
‘ont ell telat i onsiesed to have ino wave lng, ince
‘very rit all hes the ame thing atthe sate time. Te his tae hint
Dosa frequency since the pattern of motion invalves diretly oppose
‘laa of nearest neighbors. I tae motion is moe so 25 to have it