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Philips J. Res. 41, 343-384, 1986 R1l34

AN EFFICIENT SILICON COLD CATHODE FORHIGH "CURRENT DENSITIES

Il. Comparison with theory and discussion

by G. G. P. VAN GORKOM and A. M. E. HOEBERECHTSPhilips Research Laboratories, 5600 JA Eindhoven, The Netherlands

AbstractThe efficiency and energy spread of electrons emitted by very shallow p-njunctions biased in avalanche breakdown are discussed in detail. Electrontemperatures are deduced from total energy distribution (TED) measure-ments. Structure in the TEDs is explained as due to peaks in the density ofstates of silicon at 1.9 eV and 3.1 eV above the bottom of the conductionband. The values obtained for kT., k'I; = (0.18 ± 0.01) eV for emitterswith breakdown voltage Vbv = 16 V and k'T; = (0.38 ± 0.01) eV for emit-ters with Vbv = 5.2 V, are compared with values obtained from existingtheories, especially the theory based on the Boltzmann transport equation,and reasonable agreement is found. The emission efficiency tl tn ::::2 x 10-"for bare silicon emitters and n = 1.5 X 10-2 for cesiated emitters, withVbv = 5.2 V) and the dependence of n on the lattice temperature and on thework function f/J have been measured. To compare with theory, a know-ledge of the electron generation rate as a function of the position in thedepletion layer is necessary. Existing data of electron and hole multiplica-tion rates have been used and it is shown that there is a non-negligible con-tribution from the "tunnelling process to the 'electron and hole generationrates. The theory based on the Boltzmann transport equation gives a cor-rect description of the (nearly) exponential dependence of n on t/J but thepre-exponential factor is a factor of -5 too low. The usefulness of thelucky electron model has been tested. It is found that the model works welland an effective mean free path of 4.6 nm is obtained.

PACS numbers: 79.90. +b, 85.30.Kk.

1. IntroduetionThis paper is the second in a series of three, describing the emission proper-

ties of silicon p-n junction cold cathodes. In the first paper 1) we gave experi-mental data and main results, showing that silicon cold cathodes with adiameter of 1 urn are capable of delivering very high current densities (inexcess of 1000 A· cm-2) into vacuum at an effective electron temperature ofabout 0.5 eV. The factors that ultimately limit the performance of suchcathodes were discussed elsewhere").In the'next section of this paper we present additional experimental results

and in the third section we compare the new and earlier measurements withexisting theories. From this comparison we draw conclusions on the inter-

Philip. Journalof Research Vol. 41 No. 4 1986 343

G. G. P. van Gorkom and A. M. E. Hoeberechts

pretation in terms of the relevant model. In the following, we shall confineourselves to the physics of the emitters and so deal only with the efficiency ofthe emission and the energy spread of the emitted electrons. Other importantcathode properties such as current density and total emitted current, whichdepend among other things on the size and the geometry of the emitters, willbe discussed elsehwere"),

2. Experimental results

2.1. Introduetion

Device technology 4) and the experimental conditions 1) have been describedbefore, and so will not be repeated here.: Some new experiments have sincebeen performed at low temperature, close to liquid nitrogen temperature.Another new experiment is the dependence of the efficiencyon the work func-tion of the surface. This will be described in sec. 2.4.

In addition to the results of the new experiments, the results of older experi-ments will be discussed as well. This involves the value of the efficiency and theenergy spread of the emitted electrons. For the efficiency 11,

ivac ivae11=. .:::::: -.-,

lvae + Id Id(1)

where ivae is the emitted current and id the (non-emitted) diode current, wealready found 1>', at room temperature,

tÎ ::::::1 to 5 x 10-6 bare SitÎ ::::::1 to 2 X 10-2 Si-Cs, (2)

where tÎ denotes the top (usually the starting) value of the efficiency. Thespread in the results is mainly due to a spread in residual contaminants(oxygen, carbon) on the surface. The top value tÎ could usually be maintainedfor a limited period. Due to different lifetime effects, these values decreasesomewhat in time. If the vacuum conditions are sufficiently good, a stablesituation can be obtained in which the value of 11can be anywhere in the rangeof say 0.1 tÎ to 0.9 tÎ. After following the proper procedure, to be discussedelsewhere, the stable value of 11is often about 0.7 tÎ and the measurementsdescribed in this paper were normally done in this stable condition.The energy spread of the emitted electrons (for diodes with a breakdown

voltage of about 5.2 V) was found to bel)

(dE)FWHM = 0.93 eV bare Si(dE)FWHM = 1.20 eV Si-Cs,

where (dE)FwHM is the full width at half maximum.(3)

344 Philip. Journalof Research Vol.4l No.4 1986

300 K: Vbv == 5.21 V80 K: Vbv = 5.09 V, (4)

An efficient silicon cold cathode for high current densities

2.2. id-Vd characteristic and efficiency at 300 K and at 80 K

The reason for measuring the id-Vd, characteristic of the p-n emitter at dif-ferent temperatures is to establish whether the breakdown is (mainly) due tothe avalanche process or to the tunnelling process. There is reason for doubtbecartse the breakdown voltage (5.2 V) is in the range of mixed breakdown 5).The 'measurements were performed by immersing a simple tube containing ap-n emitter and an anode in a diode configuration (and also a cesium dispenserand a suitable getter) in liquid nitrogen. The actual temperature of the emitterwas not measured but it could not differ much from 80 K and this temperaturewill be assumed in the following.

The result of the measurements is given in fig. 1. It is seen that the break-down voltage shifts to lower values upon lowering the temperature, proving 5)

Vc! (V)-__~~5.~5~~~~~?-~~~4T,5~1~0

500

1000 !

1500

Fig. 1. id-Vd characteristics at T = 80 x and T = 300 K of a pon emitter with an acceptor concen-tration -1 x 1018 cm" and with a diameter of -6 urn.

that the breakdown is mainly due to the avalanche process. Extrapolating toid = 0 the more or less linear part e;f~he curves of fig. 1 (the curvature is pro-bably due to the tunnel contribution to the diode current, and not due tomicroplasmas, because no microplasmas are observed when the diodes areexamined under the optical microscope while in breakdown) gives

Philip. Journalof Research Vol. 41 No,4 1986 345

and so we obtain, taking into account the measuring accuracy

I::..Vbv = +0.12 ± 0.02 V (5)


and

I::..T

where I::.. Vbv = Vbv (300K) - Vbv (80K) and I::..T = 300 - 80 = 220K.Also' measured where the efficiencies at 300K and at 80 K of four different

emitters (emitters numbered 1 and 4 were from the same wafer, emitters 2 and3 each from different wafers, and from different 'batches'; so the spread in themeasured values give some idea of reproducibility) and the results are given intable 1.

+(5.5 ± l)xl0-4 V· K-l, (6)

TABLE I

1::..11 1 1::..1'/emitter 1'/ (T= 80 K) 1'/ (T= 300K) --

!::...T ij I::..T"

1 (no Cs) 3.9 X 10-5 2.3 X 10-5 -8 X 10-8 -2.6x 10-82 (no Cs) 1.8 X 10-5 1.0 xlO-5 - 3.7 X 10-8 -2.6x 10-8

3 (Cs) 8.94x lO-s 8.70x lO-a -1.1 X 10-6 -1.2x 10-44 (Cs) 6.68X lO-a 7.02 X lO-a + 1.5 X 10-6 +2.2x 10-4

It can be seen from table I that the efficiencyof the bare silicon emitters in-creased by a factor of 1.8 when the temperature was lowered from 300K to80 K. The cesiated emitters were much less sensitive, or perhaps even insen-sitive to the temperature (at least in this range); see table 1. Here ij is the meanvalue, Le.

ij = ~(1'/aoo + 1'/80). (7)

So for the non-cesiated emitters we have (including measuring accuracies)

1 1::..1'/- - = (-2.6 + 0.2)xlO-a K-1ij I::..T - (8a)

and for the cesiated emitters

1 1::..1'/- - ='(+0.1 ± 0.2)xl0-S K-1•ij I::..T

(8b)

2.3. Work function and energy distribution measurements

In order to be able to compare the measurements with the theory we have todetermine the work function of the electron emitting area in operating condi-

346 Phlllps Journalof Research Vol.41 No.4 1986

Phlllps Journalof Research Vol.4l No. 4 1986 347


tions. Usual methods to do so, e.g. the contact potential difference method,fail because the measured area is then much larger than the active area of thep-n emitter (usually a few square microns). Nevertheless, we need to measurethe active area only, because there may be a difference in work function be-tween this area and the rest of the chip. For this purpose, we have used ourspherical retarding field analyser 6). As this system has spherical symmetry,the so-called total energy distribution (TED) 7) is measured. The measured sig-nal") is, in fact, aicou/aV sw, where icou is the current on the collector sphere andVsw is the voltage difference between emitting n+ area and collector, which isslowly varied, and on which a small ac voltage is superimposed 6). This signalis recorded as a function of Vsw. It is easily seen that aicou/aVsw is proportionalto ajvac/aE, where jvac is the emitted current density and E the energy of theemitted electrons. This means that wemeasure, in fact, ajvac/aE vs. E which isthe wanted TED, see sec. 3.2.1. Furthermore, it can easily be seen that thelow-energy onset of such a TED gives us directly the work function of theactive area, relative to the work function of the collector. In our case, the col-lector was made of gold-plated stainless steel. This gold is not atomically cleanbut somewhat contaminated, due to repeated cycling to the atmosphere. Thework function of this collector was found to be (ref. 8) f/Jcoll = 4.3 ± 0.1 eV,which is seen to be different from atomically clean gold, which has f/JAu,clcan =5.1 eV (ref. 9).Using this method, we found that our non-cesiated emitters usually have

f/J = 4.3 eV, and that the spread in efficiencywas at least partially caused by aspread in this value of f/J. The cesiated emitters were found to have f/J = 1.7 eV,but the spread in fi of between -1X 10-2 to 2 X 10-2 of such emitters did notshow much correlation with a spread in f/J, and so it seems likely that a 'pre-exponential' factor is of some importance too, as will be discussed more fullylater.

Measured energy distributions of bare and cesiated emitters have alreadybeen published 1,6). The energy distribution of uncesiated emitters was foundto be nearly Maxwellian (in the sense to be discussed later), but the cesiatedemitters showed distributions with structure; see fig. 2a. To study this in moredetail we also measured the TED at intermediate values of the work function,and a characteristic result is given in fig. 2b. In this case, the work functionwas increased (starting from the cesiated value) by adsorption of H20 on thesurface, which was accomplished by admitting H20 vapour into the UHV sys-tern via a well controlled leak value. The same structure is observed when e.g.O2 is used to increase the work function, and also when only a fraction of amonolayer of cesium (and of course in that case no 02 or H20) is applied (thiscase is, however, difficult to m~asure because such a fraction of a monolayer


15

dkocdE

. (arbr. units)

10

5

o

-- expo TED

-0--0- Maxwell curveTe =0.4geV

4- E leV)

Pt' p10

! 3

!djvocdE(arbr.units)

t5

o

ot

~=1.7

2 3t

fJc=/'·3

3 4- E{eV)

ot

{J =2.7

2t

(Ic =/'.3

Fig.2. Total energy distributions, ajvac/aE versus E, with E as excess energy (i.e. above rfJ). Thework function of the collector (rfJc = 4.3 eV, see text) is used as a reference to determine rfJ as beingequal to the energy shift of the low energy onset relative to rfJc. a) Cesiated emitter, rfJ = 1.7 eV.The 'peaks' Ph P2 and P~ refer to structure in the drawn (the measured) curve. The broken curveis the theoretical 'Maxwellian' shape which gives the best fit to the experimental TED. b) Cesiatedemitter with some H20 (possibly as OH-) present on the surface, such that rfJ = 2.7 eV.

348 PhIlIp, Journalof Research Vol.41 No. 4 1986

An efficient silicon cold cathode/or high current densities

proved to be unstable, probably due to drift of the cesium particles in the elec-tric field parallel to the surface 2». It seems very likely, therefore, that theobserved structure (Le. the deviation from the Maxwell shape) is really a pro-perty of the silicon and not some artefact due to the adsorbed surface layer.However, the relative strength of the 'peaks' in the structure may still dependon the way the particular work function was obtained (due to possible relaxa-tion of selection rules, see sec. 3.2.1).

To end this subsection, we have to make a remark concerning the energyaxis (or more precisely, the zero of the energy axis) in fig. 2. We have chosenthe low-energy onset of the distribution (Le. corresponding to the actual valueof f/J) as the zero of energy. This is done to facilitate the comparison withtheory, because in the theory the zero of energy is chosen as the vacuumenergy just outside the silicon.

2.4. Experimental relation between work function and efficiencyIn order to establish this important relationship, we performed the follow-

ing measurements. First the emitter was cesiated and its work function andefficiency determined. Then a small amount of H20 vapour was let into the

, I ,', , , I ,', , , I , , , , I . , , , I , , , , I , , , , I , ,

r 1.5 2.0 2.5 3.0 3.5 '4.0 45. - rp (eV)

Fig. 3. Dependence of the efficiency 'I on the work function tIJ.

-4

-5 0

In TI

f -6

-7

-8

-9

-10

-11

-12

'-13

>9-possible errors

06 -tf)/O.41o TI= . e

o

Phlllps Journal of Research Vol. 41 No.4 1986 349


UHY system and r/J and 1'/ were measured again and so on. The result is givenin fig. 3. It shows a nearly exponential dependence of 1'/ on r/J. In fact, the expe-rimental points can be represented approximately by

1'/ = 0.6 exp (- _!/!_),0.41

(9)

which is the drawn line in fig. 3. Eq. (9) suggests, anticipating later results, anelectron temperature of 0.41 eY.However, there is definitely a deviation fromthe linear behaviour (i.e. linear on the logarithmic scale), which could be inter-preted as a dependence of the electron temperature on r/J or, more physically,on the energy of the emitted electrons relative to the bottom of the conductionband. If interpreted so, it would mean that the electrons have a higher tem-perature at lower energy.

3. Comparison with theory and discussion of the results

3.1. Introduetion

In the following we shall compare our experimental results with theory ormodel. Before we can do so we need a description of such matters as electricfield and potential energy distribution within the silicon, electron generationas a function of position within the depletion layer and the state of the surface(band bending, work function). These will be treated in the next subsectionsfollowed by a short discussion of electron-electron interactions.

3.1.1. Electric field and potential energy within the silicon

The electric field F and the potential V within the diode follow from Pois-son's equation, and using the abrupt approximation, we have")

d2V dF e- = -"'=' -ND(x)dx2 dx es

for 0 <x~xn (lOa)

d2V dF--=-"'='

'dx2 dxe

- -NA(x) for -Xp ~ X < 0es

(lOb)

where x = 0 is at the metallurgical junction and is positive towards the surface(see fig. 4), NA and ND are the acceptor and donor concentrations (in crrr"),e is the elementary charge (+ 1.6 X 10-19 C) and es is the permittivity of silicon(es"'=' 10-12A· s . y-l '.cm-I).

In order to calculate the field and the potential, the dope concentrationshave to be known.' As discussed before"), the doping of the active area wasdone using ion implantations followed by suitable annealing, sometimesunder oxidizing conditions (which, in the case of boron leads to a segregation

350 PhllIps Journalof Research Vol.41 No. 4 1935

I IIrXn I.. I~. I

I1 I

W lt-xjlI""'..t-----I"~I

11H-Xt

metall. junction


Si .....t-t--: _. vacuumI

.1

..

I1 ,... vacuumI II I~X

a) b)meta/l. junction

Fig. 4. a) Energy diagram of the p-n emitter. x4>is the distance between the surface and the point inthe depletion layer where eV(x4»= !/J. b) The electric field in the depletion layer, calculated for thesituation as drawn in fig. 5b.

of boron and so to some boron depletion close to the surface). The net profilesand concentrations are presently not accurately known. The result of the com-puter simulation of the process is given in fig. Sa. We have at present no meas-urements to verify these profiles, so they must be considered as somewhat un-certain. As will be discussed later, our knowledge of the electron (and hole)

dope conc.(log.scale)

tdope conc.(log. scale)

NX ~-~-------

- depth-depthXj = 10nm below surface

a) b)

Fig. 5. a) Impurity concentration as a function of depth below the surface, as given by the com-puter simulation of the manufacturing process. b) Approximation used in the calculation of theelectric field and potential energy in the depletion layer.

Phlllps Joumul of Research Vol.41 No.4 1986 351


generation rates at very high electric fields (~ 1 X 106 V· cm-I) is also ratherpoor and therefore it seems reasonable to use a suitable approximation for theprofiles and concentrations. This is shown in fig. 5b.

The arsenic, used to produce the very shallow n" layer, shows virtually notendency to segregate upon annealing and so the value NDo =::: 5 X 1019 cm:"seems reasonably certain. This is not the case for the value NA 0, due to thementioned segregation effects and therefore we will use NA 0 as an adjustableparameter (but, of course, not too different from -1 X 1018 crn": see fig. 5a),in such a way that we obtain the correct, measured, breakdown voltage (seethe next subsection).It is now easy to calculate the electric field and potential from eq. (10), see

e.g. ref. 5, and the result is

where Xp is width of the depletion layer in the p+ area (xp is a positive quantity)and Xn the width of the depletion layer in n++ layer, and

I l_eNDOXn_eNAOxpFm - - ---'-

es es (12)

F(x) = - IFml (x+xp)Xp

if -Xp ~ X< 0 (l1a)

IFmlF(x) = -- (x - xn)

Xnif 0 <x~xn, (l1b)

is the maximum electric field; see fig. 4b. The total depletion layer width Wisgiven by

(13)

where Vbi is the built-in voltage, which in our case is Vbi =::: 1.0 V, and V is thepotentialover the total depletion layer. Normally, we use the diode in thebreakdown condition and then V = Vbv• The potential V(x) within the diode isgiven by

V(x) = IFml (X+Xp)2 - ~IFml W2xp

. IFmlV(x) = - -- (x - Xn)2

2xn

if _·xp~x<O (14a)

(14b)

where the zero of the potential has been chosen at x.: Eq. (14) has been used todraw the energy diagram, fig. 4a (note that the potential energy of the elec-trons is, of course, - eV(x)).


I I - 2 (Vbv + Vbi)Fm - W . (15)


The relation between IFm I and Vbv is seen to be

The voltage drop over the n++ depletion layer (xn) is

(16)

As we have NDo>NAO (about 50 times larger), we have xp> x; andilVn++ -e; Vbv• For example, if NAo = 1X 1018 cm:" we find Vbv = 5 V andVn++ = 0.1 V. Therefore, to a good approximation both Xn and the voltagedrop over the n++ layer can be neglected and as this simplifies the calculationsconsiderably, we shall do so in the following.As we shall see, an important quantity is Xrp,which is the distance between

the surface and the position in the depletion layer where eV(xrp) = Eb = l/J (seesec. 3.1.4), where Eb is the energy barrier to the vacuum as seen by the elec-trons, see fig.4a. Using eqs (13), (14) and (15) it is easily calculated that (in theapproximation mentioned)

(17)

where Xj is the depth of the p-n junction.In order to proceed, we have to know the value of NA 0, because then we

shall be able to calculate F(x), V(x) and xrp(l/J)explicitly.

3.1.2. Maximum electric fie ld , breakdown and electron genera-tion rates

As mentioned in the previous subsection, we now need the actual value ofNA ° and this will be determined using the measured value of the breakdownvoltage (Vbv = 5.2 V at 300 K). To do so, it is necessary to know the electrongeneration rates as a function of the electric field. At the very strong electricfields we are using, electrons (and holes, of course) can be generated (apartfrom the normal thermal generation which causes leakage currents) by theimpact ionization process, leading to avalanche breakdown, and by the tun-nelling process, eventually leading to Zener breakdown.

There is still another reason why we want to know the electron and holegeneration rates as a function of the electric field, and hence as a function ofthe position in the depletion layer, because then we are able to calculate Grp asa function of l/J, Grp being the fraction of the number of electrons which are

..Phlllps Journalof Research Vol.41 No. 4 1986 353

an = An exp [ - (~)]

ap=Apexp[ - (~)J.

(18a)

G. G. P. van Gorkom and A. M. E. Hoeberechts,

generated at a potential energy equal to or higher than the barrier to thevacuum, or in other words generated at a distance equal to or larger than xqJfrom the surface. Only these electrons may escape, provided they do not losetoo much energy on their way to the surface. All other electrons, Le. the frac-tion 1 - eqJ, can never escape to the vacuum, that is to say, if they are gener-ated with negligible excess energy (which is presumably the case in avalancheand/or tunnelling breakdown) and if electron-electron interactions can be neg-lected, which is argued to be the case in sec. 3.1.5.

Let us first focus our attention on electron and hole generation by im-pact ionization. Unfortunately, these rates are not well known, differentauthors 10-12) give different results. Moreover, the measurements from whichthe electron and hole generation rates were deduced, were done with electricfields -::;7 X 106 V· cm-I. This means that for the eléctric fields we are using(IFml::;::: 1X 106 V· cm-I, see later) no measurements are available and so weshall have to extrapolate the data' obtained at lower fields. This may lead, ofcourse, to errors. In the following we shall do our calculations for two sets ofparameters; those of Lee et al.!') and those of Grant 10). We shall see thatthe electron and hole ionization rates as given by Lee et al. will lead toIFm I = 7.5 X 106 V . cm':' and this is low enough for the tunnel contributionto be negligible. This is no longer the case for the Grant parameters, and therethe tunnel contribution to the electron and hole generation rates will be takeninto account.

a) Lee parameters 11)

.The electron ionization rate an is defined as the number of electron-holepairs generated by an electron per unit distance travel, and a similar definitionholds for apo For silicon these can be written as

(18b)

According to Lee et al.:

An = 3.8 X 106 cm-I, b; = 1.75 X 106 V· cm"Ap ~ 2.25 X 107 cm-I, bp = 3.26x 106 V· cm-I. (19)

The increase in electron current density due to the multiplication process isgiven by")

354 Philip. Journalof Research Vol.41 No. 4 1986

(22)


d'J« ( . ) . J .dx = an - ap }n + ap d, (20)

where J« is the total (diode) current density, J« =l« +i».The fraction of the electrons generated at position x is defined by.

e(x) = jn(X)i,

and as h is constant throughout the depletion layer, we have

de- = (an - ap) e + apodx

(21)

In this subsection we choose, for reasons of simplicity, x = 0 at the edge of thedepletion layer at the p" side, Le. at - Xp (fig. 4), and increasing towards thesurface. Eq. (22) has been integrated numerically with boundary conditione(O) = 10-6• This value is somewhat arbitrary and corresponds to some leakagecurrent present at x = O. In fact, the results were found to be very insensitiveto this value and e(O) could be chosen in the range 0 to 10-4 without affectingthe results significantly.

The procedure was then as follows: a starting value for NAO was chosen andWand Fm were calculated using eqs (12) and (13) with the mentioned approxi-mations (and thus W=xp) for V= Vbv = 5.2 V. So we then haveF(x), eq. (11)and an(x) and ap(x) from eqs (18) and(19). Eq, (22) is then integrated numeric-ally between x = 0 and x = W. This yields an end value e(W), which should beone, of course. If not, another value for NAO is chosen and this is repeateduntil e(W) = 1.000 which is the breakdown condition. In other words, thisprocedure finds the acceptor concentration needed to get a certain value forthe breakdown voltage. With Vbv= 5.2 V we obtained NAO = 2.9 X 1017 cm-3

and IFml = 7.5 x 105 V -crrr". Formerly used emitters") had Vbv= 16 V andthe same procedure then gives NAO = 5.8x1016 cm-3 and IFml = 5.6x105

V· cm-I. These values are in agreement with those calculated in ref. 5, usingthe same parameters of Lee et al.") (note that, to compare correctly, thebreakdown voltages as given in ref. 5 include the built-in voltage). We havenow found e as a function of x and hence as a function of V(x). This gives usdirectly e~,which is the value obtained when V(x) = (/J, as a function of (/J. Theresult for Vbv = 5.2 V is given in fig. 6.We observe that the value obtained for IFml = 7.5 X 105 V -cm " in the case

Vbv= 5.2 V is still small enough for tunnel generation to be neglected. This isno longer the case when we use Grant's electron multiplication 10) as we shallsee now.



appr. Ep = 1.1e-pI2.6

-1

-4

-5Grant parametersincl. tunnel ing

-6

-7

Lee parameters-8

o 2 3 4 5 6- ~ rev;

Fig. 6. Dependence of the fraction of the electrons generated at a potential energy equal to orhigher than the work function rp, as a function of (/J. 8", has been calculated using the Leeparameters 11) and the Grant parameter set 12), see text. The dashed line is an approximation usedin sec. 3.1.1.

b) Grant parameters'Fï

Grant determined another set of parameters to be used in eq. (18). These are

An = 6.2 X 105 cm-I, bn = 1.08 X106 V· cm-Iif 2.4x 105 V· cm-I <F < 5.3 X 105 V· cm-I;

Ap = 2.0 X 106 cm-I, bp = 1.97X 106 V . cm-Iif 2.0 X 105 V . cm-I <F < 5.3 X 105 V . cm-I;

An = 5.0 X 105 cm-I, b; = 0.99 X106 V . cm-I

andAp = 5.6 X 105 cm-I, bp = 1.32 X106 V· cm-I

if F> 5.3 X 105 V· cm-I. (23)

It is seen that these differ strongly from the Lee parameters. However, atrelatively high breakdown voltages (Vbv:;::: 10 V) the two sets give similarresults, but at lower Vbv, and hence higher dope levels, the results are rather

356 Phillps Journal of Research Vol.41 No.4 1986

.---------------~------~- -~-- -


different, and the Lee parameters then give too low breakdown voltages, whencompared with the experiments 10).

We have integrated eq. (22) again using the parameter set eq. (23). We foundthat no avalanche breakdown is possible below 5.8 V. In fact the calculatedbreakdown voltage does not decrease monotonically with increasing NAo buthas a minimum at NAO ::::::2 X 1018 cm'", with Vbv = 5.8 V. This is caused by thefact that the multiplication factors, eq. (23), show some saturation at veryhigh electric fields, while the depletion layer continues to decrease and thisleads to the described behaviour. It could be that this is not a genuine effectbut is due to an incorrect extrapolation of the electron and hole multiplicationfactor to the very high fields which are present in our diodes. However, thereis another point to consider, namely the tunnel contribution to the electronand hole currents. The maximum electric field at the NAo values where theminimum occurs is about 2 X 106V.cm -1, and so tunnelling can no longer beneglected.

Unfortunately, the tunnel generation term is not well known either. For onereason, this is due to the fact that the shape of the energy barrier for tunnellingis quite uncertain 13), and in the literature two shapes are treated: a triangularbarrier and a parabolic barrier (see e.g. ref. 5). From ref. 13 it can be inferredthat the tunnel generation term can be written as

where

djn,t A F2 (Bt,)~= t exp -Ji'

(m*)! EgiBt= I;-~_____;:'-en

(24)

(25)

and, in the case of a semiconductor with a direct band gap

(m*)t e3

At = na 2 V2 fz2 Egi ' (26)

where I; = n/(2 Vi) = 1.11 for the parabolic barrier and I; = (4V2)/3 = 1.88for the triangular barrier. Note that SI units have to be used in eqs (25) and(26). Apart from the uncertainty in I; there is another problem in using eq.(24). This is due to the fact that Fis a function of x, while eqs (25) and (26) arederived for constant field. The tunnel distance Li IJ

EgLit =-

F(27)

is of the order of -7 to. 8 nm (at maximum F - 1.5X 106V· cm -1) and the fieldis already varying appreciably over such a distance. So eq. (25) and (26) must

PhllIps Journal-of Research Vol. 41 No. 4 1986 357

be considered as approximations to the real situation only. Furthermore, thereis a problem because the electrons and holes which are generated by the tun-nelling process appear separated (by a distance LIt) in space. However, forreasons of simplicity we shall ignore this effect and treat the tunnel generationas a local effect.There is still another complication in our case because silicon have an in-

direct band gap. In that case m* and Eg in eq. (25) have to be replaced by areduced effective mass m,* and by Eg - Eph, where Eph is the energy of thephonon that provides the momentum for the indirect transition (see ref. 14).Furthermore, the expression for At, eq. (26), becomes much more com-plicated 14) and contains electron-phonon coupling terms, which are ratheruncertain. In this case the value of At is much less than in the case of directtunnelling. The situation is' different again when the impurity concentrationis very high, as in our case, and impurity scattering can also provide themomentum necessary for indirect tunnelling. Moreover, we have, presum-ably, a mixed breakdown in which hot electrons with large k-vectors arepresent and this provides yet another means of momentum transfer for in-direct tunnelling.The conclusion of this discussion must be that in our case the value of At is

to a very large extent uncertain, but B, is probably rather well described by

(mr*)t (Eg - Eph)iB, = I; ----'----- en (28)


To find out whether a tunnelling contribution to the electron and hole gener-ation rates is really capable of giving a breakdown voltage of 5.2 V, we pro-ceed as follows.The increase in electron current density over a distance dx, divided by dx,

is now

(29)

where we take an, ap as given by eqs (18) and (23). As At is very uncertain, anyguess is as good as the other, and we simply take the value of At that is appro-priate for direct tunnelling, which isAt :::::40 A/V2. cm (taking m* = 0.19 mei),and for B,

. (30)

where we take Eg = 1.1 eV,Eph = 0.06 eV,mrlm« = 0.14. .

358 Philip. Journalof Research Vol.41 No.4 1986


Bq. (29) is then integrated numerically, taking J« = 104 A· crrr", which isa reasonable value corresponding to id = 1 mA for a diode with active area10-7 cm". The boundary condition used was jn(O) = 10-2 A· cm'", corres-ponding to e(O) = 10-6 as used before: and again the results were found to bevery insensitive to the actual value of jn(O) in the range 0 <jn(O) < 1A· cm'",The same iterative procedure was used as described earlier, with Vbv = 5.2 V.It turned out that for NAO = 1.2 X 1018 cm" the breakdown condition was ful-filled, and the maximum electric field was IFml = 1.5x 106 V· cm-I.

So indeed, the combined action of the avalanche and tunnelling processesresults in a breakdown voltage of 5.2 V. A separate calculation showed that,in the absence of the impact ionization process, Le. only tunnelling present,the generated diode current density is less than 10070 of Je, and so we can con-clude that the breakdown is still mainly avalanche breakdown, in agreementwith the experimental fact that the breakdown is mainly avalanche breakdown(see sec. 2.2).

The computer calculation again gives us, of course, jn(X) and hence e(x), seeeq. (21). The resulting eifJas a function of ifJ is shown in fig. 6 too.

We conclude this subsection with two remarks:(i) In the case of a mixed breakdown, as we presumably have, the resulting

e(x) depends somewhat on the chosen value of J«. This is obvious becausethe avalanche and tunnelling generation terms have different dependencieson the electric field. However, the overall effect on the final result is rela-tively small and so the result for J« = 104 A· cm-2 is very representative.

(ii) We have chosen a particular value for Ah and this has been done ratherarbitrarily. If we had taken another value for Ah the value for NA

O (andIFm I)necessary to have the breakdown at 5.2 V, would have been different.This would have resulted in different e(x) values. So eifJas a function of ifJ asgiven in fig. 6 should be regarded as somewhat uncertain. Nevertheless, asthe value obtained for NA

o = 1.2X 1018 cm:" is close to what was expectedfrom the process parameters, the eifJas a function of ifJ data (fig. 6), may berather close to reality. We shall assume so, see also the next subsection.

3.1.3. Comparison of Lee and Grant casesSummarizing the results of the previous subsection, we note that if the para-

meter set of Lee et al.!'), eq. (19), is correct we need NAO = 2.9x 1017 cm:"

(leading to IFml = 7.5 X 105 V· cm-I) to obtain a breakdown voltage of 5.2 V.In that case F is small enough for tunnelling to be neglected. If, however,the parameters of Grant 10), eq. (23), are correct, the tunnelling contribu-tion cannot be neglected and in order to have Vbv = 5.2 Va value for NA

O =1.2 X 1018 cm:" (leading to IFml = 1.5X i06 V· cm-I) is calculated.

Philip. Joumol of Research Vol. 41 No. 4 1986 359

a G. P. van Gorkom and A. M. E. Hoeberechts

As discussed before, the manufacturing process parameters were such thatNAo "'" 1X 1018 cm:" was expected, which already suggests that the Grant para-meters are correct and not the Lee parameters (note again, that this is only sofor our case with Vbv = 5.2 Y; at Vbv ~ 10 Y the two sets give similar results).We have yet another experimental quantity, namely the temperature coefficientof the breakdown voltage, which was found to be

~Vbv-- = +(5.5 ± 1)xl0-4Y·K-1•~T . (31)

This should be compared with the results of Chang et al.15). They found

(aVbV)-- :::::2 X 10-3 Y . K-1aT 200K .

if Nimp = 3 X 1017 cm:"

and

(aVbV)-- :::::5 X 10-4 Y . K-1aT 200 K

where we have taken their values at T = 200 K, being intermediate between80 K and 300 K."Comparing eqs (31) and (32) we observe that there is strong evidence that

indeed NAO :::::1X 1018 cm".So we may safely conclude that the Grant parameter set is approximately

correct and that our calculated values for NAo = 1.2xl018 cm+, IFml =1.5 x 106 Y . cm:" and etfJ as a function of <IJ will be reliable enough to be used inthe following.

Nevertheless, when calculating efficiencies in the lucky electron model, seesec. 3.3.2, we will use both parameter sets, because it gives some insight intothe influence of the values of the electron generation rates on the efficiency.And again we shall observe then that the Grant parameter set is able to explainthe dependence of the "efficiencyon <IJ satisfactorily, using reasonable valuesfor the effective mean free path, and that the Lee parameter set is not capableof doing so.

(32)

3.1.4. Surface states and band bending

. In this subsection we will discuss briefly some well known facts concerningsurface states and band bending of bare and cesiated silicon surfaces. This isnecessary in order to understand the emission parameters of p-n emitters andfor later discussions of the results.The band structure at the surface of a semiconductor is given in fig. 7. The

energy barrier to the vacuum is usually drawn as a potential step (the brokenline) but the image force potential energy has to be taken into account, of

360 Philip. Joumnl of Research Vol.41 No. 4 1986


--------~---------r--- - ---:-:;;.=--.---III

X I fJIII

fJ = fEe-EFl + ell's + Xsurface states,,,

! Wsj ...

Fig. 7. Surface states and surface band bending in silicon. All figures are for heavily doped n++,(lOO) oriented silicon. The surface denoted 'oxygen contaminated' may, in fact, also containsome carbon.

I/J (eV) X (eV) eu/, (eV) W.(nm)

clean Si 4.9 4.0 0.9 4.7ox. cont. Si 4.3 4.0 0.3 2.7ox. cont. Si + Cs 1.7 - - -clean Si + Cs 1.5 1.4 0.1 1.6

course, and therefore the potential barrier is in fact a smooth I/x curve (thedrawn curve outside the surface). The energy barrier to the vacuum which

, must be overcome by the hot electrons to be emitted is seen to be .(33)

where Ws is the surface potential ") and X the electron affinity of the surface.The work function cp is given by

(34)

where E; - EF is the energy difference between the bottom of the conductionband and the Fermi level, in the bulk.

For our case with ND = 5 X 1019 cm:" it is easily calculated that E; - EF =- 0.01 eV (at room temperature), and this will be neglected. Hence

(35)

The surface potential w, depends strongly on the nature of and the density ofsurface states, and is therefore very sensitive to adsorbates. Associated with WSis.a surface depletion layer, width w., which follows from'

..

Ws= V2esWs. (36)eND

Phlllps Journal of Research Vol. 41 No. 4 1986 361

ii = a exp (- ~)

with a = 1 (but somewhat contamination dependent) and b = 0.4 eV,in agree-ment with eq. (9).

(37)


I·

Values for W.are also given in fig.7. This shows, for instance, that the clean sur-face has W. = 5 nm, and so one has to be careful not to make the n++ layer toothin, otherwise it will be fully depleted and the breakdown will occur e1sewhere.

Combining data from the literature 5,16-20) and in-house measurements 21)

(for heavily doped n++, (lOO) silicon surfaces) we arrive at the values for lP, X,IJIs and Ws as given in fig. 7. We see that the clean Si surface has lP = 4.9 eV.However, we usually have a surface which is contaminated with some oxygenand carbon. This surface was' found to have lP = 4.3 eV (see sec. 2.3) andassuming that still X = 4.0 eV (though it is unknown whether this is correct)we have eu/, = 0.3 eV. This seems to be an acceptable value because it isknown that in the case of a thermally oxidized Si surface ('device grade') allsurface states have disappeared, resulting in a flat band condition and so.e IJIs = O.Thus, in the case of a partly oxidized surface, a value 0< e IJIs < 1 eVshould result and e IJIs = 0.3 eV seems to be a reasonable figure.

Cesiating the emitters lowers lP and X and makes IJIs close to O.This meansthat W. is very small too in this case and so a lower series resistance of the n++

layer can be expected. This is observed indeed, the series resistance of thediode is found to be lowered significantly upon Cs adsorption.To conclude this subsection, the four cases as described in fig. 7 really lead

to different values for the efficiencies: our usual, slightly contaminated p-nemitters (with Vbv = 5.2 V) have the following values:

non-cesiated ii = 2.5 X 10-5cesiated ii = 1.5 X 10-2

while those cleaned in situ by electron bombardment have."):non-cesiated ii = 6 X 10-6cesiated ii = 5 X 10-2•

It is seen that these results can be understood from

3.1.5. Electron-electron interactions within the silicon

As the p-n emitters are operated at large (internal) current densities (in thedepletion layer, perpendicular to the surface j d.L = 5 X 104 A· cm-2, see refs 1and 2) the question arises whether electron-electron (or electron-hole) inter-actions play asignificant role in the emission process. We can estimate thiseffect as follows. It is easily seen that the number of electrons per cm" in thedepletion layer due to the (vertical) diode current density h.L is given by

362 Phlllps Journalof Research Vol.41 -No, 4 1986


hlne = --, (38)eVdl

where ea; is the drift velocity, which is about 107cm- S-l. So withhl = 5 X 104

A· crrr" we findne = 3 X 1016 cnr",

and this is much smaller than the ionized impurity density (= 1X 1018 cnr")in the depletion layer. So electron-electron scattering will occur much less fre-quently than electron-impurity scattering. As it is probable that electron-phonon scattering is even more important than electron-impurity scattering,see later, we may safely conclude that when discussing e.g. electron mean freepaths, the electron-electron scattering can be neglected. This conclusion isstrengthened by the results of some experiments: both the efficiency and theenergy spread were found to be almost independent oî l-«, in the rangejvac =0.1 A· cm'? up to jvac= 1000 A· cm'? (corresponding to current densities in .the diode normal to the surface of -10 A· cm? to -105 A· cm:"). Further-more, only a relatively small amount of electrons with energy in the vacuum inexcess of Vbv+ few times kT was observed. Such electrons acquire their extraenergy from electron-electron interactions. So this effect is just observable,but unimportant for the description of electron mean free paths, energy distri-butions and efficiencies. This means that electrons generated at a potentialsmaller than V(Xqi) can never be emitted.

One might raise the question whether in the absence of appreciable inter-actions between the electrons an electron temperature can be defined. In thestatistical treatment of many-particle systems such an interaction is needed toattain an equilibrium state with a Maxwellian distribution of energies with awell defined temperature. In our case, as will be discussed later, the distribu-tion of the electrons over the-different energy states is governed by other inter-actions, such as electron-phonon and electron-impurity interactions. In gen-eral, this will not lead to a Maxwellian distribution, and so, strictly speaking,electron temperatures do not exist. However, in a limited energy range, thedistribution will be nearly Maxwellian (see later) and so an effective electrontemperature can be defined and used.

3.2. Energy distributions·

3.2.1. Theoretical form of total energy distributions and de-duction-of (effective) electron temperatures

Al>discussed previously"), the electrons are 'heated' inside the silicon by theelectric field in the depletion layer and are 'cooled down' by collisions. Thebalance between these effects produces (assuming that ~ steady state really

PhllIps Journalof Research Vol.41 No. 4 1986363

djvac E- = f N{E, El.) T{EJ.)del.,de -eVo

(39)


exists) a hot-electron distribution which can, presumably, be represented byan (effective) electron temperature Te as discussed in the previous subsection.Part of the electrons with energies above the work function (in other words:the high energy tail of the internal energy distribution) can be emitted and issubject to measurement. We measured the total amount of emitted electronsand total energy distributions (TEDs). Now we would like to discuss whatinformation can be obtained from such TED measurements. To do so we shallfollow the treatment of Gadzuk and Plummer ê"); see also ref. 7. They discussthe TEDs of field emitters, but in several ways our case is similar.It can easily be shown 23) that the TED can be written as

where E is the total energy of the emitted electrons; E = 0 is the energy of thevacuum just outside the silicon, and so electrons emitted with energy E = 0have no kinetic energy when entering the vacuum (note that this choice ofE = 0 differs from that of sec. 3.1.1), El. is the normal energy7,23), T{EJ.) isthe probability of emission over the work function barrier, N{E, El.) is asupply function and djvac is e times the number of electrons per ern" per secwith total energy between E and E + de, and eVo is the depth of the potentialenergy well").The supply function N{E, El.) is given by the product of a Fermi function

times the group velocity (in the direction of the surface, x) times a density ofstates (DOS) e

N{E, El.) = f{E, B) v-o, (40)where

v = 11-1 (aE) (41)x Bk,

(}{E) = f f (VkEtl dS, (42)

where the integration is over the constant energy surface E. If the electrons arein equilibrium with the lattice at temperature T, and if no electric fields arepresent, we have

(43)

which, in the cases of interest to us (if> ~ k'T), reduces to

f{E) ",. exp (- :T) .exp (- :r). (44)

364Phlllps Journalof Research Vol.41 No. 4 1986


Eqs (43) and (44) can be used, of course, to describe the thermionic propertiesof metals in the free electron gas model. In our case this cannot be done with-out modifications. In the first place, this is due to the fact that the hot electronsare, of course, not in thermal equilibrium with the lattice at temperature T. Ifthe depletion layer is large enough (much larger than the collision mean freepath) then a steady state energy distribution of the hot electrons may existwith associated electron temperature Te ~ T. In our specific case, even this issomewhat questionable because the depletion layer is only -50 nm. Further-more, the electric field varies rapidly within this layer, perhaps even on thescale of the mean free path. As a result transient effects (e.g. velocityover-shoot) may be of some importance. Another complication arises because ofthe fact that in the presence of a (strong) electric field there will be an asym-metry in the occupancy of the energy levels, i.e. there will be a higher pro-bability that an energy level with associated k vector more or less in the direc-tion of the field is occupied than a level (with the same E) with k direction faraway from the direction of the field. This effect is usually treated by the clas-sical Boltzmann transport equation formalism 13,24,25).

To be able to proceed we shall assume that an equilibrium distribution witheffective temperature Te is established among the hot electrons and that theangular dependence of f(E, e) can be written as done in the mentioned formal-ism. As still rIJ ~ kTe, we then have

f(E, e) = const- exp (- _!L) . exp (- ~) . f'(e) (45)n; n:with

t.f'(e) = 1 + fo cos e, (46)

where e is the angle between k and the x-axis (F is in the x direction), and foandfl are the zero- and first-order terms in theexpansion offin spherical har-monics 24), and as they depend on x we have to take their values at the surface.We then have

djvac ( rIJ) (E ) iJr eas«,-- = const· exp - - . exp - - f'(e) T(E, kt) -I--I dS.dE n: n: VkE (47)

To calculate the transmission factor T, it is assumed that the periodicityparallel to the surface (and right Up to the very surface) is preserved, so thatwe have the well-known selection rule for the transverse kt:

(48)

where the subscript i and v mean internal and vacuum, and gt is a two-dimen-

Phlllps Journalof Research Vol. 41 No,4 1986 365

·...".


sional reciprocal lattice vector. As the energy of the emitted electrons invacuum is

(49)

it simply follows, neglecting quantum mechanical reflections at the surfacepotential 'step', that

T=O (50)

In our case, the selection rules may be relaxed (somewhat) because the surfacewill not be well 'crystalline', due to the presence of adsorbates.

In the case of a spherical distribution (f1 = 0), and using the free electrongas model in which the constant appearing in eq. (47) equals2S) 2e/(2n)3 h it isstraightforward to calculate the TED from eq. (47), using eq. (48) and eq. (50).The result is

djvac 4n me ( f/J) (E )-- = exp _-- Eexp _-dE lis n: n; ' (51)

. which is the form originally obtained by Young."). Integration of eq. (51) be-tween E = 0 and 00 yields the well known Richardson equation. The relation-ship between djvac/dE and E is schematically drawn in fig. 8.Now, what can we expect to measure in the situation we have: a semicon-

ductor with a complicated band structure and density of states. As can be seenfrom eq. (47), apart from thef'(O) factor which is expected to vary smoothlywith energy and the T factor (which can only be 0 or 1), the value of the inte-gral is determined by the density of states times the group velocity. It cansimply be shown 26) that in the one-dimensional case there is an apparent can-cellation between the group velocity and the one dimensional DOS, so that thestructure in the DOS would be unobservable in a TED measurement. How-ever, as discussed by Gadzuk and Plummer "), in the real, three-dimensionalcase, the cancellation is not complete and some DOS information will bepresent in the TED. Not all the peaks in the DOS will be measurable, e.g. ifsuch a peak is due to electrons with k vectors in the <111) direction and closeto the edge of the Brillouin zone, and with E somewhat, but not much, largerthan f/J, then this peak cannot be observed due to the T(E, kt) factor.

366 Philip. Journalof Research Vol.41 ..No. 4 1986


10

djvac=sr:(arbr. units)

r 5

kTe•

o

Fig. 8. Theoretical, 'Maxwellian', TED curve. The curve is, in fact, given bydj... ( E)de = const. x E x exp - kT. '

where E is relative to tIJ, i.e. E is the (kinetic) energy of the emitted electrons just outside theemitter.

We conclude, therefore, that in a first-order approximation, eq. (51) willgive a good description of the TEDs, unless one or more very pronounced,peaks in the DOS occur in the energy range measured. The DOS of silicon ascalculated by Chelikowsky and Cohen 27) is shown in fig. 9. It can be seen thatin the energy range -4 to 5 eV above the bottom of the conduction band little

o 2 3 !,-E(eV}

1.0#1 ....#2

DOS

t 0.5

0 -14

Fig. 9. Density of states (DOS) of silicon according' to Chelikowsky and Cohen 27). The energyscale is shifted such that the bottorn of the conduction band is at E = O. Note that this zero ofenergy is different from the one in figs 2, 8 and 10. These figure is reproduced with permission ofthe authors. . :'

Phillps Journal of Research Vol. 41 -No. 4 ' 1986 367


or no structure is present, and this is the energy range measured when usinguncesiated emitters. Indeed, the measured TEDs can be represented ratherwell by eq. (51), which may be called a Maxwellian form, although the 'true'Maxwell shape is given by El. exp( - EIkTe). We refer to fig. 6 of ref. 6, wherean almost perfect fit to the 'Maxwellian' shape is found for emitters withVbv = 16 V. For our present emitters with Vbv = 5.2 V, the fit is still good (seefig. 10), and only the high-energy tail lies below the theoretical tail. This is

djvacdE

krtu: units)

15

10

5

o

-- experimentaLcurve

-0--0- Maxwell curveTe =0.38eV

o 1 2 3 t.Jt - E reV)

tfl =4.1eV tflc=4.3eV

Fig. 10. Total energy distribution of an emitter with Vbv = 5.2 V and ë :::: 4.1 eV. This is slightlylower than usual for bare Si emitters, indicating that in this case a slight (-fewOJo of a monolayer)unintended cesium 'contamination' was present. The shape of this TED is, however, almost iden-tical to the case with !/J :::: 4.3. eV. The broken curve is the Maxwellian distribution with k'T; =0.38 eV.

caused by the law of energy conservation, because the applied voltage is 5.2 Vand therefore, in the absence of appreciable electron-electron interactions, asdiscussed in sec. 3.1.5, not many electrons with energy above -1.5 eVoccur.Using eq. (51), and see fig. 8, the following electron temperatures (using(~E)FwHM = 2.45 kT.) have been obtained (but note that this is only valid inthe energy range -4 to 5 eV above the bottom of the conduction band):

Vbv = 16 V -IFml = 5.6 X 105 V· cm-1_ k'I; = (0.18 ± 0.01) eV

Vbv = 5.2 V -IFml = 1.5 X 106 V· cm'" - kT., = (0.38 ± 0.01) eV. (52)

368 Philip. Journol of Research Vol.41 No.4 1986


For cesiated emitters the shape of the TED is not a good 'Maxwellian', ascan be seen in fig. 20, where theTED of an emitter with Vbv = 5.2 V is given.We are here certainly in the energy range with pronounced peaks in the DOS;see fig. 9. However, due to the partial cancellation effects, these do not lead topronounced peaks in the TEDs, and so the deviation from the Maxwellianshape is not too large, see fig. 20. So, in first order, we can still use eq. (51) tofind an effective electron temperature (valid in the energy range -1.7 to 4 eVabove the bottom of the conduction band). We find

(kTe)eff = (0.49 ± 0.02) eV (53)

and this value can be used e.g. for electron-optical trajectory calculations.However, besides the main peak (P2) in fig. 20, two 'humps' can be observed(PI and Ps) and we interpret this structure as being caused by the structure inthe DOS. The same can be said about the structure seen at another value of rp,shown in fig. 2b. Let us first discuss fig. 20. Electrons present in 'peak' PI haveE ::::::0.5 eV and as this is close to k'I; we believe that PI corresponds to themaximum of the function represented by eq. (51). The peak P2 is then ascribedto the strong peak, labelled # 1, in the DOS (see fig. 9) and P« is ascribedtoo peak # 2 in the DOS. 'Peak' P« is, apart from the mentioned partial can-cellation, diminished in strength due to the exp(-E/k1'e) factor. Then, afterincreasing the work function to rp = 2.7 eV, such that we have the situation offig.2b, peak P2 can no longer be observed, and now we have p{ as the maxi-mum in the function represented by eq. (51). Ps is still present, but now, ofcourse, relatively stronger.To compare fig. 2 more quantitatively with fig. 9, we observe that the ener-

gies of P2 and P«E(P2) = 2.8 eVE(Ps) = 3.75 eV, (54)

relative to the bottom of the conduction band E« are somewhat higher thanthe energies of peaks # 1 and # 2 in fig. 9:

E(# 1) = 1.9 eVE(# 2) = 3.1 eV, (55)

again relative to Ec. This is due to the fact that we do not measure {! itself but{! times the group velocity. As the peaks # 1 and # 2 are mainly due to k ::::::0states, for which the group velocity is zero, the maximum in {! .aE/akx will beshifted to somewhat higher or lower energy as compared with the maximumin ç, depending on whether the band is convex or concave. It can be seen thatpeak # 2 (fig. 9) is mainly due to a band which is convex at k = 0, leading to



a higher energy peak, 'as observed, while peak # 1 is due to two bands, oneconvex again leading to a higher energy of the peak, the other concave leadingto a lower energy. This lower-energy peak is at an energy less than ifJ andcannot be observed.It is clear that measurement of TEDs is not a good means of studying the

DOS. Much more pronounced effects can be expected if only a small angle ofemission is selected. Such angular resolved TED measurements on p-n emit-ters are being performed now, and indeed very interesting structures havebeen observed 28).Now that we have determined electron temperatures from the TED meas-

urements, let us discuss these results and compare with theory.

3.2.2. Electron tem per at ur es: comparison with theory

As mentioned, the balance between heating and cooling down of the elec-trons produces a steady state energy distribution. In general, this distributionwill not be Maxwellian 13),but under certain restrictions ") the solution oftheBoltzmann transport equation is indeed a Maxwellian distribution with some(effective) electron temperature T.. The most prominent approximations madein that case is the constancy (i.e. independent of energy) of the electron meanfree path. Whether such approximations are realistic can only be judged bycomparison with experiments. As shown in the previous subsection, a Max-well distribution is found indeed in the case of uncesiated emitters, hence theassumption seems to be not bad at all and therefore it is worthwhile to cal-culate electron temperatures this way and compare these with the values ob- ,tained from the experiments. If" furthermore, it is assumed that no impactionization occurs, then the electron temperature can be deduced easily fromenergy conservation considerations 29,13)

(56)

where F is the electric field, assumed to be constant in the depletion layer, Irthe mean free path between phonon collisions, and Er the energy of the TOphonon (= 0.063 eV for Si). The relationship between k'I; and F, as given byeq. (56), is sketched in fig. 11, the broken curve, with Ir = 6.0 nm. It should beremarked that eq. (56) can only hold in a restricted range of F values

1 X 105 V· cm-I:::;: F:::;: 2 to 3 X 105 V· cm-I. (57)

The lower limit in eq. (57) has been discussed by MoIl13), the higher limit isdue to the occurrence of impact ionizations.


x


0.8

0.7«t;(eV) 0.6

t 0.5

0.4

0.3

0.2

0.1

00

non-cesiated Si

o meas p - n emit tersx Barte/ink e f al.

.'

15 x105

--+ IFI (V/cm)

Fig. 11. Electron temperature as a function of the electric field. Broken curve: simple theory withno ionization, valid if 1x 106 V· cm-I -;:;.F -;:;.2 to 3 X 106V' cm-I ..Full curve: theory of Barrelinket aI.24), valid if F':?: 2 to 3 X 106V -cm, The parameters used are: Ir = 6.0 nm, /, = 19.0 nm, Er =0.06 eV. Measured temperatures (for non-cesiated emitters) are indicated by open circles. Thecross indicates the value of kT. as measured by Bartelink et al. 24): The horizontal error bar is notan experimental error but is due to uncertainties in taking a suitable average electric field (see text).

Bartelink et al.24) solved the Boltzmann transport equation for the moregeneral case, with ionization present (and this was, treated as an absorptionmechanism). They too assume a constant electric field in the depletion layer,free electron gas model and isotropic phonon scattering with constant meanfree path, and arrive at a solution which is Maxwellian again, with an electrontemperature

k'I; = 1 (1 Eo)i'-+ -+-2 4 r Er

(58)

where

. Eo = (59)

(r + 1)3 -- Err .

and[;

r= -Ir '

(60)

Philips Journalof Research Vol. 41 No. 4 1986371

372 Phlllps Journalof Research Vol.41 -No, 4 1986


where li is the mean free path for ionization.It is easily seen that, in the limit of no ionization, r--+ 00 (which will occur

actually at relatively low fields), eq. (58) indeed goes over into eq. (56); li is as-'sumed to be constant, but eventually has to go to 00 if F::; 2 to 3x 106 V. cm-1,so eq. (58) is valid only if F ~ 2 to 3 X 106 V. cm -1.

Experiments by Bartelink et al. 24) yielded Ir = 6.0 nm and I; = 19.0 nm.Using these values electron temperatures have been calculated from eq. (58),and the result is shown by the full curve in fig. 11.

When comparing the experimental results with theory, a complication arisesbecause the theory was developed assuming a constant electric field in the de-pletion layer. In our case (and also in Bartelink's case 24» the field is not con-stant, of course, see fig. 4. So in order to compare we are forced to use someaverage field. It seems reasonable to us to take the average at 0.75 Fm (al-though this is of course somewhat arbitrary, and in fact Bartelink et al. 24)take as an average of 0.5 Fm) with 'uncertainty' ± 0.25 Fm. In this way ourexperimental points (uncesiated emitters) are represented in fig. 11. Also givenin fig. 11 is the value of k'I; as determined by Bartelink et al. for a somewhatmore heavily doped sample. It can be seen that there is a reasonable agreementbetween the theoretical curve and the experimental points, which is best atrelatively low fields, but at the highest fields the theory overestimates the elec-tron temperature. Most probably, this is mainly due to assuming indepen-dence of the mean free path values Ir and I; on the energy of the electrons, andneglecting the band structure. In fact, the agreement with experiment wouldbe better if it is assumed that Ir becomes smaller at higher energies. The con-clusion that such dependencies should be incorporated in the theory deal-ing with the transport properties of hot electrons in semiconductors was, ofcourse, reached before and has led to the quantum mechanical Monte Carlocalculation method 30,31).

Anyway, the fact that the mean free paths are energy-dependent will makethe energy distribution non-Maxwellian when taken over the full energy range.As discussed before, in such case an electron temperature cannot be defined,but when measuring a smaller range of energies, the distribution will be nearly-Maxwellian and an effective, energy-dependent, electron temperature can beobtained. As the mean free paths decrease with energy, higher effective elec-tron temperatures can be expected at lower energies. Our measurements doindicate that this is the case: electrons with relatively low energy, as measuredwhen ifJ == 1.7 eV, show a higher effective temperature k'I; == 0.49 eV thanthose with high energy, measured whene == 4.2 eV,which have k'I; = 0,.38 eV.Due to the influence of structure in the DOS on the TEDs when ifJ == 1.7 eV,some uncertainty remains whether this electron temperature difference is real.


However, as discussed in sec. 2.4, the deviation from the straight line in fig. 3could also be explained by assuming an energy-dependent effective electrontemperature, and again with higher temperatures at lower energies.The most recent method used to calculate electron transport properties in

semiconductors is the Monte Carlo technique, including quantum effects30,31).It takes full account of the band structure and of a variety of scattering mech-anisms. Quite naturally it then follows that the mean free paths, or scatteringrates, are energy-dependent, because the DOS rises sharply from zero at thebottom of the conduction band. This means that at higher energy more finalstates are available for scattering and so the scattering rate increases fast withincreasing energy of the electrons, and so the mean free path decreases. Typic-ally, at an electron energy of -2 eV, scattering rates are -2 X 1014 g-1 and soscattering times Tsc = 4 X 10-15 S 31). One consequence of this very smalllife-time is the occurrence of energy level broadening due to the uncertainty prin-ciple, which in this case reads 31)

2ft!l.E = -

Tsc(61)

and so, if Tsc = 4 X 10-15 s, bE = 0.3 eV. This already gives an appreciablecontribution to the energy spread, which is of course not included in classiéaltheories. The Monte Carlo technique has been applied successfully to explainionization rates in GaAs30) and in Si31). Some caution seems to be calledfor, however, because the results of the calculations have been doubted bysome.").The method has not yet been used to calculate energy distributions in silicon

at the very large fields we are using. Calculations 33) at lower fieldsF = 2 x 105

V. cm'" (below breakdown) show internal distributions with (!l.E)pwHM =0.65 eV and 0.92 eV (depending on the parameters used, see figs 9 and 10 ofref. 34). Although this is not the same as the external (in a TED experimentmeasurable) distribution, the associated effective electron temperature wouldbe 0.26 eV, and 0.37 eV, respectively. Comparing this with the experimentalpoints, see fig. 11, it is seen that these values are somewhat high. This could bedue to the. fact that the measured points in fig. 11 apply to emitted electronswith energy -4 to 5 eV relative to E; and the Monte Carlo calculations applyin this case to electrons with -1.5 eV above E«, where, as discussed before,somewhat higher effective 'electron temperatures (at the same electric fieldstrength) are to be expected. Therefore, in order to compare better with theexperiments, Monte Carlo calculations for electrons with higher energies, andat higher electric fields, are necessary.

PhUipsJournalof Research Vol. 41 No.4 1986 373


3.3. Efficiency

3.3.1. Comparison with the theory based on the Boltzmann trans-port equation

The theory 24) based on the Boltzmann transport equation, which has beenused in sec. 3.2.2 to describe electron temperatures can also be used to cal-culate efficiencies. This has been done by Bulucea 35) for the case of injectionof hot electrons in Si02• We note, however, that there is a mistake in Bulucea'streatment of the electron temperature. The result of his calculation of Te (seefig.50f ref. 35) is such that, at a certain value of Ir and the electric field, theelectron temperature goes to zero if [j -+- co. This can never be true, for physicalreasons: in the limit of no ionization (/i -+- co) there is an energy loss mechanismless, and so, at the same electric field, the electron temperature has to be some-what higher instead of going to zero! However, the analysis of the efficiencyby Bulucea is correct and so we will use these calculations for comparison withour experimental results, but of course we will use the correct electron tem-peratures, taken from eq. (58).

The efficiency is calculated by taking the number of electrons with energy inexcess of f/J; and with their momentum in the right direction (i.e. a cone24,35)with axis normal to the surface) and divide this by the total number of elec-trons in the whole distribution, In fact, this calculation is analogous to that ofYoung 7) for metals, the difference being that Young takes, of course, a Fermi-Dirac distribution with energies relative to the bottom of the potential well,and Bulucea takes a Boltzmann distribution, starting at the bottom of the con-duction band (which is permitted because the number of electrons' in the con-duction band in the steady state is low enough for degeneracy effects to beneglected and so a Boltzmann distribution can be used).Bulucea 35) treats the case that all hot electrons participate in the emission

process (i.e. e = 1) and finds

(0.93 f/J)tJ = aSoexp - -- ,

kTe (62)

where a = 1.87 and So is the 'loss' factor, which is present due to the factthat only part of the electrons have their momentum directed within the rightcone, and where kT; follows from eq. (58). So is somewhat dependent on f/J/kTeand onidio (see sec. 3.2.1 for definitions oî f« andfI). The variation in Sobetween f/J/kTe = 4 (f/J = 1.7 eV) and f/J/kTe = 10 (f/J = 4.3 eV), assuming areasonable value ") for idio = 0.8, is only a factor of 2, and so we can takeas an approximation a mean value So which follows from fig. 7 of ref. 34,So = 6x 10-2•

374 Philip. Journal of Research Vol.41 No. 4 1986

{ (0.93 I)}17= 0.12 exp - l/J -- + - .ia: 2.6

(65)


In our case we have to take into account the fact that all the electrons arenot yet generated at an internal potential energy equal to or higher than l/J. Sowe must have

17= aSoet/!exp (_ 0.93l/J).kT.

From fig. 6 we see that in the range of interest (1.5 ~ l/J ~ 4.3), for the para-meter set of Grant, et/! can be written as

(63)

et/! ::::: 1.1 exp ( - ..!P_),2.6

(64)

so that we find

If we now take k'I; from fig. 11, at Fm = 1.05 X 106 V· cnr" (see sec. 3.1.2):k'I; = 0.47 eV (which is, in fact close to the experimental value), we finallyobtain

17:::::0.12 exp (- _j_).0.42

(66)

This we have to compare with the experimental results as shown in fig. 3 andrepresented by eq. (9). We see that, within the limits of accuracy, the exponen-tial factor is correctly predicted by the theory, but the pre-exponential factoris a factor -5 too low. At present, the cause ofthis discrepancy is unclear. Weshall come back on this in sec. 3.3.3.

To end this subsection we would like to make two remarks:i) The treatment of Bulucea 35) is, in fact, completely analogous to that of

Young"), leading to the Richardson equation for thermionic emitters, theonly difference being that Bulucea treats the non-spherical (fl =1= 0) case.So we might call eq. (63) or eq. (66) the 'Richardson equation for p-n emit-ters' .

ii) The dependence of the efficiency on the lattice temperature can be under-stood qualitativey as follows. Upon lowering the lattice temperature therewill be fewer phonons present and therefore the mean free path due tophonon scattering will be increased slightly. This leads to a slightly higherelectron temperature as follows from eqs (56) and (58), and this, in turn,results in a somewhat higher efficiency (if l/J is independent of the latticetemperature), see eq. (62) or (66). However, we have insufficient quanti-tative knowledge of this effect and so we will leave this point here, but itwill be referred to again at the end of the next subsection.

Philip. Journal of Research Vol.41 No.4 1986 375


3.3.2. The lucky-electron model

The lucky-electron model ê") has been very successful in describing theinjection efficiency of hot electrons into Si02 87), and as such it has been usedto predict such effects in ever smaller MOS transistor structures. It is not atheory in the normal, physical, sense ofthe word but as a model it is very use-ful for practical purposes. Therefore, we want to determine whether thismodel can be used in our case too, and if so, what the values of the differentparameters then have to be.

First, we will repeat briefly the essential features of the lucky-electronmodel. It simply states that an electron at the bottom of the conduction bandat a distance xtfJ from the interface (and this is again the position in the deple-tion layer where the potential energy of an electron equals the energy barrierto the Si02) has a chance of reaching the interface without being scattered(hence the name 'lucky' electron model), and this chance, which equals theefficiency, is given by

n = A exp (- "7). (67)

Ning et al. 87) studied electron injection in Si02 using a MOSFET structurewith an inverted interface and found, at room temperature, A = 2.9 andÀ. = 9.1 nm. Furthermore, the temperature dependence of À. was found to begiven by

À. = .10 tanh (J!!_),2kT (68)

where .10 = 10.8 nm, Er = 0.063 eV (the TO phonon energy in Si) and T = lat-tice temperature. Eq. (68) strongly suggests that the scattering in Ning's casewas dominated by phonon scattering.

Clearly, À. can be interpreted as being some effective mean free path. Itsvalue of -10 nm is, however, surprisingly large, especially in the light of theMonte Carlo calculations 81), which yield true mean free paths of the order of-2 nm only. As already pointed out by Tang and Hess "), this discrepancybetween the lucky-electron model and the Monte Carlo calculations can beunderstood as follows. The electrons at a distance xtfJ from the interface arecertainly not all at the bottom of the conduction band but, on the contrary,are distributed over a fairly large range of energies. Consequently most of theelectrons may lose some energy on their way to the surface and can still beemitted. As this effect is not taken into account in the lucky-electron model,an effective mean free path results which is considerably larger than the truemean free path. This means, provided the results of the Monte-Carlo calcula-

376 PhiIJps Journalof Research. Vol.41 No.4 1986

.-----------~ ~-~ ~~~,~


tion are corrected; that the emitted electrons are not 'lucky' electrons at alland so eq. (67) should be regarded merely as an equation to fit to the experi-mental results. Still, it can be expected that the value of À is related to the truemean free path, viz. if the true mean free path decreases (e.g. due to enhancedimpurity scattering) also a decrease in À should result.

We will now analyse our experimental results, the efficiency versus </J (seefig. 3) and the temperature dependence of the efficiency (sec. 2.2) using thelucky-electron model. In doing so, we have to take into account the differencesbetween our case and Ning's case (apart from the fact that we have injectioninto vacuum instead of into Si02, but it seems reasonable to assume that thisdifference is irrelevant for the model):a) In Ning's case, the electrons travel through p type silicon only, with dope

levels ~ 2 x 1016 cm:": we have, in addition to the more heavily doped p+layer (-1 x 1018 cm-B)also a very heavily doped n++ layer (-5 x 1019 cm-B)at the surface. This means that in our case impurity scattering may be im-portant and smaller values for À can be expected.

b) Unlike Ning's case, at a distance xrp from the surface, only part of the elec-trons, a fraction Brp, is created.Incorporating these two facts in the model we now write

y/ = BrpA exp (- ;), (69)

whereÀi"l, = À~~ + Àï,!p, (70)

and Àph is identical to À in Ning's case.Comparing with the experimental results we observe, first of all, that the

temperature dependence of the efficiency is much less in our case than inNing's case: going from room temperature to 80 K gives an increase in y/ by afactor of 1.8 for non-cesiated emitters and no increase at all for cesiated emit-ters (see table I), whereas Ning found an increase by a factor of -6.at xrp =

12X 10-6 cm' to as much as -60 at xrp = 22 X 10-6 cm. This fact strongly sug-gests that in our case we have indeed a significant contribution of impurityscattering to Àt which is almost independent of the lattice temperature.

Now let us try to fit eq. (69) to the experimental data as given in fig. 3, andredrawn in fig.12. We then need xrp and Brp as a function of </J. The first is givenby eq. (17), where Xj is the distance between junction and surface, which equalsthe channel thickness (to be exact, the non-depleted part of it, Le. the part dueto Vbv; depletion due to surface band bending is irrelevant here). Taking intoaccount some loss of silicon due to the cleaning of the surface, which includesan HF dip to remove oxide froin the chip 4), we have Xj = 8 nm. The depen-dence of Brp on </J has been calculated in sec. 3.1.2 (see fig. 6). It is now easy to

Philip. Joumolof Research Vol. 41 No. 4 1986377


InT(

t -5

-6

-7

-8

-9

-10

-11

-12

-13

"",,," ,",,,,

o ,,,,,,,,,

-e- meas. points

Grantincl. tunneling

'\,\\\\\\

/\Lee \

\\\\\

1.5 2.0 2.5 3.0 3.5 4.0 4.5- t/J (eV)

Fig. 12. Dependence of the efficiency 1'/ on the work function I/J in the lucky-electron model.Broken curve: parameters of Lee, full curve: parameters of Grant, including tunnelling. Themeasurements are represented by open circles.

calculate 11as a function of f/J and the result, using the Lee parameter case andhence NAo = 2.9 X 1017 cm-3 and A = 2.9, At = À = 9.1 nm (as obtained byNing), is given by the broken curve in fig. 12. We see that the fit to the experi-mental points is not good, the dependence on f/J is too strong. Of course, wecould obtain a better fit by adjusting the value of .11> but it turned out that avalue À.t ;"" 15' nm is needed to obtain a satisfactory agreement. It is highlyunlikely that this value is correct because due to the presence of impurity scat-tering (which has to be present to understand the temperature dependence ofthe efficiency), we can only expect a smaller value than 9.1 nm for .11> not alarger one.Next we will try to fit eq. (69) to the experimental points by using the

Grant parameters, including tunnelling, see sec. 3.1.2. Again using A = 2.9,Àt = 9.1 nm it turned out that the fit was bad, the calculated efficiencies were



too high and the r/J dependence was not good (it was too weak), and so asmaller value of Àt had to be used, as expected. A value of Àt = 4.6 nm gavethe correct r/J dependence, but A = 2.9 was too high. The best fit, with A = 1and Àt = 4.6 nm is shown by the full curve in fig.12. We see that a satisfactoryagreement between theory and experiment is obtained, and that even thedeviation from the straight line (on the logarithmic scale) of the experimentalpoints, as discussed earlier, is reproduced correctly by our extended lucky-electron model. The value of A = 1 in our case is somewhat smaller thanNing's value, but the difference is not considered relevant because this value issample-dependent (sec. 2.1). The value for Àt is considerably smaller than inNing's case and we attribute this to significant impurity scattering. TakingÀph = À (Ning) = 9.1 nm we obtain from eq. (70): Àimp = 9.3 nm. The value ofÀt as obtained by us can be compared with the value obtained by Bartelink et

. al. 24), that is to say if we interpret Bartelink's ivac versus distance from thesurface data (fig. 5 of ref. 24) using the lucky-electron model. Then a valueÀt = 4.5 nm is obtained (for emitters having comparable dope levels as oursamples) which is very close to our value. As Bartelink's experiment is ratherdifferent from ours (in fact Bartelink varies xifJ by etching silicon from the puresilicon surface, whereas we vary xifJ by increasing the work function of acesiated surface by H20 adsorption) the agreement is very satisfactory.

Finally, we will discuss the temperature dependence of n more quantita-tively. Differentiating eq. (69) with respect to T gives

1 an 1 aA 1 aSifJ 1 aXifJ xifJ aÀt-;;- ar = A ar + sifJ ar - Át ar +Uer (71)

From eq. (70) we obtain

(72)

Furthermore, we haveaSifJ aSifJ aVbv iisifJ ar/Jar = avbv ar + ar/J ar

and a similar expression for aXifJlar.Differentiating eq. (17) with respect to Vbv and r/J results in

aXifJ ( Ss )1avbv = 2 eNA (Vbv + Vbi)

. {I _ (1 _ r/J )1 _ r/J.}. Vbv + Vbi (Vbv + Vbi - rIJ)! (Vbv + Vbi)l

(73)

(74)

Pbillps Journalof Research Vol.41 No. 4 1986379

axq, ( es )1al/> = -2 e NA (Vbv + Vbi - l/» •

The parameters aeq,/aVbv and aeq,/al/>have been calculated using the same com-puter program as used before (sec. 3.1.2), and the result is, at l/>= 4.3 eV (andwith NA = 1.2 X 1018 cm:")

( aeq, ) = 0.038 V-Iavbv q, = 4.3

(75)


and

(aeq,) = -0.14 V-I.al/> q, = 4.3

Furthermore, we have eq, (l/>= 4.3 eV) = 0.17, and from eqs (74) and (75) wefind

and( axq,) _ -5.2 nm- V-Iavbv q, = 4.3, Vb' = 5.2 -

(:;) q, = 4.3, Vb' = 5.2+ 11.7 nm . V-I.

Ning 37) obtained

Now, taking_1_ aÀph __ 1_ dÀph

Àph2 er - Àph2 srand Àt = 4.6 nm, À~h = 9.1 nm, we find (assuming aA/aT"" 0)

.!_ al1 = 1.3 avbv _ 3.4 ~ _ 3.4x 10-3.11 er er er

Furthermore, we take avbv/a T "'"d Vbv/d T, and so (sec. 2.2)

avbv .-_ "'"5.5x 10-4 V· K-laT

(76)

Fischer 17) measured no difference in the work function of a clean Si surfacebetween room temperature and 8q K and so we will take al/>/aT = O.So finally, we obtairi .

1 a1'/'" - - "'" -2.7 X 10-3 K-l, (77)

1'/ aT

380 Philip. Journal of Research Vol.4l No. 4 1986

Pr'liL;~3, H~E~;t."r\ftCHLABS~UF;r.'::"'\i' WY - 1P 0 1:".'( 8U Of,O

5GCO 'JÀ L'~INDHOVENTfif; NETHERLANDS


which is close to the experimental value (sec. 2.2.)

1 iltl-; il T = - (2.6 ± 0.2) x 10-3 K-1, (78)

We may thus conclude that the lucky-electron model gives the correct tem-perature dependence of-the efficiency, if Àt = 4.6 nm is used. This in turn gives

. further support for the obtained value of Àt.

If we follow the same procedure for ifJ :::= 1.7 eV we find a calculated value

1 a17- - = - 1.5 X 10-3 K-l,ti aT

(79)

while the experimental value (sec. 2.2) is

1 iltl- - = +(0.1 ± 0.2) x 10-3 K-l,ti ilT

(80)

So in this case the agreement between the model and experiment is less, whichcould be caused by the fact that we used aifJ/aT = 0 again, which is question-able. Fischer ") obtained for Cs on (lOO) silicon

ilifJ = 0.2 = + 9 X 10-4 eV . K-1sr 220(74)

and if we use this figure in our calculations, the agreement becomes evenworse. However, we do not have a surface of cesium on clean silicon but, asdiscussed before, contaminants (oxygen, carbon) are present, and such a sur-face may still have another value of aifJ/aT. In fact, for the Si-O-Cs surface,Martinelli 38) found

~ = - 3 X 10-4 eV . K-1•aT

If this figure, which is probably more pertinent for our surface than (74), isused in our calculation, then the agreement with experiment is improved.

3.3.3. Monte Carlo calculationsWe have seen that the theory based on the Boltzmann transport equation is

able to give the right dependence of 17on ifJ but the pre-exponential factor wasfound to be a factor of -5 lower than the experimentalone. The lucky-elec-tron, on the other hand, is a very useful model to describe and even predictefficiencies, once Àt is known, but the drawback is that it contains parameterswhich have no direct physical significance.

PbllIp. Journal of Researcb Vol, 41 No.4 1986 381


Thus, although both theories give useful results or at least some insight intothe physics, we believe that in order to get a full account of all emission para-meters a first principle calculation is necessary. This is done in the quantum-mechanical Monte Carlo method. In one instance ") it has already beenapplied successfully to calculate the injection efficiency of hot electrons intoSi02• Clearly, there is a need to apply the Monte Carlo method to calculatethe emission properties of our p-n emitters, and we hope that this will be donein the future.

382 Phlllps Journalof Research Vol.41 No. 4 1986

4. Conclusions

In this paper we have discussed in detail the most important emission pro-perties of silicon cold cathodes, the efficiency and the energy spread. It hasbeen shown that measurements of the total energy distributions (TEDs) pro-vide information on electron temperatures and density of states (DOS) in theparticular energy range measured. This energy range is -4 to 5 eV above thebottom of the conduction band, when using bare silicon emitters and, as thereare no prominent peaks in the DOS in this range 27), these measurements yielda nearly 'Maxwellian' distribution from which we determined the following(effective) electron temperatures: .

ta; = (0.18 ± 0.01) eV if Vbv = 16 V

ia: = (0.38 ± 0.01) eV if Vbv = 5.2 V.

Cesiated emitters show non-Maxwellian.TEDs. We interpret the structure inthese TEDs as due to pronounced peaks in the DOS in the energy range -1.7to 4 eV. However, due to partial-cancellation effects, the deviation from theMaxwellian distribution is not too large and an effective k'I; = (0.49 ± 0.01)eV, for emitters with breakdown voltage Vbv = 5.2 V, has been found. This issomewhat higher than measured on non-cesiated emitters, which could becaused by a dependence of T. on the energy of the electrons. The measuredtemperatures have been compared with the theory based on the Boltzmanntransport equation 24) and reasonable agreement has been found, although athigher energies this theory overestimates the electron temperature. This ismost probably due to a dependence of the electron mean free path on theenergy, decreasing with increasing energy, a fact that follows quite naturallyin the quantum mechanical Monte Carlo calculation method.

In order to compare the value of the efficiency n and the dependence of tfon T (lattice temperature) and 1> (the work function) with theory, knowledgeof the internal electric field and the potential energies on which the electronsare generated is needed. It is shown that calculations based on the electron andhole multiplication rates of Lee et al. 11) lead to incorrect results for our most


highly doped emitters with Vbv = 5.2 V. Using the parameters of Grant 10), andadding a tunnelling contribution to the electron generation rate leads to abetter agreement with the experiments. Comparing with the theory based onthe Boltzmann transport equation 35) a satisfactory account of the exponentialdependence of n on rp is obtained, but the pre-exponential factor is found tobe a factor of -5 too low. We have also analysed the experiments using thelucky-electron model37) and an effective mean free path parameter Àt =4.6 nm was found. This differs considerably from À = 9.1 nm as obtained byNing et al. 38) for the injection of hot electrons in Si02, which is ascribed to asignificant contribution from impurity scattering. This conclusion is supportedby the result of the analysis of the temperature dependence of the efficiency.

Finally, we would like to remark that, although the classical theories arecapable of explaining the main features of the emission process satisfactorily,quantum-mechanical Monte Carlo calculations will be necessary in order toget a full account of all details.

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tion.2S) J. W. Gadzuk and E. W. Plummer, Rev. Mod. PhyS. 45, 487 (1973).24) D. J. Bartelink, 1. L. Moll and N. I.Meyer, Phys. Rev. 130, 972 (1963).26) P. A. Wolff, Phys. Rev: 95, 1415 (1954).26) W. A. Harrison, Phys. Rev. 123, 85 (1961).27) J. R. Chelikowsky and M. L. Cohen, Phys. Rev. B14, 556 (1976).28) P. A. M. v.d. Heide, Philips Research Labs. Eindhoven, to be published.29) W. Shockley, Sol. St. Electr. 2, 35 (1961).SO) H. Sh ich ijo and K. Hess, Phys. Rev. B23, 4197 (1981).

Phllips Journalof Research Vol. 41 . No. 4 1986 383


31) J. Y. Tang and K. Hess, J. Appl. Phys. 54, 5139 (1983);V. M. Robbins, T. Wang, K. F. Brennan, K. Hess and G. E. Stillman, J. Appl. Phys.58, 4614 (1985).

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