R 167 Philips Res. Rep. 6, 211-223, 1951

MICROWAVE DIODE CONDUCTANCE IN THEEXPONENTIAL REGION OF THE CHARACTERISTIC

by G. DIEMER 621.385.2.011.22

Summary

A transit-time theory is expounded for the conductance of a diode inthe exponential region of its characteristic. A linear retarding field isassumed, and mutual.electron interaction is ignored. For diodes withshort distance between cathode and anode the value of this conduct-ance in the voltage range - 2 < Va < - I may considerably exceedthat of the total-emission conductance even in the microwave band.Application of the theoretical results to earlier measurements on"short-distance" diodes disposes of part of the former discrepancybetween theory and experiment.

Résumé

Dans eet article on expose une théorie de la conductance d'une diodedans la rëgion exponentielIe de sa caractëristique en prenant gardeau temps de transit. Le champ électrique est supposé linéaire et I'in-

- teraction des électrons est négligée. Pour les diodes à faible distancecathode-anode, la valeur de cette conductance pour les tensionscomprises entre - 2 et - 1 volt peut dëpasser considërablementcelle de la conductance d'émission totale même dans la gamme desmicro-ondes. L'application des rësultats théoriques à des mesuresprëcëdentes sur les diodes "à faible distance" explique en partie lesdivergences antërieures entre la théorie ct l'expérience.

Zusammenfassung

Es wird eine Laufzeittheorie für die Konduktanz einer Diode irnexponentiellen Gebiet ihrer Kermlinie entwickelt. Ein linearesVerzögerungsfeld wird angenommen, unter Vernachlässigung von ge-genseitiger Elektronenwirkung. Bei Dioden mit kurzem Abstandzwischen Kathode und Anode kann der Wert dieser Konduktanz irn-Spannungsgebiet - 2 < Va < - 1 denjenigen der Gesamtemissions-konduktanz sogar im Mikrowellenband beträchtlich überschreiten.Die Anwendung der theoretischen Ergebnisse auf frühere Messungenan "Kurzabstand"-Dioden beseitigt zum Teil den früheren Wider-sp~ch zwischen Theorie und Experiment.

1. Introduetion

Microwave trio des and diodes are being made with ever shorter dis-tances between cathode and control electrode. In modern plan-parallelconstructions this distance is often of the same order as the distancebetween the Epstein minimum and the cathode, which means that thenormal work point of the valve is only slightly removed from the exponen-tial part of the anode-current characteristic or sometimes even lies 'withinthat region. It is therefore of interest to investigate how the electronicconductance, which for low frequencies is in this region of the characteristicequal to the slope, behaves in the case of ultra-high frequencies.

212 G. DIEMER

For this theoretical investigation the Maxwellian velocity spectrumof the cathode emission of the plane diode has been divided into threeparts (see fig. la).

N(£)

t '

£(Voe) 66407

Fig. la. Velocity distribution at the cathode surface. E is the kinetic energy of the elec-trons; N(E) dE is the number of electrons in the interval (E, E + dE).

(a) The part where the electrons have such a low initial velocity thatthey are turned back before reaching the anode; the çontzibution ofthese electrons to the conductance is called total-emissi~n conductance.Begovich 1) and Van der Ziel2) have calculated this conductance gT forthe case where all electrons are returned and the retarding field is linear.(b) The part where the electrons are in a small region around an initialvelocity that, in the static case, just enables them to reach the anodewith a final velocity zero. Here the word velocity refers only to the com-ponent perpendicular to the surfaces; other components are of no interestfor our present considerations. At low frequencies it is these electronswhich give rise to the slope of the characteristic in the exponential regionand it is their behaviour (in particular the conductance ge caused by them)~vhich forms the main subject of this report.(c) .The part where the electrons have velocities higher than those ofthe electrons in the region (b), so that, with low anode a.c. voltages, theywill certainly all reach the anode. The conductance ga due to such electronshas been calculated by Bakker and De Vries 3) under the assumption thatall the electrons have the same initial velocity.

Below it will he explained how the electronic u.h.f. conductance g isbuilt up from the foregoing three contributions.

2.. Theory

Van der Ziel+) and Freeman 5) have performed calculations upon thestatic potential distribution in a plan-parallel diode for the exponentialregion of the characteristic. Although diodes with distances of the order

MICROWA~E DIODE CONDUCTANCE

of lOlL have not yet been discussed, from their calculations one may.conclude that for these diodes the actual field strength differs less than10% from that belonging to a linear field (see, e.g., Van der Ziel's eq.(~, 4')). Therefore in our calculations a linear retarding field is assumed.We will now consider the theory fo~ the three regions (a), (b) and (c),

mentioned in the introduction.(a) The theoretical value of the total-emission conductance gT accord-

ing to Begovich 1) and Van der Ziel 2) can be expressed in the followingformula

where Is is the total-emission current ("saturation current"), VT themean initial energy of the electrons in electronvolts, Va the anode voltageand PT a transit-time function. Figs 2 and 3 show the variation of thistransit-time function as a function of aT (aT is the transit-time angleWiT' in which iT is the transit-time from the cathode to the reversingpoint, for electrons with an initial velocity equal to the mean velocityof the Maxwellian distribution). These figures also show for comparisonthe variation of the corresponding transit-time function PI found whenall electrons leave the cathode with a uniform initial velocity equal tothe mean value of the actual velocity distribution (according to Knoland others 6)). As is seen, for not too large transit-time angles the refinementintroduced by Begovich and Van der Ziel results in an increase of theconductance by a constant factor of about 1 : 5. Another difference isthat with large values of aT the conductance nowhere becomes negative,as is to be expected also on thermodynamic grounds.The object, however, is to investigate the energy gain of all electrons.

We shall tackle this problem by taking as a first contribution the total-emission conductance according to Begovich and Van der Ziel for allelectrons, thus as if all electrons were returned. Later we shall have toapply a small correction, owing to the fact that a small fraction of theelectrons .does not return to the cathode (see below).

(b) The electrons of group (b) will now be considered. It is assumedthat there is a linear retarding field between cathode and anode and thatthese electrodes are plan-parallel at a distance d apart (fig. lb), whilst thefield strength is represented by -Fo ~1+ p cos (wt + rpH, where Fo ispositive, -Fod is the static anode d.c. voltage and -Fodp cos (wt + rp)is the anode a.c. voltage; p <{;:: 1 (see fig. lb). The electrons leave the cathodeat a moment t = 0 with a velocity component Vo perpendicular to thesurface; the distance to the cathode is in general x (rationalized MKS unitsare employed). For the present the position of the anode is disregarded

213

(1)

214 G. DIEMER

.and the electron paths are considered in an infinitely extended field ofthe strenght given above.

~ 1·=-·d{l+P'~(~t+,'J~_g_.2 66408

Fig. lb. Plan-parallel diode.

In the absence of an alternating voltage (p - 0) the transit time upto the reversing point is then

mvotoe = -F· , (2)

e 0

where e= electronic charge, m = electronic mass. Here only the velocitycomponent perpendicular to the surface is considered, and the interactionbetween the electrons is disregarded.

The general solution of the equation of motion (see Knol and others 6))is then

V [t2 P ]X = vot - ~ - +- ~cos(wt + cp) + wt sin cp - cos cp~ •toe 2 w2

(3)

The velocity at the moment t is

dxd= Vo - V

o [t + !!. ~- sin (wt + cp) + sin cp~] • (4)t toe w

For the transit-time up to the reversing point (dxjdt = 0) in thepresence of an alternating voltage we put the series

t = toe + clP + c2P2 + ....By substitution in (4), with dxjdt = 0, we fi~d for Cl and C2:

(5)

1 ... Cl = - ~sin (wtoe + cp) - sin Cp~; C2 = Cl cos (wtoe + cp). (6)

t»

By substituting (5) in (3) we find for the reversing point x - Xm to asecond order of approximation

~

-vo Vo Vo ~Xm = !votoe + p -- cos (wtoe + cp) - - sin cp + -2 - cos cp +

w~ w w~

~

Vo 2 Vo VoCI• )+ p2 -t - Cl +- Cl sin (wtoe + cp) - - SIll CPl'toe wtoe wtoe)

(7)

lIIICROWAVE DIODE CONDUCTANCE

To a first approximation Xm appears to be independent of Cl and c2whilst to a second approximation it is dependent only upon Cl'

The reversing point is now fixed at a distance Xm = d, and for the cor-responding initial velocity the following series is introduced

wherefmvoe2 = eFod,

are independent of p.

0

Ir...Wri V/,I

Rp, 1\/

I t1//

5 /

2

0.5

0.2

O.

0.00.01 0.05 0.1 0.2 0.5 I 2 5 la

- ()(r 66409

Fig. 2. Transit-time functions of the total-emission conductance; ':PT after Begovich andVan der Ziel, ':P1after Knol and others. Double logarithmic scale. -.-.- Square-law curve.

Equation (8) is now substituted in (7) with Xm = d and the left-handand right-hand members are made identically equal. By giving each ofthe coefficients of p, p2, etc., separately the value zero we find bl, b2,etc. For our purpose we need only bl, for which we find

, eFo ~ eFo Voe . eFo ~bl = -- --2 cos (ae+ cp) + - sin cp--- cos cp ,mVoe mw w mw2

in whichaB = wtoe.

The increase in the kinetic energy E(vo) with which the electrons haveto start so as to be able just to reach the anode in the presence of an alter-nating voltage ~s .

LiE = ,E(vo) - E(voe) = mvoeblP + terms of higher order. (12)

This resolves, to a first approximation, into

2LiE = E(voe)p"2 ~cos(ae + cp) + aesin cp- cos cp~. (13)

ae

215

(8)

(9)

(10)

(11)

216 G. DIEMER

We shall now define exactly the limits of the energy interval belongingto ,group (b); we take for these E(voe) - LJEm and E(voe) + LlEm, inwhich LJEm is the maximum value of, LJE according to (13). So LlE~ isindependent of cp (see fig. la). The a.c. energy, transmitted to those elec-trons of group (b) that are able to reach the anode, is given by

,,=d dxf -eFoP cos (wt + cp) -d dt,,,=0 t

in which dxJdt according to (4) has to be substituted. It can easily beseen that to a first order of approximation this integral has the samevalue for all electrons of group (b). Hence, its value is equal to LJE asgiven by (13).

4~---r---r7--,---~---r-~

~l~,~+-~ __~~'~ ~ +- +-~

66410

Fig. 3. The same as fig. 2 on a linear scale.

Now we have to consider the electronic velocity distribution at thecathode (see also fig. la).

Is (-E)N(E)dE = e2VTexp eV

TdE, (14a)

where N(E) dE is the number of electrons emitted per second in the energyinterval (E, E + dE) and Is is the total-emission current (saturationcurrent); again only the components in the direction perpendicular to thesurface are considered.We fust consider the electrons in the energy region between E(voe) - LJE

and E(voc) + LJEm. As may be seen from (7) for small p-values Xm isincreasing with increasing vo' so all these electrons are able to reach the

~nCRoWAVE DIODE CONDUCTANCE

anode. The total power, transmitted to them, is to a second order in p

Pa = N(E(voe» (LlEm + LIE) LIE (15a)

= number of electrons times energy gain per electron.The rest of the electrons, belonging to group (b), have energies between

E(vo) - LlEm and E(vo) - LIE, so they are not able to reach the anodeand all return to the cathode. According to Knol and others 6) the a.c.energy transmitted to such electrons is per electron (see their equations4 and 5)

4eVe = pE 2 (sin ae - ae cos ae) sin (ae + rp).

ae .

With the same approximation as for Wa, we find for the total a.c. powertransmitted to the returning electrons

P; = N(E(voe» (LlEm - LIE) eVe. (15b)

Introducing the anode current

in (15a) and (15b) for JY(E(voe» we.may write, according to (14a),

IaN(E(voe» = -. (14b)

e2VT

Therefore the total power to be reckoned with for the electrons of group(b) is

Pe(rp) = Pa + P; . (15c)

When taking again both the number of electrons and the energy absorbedin a first approximation only, in (16) we may take E = E(voe) = eVa.

According to (13), (14b) and (16) we find, except for a term whoseaverage over rp is zero,

Ia lVal 2} lPe(rp) = - -v. P 2 ?cos(ae+ rp)+ ae sin rp- cos rpç.e T ae

2pE~voe) ~cos(ae+rp) + aesinrp- cosrp__:2sinaesin(ae+rp) + 2ae cosn, sin(ae+rp)~. (18)ae

By taking in (18) the average over rpit is possible to calculate the cor-responding electronic conductance ge from the equation:

energy given off by the alternating field = energy absorbed by theelectrons,

or

217

(16)

(19)

218 G. DIEMER

a'he resolution of Pc(q;) according to (18) yields

Iage = V

TPe(ae) = gc(O) Pe(ae) (20)

with

and ge(O) = la/VT = the theoretical value of the static slope of thecharacteristic in the exponential region.When aB is small

(22)

and when ae is large

(23)

From (22) it is seen that for small transit-time angles the electronicconductance of group (h) approaches the static value ge(O), as was to he ex-pected. According to (23) for large transit-tm;.e angles the contributionapproaches zero roughly in inverse proportion to the square of thetransit-time angle.

/ ~/ \

1/ \

/ 1\,/ \

\\r-,

r-- r-, --f--oo 1f4 2x4 3li4 1( 5TY.. 61tt~ 7Tli~ 21( 91f4 ,o19~ "19. 31(-O(e

66411

Fig. 4. Transit-time function of the exponential conductance.

In fig. 4, Pe(ae} is plotted as a function of aB' It appears that Pc isnowhere negative and that with small aB it at first rises proportional toa~ and attains a maximum value approximately equal to 2·3. Not untila > 2:n;does Pe rapidly decrease (at aB = 2:n;,PB ~ 0·3).

(c) It will now he shown that, provided we do not get too close to thespace-charge region of the characteristic (thus in the actual exponentialregion), the energy transmission to the electrons of group (c) (which

MICROWAVE DIODE CONDUCTANCE

always reach. the anode when p is small) is negligible with respect tothe total-emission conductance gT and the exponential conductance g~.From the calculations of Bakker and De Vries3) (see their equation 21)the conductance due to these electrons can be derived as follows:·

Ia ( vaoa) 2ga = -I -I 1 - -- 2" (2 - 2 cos aa - aa sin aa) ,Va d aa

where Va is the velocity with which the electrons reach the anode whenp = 0, oa is the corresponding static transit-time and a = WOa the statictransit-time angle; here it is assumed that all electrons start with thesame initial energy tmv~ + el Val, which in our case is not quite true,but since we are only trying to reach an estimation it will be taken thatall electrons passing across start with an initial energy e(lVal + VT),which is a fairly good approximation considering the rapid decline of.the Maxwellian velocity distribution (14a). So long as IVT/Val~ 1 (i.e. inthe actual exponential region of the characteristic) then aa R:I ae• In thatcase vaoa/d = 2(VT/Va)2 is also very small.With small IVT/Val we get approximately (first order of approximation

in VT/Va)

with2

Pa(ae) = 2" (2 - 2 cos ae - ae sin ae).aa

Figure 5 gives Pa as a function of ~e. It is seen that everywhere IPal<l.Comparing, therefore, ga and ge (eqs (25) and (20), respectively, figs 5and 4), we may conclude that for ae <2n (i.e. for all cases of practicalimportance; see also section 4) ga is negligible with respect to ge'

1f'a

t2

11f'a

./ --...../ <, -0

1

3o 1Ij2 7T

Fig. 5. Transit-time function of the conductance, due to those electrons that in thepresence of a small a.c. anode voltage still always reach the anode.

219

(24)

(25)

(26)

220 G. DIEMER

. A small correction has to be made, because in calculating the tot al-emission conductance it has been assumed that also the electrons ofgroup (c) turn back. This correction is estimated for IVT/Val~ 1 by againmaking use of the expression for eVs according to Knol and others 6) ;see eq. (16). The amount of energy calculated in excess for the electronspassing across is of the order

(27)

with8

PI (as) = "2 sin ae (sin ae - ae cos as) •as

(28)

For the behaviour of PI see fig. 2. The maximum value of PI is about·3·3. From (27) it is seen that gal is even small of a higher order than gaand therefore gal is certainly negligible with respect to ga if ae < 2n and

IVT/Val ~ l.Summarizing, it may be said that for those cases where the retarding

field can be regarded as being linear and the intereetion of the electronsis negligible, the conductance in the exponential region of the charac-teristic can be described theoretically by the sum of the total-emissionconductance g; according to Begovich and Van der Ziel, which is propor-tional to the saturation current Is, and the exponential conductance ge,which is proportional to the static conductance. In section 4 it will beinvestigated how under practical conditions the ratio of the contributionsdepends upon the anode voltage.

3. Relation of total-emission conductance to exponential conductance

Here an approximate expression will be derived for the ratio of thetotal-emission conductance gT to the exponential conductance ge for smallvalues of aT (in most cases even in the centimetre-wave region aT < 1).

From eqs (1), (17) and (20) it follows that

(29)

(30)

where ao is the transit-time angle that would be obtained in the case ofa linear retarding field when IVal = VT. For normal cathode temperatures

MICROWAVE DIODE CONDUCTANCE

and distances between the electrodes of the order of 10 fJ., Qois of the orderof 1 for a wavelength of 10 cm, and Qois proportional to w2d2 (for d = lOfJ.,T = 1050 oe and A = 10 cm; Qo= 2).Using the abbreviation 1]= IVa/VTI for 1]~ Qowe find approximately

gT 4 2- = - Qoexp1].ge 1]6

In fig. 6, gT/gea~ has been plotted as a function of 1]. It is seen thataround 1] ~ 6 (anode voltage ~ - 0·6 V) there is a region where, for nottoo large values of Qo,the exponential conductance predominates strongly.In the minimum the ratio gT/ge = 0·03 a~; for T = 1050 oe this figure is

it - 10 cm, d 30 fJ., gT/ge ~ 1,A - 10 cm, d - 10 fJ., gT/ge ~ 0,1,A 3 cm, d - 30 fJ., gT/ge f>.:::! 10,it 3 cm, d = 10 fJ., gT/ge ~ 1.

!J

0V

0

5

2 17

\ Ij1

1\~ V

1

0.5

0.2

O.

0.05

0.02o 2 6 8 10 12 11,. 16 18 20-'I

2266413

Fig. 6. Ratio of total-emission conductance to exponential conductance times ljao2 versusanode voltage (1] =;= IVajVTI; ao is a characteristic transit-time angle proportional to wd).

In the last-mentioned case in the exponential region (where for mostpurposes the work point actually comes to lie) the total-emission conduct-ance becomes approximately equal to the exponential conductance.With a wavelength of 10 cm, however, a distance of 10 microns betweenthe electrodes is sufficient to keep the total-emission conductance in theexponential region low.

221

(31)

222 G. DIEMER

4. Theoretical results checked with an experimental diode

Diemer and Knol 7) have reported measurements for the electronicconductance of a "short-distance" diode with L-cathode S) on wavelenghtsof 15 and 35 cm. It is very difficult to make a cathode surface sufficientlyflat. It is, therefore, seldom justified to apply to it the foregoing theoryof a plan-parallel construction, especially as regards the total-emissionconductance. When the. valve is far beyond cut-off the electrons turnback at a very short distance from the cathode. For Va = ~1 V the averagepath is only a few microns; owing to the irregularities in the cathodesurface being of the same order of size, the electrons no longer travel ina homogeneous field and also their outward and return paths are not ofthe same length. Many of the paths traversed may be as represented infig. 7. The structure ofthe cathode surface has been taken from an analysis

66414

Fig. 7. Diagrammatic representation of the actual cathode surface and of some asym-metrical electron paths (very strongly enlarged).

of the actual cathode surface used for the measurements. For such asym-metrical paths the absorption of energy in the high-frequency alternatingfield may be much greater than in the case of symmetrical paths startingand ending perpendicular to a flat surface, as is assumed in the theory.Probably this is the main reason why in Diemer and Knol's publicationit was-found that for Va <- 2 V the total-emission conductance was muchgreater than the theoretical value to be expected.We shall now use these measurements for the exponential part ofthe

characteristic and see in how far the theory in section 2 holds. Only themeasurements with Va between -3·5 V and -0·5 Vwill be taken (correctedfor contact potential), these being given in figs 8 and 9 for wavelengthsof 15 and 35 cm respectively and for three values of Is, viz. 35, 15and 4 mA.In addition to the measured values of the conductance these diag~amsgive also the total-emission conductance gT according to Begovich andVan der Ziel (see section 1), the exponential conductance ge accordingto formula (20) (here the experimental value of ge (0) has been inserted;as is usually the case with practical diodes, this was not equal to Ia/Vr)and finally the total theoretical conductance g = gT + ge'From these figures the following conclusions are drawn.

(1) In the exponential part of the characteristic the exponential conduct-ance strongly predominates over the total-emission conductance as soonas a perceptible anode current begins to flow.

MICROWAVE DIODE CONDUCTANCE.

(2) In the real exponential region (ahout-l V anode voltage) the measuredvalues of the electronic conductance agree fairly well with the theoreticalresults.(3) The deviations in the case of more strongly negative anode voltageswill he due to irregularities in the cathode surface.

pA/V/0'

Cl 5u~ 21:103-g 5ou 3

l'052105

.1.-15cm

ls-35m) F==H

1// 1/

Ts~4,.mA-11 Id

2 91/7ge J' /

.' I"" " /I.IIT

2 rt; /geI

I-3 -2 -I

5

2IÖ-t. -~I

Fig. 8. Comparison of earlier measurementson 11 plan-parallel diode with theoreticalresults at 15 cm wavelength .

. Points measured by Diemer and Knol.gT: total-emission conductance (theoreti-cal); go: exponential conductance (theoreti-cal); g = gT + go; Va' = anode voltagecorrected for contact potential difference.

pA/V10

~ 5c:~ 2-5,038 5

1 210

À,-35cm4

Is-35mA,f::I,- I/Is·'5m"A-:

71 /1

2 Ir! [;9'9.

5

91/9. :t 19rt05

2.../...... ./... r9T

I5

11

2

M4~5

-1 0--+-Va'

66416

Fig. 9. The same as fig. 8 for 35 cmwavelength.

Eindhoven, December 1950

REFERENCES

1) N. A. Begovich, J. appl. Physics 20, 457-462, 1949.2) A. van der Ziel, Proc. Inst. Radio Engrs 38, 562, 1950.3) C. J. Bakker and G. de Vries, Physica 2, 683-697,1935.4) A. van der Ziel, Philips Res. Rep. 1, 97-118, 1946.5) J. J. Freeman, J. Res. Bur. Standards 42, 75-88, 1949.G) K. S. Knol, M. J. O. Strutt and A. van der Ziel, Physica 5, 325-334, 1938_t) G. Diemer and K. S. Knol, Philips Res. Rep. 4, 321-333, 194.9.q) H. J. Lemmens, M. J. Jansen and R. Loosj es, Philips tech. '{tw. U ,11,1.'1'\<), ,\Q'lO

I

223