Resolution and frequency response in secondary emission devices

IEEE TRANSACTIONS ON ELECTRON DEVICES. VOL. ED-14, NO. 12, DECEMBER 1967 829

denum-silicon Schottky barrier,” J . Appl. Phys. , vol. 37, pp. 2985- 151 G. Kano, M. Inoue, J. Matsuno, and S. Takayanagi, “Molyb-

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England: Clarendon Press, 1957. [el H. K. Henisch, Rectqyying Semiconductor Contacts. Oxford,

L7l S. M. Sze, C. R. Crowell, and D. Kahnn, “Photoelectric deter- mination of the. image force dielectric constant for hot electrons in Schottky barriers,” J. Appl. Phys., vol. 35, pp. 2534-2536, 1964.

Schottky barrier, Solid-State Electronics, vol. 6, pp. 281-295, 1963. [*I D. Kahng:, “Conduction properties of the Au-n-type Si

conductor Schottky-barrier diodes,” Trans. Met. Sac. A I M E , 191 C. R. Crowell, J. C. Sarace, and S. M. Sze, “Tungsten-semi-

vol. 233, pp. 47&481, 1965. [lo] K. G. Mckay, “Avalanche breakdown in silicon,” Phys. Rev.,

Pll S. M. Sze and G. Gibbons, “Avalanche breakdown voltages of abrupt and linearly graded p-n junctions in Ge, Si, GaAs, and Gap,” AppZ. Phys. Lett., vol. 8, pp. 111-113, 1966.

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[l3] C. S. Fuller and J. A. Ditzenberger, “Diffusion of donor and acceptor in silicon” J . Appl. Phys. , vol. :?, pp. 544-553, 1956.

[I4] N. I. Meyer and T. Guldbransen, Method for measuring im- purity distributions in semiconductor crystals,” Proc. I E E E , vol. 51,

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des Transistors,” 2. Phys., vol. 134, pp. 435-450, 1953.

Resolution and Frequency Response in Secondary Emission Devices

ALVISE BRAGA-ILLA

Absfracf-The resolution of a secondary emission tube is char- acterized by an equivalent, or effective, current distribution in the beam. This current distribution is defined as that which would cause the actual resolution degradation when landing on the target surface in an ideal way, i.e., without redistribution nor charge-to-potential mapping on the target. A “frequency response” technique is then introduced to measure the equivalent current distribution, and the theoretical responses for two distributions assumed a priori as possible, i.e., the Gaussian and the cosinusoidal, are derived. These h d - ings are compared with experimental results which show that (1) it is legitimate to characterize the resolution of a storage tube by means of its effective distribution, and (2) both the Gaussian and the cosine distribution are good models for the real situation, the former giving a better fit for somewhat defocused beams. The formulas and experimental techniques describe the tube behavior for all values of scanning speed during both the writing and reading cycles, and for al l input frequencies. The results can also be used to determine spatial resolution when discontinuous scanning is used.

M I. INTRODUCTION

ANY MEMORY and image processing devices use the principle of secondary emission of electrons to deposit or remove charge from an in-

sulator. An important characteristic of these devices is their resolution, which can be loosely defined as the ability to record and retain fine detail, and is thus an indication of the amount of information which can be stored or handled.

Manuscript received May 23, 1967; revised July 31, 1967. This paper is part of a dissertation submitted to the School of Engineering, University of Pittsburgh, in partial fulfillment of the requirements for the Ph.D. degree.

tute of Technology, Lexington, Mass. (Operated with support from The author is with the Lincoln Laboratory, Massachusetts Insti-

the U. S. Air Force.)

This paper develops a theory of the resolution of devices in which the energy of the bombarding electron beam is always sufficiently high to cause the insulator secondary emission coefficient to exceed unity. More specifically, the experimental verification of the theory has been carried out on tubes of the so-called “barrier grid” and “mesh collector’’ types, whose electrode configuration is shown in Fig. l(a) and (b), respectively. I t is possible that the methods described here may be extended to other components, with appropriate modi- fications.

In a typical memory device, the exploring or primary beam, modulated and deflected by the tube electrodes, deposits a charge pattern on the insulator, which is held originally at a constant electrostatic potential, lower than the collector’s. Since the secondary emission coefficient is greater than unity, the potential of the bombarded areas rises towards the collector’s. If the potential of the whole insulator is now raised by a constant amount by applying a voltage step on the support plate (Fig. l ) , the bombarded areas retain their charges and become more or less positive with respect to the insulator. The electrons liberated by a beam impinging for the second time on the insulator reach the collector or not, depending on their emission energy and the potential behavior in the insulator-collector region. The insulator is thus discharged to an equilibrium, or steady-state potential, while the current modulation at the collector is a convolutional replica of the charge pattern stored originally and of the exploring beam current density.

830 IEEE TRANSACTIONS ON ELECTRON DEVICES, DECEMBER 1967

TARGET CONTROL

BARRIER GRID

(a)

PLATE CONTROL

GRID

E

FILAMENT

HODE COLL~CTOR DEFL~CTION FOCU~ING

PLATES LENSES

(b)

Fig. 1. Barrier grid and mesh collector storage tubes.

The operation of these tubes has been studied by several authors.r1l-[51 One assumption typically made is that those electrons which are not collected return to the point of emission on the target. The case in which this does not happen, i.e., when the electrons ‘(redis- tribute,” has received limited attention.r11;[61

The limitations to the tube resolution have two ma- jor causes: the redistribution of emitted electrons and the finite dimensions of the exploring beam. This is in some way analogous to what is found in CRT’s, where the beam diameter and the excitation and diffusion phenomena in the phosphor are the limiting factors.

Another cause of resolution loss has been discussed more recently.I71 The charge distribution on the surface of the insulator originates a potential distribution, which actually controls the number of electrons reaching the collector a t a given time. Since an infinitely sharp charge transition does not produce an equally sharp potential transition, the relationship between the charge on the target and the generated potential can be considered a low-pass transfer function. This effect is due to capacitive linking between adjacent areas of the insulating material and becomes noticeable in low capac- ity targets. In the author’s experience, this effect is shadowed by the more important contributions of beam size and redistribution for tubes having practical values of target capacitance.1

The current distribution obtained inside a focused beam, a problem of great importance, for instance, in

The effect does become significant when very high resolution

considered. tubes (spatial frequencies greater than 100 cycles per mm) are

cathode-ray tubes and in television, has been studied extensively. The more recent contributions are by Herr- mann[*l and Ambos~ , [~ ] who concentrate on the design of traveling wave tubes, and by Kirstein.[lo1 The current density profile of the beam that reaches the target in a storage tube is influenced by very many factors per- taining to the design of the gun and the focusing structure, and by the potentials or currents in the electrostatic or magnetic lenses in’the tube. In all tubes, there are interactions between the focusing and the deflection sections,Ill] so that, for a given setting of the focusing potentials or currents, the beam has dissimilar current distributions when scanning different parts of the target. This well-known phenomenon can be compensated for by using (‘dynamic focusing,” i.e., by changing the focusing potentials or currents as a function of the beam position on the target.

The dependence of the deposited charge configuration on the beam current density profile, on the energy of emission of the secondary electrons, on the potential distribution in the neighboring areas of the insulator, and on the target capacitance makes the problem of resolution very difficult to analyze in a complete way. The approach which is presented here is based on a model which has received experimental confirmation for the two types of tubes mentioned earlier.

Several definitions of resolution and methods of measurement are in current usage and have been recommended by the IEEE.[l31 In television practice, resolution is measured at a known and constant scanning speed, which is determined by the scanning standards. I t is customary to quote the “extinction” resolution as the number of black and white bars, or lines, which can

BRAGA-ILLA: SECONDARY EMISSION DEVICES 83 1

be barely seen by an observer on the display. More ac- curately, one can determine the number of lines which correspond to a certain contrast ratio between the white and the black bar. This terminology has been carried over to storage tubes, so that resolution is often measured in “television lines per target diameter,” when the target is circular, or per unit length of target. An alternative and equivalent unit is the fundamental frequency of the Fourier expansion of the output pattern; the fineness of the bar structure is increased until the desired contrast ratio is obtained and the number of cycles in a given target length is determined. Clearly the number of cycles equals one-half the number of television lines over the same length, at any contrast ratio .

T o measure resolution in number of lines, one often uses the “shrinking raster” method: the deflection voltage or current is decreased, so that the pattern is written in a smaller and smaller area. The method which is recommended by the IEEE is closely related to the computer applications of the storage tube. In these uses, the scanning takes place’ in steps. The distance between digits, or inscribed target elements, is reduced until the contrast ratio at the output decreases by the desired amount. In both methods, the length of the target on which the bars or digits are written is measured in- directly by recording the scanning waveforms and by performing a separate measurement of the deflection sensitivity.

The difficulty with both approaches is that they do not characterize the response of the tube fully; the scanning speeds and the bandwidth of the electrical writing signal do not appear explicitly in any of the previous procedures of measurement. Although these as- pects can be taken into account separately, it is of interest to have a unified theory in which the important parameters are clearly emphasized. In tubes which have an electrical input and an electrical output, it is par- ticularly desirable to have a theory and a method of measurement which relate the input and output signals in the time and frequency domain.

The “frequency response” of a storage tube will be defined in the following discussion as the ratio of the peak-to-peak output current during the reading mode to the peak-to-peak value of the sinusoidal beam current during the writing mode, for given scanning speeds. The term “frequency response” is used here in a rather special sense, since obviously the phase of the signal has no significance in a memory device. As will be seen, the advantage of using the tube frequency response to evaluate the resolution is that its dependence on the scanning speed during the writing mode is evidenced.

11. THE MODEL FOR RESOLUTION As has been mentioned in the Introduction, the fac-

tors which affect the tube resolution fall into two categories:

1) nonzero size of the electron beam; 2) landing phenomena, such as redistribution and

charge-to-potential mapping.

The combination of these factors generates the over- all potential configuration in the collector-target region after the electron beam has landed on the target. From an external, or black-box, point of view, the final potential configuration and not the manner in which it has been generated is of interest. One can then proceed as if a nonideal fictitious beam, having an appropriate electron distribution in the plane perpendicular to the direction of beam motion, landed on the insulating target in an ideal way. By ideal landing i t is meant that the secondary electrons generated at one point either return to i t or are collected; in other words, there is no redistribution. I t is also meant that the potential distribution on the target is a scaled replica of the charge distribution.

I t is clear that one could equally well use the converse model to treat the resolution limitations analytically, Le., one could attribute them to an ideal beam landing nonideally on the target. The first point of view is chosen here, and the ‘<effective current distribution in the beam” is defined as the distribution which the current should have in order to cause, when landing ideally, the limitations to the resolution which are found in the real tube. In order to remain as close as possible to the actual physical situation, the distinction between the writing and the reading mode will not be dropped.

A . The Charge Distribution in the Wri t ing Mode The charge distribution in the writing mode will be

derived in terms of an effective current distribution in two dimensions j (u , v). The u direction is paralleI to the direction of motion of the beam; the v direction is normal to u and parallel to the target plane. The reference system, u v moves rigidly with the electron beam and, for convenience, its origin is the center of gravity of the beam.

I t is assumed that, when the total current changes as a function of the control grid potential, the current distribution is scaled in proportion, while preserving its functional form, an assumption which is verified for small excursions of the beam current.

One can write for the variable component of a sinusoidal beam current,

and

where the subscript w indicates the writing mode, and o is the radian frequency of the sinusoidal writing current. From these

832 IEEE TRAXSACTIOXS OX ELECTROS DEVICES, DECEMBER 1967

iw = sin (u t ) J-lJ-Ijw(u, v) dudv. (3)

If the target is scanned a t a speed uw, an appropriate choice of the time origin gives

x0 = vwt, (4)

where x. is the position of the center of gravity of the beam with respect to a reference system x, y rigidly con- nected to the tube target. The x-axis is taken parallel to the direction of motion of the beam and the y-axis perpendicular to it .

At the beginning of the writing mode, the target potential differs from the equilibrium potential, so that electrons can be deposited or removed from the insulator.

If I, is small, the deposited charge is proportional to the impinging charge p w ( x , y ) and may be expressed as

pw(x, y) = k J sin (mt)jw[u(t), v]d t , (5)

where T is the time in which a line of length L is scanned and the position variables u, v, are related to

0

x, Y by u(t) = x - x0 = x - vwt (6)

and

v = y.

Then (5) becomes

Equation (8) assumes that there is enough distance between parallel scan lines SO that no interaction occurs between them. This assumption was verified in the experiment described in the following paragraphs. The problem of bidimensional resolution can then be treated on the basis of the results on the effective distribution obtained in this

By developing the sine term and splitting j,(u, y) into odd and even parts, jW1 and jwz, (8) may be further expressed as

- 5 cos (E .) J x sin (E u)i.l(u, y)du. (9) vw 2-L

I t is seen, therefore, that the deposited charge has the form

pw(x , y ) = K1sin (E .) - Kz cos (;x) (10)

where K1 and KZ are defined in (9).

especially Barbier[’] and Braga-Illa.[51 2 For details on the equilibrium potential, see the references,

In general, K1 and Kz are functions of x, y ; however, if one considers the middle of the target, i.e., neglects the edge effects, and very reasonably assumes that the distribution j,(u, y ) decays to zero for values of I ul - ( ( L / 2 ) , then the limits of integration in (9) can be replaced with limits of f 00 without introducing any appreciable error. The latter assumption is clearly verified in any device of practical interest.

Then K1 and KZ in (10) become independent of x, and for any y the deposited charge has “spatial frequency”

W w = - .

vw

The coefficients K1 and Kz are given by

These results are valid for points of the target whose distance from the edges is a t least two beam diameters, if the beam diameter is defined as the diameter of a circle in which 99 percent of the current falls.

B. The Charge Distribution in the Reading Mode During the reading mode, the signal current is pro-

portional to the deposited charge151 for the small signals which are considered here.

The beam law of motion is now

x0 = vrt, (14)

where Y, is the scanning speed during the reading. The signal output at each time t is obtained by integrating the contributions from the constant current beam as i t scans the deposited charge, and may be expressed as

i ( t ) = kt! mJoLjT(x - v T t , y)pw(x, y) dxdy. (15) --m

I t is possible that the current distribution in the reading mode beam differs from that in the writing mode and, therefore, the subscript is used for this distribution.

Once more one can study the output from points of the target far away from the edges. Then the function j r ( x - v T t , y ) computed for x<O and x>L is zero, and one can change the integration limits in (15) to f m .

Changing again the variable of integration by means of (6) and substituting from (10) for the value of charge distribution results in

L V W JI

BRAGA-ILLA: SECONDARY EMISSION DEVICES 833

By developing the sine and cosine terms, one can see that the output current is of the form

i.e., is sinusoidal with output radian frequency wn given by

V ,

vw wo = -w;

the constants K3 and K4 are given by

If the writing and reading current distributions have even symmetry in u, i.e., jwl(u, y ) = O and jrl(u, y ) = O , one has from ( 1 3 ) that K 2 ( y ) = 0 .

Then, from ( 1 9 ) and (20)

sin - u dudy = 0, (22) t o )

and the output current has the simpler form

andj,(u, y) even in u. (23)

AS seen from (12) and ( 1 3 ) , Kl and Kz are functions of w.

By definition, the frequency response is then

I(@) - dK32 + K42 -- I(0) K30

For current distributions with even symmetry in U , one has simply

where Ka is given by (21) .

C. T h e Gaussian Hypothesis In this section it is assumed that the effective current

distribution is of Gaussian type with circular symmetry, l.e.,

The constant u represents the “spread” of the Gaussian effective distribution. This distribution is assumed to hold both during the writing and the reading modes.

Using (2) and ( 3 1 ) , j o can be eliminated to give

Sincej(u, v) has even symmetry, K 2 ( y ) is zero. Remem- bering (12) and ( 3 2 )

Using the definite integra1

the previous expression becomes

k I , exp [ - g] exp [ - -531. Y 2 (35) K1(y) = - --

v, d 2 T . U

The last equation shows that K l ( y ) , which represents the peak value of the deposited charge, decreases ex- ponentially with the square of w/v,, i.e., of the spatial frequency defined by (1 1). The value of K 3 is obtained from (21) :

k’k I, vwv, d 2 T u

K3=--

.exp [ - S] J-1 J-IeXP [ - $1 IR u2 + y2

*cos (; .) dudy.

834 IEEE TRA?\TSACTIQXS QK ELECTROX DEVICES, DECEMBER 1967

Executing the double integral, (36) becomes

-20 1 1 1 1 1 1 1 1 1 O I I I .-

0.5 4 .o 4.5

NORMALIZED FREQUENCY (kHz)

Fig. 2. Theoretical frequency response of storage tubes. and the frequency response in the Gaussian case is

1). The Cosinusoidal Hypothesis In this section, the theoretical frequency response is

obtained under the assumption that the beam density is cosinusoidal of the type If one considers the ratio of the peak signal values at

two different radian frequencies wz>wl, the scanning speed during the writing being kept constant,

I t is then clear that by measuring the frequency response according to (39) or (40), one obtains an indirect measurement of u,

This function represents a three-dimensional distribution in which the sections with planes parallel to the j , u and j , v planes are cosinusoidal, but which is not a revolution surface. The cosinusoidal revolution surface having an equation of the type

or, if w1 is not small compared to w2,

for u2 + v 2 5 - d2 (45) 4 d2

for g2 + v2 2 - 4

The measurement can be repeated for different frequencies and scanning speeds, so that a statistical analysis of the results can be made to improve the estimate.

I t must be noticed that, in applying this technique, the scanning speed during the reading v, cannot be made arbitrarily high; the constant K’ in (15) in fact decreases with v7, since the charge density per unit length decreases. Thus, the limitations to increasing v, are set by the noise in the system and by the detection se-nsitivity. Also, the scanning speed during the writing should be chosen to give reasonable values of deposited charge in relation to the beam current used. Once practical values of v, and v, are chosen, the results are independent of vT and depend on v, as indicated by (39).

The frequency response can be expressed in dB, as

leads to equations which cannot be integrated in closed form. Equation (44) is, however, close to (4.9, the differences being small where j (u , v) is large.

From (2) and (44), one obtains easily

I, = jOd2. (46)

Using a familiar trigonometric relationship and (46), the distribution (44) can be rewritten as

/2 I- cos2 (: u) cos2 (: v) I d2 I

j(u, a) = 1

0 i

(43)

The response decays very rapidly, as seen from (39) or (43). The frequency response as a function of the normalized variable w,/v, is shown in Fig. 2. Since this function is even in u,


Kl(y) = k “‘2 cos (: u) $ cos2 (+ u) --dl2

. cos2 (: y) du.

This integral is of the type

G = cos (au) cos2 (bu)da

which can be integrated in closed form to give

1 b sin (au - 2bu) U a 2(a - 2b)

G = - sin (au) cos2 (bu) $. -

b sin (au + 2bu) s-

a 2(a + 2b)

Letting a =w/vw and b = w / d and setting the limits of integration, the following expression results, after some simplifications:

The function

sin (E) F =

1 wd

wd[l-P(zY1 2vw

tends to + when (ud/2v,)-+w, and to 0 when a+ 00. At wd/2vw = 2 w , the deposited charge becomes uni-

form. This case corresponds to very large values of w , or very small writing speeds.

K 3 is computed from (21) and (52), assuming that the beam distribution during the reading is unchanged.

The double integral, upon substitution of (47) and separation of the variables, is found to equal

Simplifying and substituting in (53), one obtains the expression for the peak value of the output signal

One now obtains easily

3 IUJI?. = lim K3 = - K‘k __ .

W-+O 8 d

Therefore, the case is

2

. ( 5 5 )

frequency response in the cosinusoidal

From the measured frequency response, the param- eter d can now be determined. Since (57) cannot be solved for the normalized frequency, one proceeds graphically or numerically. The experimental curve [ I (w/um)/I (o) ] is plotted; then d / 2 is the scale factor to be applied to the abscissa axis to match the theoretical curve. The frequency response can be expressed in decibels, although the form of the equation is not as compact as in the Gaussian case. The behavior of this function is shown in Fig. 2 as a function of the normalized frequency wd/2.84 vw. The scale factor which permits the comparison of the response curves in the Gaussian and cosinusoidal cases has been obtained by requiring that the 6-dB points of the two curves coin- cide. Fig. 2 shows that the differences between the two response curves are very small in the range of interest. The cosinusoidal response is larger than the Gaussian a t low frequency, and smaller at high frequency, but the difference is always less than 1 dB.

I I I. EXPERIMENTAL RESULTS In order to test the predictions of the theory, fre-

quency response measurements have been performed on several storage tubes.

The experimental instrumentation is shown in Fig. 3 in block diagram form. The storage tubes were scanned in the horizontal direction and a sinusoidal voltage of small amplitude was applied to the control grid of the tube during the writing mode. Only one horizontal line at a time was scanned. The scanning speed during the reading and writing modes could be varied independently.

As known from the tube theory, the plate potential must be different during the writing and reading modes. This condition was obtained by applying a square wave voltage to the da te of the tube. A sinusoidal osr;llatnr

836 TEEE TRANSACTIOXS ON ELECTRON DEVICES, DECEMBER 1967

SQUARE WAVE -

GENERATOR i- INTEGRATOR X PLATES

[ V I I

OSCILLATOR BLANKING INPUT AMPLIFIER

I I 7

I 1 I I I STORAGE TUBE I I I I

I I

OUTPUT OSCILLOSCOPE

OUTPUT PREAMPLIFIER AMPLIFIER b

Fig. 3. Block diagram of the experiment.

generated the writing voltage waveform. The sinusoidal signal was blanked out by a gating pulse during the reading, controlled by the same square wave. The dc bias on the tube control grid could be modified to per- mit varying the average beam current magnitude and to experiment at different current densities. The blanking level in the writing waveform generator could also be changed, so that the reading current could be ad- justed independently of the average writing current. Focusing, astigmatism, and cathode voltages were pro- vided by networks which are not shown in the block diagram. The output signal was taken from the tube collector. A low-noise preamplifier was used to detect the reading signal. The scanning speed was computed from measurements of the deflection waveform and the deflection sensitivity.

Typical experimental curves for the frequency response of a barrier grid storage tube are shown in Fig. 4, which also shows the comparison of the Gaussian theory to the experiment results. I t is seen that the cosinusoidal response would give a slightly better fit in the case of optimum focusing, while the Gaussian hypothesis fits almost perfectly the results obtained in the case of a defocused beam. This is quite reasonable, since the focusing tends to compress the beam towards the geo- metrical center. Insofar as the actual beam current distribution is concerned, this result is not unexpected.

IV. CONCLUSIONS I t can be concluded that an effective beam current

distribution of Gaussian or cosinusoidal type can characterize suitably the resolution of secondary emission storage tubes. The parameters of the effective beam current can be obtained from frequency response measurements.

Using the results of this theory, one can determine the spatial resolution of the tube, i.e., the resolution obtained when the scanning takes place in steps and the writing is done with a stationary beam. The Gaussian

-<2 SCANNING SPEED: O. i7 rn rn /w

- i 6 I I 1 1 1 1 1 l I I I I I I I I I AVERAGE CATHODE CURRENT:Oii rnA

io 80 50 io0 200 500 4000 FREQUENCY (kHz1

Fig. 4. Experimental frequency response of a barrier grid storage tube.

effective distribution and the superposition assumption, valid for small signals, have been used[121 to study tradeoffs between packing density and number of signal levels in digital applications with step and continuous scanning.

Thus this theory 1) gives the relationship between tube resolution, frequency of the writing signal, and scanning speed; 2) allows one to determine experimen- tally the effective beam distribution; and 3 ) leads, therefore, to a discussion of the spatial resolution of the tube which permits the maximization of the amount of information stored in the device. [l2] The completeness of the theory appears to be an advantage over the other approaches, which have been briefly described in Sec- tion I. I t would be interesting to see whether a similar approach could be useful in studying the resolution of other types of memory devices which also require scanning.

ACKNOWLEDGMENT The author wishes to thank Prof. A. W. Revay of the

Department of Electrical Engineering of the University of Pittsburgh for his helpful advice and criticism. Thanks are also due to Dr. G. C. Sziklai, G. F. Newell, and K. E. Farr of the Westinghouse Research and De- velopment Center for their encouragement and support during the first phase of this work.


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Documents

Resolution and frequency response in secondary emission devices