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Mathematical model of electron gun on the field emission electron cathode basis
Vacuum 57 (2000) 267}281
Mathematical model of electron gun on the "eld emission electron cathode basis N.V. Egorov*, E.M. Vinogradova Faculty for Applied Mathematics, St. Petersburg State University, Bibliotechnaya pl. 2, St-Petersburg, Petrodvorets, 198904, Russia Received 2 May 1999; accepted 4 February 2000
Abstract In this paper the "eld emission gun is under investigation. A mathematical model is presented to describe the cathode lens in a rotationally symmetrical electron gun with a thin tip on a #at substrate as a "eld emission cathode and a system of circular apertures as focusing electrodes. The tip shape may be various. The number of the apertures may vary too. The e!ect of space charge is neglected. The potential distribution is found for the whole region of the electron gun. All geometrical dimensions of the system and the electrodes potentials are the parameters of this method. 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction In this article an attempt is undertaken to solve some problems related to the formation of the electron beam and its control on the basis of our previous results. It deals with the systems of beam formation and control in the electron guns with "eld emission electron cathodes (FEEC). The problems which are related to the formation of thermionic cathodes system are well elaborated (for example [1]). It is necessary to note that the permanent increase of requirements to the electrovacuum instruments quality leads to the necessity of the more complete calculation of all set of processes which have an in#uence on the beam formation and control systems operation. For a successful application of calculation methods for these systems (in particular, for the numerical models and numerical experiments) the modern requirements mean the three-dimensional problem solution. In spite of the publications (for example [2}8]) which show the urgency of
* Corresponding author. 0042-207X/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 2 - 2 0 7 X ( 0 0 ) 0 0 1 4 0 - 8
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the problem, these problems and the successes in using numerical methods for calculations of the physical "elds the solution of the three-dimensional problems in su$ciently general form is still a rather complicated and inadequately developed problem especially when the speci"c problems related with the engineering design are under consideration. There is an evident lack of the available software for technical use. The reason is in the high requirements of the computer hardware but mainly in the sophisticated implementation of the computational experiment. In this article the main attention is paid to the beam formation systems (guns) investigation on the basis of the electrostatic lenses. This research is initiated by the practical needs, so that its practical implementation is not related to the `irresistible di$cultiesa surmounting. Production cost of these systems is signi"cantly lower than that of the magnetic systems. At the same time the electrostatic lens calculation, its modelling and the numerical experiment for the electron guns with the "eld cathodes are more di$cult than for the magnetic lenses. It is stipulated mainly by the speci"c features of the "eld electron emission (FEE) process and namely, by the principal fact that the FEE is excited by the electric "eld and the FEE current density is strongly (exponentially) dependent on the electric "eld strength value near the FEEC surface. For electric "eld existence it is necessary to have another (except cathode) electrode. Usually, one electrode is insu$cient for the generation of the beam with the parameters required, therefore, the additional electrodes are introduced into the beam formation system. Specifying the shapes and the potentials of the electrodes allows to perform the beam excitement, its transport, focusing as well as e!ective control. But in contrast to the systems (guns) with thermionic cathodes, in which any electrode potential change has no in#uence on the cathode emission characteristics, any insigni"cant potential changes can lead to the signi"cant emission changes for the electron guns with the FEEC. But the cathode emission changes have an instantaneous in#uence on the electron beam characteristics and, therefore, lead to the beam transport and change in the focusing conditions. Thus, the calculation of the electron gun with the FEEC must take into account the FEEC in#uence on the beam focusing and transport conditions and the other electrodes in#uence both on the cathode emission ability and on the characteristics of the formation and control systems. Thus, the signi"cant di$culties arise. The absence of any considerable successes in the calculation of the guns and systems with the FEEC is related with these di$culties.
2. Problem background (formulation) Let us consider a physical model of the gun as an axially symmetrical electron}optical system which consists of a cathode, i.e. axially symmetrical thin tip on a #at metal substrate and a system of round apertures as the focusing electrodes. The tip shape may be various. The number of the apertures may be various too. The e!ect of the space charge is neglected. The potential of the tip is equal to the substrate potential and is assumed to be zero without the loss of general character of the problem. The parameters are: r (z) * tip surface shape, N * number of apertures, (R , Z ) * apertures coordinates, G G ; * apertures potentials, R * the tip curvature radius, and ¸ * the tip length. G
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3. Mathematical model For this electron}optical system with a tip, the rotationally symmetrical potentials <(r, z) without space-charge satisfy Laplace's equation. So we have to solve the boundary-value problem *<(r, z)"0 (r*0, z*0), <(r, z)"
PP X <(r,0)"0,
"0,
<(r, Z )" G "; , (1) G PV0 G where (r, z) are the cylindrical coordinates. According to the antisymmetry principle this boundary-value problem may be extended on the entire space (!R)z)#R): <(r, z)"!<(r,!z). Then the next boundary-value problem may be solved and the solution of this problem is congruent with the solution of problem (1) for z*0: *<(r, z)"0 (r*0,"z"(#R), <(r, z)"XK "0, <(r,$Z )" G "$; , G PV0 G where
(2)
r"r (z), z*0, r"r (!z), z(0. The solution of the boundary-value problem (2) (and boundary-value problem (3)) ;(r, z) can be expressed as a sum of three terms [9] XK :
<(r, z)"< (r, z)#< (r, z)#< (r, z), where
< "
*B
o(z )[r#(z!z )]\ dz .
(3)
(4)
\* o(z) is the charge density on the tip axis [!(¸!d),(¸!d)] and o(z)"!o(!z), d"R , and ¸ "¸!d. B < (r, z) is the solution of the boundary-value problem without taking the tip into account *< (r, z)"0, (5) < (r,$Z )" G "$; . G G PV0 The function < (r, z) can be represented as a next boundary-value problem solution *< (r, z)"0, (6) < (r,$Z )" G "!< (r,$Z )" G . G PV0 G PV0 B
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Let us introduce the new function u (r, z;z ) *B < " u (r, z; z )o(z ) dz , \*B where the kernel u (r, z; z ) from (6) to (7) is the boundary-value problem solution: *u (r, z; z )"0, 1 . u (r,$Z )" G "! G PV0 (r#($Z !z ) G The function o(z) is de"ned by the series [9]
o(z)" o (z), I I
o (z)"!< (0, z) ln
o (z)"! ln I>
(7)
(8)
4z(¸!z) \ , r(z)
4z(¸!z) \ *B o (z )!o (z) I I # u (0, z; z )o (z ) dz . I "z !z" r(z) \*B
(9)
4. The solution of the boundary-value problem (5) Let us consider the boundary-value problem (5): *< (r, z)"0, < (r,$Z )" G "$; . G G PV0 From < (r, z)"!< (r,!z) it is expedient to solve the problem for z*0. Thus, the boundary-value problem will be considered as *< (r, z)"0, < (r,0)"0, < (r, Z )" G "; . G G PV0 The functions < (r, z) can be represented as a Hankel transform [10] ; !; sh j(Z !z) G (z!Z )# G> < (r, z)"; # G> F (j) G G Z !Z sh j(Z !Z ) G G> G G> G sh j(z!Z ) G F (j) J (jr) dj, Z (z(Z , # (10) G G> sh j(Z !Z ) G> G> G where J is the zeroth-order Bessel function. Thus, we have to "nd the distribution of potentials for z*0 with N round apertures (; "<(z, r)" "0). The whole region of this electron-optical system is divided into N#1 X
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subregions: Z )z)Z . The boundary conditions and the continuity conditions of the electriG G> cal "eld on the planes z"Z for r)R can be written as G G < (r,0)"0,(r, Z ), r*0, G G <,(r,R)(R, < (r, Z )"; , r*R , G G G *(r, z) " , z"Z , r(R , (11) G G *z *z where (r, z)"< (r, z)" G , i"0, N. 8 WXW8G> Thus, with the Hankel transform (10) "rstly, the Laplace equation is satis"ed; secondly, the condition at z"0 is satis"ed for F (j)"0; thirdly, the limit condition < (r, z)(R is satis"ed for zPR; fourthly, the potential continuty requirement is satis"ed. The boundary conditions and the normal derivative continuity at the electrode's surfaces from (11) lead to the system of the dual integral equations
F (j)J (jr)"0, R )r(R, G G
ch j(Z !Z ) ch j(Z !Z ) 1 G G\ # G> G F (j) F (j)# j ! sh j(Z !Z ) sh j(Z !Z ) G sh j(Z !Z ) G\ G G\ G G\ G> G 1 F (j) J (jr) dj ! sh j(Z !Z ) G> G> G ; !; ; !; G! G G\ , 0)r(R . " G> G Z !Z Z !Z G> G G G\
(12)
Let 1 , (j)"! sh j(Z !Z ) G G\ ch j(Z !Z ) ch j(Z !Z ) G G\ # G> G , g (j)" G G sh j(Z !Z ) sh j(Z !Z ) G G\ G> G ; !; ; !; G\ . G! G < " G> G Z !Z Z !Z G G\ G> G Then system (12) can be written as g
G G\
(13)
F (j)J (jr)"0, R )r(R, G G j[g (j)F (j)#g (j)F (j)# g (j)F (j)]J (jr) dj"< , 0)r(R . G\ G G\ G G G G> G G> G G
Let gH (j)"1!CG ) g (j), G G G G
(14)
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where the constants CG are determined from lim gH (j)"0. G G H Then g (j)"[1!gH (j)]/CG . G G G G From (12), lim g (j)"2, then CG ". H G G Therefore,
exp(!j(Z !Z )) 1 exp(!j(Z !Z )) G> G # G G\ . gH (j)"! G G sh j(Z !Z ) sh j(Z !Z ) 2 G> G G G\ The system of the dual integral equations (14) can be represented as
F (j)J (jr)"0, R )r(R, G G j[g (j)F (j)#2(1!gH (j))F (j)# g (j)F (j)]J (jr) dj"< (r), 0)r(R . G\ G G\ G G G G> G G> G G (15)
Let F (j) have the form of an integral transform: G 0G F (j)" (t) sin jt dt. G G Making use of the discontinuity Weber integral
(16)
0, 0)t(r, J (jr) sin jt dj" (t!r)\ , 0(r(t, it may be shown that the "rst equations from system (15) with the substitutions (16) are satis"ed. F (j)J (jr) dj" G
0G
(t) dt G
J (jr) sin jt dj,0, r'R . G
Substituting (16) in the second system equations (15) we have
jg (j)J (jr) dj G G\
0G\
(t) sin jt dt#2 G\
0G>
(t) sin jt dt"< (r). G> G We have to multiply Eq. (17) by r and to integrate by r. From #
jg (j)J (jr) dj G> G
d jrJ (jr)" [rJ (jr)], dr
j(1!gH (j))J (jr) dj G G
0G
(t) sin jt dt G (17)
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Eqs. (17) can be represented as
r
0G\
(t) dt G\
! 2r #r
0G
0G>
g (j)J (jr) sin jt dt dj#2r G G\
(t) dt G
0G
(t) dt G
J (jr) sin jt dt dj
gH (j)J (jr) sin jt dt dj G G
(t) dt G>
g (j)J (jr) sin jt dt dj"= (r), r(R , G G\ G G
(18)
where = (r)"P t< dt. G G Making use of another discontinuity Weber integral
0, 0)r(t J (jr) sin jt dj" R , r't P(P\R and the integral transform of Bessel's function
(19)
2 p J (jr)" sin H sin(jr sin H) dH, p Eqs. (18) may be represented from (19) and (20) as
(20)
p P t (t) 2 0G\ G
(t) dt dt g (j) sin jt dt dj sin H sin(jr sin H) dH # 2 G\ G G\ p (r!t p 2 0G !2
(t) dt gH (j) sin jt dt dj sin H sin(jr sin H) dH G G G p p 2 0G> g (j) sin jt dt dj sin H sin(jr sin H) dH"= (r), r(R .
(t) dt # G G\ G G G> p (21)
From the transposition t"r sin H to P [t (t)/(r!t] dt: G P t (t) p r sin H (r sin H) p G G dt" r cos H dH"
(r sin H) r sin H dH. G (r!t (r!rsint With (22) Eqs. (21) can be written as
2 p 2 0G\
(t) dt g (j) sin jt sin(jr sin H) dj# 2 (r sin H) dt G\ G G\ G p p 4 0G
(t) dt gH (j) sin jt sin(jr sin H) dj ! G G G p 2 0G> g (j) sin jt sin(jr sin H) dj r sin H dH"= (r), r(R .
(t) dt # G G\ G> G G p
(22)
(23)
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Every equation of (23) is the Shlemilgh equation. Thus, the functions (t) are the solutions of the G Fredholm's integral equations of the second kind
1 0G\ 1 0G ! K (x, t) (t) dt# 2 (x)# K (x, t) (t) dt G G\ G\ G GG G p p 1 0G> K (x, t) (t) dt"U (x), ! G> G G> G p where
(24)
2 V y< (y) G U (x)" dy, G p (x!y
K (x, t)"2 G G\
K (x, t)"4 GG
g (j) sin jx sin jt dj, G G\
gH (j) sin jx sin jt dj. G G
(25)
Then
1 x!t x#t exp(!ja) a#b a#b sin jx sin jt dj" R t #i !t #i 2b 2b 2b sh jb 2b 2b C (a#ib) t(a#ib)" , C(a#ib)
,
(26)
where C is the gamma function. The kernels K (x, t) are symmetrical and can be written in an explicit form (13) and (25): GH 1 sin jx sin jt dj, K (x, t)"!2 G G\ sh j(Z !Z ) G G\ exp(!j(Z !Z )) exp(!j(Z !Z )) G> G # G G\ sin jx sin jt dj (27) K (x, t)"!2 GG sh j(Z !Z ) sh j(Z !Z ) G> G G G\ or from (26).
1 R K (x, t)"! G G\ (Z !Z ) G G\ 1 R t K (x, t)"! GG (Z !Z ) G> G 1 ! R t (Z !Z ) G G\
1 1 x!t x#t #i !t #i . 2 2 2(Z !Z ) 2(Z !Z ) G G\ G G\ 1 1 x!t x#t #i !t #i 2 2 2(Z !Z ) 2(Z !Z ) G> G G> G 1 1 x!t x#t #i !t #i . 2 2 2(Z !Z ) 2(Z !Z ) G G\ G G\
t
(28)
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The next integrals may be represented as
1 sin jx sin jt dj sh j(Z !Z ) G G\ " !4(Z !Z )xt G G\
k 1 , ((k(Z !Z ))#(t!x)) ((k(Z !Z ))#(t#x)) 2 G G\ G G\ I (29)
exp(!j(Z !Z )) G G\ sin jx sin jt dj sh j(Z !Z ) G G\
k 1 . ((k(Z !Z ))#(t!x)) ((k(Z !Z ))#(t#x)) 2 G G\ G G\ I Then from (27) and (29) the kernels K (x, t) can be represented also GH k(Z !Z ) 1 G> G K "8xt GG ((k(Z !Z ))#(t!x)) ((k(Z !Z ))#(t#x)) G> G G> G I 2 k(Z !Z ) 1 G G\ # , (30) ((k(Z !Z ))#(t!x)) ((k(Z !Z ))#(t#x)) 2 G G\ G G\ I k(Z !Z ) 1 G G\ K (x, t)"8xt . G G\ ((k(Z !Z ))#(t!x)) ((k(Z !Z ))#(t#x)) 2 G G\ G G\ I If the aperture radii are much less than the distances between apertures the kernels K (x, t) can GH be written as " !4(Z !Z )xt G G\
1 K (x, t)+C xt , G G\ (Z !Z ) G G\ 1 1 K (x, t)+C xt ! , GG (Z !Z ) (Z !Z ) G G\ G> G where
1 +1.0517998, C " k 2 1 C " +0.1502571. k 2 Consider U (x). Making use of the functions < (13) and substituting them into (25) we obtain that G G functions U (x) are G 2 ; !; ; !; G> G! G G\ x. U (x)" (31) G p Z !Z Z !Z G> G G G\
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Thus, to solve the boundary-value problem (5) we have to solve the system of the Fredholm's integral equations of the second kind (24) with the symmetrical kernels K (x, t) (28) or (30) and GH functions U (x) (31). G 5. The calculation of the function u (r, z; z ) To "nd function u (r, z;z ) we have to solve the boundary-value problem (8) *u (r, z; z )"0, 1 , u (r,$Z ; z )" G "! G PV0 (r#($Z !z ) G where z is a parameter. For convenience the new apertures' coordinates (RK ,ZK ) are introduced G G ZK "!Z , RK "R , i"1, N, G ,>\G G ,>\G ZK "!Z , RK "R , i"N#1, 2N. G G\, G G\, As for function < (r, z) the method of dual equation is applied. The region z is divided into 2N#1 subregions: ZK )z)ZK and, as in the previous section, it is supposed that G G> . uG>(r, z;z )"u (r, z;z )" K G 8 WXW8K G> The boundary conditions and the normal derivative continuity at the electrode surfaces z"ZK , G r)RK , (i"1,2N) are G uG>(r, ZK ; z )"uG (r, ZK ; z ), r*0, G G u (r, ZK ; z )"![r#(ZK !z )]\ , r*RK , G G G *uG> *uG " , z"ZK , r(RK , i"1, 2N, G G *z *z u (r,!R;z )(R, u,>(r,R;z )(R. Represent the function u (r, z;z ) as the distribution:
(32)
sh j(ZK !z) G> (F (j)#exp(!j"ZK !z ")) G sh j(ZK !ZK ) G G> G sh j(z!ZK ) G (F (j)#exp(!j"ZK u (r, z; z )"! # !z ")) J (jr) dj, G> sh j(ZK !ZK ) G> G> G ZK )z)ZK , i"1, 2N. G G>
(33)
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It is obvious that with (33) the Laplace's equation is satis"ed; the limitation conditions for function "u (r, z; z )"(R are satis"ed for zP$R; and the continuity of the function u (r, z; z ) is satis"ed. Then conditions (32) lead to the system of the dual integral equations:
(F (j)#exp(!j"ZK !z ")) J (jr) dj"[r#(Z !z )]\ , r*RK , G G G G
1 (F (j)#exp(!j"ZK !z ")) j ! G\ sh j(ZK !ZK ) G\ G G\
!
!ZK ) ch j(ZK !ZK ) ch j(ZK G> G (F (j)#exp(!j"ZK !z ")) G G\ # G G !ZK ) sh j(ZK !ZK ) sh j(ZK G> G G G\
#
1 sh j(ZK !ZK ) G> G
(F (j)#exp(!j"ZK !z ")) J (jr) dj"0, r(RK . G> G> G
(34)
Making use of the integral
exp(!j"ZK !z ")J (jr) dj" [r#(Z !z )]\ . G G
System (34) can be written as
F (j)J (jr) dj"0, r*RK , G G
1 !ZK ) ch j(ZK !ZK ) ch j(ZK G> G F (j) G G\ # F (j)# j ! sh j(ZK !ZK ) G\ !ZK ) G sh j(ZK !ZK ) sh j(ZK G G\ G> G G G\
1 F (j) J (jr) dj"< (r), r(RK , ! G G sh j(ZK !ZK ) G> G> G
(35)
where
sh j(z !ZK ) G\ J (jr) dj, ZK )z )ZK , G\ G sh j(ZK !ZK ) G G\ sh j(ZK !z ) G> j J (jr) dj, ZK )z )ZK , < (r)" 2 G G> G sh j(ZK !ZK ) G> G 2
0,
j
z ,[ZK ,ZK ]. G\ G>
(36)
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So, as "z "(Z then for di!erent values z the functions < (r) can be written as G !Z )z )!Z : sh j(Z #z ) < (r)"2 J (jr) dj, j ,\ sh j(Z !Z ) sh j(Z #z ) J (jr) dj, < (r)"!2 j , sh j(Z !Z ) < (r)"0, iON!1, N. G !Z )z )Z : sh j(Z !z ) < (r)"!2 j J (jr) dj, , sh j2Z sh j(Z #z ) j J (jr) dj, < (r)"!2 ,> sh j2Z < (r)"0, iON, N#1; G Z )z )Z : sh j(Z !z ) J (jr) dj, < (r)"!2 j ,> sh j(Z !Z ) sh j(Z !z ) J (jr) dj, j < (r)"2 , sh j(Z !Z ) < (r)"0, iON#1, N#2. G Let
1 , (j)"! sh j(ZK !ZK ) G G\ ch j(ZK !ZK ) ch j(ZK !ZK ) G G\ # G> G . g (j)" G G sh j(ZK !ZK ) sh j(ZK !ZK ) G G\ G> G Then system (35) can be written as g
G G\
F (j)J (jr) dj"0, G
(37)
r)RK ; G
j(g (j)F (j)#g (j)F (j)#g (j)F (j)) J (jr) dj"< (r), r(RK . (38) G\ G G\ G G G G> G G> G G By comparing (38) with (14) it may be noted that these systems are di!ering only with the right parts. Introducing the new functions (t): G 0K G F (j)" (t) sin jt dt, i"1, 2N (39) G G
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and making use of calculations (15)}(23) the system of Fredholm's integral equations of the second kind for the functions (t) can be obtained: G !
1 0K G\ 1 0K G K (x, t) (t) dt# 2 (x)# K (x, t) (t) dt G G\ G\ G GG G p p
!
1 0K G> K (x, t) (t) dt"U (x). G> G G> G p
(40)
From (25), (26) and (29), (37):
1 1 x!t x#t 1 R t #i ! t #i K (x, t)"! GG 2 2 2(ZK 2(ZK !ZK ) !ZK ) (ZK !ZK ) G> G G> G G> G !
1 1 1 x!t x#t R t #i ! t #i 2 2 (ZK !ZK 2(ZK !ZK 2(ZK !ZK ) ) ) G G\ G G\ G G\
.
1 1 1 x!t x#t K (x, t)" ! R t #i ! t #i G G\ (ZK !ZK ) 2 2 2(ZK !ZK ) 2(ZK !ZK ) G G\ G G\ G G\
(41)
or
k(ZK !ZK ) 1 G> G ((k(Z K !Z K ))#(t!x)) ((k(Z K !Z K ))#(t#x)) G> G G> G I 2
K (x, t)"8xt GG
#
k(ZK !ZK ) 1 G G\ , ((kK (Z !ZK ))#(t!x)) ((k(ZK !ZK ))#(t#x)) 2 G G\ G G\ I
K (x, t)"8xt G G\
k(ZK !ZK ) 1 G G\ . ((k(ZK !ZK ))#(t!x)) ((k(ZK !ZK ))#(t#x)) G G\ G G\ I 2
(42)
For the functions U (x) from (25): G
2 V y< (y) G dy U (x)" G n (x!y and
V
y
(x!y
x exp(!ja)jJ (jy) dj dy" a#x
(43)
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and from (36), (43) the functions U (x) can be represented in an explicit form: G
4 x p I ((ZK G !ZK G\ )(2k#1)!(z !ZK G\ ))#x x ! , ZK )z )ZK , G\ G ((ZK !ZK )(2k#1)#(z !ZK ))#x G G\ G\ x U (x)" 4 G ((ZK !ZK )(2k#1)!(ZK !z ))#x p G> G G> I x ! , ZK )z )ZK , G G> ((ZK !ZK )(2k#1)#(ZK !z ))#x G> G G> z , [ZK , ZK ] G\ G>
0,
(44)
or
8(ZK !ZK )(ZK !z )k 4 G G\ G !x (((ZK !ZK )2k#(ZK !z ))#x) p G G\ G I
1 x ; # , (((ZK !ZK (ZK !z )#x )2k!(ZK !z ))#x) G G\ G G
ZK )z )ZK , G\ G (45)
8(ZK !ZK )(z !ZK )k U (x)" 4 G> G G !x G (((ZK !ZK )2k#(z !ZK ))#x) p G> G G I
1 x ; # , ZK )z )Z , G G> (((ZK !ZK )2k!(z !ZK ))#x) (z !ZK )#x G> G G G 0, z ,[Z , Z ]. G\ G>
So, the function u (r, z;z ) is represented with (33), functions F (j) are calculated with the functions G
(t) (39), which are the solutions of the Fredholm's integral equations of the second kind (40) with G the symmetrical kernels (41)}(42) and functions (44)}(45).
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6. Conclusion In the presented mathematical model the distribution of the potential is found as a three functions superposition < , < , < . < is de"ned through the charge density on the tip axis. < is the solution of the boundary-value problem with apertures without taking the tip into account. < is de"ned as satisfying the boundary conditions at the apertures. Thus, if we have a solution of systems (24), (40) we have the solution of the boundary-value problem (1), i.e. the distribution of potentials for whole region of the electron}optical system with a thin tip on a #at substrate and a system of round apertures as the focusing electrodes. All geometrical dimensions of the system and the electrodes' potentials are the parameters of this method.
References [1] Kingham DR, Swanson LW. J Appl Phys 1986;41(2):157. [2] Binh VT, Purcell ST. Proceedings of the International Vacuum Electron Sources Conference 1996, Eindhoven, The Netherlands. July 1}4, 1996. Appl Surf Sci 1997; 111:157. [3] Boswell EC, Hug SE, Huang M, et al. J Vac Sci Technol B 1996;14(3):1910. [4] Chmelik J, Veneklasen L, Marx G. Optics 1989;83(5):155. [5] Herrmannsfeldt WB. Nuclear Instr and Meth 1981;187:245. [6] Hill DN, Ohlinger WL, Cochran Jr. JK, et al. J Vac Sci Technol B 1993;11(4):1743. [7] Jensen KL, Zaidman EG. J Appl Phys 1995;77(7):3569. [8] Kazuyuki OK, Toshimi O, Nobuaki Tamura, Yoshiki Uchikawa. J Electron Microsc 1981;30(4):281. [9] Egorov NV, Almazov AA. Vacuum 1999;52:295. [10] U#iand J. Dual equations method in the mathematical physics problems. Leningrad, 1977. 220 pp [in Russian].
Mathematical model of electron gun on the "eld emission electron cathode basis N.V. Egorov*, E.M. Vinogradova Faculty for Applied Mathematics, St. Petersburg State University, Bibliotechnaya pl. 2, St-Petersburg, Petrodvorets, 198904, Russia Received 2 May 1999; accepted 4 February 2000
Abstract In this paper the "eld emission gun is under investigation. A mathematical model is presented to describe the cathode lens in a rotationally symmetrical electron gun with a thin tip on a #at substrate as a "eld emission cathode and a system of circular apertures as focusing electrodes. The tip shape may be various. The number of the apertures may vary too. The e!ect of space charge is neglected. The potential distribution is found for the whole region of the electron gun. All geometrical dimensions of the system and the electrodes potentials are the parameters of this method. 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction In this article an attempt is undertaken to solve some problems related to the formation of the electron beam and its control on the basis of our previous results. It deals with the systems of beam formation and control in the electron guns with "eld emission electron cathodes (FEEC). The problems which are related to the formation of thermionic cathodes system are well elaborated (for example [1]). It is necessary to note that the permanent increase of requirements to the electrovacuum instruments quality leads to the necessity of the more complete calculation of all set of processes which have an in#uence on the beam formation and control systems operation. For a successful application of calculation methods for these systems (in particular, for the numerical models and numerical experiments) the modern requirements mean the three-dimensional problem solution. In spite of the publications (for example [2}8]) which show the urgency of
* Corresponding author. 0042-207X/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 2 - 2 0 7 X ( 0 0 ) 0 0 1 4 0 - 8
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the problem, these problems and the successes in using numerical methods for calculations of the physical "elds the solution of the three-dimensional problems in su$ciently general form is still a rather complicated and inadequately developed problem especially when the speci"c problems related with the engineering design are under consideration. There is an evident lack of the available software for technical use. The reason is in the high requirements of the computer hardware but mainly in the sophisticated implementation of the computational experiment. In this article the main attention is paid to the beam formation systems (guns) investigation on the basis of the electrostatic lenses. This research is initiated by the practical needs, so that its practical implementation is not related to the `irresistible di$cultiesa surmounting. Production cost of these systems is signi"cantly lower than that of the magnetic systems. At the same time the electrostatic lens calculation, its modelling and the numerical experiment for the electron guns with the "eld cathodes are more di$cult than for the magnetic lenses. It is stipulated mainly by the speci"c features of the "eld electron emission (FEE) process and namely, by the principal fact that the FEE is excited by the electric "eld and the FEE current density is strongly (exponentially) dependent on the electric "eld strength value near the FEEC surface. For electric "eld existence it is necessary to have another (except cathode) electrode. Usually, one electrode is insu$cient for the generation of the beam with the parameters required, therefore, the additional electrodes are introduced into the beam formation system. Specifying the shapes and the potentials of the electrodes allows to perform the beam excitement, its transport, focusing as well as e!ective control. But in contrast to the systems (guns) with thermionic cathodes, in which any electrode potential change has no in#uence on the cathode emission characteristics, any insigni"cant potential changes can lead to the signi"cant emission changes for the electron guns with the FEEC. But the cathode emission changes have an instantaneous in#uence on the electron beam characteristics and, therefore, lead to the beam transport and change in the focusing conditions. Thus, the calculation of the electron gun with the FEEC must take into account the FEEC in#uence on the beam focusing and transport conditions and the other electrodes in#uence both on the cathode emission ability and on the characteristics of the formation and control systems. Thus, the signi"cant di$culties arise. The absence of any considerable successes in the calculation of the guns and systems with the FEEC is related with these di$culties.
2. Problem background (formulation) Let us consider a physical model of the gun as an axially symmetrical electron}optical system which consists of a cathode, i.e. axially symmetrical thin tip on a #at metal substrate and a system of round apertures as the focusing electrodes. The tip shape may be various. The number of the apertures may be various too. The e!ect of the space charge is neglected. The potential of the tip is equal to the substrate potential and is assumed to be zero without the loss of general character of the problem. The parameters are: r (z) * tip surface shape, N * number of apertures, (R , Z ) * apertures coordinates, G G ; * apertures potentials, R * the tip curvature radius, and ¸ * the tip length. G
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3. Mathematical model For this electron}optical system with a tip, the rotationally symmetrical potentials <(r, z) without space-charge satisfy Laplace's equation. So we have to solve the boundary-value problem *<(r, z)"0 (r*0, z*0), <(r, z)"
PP X <(r,0)"0,
"0,
<(r, Z )" G "; , (1) G PV0 G where (r, z) are the cylindrical coordinates. According to the antisymmetry principle this boundary-value problem may be extended on the entire space (!R)z)#R): <(r, z)"!<(r,!z). Then the next boundary-value problem may be solved and the solution of this problem is congruent with the solution of problem (1) for z*0: *<(r, z)"0 (r*0,"z"(#R), <(r, z)"XK "0, <(r,$Z )" G "$; , G PV0 G where
(2)
r"r (z), z*0, r"r (!z), z(0. The solution of the boundary-value problem (2) (and boundary-value problem (3)) ;(r, z) can be expressed as a sum of three terms [9] XK :
<(r, z)"< (r, z)#< (r, z)#< (r, z), where
< "
*B
o(z )[r#(z!z )]\ dz .
(3)
(4)
\* o(z) is the charge density on the tip axis [!(¸!d),(¸!d)] and o(z)"!o(!z), d"R , and ¸ "¸!d. B < (r, z) is the solution of the boundary-value problem without taking the tip into account *< (r, z)"0, (5) < (r,$Z )" G "$; . G G PV0 The function < (r, z) can be represented as a next boundary-value problem solution *< (r, z)"0, (6) < (r,$Z )" G "!< (r,$Z )" G . G PV0 G PV0 B
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Let us introduce the new function u (r, z;z ) *B < " u (r, z; z )o(z ) dz , \*B where the kernel u (r, z; z ) from (6) to (7) is the boundary-value problem solution: *u (r, z; z )"0, 1 . u (r,$Z )" G "! G PV0 (r#($Z !z ) G The function o(z) is de"ned by the series [9]
o(z)" o (z), I I
o (z)"!< (0, z) ln
o (z)"! ln I>
(7)
(8)
4z(¸!z) \ , r(z)
4z(¸!z) \ *B o (z )!o (z) I I # u (0, z; z )o (z ) dz . I "z !z" r(z) \*B
(9)
4. The solution of the boundary-value problem (5) Let us consider the boundary-value problem (5): *< (r, z)"0, < (r,$Z )" G "$; . G G PV0 From < (r, z)"!< (r,!z) it is expedient to solve the problem for z*0. Thus, the boundary-value problem will be considered as *< (r, z)"0, < (r,0)"0, < (r, Z )" G "; . G G PV0 The functions < (r, z) can be represented as a Hankel transform [10] ; !; sh j(Z !z) G (z!Z )# G> < (r, z)"; # G> F (j) G G Z !Z sh j(Z !Z ) G G> G G> G sh j(z!Z ) G F (j) J (jr) dj, Z (z(Z , # (10) G G> sh j(Z !Z ) G> G> G where J is the zeroth-order Bessel function. Thus, we have to "nd the distribution of potentials for z*0 with N round apertures (; "<(z, r)" "0). The whole region of this electron-optical system is divided into N#1 X
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subregions: Z )z)Z . The boundary conditions and the continuity conditions of the electriG G> cal "eld on the planes z"Z for r)R can be written as G G < (r,0)"0,
F (j)J (jr)"0, R )r(R, G G
ch j(Z !Z ) ch j(Z !Z ) 1 G G\ # G> G F (j) F (j)# j ! sh j(Z !Z ) sh j(Z !Z ) G sh j(Z !Z ) G\ G G\ G G\ G> G 1 F (j) J (jr) dj ! sh j(Z !Z ) G> G> G ; !; ; !; G! G G\ , 0)r(R . " G> G Z !Z Z !Z G> G G G\
(12)
Let 1 , (j)"! sh j(Z !Z ) G G\ ch j(Z !Z ) ch j(Z !Z ) G G\ # G> G , g (j)" G G sh j(Z !Z ) sh j(Z !Z ) G G\ G> G ; !; ; !; G\ . G! G < " G> G Z !Z Z !Z G G\ G> G Then system (12) can be written as g
G G\
(13)
F (j)J (jr)"0, R )r(R, G G j[g (j)F (j)#g (j)F (j)# g (j)F (j)]J (jr) dj"< , 0)r(R . G\ G G\ G G G G> G G> G G
Let gH (j)"1!CG ) g (j), G G G G
(14)
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where the constants CG are determined from lim gH (j)"0. G G H Then g (j)"[1!gH (j)]/CG . G G G G From (12), lim g (j)"2, then CG ". H G G Therefore,
exp(!j(Z !Z )) 1 exp(!j(Z !Z )) G> G # G G\ . gH (j)"! G G sh j(Z !Z ) sh j(Z !Z ) 2 G> G G G\ The system of the dual integral equations (14) can be represented as
F (j)J (jr)"0, R )r(R, G G j[g (j)F (j)#2(1!gH (j))F (j)# g (j)F (j)]J (jr) dj"< (r), 0)r(R . G\ G G\ G G G G> G G> G G (15)
Let F (j) have the form of an integral transform: G 0G F (j)" (t) sin jt dt. G G Making use of the discontinuity Weber integral
(16)
0, 0)t(r, J (jr) sin jt dj" (t!r)\ , 0(r(t, it may be shown that the "rst equations from system (15) with the substitutions (16) are satis"ed. F (j)J (jr) dj" G
0G
(t) dt G
J (jr) sin jt dj,0, r'R . G
Substituting (16) in the second system equations (15) we have
jg (j)J (jr) dj G G\
0G\
(t) sin jt dt#2 G\
0G>
(t) sin jt dt"< (r). G> G We have to multiply Eq. (17) by r and to integrate by r. From #
jg (j)J (jr) dj G> G
d jrJ (jr)" [rJ (jr)], dr
j(1!gH (j))J (jr) dj G G
0G
(t) sin jt dt G (17)
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Eqs. (17) can be represented as
r
0G\
(t) dt G\
! 2r #r
0G
0G>
g (j)J (jr) sin jt dt dj#2r G G\
(t) dt G
0G
(t) dt G
J (jr) sin jt dt dj
gH (j)J (jr) sin jt dt dj G G
(t) dt G>
g (j)J (jr) sin jt dt dj"= (r), r(R , G G\ G G
(18)
where = (r)"P t< dt. G G Making use of another discontinuity Weber integral
0, 0)r(t J (jr) sin jt dj" R , r't P(P\R and the integral transform of Bessel's function
(19)
2 p J (jr)" sin H sin(jr sin H) dH, p Eqs. (18) may be represented from (19) and (20) as
(20)
p P t (t) 2 0G\ G
(t) dt dt g (j) sin jt dt dj sin H sin(jr sin H) dH # 2 G\ G G\ p (r!t p 2 0G !2
(t) dt gH (j) sin jt dt dj sin H sin(jr sin H) dH G G G p p 2 0G> g (j) sin jt dt dj sin H sin(jr sin H) dH"= (r), r(R .
(t) dt # G G\ G G G> p (21)
From the transposition t"r sin H to P [t (t)/(r!t] dt: G P t (t) p r sin H (r sin H) p G G dt" r cos H dH"
(r sin H) r sin H dH. G (r!t (r!rsint With (22) Eqs. (21) can be written as
2 p 2 0G\
(t) dt g (j) sin jt sin(jr sin H) dj# 2 (r sin H) dt G\ G G\ G p p 4 0G
(t) dt gH (j) sin jt sin(jr sin H) dj ! G G G p 2 0G> g (j) sin jt sin(jr sin H) dj r sin H dH"= (r), r(R .
(t) dt # G G\ G> G G p
(22)
(23)
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Every equation of (23) is the Shlemilgh equation. Thus, the functions (t) are the solutions of the G Fredholm's integral equations of the second kind
1 0G\ 1 0G ! K (x, t) (t) dt# 2 (x)# K (x, t) (t) dt G G\ G\ G GG G p p 1 0G> K (x, t) (t) dt"U (x), ! G> G G> G p where
(24)
2 V y< (y) G U (x)" dy, G p (x!y
K (x, t)"2 G G\
K (x, t)"4 GG
g (j) sin jx sin jt dj, G G\
gH (j) sin jx sin jt dj. G G
(25)
Then
1 x!t x#t exp(!ja) a#b a#b sin jx sin jt dj" R t #i !t #i 2b 2b 2b sh jb 2b 2b C (a#ib) t(a#ib)" , C(a#ib)
,
(26)
where C is the gamma function. The kernels K (x, t) are symmetrical and can be written in an explicit form (13) and (25): GH 1 sin jx sin jt dj, K (x, t)"!2 G G\ sh j(Z !Z ) G G\ exp(!j(Z !Z )) exp(!j(Z !Z )) G> G # G G\ sin jx sin jt dj (27) K (x, t)"!2 GG sh j(Z !Z ) sh j(Z !Z ) G> G G G\ or from (26).
1 R K (x, t)"! G G\ (Z !Z ) G G\ 1 R t K (x, t)"! GG (Z !Z ) G> G 1 ! R t (Z !Z ) G G\
1 1 x!t x#t #i !t #i . 2 2 2(Z !Z ) 2(Z !Z ) G G\ G G\ 1 1 x!t x#t #i !t #i 2 2 2(Z !Z ) 2(Z !Z ) G> G G> G 1 1 x!t x#t #i !t #i . 2 2 2(Z !Z ) 2(Z !Z ) G G\ G G\
t
(28)
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The next integrals may be represented as
1 sin jx sin jt dj sh j(Z !Z ) G G\ " !4(Z !Z )xt G G\
k 1 , ((k(Z !Z ))#(t!x)) ((k(Z !Z ))#(t#x)) 2 G G\ G G\ I (29)
exp(!j(Z !Z )) G G\ sin jx sin jt dj sh j(Z !Z ) G G\
k 1 . ((k(Z !Z ))#(t!x)) ((k(Z !Z ))#(t#x)) 2 G G\ G G\ I Then from (27) and (29) the kernels K (x, t) can be represented also GH k(Z !Z ) 1 G> G K "8xt GG ((k(Z !Z ))#(t!x)) ((k(Z !Z ))#(t#x)) G> G G> G I 2 k(Z !Z ) 1 G G\ # , (30) ((k(Z !Z ))#(t!x)) ((k(Z !Z ))#(t#x)) 2 G G\ G G\ I k(Z !Z ) 1 G G\ K (x, t)"8xt . G G\ ((k(Z !Z ))#(t!x)) ((k(Z !Z ))#(t#x)) 2 G G\ G G\ I If the aperture radii are much less than the distances between apertures the kernels K (x, t) can GH be written as " !4(Z !Z )xt G G\
1 K (x, t)+C xt , G G\ (Z !Z ) G G\ 1 1 K (x, t)+C xt ! , GG (Z !Z ) (Z !Z ) G G\ G> G where
1 +1.0517998, C " k 2 1 C " +0.1502571. k 2 Consider U (x). Making use of the functions < (13) and substituting them into (25) we obtain that G G functions U (x) are G 2 ; !; ; !; G> G! G G\ x. U (x)" (31) G p Z !Z Z !Z G> G G G\
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Thus, to solve the boundary-value problem (5) we have to solve the system of the Fredholm's integral equations of the second kind (24) with the symmetrical kernels K (x, t) (28) or (30) and GH functions U (x) (31). G 5. The calculation of the function u (r, z; z ) To "nd function u (r, z;z ) we have to solve the boundary-value problem (8) *u (r, z; z )"0, 1 , u (r,$Z ; z )" G "! G PV0 (r#($Z !z ) G where z is a parameter. For convenience the new apertures' coordinates (RK ,ZK ) are introduced G G ZK "!Z , RK "R , i"1, N, G ,>\G G ,>\G ZK "!Z , RK "R , i"N#1, 2N. G G\, G G\, As for function < (r, z) the method of dual equation is applied. The region z is divided into 2N#1 subregions: ZK )z)ZK and, as in the previous section, it is supposed that G G> . uG>(r, z;z )"u (r, z;z )" K G 8 WXW8K G> The boundary conditions and the normal derivative continuity at the electrode surfaces z"ZK , G r)RK , (i"1,2N) are G uG>(r, ZK ; z )"uG (r, ZK ; z ), r*0, G G u (r, ZK ; z )"![r#(ZK !z )]\ , r*RK , G G G *uG> *uG " , z"ZK , r(RK , i"1, 2N, G G *z *z u (r,!R;z )(R, u,>(r,R;z )(R. Represent the function u (r, z;z ) as the distribution:
(32)
sh j(ZK !z) G> (F (j)#exp(!j"ZK !z ")) G sh j(ZK !ZK ) G G> G sh j(z!ZK ) G (F (j)#exp(!j"ZK u (r, z; z )"! # !z ")) J (jr) dj, G> sh j(ZK !ZK ) G> G> G ZK )z)ZK , i"1, 2N. G G>
(33)
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It is obvious that with (33) the Laplace's equation is satis"ed; the limitation conditions for function "u (r, z; z )"(R are satis"ed for zP$R; and the continuity of the function u (r, z; z ) is satis"ed. Then conditions (32) lead to the system of the dual integral equations:
(F (j)#exp(!j"ZK !z ")) J (jr) dj"[r#(Z !z )]\ , r*RK , G G G G
1 (F (j)#exp(!j"ZK !z ")) j ! G\ sh j(ZK !ZK ) G\ G G\
!
!ZK ) ch j(ZK !ZK ) ch j(ZK G> G (F (j)#exp(!j"ZK !z ")) G G\ # G G !ZK ) sh j(ZK !ZK ) sh j(ZK G> G G G\
#
1 sh j(ZK !ZK ) G> G
(F (j)#exp(!j"ZK !z ")) J (jr) dj"0, r(RK . G> G> G
(34)
Making use of the integral
exp(!j"ZK !z ")J (jr) dj" [r#(Z !z )]\ . G G
System (34) can be written as
F (j)J (jr) dj"0, r*RK , G G
1 !ZK ) ch j(ZK !ZK ) ch j(ZK G> G F (j) G G\ # F (j)# j ! sh j(ZK !ZK ) G\ !ZK ) G sh j(ZK !ZK ) sh j(ZK G G\ G> G G G\
1 F (j) J (jr) dj"< (r), r(RK , ! G G sh j(ZK !ZK ) G> G> G
(35)
where
sh j(z !ZK ) G\ J (jr) dj, ZK )z )ZK , G\ G sh j(ZK !ZK ) G G\ sh j(ZK !z ) G> j J (jr) dj, ZK )z )ZK , < (r)" 2 G G> G sh j(ZK !ZK ) G> G 2
0,
j
z ,[ZK ,ZK ]. G\ G>
(36)
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So, as "z "(Z then for di!erent values z the functions < (r) can be written as G !Z )z )!Z : sh j(Z #z ) < (r)"2 J (jr) dj, j ,\ sh j(Z !Z ) sh j(Z #z ) J (jr) dj, < (r)"!2 j , sh j(Z !Z ) < (r)"0, iON!1, N. G !Z )z )Z : sh j(Z !z ) < (r)"!2 j J (jr) dj, , sh j2Z sh j(Z #z ) j J (jr) dj, < (r)"!2 ,> sh j2Z < (r)"0, iON, N#1; G Z )z )Z : sh j(Z !z ) J (jr) dj, < (r)"!2 j ,> sh j(Z !Z ) sh j(Z !z ) J (jr) dj, j < (r)"2 , sh j(Z !Z ) < (r)"0, iON#1, N#2. G Let
1 , (j)"! sh j(ZK !ZK ) G G\ ch j(ZK !ZK ) ch j(ZK !ZK ) G G\ # G> G . g (j)" G G sh j(ZK !ZK ) sh j(ZK !ZK ) G G\ G> G Then system (35) can be written as g
G G\
F (j)J (jr) dj"0, G
(37)
r)RK ; G
j(g (j)F (j)#g (j)F (j)#g (j)F (j)) J (jr) dj"< (r), r(RK . (38) G\ G G\ G G G G> G G> G G By comparing (38) with (14) it may be noted that these systems are di!ering only with the right parts. Introducing the new functions (t): G 0K G F (j)" (t) sin jt dt, i"1, 2N (39) G G
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and making use of calculations (15)}(23) the system of Fredholm's integral equations of the second kind for the functions (t) can be obtained: G !
1 0K G\ 1 0K G K (x, t) (t) dt# 2 (x)# K (x, t) (t) dt G G\ G\ G GG G p p
!
1 0K G> K (x, t) (t) dt"U (x). G> G G> G p
(40)
From (25), (26) and (29), (37):
1 1 x!t x#t 1 R t #i ! t #i K (x, t)"! GG 2 2 2(ZK 2(ZK !ZK ) !ZK ) (ZK !ZK ) G> G G> G G> G !
1 1 1 x!t x#t R t #i ! t #i 2 2 (ZK !ZK 2(ZK !ZK 2(ZK !ZK ) ) ) G G\ G G\ G G\
.
1 1 1 x!t x#t K (x, t)" ! R t #i ! t #i G G\ (ZK !ZK ) 2 2 2(ZK !ZK ) 2(ZK !ZK ) G G\ G G\ G G\
(41)
or
k(ZK !ZK ) 1 G> G ((k(Z K !Z K ))#(t!x)) ((k(Z K !Z K ))#(t#x)) G> G G> G I 2
K (x, t)"8xt GG
#
k(ZK !ZK ) 1 G G\ , ((kK (Z !ZK ))#(t!x)) ((k(ZK !ZK ))#(t#x)) 2 G G\ G G\ I
K (x, t)"8xt G G\
k(ZK !ZK ) 1 G G\ . ((k(ZK !ZK ))#(t!x)) ((k(ZK !ZK ))#(t#x)) G G\ G G\ I 2
(42)
For the functions U (x) from (25): G
2 V y< (y) G dy U (x)" G n (x!y and
V
y
(x!y
x exp(!ja)jJ (jy) dj dy" a#x
(43)
280
N.V. Egorov, E.M. Vinogradova / Vacuum 57 (2000) 267}281
and from (36), (43) the functions U (x) can be represented in an explicit form: G
4 x p I ((ZK G !ZK G\ )(2k#1)!(z !ZK G\ ))#x x ! , ZK )z )ZK , G\ G ((ZK !ZK )(2k#1)#(z !ZK ))#x G G\ G\ x U (x)" 4 G ((ZK !ZK )(2k#1)!(ZK !z ))#x p G> G G> I x ! , ZK )z )ZK , G G> ((ZK !ZK )(2k#1)#(ZK !z ))#x G> G G> z , [ZK , ZK ] G\ G>
0,
(44)
or
8(ZK !ZK )(ZK !z )k 4 G G\ G !x (((ZK !ZK )2k#(ZK !z ))#x) p G G\ G I
1 x ; # , (((ZK !ZK (ZK !z )#x )2k!(ZK !z ))#x) G G\ G G
ZK )z )ZK , G\ G (45)
8(ZK !ZK )(z !ZK )k U (x)" 4 G> G G !x G (((ZK !ZK )2k#(z !ZK ))#x) p G> G G I
1 x ; # , ZK )z )Z , G G> (((ZK !ZK )2k!(z !ZK ))#x) (z !ZK )#x G> G G G 0, z ,[Z , Z ]. G\ G>
So, the function u (r, z;z ) is represented with (33), functions F (j) are calculated with the functions G
(t) (39), which are the solutions of the Fredholm's integral equations of the second kind (40) with G the symmetrical kernels (41)}(42) and functions (44)}(45).
N.V. Egorov, E.M. Vinogradova / Vacuum 57 (2000) 267}281
281
6. Conclusion In the presented mathematical model the distribution of the potential is found as a three functions superposition < , < , < . < is de"ned through the charge density on the tip axis. < is the solution of the boundary-value problem with apertures without taking the tip into account. < is de"ned as satisfying the boundary conditions at the apertures. Thus, if we have a solution of systems (24), (40) we have the solution of the boundary-value problem (1), i.e. the distribution of potentials for whole region of the electron}optical system with a thin tip on a #at substrate and a system of round apertures as the focusing electrodes. All geometrical dimensions of the system and the electrodes' potentials are the parameters of this method.
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