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Mechanisms responsible for instability of a field emission cathode surface morphology
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Surface Science 266 (1992) 163-164 North-Holland
surface science :
i
Mechanisms responsible for instability of a field emission cathode surface morphology D.V. E r e m c h e n k o a n d V. T. M a k h o v Zelenogrud Research Institute of Physical Problems, 103460 Moscow, Russia Received 5 August 1991; accepted for publication 9 August 1991
Two possible mechanisms of instability which depend on dimensions of the emitting area have been investigated. The first mechanism indicates diffusive instability of surface microprotuberances in a particular range of wave numbers, whereas the second is indicative of dimension effect at Joule heating of emitting area.
The stability of surface morphology of the emitting surface to a great extent accounts for stability and life time of a field emission cathode (FEC). Surface morphology, stability is d e p e n d e n t upon electrophysical parameters values (electrical field intensity on the emitter, t e m p e r a t u r e of emitting medium, etc.), as well as upon their derivative along the surface, since the phenomena of thermal and mass transfer, accounting for various processes of emitter erosion are determined by the gradients of thermodynamic value~ (chemical potential of a surface atomic layer, medium t e m p e r a t u r e and so on). Taking these facts into account one can assume that the most critical p a r a m e t e r characterizing surface morphology is the characteristic size of microprotuberances existing on emitting areas. In order to assess in what way the geometry of microprotuberances affect the stability of F E C operation, one should examine the possible instability mech-. morphoiogy paanisms which d e p e n d on -,~Lat.c - ¢ rameters. We shall consider two types of ~.lC51Lil . . . . v i"lization: (1) The first type of dcstabitiz:~fion arises from diffusive instability of microprotuberances vn the surface. It manifests itself only when the electrostatic tension gets greater than the surface tension and the difference between them grows larger when the dimensions of microprotuberances increase. This type of instabi~.ity was for the first
z = l: s ink x, ( k
<<|
Fig. 1. A surface displacement duc to sclf-dlflu:~ion.
time described by ri'orns and Ffenkcl {t935t mr the charged surface of a liquid conductor. T h e n in 1960 Dyke et al, investiga~.cd the displacement of the solid tips due to diffusive mass transfer and revealed that this p h e n o m e n o n was also determined by the difference between electrostatic and surface tension, Theoretically we can obtain surface displacem e n t duc to self-diffusion for small-size protuberances on a sotid surface (fig, 1) and develop a theory of diffusive instability of a charged surface analogous to T o r n s - F r e n k e t ~hcory. The task i.s to calculate the divergence of a diffusive flow of particles on a surface with micwprotuberanccs and to find the condition of amplitude gain from the equation which is analogous to the non-discontinuity equation --+a2 Ot
~S
=0,
(~(~39-6028/92/'$I)5.00 ,.~) t992 - Elsevier Science Publishers B.V. All rights reserved
(1.])
D.V. Eremchenko, I~.LMakhov / Instability of field c:m~sion cathode surface morphology
164
where 0 is the bulk nf the atom, j = - B Ol~/as is the diffusion flow density, B is the constant dependent on temperature, tz is the chemical potential of the surface layer and Z = ~ sin ka- is the surface displacement. By using for tb'~ purpose Herring thermodynamic potential =/-to + O ( v K - E 2 / 8 ~ ' ) ,
.
(1.3)
ClTI.
(2) The second type of instability that will occur when the size of emitting area gets smaller results from the specific character of temperature field due to emitter Joule heating, It is caused by the fact that when the heating source (the emitting area) size decreases, the temperature gradient within the bulk of emitting material at a given temperature value increases, thus making the heat removal more intensive. Let us take a strip line ot ICllgtll
]-~.'1 ~ a a s a B ~lllllttllll~ a r e a
Fig. 2. A strip line of infinite length [xl ~ a as an emitting area.
thermal conductance coefficient h can be converted to a simple (one-dimensional) equation: O2T
A -~-~ + a2j~pT= O,
In the right part of this equation we have by definition the increment of amplitude gain ~:, which determines the condition of microprotuberances stability (1/~.)(O~/Ot ~ < 0. It can be seen that in the wave values domain k < E2/47ry the amplitude gain value ~ is proportional to time and has a maximum value of gain increment at k=3E2/16rry. When k increases (when the scale of microprotuberances decreases) the increment value gets smaller and at k > E2/4rcy the size of microprotuberances does not become larger. At E = 5 X 10 7 V / c m and y = 10 3 din/era, the borderline between the area~ with stable and un,~tabte geometries of microprotuberances lies at k = IW' cm -t, that is the critical size of stable protuberances is of the order of 5 × 10 -~
lllllliltC
T=T0
(1.2)
where P-o is the chemical potential of uncharged surface, 3' i~ the coefficient of surface tension, K is the surface curvature, E2/8~: is the electrostatic tension, we can obtain the following [1] equation:
1 a~ =-B.(~Zk s y k - ~ ¢ 0t
,7[, (2.1)
where U is the elliptical coordinate. The solution of the equation with finite condition T = 7o (T o is the thermostat temperature) on an elliptical cylinder with U = 1 (placed from the emitting area at a distance of the order of dimension a) can be explessed by the equality
TO cos(UajofO--~) T=
cos(aj0x~-~)
(2.2)
On emitting surface we have then:
To '~m = c o s ( a j o ~ - ~ ) .
(2.3)
Dependencc of emitter temperature on the product ajo testifies to the valitiity of the assumptions made before. On this account we can make a conclusion that with reducing dimension value a the temperature of the emitting area at a given current density value i0 also reduces. It testifies to the fact that the maximum current density value, which corresponds to the maximum admittable temperature Tom witl increase with reduction of dimension a.
x-s.
2). The emission taking place on its surface is constant and equal to ,i~. For such a model the stationary equation of heat conductance in emitting medium at the resistivity pT and a medium
Refer@~ces [1] D.\'. Eremtchenko, V.1. Makhov, V.A Fcdirko and V! R;-shii, Surface Pbyzika, Chimija & Mechanika 12 (1986) l.~0.
Surface Science 266 (1992) 163-164 North-Holland
surface science :
i
Mechanisms responsible for instability of a field emission cathode surface morphology D.V. E r e m c h e n k o a n d V. T. M a k h o v Zelenogrud Research Institute of Physical Problems, 103460 Moscow, Russia Received 5 August 1991; accepted for publication 9 August 1991
Two possible mechanisms of instability which depend on dimensions of the emitting area have been investigated. The first mechanism indicates diffusive instability of surface microprotuberances in a particular range of wave numbers, whereas the second is indicative of dimension effect at Joule heating of emitting area.
The stability of surface morphology of the emitting surface to a great extent accounts for stability and life time of a field emission cathode (FEC). Surface morphology, stability is d e p e n d e n t upon electrophysical parameters values (electrical field intensity on the emitter, t e m p e r a t u r e of emitting medium, etc.), as well as upon their derivative along the surface, since the phenomena of thermal and mass transfer, accounting for various processes of emitter erosion are determined by the gradients of thermodynamic value~ (chemical potential of a surface atomic layer, medium t e m p e r a t u r e and so on). Taking these facts into account one can assume that the most critical p a r a m e t e r characterizing surface morphology is the characteristic size of microprotuberances existing on emitting areas. In order to assess in what way the geometry of microprotuberances affect the stability of F E C operation, one should examine the possible instability mech-. morphoiogy paanisms which d e p e n d on -,~Lat.c - ¢ rameters. We shall consider two types of ~.lC51Lil . . . . v i"lization: (1) The first type of dcstabitiz:~fion arises from diffusive instability of microprotuberances vn the surface. It manifests itself only when the electrostatic tension gets greater than the surface tension and the difference between them grows larger when the dimensions of microprotuberances increase. This type of instabi~.ity was for the first
z = l: s ink x, ( k
<<|
Fig. 1. A surface displacement duc to sclf-dlflu:~ion.
time described by ri'orns and Ffenkcl {t935t mr the charged surface of a liquid conductor. T h e n in 1960 Dyke et al, investiga~.cd the displacement of the solid tips due to diffusive mass transfer and revealed that this p h e n o m e n o n was also determined by the difference between electrostatic and surface tension, Theoretically we can obtain surface displacem e n t duc to self-diffusion for small-size protuberances on a sotid surface (fig, 1) and develop a theory of diffusive instability of a charged surface analogous to T o r n s - F r e n k e t ~hcory. The task i.s to calculate the divergence of a diffusive flow of particles on a surface with micwprotuberanccs and to find the condition of amplitude gain from the equation which is analogous to the non-discontinuity equation --+a2 Ot
~S
=0,
(~(~39-6028/92/'$I)5.00 ,.~) t992 - Elsevier Science Publishers B.V. All rights reserved
(1.])
D.V. Eremchenko, I~.LMakhov / Instability of field c:m~sion cathode surface morphology
164
where 0 is the bulk nf the atom, j = - B Ol~/as is the diffusion flow density, B is the constant dependent on temperature, tz is the chemical potential of the surface layer and Z = ~ sin ka- is the surface displacement. By using for tb'~ purpose Herring thermodynamic potential =/-to + O ( v K - E 2 / 8 ~ ' ) ,
.
(1.3)
ClTI.
(2) The second type of instability that will occur when the size of emitting area gets smaller results from the specific character of temperature field due to emitter Joule heating, It is caused by the fact that when the heating source (the emitting area) size decreases, the temperature gradient within the bulk of emitting material at a given temperature value increases, thus making the heat removal more intensive. Let us take a strip line ot ICllgtll
]-~.'1 ~ a a s a B ~lllllttllll~ a r e a
Fig. 2. A strip line of infinite length [xl ~ a as an emitting area.
thermal conductance coefficient h can be converted to a simple (one-dimensional) equation: O2T
A -~-~ + a2j~pT= O,
In the right part of this equation we have by definition the increment of amplitude gain ~:, which determines the condition of microprotuberances stability (1/~.)(O~/Ot ~ < 0. It can be seen that in the wave values domain k < E2/47ry the amplitude gain value ~ is proportional to time and has a maximum value of gain increment at k=3E2/16rry. When k increases (when the scale of microprotuberances decreases) the increment value gets smaller and at k > E2/4rcy the size of microprotuberances does not become larger. At E = 5 X 10 7 V / c m and y = 10 3 din/era, the borderline between the area~ with stable and un,~tabte geometries of microprotuberances lies at k = IW' cm -t, that is the critical size of stable protuberances is of the order of 5 × 10 -~
lllllliltC
T=T0
(1.2)
where P-o is the chemical potential of uncharged surface, 3' i~ the coefficient of surface tension, K is the surface curvature, E2/8~: is the electrostatic tension, we can obtain the following [1] equation:
1 a~ =-B.(~Zk s y k - ~ ¢ 0t
,7[, (2.1)
where U is the elliptical coordinate. The solution of the equation with finite condition T = 7o (T o is the thermostat temperature) on an elliptical cylinder with U = 1 (placed from the emitting area at a distance of the order of dimension a) can be explessed by the equality
TO cos(UajofO--~) T=
cos(aj0x~-~)
(2.2)
On emitting surface we have then:
To '~m = c o s ( a j o ~ - ~ ) .
(2.3)
Dependencc of emitter temperature on the product ajo testifies to the valitiity of the assumptions made before. On this account we can make a conclusion that with reducing dimension value a the temperature of the emitting area at a given current density value i0 also reduces. It testifies to the fact that the maximum current density value, which corresponds to the maximum admittable temperature Tom witl increase with reduction of dimension a.
x-s.
2). The emission taking place on its surface is constant and equal to ,i~. For such a model the stationary equation of heat conductance in emitting medium at the resistivity pT and a medium
Refer@~ces [1] D.\'. Eremtchenko, V.1. Makhov, V.A Fcdirko and V! R;-shii, Surface Pbyzika, Chimija & Mechanika 12 (1986) l.~0.