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Numerical simulation of electron beam transport in magnetosensitive vacuum microcell
Nuclear Instruments and Methods in Physics Research A 427 (1999) 17}21
Numerical simulation of electron beam transport in magnetosensitive vacuum microcell V.A. Fedirko *, N.G. Belova Moscow State University of Technology **Stankin++ 3a Vadkovski per., Moscow 101472, Russia FTIRAN, 25 a Krasikov Str., Moscow 117218, Russia
Abstract A cylindrical multielectrode vacuum microcell with a circular emitter and a split plate anode is suggested as a "eld emitter array magnetosensitive element. Circular grid electrodes to control the electron beam and a plate target electrode serving as secondary electron emission source are included in the model. Electron beam transport in the self-consistent electric "eld and in an external magnetic "eld has been simulated using the cylindrical particle mesh code with Monte Carlo procedure. The split-anode di!erence current dependence on the intensity of a magnetic "eld parallel to the cell axis is discovered. Static characteristics and dynamic response of the element are calculated. The in#uence of the secondary electron emission is studied. The results enable to optimise the cell structure for the maximum magnetosensitivity 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Miniature electron beam systems are now being worked out for various applications such as #at displays, vacuum microelectronics and microwave devices, micro- and nano-lithography instruments, and sensors (see e.g., Ref. [1]). Magnetic sensors based on vacuum microelectronics technology have been reported recently [2,3]. We showed [4] that a cylindrical microcell (CM) with a circular wedge "eld emitter is a potentially advanced basic element of "eld emitter array for various applications. In [5] we suggested a CM with a split plate anode
* Corresponding author. Fax: 7-095-973-3917. E-mail address: [email protected]. (V.A. Fedirko)
as a magnetosensitive element. In this paper we present the model for numerical simulation of electron beam transport in multi-electrode CM and the results of CM magnetosensor numerical analysis.
2. Model and equations The cross-section of the CM is shown in Fig. 1. Two grid electrodes, (g ) and (g ), create high elec tric "eld at the "eld emission cathode (c). Grid electrodes are also used to control the electron beam. The anode plane electrode at the foot of the cylinder may be split into two coaxial plates a and a . The upper plate, (t), (we call it `targeta) serves as a screen and can also be a source of secondary electrons. The external magnetic "eld H parallel to
0168-9002/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 8 ) 0 1 5 4 0 - X
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V.A. Fedirko, N.G. Belova / Nuclear Instruments and Methods in Physics Research A 427 (1999) 17}21
Fig. 1. Cross-section of the cylindrical vacuum microcell.
the CM axis can be detected by comparing the current I through the anode a with the current I through the anode a . Neglecting rare scattering events we start from Vlasov's equation for the electron distribution function f (t, r, *). The space charge of the electrons is taken into account through the self-consistent electric potential u(t, r) obeying the Poisson's equation. The electrode potentials serve as natural boundary conditions for the electric potential: u(t, r)"u "0 at the cathode, u(t, r)"u and u(t, r)"u at both grids, u(t, r)"u , at the target u , and u(t, r)"u at the anode. We put *u/*r"0 at the axis for the symmetry reason and at the isolating parts of the walls where we neglect the surface charge. For the distribution function we admit free electron transition through the anode and grid electrodes, and adopt the specular boundary condition at the axis and at the isolating parts of the walls. The electron emission condition is stated for f (t, r, *) at the cathode. Isotropic Maxwellian with the characteristic energy ¹ seems a good approximation for it [6] as the typical potential di!erence in the cell is much higher than ¹ /e. Field emission really takes place from small chaotic spots on the cathode edge where the electric "eld is high enough. We neglect the nanostructure of the emitter edge and consider the emission averaged over the polar angle and over the edge width. As the whole area of the emitting spots is usually no more than about 1% of the cathode area, we assume that the distribution function of injected electrons di!ers from zero only within the narrow strip at the center of the cathode. At the target secondary electron emission is taken into account, the empiri-
cal net secondary electron current ratio d and the elastic scattering current ratio R being used as parameters. The secondary electron distribution is also supposed Maxwellian with the characteristic energy ¹ and cosine dependent on the angle of the electron velocity vector with the normal, which is consistent with the experimental data [7,8]. To analyse the dynamic of electron transport in the cell the particle technique modi"ed for cylindrical structures [9] is used. The electron distribution in the cell is modelled by the ensemble of the charged particles, having the form of coaxial rings with the rectangular cross-section = in the (r, z) plane of the cylindrical coordinates (r, z, h), with standard charge-to-mass ratio. The above model results in the following set of dynamic equations for the particles:
dv e *u v ) H v " X # F ! F dt m *r c r
(1)
dr "v dt
(2)
dv e *u X" ) m *z dt
(3)
dz "v X dt
(4)
e P "rv ! ) rA F F mc F
(5)
here (v , v , v ) are the cylindrical coordinates of the P X F velocity, e and m are the electron charge and electron mass respectively, P is the generalized particle F momentum, and the magnetic vector potential component A is de"ned by F 1 *(rA ) F . H" ) X r *r As P "const, one can consider the particle F movement in the (r, z)-plane only, which makes the method similar to the well-known Cloud-In-Cell technique [10]. The permanent di!erence mesh *r, *z is used in the computational domain and the constant particles' form-factor W is put ="*r; *z. The charge proportional to the polar radius of
V.A. Fedirko, N.G. Belova / Nuclear Instruments and Methods in Physics Research A 427 (1999) 17}21
a launching point is being ascribed to an injected particle, which enables to diminish the number of particles. To avoid the singularity problem at the axis we determined the space charge density by dividing the total charge in every mesh by the mesh volume. To solve the particle dynamic Eqs. (1) } (5) we use the di!erence scheme of the second order in time. New r, z coordinates of a particle and its v , v , v ! P X F velocity components at every time step are calculated with the centered di!erence scheme with the intermediate correction. Monte Carlo procedure is applied for electron emission modelling to simulate the initial velocity value and angular distribution. Poisson's equation is approximated by the second order "nite di!erence equation which is solved by the alternating direction implicit iteration procedure using tridiagonal inversion. The electric "eld for each particle is then found by weightening in accordance with the Cloud-In-Cell algorithm. For more details of the model and mathematical procedure see Refs. [9}11].
3. Numerical results We have studied numerically the electron transport in the CM of Fig. 1. The size of the computational domain is (l ;l )"(6.2;3.15) lm. The grid P X
19
electrode position and their potentials, the value of the magnetic "eld, and the secondary emission coe$cient d have been varied. The size of the cathode and the grids d have also been varied in some extent. The potential at the cathode u "0, at the target } u "10 V, and at the anode plates } u "u "50 V. The characteristic energy for the electrons emitted from the cathode ¹ "0.26 eV, and for the secondary electrons ¹ "0.75 eV, elastic re#ection coe$cient R"0.4. The "nite di!erence mesh is (*r;*z)" (0.2;0.05) lm, the time step *t"0.0025 pc. We put the cathode emission con"ned within the only mesh *z at center of the cathode according to the arguments discussed in the previous paragraph, with the mean emission current density j "2;10 A/cm. That j value seems reasonable for the model calculation though it rather should be considered as a "tting parameter for a real structure simulation. Fig. 2 represents the steady state electron distribution within the cells with di!erent grids position for u "5 V, u "25 V and d"5. One can eas ily see that the displacement of the grid electrodes markedly changes the electron beam and thus enables to achieve its e!ective control by varying the grids' potentials. The shift of the lateral electrodes closer to the target leads to more e!ective secondary emission. Current gain about 1.5 is thus
Fig. 2. Electron distribution in the (r,z)-plane at the stationary state: (a) H"0, (b) H"10 kG.
20
V.A. Fedirko, N.G. Belova / Nuclear Instruments and Methods in Physics Research A 427 (1999) 17}21
Fig. 3. Distribution of the current density j at the anode vs. radius for H"0 (left), and H"10 kG (right).
achieved, its value correlates with the calculated number of secondary electrons. The magnetic "eld de#ects the electron beam from the axis as seen in Fig. 2. Fig. 3 illustrates the distribution of the steady state current density over the anode for the electrode positions of Fig. 2b and d"5. It is highly inhomogenious for both H"0, and H"10 kGs and changes drastically in the presence of a magnetic "eld though the mean anode current density is about 200 A/cm for both cases. The current density peak shifts from the axis in a magnetic "eld. Fig. 4 shows the di!erence between the current I through the anode a and the current I through the anode a for various values of the magnetic "eld and di!erent d where we put r "0.8 lm for a and r "1.3 lm for a . We sug gest that a good magnetosensitivity can be realized
in an external circuit using the H- dependent anode di!erence current I(H)"I !I . In Fig. 5 the transient averaged anode current density is shown, after the pulsed voltages are applied to all the electrodes at t"0. The current reaches its steady-state value after long time oscillations resulting apparently from the space charge relaxation. The time delay and the oscillations period correlate with the transit time q+l (eu /(m+2 pc. Secondary electron emis sion from the target leads to the increase of the space charge density, so the higher is the net secondary emission ratio d the longer is the relaxation time. Marked current enhancement due to the secondary emission is observed while the magnetic "eld does not e!ect the mean anode current noticeably.
Fig. 4. Anode di!erence current I(H)"I !I vs. magnetic "eld.
V.A. Fedirko, N.G. Belova / Nuclear Instruments and Methods in Physics Research A 427 (1999) 17}21
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Fig. 5. Time dependence of the transit anode current. (a) H"O, u "u "5 V; d"0; (b) H"5 kGs, u "5 V, u "25 V; (1) d"0, (2) d"3, (3) d"5.
4. Conclusion
References
We have worked out the particle model for numerical simulation of electron beam transport in a multielectrode cylindrical vacuum microcell. The model has been proved e!ective for the analysis and optimisation of electron beam control in the cell. We predict the dependence of the di!erence current trough a split anode on an axial magnetic "eld. The current sensitivity of a magnetosensitive element based on that e!ect has been simulated. Transient response of the anode current is found to be dominated by electron transit but the space charge can result in the long time oscillations. Anode current enhancement can be obtained by using secondary emission.
[1] Papers from the 9th Int. Vacuum Microelectronic Conf., J. Vacuum Sci. Technol. B 15 (1997) 383. [2] Y. Sugiyama, J. Itoh, S. Kanemaru, Digest of Technical Papers of 7th Int. Conf. on Solid-State Sensors and Actuators, Yokohama, Japan, 1993, p. 884. [3] H.-X. Wang, C.-C. Zhu, J.-H. Liu, X.-P. Lee, Technical Digest of 8th Int. Vacuum Microelectronic Conf., Portland, Oregon, USA, 1995, p. 308. [4] V.A. Fedirko, N.G. Belova, V.I. Makhov, Revue `Le Vide, les Couches Mincesa 271 (Suppl.) (1994) 155. [5] V.A. Fedirko, N.G. Belova, Techn. Digest 9th Int. Vacuum Microelectronic Conf., St. Petersburg, Russia, 1996, p. 81. [6] R.D. Yong, E.W. Muller, Phys.Rev. 113 (1959) 115. [7] R. Kollath, Ann. Phys. 45 (1947) 387. [8] J.L.H. Jonker, Philips Res. Rep. 6 (1959) 372. [9] N.G. Belova, V.I. Karas', Sov. J. Plasma Phys. 16 (1990) 115. [10] R. Hockney, J. Eastwood, Computer Simulation using Particles, McGraw-Hill, London, 1981. [11] V.A. Fedirko, N.G. Belova, Russian J. Math. Modelling 7 (1995) 3.
Acknowledgements The work is supported by Russian Foundation for Basic Research, project No. 97-01-00070.
Numerical simulation of electron beam transport in magnetosensitive vacuum microcell V.A. Fedirko *, N.G. Belova Moscow State University of Technology **Stankin++ 3a Vadkovski per., Moscow 101472, Russia FTIRAN, 25 a Krasikov Str., Moscow 117218, Russia
Abstract A cylindrical multielectrode vacuum microcell with a circular emitter and a split plate anode is suggested as a "eld emitter array magnetosensitive element. Circular grid electrodes to control the electron beam and a plate target electrode serving as secondary electron emission source are included in the model. Electron beam transport in the self-consistent electric "eld and in an external magnetic "eld has been simulated using the cylindrical particle mesh code with Monte Carlo procedure. The split-anode di!erence current dependence on the intensity of a magnetic "eld parallel to the cell axis is discovered. Static characteristics and dynamic response of the element are calculated. The in#uence of the secondary electron emission is studied. The results enable to optimise the cell structure for the maximum magnetosensitivity 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Miniature electron beam systems are now being worked out for various applications such as #at displays, vacuum microelectronics and microwave devices, micro- and nano-lithography instruments, and sensors (see e.g., Ref. [1]). Magnetic sensors based on vacuum microelectronics technology have been reported recently [2,3]. We showed [4] that a cylindrical microcell (CM) with a circular wedge "eld emitter is a potentially advanced basic element of "eld emitter array for various applications. In [5] we suggested a CM with a split plate anode
* Corresponding author. Fax: 7-095-973-3917. E-mail address: [email protected]. (V.A. Fedirko)
as a magnetosensitive element. In this paper we present the model for numerical simulation of electron beam transport in multi-electrode CM and the results of CM magnetosensor numerical analysis.
2. Model and equations The cross-section of the CM is shown in Fig. 1. Two grid electrodes, (g ) and (g ), create high elec tric "eld at the "eld emission cathode (c). Grid electrodes are also used to control the electron beam. The anode plane electrode at the foot of the cylinder may be split into two coaxial plates a and a . The upper plate, (t), (we call it `targeta) serves as a screen and can also be a source of secondary electrons. The external magnetic "eld H parallel to
0168-9002/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 8 ) 0 1 5 4 0 - X
18
V.A. Fedirko, N.G. Belova / Nuclear Instruments and Methods in Physics Research A 427 (1999) 17}21
Fig. 1. Cross-section of the cylindrical vacuum microcell.
the CM axis can be detected by comparing the current I through the anode a with the current I through the anode a . Neglecting rare scattering events we start from Vlasov's equation for the electron distribution function f (t, r, *). The space charge of the electrons is taken into account through the self-consistent electric potential u(t, r) obeying the Poisson's equation. The electrode potentials serve as natural boundary conditions for the electric potential: u(t, r)"u "0 at the cathode, u(t, r)"u and u(t, r)"u at both grids, u(t, r)"u , at the target u , and u(t, r)"u at the anode. We put *u/*r"0 at the axis for the symmetry reason and at the isolating parts of the walls where we neglect the surface charge. For the distribution function we admit free electron transition through the anode and grid electrodes, and adopt the specular boundary condition at the axis and at the isolating parts of the walls. The electron emission condition is stated for f (t, r, *) at the cathode. Isotropic Maxwellian with the characteristic energy ¹ seems a good approximation for it [6] as the typical potential di!erence in the cell is much higher than ¹ /e. Field emission really takes place from small chaotic spots on the cathode edge where the electric "eld is high enough. We neglect the nanostructure of the emitter edge and consider the emission averaged over the polar angle and over the edge width. As the whole area of the emitting spots is usually no more than about 1% of the cathode area, we assume that the distribution function of injected electrons di!ers from zero only within the narrow strip at the center of the cathode. At the target secondary electron emission is taken into account, the empiri-
cal net secondary electron current ratio d and the elastic scattering current ratio R being used as parameters. The secondary electron distribution is also supposed Maxwellian with the characteristic energy ¹ and cosine dependent on the angle of the electron velocity vector with the normal, which is consistent with the experimental data [7,8]. To analyse the dynamic of electron transport in the cell the particle technique modi"ed for cylindrical structures [9] is used. The electron distribution in the cell is modelled by the ensemble of the charged particles, having the form of coaxial rings with the rectangular cross-section = in the (r, z) plane of the cylindrical coordinates (r, z, h), with standard charge-to-mass ratio. The above model results in the following set of dynamic equations for the particles:
dv e *u v ) H v " X # F ! F dt m *r c r
(1)
dr "v dt
(2)
dv e *u X" ) m *z dt
(3)
dz "v X dt
(4)
e P "rv ! ) rA F F mc F
(5)
here (v , v , v ) are the cylindrical coordinates of the P X F velocity, e and m are the electron charge and electron mass respectively, P is the generalized particle F momentum, and the magnetic vector potential component A is de"ned by F 1 *(rA ) F . H" ) X r *r As P "const, one can consider the particle F movement in the (r, z)-plane only, which makes the method similar to the well-known Cloud-In-Cell technique [10]. The permanent di!erence mesh *r, *z is used in the computational domain and the constant particles' form-factor W is put ="*r; *z. The charge proportional to the polar radius of
V.A. Fedirko, N.G. Belova / Nuclear Instruments and Methods in Physics Research A 427 (1999) 17}21
a launching point is being ascribed to an injected particle, which enables to diminish the number of particles. To avoid the singularity problem at the axis we determined the space charge density by dividing the total charge in every mesh by the mesh volume. To solve the particle dynamic Eqs. (1) } (5) we use the di!erence scheme of the second order in time. New r, z coordinates of a particle and its v , v , v ! P X F velocity components at every time step are calculated with the centered di!erence scheme with the intermediate correction. Monte Carlo procedure is applied for electron emission modelling to simulate the initial velocity value and angular distribution. Poisson's equation is approximated by the second order "nite di!erence equation which is solved by the alternating direction implicit iteration procedure using tridiagonal inversion. The electric "eld for each particle is then found by weightening in accordance with the Cloud-In-Cell algorithm. For more details of the model and mathematical procedure see Refs. [9}11].
3. Numerical results We have studied numerically the electron transport in the CM of Fig. 1. The size of the computational domain is (l ;l )"(6.2;3.15) lm. The grid P X
19
electrode position and their potentials, the value of the magnetic "eld, and the secondary emission coe$cient d have been varied. The size of the cathode and the grids d have also been varied in some extent. The potential at the cathode u "0, at the target } u "10 V, and at the anode plates } u "u "50 V. The characteristic energy for the electrons emitted from the cathode ¹ "0.26 eV, and for the secondary electrons ¹ "0.75 eV, elastic re#ection coe$cient R"0.4. The "nite di!erence mesh is (*r;*z)" (0.2;0.05) lm, the time step *t"0.0025 pc. We put the cathode emission con"ned within the only mesh *z at center of the cathode according to the arguments discussed in the previous paragraph, with the mean emission current density j "2;10 A/cm. That j value seems reasonable for the model calculation though it rather should be considered as a "tting parameter for a real structure simulation. Fig. 2 represents the steady state electron distribution within the cells with di!erent grids position for u "5 V, u "25 V and d"5. One can eas ily see that the displacement of the grid electrodes markedly changes the electron beam and thus enables to achieve its e!ective control by varying the grids' potentials. The shift of the lateral electrodes closer to the target leads to more e!ective secondary emission. Current gain about 1.5 is thus
Fig. 2. Electron distribution in the (r,z)-plane at the stationary state: (a) H"0, (b) H"10 kG.
20
V.A. Fedirko, N.G. Belova / Nuclear Instruments and Methods in Physics Research A 427 (1999) 17}21
Fig. 3. Distribution of the current density j at the anode vs. radius for H"0 (left), and H"10 kG (right).
achieved, its value correlates with the calculated number of secondary electrons. The magnetic "eld de#ects the electron beam from the axis as seen in Fig. 2. Fig. 3 illustrates the distribution of the steady state current density over the anode for the electrode positions of Fig. 2b and d"5. It is highly inhomogenious for both H"0, and H"10 kGs and changes drastically in the presence of a magnetic "eld though the mean anode current density is about 200 A/cm for both cases. The current density peak shifts from the axis in a magnetic "eld. Fig. 4 shows the di!erence between the current I through the anode a and the current I through the anode a for various values of the magnetic "eld and di!erent d where we put r "0.8 lm for a and r "1.3 lm for a . We sug gest that a good magnetosensitivity can be realized
in an external circuit using the H- dependent anode di!erence current I(H)"I !I . In Fig. 5 the transient averaged anode current density is shown, after the pulsed voltages are applied to all the electrodes at t"0. The current reaches its steady-state value after long time oscillations resulting apparently from the space charge relaxation. The time delay and the oscillations period correlate with the transit time q+l (eu /(m+2 pc. Secondary electron emis sion from the target leads to the increase of the space charge density, so the higher is the net secondary emission ratio d the longer is the relaxation time. Marked current enhancement due to the secondary emission is observed while the magnetic "eld does not e!ect the mean anode current noticeably.
Fig. 4. Anode di!erence current I(H)"I !I vs. magnetic "eld.
V.A. Fedirko, N.G. Belova / Nuclear Instruments and Methods in Physics Research A 427 (1999) 17}21
21
Fig. 5. Time dependence of the transit anode current. (a) H"O, u "u "5 V; d"0; (b) H"5 kGs, u "5 V, u "25 V; (1) d"0, (2) d"3, (3) d"5.
4. Conclusion
References
We have worked out the particle model for numerical simulation of electron beam transport in a multielectrode cylindrical vacuum microcell. The model has been proved e!ective for the analysis and optimisation of electron beam control in the cell. We predict the dependence of the di!erence current trough a split anode on an axial magnetic "eld. The current sensitivity of a magnetosensitive element based on that e!ect has been simulated. Transient response of the anode current is found to be dominated by electron transit but the space charge can result in the long time oscillations. Anode current enhancement can be obtained by using secondary emission.
[1] Papers from the 9th Int. Vacuum Microelectronic Conf., J. Vacuum Sci. Technol. B 15 (1997) 383. [2] Y. Sugiyama, J. Itoh, S. Kanemaru, Digest of Technical Papers of 7th Int. Conf. on Solid-State Sensors and Actuators, Yokohama, Japan, 1993, p. 884. [3] H.-X. Wang, C.-C. Zhu, J.-H. Liu, X.-P. Lee, Technical Digest of 8th Int. Vacuum Microelectronic Conf., Portland, Oregon, USA, 1995, p. 308. [4] V.A. Fedirko, N.G. Belova, V.I. Makhov, Revue `Le Vide, les Couches Mincesa 271 (Suppl.) (1994) 155. [5] V.A. Fedirko, N.G. Belova, Techn. Digest 9th Int. Vacuum Microelectronic Conf., St. Petersburg, Russia, 1996, p. 81. [6] R.D. Yong, E.W. Muller, Phys.Rev. 113 (1959) 115. [7] R. Kollath, Ann. Phys. 45 (1947) 387. [8] J.L.H. Jonker, Philips Res. Rep. 6 (1959) 372. [9] N.G. Belova, V.I. Karas', Sov. J. Plasma Phys. 16 (1990) 115. [10] R. Hockney, J. Eastwood, Computer Simulation using Particles, McGraw-Hill, London, 1981. [11] V.A. Fedirko, N.G. Belova, Russian J. Math. Modelling 7 (1995) 3.
Acknowledgements The work is supported by Russian Foundation for Basic Research, project No. 97-01-00070.