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Estimation of the field emission current density drawn from ultra sharp field emitters
Surface Science 70 (1978) 181-185 0 North-Holland Publishing Company
ESTIMATION OF THE FIELD EMISSION CURRENT DENSITY DRAWN FROM ULTRA SHARP FIELD EMITTERS
A. RABINOVICH Zenith Radio Corporation, Melrose Park, Illinois 60160,
USA
Received 13 April 1977; manuscript received in final form 3 June 1977
The possibility of obtaining some information about the current density of field emission drawn from ultra sharp field emitters in using Fowler-Nordheim plots is considered. Notions of equivalent and effective field emitters obeying FN law are introduced in this consideration. The conclusion is that the current density obtained from experimental FN plots is the lower bound of an unknown true value of current density.
1. Introduction
Common needle field emitters have tip radii of the order of 10-4-10-5 cm. E.W. Miiller was the first to use such tungsten needles in the field electron and field ion microscopes of his invention [ 1,2]. By ultra sharp field emitters we shall mean tips of radii of the order of lo-’ cm. Such tips are usually manufactured by a thermofield (TF) treatment of common needle field emitters [3-S]. As a result of the TF treatment the tip of the needle acquires a polyhedral shape with small microprotrusions at the corners of the polyhedron. Emission properties of this type of field emitters were systematically investigated in [6]. Current-voltage characteristics were taken and Fowler-Nordheim (FN) plots were drawn. It turned out that FN plots of such field emitters exhibited an appreciable linear portion similarly to FN plots of common field emitters. The FN plots contain information about current density and emitting area. To get a true v&e of current density from FN plots, one should know an exact law of field emission for a given field emitter. The exact law of field emission for any real field emitter is not known. In practice, the FN law of field emission is used for reduction of current density from the FN plots. The FN law was derived under the assumption of a uniform external electric field. No real field emitter obeys the FN law because the external electric field at the surface of the real emitter is never uniform. Therefore the values of current density reduced from FN plots by using FN law are never true. However, in the case of common field emitters deviations of the reduced values from true ones are not significant. In this case the external electric field in the immediate vicinity of the emitter surface (within the thickness of the potential barrier) is fairly close to uni181
182
A. Rabinovich /Estimation
of FE current density
form [7]. In the case of ultra sharp field emitters the external electric field is distinctively non-uniform even within the potential barrier thickness. Thus, one should expect significant deviation of the reduced values of current density from the true ones. The absolute value of the deviation is unknown as well as the sign of it. The purpose of this paper is to determine the sign of the deviation, that is, the relation of the reduced from FN plots values of current to true ones.
2. The equivalent field emitter As it is well known, the transmission coefficient of the potential barrier controls the field emission current density drawn from the emitter at a given applied voltage I’. The potential barrier in the case of ultra sharp field emitters is depicted by curve 1 in fig. 1. (We do not take in account the image force potential for the sake of simplicity. This does not affect our final conclusion.) Let us compare the potential barriers of the real emitter and of some emitter which differs from real one only by exhibiting a uniform electric field, so its potential curve would be as depicted by curve 2, fig. 1. That is, we want to replace the real emitter with an equivalent one, which would give the same current and current density at the given V, but obey the FN law because of the uniform external electric field. To assure that the two emitters exhibit the same current density we have to require the equivalent barriers to have equal transmission coefficients. It is easily seen from fig. 1 that this can be done if the potential curves intersect each other at some point above the Fermi level. The slope of the potential curve at the emitter surface gives the electrical strength Fe at the emitter surface. This is related to V through so-called field factor k, F,=kV.
(1)
At the given V, the equivalent emitter with field factor k,, corresponds to real emitter with field factor k,,. The question then arises as to whether the equivalence
x
Fig. 1. Potential barriers at emitter surface Dotted lines correspond to higher V.
for real (curve 1) and equivalent
(curve 2) emitters.
A. Rabinovich /Estimation of FE current density
183
is conserved as Y changes. That is, can we use the same equivalent emitter with the same k,, in some range of V, or do we need some other equivalent emitter with a different k,, at every value of V. To answer this question we have to consider an expression for the potential U as a function of V and of the distance from emitter surface
(I=jlF(x)dr=SXLFOf(X)dX=SX1kVf((~)dX=kVj’~(X)dX; 0 0 0
(2) 0
here F(x) is the electrical strength at a distance x from the emitter surface, Fc is electrical strength at the emitter surface, and f(x) = F(x)/Fe is the geometrical factor which depends primarily on the shape of the emitter and does not depend on V. We have for the real and equivalent emitters respectively:
(3)
(4)
where “re” and “eq” mean real and equivalent of the potential curves, (I,, = U,, . Hence,
values. At the point of intersection
This ‘equation determines xi. This determination does not contain dependencies on V. Therefore, As V changes the point of intersection of the potential curves moves along vertical line with the same coordinate value x1. As V rises, the point (x1, U,) moves down to the Fermi level. At some value of V it reaches the Fermi level (fig. 1, dotted lines). It is obvious from fig. 1 that in this case there is no equivalence of the transmission coefficients. To get equivalence again, we have to draw the equivalent potential curve steeper. Therefore, we need a different, sharper, equivalent emitter with a larger k,,. Thus, as I’ rises, a sharper equivalent emitter with a bigger field factor, k,,, corresponds to the same real emitter. Let us consider FN plots of equivalent emitters. An analytical expression for field emission current drawn from a field emitter which obeys the FN law is presented in eq. (5). Sufficient accuracy over a wide range of current density is made [8]: Z = X4 k2 V2 exp(-B/k
v) ,
(5)
where Z is the field emission currents, S is the emitting area, A and B are constants.
184
As it is well known write,
A. Rabinovich /Estimation
of FE current density
that FN plots are drawn on a semilogarithmic
log(1/V2) = f( I’-‘) = log(X4k2) - B/kV .
scale, we can
(6)
The slope of FN plot is d [log(Z/V’)] /d( V- ‘) = -B/k
.
(7)
All equivalent emitters must have the same values of S, A, and B, and differ only by value of k,,. Therefore, the FN plots of the equivalent emitters differ by their slopes in accordance with (7). At first approximation we can neglect the dependence of the preexponential factor in (5) on k. This simplifies consideration without an affect on the final result. Then all equivalent plots go from some point on the ordinate axis. So far as the equivalent emitters give the same current at a given V as the real emitter, FN plots of the equivalent emitters have to intersect the real plot (fig. 2). The real FN plot intersects the ordinate axis at some point C,,. We do not know the location of the point C,, from which the equivalent plots start. But, there are only two principally different possibilities which exist to place Ceq: (1) above C,,, and (2) below C,,. In case (1) the equivalent plots which intersect the real plot at higher V have bigger absolute value of slope and, therefore, less k,, . It is in contradiction with our conclusion which we obtained from the consideration of the potential barriers that k,, has to become bigger at higher V. It is only case (2) which is in agreement with that conclusion. It means that C,, is below C,,. Thus, we know the location of the equivalent FN plots relatively to the real FN plot.
3. Effective field emitter
Not every linear FN plot corresponds to the emitter which obeys the FN law. When we apply the FN law to reduce field emission current density from linear portion of the FN plot, we really obtain the current density of some not real emitter which obeys the FN law and exhibits the same FN plot as the real emitter. In ref. [6] this emitter was called “effective emitter”. The effective emitter gives the same current, at any given V, as the real emitter, but, of course, not the same current density. We need to introduce the effective emitter in the consideration because we can analyse and compare only FN plots of those emitters which obey the FN law. Let us compare the FN plots of the equivalent emitters and the effective one. It is seen from fig. 2, solid lines, that absolute values of slopes of all equivalent FN plots are less than one of the real, that is, effective FN plot. In accordance with the FN law the less the absolute value of the slope is the more the current density at a given V will be. Therefore, true current density is higher than the effective one which is reduced from real FN plot.
185
A. Rabinovich /Estimation of FE current density
-L
V Fig. 2. Fowler-Nordheim
plots for real emitter
(a), and for equivalent
emitters
(b,c).
4. Conclusion Thus, the value of current density we can reduce from the FN plot of ultra sharp field emitter is the lower bound for unknown true value of current density. We would like to notice now that our consideration is principally valid not only for the ultra sharp field emitters but for any real field emitter. There are no field emitters with uniform external electric field. Therefore, the value of current density we obtain from experimental FN plots in using of the FN law is always not the true, but just the effective one. However, only in the case of the ultra sharp field emitters do we have to expect significant differences between these two values.
Acknowledgement I would like to express my gratitude to Professor E.W. Miiller for giving me the opportunity of reading this paper at the 23rd International Field Emission Symposium, and to Professors P.H. Cutler and T.T. Tsong for proposing that I contribute this paper to the special issue.
References [l] [2] [3] [4] [5] [6]
E.W. Mutter, Z. Physik 106 (1937) 172. E.W. Mtiller, Z. Physik 131 (1951) 136. N.A. Gorbatyi and G.N. Shuppe, Zh. Tekhn. Fiz. 25 (1955) 1364. I.L. Sokolskaya, Zh. Techn. Fiz. 26 (1956) 1177. M. Drechsler, Z. Electrochem. 61 (1957) 48. V.G. Pavlov, A.A. Rabinovich and V.N. Shrednik. Zh. Tekhn. Fiz. 45 (1975) 2126; Soviet Phys.-Tech. Phys. 20 (1975) 1337. [7] V.N. Shrednik, in: Nenakalivaemye Catody (Sovetskoe Radio, Moscow, 1974) pp. 176177. [8] Ref. [7], p. 169.
ESTIMATION OF THE FIELD EMISSION CURRENT DENSITY DRAWN FROM ULTRA SHARP FIELD EMITTERS
A. RABINOVICH Zenith Radio Corporation, Melrose Park, Illinois 60160,
USA
Received 13 April 1977; manuscript received in final form 3 June 1977
The possibility of obtaining some information about the current density of field emission drawn from ultra sharp field emitters in using Fowler-Nordheim plots is considered. Notions of equivalent and effective field emitters obeying FN law are introduced in this consideration. The conclusion is that the current density obtained from experimental FN plots is the lower bound of an unknown true value of current density.
1. Introduction
Common needle field emitters have tip radii of the order of 10-4-10-5 cm. E.W. Miiller was the first to use such tungsten needles in the field electron and field ion microscopes of his invention [ 1,2]. By ultra sharp field emitters we shall mean tips of radii of the order of lo-’ cm. Such tips are usually manufactured by a thermofield (TF) treatment of common needle field emitters [3-S]. As a result of the TF treatment the tip of the needle acquires a polyhedral shape with small microprotrusions at the corners of the polyhedron. Emission properties of this type of field emitters were systematically investigated in [6]. Current-voltage characteristics were taken and Fowler-Nordheim (FN) plots were drawn. It turned out that FN plots of such field emitters exhibited an appreciable linear portion similarly to FN plots of common field emitters. The FN plots contain information about current density and emitting area. To get a true v&e of current density from FN plots, one should know an exact law of field emission for a given field emitter. The exact law of field emission for any real field emitter is not known. In practice, the FN law of field emission is used for reduction of current density from the FN plots. The FN law was derived under the assumption of a uniform external electric field. No real field emitter obeys the FN law because the external electric field at the surface of the real emitter is never uniform. Therefore the values of current density reduced from FN plots by using FN law are never true. However, in the case of common field emitters deviations of the reduced values from true ones are not significant. In this case the external electric field in the immediate vicinity of the emitter surface (within the thickness of the potential barrier) is fairly close to uni181
182
A. Rabinovich /Estimation
of FE current density
form [7]. In the case of ultra sharp field emitters the external electric field is distinctively non-uniform even within the potential barrier thickness. Thus, one should expect significant deviation of the reduced values of current density from the true ones. The absolute value of the deviation is unknown as well as the sign of it. The purpose of this paper is to determine the sign of the deviation, that is, the relation of the reduced from FN plots values of current to true ones.
2. The equivalent field emitter As it is well known, the transmission coefficient of the potential barrier controls the field emission current density drawn from the emitter at a given applied voltage I’. The potential barrier in the case of ultra sharp field emitters is depicted by curve 1 in fig. 1. (We do not take in account the image force potential for the sake of simplicity. This does not affect our final conclusion.) Let us compare the potential barriers of the real emitter and of some emitter which differs from real one only by exhibiting a uniform electric field, so its potential curve would be as depicted by curve 2, fig. 1. That is, we want to replace the real emitter with an equivalent one, which would give the same current and current density at the given V, but obey the FN law because of the uniform external electric field. To assure that the two emitters exhibit the same current density we have to require the equivalent barriers to have equal transmission coefficients. It is easily seen from fig. 1 that this can be done if the potential curves intersect each other at some point above the Fermi level. The slope of the potential curve at the emitter surface gives the electrical strength Fe at the emitter surface. This is related to V through so-called field factor k, F,=kV.
(1)
At the given V, the equivalent emitter with field factor k,, corresponds to real emitter with field factor k,,. The question then arises as to whether the equivalence
x
Fig. 1. Potential barriers at emitter surface Dotted lines correspond to higher V.
for real (curve 1) and equivalent
(curve 2) emitters.
A. Rabinovich /Estimation of FE current density
183
is conserved as Y changes. That is, can we use the same equivalent emitter with the same k,, in some range of V, or do we need some other equivalent emitter with a different k,, at every value of V. To answer this question we have to consider an expression for the potential U as a function of V and of the distance from emitter surface
(I=jlF(x)dr=SXLFOf(X)dX=SX1kVf((~)dX=kVj’~(X)dX; 0 0 0
(2) 0
here F(x) is the electrical strength at a distance x from the emitter surface, Fc is electrical strength at the emitter surface, and f(x) = F(x)/Fe is the geometrical factor which depends primarily on the shape of the emitter and does not depend on V. We have for the real and equivalent emitters respectively:
(3)
(4)
where “re” and “eq” mean real and equivalent of the potential curves, (I,, = U,, . Hence,
values. At the point of intersection
This ‘equation determines xi. This determination does not contain dependencies on V. Therefore, As V changes the point of intersection of the potential curves moves along vertical line with the same coordinate value x1. As V rises, the point (x1, U,) moves down to the Fermi level. At some value of V it reaches the Fermi level (fig. 1, dotted lines). It is obvious from fig. 1 that in this case there is no equivalence of the transmission coefficients. To get equivalence again, we have to draw the equivalent potential curve steeper. Therefore, we need a different, sharper, equivalent emitter with a larger k,,. Thus, as I’ rises, a sharper equivalent emitter with a bigger field factor, k,,, corresponds to the same real emitter. Let us consider FN plots of equivalent emitters. An analytical expression for field emission current drawn from a field emitter which obeys the FN law is presented in eq. (5). Sufficient accuracy over a wide range of current density is made [8]: Z = X4 k2 V2 exp(-B/k
v) ,
(5)
where Z is the field emission currents, S is the emitting area, A and B are constants.
184
As it is well known write,
A. Rabinovich /Estimation
of FE current density
that FN plots are drawn on a semilogarithmic
log(1/V2) = f( I’-‘) = log(X4k2) - B/kV .
scale, we can
(6)
The slope of FN plot is d [log(Z/V’)] /d( V- ‘) = -B/k
.
(7)
All equivalent emitters must have the same values of S, A, and B, and differ only by value of k,,. Therefore, the FN plots of the equivalent emitters differ by their slopes in accordance with (7). At first approximation we can neglect the dependence of the preexponential factor in (5) on k. This simplifies consideration without an affect on the final result. Then all equivalent plots go from some point on the ordinate axis. So far as the equivalent emitters give the same current at a given V as the real emitter, FN plots of the equivalent emitters have to intersect the real plot (fig. 2). The real FN plot intersects the ordinate axis at some point C,,. We do not know the location of the point C,, from which the equivalent plots start. But, there are only two principally different possibilities which exist to place Ceq: (1) above C,,, and (2) below C,,. In case (1) the equivalent plots which intersect the real plot at higher V have bigger absolute value of slope and, therefore, less k,, . It is in contradiction with our conclusion which we obtained from the consideration of the potential barriers that k,, has to become bigger at higher V. It is only case (2) which is in agreement with that conclusion. It means that C,, is below C,,. Thus, we know the location of the equivalent FN plots relatively to the real FN plot.
3. Effective field emitter
Not every linear FN plot corresponds to the emitter which obeys the FN law. When we apply the FN law to reduce field emission current density from linear portion of the FN plot, we really obtain the current density of some not real emitter which obeys the FN law and exhibits the same FN plot as the real emitter. In ref. [6] this emitter was called “effective emitter”. The effective emitter gives the same current, at any given V, as the real emitter, but, of course, not the same current density. We need to introduce the effective emitter in the consideration because we can analyse and compare only FN plots of those emitters which obey the FN law. Let us compare the FN plots of the equivalent emitters and the effective one. It is seen from fig. 2, solid lines, that absolute values of slopes of all equivalent FN plots are less than one of the real, that is, effective FN plot. In accordance with the FN law the less the absolute value of the slope is the more the current density at a given V will be. Therefore, true current density is higher than the effective one which is reduced from real FN plot.
185
A. Rabinovich /Estimation of FE current density
-L
V Fig. 2. Fowler-Nordheim
plots for real emitter
(a), and for equivalent
emitters
(b,c).
4. Conclusion Thus, the value of current density we can reduce from the FN plot of ultra sharp field emitter is the lower bound for unknown true value of current density. We would like to notice now that our consideration is principally valid not only for the ultra sharp field emitters but for any real field emitter. There are no field emitters with uniform external electric field. Therefore, the value of current density we obtain from experimental FN plots in using of the FN law is always not the true, but just the effective one. However, only in the case of the ultra sharp field emitters do we have to expect significant differences between these two values.
Acknowledgement I would like to express my gratitude to Professor E.W. Miiller for giving me the opportunity of reading this paper at the 23rd International Field Emission Symposium, and to Professors P.H. Cutler and T.T. Tsong for proposing that I contribute this paper to the special issue.
References [l] [2] [3] [4] [5] [6]
E.W. Mutter, Z. Physik 106 (1937) 172. E.W. Mtiller, Z. Physik 131 (1951) 136. N.A. Gorbatyi and G.N. Shuppe, Zh. Tekhn. Fiz. 25 (1955) 1364. I.L. Sokolskaya, Zh. Techn. Fiz. 26 (1956) 1177. M. Drechsler, Z. Electrochem. 61 (1957) 48. V.G. Pavlov, A.A. Rabinovich and V.N. Shrednik. Zh. Tekhn. Fiz. 45 (1975) 2126; Soviet Phys.-Tech. Phys. 20 (1975) 1337. [7] V.N. Shrednik, in: Nenakalivaemye Catody (Sovetskoe Radio, Moscow, 1974) pp. 176177. [8] Ref. [7], p. 169.