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As already mentioned in the previous section, field emission rarely takes place from planar surfaces but usually from pin-pointed emitters which can enhance the electric field around them by preferentially collecting electric field lines at their apex. The most common used model (but not the most accurate one) is the so called the hemisphere on a post model shown in figure 6.10. The emitter is simulated as having a semispherical tip of radius R and a main cylindrical body of height h. The emitter is placed on the lower plate of a parallel plate capacitor on the upper plate of which a voltage V > 0 is applied. It is assumed that the distance between the plates is much larger than h. This model does not accurately simulate the shape of an emitter but retains all the characteristics necessary to obtain a reliable estimate of the enhancement factor y, defined by the ratio of the local field 6’, at the apex of the emitter to the macroscopically applied field £м -V Id where d is the distance between the plates.

Numerical calculations of the Poisson equation by the present author [3] and Egdcombe and Valdre [4] for the enhancement of the field у show that

where £; above stands for the local electric field at the apex of the emitter, so the important quantities, as far as enhancement of the field is concerned, are the height h and the curvature at the apex. Obviously different structures will have different enhancement factors, so equation 6.50 above shall only be used as a guide for the order of magnitude to be expected from a given emitting structure. For example, a typical emitter may have a radius of curvature R at the apex of 10nm and a typical height of 1 micron so that we expect to obtain experimentally у = 70. Indeed у is most of the times treated as a parameter in the theory and is determined experimentally by the Fowler Nordheim plots, provided that the workfunction is known.

A pointed emitter of height h inside a parallel plate capacitor of distance d between the plates attracts the electric field lines at its apex, thus enhancing the electric field

FIGURE 6.10 A pointed emitter of height h inside a parallel plate capacitor of distance d between the plates attracts the electric field lines at its apex, thus enhancing the electric field.

The Fowler-Nordheim (FN) theory for planar surfaces can readily be taken over to the theory of field emission from spherical surfaces if the radius of curvature R at the apex is large enough. What does “large enough” mean? The length of the tunneling region xtuvaries from 1-2та, so if R » xlu the electric field lines near the apex, where most of the current emanates, can be considered parallel to each other, i.e. quasi-l-dimensional. In that case the potential barrier preserves its form discussed in the previous section. Then we can use for the emitted current density at the apex /(0 = 0)- where 6 = angle to the emitter axis the FN equation with just £ substituted for £ in 6.38, namely

As one moves away from the apex, i.e. as 0 increases, the normal to the surface electric field diminishes. A very simple way of looking at this is to observe that the highest concentration of electric field lines occurs at the apex. So we can write

where 0(0) is a decreasing function of 0.

The total current will be given by

where, contrary to equation 6.33 and because of the spherical geometry here, Aeg is an effective area which includes also corrections due to the fact that J is not constant over the surface of the emitter. If the radius of curvature of the emitter R is not much greater than x,un (practically if R < 20 nanometers) the above simple approximation of using the FN equation with £ as the real field no longer holds. Complications then arise in both the form of the tunnelling potential and in the evaluation of the paths of the electrons that are no longer straight lines. We consider the changes in the form of the tunnelling potential fist. The result we are going to give holds for all kind of emitting surfaces and is not limited to the hemisphere on a past problem. Figure 6.11 portrays an arbitrary shape tip with the inscribed sphere at its apex.

The surface around the apex may be approximated by the surface of the inscribed sphere. Assume spherical coordinates (г,ф,0) with their centre at the centre of the inscribed sphere. The system has rotational symmetry so cp does not play a role. The potential energy term that will enter the calculation of the Gamow coefficient (equation 6.25) can always be written in the three-dimensional case in the form

where uL (r,0) is the solution of the Laplace equation (having the dimension of volts)

Geometry for the calculation of the emitted current at the apex of a nanoemitter, and

FIGURE 6.11 Geometry for the calculation of the emitted current at the apex of a nanoemitter, and

is the image potential of a sphere. Decomposition 6.54 is valid for any surface. The minus sign in front of the uL term comes from the charge of the electron. In the case of a planar surface uL = Ex and и,т (x) = e2 /I6xn.

A general but simple expression for uL (r,0 = 0) near the surface of the emitter to second order in r can be obtained as follows. The Laplace equation in spherical coordinates is

The surface of the emitter however is an equipotential surface, hence all tangential fields are zero on it. Therefore all derivatives with respect to 0 vanish. We then get

If we now make a Taylor expansion of ul up to second order with respect to z = r-R at 0 = 0 we get

Potential energy distribution outside a nanoemitter for several values of the radius of curvature R of the nanoemitter. The field at the apex of the tip was kept constant at 5V/nm

FIGURE 6.12 Potential energy distribution outside a nanoemitter for several values of the radius of curvature R of the nanoemitter. The field at the apex of the tip was kept constant at 5V/nm.

A more rigorous proof of 6.59 is given in Kyritsakis and Xanthakis [5] for any curved surface. The electrostatic potential uL(z) leads into a tunneling potential energy V,„„(z) that an electron sees which is of the form

The variation of this potential energy with respect to к at a constant electric field at the emitter apex is shown in figure 6.12. It can be seen that as R decreases the extent of the barrier increases, thus making tunneling more difficult. Furthermore, it was found that the accuracy of the above expansion becomes inadequate as R approaches the value R = 5nm.

The current density along the axis of the tip /(0 = 0), can now be calculated provided that the evaluation of the awkward integral entering the Gamow exponent can be performed. This is not possible in the general case but a very good approximation can be obtained if the last term of the expansion 6.60 is treated as a perturbation using the Leibnitz integral. This is beyond the scope of this book. The interested reader is referred to reference [5]. It must be made clear, however, that the whole procedure is purely mathematical and no further physical principles are involved. Here we only quote the rather long but in closed form result

where w(y) and t|/(y) are new functions defined in terms of v(y) and у is the usual variable of FN theory. It can be seen that equation 6.61 is of a modified FN type: the exponential and pre-exponential FN terms are present in 6.61 but the curvature of the emitter has introduced correction factors in both the pre-exponential and exponential terms given

Fowler-Nordheim plots of a nanoemitter for several values of the radius of curvature R

FIGURE 6.13 Fowler-Nordheim plots of a nanoemitter for several values of the radius of curvature R: solid lines denote theoretical values and isolated points experimental data.

respectively by i (y) and w(y). The current can now be calculated by a slight generalization and improvement of equation 6.53

where a is a numerical factor that corrects the supply function to take into account that the latter is made out of Bloch functions and not made out of simple plane waves.

The FN diagrams corresponding to equation 6.62 are shown in figure 6.13. The plots are not straight lines but have a slight curvature which is mainly due to the last term (the second order term) in 6.60. From this curvature of the FN plot, the radii of curvature R of the emitters can be obtained by reverse engineering and are shown in figure 6.13. Note that on the logarithmic scale the term oAeg will appear as the intercept on the ordinate axis. This is unknown from theory—for each particular plot in figure 6.13, it has been obtained from experiment. Continuous lines in this figure denote theoretical results, isolated points experimental values, see [5] for more details.

The above theory of curved surfaces (not necessarily spherical) is good enough if one is interested in the total amount of current emitted. If one is interested in the distribution of current in space and the trajectories of the electrons in space or the evaluation of Ae,j one has to abandon the 1-dimensional WKB and use the 3-dimensional one. One has then to revert to numerical calculations and abandon any hope of an algebraic, analytic manipulation. The main difficulty lies in the evaluation of the paths of the electrons and the corresponding evaluation of the line integrals defining the transmission coefficient, see equations 6.48, 6.49. Of course, no problems occur if these line integrals are evaluated numerically. We must emphasize that paths are not defined in Quantum Mechanics.

In fact, according to the Feynmann formulation of Quantum Mechanics, a particle travels along many paths simultaneously, with each path being assigned a probability. In the region of validity of the WKB approximation, also called the semi-classical approximation, all the paths collapse into a single one or more precisely all are very close to each other and to the classical trajectory. But in the forbidden region there is no classical path. How do we calculate the WKB quantum path then?

A solution to this very difficult problem was given by Kapur and Peierls [6] as early as 1937. The procedure is simple: the barrier V(r)-E is inverted into £ —V(r) to create a well instead of a barrier; a classical path now exists if an electron falls normally into the well with zero velocity. This is the electron’s quantum path in the forbidden region and the path along which the transmission coefficient is calculated. The 3-dimensional WKB could be very useful in situations where the width of the emanated beam is of importance as in spectroscopy or lithography. The application of the method in practice is as follows.

In figure 6.14 we show the apex area of an emitting tip of arbitrary shape (not necessarily spherical). The tunnelling potential outside the emitter is obtained by first obtaining ul by solving the Laplace equation either numerically or algebraically and then adding the image potential. Then the two equipotential surfaces where V = E can be calculated and these define the tunnelling or forbidden region. It is shown as the line-shaded area in figure 6.14. The emitter is the inner all shaded semi-circle. For any entry point л on the inner boundary of the forbidden region, the point of exit r2 on the outer boundary of the tunnelling region is obtained by calculating the entire path in the tunnelling region as described above. The transmission coefficient from rx to r2 is then calculated according to equation 6.49 and then the normally emitted current density /(r2) is obtained by the standard methods of chapter 5, i.e. by multiplying the supply function by the transmission coefficient and integrating over the normal energy. The assumption behind this choice is a) that the

2- or 3-dimensional problem can be reduced to a set of 1-dimensional problems and b) that locally at each surface element dS a plane wave suppy function is incident on dS. The procedure can be repeated for every point, or more precisely for every surface element dS, on the

The tunnelling region

FIGURE 6.14 The tunnelling region (line-shaded area) around a nanoemitter. If the curvature of the emitter at its apex is high, an electron entering the tunnelling region at r, will come out at r2, where vectors q and r2 do not necessarily lie on the same line.

Two emitters with different R. The emitter with the smaller R emits electrons enclosed within a smaller angle of emission

FIGURE 6.15 Two emitters with different R. The emitter with the smaller R emits electrons enclosed within a smaller angle of emission.

inner boundary of the forbidden region, and then /(r2) can be integrated (over the outer boundary) to obtain the total current I instead of using an Аед obtained from experiment.

Furthermore, the paths in the classically allowed regions can be equally calculated and all of the paths of the electrons can thus be obtained. An important characteristic which arises from such calculations is that the emitting area depends on the apex shape of the emitter. Figure 6.15 shows schematically the beam emanated from an emitter which is nearly spherical at its apex and the beam emanated from a surface which is sharper (say ellipsoidal) at its apex. It can clearly be seen that the beam which emanates from the ellipsoidal surface has less angular spread, a characteristic very useful in practice. This is a general characteristic of surfaces: the sharper is a surface, the less the angular spread of the emitted electrons.

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