 Home Mathematics  # FIELD EMISSION FROM PLANAR SURFACES

All objects being finite end up with a surface facing air or near vacuum. We have not discussed surfaces so far but they are not very different from the interfaces we have discussed. Consider a finite 1-dimensional array of atoms as in figure 6.2. Every electron in the bulk of the solid (i.e. in the array) experiences electrostatic interactions from the electrons of the neighbouring atoms from both left and right giving rise to the crystalline potential shown in the figure. In chapter 2 we considered the crystalline potential to be infinitely periodic. But in the case of a finite crystal, simplistically portrayed in figure 6.2, the end atom on the far right has no atom to its right and therefore its electron sees no electrostatic push from the right. Therefore, its electrons do not experience the potential seen by other electrons in the bulk but an increased potential in the form of a surface barrier as shown in figure 6.2. FIGURE 6.2 Schematic form of the crystalline potential in the bulk of a solid and near its surfaces. Near the surface, a potential barrier is created. Also shown are the Bloch state energies and the Fermi level.

The same argument holds for the electrons of the last atom on the far left. The minimum energy needed to take an electron out of the solid is the energy needed to overcome this surface barrier and is called workfunction. It is denoted by W in figure 6.2. We initially introduce briefly this barrier in connection with the Schottky diode in chapter 3. This barrier is similar in nature to the barrier between a metal and a semiconductor in the corresponding interface in the sense that it derives from an asymmetry of the nearby atoms. The above description is not complete. Since the electrons in the surface atoms no longer see a symmetric potential, they will tend to spill out towards the vacuum side leaving a positive charge behind and thus creating a dipole layer near the surface. This is shown in figure 6.3.

It is customary to simplify the above picture by using the Sommerfield model. Then for metals we only need to show the conduction band, the corresponding Fermi level, and the surface barrier we have just discussed. The energy band diagram which takes into FIGURE 6.3 A dipole layer is created near the surface of a solid because the Bloch eigenstates “spill out” of the solid surface. FIGURE 6.4 The actual surface barrier shown in (a) may be approximated as having zero width as shown in (b).

account this simplification is shown in figure 6.4a. Usually the surface barrier is further approximated by a vertical line extending up to the vacuum level. Then the energy band diagram of figure 6.4a is further simplified into the band diagram of figure 6.4b. Now let us assume a parallel plate capacitor geometry where two parallel metal plates with a distance d between them, are connected by a voltage V, giving rise to a constant electric field £ = V Id and an electrostatic energy Vei = -e£x in the space between the plates. The energy diagram corresponding to this situation with the surface barrier simplified as in 6.4b is shown in figure 6.5. Assume that temperature is low and the electric field £ is in the range of a few V/пт . Then tunnelling occurs as shown by the arrow and electrons are transported from the left to the right metal plate. The phenomenon is called Field (assisted electron) Emission.

Field Emission has many applications notably in Microscopy, Lithography, and micro- wave production and amplification. In all these cases a beam of electrons is the necessary input and this is provided by Field Emission. In Microscopy, the electron beam (presently of nanometric width) is used to probe the structure of a specimen, whereas in Lithography the beam is used to “carve” characteristics on semiconductors which transform them into FIGURE 6.5 The barrier for emission under the action of an electric field £ is triangular in the approximation of zero width barrier.

devices. Microwaves, on the other hand, are produced by accelerating a beam of electrons either linearly or circularly In older technologies that beam was created thermionically, i.e. heating the emitter to very high temperatures (such that кцТ ~ W) so that electrons could be emitted by climbing over the barrier of the workfunction. Nowadays, however, a large proportion of such technologies use field emission which does not require a heating circuit and, as we will see later, has the advantage of producing beams of small angles. Last but not least we mention the revival of the vacuum valve in solid state form called “vacuum transistor”. We will allocate a separate section to it in this chapter.

When an electron is emitted from the surface of a metal it experiences the force due to the surface barrier potential which extends only up to roughly 2 A from the surface. Although this potential energy is created by the complicated electron-electron interactions at the surface a very good approximation can be obtained if the image potential of electrostatics is used for it. Therefore, the potential energy that an electron will see on its way to vacuum, called tunnelling potential for short Vlun> is (choosing the x direction as the one perpendicular to the surface) where the (-) sign of the third term of the first line of 6.21 comes from the charge of the electron and that of the fourth term from the fact that the interaction is attractive. The graph of Vuw(x) is shown in figure 6.6.

The above potential energy will enter the effective mass equation which will result in the x direction after the separation of variables. In particular, if the envelop function is assumed to be of the form  FIGURE 6.6 The potential barrier for emission when the image potential is used to represent the surface.

with fey denoting the wavevector parallel to the surface we will get after separation of variables in the x direction with Ex = E-ti2k2m, the energy along x. The current that comes out of the barrier У,ци(х) can now be calculated in the same manner that the current from a resonant tunnelling diode was calculated. In fact, equation 5.69 is directly applicable here because the applied potential may be considered large enough, so that only a Left to Right current exists. Rewriting equation 5.69 we have where we have replaced the bottom of the conduction band of an RTD emitter by the bottom of the conduction band of the metal Eb and have written E? for E° +eV. Energies are measured from the bottom of the band (so we can set Eb = 0).

Certain points should be made clear. Just as in the case of the RTD (section 5.6), the bottom of the conduction band Eb is assumed to be flat, i.e. there is no variation with x inside the metal. This is due to the fact that the electric field £ terminates on the metal surface. (Actually electric fields penetrate metals by approximately half a monolayer, but this is always neglected). On the other hand, electric fields penetrate semiconductors substantially (i.e. by nanometers) and hence the In term, called the supply function, has to be evaluated for semiconductors right at the surface. A second point is that, due to the complexity of the barrier, the transmission coefficient T(£x) can not be evaluated exactly and hence the WKB approximation has to be used, i.e. equation 6.20.

Now an inspection of figure 6.5 and an examination of equation 6.20 show that T(£v) is a rapidly decaying function as Ex decreases. This is easily seen if we write compactly where G(EX) is called the Gamow exponent and is equal to The above equation is the outcome of substituting equation 6.21 in the expression for the transmission coefficient, equation 6.20, and of using the vacuum mass of the electron instead of the effective mass. Note that Xj does not coincide with the emitting surface but is slightly above it, As Ex decreases, i.e. moves away from either Eb or Evac, the above integral increases and hence T(EX) rapidly decreases. It makes sense therefore to expand G(£x) in a Taylor expansion and keep the first two terms only. The expansion is performed around the energy point Ex = EF, i.e. where the normal to the barrier component of energy is equal to Ef. With AEx = Ex - EF, we have where If equation 6.26 is substituted into equation 6.24 we get a very good approximation for the current density / One should not confuse the thermal energy KT with the transmission at the Fermi energy T(£f ). The integral in 6.27 can be evaluated analytically. The steps are purely mathematical so we relegate the proof to Appendix E. The result is As can be seen from 6.28, dF has the dimensions of (energy) ' and denotes the energy distance one has to go below £f so that T(£) becomes negligible. It is usually of the order of leV. On the other hand kT at room temperature is approximately MAOeV so that the term dpkT is much smaller than 1. We may expand to second order the term lsin(ndpkT) to get finally It is worthwhile pointing that only a small portion of the whole fc-space contributes to emission. This portion is shown in figure 6.7, where we represent the £(/c) space in contracted 2 dimensions, one for the dimension Ex, the normal energy, and the other for the energy £ц, the energy parallel to the barrier. Since the transmission coefficient is a function FIGURE 6.7 In a phase diagram with the parallel to the barrier energy on the x-axis and the total energy on the у-axis most of the field-emitted current comes from the small shaded upper triangle.

of Ex only, any increase £ц, which corresponds to a decrease of Ex, will also lower the transmission coefficient. Hence emission comes from the states which form the shaded triangle in figure 6.7. It is customary to call the expression in the brackets in equation 6.29 as 0(T) so that we may write where T(£f,0) is the transmission coefficient at the Fermi energy at zero temperature. Obviously the expression before the factor 0(T) is the zero temperature current density.

All that remains now is to calculate T(£f), the transmission coefficient at the Fermi energy. Once the Gamow exponent is known, T(£f) immediately follows. This is given by equation 6.25a. It is customary to perform the calculation in two stages: we first ignore the image potential. This is actually what has happened historically—the researchers of the 1930s tried the triangular barrier as a first approximation and recognized that this choice was inadequate because more than the calculated current was coming out of their emitters. We will only follow this historical path because it leads us smoothly to a complicated result.

If the image potential is neglected, the integral in 6.25a is easily calculated since the integrand is of the form -Ja-bx and the limits of integration are x,=0 and x2=(Evac - Ex)/(e£). Denoting by the superscript (tr), the results for the simple triangular barrier we get by a direct calculation from which we have and Substituting the above expression, 6.31a into 6.25 to obtain T,r(£f ,0) and then the latter and dp, into the expression for the current, equation 6.30, we get This is the Fowler-Nordheim equation uncorrected for the image potential. It relates the current density to the applied electric field £. Of course, experiments measure the current I, not the current density J. In this simple case of planar geometry, we have where A is the emitting area. So if a plot of ln[ll£2^v is made, see figure 6.8, one can

£

obtain the workfunction from the slope of this curve. This equation was obtained by Fowler and Nordheim in their first paper on Field Emission. As mentioned, experiments at that time indicated the inadequacy of this equation, so Nordheim proceeded with the inclusion of the image potential in the Gamow exponent.

When this is done the resulting integrals are much more complex than the simple case of the triangular barrier, they become linear combinations of the elliptic functions of the first and second kind. Flowever, the mathematics are quite standard. Adopting the superscript FIGURE 6.8 The FN plot for the exact triangular barrier is linear and for the image rounded barrier almost linear: the latter always gives a higher current.

FN for the image corrected triangular barrier, as is common, after a rather long set of transformations of variables we have where and Actually v(y) is an elliptic function defined by an integral equation but the above approximation obtained by Forbes  is fairly accurate and good enough for our purposes.

From equations 6.34 and 6.35a we deduce that where t (y) = v(y) -213 yv(y) with у evaluated at the Fermi level, у = yF and Substituting into the basic equation 6.28 we get It can be observed that equation 6.38 has the same structure as the corresponding equation for the simple triangular barrier with correction factors v(yp) and t2 (yp) inserted in the exponent and the preexponential terms respectively. Values for these functions, v(y) and t(y), are available so that again a plot of ln(l£2 jvs l/£ - (usually abbreviated “FN plot”) can be obtained. Note that у depends on £ so that the FN plot, based on equation 6.38, is not exactly straight but almost straight as an inspection of figure 6.8 shows because both v(y) and f(y) are slowly varying functions of y. Nevertheless the procedure for obtaining the workfuction W, as outlined for the simple triangular barrier, can be carried over to the FN barrier with adequate precision.

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