THE 3-DIMENSIONAL WKB PROBLEM
The 3-dimensional WKB problem remains, still, a difficult quantum mechanical problem that has not as yet been solved in its generality. Specific approaches to the problem exist most of which have taken place relatively recently (1980s). The critical reader may ask why we should bother with such a theory when the theory of emission from planar surfaces is adequate. The answer is that emitters nowadays are not planar but are pin-like objects possessing a sharp apex with a radius of curvature R at the top. The radius R was of the order of microns in the 1950s but has continued decreasing ever since and in many applications it is now of the order of nanometers. The prime reason for having pin-pointed emitters is to enhance the applied electric field because sharp metallic features attract electric field lines. Although surfaces of radius of curvature of microns can be considered locally flat, this is no longer possible with an emitter of R in the range of nanometers and a 3-dimensional calculation of the current density is necessary.
Consider a region of space (B), where E - V < 0, sandwiched between two regions (A) and (C) where V -E> 0, see figure 6.9. As explained in 6.2, any incidence of electrons with an angle away from the normal decreases the transmission coefficient exponentially with that angle and we therefore consider only normal incidence to this barrier, see figure 6.9. In our presentation we follow closely the analysis of Das and Mahanty .
Again as in section 6.1, we write for the wavefunction
where Plank’s constant Ь has now been introduced to make the argument of the exponential a pure number. Then, following exactly the steps of section 6.1, we get the 3-dimensional equivalent of equation 6.8
FIGURE 6.9 Region В is a forbidden region sandwiched between allowed regions A and C. A mobile enters region В normal at r, and leaves region В at r2. See text for the calculation of the path.
where E, V(r) are the total energy and potential energy terms respectively of the Schroedinger equation
Assuming again that V(r) is slowly varying, in the sense of section 6.1, we can ignore as small the second derivative of a and obtain an approximate zeroth order solution o0
The formal solution of 6.41 is a line integral of the form
The path of integration is the curve of the steepest gradient (descend) of G0(r). This is also equal to the path of an electron since a(r) is the action of the electron. However, contrary to the 1-dimensional case, the path of the electron or particle in general is not known and this constitutes the major difficulty of the multi-dimensional WKB. Such a path going from the classically accessible region (A) into the classically forbidden region (B) and reemerging from such a region in (C) is shown in figure 6.9. Let rx and r2 be the points of entry to and exit from В respectively. We will come back to the problem of the evaluation of an electron’s path, but first let us write, according to 6.42, the wavefunctions of an electron propagating exclusively inside the classically accessible region A and the wavefunction of an electron having gone through the forbidden region В and reemerging in region C. So for an electron propagating from point r0 to point r; we have (see figure 6.9) for its wavefunction
where dl denotes an infinitesimal length along the electron’s path. For the wavefunction of an electron at point r, inside region C, having gone through B, we have
Note that in the second integral of equation 6.44 we have replaced the negative term (E - V) by the positive [V- E) and have taken outside of the square root the factor (/). Furthermore we have neglected multiple reflections inside region (В) which will exponentially decrease the transmission coefficient.
Our basic equation for the current density /, equation 1.60b of chapter 1, can be written more compactly as
Substituting equation 6.43 in 6.45 we get, after some manipulations, that the current density JA in region (A), is
where t (r) is the tangent vector of unit magnitude along the trajectory at r. Similarly using equation 6.44 in equation 6.45 for region (C) we get
and the line integration from l to /2 is in region (B). If we now take the ratio of |/с (^2 )| to ]a (n )| we run into infinities because at the points rx and r2 E = V. These infinities can be lifted if we linearize the potential near the turning points (or use L’ HopitaTs rule) so that we finally get that the transmission coefficient T is of the form
Equation 6.49 does not seem to be a correct generalization of the 1-D case. The only difference from the 1-dimensional (1-D) case is the pre-exponential term which is unity in the 1-D case. The pre-exponential term is usually in the range of 1-3 whereas the exponential term (a genuine generalization of the 1-D case) may vary by orders of magnitude. We will therefore accept 6.49 as a generalization of the 1-D case because the numerical difference from including or not including the awkward pre-exponential factor is insignificant. The cause of this discrepancy is the fact that the WKB expressions for the current density cannot be used at the turning points.