Appendix H

Direct Tunneling Rate in Spatially Variable Fields

In this appendix, an approximate expression for the number of the electron-hole pairs generated by internal field emission in a linearly variable electric field is derived, assuming direct tunneling. The expression is based on the two-band model and the Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) approximation.

We consider direct tunneling process. Direct tunneling is important in direct gap semiconductors like InSb, HgCdTe, GaAs, GaSb, GaAlAs, InGaAs, InAlGaAs, PbSeTe [518][55][54][7]. In Si direct tunneling is negligible in comparison with indirect (phonon-assisted) tunneling for all practically interesting fields ( ) and temperatures. This is a consequence of very large direct band gap of Si in comparison with the (indirect) band gap [86]. In Ge, both, direct and indirect tunneling are important because of low direct band gap in comparison with the indirect band gap ; indirect tunneling is important at low fields and direct tunneling at high fields. Although direct tunneling is negligible in Si-devices in practice, we think that the conclusions obtained in Section 4.2 by using the direct tunneling model which is presented in this appendix are qualitatively applicable to dominant phonon-assisted tunneling.

In order to model band-to-band tunneling in MOSFETs either some result from the junction-tunneling theory [430][237][88][7] or from the theory of internal field emission in an infinite medium [525][430][244][236][212][13] has been applied in the literature. In the junction-tunneling theory, the flux of the quasi-free moving valence band electrons which are described by the Bloch wave functions incidents on the potential barrier (junction). The main quantity of interest is the transmission probability of one electron from the valence to the conduction band, . Dependent on the approach employed, the transmission probability is either a function of the energy for the motion in the tunneling direction , of the energy associated with the motion perpendicular to the tunneling direction , or of the both energies [109]. On the other hand, in the internal field emission theory the electrons oscillate across the valence band due to applied field. The basic quantity of interest is the probability per unit time of the transition of an electron from the valence to the conduction band :

Expression H.1 originates from the Fermi Golden Rule. is the interband matrix element for the transition between the states of equal energy in the valence and the conduction band, is the electric field strength and is the width of the intersection of the I Brillouin zone in the field direction (here we assume that is parallel to -direction). The total number of generated pairs follows after an integration over the normal impulse component

is the number of the valence band electrons per unit cell interesting for tunneling ( for simple cubic lattice, due to spin). Note that we consider the planar problem in three dimensions.
In our opinion, neither the expressions known from the junction tunneling problem nor those in the internal field emission theory can be applied to the band-to-band tunneling problem in MOSFETs because of the vicinity of the oxide/bulk interface. There is a strong quantization of the motion in direction perpendicular to the gate/oxide interface in the potential well induced by the high transversal field. For tunneling in the gate/drain overlap region in -channel MOSFETs the initial states are strongly quantized, while the final states are like those in the diode-tunneling problem. The opposed conditions hold in -channel devices. This effect is not accounted for, neither in the literature nor in the present derivation. We think, however, that a proper model of tunneling in MOSFETs must include the interface-vicinity effect. In the following, the internal field emission theory for an infinite medium is applied.

Let us assume that the electric field changes according to

where is the tunneling length (Figure H.1). The interband matrix element can be derived by employing the time-independent perturbation theory, as done in [236] for uniform field. It reads

where

is the interband matrix element for the potential energy operator and ( stands for or ) are Bloch wave functions. For field H.3, the potential energy operator in the impulse representation becomes , where we substituted the -position operator . The , with and , are eigenfunctions in the conduction and the valence band in the presence of the electric field. From the time-independent Schrödinger equation

it follows

width

being the intraband correction of the band-edge energy. Of course, is fulfilled for the unperturbed Hamiltonian and the Bloch wave functions .
After adopting a band model, the dispersion relation is known, as well as . Consequently, , , and can be calculated. The eigenfunctions follow from H.7 by neglecting the interband term, i.e. the second term on the left-hand-side, after applying the periodic boundary conditions at the edges of the I Brillouin zone. Finally, can be calculated by using H.4.
Following this standard approach ([236][109][13]), the authors have derived in [457][456][56] the interband matrix element for linearly variable field H.3, assuming two-band model. It has been obtained that the field variation (the parameter ) does not influence the exponent in the expression for the number of the generated pairs . The generation rate was only weakly influenced by due to the preexponential dependence. This result seems to be not justified from the physical point of view, because the field variation directly changes the tunneling length. Since the wave functions decay exponentially within the forbidden gap, the change in the tunneling length should influence in an exponential manner. The erroneous result in these papers we attribute to the assumption that the eigenfunctions and do not change explicitly with the electric field strength along the tunneling path. A small field variation has been implicitly adopted in the derivation. This assumption is not justified. In addition, the derivations in [457][456][56] employ the result of an analytical integration over the parallel impulse component , which has been proposed in [54] while solving the uniform field problem. We have proven that this analytical result is incorrect. Moreover, we have obtained that Kane approximation is an accurate solution to the problem [236].

In order to overcome the difficulties in analytical derivation of the rigorous result, we apply the WKBJ method with an appropriate barrier for the variable field to derive an approximate model. We follow the quasi-classical interpretation of the internal field emission result by introducing a quantity ([13])

is the probability of the penetration of an incident electron from the valence band to the conduction band and is the period of oscillations of the valence band electrons. Relationship H.9 assumes a homogenous field. The total number of pairs generated by internal field emission results by replacing H.9 in H.2

Expression H.10 may be written in the form , where is the field-dependent oscillation frequency, is the concentration of the valence band electrons which are active in tunneling and is the total averaged penetration probability per particle ([525][302]). Let us introduce the transmission probability per particle calculated by WKBJ approximation, . As well known, the transmission probability reads

where is the imaginary part of the wave vector in direction of the electric field. The and denote the classical turning points. When is calculated from by using H.9 where follows from the time-independent perturbation theory assuming the band model and a constant field , the following relationship can be established [237]

Consequently, in a uniform field the WKBJ yields the same result as the rigorous approach, with exception of the preexponential factor . Motivated by this fact and the discussion in [237] we heuristically assume

for a spatially variable field.
For the band model, a two-band model is adopted, for which it is known to result in a dispersion relation

for the conduction band and

for the valence band, where is the free electron mass in vacuum, is the reduced mass in tunneling and is the matrix element for the interaction between both bands. We further consider the kinetic energy in the conduction band H.14 ( is replaced by ), which yields for the factor

The physical solution with ``+'' sign is chosen so that in the limit , is fulfilled. For a typical condition a very accurate approximation of H.16 may follow

In the conditions relationship H.17 reduces to

which also follows by neglecting the ``free-kinetic energy'' term in H.14. Expression H.18, known as Franz's dispersion relation [91], is sometimes a crude approximation of H.16, but is very convenient for analytical handling and will be assumed henceforward. The required for H.11 follows from H.18, with benefit of

where is the energy associated with the motion in direction perpendicular to the field.

A particle with an incident energy changes its impulse while penetrating into the forbidden gap according to the variation in the relative potential energy due to the field

At the incident point and , while and hold at the endpoint, Figure H.1. The equivalent tunneling barrier has an interesting property

Therefore, the particles properly feel the forces and at the band edges.
Field H.3 yields . The barrier has a maximum of at . Henceforward we replace with . It follows

The barrier is shown in Figure H.2 for different field variation as a parameter. The classical turning points in the WKBJ approach, including the normal-impulse energy , are given by

where . Further useful relationships are , and .

After a replacement of H.22 in H.19, transmission probability H.11 reads

where

The subintegral function in posses a restriction

which also follows from the criterium that in H.23 is a finite real number. The integral can be transformed

with

Finally,

In a constant field it follows , , and . Relationship H.29 yields

This expression is identical to the Kane formula in [236], with exception for the prefactor . Relationship H.29 is replaced in H.10. We integrate over the energy instead of , with benefit of . The integration interval ranges from to given by H.26. In order to simplify the expression let us observe that is a rapidly convergent series expansion, because of restriction H.26 which means . We truncate this expansion after the second term with and keep the first three terms in the exponent of H.29 til the first power of . In order to integrate H.10, it is assumed that . The resulting reads

It is assumed that the oscillation frequency depends on the field at the starting point . In a uniform field , where is the Kane-Keldysh result (expression (39) in [236]).
The replacement is justified if the exponential term for is larger than about . It follows

where we benefit from H.26. It is adopted that the characteristic field is much larger than the field of interest . For e.g. and it follows , whereas for we have . Consequently, for common electric field strengths the proposed expression holds for the field variations of practical interest.

In Figure H.3 we compared the expression for with and assuming a simple one-dimensional analytical model for the potential and field distributions in the totally depleted uniformly doped bulk. The tunneling rate from the surface is plotted against the surface field which is also the starting field in tunneling.