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THE NATURE OF THE ELECTRON IMPACT IONIZATION ENHANCEMENT IN A
MULTIQUANTUM WELL STRUCTURE : A NUMERICAL AND ANALYTICAL APPROACH
K. Brennan, Yang Wang
To cite this version:
K. Brennan, Yang Wang. THE NATURE OF THE ELECTRON IMPACT IONIZATION EN- HANCEMENT IN A MULTIQUANTUM WELL STRUCTURE : A NUMERICAL AND AN- ALYTICAL APPROACH. Journal de Physique Colloques, 1987, 48 (C5), pp.C5-277-C5-280.
�10.1051/jphyscol:1987560�. �jpa-00226764�
JOURNAL DE PHYSIQUE
C o l l o q u e C 5 , s u p p l 6 m e n t au n o l l , T o m e 48, n o v e m b r e 1987
THE NATURE OF THE ELECTRON IMPACT IONIZATION ENHANCEMENT I N A MULTIQUANTUM WELL STRUCTURE : A NUMERICAL AND ANALYTICAL APPROACH
K. BRENNAN a n d YANG WANG
School of Electrical Engineering and Microelectronics Research Center, Georgia Institute of Technology, Atlanta, G A 3 0 3 3 2 , u.S.A.
We present both numerical and analytical calculations verifying the electron impact ionization enhancement in multiquantum well structures. Owing to the inherent nonlinearity of the impact ionization process and the existence of a threshold energy, the electron ionization rate is always enhanced within the narrow gap layer over its corresponding bulk rate. Depending upon the extent of the enhancement, the applied field strength, and the dev ice geometry, the net super- lattice ionization- rate can be greater than the weighted average of the constituent bulk rates.
In most bulk 111-V semiconductor materials, the electron to hole ionization rates ratio approaches unity [ll over most applied electric field strengths. As is well known, avalanche photodetectors exhibit poor performance in terms of maximum gain, gain to noise ratio, bandwidth, and bit error rate under these conditions [2- 41, The electron to hole ionization rates ratio can be artificially enhanced via band edge engineering [5,6J. In order to enhance the electron ionization rate over both its bulk and the corresponding hole ionization rates, the electron distribution must be selectively heated. Numerous device schemes have been suggested which provide an enhanced electron ionization rate [7-121 by selectively heating the electron distribution via a built-in field arising from a fully depleted p-i-n [8,9]
or p-n [10,111 junction. It has been determined, theoretically, that the electron ionization rate can be enhanced by roughly four orders of magnitude over the hole ionization rate in these structures [9-131.
The simplest structure which provides enhancement is the simple multiquantum well device [5,6,141 which consists of alternating layers of narrow and wide band gap materials such as GaAs and AlGaAs. It at first appears surprising that the electron ionization rate in these structures is enhanced over its bulk value since the average energx gain from the structure is zero. As we show below, the ioniza- tion rate is enhanced owing to its inherent nonlinearity and the existence of a threshold energy. We present both numerical and analytical verification of the enhancement effect.
The superposition of a spatially periodic electric field and a constant overall bias field results in local heating of the carrier distribution. If the local heating (arising from the band edge discontinuity) is sufficiently large such that the carriers are heated above the threshold energy, impact ionization will occur.
The enhancement can be explained quantitatively I151 by realizing that the ionization rate depends exponentially upon the field, F, both the constant bias and
periodic components as,
-
where E is the average electron energy, and I/T;:~(E) is the energy dependent - - -
phonon scattering rate. If the periodic component of the field is assumed to be
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987560
J O U R N A L D E PHYSIQUE
cosinusoidal, then the average impact ionization rate can be calulated by integrating equation (1) with respect to z as,
where Vo is the conduction band edge discontinuity, and C is the z independent part
of the exponent, ,7
Equation (2) can be readily evaluated using the definition of the modified zeroth order Bessel function as 1161.
where B = 2 V c / ~ . F ~ The modified Bessel function is always greater than one for
0 0-
positive argument. Therefore, it is found that the ionization rate within a periodic potential structure is always enhanced over the bulk rate.
The enhancement arises from the fact that an exponential function of the periodic field, and not the periodic field itself, is integrated over the period of the structure. If the periodic field alone is integrated over a full period, as in the case for determining the average field, it averages to zero. However, the exponential of the field does not average to zero over the full period, but is always greater than zero, thereby leading to an enhancement.
In addition, we present numerical calculations of the electron ionization rate in a simple multiquantum well structure in both Figures 1 and 2 . The numerical calculations are performed using an ensemble Monte Carlo program which is particu- larly well adapted to high energy, high electric field transport, i.e., it includes the full details of the band structures, phonon scattering rates, and the potential structure [131. Brennan et al. [I71 first demonstrated, using the Monte Carlo method, that the conduction band edge discontinuity at the GaAs/AlGaAs interface can produce an enhancement in the ionization rate. If the ionization rate is calculated by averaging over many periods of the multiquantum well structure, the net ioniza- tion rate is found from the weighted average of the rates within the GaAs and AlGaAs layers as,
It is important to realize that the ionization rates in each layer are not necessarily the bulk ionization rates. As we have shown above, the ionization rate within the GaAs layer is always enhanced over its corresponding bulk value.
Depending upon the extent of this enhancement, the net ionization rate within the total structure can be larger than the corresponding bulk GaAs rate. Clearly, as the width of the GaAs layer approaches zero, the net ionization rate must approach the bulk AlGaAs rate. Conversely, as the width of the AlGaAs layer approaches zero, the net rate must approach that of bulk GaAs. Interestingly, as shown in Figure 1 , the net ionization rate can be larger than either the bulk GaAs or bulk AlGaAs rates depending upon the applied electric field and geometry, barrier and well widths.
As thc electric field strength increases, the net superlattice rate decreases below the calculated bulk rate (Figure 7 ) . At high electric field strengths, the net ionization rate approaches that equal to the weighted average of the constituent bulk rates. This is due to two effects, the ionization rate increases in the AlGaAs layers and, more importantly, the band edge discontinuity provides a less signifi- cant fraction of the carrier heating.
At low applied electric fields, the potential step significantly alters the ionization rate by providing a larger proportion of the energy needed to reach threshold. Figure 1 clearly shows that th.e electron ionization rate is strongly enhanced over the bulk GaAs rate at low electric fields. Notice that in order for the net superlattice rate to exceed the bulk GaAs rate, since no ionization occurs within the AlGaAs layers, the enhancement must be very large. This is obvious since the enhancement must compensate for the "dead space" effect of the AlGaAs layers.
Clearly, from equation (5), if the electron ionization rate is simply the bulk GaAs rate, then in a symmetric structure (equal barrier and well widths) the net ioniza- tion rate would be one-half the corresponding bulk GaAs rate. What is observed, however, is that the ionization rate, at low applied fields, is greatly enhanced over the bulk rate.
Figure 2 illustrates the effect of the layer widths on the net ionization rate as a function of the applied electric field. At low applied electric fields, the electron ionization rate is most strongly influenced by the presence of the poten- tial step. Again as the field increases, the net ionization rate simply approaches the weighted average of the bulk rates. Notice that in the high field case, the ionization rate increases linearly as the width of the GaAs layer increases, while at low fields, the ionization rate peaks away from either limit; mainly GaAs or mainly AlGaAs.
In summary, the electron ionization rate enhancement in a simple multiquantum well structure is shown both analytically and numerically to arise from the nonlinear aspects of the ionization process, i.e., the exponential dependence upon the field. It is found that the ionization rate within the narrow gap layer is always enhanced by the potential discontinuity over the corresponding bulk rate, Depending upon the applied electric field and the device geometry, the net super- lattice rate can exceed either bulk rate of the constituent materials.
Acknowledgements
The authors gratefully acknowledge the technical assistance of Peggy Knight and Diana Fouts at the Georgia Institute of Technology.
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ELECTRIC FIELD (kVlcm) Figure 1. Calculated electron impact ionization rate as a funcrion of inverse electric field for both bulk GaAs and a 500 A wide well and barrier superlattice. The experimental measure- ments are due to Bulman et. al. [18].
