INTRODUCTION
In recent years, advances in technologies such as MBE have made it possible
to fabricate very small devices with dimensions comparable to the Debye’s
length. As a consequence of this small dimensions, the I/V characteristics
of such devices must be calculated by quantum theories. Among these devices,
there are devices such as Resonant Tunnelling Diodes (RTD) which due to
long dwelling time of their electrons inside the device, the scattering
phenomena have a major rule in their operation and as a consequence, a
ballistic quantum transport theory will not be adequate.
Unfortunately, the problem of quantum electron transport in presence
of scattering phenomena is a hard problem and there is not yet an exact
formulation for it. In this situation, a simple approximate formulation
known as optical model attracted much attention. In the optical model,
the effects of scattering phenomena are introduced by an imaginary potential
into the Schrödinger equation (Schiff, 1968):
In this formula, V is the real and w is the imaginary potential. This
method has been used by many authors for simulation of RTD. Some of them
(Hu and Stapleton, 1993) have used it directly through the Schrödinger
equation as explained in the Eq. 1 . Others have used
it via a path integral method (Zohta, 1990a, b; Zohta and Ezawa, 1992)
or in conjunction with the FebryPerot resonator (Furuya et al.,
1994) or by transfer matrix or scattering matrix method (Yuming, 1988;
Zohta, 1993; Zohta and Ezawa, 1992). All these approaches were successful
and ended into similar results but there are two ambiguities common to
all of them.
The first ambiguity is related to the amount of the imaginary potential
that is needed in the calculations. The second ambiguity is related to
the incoherent current which eventually flows through the device. The
optical model itself says nothing about these two problems and we must
use additional theories and models to clarify them.
PHYSICBASED IMAGINARY POTENTIAL FOR RTD
Determining the imaginary potential profile is the first step in the
usage of optical model. Nevertheless, many papers have taken it as an
unknown parameter and simply introduced their results for some different
values (Hu and Stapleton, 1991; Zohta, 1990b; Hu and Stapleton, 1993).
Some other papers that have tried to calculate it on a physical base,
considered it simply as a scalar (a single number) (Zohta and Tanamoto,
1993). Here we will consider it in more detail and calculate it in its
complete form with complete dependency of its bias, energy, position and
temperature.
A simple form of imaginary potential can be obtained from mobility. Let
us consider this simple form before going into details. By calculating
the divergence of coherent current density (the current density from the
Schrödinger equation when there exists an imaginary potential in
the equation) using Eq. 1 and its complex conjugate,
we have:
Comparing this term with the classical formula for ∇.j in the case
G = 0 yields (R is the recombination and G is the generation term):
Considering the above relation and the relation existing between the
carrier’s lifetime and the recombination term, R = n/τ, we get w
= ħ/2τ (n is the carrier density which is equal to Ψ*Ψ
and τ is the carrier’s lifetime). Now if we use the mobility formula,
μ = qτ/m*, to substitute the carrier’s lifetime, we reach the
desired relation between the imaginary potential and mobility:
As a numerical example, for GaAs with m* = 0.067m0 and μ = 7500
at 300k the imaginary potential got from the above mentioned procedure
will be equal to 1.8E22. Nevertheless, the resulted imaginary potential
is a scalar and is over simplified because mobility is only a low field
averaged quantity and as it will be seen later, the real situations are
more complicated.
To obtain a better estimation for the imaginary potential, we can use
the scattering rates to estimate the carrier’s lifetime instead of using
the mobility:
In the above formula, the summation is over the scattering rates of various
phenomena involved in the motion of electron and E_{tot} is the
electron’s total energy. For electrons in the RTD’s well, it is sufficient
to include the scattering rates from the absorption and emission of polar
optical phonons that cause electrons to scatter inside the gamma valley
or from gamma valley to L valleys and the scattering rate from the acoustic
phonons (totally 5 scattering phenomena). Scattering rates formulas are
so long that we do not want to repeat them here. The reader interested
in this issue may refer to appendix M of the reference (Singh, 1993).
Figure 1 shows the results of those that have calculated
for GaAs at 300 K. It shows the five mentioned scattering rates as a function
of the electron’s total energy. The total energy can be calculated using
the following formula:

Fig. 1: 
Electron’s scattering rates in the gamma valley of GaAs
at T = 300K as a function of its total energy 

Fig. 2: 
Imaginary potential obtained from the Eq.
7 (black circles) as a function of position along the RTD’s well
for T = 300K at different biases (from 0V to 0.5V) and different wavevectors
(corresponds to different electrons at the contact) in comparison
with the imaginary potential obtained from the mobility (solid line) 
In this formula, the first term is the kinetic energy of electron at
the contact. The second term is due to two electron’s transverse degrees
of freedom and the third term is due to the potential energy. The imaginary
potential finally becomes:
The imaginary potential term calculated in this way has its full dependency
on bias (via third term in Eq. 6), position (also via
the third term of Eq. 6) and temperature (via second
term of Eq. 6) and obviously is more reliable than a
simple scalar term.
To show the scale of changes of the imaginary potential made by our model,
we drew Fig. 2. In this figure, the horizontal axis
is the position axes along the RTD’s well. The imaginary potential for
different biases in the range of 0 V to 0.5 V and for different values
of wave vector at the contact has been calculated and drawn on the figure.
Also on the figure, the value obtained from mobility (Eq.
4) has been shown. We see the imaginary potential is not at all a
single value and may vary very widely in different cases. We also see
that the estimation obtained from the mobility is a very poor estimation.
INCOHERENT ELECTRON CURRENT IN RTD
When the imaginary potential term is added to the Schrödinger equation,
the divergence of the current density is no longer equal to zero (Eq.
2) and the sum of the squared terms of transmission coefficient and
reflection coefficient on the left side of the device is no longer equal
to unity (Zohta, 1993):
The difference is due to the scattering phenomena which take some of
the incoming electrons from their inphase (coherent) wave functions and
scatter them into random phase (incoherent) states. These scattered electrons
eventually make the incoherent current term that different opinions have
been presented about it.
Some authors (Yuming, 1988; Hu and Stapleton, 1991) take the coherent
current on the left side of the device as the total electron current (coherent
current plus incoherent current). They have actually supposed that all
scattered electrons finally go out from the well through only the right
barrier and towards the right direction. Other authors (Zohta, 1993; Zohta
and Ezawa, 1992; Zohta and Tanamoto, 1993) take not all but a portion
of the term
as the incoherent current term. They have actually supposed that a portion
of the scattered electrons in the well eventually go through the right
barrier and others go through the left one.
In a previous study (Sharifi, 1999), we explained that the scattering
phenomena cause the electrons to become incoherent with the incoming electrons,
but they don’t cause them to become classical particles having no phase
and wave nature. In other words, the scattered electrons must be considered
as quantum particles either. Therefore assigning any property to the squared
terms of the transmission coefficients of the right barrier and the left
barrier separately as is the case in the references (Zohta and Ezawa,
1992; Zohta and Tanamoto, 1993), is not a correct assignment because this
assignment will erase any interfering term between the two barriers.
Therefore we suggest to use the metastable states for modelling of the
scattered electrons. Metastable states are the states of a system when
its boundary conditions are set in such a way that they show only outgoing
electrons from all the boundaries of the system. Therefore, the metastable
states may model the trapped electrons in the well of RTD which are gradually
going out from both two barriers. Another point that supports the usage
of the metastable states, is the fact that the density of states in the
well of RTD has a big peak at the energy of these states; therefore, we
expect most of the scattered electrons go to these states.
Further, we will calculate these states for RTD and in the section after
that we will use them for calculation of the incoherent current.
AN APPROXIMATE ANALYTICAL METHOD FOR METASTABLE STATES OF RTD
In a previous work (Sharifi and Adibi, 1999), we had introduced a numerical
method for calculating metastable states of RTD. Here we introduce an
approximate analytical method. We begin by introducing an approximate
potential profile using a WKB concept. This potential profile is constructed
by replacing the two barriers with two impulse functions having the same
area (Fig. 3).
The d1 and d2 are the powers of the two impulse functions replaced the
left and the right barriers, respectively. Bh is the barrier’s height.
B_{w} and W_{w} are the barrier’s width and the well’s
width, respectively and Vb is the applied bias voltage. For the wave function,
we assume three proper combinations of exponential terms in the three
regions (Fig. 3) (all ks are complex, E is complex too).
Let us to consider the zero bias condition at first (k1 = k2 = k3 = k).
By equating the corresponding wave functions at the two sides of the two
barriers, respectively and equating the difference of gradients of corresponding
wave functions at two sides of two barriers with the corresponding integrals
of the potential profile, we get the following relations:

Fig. 3: 
Potential profile of RTD (uppergraph) and its approximation
by a stair shaped one with two impulse functions; d1 and d2 (lowergraph) 
One of the four constants in the above four relations can be omitted
from normalization concept. We selected it to be B and take it to be e^{ikWw/2}.
Three other constants can then be calculated from the first three relations:
Inserting these constants into the last relation will give us the desired
equation for k:
Table 1: 
A few first metastable states energy’s Eigenvalue
for RTD at zero bias (The RTD’s parameters are the same as Fig.
4) 


Fig. 4: 
The real part (left plot) and the imaginary part (right
plot) of Eigenvalue of the first metastable state of RTD as a function
of bias voltage calculated with the Eq. 15. The
parameters are: W _{w} = 50A, B _{w} = 24A, Bh = 1eV 
This is a nonlinear multipleanswer algebraic equation which may be solved
by a computer to get the answers for k. The Eigenvalues for energy can
then be calculated using
Table 1 shows a few first energy Eigenvalues. Any
of these Eigenvalues has a negative imaginary part. This negative imaginary
part causes the corresponding probability density function to decay in
time as expected of a metastable state.
Now the procedure can extend to nonzero bias condition. It is lengthier
but is straightforward and ends into the following equation.
The k1, k2 and k3 may be replaced with their definitions in the Eq.
10 to obtain an equation for energy. Again, the equation is a nonlinear
multipleanswer complex equation which can be solved using a computer. Figure
4 shows the result for the first metastable state energy’s Eigenvalue,
as a function of bias voltage. We see the real part has almost a linear
functionality of the bias but the imaginary part has a more complex functionality.
After calculating the complex Eigenvalue; E, the complex wave vectors,
k1, k2 and k3 may be calculated from Eq. 10 and then
the constants A, C and D may be calculated from an extension form of the
Eq. 12. We need these quantities for calculating the
incoherent current at the next section.
INCOHERENT CURRENT MODEL USING METASTABLE STATES
For incoherent current, we suggest the following formula (in the formula
0 is the left hand side of the device, L is the right hand side and x
is a middle point inside the device.):
j_{incoh }(x) = (j_{coh
}(0)  j_{coh} (x)) T_{sr}  (J_{coh}
(x)  j_{coh} (L))T_{sl} 
(15) 
This formula has a clear interpretation. As mentioned before in optical
model, the divergence of coherent current density is related to the scattering
rate of electrons (Eq. 2). Therefore the integral of
that term from 0 to x which is the coherent current density
itself is equal to the total scattering rate in the interval [0, x]. In
Eq. 15, this integral is multiplied by Tsr, which
is the transmission to right coefficient to yield the positive term of
our incoherent current density. In the same manner, the total electrons
scattered in the interval [x, L] is multiplied by Tsl (the transmission
to left coefficient) to yield the negative term of the incoherent current.
The two parameters, Tsr and Tsl, may be calculated by the following relations
using the before mentioned metastable states:
Tsr and Tsl as calculated above are no longer local parameters due to the transmission
coefficients of the two barriers separately, but are global parameters that
preserve all interference terms that may exist between the two barriers. Therefore
the introduced model preserves the wave nature of incoherent electrons as well.
Figure 5 shows the calculated Tsr as a function of bias. In
the figure, the lines Tsr = 1 and Tsr = 0.5 have also been plotted which the
first line corresponds to those papers (Yuming, 1988) that supposed all scattered
electrons will eventually go to the right. The second line corresponds to those
(Zohta, 1993) that supposed a half of scattered electrons will go to the right.
From the figure, we see that the result obtained from metastable states is
close to the results of the first papers at high biases and to the second papers
at low biases. This is a very good behaviour that matches our expectations and
now is obtained nicely from a firm physical base.

Fig. 5: 
The transmission to right coefficient, Tsr, of the incoherent
electrons as a function of bias voltage calculated from the metastable
states ( Eq. 16) in comparison with the two previous
models (RTD’s parameters are the same as Fig. 4) 
The incoherent current as suggested by Eq. 15 has an
interesting aspect. Divergence of total current, if its incoherent term
is calculated by that equation, will be equal to zero in agreement with
the particle conservation concept.
THE ALGORITHM AND THE RESULTS
In our model, the currentvoltage characteristics of RTD can be calculated
in the following steps:
• 
Calculate the Fermi level and then the potential profile
by solving the Poisson’s equation using the impurity profile of all
device layers including buffer, spacer and contact layers. 
• 
Calculate the imaginary potential as a function of position for
any bias and any (longitudinal) wave vector at the contact (Eq.
7). 
• 
Insert the potential profile from the first step and the imaginary
potential from the second step into the Schrödinger equation.
Solve it and calculate the coherent current profile for any wave vector
at any bias point. 

Fig. 6: 
The coherent, incoherent and total current of RTD at
T = 300K in conjunction with the ballistic current. The brief RTD’s
parameters are: Well: 50 A GaAs; Barriers: 17 A AlGaAs; Barrier height:
0.65 eV; Contact doping: 1e18/cm^{3} 

Fig. 7: 
The results of our model for the total current of RTD
in comparison with the two previous models (RTD’s parameters are the
same as Fig. 6) 
• 
Calculate the metastable states and then Tsr and Tsl
at any bias point using Eq. 9. 
• 
Calculate the incoherent current for any wave vector at any bias
point using Eq. 15. 
• 
Calculate the total coherent, the total incoherent and the total
current at any bias point by a summation over the corresponding the
above partial terms weighted by density of states (Sharifi and Adibi,
1999). 
Figure 6 shows the total coherent, incoherent and total
current in conjunction with the ballistic current (current, when there
is no scattering). Figure 7 compares the results of
our model for total current with the two previous models as mentioned
before. The first previous model (No. 1) had supposed that all scattered
electrons go eventually through the right barrier (Yuming, 1988). The
second previous model (No. 2), on the other hand, had supposed that half
of scattered electrons go through the right barrier (Zohta, 1993). We
see the new results are closer but not equal to the model number 1.
CONCLUSION
In this study, we introduced two physicbased models, one for the imaginary
potential and another for the incoherent current based on the metastable
states. We then used these two models through the optical model for simulation
of RTD and compared our results with the previous results. The two models
introduced are not restricted to RTD and can be used in other quantum
devices as well. Beside of the incoherent current, there is another important
quantity that can be calculated from the metastable states too. This
quantity is the incoherent electron density which its determination enables
us to do a truly self consistent calculation. We believe by these two
models, the optical model has gotten a better situation and in the future
it will be used more than past.