A distinct coordinate system
is defined for each lead
s, as shown in the figure. This is just a matter of convenience as the same derivations/expressions become applicable for all the leads. The potential is assumed to be constant along the direction
while the potential profile along
, is set equal to that along the lead-device interface. The wavefunction outside the system of leads and device is considered zero. Keeping these assumptions in mind, the wavefunctions within the leads can be broken into two independent components - traveling waves along
and wavefunctions with a discretized energy spectrum along
due to the quantum confinement in that direction. The latter part is computed by solving the 1-D Schrödinger equation along
,
where
is the electron potential energy along
in lead
s and
is the quantization mass. Eq. (
20) becomes an eigenvalue problem as a consequence of the zero-value Dirichlet boundary conditions imposed by the lead edges.
and
represent the
eigenstate and eigen-energy, respectively, in lead
s of the one-dimensional (1-D) eigenvalue problem which can be written as,
where the 1-D Hamiltonian matrix
is a
tridiagonal matrix, built as follows:
The computed eigenstates are normalized along the
direction and the total wavefunction inside lead
s is then given by:
Here, the expression inside the first summation represent
traveling waves with energy (along the
x and
z directions)
, going into and reflecting out of the device, respectively, through the lead
s. The index
denotes the wavefunctions
and the corresponding energies
in the conduction band valley
v. The expression within the second summation represents the gradually decaying modes with energy
. The coefficients
’s are chosen as inputs for the different waves traveling into the device, while the coefficients
need to be determined. For all other leads
, the injection amplitudes
. The wavevectors
for the traveling modes are given by:
and, for the evanescent modes, by:
. The energy
will be referred to as the `injection energy’. Note that
depending on the left or right contact, respectively.
The boundary conditions at the interface
between the device and lead
s dictate the continuity of both the wavefunction at the interface,
, and, the normal derivative,
for all
. Here
denotes the wavefunction inside the device. Using Eq. (
23) and combining the aforementioned boundary conditions together, we obtain:
Using finite-differences to discretize the right hand side of the above equation, we obtain
Now, multiplying both sides of Eq. (
23) by
and integrating over
in the range
, where
is the vertical height of lead
s, we obtain for
,
The wavefunctions
are eigenstates of a Hermitian operator (
) and so are mutually orthogonal,
. Using this relation and replacing
with
, observing continuity of the wavefunction at the interface, Eq. (
26) gives us
Note that since evanescent waves are not injected, the above relation will not contain the
term for
. Eq. (
25) can be re-written as:
In practice the wavefunctions
are calculated separately for each injected traveling wave and thus they depend on
m as well (a superscript
m will be used to denote this dependence henceforth). In our problem it has been seen that the extent of numerical errors is reduced with the introduction of more discretization into the system. For this reason, we also use a discretized version of the wave vectors
. The deduction of the same is given below in brief. Our starting point is the Schrödinger equation for the forward traveling wave
,
Using centered differences method, the second order derivative can be written as:
Using basic trigonometric identities, the discretized wave vector can be retrieved as,
Similarly for the evanescent waves
, the wave vector can be written as:
The QTBM Hamiltonian used to calculate the wavefunctions
is built using the closed system Hamiltonian
, described in Sec.
2.1. The matrix elements corresponding to drain and source ends of the device are modified to capture the QTBM boundary conditions. Let
and
be mesh indices of the
(
) and
(
) points in the range
, representing the left and right contacts, respectively. Two
matrices
and
are built which include the reflected and transmitted (both traveling and evanescent) waves traveling into and out of the device, respectively, shown below,
where the
A’s are defined in the same way as for Eq. (
4) and
Here,
p and
q represent the mesh indices
(or
),
i (or
j) being an integer in the range
(or
, respectively). We also define a
matrix
, expressing the component of the wave (traveling) injected into the device:
where
. The resulting linear system can be written as: