nextnano^{3}  Tutorial
next generation 3D nano device simulator
1D Tutorial
Ballistic current in a resonant tunneling diode (RTD)
Author:
Stefan Birner,
Tillmann
Kubis
If you are interested in the input files that are used within this tutorial, please contact stefan.birner@nextnano.de.
1) > 1D_RTD_AlGaAs_GaAs_green.in
==> Please see also this
RTD tutorial.
Ballistic current in a resonant tunneling diode (RTD)
This input file for the Resonant Tunneling Diode (RTD) is based on the paper of
Selfconsistent quantum transport theory: Applications and assessment of
approximate models
T. Kubis, P. Vogl
Journal of Computational Electronics 6, 183 (2007)
The resonant tunneling structure has the following properties:
 device length: 40 nm
 two 3 nm Al_{0.3}Ga_{0.7}As barriers separated by a 4 nm
GaAs quantum well in the center
 16 nm intrinsic region:
 3 nm left and right of the barriers, 3 nm + 3 nm in barriers
and 4 nm in well
 12 nm ndoped region left and right of the intrinsic region with n = 2 *
10^{17} cm^{3}
 conduction band offset: CBO(GaAs/Al_{0.3}Ga_{0.7}As) =
230 meV
 electron masses:
 GaAs:
m_{e} = 0.067 m_{0}
 Al_{0.3}Ga_{0.7}As: m_{e} = 0.0919 m_{0}
 temperature: 300 K
 grid resolution: 1 nm (homogeneous grid)
The method of calculating the carrier transport is defined as "fully
selfconsistent nonequilibrium Green's function (NEGF) approach for vertical
quantum transport in open quantum devices with contacts". For the purpose of
this tutorial (to make the calculations orders of magnitudes faster), we do not
include scattering, thus the method of calculating the current is termed
"ballistic transport".
The electrons are described within a oneband model with a variable effective
mass, i.e. a spatially dependent (= material dependent) effective electron mass
m_{e}(z).
The following figure shows the IV characteristics of the resonant tunneling diode.
The resonance peak (resonant tunneling) is at an applied voltage of 170
mV.
This currentvoltage (IV) characteristic shows a negative differential conductivity
which can be used in amplifiers or oscillators.
The applied bias modifies the Fermi levels, shifts the energy of the resonant
state and changes the conduction band profile due to charge redistributions
(Poisson equation), and thus alters the transmission properties of the barriers.
An important quantity of RTDs is the "peaktovalley" ratio of the current which
is defined as the current density in resonance j_{r} divided by the
current density in the valley j_{v}. For our ballistic simulation we
obtain a peaktovalley ratio of j_{p}/j_{v} = 6.
Schrödinger equation with Dirichlet boundary conditions at 0 nm and 40 nm
The following figure shows the selfconsistently calculated conduction band
profile of the GaAs/AlGaAs resonant tunneling structure plotted at the resonant
bias voltage of 170 mV including the wave functions (psi^{2}). Note that
these wave functions do not necessarily correspond to reality as we imposed
artificially high (i.e. infinite) barriers to the left and right side of the
sample region by applying Dirichlet boundary conditions to the Schrödinger
equation (i.e. psi = 0 at 0 nm and at 40 nm). We also note that these
wave functions are not used in the NEGF formalism. They are only shown for a
qualitative understanding of the structure.
Nevertheless, one can see the two states that are interesting in understanding
the device a bit more clearly.
In front of the left barrier of any RTD under forward bias, a
triangular quasibound is formed.
In the ballistic case, the triangular
quasibound state is below the left contact and is thus irrelevant
for the conductance whereas the resonance state
is responsible for the peak in the IV characteristic. If one increases the
voltage further, the resonance state will move below the energy of the left
contact and will then become irrelevant for the conductance, similar to the
triangular quasibound state. In that case, the current drops into the valley of
the IV characteristic.
The right figure shows the local density of states LDOS(z,E) of the simulated
structure. Note that the Schrödinger equation for this structure was simulated
with Dirichlet boundary conditions (psi = 0) at z = 0 nm and z = 40 nm. This
will lead to artificially confined states that are not present in a simulation
where "open boundaries" are used.
(Note: The LDOS figure does not include the factor of 2
due to spin degeneracy. The code has been modified meanwhile. The factor of 2
is now included in the LDOS outout.)
Schrödinger equation with Dirichlet boundary conditions at 300 nm and 340
nm
A more realistic picture of the wave functions can be obtained by calculating
the wave functions for the same structure but this time including a flatband
region of 300 nm region to the left and to the right of the structure mimicking
some kind of "open boundaries".
One can still see the triangular quasibound state
which looks nearly identical to the structure with the Dirichlet boundary
conditions at 0 nm and 40 nm.
The resonance states, however, look
differently. The resonance states couple to the states in the contacts more
strongly and now there are several of them. This corresponds to a broadening of
the resonance energy.
This explains why a model that takes into account "open boundary conditions"
(like our NEGF formalism) is very important in modeling transport.
The right figure shows the local density of states LDOS(z,E) of the simulated
structure. Note that the Schrödinger equation for this structure was simulated
with Dirichlet boundary conditions (psi = 0) at z =  300 nm and z =
340 nm. This large quantum cluster mimicks open boundary conditions, i.e. the
states in the central RTD region are not influenced significantly by the
boundary conditions. The units of the LDOS are [nm^{1 }eV^{1}] .
(Note: The LDOS figure does not include the factor of 2
due to spin degeneracy. They code has been modified meanwhile. The factor of 2
is now included in the LDOS outout.)
Results of ballistic NEGF calculations
Energy resolved electron density and energy resolved local density of states
(LDOS)
The following figure shows the selfconsistently calculated conduction band
profile of the GaAs/AlGaAs resonant tunneling structure plotted at the resonant
bias voltage of 170 mV together with
 (left figure) the energy resolved electron density
n(z,E) as a contour plot in units of [eV^{1} Angstrom ^{3}]
= [eV^{1 }10^{24} cm^{3}] .
==> EnergyResolvedDensity_avs.fld , *.coord, *.dat
Note that the horizontal legend is relevant for the color map
of the energy resolved electron density.
 (right figure) the spectral function (energy resolved local density of states
LDOS(z,E)) is shown as a contour plot in units of [eV^{1} Angstrom ^{1}] .
==> spectral_real_avs.fld, *.coord, *.dat
Note that the horizontal legend is relevant for the color map
of the energy resolved local density of states.
The vertical legend refers to the color map of the conduction band profile. The
maximum value of the conduction band edge can be read directly from this legend.
The right most point is at 0 eV.
The right figure shows that the triangular quasibound
state has a high local density of states. As expected and which can
be seen in the left figure, this state is not occupied with electrons. It can be
nicely seen that the states from the left contact are in resonance with the
quantum well resonance states.
Further below are the same plots at different biases: 70 mV, 200 mV (voltage of valley of
IV characteristics), 400 mV
Including scattering
T. Kubis and P. Vogl showed in their above cited paper, that due to
scattering, the triangular quasibound state
will be filled by electrons that have a "high" energy at the left contact, and
then lose energy by scattering events and thus fill this triangular boundstate.
The peak current in the IV characteristic then occurs at a bias voltage where
this quasibound state is aligned with the resonant quantum well state.
Assuming ballistic transport, the electrons in the contact fly towards the
barrier, maintaining their (high) energy and therefore hit the resonance energy
in the well at a significantly lower bias as compared to a calculation including
scattering where the electrons lose energy to occupy the triangular quasibound
state. In the ballistic case, the IV characteristic is then entirely determined
by the properties of the contact rather than by the doping of the device. A
careful inclusion of all inelastic processes is therefore necessary to capture
the physical mechanisms that determine the resonance bias and peak (and valley)
value of the RTD.
For the results of the calculations including scattering (and a discussion of
it), we refer to the section 4 of the above mentioned paper:
Selfconsistent quantum transport theory: Applications and assessment of
approximate models
T. Kubis, P. Vogl
Journal of Computational Electronics 6, 183 (2007)
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