Nanoelectronic Modeling (NEMO): A New Quantum Device Simulator

Dan Blanks, Gerhard Klimeck, Roger Lake, \*R. Chris Bowen, \*\*William
R. Frensley,\
\*\*Manhua Leng, and \*Chenjing L. Fernando

Raytheon TI Systems, Dallas, TX 75265

\*Texas Instruments Inc., Dallas, TX 75265

\*\*School of Engineering, The University of Texas at Dallas,
Richardson, TX 75083

**Abstract**

*Device simulation is a standard practice in the design of
state-of-the-art semiconductor devices. These simulations are based on
semi-classical models that break down as device geometries approach
atomic dimensions. Conventional device simulators may include effects
such as tunneling and quantum confinement in an* ad hoc *manner, but
such approaches are applicable only over a narrow range of device
designs and operating conditions. To address this problem, we developed
NEMO (NanoElectronic MOdeling), a new quantum device design tool that
simulates a wide variety of quantum devices including RTDs, HEMTs, HBTs,
superlattices, and Esaki diodes. NEMO uses a non-equilibrium Green\'s
function formalism to treat the effects of quantum charging,
bandstructure, and incoherent scattering. A collection of models allows
the user to trade off simulation accuracy with speed and memory
requirements. This approach addresses the diverse needs of the engineer
who desires a black-box design tool and the theorist who is interested
in a detailed investigation of the physics. Here we present an overview
of the theoretical methodology, graphical user interface features, and
simulations for various devices.*

# Introduction

NASA and the semiconductor industry have a long history of mutually
beneficial interaction. The race for the Moon in the 1960\'s spurred the
need for on-board electronics with reduced weight and greater
functionality. Prior to the development of the commercial IC market, the
space program along with the military provided major incentives for
government funding of semiconductor devices far beyond the immediate
needs of the marketplace. Consequently, the eventual triumph of the
Apollo mission was also a victory for the semiconductor industry and
played a significant factor in today\'s dominance of US industry in the
world commercial semiconductor market.

Just as microelectronics was the key to the early days of space
exploration, nanoelectronics will be essential for its future. Declining
budgets have only increased the need for smaller and better electronics.
In addition to the obvious benefits of reduced weight and size,
nanoelectronic devices possess functionality not available in
conventional devices. For example, resonant tunneling devices (RTD)
exhibit negative differential resistance (NDR) with the possibility of
Terahertz switching speeds. Since they are majority-carrier devices,
RTD\'s can operate from cryogenic temperatures to room temperature. The
combination of smaller size and majority-carrier operation also enhances
resistance to radiation, a key problem in space-based systems. Even more
advanced structures such as quantum dots and single-electron transistors
(SET) hold the potential for extremely dense three-dimensional arrays of
nanoelectronic devices.

Despite these exciting possibilities, nanoelectronic devices are still
in their relative infancy. The expense and difficulty of device
fabrication precludes simply building and testing vast arrays of quantum
devices. To focus efficiently on the best design, engineers need a tool
that predicts electronic characteristics as a function of the device
geometry and composition. In a more scientific mode, such a simulator
would greatly enhance the understanding of quantum effects that drive
the transport process and provide a means to investigate new device
concepts.

Even conventional semiconductor devices require a correction for quantum
effects associated with the smaller device features. MOS devices, for
example, exhibit electron confinement effects in the inversion layer.
This phenomena is a function of decreasing oxide thickness rather than
the overall size of the device. Quantum effects becomes important as the
oxide layer thickness decreases below the 30 Å, which will soon be a
standard for manufactured integrated circuits. Problems of this nature
will become more prevalent as device geometries continue to shrink.

Nanoelectronic device modeling requires a fundamental quantum-mechanical
approach. State-of-the-art semiconductor device simulators based on
drift-diffusion equations (e.g. PISCES) or Boltzmann\'s equation (e.g.,
DAMOCLES \[1\]) can only include these effects in an *ad hoc* manner.
Therefore, the semiconductor industry needs a new fully
quantum-mechanically based TCAD tool.

To address this problem, we developed a general purpose 1-D quantum
device simulator called NEMO (NanoElectronic MOdeling). NEMO can
simulate a wide variety of nanoelectronic devices, including RTD, HEMTs,
Esaki diodes, and three terminal devices like the bipolar quantum
tunneling transistor. NEMO encompasses the diverse needs of the engineer
who wants a black-box design tool and the theorist interested in a
detailed investigation of the physics. It has a collection of models
that allows the user to trade off between calculation speed and
accuracy. NEMO also includes a graphical user interface (GUI) that
enables parameter entry, calculation control, display of calculation
results, and *in-situ* data analysis methods.

This paper reviews the capabilities of NEMO, summarize the theoretical
approach, and give examples of several NEMO simulations.

# NEMO Theory

## Overview

A full discussion of the NEMO theory is beyond the scope of this
article. Here we highlight the theoretical framework and the available
transport models. A comprehensive description of the NEMO theory is in
reference \[2\] with a recent amendment in reference \[3\]. For a list
of references to work on resonant tunneling diodes we refer the reader
to the list provided in reference \[2\]. The first examples of our
generalized boundary conditions that allow us to treat realistically
long devices are in reference \[4\]. Examples of RTD simulations
including effects due to scattering and bandstructure \[5-9\] are also
in the literature. Numerical tools developed for this project are
discussed in references \[9, 10\]. Boykin \[11-13\] provided the sp3s\*
full band parameters for typical III-V materials and developed numerical
tools that allow for the numerical treatment of full bandstructure
effects.

Prior to NEMO, most quantum device simulations solved the single
particle effective mass Schrödinger equation coupled with Poisson's
equation. To make the problem tractable, these simulations employed one
or more stringent approximations, including low temperature, low bias,
simplified bandstructure, or no scattering. The calculated results were
usually limited to a specific type of device operating over a small
range of conditions. The transition from research to production requires
a more comprehensive TCAD tool that simulates a variety of devices under
a wide range of conditions, particularly for room temperature and high
voltage operation.

Although the Schrödinger equation approach is widely used, it cannot
properly treat incoherent scattering or the electron reservoirs in the
contacts. These limitations arise because Schrödinger\'s equation only
models the states of the system for a single electron (dynamics) and not
the carrier transport (kinetics). Modifications of the states due to
interactions with other particles (electrons, phonons, etc.) must be
patched in as effective potentials while another independent equation
treats state occupation and carrier transport. This method de-couples
the carrier dynamics from the carrier kinetics. In reality, scattering
events redistribute the occupation of quantum states, break their
coherence, and shift their resonant energies. In addition, the Pauli
exclusion principle forbids the scattering into occupied states. All of
these phenomena couple the dynamic and kinetic processes. Moreover, the
Schrödinger equation-based models assume flat band profiles for the
electron reservoirs whereas realistic devices exhibit complex band
bending due to carrier accumulation at the interface.

The non-equilibrium Green function formalism (NEGF) provides a
comprehensive framework that couples dynamics and kinetics in a
Hamiltonian-based system. It also can apply different approximations in
different areas of the device. To use this capability, NEMO divides the
device into non-equilibrium and reservoir regions. The non-equilibrium
region encloses the quantum well and barrier structures. The remaining
reservoir regions encompasses the device contacts and the adjacent
layers which act as current sources and sinks. The reservoirs typically
have a high electron density and are adequately modeled with a
semi-classical relaxation time approximation coupling quantum mechanical
states. In the non-equilibrium region, NEMO uses the complete quantum
theory to explicitly treat the scattering processes. By concentrating
the full quantum calculation on the non-equilibrium region and using a
relaxation time approximation elsewhere, NEMO can simulate realistically
long structures with spatially non-uniform electron reservoirs.

The essential breakthrough in modeling RTDs is the proper treatment of
the reservoirs (relaxation time approximation scattering) and the proper
band structure. For example, we find that NEMO simulations that neglect
scattering in the non-equilibrium region are in excellent agreement with
experimental measurements for a large set of devices \[7\], including
current vs. voltage (I-V) data of RTDs at room temperature and tunnel
current through Si/SiO2/Metal \[14\] structures. Usually, under
high-temperature/high-current conditions, most of the current flows
coherently through the resonant states rather than via incoherent
processes. Only RTDs operated at low temperature or with low current
density require explicit treatment of scattering in the non-equilibrium
region.

Because of the complexity and relatively undeveloped nature of the
theory, the current NEMO simulator only applies to 1-dimensional
devices. We have prototyped a 2-D simulation tool based on NEGF which
can be used to analyze ultra-scaled FETs.

The main simulation options can be distinguished by the choice of
potential, bandstructure and scattering models. The following sections
summarize these options.

## Potential and Charge Models

The two major potential models available in NEMO are Thomas-Fermi and
Hartree self-consistent charge calculation. The Thomas-Fermi model uses
a semi-classical approximation to calculate charge in the reservoir
regions. Charge in the non-equilibrium region is either set to zero or
calculated from a quasi Fermi-level linearly interpolated between the
contact Fermi levels. The final step uses Poisson\'s equation to
calculate the electrostatic potential and semi-classical charge
self-consistently throughout the device.

At best, the Thomas-Fermi model can only give an approximate estimate of
charge in the non-equilibrium region and the reservoirs containing
quantized states. This is inadequate for devices with appreciable charge
build-up in these regions. The Hartree model is more rigorous in that it
calculates quantum charging effects self-consistently in the
non-equilibrium region as well as the terminals. The LDA approximation
\[15\] takes exchange and correlation effects into account. While the
Hartree calculation may be computed over the entire device, we usually
only apply it to the region of the device that contains quantum states
and use the semi-classical approximation elsewhere. Limiting the
calculation range in the manner can dramatically reduce computation time
with little effect on the calculation accuracy.

The Hartree model is usually more accurate than the Thomas-Fermi
approach, particularly in the quantum well region where charge buildup
can have a dramatic influence on device behavior. Alternatively, the
Thomas-Fermi calculation is much faster and yields a potential profile
very close to the Hartree model. In fact, the Thomas-Fermi calculation
is so fast that it can generate band profile curves as we change the
applied bias in real-time. The Resonances mode of the NEMO GUI uses this
capability.

Another potential model available in NEMO is the linear potential drop
in the non-equilibrium region. This models assumes that the free charge
is identical to the doping in the terminals such that the potential is
flat in each terminal and drops linearly in the non-equilibrium region.

## Band Structure Models

Detailed understanding of the band structure is even more critical for
quantum devices than for most conventional semiconductor devices. Charge
can conduct through resonant states at very high energies compared to
the band edge of the well material. Simultaneously, the same carriers
can tunnel through the band gap of the barrier material where there is
an effective imaginary band structure. In both cases, the carriers
sample portions of the band structure rarely accessed by semi-classical
devices. Even if conduction takes place primarily in a single band, a
multiple-band calculation may be required to obtain the proper shape of
that band, especially for energies high above the band edge. Thus, NEMO
provides a large array of band structure models, including:

-   Single band effective mass

-   Multiple de-coupled single bands (no transport between the bands,
    but electrostatic coupling via Poisson\'s Equation).

-   Parameterized single-band which includes band-non-parabolicity
    derived from several sources of bandstructure data (sp3s\*, k•p,
    tables)

-   k•p nearest neighbor 2-bands

-   sp^3^s^\*^ nearest neighbor 10-bands with and without spin orbit
    coupling and with and without explicit spin basis.

The appropriate model depends on the details of the device structure and
the operating conditions. NEMO allows the user to select among these
models to trade off accuracy with speed and memory usage.

An internal database provides all relevant band structure parameters for
any standard III-V compound. The user can insert customized parameter
values for any material system. The Band Structure dialog discussed in
Sec.4.3 provides a user-friendly method to examine these models.

## Scattering Processes

Scattering processes often dominate conduction at higher voltages,
particularly in RTDs where scattering is the ultimate limit of the
valley current. For the single band models NEMO includes models for the
following scattering mechanisms:

-   Acoustic phonon

-   Polar optical phonon

-   Interface roughness

-   Alloy disorder

# NEMO GUI![](media/image1.emf){width="4.073611111111111in" height="2.9722222222222223in"}

## Overview

Another important goal of the NEMO project was to make a user-friendly
simulator that provides as much control as possible over every aspect of
the simulation. Flexibility and ease-of-use are difficult to achieve
simultaneously, but given the complexity of quantum device simulations
it became clear that both criteria were vital to the program success.
Consequently, graphical user interface (GUI) development was a major
part of the NEMO program. Here we present an overview of the most
important GUI features, including:

-   Main Screen

-   Device Structure Dialog

-   Material Parameters Dialog

-   Simulation Parameters Dialog

-   Simulation Library Dialog

-   Band Structure Dialog

-   Plot Features

## Main Screen

The Main Screen shown in Fig. 1 is the central location where the user
controls the NEMO simulation. From the Main Screen, you can:

-   Raise all of the major NEMO windows.

-   Switch between Current vs. Voltage and Resonances mode.

-   Enter test conditions (bias, temperature, etc.).

-   Initiate and abort a calculation.

-   Interrupt a calculation and decide whether to abort or continue.

-   Open and save NEMO parameter files

-   Open and save files with NEMO calculation results.

The Resonances calculation mode is shown in Fig. 2. Energy resonances
refer to the quasi-bound states that form in a quantum well structure.
The location of these states in energy and position is critical to the
operation of a quantum device since most of the device current flows
through these states.

When in the Resonances calculation mode, NEMO displays a calculated band
profile with horizontal cursors located at the Fermi levels in each
device contacts. The user can directly enter the bias into the Test
Conditions box or use the mouse pointer to select the emitter Fermi
level cursor. A Thomas-Fermi approximation calculates the band profile
so quickly that the band profile follows the Fermi level as the user
moves the cursor.

## Device structure dialog![](media/image2.png){width="3.96875in" height="3.1180555555555554in"}

The Device Structure window in Fig. 3 contains a Layer Arrangement
spreadsheet for entry of the device layer information. Each row
represents one layer in the device and defines layer properties such as
composition and thickness.
![](media/image3.png){width="4.705555555555556in"
height="3.0215277777777776in"}

The Materials list displays compounds that are consistent with the
Substrate selection in the Material Options panel. Double-clicking the
mouse pointer on an item in the Materials list creates a corresponding
entry in the Layer Arrangement table. In this manner, the user can
quickly enter very complicated layer arrangements with a minimum of
typing. Figure 3 shows the Device Structure dialog for a 2-barrier RTD
device.

## Material parameters dialog

Material parameters include any intrinsic properties relevant to the
electronic properties of the compound. Examples include the energy gap,
effective mass, and dielectric constant. The proper choice of material
parameters is essential to accurate quantum device calculations,
particularly for quantum devices where tunneling processes are extremely
sensitive functions of material parameter values. NEMO furnishes default
parameters for a wide variety of materials and displays them in the
Material Parameters dialog in Fig. 4. If the default parameters suffice
for the devices of interest, the user never needs bring up the Material
Parameters window. However if the user wants to provide customized
parameters to the simulation NEMO allows the user to enter his own
parameter values for any material. You can even enter a user-defined
material with your own customized parameter values.

# Simulation parameters![](media/image4.emf){width="5.2347222222222225in" height="2.854861111111111in"}

## Overview

Simulation parameters control the selection of theoretical models used
in the NEMO calculation. NEMO includes different models to calculate the
current, potential, charge, scattering processes, and the band
structure. The simulation parameters also specify the desired
calculation outputs options. To simplify the parameter input process,
NEMO uses a hierarchical approach for the display and entry of
simulation parameter values. In addition to traditional entry methods
(e.g., text boxes, option lists, etc.) NEMO also provides graphical
methods to enter regional parameters that depend on relative positions
of the device barriers and wells. The next few sections discuss these
features.

### Simulation parameters \-- hierarchical methodology 

NEMO has a rich variety of simulation models. While this provides the
maximum flexibility in terms of applicability to different types of
devices and test conditions, the downside is that NEMO requires over 100
simulation parameters. Traditional device simulators force the users to
familiarize themselves with all available simulation parameters and
ensure that they are set correctly.

To minimize this burden for the users, NEMO employs an innovative
hierarchical approach to input simulation parameters. The top level of
this hierarchy specifies the highest level calculation option (Current
or Resonances). Subsequent levels contain more detailed options such as
the choice of potential model and scattering processes.

For parameter entry, NEMO creates text boxes, radio boxes, and option
lists that correspond to the simulation parameters for a single
hierarchy level and display them in the Simulation Parameter table as
shown in Fig. 5. Each time the user presses Next, NEMO goes through the
following process:

-   Accept parameter settings for the latest displayed hierarchy level.

-   Determine the parameters needed for the next level.

-   Create new GUI elements.

-   Display new level.

NEMO only displays parameters required by the prior selections. For
example, different potential models require different parameter sets.
Once the user selects a potential model, NEMO only shows parameters that
are consistent with that model. Parameters unique to other models are
either not created or are disabled from input.

This method of limiting the displayed parameter set greatly diminishes
the burden on the user. For most simulations, the user need only specify
high level options in the hierarchy. NEMO provides default settings that
usually suffice for the lower-level parameter values. In practice, the
user can set the higher level options of interest and then select the
Finish button to accept the remaining default parameter settings. NEMO
arranges the simulation parameter levels in a scrollable table such that
the user can examine the parameter values. The Prev button moves
backwards to the previous simulation level. The Start button restarts
the input procedure at the top level.

### ![](media/image5.emf){width="3.25625in" height="3.39375in"}

While current vs. voltage is the most commonly viewed output, NEMO can
calculate a large range of device characteristics. The Output Options
panel shown in Fig. 7 displays selection toggle buttons to select any
combination of outputs. If an output requires more specific information
that needed for the original calculation, NEMO will create the
corresponding hierarchical level and display it in a subsequent panel.
![](media/image6.emf){width="2.8534722222222224in"
height="3.2680555555555557in"}

### Simulation parameters \-- graphical parameter entry

One of the key advantages of the NEGF formalism is that it can apply
different theoretical algorithms in different regions of the device. To
reduce the memory and calculation time, we usually limit the full
Green\'s function calculation to the region surrounding the quantum
wells and barriers and use the standard semi-classical approximations
elsewhere. The same approach is used to limit the calculations of energy
resonances and quantum charge to the areas of interest. These regions
are not necessarily coincident, but some regions must be contained
within other regions.

To set the proper location for these regions, NEMO displays the band
profile with pairs of movable cursors to indicate the boundary locations
(Fig. 6). The Fermi level is a movable cursor just as it is in the Main
Screen when set to Resonances mode. The real-time band profile
calculation provides visual feedback of the behavior of the bands with
bias. This capability is essential to estimate the best locations of the
region boundaries.

## Simulation library![](media/image7.emf){width="4.450694444444444in" height="3.479861111111111in"}![](media/image8.emf){width="4.893055555555556in" height="4.190972222222222in"}

The Simulation Library is a collection of simulations for a variety of
devices. The Simulation Library window (Fig. 8) displays a hierarchy of
material systems, devices, and models. The Comments text box explains
the application for the simulation selected in the Files list. Users are
requested to enter comments into every input file. The comments can be
browsed by the library dialog box and can be used to describe more
carefully the intended simulation. The library simulations provide an
excellent starting point for NEMO calculations. It is usually easier to
load one of the library examples and alter the device structure than to
create a device and input simulation parameters from scratch.

## Band structure dialog

As discussed earlier in this paper, NEMO includes a collection of band
structure models. To examine these models in detail, the Band Structure
dialog displays plots of band structure parameters in several forms,
including the energy bands, effective mass, and electron density as a
function of Fermi level. The example in Fig. 9 illustrates the
conduction, heavy-hole, light-hole and split-off bands for GaAs in the
\<100\> (positive k) and \<111\> (negative k) direction derived from an
sp^3^s^\*^ model including spin-orbit interaction.

![](media/image9.emf){width="5.51875in" height="4.800694444444445in"}

## Plot Features

### Plot Slicers

Most of the NEMO calculations generate a 2-D set of data for a given
bias point. This data is typically either plotted against position in
the device (e.g., band profile) or versus energy (e.g., transmission
coefficients). During a current vs. voltage scan, NEMO stores the data
for any given output into graphical tool called a Plot Slicer. The Plot
Slicer retains the calculation for each bias point. After the scan
finishes, the user can move the horizontal slide bar to view the
calculation results for any bias point. In effect, the Plot Slicer
displays cross-sections of a 3-dimensional data set. Calculated outputs
that vary with position can also be switched to show cross-sections in
which the slide bar moves between different positions and the data is
plotted vs. the applied bias.

The Plot Linker connects all of the active Plot Slicers to a single
slide bar. A cross-hairs cursor displays the corresponding position in
the current vs. voltage scan. As the main slide bar moves, all of the
other Plot Slicers move in unison to the same bias. The buttons marked
with \"\<\<\" and \"\>\>\" position the slide bar to voltages at local
minima and maxima of the current calculation. The Plot Linker is
extremely useful for analyzing device characteristics. Figure 10 shows
examples of Plot Slicers for charge and energy resonances of a 2-barrier
RTD linked to the I-V calculation.

### 3-D plots![](media/image10.emf){width="6.185416666666667in" height="2.9159722222222224in"}

Multiple 2-D PLOTS can be assembled into a 3-D plot. The 3-D plot tool
displays a surface plot of most of the NEMO data sets. The user has
control over many display attributes such as colors, grid density,
number of contours, etc. Figure 11(a) shows the band profile as a
function of position and voltage.

### Contour plots

The 3-D plot tool can also display contour plots of the same 3-D surface
data. Most of the same format controls for the 3-D tool apply to the
contour plot.

This sample plot in Fig. 11(b) shows the density of states for a
2-barrier RTD device as a function of energy and position.

# NEMO Simulations

The RTD device was chosen as an ideal \"straw man\" for the comparison
of calculation to experiment. An RTD utilizes quantum resonant states as
the basis of its operation, is relatively simple to construct, has as a
reasonably 1-dimensional geometry, and can operate at room temperature.
However, we also stressed that NEMO had to operate on any general
1-dimensional device. In the course of the NEMO program, we have
simulated HEMT, MOS, and Esaki diodes.

As examples we consider in the following sections high performance
room-temperature InP based RTDs, examine a scattering limited RTD in the
GaAs/AlAs material system and look at several MOS device simulations.

## High Performance Room Temperature InP-based RTDs

State-of-the-art InP-based RTDs are being developed for circuit
applications such as low power memory cells \[16\], high speed adders
\[17\] and high speed logic \[18\]. Depending on the application RTDs
must be designed to provide either a low \[16\] or a high \[17\] current
density. Quantitative modeling of such devices is expected to reduce the
device development cycle time significantly \[19\].
![](media/image15.emf){width="2.254861111111111in"
height="1.5833333333333333in"}![](media/image15.emf){width="2.267361111111111in"
height="1.5833333333333333in"}![](media/image15.emf){width="2.267361111111111in"
height="1.5833333333333333in"}![](media/image15.emf){width="2.267361111111111in"
height="1.5833333333333333in"}![](media/image15.emf){width="2.267361111111111in"
height="1.5833333333333333in"}![](media/image15.emf){width="2.267361111111111in"
height="1.5833333333333333in"}

We demonstrate NEMO\'s ability to quantitatively model pseudomorphic
InP-based RTDs which can include lattice matched In~0.53~Ga~0.47~As,
In~0.52~Al~0.48~As, and pseudomorphic AlAs. We present quantitative
simulations of strained and unstrained InP-based RTDs that include
quantum charge self-consistency (Hartree) in a full band (sp^3^s^\*^)
model. The quantitative modeling of these devices requires accurate
modeling of the bandstructure for each material. The strained AlAs
barriers and InAs wells have altered bandgaps, band-offsets, effective
masses and relative line-ups of Γ and X valley from their respective
bulk values. Important bandstructure features to be modeled are the
non-parabolicity of the In~0.53~Ga~0.47~As and InAs and the complex band
wrapping in the In~0.52~Al~0.48~As and AlAs barriers. All these features
combined required us to develop sp^3^s^\*^ bandstructure parameters
optimized for the conduction and valence bands including spin-orbit
interactions \[13\]. Not only is accurate bandstructure necessary, but a
self-consistent calculation of the electrostatic potential with the
quantum charge in both the quantized emitter and well states is required
for predictive accuracy. We find quantitative agreement between
simulation and experiment for a test matrix of unstrained InGaAs/InAlAs
(FIG. 12) and strained InGaAs/AlAs (FIG. 13) high current density RTDs
in which barrier, well, and spacer widths are varied systematically. The
material parameters are fixed and only structural changes from one
device to the next are changed for all the simulations. Our charge
self-consistency model is evaluated with structures which have
intentional barrier asymmetry. We quantitatively model the charge
accumulation/depletion inside an RTD in the forward/reverse bias
direction and achieve good agreement between theory and experiment.

Our comparisons between experimental and calculated I-Vs show that the
room temperature valley current of InGaAs/InAlAs and InGaAs/AlAs RTDs is
determined by thermionic emission through the first excited state rather
than incoherent scattering. Preliminary calculations of state-of-the-art
RTDs with the InAs notched well indicate that the regime has been
reached in which the room temperature valley current is determined by
incoherent scattering processes rather than thermionic emission. NEMO
can now be used to design devices quantitatively to address issues like
the improvement of the
peak-to-valley-ratio.![](media/image15.emf){width="2.267361111111111in"
height="1.5833333333333333in"}![](media/image15.emf){width="2.267361111111111in"
height="1.5833333333333333in"}![](media/image15.emf){width="2.267361111111111in"
height="1.5833333333333333in"}![](media/image15.emf){width="2.254861111111111in"
height="1.5833333333333333in"}![](media/image15.emf){width="2.254861111111111in"
height="1.5833333333333333in"}![](media/image15.emf){width="2.267361111111111in"
height="1.5833333333333333in"}

## Scattering-limited low temperature GaAs / AlAs RTD

RTD device performance is ultimately limited by the amplitude of the
off-resonant valley current due to incoherent scattering processes such
as polar optical phonon (POP), acoustic phonon, interface roughness (IR)
and alloy disorder scattering. The scattering mechanisms that give rise
to valley currents in RTDs have been studied using a number of different
approaches (for representative examples, see references \[20-22\]). Our
approach is based on a non-equilibrium Green function formalism which
includes scattering effects through self-energy terms in a truncated
self-consistent Born approximation \[2, 7\]. This algorithm is similar
to the multiple sequential scattering algorithm described by Roblin and
Liou \[21\]. We model interface roughness as a single layer of alloy
where the cations of a single species cluster into islands \[2, 8\]. A
screened bulk Fröhlich Hamiltonian is used to model the POP interaction
\[2\]. Since the self-energies have a strong momentum dependence, we
numerically integrate over the incident transverse momenta.

The GaAs-AlAs RTD simulated here was grown by molecular-beam-epitaxy
with 19.5 nm intrinsic GaAs spacer layers, 3.1 nm AlAs barriers, and a
6.2 nm GaAs well. The n^+^ contacts are Si doped at 10^18^ cm^-3^.
Measurements are made at a temperature of 4.2 K to eliminate tunneling
through X states in the AlAs barriers. The valley current in the region
of the I-V past the current turn-off is dominated by a
polar-optical-phonon assisted tunneling process illustrated in the inset
of Fig. 14(a). If we neglect polar optical phonon and interface
roughness scattering, the calculation of the valley current is too low
and the peak-to-valley ratio is too high by 3 orders of magnitude.
![](media/image16.wmf){width="3.0430555555555556in"
height="2.984027777777778in"}![](media/image17.wmf){width="3.0368055555555555in"
height="3.0125in"}

## MOS simulations

For the year 2010, the Semiconductor Industry Association Roadmap shows
a minimum gate length of 0.07 µm. Operating MOS devices with gate
lengths of 45 nm have recently been demonstrated \[23\]. As dimensions
are reduced, quantum mechanical effects influence the device
performance. The reduction in gate oxide thickness is limited by the
tunneling current through the oxide. For ultra-thin oxides, the quantum
charge distribution affects the gate capacitance and the threshold
voltage. Tunneling through the gate depletion region is a significant
contributor to the current for devices with ultra-short gate lengths of
25-30 nm.

Recent experimental work \[14, 24\] has been undertaken to characterize
the effective mass and barrier height of ultra-thin (1.6-3.5 nm) layers
of SiO~2~. The experimentally measured tunnel currents versus applied
voltage (I-V\'s) are shown in Fig. 15 for 4 different barrier
thicknesses. The simulated I-V\'s are also plotted. The simulations
accurately predict the tunnel current over a range of 10 orders of
magnitude through the Al / SiO~2~ / Si diodes.

In an inversion layer near a barrier such as SiO~2~, the actual charge
distribution can be significantly different from the semi-classically
calculated charge distribution. The difference is a quantum mechanical
effect. The actual charge is shifted away from the interface. This shift
increases the effective barrier width and reduces the gate capacitance.
The charge distributions calculated semi-classically and
quantum-mechanically are shown in Fig. 16. Detailed results can be found
in reference \[14\]. The distributions are significantly different in
the inversion layer. The semi-classical charge distribution of NEMO has
been compared for this structure to a calculation using MEDICI and
identical results have been
obtained.![](media/image18.wmf){width="3.102777777777778in"
height="2.7152777777777777in"}

The quantum effect on the charge distribution reduces the gate
capacitance in inversion since the charge is shifted away from the
barrier. Figure 17 shows the NEMO calculation of the capacitance for the
structure shown in Fig. 16 for the semi-classical and quantum-mechanical
charge distributions.

# Summary![](media/image19.wmf){width="3.063888888888889in" height="3.063888888888889in"}

In conclusion, we have created a comprehensive, versatile, and
user-friendly quantum device modeling tool. The tool has predictive
capability that we have tested against a number of different devices and
materials.

# References

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