1. Poisson and Schrödinger simulation of a nanowire quantum dot

1.1. Requirements

1.1.1. Software components

  • QTCAD

  • Gmsh

1.1.2. Geometry file

  • qtcad/examples/tutorials/meshes/nanowire.geo

1.1.3. Python script

  • qtcad/examples/tutorials/nanowire.py

1.1.4. References

1.2. Briefing

In this tutorial, we explain how to perform a basic electrostatic calculation for conduction electrons at equilibrium in a nanowire quantum dot. It is assumed that the user is familiar with basic Python. It is also assumed that the user has access to the Gmsh geometry file describing the nanowire device to simulate. Geometry files for all the QTCAD tutorials are stored in the folder examples/tutorials/meshes.

The full code may be found at the bottom of this page, or in qtcad/examples/tutorials/nanowire.py.

Do not hesitate to look into the API source code for detailed descriptions of the inputs and outputs of each class, method, or function used in the tutorials. The source code also contains documentation for many other useful features not described here.

1.3. Mesh generation

The mesh is generated from the geometry file qtcad/examples/tutorials/meshes/nanowire.geo by running Gmsh. This can be done in the terminal with

gmsh nanowire.geo

This wil produce a .msh file in the same directory as the geometry file. See Arbitrary geometry definition with Gmsh for more information about interfacing with Gmsh.

1.4. Setting up the device and solving the electrostatic problem

1.4.2. Input parameters

# Define some device parameters
Vgs = 0.5 # Gate-source bias
Ew = mt.Si.chi + mt.Si.Eg/2 # Metal work function
Ltot = 20e-9 # Device height in m
radius = 2.5e-9 # Device radius in m

The first line defines the gate-to-source bias \(V_{GS} = V_G - V_S\), which is applied at the device boundary wrapped around the oxide shell. The next line defines the work function of the metal forming the gate contact surrounding the transistor. This work function is chosen to be equal to the work function of intrinsic silicon at zero temperature. Finally, the last two parameters are the device height and radius; these quantities are not used as inputs to the simulation (the geometry being entirely defined in nanowire.geo), but are defined for later plotting purposes.

In addition to the device parameters above, paths to various input and output files are defined, which will be used below.

# Paths to mesh file and output files
script_dir = pathlib.Path(__file__).parent.resolve()
path_mesh = script_dir / "meshes" / "nanowire.msh"
path_hdf5 = script_dir / "output" / "nanowire.hdf5"
path_vtu = script_dir / "output" / "nanowire.vtu"
path_energies = script_dir / "output" / "nanowire_energies.txt"
path_linecuts_z = script_dir / "output" / "nanowire_linecuts_z.txt"
path_linecuts_x = script_dir / "output" / "nanowire_linecuts_x.txt"

1.4.3. Defining the mesh

The mesh is defined by loading the mesh file produced by Gmsh and setting units of length

# Load the mesh
scaling = 1e-9
mesh = Mesh(scaling, str(path_mesh))

Note the two arguments of the constructor of the Mesh class: scaling and str(path_mesh). While str(path_mesh) is simply the string pointing at the mesh file to use, scaling indicates a number by which all coordinates in the mesh file should be multiplied. Since the code uses SI units, the value chosen here (1e-9) indicates that lengths in the mesh file are nanometers.

The mesh may also be displayed using:

# Plot the mesh
mesh.show()

This will call the Visualization ToolKit (VTK) through its Python wrapper (PyVista) to produce a 3D representation of the mesh, like this:

Mesh illustration

Fig. 1.4.1 Mesh illustration

1.4.4. Creating the device

The device to simulate is then initialized on the mesh defined above:

# Create device from mesh and set statistics.
# It is important to set statistics before creating boundaries since
# they determine the boundary values. Failing to do so will produce
# inconsistent results.
d = Device(mesh)
d.statistics = "FD_approx"  # Analytic approximation to Fermi-Dirac statistics (default)

The first line calls the constructor of the device class with the mesh object as an argument. The second line specifies which statistical model is used to calculate classical carrier densities at equilibrium. Choices are "MB" (Maxwell–Boltzmann), "FD" (Fermi–Dirac), and "FD_approx" (approximate Fermi–Dirac). The first choice ("MB") is simple and efficient but inaccurate, and leads to poor convergence at strong doping densities. The second choice ("FD") uses Fermi–Dirac statistics by calculating Fermi–Dirac integrals numerically. This is the most accurate choice, but imposes a significant computational overhead. The third choice ("FD_approx", which is also the default value) considers Fermi–Dirac statistics, but uses analytic approximations of the Fermi–Dirac integrals that provide a relative error \(\leq 0.4 \%\) on carrier densities to speed up calculations. This reduced accuracy is rarely a limitation in practice. For more information on analytic approximations of the Fermi–Dirac integrals, see

    1. Gao et al. J. Appl. Phys. 114, 164302 (2013).

    1. Bednarczyk and J. Bednarczyk, Phys. Lett. A 64, 409 (1978).

    1. Van Halen and D. L. Pulfrey. J. Appl. Phys. 57, 5271 (1985).

Note

Since we have not specified any temperature here, the device temperature will take its default value of \(300~\mathrm{K}\). Temperature may be set with the set_temperature method of the Device class. Note that in all QTCAD simulations, temperature is assumed to be uniform.

Note

Upon instantiation of the Device object, the following warning is printed: The attribute conf_carriers of the device object was not specified; conf_carriers will be set to 'e'. The Device attribute conf_carriers specifies which charge carriers (electrons or holes) are to be treated quantum-mechanically by the various quantum solvers of QTCAD, e.g., the Schrödinger equation solver. By default, conf_carriers is set to 'e', meaning that electrons but not holes are treated quantum-mechanically. This warning may be turned off by instantiating the Device object d through d = Device(mesh, conf_carriers="e").

1.4.5. Specifying physical regions and boundaries

It is now time to specify the value of device parameters (doping density, effective mass, permittivity, etc.) at each point of the mesh over which the device is build. This may be done explicitly for each node of the mesh; however this approach is tedious and produces code that is prone to bugs and difficult to read. A better way is to leverage physical volumes and surfaces defined in the geometry file to define regions and boundaries.

Regions are a collection of elements that belong to a given 3D physical group. A physical group is a set of elements associated with an integer or string label in the geometry file. It is possible to define regions based on physical groups, and specify a material for such a region. In the present example, this is done as follows

# Create regions first.
# Make sure each element is assigned to a region, otherwise default
# (silicon) parameters are used
d.new_region('channel_ox',mt.SiO2)
d.new_region('source_ox',mt.SiO2)
d.new_region('drain_ox',mt.SiO2)
d.new_region('channel', mt.Si)
d.new_region('source', mt.Si, pdoping=5e20*1e6, ndoping=0)
d.new_region('drain', mt.Si, pdoping=5e20*1e6, ndoping=0)

The first three lines define the oxide that surrounds the channel, source, and drain regions, respectively, using their string labels defined in the geometry file (and stored in the .msh file). The material used is silicon dioxide (mt.SiO2). The last three lines define the semiconductor (silicon) regions, and set the source and drain to be positively doped with a doping density of \(5 \times 10^{20} \textrm{ cm}^{-3}\).

Remark: When writing geometry files, it is important to make sure that all elements in a given mesh belong to one and only one region. If an element belongs to two or more regions at the same time, an exception will be raised. If an element does not belong to any region, then its nodes will be given default properties (those of intrinsic silicon).

In addition to 3D physical groups (physical volumes), Gmsh supports 2D physical groups (physical surfaces), which also associate certain two-dimensional elements to a label. These may be leveraged to define boundaries as follows:

# Then create boundaries
d.new_ohmic_bnd('source_bnd')
d.new_ohmic_bnd('drain_bnd')
d.new_gate_bnd('gate_bnd',Vgs,Ew)

This creates two Ohmic boundaries located at the ends of the source and drain regions, respectively, along with a gate boundary at the surface that wraps around the channel. As in defining regions, the first argument is the string label used in Gmsh. For the gate boundary, the second argument is the applied potential (which is added to the built-in potential calculated internally using the relevant physical model). The method that creates a gate boundary also requires a third argument, i.e. the metal work function.

1.4.6. Creating and running the Poisson solver

Once the device and its parameters are defined, a non-linear Poisson solver can be instantiated and executed. In the header of the current file, we have imported the Solver class of the poisson module. We have also imported the SolverParams class of the poisson module, which enables us to configure the Poisson solver. A solver object is then instantiated with

# Create and configure the non-linear Poisson solver
params_poisson = PoissonSolverParams()
params_poisson.tol = 1e-5 # Convergence threshold (tolerance) for the error
s = PoissonSolver(d, solver_params=params_poisson)

The first line instantiates an object used to define the parameters of the non-linear Poisson solver. The second line sets the error tolerance on the electric potential in volts, i.e. the error threshold below which the non-linear Poisson self-consistent loop is considered to have converged. The last line instantiates the Poisson solver, with parameters specified using the two previous lines.

Note that the solver is instantiated with the device d as input argument, enabling the solver to correctly calculate the potentials and charge densities appropriate to the geometry, mesh, materials, and doping levels stored in this object. This also indicates that results of the calculations will be stored in this device.

Once the non-linear Poisson solver is configured, the problem is simply solved with

# Self-consistent solution
s.solve()

This will launch the following self-consistent loop: 1. The electric potential is set to the default initial guess (charge-neutral configuration). 2. The equilibrium electron and hole distribution resulting from this electric potential is calculated following the carrier statistics model chosen above ("FD_approx"). 3. The electric potential is calculated again by solving the Poisson equation resulting from the charge distribution corresponding to appropriate electron, hole, and ionized dopant concentrations. 4. Steps 2 and 3 are iterated until the potential varies by less than the tolerance between successive iterations.

1.5. Saving and visualizing results

1.5.1. Spatially-dependent variables

We may save the value of spatially-dependent variables (carrier densities) at each mesh point as follows

# To save throughout the device: n, p, phi, EC, EV
arrays_dict = {"n": d.n/1e6, "p": d.p/1e6, "phi": d.phi,
    "EC": d.cond_band_edge()/ct.e, "EV": d.vlnce_band_edge()/ct.e}
io.save(str(path_hdf5), arrays_dict)

The first line creates a dictionary of arrays to save, i.e. a set of objects labeled by strings. Each object is an array giving the value of a variable at each local node in the mesh. Variables considered here are, respectively, the electron density, the hole density, the electric potential, the conduction band edge, and the valence band edge (we convert units of density into \(\textrm{cm}^{-3}\), and units of energy into \(\textrm{eV}\)). Note that the format in which data is to be saved is determined automatically from the extension of the file in the input path, here .hdf5, as dictated by the path_hdf5 variable.

The second line saves all these quantities in the file whose path is path_hdf5. Specific variables stored in a Python array (e.g. the electric potential) may subsequently be loaded from this data file simply with

# A given variable may then be loaded with
phi = io.load(str(path_hdf5), "phi")

As will be seen later, it is possible to visualize results directly in QTCAD. However, for more extensive 3D data visualization with a Graphical User Interface, it is useful to export the data into a .vtu file, which may then be loaded in ParaView, a free, open-source, and multi-platform data analysis and visualization application. Saving into .vtu format may be done with:

# These variables may also be saved in vtu format for external visualization
io.save(str(path_vtu), arrays_dict, mesh)

When saving into .vtu format, it is important to specify the mesh, since the files will contain information about the geometry. In addition, because ParaView does not easily handle distances with very small numerical values (~1e-9), distances in the .vtu file are rescaled to be expressed with respect to a chosen unit of length, which may be set using the keyword argument length_unit. Available values for this argument are 'nm', 'um', 'mm', and 'm'. The default unit of length in output .vtu files is 'nm'.

1.5.2. Linecuts

For 3D devices, it is often useful to consider variables along specific linecuts. The following code block shows how to save the variables defined above along two linecuts: one that corresponds to the symmetry axis (along \(z\)) of the nanowire, and one that cuts through the device along the direction perpendicular to the symmetry axis in the middle of the channel.

# Linecuts along z with x=y=0 for the same quantities
# Save n, p and phi throughout the device and along z axis
arrays = [d.n, d.p, d.phi]
array_names = ["n", "p", "phi"]
analysis.save_linecut(str(path_linecuts_z), mesh, arrays,
    (0,0,0), (0,0,Ltot), array_names=array_names)

# Linecuts along x with z=L/2 and y=0 for the same quantities
analysis.save_linecut(str(path_linecuts_x), mesh, arrays,
    (-radius,0,Ltot/2.), (radius,0,Ltot/2.), array_names=array_names)

1.5.3. Plotting linecuts of the results

The data saved in .vtu format may easily be imported and plotted in ParaView. In addition, for 2D plots, it is straightforward to use the standard Python library Matplotlib. However, QTCAD also includes some simple visualization functions that are useful to plot results on the fly.

For example, a linecut of the electric potential may be directly plotted with:

# Plot linecut of the potential along the z axis
analysis.plot_linecut(mesh,d.phi,(0,0,0),(0,0,Ltot),
    title='Electric potential (V)')

which produces:

Electric potential

Fig. 1.5.1 Electric potential

The following code also enables to plot the band diagram across a given linecut

# Plot band diagram
analysis.plot_bands(d, (0,0,0), (0,0,Ltot))

The result is

Band diagram

Fig. 1.5.2 Band diagram

1.6. Solving Schrödinger’s equation

1.6.1. Setting up the confinement potential and the Schrödinger solver

We first specify which charge carrier (electrons: 'e', holes: 'h') shall be treated as confined in this simulation. Looking at the above band diagram, it is clear that the electrons are the confined carriers since there is a potential well in the conduction band edge for electrons, but a potential barrier in the valence band edge for holes.

In addition, we must retrieve the confinement potential for the chosen charge carrier resulting from the electric potential calculated with the Poisson solver.

The above two actions are completed with

# Calculate the confinement potential from the electric potential
# found above, and set the confined carriers to be electrons
# The confinement potential is stored in d.V
d.conf_carriers = 'e'
d.set_V_from_phi()

The next step is to instantiate the Schrödinger solver and its parameters. This is very similar to what was done above for the Poisson solver.

# Then instantiate the Schrodinger solver and its parameters
params_schrod = SchrodingerSolverParams()
params_schrod.tol = 1e-5        # Tolerance on energies in eV
params_schrod.num_states = 5    # Number of states to consider
s = SchrodingerSolver(d, solver_params=params_schrod)

In the first line, we instantiate a SolverParams object appropriate to Schrödinger solvers. The second line sets the tolerance on energies in electron-volts. The third line specifies the number of eigenstates and energies to consider (starting from and including the ground state) in the diagonalization of the device Hamiltonian. The fourth line instantiates the Schrödinger solver based on the device object (which contains the confinement potential in the attribute d.V) and on the solver parameters.

1.6.2. Solving Schrödinger’s equation and viewing results

We then solve Schrödinger’s equation with

# Solve Schrodinger's equation
s.solve()

As before, the results of the calculation are stored in relevant device attributes, here d.energies and d.eigenfunctions. The attribute d.energies contains all the eigenenergies in SI units in ascending order from the ground state. The attribute d.eigenfunctions is a 2D array whose first index (rows) corresponds to global nodes in the mesh, and whose second index (columns) corresponds to the eigenenergies, in ascending order.

The energy levels may be printed on the screen and saved in a text file using

# Output and save energy levels
d.print_energies()
d.save_energies(str(path_energies))

For the current example, this yields

Energy levels (eV)
[0.54798018 0.61320023 0.68496648 0.7631776  0.84786669]

The energies are expressed with respect to the Fermi level (set to zero throughout the device at equilibrium).

In addition, we may compute and print the geometric properties of the quantum dot using the analyze_dot function of the analysis module:

# Get and print the quantum-dot geometric properties
dot_props = analysis.analyze_dot(d, verbose=True)

These properties are stored in the dictionary dot_props. If the verbose keyword argument is set to True, the function will also print the properties on the screen. For the current example, this yields:

--------------------------------------------------------------------------------
Geometric properties of the quantum dot:
--------------------------------------------------------------------------------
position        :  [ 2.70608902e-13 -4.88703648e-13  9.99954680e-09]
std             :  [5.84702875e-10 5.84499147e-10 9.32973459e-10]
size            :  [2.33881150e-09 2.33799659e-09 3.73189383e-09]
--------------------------------------------------------------------------------

Here, the length units are in meters. Definitions of dot position and size are given in the API reference of the analyze_dot function.

1.7. Visualizing results in 3D

For detailed and flexible 3D visualization, it is best to export the quantities to be visualized in .vtu format (see above), and open them in ParaView (see Visualizing QTCAD quantities with ParaView).

However, it is often convenient to produce quick and dirty plots directly from the QTCAD Python API at the end of a simulation. This may be done through functions stored in the analysis module, which wrap relevant PyVista functions.

For example, density plots of each variable of interest may be displayed on orthogonal planar slices inside the device with:

# Plot p and phi (with n and p converted to cm^-3)
# on orthogonal planar slices inside the device
analysis.plot_slices(mesh,d.p/1e6,title='p (cm^-3)')
analysis.plot_slices(mesh,d.phi,title='phi (V)')

Above, we multiply the densities by 1e-6 to convert from SI units to CGS.

The output for the hole density looks like this:

Classical hole density

Fig. 1.7.1 Classical hole density

We may then plot the first few electron envelope functions with

# Plot ground and first few excited state wavefunctions along the same slices
analysis.plot_slices(mesh,d.eigenfunctions[:,0],title="Ground state")
analysis.plot_slices(mesh,d.eigenfunctions[:,1],title="First excited state")
analysis.plot_slices(mesh,d.eigenfunctions[:,2],title="Second excited state")

This results in the three following figures.

Ground state

Fig. 1.7.2 Ground state

First excited state

Fig. 1.7.3 First excited state

Second excited state

Fig. 1.7.4 Second excited state

In Quantum transport—Master equation, we will see how to use the confinement potential obtained here to define a many-body problem, and compute a charge-stability diagram for quantum transport by sequential tunneling through the nanowire system.

1.8. Full code

__copyright__ = "Copyright 2024, Nanoacademic Technologies Inc."

from qtcad.device import constants as ct
from qtcad.device.mesh3d import Mesh
from qtcad.device import io
from qtcad.device import analysis
from qtcad.device import materials as mt
from qtcad.device import Device
from qtcad.device.poisson import Solver as PoissonSolver
from qtcad.device.schrodinger import Solver as SchrodingerSolver
from qtcad.device.poisson import SolverParams as PoissonSolverParams
from qtcad.device.schrodinger import SolverParams as SchrodingerSolverParams
import pathlib

# Define some device parameters
Vgs = 0.5 # Gate-source bias
Ew = mt.Si.chi + mt.Si.Eg/2 # Metal work function
Ltot = 20e-9 # Device height in m
radius = 2.5e-9 # Device radius in m

# Paths to mesh file and output files
script_dir = pathlib.Path(__file__).parent.resolve()
path_mesh = script_dir / "meshes" / "nanowire.msh"
path_hdf5 = script_dir / "output" / "nanowire.hdf5"
path_vtu = script_dir / "output" / "nanowire.vtu"
path_energies = script_dir / "output" / "nanowire_energies.txt"
path_linecuts_z = script_dir / "output" / "nanowire_linecuts_z.txt"
path_linecuts_x = script_dir / "output" / "nanowire_linecuts_x.txt"

# Load the mesh
scaling = 1e-9
mesh = Mesh(scaling, str(path_mesh))

# Plot the mesh
mesh.show()

# Create device from mesh and set statistics.
# It is important to set statistics before creating boundaries since
# they determine the boundary values. Failing to do so will produce
# inconsistent results.
d = Device(mesh)
d.statistics = "FD_approx"  # Analytic approximation to Fermi-Dirac statistics (default)

# Create regions first.
# Make sure each element is assigned to a region, otherwise default
# (silicon) parameters are used
d.new_region('channel_ox',mt.SiO2)
d.new_region('source_ox',mt.SiO2)
d.new_region('drain_ox',mt.SiO2)
d.new_region('channel', mt.Si)
d.new_region('source', mt.Si, pdoping=5e20*1e6, ndoping=0)
d.new_region('drain', mt.Si, pdoping=5e20*1e6, ndoping=0)

# Then create boundaries
d.new_ohmic_bnd('source_bnd')
d.new_ohmic_bnd('drain_bnd')
d.new_gate_bnd('gate_bnd',Vgs,Ew)

# Create and configure the non-linear Poisson solver
params_poisson = PoissonSolverParams()
params_poisson.tol = 1e-5 # Convergence threshold (tolerance) for the error
s = PoissonSolver(d, solver_params=params_poisson)

# Self-consistent solution
s.solve()

# To save throughout the device: n, p, phi, EC, EV
arrays_dict = {"n": d.n/1e6, "p": d.p/1e6, "phi": d.phi, 
    "EC": d.cond_band_edge()/ct.e, "EV": d.vlnce_band_edge()/ct.e}
io.save(str(path_hdf5), arrays_dict)

# A given variable may then be loaded with
phi = io.load(str(path_hdf5), "phi")

# These variables may also be saved in vtu format for external visualization
io.save(str(path_vtu), arrays_dict, mesh)

# Linecuts along z with x=y=0 for the same quantities
# Save n, p and phi throughout the device and along z axis
arrays = [d.n, d.p, d.phi]
array_names = ["n", "p", "phi"]
analysis.save_linecut(str(path_linecuts_z), mesh, arrays,
    (0,0,0), (0,0,Ltot), array_names=array_names)

# Linecuts along x with z=L/2 and y=0 for the same quantities
analysis.save_linecut(str(path_linecuts_x), mesh, arrays,
    (-radius,0,Ltot/2.), (radius,0,Ltot/2.), array_names=array_names)

# Plot linecut of the potential along the z axis
analysis.plot_linecut(mesh,d.phi,(0,0,0),(0,0,Ltot),
    title='Electric potential (V)')

# Plot band diagram
analysis.plot_bands(d, (0,0,0), (0,0,Ltot))

# Calculate the confinement potential from the electric potential
# found above, and set the confined carriers to be electrons
# The confinement potential is stored in d.V
d.conf_carriers = 'e'
d.set_V_from_phi()

# Then instantiate the Schrodinger solver and its parameters
params_schrod = SchrodingerSolverParams()
params_schrod.tol = 1e-5        # Tolerance on energies in eV
params_schrod.num_states = 5    # Number of states to consider
s = SchrodingerSolver(d, solver_params=params_schrod)

# Solve Schrodinger's equation
s.solve()

# Output and save energy levels
d.print_energies()
d.save_energies(str(path_energies))

# Get and print the quantum-dot geometric properties
dot_props = analysis.analyze_dot(d, verbose=True)

# Plot p and phi (with n and p converted to cm^-3)
# on orthogonal planar slices inside the device 
analysis.plot_slices(mesh,d.p/1e6,title='p (cm^-3)')
analysis.plot_slices(mesh,d.phi,title='phi (V)')

# Plot ground and first few excited state wavefunctions along the same slices
analysis.plot_slices(mesh,d.eigenfunctions[:,0],title="Ground state")
analysis.plot_slices(mesh,d.eigenfunctions[:,1],title="First excited state")
analysis.plot_slices(mesh,d.eigenfunctions[:,2],title="Second excited state")