3.3. Tutorial 3: Two-dimensional pn junction with a grain boundary¶

In this tutorial we show how to build a two-dimensional $pn^{+}$ junction containing a grain boundary.

3.3.1. Building a two-dimensional system¶

We want to simulate a two-dimensional $pn^{+}$ junction (homojunction) with a columnar grain boundary as depicted below.

As usual, we start by importing the sesame and numpy packages. We construct the mesh of the system and make an instance of the Builder(). Notice that in this case we provide the Builder() function with both x and y grids; this automatically tells the code to build a two-dimensional system:

import sesame
import numpy as np

# dimensions of the system
Lx = 3e-4 # [cm]
Ly = 3e-4 # [cm]

# position of p-n junction [cm]
junction = .1e-4

# Mesh
x = np.concatenate((np.linspace(0, .2e-4, 30, endpoint=False),      # mesh near the contact
                np.linspace(0.2e-4, 1.4e-4, 50, endpoint=False),    # mesh in depletion region
                np.linspace(1.4e-4, 2.7e-4, 50, endpoint=False),    # mesh in bulk
                np.linspace(2.7e-4, 2.98e-4, 30, endpoint=False),   # mesh near the GB end point
                np.linspace(2.98e-4, Lx, 10)))                    # mesh near the contact

y = np.concatenate((np.linspace(0, 1.25e-4, 50, endpoint=False),
                np.linspace(1.25e-4, 1.75e-4, 50, endpoint=False),  # mesh near the GB core
                np.linspace(1.75e-4, Ly, 50)))

# Create a system
sys = sesame.Builder(x, y)

We define and add a material as before:

# Dictionary with the material parameters
mat = {'Nc':8e17, 'Nv':1.8e19, 'Eg':1.5, 'affinity': 3.9, 'epsilon':9.4,
        'mu_e':200, 'mu_h':200, 'tau_e':10e-9, 'tau_h':10e-9, 'Et':0}

# Add the material to the system
sys.add_material(mat)

We next define functions delimiting the regions with different doping values. Because the model is 2-dimensional, the function input argument pos is a tuple of the form (x,y). In the functions below, we first “unpack” the x and y coordinate, and determine the location of the x-coordinate relative to the junction:

# Add the donors
def n_region(pos):
    x, y = pos
    return x < junction

nD = 1e17  # [cm^-3]
sys.add_donor(nD, n_region)

# Add the acceptors
def p_region(pos):
    x, y = pos
    return x >= junction

nA = 1e15  # [cm^-3]
sys.add_acceptor(nA, p_region)

We specify contacts as before:

# Use perfectly selective Ohmic contacts
sys.contact_type('Ohmic', 'Ohmic')
Sn_left, Sp_left, Sn_right, Sp_right = 1e7, 0, 0, 1e7
sys.contacts(Sn_left, Sp_left, Sn_right, Sp_right)

Note

Sesame assumes that the contact properties (e.g. recombination velocity, metallic work function) are uniform along the y-direction.

3.3.2. Adding a grain boundary¶

Now we add a line of defects to simulate a grain boundary using the sys method add_line_defects(). The necessary inputs are the grain boundary defect’s electrical properties (e.g. capture cross sections, energy level, defect density, and charge states), and the endpoints defining the grain boundary location (recall a grain boundary is represented by a line in a 2-dimensional simulation). Below we show code defining these properties for our example, and adding the grain boundary to the simulation:

# gap state characteristics
rho_GB = 1e14               # defect density [1/cm^2]
S_GB = 1e-15                # trap capture cross section [cm^2]
E_GB = 0.4                  # energy of gap state (eV) from intrinsic energy level


# Specify the two points that make the line containing additional charges
p1 = (.1e-4, 1.5e-4)   # [cm]
p2 = (2.9e-4, 1.5e-4)  # [cm]

# Add the line of defects to the system
sys.add_line_defects([p1, p2], rho_GB, S_GB, E=E_GB, transition=(1/-1))

The type of the charge transition $\alpha/\beta$ is specified by assigning the transition input value as shown above. In our example we chose a mixture of donor and acceptor at energy E. An acceptor would be described by (-1,0) and a donor by (1,0).

Note

Avoid adding charges on the contacts of the system, as these will not be taken into account. The code is not equiped to deal with such boundary conditions.
In order to add another gap state at a different energy at the same location, one repeats the exact same process.
Here we assumed equal electron and hole surface recombination velocities. The function add_line_defects() takes two surface recombination velocities as argument. The first is for electrons, the second for holes. To use different values write
```
sys.add_line_defects([p1, p2], rho_GB, Sn_GB, Sp_GB, E=E_GB)
```
A continuum of states can be considered by omitting the energy argument above. The density of states can be a callable function or a numerical value, in which case the density of states is independent of the energy.

3.3.3. Computing the IV curve¶

The computation of the IV curve proceeds as in the previous tutorials. We show the code below:

# Solve equilibirum problem first
solution = sesame.solve(sys, 'Poisson')


# define a function for generation profile
f = lambda x, y: 2.3e21*np.exp(-2.3e4*x)
# add generation to the system
sys.generation(f)

# Specify applied voltages
voltages = np.linspace(0, .9, 10)
# Compute IV curve
j = sesame.IVcurve(sys, voltages, '2dGB_V', guess=solution)
# rescale to dimension-ful current
j = j * sesame.scaling.current

# Save the computed IV data
result = {'voltages':voltages, 'j':j}
np.save('2dGB_IV', result)

3.3.4. Plotting system variables¶

The 2-dimensional solutions can be plotted with tools we describe more fully in tutorial 4. As a preview, we list the commands for loading and plotting the electrostatic potential:

sys, results = sesame.load_sim('2dGB_V_0.gzip')
sesame.plot(sys, results['v'])

The output is shown below:

Note

As discussed more fully in Tutorial 4, quantities in Sesame are dimensionless by default. The electrostatic potential shown above is dimensionless, scaled by the thermal voltage. The scaling field of sys provides the relevant quantites needed to rescale quantities to dimension-ful form.

3.3. Tutorial 3: Two-dimensional pn junction with a grain boundary¶

3.3.1. Building a two-dimensional system¶

3.3.2. Adding a grain boundary¶

3.3.3. Computing the IV curve¶

3.3.4. Plotting system variables¶

3.3.5. Spatial variation of material parameters¶

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3.3. Tutorial 3: Two-dimensional pn junction with a grain boundary¶

3.3.1. Building a two-dimensional system¶

3.3.2. Adding a grain boundary¶

3.3.3. Computing the IV curve¶

3.3.4. Plotting system variables¶

3.3.5. Spatial variation of material parameters¶

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