You are viewing the documentation for an older COMSOL version. The latest version is available here.
PDF
Double Barrier 1D
This tutorial uses a simple 1D model of a double barrier structure to illustrate the essential steps to perform quantum mechanical simulations. Topics discussed include the eigenenergies and time evolution of quasi bound states, resonant tunneling phenomenon, and transmission versus energy curves. The numerical results agree very well with analytical solutions.
Introduction
The double barrier structure is of interest both because of its pedagogical function as an example to demonstrate the concepts of tunneling and quasi bound stats to students of quantum mechanics, as well as its practical application in semiconductor devices such as resonant-tunneling diodes (Ref. 1).
To keep the relevance to semiconductor physics without losing generality, the model is assumed to be constructed from layers of GaAs and AlGaAs. The envelope function approximation (Chap. 1, Sec. 1.7 in Ref. 2) can be employed to justify the use of the single-particle Schrödinger equation. Note that not only the electron potential energy, but also the effective mass, is a function of material composition. To properly take care of the discontinuity in the effective mass, the Schrödinger Equation physics interface applies the BenDaniel-Duke boundary condition (Chap. 2, Sec. 2.6 in Ref. 2) by default.
Model Definition
The fraction of Al in the AlGaAs material is assumed to be 0.32. The width of the well is assumed to be 10 nm and the width of the barriers 5 nm. The modeled domain has a width of 40 nm.
Four studies are set up to discuss different aspects related to the double barrier structure. First, the eigenenergies are solved for the quasi bound states using an eigenvalue study, with open boundary conditions for outgoing waves at both ends of the modeling domain. Then, the time evolution of one of the quasi bound states is solved in a time-dependent study. Next, the resonant tunneling condition is solved in an eigenvalue study, with a special type of open boundary condition for incoming waves on one end of the modeling domain. Finally, the transmission and reflection coefficients are computed using a stationary study, with regular open boundary conditions and a prescribed incoming wave from one end of the modeling domain.
Results and Discussion
Figure 1 summarizes the result for the quasi bound states: the electron potential energy (black solid line), the real part of the eigenenergies (black dotted line), and the real and imaginary parts of the wave functions shifted vertically to center around their eigenenergies (color solid and dashed curves).
Figure 1: Summary plot for the quasi bound states, including the electron potential energy (black solid line), the real part of the eigenenergies (black dotted line), and the real and imaginary parts of the wave functions shifted vertically to center around their eigenenergies (color solid and dashed curves).
The eigenenergies can be computed analytically using the transfer matrix method (Chap. 2, Sec. 2.10 in Ref. 2). We found that the numerical solution obtained using the eigenvalue study agree with the analytical solution to at least 6 significant digits for both the real and the imaginary part of the eigenenergies.
The time evolution of the quasi bound states can be explicitly solved using a time-dependent study with one of the quasi bound states’ wave function as the initial condition. The result shows the wave function leaking out of the modeling domain as one would expect. It is best visualized with an animated movie as discussed in the Modeling Instructions below.
The decay of the total probability over time can be plotted as in Figure 2. The computed decay curve matches well with the analytical curve calculated from the imaginary part of the analytical eigenenergy.
Figure 2: The computed decay of the total probability over time matches well with the analytical solution.
Figure 3 shows the summary of the resonant tunneling condition. The wave functions and the eigenenergies closely resemble those of the quasi bound states, as one would expect. In this symmetric double barrier structure, the transmission is 100% under the resonant tunneling condition, and the eigenenergies are real valued. The numerical solution agrees with the analytically solution to at least 7 digits for the eigenenergies.
Figure 3: Summary plot of the resonant tunneling condition.
The final stationary study solves for the transmission versus energy curve by sweeping the total energy in the stationary Schrödinger equation. Figure 4 summarizes the wave functions for five different energies. The first, third, and fifth one are at resonant tunneling condition, and the wave functions resemble the ones from the previous study as expected.
The second and fourth one are at in-between energies, where the transmission probability is very small and most of the wave function is reflected back to the left-hand side. This is clearly seen in the large amplitude of the wave function on the left and very small amplitude on the right.
Figure 4: Stationary study solutions of five selected energies.
Figure 5 shows the computed transmission and reflection coefficients versus energy in colored curves and the analytical transmission coefficients in red circles. The agreement is very good.
Figure 5: Transmission curve matches very well with analytical result.
References
1. S. M. Sze and K. K. Ng, Physics of Semiconductor Devices, 3rd ed., John Wiley & Sons, Inc., 2007, Chap. 8, Sec. 8.4.
2. P. Harrison, Quantum Wells, Wires and Dots, 3rd ed., John Wiley & Sons, Inc., 2009.
Application Library path: Semiconductor_Module/Verification_Examples/double_barrier_1d
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  1D.
2
In the Select Physics tree, select Semiconductor>Schrödinger Equation (schr).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select Preset Studies for Selected Physics Interfaces>Eigenvalue.
6
Click  Done.
Geometry 1
The Model Wizard starts the COMSOL Desktop with the Geometry node selected. We can take the chance to set the length unit to a more convenient one.
1
In the Model Builder window, under Component 1 (comp1) click Geometry 1.
2
In the Settings window for Geometry, locate the Units section.
3
From the Length unit list, choose nm.
Now set up some global parameters.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file double_barrier_1d_param.txt.
Next draw the geometry for the double barrier structure.
Geometry 1
Interval 1 (i1)
1
In the Model Builder window, under Component 1 (comp1) right-click Geometry 1 and choose Interval.
2
In the Settings window for Interval, locate the Interval section.
3
From the Specify list, choose Interval lengths.
4
In the Left endpoint text field, type -L/2.
5
In the table, enter the following settings:
Lengths (nm)
lw
lb
L-2*lw-2*lb
lb
lw
6
Click  Build All Objects.
Now we set up the physics. First enter the eigenvalue scale, which has the unit of energy. The eigenvalue returned by the eigenvalue study step is unitless. The eigenenergy is then given by the product of the eigenvalue and the eigenvalue scale.
Schrödinger Equation (schr)
1
In the Model Builder window, under Component 1 (comp1) click Schrödinger Equation (schr).
2
In the Settings window for Schrödinger Equation, locate the Model Properties section.
3
Find the Eigenvalue study subsection. In the λscale text field, type lmbd0.
Then enter the effective mass and potential energy in the well regions. The selections are default to all domains but we will modify the barrier regions in later steps.
Effective Mass 1
1
In the Model Builder window, under Component 1 (comp1)>Schrödinger Equation (schr) click Effective Mass 1.
2
In the Settings window for Effective Mass, locate the Effective Mass section.
3
In the meff,e,11 text field, type mw.
Electron Potential Energy 1
1
In the Model Builder window, click Electron Potential Energy 1.
2
In the Settings window for Electron Potential Energy, locate the Electron Potential Energy section.
3
From the Ve list, choose User defined. In the associated text field, type 0.
Now we enter the effective mass and potential energy in the barrier regions.
Effective Mass 2
1
In the Physics toolbar, click  Domains and choose Effective Mass.
2
Select Domains 2 and 4 only.
3
In the Settings window for Effective Mass, locate the Effective Mass section.
4
In the meff,e,11 text field, type mb.
Note the red triangle appearing on the Effective Mass 1 node where we set the effective mass in the well regions to all domains in an earlier step. This red triangle indicates that some selections in that node (Effective Mass 1) are overridden by the current node (Effective Mass 2).
We can see this by clicking on the Effective Mass 1 node and observe that domains 2 and 4 in the selection box are now appended with "(overridden)".
Effective Mass 1
Also note the red triangle appearing on the Effective Mass 2 node, indicating the source of the overriding feature.
Electron Potential Energy 2
1
In the Physics toolbar, click  Domains and choose Electron Potential Energy.
2
Select Domains 2 and 4 only.
3
In the Settings window for Electron Potential Energy, locate the Electron Potential Energy section.
4
From the Ve list, choose User defined. In the associated text field, type Vb.
Note the orange circle now appearing on the Electron Potential Energy 1 node, indicating that the potential energy entered at the current node will be added to that node. This is called a "contributing" feature in COMSOL. In brief, an "overriding" feature (such as the Effective Mass node) replaces the value of the physical quantity being entered, and a "contributing" feature (such as the Electron Potential Energy node) adds to the value of the physical quantity being entered. Thanks to the "contributing" nature of the Electron Potential Energy node, a complicated potential profile can be conveniently built by using multiple nodes with overlapping selections.
For quasi bound states, we look for solutions with wave functions leaking out of the modeling domain without reflection back from the end points of the modeling domain. This is done by using the Open Boundary condition.
Open Boundary 1
1
In the Physics toolbar, click  Boundaries and choose Open Boundary.
2
In the Settings window for Open Boundary, locate the Boundary Selection section.
3
From the Selection list, choose All boundaries.
This boundary condition only applies to exterior boundaries, so we just selected "All boundaries" for convenience.
For verification purposes, we set up a fine mesh for better accuracy in the solution.
Mesh 1
Edge 1
In the Mesh toolbar, click  Edge.
Distribution 1
1
Right-click Edge 1 and choose Distribution.
2
In the Settings window for Distribution, locate the Distribution section.
3
In the Number of elements text field, type 50.
Now we set up the eigenvalue study. The Open Boundary condition introduces nonlinearity in the eigenvalue problem. This usually requires an iterative process, where the solution from a previous eigenvalue study step is entered as the linearization point for the next iteration, until the eigenvalue is converged. Here we take a shortcut by feeding the analytical solution as the linearization point.
Study 1
Step 1: Eigenvalue
1
In the Model Builder window, under Study 1 click Step 1: Eigenvalue.
2
In the Settings window for Eigenvalue, locate the Study Settings section.
3
In the Desired number of eigenvalues text field, type 1.
4
In the Search for eigenvalues around text field, type reE_anal/lmbd0.
As mentioned earlier, the eigenenergy is given by the product of the eigenvalue and the eigenvalue scale. So here the analytical eigenenergy is divided by the eigenvalue scale to obtain the guess for the eigenvalue.
Show default solver to access the setting for the linearization point.
5
In the Model Builder window, click Study 1.
6
In the Settings window for Study, type Study 1 Quasi bound states in the Label text field.
Solution 1 (sol1)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 1 (sol1) node, then click Eigenvalue Solver 1.
3
In the Settings window for Eigenvalue Solver, locate the General section.
4
In the Value of eigenvalue linearization point text field, type reE_anal/lmbd0.
Use a parametric sweep to solve for all three quasi bound states in a sequence.
Parametric Sweep
1
In the Study toolbar, click  Parametric Sweep.
2
In the Settings window for Parametric Sweep, locate the Study Settings section.
3
Click  Add twice.
4
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
reE_anal (Sweeping parameter for quasi bound states)
reE1anal reE2anal reE3anal
J
imE_anal (Sweeping parameter for quasi bound states)
imE1anal imE2anal imE3anal
J
The parameter imE_anal is not used in the solution process. The purpose of including it in the parametric sweep is to easily compare with the analytical solution once the numerical solution is obtained.
5
In the Study toolbar, click  Compute.
Results
Normalized Wave Function (schr)
The modeling result for the quasi bound states is summarized in the 1D plot group Potential Energy, Eigenenergy, and Wave Function.
Quasi bound state summary plot
1
In the Model Builder window, under Results click Potential Energy, Eigenenergy, and Wave Function (schr).
2
In the Settings window for 1D Plot Group, type Quasi bound state summary plot in the Label text field.
3
Click to expand the Title section. From the Title type list, choose Manual.
4
In the Title text area, type Summary plot for the quasi bound states.
5
Locate the Plot Settings section.
6
Select the x-axis label check box. In the associated text field, type X (nm).
7
Select the y-axis label check box. In the associated text field, type Energy (eV).
8
Locate the Legend section. Clear the Show legends check box.
9
In the Quasi bound state summary plot toolbar, click  Plot.
We can also perform some global evaluations to compare the result with analytical solutions.
Eigenvalue
1
In the Model Builder window, expand the Results>Derived Values node, then click Eigenvalue.
2
In the Settings window for Global Evaluation, locate the Expressions section.
3
In the table, enter the following settings:
Expression
Unit
Description
real(schr.Ei)/reE_anal-1
1
error of real part
imag(schr.Ei)/imE_anal-1
1
error of imaginary part
4
Click  Evaluate.
Table
1
Go to the Table window.
We see that both the real and the imaginary part of the eigenenergy agree with the analytical solution to 6 or more significant digits.
The wave functions of the quasi bound states leak out of the double-barrier quantum well over time, and the probability density diminishes correspondingly. We can see this behavior explicitly by running a transient study. Add a time-dependent study with the initial condition set to the normalized wave function of the third quasi bound state.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select General Studies>Time Dependent.
4
Click Add Study in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 2
Step 1: Time Dependent
1
In the Settings window for Time Dependent, click to expand the Values of Dependent Variables section.
2
Find the Initial values of variables solved for subsection. From the Settings list, choose User controlled.
3
From the Study list, choose Study 1 Quasi bound states, Eigenvalue.
4
In the Model Builder window, click Study 2.
5
In the Settings window for Study, type Study 2 Time evolution of the 3rd quasi bound state in the Label text field.
Schrödinger Equation (schr)
Initial Values 2 for time dependent study
1
In the Physics toolbar, click  Domains and choose Initial Values.
2
In the Settings window for Initial Values, type Initial Values 2 for time dependent study in the Label text field.
3
Locate the Domain Selection section. From the Selection list, choose All domains.
4
Locate the Initial Values section. In the psi text field, type schr.Psi.
The periodic cycle time T3 and the decay time constant tau3 for the 3rd quasi bound state have been computed in the global parameters table. Set up the time steps accordingly, so that the time increment is small enough to show the oscillatory behavior of the wave function and the total simulation time is long enough to show the decay of the quasi bound state.
Study 2 Time evolution of the 3rd quasi bound state
Step 1: Time Dependent
1
In the Model Builder window, under Study 2 Time evolution of the 3rd quasi bound state click Step 1: Time Dependent.
2
In the Settings window for Time Dependent, locate the Study Settings section.
3
In the Output times text field, type range(0,T3/10,tau3/3).
4
In the Home toolbar, click  Compute.
Results
Wave Function (schr)
The time evolution of the wave function and the probability density can be easily shown with COMSOL’s animation functionality.
Animation 1
1
In the Wave Function (schr) toolbar, click  Animation and choose Player.
2
In the Settings window for Animation, locate the Frames section.
3
From the Frame selection list, choose All.
4
Click the  Play button in the Graphics toolbar.
The outward propagating (leaking) of the wave function is clearly seen in the animation.
Animation 2
1
Right-click Animation 1 and choose Duplicate.
2
In the Settings window for Animation, locate the Scene section.
3
From the Subject list, choose Probability Density (schr) 1.
4
Click the  Play button in the Graphics toolbar.
The decay of the probability density is also clearly seen in the animation.
We can also compare the decay of the total probability using a global plot. First remove some default plots to shorten the list of plot nodes in the Model Builder tree.
Potential Energy (schr) 1
In the Model Builder window, under Results right-click Potential Energy (schr) 1 and choose Delete.
Effective Mass (schr) 1
In the Model Builder window, right-click Effective Mass (schr) 1 and choose Delete.
Compare decay of total probability
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type Compare decay of total probability in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 2 Time evolution of the 3rd quasi bound state/Solution 6 (sol6).
4
Locate the Title section. From the Title type list, choose Manual.
5
In the Title text area, type Decay of the total probability.
6
Locate the Plot Settings section.
7
Select the y-axis label check box. In the associated text field, type Total probability.
Global 1
1
Right-click Compare decay of total probability and choose Global.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
schr.int(schr.Pr)
1
computed decay curve
exp(-t/tau3)^2
 
analytical decay curve
4
Click to expand the Coloring and Style section. Find the Line style subsection. From the Line list, choose Cycle.
5
In the Compare decay of total probability toolbar, click  Plot.
In the next study, we look for the resonant tunneling solution. Unlike the quasi bound states, where the wave function propagates outward on both open boundaries, for the resonant tunneling condition, the wave function propagates inward from one boundary and goes out of the other boundary. We will modify the boundary condition accordingly by overriding the left open boundary condition.
Schrödinger Equation (schr)
Open Boundary 2 for resonant tunneling study
1
In the Physics toolbar, click  Boundaries and choose Open Boundary.
2
In the Settings window for Open Boundary, type Open Boundary 2 for resonant tunneling study in the Label text field.
3
Select Boundary 1 only.
To reverse the wave function propagation direction across the open boundary, first turn on Advanced Physics Options, since this option is rarely used and normally hidden.
4
Click the  Show More Options button in the Model Builder toolbar.
5
In the Show More Options dialog box, in the tree, select the check box for the node Physics>Advanced Physics Options.
6
Click OK.
Now we can change the propagation direction.
7
In the Model Builder window, click Open Boundary 2 for resonant tunneling study.
8
In the Settings window for Open Boundary, click to expand the Open Boundary Type section.
9
From the list, choose Incoming.
Set up an eigenvalue study to solve for the resonant tunneling energies, following a procedure similar to the one for Study 1.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select Preset Studies for Selected Physics Interfaces>Eigenvalue.
4
Click Add Study in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 3
Step 1: Eigenvalue
1
In the Settings window for Eigenvalue, locate the Study Settings section.
2
In the Desired number of eigenvalues text field, type 1.
3
In the Search for eigenvalues around text field, type E_anal/lmbd0.
4
Locate the Physics and Variables Selection section. Select the Modify model configuration for study step check box.
5
In the tree, select Component 1 (comp1)>Schrödinger Equation (schr)>Initial Values 2 for time dependent study.
6
Click  Disable.
7
In the Model Builder window, click Study 3.
8
In the Settings window for Study, type Study 3 Resonant tunneling in the Label text field.
Solution 7 (sol7)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 7 (sol7) node, then click Eigenvalue Solver 1.
3
In the Settings window for Eigenvalue Solver, locate the General section.
4
In the Value of eigenvalue linearization point text field, type E_anal/lmbd0.
Parametric Sweep
1
In the Study toolbar, click  Parametric Sweep.
2
In the Settings window for Parametric Sweep, locate the Study Settings section.
3
Click  Add.
4
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
E_anal (Sweeping parameter for transmission=1 solutions)
E1anal E2anal E3anal
J
5
In the Study toolbar, click  Compute.
Results
Normalized Wave Function (schr) 1
Remove some default plots and examine the main summary plot.
1
Right-click Results>Normalized Wave Function (schr) 1 and choose Delete.
Effective Mass (schr) 1, Potential Energy (schr) 1, Probability Density (schr) 2
1
In the Model Builder window, under Results, Ctrl-click to select Probability Density (schr) 2, Potential Energy (schr) 1, and Effective Mass (schr) 1.
2
Right-click and choose Delete.
Resonant tunneling summary plot
1
In the Model Builder window, under Results click Potential Energy, Eigenenergy, and Wave Function (schr).
2
In the Settings window for 1D Plot Group, type Resonant tunneling summary plot in the Label text field.
3
Locate the Title section. From the Title type list, choose Manual.
4
In the Title text area, type Summary plot for the resonant tunneling condition.
5
Locate the Plot Settings section.
6
Select the x-axis label check box. In the associated text field, type X (nm).
7
Select the y-axis label check box. In the associated text field, type Energy (eV).
8
Locate the Legend section. Clear the Show legends check box.
9
In the Resonant tunneling summary plot toolbar, click  Plot.
As in the case of quasi bound states, here we can also perform some global evaluations to compare the result with analytical solutions.
Eigenvalue 1
1
In the Model Builder window, under Results>Derived Values click Eigenvalue 1.
2
In the Settings window for Global Evaluation, locate the Expressions section.
3
In the table, enter the following settings:
Expression
Unit
Description
real(schr.Ei)/E_anal-1
1
error of real part
imag(schr.Ei)/E_anal
1
error of imaginary part (exact imaginary part is zero)
4
Click  Evaluate.
Table
1
Go to the Table window.
We see that the real part of the eigenenergy agrees with the analytical solution to 7 or more significant digits. The imaginary part is essentially zero within numerical precision, as one would expect from the fact that the exact solution is real valued.
The resonant tunneling condition discussed above are at specific energies with 100% transmission of the incoming wave function through the double barrier. To obtain the general curve of the transmission versus the energy, we solve the Schrödinger equation using a stationary study. First parameterize the energy of the incoming wave function.
Schrödinger Equation (schr)
1
In the Model Builder window, under Component 1 (comp1) click Schrödinger Equation (schr).
2
In the Settings window for Schrödinger Equation, locate the Model Properties section.
3
Find the Stationary study subsection. In the E text field, type E0.
Then set up an ordinary open boundary condition (outgoing type by default). Add an incoming wave with a given amplitude psi0.
Open Boundary 3 for transmission vs. energy study
1
In the Physics toolbar, click  Boundaries and choose Open Boundary.
2
In the Settings window for Open Boundary, type Open Boundary 3 for transmission vs. energy study in the Label text field.
3
Select Boundary 1 only.
4
Locate the Open Boundary section. Select the Incoming wave check box.
5
In the ψ0 text field, type psi0.
Now set up a stationary study with an auxiliary sweep over the energy parameter E0. The list of values for E0 is chosen to capture the very sharp peaks in transmission around the resonant tunneling energies.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select General Studies>Stationary.
4
Click Add Study in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 4
Step 1: Stationary
1
In the Settings window for Stationary, locate the Physics and Variables Selection section.
2
Select the Modify model configuration for study step check box.
3
In the tree, select Component 1 (comp1)>Schrödinger Equation (schr)>Initial Values 2 for time dependent study.
4
Click  Disable.
5
Click to expand the Study Extensions section. Select the Auxiliary sweep check box.
6
Click  Add.
7
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
E0 (Energy for stationary study)
E1anal*(1-0.5*10^range(0,-0.2,-3)) E1anal E1anal+0.5*(E2anal-E1anal)*10^range(-3,0.2,-0.1) E2anal-0.5*(E2anal-E1anal)*10^range(0,-0.2,-3) E2anal E2anal+0.5*(E3anal-E2anal)*10^range(-3,0.2,-0.1) E3anal-0.5*(E3anal-E2anal)*10^range(0,-0.2,-3) E3anal E3anal*(1+0.1*10^range(-3,0.2,0))
J
8
In the Model Builder window, click Study 4.
9
In the Settings window for Study, type Study 4 Transmission vs. energy in the Label text field.
10
In the Home toolbar, click  Compute.
Results
Wave Function (schr) 1
Delete some default plots and examine the main summary plot at a few different energies.
1
Right-click Results>Wave Function (schr) 1 and choose Delete.
Effective Mass (schr) 1, Potential Energy (schr) 1, Probability Density (schr) 2
1
In the Model Builder window, under Results, Ctrl-click to select Probability Density (schr) 2, Potential Energy (schr) 1, and Effective Mass (schr) 1.
2
Right-click and choose Delete.
Transmission summary plot
1
In the Model Builder window, under Results click Potential Energy, Energy, and Wave Function (schr).
2
In the Settings window for 1D Plot Group, type Transmission summary plot in the Label text field.
3
Locate the Data section. From the Parameter selection (E0) list, choose Manual.
4
In the Parameter indices (1-97) text field, type 17 33 49 65 81.
5
Locate the Title section. From the Title type list, choose Manual.
6
In the Title text area, type Summary plot for the scattering problem.
7
Locate the Plot Settings section.
8
Select the x-axis label check box. In the associated text field, type X (nm).
9
Select the y-axis label check box. In the associated text field, type Energy (eV).
10
Locate the Legend section. Clear the Show legends check box.
11
In the Transmission summary plot toolbar, click  Plot.
The first, third, and last plotted energies are chosen to be the three resonant tunneling energies. It can be seen that the wave functions at these energies resemble the eigenfunctions from the previous study, which they should. The second and fourth plotted energies are between the resonant tunneling energies. From the profiles of the wave functions it can be seen that the transmission coefficient is very small at these energies.
We can also plot the transmission and reflection coefficients as functions of the energy, and compare with analytical results.
Reflection & Transmission vs. Energy
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type Reflection & Transmission vs. Energy in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 4 Transmission vs. energy/Solution 12 (sol12).
Global 1
1
Right-click Reflection & Transmission vs. Energy and choose Global.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
abs(schr.open3.int(psi/schr.psiI)-1)^2
1
Reflection coeff.
abs(schr.open1.int(psi)/psi0)^2
1
Transmission coeff.
4
Locate the x-Axis Data section. From the Unit list, choose eV.
5
In the Reflection & Transmission vs. Energy toolbar, click  Plot.
6
Click the  y-Axis Log Scale button in the Graphics toolbar.
Reflection & Transmission vs. Energy
1
In the Model Builder window, click Reflection & Transmission vs. Energy.
2
In the Settings window for 1D Plot Group, locate the Title section.
3
From the Title type list, choose Manual.
4
In the Title text area, type Reflection and transmission coeff. vs. energy.
5
Locate the Plot Settings section.
6
Select the x-axis label check box. In the associated text field, type Energy (eV).
7
Select the y-axis label check box. In the associated text field, type Reflection and transmission coeff..
8
Locate the Legend section. From the Position list, choose Lower right.
The very sharp peaks in transmission corresponding to the resonant tunneling condition are clearly seen in the plot.
Now import the analytical transmission coefficients and plot them on the same graph to compare with the numerical results.
Analytical transmission coefficients
1
In the Results toolbar, click  Table.
2
In the Settings window for Table, type Analytical transmission coefficients in the Label text field.
3
Locate the Data section. Click Import.
4
Browse to the model’s Application Libraries folder and double-click the file double_barrier_1d_anal.csv.
5
Locate the Column Headers section. In the table, enter the following settings:
Column
Header
1
E0 (eV)
2
Analytical transmission coeff.
Table
1
Go to the Table window.
2
Click Table Graph in the window toolbar.
Results
Table Graph 1
In the Model Builder window, under Results>1D Plot Group 14 right-click Table Graph 1 and choose Copy.
Table Graph 1
In the Model Builder window, right-click Reflection & Transmission vs. Energy and choose Paste Table Graph.
1D Plot Group 14
In the Model Builder window, under Results right-click 1D Plot Group 14 and choose Delete.
Table Graph 1
1
In the Model Builder window, under Results>Reflection & Transmission vs. Energy click Table Graph 1.
2
In the Settings window for Table Graph, locate the Coloring and Style section.
3
Find the Line style subsection. From the Line list, choose None.
4
Find the Line markers subsection. From the Marker list, choose Circle.
5
Click to expand the Legends section. Select the Show legends check box.
6
In the Reflection & Transmission vs. Energy toolbar, click  Plot.
It can be seen that the numerical and analytical transmission coefficients match very well.
This tutorial has been built incrementally, with new boundary and/or domain conditions added for subsequent studies. Therefore at this point if we try to solve one of the previous studies, it may not solve or may not give the correct answer, due to the change in boundary and/or domain conditions. This can be fixed by disabling the unwanted new boundary and/or domain conditions in each previous study step.
Study 1 Quasi bound states
Step 1: Eigenvalue
1
In the Model Builder window, under Study 1 Quasi bound states click Step 1: Eigenvalue.
2
In the Settings window for Eigenvalue, locate the Physics and Variables Selection section.
3
Select the Modify model configuration for study step check box.
4
In the tree, select Component 1 (comp1)>Schrödinger Equation (schr)>Initial Values 2 for time dependent study.
5
Click  Disable.
6
In the tree, select Component 1 (comp1)>Schrödinger Equation (schr)>Open Boundary 2 for resonant tunneling study.
7
Click  Disable.
8
In the tree, select Component 1 (comp1)>Schrödinger Equation (schr)>Open Boundary 3 for transmission vs. energy study.
9
Click  Disable.
Study 2 Time evolution of the 3rd quasi bound state
Step 1: Time Dependent
1
In the Model Builder window, under Study 2 Time evolution of the 3rd quasi bound state click Step 1: Time Dependent.
2
In the Settings window for Time Dependent, locate the Physics and Variables Selection section.
3
Select the Modify model configuration for study step check box.
4
In the tree, select Component 1 (comp1)>Schrödinger Equation (schr)>Open Boundary 2 for resonant tunneling study.
5
Click  Disable.
6
In the tree, select Component 1 (comp1)>Schrödinger Equation (schr)>Open Boundary 3 for transmission vs. energy study.
7
Click  Disable.
Study 3 Resonant tunneling
Step 1: Eigenvalue
1
In the Model Builder window, under Study 3 Resonant tunneling click Step 1: Eigenvalue.
2
In the Settings window for Eigenvalue, locate the Physics and Variables Selection section.
3
In the tree, select Component 1 (comp1)>Schrödinger Equation (schr)>Open Boundary 3 for transmission vs. energy study.
4
Click  Disable.