Figure 2.2

Figure 2.3: This is a typical current density versus bias curve for a Double Barrier Resonant Tunneling Diode (DBRTD). Here a load line is shown as well. This is not quantitatively the load line used in this measurement 13.

Figure 2.4: This is a memory cell based upon a RTD using load line switching15.

Figure 2.5: These curves show several read write cycles of a QSD. The curves are grouped into states "1" and "2". Application of about 1.2 volts switches the device from curve "1" to curve "2". Application of about -1.2 volts switches the device from curve "2" to curve "1".

Figure 3.1: This is the two dimensional discretization scheme. dz and dy are node spacings in z and y, respectively. The model space is indexed in i along z and j along j. z_i,j is a solution at the node location (i,j).

Figure 3.2: This is the symmetric Lanczos algorithm34.

Figure 3.3: This is a flow chart of the process used to determine eigenvalue and eigenvectors.

Figure 3.4: This is the sparse matrix element structure.

Figure 3.5: This is the density of states (DOS) and transmission coefficint spectrum (t) at several locations in the DBRTD (Double Barrier Resonant Tunneling Diode) device shown above the graph. Curve 1 corresponds the beginning of the device at the contact, curve 2 corresponds to the end of the N+ region, curve 3 corresponds to the N- region adjacent to the barrier and curve 4 corresponds to the heterostructure quantum well. Note that the transmission coefficient in curve 5 peaks at about 0.2 eV. This coincides with the peak in the DOS spectrum of curve 4 which is the heterostructure quantum well. All other curves show a minimum at this energy indicating the electron lifetime is small except in the well. The other maxima and minima particularly in curve 1 are due to interference between incident wave and the wave reflected from the barrier. Here DOS is defined as G*G.

Figure 3.6: On the left is a self-consistent solver flow chart and on the right is an illustration of the convergence of the space charge and maximum potential update versus iteration. Positive space charge errors are symbolized by boxes and negative errors by circles. The + and - symbols show maximum potential update on each iteration. Note that after about 10 iterations the space charge error is ±10¹⁵ and the potential update is near zero. In each case there is some oscillation between negative and positive values.

Figure 3.7: The electron concentration profile of a wide DBRTD. Here the concentration on either end is in the contact region and in between concentration is in the heterostructure quantum well. This is a wide model with 565Å between nodes. The solution is similar to independent solutions at 565Å spacing across the device.

Figure 3.8: This is the structure of the two dimensional DBRTD model.

Figure 3.9 This is the concentration profile in a very narrow DBTRD. A barrier is used on the sides to simulate Fermi level pinning. The high concentration on either end is in the contact region. The N++ regions show lateral interference effects.

Figure 3.10: This is the self-consistent potential profile.

Figure 4.1: This is the zinceblend nearest neighbor structure. The light sphere is an anion ( a ) and the dark spheres are cations ( c ).

Figure 4.2 This is the bandstructure of the first conduction band in GaAs and AlAs. A valence band offset of 0.54 eV is used.

Figure 4.3: This is a plot of the transmission coefficient versus k|| and energy. k|| is varied from 0 on the left to 2/a_L on the right where a is the node spacing. Energy is varied from 0 in front to about 20kT (0.518 eV) in the back. The transmission coefficient is highly dependent on k||.

Figure 4.4: This is a comparison between the self-consistent simulations using the tight binding and effective mass approximation. These are two curves, dark for tight binding and light for effective mass. These two curves are nearly identical making the separate curves difficult to distinguish. No adjustable parameters are used to achieve a match beyond reasonable band and effective mass parameters. Here the density of states (DOS) of the first conduction band is used. The calculation using the total DOS gives the same results.

Figure 4.5: This concentration profile shows a comparison between tight binding and effective mass approximation simulations of a 100 Å AlAs barrier. The solid line shows the tight binding concentration which is about 7x10¹⁶ cm^-3 in the barrier. Since the effective mass waves are evanescent in the barrier, there is very little concentration in the dashed curve.

Figure 4.6: This is a single barrier device structure in the GaAs/AlAs materials system. Here the AlAs barrier is 14 Å or about 5 ML.

Figure 4.7: Potential profiles are shown on the left at zero and 0.3 volts bias. Electron concentration is shown to the right for these two cases. In both cases the solid curve is for the 0.3 volt bias case. Note concentration increase on upwind side of the barrier at position 85.

Figure 4.8: Current density versus bias is shown for 5 ML and 7 ML AlAs barriers. The dots are intermediate points as convergence occurs. These simulations are done at 77K.

Figure 4.9: This is the DBRTD device structure. It is a symmetric structure with a 50 Å heterostructure quantum well with 17Å AlAs barriers.

Figure 4.10: This is the density of states (DOS) and transmission (t) at several locations in the DBRTD device shown above the graph. Curve 1 corresponds to the beginning of the device at the contact, curve 2 corresponds to the end of the N++ region, curve 3 corresponds to the N- region adjacent to the barrier and curve 4 corresponds to the heterostructure quantum well. Note that the transmission coefficient in curve 5 peaks at about 0.18 eV. This coincides with the peak in the DOS spectrum of curve 4 which is the heterostructure quantum well. All other curves show a minimum at this energy indicating the electron lifetime is small except in the well. The other maxima and minima particularly in curve 1 are due to interference between incident wave and the wave reflected from the barrier. The transmission coefficient is larger than in Figure 3.5.

Figure 4.11: This figure shows the potential and concentration profile for this DBRTD. The solid curve is the tight binding approximation and the dashed curve is the effective mass approximation. Note that the concentration is very similar except in the barrier region where the tight binding concentration is larger, as expected. As a consequence the potential profile from the tight binding simulation is about 13% larger.

Figure 4.12: These are the tight binding simulation potential and concentration profiles at zero and 0.30 volts bias. Note the upwind potential barrier at position 40.

Figure 4.13: This is the DBRTD transmission spectrum. The resonance at about 0.18 eV is shown as well as at 0.36 eV and 0.42 eV. The resonance at 0.36 eV is a resonance-antiresonance pair (a Fanno resonance) caused by interference between G-G and G-X waves. This is confirmation of G-X band mixing.

Figure 4.14: This is a plot of current density versus bias voltage for this DBRTD using several assumptions. Curve 1 is a self-consistent simulation based on the effective mass approximation. Curve 2 is a non self-consistent tight binding simulation assuming a straight line potential approximation. Curve 3 is a non self-consistent tight binding simulation assuming a better potential approximation. Curve 4 is a self-consistent tight binding simulation.

Figure 4.15: This is a delta doped MODFET device structure.

Figure 4.16: The potential profile is shown to the left and the concentration profile is shown to the right. Two curves are shown. The dashed one is a Thomas Fermi simulation and the solid one is a tight binding simulation. Note the interference minimum that is located in the vacinity of the pulse doped region.

Figure 5.1: This is a concentration profile comparison between tight binding and coupled effective mass simulations. On the left the tight binding and coupled effective mass simulation with S_G,X = 0.42 are shown. On the right a logarithmic blow up of the concentration in the barrier region calculated by these two methods is shown. Coupled effective mass approximation based concentrations with S_G,X ranging from 0.1 to 0.5 are compared to concentration calculations using the tight binding approximation.

Figure 5.2: This is the potential profile of the coupled and tight binding simulations. The arrow shows the two best matches where the dark curve is the tight binding simulation and the light curve is the coupled effective mass simulation with S_G,X = 0.42.

Figure 5.3: This is a comparison between the tight binding and coupled effective mass simulations for an AlAs barrier of 17Å.

Figure 5.4: Using the coupling parameter S_G,X = 0.35 the concentration in the barrier region is a fairly good match. In the heterostructure itself the concentration is flat.

Figure 5.5: The potential profile using coupling is between the single valley effective mass and tight binding curves.

Figure 5.6: This is a comparison between transmission spectrum from the tight binding and coupled valley effective mass approximations. The light curve is from the tight binding simulation and the dark curves are coupled effective mass simulations. Note compared to the single valley effective mass spectrum these coupled effective mass simulations show transmission peaks related to G-X mixing. The low energy transmission spectrum is much lower despite the coupling parameters ranging from S_G,X = 0.1 to 0.25 tried. Note in addition that the transmission peaks are at a lower energy for the coupled effective mass case.

Figure 6.1: A QSD structure is shown on the left and currents from Schrödinger Poisson self-consistent simulations of this structure are shown on the right.

Figure 6.2: The Schrödinger Poisson conduction band edges are shown on the left with one potential solution about 0.53 eV below the other. This solution is below the contacts, resulting in quasi-bound states. The concentration in this region is 6.9x10¹⁷ cm^-3 lower. The difference in concentration in this region is 1.3x10¹³ cm^-3. For this zero bias case both solutions are globally space charge neutral so that the difference in concentration is made up elsewhere in the device.

Figure 6.3: This parameter study shows the relationship between the N++ layer concentration and the ratio between the currents of the two solutions. The concentrations for the bar chart are from left to right 1e18, 2e18, 4e18, 5e18, 6e18, and 8e18. The 4e18 is the best choice below the solid solubility limit in GaAs. This suggests other materials that support greater concentrations might have larger current ratios.

Figure 6.4: A parameter study is done to determine the width of the N+ and N- regions. Current ratios are shown in the bar chart for N+ and N- widths of 170Å, 136Å, 101Å, 51Å, and 34Å. Best current ratios are shown for N+ width of 101Å and width of 34Å to 50Å.

Figure 6.5: This is a suggested device structure with 100Å N++ region.

Figure 6.6: The potential profile in the graph on top shows three self-consistent solutions. The solutions with lower potentials in the N++ regions around positions 80 and 140 might be assumed to have lower energy. The lower potential solutions have more oscillatory concentration profiles in these regions. The character of the solutions in the N++ regions suggest a function of DeBrogle wavelengths62.

Figure 6.7: The plot on the left shows the potential profile for the device structure in Figure 6.4. The switched state is shown on the right. The resonant structures are very similar.

Figure 6.8: The second solution shown on the right side of Figure 6.7 contains quasi-bound states. The quasi-bound states occur at about -0.02 ev. The concentration in those quasi-bound states is about 8.9x10¹⁷ cm^-3 as shown above. The two curves, solid and dashed, are from eigenvectors corresponding to slightly different eigenvalues for the two layers.

Figure 6.9: The graph on the left shows the density of states (DOS) spectrum for the N++ layer and the heterostructure quantum well (HQW). The transmission coefficient (t) spectra is also shown. The transmission resonance peak is at about 0.18 eV for the solution on the left and 0.16 eV for the solution on the right. These correspond to the left and right solutions above, respectively. At these transmission resonances there is a DOS node elsewhere in the device. The resonance at 0.038 eV in the solution on the left peaks in the N++ regions and is diminished but still a resonance elsewhere. Resonances also occur at about 0.078 eV in the left solution and at about 0.08 eV on the right solution and are diminished in the heterostructure quantum well.

Figure 6.10: G, X, and L log concentration profiles are shown. Using this coupling parameter the potential profile is very similar to the G only case.

Figure 6.11: This is the tight binding DOS and transmission spectra for a tight binding simulation. These curves are very similar to those shown with the effective mass approximation in Figure 6.9. Transmission coefficients are generally higher at low energy than with the effective mass approximation. The solution on the left has similar resonances except that the HQW peak is at about 0.2 eV which is about 0.02ev above the corresponding solution in Figure 6.9. This is due to the higher potential in that portion of the device for the tight binding solution. The resonances on the right are shifted down about 0.015 eV from those on the right side in Figure 6.9.

Figure 6.12: This is a memory circuit implemented with a three terminal QSD (TQSD). 63

Figure 6.13: This plot shows the advantage of on/off resistance ratio in the performance of a resistance based memory cell.

Figure 6.14: This is a three terminal triple barrier device structure.

Figure 6.15: This is a triple barrier device structure. It is similar to the read terminal structure in the figure above.

Figure 6.16: These are current ratios between solutions for a range of device structure parameters. This suggests AlAs barriers of 17Å and QW width of 50Å or 100 Å.

Figure 6.17: On the left are potential profiles from self-consistent Schrödinger Poisson solutions and on the right are concentration profiles. Curve 1 is a solution with quasi-bound state solutions and curve 2 is a solution without quasi-bound state solutions. The top two graphs are based on the effective mass approximation and the bottom two are based on the tight binding approximation.

Figure 6.18: The current density versus bias voltage plot shows two solutions. The inset region on the box in the curve on the left is shown in the curve on the right. Curve 1 is the quasi-bound state solution, curve 2 is the solution not supporting quasi-bound states. Curves 3 and 4 are tight binding solutions.

Figure 6.19: Current density versus bias curves measured in the laboratory show two solutions. Both solutions show inflections suggesting resonance peaks and valleys. A discontinuity is also shown at about 0.6 volts bias.

Table 4.2: The valence band offset at the GaAs/AlAs heterostructure interface at room temperature using the simple ratio.