Support

Support Options

Submit a Support Ticket

 

AQME Advancing Quantum Mechanics for Engineers

by Tain Lee Barzso, Dragica Vasileska, Gerhard Klimeck

Version 116
by Gerhard Klimeck
Version 190
by Michael Anderson

Deletions or items before changed

Additions or items after changed

1 -
== AQME ==
+
[[Image(aqme_large.gif, class=align-center)]]
2 +
== Introduction to Advancing Quantum Mechanics for Engineers and Physicists ==
3
4 -
[[Image(aqme_small.png, 450 class=align-center)]]
+
"Advancing Quantum Mechanics for Engineers" (AQME) toolbox is an assemblage of individually authored tools that, used in concert, offer educators and students a one-stop-shop for semiconductor education. The AQME toolbox holds a set of easily employable nanoHUB tools appropriate for teaching a quantum mechanics class in either engineering or physics. Users no longer have to search the nanoHUB to find the appropriate applications for discovery that are related to quantum mechanics; users, both instructors and students, can simply log in and take advantage of the assembled tools and associated materials such as homework or project assignments.
5
6 +
Thanks to its contributors, nanoHUB users and AQME’s toolbox have benefited tremendously from the hard work invested in tools development. Simulation runs performed using the AQME tools are credited to the individual tools, and count toward individual tool rankings. Uses of individual tools within the AQME tool set are also counted, to measure AQME impact and to improve the tool. On their respective pages, the individual tools are linked to the AQME toolbox.
7
8 +
Participation in this open source, interactive educational initiative is vital to its success, and all nanoHUB users can:
9
10 -
Introduction to Quantum Mechanics for Physicists and Engineers with AQME
+
*[http://www.nanohub.org/contribute/ Contribute content] to AQME by uploading it to the nanoHUB. (See “Contribute>Contribute Content” on the nanoHUB mainpage.) Tagging contributions with “AQME” will effect an association with this initiative and, because the toolbox is actively managed, such contributions may also may be added to the toolbox.
11 -
Nanotechnology has yielded a number of unique structures that are not found anywhere in nature. Most demonstrate an essential quality of Quantum Mechanics known as quantum confinement. The idea behind confinement is all about keeping electrons trapped in a small area. The sizes we're talking about here for confinement have to be less than 30 nm for effective confinement. Quantum confinement comes in several flavors. 2-D confinement is only restricted in one dimension, and the result is a quantum well (or plane). These are what most lasers are currently built from. 1-D confinement occurs in nanowires. 0-D confinement is found only in the quantum dot.
+
12
13 +
* Provide feedback for the items you use in AQME and on nanoHUB.org through the review system. (Please be explicit and provide constructive feedback.)
14
15 -
One is probably wondering why confinement is so important. For one thing, it leads to new electronic properties that are not present in today's semiconductor devices. Consider the quantum dot. The typical quantum dot is anywhere between 3-60 nm in diameter. That's still 30 to 600 times the size of a typical atom. A quantum dot exhibits 0-D confinement, meaning that electrons are confined in all three dimensions. The only things in nature that have 0-D confinement are atoms. So a quantum dot can be loosely described as an 'artificial atom'. This is vitally important because we can't readily experiment on regular atoms. They're too small and too difficult to isolate in an experiment. Quantum dots, on the other hand, are large enough to be manipulated by magnetic fields and can even be moved around with an STM or AFM. We can deduce many important atomistic characteristics from a quantum dot that would otherwise be impossible to research in an atom.
+
* Let us know when things do not work by filing a ticket via the nanoHUB "Help" feature on every page.
16
17 -
Confinement also increases the efficiency of today's electronics. The laser is based on a 2-D confinement layer that is usually created with some form of epitaxy like Molecular Beam Epitaxy or Chemical Vapor Deposition. The bulk of modern lasers created with this method are highly functional, but ultimately inefficient in terms of energy consumption and heat dissipation. Moving to 1-D confinement in wires or 0-D confinement in quantum dots allows for higher efficiencies and brighter lasers. Quantum dot lasers are currently the best lasers available though their fabrication is still being worked out.
+
* Finally, let us know what you are doing and submit[http://www.nanohub.org/feedback/suggestions/ your suggestions] improving the nanoHUB by using the "Feedback" section, which you can find under "[http://www.nanohub.org/support/ Support]"
18 -
Confinement is just one manifestation of quantum mechanics in nanodevices. Tunneling and quantum interference are the other two manifestations of quantum mechanics in the operation of, for example, scanning tunneling microscopes and resonant tunneling diodes, respectively.
+
19
20 -
Because of the importance of understanding quantum mechanics to understand the operation of nanoscale devices, almost every Electrical Engineering department in which there is a strong nanotechnology experimental or theoretical group and all Physics departments teach the fundamental principles of quantum mechanics and its application to nanodevice research. Within these courses one is first introduced to the concept of particle-wave duality (the photoelectric effect and the double-slit experiment), the solutions of the time-independent Schrodinger equation for open systems (piece-wise constant potentials), tunneling and bound states. The description of the solution of the Schrodinger equation for periodic potentials (Kronig-Penney model) naturally follows from the discussion of double well, triple well and n-well structures. This leads the students to the concept of energy bands and energy gaps and the concept of the effective mass that can be extracted from the precalculated bandstructure by fitting the curvature of the bands. The Tsu-Esaki formula is then derived so that having calculated the transmission coefficient one can calculate the tunneling current in resonant tunneling diode and Esaki diode. After establishing basic principles of quantum mechanics, the harmonic oscillator problem is then discussed in conjunction with understanding vibrations of a crystalline lattice and the concept of phonons is introduced as well as the concept of creation and annihilation operators. The typical quantum mechanics class for undergraduate/first year graduate students is then completed with the discussion of the stationary and time dependent perturbation theory and the derivation of the Fermi Golden Rule which is used as a starting point of a graduate level class in semiclassical transport. Yet another issue that is discussed sometimes in a typical quantum mechanics class is the concept of Coulomb Blockade.
21
22 -
AQME assembles a set of nanoHUB tools that we believe are of immediate interest for the teaching of quantum mechanics class for both Engineers and Physicists. Users no longer have to search the nanoHUB to find the appropriate applications for this particular purpose. This curated page provides a “on-stop-shop” access to associated materials such as homework or project assignments.
+
Finally, be sure to share AQME and other nanoHUB success stories; the nanotechnology community and its supporters need to hear of nanoHUB's impact.
23 -
We invite you to participate in this open source, interactive educational initiative:
+
24
25 -
* Contribute your content by uploading it to the nanoHUB. (See “Contribute Content”) on the nanoHUB mainpage. If you tag your contribution with “AQME” we will easily associate your contribution to this tool and may include it into this curated page.
+
'''Discovery that is Possible through Quantum Mechanics'''
26 -
* Provide feedback for the items you use on the nanoHUB through the review system. (Please be explicit and provide constructive feedback.)
+
27 -
* Let us know when things do not work for you – file a ticket through the nanoHUB “Help” feature on every page
+
28 -
* Finally, let us know what you are doing and your suggestions improving the nanoHUB by using the “Feedback” section, which you can find under “Support”
+
29
30 -
Thank you for using the nanoHUB, and be sure to share your nanoHUB success stories with us. We like to hear from you, and our sponsors need to know that the nanoHUB is having impact.
+
Nanotechnology has yielded a number of unique structures that are not found readily in nature. Most demonstrate an essential quality of Quantum Mechanics known as quantum confinement. Confinement is the idea of keeping electrons trapped in a small area, about 30 nm or smaller. Quantum confinement comes in several dimensions. 2-D confinement, for example, is restricted in only one dimension, resulting in a quantum well (or plane). Lasers are currently built from this dimension. 1-D confinement occurs in nanowires, and 0-D confinement is found only in the quantum dot.
31
32 -
=== Particle-Wave Duality ===
+
The study of quantum confinement leads, foremost, to electronic properties not found in today’s semiconductor devices. The quantum dot works well as a first example. The typical quantum dot is anywhere between 3-60 nm in diameter. That’s still 30 to 600 times the size of a typical atom. A quantum dot exhibits 0-D confinement, meaning that electrons are confined in all three dimensions. In nature, only atoms have 0-D confinement; thus, a quantum dot can be described loosely as an ‘artificial atom.’ This knowledge is vitally important, as atoms are too small and too difficult to isolate in experiments. Conversely, quantum dots are large enough to be manipulated by magnetic fields and can even be moved around with an STM or AFM. We can deduce many important atomistic characteristics from a quantum dot that would otherwise be impossible to research in an atom.
33
34 -
==== Diagram 1 ====
+
Confinement also increases the efficiency of today’s electronics. The laser is based on a 2-D confinement layer that is usually created with some form of epitaxy such as Molecular Beam Epitaxy or Chemical Vapor Deposition. The bulk of modern lasers created with this method are highly functional, but these lasers are ultimately inefficient in terms of energy consumption and heat dissipation. Moving to 1-D confinement in wires or 0-D confinement in quantum dots allows for higher efficiencies and brighter lasers. Quantum dot lasers are currently the best lasers available, although their fabrication is still being worked out.
35
36 -
[[Image(pic1_duality.png, 200 class=align-left)]] Publicized early in the debate about whether light was composed of particles or waves, a wave-particle dual nature soon was found to be characteristic of electrons as well. The evidence for the description of light as waves was well established at the turn of the century when the photoelectric effect introduced firm evidence of a particle nature as well. On the other hand, the particle properties of electrons was well documented when the !DeBroglie hypothesis and the subsequent experiments by Davisson and Germer established the wave nature of the electron.
+
Confinement is just one manifestation of quantum mechanics in nanodevices. Tunneling and quantum interference are two other manifestations of quantum mechanics in the operation of scanning tunneling microscopes and resonant tunneling diodes, respectively. For more information on the theoretical aspects of Quantum Mechanics check the following resources:
37
38 -
== Solution of the Time-Independent Schrodinger Equation ==
+
[[Resource(4920)]]
39
40 -
=== Piece-Wise Constant Potential Barrier Tool – Open Systems ===
+
[[Resource(5164)]]
41
42 +
Because understanding quantum mechanics is so foundational to an understanding of the operation of nanoscale devices, almost every Electrical Engineering department (in which there is a strong nanotechnology experimental or theoretical group) and all Physics departments teach the fundamental principles of quantum mechanics and their application to nanodevice research. Several conceptual sets and theories are taught within these courses. Normally, students are first introduced to the concept of particle-wave duality (the photoelectric effect and the double-slit experiment), the solutions of the time-independent Schrödinger equation for open systems (piece-wise constant potentials), tunneling, and bound states. The description of the solution of the Schrödinger equation for periodic potentials (Kronig-Penney model) naturally follows from the discussion of double well, triple well and n-well structures. This leads the students to the concept of energy bands and energy gaps, and the concept of the effective mass that can be extracted from the pre-calculated band structure by fitting the curvature of the bands. The Tsu-Esaki formula is then investigated so that, having calculated the transmission coefficient, students can calculate the tunneling current in resonant tunneling diode and Esaki diode. After establishing basic principles of quantum mechanics, the harmonic oscillator problem is then discussed in conjunction with understanding vibrations of a crystalline lattice, and the idea of phonons is introduced as well as the concept of creation and annihilation operators. The typical quantum mechanics class for undergraduate/first-year graduate students is then completed with the discussion of the stationary and time-dependent perturbation theory and the derivation of the Fermi Golden Rule, which is used as a starting point of a graduate level class in semiclassical transport. Coulomb Blockade is another discussion a typical quantum mechanics class will include.
43
44 -
[[Image(pic2_tunnel.png, 300 class=align-right)]] The [[Resource(4826)]] (PCPBT) allows calculation of the transmission and the reflection coefficient of arbitrary five, seven, nine, eleven and 2n-segment piece-wise constant potential energy profile.
+
== Particle-Wave Duality ==
45 +
46 +
[[Image(pic1_duality.png, 200, class=align-left)]] A wave-particle dual nature was discovered and publicized in the early debate about whether light was composed of particles or wave. Evidence for the description of light-as-waves was well established at the turn of the century when the photoelectric effect introduced firm evidence of a light-as-particle nature. This dual nature was found to also be characteristic of electrons. Electron particle nature properties were well documented when the DeBroglie hypothesis, and subsequent experiments by Davisson and Germer, established the wave nature of the electron.
47 +
48 +
[[Resource(4916)]]
49 +
50 +
This movie helps students to better distinguish when nano-things behave as particles and when they behave as waves. The link below connects to an exercise on these concepts.
51
52 [[Div(start, class=clear)]][[Div(end)]]
53 +
54 +
[[Resource(4918)]]
55
56 -
==== Scanning Tunneling Microscope (STM) ====
+
[[Div(start, class=clear)]][[Div(end)]]
57
58 -
[[Image(pic3_stm.png, 200 class=align-left)]] For the case of multi-well structure it also calculates the quasi-bound states so it can be used as a simple demonstration tool for the formation of energy bands. Also, it can be used in the case of stationary perturbation theory exercises to test the validity of, for example, the first order and the second order correction to the ground state energy of the system due to small perturbations of, for example, the confining potential. The PCPBT tool can also be used to test the validity of the WKB approximation for triangular potential barriers.
+
59 +
== Solution of the Time-Independent Schrödinger Equation ==
60 +
=== [/tools/aqme/ Piece-Wise Linear Barrier Tool in AQME] – Open Systems ===
61
62 [[Div(start, class=clear)]][[Div(end)]]
63
64 -
==== Tunneling: Limiting Device Miniaturization ====
+
[[Image(pcpbt1.bmp, 120, class=align-left)]]
65 +
[[Image(pcpbt2.bmp, 120, class=align-left)]]
66 +
[[Image(pcpbt3.bmp, 120, class=align-left)]]
67
68 -
[[Image(pic4_tunlim.png, 200 class=align-left)]]
+
[[Div(start, class=clear)]][[Div(end)]]
69 +
70 +
71 +
The [/tools/aqme/ Piece-Wise Linear Barrier Tool in AQME] allows calculation of the transmission and the reflection coefficient of arbitrary five, seven, nine, eleven and 2n-segment piece-wise constant potential energy profile. For the case of multi-well structure it also calculates the quasi-bound states so it can be used as a simple demonstration tool for the formation of energy bands. Also, it can be used in the case of stationary perturbation theory exercises to test the validity of, for example, the first order and the second order correction to the ground state energy of the system due to small perturbations of, for example, the confining potential. The [/tools/aqme/ Piece-Wise Linear Barrier Tool in AQME] can also be used to test the validity of the WKB approximation for triangular potential barriers.
72
73 [[Div(start, class=clear)]][[Div(end)]]
74
75 -
=== Bound States Lab ===
+
Available resources:
76
77 -
[[Image(pic5_boundst.png, 200 class=align-left)]] The [[Resource(2531)]] Calculation Lab determines the bound states and the corresponding wavefunctions in a square, harmonic and triangular potential well. Maximum number of eigenstates that can be calculated is 100. Students clearly see the nature of the separation of the states in these three prototypical confining potentials with which we can approximate realistic quantum potentials that occur in nature.
+
* [[Resource(4831)]]
78 +
* [[Resource(4833)]]
79 +
* [[Resource(4853)]]
80 +
* [[Resource(4873)]]
81 +
* [[Resource(5319)]]
82 +
* [[Resource(4849)]]
83 +
* [[Resource(5102)]]
84 +
* [[Resource(5130)]]
85
86 [[Div(start, class=clear)]][[Div(end)]]
87
88 +
=== [/tools/aqme/ Bound States Lab in AQME] ===
89 +
90 +
[[Div(start, class=clear)]][[Div(end)]]
91
92 -
Energy eigenstates of a harmonic oscillator (left panel). Probability density of the ground state that demonstrates purely quantum-mechanical behavior (middle panel). Probability density of the 20th subband where we start to see more classical behavior (right panel) as the well opens.
+
The [/tools/aqme/ Bound States Lab in AQME] determines the bound states and the corresponding wavefunctions in a square, harmonic, and triangular potential well. The maximum number of eigenstates that can be calculated is 100. Students clearly see the nature of the separation of the states in these three prototypical confining potentials, with which students can approximate realistic quantum potentials that occur in nature.
93
94 -
[[Image(pic6_state1top.png, 120 class=align-left)]]
+
The panel below (left) shows energy eigenstates of a harmonic oscillator. Probability density of the ground state that demonstrates purely quantum-mechanical behavior is shown in the middle panel below. Probability density of the 20th subband demonstrates the more classical behavior as the well opens (right panel below).
95 -
[[Image(pic7_state2left.png, 140 class=align-left)]]
+
96 -
[[Image(pic8_state3right.png, 125 class=align-left)]]
+
97
98 [[Div(start, class=clear)]][[Div(end)]]
99
100 -
== Energy Bands and Effective Masses ===
+
[[Image(pic6_state1top.png, 120, class=align-left)]]
101 +
[[Image(pic7_state2left.png, 140, class=align-left)]]
102 +
[[Image(pic8_state3right.png, 125, class=align-left)]]
103
104 [[Div(start, class=clear)]][[Div(end)]]
105
106 -
=== Periodic Potential Lab ===
+
Available resources:
107
108 -
[[Image(pic10_perpot2.png, 150 class=align-right)]] [[Image(pic9_perpot1.png, 160 class=align-right)]] The [[Resource(3847)]] solves the time independent Schroedinger Equation in a 1-D spatial potential variation. Rectangular, triangular, parabolic (harmonic), and Coulomb potential confinements can be considered. The user can determine energetic and spatial details of the potential profiles, compute the allowed and forbidden bands, plot the bands in a compact and an expanded zone, and compare the results against a simple effective mass parabolic band. Transmission is also calculated. This Lab also allows the students to become familiar with the reduced zone and expanded zone representation of the dispersion relation (E-k relation for carriers).
+
* [[Resource(4884)]]
109 +
* [[Resource(4976)]]
110
111 [[Div(start, class=clear)]][[Div(end)]]
112
113 -
=== Bandstructure Lab ===
114
115 +
== Energy Bands and Effective Masses ==
116
117 +
[[Div(start, class=clear)]][[Div(end)]]
118 +
119 +
=== [/tools/aqme/ Periodic Potential Lab in AQME] ===
120 +
121 +
[[Image(pic10_perpot2.png, 150, class=align-right)]] [[Image(pic9_perpot1.png, 160, class=align-right)]] The [/tools/aqme/ Periodic Potential Lab in AQME] solves the time-independent Schrödinger Equation in a 1-D spatial potential variation. Rectangular, triangular, parabolic (harmonic), and Coulomb potential confinements can be considered. The user can determine energetic and spatial details of the potential profiles, compute the allowed and forbidden bands, plot the bands in a compact and an expanded zone, and compare the results against a simple effective mass parabolic band. Transmission is also calculated. This lab also allows the students to become familiar with the reduced zone and expanded zone representation of the dispersion relation (E-k relation for carriers).
122 +
123 +
Available resources:
124
125 -
[[Image(pic12_band2.png, 160 class=align-right)]] [[Image(pic11_band1.png, 160 class=align-right)]] Bandstructure of Si (left panel) and !GaAs (right panel).
+
[[Resource(4851)]]
126
127 [[Div(start, class=clear)]][[Div(end)]]
128
129 -
In solid-state physics, the electronic band structure (or simply band structure) of a solid describes ranges of energy that an electron is "forbidden" or "allowed" to have. It is due to the diffraction of the quantum mechanical electron waves in the periodic crystal lattice. The band structure of a material determines several characteristics, in particular its electronic and optical properties. The [[Resource(1308)]] tool enables the study of bulk dispersion relationships of Si, !GaAs, !InAs. Plotting the full dispersion relation of different materials, students first get familiar with a band-structure of direct band-gap (!GaAs, !InAs) and indirect band-gap semiconductors (Si). For the case of multiple conduction band valleys one has to determine first the Miller indices of one of the equivalent valleys and from that information it immediately follows how many equivalent conduction bands one has in Si and Ge, for example. In advanced applications, the users can apply tensile and compressive strain and observe the variation in the bandstructure, bandgaps, and effective masses. Advanced users can also study bandstructure effects in ultra-scaled (thin body) quantum wells, and nanowires of different cross sections. Bandstructure Lab uses the sp3s*d5 tight binding method to compute E(k) for bulk, planar, and nanowire semiconductors.
+
=== [/tools/aqme/ Band Structure Lab in AQME] ===
130
131 [[Div(start, class=clear)]][[Div(end)]]
132
133 -
[[Image(diamond.png, 140 class=align-left)]] First Brillouin zone of FCC lattice that corresponds to the first Brillouin zone for all diamond and Zinc-blende materials (C, Si, Ge, !GaAs, !InAs, !CdTe, etc.). There are 8 hexagonal faces (normal to 111) and 6 square faces (normal to 100 ). The sides of each hexagon and square are equal.
+
[[Image(pic12_band2.png, 160, class=align-right)]] [[Image(pic11_band1.png, 160, class=align-right)]] Band structure of Si (left panel) and !GaAs (right panel).
134
135 [[Div(start, class=clear)]][[Div(end)]]
136
137 -
Supplemental Information: Specification of High-symmetry points
+
In solid-state physics, the electronic band structure (or simply band structure) of a solid describes ranges of energy that an electron is "forbidden" or "allowed" to have. It is due to the diffraction of the quantum mechanical electron waves in the periodic crystal lattice. The band structure of a material determines several characteristics, in particular, its electronic and optical properties. The [/tools/aqme/ Band Structure Lab in AQME] enables the study of bulk dispersion relationships of Si, !GaAs, !InAs. Plotting the full dispersion relation of different materials, students first get familiar with a band structure of a direct band gap (!GaAs, !InAs), as well as indirect band gap semiconductors (Si). For the case of multiple conduction band valleys, students must first determine the Miller indices of one of the equivalent valleys, then, from that information they can deduce how many equivalent conduction bands are in Si and Ge, for example. In advanced applications, the users can apply tensile and compressive strain and observe the variation in the band structure, band gaps, and effective masses. Advanced users can also study band structure effects in ultra-scaled (thin body) quantum wells, and nanowires of different cross sections. Band Structure Lab uses the sp3s*d5 tight-binding method to compute E(k) for bulk, planar, and nanowire semiconductors.
138 +
139 +
Available resource:
140 +
141 +
[[Resource(5201)]]
142
143 [[Div(start, class=clear)]][[Div(end)]]
144
145 +
[[Image(diamond.png, 140, class=align-left)]] The figure on the left illustrates the first Brillouin zone of FCC lattice that corresponds to the first Brillouin zone for all diamond and Zinc-blende materials (C, Si, Ge, !GaAs, !InAs, !CdTe, etc.). There are 8 hexagonal faces (normal to 111) and 6 square faces (normal to 100). The sides of each hexagon and each square are equal.
146 +
147 +
[[Div(start, class=clear)]][[Div(end)]]
148 +
149 +
Supplemental Information: Specification of High-Symmetry Points
150 +
151 +
[[Div(start, class=clear)]][[Div(end)]]
152
153 ||Symbol||Description||
154 ||Γ||Center of the Brillouin zone||
155 ||Simple Cube||
156 ||M||Center of an edge||
157 -
||R||Corner Point||
+
||R||Corner point||
158 ||X||Center of a face||
159 -
||Face-Centered cubic||
+
||Face-Centered Cubic||
160 ||K||Middle of an edge joining two hexagonal faces||
161 ||L||Center of a hexagonal face||
162 ||U||Middle of an edge joining a hexagonal and a square face||
163 ||W||Corner point||
164 ||X||Center of a square face||
165 ||Body-Centered Cubic||
166 ||H||Corner point joining four edges||
167 ||N||Center of a face||
168 ||P||Corner point joining three edges||
169 ||Hexagonal||
170 ||A||Center of a hexagonal face||
171 -
||H||Corner POint||
+
||H||Corner point||
172 ||K||Middle of an edge joining two rectangular faces||
173 ||L||Middle of an edge joining a hexagonal and a rectangular face||
174 ||M|| Center of a rectangular face||
175 -
176
177 [[Div(start, class=clear)]][[Div(end)]]
178
179
180 == Real World Applications ==
181
182 [[Div(start, class=clear)]][[Div(end)]]
183
184 -
=== Schred ===
+
=== [/tools/aqme/ Schred Tool in AQME] ===
185
186 -
[[Image(pic13_schred1.png, 140 class=align-left)]] [[Image(pic14_schred2.png, 155 class=align-left)]] [[Image(pic15_schred3.png, 140 class=align-left)]]
+
[[Image(pic13_schred1.png, 140, class=align-left)]] [[Image(pic14_schred2.png, 155, class=align-left)]] [[Image(pic15_schred3.png, 140, class=align-left)]]
187
188 [[Div(start, class=clear)]][[Div(end)]]
189
190 -
[[Resource(221)]] calculates the envelope wavefunctions and the corresponding bound-state energies in a typical MOS (Metal-Oxide-Semiconductor) or SOS (Semiconductor-Oxide- Semiconductor) structure and a typical SOI structure by solving self-consistently the one-dimensional (1D) Poisson equation and the 1D Schrodinger equation. The Schred tool is specifically designed for Si/!SiO2 interface and takes into account the mass anisotropy of the conduction bands as well as different crystallographic orientations.
+
The [/tools/aqme/ Schred Tool in AQME] calculates the envelope wavefunctions and the corresponding bound-state energies in a typical MOS (Metal-Oxide-Semiconductor) or SOS (Semiconductor-Oxide- Semiconductor) structure and in a typical SOI structure by solving self-consistently the one-dimensional (1-D) Poisson equation and the 1D Schrödinger equation. The Schred tool is specifically designed for Si/!SiO2 interface and takes into account the mass anisotropy of the conduction bands, as well as different crystallographic orientations.
191
192 -
[[Div(start, class=clear)]][[Div(end)]]
+
Available resources:
193 +
194 +
* [[Resource(4900)]]
195 +
* [[Resource(4904)]]
196
197 -
==== This is schematically shown in the figure below. ====
198
199 [[Div(start, class=clear)]][[Div(end)]]
200
201 -
[[Image(pic16_bandschem.png, 140 class=align-left)]] Right panel - Potential diagram for inversion of p-type semiconductor. In this first notation Εij refers to the j-th subband from either the Δ2-band (i=1) or Δ4-band (i=2). Left panel - Constant-energy surfaces for the conduction-band of silicon showing six conduction-band valleys in the <100> direction of momentum space. The band minima, corresponding to the centers of the ellipsoids, are 85% of the way to the Brillouin-zone boundaries. The long axis of an ellipsoid corresponds to the longitudinal effective mass of the electrons in silicon, , while the short axes correspond to the transverse effective mass, . For <100> orientation of the surface, the Δ2-band has the longitudinal mass (ml) perpendicular to the semiconductor interface and the Δ4-band has the transverse mass (mt) perpendicular to the interface. Since larger mass leads to smaller kinetic term in the Schrodinger equation, the unprimed lader of subbands (as is usually called), corresponding to the Δ2-band, has the lowest ground state energy. The degeneracy of the unprimed ladder of subbands for <100> orientation of the surface is 2. For the same reason, the ground state of the primed ladder of subbands corresponding to the Δ4-band is higher that the lowest subband of the unprimed ladder of subbands, The degeneracy of the primed ladder of subbands for (100) orientation of the interface is 4.
+
[[Image(pic16_bandschem.png, 140, class=align-left)]] Right panel - Potential diagram for inversion of p-type semiconductor. In this first notation Εij refers to the j-th subband from either the Δ2-band (i=1) or Δ4-band (i=2). Left panel - Constant-energy surfaces for the conduction-band of silicon showing six conduction-band valleys in the direction of momentum space. The band minima, corresponding to the centers of the ellipsoids, are 85% of the way to the Brillouin-zone boundaries. The long axis of an ellipsoid corresponds to the longitudinal effective mass of the electrons in silicon, while the short axes correspond to the transverse effective mass . For orientation of the surface, the Δ2-band has the longitudinal mass (ml) perpendicular to the semiconductor interface and the Δ4-band has the transverse mass (mt) perpendicular to the interface. Since larger mass leads to smaller kinetic term in the Schrodinger equation, the unprimed lader of subbands (as is usually called), corresponding to the Δ2-band, has the lowest ground state energy. The degeneracy of the unprimed ladder of subbands for orientation of the surface is 2. For the same reason, the ground state of the primed ladder of subbands corresponding to the Δ4-band is higher that the lowest subband of the unprimed ladder of subbands, The degeneracy of the primed ladder of subbands for (100) orientation of the interface is 4.
202
203 [[Div(start, class=clear)]][[Div(end)]]
204
205 -
[[Image(pic17_moreschred.png, 140 class=align-left)]] The Schred tool is a very important tool for the design of conventional, FD and dual-gate MOSFET devices shown to the right on the alternative device structures map.
+
=== [/tools/aqme/ 1-D Heterostructure Tool AQME] ===
206 +
207 +
The [/tools/aqme/ 1-D Heterostructure Tool AQME] simulates confined states in 1-D heterostructures by calculating charge self-consistently in the confined states, based on a quantum mechanical description of the one dimensional device. The greater interest in HEMT devices is motivated by the limits that will be reached with scaling of conventional transistors. The [[Resource(5203)]] in that respect is a very valuable tool for the design of HEMT devices as one can determine, for example, the position and the magnitude of the delta-doped layer, the thickness of the barrier and the spacer layer for which one maximizes the amount of free carriers in the channel which, in turn, leads to larger drive current. This is clearly illustrated in the examples below.
208
209 [[Div(start, class=clear)]][[Div(end)]]
210
211 -
=== 1D Heterostructure Tool ===
+
[[Image(1dhet1.png, 120, class=align-left)]]
212 +
[[Image(1dhet2.png, 120, class=align-left)]]
213
214 -
The [[Resource(5203)]] simulates confined states in 1D heterostructures by calculating charge self-consistently in the confined states, based on a quantum mechanical description of the one dimensional device. The Heterostructure tool is very important for the design of High Electron Mobility Transistors. The greater interest in HEMT devices is motivated by the limits that will be reached with scaling of conventional transistors.
+
[[Div(start, class=clear)]][[Div(end)]]
215
216 -
Mobile communication and information systems undergo a rapid and highly sophisticated technological change. In the recent past, mobile phone systems have undergone an exponential rise of distribution. At the same time desktop computers have been provided with a steady growth of computational power, and saturation effects for the simple scaling of dynamic random access memories (DRAM) and processors become visible. The mobile broadband access to the internet data communication makes that mobile cell phone communication merges with data communication features. This encourages the development of mixed signal technologies due to the increased signal processing of digital data communication and the increased demands towards the analog RF transmission. The overall development promotes a merge of the desktop computer world and the mobile communication world, which so far have been regarded as separate areas. Being exposed to a mass market, every function of a cell phone, whether digital or analog, is subject to an aggressive selection process with respect to cost and functionality. This represents a new challenge to III-V semiconductor components which were initially developed for high prize niche markets.
+
Available resources:
217
218 -
At the same time, applications in higher frequency bands also develop into markets: The frequency range of 27 GHz-42 GHz has been allocated all over the world in order to provide broadband internet access services. Furthermore, for higher mm-wave frequencies, i.e. at 77 GHz, automotive applications such as collision avoidance radars await their introduction from niche into mass markets. Up to now there is no consensus on what the best technology for which application will be. Circuits realized on silicon are now operating up to 60 GHz but the power capabilities and the efficiency is much lower than in other technologies [ ]. Devices realized in other material systems like !AlGaN/GaN ,show very good power capabilities and high frequency performance and thus are promising technologies for very high power applications in the future. But the most important key for success is to realize a function as cheap as possible. Sales of discrete devices stagnate whereas the total volume of non silicon semiconductors increases dramatically [ ]. This shows that the integration level is increasing and more functions previously realized by hybrid systems are integrated on chip.
+
* [[Resource(5231)]]
219 +
* [[Resource(5233)]]
220
221 -
The most commonly used semiconductor devices for applications in the GHz range now are !GaAs based MESFETs, HEMTs and HBTs [5]. Although MESFETs are the cheapest devices because they can be realized with bulk material, i.e. without epitaxially grown layers, HEMTs and HBTs are promising devices for the near future. The advantage of HEMTs and HBTs is a factor of 2 to 3 higher power density compared to MESFETs which leads to significantly smaller chip size. In the figure below the gain per amplifier stage and output power is shown versus the operation frequency for different FETs, HBTs, and HEMTs. It clearly demonstrates advantages of HEMTs for operation frequencies above 10 GHz and very good competitiveness for lower frequencies. Since it is quite expensive to keep different technologies established in a production line it is favorable to cover many applications with a single technology. This represents a major advantage of HEMTs if applications in the whole frequency range from 1 GHz to 100 GHz are requested.
+
222 +
The most commonly used semiconductor devices for applications in the GHz range now are !GaAs based MESFETs, HEMTs and HBTs. Although MESFETs are the cheapest devices because they can be realized with bulk material, i.e. without epitaxially grown layers, HEMTs and HBTs are promising devices for the near future. The advantage of HEMTs and HBTs is a factor of 2 to 3 higher power density compared to MESFETs which leads to significantly smaller chip size.
223
224 [[Div(start, class=clear)]][[Div(end)]]
225
226 -
[[Image(hetero1.png, 120 class=align-left)]] Gain per amplifier stage and output power of amplifiers with HEMTs, HBTs, and other FETs.![4]
+
HEMTs are field effect transistors where the current flow between two ohmic contacts, Source and Drain, and it is controlled by a third contact, the Gate. Most often the Gate is a Schottky contact. In contrast to ion implanted MESFETs, HEMTs are based on epitaxially grown layers with different band gaps Eg. A schematic cross section of a HEMT is shown below.
227
228 [[Div(start, class=clear)]][[Div(end)]]
229
230 -
HEMTs are field effect transistors where the current flow between two ohmic contacts, Source and Drain, and it is controlled by a third contact, the Gate. Most often the Gate is a Schottky contact. In contrast to ion implanted MESFETs, HEMTs are based on epitaxially grown layers with different band gaps Eg. A schematic cross section of a HEMT is shown below.
+
[[Image(hetero2.png, 120, class=align-left)]] Schematic cross section of a High Electron Mobility Transistor (HEMT).
231
232 [[Div(start, class=clear)]][[Div(end)]]
233
234 -
[[Image(hetero2.png, 120 class=align-left)]] Schematic cross section of a High Electron Mobility Transistor (HEMT) used in this work.
+
[[Image(hetero4.png, 150, class=align-right)]] If two semiconductors with different band gap energies are joined together, the difference is divided up into a band gap offset in the valence band ΔEV and a band gap offset in the conduction band ΔEC. One of the most common assumptions made for the AlGaAs/InGaAs material system is 40% valence band offset and 60% conduction band offset. This is only valid for Al contents below about 45%. For higher Al contents the bandgap of AlGaAs changes from direct to indirect. In the figure below such an AlGaAs/InGaAs HEMT with a delta doped upper barrier layer is shown. The conduction band energy under the gate along the cutting line A-A' is shown to the right.
235
236 [[Div(start, class=clear)]][[Div(end)]]
237 -
238 -
Near a semiconductor heterojunction electrons are transferred from the material with the higher conduction band energy EC to the material with the lower EC where they can occupy a lower energy state. This can be a large number of electrons especially if the semiconductor with the high EC barrier is doped. Near the interface, a two dimensional electron gas (2DEG), the channel, is created. This way it is possible to separate the electrons in the channel from their donor atoms which reduces Coulomb scattering and hence increases the mobility of the conducting electrons. If the channel is built only by a single heterojunction the electrons are penetrating into the buffer under the channel very easily where their mobility is usually lower and the control of the gate is poor. To keep the electrons in the channel a second energy barrier below the channel can be introduced by a material with a higher EC than the channel material. In the figure below the band gaps of the most important III-V semiconductors and the available substrates are shown.
239
240 [[Div(start, class=clear)]][[Div(end)]]
241
242 -
[[Image(hetero3.png, 150 class=align-left)]] Lattice constant versus band gap of the most important III-V compound semiconductors
+
243 +
=== [/tools/aqme/ Resonant Tunneling Diode Lab in AQME] ===
244
245 +
[[Div(start, class=clear)]][[Div(end)]]
246 +
[[Image(rtd1.png, 140, class=align-center)]]
247 +
[[Image(rtd2.png, 140, class=align-center)]]
248 [[Div(start, class=clear)]][[Div(end)]]
249
250 -
+
Put a potential barrier in the path of electrons, and it will block their flow; but, if the barrier is thin enough, electrons can tunnel right through due to quantum mechanical effects. It is even more surprising that, if two or more thin barriers are placed closely together, electrons will bounce between the barriers, and, at certain resonant energies, flow right through the barriers as if there were none. Run the [/tools/aqme/ Resonant Tunneling Diode Lab in AQME], which lets you control the number of barriers and their material properties, and then simulate current as a function of bias. Devices exhibit a surprising negative differential resistance, even at room temperature. This tool can be run online in your web browser as an active demo.
251 -
[[Image(hetero4.png, 150 class=align-right)]] If two semiconductors with different band gap energies are joined together, the difference is divided up into a band gap offset in the valence band ΔEV and a band gap offset in the conduction band ΔEC. One of the most common assumptions made for the AlGaAs/InGaAs material system is 40% valence band offset and 60% conduction band offset. This is only valid for Al contents below about 45%. For higher Al contents the bandgap of AlGaAs changes from direct to indirect. In the figure below such an AlGaAs/InGaAs HEMT with a delta doped upper barrier layer is shown. The conduction band energy under the gate along the cutting line A-A' is shown to the right.
+
252
253 [[Div(start, class=clear)]][[Div(end)]]
254
255 -
Conduction band diagram of a delta doped HEMT. The Fermi level EF and the quantum energy level Ee of the electrons in the channel are indicated by the dashed line.
+
[[Image(pic18_restunn.png, 120, class=align-left)]] [[Image(pic19_restun2.png, 120, class=align-left)]]
256 -
The conduction band of the channel relative to the Fermi level EF is determined by ΔEC,, the doping level ND, the barrier height of the Schottky contact qΦB, the gate to channel separation dGC, and the applied voltage on the gate VGS. To obtain high drain currents ID and high transconductance gm it is favorable to maximize qΦB, ND, ΔEC, and to minimize dGC. If a homogeneously doped upper barrier layer is used qΦB, ND, and dGC are directly related to each other. A decrease in dGC reduces the total doping in the barrier layer which shifts the threshold voltage (VT) to more positive values and thus reduces ID max. If ND is increased ΦB of the Schottky contact is reduced.
+
257
258 -
This direct trade-off can be overcome if a delta doping is used. A delta doping in an (Al)!GaAs layer can be realized by growing pure silicon for a short period of time within the growth of an undoped !AlGaAs/InAlAs layer. This way ND is not reduced by a reduction of dGC. The sheet doping concentration can be adjusted by the amount of silicon incorporated in the crystal and the activation of the dopands. The activation depends on various parameters of MBE growth such as substrate temperature. The upper limit of activated sheet doping concentration is in the order of 4E12 cm-2. Φ_ is about 650 meV and decreases if the separation between the delta doping and the gate gets below 10 nm. The achievable minimum dGC highly depends on the applied technology. Tight process control can yield to . The goal of the channel is to provide a high current density. The electron concentration is mainly determined by ΔEC , as well as the doping concentration in the barrier layers and its distance to the channel. To increase ΔEC it is favorable both to reduce Eg of the channel material and to increase Eg of the barrier layers.
+
[[Div(start, class=clear)]][[Div(end)]]
259
260 -
The 1D Heterojunction lab in that respect is a very valuable tool for the design of HEMT devices as one can determine, for example, the position and the magnitude of the delta-doped layer, the thickness of the barrier and the spacer layer for which one maximizes the amount of free carriers in the channel which, in turn, leads to larger drive current. This is clearly illustrated in the example below.
+
Available resources:
261 +
262 +
* [[Resource(891)]]
263 +
* [[Resource(3949)]]
264
265
266 [[Div(start, class=clear)]][[Div(end)]]
267
268 -
=== Resonant Tunneling Diode Lab ===
+
=== [/tools/aqme/ Quantum Dot Lab in AQME] ===
269
270 -
Put a potential barrier in the path of electrons, and it will block their flow. But if the barrier is thin enough, electrons can tunnel right through due to quantum mechanical effects. Even more surprising, if two or more thin barriers are placed closely together, electrons will bounce between the barriers, and at certain resonant energies, flow right through the barriers as if they were not there! Check out the [[Resource(230)]] lab, which lets you control the number of barriers and their material properties, and then simulate current as a function of bias. Devices exhibit a surprising negative differential resistance, even at room temperature! Run this tool online, right in your web browser! View a demo of this tool in action.
+
Individual quantum dots can be created from two-dimensional electron or hole gases present in remotely doped quantum wells or semiconductor heterostructures. The sample surface is coated with a thin layer of resist. A lateral pattern is then defined in the resist by electron beam lithography. This pattern can then be transferred to the electron or hole gas by etching, or by depositing metal electrodes (lift-off process) that allow the application of external voltages between the electron gas and the electrodes. Such quantum dots are mainly of interest for experiments and applications involving electron or hole transport, i.e., an electrical current. The energy spectrum of a quantum dot can be engineered by controlling the geometrical size, the shape, and the strength of the confinement potential. Also, in contrast to atoms, it is relatively easy to connect quantum dots by tunnel barriers to conducting leads, which allows the application of the techniques of tunneling spectroscopy for their investigation. Confinement in quantum dots can also arise from electrostatic potentials (generated by external electrodes, doping, strain, or impurities).
271
272 [[Div(start, class=clear)]][[Div(end)]]
273
274 -
[[Image(pic18_restunn.png, 120 class=align-left)]] [[Image(pic19_restun2.png, 120 class=align-left)]]
+
[[Image(pic25_qdot.png, 140, class=align-left)]] [/tools/aqme/ Quantum Dot Lab in AQME] computes the eigenstates of a particle in a box of various shapes, including domes and pyramids.
275
276 [[Div(start, class=clear)]][[Div(end)]]
277
278 -
A resonant tunnel diode (RTD) is a device which uses quantum effects to produce negative differential resistance (NDR). As an RTD is capable of generating a terahertz wave at room temperature, it can be used in ultra high-speed circuitry. Therefore The RTD is extensively studied.
+
Available resources:
279 -
RTDs are formed as a single quantum well structure surrounded by very thin layer barriers. This structure is called a double barrier structure. Carriers such as electrons and holes can only have discrete energy values inside the quantum well. When a voltage is placed across an RTD, a terahertz wave is emitted which is why the energy value inside the quantum well is equal to that of the emitter side. As voltage increased, the terahertz stops because the energy value in the quantum well is outside the emitter side energy. This structure can be grown by molecular beam heteroepitaxy. !GaAs and !AlAs in particular are used to form this structure. !AlAs/InGaAs or !InAlAs/InGaAs can be used.
+
280 -
In quantum tunneling through a single barrier, the transmission coefficient, or the tunneling probability, is always less than one (for incoming particle energy less than the potential barrier height). Consider a potential profile which contains two barriers (which are located close to each other), one can calculate the transmission coefficient (as a function of the incoming particle energy) using any of the standard methods. It turns out that, for certain energies, the transmission coefficient is equal to one, i.e. the double barrier is totally transparent for particle transmission. This phenomenon is called resonant tunneling ![1]. It is interesting that while the transmission coefficient of a potential barrier is always lower than one (and decreases with increasing barrier height and width), two barriers in a row can be completely transparent for certain energies of the incident particle.
+
281
282 -
[[Div(start, class=clear)]][[Div(end)]]
+
* [[Resource(189)]] is a nano 101, introductory lecture that starts from particle-wave duality and explores the concepts of quantum dots.
283 +
* [[Resource(4194)]] (by Lee, Ryu, Klimeck)
284 +
* [[Resource(2846)]] (by Fodor, Guo)
285 +
* [[Resource(4203)]]
286 +
287
288 -
[[Image(pic20_restun3.png, 140 class=align-left)]] [[Image(pic21_restun4.png, 120 class=align-left)]] [[Image(pic22_restun5.png, 140 class=align-left)]]
289
290 [[Div(start, class=clear)]][[Div(end)]]
291
292 -
The potential profiles required for resonant tunneling can be realized in semiconductor system using heterojunctions which utilize semiconductors of different types to crease potential barriers or wells in the conduction band or the valence band. It is worth noting that resonant tunnel diodes are intraband tunnel diodes.
+
== Scattering and Fermi's Golden Rule ==
293
294 +
[[Image(scattering.png, 250, class=align-left)]]
295 [[Div(start, class=clear)]][[Div(end)]]
296
297 -
[[Image(pic23_restun6.png, 150 class=align-left)]] [[Image(pic24_restun7.png, 150 class=align-left)]]
+
Scattering is a general physical process whereby some forms of radiation, such as light, sound, or moving particles are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, scattering also includes deviation of reflected radiation from the angle predicted by the law of reflection. Reflections that undergo scattering are often called diffuse reflections, and unscattered reflections are called specular (mirror-like) reflections. The types of non-uniformities (sometimes known as scatterers or scattering centers) that can cause scattering are too numerous to list, but a small sample includes particles, bubbles, droplets, density fluctuations in fluids, defects in crystalline solids, surface roughness, cells in organisms, and textile fibers in clothing. The effects of such features on the path of almost any type of propagating wave or moving particle can be described in the framework of scattering theory. In quantum physics, Fermi's golden rule is a way to calculate the transition rate (probability of transition per unit time) from one energy eigenstate of a quantum system into a continuum of energy eigenstates, due to a perturbation. The [/tools/aqme/ Bulk Monte-Carlo Lab in AQME] calculates the scattering rates dependence versus electron energy of the most important scattering mechanisms for the most commonly used materials in the semiconductor industry, such as Si, Ge, !GaAs, !InSb, !GaN, !SiC. For proper parameter set for, for example, 4H SiC please refer to the following article.
298
299 -
[[Div(start, class=clear)]][[Div(end)]]
+
Available Resources:
300
301 -
==== III-V Resonant Tunnel Diodes ====
+
* [[Resource(5019)]]
302 +
* [[Resource(5277)]]
303
304 -
Resonant tunnel diodes are typically realized in III-V compound material systems, where heterojunctions made up of various III-V compound semiconductors are used to create the double or multiple potential barriers in the conduction band or valence band. Reasonably high performance III-V resonant tunnel diodes have been realized. But such devices have not entered mainstream applications yet because the processing of III-V materials is incompatible with Si CMOS technology and the cost is high.
305
306 -
==== Si/!SiGe Resonant Tunnel Diodes ====
+
== Coulomb Blockade ==
307
308 -
Resonant tunnel diodes can also be realized using the Si/!SiGe materials system. But the performance of Si/!SiGe resonant tunnel diodes was limited due to the limited conduction band and valence band discontinuities between Si and SiGe alloys. Resonant tunneling of holes through Si/!SiGe heterojunctions was attempted first because of the typically relatively larger valence band discontinuity in Si/SiGe heterojunctions than the conduction band discontinuity. This has been observed, but negative differential resistance was only observed at low temperatures but not at room temperature ![3]. Resonant tunneling of electrons through Si/!SiGe heterojunctions was obtained later, with a limited peak-to-valley current ratio (PVCR) of 1.2 at room temperature ![4]. Subsequent developments have realized Si/!SiGe RTDs (electron tunneling) with a PVCR of 2.9 with a PCD of 4.3 kA/cm2 ![5] and a PVCR of 2.43 with a PCD of 282 kA/cm2 at room temperature ![6].
+
In physics, a Coulomb blockade, named after Charles-Augustin de Coulomb, is the increased resistance at small bias voltages of an electronic device comprising at least one low-capacitance tunnel junction. According to the laws of classical electrodynamics, no current can flow through an insulating barrier. According to the laws of quantum mechanics, however, there is a non-vanishing (larger than zero) probability for an electron on one side of the barrier to reach the other side. When a bias voltage is applied, this means that there will be a current flow. In first-order approximation, that is, neglecting additional effects, the tunneling current will be proportional to the bias voltage. In electrical terms, the tunnel junction behaves as a resistor with a constant resistance, also known as an ohmic resistor. The resistance depends exponentially on the barrier thickness. Typical barrier thicknesses are on the order of one to several nanometers. An arrangement of two conductors with an insulating layer in between not only has a resistance, but also a finite capacitance. The insulator is also called dielectric in this context, the tunnel junction behaves as a capacitor.
309
310 +
[[Div(start, class=clear)]][[Div(end)]]
311 +
[[Image(cb.png, 140, class=align-center)]]
312 [[Div(start, class=clear)]][[Div(end)]]
313
314 -
=== Quantum Dots Lab ===
+
The [/tools/aqme/ Coulomb Blockade Lab in AQME] allows simulation of non-linear current-voltage (I-V) characteristics through single and double quantum dots and as such illustrates various single electron transport phenomena.
315
316 [[Div(start, class=clear)]][[Div(end)]]
317
318 -
[[Image(pic25_qdot.png, 140 class=align-left)]] [[Resource(qdot)]] computes the eigenstates of a particle in a box of various shapes including domes and pyramids.
319
320 -
[[Div(start, class=clear)]][[Div(end)]]
+
Available resources:
321 +
322 +
* [/site/resources/2008/03/04233/cb_lab_descriptions_v2.doc Coulomb Blockade Exercises]
323 +
* Coulomb blockade design exercises that uses Quantum Dot Lab
324 +
325 +
Users no longer have to search the nanoHUB to find the appropriate applications for discovery that are related to quantum mechanics; users, both instructors and students, can simply log in to the AQME toolbox and take advantage of the assembled tools and resources, such as animations, exercises or podcasts.
326
327 -
=== References ===
+
=== AQME Constituent Tools ===
328
329 -
----
+
==== [[Resource(pcpbt)]] ====
330 -
i. Yoshida, M. Katsueda, Y. Maruyama, I. Kohjiro, "A Highly Efficient 1.9 GHz Si High Power MOSFET," ESSDERC' 96 - 26th European Solid-State Device Research Conference, (G. Baccarani and M. Rudan, Eds.) Gif-sur-Yvette Cedex, France: Editions Frontiers, pp. 405¬408, 1996. [[BR]]
+
==== [[Resource(bsclab)]] ====
331 -
ii. K. K. Chu, M. J. Murphy, J. Burm, W. J. Schaff , L. F. Eastman, "High Speed-High Power AlGaN/GaN Heterostructure Field Effect Transistors with Improved Ohmic Contacts," 24th International Symposium on Compound Semiconductors, WA2, 1997. [[BR]]
+
==== [[Resource(1308)]] ====
332 -
iii Y.¬F. Wu, D. Kapolnek, P. Kozodoy, S. Keller, B. P. Keller, S. P. Denbaars, U. K. Mishra, "AlGaN/GaN MODFETs with Low Ohmic Contact Resistance Through Source-Drain n+ Re-growth," 24th International Symposium on Compound Semiconductors, WA3, 1997. [[BR]]
+
==== [[Resource(3847)]] ====
333 -
iv E. J. Lum, "GaAs Technology Rides the Wireless Wave," IEEE GaAs IC Symp. Techn. Digest , 1997, pp. 11¬13. [[BR]]
+
==== [[Resource(221)]] ====
334 -
v H. Brech. Optimization of GaAs based High Electron Mobility Transistors
+
==== [[Resource(5203)]] ====
335 -
by Numerical Simulations. Doctoral dissertation. München. March 1998. [Online]. Available: http://www.iue.tuwien.ac.at/phd/brech/diss.htm#contents [[BR]]
+
==== [[Resource(230)]] ====
336 +
==== [[Resource(qdot)]] ====
337 +
==== [[Resource(bulkmc)]] ====
338 +
==== [[Resource(2925)]] ====

nanoHUB.org, a resource for nanoscience and nanotechnology, is supported by the National Science Foundation and other funding agencies. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.