AQME Advancing Quantum Mechanics for Engineers
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 Version 190
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1  [[Image(aqme_large.gif, class=aligncenter)]]  

2  +  == Introduction to Advancing Quantum Mechanics for Engineers and Physicists ==


3  
4    +  "Advancing Quantum Mechanics for Engineers" (AQME) toolbox is an assemblage of individually authored tools that, used in concert, offer educators and students a onestopshop 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  
11  +  *[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.


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    +  * Let us know when things do not work by filing a ticket via the nanoHUB "Help" feature on every page.


16  
17    +  * 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  
19    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 particlewave duality (the photoelectric effect and the doubleslit experiment), the solutions of the timeindependent Schrodinger equation for open systems (piecewise constant potentials), tunneling and bound states. The description of the solution of the Schrodinger equation for periodic potentials (KronigPenney model) naturally follows from the discussion of double well, triple well and nwell 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 TsuEsaki 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.


20  
21    AQME 
+  Finally, be sure to share AQME and other nanoHUB success stories; the nanotechnology community and its supporters need to hear of nanoHUB's impact.

22    +  
23  
24    +  '''Discovery that is Possible through Quantum Mechanics'''


25  
26    +  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. 2D confinement, for example, is restricted in only one dimension, resulting in a quantum well (or plane). Lasers are currently built from this dimension. 1D confinement occurs in nanowires, and 0D confinement is found only in the quantum dot.


27  
28    +  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 360 nm in diameter. That’s still 30 to 600 times the size of a typical atom. A quantum dot exhibits 0D confinement, meaning that electrons are confined in all three dimensions. In nature, only atoms have 0D 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.


29  
30    +  Confinement also increases the efficiency of today’s electronics. The laser is based on a 2D 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 1D confinement in wires or 0D 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.


31  
32    +  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:


33  +  
34  +  [[Resource(4920)]]


35  +  
36  +  [[Resource(5164)]]


37  +  
38  +  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 particlewave duality (the photoelectric effect and the doubleslit experiment), the solutions of the timeindependent Schrödinger equation for open systems (piecewise constant potentials), tunneling, and bound states. The description of the solution of the Schrödinger equation for periodic potentials (KronigPenney model) naturally follows from the discussion of double well, triple well and nwell 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 band structure by fitting the curvature of the bands. The TsuEsaki 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/firstyear graduate students is then completed with the discussion of the stationary and timedependent 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.


39  
40  == ParticleWave Duality ==  
41  
42    [[Image(pic1_duality.png, 200 class=alignleft)]] 
+  [[Image(pic1_duality.png, 200, class=alignleft)]] A waveparticle dual nature was discovered and publicized in the early debate about whether light was composed of particles or wave. Evidence for the description of lightaswaves was well established at the turn of the century when the photoelectric effect introduced firm evidence of a lightasparticle 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.

43  
44  [[Resource(4916)]]  
45  
46    This movie helps students to better 
+  This movie helps students to better distinguish when nanothings behave as particles and when they behave as waves. The link below connects to an exercise on these concepts.

47  
48  [[Div(start, class=clear)]][[Div(end)]]  
49    
50    Exercises:


51  
52    +  [[Resource(4918)]]


53  
54  [[Div(start, class=clear)]][[Div(end)]]  
55  
56  
57    == Solution of the TimeIndependent 
+  == Solution of the TimeIndependent Schrödinger Equation ==

58    +  === [/tools/aqme/ PieceWise Linear Barrier Tool in AQME] – Open Systems ===


59    === PieceWise 
+  
60  
61  [[Div(start, class=clear)]][[Div(end)]]  
62  
63    +  [[Image(pcpbt1.bmp, 120, class=alignleft)]]


64  +  [[Image(pcpbt2.bmp, 120, class=alignleft)]]


65  +  [[Image(pcpbt3.bmp, 120, class=alignleft)]]


66  
67  [[Div(start, class=clear)]][[Div(end)]]  
68  
69    Exercises:


70  
71    +  The [/tools/aqme/ PieceWise Linear Barrier Tool in AQME] allows calculation of the transmission and the reflection coefficient of arbitrary five, seven, nine, eleven and 2nsegment piecewise constant potential energy profile. For the case of multiwell structure it also calculates the quasibound 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/ PieceWise 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    +  Available resources:


76  
77    * [[Resource( 
+  * [[Resource(4831)]]

78    +  * [[Resource(4833)]]


79    * 
+  * [[Resource(4853)]]

80    +  * [[Resource(4873)]]


81    * [[Resource(4849)]]

+  * [[Resource(5319)]]

82    +  * [[Resource(4849)]]


83    * [[Resource(5102)]]

+  * [[Resource(5102)]]

84    +  * [[Resource(5130)]]


85    * [[Resource(5130)]]

+  
86  
87  [[Div(start, class=clear)]][[Div(end)]]  
88  
89    === Bound States Lab ===

+  === [/tools/aqme/ Bound States Lab in AQME] ===

90  
91  [[Div(start, class=clear)]][[Div(end)]]  
92  
93    The [ 
+  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.

94  
95    +  The panel below (left) shows energy eigenstates of a harmonic oscillator. Probability density of the ground state that demonstrates purely quantummechanical 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).


96  
97  [[Div(start, class=clear)]][[Div(end)]]  
98  
99    [[Image(pic6_state1top.png, 120 class=alignleft)]]

+  [[Image(pic6_state1top.png, 120, class=alignleft)]]

100    [[Image(pic7_state2left.png, 140 class=alignleft)]]

+  [[Image(pic7_state2left.png, 140, class=alignleft)]]

101    [[Image(pic8_state3right.png, 125 class=alignleft)]]

+  [[Image(pic8_state3right.png, 125, class=alignleft)]]

102  
103  [[Div(start, class=clear)]][[Div(end)]]  
104  
105    +  Available resources:


106    +  
107    +  
108  
109    * [[Resource(4976)]]

+  * [[Resource(4884)]]

110  +  * [[Resource(4976)]]


111  
112  [[Div(start, class=clear)]][[Div(end)]]  
113  
114  
115  == Energy Bands and Effective Masses ==  
116  
117  [[Div(start, class=clear)]][[Div(end)]]  
118  
119    === Periodic Potential Lab ===

+  === [/tools/aqme/ Periodic Potential Lab in AQME] ===

120  
121    [[Image(pic10_perpot2.png, 150 class=alignright)]] [[Image(pic9_perpot1.png, 160 class=alignright)]] The [ 
+  [[Image(pic10_perpot2.png, 150, class=alignright)]] [[Image(pic9_perpot1.png, 160, class=alignright)]] The [/tools/aqme/ Periodic Potential Lab in AQME] solves the timeindependent Schrödinger Equation in a 1D 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 (Ek relation for carriers).

122  
123    +  Available resources:


124  
125    +  [[Resource(4851)]]


126  
127  [[Div(start, class=clear)]][[Div(end)]]  
128  
129    === 
+  === [/tools/aqme/ Band Structure Lab in AQME] ===

130  
131  [[Div(start, class=clear)]][[Div(end)]]  
132  
133    [[Image(pic12_band2.png, 160 class=alignright)]] [[Image(pic11_band1.png, 160 class=alignright)]] 
+  [[Image(pic12_band2.png, 160, class=alignright)]] [[Image(pic11_band1.png, 160, class=alignright)]] Band structure of Si (left panel) and !GaAs (right panel).

134  
135  [[Div(start, class=clear)]][[Div(end)]]  
136  
137    In solidstate 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 [ 
+  In solidstate 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 ultrascaled (thin body) quantum wells, and nanowires of different cross sections. Band Structure Lab uses the sp3s*d5 tightbinding 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=alignleft)]] 
+  [[Image(diamond.png, 140, class=alignleft)]] The figure on the left illustrates the first Brillouin zone of FCC lattice that corresponds to the first Brillouin zone for all diamond and Zincblende 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 
+  Supplemental Information: Specification of HighSymmetry Points

150  
151  [[Div(start, class=clear)]][[Div(end)]]  
152  
153  SymbolDescription  
154  ΓCenter of the Brillouin zone  
155  Simple Cube  
156  MCenter of an edge  
157    RCorner 
+  RCorner point

158  XCenter of a face  
159    FaceCentered 
+  FaceCentered Cubic

160  KMiddle of an edge joining two hexagonal faces  
161  LCenter of a hexagonal face  
162  UMiddle of an edge joining a hexagonal and a square face  
163  WCorner point  
164  XCenter of a square face  
165  BodyCentered Cubic  
166  HCorner point joining four edges  
167  NCenter of a face  
168  PCorner point joining three edges  
169  Hexagonal  
170  ACenter of a hexagonal face  
171    HCorner 
+  HCorner point

172  KMiddle of an edge joining two rectangular faces  
173  LMiddle of an edge joining a hexagonal and a rectangular face  
174  M Center of a rectangular face  
175  
176  [[Div(start, class=clear)]][[Div(end)]]  
177  
178  
179  == Real World Applications ==  
180  
181  [[Div(start, class=clear)]][[Div(end)]]  
182  
183    === Schred ===

+  === [/tools/aqme/ Schred Tool in AQME] ===

184  
185    [[Image(pic13_schred1.png, 140 class=alignleft)]] [[Image(pic14_schred2.png, 155 class=alignleft)]] [[Image(pic15_schred3.png, 140 class=alignleft)]]

+  [[Image(pic13_schred1.png, 140, class=alignleft)]] [[Image(pic14_schred2.png, 155, class=alignleft)]] [[Image(pic15_schred3.png, 140, class=alignleft)]]

186  
187  [[Div(start, class=clear)]][[Div(end)]]  
188  
189    [ 
+  The [/tools/aqme/ Schred Tool in AQME] calculates the envelope wavefunctions and the corresponding boundstate energies in a typical MOS (MetalOxideSemiconductor) or SOS (SemiconductorOxide Semiconductor) structure and in a typical SOI structure by solving selfconsistently the onedimensional (1D) 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.

190  
191    +  Available resources:


192  
193    * [[Resource(4900)]]

+  * [[Resource(4900)]]

194    +  * [[Resource(4904)]]


195    * [[Resource(4904)]]

+  
196  
197  
198  [[Div(start, class=clear)]][[Div(end)]]  
199  
200    [[Image(pic16_bandschem.png, 140 class=alignleft)]] Right panel  Potential diagram for inversion of ptype semiconductor. In this first notation Εij refers to the jth subband from either the Δ2band (i=1) or Δ4band (i=2). Left panel  Constantenergy surfaces for the conductionband of silicon showing six conductionband valleys in the direction of momentum space. The band minima, corresponding to the centers of the ellipsoids, are 85% of the way to the Brillouinzone boundaries. The long axis of an ellipsoid corresponds to the longitudinal effective mass of the electrons in silicon, 
+  [[Image(pic16_bandschem.png, 140, class=alignleft)]] Right panel  Potential diagram for inversion of ptype semiconductor. In this first notation Εij refers to the jth subband from either the Δ2band (i=1) or Δ4band (i=2). Left panel  Constantenergy surfaces for the conductionband of silicon showing six conductionband valleys in the direction of momentum space. The band minima, corresponding to the centers of the ellipsoids, are 85% of the way to the Brillouinzone 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 Δ2band has the longitudinal mass (ml) perpendicular to the semiconductor interface and the Δ4band 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 Δ2band, 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 Δ4band 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.

201  
202  [[Div(start, class=clear)]][[Div(end)]]  
203  
204    === 
+  === [/tools/aqme/ 1D Heterostructure Tool AQME] ===

205  
206    The [ 
+  The [/tools/aqme/ 1D Heterostructure Tool AQME] simulates confined states in 1D heterostructures by calculating charge selfconsistently 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 deltadoped 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.

207  
208  [[Div(start, class=clear)]][[Div(end)]]  
209  
210    [[Image(1dhet1.png, 
+  [[Image(1dhet1.png, 120, class=alignleft)]]

211    [[Image(1dhet2.png, 
+  [[Image(1dhet2.png, 120, class=alignleft)]]

212  
213  [[Div(start, class=clear)]][[Div(end)]]  
214  
215    +  Available resources:


216  
217    * [[Resource(5231)]]

+  * [[Resource(5231)]]

218    +  * [[Resource(5233)]]


219    * [[Resource(5233)]]

+  
220  
221  
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  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    [[Image(hetero2.png, 120 class=alignleft)]] Schematic cross section of a High Electron Mobility Transistor (HEMT).

+  [[Image(hetero2.png, 120, class=alignleft)]] Schematic cross section of a High Electron Mobility Transistor (HEMT).

231  
232  [[Div(start, class=clear)]][[Div(end)]]  
233  
234    [[Image(hetero4.png, 150 class=alignright)]] 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 AA' is shown to the right.

+  [[Image(hetero4.png, 150, class=alignright)]] 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 AA' is shown to the right.

235  
236  [[Div(start, class=clear)]][[Div(end)]]  
237  
238  [[Div(start, class=clear)]][[Div(end)]]  
239  
240  
241    === Resonant Tunneling Diode Lab ===

+  === [/tools/aqme/ Resonant Tunneling Diode Lab in AQME] ===

242    +  
243    +  
244  
245  +  [[Div(start, class=clear)]][[Div(end)]]


246  +  [[Image(rtd1.png, 140, class=aligncenter)]]


247  +  [[Image(rtd2.png, 140, class=aligncenter)]]


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  
252  [[Div(start, class=clear)]][[Div(end)]]  
253  
254    +  [[Image(pic18_restunn.png, 120, class=alignleft)]] [[Image(pic19_restun2.png, 120, class=alignleft)]]


255  +  
256  +  [[Div(start, class=clear)]][[Div(end)]]


257  
258    +  Available resources:


259  
260    * [[Resource(3949)]]

+  * [[Resource(891)]]

261  +  * [[Resource(3949)]]


262  
263  
264  [[Div(start, class=clear)]][[Div(end)]]  
265  
266    === Quantum 
+  === [/tools/aqme/ Quantum Dot Lab in AQME] ===

267  
268    Individual quantum dots can be created from twodimensional 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 (liftoff 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, 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).

+  Individual quantum dots can be created from twodimensional 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 (liftoff 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).

269  
270  [[Div(start, class=clear)]][[Div(end)]]  
271  
272    [[Image(pic25_qdot.png, 140 class=alignleft)]] [ 
+  [[Image(pic25_qdot.png, 140, class=alignleft)]] [/tools/aqme/ Quantum Dot Lab in AQME] computes the eigenstates of a particle in a box of various shapes, including domes and pyramids.

273  
274  [[Div(start, class=clear)]][[Div(end)]]  
275  
276    +  Available resources:


277  
278    * [[Resource(2846)]]

+  * [[Resource(189)]] is a nano 101, introductory lecture that starts from particlewave duality and explores the concepts of quantum dots.

279  +  * [[Resource(4194)]] (by Lee, Ryu, Klimeck)


280  +  * [[Resource(2846)]] (by Fodor, Guo)


281  +  * [[Resource(4203)]]


282  +  
283  
284  
285  [[Div(start, class=clear)]][[Div(end)]]  
286  
287  == Scattering and Fermi's Golden Rule ==  
288  
289    +  [[Image(scattering.png, 250, class=alignleft)]]


290  +  [[Div(start, class=clear)]][[Div(end)]]


291  
292    +  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 nonuniformities 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 (mirrorlike) reflections. The types of nonuniformities (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 MonteCarlo 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.


293  
294    +  Available Resources:


295  
296    * 
+  * [[Resource(5019)]]

297    +  * [[Resource(5277)]]


298    +  
299    +  
300    * 
+  
301  
302  
303  == Coulomb Blockade ==  
304  
305    In physics, a Coulomb blockade, named after CharlesAugustin de Coulomb, is the increased resistance at small bias voltages of an electronic device comprising at least one lowcapacitance 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 
+  In physics, a Coulomb blockade, named after CharlesAugustin de Coulomb, is the increased resistance at small bias voltages of an electronic device comprising at least one lowcapacitance 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 nonvanishing (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 firstorder 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.

306  
307  +  [[Div(start, class=clear)]][[Div(end)]]


308  +  [[Image(cb.png, 140, class=aligncenter)]]


309  [[Div(start, class=clear)]][[Div(end)]]  
310  
311  +  The [/tools/aqme/ Coulomb Blockade Lab in AQME] allows simulation of nonlinear currentvoltage (IV) characteristics through single and double quantum dots and as such illustrates various single electron transport phenomena.


312  
313  [[Div(start, class=clear)]][[Div(end)]]  
314  
315    The [[Resource{2925)]] allows simulation of nonlinear currentvoltage (IV) characteristics through single and double quantum dots and as such illustrates various single electron transport phenomena.


316  
317    +  Available resources:


318  
319  +  * [/site/resources/2008/03/04233/cb_lab_descriptions_v2.doc Coulomb Blockade Exercises]


320  +  * Coulomb blockade design exercises that uses Quantum Dot Lab


321  
322    +  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.


323  
324    +  === AQME Constituent Tools ===


325  
326    +  ==== [[Resource(pcpbt)]] ====


327  +  ==== [[Resource(bsclab)]] ====


328  +  ==== [[Resource(1308)]] ====


329  +  ==== [[Resource(3847)]] ====


330  +  ==== [[Resource(221)]] ====


331  +  ==== [[Resource(5203)]] ====


332  +  ==== [[Resource(230)]] ====


333  +  ==== [[Resource(qdot)]] ====


334  +  ==== [[Resource(bulkmc)]] ====


335  +  ==== [[Resource(2925)]] ==== 