DOCSLIB.ORG

• Quantum Wells and Superlattices Quantum Wells • Quantum Wells

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

2•. QuantumQuantum W wellsell Sta andtes ( QsuperlatticesWS) and Quantum Size Effects Qualitative explanationQuantum… wells k zz •2D conducting system (x-y plane), size - y quantization in z, k y Dd d ! λF x kx Metals: λ 0.5 nm • F ! Electronic structure •Semiconductors: λF 10 100 nm in parabolic subbands ! − ikx x ik y y "(x, y, z) = !n (z)e e 2 2 2 2 ! n k x + k y E(n•,Filmk ,k ) = deposition + technology (e.g. molecular-beam epitaxy) ! x y 2D2 2 quantum wells with c-Si, GaAs 4 26 • Quantum wells and superlattices •2D electron density of states (per unit area): (2m! m!)1/2 DOS (!) = g g x y 2D s v 2π 2 ! effective masses fo x and y motion gs spin degeneracy, 2, Si, GaAS gv valley degeneracy (degeneracy of states at bottom of conduction band); 1 for GaAs (single valley); 2 or 4 for Si (indirect gap; depending on number of valleys involved) GaAS: ! ! ! 3D mx = my = ml = 0.067me direct gap semiconductor Si{100}: indirect gap semiconductor, more complicated m! 0.19 m transverse t ! e m! 0.92 m longitudinal l ! e 27 • Quantum wells and superlattices •Calculate λF Electron density (areal) na ! ! 1/2 (mxmy) !F na = DOS2D(!F )!F = gsgv 2π!2 2π 4πn 1/2 k = = a ⇒ F λ g g F ! s s " GaAS {100}: λF = 40 nm Si {100}: λ = 35 110 nm (depending on n ) F − a 28 • Quantum wells and superlattices •Square-well potential Infinite , z > d, z < 0 • ∞ V (z) = 0 , 0 < z < d 1D particle in a box Ψz(0) = Ψz(d) = 0 nπ Ψ = A sin k z , k = z z z d •Finite Maximum number of bound states: V0 2m! V d2 1/2 n = 1 + Int z| 0| d max 2π2 !" ! # $ 29 • Quantum wells and superlattices •Electron energy: 2 2 ! 2 ! 2 ! = !n(z) + ! kx + ! ky 2mx 2my - quantized kinetic energy for 2D free e motion energy levels ! ! mx, my effective masses for (x,y) 2 2 ! π 2 n = 1, 2, 3... !n(z) = ! 2 n 2mzd effective mass for motion in z direction ! (z) 50 meV (d=10 nm) • GaAs, 1 ! • ! n2 n ∝ 30 2. Quantum Well States (QWS) and Quantum Size • QuantumEffects wells and superlattices Qualitative explanation… kz n = 1, 2, 3... !n subbands k y D kx Electronic structure in parabolic subbands ikx x ik y y "(x, y, z) = !n (z)e e 2 2 2 2 ! n k x + k y E(n,k ,k ) = + finite well x y 2D2 2 zero point4 energy • No allowed state until ! = !1(z) infinite well • DOS constant until ! = ! 2 ( z ) subband starts to be populated infinite well 31 • Quantum wells and superlattices Sandwitch thin layer GaAs (d 5 nm, ! 1.5 eV) • ! g ! between thick layers AlxGa1-xAs (! 1.42 + 1.26x eV 2 eV for x 0.45) electrons g ! ! ! •Similar unit-cell parameters! possible epitaxial growth one over the other. conduction •The band gap of AlGaAs (large gap) bands straddles that of GaAs (smaller gap). valence •Electrons in conduction band of GaAs layer confined in z direction, by forbidden energy gap of AlGaAs in both sides holes 32 • Quantum wells and superlattices •Triangular potential well Si-SiO2 junction in MOS Triangular wells - infinite Triangular potential wells are quite common in real devices. Examples: Si MOSFETs; single-interface GaAs/Al0.3Ga0.7As heterojunctions. Idealization: GaAS-n-AlGaAs , z < 0 ∞ V (z) = Ez , z > 0 Linear variation of confining electrostatic potential Quantized energy levels 2 1/3 3 2/3 E 3 2/3 !n(z) ( π!e) (n + ) ; n = 0, 1, 2, ... ! 2 2m! 4 ! z " 33 • Quantum wells and superlattices •Optical absorption: • Bulk ! absorption involves e- states at conduction band minimum and valence band maximum. bulk • QW! quantized levels n2 ! square well n ∝ d2 2/3 !n E triangular well Lowest allowed∝ level, n=1 in square well and n=0 in absorption triangular well threshold •GaAs, "F=10 nm • The threshold energy for optical absorption increases ("100 meV) !sharp exciton (bound e-h pair) peak • GaAs, d=21 nm ! max. number bound states nmax=4 ! 4 peaks d=14 nm ! max. number bound states nmax=3 ! 3 peaks •Absorption profile: exciton peaks occurring at each discontinuity in DOS (peak when a new subband starts to be populated) 34 • Quantum wells and superlattices • Thinest QW !GaAs d=5nm Fine structure: splitting in exciton peak one bound state n=, in valence band split into two 35 • Quantum wells and superlattices 1 j=3/2 states split into two ! conf ∝ m! heavy holes light holes Bulk QW heavy!flat band The light is the heavy now 36 • Quantum wells and superlattices Metals, example Pb islands Building island heights Number of islands Covered area histograms 37 Courtesy: Rodolfo Miranda R. Otero, A. L. Vázquez de Parga and R. Miranda PRB 66, 115401 (2002) • Quantum wells and superlattices Superlattices • Molecular beam-epitaxy ! ordered arrays of heterostructures or homojunctions • d1 material (1) + d2 material (2) ! new period d= d1 + d2 Periodic array of QW’s 38 • Quantum wells and superlattices • Periodic array of semiconductor QW’s. •d1 : thickness of well •d2 : thickness of barrier • Two possibilities: 1- Isolated QW’s, d2 >> d1 # "F Multiple quantum wells (no tunnelling of electrons through barriers) 2- Interacting QW’s, d1 # d2 Superlattice (very different properties from bulk material) 39 • Quantum wells and superlattices • Electronic structure of semiconductor superlattices gaps at nπ k = d energy-width • Tight-binding model ! Bloch function, array N QW’s 1/2 iknd Enveloppe function of nth well centerd Ψ (z) = N − e ψ (z nd) k n − at z=nd, overlap with the two n neighbours. Equivalent to atomic orbitals ! or Wannier functions • Analogy with normal crystals ! energy of electron motion in superlattice !(k) = ! α 2β cos kd i − i − i i-th level of a self-energy overlap cos-like dependence, miniband 40 well, i=1,2,3... integral, >0 integral, >0 • Quantum wells and superlattices 41 • Quantum wells and superlattices Density of states ! Nm 1 (! !i + α) g(!) = cos− − − 2π2 2β ! ! i " , ! ! + α < 2 β | − i i| | i| Smearing corresponding to band width 42 • Quantum wells and superlattices π Gaps at ∓ d nπ • Forbidden gaps at k = (d >> a) d v (k) ! smaller than BZ !no periodic extended state g along z • Bloch oscillations: E d.c. electric field ! oscillatory electron velocity, a.c current in superlattice. !(k) = !0 !1 cos kd ! group velocity of electron: − 1 ∂" "1 vg(k) = = sin kd ! ∂k ! 43 • Quantum wells and superlattices (previous page figure) • Consider ! ( k ) = ! 0 ! 1 cos k d − 1 ∂" "1d ! group velocity of electron: vg(k) = = sin kd ! ∂k ! π Periodic, v = 0 at k = , Bragg scattering g | | d ! ˙ • Apply E !k = eE k = ko eEt/! ⇒ − ⇒ − ! Period for motion in reciprocal space between k = π/d ± 2π ! T = d eE Bloch oscillatrions observed if T < τ (scattering relaxation time) , not possible to observe in bulk crystals (d=a) ; but superlattice d 10 100a ! − 44 • Quantum wells and superlattices • Stark ladder : applied E field prevents minibands !no Bloch oscillations (no wave function overlap) If eEd 4 β ! | i| miniband field induced width potential displacement 45 • Quantum wells and superlattices • Effects on phonon propagation !acoustic phonons folded back at k = π /d ± (Phonon dispersion Brillouin new boundaries (d: new period superlattice) in a crystal) - Optical phonons ! confined to one or other layer of material: confined modes • Doping superlattice !periodic array of homojunctions or p-n junctions (nipi structure) intrinsic layer - Band-edge modulations determined by doping level: between n and p $ dopant concentration ! $ Eo modulation energy (next page figure) !eff = ! E ! “indirect” minimum gap in real space g g − 0 46 • Quantum wells and superlattices E N , doping concentration Nipi structure o ∝ d spatial variation of n- and p- doping spatial variation (in z) of conduction and valence bands nipi semimetallic 47 • Quantum wells and superlattices Nipi structure • To estimate the modulation consider the homojunction as a capacitor: semiconductor !! dielectric constant ! C/A 0 ! d/2 Charge per unit area, Q/A = Nddn = Nadp widths of p-type and n-type regions (assume the same for - and + regions) 2 2 2 e Nddnd e Ndd C = Q/V E0 = (dn = dp = d/2) ⇒ " 2!!0 4!!0 electrostatic potential difference=E0/e 48 • Quantum wells and superlattices Nipi structure • Very high doping ! modulation very pronounced, nipi semimetal Band modulation and ef f can be controlled varying charge • !g densities in n- and p- layers ! injection of excess e- or holes electrically or optically reduce E0 by neutralization of charge donors or acceptors. e- -hole pairs after absorption of photons eff Light intensity $ ! ! g $ 49.
Recommended publications

  • Accomplishments in Nanotechnology

    U.S. Department of Commerce Carlos M. Gutierrez, Secretaiy Technology Administration Robert Cresanti, Under Secretaiy of Commerce for Technology National Institute ofStandards and Technolog}' William Jeffrey, Director Certain commercial entities, equipment, or materials may be identified in this document in order to describe an experimental procedure or concept adequately. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the materials or equipment used are necessarily the best available for the purpose. National Institute of Standards and Technology Special Publication 1052 Natl. Inst. Stand. Technol. Spec. Publ. 1052, 186 pages (August 2006) CODEN: NSPUE2 NIST Special Publication 1052 Accomplishments in Nanoteciinology Compiled and Edited by: Michael T. Postek, Assistant to the Director for Nanotechnology, Manufacturing Engineering Laboratory Joseph Kopanski, Program Office and David Wollman, Electronics and Electrical Engineering Laboratory U. S. Department of Commerce Technology Administration National Institute of Standards and Technology Gaithersburg, MD 20899 August 2006 National Institute of Standards and Teclinology • Technology Administration • U.S. Department of Commerce Acknowledgments Thanks go to the NIST technical staff for providing the information outlined on this report. Each of the investigators is identified with their contribution. Contact information can be obtained by going to: http ://www. nist.gov Acknowledged as well,
  • Federico Capasso “Physics by Design: Engineering Our Way out of the Thz Gap” Peter H

    Federico Capasso “Physics by Design: Engineering Our Way out of the Thz Gap” Peter H

    6 IEEE TRANSACTIONS ON TERAHERTZ SCIENCE AND TECHNOLOGY, VOL. 3, NO. 1, JANUARY 2013 Terahertz Pioneer: Federico Capasso “Physics by Design: Engineering Our Way Out of the THz Gap” Peter H. Siegel, Fellow, IEEE EDERICO CAPASSO1credits his father, an economist F and business man, for nourishing his early interest in science, and his mother for making sure he stuck it out, despite some tough moments. However, he confesses his real attraction to science came from a well read children’s book—Our Friend the Atom [1], which he received at the age of 7, and recalls fondly to this day. I read it myself, but it did not do me nearly as much good as it seems to have done for Federico! Capasso grew up in Rome, Italy, and appropriately studied Latin and Greek in his pre-university days. He recalls that his father wisely insisted that he and his sister become fluent in English at an early age, noting that this would be a more im- portant opportunity builder in later years. In the 1950s and early 1960s, Capasso remembers that for his family of friends at least, physics was the king of sciences in Italy. There was a strong push into nuclear energy, and Italy had a revered first son in En- rico Fermi. When Capasso enrolled at University of Rome in FREDERICO CAPASSO 1969, it was with the intent of becoming a nuclear physicist. The first two years were extremely difficult. University of exams, lack of grade inflation and rigorous course load, had Rome had very high standards—there were at least three faculty Capasso rethinking his career choice after two years.
  • Bandgap-Engineered Hgcdte Infrared Detector Structures for Reduced Cooling Requirements

    Bandgap-Engineered Hgcdte Infrared Detector Structures for Reduced Cooling Requirements

    Bandgap-Engineered HgCdTe Infrared Detector Structures for Reduced Cooling Requirements by Anne M. Itsuno A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Electrical Engineering) in The University of Michigan 2012 Doctoral Committee: Associate Professor Jamie D. Phillips, Chair Professor Pallab K. Bhattacharya Professor Fred L. Terry, Jr. Assistant Professor Kevin P. Pipe c Anne M. Itsuno 2012 All Rights Reserved To my parents. ii ACKNOWLEDGEMENTS First and foremost, I would like to thank my research advisor, Professor Jamie Phillips, for all of his guidance, support, and mentorship throughout my career as a graduate student at the University of Michigan. I am very fortunate to have had the opportunity to work alongside him. I sincerely appreciate all of the time he has taken to meet with me to discuss and review my research work. He is always very thoughtful and respectful of his students, treating us as peers and valuing our opinions. Professor Phillips has been a wonderful inspiration to me. I have learned so much from him, and I believe he truly exemplifies the highest standard of teacher and technical leader. I would also like to acknowledge the past and present members of the Phillips Research Group for their help, useful discussions, and camaraderie. In particular, I would like to thank Dr. Emine Cagin for her constant encouragement and humor. Emine has been a wonderful role model. I truly admire her expertise, her accom- plishments, and her unfailing optimism and can only hope to follow in her footsteps. I would also like to thank Dr.
  • Optimizing Fabrication and Modeling of Quantum Dot Superlattice for Fets and Nonvolatile Memories Pial Mirdha University of Connecticut - Storrs, Pialmirdha@Gmail.Com

    Optimizing Fabrication and Modeling of Quantum Dot Superlattice for Fets and Nonvolatile Memories Pial Mirdha University of Connecticut - Storrs, [email protected]

    University of Connecticut OpenCommons@UConn Doctoral Dissertations University of Connecticut Graduate School 12-12-2017 Optimizing Fabrication and Modeling of Quantum Dot Superlattice for FETs and Nonvolatile Memories Pial Mirdha University of Connecticut - Storrs, [email protected] Follow this and additional works at: https://opencommons.uconn.edu/dissertations Recommended Citation Mirdha, Pial, "Optimizing Fabrication and Modeling of Quantum Dot Superlattice for FETs and Nonvolatile Memories" (2017). Doctoral Dissertations. 1686. https://opencommons.uconn.edu/dissertations/1686 Optimizing Fabrication and Modeling of Quantum Dot Superlattice for FETs and Nonvolatile Memories Pial Mirdha PhD University of Connecticut 2017 Quantum Dot Superlattice (QDSL) are novel structures which can be applied to transis- tors and memory devices to produce unique current voltage characteristics. QDSL are made of Silicon and Germanium with an inner intrinsic layer surrounded by their respective oxides and in the single digit nanometer range. When used in transistors they have shown to induce 3 to 4 states for Multi-Valued Logic (MVL). When applied to memory they have been demonstrated to retain 2 bits of charge which instantly double the memory density. For commercial application they must produce consistent and repeatable current voltage characteristics, the current QDSL structures consist of only two layers of quantum dots which is not a robust design. This thesis demonstrates the utility of using QDSL by designing MVL circuit which consume less power while still producing higher computational speed when compared to conventional cmos based circuits. Additionally, for reproducibility and stability of current voltage characteristics, a novel 4 layer of both single and mixed quantum dots are demonstrated.
  • Electronic Supplementary Information: Low Ensemble Disorder in Quantum Well Tube Nanowires

    Electronic Supplementary Information: Low Ensemble Disorder in Quantum Well Tube Nanowires

    Electronic Supplementary Material (ESI) for Nanoscale. This journal is © The Royal Society of Chemistry 2015 Electronic Supplementary Information: Low Ensemble Disorder in Quantum Well Tube Nanowires Christopher L. Davies,∗a Patrick Parkinson,b Nian Jiang,c Jessica L. Boland,a Sonia Conesa-Boj,a H. Hoe Tan,c Chennupati Jagadish,c Laura M. Herz,a and Michael B. Johnston.a‡ (a) 3.950 nm (b) GaAs-QW 2.026 nm 2.181 nm 4.0 nm 3.962 nm 50 nm 20 nm GaAs-core Fig. S1 TEM image of top of B50 sample Fig. S2 TEM image of bottom of B50 sample S1 TEM Figure S1(a) corresponds to a low magnification bright field TEM image of a representative cross-section of the sample B50. The thickness of the GaAs QW has been measured in different regions. S2 1D Finite Square Well Model The variations in thickness are found to be around 4 nm and 2 For a semiconductor the Fermi-Dirac distribution for electrons in nm in the edges and in the facets, respectively, confirming the the conduction band and holes in the valence band is given by, disorder in the GaAs QW. In the HR-TEM image performed in 1 one of the edges of the nanowire cross section, figure S1(b), the f = ; (1) e,h exp((E − Ec,v) ) + 1 variation in the QW thickness (marked by white dashed lines) f b between the edge and the facets is clearly visible. c,v where b = 1=kBT, T is the electron temperature and Ef is the Figures S2 and S3 are additional TEM images of sample B50 Fermi energies of the electrons and holes.
  • Optical Pumping: a Possible Approach Towards a Sige Quantum Cascade Laser

    Optical Pumping: a Possible Approach Towards a Sige Quantum Cascade Laser

    Institut de Physique de l’ Universit´ede Neuchˆatel Optical Pumping: A Possible Approach towards a SiGe Quantum Cascade Laser E3 40 30 E2 E1 20 10 Lasing Signal (meV) 0 210 215 220 Energy (meV) THESE pr´esent´ee`ala Facult´edes Sciences de l’Universit´ede Neuchˆatel pour obtenir le grade de docteur `essciences par Maxi Scheinert Soutenue le 8 octobre 2007 En pr´esence du directeur de th`ese Prof. J´erˆome Faist et des rapporteurs Prof. Detlev Gr¨utzmacher , Prof. Peter Hamm, Prof. Philipp Aebi, Dr. Hans Sigg and Dr. Soichiro Tsujino Keywords • Semiconductor heterostructures • Intersubband Transitions • Quantum cascade laser • Si - SiGe • Optical pumping Mots-Cl´es • H´et´erostructures semiconductrices • Transitions intersousbande • Laser `acascade quantique • Si - SiGe • Pompage optique i Abstract Since the first Quantum Cascade Laser (QCL) was realized in 1994 in the AlInAs/InGaAs material system, it has attracted a wide interest as infrared light source. Main applications can be found in spectroscopy for gas-sensing, in the data transmission and telecommuni- cation as free space optical data link as well as for infrared monitoring. This type of light source differs in fundamental ways from semiconductor diode laser, because the radiative transition is based on intersubband transitions which take place between confined states in quantum wells. As the lasing transition is independent from the nature of the band gap, it opens the possibility to a tuneable, infrared light source based on silicon and silicon compatible materials such as germanium. As silicon is the material of choice for electronic components, a SiGe based QCL would allow to extend the functionality of silicon into optoelectronics.
  • Sankar Das Sarma 3/11/19 1 Curriculum Vitae

    Sankar Das Sarma 3/11/19 1 Curriculum Vitae

    Sankar Das Sarma 3/11/19 Curriculum Vitae Sankar Das Sarma Richard E. Prange Chair in Physics and Distinguished University Professor Director, Condensed Matter Theory Center Fellow, Joint Quantum Institute University of Maryland Department of Physics College Park, Maryland 20742-4111 Email: [email protected] Web page: www.physics.umd.edu/cmtc Fax: (301) 314-9465 Telephone: (301) 405-6145 Published articles in APS journals I. Physical Review Letters 1. Theory for the Polarizability Function of an Electron Layer in the Presence of Collisional Broadening Effects and its Experimental Implications (S. Das Sarma) Phys. Rev. Lett. 50, 211 (1983). 2. Theory of Two Dimensional Magneto-Polarons (S. Das Sarma), Phys. Rev. Lett. 52, 859 (1984); erratum: Phys. Rev. Lett. 52, 1570 (1984). 3. Proposed Experimental Realization of Anderson Localization in Random and Incommensurate Artificial Structures (S. Das Sarma, A. Kobayashi, and R.E. Prange) Phys. Rev. Lett. 56, 1280 (1986). 4. Frequency-Shifted Polaron Coupling in GaInAs Heterojunctions (S. Das Sarma), Phys. Rev. Lett. 57, 651 (1986). 5. Many-Body Effects in a Non-Equilibrium Electron-Lattice System: Coupling of Quasiparticle Excitations and LO-Phonons (J.K. Jain, R. Jalabert, and S. Das Sarma), Phys. Rev. Lett. 60, 353 (1988). 6. Extended Electronic States in One Dimensional Fibonacci Superlattice (X.C. Xie and S. Das Sarma), Phys. Rev. Lett. 60, 1585 (1988). 1 Sankar Das Sarma 7. Strong-Field Density of States in Weakly Disordered Two Dimensional Electron Systems (S. Das Sarma and X.C. Xie), Phys. Rev. Lett. 61, 738 (1988). 8. Mobility Edge is a Model One Dimensional Potential (S.
  • Quasicrystalline 30 Twisted Bilayer Graphene As an Incommensurate

    Quasicrystalline 30 Twisted Bilayer Graphene As an Incommensurate

    Quasicrystalline 30◦ twisted bilayer graphene as an incommensurate superlattice with strong interlayer coupling Wei Yaoa,b, Eryin Wanga,b, Changhua Baoa,b, Yiou Zhangc, Kenan Zhanga,b, Kejie Baoc, Chun Kai Chanc, Chaoyu Chend, Jose Avilad, Maria C. Asensiod,e, Junyi Zhuc,1, and Shuyun Zhoua,b,f,1 aState Key Laboratory of Low Dimensional Quantum Physics, Tsinghua University, Beijing 100084, China; bDepartment of Physics, Tsinghua University, Beijing 100084, China; cDepartment of Physics, The Chinese University of Hong Kong, Hong Kong, China; dSynchrotron SOLEIL, L’Orme des Merisiers, Saint Aubin-BP 48, 91192 Gif sur Yvette Cedex, France; eUniversite´ Paris-Saclay, L’Orme des Merisiers, Saint Aubin-BP 48, 91192 Gif sur Yvette Cedex, France; and fCollaborative Innovation Center of Quantum Matter, Beijing 100084, P. R. China Edited by Hongjie Dai, Department of Chemistry, Stanford University, Stanford, CA, and approved May 25, 2018 (received for review December 1, 2017) The interlayer coupling can be used to engineer the electronic Although the electronic structures of commensurate het- structure of van der Waals heterostructures (superlattices) to erostructures have been investigated in recent years (13–17), obtain properties that are not possible in a single material. So research on incommensurate heterostructures remains limited far research in heterostructures has been focused on commen- for two reasons. On one hand, incommensurate heterostruc- surate superlattices with a long-ranged Moire´ period. Incom- tures are difficult to be stabilized, and thus, they are quite mensurate heterostructures with rotational symmetry but not rare under natural growth conditions. On the other hand, translational symmetry (in analogy to quasicrystals) are not only it is usually assumed that the interlayer interaction is sup- rare in nature, but also the interlayer interaction has often been pressed due to the lack of phase coherence.
  • Optical Physics of Quantum Wells

    Optical Physics of Quantum Wells

    Optical Physics of Quantum Wells David A. B. Miller Rm. 4B-401, AT&T Bell Laboratories Holmdel, NJ07733-3030 USA 1 Introduction Quantum wells are thin layered semiconductor structures in which we can observe and control many quantum mechanical effects. They derive most of their special properties from the quantum confinement of charge carriers (electrons and "holes") in thin layers (e.g 40 atomic layers thick) of one semiconductor "well" material sandwiched between other semiconductor "barrier" layers. They can be made to a high degree of precision by modern epitaxial crystal growth techniques. Many of the physical effects in quantum well structures can be seen at room temperature and can be exploited in real devices. From a scientific point of view, they are also an interesting "laboratory" in which we can explore various quantum mechanical effects, many of which cannot easily be investigated in the usual laboratory setting. For example, we can work with "excitons" as a close quantum mechanical analog for atoms, confining them in distances smaller than their natural size, and applying effectively gigantic electric fields to them, both classes of experiments that are difficult to perform on atoms themselves. We can also carefully tailor "coupled" quantum wells to show quantum mechanical beating phenomena that we can measure and control to a degree that is difficult with molecules. In this article, we will introduce quantum wells, and will concentrate on some of the physical effects that are seen in optical experiments. Quantum wells also have many interesting properties for electrical transport, though we will not discuss those here.
  • Schrödinger Equation: (Time Independent) Hψ = Eψ This Is a Differential Eigenvalue Equation

    Schrödinger Equation: (Time Independent) Hψ = Eψ This Is a Differential Eigenvalue Equation

    Physics and Material Science of Semiconductor Nanostructures PHYS 570P Prof. Oana Malis Email: [email protected] Lecture 9 Review of quantum mechanics, statistical physics, and solid state Band structure of materials Semiconductor band structure Semiconductor nanostructures Ref. Davies Chapter 1 Quantum Mechanics (QM) • The Schrödinger Equation: (time independent) Hψ = Eψ This is a differential eigenvalue equation. H Hamiltonian operator for the system (energy operator) E Energy eigenvalue, ψ wavefunction Particles are QM waves! |ψ|2 probability density; ψ is a function of ALL coordinates of ALL particles in the problem! One Page Elementary Quantum Mechanics & Solid State Physics Review • Quantum Mechanics of a Free Electron: 2 – The energies are continuous: E = (k) /(2mo) (1d, 2d, or 3d) – The wavefunctions are traveling waves: ikx ikr ψk(x) = A e (1d) ψk(r) = A e (2d or 3d) • Solid State Physics: Quantum Mechanics of an Electron in a Periodic Potential in an infinite crystal : – The energy bands are (approximately) continuous: E= Enk – At the bottom of the conduction band or the top of the valence band, in the effective mass approximation, the bands can be written: 2 Enk (k) /(2m*) – The wavefunctions are Bloch Functions = traveling waves: ikr Ψnk(r) = e unk(r); unk(r) = unk(r+R) QM Review: The 1d (infinite) Potential Well (“particle in a box”) In all QM texts!! Consider the case of a particle in a 1-D potential well, with width L e infinite barriers V(x) = 0 for 0 x L V(x) = for x<0, x>L Schrödinger equation Inside the well
  • Stationary States in a Potential Well- H.C

    Stationary States in a Potential Well- H.C

    FUNDAMENTALS OF PHYSICS - Vol. II - Stationary States In A Potential Well- H.C. Rosu and J.L. Moran-Lopez STATIONARY STATES IN A POTENTIAL WELL H.C. Rosu and J.L. Moran-Lopez Instituto Potosino de Investigación Científica y Tecnológica, SLP, México Keywords: Stationary states, Bohr’s atomic model, Schrödinger equation, Rutherford’s planetary model, Frank-Hertz experiment, Infinite square well potential, Quantum harmonic oscillator, Wilson-Sommerfeld theory, Hydrogen atom Contents 1. Introduction 2. Stationary Orbits in Old Quantum Mechanics 2.1. Quantized Planetary Atomic Model 2.2. Bohr’s Hypotheses and Quantized Circular Orbits 2.3. From Quantized Circles to Elliptical Orbits 2.4. Experimental Proof of the Existence of Atomic Stationary States 3. Stationary States in Wave Mechanics 4. The Infinite Square Well: The Stationary States Most Resembling the Standing Waves on a String 3.1. The Schrödinger Equation 3.2. The Dynamical Phase 3.3. The Schrödinger Wave Stationarity 3.4. Stationary Schrödinger States and Classical Orbits 3.5. Stationary States as Sturm-Liouville Eigenfunctions 5. 1D Parabolic Well: The Stationary States of the Quantum Harmonic Oscillator 5.1. The Solution of the Schrödinger Equation 5.2. The Normalization Constant 5.3. Final Formulas for the HO Stationary States 5.4. The Algebraic Approach: Creation and Annihilation Operators 5.5. HO Spectrum Obtained from Wilson-Sommerfeld Quantization Condition 6. The 3D Coulomb Well: The Stationary States of the Hydrogen Atom 6.1. The Separation of Variables in Spherical Coordinates 6.2. The Angular Separation Constants as Quantum Numbers 6.3. Polar andUNESCO Azimuthal Solutions Set Together – EOLSS 6.4.
  • Twisted Bilayer Graphene Superlattices

    Twisted Bilayer Graphene Superlattices

    Twisted Bilayer Graphene Superlattices Yanan Wang1, Zhihua Su1, Wei Wu1,2, Shu Nie3, Nan Xie4, Huiqi Gong4, Yang Guo4, Joon Hwan Lee5, Sirui Xing1,2, Xiaoxiang Lu1, Haiyan Wang5, Xinghua Lu4, Kevin McCarty3, Shin- shem Pei1,2, Francisco Robles-Hernandez6, Viktor G. Hadjiev7, Jiming Bao1,* 1Department of Electrical and Computer Engineering University of Houston, Houston, TX 77204, USA 2Center for Advanced Materials University of Houston, Houston, TX 77204, USA 3Sandia National Laboratories, Livermore, CA 94550, USA 4Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 5Department of Electrical and Computer Engineering Texas A&M University, College Station, Texas 77843, USA 6College of Engineering Technology University of Houston, Houston, TX 77204, USA 7Texas Center for Superconductivity and Department of Mechanical Engineering, University of Houston, Houston, TX 77204, USA *To whom correspondence should be addressed: [email protected]. 1 Abstract Twisted bilayer graphene (tBLG) provides us with a large rotational freedom to explore new physics and novel device applications, but many of its basic properties remain unresolved. Here we report the synthesis and systematic Raman study of tBLG. Chemical vapor deposition was used to synthesize hexagon- shaped tBLG with a rotation angle that can be conveniently determined by relative edge misalignment. Superlattice structures are revealed by the observation of two distinctive Raman features: folded optical phonons and enhanced intensity of the 2D-band. Both signatures are strongly correlated with G-line resonance, rotation angle and laser excitation energy. The frequency of folded phonons decreases with the increase of the rotation angle due to increasing size of the reduced Brillouin zone (rBZ) and the zone folding of transverse optic (TO) phonons to the rBZ of superlattices.