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Part 5: the Bose-Einstein Distribution
PHYS393 – Statistical Physics Part 5: The Bose-Einstein Distribution Distinguishable and indistinguishable particles In the previous parts of this course, we derived the Boltzmann distribution, which described how the number of distinguishable particles in different energy states varied with the energy of those states, at different temperatures: N εj nj = e− kT . (1) Z However, in systems consisting of collections of identical fermions or identical bosons, the wave function of the system has to be either antisymmetric (for fermions) or symmetric (for bosons) under interchange of any two particles. With the allowed wave functions, it is no longer possible to identify a particular particle with a particular energy state. Instead, all the particles are “shared” between the occupied states. The particles are said to be indistinguishable . Statistical Physics 1 Part 5: The Bose-Einstein Distribution Indistinguishable fermions In the case of indistinguishable fermions, the wave function for the overall system must be antisymmetric under the interchange of any two particles. One consequence of this is the Pauli exclusion principle: any given state can be occupied by at most one particle (though we can’t say which particle!) For example, for a system of two fermions, a possible wave function might be: 1 ψ(x1, x 2) = [ψ (x1)ψ (x2) ψ (x2)ψ (x1)] . (2) √2 A B − A B Here, x1 and x2 are the coordinates of the two particles, and A and B are the two occupied states. If we try to put the two particles into the same state, then the wave function vanishes. Statistical Physics 2 Part 5: The Bose-Einstein Distribution Indistinguishable fermions Finding the distribution with the maximum number of microstates for a system of identical fermions leads to the Fermi-Dirac distribution: g n = i . -
The Debye Model of Lattice Heat Capacity
The Debye Model of Lattice Heat Capacity The Debye model of lattice heat capacity is more involved than the relatively simple Einstein model, but it does keep the same basic idea: the internal energy depends on the energy per phonon (ε=ℏΩ) times the ℏΩ/kT average number of phonons (Planck distribution: navg=1/[e -1]) times the number of modes. It is in the number of modes that the difference between the two models occurs. In the Einstein model, we simply assumed that each mode was the same (same frequency, Ω), and that the number of modes was equal to the number of atoms in the lattice. In the Debye model, we assume that each mode has its own K (that is, has its own λ). Since Ω is related to K (by the dispersion relation), each mode has a different Ω. 1. Number of modes In the Debye model we assume that the normal modes consist of STANDING WAVES. If we have travelling waves, they will carry energy through the material; if they contain the heat energy (that will not move) they need to be standing waves. Standing waves are waves that do travel but go back and forth and interfere with each other to create standing waves. Recall that Newton's Second Law gave us atoms that oscillate: [here x = sa where s is an integer and a the lattice spacing] iΩt iKsa us = uo e e ; and if we use the form sin(Ksa) for eiKsa [both indicate oscillations]: iΩt us = uo e sin(Ksa) . We now apply the boundary conditions on our solution: to have standing waves both ends of the wave must be tied down, that is, we must have u0 = 0 and uN = 0. -
Density of States Information from Low Temperature Specific Heat
JOURNAL OF RESE ARC H of th e National Bureau of Standards - A. Physics and Chemistry Val. 74A, No.3, May-June 1970 Density of States I nformation from Low Temperature Specific Heat Measurements* Paul A. Beck and Helmut Claus University of Illinois, Urbana (October 10, 1969) The c a lcul ati on of one -electron d ensit y of s tate va lues from the coeffi cient y of the te rm of the low te mperature specifi c heat lin ear in te mperature is compli cated by many- body effects. In parti c ul ar, the electron-p honon inte raction may enhance the measured y as muc h as tw ofo ld. The e nha nce me nt fa ctor can be eva luat ed in the case of supe rconducting metals and a ll oys. In the presence of magneti c mo ments, add it ional complicati ons arise. A magneti c contribution to the measured y was ide ntifi e d in the case of dilute all oys and a lso of concentrated a lJ oys wh e re parasiti c antife rromagnetis m is s upe rim posed on a n over-a ll fe rromagneti c orde r. No me thod has as ye t bee n de vised to e valu ate this magne ti c part of y. T he separati on of the te mpera ture- li near term of the s pec ifi c heat may itself be co mpli cated by the a ppearance of a s pecific heat a no ma ly due to magneti c cluste rs in s upe rpa ramagneti c or we ak ly ferromagneti c a ll oys. -
Phys 446: Solid State Physics / Optical Properties Lattice Vibrations
Solid State Physics Lecture 5 Last week: Phys 446: (Ch. 3) • Phonons Solid State Physics / Optical Properties • Today: Einstein and Debye models for thermal capacity Lattice vibrations: Thermal conductivity Thermal, acoustic, and optical properties HW2 discussion Fall 2007 Lecture 5 Andrei Sirenko, NJIT 1 2 Material to be included in the test •Factors affecting the diffraction amplitude: Oct. 12th 2007 Atomic scattering factor (form factor): f = n(r)ei∆k⋅rl d 3r reflects distribution of electronic cloud. a ∫ r • Crystalline structures. 0 sin()∆k ⋅r In case of spherical distribution f = 4πr 2n(r) dr 7 crystal systems and 14 Bravais lattices a ∫ n 0 ∆k ⋅r • Crystallographic directions dhkl = 2 2 2 1 2 ⎛ h k l ⎞ 2πi(hu j +kv j +lw j ) and Miller indices ⎜ + + ⎟ •Structure factor F = f e ⎜ a2 b2 c2 ⎟ ∑ aj ⎝ ⎠ j • Definition of reciprocal lattice vectors: •Elastic stiffness and compliance. Strain and stress: definitions and relation between them in a linear regime (Hooke's law): σ ij = ∑Cijklε kl ε ij = ∑ Sijklσ kl • What is Brillouin zone kl kl 2 2 C •Elastic wave equation: ∂ u C ∂ u eff • Bragg formula: 2d·sinθ = mλ ; ∆k = G = eff x sound velocity v = ∂t 2 ρ ∂x2 ρ 3 4 • Lattice vibrations: acoustic and optical branches Summary of the Last Lecture In three-dimensional lattice with s atoms per unit cell there are Elastic properties – crystal is considered as continuous anisotropic 3s phonon branches: 3 acoustic, 3s - 3 optical medium • Phonon - the quantum of lattice vibration. Elastic stiffness and compliance tensors relate the strain and the Energy ħω; momentum ħq stress in a linear region (small displacements, harmonic potential) • Concept of the phonon density of states Hooke's law: σ ij = ∑Cijklε kl ε ij = ∑ Sijklσ kl • Einstein and Debye models for lattice heat capacity. -
Intermediate Statistics in Thermoelectric Properties of Solids
Intermediate statistics in thermoelectric properties of solids André A. Marinho1, Francisco A. Brito1,2 1 Departamento de Física, Universidade Federal de Campina Grande, 58109-970 Campina Grande, Paraíba, Brazil and 2 Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008, 58051-970 João Pessoa, Paraíba, Brazil (Dated: July 23, 2019) Abstract We study the thermodynamics of a crystalline solid by applying intermediate statistics manifested by q-deformation. We based part of our study on both Einstein and Debye models, exploring primarily de- formed thermal and electrical conductivities as a function of the deformed Debye specific heat. The results revealed that the q-deformation acts in two different ways but not necessarily as independent mechanisms. It acts as a factor of disorder or impurity, modifying the characteristics of a crystalline structure, which are phenomena described by q-bosons, and also as a manifestation of intermediate statistics, the B-anyons (or B-type systems). For the latter case, we have identified the Schottky effect, normally associated with high-Tc superconductors in the presence of rare-earth-ion impurities, and also the increasing of the specific heat of the solids beyond the Dulong-Petit limit at high temperature, usually related to anharmonicity of interatomic interactions. Alternatively, since in the q-bosons the statistics are in principle maintained the effect of the deformation acts more slowly due to a small change in the crystal lattice. On the other hand, B-anyons that belong to modified statistics are more sensitive to the deformation. PACS numbers: 02.20-Uw, 05.30-d, 75.20-g arXiv:1907.09055v1 [cond-mat.stat-mech] 21 Jul 2019 1 I. -
Arxiv:1608.06287V1 [Hep-Th] 22 Aug 2016 Published in Int
Preprint typeset in JHEP style - HYPER VERSION Surprises with Nonrelativistic Naturalness Petr Hoˇrava Berkeley Center for Theoretical Physics and Department of Physics University of California, Berkeley, CA 94720-7300, USA and Theoretical Physics Group, Lawrence Berkeley National Laboratory Berkeley, CA 94720-8162, USA Abstract: We explore the landscape of technical naturalness for nonrelativistic systems, finding surprises which challenge and enrich our relativistic intuition already in the simplest case of a single scalar field. While the immediate applications are expected in condensed matter and perhaps in cosmology, the study is motivated by the leading puzzles of funda- mental physics involving gravity: The cosmological constant problem and the Higgs mass hierarchy problem. This brief review is based on talks and lectures given at the 2nd LeCosPA Symposium on Everything About Gravity at National Taiwan University, Taipei, Taiwan (December 2015), to appear in the Proceedings; at the International Conference on Gravitation and Cosmology, KITPC, Chinese Academy of Sciences, Beijing, China (May 2015); at the Symposium Celebrating 100 Years of General Relativity, Guanajuato, Mexico (November 2015); and at the 54. Internationale Universit¨atswochen f¨urTheoretische Physik, Schladming, Austria (February 2016). arXiv:1608.06287v1 [hep-th] 22 Aug 2016 Published in Int. J. Mod. Phys. D Contents 1. Puzzles of Naturalness 1 1.1 Technical Naturalness 2 2. Naturalness in Nonrelativistic Systems 3 2.1 Towards the Classification of Nonrelativistic Nambu-Goldstone Modes 3 2.2 Polynomial Shift Symmetries 5 2.3 Naturalness of Cascading Hierarchies 5 3. Interacting Theories with Polynomial Shift Symmetries 6 3.1 Invariants of Polynomial Shift Symmetries 6 3.2 Polynomial Shift Invariants and Graph Theory 6 3.3 Examples 7 4. -
Quantum Phases and Spin Liquid Properties of 1T-Tas2 ✉ Samuel Mañas-Valero 1, Benjamin M
www.nature.com/npjquantmats ARTICLE OPEN Quantum phases and spin liquid properties of 1T-TaS2 ✉ Samuel Mañas-Valero 1, Benjamin M. Huddart 2, Tom Lancaster 2, Eugenio Coronado1 and Francis L. Pratt 3 Quantum materials exhibiting magnetic frustration are connected to diverse phenomena, including high Tc superconductivity, topological order, and quantum spin liquids (QSLs). A QSL is a quantum phase (QP) related to a quantum-entangled fluid-like state of matter. Previous experiments on QSL candidate materials are usually interpreted in terms of a single QP, although theories indicate that many distinct QPs are closely competing in typical frustrated spin models. Here we report on combined temperature- dependent muon spin relaxation and specific heat measurements for the triangular-lattice QSL candidate material 1T-TaS2 that provide evidence for competing QPs. The measured properties are assigned to arrays of individual QSL layers within the layered charge density wave structure of 1T-TaS2 and their characteristic parameters can be interpreted as those of distinct Z2 QSL phases. The present results reveal that a QSL description can extend beyond the lowest temperatures, offering an additional perspective in the search for such materials. npj Quantum Materials (2021) 6:69 ; https://doi.org/10.1038/s41535-021-00367-w INTRODUCTION QSL, both from experimental and theoretical perspectives13,18–21, The idea of destabilizing magnetic order by quantum fluctuating but some alternative scenarios have also been proposed, such as a 1234567890():,; resonating valence bonds (RVB) originated with Anderson in Peierls mechanism or the formation of domain wall networks, 22–24 19731. Systems with frustrated interactions are particularly among others . -
Multicritical Nambu-Goldstone Modes and Nonrelativistic Naturalness
1 Multicritical Nambu-Goldstone Modes and Nonrelativistic Naturalness Petr Hoˇrava Berkeley Center for Theoretical Physics Bay Area Particle Theory Seminar SFSU, San Francisco, CA October 10, 2014 work with Tom Griffin, Kevin Grosvenor, Ziqi Yan 2 Puzzles of Naturalness Some of the most fascinating open problems in modern physics are all problems of naturalness: • The cosmological constant problem • The Higgs mass hierarchy problem • The linear resistivity of strange metals, the regime above Tc in high-Tc superconductors [Bednorz&M¨uller'86; Polchinski '92] In addition, the first two •s { together with the recent experimental facts { suggest that we may live in a strangely simple Universe ::: Naturalness is again in the forefront (as are its possible alternatives: landscape? :::?) If we are to save naturalness, we need new surprises! 3 What is Naturalness? Technical Naturalness: 't Hooft (1979) \The concept of causality requires that macroscopic phenomena follow from microscopic equations." \The following dogma should be followed: At any energy scale µ, a physical parameter or a set of physical parameters αi(µ) is allowed to be very small only if the replacement αi(µ) = 0 would increase the symmetry of the system." Example: Massive λφ4 in 3 + 1 dimensions. p λ ∼ "; m2 ∼ µ2"; µ ∼ m= λ. Symmetry: The constant shift φ ! φ + a. \Pursuing naturalness beyond 1000 GeV will require theories that are immensely complex compared with some of the grand unified schemes." 4 Gravity without Relativity (a.k.a. gravity with anisotropic scaling, or Hoˇrava-Lifshitz gravity) Field theories with anisotropic scaling: xi ! λxi; t ! λzt: z: dynamical critical exponent { characteristic of RG fixed point. -
The Debye Model Professor Stephen Sweeney
Solid State Physics Lecture 8 – The Debye model Professor Stephen Sweeney Advanced Technology Institute and Department of Physics University of Surrey, Guildford, GU2 7XH, UK [email protected] Solid State Physics - Lecture 8 Recap from Lecture 7 • Concepts of “temperature” and thermal Dulong-Petit equilibrium are based on the idea that individual particles in a system have some form of motion • Heat capacity can be determined by considering vibrational motion of atoms • We considered two models: • Dulong-Petit (classical) • Einstein (quantum mechanical) • Both models assume atoms act independently – this is made up for in the Debye model (today) Solid State Physics - Lecture 8 Summary of Dulong-Petit and Einstein models of heat capacity Dulong-Petit model (1819) Einstein model (1907) • Atoms on lattice vibrate • Atoms on lattice vibrate independently of each independently of each other other • Completely classical • Quantum mechanical • Heat capacity (vibrations are quantised) independent of • Agreement with temperature (3NkB) experiment good at very • Poor agreement with high (~3NkB) and very low experiment, except at (~0) temperatures, but high temperatures not inbetween Solid State Physics - Lecture 8 A more realistic model… • Both the Einstein and Dulong-Petit models treat each atom independently. This is not generally true. • When an atom vibrates, the force on adjacent atoms changes causing them to vibrate (and vice-versa) • Oscillations can be broken down into modes 1D case Nice animations here: http://www.phonon.fc.pl/index.php -
Mixed Valence Driven Heavy-Fermion Behavior and Superconductivity in Kni2se2
Mixed Valence Driven Heavy-Fermion Behavior and Superconductivity in KNi2Se2 James R. Neilson,1,2† Anna Llobet,3 Andreas V. Stier,2 Liang Wu,2 Jiajia Wen,2 Jing Tao,4 Yimei Zhu,4 Zlatko B. Tesanovic,2 N. P. Armitage,2 and Tyrel M. McQueen1,2* 1 Department of Chemistry, Johns Hopkins University, Baltimore MD. 2 Institute for Quantum Matter, and Department of Physics and Astronomy, Johns Hopkins University, Baltimore MD. 3 Lujan Neutron Scattering Center, Los Alamos Neutron Science Center, Los Alamos National Laboratory, Los Alamos, NM. 4 Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, NY. †[email protected] *[email protected] Based on specific heat and magnetoresistance measurements, we report that a “heavy” electronic state exists below T ≈ 20 K in KNi2Se2, with an increased carrier mobility and enhanced effective electronic band mass, m* = 6mb to 18mb. This “heavy” state evolves into superconductivity at Tc = 0.80(1) K. These properties resemble that of a many-body heavy-fermion state, which derives from the hybridization between localized magnetic states and conduction electrons. Yet, no evidence for localized magnetism or magnetic order is found in KNi2Se2 from magnetization measurements or neutron diffraction. Instead, neutron pair-distribution-function analysis reveals the presence of local charge-density-wave distortions that disappear on cooling, an effect opposite to what is typically observed, suggesting that the low-temperature electronic state of KNi2Se2 arises from cooperative Coulomb interactions and proximity to, but avoidance of, charge order. Introduction Methods KNi Se was synthesized from the elements as a Many-body states, in which complex phenomena 2 2 polycrystalline, lustrous, purple-red powder, as previously arise from simple interactions, often exhibit rich phase described12; the same batch of sample was used for all diagrams and lack formal predictive theories. -
Phonons – Thermal Properties
Phonons – Thermal properties Till now everything was classical – no quantization, no uncertainty.... WHY DO WE NEED QUANTUM THEORY OF LATTICE VIBRATIONS? Classical – Dulong Petit law All energy values are allowed – classical equipartition theorem – energy per degree of freedom is 1/2kBT. Total energy independent of temperature! Detailed calculation gives C = 3NkB In reality C ~ aT +bT3 Figure 1: Heat capacity of gold Need quantum theory! Quantum theory of lattice vibrations: Hamiltonian for 1D linear chain Interaction energy is simple harmonic type ~ ½ kx2 Solve it for N coupled harmonic oscillators – 3N normal modes – each with a characteristic frequency where p is the polarization/branch. Energy of any given mode is is the zero-point energy th is called ‘excitation number’ of the particular mode of p branch with wavevector k PH-208 Phonons – Thermal properties Page 1 Phonons are quanta of ionic displacement field that describes classical sound wave. Analogy th with black body radiation - ni is the number of photons of the i mode of oscillations of the EM wave. Total energy of the system E = is the ‘Planck distribution’ of the occupation number. Planck distribution: Consider an ensemble of identical oscillators at temperature T in thermal equilibrium. Ratio of number of oscillators in (n+1)th state to the number of them in nth state is So, the average occupation number <n> is: Special case of Bose-Einstein distribution with =0, chemical potential is zero since we do not directly control the total number of phonons (unlike the number of helium atoms is a bath – BE distribution or electrons in a solid –FE distribution), it is determined by the temperature. -
Thermodynamics of the Magnetic-Field-Induced "Normal
Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2010 Thermodynamics of the Magnetic-Field- Induced "Normal" State in an Underdoped High T[subscript c] Superconductor Scott Chandler Riggs Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] THE FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES THERMODYNAMICS OF THE MAGNETIC-FIELD-INDUCED “NORMAL” STATE IN AN UNDERDOPED HIGH TC SUPERCONDUCTOR By SCOTT CHANDLER RIGGS A Dissertation submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy Degree Awarded: Summer Semester, 2010 The members of the committee approve the dissertation of Scott Chandler Riggs defended on March 27, 2010. Gregory Scott Boebinger Professor Directing Thesis David C. Larbalestier University Representative Jorge Piekarewicz Committee Member Nicholas Bonesteel Committee Member Luis Balicas Committee Member Approved: Mark Riley, Chair, Department of Physics Joseph Travis, Dean, College of Arts and Sciences The Graduate School has verified and approved the above-named committee members. ii ACKNOWLEDGMENTS I never thought this would be the most difficult section to write. From Tally, I’d have to start off by thanking Tim Murphy for letting me steal...er...borrow way too much from the instrumentation shop during the initial set-up of Greg’s lab. I still remember that night on the hybrid working with system ”D”. After that night it became clear that ”D” stands for Dumb. To Eric Palm, thank you for granting more power during our specific heat experiments. We still hold the record for most power consumed in a week! To Rob Smith, we should make a Portland experience every March.