DOCSLIB.ORG

Crystal Physics the Properties of Solids

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Crystal Physics the Properties of Solids Crystal Physics The properties of solids 2222 2 22ZeAABe ZZe H iA iAiAijAB22mme A, 444000 r iA r ij r AB electronic band structure E vs. k structure bond potentials density of states phonon band structure vs. k optical absorption Boltzmann transport density of states equilibrium properties optical properties cv, free energies, bulk modulus,... Calculating free energies Electronic component Phonon component Gibbs free energy G(T, , H, E, ) GUTSN ii ij ij EPHMkK l l G G G S N i ij T ,,EH , ij i TEH,, , TEH,, , G G Pk M l E H k TH,, , l TE,,, http://www.iucr.org/education/pamphlets/18 Groups Crystals can have symmetries: translation, rotation, reflection, inversion,... x 10 0 x yy 0cossin z 0sincos z Point symmetries can be represented by matrices. All such matrices that bring the crystal into itself form the group of the crystal. A, B G AB G 32 point groups (one point remains fixed during transformation) 230 space groups Cyclic groups C2 C4 http://en.wikipedia.org/wiki/Cyclic_group 2G Pyroelectricity i EiT Pyroelectricity is described by a rank 1 tensor P i i T 10 0 x xx 01 0 yy y 00 1 zz 0 10 0xx 0 010 0 yy 00 1zz 0 Pyroelectricity example Turmalin: point group 3m Quartz, ZnO, LaTaO 3 for T = 1°C, E ~ 7 ·104 V/m Pyroelectrics have a spontaneous polarization. If it can be reversed by an electric field they are called Ferroelectrics (BaTiO3) Pyroelectrics are at Joanneum research to make infrared detectors (to detect humans). 10 Pyroelectric crystal classes: 1, 2, m, mm2, 3, 3m, 4, 4mm, 6, 6mm Rank 2 Tensors Electric susceptibility Dielectric constant Magnetic susceptibility Thermal expansion Electrical conductivity Thermal conductivity Seebeck effect Peltier effect 2G Electric susceptibility ij EijE PEiijj PEx xx xy xz x PE yyxyyyzy PEzzxzyzzz Transforming P and E by a crystal symmetry must leave the susceptibility tensor unchanged UP UE UUPU11 UE = U-1U If rotation by 180 about the z axis is a symmetry, 100 100 xxxyxz 1 1 U 010 UU 010 UU yxyyyz 001 001 zx zy zz xz = yz = zx = zy = 0 Cubic crystals All second rank tensors of cubic crystals reduce to constants 216: ZnS, GaAs, GaP, InAs 221: CsCl, cubic perovskite 225: Al, Cu, Ni, Ag, Pt, Au, Pb, NaCl 227: C, Si, Ge, spinel 229: Na, K, Cr, Fe, Nb, Mo, Ta Multiferroics simultaneously ferroelectric and ferromagnetic BiFeO3 If two magnetic sublattices have different charge, changing the magnetic field can change the polarization and changing the electric field can change the magnetization. Rank 3 Tensors Piezoelectricity Piezomagnetism Hall effect Nerst effect Ettingshausen effect Nonlinear electrical susceptibility Tensor notation We need a way to represent 3rd and 4th rank tensors in 2-d. 1 1 1 1 2 6 1 3 5 2 2 2 2 3 4 3 3 3 rank 3 rank 4 gg gg36 312 14 1123 Piezoelectricity average position G Pk + is E average position - k P 2G average position k d ijk + not ijE k ij average position - Piezoelectricity (rank 3 tensor) AFM's, STM's Quartz crystal oscillators Surface acoustic wave generators No inversion symmetry Pressure sensors - Epcos Fuel injectors - Bosch Inkjet printers lead zirconate titanate (Pb[ZrxTi1−x]O3 0<x<1) —more commonly known as PZT barium titanate (BaTiO3) lead titanate (PbTiO3) potassium niobate (KNbO3) lithium niobate (LiNbO3) lithium tantalate (LiTaO3) sodium tungstate (Na2WO3) Ba2NaNb5O5 Pb2KNb5O15 Piezoelectric crystal classes: 1, 2, m, 222, mm2, 4, -4, 422, 4mm, -42m, 3, 32, 3m, 6, -6, 622, 6mm, -62m, 23, -43m Nonlinear optics Period doubling crystals no inversion symmetry PEE12 23 3 E 23GG1 PEij EE jk EEij2 EE ijk E 2 1 costt 2 1 cos(2 ) 806 nm light : lithium iodate (LiIO3) 860 nm light : potassium niobate (KNbO3) 980 nm light : KNbO3 1064 nm light : monopotassium phosphate (KH2PO4, KDP), lithium triborate (LBO). 1300 nm light : gallium selenide (GaSe) 1319 nm light : KNbO3, BBO, KDP, lithium niobate (LiNbO3), LiIO3 Birefringence (Doppelbrechung) Two indices of refraction Calcite http://en.wikipedia.org/wiki/Birefringent k Birefringence http://en.wikipedia.org/wiki/Crystal_optics Birefringence Calcite Rhombohedral = Trigonal Optical effects 22 0 PPii P i PPii E j EE jk EEE jkl EEEEEEjjkjkl spontaneous Ek DC polarization Ek,El DC Pockels effect Kerr effect 22 PPii PEHEHHijkjkl EHjk EHH jkl H ,H DC Hk DC k l Faraday effect Cotton–Mouton effect Electrostriction P 2G k ij d ijk ijEE k k ij piezoelectricity Electrostriction Rank 4 Tensors Stiffness tensor Compliance tensor Piezoconductivity Electrostriction Magnetostriction How the Seebeck effect depends on stress How the electric susceptibility depends on stress How the magnetic susceptibility depends on stress Nonlinear electric susceptibility Nonlinear magnetic susceptibility Symmetric and asymmetric tensors 22GGP P k j kj jk EEjk E j EE k j E k Symmetric Asymmetric electric susceptibility Seebeck effect magnetic susceptibility Peltier effect electrical conductivity piezoconductivity thermal conductivity stiffness tensor.
Load more

Recommended publications
  • Physics 112: Classical Electromagnetism, Fall 2013 Birefringence Notes
    Physics 112: Classical Electromagnetism, Fall 2013 Birefringence Notes 1 A tensor susceptibility? The electrons bound within, and binding, the atoms of a dielectric crystal are not uniformly dis- tributed, but are restricted in their motion by the potentials which confine them. In response to an applied electric field, they may therefore move a greater or lesser distance, depending upon the strength of their confinement in the field direction. As a result, the induced polarization varies not only with the strength of the applied field, but also with its direction. The susceptibility{ and properties which depend upon it, such as the refractive index{ are therefore anisotropic, and cannot be characterized by a single value. The scalar electric susceptibility, χe, is defined to be the coefficient which relates the value of the ~ ~ local electric field, Eloc, to the local value of the polarization, P : ~ ~ P = χe0Elocal: (1) As we discussed in the last seminar we can `promote' this relation to a tensor relation with χe ! (χe)ij. In this case the dielectric constant is also a tensor and takes the form ij = [δij + (χe)ij] 0: (2) Figure 1: A cartoon showing how the electron is held by anisotropic springs{ causing an electric susceptibility which is different when the electric field is pointing in different directions. How does this happen in practice? In Fig. 1 we can see that in a crystal, for instance, the electrons will in general be held in bonds which are not spherically symmetric{ i.e., they are anisotropic. 1 Therefore, it will be easier to polarize the material in certain directions than it is in others.
    [Show full text]
  • 4 Lightdielectrics.Pdf
    4. The interaction of light with matter The propagation of light through chemical materials is described by a wave equation similar to the one that describes light travel in a vacuum (free space). Again, using E as the electric field of light, v as the speed of light in a material and z as its direction of propagation. !2 1 ! 2E ! 2E 1 ! 2E # n2 & ! 2E # & . 2 E= 2 2 " 2 = % 2 ( 2 = 2 2 !z c !t !z $ v ' !t $% c '( !t (Read the variation in the electric field with respect distance traveled is proportional to its variation with respect to time.) The refractive index, n, (also represented η) describes how matter affects light propagation: through the electric permittivity, ε, and the magnetic permeability, µ. ! µ n = !0 µ0 These properties describe how well a medium supports (permits the transmission of) electric and magnetic fields, respectively. The terms ε0 and µ0 are reference values: the permittivity and permeability of free space. Consequently, the refractive index for a vacuum is unity. In chemical materials ε is always larger than ε0, reflecting the interaction of the electric field of the incident beam with the electrons of the material. During this interaction, the energy from the electric field is transiently stored in the medium as the electrons in the material are temporarily aligned with the field. This phenomenon is referred to as polarization, P, in the sense that the charges of the medium are temporarily separated. (This must not be confused with the polarization, which refers to the orientation or behavior of the electric field.) This stored energy is re-radiated, but the beam travel is slowed by interaction with the material.
    [Show full text]
  • EM Dis Ch 5 Part 2.Pdf
    1 LOGO Chapter 5 Electric Field in Material Space Part 2 iugaza2010.blogspot.com Polarization(P) in Dielectrics The application of E to the dielectric material causes the flux density to be grater than it would be in free space. D oE P P is proportional to the applied electric field E P e oE Where e is the electric susceptibility of the material - Measure of how susceptible (or sensitive) a given dielectric is to electric field. 3 D oE P oE e oE oE(1 e) oE( r) D o rE E o r r1 e o permitivity of free space permitivity of dielectric relative permitivity r 4 Dielectric constant or(relative permittivity) εr Is the ratio of the permittivity of the dielectric to that of free space. o r permitivity of dielectric relative permitivity r o permitivity of free space 5 Dielectric Strength Is the maximum electric field that a dielectric can withstand without breakdown. o r Material Dielectric Strength εr E(V/m) Water(sea) 80 7.5M Paper 7 12M Wood 2.5-8 25M Oil 2.1 12M Air 1 3M 6 A parallel plate capacitor with plate separation of 2mm has 1kV voltage applied to its plate. If the space between the plate is filled with polystyrene(εr=2.55) Find E,P V 1000 E 500 kV / m d 2103 2 P e oE o( r1)E o(1 2.55)(500k) 6.86 C / m 7 In a dielectric material Ex=5 V/m 1 2 and P (3a x a y 4a z )nc/ m 10 Find (a) electric susceptibility e (b) E (c) D 1 (3) 10 (a)P e oE e 2.517 o(5) P 1 (3a x a y 4a z ) (b)E 5a x 1.67a y 6.67a z e o 10 (2.517) o (c)D E o rE o rE o( e1)E 2 139.78a x 46.6a y 186.3a z pC/m 8 In a slab of dielectric
    [Show full text]
  • Notes 4 Maxwell's Equations
    ECE 3317 Applied Electromagnetic Waves Prof. David R. Jackson Fall 2020 Notes 4 Maxwell’s Equations Adapted from notes by Prof. Stuart A. Long 1 Overview Here we present an overview of Maxwell’s equations. A much more thorough discussion of Maxwell’s equations may be found in the class notes for ECE 3318: http://courses.egr.uh.edu/ECE/ECE3318 Notes 10: Electric Gauss’s law Notes 18: Faraday’s law Notes 28: Ampere’s law Notes 28: Magnetic Gauss law . D. Fleisch, A Student’s Guide to Maxwell’s Equations, Cambridge University Press, 2008. 2 Electromagnetic Fields Four vector quantities E electric field strength [Volt/meter] D electric flux density [Coulomb/meter2] H magnetic field strength [Amp/meter] B magnetic flux density [Weber/meter2] or [Tesla] Each are functions of space and time e.g. E(x,y,z,t) J electric current density [Amp/meter2] 3 ρv electric charge density [Coulomb/meter ] 3 MKS units length – meter [m] mass – kilogram [kg] time – second [sec] Some common prefixes and the power of ten each represent are listed below femto - f - 10-15 centi - c - 10-2 mega - M - 106 pico - p - 10-12 deci - d - 10-1 giga - G - 109 nano - n - 10-9 deka - da - 101 tera - T - 1012 micro - μ - 10-6 hecto - h - 102 peta - P - 1015 milli - m - 10-3 kilo - k - 103 4 Maxwell’s Equations (Time-varying, differential form) ∂B ∇×E =− ∂t ∂D ∇×HJ = + ∂t ∇⋅B =0 ∇⋅D =ρv 5 Maxwell James Clerk Maxwell (1831–1879) James Clerk Maxwell was a Scottish mathematician and theoretical physicist.
    [Show full text]
  • The Response of a Media
    KTH ROYAL INSTITUTE OF TECHNOLOGY ED2210 Lecture 3: The response of a media Lecturer: Thomas Johnson Dielectric media The theory of dispersive media is derived using the formalism originally developed for dielectrics…let’s refresh our memory! • Capacitors are often filled with a dielectric. • Applying an electric field, the dielectric gets polarised • Net field in the dielectric is reduced. Dielectric media + - E - + E - + - + LECTURE 3: THE RESPONSE OF A MEDIA, ED2210 2016-01-26 2 Polarisation Electric field - Polarisation is the response of e.g. - - - • bound electrons, or + - • molecules with a dipole moment - - Emedia - In both cases the particles form dipoles. Electric field Definition: The polarisation, �, is the electric dipole moment per unit volume. Linear media: % � = � �'� where �% is the electric susceptibility LECTURE 3: THE RESPONSE OF A MEDIA, ED2210 2016-01-26 3 Magnetisation Magnetisation is due to induced dipoles, either from Induced current • induced currents B-field • quantum mechanically from the particle spin + Definition: The magnetisation, �, is the magnetic dipole Bmedia moment per unit volume. Linear media: Spin + � = � �/�' B-field Magnetic susceptibility : �+ NOTE: Both polarisation and magnetisation fields Bmedia are due to dipole fields only; not related to higher order moments like the quadropoles, octopoles etc. LECTURE 3: THE RESPONSE OF A MEDIA, ED2210 2016-01-26 4 Field calculations with dielectrics The capacitor problem includes two types of charges: • Surface charge, �0, on metal surface (monopole charges) • Induced dipole charges, �123, within the dielectric media Total electric field strength, �, is generated by �0 + �123, The polarisation, �, is generated by �123 The electric induction, �, is generated by �0, where % �: = �'� + � = � + � �'� = ��'� = �� Here � is the dielectric constant and � is the permittivity.
    [Show full text]
  • Two New Experiments on Measuring the Permanent Electric
    Clear evidence of charge conjugation and parity violation in K atoms from atomic permanent electric dipole moment experiments Pei-Lin You1 Xiang-You Huang 2 1. Institute of Quantum Electronics, Guangdong Ocean University, Zhanjiang Guangdong 524025, China. 2. Department of Physics , Peking University, Beijing 100871,China. Quantum mechanics thinks that atoms do not have permanent electric dipole moment (EDM) because of their spherical symmetry. Therefore, there is no polar atom in nature except for polar molecules. The electric susceptibilityχe caused by the orientation of polar substances is inversely proportional to the absolute temperature T while the induced susceptibility of atoms is temperature independent. This difference in temperature dependence offers a means of separating the polar and non-polar substances experimentally. Using special capacitor our experiments discovered that the relationship betweenχe of Potassium atom vapor and T is justχe=B/T, where the slope B≈283(k) as polar molecules, but appears to be disordered using the traditional capacitor. Its capacitance C at different voltage V was measured. The C-V curve shows that the saturation polarization of K vapor has be observed when E≥105V/m and nearly all K atoms (more than 98.9％) are lined up with the field! The ground state neutral K atom is polar atom -9 with a large EDM : dK >2.3×10 e.cm.The results gave clear evidence for CP (charge conjugation and parity) violation in K atoms. If K atom has a large EDM, why the linear Stark effect has not been observed? The article discussed the question thoroughly. Our results are easy to be repeated because the details of the experiment are described in the article.
    [Show full text]
  • Quantum Mechanics Polarization Density
    Quantum Mechanics_polarization density In classical electromagnetism, polarization density (or electric polarization, or simplypolarization) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is placed in an external Electric field, its molecules gain Electric dipole moment and the dielectric is said to be polarized. The electric dipole moment induced per unit volume of the dielectric material is called the electric polarization of the dielectric.[1][2] Polarization density also describes how a material responds to an applied electric field as well as the way the material changes the electric field, and can be used to calculate the forces that result from those interactions. It can be compared to Magnetization, which is the measure of the corresponding response of a material to a Magnetic field in Magnetism. TheSI unit of measure is coulombs per square metre, and polarization density is represented by a vectorP.[3] Definition The polarization density P is defined as the average Electric dipole moment d per unit volume V of the dielectric material:[4] which can be interpreted as a measure of how strong and how aligned the dipoles are in a region of the material. For the calculation of P due to an applied electric field, the electric susceptibility χ of the dielectric must be known (see below). Polarization density in Maxwell's equations The behavior of electric fields (E and D), magnetic fields (B, H), charge density (ρ) and current density (J) are described by Maxwell's equations in matter. The role of the polarization density P is described below.
    [Show full text]
  • Active Metamaterials with Negative Static Electric Susceptibility
    Active metamaterials with negative static electric susceptibility F. Castles1,2*, J. A. J. Fells3, D. Isakov1,4, S. M. Morris3, A. Watt1 and P. S. Grant1* 1 Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK. 2 School of Electronic Engineering and Computer Science, Queen Mary University of London, Mile End Road, London E1 4FZ, UK. 3 Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK. 4 Warwick Manufacturing Group, University of Warwick, Coventry CV4 7AL, UK. * email: [email protected]; [email protected] 1 Well-established textbook arguments suggest that static electric susceptibility must be positive in “all bodies”1. However, it has been pointed out that media that are not in thermodynamic equilibrium are not necessarily subject to this restriction; negative static electric susceptibility has been predicted theoretically in systems with inverted populations of atomic and molecular energy levels2,3, though this has never been confirmed experimentally. Here we exploit the design freedom afforded by metamaterials to fabricate active structures that exhibit the first experimental evidence of negative static electric susceptibility. Unlike the systems envisioned previously— which were expected to require reduced temperature and pressure4—negative values are readily achieved at room temperature and pressure. Further, values are readily tuneable throughout the negative range of stability − (0) , resulting in magnitudes that are over one thousand times greater than predicted previously4. This opens the door to new technological capabilities such as stable electrostatic levitation. Although static magnetic susceptibility may take positive or negative values in paramagnetic and diamagnetic materials respectively, a standard theoretical argument by Landau et al.
    [Show full text]
  • Electromagnetic Optics
    Electromagnetic optics • Optics IS a branch of electromagnetism • Electromagnetic optics include both – Wave optics (scalar approximation) – Ray optics (limit when λ → 0) • Can also describes the propagation of wave in media characterized by their e.m. properties. P. Piot, PHYS 630 – Fall 2008 Maxwell’s equations • Macroscopy Maxwell’s equations P. Piot, PHYS 630 – Fall 2008 Maxwell’s equations II • Consider a linear, homogenous, isotropous, and non dispersive medium then Maxwell’s equations simplify to • For now on we consider non-magnetic media (M=0) and µ=µ0. P. Piot, PHYS 630 – Fall 2008 Wave equation (revisited) • From Faraday’s law • From Ampere’s law • So both equation can be cast in the form P. Piot, PHYS 630 – Fall 2008 Dielectric Media • A dielectric medium is characterized by the medium equation, i.e. the relation between polarization and electric field vectors. • Can also be viewed as a system with E as input and P as output characterized by an impulse-response function h E(r,t) P(r,t) h(r,t) • The E-field generates electron oscillations in the medium that are responsible for polarization (and medium-induced E-fields) P. Piot, PHYS 630 – Fall 2008 Dielectric Media: some definitions • Linear media: medium equation is of the form • Nondispersive media: P(t) depends on E(t) [at the SAME time]: the time-history of E is not involved (assumed infinitely fast material, this is an idealization, but good approximation • Homogenous media: the medium equation P=f(E) is independent of the position r • Isotropic media: the medium P=f(E) equation is independent of the direction • Spatially nondispersive: the medium equation is local P.
    [Show full text]
  • Electrostatics in Material
    Electrostatics in material Bound charges Laplacean methods and dielectrics θ q zˆ ˆ z ˆ η zˆ Z ηˆ PPz= 0 ε 0 σ R E zˆ R s b ε 0 α ε Image methods Separation of variables ε I D top σ Displacement f Q σb < 0 field V r Stored E ε d energy in a dielectric Dbottom σb > 0 −σ above f E ε 0 Dielectric below E ε ε 0 F Forces d ε H Dielectric boundary F conditions 1 x − x xˆ From Physics 212, one might get the impression that going from electrostatics in vacuum to electrostatics in a material is equivalent to replacing epsilon_0 to epsilon > epsilon_0. This is more-or-less true for some dielectric materials such as Class A dielectrics but other types of materials exist. For example, there are permanent electrets which are analogous to permanent magnets. Here the electric field is produced by “bound” charges created by a permanently “frozen-in” polarizability. The polarizability is the electric dipole moment per unit volume that is often induced in the material by an external electric field. We will introduce the displacement field (or D-field) which obeys a Gauss’s Law that only depends on free charges. Free charges are the charges controllable by batteries, currents and the like. To a large extent one cannot totally control bound charges. We discuss the boundary conditions that E-fields and D-fields obey across a dielectric boundary. We turn next to a discussion of Laplace’s Equation in the presence of dielectrics.
    [Show full text]
  • ELECTRIC DISPLACEMENT If You Put a Dielectric in an External Field
    ELECTRIC DISPLACEMENT If you put a dielectric in an external field Eext, it polarizes, adding a new field, Einduced (from the bound charges). These superpose, making a total field Etot. What is the vector equation relating these three fields? A) Etot + Eext + Einduced = 0 B) Etot = Eext − Einduced C) Etot = Eext + Einduced D) Etot = −Eext + Einduced E) Something else! We define "Electric Displacement" or "D" field: D =ε0E + P. If you put a dielectric in an external field Eext, it polarizes, adding a new field, Einduced (from the bound charges). These superpose, making a total field Etot. Which of these three E fields is the "E" in the formula for D above? A) Eext B) Einduced C) Etot Linear Dielectric: P = ε0 χeE χe is the “Electric Susceptibility (Usually small, always positive) Linear Dielectric: P = ε0 χeE χe is the “Electric Susceptibility” D = ε0E + P = ε0E +ε0 χeE = ε0 (1+ χe )E ≡ ε0εrE εr is the dielectric constant ε ≡ ε0εr is the permittivity We define D =ε E + P, with 0 D⋅da = Q ∫ free, enclosed A point charge +q is placed at the center of a dielectric sphere (radius R). There are no other free charges anywhere. What is |D(r)|? R + +q A) q/(4 π r2) everywhere 2 B) q/(4 π ε0 r ) everywhere 2 2 C) q/(4 π r ) for r<R, but q/(4 π ε0 r ) for r>R D) None of the above, it's more complicated E) We need more info to answer! A solid non-conducting dielectric rod has been injected ("doped") with a fixed, known charge distribution ρ(s).
    [Show full text]
  • "Wave Optics" Lecture 21 (PDF)
    Lecture 21 Reminder/Introduction to Wave Optics Program: 1. Maxwell’s Equations. 2. Magnetic induction and electric displacement. 3. Origins of the electric permittivity and magnetic permeability. 4. Wave equations and optical constants. 5. The origins and frequency dependency of the dielectric constant. Questions you should be able to answer by the end of today’s lecture: 1. What are the equations determining propagation of the waves in the media? 2. What are the relationships between the electric field and electric displacement and magnetic field and magnetic induction? 3. What is the nature of the polarization? How can we gain intuition about the dielectric constant for a material? 4. How do Maxwell’s equations lead to the wave nature of light? 5. What is refractive index and how it relates to the dielectric constant? How does it relate to the speed of light inside the material? 6. What is the difference between the phase and group velocity? 1 Response of materials to electromagnetic waves – propagation of light in solids. We classified materials with respect to their conductivity and related the observed differences to the existence of band gaps in the electron energy eigenvalues. The optical properties are determined to a large extent by the electrons and their ability to interact with the electromagnetic field, thus the conductivity classification of materials, which was a result of the band structure and band filling, is also useful for classifying the optical response. The most widely used metric for quantifying the optical behavior of materials is the index of refraction, which captures the extent to which the electromagnetic wave interacts with the material.
    [Show full text]