ELECTRONICS DEVICES AND MATERIALS

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1 2-2-2 ELECTRONICS DEVICES AND MATERIALS Atsunori KAMEGAWA SYLLABUS! Introduction to materials structure and dielectric physics (04/27)! Ferroelectricity involved in structural phase transitions (05/25) Material design of dielectrics and introduction to metamaterials (06/0)! Ferroelectric devices (06/08) LAST TIME: DIELECTRIC DISPERSION ~ DIELECTRIC SPECTROSCOPY ~ 2-3 LAST TIME: PIEZOELECTRICITY, PYROELECTRICITY AND FERROELECTRICITY 2-4 Space Charge Polarization Dipolar Polarization Ionic Polarization Electronic Polarization Vacuum permittivity= Relaxation type Hz Frequency Resonance type Piezoelectricity Piezoelectricity: the ability of some Pyroelectricity materials to generate an electric potential in response to applied mechanical stress Ferroelectricity 20 point groups lacking in a center of symmetry Pyroelectricity: the ability of certain materials to generate an electrical potential when they are heated or cooled (having spontaneous polarization) 0 point groups including polar vectors Ferroelectricity: Physical property of a material whereby it exhibits a spontaneous polarization, the direction of which can be switched between equivalent states by the application of an external electric field

FERROELECTRICITY INVOLVED IN STRUCTURAL PHASE TRANSITIONS. Ferroelectricity 2. Phase Transition 2-5 FERROELECTRICS: 2-6 POLARIZATION REVERSAL (POLARIZATION SWITCHING) Dielectric P Ferroelectric P 3.

2 FERROELECTRICITY INVOLVED IN STRUCTURAL PHASE TRANSITIONS. Ferroelectricity 2. Phase Transition 2-5 FERROELECTRICS: 2-6 POLARIZATION REVERSAL (POLARIZATION SWITCHING) Dielectric P Ferroelectric P 3. First-order and Second-order Phase Transitions! Ehrenfest Classification! Landau s Theorem 4. Ferroelectricity Involved in Structural Phase Transitions E E 5. Classification of Ferroelectrics! Displacive Type! Order-Disorder Type Linear response Strongly nonlinear field dependence polarization hysteresis loop POLARIZATION REVERSAL (POLARIZATION SWITCHING) FERROELECTRIC DOMAINS P P E P P Single domain crystal: energetically non-favorable Fabrication -heating above T c -cooling in an external E-field Vertical planes no 2-fold rotation (C 2 ) no mirror (m) Polar Crystal Structure P Polarization Reversal (Single domain crystal) Multi-domain crystal: energetically favorable Controlled fabrication via E-field poling

3 FERROELECTRICS: CURIE TEMPERATURE 2-9 PHASE TRANSITION 2-0 Ferroelectricity disappears above certain temperature called the transition temperature T C (Curie temperature). ferroelectricity paraelectricity Phase Transision: the transformation of a thermodynamic system from one phase to another with a change in a variable such as the temperature, pressure, external magnetic field. BaTiO 3 rhombohedral orthorhombic tetragonal T C cubic Above T C the crystal is in a paraelectric phase where e > and e(t) decreases rapidly with increasing T, P(E=0)=0. Pressure Solid Phase Liquid Phase Critical Point vapor pressure curve Magnetization Gaseous Phase Triple point Temperature Schematic Phase Diagram of water ferromagnetism H=0 paramagnetism H>0 Temperature Phase Transition of ferromagnetism THERMODYNAMICS OF PHASE TRANSITION ~EHRENFEST CLASSIFICATION ~ 2- REVIEW OF THERMODYNAMICS 2-2 Ehrenfest s definition of phase transition: In an n-th order phase transition, the n-th and higher thermodynamic derivatives of Gibbs free energy G with respect to T and P show discontinuities. Gibbs free energy =H-TS=U+pV-TS Total differentiation d=du+pdv+vdp-tds-sdt Ehrenfest Classification (First law of thermodynamics: du =TdS-pdV)! first order phase transtion () d=vdp-sdt! second order phase transtion ()! n-th order phase transtion () G T =-S p G p =V T S, V = the first derivatives of Gibbs energy

4 THERMODYNAMICS OF PHASE TRANSITION ~EHRENFEST CLASSIFICATION ~ 2-3 THERMODYNAMICS OF PHASE TRANSITION ~EHRENFEST CLASSIFICATION ~ 2-4 When entropy changes discontinuously, The st derivative of G with respect to T The transition occurs at T trns under constant pressure The value of entropy jumpδ trns S Δ trns H = T trns Δ trns S dh =TdS+Vdp The enthalpy change Latent heat of st order phase transtion notable characteristic of st order phase transition definition of constant-pressure specific heat capacity Cp= Cp=T H T p S T p =-T 2 G T 2 p dh =TdS+Vdp Cp = 2nd derivative of G with respect to T Cp also shows discontinues in st order phase transition THE FIRST-ORDER PHASE TRANSITION Gibbs free energy Entropy, S Specific heat, Cp Stable low temp. supercooling state supercooling state superheating state Stable high temp. Temperature S discontinuity Temperature undefined at phase transition superheating state Temperature Characteristic of st order phase transition " coexsisting of high and low temp. phase ex) ice melting =0 C: coexisting of water and ice " temperature hysteresis of phase transition ex) supercooling phenomenon 2-5 THE SECOND-ORDER PHASE TRANSITION Gibbs energy Entropy, S Specific heat, Cp low temperature phase high temperature phase temperature temperature phase transition temperature Gibbs free energy (G): continuous & smooth change Entropy: st derivative of G continuous change Specific heat: 2nd derivative of G discontinuous change no associated latent heat!! 2-6

5 EHRENFEST CLASSIFICATION Gibbs energy Entropy, S Specific heat, Cp Stable low temp. supercooling state st order Stable temp. temperature temperature superheating state temperature Gibbs energy Entropy, S Specific heat, Cp low temperature phase 2nd order high temperature phase temperature temperature temperature 2-7 MODERN CLASSIFICATION OF PHASE TRANSITIONS Weakness of Ehrenfest scheme Ehrenfest scheme does not take into account the case where a derivative of free energy diverges (which is only possible in the thermodynamic limit). ex) the specific heat in ferromagnetic transition diverges to infinity. Modern classification of Phase Transitions The second-order phase transition (continuous phase transition) No latent heat 2-8 SYMMETRY-BREAKING TRANSITION ~ INTRODUCTION TO LANDAU S THEORY~ 2-9 LANDAU PHENOMENOLOGICAL THEORY OF PHASE TRANSITIONS 2-20 In general thermodynamics, phase transition indicates the transformation from one phase to another. In a narrow sense, phase transition take place between phases with different symmetry and without long-range atomic diffusion. The terminology phase transtiion is used as this narrow sense in ferroelectric physics. Symmetry-breaking process: the transition from the more symmetrical phase to the less symmetrical one " Thermodynamic potential density of free energy " Quantity exists which could be used as a measure of order which appears in the ordered phase #order parameter " Landau-Devonshire theory #order parameter spatially constant (spatial fluctuations are neglected, ferroelectrics P F(m) ) Free energy of Landau Theory F(m)=F 0 + am 2 + bm 4 + cm 6 +- HM 2 4 6

6 ORDER PARAMETER Partial list of transitions and order parameters 2-2 FREE ENERGY OF FERROELECTRICITY IN LANDAU TREATMENT 2-22 Transition Liquid-gas Ferromagnetic Ferroelectric Superconductors Siperfluid Phase Separation Order parameter density magnetization polarization complex gap parameter condensate wave function concentration Order parameter of ferroelectricity: polarization, P F(m)=F 0 + ap 2 + bp 4 + cp 6 +- EP Simplifications: we neglect dependence of parameters on p and for simplicity only a = f(t) = a 0 (T-T 0 ) L. P. Kadanoff et al., Rev. Mod. Phys., 39 (967), 395. SECOND ORDER TRANSITION OF LANDAU THEORY F(m)=F 0 + ap 2 + bp 4 + cp 6 +- EP b 0 # second order transition minimization of F # equilibrium order parameter P F equation of state P =ap + bp3 + -E=0 If E=0: a a a+bp 2 =0 # P= b # P= 0 b (T 0 -T) Polarization E: external electric field E=0 T 0 = E>0 Temperature 2 F P s Nontrivial solution; only if T<T0, for T>T0 #P0 T>T 0 T<T 0 P If E 0: cubic equation to solve #P 0 above T c 2-23 For your understanding of the Landau s 2nd order transition ORDER PARAMETER AND THE SECOND-ORDER PHASE TRANSITION β-brass(cuzn) order structure (B2) Cu Zn random occupation disorder structure (A2) order parameter 0 Occupation fraction Cu on body-centerω(cu) Zn on body-centerω(zn) When the occupation is random, ω(cu)= ω(zn)=0.5 Order parameter: m ω(cu) - ω(zn) m= ω(cu) + ω(zn) m= for a fully ordered state m=0 for a completely random state Temperature 2-24

7 F FIRST ORDER TRANSITION OF LANDAU THEORY b < 0 # first order transition We need term + cp 6 6 in order to keep F positive. If E=0: F P =0 # ap b P3 +cp 5 = 0# trivial solution P=0 nontrivial solution $# a b P 2 +cp 4 = 0 T I >T c P s (T c ) P P s (T III ) T II =T c T III <T C minimum of F appears at P 0 # discontinuous jump of P from 0 to some value C E< E2< E3 Polarization E=0 E E2 E3 Temperature If E 0: at some critical E C discontinuous transition becomes continuous critical point defined by (T C,E C ) 2-25 Polarization PHASE TRANSITION OF FERROELECTRISITY st order transition spontaneous polarization,ps T0 ε /ε E< E2< E3 E=0 E E2 E3 Temperature Temperature Polarization 2nd order transition spontaneous polarization,ps E=0 ε Temperature E>0 /ε E: external electric field 2-26 Temperature WHICH DOES BARIUM TITANATE BEHAVE ST OR 2ND ORDER PHASE TRANSITION? 2-27 CLASSIFICATION OF FERROELECTRICS ~ACCORDING TO THE NATURE OF THE FERROELECTRIC PHASE TRANSITION ~ 2-28 BaTi0 3 a-axis a-axis! Displacive Type! Order-Disorder Type Temperature ( C) rhombohedral (C 3v ) orthorhombic(d 2v ) tetragonal(c 4v ) cubic(o h )

8 DISPLACIVE TYPE 2-29 SUCCESSIVE PHASE TRANSITION OF BARIUM TITANATE 2-30 Perovskite-type Oxide, ABO 3 ex) BaTiO 3, PbTiO 3 Cubic Perovskite Structure (Oh) Projection of the ABO 3 structure on to the (00) plane a-axis c a-axis Temperature ( C) A(Ba) B(Ti) O a Dsiplacement of atoms rhombohedral (C 3v ) orthorhombic(d 2v ) tetragonal(c 4v ) cubic(o h ) 3m mm2 4mm m3m ORDER-DISORDER TYPE Sodium Nitrite, NaNO2 Paraelectric phase (D 2h ) Ferroelectric phase (C 2v ) ORDER-DISORDER TYPE TGS; tri-glycine sulfate, (NH 2 CH 2 COOH) 3 H 2 SO 4 c b N O Na NO 2 - b c a Dielectric constant, Temperature ( C) Temperature ( C) Inverse of dielectric constant, - Spontaneous Polarization Inverse of dielectric constant

Introduction to solid state physics

PHYS 342/555 Introduction to solid state physics Instructor: Dr. Pengcheng Dai Professor of Physics The University of Tennessee (Room 407A, Nielsen, 974-1509) Chapter 13: Dielectrics and ferroelectrics