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					                      What is Ferroelectric?

Ferroelectrics are materials which possess a “spontaneous”
electric polarization Ps which can be reversed by applying a
suitable electric field E.

This process is known as “switching”, and is followed by “hysteresis”.

Ferroelectrics are electrical analogues of “ferromagnetics” (P-E and M-H
relations).
          Ferroelectric Characteristics
    Three important characteristics of ferroelectrics:
• Reversible polarization
• “Anomalous” properties (i.e. ferroelectric disappears
  above a temperature Tc known as “Curie Point”
• Non-linearities
             Ferroelectric Characteristics
• “Anomalous” properties (i.e. ferroelectric disappears
  above a temperature Tc known as “Curie Point”
• Above Tc, the anomaly is frequently of the “ Curie-Weiss”
  form: (Curie-Weiss Relation)

                     e = C / (T-T0)

     C ~ Curie-Weiss constant
     T0 is called “Curie-Weiss Temperature”
     T0 < Tc in materials with first-order transitions
     T0 = Tc in materials with second-order transitions

     !! Some materials do not follow Curie-Weiss Relation !!
     Ferroelectric Characteristics
       • Dielectric non-linearities
Measured dielectric permittivity changes with
        change of applied (bias field)
              What is Piezoelectricity?
                    Piezoelectrics are
 materials which acquire electric polarization under
external mechanical stresses (Direct Effect),
                          OR
 materials that change size or shape when subject to
external electric field E (Converse Effect).
             ! (Piezo ~ Pressure or Stress) !

 Many piezoelectric materials are NOT ferroelectric
        All ferroelectrics are piezoelectric
Above T0, some ferroelectrics are STILL piezoelectric
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                      ้
ตัวอย่ างได้ รับการเอือเฟื้ อจาก Prof. Kenji Uchino, Penn State University, USA
                                 ้
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                       ้
 ตัวอย่ างได้ รับการเอือเฟื้ อจาก Prof. Kenji Uchino, Penn State University, USA
      การประยุกต์ ใช้ งานในอัลตร้ าโซนิกมอเตอร์ แบบท่ อ




                      ้
ตัวอย่ างได้ รับการเอือเฟื้ อจาก Prof. Kenji Uchino, Penn State University, USA
                     Structural Symmetry
Crystals in Nature            7 Crystal Systems                    14 Bravais Unit Cells
                       Triclinic, Monoclinic, Orthorhombic,
                      Tetragonal, Trigonal, Hexagonal, Cubic



                             Symmetry Elements
                       1,1,2, 2,3, 3,4, 4,6, 6, m, translation

                      230 Space Groups (Microscopic)


                     IF Translation Symmetry Removed


                     32 Point Groups (Macroscopic)
                    Structural Symmetry
  Crystal       Point Groups             Centro              Non-Centrosymmetry
 Structure                              Symmetry
                                                         Piezoelectric      Pyroelectric
  Triclinic             1, 1                  1                  1               1

 Monoclinic         2, m, 2/m                2/m                2, m            2, m

Orthorhombic    222, mm2, mmm               mmm              222, mm2           mm2

 Tetragonal    4, 4, 4/m, 422, 4mm,    4/m, (4/m)mm       4, 4, 422, 4mm,      4, 4mm
                  42m, (4/m)mm                                   42m
  Trigonal       3, 3, 32, 3m, 3m           3, 3m            3, 32, 3m         3, 3m

 Hexagonal     6, 6, 6/m, 622, 6mm,    6/m, (6/m)mm       6, 6, 622, 6mm,      6, 6mm
                  6m2, (6/m)mm                                   6m2
   Cubic        23, m3, 432, 43m,         m3, m3m             23, 43m          None
                      m3m

                 Point Groups for Seven Crystal Systems
               Note that: underlined numbers represent inversion symmetry
                 Structural Symmetry
                                  32 Crystal Classes
                   (all crystalline materials are electrostrictive)


              11 Classes                                21 Classes
           Centro-Symmetric                        Non-Centrosymmetric



                1 Class                                   20 Classes
            Non-Piezoelectric                            Piezoelectric


               10 Classes
                                                        10 Classes
            Unique Polar Axis
                                                    NO Unique Polar Axis
             (Pyroelectric)


1, 2, m, 2mm, 4, 4mm, 3, 3m, 6, 6mm

    Pyroelectrics: Spontaneous polarization upon heating or cooling
    Ferroelectrics: Reversible or re-orientable spontaneous polarization
    Ferroelectrics are a subgroup of the polar materials and
                   are BOTH pyroelectric and piezoelectric
                    Polarization (P)
Polarization (P) = Values of the dipole moment per unit volume
                = Values of the charge per unit surface area


           P = Nm / V = Nqd/Ad = Nq/A

      N = number of dipole moment per unit volume

      m = dipole moment = qd
      q = charge
      d = distance between positive and negative charges

      V = volume       AND        A = surface area
            Spontaneous Polarization (Ps)
Spontaneous polarization (Ps) exists in 10 classes of polar crystals
    with a unique polar axis (out of 20 piezoelectric classes)

                        BaTiO3 Single Crystal




       Cubic (T > Tc)                     Tetragonal (T < Tc)
          Ps = 0                                Ps  0
                    Pyroelectric Effect




          BaTiO3                     Triglycine Sulfate (TGS)
Pyroelectric Effect = Change of spontaneous polarization
(Ps) with temperature (T) (Discovered in Tourmaline by
Teophrast (314 B.C.) and named by Brewster in 1824);

        p = pyroelectric coefficient = Ps/T
Notice that BaTiO3 and TGS (and most crystals) has a negative
pyroelectric coefficient
Spontaneous Polarization (Ps) Re-Orientation
  Ceramics  a large number of randomly oriented crystallites 
          polarization re-orientation “Poling Process”




                               E

       Unpoled                                  Poled
Changes in Ps-directions require small ionic movements
 Larger number of possible directions of polar axes 
     Closer to poling direction  Easily poled
       Tetragonal 4mm  6 possible polar axes
      Rhombohedral 3m  8 possible polar axes
             better alignment (poled)
                        Ferroelectric Domains
Ferroelectric Domains = A region with uniform alignment (same direction)
                    of spontaneous polarization (Ps)
        Domain Walls = The interface between the two domains
                    very thin ( < a few lattice cells)




            A ferroelectric single crystal, when grown, has multiple ferroelectric domains
                                                    
                                   Applying appropriate electric field
                                                    
                         Possible single domain through domain wall motion
                                                    
                                         Too large electric field
                                                    
                   Reversal of the polarization in the domain “domain switching”
                                                    
                                     Hysteresis Loop
                            Ferroelectric Hysteresis Loop
                                                    D




                                           Hysteresis Loop
                          Starting from very small E-field  Linear P-E relationship (OA)
                         E  leads to domain re-alignment in the positive direction along E
                         rapid increase in P (OB) until it reaches the saturation value (Psat)
E  results in  P, but NOT all to Zero P as E = 0 (BD) because some domains remain aligned in positive direction
                                     Remnant OR Remanent Polarization (Pr) 
              Certain opposite E is needed to completely depolarize the domain  Coercive Field (Ec)
                               As E  in negative direction  direction of domains flip
                                                 Hysteresis Loop 
                         Spontaneous Polarization (Ps) is obtained through extrapolation 
                             Hysteresis Loop is observed by a Sawyer-Tower Circuit 
Ferroelectric Curie Point and Phase Transitions
    Curie Point (Tc) = Phase transition temperature between
           non-ferroelectric and ferroelectric phases

                  T < Tc = Ferroelectric Phase
         T > Tc = Paraelectric (Non-ferroelectric) Phase

  Transition Temperature = Other phase transition temperature
            between one ferroelectric phase to another
Ferroelectric Curie Point and Phase Transitions
 Near Curie Point (Tc)  Thermodynamic properties
         (dielectric, elastic, optical, thermal)
      show “ anomalies” and structural changes
Ferroelectric Curie Point and Phase Transitions
In most ferroelectrics, er above Curie Point (Tc) obeys
                Curie-Weiss Relation

                 e = e0 + C/(T-T0)
               C = Curie-Weiss constant
            T0 = Curie-Weiss Temperature
           (different from Curie Point Tc)
        T0 < Tc for first-order phase transition
       T0 = Tc for second-order phase transition

Tc = actual temperature when crystal structure changes
    T0 = formula constant obtained by extrapolation
     (Usually e0 term is neglected because e0 << e near T0)
   Ferroelectric Curie Point and Phase Transitions

In relaxor ferroelectrics, such
 as Pb(Mg1/3Nb2/3)O3 (PMN),
  and Tungsten-Bronze type
         compounds,
   such as (Sr1-xBax)Nb2O6,
      er does NOT obey
    Curie-Weiss Relation
    (1/e) – (1/em) = C’/(T-Tm)n

       C’ = constant
 Tm = Temperature with em
em = Maximum dielectric constant
           1<n<2
            Equilibrium Properties of Crystals
                          Heckmann’s Diagram


                                     (1)




                                     (1)




                               (2)         (0)




                    (2)                              (0)




Relations Between Thermal, Electrical, and Mechanical Properties of Crystals
                     (Rank of Tensors in Parenthesis)
               Equilibrium Properties of Crystals
                          Heckmann’s Diagram
 Three Outer Corners: Temperature (T), Electric Field (Ei), and Stress (ij)  “Forces”

Three Inner Corners: Entropy (S), Electric Displacement (Di), and Strain (ij)  “Results”



                           Lines Joining These Corner Pairs
                                          
                                  “Principal Effects”




Relations Between Thermal, Electrical, and Mechanical Properties of Crystals
       Equilibrium Properties of Crystals
I.   An increase of temperature produces a change of entropy dS:

                             dS = (C/T)dT

     where         C ( a scalar) is the “heat capacity per unit volume”
                   T is the absolute temperature

II. A small change of electric field dEi produces a change of electric
    displacement dDi

                             dDi = ijdEj

     where          ij is the “permittivity” tensor

III. A small change of stress dkl produces a change of strain dxij

                             dxij = sijkl dkl

     where         sijkl is the “elastic compliance”
      Equilibrium Properties of Crystals
Coupled Effects : Lines joining pairs not on the same corner
              Bottom : Thermoelastic Effects
   Right : Electrothermal Effects (Pyroelectric Effects)
  Left : Electromechanical Effects (Piezoelectric Effects)
     Direct and Converse Piezoelectric Effects
              (Third-Rank Tensors)
                dDi = dijkdjk  “Direct Effect”

              dxij = dijkdEk  “Converse Effect”

           where dijk is the “piezoelectric coefficient ”


        Thermoelastic Effects : dxij = jjdT

          Pyroelectric Effects : dDi = pidT

				
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