Introduction to solid state physics
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1 PHYS 342/555 Introduction to solid state physics Instructor: Dr. Pengcheng Dai Professor of Physics The University of Tennessee (Room 407A, Nielsen, ) Chapter 13: Dielectrics and ferroelectrics Lecture in pdf format will be available at: 1
2 Polarization The polarization P is defined as the dipole moment per unit volume, averaged over the volume of a cell. The total dipole moment, p q, nrn The electric field at a point r from a dipole moment p is: E 2 3( pr) r r p ( r) 5 r Dai/PHYS 342/555 Spring 2012 Chapter 13-2
3 Dai/PHYS 342/555 Spring 2012 Chapter 13-3
4 Dai/PHYS 342/555 Spring 2012 Chapter 13-4
5 Dai/PHYS 342/555 Spring 2012 Chapter 13-5
6 Dai/PHYS 342/555 Spring 2012 Chapter 13-6
7 Dai/PHYS 342/555 Spring 2012 Chapter 13-7
8 Dai/PHYS 342/555 Spring 2012 Chapter 13-8
9 Dai/PHYS 342/555 Spring 2012 Chapter 13-9
10 Dai/PHYS 342/555 Spring 2012 Chapter 13-10
11 Dai/PHYS 342/555 Spring 2012 Chapter 13-11
12 Dai/PHYS 342/555 Spring 2012 Chapter 13-12
13 Dai/PHYS 342/555 Spring 2012 Chapter 13-13
14 Dai/PHYS 342/555 Spring 2012 Chapter 13-14
15 Dai/PHYS 342/555 Spring 2012 Chapter 13-15
16 Dai/PHYS 342/555 Spring 2012 Chapter 13-16
17 Dai/PHYS 342/555 Spring 2012 Chapter 13-17
18 Dai/PHYS 342/555 Spring 2012 Chapter 13-18
19 Dai/PHYS 342/555 Spring 2012 Chapter 13-19
20 Dai/PHYS 342/555 Spring 2012 Chapter 13-20
21 Dielectric constant and polarizability The dielectric constant of an isotropic medium relative to vacuum is defined as E 4 P =1+4 E The polarizability of an atom is defined in terms of the local electric field at the atom: p E local The polarization of a crystal is then P N jpj= N j jelocal( j). j j Dai/PHYS 342/555 Spring 2006 Chapter 13-21
22 If the local field is given by the Lorentz relation, then 4 ( )= ( 4 P N j j E P N j j E P). 3 3 j P N j j =. E 4 1 N j j 3 Dai/PHYS 342/555 Spring 2006 Chapter 13-22
23 The total polarizability can be separated into three parts: 1. electronic: arises from the displacement of the electron shell relative to a nucleus. 2. ionic: comes from the displacement of a charged ion with respect to other ions. 3. dipolar: from molecules with a permanent electric dipole moment that can change orientation in an applied electric field. Dai/PHYS 342/555 Spring 2006 Chapter 13-23
24 Dai/PHYS 342/555 Spring 2012 Chapter 13-24
25 Dai/PHYS 342/555 Spring 2012 Chapter 13-25
26 Dai/PHYS 342/555 Spring 2012 Chapter 13-26
27 Dai/PHYS 342/555 Spring 2012 Chapter 13-27
28 Dai/PHYS 342/555 Spring 2012 Chapter 13-28
29 Dai/PHYS 342/555 Spring 2012 Chapter 13-29
30 Dai/PHYS 342/555 Spring 2012 Chapter 13-30
31 Dai/PHYS 342/555 Spring 2012 Chapter 13-31
32 Dai/PHYS 342/555 Spring 2012 Chapter 13-32
33 Dai/PHYS 342/555 Spring 2012 Chapter 13-33
34 Dai/PHYS 342/555 Spring 2012 Chapter 13-34
35 Dai/PHYS 342/555 Spring 2012 Chapter 13-35
36 Dai/PHYS 342/555 Spring 2012 Chapter 13-36
37 Dai/PHYS 342/555 Spring 2012 Chapter 13-37
38 Dai/PHYS 342/555 Spring 2012 Chapter 13-38
39 Dai/PHYS 342/555 Spring 2012 Chapter 13-39
40 Dai/PHYS 342/555 Spring 2012 Chapter 13-40
41 Dai/PHYS 342/555 Spring 2012 Chapter 13-41
42 Dai/PHYS 342/555 Spring 2012 Chapter 13-42
43 Dai/PHYS 342/555 Spring 2012 Chapter 13-43
44 Dai/PHYS 342/555 Spring 2006 Chapter 13-44
45 Dai/PHYS 342/555 Spring 2012 Chapter 13-45
46 Dai/PHYS 342/555 Spring 2012 Chapter 13-46
47 Dai/PHYS 342/555 Spring 2012 Chapter 13-47
48 Dai/PHYS 342/555 Spring 2012 Chapter 13-48
49 Show that the polarizability of a conducting metallic sphere 3 of radius a is a. Dai/PHYS 342/555 Spring 2012 Chapter 13-49
50 Structural phase transitions The stable structure at a temperature T is determined by the minimum of the free energy F U TS. Ferroelectric crystals A ferroelectric state is a state where the center of positive charge of the crystal does not coincide with the center of negative charge. Ferroelectricity disappears above a certain temperature, where the crystal is in the paraelectric state. Dai/PHYS 342/555 Spring 2012 Chapter 13-50
51 Ferroelectricity Ferroelectricity ti it is an electrical ti lphenomenon whereby certain materials may exhibit a spontaneous dipole moment, the direction of which can be switched between equivalent states by the application of an external electric field. The internal electric dipoles of a ferroelectric material are physically tied to the material lattice so anything that changes the physical lattice will change the strength of the dipoles and cause a current to flow into or out of the capacitor even without the presence of an external voltage across the capacitor. Dai/PHYS 342/555 Spring 2012 Chapter 13-51
52 Ferroelectric properties Most ferroelectric materials undergo a structural phase transition from a high-temperature nonferroelectric (or paraelectric) phase into a low-temperature ferroelectric phase. The paraelectric phase may be piezoelectric i or nonpiezoelectric and is rarely polar. The symmetry of the ferroelectric phase is always lower than the symmetry of the paraelectric phase. Dai/PHYS 342/555 Spring 2006 Chapter 13-52
53 The temperature of the phase transition is called the Curie point, T C. Above the Curie point the dielectric permittivity falls off with temperature according to the Curie Weiss law where C is the Curie constant, T 0 (T 0 T C ) is the Curie Weiss temperature. Some ferroelectrics, such as BaTiO 3, undergo several phase transitions into successive ferroelectric phases. Dai/PHYS 342/555 Spring 2006 Chapter 13-53
54 A naïve picture Dai/PHYS 342/555 Spring 2006 Chapter 13-54
55 Dai/PHYS 342/555 Spring 2006 Chapter 13-55
56 A naïve picture Dai/PHYS 342/555 Spring 2006 Chapter 13-56
57 A naïve picture Dai/PHYS 342/555 Spring 2006 Chapter 13-57
58 A naïve picture Dai/PHYS 342/555 Spring 2006 Chapter 13-58
59 A naïve picture The local alignment of dipoles can exist over any length scale. Different regions may exist with different polarisation orientations: Call these domains in line with magnetic materials. In contrast with magnetism, domain walls are abrupt. Dai/PHYS 342/555 Spring 2006 Chapter 13-59
60 Applied field Suppose we now apply an electric field, horizontal in the figure. If it is of sufficient strength, the small ions will be able to overcome the barrier and dipoles will switch direction The dipoles are polarised by the applied field. Domain walls move. Dai/PHYS 342/555 Spring 2006 Chapter 13-60
61 Polarisation vs. E-field Suppose we start with a material where there are many domains which are aligned randomly. What is the initial polarisation? Dai/PHYS 342/555 Spring 2006 Chapter 13-61
62 Polarisation vs. E-field If we apply a small electric field, such that it is not able to switch domain alignments, then the material will behave as a normal dielectric: P E As E is increased, we start to flip domains and rapidly increase P. When all domains are switched, we reach saturation. What happens if the E-field is now removed? Dai/PHYS 342/555 Spring 2006 Chapter 13-62
63 Polarisation vs. E-field The value at zero field is termed the remnant polarisation. The value of P extrapolated back from the saturation limit is the spontaneous polarisation. Reversal of the field will eventually remove all polarisation The field required is the coercive field. Further increasing the reverse field will completely l reverse the polarisation, and so a hysteresis loop is formed Dai/PHYS 342/555 Spring 2006 Chapter 13-63
64 Dai/PHYS 342/555 Spring 2006 Chapter 13-64
65 Polarisation hysteresis The essential feature of a ferroelectric is not that there is a spontaneous polarisation, but that the spontaneous polarisation can be reversed by the application of an electric field. Now, E is small enough for the applied field to reverse the direction of the dipoles, i.e. move the atoms within the crystal, what else might affect this change? What are the relative sized of E and k B T? What will happen if E k B T? Dai/PHYS 342/555 Spring 2006 Chapter 13-65
66 Curie temperature Above a critical temperature the spontaneous polarisation will be lost due to one of two effects: A change of structure such that there is a single minimum in the energy mid-way between sites The rate that the small ions hop is so high h that t on average there is no net polarisation Dai/PHYS 342/555 Spring 2006 Chapter 13-66
67 Domain Wall Movement Dai/PHYS 342/555 Spring 2006 Chapter 13-67
68 Piezoelectric effect The application of an electric field induces a geometrical change. Alternatively, a distortion ti of the material induces a potential difference. Used in many electrical devices, e.g. sound-to-electricity Dai/PHYS 342/555 Spring 2006 Chapter 13-68
69 Dai/PHYS 342/555 Spring 2006 Chapter 13-69
70 Dai/PHYS 342/555 Spring 2006 Chapter 13-70
71 Crystallography y and ferroelectrics Th t l l ifi ti f T T h OT d O h The crystal classification of a material has immediate implications for ferroelectric effects There are 32 crystal classes 11 of them have a centre of symmetry (centrosymmetric) and cannot support ferroelectricity Of the remaining 21, the O- point group (432) also excludes ferroelectricity. The remaining 20 classes all exhibit the piezoelectric effect Of these, 10 have a unique polar direction. C 4 S 4 C 4h D 4 C 4v D 2d D 4h D 2 C 2v D 2h C 2 C s C 2h C 1 C i C 3 S 6 D 3 C 3v D 3d C 6 C 3h C 6h D 6 C 6v D 3h D 6h Dai/PHYS 342/555 Spring 2006 Chapter 13-71
72 Classification of ferroelectrics i. Crystal-chemical : l hydrogen-bonded d d (e.g. KH 2 PO 4 ) or otherwise (e.g. double oxides). ii. No. of polarisation directions: single direction (e.g. PbTa 2 O 6 ), several equivalent directions (e.g. BaTiO 3 ). iii. Centrosymmetric non-polar phase: E.g. Rochelle salts exhibit piezoelectric phase above T c, whereas BaTiO 3 is centrosymmetric. iv. First vs. second order phase change at T c. It turns out this corresponds to the value of the Curie constant t (C), one group being of the order of 10 3 K, and the other 10 5 K. These four classifications do not necessarily coincide. Dai/PHYS 342/555 Spring 2006 Chapter 13-72
73 Antiferroelectrics If the free energy of an antipolar phase is comparable to the polar state then the material is termed antiferroelectric. If a material exhibits ferroelectric effects in one polar direction, and antiferroelectric effects perpendicular, it may be termed ferrielectric. Antipolar Polar Dai/PHYS 342/555 Spring 2006 Chapter 13-73
74 Perovskites Perovskite is a naturally occurring mineral with chemical formula CaTiO 3. This is a prototype for many ABO 3 materials which are very important in ferroelectrics. These materials may be envisaged by consideration of a non-polar polar, cubic basic building block Dai/PHYS 342/555 Spring 2006 Chapter 13-74
75 Perovskites Dai/PHYS 342/555 Spring 2006 Chapter 13-75
76 Perovskites Below the Curie temperature, these crystals undergo symmetry lowering distortions. We ll initially focus up the distortions of BaTiO 3. There are three phase transitions in order of decreasing temperature: 120 o C, 5 o C, and -90 o C. Dai/PHYS 342/555 Spring 2006 Chapter 13-76
77 BaTiO 3 Above 120 o C BaTiO 3 is cubic (non-polar) Ti Ba Dai/PHYS 342/555 Spring 2006 Chapter 13-77
78 BaTiO 3 From 120 o C down to ~5 o C, there is a distortion to a tetragonal phase. All of the cube directions can undergo this type of distortion: this leads to complexity in domain formation. Dai/PHYS 342/555 Spring 2006 Chapter 13-78
79 BaTiO 3 From 5 o C down to around -90 o C the structure is orthorhombic by dilation along [110] directions and contraction along [1-10]. Dai/PHYS 342/555 Spring 2006 Chapter 13-79
80 BaTiO 3 Finally the lowest temperatures yield rhombohedra (distortions along the body diagonal). There are therefore 8 equivalent distortion directions Dai/PHYS 342/555 Spring 2006 Chapter 13-80
81 directions directions directions The phase transition sequence in perovskites Dai/PHYS 342/555 Spring 2006 Chapter 13-81
82 PbTiO 3 internal structure The foregoing analysis of BaTiO 3 focuses on the shape of the unit cell. However, for the ferroelectric effect, we also require internal structural changes. In light of current interest in Cu-related defects in lead titanium tri-oxide, we ll review this tetragonal system at room temperature Dai/PHYS 342/555 Spring 2006 Chapter 13-82
83 PbTiO 3 internal structure X-ray and neutron scattering yield the internal structure: a=3.904å, c=4.150å, so that c/a= Taking the Pb site as the origin, i the displacements are z Ti = z OI = z OII = Here OI are the polar O-sites, and OII are the equatorial. Thus, since the oxygen atoms are all displaced by the same amount, the oxygen cage remains intact, and shifts relative to the Ti and Pb sites. Dai/PHYS 342/555 Spring 2006 Chapter 13-83
84 PbTiO 3 Ti Pb Dai/PHYS 342/555 Spring 2006 Chapter 13-84
85 PbTiO 3 It should be clear that this system has a net dipole in each unit cell, and furthermore that the distortion can be along any of the x, y and z directions. In real PbTiO 3, there is a non-trivial role for point defects, especially O-vacancies. Dai/PHYS 342/555 Spring 2006 Chapter 13-85
86 Classification of ferroelectric crystals Ferroelectric crystals can be classified into two main groups: order-disorder and displacive transition. Dai/PHYS 342/555 Spring 2012 Chapter 13-86
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