ionionion eN µσ =
TkD
B
eion =µ
TkxxQ
TkQ
DDeDD
BBo
eTkQoe
B1)ln(/ =⋅−=−=→= −
Ionic conduc)on: Under the influence of an external electric field, ions (nega)vely or posi)vely charged) can move. Ionic conduc)on takes place. For example, ions move to a vacancy or ions move inters))ally. The conduc)vity is: The mobility depends on the vacancy concentra)on which makes it possible for ions to jump to another loca)on. The vacancy concentra)on in pure materials is predicted by the Einstein Rela)on Where T the temperature, kB the Boltzmann constant, and De is the diffusion coefficient The parameter Q is the ac)va)on energy for the diffusion process. Do is the pre-‐experimental factor. It depends on vibra)on and frequency. Now is:
σ
σTkDeNee
TkDeN
B
oo
TkQoion
TkQ
B
oion
BB ⋅==⋅= −− σσσσ //
2
0
-‐2
-‐4
)ln(σ (Sm-‐1)
1 2 3 )/1(/103 KT
-‐AgI α
-‐AgI β
-‐Al2O3 β-‐Al2O3 ''β
(ZrO2)9(Y2O3)1
CaF2
AgCl LiI
Plo`ng an Arrhenius plot of the natural log of the data (ln(y)) as a func)on of the inverse temperature (1/T) easily reveals the ac)va)on energy as the slope of a straight line. The pre-‐experimental factor becomes the intercept at the y-‐axis. Somewhat hidden is the 1/T dependence in Two temperature ranges are visible at least for some samples. There is an extrinsic range and an intrinsic range
oσ
TkDeN
TkQ
B
oo
Boion ⋅=−= σσσ )ln()ln(
Temperature rises
oσLog(conduc)vity)
1/Temperature
extrinsic
concentra)on
RTB
oRTo
TkQRTRToion Tk
eNDTeTTT B == − )()( / σσσ
Extrinsic: ac)va)on energy Ea is small Intrinsic: added on thermal ac)va)on Steeper slope
RTB
oRTo
TkERTRToion Tk
eNDTeTTT Ba == − )()( / σσσ
( )TkHTkERTRToion
BsBa eeTTT 2//)( Δ−− ⋅=σσ
( ) Bsa kHEslope /5.0 Δ+−=
1. For low (extrinsic) temperatures: extrinsic defects (doping) determines the mobility of ions . 2. At high (intrinsic) temperatures: thermally formed (intrinsic) defects (such as vacancies) take over and determine the mobility. Vacancy-‐inters))al pairs are generated. These are Frenkel defects. Overall: charge neutrality is maintained. While it “costs” energy to form defects, there is a gain in entropy , there is a sweet spot for a defect concentra)on with minimum free energy .
sHΔST Δ⋅−
GΔ
Example: Replacing a mono-‐valent metal with a di-‐valent metal in an ionic crystal. Add Mg2+ to replace Na+ in NaCl. The results are missing ca)ons, i.e. Na vacancies. (one Mg2+ replaces 2 Na+, or requires one extra Cl-‐ to neutralize charge). This generates Schofky defects (only vacancy, no inters)tal) Ques)ons: Ionic single crystals are befer conductors than polycrystalline materials. Why? The type of compound determines the majority carrier (posi)ve or nega)ve) type. Back to basics about ionic crystal vacancy forma)on. 1) Ca)on and anion vacancies are formed 2) Frenkel defects are a pair of a vacancy and an inters))al 3) Adding di-‐valent metals into a monovalent metal compound results in vacancies. 4) Non-‐stoichiometric compositce contain high concentra)ons of vacancies.
Metal oxides can be ionic conductors, semiconductors or even metals. Titanium oxides are an example. TiO2 is an insulator, while TiO is a metal.
conduc&on Material Conduc&vity (S/m)
Carriers & media&on
Ionic Ionic crystals <10-‐16 to 10-‐2 Ions, defects
Solid electrolytes 10-‐1 to 103
Liquid electrolytes 10-‐1 to 103
Electronic Insulators 10-‐10 Electrons, holes
Semiconductors 10-‐3 to 104 Electrons, holes
Metals 103 to 107 electrons
Orders of magnitude:
Piezoelectric materials: Mechanical strain generates a dielectric polariza)on resul)ng in an overall electric bias. This occurs in crystals which lack a center of symmetry. Symmetric crystals under stain: change in dimensions follow. Unsymmetric crystals under strain: electric dipoles are formed as the bond length between ca)ons and anions is changed Examples are Quartz and ZnO Ferroelectric materials are a subset of piezoelectric materials. In contrast to piezoelectrics they polarize spontaneously (pyroelectricity) and have two stable polariza)ons (ferrorelectric). The reverse effect is also possible: An applied electric field changes the dimensions of the sample. The response strongly depends on the frequency of the field. This is used in quartz oscillators. Their overall dimensions determine the resonant frequency and can stabilize clocks for example.
TQeDTk
eNeNTkQ
o
B B ∝⇒=⇒⋅⋅=== ρµσ
ρ 0111 /
Metals: Electrical resis)vity of a metal is temperature dependent. The number density of electrons is constant:
Magne)c materials: Iron (ferrum) was the first material known to be magne)c. The same observed trends all inherited the name ferro…. Back from ionic crystals via dipoles to metals of a special kind.
Origins of magne)sm: Moving charge: A charge in mo)on generates a magne)c field. Electromagnets: current in a coil generates a magne)c dipole field. Electrons in an atom: these electrons move (have momentum) which generates a magne)c field that is manifested as orbital magne)c moments. Spin of electrons (and other charged fermions): they too have a magne)c moment. Placed in an external magne)c field materials respond in different ways. Why? The causes can be found in the electronic structure of the atoms (as built up from the simple hydrogen atom model). The key parameter is the pairing of electrons: unpaired vs. paired ones.
The key parameter is the pairing of electrons: unpaired vs. paired ones. The spins of paired electrons line up in opposite direc)on. Their respec)ve magne)c moments sum in opposite direc)on and cancel each other. In the unpaired case, the spins of the unpaired electrons can add up and result in a net magne)c moment for each atom and in cases for all atoms in the macroscopic material.
Some defini)ons and concepts A magne)c field H in vacuum results in a magne)c induc)on B by the permeability of free (vacuum) space In S.I. Units: When generated by a solenoid, i.e. a (infinitely) long spiral of a conductor with current I and length :
Solenoid: Once material is present within the magne)c field, it can be magne)zed. The magne)za)on M and the field H add to the induc)on B:
mHenryHB oo
7104 −⋅=⋅= πµµ
)( mANTTeslaB ==
mAH = m
AT
mHenry
o ==µ
nIH ⋅=
( )MHB o +⋅= µ
Once material is present within the magne)c field, it can be magne)zed. The magne)za)on M and the field H add to the induc)on B: The magne)za)on itself is caused by the magne)c field. The material property is the suscep)bility of the material Xm which reflects how sensi)ve (suscep)ble) the material is to the magne)c field Both combined give where is the rela)ve permeability that relates B to H in the presence of material (non vacuum). As noted above the S.I. units standard was used above. In many books the older cgs system was used. Conven)ons are somewhat different in the two systems: SI units (as above) cgs units Just like the polarizebility described dielectric materials here the suscep)bility Xm is the classifying property for magne)c materials.
( )MHB o +⋅= µ
HM mΧ=
( ) ( ) HHHB momo ⋅Χ+⋅=Χ+⋅= 1µµ
HB ro ⋅= µµ
( )mr Χ+= 1µ
( ) ( ) HMHB mooo ⋅Χ+⋅=+⋅=⋅= − 1104 7 µµπµ
( ) HMHB mo ⋅Χ⋅+=⋅+== ππµ 4141
Be careful when using magne&c parameters. Check the tables
Several magne)c behaviors are observed: 1) Diamagne)c 2) Paramagne)c 3) Ferromagne)c 4) An)ferromagne)c 5) Ferrimagne)c
materials
Diamagne)c materials: The suscep)bility is nega)ve. Diamagne)c materials have no magne)c moment of their own. In the presence of an external field they respond in a “non-‐coopera)ve” way and weaken the induc)on B. The Suscep)bility is small and nega&ve and independent of temperature. All materials have a diamagne)c effect. However, it is masked by other effects such as paramagne)sm or ferromagne)sm. Examples of Xm are Au: -‐2.74 10-‐6 Cu: -‐0.77 10-‐6
0<Χm
HM mΧ=
Paramagne)c materials: Unpaired electrons of par)ally filled orbits of some of the atoms and ions in a material are typically randomly oriented. An external magne)c field will line them up in the same direc)on and enhance the overall induc)on. Xm is small and posi)ve and drops with temperature.
0>Χm
Curie-‐Weiss law: of temperature dependence In classically paramagne)c materials i.e. the Curie law.
θ−=ΧTConst
m
K0=θ
Paramagne)c origins: The spin of unpaired electrons contributes (visualize (not actually) electrons as spheres spinning around their axis). External fields try to align the spinning direc)on. The electron orbits contribute to paramagne)sm. It occurs in rare earth metals and dilute gasses. For this the atoms have to align, which is harder to accomplish in a solid where the crystal field “)es down” the outer shell electrons. However, in rare earth elements the 4f electrons are not the outer shell and are shielded from the crystal fields.
Ferromagne)c materials: Exhibit hysteresis. The magne)za)on is not linear with the magne)c field but also depends on the preceeding history. Xm is much larger than in the other cases and not constant. Above a cri)cal Curie temperature ferromagne)sm vanishes. The much smaller paramagne)c behavior remains.
An)ferromagne)c materials: for example Hema)te Fe2O3 The spins of the electrons align in alterna)ng opposite direc)ons. Overall they cancel. In an applied field, some of the cancella)on is removed and magne)za)on occurs. Above the Neel temperature this vanishes. Paramagne)sm remains. Ferrimagne)c materials: for example magne)te Fe3O4 Just like above, the spins align in opposite direc)ons but are of different magnitudes. A net magne)za)on results. With temperature the material changes from magne)c (low temp) to a region with the moments cancel (2), and then above the Curie temperature to paramagne)c.
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