1 Objective Grain Size Varistors Hall- Petch Creep Microstructure-Properties: I Lecture 5B The Effect of Grain Size on Varistors 27-301 October, 2007 A. D. Rollett
1
Objective
Grain Size
Varistors
Hall-Petch
Creep
Microstructure-Properties: ILecture 5B
The Effect of Grain Sizeon Varistors
27-301October, 2007A. D. Rollett
2
Objective
Grain Size
Varistors
Hall-Petch
Creep
Objective• This lecture is concerned with the effects of grain
size on properties.• This is the second of two examples:
the effect of grain size on resistance in ceramicsused for varistors (e.g. in surge protectors).
• The previous example was the effect of grain sizeon mechanical properties, namely the Hall-Petcheffect, and Nabarro-Herring creep.
• Similar considerations apply to magnetic hardnessalso.
3
Objective
Grain Size
Varistors
Hall-Petch
Creep
Key Concepts
• Grain boundaries (effectively) have properties that differ fromthe matrix.
• Properties of polycrystal depend on the content of planardefects, i.e. grain boundaries, i.e. grain size.
• Grain boundaries in semiconductors used to make varistorshave a one-way voltage barrier.
• The Hall-Petch effect quantifies the trend of increasingstrength and toughness with decreasing grain size.
• Creep rates (Coble creep) increase with increasing grainboundary area (per unit volume), hence decreasing grain size.
• Low temperature service optimized by fine grain size, but hightemperature service optimized by use of single crystals.
4
Objective
Grain Size
Varistors
Hall-Petch
Creep
SurgeProtectors
• Surge protectionmeans inserting acomponent into acircuit that preventsthe voltage fromrising above a certainvalue.
• Note the diagramshowing varistors inparallel with the load
http://www.sosnet.com/StaticPages/how_surge_protectors_work.html
5
Objective
Grain Size
Varistors
Hall-Petch
Creep
Varistors• Varistor = variable resistor, i.e. a circuit element whose
resistance varies with the voltage applied.• As typically fabricated, they have highly non-linear response
and are useful as voltage limiters.• They operate by retaining high resistance to some voltage,
above which their resistance drops rapidly.• For short times they can pass large currents thereby
preventing the voltage from rising much above the breakdownvoltage.
• Varistors can therefore function as self-reseting circuitbreakers (actually shunts, not breakers!).
• Their electrical properties depend on the electrical propertiesof their grain boundaries. For example, the breakdown voltageof a varistor is roughly proportional to the number of grainboundaries between the electrodes, i.e. inversely proportionalto grain size.
6
Objective
Grain Size
Varistors
Hall-Petch
Creep
NotationVaristors:• n Carrier concentration• µ Carrier mobility• e Carrier charge• E Potential gradient• V Electric Potential• x Distance• ε Permettivity
Yield Strength, Creep Strength:• σy Yield Strength• σ0 Friction stress• k constant in Hall-Petch Eq.• d Grain size• τ Shear stress• D Diffusion coefficient• Q Activation energy• R Gas Constant• T Temperature• Ω Atomic Volume• J Vacancy Flux
7
Objective
Grain Size
Varistors
Hall-Petch
Creep
Examples ofVaristorCircuit
Components
Electroceramics
8
Objective
Grain Size
Varistors
Hall-Petch
Creep
Macrostructure of a Surge Arrester• The size and structure of the device depends on the
application, e.g. at what voltage it is designed to limit to, andhow much current it must be able to pass in a given surge.
Electroceramics
9
Objective
Grain Size
Varistors
Hall-Petch
Creep
Current-voltage characteristic• At low voltages, the response is ohmic, i.e. the current is
proportional to the voltage. At higher voltages the response ispower-law, with a large exponent (compare this to the power-law relationship for plastic flow!). The better the device, thelarger the exponent. The typical breakdown voltage rangesfrom tens to hundreds of volts.
Electroceramics
10
Objective
Grain Size
Varistors
Hall-Petch
Creep
Varistor application• A varistor (“VDR” in the figure) is typically included
in parallel with the load so that the latter never seesanything above some maximum voltage.
Electroceramics
11
Objective
Grain Size
Varistors
Hall-Petch
Creep
Material, microstructure
• Varistors can be made from a range ofsemiconducting ceramics: SiC, ZnO, TiO2and SrTiO3.
• ZnO with Bi dopant and other oxides (Co,Sb, Fe) is standard material.
• Critical feature is the segregation of thedopant to the grain boundaries.
12
Objective
Grain Size
Varistors
Hall-Petch
Creep
Varistor microstructure• The real microstructure contains a range of grain
sizes and shapes (left). For the purposes ofunderstanding varistor behavior, one can idealizethe microstructure as a “brick” structure, i.e. aregular lattice of cubical grains.
Electroceramics
13
Objective
Grain Size
Varistors
Hall-Petch
Creep
ZnO
• ZnO has a 3.2eV band gap and so thepresence of electron donor additions suchas Co, Sb, Fe to make it an n-type extrinsicsemiconductor are vital. The presence ofthe donor sites makes the grain interiorsconductive.
• The Bi segregates strongly to grainboundaries (and other interfaces) where itprovides acceptor states. The presence ofthe acceptor states locally depresses theFermi level in the grain boundary.
14
Objective
Grain Size
Varistors
Hall-Petch
Creep
ZnO, contd.• A typical ZnO compact has grain size 10-50µm,
with an intergranular phase of thickness 1-1000nm.• The high Bi-content intergranular phase has high
resistance, ~ 106 Ωm.• Heating to high temperatures (typical = 1250°C)
drives off oxygen, leaving vacancies on the oxygensub-lattice (wurtzite structure). Thermal activationcan ionize these vacancies, thereby releasingelectrons into the conduction band (giving n-typeconduction).
• Typical compositions include ~1mol% dopants:96.5ZnO-0.5Bi2O3-1.0CoO-0.5MnO-1.0Sb2O3-0.5Cr2O3.
15
Objective
Grain Size
Varistors
Hall-Petch
Creep
Basic Explanation• The most basic explanation is as follows.• Each grain boundary in a varistor material is effectively a pair of
back-to-back semiconductor diodes (p-n junctions).• At each p-n junction, the electrons on the n-doped side flow into
the p-doped side, thereby setting up a depletion zone, in whichthe carrier concentration is low and resistance is high.
• When you apply a voltage across the varistor, there is apotential across each grain boundary. This potential biaseseach of the diodes, one forwards and the other backwards.
• The forward biased diode will conduct more easily but thebackward biased diode will have an enlarged depletion zoneand its barrier increases, thereby blocking the flow of current.
• At a high enough voltage across each grain boundary,however, the carriers can tunnel through the barrier of thereverse-biased diode and current can then flow.
16
Objective
Grain Size
Varistors
Hall-Petch
Creep
p-n diode junctions (silicon)• It is useful to go back to basics and consider how to
form a p-n diode in terms of doped semiconductors.• Consider a block of Si with two (adjacent) regions of
doping - one p-type and one n-type.• p-type means that conduction is hole-dominated
(acceptor dopant atoms). n-type means electron-dominated conduction (donor dopant atoms).
n-typep-type
17
Objective
Grain Size
Varistors
Hall-Petch
Creep
Fermi levels• For acceptor dopants (e.g. boron), the Fermi level is
low in the gap. For donor dopants (e.g.phosphorus, arsenic) the Fermi level is high in thegap.
BandGap
BandGap
Ee Ee
Electron Energy
Electron Energy
p-type n-type
Conduction band
valence band
18
Objective
Grain Size
Varistors
Hall-Petch
Creep
Electron energies at junction• When we join the p-type to the n-type, the rule is
that the Fermi level is constant throughout thematerial (otherwise there would be a net flow ofelectrons in the material). The result is a bending ofthe energy levels in the junction region.
BandGap
Ee
Electron Energy
Junction of p- & n-types
p-type n-type
19
Objective
Grain Size
Varistors
Hall-Petch
Creep
Potential vs. electron energy• Electric potential (voltage) is the opposite of
electron energy (from the change in sign).• Holes move down gradients in electric potential:
electrons move down gradients in electron energy.• By equilibrating Fermi levels, no net electron (or
hole) flow will occur between the p- and n-typeregions.
Potential (V)+
-
p-type n-type
~0.8V
Ee
20
Objective
Grain Size
Varistors
Hall-Petch
Creep
Junction Region• In addition to the gradients in electron energy and
potential, there is some flow of electrons from the n-type into the p-type region with recombination of thecarriers.
• This depletes the concentrations of holes andelectrons on either side of the junction.
• Carrier depletion obviously decreases conductivity.• Conduction: the conductivity depends (linearly) on
the carrier concentration, n, mobility, µ, andcharge,e;
σ = n e µ
21
Objective
Grain Size
Varistors
Hall-Petch
Creep
Conductivity in a semiconductor• Typical values for n-type doped silicon (subscript “n”
denotes quantity in n-type): majority carrier concentration, nn, = 1022 electrons.m-3
mobility, µn, = 0.35 m2V-1s-1
and charge,e, = -1.6.10-19C.
minority carrier concentration, pn, = 2.3.1010 holes.m-3
mobility, µh, = 0.044 m2V-1s-1
and charge,e, = +1.6.10-19C.• Remember: electric field = -1*gradient of potential;
E = -dV/dx
22
Objective
Grain Size
Varistors
Hall-Petch
Creep
Junction region
• The local electric fieldrepels electrons on then-type side, and repelsholes on the p-type side.
• Only minority carriers oneither side of thejunction are available tocarry current.
p-type n-type
[Electronic Materials]
23
Objective
Grain Size
Varistors
Hall-Petch
Creep
Biasing a p-n junctionNow we consider what
happens when we applyan external voltage(electric potential) to thesystem and require acurrent to flow throughthe junction.
Forward bias: increase thepotential on the p-typeside, which is equivalentto decreasing theelectron energy; thisdecreases the differencein energy between thetwo materials. You canalso think of making then-type more negative,which increases thedensity of electrons, andthe p-type more positive,which increases thedensity of holes.
p-typen-type
- + Forward bias
Forward biasReverse bias
24
Objective
Grain Size
Varistors
Hall-Petch
Creep
Biasing, contd.• Forward bias = lowers the potential (voltage) on the
n-type side, and raises it on the p-type side. Thistends to diminish the depletion zone (from bothsides).
• Reverse bias = as expected, this raises thepotential (voltage) on the n-type side, and lowers iton the p-type side. This tends to widen thedepletion zone (from both sides).
25
Objective
Grain Size
Varistors
Hall-Petch
Creep
Biasing: minoritycarrier conc.
• Bias voltage changes thedensity of minority carriers atthe edge of the depletion zoneand thus the current that canbe carried across the zone.
• Increasing forward biasincreases the number ofmajority carriers (holes) in thep-type side which flow into then-type side, raise the (minoritycarrier) level on that side andincrease current capacity. Thedensity is proportional to theexponential of the voltageacross the junction.
[Electronic Materials]
26
Objective
Grain Size
Varistors
Hall-Petch
Creep
Grain boundary electric double layer
Electroceramics
• The electronic structure at a grain boundary in a ceramic isunderstood as having acceptor states (not well understood!)that cause a local increase in the electron energy. Thisconstitutes a barrier to electron motion through the material.
• The thickness of the transition layer is of order √ (2εV/eN),where ε is the permettivity (in Si, 1.08.10-10 F.m-1), V is thevoltage across the layer, e is the electron charge (1.6.10-19C),and N is the density of carriers in whichever is the more lightlydoped region. A typical value might be of order 1 µm.
27
Objective
Grain Size
Varistors
Hall-Petch
Creep
Band Structure at a Grain Boundary• Equilibration of the chemical
potential of electrons throughoutthe solid equalizes the Fermilevels inside and outside theboundaries. Chargeredistribution occurs.Conduction electrons aredepleted from the boundaryvicinity (and go into the acceptorstates in the boundary).
• A potential energy barrier at theboundary is created.
• Applying a voltage across thematerial tilts the energy levelsuntil breakdown occurs.
Electroceramics
28
Objective
Grain Size
Varistors
Hall-Petch
Creep
Grain Boundary control• As a consequence,the electrical properties depend on (a) the
doping of the grain boundaries and (b) the microstructurethrough the number and arrangement of the boundaries.
• Chiang gives an example of estimating the breakdown voltagebased on a 3V breakdown for an individual boundary. For a1mm thick device with a 10µm grain size, one expects about100 boundaries through the thickness, which predicts abreakdown voltage of ~300V.
• Thus for a constant grain size, the breakdown voltage isproportional to the size of the specimen. Alternatively, if oneis designing to a specified breakdown voltage, then smallergrain sizes allow smaller device sizes.
• The finite width of the depletion layer at a grain boundary,however, limits the extent to which the grain size can bereduced.
29
Objective
Grain Size
Varistors
Hall-Petch
Creep
Voltage-Current Characteristic• This is the characteristic that one can observe
across a polycrystal, i.e. a breakdown voltage ofabout 300V. The inverse slope, α, is a measure ofvaristor quality.
Electroceramics
30
Objective
Grain Size
Varistors
Hall-Petch
Creep
Relation to Diodes• Each boundary can be regarded as a pair of back-
to-back Schottky barriers, i.e. metal-semiconductorjunctions.
• Chemistry of the boundaries is not well understood.Bi3+ is an electron donor solute, so it is not clearhow it functions as an acceptor in the boundary!
• The oxidation state is important: quenched samplesof ZnO exhibit little or no breakdown. Apparently,oxidation of the grain boundaries during post-sintering cool-down is important for development ofthe critical properties.
31
Objective
Grain Size
Varistors
Hall-Petch
Creep
Typical Varistor Application
Electroceramics
32
Objective
Grain Size
Varistors
Hall-Petch
Creep
Summary• Grain size is a critically important aspect of polycrystalline
materials.• In the case of varistors, a special electronic structure in the
grain boundary layer produces a back-to-back diode that hasa well-defined breakdown voltage. The electricalcharacteristics of the device are directly related to theelectrical properties of the boundary and the grain size.
• In the case of the Hall-Petch effect, in most materials, both thestrength and the toughness increase as the grain size isreduced. This effect can be explained by the resistance of theboundaries to plastic flow (in the case of strength) and/or thedecreased microcrack size in the case of fracture.
• Grain size can play a major role in controlling creepresistance. Larger grain size increases creep resistance -hence the use of single crystals where feasible, especially forsuperalloys.
33
Objective
Grain Size
Varistors
Hall-Petch
Creep
Bibliography• Electroceramics, A.J. Moulson & J.M. Herbert, Chapman &
Hall, ISBN 0-412-29490-7, 621.381/M92e• Physical Ceramics (1997), Y.-T. Chiang, D.P. Birnie III, W.D.
Kingery, Wiley, New York, 0-471-59873-9.• Mechanical Behavior of Materials (1966), F. McClintock and
A. S. Argon, Addison Wesley.• Electronic Materials (1990), edited N. Braithwaite & G.
Weaver, (The Open University) Butterworths.• Mechanical Behavior of Materials, T.H. Courtney, McGraw-
Hill, ISBN 0-07-013265-8, 620.11292,C86M• Microstructure and Properties of Materials, J.C.M. Li, editor,
World Scientific, ISBN 981-02-2403-6
34
Objective
Grain Size
Varistors
Hall-Petch
Creep
Example Problem for Varistors• How much Bismuth oxide must I add to ZnO
(proportions by weight) in order to dope the grainboundaries to the desired level?
• Can estimate a minimum amount by assuming thatwe need, say, 2 layers of Bi atoms along everyboundary in order to accomplish the requireddoping.
• From here on, it is college chemistry to make theestimate.
• Suppose that the grain size in the ZnO is 12µm (asin the exam question). That is, 3V grain boundarybreakdown voltage, 250V device breakdown, 1mmthick.
35
Objective
Grain Size
Varistors
Hall-Petch
Creep
Bi doping levels in ZnO• Step 1: grain boundary area per unit volume, AV, =
1/d = 8.3 104 m2/m3.• Step 2: atomic area, Aatom ~ 0.32 10-18 m2
• Step 3: no. of atoms per unit volume = AV/Aatom =8.3/9 104/10-20 ~ 1024 atoms/m3.
• Step 4: moles(Bi)/m3 = 1024/Nav = 1.7 moles/m3.• Step 5: molecular weight of Bi2O3=464 gms• Step 6: weight(Bi2O3) per m3 = 770 gms• Step 7: density of ZnO = 5.6 Mg/m3.• Step 8: weight proportions are 1:7200, Bi2O3:ZnO.