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Microstructure-Properties: ILecture 5B

The Effect of Grain Sizeon Varistors

27-301October, 2007A. D. Rollett

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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.

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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.

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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

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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.

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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

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Examples ofVaristorCircuit

Components

Electroceramics

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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

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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

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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

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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.

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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

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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.

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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.

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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.

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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

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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

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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

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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

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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

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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

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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]

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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

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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).

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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]

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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.

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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

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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.

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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

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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.

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Typical Varistor Application

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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.

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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

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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.

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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.