Download Microstructure-Properties: I Lecture 5B The Effect of Grain Size on

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Objective
Microstructure-Properties: I
Lecture 5B
The Effect of Grain Size
on Varistors
Grain
Size
Varistors
HallPetch
Creep
27-301
October, 2007
A. D. Rollett
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Grain
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Varistors
HallPetch
Creep
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• 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 ceramics
used for varistors (e.g. in surge protectors).
• The previous example was the effect of grain size
on mechanical properties, namely the Hall-Petch
effect, and Nabarro-Herring creep.
• Similar considerations apply to magnetic hardness
also.
Key Concepts
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Grain boundaries (effectively) have properties that differ from
the matrix.
Properties of polycrystal depend on the content of planar
defects, i.e. grain boundaries, i.e. grain size.
Grain boundaries in semiconductors used to make varistors
have a one-way voltage barrier.
The Hall-Petch effect quantifies the trend of increasing
strength and toughness with decreasing grain size.
Creep rates (Coble creep) increase with increasing grain
boundary area (per unit volume), hence decreasing grain size.
Low temperature service optimized by fine grain size, but high
temperature service optimized by use of single crystals.
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Size
Varistors
HallPetch
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Surge
Protectors
• Surge protection
means inserting a
component into a
circuit that prevents
the voltage from
rising above a certain
value.
• Note the diagram
showing varistors in
parallel with the load
http://www.sosnet.com/StaticPages/how_surge_protectors_work.html
Varistors
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Varistors
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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 breakdown
voltage.
Varistors can therefore function as self-reseting circuit
breakers (actually shunts, not breakers!).
Their electrical properties depend on the electrical properties
of their grain boundaries. For example, the breakdown voltage
of a varistor is roughly proportional to the number of grain
boundaries between the electrodes, i.e. inversely proportional
to grain size.
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Notation
Varistors:
• n Carrier concentration
• µ Carrier mobility
• e Carrier charge
Objective
• E Potential gradient
Grain
• V Electric Potential
Size
• x Distance
Varistors
• ε Permettivity
HallPetch
Creep
Yield Strength, Creep Strength:
• σy Yield Strength
• σ0 Friction stress
• k constant in Hall-Petch Eq.
• d Grain size
•
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•
•
•
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τ Shear stress
D
Q
R
T
Ω
J
Diffusion coefficient
Activation energy
Gas Constant
Temperature
Atomic Volume
Vacancy Flux
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Size
Varistors
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Examples of
Varistor
Circuit
Components
Creep
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, and
how much current it must be able to pass in a given surge.
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Electroceramics
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Current-voltage characteristic
•
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At low voltages, the response is ohmic, i.e. the current is
proportional to the voltage. At higher voltages the response is
power-law, with a large exponent (compare this to the powerlaw relationship for plastic flow!). The better the device, the
larger the exponent. The typical breakdown voltage ranges
from tens to hundreds of volts.
Varistors
HallPetch
Creep
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 sees
anything above some maximum voltage.
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Electroceramics
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Material, microstructure
• Varistors can be made from a range of
semiconducting ceramics: SiC, ZnO, TiO2
and SrTiO3.
• ZnO with Bi dopant and other oxides (Co,
Sb, Fe) is standard material.
• Critical feature is the segregation of the
dopant 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 of
understanding varistor behavior, one can idealize
the microstructure as a “brick” structure, i.e. a
regular lattice of cubical grains.
Varistors
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Electroceramics
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ZnO
• ZnO has a 3.2eV band gap and so the
presence of electron donor additions such
as Co, Sb, Fe to make it an n-type extrinsic
semiconductor are vital. The presence of
the donor sites makes the grain interiors
conductive.
• The Bi segregates strongly to grain
boundaries (and other interfaces) where it
provides acceptor states. The presence of
the acceptor states locally depresses the
Fermi 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 oxygen
sub-lattice (wurtzite structure). Thermal activation
can ionize these vacancies, thereby releasing
electrons into the conduction band (giving n-type
conduction).
• Typical compositions include ~1mol% dopants:
96.5ZnO-0.5Bi2O3-1.0CoO-0.5MnO-1.0Sb2O30.5Cr2O3.
Basic Explanation
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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 which
the carrier concentration is low and resistance is high.
When you apply a voltage across the varistor, there is a
potential across each grain boundary. This potential biases
each of the diodes, one forwards and the other backwards.
The forward biased diode will conduct more easily but the
backward biased diode will have an enlarged depletion zone
and 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 the
reverse-biased diode and current can then flow.
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Size
Varistors
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 electrondominated conduction (donor dopant atoms).
HallPetch
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p-type
n-type
Fermi levels
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• 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 the
gap.
Ee
Electron
Energy
Varistors
HallPetch
Ee
Electron
Energy
Conduction band
Band
Gap
Band
Gap
Creep
valence band
p-type
n-type
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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 the
material (otherwise there would be a net flow of
electrons in the material). The result is a bending of
the energy levels in the junction region.
Junction of p- & n-types
Ee
Electron
Energy
Band
Gap
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-type
regions.
Potential (V)
-
+
~0.8V
Creep
Ee
p-type
n-type
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Junction Region
• In addition to the gradients in electron energy and
potential, there is some flow of electrons from the ntype into the p-type region with recombination of the
carriers.
• This depletes the concentrations of holes and
electrons on either side of the junction.
• Carrier depletion obviously decreases conductivity.
• Conduction: the conductivity depends (linearly) on
the carrier concentration, n, mobility, µ, and
charge,e;
σ=neµ
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Conductivity in a semiconductor
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• 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
p-type
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• The local electric field
repels electrons on the
n-type side, and repels
holes on the p-type side.
• Only minority carriers on
either side of the
junction are available to
carry current.
[Electronic Materials]
n-type
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Biasing a p-n junction
Now we consider what
happens when we apply
an external voltage
(electric potential) to the
system and require a
current to flow through
the junction.
Forward bias: increase the
potential on the p-type
side, which is equivalent
to decreasing the
electron energy; this
decreases the difference
in energy between the
two materials. You can
also think of making the
n-type more negative,
which increases the
density of electrons, and
the p-type more positive,
which increases the
density of holes.
Reverse bias
n-type
p-type
-
+
Forward bias
Forward 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. This
tends to diminish the depletion zone (from both
sides).
• Reverse bias = as expected, this raises the
potential (voltage) on the n-type side, and lowers it
on the p-type side. This tends to widen the
depletion zone (from both sides).
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Biasing: minority
carrier conc.
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Bias voltage changes the
density of minority carriers at
the edge of the depletion zone
and thus the current that can
be carried across the zone.
Increasing forward bias
increases the number of
majority carriers (holes) in the
p-type side which flow into the
n-type side, raise the (minority
carrier) level on that side and
increase current capacity. The
density is proportional to the
exponential of the voltage
across the junction.
[Electronic Materials]
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Grain boundary electric double layer
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The electronic structure at a grain boundary in a ceramic is
understood as having acceptor states (not well understood!)
that cause a local increase in the electron energy. This
constitutes 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 the
voltage across the layer, e is the electron charge (1.6.10-19C),
and N is the density of carriers in whichever is the more lightly
doped region. A typical value might be of order 1 µm.
Varistors
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Electroceramics
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Band Structure at a Grain Boundary
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Equilibration of the chemical
potential of electrons throughout
the solid equalizes the Fermi
levels inside and outside the
boundaries. Charge
redistribution occurs.
Conduction electrons are
depleted from the boundary
vicinity (and go into the acceptor
states in the boundary).
A potential energy barrier at the
boundary is created.
Applying a voltage across the
material tilts the energy levels
until breakdown occurs.
Electroceramics
Grain Boundary control
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As a consequence,the electrical properties depend on (a) the
doping of the grain boundaries and (b) the microstructure
through the number and arrangement of the boundaries.
Chiang gives an example of estimating the breakdown voltage
based on a 3V breakdown for an individual boundary. For a
1mm thick device with a 10µm grain size, one expects about
100 boundaries through the thickness, which predicts a
breakdown voltage of ~300V.
Thus for a constant grain size, the breakdown voltage is
proportional to the size of the specimen. Alternatively, if one
is designing to a specified breakdown voltage, then smaller
grain 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 be
reduced.
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Voltage-Current Characteristic
• This is the characteristic that one can observe
across a polycrystal, i.e. a breakdown voltage of
about 300V. The inverse slope, α, is a measure of
varistor quality.
Grain
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Electroceramics
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Relation to Diodes
• Each boundary can be regarded as a pair of backto-back Schottky barriers, i.e. metal-semiconductor
junctions.
• Chemistry of the boundaries is not well understood.
Bi3+ is an electron donor solute, so it is not clear
how it functions as an acceptor in the boundary!
• The oxidation state is important: quenched samples
of ZnO exhibit little or no breakdown. Apparently,
oxidation of the grain boundaries during postsintering cool-down is important for development of
the critical properties.
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Typical Varistor Application
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Electroceramics
Summary
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•
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HallPetch
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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 has
a well-defined breakdown voltage. The electrical
characteristics of the device are directly related to the
electrical properties of the boundary and the grain size.
In the case of the Hall-Petch effect, in most materials, both the
strength and the toughness increase as the grain size is
reduced. This effect can be explained by the resistance of the
boundaries to plastic flow (in the case of strength) and/or the
decreased microcrack size in the case of fracture.
Grain size can play a major role in controlling creep
resistance. Larger grain size increases creep resistance hence the use of single crystals where feasible, especially for
superalloys.
Bibliography
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Size
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Varistors
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Creep
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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, McGrawHill, 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 grain
boundaries to the desired level?
• Can estimate a minimum amount by assuming that
we need, say, 2 layers of Bi atoms along every
boundary in order to accomplish the required
doping.
• From here on, it is college chemistry to make the
estimate.
• Suppose that the grain size in the ZnO is 12µm (as
in the exam question). That is, 3V grain boundary
breakdown voltage, 250V device breakdown, 1mm
thick.
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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.