1 Microstructure-Properties: I Lecture 5B The Effect of on 27-301 October, 2007 A. D. Rollett
2 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 effect, and Nabarro-Herring creep. Similar considerations apply to magnetic hardness also.
3 Key Concepts 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. boundaries in semiconductors used to make varistors have a one-way voltage barrier. The effect quantifies the trend of increasing strength and toughness with decreasing grain size. 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.
4 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
5 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. 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.
6 Notation : n Carrier concentration µ Carrier mobility e Carrier charge E Potential gradient V Electric Potential x Distance ε Permettivity Yield Strength, Strength: σ y Yield Strength σ 0 Friction stress k constant in Eq. d size τ Shear stress D Diffusion coefficient Q Activation energy R Gas Constant T Temperature Ω Atomic Volume J Vacancy Flux
7 Examples of Varistor Circuit Components Electroceramics
8 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. Electroceramics
9 Current-voltage characteristic 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. Electroceramics
10 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. Electroceramics
11 Material, microstructure can be made from a range of semiconducting ceramics: SiC, ZnO, TiO 2 and SrTiO 3. 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.
12 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. Electroceramics
13 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.
14 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, ~ 10 6 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-0.5Bi 2 O 3-1.0CoO-0.5MnO-1.0Sb 2 O 3-0.5Cr 2 O 3.
15 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 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.
16 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). p-type n-type
17 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 the gap. E e Electron Energy Band Gap Conduction band valence band E e Electron Energy Band Gap p-type n-type
18 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. Electron Energy E e Junction of p- & n-types Band Gap p-type n-type
19 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 E e p-type n-type
20 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 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 = n e µ
21 Conductivity in a semiconductor Typical values for n-type doped silicon (subscript n denotes quantity in n-type): majority carrier concentration, n n, = 10 22 electrons.m -3 mobility, µ n, = 0.35 m 2 V -1 s -1 and charge,e, = -1.6.10-19 C. minority carrier concentration, p n, = 2.3.10 10 holes.m -3 mobility, µ h, = 0.044 m 2 V -1 s -1 and charge,e, = +1.6.10-19 C. Remember: electric field = -1*gradient of potential; E = -dv/dx
22 Junction region p-type n-type 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]
23 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. n-type - + p-type Forward bias Forward bias Reverse bias
24 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).
25 Biasing: minority carrier conc. 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]
26 boundary electric double layer 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 (1.6.10-19 C), and N is the density of carriers in whichever is the more lightly doped region. A typical value might be of order 1 µm. Electroceramics
27 Band Structure at a Boundary 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
28 Boundary control 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.
29 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. Electroceramics
30 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. Bi 3+ 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.
31 Typical Varistor Application Electroceramics
32 Summary 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 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. 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.
33 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 Example Problem for 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.
35 Bi doping levels in ZnO Step 1: grain boundary area per unit volume, A V, = 1/d = 8.3 10 4 m 2 /m 3. Step 2: atomic area, A atom ~ 0.3 2 10-18 m 2 Step 3: no. of atoms per unit volume = A V /A atom = 8.3/9 10 4 /10-20 ~ 10 24 atoms/m 3. Step 4: moles(bi)/m 3 = 10 24 /N av = 1.7 moles/m 3. Step 5: molecular weight of Bi 2 O 3 =464 gms Step 6: weight(bi 2 O 3 ) per m 3 = 770 gms Step 7: density of ZnO = 5.6 Mg/m 3. Step 8: weight proportions are 1:7200, Bi 2 O 3 :ZnO.