Structural devices

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# 2.4 Thermal and electrical conductance

Thermal conductance, Gt, is analogous to electrical conductance, Ge. The longer a conductor is and the smaller its cross section, the lower its conductance will be.

Thermal conductance is given by:

and electrical conductance by:

where K and σ are the thermal and electrical conductivities respectively. These are material properties that are independent of geometry.

The independence of K and σ from geometry holds true for bulk materials. In thin films, however, this cannot be assumed always to be so, and their values can change by tens of per cent as film thicknesses get into the sub-micrometre range.

## SAQ 1

• (a) Explain in a few words why a design with low thermal conductance along the beam increases the sensitivity as a gas pressure sensor.

• (b) Explain in a few words why a design with a beam 1000 times longer than its thickness promotes sensitivity as a gas pressure sensor (dimensions of beam: 1µm × 25µm × 1000µm).

• (a) The operating principle is based on gas cooling of the metal. It is therefore important to impede the loss of heat by conduction along the beam. The conductance across the beam needs to be high to allow effective cooling via the back surface as well as the top. The conductance along the beam needs to be low.

• (b) For heat flow front to back, the effective area here is 25 µm × 1000 µm and the path length for the heat flow is 1 µm. For heat flow along the beam, the relevant area is 25 µm × 1 µm and the path length is 1000 µm (i.e. one million times greater).

Figure 3(e) shows the silicon nitride patterned by etching a slot broad enough to straddle the two long edges of the polysilicon rectangle. Note also that the nitride layer coats the underside of the wafer. This is easy to do, because the nitride condenses on the wafer out of a precursor gas mixture in a furnace. That is to say, the wafer surface is a place where the different gases in the mixture prefer to react with one another to form the solid silicon nitride as a product. The nitride formed at the same time on the back of the wafer is important because it will protect it from being attacked by an aggressive etching process towards the end of the making of the sensor.

The next layers on the final device are two metals. Because one of these metals is platinum, which is too unreactive to be etched in the normal way, another patterning technique is used – the lift-off method. This involves using photoresist material as a sacrificial layer (see Box 4). It is spun on and patterned – resist is removed from the places where the metal is to remain – before the metals are sputtered on (see Figure 3(f)). Figure 3(g) shows the two metal layers, chromium and platinum, on top of the patterned photoresist. Why have these metals been used? The only function of the chromium here is as a means of allowing the platinum to stick to the silicon nitride. Platinum does not stick well to silicon nitride, but it does to chromium; and chromium does bond well to silicon nitride. This technique of using intermediate layers of material between others that are otherwise incompatible with one another is commonly used in microengineering.

## Box 4 Photoresists

Photoresist, a lacquer-like material, is applied in liquid form. It is formed into a precisely controlled thin layer by putting a few drops of the resist onto the centre of the wafer, which is then spun for a minute or so at a few thousand rpm. ‘Patterning’ is the word used to describe the process of photographically exposing and developing this layer, such that defined areas of the wafer are left clear of photoresist.

So why have the platinum at all? The main reason is the inertness that makes it so troublesome to etch. This means that it will not become oxidised when it is being heated in its role in the Pirani sensor. Surface reactions such as oxidation become enormously significant in thin-film materials. When (as is the case here) the layer is only 200 nm thick, if a material oxidises at all it doesn't take long for the film properties to be noticeably affected. Chemical inertness, especially in hot metals, is essential when we want to detect small changes in electrical resistivity.

Another thing that happens to thin films when they are heated is that they tend to diffuse (soak) into the underlying material. This is a further reason for choosing silicon nitride as that underlying layer. It is an excellent barrier to diffusion (see Box 5). So, we can be confident that the platinum will remain in its elemental form, and stay on the surface. This will help to maintain its electrical stability – if its resistance varies significantly over time, it won't be much use as a sensor.

## Box 5 Diffusion

Diffusion (which can generally be defined as the smoothing-out of differences, whether in temperature or material composition) is a process that occurs through the jiggling of atoms. Above a temperature of absolute zero, all atoms – whether in the solid, liquid or gaseous state – move. In solids this movement is in the form of vibration around an average position, and the amplitude of this vibration increases with temperature – actually, it is an expression of temperature. The more the atoms jiggle, the more likely it is that they will jiggle hard enough to move into a vacant position nearby, or swap places with a neighbour.

Figure 3(h) shows the chromium/platinum track on top of the nitride. The unwanted metal floated off as the photoresist below it was dissolved away in acetone. The metal was well enough bonded to the nitride and, at 200 nm, thin enough to tear neatly at the edges of the pattern in the photoresist.

The final step before the wafer is sawn up to release the devices is an anisotropic etch (see Box 6) – Figure 3(i) and Figure 3(j).

## Box 6 Anisotropic etch

This is a special example of the etching processes described in Section 8 Etching. The etch used at this point in the making of this device is an anisotropic wet etch. That is to say, it is done with a liquid chemical. These anisotropic wet etches make use of the crystallography of the single-crystal silicon. When exposed to the chemical, the silicon is eaten away preferentially in certain directions in its crystal lattice. In particular, the lattice orientation called the (111) planes are very resistant to attack by the chemicals. This enables the designer to produce distinctive shapes in the silicon and, more importantly, to exercise great control over the final dimensions of the etched feature without having to be very careful in timing the etching process itself.

The whole wafer is immersed in the etchant solution. Because the wafer was specified to have the right crystal orientation, and because the patterns in the surface coatings were properly aligned with respect to the crystal planes in the wafer, this etch results in a long V-shaped trench appearing underneath the nitride/chromium/platinum sandwich, leaving it suspended but anchored at both ends. The polysilicon (the first layer that was put down, if you can remember that far back!) is very quickly etched away during this process, as the multiplicity of grain boundaries and crystal orientations in the layer present lots of easy routes for the etchant to attack the material. Once the sacrificial layer has gone, the silicon wafer is exposed to the etchant across the whole width of the bridge that is being undercut. The nature of this etch is that it is self-limiting: once it has progressed to the point where the sloping sidewalls are fully formed and meet at the bottom, it stops, as shown in Figure 3(j). All the exposed planes of silicon atoms are densely packed and etch only very slowly. This is very useful when it is important to have good uniformity from device to device and wafer to wafer. This is certainly the case here, as the cool sidewalls of the trench are the main sink to which the gas whose pressure is being measured will carry the heat from the platinum track.

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