Why does a thermoelectric generator need both p and n elements? A thermoelectric (peltier) generator turns heat flow into DC electrical power. Here is a typical schematic:

(image from here). Notice that there are two thermoelectric materials required: p-type (red) and n-type (yellow).
Why can't you make a thermoelectric generator with just one material (say, n-type), with a wire to return the current? As follows:

If the wire is long and thin, ~100% of the heat will travel through the thermoelectric, not the wire. It seems to me that this is simpler, equally efficient, and has the advantage that you only need to find one good thermoelectric material instead of two.
Since nobody does this, I figure there must be some problem with this design. What is it?
 A: You don't.
However, if you use a wire to return the current, your efficiency will drop. The efficiency of a thermometric generator increases with:


*

*the difference in Seebeck coefficient between your two element types (Note you could even use two N types, or two P types as long as their magnitude is different)

*the thermal resistance, or insulation value between the hot side and cold side.

*the electrical conductivity of the circuit

*the temperature difference


Using metal for one of the elements would reduce the difference in the Seebeck coefficient, as well as the thermal resistance, though it would increase the electrical conductivity. Overall it usually reduces the efficiency.
However, it some instances it is much easier to manufacture devices with only one type, for example, printing on flexible substrates. In these cases metals are used for their excellent electrical conductivity.
A: When a material is heated at one end, the atoms at the 'hot' end have more kinetic energy than the atoms at the 'cold' end.
For an n-type material, free electrons are able to diffuse from the hot side to the cold side, thereby generating a potential difference (voltage) across the material which is proportional to the temperature difference across the device.

A similar thing happens to an p-type material, however the signs of the charge carriers and hence voltages are opposite. 
The charge carriers in both materials, in effect, carry heat with them from the hot end to the cold end, whilst setting up a voltage at the same time.
If you were to connect an external circuit across the n-type material, heat will be conducted away from the hot junction by the wire itself, so no electricity will flow.
To create a thermoelectric generator, what is needed is to connect both the n-type and p-type material 'back-to-back' with the hot junction, so their voltages are added together, whilst the wires to the external circuit are both connected to terminals at the cold junction, thereby allowing electricity to flow through the external circuit. As long as there is a temperature gradient across the device, there will be a voltage given out across the terminals at the cold junction. Of course, if an external circuit is connected and an electric current flows, heat will need to be added to the hot junction to maintain the flow of electric current.

In practice, there may be hundreds of such junctions connected (electrically in series, thermally in parallel) to generate a 'useable' quantity of electricity.

A: Current flows only when there is difference in electric potential or commonly reffered to as voltage, First the second part of your Question, about thickness of wire, if wire is too thin then according to the following relation [(R=ρxl/A), where R is the resistance of wire, ρ is the resistivity of the wire, l is the length of the wire, and A is the cross-sectional area of the wire.] if cross-sectional area is reduced the resistance increases a lot, which hinders the current flow.
Now the first part of your Question, Why 2 different Semiconductors are used. To make the current flow, there needs to be some non-uniformity in the distribution of electrons to cause a potential difference(which is very essential for current to flow),which is provided by two differently doped silicon or germanium substrates, where n-type is at a relative higher potential and p-type is at a relative lower potential. After these 2 semiconductors are bought into contact, the electrons begin to flow intrinsically and after few microseconds a potential barrier is formed which prevents further flow of electrons. To cross this potential barrier we require biasing voltage which depends on semiconductor being used and the component being made(e.g, for silicon diodes it is nearly 0.7 Volts and for Germanium diodes it is 0.3 Volts, while as for Transistors it is even lower.). This biasing Voltage can be attained by many ways, by using electric field, by using photons in photo-voltaics, or by using heat as an energy for electrons in n-region to allow them to move in an external circuit and then cross the potential barrier.  Which is actually the flow of current.
If you would use just n-type or p-type, current will still flow but ntype is what actually holds surplus electrons, which aid in conduction and similary ptype helps electrons flow faster increasing the rate of flow of charge or simply increasing current.
Here current I refer to as recognisable current above 20mA, otherwise the electrons are enoughly motile at room temperature in any of the semiconductors whether doped or not to move in an external circuit to make current happen, but that would be very very small, and that's currently of no use in my opinion.
