We are researching thermoelectric generation (Seebeck effect) for a science fair project. We remain confused about how the energy differential between the hot and cold junctions is converted to electricity. Do the valence electrons and holes in the N and P doped pellets move away from the heat towards the cold (faster than movement in the opposite direction) because the energy is lower in the cooler part – looking to find an energy equilibrium? So now that the electrons have moved to the cooler side that end is more negatively charged than the warm side? It is at this point that we have electric potential for a circuit?
Also, in a Peltier tile the p and n doped pellets do not touch each other but are connected by another conductor on either side. Are the valence electron and hole trying to find each other through the mutual conductive connection – is that attraction creating the electric potential?
The simplest application of electrical doping is a diode. A diode is a semiconducting crystal (for example, silicon or germanium) with a small amount of some other element introduced. Each silicon atom in the crystal has four valence electrons. To form the crystal, each silicon atom shares one valence electron with each of four "neighbor" atoms, and the neighbors in turn each share an electron back. With the electron sharing, each silicon atom now has access to eight electrons. Eight is the magic number of outer-shell electrons that forms the noble-gas column of the periodic table, which other elements try to reach through chemical bonds.
The dopant (say phosphorus) replaces some of the silicon atoms in the crystal lattice. Instead of four valence electrons, phosphorus has five. When the phosphorus atom bonds with its silicon neighbors, it shares four of its valence electrons, but the fifth is unbonded and free to move. The charge carrier is the free electron, and the doped silicon is an "n-type crystal," n for negative.
Now imagine another silicon crystal with a dopant (say aluminum) that has only three valence electrons. One of the silicon neighbors of the aluminum atom gets shorted an electron. The missing electron that should have filled the outer shell of the silicon atom is called a hole. Now we have made a "p-type crystal," p for positive, and the holes are the current carriers. Though the aluminum atoms cannot move, electrons that come to fill their holes leave behind a vacancy that must be filled, so holes appear to move.
To make a diode, we put the p-type and n-type crystals together. Free electrons from the n-type crystal immediately move across the "p-n junction" to fill the holes in the p-type crystal. However, the electron flow soon stops. The problem is that a negative charge is building up in the p-type crystal, and a positive charge in the n-type crystal. Remember that the original crystals were perfectly charge-balanced: the only difference between them was how the electrons were distributed. So the negative charge that has built up in the p-crystal repels electrons trying to flow in, while the positive charge in the n-type crystal starts to keep hold of its electrons.
The end result is a kind of switch. We can make a simple circuit by connecting the ends of the diode to a resistor and a battery. If the battery's positive terminal is connected to the p-type crystal where extra electrons have piled up, those electrons will flow easily around the wire and into the n-type crystal. In fact, the diode's extra free electrons will amplify whatever current would naturally occur in the wire if the diode weren't present. However, if you turn the battery around and connect the positive terminal to the n-type crystal, almost no current flows.
Your thermoelectric generator picture is similar to a diode, except the n- and p-type crystals are not in direct contact. Instead, their ends are connected at a heated junction. Heat is just the motion of particles, so the particles at the hot side move faster than particles at the cold side. There is then a net movement of particles from the hot side to the cold side. To understand this, think of two groups of kids playing tag: the two-year-olds are confined by their parents at one end of the football field, and the ten-year-olds are allowed to range freely starting at the other end. Over time, the ten-year-olds get more spread out than the two-year-olds, so the concentration of kids is higher at the the two-year-old end of the field.
With more electrons in the cold end of the n-type crystal, you get an electrostatic force just like in the diode. Similarly, the p-type crystal ends up with more holes in the cold end. Unlike a diode, the crystals in the thermoelectric generator aren't in contact and the free electrons in the n-type can't "see" the holes in the p-type. Instead, the physical buildup of charges at the cold end provides the electromotive force. Electrons flow from the cold end of the n-type crystal, through the load-bearing circuit, and back up through the p-type crystal to the hot junction. (Note: in most circuit diagrams, including the one linked above, arrows point in the direction that positive charges would travel, so electrons move in the opposite direction of the arrows.)