(resolved) Seebeck effect: why can't electron+hole combine at cold end?

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In the Seebeck effect, when one end of a conductor is heated, electrons and holes migrate toward the cooler end, creating a potential difference. The discussion questions why these charge carriers do not recombine at the cold end instead of traveling across the conductor to do work, such as powering a bulb. The underlying reason is that the energy levels and spatial separation of electrons and holes prevent immediate recombination at the cold end. This movement is essential for generating a usable electrical current through a load. Understanding this process is crucial for optimizing thermoelectric materials and devices.
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Hi,
In seebeck effect:
Ref:
1572843145788.png

Once the red region is hot, the electrons and holes move toward cooler region i.e. bottom. So there is a positive potential toward right side of cold side and negative region toward left side of cold side. So why can't the electrons and holes combine here only instead of going all the way across the wire and meet with a resistance (load of bulb). Why can't they just recombine in the cold side?
 
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Sorry for question. I found a more appropriate diagram:
1572844375255.png

Ref:
 

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