| Summary: | Abstract The classical problem of best thermoelectrics, which was believed originally solved by Mahan and Sofo [Proc. Natl. Acad. Sci. USA 93, 7436 (1996)], is revisited and discussed in the quantum limit. We express the thermoelectric figure of merit (z T) as a functional of electronic transmission probability $${{{\mathcal{T}}}}$$ T by the Landauer–Büttiker formalism, which is able to deal with thermoelectric transport ranging from ballistic to diffusive regimes. We also propose to apply the calculus of variations to search for the optimal $${{{\mathcal{T}}}}$$ T giving the maximal z T. Our study reveals that the optimal transmission probability $${{{\mathcal{T}}}}$$ T is a boxcar function instead of a delta function proposed by Mahan and Sofo, leading to z T exceeding the well-known Mahan–Sofo limit. Furthermore, we suggest realizing the optimal $${{{\mathcal{T}}}}$$ T in topological material systems. Our work defines the theoretical upper limit for quantum thermoelectrics, which is of fundamental significance to the future development of thermoelectrics.
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