The best thermoelectrics revisited in the quantum limit

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 (zT) as a functional of electronic transmission probability T by the Landauer-Buttiker 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 T giving the maximal zT. Our study reveals that the optimal transmission probability T is a boxcar function instead of a delta function proposed by Mahan and Sofo, leading to zT exceeding the well-known Mahan-Sofo limit. Furthermore, we suggest realizing the optimal 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.