Thermal conductivity and enhanced thermoelectric efficiency of composition-graded $${\hbox {Si}}_c{\hbox {Ge}}_{1-c}$$ alloys

We explore the efficiency of a thermoelectric energy converter constituted by a Si/Ge nanowire of length L. A constitutive equation of thermal conductivity as function of composition and temperature is derived in accordance with experimental data obtained at the constant temperatures $$T=300\,\hbox {K}$$, $$T=400\,\hbox {K}$$, and $$T=500\,\hbox {K}$$ by a nonlinear regression method. A thermodynamic model of thermoelectric energy converter is developed in accordance with second law of thermodynamics. Then, we investigate the thermoelectric efficiency of such system as function of the composition, and of both composition and temperature gradients applied at its ends. For each temperature, we calculate the values of composition and heat conductivity giving the optimal efficiency of the thermoelectric energy conversion. A series of constraints on the material functions entering the model equations, which are necessary and sufficient to guarantee the optimal efficiency of the system, are determined and discussed.