Uncertainty analysis of microscopic parameters obtained from the single parabolic band modeling of thermoelectric materials

The single parabolic band (SPB) model is a popular, easy-to-use, and somewhat powerful tool to describe thermoelectric transport and is often employed for material optimization and the extraction of band structure parameters. While the fundamental limitations of the model are well-known and usually recognized, a discussion on the uncertainty in the derived parameters when the model is valid is missing, but indispensable for a qualified comparison and analysis of reported values. In this work, we derive analytical expressions for the uncertainty in the modeling parameters—the reduced Fermi level, density of states (DOS) mass, carrier concentration (), mobility parameter ( <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi>μ</a:mi> <a:mn>0</a:mn> </a:msub> </a:math> ), Lorenz number (), and electronic thermal conductivity ( <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msub> <c:mi>K</c:mi> <c:mrow> <c:mrow> <c:mi mathvariant="normal">E</c:mi> </c:mrow> </c:mrow> </c:msub> </c:math> ) based on experimental uncertainty in the Seebeck coefficient (), electrical conductivity (), and Hall coefficient ( <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:msub> <f:mi>R</f:mi> <f:mrow> <f:mrow> <f:mi mathvariant="normal">H</f:mi> </f:mrow> </f:mrow> </f:msub> </f:math> ). As the uncertainty in the SPB-derived quantities resulting from the Seebeck coefficient depends not only on the uncertainty of the Seebeck coefficient but also on the Seebeck coefficient value itself, it is discussed in detail. The uncertainty in the most cited parameter derived from the SPB model, i.e., the DOS mass, is dominated by the uncertainty arising due to . For a relative uncertainty of 5% in alone, the uncertainty in the DOS mass ranges from 5% to 15% for 50–400 µV/K, whether scattering of electrons by acoustic phonons, polar optical phonons, ionized impurities, or neutral impurities is assumed. Associated challenges to accurate and insightful modeling of the thermoelectric (TE) materials are discussed and exemplified by an analysis of DOS masses for <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msub> <i:mi>Mg</i:mi> <i:mn>2</i:mn> </i:msub> <i:mi>Si</i:mi> </i:math> from various literature sources, where the observed large scatter can be rationalized by large but realistic uncertainties in and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msub> <k:mi>R</k:mi> <k:mrow> <k:mrow> <k:mi mathvariant="normal">H</k:mi> </k:mrow> </k:mrow> </k:msub> </k:math> , thereby highlighting the need for better measurement protocols.