  In general, it is difficult to access the thermal conductivity of thin
insulating films experimentally just by electrical means. Here, we present a
new approach utilizing the tunnel magneto-Seebeck effect (TMS) in combination
with finite-element modeling (FEM). We detect the laser-induced TMS and the
absolute thermovoltage of laser-heated magnetic tunnel junctions with 2.6 nm
thin barriers of MgAl$_2$O$_4$ (MAO) and MgO, respectively. A second
measurement of the absolute thermovoltage after a dielectric breakdown of the
barrier grants insight into the remaining thermovoltage of the stack. Thus, the
pure TMS without any parasitic Nernst contributions from the leads can be
identified. In combination with FEM via COMSOL, we are able to extract values
for the thermal conductivity of MAO ($0.7$ W/(K$\cdot$m)) and MgO ($5.8$
W/(K$\cdot$m)), which are in very good agreement with theoretical predictions.
Our method provides a new promising way to extract the experimentally
challenging parameter of the thermal conductivity of thin insulating films.
