A tunneling magnetoresistance (TMR) device, like a TMR read head for a magnetic recording disk drive, has low magnetic damping, and thus low mag-noise, as a result of the addition of a ferromagnetic backing layer to the ferromagnetic free layer. The backing layer is a material with a low Gilbert damping constant or parameter α, the well-known dimensionless coefficient in the Landau-Lifshitz-Gilbert equation. The backing layer may have a thickness such that it contributes up to two-thirds of the total moment/area of the combined free layer and backing layer. The backing layer may be formed of a material having a composition selected from (CoxFe(100-x))(100-y)Xy, (Co2Mn)(100-y)Xy and (Co2FexMn(1-x))(100-y)Xy, where X is selected from Ge, Al and Si, and (Co2Fe)(100-y)Aly, where y is in a range that results in a low damping constant for the material.