Spin-flip scattering engendered negative $Δ_T$ noise

$\Delta_T$ noise generated due to temperature gradient in the absence of charge current has recently attracted a lot of interest. In this paper, for the first time, we derive spin-polarized charge $\Delta_T$ noise and spin $\Delta_T$ noise along with its shot noise-like and thermal noise-like contributions. Introducing a spin flipper at the interface of a bilayer metal junction with a temperature gradient, we examine the impact of spin-flip scattering. We do a detailed analysis of charge and spin $\Delta_T$ noise in four distinct setups for two distinct temperature regimes: the first case of one hot \& the other cold reservoir and the second case of reservoirs with comparable temperatures, and also two distinct bias voltage regimes: the first case of zero bias voltage and second case of finite bias voltage. In all these regimes, we ensure that the net charge current transported is zero always. We find negative charge $\Delta_T$ noise for reservoirs at comparable temperatures while for the one hot \& another cold reservoir case, charge $\Delta_T$ noise is positive. We also see that spin $\Delta_T$ noise and spin $\Delta_T$ thermal noise-like contributions are negative for one hot and the other cold reservoir case. Recent work on the general bound for spin $\Delta_T$ shot noise with a spin-dependent bias suggests it is always positive. In this paper, we see spin $\Delta_T$ shot noise-like contribution to be negative in contrast to positive charge $\Delta_T$ shot noise contribution, although in the absence of any spin-dependent bias. Spin-flip scattering exhibits the intriguing effect of a change in sign in both charge and spin $\Delta_T$ noise, which can help probe spin-polarized transport.