Given a family $(μ_λ,λ\geq0)$ of integrable mean-zero probability measures such that, for every $λ\geq0$, $μ_λ$ is the image of $μ_1$ under the homothety $y\longmapsto\sqrtλy$, we provide a necessary and sufficient condition on $μ_1$ under which the Root embedding algorithm yields a self-similar martingale with one-dimensional marginals $(μ_λ,λ\geq0)$. Precisely, if $τ_λ$ and $R_λ$ denote the Root solution to the Skorokhod embedding problem (SEP) and the Root regular barrier for $μ_λ$ respectively, then this condition is equivalent to the property that $(R_λ,λ\geq0)$ is non-increasing in the sense of inclusion, which in turn is equivalent to the assertion that $(τ_λ,λ\geq0)$ is non-decreasing a.s. We show that there are many examples for which this result applies and we provide some numerical simulations to illustrate the monotonicity property of regular barriers $(R_λ,λ\geq0)$ in this case.