Table 1 T dependence of σSH

From: Critical enhancement of the spin Hall effect by spin fluctuations

T regime

\({\sigma }_{{{{\rm{AFM,SH}}}}}^{{{{\rm{side}}}}\,{{{\rm{jump}}}}}\)

\({\sigma }_{{{{\rm{AFM,SH}}}}}^{{{{\rm{skew}}}}\,{{{\rm{scatt.}}}}}\)

\({\sigma }_{{{{\rm{FM,SH}}}}}^{{{{\rm{side}}}}\,{{{\rm{jump}}}}}\)

\({\sigma }_{{{{\rm{FM,SH}}}}}^{{{{\rm{skew}}}}\,{{{\rm{scatt.}}}}}\)

I

T > TN,C

\(\propto {\tau }_{{{{\bf{k}}}}}{T}^{2}/\sqrt{T-{T}_{{{{\rm{N}}}}}}\)

\(\propto {\tau }_{{{{\bf{k}}}}}^{2}{T}^{6}/(T-{T}_{{{{\rm{N}}}}})\)

 ∝ τkT3/(T − TC)

\(\propto {\tau }_{{{{\bf{k}}}}}^{2}{T}^{6}/{(T-{T}_{{{{\rm{C}}}}})}^{2}\)

II

T < TN,C

\(\propto {\tau }_{{{{\bf{k}}}}}{T}^{2}/\sqrt{{T}_{{{{\rm{N}}}}}-T}\)

\(\propto {\tau }_{{{{\bf{k}}}}}^{2}{T}^{6}/({T}_{{{{\rm{N}}}}}-T)\)

∝ τkT3/(TC − T)

\(\propto {\tau }_{{{{\bf{k}}}}}^{2}{T}^{6}/{({T}_{{{{\rm{C}}}}}-T)}^{2}\)

III

T ~ 0 (TN,C → 0)

∝ τkT3/2

\(\propto {\tau }_{{{{\bf{k}}}}}^{2}{T}^{9/2}\)

∝ τkT3/2a

\(\propto {\tau }_{{{{\bf{k}}}}}^{2}{T}^{3}\)b

  1. An additional T dependence appears via the carrier lifetime τk. Note that the scaling law breaks down in the vicinity of the transition temperature TN,C as indicated by shades in Figs. 4 and 5. See the main text for details.
  2. a In the absence of disorder or impurity scattering, this temperature dependence is modified as ∝ τkT5/3.§
  3. b In the absence of disorder or impurity scattering, this temperature dependence is modified as \(\propto {\tau }_{{{{\bf{k}}}}}^{2}{T}^{10/3}\S\).
  4. § Not considered in the main text, but briefly discussed in Supplementary Note 5.
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