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A related discussion of an intrinsic spin Hall effect for hole-doped bulk semiconductors came to our attention as this work was nearing completion. S. Murakami, A. Nagaosa, and S.-C. Zhang (private communication). Their argument is based on the quantum holonomy of hole-doped bulk semiconductors.
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85032427082
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note
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The electron group velocity includes a term proportional to spin [see Eq. (10)] which is dropped because in Eq. (6) it is the spin current which is included This spin current is formed by the anticommutator of the spin and the group velocity, and the second term hence vanishes.
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24
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J. Sinova, A. H. MacDonald, and Q. Niu, et al. (unpublished).
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D. Culcer et al. (to be published).
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26
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John Wiley & Sons, Inc., New York
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The derivation of this expression is identical to the derivation of the frequency-dependent Hall conductivity expression [see, for example, M. Marder, Condensed Matter Physics (John Wiley & Sons, Inc., New York, 2000), pp. 573-574] except that the x component of the charge current is replaced by the x component of the spin current.
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