Orbital angular momentum induced intrinsic spin Hall effect

Wonsig Jung, 1 Dongwook Go, 2 Hyun-woo Lee, 2 and Changyoung Kim 1

1 Institute of Physics and Applied Physics, Yonsei University, Seoul 120-749, Korea2 Department of Physics, Pohang University Science and Technology, Pohang 790-784, Korea

[email protected]

Orbital angular momentum driven intrinsic spin Hall effect

We propose a mechanism of intrinsic spin Hall effect (SHE). In this mechanism, local orbital angular momentum

(OAM) induces electron position shift and couples with the bias electric field to generate orbital Hall effect (OHE).

SHE then emerges as a concomitant effect of OHE through the atomic spin-orbit coupling. Spin Hall conductivity

due to this mechanism is estimated to be comparable to experimental values for heavy metals. This mechanism

predicts the sign change of the spin Hall conductivity as the spin-orbit polarization changes its sign, and also

correlation between the spin Hall conductivity and the splitting of the Rashba-type spin splitting at surfaces.

The 6th SRC summer workshop on topological matter

OAM-induced charge polarization

S. R. Park et al., PRL (2011)

Electron momentum is coupled to OAM of atomic

orbitals and induces charge polarization : 𝑷 = 𝛼𝐾𝒌 × 𝑳,thus proper position operator should be

𝑹(𝒌) = 𝒓 + 𝛿𝒓 𝒌 = 𝐫 +𝛼𝐾𝑒𝒌 × 𝑳

Model Hamiltonian and EOM

𝐻 =ℏ2𝒌2

2𝑚+ 𝛼𝐿𝑆𝑳 ⋅ 𝑺 − 𝑒𝑬 ⋅ 𝑹(𝒌)

𝑑𝑅𝑖𝑑𝑡

=ℏ𝑘𝑖𝑚

+ ℱ𝑖𝑗𝑑𝑘𝑗

𝑑𝑡,

𝑑𝑘𝑖𝑑𝑡

= −𝑒𝐸𝑖ℏ

,

where ℱ𝑖𝑗 = 𝜕𝑘𝑖𝐴𝑗 − 𝜕𝑘𝑗𝐴𝑖 + 𝑖 𝐴𝑖 , 𝐴𝑗and 𝐴𝑖 = 𝛿𝑟𝑖 = (𝛼𝐾/𝑒)𝜖𝑖𝑗𝑘𝑘𝑗𝐿𝑘

Non-commutative geometry of position

space introduces anomalous velocity which

is proportional to the non-abelian gauge

field. It produces orbital Hall effect (OHE).

Combined with atomic L-S coupling, spin

Hall effect (SHE) emerges.

Summary of Calculation Discussion

References

For J=3/2 band,

𝑗𝛼𝛽𝑠𝑝𝑖𝑛

= +2

9𝜋2𝜖𝛼𝛽𝛾𝐸𝛾𝑒𝛼𝐾𝑘𝐹

3 2 −3

10

𝛼𝐾ℏ𝑘𝐹2

𝑒−5

3

and, for J=1/2 band,

𝑗𝛼𝛽𝑠𝑝𝑖𝑛

= −2

9𝜋2𝜖𝛼𝛽𝛾𝐸𝛾𝑒𝛼𝐾𝑘𝐹

3 2 −3

10

𝛼𝐾ℏ𝑘𝐹2

𝑒−1

3

anomalous

velocityoccupation change

due to relaxation

𝛼𝐾 can be measured from the magnitude of

Rashba-type spin-orbit coupling. In large L-S

coupling regime,

𝛼𝑅 ≈𝛼𝐾 𝑬int

ℏ.

Thus, spin Hall conductivity for heavy metal

is estimated as 𝜎𝑆𝐻 ~ 104 − 105 Ωm−1,which is comparable to experimental values.

This mechanism also predicts the sign change

of the spin Hall conductivity as the spin-orbit

polarization changes its sign.

1. S. R. Park, C. H. Kim, J. H. Han, and C. Kim,

Phys, Rev. Lett. 107, 156803 (2011)

2. W. Jung, D. Go, H.-W. Lee, and C. Kim,

arXiv:1407.3446v1 (2014)