hidden spin polarization in inversion-symmetric bulk crystals€¦ · hidden spin polarization in...

Correction notice Nature Phys. http://dx.doi.org/10.1038/nphys2933 (2014)

Hidden spin polarization in inversion-symmetric bulk crystals Xiuwen Zhang, Qihang Liu, Jun-Wei Luo, Arthur J. Freeman and Alex Zunger In the version of this Supplementary Information file originally published, there were typographical errors in the captions of Figs 1 and 2, the caption of Table 1 and on page 8. On page 9, the compensation parameter values should have been 0.9 and 0.7. These errors have been corrected in this file 15 April 2014.

© 2014 Macmillan Publishers Limited. All rights reserved.

SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHYS2933

NATURE PHYSICS | www.nature.com/naturephysics 1

Hidden spin polarization in inversion-symmetric bulk crystals

1

Supplementary materials: Hidden spin polarization in

inversion-‐symmetric bulk crystals

Xiuwen Zhang1,2,3,*, Qihang Liu1,4,*, Jun-Wei Luo3,#, Arthur. J. Freeman4 and Alex

Zunger1,# 1University of Colorado, Boulder, CO 80309, USA 2Colorado School of Mines, Golden, CO 80401, USA 3National Renewable Energy Laboratory, Golden, CO 80401, USA 4Deprtment of Physics and Astronomy, Northwestern University, Evanston, IL 60208,

USA *These authors contributed equally to this work. #E-mail: [email protected]; [email protected]

A. ZrCoBi with D-1 effect

The D-1 effect originates from the combination of non-centrosymmetric global

space group (e.g. F-43m for ZrCoBi) with site inversion asymmetry (e.g. Td site point

group for all atomic sites in ZrCoBi). Fig. S1a shows the crystal structure of the

half-Heusler material ZrCoBi. The calculated band structure shown in Fig. S1b

reveals that D-1 effect lifts the spin degeneracy between CB1 and CB2 and between

VB1 and VB2 creating the spin splitting. The corresponding spin textures obtained

from all atoms in the unit cell are shown in Figs. S1c and S1d and exhibit strong spin

polarization.


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2

FIG S1: ZrCoBi (F-43m) with D-1 effect. a, Crystal structure and the site point

group of each atomic site. b, Low-energy band structure around Γ point along the Γ-K

and Γ-W directions in the Brillouin zone. The two lowest conduction (CB1 and CB2)

and two highest valence bands (VB1 and VB2) are shown in red. c, d, Spin

polarization indicated by brown arrows near Γ point for CB1+CB2 and VB1+VB2

band pairs in the small range k-plane indicated by solid boxes in b containing the Γ-K

and Γ-W directions, respectively.





3

B. Silicon with D-2 effect

The D-2 effect originates from the combination of centrosymmetric global space

group (e.g. Fd-3m for silicon) where all atoms have non-polar site point groups and,

at least one site point group has inversion asymmetry (e.g. Td site point group for all

atomic sites in silicon). The two Si layers connected by inversion are indicated in Fig.

S2a as α-sector and β-sector. The calculated band structure as shown in Fig. S2b

illustrates band splitting in the vicinity of Γ point in the Brillouin zone between CB1

and CB2 and between VB1 and VB2. The corresponding spin texture, projected on

the real-space sectors α and β, forming the inversion partners, are shown as brown and

green arrows, respectively in Figs. S2c and S2d. We see that although the net spin

polarization vanishes since the material is centrosymmetric, each sector (α or β) has

its own local spin polarization due to the non-centrosymmetric site point group Td of

Si atoms. While it is possible that the observed large spin splitting in Si/Ge quantum

well1 or thin film Si2 is indeed a surface D-1 effect as previously surmised, the

possibility of a D-2 contribution found here was never contemplated before, and

should be assessed.

The VB1 and VB2 have a band crossing at Γ point since they belong to the same Γ8

state, while the CB1 and CB2 don’t have band crossing at Γ point, illustrating that

band crossing is not necessary for compensated spin polarization. Whereas, the band

crossing at L point for NaCaBi as shown in Fig. 4 and that at X point for LaOBiS2 as

shown in Fig. 5 in the main text is due to the fractional translation operations.





4

FIG S2: Silicon (Fd-3m) with D-2 effect. a, Crystal structure and the site point group

of each atomic site. b, Lowe-energy band structure near Γ point along the Γ-L and Γ-X

directions in the Brillouin zone. The two lowest conduction (CB1 and CB2) and two

highest valence bands (VB1 and VB2) are shown in red. c, d, Spin polarization

indicated by green and brown arrows near Γ point for CB1+CB2, VB1+VB2 band

pairs in the small range k-plane indicated by solid boxes in b containing Γ-L and Γ-X

directions, respectively. The green (brown) arrows are the spin polarizations projected

on the α-sector (β-sector) as shown in a.





5

C. Classification of electric polarization and second harmonic generation (SHG)

in bulk materials.

FIG S3: Classification of electric polarization in bulk materials. The point groups

are given in Schoenflies notation. Site point group (SPG) refers to the set of point

group operations that transform the atomic site into itself. Polar point groups are the

point groups containing a unique anisotropic axis that has a dipole on site. CS:

centro-symmetric.





6

FIG S4: Classification of second harmonic generation in bulk materials. The point

groups are given in Schoenflies notation. Site point group (SPG) refers to the set of

point group operations that transform the atomic site into itself. Polar point groups are

the point groups containing a unique anisotropic axis that has a dipole on site. CS:

centro-symmetric. “IA-SHG-1” and “DF-SHG-1” denote the net SHG effects induced

by site inversion asymmetry (IA) and site dipole field (DF), e.g. in GaAs and BiTeI,

respectively. “IA-SHG-2” and “DF-SHG-2” are the compensated SHG effects, which

can be observed in experiment if the photonic field has a small penetration depth in

the bulk materials.

D. Characterization of site dipole field

In this study, we find that since SOC is a relativistic effect that anchored on

particular nuclear sites in the bulk, the origin of SOC-induced spin polarization in

bulk materials is the site effect, e.g., site dipole field. We characterize the strength of

site dipole field by calculating the gradient of the self-consistent electrostatic potential

(ionic plus Hartree potentials) at atomic sites, from density functional theory. Table

S1 lists the calculated site dipole field in LaOBiS2 and α-SnTe for all the atomic sites

in the primitive cell. Note that since Bi and S atoms contribute dominantly to the 6

FIG S4: Classification of second harmonic generation in bulk materials. The point

groups are given in Schoenflies notation. Site point group (SPG) refers to the set of

point group operations that transform the atomic site into itself. Polar point groups are

the point groups containing a unique anisotropic axis that has a dipole on site. CS:

centro-symmetric. “IA-SHG-1” and “DF-SHG-1” denote the net SHG effects induced

by site inversion asymmetry (IA) and site dipole field (DF), e.g. in GaAs and BiTeI,

respectively. “IA-SHG-2” and “DF-SHG-2” are the compensated SHG effects, which

can be observed in experiment if the photonic field has a small penetration depth in

the bulk materials.

D. Characterization of site dipole field

In this study, we find that since SOC is a relativistic effect that anchored on

particular nuclear sites in the bulk, the origin of SOC-induced spin polarization in

bulk materials is the site effect, e.g., site dipole field. We characterize the strength of

site dipole field by calculating the gradient of the self-consistent electrostatic potential

(ionic plus Hartree potentials) at atomic sites, from density functional theory. Table

S1 lists the calculated site dipole field in LaOBiS2 and α-SnTe for all the atomic sites

in the primitive cell. Note that since Bi and S atoms contribute dominantly to the





7

low-energy spectrum of LaOBiS2, the R-2 effect shown in Fig. 5b are not related to

La sites despite they also have a large dipole field. In the two BiS2 layers (α-sector

and β-sector) as shown in Fig. 5a, the Bi sites with heavy masses contribute

dominantly to the spin polarization. We find that the Bi sites in the α-sector (BiS2

layer) have the same site dipole field perpendicular to the layer, while the β-sector has

similar situation but opposite site dipole field. TABLE S1: Strength of site dipole field of LaOBiS2 and α-SnTe. There are 10

atomic sites in the primitive cell of LaOBiS2, and the two Bi sites are inversion

partners, so as the two La sites, the two O sites, the first two S sites (S1 and S2), as

well as the second two S sites (S3 and S4). The site dipole fields in LaOBiS2 are all

along [001] direction. There are 2 atomic sites in the primitive cell of α-SnTe. The site

dipole fields in α-SnTe are all along [111] direction. The site point groups of each

atomic site are also provided.

Atomic site Site point group Site dipole field (105 KV/cm)

LaOBiS2 (P4/nmm) La1 C4v -2.6 La2 C4v 2.6 O1 S4 0 O2 S4 0 Bi1 C4v 1.8 Bi2 C4v -1.8 S1 C4v 0.4 S2 C4v -0.4 S3 C4v -1.1 S4 C4v 1.1

α-SnTe (R3m) Sn C3v 0.4 Te C3v -0.2





8

E. Simple model Hamiltonian description on the measurability of the

compensated spin polarization

The basic idea of how to measure the residual spin polarization is that a probe

signal (ARPES and STM) exponentially decays from the top atomic layer into bulk.

First we use model Hamiltonian to derive the residual spin polarization in a local

asymmetric sector. For simplicity while maintaining generality, we take R-2 effect as

an example. Considering the Hamiltonian of an otherwise free electron in this sector,

𝐻𝐻 = − ℏ!

!!∇! + 𝝀𝝀𝒌𝒌 ⋅ 𝝈𝝈, (S1)

where 𝝈𝝈 denotes Pauli matrix vector and 𝝀𝝀𝒌𝒌 the linear-k term caused by local

inversion asymmetry. For classic Rashba and Dresselhaus effect, we have 𝜆𝜆 𝒌𝒌 =

𝛼𝛼! 𝑘𝑘! ,−𝑘𝑘! , 0 and 𝛼𝛼! 𝑘𝑘! , 𝑘𝑘! , 0 , respectively.

Next we consider a material consisting of two symmetric sectors, 𝛼𝛼 and 𝛽𝛽, and

the interaction of the two sectors is tk. Therefore, regarding the global inversion

symmetry of the system, the model Hamiltonian of the material is simplified as

(𝜖𝜖!! = 𝜖𝜖!! = 𝜖𝜖𝒌𝒌 for denoting different layers)

𝐻𝐻!"# =

𝜖𝜖!! (𝝀𝝀𝒌𝒌 ⋅ 𝝈𝝈)𝟏𝟏𝟏𝟏 𝑡𝑡! 0(𝝀𝝀𝒌𝒌 ⋅ 𝝈𝝈)𝟏𝟏𝟏𝟏∗ 𝜖𝜖!! 0 𝑡𝑡!

𝑡𝑡!∗ 0 𝜖𝜖!! −(𝝀𝝀𝒌𝒌 ⋅ 𝝈𝝈)𝟏𝟏𝟏𝟏

0 𝑡𝑡!∗ −(𝝀𝝀𝒌𝒌 ⋅ 𝝈𝝈)𝟏𝟏𝟏𝟏∗ 𝜖𝜖!!

. (S2)

By diagonalizing the matrix in Eq. (S2), we get the two-fold degenerate eigenvalues

𝐸𝐸± 𝒌𝒌 = 𝜖𝜖𝒌𝒌 ± 𝝀𝝀𝒌𝒌 ! + 𝑡𝑡!!. (S3)

and the corresponding wavefunctions

|𝑘𝑘, ↑ = (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐! + 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝜑𝜑!)⨂|↑ (S5)

|𝑘𝑘, ↓ = (𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠! + 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝜑𝜑!)⨂|↓ , (S6)

where |𝑘𝑘, ↑ and |𝑘𝑘, ↓ denote two hybrid states in two-fold degenerate bands, while

𝜑𝜑! and 𝜑𝜑! denote the wavefunction of individual sectors 𝛼𝛼 and 𝛽𝛽, respectively,

which can be solved by (S1). Therefore, the spin expectation is written as

𝑘𝑘, ↑ 𝝈𝝈 𝑘𝑘, ↑ = 𝑐𝑐𝑐𝑐𝑐𝑐!𝜃𝜃 𝛔𝛔 ! + 𝑠𝑠𝑠𝑠𝑠𝑠!𝜃𝜃 𝝈𝝈 ! (S7)

𝑘𝑘, ↓ 𝝈𝝈 𝑘𝑘, ↓ = −𝑠𝑠𝑠𝑠𝑠𝑠!𝜃𝜃 𝝈𝝈 ! − 𝑐𝑐𝑐𝑐𝑐𝑐!𝜃𝜃 𝝈𝝈 !. (S8) 8

E. Simple model Hamiltonian description on the measurability of the

compensated spin polarization

The basic idea of how to measure the residual spin polarization is that a probe

signal (ARPES and STM) exponentially decays from the top atomic layer into bulk.

First we use model Hamiltonian to derive the residual spin polarization in a local

asymmetric sector. For simplicity while maintaining generality, we take R-2 effect as

an example. Considering the Hamiltonian of an otherwise free electron in this sector,

𝐻𝐻 = − ℏ!

!!∇! + 𝝀𝝀𝒌𝒌 ⋅ 𝝈𝝈, (S1)

where 𝝈𝝈 denotes Pauli matrix vector and 𝝀𝝀𝒌𝒌 the linear-k term caused by local

inversion asymmetry. For classic Rashba and Dresselhaus effect, we have 𝜆𝜆 𝒌𝒌 =

𝛼𝛼! 𝑘𝑘! ,−𝑘𝑘! , 0 and 𝛼𝛼! 𝑘𝑘! , 𝑘𝑘! , 0 , respectively.

Next we consider a material consisting of two symmetric sectors, 𝛼𝛼 and 𝛽𝛽, and

the interaction of the two sectors is tk. Therefore, regarding the global inversion

symmetry of the system, the model Hamiltonian of the material is simplified as

(𝜖𝜖!! = 𝜖𝜖!! = 𝜖𝜖𝒌𝒌 for denoting different layers)

𝐻𝐻!"# =

𝜖𝜖!! (𝝀𝝀𝒌𝒌 ⋅ 𝝈𝝈)𝟏𝟏𝟏𝟏 𝑡𝑡! 0(𝝀𝝀𝒌𝒌 ⋅ 𝝈𝝈)𝟏𝟏𝟏𝟏∗ 𝜖𝜖!! 0 𝑡𝑡!

𝑡𝑡!∗ 0 𝜖𝜖!! −(𝝀𝝀𝒌𝒌 ⋅ 𝝈𝝈)𝟏𝟏𝟏𝟏

0 𝑡𝑡!∗ −(𝝀𝝀𝒌𝒌 ⋅ 𝝈𝝈)𝟏𝟏𝟏𝟏∗ 𝜖𝜖!!

. (S2)

By diagonalizing the matrix in Eq. (S2), we get the two-fold degenerate eigenvalues

𝐸𝐸± 𝒌𝒌 = 𝜖𝜖𝒌𝒌 ± 𝝀𝝀𝒌𝒌 ! + 𝑡𝑡!!. (S3)

and the corresponding wavefunctions

|𝑘𝑘, ↑ = (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐! + 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝜑𝜑!)⨂|↑ (S5)

|𝑘𝑘, ↓ = (𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠! + 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝜑𝜑!)⨂|↓ , (S6)

where |𝑘𝑘, ↑ and |𝑘𝑘, ↓ denote two hybrid states in two-fold degenerate bands, while

𝜑𝜑! and 𝜑𝜑! denote the wavefunction of individual sectors 𝛼𝛼 and 𝛽𝛽, respectively,

which can be solved by (S1). Therefore, the spin expectation is written as

𝑘𝑘, ↑ 𝝈𝝈 𝑘𝑘, ↑ = 𝑐𝑐𝑐𝑐𝑐𝑐!𝜃𝜃 𝛔𝛔 ! + 𝑠𝑠𝑠𝑠𝑠𝑠!𝜃𝜃 𝝈𝝈 ! (S7)

𝑘𝑘, ↓ 𝝈𝝈 𝑘𝑘, ↓ = −𝑠𝑠𝑠𝑠𝑠𝑠!𝜃𝜃 𝝈𝝈 ! − 𝑐𝑐𝑐𝑐𝑐𝑐!𝜃𝜃 𝝈𝝈 !. (S8)





9

We can clearly see that the total spin is projected on both sectors. To extract the

residual spin polarization of 𝛼𝛼 sector, we combine Eq. (S7) and Eq. (S8) and have

𝑃𝑃! = −𝑃𝑃! = 𝑐𝑐𝑐𝑐𝑐𝑐!𝜃𝜃 𝝈𝝈 ! − 𝑠𝑠𝑠𝑠𝑠𝑠!𝜃𝜃 𝝈𝝈 ! = 𝑐𝑐𝑐𝑐𝑐𝑐2𝜃𝜃 𝝈𝝈 !, (S9)

where 𝑐𝑐𝑐𝑐𝑐𝑐2𝜃𝜃 = 1/ 1+ 𝑡𝑡!!/ 𝝀𝝀𝒌𝒌 ! is the compensation parameter of the local spin

polarization which closely related to the interaction between the two sectors. If 𝑡𝑡! is

very small, 𝑐𝑐𝑐𝑐𝑐𝑐2𝜃𝜃 ≈ 1, and thus the local spin polarization is almost as large as if

only sector 𝛼𝛼 (or 𝛽𝛽) exists, i.e. 𝝈𝝈 ! (or 𝝈𝝈 ! ), which can be detected by the

penetrating probe into the material. For example, the compensation parameter of

LaOBiS2 and NaCaBi according to our calculation is about 0.9 and 0.7, respectively,

indicating strong local spin polarizations. In addition, Eq. (S9) also shows that the

total spin polarization is cancelled by the global inversion symmetry, in agreement

with our conclusion.

For a probe penetrating into the sample, the signal from the first atomic layer is

stronger than from the second atomic layer and therefore the spin polarization in

centrosymmetric crystals is not fully compensated. Therefore, we can expect the final

results of the probe measurement to be (−1)!!!𝑒𝑒!!!/!!!! 𝝈𝝈 , where 𝐿𝐿!, 𝐿𝐿 and

𝝈𝝈 are the position of the n-th layer, probe depth and the local spin polarization

projected on the first layer, respectively. For NaCaBi in which every CaBi layer is

equally arranged with the distance Δ𝐿𝐿 = 0.34 𝑛𝑛𝑛𝑛, we can expect the measured spin

polarization with a probe depth 𝐿𝐿 = 0.5 𝑛𝑛𝑛𝑛 to be !!!!!!!/!

𝝈𝝈 ≅ 0.66 𝝈𝝈 . Such a

strong signal should be detected by the advanced experimental technique in the future.

References 1. Nestoklon, M. O., Ivchenko, E. L., Jancu, J. M. & Voisin, P. Electric field effect on electron spin splitting in SiGe/Si quantum wells. Phys. Rev. B 77, 155328 (2008). 2. Gierz, I., Suzuki, T., Frantzeskakis, E., Pons, S., Ostanin, S., Ernst, A., et al. Silicon Surface with Giant Spin Splitting. Phys. Rev. Lett. 103, 046803 (2009).



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