Unified Model of Intrinsic Spin-Hall Effect in Spintronic, Optical,

and Graphene Systems

Takashi FUJITA1;2, Mansoor Bin Abdul JALIL1;3, and Seng Ghee TAN2;3

1Information Storage Materials Laboratory, Electrical and Computer Engineering Department,

National University of Singapore, 4 Engineering Drive 3, Singapore 1175762Data Storage Institute, A*STAR (Agency for Science, Technology and Research),

DSI Building, 5 Engineering Drive 1, Singapore 1176083Computational Nanoelectronics and Nano-device Laboratory, Electrical and Computer Engineering Department,

National University of Singapore, 4 Engineering Drive 3, Singapore 117576

(Received May 28, 2009; accepted August 10, 2009; published October 13, 2009)

A semiclassical model of the intrinsic spin-Hall effect (SHE) is presented which is relevant for a wideclass of systems, including the two-dimensional electron gas with Rashba SOC. We start with a fullyquantum mechanical treatment, introducing a gauge field formulation of the effect. Using thisformulation as a basis, we construct an intuitive picture of the SHE, and derive general expressions forthe spin-Hall current and conductivity. Finally, we propose an analogous effect in bilayer graphenesystems.

KEYWORDS: spin-Hall effect, spin–orbit coupling, spintronicsDOI: 10.1143/JPSJ.78.104714

1. Introduction

The spin-Hall effects1–4) (SHEs) entail a collection ofphenomena in which a pure, transverse spin current is gen-erated in response to a longitudinally applied electric field.Unlike extrinsic versions of the effect,5,6) the intrinsic SHEsare finite in the absence of impurities, and originate from thespin–orbit coupling (SOC) present in the band structure ofthe host system. The rich study of the SHE has been largelyfueled by its potential application as a spin current source inthe emergent technology called spintronics.7,8)

The intrinsic SHE has been studied in many systems, mostnotably in semiconductor spintronics. The two essentialingredients for the intrinsic SHE in semiconductors are SOCand an electric field. An external magnetic field is notnecessary; the spin polarization in the SHE does not violatetime-reversal (TR) symmetry since it is momentum-depend-ent. When discussing the intrinsic SHE in semiconductors,the two seminal examples which come to light are the SHEof holes in p-doped bulk semiconductors discovered byMurakami et al.,9,10) and in two-dimensional electron gases(2DEGs) with Rashba SOC discovered by Sinova et al.11)

Remarkably, the spin-Hall conductivity (SHC) in the Rashbasystem was found to be universal (i.e., independent ofsystem parameters) whenever the Fermi level intersects bothsubbands of the Rashba Hamiltonian.11) This, and the factthat the spin-Hall current can be dissipationless due to its TRsymmetry,9) has attracted considerable interest in the SHE,from both theoretical and experimental points of view.12–15)

From a theoretical standpoint, the SHE is characterized bythe SHC, which is typically calculated as the linear Kuboresponse of the transverse spin current to a charge currentexcitation. However, this treatment reveals little of theunderlying physics which drives the effect. In fact, basedon heuristic grounds, the driving mechanisms of the twoaforementioned SHEs are rather distinct (they are oftenconfused with each other in the literature).16,17) The formerSHE9,10) arises from the topological Berry curvature inmomentum space,18,19) which results in spin-dependentequations of motion.20) Here, the finite curvature in reciprocal

space represents an analog of an ordinary magnetic field, butwhich is two-valued depending on the sign of the spin — tobe more precise, under the adiabatic approximation, thesystem Hamiltonian can be decomposed into two copies,representing the light-hole and heavy-hole bands, each withan SU(2) gauge field giving rise to a two-valued Berrycurvature on each band. Physically, the effect of the spin-dependent curvature is to push opposite spin species alongopposite transverse directions, leading to spin separation.Following the discovery of this effect in p-doped semi-conductors, there have been theoretical studies of analogouseffects in other systems. Some examples are the SHE ofphotons,21–24) phonons,25) and excitons.16,26) On the otherhand, the SHE in Rashba systems11) occurs as a result of adistinct mechanism, namely the time-resolved spin dynamicsinduced by the applied electric field.17) Henceforth, we shallrestrict our discussions to this latter type of intrinsic SHE.

In this article, we examine the SHE in the Rashba 2DEGsystem, focusing on the physical mechanism which drivesthe effect. Subsequently, we apply the same formalism toseveral other intrinsic SHEs studied in both semiconductorand optical systems, thus unifying those SHEs into a singleclass. To facilitate application of our model to varioussystems, we begin with a general spin–orbit model inthe presence of an electric field. From a fully quantummechanical analysis, we show that the electric field inducesa gauge field in the general model, which has the physicalinterpretation of an effective magnetic field. This gauge fieldis akin to that arising in the p-doped bulk systems, but itis defined in time rather than momentum space. We thenprovide a semiclassical model of the intrinsic SHE, explain-ing clearly how it is driven by the effective field. Inparticular, we clarify that the SHE is an adiabatic effect,resulting from the relaxation of spins to the effective field,rather than their precession about the Rashba field. Thesimple, heuristic nature of our model allows us to constructan intuitive expression for the spin-Hall current and SHC.Our general expressions are found to reproduce the intrinsicSHC in a wide class of systems without recourse to linearresponse theory, e.g. in 2DEGs with combined Rashba and

Journal of the Physical Society of Japan

Vol. 78, No. 10, October, 2009, 104714

#2009 The Physical Society of Japan

104714-1

linear Dresselhaus SOC,27–33) bulk semiconductors withcubic Dresselhaus SOC,34) and 2D hole systems with RashbaSOC.35–38) Thus, our model unifies the SHE across varioussystems in a transparent manner. Our closed expression forthe SHC also allows one to conveniently analyze the SHE ingeneral SOC systems using straightforward vector algebra,without the need for cumbersome operator algebra. Finally,we propose an analogous effect in bilayer graphene systems.

Before moving to the paper proper, we briefly discussthe important topic of impurity scattering in the SHE. Ourcalculations in this paper for the SHC are based on afully ballistic transport model, which exclude the effects ofscattering by impurities. Previous studies have shown thatthe SHC in infinite Rashba (and Dresselhaus) systemsvanishes when vertex corrections are introduced to modelthe effects of impurity scattering.39,40) However, this occursdue to the special form of the Rashba Hamiltonian,41) anddoes not apply to general SOC systems, for example in bulksemiconductors with cubic Dresselhaus SOC34) which is alsoanalyzed in this paper.

2. Theory

2.1 Time-dependence of spin–orbit coupling systemsWe consider the general spin–orbit Hamiltonian in the

presence of an electric field E,

H ¼p2

2m� �� ��ðkÞ þ eE � r; ð1Þ

where p ¼ h�k is the momentum, m the effective mass, � isthe SOC strength, � is the vector of Pauli matrices, and �ðkÞis a momentum-dependent effective field. We study the time(t) evolution of the above quantum system. To incorporatethe explicit t-dependence of the system quantum mechan-ically, we switch to the interaction quantum picture,42)

splitting H into two parts, H ¼ H0 þH1, where

H0 ¼ eE � r ð2Þ

governs the time evolution of the operators, and

H1 ¼p2

2m� �� ��ðkÞ ð3Þ

governs the time evolution of the states in the new picture.The momentum operator in the interaction picture (subscriptI) is found to be

pIðtÞ ¼ eiH0t=h� pe�iH0t=h� ¼ p� eEt; ð4Þ

i.e., with the expected linear t-dependence due to E. Statevectors j ðtÞi in the Schrodinger picture correspondinglytransform as j IðtÞi ¼ eiH0t=h� j ðtÞi, and evolve according tothe new ‘‘Schrodinger equation’’, HIðtÞj IðtÞi ¼ ih�@tj IðtÞi,where the Hamiltonian HIðtÞ is computed as

HIðtÞ ¼p2

I

2m� �� ��ðkIðtÞÞ; ð5Þ

where

�ðkIðtÞÞ ¼ ��eEit

h�

@�

@kiþ

e2EjElt2

2h�2

@2�

@kl@kj� � � � ;

and summation over repeated indices is implied. TheHamiltonianHIðtÞ is that of a particle subject to an explicitlyt-dependent spin–orbit field, �ðtÞ. We consider diagonaliz-

ing the above Schrodinger equation at time t, by applying aunitary rotation UðtÞ which aligns the reference zz-axis alongthe instantaneous field �ðtÞ, i.e.

UðtÞHIðtÞUyðtÞ ¼ UðtÞ ih�@tð ÞUyðtÞ;p2

I

2m� ��zj�ðtÞj ¼ ih�@t þ ih�UðtÞ _UUyðtÞ;

¼ �� h�A0ðtÞ: ð6Þwhere � ¼ ih�@t is the energy operator. On the right-hand-side, a gauge field A0ðtÞ � �iU _UUy appears from thet-dependence of U. The energy term h�A0ðtÞ can be writtenas a Zeeman-like term,17) indicating the presence of aneffective magnetic field in the rotated frame. Since _UU ¼ð1=2ÞiU� � !,43) where ! ¼ !ðtÞ is the instantaneous angularvelocity of the rotating frame, the effective magnetic field asseen from the laboratory frame is �!. For the unit vectornðtÞ ¼ �ðtÞ=j�ðtÞj, we also have that _nn ¼ !� n or, equiv-alently, ! ¼ n� _nnþ ð! � nÞn. Although the latter compo-nent depends on the choice of ! (! is not unique, as there isa rotational freedom about n itself), the former component,corresponding to an effective field �? ¼ _nn� n, dependsonly on the t-dependence of �ðtÞ. This magnetic fieldcan also be derived classically, by comparing vectors inadjacent time-dependent frames.17,44) �? represents aphysical magnetic field which couples to the electronspins,44,45) and, as we show below, is precisely thecomponent which underpins the SHE in the Rashba systemas well as various other spintronic and optical systems.

2.2 Analogy with the momentum space Berry curvatureBefore proceeding, we note that the SHE in p-doped

bulk semiconductors discovered by Murakami et al.9,10)

cannot be described by the theory introduced above, asit arises from a distinct mechanism, namely the Berrycurvature in momentum space. However, there is a remark-able analogy between the two mechanisms in terms of thegauge fields involved. To obtain the Berry curvature, wediagonalize the general spin–orbit Hamiltonian (1) withrespect to the momentum k, by applying a unitary trans-formation UðkÞ which aligns the reference zz-axis along thedirection of �ðkÞ, i.e.,

UðkÞHUyðkÞ ¼p2

2m� ��zj�ðkÞj þ UðkÞðeE � rÞUyðkÞ

¼ �; ð7Þwhere the energy operator � remains unchanged, as themomentum k is t-independent in the Schrodinger quantumpicture. In the last term of (7), the position operator r ¼ irkacts as a partial derivative in momentum space, and weobtain from the k-dependence of U:

UðkÞðeE � irkÞUyðkÞ ¼ eE � ðrþ iUðkÞrkUyðkÞÞ;¼ eE � ðr�AðkÞÞ; ð8Þ

where AðkÞ � �iUðkÞrkUyðkÞ is a gauge field in k-space.Taking the curvature of AðkÞ (after applying the adiabaticapproximation9)), we obtain the momentum space Berrycurvature. Thus, the intrinsic SHEs in p-doped semiconduc-tors and 2DEGs with Rashba SOC can both be described bythe action of gauge fields, although they are defined indifferent spaces (k-space and time, respectively). This linkwas further elucidated by us elsewhere.17)

J. Phys. Soc. Jpn., Vol. 78, No. 10 T. FUJITA et al.

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2.3 Calculation of spin-Hall current and conductivityWe consider the DC response of a SOC system to a charge

current ji due to an electric field Ei. In particular, wecalculate the spin current

jls; j ¼s

2fvj; �lg; ð9Þ

where s is the value of the spin angular momentum and f�gdenotes the anticommutator, and also the SHC �lij � jls; j=Ei.The electric field results in a t-dependent spin–orbit field�ðtÞ, which, as discussed above, is necessarily accompaniedby an additional field �? ¼ _nn� n (Fig. 1). Assuming thatthe spins align themselves to the net magnetic field ��, i.e.the sum of � and the t-dependent correction, the classicalspin vector is

s ¼ �s��

j��j; ð10Þ

where + (�) represents spins aligned parallel (anti-parallel)to ��. Along the ll-coordinate, the component of the spinvector arising from �? is

sl ¼ �sð _nn� nÞlj��j

; ð11Þ

where �� is expressed in terms of its equivalent angularvelocity. Of course, the spin may well have a componentalong ll due to � itself, but it is not proportional to theelectric field and does not contribute to the spin-Hallcurrent. We consider now the adiabatic limit, i.e., wherej�j � j�?j, so that the spins are primarily aligned to � butwith a small component along �?. The adiabatic spinpolarization along ll is then

sl � �sh�

2�j�j3�lnq _��n�q; ð12Þ

where �lnq is the Levi–Civita symbol. To obtain the spin-Hall(sH) current, we consider the velocity operator in thepresence of SOC, which reads

vj ¼@H@pj¼

pj

m� �h��1� �

@�

@kj: ð13Þ

Substituting this expression into the spin current operator(9), we obtain jls; j ¼ sðpj�l=m� �@�l=@kjÞ. The spin currenttherefore has two contributions; the first due to the spin-polarization of the states, and second due to the variation ink of �ðkÞ. The sH current is the part of jls; j proportional to Ei,which corresponds only to the first contribution, i.e.

jls; jðsHÞ ¼pjs

l

m: ð14Þ

The total sH current can be calculated semiclassically bysumming the expectation value of the jls; jðsHÞ operator overall states up to the Fermi level,

jls; jðsHÞ ¼Z

dDk

ð2�ÞDpjðkÞm

slðkÞ; ð15Þ

where D is the dimension of the system. We carefully notethat the expression for the spin polarization slðkÞ hasopposite signs for spins pointing parallel and antiparallelto �ðkÞ (12). These correspond to the two spin-split bands ofthe spin–orbit Hamiltonian (1); namely, electrons pointingparallel (antiparallel) to � with energies Eþð�Þ ¼ p2=2m�ðþÞ�j�ðkÞj. Since the Fermi level usually far exceeds theband-splitting, EF � 2j�j, both bands are occupied withtheir Fermi wavevectors fulfilling the relation:

EF ¼ðh�k�F Þ

2

2m �j�ðk�F Þj; ð16Þ

where kþF > k�F . In the region where the two Fermi surfacesoverlap (i.e., k < k�F ), there is a complete cancellation ofslðkÞ. The finite contribution to the sH current comesfrom the states in the annular region k�F < k < kþF of theFermi surface occupied only by the ground state band, Eþ.Noting this, and writing the time derivative of �n in theexpression for sl (12) in terms of the wavevector ki, weobtain

jls; jðsHÞ ¼Z

dDk

ð2�ÞDpjðkÞm

sh�

2�j�j3�lnq �

@�n

@ki

eEi

h�

� ��q; ð17Þ

where the limits of integration in k goes from k�F to kþF .Dividing both sides through by the electric field Ei, weobtain an expression for the intrinsic SHC,

�lij �jls; jðsHÞEi

¼esh�

2�mð2�ÞD

ZdDk

kj

j�j3�lnq�n

@�q

@ki: ð18Þ

Using a semiclassical approach, we have constructed ageneral expression for the intrinsic SHC which makestransparent the physical mechanism driving the phenomena.Our analysis is relevant for describing the SHE in a wideclass of systems as discussed below.

3. Results and Discussion

In Table I we list several spintronic systems which can berepresented by the general spin–orbit Hamiltonian in eq. (1).For each system, we list the expression for the systemHamiltonian H, the system dimension D, the effectivemagnetic field �ðkÞ, and compute the zz-spin polarization,szðkÞ, using eq. (12) and assuming an electric field appliedalong the xx-direction. Since the szðkÞ are odd in ky, carrierstraveling in opposite yy-directions become polarized alongopposite zz-directions. This is the physical origin of theintrinsic SHE. Below we explicitly determine the SHC inspecific spintronic systems, before examining analogouseffects in optical and bilayer graphene systems.

3.1 Combined Rashba and Dresselhaus SOCThe out-of-plane spin polarization driven by the electric

field in the combined Rashba–Dresselhaus (RD) system is

B(t) B(t)

Fig. 1. (Color online) In the presence of a time-dependent magnetic field,

BðtÞ ¼ jBðtÞjnðtÞ, an additional magnetic field B? ¼ _nn� n (green,

vertical arrow) is seen by spins. The net instantaneous magnetic field

felt by spins is the vector sum of BðtÞ and B?, denoted by the dashed,

black arrow.

J. Phys. Soc. Jpn., Vol. 78, No. 10 T. FUJITA et al.

104714-3

given in Table I. It vanishes when j�j ¼ j�j, in agreement with the well-known result that the net sH current is zero under thiscondition. When j�j 6¼ j�j, using eq. (18), and the relation kþF � k�F ¼ 2mj�RDj=h� 2k, the SHC is

�zxy ¼ �eh�

2ð�2 � �2Þ4mð2�Þ2

Z 2�

0

Z kþF

k�F

dk d�sin2 �

ð�2 þ �2 þ 2�� sin 2�Þ3=2

¼ �eð�2 � �2Þ

8�2

Z 2�

0

d�sin2 �

�2 þ �2 þ 2�� sin 2�

¼ �e

8�

�2 � �2

j�2 � �2j; ð19Þ

in agreement with previous studies.29,32,33) In particular, for the � ¼ 0 or � ¼ 0 cases, the universal value of j�zxyj ¼ e=8� isreproduced.11,27,29)

3.2 n-doped bulk semiconductorsFor conduction electrons under the influence of k3-Dresselhaus SOC, the SHC (18) is

�zxy ¼eh�

2

4mð2�Þ3

Z 2�

0

Z �

0

Z kþF

k�F

dk d d� k2 sin k2y ðk2

z � k2y Þðk2

x þ k2z Þ

j�Dj3

!; ð20Þ

where �D is the k3-Dresselhaus field, and ð; �Þ are spherical angles in k-space. Using the interband relation valid for small ,kþF � k�F � 2mj�DðkFÞj=h� 2kF, where kF ¼ ðkþF þ k�F Þ=2, the above yields a SHC of

�zxy ¼ �ekF

12�2; ð21Þ

in agreement with results obtained previously.34)

3.3 Holes in III–V semiconductor quantum wells withRashba SOC

The SHC (18) is (s ¼ 3=2),

�zxy ¼ �9eh�

2

4�mð2�Þ2

Z 2�

0

Z kþF

k�F

dk d�sin2 �

k2;

¼9eh�

2

16��m

1

kþF�

1

k�F

� �; ð22Þ

where � is the azimuthal angle in k-space. In the limit ofsmall Rashba coupling, the band resolved Fermi wave-vectors are k�F �

ffiffiffiffiffiffiffiffi2�np

� ð2�m=h� 2Þ�n,35) where n is thehole density, giving a universal SHC of

�zxy ¼ �9e=8� ð23Þ

in agreement with previous studies.35–37)

3.4 Bilayer grapheneWe propose an analogous effect in bilayer graphene

(BLG). The BLG system is modeled as two coupledhoneycomb lattices, with each layer having two inequivalentlattice sites ~AA; ~BB and A;B in the top and bottom layersrespectively. We assume the Bernal stacking ( ~AA–B) config-uration. In the low energy limit, electrons in the BLG systemare described by an effective 2-by-2 Hamiltonian,46)

HBLG ¼ �h�

2

2m� ��BLG; ð24Þ

where �BLG ¼ ðk2x � k2

y ; 2kxky; 0Þ. � here is the vector ofPauli operators acting on the pseudospin, rather than theactual electron spin. The symmetry of the lattice supportstwo inequivalent, degenerate points, K and ~KK, on thehexagonal Brillouin zone. In the ~KK-valley, the two-compo-nent BLG eigenstates are of the form �� ¼ ð �ð ~BBÞ; �ðAÞÞ,i.e., describing the electronic amplitude of electrons on thetwo layers, with eigenvalues E� ¼ ðh� 2=2mÞj�BLGj. Thecontrolled doping of BLG can shift the Fermi level intoeither the conduction (E� state) or valence band (Eþ),47)

Table I. List of systems and their Hamiltonians H in which the spin-Hall effect is analyzed. D is the system dimension, �, �, , and � are the respective

SOC strengths, �l (l ¼ x; y; z) are the Pauli operators, kl are the wavevectors, �� ¼ �x � i�y and k� ¼ kx � iky. �ðkÞ is the momentum-dependent

effective magnetic field, and sz is the zz-spin polarization of electrons resulting from an electric field applied in the xx-direction, obtained from eq. (12).

For the case of bilayer graphene, �z represents the pseudospin polarization.

System (D) H �ðkÞ szðkÞ for E ¼ Exxx

Rashba–Dresselhaus (2)

h�2k2

2mþ �ðkx�y � ky�

xÞ �ky þ �kx��kx � �ky

0

0@

1A

eExh�kyð�2 � �2Þ4j�ðkÞj3þ �ðky�y � kx�

xÞ

k3-Dresselhaus (3)

h�2k2

2mþ ½kxðk2

y � k2z Þ�x

�kxðk2y � k2

z Þ�kyðk2

z � k2x Þ

�kzðk2x � k2

y Þ

0B@

1CA

eExh�kyðk2x þ k2

z Þðk2y � k2

z Þ4j�ðkÞj3

þ kyðk2z � k2

x Þ�y þ kzðk2x � k2

y Þ�z

Heavy holes in QW (2)h�

2k2

2mþ i�

2ðk3��þ � k3

þ��Þk3y � 3k2

x ky

k3x � 3kxk

2y

0

0@

1A

9eExh�ky

4�k5

Bilayer graphene (2) �h�

2

2m

0 k2�

k2þ 0

� � k2x � k2

y

2kxky0

0@

1A �z ¼

2meExky

h�2k4

J. Phys. Soc. Jpn., Vol. 78, No. 10 T. FUJITA et al.

104714-4

defining which band contributes to the Hall transport. Whenan electric field E ¼ Exxx is applied to the system, anadditional out-of-plane component accompanies the strictlyin-plane pseudospin field �BLG. In the adiabatic limit, theout-of-plane component induces a pseudospin polarization �z

along �zz (see Table I), which corresponds to charge transferbetween the two layers. This is an essential ingredient for thetechnology known as pseudospintronics,48) in which binarystates are encoded by relative charge densities on the twomonolayers. The physical effect of this polarization for theground state is illustrated in Fig. 2: electrons with py > 0

(py < 0) are separated to the bottom (top) graphene mono-layers of the BLG system. For the K-valley, the BLGeigenstates are of the form �� ¼ ð �ðAÞ; �ð ~BBÞÞ, i.e.the order of the components is reversed, and thus thepseudospin polarization affects electrons in the oppositemanner compared to the ~KK-valley; electrons with py > 0

(py < 0) are transferred to the top (bottom) layer. Thus, thenet effect is expected to vanish when the contributions fromboth valleys are taken into consideration. Recently in theliterature, the idea of producing an imbalance of electronstates depending on their valley degree of freedom has cometo light. The effect we propose would then be finite, afterpassing electrons through such valley filters.49,50) Further-more, we assume that the applied electric field is sufficientlyweak such that intervalley transitions can be neglectedwithin the BLG system.51) Our proposed effect in BLG iscompletely analogous to the SHE with replacements � $ �.It may therefore be termed the intrinsic pseudospin-Halleffect. Such an effect should be of interest to the field ofpseudospintronics.

Finally, we draw attention to an analogous effect inoptics, when polaritons undergo Rayleigh scattering in asemiconductor microcavity.52) The polariton polarization isrepresented by a pseudospin �, where the pseudospin fieldis exactly that of bilayer graphene �BLG, above. Uponscattering (which changes the wavevector k akin to theE-field effect in the above cases), polaritons acquire a finite�z component, corresponding to circular polarization, whosesign depends on the initial momenta. The manifestation ofthis effect has been observed in experiments,53–55) which isa promising indication for the detection of the effect inspintronic and graphene systems.

4. Summary

We presented a simple, semiclassical model for the SHEin Rashba systems. The heuristic and direct nature of ourmodel allows one to construct an intuitive expression for the

intrinsic spin-Hall conductivity, maintaining a clear physicalpicture of the effect. In particular, we clarify that the SHEis an adiabatic effect, in the sense that it results from therelaxation of spins to an out-of-plane magnetic field, ratherthan the spin precession about the in-plane Rashba field. Theout-of-plane magnetic field component was derived froma quantum mechanical gauge field approach, which drawsparallels between the SHE in the Rashba and bulk Luttingersystems. Finally, we proposed an analogous effect in bilayergraphene systems.

Acknowledgments

The authors would like to thank the Agency for Science,Technology and Research (A*STAR) of Singapore, theNational University of Singapore (NUS) Grant No. R-398-000-047-123 and the NUS Nanoscience and Nanotechnol-ogy Initiative for financially supporting their work.

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