Direct Mapping of Spin and Orbital Entangled Wavefunction under Interband Spin-Orbit coupling of Rashba Spin-Split Surface States

2 MB·

by  Ryo Noguchi

Direct Mapping of Spin and Orbital Entangled Wavefucntion under Interband
Spin-Orbit coupling of Rashba Spin-Split Surface States
Ryo Noguchi,1 Kenta Kuroda,1,∗ K. Yaji,1 K. Kobayashi,2 M. Sakano,1
A. Harasawa,1 Takeshi Kondo,1 F. Komori,1 and S. Shin1
1Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan
2Department of Physics, Ochanomizu University, Bunkyo-ku, Tokyo 112-8610, Japan
(Dated: January 26, 2017)
We use spin- and angle-resolved photoemission spectroscopy (SARPES) combined with
polarization-variable laser and investigate the spin-orbit coupling effect under interband hybridiza-
7 tion of Rashba spin-split states for the surface alloys Bi/Ag(111) and Bi/Cu(111). In addition to
1 theconventionalbandmappingofphotoemissionforRashbaspin-splitting,thedifferentorbitaland
0 spin parts of the surface wavefucntion are directly imaged into energy-momentum space. It is un-
2 ambiguouslyrevealedthattheinterbandspin-orbitcouplingmodifiesthespinandorbitalcharacter
of the Rashba surface states leading to the enriched spin-orbital entanglement and the pronounced
n
momentum dependence of the spin-polarization. The hybridization thus strongly deviates the spin
a
J and orbital characters from the standard Rashba model. The complex spin texture under inter-
band spin-orbit hybridyzation proposed by first-principles calculation is experimentally unraveled
5
by SARPES with a combination of p- and s-polarized light.
2

PACSnumbers:
]
i
c
s
Arealizationoffunctionalcapabilitiestogeneratespin- systemBiCu /Cu(111)asshowninFig.1(b)[14,20,21].
- 2
l splitting of electronic states without any external mag- Whilethepresenceofthespin-polarizedelectronicbands
r
t netic field is a key subject in the research of spintron- has been demonstrated by spin and angle-resolved pho-
m
ics [1]. A promising strategy exploits the influence of toemission spectroscopy (SARPES) [21–23] and inverse-
.
t spin-orbit (SO) interaction that can give rise to the lift- SARPES [8], the SO entanglement and particularly the
a
ingofspindegeneracyunderbrokenspaceinversionsym- spintextureduetotheinterbandSOcouplingisstillun-
m
metry, the so-called Rashba effect [2]. In the conven- derdiscussion, becauseofthelackoftheorbitalselectiv-
-
d tional Rashba model, an eigenstate of the SO-induced ityinthepreviousexperiments. Uptonow, thecomplex
n spin-splitting is treated with an assumption of a pure spintextureofthesesurfacestateswasonlyindirectlyde-
o spin state fully chiral spin-polarized which protects elec- tected by quantum interference mapping through scan-
c
trons from backscattering [2–5]. However, in real mate- ning tunneling spectroscopy [9, 24, 25]. Thus, no con-
[
rials, the assumption can be usually broken because the clusiveunderstandingofthephenomenonapartfromthe

1 SO coupling mixes different states with different orbitals conventional Rashba model has been achieved yet.
v
and orthogonal spinors in a quasiparticle eigenstate [6–
9 In this Rapid Communication, we directly investi-
8]. The SO entanglement can permit the spin-flip elec-
6 gate the SO entanglement in the Rashba surface states
2 tron backscattering [9] and moreover orbital mixing in of BiAg /Ag(111) surface alloy by using a combina-
2
7 theeigenstatecanplayasignificantroleinanemergence
tion of polarization variable laser with SARPES (laser-
0 of the large spin-splitting [10–14]. Therefore, it is essen-
SARPES) and compare to those of BiCu /Cu(111) as
. 2
1 tially important to experimentally explore the SO cou- a simple case. In contrast to the previous experi-
0 pling not only in the lifting spin degeneracy but also in
ments [8, 21–23], our laser-SARPES deconvolves the or-
7 the spin and orbital wavefunction as eigenstates.
bital wavefucntion and the coupled spin, and the surface
1
: wavefunctionsaredirectlyimagedintomomentum-space
v BeyondtheconventionalRashbamodel,awell-ordered
throughorbital-selection rule. Itis shownthat theinter-
i surface alloy BiAg grown on Ag(111) provides an ideal
X 2 bandSOcouplingmodifiesthespinandorbitalcharacter
case to study the SO entanglement in Rashba surface
r of the Rashba surface states leading to the spin-orbital
states. In the surface alloy, an occupied sp -like band
a z
entanglementandthek dependence. Theresultingspin
and a mostly unoccupied p -like band show significant ||
xy
texture thus shows a large deviation from the conven-
Rashbaspinsplitting[15,16]andcrosseachotheratthe
tional Rashba model. The full spin information is ex-
specific k [17–19] as shown in Fig. 1 (b). In particular,
||
perimentally unraveled only by a combination of p- and
densityfunctionaltheory(DFT)calculationsshowedthe
s-polarized light in accordance with a view of the SO
strong SO entanglement [8, 9] and predicted the com-
entanglement.
plex spin texture that is significantly different from the
conventional Rashba model; the sp band switches spin- Thelaser-SARPESmeasurementwasperformedatthe
z
polarization at the crossing through SO-induced inter- Institute for Solid State Physics (ISSP), the Univesity of
bandhybridization[19]whichisincontrasttothesimilar Tokyo, with a high-flux 6.994-eV laser and ScientaOmi-
2
(a) light Dete=cMtiiornro pr lpalnaene photoelectrons Surface Brillouin zone ptalel etneemrgpyera(atunrgeulawra)srekseopltutaiotn∼s 1o5f tKh.e Tsehteupinisstr2ummeenV-
p-pol.
s-pol. (0.3◦)and20meV(0.7◦)forspin-integratedARPESand
Spin-up
z 50 Γ Κ SARPES,respectively. TheBiAg2 andBiCu2 surfaceal-
y x Crystal surface ky kx Spin-down lloityesrawtuerreeso[1b4ta,i1n5e]d. Lboywt-hene√eprgroyce√eldeuctrreosnpdrieffseranctteidoninmtehae-
(b) BiAg pxy BiCu pxy surements confirmed the ( 3× 3)R30◦ reconstruction
2 2
of the surface alloys.
E E E
F F
First-principles calculations were performed using the
k
sp sp VASP code [27]. The projector augmented wave
(c) 0BiAg2/Ag(111) z pxy 0 BiCu2/Cu(111)z spz pxy method [28] is used in the plane-wave calculation. The
generalized gradient approximation by Perdew, Burke,
eV) -0.3 -0.2 High and Ernzerhof [29] is used for the exchange-correlation
(
EF potential. The spin-orbit interaction is included. The
E- -0.6 -0.4
spz atom positions of BiAg2 are optimized. Those of BiCu2
-0.9 -0.2 -0.1 0 0.1 0.2 -0.6-0.2 -0.1 0 0.1 0.2 0.3 Low are taken from the experimental data of Ref. [30].
kx (Å-1) kx (Å-1)
Let us start with showing a brief overview of observed
(d) Low High Low High Odd Even
electronicstructureofBiAg andBiCu surfacealloysin
sp p sp p sp p 2 2
z xy z xy z xy Fig. 1(c). The sp -derived bands and the higher-lying
0.0 z
V) p bands disperse downwards in energy with a large
(eEF-0.2 Rxayshba spin-splitting in the both materials. Compared
E--0.4 withthesurfacebandsinBiAg ,themostpartofthesur-
2
-0.6 BiAg2 BiAg2 BiAg2 facebandsinBiCu2 isabovetheFermilevel(EF). These
0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3 results are in good agreement with previous works [14–
(e)
sp p sp p sp p 17].
z xy z xy z xy
0 Considering the orbital selection rule in the dipole ex-
(eV)-0.2 citation[31],(cid:15)p and(cid:15)s enableustodrawdifferentorbital
E-EF-0.4 spyomsemdeotfryB.i 6Ssinacnedt6hpeosrubriftaacles [s1t9a]t,e(cid:15)s nseealerctEivFelayrdeecteocmts-
p
BiCu BiCu BiCu
-0.6 2 2 2 weight of even-parity orbital with respect to the mirror
0.1 0.2 0.3 0.1 0.2 0.3 0.1 0.2 0.3
kx (Å-1) kx (Å-1) kx (Å-1) plane, mainly from s, pz and px components, while (cid:15)s is
sensitive to the odd-parity orbital mainly derived from
FIG.1: (Coloronline)(a)Theexperimentalconfigurationfor p .
y
p- or s-polarization with an angle incidence of 50 ◦. A mir-
Figure 1(d) summarizes the linear polarization depen-
ror plane of the surface coincides with the plane of incidence
dence in BiAg . For the result obtained by (cid:15) [see the
(x-z plane). Surface Brillonin zone of the surface alloys [un- 2 p
derlyingfcc(111)substrateofAg(111)andCu(111)]isshown left panel], we observe the strong intensity for the spz
by solid (dashed) line. (b) Schematic of the energy disper- and inner p bands. The data particularly displays
xy
sion of Rashba spin-split bands in (left) BiAg2 and (right) the band crossing of the outer spz and inner pxy bands
BiCu2 surface alloys. (c) ARPES intensity maps for BiAg2 aroundk =0.15˚A−1[Fig.1(d)],wherethespectralinten-
andBiCu surfacealloyswithp-polarizationalonghighsym- ||
metry Γ¯-K¯2 line of the surface Brillouin zone. The dashed sityoftheouterspz isstronglysuppressed. Surprisingly,
lines indicate the edge of the projected bulk bands. (d) and switching the light polarization (cid:15)p to (cid:15)s, the spectral in-
(e) The magnified ARPES intensity maps with (left) p- and tensity is dramatically changed [see the middle panel in
(middle) s-polarization, and (right) the differential intensity Fig. 1(d)]. The dispersion of the outer sp is clearly ob-
z
maps, which are obtained by Ip−Is where Ip and Is are the servedevenatlagerk >0.15˚A−1 togetherwiththeouter
photoelectron intensity obtained by p- and s-polarized light ||
p band. Consequently, the overall parabolic dispersion
without normalization, respectively. xy
of the outer sp band is clearly seen, which was absent
z
in previous experiments [14, 15, 22]. The right panel of
Fig. 1(d) shows the differential intensity map [see figure
cron DA30L photoelectron analyzer [26]. The experi- captionofFig.1]. Thered-bluecolorcontrastreflectsthe
mental configuration is shown in Fig. 1(a). The p- and contribution of the even- and odd-orbital components in
s-polarizations ((cid:15) and (cid:15) , respectively) were used in the the surface wavefunction. It is immediately found that
p s
experiment. The photoelectrons were detected along Γ¯- theorbitalcharacterofspz bandchangestheorbitalchar-
K¯ line of the surface Brillouin zone. The spectrometer acter at the band crossing.
resolved the spin component along y, which is perpen- In contrast, the result for BiCu is found to be simple
2
dicular to the mirror plane of the surface. The sam- [see Fig. 1(e)]: the sp and p bands comprise mainly
z xy
3
BiAg BiCu
2 2
(a) p-polarization (b) s-polarization (c) p-polarization (d) s-polarization
kx (Å-1) kx (Å-1)
nits) -0.03 nits) 0.07
b. u 0.00 b. u 0.10
ar 0.03 ar 0.13
y ( y (
sit 0.06 sit 0.16
n n
nte 0.09 nte 0.19
d i 0.12 d i 0.22
e e
v v
ol 0.15 ol 0.25
s s
e e
n-r 0.18 n-r 0.28
pi pi
S 0.21 S 0.31
0.24 0.34
-0.6 -0.4 -0.2 0 -0.6 -0.4 -0.2 0 -0.6 -0.4 -0.2 0 -0.6 -0.4 -0.2 0
E-E (eV) E-E (eV) E-E (eV) E-E (eV)
F F F F
(e) p-polarization (f) s-polarization (g) p-polarization (h) s-polarization
Sy Sy
V) 0 1 0 1
(e-0.2 V)
E-EF-0.4 0 E(eF-0.2 0
E--0.4
-0.6 -1 -1
Low High Low High
-0.6
0 0.1 0.2 0.3 0 0.1 0.2 0.3 0.1 0.2 0.3 0.1 0.2 0.3
-1 -1 -1 -1
kx (Å ) kx (Å ) kx (Å ) kx (Å )
FIG. 2: (Color online) (a)-(d) SARPES spectra with spin quantum axis along y for BiAg and BiCu by using (cid:15) and (cid:15) .
2 2 p s
Spin-upandspin-downspectraareplottedwithredandbluelines. Thepeakpositionsinthespin-resolvedspectraforsp and
z
p bands are indicated by open and closed triangles, respectively. (e)-(h) The corresponding spin-polarization and intensity
xy
maps with the two-dimensional color codes [32]. The dashed lines indicate the edge of the projected bulk bands.
even- and odd-parity orbitals, respectively. These two (cid:15) and (cid:15) . Figures 2(a)-(d) show SARPES spectra ob-
p s
bands in BiCu are separated in momentum-space away tained by using (cid:15) and (cid:15) for BiAg and BiCu . The
2 p s 2 2
from the band crossing. Nevertheless, we see the reduc- corresponding spin-polarization and intensity maps [32]
tion of the spectral intensity of the outer sp band when are shown in Figs. 2(e)-(h). For (cid:15) , the inner and outer
z p
itoverlapswiththeprojectedbulk-band,asobservedalso sp -bands around Γ¯ point in the both materials show
z
in the even-part of the outer sp band in BiAg [see the negative and positive spin-polarization, respectively, dis-
z 2
left panel in Fig. 1(d)]. This common feature suggests playingaconventionalRashba-typespin-texture[22,23].
the interaction of the the outer sp band with the bulk Theobservedspinpolarizationisfoundtobelargeupto
z
sp projection from the substrate [14, 16] modifies the nearly 80 and 60 % for BiAg and BiCu , respectively.
z 2 2
spectral weight of the orbital wavefunction particularly Most remarkably, we find that the sign of the spin
for the even-orbitals. polarization sensitively depends on the linear polariza-
Apparently, the significant k dependence of the or- tion [33]. This can be seen in the SARPES spectra par-
||
bital symmetry is unique for the outer spz of BiAg2. ticularly for kx=0.12 ˚A−1 in BiAg2. The spectral weight
ThisresultindicatesthepresenceoftheinterbandSOhy- of the spin-up is considerably larger than the spin-down
bridization that allows to mix the even- and odd-orbital for (cid:15) and achieves nearly +80 % spin-polarization. For
p
components in the surface wavefunction of the outer spz (cid:15)s,theintensityrelationturnstotheoppositeandthere-
band. Previously,ithasbeenbelievedthatthehybridiza- sultingspin-polarizationisfoundtobe−70%. Sincethe
tion associates with gap opening [17–19, 25]. However, linearpolarizationissensitivetodifferentorbitalsymme-
in our laser-SARPES experiment, there is no clear gap try, our laser-SARPES umambiguously reveals that the
observed around the crossing point. spin direction strongly depends on the orbital character.
To get further insight into the influence of the inter- Figures 3(a) and (b) represent the calculated spin tex-
band SO coupling, we carried out laser-SARPES mea- ture for the both materials, which is consistent with pre-
surements collaborated with the orbital selection rule of vious theoretical results [14, 19, 20]. In BiAg , the hy-
2
4
(a) (c) (a) (b)
Even Odd p-pol.+s-pol. p-pol.+s-pol.
Spin-texture
BiAg BiCu
V) 0 0 0 2 0 2 Sy
(e-0.3 -0.3 V) 1
F e-0.2 -0.2
E (
-E--00..96BiAg2 --00..96 BiAg2 E-EF-0.4 -0.4 0
0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3
-1 -1 -0.6 -1
kx (Å ) kx (Å ) Low High
-0.6
0 0.1 0.2 0.3 0.1 0.2 0.3
(b) (d) -1 -1
Spin-texture Even Odd kx (Å ) kx (Å )
0.8 0.8
V) 0.4 0.4 FIG. 4: (Color online) (a) and (b) Experimentally obtained
e
( orbital-integrated spin-texture of the Rashba surface states
F 0 0
-E-0.4 -0.4 in BiAg2 and BiCu2, which is in good agreement with that
E theoreticallypredictedasshowninFigures3(a)and(b). The
BiCu BiCu
-0.8 2 -0.8 2 color arrangement indicates the total photoelecton intensity
0 0.2 0.4 0 0.2 0.4 0 0.2 0.4 and spin-polarization [32].
-1 -1
kx (Å ) kx (Å )
FIG. 3: (Color online) (Color online) (a) and (b) Calculated
InBiAg [Figs.3(a)and(c)],thespin-polarizationcou-
totalspintextureforBiAg andBiCu surfacealloys,respec- 2
2 2
tively. (c) and (d) The coupled spin textures decomposed pledtotheeven-orbitalcomponentpredominatesthespz
into (left) even- and (right) odd-orbital characters. The size state around the Γ¯ point, and finally the opposite spin
of the circles is proportional to the total spin polarizations. coupledtotheodd-orbitalcomponentbecomesdominant
Theredandbluecolorindicatespin-upandspin-downquan- at higher k . This indicates that the weight of the even-
||
tized along y, respectively.
and odd-orbital components in the surface wavefunction
play a significant role in the total spin texture through
the SO entanglement [Fig. 3(a)]. Our experiment indeed
bridization of the sp and p bands is found in a gap
z xy
demonstratesthesignificantk -dependenceoftheorbital
opening at the band crossing where the spin polariza- ||
wavefunction [Fig. 1(d)], which shows a good agreement
tion changes its sign [Fig. 3(a)]. Due to the hybridiza-
with DFT calculation [42]. Therefore, the hybridization
tion, one may not assign whether the two branches at
through the interband SO coupling modifies the orbital
thegappedpointoriginatefromeitherthesp orthep
z xy
component and induces the SO entanglement in the sp ,
derived states. Nevertheless let us refer the sp and p z
z xy
which considerably deviates the spin texture from the
bands,sinceourexperimentalresultshowsthehybridiza-
conventional Rashba model.
tion avoids the gap opening.
Apparently, the mapping of the spin in our experi-
The complex spin textures are decomposed into the
ment [Fig. 2(e)-(h)] does not reflect the predicted spin
even and odd orbital contributions in Figs. 3(c) and (d).
texture [Figs. 3(a) and (b)]. This is because photomie-
TheseresultsclearlyshowtheSOentanglementinwhich
sionmeasurementbyusinglinearlypolarizedlightinour
the different orbital components are coupled with op-
experimental set-up selects the specific orbital symme-
posite spin. The calculated SO entangled texture re-
try [8]. Indeed, the orbital-dependent spin texture in
produces our experimental results for the spin mapping
theory [Figs. 3(c) and (d)] shows good agreement with
of the surface wavefunction (see Fig. 2). By general
our laser-SARPES results for (cid:15) and (cid:15) .
group-theoretical analysis for the mirror symmetry [6], p s
We now show that the total spin information can be
the wavefucntion under SO coupling is generally repre-
tracedbackonlybyusingacombinationofthespinmap-
sented as:
ping with (cid:15) and (cid:15) lasers. Owing to selective detection
p s
|Ψ >=|even,↑(↓)>+|odd,↓(↑)>, (1) of the pure orbital-symmetry in our experimental set-up
±i
of Fig. 1(b), the orbital-dependence in the spin polar-
wherethespinors|↑>,|↓>arequantizedalongy,which
ization is eliminated by integrations of spin-polarization
is perpendicular to the mirror plane, and the index ±i
maps (P ) in Figs. 4(a) and (b) as follows:
total
is representation for the mirror symmetry. This explains
not only that the even- and odd-parity orbitals couple (I +I )−(I +I )
P = ↑,p ↑,s ↓,p ↓,s , (2)
with opposite spins but also that the SO entanglement total (I +I )+(I +I )
↑,p ↑,s ↓,p ↓,s
is a general consequence of the SO coupling. Indeed
the similar SO-coupled states have been recently con- whereI (I )andI (I )indicatesthespin-upand
↑,p ↑,s ↓,p ↓,s
firmed in surface states of topological insulators [34–39] spin-down intensities obtained by (cid:15) ((cid:15) ). The mapping
p s
and Rashba states in BiTeI [40, 41]. of P clearly demonstrates the complex spin texture
total
5
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z
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Aid for Scientific Research (B) through Projects No. 156805 (2010).
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26287061 and for Young Scientists (B) through Projects
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