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arXiv:1401.6515

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Spin"textures"in"momentum"space• Structural"inversion"asymmetry"(SIA)"spinIorbit"eﬀect– Rashba"eﬀect[1],"Dresselhaus"eﬀect[2]"""in"polar"semiconductor"and"its"heterostructure"– Noncentrosymmetric"superconductors[3]""– CurrentIdriven"spin"transfer"torque[4]"– Surface"states,"Topological"insulators""[1]E."I."Rashba:"Sov."Phys."Solid"State"2"(1960)"1109."[2]G."Dresselhaus:"Phys."Rev."100"(1955)"580."[3]P."A."Frigeri,"D."F."D."Agterberg,"A."A."Koga,"and"M."M."Sigrist:"Phys."Rev."Le\."92"(2004)"097001."[4]I."M."Miron,"G."Gaudin,"S."Auﬀret,"B."Rodmacq,"A."Schuhl,"S."Pizzini,"J."Vogel,"and"P."Gambardella:"""Nature"Materials"9"(2010)"230."

GK
1) ZnO5,"49 7]"2) LaAlO3/SrTiO3De5"3)"Zc SR,"&\6¢*"4)" ¤2A(σxy)¤ .!"{{:¡YM"(arXiv:1402.0535)"5)""B20CoSi"Weyl¢(arXiv:1401.6515)""

ashba$type$(induced$by$electric$ﬁeld)

spin:orbit$interac=on[1]$arise$due$to$the$non:centrosymmetric$poten=al.$
e$to$the$electric$ﬁeld$applied$perpendicular$to$the$two:dimensional$electron$system.$







:v





Introducgon"IRashba"eﬀectI





J



Eﬀec=ve$$
v
:e
:e
:e SOI$of$Rashba$type$(induced$by$electric$ﬁeld)
magne=c$ﬁeld:v   J  Rashba$spin:orbit$interac=on[1]$arise$due$to$the$non:centrosymmetric$poten=al.$That$case$to$the$electric$ﬁeld$applied$perpendicular$to$the$two:dimensional$electron$system.$
The"Rashba"eﬀect[1]"is"one"of"the"spinIorbit"interacgon"(SOI)."
We$can$think$that$electric$ﬁeld$is$changed$to$magne=c$ﬁeld.

e 2  :vH = α k σ − k σ
RSOx yy x
2 2 σ ⋅ ∇V (r) × k4m c∇V (r) = −E=(0,0,−E)
22zv
e :e
Electric$ﬁeld :eJ())The"Rashba"parameter"involves"electric"ﬁeld."
Eﬀec=ve$$:emagne=c$ﬁeld
α =−E"4m c
If$we$assume$that$α$>$0:v
α{¨Rashba"parameter −k σJ  R
α$is$aﬀected$by$the$strength$of$the$electric$ﬁeld.$
H=
αkσ€
RRx yy xy
k{¨component"of"bloch"wave"vector"€ € We$can$think$that$electric$ﬁeld$is$changed$to$magne=c$ﬁeld.
iek
H RSO k=α ( kxσ y − kyσ x )σ{¨spin
i
H =σ⋅ (∇Vx(r) × k )(2 2(z)2alue

SO

Electric$ﬁeld

(k = k + k )2x2y€∇V (r) = −E = (0,0,−E z )If$we$assume$that$α$>$0

E

e$of$spin

m*>0 €)in θ k ,cos θ k ,0unterclockwise$EFEnergy$eigenvalue)θ k ,−cos θ k ,0
2k 2
Clockwise$
E=± αk2m*(yy4m 2c 2m*<0k
k = k x2 + k y2)

Expected$value$of$spin

()ϕk+ σ ϕk+ € = −sin θ k ,cos θ k ,0Counterclockwise$y

e 2 2E4m 2c 2 zkxkx
α$is$aﬀected$by$the$strength$of$the$electric$ﬁeld.$

α =−

€€
Conduc=on$bandValence$bandkyx
Direc=on$of$spin$is$reversed$to$reverse$the$direc=on$of$the$electric$ﬁeld.$

Electric$ﬁeld

E

HR =kx[1]$E.I.$Rashba,$Sov.$Phys.$:$Solid$State$2$(1960)$1109.
m*>0

EF

E

ky

eE · (kkx)¡¤~¢"{{{{¢]0Conduc=on$bandValence$band

To"invesggate"the"Rashba"eﬀect"is"important"for"understanding"the"interface"
Direc=on$of$spin$is$reversed$to$reverse$the$direc=on$
m*<0
ϕ σ ϕ = (sin θ ,−cos θ ,0)of$the$electric$ﬁeld.$
states."k
Clockwise$
What’s"problem?"[1]$E.I.$Rashba,$Sov.$Phys.$:$Solid$State$2$(1960)$1109.
−k−
kk

k

[1]"E."Rashba,"Sov."Phys."Solid."State.,"2,1109,"(1960)

Submitted to Applied Physics Express

Tunable"Rashba"eﬀect"on"strained"ZnO":"FirstIprinciple"densityIfuncgonal"study"""Applied'Physics'Express,'in'press.

Fig. 1. Crystal structure (a) and the optimized structural parameter (c/a, u) as a function of biaxialstrain (b).

α ="2.19"meV"̊A
R

α ="1.15"meV"̊A
R

α ="4.41"meV"̊A
R"
MgZnO/ZnO""
interface"(α is"0.70"meV"̊A)"
R"
Fig. 2.

3 Band structures of the lowest conduction band and spin textures. The results for TBS(!xx = 4%) (a), no strain (!xx = 0%)(b), and CBS (!xx = −4%) (c) are shown. Spin textures arecalculated for the energy which is 3.5 meV above the conduction band minimum.

Y."Kozuka,"et"al.:"Phys."Rev."B"87"(2013)"205411.

Fig. 3.

Fig. 3.

Fig. 4.

Schematic view of the strain and the Rashba parameter.

Schematic view of the strain and the Rashba parameter.

Electric polarization difference ∆P under the biaxial strain. Calculations are carried out by

using the Berry phase (BP) method and point charge model (PCM).

FirstIprinciples"calculagon"of"""perovskiteItype"superlapce"(LaAlO3)2/(SrTiO3)2"Introducgon"
Superlapce"can"realize"novel"physical"properges"
not"found"in"the"bulk"system.""LaAlO
3/SrTiO3""
Bulk"LaAlO3"and"SrTiO3"are"insulators.""Fig.1"Schemagc"of"LaAlO3/SrTiO3"[1].
LaAlO/SrTiO
33""a"high"mobility"twoIdimensional"electron"gas"(2DEG)[1]""The"peculiar"phenomenon"originates"from"the"interface."
What"phenomenon"is"happening"at"the"interface?

Fig.2"The"temperature"dependence"of"the"Hall"mobility"μH(T)"[1].

[1]"A.Ohtomo"and"Y.Hwang,"Nature"427,"423"(2004).

Computagonal"Model
••••We"assumed"SrTiO3"substrate."
Lapce"constant"is"a=b=3.905"s,c="14.944s.""(SrTiO3:"a=b=c=3.905s[3],"LaAlO3:"a=b=c=3.78s[4])"The"volume"is"equivalent"to"bulk.

a0

LaAlO3

a0+

14.944s

L0

SrTiO3L0
3.905sa0!>!a’a0!<!L0
LaAlO3

SrTiO3L0

[3]"J."H.Haeni"et"al,"nature."430,"7001,"758(2004).""[4]"Karolina"Janicka"et"al,"J."Appl."Phys."103,"07B508"(2008)."

Computagonal"Methods
FirstIprinciple"calculagon"using"density"funcgonal"theory"(code":"OpenMX)"
Pseudopotengal":"normIconserving"pseudopotengal"
Exchange"correlagon"potengal":"GGA"
Energy"cutoﬀ":"200Ry"
KIpoint":"10×10×1"
Radial"wave"funcgon"and"Basis"funcgon"
La":"6.0Is2p2d1f1,"Al":"7.0Is3p3d3,"Sr":"10.0Is2p2d1,"Ti":"7.0Is3p3d1,"O":"5.0Is3p3"
Analysis of the spin distribution [5](R )iR ⋅k $!
!ψ α ( k, µ, r) $ # ∑ cµiα φi ( r) e &nin&=#
ψ ( k, µ, r) = ## β&&(Rn )iR n ⋅kψk,µ,r) % #∑ cµiβ φi (r) e &" (" i%The spin direction ateach k point.

! P (k, µ ) P (k, µ ) $αααβφi&P(k, µ ) = #αβ# Pβα (k, µ ) Pββ (k, µ ) &VXC"% µ{¨band index  φ{¨basis function
ωαβi(R)∗(R)iR⋅kα,βαβnnn(*{¨overlapmatrix
VSXC{¨spin index
Pαβ (k, µ ) = ∑)cµiα cµ jβ Sij + eij
cµiαijωαβ{¨expansion coefficientR{¨periodicnvector Sijz
θP

ϕPx

Py

[5]"H."Kotaka,"F."Ishii,"and"M."Saito,"Jpn."J."Appl."Phys.,"52,"035204(2013)."Band"Structure"of"(LaAlO3)2/(SrTiO3)2"
-0.15

-0.15

0.05

-0.15

-0.01

xX
-0.153MXy

Energy [eV]0.01
-0.152

1

Energy [eV]-0.151

0.03

Energy [eV]

Energy [eV]

Energy [eV]

2

0-1
-2

-0.151-0.151-0.152-0.152-0.153xX K-0.153

K
K

X

M

X

K

M

My

K

(LaAlO3)2/(SrTiO3)2"shows"a"metallic"property."
yThis"result"corresponds"to"the"experimentally"observed"
metallic"conducgvity"in""LaAlO3"/(SrTiO3)n"(n">=2)[7]"We"can"see"the"band"splipng"around"Γ"point"and"M"point."

We"focus"on"the"band"splipng."
[7]"C.W.Bark"et"al.,"PNAS,"108,"4720"(2011).

Atom"decomposigons"of"states"at"Fermi"level
Around"Γ"point

Around"M"point

1"
0.8"
0.6"
0.4"

hkig
oprmqnklm

0.2"
0"

0.7"
0.6"
0.5"
0.4"
0.3"
0.2"
0.1"
0"

jqig
oprmqnklm

SrO"TiO2"SrO"TiO2"LaO"AlO2"LaO"AlO2"

SrO" TiO2"SrO" TiO2"LaO"AlO2"LaO"AlO2"Ti"dIorbitals"next"to"LaO"interface.

O"pIorbitals"in"AlO2"layer"next"to""SrO"interface.

We"conﬁrmed"that"metallic"state"originates"from"interface."
Electron!doping!to"TiO2"layer"and"hole!doping!to"AlO2"layer"through"the"interface."
What"are"spin"textures?"Spin"textures
The"spin"texture"of"hole"pocket"bands"in"the"ﬁrst"Brillouin"zone."• The"spin"component"of"the"z"direcgon"is"almost"0."• (b)"shows"the"states"lower"energy"than"(a)."Energyz ⊗ z ⊗0.5(a)'M 0.4

'0.2
'0.1
'X 0
ky'
-0.1
'-0.2
'-0.3
-0.4
'-0.5
'-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5''''kx

2
1.5

0.3

10.5
0-0.5

clockwise

0.5(b)

'M 0.4

'
0.2
'
0.1
'
X 0ky'
-0.1
'-0.2
'
-0.3
-0.4
'
-0.5
'
-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5''''kx

2
1.5

0.3

k
We"conﬁrmed"Rashba"eﬀect"around"the"t"point.
What"is"direcgon"of"the"electric"ﬁeld?

angIclockwise

1
0.5
0-0.5





:v







JEﬀec=ve$$
v
:e
:e
:e SOI$of$Rashba$type$(induced$by$electric$ﬁeld)
magne=c$ﬁeld

Electric$ﬁeld:v   J  Rashba$spin:orbit$interac=on[1]$arise$due$to$the$non:centrosymmetric$poten=al.$That$case$to$the$electric$ﬁeld$applied$perpendicular$to$the$two:dimensional$electron$syste
We$can$think$that$electric$ﬁeld$is$changed$to$magne=c$ﬁeld.

()()
Rashba"eﬀect"and"Electric"ﬁeld
e 2  :vH = α k σ − k σ
H SO =RSOx yy x2 2 σ ⋅ ∇V (r) × k
4m c∇V (r) = −E= (0,0,−E z )
2 2v
e :e

Electric$ﬁeld :e

If$we$assume$that$α$>$0  €€

J:eEﬀec=ve$$magne=c$ﬁeld
E
4m 2c 2 z:v   J
α$is$aﬀected$by$the$strength$of$the$electric$ﬁeld.$
α =−y
The""band"of"angIclockwise"spin"texture"is"lower"energy"than"that"of"clockwise."€ € We$can$think$that$electric$ﬁeld$is$changed$to$magne=c$ﬁeld.
e 2x
H SO =2 2 σ ⋅ ∇V (r) × k
4m c(Energy$eigenvalue

H R = α R ( kxσ yE=)

2k 2k = k x2 + k y2* ± αk2m yx−k σ (Electric$ﬁeld)e 2 2E4m 2c 2 zkxkx
α$is$aﬀected$by$the$strength$of$the$electric$ﬁzm*>0 €€yy

If$we$assume$that$α$>$0

E

Expected$value$of$spin) ∇V (r) =k−E = (0,0,−E ) H RSO k=α ( kxσ y − kyσ x )α =−y()€ €
α{¨Rashba"parameterE€Conduc=on$bandValence$bandCounterclockwise$Rkx
Direc=on$of$spin$is$reversed$to$reverse$the$direc=on$
Energy$eigenvaluem*<0
ϕ σ ϕ = (sin θ ,−cos θ ,0)k{¨component"of"bloch"wave"vector"
of$the$electric$ﬁeld.$
iElectric$ﬁeld
kk
€
k=k+
kClockwise$
E=±αk()
σ{¨spin2mE
ik
[1]$E.I.$Rashba,$Sov.$Phys.$:$Solid$State$2$(1960)$1109.
+k

ϕ σϕ+k€= −sin θ k ,cos θ k ,0Fy−k−kk2

k22x*2yExpected$value$of$spin€ky()

ϕk+ σ ϕk+ € = −sin θ k ,cos θ k ,0€Counterclockwise$

m*>0

EFxk

Conduc=on$bandValence$band
Direc=on$of$spin$is$reversed$to$reverse$the$dirof$the$electric$ﬁeld.$

m*<0
The"electric"ﬁeld"is"directed"LaAlO
ϕ σ ϕ 3"to"SrTiO= (sin θ ,−cos θ,0)
3"at"the"LaO"interface.""k€
Clockwise$This"is"consistent"with"simple"understanding"by"electric"ﬁeld"at"the"interface"[1]$E.I.$Rashba,$Sov.$Phys.$:$Solid$State$2$(1960)$1
induced"by"charge"transfer"as"(LaO)+wand"(TiO2)Iw."−k€

Rashba"coeﬃcient?

−kkk

Rashba"coeﬃcient
We"esgmate"the"Rashba"coeﬃcient"by"using"freeIelectron"model[8].
EnergyΔk0
αR =ER

4 ERΔk0k
In"the"bands"around"the"t"point",""uk0"is"0.0006s=1!and"ER""is"0.01meV.!
"Then"our"evaluated"αR"is"about"10meV£s."This"αR"is"good"agreement"with"experimental"value"12meV£sg[9].
[8]"K."Ishizaka"et"al.,"nature"materials"10,"521"(2011)."[9]"A.D.Caviglia"et"al.,"Phys.Rev.Le\."104,"126803"(2010).Summary• We"performed"ﬁrstIprinciples"calculagon"of"{{g(LaAlO3)2/(SrTiO3)2."• Calculated"band"structures"show"interface"induced"metallic"states"due"to"carrier"doping"through"interface.""• We"found"that"electron"doping"at"TiO2"layer"at"LaO"interface"and"hole"doping"AlO2"layer"at"SrO"interface.""• The"interface"states"shows"spin"splipng"and"voltexIlike"spin"
textures"ygRashba"eﬀect.""• Calculated"Rashba"coeﬃcient"vR,"10meV£s"is"in"good"
agreement"with"experimental"value.""• The"clockwise/angIclockwise"of"spinIvortex"is"consistent"
with"Rashba"Hamiltonian"when"we"use"the"sign"of""vR"as"same"as"electricIﬁeld"induced"in"the"interface.