Table Of ContentEffects of surface-bulk hybridization in 3D topological ‘metals’
Yi-Ting Hsu,1 Mark H. Fischer,1,2 Taylor L. Hughes,3 Kyungwha Park,4 and Eun-Ah Kim1
1Department of Physics, Cornell University, Ithaca, New York 14853, USA
2Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
3Department of Physics, University of Illinois, 1110 West Green St, Urbana IL 61801, USA
4Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA
(Dated: July 21, 2014)
Identifying the effects of surface-bulk coupling is a key challenge in exploiting the topological
nature of the surface states in many available three-dimensional topological ‘metals’. Here we
combine an effective-model calculation and an ab-initio slab calculation to study the effects of the
4 lowest order surface-bulk interaction: hybridization. In the effective-model study, we discretize an
1 establishedlow-energyeffectivefour-bandmodelandintroducehybridizationbetweensurfacebands
0 andbulkbandsinthespiritoftheFanomodel. Wefindthathybridizationenhancestheenergygap
2 between bulk and Dirac surface states and preserves the latter’s spin texture qualitatively albeit
with a reduced spin-polarization magnitude. Our ab-initio study finds the energy gap between the
l
u bulkandthesurfacestatestogrowuponanincreaseintheslabthickness,verymuchinqualitative
J agreementwiththeeffectivemodelstudy. Comparingtheresultsofourtwoapproaches,wededuce
7 that the experimentally observed low magnitude of the spin polarization can be attributed to a
1 hybridization-type surface-bulk interaction. We discuss evidence for such hybridization in existing
ARPES data.
]
l
l
a I. INTRODUCTION in the metallic regime15–17.
h
-
s
Many discrepancies between experimental measure-
e
m ments and theoretical predictions of ideal topological in- OurstartingpointistheobservationmadebyBergman
sulators (TI’s) are attributed to the fact that the chem- and Refael18 that the lowest order electron-electron in-
.
t ical potential lies in the conduction band and the bulk teraction term between the surface state and the bulk
a
m band interferes with measurements1–4. That is, many states can be viewed as a hybridization term in the
available TI materials are actually metallic. Recent de- Fano model19. The key effect of hybridization in this
-
d velopments in thin-film experiments4–8 further call for low-energy effective theory is to spectroscopically sepa-
n studiesofsurface-bulkelectroninteractioninfilms. How- rate surface states localized on the surface from the ex-
o ever, explicit first-principles calculations on TI films are tendedmetallicbands. Buildingontheprinciplesunder-
c limited to very thin slabs with thickness less than 10 lying this low-energy effective theory, we study the hy-
[
nm due to the computational cost. Therefore, there is a bridization effects with a microscopic model of 3D time-
2 need for a simple microscopic model which incorporates reversalinvariantstrongtopologicalinsulatorstoaddress
v surface-bulk interactions that can be used to study how the thickness dependence of physical quantities and con-
8 physical properties depend on the film thickness. nect the results to ab-initio slab calculations. Specifi-
7
cally, we study TI slabs with finite thickness in the pres-
3 Oneimportantquestionsuchamodelshouldaddressis
enceofsurface-bulkinteractionfromtwocomplementary
3 the effect of surface-bulk interaction on the spin-texture.
. The surface spin-texture is a key physical characteris- perspectives: a simple microscopic model including the
1
lowestordersurface-bulkinteractionandanab-initiocal-
0 tic of topological surface states in ideal TIs. While low-
culation of a few-QL Bi Se . We focus on how the spec-
4 energy effective theories guided by symmetries predict 2 3
troscopicpropertiesandthespintextureevolveasafunc-
1 perfect spin-momentum locking for topological surface
: stateswithinthebulkgap, spin-andangle-resolvedpho- tion of film thickness.
v
i toemission spectroscopfy (SARPES) data show lower in-
X plane spin polarization and total spin magnitude9–11.
r On the other hand, an ab-initio calculation on a few- The paper is structured as follows. In section II we
a
quintuple-layer (QL) slab11 found both the spin polar- constructalatticemodelforaslabwithsurface-bulkhy-
izationandthetotalspinmagnitudetobemuchsmaller. bridization (S-B hybridization) and study the spectro-
Although reduction in spin polarization and total spin scopic properties of the model as well as the effects of
magnitude are to be anticipated at large surface Fermi- the S-B hybridization on the spin texture. In section III
momenta where hexagonal warping manifests12–14, lit- we present an ab-initio study on 4,5,6-QL Bi Se using
2 3
tle is understood about the reduction observed at small density functional theory (DFT) and discuss the insight
Fermi-momenta and how the surface-bulk interaction af- the simple hybridization model offers in understanding
fects the spin texture. However, such understanding the ab-initio results. We then conclude in section IV
is crucial for pursuing technical applications of spin- with discussions of implications of our results and open
momentum locking in thin films of topological insulators questions.
2
II. LATTICE MODEL FOR A SLAB WITH S-B (↑,↓)and(P ,P )spaces,respectively. a ,a anda are
1 2 x y z
HYBRIDIZATION the lattice constants in x, y, z directions respectively.
We use the parameters for Bi Se obtained by fitting
2 3
A. The Model the continuum model to ab-initio calculations21: m =
0.28 eV, A = 2.2 eV˚A, A = 4.1 eV˚A, B = 10 eV˚A2,
1 2 1
B = 56.6 eV˚A2, C = −0.0068 eV, D = 1.3 eV˚A2 and
In order to introduce surface-bulk hybridization as a 2 1
perturbation in the spirit of the Fano model18 and study D2 =19.6 eV˚A2.
We model a slab of Bi Se by imposing open bound-
its effects on a slab with finite thickness, we first need 2 3
a lattice model for a slab. For this, we discretize20 the ary conditions at the top and the bottom surfaces which
effective continuum model by Zhang et al.21, which is a breakthetranslationalsymmetryalongthez-axis. Since
k is no longer a good quantum number, while k still
four band k·p Hamiltonian guided by symmetries and z (cid:107)
first-principle-calculationresults. Wethenmakethesys- is, we substitute ck(cid:107),kz = √1N (cid:80)jeikzjazck(cid:107),j in Eq. (1)
temsizefinitealongtheverticalaxis,i.e.,thefilmgrowth and label the four-spinor operators by the in-plane mo-
direction. mentum k =(k ,k ) and the index of layers j stacking
(cid:107) x y
Thelow-energyeffectivefour-bandmodelinRef.21de- in the z direction. Now the Hamiltonian for a slab with
scribes a strong 3D TI with rhombohedral crystal struc- N layers is
ture such as Bi Se . In this effective model each QL is
2 3 (cid:88)
treated as a layer since the inter-QL coupling is weak, H (N)= H (k ,N), (5)
0 0 (cid:107)
and the four lowest-lying spin-orbital bands come from k(cid:107)
the mixing of the two P atomic orbitals from Bi and Se
z
where
referred to as P and P and the two spins ↑, ↓. We take
1 2
this effective model written in terms of 4×4 Γ-matrices
N
(cid:88)
anddiscretizeitfollowingthediscretizationschemeused H (k ,N)= Mc† c +Tc† c +T†c† c
for 2D TIs in Ref. 22 such that the lattice Hamiltonian 0 (cid:107) k(cid:107),j k(cid:107),j k(cid:107),j+1 k(cid:107),j k(cid:107),j k(cid:107),j+1
j=1
reduces to the low-energy effective k·p Hamiltonian in (6)
Ref.21inthelimit|k|→0. Theresultinglatticemodelin and the 4×4 matrices T and M are defined as
the rotated spin-orbital basis {|P ,↑(cid:105), −i|P ,↑(cid:105), |P ,↓(cid:105),
1 2 1
D B iA
i|P2,↓(cid:105)} is T≡−a21I4×4+ a21Γ5− 2a1Γ1, (7)
z z z
(cid:88)
H = h (k ,k )c† c , (1)
0 0 (cid:107) z k(cid:107),kz k(cid:107),kz and
k(cid:107),kz
A A
M≡(cid:15) (k )I + 2 sin(k a )Γ3+ 2 sin(k a )Γ4+M (k )Γ5
where c† is an operator creating a four spinor in the 2 (cid:107) 4×4 a x x a y y 2 (cid:107)
rotatedks(cid:107)p,kinz-orbital basis with an in-plane momentum x y (8)
k = (k ,k ) and perpendicular momentum k , and the where
(cid:107) x y z
lattice Hamiltonian23 is 2D 2D 2D
(cid:15) (k )≡C+ 1+ 2[1−cos(k a )]+ 2[1−cos(k a )]
h (k ,k )=(cid:15) (k ,k )I + A1 sin(k a )Γ1 (2) 2 (cid:107) a2z a2x x x a2y y y
0 (cid:107) z 1 (cid:107) z 4×4 a z z (9)
z
and
A A
+ 2 sin(k a )Γ3+ 2 sin(k a )Γ4+M (k ,k )Γ5,
ax x x ay y y 1 (cid:107) z M (k )≡m−2B1−2B2[1−cos(k a )]−2B2[1−cos(k a )].
2 (cid:107) a2 a2 x x a2 y y
z x y
with
(10)
2D 2D For the sake of simplicity, the results presented in
(cid:15) (k ,k )≡C+ 1[1−cos(k a )]+ 2[1−cos(k a )]
1 (cid:107) z a2 z z a2 x x the remainder of this section are calculated with
z x a =a =a =1˚A.
2D x y z
+ 2[1−cos(k a )], (3)
a2 y y
y We can diagonalize H (k ,N) as
0 (cid:107)
4N−4
M1(k(cid:107),kz)≡m− 2aB21[1−cos(kzaz)]− 2aB22[1−cos(kxax)] H0(k(cid:107),N)= (cid:88) EB0,α(k(cid:107))b0α†,k(cid:107)b0α,k(cid:107)
z x α=1
(11)
− 2B2[1−cos(k a )]. (4) (cid:88)4
a2 y y + E0 (k )d0† d0 ,
y D,β (cid:107) β,k(cid:107) β,k(cid:107)
β=1
The Gamma matrices are defined as Γ1 =σz⊗τx, Γ2 =
−I2×2 ⊗ τy, Γ3 = σx ⊗ τx, Γ4 = σy ⊗ τx, and Γ5 = where b0α,k(cid:107) and d0β,k(cid:107) are four-spinor annihilation op-
I ⊗τz, where σj and τj are Pauli matrices acting on erators for bulk and surface states (henceforth referred
2×2
3
to as the “Dirac” states) respectively in the absence of After diagonalizing the full Hamiltonian in the tight-
hybridization, E0 and E0 are their corresponding binding spin-orbital bases, we can write
B,α D,β
eigenenergies. Here, α and β label the unhybridized
bulkandDiracstatesrespectively. Allenergyeigenstates 4N−4
(cid:88)
are two-fold degenerate as required by inversion (P) and H(k(cid:107))= EB,α(k(cid:107))b†α,k(cid:107)bα,k(cid:107)
time-reversal(T)symmetries. Foreachin-planemomen- α=1
(15)
tumk ,fourenergyeigenstateswiththeirenergiesclosest 4
(cid:107) (cid:88)
to the Dirac point are labeled to be valence (β = 1,2) + ED,β(k(cid:107))d†β,k(cid:107)dβ,k(cid:107),
and conduction (β = 3,4) Dirac states. α label the re- β=1
maining 4N −4 bulk states. A natural choice for the
labeling is to let α = 1,··· ,2N −2 denote valence bulk where bα,k(cid:107), dβ,k(cid:107), EB,α and ED,β are defined similarly
statesandletα=2N−1,··· ,4N−4denoteconduction to the corresponding symbols with a superscript 0 in
bulk states with the eigenenergies increasing monotoni- Eq. (11) but in the presence of hybridization. EB,α and
cally with α. As the model is derived from a low-energy ED,β are again two-fold degenerate as H(k(cid:107)) preserves
effective model near the Dirac point, it will break down parity and time-reversal symmetries by design. α and
at large energies. However, we expect qualitatively cor- β label the bulk and Dirac states for H(k(cid:107)), where the
rectresultswhenitcomestotrendsofphysicalproperties termsbulkandDiracstatesaredefinedinthesamefash-
over the hybridization strength and film thickness which ion as for H0(k(cid:107),N) in the absence of hybridization.
onlyrequireknowledgeofthelow-energyphysicsnearthe
insulating gap.
Now we introduce the S-B hybridization term that is
B. Topological Metal Regime
allowed by symmetries in the spirit of Fano model19.
The Fano model is a generic model describing the mix-
We begin our numerical study with no hybridization.
ing between extended states c with energy (cid:15) and a
k k Intheabsenceofhybridization,dependingonthechemi-
localized state b with energy (cid:15) through Hamiltonian
H =(cid:15)b†b+(cid:80) [(cid:15) c†c +A (c†b+b†c )],whereA repre- cal potential µ, we now define three regimes: topological
F k k k k k k k k insulator(TI), metal(M), and topological metal(TM)(see
sentsscatteringstrength. BergmanandRefael18 pointed
Fig.1(a)). Thefamiliartopologicalinsulator(TI)regime
out that H can be used to describe the lowest order in-
F is where the chemical potential lies within the bulk gap
teraction between a helical surface state and a metallic
and the system is actually a bulk band insulator, i.e.,
bulkband,i.e. hybridization. Theystudiedeffectsofhy-
E ≤ µ ≤ E with E being the bottom of the conduc-
bridization in a field theoretic approach. Here we use a v c c
tion band and E the top of the valence band. Within
symmetry-preserving form of the surface-bulk hybridiza- v
theTIregime,theDiracstatesfeatureRashba-typespin-
tion term in the spirit of H for the microscopic model
F momentum locking and a spatial profile localized on the
ofBi Se showninEq.(11)tostudytheeffectsofmixing
2 3 surfaces. Of our particular interest is the distinction we
between Dirac states and bulk states.
will draw between M and TM regimes based on whether
For simplicity we consider the case where the hy-
the Dirac states retain the spin-momentum locking and
bridization strength preserves in-plane momenta k and
(cid:107) the surface localization when being away from the TI
is independent of energy and k , i.e.
(cid:107) regime.
h(cid:48)(k )=(cid:88)g b0† d0 +H.c.. (12) In order to examine the above two properties of
(cid:107) α,k(cid:107) β,k(cid:107) Dirac states, we define |ψ (cid:105) ≡ d† |0(cid:105) (d0† |0(cid:105)) and
α,β D,k(cid:107) 3,k(cid:107) 3,k(cid:107)
We then impose T and P symmetries on the full hy- |ψ˜D,k(cid:107)(cid:105) ≡ d†4,k(cid:107)|0(cid:105) (d04†,k(cid:107)|0(cid:105)) in the presence(absence) of
bridization perturbation H(cid:48)(k ) by constructing H(cid:48)(k ) hybridization to represent the pair of degenerate Dirac
(cid:107) (cid:107)
through states above the Dirac point. Now the spatial pro-
file of the conduction Dirac states is |Ψ (z)|2 ≡
H(cid:48)(k )=h(cid:48)(k )+Ph(cid:48)(k )P†+Th(cid:48)(k )T† D,k(cid:107)
(cid:107) (cid:107) (cid:107) (cid:107) |ψ (z)|2 +|ψ˜ (z)|2 which is a function of z mea-
+PTh(cid:48)(k )(PT)†, (13) D,k(cid:107) D,k(cid:107)
(cid:107) suredfromthebottomoftheslabalongthefinitedimen-
where the representations for T and P symmetry opera- sion of the slab. This quantity will show whether the
torswiththespatialinversioncenteratthemiddlepoint Dirac states are localized on the surfaces or not. Let us
of the slab in the current 4N tight-binding spin-orbital identify a particular k(cid:107) of interest for a given value of
basis are T = K iσy ⊗ I ⊗ I with k ↔ −k , chemical potential as the in-plane momentum at which
2×2 N×N (cid:107) (cid:107)
and P = I ⊗ τz with z : [0,N/2] ↔ [N/2,N] and the chemical potential µ intersects the Dirac branch; we
2×2
k(cid:107) ↔ −k(cid:107), respectively. Here, K is the usual complex denote such in-plane momentum by k(cid:107),µ. To illustrate
conjugation operator. Finally, the full Hamiltonian in- thefeaturesdefiningthethreeregimes,wewillnowshow
cluding hybridization at a given in-plane momentum k thespatialprofilesandthespinpolarizationsoftheDirac
(cid:107)
reads states in the three regimes in the absence of hybridiza-
tion. In the next section we will add the hybridization
H(k )=H (k ,N)+H(cid:48)(k ). (14)
(cid:107) 0 (cid:107) (cid:107) and examine its effects.
4
(a) (b) define the system to behave as a topological metal(TM)
when the chemical potential lies within this window, i.e.
E <µ<E , represented by µ .
c M TM
AdetectablecharacteristicofTIandTMregimesisthe
spin-momentum locking. One measure to quantify spin-
momentumlockingatthesurfaceisthroughtheso-called
“spin polarization” which is the expectation value of the
spincomponentperpendiculartothein-planemomentum
of a Dirac state, i.e.,
(c) (d) (cid:104)S (cid:105)(k )≡(cid:104)ψ |S |ψ (cid:105) (16)
nˆ (cid:107) D,k(cid:107) nˆ D,k(cid:107)
with nˆ·k = 0. Here the nˆ component of the quantum
(cid:107)
spin operator is defined as S ≡ (cid:126)(cid:126)σ ·nˆ ⊗I ⊗I ,
nˆ 2 2x2 NxN
where(cid:126)σ =(σ ,σ ,σ ) are Pauli matrices acting on spin,
x y z
I actsontheorbitaldegreeoffreedom,andI acts
2x2 NxN
onthelayerindex. (cid:104)S (cid:105)(k xˆ)isevaluatedatk xˆ=k xˆ
y x x x,µ
and shown for µ = µ ,µ ,µ in Fig. 2(a). We see
TI TM M
here that, in the absence of hybridization(g=0), the spin
polarization stays maximal in the TI and TM regimes
(e) (f)
whilerapidlydroppinguponenteringtheMregime. An-
other quantity of experimental interest is the total spin
magnitudeassociatedwiththeDiracstateswithin-plane
momentum k defined in terms of spin polarization as
(cid:107)
(cid:115)
(cid:88) (cid:2) (cid:3)2
S(k )≡ (cid:104)S (cid:105)(k ) . (17)
(cid:107) i (cid:107)
i=x,y,z
Fig.2(a)and2(b)showthatthespin-momentumlocking
quantified using these measures clearly distinguishes the
FIG. 1. (a) Spectra of the model on a 300-layer-thick slab. TM regime from the ordinary metal regime (M) in the
The three chemical potentials µ , µ and µ are taken absence of hybridization.
TI TM M
as representative points for the three regimes TI(µ≤E ), M
c
(µ (cid:38) E ), and TM(E ≤ µ (cid:46) E ) as defined in the text,
M c M (a) (b)
respectively. (b)-(d)The corresponding unhybridized(dashed,
blue)andhybridized(solid,green)spatialprofilesofthepairof
degenerate conduction Dirac states |Ψ (z)|2 at k =k
D,k(cid:107) (cid:107) (cid:107),µ
with µ = µ , µ , and µ , respectively. (e) Effect of the
TI TM M
hybridization on the spectra. (f)ARPES data on Bi Se 3.
2 3
Inspecting |Ψ (z)|2 at k = k at the represen-
D,k(cid:107) (cid:107) (cid:107),µ
tative values of chemical potential for the three regimes
FIG. 2. The effect of hybridization on the degree of spin-
µ ,µ ,µ shown in Fig. 1, we find that the spatial
TI M TM momentumlockingindifferentregimes. Thespinexpectation
profile of the Dirac states |Ψ (z)|2 indicates surface
D,k(cid:107) valuesarecalculatedfora300-layer-thickslabusingaconduc-
localized states of the slab in the TI regime as expected tionDiracstate|ψ (cid:105)withk =k atdifferentrepresen-
D,kxxˆ x x,µ
(see Fig. 1(b)). On the other hand, in the M regime, tativechemicalpotentialsµ=µ ,µ andµ . (a)Spinpo-
TI TM M
where the chemical potential is well within the bulk con- larization(cid:104)S (cid:105)(k xˆ). (b)TotalspinmagnitudeS(k xˆ)(defined
y x x
duction band, |Ψ (z)|2 is fully delocalized over the in the text).
D,k(cid:107)
entireslab(seeFig.1(c)). Inthisregime,thesystemcan-
not be distinguished from an ordinary metal. However,
even with µ>E there is an energy window between E
c c
and a crossover energy scale E , where the Dirac states C. Effects of S-B Hybridization
M
are still spatially localized on the surfaces in the sense
that |ΨD,k(cid:107)(z)|2 is peaked on each surface of the slab We now turn to the effects of hybridization. One ef-
and decays away from the surfaces (see Fig. 1(d)). The fectofhybridizationthatismanifestintheexperimental
crossover energy scale E is a threshold energy, where, detection of Dirac surface states in the TM regime is an
M
within the regime µ (cid:38) E (regime M), wavefunctions increase in the bulk-Dirac state energy gap. We quan-
M
for all states at in-plane momentum k delocalize. We tify this energy gap, for a given chemical potential µ,
(cid:107),µ
5
using the energy difference between a Dirac state above (a) (b)
the Dirac point and the energetically closest bulk state
defined by
∆ (µ)≡E(0) (k )−E(0)(k ) (18)
DB B,2N−1 (cid:107),µ D,3 (cid:107),µ
in the presence(absence) of hybridization. Comparing
Fig. 1(a) to (e), we find that the key effect of hybridiza-
tionthatisspectroscopicallydetectableistheincreasein
∆ (µ) in both TM and M regimes compared to the TI
DB
regime. Otherwisethespectraintheabsenceorpresence
FIG. 3. Thickness dependence of the hybridization effects.
of hybridization look similar. Note that most ARPES
(a)Dimensionless measure of bulk-Dirac energy gap r =
data on 3D TIs exhibit a clear energy gap between the ∆ /∆ (definedinthetext)atdifferentslabthicknessfor
DB BB
Dirac branch and the bulk states at a chemical poten- µ = µ . (b)The spin polarization of a conduction Dirac
TM
tial well into the bulk band as shown in Fig. 1(f). This state (cid:104)S (cid:105)(k xˆ) at µ=µ .
y x TM
experimental trend hints at the possibility that a sizable
hybridizationbetweenDiracstatesandthebulkstatesis
commonin3DTImaterials. Inordertodemonstratethe measured at the same in-plane momentum k where
(cid:107),µ
effect of hybridization, we choose a value of g = 5meV ∆ (µ) is calculated. This dimensionless quantity r(µ)
DB
thatissubdominanttoallthehoppingtermsyetsubstan- allowsustocompensateforfinitesizeeffectsthough∆
BB
tial in this paper. However, key effects of hybridization would be hard to measure experimentally for realistic
do not depend qualitatively on the value of g. bulk samples due to the lack of the required energy res-
AnothereffectofhybridizationistobroadentheDirac olution. Fig. 3(a) shows that the hybridization induced
state wavefunctions in the TI and TM regimes. The de- enhancementinthebulk-Diracenergygapbecomesmore
gree of broadening depends on the chemical potential µ, prominent with increasing slab thickness. Comparing
hybridizationstrengthg,andtheslabthicknessN. How- the existing ARPES data on bulk samples3 and on thin
ever, as long as g is the smallest energy scale in the total films5, we find the Dirac branch to be better separated
Hamiltonian as is the case for Figs. 1(b-d), the Dirac fromthebulkstatesinthebulksamplesthaninthethin
states in the TI and TM regimes remain localized on the films, which is consistent with the hybridization effect
surfaces. A tangible consequence of the wavefunction shown in Fig. 3(a). Finally Fig. 3(b) shows that the re-
broadening is the quantitative suppression of the spin- ductioninspin-polarizationmagnitude|(cid:104)S (cid:105)(k xˆ)|isalso
y x
momentumlocking. Asmentionedearlier,intheabsence intensified with increasing thickness. Such an enhance-
of hybridization the Dirac states of TI and TM exhibit ment in the impact of hybridization with the increase
a maximal degree of spin-momentum locking. However, in slab thickness can be explained from the fact that a
hybridization rotates the spin vectors of different atomic thicker slab implies a larger number of bulk states that
orbitals and layers away from the direction perpendicu- mix with a fixed number of Dirac surface states for a
lar to the in-plane momentum. Hence, both measures of given strength of hybridization g.
spin-momentum locking shown in Fig. 2 show quantita-
tive reduction upon hybridization. This is in qualitative
agreement with the low values of spin polarization and
III. DFT CALCULATIONS OF THIN BI SE
totalspinmagnitudefoundinafirst-principlecalculation 2 3
SLABS
of a thin slab in a previous work11 and our DFT results
inthenextsection. Notethatthehybridizationstillpre-
Now we turn to an ab-initio study of thin slabs to
serves the spin-texture winding despite the quantitative
compare with the simple phenomenological model of hy-
reduction in the spin polarization.
bridization we explored in the previous section. The ap-
Finally, we study how the effects of hybridization
proach of the previous section is limited, in the sense
on the two experimentally accessible characteristics of
that it builds on a low-energy effective description of the
the TM regime, namely how the bulk-Dirac energy gap
band structure, and that there is no detailed knowledge
∆ (µ) and the spin polarization (cid:104)S (cid:105)(k ) of a Dirac
DB nˆ (cid:107)
of the hybridization strength g which could in princi-
state, vary with the slab thickness. Since the quantized
ple be k -dependent. On the other hand, the DFT ap-
energyspacingsduetofinitesizeeffectsdecreaseswithin- (cid:107)
proach on slabs, which does not require calculating sur-
creasingslabthickness,weconsideradimensionlessmea-
face and bulk separately as in the calculations of semi-
sure that quantifies the bulk-Dirac energy gap:
infinite systems21, is limited to very thin films of several
r(µ)≡∆ (µ)/∆ (µ), (19) QLs due to computational limits. By combining the two
DB BB
approaches, we extract a more robust understanding of
where ∆ (µ) ≡ E(0) (k )−E(0) (k ) is the the effects of hybridization in the TM regime and impli-
BB B,2N+1 (cid:107),µ B,2N−1 (cid:107),µ
energy spacing in the presence(absence) of hybridization cations on their trends over film thickness.
between the two lowest lying conduction bulk branches We calculate the electronic structure of Bi Se (111)
2 3
6
slabs of 4-6 QLs using the VASP code24,25 with ing reduction in the spin polarization and the total spin
the projector-augmented-wave method26, within the magnitude29.
generalized-gradientapproximation(GGA)27. Spin-orbit Now we discuss the thickness dependence in the bulk-
couplingisincludedself-consistently. Weuseexperimen- Dirac energy gap measure and the spin polarization. We
tallatticeconstants28andanenergycutoffof420eVwith calculate the dimensionless measure of bulk-Dirac en-
a31×31×1k-pointgrid. OurDFTcalculationsarelim- ergy gap r = ∆ /∆ in the TM regime at the k
DB BB (cid:107)
ited up to 6 QLs. For 5-6 QLs, the overlap between top point where the Dirac surface state branch has slightly
and bottom surface states is already very small yielding higher energy than the bottom of the conduction band
an energy gap of the order of meV at Γ. Expectation E , as indicated in Fig. 4(a). We find that the ra-
c
valuesofspincomponents(cid:104)S (cid:105),(cid:104)S (cid:105),(cid:104)S (cid:105)arecalculated tio ∆ (N )/∆ (N ) is close to (N /N )2 at the k
x y z BB 1 BB 2 2 1 (cid:107)
from the summation of the expectation values of each point of interest as expected of finite-size-effect origin
atom. of the scale ∆ (N). Surprisingly, despite the small
BB
range of thickness accessible to the slab DFT calcula-
tion,thedimensionlessmeasureofbulk-Diracenergygap
r = ∆ /∆ in Fig. 5(a) shows a significant increase
DB BB
upon an increase in the slab thickness. This is quali-
tatively consistent with observations from the effective
model and hybridization effects in Sec. II. On the other
hand, the range of thickness in the present calculation
appears to be too small to show any change in the (cid:104)S (cid:105)
y
as a function of slab thickness [Fig. 5(b)].
IV. CONCLUSION
FIG. 4. (a) DFT-calculated band structure of a 6-QL slab
of Bi Se . (b)DFT-calculated spin expectation values of the
2 3 WecombinedaFano-typehybridizationmodelcalcula-
conduction Dirac state (cid:104)S (cid:105)(k xˆ) for a 6-QL Bi Se slab.
i x 2 3 tionwithanab-initioslabcalculationtostudythelowest
ordereffectsofsurface-bulkinteractionintopologicalin-
sulators with a particular focus in the TM regime. We
defined the TM regime of a topological insulator to be
wheretheDiracsurfacestatesandbulkstatescoexistand
interact, yet the spin-winding is preserved albeit with a
reduced spin-polarization magnitude. The hybridization
model presented in Sec. II captures the spin-polarization
reduction of the Dirac states originating from the hy-
bridization with bulk states. Given the metallic behav-
ior of most TIs, and the experimental evidence of re-
duced spin polarization, our simple model offers a useful
starting point for applications of TIs which need to take
real materials in the TM regime into account. Moreover,
FIG. 5. (a) Ratio r = ∆ /∆ within the TM regime, the hybridization-driven bulk-Dirac energy gap explains
DB BB
calculated at a fixed k(cid:107). (b) (cid:104)Sy(cid:105) of the conduction Dirac why the Dirac branch shows up so well separated from
statecalculatedusingDFTasafunctionofslabthicknessN. bulk states in ARPES experiments. Note that this en-
ergy gap and the suppression of total spin magnitudes
Figure4showstheDFT-calculatedbandstructureand are both experimentally observed phenomena that can-
spin expectation values (cid:104)S (cid:105)(k xˆ) of a 6-QL slab. The not be accessed by the typical approach of coupling a
i x
surface states are doubly degenerate and have a Dirac single“surfacelayer”toabulkelectronicstructuretoin-
dispersion and we show five confined states in the bulk cludesurfacestatesinsemi-infinitesystemsasinRef.21.
conductionbandregion[Fig.4(a)]. Forsmall|k |values, We propose SARPES experiments for films of varying
x
(cid:104)S (cid:105)ofaDiracconductionstateisclearlydominantover thickness to test our predictions for hybridization-driven
y
other components and exhibits spin-momentum locking suppressionofspinpolarizationforfurthervindicationof
[Fig. 4(b)]. As |k | increases, a small z component of the model.
x
spin expectation value develops. However, over the en- PromisingfuturedirectionsincludeDFTtoolstostudy
tirerangeofk ,(cid:104)S (cid:105)ismuchlessthanthemaximalvalue, slightly thicker systems. This might reveal thickness de-
x y
in agreement with previous DFT study11. A comparison pendence in spin polarization and compared to the re-
betweenFig.2andFig.4(b)indicatesthatourhybridiza- sults of the simple model. Also this would reveal more
tion model is an effective way to capture the broadening detailed knowledge of the magnitude and k -dependence
(cid:107)
of the Dirac surface state wavefunction and the result- ofthehybridizationstrengthg. PreliminaryDFTresults
7
show that g(k ) has a significant k -dependence. An- 3D TIs.
(cid:107) (cid:107)
other interesting direction will be to study consequences Acknowledgements. We thank S. Oh for stimu-
ofthehybridizationeffectontransportproperties. Many lating discussions. E.-A.K., Y.-T.H., and M.H.F. were
puzzling aspects of transport experiments4,6–8 have been supported in part by NSF CAREER grant DMR-095582
attributed to the presence of bulk states or surface-bulk andinpartbytheCornellCenterforMaterialsResearch
interaction. There is growing theoretical interest on the with funding from the NSF MRSEC program (DMR-
transportpropertiesoftopologicaledgestatesinthepres- 1120296). K.P. was supported by NSF DMR-0804665,
enceofmetallicbulkstates30–33 aswell. Ourmicroscopic DMR-1206354, and San Diego Supercomputer Center
model offers a simple starting point to theoretically ad- (DMR-060009N). TLH was supported by US DOE un-
dress effects of surface-bulk interaction on transport in der QMN DE-FG02-07ER46453.
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