Table Of ContentMott scattering at the interface between a metal and a topological insulator
Erhai Zhao,1 Chun Zhang,2 and Mahmoud Lababidi1
1Department of Physics and Astronomy, George Mason University, MS 3F3, Fairfax, VA 22030
2 Department of Physics and Department of Chemistry,
National University of Singapore, 2 Science Drive 3, Singapore 117542
(Dated: January 18, 2011)
We compute the spin-active scattering matrix and the local spectrum at the interface between
a metal and a three-dimensional topological band insulator. We show that there exists a critical
incident angle at which complete (100%) spin flip reflection occurs and the spin rotation angle
jumps by π. We discuss the origin of this phenomena, and systematically study the dependence of
1
spin-flip and spin-conserving scattering amplitudes on the interface transparency and metal Fermi
1
surface parameters. The interface spectrum contains a well-defined Dirac cone in the tunneling
0
limit, and smoothly evolves into a continuum of metal induced gap states for good contacts. We
2
also investigate the complex band structure of Bi Se .
2 3
n
a PACSnumbers: 73.20.-r,75.70.Tj,85.75.-d
J
7
1 I. INTRODUCTION cantly from its two dimensional analog, the interface be-
tween a metal and a quantum spin Hall (QSH) insulator
] studied by Tokoyama et al13. They predicted a giant
l Recently discovered three dimensional topological
al bandinsulators1–3,suchasBi1−xSbx4 andBi2Se35–7,are spin rotation angle α ∼ π and interpreted the enhance-
h spin-orbitcoupledcrystalsolidswithabulkgapbutpro- ment as resonance with the one-dimensional helical edge
- tected gapless surface states. The low energy excitations modes. Bycontrast,forM-TIinterfacewepredictacrit-
s
icalincidentangleatwhichcompletespinflippingoccurs
e at the surface are helical Dirac fermions, i.e., their spin
m andmomentumareentangled(locked)8. Thechargeand and the spin rotation angle jumps by π. We will explain
its origin, in particular its relation to the surface helical
. spin transport on the surface of a topological insulator
at are intrinsically coupled9. This makes these materials Dirac spectrum, and discuss its spintronic implications.
m a promising new platform for spintronics. In addition, This paper is organized as follows. We will first com-
putethescatteringmatrixusingak·pcontinuummodel
heterostructures involving topological insulator, super-
-
d conductor, and/or ferromagnet have been predicted to by matching the envelope wave functions at the M-TI
n show a remarkable array of novel spectral and transport interface. This simple calculation is easy to understand,
o properties (for review see Ref.10–12). anditbringsoutthemainphysicsofourproblem. Along
c the way, we will discuss the complex band structure of
[ Electronic or spintronic devices based on topological
insulators will almost inevitably involve metal as mea- Bi2Se3,whichdescribesthedecaying(ratherthanpropa-
2 surement probes or functioning components13. This mo- gating Bloch wave) solutions of the crystal Hamiltonian.
v The various caveats of this calculation are then reme-
tivates us to study the local spectrum near the interface
4
died by considering a much more general lattice model.
between a metal (M) and a topological insulator (TI).
6
Mostimportantly, itenablesustotrackhowthescatter-
0 For a metal-ordinary semiconductor junction with good
ing matrix and interface spectrum change with interface
3 contact, it is well known that the metallic Bloch states
. penetrate into the semiconductor as evanescent waves transparency. It also sheds light on the origin of perfect
5
spin-flip scattering at the critical angle. We will show
0 localized at the interface (for energies within the band
that the results obtained from these two complementary
0 gap). Such interface states are known as metal induced
1 gap states (MIGS)14,15. They play an important role methods are consistent with each.
: in controlling the junction properties, e.g., by pinning
v
i thesemiconductorFermileveltodeterminetheSchottky
X barrier height16, a key parameter of the junction. II. MODEL HAMILTONIAN AND COMPLEX
BAND STRUCTURE
r The local spectrum at the M-TI junction is intimately
a
relatedtothespin-activescatteringofelectronsattheM-
We consider Bi Se as a prime example of 3D strong
TI interface. In this paper, we systematically study the 2 3
topological insulators. Its low energy k·p Hamiltonian
evolutionofthescatteringmatrixandtheinterfacespec-
was obtained by Zhang et al6,
tra with the junction transparency and metal Fermi sur-
face parameters. The scattering matrix17 we obtain here
3
alsoformsthebasistoinvestigatethedetailsofthesuper- Hˆ (k)=(cid:15) (k)ˆ1+(cid:88)d (k)Γˆ .
TI 0 µ µ
conducting proximity effect near the superconductor-TI
µ=0
interface18, which was shown by Fu and Kane to host
Majorana fermions19. Here d0(k) = M −B1kz2 −B2(kx2 +ky2), d1(k) = A2kx,
The scattering at the M-TI interface differs signifi- d (k) = A k , d (k) = A k , and (cid:15) (k) = C +D k2 +
2 2 y 3 1 z 0 1 z
2
(a) (b)
y
M TI 0.2 0.2
TI
z E0.0 E0.0
E
k F
i (cid:45)0.2 (cid:45)0.2
! x 0.1 0.1
k (cid:45)0.1 0.0Imkz (cid:45)0.1 0.0Imkz
o 0.0 (cid:45)0.1 0.0 (cid:45)0.1
Rekz 0.1 Rekz 0.1
FIG. 2: The complex band structure of topological insulator
described by Hˆ (k) for k = 0, k = 0.02 (left) and 0.04
FIG. 1: (a) Scattering geometry at a metal (M)-topological TI y x
(right). E is measured in eV, and k in ˚A−1. Subgap states
insulator (TI) interface. (b) Schematic band structure of the
metal (modeled by Hˆ ) and topological insulator. withcomplexkz representevanescentwaves. Thetopologyof
M real lines25 changes as k is increased.
Thursday, November 11, 2010 x
D (k2+k2). ThenumericalvaluesofM,A,B,C,D are
2 x y ary condition assumes good atomic contact between two
given in Ref.6. We choose the basis (|+ ↑(cid:105), |+ ↓(cid:105), |− ↑ materials.
(cid:105),|− ↓(cid:105)), where ± labels the hybridized pz orbital with WeareinterestedinenergiesbelowthebandgapofTI,
even (odd) parity6. The Gamma matrices are defined so Φˆ is evanescent in nature and only penetrates into
TI
as Γˆ0 = τˆ3 ⊗ ˆ1, Γˆi = τˆ1 ⊗ σˆi, with τˆi (σˆi) being the TIforafinitelength. Suchlocalized(surfaceorinterface)
Pauli matrices in the orbital (spin) space. The chemical states inside topological insulator can be treated within
potential of as-grown Bi2Se3 crystal actually lies in the the k·p formalism using the theory of complex band
conduction band8. By hole doping8 or applying a gate structures, pioneered by Kohn23, Blount24, and Heine25
voltage20, thechemicalpotentialcanbetunedinsidethe et al. The main idea is to allow the crystal momentum
gap. The system is well described by HTI (note that to be complex and analytically continue HTI(k) to the
energy zero is set as in the middle of the band gap). complex k plane. While the extended Bloch waves are
In this section, we first adopt a rather artificial model the eigen states of H (k) for real k, eigen functions of
TI
for metals with negligible spin-orbit coupling. It is ob- H (k)forcomplexkdescribelocalizedstates. Together
TI
tained by turning off the spin-orbit interaction (setting they form a complete basis to describe crystals of finite
dµ = 0 for µ=1,2,3) in HTI and shifting the Fermi level dimension.
into the conduction band. The result is spin-degenerate In our scattering problem, we have to find all eigen
two-band Hamiltonian states of H (k) with energy E and wave vector k =
TI
(k ,k ,k˜ ),wherek andk aregivenandreal,butk˜ is
Hˆ (k)=[(cid:15) (k)−E ]ˆ1+d (k)Γˆ . x y z x y z
M 0 F 0 0 complex and unknown. For a general k·p Hamiltonian
suchasHˆ ,wefollowChangandSchulman26 torewrite
Its band structure, schematically shown in Fig. 1(b), TI
it as
consistsoftwooppositelydispersingbands(thesolidand
dashline). EF istunedtobemuchhigherthantheband Hˆ =hˆ (k ,k )+hˆ k˜ +hˆ k˜2,
TI 0 x y 1 z 2 z
crossing point, so the scattering properties of low energy
electrons near the Fermi surface are insensitive to the where hˆ = A Γˆ , and hˆ = −B Γˆ . Then the eigen
1 1 3 2 1 0
bandcrossingathighenergies. Thisclaimwillbeverified equation (Hˆ −Eˆ1)φˆ = 0 can be reorganized into an
TI
laterusingamoregenericmodelforthemetal. Asimilar eigen value problem for k˜ ,
model was used in the study of metal-QSH interface13. z
Matching the wave functions of two dissimilar mate- (cid:18)0 1 (cid:19)(cid:18)φˆ (cid:19) (cid:18)φˆ (cid:19)
rials (such as Au and Bi2Se3) at interface is in general −hˆ−1(hˆ −Eˆ1) −hˆ−1hˆ φˆ(cid:48) =k˜z φˆ(cid:48) .
complicated within the k·p formalism, because the en- 2 0 2 1
velope wave functions on either side are defined using Then all possible values of k˜ can be obtained for given
different basis (see Ref. 21 and reference therein). For z
incident parameter E, k , and k . For the anisotropic
the particular model H , however, such complication is x y
M Dirac Hamiltonian H (k), the energy eigenvalues can
circumvented. Then, then wave functions at the metal- TI
beobtainedanalytically28,whichallowsforananalytical
TI interface (z = 0) satisfy the Ben-Daniel and Duke
solution of the complex band structure.
boundary condition22,
For E within the gap, there are in general 4 pairs of
Φˆ =Φˆ , vˆ Φˆ =vˆ Φˆ . complex solution of k˜z, for if k˜z is a solution so is k˜z∗.
M TI M M TI TI We label those with positive imaginary parts with {k˜ν},
z
HereΦˆ isthefour-componentwavefunction,andtheve- and the corresponding wave function {φˆν}, ν =1,2,3,4.
i
locity matrix vˆ = ∂Hˆ /∂k , i ∈ {M,TI}. Such bound- They are decaying solutions in the half space z > 0.
i i z
3
In our model, k˜z turns out to be doubly degenerate, as up to the trivial ei(kxx+kyy) and renormalization factor.
shown in Fig. 2. The wave function inside TI (z > 0) Here k = zˆ·k, and {r } are the reflection amplitudes.
z i
then has the form We identify the spin flip amplitude f =r and the spin-
4
conservingamplitudeg =r . Notethatthereisnoprop-
ΦˆTI =(cid:88)tνeik˜zνzφˆν. agating mode at energy E 3available in the valence band
ν for the reflected electron. So k(cid:48) is purely imaginary. At
z
such energy, there is no propagating mode available in
TI.WehavediscussedtheevanescentwavefunctionΦˆ
TI
III. SCATTERING MATRIX FROM in the previous section. With Φˆ and Φˆ , we solve the
WAVE-FUNCTION MATCHING M TI
boundary condition at z = 0 to obtain r ,t and the
ν ν
scattering matrix S.
Tosetthestagefordiscussingscatteringoffatopolog-
Fig. 3 shows the magnitude and phase of f and g ver-
ical insulator, it is instructive to recall the generic fea-
sustheincidentangleθ forE =0.1eV,withE settobe
F
turesofelasticscatteringofelectronsbyaheavyionwith
0.28eV. At normal incidence, θ = 0, spin flip scattering
spin-orbit interaction. This classical problem was solved
is forbidden as in the single-ion Mott scattering. With
by Mott, and known as Mott scattering. The scattering
increasing θ the magnitude of g drops continuously. At
matrix has the general form27
a critical angle θ , |g| drops to zero and we have perfect
c
(100%) spin flip reflection. At the same time, the spin
Sˆ =uˆ1+wσˆ ·(k ×k ),
Mott i o rotation angle α (the relative phase between f and g)
jumps by π.
where k and k are the incident and outgoing momen-
i o
It is tantalizing to think of what happens at θ as res-
tum respectively, σˆ is the Pauli matrix, and u,w de- c
onant scattering with the helical surface mode of the TI.
pend on the scattering angle. It is customary to de-
This however is problematic. We are considering good
finethespin-flipamplitudef =S , andspin-conserving
21
contacts at which the wave functions of the two mate-
amplitude g = S . Both f and g are complex num-
11
rials hybridize strongly. Surface mode is preempted by
bers, their relative phase defines the spin rotation angle
MIGS. Indeed, we checked that the corresponding crit-
α=Arg(g∗f). One immediately sees that for back scat-
ical transverse momentum k depends only weakly on
tering, Sˆ =uˆ1, so there is no spin flip, f =0. As we (cid:107)
Mott E. This is at odds with the linear dispersion of the TI
willshowbelow,thisalsoholdstrueforscatteringoffTI.
surface mode, E = A k 6. To gain better understand-
Now consider an electron coming from the metal with 2 (cid:107)
ing, we now switch to a lattice model to systematically
momentum k incident on the M-TI interface located at
study the role of interface transparency and metal Fermi
z = 0, as schematically shown in Fig. 1(a). We assume
surface parameter (E ,k ,v ) on the scattering matrix.
theinterfaceistranslationallyinvariant,sothetransverse f f f
momentum k =(k ,k ) is conserved, and the energy E
(cid:107) x y
oftheelectronlieswithinthebandgapofTI.Then,only
IV. INTERFACE SPECTRUM AND
totalreflectionispossible,butthespin-orbitcouplingin-
SCATTERING MATRIX FROM LATTICE
side TI acting like a k-dependent magnetic field rotates
GREEN FUNCTION
the spin of the incident particle. The scattering (reflec-
tion) matrix has the form
We consider a simple lattice model for the M-TI junc-
(cid:18)g f¯(cid:19) tion. The topological insulator is modeled by a tight
Sˆ(k)= , binding Hamiltonian on cubic lattice,
f g¯
where |g|2+|f|2 =1. Our goal is to find the dependence HR = (cid:88)(cid:110)ψˆk†(cid:107),n(b1Γˆ0−ia21Γˆ3)ψˆk(cid:107),n+1+h.c.
of the scattering amplitudes f,g on k, or equivalently, k(cid:107),n
on energy E and incident angle θ. From time-reversal (cid:104) (cid:105) (cid:111)
symmetry, f¯(E,θ) = f(E,−θ) and g¯(E,θ) = g(E,−θ). + ψˆk†(cid:107),n d(k(cid:107))Γˆ0+a2(Γˆ1sinkx+Γˆ2sinky) ψˆk(cid:107),n .
We shall show that f(k ) = −f(−k ), g(k ) = g(−k ).
(cid:107) (cid:107) (cid:107) (cid:107)
Sof isanoddfunctionofθ,whilegiseveninθ. Sinceour Here ψˆ = (ψ ,ψ ,ψ ,ψ )T is the annihilation op-
+↑ +↓ −↑ −↓
problem can be viewed as coherent multiple scattering erator, d(k ) = M −2b +2b (cosk +cosk −2) with
(cid:107) 1 2 x y
from a lattice array of Mott scatters occupying half the k measured in 1/a. The cubic lattice consists of layers
space,wewillrefertospin-activescatteringatthemetal- of square lattice stacked in the z direction, n is the layer
TI interface as Mott scattering. index, and k is the momentum in the xy plane. The
(cid:107)
Consider a spin up electron from the conduction band isotropicversionofH , witha =a , b =b , wasstud-
R 1 2 1 2
of the metal with momentum k and energy E =(cid:15) (k)− ied by Qi et al as a minimal model for 3D topological
0
E −d (k) lying within the band gap of TI. The wave insulators28. To mimic Bi Se , we set the lattice spacing
F 0 2 3
function inside the metal (z <0) has the form a = 5.2˚A, which gives the correct unit cell volume, and
a =A /a,b =B /a2 fori=1,2. Althoughacrudecar-
i i i i
ΦˆM =(r1e−ikz(cid:48)z,r2e−ikz(cid:48)z,eikzz+r3e−ikzz,r4e−ikzz)T, icature of the real material, HR yields the correct gap
4
(cid:200)g(cid:200)
1.0 g 1 0.5 Α(cid:144)Π
0.8
0.6
0.5 0
0.4
! !
f
0.2
Θ Θ(cid:45)0.5 Θ
1.4 A0.1rg g0.2 0.3 0.4 0.5 Π 0.1 0.2 0.3 0.4 0.5Π 0.1 0.2 0.3 Π
1.2
Π ! ! FIG. 5: The spin-conserving reflection amplitude |g| and
1.0 spin rotation angle α versus the incident angle θ for increas-
0.8 ing contact transparency, J/tM = 0.25,1,1.5,2 (from left to
right). t = 0.18eV, µ = −4t , E = 0.05eV, k = 0.
0.6 " # Arg f |f|2 =1−M|g|2. M M y
0.4
Π
0.2
Θ between metal and TI is described by hopping,
0.1 0.2 0.3 0.4 0.5 Π
" # H =− (cid:88) J ψ† φ +h.c.
LR (cid:96) k(cid:107),n=1,(cid:96),σ k(cid:107),n=0,σ
FIG.3: Themagnitudes(upperpanel)andthephases(lower k(cid:107),(cid:96),σ
panel)ofthespin-flipamplitudef andspin-conservingampli-
tude g versus the incident angle θ. E = 0.1eV, EF=0.28eV. J(cid:96) is the overlap integral between the p-orbital (cid:96) = ±
|g|2+|f|2 = 1. Arg(g) and Arg(f) are shifted upward by π of TI and the s-like orbital of metal. For simplicity, we
for clarity. assume J is independent of spin. Then, J =−J =J.
(cid:96) + −
J can be tuned from weak to strong. Small J mimics a
large tunneling barrier between M and TI, and large J
(comparable to t or B ) describes a good contact.
M 2
The lattice Green function of the composite system
is computed via standard procedure by introducing the
inter-layer transfer matrix and the method of interface
Greenfunctionmatching29. Fig. 4showstwoexamplesof
the local spectral function (momentum-resolved density
of states) at the interface,
N(E,k )=− (cid:88) ImTrGˆ(E,k ) ,
(cid:107) (cid:107) n,n
n=0,1
FIG. 4: The spectral function N(E,k ,k = 0) at the inter-
face of metal and topological insulatoxr. Lyeft: good contact, where Gˆ(E,k(cid:107))n,n(cid:48) is the local Green function at the in-
J =t , showing the continuum of metal induced gap states. terfacewithn,n(cid:48) =0,1,andthetraceisoverthespinand
M
Right: poor contact with low transparency, J = 0.2tM, orbital space. In the tunneling (weak coupling, small J)
showing well defined Dirac spectrum as on the TI surface. limit, the interface spectrum includes a sharply defined
tM =0.18eV, µM =−4tM, a is lattice spacing. DiracconeasonthesurfaceofTI.AsJ isincreased, the
linearly dispersing mode becomes ill defined and eventu-
allyreplacedbyacontinuumofmetalinducedgapstates.
size and surface dispersion, it also reduces to the contin-
Once the lattice Green function is known for given in-
uum k·p Hamiltonian Hˆ in the small k limit, aside
TI cident E and k , the scattering (reflection) matrix can
(cid:107)
from the topologically trivial (cid:15) (k) term.
0 be constructed from Gˆby29,
Asagenericmodelformetal,weconsiderasingleband
tight binding Hamiltonian on cubic lattice, Sˆ(E,k )=Gˆ(E,k ) g−1(E,k )−ˆ1
(cid:107) (cid:107) 0,0 M (cid:107)
(cid:88)
H = [h(k )n −t φ† φ +h.c.] where g is the spin-degenerate bulk Green function of
L (cid:107) k(cid:107),n,σ M k(cid:107),n,σ k(cid:107),n+1,σ M
k(cid:107),n,σ metal. Fig. 5 shows the evolution of |g(θ)| and α(θ)
for increasing J, where a level broadening of E/10 is
where h(k ) = −2t (cosk +cosk )−µ . The Fermi used. Mostimportantly,weobservethattheexistenceof
(cid:107) M x y M
surface parameters of the metal can be varied by tuning acriticalangelθ , wherecompletespin-flipoccursandα
c
t and µ . The metal occupies the left half space, n≤ jumps by π, is a robust phenomenon. It is independent
M M
0, and the TI occupies the right half space n ≥ 1. The of the details of the contact, the metal Fermi surface, or
interface domain consists of layer n=0,1. The coupling other high energy features in the band structure.
5
To understand the perfect spin flip, we first focus on by ARPES (or scanning tunneling microscope) experi-
the tunneling limit, J (cid:28) t . In this limit, the local ments on metal film coated on a topological insulator.
M
spectrum at layer n = 1 as shown in the right panel of Our results also suggest that a topological insulator can
Fig. 4 approaches the TI surface spectrum, namely the serveasaperfectmirrortofliptheelectronspininmetal.
helical Dirac cone. An incident up spin tunneling across Suchspin-activescatteringattheM-TIinterfacemaybe
the barrier will develop resonance with the helical mode, exploited to make novel spintronic devices. The magni-
whichisaquasi-stationarystatewithlonglifetime,ifits tude of g or f can be measured by attaching two fer-
momentum and energy satisfy k = E/A . Moreover, it romagnetic leads to a piece of metal in contact with
(cid:107) 2
has to flip its spin, since only down spin can propagate TI, forming a multi-terminal device. One of the ferro-
in the k direction (suppose k =0). The π jump in the magneticleadsproducesspin-polarizedelectronsincident
x y
phaseshiftisalsocharacteristicoftheresonance. Indeed, on the M-TI interface at some angle, while the other
we have checked that precisely at θ the resonance cri- lead detects the polarization of reflected electron, as in
c
terion, k sinθ = E/A , is met. We also varied µ for a giant magneto-resistance junction. The spin rotation
f c 2 M
fixed J and t , bigger µ yields a bigger Fermi surface angle α can be measured indirectly by comparing the
M M
and a smaller θ . This is consistent with the resonance predicted current-voltage characteristics of M-TI-M or
c
criterion above. Superconducto-TI-Superconductor junctions, which are
As J is increased, the width of the resonance grows sensitive the phase shift α. It can also be inferred from
and eventually it is replaced by a broad peak (dip) in the spin transport in a TI-M-TI sandwich, as discussed
|f| (|g|), but the vanishing of |g| and π shift in α at θ for QSH insulator in Ref.13. Detailed calculations of the
c
persist to good contacts, even though in this limit the transport properties of these structured, using the scat-
interface is flooded by MIGS (left panel of Fig. 4) and tering matrix obtained here, will be subjects of future
bears little resemblance to the Dirac spectrum. With all work.
other parameters held fixed, θ increases with J. Quali-
c
tatively, coupling to TI renormalizes the metal spectrum
neartheinterface,producingasmallereffectivek (hence
f
a larger θ ) compared to its bulk value. It is remarkable
c
that perfect spin flip at the critical angle persists all the
way from poor to good contacts. Indeed, the main fea-
turesobservedhereforforgoodcontactsusingthelattice
model agree well with the results obtained in previous
section by wave function matching. VI. ACKNOWLEDGEMENTS
V. DISCUSSIONS We thank Liang Fu, Parag Ghosh, Predrag Nikolic,
Indu Satija, and Kai Sun for helpful discussions. This
We now discuss the experimental implications of our work is supported by NIST Grant No. 70NANB7H6138
results. The M-TI interface spectrum can be measured Am 001 and ONR Grant No. N00014-09-1-1025A (EZ).
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