Table Of ContentProximity band structure and spin textures on both sides of
topological-insulator/ferromagnetic-metal interface and their transport probes
J. M. Marmolejo-Tejada,1,2 K. Dolui,1 P. Lazi´c,3 P.-H. Chang,4
S. Smidstrup,5 D. Stradi,5 K. Stokbro,5 and B. K. Nikoli´c1,∗
1Department of Physics and Astronomy, University of Delaware, Newark, DE 19716-2570, USA
2School of Electrical and Electronics Engineering,
Universidad del Valle, Cali, AA 25360, Colombia
3Rudjer Boˇskovi´c Institute, PO Box 180, Bijeniˇcka c. 54, 10 002 Zagreb, Croatia
4Department of Physics and Astronomy, University of Nebraska Lincoln, Lincoln, Nebraska 68588, USA
7
5QuantumWise A/S, Fruebjergvej 3, Box 4, DK-2100 Copenhagen, Denmark
1
0 The control of recently observed spintronic effects in topological-insulator/ferromagnetic-metal
2 (TI/FM) heterostructures is thwarted by the lack of understanding of band structure and spin
r texture around their interfaces. Here we combine density functional theory with Green’s function
a techniques to obtain the spectral function at any plane passing through atoms of Bi Se and Co
2 3
M
or Cu layers comprising the interface. In contrast to widely assumed but thinly tested Dirac cone
gapped by the proximity exchange field, we find that the Rashba ferromagnetic model describes
8 the spectral function on the surface of Bi Se in contact with Co near the Fermi level E0, where
2 3 F
2 circularandsnowflake-likeconstantenergycontourscoexistaroundwhichspinlockstomomentum.
The remnant of the Dirac cone is hybridized with evanescent wave functions injected by metallic
] layers and pushed, due to charge transfer from Co or Cu layers, few tenths of eV below E0 for
l F
l both Bi Se /Co and Bi Se /Cu interfaces while hosting distorted helical spin texture wounding
a 2 3 2 3
h arounda singlecircle. Thesefeatures explainrecent observation[K.Kondou et al., Nat. Phys. 12,
- 1027 (2016)] of sensitivity of spin-to-charge conversion signal at TI/Cu interface to tuning of EF0.
s Interestingly, three monolayers of Co adjacent to Bi Se host spectral functions very different from
2 3
e
thebulkmetal,aswellasin-planespintexturessignifyingthespin-orbitproximityeffect. Wepredict
m
thatout-of-planetunnelinganisotropicmagnetoresistanceinverticalheterostructureCu/Bi Se /Co,
2 3
. where current flowing perpendicular to its interfaces is modulated by rotating magnetization from
t
a paralleltoorthogonaltocurrentflow,canserveasasensitiveprobeofspintextureresidingatE0.
F
m
-
d Therecentexperimentsonspin-orbittorque(SOT)[1, (a) Bi2Se3 Lead Bi2Se3 (6 QL) Co (3 ML)
n 2] and spin-to-charge conversion [3, 4] in topological-
o Vacuum
insulator/ferromagnetic-metal(TI/FM)heterostructures
c 2 1
[ have ignited the field of topological spintronics. In these (b) x
2 devices, giant non-equilibrium spin densities [5–8] are Cu Lead Bi2Se3 (6 QL) Co Lead mϕ m
expected to be generated due to strong spin-orbit cou- θ z
v
2 pling (SOC) on metallic surfaces of three-dimensional y
2 1 3 4
6 (3D) TIs and the corresponding (nearly [9]) helical spin-
V
4 momentum locking along a single Fermi circle for Dirac b
0 electrons hosted by those surfaces [10]. Such strong in- FIG. 1. Schematic view of TI-based heterostructures where:
0
terfacial SOC-driven phenomena are also envisaged to (a) semi-infinite Bi Se layer is attached to n monolayers of
. 2 3
1 underlie a plethora of novel spintronic technologies [11]. Co(0001);(b)6QLsofBi2Se3 aresandwichedbetweensemi-
0 infinite Cu(111) layer and semi-infinite Co(0001) layer. Both
7 These effects have been interpreted almost exclusively heterostructures are infinite in the transverse direction, so
1 using simplistic models, such as the Dirac Hamiltonian that the depicted supercells are periodically repeated within
v: for the TI surface with an additional Zeeman term de- the xy-plane. The magnetization m of the Co layer is fixed
i scribingcouplingofmagnetizationoftheFMlayertothe along the z-axis in (a), or rotated within the xy-plane or the
X surface state spins [10], HˆDirac =v (σˆ ×pˆ) −∆m·σˆ, xz-plane in (b). Applying the bias voltage Vb to the vertical
r F z heterostructureinpanel(b)leadstoachargecurrentflowing
wherepˆ isthemomentumoperator,σˆ isthevectorofthe
a perpendicularlytobothBi Se /CuandBi Se /Cointerfaces.
2 3 2 3
Pauli matrices, m is the magnetization unit vector and
v istheFermivelocity. Thus,theonlyeffectofFMlayer
F
captured by HˆDirac is proximity effect-induced exchange
coupling ∆ which opens a gap in the Dirac cone energy- cannotbecapturedbysimplisticmodelslikeHˆDirac. The
momentum dispersion [10], thereby making Dirac elec- propertiesofTI/FMinterfacesarefarmorecomplexdue
trons massive. On the other hand, recent first-principles to injection of evanescent wave functions from the FM
calculations [12, 13] demonstrate that band structure of layer into the bulk gap of the TI layer, which can hy-
even TI/ferromagnetic-insulator (TI/FI) bilayers, where bridizewithsurfacestateofTIandbluritsDiraccone(as
hybridizationbetweenTIandFIstatesislargelyabsent, already observed in tight-binding models of TI/FM in-
2
(a) n = 0 (b) n = 1 (c) n = 2 (d) n = 3
1 1 1 1 high
0.4
0.2
)
V
e
( 0
0F
E
-
E-0.2
-0.4
low
(e) (f) (g) (h)
2 2 2 2 high
0.4
0.2
)
V
e
( 0
0F
E
-
E-0.2
-0.4
low
X Γ Y X Γ Y X Γ Y X Γ Y
FIG. 2. Spectral function, defined in Eq. (2), at plane 1 for panels (b)–(d) or plane 2 for panels (f)–(h) within Bi Se /Co(n
2 3
ML) heterostructure in Fig. 1(a) with m(cid:107)zˆ. For comparison, panels (a) and (e) plot the spectral function at planes 1 (akin
to Ref. [15]) and 2, respectively, within semi-infinite Bi Se crystal in contact with vacuum (i.e., n=0). From Γ to Y we plot
2 3
A(E;k =0,k ;z∈{1,2}), while from Γ to X we plot A(E;k ,k =0;z∈{1,2}).
x y x y
terfaces [7, 14]), as well as related charge transfer. Thus, the TI surface [18].
thekeyissuefortopologicalspintronics[1–4,11]istoun-
An attempt [20] to obtain the spectral function,
derstandbandstructureandspintextures(includingthe
A (E;k) = (cid:80)i∈QLjwi δ(E −ε ), directly from DFT
fate of the Dirac cone and its spin-momentum locking) j n,i nk nk
computed energy-momentum dispersion ε (n is the
in hybridized TI with FM or normal metal (NM) [4] lay- nk
band index and k is the crystal momentum) and site-
ers at nanometer scale around the interface where they
projected character wi of the corresponding eigenfunc-
arebroughtintocontact,wherepropertiesofbothTIside nk
tions for TI/FM supercells has produced ambiguous re-
andFMorNMsideoftheinterfacecanbequitedifferent
sults. Thisisduetoarbitrarinessinbroadeningthedelta
from the properties of corresponding bulk materials.
functionδ(E−ε ),aswellasduetousageofatomicsites
nk
iwithinthewholej quintuplelayer(QL )ofBi Se (one
For example, computational searches [15] for new ma- j 2 3
QL consists of three Se layers strongly bonded to two Bi
terials realizing 3D TIs (or other topologically non-
layersinbetween)whicheffectivelyaveragesthespectral
trivial electronic phases of matter like Weyl semi-
function over all geometric planes within QL . Similar
metals [16] and Chern insulators [17]) have crucially j
ambiguities (such as setting the amount of electron den-
relied on first-principles calculations of spectral func-
sity which is localized on the surface or within the whole
tion on their boundaries and its confirmation by spin-
interfacial QL) plague interpretation of projected DFT
and angle-resolved photoemission spectroscopy (spin-
band structure of TI/FI [13] and TI/FM bilayers [21].
ARPES) [18]. A standard density functional theory
(DFT)-based framework developed for this purpose— Herewedevelopaframeworkwhichcombinesthenon-
where DFT band structure around the Fermi level E0 collinearDFTHamiltonianHDFT,representedinabasis
F
is reconstructed using the Wannier tight-binding Hamil- of variationally optimized localized atomic orbitals [22],
tonian [19] used to obtain the retarded Green’s function with retarded GF calculations from which one can ex-
(GF) of semi-infinite homogeneous crystal and the spec- tract the spectral function and spin textures at an arbi-
tral function on its surface in contact with vacuum [15– trary geometric plane of interest within a junction com-
17]—is difficult to apply to complicated inhomogeneous bining TI, FM and NM layers. It also makes it possible
systems like TI/FM bilayers due to strongly entangled to compute their spin and charge transport properties in
bands in the region of interest around E0. Also, spin- the linear-response regime or at finite bias voltage. We
F
ARPES experiments cannot probe buried interfaces be- apply this framework to two Bi Se -based heterostruc-
2 3
lowtoomanymonolayers(e.g.,penetrationdepthoflow- tures whose supercells are depicted in Fig. 1, where we
energy photons is 2–4 nm) of FM or NM deposited onto assume that those supercells are periodically repeated in
3
(a) (b) E-E0= 0 eV (c) E-E0= 0.05 eV (d) E-E0= -0.35 eV
1 F F F high1x10-3
0.2
0.5
0.1
)
V
e
0(F 0 kx 0 zS
E
-
E −0.1
-0.5
−0.2
-1 low-1x10-3
X Γ Y −0.2 −0.1 0 0.1 0.2 −0.2 −0.1 0 0.1 0.2 −0.2 −0.1 0 0.1 0.2
k k k
y y y
(e) (f) E-E0= 0 eV (g) E-E0= 0.05 eV (h) E-E0= -0.35 eV
1 0.2 F F F high1x10-9
0.5 0.1
)
V
e
0(F 0 kx 0 zS
E
-
E
-0.5 −0.1
-1 −0.2 low-1x10-9
X Γ Y −0.2 −0.1 0 0.1 0.2 −0.2 −0.1 0 0.1 0.2 −0.2 −0.1 0 0.1 0.2
k k k
y y y
FIG. 3. Spectral function at: (a)–(d) plane 1 in Fig. 1(b) which is passing through Se atoms at the Bi Se /Co interface
2 3
with m(cid:107)zˆ; and (e)–(h) plane 2 in Fig. 1(b) which is passing through Se atoms at Bi Se /Cu interface, where we re-
2 3
move Co layer to make Bi Se semi-infinite along the z-axis. In panels (a) and (e), we plot A(E;k = 0,k ;z ∈ {1,2})
2 3 x y
from Γ to Y and A(E;k ,k = 0;z ∈ {1,2}) from Γ to X. Panels (b)–(d) and (f)–(h) plot constant energy contours of
x y
A(E−E0 ∈{0.0 eV,0.05 eV,−0.35 eV};k ,k ;z∈{1,2}) at three energies marked by horizontal dashed lines in panels (a)
F x y
or (e), respectively, as well as the corresponding spin textures where the out-of-plane S component is indicted in color (red
z
for positive and blue for negative). The units for k and k are 2π/a where a is the lattice constant of a common supercell
x y
combining two unit cells of the two layers around the corresponding interface.
the transverse xy-direction. The heterostructure in Fig. 1(b) consists of semi-
The heterostructure in Fig. 1(a) consists of Bi Se , infinite Cu and Co leads sandwiching a Bi Se layer of
2 3 2 3
chosenastheprototypical3DTI[9,10],whosesurfaceis finite thickness, where we choose Cu as the NM layer
covered by n monolayers (MLs) of Co. The retarded GF similar to the very recent spin-to-charge conversion ex-
of this heterostructure is computed as periment of Ref. [4]. Such a heterostructure is termed
vertical or current-perpendicular-to-plane in spintronics
G (E)=[E−HDFT−ΣBi2Se3(E)]−1, (1)
k(cid:107) k(cid:107) k(cid:107) terminology since applying bias voltage Vb drives a cur-
rentperpendicularlytotheTI/FMinterface. Itsretarded
where k = (k ,k ) is the transverse k-vector,
(cid:107) x y
GF is computed as
ΣBi2Se3(E) is the self-energy [23] describing the semi-
inkfi(cid:107)nite Bi2Se3 lead and HDk(cid:107)FT is the Hamiltonian of the Gk(cid:107)(E)=[E−HDk(cid:107)FT−ΣCk(cid:107)u(E)−ΣCk(cid:107)o(E)]−1, (3)
active region consisting of n MLs of cobalt plus 6 QLs of
where HDFT describes the active region consisting of 6
Bi2Se3 to which the lead is attached. We choose n=1–3 k(cid:107)
QLs of Bi Se plus 4 MLs of Cu and 4 MLs of Cu. Its
since ultrathin FM layers of thickness (cid:39)1 nm are typi- 2 3
linear-responseresistanceRisgivenbytheLandauerfor-
cally employed in SOT experiments [24] in order to pre-
mula
serveperpendicularmagneticanisotropy(notethatmag-
netocrystalline anisotropy does favor out-of-plane m in 1 e2 (cid:90) (cid:90) (cid:18) ∂f (cid:19)
= dk dE − Tr[ΓCoG ΓCuG† ],
Bi2Se3/Co bilayers [21]). The spectral function (or lo- R hΩBZ BZ (cid:107) ∂E k(cid:107) k(cid:107) k(cid:107) k(cid:107)
cal density of states) at an arbitrary plane at position z (4)
within the active region is computed from where we assume temperature T = 300 K in the Fermi-
Dirac distribution function f(E), Γα =i(Σα −[Σα ]†)
A(E;kx,ky,z)=−Im[Gk(cid:107)(E;z,z)]/π, (2) andΩ istheareaofthetwo-dimenks(cid:107)ional(2kD(cid:107))Brillko(cid:107)uin
BZ
where the diagonal matrix elements G (E;z,z) are ob- zone (BZ) within which k vectors are sampled.
k(cid:107) (cid:107)
tained by transforming Eq. (1) from orbital to a real- The spectral function of the heterostructure in
space representation. Fig. 1(a) computed at planes 1 and 2 within the Bi Se
2 3
4
(a) (b) E-E0= 0 eV (c) E-E0= 0.05 eV (d) E-E0= -0.35 eV
1 F F F high1x10-1
0.2
0.5
0.1
)
V
e
0(F 0 kx 0 zS
E
-
E −0.1
-0.5
−0.2
-1 low-1x10-1
X Γ Y −0.2 −0.1 0 0.1 0.2 −0.2 −0.1 0 0.1 0.2 −0.2 −0.1 0 0.1 0.2
k k k
y y y
(e) (f) E-E0= 0 eV (g) E-E0= 0.05 eV (h) E-E0= -0.35 eV
1 y F F F high1x10-2
0.2
0.5
0.1
)
V
e
0(F 0 kx 0 zS
E
-
E −0.1
-0.5
−0.2
-1 low-1x10-2
X Γ Y −0.2 −0.1 0 0.1 0.2 −0.2 −0.1 0 0.1 0.2 −0.2 −0.1 0 0.1 0.2
k k k
y y y
FIG. 4. Spectral function at: (a)–(d) the surface of a semi-infinite Co layer in contact with vacuum where m is perpendicular
to the surface; and (e)–(h) plane 3 in Fig. 1(b) which is passing through Co atoms at the Bi Se /Co interface with m(cid:107)zˆ. In
2 3
panel (e), we plot A(E;k = 0,k ;z ∈ 3) from Γ to Y and A(E;k ,k = 0;z ∈ 3) from Γ to X. Panels (b)–(d) and (f)–(h)
x y x y
plot constant energy contours of the spectral function at three energies marked by horizontal dashed lines in panels (a) or (e),
respectively, as well as the corresponding spin textures where the magnitude of the out-of-plane S component is indicted in
z
color (red for positive and blue for negative). The units for k and k are 2π/a where a is the lattice constant of a common
x y
supercell combining two unit cells of the two layers around the corresponding interface.
layer is shown in Figs. 2(b)–(d) and 2(f)–(h), respec- For infinitely many MLs of Co attached to 6 QLs
tively, where plane 1 is passing through Se atoms on the of Bi Se within the Cu/Bi Se /Co heterostructure in
2 3 2 3
Bi Se surface in contact with Co layer and plane 2 is Fig. 1(b), the remnant of the Dirac cone from the TI
2 3
three QLs (or (cid:39)2.85 nm) away from plane 1. For com- surface can be identified in Fig. 3(a) at around 0.5 eV
parison,wealsoshowinFigs.2(a)and 2(b)thespectral below E0 while it is pushed even further below in the
F
function at the same two planes within the semi-infinite case of Cu/Bi Se interface in Fig. 3(e). The differ-
2 3
Bi Se layer in contact with vacuum, thereby reproduc- ence in work functions Φ =5.0 eV or Φ =4.7 eV
2 3 Co Cu
ing the results from Ref. [15] by our formalism. While andelectronaffinityχ =5.3 eVdetermines[20]the
Bi2Se3
the Dirac cone at the Γ-point is still intact in Fig. 2(b) bandalignmentandthestrengthofhybridization, where
for n = 1 ML of Co, its Dirac point (DP) is gradually n-type doping [see also Figs. 6(c) and 6(d)] of the
pushedintothevalencebandofBi Se withincreasingn Bi Se layer pins E0 of the whole Cu/Bi Se /Co het-
2 3 2 3 F 2 3
because of charge transfer from metal to TI. The charge erostructure in the conduction band of the bulk Bi Se .
2 3
transfer visualized in Figs. 6(c) and 6(d) is relatively The remnant of the Dirac cone is quite different from
small, but due to small density of states (DOS) at the the often assumed [10] eigenspectrum of HˆDirac be-
DP it is easy to push it down until it merges with the cause of hybridization with the valence band of Bi Se ,
2 3
larger DOS in the valence band of the TI. Adding more as well as with states injected by the Co or Cu lay-
MLs of Co in Figs. 2(c) and 2(d) also introduces addi- ers whose penetration into TI is visualized by plot-
tional bands within the bulk gap of Bi Se due to injec- ting position- and energy-dependent spectral function
2 3
tion of evanescent wave functions which hybridize with A(E;z) = 1 (cid:82)dk dk A(E;k ,k ;z) in Fig. 5(a). On
ΩBZ x y x y
the Dirac cone. The metallic surface states of Bi Se it- the other hand, the energy-momentum dispersion in the
2 3
selfpenetrateintoitsbulkoveradistanceof2QLs[6],so vicinity of E0 and for an interval of k vectors around
F (cid:107)
thatinFig.2(e)thespectralfunctiononplane2vanishes theΓ-pointissurprisinglywell-describedbyanothersim-
insidethegapofthesemi-infiniteBi Se layerincontact plistic model—ferromagnetic Rashba Hamiltonian [25].
2 3
with vacuum, while the remaining states inside the gap
in Figs. 2(f)–(h) can be attributed to the Co layer. In Figs. 3(b)—(d) and Figs. 3(f)—(h) we show con-
stant energy contours of the spectral function at three
5
selected energies E denoted in Figs. 3(a) and 3(e) by (a) 1Cu Bi2Se3 Cohigh (b)
dashed horizontal lines. Instead of a single circle as 60 %)1
tHttˆhohDeehierecaixocga,negnsootrsanpsnaeitnlctgwerlnueaemrhpregioxnyfaggHc[oo9Dnn])FtooTsruusroffinfofctowiherflentatihlksyoeel-aalewitkiegaedeycnBofsrnpio2temSoceut3rrDuslPma(ydfeuoorerf, 0E−E(eV)F−00..055 TAMR (%)out23450000 TAMR (in-010 9EEE0AFFn---gEEEl000eFF1 8===ϕ0 (-00d0e.e0.gV325)75e0eVV360
here we find multiple circular and snowflake-like con- 10 F F
tours close to the Γ-point. The spin textures within the −1 low 0
10 20 30 40 50 60 70 -90 -60 -30 0 30 60 90
constant energy contours are computed from the spin- z (Å) Angle θ (deg)
resolvedspectralfunction. ForenergiesnearE =E0,the
F
spin textures shown in Figs. 3(b) and 3c) are quite dif- FIG.5. (a)Theposition-andenergy-dependentspectralfunc-
tion A(E;z) = 1 (cid:82)dk dk A(E;k ,k ;z) from the left Cu
ferent from the helical ones in isolated Bi2Se3 layer [15]. ΩBZ x y x y
lead, across Bi Se tunnel barrier, toward the right Co lead
Nevertheless, Fig. 3(d) shows that the remnant Dirac 2 3
for the heterostructure in Fig. 1(b). (b) The out-of-plane
cone still generates distorted helical spin texture wound-
TAMR (θ) ratio defined in Eq. (5) as function of angle θ
out
ing along a single circle but with out-of-plane S compo-
z betweenthemagnetizationmandthedirectionofcurrentin-
nent due to the presence of Co layer. jected along the z-axis in Fig. 1(b). Inset in panel (b) shows
The envisaged applications of TIs in spintronics are angular dependence of the in-plane TAMR (φ) ratio. In or-
in
based[1–5,7,8]onspintexturesliketheoneinFig.3(d) dertoconvergetheintegrationoverthetransversewavevector
k inEq.(4),weemployauniformgridof101×101k-points
since it maximizes [5, 6] generation of nonequilibrium (cid:107)
for TAMR (θ) and 251×251 k-points for TAMR (φ).
spin density when current is passed parallel to the TI out in
surface. However, utilizing spin texture in Fig. 3(d) in
lateralTI/FMheterostructureswouldrequiretoshiftE
F by the charge distribution at the metal/vacuum inter-
(by changing the composition of TI [4] or by applying
face and thereby confine wave functions into a Rashba
a gate voltage [2]) by few tenths of eV below E0 of
F spin-split quasi-2D electron gas [28]. Thus, Figs. 4(a)–
the undoped heterostructures in Fig. 3(a). For example,
(d) explains the origin of recently observed [29] SOT in
extreme sensitivity of spin-to-charge conversion was re-
theabsenceofanyadjacentheavymetalorTIlayersince
centlyobservedRef.[4]onthesurfaceof(Bi Sb ) Te
1−x x 2 3 passingcurrentparalleltoMLsofCohostingnonzeroin-
TIcoveredbya8nmthickCulayerasE oftheTIlayer
F plane spin textures will generate a nonequilibrium spin
was tuned, which is difficult to explain by assuming that
density[6]S andspin-orbittorque∝S ×m[8,26].
theDiracconeontheTIsurfaceremainsintactafterthe neq neq
Finally,weproposeapurelychargetransportmeasure-
deposition of the Cu layer (e.g., Ref. [4] had to invoke
ment that could detect which among the spin-textures
“instability of the helical spin structure”). On the other
shown in Figs. 3(b)–(d) resides at the Fermi level of
hand, it is easy to understand from Figs. 3(f)–(h) in-
TI/FMinterface. Ourschemerequirestofabricateverti-
terface demonstrating how spin textures at Bi Se /Cu
2 3 calheterostructureinFig.1(b)andmeasureitstunneling
interface change dramatically as one moves E (even
F anisotropic magnetoresistance (TAMR). The TAMR is a
slightly) below or above E0. Comparing Figs. 3(a)–(d)
F phenomenon observed in magnetic tunnel junctions with
with 3(e)–(h) makes it possible to understand the effect
a single FM layer [7, 27, 30, 31], where SOC makes the
of the magnetization of the Co layer, which modifies [25]
band structure anisotropic so that the resistance of such
RashbadispersionaroundE0 andthecorrespondingspin
F junctions changes as the magnetization m is rotated by
textures (particularly the out-of-plane S component).
z angle θ or φ in Fig. 1(b). The resistance change is quan-
The theoretical modeling of SOT in TI/FM [1, 8] or
tified by the TAMR ratio defined as [27, 31]
heavy-metal/FM [26] bilayers is usually conducted by
starting from strictly 2D Hamiltonians, such as HˆDirac R(α)−R(0)
TAMR (α)= . (5)
or the Rashba ferromagnetic model [25], respectively, out(in) R(0)
so that the FM layer is not considered explicitly. Fig-
ures 4(e)–(h) show that this is not warranted since the Here α ≡ θ for TAMR where magnetization in
out
Bi Se layerinducesproximitySOCandthecorrespond- Fig.1(b)rotatesintheplaneperpendiculartotheTI/FM
2 3
ing in-plane spin texture on the first ML of Co, which interface,andα≡φforTAMR wheremagnetizationin
in
decays to zero only after reaching plane 4 in Fig. 1(b). Fig.1(b)rotateswithintheplaneoftheTI/FMinterface.
In fact, we find non-trivial in-plane spin texture even In the case of TAMR (θ), R(0) is the resistance when
out
on the surface of Co in contact with vacuum, as shown m (cid:107) zˆ in Fig. 1; and in the case of TAMR (φ), R(0) is
in
in Figs. 4(b)–(d), which is nevertheless quite different the resistance when m(cid:107)xˆ in Fig. 1. Thus, TAMR (θ)
out
from those in Figs. 4(f)–(h). The in-plane spin texture changes due to the different orientations of the magneti-
in Figs. 4(b)–(d) is a consequence of the Rashba SOC zation with respect to the direction of the current flow,
enabled by inversion asymmetry due to Co surface [27] while the situation becomes more subtle for TAMR (φ)
in
where an electrostatic potential gradient can be created where the magnetization remains always perpendicular
6
(a) (b) (c) (d)
(e/Å3)
FIG. 6. (Color online). Top and side view of common unit cells for (a) Bi Se /Cu(111) and (b) Bi Se /Co(0001) bilayers.
2 3 2 3
Panels (c) and (d) show charge rearrangement around the interface of bilayers in panels (a) and (b), respectively.
to the current flow. Figure 5(b) demonstrates that the In order to avoid interaction with periodic images of the
largest TAMR (θ = ±90◦) is obtained by tuning the bilayer, 18 ˚A of vacuum was added in the z–direction.
out
Fermi level to E −E0 =−0.35 eV so that nearly heli-
F F ForthecaseofBi Se onCo(0001),themostfavorable
cal spin texture in Fig. 3(d) resides at the Fermi level. 2 3
positionyieldsabindingenergyof460meVperCoatom.
Another signature of its presence is rapid increase of
Both ML of Co and QL of Bi Se in direct contact gain
TAMR (θ)whentiltingmbysmallanglesθ awayfrom 2 3
out somecorrugation, roughlyaround(cid:39)0.1 ˚A, whiletheav-
the current direction. The in-plane TAMR (φ) shown
in erage z–distance between them is 2.15 ˚A. The average
in the inset of Fig. 5(b) is much smaller (and difficult to
distance between the ML of Cu and QL of Bi Se in di-
converge in the number of transverse k-points) quantity 2 3
rect contact is around 2.26 ˚A with smaller corrugation
whichdoesnotdifferentiatebetweenspintexturesshown
than in the case of Co(0001), while the binding energy
in Figs. 3(b)–(d).
is 294 meV per Cu atom. For other relative positions of
Bi Se layerwithrespecttoCu(111)andCo(0001)layers
2 3
the difference in binding energy is very small. Binding
METHODS
energies in both cases are rather small, thereby signal-
ingthedominantvdWforces. Nevertheless,somecharge
We employed the interface builder in the VNL [39] and rearrangement does occur at the interface due to push
CellMatch[32]packagestoconstructacommonunitcells back/pillow effect [37], as shown in Figs. 6(c) and 6(d)
for: (a) Bi Se /Cu(111) bilayer, where the common unit
2 3 where charge rearrangement is more pronounced for the
cell is 5 × 5 in size compared to the smallest possible
case of Bi Se /Cu(111) interface.
2 3
Cu(111) slab cell and copper is under compressive strain
ThecalculationoftheretardedGFinEqs.(1)and (3)
of 0.9 % while Bi Se lattice constant is unchanged; (b)
2 3
requires HDFT represented in the linear combination of
Bi2Se3/Co(0001) bilayer where Co(0001) has the same k(cid:107)
lattice constant as Bi Se , so the same unit cell as for atomic orbitals (LCAO) basis set which makes it possi-
2 3
Cu(111) is used without any strain on Co(0001). These bletospatiallyseparatesystemintotheactiveregionat-
two unit cells are illustrated in Figs. 6(a) and 6(b), re- tached to one or two semi-infinite leads, as illustrated in
spectively. In order to determine the best stacking of Figs. 1(a) and 1(b), respectively. We employ ATK pack-
atomic layers and the distance of Bi Se atoms with re- age [38] for pseudopotential-based LCAO noncollinear
2 3
spect to surfaces of Cu(111) and Co(0001), we use DFT DFT calculations yielding HDFT, from which we obtain
k(cid:107)
calculations as implemented in the VASP package [33]. retarded GFs and the corresponding spectral functions,
The electron core interactions are described by the pro- as well as the resistance in Eq. (4). In ATK calculations,
jector augmented wave (PAW) method [34], and vdW- we use Perdew-Burke-Ernzerhof (PBE) parametrization
DF [35] with optB88 is used as density functional [36] of generalized gradient approximation for the exchange-
inordertodescribevanderWaals(vdW)forcesbetween correlationfunctional;norm-conservingpseudopotentials
QLsofBi Se orbetweenBi Se andmetalliclayers. The for describing electron-core interactions; and LCAO ba-
2 3 2 3
cutoffenergyfortheplanewavebasissetis520eVforall sis set generated by the OpenMX package [22, 40] which
calculations, while k-points were sampled at 3×3 surface consists of s2p2d1 orbitals on Co, Cu and Se atoms, and
mesh. We use Cu and Co layers consisting of 5 MLs, s2p2d2 on Bi atoms. These pseudoatomic orbitals were
where 3 bottom MLs are fixed at bulk positions while generated by a confinement scheme [22] with the cutoff
the top two metallic MLs closest to Bi Se are allowed radius7.0and8.0a.u. forSeandBiatoms, respectively,
2 3
tofullyrelaxuntilforcesonatomsdropbelow1 meV/˚A. and 6.0 a.u. for Co and Cu atoms. The energy mesh
7
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