Spin-Charge Locking and Tunneling into a Helical Metal

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by  P. Schwab

Spin-Charge Locking and Tunneling into a Helical Metal
P. Schwaba, R. Raimondib, C. Gorinic
aInstitut fu¨r Physik, Universit¨at Augsburg, 86135 Augsburg, Germany
bCNISM and Dipartimento di Fisica ”E. Amaldi”, Universita` Roma Tre, 00146 Roma, Italy
cInstitut de Physique et Chimie des Mat´eriaux de Strasbourg (UMR 7504),
CNRS and Universit´e de Strasbourg, 23 rue du Loess, BP 43, F-67034 Strasbourg Cedex 2, France
(Dated: February 1, 2011)
We derive a kinetic equation for the electrons moving on the surface of a three-dimensional
topologicalinsulator. Duetothehelicalnatureoftheexcitationsbackwardscatteringissuppressed
in the collision integral, and the spin dynamics is entirely constrained by that of the charge. We
1 further analyze the tunneling between the helical and a conventional metal or ferromagnet. We
1 find that the tunnel resistance strongly depends on the angle between the magnetization in the
0 ferromagnet and the current in the helical metal. A nonmagnetic layer on top of the helical metal
2 amplifies the current-induced spin polarization.

n
a
J Topological insulators1,2 have recently attracted con- to remain fully occupied and thus not relevant for the
siderableinterest,especiallyaftertheirexperimentaldis- dynamics of the system.
0
3 covery in two-3 and three dimensions4–7. While insu- Becauseofthehelicalnatureoftheexcitations, sucha
lating in the bulk, such materials possess gapless helical surface conductor is called a helical metal and presents

] edgestateswhoseexistencedependson–andisprotected novel and interesting properties. For instance the veloc-
l
l by – time reversal invariance8–14. This makes the lat- ityoperatorisgivenbyx˙ =v e ×σ,sothattheparticle
a F z
terrobustagainsttime-reversalsymmetricperturbations current becomes j = 2v e ×s, s being the spin polar-
h F z
- (such as impurity scattering) and at the same time very ization. This means that the particle current is entirely
s
sensitive to time-reversal breaking ones (such as mag- constrained by the spin density or that, vice versa, the
e
m netic fields). When the topological insulator is a three- in-plane components of the spin density are constrained
dimensional system, the gapless excitations are confined by the particle current.
.
t to its surface and form a two-dimensional conductor and Such a constraint will also become apparent later in
a
m presentsnovelandinterestingproperties,seeforexample the kinetic equation for a disordered helical metal. The
Ref. [15]forarecentsummary. Inparticular,Burkovand equation will be valid when the Fermi energy is far from
-
d Hawthorn16 considered the problem of spin-charge cou- the Dirac point and satisfies the condition (cid:15)F (cid:29) 1/τ,
n pled transport on a helical metal and derived diffusion with τ the scattering time. We will follow a procedure
o equations for charge and spin. They predicted a distinc- similar to what was done for graphene in Ref. [18]. The
c tive magnetoresistance effect when the helical metal is starting point is the retarded Green function which in
[
placed between a ferromagnet and a normal metal. In the absence of disorder reads

1 thispaperweextendtheirworkinseveralways. Wefirst
v derive a kinetic equation which is valid even beyond the GRss(cid:48) =GR0[σ0]ss(cid:48) +GR·[σ]ss(cid:48), (2)
2 diffusive regime. In this latter regime we obtain a dif-
0 where
fusion equation which agrees with that of Burkov and
58 Hawthorn as far as the charge component is concerned. GR = 1(cid:0)GR+GR(cid:1) (3)
. On the other hand for the spin density we find a differ- 0 2 + −
1
ent behaviour, namely the spin dynamics is constrained
0 and
tofollowthechargeone. Secondly,weconsidertheeffect
1
1 of bringing the helical metal in contact with a ferromag- GR = 1kˆ×e (cid:0)GR−GR(cid:1), (4)
: net and discuss its unconventional magnetoresistance. 2 z + −
v
ForthesimplestcasetheeffectiveHamiltoniandescrib-
i with
X ing the surface states of a topological insulator has the
form2,17 GR =(cid:0)(cid:15)∓v k+µ+i0+(cid:1)−1 (5)
r ± F
a
H =vFk×ez·σ, (1) andkˆ beingtheunitvectorinthek-direction. Forclarity
we included in Eq. (2) the spin indices s and s(cid:48). The
where the parameter v is the velocity of the gapless ex-
F GreenfunctionGˇ hasthetwo-by-twomatrixstructureof
citations,e isaunitvectorperpendiculartothesurface,
z the Keldysh formalism
k is the two-dimensional momentum operator, σ are the
Pauli matrices, and units of measure such that (cid:126) = 1 (cid:18)GR(R,T;k,(cid:15)) G(R,T;k,(cid:15)) (cid:19)
have been used. The eigenstates of H form two bands Gˇ(R,T;k,(cid:15))= 0 GA(R,T;k,(cid:15)) , (6)
with linear dispersion, (cid:15) = ±v k, and we will assume
± F
in the following that the Fermi energy is located deep where R and T are the center-of-mass space and time
enough in the upper band for states in the lower band coordinates, while k and (cid:15) are the Fourier transformed
2
variables of the relative coordinates. The left-right sub- which is just the standard kinetic equation for a metal
tracted Dyson equation reads with an angle dependent scattering potential19,20. Here,
∂ Gˇ+ vF (cid:8)e ×σ·∂ ,Gˇ(cid:9)+i(cid:2)H,Gˇ(cid:3)=−i(cid:2)Σˇ,Gˇ(cid:3), (7) starting from a short range potential, we find
T 2 z R
1 1
W(ϕ,ϕ(cid:48))= (1+kˆ·kˆ(cid:48))= (1+cos(ϕ−ϕ(cid:48))), (15)
where [,] and {,} are the commutator and anticommu- τ τ
tator. On the right-hand-side of the equation the self-
energy Σˇ appears. For a delta-correlated impurity po- where the cosine term accounts for the absence of
tential with (cid:104)V(x)V(x(cid:48))(cid:105) = u2δ(x−x(cid:48)) and within the backscattering. The spin-dependent contributions to g
can be reconstructed from g , cf. Eq.(13):
Bornapproximationtheself-energyisproportionaltothe 0
Green function, integrated over the momentum
g ≈kˆ g , and g ≈−kˆ g . (16)
x y 0 y x 0
(cid:90) d2k
Σˇ(R,T;(cid:15))=u2 Gˇ(R,T;k,(cid:15)). (8)
(2π)2 Thegz componentisnonzeroonlytosubleadingorderin
1/v k and is after some algebra determined as
F F
We define a quasiclassical Green function as
(cid:32) (cid:33)
i (cid:90) 1 kˆ kˆ
gˇ(R,T;kˆ,(cid:15))= dξGˇ(R,T;k,(cid:15)), (9) g ≈ x(cid:104)kˆ g (cid:105)− y(cid:104)kˆ g (cid:105)+v kˆ ∂ g −v kˆ ∂ g .
π z v k τ y 0 τ x 0 F y x 0 F x y 0
F F
where ξ = v k−µ and the integration is performed in (17)
F
the vicinity of the Fermi energy. From Eqs. (2)-(5), one Equations(14), (16)and(17)areoneofthemainresults
obtainstheretardedcomponentofgˇforenergiescloseto obtained in this paper.
the Fermi level (|(cid:15)|(cid:28)v k ) as It is instructive to study the angular average of the
F F
kinetic equation, since the latter is closely related to the
1 1
gR = + kˆ×e ·σ, (10) continuity equation for the observables. Recall, for ex-
2 2 z ample, that the particle and spin densities are obtained
i.e. gR is a projector on the upper band of the Hamilto- from the quasiclassical Green function by taking the an-
nian (1). The self-energy reads gle average and integrating over the energy19,
i N (cid:90) N (cid:90)
Σˇ =− (cid:104)gˇ(cid:105), (11) n=− 0 d(cid:15)(cid:104)g (cid:105)+N eφ, s=− 0 d(cid:15)(cid:104)g(cid:105), (18)
τ 2 0 0 4
where 1/τ = πN u2, N = k /(2πv ) is the single-
0 0 F F where φ is the scalar electrical potential. From Eq. (14)
particle density of states at the Fermi energy and (cid:104)gˇ(cid:105)
we obtain the continuity equation for the density as
is the average of the quasiclassical Green function over
the Fermi surface. Finally, for the Keldysh component ∂ (cid:104)g (cid:105)+∂ ·(cid:104)v kˆg (cid:105)=0. (19)
T 0 R F 0
of the Green function we find from Eq. (7) the kinetic
equation Using Eq. (16) we find v (cid:104)kˆ g (cid:105) = −v (cid:104)g (cid:105) and
F x 0 F y
∂ g+ vF {e ×σ·∂ ,g}+iv k (cid:104)kˆ×e ·σ,g(cid:105) vF(cid:104)kˆyg0(cid:105) = vF(cid:104)gx(cid:105), i.e. we verify the general relations
T 2 z R F F z between the particle current and the in-plane spin den-
=−1g+ 1(cid:104)g(cid:105)+ 1 (cid:110)kˆ×e ·σ,(cid:104)g(cid:105)(cid:111). (12) sity stated already below Eq. (1). For the spin density
τ τ 2τ z we find
Notice that this is a matrix equation in spin space, v 1
gss(cid:48) = g0[σ0]ss(cid:48) +g·[σ]ss(cid:48). However, its structure can ∂Tsx+ 4F∂yn+ 2τsx =0 (20)
be considerably simplified. Equation (12) is derived un-
v 1
der the assumption that vFkF is much larger than all ∂Tsy− 4F∂xn+ 2τsy =0, (21)
other energy scales in the problem. Therefore, the lead-
ingcontributiontog mustcommutewithkˆ×e ·σ,orin fromwhichweidentify2τ asthespinrelaxationtime. In
z
(20) and (21) we ignored the scalar electric potential, φ.
other words, g is diagonal in the eigenstates of H. Since
In order to compare with Ref. [16] we will now dis-
thelowerbandhasnodensityofstatesattheFermilevel,
cuss the diffusive limit. The particle density obeys the
only the upper band contributes to g which is then pro-
diffusion equation
portional to the upper-band projector, i.e.
gss(cid:48) =g0(cid:16)[σ0]ss(cid:48) +kˆ×ez·[σ]ss(cid:48)(cid:17). (13) ∂ n−D∂2n=0, D = vF2 τ ,τ =2τ, (22)
T R 2 tr tr
From the spin-trace of Eq. (12) we obtain then a kinetic
i.e., the particle current is j = −D∂ n. The transport
equation for g , R
0 time τ being twice as long as the scattering time τ
tr
(cid:90) dϕ(cid:48) stems from the absence of backscattering in the helical
∂ g +v kˆ·∂ g =− W(ϕ−ϕ(cid:48))[g (ϕ)−g (ϕ(cid:48))],
T 0 F R 0 2π 0 0 metal. Equation (22) is consistent with Ref. [16], notice
(14) howeverthe different definition of thediffusion constant.
3
Sincethespinrelaxationtimeisveryshort(itequalsthe
transportscatteringtime)thespindynamicsisnotdiffu-
sive. However, Eqs. (20) and (21) are still valid. In the
diffusive limit the time derivative of the spin-density is ly
small compared to the spin relaxation term, i.e. the spin
density is given by the spatial derivative of the charge I
density so again we identify the general relation between
particle current and spin density.
We will now analyze the transport through the heli-
cal metal when it is contacted via a tunnel junction to a
lx
ferromagnet. We will find an unconventional magnetore-
sistance effect that arises since the tunneling probability
FIG.1. Schematicviewofsystemunderconsideration: elec-
betweenthehelicalmetalandtheferromagnetisstrongly trons can tunnel between a ferromagnet (top) and a helical
angle dependent: the overlap of two spinors with polar- metal (bottom). The size of the contact is l times l , the
x y
ization in mˆ- and mˆ(cid:48)-directions depends on the angle current is assumed to flow only in x-direction.
between the two vectors, |(cid:104)mˆ|mˆ(cid:48)(cid:105)|2 = 1 + 1mˆ ·mˆ(cid:48). The
2 2
statesintheconductionbandofthehelicalmetalarepo-
larized as mˆ(cid:48) =kˆ×e , so that the tunneling probability tunneling between the helical metal and the spin up or
z
from a state mˆ in the ferromagnet into a state in the downbandoftheferromagnet. Asanticipatedabove,the
helical metal with momentum k depends on the angle tunneling probability between the helical metal and the
between mˆ and k. ferromagnetisakˆ-dependentfunction. Aftertheangular
In order to make these considerations more formal we average and integrating over the energy we obtain the
introduce the tunneling Hamiltonian following continuity equation for the charge density, ρ=
(−e)n,
(cid:90)
(cid:88)
H = d2xt(x)ψ†(x)ψ (x)+c.c., (23)
t s F,s ∂ ρ+∂ ·j=−δ(R)[G (U −U )+G (U −U )]
T R ↑ ↑ ↓ ↓
s
G −G
−δ(R) ↑ ↓eˆ ×mˆ ·j, (28)
whereψ†(x)isthefieldoperatorforanelectronwithspin e2N v z
s 0 F
s in the helical metal and ψ (x) that for the ferromag-
F,s having introduced the tunneling conductances
net. For a point-like (on the quasiclassical scale) contact
at x=0 the tunneling amplitude is
G =πe2|t|2N N (29)
↑,↓ 0 ↑,↓
t(x)=tδ(x). (24)
and the integrals
Eq. (23) leads then to an additional contribution to the 1 (cid:90) 1 (cid:90)
self-energy of the form U = d(cid:15)(cid:104)g (cid:105), U = d(cid:15)(cid:104)g (cid:105). (30)
2e 0 ↑,↓ 4e ↑↓
(cid:90) d2k
Σ (R,T;(cid:15))=|t|2δ(R) Gˇ (R,T;k,(cid:15)) (25) The latter have the meaning of a voltage, or more pre-
t (2π)2 F
cisely of a magneto-electrochemical potential21, cf. Eq.
(18). Sources and sinks for the charge density in the he-
where Gˇ is the Green function of the ferromagnet. We
F lical metal appear on the right hand side of the equation
assume that the conduction electrons in the ferromagnet
due to tunneling. One observes that a tunnel current
can be described in terms of an incoherent superposition
cannot only be driven by a voltage across the contact,
of majority and minority carriers, so we write
but also by a current bias in the helical metal.
Gˇ =Gˇ P +Gˇ P , (26) For simplicity we will assume in the following equal
F F,↑ ↑ F,↓ ↓
magneto-electrochemicalpotentialsforspinupanddown
where P↑,↓ = (σ0 ±mˆ ·σ)/2 projects on states parallel electrons in the ferromagnet, U↑↓ = U↑ = U↓. Further-
antiparallel to mˆ. The kinetic equation (14) becomes more, instead of a point-like tunnel contact we will con-
sider an extented tunneling region as depicted in Fig. 1.
1 1 (cid:16) (cid:17)
(∂ +v kˆ·∂ )g =− g + (cid:104)g (cid:105)+kˆ·(cid:104)kˆ(cid:48)g (cid:105) This is achieved by replacing the δ-function on the right
T F R 0 τ 0 τ 0 0 handsideofEq. (28)byanappropriatefunctioncharac-
−π|t|2δ(R)N (1+mˆ ·kˆ×eˆ )(g − 1(cid:104)g (cid:105)) terising the shape of the contact. In particular, we will
↑ z 0 2 ↑ study in detail a junction with a long extension l in y-
y
−π|t|2δ(R)N (1−mˆ ·kˆ×eˆ )(g − 1(cid:104)g (cid:105)), (27) direction22. We replace then the δ-function in Eq. (28)
↓ z 0 2 ↓ by
whereg↑,↓ andN↑,↓ arethequasiclassicalGreenfunction (cid:40)
l−1δ(x) for−l /2<y <l /2
and the density of states in the ferromagnet. The terms δ(R)→ y y y , (31)
in the second and third line of this equation describe 0 otherwise
4
and we assume that current flows only in x-direction. the spin-density in the normal metal we arrive at
The continuity equation thus becomes one-dimensional.
(cid:126)/2e2 1
Duetotheremainingone-dimensionalδ-functionthecur- ∂ sa+∂ ·ja =− δ(R)sa
rent density jumps at x = 0. The size of this jump is t R Rt N↑↓
determined by integrating the continuity equation with
(cid:126)/2e2 δ(R)
respect to the x-coordinate for a region close to the tun- − (j×e ) , (35)
R eN v z a
neling contact with the result t 0 F
where ja is the spin current in the normal metal and
G +G G −G j +j j the charge current in the helical metal. Whereas the
j −j =− ↑ ↓(U −U )+ ↑ ↓ + −mˆ ,
+ − l ↑↓ l N v e2 2 y tunnelingofchargeiscontrolledbythevoltageacrossthe
y y 0 F
(32) tunneljunction,thetunnelingofspiniscontrolledbythe
where j = j(x = ±0) is the current right and left currentdensityinthehelicalmetal,i.e. acurrentflowing
±
fromthetunnelcontactandweassumedthatδ(x)j(x)= parallel to the junction. Thus injection of a pure spin
δ(x)(j + j )/2. When we fix the current left to the currentintothenormalmetalispossible. Furthermore,if
+ −
contact to zero we determine the tunnel resistance as spintunnelinginandoutarebalancedsothatthereisno
spin current injection, there is a steady state determined
1 (cid:126) λ mˆ G −G by
R = − F y ↑ ↓, (33)
t G↑+G↓ e2 2ly G↑+G↓ 1 1
s=− j×e , (36)
N eN v z
↑↓ 0 F
where for clarity we put back (cid:126). Through mˆ the tunnel
y
resistance depends on the orientation of the ferromagnet which is equivalent to
with respect the the x-axis, similar to what was found
(cid:12) 2N (cid:12)
by Burkov and Hawthorn16. The relative size of the s(cid:12) = ↑↓s(cid:12) . (37)
magnetoresistance effect increases with increasing tun- (cid:12)normalmetal N0 (cid:12)helicalmetal
nel conductance being controlled by the dimensionless
Anormalmetalontopofthehelicalmetalthusamplifies
parameters (G − G )/(e2/(cid:126)) and λ /l . Apparently
↑ ↓ F y the current-induced spin polarization.
thetunnelresistancemayevenbecomenegativeforlarge
We finally turn to the question of how robust the ef-
enough values of these two parameters. However, as we
fects found are, having in mind that the effective Hamil-
will see below, when this happens equation (33) is no
tonian (1) is only valid in the vicinity of the Dirac point,
longer valid and a more careful treatment which takes
whereas the interpretation of the experimentally ob-
intoaccountafinitecontactareaofthetunneljunctionis served Fermi surfaces4,5,7 requires at finite doping terms
needed. To analyze a junction with a finite width ly and thatarequadraticandevencubicink23. Forexample,to
length l we make the following replacement in Eq. (28):
x thirdorderink,theeffectiveHamiltonianforthesurface
δ(R) → (lxly)−1 when R is inside the tunnel junction statesofthetopologicalinsulatorsBi Te andBi Se is23
2 3 2 2
and δ(R) → 0 otherwise. This leads to the tunnel resis-
tance k2 λ
H =vk×e ·σ+ + (k3 +k3)σ , (38)
z 2m∗ 2 + − z
(cid:18) (cid:19)
1 λ G −G
Rt = G +G f lF ↑e2/(cid:126) ↓mˆy , (34) where the Dirac velocity contains a second order correc-
↑ ↓ y tion, v = v (1+αk2), k = k ±ik , and α, m∗,λ are
F ± x y
the parameters characterizing the strength of the higher
withthefunctionf(x)=x/(ex−1). Forsmallx,f(x)≈ ordercorrections. Inordertounderstandtheeffectthese
1−x/2andonerecoversEq.(33). Whenxincreasesand extratermsintheHamiltonianhaveonthemagnetoresis-
becomesoforderone,theresistanceremainspositive,but tanceandthespininjection,recallthattheoriginofboth
depends strongly on the sign of x, i.e., the orientation isthehelicityoftheconductionelectrons. Thequadratic
of the ferromagnet. For x > 0, the resistance decreases termintheHamiltonian(38)hasnospinstructure, does
exponentiallyintheparameter(G↑−G↓)/(e2/(cid:126))(λF/ly), not affect the helicity of the eigenstates, and therefore
while it increases linearly for x<0. The tunnel junction cannot qualitatively change our results. The cubic term
is then acting as a spin-diode. ontheother handdisturbsthehelicityof theeigenstates
In this final part of the paper we consider tunnel- (the angle between the velocity and the spin now de-
ing from the helical metal into a normal, nonmagnetic pends on the position on the Fermi surface), so for a
metal. The tunnel resistance for the charge becomes strong cubic term we leave the region where our results
R =1/(G +G ) since spin up and down have both the are reliable.
t ↑ ↓
samedensityofstates,N =N =N andalsoidentical In conclusion we have derived a kinetic equation for a
↑ ↓ ↑↓
tunnel conductance. Nevertheless tunneling into a nor- helical model and have shown that the spin dynamics is
malmetalisofinterestsince,aswewilldemonstrate,the constrained to follow that of the charge. In the diffusive
helical metal injects spin into the normal metal. Going regime for charge, the spin density is described in terms
through the same steps as in Eqs. (25)–(28) but now for of the charge density gradient. Furthermore, when the
5
helical metal is placed in contact with a ferromagnetic ratethatcanbecontrolledindependentlyfromtheinjec-
metal, the tunneling current depends on the relative ori- tion of charge carriers.
entation between the current and the polarization in the We thank the Deutsche Forschungsgemeinschaft
ferromagnet. Inthelimitoflargetunnelingconductance, (SPP1285) and the French Agence Nationale de la
thedeviceactsasaspin-diode. Ahelicalmetalincontact Recherche (grant no. ANR-08-BLAN-0030-02) for finan-
with a normal metal injects spin into the latter, with a cial support.
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