Table Of ContentSpin Helix of Magnetic Impurities in Two-dimensional Helical Metal
Fei Ye1,2, Guo-Hui Ding3, Hui Zhai2 and Zhao-Bin Su4
1. College of Material Science and Optoelectronics Technology,
Graduated University of Chinese Academy of Science, Beijing 100049, P. R. China
2. Institute for Advanced Study, Tsinghua University, Beijing 100084, P. R. China
3. Department of Physics, Shanghai Jiaotong University, Shanghai 200240, P. R. China and
4. Institute of Theoretical Physics, Chinese Academy of Science, Beijing 100089, P. R. China
We analyze the Ruderman-Kettel-Kasuya-Yosida(RKKY) interaction between magnetic impuri-
tiesembeddedinthehelicalmetalonthesurfaceofthree-dimensionaltopologicalinsulators. Apart
fromtheconventionalRKKYterms,thespin-momentumlockingofconductionelectronsalsoleads
0 to a significant Dzyaloshinskii-Moriya (DM) interaction between impurity spins. For a chain of
1 magneticimpurities,theDMtermcanresultinsingle-handedspinhelixonthesurface. Thehand-
0 edness of spin helix is locked with the sign of Fermi velocity of the emergent Dirac fermions on the
2 surface. We also show the polarization of impurity spins can be controlled via electric voltage for
dilute magnetic impurity concentration.
n
a
PACSnumbers: 72.25.Dc,73.20.-r,75.30.Hx,85.75.-d
J
1
3
It is now known that a three-dimensional(3D) insula- symmetry analysis and perturbation calculation, which
tors with time reversal symmetry can be classified into is usually absent in systems like graphene[20–22]. Since
]
i two categories: ordinary and topological insulator(TI) theemergentDiracHamiltonianofhelicalmetalisapure
c
s [1–5]. Thoughbothofthemarefullygappedinthebulk, spin-orbit coupling, the magnitude of DM term is of the
- TI has metallic surface states robust against any time same order of the conventional RKKY one. Secondly,
l
r reversal invariant perturbations. These metallic surface the DM term can lead to single-handed spin helix if the
t
m states dubbed as “helical metal” can be described by a impurities are aligned into a chain, of which the handed-
t. two-dimensional(2D) massless Dirac equation. Compar- ness depends on the sign of the Fermi velocity vF of the
a ingtootheremergentrelativisticmaterialssuchassingle- helical metal. For strong TIs with a single Dirac cone
m
layergraphene,thesurfacestatesofTIshavetwounique attached to each surface, v is fixed, therefore the cor-
F
- features. One is that the number of Dirac nodes is odd. responding spin helix must be single-handed. The spin
d
Infact, accordingtoano-gotheorem[6], theDiracnodes helix is illustrated in Fig. 1, where two types of Dirac
n
alwaysappearinpairsintheconventional2Dlatticesuch fermions are considered and the spin helix are perpen-
o
c as graphene. A surface state of TI can invalidate no- dicular(Fig. 1a) and parallel (Fig. 1b) to impurity chain,
[ go theorem because a pair of Dirac nodes are separated respectively. ThisisahallmarkofastrongTI,andcould
1 onto two opposite surfaces. TI with single Dirac cone bepossiblydetectedbyspin-polarizedscantunnelingmi-
v has been theoretically studied and experimentally ob- croscopy, or optical measurement like Kerr effect.
1 served by angle-resolved photo-emission spectroscopy[7–
1 11]. The other feature is that electron spin and mo- (a) z E (b) z E
1 y y
mentum are intimately locked in helical metal, origi-
0
2. nated from strong spin-orbit coupling. Evidence of the kx ky kx ky
spin-momentumlockinghasalsobeenobservedinrecent
0
experiments[12–15].
0
1 The effect of magnetic impurities in a metal is a cen-
: tral issue in condensed matter physics, since the cou- x x
v
pling between impurity and conduction electrons can
i
X lead to many intriguing phenomena such as Kondo ef-
r fectandRuderman-Kettel-Kasuya-Yosida(RKKY)inter- FIG. 1. Illustration of spin helix of a chain of magnetic im-
a action. The analogy of these effect in a helical metal of- purities. Two types of Dirac Hamiltonian, Hˆa and Hˆb(see
0 0
ten reveals intrinsic property of a TI, which has received Model section), are considered, which lead to spin helix (ma-
genta vectors) perpendicular(a) and parallel(b) to the chain
considerable attention recently [16–19]. In this letter we
direction, respectively. The insets show spin orientation of
point out a novel feature in RKKY interaction between
conduction electron in momentum space.
impurity spins in helical metal, which manifests directly
the two features of TI mentioned above. Firstly, due
Model: We start from the Hamiltonian of the conduc-
to the spin-momentum locking of conduction electron,
tion electrons with only one Dirac cone
we show the induced RKKY interaction must contain
ananisotropicDzyaloshinskii-Moriya(DM)termbyboth Hˆa =(cid:88)cˆ†((cid:126)v k·σ−E 1)cˆ , (1)
0 k F F k
k
2
withFermienergyE andPaulimatrixσ. Viaaunitary the even order terms for time reversal invariant system.
F √
transformation aˆks = [cˆk↑+sgn(vF)se−iθkcˆk↓]/ 2 with To the lowest order, the most general form of the in-
θ the angle of vector k and s=±1, it can be diagonal- teraction is Hˆ (r ) = (cid:80) Γ (r )Sˆα1Sˆα2 with
izked as Hˆa = (cid:80) (s(cid:126)|v |k−E )aˆ† aˆ . In practice, r ≡r −r . Wrkkiytho1u2tspin-oαr1bαi2tcoαu1αp2ling12asi1nth2eusual
0 s=±1 F f ks ks 12 1 2
the recently discovered TI material, e.g., Bi Se has sur- case, only terms with α = α can exist due to a rota-
2 3 1 2
face state described by the Rashba type of Hamiltonian tional symmetry in spin space. However, the situation is
Hˆb =(cid:80) cˆ†[(cid:126)v (k×σ) −E ]cˆ [8, 9], which is related quite different for the helical metal, since Hˆa is invari-
0 k k F z f k 0
to Hˆa by a rotation of k around z axis by π/2, hence we ant only under a joint rotation in both spin and orbital
0
can focus on Hˆa in the following. space, which allows more terms as soon become clear.
0
Suppose there are N impurity spins located at r Letusconsideraglobalrotationaroundz axisbyϕwith
imp n
denoted by Sn, which interact with conduction electrons the form Rˆez,ϕ ≡ exp[iϕez ·ˆscond] × exp[iϕez · (Sˆ1 +
via the following spin-spin coupling Sˆ2)]×exp[iϕez ·Lˆ] with total spin of conduction elec-
trons ˆs = (cid:82) drcˆ†(r)σcˆ(r)/2 and orbital angular mo-
cond
N(cid:88)imp mentum Lˆ ≡ (cid:82) drcˆ†(r)(r×p)cˆ(r). Under this rotation,
Hˆ = λ sˆz(r )Sˆz +λ [sˆx(r )Sˆx+sˆy(r )Sˆy],(2)
I z n n ± n n n n Hˆa is obviously invariant, but Hˆ (r ,r ) → Hˆ (r(cid:48),r(cid:48))
n=1 0 I 1 2 I 1 2
with r(cid:48) ≡ e−iϕ(xpy−ypx)reiϕ(xpy−ypx), which breaks the
where sˆx,y,z(r) are the spin operators of the conduction rotational symmetry. However since the energy behaves
electron at r. The coupling constants are assumed to be like a scalar, one expects Hˆ (r ) = Hˆ (r(cid:48) ), then
rkky 12 rkky 12
isotropic in xy plane λx =λy ≡λ±, but may differ from whatweareleftwithistoconstructRˆ -invariantswith
e,ϕ
the z-component. The total Hamiltonian is simply the vectors e (≡r /r ), Sˆ and Sˆ .
sum Hˆ =Hˆa,b+Hˆ . 12 12 12 1 2
0 I There are many such Rˆ -invariants terms, most of
Before proceeding with the discussion of the effective ez,ϕ
which can be further eliminated by the following two
interactionbetweenimpurityspins,wefollowRKKY[23–
symmetries. The first one is exchanging r and r which
25] to divide Hˆ into two parts: Hˆ and Hˆ . Hˆ con- 1 2
I I0 I1 I0 obviouslyleavesHˆ invariant. Thustermslikee ·(Sˆ ×Sˆ )
sists of the diagonal term of aˆ† aˆ and can be written z 1 2
(cid:16) ks ks(cid:17) are not allowed. The second one is a global rotation
as HˆI0 = λ±(cid:80)k(ek ·ˆsk) ek·Sˆimp , where ek = k/k, Rˆe12,π around axis e12 by π, which rules out invariants
ˆs = cˆ†σcˆ , and Sˆ = V−1(cid:80)NimpSˆ is the density like [e12 ×(S1 ×S2)]z that changes sign under rotation
k k k imp n=1 n R . Hence, we are left with the following two terms
of impurity spin. And the remaining part is off-diagonal e12,π
denotedbyHˆ . NoticethatforthehelicalHamiltonian, in addition to the conventional ones, e12·(Sˆ1×Sˆ2) and
I1
theelectriccurrentoftheconductionelectronsispropor- (e12 · Sˆ1)(e12 · Sˆ2). One then constructs the effective
tionaltotheirspins,forinstance,Jˆ =ev (cid:80) ˆs forHˆa, Hamiltonian of impurity spins as
F k k 0
therefore HˆI0 actually implies a direct coupling between Hˆ (r )=Γ (r )[λ2(SˆxSˆx+SˆySˆy)+λ2SˆzSˆz]
the electric current and the impurity spins, which pro- rkky 12 0 12 ± 1 2 1 2 z 1 2
videsamechanismtocontroltheimpurityspinsthrough +Γ1(r12)λ2±[(e12·Sˆ1)(e12·Sˆ2)]
the electric voltage for dilute impurity concentration. +Γ (r )λ λ [e ·(Sˆ ×Sˆ )], (3)
DM 12 z ± 12 1 2
RKKYinteraction: Forsimplicitywefocusonthecase
of two impurities at r1 and r2, respectively. The results wheretheΓ0-termisjusttheconventionalRKKYterms.
given below via the symmetry analysis are also valid for Derivation of Γ0,1,DM: The coefficients Γ0,1,DM in
many impurity case as long as we consider two-body in- Eq. (3) can be computed following RKKY’s second or-
teractions. The RKKY interaction can be obtained by der perturbation[23–25]. Let us consider the following
integrating out the degree of freedom of conduction elec- generalized partition function which is partially traced
trons,followedbyanexpansiontothesecondorderofthe over conduction electrons, Z = Trconde−β(Hˆ0+HˆI). To
coupling parameters λz,±[23–25]. The effective interac- the quadratic order of λz,±, we obtain
tion between impurity spins thus obtained contains only
Z ≈exp β (cid:88) (cid:88)(cid:88)fk1s1(1−fk2s2)[ (cid:88) λ Fa Sˆa][ (cid:88) λ Fb Sˆb ]ei(rm−rn)(k1−k2) (4)
Z V2 ξ −ξ a k1s1;k2s2 n b k2s2;k1s1 m
0 k1s1k2s2n,m k2s2 k1s1 a=x,y,z b=x,y,z
where Z is the partition function of conduction elec- tion function, and the three F-functions take the forms
0
trons, fks =[1+eβ(s(cid:126)|vF|k−Ef)]−1 is the Fermi distribu- Fkx1s1;k2s2 = 14sgn(vF)(s1e−iθk1 +s2eiθk2), Fky1s1;k2s2 =
3
4isgn(vF)(s1e−iθk1 − s2eiθk2) and Fkz1s1;k2s2 = 14(1 − 1 Γ0(r) 1 Γ0(r)
s1s2e−i(θk1−θk2)), which depend only on θk1,2, the angle 0.5 ΓDΓM1((rr)) 0.5 ΓDΓM1((rr))
of k . By comparing with Eq. (3), one can extract the
1,2
coefficientsΓ fromEq.(4)straightforwardly. After 0 0
0,1,DM
integrating over θk1,2, we have Γ0(r)=−2[A(r)−B(r)], -0.5 kFa0 =1.8 -0.5 kFa0 =0.5
Γ (r)=−4B(r)andΓ (r)=−2C(r),wherethefunc-
1 DM -1 -1
tions A, B and C read in the zero temperature limit 1 2 3 4 1 2 3 4
(cid:90) (cid:90) r r
1 (cid:88)
A(r)= x dx x dx J (x )J (x )
8(cid:126)|vF|r3 s1s2 1 1 2 2 0 1 0 2 FIG. 2. Plot of Γ0,1,DM as functions of the distance between
impurities r, k a = 1.8 (left) and k a = 0.5 (right). The
θ(x −x )θ(x −x ) F 0 F 0
× F 1 2 F distance between impurities is in unit of lattice spacing a0.
(cid:90)s2x2−s1(cid:90)x1 The coefficient 8(cid:126)vFa30 is taken as one for convenience.
1 (cid:88)
B(r)= x dx x dx J (x )J (x )
8(cid:126)|v |r3 1 1 2 2 1 1 1 2
F
s1s2 the amplitude fluctuation along x-axis, which satisfy the
×θ(xF −x1)θ(x2−xF) commutationrelation[θˆn,ρˆm]=iδm,n,andarerelatedto
s2x2−s1x1 Sˆ throughSˆx =ρˆ, Sˆy+iSˆz =eiθˆ[(S−ρˆ)(S+ρˆ+1)]1/2
(cid:90) (cid:90) n n n n
C(r)= 8s(cid:126)g|nv(v|Fr)3 (cid:88) x1dx1 x2dx2 and Sˆny −iSˆnz = [(S −ρˆ)(S +ρˆ+1)]1/2e−iθˆ. One may
F verify these equations satisfying the angular momentum
s1s2
[s J (x )J (x )−s J (x )J (x )] algebra, and Sˆ2 = S(S + 1). In continuum limit the
1 1 1 0 2 2 0 1 1 2
effective Hamiltonian for the dynamics of impurity spins
θ(x −x )θ(x −x )
× F 1 2 F (5) can be written as
s x −s x
2 2 1 1 (cid:90)
where x ≡ kr, xF = kFr, θ(x) is Heaviside step func- Hˆimp = dx[−D1cos(d∂xθˆ)+D2cos(2θˆ)
tion, and J (x) are the first and second order Bessel
0,1 +D sin(d∂ θˆ)+D ρˆ2] (7)
functions,respectively. NotethatΓ (r)hasalreadybeen 3 x 0
0
obtained in Ref.[18], which agrees with ours. It is the Γ1 with [θˆ(x),ρˆ(x(cid:48))]=iδ(x−x(cid:48)). The parameters are given
andΓ termsthathavenotbeenaddressedbefore,and
DM by D =J d, D =(J +J )S(S+1)/(2d), D =(J −
the physics discussed in this letter is essentially coming 0 x 1 z y 2 z
J )S(S+1)/(2d) and D =J S(S+1)/d.
from them. y 3 DM
We first consider the isotropic case, i.e., J =J . The
The numerical results of Γ (r) are plotted in z y
0,1,DM θˆ(x) field has a helical background(see Fig. 1(a)), with
Fig.3whereacutoffissetforxandtheCauchyprinciple
helicalangleηsatisfyingtan(ηd)=2J /(J +J ). The
integration is understood when singularities are encoun- DM z y
sign of η is the same as that of J , and its amplitude
teredinEq.(5). Despiteoftheoscillationoftheconven- DM
is saturated to π/2d as J is large enough as shown in
tional RKKY interaction with respect to r for large k , DM
F the lower panel in Fig. 3. Replacing θˆ(x) in Eq. (7) with
it is interesting to note that the DM interaction is of the
same order of the conventional ones, and depends on the ηx+θˆ(x),wehaveHˆimp =(cid:82) dx[−(cid:112)D12+D32cos(d∂xθˆ)+
signofvF asseenfromtheexpressionofC(r)inEq.(5). D0ρˆ2] describing the quantum fluctuation upon the he-
Single-handed Spin Helix: To reveal the effect of DM lical background. The low energy excitations can be
interaction,wefocusonasimplecaseofone-dimensional obtained by expanding cos(d∂xθˆ) to the second order
chain of impurities aligned with equal spacing d along of d, which leads to a linear spectrum in k as ωk =
x-axis, and the corresponding Hamiltonian reads d(2D0)1/2(D12+D32)1/4|k|.
Next we consider the effect of spin anisotropy, i.e.
(cid:88)
Hˆ = (−J SˆzSˆz −J SˆySˆy +J SˆxSˆx ) J (cid:54)=J . Withoutlossofgenerality,weassumeJ >J (if
imp z n n+1 y n n+1 x n n+1 z y z y
n J < J , one can shift θˆ → θˆ + π/2 to get posi-
z y
+JDM(SˆnzSˆny+1−SˆnySˆnz+1) (6) tGivoerdDon2)(.SGIn) mthoisdeclaHsˆe the≈H(cid:82)amdxiltuo[nKia−n1(b∂ecθˆo)m2+esKaρˆs2in]e+-
where J = −Γ (d)λ2, J = −Γ (d)λ2, J = [Γ (d)+ √ imp 2 x
z 0 z y 0 ± x 0 h∂ θˆ+D cos( 8πθˆ), which is written in the standard
Γ (d)]λ2, and J =−Γ (d)λ λ . Here we consider x 2
1 ± DM DM z ± form by rescaling θˆand ρˆin order to use the well known
the case for k close to the Dirac point, where we can
F
results in literature, e.g. in Refs.[26, 27]. The coeffi-
simplyassumeJy,z >Jx >0ascanbereadoffinFig.2b (cid:112)
cients take the form u = d J (J +J )S(S+1), K =
forr ≈1. Theothersituationsincludingtwodimensional x y √z
(cid:112)
π−1 J /(J +J )S(S+1) and h = 2πJ S(S+1).
arrays of magnetic impurities are left for future studies. x z y DM
In case K > 1, the cosine term is irrelevant, so that the
In this case, the spins are lying in yz plane in the
system is massless and the spin helix takes place for any
classicallimit. Therefore,wecanintroducetwovariables
θˆ and ρˆ to describe the spin rotation in yz-plane and finite h. However in case K < 1, the theory turns out
n n
4
Jz(cid:0)Jy particles.
WethankC.X.Liu,D.Qian,J.WangandY.Y.Wang
non-helicalstate
for stimulating discussions. FY is financially supported
by NSFC Grant No. 10904081. HZ is supported by
theBasicResearchYoungScholarsProgramofTsinghua
lefthelix righthelix University, NSFC Grant No. 10944002 and 10847002.
(0;0) J
DM
helicalangle
(0;0)
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with the sign of Fermi velocity v of the emergent Dirac
F
