Classical spins in topological insulators

Abstract

Following the recent theoretical proposal and experiment on quantum spin Hall effect in HgTe/CdTe quantum wells, we consider a single magnetic impurity localized in the bulk of the system, which we treat as a classical spin. It is shown that there are always localized excited states in the bulk energy gap for arbitrarily strong impurity strength in inverted region, while the localized excited states vanish for very strong impurity strength in normal region. Similar conclusion also applies to three-dimensional topological insulators. This distinct difference serves as another novel criterion for the conventional and topological insulating phases when the time-reversal symmetry is broken, and can be easily experimentally observed through the STM and/or ARPES experiments. Comment: 5 pages, 4 figures.

Full text

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arXiv:1005.0299v1 [cond-mat.mtrl-sci] 3 May 2010
Classical spins in topological insulators
Qin Liu1and Tianxing Ma2
1Department of Physics, Fudan University, Shanghai 200433, China
2Max-Planck-Institut f¨ ur Physik Komplexer Systeme,
N¨ othnitzer Strasse 38, 01187 Dresden, Germany
(Dated: May 4, 2010)
Following the recent theoretical proposal and experiment on quantum spin Hall effect in
HgTe/CdTe quantum wells, we consider a single magnetic impurity localized in the bulk of the
system, which we treat as a classical spin. It is shown that there are always localized excited states
in the bulk energy gap for arbitrarily strong impurity strength in inverted region, while the localized
excited states vanish for very strong impurity strength in normal region. Similar conclusion also
applies to three-dimensional topological insulators. This distinct difference serves as another novel
criterion for the conventional and topological insulating phases when the time-reversal symmetry is
broken, and can be easily experimentally observed through the STM and/or ARPES experiments.
PACS numbers: 75.30.Hx 73.20.At 72.25.Dc
The quantum spin Hall (QSH) effect is a state of mat-
ter with topological properties distinct from those of con-
ventional insulators. [1–3] The first proposal of experi-
mental realization of this effect is given in the work by
Bernevig et al. [3] where they consider the HgTe/CdTe
semiconductor quantum wells (QWs), and show that
when the thickness of the QW is varied, the electronic
states change from a normal to an inverted type at a crit-
ical thickness dc. This transition is a topological quan-
tum phase transition between a conventional insulating
phase and a phase exhibiting the QSH effect with a sin-
gle pair of helical edge states. This phase transition can
be understood by the relativistic Dirac model in (2+1)
dimensions, which mimic the electronic states near the Γ
point. At the quantum phase transition point, d=dc, the
mass term in the Dirac equation changes sign, leading
to two distinct U(1)-spin and Z2topological numbers on
either side of the transition. Recently, the QSH phase in
HgTe/CdTe QWs has been observed in the transport ex-
periments, [4] which confirms Bernevig et al.’s theoretical
predictions.[3]
Following this pioneer work, there emerge various dis-
cussions on the novel properties of the topological in-
sulating phase in both two- and three-dimensional (2-
, 3-D) systems, [5–8] however, most of these are con-
sidered within the framework of the preservation of the
time-reversal symmetry (TRS), among which, we notice
that two of them show that the novel properties of this
topological system can also be manifested by breaking
the TRS on the surface through the so-called topological
magneto-electric effect [9] or local charge and spin den-
sity of states. [10] In the meanwhile, we notice that the
system with Mn doped impurities in the bulk of the HgTe
QWs has been discussed in Ref.[11], where by breaking
the TRS in the bulk, the quantum anomalous Hall effect
is realized. On the other hand, it is well known that a
single magnetic impurity in a superconductor breaks the
TRS and induces low energy bound states in the super-
conducting gap. [12, 13]
Motivated along this line, we discuss the presence of a
single magnetic impurity located in the 2D bulk of the
HgTe/CdTe QWs, which we treat as a classical spin in
both normal and inverted regimes. Similar to the discus-
sions in BCS superconductors by Shiba in 1968 [13], we
show that in the inverted regime of the HgTe/CdTe QWs,
there are always localized excited states (LES) in the
bulk energy gap for arbitrarily strong impurity strength,
while in normal regime, the LES vanish into the bulk for
very strong impurity strength. This distinct difference
of the response to the single magnetic impurity in bulk
serves as another novel criteria for the conventional and
topological insulating phases when the TRS is broken,
and can be experimentally observed through the STM
and/or ARPES measurements.
The starting point of this paper is the effective four-
band model[3] H0(?k) = εkσ0τ0+ Mkσ0τ3+ Akxσ3τ1+
Akyσ0τ2in HgTe/CdTe QWs around the Γ point in the
basis of |E1,+?, |H1,+?, |E1,−?, |H1,−?, plus that
of a short-range single magnetic impurity located at the
origin. The exchange interaction in Mn doped HgTe QWs
has been discussed in several literatures, [11, 14] where it
is established that the s-band and p-band electrons have
different sp-d exchange coupling strength. To focus on
the physical picture, we first consider the isotropic case
where Js= Jp= J, then the full Hamiltonian takes the
form
H =
?
k
c†
kH0(?k)ck+J
2
?
kk′
c†
k(?S ·? σ)τ0ck′
(1)
Here?S is the spin vector of the magnetic impurity, and
the Pauli matrix σi’s act on spin space while τi’s act on
the two electric subbands space, σ0 and τ0 are both 2
by 2 unit matrices. We will show later that our result is
robust for a general form of the exchange coupling. The
full Green’s function (GF) of Hamiltonian (1) is obtained
through the equation of motion formulation [13] as
Gkk′(ω) = G0
k(ω)δkk′ + G0
k(ω)t(ω)G0
k′(ω)(2)

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where the t-matrix takes the form
t(ω) =
?JS
2
?2F (ω) +J
1 −?JS
2
??S ·? σ
?
τ0
2F (ω)?2
(3)
with S2= S2
a F-function as
x+S2
y+S2
z. In the above, we have introduced
F(ω) =
1
N
?
k
G0
k(ω) = diag(FeFhFeFh) (4)
where the diagonal elements are
Fe(h)(ω) =
1
N
?
k
ω − εk+ (−)Mk
Dk
(5)
In the tight-binding model, εk = C − 2D(2 − coskx−
cosky), Mk = M − 2B(2 − coskx− cosky) and Dk =
(ω − εk− Mk)(ω − εk+ Mk) − A2(sin2kx + sin2ky),
where M, A, B, C ,D are material parameters intro-
duced in the effective four-band model in Ref. [3]. Then
the eigen-energies for the excited states are obtained by
finding the poles of the GF in the bulk energy gap as
JS
2Fe(h)(ω) = ±1 (6)
which is consisted of four equations.
To characterizethe momentum integration in both nor-
mal and inverted regimes, we notice that each diagonal
block in H0(?k) describes a quantum anomalous Hall sys-
tem, [15] with the two forming a time-reversal conju-
gate pair. Using the result obtained by Qi et al.,[15] the
topological behavior of this system is totally governed by
two key parameters. In our case the correspondences are
e = −M
2B+ 2, which is related to the mass term in the
(2+1)-D Dirac model, and c = −2B
the sign of the Chern number. Therefore in the inverted
regime we have |e| < 2 corresponding to d > dc, other-
wise it is topological trivial corresponding to a normal
insulating phase. Furthermore, we set εk = 0 without
loss of generality to obtain a particle-hole symmetric sys-
tem, as we know that the quadratic kinetic term has no
contribution to the topology of this system. [15]
We therefore rewrite Eq.(5) in terms of e and c, the
result of which with ω in the bulk energy gap at several
values of e-parameter is shown in Fig.1, where the topo-
logical nontrival regime with |e| < 2 is given on the left
column, while the trivial regime is plotted in the right
column with |e| > 2. In each panel, the contribution
from the electron subband, Fe(ω), is shown in red and
that from the hole subband, Fh(ω), is shown in cyan. It is
clear to see that the resonant condition, Eq.(6), is always
satisfied for any given impurity strength J in the inverted
regime, and there are four LES in the bulk gap, two come
from the electron subband and two from the hole sub-
band. Moreover, the stronger the impurity strength is,
the nearer the LES are to the middle of the bulk gap,
Awhich determines
FIG. 1: (Color online) F-function versus ω in the bulk energy
gap with several values of e and c2= 1.
nontrivial regime with |e| < 2 is shown on the left column,
while the topological trivial regime with |e| > 2 is shown on
the right column of the figure. Fe(e,ω) is plotted as red lines
while Fh(e,ω) is plotted as cyan lines. The dotted lines are
guidelines for eyes to indicate the boundary values of the F-
functions.
The topological
FIG. 2: (Color online) Energy spectrum obtained by direct
diagonalization of Hamiltonian (1) with ǫk = 0 as a func-
tion of impurity strength in both (a) inverted and (b) normal
regimes. The localized states are shown in red.

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FIG. 3: (Color online) F-function versus ω with ω in the
bulk energy gap. The F-functions are calculated numerically
from Eq. (5) with the parameters taken from Ref. [3] for
HgTe/CdTe QW at d = 58˚A and d = 70˚
as red lines while Fh(ω) are shown as cyan lines. Inset: the
enlarged structures for the nontrivial regime.
A. Fe(ω) are shown
and no matter how strong the impurity strength is, there
are always LES in the bulk gap which appear as peaks
in the density of states (DOS).
In contrast, for the normal regime, we see that the
resonant condition is not always satisfied for any impu-
rity strength. For weak impurity strength, there could
exist two LES near the gap edge, however, as we in-
crease the impurity strength, these LES merge quickly
into the bulk and vanish finally.
in the DOS in the bulk gap are not stable against the
strong impurity strength for this case. This distinct dif-
ference of the response to the single magnetic impurity in
bulk serves as another novel criteria for the conventional
and topological insulating phases when the TRS is bro-
ken. Note that this result is robust to the explicit form
of the exchange interaction. For the Mn doped HgTe
QWs [11, 14], we consider a form of exchange interac-
tion, Js(p)
z
Szσz+Js(p)
?
(Sxσx+Syσy), in electron and hole
bands separately. It turns out that the only difference
with the isotropic model is to replace JS in Eq.(6) by
?
are still four persistent LES in the gap, which never merge
into the bulk.
To justify the above results obtained from t-matrix
method, we directly diagonalize the Hamiltonian (1) on
a square lattice in both inverted and normal regimes by
That is, the peaks
(Js,p
z )2S2
z+ (Js,p
? )2S2
?, and in the inverted region there
FIG. 4: Effective impurity concentration
of effective impurity strength (1−ξ2)2/(1+ξ2)2. The shaded
area indicates the range of the impurity concentration where
the impurity band and the continuum are separated.
1
τsMas a function
taking e = 0.5 and 3.5 for example. The obtained energy
spectrum is plotted versus impurity strength in Fig.2.
The four persistent LES (red lines) in the bulk energy
gap are clearly shown for the nontrivial case where we see
that they approach the middle of the gap as the increas-
ing of J and do not vanish. While in the trivial regime,
there are only two LES for small J and merge into the
bulk for large J. This result is in perfect agreement with
the above analysis through GF discussions.
By using the real parameters of the HgTe/CdTe QWs
system given in Ref. [3], we plot the F-function at the
quantum well width d = 58˚
are shown in Fig.3. We see that for d > dc where the
QSH effect is predicted, there are always LES in the bulk
energy gap for arbitrarily strong impurity strength; while
for d < dc, which is a normal insulator, the LES vanish
for strong impurity strength.
Considering that in real materials there is always a fi-
nite concentration of magnetic impurities, under which
the localized excited states grow into impurity band, we
estimate the critical concentration of magnetic impuri-
ties
τsMin the topological nontrivial regime as a func-
tion of impurity strength (1 − ξ2)2/(1 + ξ2)2, at which
the impurity band can not be distinguished from the con-
tinuum, and the result is shown in Fig.4. Here we have
denoted ξ =
2πNF as the effective impurity strength
and
M
However we suggest that experimentally the actual con-
centration should be much lower than the critical values,
so that not only its influence on the exchange coupling
J is negligible, [14] but also the impurities can be con-
sidered isolated and their coupling, such as RKKY inter-
actions, can be ignored. Furthermore, we speculate that
our results should be still valid even within a complete
quantum treatment of the magnetic impurity. [16]
Interestingly, the existence of LES in the bulk energy
gap of a topological insulator is not special in two di-
A and d = 70˚A, which
1
JS
1
τsM= niJS
ξ
(1+ξ2)2 as the effective concentration.

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mensions, but is also true for 3D strong topological in-
sulators (STI). As an example, we consider the strained
HgTe which is believed to be a STI. [9, 17] The effect
of magnetic impurities on the surface states of strained
HgTe has been discussed by one of the authors, [10] here
we focus on its effect in bulk. The model Hamiltonian
describing strained HgTe with time-reversal as well as
inversion symmetries takes the form [9, 17]
H3D(?k) = MkΓ1+ A⊥kxΓ5+ A⊥kyΓ2+ A?kzΓ3
(7)
where Mk = M0− M1(k2
resentation for Gamma matrices is chosen in such a way
that they are invariant under the joint transformations
of inversion and time-reversal symmetries.
features are worth noticing about this model Hamilto-
nian. Firstly, by setting A? = 0, Eq.(7) recovers the
2D HgTe/CdTe QW model.
the above, there are always LES in the inverted regime
for arbitrarily strong impurity strength.
comparing Eq.(7) with the Kane model [19] in the ba-
sis of
??E,1
that they have exactly the same form. Therefore the ma-
trix form for Kondo-like sp-d exchange term, [11, 14, 19]
Hex(? r) = −?
be similarly extracted as
x+ k2
y) − M2k2
z, and the rep-
[18] Two
As we have discussed in
Secondly, by
2
?,
??LH,−1
2
?,
??E,−1
2
?,
??LH,1
2
?, we observe
mJ(? r −?Rm)?Sm·? σ, in the same basis can
Hex =




−∆sez
0
−∆se+
0
0−∆se−
0
∆sez
0
0
1
3∆pez
0
−2
3∆pe−
−2
3∆pe+
0
−1
3∆pez



(8)
where ∆s= yN0Js?S?, ∆p= yN0Jp?S?, [14] and ? e is the
unit vector along the direction of impurity spin.
Following the methods developed by Fu et. al on 3D
topological insulators with inversion symmetries, [7, 20]
the STI phase characterized by odd number of Dirac
points (kx,ky) in the 2D surface states of Hamiltonian (7)
can be analyzed through two parameters r = M2/M1and
r1= M0/4M1. Though we wouldn’t elaborate the results
in general, some specific examples are listed below. On
a square lattice, for r = 1.5, there is one Dirac point
at (0,0) for r1= 0.75; three Dirac points at (0,0), (0,π)
and (π,0) for r1= 1.25; the three Dirac points then move
to (0,π), (π,0) and (π,π) for r1= 2.25; while for r1= 3
there is only one Dirac point again at (π,π). For even
larger r1the system evolves out of the STI phase.
Using the same formulation, [13] the full GF for the
system H =?
by finding the t-matrix as
kc†
kH3D(?k)ck+?
kk′c†
kHexck′ is obtained
t3D(ω) =
Hex
1 − HexF (ω)
(9)
where again the results in Eqs.(4) and (5) are recovered
[ǫk = 0 automatically here since there are no kinetic
term in the Dirac Hamiltonian (7)] with Mk = M0−
2M1(2 − coskx− cosky) − 2M2(1 − coskz) and Dk =
ω2−?M2
in the 3D case. The resonant conditions for LES are ob-
tained similarly by finding the poles of the full GF in the
bulk energy gap as
k+ 2A2
⊥(2 − coskx− cosky) + 2A2
?(1 − coskz)?
∆sFe(ω) = ±1,
∆p
?4 − 3e2
3
z
Fh(ω) = ±1 (10)
By numerically plotting Fe(h)as a function of ω in the
bulk energy gap, similar behavior as shown in Fig.1 is
found respectively for STI and non-STI phases using the
values of r and r1illustrated above. We see that the same
conclusion applies to 3D topological insulators. In the
STI phase there are always LES in the bulk energy gap
for arbitrary exchange interaction strength, while when
out of STI phase, the LES exist only for very weak ex-
change interaction strength. Therefore we believe that
the existence of nonvanishing LES in the bulk energy
gap for arbitrary impurity strength plays the role of a
general characterization for topological insulators.
Experimentally we suggest to detect this effect in the
recently achieved HgTe/CdTe QWs [3, 4] by doping a
small concentration of Mn+2ions in bulk.
LES evolve with the combination of exchange coupling
strength and the magnetic moments (at the mean-field
level), though it is hard to adjust the impurity exchange
coupling strength, it may possible to tune the Mn mo-
ments by a small magnetic field. [11, 14]. When the Mn
moments are larger than some critical value, there will
be four peaks in the DOS spectrum in STM measure-
ments for the QW width d > dc, which persist for even
larger polarization. When d < dc, the peaks in the DOS
spectrum will be broadened and vanish as the increasing
of the polarization. However, for 3D STI systems with
impurities doped deep in bulk, ARPES measurements
will be more appropriate. We suggest to use this distinct
signal to differentiate experimentally the topological and
the conventional insulating phases.
Since the
The authors thank Prof. Shou-Cheng Zhang and Dr.
Chao-Xing Liu for many illuminating discussions. Q. Liu
acknowledges the support of China Scholarship Council
for support. This work is supported by HKSAR RGC
Project No. CUHK 401806.
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