Download Quantum Spin Hall Effect and their Topological Design of Devices

JOURNAL ON PHOTONICS AND SPINTRONICS
ISSN 2324 - 8572 (Print) ISSN 2324 - 8580 (Online)
VOL.4 NO.2
MAY 2015
http://www.researchpub.org/journal/jps/jps.html
Quantum Spin Hall Effect and their Topological
Design of Devices
S. Humeini, PhD, Kuwait University

rotational symmetry of the magnetic field is broken.
Abstract—Through consider the quantum spin Hall
effect as an important effect that characterizes 2dimensional semiconductors 1 are designed and discussed
many spintronic devices on the basis of three classes of the
topological insulators and the manager of the charge
conservation symmetry and spin- S z conservation
symmetry obtaining some designs of devices on new matter
states and possibly going non-conventional conductors and
topological insulators.
In other phenomena as superconducting, the symmetry broken
is the gauge symmetry, obtaining a photonic condensation
required to the superconducting, for example, in magnetic
levitation or reactors to different works.
A quantum effect that can to help obtain these non-trivial states
of matter is the Quantum Hall State, which topologically
characterize the conductors, insulators, crystals, magnets, or
any other components to give these nontrivial matter states.
Keywords —quantum spin Hall Effect, Spintronic devices,
Topological insulators.
We consider the spin Hall conductance in the plane XY , given
by
 xy  n
I. INTRODUCTION
T
searched of new states of matter have established in
new research fields the possibility of the use of quantum
properties of the metals, insulators, superconductors, magnets,
etc, bringing that these new states are differentiated by the
broken symmetry. For example, the atomic net of nano-crystals
of some metals can be modified such that their symmetry is
broken having quantum special properties very useful to
condensed matter, for example, to permanent superconductors
or the modification or the magnetic field through special
magnets (see the figure 1) to re-directing magnetic fields and
modified in intensity.
A)
B)
HE
e2
,

(1)
where n, is the first Chern number in the topological
characterizing to n  terminal conductance (see the figure 2),
having that
n
d 2k 
 F  (k ),
(2 ) 2
(2)
Then the topological states of matter are defined and described
by the topological field theory2 [1]:
2
The known topological field theories fall into two general classes:
Schwarz-type TQFTs and Witten-type TQFTs. Witten TQFTs are also
sometimes referred to as cohomological field theories. Likewise, in
Schwarz-type TQFTs, the correlation functions (as for example the
conductance sxy) or partition functions of the system are computed by the path
integral of metric independent action functionals. For instance, in the BF
model, the spacetime is a two-dimensional manifold M , the observables are
Fig. 1. A) Metallic Nano-crystal whose inner net is modified and their
symmetry is broken involving new elements in alloys. In this case we say that
the translational symmetry is broken. B) Magnets in imam. In this case the
constructed from a two-form
F , an auxiliary scalar B, and their derivatives.
The action (which determines the path integral) is
S
Soltan Humeini, is a Emeritus profesor of QED Laboratory of Kuwait
University (e-mail: [email protected]).
1
The quantum spin Hall state is a state of matter proposed to exist in
two-dimensional semiconductors that have a quantized spin-Hall conductance
and a vanishing charge-Hall conductance. The quantum spin Hall state of
matter is the cousin of the integer quantum Hall state, and both states can be
realized on lattice which not require the application of a prolonged magnetic
field.
 BF ,
M
The space-time metric does not appear anywhere in the theory, so the theory
is explicitly topologically invariant. Likewise we have the Schwarz’s
functional:
S
 A  dA,
M
14
JOURNAL ON PHOTONICS AND SPINTRONICS
ISSN 2324 - 8572 (Print) ISSN 2324 - 8580 (Online)
S eff 
 xy
2
d
2
VOL.4 NO.2
MAY 2015
http://www.researchpub.org/journal/jps/jps.html
xdt   A  A ,
A very important achievement was the realization that the
quantum spin Hall state remain to be non-trivial even after the
introduction of spin-up spin-down scattering,[3] which destroy
the quantum spin Hall effect. In order experiment was
introduced a topological Z 2 , invariant who characterizes a state
as trivial or non-trivial band insulator (regardless if the state
exhibits or does not exhibit a quantum spin Hall Effect). Further
stability studies of the edge liquid (see the figure 4) through
which conduction takes place in the quantum spin Hall state
proved, both analytically and numerically that the non-trivial
state is robust to both interactions and extra spin-orbit coupling
terms that mix spin-up and spin-down electrons. Such a
non-trivial state (exhibiting or not exhibiting a quantum spin
Hall Effect) is called a topological insulator, which is an
example of symmetry protected topological order protected by
charge conservation symmetry and time reversal symmetry.
A)
B)
(3)
We want establish (pure) spin Hall effect, to the design of the
different spintronic devises.
Fig. 2. Topological characterizing of
n  terminal conductance.
To it is necessary generalize the ordinary Hall effect with
magnetic field.
Fig. 4. A) The Chiral QHE liquids in
D  1, B) The helical (QSHE)
liquids in D  1. Spatially the QHE separates the two chiral states of a
spinless 1D liquid. The QSHE state spatially separates the four chiral states of
a spinful 1D liquid.
II. GENERALIZATION AND REVERSAL SYMMETRY IN
QUANTUM MECHANICS
Then in this particular, we cannot to go to the theorems of the
topological field theory, since chirality and helically states can
never to be constructed microscopically from a purely
1D model [4], only to helical liquid (1 / 2) D, or 1D Fermi
liquid.
Then is required a 2  dimensional theory which permits a
time reversal symmetry in quantum mechanics.
In this theory the wave function of a particle with integer
spin changes by 1 , under 2 , rotation. For other side the
wave function of a half-integer spin changes by  1, under
The generalizations of the Hall Effect conduces us to the
observation of the three cases (see figure 3) that let us to see
two physical effects that are the appearing in some cases of Hall
voltage and also in other cases the spin accumulation, under
certain considerations as polarization in the case the Hall
Effects is observed under magnetization.
2 , rotation.
For other side, in the Kramers theorem3 in a time reversal
invariant system with half-integer spins  2  1, then all states
are changed for degenerate doublets. Other interest aspect
Fig. 3. Theoretical predictions of the spin Hall Effect [2].
3
In quantum mechanics, the Kramers degeneracy theorem states that for
every energy eigenstate of a time-reversal symmetric system with half-integer
total spin, there is at least one more eigenstate with the same energy. In other
words, every energy level is at least doubly degenerate if it has half-integer
spin. In theoretical physics, the time reversal symmetry is the symmetry of
physical laws under a time reversal transformation:
For other way, the spin effects in their chirality and helically
could to bring the step of one case in other under change of the
regime, let magnetic field or magnetization or nothing of the
two. However, the manager of spin is not easy, if we not have
some topological considerations to the manager of the
scattering effects, which contemplates the necessity of a
topological surface theory based in certain symmetry respect to
Z 2 , invariant which characterizes to a state as trivial or
non-trivial when there is certain insulator component.
 : t  t ,
If the Hamiltonian
H,
operator commutes with the time-reversal operator,
that is
[ H ,  ]  0,
then for every energy eigenstate n , the time reversed state  n , is also
an eigenstate with the same energy. Of course, this time reversed state might be
identical to the original state, but that is not possible in a half-integer spin
system since time reversal reverses all angular momenta, and reversing a
half-integer spin cannot yield the same state (the magnetic quantum number is
never zero).
Another more famous example is Chern–Simons theory, which can be used
to compute knot invariants. In general partition functions depend on a metric
but the above examples are shown to be metric-independent.
15
JOURNAL ON PHOTONICS AND SPINTRONICS
ISSN 2324 - 8572 (Print) ISSN 2324 - 8580 (Online)
VOL.4 NO.2
MAY 2015
http://www.researchpub.org/journal/jps/jps.html
observed is in condensed matter physics, where in the
Andersons’s
theorem
is
established
that
And
the
general
pairing
BCS pair  (k , up)  (-k , down).
between Kramers doublets is established.
QSH.
Then an effective tight-binding model is the obtained
considering the square lattice with 4  orbitals per site, to
know,
s,  , s,  , ( p x  ip y , )  ,  ( p x  ip y , )  ,
(4)
Nearest neighbor hopping integrals. Mixing matrix elements
between the s, and the p, states must be odd in k . Then the
effective Hamiltonian matrix is
 h( k )
H eff (k x , k y )  
 0
0 
,
h' (k ) 
(5)
where
m( k )

h(k )  
 A(sin k x  i sin k y )
A(sin k x  i sin k y ) 
  d a (k ) a ,

 m( k )

(6)
Fig. 5. 2-dimensional topological surface control.
then
 m  Bk 2

 A(k  ik )
x
y

III. INSULATORS AND QSH ISULATORS
A(k x  ik y ) 
  0,
 m  Bk 2 
(7)
is the relativistic Dirac equation in 2  1 dimensions with a
mass term tunable by the sample thickness d , with m  0, for
d  d c ' . The mass domain wall is formed cutting the Hall bar
It’s necessary establish distinctions between a conventional
insulator and QSH insulator. An preliminary study [5-7],
establish that the band diagram of a conventional insulator,
conventional insulator with accidental surface states, and QSH
insulator are differentiated as can be viewed considering the
blue and red color code for up and down spins (see figure 6).
along the y  direction. The domain-wall structure appears in
the band structure mass term. This leads to states localized on
the domain wall which still disperse along the x  direction
(see the figure 7).
Fig. 6. Band diagrams of three classes of insulators.
Fig. 7. The mass domain wall.
From a point of view of the chemistry with topology the
searching of the QSH state has been made through Graphene,
where the spin-orbit coupling only has been calculed about
10-3meV. Not realized in experiments.
The spectral studies realized on several substances and
chemical composites has given that in the type III quantum
wells work, for example HgTe, has a negative band gap [8].
Also a tuning the thickness of the HgTe / CdTe, has a quantum
well leads to a topological quantum phase transition into the
To experimental level the fabrication of several alloys
sample of HgTe / CdTe, quantum wells have given doped
regimes, since several meV , can to produce a gate system from
n, to p, doped regimes. two tuning parameters, the thickness
d , of the quantum well, and the gate voltage, are controlled in
the experimental setup (see the figure 8).
16
JOURNAL ON PHOTONICS AND SPINTRONICS
ISSN 2324 - 8572 (Print) ISSN 2324 - 8580 (Online)
VOL.4 NO.2
MAY 2015
http://www.researchpub.org/journal/jps/jps.html
Fig. 9.
Fig. 8. High mobility samples of
  k , in two regimes.
HgTe / CdTe, as best candidate to the
fabrication of insulators [9].
Then can be given the following predictions:
Scolium. 3. 1. In the normal regime d  d c ' , the E gap , length
 e2 
G LR    0. In the inverted regime d  d c ' , the E gap , has
  
e2 
height in G 

  2 (this is the case when could have the helical
LR 
  
(QSHE) liquid with D  1 ). (see the figure 9).
Fig. 10. Crossing to the magneto-conductance zone.
has gate
Then the theoretical predictions meet with an evidence in the
QSH state of HgTe, analysis (see the figure 11).
In the case   k , the edge (punctured line in the figure 9) is
the limit to cross to the magneto-conductance. The crossing of
the helical edge states is protected by the TR symmetry. TR
breaking term such as the Zeeman magnetic field causes a
singular perturbation and will open up a full insulating gap (see
the figure 10):
E gap  g B ,
(8)
Then the conductance now takes the activated form:
  f ( )e  g B k ,
(9)
Fig. 11. Experimental evidence.
The graph to the crossing to the magneto-conductance given
in the figure 11, meets the experimental evidence in the
dependency of the magnetic field with the residual conductance
(see the figure 12).
17
JOURNAL ON PHOTONICS AND SPINTRONICS
ISSN 2324 - 8572 (Print) ISSN 2324 - 8580 (Online)
VOL.4 NO.2
MAY 2015
http://www.researchpub.org/journal/jps/jps.html
TABLE I
TOPOLOGICAL INSULATOR DEFINITIONS: ACTIONS, PERIODICITY AND TIME
REVERSAL SYMMETRY
Symbol
S0
Effective Action
Electromagnetic
Action: Relates many
Axion physics
Action to Periodic
S
e2
Systems:  
c
Integral
S0 
1
8
d
3


1
xdt  E 2  B 2 



     3
S  

 d xdtE  B
 2  2 
Fig. 12. Magnetic field dependence of the residual conductance.
What happen with the QSH state in InAs / GaSb, type II,
quantum wells?
For one side HgTe, is not a material that can be easily
fabricated. Our researches treat of obtain new semi-conductor
materials which can lead to QSH. For other side, in HgTe, the
band inversion occurs intrinsically in the material. However in
InAs / GaSb, quantum wells, a similar inversion can occur,
since the valance band edge of GaSb, lies above the
conduction band edge of InAs. The theoretical work shows that
the QSH can occur in InAs / GaSb, quantum wells. This
material can be fabricated commercially in many places around
the world.
the system is time reversal symmetric only when   0,
having a trivial insulator. In the case when    , then we
have a non-trivial insulator. Their action can be seen in the table
1. The insulator component device can be seen as the figure 13.
A)
B)
Fig. 13. Periodic system insulator component device.
IV. TOPOLOGICAL INSULATORS AND SPINTRONIC DEVICES
Considering an analog system of a periodic ring as described
in the figure 13 B), but with the following characteristics of the
flux enters:
2  Dimensional semiconductors are designed on the basis of
three classes of the topological insulators and the manager of
the charge conservation symmetry and spin- S z conservation
symmetry, which establish certain behavior of the manager
scopes to the time reversal symmetry, relating the periodicity
with the time reversal symmetry of these insulator design. As
was mentioned in the section the design of the topological
insulator must contemplate the necessity of a topological
surface theory based in certain symmetry respect to invariant
which characterizes to a state as trivial or non-trivial when there
is certain insulator component. The best 2-dimensional
topological band invariant is the given by S z , Topological
band invariant in the momentum space based on single particle
states [10].
   dx  A  ,
  e i /  ,
0
(10)
the physics is completely invariant under the shift of , then
also is completely invariant to   2n. Under time reversal,
 , implies   , therefore the time reversal is recovered for two
special values of   0, and    . The ME term is a total
derivative, independent of the bulk values of the fields:
     3


S  

  d xdt   F F
2

16




     3

 


(11)

  d xdt (  A  A ),
 2  16 
Integrated over a spatially and temporally periodic system,
For other side, considering the topological field theory term
in the effective action we can have design valid to interacting in
disordered systems, directly measurable physically. This can
relates axion physics (See the table 1). For a periodic system,
18
JOURNAL ON PHOTONICS AND SPINTRONICS
ISSN 2324 - 8572 (Print) ISSN 2324 - 8580 (Online)
 cdtd
3
VOL.4 NO.2
MAY 2015
http://www.researchpub.org/journal/jps/jps.html
xE  B   dxdyB z  cdtdz  t Az  n ,
2
0
  D  4 ,
  B  0,
1 B
E
,
c t
4
1 D
H 
j´
,
c
c t
D  E  4P  2 P3B,
(12)
Their contribution to partition function is given by e in .
Therefore the partition function is invariant under the shift,  ,
where   2n. The time reversal symmetry is recovered at
  0, and    .
H  B  4M  2 P3E,
(13)
(14)
(15)
(16)
(17)
predict the robust TME effect. In the equations (16) and (17)
the term P3   / 2 , is the electromagnetic polarization,
microscopically given by the Chern-Simons term over the
momentum space or k  p  space (see the figure 16).
Fig. 14. Behavior of the Chern number versus with the behavior of the
conductance and polarization.
Fig. 16. TME effect: a). It’s had
4P   / 2B .
We can affirm, under the effective action described in (12)
that from    , implies that QHE, on the boundary has
conductance (1) with n  1 / 2. But to a sample with boundary,
this is only insolating when a small   breaking field (see the
figure 14 and figure 15) is applied to the boundery. Then the
surface theory is a CS term, describing the half QH.
4M   / 2E , b) It’s had
V. RESULTS
We can give the following proposition on the discussed in the
sections II, III, and IV, and considering the advantages of the
topological field theory on topological Z 2 , invariant symmetry
in the design of topological insulators, considering that
non-interacting topological insulators are characterized by the
index ( Z 2 , topological invariants) similar to the genus in
topology. Then we can enounce a proposition sufficiently
general that involves the insulator classes that manipulate the
"protected" conducting states in the surface insulator and which
are required by time-reversal symmetry and the band structure
of the material. The states cannot be removed by surface
passivation if it does not break the time-reversal symmetry.
Fig. 15. Small breaking field is applied to the boundary of the band
insulator.
Proposition. 5. 1. Generalizing the topological field theory
of the QHE and TI, and applying the action functional given by
2
Schwartz with conductance  xy  n e , we have in general

the actions:
2
Then each Dirac cone contributes  xy  1 e , to the QH.
2 
Therefore,    , implies an odd number of Dirac cones on the
surface! The surface of TI, in usual technologies has a 1 / 4, of
grapheme.
S1   d 2 kda(k )) d 3 xA( x)  dA( x),
(18)
S 2   d 3 k (a(k )  da(k )  a(k )  da(k )  d 2 a(k )  ) 
The equations of axion electrodynamics given by
  d 4 xdA( x)  dA( x),
19
(19)
JOURNAL ON PHOTONICS AND SPINTRONICS
ISSN 2324 - 8572 (Print) ISSN 2324 - 8580 (Online)
VOL.4 NO.2
MAY 2015
http://www.researchpub.org/journal/jps/jps.html
Proof. Through axioms that have been used by Schwarz-type
QFTs, to explain topologically invariant in our insulator class,
we have that
S   BF ,
(20)
For example, under low frequency Faraday and their relation
with the Kerr rotation the adiabatic requirement to surface gap
must be   E gap , and the “topological angle”, that is to
say the angle to our Kerr rotation under the low Faraday
Frequency  TOPO , can be in general described as:
M
and using the 3-dimensional Chern-Simons form,
1


Tr F  A  A  A  A, where “traces” are “integrals” to
3


define the action we have 4 that to the electromagnetic
polarization (which is given by the Chern-Simons term over the
momentum space):


(2n  1)
,

  /    ' / ' 
 ( B)  uB  sgn Arc tan
where
uB,
is
the
normal
contribution
(23)
and


(2n  1)
 , is is the topological
sgn Arc tan
1
1
 /    ' /  ' 


3

3
ijk

P3 (0 )   d kV 
d k  Tr{ fij  [ai , a j ]  ak , (21)
2 
16

3

contribution (see the figure 19, and the figure 20), that is to say,
the angle  TOPO , which is of order:  TOPO  3.6  10 3 rads.
Then applying some electrical tension producing a
microscopic conductance (see the figure 17) given by:
I  (t ) 

    ( )E (t ) 
  x, y, z
u
dte   du



 x, y, z 0
i t
1
0
lim
 
1
Tr{I (iu1 ) I  (t )},

(22)
having that An electron-monopole dyon becomes an anyon!
Fig. 18. Family three TI.
One important note is not treat to obtain a theorem to the
topological stability of the surface states, that is to say, it is not
possible to construct a 2D-model with an odd number of Dirac
cones, in a system with TR-symmetry. In this 2-dimensional
case surface states of a TI with is a holographic liquid [11].
Fig. 17. Application of the electrical tension of the electric field oscillates
with a frequency E (t )  E e it , can produce an angle polarization
  2 2 P3 , that can have an identification with a functioning to a periodic
system insulator component device, discussed in the past section IV.
The action S 1 , and S 2 , are obtained when the term is the
electromagnetic polarization, microscopically given by the
Chern-Simons term over the momentum space or k  p 
space each identified as the family three TI (see the figure 18).
Then finally can be demonstrated the proposition.
Now, we need have the following considerations in our
qualitative analysis to the study of design of spintronic devices.
4
The action S of Chern–Simons theory is proportional to the integral of the
Chern–Simons 3-form
k
S
4
Fig. 19. Topological contribution of angle
2


M Tr A  dA  3 A  A  A,
 TOPO , considering the function: y
= 4x +sign(x)acot((2x-1)/\sqrt(3) + \sqrt(1/3))
20
JOURNAL ON PHOTONICS AND SPINTRONICS
ISSN 2324 - 8572 (Print) ISSN 2324 - 8580 (Online)
VOL.4 NO.2
MAY 2015
http://www.researchpub.org/journal/jps/jps.html
Fig. 20. A) Energy gap, B) Total angule curve.
Fig. 23. Distribution of the saturated surface energy states in large sample in
d  d c , normal regime.
In the special case in which M , is the 3  sphere, as could be in
the case of the insulator device component (see the figure 13),
we can to apply certain normalized correlation functions, as
Witten has shown, since these normalized correlation functions
are proportional to known knot polynomials. For example, in
G  U (1), Chern–Simons theory at level k , the normalized
correlation function is, up to a phase, equal to
sin( /( k  N )) (see the figure 23 and 24)
,
sin(N /( k  N ))
Accord to the topological field theory and using the physical
modes to CS given by sin( /( k  N )) , we have a modeling
sin(N /( k  N ))
of the saturated topological surface states (Figure 21) which can
establish a periodic behavior of the Dirac cones existing when
is controlled the electrical field in an topological insulator.
Fig. 21. Periodic behavior of the Dirac cones due to the electrical field in an
topological surface insulator.
But this also can be measured through of the actions realized by
S  , considering radial polarized field curves, which are accord
with the surface of the figure 21, to saturated topological
surface states (see the Figure 22).
Fig. 24. STM probe of the topological surface states. 3D-model of the local
behavior of the E gap , when is considered d  d c , inverted regime.
Then we can to say finally that indeed, the fundamental field
equations of the Standard Model (of Einstein, Maxwell,
Yang-Mills) are all geometrical field equations, from which can
be deduced topological field equations having importance only
the topological term within the Standard Model. This term
defines and described the TI. Here we remember the words
given by Frank Wilzcek5: “Topological insulator is a window
into the universe!” [12].
Fig. 22. Radial projection of the periodic behavior established between fields
F  , F  and F .
5
21
Nobel Prize of Physics 2004.
JOURNAL ON PHOTONICS AND SPINTRONICS
ISSN 2324 - 8572 (Print) ISSN 2324 - 8580 (Online)
VOL.4 NO.2
MAY 2015
http://www.researchpub.org/journal/jps/jps.html
[11] C. Wu, B. A. Bernevig, and S. C. Zhang, Phys. Rev. Lett. 96, 106401
(2006).
[12] F. Wilczek, A theoretical physicist examines exotic particles lurking in
new materials, Nature, 458 129 (2009). [MIT-CTP 4017].
VI. CONCLUSIONS
The design of the new insulators to components and spintronic
devices must be under topological criteria of field to conform
the topological insulators based in the geometrical surface
design with the enough satured topological surface states
2
accord with the conductance  xy  n e , with symmetry

defined in time reveral symmetry to a polarization related with
angle   2 2 P3 , when P3  P3 ( ). The their conductivity
 , is that given as

e 1
M 8 3
d
3
1


k ijk Tr{ f ij  [a i , a j ]  a k ,
3


(24)
The family of materials that comply this spectrum are the given
in the superior levels (above of the punctured line in the figure
18), satisfying the QHE, and TI-Chern-Simmons theory as was
demonstrated in the proposition 5. 1.
ACKNOWLEDGMENTS
I am very grateful with the JPS, for their help and advice to the
quality of this paper.
Abbreviators
TI  Topological insulators.
QSHE  Quantum spin Hall Effect.
QHE  Quantum Hall Effect.
CS  Chern-Simons class.
QFT  Quantum field theory.
TR  Time Reversal (symmetry).
TME  Topological magneto-electric (Effect).
ME  Magneto-electric.
References
[1]
Witten, Edward (1988a), "Topological quantum field theory",
Communications in Mathematical Physics 117 (3): 353–386.
[2] S. Murakami, N. Nagaosa, and S. C. Zhang, Science 301, 1348 (2003).
[3] C.L. Kane and E.J. Mele, Z2 Topological Order and the Quantum Spin
Hall Effect, Physical Review Letters 95, 146802 (2005).
[4] C Wu, BA Bernevig, SC Zhang, “Helical liquid and the edge of quantum
spin Hall systems,” Physical review letters 96 (10), 106401.
[5] C. L. Kane and E. J. Mele, “Quantum Spin Hall Effect in Graphene,”
Phys. Rev. Lett. 95, 226801.
[6] J. Maciejko, C. Liu, Y. Oreg, XL. Qi, C. Wu, SC. Zhang, “Kondo effect in
the helical edge liquid of the quantum spin Hall state,” Physical review
letters 102 (25), 256803
[7] C. Xu, J. E. Moore, “Stability of the quantum spin Hall effect: Effects of
interactions, disorder, and Z2 topology,” Phys. Rev. B 73, 045322.
[8] BA. Bernevig, TL. Hughes, SC. Zhang, “Quantum spin Hall effect and
topological phase transition in HgTe quantum wells,” Science. 2006 Dec
15; 314(5806):1757-61.
[9] M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W.
Molenkamp,, X-L Qi, S-C Zhang “Quantum Spin Hall Insulator State in
HgTe Quantum Wells,” Vol. 318 no. 5851 pp. 766-770, DOI:
10.1126/science.1148047.
[10] P. Bhattacharya, R. Fornari, H. Kamimura, Comprehensive
semiconductor science and technology, Vol 1 Set: ELSEVIER, 2011.
22