quantum spin hall effect and their topological design of...

JOURNAL ON PHOTONICS AND SPINTRONICS VOL.4 NO.2 MAY 2015

ISSN 2324 - 8572 (Print) ISSN 2324 - 8580 (Online) http://www.researchpub.org/journal/jps/jps.html

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Abstract—Through consider the quantum spin Hall

effect as an important effect that characterizes 2-

dimensional semiconductors1 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-zS conservation

symmetry obtaining some designs of devices on new matter

states and possibly going non-conventional conductors and

topological insulators.

Keywords —quantum spin Hall Effect, Spintronic devices,

Topological insulators.

I. INTRODUCTION

HE 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)

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

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.

rotational symmetry of the magnetic field is broken.

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.

We consider the spin Hall conductance in the plane ,XY given

by

)1(,2

enxy

where ,n is the first Chern number in the topological

characterizing to n terminal conductance (see the figure 2),

having that

)2(),()2( 2

2

kFkd

n

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

constructed from a two-form ,F an auxiliary scalar ,B and their derivatives.

The action (which determines the path integral) is

M

BFS ,

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:

M

dAAS ,

Quantum Spin Hall Effect and their Topological

Design of Devices

S. Humeini, PhD, Kuwait University

T



15

)3(,2

2

AAxdtdS

xy

eff

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.

II. GENERALIZATION AND REVERSAL SYMMETRY IN

QUANTUM MECHANICS

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.

Fig. 3. Theoretical predictions of the spin Hall Effect [2].

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

,2Z invariant which characterizes to a state as trivial or

non-trivial when there is certain insulator component.

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.

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 ,2Z 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)

Fig. 4. A) The Chiral QHE liquids in ,1D B) The helical (QSHE)

liquids in .1D Spatially the QHE separates the two chiral states of a

spinless D1 liquid. The QSHE state spatially separates the four chiral states of

a spinful D1 liquid.

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

D1 model [4], only to helical liquid ,)2/1( D or D1 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

,2 rotation.

For other side, in the Kramers theorem3 in a time reversal

invariant system with half-integer spins ,12 then all states

are changed for degenerate doublets. Other interest aspect

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:

,: tt

If the Hamiltonian ,H operator commutes with the time-reversal operator,

that is

,0],[ H

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).



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observed is in condensed matter physics, where in the

Andersons’s theorem is established that

down). ,(-up) ,( pair BCS kk And the general pairing

between Kramers doublets is established.

Fig. 5. 2-dimensional topological surface control.

III. INSULATORS AND QSH ISULATORS

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).

Fig. 6. Band diagrams of three classes of insulators.

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

10-3

meV. 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 ,/ CdTeHgTe has a quantum

well leads to a topological quantum phase transition into the

QSH.

Then an effective tight-binding model is the obtained

considering the square lattice with 4 orbitals per site, to

know,

)4(,),(,),(,,,, yxyx ippippss

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

)5(,)('0

0)(),(

kh

khkkH yxeff

where

)6(

,)()()sin(sin

)sin(sin)()( a

a

yx

yxkd

kmkikA

kikAkmkh

then

)7(,0)(

)(2

2

BkmikkA

ikkABkm

yx

yx

is the relativistic Dirac equation in 12 dimensions with a

mass term tunable by the sample thickness ,d with ,0m for

.'cdd The mass domain wall is formed cutting the Hall bar

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. 7. The mass domain wall.

To experimental level the fabrication of several alloys

sample of ,/ CdTeHgTe 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).



17

Fig. 8. High mobility samples of ,/ CdTeHgTe as best candidate to the

fabrication of insulators [9].

Then can be given the following predictions:

Scolium. 3. 1. In the normal regime ,'cdd the ,gapE length

has gate .02

eGLR

In the inverted regime ,'cdd the ,gapE has

height in 22

eGLR

(this is the case when could have the helical

(QSHE) liquid with 1D ). (see the figure 9).

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):

)8(,BgEgap

Then the conductance now takes the activated form:

)9(,)(

kBg

ef

Fig. 9. ,k in two regimes.

Fig. 10. Crossing to the magneto-conductance zone.

Then the theoretical predictions meet with an evidence in the

QSH state of ,HgTe analysis (see the figure 11).

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).



18

Fig. 12. Magnetic field dependence of the residual conductance.

What happen with the QSH state in ,/ GaSbInAs 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

,/ GaSbInAs 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 ,/ GaSbInAs quantum wells. This

material can be fabricated commercially in many places around

the world.

IV. TOPOLOGICAL INSULATORS AND SPINTRONIC DEVICES

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-zS 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 zS , Topological

band invariant in the momentum space based on single particle

states [10].

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,

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.

Considering an analog system of a periodic ring as described

in the figure 13 B), but with the following characteristics of the

flux enters:

Adx , )10(,0/

i

e

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:

)11(),(162

162

3

3

AAxdtd

FFxdtdS

Integrated over a spatially and temporally periodic system,

TABLE I

TOPOLOGICAL INSULATOR DEFINITIONS: ACTIONS, PERIODICITY AND TIME

REVERSAL SYMMETRY

Symbol Effective Action Integral

0S

Electromagnetic Action: Relates many

Axion physics

223

0

1

8

1BE

xdtdS

S

Action to Periodic

Systems:

c

e

2

BE

xdtdS 3

22



19

)12(,2

0

3 nAcdtdzdxdyBxcdtd ztzBE

Their contribution to partition function is given by .nie

Therefore the partition function is invariant under the shift, ,

where .2 n The time reversal symmetry is recovered at

,0 and .

Fig. 14. Behavior of the Chern number versus with the behavior of the

conductance and polarization.

We can affirm, under the effective action described in (12)

that from , implies that QHE, on the boundary has

conductance (1) with .2/1n 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.

Fig. 15. Small breaking field is applied to the boundary of the band

insulator.

Then each Dirac cone contributes ,2

1 2

exy to the QH.

Therefore, , implies an odd number of Dirac cones on the

surface! The surface of TI, in usual technologies has a ,4/1 of

grapheme.

The equations of axion electrodynamics given by

)17(,24

)16(,24

)15(,1

´4

)14(,1

,0

)13(,4

3

3

EMBH

BPED

DjH

BE

B

D

P

P

tcc

tc

predict the robust TME effect. In the equations (16) and (17)

the term ,2/3 P is the electromagnetic polarization,

microscopically given by the Chern-Simons term over the

momentum space or pk space (see the figure 16).

Fig. 16. TME effect: a). It’s had EM 2/4 , b) It’s had

BP 2/4 .

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 ,2Z invariant symmetry

in the design of topological insulators, considering that

non-interacting topological insulators are characterized by the

index ( ,2Z 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.

Proposition. 5. 1. Generalizing the topological field theory

of the QHE and TI, and applying the action functional given by

Schwartz with conductance ,2

enxy we have in general

the actions:

)18(),()())( 32

1 xdAxxAdkkdadS

)19(,)()(

))()()()()((

4

23

2

xdAxxdAd

kadkdakakdakakdS



20

Proof. Through axioms that have been used by Schwarz-type

QFTs, to explain topologically invariant in our insulator class,

we have that

)20(,M

BFS

and using the 3-dimensional Chern-Simons form,

,3

1Tr

AAAAF where “traces” are “integrals” to

define the action we have4

that to the electromagnetic

polarization (which is given by the Chern-Simons term over the

momentum space):

)21(,],[3

1{Tr

16

1)( 3

2

3

03

kjiij

ijk aaafkdkdP

V

Then applying some electrical tension producing a

microscopic conductance (see the figure 17) given by:

)22()},()({Tr1

lim

)()()(

1

,, 0

1

0

,,

tIiuIdudte

tEtI

zyx

u

ti

zyx

having that An electron-monopole dyon becomes an anyon!

Fig. 17. Application of the electrical tension of the electric field oscillates

with a frequency ,)( tieEtE can produce an angle polarization

,2 3

2 P that can have an identification with a functioning to a periodic

system insulator component device, discussed in the past section IV.

The action ,1S and ,2S are obtained when the term is the

electromagnetic polarization, microscopically given by the

Chern-Simons term over the momentum space or pk

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

,3

2Tr

4M

AAAdAA

kS

For example, under low frequency Faraday and their relation

with the Kerr rotation the adiabatic requirement to surface gap

must be ,gapE 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:

)23(,'/'/

)12(tansgn)(

nArcuBB

where ,uB is the normal contribution and

'/'/

)12(tansgn

nArc , is is the topological

contribution (see the figure 19, and the figure 20), that is to say,

the angle ,TOPO which is of order: .106.3 3 radsTOPO

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. 19. Topological contribution of angle ,TOPO considering the function: y

= 4x +sign(x)acot((2x-1)/\sqrt(3) + \sqrt(1/3))



21

Fig. 20. A) Energy gap, B) Total angule curve.

Accord to the topological field theory and using the physical

modes to CS given by ,))/(sin(

))/(sin(

NkN

Nk

we have a modeling

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. 22. Radial projection of the periodic behavior established between fields

FF , and .F

Fig. 23. Distribution of the saturated surface energy states in large sample in

,cdd 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

),1(UG Chern–Simons theory at level k , the normalized

correlation function is, up to a phase, equal to

,))/(sin(

))/(sin(

NkN

Nk

(see the figure 23 and 24)

Fig. 24. STM probe of the topological surface states. 3D-model of the local

behavior of the ,gapE when is considered ,cdd 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].

5 Nobel Prize of Physics 2004.



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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

accord with the conductance ,2

enxy with symmetry

defined in time reveral symmetry to a polarization related with

angle ,2 3

2 P when ).(33 PP The their conductivity

, is that given as

)24(,],[3

1{Tr

8

1 3

3

kjiij

ijk aaafkdM

e

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

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Communications in Mathematical Physics 117 (3): 353–386.

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Hall Effect, Physical Review Letters 95, 146802 (2005).

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quantum spin hall effect and their topological design of...

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