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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 2n. 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 4P 2 P3B, (12) Their contribution to partition function is given by e in . Therefore the partition function is invariant under the shift, , where 2n. The time reversal symmetry is recovered at 0, and . H B 4M 2 P3E, (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 4P / 2B . 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. 4M / 2E , 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 it , 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