Edouard Sonin

Quantum spin Hall effect in topological insulators

Racah Institute of Physics

Hebrew University of Jerusalem

Content

Spin Hall effect Topological Insulator Quantum spin Hall effect: (i)  Was it really observed? (ii)  If not, how and whether

could it be observed?

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Spin Hall effect

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Spin balance

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Rashba model:

is the group-velocity operator

Spin:

Torque:

Spin current:

Sonin"Adv. Phys. 59, 181 (2010) "

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Spin balance

{ }

Rashba model:

is the group-velocity operator

Spin:

Torque:

Spin current:

{ }

{ } { }

Spin balance

{ }

Rashba model:

is the group-velocity operator

Spin:

Torque:

Spin current:

{ }

{ } { }

Spin balance

{ }

Rashba model:

is the group-velocity operator

Spin:

Torque:

Spin current:

Are spin currents observable?

A moving magnetic moment generates an electric field

Electrodynamics (Griffiths, Introduction to Electrodynamics, Prentice–Hall,1999, Problem 12.62)

Inverse spin Hall e!ect:Valenzuela and Tinkham, Nature, 442,176 (2006) - spin diffusion currentJungfleisch et al., arXiv: 1011.0889 - magnon spin current

Vacuum:

Rasba medium:

Spin accumulation by a spin current(Dyakonov & Perel, 1971)

Bulk spin current:

Bulk spin currentSpin diffusion current:

Spin diffusion current

Spin balance equation:

At the border:

Experimental detection of the spin Hall effect: spin accumulation at sample edges

Experimental Observation of the Spin-Hall Effect in a Two-DimensionalSpin-Orbit Coupled Semiconductor System

J. Wunderlich,1 B. Kaestner,1,2 J. Sinova,3 and T. Jungwirth4,5

1Hitachi Cambridge Laboratory, Cambridge CB3 0HE, United Kingdom2National Physical Laboratory, Teddington T11 0LW, United Kingdom

3Department of Physics, Texas A&M University, College Station, Texas 77843-4242, USA

4Institute of Physics ASCR, Cukrovarnicka 10, 162 53 Praha 6, Czech Republic5School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom

PRL 94, 047204 (2005)P HY S I CA L R EV I EW LE T T ER S week ending

4 FEBRUARY 2005

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Weak disorder, extrinsic Hall effect:

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Weak disorder, intrinsic spin Hall effect:

Experimental detection of the spin Hall effect: spin accumulation at sample edges

Experimental Observation of the Spin-Hall Effect in a Two-DimensionalSpin-Orbit Coupled Semiconductor System

J. Wunderlich,1 B. Kaestner,1,2 J. Sinova,3 and T. Jungwirth4,5

1Hitachi Cambridge Laboratory, Cambridge CB3 0HE, United Kingdom2National Physical Laboratory, Teddington T11 0LW, United Kingdom

3Department of Physics, Texas A&M University, College Station, Texas 77843-4242, USA

4Institute of Physics ASCR, Cukrovarnicka 10, 162 53 Praha 6, Czech Republic5School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom

PRL 94, 047204 (2005)P HY S I CA L R EV I EW LE T T ER S week ending

4 FEBRUARY 2005

!"#$%& '()$#'*#+ *,"#(%)(- ./00 *11*-"2

Weak disorder, extrinsic Hall effect:

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?11*-" (% !*@(-$%'A-"$#)

B/"$+ CD*#)+ E$))/#'+ /%' FG)=/0$@

Is spin accumulation is a good probe of the spin current?

Spin current without spin accumulation:

Equilibrium spin currents, spin torque instead of spin accumulation

Spin accumulation without spin current:

Intrinsic spin Hall effect in the Rashba model

Sonin, PRL 99, 266602 (2007)

Topological insulators

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BHZ model [Bernevig, Hughes, and Zhang, Science 314, 1757 (2006)]:

Kane and Mele, PRL 95, 146802; ibid., 226801 (2005).

Eigenstates:

kp theory

Edge states

Boundary condition:

What «spin» (moment) are we going to study?

s-type conduction band:

Mechanical moment:

Operator of the effective moment:

Operator of the group velocity:

Moment current:

Magnetic moment:

p-type valence band:

Transverse moment currents induced by an electric field

Plane-wave presentation

Corrected plane-wave eigenstate:

Transverse moment current:

The total transverse moment current is proportional to the Chern numbr:

Topological term:

Chern number

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Quantum Spin Hall Insulator Statein HgTe Quantum WellsMarkus König,1 Steffen Wiedmann,1 Christoph Brüne,1 Andreas Roth,1 Hartmut Buhmann,1Laurens W. Molenkamp,1* Xiao-Liang Qi,2 Shou-Cheng Zhang2

AUTHORS’ SUMMARY

The discovery more than 25years ago of the quantumHall effect (1), in which the

“Hall,” or “transverse electrical” con-ductance of a material is quantized,came as a total surprise to the physicscommunity. This effect occurs inlayered metals at high magneticfields and results from the forma-tion of conducting one-dimensionalchannels that develop at the edgesof the sample. Each of these edgechannels, in which the current movesonly in one direction, exhibits a quan-tized conductance that is character-istic of one-dimensional transport. Thenumber of edge channels in the sam-ple is directly related to the value ofthe quantumHall conductance.More-over, the charge carriers in these chan-nels are very resistant to scattering.Not only can the quantum Hall effect be observed in macroscopic samplesfor this reason, but within the channels, charge carriers can be transportedwithout energy dissipation. Therefore, quantum Hall edge channels may beuseful for applications in integrated circuit technology, where power dis-sipation is becomingmore andmore of a problem as devices become smaller.Of course, there are some formidable obstacles to overcome—the quantumHall effect only occurs at low temperatures and high magnetic fields.

In the past few years, theoretical physicists have suggested thatedge channel transport of current might be possible in the absence of amagnetic field. They predicted (2–4) that in insulators with suitableelectronic structure, edge states would develop where—and this isdifferent from the quantum Hall effect—the carriers with oppositespins move in opposite directions on a given edge, as shown sche-matically in the figure. This is the quantum spin Hall effect, and itsobservation has been hotly pursued in the field.

Although there are many insulators in nature, most of them do not havethe right structural properties to allow the quantum spin Hall effect to beobserved. This is where HgTe comes in. Bulk HgTe is a II-VI semi-conductor, but has a peculiar electronic structure: In most such materials,the conduction band usually derives from s-states located on the group IIatoms, and the valence band from p-states at the VI atoms. In HgTe thisorder is inverted, however (5). Using molecular beam epitaxy, we cangrow thin HgTe quantum wells, sandwiched between (Hg,Cd)Te barriers,that offer a unique way to tune the electronic structure of the material: Whenthe quantum well is wide, the electronic structure in the well remainsinverted. However, for narrow wells, it is possible to obtain a “normal”alignment of the quantumwell states. Recently, Bernevig et al. (6) predicted

theoretically that the electronicstructure of inverted HgTe quan-tum wells exhibits the propertiesthat should enable an observationof the quantum spin Hall insula-tor state. Our experimental obser-vations confirm this.

These experiments only be-came possible after the devel-opment of quantum wells ofsufficiently high carrier mobility,combined with the lithographictechniques needed to pattern thesample. The patterning is espe-cially difficult because of the veryhigh volatility of Hg. Moreover,we have developed a special low–deposition temperature Si-O-Ngate insulator (7), which allowsus to control the Fermi level (theenergy level up to which all

electronics states are filled) in the quantum well from the conduction band,through the insulating gap, and into the valence band. Using both electronbeam and optical lithography, we have fabricated simple rectangularstructures in various sizes from quantum wells of varying width andmeasured the conductance as a function of gate voltage.

We observe that samples made from narrow quantum wells with a“normal” electronic structure basically show zero conductance when theFermi level is inside the gap. Quantum wells with an inverted electronicstructure, by contrast, show a conductance close to what is expected for theedge channel transport in a quantum spin Hall insulator. This interpretationis further corroborated by magnetoresistance data. For example, high–magnetic field data on samples with an inverted electronic structure show avery unusual insulator-metal-insulator transition as a function of field,which we demonstrate is a direct consequence of the electronic structure.

The spin-polarized character of the edge channels still needs to beunequivocably demonstrated. For applications of the effect in actualmicroelectronic technology, this low-temperature effect (we observe itbelow 10 K) will have to be demonstrated at room temperature, which maybe possible in wells with wider gaps.

Summary References1. K. v. Klitzing, G. Dorda, M. Pepper, Phys. Rev. Lett. 45, 494 (1980).2. S. Murakami, N. Nagaosa, S.-C. Zhang, Phys. Rev. Lett. 93, 156804 (2004).3. C. L. Kane, E. J. Mele, Phys. Rev. Lett. 95, 146802 (2005).4. B. A. Bernevig, S.-C. Zhang, Phys. Rev. Lett. 96, 106802 (2006).5. A. Novik et al., Phys. Rev. B 72, 035321 (2005).6. B. A. Bernevig, T. L. Hughes, S.-C. Zhang, Science 314, 1757 (2006).7. J. Hinz et al., Semicond. Sci. Technol. 21, 501 (2006).

RESEARCHARTICLES

Conductance channel withdown-spin charge carriers

Conductance channel withup-spin charge carriers

Quantumwell

Schematic of the spin-polarized edge channels in a quantum spin Hallinsulator.

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Quantum spin Hall effect:

Quantum spin conductivity:

Quantum conductance of a ballistic conductor (Landauer-Buttiker theory):

Quantum charge conductance:

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Moment accumulation

without bulk moment current:

If there are magnetic impurities at edges:

Accumulation at edge states !"!

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without bulk moment current:

If there are magnetic impurities at edges:

Accumulation at edge states electrode

electrode

Moment accumulation

without bulk moment current:

If there are magnetic impurities at edges:

Accumulation at edge states

Taking into account the boundary condition

Transverse moment current violates the charge conservation law!

Transverse moment current:

Transverse moment current is proportional to another invariant:

Superposition of an incident and a reflected wave:

Sonin, Phys. Rev. B 82, 113307 (2010).

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conventional insulator� topological insulator�

Is spin accumulation is a good probe of the spin current?

Spin current without spin accumulation:

Equilibrium spin currents, spin torque instead of spin accumulation

Spin accumulation without spin current:

Intrinsic spin Hall effect in the Rashba model

Conclusions

Question:

Up to now there is no experimental evidences of the quantum spin Halleffect

Edge spin accumulation (if and when experimentally detected) will not provide such evidences

New ideas and new experimental set ups are wanted for real detectionof the quantum spin Hall effect

Is «Quantum spin Hall state» a good name for a topological insulator???