Quantum Spin Hall Effect in Topological Insulators

Racah Institute of Physics Hebrew University of Jerusalem

Quantum spin Hall effect in topological insulators

Edouard Sonin

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Spin Hall effect

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

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@S h + J = G Spin: SpinS = balance( y ) @t ri i 2 h Spin current: @S h Ji = ( y{ vi + vi } ) Spin: S = ( y ) 4 + iJi = G 2 Spin@t balancer Spin balance ~v is the group-velocity operator h Spin current: J = ( y{ vSonin+ v } ) @S i h i i @S Adv.4 Phys. 59, 181 (2010)h + iJ = G Rashba model: Spin: S = Spin:( y S ) = ( y ) i + iJi = G 2 @t r h @t r ~v is the group-velocity operator 2 ~v = ( i~ +h [^z ~]) Spin current: h mSpin Jcurrent:ir= ( y{ Jvi =+ vi( }y {) Rashbav + v model:} ) 4 i 4 i i i h2 h Torque: G~ v =is the group-velocity y{[~ operator[^z~v = ~ ]]( i}~ + [^{z[[~ ~])z^] ~] y} 2m ~v is the group-velocity rm roperator r n o Rashba model: 2 Rashbai h model: ~ ~ 2 h Torque: G = 2 y{[~ [^z ]] } {[[ z^] ~] y} ih ~ h h 2m r r Ji = ~v(= y ( i +~v =[^iz y(~ ]) i~) + [^z ( ~y])n{ [~ z^]i +[~ z^]i } ) o 4m m r r r m r 4m 2 i h i ihh22 h2 G = y{[~ [^z ~ ]] } {[[~ z^] ~] y} Torque: Torque: G Ji== ( yy { [~i [^z i ~ ]]y }) { [[(~ y{z^] [~~] z^ ]iy+[} ~ z^]i } ) 2m 24mm r r r rr 4m r n n o o

ih2 2 h2 2 ih h Ji = ( Jy = i ( i yy ) (y y{ ) [~ z^](i +[y{~ [~z^]iz^ ]} +[) ~ z^] } ) 4m i r 4mr ri r4mi 4m i i Are spin currents observable?

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

A moving magnetic moment generates an electric ﬁeld

Vacuum:

Rasba medium:

Inverse spin Hall eﬀect: Valenzuela and Tinkham, Nature, 442,176 (2006) - spin diffusion current Jungfleisch et al., arXiv: 1011.0889 - magnon spin current Spin accumulation by a spin current (Dyakonov & Perel, 1971) Spin diffusion current Bulk spin current:

Spin diffusion current: Bulk spin current Spin balance equation:

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

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WeakWeak disorder, disorder, intrinsic extrinsic spin Hall Hall effect: effect:

PHYSICAL REVIEW LETTERS week ending PRL 94, 047204 (2005) 4 FEBRUARY 2005

Experimental Observation of the Spin-Hall Effect in aTwo-Dimensional Spin-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 Kingdom 2National 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 Republic 5School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom 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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- 1.5 -1.0 - 0.5 0.0 0.5 1.0 1.5 - 2 -1 0 1 2 Kane and Mele, PRL 95, 146802; ibid., 226801 (2005). BHZ model [Bernevig, Hughes, and Zhang, Science 314, 1757 (2006)]:

kp theory

Eigenstates: Edge states

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

p-type valence band:

s-type conduction band:

Magnetic moment: Mechanical moment:

Operator of the effective moment:

Moment current:

Operator of the group velocity: Transverse moment currents induced by an electric ﬁeld Plane-wave presentation

Corrected plane-wave eigenstate:

Transverse moment current:

Topological term:

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

!"#$% %&'(#$ )* !"#$% %&'(#$ )+ RESEARCH ARTICLES

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

AUTHORS’ SUMMARY

he discovery more than 25 Conductance theoretically that the electronic years ago of the quantum channel with structure of inverted HgTe quan- THall effect (1), in which the up-spin charge tum wells exhibits the properties “Hall,” or “transverse electrical” con- carriers that should enable an observation ductance of a material is quantized, of the quantum spin Hall insula- came as a total surprise to the physics tor state. Our experimental obser- community. This effect occurs in vations confirm this. layered metals at high magnetic These experiments only be- fields and results from the forma- came possible after the devel- tion of conducting one-dimensional opment of quantum wells of channels that develop at the edges sufficiently high carrier mobility, on July 15, 2010 of the sample. Each of these edge combined with the lithographic channels, in which the current moves techniques needed to pattern the only in one direction, exhibits a quan- sample. The patterning is espe- tized conductance that is character- Conductance cially difficult because of the very istic of one-dimensional transport. The channel with high volatility of Hg. Moreover, number of edge channels in the sam- Quantum down-spin we have developed a special low– ple is directly related to the value of well charge carriers deposition temperature Si-O-N the quantum Hall conductance. More- Schematic of the spin-polarized edge channels in a quantum spin Hall gate insulator (7), which allows over, the charge carriers in these chan- us to control the Fermi level (the insulator. www.sciencemag.org nels are very resistant to scattering. energy level up to which all Not only can the quantum Hall effect be observed in macroscopic samples electronics states are filled) in the quantum well from the conduction band, for this reason, but within the channels, charge carriers can be transported through the insulating gap, and into the valence band. Using both electron without energy dissipation. Therefore, quantum Hall edge channels may be beam and optical lithography, we have fabricated simple rectangular useful for applications in integrated circuit technology, where power dis- structures in various sizes from quantum wells of varying width and sipation is becoming more and more of a problem as devices become smaller. measured the conductance as a function of gate voltage. Of course, there are some formidable obstacles to overcome—the quantum We observe that samples made from narrow quantum wells with a

Hall effect only occurs at low temperatures and high magnetic fields. “normal” electronic structure basically show zero conductance when the Downloaded from In the past few years, theoretical physicists have suggested that Fermi level is inside the gap. Quantum wells with an inverted electronic edge channel transport of current might be possible in the absence of a structure, by contrast, show a conductance close to what is expected for the magnetic field. They predicted (2–4) that in insulators with suitable edge channel transport in a quantum spin Hall insulator. This interpretation electronic structure, edge states would develop where—and this is is further corroborated by magnetoresistance data. For example, high– different from the quantum Hall effect—the carriers with opposite magnetic field data on samples with an inverted electronic structure show a spins move in opposite directions on a given edge, as shown sche- very unusual insulator-metal-insulator transition as a function of field, matically in the figure. This is the quantum spin Hall effect, and its which we demonstrate is a direct consequence of the electronic structure. observation has been hotly pursued in the field. The spin-polarized character of the edge channels still needs to be Although there are many insulators in nature, most of them do not have unequivocably demonstrated. For applications of the effect in actual the right structural properties to allow the quantum spin Hall effect to be microelectronic technology, this low-temperature effect (we observe it observed. This is where HgTe comes in. Bulk HgTe is a II-VI semi- below 10 K) will have to be demonstrated at room temperature, which may conductor, but has a peculiar electronic structure: In most such materials, be possible in wells with wider gaps. the conduction band usually derives from s-states located on the group II atoms, and the valence band from p-states at the VI atoms. In HgTe this Summary References order is inverted, however (5). Using molecular beam epitaxy, we can 1. K. v. Klitzing, G. Dorda, M. Pepper, Phys. Rev. Lett. 45, 494 (1980). grow thin HgTe quantum wells, sandwiched between (Hg,Cd)Te barriers, 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). SCIENCE

that offer a unique way to tune the electronic structure of the material: When = the quantum well is wide, the electronic structure in the well remains 4. B. A. Bernevig, S.-C. Zhang, Phys. Rev. Lett. 96, 106802 (2006). 5. A. Novik et al., Phys. Rev. B 72, 035321 (2005). BICKEL

inverted. However, for narrow wells, it is possible to obtain a “normal” 6. B. A. Bernevig, T. L. Hughes, S.-C. Zhang, Science 314, 1757 (2006). : alignment of the quantum well states. Recently, Bernevig et al.(6)predicted 7. J. Hinz et al., Semicond. Sci. Technol. 21, 501 (2006). CREDIT: C

766 2 NOVEMBER 2007 VOL 318 SCIENCE www.sciencemag.org Quantum spin Hall effect:

Quantum spin conductivity:

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

Quantum charge conductance: !"!#$%electrode&! If there are magnetic impurities at edges: edges: at edges: at at If there aremagneticimpurities If there aremagneticimpurities If there aremagneticimpurities < < < Accumulation atedgestatesAccumulation atedgestatesAccumulation atedgestates

!"!#$%electrode&! without bulkmomentcurrent: without bulkmomentcurrent: without bulkmomentcurrent: Moment accumulation Moment accumulation Moment accumulation Taking into account the boundary condition Sonin, Phys. Rev. B 82, 113307 (2010).

Transverse moment current violates the charge conservation law! Superposition of an incident and a reflected wave:

Transverse moment current:

Transverse moment current is proportional to another invariant: conventional insulator topological insulator

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0.0 ! 0.4 ! 0.2 0.0 0.2 0.4 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

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

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

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

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