Electrical Control of the Kondo Effect at the Edge of a Quantum Spin Hall System

Transcription

1 Correlations and coherence in quantum systems Évora, Portugal, October Electrical Control of the Kondo Effect at the Edge of a Quantum Spin Hall System Erik Eriksson (University of Gothenburg) Anders Ström (University of Gothenburg) Girish Sharma (École Polytechnique) Henrik Johannesson (University of Gothenburg) supported by the Swedish Research Council

2 Outline Quantum spin Hall effect... some basics At the edge: A new kind of electron liquid Adding a magnetic impurity and a Rashba spin-orbit interaction Electrical control of the Kondo effect! Summary and outlook

3 Quantum spin Hall effect... some basics

4 Quantum spin Hall effect

5 Integer Quantum Hall effect B skipping currents quantization chiral edge states B. I. Halperin, PRB 25, 2185 (1982) no channel for backscattering ballistic transport along the edge, accounts for the quantization of the Hall conductance M. Büttiker, PRB 38, 9375 (1988)

6 Can a system be stable against local perturbations without breaking time-reversal invariance?

7 Consider a Gedanken experiment... B. A. Bernevig and S.-C. Zhang, PRL 96, (2006) > uniformly charged cylinder with electric field E = E(x, y, 0) spin-orbit interaction (E k) σ = Eσ z (k y x k x y) > B = A cf. with the IQHE in a symmetric gauge A = B 2 (y, x, 0) > Lorentz force A k eb(k y x k x y)

8 Quantum spin Hall (QSH) system single Kramers pair Two copies of a IQH system, bulk insulator with helical edge states > > First proposed by Kane and Mele for graphene (2005) too weak spin-orbit interaction, doesn t quite work... Bernevig et al. proposal for HgTe quantum wells (2006) Experimental observation by König et al. (2007)

9 Bernevig et al. proposal for HgTe quantum wells (2006) Experimental observation by König et al. (2007) from M. König et al., Science 318, 766 (2007)

10 Are the helical edge states stable against local perturbations? Yes! As long as the perturbations are time-reversal invariant! Look at the band structure of an HgTe quantum well... strong spin-orbit interactions in atomic p-orbitals create an inverted band gap (p-band on top of s-band) supports a single Kramers pair of helical edge states inside the inverted gap Kramers degeneracy at k=0 protects the stability of the edge states ballistic transport B. A. Bernevig et al., PRL 95, (2005) G = 2e2 h 4-terminal measurement, equilibration in contacts

11 At the edge: A new kind of electron liquid # of Kramers pair in a nontrivial (trivial) helical liquid 2D topological insulator (a.k.a. quantum spin Hall system) NK = 1 mod 2 = 0 mod 2 ν =0, 1 is a Z2 topological invariant and can be calculated from the band structure of the bulk ( bulk-edge correspondence ) L. Fu and C. L. Kane, PRB 76, (2007)

12 At the edge: A new kind of electron liquid # of Kramers pair in a nontrivial (trivial) helical liquid NK = 1 mod 2 = 0 mod 2 ν =0, 1 is a Z2 topological invariant and can be calculated from the band structure of the bulk ( bulk-edge correspondence ) 2D topological insulator (a.k.a. quantum spin Hall system) L. Fu and C. L. Kane, PRB 76, (2007) What if time-reversal symmetry is broken...?

13 ... for example, by the presence of a magnetic impurity? case study: Mn 2+ large and positive single-ion anisotropy (S z ) 2 from M. König et al., Science 318, 766 (2007) S =5/2 S eff =1/2 low T anisotropic spin exchange with the edge electrons R. Zitko et al., PRB 78, (2008) H K = Ψ (0) J (σ + S eff + σ S + eff )+J zσ z Seff z Ψ(0), Ψ T = ψ, ψ

14 Adding a magnetic impurity... The Kondo interaction is time-reversal invariant! Could it still cause a spontaneous breaking of time reversal invariance and collapse the QSH state?

15 Adding a magnetic impurity... Recall the Kondo effect One-loop RG equations: P. W. Anderson, J. Phys. C 3, 2436 (1970) J D = νj J z +... J z D = νj strong-coupling physics for T<<T K T K = D 0 exp( const./j 0 ) J 0 max(j,j z ) D=D0 formation of impurity-electron singlet ( Kondo screening )

16 Adding a magnetic impurity... T=0 insulator! Pauli principle: punctured 1D lattice

17 Adding a magnetic impurity... Does this really happen for the helical liquid? To find out, first add e-e interactions... important in 1D! bulk k F k F k + local (at impurity site) + + ( ) from Kondo 0 if U(1) spin symmetry is broken T. L. Schmidt et al., PRL 108, (2012)

18 Adding a magnetic impurity... Adding the kinetic energy and bosonizing... H = (v/2) dx ( x ϕ) 2 +( x ϑ) 2 + A κ cos( 4πKϕ)+ B κ sin( 4πKϕ)+ C K x ϑ Luttinger liquid parameter functions of Kondo couplings + + g U 2(πκ) 2 cos( 16πKϕ) g ie 2π 2 K :( 2 xϑ) cos( 4πKϕ) :...perturbative RG and linear response J. Maciejko et al., PRL 102, (2009)

19 Adding a magnetic impurity... no correction to the edge conductance at T = 0! G =2e 2 /h weak e-e interaction strong e-e interaction from J. Maciejko et al., PRL 102, (2009)

20 Adding a magnetic impurity... Due to its topological nature, the QSH state follows the new shape of the edge. Weak coupling QSH states are robust against local breaking of time-reversal symmetry! J. Maciejko et al., PRL 102, (2009)

21 But... one important thing is missing from the analysis!

22 Rashba spin-orbit interaction! z 2DEG d semiconductor heterostructure d z Spatial asymmetry of band edges mimics an E-field in the z-direction H R = α(k x σ y k y σ x ) Yu. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039 (1984)

23 Rashba spin-orbit interaction! z d x semiconductor heterostructure d z H R = α k x σ y doesn t conserve spin x

24 reversal invariance electron transport then becomes balwhich sets the scale below which the electrons screen the listic, provided that the electron-electron (e-e) interaction impurity can be controlled by varying the strength is sufficiently well-screened so that higher-order scatterof the Rashba interaction. Surprisingly, for a strongly?? ing processes do not come into play. anisotropic Kondo exchange, a non-collinear spin inter action mediated by the Rashba coupling becomes rel The picture gets an added twist when including effects breaks the locking of spin to momentum. However, stillcompetes with the evant (in the sense there of RG)isand from... magnetic impurities, contributed by dopant ions or Kondo screening. This challenges the expectation that electronsatrapped potential inhomogeneities. an edge, singlebykramers pair on thesince QSH and this is all that matters! the Kondo effect is stable against time-reversal invari edge electron can backscatter from an impurity via spin L. Kane and E.?J. Mele, PRL 95, (2005) ant C. perturbations. Moreover, we show that the im exchange, time-reversal invariance no longer protects the purity contribution to the dc conductance at tempera helical states from mixing. In addition, correlated twotures T > TK can be switched on and off by adjusting electron? and inelastic single-electron processes?? must kinetic term Rashba 1 coupling. With the Rashba coupling being now also be accounted for. As a result, at high temper- the Rashba tunable by a gate voltage, this suggests a new inroad to atures T electron scattering off the impurity leads to a z y? H = vf of dx Ψ (x) [ iσat low Ψ(x) + α control dx Ψcharge (x) [ iσ x ]atψ(x) transport the edge of a QSH device. ln(t ) correction the conductance x ] frequencies Model. To model the ω, which, however, vanishes? in the dc limit ω 0. At edge electrons, we introduce T 2 2 two-spinors Ψ = ψ, ψ, where ψ (ψ ) annihi low T, for weak e-e interactions, the quantized vedge vf + αthe α = conlates a right-moving (left-moving) electron with ductance G0 = e2 /h is restored as T 0 with power 1 spin-up x iσ(spin-down) θ/2 laws distinctive of a helical edge liquid. For strong Ψx inter=e Ψ along the growth direction of the quantum well. Neglecting e-e interactions, the edge Hamiltonian actions the edge liquid freezes into an insulator at T = 0, 2 + α2 cos θ = v /v v = v F α α F can then be written as with thermally induced transport via tunneling of frac tionalized charge excitations through the impurity?. 1 in σ H = vf dx Ψ (x) [ iσ z x ] Ψ(x) + A more complete description of edge H transport ay R = α k x H =must vα include dx Ψ iσ z ofx a sin Ψ QSH system also(x) the presence Rashba θ(x) = α/vα cos θ = F /vα z v z spin-orbit interaction. This interaction, which can be +α dx Ψ (x) [ iσ y x ] Ψ(x) + (1) tuned by an external gate voltage, is a built-in feature T? x Ψ = ψ, ψ of a quantum well. In fact, HgTe quantum wells ex +Ψ (0) [Jx σ x SH += Jyv σ y S y + dx Jz σ z Ψ S z ] Ψ(0), T K (x) [ θ Ψ = ψ, ψ sin θ = α/vα F hibit some of the largest known Rashba couplings of any semiconductor heterostructures?the. As Rashba-rotated a consequence, spinor with vf the Fermi velocity parameterizing the linear ki isaassumed Kramers spin is no longer conserved, contrary to defines what second term encodes the Rashba inter pairnetic energy. still The? 2 2 z 2(.dx T K λ/2) action strength 2 2( K+λ/2) 1 the 2 term 2 in the minimal model of However, of α, with being 1 iσ z =y )v2= Ψ (x) Ψ (x) αt v x Ψ = ψ, ψ γ0 a QSH (JxH +system J, γ (J J ) T, γ0e third JN T 2K 1, γ 0Ean anti JN 2 2 H dx Ψ (x) iσ Ψ (x) xx α 0 y C = (v/2) dxtime-reversal ( x ϕ) + in( x ϑ)ferromagnetic Kondo interaction between electrons (withc since the Rashba H interaction preserves variance, Kramers theorem guarantees that the edge Pauli matrices σ i, i = x, y, z) and a spin-1/2 magnetic im A B C z + cos( 4πKϕ)+ sin( 4πKϕ)+ ϑ H = v Idx=ΨG (x) V iσ δix Ψ (x) Adding the Rashba interaction...

25 Adding the Rashba interaction... e-e interaction is invariant under Ψ Ψ Kondo interaction H K = Ψ (0) J (σ + S eff + σ S + eff )+J zσ z S z eff Ψ(0), Ψ =e iσx θ/2 Ψ S =e isx θ/2 Se isx θ/2 H K = Ψ (0)[J x σ x S x + J yσ y S y + J zσ z S z + J NC (σ y S z + σ z S y )]Ψ (0) XYZ Kondo Non-Collinear term depend on the Rashba coupling controllable by a gate voltage α...bosonization and perturbative RG E. Eriksson et al., PRB 86, (R) (2012)

26 Electrical control of the Kondo temperature via the Rashba angle θ ~ gate voltage easy-plane Kondo J x = J y = 20 mev, J z = 10 mev easy-axis Kondo J x = J y =5meV, J z = 50 mev J NC Region where dominates the RG flow obstruction of Kondo screening! challenges the Meir-Wingreen conjecture that the Kondo effect is blind to time-reversal invariant perturbations Y. Meir and N.S. Wingreen, PRB 50, 4947 (1994)

27 Low-temperature transport, T T K (away from the dome ) weak e-e interaction K>1/4 T =0 G = 2e2 h

28 Low-temperature transport, T T K (away from the dome ) weak e-e interaction G = 2e2 h δg 1/4 <K<2/3 K>2/3 (T/T K ) 8K 2 (T/T K ) 2K+2 signature of Rashba!

29 Low-temperature transport, T T K (away from the dome ) weak e-e interaction G = 2e2 h δg 1/4 <K<2/3 K>2/3 (T/T K ) 8K 2 (T/T K ) 2K+2 signature of Rashba! strong e-e interaction T =0 G =0 K<1/4

30 Low-temperature transport, T T K (away from the dome ) weak e-e interaction G = 2e2 h δg 1/4 <K<2/3 K>2/3 (T/T K ) 8K 2 (T/T K ) 2K+2 signature of Rashba! strong e-e interaction K<1/4 G (T/T K ) 2(1/4K 1) blind to Rashba from instanton processes J. Maciejko et al., PRL 102, (2009)

31 High-temperature transport, T T K I = G 0 V δi J,z ω T

32 High-temperature transport, T T K I = G 0 V δi J,z ω T conductance correction in dc limit: γ 0 (J x + J y) 2 T 2( K λ/2) 2 1, γ 0 (J x J y) 2 T 2( K+λ/2) 2 1, γ E 0 J 2 NCT 2K 1, γ E 0 J 2 NCT

33 Summary and outlook E. Eriksson et al., PRB 86, (R) (2012) Magnetic impurity in a helical edge liquid: Rashba coupling allows electrical control of Kondo temperature and IV-characteristics blocking of Kondo screening!? Rashba-induced impurity correction accessible in experiment? Interesting open problems: Two-loop RG, thermal transport, effects from Dresselhaus spin-orbit interactions, Kondo lattice in a helical liquid, higher-spin impurities... and more!