Protected/non-protected edge states in two-dimensional lattices

Transcription

1 Protected/non-protected edge states in two-dimensional lattices (a review of edge states behavior) A. Aldea, M. Niţa and B. Ostahie National Institute of Materials Physics Bucharest-Magurele, Romania MESO-III, Busteni, sept.217 p. 1

2 foreword generic surface/edge states : Tamm, Shockley 193, catalysis new context: support for topological effects in condensed matter: - robust against disorder cannot be scattered back by an impurity (non-dissipative) protected by specific symmetry. - non-protected : backscattering allowed : dissipative states electrical resistance! both types may appear in the same material (absence of backscattering) (edge current: tight-binding calc.) p. 2

3 outline 1. classical case - 2DEG in strong magnetic field (IQHE) 2. phosphorene : dissipative edge states 3. Lieb lattice: unconventional edge states 4. hexagonal lattice (graphene): spin-orbit int. (QSHE) 5. short-cut edge states (electrical manipulation; intermediate quantum Hall plateaus) p. 3

4 introduction to edge states Topological phenomena in electron systems: - characterized by integer numbers and protection ensured by sym. Two complementary descriptions: Chern numbers (infinite systems, bulk properties - top.ivariant) Edge states (confined systems) bulk-edge correspondence (Hatsugai) Topological insulators (2D,3D): - insulating in the bulk, metallic at the surface prototype: graphene (with spin-orbit: helical states: edge states carrying opposite spins in opposite directions QSHE starting point : 2D electron gas + strong magnetic field : Landau levels: E f in the gaps Hall resistance R H = integer,r L = (no dissip.) IQHE p. 4

5 I. introduction to IQHE Kubo approach (QM conductivity formula, V ) Pavel Streda + TKNN (1982) : transverse (Hall) conductivity σ xy =... = integer if Fermi levele F gap σ xx =, absence of dissipation recent description: Chern number σ xy = e2 d kb( k) = e2n h BZ h Chern, (Berry curvature) B n (k) = k A n (k) (Berry connection) A n (k) = i < u n (k) k u n (k) > (Bloch fncn) Ψ n (k) = exp(ikr)u n (k)? how do electrons circulate in the Hall device? p. 5

6 electron transmission in a Hall device finite plaquette + 4 leads (in cross) + perpendicular magnetic field B. T α,β - transmission coefficient (α,β=1,..,4). I α = e2 h β T αβv β (Landauer-Buettiker, wan der Pauw) can be calculated in Green function formalism T αβ! assume : T α,α+1 = 1, all others vanish (including T α+1,α ) (just one perfectly conducting channel) then inverting matrix T αβ R H = 1(h/e 2 ), R L =, i.e. IQHE? would such a scenario be possible?? what would be the physical support? p. 6

7 spectral properties of 2D confined system tb Hamiltonian in magnetic field: H = i,j t ije iφ(i,j) a i a j 6 4 spectrum placheta 2x2 [211/ sp2_anulare.for] sp2_anulare.otp Eigenenergies Magnetic Flux Eigen_energies Magnetic flux Hofstadter-type : phoshorene (left) and 2D electron gas (right) effect of confinment (b.c.) and broken time-reversal symmmetry: gaps filled with states localized around perimeter (edge states skipping orbits) carry current along edgesde n /dφ ; definite sign ( ) de n dφ (chirality) no back scattering: thre are no states with opposite chirality in the vecinity. p. 7

8 ! remember scenario : properties of edge states fulfill requirements! remember Streda (E f in gap) edge states responsible for IQHE.? question left: why the injected electron gets out at the next lead and does not travel round about the plaquette?! edge states are generated and protected by the magnetic field.? is it possible to get quantum Hall effect without external B? Haldane : AQHE (anomalous) p. 8

9 II. edge states in phosphorene tight-binding Hamiltonian: H = n,m t n,me iφ mn a na m, (n,m=site index), φ nm =Peierls phase if magnetic field puckered structure two atoms A/B per cell five hopping integrals (t 1,t 3 - connect atoms in the same plane, t 2,t 4,t 5 -different planes) bipartite lattice symmetry: broken by t 4, broken electron-hole sym anisotropy Ox Oy,k x k y two types of margins: zig-zag (support edge states), arm-chair p. 9

10 ribbon confinment periodic boundary conditions, Fourier transform, diagonalization semiconductor hybrid Dirac spectrum ribbon geometry: vanishing b.c. along zig-zag edges quasi-flat band in the gap edge states at anyk x degeneracy lifted B E(k x ) = 4 ( t 4 t 1 /t 2 ) (1+coskx ), Ezawa 214, Ostahie-Aldea 216 p. 1

11 phosphorene plaquette in magnetic field transport properties! expect two types of edge states: flat band + chiral in Landau gaps Eigenenergies Magnetic Flux i) Hofstatder (double) butterfly G Hall (e 2 /h) ii) IQHE: usual steps in the range of chiral edge states iii) edge states in the quasi-flat band: do not support QHE (G H = ), dissipative (G L ) non-robust/non-protected iv) why only 6 peaks in longitudinal conductance? p. 11 V gate G Long (e 2 /h)

12 superradiance of flat edge states non-hermitian Hamiltonian in 1D systems : complex eigenvalues overlapping and segregation with τ : superradiative effect. edge states 1D Hall device= open system : plaquette(h S )+leads(h L )+coupl (τ) non-hermitian description H S Heff S - complex eigenvalues zig zag edge Im E T read out the edge V Re E gate Note: lead configuration: all on the same zig-zag edge Note: many edge states with ImE = Note: 11 edge states with ImE segregation? how the coupling strength affects the transport? p Dos (arb. units)

13 three conducting regimes of the edges states (! general question, back to 2DEG) depending on the coupling strength τ: I. resonant (τ/t << 1) - sharp peaks of the transmission coefft II. quantum Hall (τ/t 1) - plateaus III. superradiance (τ/t > 1) - re-entrance of the oscillatory regime. transmission coefficient T_ V_gate (! V gate - gate potential spans the energy range of edge states) 26-tau4 u 1:3 26-tau15 u 1:3 26-tau24 u 1:3 II I? how does the electron propagate along edges in each case? p. 13

14 scattering wave function lead wave fcn Φ L + edge state in plaquette = scattering wave Ψ α = Φ L α +GV Φ L α Lipmann-Schwinger (V = H SL,G =..) projection of Ψ α on the plaquette sites: n Ψ α 2 = τ 2 n 1 V gate H S eff α 2, n plaquette,v gate gap (a) - regime I - resonant (c) - regime II - QHE (d) - regime III - superrad Answer to question in QHE scenario why T 12 = 1, butt 13 =. Non-Hermitian approach of edge states.. B.Ostahie, M.Nita, A.Aldea, Phys.Rev.B 94(216) p. 14

15 III. Lieb lattice: unconventinal edge states Lieb lattice: 2D, line-centered square, 3atoms/unit cell! note different connectivity of A,B,C atoms Y C (n,m+1) (n+1,m+1) (n+2,m+1) (m+1)a A C (n,m) (n+1,m) (n+2,m) ma t y t x A B A B O na (n+1)a (n+2)a X! energy spectrum: 3 bands, symmetric, 2 Dirac cones flat band (macroscopically degenerate) look for edge states in finite Lieb lattice in magnetic field p. 15

16 magnetization, transmission coefficient 2 1 Type-II edge states (a) Twisted edge states 2 (a) 2 (b) M α -1-2 Conventional edge states T12, T T 12 RH, RL, T R H T 12 R L Pedge (b) T Energy Energy - range of interest E α diamgnetic moment: M α = < α dh dφ α > sign given by chirality of state α. edge localization: P edge α = i edge < Ψ α i > 2 edge states with alternating chirality transport : a)t 12 = T 21, b)r H =,R L dissipation and oscillation p. 16

17 Lieb confined system: energy spectrum twisted Eigenenergies conventional (a) Eigen-energies magnetic flux magnetic flux.8 1 (a) unconventional: generated by magnetic field, but do not support IQHE, pairs with opposite chirality, changing chirality de dφ -.5 (b) dissipative transport at B Eigenenergies non-robust against disorder magnetic flux Spectral and transport properties of the Lieb lattice, M.Nita, B.Ostahie, A.Aldea, PRB 87 p. 17

18 IV. graphene as topological insultor time-reversal protected edge states edge states in graphene: zig-zag/ arm chair how to evidentiate: calculation of the wave functions (analitycally or numerically), UHV STM (9nm 9nm)- Y.Kobayashi (Phys.Rev.B 71,19346 (25))- signal only along zig-zag edges. notable : only zig-zag edges support edge states. remember: magnetically induced edge states -all around the perimeter Kane & Mele: Haldane model with spin, specific phases of the NNN hopping, specific SOI = intrinsic, without spin-flip. p. 18

19 role of confinment: graphene ribbon ribbon geometry with zig-zag edges, we are left with k x only : 3 graphene zigzag :Nlat=2,flux= [netto_test.f] zigzag_2_.otp 2 eigen-energies 1-1 Wakabayashi et al, k_x i) gapless energy spectrum ii) for any k x between K x,k x there are 2 states ate =, which prove to be edge states localized near zigzag edges of the cylinder iii) no dispersion (de/dk x =, non-conducting) iv) ribbon with arm-chair edges : surface states missing Note : no spin here, everything degenerated with respect to the spin p. 19

20 Kane & Mele: role of spin-orbit coupling intrinsic spin-orbit interaction: preserves the time reversal symmetry, conserves the spin (no spin-flip l z s z ) opens a gap: semi-metal (topological) insulator 3 pxg.otp eigenenergies k_x dispersive edge states (helical) E(k, ) = E( k, ) any E gap, pair of states carrying spin in opposite directions. total spin conductance G +G = 2(e/4π) [no charge current] robust against (nonmagnetic) impurities due absence of spin flip typical topological effect - QSHE (Z 2 ) p. 2

21 SOI+magnetic field: competing symmetries plaquette geometry Valori Proprii effect I : spin imbalance of chiral edge states (in Landau gaps) unconventional QSHE, IQHE second relativistic gap first realtivistic gap second relativistic gap topological gap imbalanced gap first realtivistic gap Conductance <spinimbalanced gap> intermediate plateaus DOS G H S <topolog. gap> G H Q DOS Spin-resolved DOS (arb.units) imbalanced gap Energy effect II : closing of the SO gap Topological properties of mesoscopic graphene B.Ostahie, M.Nita, A.Aldea, PRB topological. gap Magnetic Flux p. 21

22 V. electrical manipulation of edge states another competition in graphene: perpendicular magnetic + in-plane electric fields shortcut of edge states in graphene plaquette changes: current circulation, matrix of the transmission coefficients, plateaus of IQHE p. 22

23 IQHE with shortcut edge states R Hall / / Energy - conventional plateaus forr H : 1, 1/3, 1/5 (spinless) Novoselov - additional intermediate plateaus due to shortcut : 2/3, 4/15,.. Electrical manipulation of the edge states in graphene and the effect on the quantum Hall transport, B.Ostahie, M.Nita,A. Aldea,PRB 91 p. 23

24 Different kinds of edge states Ψ 2 zig-zag edge state: E = (no matter B = orb ) helical state in the spin-orbit gap zig-zag : E, B chiral: 2DEG, Landau gap (B ) p. 24

25 summary topological edge states - insensitive to disorder due to protecting symmetry - can be manipulated by electric field or magnetic field (spin imbalance in presence of SOI ) new plateaus of IQHE - killed by competing symmetries dissipative edge states - sensitive to disorder (phosphorene, Lieb lattice) electrical resistance coupling to contacts : three conducting regimes of the edge states (resonant, QHE, superradiative) p. 25