High-mobility electron transport on cylindrical surfaces

Transcription

1 High-mobility electron transport on cylindrical surfaces Klaus-Jürgen Friedland Paul-Drude-nstitute for Solid State Electronics, Berlin, Germany Concept to create high mobility electron gases on free standing semiconductor heterostructures -Experiment Electronic effects on a cylinder surface - Adiabatic transport transport nontrivial trajectories Landauer Büttiker using open orbits - Quantum Hall effect on a cylinder surface Landauer Büttiker fails take self- consistent screening - Commensurable resistance oscillations for tangentially directed magnetic fields Calculation of skipping orbits

2 Contributions A. Riedel, R. Hey, H. Kostial PD U. Jahn,. Höricke, A. Siddiki University ugla, Turkey D. K. aude. HFL CNRS, France

3 n X Ga -X As Stressor x n h (nm) h (nm) #A #B Rolling-up a Heterostructure (Al,Ga)As with DEG, h (n,ga)as stressor, h (AlAs) release layer ε r B B = B cos(ϕ) r = h + χh 3 h + 6χh 6εχ( + υ )h ( h + h ratio of Young s moduli χ, Poisson ratio ν and strain ε h h + χh h 3 ) + χ h - Strain gradient ε % ε - agnetic field gradient T/µm

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5 Heterostucture tubes containing a Hall bar -Shallow mesa- etching and Ohmic contacts on the flat surface -Etching from a starting line, rolling-up a tube r µm A. B. Vorob'ev et al., Physica E,. Drawback for (Al,Ga)As System: Surface states at the new surface depletion of carriers, decrease mobility S. endach, et a., Physica E,.

6 New surface Structure SPSL QW SPSL Concept for high-mobility DEGs on cylindrical surfaces (mev) potential, V Γ,X -E f δ δ E X E Γ E Γ X X Γ density ( 8 ) cm -3 Concept for freestanding heterostructures: Screening of potential fluctuations also from surface charges of the new surface DEGs on tubes: obility of up to m /Vs at n S.7x 6 m position (nm) K.-J. Friedland et al., Phys. Rev.Lett. 996.

7 Adiabatic transport on cylindrical surface ean free path compares with the rolling radius: l mfp µm r ϕ = low gradient Negative bend resistance in a wide cross junction R at zero magnetic field h e T F 3 T R R L F B = R,3 = ) < T T T D, bend resis stance (Ω) - F T > T B ϕ R,3 -,3 -, -,,,,,3 L T R B (T) K.-J. Friedland et al., Phys. Rev.B 7.

8 Extended Ballistic transport on cylindrical surface ean free path compares with the rolling radius: l mfp r, ϕ = 9 δb /B 3% trochoid-like trajectories (ETT) move oppositely to guided trajectories (GT) R 3 ETT e GT y B GT ETT h T T R R L B =, 3 = ) D, bend res sistance (Ω) 6 - R,3 R, B <, B >, T T GT R GT R >, =, B (T) T T GT R GT R T T 3 B ETT L ETT L >, = K.-J.Friedland et al., Phys. Rev.B 7

9 µ + K K B K + µ µ µ 3 µ R 3 µ µ = µ µ µ = 3 K K e h, K ) ( K e h + = µ µ 3, e h = µ µ µ 3 = µ

10 µ 3 µ R = * B µ 3 µ T µ K µ h = e K + K K K T µ µ µ µ _ 3 µ µ h µ = µ 3 = µ, =, e K + B=: Negative bend resistance =, µ 3 µ h e T

11 Quantum Hall effect one dimensional Landau states + µ µ µ 3 µ y, B K -K -K E LL3 E LL E F E LL y µ µ µ µ = 3 K K K ) ( K K K ) ( e h 3 3 = µ = µ µ = µ µ = µ µ, e h µ 3 Quantum Hall effect dominated by - one-dimensional Landau states (DLS) at the low magnetic field side with - maximum number of states Landauer Büttiker approach:

12 µ 3 ETT QHE y, B µ -K K µ 3 -K y B µ bend resis stance (kω),,5,,5 (a) R,3 E LL3 E LL E LL y 3 R 3, E F 8 6 Hall resist tance (kω) ETT GT, agnetic field B (T)

13 Quantum Hall effect X= Landauer-Büttiker approach - One-dimensional Landau states - Bulk insulator R(kΩ) Hall resistance (kω) a) b) ν R 37 ν R 8 ν ν ν -ν R 78 R 37 -R agnetic Field (T)

14 D Landau states 6 R Landauer Büttiker R (k kω) R 78 6 R B (T)

15 Quantum Hall effect X= Landauer-Büttiker approach - One-dimensional Landau states - Bulk insulator Hall resistance (kω) R L (Ω) R 37 R 6 6 R 3 5 5? agnetic Field (T)

16 D Landau states 6 R R (k kω) R 78 6 Landauer Büttiker R B (T)

17 k y, E y j x B x, E x j j x y = σ xx = σ E xy x E + σ x xy + σ E xx y E y () σ xy j y =, Ey = Ex = µ BE σ xx x () divj = δ jx divj = δx σ xy δ E σ x xy δ Φ = ( σ xx + ) = ( σ xx + ) = σ δx σ δx xx xx General solution Φ = C ( x + C( where E = Φ( x, δφ δφ δc( δc( From (): = µ B x + = µ BC( δy δx δy δy Field gradient along the current : B( x) = Bkx C = Ce µb ky, C = µ B ky µ Bky µ Bky Ey = Cµ Bkye, Ex = Ce jx = σ xxce

18 Asymmetry: V H V H V L - V L = V H - V H Skin channel for the current: A.V. Chaplik, JETP Lett. () L Skin = µ B (L Skin µm at B=,T), R 78 (Ω) Static Skin effect Chaplik's theory R =6.3, B =.5T R fit = R B e B / B B B / e B B B =B cos(ϕ ϕ ) B (T) V L V H V H 5 ϕ -ϕ = 3 V L V H L Skin ρ L (Ω) V L V L 5 R B (T)

19 Quantum Hall effect X= Landauer-Büttiker approach - One-dimensional Landau states - Bulk insulator Hall resistance (kω) R L (Ω) R 37 R 6 6 R agnetic Field (T)

20 Self-consistent calculation of the density and current distribution Total electrostatic potential energy Vtot ( x, = Vbg ( x, + Vext ( x, + VH ( x, V bg ( x, background potential generated by the donors V ext ( x, external potential from the gates (which will be used to simulate the filling factor gradient) V H ( x, Hartree potential to describe the mutual electronelectron interaction Electron density D( E, x, f ( E) = /[exp( E / kbt )) + ] * µ n (local) density of states Fermi function electrochemical potential * ( x, = D( E, x, f ( E + V ( x, µ de el tot ) Hartree potential explicitly depends on the electron density via e V = H ( x, K( x, y, x, y ) nel ( x, y ) dx dy κ K ( x, y, x, y ) solution of the D Poisson equation satisfying the periodic boundary conditions, we A

21 Screening theory in the QHE R. Gerhardts, K. Güven A. Siddiki (3-7) r r r E B Classical and quantum mechanical drift velocities, v D = c B Current flows along the ncompressible Stripes v y eex = mω c S CS - Wave Guide on a Cylinder Surface - Self consistent calculation of carrier and current distribution.8 A C E B ν(x,< D ν(x,> F -quantized conductance at average filling factors ν > Calculations A. Siddiki

22 Hall resistance (kω) Quantum Hall effect 5 5 R 37 R 6 R 8 ν =.5 x (µm).8. A C.8 B D ν(x,< E F ν(x,> 8 6 R 3 ν =. x (µm). ν(x,= R L (Ω) 5 5 agnetic Field (T) ν =.9.8. SSE y (µm) K.-J. Friedland et al., Phys. Rev.B 9.

23 Resistance oscillations for tangentially directed magnetic fields resistance ρ (Ω) ρ ρ 7 6 B = x ρ B 7(+) 6(-) R B = ρ 3 3 magnetic field (T)

24 Resistance oscillations for tangentially directed magnetic fields resistance ρ (Ω) ρ ρ 7 6 B = x ρ B ( T)..5. min max W = µm W = µm.5 ρ magnetic field (T). 3 q

25 Oscillations with free electron states Calculation of the effective potential: -Free electron states, classically Snake-like orbits - SLO B B =B cos(ϕ), ϕ 9 B L free - stripe of width L free e B L E F = hk X m (99) free J. E. üller, Phys. Rev. Lett. + R DLC x resistance ρ (Ω) ρ 7 6 x ρ 3 3 magnetic field (T) 3 5

26 Oscillations with free electron states Commensurability h L = free ( kf k X ) R eb SLO period wave-guide width W B B =B cos(ϕ), ϕ 9 B + R L free DLC W B ( T) W L free min max = C B = πq, W = µm 3 q W = µm

27 Calculation adiabatic skipping orbits ( Snake -like orbits) Local cyclotron radius: R cycl hkf (Y,B ) = = eb L SLO F SLO eb L SLO =.7 Y,k X Y 5 Y B L B Y = B R hk = R -5 - L SLO =. 5 5 X

28 Calculation rough boundary scattering a) 5 L SLO =3.9 / Y =3 k X =-3 b) 5 Y =-3, k X =-3 Y -5 k X = 3-5 Y = 3, k X = X Compensation of skipping orbits with statistical scattering at the rough wave-guide boundary X Need a preferential momentum directions k y k x pre-selection

29 Calculation - k y k x pre-selection c) d) 5 L SLO =.6, k X =3 5 L SLO =3.9, k X =3 Y Adiabatic transfer with opposite sign into opposite edge terminals with preferential momentum directions k y k x pre-selection X - Low k X L L / ( ) SLO free X

30 Transverse force due to torque from induced magnetic moment = B m B m DLC W

31 zitterbewegung of a D Skin -channel B σ p H so [ r k r σ ] zitterbewegung due to spin precession J. Schliemann Phys.Rev.B (6)!! All electrons along L free have the chance to acquire an orthogonal momentum p x k Y σ k X Spin de-phasing length L so = h.6µ m mβ using Dresselhaus term β for 3 nm wide GaAs QW K.-J. Friedland et al., phys. stat. sol., 8

32 QHE : Summary Transport theory beyond Landauer Büttiker - Sequential transport along incompressible/compressible regions - Screening theory in nontrivial geometries to model lateral position of incompressible stripes quantized in R l and R H Oscillations in tangentially oriented fields: Commensurability of free-electron length L free with the wave guide width W Snake -like trajectories compensate by scattering at rough boundaries - Torque from induced magnetic moment and/or - Spin precession in a one-dimensional skin-channel allow to acquire the necessary orthogonal momentum k X