Christian Scheller 23.09.2016
J Schlappa et al. Nature 485, 82 (2012) 1D excitations Free electron: Spin & Charge Bound to nucleus: Spin & Charge & angular momentum (orbitals) 1D system: Electron breaks up into fundamental excitations Spinon Charge Orbital degree Spin Holon Orbiton Detection? Measure dispersion (Spectroscopy) need separate E & k control Detection schemes? Tunneling between 1D wires ( E&k control: Bias & B-field ) X-ray scattering J Schlappa et al. Nature 485, 82 (2012) 1D Mott ins. Sr2CuO3
Device 200mm Separate contacts to upper / lower quantum well By means of surface gate depletion technique: SG: negative bias => deplete both wells MG: positive bias => recover conducting stripe in upper well BG,CG: negative bias => pinch off upper system UW Ohmic LW Ohmic Various surface gates: SG (split gate) MG (mid-line gate) BG (bar gate) CG (cut-off gates) WG (wire gates) Ohmics & tunneling configuration Efficient gating for wire region in UW only
Device (zoom-in) Before bridge fabrication After bridge fabrication Lead regions: Lc,Lb,La,Ld Large => 2D DOS Wire regions: W In-between gates Separated wires in UW Continuous 2DEG in LW Interconnecting wires: Air bridge fabrication 2mm 1mm Array of surface gates: 20*300 6000 wires Lwire = 1mm 5mm 0.6mm Before bridge fabrication After bridge fabrication 1mm 0.3mm 0.19mm
Device & tunneling BY Sample geometry + tunneling 14nm AlGaAs 18nm GaAs Tunneling distance Sample geometry + tunneling 40nm AlGaAs spacer 40nm AlGaAs Si-doped + + + + + + + + + + + + + + + + + 20nm AlGaAs spacer 20nm GaAs cap 0.83nm AlGaAs 0.56nm GaAs *10 Tunnel barrier Momentum kick: Lorentz force during tunneling distance Tunneling distance: Wavefunction center of mass relevant Tunneling probability: Wavefunction overlapp 2DEG depth 90nm (120nm) for UW (LW) BY F = -e*(Ds/Dt)*BY = ħ*Dk/Dt => Dk = -e*Ds*BY/ħ Ds Tunneling + Lorentz force Mode structure + B-field (momentum kick) Density calculation 5 5
Spectroscopy, E&k control Match e.g. left Fermi point of UW with dispersion of LW 2 dispersions (UW/LW) Each with 2 Fermi-points Total of 4 resonance curves Dispersions symmetric in momentum B-field symmetry Dispersions not symmetric in energy No symmetry in bias +(-) for right (left) Fermi points Differential conductance => resonances correspond to onset of current (not to lines of enhanced current)
Tunneling spectroscopy Three mode regime Two mode regime Single mode regime “leads” Upper crossing (+) corresponds to sum of densities (upper/lower system) “leads” are not gated, i.e. weak dependence on Vwg Density in wire strongly gate dependent => parabolas Finite size effects from leads (orange dashed lines)
Densities 1D density 2D density 2mm Wire region (W): strong gate dependence in UW Constant in LW Lead region Density almost gate independent 1DEG: n1D*p/2 = kF = 2pn2D :2DEG => n2D=(n1D)2*p/8 ; 61/mm <=> 1.45e15/m2
Spectroscopy, single mode 2D – 2D tunneling 2D – 1D tunneling (showing 2D dispersion, 1D as spectrometer) h+ holon band s- spinon band s+ spinon (shadow) band 2D as spectrometer (LW, W-region)
Aharonov Bohm 𝐵 d L Enclosed flux: f=B*L*d=n*f0 Oscillations with period DB=f0/(L*d) Using DB=0.26T => L=0.5mm (lithographic length = 0.6mm) Consistent with soft confinement (inflection point relevant, not lithographic width)
Spectroscopy, 3 modes Luttinger Liquid: “simple” model for 1D (dispersion E(kkF)=ak, i.e. linear instead of parabolic) => analytic treatment of collective excitations (bosonization) @ arbitrary interaction strength Interaction parameter: Kc,Ks for charge and spin, typically Ks=1 and Ks=<1(>1) for repulsive (attractive) interaction Renormalization: m* m*Kc => vc = vF/Kc >vF Group velocity: Dispersions:
Summary Tunneling spectroscopy in surface gated quantum wire array Momentum control through bias, B-field Tuneable 1D density (gate voltage vs B-field), 2D density not affected Finite size effects in tunnelling (Aharonov Bohm) Holon and spinon band (+shadow band) Spin charge separation: different curvature (Ks,Kc) of spinon, holon branch