1. Christian Scheller

2. 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

3. 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

4. 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

5. 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

6. 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)

7. 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)

8. 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

9. 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)

10. 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)

11. Spectroscopy, 3 modes
Luttinger Liquid: “simple” model for 1D (dispersion E(kkF)=ak, i.e. linear instead of parabolic)
=> analytic treatment of collective excitations 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:

12. 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