Secondary electron interference from trigonal warping in clean - PowerPoint PPT Presentation

Secondary electron interference from trigonal warping in clean carbon nanotubes A. Dirnaichner et al. , PRL 117 , 166804 (2016) Dr. Andreas K. H¨ uttel Institute for Experimental and Applied Physics University of Regensburg 28th International Conference on Low Temperature Physics, G¨ oteborg

overgrown, “ultraclean” carbon nanotube device Ti/Pt CNT SiO 2 • CNT growth in situ over p Si + gate Ti/Pt electrodes • V g � 0 − → hole conduction • no Coulomb blockade 5 • transparent contacts, weak scattering A. Dirnaichner et al. , PRL 117 , 166804 (2016)

a carbon nanotube as Fabry-P´ erot interferometer "semitransparent "semitransparent mirror" mirror" weakly scattered electron wave • strong coupling of nanotube and contacts, no charge quantization • weak scattering − → Fabry-P´ erot interferometer for electrons A. Dirnaichner et al. , PRL 117 , 166804 (2016)

the initial observation W. Liang et al. , Nature 411, 665 (2001) • large conductance, oscillating in gate voltage V g , bias voltage V sd • fixed interferometer geometry; we tune the electron wave vector • dominant frequency corresponds to distance between contacts

our data — much larger energy range ∆ E ≃ 0 . 4eV • narrow oscillation ( ↔ interferometer length) • frequency doubling / beat • slow modulation of the averaged conductance − → nanotube is not just a one-channel system; valley degeneracy, dispersion relation! A. Dirnaichner et al. , PRL 117 , 166804 (2016)

our data — much larger energy range ∆ E ≃ 0 . 4eV • narrow oscillation ( ↔ interferometer length) • frequency doubling / beat • slow modulation of the averaged conductance − → nanotube is not just a one-channel system; valley degeneracy, dispersion relation! A. Dirnaichner et al. , PRL 117 , 166804 (2016)

our data — much larger energy range ∆ E ≃ 0 . 4eV • narrow oscillation ( ↔ interferometer length) • frequency doubling / beat • slow modulation of the averaged conductance − → nanotube is not just a one-channel system; valley degeneracy, dispersion relation! A. Dirnaichner et al. , PRL 117 , 166804 (2016)

f -8 -6 -2 -10 -12 -14 impurity scattering? no! • discrete Fourier transform of interference pattern (apply sliding window to G ( V g ) , plot transform as function of window position) • only one fundamental frequency and its harmonics − → no impurities that subdivide the nanotube − → interference effects must be due to intrinsic nanotube structure • from decay of harmonics, extract mean path of electrons − → ℓ = 2 . 7µm ≃ 2 . 7 L A. Dirnaichner et al. , PRL 117 , 166804 (2016)

structure of single wall carbon nanotubes armchair armchair (n,n) T zigzag a 2 (4,2) a 1 C � chiral zigzag (0,0) (n,0) • typically, classification into armchair, zigzag, chiral • chiral nanotubes can be further subdivided into armchair-like, zigzag-like A. M. Lunde et al., PRB 71, 125408 (2005), M. Marga´ nska et al., PRB 92, 075433 (2015) • let’s discuss the interferometer behaviour of these four groups • band structure & symmetry, real-space tight binding calculations A. Dirnaichner et al. , PRL 117 , 166804 (2016)

interference in a zigzag nanotube ε a ε b k a,l k a,r k b,l k b,r k || k || a a b b 0 0 zig zag (12,0) 4 T 0 ε 0.22 (eV) 0.16 0.10 zigzag ( θ = 0 ◦ , (n,0)): • Dirac cones around k ⊥ = ± K ⊥ , k � = 0 • angular momentum conservation − → only backscattering within cone • two channels, identical accumulated phase − → looks like one channel A. Dirnaichner et al. , PRL 117 , 166804 (2016)

interference in a zigzag-like nanotube κ > κ < ε a k a,l k a,r k b,l k b,r k || k || 0 0 a a b b zig zag like (6,3) 4 T 0 ε 0.22 (eV) 0.16 0.10 zigzag-like (0 ◦ < θ < 30 ◦ , n − m 3gcd ( n , m ) / ∈ Z ): • asymmetric Dirac cones around k ⊥ = ± K ⊥ , k � = 0 • angular momentum conservation − → only backscattering within cone • two channels, identical accumulated phase − → looks like one channel A. Dirnaichner et al. , PRL 117 , 166804 (2016)

interference in an armchair nanotube κ > κ < ε k a,r k a,l k b,l k b,r 0 k || b a a b armchair (7,7) 4 T 0 ε 0.22 0.16 0.10 (eV) armchair ( θ = 30 ◦ , (n,n)): • Dirac cones at k ⊥ = 0, k � = ± K � • parity symmetry − → only backscattering within a / b branch • two channels, different accumulated phase, beat; T constant A. Dirnaichner et al. , PRL 117 , 166804 (2016)

interference in an armchair-like nanotube κ > κ < ε k a,r k a,l k b,r k b,l 0 k || b a a b E n 4 armchair like (10,4) T 0 ε 0.22 (eV) 0.16 0.10 armchair-like (0 ◦ < θ < 30 ◦ , n − m 3gcd ( n , m ) ∈ Z ): • Dirac cones at k ⊥ = 0, k � = ± K � • NO parity − → two channels, different phase, mixing of channels • beat plus slow modulation of T A. Dirnaichner et al. , PRL 117 , 166804 (2016)

meaning of the average conductance maxima • armchair-like CNT: phase difference of Kramers modes � � ∆ φ θ ( E ) = | φ θ a ( E ) − φ θ κ θ > − κ θ b ( E ) | = 2 L < >,< : longitudinal wave vectors measured from K / K ′ points κ θ • averaged conductance has maximum when ∆ φ θ ( E ) = 2 π n • relevant parameter: chiral angle θ − → use this for chiral angle determination! • extract from data maxima positions V n g of G ( V g ) • convert V n g from gate voltage to energy • compare with calculated maxima positions for given θ A. Dirnaichner et al. , PRL 117 , 166804 (2016)

chiral angle determination 10 30° 22° 8 15° 6 9° 4 2 4° 1° 0 0.0 0.1 0.2 0.3 0.4 E (eV) result for our device: 22 ◦ ≤ θ < 30 ◦ solution of a hard problem — chirality determination from transport A. Dirnaichner et al. , PRL 117 , 166804 (2016)

error sources mainly: conversion of G maxima positions from gate voltage to energy • band gap at V g > 0, energy offset ∆ E • lever arm α ( V g ) hard to determine, varies strongly close to band gap → 55meV < ∆ E < 60meV → error bars A. Dirnaichner et al. , PRL 117 , 166804 (2016)

contact area: rotational symmetry broken broken rotational symmetry at contacts • at contacts, rotational symmetry broken − → argument for angular momentum conservation breaks down • integrate this into tight-binding model: differing on-site energies for top and bottom of nanotube • result: slow oscillations of G also recovered for zigzag-like nanotube! • same evaluation of the chiral angle possible! A. Dirnaichner et al. , PRL 117 , 166804 (2016)

conclusions • complex Fabry-P´ erot interference observed over a large energy range • theoretical analysis for different nanotube types, confirmed by real-space tight binding calculations • interference pattern is due to trigonal warping of dispersion relation and mixing of Kramers channels • slow modulation of averaged conductance G — robust, easily extracted • G depends on chiral angle θ of the nanotube • approach towards a hard problem — chirality determination from low-temperature transport A. Dirnaichner et al. , PRL 117 , 166804 (2016)

Thanks Alois Dirnaichner Miriam del Valle Karl G¨ otz Felix Schupp Nicola Paradiso Milena Grifoni Christoph Strunk A. Dirnaichner et al. , PRL 117 , 166804 (2016)

Thank you! — Questions? 10 30° 22° 8 15° 6 4 9° 2 4° 1° 0 0.0 0.1 0.2 0.3 0.4 E (eV) A. Dirnaichner et al. , PRL 117 , 166804 (2016)