Electron confinement in metallic nanostructures Pierre Mallet LEPES-CNRS associated with Joseph Fourier University Grenoble (France) Co-workers : Jean-Yves Veuillen, Stéphane Pons http://lepes.polycnrs-gre.fr/
Introduction Properties of solids : mainly related to their electronic structure At surfaces, electronic properties are altered! Surface states, confined in the direction perpendicular to the surface. Such surface states exist also for metals! Theoretical predictions (193-) Experimental observation : UHV + electron spectroscopy D surface state 1D quantum wire 0D quantum dot
1981 : Scanning Tunneling Microscope (STM) is born!!! Binnig & Rohrer Nobel Price in 1986 subnanometric lateral resolution : Topographic informations (morphology, growth, size distribution ) Spatially resolved electron spectroscopy inside a single nanostructure!!!! Ni island grown on Cu(111) Empty electronic states (+0.5 ev above the Fermi level) UHV STM conductance image at 40K nm
Topics 1. Shockley Surface states. Low Temperature STM/STS 3. Quantum interferences of Shockley surface state electrons 4. Quantum resonators 5. Interaction between an adsorbate and a D electron gas
1.0 Shockley Surface State 1. Shockley Surface State H. Lüth, Surfaces & interfaces of solid materials, Springer (1995) N. Memmel, Surface Science Reports 3, 91 (1998)
1.1 Shockley Surface State Theoretical description: Semi-infinite Chain in the Nearly-Free Electron model. Electron-electron interaction is neglected. 1D model. V(z) (Effective potential induced by the cristal) has the following shape: Potential energy Crystal z<0 V 0 Vacuum z>0 z<0 V(z) = V g cos gz (with g = π/a) z>0 V(z) = V 0 0 z=0 z We have to solve the single-electron Schrödinger equation: m z + V ψ ( z) = Eψ( z)
1. Shockley Surface State Bulk states solutions (z<0) Near the center of Brillouin zone: plane wave and parabolic dispersion Near the Brillouin zone boundary (k=g/), solutions are well known : ± iκz iπz/ a ψ ( z) [ ( i z/ a B = C e e + πκ ± πκ π + 1 ) e ] mav G mav G 4 ± (( g/) + κ ) E = ± V g m g+ κ 4 m with κ = k g/ In this 1d model, a gap Vg is found at κ = 0, which separates the allowed bulk states. k
1.3 Shockley Surface State Because the chain is semi-infinite, the wave function is evanescent in the vacuum: ψ ( z> 0) V = D e m( Vo E) Matching of the wave functions and their derivative at z=0 (for each energy eigenvalue E within the allowed band). z 0 Standing Block wave matched to an exponentially decaying wave function in vacuum.
1. 4 Shockley Surface State Surface solutions Since the crystal is not infinite, solutions with purely imaginary κ are also possible, with no divergence of ψ : κ = -iq E = (( g/) m q ) ± V g 4 4 m g q E has to be real : In that case, new states are found inside the bulk gap: qz iδ ψ S( z< 0) = D e e isin( zg/ + δ) qz + iδ ψ S( z< 0) = D e e cos(zg/ δ) 0 < q < m V g /għ with sin(δ) = Matching of ψ S and ψ S / z at z = 0 gives only 1 possible surface state (i.e. one value of E inside the gap) The solution is a standing wave with an exponentially decaying amplitude. πq mav g
1.5 Shockley Surface State Shockley surface state : Example Cu(111) at Γ: ψ 0 3D generalization D translational symmetry parallel to the surface: thewavefunctionshave DBloch waves component Energy eigenvalues for the surface states become : E S (( g/) + k q q k // ) (, //) = ± m V g 4 4 m g q Matching conditions at the surface will give one energy value E S for a given k // D band structure for the surface state : E S (k // )
1. 6 Shockley Surface State Photoemission on Cu N. Memmel, Surface Science Reports 3, 91 (1998)
.0 Low temperature STM/STS. STM/STS
.1 Low temperature STM/STS - principle Basic principle of STM Feedback loop I Z (x,y) I Piezoelectric ceramic + - R I 3D positioning at sub-nanometer scale I Metallic tip d ~ 1 nm V Sample
. Low temperature STM/STS Constant current image Constant current mode, feedback on : topography
.3 Low temperature STM/STS beyond topography Beyond topography Tersoff & Hamann + Lang extension I( V, T, r) = + TR( EV,, z) ρ ( E, r) ρ ( E ev) [ f(e-ev,t) - f(e,t) ]de s t At low temperature at low bias V (lower than work functions) with the assumption ρ t (E) ρ(e F ) ( V, r) s( E ev, r) dv di ρ F + V, r constant Scanning Tunneling Spectroscopy V constant, Dynamic conductance imaging r
.4 Low temperature STM/STS STS and superconductivity Scanning Tunneling Spectroscopy dv di (V) Numerical derivative I (t) I + - R I I r fixed V (t) feedback loop OPEN Example: STS on superconducting Nb Direct probe of the gap in the quasiparticles LDOS S.H. Pan et al., Appl. Phys. Lett. 73, 99 (1998)
.5 Low temperature STM/STS conductance imaging Conductance imaging : di/dv (r) at fixed V cos(ωt-ϕ) I (t) Lock-in detection dv di (r) image Feedback on z I (t) I + - R I I r 100 nm V + v 0 cosωt Frequency of the bias modulation higher than the band pass of the feedback loop!!! Example of di/dv map of a type II superconductor (NbSe ) at V = 1,3 mv, with an external magnetic field of 1T. H.F. Hess et al., Physica B 169, 4 (1991)
.6 Low temperature STM/STS experimental set-up at LEPES Home made Low temperature STM Beetle design Ultra-High Vacuum Chambers
.7 Low temperature STM/STS Surface state of Cu(111) Tunneling spectroscopy of Cu(111) Shockley surface state k // (Å -1 ) Photoémission S.D. Kevan et al., PRL 50, 56 (1983) E (ev) 3 x 3 nm Cu(111) 40 x 40 nm Constant current image Normalized di/dv Spectroscopie tunnel T = 100K 1. 1 0.8 0.6 0.4 R T (G Ω ) 0.4 0.4 0.4 0.75 1 0. 0-0.8-0.6-0.4-0. 0 0. 0.4 0.6 0.8 V (V) sample Série SUBL
3.0 Quantum interferences 3. Scattering of surface state electrons
3.1 Quantum interferences Cu(111) 0 x 0 nm STM images of Cu(111) T = 100K Topography : terraces separated by a monoatomic step di/dv map V = +00mV di/dv map shows nice standing waves
3. Quantum interferences Eigler and Avouris Standing waves on Cu(111) First STM observation in june 1993 by Eigler s group (IBM San-José, California) nature N 363 (june 93) Same month, same phenomena observed on Au(111) by Hasegawa and Avouris (IBM, New York) Origin of these waves? Shockley-like surface state : a quasi D free-electron gas lies in the surface plane. Part of the electrons are scattered coherently by the surface defects (step, adsorbate, vacancy), generating quantum interferences. Spatial modulation of the LDOS is imaged by STM
3.3 Quantum interferences model a Surface LDOS : ρ ( E, x, y) = ψ ( x, y) δ(e-e ) k y kx, ky for a parabolic dispersive surface state: 0 k k E = const. k x k x k E E F 0 E D 0 E k = + D E0 m k * E0 D : band edge energy : effective mass m* k k L 1 ρ ( E, x, y) = 0 ψ ( x, y) dk x π k 0 k k x L 0 : LDOS of a D electron gas without any scattering L * 0 = m π
3.4 Quantum interferences scattering by a 1D step edge_ model b In the vicinity of a step edge : side view Coherent elastic processes E is conserved k = k y x k k Top view Problem invariant under translation along y k y = k y and k x = -k x The step is modelled by a coherent reflection (amplitude : r(k x ) and phase: ϕ(k x )) The incoming plane wave has to be superimposed coherently by the reflected plane wave : Electrons may also be : x y e ikx x iϕ r k e kx) ψ e ikx x x k (, ) = ( + ( ) ( ) e transmitted in the surface state of the adjacent terrace : prob. t(k x ) absorbed at the step (scattering into bulk states) :prob. a(k x ) Particle conservation : r + t + a = 1 ik yy Incoherent processes at the step!
3.5 Quantum interferences coherent reflection model c Surface LDOS is then obtained using the previous integral: (*) Which is rewritten as ρ k L 1 ρ ( E, x, y) = 0 ( ψ ( x, y) + t e ) dk π k x 0 k k + k L 1 r(k ) cos(k ( )) (, ) 0 + x+ k E x = x x ϕ ρ x dk x π 0 k k For ϕ(k x ) π(see part 4), and r(k x ) dominated by r(k) in the integral, an analogic solution is found: ( E, x ) = L 0 ( 1 r(k) J 0 ( kx x Incoherent summation x Electrons transmitted from the adjacent terrace )) Electrons emitted from bulk to surface states (a =e ) This model of coherent reflection on a step explains the LDOS oscillations but shows also a decay due to the summation in k space. r 1 (inelastic processes at the step) reduces homogeneously the LDOS. L 0 0 x - decay λ = π / k ρ (x) x
Wavelength of the standing waves versus energy Ag(111), T=5K 55 x 55 nm di/dv maps http://omicron-instruments.com/products/lt_stm/r_ltstmm.html Data courtesy : H. Hovel et al., Dortmund university, Germany
3.6 Quantum interferences E(k) ρ E, x) = L0 (1 r(k)j (kx)) ( 0 Parabolic dispersion of the surface state Ag(111), T=5K TOPO z (Å) 00 00 di/dv (V) E (mev) 100 0 E F E (mev) 100 0 di/dv (+150 mev) -100 0 100 00 x (Å) -100 Accurate determination of the surface state parameters: k (Å -1 ) E 0 D = (-65±3) mev m* = (0.40 ±0.01) m e O. Jeandupeux, L. Bürgi, A. Hirstein, H. Brune and K. Kern, Phys. Rev. B 59, 1596 (1999)
3.7 Quantum interferences FS mapping Mapping of the Fermi Surface Constant current image at +5mV 4,5 x 55 nm FFT Cu(111), T=150K L. Petersen, E.W. Plummer, Phys. Rev. B 57, R6858 (1998)