Electron confinement in metallic nanostructures

Pierre MalletLEPES-CNRS

associated with Joseph Fourier UniversityGrenoble (France)

http://lepes.polycnrs-gre.fr/

Co-workers : Jean-Yves Veuillen, Stéphane Pons

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 (1932-)Experimental observation : UHV + electron spectroscopy

2D 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 !!!!

Empty electronic states (+0.25 eV above the Fermi level)UHV – STM conductance image at 40K 2 nm

Ni island grown on Cu(111)

Topics

1. Shockley Surface states

2. Low Temperature STM/STS

3. Quantum interferences of Shockley surface state electrons

4. Quantum resonators

5. Interaction between an adsorbate and a 2D 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 32, 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:

z<0 V(z) = 2 Vg cos gz (with g = 2π/a)

z>0 V(z) = V0Pot

entia

l ene

rgy

Vacuumz>0

Crystalz<0

0 z

z=0

V0

We have to solve the single-electron Schrödinger equation:

)()(2 2

22zEzV

zmψψ =

+

∂∂−

1.2 Shockley Surface State

Bulk states solutions (z<0)

Near the Brillouin zone boundary (k=g/2), solutions are well known :

] ) 1 ( [ )( /2 22/ eVmaVmaeeCz azi

GG

aziziB

ππκ πκπκψ −± +

±−+=

Near the center of Brillouin zone: plane wave and parabolic dispersion

κκ 222

42222

42))2/(( g

mVm

gE g+±+=±with κ = k – g/2

In this 1d model, a gap 2Vg is found at κ = 0, which separates the allowed bulk states.

k

1.3 Shockley Surface State

ez zEVomV

)(2D )0(

−−=>ψ

Because the chain is semi-infinite, the wave function is evanescent in the vacuum:

Matching of the wave functions and their derivative at z=0 (for each energy eigenvalue E within the allowed band).

Standing Block wave matched to an exponentially decaying wave function in vacuum.

0

1. 4 Shockley Surface State

Surface solutionsSince the crystal is not infinite, solutions with purely imaginary κκκκare also possible, with no divergence of |ψψψψ|2 :

κ = -iq qgm

VmqgE g

222

42222

42))2/(( −±−=

In that case, new states are found inside the bulk gap:) 2/sin(2 )0( δψ δ +=< − zgieeDz iqz

S) os(zg/2 )0( δψ δ −=< + ceeDz iqz

SVma

qg

πδ2

)2sin(with −=

Matching of ψS and ∂ ψS / ∂z at z = 0 gives only 1 possible surface state(i.e. one value of E inside the gap)

The solution is a standing wavewith an exponentially decaying amplitude.

0 < q < 2m|Vg|/għ2E has to be real :

1.5 Shockley Surface State

3D generalization2D translational symmetry parallel to the surface: the wave functions have 2DBloch waves component

Matching conditions at the surface will give one energy value ES for a given k//

2D band structure for the surface state : ES(k//)

Shockley surface state :

0

qgm

VmqkgkqE gS

222

4222

//22

// 42) )2/(() , ( −±−+=

Energy eigenvalues for the surface states become :

Example Cu(111) at ΓΓΓΓ:

| ψψ ψψ|2

1. 6 Shockley Surface StatePhotoemission on Cu

N. Memmel, Surface Science Reports 32, 91 (1998)

2.0 Low temperature STM/STS

2. STM/STS

2.1 Low temperature STM/STS - principle

Piezoelectric ceramic

3D positioning at sub-nanometer scale

Metallic tip

Sample

I

Feedback loop

V

d ~ 1 nm

I

I

∆Z (x,y)

I

+ - R

Basic principle of STM

2.2 Low temperature STM/STS – Constant current image

Constant current mode, feedback on : topography

2.3 Low temperature STM/STS – beyond topography

Beyond topography

[ ]dEeVErEzVETRrTVI ts T)f(E, - T)eV,-f(E )( ),( ),,( ),,( ×−=∞+

∞−ρρ

Tersoff & Hamann + Lang extension

At low temperatureat low bias V (lower than work functions)

with the assumption ρt(E) ≈ ρ(EF)) , ( ),( s reVErVdV

dIF +∝ ρ

Scanning Tunneling Spectroscopy

constant r , V rconstant,VDynamic conductance imaging

2.4 Low temperature STM/STS – STS and superconductivity

Example: STS on superconducting Nb

Direct probe of the gap in the quasiparticles LDOS

S.H. Pan et al., Appl. Phys. Lett. 73, 2992 (1998)

Scanning Tunneling Spectroscopy

feedback loopOPENV (t)

fixedrI

I

I

I (t)

)(VdVdI

R

Numerical derivative

+ -

2.5 Low temperature STM/STS – conductance imaging

Conductance imaging : dI/dV (r) at fixed V

Frequency of the bias modulation higher than the band pass of the

feedback loop !!!

Example of dI/dV map of a type II superconductor (NbSe2) at

V = 1,3 mV, with an external magnetic field of 1T.

H.F. Hess et al., Physica B 169, 422 (1991)

100 nm

I

I

I

+ - R

V + v0cosωωωωt

I (t)

r

image )(rdVdI

Feedback on

cos(ωωωωt-ϕϕϕϕ) Lock-indetectionI (t)

∆∆∆∆z

2.6 Low temperature STM/STS –experimental set-up at LEPES

Ultra-High Vacuum Chambers

Home madeLow temperature STM

Beetle design

Cu(111) 40 x 40 nm2

Constant current image

2.7 Low temperature STM/STS – Surface state of Cu(111)

Tunneling spectroscopy of Cu(111) Shockley surface state

3 x 3 nm2

0

0 .2

0 .4

0 .6

0 .8

1

1 .2

-0 .8 -0 .6 -0 .4 -0 .2 0 0 .2 0 .4 0.6 0 .8

0 .4

0.4

0 .4

0 .7 5

1

Nor

mal

ized

dI/d

V

Vsam ple

(V )

RT

( G )ΩΩΩΩ

Série SUBL

Spectroscopie tunnelT = 100K

PhotoémissionS.D. Kevan et al.,

PRL 50, 526 (1983)

E (eV)

k// (Å-1)

3.0 Quantum interferences

3. Scattering of surface state electrons

3.1 Quantum interferences – Cu(111)

20 x 20 nm2 STM images of Cu(111)

Topography :2 terraces separated by

a monoatomic step

dI/dV map shows nice standing waves…

dI/dV mapV = +200mV

T = 100K

3.2 Quantum interferences – Eigler and Avouris

Standing waves on Cu(111)

First STM observation in june 1993 by Eigler’s group (IBM San-José, California)

N° 363 (june 93)

nature

Origin of these waves?Shockley-like surface state : a quasi 2D free-electron gas

lies in the surface plane.

Same month, same phenomena observed onAu(111) by Hasegawa and Avouris

(IBM, New York)

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 :=

ky , kxk

2 )E-E( ),( ),,( δψρ yxyxE k

dk ),( k

1 L ),,( xk

0

222

0 ⋅−

= yxk

yxE kx

ψπρL0 : LDOS of a 2D electron gas without any scattering

20*

πmL =

E = const.

kx

ky

k

kx0*2 222

0 mkEE D

k +=

20E D

*m: band edge energy: effective mass

E

EF

k 2

0E D

0

for a parabolic dispersive surface state:

3.4 Quantum interferences – scattering by a 1D step edge_ model b

In the vicinity of a step edge :

The step is modelled by a coherent reflection (amplitude : r(kx) and phase: ϕϕϕϕ(kx))

Coherent elastic processes E is conserved |k|=|k’|

Problem invariant undertranslation along y

ky = k’y and kx = -k’xx

k’k

y Top view

side view

Electrons may also be :

transmitted in the surface state of the adjacent terrace : prob. t(kx)2

absorbed at the step (scattering into bulk states) :prob. a(kx)2

Particle conservation : r2 + t2 + a2 = 1

Incoherent processes

at the step !

eeekreyx yk yixkxikxix

xkxik ))((),( )( −+= ϕψ

The incoming plane wave has to be superimposed coherently by the reflected plane wave :

3.5 Quantum interferences – coherent reflection model c

For ϕ(kx) ≈ −π (see part 4), and r(kx) dominated by r(k) in the integral, an analogic solution is found:

))2(Jr(k)1(L ),( 00 kxxE −=ρ

dk

))(cos(2k)r(k1 2L ),( xk

0 220

−++=

kkkxxE

x

xxx ϕπρ

Which is rewritten as

x

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.

L0

0

x-2 decay

λ = π / k

ρρρρ (x)

Surface LDOS is then obtained using the previous integral:

dk )e t ),( ( k

1 L ),,( x22k

0

222

0 ++⋅−

= yxk

yxE kx

ψπρ

Electrons transmitted from

the adjacent terrace

Electrons emitted from bulk to surface states

(a2=e2)

Incoherent summation

(*)

http://omicron-instruments.com/products/lt_stm/r_ltstmm.htmlData courtesy : H. Hovel et al., Dortmund university, Germany

Ag(111), T=5K55 x 55 nm2 dI/dV maps

Wavelength of the standing waves versus energy…

3.6 Quantum interferences – E(k)

Ag(111), T=5K

O. Jeandupeux, L. Bürgi, A. Hirstein, H. Brune and K. Kern, Phys. Rev. B 59, 15926 (1999)

Parabolic dispersion of the surface state

EF

TOPO

dI/dV (V)

dI/dV(+150 meV)

0

100

200

-100

x (Å)0 100 200

z (Å)

E (m

eV)

-100

0

100

200

E (m

eV)

k (Å-1)

))2(Jr(k)1(L),( 00 kxxE −=ρ

E02D = (-65±3) meV

m* = (0.40 ±0.01) me

Accurate determination of the surface state parameters:

3.7 Quantum interferences – FS mapping

Mapping of the Fermi Surface

Cu(111), T=150K

Constant currentimage at +5mV

42,5 x 55 nm2

FFT

L. Petersen,… E.W. Plummer, Phys. Rev. B 57, R6858 (1998)