Short description Theory of 1-D tunneling Actual 3D barriers tip modeling atomic resolution Hardware...

Short description

Theory of 1-D tunneling

Actual 3D barrierstip modelingatomic resolution

Hardware

Examples

Scanning Tunneling Microscopy (STM)

Bibliography

• Scanning Probe Microscopy and Spectroscopy (Wiesendanger, Cambridge UP)

• Scanning Probe Microscopies: Atomic Scale Engineering by Forces and Currents

Distance Controland Scanning Unit

TunnelingCurrent Amplifier

Data Processingand Display

Tunneling Voltage

Pie

zole

lectr

ic T

ub

ew

ith

Ele

ctr

od

es

Sample

Sample

Tip

Scanning Tunneling Microscopy (STM)

Electron tunneling

Fundamental process:

Electron tunneling

Typical quantum phenomenon

Wave-particle impinging on barrier

Probability of finding the particle beyond the barrier

The particle have “tunneled” through it

Tunneling definition

Role of tunneling in physics and knowledge development

• Field emission from metals in high E field ( Fowler-Nordheim 1928)• Interband tunneling in solids (Zener 1934)• Field emission microscope (Müller 1937)• Tunneling in degenerate p-n junctions (Esaki 1958)• Perturbation theory of tunneling (Bardeen 1961)• Inelastic tunneling spectroscopy (Jaklevic, Lambe 1966)• Vacuum tunneling (Young 1971)• Scanning Tunneling Microscopy (Binnig and Rohrer 1982)

Electron tunneling

Elastic

Energy conservation during the processIntial and final states have same energy

Inelastic

Energy loss during the processInteraction with elementary excitations

(phonons, plasmons)

1D 3D

Planar Metal-Oxide-Metal junctions Scanning Tunneling Microscopy

Rectangular barriers 3D

Planar Metal-Oxide-Metal junctions Scanning Tunneling Microscopy

Time independent

Matching solutions of TI Schroedinger eq

Time-dependent

TD perturbation approach:(t) + first order pert. theory

ikzikze Re1

22 2

mE

k

xzxz BeAe 2

202 )(2

EVmx

ikzTe3

Electron tunneling across 1-D potential barrier

EV

dzd

m 0

2

22

2

Plane-wave of unit amplitude traveling to the right+plane-wave of complex amplitude R traveling to the left

Region 1

Region 2

exponentially decaying wave

Region 3 plane-wave of complex amplitude T traveling to the right.

The solution in region 3 represents the “transmitted” wave

yprobabilit ontransmissi 2 T

Time independent

iksixsixs

iksixsixs

kTeBeAex

TeBeAe

BAxRk

BAR

1

1

xsxkxk

xsxkR

222222

22222

sinh4

sinh


Continuity conditions on and d/dz give

At z=0

At z=s

xsxkxk

xkT

222222

222

sinh4

4

yprobabilit ontransmissi 2 Typrobabilit reflection

2 R

1 22 TR

Time independent


A square integrable (normalized) wave function has toremain normalized in time

In a finite space region this conditions becomes

0, 2

dztz

dtd

02

**2

zzmi

dzdtd

02

**2

b

a

b

a zzmi

dzdtd

zzm

itzj

**

2,

0,,, tajtbjP

dtd

ba

Probability conservation

Probability current

Time independent


Applying toour case

zzm

itzj

**

2,

Region 1

Region 3

22

1 112

, RvRmk

tzj

22

2, TvT

mk

tzjT

vtzj i ,

2T

j

j

T

i T

tcoefficien ontransmissi T )4()(

)(sinh1

1

22

2222

xkxk

xs

T

Time independent


Exponential is leading contribution

20 )(2

EVmsxs

For strongly attenuating barriers xs >> 1

xsexkxk 2

222

22

)(16

T

Barrier width s = 0.5 nm, V0 = 4 eV T ~ 10-5

Barrier width s = 0.4 nm, V0 = 4 eV T ~ 10-4

Extreme sensitivity to z

The transmission coefficientdepends exponentially on barrier width

Large barrier height (i.e. small )

2222

22

22

22222

22

2222 )(

16

)4()(

42

1

)4()(

)(sinh1

1xke

xk

xkxkee

xkxk

xsxsxsxs

T

2

2

2)(sinh

xsxs eexs

Time independent

Exponential dependence of tunneling current


At the surface the wavefunction is very complicated to calculate

If barrier transmission is small, use perturbation theoryBut no easy way to write a perturbed Hamiltonian

Approximate solutions of exact Hamiltonian within the barrier region

szforbez

zforaezkz

r

kzl

)(

0 )(

l has to be matched with the correct solution of H for z 0

r has to be matched with the correct solution of H for z 0

Ideal situation:incident state from left has some probabilityto appear on right… And we can calulate it…

Real situation:

Different approach

Time-dependent


l,r = electron states at the left and right regions of the barrier

HT = transfer Hamiltonian

rrr

lll

TTrl

EH

EH

HHHHHHdt

tditH

0

0

0)(

)()(

Time-dependent

With the exact hamiltonian on left and right,we add a term HT representing the transition rate from l to r.

HT is the term allowing to connect the right and left solutions


Choose the wavefunction

Put into hamiltonian

tiE

r

tiE

l

rl

edec

dttd

itH)(

)(

dt

edecd

i

edecHedecHH

tiE

r

tiE

l

tiE

r

tiE

lT

tiE

r

tiE

lrl

rl

rlrl

)(

Time-dependent


tiE

rr

tiE

ll

tiE

r

tiE

lT

tiE

rr

tiE

ll

rl

rlrl

edEecE

edecHedEecE

0

tiE

r

tiE

lT

rl

edecH

The total probability over the space is

1

1

**

*

dzedecHedec

dzH

tiE

r

tiE

lT

tiE

r

tiE

l

T

rlrl

Time-dependent


So the tunneling matrix element

1

1**

dzHdzH

MM

lTrrTl

rllr

Using the Fermi golden rule to obtain the transmitted current

rrlt dE

dNMj

22

Density of states of the final state

Mlr = Probability of tunneling from state l to state m

In general, the tunneling current contains information on thedensity of states of one of the electrodes, weighted by M

But ………… each case has to calculated separately

z

dzzixez 0

''

0

2

2 )(2

zVEmx

Electron tunneling across “real” 1-D potential barrier

EzV

dzd

m

2

22

2

Try a solution

Time independent

zzVEm

dzzd

22

2 2

zxdz

zd 2

2

2

V(z) = slowly varying potential

particle moving to the right with continuously varying wave-number (x)

zzixdz

zd zzxz

dzzdx

idz

zd 2

2

2

Introduce a more real potential: how to represent it?

Electron tunneling across “real” 1-D potential barrier

This is true only if the first term is negligible, i.e.

Time independent

zxdz

zd 2

2

2

zzxz

dzzdx

idz

zd 2

2

2

2xdz

zdx

but

1 x

dzzdx

x

variation length-scale of x(z)(approximately the same as the variation length-scale of V(z)) must be much greater than the particle's de Broglie wavelength

WKB approximationWenzelKramerBrillouin

2

2 )(2

zVEmx

For E > V(z), x is real and the probability density is constant 2

02 z

Electron tunneling across “real” 1-D potential barrierTime independent

Suppose the particle encounters a barrier between 0 < z1 < z2

so E < V(z) and x is imaginary

z

z

z

z

zdzzxdzzxdzzix

eeez 11

1

0''

1

''''

0

the probability density inside the barrier is

z

zdzzx

ez 1''22

12

Inside the barrier

the probability density at z1 is2

1

the probability density at z2 is

2

1''22

1

2

2

z

zdzzx

e

z

dzzixez 0

''

0Neglect the exp growing part

Electron tunneling across “real” 1-D potential barrierTime independent

So the transmission coefficient becomes

Tunneling probability very small

2

1

2

2

2

1

2

1')'(2

2''2

2

1

2

2z

z

z

zdzzVEmdzzx

ee

T

The wavenumber is continuosly varyingdue to the potential: more real

2

2 )(2

zVEmx

reasonable approximation for the tunneling probability if the incident << z (width of the potential barrier)


Square barrierplane wave

xsexkxk 2

222

22

)(16

T Exponential dependence

of the transmission coefficient

rrlt dE

dNMj

22

Square barrierelectron states

current depends on transfer matrix elements (containing exp. dependence)and on DOS

Real barrierPlane waves

2

1')'(2

2 z

zdzzVEm

e TTrue barrier representation if <<zVarying exponential dependenceof the transmission coefficient

Electron tunneling across 1-D metal electrodes

Planar tunnel junctions

U=Bias voltage

Similar free electron like electrodes

At equilibrium there is no net tunneling currentand the Fermi level is aligned

What is the net current if we apply a bias voltage?

)()(1

zEzV F

We must consider the Fermi distribution of electrons

insulator = vacuumThe insulator defines the barrier

maxmax

001 )()(

1)()(

E

zzz

v

zzzz dEEvnm

dvEvnvN TT

vz = electron speed along zn(vz)dvz = number of electrons/volume with vz

T(Ez) = transmission coefficient of e- tunneling through V(z) e- with energy Ez =mvz

2/2f(E) = Fermi Dirac distribution

0

032

2

1 )()(2

max

r

E

zz dEEfdEEm

N T

KT

EE F

e

Ef

1

1)(


n(vz)dvz = number of electrons/volume with vz

032

3

33

4

)(2

)(4 ryxz dEEf

mdvdvEf

mvn

zyxzyx dvdvdvEfm

dvdvdvvn )(4 33

4

2

222

2 rr

yxr

vm

E

vvv

Flux from electrode 1 to electrode 2

003

2

021 )()(

2)(

max

rz

E

zz dEeUEfdEEfm

dEENNN

T

Total number of electrons tunneling across junction


0

032

2

1 )()(2

max

r

E

zz dEEfdEEm

N T

Flux from electrode 2 at positive potential U to electrode 1

0

032

2

2 )()(2

max

r

E

zz dEeUEfdEEm

N T

032

2

1 )(2 rdEEfem

032

2

2 )(2 rdEeUEfem

max

021)(

E

zz dEEJ T tunneling current across junction

The current depends on electron distribution

2

1 1)(

22

)(s

szF dzEzE

m

z eE

T


)()(1

zEzV F

T is small when EF-Ez is large

e- close to the Fermi level of the negatively biased electrodecontribute more effectively to the tunneling current

since

max

021)(

E

zz dEEJ T

For positive U 2 is negligible so the net current flows from 1 to 2

zF EEA

z eE

1)(T


dzzs

s

s 1

1

1

To perform the integration over the barrier

1

1

1

1

1 1

0322F

F

zFF zFE

eUE z

EEA

zF

eUE

z

EEAdEeEEdEeeU

emJ

ms

A22

define

By integration it can be shown that

At 0 K zF EEem

1321 2

eUEEem

zF 1322 2

Applications of tunnel equation

1

111

1

0

0

2 32

Fz

FzFzF

Fz

EE

EEeUEEE

eUEEeVem

hence


eUAA eeUes

eJ

4 22

Current density flowing from electrode 1 to electrode 2 and vice versa

If V = 0 dynamic equilibrium: current density flowing in either direction

Aes

eJ

4 221 eUAeeUs

eJ

4 222

dzzs

s

s 1

1

1

ms

A22

For positive U 2 is negligible sothe net current flows from 1 to 2

integrating


eUAA eeUes

eJ

4 22

Low biases

eU

A

eUA

eeeUs

eJ

42

22

2211

eUA

AeU

AeU

AeUA eeeee

ab

kaab

aba kk

kk 11

Electron tunneling across 1-D metal electrodesLow biases eV

A

A

A

eA

eUs

e

eeUeU

As

e

eeU

AeUs

eJ

12

4

2

4

2

1 4

22

22

22

AeA

eUs

eJ

24 22

ms

A22

12

A

AeUs

meJ

42

22

2

At low biases the current varies linearly with applied voltage, i.e.Ohmic behavior

Neglect second order contributions in U

since


0

23

023

0

212

96.24

0

296.24

02

23

2

1 16

2.2

eUm

eFmeF e

eUe

FeJ

High biases eU

eUs

s 02

0

sU

F

eUAA eeUes

eJ

4 22

Put into general eq.

Electric field strength

evaluating a numerical factor (not included in eq)

eUFor this condition Second term of eq is negligible

Electron tunneling across 1-D metal electrodesHigh biases

The situation is reversed for e- tunneling from 1 to 2: all available levels are emptyanalogous to field emission from a metal electrode: Fowler-Nordheim regime

EF2 lies below the bottom of CB1

Hence e- cannot tunnel from 2 to 1there are no levels available

Uconst

eUJ

2

23

0296.24

02

23

16

2.2

meFe

FeJ


Square barrier,plane wave

xsexkxk 2

222

22

)(16

T Exponential dependence

of the transmission coefficient

rrlt dE

dNMj

22


current depends on transfer matrix elements (containing exp. dependence)and on DOS

Real barrierPlane waves

2

1')'(2

2 z

zdzzVEm

e TVarying exponential dependenceof the transmission coefficient

Real barrierMetal electrodes

Tunneling is most effective for e- close to Fermi level

Low biases: Ohmic behavior

High biases: Fowler-NordheimCurrent flows from – to + electrode

003

2

021 )()(

2)(

max

rz

E

zz dEeUEfdEEfm

dEENNN

T

3-D potential barrier

rrlt dE

dNMj

22


003

2

021 )()(

2)(

max

rz

E

zz dEeUEfdEEfm

dEENNN

TReal barrierMetal electrodes

Join and extend the expression to have the equation for the tunneling currentbetween a tip and a metal surface

dzHM rTlrl *2

Consider two many particle states of the sytem 0,

= state with e- from state in left to state in right side of barrier

0, are eigenstates given by the WKB approximation

z

dzzixez 0

''

0

Trick: both are good on one side only and insidethe barrier but not on the other side of the barrier

1) Matrix element

0


Applyng a step function along z that is 1 only over barrier region

is linear combination of one intial state 0 and numerous final states

Put into Schroedinger equation and get a matrix with elements like

dSdzHHm

M

*0

*0

2

2

tiEtiE ebea 00

**2

2

dSm

M

The tunneling current depends on the electronic states of tip and surface

Problem: calculation of the surface AND tip wavefunctions

the tunneling matrix element can be evaluated by integratinga current-like operator over a plane lying in the insulator slab


f(E) = Fermi functionU = bias voltage applied to the sampleM = tunneling matrix element = unperturbed electronic states of the surface = unperturbed electronic states of the tipE (E) = energy of the state () in the absence of tunneling, are not eigenfunctions of the same H

rrlt dE

dNMj

22


003

2

021 )()(

2)(

max

rz

E

zz dEeUEfdEEfm

dEENNN

TReal barrierMetal electrodes

Join the expression to have the equation for the tunneling currentbetween a tip and a metal surface

EEMeUEfEf

eI

22

2) Current density

Not the many particle states

EEMEfeUEfeUEfEf

eI

211

2


EEMeUEfEf

eI

21

2

At low T one can consider only one directional tunneling

EEMeUEfEf

eI

22

eUEfEf

Low T + small (10 meV) applied bias voltage (U)

FF EEEEMU

eI

222

EEMeUEfEf

eI

21

2

KT

EE F

e

Ef

1

1)(

EE

EfeUEfeUEf

)()()(

For the Fermi function )()(

EEEf

)()( FEEeUEf

FF

F

EEEEMeUEfEEMEfEf

EEMEEeUEfEf

EEMeUEfEf

)()(1

)()(1

1

22

2

2

Low T + small (10 meV) applied bias voltage (U), E <EF 1)( Ef

Tip modeling

Point like tip (unphysical)

The matrix element is proportional to the probability density of surface states measured at r0

i.e. the local density of states at the Fermi level

Low T + small applied bias voltage (U)

FF EEEEMU

eI

222

FEEI2

0r

The image represents a contour map ofthe surface DOS at the Fermi level

Tip modeling

FxR

Ft EEreEnUI2

02 )(

tip with radius Rs-type only (quantum numbers l 0 neglected) wave functionswith spherical symmetry to calculate the matrix element

Surface local density of states (LDOS) at EF

measured at r0

EF = Fermi energyr0 = center of curvature of the tipx = (2m)1/2/ ħ = decay rate = effective potential barrier height

Low T + small applied bias voltage (U)

FF EEEEMU

eI

222

nt(EF) = density of states at the Fermi level for the tip

FF EErEr2

00 )(,

Tip modeling

STM is imaging the LDOS at the tip position

Multiplied by the tip DOS

The matrix element is integrated in a point of the barrier region sSo the value of at r0 is no physically relevant, but it represents the lateralaveraging due to finite tip size

FxR EreI , 0

2

)(22

0)( Rsxer

FxR

Ft EEreEnUI2

02 )(

The exponential dependence comes from the matrix element

Fxs EreI , 0

2

Tip modeling

Surface local density of states (LDOS) at EF

measured at r0

Sample wavefunctions haveexponential decay in the z directionso little corrugation at s from surface

)(22

0)( Rsxer

STM is imaging the LDOS at the tip position

Calculated LDOS for Au(111)

Tip center position

Au latticeparameter

Multiplied by the tip DOS Low T + small applied bias voltage (U)

STM: atomic resolution

We observe features with a spatial resolution better than 0.1 nmmuch lower of the tip curvature radiusSmaller than spherical approximation of the tip wavefunctions (0.8 nm)

Model failing to explain the most important featureof the STM: atomic resolution

1.0Å

STM: atomic resolutionWhy?

1.0Å

Accuracy of perturbation theory:depends critically on the choice of the unperturbed wavefunctions, or the unperturbed Hamiltonians.

For 3D tunneling the choice of unperturbed Hamiltonians is not unique. This is especially true for higher biases, in which the potential in the tunneling gap is not flat.Solution

the unperturbed wave functions of sample and tip has to be different in the gap region

02

02

22

22

EUm

EUm

Te

Se

•This unperturbed Hamiltonian minimizes the error introduced by neglecting the higher terms in the perturbation series.•The tip states are invariant as the bias changes, simplify calculations. •Easier estimation of bias distortion because the bias only affects the sample wave function, thus can be treated perturbatively


2

sin

cos

,,

z

y

x

To calculate I, the of the acting atom is expanded in terms of a complete set of eigenfunctions.

Two choices:spherical coordinatesparabolic coordinates

Spherical coordinates are appropriatefor describing atom loosely bonded on the tip

Parabolic coordinates are appropriatefor describing atom tightly bonded to the tip body.

dd

h

hh

mM

0

**2

2

Calculated on the paraboloid


dd

h

hh

mM

0

**2

2

Differences to Bardeen expression

the wave functions are the eigenfunctionsof tip and sample unperturbed Hamiltonianswhich are different in the gap region.

It is valid only on the paraboloid that is the boundaryof the tip body, not in the entire barrier region

what is needed for calculating the tunneling matrix elementsis the wave functions on the boundary of the tip

STM: atomic resolutionOn and outside boundary the tip satisfies the free electron

Schroedinger equation decaying exponentially

022 22 2

mE

expand in term of the parabolic eigenfunctionswith boundary conditions to be regular at r

The contribution of the tip wave function is determined only by its asymptotic values.

The details of the tip wave functions near the center of the acting atom are not important

unperturbed wave functions of sample and tip different in the gap region

On and inside boundary the sample satisfies the free electronSchroedinger equation decaying exponentially

022

22 2

mE

expand in term of the parabolic eigenfunctionswith boundary conditions to be regular at center of the acting atom

The contribution of the sample wave function is determined only by the values of the sample wave function in the vicinity of the center of the acting atom.

The details of the sample wave functions outside the tip body are not important

,,

lmml

llm ri

,,

lmml

llm rk


dd

h

hh

mM

0

**2

2

So M has to be integrated using orthonormal wavefunctions

That leads to determine only the coefficients of the tip andsample expansion on orthonormal wavefunctions.

The coefficients are determined by calculating the derivatives of the at the center of the acting atom

rkri ll

lm

,

,

,,

lmml

llm ri

Bessel functions

Spherical harmonics

mllmlmm

M,

2

2

M gives the correspondence between tip and sample wavefunctions

STM: atomic resolution ,

,lm

mlllm ri

ml

lmlmmM

,

2

2

For a choosen tip state, M changes and defines the relation to the coeffiecients of the surface

Tip states

3

3

22

2

2

4

8

4

4

2

22

rxye

reyx

rye

rxer

e

r

xy

r

yx

r

y

r

x

r

s

p

d

xy

yx

y

x

e

xy

yx

y

x

r

2

2222

0

0

0

0

2

2

2

2

2

0

r

r

r

r

yx

yx

y

x

r

M

The tunneling matrix elements are related to the sample wavefunction derivatives

So the atomic resolution is given by the l 0 wave functions


The approximation on s state only is wrongthe surface state of a real W tip extends into vacuum more than s and d states

It is the most protruding electronic states that provides the JNot only the electron states at the Fermi level


Reciprocity principle

Is a basic microscopic symmetry ofSTM

If the "acting" electronic state of the tip and the sample state are interchanged, the image should be the same.

An image of microscopic scale may be interpreted either as by probing the sample state with a tip state or by

probing the tip state with a sample state

Band structure effects

The electron energy in a solid depends on the band structure

E

eUmRs st

eeUET 222

)(22

),(

dEeUETEEeUI s

eU

t ),( 0

)(kEE k is such that k+G=k

The surface and tip define the direction z

z

kkk EEEE z

This may results in tunneling from surface or bulk states depending on their spatial extension

Also T is changing as a function of E

kk EE

eUmx z

st

222

2

Electrons in states with large parallel wavevector tunnel less effectively

Unchanged

Tunneling

Current (nA)

Lower

Tunneling

Current (nA)

Constant height imaging

z

Constant current imaging

Higher

Tunneling

Current (nA)

Unchanged

Tunneling

Current (nA)

Typical working mode

Applied only on very flat regions

Imaging: spatial configuration and energy dependence of electron states (LDOS)need not to correspond in any simple way to the atomic positions

At the Bragg reflection the potentialgives rise to a forbidden energy region

The band gap

Example: linear lattice Si and Ge (111) cleaved surfaces

ax

iee

ax

ee

ax

iax

i

ax

iax

i

sin2

cos2


Imaging: spatial configuration and energy dependence of electron states (LDOS)need not to correspond in any simple way to the atomic positions

Charge density ON atomic positions

axax

22

22

sin

cos

Charge density BETWEEN atomic positions

In the image always topographic AND electronic features


Finite bias

But for eU about 1 eV?

Larger distortion of tip and sample wavefunctions

The sum has to be done on many different states

Approximation

Use undistorted tip and sample wavefunctions also at finite bias

dEEEeUI s

eU

t 00

, r

FxR

Ft EEreEnUI2

02 )(

DOStip DOSsample

eU = 0.01 eV

FxR

Ft EEreEnUI2

02 )(

Integral over all e- statesup to eU from Fermi levelat the tip position

Finite bias

But DOS sample decays into vacuum depending on barrier defined by the tip-sample distance so use WKB approximation

),(, 222

)(2

0

2

eUETEeEE s

EeUm

Rs

ss

st

r

dEeUETEEeUI s

eU

t ),( 0

Integral over all electronic statesup to eU from Fermi levelImaging occupied or unoccupied states

dEEEeUI s

eU

t 00

, r

The M now appears as DOS but the effects of finite biases are

included as modified x

Finite bias

What does it means imaging occupied or unoccupied electronic states?

dEeUETEEeUI s

eU

t ),( 0

At constant current means tunneling from all sample occupied states into alltip unoccupied statesAll is defined by bias voltage

Occupied

At constant current means tunneling from all tip occupied states into allsample unoccupied statesAll is defined by bias voltage

Unoccupied

Finite bias

Integral over all electronic states from Fermi level up to eU The information is geometric and electronic and is convoluted

The two states give different TOTAL intensity in the image

To separate the two one can collect images at different biases

dEeUETEEeUI s

eU

t ),( 0

Tunneling Spectroscopy

For metals the dI/dU is proportional to DOS at a given energy (low eU)

Integral over all electronic statesup to eU from Fermi level

The current is proportional to the occupied or unoccupied integral DOS

dEeUETEEeUI s

eU

t ),( 0

dE

dUeUEdT

EEeUeUeUTeUdUdI

s

eU

tst

, , 0

0

background

However this cannot be measured at constant current with feedback loop on

Large voltage dependent background due to T

Solution: dI/dU at constant separation (feedback loop off)


dE

dUeUEdT

EEeUeUeUTeUdUdI

s

eU

tst

, , 0

0

For e- injection into semiconductor unoccupied stateThe e- come mainly from EF so the I is mainly due to sample DOS

For e- injection into tip unoccupied stateThe e- come mainly from lowest lying levels of semiconductor so:problem: the I is mainly due to tip DOS?

For now, consider the tip DOS as constant so

,,

0

eU

ss dEdU

eUEdTEeUeUTeU

dUdI


DOS

For semiconductors no low voltage approximation: I needs to be normalized

For U > 0 T(E,eU) < T(eU,eU) and maximum transmission occurs at E = eU

eU

s

eUs

s

dEeUeUTeUET

EeU

dEdU

eUEdTeUeUT

EeU

UI

dUdI

0

0

,,1

,,

background

Normalization term

E

eUmRs tt

eeUET 222

)(22

),(

For U < 0 T(E,eU) > T(eU,eU) and maximum transmission occurs at E = 0

The terms have same order of magnitude

The background and denominator terms have same order of magnitude

Larger than sample DOS

Tunneling SpectroscopyAcquiring STS spectra

Sample and hold technique

Stop the tip on a locationDisable feedbackScan V and monitor I

Si(111)-(2x1)

Taken at different initialmeasuring conditions, i.e.different tip-sample distances

Tunneling SpectroscopyAcquiring STS spectra

Measuring at the same time the dI/dV one obtainsthe normalized conductance, independent ofTip-sample distance

occupied empty

Bulk DOS

-bonded chain

Data show that the normalized conductance

does not depend on tip-sample distance

Tunneling SpectroscopyBand structure effects

Measured voltage dependence of x

But what about the increase below 1 eV?

Close to the maximum wavevector at the edge of SBZ

The data allow to get (about 4.2 eV)and gives x = 22 nm-1

2

22

22k

meUm

x

Using this with the data one gets

1

2 A 2.22

mx Minimum value

22

22

22kk z

meUmx

1

A 1.1

k

At low bias the current is dominated by states at the edge of SBZ

Tunneling SpectroscopyObtaining STS images

dI/dV with lock inCurrent-imaging tunneling spectroscopy(CITS)

Voltage-dependent imaging

Apply modulationCollect dI/dV while scanningsimultaneously at each point

DOS at the set point of imaging conditionEmphasize one statePossible only in stabletunneling conditions(not in band gap)

Feedback on only 30% of the timeCollect dI/dV at fixed separation

-0.35 V

-0.8 V

-1.7 V

Need to be done at V followingtopography of nuclei

Integrate over an energyinterval at state onset

Spatial relationship between occupied and unoccupied states

-0.7 V

+0.7 V

Scanning Tunneling Microscopy (STM)Design and instrumentation

Approach mechanism

Enables the STM tip to be positioned within tunneling distance of the sample

High precision scanning mechanism

Enables the tip to be rastered above the surface

Control electronics

Control tip-surface separationDrive the scanning elements Facilitate data acquisition.

Vibration isolation

The microscope must be designed to be insensitive or isolated from ambient noise and vibrations.

Review of Scientific Instruments 60 (1989) 165Surface Science Reports 26 (1996) 61


Vibration isolation

It is essential for successful operation of tunneling microscopes.

This stems from the exponential dependence of the tunneling current on the tip-sample separation.

Typical surface corrugation is 0.1 0.01 nm or less

tip - sample distance must be maintained with an accuracy of better than 0.001 nm = 1 pm

z

STM sensitivity to external and internal vibrational sources:

Structural rigidity of the STM itselfProperties of the vibrational isolation system

Nature of the external and internal vibrational sources

Design criteria: The system response to external vibrations and internal driving signals is

less than the desired tip sample gap accuracy throughout the bandwidth of the instrument.


Floor vibrations

Damped with table

1-20 Hz Low-frequency floor vibration(amplitude several m)~ 8 Hz ventilation~ 29 Hz motors~ 60 Hz transformers

Isolation system scheme For a spring and a single viscous damping systemthe vibration amplitude transfer is

22

2

2

2

21

21

nn

nST

= external excitation frequencyn = 5/L system resonance frequencyL = spring elongation with mass loaded = / c damping ratio = system damping coefficient c = 4mn critical damping coefficient

viscous Damping system

spring


22

2

2

2

21

21

nn

nT

Viton (most effective against amplitude shock) = 0.3 – 0.05Problem: when strained under compression their spring constant is large, resulting inresonance frequency > 10-100 Hz

Damping materials

Metal springshave smaller spring constantsyielding resonance frequencies as low as0.5 Hz but they provide little damping

Single isolation system

< n, complete amplitude transfer with TS ( ) ~ 1 = n, amplification at the resonance frequency > n, damping viscous damping reduces T at n but increases T at > n

i.e. the decrease rate is reduced for heavily damped systems

a single spring system with extension of 25 cm is required for a vn of 1 Hz.

Two stage system isolation two sets of springsSprings + table


Other solution: a rigidly constructed STM does not require many stages of vibration isolation

Piezo drivers with m up to 100 kHz can be madebut• joints tightened by screws• epoxy junctions• three-point contacts • walker resonance• loose spring connectorsoften reduce this to 1-5 kHz

22

2

2

2

22

2

2

2

'121

21

Q

xT

mm

m

nn

nTOTAL

System with one stage vibration isolation and structural damping with m >> n the resultant T is

Damping system Rigid microscope design

Q’ = (m/2) tip-sample junction quality factor

For excitation amplitude of 1 m, a stability of better than 0.001 nm requires a vibration isolation-microscope system with an overall amplitude transfer function T() of better than 10-6

22

2

2

2

'1

Q

T

mm

mM

Microscope vibration amplitude transfer


Solid line: m = 2 KHz, n = 2 Hz= 0.4 Q’ =10Floor vibration amplitude of afew hundred nm, the gap stability will be worse than 0.1 nm

dotted line: Very rigid STMm = 12 KHz, n = 2 Hz = 0.4 Q’ =50the amplitude transfer is worse than 0.1 nm at 200 Hz

dashed line: Very rigid STM + vibration isolation tablem = 12 KHz, n = 1 Hz = 0.4 Q’ =50the amplitude transfer is 0.001 nm at 200 Hz

Dash-dotted line: two-stage vibration isolation:internal spring system (n = 1 Hz, = 0.4 )external table (n = 1.1 Hz, = 0.5 )Structural damping of STM assembly m = 2 kHz and Q' = 10estimated vibration amplitude is ~ 0.0001 nm in most of the frequency range,

Q’ = (m/2)


Approach mechanism

Enables the STM tip to be positioned within tunneling distance of the sample

Coarse motion devices to bring the tipand the sample into tunneling range

Inchworm stepper motor

Compact dimensions and high m,Vacuum compatibilityReliabilityHigh mechanical resolution.


Operating principleThree piezoelectric elementsOuter elements 1 and 3 contract and clamp the motor to the shaft The center element 2 contracts along the shaft direction These elements operate independently

the motor can move relative to the shaft if the shaft is fixedthe shaft can be moved relative to the motor if the motor is fixed

In this example the motor is held fixed and the shaft is moved

To move the shaft one step towards the right3 is clamped and 1 is unclamped2 contracts and the shaft is then moved towards the right1 is then clamped and element 3 is unclamped2 is extended to its original length

Similar to those used to climb a rope.


High precision scanning mechanism

Enables the tip to be rastered above the surface

Typical piezoelectric ceramic is PZT-5H (lead zirconate titanate)Large piezoelectric response (~ 0.6 nm/V).

Tube better than tripodes due to higher m

in-plane tip motionthe outer electrode is sectioned in 4 equal segments x and y directions given by applying differential scan signals (Vx

+, Vx-= - Vx

+; Vy+, Vy

- = - Vy

+)

Z- motioncommon mode signals(Vx

+ = Vx-; Vy

+ = Vy-)

applied to the electrodes allows extension of the tube in the z direction

The voltages are referenced to theconstant potential applied to the electrode located on the inner surface of the tube.


Piezoelectric equation

kEijij du Deformation tensorE field components

Piezotensor

3331

13

0

000

00

dd

d

dpiezo

Piezo ceramics are made such as

dd

d

dpiezo

0

000

00

0rE

lx

durr

For a cylinder

lenght l0Thickness h

E

Vhl

dx 0


Bimorph cells

Two plates of piezoelectric material glued together with opposite polarization vectors

Applying V one plate will extend, the other will be compressed, resulting in a bend of the whole element

Four sectors for electrodes Allow to move along the Z axis and in the X, Y plane using a single bimorph element


The resonance frequency of the scanning element is an important factor in determining the data acuisition speed data, since it has its own TFor scan < se the scanner responds uniformly to the drive voltage.For scan ~ se the amplitude of the scanners motion may increase dramaticallyFor scan > se the mechanical response falls off.

se of the scanning element may be as high as 100 kHzm is usually substantially lower (1-10 kHz)So scanning speed is limited much below 1 kHz1 frame: 400 lines2 lines /s = 0.5 HzTotal 200 s

Limits: feedback loop gain

Scanning Tunneling Microscopy (STM)Design and instrumentationControl electronics

Control tip-surface separationDrive the scanning elements Facilitate data acquisition.

The preamp is located as close to the tip as possible to minimize noise

I is measured by a preamplifierwith a variable gain of 106-109 V/Aand variable c to limit the bandwidth below the primary mechanical m

The tunneling current is linearized by a logarithmic amplifier

The tunneling current is then compared to a set-point, with the difference signal fed into a feedback amplifier that has an integrating amplifier withvariable time constant.The feedback signal is then amplified by a high voltage amplifier, the output of which is applied to the z-piezo to maintain the tunneling current at the desired set-point.

The x- and y-piezos are connected to high voltage amplifiers, which amplify slow scan (x) and fast scan (y) sweep signals generated by PC controlled DACs.

Scanning Tunneling Microscopy (STM)Examples of STM Apparatus

STM scanner

10 nm

1D WIRESK on InAs(110)

1D systems

C60/Ge

7.4 x 7.4 nm2

C60 Molecular Orbitals

Same orientation: hexagon facing up

141313 R

C60 - C60 = 1.44 nm

C60/Ag(100)

J Chem Phys, 117, 9531 (2002)

C60/Au(111)

PRB 69, 165417 (2004)

STM simulation

V = + 2.0 VI = 1.8 nA

Obtained afterannealing at 620 °C

STM

STM/STSCarbon Nanotubes

Nanotubes can be either metallic or semiconducting depending on small variations in the chiral winding angle or diameter

Si(111)-(7x7) Surface

Sticks-and-Balls Model

STM Image

Surface Reconstruction

Pt-Ni Alloy (100) surface

NanomanipulationQuantum Corrals

Fe atoms on Cu(111)

NanomanipulationQuantum Corrals are fabricated by manipulating atoms adsorbed at a solid surface to give a specific shape to the corral.

The STM tip is used to lift and put down the atomic units.

Peculiar effect related to Quantum Corrals Formation of a two-dimensional electronic gas (standing waves)confined within the corral.

In general the standing waves are particular modes of vibrations in extended objects like strings.

These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves.

The waves must change phase upon reflection. Under these conditions, the medium appears to vibrate in regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

Standing waves

24.7 x 13.8 nm2

Here the is electronic eigenstate of the surfaceand in particular we consider a 2-D electron gaswith functions similar to free-particle states

rkr )( ie

The waves are scattered at the step edgesand we observe the interference patternof the incident and scattered wave

Observing standing waves on metal surfaces

)()(),(2

EEELDOS rr

r = reflection amplitude

ei = phase shift

T ~ 4 K Cu(110)

5 nm

2 nm

Pentacene on Cu

TiOx clusterOn HOPG

Lattice distortionOr charge transfer effect?

150 x 150 nm2

Cu on Cu(111)

InstabilityAnd

Diffusioncoefficient

200 x 200 nm2

Cu on Cu(111)

Diffusion coefficients

TiO2 might form a vacancy of O, that is moving perpendicular to the rows.

(A) Ball model of the TiO2(110) surface (see text for explanations). A bridging O vacancy is marked by a circle. The arrow denotes the observed vacancy diffusion pathway.(B) and (C) Two consecutive STM images extracted from movie S1 (~8.5 s/frame).(D) Difference image, in which C) is subtracted from B). Bright protrusions indicate the presence of vacancies in B), whereas dark depressions indicate the new vacancy positions in C). (E) Displacement-vector density plot of oxygen vacancies as in D).(F) Observed frequency of O vacancy diffusion events as function of O2 exposure.

Vacancies are more mobile on the TiO2(110) surfaces after O2 exposure

(A) STM images showing the four different initial/final configurations resulting from the encounter between oxygen vacancies and O2 molecules. To each of these corresponds an atomistic pathway shown in B). The squares denote vacancy positions and the arrows indicate the diffusion path of O2 molecules. (B) Atomistic ball model illustrating the four adsorbate-mediated diffusion pathways.

Oxigen vacancies on TiO2(110)

Activity of surface for catalysis

• Operating principle

• Cantilever response modes

• Short theory of forces

• Force-distance curves

• Operating modes• contact• tapping –non contact• AM-AFM• FM-AFM

• Examples

Atomic Force Microscopy (AFM)

Basic idea:Surface-tip interaction

Response of the cantilever

Contact Mode Tapping Mode Non-Contact Mode

AFM basics

The AFM working principleMeasurement of the tip-sample interaction forceProbes: elastic cantilever with a sharp tip on the end

The applied force bends the cantilever

By measuring the cantilever deflection it is possible to evaluate the tip–surfaceforce.

How to measure the deflection

4 quadrantphotodiode

AFM basics

Two force components:FZ normal to the sample surface FL In plane, cantilever torsion

I01, I02, I03, I04, reference values ofthe photocurrent

I1, I2, I3, I4, values after change of cantilever position

Differential currents ΔIi = Ii - I0i will characterize thevalue and the direction of the cantilever bending or torsion.

ΔIZ = (ΔI1 + ΔI2) − (ΔI3 + ΔI4) ΔIL = (ΔI1 + ΔI4) − (ΔI2 + ΔI3)

ΔIZ is the input parameter in the feedback loopkeeping ΔIZ = constant in order to make the bending ΔZ = ΔZ0 preset by the operator.

AFM basics

Interaction forces cause the cantilever to bend while scanning

z

y

x

zzzyzx

yzyyyx

xzxyxx

F

F

F

ccc

ccc

ccc

z

y

x

The deflection vector is linearly dependent on applied force according to Hooke’s law

Cantilever responseThe tip is “in contact” with the surface

l = cantilever lenghtw = cantilever widtht = cantilever heightltip = tip height

z = cantilever deflection along yz = cantilever deflection along z = deflection angle

Vertical force Fz applied at the end induces the cantilever bending

Cantilever response

yz

tg

cantilever deflection anglearound setpoint

zzz

zyz

zxz

Fcz

Fcy

Fcx

Assume a bending with radius R

Iz= momentum of inertiawrt neutral axis

Y = Young modulusNeutral axis

Section

At any section S there is a torque wrt neutral axis

Rz

LL

Hooke’s law

Resulting force acting on dS

dSdF

YLL

RdS

Yz

LL

dSYdF

dF

zSSS

z IRY

dSzRY

RdS

YzzdFM 22

2

21dy

udR

yLFM z yLFdy

udYI zz 2

2

For any point along the cantilever y direction

Cantilever response

u(y) = deflection along z of a cantilever point at the distance y from the fixed end

For small angles zYIdy

udM 2

2

but

12

3wtI z

the longitudinal extension L is proportional to the distance z from the neutral plane

Soft cantilever

The feedback keeps a constant cantilever deflection, obtaining a constant force surface image The variation in the force while scanning leads to changes in z, providing the topography. Force setpoint: the force intensity exerted by the tip on the surface when approached. ~ 0.1 nN

3

3

4LYwt

kc

yLYIF

dyud

z

z 2

2

2

2yLy

YIF

dydu

z

z

integration

integration

zzzz

FwYt

LF

wYtL

FYIL

z 3

3

3

33 4312

3

zkF cz

The deflection is proportional to measured signal

Cantilever response yLF

dyud

YI zz 2

2

62

32 yLyYIF

uz

z

z

zLy YI

LFuz

3

3

12

3wtI z

Interatomic force constants in solids: 10 100 N/mIn biological samples ~ 0.1 N/m.Typical values for k in the static mode are 0.01–5 N/m.

Lz32

zLL

YIzYIL

YIFL

dydu

tg z

zz

z

Ly

233

22 3

22

Cantilever response

zzz

zyz

zxz

Fcz

Fcy

Fcx

zFwYt

Lz 3

34

cYIL

wYtL

cz

zz 3

4 3

3

3

The magnitude characterizes the cantilever stiffness

coefficient of inverse stiffness

It is the largest among the tensor cij

ztip

tip FwYt

LLLy 3

3

Lz32

zL

LLy tiptip

2

3

zztip

yz cL

Lc

2

3 0xzc

z

ztip

cFz

cFL

Ly

x

2

3

0

zYIL

c3

3

Force spectroscopy at fixed location

Ltip << L so cyz can be neglected

z = cantilever deflection along yz = cantilever deflection along z = deflection angle

Longitudinal force Fy applied at the end induces the cantilever bending

Cantilever response

yz

tg

cantilever deflectionangle around setpoint

yzy

yyy

yxy

Fcz

Fcy

Fcx

zYIdy

udM 2

2

tipz LFdy

udYI 2

2

tipyLFM

z

tipz

YI

LF

dyud

2

22

2y

YI

LFu

z

tipz

Similaly to previous case

Longitudinal force Fy applied at the end results in a torque


Cantilever response

2

2y

YI

LFu

z

tipz2

2L

YI

LFuz

z

tipz

Ly

zYIL

c3

3

ztip

Ly cFL

Luz

2

3

cL

L

YI

LLc tip

z

tipzy 2

3

2

2

Lz

cFL

LF

YI

LL

dydu

ytip

yz

tip

Ly

232

The deflection is proportional to measured signal

the axial force results in the tip deflection in vertical direction


Cantilever response

cL

Lc

L

Lc tip

zytip

yy 2

232

Lz

2

the axial force results in the tip deflection not only in the vertical but also in longitudinal direction

tipLy

zL

Ly tip

2

Very small compared to c

ytip

ytip

cFL

Lz

cFL

Ly

x

2

3

3

0

2

2

All these deflections are small compared to the main bending in the z axis

Transverse force Fx

Cantilever response

xzx

xyx

xxx

Fcz

Fcy

Fcx

simple bending

The simple bending is similar to the vertical bending of z-type Exchange the beam width (w) with thickness (t)

cwt

tYwL

cbend 2

2

3

34

twisting

Cantilever response

Twisting

The torsion is directly related to beam deflecton angle

LGwt

M3

3

G= Shear modulus ~ 3Y/8

The torque by Fx is tipx LFM

The lateral deflection istiptors Lx

torsxtors xFc

cL

L

Ywt

LL

Gwt

LL

M

L

Fx

c tiptiptiptip

x

torstors 2

2

3

2

3

22 283

cwt

tYwL

cbend 2

2

3

34 xxxxtorsbendtorsbend FcFccxxx

cL

L

wt

c tipxx

2

2

2

2 2

Cantilever response

xxxxtorsbendtorsbend FcFccxxx cL

L

wt

c tipxx

2

2

2

2 2

0

0

22

2

2

2

z

y

cFL

L

wt

x xtip

The deflections in y and z are of the second order with respect to x deflection

L = 90 mLtip = 10 mw = 35 mt = 1 m

192.1 mNc

12

2

0016.01220

1 mNccwt

cbend

12

2

05.04012 mNcc

L

Lc tip

tors

132.061

2

3 mNccL

Lc tip

yz

12

2

071.02713 mNcc

L

Lc tip

yy

132.061

2

3 mNccL

Lc tip

zy

Dominant distortions czz, cyz, czy

Lateral distortions are much smaller

Simplebending

twisting

Fixed end

Cantilever effective mass and eigenfrequency

l = cantilever lenghtw = cantilever widtht = cantilever heightltip = tip height

Cantilever is vibrating along z

3

3

4LYwt

kc

u(t,y) = deflection along z of a cantilever point at the distance y from the fixed end

y

dy

u(y)

L

mdyytudEk 2

, 2Kinetic energy

62

32 yLyYIF

uz

z

z

zLy YI

LFu

3

3

3

3

2

2

3

32

3

3262

3,

Ly

Ly

L

uyLyL

uytu Ly

Ly

Cantilever effective mass and eigenfrequency

20

2

0,

214033

2, Ltu

mdy

Lm

ytudELL

k Kinetic energy

3

3

2

2

3

32

,Ly

Ly

L

uytu Ly

Lmdy

ytudEk 2, 2

Potential energy c

Ltudu

cu

FduELtuLtu

P 2,2,

0

,

0

c

LtuLtu

mET 2

,,

214033 2

2

0

,,

14033

c

LtuLtu

m Equation of motion

14033

*m

m

0* cu

um

Y

Lt

cm 20

029.1*

1 Cantilever eigenfrequency

The cantilever eigenfrequency must be as high as possible to avoid excitation of natural oscillations due to the probe trace-retrace move during scanning or due to external vibrations influence

Tip-surface interactionOrigin of forces

Tip-surfaceSeparation (nm)

0

1

10

100

1000

Interatomic forces (adhesion)

Van der Waals(Keesom,Debye,London)

Electric, magnetic,capillary forcesNon contact

Intermittentcontact

Contact

Born repulsive interatomic forces

Origin: large overlap of wavefunction of ion cores of different moleculesPauli and ionic repulsion

Cgs/esu

122

rC

UR

-

+

-

+

041

Origin of forces

Elastic forces in contact

Origin: object deformation when in contact

Isotropic cantilever and sample two parameters to describe elastic propertiesY = young modulus = Poisson ratio

Close to the contact point the undeformed surfaces are described by two curvature radii

Deformations are small compared to surfaces curvature radii

deformationand penetration

the contact pressure is higher for stiffer samples

2

23

132

RYhF

Assumptions

Hertz problem solution:allows to find the contact area radius R and penetration depth h as a function of applied load

contact area radius : up to 10 nmPenetration depth : up to 20 nm contact pressure : up to 10 GPa.

Origin of forces

Potential energy of the dipole moment in an electric field E

Keesom Dipole forces

q- q+

d

2rq

EU

1cos3 23

rE

Field intensity produced by the dipole

Origin: fluctuation (~10-15 s) of the electronic clouds around a moleculeDipole formation

Cgs/esu

221

rqq

fC rqq

drfU C21

Coulomb forcebetween point charges Coulomb potential energy

2rq

E

Electric field

= qd = dipole moment

is the angle between dipole and r

041

For r >> d

Origin of forces

Potential energy of the interacting dipole moments

1

q- q+

d

cossinsincoscos2 21213

21 r

U

Maximum attraction for 1= 2 = 0°

When two atoms or molecules interacts2

q- q+

d

Maximum repulsion for 1= 2 = 90°3

2

max

2r

U

Origin of forces


3

2

max

2r

U

In a gas thermal vibrations randomly rotates dipoleswhile interaction potential energy aligns dipoles


6

4 132

rTkU

BAV

Orientational interaction

Total orientation potential is obtained by statistically averaging over all possible orientations of molecules pair

de

dUeU

TkU

TkU

AV

B

B

For U << KBTTk

Ue

B

TkU

B

1

Udd

dTk

UUd

U BAV

2

0 Ud

Origin of forces

Potential energy of the interacting dipole moments

q- q+

d

Induced dipole moment

Debye Dipole forces

Origin: fluctuation of the electronic clouds around a moleculedipole formation, interaction of the dipole with a polarizable atom or molecule

Eind

q- q+

d

2

2

0

EdEU

E

ind

1cos3 23

rE

For r >> d6

2 1r

U ind

Induction interaction

The induced dipole is “istantaneous” on time scale of molecular motionSo one can average on all orientations

Origin of forces

Potential energy of atom 1in the field due to dipole 2

Field induced by atom 2

London Dipole forcesOrigin: fluctuation of the electronic clouds around the nucleus dipole formation with the positively charged nucleus interaction of the dipole with a polarizable atom

2

21

0

EdEU

E

ind

322

rE

-

+

-

+dipole Polarizable

atom2 1

RMS dipole moment forfluctuating electron-nucleus

i

i 2

ih

2

22

Ionization energy6

21 14

3r

hU i

32

32

22r

h

rE i

The dipole formation of atom 2 is given by the polarizability

Origin of forces

Fluctuation of the electronic clouds around the nucleus.dipole formation with the positivecharge of nucleus.interaction of the dipole witha polarizable atom

621

43

rh

U i

6

4

32kTr

U

KeesomFluctuation of the electronicclouds around a molecule.dipole formation

Origin Potential energy

Fluctuation of the electronicclouds around a molecule.dipole formation.interaction of the dipole witha polarizable atom or molecule

6

2

rU ind

Debye

London

122

rC

UR Large overlap of core wavefunctionof different molecules

Born

Origin of forces

van der Waals dipole forces between two molecules

6212

4

6

2

621

6

4 143

32

43

32

rh

kTrrh

kTrU iind

indi

61

rC

U

61

122

rC

rC

U

Total potentials between two molecules

Lennard-Jones potential

Origin of forces

van der Waals dipole forces between macroscopic objects

Additivity: the total interaction can be obtained by summationof individual contributions.

Continuous medium: the summation can be replaced by an integration overthe object volumes assuming that each atom occupies a volume dV with a density ρ.

Uniform material properties: ρ and C1 are uniform over the volume of the bodies.

61

rC

U Urf )(

The total interaction potential between two arbitrarily shaped bodies

1 2

2121 )()(v v

dVdVrfrU

1 2

2162112 1

)(v v

dVdVr

CrU 2112 CH

Hamaker constant

Origin of forces

The force must be calculated for each shape

For a pyramidal tip at distance D from surface

DH

DF

3tan2

)(2

2112 CH

Hamaker constant

Same role as the polarizabilityDepends on material and shape

Origin of forces

The force must be calculated for each shape

Nxh

CF

15

2211

2

103.16

tan

Conical probe Pyramidal probe

Nxh

CF

15

2211

2

102.53

tan2

Tip radius r << h Tip radius r << h

Conical proberounded tip

NxF 13101.1

Nxh

RCF

9

2211

2

103.36

For r >> h

Origin of forces

Adhesion forces

Middle range where attraction forces (-1/r6) and repulsive forces (1/r12) act

adhesionIt originates from the short-range molecular forces.

two types - probe-liquid film on a surface (capillary forces) - probe-solid sample (short-range molecular electrostatic forces)

electrostatic forces at interface arise from the formation in a contact zone of an electric double layer

Origin for metals- contact potential- states of outer electrons of a surface layer atoms - lattice defects

Origin for semiconductors- surface states- impurity atoms

Origin of forces

Capillary forces

Similar to VdW forceNF 910

Cantilever in contact with a liquid film on a flat surfaceThe film surface reshapes producing the "neck“

The water wets the cantilever surface:The water-cantilever contact (if it is hydrophilic) is energetically favored as compared to the water-air contact

Consequence: hysteresis in approach/retraction

Origin of forces

How to obtain info on the sample-tip interactions?

The sample is ramped in Z

and deflection c is measured

Force-distance curves




c or z

The deflection of the cantilever is obtained by the optical lever technique

PSD = position sensitive detector

When the cantilever bends the reflected light-beam moves by an angle

zLL

YIzYIL

YIFL

dydu

tg z

zz

z

Ly

233

22 3

22

zL

23

d = detector - cantilever distance laser spot movement

ddPSD 2tan2 d

Lz PSD

3

High sensitivity in z is obtained by L << d

Vertical resolution depends on the noise and speed of PSD m

t

1310

T = 0.1 ms z ~ 0.01 nm

Measured quantities: Z piezo displacement, PSD i.e. I or V

The sample is ramped in Z and deflection c is measured D = Z –(c + s)

Force – displacementcurve

AFM force-displacement curve does not reproduce tip-sample interactions,but is the result of two contributions: the tip-sample interaction F(D) and the elastic force of the cantilever F = -kcc


Must be converted to D and F

D = tip-sample distancec = cantilever deflections = sample deformationZ = piezo displacement



D = Z –(c + s)


Must be converted to D and F

In non-contact D = Z (c = 0 so F(D)=0)In contact Z = c and D = 0 so F(D)=kc

a) Infinitely hard material (s=0), no surface forces

PSD-Z curve: two linear parts

zero force linedefines zero deflectionof the cantilever

Linearregime

sensitivity IPSD/ Z c = IPSD/(IPSD/ Z)

F-D curve

F(Z) = kc

D=Z-c

F(D) = k IPSD(Z)/(IPSD/Z)

Z = 0 at the intersection point

Z > 0 if surface isretracted from tip

Conversion between PSD and Z



D = Z –(c + s)


In contact Z = c and D = 0

b) Infinitely hard material (s=0)

PSD-Z curve

zero force line =0 deflection at large distance

Linearregime sensitivity IPSD/ Z

from the linear part

c = IPSD/(IPSD/ Z)

F-D curveF(Z) = kc D=Z-c

Z = 0 at the intersection point (extrapolated)


long-range exponential repulsive force

Accuracy: force curves from a large distanceApply a relatively hard force to get to linear regimeThe degree of extrapolation determines the error in zero distance.

In non-contact D = Z - c

F(D) = k IPSD(Z)/(IPSD/Z)

s

s


D = Z –(c + s)Force-distance curves

c) Deformable materials without surface force

PSD-Z curve

F=0 line

F-D curve


If tip and/or sample deform the contact part of PSD-Z curve is not linear anymore

In non-contact D = Z (c = 0)

Hertz model: elastic tip radius Rplanar sample of the same material (Y) 213

223

RYF s

s = indentation

For many inorganic solids s << c For high loads c~F/kc

sensitivity IPSD/ Z from the linear part

the force curves have to be modeled to describe indentation =‘‘soft’’ samples: cells, bubbles, drops, or microcapsules.

But: indentation and contact area are still changing with the loadIt is more appropriate to use indentation rather than distance after contactthe abscissa would show two parameters: D before contact and s in contact

If s 0 ‘‘zero distance’’ (Z=0) must be defined

If s ~ c

In contact the distance equals an interatomic distance

s

s

Force-distance curvesc) Deformable materials with surface force

At some distance the gradient of the attraction exceeds kc and the tip jumps onto the surface.

- very soft materialssurface forces are a problemleading to a significantdeformation even before contact

- relatively hard materialsDue to attractive and adhesion forcesit is practically difficult to precisely determine where contact is established

Tip approaching a solid surface attracted by van der Waals forces

In this case it is practically impossible to determine zero distance and one can only assume thatthe indentation caused by adhesion is negligible.

Adhesion forces add to the spring force and can cause an indentation

D = Z –(c + s) Tip-sample force Fc = -kcc

Tip-sampleinteraction

Force – displacement curve

Lennard-Jones force, F(D)= -A/D7 + B/D13

Elastic force of the cantilever for different c

At each distance the cantilever deflects until Fc=F(D) so that the system is in equilibriumThe equilibrium points are a, b, c

The corresponding distances are not D but Z i.e. the sample and the cantilever rest position separationthat are given by the intersections between lines and thehorizontal axis (,,)

Because we measure Z = the sample and the cantilever rest position separation


D = Z –(c + s)Total potential of cantilever-sample system Utot = Ucs(D) + Uc(c) + Us(s)

Ucs(D) = tip - sample interaction potential

Uc(c) = cantilever elastic potential

Us(s) = sample deformation potential

assume

The relation between Z and c is obtained by forcing the system to be stationary

0

c

tot

s

tot UU

DUU cs

s

cs

And since

ncs

cccc

ssss

DC

DU

kU

kU

2

22

2

sscc kk

ncscc Z

Ck

The measured force- displacement curve can be converted into the force-distance curve



two characteristic features of force-displacement curves:discontinuities BB’ and CC’hysteresis between approach and withdrawal curve

jump-to-contact

jump-off-contact

In the region between b' and c' each line has three intersections = three equilibrium positions.Two (between c’ and b and between b' and c) are stableOne (between c and b) is unstable

During approach the tip follows the trajectory from c’ to band then "jumps" from b to b‘During retraction, the tip follows the trajectory from b' to cand then jumps from c to c’


The slope of the lines 1-3 is the elastic constant of the cantilever kc.for high kc, the unsampled stretch b-c becomes smaller, the jump-to-contact firstincreases with kc and then, for high kc, disappears.

The jump-off-contact always decreases, so that thetotal hysteresis diminishes with kc. When kc is greater than the greatest value of the tip-sample force gradient, hysteresis and jumps disappear and the entire curve is sampled

To obtain complete force-displacement curvesone should employ stiff cantileversStiff cl have a reduced force resolutionTherefore it is necessary to reach a compromise

The cantilever isforced to oscillate

Operation modes

Tapping: Amplitude modulation (AM-AFM)Non-contact: Frequency modulation (FM-AFM)

AM-AFM: a stiff cantilever is excited at free resonance frequencyThe oscillation amplitude depends on the tip-sample forces Contrast: the spatial dependence of the amplitude change is used as a feedback

to measure the sample topography Image = profile of constant amplitude

FM-AFM: the cantilever is kept oscillating with a fixed amplitude at resonance frequencyThe resonance frequency depends on the tp-sample forcesContrast: the spatial dependence of the frequency shift, i.e. the difference between

the actual resonance frequency and that of the free cantileverImage = profile of constant frequency shift.

Contact Mode

Tapping Mode

Non-Contact Mode

Static cantilever

Experiments in UHV: FM-AFM experiments in air or in liquids: AM-AFM

Operation in non-contact or intermittent contact mode is not exclusive of a given dynamic AFM method

Drawbacks:The download force of the tip may damage the sample (expecially polymers and biological samples)

Under ambient conditions the sample is always coveredby a layer of water vapour and contaminants, and capillary forces pull down the tip, increasing the tip-surface forceand add lateral dragging forces

Equiforce surfaces are measured

Contact mode

The tip is brought “to contact” with the surfaceuntil a preset deflection is obtained.Then the raster is performed keeping deflectionconstant.

Info on lateral dragging forces can be obtained

Operation modes: AM

l defines z = 0

0llkFel

l0 = spring at restl = spring extension + mass = m*k = cantilever spring constantm = cantilever mass

Cantilever = spring with k and pointless mass m

14033

*m

m

Y

Lt

cmmk

20

029.1*

1*

020 zz

00cos tZtz

00

00

20

202

0

arctan

zv

vzZ

Harmonic oscillator

Operation modes: AM

Frictional force vFfr

02 20 zzz

ttt BeAeetz

damped harmonic oscillator

*2m

Three solutions

0

Aperiodic motion

20

2

0

0cos tZetz t

0

00

2002

0

arctanzv

zvzZ

0

Critical damping

BtAZetz t

Qualityfactor

0For small damping

)()()(

2)(

2TtEtE

tEEtE

QT

Stored energy

teEtE 20)(

220

TQ

Q characterizes the rate of the energy transformationQ is the number of a system oscillations over its characteristic damping time 1/

Energy loss /period

Under damping

Operation modes: AM

tAzz cos020

Forced harmonic oscillator tFtF cos0

*0

0 mF

A

For 0

tZtZtz coscos 000

2

10

0

0222

0

001

22

2122

0

00

arctan

*

CC

vC

mF

xC

CCZA

Z

Drivingoscillation

freeoscillation

For 0 =

ttA

tZtz 00

000 sincos

resonance

Operation modes: AM

tAzzz cos2 020

Forced damped harmonic oscillator tFtF cos0

*0

0 mF

A

tZtztz dho cos0

222220

00

4

AZ Amplitude

As is decreased the Z0 becomes more peaked at 0 (resonance) when 0 A weakly damped oscillator can be driven to large amplitude by a relatively small amplitude external driving force

*2m

For t > 1/ only forced oscillations will be present

2

0Q

in phase (~0) for < 0

in phase quadrature (=/2) at 0

in antiphase (=) for > 0

22

0

2arctan

Phase

/0 /0

0

0

AZ

resonant amplificationfactor of the oscillator

Q=16

Q=8

Q=4

Excitation

Q=16

Q=8

Q=1

zdho(t) = solutions for damped harmonic oscillator

Operation modes: AM

Amplitude

For light damping Z0 becomes more peaked at 0

Lorentzian

Phase

/0 /0

0

0

AZ

2

220222

0

00

Q

AZ

22

0

0arctan

Q

Q0

The more is Q, the less is the resonance peak width.

2022

0 21

12QR

Resonant frequency for damped forced h.o.

The cantilever oscillation amplitude depends on - Driving amplitude A0

- Value of driving frequency with respect to 0

Resonant width

Operation modes: AM

• F(z) does not depend on time• Qualitative behavior is the same as before• Change of the oscillator equilibrium position

For small oscillations expand F(z) around equilibrium position z0

*/cos2 020 mzFtAzzz

Forced damped harmonic oscillator + External force

0ztztz *0

020 m

Fz z

tAm

Fzz

dzzdF

mzz z

z

cos

**1

2 0020

20

0

0

...0

0 tz

dzzdF

FzFz

z

mk

dzzdF

kkz

0

tFtF cos0

*0

0 mF

A *2m

Excitation

z0 is given by

tAzzz cos2 02

0*

120

2

zdzzdF

m

effective spring constant

the resonance frequency of a weakly perturbed ho depends on the gradient of the interaction.

effective resonance frequency

Operation modes: AM

2

220222

00

Q

AZ

The force gradient gives a shift of resonant frequency

0

202

zRR dz

zdFk

2

220222

0

00

Q

AZ

0*

120

2

zdzzdF

m

2022

0 21

12QR

000

202222

022

*1

2*

12

zR

zR

zR dz

zdFkdz

zdFmdz

zdFm

Resonant frequency ofdamped forced h.o. + force

110

0

2

20

202

zRR

Rz

RRR

dzzdF

k

dzzdF

k

02

0

zdzzdF

k

The change of the resonant frequency can be used to measure the force gradient

Operation modes: AM

22

0arctan

QPhase shift

0*

120

2

zdzzdF

m

The force gradient gives a frequency shift resulting also in a phase shift wrt curve at 0

The change of the phase shift and amplitude can be used to measure the force gradient

The force gradient gives a frequency shift resulting also in anamplitude change wrt curve at 0

Tapping: Amplitude modulation (AM-AFM)

AM-AFM: the cantilever is excited at free resonance frequency 0

The oscillation amplitude A is used as a feedback parameter to measure the sample topography Other signals : phase shift between the driving excitation and the tip oscillation signal

: frequency shift between the effective frequency and the tip oscillation frequency

Operation modes: AM

AFM gives 3D images of the sample surface

two different (not always independent) resolutions should be distinguished:lateral and vertical

Vertical resolution

noise from the detection system

mt

1310

T = 0.1 ms z ~ 0.01 nm

thermal fluctuations of the cantilever

limited by

nmkk

TKz B 074.0

3

4

K = 40 nm, T = 295 Kz ~ 0.01 nm

Operation modes: AM

Analogy with optical microscopy

Calculated values

Lateral resolution

the lateral resolution of AFM is defined as the minimum detectable distance d between two sharp spikes of different heights

R= tip radius z ~ vertical resolution hzzRd 2

Atomic resolution with radius of 0.2 0.5 nm, i.e. almost a single atom

Operation modes: AM

Convolution effects: width larger than real values

DNA chain

Lateral resolution

Sample deformations due to high loads resultsin smaller apparent height

factors influencing image resolution in AM-AFM

• sample type • depend on the nature and geometry of the tip (radius and k)• operational parameters (average and maximum forces)

Operation modes: AM

Limits of AM-AFM

The download force of the tip may damage the sample (expecially polymers and biological samples)

Sample deformations

Under ambient conditions the sample is always covered by a layer of water vapour and contaminants, and capillary forces pull down the tip, increasing the tip-surface force and add lateral dragging forces

To obtain atomic resolution one has to increase the sensitivity to amplitude changes

QB

s B = bandwidth

10 x 10 nm2 image256 x 256 pixelScanning speed 2 lines/s (20 nm/s)B= 2 x 256 = 512 Hz

Increasing Q might be an option to obtain higher s

But increasing Q reduces B, i.e. the response time of the system

2

0Q

0

21

Q /2 transient decay

transient beat

Moving the tip to a new position means perturbing the system that respond with

In UHV Q = 50000

0 = 50 kHz = 2 s Too long acquisition time

Operation modes: FM

Frequency modulation

The driving signal of the cantilever oscillation is generated through a feedback loop where the a.c. signal coming from the PSD is amplified and used as the excitation signal

An automatic gain controller keeps the vibration amplitude constant

The signal used to produce the image comes from the direct measurement of cantilever resonance frequency (that depends on the tip–surface interaction)

from AM mode, the cantilever is kept oscillating at its current resonant frequency (different from 0 due to the tip–sample interaction) with a constant amplitude A0

In FM-AFM, the spatial dependence of the induced in the cantilever motion by the tip-sample interaction is used as the source of contrast

During the scan, the tip–sample distance is varied in order to achieve a set value for .

The topography represents a map of constant frequency shift over the surface.

Operation modes: FM

The dynamics of the cantilever is that of a self-driven oscillator, different in many aspects (in particular the approach to the steady-state) from AM-AFM

QB

s

B does not depend on Q

In FM the sensitivity can be increased by using a very high QThe frequency detection is not affected by the transient terms in the amplitude that limit the AM detection mode

B is set only by the characteristics of the FM demodulatorThe original demodulator measured a frequency shift of 0.01 Hz at 50 kHz with B=75 Hz

Now: 5 mHz with 500 Hz bandwidth

FM and AM modes have essentially the same sensitivity if the same set of parameters are used

Operation modes: FMCantilever dynamics = self-driven oscillatordifferent from AM-AFM in particular the approach to the steady-state

Assume that the tip-sample forces Fts(z) are known

Why it is complicate to find a description of cantilever dynamics under the influence of Fts(z)i.e. a relation between the frequency shifts and Fts(z) ?

- Intrinsic anharmonicity of Fts(z)

- Effective non-local character of the interaction:in FM-AFM vibration amplitudes are much larger than interaction range of Fts(z)cantilever is sensing the interaction just for a very small part of its oscillation cycle

tFmzFzzz excts */2 20

- Fexc(t) is no longer a pure harmonic driving force with constant Aexc and constant exc

but a function describing the oscillator control amplifier

The amplifier takes the input signal from the PSDmodifies its amplitude to force the system to oscillate at the set amplitude A0

This feedback loop keeps the cantilever always vibrating at its current resonance frequency with the same constant amplitude A0

Operation modes: FM

tFmzFzzz exc */2 20

The feedback loop keeps the cantilever always vibrating at its current resonance frequency with the same constant amplitude A0

The feedback loop assures that the energy losses (intrinsic to cantilever + due to tip–surface interaction) are exactly compensated by the excitation dynamically in order to keep the amplitude constant

Under these conditions, both the excitation and damping terms can be neglected

0 zkkzm ts

0z

tsts dz

zdFk

0* 0 zzFkzzm ts

For small oscillations the tip-sample interaction can be developed linearly around z0

02

0

zdzzdF

k

0

0 20

zz dz

zdFk

To be solved numerically

Frquency shift vs force gradient

Operation modes: FM

0* 0 zzFkzzm tsFor large oscillations

For typical oscillation amplitudes A0 = 20 nm and a stiff cantilever k = 30 N/m

Cantilever is weakly perturbed harmonic oscillator

But how comes atomic resolution?

elastic energystored in cantilever

Restoring force at turning point close to the surface

2

20kA tip-surface

interaction energy

00 Azdts

cFkA

tip-surface interaction force

3.75 x104 eV

600 nN

1- 10 eV

1-10 nN

2

0 000

0200 coscos

41

,,, dAAdFkA

Akd ts

Relation between the frequency shift and the average of Fts in a full harmonic cycle:

depends on the operation conditions and the forces at distance of closest approach d

Operation modes: FM

Example: LR vdW interaction Not dominated by the interaction of the tip atoms closer to surface Depends on the macroscopic shape of the tipHow could provide the lateral resolution needed?

Atomic contrast relies on a significant lateral variation of the tip–surface interaction on an atomic length scale

This can only be provided by short-range (SR) interactions.

• long-range (LR) electrostatic• vdW attractive• repulsive contact Hertzian

Appropriate to explain main features of AFMbut cannot provide an explanation for the atomic resolution

Operation modes: FM

Insulators (alkali halides and oxides)and oxidised tipsAll the dangling bonds are saturated

Confined microscopic electric field around the oxygen tip apex provides the key to the lateral variation of the interactionIt is due to electrons taken from the surrounding Si atoms located on the strongly localised O 2p wavefunctions

The normal displacements of the surface ions + the related surface polarisation due to thestrongly localised electric field are responsible for the atomic resolution in these materials

Covalent interaction: approximated by exponentially decaying potentials The dependence on orbital overlap explains the exponential variation from d and makes it suitable for atomic resolution

Semiconductor surfaces candidate force: covalent bonding interactions

In semiconductor surfaces and in the tip there are undercoordinated atoms with unsaturated bondsThey can contribute to the total tip–surface interactionand provide atomic resolution.

Different from vdW interaction: related to the overlap of the atomic wavefunctions

Origin of atomic resolution

Operation modes: FMExample: Si(111)-(7x7) FM AFM in UHV

“Force”-distance curvesType 1 Type 2: after accidental tip modification

This defines the zero of tip-sample d

Frequency is changing well before amplitude

But amplitude can decrease even for attractive forcesSo A’ might not be the true contact distance

The discontinuity in the is interpreted asformation of chemical bond with the surface

Operation modes: FMExample: Si(111)-(7x7) FM AFM in UHV

Type 1Type 2: after accidental tip modification

Cantilever vibration amplitude: 16 nm = 1.1 Hz

The interaction of the tip having the discontinuity in the frequency shift gives a very significant contribution to the image contrast of the noncontact AFM images

Cantilever vibration amplitude: 14.8 nm = 13 Hz

Operation modes: FMExample: Si(111)-/7x7) FM AFM in UHV

Unequivalent adatoms

Possible origins of the contrast between inequivalent adatoms- the true atomic heights corresponding to the adatom core positions- the stiffness of interatomic bonding with the adatoms- the amount of charge of adatom- the chemical reactivity of adatoms

Calculations exlude the first two

Adatom charge vdW or electrostatic

Chemical reactivity covalent bonding formation

Chemical bonding is responsible of the resolution

Cantilever k=41 N/m, = 172 kHzTip apex radius 5–10 nm; Q ~ 38 000 in UHV

Short description Theory of 1-D tunneling Actual 3D barriers tip modeling atomic resolution Hardware...

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Transcript of Short description Theory of 1-D tunneling Actual 3D barriers tip modeling atomic resolution Hardware...