Transcript Electron tunneling across 1

Scanning Tunneling Microscopy (STM)
Short description
Theory of 1-D tunneling
Actual 3D barriers
tip modeling
atomic resolution
Hardware
Examples
Bibliography
• Scanning Probe Microscopy and
Spectroscopy (Wiesendanger, Cambridge UP)
• Scanning Probe Microscopies: Atomic Scale
Engineering by Forces and Currents
Piezolelectric Tube
with Electrodes
Scanning Tunneling
Microscopy (STM)
Tunneling
Current Amplifier
Sample
Distance Control
and Scanning Unit
Tunneling Voltage
Data Processing
and Display
Tip
Sample
Fundamental process:
Electron tunneling
Electron tunneling
Typical quantum phenomenon
Tunneling definition
Wave-particle impinging on barrier
Probability of finding the particle beyond the barrier
The particle have “tunneled” through it
Role of tunneling in physics and knowledge development
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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 process
Intial and final states have same energy
1D
Planar Metal-Oxide-Metal junctions
Rectangular barriers
Planar Metal-Oxide-Metal junctions
Time independent
Matching solutions of TI Schroedinger eq
Inelastic
Energy loss during the process
Interaction with elementary excitations
(phonons, plasmons)
3D
Scanning Tunneling Microscopy
3D
Scanning Tunneling Microscopy
Time-dependent
TD perturbation approach:
(t) + first order pert. theory
Electron tunneling across 1-D potential barrier
Time independent
Region 1
2mE
k 
2
2
Region 2
x2 
Region 3
2 d 2

V0  E
2m dz 2
 1  e ikz  Reikz
Plane-wave of unit amplitude traveling to the right+
plane-wave of complex amplitude R traveling to the left
 2  Ae xz  Be xz
2m(V0  E )
2
exponentially decaying wave
 3 Te ikz
plane-wave of complex amplitude T traveling to the right.
The solution in region 3 represents the “transmitted” wave
T
2
 transmissi on probability
Electron tunneling across 1-D potential barrier
Time independent
Continuity conditions on  and d/dz give
At z=0
1R A B
k 1  R   x A  B 
At z=s
Ae ixs  Be ixs Te iks
x Ae ixs  Be ixs   kTe iks
R
2

R
k
2
 x 2  sinh 2 xs 
2
4k x  k  x
2
2
2
2
 sinh
2 2
2
T
xs 
T
 reflection probability
R
2
2
T
2
1

2
4k 2x 2
4k 2x 2  k 2  x 2  sinh 2 xs 
2
 transmissi on probability
Electron tunneling across 1-D potential barrier
Time independent
A square integrable (normalized) wave function has to
remain normalized in time
d
dt



 z ,t  dz  0
2

d
i  * 
 * 
2
 dz 


 0
dt 
2m  z
z  

In a finite space region this conditions becomes
d
dt
b

a
2
 dz 
b
i  * 
 




 0
2m  z
z  a
d
Pa ,b  j b,t   j a ,t   0
dt
Probability conservation
*
i  * 
 * 


j z ,t  

2m 
z
z 
Probability current
Electron tunneling across 1-D potential barrier
Time independent
Applying to
our case
Region 1
i  * 
 * 


j z ,t  

2m 
z
z 
j1 z ,t  

k
1 R
2m
2
  v 1  R 
2
ji z ,t   v
Region 3
jT z ,t  
ji
T
T
jT
k
T
2m
2
 vT
2
2
T  transmissi on coefficien t
T
1
2
2 2
(
k

x
)
1  sinh 2 (xs )
( 4k 2x 2 )
Electron tunneling across 1-D potential barrier
Time independent
2m(V0  E )
xs  s
2
For strongly attenuating barriers xs >> 1
Large barrier height (i.e. small )
 e xs  e xs
sinh (xs )  
2

2
T
1
(k 2  x 2 )2
2
1  sinh (xs )
( 4k 2x 2 )

1
 e 2xs  2  e 2xs

4

Exponential is leading contribution
16k 2x 2
 2xs 2
2
2 2
e (k  x 2 )2
 (k  x )

2 2
 ( 4k x )
16k 2x 2
2xs
T 2
e
(k  x 2 )2
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 coefficient
depends exponentially on barrier width
2
Exponential dependence of tunneling current
Electron tunneling across 1-D potential barrier
Time-dependent
Ideal situation:
incident state from left has some probability
to appear on right
… And we can calulate it…
Real situation:
At the surface the wavefunction is very complicated to calculate
Different approach
If barrier transmission is small, use perturbation theory
But no easy way to write a perturbed Hamiltonian
Approximate solutions of exact Hamiltonian within the barrier region
 l (z )  ae kz for z  0
 r (z )  be kz for z  s
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
Electron tunneling across 1-D potential barrier
Time-dependent
With the exact hamiltonian on left and right,
we add a term HT representing the transition
rate from l to r.
HT = transfer Hamiltonian
d (t )
dt
H (Hl  Hr )  HT  H 0  HT
H (t )  i
H 0 l  El  l
H 0 r  E r r
l,r = electron states at the left and
right regions of the barrier
HT is the term allowing to connect the right and left solutions
Electron tunneling across 1-D potential barrier
Time-dependent
Choose the wavefunction
Put into hamiltonian
  c l e
iEl t

H (t )  i
iEl t
iErt

(Hl  Hr )c l e   d r e 

iEl t
iErt

d c l e   d r e 
i 
dt




 d r e
iErt

d (t )
dt
iEl t
iErt


  HT c l e   d r e 








Electron tunneling across 1-D potential barrier
Time-dependent
cEl  l e
cEl  l e
iEl t

iEl t

 dE r r e
 dE r r e
iErt

iEl t
iErt

 HT c l e   d r e 

iErt

iEl t
iErt

HT c l e   d r e 


0


The total probability over the space is
*

 HT dz  1
iErt
 * iEl t
*


c

e

d

e
r
l
 
iEl t
iErt
 
HT c l e   d r e 
 
 

dz  1






Electron tunneling across 1-D potential barrier
So the tunneling matrix element
Mlr  Mrl  1
*
*

H

dz


 l T r
 r HT  l dz  1
Mlr = Probability of tunneling from state l to state m
Using the Fermi golden rule to obtain the transmitted current
jt 
2 dN
2
Mrl

dE r
Density of states of the final state
In general, the tunneling current contains information on the
density of states of one of the electrodes, weighted by M
But ………… each case has to calculated separately
Electron tunneling across “real” 1-D potential barrier
Time independent
Introduce a more real potential: how to represent it?
2 d 2

V z   E
2m dz 2
V(z) = slowly varying potential
d 2 z 
2m
  2 E V z  z 
dz 2

d 2 z 
 x 2 z 
dz 2
x2 
z
Try a solution
d z 
 ix z  z 
dz
ix z 'dz '

0
 z    0e
2m(E V z )
2
particle moving to the right with
continuously varying wave-number (x)
d 2 z 
dx z 
i
 z   x 2 z  z 
dz 2
dz
Electron tunneling across “real” 1-D potential barrier
Time independent
but
d 2 z 
 x 2 z 
dz 2
d 2 z 
dx z 
i
 z   x 2 z  z 
dz 2
dz
This is true only if the first term is negligible, i.e.
dx z 
 x 2
dz
WKB approximation
Wenzel
Kramer
Brillouin
x
 x 1
dx z 
dz
x2 
2m(E V z )
2
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
For E > V(z), x is real and the probability density is constant
 z    0
2
2
Electron tunneling across “real” 1-D potential barrier
Time independent
Suppose the particle encounters
a barrier between 0 < z1 < z2
so E < V(z) and x is imaginary
z
 z    0e 
0
Inside the barrier
Neglect the exp growing part
z
z1
ix z 'dz ' z

0
 z    0e
e 1
 x z ' dz '
z
  1e
z1  x z ' dz '
the probability density inside the barrier is
2
 z    1 e
2
the probability density at z1 is
the probability density at z2 is
1
z
z1 2 x z ' dz '
2
2
2
2  1 e
z2
ix z 'dz '
z1 2 x z ' dz '
Electron tunneling across “real” 1-D potential barrier
Time independent
2
2  1
2
x2 
So the transmission coefficient becomes
T
2
1
2
2
e
z2
z1
2 x z ' dz '
e

2

z2
z1
2m (E V z ')dz '
Tunneling probability very small
The wavenumber is continuosly varying
due to the potential: more real
reasonable approximation for the tunneling probability
if the incident  << z (width of the potential barrier)
2m(E V z )
2
Electron tunneling across 1-D potential barrier
Square barrier
plane wave
Square barrier
electron states
Real barrier
Plane waves
16k 2x 2
2xs
T 2
e
(k  x 2 )2
2 dN
2
jt 
Mrl

dE r
Te
2

z2
z1
2m (E V z ')dz '
Exponential dependence
of the transmission coefficient
current depends on transfer matrix
elements (containing exp. dependence)
and on DOS
True barrier representation if <<z
Varying exponential dependence
of the transmission coefficient
Electron tunneling across 1-D metal electrodes
Planar tunnel junctions
insulator = vacuum
The insulator defines the barrier
Similar free electron like electrodes
At equilibrium there is no net tunneling current
and the Fermi level is aligned
V (z )  EF1   (z )
U=Bias voltage
What is the net current if we apply a bias voltage?
We must consider the Fermi distribution of electrons
Electron tunneling across 1-D metal electrodes
vz = electron speed along z
n(vz)dvz = number of electrons/volume with vz
T(Ez) = transmission coefficient of e- tunneling
through V(z) e- with energy Ez =mvz2/2
f(E) = Fermi Dirac distribution
1
f (E ) 
1e
v max
1
N1  v z n (v z )T(E z )dv z 
m
0

m3
f (E )dv x dv y  2 23

m2
N1 
2 23
E max
 T(E
0


0
z
)T(E z )dE z
0
f (E )dE r
v r2  v x2  v y2
Er 

z
 n (v
m4
n v dv x dv y dv z 
f (E )dv x dv y dv z
4 33
n(vz)dvz = number of electrons/volume with vz
m4
n v z  
4 33
E max
E E F
KT
m
2v r2
)dE z  f (E )dE r
0
Flux from electrode 1 to electrode 2
Electron tunneling across 1-D metal electrodes
N1 
N2 
m
E max
2
 T(E
2 
2 3
m
2
0
E max
2 
2 3
 T(E
0

z
)dE z  f (E )dE r
0

z
)dE z  f (E  eU )dE r
0
Flux from electrode 2 at positive potential U to electrode 1
E max
 m2
N  N1  N2   T(E z )dE z 
3
2



0



0 f (E )dE z  0 f (E  eU )dE r 


Total number of electrons tunneling across junction
em 2
1  2 3
2 
J 


0
f (E )dE r
E max
 T(E )
z
0
1
 2 dE z
em 2
2  2 3
2 


0
f (E  eU )dE r
tunneling current across junction
The current depends on electron distribution
Electron tunneling across 1-D metal electrodes
since
V (z )  EF1   (z )
T(E z )  e

2 2m

s2
s1
EF1  ( z ) E z dz
J 
E max
 T(E )
z
0
T is small when EF-Ez is large
e- close to the Fermi level of the negatively biased electrode
contribute more effectively to the tunneling current
For positive U 2 is negligible so the net current flows from 1 to 2
1
 2 dE z
Electron tunneling across 1-D metal electrodes
Applications of tunnel equation
To perform the integration over the barrier
1

s
define
A

s1
s1
 z dz
2s 2m

By integration it can be shown that
T(Ez )  e
1 
At 0 K

em
E F1  E z
2 3
2 

 eV
em 
  2 3 E F1  E z
2  
 0
hence
J 
em
2 2 3
eU EF1 eU e A
 0

2 
A EF1  E z

em
E F1  E z  eU
2 3
2 

0  E z  E F1  eU
E F1  eU  E z  E F1
E z  E F1
E F1  E z
EF
dE z  
1
E F1 eU
E
F1

 Ez e
A E F1  E z
dE z 


Electron tunneling across 1-D metal electrodes
integrating
J 

e
A

e
4 2s 2

   eU  e A
 eU

Current density flowing from electrode 1 to electrode 2 and vice versa

1
s
A

s1
s1
 z dz
2s 2m

If V = 0 dynamic equilibrium: current density flowing in either direction
J1 
e
A

e
4 2s 2

J2 
e

  eU  e A
2
2
4 s
 eU
For positive U 2 is negligible so
the net current flows from 1 to 2
Electron tunneling across 1-D metal electrodes
Low biases
eU  
a
 b
k

e
A
J 

e
4 2s 2

   eU  e A
 eU

k
b
b


 a 1    a k 1  k 
a
a


k
e A
 eU
e
A 
1
eU

e
 eU
A  1
2

eU
A


e
A
2 




J 




eU
e
e
4 2s 2 



 


 e A  e
A
eU
2 
Electron tunneling across 1-D metal electrodes
eV  
Low biases
Neglect second order contributions in U


e
 1  A eU



J 




eU

4 2s 2 
2 



 A 
e
eU
 eU e
     A
2
2
4 s 
2 

e
 A
 A 
eU


1
e
2
2


4 s
 2

since
A
2
  1
2s 2m
A

J 
e
A
eU
4 2s 2
2
 A
 e





 e A
e 2 2m 
A
J 
U
e
4 2 2 s


At low biases the current
varies linearly with
applied voltage, i.e.
Ohmic behavior
Electron tunneling across 1-D metal electrodes
eU  
High biases

0
2
s 
s0
eU
F 
U
s
Electric field strength
Put into general eq.
J 

e
A

e
4 2s 2

   eU  e A
 eU

evaluating a numerical factor (not included in eq)

4

2.2e F 
J 
 e 2.96eF
2
16 0 

3
For this condition
2
eU  
3
2
2m0

2eU
  1 
0


e


4

2.96eF
3
2
2m0
1
2eU
0
Second term of eq is negligible





Electron tunneling across 1-D metal electrodes
High biases
J 
2.2e F
e
2
16 0
3
2
4

2.96eF
J  U 2e

3
2
2m  0
const
U
EF2 lies below the bottom of CB1
Hence e- cannot tunnel from 2 to 1
there are no levels available
The situation is reversed for e- tunneling from 1 to 2: all available levels are empty
analogous to field emission from a metal electrode: Fowler-Nordheim regime
Electron tunneling across 1-D potential barrier
Square barrier,
plane wave
Square barrier
electron states
Real barrier
Plane waves
16k 2x 2
2xs
T 2
e
(k  x 2 )2
2 dN
2
jt 
Mrl

dE r
Te
2

z2
z1
current depends on transfer matrix
elements (containing exp. dependence)
and on DOS
2m (E V z ')dz '
E max
Real barrier
Metal electrodes
Exponential dependence
of the transmission coefficient
 m2
N  N1  N2   T(E z )dE z 
3
2
0
Varying exponential dependence
of the transmission coefficient
 f (E )dE

0
z



  f (E  eU )dE r 
0

Tunneling is most effective for e- close to Fermi level
Current flows from – to + electrode
Low biases: Ohmic behavior
High biases: Fowler-Nordheim
3-D potential barrier
Square barrier
electron states
Real barrier
Metal electrodes
jt 
2 dN
2
Mrl

dE r
E max
 m2
N  N1  N2   T(E z )dE z 
3
2
0
 f (E )dE



  f (E  eU )dE r 
0


0
z
Join and extend the expression to have the equation for the tunneling current
between a tip and a metal surface
1) Matrix element
Mrl
2
  l*HT  r dz
Consider two many particle states of the sytem
 = state with
e-
0, 
from state  in left to state  in right side of barrier
0
0,  are eigenstates given by the WKB approximation
z
ix z 'dz '
 z    0e 0
Trick: both  are good on one side only and inside
the barrier but not on the other side of the barrier

3-D potential barrier
 is linear combination of one intial state 0 and numerous final states 
  a 0e iE t   b e iE t
0
Put into Schroedinger equation and get a matrix with elements like
M
 2
*
*


H



H



0
0 dz dS

2m


Applyng a step function along z that is 1 only over barrier region
M
 2
*
*

dS













2m


the tunneling matrix element can be evaluated by integrating
a current-like operator over a plane lying in the insulator slab
The tunneling current depends on the electronic states of tip and surface
Problem: calculation of the surface AND tip wavefunctions
3-D potential barrier
Square barrier
electron states
Real barrier
Metal electrodes
jt 
2 dN
2
Mrl

dE r
E max
 m2
N  N1  N2   T(E z )dE z 
3
2
0
 f (E )dE

0
z



  f (E  eU )dE r 
0

Join the expression to have the equation for the tunneling current
between a tip and a metal surface
2) Current density
I 
2e

 f E   f E



 eU M  E  E  
2
f(E) = Fermi function
U = bias voltage applied to the sample
M = tunneling matrix element
Not the many particle states
 = unperturbed electronic states of the surface
 = unperturbed electronic states of the tip
E (E) = energy of the state  () in the absence of tunneling
,  are not eigenfunctions of the same H
3-D potential barrier
I 
2e

 f E   f E



 eU  M  E  E  
2
 f E  f E  eU 
I 
2e

 f E 1  f E



 eU   f E  eU 1  f E   M  E  E  
2
At low T one can consider only one directional tunneling
2e
I 

f E  1  f E  eU  M  E  E  


2
I 
2e

 f E 1  f E



 eU  M  E  E  
2
1
f (E ) 
1e
Low T + small (10 meV) applied bias voltage (U)
f (E )
f (E  eU )  f (E )  eU

E E
For the Fermi function
 f (E )  eU (E  EF )
f E 1  f E  eU  M  E  E  
f E 1  f (E )  eU (E  E ) M  E  E  
f E 1  f (E )M  E  E   f E eU M  (E
f (E )
  (E )
E
2





2



F



2



2




Low T + small (10 meV) applied bias voltage (U), E <EF
2
2e 2
I 
U   M  E  E F  E   E F 




 E F ) E   E F 
f (E )  1
E E F
KT
Tip modeling
Low T + small applied bias voltage (U)
2
2e 2
I 
U   M  E  E F  E   E F 


Point like tip (unphysical)
I     r0   E  EF 
2

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
The image represents a contour map of
the surface DOS at the Fermi level
Tip modeling
Low T + small applied bias voltage (U)
2
2e 2
I 
U   M  E  E F  E   E F 


tip with radius R
s-type only (quantum numbers l  0 neglected) wave functions
with spherical symmetry to calculate the matrix element
I  U  nt EF   e 2xR
  (r0 )  E  EF 
2

nt(EF) = density of states
at the Fermi level for the tip
 r0 , EF     (r0 )  E  EF 
2

Surface local density of states (LDOS) at EF
measured at r0
EF = Fermi energy
r0 = center of curvature of the tip
x = (2m)1/2/ ħ = decay rate
 = effective potential barrier height
Tip modeling
I  U  nt EF   e 2xR
  (r0 )  E  EF 
2

The matrix element is integrated in a point of the barrier region s
So the value of  at r0 is no physically relevant, but it represents the lateral
averaging due to finite tip size
I  e 2xR  r0 , E F 
The exponential dependence comes from the matrix element
2
  (r0 )  e 2x (s R )
I  e 2xs  r0 , E F 
STM is imaging the LDOS at the tip position
Multiplied by the tip DOS
Tip modeling
Sample wavefunctions have
exponential decay in the z direction
so little corrugation at s from surface
Tip center
position
Surface local density
of states (LDOS) at EF
measured at r0
2
  (r0 )  e 2x (s R )
Au lattice
parameter
Calculated LDOS for Au(111)
STM is imaging the LDOS at the tip position
Multiplied by the tip DOS
Low T + small applied bias voltage (U)
STM: atomic resolution
1.0Å
We observe features with a spatial resolution better than 0.1 nm
much lower of the tip curvature radius
Smaller than spherical approximation of the tip wavefunctions (0.8 nm)
Model failing to explain the most important feature
of the STM: atomic resolution
STM: atomic resolution
Why?
Accuracy of perturbation theory:
depends critically on the choice of the unperturbed wave
functions, 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
 2 2

 
  US  E     0
 2me

 2 2

 
  UT  E    0
 2me

•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
1.0Å
STM: atomic resolution
To calculate I, the  of the acting atom is expanded in
terms of a complete set of eigenfunctions.
Two choices:
spherical coordinates
parabolic coordinates
Spherical coordinates are appropriate
for describing atom loosely bonded on the tip
Parabolic coordinates are appropriate
for describing atom tightly bonded to the tip body.
 , , 
x 
y 
z 
 cos 
 sin 
   
2
M
 2

2m
   h h
 *
*
 0        h dd
Calculated on the paraboloid
STM: atomic resolution
M
 2

2m
   h h
 *
*
 0        h dd
Differences to Bardeen expression
the wave functions are the eigenfunctions
of tip and sample unperturbed Hamiltonians
which are different in the gap region.
It is valid only on the paraboloid that is the boundary
of the tip body, not in the entire barrier region
what is needed for calculating the tunneling matrix elements
is the wave functions on the boundary of the tip
STM: atomic resolution
On and outside boundary the tip  satisfies the free electron
Schroedinger equation decaying exponentially

2
    0
2
2  
expand  in term of the parabolic eigenfunctions
with boundary conditions to be regular at r 
    lm il r lm  ,  
2mE
2
unperturbed wave functions
of sample and tip different
in the gap region
l ,m
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
On and inside boundary the sample  satisfies the free electron
Schroedinger equation decaying exponentially
expand  in term of the parabolic eigenfunctions
with boundary conditions to be regular at center of the acting atom

2
  2    0
2  
2mE
2
   lm kl r lm  ,  
l ,m
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
STM: atomic resolution
M
 2

2m
   h h
 *
*
 0        h dd
So M has to be integrated using orthonormal wavefunctions
lm  ,  
Spherical harmonics
il r , kl r 
M
 2

2m

l ,m
lm
Bessel functions
lm
That leads to determine only the coefficients of the tip and
sample expansion on orthonormal wavefunctions.
The coefficients are determined by calculating the derivatives
of the  at the center of the acting atom
    lm il r lm  ,  
l ,m
M gives the correspondence between tip and sample wavefunctions
STM: atomic resolution
M
 2

2m

l ,m
lm
    lm il r lm  ,  
lm
l ,m
For a choosen tip state, M changes and defines
the relation to the coeffiecients of the surface 
Tip states
s
p
d
e r
 
2r
xe r
x 
4r 2
ye r
y 
4r 2

x 2  y 2 e r
 x 2 y 2  
8r 3
xye r
xy 
4r 3
M 
  e r
 
 r0 
 x x 

x
 y y 

y
  x
 2  2
 2
2
x
y
2
y
2
 x
2 xy xy
2
 y 2
r0
r0
 2
xy
r0
r0
The tunneling matrix elements are related to the sample wavefunction derivatives
STM: atomic resolution
The approximation on s state only is wrong
the surface state of a real W tip
extends into vacuum more than s and d states
So the atomic resolution is given by the l  0 wave functions
It is the most protruding electronic states that provides the J
Not only the electron states at the Fermi level
STM: atomic resolution
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
I 
Band structure effects
eU
   eU  E   E T (E , eU )dE
t
s
0
T (E , eU )  e
2( s R )
2m  t s eU


E 
2 
2
  2

The electron energy in a solid depends on the band structure
E  E (k)
k is such that k+G=k
The surface and tip define the direction z
 
E  E k   E kz   E k
This may results in tunneling from surface or bulk states depending on their spatial extension
Also T is changing as a function of E
x 
 
2m t  s eU





E
k

E
k
z

2  2
2
Electrons in states with large parallel wavevector tunnel less effectively
z
Constant current imaging
Unchanged
Tunneling
Unchanged
Current (nA)
z
Tunneling
Current (nA)
Typical working mode
Constant height imaging
Higher
Tunneling
Current (nA)
Applied only on very flat regions
Lower
Tunneling
Current (nA)
Constant current imaging
Imaging: spatial configuration and energy dependence of electron states (LDOS)
need not to correspond in any simple way to the atomic positions
Example: linear lattice
Si and Ge (111) cleaved surfaces
 e
i
x
a
 e
i
x
a
e
i
x
a
e
i
x
a
x
 2 cos   
 a
x
 2i sin  
 a
At the Bragg reflection the potential
gives rise to a forbidden energy region
The band gap
Constant current imaging
Imaging: spatial configuration and energy dependence of electron states (LDOS)
need not to correspond in any simple way to the atomic positions
 

 2
   
2
x
 cos 2   
 a
x
 sin 2   
 a
Charge density ON atomic positions
Charge density BETWEEN atomic positions
In the image always topographic AND electronic features
Finite bias
eU = 0.01 eV
I  U  nt EF   e 2xR
  (r0 )  E  EF 
2

But for eU about 1 eV?
The sum has to be done on many different states
Larger distortion of tip and sample wavefunctions
Approximation
Use undistorted tip and sample wavefunctions also at finite bias
I  U  nt EF   e 2xR
I 
  (r0 )  E  EF 
2

eU
   eU  E   E , r dE
t
s
0
DOStip
0
DOSsample
Integral over all e- states
up to eU from Fermi level
at the tip position
Finite bias
I   t  eU  E  s E , r0 dE
eU
0
But DOS sample decays into vacuum depending on barrier
defined by the tip-sample distance so use WKB approximation
s E , r0   s E e
2( s R )
2m  t s eU


E 
2 
2
  2

 (E , eU )
 s E T
The M now appears as DOS but
the effects of finite biases are
included as modified x
I 
eU
   eU  E   E T (E , eU )dE
t
s
0
Integral over all electronic states
up to eU from Fermi level
Imaging occupied or unoccupied states
Finite bias
I 
eU
   eU  E   E T (E , eU )dE
t
s
0
What does it means imaging occupied or unoccupied electronic states?
Occupied
At constant current means tunneling
from all sample occupied states into all
tip unoccupied states
All is defined by bias voltage
Unoccupied
At constant current means tunneling
from all tip occupied states into all
sample unoccupied states
All is defined by bias voltage
I 
Finite bias
eU
   eU  E   E T (E , eU )dE
t
s
0
Integral over all electronic states from Fermi level up to eU
The information is geometric and electronic and is convoluted
To separate the two one can collect images at different biases
The two states give different TOTAL intensity in the image
Tunneling Spectroscopy
I 
eU
 t  eU  E  s E T (E , eU )dE
0
Integral over all electronic states
up to eU from Fermi level
The current is proportional to the occupied or unoccupied integral DOS
For metals the dI/dU is proportional to DOS at a given energy (low eU)
dI
 eU , eU  
 t 0  s eU T
dU
dT E , eU 
0 t  eU  E  s E  dU dE
eU
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)
Tunneling Spectroscopy
dI
 eU , eU  
 t 0  s eU T
dU
eU
 t  eU  E  s E 
0
dT E , eU 
dE
dU
For e- injection into semiconductor unoccupied state
The e- come mainly from EF so the I is mainly due to sample DOS
For e- injection into tip unoccupied state
The 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
dI
 eU , eU  
 s eU T
dU
eU

0
s E 
dT E , eU 
dE
dU
Tunneling Spectroscopy
background
DOS
s E  dT E , eU 
dI
s eU   
dE
T eU , eU 
dU
dU 
0
eU
I
1
T E , eU 
s E 
dE

U


eU
T eU , eU
Normalization term
eU
0
For semiconductors no low voltage approximation: I needs to be normalized
T (E , eU )  e
2( s R )
2m  t t eU


E 

2
2  2

For U > 0 T(E,eU) < T(eU,eU) and maximum transmission occurs at E = eU
The terms have same order of magnitude
For U < 0 T(E,eU) > T(eU,eU) and maximum transmission occurs at E = 0
The background and denominator terms have same order of magnitude
Larger than sample DOS
Tunneling Spectroscopy
Acquiring STS spectra
Sample and hold technique
Stop the tip on a location
Disable feedback
Scan V and monitor I
Taken at different initial
measuring conditions, i.e.
different tip-sample distances
Si(111)-(2x1)
Tunneling Spectroscopy
Acquiring STS spectra
Measuring at the same time the dI/dV one obtains
the normalized conductance, independent of
Tip-sample distance
-bonded chain
Data show that
the normalized conductance
does not depend on tip-sample distance
Bulk DOS
occupied
empty
Tunneling Spectroscopy
x 
2m 2meU
2


k
2
2
Band structure effects
x 
Measured voltage dependence of x
 1
2m
 2.2 A
2

Minimum value
The data allow to get
 (about 4.2 eV)
and gives x = 22 nm-1
But what about the increase below 1 eV?
Using this with the data one gets
x 
2m 2meU

 kz
2
2
2
k
 1
k  1.1 A
Close to the maximum wavevector at the edge of SBZ
At low bias the current is dominated by states at the edge of SBZ
2
Tunneling Spectroscopy
Obtaining STS images
dI/dV with lock in
Apply modulation
Collect dI/dV while scanning
simultaneously at each point
Current-imaging
tunneling spectroscopy
(CITS)
Feedback on only 30% of the time
Collect dI/dV at fixed separation
Voltage-dependent imaging
Integrate over an energy
interval at state onset
-0.35 V
+0.7 V
-0.7 V
-0.8 V
DOS at the set point
of imaging condition
Emphasize one state
Possible only in stable
tunneling conditions
(not in band gap)
-1.7 V
Need to be done at V following
topography of nuclei
Spatial relationship between
occupied and unoccupied states
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 separation
Drive 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) 165
Surface Science Reports 26 (1996) 61
Scanning Tunneling Microscopy (STM)
Design and instrumentation
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
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.
STM sensitivity to external and internal vibrational sources:
Structural rigidity of the STM itself
Properties of the vibrational isolation system
Nature of the external and internal vibrational sources
z
Scanning Tunneling Microscopy (STM)
Design and instrumentation
Floor vibrations
1-20 Hz Low-frequency floor vibration
(amplitude several m)
~ 8 Hz ventilation
~ 29 Hz motors
~ 60 Hz transformers
Isolation system scheme
Damped with table
For a spring and a single viscous damping system
the vibration amplitude transfer is
2
TS 
spring
viscous
Damping
system

 

1   2

n 

2


2 

 1  2    2
n 

 n



2
 = external excitation frequency
n = 5/L system resonance frequency
L = spring elongation with mass loaded
 = / c damping ratio
 = system damping coefficient
c = 4mn critical damping coefficient
Scanning Tunneling Microscopy (STM)
Design and instrumentation
Single isolation system
2
T 

 

1   2
 n 
2


2 

 1  2    2
n 

 n



2
Damping materials
Viton (most effective against amplitude shock)
 = 0.3 – 0.05
Problem: when strained under compression their
spring constant is large, resulting in
resonance frequency > 10-100 Hz
Metal springs
have smaller spring constants
yielding resonance frequencies as low as
0.5 Hz but they provide little damping
 < 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 springs
Springs + table
Scanning Tunneling Microscopy (STM)
Design and instrumentation
Other solution: a rigidly constructed STM does not require many stages of vibration isolation
Microscope vibration amplitude transfer
Piezo drivers with m up to 100 kHz can be made
but
• joints tightened by screws
• epoxy junctions
• three-point contacts
• walker resonance
• loose spring connectors
often reduce this to 1-5 kHz
TM 


 m

2
 1  2
m

2



2



  
Q '

 m

2
Q’ = (m/2)
tip-sample junction quality factor
System with one stage vibration isolation and structural damping with m >> n the resultant T is
2
TTOTAL 

 

1   2
 n 
2


2 

 1  2    2
n 

 n
Damping system



2
x


 m

2
 1  2
m

2



2



  
Q'

 m

2
Rigid microscope design
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
Scanning Tunneling Microscopy (STM)
Design and instrumentation
Solid line: m = 2 KHz, n = 2 Hz
= 0.4 Q’ =10
Floor vibration amplitude of a
few hundred nm, the gap stability
will be worse than 0.1 nm
dotted line: Very rigid STM
m = 12 KHz, n = 2 Hz
 = 0.4 Q’ =50
the amplitude transfer is worse than
0.1 nm at 200 Hz
dashed line: Very rigid STM +
vibration isolation table
m = 12 KHz, n = 1 Hz
 = 0.4 Q’ =50
the 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' = 10
estimated vibration amplitude is ~ 0.0001 nm in most of the frequency range,
Q’ = (m/2)
Scanning Tunneling Microscopy (STM)
Design and instrumentation
Approach mechanism
Enables the STM tip to be positioned
within tunneling distance of the sample
Coarse motion devices to bring the tip
and the sample into tunneling range
Inchworm stepper motor
Compact dimensions and high m,
Vacuum compatibility
Reliability
High mechanical resolution.
Scanning Tunneling Microscopy (STM)
Design and instrumentation
Operating principle
Three piezoelectric elements
Outer 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 fixed
the 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 right
3 is clamped and 1 is unclamped
2 contracts and the shaft is then moved towards the right
1 is then clamped and element 3 is unclamped
2 is extended to its original length
Similar to those used to climb a rope.
Scanning Tunneling Microscopy (STM)
Design and instrumentation
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 motion
the 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- motion
common mode signals
(Vx+ = Vx-; Vy+ = Vy-)
applied to the electrodes allows extension of the
tube in the z direction
The voltages are referenced to the
constant potential applied to the electrode
located on the inner surface of the tube.
Scanning Tunneling Microscopy (STM)
Design and instrumentation
Piezoelectric equation
Deformation tensor
uij  dij Ek
E field components
Piezo ceramics are made such as
Piezo
tensor
d piezo
E
d piezo
For a cylinder
lenght l0
Thickness h
urr  d Er 
x
l0
0
 0
d
0 d
0 0
0 d
l0
x  d  V
h
0
 0
d31
0 d13
0 0
0 d33
Scanning Tunneling Microscopy (STM)
Design and instrumentation
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
Scanning Tunneling Microscopy (STM)
Design and instrumentation
The resonance frequency of the scanning element is
an important factor in determining the data acuisition
speed data, since it has its own T
For scan < se the scanner responds uniformly to the
drive voltage.
For scan ~ se the amplitude of the scanners motion
may increase dramatically
For scan > se the mechanical response falls off.
se of the scanning element may be as high as 100 kHz
m is usually substantially lower (1-10 kHz)
So scanning speed is limited much below 1 kHz
1 frame: 400 lines
2 lines /s = 0.5 Hz
Total 200 s
Limits: feedback loop gain
Scanning Tunneling Microscopy (STM)
Control electronics
Design and instrumentation
Control tip-surface separation
Drive the scanning elements
Facilitate data acquisition.
I is measured by a preamplifier
with a variable gain of 106-109 V/A
and variable c to limit the bandwidth
below the primary mechanical m
The preamp is located as close to
the tip as possible to minimize noise
The tunneling current is linearized
by a logarithmic amplifier
The tunneling current is then compared to a setpoint, with the difference signal fed into a feedback
amplifier that has an integrating amplifier with
variable 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
K on InAs(110)
1D WIRES
10 nm
1D systems
C60/Ge
STM
13  13R14
C60/Ag(100)
J Chem Phys, 117, 9531 (2002)
STM simulation
7.4 x 7.4 nm2
Obtained after
annealing at 620 °C
C60 - C60 = 1.44 nm
V = + 2.0 V
I = 1.8 nA
C60 Molecular Orbitals
Same orientation: hexagon facing up
C60/Au(111)
PRB 69, 165417 (2004)
STM/STS
Carbon Nanotubes
Nanotubes can be
either metallic or
semiconducting
depending on small
variations in the
chiral winding angle
or diameter
Surface Reconstruction
Si(111)-(7x7) Surface
Sticks-and-Balls Model
STM Image
Pt-Ni Alloy (100) surface
Nanomanipulation
Quantum Corrals
Fe atoms on Cu(111)
Nanomanipulation
Quantum 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.
Standing waves
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".
Observing standing waves on metal surfaces
24.7 x 13.8
LDOS   (r, E ) 
nm2
2
   (r)  (E  E )

Here the  is electronic eigenstate of the surface
and in particular we consider a 2-D electron gas
with functions similar to free-particle states
The waves are scattered at the step edges
and we observe the interference pattern
of the incident and scattered wave
 (r )  e i k  r
r = reflection amplitude
ei = phase shift
T ~ 4 K
Cu(110)
Pentacene on Cu
5 nm
2 nm
TiOx cluster
On HOPG
Lattice distortion
Or charge transfer effect?
Cu on Cu(111)
Instability
And
Diffusion
coefficient
150 x 150 nm2
Cu on Cu(111)
200 x 200 nm2
Diffusion coefficients
Vacancies are more mobile on the TiO2(110) surfaces after O2 exposure
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.
(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
Atomic Force Microscopy (AFM)
• Operating principle
• Cantilever response modes
• Short theory of forces
• Force-distance curves
• Operating modes
• contact
• tapping –non contact
• AM-AFM
• FM-AFM
• Examples
AFM basics
Basic idea:
Surface-tip interaction
Response of the cantilever
Contact Mode
Tapping Mode
Non-Contact Mode
AFM basics
The AFM working principle
Measurement of the tip-sample interaction force
Probes: 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–surface
force.
How to measure the deflection
4 quadrant
photodiode
AFM basics
Two force components:
FZ normal to the sample surface
FL In plane, cantilever torsion
I01, I02, I03, I04,
reference values of
the photocurrent
I1, I2, I3, I4,
values after change of
cantilever position
Differential currents
ΔIi = Ii - I0i
will characterize the
value 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 loop
keeping ΔIZ = constant in order to make the bending ΔZ = ΔZ0 preset by the operator.
Cantilever response
The tip is “in contact” with the surface
Interaction forces cause the cantilever to bend while scanning
l = cantilever lenght
w = cantilever width
t = cantilever height
ltip = tip height
The deflection vector  is linearly dependent on applied force according to Hooke’s law
 x

 y
 z

 cxx
 
   cyx
 c
  zx
cxy
cyy
czy
cxz
cyz
czz
 Fx

 Fy

 Fz





Cantilever response
Vertical force Fz applied at
the end induces
the cantilever bending
x  cxz Fz
y  cyz Fz
z  czz Fz
z = cantilever deflection along y
z = cantilever deflection along z
 = deflection angle
  tg 
z
y
cantilever deflection angle
around setpoint
Cantilever response
Assume a bending with radius R
the longitudinal extension L is proportional
to the distance z from the neutral plane
L z

L
R
Neutral axis
dF
dF  Y
Section
Hooke’s law
L
dF
Y 
L
dS
dS
dS
 Yz
L
R
L
Y = Young modulus
Resulting force acting on dS
At any section S there is a torque wrt neutral axis
Mz   zdF  Yz 2
S
S
dS Y

R
R
2
 z dS 
S
Y
Iz
R
Iz= momentum of inertia
wrt neutral axis
u(y) = deflection along z of a cantilever point at the distance y from the fixed end
For small angles
but
1
d 2u

R
dy 2
M  Fz L  y 
For any point along the cantilever y direction
d 2u
M 
YI z
dy 2
d 2u
YI z
 Fz L  y 
2
dy
t 3w
Iz 
12
Cantilever response
d u
YI z
 Fz L  y 
2
dy
2
Fz
d 2u
L  y 

2
dy
YI z
y2 
Fz 
du
 Ly 


integration
dy YI z 
2 
Fz L3
z  u y L 
3YI z
L3
12L3
4L3
z 
Fz 
Fz 
Fz
3
3
3YI z
3Yt w
Yt w
du
  tg 
dy
y L
z 
integration
 Ly 2 y 3 



6 
 2
F
u  z
YI z
t 3w
Iz 
12
L2Fz
L2 z 3YI z
3



z
2YI z
2YI z
L3
2L
2
L
3
Ywt 3
kc 
4L3
Fz  kc z
The deflection is proportional to measured signal
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
Interatomic force constants in solids: 10  100 N/m
In biological samples ~ 0.1 N/m.
Typical values for k in the static mode are 0.01–5 N/m.
Soft cantilever
Cantilever response
z 
czz
x  cxz Fz
4L
Fz
3
Yt w
3
y  cyz Fz
z  czz Fz
4L3
L3


c
3
Yt w
3YI z
coefficient of inverse stiffness
The magnitude characterizes the cantilever stiffness
It is the largest among the tensor cij
y  Ltip 
2
z  L
3
cyz 
3Ltip
2L
Ltip L3
Yt w
czz
3
Fz
y  Ltip 
cxz  0
Ltip << L so cyz can be neglected
Force spectroscopy at fixed location
3Ltip
2L
z
x  0
y 
3Ltip
2L
z  cFz
cFz
L3
c 
3YI z
Cantilever response
Longitudinal force Fy applied at
the end induces the cantilever bending
x  cxy Fy
z = cantilever deflection along y
z = cantilever deflection along z
 = deflection angle
y  cyy Fy
  tg 
z  czy Fy
Longitudinal force Fy applied at the end results in a torque
d 2u
M 
YI z
dy 2
d 2u
YI
 Fz Ltip
2
dy
M  Fy Ltip
cantilever deflection
angle around setpoint
Similaly to previous case
Fz Ltip
d 2u

2
dy
YI z
u 
Fz Ltip
2YI z
z
y
y2
Cantilever response
Longitudinal force Fy applied at
the end induces the cantilever bending
u 
Fz Ltip
2YI z
y
z  u
czy 
z  u
2
y L
Ltip L2
2YI z


3Ltip
2L
3Ltip
2L
y L

Fz Ltip
2YI z
L3
c 
3YI z
L
2
cFz
c
 
du
dy

y L
Ltip L
YI z
Fy 
3Ltip
L
2
cFy 
2z
L
The deflection is proportional to measured signal
the axial force results in the tip deflection in vertical direction
Cantilever response
Longitudinal force Fy applied at
the end induces the cantilever bending
y  Ltip
2z
 
L
cyy 
2Ltip
L
czy 
y 
3L
2
tip
2
L
2Ltip
L
z
x  0
y 
c
z 
Very small compared to c
2
3Ltip
L2
3Ltip
2L
cFy
cFy
the axial force results in the tip deflection not only in the vertical but also in longitudinal direction
All these deflections are small compared to the main bending in the z axis
Cantilever response
simple bending
Transverse force Fx
x  cxx Fx
y  cyx Fx
z  czx Fx
The simple bending is similar to the vertical bending of z-type
Exchange the beam width (w) with thickness (t)
cbend
4L3
t2

 2c
3
Yw t
w
twisting
Cantilever response
Twisting
The torsion is directly related to beam deflecton angle 
Gwt 3
M 
3L
G= Shear modulus ~ 3Y/8
M  Fx Ltip
The torque by Fx is
The lateral deflection is
ctors
xtors   Ltip
ctors Fx  xtors
2
2
2
2
Ltip
3Ltip
L
8Ltip
L 2Ltip
xtors




 2 c
Fx
M
Gwt 3 Ywt 3
L
cbend
cxx
4L3
t2

 2c
Yw 3t
w
2
t2
2Ltip
 2  2
w
L


c


x  xbend  xtors  cbend  ctors Fx  cxx Fx
Cantilever response
x  xbend  xtors  cbend  ctors Fx  cxx Fx
Simple
bending
cxx
2
t2
2Ltip
 2  2
w
L

2
t2
2Ltip
x   2  2
w
L

y  0
The deflections in y and z are of the second
order with respect to x deflection

cFx



c


twisting
z  0
L = 90 m
Ltip = 10 m
w = 35 m
t = 1 m
c  1.92mN 1
3Ltip
1
cyz 
c  c  0.32mN
2L
6
Dominant distortions czz, cyz, czy
3Ltip
1
czy 
c  c  0.32mN 1
2L
6
2
3Ltip
1
cyy  2 c 
c  0.071mN 1
L
27
ctors 
cbend
2
2Ltip
L2
c 
1
c  0.05mN 1
40
t2
1
 2c 
c  0.0016mN
w
1220
1
Lateral distortions are much smaller
1
Cantilever effective mass and eigenfrequency
Cantilever is vibrating along z
Fixed end
y
u(y)
l = cantilever lenght
w = cantilever width
t = cantilever height
ltip = tip height
dy
Ywt 3
kc 
4L3
u(t,y) = deflection along z of a cantilever point at the distance y from the fixed end
dE k  ut , y 2
Kinetic energy
u
y L
u 
Fz L3

3YI z
Fz
YI z
 Ly
y 



2
6


2
3
mdy
2L
u t , y   u
y L
3  Ly 2 y 3  u y L

 

3 
L  2
6 
2L3
 3y 2 y 3 
 2  3 
L 
 L
Cantilever effective mass and eigenfrequency
u
 3y 2 y 3 
 2  3 
u t , y  
2L  L
L 
y L
3
Kinetic energy

L
0
dE k 
Potential energy
EP 
ET
Equation of motion
dE k  ut , y 2
L
2
0 ut , y 
u t ,L 

0
mdy
2L
m
33 m
ut , L 2
dy 
2L
140 2
Fdu 
u t ,L 

0
u
u 2 t , L 
du 
c
2c
33 m
u 2 t , L 
2

u t , L  

140 2
2c
33m
u t , L 
ut , L  
0
140
c
m * u 
0 
u
0
c
1
1.029t

cm *
L2
Y

m* 
33m
140
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 interaction
Origin of forces
Tip-surface
Separation (nm)
1000
Non contact
Intermittent
contact
100
10
Electric, magnetic,
capillary forces
Van der Waals
(Keesom,Debye,London)
1
Contact
0
Interatomic forces (adhesion)
Origin of forces
Cgs/esu
Born repulsive interatomic forces
Origin: large overlap of wavefunction of
ion cores of different molecules
Pauli and ionic repulsion
-
-
+
+
1  40
C2
UR  12
r
Origin of forces
Elastic forces in contact
Origin: object deformation when in contact
Assumptions
Isotropic cantilever and sample  two parameters to describe elastic properties
Y = 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
deformation
and penetration
F 
3
2
h 2Y R
31   2 
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 nm
Penetration depth : up to 20 nm
contact pressure : up to 10 GPa.
the contact pressure is higher for stiffer samples
Origin of forces
Cgs/esu
Keesom Dipole forces
Coulomb force
between point charges
1  40
Coulomb potential energy
Electric field
q1q2
U    fC dr 
r
qq
fC  1 22
r
E 
q
r2
 = qd = dipole moment
Origin: fluctuation (~10-15 s) of the electronic clouds around a molecule
Dipole formation
d
q-

q+
For r >> d
Potential energy of the dipole moment in an electric field E
Field intensity produced by the dipole
E 

r3
3 cos 2   1
 is the angle between dipole and r
U  E 
q
r2
Origin of forces
Keesom Dipole forces
When two atoms or molecules interacts
1
2
d
d
q-
q+
q-
q+
Potential energy of the interacting dipole moments
U 
1 2
r
3
2 cos 1 cos 2  sin 1 sin 2 cos  
Maximum attraction for 1= 2 = 0°
Maximum repulsion for 1= 2 = 90°
Umax
2 2
 3
r
Origin of forces
Umax
2 2
 3
r
Keesom Dipole forces
In a gas thermal vibrations randomly rotates dipoles
while interaction potential energy aligns dipoles
Total orientation potential is obtained by statistically
averaging over all possible orientations of molecules pair
For U << KBT
UAV 
e

U
kBT
U
1
kBT
U2
Ud   kBT d
Ud  0
 d  Ud
UAV
2 4 1

3kBT r 6
Orientational interaction
UAV
Ue


e


U
kBT
U
kBT
d
d
Origin of forces
Debye Dipole forces
Origin: fluctuation of the electronic clouds around a molecule
dipole formation, interaction of the dipole
with a polarizable atom or molecule

d
q-
ind  E
Induced dipole moment
q+
E
E 2
0
2
U   ind dE  
Potential energy of the interacting dipole moments
The induced dipole is “istantaneous” on time scale of molecular motion
So one can average on all orientations
For r >> d
E 

r3
3 cos 2   1
2
U  ind
1
r6
Induction interaction
Origin of forces
London Dipole forces
Origin: fluctuation of the electronic clouds around the nucleus
dipole formation with the positively charged nucleus
interaction of the dipole with a polarizable atom
dipole
2
-
-
+
+
Field induced by atom 2
E
Polarizable
atom
U   ind dE  
0
1
E 
2 2
r3
Potential energy of atom 1
in the field due to dipole 2
2 

i
i
RMS dipole moment for
fluctuating electron-nucleus
The dipole formation of atom 2 is given by the polarizability 
2

2  2
h i
Ionization energy
E 
1E 2
22 2 h i 2

3
r
r3
3h i 12 1
U 
4
r6
2
Origin of forces
Origin
Fluctuation of the electronic
clouds around a molecule.
dipole formation
Fluctuation of the electronic
clouds around a molecule.
dipole formation.
interaction of the dipole with
a polarizable atom or molecule
Fluctuation of the electronic
clouds around the nucleus.
dipole formation with the positive
charge of nucleus.
interaction of the dipole with
a polarizable atom
Large overlap of core wavefunction
of different molecules
Potential energy
2 4
U 
3kTr 6
U 
U 
2
ind
r6
3 h i 12
4 r6
C2
UR  12
r
Keesom
Debye
London
Born
Origin of forces
van der Waals dipole forces between two molecules
2
 2 4
 1
2 4
3 h i 12 ind
3
2
U 

 6  
 ind  h i 12  6
6
6
3kTr
4 r
r
4
 3kT
r
C1
U  6
r
Total potentials between two molecules
C 2 C1
U  12  6
r
r
Lennard-Jones potential
Origin of forces
van der Waals dipole forces between macroscopic objects
C1
U  6
r
f (r )  U
Additivity: the total interaction can be obtained by summation
of individual contributions.
Continuous medium: the summation can be replaced by an integration over
the 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.
The total interaction potential between two arbitrarily shaped bodies
U (r )  1 2 
 f (r )dV dV
v1 v 2
1
U (r )   C 1 1 2   6 dV1dV2
v1 v 2 r
2
1
2
H   2C 1 1 2
Hamaker constant
Origin of forces
The force must be calculated for each shape
For a pyramidal tip at distance D from surface
2H tan 2 
F (D )  
3D
H   2C 1 1 2
Hamaker constant
Same role as the polarizability
Depends on material and shape
Origin of forces
The force must be calculated for each shape
Conical probe
F 
 2C1 1 2 tan 2 
6h
 1.3x 10 15N
Tip radius r << h
Pyramidal probe
2 2C 1 1 2 tan 2 
F 
3h
 5.2x 10 15N
Tip radius r << h
Conical probe
rounded tip
F  1.1x 10 13N
 2C 1 1 2R
F 
6h 2
 3.3x 10 9 N
For r >> h
Origin of forces
Adhesion forces
Middle range where attraction
forces (-1/r6) and repulsive forces (1/r12) act
adhesion
It 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
Cantilever in contact with a liquid film on a flat surface
The 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
F  10 9 N
Consequence: hysteresis in approach/retraction
Similar to VdW force
Force-distance curves
How to obtain info on the sample-tip interactions?
Force-distance curves
The sample is ramped in Z
and deflection c is measured
Force-distance curves
Force-distance curves
The deflection of the cantilever is
obtained by the optical lever technique
When the cantilever bends the reflected light-beam
moves by an angle
du
  tg 
dy
y L
c
L2Fz
L2 z 3YI z
3



z
3
2YI z
2YI z
L
2L
 
3
z
2L
d = detector - cantilever distance
laser spot movement
or z
PSD = position sensitive detector
PSD  2d tan   2d
z 
 PSD L
3d
High sensitivity in z is obtained by L << d
Vertical resolution depends on
the noise and speed of PSD
10 1 3
m
t
T = 0.1 ms
z ~ 0.01 nm
Force-distance curves
Measured quantities: Z piezo displacement, PSD i.e. I or V
Must be converted to D and F
The sample is ramped in Z and deflection c is measured
D = Z –(c + s)
D = tip-sample distance
c = cantilever deflection
s = sample deformation
Z = piezo displacement
Force – displacement
curve
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 = -kcc
Force-distance curves
Measured quantities: Z piezo displacement, PSD i.e. I or V
Must be converted to D and F
D = Z –(c + s)
a) Infinitely hard material (s=0), no surface forces
Linear
regime
PSD-Z curve: two linear parts
zero force line
defines zero deflection
of the cantilever
Z = 0 at the
intersection point
F-D curve
sensitivity IPSD/ Z
F(Z) = kc
D = tip-sample distance
c = cantilever deflection
s = sample deformation
Z = piezo displacement
Conversion between PSD and Z
c = IPSD/(IPSD/ Z)
F(D) = k IPSD(Z)/(IPSD/Z)
D=Z-c
Z > 0 if surface is
retracted from tip
In non-contact D = Z (c = 0 so F(D)=0)
In contact Z = c and D = 0 so F(D)=kc
Force-distance curves
Measured quantities: Z piezo displacement, PSD i.e. I or V
D = Z –(c + s)
b) Infinitely hard material (s=0)
long-range exponential repulsive force
Linear
regime
PSD-Z curve
sensitivity IPSD/ Z
from the linear part
c = IPSD/(IPSD/ Z)
D = tip-sample distance
c = cantilever deflection
s = sample deformation
Z = piezo displacement
zero force line =
0 deflection at large distance
Z = 0 at the intersection point (extrapolated)
F-D curve
Accuracy: force curves from a large distance
Apply a relatively hard force to get to linear regime
The degree of extrapolation determines
the error in zero distance.
F(Z) = kc
D=Z-c
F(D) = k IPSD(Z)/(IPSD/Z)
In contact Z = c and D = 0
Z > 0 if surface is
retracted from tip
In non-contact D = Z - c
Force-distance curves
c) Deformable materials without surface force
s
D = Z –(c + s)
Z > 0 if surface is
retracted from tip
PSD-Z curve
In non-contact D = Z (c = 0)
F=0 line
D = tip-sample distance
c = cantilever deflection
s = sample deformation
Z = piezo displacement
If tip and/or sample deform
the contact part of PSD-Z curve
is not linear anymore
F-D curve
Hertz model: elastic tip radius R
planar sample of the same material (Y)
F 
3
2
 s 2Y R
31   2 
s = indentation
For many inorganic solids s << c
For high loads c~F/kc
s
If s ~ c
the force curves have to be modeled to describe indentation =
‘‘soft’’ samples: cells, bubbles, drops, or microcapsules.
If s  0 ‘‘zero distance’’ (Z=0) must be defined
In contact the distance equals an interatomic distance
But: indentation and contact area are still changing with the load
It is more appropriate to use indentation rather than distance after contact
the abscissa would show two parameters: D before contact and s in contact
sensitivity IPSD/ Z
from the linear part
Force-distance curves
c) Deformable materials with surface force
Tip approaching a solid surface
attracted by van der Waals forces
- very soft materials
surface forces are a problem
leading to a significant
deformation even before contact
At some distance the gradient
of the attraction exceeds kc
and the tip jumps onto the surface.
- relatively hard materials
Due to attractive and adhesion forces
it is practically difficult to precisely
determine where contact is established
Adhesion forces add to the
spring force and
can cause an indentation
s
s
In this case it is practically impossible
to determine zero distance and
one can only assume that
the indentation caused by
adhesion is negligible.
Force-distance curves
Because we measure Z = the sample and the cantilever rest position separation
D = Z –(c + s)
Tip-sample force Fc = -kcc
Force – displacement curve
Elastic force of the cantilever for different c
At each distance the cantilever deflects until Fc=F(D)
so that the system is in equilibrium
The equilibrium points are a, b, c
The corresponding distances are not D but Z
i.e. the sample and the cantilever rest position separation
that are given by the intersections between lines and the
horizontal axis (,,)
Tip-sample
interaction
Lennard-Jones force, F(D)= -A/D7 + B/D13
Force-distance curves
D = Z –(c + s)
Total potential of cantilever-sample system
Utot = Ucs(D) + Uc(c) + Us(s)
assume
Ucs(D) = tip - sample interaction potential
Uc(c) = cantilever elastic potential
Us(s) = sample deformation potential
ks  s2
Us  s  
2
kc  c2
Uc  c  
2
C
Ucs D    n
D
The relation between Z and c is obtained by forcing the system to be stationary
Utot
Utot

0
 s
 c
And since
Ucs
Ucs

 s
D
kc c  ks  s
kc  c 
Z
C
 s  c 
n
The measured force- displacement curve can be converted into the force-distance curve
Force-distance curves
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 stable
One (between c and b) is unstable
During approach the tip follows the trajectory from c’ to b
and then "jumps" from b to b‘
During retraction, the tip follows the trajectory from b' to c
and then jumps from c to c’
Force-distance curves
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 first
increases with kc and then, for high kc, disappears.
The jump-off-contact always decreases, so that the
total 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 curves
one should employ stiff cantilevers
Stiff cl have a reduced force resolution
Therefore it is necessary to reach a compromise
Operation modes
Contact Mode
Tapping Mode
Non-Contact Mode
Static cantilever
The cantilever is
forced to oscillate
Tapping: Amplitude modulation (AM-AFM)
Non-contact: Frequency modulation (FM-AFM)
AM-AFM: a stiff cantilever is excited at free resonance frequency
The 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 frequency
The resonance frequency depends on the tp-sample forces
Contrast: the spatial dependence of the frequency shift, i.e. the difference between
the actual resonance frequency and that of the free cantilever
Image = profile of constant frequency shift.
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
Contact mode
The tip is brought “to contact” with the surface
until a preset deflection is obtained.
Then the raster is performed keeping deflection
constant.
Equiforce surfaces are measured
Info on lateral dragging forces can be obtained
Drawbacks:
The download force of the tip may damage the sample
(expecially polymers and biological samples)
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
Operation modes: AM
Cantilever = spring with k and pointless mass m
l0 = spring at rest
l = spring extension + mass = m*
k = cantilever spring constant
m = cantilever mass
m* 
33m
140
Fel  k l  l 0 
l defines z = 0
z  02z  0
0 
k

m*
1
1.029t

cm *
L2
z t   Z cos 0t  0 
Harmonic oscillator
Y

Z 
z
2
0

v 02
02
 v0
 z 00
 0   arctan 




Operation modes: AM
damped harmonic oscillator
Frictional force
z  2z  02z  0
 
Three solutions
z t   e t Ae t  Be t 
Aperiodic motion
2m *
  0
  0
  0
   2  02

Ffr   v
z t   Ze t cos t  0 
Z 
z
2
0
v  z 0 
 0




z t   Ze t A  Bt 
2
 v0
 
 
 z 0  
0   arctan 
Critical damping
Under damping
Stored energy
E (t )
E (t )
Quality
Q

2


2

factor
ET
E (t )  E (t T )
For small damping
Q 
0  




 0
T
2
2
Energy loss /period
E (t )  E 0e 2t
Q characterizes the rate of the energy transformation
Q is the number of a system oscillations over its characteristic damping time 1/
Operation modes: AM
F t   F0 cost 
Forced harmonic oscillator
z  02z  A0 cos t 
For 0  
Z0 
A0
  2
C1  x 0 
Driving
oscillation
Z 
2
0
F0
m *   
2
0

 0   arctan  

C1
C2




2

F0
m*
For 0 = 
z t   Z cos 0t  0   Z 0 cos t
free
oscillation
A0 
C 12  C 22
C2 
v0
0
z t   Z cos 0t   0  
resonance
A0t
0
sin 0t
Operation modes: AM
F t   F0 cost 
Forced damped harmonic oscillator
z  2z  02z  A0 cos t 
 
z t   zdho t   Z 0 cost   
Z0 

2
0


2m *
A0 
zdho(t) = solutions for damped harmonic oscillator
 4 22
Amplitude
Phase
 2 

  arctan  2
2 



 0

Q=16
Z0
A0
F0
m *
For t > 1/ only forced oscillations will be present
A0
2 2

Excitation
Q=8
Q=4
Q=8
resonant amplification
factor of the oscillator
Q 
/0
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
0
2
Q=1
Q=16
/0
in phase (~0) for  < 0
in phase quadrature (=/2) at 0
in antiphase (=) for  > 0
Operation modes: AM
Z0
A0
Amplitude
Phase
/0
/0
For light damping Z0 becomes more peaked at 0
Lorentzian
A0
Z0 
022
Resonant width
2
2 2
0  

2
Q
0


 
Q
The more is Q, the less is the resonance peak width.
R 
02  2 2  0 1 
1
2Q 2
Resonant frequency for damped forced h.o.
The cantilever oscillation amplitude depends on
- Driving amplitude A0
- Value of driving frequency  with respect to 0


0 

  arctan 
2
2 


Q



0


Operation modes: AM
F t   F0 cost  Excitation
Forced damped harmonic oscillator + External force
z  2z  02z  A0 cos t   F z  / m *
 
• F(z) does not depend on time
• Qualitative behavior is the same as before
• Change of the oscillator equilibrium position

A0 
2m *
z0 is given by
For small oscillations expand F(z) around equilibrium position z0
F z   F
z0
dF z 

z t   ...
dz z 0
 2
1 dF z 



z   2z    0 

m * dz z 0

F0
m *
z t   z t   z 0
 z0 
2
0
F
z0
m*
F z

0
z    2z 
 A0 cos t 
0 0

m*

z  2z   2z   A0 cos t 
 2  02 
1 dF z 
m * dz z 0
k  k 
dF z 
dz z 0
effective spring constant
 
k
m
effective resonance frequency
the resonance frequency of a weakly perturbed ho depends on the gradient of the interaction.

2
0
 2  
2
2
0
 
A0
Z0 
 
2
R 
Operation modes: AM
A0
Z0 
   2
2
Q
2
 2  
2
2

R 
2
2
0
 
 2  02 
2
Q2
1 dF z 
 
 2 2 
m * dz z 0
2
0
Resonant frequency of
damped forced h.o. + force
02  2 2  0 1 
R 

2
R


2
R
1 dF z 


m * dz z 0

2
R
1
2Q 2
1 dF z 
m * dz z 0

02 dF z 
k
dz
02 dF z 
k
dz
z0
The force gradient gives a shift of resonant frequency
  R  R 
2R 
02 dF z 
k
dz
z0
 R 


02 dF z 

R
1 2
 1


R k dz
z0


 
0 dF z 
2k
dz
z0
The change of the resonant frequency can be used to measure the force gradient
z0

0 


  arctan 
2
2 



Q





1 dF z 
   
m * dz z 0
2
2
0
Operation modes: AM
Phase shift
The force gradient gives
a frequency shift
resulting also in a
phase shift wrt
curve at 0
The force gradient gives
a frequency shift
resulting also in an
amplitude change wrt
curve at 0
The change of the phase shift and amplitude can be used to measure the force gradient
Operation modes: AM
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
AFM gives 3D images of the sample surface
two different (not always independent) resolutions should be distinguished:lateral and vertical
Vertical resolution
limited by
noise from the detection system
10 1 3
m
t
T = 0.1 ms
z ~ 0.01 nm
thermal fluctuations of the cantilever
z 
4KBT
0.074

nm
3k
k
K = 40 nm, T = 295 K
z ~ 0.01 nm
Operation modes: AM
Lateral resolution
Analogy with optical microscopy
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
d 
2R

z 
z  h

Calculated values
Atomic resolution with radius of 0.2 0.5 nm, i.e. almost a single atom
Operation modes: AM
Lateral resolution
Convolution effects: width larger than real values
DNA chain
Sample deformations due to high loads results
in 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
s 
B
Q
B = bandwidth
Increasing Q might be an option to obtain higher s
10 x 10 nm2 image
256 x 256 pixel
Scanning speed 2 lines/s (20 nm/s)
B= 2 x 256 = 512 Hz
But increasing Q reduces B, i.e. the response time of the system
Q 
0
2
 
1


2Q
0
/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
=2s
Too long acquisition time
Operation modes: FM
Frequency modulation
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
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
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
B does not depend on Q
s 
B
Q
FM and AM modes have essentially the same sensitivity
if the same set of parameters are used
In FM the sensitivity can be increased by using a very high Q
The 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 demodulator
The original demodulator measured a frequency shift of 0.01 Hz at 50 kHz with B=75 Hz
Now: 5 mHz with 500 Hz bandwidth
Operation modes: FM
Cantilever dynamics = self-driven oscillator
different 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
z  2z  02z  Fts z  / m *  Fexc t 
- 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 PSD
modifies 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
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
z  2z  02z  F z  / m *  Fexc t 
m * z  kz  Fts z 0  z   0
To be solved numerically
For small oscillations the tip-sample interaction can be developed linearly around z0
mz  k  kts z  0
 
0 dF z 
2k
dz
z0
 z 0 
kts  
0 dF z 
2k
dz
z0
Frquency shift vs force gradient
dFts z 
dz
z0
Operation modes: FM
But how comes atomic resolution?
For large oscillations
m * z  kz  Fts z 0  z   0
For typical oscillation amplitudes A0 = 20 nm and a stiff cantilever k = 30 N/m
Restoring force at
turning point close
to the surface
elastic energy
stored in cantilever
kA02

2
3.75 x104 eV
tip-surface
interaction energy
kA0  Fts
d z c A0
1- 10 eV
600 nN
1-10 nN
tip-surface
interaction force
Cantilever is weakly perturbed harmonic oscillator
Relation between the frequency shift and the average of Fts in a full harmonic cycle:
d , k , 0 , A0  
1 0
4 2 kA0

2
0
Fts d  A0  A0 cos  cos d
depends on the operation conditions and the forces at distance of closest approach d
Operation modes: FM
• long-range (LR) electrostatic
• vdW attractive
• repulsive contact Hertzian
Appropriate to explain main features of AFM
but cannot provide an explanation for the atomic resolution
Example: LR vdW interaction
Not dominated by the interaction of the tip atoms closer to surface
Depends on the macroscopic shape of the tip
How 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.
Origin of atomic resolution
Operation modes: FM
Semiconductor surfaces
candidate force: covalent bonding interactions
In semiconductor surfaces and in the tip there are
undercoordinated atoms with unsaturated bonds
They can contribute to the total tip–surface interaction
and provide atomic resolution.
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
Different from vdW interaction: related to the overlap of the atomic wavefunctions
Insulators (alkali halides and oxides)
and oxidised tips
All the dangling bonds are saturated
Confined microscopic electric field around the oxygen tip apex
provides the key to the lateral variation of the interaction
It 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 the
strongly localised electric field are responsible for the atomic resolution in these materials
Operation modes: FM
Example: Si(111)-(7x7)
This defines the zero
of tip-sample d
Type 1
FM AFM in UHV
“Force”-distance curves
But amplitude can decrease even for attractive forces
So A’ might not be the true contact distance
Frequency is changing well before amplitude
Type 2: after accidental
tip modification
The discontinuity in the  is interpreted as
formation of chemical bond with the surface
Operation modes: FM
Example: Si(111)-(7x7)
Type 1
Cantilever vibration amplitude: 16 nm
 = 1.1 Hz
FM AFM in UHV
Type 2: after accidental
tip modification
Cantilever vibration amplitude: 14.8 nm
 = 13 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
Operation modes: FM
Example: 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 kHz
Tip apex radius 5–10 nm; Q ~ 38 000 in UHV