short description theory of 1-d tunneling actual 3d barriers tip modeling atomic resolution hardware...
TRANSCRIPT
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
Electron tunneling across 1-D potential barrier
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
Electron tunneling across 1-D potential barrier
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
Electron tunneling across 1-D potential barrier
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
Electron tunneling across 1-D potential barrier
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
Electron tunneling across 1-D potential barrier
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
Electron tunneling across 1-D potential barrier
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
Electron tunneling across 1-D potential barrier
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
Electron tunneling across 1-D potential barrier
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
Electron tunneling across 1-D potential barrier
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)
Electron tunneling across 1-D 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)(
Electron tunneling across 1-D metal electrodes
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
Electron tunneling across 1-D metal electrodes
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
Electron tunneling across 1-D metal electrodes
)()(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
Electron tunneling across 1-D metal electrodes
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
Electron tunneling across 1-D metal electrodes
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
Electron tunneling across 1-D metal electrodes
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
Electron tunneling across 1-D metal electrodes
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
Electron tunneling across 1-D potential barrier
Square barrier,plane 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 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
Square barrierelectron states
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
3-D potential barrier
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
3-D potential barrier
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
Square barrierelectron states
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
3-D potential barrier
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
STM: atomic resolution
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
STM: atomic resolution
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
STM: atomic resolution
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
STM: atomic resolution
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
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
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
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
Charge density ON atomic positions
axax
22
22
sin
cos
Charge density BETWEEN atomic positions
In the image always topographic AND electronic features
Constant current imaging
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)
Tunneling Spectroscopy
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
Tunneling Spectroscopy
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
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
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.
Scanning Tunneling Microscopy (STM)Design and instrumentation
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
Scanning Tunneling Microscopy (STM)Design and instrumentation
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
Scanning Tunneling Microscopy (STM)Design and instrumentation
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
Scanning Tunneling Microscopy (STM)Design and instrumentation
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)
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 tipand the sample into tunneling range
Inchworm stepper motor
Compact dimensions and high m,Vacuum compatibilityReliabilityHigh mechanical resolution.
Scanning Tunneling Microscopy (STM)Design and instrumentation
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.
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 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.
Scanning Tunneling Microscopy (STM)Design and instrumentation
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
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 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
Longitudinal force Fy applied at the end induces the cantilever bending
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
Longitudinal force Fy applied at the end induces the cantilever bending
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
Keesom Dipole forces
3
2
max
2r
U
In a gas thermal vibrations randomly rotates dipoleswhile interaction potential energy aligns dipoles
Keesom Dipole forces
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
Force-distance curves
Force-distance curves
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
Force-distance curves
Must be converted to D and F
D = tip-sample distancec = cantilever deflections = sample deformationZ = piezo displacement
Measured quantities: Z piezo displacement, PSD i.e. I or V
D = tip-sample distancec = cantilever deflections = sample deformationZ = piezo displacement
D = Z –(c + s)
Force-distance curves
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
Measured quantities: Z piezo displacement, PSD i.e. I or V
D = tip-sample distancec = cantilever deflections = sample deformationZ = piezo displacement
D = Z –(c + s)
Force-distance curves
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)
Z > 0 if surface isretracted from tip
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 = tip-sample distancec = cantilever deflections = sample deformationZ = piezo displacement
D = Z –(c + s)Force-distance curves
c) Deformable materials without surface force
PSD-Z curve
F=0 line
F-D curve
Z > 0 if surface isretracted from tip
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
Force-distance curves
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
Force-distance curves
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 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’
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 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