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

Atomic Force Microscopy (AFM)

Bibliography

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

• NT-MDT slides

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:

F_Z normal to the sample surface F_L 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= I_i - I_0i

will characterize the

value and the direction of the cantilever bending or torsion.

ΔI_Z = (ΔI₁ + ΔI₂) − (ΔI₃ + ΔI₄) ΔI_L = (ΔI₁ + ΔI₄) − (ΔI₂ + ΔI₃)

ΔI_Z is the input parameter in the feedback loop

keeping ΔI_Z = constant in order to make the bending ΔZ = ΔZ₀ preset by the operator.

AFM basics

Interaction forces cause the cantilever to bend while scanning

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

Cantilever response

The tip is “in contact” with the surface

l = cantilever lenght w = cantilever width t = cantilever height l_tip = tip height

































 





















z y x

zz zy

zx

yz yy

yx

xz xy

xx

F F F

c c

c

c c

c

c c

c

z y x

y = cantilever deflection along y

z = cantilever deflection along z

 = deflection angle Vertical force F_z applied at

the end induces

the cantilever bending

Cantilever response

cantilever deflection angle around setpoint

Consider vertical force only

Vertical forces

y tg z



 

 



z zz

z yz

z xz

F c

z

F c

y

F c

x













Assume a bending with radius R

Y = Young modulus Neutral axis

Section

Hooke’s law

Resulting force acting on dS

dF

Cantilever response

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

inner edge -> compressed.

outer edge -> stretched.

there is a neutral plane of zero stress between the two surfaces

Rz LL _



dS Y dF

LL _



R Yz dS LL

Y dS

dF ^ _ ^

I_z= momentum of inertia of the section wrt neutral axis Section

At any section S there is a torque wrt neutral axis Resulting force acting on dS

Cantilever response

u(y) = deflection along z of a cantilever point at the distance y from the fixed end The point trajectory follows a curve with curvature

For small angles

y R

Yz dS LL

Y dS

dF ^ _ ^

z S

S S

z I

R dS Y

R z Y R

Yz dS zdF

M ^  ^  ^{2 }  ^{2 }

2

1 2

dyu d

R ^{ Mz} _dy^{d u2YIz}

 2

12

3w I_z ^ t

For any point along the cantilever y direction

Cantilever response

For small angles y

integration

integration





F

M_z ^ _z ^ F





dyu

YI_z d ²₂ ^ _z ^

z

z YI

dy u M ^ d²₂



 



 

 2

y 2

YIF Ly dydu

z z



 



 

 2 6

3

2 y

Ly YIF

u

z z

z L z

y YI

L u F

z 3

 3



 _

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

The deflection is proportional to measured signal

Cantilever response

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.

3 3

4L k_c ^ Ywt

z z

z z

w F YtL w F

YtL YIL F

z ³ ₃^{3 4}₃³

3 12

3  





z k F_z ^ _c^ 12

3w I_z  t

L z 3

 2



L z LYI

z YIL

YIF L dydu

tg ^z

z z

z L

y



 









 2

3 3

2

2 ³

2

 2



z L z

y FYIL

u

z 3

 3



 _

Cantilever response

The magnitude characterizes the cantilever stiffness coefficient of inverse stiffness

It is the largest among the tensor c_ij

Force spectroscopy at fixed location L_tip << L so c_yz can be neglected

z zz

z yz

z xz

F c z

F c y

F c x













Fz

w YtL z ^{ 4}₃³



YIL c w

YtL c

zz   z 

3

4 ³

3 3

tip z

tip F

w Yt

L L L

y ^{ } ₃ ³

 

L z 3

 2

 z

L L L

y ^ _tip ^{ tip }

 2

 3

tip zz

yz c

L c L

2

 3 c_xz  0

z tip z

cF z

L cF y L

x













2 3 0

YIz

c L

3

 3

z = cantilever deflection along y

z = cantilever deflection along z

 = deflection angle Longitudinal force F_y applied at

the end induces the cantilever bending:

Y-type bending

Cantilever response

cantilever deflection angle around setpoint

Similarly to previous case

Longitudinal force F_y applied at the end results in a torque Longitudinal forces

y tg z



 

 

y 

zy y yy

y xy

F c z

F c y

F c x













YIz

dy u M ^ d ²₂

tip y

z F L

dy u YI d ²₂ ^

tip yL F M ^

z tip y

YI L F dy

u

d _

2

2 2

2 y

YI L u F

z tip

 y

Longitudinal force F_y applied at

the end induces the cantilever bending

Cantilever response

The deflection is proportional to measured signal

the axial force results in the tip deflection in vertical direction

2 but

2 y

YI L u F

z y tip

 ²

2 L

YI L u F

z

z y tip L

y 



 _

YIz

c L

3

 3

tip y L

y cF

L u L

z 2

 3



 _

L c c_zy L^tip

2

 3 _L ^cF _L^z

F L YI

L L dy

du tip y

z y tip L

y

 









3 2

 2

Longitudinal force F_y applied at

the end induces the cantilever bending

Cantilever response

the axial force results in the tip deflection not only in the vertical but also in longitudinal direction Very small compared to c

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

L c c L

L

c_yy ^{ 2}L^tip _zy ^{ 3 tip}₂²

Lz

 2



Ltip

y  

 L z

y ^ L^{tip }

 2

tip y tip y

L cF z L

L cF y L

x

2 3 3 0

2 2













Transverse force F_x

Cantilever response

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

torsion Transverse forces

x zx

x yx

x xx

F c

z

F c

y

F c

x













wt c t

YwL

c_bend ^{ 4} ₃^{3  2}₂

Cantilever response

Torsion

The torsion is directly related to beam deflecton angle 

G= Shear modulus ~ 3Y/8

The torque by F_x is The lateral deflection is

L M Gwt

3

 3



xLtip

F M ^

tors Ltip

x  

 c_torsF_x ^{ }x_tors

L c L Ywt

L L Gwt

L L M

L F

c x ^{tip tip tip tip}

x

tors tors 2

2 3

2 3

2

2 3 8 2







 

 

wt c

c_bend ^{ 2}₂ ^x ^{ }x_bend ^{ }x_tors ^





L c L w

c_xx t ^tip _^









 

 ₂

2

2

2 2

L c c_tors ^{ 2}L^tip₂²

Cantilever response

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

Simple bending

torsion





bend xtors c c F c F

x

x ^{      }

 c

L L wt

c_xx ^tip _^









 

 ₂

2 2

2 2

0 0

2

2 2 2

2



















 





z y

L cF L wt

x ^tip _x

YIz

c L

3

 3























 

 

















2 1 0 3

2 3 0 3

0 2 0

2 2 2

2 2

2

L

L L

L L

L L L wt

c c

c c

c c

c

c c

c

tip

tip tip

tip

zz zy

zx

yz yy

yx

xz xy

xx

C

Cantilever response

L = 90 m L_tip = 10 m w = 35 m t = 1 m

Dominant distortions c_zz, c_yz, c_zy

Lateral distortions are much smaller

92 1

.

1 ^

 mN

c

2 1

2 0.0016

1220

1  



 c c mN

wt c_bend

1 2

2

05 . 40 0

2 1 _





 c c mN

L c_tors L^tip

32 1

. 6 0

1 2

3 _





 c c mN

L c_yz L^tip

1 2

2

071 . 27 0

3 1 _





 c c mN

L c_yy L^tip

32 1

. 6 0

1 2

3 _





 c c mN

L c_zy L^tip

YIz

c L

3

 3

Fixed end

Cantilever effective mass and eigenfrequency

l = cantilever lenght w = cantilever width t = cantilever height l_tip = tip height

Cantilever is vibrating along z

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

dy

u(y)

Kinetic energy

u(t,L) = deflection along z of cantilever free end

3 3

4L k_c ^Ywt

 

 

L y mdy t

u dE_k

, ² 2

 



 



 

 2 6

3

2 y

Ly YIF

u

z z

 

z

YIzL L F

t

u ,  3 ³

     



 



 

 



 



 

 ₃ ^{2 3} 3 ₂² ₃³

2 , 6

2 , 3

, L

y L

L y t y u

Ly L L

t u y

t u

Cantilever effective mass and eigenfrequency

Kinetic energy

Potential energy

Equation of motion

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

work done to move the

beam end the distance u(t,L)

 

  dy m u t L

mL y

t u dE ^L

L

k ,

2 140

33

, 2 ²

0

2

0     



 

 

L y mdy t

u dE_k

, ² 2

 

     

cL t du u

c Fdu u

E_P ^{u t L u t L}

2 ,

, 2 0 ,

0  

  

 

   

c L t L u

t m u

E_T

2 , ,

2 140

33 2  ²

 

 ,   ,  ₀

140

33  

c L t L u

t m u_

140

* 33m m 

0

*   c u u m 

 Y

L t

cm ²

0 1.029

* 1 



  ^{ }



 



 

 3 ₂² ₃³

2 , ,

L y Ly

L t y u

t u

Tip-surface interaction Origin of forces

Tip-surface Separation (nm)

0 1 10 100 1000

Interatomic forces (adhesion, Hertz problem) Van der Waals

(Keesom,Debye,London) Electric, magnetic, capillary forces Non contact

Intermittent contact

Contact

Born repulsive interatomic forces

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

Cgs/esu

- +

- +

Origin of forces

Contact

122

r U

^ C

4 0

1  

Elastic forces in contact

Origin: object deformation when in contact Hertz problem: determination of deformation

• 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

• Two spheres r₁ = r₂

deformation and penetration

the contact pressure is higher for stiffer samples Assumptions

Hertz problem solution:

allows to find the contact area radius a

and penetration depth h as a function of applied force for a surface (r’=) and a tip (r=R)

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

Origin of forces

Contact

Same materials a² = hR





2 3

1 3

2



 h Y R F

Potential energy of the dipole moment in an electric field E

Keesom Dipole forces

q^- q⁺

d 

Field intensity produced by the dipole

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

Cgs/esu

Coulomb force

between point charges Coulomb potential energy Electric field

 = qd = dipole moment

 is the angle between dipole and r

For r >> d

Origin of forces

Intermittent contact

r2

E q

U    

1 cos

3 ²

3 

  

E r

22

r1

q

f_C ^ q r

q dr q

f

U ^{ }



4 0

1  

Potential energy of the interacting dipole moments

₁

q^- q⁺

d

Maximum attraction for ₁= ₂ = 0°

When two atoms or molecules interacts

₂

q^- q⁺

d

Maximum repulsion for ₁= ₂ = 90°

Origin of forces Keesom Dipole forces

Intermittent contact







¹ ₃² 2cos ₁ cos ₂  sin ₁ sin ₂ cos

 r U

3 2

max 2

U _{ } r

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

Keesom Dipole forces

Orientational interaction

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

For U << K_BT

Origin of forces

Intermittent contact

3 2

max 2

U _{ } r

6

4

1

3 2

r T U k

AV B

 















 d e

d U Ue

T k

U T k

U

AV

B B

T k e U

B T

kU

B  

 1













  





Ud d

T d kU Ud

U_AV ^B

2

0



Potential energy of the interacting dipole moments



q^- q⁺

d

Induced dipole moment

Debye Dipole forces

Origin: fluctuation of the electronic clouds around a molecule dipole formation, interaction of the dipole

with a polarizable atom or molecule



q^- q⁺

d

For r >> d

Induction interaction

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

Origin of forces

Intermittent contact

ind E

 

2

2 0

dE E

U 



1 cos

3 ²

3 

  

E r ⁶

2 1

U  _ind r

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

Field induced by atom 2

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 + Polarizable

2 1 atom

RMS dipole moment for fluctuating electron-nucleus

Ionization energy

The dipole formation of atom 2 is given by the polarizability 

Origin of forces

Intermittent contact

2

1 2 0

dE E

U ^E _ind 

  





32

2 E r

 ^



i i

₂

hi

 

2 2  2

2 6

1 1

4 3

r U hⁱ 



3 2 

32 2 2

r h E r ⁱ





Fluctuation of the electronic clouds around the nucleus.

dipole formation with the positive charge of nucleus.

interaction of the dipole with a polarizable atom

Keesom

Fluctuation of the electronic clouds around a molecule.

dipole formation

Origin Potential energy

Fluctuation of the electronic clouds around a molecule.

dipole formation.

interaction of the dipole with

a polarizable atom or molecule

Debye

London

Large overlap of core wavefunction of different molecules

Born

Origin of forces

61 2

4 3

r U hⁱ 





6 4

3 2 U _{ } kTr

6 2

U r^ind





122

r

U

^ C

van der Waals dipole forces between two molecules

Total potentials between two molecules

Lennard-Jones potential

Origin of forces

2 6 2 1

4 6

2 61 2

6

4 1

4 3 3

2 4

3 3

2

h r kT

r r

h

U kTr ^{i ind} _{ind i }



 



  











          

61

r U ^{ } C

61 122

r C r

U ^ C ^

van der Waals dipole forces between macroscopic objects

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

Hamaker constant

Origin of forces

61

r

U ^{ } C f ( r ) ^{ } U

 



1 2 1 2

2

1 ( )

)

(

r

f r dV dV

U  

 



1 2 6 1 2

2 1

2 1

1

)

(

dV dV

C r r

U    H ^ 

C





The force must be calculated for each shape

For a pyramidal tip at distance D from surface

Hamaker constant

Same role as the polarizability Depends on material and shape

Origin of forces

D D H

F 

 3

tan ) 2

(  

2 1 2 1

 

 C

H ^

The force must be calculated for each shape

Conical probe Pyramidal probe

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

Conical probe rounded tip

For r >> h

Origin of forces

N

x h

F C

15 2 2 1 2 1

10 3 . 1

6

tan

 

    

N x

h F C

15 2 2 1 2 1

10 2 . 5

3

tan 2

 

     F 1.1x10^13N

N x

h R F C

9 2 2 1 2 1

10 3 . 3

6

 

   

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

Similar to VdW force 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

Consequence: hysteresis in approach/retraction

Origin of forces

N F 10^9

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

d = detector - cantilever distance

_PSD = laser spot movement

High sensitivity in z is obtained by L << d

Vertical resolution depends on

the noise and speed of PSD t= 0.1 ms

z ~ 0.01 nm

B. Cappella, G. Dietler, Surface Science Reports 34 (1999) 1-104

time for measuring a point of the force curve

L z LYI

z YIL

YIF L dydu

tg ^z

z z

z L

y



 









 2

3 3

2

2 ³

2

 2



L ^z

 2

 3



 d

PSD  2d tan  2



d z ^PSDL

3

 



t m 10^13

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 – 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 = -k_c_c

Force-distance curves

Must be converted to D and F

D = tip-sample distance

_c = cantilever deflection

_s = sample deformation Z = piezo displacement

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

D = tip-sample distance

_c = cantilever deflection

_s = sample deformation Z = 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)=k_c a) Infinitely hard material (_s=0), no surface forces

PSD-Z curve: two linear parts

zero force line

defines zero deflection of the cantilever

Linear regime

sensitivity I_PSD/ Z _c = I_PSD/(I_PSD/ Z) F-D curve

F(Z) = k_c

D=Z-_c

F(D) = k I_PSD(Z)/(I_PSD/Z)

Z = 0 at the intersection point

Z > 0 if surface is retracted from tip

Conversion between _PSD and Z

H.-J. Butt et al. / Surf. Sci. Rep. 59 (2005) 1

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

D = tip-sample distance

_c = cantilever deflection

_s = sample deformation Z = 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 Linear

regime sensitivity I_PSD/ Z

from the linear part

_c = I_PSD/(I_PSD/ Z)

F-D curve

F(Z) = k_c D=Z-_c Z = 0 at the intersection point (extrapolated)

Z > 0 if surface is retracted from tip

long-range exponential repulsive force

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.

In non-contact D = Z - _c

F(D) = k I_PSD(Z)/(I_PSD/Z)

s

s

D = tip-sample distance

_c = cantilever deflection

_s = sample deformation Z = 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 is retracted 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 R

planar sample of the same material (Y)

_s = indentation For many inorganic solids _s << _c For high loads _c~F/k_c

sensitivity I_PSD/ 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 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 If _s  0 ‘‘zero distance’’ (Z=0) must be defined

If _s ~ _c

In contact the distance equals an interatomic distance





3 2

2 3





 Y R

F ^s

s

s

Force-distance curves

c) Deformable materials with surface force

At some distance the gradient of the attraction exceeds k_c

and the tip jumps onto the surface.

- very soft materials

surface forces are a problem leading to a significant

deformation even before contact

- relatively hard materials

Due to attractive and adhesion forces it 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 that the indentation caused by adhesion is negligible.

Adhesion forces add to the spring force and

can cause an indentation

B. Cappella, G. Dietler, Surface Science Reports 34 (1999) 1-104