PY5020 Nanoscience Scanning probe microscopy Nanoscience... · Scanning probe microscopy . Outline Scanning tunnelling microscopy (STM) ... The Lennard-Jones potential

PY5020 Nanoscience

Scanning probe microscopy

Outline

Scanning tunnelling microscopy (STM)

- Quantum tunnelling

- STM tool

- Main modes of STM

Contact probes

Vbias

Use the point probes to measure the local I-V curves at different locations on

the surface.

Schematic of a tunnelling gap with a

square barrier

Φ is the work function; ~ 5 eV for metals

Ef is the Fermi energy

Schrödinger equation can be solved for

each area square barrier

EΨΨUd

Ψd

m )z(

z2 2

2Schrödinger equation:

Solutions for different areas:

zz

1

ikikAeeΨ

zz

2

CeBeΨ )-z(

3

dikDeΨ

0(z) U 0(z) VU 0(z) U

2

2 2

mEk

2

2 2

mEk

2

02)(2

EVm

Probability current density through

the barrier Transmitted current:

Transition Probability:

Incident current:

Potential is complicated by potential

wells at atom locations

Similar approach is applied as

in the square barrier problem Search for wave functions satisfying:

Probability of finding an electron at a distance z into the barrier:

Apply positive bias ΔV to the

substrate to make current flow

eΔV

Local density of states (LDOS) is defined as:

2

(z)1

),z(

E

E nE

Ψ

ervalintenergy

volumeunitelectronsN /

At a distance z at energy E

d

F

nne

E

VeE

ΨI22

(0)

I is linear with voltage for small ΔV

Current I through the tunnel barrier:

Or in simple form

Since =

Comparing these equations:

Scanning electron microscope

http://fys.kuleuven.be

We can raster x and y while measure z, i.e. have a 3D microscope

Topography scan

Controling the tunnelling gap

A

Ultra high vacuum

STM images

7x7 reconstraction on Si(111)

Takayanagi K et. al. Surf. Sci.

1985. V. 164. P. 367.

HOPG honey comb structure

Courtesy of Alexander Chaika

Scanning tunnelling spectroscopy (STS)

http://pubs.rsc.org/en/content/articlehtml/2015/cs/c4cs00204k

dVTeVEEeVEfEfe

I FsubstrateFtipFF ),()()()()(4

)(Ef

)(Etip

)(Esubstrate

),( VT

is the Fermi-Dirac distribution function

is the local density of states of tip (LDOS)

is the LDOS of substrate

is the e transmission probability at energy ε and applied voltage V

Fermi-Dirac distribution function

Varies between 1 and 0. Tells probability of occupation of a certain election state as

a function of energy with respect to a Fermi level.

http://ph.qmul.ac.uk/~anthony/spfm/17.html

E-Ef

T=5 K

T=1000 K

T=5000 K

Assumptions of STS

0

)()(4

eV

FsubstrateFtip deVETEe

dVTeVEEeVEfEfe

I FsubstrateFtipFF ),()()()()(4

)(| eVEconstV

IFsubstrateV

• At room temperature kT = 25 meV

• For V>~2kT/e, f(E) can be well approximated by a step function

• LDOS of tip does not change significantly with E

Current Imaging Tunnelling Spectroscopy (CITS)

Acquire a topography image and measure I-V curve at each point

Apparent tunnel barrier height

What is the force on lead atom on

the tip

zkF

http://analyticalprofessional.blogspot.ie

Cantilever can be used to measure

fine deflections

F

STM Tip

The force on the tip atom

zkF

k is the spring constant

• Atoms interact via the close

and long range forces that

are electrostatic in nature

• Lead atom of the tip

interacts with the lead atom

on the substrate (atomic

interactions – short range –

atomic bond).

• Groups of atoms on the tip

and substrate interact

collectively via the

WanderVaals forces (long

range of the order of nm)

• What spring constant of a

cantilever should be?

mNmkkgH /110m z10~ 23015

Tip-sample forces

Set point ~ 0.5 nm

Surface

d

The method of choice today

Adv. Mater. (2009) 21, 1187–1192

Normal force

A+B = Up

C+D=Down

Measure the difference

Magnitude of voltage tells by

how much the tip moved

Polarity of the voltage gives

Up or Down

Lateral force

A+C=Left

B+D=Right

A B

C D

http://www.parkafm.com/AFM_guide/true_non_contact_mode_1.php

Ref. Voltage

(set point)

Ph

oto

dio

de

How to maintain constant height

http://www.cmbi.ru.nl/redock/Glossary.php

The Lennard-Jones potential

- Captions dispersion and Pauli repulsive interactions

- U* is depth of potential; at r = σ U(r) goes to zero

612

***4)(

r

r

r

rUrU

-2.0

2.0

0

σ σ

Electrostatic interatomic and intermolecular forces

https://www.pinterest.com/pin/428897564488110883/

2

04

)cos()(

r

QprU

3

0

212121

4

)cos()sin()sin()cos()cos(2)(

r

pprU

Angle averaged dipole-dipole (Keesom)

Angle averaged dipole induced

Polarisation (Debye)

Dipole-dipole

Charge-dipole

- Ion-ion r

QQrU

0

21

4)(

Dispersion interatomic

Intermolecular interaction (London)

62

0

2

2

2

1

)4(3)(

Trk

pprU

B

21

21

62

0

0201

)4(2

3)(

II

II

rrU

62

0

01

2

202

2

1

)4()(

r

pprU

Van der Waals forces

Van der Waals is a sum of forces which interaction potential varies as 1/r6:

Orientation or Keesom force: thermally angle averaged dipole-dipole

interaction between two atoms or molecules

Debye force: thermally angle averaged dipole-induced dipole interaction

London or Dispersion force: dispersion force between acting between

molecules and atoms irrespective of their polarisation

62

0

2

2

2

1

)4(3 Trk

pp

B

62

0

01

2

202

2

1

)4( r

pp

6

21

21

62

0

0201

)4(2

3

r

C

II

II

r

)()()()( rUrUrUrU LondonDebueKeesom

From interaction to tip-sample interaction – simple

theory

https://nanohub.org/resources/7698

From interaction to tip-sample interaction – simple

theory

https://nanohub.org/resources/7698

Surface – surface interactions

https://nanohub.org/resources/7698

Derjaguin approximation

By measuring the lift off force of tip from the substrate it is

possible to correlate the tip sample force to the values of

surface energy. From the previous slides:

2

22

6)(

d

RC

d

WdF

tipsphereplane

2

2

12)(

d

CdW planeplane

By comparing these equation, the Derjaguin approximation:

planeplanetipplanesphere dWRdF )(2)(

http://nanohub.org/resources/7320

From F-z to F-d curves

Cantilever can be used to measure

fine deflections

F

12

3bh

I

l

h b

www.atomic-force.com

δ

δ

Cantilever equilibrium position

http://nanohub.org/resources/7320

http://www3.physik.uni-greifswald.de/method/afm/eafm.htm

δ =

F-d to F-z conversion

http://www.rsc.org/chemistryworld/News/2007/February/28020702.asp

Chemically resolved AFM