Scanning Tunneling Microscopy (STM)

Short description

Theory of 1-D tunneling Actual 3D barriers

tip modeling

atomic 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 Control and Scanning Unit Tunneling

Current Amplifier

Data Processing and Display

Tunneling Voltage Piezolelectric Tube with Electrodes

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 process Intial and final states have same energy

Inelastic

Energy loss during the process

Interaction 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

ikz

eikz ^{ }

 Re

1

2

2 2

 k ^ mE

xz

xz Be

Ae ^{ }

2 



02

2 2 ( )

 E V

x ^ m ^

Teikz 3 



Electron tunneling across 1-D potential barrier



 V  E dz

d

m ^{ }

 ₀

2 2 2

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

y probabilit on

transmissi

2  T

Time independent





iks ixs

ixs

kTe Be

Ae x

Te Be

Ae





















k

B A

R











 1 1

 





x k

xs x

R ₂ k_{2 2 2 2}² ₂

2 2 2 2

sinh 4

sinh





 

Electron tunneling across 1-D potential barrier

Continuity conditions on  and d/dz give At z=0

At z=s





x k

x

T ² _{2 2 2} k ² ₂²_{2 2} sinh 4

4



 

y probabilit on

transmissi

2  T

y probabilit reflection

2  R

2 1

2 T 

R

Time independent

Electron tunneling across 1-D potential barrier

A square integrable (normalized) wave function has to remain normalized in time

In a finite space region this conditions becomes

 ,  ²  0



d 

2 0

* *

2  



 







 



 















dtd  ^    

2 0

* *

2  



 







 



 



b

a b

a dz im z z

dtd  ^    

  _



 







 



 

z z

m t i

z

j ^{* *}

, 2^    





,  j b t  j a t  dt P

d

b a

Probability conservation

Probability current

Time independent

Electron tunneling across 1-D potential barrier

Applying to

our case   _



 







 



 

z z

m t i

z

j ^{* *}

, 2^    

Region 1

Region 3

 



 



1 1 1

, 2 R v R

mk t

z

j ^{    }

  ^{2 2}

, 2 T vT

m t k

z

j_T ^{  }

z t  v j_i , ^

T

j j

T

i 

T 

t coefficien on

transmissi

T  (4 )

) )(

( sinh 1

1

2 2

2 2 2 2

x k x xs k ^

 T 

Time independent

Electron tunneling across 1-D potential barrier

Exponential is leading contribution

02 )

( 2



E V

s m

xs ^{ }

For strongly attenuating barriers xs >> 1

e

x k

x

k

2 2 2

2 2

) (

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 coefficient

depends exponentially on barrier width Large barrier height (i.e. small )

2 2 2

2

2 2

2 2

2 2 2

2 2

2 2

2 2

2 2 ( )

16 )

4 (

) (

4 2

1 )

4 (

) )(

( sinh 1

1

x k

e k x x

k x k

e x e

k x

xs k ^{xs xs xs }

 



 



  

 



 _

T

2 2

) 2 (

sinh 



 



 

 e^xs e^xs xs

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 theory But no easy way to write a perturbed Hamiltonian

Approximate solutions of exact Hamiltonian within the barrier region

s z

for be

z

z for ae

z

r kz l kz







 ^

) (

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 probability to 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

H_T = transfer Hamiltonian

r r r

l l l

T r T

l

E H

E H

H H

H H

H

H t i ddtt H









 

















0 0

) 0

(

) ) (

( 

Time-dependent

With the exact hamiltonian on left and right, we add a term H_T representing the transition rate from _l to _r.

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

Electron tunneling across 1-D potential barrier

Choose the wavefunction

Put into hamiltonian





t iE r

t iE l

r

l

d e

e

c

^

  



dt t i d

t

H( )  _  ^{( )}

dt

e d

e c

d i

e d

e c

H e

d e

c H

H

t iE r

t iE l

t iE r

t iE T l

t iE r

t iE r l

l

r l

r l

r l



 



 

 







 

 







 



 

 

 























 ) 

(

Time-dependent

Electron tunneling across 1-D potential barrier













t iE r

r t

iE l

l

t iE r

t iE l

T t

iE r

r t

iE l

l

r l

r l

r l

e dE

e cE

e d

e c

H e

dE e

cE

 

 

 



 







 

















0

 







  ^







t iE r

t iE T l

r

l d e

e c

H  

The total probability over the space is

1 1

*

*

*

 







 







 



 





dz e

d e

c H

e d

e c

dz H

t iE r

t iE T l

t iE r t

iE l

T

r l

r l







   







Time-dependent

Electron tunneling across 1-D potential barrier

So the tunneling matrix element

1 1

*

*  









M M

T l r T r

l

rl lr









Using the Fermi golden rule to obtain the transmitted current

rl r

t dE

M dN

j ^{2 2}



 

Density of states of the final state M_lr = Probability of tunneling from state l to state m

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

But ………… each case has to calculated separately

 

zix z dz

e

z ₀ ^{0 ' '}



 

2

2 2 ( )



z V E x ^ m ^

Electron tunneling across “real” 1-D potential barrier

  

 V z E

dz d

m ^{ }



2 2 2

2

Try a solution

Time independent

  m





d  _{  } 

2 2

2 2



  x  z dz

z

d  ₂

2

2  

V(z) = slowly varying potential

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

  ix   z z dzz

d 

      z x    z z

dz z i dx dz

z

d   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

  x  z dz z

d  ₂

2

2        z x    z z

dzz i dx dz z

d   ₂ 

2

2  

  _{x 2}

dz z

dx _

but

  ^{ x 1}

dzz dxx

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 approximation

Wenzel Kramer Brillouin

 

2

2 2 ( )

 z V E

x m 



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

 

Electron tunneling across “real” 1-D potential barrier

Time independent

Suppose the particle encounters a barrier between 0 < z₁ < z₂ so E < V(z) and x is imaginary

  z ^ 

e ^

e ^

^ 

e ^



the probability density inside the barrier is

   ^

z

z x z dz

e

z





Inside the barrier

the probability density at z₁ is 2



the probability density at z₂ is

 

2

1 2 ' '

2 1 2

2

z

z x z dz

e





 z ₀e^{0zix z'dz'}



Neglect the exp growing part

Electron tunneling across “real” 1-D potential barrier

Time independent

So the transmission coefficient becomes

Tunneling probability very small

2 1 2

2



 

 

 

 



2 1 2

1 2 ' ' 2 2 ( ' ) '

2 1

2 2

z z z

z x z dz m E V z dz

e

e

 T 

The wavenumber is continuosly varying due to the potential: more real

 

2

2 2 ( )

 z V E

x  m 

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

Electron tunneling across 1-D potential barrier

Square barrier plane wave

e

x k

x

k

2 2 2

2 2

) (

16 

 

T Exponential dependence

of the transmission coefficient

rl r

t dE

M dN

j ^{2 2}



 

Square barrier

electron states current depends on transfer matrix

elements (containing exp. dependence) and on DOS

Real barrier

Plane waves

 

 2

1 2 ( ' ) '

2 ^z

z m E V z dz

e

T

True barrier representation if <<z Varying exponential dependence

of 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 current and the Fermi level is aligned

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

) ( )

(z E ₁ z V  _F  

We must consider the Fermi distribution of electrons

insulator = vacuum

The insulator defines the barrier





 ^{max max}

0 0

1 ^v _z ( _z ) ( _z ) _z 1 ^E n(v_z ) (E_z )dE_z dv m

E v

n v

N T T

v_z = electron speed along z

n(v_z)dv_z = number of electrons/volume with v_z T(E_z) = transmission coefficient of e^- tunneling through V(z) e^- with energy E_z =mv_z²/2 f(E) = Fermi Dirac distribution





 0

3 0 2

2

1 ( ) ( )

2

max

r E

z

z dE f E dE

m E

N T

 

KT E E _F

e E

f _



 1 ) 1 (

Electron tunneling across 1-D metal electrodes

n(v_z)dv_z = number of electrons/volume with v_z

  

 



3 0 2

3 3

3

4 ( )

) 2

4 ( ^{x y r}

z m f E dv dv m f E dE

v

n    

 _{v dvxdvydvz} ^m _{f E dvxdvydvz}

n ( )

4 ^{3 3}

4

 



2 2 2 2

2 _r

r

y x r

v E m

v v v







Flux from electrode 1 to electrode 2

 







  















2

0 2

1 ( ) ( )

) 2

max (

r z

E

z

z dE m f E dE f E eU dE

E N

N

N ^T 

Total number of electrons tunneling across junction

Electron tunneling across 1-D metal electrodes





 0

3 0 2

2

1 ( ) ( )

2

max

r E

z

z dE f E dE

m E

N T

 

Flux from electrode 2 at positive potential U to electrode 1





 0

3 0 2

2

2 ( ) ( )

2

max

r E

z

z dE f E eU dE

m E

N T

 



 2 3 0 2

1 ( )

2em f E dE^r

 

 ^ 2 3



2

2 ( )

2em f E eU dE^r

 



 



 ^max

0

2

) 1

(

E

z

z dE

E

J _T   tunneling current across junction

The current depends on electron distribution

 ^  ^{ }

2

1 1 ( )

2

) 2

(

s

s EF z Ezdz

m

z e

E ^{ }

T

Electron tunneling across 1-D metal electrodes

) ( )

(z E ₁ z

V  _F 

T is small when E_F-E_z is large

e^- close to the Fermi level of the negatively biased electrode contribute more effectively to the tunneling current

since

 



 ^max

0

2

) 1

(

E

z

z dE

E

J T  

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

F Ez

E A

z e

E ) ^{  1} T(

Electron tunneling across 1-D metal electrodes

 _{z dz}

s

s



  ¹

1

1 



To perform the integration over the barrier

 







  



 





    ₁ 

1

1 1

1 1

3 0

2 2

F F

z

F F z E F

eU

E z

E E

A F z

eU E

z E

E

A dE E E e dE

e em eU

J ^{ }

 

 m A  ^2s ²

define

By integration it can be shown that

At 0 K _ em





3 1

1 2 2

  ^em





3 1

2 2 2



Applications of tunnel equation

























1

1 1

1

1

0

0 2 ^{2 3}

z F

z F z F

F

F z

E E

E E

eU E

E E

eU E

E em eV

 

 hence

Electron tunneling across 1-D metal electrodes

 





s

J e ^{    }

   ^  ^

4² ²

Current density flowing from electrode 1 to electrode 2 and vice versa If V = 0 dynamic equilibrium: current density flowing in either direction

 

 e ^A

s

J e ^

 

4 ^{2 2}

1 





s

J e ^{  }

   ^

4^{2 2}

2 

 z dz s

s

s

  ¹

1

1 



 m A ^2s ²

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

Electron tunneling across 1-D metal electrodes

 





s

J e ^{    }

   ^  ^

4² ²

Low biases

 eU 





  ^A

AeU

e e

s eU

J e ^{ }











  

 

4 ² ^{2 2}



 



 

 1 1 2 A2eU

A A eU

A eU eU

A

e e e e

e

 



 





 

   

  



 

 

 



 

 



 a k ba

a a b

b

a ^{k k} 1 ^{k k} 1

Electron tunneling across 1-D metal electrodes

Low biases eV  

 







 

 



 

 

 

A

A

A

A e s eU

e

e eU eU

s A e

eU e A s eU

J e









 



 











   



 























 



 



2 1 4

4 2

1 2 4

2 2

2 2

2 2







 

 eU A e ^A

e s

J ^

 

2 4 ² ²

 m A  2s 2

2   1 A





 U e _A s

m

J e ^

 

4

2

2 2 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















 



 



 ^{  2  0}

3 2 0

3 0

1 2 96 2

. 2

4

0 96 2

. 2

4

2 0

2

3 1 2

16 2 .

2   

 

 





m eU m eF

eF eU e

F e

J e ^{ }



High biases eU  

eU s s⁰

 2⁰ 

  

s F  U

 





s

J e ^{    }

   ^  ^

4² ²

Put into general eq.

Electric field strength

evaluating a numerical factor (not included in eq)

 eU 

For this condition Second term of eq is negligible

Electron tunneling across 1-D metal electrodes

High biases

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

E_F2 lies below the bottom of CB₁ Hence e^- cannot tunnel from 2 to 1 there are no levels available

U const

e U

J ^{ 2 }

2 3

2 0

96 . 2

4

2 0

2 3

16 2 .

2  ^{ }





eF m

F e

J e ^



 

Electron tunneling across 1-D potential barrier

Square barrier, plane wave

e

x k

x

k

2 2 2

2 2

) (

16 

 

T Exponential dependence

of the transmission coefficient

rl r

t dE

M dN

j ^{2 2}



 

Square barrier

electron states current depends on transfer matrix

elements (containing exp. dependence) and on DOS

Real barrier

Plane waves

 

 2

1 2 ( ' ) '

2 ^z

z m E V z dz

e

T

Varying exponential dependence of the transmission coefficient

Real barrier

Metal electrodes

Tunneling is most effective for e^- close to Fermi level

Low biases: Ohmic behavior High biases: Fowler-Nordheim Current flows from – to + electrode

 







  







  3 0^ 0^

2

0 2

1 ( ) ( )

) 2

max (

r z

E

z

z dE m f E dE f E eU dE

E N

N

N ^T 

3-D potential barrier

rl r

t M dEdN

j ^{2 2}



 

Square barrier electron states

 







  







  3 0^ 0^

2

0 2

1 ( ) ( )

) 2

max (

r z

E

z

z dE m f E dE f E eU dE

E N

N

N ^T 

Real barrier Metal electrodes

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



 H dz

M_rl ² _l^* _T_r

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 ^₀e^{0zix z'dz'}



Trick: both  are good on one side only and inside the 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





M^{  }₂m^2





 

  _{ }_ ^

 a ₀e ^iE0t b e ^{iE t}





2

2 ^{   }

  _m



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 integrating a current-like operator over a plane lying in the insulator slab