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Lecture 4 - Coulomb blockade & SET
Fulton TA and Dolan GJ, Phys.
Rev. Lett. 59 (1987) 109.
Circuit for SET transistor 1
V1
C1
R1
VC
Q
Vg
nanocluster
Cg
C2
tunnelling
junction
R2
tunnelling
junction
When an electron tunnels
onto or off the cluster
already containing n
excess electrons, the
change in electrostatic
energy is:
U  
V2

1
 ne 2  21 e 2
C

Algebra – first calculate charge on middle island:
Q   Qi  C1 VC  V1   Cg VC  Vg   C2 VC  V2 
i
1
Q  VC C   CiVi   ne C  C1  Cg  C2
i
Rearrange2
VC 
Q   CiVi
i
C
Electrostatic
energy
Circuit 2for SET transistor 2
Qi
U 
i 2Ci
expand Qi as CiVi and substitute in (2)
1Q
1
1 1 
2
2
U
  CiVi 
CiVi 


2 C 2 i
2 C  i

2
2
Only term that varies as the charge on the
cluster varies.
Consider process when 1 charge tunnels onto or off the cluster.
n  n 1
2
2 2


1
n

1
e
1
n
e
1

U (n) 


 ne 2  21 e 2
2
C
2 C
C
2


3
Circuit for SET transistor
V1
C1
R1
VC
Q
Vg
tunnelling
junction
nanocluster
Cg
C2
When an electron tunnels
onto or off the cluster
already containing n
excess electrons, the
change in electrostatic
energy is:
R2
tunnelling
junction

1
U 
 ne 2  21 e 2
C

V2
The work done by the voltage sources when an electron tunnels
through the jth junction is:
e
Wi   CiVi
C

i j
e
W   eV j 
C jV j
C

j

Circuit for SET transistor
0
C1
R1
VC
Q
Vg
enanocluster
Cg
C2
tunnelling
junction
R2
V
Q0  CgVg  Q00
tunnelling
junction
Enthalpy change
e e

E 
 ne  Q0   C2V 

C  2


1
e e



E 

ne

Q

C
V
0
1

C  2

2
Coulomb staircase
Electrons fill island until
Tunnelling current
E1 
e e




ne

Q

CV
 0
0


C  2

 E2
1 
e  Q0 1 
I

V

n
 


eR2
2 R2 
C
e 2 
Single electron transistor
blockade
Experimental Coulomb staircase
Wilkins R et al., Phys. Rev. Lett.
63 (1989) 801.