Coulomb Blockade and 'Orthodox' Theory of Single-electron Device

Coulomb Blockade and
Coulomb staircase
Zhang Jun
01/06/2004

Introduction
Single-electron effect in mesoscopics
world
Dimension less than ~30nm
Main component of single-electronics:
tunnel junction with a very small
capacitance and a resistance

Realization of these
junctions
Metal-insulator-metal structure
GaAs quantum dots
Silicon structures
Large molecules with conducting cores

Coulomb island

Change of voltage
across the junction
The tunneling of only one electron may
produce a noticeable change e/C of the
voltage across the junction.
Coulomb blockade: suppression of
tunneling at voltages |V|

Coulomb blockade
Capacitor charge
Q1=C1*V1
Q2=C2*V2
The charge on the
island,
Q=Q2-Q1= -ne
n=n1-n2

Voltage drops of the
tunneling junction
V1=(C2Va+ne)/Ceq
V2=(C1Va-ne)/Ceq
Ceq=C1+C2

Electrostatic energy
stored in the capacitors
Es=Q 12
/2C 1
+Q 22 /2C 2
Es=(C 1
C 2
Va 2 +Q 2 )/2C eq

The work done by the
voltage source
Ws=∫dtVaI(t)=VaΔQ
One electron tunneling through 2
so that n2’=n2+1, n’=n-1, Q’=Q+e,
V1’=V1-e/Ceq, ΔQ = -eC1/Ceq
Result: Ws(n2)=-n2eVaC1/Ceq
Ws(n1)=-n1eVaC2/Ceq

The total energy
E(n1,n2)
=Es-Ws
=(C 1
C 2
Va 2 +Q 2 )/2C eq
+eVa(C1n2+C2n1)/Ceq

The change in energy of
the system with a
particle tunneling
ΔE 2+ =E(n 1
,n 2
)-E(n 1
,n 2
-1)
=Q 2 /2Ceq-(Q+e) 2 /2Ceq+eVaC 1
/Ceq
=e/Ceq[-e/2+(en+VaC 1
)]

When the island is
initially neutral (n=0)
ΔE2= -e 2 /2Ceq+eVaC1/Ceq >0
For C1=C2=C, the requirement
becomes simply |Va|>e/Ceq

Band diagram

Coulomb staircase

Thank You!