1. Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)
COULOMB BLOCKADE PH4101
Presented by:
AMIT KUMAR (09MS 086)
VIVEK SINHA (09MS 066)
DPS, IISER-K
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

2. WHAT IS COULOMB BLOCKADE
The phenomenon that tunneling through a metallic grain with small capacitance may be inhibited at low temperatures and small applied voltages.
why?
The addition of a single electron to system requires an electrostatic charging energy of order , where C is its capacitance, T the temperature and V the applied voltage.
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

3. Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)
Coulomb Blockade
Addition energy considering elcctrostatic effects
is the ionic positive charge compensating the electronic charge eN.
Valid only for N, N0 >>1
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

4. SINGLE BARRIER TUNNELING
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

5. SINGLE BARRIER TUNNELING
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

6. Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)
Boundary Condition
Transmission Matrix and Scattering matrix
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

7. Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)
DOUBLE BARRIER
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

8. Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)
Double Barrier
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

9. Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)
Double Barrier
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

10. CB in Single Electron Tunneling
When an electron tunnels into the quantum system, it does so by overcoming the charging energy. The removal of an electron increases the charging energy by discrete amount of
But this in turn raises the charging energy barrier for sequential tunneling. The net effect is that one electron tunnels at one time.
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

11. Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)
SINGLE ELECTRON BOX
Remember charge quantization!
tries to change the charge by
Two charges and created at capacitance of tunneling contact and gate capacitance respectively.
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

12. Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)
SEB
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

13. SINGLE ELECTRON TRANSISTOR
Let us consider a system between two leads which is coupled by two weak tunneling contacts. If a third gate is also present, the set-up is what we call single-electron transistor
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

14. Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)
SET
y axis is and x axis is
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

15. Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)
SET
Gate charge
Total Capacitance
Addition energy
Bias voltage
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

16. CONDITIONS FOR COULOMB BLOCKADE
Tunnel Resistance for single electron charging
Weak Coupling means
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

17. CONDITIONS FOR COULOMB BLOCKADE
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

18. AN INTERLUDE: ANDERSON HUBBARD AND CONSTANT INTERACTION MODELS
General Hamiltonian
Second term describes all possible electron interaction. Equivalent to real space Hamiltonian
With
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

19. ANDERSON HUBBARD AND CONSTANT INTERACTION MODELS
For states of definite spin, matrix elements of pair potential given by
If overlap between two different states is weak, one can replace the interacting part by Anderson-Hubbard Hamiltonian
ψα(ξ) are the basis single-particle functions, we remind, that spin quantum
numbers are included in α, and spin indices are included in ξ ≡ r, σ as variables.
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

20. ANDERSON HUBBARD AND CONSTANT INTERACTION MODELS
In the CIM, many states with similar energies interact. One can assume
This is equivalent to charging energy for large N.
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

21. TUNNELING TRANSITION RATES
Determined by GOLDEN RULE expression.
Consider transition from a state to a state due to coupling to the left lead
transition determined by the full probability of tunneling of one electron from any state in the left lead to any single particle state, say
Summation in Sense of CIM over initial and final states.
the transition from the state |n to the state |n + 1 due to the coupling to
the left lead, is determined by the full probability of tunneling of one electron
from any state |k in the left lead to any single-particle state |α:
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

22. TUNNELING TRANSITION RATES
Summation over all states in the left lead and all possible single particle states in the system in the sense of the CIM.
describes the coupling to the left lead.
and are Fermi-distribution functions that respectively describe state to be occupied and state to be vacant.
The state may be interpreted as a mixture of many single-particle states
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

23. TUNNELING TRANSITION RATES
Simplifying we get the conductance and tunneling transition rates as
are DOS and
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

24. TUNNELING TRANSITION RATES
The master equation of probability to find a state with n electrons may be written as
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

25. TUNNELING TRANSITION RATES
The first two terms are related to finding a state starting with a state
Last term indicates the reverse process
Current from any lead to system is
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

26. Coulomb blockade oscillations
conductance versus gate voltage
The upper graph is for higher temperature
Obtained by numerical solution of
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

27. Coulomb blockade oscillations
The conductance is maximum at degeneracy points where
The addition energy is given by
This vanishes at degeneracy points.
The conductance is low between these points at low T.
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

28. Coulomb Blockade Staircase
Current-voltage curve.
Coulomb staircase: voltage-current curves of asymmetric and symmetric junctions respectively(left is assymetric) at low temperature at different gate voltages. Dashed line shows the change at higher temperature .Voltage is in units of
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

29. Coulomb Blockade Staircase
Coulomb gap seen at low T and low V
Gap closes at degeneracy points.
In assymetric junction, Coulomb stair is more prominent.
Observed due to participation of higher charged states in transport at higher voltage.
Threshold voltage of nth step is
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

30. Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)
Coulomb Diamonds
Contour plots of J(V, VG ) and
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

31. Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)
Coulomb Diamonds
The white rhombic-shaped regions are the regions of Coulomb blockade.
At zero temperature there is no current, one state with n electrons is stable with respect to tunneling. For example, near the point V = VG = 0 the state n = 0 is stable. If one changes the gate voltage, the other charge states become stable.
At large enough bias voltage the stability is lost and sequential tunneling events produce a finite current. The contour plots, as well as a schematic representations of the boundaries of the stability region in the (V, VG) plane,are often called stability diagrams.
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

32. Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)
Coulomb Diamonds
Stability condition
Implications
These inequalities show that if the system leaves the stability region at finite bias voltage V , the new state with n = n ± 1, which occurs as a result of tunneling through one junction, is also unstable with respect to tunneling through the other junction. As a result, the systems returns back to the state n, and the cycle is started again, thus the current flows through the system.
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

33. Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)
APPLCIATIONS
Coulomb Blockade Thermometer
Application to Quantum Dots
Memory Devices etc.
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

34. Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)
THANK YOU
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)