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COULOMB BLOCKADE PH4101

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

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COULOMB BLOCKADE PH4101

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  1. COULOMB BLOCKADEPH4101 • 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. 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. Boundary Condition • Transmission Matrix and Scattering matrix Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

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

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

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

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

  31. 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,areoften called stability diagrams. Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)

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

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

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