Single Electron Phenomena

SINGLE ELECTRON PHENOMENA

Theory

In electronics, the transistor is a king. Computers uses transistors to compute. Also, transistors are used as tiny switches for tunning on and off. It is also used in transferring and amplifying signals, making logical decisions, etc.

Today, microchips have a billion transistors, each one turning on and off a billion times every second. These chips require manufacturing processes with roughly 100‒nm resolution. Every year this resolution drops, manufacturing even small transistors. Thus, each transistor is reduced to a few atoms or even a single atom.

In 1970, silicon transistors required about 10 million electrons. Current transistors requires closer to 10,000 electrons.

In fact, we here already built single‒atom and single‒electron transistors. We can use single‒electron transistors to make sensitive amplifiers, electrometers, oscillators and other digital electron circuits. All these instruments will be operated by using single electrons or quantum dots.

Condition for single electron phenomena to occur

For single electron phenomena to occur, we have to keep the single electron or quantum dot in isolation.

If any electron on one side of the barrier could just junnel across it, there would not be any isolation. The dot would not be a quantum dot because it would still essentially be part of the bulk, so we need to control the addition and subtraction of electrons.

Rules for single electron phenomena to occur

There are two rules for preventing electrons from tunneling back and forth from a quantum dot. When we follow these rules, they help to ensure that the dot remains isolated and quantized.

The rules are

Rule 1: The coulomb Blockaded

Rule 2: Overcoming uncertainty

Rule 1: The coulomb Blockade

We know that the coulomb blockade can prevent unwanted tunneling. Hence we can keep the quantum dots isolated. The condition for this is given by

  E_C =  e² / 2C_dot >> K_BT         ... (1)

Rule 2: Overcoming uncertainity

For the second condition, to keep quantum dots electronically isolated, we look to the uncertainty principle.

According to uncertainty principle

 ΔΕ_{C .} Δt = h         …………(2)

Energy uncertainty

 ΔΕ_C = h/Δt                ………(3)

Here, h is the Planck's constant and ∆t is the measurement time, quantum dot is a tiny capacitor then the measurement time ∆t is capacitor's time constant.

The time constant for a capacitor is RC, where R is the resistance and C is the capacitance.

 We can write the time constant as

 Δt = R_tC_dot          ……… (4)

Substituting Eqn. (4) in Eqn (3), we et

 ΔΕ_C = h / R_tC_dot        ……… (5)

Where R_t = tunneling resistance and

C_dot  = capacitance dot.

Our aim is to keep electrons from tunneling freely back and forth to and from the dot. To ensure this, the uncertainty of the charging energy must be less than the charging energy itself.

For maintaining electron isolation in quantum dot, we need

AE_C < E_C       …………(6)

Substituting Eqn.(3) and Eqn. (1) in equation (6), we get 2

 h/∆t < e² / 2C_dot         ... (7)

 Substitute Eqn.(4) in Eqn.(7), we get

       ……….(8)

In other word we can write the Eqn. (8) as

 R_t >> h/e²        ………(9)

Substituting the values for h = 6.625 × 10^‒34 Js and E = 1.6× 10^‒19 C, we get

 h/e² = 25878 Ω is the resistance quantum.

This high resistance value is like a thick insulating material surrounding the quantum dot.

Thus, we keep the quantum dots electronically isolated.

Conditions for tunneling

The two tunneling conditions are

 When these conditions are met and the voltage accross the quantum dot is scanned, then the current jumps in increments everytime the voltage changes by the value of equation ∆V= e / C_dot.

This is called a coulomb blockade because the electrons are blocked from tunneling except at the discrete voltage change positions.

The two conditions or rules which explain the single electron phenomenon will help to build a single electron transistor,