Quantum Resistance and Quantum Conductance

QUANTUM RESISTANCE AND QUANTUM CONDUCTANCE

Definition of quantum conductance 

The quantum conductance or conductance quantum, denoted by the symbol G₀ is the quantized unit of electrical conductance.

It is defined as

Definition of quantum Resistance

The reciprocal of quantum conductance is Quantum Resistance or Resistance quantum.

Where

e → electron charge and

h → Planck's constant

Derivation

A simple derivation of the formula for G₀ is given below.

A one dimensional (1D) quantum wire [see Fig. 5.28] connects adiabatically two reservoirs with chemical potential μ₁ and μ₂. The connections are assumed to be non-reflecting.

Fig. 5.28 one dimensional quantum wire connecting adiabatically two reservoirs with chemical potential μ₁, and μ2.

It is also assumed that the wire is sufficiently narrow so that only the lowest transverse mode in the wire is below the Fermi energy (E_F)

The current density is given by

J = -nev_d …….(1)

where

n = Density of electrons

e = Charge of the electron

V_d = Drift Velocity of the electron

The density of electrons is determined by

………..(2)

Substituting in eqn. (2) in equ. (1) we get

 ………(3)

where dn/dE is the density of states

we know

E=Nhv ……..(4)

Where N → number of electrons

We know

N = nAl……..(5)

Substituting Eqn (5) in Eqn. (4) we get

E=nhvAl

Differentiating we get

dE = dn hv Al

……….(6)

Accounting spin dengeneracy, multiply by ‘2' in Eqn (6) we get

……..(7)

If V is the voltage between two reservoirs, then we can write

μ₁ – μ₂ = -eV ………..(8)

Substitute Eqn.(7) and Eqn.(8) in Eqn. (3) we get

………..(9)

We know velocity = distance/time

V_d = l/t …………(10)

We also know frequency=1/time

V = l/t ……….(11)

Substituting eqn. (10) and Eqn. (11) in equation (9) we get,

……(12)

………(13)

Therefore, Eqn (13) can also be written as

…….(14)

…....(15)

….....(16)

It is important to note in this derivation, nothing is mentioned about the material properties of the conductor or its dimensions, therefore G is a truly fundamental unit.

If there are N electronic channels, then equation (15) becomes

………..(17)

Equation (17) is called Landauer formula

 (i.e) Conductance ………(18)

Similarly we can write equation (16) as

Resistance …………(19)

Conclusion

As the number of channels increases, conductance increases and resistance decreases. The classical theory also predicts this behavior, although the quantum theory shows that this happen in discrete steps, as the number of electron channels increases.

As N gets very large, the electron channels essentially form a continuum, and the quantum theory tends towards the classical limit.