Density of States (b) in Quantum Wire (One Dimension), Quantum Dot (Zero Dimension)

DENSITY OF STATES

3. DENSITY OF STATES IN QUANTUM WIRE (OR) DENSITY OF STATES IN ONE DIMENSION

Consider the one dimensional system, the quantum wire, in which only one direction of motion is allowed. e.g along x-direction.

Explanation

In one dimension, such as for a quantum wire, the density of states is defined as the number of available states per unit length per unit energy around an energy E.

The electron inside the wire are confined in a one-dimensional infinite potential well with zero potential inside the wire and infinite potential outside the wire.

At x = 0; V(x) = 0 for an electron inside the wire

At x = L; V(x) = ∞ for an electron outside the wire.

The reduced phase space now consists only the x plane and nx coordinates as shown in Fig. 5.14

In one-dimensional space n2 = nx2

Derivation

The number of available energy states lying in an interval of length is

Z'(E)dE=n+dn - n = dn

Substitute Eqn. (5) in the above equation

….....(15)

According to Pauli's exclusion principle each energy level can occupy 2 electrons of opposite spin.

 Multiplying Eqn. (15) by 2 we get

Number of quantum states per unit length and unit energy is

….....(16)

The density of states in one dimensional (quantum wire) is given by

……..(17)

If the electron has potential energy E0, we have

……….(18)

Inference

From Equ. (18), the density of states in one dimensional system has a functional dependence on energy.

i.e.,

For more than one quantized state, the one dimensional density of states is given by

.......(19)

Where En are the energies of the quantized states of the wire and σ (E-En) is the step function.

The density of states in quasi-continum (or) quantum wire is shown in Fig. 5.15. The discontinuities in the density of states, are known as Van Hove Singularities.

4. DENSITY OF STATES IN QUANTUM DOT (or) DENSITY OF STATES IN ZERO DIMENSION

In a zero dimensional system (such as quantum dot), the density of states are truly discrete and they don't form a quasi continuum.

Explanation

In zero dimensional system [quantum dot], the electron is confined in all three spatial dimensions and hence no motion of electron is possible. Each quantum state of a zero dimensional system can therefore be occupied by only two electrons. So, the density of states for a quantum dot is merely a delta function.

…......(20)

Here, the factor 2 accounts for spin.

For more than one quantum state, the density of sates is given by

………(21)

The density of states for quantum dot is shown in Fig 5.16

Table 5.2 Summary of the density of states with various degrees of freedom.