Page 105 - Principles and Applications of NanoMEMS Physics

3. NANOMEMS PHYSICS: Quantum Wave Phenomena 93

lowest energy eigenvalue (ground state energy) is given by the sum of the
one-electron energies up to a maximum energy level denoted by E and
F
called Fermi energy. This is obtained when N electron states with energy less
than E are occupied, and all states with energy greater than E are
F F
unoccupied. To obtain an expression for E , one pictures the states in (38)
F
as a grid of point in k k k space where they form a fine three-dimensional
x y z G
grid of spacing π L , such that a sphere centered at k = 0 would contain
2
4π 1 Vk 3 Vk 3 G
k ⋅ = ⋅ 2 = points of the grid when its radius is k ,
3
3 § 2π · 3 6π 2 3π 2
¨ ¸
© L ¹
including spin. Since each point in the grid represents one electron, the
number of grid points contained in a sphere with the largest radius, k ,
F
corresponding to E must equal N,
F
Vk 3
F = N . (39)
3π 2

Thus, the largest electron momentum is,

3π 2 N ) 1 / 3
k = 1 / 3 = (3π 2 n , (40)
F
V
where n is the electron density in the metal, and the Fermi energy is,

2
= § 3π 2 N · 2 / 3 = 2 ) 1 / 3
E = ¨ ¨ ¸ ¸ = (3π 2 n . (41)
F
2m © V ¹ 2m
At absolute zero, all levels are filled up to E . For an arbitrary energy E,
F
less than E , the total number of electrons with energy less than E is given
F
by,

3 / 2
N = V § ¨ 2m · ¸ , (42)
3π 2 © = 2 ¹

from where the density of states is given by,