Page 106 - Principles and Applications of NanoMEMS Physics

94 Chapter 3

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dN V § 2 m ·
E
D () ≡ = 2 ¨ 2 ¸ E . (43)
dE 2π © = ¹
Excitation of the ground state of a Fermi gas requires, due to Pauli exclusion
principle constraints, the addition of particles with momentum greater
G
G
k > k , or the destruction of a particles (creation of holes) with k < k .
F F
However, if these particles came from outside the system, then the total
number of particles N would change and we would have a different system.
When one insists on inducing excitations that conserve the number of
G
G
particles, then creating a particle with k > k = k , is accompanied by
F
G G
creating a hole with k < k = k′ , i.e., particle-hole excitations which can
F
G G
be identified by two quantum numbers k, k′ are created.
These excitations may be caused by a number of influences, in particular,
a rise in temperature or the application of a magnetic field k . Since, under
F
no interaction, all states are occupied up to k , electrons closest to E will
F F
require the minimum energy to excite. Thus, the energy necessary to excite
2
F
an electron of momentum k , for instance, is E = = 2 (k − k 1 ) .
1 Excitation
2 m
Temperature-induced excitations of the Fermi gas are captured by the
specific heat, given by [28],
∂ E π 2
C el = = D (E F ) Tk B 2 = T γ , (44)
∂ T 3

where k is Boltzmann’s constant, and magnetic field-induced excitations
B
are captured by the magnetic susceptibility given by,

χ = M = 2D (E )µ , (45)
2
B F B
where µ is the Bohr magneton. Clearly, these quantities involve the
B
density of states evaluated at one point, namely, the Fermi energy. This fact,
coupled to the circumstance that, as long as one is dealing with a non-
interacting free electron gas (ED ) will have the same value, suggests that
F
solving both (44) and (45) for D (E ) and taking the ratio of the resulting
F
quantity must be equal to one. This ratio, called the Wilson ratio, is given by
[133],