Page 113 - Principles and Applications of NanoMEMS Physics

3. NANOMEMS PHYSICS: Quantum Wave Phenomena 101

unraveled by solving the interacting electron problem. The Hamiltonian in
question, given by (see Appendix B),

° § 1
+
H 0 = 2 π v ­ ¨ + g · ¸ ¦ ¸ ª 1 N ˆ 2 v + ¦ n q b qv b qv º »
«
® ¨
L 2 ° © g ¹ v = L , R ¬ 2 q ¼ , (63)
¯
§ 1 ª · + + º ½
°
+ ¨ ¨ − g ¸ N ˆ L N ˆ R − ¦ n q (b qR b qL + b qR b qL ) ¾
»
«
¸
© g ¬ ¹ q ¼ °
¿
must be diagonalized to determine the pertinent types of solutions holding in
one dimension. This Hamiltonian diagonalization is facilitated by the
procedure of bosonization [137]-[139] discussed in detail in Appendix B. In
essence, one-dimensional bosonization transforms a nondiagonal fermionic
Hamiltonian into a diagonal bosonic one, with the assumption that the one-
dimensional dispersion relatonship is linear, and given by
E () vk = = k − k [134]. The nature of this dispersion relation gives rise
F F
to the transport characterization in terms of spinless left- and right-moving
electrons with respective electron densities N and N , the parameter g,
L R
which captures the electron-electron interaction strength in the problem, and
the Fermi velocity v . Kane and Fisher [140] have captured this
F
phenomenology with he following set of expressions. The Hamiltonian (63)
is rewritten as [140], [141],

H = v π [N + N + 2 λ N N ], (64)
2
2
0 0 R L R L
with

( ª g −1 + g )º ( 1 g− 2 )
v = v , λ = , (65)
0 « » 2
+
¬ 2 ¼ 1 g
with λ as the interaction strength parameter between the left- and right-
moving electron species, and g, called the Lüttinger parameter. For g = 1
the interaction is zero, and the Hamiltonian then captures the behavior of a
noninteracting electron gas with velocity equal to the Fermi velocity
v = v . From (65) it is seen that repulsive interactions, which per (64)
0
F
imply λ > 0 , lead to g < 0 , and the opposite is true for attractive
interactions. In terms of the two-particle interaction potentials, V and V ,
2 4
between fermions moving in opposite directions, namely, left and right, and
either both left- or both right-moving, respectively, v and g are given by,