A.4 Density of States

When going from small electron systems, such as atoms and molecules, to large electron systems, solids for instance, the number of available electron states reaches high values so that it is best expressed in terms of a density per energy and volume. In solid state theory the assumption of periodic boundary conditions is frequently employed and delivers simple first-order approximation of the density of states (DOS). However, in a few cases, such as tunneling, DOS decomposed in one energy component perpendicular (Ex  ) and one parallel (Eyz  ) to a certain plain is required. The corresponding derivation will be outlined in the following.

In each direction with periodic boundary conditions, each quantum number n  is directly related to one wavevector k  :

k =---n ,                        (A.37)
If the length of the periodicity L  is increased, the single quantum states narrows in the k  space and k  becomes an continuous quantity. Then the summation over the single states can be replaced by an integral.
∑     +∫∞      L  +∫∞
   =     dn = 2π   dk                   (A.38)
 n   -∞         -∞
Making use of the relation
     2 2
E = ℏ-k-                         (A.39)
one obtains the number of states in the x  -direction
               +∫∞         +∫∞   ∘ ----
     ∑     Lx-               1-- -me-
Nx =    = 22π    dkx = Lx    πℏ  2Ex dEx .        (A.40)
     nx        0          0 ◟---◝◜---◞
For periodic boundaries in two dimensions, an integral transformation to polar coordinates is carried out in order to obtain the number of states in the yz  -plane.
       ∑     LyLz +∫ ∞+∫ ∞             +∫ ∞ me
Nyz =      = (2π)2      dkydkz = LyLz   2πℏ2 dEyz     (A.41)
      ny,nz        0  0          ◟=◝A◜y◞z 0 ◟ ◝◜-◞
with Eyz  being the energy in the yz  -plane. Combining both solutions yields
                         +∫∞ +∫∞     ∘ ----
Nxyz = ∑         = LxAyz       me--  me-dEyzdEx  , (A.42)
        nx,ny,nz                π2ℏ3  2Ex
                         0+∞ 0E
                         ∫ ∫   me  ∘ me--
                 = LxAyz      2π2ℏ3  2Ex-dExdE ,   (A.43)
                         0 0
where an integral transformation from (Eyz,Ex )  to (E, Ex)  with the constraint E = Ex + Eyz  is performed. The split one/two dimensional DOS is defined as
D1D+2D(Ex)  = 2mπe2ℏ3  m2Eex-                (A.44)
while the commonly known three dimensional DOS for a free electron gas reads
           ∫E     ∘ ----
D   (E)  =   -me--  -me-dE  ,            (A.45)
  3D         2π2ℏ3  2Ex   x
           0m   ∘ -----
         = --2e3  2meE  ,                (A.46)
           2π ℏ
which is usually found in textbooks [129].

For the case that the electrons are confined in the x  -direction, the number of states are counted in the following way:

     ∑                +∫∞∑
Nx =    θ(Ex - Enx) =      δ(Ex - Enx)dEx         (A.47)
      nx              0  nx
Here, E
  nx  denotes the quasi-bound states with the quantum number n
 x  . In order to refer N
  x  to the unit volume, one must introduce the the square of the wavefunction.
∑               +∫∞ ∑
   θ(Ex - Enx) =       δ(Ex - Enx)dEx            (A.48)
nx              0  n◟x----◝◜-----◞
When Dc1D(Ex)  is multiplied with D2D  and the same integral transformation as for the derivation of D3D(E )  is performed, one obtains the DOS for an one-dimensionally confined electronic system:
           ∫E    ∑
Dc3D(E)  =   D2D    δ(Ex - Enx )dEx ,       (A.49)
           0      nx
         = D2D    Θ(E - Enx)                (A.50)
Note that in this derivation the spin degeneracy introducing a factor 2  has been neglected.