A.3 WKB Formulas for Different Shapes of Energy Barriers

In the classical forbidden region, the shape of the wavefunction is dominated by the exponential term in equation (A.22).

           (    x∫2                 )
˜          (  1-  ∘ -------------  )
ψ(x)  ≈ exp  -ℏ     2me(ϕ(x)- E ) dx         (A.23)
           (    x1                )
              √2m---x∫2∘ --------
      ≈ exp( ------e    ϕ(x)- E dx)          (A.24)
                ℏ   x1
x1  and x2  stand for the classical turning point at the semiconductor-dielectric interface and the position of the trap, respectively. Supposing that only a negligible amount of charges is located in the dielectric, the potential energy ϕ(x)  can be expressed as
ϕ(x)  = ϕ + ϕ2---ϕ1 (x- x ).              (A.25)
         1  x◟2--◝◜-x1◞      1
             =q0Fox
For a trapezoidal barrier (see Fig. A.2), ψ˜(x)  simplifies to
           (                                   )
               √2me-∫x2∘ ---------------------
ψ˜(x)  ≈ exp( - --ℏ--    ϕ1 + q0Fox (x- x1)- E dx)  , (A.26)
                    x1
           (   2√2me-                       3x )
      ≈ exp  -3ℏq-F-- (ϕ1 + q0Fox (x- x1)- E )2|x21 , (A.27)
           (    √0-ox (                    ))
      ≈ exp  --2-2me-  (ϕ2 - E )32 - (ϕ1 - E )32 .    (A.28)
              3ℏq0Fox
If tunneling occurs through a triangular barrier (see Fig. A.2), the classically forbidden region extends to
         E - ϕ
x0 = x1 +-----1.                     (A.29)
         q0Fox
For negative electric fields (ϕ2 < E < ϕ1  ), one obtains
           (                       )
              1 x∫0∘ -------------
˜ψ(x) ≈ exp (- ℏ-    2me(ϕ(x)- E) dx)  ,            (A.30)
               x1
           (  2√2me--                      3 x)
     ≈ exp  - 3ℏq-F--(ϕ1 + q0Fox (x - x1)- E)2|0x1 , (A.31)
           (  √--0-ox        )
     ≈ exp  -2-2me- (ϕ1 - E)32  .                    (A.32)
            3ℏq0Fox
while positive electric fields (ϕ1 < E < ϕ2  ) results in
           (    x∫2                 )
˜          (  1-  ∘ -------------  )
ψ(x) ≈ exp  - ℏ     2me(ϕ(x)- E) dx   ,            (A.33)
           (   x√0----                         )
              2--2me-                      32 x2
     ≈ exp  - 3ℏq0Fox (ϕ1 + q0Fox (x - x1)- E) |x0 , (A.34)
           (  2√2me--        3)
     ≈ exp  - 3ℏq-F--(ϕ2 - E )2 .                  (A.35)
                 0 ox
The these two cases are commonly known as the Fowler-Nordheim formulas [186]. For a rectangular barrier with E < ϕ1 = ϕ2 = ϕ  , the integral in equation (A.23) simplifies to a multiplication.
           (  √2me(ϕ-E)       )
˜ψ(x) ≈ exp  - ----ℏ----(x2 - x1)             (A.36)

PIC

Figure A.2: Schematic representation of a trapezoidal (left) and a triangular barrier for a negative (middle) and a positive (right) gate bias. ϕ(x)  displays the shape of the potential energy and takes the values ϕ1  and ϕ2  for the semiconductor-dielectric interface at x1  and the trap at x2  . For the case of triangular barriers, the classical forbidden region is decreased to the point x0  .