WKB Formulas for Different Shapes of Energy Barriers

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)

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 $x0$ .

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