A.1 Fermi’s Golden Rule

Fermi’s golden rule provides one way to calculate the transition rate between two certain quantum mechanically defined states. Due to its generality, it has various applications in the field of atomic, nuclear, and solid-state physics. In the case of NBTI, it is of most interest for charge transfer reactions and electron tunneling in particular. In the following, Fermi’s golden rule is derived for electron tunneling from the substrate into an oxide defect as illustrated in Fig. A.1.


Figure A.1: Schematic representation of the channel, the barrier, and the trap region. Interactions between the channel and the trap system are neglected to first order so that both systems are characterized by a separate Hamiltonian Hch(x)  and Htr(x)  , respectively. This means that the attractive trap potential Vtr(x)  is not accounted for in Vch(x)  so that Vch(x ) = V0  in the trap region. Vice versa, the channel potential Vch(x)  is omitted in Vtr(x)  , which consequently takes the value V0  in the channel region.

The system is divided into three separate regions, namely the channel, the insulator barrier, and the trap region. The electron wavefunctions ψl(x)  and ψr(x)  extend into the classically forbidden barrier region. Their overlap actually leads to a mutual influence between the channel and the trap system. However, this influence is assumed to be negligible so that both systems can be treated independently to first order. This justifies the assumption that in a first approximation the channel and the trap system can be described by their own Hamiltonians Hch(x)  and Htr(x)  . For the derivation of the tunneling rate, the Hamiltonian of the common system is taken as a starting point.

H (x)  = Hch(x)+ H ′(x)                 (A.1)
Hch(x ) = - 2m-Δ  +Vch(x)               (A.2)
  ′          e
H (x)  = Vtr(x)                         (A.3)
H  (x)  is viewed as the time-dependent perturbation that triggers the scattering from the band states ψl(x)  into the trap states ψr(x)  . The solution ψ(x,t)  of the common system H (x)  can be written as a linear combination of the eigen wavefunctions ψl(x)  of the unperturbed system Hch(x )  .

        ∑         (   El )
ψ(x,t) =    al(t) exp - i-ℏ t ψl(x)             (A.4)
This expansion of the wavefunction is inserted into the time-dependent Schrödinger equation
H(x) ψ(x,t) = iℏ ∂tψ (x,t)                  (A.5)
and leads to
                          (      )                          (      )
(Hch(x )+ Vtr(x)) ∑  al(t) exp - iElt ψl(x) = iℏ ∑ ((∂tal(t)) exp - iElt  ψl(x)
                 l            ℏ               l                 ℏ
                                                (   E )     (   E )                  (A.6)
                                          +al(t)   - i-l  exp  - i-lt  ψl(x)) .
                                                    ℏ           ℏ
Due to Hch(x )ψl(x) = Elψl(x)  , the above equation simplifies to
      ∑         (   El )            ∑             (   E )
Vtr(x)    al(t) exp - i-ℏ t ψl(x) = iℏ   l(∂tal(t)) exp - iℏlt ψl(x) .       (A.7)
Multiplying both sides by      *     E
ψr(x) exp(iℏrt)  from the left and integrating over space yields
           i    (   El - Er ) ∫
∂tar(t) = - ℏ-exp  - i--ℏ---t     ψ*r(x) (H  - El) ψl(x) dx ,     (A.8)
                (          )  ∫
       = - i-exp  - iEl --Er-t   ψ*r(x) (H  - Hch ) ψl(x) dx ,   (A.9)
           ℏ    (      ℏ   )  ∫
           i-       El --Er       *
       = - ℏ exp  - i  ℏ   t     ψr(x) Vtr(x ) ψl(x ) dx ,      (A.10)
where M
  lr  is referred to as the matrix element. |a (t)|2
 r  gives the transition probability P  (t)
 lr  that an electron initially located in the state ψ (x)
 l  , evolves into the final states ψ (x)
 r  after a time t. Therefore, it must be divided by the time t  in order to yield the transition rate r
 lr  .
                           ||  ∫t   (i(E-E )t′)   ||2
     P (t)  |a (t)|2        ||1ℏ   exp ---lℏ-r-  dt′||
rlr = -lr---= --r--- = |Mlr|2 ---0-------------------    (A.11)
       t       t                      t
The integrand is sharply peaked at El = Er  and can be approximated as a δ  -function.
|                        |
||   ∫t   (          ′)   ||2
||1-   exp  i(El --Er-)t dt′|| ≈ 2πtδ(El - Er)        (A.12)
|ℏ  0          ℏ         |    ℏ
Substituting the integral in rate expression (A.11), one finally obtain ‘Fermi’s golden rule’.
rlr = 2π-|Mlr|2 δ(El - Er )               (A.13)