7.1 Transition Rates according to the NMP Theory

The transition rates of charge transfer reactions must be discussed on the basis of configuration coordinate diagrams. One such a diagram is depicted in Fig. 7.1 for the case of hole trapping. The adiabatic potentials Ui(q)  and Uj(q)  in the configuration coordinate diagrams are approximated by the parabolas around their respective minima Ui  and Uj  assuming the harmonic approximation:

        1    2      2
Ui(q)  = 2 M ωi(q - qi) + Ui                (7.1)
        1    2      2
Uj(q) = 2 M ωj(q - qj) + Uj                (7.2)
ωi  and ωj  denote the vibrational frequency of the oscillator potential when the defect is in the charge state i  and j  , respectively. It is stressed that these oscillator potentials of both charge states are assumed to have different curvatures ωi ⁄= ωj  in this derivation. The transition barriers ΔUb,ij  and ΔUb,ji  differ by the energy Uj - U(qi)  , which can be expressed as
Uj - U(qi) = Uj - Ui + Ui - U (qi)
           = Ev - Et - ΔE                   (7.3)
using the relations
 Ui - Uj  = Et - Ev ,                    (7.4)

U(qi)- Ui = Ev - E = ΔE  .               (7.5)


Figure 7.1: The configuration coordinate diagram for a hole trapping process. The left parabola corresponds to the case when the defect is neutral and the holes reside in the substrate valence band. Then the hole can be thermally excited by an energy ΔE  , accompanied by an upwards shift of the left parabola from Ui(q)  (solid) to U (q)  (dashed). By contrast, when the defect is positively charged (right parabola), the whole system including the defect and the substrate is represented by the parabola Uj(q)  (solid). In general, the curvature of Ui(q)  and Uj(q)  do not need to be equal. As a consequence, both adiabatic potentials are characterized by their own oscillator frequency (ωi  , ωj  ) and in further consequence their own Huang Rhys factor (Si  , Sj  ).

Ui - Uj  corresponds to the separation of the trap level from the valence band edge and U(qi)- Ui  gives the kinetic energy of the substrate hole. Making use of expression (7.3), the difference between the transition barriers can be expressed as:

ΔUb,ij - ΔUb,ji = Uj - U(qi)
               = - ΔE - Et + Ev             (7.6)
Using the identity (2.62) and equation (7.6), one obtains the following relation:
exp(- βΔUb,ji)fn(E ) = exp(- β ΔUb,ij)exp(- β(Et - Ef))fp(E ) (7.7)
Analogously to the equation (2.59), the trapping dynamics are governed by the rate equation
∂f   =  E∫v((1- f )eNMP (E)exp(- βΔU    )f (E )- fcNMP (E )exp(- βΔU    )f (E))D  (E)dE         (7.8)
 tt    -∞       t p               b,ji n       tp               b,ij p      p
eNpMP (E)  = eNpM,0P exp(- βΔUb,ji) ,             (7.9)
NMP         NMP
cp   (E)  = cp,0 exp(- βΔUb,ij) .            (7.10)
Making use of (7.6), the above rate equation can be simplified to
       (                                   ) E∫v
∂tft  =  (1- ft)eNpM,0P exp(- β(Et - Ef)) - ftcNpM,0P    exp(- βΔUb,ij)fp(E)Dp (E )dE .       (7.11)
                                            - ∞
In order to evaluate the integral in the above equation, analytic expressions for ΔUb,ij  and Dp (E)  are required. The former is defined as the energy difference between Ui  and the intersection point IP in the configuration coordinate diagram. The position of of this point can be derived from the condition
Ui(q) = Uj(q)                       (7.12)
and reads
                   ∘ -----------------------------
            qj-qi     (qj-qi)2  (Uj-Ui)∕(12Mω2i)+(qj-qi)2
(q- qi) = - Ri-1    (Ri-1)2 +        Ri-1             (7.13)
Ri  = ωωij .                        (7.14)
Inserting the expression (7.13) into equation (7.1), one obtains
ΔUb,ij  = Ui(q)- Ui
           S ℏω        ∘ -----------U---U--
       = --2i--i2(1- Ri  1 + (R2i - 1)-j---i)2 ,    (7.15)
         (Ri - 1)                    Siℏωi
where the Huang Rhys factor Si  is defined by the equation
1    2      2
2M ω i(qj - qi) = Siℏωi .                 (7.16)
A second order expansion of equation (7.15) delivers
ΔUb,ij ≈  S(1i+ℏRωii)2-+ 1R+iRi (Uj - Ui)+ 4SRiiℏωi(Uj - Ui)2 .    (7.17)
According to equation (7.14), the oscillator frequencies ω
  i  and ω
 j  differ and thus the quantity R
  i  deviates from unity. Since R
  i  enters the above expression for the barrier height, the oscillator frequencies have a strong impact on the transition rates. When the kinetic energy of the substrate hole is taken into account, U
  i  must be replaced by U(q )
   i  and equation (7.17) can be rewritten as
         -Siℏωi   -Ri-                    --Ri-                  2
ΔUb,ij  ≈ (1+Ri)2 + 1+Ri((- ΔE )+ (Ev - Et))+ 4Siℏωi((- ΔE ) +(Ev - Et)) .     (7.18)
In the case of strong electron-phonon coupling, Siℏωi ≫ |ΔE +Et - Ev| holds and the third term can be neglected. Assuming parabolic bands (see Appendix A.4), the valence band density of states can be expressed as
           √ ----
Dp(E) = Dp,0  ΔE                       (7.19)
Dp,0 = √m-e---.                       (7.20)
Using (7.19), the integral in equation (7.11) can be evaluated as
   exp(- βΔU   )f (E )D (E)dE =
            b,ij  p    p
-∞    (           )    (                 )                     (      )                    (7.21)
          -Siℏωi--          -Ri---                              ---β-- - 3∕2
= exp  - β (1+ Ri)2 exp  - β 1+ Ri(Ev - Et) Dp,0 exp (β (Ev - Ef)) 1 + Ri    Γ (3∕2)
and simplifies to
                                    (           )    (                  )
 E∫v                                      -Siℏωi--         -Ri---                3∕2
- ∞ exp(- βΔUb,ij)fp(E)Dp (E )dE  = exp  - β (1 + Ri)2 exp  - β 1+ Ri(Ev - Et) (1+ Ri) p
                               = exp(- β ΔUb,ij|ΔE=0)(1+ Ri) p                            (7.22)
p = Dp,0 exp(β(Ev - Ef)) β   Γ (3∕2) .         (7.23)
Here, Γ (x)  denotes the Gamma function, which is defined by
Γ (x)  =   tx- 1exp(- t)dt .                 (7.24)
With (7.22), the compact form of the rate equation (7.11) can be rewritten as
∂f   = ((1- f )eNMP exp(- β(E - E )) - f cNMP ) exp (- βΔU |    ) (1 + R )3∕2p .       (7.25)
 tt          t p            t   f    t p              b,ijΔE=0       i
From this, it follows that
          NMP                         3∕2
1∕τcap  = cp,0 exp(- βΔUb,ij|ΔE=0)(1+ Ri)   p ,                    (7.26)
1∕τem   = eNp,M0Pexp(- β(Et - Ef))exp(- β ΔUb,ij|ΔE=0)(1+ Ri)3∕2p .   (7.27)
Just as in the standard SRH theory (see Section 2.5), the right-hand side of equation (7.26) can be simplified using the definition
NMP                          3∕2     NMP
cp,0 exp(- β ΔUb,ij|ΔE=0)(1+ Ri)   = σp   vth,p       (7.28)
σp   = σp,0 exp(- βΔUb,ij|ΔE=0) .              (7.29)
Analogously to Section 2.5, thermal equilibrium can be assumed so that the trap occupation follows Fermi-Dirac statistics and detailed balance applies for the rate equation (7.25). Then cNpMP  equals eNpMP  and the hole capture and emission time constant reads
              NMP                         3∕2
1∕τcap  = vth,pσp,0 exp(- βΔUb,ij|ΔE=0)(1+ Ri)   p                      (7.30)
1∕τem   = vth,pσNpM,P0 exp(- β(Et - Ef))exp(- βΔUb,ij|ΔE=0)(1+ Ri)3∕2p .   (7.31)
It is emphasized that the barrier heights are correctly calculated by determining the crossing point of two parabolas. Thereby, one avoids the artificial differentiation, whether E
 t  is located above or below E
  v  , as it has been the case in equation (6.10) of the TSM. Additionally, the NMP barriers have not been assumed to be independent of the energy of the hole in contrast to the Kirton model and the TSM.