Transition Rates according to the NMP Theory

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

with

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)

with

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)

with

3∕2 Dp,0 = √m-e---. (7.20) 2π2ℏ3

Using (7.19), the integral in equation (7.11) can be evaluated as

∫Ev 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 3∕2 = exp(- β ΔUb,ij|ΔE=0)(1+ Ri) p (7.22)

with

-3∕2 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) 0

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)

with

NMP NMP 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.

[next] [front] [up]