2.5 Effective Rates into Single Traps

Up to this point, trapping has been defined for transitions from one band state in the substrate into one trap state or vice versa. Each of the possible transitions is associated with one rate entering the trapping dynamics. In order to reduce the set of rate equations, compact analytical expressions for the overall trapping rates into one defect are required. In this section, a derivation of the sought expressions will be presented for the case of elastic as well as inelastic tunneling.

2.5.1 Elastic Electron Tunneling

The derivation below follows the approach of Tewksbury [23] with some slight modifications. For a proper description of charge trapping, Fermi’s golden rule is taken as a starting point. This fundamental law (see Appendix A.1) gives the rate for a transition between a certain initial and final state. In its most general form it reads:

rtb = 2π|Mtb|2δ(Et - Eb)                   (2.33)
The subscripts t  and b  denote a trap or a band state, respectively, and Mtb  represents the tunneling matrix element. The δ  -function indicates that the electron energy before and after the transition must be conserved as it is required for elastic tunneling. In semiconductor devices, the charge carriers captured in the dielectric can originate from several different energy levels of the substrate valence or conduction band. In order to account for their contribution to the whole tunneling rate, the sum over all k  states has to be taken.
   ∑   2π    2
r =    ℏ-|Mtk|δ (Et - E(k))                 (2.34)
Here, Et  and E(k)  is the trap level and the electron energy in the substrate, respectively. The subscript t  in equation (2.33) has been omitted since only tunneling into and out of a certain trap is considered. The matrix element Mtk  involves the trap wavefunction, whose exact form is in general unknown. Often, the calculation of the matrix element is simplified to a one-dimensional δ  -type trap potential. Its solution [23],
Mc∕v,tkx(Ex, xt)  = Mc0∕v0,tkx(Ex,xt) ζWKB,c∕v(Ex,xt) ,    (2.35)
consists of the factors
                   ℏ2Kc∕v,x(Ex,xt)∘ kc∕v,x(Ex)
Mc0∕v0,tkx(Ex,xt) =  ----mt------ ---2Lx---         (2.36)
                     (  x∫t              )
ζWKB,c∕v(Ex,xt)  = exp  -   Kc ∕v,x(Ex,xt)dx          (2.37)
K2c∕v,x(Ex,x ) = 2mt2-|Ec ∕v,ox(x)- Ex | ,          (2.38)
 k2c∕v,x(Ex)   = 2mn2∕p|Ex - Ec∕v,sc(xif)| .        (2.39)
Note that electron and hole tunneling proceed through an energy barrier formed by the oxide conduction and valence band, respectively. The matrix elements for these cases are labeled by ‘c  ’ and ‘v  ’, accordingly. The second factor ζWKB,c∕v(Ex,xt)  of the matrix element in equation (2.35) arises from the exponential decay of the electron wavefunction, which can be derived using the WKB approximation. Since this factor shows a dependence on the carrier kinetic energy |Ex - Ec∕v,sc(x )| perpendicular to the interface, the summation over all band states must be split into a one-dimensional and a two-dimensional part (see Appendix A.4):
           (                                                 )
∑             ∞∫ ∫E                     E∫v∫Ev
   = LxAyz (      Dn,1D+2D(Ex)dExdE +       Dp,1D+2D (Ex )dExdE )  ,         (2.40)
 k           Ec Ec                    -∞ E
                      ∘ -------------
Dn ∕p,1D+2D(Ex)  = mπn2∕ℏp3  2|E--mEn∕p-(x)| .          (2.41)
                          x  c∕v,sc
In Dn∕p,1D+2D(Ex)  a factor of two has been introduced in order to account for spin degeneracy. Note that the DOS has been derived based on the parabolic band approximation, however, the potential well in the inversion layer gives rise to the formation of subbands (see Section 2.1) and in consequence a different DOS. This means that the equation (2.40) neglects quantization effects in the inversion layer. Nevertheless, inserting the expression (2.40) into equation (2.34) yields
              ∞∫ ∫E
r = LxAyz 2π-(     Dn,1D+2D (Ex ) |Mc0,tb(Ex, xt)|2 ζ2  (Ex,xt)δ(Et - E)dExdE
          ℏ                                   WKB,c
             EcEc                                                                   (2.42)
           +       Dp,1D+2D (Ex ) |Mv0,tb(Ex,xt)|2 ζ2WKB,v(Ex,xt)δ(Et - E)dExdE ) ,
            - ∞ E
which can be simplified to
r = L A  2π-(  D       (E ) |M    (E ,x )|2 ζ2    (E  ,x )dE
     x yz ℏ      n,1D+2D  x    c0,tb  x  t    WKB,c  x  t  x
             Ec                                                      (2.43)
           +   Dp,1D+2D(Ex) |Mv0,tb(Ex,xt)|2 ζW2KB,v (Ex, xt)dEx) .
The factor ζWKB,v (Ex,xt)  enters the above equation as a square, which corresponds to a transmission coefficient of an electron through an energy barrier [126], and will be referred to as the ‘WKB factor’ in the following.

In this derivation, the calculation of the matrix element has been reduced to a one-dimensional problem in favor of a compact and analytical expression. In order to correct for this approximation, the term σn∕p,xy∕Ayz  must be introduced following Freeman’s approach [103].

r(Et)  = re(Et) + rh(Et)                                                 (2.44)
            2π-                                 2  2
re(Et)  = Lx  ℏ    Dn,1D+2D(Ex) σn,yz |Mc0,tb(Ex,xt)| ζWKB,c(Ex,xt)dEx      (2.45)
            2π ∫Ev
rh(Et)  = Lx ---   Dp,1D+2D(Ex) σp,yz |Mv0,tb(Ex,xt)|2 ζ2WKB,v(Ex,xt)dEx    (2.46)
             ℏ Et
Keep in mind that these equations describe elastic tunneling, meaning that a trap can only exchange charge carriers with those bound states, whose energy E  coincides with the trap level E
  t  — even though E  can have different components E
  x  . In this derivation, the cross-sections σ
 n∕p,yz  correspond to fitting parameters but can be estimated by analytical expressions presented in [23]. Since the values obtained by these expressions range around 10-16cm2  and are subject to small variations, the cross-sections are assumed to be constant throughout this thesis.

2.5.2 Shockley-Read-Hall Theory

Compact expressions for inelastic transitions are provided by the framework of the Shockley-Read-Hall (SRH) theory [127]. It has been developed to describe recombination centers in bulk but has also been extended for the case of electron or hole trapping into dielectrics [128]. The following derivation of the SRH rates is generalized to NMP transitions, presuming the case of hole trapping, and starts from equation (2.27).

r (E ) = A   fLSF(E ,E )                  (2.47)
 tb        tb  tb    t  b
The trap and the band state involved in the NMP transition are labeled by the subscripts t  and b  , respectively. Et  denotes the thermodynamic trap level and Eb  an arbitrary state in the valence band. Since the NMP theory assumes thermal transitions, the energies of the states t  and b  may differ, meaning that the trap can in principle exchange charge carriers with the entire valence band. In order to account for this fact, a sum over all band states has to be carried out.
r(E ) = ∑  A   fLSF(E ,E )                (2.48)
 t        b  tb  tb    t  b
Employing to the parabolic band approximation (see Section A.45), the sum over all band states b  can be approximated by an integral over the valence band DOS Dp (E )  , where E  can be identified with the energy Eb  of the valence band state b  (see Appendix A.4).
∑   = L  A   E∫vD  (E ) dE                 (2.49)
 b      x yz-∞   p
Using the expression (2.49), the transition rate can be rewritten as
                E∫v       LSF
rtb(E )  = LxAyz-∞ Atb(E)ftb (E)Dp (E ) dE .        (2.50)
For a hole capture process, one must account for the joint probability that the trap must be occupied by an electron while the band state with an energy E  is empty. The first condition is considered by the trap occupancy ft  . The second one can be expressed by
fp = 1- fFD                         (2.51)
with fFD  being the Fermi-Dirac distribution
fFD(E ) =  -------1---------,             (2.52)
           1+ exp(β(E - Ef))
   β    =  -1--.                          (2.53)
With this joint probability, one obtains the rate equation for hole capture,
∂tft  = - LxAyz ∫v ft Atb(E)fLSF(E)Dp(E)fp(E)dE ,    (2.54)
               -∞          tb
where the ‘minus’ sign on the right-hand side in the above equation reflects the fact that the trap occupancy is decreased by a hole capture event. An analogous argumentation for hole emission yields
∂f   = L A   E∫v(1- f ) A (E)fLSF(E)D  (E )f (E )dE ,    (2.55)
 tt     x  yz- ∞     t   bt    bt      p    n
where fn  equals the Fermi-Dirac distribution fFD  . Combining the rate equations for hole capture (2.54) and emission (2.55) and using the shorthands
cp(E ) = LxAyzAtb(E )fLtSbF(E ) ,              (2.56)
e(E ) = L  A  A  (E )fLSF (E ) ,              (2.57)
p         x yz bt   bt
one obtains
∂tft  =    ((1- ft)ep(E )fn(E )- ftcp(E)fp(E))Dp(E )dE        (2.58)
        ∫v (       ep(E )             )
      =     (1- ft)cp(E-)fn (E )- ftfp(E )  cp(E )Dp (E)dE .   (2.59)
If thermal equilibrium is assumed [127], the trap level Et  is occupied according to the Fermi-Dirac statistics
ft(E)  = fFD(Et) = 1+exp(β(Et-Ef))             (2.60)
and detailed balance
∂tft  = 0                         (2.61)
must be employed. Using the identity
fn(E ) = exp(- β(E - Ef))fp(E)              (2.62)
one obtains
ep(E)  = exp(- β(Et - E)) .                (2.63)
Inserting this result back into equation (2.59) yields
∂tft  = ((1- ft)exp(- β(Et - Ef))- ft) ∫vcp(E)Dp (E )fp(E )dE  .       (2.64)
The hole capture time constant can be expressed as
          E∫vc (E)D  (E )f(E )dE
--1--    -∞--p-----p---p------
τcap,h  =    E∫v                 p ,            (2.65)
where the first term on the right-hand side of the above equation is defined as the product of a cross section σSpRH  and the thermal velocity vth,p  . The definition of the hole emission time constant follows from equation (2.64).
               SRH                          SRH
∂tft = (1- ft)σ◟p--vth,pp exp◝◜(--β(Et --Ef))◞- ftσ◟p-◝v◜th,pp◞
                       =1∕τem,h              =1∕τcap,h
For the ratio of the time constants, one obtains the well-known relation
τcap,h(E)  = exp(- β (Et - Ef)) ,             (2.66)
 τem,h(E )
which is frequently invoked in the context of charge trapping — in particular for NMP models [5512156124125]. Note that the quantity σSRpH  in equation (2.65) contains the matrix element Atb  and the Frank-Condon factor fLtSbF(Et,Eb)  . The former is associated with an electron tunneling process and thus often approximated by a WKB factor. The latter is strongly determined by the barrier height of an NMP transition (see Section 2.4). This suggests that also the cross section σSpRH  in the SRH theory should somehow reflect this barrier dependence. For a hole capture process with Et < Ev  (cf. Fig. 2.5), the barrier height can be split into two components, namely |ΔEt | and ΔEb  . The former is defined as
ΔEt  = Et,0 - Ev,0 ,                  (2.67)
where Et,0  and Ev,0  are the trap level and the valence band edge in the absence of an electric field. The value of |ΔEt| corresponds to minimal energy required for a transition. The latter, that is ΔEb  , represents only the remainder to the overall barrier |ΔEt |+ ΔEb  . For the Et > Ev  , the component |ΔEt | vanishes and only ΔEb  remains. Using the above definitions of the barriers, the cross section can be written as
 SRH     SRH               exp(- β ΔEb) ,                 Et > Ev
σp    = σp,0  exp(- xt∕xp,0) exp(- β ΔEb) exp(- β(Ev - Et)), Et < Ev ,      (2.68)
where  SRH
σp,0  and xp,0  denote a temperature-independent cross section and the characteristic tunneling length. It is remarked at this point that the transition barriers has been assumed to be independent of the energy of the holes.
                                            {1,                 E  > E
1∕τcap,h  = σSpR,0Hvth,pp exp(- xt∕xp,0) exp(- βΔEb )                    t   v ,          (2.69)
                                             {exp(- β (Ev - Et)), Et < Ev
            SRH                                exp(- β(Et - Ef)), Et > Ev
1∕τem,h = σ p,0 vth,pp exp(- xt∕xp,0) exp(- βΔEb ) exp(- β(E - E )), E < E  .          (2.70)
                                                       v    f     t    v


Figure 2.5: The transition barrier for a hole capture process with Et < Ev  . The hole is initially located in the bulk at an energy level ~ Ev  . Its capture can be imagined to proceed over a downwards directed barrier of the height |ΔEt|+ ΔEb  . For this process, an minimum energy of ΔEt  is required to push down the hole from Ev  to Et  . As a consequence, the forward barrier is higher than its reverse counterpart by a value of |ΔEt | . The rest of the barrier height is accounted for by ΔEb  , which is assumed to have a finite value for generality. It is noted here that the SRH theory [127] has been derived without assuming thermal barriers. Therefore, this theory is usually considered for processes proceeding without or with a negligible thermal barrier.

Making use of

 p  = Nv  exp (- β(Ef - Ev )) ,            (2.71)
Ev  = Ev,0 - q0φs ,                      (2.72)

Et  = Et,0 - q0φs - q0Foxxt              (2.73)
with Et,0  and Nv  being the trap level in the flat band case and the effective valence band weights, respectively, the rates (2.69) and (2.70) can be rewritten as
1∕τcap,h  = -SRpH--- exp(- xt∕xp,0) exp(- β ΔEb) 1,                       Et > Ev ,          (2.74)
          τp,0  Nv                           exp(βΔEt )exp(- βq0Foxxt), Et < Ev
            1                              {exp (- βΔE  )exp(βq F x ),  E > E
1∕τem,h  = -SRH-exp (- xt∕xp,0) exp (- βΔEb )              t       0 ox t    t    v           (2.75)
          τp,0                               1,                         Et < Ev
-1---= Nvvth,pσSpR,0H .                    (2.76)
The corresponding expression for electron trapping from the conduction band can be derived in an analogous manner and reads
1∕τ     = ---n--- exp (- x ∕x ) exp (- βΔE )  exp(- βΔEt )exp(βq0Foxxt), Et > Ec ,        (2.77)
   cap,e    τSnR,0HNc        t  p,0           b   1,                        Et < Ec
1∕τem,e  = --1--exp(- xt∕xp,0) exp(- βΔEb )   1,                        Et > Ec           (2.78)
          τSnR,0H                              exp(βΔEt )exp(- βq0Foxxt), Et < Ec
-1---= Ncvth,nσSRH .                    (2.79)
τnSR,H0           n,0
Finally, it should be mentioned that the conventional SRH theory as established in [127] does not account for charge carrier tunneling and the possible presence of thermal barriers.