Analytics Derivation of the Capture and Emission Time Constants

7.4 Analytics Derivation of the Capture and Emission Time Constants

In order to promote the understanding of the eNMP model, $τcap$ and $τem$ will be derived analytically in the following. The time constants observed in TDDS can be calculated on the basis of first passage times of a two-step process (see Fig. 7.5).

r + r + r τ = -AB----BC----BA (7.52) rABrBC = --1-+ --1-+ -1-rBA- (7.53) rAB rBC rBC rAB

This quantity corresponds to the mean time it takes the considered system to arrive at the state $C$ , provided that it was in the state $A$ but not in state B at the beginning. In the eNMP model, one is only interested in the transition times between the stable states $1$ and $2$ , in which the defect dwells most of the time. Since the metastable states $′ 1$ and $′ 2$ are energetically higher than their corresponding stable counterparts $1$ and $2$ , the defect only remains temporarily in these metastable states. This is in agreement with the condition for the first passage time that the system must not be in state B at the beginning. As a result, the transition rates between the states $1$ and $2$ can be reasonably described as the inverse of first passage times.

Figure 7.5: The state diagram for a two-step process from the state $A$ to $C$ . The first passage time of such a process is calculated by equation (7.52). Consider that the transition rate $rCB$ , indicated by the dashed arrow, does not enter this equation.

The various transition pathways allowed in the eNMP model are summarized in the state diagrams of Fig. 7.6. The corresponding first passage times for the hole capture or emission read

τ2c′ap = -1- + -1- + -1-r2′1, (7.54) r12′ r2′2 r2′2r12′ τ1′ = -1- + -1- + -1-r1′1, (7.55) cap r11′ r1′2 r1′2r11′ 2′ -1- -1- -1-r2′2 τem = r22′ + r2′1 + r2′1r22′ , (7.56) 1′ -1- -1- -1-r1′2 τem = r21′ + r1′1 + r1′1r21′ . (7.57)

Figure 7.6: A simplified state diagrams of hole capture and emission over the metastable states $1′$ and $2′$ . The superscript of $τ$ denotes the intermediate state, which has been passed through during a complete capture or the emission event. Note that there exist two competing pathways for a hole capture event, namely one over the intermediate state $1′$ and one over $2′$ . Of course, the same holds true for a hole emission event.

For studying the field dependence of these capture and emission times, the definition ( $2.67$ ) is used for $ΔEt$ and $ΔE ′t$ in the expression for the NMP barriers (7.17):

ε12′ ≈ --S1ℏω1- + R1(Ev---Et +-εT2′) (1 +R1 )2 1 + R1 --S1ℏω1- R1(ΔEt---εT2′) R1q0xtFox = (1 +R1 )2 - 1+ R1 + 1+ R1 (7.58) S ′ℏω ′ R ′(E - E′) ε1′2 ≈ ---1--1′2 + -1---v--′t-- (1 +R1 ) 1+ R1′ = --S1′ℏω1-- - R1′ΔE-t-+ R1′q0xtFox- (7.59) (1 +R1 ′)2 1+ R1′ 1 + R1′

Using the definitions

2′ τc,min = 1∕r2′2 , (7.60) τ2e′,min = 1∕r22′ , (7.61) 1′ τc,min = 1∕r11′ , (7.62) τ1e′,min = 1∕r1′1 , (7.63) 1 τNpM,0P = -NMP------- , (7.64) σp,0 vth,pNv

the mean time constants (7.54)-(7.57) can be expressed as

′ N2 ( R1q0xtFox) ′ ( N1 ) τ2cap = τ0p--exp β--1+-R--- + τc2,min 1+ -p- exp (βq0xtFox) , (7.65) ( 1 ) τ1′cap = τ1c,′min + τ0N3exp βR1′q0xtFox , (7.66) p ( 1+ R1′) 2′ 2′ q0xtFox- τem = τe,min + τ2′ exp - β1 + R1 , (7.67) ′ ( q0xtFox) ′ τ1em = τ1′ exp - β1-+-R-′ +τ1e,min(1+ exp (β (E ′t - Ef))) (7.68) 1

with

N = N exp(β(ε ′ - ΔE )) , (7.69) 1 v T2 ( t ) ( ) N2 = ----Nv---- exp β -S1ℏω1-2 exp - β R1(ΔEt---εT2′) , (7.70) (1+ R1 )3∕2 ( (1 + R1) ) ( 1 + R1 ) ----Nv---- -S1′ℏω1′- -R1′--- ′ N3 = (1+ R1′)3∕2 exp β(1+ R1′)2 exp - β 1+ R1′ΔEt (1+ exp (β(ΔE t - ΔEt ))) , (7.71) τNMP ( S ℏω ) ( ΔE - ε ′) τ2′ = ---p,0-3∕2 exp β --1--1-2 exp β---t---T2- (1+ exp(βεT2′)) , (7.72) (1+ R1 ) ( (1 + R1) ) ( 1+ R1) τ′ = ---τNp,M0P--- exp β-S1′ℏω1′- exp β-ΔE-′t-- . (7.73) 1 (1+ R1′)3∕2 (1+ R1′)2 1 +R1 ′

Recall that the hole capture process can proceed from state $1$ over one of the metastable states $2′$ or $1′$ to the final state $2$ according to the state diagram of Fig. 7.3. The corresponding capture time constants are denoted as $τ2c′ap$ and $τ1c′ap$ , respectively, and will be discussed in the following. When the transition pathway $T1→2′→2$ is preferred, the capture time constant in the form of (7.52) is given by

τ2c′ap = r12′ +-r2′1 +-r2′2 . (7.74) r12′r2′2

Each of the summands in the nominator can be dominant so that $′ τ2cap$ is characterized by three distinct regimes, namely B, C, and D in Fig. 7.7.

At extremely high negative oxide fields (regime D), $r ′ 12$ is the dominant rate meaning that the transition² $T ′ 1→2$ proceeds much faster than $T ′ 2→2$ (cf. Fig. 7.8). Thus the pace of the complete capture process ( $T ′ 1→2 →2$ ) is determined by the second transition $T′ 2→2$ , which is much slower and has a time constant of $τ2′ c,min$ . Since this second step is only thermally-activated, $2′ τcap$ does not depend on the oxide field. This is consistent with equation (7.65) at extremely high negative oxide fields, at which both exponential terms become negligible compared to $2′ τc,min$ .
At high negative oxide fields (regime C), the rate $r12′$ approaches the order of magnitude of $r2′1$ and even falls below $r2′2$ . Then the transition $T2′→2$ over the thermal barrier $ε2′2$ is undergone immediately after the defect has changed from the state $1$ to $2′$ . Thus the kinetics of the hole capture process are governed by the forward rate of the NMP process $T1→2′$ . As a result, $τ2ca′p$ shows an exponential oxide field dependence, which is reflected in the first term of equation (7.65). Note that the second term is negligible due to its steeper exponential slope within this regime.
At low negative oxide fields (regime B), $r12′$ is already outbalanced by its reverse rate $r2′1$ (see Fig. 7.8) and the ratio of both rates determines the oxide field dependence. This gives an increased exponential slope originating from the second term of equation (7.65).

The transitions between these three regimes are smooth so that a curvature appears in the time constant plots of $τcap$ .

Figure 7.7: Left: The calculated hole capture time constants as a function of the oxide field. The different regimes of $τcap$ (A, B, C, and D) are separated by the thin vertical lines and labeled by the green circles with the capital letters. The dotted curves $τicap$ show the capture processes over a metastable state $i$ . The field dependence of $τcap$ within a certain regime is shown by the dashed curve, which becomes constant if $τcap$ is insensitive to $Fox$ . Right: The same but for the hole emission time constants with the regimes (E, F, and G).

Figure 7.8: A schematic representation of adiabatic potentials in the regimes B, C, and D. The arrows show the possible directions of the transitions involved in the capture process. Their thicknesses indicate the magnitude of their rates, where the thinner arrows are associated with larger transitions times and thus governs the oxide field and temperature dependence of the complete capture process $T1→2′2$ . With higher oxide fields (B $→$ D) the blue potential (neutral defect) is raised relative to the red one (positive defect). This is associated with an increase of $r12′$ and a decrease of the reverse rate $r2′1$ . In contrast to the charge transfer reactions $T1→2 ′$ and $T2→′1$ , the thermal transition $T2′→2$ is not affected by the oxide field.

However, when the transition over the metastable state $1′$ is favored (regime A), the capture time constant can be again formulated as a first passage time:

τ1ca′p = r11′ +-r1′1-+r1′2 (7.75) r11′r1′2

Since the metastable state $1′$ is situated above the state $1$ by definition, $r1′1 ≫ r11′$ holds. Therefore, the expression (7.75) can be approximated by

′ r1′1 1 τ1cap ≈ r-′r-′ + r-′ , (7.76) 11 12 11

which is characterized by only two regimes (A’ and A”) now.

At negative oxide fields (regime A’), the state $1′$ is located relatively high (see Fig. 7.9) and the transition rate $r1′2$ exceeds $r11′$ . Therefore, the first term of expression (7.76) vanishes and the field-insensitive transition $T1→1′$ with a time constant of $τ1c,′min$ dominates $τ1c′ap$ .
When reducing the oxide field, the state $′ 1$ is shifted downwards in the configuration coordinate diagram, thereby decreasing the transition rate $r1′2$ . At a certain oxide field, $r1′2$ falls below $r1′1$ and the first term of the expression (7.76) becomes dominant (regime A”). As a consequence, $1′ τcap$ governed by the field-dependent transition $T1′→2$ , which causes the exponential term of the expression (7.66). Depending on the value of $τ1c′,min$ , there exists a crossing point between the curves $τ2c′ap$ and $τ1ca′p$ , marking the transition between the regime A and B. It is noted that the NMP transitions in the regimes A, B, and C also involve a nearly negligible $1∕p$ field dependence, which has already been present in the TSM for instance.

The transition between A’ and A” yields a kink, which is visible in $′ τ1cap$ of Fig. 7.7 (dotted line) but not in the overall hole capture time given by

1 ---- = -11′-+ -12′-. (7.77) τcap τcap τcap

As a result, this transition has not been observed in TDDS experiments, the regimes A’ and A” are not differentiated in Fig. 7.8.

Figure 7.9: The same as in Fig. 7.8 but for the regimes A’ and A” of the oxide field dependence of $τ1ca′p$ .

Also the hole emission process has the possibility to proceed over either the state $1′$ or $2′$ , with $τ1em′$ and $τ2e′m$ being the corresponding emission time constants (see Fig. 7.10). For the transition pathway over $2′$ the emission time constant can be expressed as:

τ2e′m = r22′ +-r2′2 +-r2′1 (7.78) r22′r2′1

Since $r2′2 ≫ r22′$ applies, $′ τ2em$ has only two regimes, labeled with the capital letters F and G in Fig. 7.7.

′ r2′2 1 τ2em ≈ r-′r′-+ r--′ (7.79) 22 2 1 22

At extremely high negative oxide fields (regime G), the state $1$ is shifted upwards so that $r22′$ dominates and the field-dependent NMP transition $T2′→1$ determines the pace of $T1→2′→2$ . The sensitivity of $T2′→1$ to the oxide field is reflected in the exponential term of equation (7.67).
At high negative oxide fields (regime F), the transition $T2′→1$ proceeds much faster than $T2→2 ′$ over the purely thermal barrier $ε22′$ . Thus, $2′ τem$ is determined by the field-insensitive transition $T2→2′$ with a time constant of $2′ τe,min$ . It is important to note that the field independence of this regime is experimentally observed in the time constant plots of ‘normal’ defects (cf. Fig. 7.4 left).

Figure 7.10: The same as in Fig. 7.8 but for the regimes E, F, and G of the oxide field dependence of $τem$ .

At a low oxide field (regime E), the state $1′$ is further shifted down, which speeds up the NMP transition $T2→1 ′$ and allows the pathway over the metastable state $1′$ . The corresponding emission time constant $τ1e′m$ is given by

τ1e′m = r21′ +-r1′2-+-r1′1 . (7.80) r21′r1′1

For a sufficiently large barrier $ε1′1$ the rate $r1′1$ is negligible compared to $r21′$ and $r1′2$ and the above equation simplifies to

′ 1 r1′2 τ1em = r-′ + r-′r′- . (7.81) 11 21 11

In this case, the state diagram reduces to a subsystem which includes the states $1′$ and $2$ and is marginally disturbed by the rate $r ′ 11$ . Then the states $1′$ and $2$ can be assumed to be in quasi-equilibrium.

f1′r1′2 = f2r21′ (7.82)

In this subsystem the condition $f1′ + f2 = 1$ holds so that the trap occupancy $f′t = f1′$ is given by

f1′ = --1r-′- (7.83) 1+ r211′2 --------1--------- = 1+ exp(β(E′t - Ef)) . (7.84)

In the above equation, it becomes obvious that the condition $r1′2 = r21′$ is equivalent to $E′= Ef t$ . Furthermore, this equation can be used to simplify the equation (7.68) as follows:

τ1′ = τ′ exp (- β q0xtFox) + τ1e,′min . (7.85) em 1 1+R1′ ft′

If $E ′t$ falls below $Ef$ at a certain relaxation voltage, the state $1′$ becomes occupied and the emission time $′ τ1em$ is determined by the field-independent transition $T1′→1$ with the time constant $′ τ1e,min$ . By contrast, if $E ′t$ is raised above $Ef$ , the state $1′$ is underpopulated thereby slowing down the hole emission process. This occupancy effect is reflected in the second term, which reacts sensitive to changes in $Ef$ .

The overall hole emission time $τem$ follows from

1 1 1 τem- ≈ τ1′ + τ2′- (7.86) em em

and is depicted in Fig. 7.7. At a certain oxide field, when the state $′ 1$ is shifted below state $2$ , $1′ τem$ reaches its minimum value and falls below $2′ τem$ . The resulting drop in $τem$ is observed as the field dependence characterizing ‘anomalous’ defects at weak oxide fields in TDDS experiments. As pointed out in Fig. 7.4, the drop of $τem$ occurs when the minimum of the state $′ 1$ passes that of state $2$ , and is thus related to the exact shape of the configuration coordinate diagram.

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