MajorUy-carrier capture cross-section determination in the farge deep-trap concentration cases

A method to determine the thermal cross section of a deep level from capacitance measurements is reported. The results enable us to explain the nonexponential behavior of the capacitance versus capture time when the trap concentration is not negligible with respect to that of the shallow one, and the Debye tail effects are taken into account. A figure of merit for the nonexponential behavior of the capture process is shown and discussed for different situations of doping and applied. bias. We have also considered the influence of the position of the trap level's energy on the nonexponentiality of the capture transient. The experimental results are given for the gold acceptor level in silicon and for the DX center in Alo.ss GIlo.4s As, which are in good agreement with the developed theory.


t. ~NTRODUCTION
The measurements of the thermal cross section and its dependence on temperature are essential to obtain a thorough picture of the deep levels.While the indirect measurements are immediately rejected due to their lack of precision, the direct methods are more reliable when the deep-trap-toshallow-concentration ratio N T/ N is small and the influence of the edge region is negligible.
below the Fermi level EF in the bulk.The band diagrams of an n-type Schottky barrier in the sequence of events during the measurement are shown in Fig. 1.
However, in most cases these conditions do not hold.When the edge-region influence is not negligible, the capture probability is nonuniform.The existence of the free-carrier tail extending from the bulk towards the junction establishes two regions of capture, fast and slow, which pose problems in the interpretation of the results.So, after each pulsed bias the trarrsient will have two components.Several authors have dealt with this problem.However, only the N T/ N < 0.1 case has generaHy been considered.1-7 Recently, some initial modds have also been carried out for the large deep-trap-todonor-concentration ratio,8,9 in which case the fast component also presents a nonexponential behavior, which becomes more important as the N T/ N ratio increases.
In the present work, we analyze the effects of a high N T/ N ratio and deduce a method for the suitable interpretation of the usual semilogarithm plot of the capacitance variation versus the capture time, We have taken into account that the capture process is originated as in the neutral.semiconductor as in the Debye tail.Besides the dependence of the capacitance with regard to the junction bias, concentration ratio, and the applied bias pulse for different capture time, it has also been analyzed and expressed using a figure of merit.The experimental values obtained for the gold acceptor level in silicon are in good agreement with the developed model, which has also been applied to deduce the electron capture cross section oftheDX center in Alo.ss Gao.4S As.Its behavior concurs with the conclusions deduced from this model.

U.ANAlYSIS
We assumed that the semiconductor contains a constant volume density N D of shallow donors and a density NT of a donor deep level with a free ionization energy ET which lies Let W 2 (0) be the depletion width and Wu (0) the crossover position of the Fermi level and the trap level in its steady state with an applied bias V 2 [Fig. 1 (a) ].When a bias pulse of height V i -V 2 is applied, the depleted region width shrinks initially to Wi and those centers between WI and some levels will also be filled in the region between Wu (At) and WI (At) due to the Oebye tail of the bulk electrons during the pulse time At.The width of this region Wu -WI = Al can be deduced by assuming that the occupancy factor at Wu is half of the value of IN (At), obtained in the neutral region which was given in a previous work;9 we take into account that the free-carrier concentration changes to keep the charge neutrality out of the space-charge layer. (1) (2) and en and C n are, respectively, the emission and capture thermal coefficients.In our experimental case en <cnN D , thus the above expressions can be approximated by (7) where In order to obtain the time evolution of the edge region from these assumptions, it is necessary to solve the detailed balance equations in the edge region with the corresponding boundary conditions given by Eq. ( 7): where n (x,At) must be the exact distribution of the free car- where V(x,At) is the potential distribution in the spacecharge region which is known by solving the Poisson equation: x (l -IN (x,At)] -n (x,At)} , (11) where E. is the permittivity of the semiconductor and q the electron charge.
The numerical calculation of these equations must be performed in an iterative form.For each time At, n(x,At) and/(x,At) are obtained from Eqs. (9) and (10) byassuming that V(x,At) has remained momentarily equal to V{x,At -c5(At)), with c5(At) being the increment oftime.WI(At) has been deduced in accordance with the results of Warner and Jindall lO from the position where qV( WI,At)lkT= -In 0.55, and Al (At) is given by the condition It should be pointed out that in the large deep-trap concentration cases, unlike the N TIN D < 0.1 case, the available free-carrier concentration to participate in the capture process at any position of the edge region changes due to the capture process in the neutral region, given by Eq. (1).At the beginning of the process the concentration offree carrier at this region is NT + N D but becomes N D in about five capture-time constants.So, initially, the free-carrier charge in the space-charge layer diminishes, and the edge-region capture becomes more difficult.Therefore, Al (At) remains very small and can even decrease when the occupancy factor IN increases, since the corresponding diminution of n (x,A.t)allows condition (12) to be accomplished with smaller values of AI' This effect is more significant as N TIN D increases.Whenl N (At) has approached its maximum value, Al begins to rise.In the large deep-trap concentration cases the charge change due to the edge-region capture is enough to force a considerable variation of the space-charge layer width WI' !n Fig. 1 (c), a band diagram after the application of the bias pulse is shown.Now W 2 (At) and Wu (At) are the new values for the depletion width and the crossover point, which are functions of the values WI (At) and A I (At).The decrease of the charge in the space-charge layer, due to the capture process, is balanced by an increase of the space-charge layer width W 2 (At).However, to obtain its value it is necessary to know WI (At) and Al (At) a priori.It can be deduced exactly by solving the above equations for the space-charge layer biased at VI' assuming constant profiles for ND and NT' In the last years several works 6 • 7 ,ll-14 have been devoted to deduce the analytical expressions ofthe time evolution of the edge-region width under different conditions.In the present work we used an alternative to the exact solution of Poisson's equation.It involves solving the capture kinetics equation at the edge region, assuming for each At a freecarrier concentration at WI' given by the solution of the capture process in the bulk.Then we obtain a simple analytical approximation which we use elsewhere IS: Once n is known, the band-bending energy qVA.(At) corresponding to the position where 1= IN/2 is deduced from Eq. (13).A good approximation is given by This result is in good agreement with approximations obtained by other authors.6.7.11-14Now WI (At) and Al (At) can be calculated.assuming that the trap occupancy is zero for x< WI -AI andIN(At) for WI -AI <x< WI' So, we and where VB is the built-in potential and Figures 2 and 3 show, respectively, the variation of A I and W, versus the capture time computed for the gold acceptor level in n-Si at 80 K for different N TIN D ratios, with Un = 7X 10-17 cm Z , ND = 1.35X 10 14 cm-3 , and VI = O.
It should be pointed out that the calculated Al values behave as we had previously predicted.WI is initial.Iy constant.It increases due to the variation of the fraction of occupied traps; afterwards, the changes are only produced by the A I variation.The total variation of WI is approximately 1 order of magnitude smaller than its initial value, for N TIN D <0.1; butwhenNTIN D increases toward unity, the variation of WI is comparable to its initial value.Obviously, this considerable rearrangement of the space-charge layer at VI determines the new value of the space-charge layer when the bias is returned to V z .
At(See) Since the bias V z is fixed, we have at any time (18) where n(x,at + t) and/(x,at + t) are, respectively, the values of the free-majority carriers and occupancy factor at the X position after being biased during t seconds at V z ; " I" is any point in the neutral region.
To eliminate the effects due to the emission of the trapped charge we consider only the initial value for t = 0; hence, the/value in the above expression correspond.s to the final value of the occupancy factor during the capture process.Like the above case for VI' Eq. ( 18) can only be solved.by a long computer process.
In order to deduce useful solutions for Eq. ( 18), we have assumed similar approximations to the NT <N Dease. 6Thus, we have considered that the filling of the traps happens in the region between W IA (at) and Wu (0) and that its occupancy factor becomes/(at).This is equivalent to saying that we replace the true occupancy factor with a new rectangular We consider that the fast decrease of the free-carrier density, at the edge region occurs like the small deep-trap concentration case.Therefore, the approximation of a rectangular occupancy factor has validity as in the small deep-trap concentration case.So, expression (18) can be approximated by (20) In spite of the fact that we do not use the depletion approximation, the high-frequency capacitance C HF = €.A /W remains rigorous, so that it can be checked by calculating directly C = dQ /dV at high frequency.Then, making straightforward calculations we obtain whereA o = W 2 (0) -Wu (0).This parameter has been deduced from the exact integration of Poisson's equation atequilibriurn: where EF -ET is the energetic separation between the majority carrier's quasi-Fermi level and the trap level.
Due to the complexity of Eq. ( 21), we have solved it taking into account three time intervals.
(a) For short capture times, we obtain lot_O' where A is the area.In this case, the denominator imposes a restrictive condition W u (0) > W u , which yields On the other hand, one observes that dC 2 / dAt I::..t _ 0 depends on the polarization through W~ (0)/( wit -W~;.) and on the doping.Moreover, when the above restrictive condition is fulfilled dC 2 /dAt I::..t-o is greater or equal to 1].This will be analyzed below when we present the figure of merit.(b) For greater capture times in expression (21), the exponential variation versus time is stronger than that of the term in bracket.It is due to the logarithmic dependence of A I on At.So, we can write Now the slope of the semilogarithmic plot of C 2 (At) -C 2 ( (0) vs at changes continually from the initial values towards lower ones.However, when M(At) depends slightly on At, the semilogarithmic plot shows a region, whose slope is near to 1].
(c) Further capture-time increases give rise to a lower dependence of dC 2 / dAt in relation to At due to greater A I influence.This, every time, increases more slowly due to the fact that the free-carrier concentration falls when A I increases.Xt gives rise to a very soft variation of the capacitance which makes it hard to obtain the saturation capture values corresponding to A I = ,1,0. 6, 7 This process becomes slower as the level is deeper.

III. RE5UlT5 AND DI5CUSSlON
From the above analysis we can deduce important aspects in relation to the fitting between the experimental data and the theoretical values of C 2 (at).
The calculated capacitance variation in the final part of the transient shows a great dependence on the energy position of the level, through theA I values, as can be seen in Fig. 4.This dependence makes it very difficult to use this part of the capture transient to determine the capture cross section ifthe energy position is not well known. 6 On the contrary, the initial part of the capture transient depends more on the doping conditions and on the capture cross section than on the energy position; but the dependence on the concentration is so strong that it is not very easy to determine the capture cross section by the fitting of the experimental capacitance variation, unless the N T/ N D ratio is known with high accuracy (Fig. 5).Accordingly, we have taken great care over the determination of the energy position of the level and the concentration values in order to obtain a whole fitting of the capture transient with our model.
The experimental study has been carried out on the DX donor center in n-Alo.ssG~.4sAs(Te)/p-Alo.3sG~.6sAs ( Ge) heterojunction and on the gold acceptor level in P + N silicon diodes.More details of the samples are given elsewhere. 9 ,1l Capacitance isothermal transients have been  100~------------------------- recorded, using a Boonton model 72BD monitored by a computer.The bias pulses have been applied using a HP 8013A and a HP 214A.
For the Alx Gal _ x As heterojunctions we have determined ND = 1.2X 10 17 cm-3 and NT = 1.75X 10 17 cm-3 -!. from C( V) measurements.At each voltage, the capacitance values for fJ..t = 0 and fJ..t = 00, corresponding to zero and the maximum electron occupancy factor, have been measured, respectively, by lowering slowly the temperature (from a high enough value for the thermal emission to take place) with the bias applied or with the sample in short circuit and by applying the reverse bias required when the low temperature has been reached.Thus, we have the situation discussed above for C z and C I in the fJ..t = 0 and fJ..t = 00 conditions.
Figure 6 shows the capacitance transients for the heterojunction at different VI values at 77 K. Their behavior corroborates the above theoretical results described.They are nonexponential even though the bias variation is small; in this case, however, the capture happens only at the edge region, since condition ( 24) is not fulfilled.For greater bias pulses VI = 0 and V z = 5 V, the above condition is fulfilled and the initial decays are faster according to condition (23).
Figure 7 shows the semilogarithmic plot of the capacitance variation versus t at different temperatures for the DX center.The continuous lines indicate the results of our model, which give us the same values for C z as the exact comput- er calcuJ.ation of Poisson's equation [Eq. (18)].This justifies the use of the equations deduced to determine C z • The dots correspond to our experimental measurements.In this case, at each temperature the experimental data have been fitted using different values of 0'". Figure 8 shows the Arrhenius plot of the best values of 0'" vs lOOO/T.The slope corresponds to an electron cross section activated with the temperature according to the Arrhenius plot of the eInlSS10n coefficient (160 ± 5 me V).Moreover, it agrees with the energy position given by other authors l6 • If we use a greater value of E T , a relatively good fitting can still be obtained, but there is no self-consistency with emission results.(c) R -+ 1 as N TI N D -+ O.However, if inequality ( 24) is not fully accomplished, R can be smaller than 1, since in this case the predominant region where the capture happens is the edge region.
(d) When VI approaches V 2 , thenonexponential behavior remains in spite of the small variation of the bias.If we consider that the effects of large NT become more important than those of the edge region, when Ao -+ 0, the figure of merit R can be extrapolated to 1 + N TIN D' It indicates that to obtain an exponential capture transient it is more significant to have a small trap density than to apply small.bias pulses.
(e) For fixed NTIN D , R increases as u is decreased, which corroborates the strong influence of NTIND and the applied bias on the capture transient form.
(f) The computer estimation of the error made on R, using approximation (8) to solve Eq. ( 6) always gives a value lower than 7 % in the worst case.

This analysis holds for the acceptor level by replacing ND byND -NT' IV. CONCLUSIONS
To summarize, we have analyzed.the capacitance variation versus capture time, and we have explained its behavior in the large deep-trap concentration case.Large values of N TIN D give rise to a strong modification of the capture transient.First, there appears a considerable increase of the slope of the logarithm of the capacitance variation versus the capture time.Mterwards, the effects of the edge region are the determining factors that produce the nonexponential behavior.These effects are enhanced for Jarge deep-trap concentration if the level is deeper.However, if inequality (24) is not fulfilled, the edge region always determines the slope.So, in agreement with the results of Wang and Sah l7 for the emission transients, the only guarantee for an exponential transient is a small concentration ratio, N TIN D < 0.1.
The corresponding theoretical equations for free carrier's capture by deep-level impurities in the space-cbarge layer and in the neutral region have also been calculated.For this capture an analytical.approximation for the edge-region width has been given.Due to its logarithmic dependence on the capture time one should note that at low temperature when the pulse bias is applied, thermal equilibrium is not reached, although the capacitance variations show a fixed value.18,19 The NTIND ratio has been found to have great infl\uence on the A 1 determination and on the WI variation.
Therefore, we have taken special.care in the concentration values determination from the C-V measurements.
The developed model provides a method to measure the thermal cross section from an easy numerical simulation of the semilogarithmic plot of the capacitance variation versus !:1t.The experimental resul:ts found for the electron cross section of the well-known gold acceptor level inp+n silicon diode agrees with that measured previously by means of other techniques, 1.2,8,9 which corroborates the developed modd.Moreover, the experimental val.ues for the DX center show a behavior corresponding to that deduced from our model.We have found for this center that the values used in the fitting follow a law: (Tn = (Too exp( -Ed1kT) , with (Too = 1.5 X 10-19 cm 2 and Ea = 125 meV.This value agrees with the results Er = 35 meV obtained from the fitting and with the activation energy deduced from the Arrhenius plot of the emission coefficients (160 ± 5 meV): Ed = 160-35 meV.
On the other hand, we have proposed a figure of merit in order to have a criterion to give us a measurement of the nonexponentiality of the capture transient under given conditions.The nonexponential behavior depends strongly on the NrlN D ratio, on the bias conditions, and on the energy position of the level through AI' One can see that the transient is only exponential (figure of merit equal to 1) if NT <ND • Finally, it should be noted that using the constant capacitance technique also presents problems of nonexponentiality.According to expression (1), the capture in the neutral region will be nonexponential.Moreover, according to expression ( 15), the charge variation at the edge region will give rise to nonexponential changes of the applied bias in order to maintain the capacitance at a constant rate.However, in this case, the relation between the voltage changes and the charge variation is a little more simple than in the constant voltage case, but its experimental setup is more complex.Generally, its utilization does not present any significant advantage.
FIG. I. Band diagrams at different time sequences: (a) Steady-state condition at V, bias.(b) Capture condition when a V I -V 2 bias pulse is applied.(c) Emission condition when the bias pulse is removed.

FIG. 2 .
FIG. 2. Edge region width A, vs the capture time calculated for the gold acceptor level in p + n silicon diode at 80 K and V, = O.Values used are u" = 7.5 X 10-17 cm+ 2 and ND = 1.35x 10'4 cm-3 for different NTIND ra, tios.
FIG. 3. Space-charge layer width vs the capture time at the same conditions of Fig. 2.
FIG. S. Theoretical capacitance change vs capture time for an acceptor level for different N TIN D ratios in a silicon p + n junction.The values used are u.=7.5XIO-17 cm 2 , ET=S48 meV, T=80 K, and ND = 1.35 X 10 ' • cm-3 •

0 ' ".
FIG. 6. Capacitance vs the capture time measured on the DX center in n-Alo.~~GIIo.4~As: P-Aio.H GIIo .•, As heterojunction at 77 K for different bias condition.No = 1.2x 10 17 cm-3 ; NT = 1.4SND • FIG. 8. Arrhenius plot of the electron cross-section values obtained for the DX center from the fitting shown in the former figure.

.
(sec) FIO. 9. Theoretical capacitance at V 2 vs the capture time at 80 K for the gold acceptor level in ap+n silicon diode at different NT/ND values.The dots indicate the experimental results.ND = 1.35 X lO 14 cm-3 andu.= 7.5 X 10-11 cm 2 • thermal.cross section of the gold acceptor level does not show dependence with the temperature.In this case the energy position of the level is well known.Therefore, the fitting is easier.The best value of Un to fit each measurement turns out to be independent of the N T/ N D ratio in good agreement with our previous work, 9 and so confirms the developed analysis.The value deduced is Un = 7.5X 10-17 cm+ 2 • Figure10shows the semilogarithmic plot of the capacitance variation versus Ilt for N T/ N D = 0.75 and 0.1.1n the last case the nonexponential behavior is only denoted at long time intervals, while, in the other case, it is observed in the whole transient.In order to know, a priori, how nonexponential the capture transient is, we have defined a figure of merit by the ratio between the time constant deduced from the initial part of the transient, supposing the single exponential model 17* and the true value 17 , It provides us with information regarding the inaccuracy of the exponential model:(29)    where the nonindependent variables r, So.SI' and u are, respectively, W 2 (0), W 2 (0) -Ww WI -WI)., and WI nor-(b) R increases as NT IN D is raised, w hicn indicates the important effects of the large deep-trap concentration case on the nonexponentiality of the capacitance capture transients.