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Theory of nonequilibrium properties of MIS capacitors including charge exchange of interface states with both bands
THEORY OF NONEQUILIBRIUM PROPERTIES OF MIS CAPACITORS INCLUDING CHARGE EXCHANGE OF INTERFACE STATES WITH BOTH BANDS U. KELBERLAU and R. KASSINC Institut ftir Angewandte Physik, 4400 Munster, West Germany (Received 25 July 1977; in revised
form
21 June 1978)
Ahstraet-An improved model of the nonequilibrium behaviour of MIS varactors is presented taking into account the charge exchange of interface states with both bands. C-V curves were calculated for donor and acceptor type interface states of typical distribution in energy. By means of the proposed model the bulk minority carrier generation time constant can be measured and related with semiconductor bulk properties. This allows one to distinguish between interface states acting with the conduction band and those interacting with the valence band. A possible experimental proof of the presented model as well as its limitations are discussed in detail.
NOTATION
states influence e, the transfer inefficiency. As it is desirable to store the information as long as possible and to transfer this information with losses as low as possible, the minority carrier generation time constant should be very large and the interface state density very low. Since there are many process steps during the device fabrication, such as diffusion and ion implantation, it is important to have lirm control of the minority carrier generation time and the interface state density after each process step. Widely used experimental techniques for measuring the density of interface states are provided by the admittance method of Nicollian and Giitzberger[l] and the quasistatic method of Kuhn(2] and Castagne[3,41. The admittance method is known to be the most accurate and comprehensive one supplying most details of the semiconductor-insulator interface. This precise and most sensitive technique, however, has proved to be unsuitable for a fast process-monitoring and for providing in-, formation about unknown bulk semiconductor properties, e.g. about the minority carrier generation time constant. Another disadvantage is that the interface state density can be measured in a small part of the energy gap only. The quasistatic method, on the other hand, offers an easy and fast technique determining the interface state distribution over large parts of the energy gap. Down to values of about 10’“cm-2V-’ this method is nearly as accurate as the admittance method[S] near mid-gap. In the quasistatic method the interface state density can be determined from the difference between the measured current response to a linear voltage ramp, being equivalent to the zero-frequency-curve, and a callNTROLNJCTION culated ideal or high-frequency-curve. The voltage ramp The MOS varactor is the basic element of one of the must be very slow in order to keep the sample in a quasi most used devices in the semiconductor industry. The equilibrium state at any time, otherwise large errors in electrical properties of the MIS varactor are mainly the obtained results may arise. This equilibrium state, determined by the shallow and deep impurity concen- however, can hardly be attained for silicon samples with trations in the semiconductor bulk and the density of large minority carrier generation time constants as used states at the semiconductor-insulator interface. In CCDs for CCD fabrication. Therefore it is necessary to underfor instance the semiconductor bulk properties, i.e. stand the nonequilibrium behaviour of MIS varactors. mainly the dopants and defects acting like deep levels, But up to now there are only a few papers on this topic. influence the refresh time constant, whereas the interface In [2,3] this problem is only phenomenologicahy treated energy gap, eV Fermi level, eV electron imref, eV hole imref, eV intrinsic level, eV gate voltage, V surface potential, stationary, V llq(& - -5). V Fermi potential, V voltagd step, V U/q temperature voltage, V width of region where generation dominates, cm equilibrium depletion layer width, cm oxide caoacitance ner unit area, F cm-’ oxide thickness suace charae within the semiconductor, C . cm-* space charge within the inversion region, C cm-’ space charge within the interface states, C . cm-* free electron, hole concentration, cm-’ hole concentration at the interface, crne3 equilibrium electron concentration when Fermi level lies at the trap level, cm-’ intrinsic electron concentration, cm-’ bulk donor density, crne3 interface state density, cm-* V-’ capture rate for holes, electrons, cm3set-’ hole, electron capture cross section, cm’ hole capture time constant, set emission time constant, set minority carrier generation time constant, set emission or hole capture time constant of interface states, set mean thermal velocity, cm set-’ dU.Jdt = ramp velocity, V set-’ dielectric constant, F cm-‘.
37
U.
38
KELBERLAU and R. KASSING
and in [6,7] quantitative statements are only given about the high-frequency-C, V curves. A general theory of the high-frequency-C, V curves and current transients was introduced by Sah and Fu [8,9], and an exact calculation of the frequency dependent admittance of the MOS-capacitance given by Temple and Shewchun[lO]. The model proposed by Zerbst [ 111,leading to the well known “Zerbst plot”, yields the generation time constant and the surface generation velocity from the time dependence of the pulse capacitance. This model was used in [12] for a critical discussion of the interpretation of the measured generation lifetime and surface generation velocity. Recently Collins and Churchill[l3] gave an exact computer calculation of the dynamic properties of the MOS-capacitance including deep bulk donor and acceptor levels neglecting, however, a continuous surface state density. It was shown in [13], that the Zerbst plot yields the correct generation time constant 7 but the wrong surface generation velocity. Comparable results with those given in this paper have been reported by Briunig and Wagemann[ 141. THEORY
For an easier understanding the employed model can be simply described as follows. If a bias is applied to an MIS varactor, voltage drops across the insulator and the semiconductor. If the corresponding band bending in the semiconductor is large enough, an inversion layer will be formed. A rapid further increase of the bias yields the situation as shown in Fig. 1 for an n-type MIS capacitor. The upper line represents the band bending at a time t = 0’ just after the application of the voltage step to a sample in a quasi equilibrium state (lower line). As can be seen from this figure the hole concentration within the inversion layer has to be enlarged to reach the new equilibrium situation. Therefore electron-hole pairs are thermally generated, the holes of which flowing into the inversion layer. A fraction of the electrons neutralizes the positively charged donors at the end of the depletion region, some flowing around the external circuit to increase the negative charge on the gate. By this effect the depletion layer width decreases and therefore the capacitance increases with time. This process stops, provided no further voltage step is applied, when the required hole concentration is reached. That is when the depletion layer width is reduced to
-qlY,+AV,l
h
Fig. 1. Schematic draw of the band bending after the application of a voltage step to a sample in quasi equilibrium state in deep depletion (lower line). X,(f), X,(r) mark the time dependent locations where the band bending equals 24a before and after the applied voltage step.
nearly the initial one since the equilibrium depletion layer width is almost independent of voltage. if strong inversion exists, or if the additional voltage drop across the semiconductor reduces to zero with time due to the electron-hole pair generation. In this model it is assumed[5] that the carrier generation (a) is controlled by a minority carrier generation time constant T” and (b) is dominating between the areas XI. Xz (in Fig. I. with X, - X, = A W(l), one dimensional model), corresponding to a band bending 2& before and after the applied voltage step. The latter indicates the onset of strong inversion under equilibrium conditions. The last assumption, seeming first of all to be arbitrary, was made since the correct local dependent recombination excess R = r(np -n?), or the distribution of the imrefs is not known. This assumption which also underlies the calculations of Heiman[6] may be justified by the following arguments. As the rearrangement of the electrons after the applied voltage step is very fast-within the dielectric relaxation time-due to the high electric field, the electrons can be regarded to be always in equilibrium with the conduction band. For bias without the occurrence of an inversion layer, only a very small amount of holes has to be generated so that up to the onset of strong inversion practically no non-equilibrium behaviour is observed. Therefore that part of the depletion layer with a band bending smaller than ~I+!I” can be assumed to be in an equilibrium state at any time. Furthermore the onset of strong inversion in the C-V curve of an MIS varactor is very sharp (see e.g. Fig. 2 the NF curve), due to the exponential increase of the inversion charge with surface potential. The presented model is suitable to describe the response of the MIS structure to a voltage step as well as to a voltage ramp which increases the inversion bias at a constant rate (Y= dIl,/dt. In the following calculations we only examine the latter case. Up to now the influence of surface states has been neglected. Usually impurity states are distinguished by their charging behavior. They are called “donors” if they change from the positive to the neutral and “acceptors” if they change from the neutral to the negative charge state. Interface states, however, may not be divided this way since always the sum of both is measured as the flat band displacement. The nonequilibrium properties of the MIS capacitor, however, allow one to distinguish between interface states interacting with the conduction band and those interacting with the valence band. Therefore in the following interface states will be called “donors” if they interact with the conduction band, i.e. if they emit their electrons into the conduction band and “acceptors”, if they interact with the valence band, i.e. emit their electrons into the valence band. Donor type states emit their electrons into the conduction band with a time constant determined by their energy level. Since the time for the rearrangement of the emitted electrons is very short they only contribute to the charge balance. Acceptor type states emit their electrons into the valence band, i.e. they capture holes. Therefore they do not only contribute to the charge balance but affect the generation current as well,
Nonequilibrium
39
properties of MIS capacitors
Fig. 2. Calculatedslowramp C-V curves for the case that no minority carriers are generated (pulse curve) and that the generation current is large enough to establish the equilibrium (NF curve). thus delaying the build up of the inversion layer. Since the capture rate of holes depends on the hole concentration the corresponding capture time constant varies with the generated hole concentration. It should be noted, that from this point of view, it depends on the energy level and the hole concentration whether the interface states act as donors or acceptors, i.e. it depends on which time constant is the smaller one, the electron emission time constant 7,” = (c.nJ’ or the hole capture time constant TV,,= (c,pJ’ [ 151. In the following the nonequilibrium C-V curves will be calculated according to the propoposed model. From balance of charge it follows
can be related to A, I,&.This can be attained by realizing that the hole current generated in the region A W(t) equals the time derivative of the semiconductor hole concentration Q,=.,. Therefore it is Len(t) = p
A W(t) =
&.,.
(3)
0
With
Wo+AW(t)=[-$32$,3+AVD(f))]"2(4) and
Cox(CJg(t)- &(t)-AV,(t))
= - Qsc(&(U,AVp(O) - Qss(rGs(0)
wo=[~2q2
(1)
where the oxide charge and the work function difference are neglected, because these are time independent and US(t), $S((t) being the gate voltage and the stationary surface potential, respectively. The semiconductor space charge Qre is given by the solution of the Poisson equation (n-type semiconductor)
(5)
it follows
AV,(t)=$AW'(t)+2AW(t)Wo)
(6)
and therefore combining eqns 4-6 with eqn 3
P
--
exp
(
-- vF VT
>
In [ 141the second term on the RHS of eqns 6 and 7 was neglected giving rise to large errors, especially for small values of A W(t) and A VP(t), respectively. The time derivative of the space charge QScs in the inversion layer can be easily obtained by differentiating eqn (2) with respect to time and regarding only the first term that describes the hole concentration
shallow donors In eqn (2) it is assumed that the whole voltage drop A V,(t) occurs across the nonequilibrium region A W(t) and that there is no considerable mobile carrier concentration. Equation (1) is a differential equation in &, &, if A V,,
The value of A V(t) enters eqn (8) since eqn (8) contains QSc itself. This was not taken into account in [14].
40
U. KELLIERLAU and R. KASYING
Combining eqns (6) and (7) with eqn (1) one obtains a differential equation in &, I+&which can be solved numerically for a given ramp velocity a = dlJ,/dr. Thus (6:(t) is known and C(r) = d(Qlc + QS,)/dU, can be calculated. The nonequilibrium behaviour of the MIS capacitor is regulated by the ratio of the charge required due to time
varying bias U,(f), and the charge produced by the minority carrier generation current jOcn- (~JT~). Figure 2 shows calculated C-V curves for the extreme situation that no minority carrier generation at all takes place (pulse curve) and that the generation current is sufficient to produce the required minority charge (low-frequency curve). In all other cases the corresponding C-V curves are lying between these two curves. The minority generation current can be influenced by the temperature due to the strong temperature dependence of ni as well as by the generation time constant TV.This is shown in Figs. 3 and 4. As can be seen, a change of 3 orders of magnitude of 7,, is equivalent to the small variation of IOOK in temperature. Furthermore it is obvious that for silicon of higher quality with 70 2. 10m5set up to room temperature
and with practical ramp velocities no equilibrium can be attained. Thus it is more necessary to get an insight into the nonequilibrium behaviour of MIS varactors. MFLUENCE
OF SURFACE
STATES
Donors
As mentioned above, in this paper interface states will be called donors, if they interact with the conductance band, independent of their charging behaviour, and they will be called acceptors if they interact with the valence band. Thus one and the same state can act as a donor as well as an acceptor depending on whether the corresponding emission or capture time constant is smaller. Donors have to be considered in the charge balance only, i.e. as a Q,,(IG;) in eqn (1). In Fig. 5 the known influence of a homogeneous distribution in energy of interface states N,, on the equilibrium C-V curves is shown for increasing values of N,, rising to 5 x 10” cm-‘V-l. As can be seen clearly, the interface states have hardly any influence on the shape of the C-V curves beyond the onset of the strong inversion, because the band bending almost stays constant in that region.
Fig. 3. Calculated slow ramp C-V curves showing the influence of temperature on the minority carrier generation current. N,, = 0, 70= IO-'sec. ND = lOI cmm3,d,. = 100 nm, (I = 60 mV/sec.
Fig. 4. Calculatedslowramp C-V curves showing the influence of the minority carrier generation time constant 7”. N,, = 0, No = 10” cm-‘, do. = 100nm, (I = 60 mV/sec, T = 300K.
Nonequilibrium
Fig. 5. Equilibrium C-V
properties of MIS capacitors
41
curves calculatedfor different values of N,,. ND = l@ cm-‘, do, = 100 mn.
If the temperature is lowered or the ramp velocity a increased, the surface states are no longer in equilibrium with the conduction band. The emission time constant of the surface states depends on their energy distance from the condution band and the capture cross section. Therefore we obtain according to the Schockley, Read, Hall mechanism E, - E
(9)
Thus for Q*(t) a relation holds
Eqn (10) describes the time-dependent change of charge of the interface states, due to the emission of excess electrons if the electron imref EFa near the surface crosses the energy level E at the time to(E). At time I = t,(E) no electrons are emitted and &(t = to(E)) = 0. For I > t,,(E) interface states with energy levels above
E = EFn begin to emit electrons with the corresponding time constant T,~(E). The electron imref EF” near the surface changes with time because the gate voltage and the band bending changes with time; see also section on “Discussion”. C-V curves for different temperatures and a given a = dU,/dt = 60 mV/sec with Ns = 10” cm-* V-’ and T,,~= 0 are shown in Fig. 6. The value T,,~= 0 means that all interface states are in equilibrium with the conduction band independent of their energy level. Thus they will be discharged as soon as the fermi level crosses their energy level. Up to about 1 V, the voltage where strong inversion occurs in equilibrium, all curves behave like the equilibrium curve for N,, = 10” cm-’ V-‘, then spreading due to the delay of the build-up of the inversion layer as the temperature is lowered. A calculation taking into account an emission time constant according to eqn (8) with T..~# 0 yields the curves shown in Fig. 7. Starting in eqn (9) with energy levels close to EC (i.e. strong majority carrier accumulation) for rsso one obtains 7,.0 = (c,NJ’,. giving a value of about lo-“set, if one assumes an energy independent value of the capture coefficient c. = u&,
Fig. 6. Influence of interface states on the temperature dependent slow ramp C-V curves calculated with N,, = 10” cm-* V-‘, ~~~0= 0, ND = 10” cm-3, do, = 100 nm, (I = 60 mV/sec.
42
U. KELBERLAUand R. KAWNG
Fig. 7. Influence of electron emissiontime constant of donor type interface states on the temperature dependent with N,, = 10” cm-* V-l, 7 - IO-‘*set ND = 10” cm-j. d,, = 100 nm, a = 60 mV/sec. “‘’
slow ramp C-V curves calculated
of about lo-’ cm3 see-‘. Figure 7 shows that now, due to the temperature dependent emission time constant, the region where the capacitance is affected by interface states is getting smaller with decreasing temperature. Only those states lying close to the conduction band are fast enough to emit electrons immediately after crossing the fermi level. At temperatures of about 200 K nearly no influence of the surface states can be observed.
DONORS AND ACCEPTORS
In the previous calculations an interaction of the interface states with the valence band was neglected. This, however, is not realistic since the interface states will emit their electrons into the valence band if there are enough holes, or strictly speaking, if the time constant of the emission of electrons into the conduction band is larger than the hole capture time. Starting from a nonequilibrium situation, the hole concentration in the inversion layer has to be enlarged by a certain amount in order to reach the equilibrium state. If there are acceptor type states capturing holes from the valence band, these holes must be generated in addition to the inversion layer charge, which depends on the band bending only. Therefore, contrary to the donor type states, acceptor type interface states have to be taken into account also in the generated current in eqn (3) yielding
interface state density near the midgap of 10” cm-’ and 2x 10’“cm-2V-’ (Fgi . 8a ) a s well as :: 10” cm-’ V-’ and 2 x 10” cm-* V-’ (Fig. 8b), respectively. As can be seen, a reincrease of the capacitance may take place in a certain temperature range which one is inclined to explain by one deep impurity level of corresponding concentration. The latter case is shown in Fig. 9, which was obtained using a distribution (13)
with Nss,,= 10” cm-’ V-’ and CT= 0.1 eV. The mentioned effect is shown more clearly, in Fig. 10 where a C-V curve of an MIS varactor with donor type states only is compared with that of a sample containing both donor and acceptor type states for T = 270 K. The reincrease of the C-V curve of an MIS varactor including acceptor type states may be understood by the following argument. The larger the band bending the more acceptor states have to emit their electrons into the valence band. But this emission requires the presence of holes which have to be produced by the slow thermal generation process in the small region AW(t), thus the emission lags behind. If however, a band bending is reached where the inversion layer charge increases considerably, an amount of holes is generated causing nearly all acceptor states to emit their electrons, thus the corresponding increase of the capacitance results. When the temperature is decreased, the generation current does not last to fill the interface states with holes so that the effect described above will disappear (see Fig. 8). Furthermore the hole capture time constant In the previous calculations for the Figs. 5-7 an energy independent interface state density N,,(E) = const. was considered. In [16], however, a typical distribution in 1 7.r.A= (12) energy of the interface states was deduced from a very CDPS large number of investigated samples. This typical instead of the electron emission time constant has to be N,,(E) distribution which is shown in the sketch, can be applied in eqn (9). The result of the corresponding cal- approached by the superposition of two exponential culations is given in Fig. 8 for a donor and acceptor type functions and a constant value. This was already taken
43
Nonequilibrium properties of MIS capacitors
ug (a)
Fig. 8. Influence of donor and acceptor type interface states on the temperature dependent slow ramp C-V curves calculated with an energy distribution of the states as given in [9]; realized by N:(E)= N,A,,,expBA(E-&)+2X 10’0cm-2V-‘, NE(E) = N%exp &,(.J$ - E)+2X 10” cm” V-’ (Fig. 8a) and N;(E) = N*r.0 cxp~~(E-E)+2~10’~cm-*V-‘, Ng(E)=ND $,o exp &(& - @ + 2 x 10cm-* V-’ (Fig. 8b). Parameters: N $, = N”:, = lOI cm-’ V-‘, PA = &, = 40 V-‘, ND = 10” cm-‘, d,, = 100nm, o = 60 mV/sec.
I
--
1
0 I+
t
i
4
Fig. 9. Calculated equilibrium slow ramp C-V curve of an N,, distribution similar to one deep impurity level. N,, = N,,a exp (E- EJu)*, N,,O= IO” cm-* V-t, (r = 0.1 eV, No = 10” cme3, d,, = 100~1, a = 60 mV/sec. The equilibrium curve with N,, = 0 is also given.
44
LJ. KELBERLAUand R. KAWNG
Fig. 10. Comparison of a C-V curve calculated for donor type states only, with a C-V curve calculated for donor and acceptor type interface states, which shows the reincrease of the capacitance if acceptor type states are present, more clearly.Parametersare the same as given in Fig. 8(b), T = 270 K.
into account in Figs. 8 and 10 where N,,(E) = NE0 exp(-/?&EC - E))+2X 10’“cm-2V-’ donors + N$exp
(- BA(E- IL))+ 2 X IO” cm-* V-’ acceptors
was used, with NZo= N:,o= lOI cm-’ V-’
and
& = /IA = 40 eV_‘.
The corresponding flat band voltage caused by the fast interface states near the band edges have been suppressed.
Fig. Il. Schematic draw of the course of imrefs in the nonequilibrium state. I&+,, & = hole, electron imref. Dotted line: EFn if electron generation current is neglected.
This may be deduced from
jsen = q$ 0 AW(t) DISCUSSIONAND PROPO6AL FOR Ah’ MpEllIMENTAL PROOF
The critical assumption of the proposed model is that the minority carrier generation dominates in a region AW(t). For a correct calculation the real shape of the imrefs has to be known. As this, however, has not been possible up to now, the shape of the imrefs and therefore the corresponding carrier concentrations can be given only approximately: If a voltage step is applied, as was discussed in the theory, and if the electrons are instantly in equilibrium with the conduction band, and if the electron generation current is neglected, the imref EI+ will go straight up to the interface, see Fig. 11 dotted line. Since the hole concentration is very high within the inversion layer the imref Erp is nearly horizontal in this region and the di!Terence EFp - Eh,, is equal to AV,(t). This course of the imrefs may be correct for smaller values of the band bending, i.e. qAV,(t) being much smaller than Es. For larger values of AV, the electron concentration calculated by means of a straight EFn is too small to carry the electron generation current.
= snp,E’F,.
Therefore a course of EFn has to be assumed as is shown in Fig. 11 by the solid line. From this follows that even in the inversion region the generation of holes has to be considered. Since the inversion layer width, however, is small compared with AW(t) no considerable error arises from this neglection. The difference EF~ - EF~ at the semiconductor-insulator interface also influences the charging of the interface states. All states between EF~and EFn have to be discharged with the appropriate time constant as well as those above EF,,.Thus E,=“(l)has to be used in eqn (10) as the lower integration limit. Since EF”, however, is unknown, for the calculations E&f) was used as the lower integration limit, because &(t) at the interface is known from AV,,(r). This is not correct, but practically no error arises because the emission time constant of the corresponding interface states is so large that they always interact with the valence band, a process for which the hole capture time constant is valid. The latter depends on the hole concentration and therefore on Et+,.
Nonequilibrium propelties of MIS capacitors In the region between the bulk and the band bending 211,, the splitting of the fermi level was neglected, because it is assumed to be very small in this region. Whether this assumption is really justified, may be experimentally proved: For a given ramp velocity and temperature the space charge region is widened up to a maximum value, when the capcitance has reached the value of about C,, in strong inversion. This means that any increase of the gate voltage drops across the insulator and that the whole gate current C,.& is now carried by the minority carrier generation in the region A W. In this case the relation holds
(14) Now assuming &, = (qni/2ro)A W(t) with A W(t) detined above, one obtains
as
~(2~B+AVP)=270 qni
(15) Thus a plot d/(2$~ +AV,) vs ~/i2Ndqe)C,,L$(l/d for different temperatures (the values of 2&+ Nh C,,, o=, rri(T) are known) yields the slope Q and an intersection of about d(24B), if the used model is correct, or about zero if for AW(t) nearly the whole depletion region has to be considered. The value of 2&r t A V, is the maximum band bending reached in strong inversion and can be obtained from the relation
This proof, however, can be made only in the small temperature range, where the oxide capacitance C,, in strong inversion is reached. The presented model can be improved taking into account surface recombination, and doping profiles of shallow impurity levels near the interface as well as deep bulk levels. In a forthcoming paper the comparison of experimental results with theory will be given.
I. E. H. Nicollian and A. Gotzberger, Bell Sysr. Tech. .I. 46, 1055(1%7). 2. M. Kuhn, Solid-St. Electron 13, 873 (1970). 3. R. Castagk and A. Vapaille, Surface Sci. 28, 157(1971). 4. R. Castagnt, C.R. Acad. SC. Paris 267,866 (1%8). 5. G. Declerck,R. van Overstraeterand G. Broux, Solid-St. Efecrron 16, 1451(1973). 6. F. P. Heimann, IEEE ED-13,781 (l%7). 7. R. F. Pierret, IEEE ED-19,869 (1972). 8. C. T. Sah and H. S. Fu, Phys. Status Solidi (a) 11,297 (1972). 9. C. T. Sah and H. S. Fu, Phys. Starus Solidi (a) 14,59 (1972). 10. V. Templeand J. Shewchun,Solid-St. Elecrron 16.93 (1973). 11. M. Zerbst, Z. Angew. Phys. 12, 30 (1966). 12. D. K. Scbroder and J. Guldberg, Solid-St. Elecrron 14, 1285 (1971). 13. T. W. Collins and I. N. Churchill, IEEE ED-zZ,% (1975). 14. D. Braunig and H. G. Wagemann, IEEE ED-21,241 (1974). 15. W. Schockiey and W. T. Read Jr., Phys. Reu. 87,835 (1952). 16. H. Flietner and Ngo Duong Sinh, Phys. Status Solidi (a) 37, 533 (1976).
form
21 June 1978)
Ahstraet-An improved model of the nonequilibrium behaviour of MIS varactors is presented taking into account the charge exchange of interface states with both bands. C-V curves were calculated for donor and acceptor type interface states of typical distribution in energy. By means of the proposed model the bulk minority carrier generation time constant can be measured and related with semiconductor bulk properties. This allows one to distinguish between interface states acting with the conduction band and those interacting with the valence band. A possible experimental proof of the presented model as well as its limitations are discussed in detail.
NOTATION
states influence e, the transfer inefficiency. As it is desirable to store the information as long as possible and to transfer this information with losses as low as possible, the minority carrier generation time constant should be very large and the interface state density very low. Since there are many process steps during the device fabrication, such as diffusion and ion implantation, it is important to have lirm control of the minority carrier generation time and the interface state density after each process step. Widely used experimental techniques for measuring the density of interface states are provided by the admittance method of Nicollian and Giitzberger[l] and the quasistatic method of Kuhn(2] and Castagne[3,41. The admittance method is known to be the most accurate and comprehensive one supplying most details of the semiconductor-insulator interface. This precise and most sensitive technique, however, has proved to be unsuitable for a fast process-monitoring and for providing in-, formation about unknown bulk semiconductor properties, e.g. about the minority carrier generation time constant. Another disadvantage is that the interface state density can be measured in a small part of the energy gap only. The quasistatic method, on the other hand, offers an easy and fast technique determining the interface state distribution over large parts of the energy gap. Down to values of about 10’“cm-2V-’ this method is nearly as accurate as the admittance method[S] near mid-gap. In the quasistatic method the interface state density can be determined from the difference between the measured current response to a linear voltage ramp, being equivalent to the zero-frequency-curve, and a callNTROLNJCTION culated ideal or high-frequency-curve. The voltage ramp The MOS varactor is the basic element of one of the must be very slow in order to keep the sample in a quasi most used devices in the semiconductor industry. The equilibrium state at any time, otherwise large errors in electrical properties of the MIS varactor are mainly the obtained results may arise. This equilibrium state, determined by the shallow and deep impurity concen- however, can hardly be attained for silicon samples with trations in the semiconductor bulk and the density of large minority carrier generation time constants as used states at the semiconductor-insulator interface. In CCDs for CCD fabrication. Therefore it is necessary to underfor instance the semiconductor bulk properties, i.e. stand the nonequilibrium behaviour of MIS varactors. mainly the dopants and defects acting like deep levels, But up to now there are only a few papers on this topic. influence the refresh time constant, whereas the interface In [2,3] this problem is only phenomenologicahy treated energy gap, eV Fermi level, eV electron imref, eV hole imref, eV intrinsic level, eV gate voltage, V surface potential, stationary, V llq(& - -5). V Fermi potential, V voltagd step, V U/q temperature voltage, V width of region where generation dominates, cm equilibrium depletion layer width, cm oxide caoacitance ner unit area, F cm-’ oxide thickness suace charae within the semiconductor, C . cm-* space charge within the inversion region, C cm-’ space charge within the interface states, C . cm-* free electron, hole concentration, cm-’ hole concentration at the interface, crne3 equilibrium electron concentration when Fermi level lies at the trap level, cm-’ intrinsic electron concentration, cm-’ bulk donor density, crne3 interface state density, cm-* V-’ capture rate for holes, electrons, cm3set-’ hole, electron capture cross section, cm’ hole capture time constant, set emission time constant, set minority carrier generation time constant, set emission or hole capture time constant of interface states, set mean thermal velocity, cm set-’ dU.Jdt = ramp velocity, V set-’ dielectric constant, F cm-‘.
37
U.
38
KELBERLAU and R. KASSING
and in [6,7] quantitative statements are only given about the high-frequency-C, V curves. A general theory of the high-frequency-C, V curves and current transients was introduced by Sah and Fu [8,9], and an exact calculation of the frequency dependent admittance of the MOS-capacitance given by Temple and Shewchun[lO]. The model proposed by Zerbst [ 111,leading to the well known “Zerbst plot”, yields the generation time constant and the surface generation velocity from the time dependence of the pulse capacitance. This model was used in [12] for a critical discussion of the interpretation of the measured generation lifetime and surface generation velocity. Recently Collins and Churchill[l3] gave an exact computer calculation of the dynamic properties of the MOS-capacitance including deep bulk donor and acceptor levels neglecting, however, a continuous surface state density. It was shown in [13], that the Zerbst plot yields the correct generation time constant 7 but the wrong surface generation velocity. Comparable results with those given in this paper have been reported by Briunig and Wagemann[ 141. THEORY
For an easier understanding the employed model can be simply described as follows. If a bias is applied to an MIS varactor, voltage drops across the insulator and the semiconductor. If the corresponding band bending in the semiconductor is large enough, an inversion layer will be formed. A rapid further increase of the bias yields the situation as shown in Fig. 1 for an n-type MIS capacitor. The upper line represents the band bending at a time t = 0’ just after the application of the voltage step to a sample in a quasi equilibrium state (lower line). As can be seen from this figure the hole concentration within the inversion layer has to be enlarged to reach the new equilibrium situation. Therefore electron-hole pairs are thermally generated, the holes of which flowing into the inversion layer. A fraction of the electrons neutralizes the positively charged donors at the end of the depletion region, some flowing around the external circuit to increase the negative charge on the gate. By this effect the depletion layer width decreases and therefore the capacitance increases with time. This process stops, provided no further voltage step is applied, when the required hole concentration is reached. That is when the depletion layer width is reduced to
-qlY,+AV,l
h
Fig. 1. Schematic draw of the band bending after the application of a voltage step to a sample in quasi equilibrium state in deep depletion (lower line). X,(f), X,(r) mark the time dependent locations where the band bending equals 24a before and after the applied voltage step.
nearly the initial one since the equilibrium depletion layer width is almost independent of voltage. if strong inversion exists, or if the additional voltage drop across the semiconductor reduces to zero with time due to the electron-hole pair generation. In this model it is assumed[5] that the carrier generation (a) is controlled by a minority carrier generation time constant T” and (b) is dominating between the areas XI. Xz (in Fig. I. with X, - X, = A W(l), one dimensional model), corresponding to a band bending 2& before and after the applied voltage step. The latter indicates the onset of strong inversion under equilibrium conditions. The last assumption, seeming first of all to be arbitrary, was made since the correct local dependent recombination excess R = r(np -n?), or the distribution of the imrefs is not known. This assumption which also underlies the calculations of Heiman[6] may be justified by the following arguments. As the rearrangement of the electrons after the applied voltage step is very fast-within the dielectric relaxation time-due to the high electric field, the electrons can be regarded to be always in equilibrium with the conduction band. For bias without the occurrence of an inversion layer, only a very small amount of holes has to be generated so that up to the onset of strong inversion practically no non-equilibrium behaviour is observed. Therefore that part of the depletion layer with a band bending smaller than ~I+!I” can be assumed to be in an equilibrium state at any time. Furthermore the onset of strong inversion in the C-V curve of an MIS varactor is very sharp (see e.g. Fig. 2 the NF curve), due to the exponential increase of the inversion charge with surface potential. The presented model is suitable to describe the response of the MIS structure to a voltage step as well as to a voltage ramp which increases the inversion bias at a constant rate (Y= dIl,/dt. In the following calculations we only examine the latter case. Up to now the influence of surface states has been neglected. Usually impurity states are distinguished by their charging behavior. They are called “donors” if they change from the positive to the neutral and “acceptors” if they change from the neutral to the negative charge state. Interface states, however, may not be divided this way since always the sum of both is measured as the flat band displacement. The nonequilibrium properties of the MIS capacitor, however, allow one to distinguish between interface states interacting with the conduction band and those interacting with the valence band. Therefore in the following interface states will be called “donors” if they interact with the conduction band, i.e. if they emit their electrons into the conduction band and “acceptors”, if they interact with the valence band, i.e. emit their electrons into the valence band. Donor type states emit their electrons into the conduction band with a time constant determined by their energy level. Since the time for the rearrangement of the emitted electrons is very short they only contribute to the charge balance. Acceptor type states emit their electrons into the valence band, i.e. they capture holes. Therefore they do not only contribute to the charge balance but affect the generation current as well,
Nonequilibrium
39
properties of MIS capacitors
Fig. 2. Calculatedslowramp C-V curves for the case that no minority carriers are generated (pulse curve) and that the generation current is large enough to establish the equilibrium (NF curve). thus delaying the build up of the inversion layer. Since the capture rate of holes depends on the hole concentration the corresponding capture time constant varies with the generated hole concentration. It should be noted, that from this point of view, it depends on the energy level and the hole concentration whether the interface states act as donors or acceptors, i.e. it depends on which time constant is the smaller one, the electron emission time constant 7,” = (c.nJ’ or the hole capture time constant TV,,= (c,pJ’ [ 151. In the following the nonequilibrium C-V curves will be calculated according to the propoposed model. From balance of charge it follows
can be related to A, I,&.This can be attained by realizing that the hole current generated in the region A W(t) equals the time derivative of the semiconductor hole concentration Q,=.,. Therefore it is Len(t) = p
A W(t) =
&.,.
(3)
0
With
Wo+AW(t)=[-$32$,3+AVD(f))]"2(4) and
Cox(CJg(t)- &(t)-AV,(t))
= - Qsc(&(U,AVp(O) - Qss(rGs(0)
wo=[~2q2
(1)
where the oxide charge and the work function difference are neglected, because these are time independent and US(t), $S((t) being the gate voltage and the stationary surface potential, respectively. The semiconductor space charge Qre is given by the solution of the Poisson equation (n-type semiconductor)
(5)
it follows
AV,(t)=$AW'(t)+2AW(t)Wo)
(6)
and therefore combining eqns 4-6 with eqn 3
P
--
exp
(
-- vF VT
>
In [ 141the second term on the RHS of eqns 6 and 7 was neglected giving rise to large errors, especially for small values of A W(t) and A VP(t), respectively. The time derivative of the space charge QScs in the inversion layer can be easily obtained by differentiating eqn (2) with respect to time and regarding only the first term that describes the hole concentration
shallow donors In eqn (2) it is assumed that the whole voltage drop A V,(t) occurs across the nonequilibrium region A W(t) and that there is no considerable mobile carrier concentration. Equation (1) is a differential equation in &, &, if A V,,
The value of A V(t) enters eqn (8) since eqn (8) contains QSc itself. This was not taken into account in [14].
40
U. KELLIERLAU and R. KASYING
Combining eqns (6) and (7) with eqn (1) one obtains a differential equation in &, I+&which can be solved numerically for a given ramp velocity a = dlJ,/dr. Thus (6:(t) is known and C(r) = d(Qlc + QS,)/dU, can be calculated. The nonequilibrium behaviour of the MIS capacitor is regulated by the ratio of the charge required due to time
varying bias U,(f), and the charge produced by the minority carrier generation current jOcn- (~JT~). Figure 2 shows calculated C-V curves for the extreme situation that no minority carrier generation at all takes place (pulse curve) and that the generation current is sufficient to produce the required minority charge (low-frequency curve). In all other cases the corresponding C-V curves are lying between these two curves. The minority generation current can be influenced by the temperature due to the strong temperature dependence of ni as well as by the generation time constant TV.This is shown in Figs. 3 and 4. As can be seen, a change of 3 orders of magnitude of 7,, is equivalent to the small variation of IOOK in temperature. Furthermore it is obvious that for silicon of higher quality with 70 2. 10m5set up to room temperature
and with practical ramp velocities no equilibrium can be attained. Thus it is more necessary to get an insight into the nonequilibrium behaviour of MIS varactors. MFLUENCE
OF SURFACE
STATES
Donors
As mentioned above, in this paper interface states will be called donors, if they interact with the conductance band, independent of their charging behaviour, and they will be called acceptors if they interact with the valence band. Thus one and the same state can act as a donor as well as an acceptor depending on whether the corresponding emission or capture time constant is smaller. Donors have to be considered in the charge balance only, i.e. as a Q,,(IG;) in eqn (1). In Fig. 5 the known influence of a homogeneous distribution in energy of interface states N,, on the equilibrium C-V curves is shown for increasing values of N,, rising to 5 x 10” cm-‘V-l. As can be seen clearly, the interface states have hardly any influence on the shape of the C-V curves beyond the onset of the strong inversion, because the band bending almost stays constant in that region.
Fig. 3. Calculated slow ramp C-V curves showing the influence of temperature on the minority carrier generation current. N,, = 0, 70= IO-'sec. ND = lOI cmm3,d,. = 100 nm, (I = 60 mV/sec.
Fig. 4. Calculatedslowramp C-V curves showing the influence of the minority carrier generation time constant 7”. N,, = 0, No = 10” cm-‘, do. = 100nm, (I = 60 mV/sec, T = 300K.
Nonequilibrium
Fig. 5. Equilibrium C-V
properties of MIS capacitors
41
curves calculatedfor different values of N,,. ND = l@ cm-‘, do, = 100 mn.
If the temperature is lowered or the ramp velocity a increased, the surface states are no longer in equilibrium with the conduction band. The emission time constant of the surface states depends on their energy distance from the condution band and the capture cross section. Therefore we obtain according to the Schockley, Read, Hall mechanism E, - E
(9)
Thus for Q*(t) a relation holds
Eqn (10) describes the time-dependent change of charge of the interface states, due to the emission of excess electrons if the electron imref EFa near the surface crosses the energy level E at the time to(E). At time I = t,(E) no electrons are emitted and &(t = to(E)) = 0. For I > t,,(E) interface states with energy levels above
E = EFn begin to emit electrons with the corresponding time constant T,~(E). The electron imref EF” near the surface changes with time because the gate voltage and the band bending changes with time; see also section on “Discussion”. C-V curves for different temperatures and a given a = dU,/dt = 60 mV/sec with Ns = 10” cm-* V-’ and T,,~= 0 are shown in Fig. 6. The value T,,~= 0 means that all interface states are in equilibrium with the conduction band independent of their energy level. Thus they will be discharged as soon as the fermi level crosses their energy level. Up to about 1 V, the voltage where strong inversion occurs in equilibrium, all curves behave like the equilibrium curve for N,, = 10” cm-’ V-‘, then spreading due to the delay of the build-up of the inversion layer as the temperature is lowered. A calculation taking into account an emission time constant according to eqn (8) with T..~# 0 yields the curves shown in Fig. 7. Starting in eqn (9) with energy levels close to EC (i.e. strong majority carrier accumulation) for rsso one obtains 7,.0 = (c,NJ’,. giving a value of about lo-“set, if one assumes an energy independent value of the capture coefficient c. = u&,
Fig. 6. Influence of interface states on the temperature dependent slow ramp C-V curves calculated with N,, = 10” cm-* V-‘, ~~~0= 0, ND = 10” cm-3, do, = 100 nm, (I = 60 mV/sec.
42
U. KELBERLAUand R. KAWNG
Fig. 7. Influence of electron emissiontime constant of donor type interface states on the temperature dependent with N,, = 10” cm-* V-l, 7 - IO-‘*set ND = 10” cm-j. d,, = 100 nm, a = 60 mV/sec. “‘’
slow ramp C-V curves calculated
of about lo-’ cm3 see-‘. Figure 7 shows that now, due to the temperature dependent emission time constant, the region where the capacitance is affected by interface states is getting smaller with decreasing temperature. Only those states lying close to the conduction band are fast enough to emit electrons immediately after crossing the fermi level. At temperatures of about 200 K nearly no influence of the surface states can be observed.
DONORS AND ACCEPTORS
In the previous calculations an interaction of the interface states with the valence band was neglected. This, however, is not realistic since the interface states will emit their electrons into the valence band if there are enough holes, or strictly speaking, if the time constant of the emission of electrons into the conduction band is larger than the hole capture time. Starting from a nonequilibrium situation, the hole concentration in the inversion layer has to be enlarged by a certain amount in order to reach the equilibrium state. If there are acceptor type states capturing holes from the valence band, these holes must be generated in addition to the inversion layer charge, which depends on the band bending only. Therefore, contrary to the donor type states, acceptor type interface states have to be taken into account also in the generated current in eqn (3) yielding
interface state density near the midgap of 10” cm-’ and 2x 10’“cm-2V-’ (Fgi . 8a ) a s well as :: 10” cm-’ V-’ and 2 x 10” cm-* V-’ (Fig. 8b), respectively. As can be seen, a reincrease of the capacitance may take place in a certain temperature range which one is inclined to explain by one deep impurity level of corresponding concentration. The latter case is shown in Fig. 9, which was obtained using a distribution (13)
with Nss,,= 10” cm-’ V-’ and CT= 0.1 eV. The mentioned effect is shown more clearly, in Fig. 10 where a C-V curve of an MIS varactor with donor type states only is compared with that of a sample containing both donor and acceptor type states for T = 270 K. The reincrease of the C-V curve of an MIS varactor including acceptor type states may be understood by the following argument. The larger the band bending the more acceptor states have to emit their electrons into the valence band. But this emission requires the presence of holes which have to be produced by the slow thermal generation process in the small region AW(t), thus the emission lags behind. If however, a band bending is reached where the inversion layer charge increases considerably, an amount of holes is generated causing nearly all acceptor states to emit their electrons, thus the corresponding increase of the capacitance results. When the temperature is decreased, the generation current does not last to fill the interface states with holes so that the effect described above will disappear (see Fig. 8). Furthermore the hole capture time constant In the previous calculations for the Figs. 5-7 an energy independent interface state density N,,(E) = const. was considered. In [16], however, a typical distribution in 1 7.r.A= (12) energy of the interface states was deduced from a very CDPS large number of investigated samples. This typical instead of the electron emission time constant has to be N,,(E) distribution which is shown in the sketch, can be applied in eqn (9). The result of the corresponding cal- approached by the superposition of two exponential culations is given in Fig. 8 for a donor and acceptor type functions and a constant value. This was already taken
43
Nonequilibrium properties of MIS capacitors
ug (a)
Fig. 8. Influence of donor and acceptor type interface states on the temperature dependent slow ramp C-V curves calculated with an energy distribution of the states as given in [9]; realized by N:(E)= N,A,,,expBA(E-&)+2X 10’0cm-2V-‘, NE(E) = N%exp &,(.J$ - E)+2X 10” cm” V-’ (Fig. 8a) and N;(E) = N*r.0 cxp~~(E-E)+2~10’~cm-*V-‘, Ng(E)=ND $,o exp &(& - @ + 2 x 10cm-* V-’ (Fig. 8b). Parameters: N $, = N”:, = lOI cm-’ V-‘, PA = &, = 40 V-‘, ND = 10” cm-‘, d,, = 100nm, o = 60 mV/sec.
I
--
1
0 I+
t
i
4
Fig. 9. Calculated equilibrium slow ramp C-V curve of an N,, distribution similar to one deep impurity level. N,, = N,,a exp (E- EJu)*, N,,O= IO” cm-* V-t, (r = 0.1 eV, No = 10” cme3, d,, = 100~1, a = 60 mV/sec. The equilibrium curve with N,, = 0 is also given.
44
LJ. KELBERLAUand R. KAWNG
Fig. 10. Comparison of a C-V curve calculated for donor type states only, with a C-V curve calculated for donor and acceptor type interface states, which shows the reincrease of the capacitance if acceptor type states are present, more clearly.Parametersare the same as given in Fig. 8(b), T = 270 K.
into account in Figs. 8 and 10 where N,,(E) = NE0 exp(-/?&EC - E))+2X 10’“cm-2V-’ donors + N$exp
(- BA(E- IL))+ 2 X IO” cm-* V-’ acceptors
was used, with NZo= N:,o= lOI cm-’ V-’
and
& = /IA = 40 eV_‘.
The corresponding flat band voltage caused by the fast interface states near the band edges have been suppressed.
Fig. Il. Schematic draw of the course of imrefs in the nonequilibrium state. I&+,, & = hole, electron imref. Dotted line: EFn if electron generation current is neglected.
This may be deduced from
jsen = q$ 0 AW(t) DISCUSSIONAND PROPO6AL FOR Ah’ MpEllIMENTAL PROOF
The critical assumption of the proposed model is that the minority carrier generation dominates in a region AW(t). For a correct calculation the real shape of the imrefs has to be known. As this, however, has not been possible up to now, the shape of the imrefs and therefore the corresponding carrier concentrations can be given only approximately: If a voltage step is applied, as was discussed in the theory, and if the electrons are instantly in equilibrium with the conduction band, and if the electron generation current is neglected, the imref EI+ will go straight up to the interface, see Fig. 11 dotted line. Since the hole concentration is very high within the inversion layer the imref Erp is nearly horizontal in this region and the di!Terence EFp - Eh,, is equal to AV,(t). This course of the imrefs may be correct for smaller values of the band bending, i.e. qAV,(t) being much smaller than Es. For larger values of AV, the electron concentration calculated by means of a straight EFn is too small to carry the electron generation current.
= snp,E’F,.
Therefore a course of EFn has to be assumed as is shown in Fig. 11 by the solid line. From this follows that even in the inversion region the generation of holes has to be considered. Since the inversion layer width, however, is small compared with AW(t) no considerable error arises from this neglection. The difference EF~ - EF~ at the semiconductor-insulator interface also influences the charging of the interface states. All states between EF~and EFn have to be discharged with the appropriate time constant as well as those above EF,,.Thus E,=“(l)has to be used in eqn (10) as the lower integration limit. Since EF”, however, is unknown, for the calculations E&f) was used as the lower integration limit, because &(t) at the interface is known from AV,,(r). This is not correct, but practically no error arises because the emission time constant of the corresponding interface states is so large that they always interact with the valence band, a process for which the hole capture time constant is valid. The latter depends on the hole concentration and therefore on Et+,.
Nonequilibrium propelties of MIS capacitors In the region between the bulk and the band bending 211,, the splitting of the fermi level was neglected, because it is assumed to be very small in this region. Whether this assumption is really justified, may be experimentally proved: For a given ramp velocity and temperature the space charge region is widened up to a maximum value, when the capcitance has reached the value of about C,, in strong inversion. This means that any increase of the gate voltage drops across the insulator and that the whole gate current C,.& is now carried by the minority carrier generation in the region A W. In this case the relation holds
(14) Now assuming &, = (qni/2ro)A W(t) with A W(t) detined above, one obtains
as
~(2~B+AVP)=270 qni
(15) Thus a plot d/(2$~ +AV,) vs ~/i2Ndqe)C,,L$(l/d for different temperatures (the values of 2&+ Nh C,,, o=, rri(T) are known) yields the slope Q and an intersection of about d(24B), if the used model is correct, or about zero if for AW(t) nearly the whole depletion region has to be considered. The value of 2&r t A V, is the maximum band bending reached in strong inversion and can be obtained from the relation
This proof, however, can be made only in the small temperature range, where the oxide capacitance C,, in strong inversion is reached. The presented model can be improved taking into account surface recombination, and doping profiles of shallow impurity levels near the interface as well as deep bulk levels. In a forthcoming paper the comparison of experimental results with theory will be given.
I. E. H. Nicollian and A. Gotzberger, Bell Sysr. Tech. .I. 46, 1055(1%7). 2. M. Kuhn, Solid-St. Electron 13, 873 (1970). 3. R. Castagk and A. Vapaille, Surface Sci. 28, 157(1971). 4. R. Castagnt, C.R. Acad. SC. Paris 267,866 (1%8). 5. G. Declerck,R. van Overstraeterand G. Broux, Solid-St. Efecrron 16, 1451(1973). 6. F. P. Heimann, IEEE ED-13,781 (l%7). 7. R. F. Pierret, IEEE ED-19,869 (1972). 8. C. T. Sah and H. S. Fu, Phys. Status Solidi (a) 11,297 (1972). 9. C. T. Sah and H. S. Fu, Phys. Starus Solidi (a) 14,59 (1972). 10. V. Templeand J. Shewchun,Solid-St. Elecrron 16.93 (1973). 11. M. Zerbst, Z. Angew. Phys. 12, 30 (1966). 12. D. K. Scbroder and J. Guldberg, Solid-St. Elecrron 14, 1285 (1971). 13. T. W. Collins and I. N. Churchill, IEEE ED-zZ,% (1975). 14. D. Braunig and H. G. Wagemann, IEEE ED-21,241 (1974). 15. W. Schockiey and W. T. Read Jr., Phys. Reu. 87,835 (1952). 16. H. Flietner and Ngo Duong Sinh, Phys. Status Solidi (a) 37, 533 (1976).