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Small signal equivalent circuit model for the metal-insulator-semiconductor tunnel diode
SMALL SIGNAL EQUIVALENT CIRCUIT MODEL FOR THE METAL-INSULATOR-SEMICONDUCTOR TUNNEL DIODE?
Department
of Engineering
V. A. K. TEMPLES and J. SHEWCHUN Physics and the Institute for Materials Research, M&faster Ontario, Canada
(Received 27 February 1976; in revised
form
20 October
University,
Hamilton,
1977)
Abstrac-The small signala.c. equivalentcircuit technique is extended to deal with MIS diodes in which there may be considerable tunneling current. Exact a.c. equivalent circuit models are derived for metal-to-surface state tunneling and metal-to-semiconductor band tunneling. In addition, the proper admittance matrix node reduction process for preelimination of surface state nodes is given. This reduction process allows for a considerable simplification of the a.c. admittance problem. NOTATION
capture cross section times thermal velocity of electrons, holes respectively total energy, transversekinetic energy energy of the semiconductor conduction, valence band edges, energy of the ith surface state level respectively electron occupation of the metal conduction band, semiconductor conduction, valence, and ith surface state bands respectively d.c. tunnel current flowing between bands “a” and “b” a.c. tunnel current flowingbetween bands “a” and “b” Boltzmann constant times the temperature in “K density of surface states in the ith surface state band electron. hole concentrations at the interface electron, hole concentrations with the Fermi level at the energy of the ith surface state band electron charge d.c. and a.c. barrier penetration factors. (These are defined more explicitly in the paper) barrier transmission probability at E, E, d.c. voltages across the diode, semiconductor respectively d.c. quasi Fermi level shifts of the conduction, valence and ith surface state band, respectively, at the semiconductor-insulator interface a.c. voltages (or a.c. quasi Fermi level shifts) associated with V,,,, Vk, V., V,,. and Vi respectively. 1. INTRODUCTION
Electrical equivalent circuit models have long been recognized as an important aid in the analysis and improved understanding of physical systems. This is espycially true in analyzing electron transport phenomena where the analogy between transport and current, and potential and voltage are more or less direct. Complicated relationships become readily apparent if the prob‘IThis work was supported by the National Research Council of Canada. *Present address: General Electric Research and Development Cenbr, Schnectady. NY, U.S.A.
lem can be transposed into an equivalent circuit form. With respect to the MIS system there are two types of equivalent circuit models. The first type is a representation of the MIS diode with as few elements as possible the aim of which is to give approximate admittance response functions[l. 21. These are sometimes very accurate in particular ranges of bias and/or frequency. The second type is a more or less exact equivalent circuit model, represented, as one would expect, by a transmission line[3,4]. Such representations are amenable to rapid (and exact) computer solution[5-71. Furthermore, it is often possible to justify approximations to the exact equivalent circuit model which enable its reduction to simpler lumped parameter forms [3,4]. This paper extends the small signal a.c. equivalent circuit technique in the MIS system to regions in whieh there may be appreciable tunnel&. Such tunnel junctions are potentially import&t for application in transistor structures(81 and as energy conversion devices[9, IO]. Specifically, this paper will present two alternate exact equivalent circuit models for direct metal to semiconductor band tunneling and two alternate exact equivalent circuit models for tunneling into surface state bands. (Which alternate is chosen depends on the type of d.c. solution which has been used.) Furthermore, in conjunction with the a.c. surface state tunneling model, this paper specifies the pioper node reduction process for the pre-elimination of all surface state nodes prior to solving the transmission line problem. It becomes apparent that this reduction process is an important shortcut in simplifying the admittance matrix when one considers that a numerical histogram of the surface state distribution may require several “surface state bands” per kT, of band-gap. Such reduction processes have already been considered, for example, by Sah[ll] and Green[lZ] for normal MIS diodes but not specifically for the tunneling (through the oxide) process. In addition, Sah [ 131has catalogued twelve processes and equivalent circuits involving tunneling mechanisms in the semiconductor-oxide interface or the semiconductor alone which can be readily incorporated into the transmission line model being described here, for completeness.
V. A.
808
K. TEMPLEand
2.MODEL FOR DIRRCT TUNNELING FROM
TIIR METAL TOTRE SEMICONDUCTORBAND
To set the problem in perspective, consider the non tunneling a.c. small signal equivalent circuit shown in Fig. l(a). Only the “bare” transmission line model is given since our interest is primarily.in the interface and insulator region. References[3, 4 or 61, give a more complete model of the semiconductor portion and define the element values. Notice, in particular, that the model of the semiconductor contains three interconnected branches which terminate at the I-S (Insulator-Semiconductor) interface at nodes “n” on the electron band conduction path, “p” on the hole band conduction path and “k” on the displacement current path. The oxide capacitance, C,,, connects node “k” to node “m”, the metal. As many nodes “i” as there are surface state bands are added (in parallel with each other) as indicated in Fig. l(b). The current sources in the hole and electron conduction paths drive the excess majority and minority carriers in the direction of the d.c. carrier flow with a strength proportional to the d.c. current. These current sources are zero for a thick oxide but must be accounted for in MIS tunnel diodes.
‘M__
(a)
NODES
__
PP
CHARllClERlZATlON
Fig. 1. (a) MIS small signal ax. equivalent circuit model (shown for the steady state but without specifying any ax. generationrecombination processes). (b) a.c. model for Shockley-Read-Hall generation-recombination at the I-S interface or in the bulk. Reference [I31 can be consulted for a virtually complete catalogue of the processes and equivalent circuit models to describe other carrier processes in the semiconductor and at the semiconductor-insulator interface.
The objective of this paper is to derive expressions for the additional admittances which, in the presence of tunneling, connect node “m” to nodes “p”, “k” and “n”. Furthermore, this paper will give the node reduction process by which each surface state node can be eliminated, in turn, by adding appropriate compensatory admittances between nodes m, n, p and k. The derivations given will result in an a-c. equivalent circuit model of the tunneling region which is not a transmission line and has no purely passive elements apart from the oxide capacitance. At this point it is appropriate to make several comments on the type of ,a.c. model required for the semiconductor portion of the MIS tunnel diode. For accurate general purpose modeling, the Sah model is by far the best since it (i) allows for the break up of the a.c. current in the semiconductor at x = 0’ (the semiconductor-insulator interface) into three separate components-the
i. SHEWCHUN
displacement current and the electron and hole band conduction currents, (ii) allows for the a.c. effect of d.c.current flowing in the semiconductor (in an accurate model a.c. current sources proportional to the d.c. band currents are needed as indicated in Fig. I), and (iii) allows for an accurate representation of the semiconductor generation-recombination process. However it is not the only model which can be used. Any of the usual lumped circuit model or parameterized models may prove helpful provided conditions (i) to (iii) are dealt with. With condition (iii), the a.c. generation-recombination currents will be larger than in the non-tunneling diode and will usually have to be taken into account. Condition (ii) may be more of a problem and will require the addition of at least one active element into any model that is being considered. Condition (i) is the most important. Clearly at least two current paths at x = 0’ are mandatory-the displacement current path and the path which carries the largest part of the conduction current at x = 0’. Frequently, however, the third current path will be required even if it carries only a small current component. To illustrate this, consider the example[9, 10, 141 of a reverse biased Al-SiO+t Si tunnel diode where the tunnel current into the electron (majority carrier) band at x = 0’ is two to three orders of magnitude huger than the tunnel current into the hole (majority carrier) band. The temptation is to omit the minority carrier band component. However, it turns out that such an omission can be justified in reverse bias only if the minority carrier tunnel current is small with respect to both the majority carrier tunnel current and the effective rate of minority carrier generation in the semiconductor. Since the a.c. model is derived by considering a small disturbance to a d.c. steady state, the elements of the ac. model can be calculated only after solving the dc. problem. Such d.c. solutions in the MIS tunnel diode are discussed, for example, in Refs.[9, 10, 151.One merely expands the d.c. expression for tunnel current in a Taylor series. The elements of the particular a.c. network that represents the expansion are related to the partial derivatives of the dc. current component with respect to small changes in the various d.c. voltages involved. The obvious general model for a.c. current flow from the metal to semiconductor bands is derived below. Any dc. expression, say, for example, for I,,,,, can be used and the following Taylor expansion performed: aJ,” =_vm+aJm”vk+aJ,.v”
Irnnav, E
(I_~v,
av,
+
av,
(1)
a,‘vk + an%
A similar equation can be written for jmp The resulting a.c. model is given in Fig. 2 and contains three current sources, as shown, for each of the two metal to semiconductor band currents. This model, identified as Model 1, has the advantages of generality and mathematical simplicity but makes no distinction between a.c. barrier effects and a.c. Fermi function effects. Most expressions available for I,,,. do, however, consist of an integrated product of electron occupation
Small signalequivalentcircuit modelfor the metal-insulator-semiconductortunnel diode
aBvk
f-z i-@-
a:v,
n
In
energies in the semiconductor. This assumption usually fails only for a very high tunnel current density combined with a low temperature. Expanding J,,. so that the a.c. current Sr,,,. (now written as j,,,“) can be considered as the sum of two components, one arising from a.c. barrier modulation through the a.c. variation of T(E, El) and one arising from the a.c. modulation in the electron occupation difference, cf. - j,,,). yields
aPI,
---I% w
a+ aVn
ah
m
I
P
Fig. 2. Additions to the a.c. model to account for direct ax. metal to semicanductorband currents, Model I.
difference and oarrier transmission probability. On this basis, it is possible to consider a second model (Model 2). In deriving Model 2 below, we adopt a particular well known expression for the d.c. tunnel current from the metal to the semiconductor conduction band[l7] and write an appropriate expansion from which the a.c. model is derived. Hence, for the metal to semiconductor electron band current we can write[l6, 17, 191
-Appropriate E range
Appropnate El range
El)
(3)
I
j~,=KldE[u,-f,)[fdE,~“~ +
-f,,,)/dE,r(E,
dE,@,
Utilizing the known dependence of the Fermi function on energy and hence voltage, we get
ah
.A,. = K /dEU.
809
E,)dE,
(2)
I
dE &%E,) ‘av,v’
m II
+K +K where II,,,, ok and 0. are the relevant a.c. potentials. Equation (4) can easily be modeled in terms of voltage driven current sources. That is, + &'vk + (Imn(vm- ok)+ &,,(l)k - 0.) lmn = ‘,,,mv,,, (5)
where a.k, anm, a,,,” and an,,, are defined by inspection using eqn (4). The schematic representing eqn (5) is given in Fig. 3, where the current flow from the metal to the where f,, is the electron occupation of the semiconductor conduction band at energy E and dE, Ed, the barrier semiconductor is defined as positive. Figure 3 also shows transmission probability. Both r(E,E,) and K(a the appropriate current components of jmP, the current constant) are given, for example, in Refs.1161 or [171. A from the metal to the- semiconductor valence band. Elements apt, upm, amp yd a,, are defined by following similar expression can be written for J,,,,. The discussion below will be given, for simplicity, for only the one through eqns (2H4) with f, replacing f., VP replacing V,, component, the expression for the other’ following, and v, replacing v.. A simplification of the a.c. model takes place logically, by inspection. Making a small change 6V, in the voltage applied to whenever it can be determined that T(E, Ed depends, not on v,,, and t)k separately, but on (v,,, - vk), Such may be the metal results in a change SV, in the v&age at the I-S the case if the tunneling is a one band rather *an a two (Insulator-Semiconductor) interface and also a change band[l6, It?] process and if the field in the barrier is SV. in the semiconductor conduction band quasi-Fermi level. The required expansion of I,,,. is in terms of SV,,,, constant and image force barrier reduction is neglected. whence an’” ( II,,, - ok) can be combined SVk and SV,. With respect to SV., there is an implicit Thena’=-a” with a:.(v ,,, -“v,; and the model reduces to a pair of assumprion made, namely that Jingle d.c. aJ B.C.quasi: current sources. Fermi shifts can be used to describe the entire conduction band. In point of fact, tunneling occurs at all ener3. MoDEIs Fat MEML To SURFACE STATE TUNNEUNC gies in the band, with different strengths at each energy. While the direct tunnel current components are usually For our implicit assumption to hold, redistribution of tunneled electrons must occur quickly enough to ap- much larger than the surface state components, there are enough interesting and important surface state effects in proximately maintain a Fermi distribution of carrier
810
V. A.
K. T@MPLE and J.
SHEWCHUN
been made previously by Freedman and Dahlke[l9] and leads to a useful a.c. model provided SR& has a simple form. It turns out that 6RL0 has the simple form of eqn (8) below whenever the effective barrier is approximately one band in nature [17, 191and the insulator electric field is nearly constant. Under these assumptions SR$O=R’T,(u, - 0,‘)
(8)
where R : I is defined in Ref. [ 191.Because we are dealing with a particular surface state energy in the semiconductor band gap, Sfi must oscillate with a.c. interface potential, ok. At the same time, the actual occupation of the state changes with a.c. quasi-Fermi level, Vi.Hence (9) Similarly &I =fm(l Fig. 3. Additions to the a.c. model to account for direct a.c. metal to semiconductor band currents, Model 2.
the MIS tunnel diode to require accurate and general models for evtiluatifig surface state tunneling currents. For a surface state tunneling model, the d.c. current from the metal into the ith surface state band, JI, is expanded in small changes in V,,,, V, and Vi. V,,, is the d.c. metal voltage, V, the d.c. voltage across the semiconductor and V, the d.c. quasi-Fermi level shift associated with the surface state. Two models are discussed, A and B. Both of these are derived by considering tunneling into surface states at a single energy and then building up a full surface state distribution by summing over a large number of such surface state levels. In Model A, a specific assumption is made as to the form of the barrier dependence on V, and Vk. In Model L?, no such assumptions are made.
-fmKuk
-
h,,)‘$kT
(10)
Inserting (8) to (10) in (7) gives an adequate expression to model, but with a little additional algebra, j,,,i can be written as jmiPqN,(fi-f,)R:l(l), +RiToqNJ,(l
-
uk)
-fmNt(m- vi)qlkT
+R:oq~i[f,(l-f~)-fi(~-fi)l(oi-~,)q/kT
(II)
This equation is modeled in Fig. 4(a). Table I(a), accompanying the figure defined the elements of Fig. 4(a) either by inspection from eqn (11) or from Refs. (4) or (6).
Surface state Model A Surface state model A utilizes the approach that the d.c. tunnel current into the ith surface state can be written as proportional to a barrier penetration factor, I?&, and an electron occupation d#erence[l9]. The equation is (6) Applying an a.c. voltage, SV,,, = u,,,,to the metal results in a.c. voltages uk and uI appearing at nodes k and i. Separating out a.c. current contriiutions by changes in h, f,,, and R&, gives e$ (7)
Equation (7) can be interpreted physically as separating the contributions of the oscillating occupation’&ertnce and a.c. barrier modulation. Thk separaiion has in fact
Fis. 4. (a) a.c. Surface state Model A including the effects of tunneling. (b) !&face state Model A upon elimination of the surface state node or nodes.
elimination in &ace state Model A Using Fig. 4(a) to write the a.c. currents from node “i” to nodes “m” , ‘VI”, V’ and “p” and then applying Kiichoff’s current law at node “i” allows one to eliminate ul from the various current expressions. The’ modei shown in Fg 4(b) results. Table I(b) defines the elements of Fig. 4(b) with respect to those of’ Fig., 4(a). Note thaf adding more surface state nodes requires only a simple sum over the proper parallel admittances of the IWe
Small signalequivalent circuit model for the metal-insulator-semiconductortunnel diode
811
Table 1. Mement valuesof Fii. 4 fsurface state IUodel A) (a) Elements of Figure 4a
G
pi
'i
(b) Elements of Figure 4b arising from elimination of one or more
- Ni p fi cpi q'/kT = Ni fi(l - fi)q2/kT
Gni.= Ni N(1 _ fi) cni q2/kT A A
pi ni
Y nk n ~ Gni SCi/~,
= Ri f, q2/kT = Ri(1 - fi) q2/kT
dere Ri, the recombination rate is given by
c I(NP - ni2) Ri = Ni 'ni D 'nitN + "oil + cpitp + poi) Elements below are from quation(l1) sj
= q2 "iCfm(l - f&f+'
si
= q R$fi
-fi)] R;,/kT
- fm, 4l
where Gi = Gmi t Gpi + Gni + sCi
G = q2 Ni fm(l - fm, RTotkT mi
- Ami - Api - Ani
reduced node model as indicated in Table I(b). This, as previously stated, considerably reduces the a.c. admittance matrix. Surface stateModel B ?he need for a second surface state model arises when SRTOof eqn (7) cannot be properly expanded in the simple manner indicated in eqn (8). The appropriate general expansion is SRio = a,,,‘~,,,+ akivk
(12)
where
Notethatifweassumcthata~=-a,‘wegetback surface state Model A. However with two-band models of the tunneling barrier[M, 181,cl: will often dilIer from -a,,,’ so that surface state Model B may be necessary. Certainly, in numerical solutions, Model B is only slightly more involved than Model A. By’ substitutin$ eqns (9), (IO) and (12) in (7). one obtains j~=a~.i(v,-vk)+q~(tll-vk)+(ami+cr*I)Vk
+q2N&(l +q*N&41
-fm)RiTdt)m - vc)/kT -fi)fr(ar -
vk)/kT.
Fii. 5. (a) P.C. Surface state Model B including the effects of tunneling. (b) Surface state Model B upon elimination of the surface state node or nodes.
S(a) is identical to that of Fig. 4(a) except for the extra current branch, akbk between nodes m and i. Node mductkm withsutface stateModel B The reduced node form of Model B is given in Fii, 5(b). Note that the reduced form of Model B does not differ greatly from the reduced form of Model A, given previously in,Fii. 4(b). The three cutrent branches h-v*, h,&k and &kuk are new, where the h parameters are dehned~in Table 2 in terms of the B and a parameters.
(13) C CONC~N
The appropriate model incorporating the jmi of eqn (13) into the normal.Shockley-Read-Hall type surface state picture is shown in Fig. S(a). The elements of model “B” are defined in Table 2(a). Note that the form of Fig.
This paper has investigated the type of a.c. equivalent circuit models required for the analysis of a.c. response in MIS tunnel diodes. Two types of models arise depending on whether or not the effect of ac. barrier
812
V. A. K.
TEMPLE and
J.
SHEWCHUN
Table 2. Element values of Fig. 5 (surface state Model B) (a)
Elements of Figure 5a Note that capitalized admittances are defined in Table la gpi ='Gpi 'i
= ci
(b) Elements of Figure 5h arising from elimination of one or more nodes '
Ymn = F gmi9niIgi Ymk = F 'ci(gmi+ ahf )/gi = s i 9mi.gpl.lgi
gni = Gni
ymp
Sni = Gmi
Y nk = F 9niscf/gj
api = A ani
.
Pl
= Ani
'pk
= z g .sc fg. i pl i 1
Y
= 4 gni9pfI9i
np
= Ami + bi ami k
a = c a .g ./gi i mi pl mp
al;li = ~6:
= z a .g ./g. i pl ml 1 apm b = f ahi(9pi - apj)/gi mp
.
aI, =c(b + dl
cakand ctiare defined by equation (12)
a mn = c i ami9ni/9i a nm = i ani9mi/Yf brim= 2 aAi(9ni - a,f)/9i i i:f apignf/9f aFn a = z ani9pi/9i i . np h = z a'(g f k nimn h = ; alkipi mp . h = ; aLsci/gi mk
a,i)/9i
api)/gi
where gi = sci + gmi + 9
pi +
- ami - api
modulation is approximated or dealt with exactly. Both metal-to-semiconductor band models have been derived. With the surface state models, a formal node reduction process is specified to simplify the handling of the admittance matrix. Such a.c. models are required to extend the exact a.c. analysis of the MIS system to diodes in which there is an appreciable tunnel current. While, this paper has primarily considered the appropriate equivalent circuit models in the insulator. region, some comments can be made with respect to models of the semiconductor region. In particular the a.c. model of the semiconductor, to be applicable in MIS tunnel diodes and to be compatible- with tunneling region models derived in this paper, should have a formulation which allows for current paths at ,the I-S interface which can be associated with the displacement current and the two individual conduction currents. Semiconductor models which neglect the minority carrier current path can be used only if the tunneling current is truly negligible with respect to both the majority band current and the semiconductor minority carrier generation rate. -cI&s
1. K. Lehovec and A. Slobodskoy, Solid-St. Eketron. 7, 59 (1964).
gni
- ani
2. E. H. Nicollian and A. Goetzberger, Bell Sysl. Tech. J 46, 1055(1967). 3. C. T. Sah, Technical Rept. No. 3. Electrical Engineering Research Lab., University of Illinois, Urbana, Ill. 4. C. T. Sah, R. F. Pierret and A. B. Tole, Solid-St Electron. 12,681 (1%9). 5. L. Forbes and C. T. Sah, IEEE Trans. ED16, 1036(1%9). 6. V. A. K. Temple and J. Shewchun, Solid-St. Electron. 16, 93 (1973). 7. J. Shewchun, V. A. K. Temple and R. A. Clarke, J. Appl. Phys. 43, 3487(1972). 8. J. Shewchun and R. A. Clarke, Solid-St. Electron. 16, 213 (1973). 9. M. A. Green, F. D. King and J. Shewchun, Solid-St.Elecfron. 17, 551 (1974). 10. J. Shewchun,M. A. Green and F. D. King, Solid-St. Electron. 17, 563 (1974). 11. C. T. Sah, Electron. Serf. 7,713 (1971). 12. M. A. Green, V. A. K. Temple and J. Shewchun. Solid-St. Electron. 15, 1027(lp12). 13. C. T. Sah, Phys. Stat. Solidi. (a) 7,541 (1971). 14. M. A. Green and J. Shewchun, Solid-St. Electron. 17. 349 (1974). 15. V. A. K. Temple, M. A. Green, and J. Shewchun, 1. Appl. Phys. 454934 (1974). 16. V. A. K. Temple and J. Shewchun, Solid-St. Elecrron. 17,417 (1974). 17. W. A. Harrison, Phys. Rev. 123,85 (l%l). 18. P. Schnupp, Phys. Stouts Sol. 31,567 (1%7). 19. L. B. Freeman and W. E. Dahlke, Solid-St. Electron. 13. 1433 (IWO).
Department
of Engineering
V. A. K. TEMPLES and J. SHEWCHUN Physics and the Institute for Materials Research, M&faster Ontario, Canada
(Received 27 February 1976; in revised
form
20 October
University,
Hamilton,
1977)
Abstrac-The small signala.c. equivalentcircuit technique is extended to deal with MIS diodes in which there may be considerable tunneling current. Exact a.c. equivalent circuit models are derived for metal-to-surface state tunneling and metal-to-semiconductor band tunneling. In addition, the proper admittance matrix node reduction process for preelimination of surface state nodes is given. This reduction process allows for a considerable simplification of the a.c. admittance problem. NOTATION
capture cross section times thermal velocity of electrons, holes respectively total energy, transversekinetic energy energy of the semiconductor conduction, valence band edges, energy of the ith surface state level respectively electron occupation of the metal conduction band, semiconductor conduction, valence, and ith surface state bands respectively d.c. tunnel current flowing between bands “a” and “b” a.c. tunnel current flowingbetween bands “a” and “b” Boltzmann constant times the temperature in “K density of surface states in the ith surface state band electron. hole concentrations at the interface electron, hole concentrations with the Fermi level at the energy of the ith surface state band electron charge d.c. and a.c. barrier penetration factors. (These are defined more explicitly in the paper) barrier transmission probability at E, E, d.c. voltages across the diode, semiconductor respectively d.c. quasi Fermi level shifts of the conduction, valence and ith surface state band, respectively, at the semiconductor-insulator interface a.c. voltages (or a.c. quasi Fermi level shifts) associated with V,,,, Vk, V., V,,. and Vi respectively. 1. INTRODUCTION
Electrical equivalent circuit models have long been recognized as an important aid in the analysis and improved understanding of physical systems. This is espycially true in analyzing electron transport phenomena where the analogy between transport and current, and potential and voltage are more or less direct. Complicated relationships become readily apparent if the prob‘IThis work was supported by the National Research Council of Canada. *Present address: General Electric Research and Development Cenbr, Schnectady. NY, U.S.A.
lem can be transposed into an equivalent circuit form. With respect to the MIS system there are two types of equivalent circuit models. The first type is a representation of the MIS diode with as few elements as possible the aim of which is to give approximate admittance response functions[l. 21. These are sometimes very accurate in particular ranges of bias and/or frequency. The second type is a more or less exact equivalent circuit model, represented, as one would expect, by a transmission line[3,4]. Such representations are amenable to rapid (and exact) computer solution[5-71. Furthermore, it is often possible to justify approximations to the exact equivalent circuit model which enable its reduction to simpler lumped parameter forms [3,4]. This paper extends the small signal a.c. equivalent circuit technique in the MIS system to regions in whieh there may be appreciable tunnel&. Such tunnel junctions are potentially import&t for application in transistor structures(81 and as energy conversion devices[9, IO]. Specifically, this paper will present two alternate exact equivalent circuit models for direct metal to semiconductor band tunneling and two alternate exact equivalent circuit models for tunneling into surface state bands. (Which alternate is chosen depends on the type of d.c. solution which has been used.) Furthermore, in conjunction with the a.c. surface state tunneling model, this paper specifies the pioper node reduction process for the pre-elimination of all surface state nodes prior to solving the transmission line problem. It becomes apparent that this reduction process is an important shortcut in simplifying the admittance matrix when one considers that a numerical histogram of the surface state distribution may require several “surface state bands” per kT, of band-gap. Such reduction processes have already been considered, for example, by Sah[ll] and Green[lZ] for normal MIS diodes but not specifically for the tunneling (through the oxide) process. In addition, Sah [ 131has catalogued twelve processes and equivalent circuits involving tunneling mechanisms in the semiconductor-oxide interface or the semiconductor alone which can be readily incorporated into the transmission line model being described here, for completeness.
V. A.
808
K. TEMPLEand
2.MODEL FOR DIRRCT TUNNELING FROM
TIIR METAL TOTRE SEMICONDUCTORBAND
To set the problem in perspective, consider the non tunneling a.c. small signal equivalent circuit shown in Fig. l(a). Only the “bare” transmission line model is given since our interest is primarily.in the interface and insulator region. References[3, 4 or 61, give a more complete model of the semiconductor portion and define the element values. Notice, in particular, that the model of the semiconductor contains three interconnected branches which terminate at the I-S (Insulator-Semiconductor) interface at nodes “n” on the electron band conduction path, “p” on the hole band conduction path and “k” on the displacement current path. The oxide capacitance, C,,, connects node “k” to node “m”, the metal. As many nodes “i” as there are surface state bands are added (in parallel with each other) as indicated in Fig. l(b). The current sources in the hole and electron conduction paths drive the excess majority and minority carriers in the direction of the d.c. carrier flow with a strength proportional to the d.c. current. These current sources are zero for a thick oxide but must be accounted for in MIS tunnel diodes.
‘M__
(a)
NODES
__
PP
CHARllClERlZATlON
Fig. 1. (a) MIS small signal ax. equivalent circuit model (shown for the steady state but without specifying any ax. generationrecombination processes). (b) a.c. model for Shockley-Read-Hall generation-recombination at the I-S interface or in the bulk. Reference [I31 can be consulted for a virtually complete catalogue of the processes and equivalent circuit models to describe other carrier processes in the semiconductor and at the semiconductor-insulator interface.
The objective of this paper is to derive expressions for the additional admittances which, in the presence of tunneling, connect node “m” to nodes “p”, “k” and “n”. Furthermore, this paper will give the node reduction process by which each surface state node can be eliminated, in turn, by adding appropriate compensatory admittances between nodes m, n, p and k. The derivations given will result in an a-c. equivalent circuit model of the tunneling region which is not a transmission line and has no purely passive elements apart from the oxide capacitance. At this point it is appropriate to make several comments on the type of ,a.c. model required for the semiconductor portion of the MIS tunnel diode. For accurate general purpose modeling, the Sah model is by far the best since it (i) allows for the break up of the a.c. current in the semiconductor at x = 0’ (the semiconductor-insulator interface) into three separate components-the
i. SHEWCHUN
displacement current and the electron and hole band conduction currents, (ii) allows for the a.c. effect of d.c.current flowing in the semiconductor (in an accurate model a.c. current sources proportional to the d.c. band currents are needed as indicated in Fig. I), and (iii) allows for an accurate representation of the semiconductor generation-recombination process. However it is not the only model which can be used. Any of the usual lumped circuit model or parameterized models may prove helpful provided conditions (i) to (iii) are dealt with. With condition (iii), the a.c. generation-recombination currents will be larger than in the non-tunneling diode and will usually have to be taken into account. Condition (ii) may be more of a problem and will require the addition of at least one active element into any model that is being considered. Condition (i) is the most important. Clearly at least two current paths at x = 0’ are mandatory-the displacement current path and the path which carries the largest part of the conduction current at x = 0’. Frequently, however, the third current path will be required even if it carries only a small current component. To illustrate this, consider the example[9, 10, 141 of a reverse biased Al-SiO+t Si tunnel diode where the tunnel current into the electron (majority carrier) band at x = 0’ is two to three orders of magnitude huger than the tunnel current into the hole (majority carrier) band. The temptation is to omit the minority carrier band component. However, it turns out that such an omission can be justified in reverse bias only if the minority carrier tunnel current is small with respect to both the majority carrier tunnel current and the effective rate of minority carrier generation in the semiconductor. Since the a.c. model is derived by considering a small disturbance to a d.c. steady state, the elements of the ac. model can be calculated only after solving the dc. problem. Such d.c. solutions in the MIS tunnel diode are discussed, for example, in Refs.[9, 10, 151.One merely expands the d.c. expression for tunnel current in a Taylor series. The elements of the particular a.c. network that represents the expansion are related to the partial derivatives of the dc. current component with respect to small changes in the various d.c. voltages involved. The obvious general model for a.c. current flow from the metal to semiconductor bands is derived below. Any dc. expression, say, for example, for I,,,,, can be used and the following Taylor expansion performed: aJ,” =_vm+aJm”vk+aJ,.v”
Irnnav, E
(I_~v,
av,
+
av,
(1)
a,‘vk + an%
A similar equation can be written for jmp The resulting a.c. model is given in Fig. 2 and contains three current sources, as shown, for each of the two metal to semiconductor band currents. This model, identified as Model 1, has the advantages of generality and mathematical simplicity but makes no distinction between a.c. barrier effects and a.c. Fermi function effects. Most expressions available for I,,,. do, however, consist of an integrated product of electron occupation
Small signalequivalentcircuit modelfor the metal-insulator-semiconductortunnel diode
aBvk
f-z i-@-
a:v,
n
In
energies in the semiconductor. This assumption usually fails only for a very high tunnel current density combined with a low temperature. Expanding J,,. so that the a.c. current Sr,,,. (now written as j,,,“) can be considered as the sum of two components, one arising from a.c. barrier modulation through the a.c. variation of T(E, El) and one arising from the a.c. modulation in the electron occupation difference, cf. - j,,,). yields
aPI,
---I% w
a+ aVn
ah
m
I
P
Fig. 2. Additions to the a.c. model to account for direct ax. metal to semicanductorband currents, Model I.
difference and oarrier transmission probability. On this basis, it is possible to consider a second model (Model 2). In deriving Model 2 below, we adopt a particular well known expression for the d.c. tunnel current from the metal to the semiconductor conduction band[l7] and write an appropriate expansion from which the a.c. model is derived. Hence, for the metal to semiconductor electron band current we can write[l6, 17, 191
-Appropriate E range
Appropnate El range
El)
(3)
I
j~,=KldE[u,-f,)[fdE,~“~ +
-f,,,)/dE,r(E,
dE,@,
Utilizing the known dependence of the Fermi function on energy and hence voltage, we get
ah
.A,. = K /dEU.
809
E,)dE,
(2)
I
dE &%E,) ‘av,v’
m II
+K +K where II,,,, ok and 0. are the relevant a.c. potentials. Equation (4) can easily be modeled in terms of voltage driven current sources. That is, + &'vk + (Imn(vm- ok)+ &,,(l)k - 0.) lmn = ‘,,,mv,,, (5)
where a.k, anm, a,,,” and an,,, are defined by inspection using eqn (4). The schematic representing eqn (5) is given in Fig. 3, where the current flow from the metal to the where f,, is the electron occupation of the semiconductor conduction band at energy E and dE, Ed, the barrier semiconductor is defined as positive. Figure 3 also shows transmission probability. Both r(E,E,) and K(a the appropriate current components of jmP, the current constant) are given, for example, in Refs.1161 or [171. A from the metal to the- semiconductor valence band. Elements apt, upm, amp yd a,, are defined by following similar expression can be written for J,,,,. The discussion below will be given, for simplicity, for only the one through eqns (2H4) with f, replacing f., VP replacing V,, component, the expression for the other’ following, and v, replacing v.. A simplification of the a.c. model takes place logically, by inspection. Making a small change 6V, in the voltage applied to whenever it can be determined that T(E, Ed depends, not on v,,, and t)k separately, but on (v,,, - vk), Such may be the metal results in a change SV, in the v&age at the I-S the case if the tunneling is a one band rather *an a two (Insulator-Semiconductor) interface and also a change band[l6, It?] process and if the field in the barrier is SV. in the semiconductor conduction band quasi-Fermi level. The required expansion of I,,,. is in terms of SV,,,, constant and image force barrier reduction is neglected. whence an’” ( II,,, - ok) can be combined SVk and SV,. With respect to SV., there is an implicit Thena’=-a” with a:.(v ,,, -“v,; and the model reduces to a pair of assumprion made, namely that Jingle d.c. aJ B.C.quasi: current sources. Fermi shifts can be used to describe the entire conduction band. In point of fact, tunneling occurs at all ener3. MoDEIs Fat MEML To SURFACE STATE TUNNEUNC gies in the band, with different strengths at each energy. While the direct tunnel current components are usually For our implicit assumption to hold, redistribution of tunneled electrons must occur quickly enough to ap- much larger than the surface state components, there are enough interesting and important surface state effects in proximately maintain a Fermi distribution of carrier
810
V. A.
K. T@MPLE and J.
SHEWCHUN
been made previously by Freedman and Dahlke[l9] and leads to a useful a.c. model provided SR& has a simple form. It turns out that 6RL0 has the simple form of eqn (8) below whenever the effective barrier is approximately one band in nature [17, 191and the insulator electric field is nearly constant. Under these assumptions SR$O=R’T,(u, - 0,‘)
(8)
where R : I is defined in Ref. [ 191.Because we are dealing with a particular surface state energy in the semiconductor band gap, Sfi must oscillate with a.c. interface potential, ok. At the same time, the actual occupation of the state changes with a.c. quasi-Fermi level, Vi.Hence (9) Similarly &I =fm(l Fig. 3. Additions to the a.c. model to account for direct a.c. metal to semiconductor band currents, Model 2.
the MIS tunnel diode to require accurate and general models for evtiluatifig surface state tunneling currents. For a surface state tunneling model, the d.c. current from the metal into the ith surface state band, JI, is expanded in small changes in V,,,, V, and Vi. V,,, is the d.c. metal voltage, V, the d.c. voltage across the semiconductor and V, the d.c. quasi-Fermi level shift associated with the surface state. Two models are discussed, A and B. Both of these are derived by considering tunneling into surface states at a single energy and then building up a full surface state distribution by summing over a large number of such surface state levels. In Model A, a specific assumption is made as to the form of the barrier dependence on V, and Vk. In Model L?, no such assumptions are made.
-fmKuk
-
h,,)‘$kT
(10)
Inserting (8) to (10) in (7) gives an adequate expression to model, but with a little additional algebra, j,,,i can be written as jmiPqN,(fi-f,)R:l(l), +RiToqNJ,(l
-
uk)
-fmNt(m- vi)qlkT
+R:oq~i[f,(l-f~)-fi(~-fi)l(oi-~,)q/kT
(II)
This equation is modeled in Fig. 4(a). Table I(a), accompanying the figure defined the elements of Fig. 4(a) either by inspection from eqn (11) or from Refs. (4) or (6).
Surface state Model A Surface state model A utilizes the approach that the d.c. tunnel current into the ith surface state can be written as proportional to a barrier penetration factor, I?&, and an electron occupation d#erence[l9]. The equation is (6) Applying an a.c. voltage, SV,,, = u,,,,to the metal results in a.c. voltages uk and uI appearing at nodes k and i. Separating out a.c. current contriiutions by changes in h, f,,, and R&, gives e$ (7)
Equation (7) can be interpreted physically as separating the contributions of the oscillating occupation’&ertnce and a.c. barrier modulation. Thk separaiion has in fact
Fis. 4. (a) a.c. Surface state Model A including the effects of tunneling. (b) !&face state Model A upon elimination of the surface state node or nodes.
elimination in &ace state Model A Using Fig. 4(a) to write the a.c. currents from node “i” to nodes “m” , ‘VI”, V’ and “p” and then applying Kiichoff’s current law at node “i” allows one to eliminate ul from the various current expressions. The’ modei shown in Fg 4(b) results. Table I(b) defines the elements of Fig. 4(b) with respect to those of’ Fig., 4(a). Note thaf adding more surface state nodes requires only a simple sum over the proper parallel admittances of the IWe
Small signalequivalent circuit model for the metal-insulator-semiconductortunnel diode
811
Table 1. Mement valuesof Fii. 4 fsurface state IUodel A) (a) Elements of Figure 4a
G
pi
'i
(b) Elements of Figure 4b arising from elimination of one or more
- Ni p fi cpi q'/kT = Ni fi(l - fi)q2/kT
Gni.= Ni N(1 _ fi) cni q2/kT A A
pi ni
Y nk n ~ Gni SCi/~,
= Ri f, q2/kT = Ri(1 - fi) q2/kT
dere Ri, the recombination rate is given by
c I(NP - ni2) Ri = Ni 'ni D 'nitN + "oil + cpitp + poi) Elements below are from quation(l1) sj
= q2 "iCfm(l - f&f+'
si
= q R$fi
-fi)] R;,/kT
- fm, 4l
where Gi = Gmi t Gpi + Gni + sCi
G = q2 Ni fm(l - fm, RTotkT mi
- Ami - Api - Ani
reduced node model as indicated in Table I(b). This, as previously stated, considerably reduces the a.c. admittance matrix. Surface stateModel B ?he need for a second surface state model arises when SRTOof eqn (7) cannot be properly expanded in the simple manner indicated in eqn (8). The appropriate general expansion is SRio = a,,,‘~,,,+ akivk
(12)
where
Notethatifweassumcthata~=-a,‘wegetback surface state Model A. However with two-band models of the tunneling barrier[M, 181,cl: will often dilIer from -a,,,’ so that surface state Model B may be necessary. Certainly, in numerical solutions, Model B is only slightly more involved than Model A. By’ substitutin$ eqns (9), (IO) and (12) in (7). one obtains j~=a~.i(v,-vk)+q~(tll-vk)+(ami+cr*I)Vk
+q2N&(l +q*N&41
-fm)RiTdt)m - vc)/kT -fi)fr(ar -
vk)/kT.
Fii. 5. (a) P.C. Surface state Model B including the effects of tunneling. (b) Surface state Model B upon elimination of the surface state node or nodes.
S(a) is identical to that of Fig. 4(a) except for the extra current branch, akbk between nodes m and i. Node mductkm withsutface stateModel B The reduced node form of Model B is given in Fii, 5(b). Note that the reduced form of Model B does not differ greatly from the reduced form of Model A, given previously in,Fii. 4(b). The three cutrent branches h-v*, h,&k and &kuk are new, where the h parameters are dehned~in Table 2 in terms of the B and a parameters.
(13) C CONC~N
The appropriate model incorporating the jmi of eqn (13) into the normal.Shockley-Read-Hall type surface state picture is shown in Fig. S(a). The elements of model “B” are defined in Table 2(a). Note that the form of Fig.
This paper has investigated the type of a.c. equivalent circuit models required for the analysis of a.c. response in MIS tunnel diodes. Two types of models arise depending on whether or not the effect of ac. barrier
812
V. A. K.
TEMPLE and
J.
SHEWCHUN
Table 2. Element values of Fig. 5 (surface state Model B) (a)
Elements of Figure 5a Note that capitalized admittances are defined in Table la gpi ='Gpi 'i
= ci
(b) Elements of Figure 5h arising from elimination of one or more nodes '
Ymn = F gmi9niIgi Ymk = F 'ci(gmi+ ahf )/gi = s i 9mi.gpl.lgi
gni = Gni
ymp
Sni = Gmi
Y nk = F 9niscf/gj
api = A ani
.
Pl
= Ani
'pk
= z g .sc fg. i pl i 1
Y
= 4 gni9pfI9i
np
= Ami + bi ami k
a = c a .g ./gi i mi pl mp
al;li = ~6:
= z a .g ./g. i pl ml 1 apm b = f ahi(9pi - apj)/gi mp
.
aI, =c(b + dl
cakand ctiare defined by equation (12)
a mn = c i ami9ni/9i a nm = i ani9mi/Yf brim= 2 aAi(9ni - a,f)/9i i i:f apignf/9f aFn a = z ani9pi/9i i . np h = z a'(g f k nimn h = ; alkipi mp . h = ; aLsci/gi mk
a,i)/9i
api)/gi
where gi = sci + gmi + 9
pi +
- ami - api
modulation is approximated or dealt with exactly. Both metal-to-semiconductor band models have been derived. With the surface state models, a formal node reduction process is specified to simplify the handling of the admittance matrix. Such a.c. models are required to extend the exact a.c. analysis of the MIS system to diodes in which there is an appreciable tunnel current. While, this paper has primarily considered the appropriate equivalent circuit models in the insulator. region, some comments can be made with respect to models of the semiconductor region. In particular the a.c. model of the semiconductor, to be applicable in MIS tunnel diodes and to be compatible- with tunneling region models derived in this paper, should have a formulation which allows for current paths at ,the I-S interface which can be associated with the displacement current and the two individual conduction currents. Semiconductor models which neglect the minority carrier current path can be used only if the tunneling current is truly negligible with respect to both the majority band current and the semiconductor minority carrier generation rate. -cI&s
1. K. Lehovec and A. Slobodskoy, Solid-St. Eketron. 7, 59 (1964).
gni
- ani
2. E. H. Nicollian and A. Goetzberger, Bell Sysl. Tech. J 46, 1055(1967). 3. C. T. Sah, Technical Rept. No. 3. Electrical Engineering Research Lab., University of Illinois, Urbana, Ill. 4. C. T. Sah, R. F. Pierret and A. B. Tole, Solid-St Electron. 12,681 (1%9). 5. L. Forbes and C. T. Sah, IEEE Trans. ED16, 1036(1%9). 6. V. A. K. Temple and J. Shewchun, Solid-St. Electron. 16, 93 (1973). 7. J. Shewchun, V. A. K. Temple and R. A. Clarke, J. Appl. Phys. 43, 3487(1972). 8. J. Shewchun and R. A. Clarke, Solid-St. Electron. 16, 213 (1973). 9. M. A. Green, F. D. King and J. Shewchun, Solid-St.Elecfron. 17, 551 (1974). 10. J. Shewchun,M. A. Green and F. D. King, Solid-St. Electron. 17, 563 (1974). 11. C. T. Sah, Electron. Serf. 7,713 (1971). 12. M. A. Green, V. A. K. Temple and J. Shewchun. Solid-St. Electron. 15, 1027(lp12). 13. C. T. Sah, Phys. Stat. Solidi. (a) 7,541 (1971). 14. M. A. Green and J. Shewchun, Solid-St. Electron. 17. 349 (1974). 15. V. A. K. Temple, M. A. Green, and J. Shewchun, 1. Appl. Phys. 454934 (1974). 16. V. A. K. Temple and J. Shewchun, Solid-St. Elecrron. 17,417 (1974). 17. W. A. Harrison, Phys. Rev. 123,85 (l%l). 18. P. Schnupp, Phys. Stouts Sol. 31,567 (1%7). 19. L. B. Freeman and W. E. Dahlke, Solid-St. Electron. 13. 1433 (IWO).