SolidState
Electronics, 1975, Vol. 18. pp. 791798.
Pergamon Press.
Printed in Great Britain
A MODEL OF OHMIC CONTACTS TO SEMICONDUCTORS
Centro
di Studio per i Metodi
B. PELLEGRINI and G. SALARDI e i Dispositivi di Radiotrasmissione del Consiglio di Pisa, Via Diotisalvi 2,561C4l Pisa, Italy (Received
6 August
Nazionale
1974; in revised form 30 December
delle Ricerche,
Universitl
1974)
AbstractA unified and detailed model of both thermionic and tunneling ohmic contacts to semiconductors is described. The modeltakes into accountthe actualprofile of the energy barrier as it is determined by the difference between the metal work function and the semiconductor electron affinity, by the ionized impurities in the semiconductor, by the interfacial quantum electric dipole and by the classical and quantum penetration of the charge carriers into the depletion layer. The current, above and below the energy barrier peak, is computed by using the Kemble generalized transmission coefficient, by taking into account the anisotropy of the effective masses as well as their dependence on impurityconcentration, and by employing the FermiDirac statistics. Good agreement between theoretical and experimental results is observed. The model enables the design of ohmic contacts to semiconductors, i.e. the determination of the semiconductor impurity concentration required to achieve preassigned features.
NOTATION
k ml, m, m,
t: ND N” n n, P 4 & s T  VL, VI. u Vh 0. V.
VI 0, “‘I
Vo VW
0”
Richardson’s constant diffusion capacitance depletionlayer capacitance function defined by Eq. (5) semiconductor energy gap metalFermi level semiconductorFermi level transverse energy of the metal electrons tunneling into the semiconductor energy gap electric field electric field at the interface limit value of the upper 3 dB frequency FermiDirac integral Planck’s constant current interval in which the IV characteristic is linear current through the contact current density which determines the linearity interval of the IV characteristic (Eq. (6)(9)) Boltzmann’s constant longitudinal effective mass of the electron transverse effective mass of the electron effective mass of the electron in the direction x normal to the contact surface rest mass of the electron acceptor concentration donor concentration density, at the interface, of the metal electrons tunneling into the semiconductor energy gap electron concentration number of equivalent minima of the conduction band function given by Eq. (15) electronic charge specific resistance of the contact contact area absolute temperature voltage range in which the IV characteristic is linear contact bias voltage capacitance builtinvoltage contact potential difference correction of vh due to the penetration of the semiconductor electrons into the depletionlayer electric potential minimum of the electric potential correction of D,, due to the metal electrons tunneling into the semiconductor energy gap electric potential at the interface electric potential equal to tl,,, when F, > 0 and equal to u0 when F,
W w X
Xd
X.
Xi X, a e 6 9 I9
A. hv
CJ @hi
X
height of energy barrier seen by the conduction band electrons of the semiconductor width of the depletionlayer abscissa of a depletionlayer section characterized by only fixed electric charges catesian abscissa taken on an axis normal to the contact surface abscissa of the depletionlayer section beyond which the electric potential becomes parabolic in shape abscissa of the depletionlayer section beyond which the electric potential becomes exponential in shape abscissa of the flex point of the electric potential abscissa of the electric potential minimum coefficient defined by Eq. (3) maximum relative error of the current semiconductor dielectric constant electron energy in the x direction normalized to kT angle between the x axis and a major principal axis of the crystal generalized Debye length penetration mean length of the metal electrons tunneling into the semiconductor height of the energy barrier metal work function semiconductor electric affinity
1. PRELIMINARIES (a)
Electric
potential
In the present work we extend the preceding theory of the rectifying metalsemiconductor interfaces [ 141 to ohmic contacts. The electric potential and the energy band diagrams which arise in the contact region between an ntype semiconductor and a low workfunction metal have the shape shown in Figs. l(a) and (b). The analytical expression for the electrical potential, w(x), depending on the interfacial quantum electric dipole and on the charge of the depletion layer, is given by equation (11A) in the Appendix. In order to evaluate the ohmic behaviour of the contact, it is necessary to determine with accuracy the “tail” of the potential curve beyond the depletion layer, since the most electrons tunneling the energy hairier start from or arrive there. As it is shown in detail in Appendix, C+(X)is given, for 791
B.
792
PELLEGRINIand
G.
SALARDI
(b) Current A general expression[6] for the current through an energy barrier, as the one of Fig. l(b), which holds when diffusion[7141 and electronphonon scattering[l5,16] are negligible, so that the transport mechanism across the contact is the thermionic emission [6,9, 17231,is given by 4”JkT i= Sa er I
where
In equations (2) and (3) S is the contact area; v, = vc  v  us; A0 = 4?rqk2mo/h3 = 120A/cm”K* is the Richardson constant; h is Plank’s constant; n, is the number of equivalent minima of the conduction band; ma is the electron rest mass; and ml and m, are the longitudinal and transverse effective masses of the electron. The mass m, is given by [6,22,23] ,
m, = m,m,/(mr cos'13 tmr sin2 IV),
1w
Fig. 1. (a) Electric potential and, (b) energy band diagrams of a contact between a metal and an ntype semiconductor.
(4)
where 19is the angle between the x axis and a major principal axis of the crystal. The function D(~T[), for 9 > 0, is defined by X,(S)
x
D(~)=$2m,)"'
ax., by
{q[v,  vi(x)]+ qkT}“‘dx, IX1(4) (5)
w=v,v2v,exp[(xx,)/&],
(1)
where the symbols have the following meanings: D, is the contact potential difference; u is the voltage biasing the contact; ve is given by (4A) in Appendix; x is the
where the abscissae xl and x2 of the turning points are the solutions of the equation vS vi(x) + TkT = 0.When the energy barrier peak is nearly parabolic in shape, such as it happens in the case of Fig. l(b), equation (5) still holds [ 17,201for 71< 0, provided that 77is replaced by 1~1 and the sign of the right hand term be changed. In Appendix the computation of ~(1~1) is shown. Equations (2)(S) take into account the shape of the energy barrier, the anisotropy of the electron effective masses, the manyvalley structure of the semiconductor energy band, the FermiDirac statistics and, lastly, the quantummechanical reflection and tunneling through the energy barrier by means of Kemble’s generalized transmission coefficient [6,17201. The energy barrier lowering, due to the image force, has been intentionally excluded because the energy barrier peak is placed in a region where a large mobile charge exists[l, 41.
abscissa taken on an axis normal to the contact surface; A, = (2eSv,/qNo)“* is a “generalized” Debye length, E$is the semiconductor dielectric constant, q is the electron charge and ND is the impurity concentration, x, = W  A, where W is the width of the depletion layer, given by (lOA) in Appendix. Other symbols appearing in Figs. l(a) and (b), have the following meaning: vm is the minimum of the electric potential and x,,, its abscissa; ua = vi(O)is the potential at the interface x = 0; cPr is the metal work function: x is the semiconductor electric affinity; ES (EM) is the semiconductor (metal) Fermilevel evaluated, upward, from the conductionband bottom; V, = oM x + Es;F0 is the electric field at x = 0’; @= a,,, x  qv, (where 2. OHMIC CONTACTS vS= u,,,when F,, > 0 and v5 = v, when F.< 0)is the height of the energy barrier, i.e. the minimum energy which must Equation (2) is quite general since it takes into account be furnished to a metal electron, belonging to EM, for the physical properties of the contact. It holds, in classical penetration into the semiconductor bulk; EG is particular, for every donor concentration ND, provided the semiconductor forbidden energy gap; k is the that the dependence of the effective masses on No be Boltzmann constant; T is the absolute temperature; qkT considered[24]. Therefore equation (2) may be employed is the electron energy in the x direction, evaluated, to compute the currentvoltage characteristics of both downward, from the peak of the energy barrier; x1 and xf rectifying[4] and ohmic contacts. are abscissae, for 77>O, of the classical turning points. We shall use equation (2) to analyse and to design
Ohmic contacts to semiconductors
ohmic contacts. We start from the McLaurin series expansion, truncated after the second term, of i(o) given by (2). We have
where 1 [ 1 t exp (D)][l t exp (  7 + @/Id’)] dq’
193
Once the contact metal has been chosen, the corresponding graphs for Rs and Jr enable ready determination of the impurity concentration, which is the only lacking quantity to accomplish the design. To deal with the dynamic behaviour of the contact, it is necessary to consider also the specific shunt capacitance which is composed of a diffusion capacitance Co and a depletionlayer capacitance CS. The last one is given, in the neighbourhood of u = 0, according to the preceding theory[l, 21, by
(7) and 1
[ ltexp (D)][l
+exp ( n t WI’)1
X
where vb is the builtinvoltage across the capacitor. The upper 3 dB frequency is lower than
(8) 1 texp(n
[email protected]/kT)1
(11)
texp(D)
Equation (6) shows that the iu characteristics of a metalsemiconductor contact can be considered linear in the voltage range for which ]v/RsJL(Q 1. Then the contact is ohmic and its specific resistance Rs is given by (7). Over the voltage range a current interval I,, Z. can be associated so that I. = G(JL I),
(9)
where E is the maximum relative current error. Now we can give t an arbitrary value and then determine, by means of equations (8) and (9), the extent of the current range within which the iu characteristic is linear apart from an error E. All semiconductor devices require ms contacts in order to electrically interconnect with an external network. The contact must satisfy the following requirements: (i) its resistance Rs/S must be much smaller than the static and dynamic impedance of the connected electron device in order to minimize the d.c. and a.c. power losses; (ii) the limit current I, must be larger than the maximum absolute value of the device current and its relative error l must be small enough to avoid non linear distortions. Statements (i) and (ii) are rules for choosing the values of the quantities Ra L, E and S which characterize an ohmic contact; these value for the starting data for contact design. Two general methods of approach can be applied: either (1) the barrier energy can be reduced to such an amount that thermally excited current over the barrier is large enough to reach the required values for Rs and Z.; or (2) the semiconductor can be doped to such a high carrier density as to allow tunneling to take place and to obtain again the required Rs and I.. Both cases may be analyzed by means of the general equations (7), (8) and (9). As examples, this analysis will be performed for several metals in contact with n and ptype silicon. The results of the computation for Rs and J, will be plotted vs impurity concentration; they agree very closely ._ . with previous experimental results [24].
We may plot fHvs impurity concentration as well, and this graph allows one to claim that for any frequency f > fHthe dynamic impedance of the contact is lower than the static resistance Rs. The dependence of Rs and .ZLon temperature[24] has not been considered here because of the unknown dependence on temperature of some input parameters of the proposed model. Therefore all theoretical and experimental results are referred here to T = 300°K. For the technological features, which are not considered here, an up to date review is performed by Rideont [25].
3. OHMIC CONTACTS TO SILICON
(a) Premises The model is used to compute, as functions of the semiconductor impurity concentration, the quantities which characterize the ohmic contacts of platinum silicide and ahuninium to silicon of both n and ptype. Such ohmic contacts are commonly used in the silicon device industry [26]. The starting physical parameters of the analysis are the work function % and the Fermi level EM of the metal; the electron affinity ,y, and the Fermi level Es (or the impurity concentration) of the semiconductor; the transverse energy ET which characterizes the metal electrons which tunnel into the semiconductor forbidden energy gap, and which, together with the FermiThomas charge of the metal, yields the interfacial electric dipole. Notice that for each metal or metallic compound the value of the parameter ET, which is used for the ohmic contact, is the one formely computed for the same low concentration rectifying contact [3,4]. The silicon imput data necessary for the computation are those reported in refs. [3], [4] and [27]. For each given value of the Fermi level Es, the impurity concentration ND and the voltage u. required for the calculation, are preliminarly computed according to the relationships[5]
,l
(12)
194
B. PELLECRINIand G. SALARDI
and 2kTP
(13)
ve = GE’
where (cp)“’ 1+ exp (cp Es/kT) d”
(14)
is the FermiDirac integral[9] and I ‘=
cp(cp”Z+Y) ltexp(cpEs/kT)
(IS)
dq,
with [S] y = 3 x 0.892977/4(kT)“‘fi I”, 6 = 8r(?m,)“*/3qhF,n
and F,,, = (2/3){2NDkT[4+ [email protected]
+ Es)/kT]/cs}“‘.
The quantities Es, v, and A, are plotted vs No in Fig. 2. The effective mass m,, which through D (see equation (5)) influences the values of RS and .I, (see equations (7) and (8)), depends on No and T. The values for M, are deduced from the graphs reported by Chang et al. [24]; they refer to the [l, 1, l] = x direction, which is the one considered here. The method and the program to compute the quantities vm,vO,vb, @ and so on, concerning the energy barrier, are the same used formerly[3,4]. The derivatives which appear in equations (8) and (lo), as well as in the Appendix, are computed as average rates of change in the voltage range of O0.02 V. The main contact parameters Rs and JL can be now computed performing the integrals, of equations (7) and (8). An analogous computation is made for ptype silicon. In this case we have three nonequivalent valence bands
characterized by the isotropic masses[27] m,, =  0.49 ma, mu2=  0.16 m. and mU3=  0.24 mo, which we consider independent from the acceptor atom concentration NA.For a given value of the Fermi level Es, evaluated in this case downward from the common valence band tops, v, remains unchanged [y = (m,/Na)1’6] whereas, according to an equation analogous to (12), N., = 0,465 ND. The specific conductance l/Rs and the current density JL are obtained by summing the conductance or the current components computed for each of the three valence bands. (b) Platinum silicide The input data for PtSi are: EM = 2.21 eV, CPM= 4.655 eV and ET = 0.08 eV. The values of E, and QM are those which have been deduced in previous work[4] by fitting theoretical reverse iv characteristics to experimental ones for ND = 3.2 x 10” cme3. The various quantities of the contact are plotted vs ND (or NA) in Figs. 37. In Fig. 3, also the experimental values of the specific resistance for the contacts to the ntype silicon obtained by Chang et al. [24] are reported together with Rs and .I, for both n and ptype silicon. Very good agreement between experimental and theoretical results was observed. Notice the large magnitude difference for Rs for the nand the ptype silicon cases. This is due to the large difference between the energy barrier heights @in the two cases (Fig. 5). As a consequence the PtSi(n)Si ohmic contact (for high ND) is dominated by tunneling, whereas the PtSi(p)Si contact is thermionicemission limited. We emphasize that both semiconductor types have been analyzed by means of the same unified approach. In Fig. 4 the frequency fH given by (11) is plotted vs No or N+
I/
\
ND,
cm
’
Fig. 2. ES, u, and A, vs silicon impurity
concentration
ND
Fig. 3. Specific resistance and limit current IJ,l vs silicon impurity concentration for ohmic contacts between PtSi and silicon of both nand ptype. The experimental values are deduced from ref. [24].
Ohmic contacts to semiconductors
101
1
11’
1
10'0
ION'
‘1’
1
1
1”
&.A!!.
‘1
to’0
ION'
195
IO*<
cm3 10”
10”
Fig. 4. Plots of the frequency fH for contacts PtSiSi.
IO"
10"
ND. cm’
Fig. 6. Diagrams of W, x., x, and x,,, vs No (seeFig. 1). I
“‘I
Fig. 7. Diagrams of
03' ION'
I"'
j 10'6
"I'
'1)' 10'9
N'.N,.
cm
"'I KY0
102'
3
Fig. 5. Diagrams of @, ua and v, for contacts PtSiSi (see Fig. 1).
In Figs. 5 and 6 the quantities which characterize the barrier are reported; here xf is the nullcharge abscissa in the depletion layer. If we set Vr = 1.LI&, the linearity of the io characteristic is assured, apart from the relative error E over the voltage range  EVL,+ QVL. In Fig. 7 VL is plotted vs ND or Na. VL is always larger than the value 2kTlq = 0.052 V which it achieves for low doped rectifying contacts. (c) Ahminium Aluminium, as well as platinum silicide, is often used in practice to construct ohmic contacts to silicon[24,26]. Such contacts are analyzed in this section by means of the same procedure used for PtSi. The only difference is that in this case the transverse energy & = 0.2 eV is obtained by fitting, for No = 2.7 x 10” cmm3,the theoretical value of the specific resistance to the experimental one obtained by Chang et al.[24]. This value of Er is slightly smaller than the values of 0.21 and 0.25 eV obtained by matching the theoretical results with the experimental data for the low dopings ND = 1.2 x lOI and ND = 8.2 x lOI cme3,
for contact P&(n)Si I
“‘I
‘\“‘I/’
No,Na,
cm’
“‘I
V, = RslJLj vs impurity concentratian PtSiSicontacts.
for
respectively[3]. The difference should be due to the fact that the last two results concern contacts built by evaporating the metal in vacuum onto vacuumcleaved silicon samples. These contacts are relatively free from contaminants. Surface contamination cannot be excluded, however, in the usual technique employed to fabricate ohmic contacts[25] and it should reduce the concentration (and hence J$) for the electrons tunneling into the semiconductor. This does not happen for the contact to PtSi, since both the rectifying contacts, characterized by low Nh and the ohmic one, characterized by high ND,are fabricated with the same technique of the solidsolid chemical reaction[24,28] occurring beneath the semiconductor surface. Once ET has been obtained we may compute the other quantities as functions of ND (or NA). In Fig. 8 we show only the primary quantities Rs and 1‘. Again the theoretical values for i%(n) agree very closely with experimental ones. The ohmic contacts of aluminium to both n and ptype silicon are tunneling dominated contacts. (d) Design example The graphs of Rs and .JLvs ND (or N*) enable one to design and analyse ohmic contacts. Here we carry out a detailed design example. Suppose we desire an ohmic contact to ntype silicon having the following features: R 4 la, I. = 30 mA with S = lo’ cm* and Qc 3 per cent.
796
B. PELLEGRINI and G. SALARDI
effects of the interfacial and penetrating states on the measurable quantities of the contact itself. The ever more accurate fabrication and measurement techniques for solidsolid intimate contacts, the numerous electric quantities which depend on interfacial phenomena and which are easily measurable with accuracy, and the vast amount of experimental data existing in the literature about these quantities, should encourage and facilitate the theoretical analysis of quantummechanical interfacial effects. That is, the study of the various types of interfacial states arising in the solidsolid contact should be helped by the experiments much more than the analogous study on the surface states relative to the vacuumsolid “interface”, since for this latter case the measurements are much more difficult to make and the experimental results are less numerous, less accurate and less reliable. Fig. 8. As in Fig. 3 for the AISi contacts. The experimental pomts
havebeendeducedfromref.[24]. We have to satisfy the conditions (a) and (b) I.&12 IO’A/cm2. Let us first try a PtSi(n)Si
contact.
Rs <
REFERENCES
B. Pellegrini, Phps. Ret’ R7, 5299 (1971). 2. B. Pellegrini, SolidSt. Electron. 17, 217 (1974). 3. B. Pellegrini, Properties of SiliconMetal Contacts L’S Metal Work Function, Silicon Impurity Concentration, and Bias Voltage. Tecnico Scientifica, Pisa (1974). 4. B. Pellegrini, SolidSt. Ebxtron. 18, 417 (1975). 5. B. Pellegrini, SolidSt. Electron. 13, 1I75 (1970). 6. S. J. Fonash, So/idSt. Electron. 15, 783 (1972). 7. W. Schottky, Naturwiss 26, 843 (1938). 8. C. R. Crowelland S. M. Sze, SolidSt. Electron. 9,1035 (1966). 9. S. M. Sze, Physics of Semiconductor Deuices, chapter 8, Wiley, New York (19693. IO. B. Pellegrini. IEEE Trans. E/e&on Denices ED16,218 (1966). I I. B. Pellegrini, Alta Freauenza 39, 133 (1970). (English tram].). I?. B. Pellegrini, AIta Frequenza 39,429 (1970). (English tram].). 13. B. Pellegrini, A&a Frequenza 40,513 (1971). (English transl.). 14. C. R. Crowell and M. Beguwala. SolidSt. Electron. 14. 1149 (1971). 15. C. R. Crowell and S. M. Sze, SolidSt. Electron. 8,673 (1965). 16. C. R. Crowell and S. M. Sze, Solid%. Electron. 8,979 (1965’. 17. E. C. Kemble, The Fundamenfal Principles of Ouuntum Mechanics with Elementary Applications,’ pp. lb9:I 12 and 575578, (2nd edn.) Dover. New York (1958). 18. S. C. Miller and R. H. Good, Phys. Rev. 91,‘174 (1953). 19. E. L. Murphy and R. H. Good, Phys. Rex 102,1464 (1956). 20. V. L. Rideout and C. R. Croweli, SolidSr. Electron. 13, 993 (1970). 21. H. A. Bethe, MITRadiation Laboratory, Report 4312 (1942). 22. C. R. Crowell, SolidSt. Elecfron. 8, 395 (1965). 23. C. R. Crowell, SolidSt. Electron. 12, 55 (1%9). 24. C. Y. Chana. Y. K. Fana and S. M. Sze. SolidSt. Electron. 14. 541 (1971).’ 25. V. L. Rideout, SolidSt. Electron. (to be published). 26. M. P. Lepselter and J. M. Andrews, Ohmic Conracts to Semiconductors, p. I59 (Edited by B. Schwartz) Electrochem. Sot. (1969). 27. H. F. Wolf, Semiconductors, pp. 33 and 36, Wiley. New York (1971). 28. J. M. Andrews and M. P. Lepselter, SolidSf. Electron. 13. 101 I (1970). I.
lO‘R x cm2
From the diagrams
of Fig. 3 the first condition is satisfied by ND 2 2.4 x 10’9cm~~‘,whereas the second one is verified by No 3 1.6 x lOI cm‘. We must take the higher value and so we set ND = 2.4 x lOI cmj. As a consequence we have an advantageous reduction of the relative error Q to 1 per cent. We have also a voltage drop V, = RI, = 30 mV and jH reaches IOOMHz. Let us now try the Al(n)Si contact. Condition (a) is verified by ND 2 1.1 x 10’9cm‘, while condition (b) is satisfied by No 2 1.6 x 10’9cm’ (see Fig. 8). This time ND = 1.6 x 10’9cm’ must be chosen, and now, as a consequence, the error E holds its value of 3 per cent but R is reduced to 0.18R, V, becomes 5.4mV, and fH reaches 550 MHz. With aluminium we attain the design features with a smaller silicon doping. 4. CONCIL’SlONS
A model of ohmic contacts to semiconductors which takes into account most of their intrinsic mechanisms and phenomena and which explains their experimental features has been proposed. The model allows one to make a detailed analysis of ohmic contacts and gives tools for their design. The model has been applied to PtSi and Al contacts to silicon but it would be easy to employ it for every other combination of metal and semiconductor. The good agreement between theoretical and experimental results supports the validity of the model itself, and is a further evidence for the general validity of the theory of the interface[ld] which enables a unified approach to both rectifying and ohmic contacts. However a better and deeper quantum mechanical analysis of the penetrating and interfacial states at the contact, i.e. a direct computation, for instance, of ET from the first principles, is desirable. The parts of the present and former theories which deal with the electric behaviour of the contact should still be valid for analysing the electric
4PPENDIX
Several contact quantities depend, through D(lr)l) and its derivative, on the shape of the electric potential U,(I) (equation (5)). It was shown in ref. [2] that the density of electrons which tunnel from the metal into the forbidden energy gap in the semiconductor near the valence and conduction band are, Nv exp ( x/A,) and NC.exp ( x/A,), respectively. Normally, except for contacts with very low workfunction metals such as Mg on Si[3], we have NC =O. Also, the concentration of the “holes” penetrating into the semiconductor energy gap is null or
791
negligible (except for the contacts of very high work function metals, such as Pt and Pd, to Si or GaAs). In these cases the electric charge,  qNv exp ( x/A,), can be neglected with respect to the charge qNo due to the ionized impurity for x > w,,. = x,  Av In(r), where we may assume r =O.l, and where the abscissa xf of the flex point of the electric potential (case of F, > 0) is given by the equation
t
x
x, = hv In (NV/No).
GA)
By integrating Poisson’s equation in the “depletion” layer, i.e. for x > w,,., we obtain for the electric field F the relationship
Iv,2
D“/ 1, 0.4)
1 N I
p=24No v,vv,es
“<.” a(s)
where n(v,) is the concentration of electrons, from the semiconductor conduction band, which penetrate in both a classical and a quantummechanical manner into the depletion layer. Its expression is reported in refs.[2] and[5]. For values of VIsmaller than a certain value v *, n ( vi) 3 0, hence from equation (2A) we have
F*= y(v<
v
v,
vt  v.),
v,6’1 v;
T7T 3 “c_v_“e
I$v26
_

43
c
__
tiz 1
tb)
where s< zN,1 I
Y<” __ n(ui) dvi.
(4A)
According to (2A), F2 = 0 and dF*/dv, = 0 for vi = we can put, for v* < vI < v,  v, F* = (v,  v  v$/h:.
V, 
v,
and
(5A) (C)
NOW, l/A.’ and v* may be computed by imposing that the curves which correspond to (3A) and (5A) be tangent to one another for vi = v*. Then we obtain Fig. 9. Diagramsof the electric potential in various
cases.
64) and v*=u c v2v
*.
(7A)
By integrating equation (5A) we have vi = v,  v  2v. exp [  (x  x.)/A. 1,
@A)
/
and x. = WA.,
(9.4)
where W is the “width” of the depletion layer given by [ 1,2] w=
[
4% 1,
2Ps (u,v)
‘I2
(see Fig. 9). Equation (8A) holds for x 2 x., whereas for 0~ x s x. the electric potential is the one determined in ref. [2] and, on the former assumption of NC = 0, is given by v,=vau.+ex(Wt)+:NvA$exp(x/A”), s (1lA) where v., = qNvAv21cs is the correction due to quantummechanical tunneling of metal electrons into the semiconductor to the builtinvoltage v, of the capacitor, given by lJ*=vC+vqvOv~.
(12A)
The method used in refs. [l] and [2] to solve Poisson’s equation in the semiconductor, which led to equations (3A), (10A) and (llA), has been here extended to the case of whichever impurity concentration giving equations (SA) and (8A). It holds if an abscissa w can be found in which the electric charge is chiefly constituted by the ionized impurity atoms, i.e. if the condition wmin< x. or, at least, the condition x, < x. is verified. Equations (8A) and (11A), which take into account in a detailed way all of the causes which yield the contact electric potential, enable one to express the functions D(ju\) and dD/dv(, _0 in a closed form, with the help of some proper approximations. To this aim the electric potential v,(x) near the interface x = 0 given by (llA), is expressed by means of approximated polynomial expressions. The x axis is divided into the three intervals (0, xd), (x.,,x.) and (x., m)(x, will be later defined). In the third interval the potential vi2is that given by (8A),whereas in the second the potential, according to (llA), may be put in the form (Fig. 9) “i2=&v
4
+4Noxw5
( >
c.3
2
In the tirst interval (0, xd) the potential u,, may be approximated in different ways according to the signs of F, and u, (Fig. 9). In the case of F,>O and v, >O (Fig. 9a) we can put VU= vd = v, = v. for x,,, c x < xd where xd is defined by uiZ(xd)= ud, i.e. by (us+ vq  v&s x‘j = (14A) q&W ’ WhenOsxcx, wecanput vi, = vo(0, vJ
(
c2
>
f.
(15A)
B. PELLEGRINI and G. SALARLX
798
Equation (14A) and (15A) hold also in the case of F,O (Fig. 9b) when we put u, = v., = uO. In the case of F. < 0 and u, < 0 (Fig. SC)the potential may be approximated in a better way by the following relationship .,=(se)x’F,x+a”,
(16A)
Equation (16A) is obtained by computing the coefficient of x’ in the McLaurin series expansion of ui, given by (llA), on the assumption that the curves which represent vi, and uizare tangent to one another in a point xd, a,,. In this case xd =2,/(F,t$W),
in which fl, = 1 [q(o,  vu)+ qkTl”‘dx,
(20A)
and /3: =
dx, j&(u, vii)+ qkT]“‘I u0
(2tA)
where j = 1, 2, 3 and the integrals deal with the jth interval. Recalling the former approximate expression for u,,, the integrals (20A) and (21A) concerninn the first and the second interval. .become of the type J (ax’ + bx + c)” (gx't fx t p) dx where m = I, 0, f. Such an integral may be expressed to a closed form. In the third interval, i.e. for u, 2~. s qkT/q s u,, we have
and ud = uil(xd)= uiZ(xd).This third case occurs in contacts with high workfunction metals and, with reference to  vi,in contacts to ptype semiconductors. The functions D(lsl) and dD/du lUzOmay be put in the form (see eauation (5))
PI = 2A.(2qu.)“*[(l  p)“*  p”* arc tan(l/p  l)“*], (22A)
(18A) D = $2rr1,)“~(& + /?z+ p,), h.q ‘I2 (2&P)
112arctanCilp
and dD ;i;;.,
 l)“*,
(23A)
where =+(2m,)“Q3: h
t&+/i;),
(19A)
P =
(8,  qkTlq)h.
(24.4)