2 92 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-31, NO. 3, MARCH 1984

Calculation of Tunneling Currents in (Hg,Cd)Te Photodiodes Using a Two-sided Junction

Potential WILLIAM A. BECK ANII NORMAN E. BYER, MEMBER, IEEE

Abstract-It has been shown previously that tunneling current CHI be- come the dominant dark current and hence the performance-limiting factor in diodes formed in narrow bandgap semiconductors, r w h as Hgl-,Cd,Te. In this paper, we calculate the tunneling current using a Kane approximation for the nonparabotic conduction band and a more realistic junction potential than has been used previously. The Iloten- tial used here is characteristic of a linearly graded n-type region inter- secting a uniformly doped p-type region and is a better approxinlation to t3e actual potential in a diode formed by ion implantation .;?to a p-type substrate. We show that significant errors sometimes arise when the abrupt junction model is used to calculate tunneling current in these structures. The effect of changes in base carrier concentration and n-side donor gradient is shown for x between 0.196 and 0.400, which corre- sponds to a photodiode spectral cutoff in the important 3- to 14-pm region.

INTRODUCTION

C ONSIDERABLE effort has recently been expended on developing narrow bandgap photodiodes for infrarel de-

tection in the 3-5- and 8-14-pm spectral regions. Much 0 ? this development effort has been directed toward identifying, and reducing the dark current through the devices so that the:r de- tectivity will be limited by background fluctuations rather than internal noise. In particular, recent experimental and theoreti- cal work has shown that tunneling can become the dom:~nant dark current mechanism in (Hg,Cd)Te diodes, especia1:y at high carrier densities or at large reverse biases.

The two basic mechanisms by which tunneling curren: can flow in a diode are 1) direct band-to-band tunneling arid 2) indirect tunneling via interband states. The primary experi- mental evidence for direct band-to-band tunneling in (HLg:,Cd) Te photodiodes is the dependence of diode reverse breakc.own voltage on temperature and bias. In particular, it has been shown that the reverse breakdown voltage decreases as the diode temperature is lowered [ 11, and that the voltage &pen- dence of the reverse current (at sufficiently high bias) can be described by simple analytical expressions for tunneling cur- rent [2], [3]. Interband-state-assisted tunneling is indicated by peaks in the forward current-voltage characteristic [4]. Both tunneling mechanisms can be enhanced near the surface; this enhancement has been used to explain experimental results

In this paper, we restrict our scope to calculation of direct

Manuscript received May 6 , 1983. The authors are w4th Martin Marietta Laboratories, Baltimore, MD

[31.

21 227.

Doping Proflie Potential Profile

t b L x

(C) Fig. 1. Doping and potential profiles used in tunneling calculations

(dashed lines show the location of the depletion region edges): (a) uniform field, (b) one-sided abrupt, and (c) present work.

band-to-band tunneling currents. Because calculation of actual wave functions in the presence of the p-n junction potential is impractical, the WKB approximation has always been used to calculate the tunneling probability and hence the tunneling current. The accuracy of the result is dependent on both the accuracy of the assumed electrostatic potential at the junction and the electron dispersion relation used within the forbidden gap. The doping and potential profiles that have been used in this and previous work are shown in Fig. 1. Sood et al. [3] used a potential with constant electric field (Fig. l(a)) and a dispersion relation with a constant effective mass. This com- bination is physically unrealistic but did yield a quantitative fit to experimental data by adjustment of the theoretical param- eters. Anderson [2], [4] used a one-sided abrupt junction potential (Fig. l(b)) and a k . p approximation for the dis- persion relation. In this paper we retain the k . p dispersion relation and most of Anderson’s technique but substitute a more realistic junction potential which approximates the n- side donor profile as a linear function of distance with adjust- able slope (Fig. l(c)). This potential is particularly useful be- cause it can be employed to calculate tunneling current in the entire range between the abrupt junction and linearly graded junction models, Thus the effect of various doping schemes on tunneling current can be estimated.

0018-9383/84/03~10-0292$01.00 0 1984 IEEE

BECK AND BYER: TUNNELING CURRENTS IN PHOTODIODES USING A JUNCTION POTENTIAL 293

First, the analysis which leads to the tunneling current ex- pressions will be presented. This will be followed by results that were computed for a range of technologically important material parameters. Finally, we will discuss the implications of our results for device design and further study.

CALCULATION PROCEDURE The direct tunneling current is given by [SI :

where x is normal to the junction, k,, k y , k , is the electron wave vectors in the three spatial directions, u, is the x com- ponent of electron velocity, T ( k ) is the tunneling probability, and F(E) is the Fermi function on the n- or p-side of the junc- tion and the integral is evaluated over all electron wave vectors at which the density of states on both sides of the junction is nonzero. As described by Anderson [6], the tunneling proba- bility in the k a p approximatfon is

where

€(X) t 2 u(x) = L

VqXJ.. A is the Planck's constant/2n, Eg is the bandgap energy, EL is the energy component of electron perpendicular to the nor- mal of the junction €(X) is the local electron energy measured with respect to the local conduction band energy. The variable m: equals an effective mass related to the interband matrix element P by

To evaluate the tunneling integral in (2a), we must first cal- culate € ( X ) and du/dx for our potential. To do this it is con- venient to divide the junction into three regions: 1) an n-side linearly graded region, 2) a p-side linearly graded region, and 3) a p-side uniformly doped region. Table I shows the expres- sions that we derived for €(X) and du/dx in each of the three regions. The conventions used are shown in Fig. 2 and de- scribed below.

1) Energy E is measured relative to the conduction band edge at the n-side edge of the depletion region.

2 ) Position x is measured relative to the junction, that is, where the graded donor concentration equals the base acceptor concentration. Positive x is on the p-side. The variables in the table are:

N A is the substrate acceptor concentration,

TABLE I EXPRESSIONS FOR € ( x ) AND dx FOR TWO-SIDED MODEL

?air?' C W l ax -2 JG GnlA)dJ

I f l Ecp

Fig. 2. Band diagram of the p-n junction showing definitions of the energy and position variables.

V is the applied junction bias, xps = NA Is is the width of the p-side graded region, V,(Vp) is the electrostatic potential variation from the

junction to the n-side (p-side) edge of the de- pletion region, and

x,, xp is the width of the n-side (p-side) depletion re- gion.

By electrostatics and overall charge neutrality V,, V,, andx, can all be written in terms of x,

if x, <- = xps N A S

S is the donor gradient,

2 94 IEEE l’;bANSACTIONS ON ELECTRON DEVICES, VOL. ED-31, NO. 3, MARCH 1984

where 4 is the electronic charge and ex is the static dieltctric constant of (Hg,Cd)Te. The value of x, is then determintd by the condition that

Vn -t V p = Vt (EFn - Ecn) - (EFp - Ecp) (4)

where ( E F ~ - E,,), (EpP - E c p ) are the local Fermi energies at the n- and p-sides of the depletion region relative to the local conduction band.

Since the doping level at each edge is determined by x , and x p , the Fermi levels can be calculated by the method of Schmit [7] which incorporates a parabolic valence band and a non- parabolic conduction band. In practice, the computer pro1;ram varies x, (and therefore V,, V p , and the local Fermi energies) until (4) is satisfied and this fixes the value of the cther parameters.

Substituting the expressions from the table into (2a) and (2b) yields

T(k) = exp { - 2 ”Eg e+ EgE1)

2x * [z lA G,(u)du t 4 Gps(u) du ( 5 )

B

3Vn A

I -‘I

1 if E > V,

-1 if E < V, - Eg

1 if E > V, t V(2‘p)

- 1 if E < V, t V ( Y p ) - Eg.

u(0)if V, - E g < E < V,

u G P ) if V, + V(2‘,) - Eg < E < V, t VGP)

where N

X, = lesser of x,. NA/a

This expression is then substituted in (3a) and (2) along with

1 aE VX = nak,.

The integral is then transformed to an integral over E and E l by a method similar to that used in Appendices B and C of [ 61 . As in that reference, the approximation 0 < E l << Eg is used and the integral over El is extended from zero to infinity. This operation results in

where U ( X n )

Base = - G,(u)du -t 2x, G,,(u)du 3 V n 3 V,

The limits of integration used were E< is the greater of (lesser of (EF, - 4kT, EF, - 3kT), 0 ) and E> = Vtotal - Eg where

Vtotal = V, t V p = total junction potential.

The integration in “base,” as well as the energy integration, is performed numerically.

RESULTS Current-voltage relationships and zero bias resistance-area

products ( R o A ) were calculated for a wide range of bandgaps, substrate acceptor concentrations, and n-side donor gradients. Since no analytical expression was available for the zero bias resistance, the RoA’s were determined by calculating the cur- rent at a bias of 10 mV. Data which are referred to as “LSS” in this section were calculated using an n-side donor gradient equal to the boron concentration gradient at the junction after a 1 X 1014 cm-2, 100-keV implant of boron, and assuming a 100-percent activation of the boron and no damage doping. The boron implant profile has been measured [8] and can be represented by

n(x) = n p exp (@;$I2)

where n p is the peak boron concentration, R, = 0.2337 pm, and

A R , = 0.1635 pm.

The boron gradient at the junction is therefore

s=-- dn(x) - - 2 In (np/NA dx ARP NA .

Note that s is rather insensitive to the peak concentration (and hence to the total dose).

The LSS conditions give the maximum value of s that can result from ion implantation of the type described and the largest tunneling currents. Recent measurements [9] have shown, however, that junctions in boron-implanted (Hg,Cd)Te are formed by electrically active damage sites which extend much deeper than the boron. Indeed, the actual donor gradient at the junction can be about 10 times less than that expected from the boron profile. Therefore, data are also presented here for s = sLss/lO or s = s L ~ s 1 5 .

Fig. 3 shows the calculated RoA at 77 K as a function of sub- strate acceptor concentration and spectral cutoff wavelength. Note that the curves for which s =sLs,/lO (damage doping) are dramatically shifted from the LSS curves.

Fig. 4(a) shows the reverse voltage at which the tunneling current density equals the background-induced current density from a 300-K, 180” field of view (FOV) background ( V B L , ~ ) . This value is essentially the maximum reverse bias at which the diode could be operated and still be performance-limited by background noise. This information is useful because diodes that are connected to MOS signal processors generally are op- erated under reverse bias [ 101 , The predictions of the abrupt junction model‘ are also shown for comparison. Fig. 4(b)

‘Calculated by using the formulation for MIS devices in SBRC final report Naval Research Lab on Contract N00173-76-C-0316 and b y setting the “surface potential” equal to Eg plus the applied bias voltage.

BECK AND BYER: TUNNELING CURRENTS IN PHOTODIODES USING A JUNCTION POTENTIAL 2 95

Fig. 3. Calculated tunneling-limited R(,A products of (Hg,Cd)Te diodes at 77 K.

1.4 - a - LSS gradient --- Abrupt junction model

2 4 6 8 10 12 14

CUTOFF WAVELENGTH lpmi

(a) CUTOFF WAVELENGTH Ipml

(b)

Fig. 4. Bias voltage at which tunneling current equals 300 K, 180 FOV background current; detector temperature = 7 7 K. Base carrier con: centration in cm-3 is indicated for each curve; (a) a = LSS gradient and abrupt junction gradient, (b) a =a~ss/10.

shows V B L , ~ for the case of s = .s~ss/lO. A large shift from the LSS and abrupt junction values is apparent. Figs. 3 and 4 can be used to select an appropriate substrate carrier concen- tration for diodes with a given cutoff wavelength and specified operating bias.

Relatively little of the published experimental data have been sufficiently detailed to test the tunneling models. An example of the problems inherent in such a test is provided in Fig. 5. The heavy solid line shows a detailed current-voltage charac- teristic measured by Sood et al. [3] for a boron-implanted

(Hg,,,,, , Cdo.213)Te diode operated at 40 K. The p-type base carrier concentration is specified between 3 X lo” cm-3 and 1 X 10l6 ~ m - ~ . The lighter solid lines in Fig. 5 show the predictions of our model for NA = 3 X 6 X and 1 X 10l6 ~ m - ~ , with s equal to the LSS value. The dashed lines represent the same calculation but with s equal to s~ss/S. Finally, the predictions of the abrupt junction model are shown as broken lines. Unfortunately, the differences due to the experimental uncertainties are nearly as large as the differences between the models.

296 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-31, NO. 3, MARCH 1984

l O - 1 1 -Honeywell data

--Abrupl junction

lc4 I I I 100 200 300 400

REVERSE B I A S (mVi

Fig. 5 . Comparison of calculated tunneling currents with experiment; base carrier concentration in ~ r n - ~ is indicated for each curve.

2 2 3.0 I\

LSS Abrupt vBLlP = 0.68~

,dl8 ,019 1020 ,021 1022 1023 ,024 I I I I I I

n - S I D E D O P I N G G R A D I E N T Icrn-')

Fig. 6. Dependence of calcuated tunneling current on n-side dcnor gradient; T = 77 K cutoff wavelength = 11.02 urn. Base carrier (!on- centration in ~ r n - ~ ' i s indicated for each curve.

It can be seen from Fig. 6 that the one-sided abrupt junction potential is a limiting case of our potential. The figure shows that for large s, V,,,, is independent of s and is equal to lhe V B L ~ ~ given by the abrupt junction model. For small values of s, V,,,, becomes independent ofNA (the curves of differ- ent NA converge); this is thus the linearly graded junction limit. The LSS values of s are also indicated.

DISCUSSION Before considering the implications of the two-sided tunnel-

ing calculations, we should estimate the accuracy of the results. First, although the linearly graded donor profile used here

better approximates the actual implanted profile than the very high step function used in the one-sided abrupt model, it still is inadequate. Since the n-side depletion widths obtained are sometimes comparable to the Gaussian width AR, of the im- planted boron profile, the linear approximation is not strictly justified even within the depletion region. Thus the value of S L S S used in the calculation, which was the donor gradient at the junction, might be a factor of two lower than the effective gradient within the n-side depletion region.

The major source of error in the calculation itself is believed to be due to the uncertainty in the interband matrix element P for (Hg, Cd)Te. A survey of the literature indicates [I 11 P = 8.2 (+ 0.6) X e V . cm, an uncertainty of 7 percent; P = 8.4 X lo-' eV . cm was used for our results. Because P ap- pears in the argument of the exponential in (3a) (through rn;), the 7-percent uncertainty produces about an order of magni- tude uncertainty in the tunneling current. There is also some uncertainty in the heavy hole effective mass. A literature survey [ l l ] indicates r n g = 0.50 (+ 0.13) rno while we used rnb = 0.55 rn, here. However, this deviation causes small errors compared to the effect of P.

The large uncertainty in the absolute tunneling current, cou- pled with the strong dependence of the tunneling current on experimental parameters such as N A , makes it is difficult to use comparisons such as those shown in Fig. 5 to test the ab- solute accuracy of the various tunneling models. For instance, the experimental data in Fig. 5 can be described either by the two-sided model used here with s = s ~ s ~ / 5 , if tunneling takes place over the entire junction, or by the uniform field model, if tunneling is assumed to occur only near the surface [ 3 ] . More extensive doping profile measurements will be needed before these effects can be separated.

The calculation is most useful for predicting critical values of the parameters upon which the tunneling is strongly depen- dent. For instance, even with the uncertainty due to P, the carrier concentration corresponding to a specified tunneling current can be calculated to within 20-30 percent. Thus for a background-limited diode with a cutoff wavelength of 12 pm at 77 K and a required reverse bias of 100 mV, the LSS curve in Fig. 4 shows that the base carrier concentration should be less than 1 X 10l6 cmF3 + 25 percent or less than 7.5 X 1015 ~ r n - ~ .

Finally, the calculation shows the need for more detailed measurements of the electrically active donor profile which results from ion implantation into (Hg,Cd)Te and subsequent anneal treatments. As shown in Fig. 6 the donor gradients which result from an activated boron implant with no damage doping (LSS) are at the bottom of the range at which the one- sided abrupt junction is valid. If damage doping occurs, the one-sided approximation breaks down and very significant errors occur from its use. Accurate values of s must be mea- sured before definitive tunneling predictions can be made.

CONCLUSIONS The tunneling current through (Hg,Cd)Te diodes formed by

ion implantation of boron into p-type substrates has been cal- culated using a two-sided junction potential model. Because the actual doping profile in such devices is intermediate be-

IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-31, NO. 3, MARCH 1984 297

tween the abrupt and linearly graded approximations, the two- sided potential yields more accurate results than the simpler potentials used previously. For implants that contain no dam- age doping, the errors introduced by use of the abrupt junction model are small compared to the experimental uncertainties. However, significant errors arise if damage doping is present.

Data have been presented which can be used to select the base carrier concentration for a diode intended to operate with a specific R o A or reverse bias. The results both for boron impurity doping and for damage doping demonstrate the need for more accurate doping profile measurements before truly accurate modeling can be done.

REFERENCES [l] P. J. McNally, “Development of (Hg,Cd)Te elevated temperature

photovoltaic detectors,” Honeywell Radiation Center, Lexington,

MA, available from NTIS (AD-762-352), 1973. W. W. Anderson, Infrared Phys., voL 20, p. 353, 1980. A. K. Sood, private communication; Honeywell Electrooptics Operations, Lexington, MA. W. W. Anderson, Appl. Phys. Lett., vol. 41, p. 1080, 1982. C. B. Duke, Tunneling in Solids. New York: Academic Press, 1969.

W. W. Anderson, Infrared Phys., vol. 14, p. 147, 1977. J. L. Schmit, J. Appl. Phys., vol. 41, no. 7, p. 2876, 1970. H. Ryssal, K. Muller, J. Biersack, W. Kmger, G. Lang, and F. Jahnel, Phys. Status Solidi (A), vol. 57, p. 619, 1980. L. 0. Bubulac, W. E. Tennant, S. H. Shin, C. C. Wang, M. Lanir, E. R. Gertner, and F. D. Marshall, Japan. J. Appl. Phys., vol. 19, p. 495, 1980. P. Felix, M. Moulin, B. Munier, J. Portmann, and J.-P. Reboul, IEEE Trans. Electron Devices, vol. ED-27, no. 1, p. 175, Jan. 1980. R. Dornhaus and G. Nimtz, Solid State Physics. New York: Springer, 1976, p. 37.

FIPQS (Full - - Isolation by Porous Oxidized - Silicon) - Technology and Its Application to LS!’s

KAZUO IMAI AND HIDEYUKI UNNO

Abstract-Processing steps of FIPOS (Full Lsolation by Porous Gxi- dized Silicon) technology and its application to LSI’s are presented. FIPOS technology realizes a siliconan-insulator structure, utilizing thick porous oxidized silicon and donors produced by proton implantation. New processing steps are proposed which provides small surface step and are suitable for LSI fabrication. Formation conditions of thick porous oxidized silicon are established by density control technique for porous silicon using a newly developed anodization system. CMOS de- vices are fabricated in isolated silicon layers and it is shown that the characteristics of n-channel and p-channel MOSFETS’s are sufficient for application to CMOS LSI’s. A FIPOS/CMOS logic array with 1.3K gate is successfully fabricated, which shows a higher speed and lower power dissipation than the gates fabricated by bulk CMOS technology.

These results indicate that FIPOS technology is very useful for realizing high-performance CMOS LSI’s.

I. INTRODUCTION

R ECENTLY, a thin single crystalline, silicon-on-insulator ,structure (SOI) has attracted a special interest since it

realizes the perfect isolation structure and low parasitic cqac i - tance. Using the SO1 structure, high density and low power LSI’s can be obtained.

Manuscript received March 2,1983;revised June 10, 1983. The authors are with Musashino Electrical Communication Labora-

tory, Nippon Telegraph and Telephone Public Corporation, Musashino- shi, Tokyo, 180 Japan.

Several approaches have been taken to realize a SO1 structure including silicon-on-sapphire [ 1 ] , oxygen implantation [2] , laser annealing [3] , and the zone melting method [4] .

Recently, one of the authors has proposed a new method to obtain a SO1 structure by using porous oxidized silicon [ 5 ] and this is called FIPOS (_Full Isolation by _Porous Oxidized

point of the isolated Si layer formation. In FIPOS, the isolated Si layer is made of a part of the bulk Si, which results in its high-quality layer. In other methods, the Si layer is grown on the insulator substrate, and defects or impurities may degrade the quality of the silicon layer, for example, aluminum contam- ination in the SOS method and grain boundary existence in laser annealing.

FIPOS technology is based on following characteristics of the porous silicon and proton implantation: 1) Only p-type Si is changed to porous Si, 2) the porous silicon oxidation rate is large enough for the formation of a thick insulator with- out wafer warpage, and 3) donor level is generated by proton implantation in annealing at 450°C and disappears after an- nealing at temperatures above 700°C.

In the FIPOS process, proton implantation for donor forma- tion and thick porous oxidized silicon formation are necessary. These two techniques are the most important in FIPOS.

- Silicon). The FIPOS method is different from others on the

0018-9383/84/0300-0297$01.00 0 1984 IEEE