Anal. Chem. 1991, 63,931-936
931
Microstep Electrodes: Band Ultramicroelectrodes Fabricated by Photolithography and Reactive Ion Etching Marie Samuelsson, Marten Armgarth, a n d Claes Nylander" Laboratory of Applied Physics, Linkoping Institute of Technology, S-581 83 Linkoping, Sweden
Photdtthography and reactive ion etching have been used to fabricate a new type of band uttramkroelectrodes: microstep electrodes. The process aikws the length/wktth ratlo of band electrodes to be maxlmized. The paper describes the new geometry, the fabrlcatlon process, and the results from chronoamperometrlc evaluation of the electrodes. Comparison of the experimental results with the theory for hemicylindrlcai band electrodes Is dlscwrsed In some detail. Also the effect of overiapplng diffusion fronts is demonstrated and dlscussed.
INTRODUCTION Ultramicroelectrodes offer new possibilities in most of the applications where voltammetricand amperometricelectrodes are being employed. They also widen the area of applications into those demanding high sweep rates, low iR drops, low capacitance, and minimized influence from redox active surface adsorbants. These, and more, advantages associated with making electrodes smaller than the typical diffusion distance in an electrolyte have been discussed by several authors (1, 2). One obvious disadvantage of extremely small electrodes is that they generate only very small currents. Currents from disk electrodes of micrometer dimensions are generally so small that preamplification and electrical screening precautions become necessary. Band electrodes, which is the subject of this paper, constitute an interesting compromise in that one of their dimensions is small enough to give rise to nonlinear diffusion, while the other dimension is large enough to generate more practical current levels. This is especially important when aiming for low voltammetric detection limits. In that context, band electrodes compare favorably with macroelectrodes in that the part of the residual current, associated with double layer charging and surface oxide formation, is reduced in proportion to the surface area of the metal, while the faradaic current is not. Alternatively, the high current possibility allows for the use of ultramicroelectrodes together with simple and inexpensive instrumentation. There is, therefore, scope for developing band electrodes where one diension is minimized, while the other dimension is maximized. It is also important, for practical and commercial applications, to develop microelectrodes that are amenable to mass production. Band electrodes have so far been fabricated in essentially two ways. One method is to glue together a glass/thin metal/glass sandwich and polish one end of the sandwich. The polished edge of the metal layer can then be used as a band electrode. Metal foils as well as vacuum deposited metal films have been utilized, the latter allowing extremely narrow electrodes to be made (3-5). Another method for making band electrodes involves photolithography. A thin metal film can be vacuum deposited onto a substrate and coated with photoresist. The resist is patterned photolithographically and the parts of the metal that are left unprotected are etched away. The remaining 0003-2700/91/0363-0931$02.50/0
photoresist is thereafter removed, leaving narrow bands of metal on the substrate surface. Also a lift-off technique can be used (6, 7), in which photoresist is deposited and patterned before metal deposition. When the photoresist is removed, all metal on top of the photoresist strips off automatically while metal between the photoresist lines stays put. The main advantage of the sandwich method is that the width of the electrode is defined by the film thickness. When vacuum deposition techniques are being used, one can readily attain widths less than approximately 10 nm. Below 10 nm, the films almost always becomes discontinuous, resulting in something of a disk electrode array. One of the main disadvantages of the glass sandwich method is the limitations of the polishing step (8). Even the finest polishing pastes are really too coarse when dealing with electrodes of nanometer dimensions. The structure is also rather limited in terms of geometrical variability. The photolithographic method has the advantage of being highly controllable and reproducible. It also allows fabrication of highly sophisticated electrode geometries, a possibility that has been utilized only to a limited extent so far. Modern high-resolution photolithography enables production of electrode structures with 0.5-pm dimensions. It has been pointed out (2) that photolithographically produced electrodes are difficult to resurface by means of polishing them. One should, however, keep in mind that the procedure of polishing electrodes between measurements is out of the question anyway when it comes to practical applications such as process monitoring, etc. The two methods of fabrication have also been combined. Thorman et al. (4) used photolithography to etch out 15500-pm-wide stripes of the metal film before they assembled a sandwich and polished the ends. The purpose of this was to divide a band electrode into an array of short band electrodes that could be addressed individually or in parallel. In this paper, we demonstrate a new geometry, microstep electrodes, the fabrication process of which is based on photolithography and reactive ion etching (RIE). We describe the basic geometry, the fabrication process, and some evaluation results. We also discuss the theoretical implications of making very narrow band electrodes and the effect of placing band electrodes very close to each other. The purpose of packing band electrodes close together is, of course, to achieve largest possible current per unit area.
GEOMETRY AND FABRICATION OUTLINE The principles of our method of fabricating microstep electrodes are as follows: (1)Start with a silicon wafer as the substrate and insulate its surface by growing a high-quality silicon dioxide. A semiconductor substrate is used instead of glass because it is perfectly flat and withstands the subsequent processing steps. (Sapphire wafers, such as the ones used in silicon-on-sapphire technology would be a possible alternative.) (2) Vacuum deposit a 10-1000 nm thick metal film on top of the silicon dioxide. Thinner metal films tend to be discontinuous, while thicker films tend to grow into whisker type 0 1991 American Chemical Society
932
ANALYTICAL CHEMISTRY, VOL. 63, NO. 9, MAY 1, 1991 S iJN4
1 ,
/Metal
\
Si02 Si
Figure 1. (a, top) Cross section of the microstep electrode. (b, bottom) Top view of the type of microstep electrode produced in the present work (see text for further details).
structures and often suffers from built-in mechanical stresses. Use plasma CVD to deposit an insulating film, e.g., Si3N4,on top of the metal. Since the metal in question, almost always gold or platinum, is very inert, it is advisable to promote the adhesion between the layers with approximately half of a nanometer of, e.g., chromium. (3) Divide the surface into two regions by depositing, exposing, and developing a layer of photoresist. Use RIE to etch away both the top insulator and the metal in the area that is not protected by the photoresist (see Figure la). The important feature of RIE patterns is that they have extremely sharp and vertical steps. In our case, the step constitutes a band electrode, the width of which being defined by the metal film thickness and the length being the length of the boundary between the two regions. Photolithography offers considerable freedom in defining the shape of the electrode. One way of making a very long electrode is obviously to shape the boundary between the two regions in the shape of a meander line (see Figure lb). The resolution of photolithography does actually make it possible to produce an up to several tens of meters long meander line within a single square centimeter. An alternative geometry to the one depicted in Figure l b is one where the fingers are separated into individual band electrodes, each having its own contact pad. Another, very different, alternative is one where holes are etched all over the surface, a structure resembling the tubular band electrodes described by Kovach et al. (9). One fact that cannot be ignored is that any two electrodes operating close to each other in an unstirred solution will inevitably affect each other sooner or later, as their diffusion fronts start overlapping. Experimental observations of this has been reported, e.g., by Bartelt et al. (IO), who referred to it as shielding. The effect becomes practically significant when the interelectrode distances are down on the submillimeter scale. It is equally important when a band electrode is folded as the one in Figure lb, in order to maximize its length within a limited surface area. In order to illustrate this point, we have made electrodes of the type shown in Figure l b with repeat distances d of 1000,100, and 10 pm, respectively. As will be shown later, the 10-pm repeat distance results in considerable overlap of the diffusion fronts and hence over-
rules the advantage of band electrodes as producing quasisteady-state currents. Such electrodes are still superior to ordinary macroelectrodes in that the active metal surface area, and hence the residual currents, are minimized. Such an electrode works essentially as a conventional voltammetric or amperometric electrode but with an improved signal-tobackground ratio and reduced double layer capacitance.
EXPERIMENTAL SECTION Standard 3-in. silicon wafers were used as substrates, the surfaces of which were electrically insulated with silicon dioxide. The wafers were first cleaned in a standard semiconductor cleaning process and thereafter oxidized in water vapor and argon at 1200 "C for 80 min. The resulting oxide layer was about 1 p m thick as verified by ellipsometry. The wafers were then metallized at lo-' Torr with 0.8-nm chromium,200-nm gold, and finally 0.8-nm chromium. All metal depositions were performed sequentially without breaking the vacuum. The two thin layers of chromium served as adhesion promoters to the silicon dioxide and the following layer of silicon nitride. Silicon nitride was chosen as the top insulator because it is known to be less permeableto water vapor and alkali ions than silicon dioxide. The silicon nitride layer was grown by means of plasma deposition from a mixture of NH3, SiH4,and N2 in a Plasma Processing 500 system (VacuTecAB). Optimal processing parameters were found to be 0.9 Torr of pressure, 10 W of power, and 350 "C substrate temperature. The nitride was grown for 45 min, resulting in 500-nm layers. Thickness and refractive index were subsequently checked by means of ellipsometry. The wafers were thereafter annealed in air at 200 "C for 30 min and spin-coated with positive photoresist (S 1400-27 Microposit Shipley). The photoresist was baked at 95 "C for 30 min, and thereafter exposed to UV light through a chromium lithographic mask in a Karl Zuss MJB 21 mask aligner. The exposed parts of the resist were removed in a resist developing bath (MF 312 Developer CD 27 Shipley). The photolithographicmask contained the patterns for eight electrodes of three different types. All three types were of the finger shape depicted in Figure l b but with different numbers of fingers, i.e., different repeat distances. The electrodes, which all covered the same total area, had repeat distances of (A) lo00 pm, (B) 100pm, and (C) 10 pm. This resulted in edges of lengths 0.19, 1.59, and 15.42 m, excluding the edge around the contact pads as well as the lead from the pad to the electrode fingers. Prior to etching the structures, the resist was hardened by baking it for 30 min at 115 "C. RIE was used to etch away both the nitride and the metal layers in the areas that were not protected by photoresist. The nitride layer was etched in 40 mTorr of CF4/02at 150 W of power, and the metals were etched in 40 mTorr of CBrF3/Ar at 350 W. The resist was subsequently removed with acetone in an ultrasonic bath, and the wafers were divided into individual electrodes. The nitride at the electrical contact pads was then removed by manually applying a droplet of HF (40%),thereby exposing a patch of metal. This process step can of course also be done photolithographically. Thereafter the electrodes were immersed in a chromium etchant (Ce[SO4I2/HNO3)to ensure that there was no chromium left on the free surfaces. Electrical contact to the electrodes was achieved simply by inserting them into a slightly modified printed circuit board edge connector. A gold wire served as the counter electrode, and a standard Ag/ AgCl reference electrode was applied for control of the potential. All experiments were perfomed with an EG&G PAR 173 potentiostat controlled by an EG&G PAR 175 universal programmer. Curves were recorded on a standard XYt chart recorder. All experiments were performed in aqueous solutions of K4Fe(CN)6with 0.5 M KN03as the background electrolyte. The chemicals were of pro analysi grade (Merck GmbH). Since diffusion coefficientsare somewhat dependent on concentration and background electrolyte as well as on temperature, we decided to determine the diffusion coefficient of Fe(CN)64experimentally in the same electrolyte as the one used for electrode evaluation. This was done chronoamperometrically using a macroelectrode fabricated by evaporatinggold onto a glass slide.
ANALYTICAL CHEMISTRY, VOL. 63, NO. 9, MAY 1, 1991
0
30
20
10
40
933
50
t (s)
Figure 3. Chronoamperograms at the same electrodes and conditions as in Figure 2. Applied potential is 0.5 V. Note that the vertical axis shows current per unit length.
The chronoamperomogram for a hemicylindrical band electrode of radius ro has been predicted theoretically by Szabo et al. (12)to follow the equation
i = TnFDlC
2 1OOmV
Figure 2. Cyclic voltammograms of 5 mM Fe(CNh'- in a 0.5 M KNO, electrolyte at gold microstep electrodes of type A and 6 . Scan rate was 100 mV/s and solution unstirred.
The chronoamperometric current at 0.5 V (vs Ag/AgCl) was plotted against t - l / * ,and the slope of the so obtained straight line was inserted into the Cottrell equation, giving a diffusion coefficient D = 0.68 X lo4 m2/s. This agrees well with the value published by von Stackelberg et al. (II), who used KCl as a background electrolyte.
RESULTS AND DISCUSSION The electrodes were evaluated for oxidation of 5 mM Fe(CN):- in a 0.5 M KNO, electrolyte. Figure 2 shows cyclic voltammograms recorded at electrodes A and B with a scan rate of 100 mV/s in unstirred solutions. (Note the different scales on the vertical axes.) Both electrodes behave as expected from ordinary band electrodes. The diffusion-limited peak currents scale rather well with the electrode lengths. These currents are 165 rA/m for electrode A and 154 pA/m for electrode B. Cyclic voltammograms were recorded only for the purpose of checking the qualitative properties of the electrodes and for establishing a suitable potential for chronoamperometry,where the current is truly limited by diffusion only. This potential was chosen to 0.5 V (vs Ag/AgCl) for all further experiments. Figure 3 shows chronoamperometric recordings on electrodes A and B. Note that the results are plotted as the current per unit length of electrode. As expected, both electrodes give roughly the same initial currents per unit length. Electrode A thereafter behaves very much as a band electrode, Le., it gives a quasi-steady-state current, while the current from electrode B drops more rapidly. This is not surprising but rather well illustrates the fact that when band electrodes are placed close to each other their diffusion, layers will inevitably start overlapping after a certain time, which is determined by the interelectrode spacing and the diffusion coefficient. Even if electrode B is one single electrode, the meander line geometry makes it behave as an array of closely spaced band electrodes held at the same potential.
I p +.q$Zj1 e-[(rDt)/r,211~2/10
(1) where 1 is the length of the electrode and r, its radius. C is the bulk concentration, and D is the diffusion coefficient. n is the number of electrons exchanged, F is Faraday's constant, and y is Euler's constant (y = 0.577 22). The geometricaldimensions of our electrodes and the time scale of our experiments are such that the first term in eq 1 can be omitted. (The error is less than Using the same argument, one can also omit the term e5/3(error approximately l % ) , ending up with the following simple expression:
anFDIC
i= In
li""
(2) -?
This equation can be rewritten as
which gives a simple way of testing the applicability of eq 2 by plotting l/i vs In t. This should give a straight line, which in our case, where C = 5 mM and D = 6.8 X m2/s, has the slope
2rnFDC = 485 m/A
(4)
The resulting plots for the data in Figure 3 are shown in Figure 4, in which electrode A gives a perfectly straight line from 0.25 to 52 s. The slope of line A is 482 m/A, which is in very good agreement with eq 4. Electrode A can therefore be regarded to be an essentially perfect band electrode on the present time scale. It should be remembered that eq 2 has been derived for a hemicylindricalband electrode. However, Deakin et al. (13) have shown that eq 2 also describes the current from a flat band electrode if the cylinder radius ro is substituted by W/4, where W is the width of the flat band electrode. Our results show that, for a certain time interval, eq 2 also holds for a plane electrode that is standing up
934
ANALYTICAL CHEMISTRY, VOL. 63, NO. 9, MAY 1, 1991
-
3
-
2
.
1
0
1
2
3
4
5
0
1
2
3
4
Flgure 4. Data from Figure 3 replotted for testing its correlation to eq 2.
perpendicular to the surface and has a 0.5-pm-thick insulating film on top. The substitution of r, for another geometrical factor (W/4 in the case of ordinary flat band electrodes) does not affect the slope of the line in plots such as the one in Figure 4, but only the position of the line. For electrode A, the results give an equivalent F, of 70 f 40 nm. The logarithmic function in eq 3 makes it very difficult to experimentally verify any attempted substitution for F,. Calculating the differential for r, from eq 2 ar, dr, dr, dr0 ar, dr, = di - dl + - d t + - dC + - dD (5) di ai at dD gives the following equation for the relative error in r,:
+
5
6
'
1
8
4 (s'9
In t
ac
Figure 5. Curve B from Figure 3 replotted in order to correlate it with Cottrell's equation. The fact that a perfectly straight line is obtained from 1.5 s and onwards indicates planar diffusion.
I
C
'D
* E
04 0
10
20
30
t
40
50
60
(s)
Flgwe 6. Chronoamperograms of the oxidation of 5 mM Fe(CNh? in an unstirred 0.5 M KNO, electrolyte at gold microstep electrodes of type C, D, and E. Applied voltage is 0.5 V. The active goH areas are 0.03, 0.91, and 1.54 cm2,respectively. Inserting the appropriate errors in current, time, length, concentration, and diffusion coefficient shows that the uncertainty in an experimental estimation of F, can be as high as 50-75%. Figure 4 also shows that electrode B follows eq 2, but only for the first 1.6 s. Thereafter the current falls off more rapidly than predicted by eq 2. The obvious explanation for this is that after approximately 1.6 s the diffusion fronts from adjacent electrode fingers start overlappingand gradually fusing into one planar diffusion front, entering into a regime where the diffusion geometry is that of a conventional macroscopic electrode. The time it takes before a band electrode array with a certain interelectrode spacing leaves the microelectrode regime can be theoretically estimated from (see Appendix)
rtey (7) 40 where is the extension of the diffusion front. In the case of electrode B with a 100-wm interelectrode spacing, the time for two diffusion fronts to meet halfway is 1.59 s, i.e., in very good agreement with the experimental results. The corresponding calculation for electrode A gives 159 s, which could also be verified by experiments, although less accurately. After the first 1.6 s, the current from electrode B follows t - l I 2 as expected from the Cottrell equation, which describes the current from a conventional planar macroelectrode: nFDAC i(t) = t=-
This time dependence was verified by plotting the data for electrode B from Figure 4 is l/i vs dt. The result is shown
in Figure 5 where we obtained a perfectly straight line from approximately 1.6 s and onward. The slope of the line is, however, higher than that calculated by inserting numbers into eq 8. One should, however, remember that the linear diffusion process in the present case has another starting point than that for an ordinary planar electrode for which eq 8 is derived. Structures with an interelectrode distance of 10 pm (type C) were also studied. Equation 7 yields, for this electrode, a time of only 15.9 ms for the diffusion fronts to meet. On our experimental time scale, this electrode behaved exactly as a conventional macroelectrode; i.e., it follows the Cottrell equation. Figure 6 shows a chronoamperogram for electrode C together with two other electrodes. One (type D)is identical with that of type C except for the fact that there is no insulator on top of the metal. Electrode E has no insulator and has gold all over the surface; Le., it is a conventional macroelectrode with the same total area as electrodes A-D. As seen in Figure 6, electrodes C, D, and E are indistinguishable on the present time scale. One conclusion to draw from this is that, for the present redox couple, the very small surface area of gold exposed to the electrolyte in electrode C does not present any current limitation whatsoever. The surface area of gold exposed to the electrolyte for electrodes C, D, and E is 0.03, 0.91, and 1.54 cm2, respectively.
CONCLUSIONS The fabrication process presented in this paper seems to be a viable route to making band electrodes in large numbers. It should, however, be pointed out that obtaining sharp vertical steps and a high-quality nitride layer is essential to the microstep electrodes. During optimization of the etching process, we obtained samples where the steps microscopically were less sharp. These behaved very differently from the ones where
ANALYTICAL CHEMISTRY, VOL. 63, NO. 9, MAY 1, 1991
i
I
I
Figure 7. Schematic illustration of how the diffusion fronts develop at the microstep electrode. Stages a and b are very short in time. During stage c, the electrode behaves very similar to a hemicylindrical electrode and during stage d very much like an ordinary planar electrode.
the steps were perfectly sharp and vertical. This is obviously due to the fact that nonvertical step electrodes are much wider than their vertical counterparts. As mentioned above, the quality of the nitride layer is essential to the process. Electrodes fabricated in batches where the nitride did not come out exactly right turned out to be unstable. In some of these cases, electrodes behaved well as long as they were operated at positive potentials only but exhibited slow accumulation effects as soon as they were brought to negative potentials. The insulating layer could in principle, be made thinner, but this is hardly advisable. The thinner the insulator is, the more sensitive are the electrodes to imperfections in the insulator. The capacitance of the insulated area is, however, not critical. An 0.5-pm-thick layer of silicon nitride gives a maximum capacitance of 7 nF/cm2. This means for the present electrodes a typical capacitive charge of only A
distinguished from true hemicylindrical diffusion. (d) When diffusion fronts from adjacent electrodes overlap, they fuse gradually into one planar diffusion front whereafter the current decreases at t-1/2,as for a conventional macroelectrode. The above time intervals are only approximative and have been calculated from eq 7. This approximative model still serves to give a feel for the effects of different geometries a t band electrodes. It is obvious from the experimental results that the time dependence of the band electrode current is well described by the equation for hemicylindrical electrodes as long as the electrode dimensions are within the submicrometerrange. For most purposes, the only alteration needed for the equation is substituting the cylinder radius ro by a factor that depends on the detailed geometry.
ACKNOWLEDGMENT Fredrik Enquist is gratefully acknowledged for advise on the processing details. We also thank Ingemar Lundstrom and Olle Inganas for their support and encouragement. APPENDIX The growth of the diffusion layer for a hemicylindrical electrode can be found from the theoretical considerations presented by Amatore et al. (14). They used the technique of conformal mapping. From eqs 30 and 29 in ref 14, we have
and
S.
The fact that chromium was applied as an adhesion promotor between layers did not seem to give any adverse effects on the electrochemical behavior of the electrodes. This is probably due to the fact that we put some effort into making the chromium layers as thin as 0.8 nm. In order to avoid any chromium being exposed to the electrolyte, we also ended the fabrication process by briefly dipping electrodes into a chromium etching solution. It is interesting to note how well the present electrodes are described by the hemicylindrical diffusion theory, even though these electrodes are perpendicular to the surface and also have a relatively thick insulator on top. The explanation lies of course in their very small dimensions. Generally speaking, detailed electrode geometry plays a minor role as long as the electrodes are sufficiently small. The only factor to worry about is the substitution for ro in eq 3. It was also concluded that the exact substitution for ro is difficult to verify experimentally because of the form of eq 3. Even very small experimental errors result in very large errors in the determination of the geometrical factor. The above conclusions are valid only for times ranging from about 0.1 s (depending on geometrical dimensions). The diffusion process at these electrodes can be summarized schematically in four steps, as illustrated by Figure 7. Note that the features in Figure 7 are drawn very much out of scale. It is very illustrative for the reader to redraw Figure 7 to scale on a large piece of paper. (a) Initially there is planar diffusion with minor edge effeds. This stage lasts for less than 0.1 ps. (b) During this stage, the diffusion front gradually becomes equal to that of a cylindrical electrode (one-quarter of a cylinder). When the insulator is 0.5 pm thick, this stage lasts for approximately 0.3 ms. (c) When the diffusion front has passed above the insulator and starts to develop into the open space of the solution, it gradually turns into half a cylinder. After 0.1 s, it has advanced approximately 10 pm and its shape can hardly be
935
e = 1 + ((1/2)
1 In r - r ) / l n P
(A21
where 6(&7) is the transformed diffusion layer width. The relation between the cylindrical (r) and the transformed (y) space coordinate is
T
is the transformed time scale given by
nFvt
r=-
RT
y = 0.577 22 is Euler's constant and fl is the so-called cylindrical factor
P=2P nFvr,2
It has been assumed that fl >> 1 (small electrode radius) and that T is not too small (>l). Although the scan rate v is included in these expressions, Amatore et al. states that the expressions A1 and A2 are valid for any controlled-potential technique. By taking the inverse transformation (eqs A2-A5 inserted into eq A l ) we obtain r, (In (4Dt/r,2) - 27j2 21n-= (A61 ro In ( m t / r , 2 ) - 37 Note that Y does not appear in eq A6. It was merely used as a dummy constant in order to obtain dimensionless variables in the conformal mapping. Equation A6 is valid if f127 >> 1 which in our case corresponds to t > 1ps. By rewriting eq A6 we obtain
936
Anal. Chem. 1991, 63,936-944
In rs = In
-
y In (4Dt/r,2) - 47
-
2 In ( 4 D t / r O 2-) 37
If 4Dt/r,2 >> e47,i.e., t >> 10 ps, the last term can be neglected, yielding
which is equivalent to eq 7 . In our case, we use microstep electrodes and not hemicylindrical electrodes. It has, however, been found that the characteristics of a band electrode can be described by those of an equivalent hemicylindrical electrode, with an equivalent radius of r, = W / T or = w / 4 , depending on the detailed conditions (15). Anyhow, this is of less importance since the growth of the depletion layer, which determines the current, with time according to eq A7 is independent of the size of the electrode, and hence of the form.
LITERATURE CITED (1) Wightman, R. M. Anal. Chem. 1981, 53, 1125A. (2) Wightman, R. M. Sc/ence 1988, 240, 415. (3) Wehmeyer, K. R.; Deakin, M. R.; Wightman, R. M. Anal. Chem. 1985. 5 7 , 1913-1916.
(4) Thorman, W.; Van den Bosch, P.; Bond, A. M., Anal. Chem. 1985, 5 7 , 2764-2770. (5) Bond, A. M.; Henderson, L. E.; Thormann, W. J. J. Phys. Chem. 1988, 90, 2911-2917. (6) Morita, M.; Longmire, M. L., Murray, R. W. Anal. Chem. 1988, 60, 2770-2775. (7) Kittlesen, G. P.; White, H. S.;Wrighton, M. S., J. Am. Chem. SOC. 1984, 106, 7389-7396. (8) . . Seibold. J. D.; Scott, E. R.; White, H. S.. J. .€/echoanal. Chem. 1989, 264, 281-289. (9) Kovach, P. M.; Caudill, L.; Peters, D. G.; Wightman, R. M., J. Nectroanal. Chem. 1985, 185, 265-295. (10) Berteit, J. E.; Deakin, M. R.; Amatore, C.; Wightman, R. M., Anal. Chem. 1988. 60. 2167-2169. (11) von Stackeiberg,‘M.; Piigram, M.; Toome, V. 2.Nectrochem. 1953, 57. 342. (12) Szabo, A.; Cope, D. K.; Taliman, D. E.; Kovach, P. M.; Wightman, R. M., J. Electroanal. Chem. 1987, 217, 417-423. (13) Deakin, M. R.; Wightman, R . M.; Amatore, C. A. J. Nectroanal. Chem. 1988, 215, 49-61. (14) Amatore, C. A.; Deakin, M. R.; Wightman, R. M. J. Electroanal. Chem. 1988, 206, 23-36. (15) Amatore, C. A.; Fosset, B.; Deakin, M. R.; Wightman, R. M. J. Nectroanal. Chem. 1987, 225, 33-48.
RECEIVED for review June 18, 1990. Revised manuscript received October 1,1990. Accepted January 28,1991. This work has been made possible by a grant from The National Swedish Board for Technical Development (STU).
Development and Optimization of Piecewise Linear Discriminants for the Automated Detection of Chemical Species Thomas F. Kaltenbach and Gary W. Small*
Department of Chemistry, The University of Iowa, Iowa City, Iowa 52242
A pattern recognition technlque based on plecewlse linear discriminant analysis (PLDA) is described. Algorithms for the calculation and optimization of piecewise h e a r dlscriminants are presented. A simplex optimization of the individual discrlmlnants Is described, and a new method to optimize a piecewise linear dlscrimlnant Is proposed and shown to produce slgntficantly improved results over the nonoptlmized method. This methodology is demonstrated through the use of a set of Fourier transform infrared interferograms collected by a remote sensor. The discriminant analysis methods produce a yeslno declsion about the presence of a target anaiyte. The results obtained from the PLDA technique are compared with previous results from single linear dlscriminants and drown to be superior wtth respect to the separatlon statistics and the signal-to-noise ratio of the response.
INTRODUCTION Pattern recognition techniques provide automated capabilities for classifying unknown observations into predefined categories or classes. In analytical chemistry applications, these techniques are used most often in qualitative analyses. In this context, the observations typically consist of digitized instrumental responses, and the categories are defined as the possible targets of the analysis being performed. The observations can thus be considered as points in a multidimensional data space. The location of a given observation in this data space is indicative of its category. As an example,
pattern recognition techniques have been widely used to interpret spectral data automatically (1-3). Here, the spectrum of an unknown compound is categorized as representative of a particular chemical structural class. One prominent pattern recognition technique is linear discriminant analysis (LDA), which allows observations to be placed into two or more classes separated by multidimensional linear surfaces. The separating surfaces are termed discriminants, as they define boundaries in the data space that allow the classes to be discriminated. For applications in which nonlinear separating surfaces are appropriate, piecewise linear discriminant analysis (PLDA) can be used. The piecewise linear discriminant consists of multiple linear discriminants that collectively form a piecewise approximation of a nonlinear separating surface. Several methods for computing piecewise linear discriminants have been reported. Duda and Fossum ( 4 ) have described several of these algorithms, while Isenhour et al. (5) have applied piecewise linear discriminants to the interpretation of mass spectral data. Chang (6) proposed a method for computing piecewise linear convex and concave surfaces. Mangasarian (7) has described a recursive method utilizing multiple parallel discriminants, and Takiyama (8)has adapted Mangasarian’s method to an iterative training procedure. Each of these methods calculates the individual discriminants in a stepwise manner. A “one-at-a-time” calculation, however, often does not produce an optimum piecewise linear discriminant, since the approximation of the separating surface utilizes the set of linear discriminants collectively.
0003-2700/91/0363-0936$02.50/0 0 1991 American Chemical Society
