Surface Science 48 (1975) 921 0 NorthHolland Publishing Company
GENERAL FORMALISM FOR QUANTITATIVE
AUGER ANALYSIS
C.C. CHANG Bell Laboratories, Murray Hill, New Jersey 07974,
USA
Auger electron spectroscopy can provide absolute, quantitative, analyses of surface chemical composition. The basic formalisms for treatment of the homogeneous surface and the layered structure are presented; their use and typical limitations are demonstrated by examining some applications.
1. Introduction A generally applicable formalism for quantitative Auger electron spectroscopy [l4] * is required to fully utilize the enormous capabilities of the Auger technique. However, it is clear that a complete, accurate, treatment is not possible at this time. This article outlines an approximate formalism and the question of accuracy is examined by (1) reviewing examples in which AES was employed primarily as an analytical technique, (2) testing the internal selfconsistency of the quantitative results and (3) comparing the AES results with those from other analytical techniques. Typical limits of error, some major difficulties, and the physics of important factors controlling the Auger current have now been investigated by many workers, which justify the use of a simple and general formalism for obtaining quick, approximate answers with the possibility of further refinement if higher accuracy is desired.
2. Formalism
for quantitative
AES
2.1. Homogeneous case The key step in obtaining a generally the derivation of the expression for the terms of the measured Auger current Ii ments are distributed homogeneously). the result for the simplest case is
applicable equation for quantitative AES is atomic concentration, Xi, of element i, in for the homogeneous surface (where all eleThis derivation has been accomplished [ 11;
* For a comprehensive review of AES and a complete bibliography up to early 1972, see ref. 1.
10
C.C. ChangjGeneral
forrmlism
for quantitative
Auger analmis
where A, is the normalized Auger signal andj runs over all elements present. The “i are inverse Auger sensitivity factors defined by
where the superscript 0 indicates the pure element and s is an arbitrarily chosen standard element. The assumption Cut= C$ implied in eq. (1) neglects certain factors generally referred to as “matrix effects”; some of these are: (a) Chemical effects. The Auger peak energy and shape may change with chemical environment, especially for valence spectra [ I] . When this gives rise to a clearly distinguishable “new” Auger peak, the problem can obviously be circumvented by introducing a new 0: for that transition (as illustrated below for B and Si in SiO,). (b) Diffraction. There are two processes: (i) diffraction of the incident electron beam [5,6] and (ii) diffraction of the escaping Auger electron at the Auger energy Ei. The second effect is the more pronounced, and is related to Kikuchi correlations [681. (c) Energy loss. Cross sections for inelastic processes (e.g., ionization [ 19,10] , plasmon excitation [l,l 11, etc.) can change with surface composition. These changes alter the effective cr by changing the ionization probability, escape probability, Auger peak shape, etc. (d) Backscattering factor. Sufficiently energetic backscattered electrons can effectively increase the primary beam intensity [I ,5,12]. This backscattering factor is generally larger for heavier elements and lower energy transitions. (e) Variations of the electron mean free path [1,2,13] D(E). Some experimental values ofD are shown in fig. 1 for pure elements. Since D depends most strongly on the density of valence electrons [I ,141 , we see in fig. 1 that the Di are fairly independent of atomic number; therefore an averaged curve through the data points of fig. 1 provides an approximate value of D(E). Probably the most convenient unit to use for D in Auger work is the monolayer which contains 1 X 10’5 atoms~cm2 and is 2.5 A thick. Alternatively, if thickness units are used, a simultaneous specification of density would be required for absolute quantitative work, but the density is ofter not known. In monolayer units, the dashed line of fig. 1 is given by: D = 0.2 @monolayers
(E in units of electron volts).
(3)
Most spectrometers collect electrons leaving the surface at various angles, and a value of 1.33 D provides a fair approximation [2,15] to the escape depth averaged over these angles. Differentiation of eq. (1) shows that an error AIi in estimating the Auger current causes an error AXi/Xi in the calculated concentration of
C.C. Char&General formalism for quantitativeAuger analysis
ENERGY (ev)
Fig. 1. Mean free path D of electrons in various smallest near 7.5 eV and increases approximately
AXi/Xi  (Ar,ll,)(l
materials (after J.C. Tracy, see ref. 1). D is as fi between 100 eV and 2000 eV.
 Xi),
(4a)
1,
(4b)
so that AXi/Xi+OasXi+ and (4c)
therefore, the error in Xi is always smaller than the error in the measurement of Zi. The above treatment allows us to define “matrix effects” as those effects which cause the (Yito deviate from CY~;the ratio ai/oP is a measure of the strength of matrix effects. Many matrix effects simply multiply all (Y’Sby a constant; such matrix effects “cancel out” in eq. (1) and do not affect Xi. Because of the “noncancelled” matrix effects, and several other sources of error [l] , the accuracy of the absolute concentrations calculated using eq. (1) cannot be expected to be better than +20%, unless precise values of Qli(Xi) are determined from known compounds with Xi close to those being investigated. The relative concentrations (i.e., comparisons of concentrations in different samples) can be determined with an accuracy of *5% since relative measurements depend primarily on the reproducibility of the Auger data. 2.2. Layer structure Layered structures can be treated by applying eq. (1) to thin layers and calculating the integrated Auger signal Ai. The physical meaning OfAi is illustrated in fig. 2, where the electron escape probability [1,2,12], e&, and the total Auger current,
12
C. C. C~ang~~enerui f~r~na~is~ for quantitative Auger unalysis SURFACE
I.5
1.0
d (DEPTH,
2.0
z/D)
Fig. 2. Electron escape probability e d and the total Auger signal 1  evd, as functions of d = z/D, where z is the distance from the surface. ‘The Auger signal from layer i is equal to Ai.
1  eSd, are shown as functions of the normalized ejection depth d = zfr;), where z monolayers is the distance from the surface. For simplicity, we consider only homogeneous layers; then the layer i of fig. 2 produces the signal d2
Aj=
Xi eg dg = Xi(eedr  ed2), (5) s dr which is only approximate because the exact expression must include the dependence of electron escape probability on the collection angle of the spectrometer [ 151, on the composition Xi, etc. Below one monolayer, the eed dependence appears to break down; however, for a thin surface “layer”, d < 1 so that
Aia
] ediedi
which is the correct relationship. Therefore eq. (5) remains valid at arbitrarily low surface concentrations. For a material consisting of layers with thicknesses zl, z2,23, ... , (total thickness zI +zz + z3 + ... we have A: =X,(1
 eedr),
Ai =X2eWd:(1
~$1
(oa)
eedz),
zX3edCd:(l
(6b) _eeds),
(6c)
etc., where the superscript on d specifies the proper D to use; i.e., d: = z1/D2, (neglecting the dependence of Di on Xi), and the superscripts on A specify the overlayers. leads to Eq. (6) shows that EmAfifiv is not normalized. Renormalization
C. C. C~an~~Gerleral for~~ism
A&g, Ai=
for q~~rira~iye Auger analysis
13
.I.
z
’
~,A&$
When eqs. (6) and (7) can be solved for the Xi and di, the absolute amount of material detected, Ci (monolayers), can be calculated as Ci = XidiDi.
(8)
Although AES is noted for the ability to detect thin surface “layers” (< 0.1 a), eq. (6) shows that material under fairly thick layers can be detected. For example, since Ai @ 0.01 is easily detectable; we can estimate the thickness dT of material T on a substrate S which will produce a signal A, = 0.01 (assum~g Cm Ahkp..’ = 1 for simplicity). Since As
=.edT
(9)
’
therefore d,
= InA,
= 4.6(DT
units).
(10)
Since typically DT < 10 monolayers, assuming 2.5 &monolayer.
film thickness to dT > 100 A can be measured,
3. Application of the formalism The most important consequence of eq. (1) is that AES can provide quantitative analyses if a table of o’s is compiled. The o’s can be estimated theoretically [ 1,9] , but are more accurately determined experimentally by use of pure elements [eq. (2)], or known compounds. Typical values of ty, normalized to (Y(Si, 92 eV) = 1 .OO, are shown in table 1. For determining these Qi, the Ii were measured using the peaktopeak heights in dN(E)/dEtype spectra from the cylindrical mirror analyzer (CMA). All of the o’s listed were determined by the author and have been tested by application to compounds; in most instances, no large variations in the Q’S(greater than 230%) were observed from sample to sample or in different materials, except for valence transitions [I ] . These a’s are in fair agreement (wit~n + 30%) with those determined from pure elements; therefore, approximate o’s can be obtained from elemental spectra (e.g., the calibrated spectra published by Palmberg et al. [16]). However, (Yvalues determined from compounds are often more accurate because the experimental conditions can be made more nearly the same for several elements in a single compound than for several samples of pure elements. In addition, the Dj, and most matrix effects, are nearly identical for all elements in a homogeneous material. When new phases form, an 01specific to an element in that phase can be used, as shown for B and Si in table 1. The Ao’s in the table are maximum deviations observed in actual
14
C.C. ChangjGeneral formalism for quantitative Auger analysis
Table 1 Inverse Auger sensitivity Element
Ei (eV)
B (SiO2) B c N 0 (SiO2 ) 0 (SiOz ) Al
166 179 273 386 495 507 66
Al Si (SiO2) Si Si P S
1396 78 92 1620 120 150
Ti
factors* pi j; 3.6 2.1 3.0 1.4 6.0 1.4 1.7
normalized Affi
+ 0.4 * 0.7
10 +3 1.9 f 0.2 1.00 10 *3 0.86 f. 0.2 0.70
418
1.1
to Si 92 eV peak Element
t’i (eV)
Fe cu Ga Ga AS AS Pd
703 920 84 1070 95 1228 330
Ta W W Pt Pt AU
1680 169 1800 168 1967 150
Au
2024
1.9 1.9 14 3.7 16.6 5.7 0.70 10 3.5 7 8 14 10 17
22 i 0.7 + 2 +2
* 0.5 i2 +2 *2
* Applicable only to CMA with 8” aperture, peaktopeak heights of dN(E)/dEtype spectra. 3 keV, 45” incident electrons, 1.0 1 Vp_p for Ei < 1000 V, 5 Vp_p for Ei > 1000 V.
experiments and have no further statistical significance; they are listed to provide an approximate idea of typical variations. Note that these 0~‘sapply only under the conditions specified at the bottom of the table. 3.1. Analysis of AZ111 photoresist
Eq. (1) was applied to data from developed AZ1 11 photoresist. The calculated chemical composition (neglecting hydrogen), and the measured values, are presented in table 2. Photoresist is an extremely difficult material to examine with AES because it is insulating and presents problems with electrostatic charging; it is also rapid& decomposed by the incident electron beam. Table 2 shows that if the measured concentrations are multiplied by 2.68, the concentrations for 0, S, and N agree with calculated values. The simplest explanation is that the photoresist decomposed at the
Table 2 AZ1 11 Photoresist
composition
Measured: Calculated: Meas. X 2.68 All values in at%.
measured
with AES
c
0
S
N
92.6 80.5 247.9
6.2 16.6 16.6
0.65 1.6 1.7
0.5 1.4 1.3

C. C. Chang/General formalism for quantitative Auger analysis
15
surface into a C layer. If that is the case, the apparent agreement between “calculated” and “measured X 2.68” values is misleading because the shorter ejection depth of the 150 eV Auger electrons for S, compared to the SO7 eV electrons for 0, should cause the S signal to appear too small compared to 0; we shall examine a numerical example of this later on. Nonetheless, these results illustrate the kind of detailed analysis that is possible in spite of the difficult nature of the material analyzed. 3.2. Bdoped SO2 B concentrations in SiOZ films, deposited at 480 *C using a mixture of SiH,, 02, BZH6, N,, and Ar, were measured with AES. The results were compared with those from electron microprobe and infrared absorption analyses. Fig. 3 displays the measured concentrations against B,H, flow rate, taken from seven specimens. Samples 2, 5 and 6 were investigated using all three techniques. Typical standard deviations from ten readings are shown for the microprobe and Auger data. The remarkable agreement among the three techniques is fortuitous since the absoZute accuracy is I
~~~~~ ABSOWTE 6 CALIBRATION 0 ep PROBE 0 IR
7
0 AUGER
BZHs FLOW RATE (CC/mlnl
Fig. 3. B concentrations in Bdoped SiO2, measured using AES, the electron microprobe, and infrared absorption. The B concentration was varied by controlling the Bz H6 flow into the reactor. Samples 2,5 and 6 were investigated using all three techniques. The error bars are standard deviations from ten readings.
16
C. C. Chang/GeneraE formalism for quantitative Auger analysis
t
GOOH
A50
A100
Fig. 4. Auger spectra from GaxAlr_x AS,, PI_Y between 50 and 150 eV, obtained while the surface was continuously cleaned by ion milling with Ar. The Ppeak at 1 at% (y = 0.98) is almost halfthe size of the As peak at 50 at%. Therefore, these spectra are ideal for measuring small P concentrations accurately using the As peak as reference. The Al peaks are somewhat smaller than they should be due to slight oxygen contamination in the AI gas; 2.5 V _ , 3.0 keV incident electrons. PP
probably no better than a factor of two for all three methods. The low absolute accuracy of the Auger results in this case is due partly to the uncertainty in the value of oB; the BKLL Auger peak shape is sensitive to chemical environment. Also, elemental B was found to segregate to the surface during electron bombardment, and the relatively thick (10 0008) and poor quality oxide caused electrostatic charging difficulties. These results not only provided an absolute c~ibration for the B concentration, but also revealed the degree of control over the doping level, and showed that the B concentration was linear with B,He flow rate.
C.C. Chang/Generalformalism for quantitativeAuger analvsis
17
100 ABSOLUTE PCALIBRATION
O
20
40
60
80
100
ATOM PEFtCENT STANDARD
Fig. 5. Calibration of the P Auger peak at 120eV. slope which passes through the origin.
3.3. Absolute concentrations
The data points
fall on a straight
line of unit
in GaAlAsP
Absolute concentrations in single crystal GaAlAsP were determined from Auger data alone, without calibrations from other analytical techniques. LOVJenergy Auger spectra from GaAlAsP and GaAlAs are shown in fig. 4; such spectra were used for measuring the P concentration. For Ga, Al, and As, however, the high energy Auger transitions (see table 1) were used, because of the severe spectral overlap seen in fig. 4. The necessary (Y’Swere obtained as follows: aGa and oAs were determined from GaAs. Then oAl was measured using several crystals of GaXAll_,As with different values of x. Note that x can be evaluated directly from the Auger data by subtraction of the Ga and As concentrations, crp was finally obtained in like manner from several samples of GaAIAs,,P1_Y with different values ofy. Fig. 5 shows the measured P concentrations in GaAIAsyP1_ ,, withy = 0.62 and 0, corresponding to P concentrations of 19% and SO%, respectively. In these experiments, P concentrations down to < 0.1 at% (Xp < 0.001) could be readily measured. 3.4. Carbon layers on SiO, Analysis of layered structures is il!ustrated by the data of table 3. Results from three oxidized Si wafers, with increasing amounts of C contamination from sample 1 to 3, respectively, are presented. The objective was an accurate determination of the absolute amount of C on the surface.
C.C. ChanglGeneral formalism for quantitative Auger analysis
18
Table 3 Quantitative Auger analysis of C layers on SQ Sample
Si(78V) Qi Di (monolayers)
1
3
1.9 1.8
0 (495 vt 6.0
4.2
C(273V) 3.0 3.1
I (mm) al (mm) Aj(%) Cj (mon$ayers) Xi (at%)
106 201 32 (0.60)* 33
126;69,’ 61
3 9 1.4 0.02 0
I (mm) al(mm) 14i (70) Ci (mon$ayers) Xi (at?&)
90 171 25 (0.58)* 32
72 432 62 (2.89)* 68
31 93 13 0.33 0
I(mm) cul (mm)
92 115 21 (0.56)* 31
81 486 58 (2.91)* 69
60 180 21 0.55 0
Ai (72) Cj (monolayers)
Xi (at%,)*
69 414
z
_ 624 100 (3.41) 100 _ 696 100 (3.80) 100 _ 841 100 (4.02) 100
* Corrected for attenuation of Auger electrons traversing the C layer (i.e., the expected vatues if the C contamination were absent).
This example is used to demonstrate a complete computation procedure. For this purpose, the values of (Yjand Di used are listed at the top of table 3. We illustrate the computation procedure using sample 3 for which the layer corrections are largest because of the large C contamination. The Auger intensities Zi are listed in the row labelled 1. The weighted Auger intensities Crjij are also listed and the sum, Cioli$, is shown under C. Next, the Ai were computed using eq. (1). The amount of carbon, Cc, was then calculated, assuming (1) a uniform C layer of thickness do and (2) that GnzAbk7... t?r = 1 (the validity of these assumptions are examined below): c, = d(+C,
(11)
where d, is given by A, = 1 _ e1.33
dC
(12)
or d, =(1/1.33)hr(l
 0.21) = 0.18,
(13)
which gives CC =
0.18 X 3.1 = 0.55 monolayer.
The substrate SiOz can be assumed to be homogeneous, given by
04) so that the amount of 0 is
CC. Chang/General formalism for qwntitative Auger analysis
Co e1.33 dC = A,D,,
(15)
where e1.33 dC is the attenuation C, = 0.58 X 4.2 exp(l.33
19
due to the C layer; therefore, X 0.55/4.2)
= 2.91 monolayers.
(16)
Similarly, Csi = 0.21 X 1.8 exp (1.33 X 0.55/l .S) = 0.56 monolayer.
(17)
These values of C0 and Csi are just the detected quantities since the amount of substrate material is effectively infinite. This is why they are enclosed in parentheses in table 3. The corresponding entries for samples 1 and 2 were calculated in a similar way. The quantitative results of table 3 provide opportunities to investigate internal selfconsistency, which is useful for examining the accuracy of the numerical parameters such as cuand D, and for examining the validity of some of the simplifying approximations, such as the neglecting of various matrix effects, and the assumption of a uniform C layer. For example, the substrate composition must correspond to SiO,. The ratio Asi/Ao in sample 1 is nearly correct, but it decreases progressively in samples 2 and 3 with increased C contamination. This decrease is caused by the difference in electron attenuation due to the smaller Di for Auger electrons from Si (78 eV) than from 0 (495 eV). The “corrected” Xi can be calculated from Ci as follows (using data of sample 3): in a layer of substrate material 4.2 monolayers thick, there are 2.91 monolayers of 0; this same 4.2 monolayers contains Csi (corrected)
=
‘siDo DSi
0.56 X 4.2 = ____ = 1.3 1 monolayers 1.8
of Si (for a total of 4.22 monolayers!). This corresponds to Xi(corrected) = 31 and 69%, respectively, for Si and 0. These Xi(corrected) are also shown for samples 1 and 2. The agreement of the Xs and X0 with the expected values of 33 and 67% respectively, indicates that the assumption of a uniform C layer is fairly valid. Because the Csi and Co have been compensated for attenuation through the carbon layer, the ~Ci’s increase progressively from samples 1 to 3; notice that these increases are slightly larger than the increases in C,. The explanation for this discrepancy is that the carbon layer contains some oxygen; note that X0 is larger than 2X,,. Therefore, the above calculation can be further refined by incorporating an amount of oxygen equal to X0  2Xa into the C layer. An astute reader would have observed that the assumption E:,Ahk~.. = 1 greatly simplified the above calculation for di. The validity of this assumption must now be examined. Again, using sample 3 of table 3, we have d, = 0.55/3.1 = 0.18, and
(19)
20
CC. Chang/Gcneral formalism for quantitative Auger analysis
x
m
Al*k,,=A: +A$ +A$ m (20)
= I 
exp(dc)
++exp+#
t$exp(c$j
= 0.98
(where d? = dCF)cl~Si, dg = dcDc/Do), which is sufficiently close to unity SO that our assumption is valid within experimental error. For thinner layers of C (samples I and 2j, this sum is even closer to unity. If the C layer had been thicker, and if the high energy Si transition at 1620 eV had been used, the sum would have deviated appreciably from unity. The resulting error, however, can be corrected by use of eq. (7) to obtain the individual Aihbgr, which can be inserted into eq. (6) to calculate a new value of d,.
4. Concluding remarks A simple, approx~ate formaIism for quantitative AES has been outlined, based on the equation for the normalized Auger signal A, [eq. (l)] . The only required parameters are the o+ and Di. At present, empirical calibrations provide the most useful values of ai> and values of Dj that are sufficiently accurate for most applications are now known. For homogeneous surfaces, Ai = Xi. Layered structures can be analyzed by use of eqs. (6) and (7). The successful application of eq. (1) (see refs. 1719 for more examples of applications) is probably telling us that, in many systems, the noncancelled matrix effects are small and the ejection depths do not deviate appreciably from some standard II(E). There is little doubt that in the near future, the detailed investigations presently being conducted in numerous laboratories will provide an abundance of quantitative characterizations of matrix effects and accurate determinations Of L)i(E) [20231 and o+ (see refs. {4,24263 for studies of Auger sensitivities). SUperior methods of data acquisition and reduction should also be forthcoming; for example, the Ii can be measured very accurately by Dynamic Background Subtraction [27], and data reduction should eventually be computerized. One parameter which was entirely neglected here is the matrix correction n which accounts for the changes in I)(Ei) with changes in composition. This factor was introduced in ref. [ 1] where it was shown that the condition Q = 1 is necessary for eq. (1) to be valid. Therefore, the formalism outlined here will provide erroneous results for that element whose qfactor deviates appreciably from unity in the particular material being studied. Preliminary results 128 ] indicate that n can provide an important correction when accuracies better than *500/o are desired.
C. C. ChanglGeneral formalism for quantitative Auger analysis
21
Acknowledgements The author is grateful to M.B. Panish for the GaAlAsP samples, to D. Wonsidler for the electron microprobe analysis, to A.C. Adams who grew the Bdoped oxide films and provided the infrared absorption results, to D.M. MacArthur who supplied the photoresist samples, and to G. Quintana who assisted in most of the Auger experiments reported here.
References [I] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [l l] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
Chuan C. Chang, in: Characterization of Solid Surfaces, Eds. P.F. Kane and G.B. Larrabee (Plenum, New York, 1974) ch. 20. M.P. Seah, Surface Sci. 32 (1972) 703. M.P. Seah, Surface Sci. 40 (1973) 595. F. Meyer and J.J. Vrakking, Surface Sci. 33 (1972) 271. J.H. Neave, C.T. Fox and B.A. Joyce, Surface Sci. 29 (1972) 441. T.W. Rusch, J.P. Bertino and W.P. Ellis, Appl. Phys. Letters 23 (1973) 359. L. McDonnell, B.D. Powell and D.P. Woodruff, Surface Sci. 40 (1973) 669. GE. Becker and H.D. Hagstrum, J. Vacuum Sci. Technol. 11 (1974) 284. H.E. Bishop and J.C. Riviere, J. Appl. Phys. 40 (1969) 1740. R.L. Gerlach and A.R. DuCharme, Surface Sci. 32 (1972) 329. P.J. Feibelman, Phys. Rev. B 7 (1973) 2305. T.E. Gallon, J. Phys. D (Appl. Phys.) 5 (1972) 822. M.L. Tarng and G.K. Wehner, J. Appl. Phys. 44 (1973) 1534. J.C. Shelton, Surface Sci. 44 (1974) 305. J.C. Shelton, J. Electron Spectroscopy Related Phenomena 3 (1974) 417. P.W. Palmberg, G.E. Riach, R.E. Weber and N.C. MacDonald, Handbook of Auger Electron Spectroscopy (Physical Electronics Ind. Inc., Edina, Minn., 1972). C.C. Chang, P. Petroff, G. Quintana and J. Sosniak, Surface Sci. 38 (1973) 341. C.C. Chang and G. Quintana, J. Electron Spectroscopy Related Phenomena 2 (1973) 363. C.C. Chang, A.C. Adams, G. Quintana and T.T. Sheng, J. Appl. Phys. 45 (1974) 252. R.G. Steinhardt, J. Hudis and M.L. Pearlman, Phys. Rev. B 5 (1972) 1016. J.C. Riviere, Contemp. Phys. 14 (1973) 513. K. Goto, K. Ishikawa, T. Toshikawa and R. Shimizu, Appl. Phys. Letters 24 (1974) 358. I. Lindau and W.E. Spicer, J. Electron Spectroscopy Related Phenomena 3 (1974) 409. R. Bouwman, L.H. Toneman and A.A. Holscher, Vacuum 23 (1973) 163. J.M. Morabito, Thin Solid Films 19 (1973) 21. J.M. Morabito, Analytical Chem. 46 (1974) 189. J.T. Grant, T.W. Haas and J.E. Houston, in: Proc. 2nd Intern. Conf. on Solid Surfaces, Kyoto, 1974, to be published in Japan. J. Appl. Phys. C.C. Chang, to be published.