On the accurate determination of absorption coefficient from reflectance and. transmittance measurements: application to Fe-doped GaN

Transcription

1 On the accurate determination of absorption coefficient from reflectance and transmittance urements: application to Fe-doped GaN David C. Look a) Semiconductor Research Center, Wright State University, Dayton, OH Air Force Research Laboratory Sensors Directorate, Wright-Patterson AFB, OH Wyle Laboratories, Inc., 601 Mission Point Blvd., Dayton, OH Jacob H. Leach Kyma Technologies Inc., 889 Midway West Road, Raleigh, NC 7617 a) Electronic mail: For light impinging normally on the surface of a double-side-polished sample of thickness d, the sample s absorption coefficient can be determined from the well-known formula for fractional transmittance: T = (1 R) exp(-d)/[1 R exp(-d)]. Here R is a fundamental property of the air/sample interface and is known as the reflectance coefficient. Often R in this equation is equated to the ured top-surface reflectance R, but such an approximation can lead to serious error. In fact, we show explicitly that R = R + R(1 R) exp(-d)/[1 R exp(-d)] and then further develop an easily solvable transcendental equation that determines both R and α from T and R. In strongly absorptive regions (αd >> 1) it turns out that R R, but in the opposite limit (αd << 1), R R /( R ). Our formulation enables accurate determinations of: (1) ε, the high-frequency dielectric constant; and () relatively weak absorbances, such as those related to defects or impurities with energy levels in the bandgap. We also compare the exact calculations of α in semi-insulating GaN:Fe with those obtained from commonly used approximations. 1

2 I. INTRODUCTION A detailed knowledge of absorptive properties is important for thin films used in displays, lightemitting-diodes, and solar-cells 1-9 and also for bulk materials used in rare-earth lasers, optical components, and photoconductive switches In all these materials, an important fundamental parameter is the absorption coefficient α, in some wavelength or energy range, and α is generally determined by uring the fractional transmittance (T ) of normally impinging light. Despite the importance of α, a scan of the literature shows that there is no universally accepted formula for its calculation. For example, one such formula is α = -(1/d)ln[T ] 4-6, where d is the sample thickness; another is α = -(1/d)ln[T /(1-R )] 7,8, where R is the ured reflectance from the top surface; and a third is α = -(1/d)ln[T /(1-R ) ] 15. However, each of these formulas is an approximation and care must be exercised in applying any one of them to a particular structure, whether a single, bulk slab or a complex arrangement of several layers. In the single-slab case (and only in this case) α and T can be related by a well-known formula that accounts for multiple reflections at both the top and bottom interfaces 15 : T d (1 R) e (1) d 1 R e where R is the so-called reflection coefficient, another fundamental parameter. But note that Eq. 1 contains two unknowns, α and R, and thus it is necessary to have a second, independent equation to be able to solve for both of them. Fortunately, by following the same procedure 15 as that used to derive Eq. 1, we can derive a second equation: R d R(1 R) e R () d 1 R e

3 From Eq., it is immediately clear that if αd >> 1, then R R, which means that the third approximation listed above, α = -(1/d)ln[T /(1-R ) ], becomes accurate. However, if αd 1, Eqs. 1 and must be solved simultaneously to determine α and R. Finally, for completeness and convenience, we also derive a third equation, for the absorbance A : A (1 R)(1 e 1 Re d d ) (3) Of course, only two of Eqs. 1-3 are independent since T + R + A = 1, as required by definition. The important benefit of these new derivations, Eqs. and 3, is that R and α can be exactly determined at any energy by simply uring T and R at that energy; i.e., no model or further information of any kind is necessary. But this possibility holds only for a single slab of material; if more than one material is involved, as with a thin film on a substrate, then some sort of model along with curve fitting will normally be required, as discussed in many sources 3. II. Accurate determination of R and α A. Theory By algebraic manipulation of Eqns. and 3, we can show that e d (1 R A (1 R RA ) ) R R R(1 R RR ) (4) The second equality in Eq. 4 (i.e., term = term 3) forms a transcendental equation that can be used to determine R; e.g., in Mathcad it can be solved by the command: R = root( nd term 3 rd term, R). Once R is known, then either the nd or 3 rd term can be equated with the first term to determine α. Although this method is quick and convenient in Mathcad the equation can also be solved by using a stable and rapidly converging iteration. By letting y = e -αd, Eq. can be written R = R 3

4 R(1 - R) y /(1 R y ). The magnitude of the second term on the right-hand side ranges from 0, at y = 0, to a maximum of 0.17, which occurs at y = 1 and R = 0.4. Thus, this term is always relatively small, even in non-absorptive regions, so we can initially set y 0 = 0 in the iteration. Then, in step 1 of the iteration, R 1 = R, and y 1 can be found from the first equality in Eq. 4: y 1 = (e -αd ) 1/ = (1 R 1 A )/(1 R 1 R 1 A ). In step, R = R R 1 (1 R 1 ) y 1 /(1 R 1 y 1 ), and y = (1 R A )/(1 R R A ). After n steps, R = R n and α = -(1/d)ln(y n ), and in our experience, only a few steps are usually necessary for reasonable convergence. In low-absorptive regions, i.e., αd << 1 (or e -αd 1), we find from Eq. that R = R /( R ). Thus, the ratio R/R can be as low as 0.5 for small values of R, giving an appreciable error to the common assumption R = R. In Fig. 1, we plot R vs αd for R = 0.1, 0.5, and 0.9. If R = 0.1, the ratio R /R is as large as 1.8, occurring near αd = 0. For R = 0.5 and 0.9, the ratios at αd = 0 are 1.33 and 1.05, respectively. On the other hand, for αd =, the ratio R /R = 1.015, 1.005, and for R = 0.1, 0.5, and 0.9, respectively. Thus, from a qualitative point of view, the assumption R R holds well for large αd at any value of R, and for small αd only if R is close to unity. We now apply the exact analysis of R and α to a semi-insulating substrate of GaN doped with Fe and compare the results with those obtained under various approximations. To generate approximation #1, we set R R in Eq. 4, giving e d 1 R A 1 R RA 1 R RR 1 R A R A (1 R T ) R T (5) 4

5 R = 0.9 R R = R + R(1 - R) e -d /(1 - R e -d ) R = R = d Figure 1 (color online). Theoretical values of ured reflectance R as a function of αd for various values of the reflection coefficient R. For approximation #, we begin with Eq. 1, which is exact, assume that R exp(-αd) << 1, and then let R = R, giving e d T (1 R e (1 R) d ) T d R e 1 (1 R) T R = R (1 R ) (6) In summary, our three equations for comparative purposes are: exact, α = -(1/d)ln[(1 R A )/( 1 R RA )]; approximation #1, α 1 = -(1/d)ln[T /{(1 R ) + R T )}], in which it is assumed that R = R ; and approximation #, α = -(1/d)ln[T /(1 R ) ], in which it is 5

6 assumed that R = R and R exp(-αd) << 1. Approximation # is used very commonly in the literature. B. Application to GaN:Fe Fe-doped GaN is useful as a substrate for GaN-based transistors and also as the active material in high-power photoconductive switches For the latter application, the interaction of light with the material is quite important, and thus a correct treatment of is essential. We will consider a 430- m-thick sample grown by hydride vapor phase epitaxy with [Fe] = 5.6 x cm -3, as ured by secondary ion mass spectroscopy 10,13. The percentage values of transmittance T and reflectance R were ured in a Perkin-Elmer Lambda 900 UV/Vis/NIR spectrometer over the wavelength range nm, and the absorbance was then calculated from A = T - R. These quantities are plotted in Fig.. From A and R, R and α are calculated R, R, T, A (%) T R 30 0 R 10 A E (ev) 6

7 Figure (color online). Comparison of the reflection coefficient R (solid red line) with the ured reflectance R (open red squares) as a function of energy E in Fe-doped GaN. Also shown are the ured transmittance T (open black triangles) and absorbance A (open blue circles). The feature at E 1.4 ev is due to a source change in the spectrometer. from Eq. 4, and R is then plotted in Fig. in order to compare with R. We see that R R for E 3. ev, but R << R at lower energies. In Fig. 3, we compare the exact calculation of α vs energy with that obtained from approximations #1 and #, described above. The most outstanding (cm -1 ) GaN:Fe x x 100 source change E (ev) Figure 3 (color online). Absorption coefficient vs energy E for Fe-doped GaN determined by the exact method (α) and two approximate methods (α 1 and α ). The, α 1, and α curves artificially saturate above 3.3 ev because the detector (photomultiplier) noise magnitude becomes larger than the T signal magnitude. 7

8 difference between the exact (α) and approximate (α 1 and α ) analyses is that both of the latter curves are negative is the low-absorptive regions, from about ev. This is clearly not physical, and renders the α 1 and α approximations useless in these regions. On the other hand, use of the exact method gives good data for absorption coefficients as low as 0.1 cm -1. For comparison, Gladkov et al 10 have used micro-ellipsometry to determine α cm -1 at.5 ev (or ~ 500 nm) in bulk GaN doped with a concentration of Fe similar to that of ours. From the data of Fig. 3 we obtain a similar result, α 3 cm -1 at.5 ev, from simple transmittance/reflectance urements. III. ACCURATE DETERMINATION OF OPTICAL CONSTANTS AND A. Theory The parameters and R, as accurately determined above, can be related to the real and imaginary parts of the index of refraction, n = + i. This is accomplished through the well-known formulas 3,15 = /c and 1 1 R (7) Here we assume normal incidence. For small, such that <<, Eq. 7 can be inverted to give (1 + R 1/ )/(1 - R 1/ ). Now, by definition, n = 1/, where is the complex dielectric constant, ε = ε r + iε i, and in the low-absorptive regions, i.e., ε i << ε r, it follows that = ε r. In semiconductor materials, the frequency dependence of ε r is often represented at low frequencies by a value ε 0, the static dielectric constant, and at high frequencies by a value ε, the high-frequency dielectric constant. Over the range of our optical spectrometer, nm ( ev), ε r () in most semiconductors can be represented by ε r () = ε + ε Drude () + ε Lorenz () where the Drude term 8

9 accounts for free electrons (plasmonic effects) and the Lorenz term, for bandgap transitions. In highly-doped materials, it often happens that ε r () ε + ε Drude () over a wide range of energies below the bandgap, and since ε Drude () is negative, ε r () will vanish at a certain value of, the plasmonic resonance frequency 3. The value of also affects electronic properties via the electronphonon interaction and, in this regard, Rode 16 has compiled a well-annotated table of for nineteen common semiconductor materials. Accurate urements of often involve microwave techniques, but in principle can also be accomplished by employing common optical spectrometers such as the one mentioned above. However, to obtain accurate values of it is critical to determine R from Eq. 4, rather than estimate it from R R, as illustrated below. B. Determination of ε in undoped, bulk ZnO, GaN, ZnSe, and GaAs The maximum wavelength of our spectrometer, 300 nm (0.39 ev), is well above the phonon energies and well below the bandgap energies for most of the semiconductors in Rode s table 16 ; for such materials, ε r (300 nm) ε. In particular, this condition holds for ZnO, GaN, ZnSe, and GaAs, materials which we have available in bulk form. For this study, we ured R and T at 300 nm for three samples of ZnO, four of GaN, two of ZnSe, and one of GaAs. We then calculated R for each, from Eq. 4, and finally calculated ε,exact = exact = (1 + R 1/ )/(1 R 1/ ). For comparison, we also approximated R R and calculated ε,approx. = approx = (1 + R 1/ )/(1 R 1/ ). As shown in Table 1, the values of ε,exact agree quite well with the literature values, whereas those of ε,approx are all in much poorer agreement. These results demonstrate the usefulness of our new 9

10 Table 1. Comparison of = 1/ from the literature (Ref. 16) with values determined from exact = (1 + R 1/ )/(1 R 1/ ) and approx = (1 + R 1/ )/(1 R 1/ ). Material Sample = 1/ (literature) (Ref. 16) exact (calc) (300 nm) Deviation % approx (calc) (300 nm) Deviation % ZnO < < GaN ZnSe GaAs Avg. Dev. 4% Avg. Dev. 8% methodology, and also suggest that reflectance/transmittance urements can often be used as a simple, fast, nondestructive method of obtaining. IV. SUMMARY We have developed an approximation-free and model-free method to determine absorption coefficient α and reflection coefficient R from urements of transmittance T and reflectance R in a single slab of material. For small values of αd, where d is the sample thickness, this method is much more accurate than typical methods that are based on T alone, or on T and R, where R is approximated by R. Thus, it is useful for investigating small impurity and defect absorbances rather than just the much larger absorbances that result from bandgap excitations. For example, below-bandgap Fe-based absorption coefficients as low as 0.1 cm -1 can be studied in Fedoped GaN, whereas the common approximate methods actually give negative values of for E 10

11 .8 ev, clearly unphysical. Our method also enables accurate determinations of ε, the highfrequency dielectric constant, for many different semiconductor materials. ACKNOWLEDGMENTS We wish to thank W. Rice, T.A. Cooper, and D.K. McFarland for technical assistance. We gratefully acknowledge support from AFOSR Lab Task 14RY07COR (G. Pomrenke), NSF Grant DMR (C. Ying), and AFRL Contract HC D-4005 (D. Tomich). 1. T. Minami, Semicond. Sci. Technol. 0, S35 (005).. F. Ruske, A. Pflug, V. Sittinger, B. Szyszka, D. Greiner, and B. Rech, Thin Solid Films 518, 189 (009). 3. D.C. Look and K.D. Leedy, Phys. Status Solidi A 1, 147 (015). 4. K. Tang, S. Gu, J. Liu, J. Ye, S. Zhu, and Y. Zheng, J. Alloys and Compounds 653, 643 (015). 5. Y. Chen, Y. Sun, X. Dai, B. Zhang, Z. Ye, M. Wang, and H. Wu, Thin Solid Films 59, 195 (015). 6. F.A. Garces, N. Budini, R.D. Arce, and J.A. Schmidt, Thin Solid Films 574, 16 (015). 7. R. Muydinov, A. Steigert, S. Schonau, F. Ruske, R. Kraehnert, B. Eckhardt, I. Lauermann, and B. Szyszka, Thin Solid Films 590, 177 (015). 8. Yu Wang, A. Capretti, and L.D. Negro, Opt. Mater. Express 5, 415 (015). 9. M. Patel, H-S. Kim, and J. Kim, Adv. Electron. Mater. 1, (015). 11

12 10. E. Gladkov, E. Hulicius, T. Paskova, E. Preble, and E.R. Evans, Appl. Phys. Lett. 100, (01). 11. Y.F. Chen, H. Lu, D.J. Chen, F.F. Ren, D. Zhou, R. Zhang, and Y.D. Zheng, J. Vac. Sci. Technol. B 33, (015). 1. D. N. Nolte, J. Appl. Phys. 85, 659 (1999). 13. J.H. Leach, R. Metzger, E.A. Preble, and K.R. Evans, Proc. SPIE 865, 8651Z (013). 14. A. Rosen and F.J. Zutavern, High-Power Optically Activated Solid-State Switches, Artech House (1994). 15. J.I. Pankove, Optical Processes in Semiconductors, Dover, New York, D.L. Rode, Semiconductors and Semimetals 10, 1 (1975). FIGURE CAPTIONS Figure 1 (color online). Theoretical values of ured reflectance R as a function of αd for various values of the reflection coefficient R. Figure (color online). Comparison of the reflection coefficient R (solid red line) with the ured reflectance R (open red squares) as a function of energy E in Fe-doped GaN. Also shown are the ured transmittance T (open black triangles) and absorbance A (open blue circles). The feature at E 1.4 ev is due to a source change in the spectrometer. Figure 3 (color online). Absorption coefficient vs energy E for Fe-doped GaN determined by the exact method (α) and two approximate methods (α 1 and α ). The, α 1, and α curves artificially saturate above 3.3 ev because the detector (photomultiplier) noise magnitude becomes larger than the T signal magnitude. 1