2010) 1( 23مجلة ابن الھیثم للعلوم الصرفة والتطبیقیة المجلد دراسة نظریة لتأثیر درجة حرارة المحیط على معامل األمتصاص ،میسون فیصل احمد *محمد عبد الرضا حسین قسم الیزر والبصریات ،الجامعة التكنلوجیة * قسم الفیزیاء، كلیة العلوم ،جامعة بغداد الخالصة معادلة ریاضیة جدیدة لوصف العالقة بین معامل الخمود او معامل االمتصاص و درجة حرارة الوسط تاقترح لقد اشتقت هذه المعادلة اعتمادا على . ة ضمن مجال االشعة تحت الحمراءلمعالمحیط والطول الموجي لبعض المواد المست قیم لارجیة مختلفة التي تصف العالقة بین قیم النفاذیة دالة للطول الموجي بعض المنحنیات العملیة المأخوذة من مصادر خ ان الدراسة المكثفة و المتعمقة لطبیعة هذه المنحنیات و اسلوب تغیرها مع كال من . من درجات حرارة المحیطعدیدة وضع معادلة تصف تغیر اینشتاین االحصائي قادت الى - واعتمادا على توزیع بوز الطول الموجي و درجة الحرارة ان االفتراض ، تغیر النفاذیة مع درجة حرارة المحیط من ثممعامل الخمود دالة للطول الموجي و درجة حرارة المحیط و االساسي في عملیة ایجاد هذه المعادلة هو اعتبار ان التغیر الرئیس في النفاذیة یعود الى التغیر في قیم معامل الخمود ان النتائج النظریة للمعادلة المقترحة اعطت تطابقا جیدا مع مجمل النتائج . حرارة الوسط الخارجينتیجة لتغیر درجة .العملیة التي تمت دراستها في هذا البحث IBN AL- HAITHAM J . FO R PURE & APPL. SC I VO L. 23 (1) 2010 Theoretical Study o n the Effect of Ambient Temperature on Absorption Coefficient M. A. Hussain and M.F. Ahmed Laser and Optoe lectronic Departme nt, Unive rsity of Technology. Physics Departme nt, Science College, Unive rsity of Baghdad. Abstract Anew mathematical formula was p rop osed to describe the behavior of the extinction coefficient as a function of ambient temp erature and wavelengths for some of infrared materials. This formula was derived depending on some exp erimental data of transmittance sp ectrum versus wavelengths for many ambient temp eratures. The extensive st udy of the sp ectrum characterist ics and depending on Bose-Einst ein distribution led to derive an equation connecting the extinction coefficient or the absorp tion coefficient with the ambient temp erature and wavelengths of t he incident rays. T he basic assump tion in deriving p rocess is the decreasing in transmittance value with the increasing temp erature which is only due to the changing in extinction coefficient values. Introduction M any infrared transp arent materials, esp ecially those that are st rongly absorbing materials in the visible region, have high indices of refraction in the infrared region [1]. This is esp ecially true for semiconductors such as Ge, InAs, InSb, which are widely used in the infrared as windows, lenses, and long wavelength p ass filters. Their high refractive indices cause large reflection losses so t hat even thin non absorbing p lates of these materials transmit only 50% or less of the incident radiation. A t ransp arent lay er of another material such as SiO or ZnS to p roduce zero reflectance can coat these materials [1]. The design of any op tical sy st em requires the selection of materials based up on knowledge of the op tical, mechanical and thermal p rop erties available. A st udy of the material characterist ics, p articularly the absorp tion and disp ersion p rocesses, is therefore essential for the selection of suitable materials for use both as substrates and evap orated lay er materials. All of the observed intrinsic absorp tion characteristics p resent in the sp ectrum of an infrared op tical material can be classified by three fundamental p rocesses involving the interaction between the material and the incident electromagnetic radiation, namely ; electronic absorp tion, latt ice or phonon absorp tion and free-carrier absorp tion. All these processes which are affected by the ambient temp erature would make the absorp tion or extinction coefficient and then the resulting transmittances decrease as the temperature increases. This decrease in transmittance with the increasing temp eratures is due to the changing in values of both refractive index and extinction coefficient or absorp tion coefficient, but since the exp erimental results illustrate that the change in the refractive index of both the coating and substrate materials is small comp aring with the change of the extinction coefficient so it can be assumed that the transmittance decreases entirely due to an increase of absorp tion coefficient or extinction coefficient and not t o refractive indices. IBN AL- HAITHAM J . FO R PURE & APPL. SC I VO L. 23 (1) 2010 In this p aper, we will p rop ose a new mathematical formula for the relationship between the extinction coefficient of some semiconductor materials and both ambient temp erature and incident wavelength. General description of absorption The electronic absorption characterist ics observed towards the higher frequency end of the infrared sp ectrum are the result of interaction between the incident radiation and the motions of electrons or holes within the material [2]. Only electromagnetic radiation with sufficient energy to cause an electron to transfer between the valence band and conduction band (hf) will be absorbed by this mechanism. The various transitions of these electrons define the p osition of the short wavelength absorption edge. The resulting sp ectrum provides information on the width of the ener gy band gap of the material, and through sp ectral anomalies, can indicate the presence of impurities [3]. The lattice absorp tion characterist ics observed at the lower frequency regions, in the middle to far-infrared wavelength range, define the long wavelength transp arency limit of the material, and are the result of the interactive coupling between the motions of thermally induced vibrations of the constituent atoms of the substrate cryst al lattice and the incident radiation. Hence, all materials are bounded by limiting regions of absorp tion caused by atomic vibrations in the far-infrared (>10µm), and motions of electrons and/or holes in the short - wave visible regions. In the inter band region, the frequency of the incident radiation has insufficient energy ( E=hf ) to transfer electrons to the conduction band and cause absorp tion; here the material is essentially loss-free. In addition to the fundamental electrons and lattice absorp tion p rocess, free carrier absorp tion in semiconductors can be p resent. This involves electronic transitions between initial and final st ates within the same energy band. The absorp tion or emission of the resulting p hotons is accomp anied with by ascattering by op tical or acoust ic-mode p honon vibrations or by charged imp urities. This ty p e of absorp tion is evident where the sp ectral p rofile of the material is highly absorbing, p roducing considerably lower transmission than otherwise exp ected. These intrinsic absorp tion prop erties of semiconductors and insulators define the transp arency of the material. To be transmitt ed in the region between the electronic and lattice absorp tion, the incident radiation must have a lower frequency than the band-gap ( ��) of the material. This is defined by the short wavelength semiconductors edge at ( � = ℎ� /��� ), p reventing electrons t ransferring to the conduction band. The generalized p rofile of the electronic edge is known as the Urbach tail (4) where the exp onentially increasing absorp tion coefficient follows the general relationship : �( �� �) � (��) = � ��� …………………………………………………………. (1) ( kB ) is Boltz mann constant, ( h ) is Blank constant, ( f ) is the incident radiation frequency, and ( T ) is t he temperature. The concept of temp erature and thermal equilibrium associated with cryst al solids are based on individual atoms in the sy st em p ossessing vibration motion. The classical theory of thermal energy by atomic vibrations, thought p roviding suitable exp lanations at elevated temp erature, has p roved unsatisfactory at reduced temp eratures. Quantum mechanics has subsequently p rovided theories based upon st atist ical p robabilities t hat have provided p ossible mechanisms to exp lain some of the observed p henomena. A sy st em of vibrating atoms in a cryst al is highly comp licated, and beyond any realizable theoretical methods of analy sis or IBN AL- HAITHAM J . FO R PURE & APPL. SC I VO L. 23 (1) 2010 calculations to verify sp ectral measurements from the total thermal energy of a cryst alline substrate. For a sy st em of distinguishable p articles, the p robability st atist ics of the energy (E) of the sy st em are described by M axwell-Boltz mann general equation[5-7]: ��� = �� � � �� � ………………………………….…………………… (2) If p articles are indist inguishable they are divided into t wo t y p es: (i) Electrons; which are subject to the Pauli Exclusion Princip le[5-7] and obey Fermi-Dirac st atist ics ��� = � � ��� � ��� �� ………………………………………………………… (3) (Ef) is Fermi energy . (ii) P Photons and p honons which are defined by Bose-Einst ein st atist ics ��� = � � ��� �� ��� ……….….…….…………….…………………… (4) (α) is a normalizing constant, adjust ed so that the total p robability is equal to unity when each function is summed over all the energy st ates available. For the electronic absorp tion edge, the effect of increasing temp erature on the forbidden energy gap reduces the energy gap shifting the edge p osition to short er wavelengths. This shift was fitted for various materials by the following emp irical relationship[5]: �� (�) = �� (0) − ��� � �� ……………..….………….…………………… (5) Where �� (0) is the value of the energy gap at z ero Kelvin and α and β are constants. Experimental results (Single-Layer Coating) In this research, we dep end on two ty p es of exp erimental data: (1) The transmission sp ectrum [1] of the semiconductor materials Ge, InAs and InSb which have refractive indices near absorp tion edge 4.1, 3.4 and 4.0, resp ectively. In the near infrared region, the exp erimental and theoretical results show that SiO film is the most suitable antireflection coating for Ge, and InAs while ZnS film was found most suitable as a single lay er antireflection coating for the (7-15) µm. The transmittances versus wavelength for different temp eratures of one lay er coatings are obtained for the following cases: a) InAs p late (thickness 0.21 mm) with coating consists of SiO with quarter wavelength thickness (at 5.75 µm) as shown exp erimentally in Fig. (1-A). b) Ge p late (thickness 2.0 mm) with coating consists of ZnS with quarter wavelength thickness (at 9.80 µm) as shown exp erimentally in Fig. (1-B). c) InSb p late (thickness 0.08 mm) with coating consists of ZnS with quarter wavelength thickness (at 10.8 µm) as shown exp erimentally in Fig. (1-C). All these figures show that the transmittance decreases st rongly with increasing ambient temp erature, while the maximum peaks of transmittance are shifted towards t he high values of wavelength. IBN AL- HAITHAM J . FO R PURE & APPL. SC I VO L. 23 (1) 2010 (2) The exp erimental st udies [8-10] of some other materials like (PbS, PbSe, and PbTe) which represent the relation between the thermal energy and the p hoton energy at minimum absorp tion as shown in Fig.(2). Both of these exp erimental data are used to evaluate a theoretical equation for the extinction coefficient as a function of ambient temp erature and incident wavelength. Theoretical Study The extensive st udy of the exp erimental curves of transmittance behaviour versus wavelengths and temp erature (Fig.1) gives: (1) The extinction coefficient for the case of constant ambient temp erature has the mathematical form: � = �� � ��(����� ) � …………………………………………………… (6) (a0, a1, a2) are constant p arameters for wavelength. (2) The extinction coefficient for the case of constant frequency or wavelength of the incident radiation has the mathematical form: � = �� � �� �� ��� � ��(������ ) � ………………………………………………. (7) (b0, b1, b2, b3) are constant p arameters for temp erature dep ending on the incident wavelength and the material nature. (3) The combination of both cases to find the general case of variable frequency and temp erature will then has the mathematical form: � = � � �� �� ��� � ��(������� ���) � …………………………..………………………. (8) (  ) is a const ant which dep ends on the material ,the p hy sical meaning of the parameter  is the ratio between the deviations in p hoton energy at maximum transmittance (minimum absorp tion coefficient) t o heat energy due to t he relation: � = − ∆(�� ) ∆(�� �) …………………………..………………………………………. (9) From the above equation, it is obvious that the minimum value of extinction coefficient (minimum absorp tion) occurs when: �� − ��� � − ℎ� = 0 ………………………………….….…….…………… (10) This result imp lies that the energy of the incident light, at maximum transmittance or minimum absorp tion depends linearly on the heat energy . This is a real fact which can be shown via the curves in (Fig.2) for the materials (PbS, PbSe and PbTe). The parameter c3 has a relationship with the melting temp erature of the substrate (Tm), it equals to: �� = 2� ��� ……………………………………………….………..….…………….. (11) By using the general form of multilay er sy st em equation [11], one can obtain the values of the extinction coefficient that fit the exp erimental data of transmittance sp ectrum, it was found that t he parameter c1 has also a relation with melting temp erature, it equals to: IBN AL- HAITHAM J . FO R PURE & APPL. SC I VO L. 23 (1) 2010 �� = 2� ��� ………………………………………………….……………………….. (12) While both of p arameter  and  depending on the ty p e of material. Table (1) shows these values for the chosen materials. The parameter c2 was found to depend on the thickness of coating lay er and gap energy of the substrate lay er, it has the form: �� = ��� ���� � ………………………………………………………….. (13) (n and d) are the refractive index and the thickness of the coating lay er, (Eg) is the energy gap of the subst rate. Then the new formula of the extinction coefficient has t he form: � = � � �� �� � �� � � …………………………………………………….…….. (14) � = 4� � �� ( �� ����� ��� ) �� � ……………………………………….……… (15) Re sults and Discussion (1) The theoretical st udy for the behaviours of the exp erimental transmittance sp ectrum of the substrate materials Ge, InAs and InSb gives a value of 1.0 for the p arameter  which means that t he heat energy equals to (kBT), while it’s p ositive value means that t he maximum p eak of transmittance (λm ) increases with t he increase of the temp erature. The theoretical studies of the materials (PbS, PbSe, and PbT e) give a value (-3.5) for the p arameter Ω, which means that the heat energy equals to ( � � ���), while the minus value means that t he maximum p eak of transmittance decreases with t he increase of the temp erature which is a different behaviour comp aring with the semiconductor materials. The maximum transmitt ance occurs at the minimum value of the extinction or absorp tion coefficient, the wavelength λm at which minimum extinction coefficient can be obtained by letting the derivative of k in equation [13] with resp ect to λ for constant t emp erature equals to zero, that will give the result: ℎ�� = 2�� �� − ��� � ………………..……….……………………. (16) From the above equation, the maximum transmittance wavelength can be obtained: �� = �� ��(������) ………………………………………………………………..….. (17) Equation (17) has imp ortant conclusion st ates that the increasing of temp erature led to increase the value of λm when  has p ositive values and decrease the value of λm when  has negative values. Since the values of temp erature are bounded between zero and Tm that means the values of λm are bounded between two values: ��(���) = �� ����(�� �) ………………………………………………… (18) IBN AL- HAITHAM J . FO R PURE & APPL. SC I VO L. 23 (1) 2010 ��(���) = �� ����� ………………………………………………….. (19) (2) Equation (13) is a new semi-emp irical equation evaluated from the exp erimental values of the transmitt ance versus wavelength for different values of temp eratures. By using the general equation of multilayer sy st em, and introducing the value of refractive index, as a const ant and extinction coefficient according to equation (13), one can obtain the theoretical transmittance. The results of theoretical transmittance are nearest to the exp erimental values within an average error less than 10.0% as shown in the case of the semiconductor substrate materials discussed above. The theoretical and exp erimental results are shown in figures (3). The agreement between the theoretical and exp erimental results are good for InSb, but there is a shift in the maximum values of transmittance for InAs, while for Ge substrate, the agreements are good excep t for the temp erature 398 K in which the theoretical transmittances are lower within less 10% comparing with the exp erimental values. (3) The melting temp eratures (Tm ) for lead comp ound (PbS, PbSe, and PbT e) are (1387, 1338, and 1190) K, and the thicknesses of the samp les are (1.25, 0.68 and 0.35) mm, resp ectively. Figures (2) represent the theoretical and exp erimental values of minimum absorp tion wavelength at different temp eratures. The agreements are good for PbS and PbT e, but for PbSe there is a shift between the theoretical and exp erimental curves due to the value of Em which is related to melting temp erature Conclusions The final semi-emp irical formula for the extinction coefficient was derived depending on transmittance sp ectrum of some semiconductor materials. The resulting formula has a good accuracy comp aring with the exp erimental results of the transmittance sp ectrum for the materials InAs (coated with SiO), Ge (coated with ZnS) and InSb (coated with ZnS), also the agreement is good for the exp erimental results of incident p hoton energy versus heat energy at minimum absorp tion coefficient for the materials PbS, PbSe and PbT e. The use of the mathematical model depends on Bose-Einst ein st atist ics and on behaviour st udy of the exp erimental curves through using fitt ing theory. The need is st ill up to st udy more materials to reach an exact equation for the extinction coefficient as a function of wavelength and ambient temp erature. Re ferences 1. Cox,J.T . and Hass. (1964) ‘Antireflection Coatings’ in Phy sics of Thin Films, New York: Academic, 2. Gary , J. Hawkins, (1998) “Sp ectral Characterization of Infrared Op tical M aterials and Filters’, PhD t hesis, university of Reading, Dep artment of Cy bernetics, 3. Sze,S.M . (1981)”Phy sics of Semiconductors Devices”, Second Edition, John Wiley and Sons 4. Urbach,F. (1953).Phy sical Review, 92:1324 5. Pankove, J.I. (1971).”Optical p rocesses in semiconductors”, Dover Publications,Inc. ISBN 0-486-60275-3, pp 35 6. Houghton,J.T . and Smith, S.D. (1966).”Infrared Phy sics”, Oxford University Press, pp 92 7. Rosenberg,H.M .” (1978). The Solid State”, Oxford University Press, pp 99, 8. Ravindra,N.M . ; Auluck, S. and Srivast ava, V.K. (1970)Phy s.State.Sol., (a)52,k151 9. Gibson,A.F. (1952 )“ The absorp tion sp ectra of single cryst al of lead sulphide solenoid and telluride”,Proc.Phy s.Soc. B65:378-387 10. Smith,R.A. (1953)”Advance in Phy sics” 2:321-368 11. Berning, P.H. (1963)” Theory and calculation of op tical thin-film”, in phy sics of thin films, G. Hass, New York: Academic 1: 69-121. IBN AL- HAITHAM J . FO R PURE & APPL. SC I VO L. 23 (1) 2010 A B C Fig. (2): Comparison betwee n the present work and experime ntal resul ts of i ncident photon ene rgy versus he at ene rgy at minimum absorption coefficie nt for: The smooth line represe nts the the oreti cal results. The dotted line represe nts the experime ntal data. IBN AL- HAITHAM J . FO R PURE & APPL. SC I VO L. 23 (1) 2010 0.000 0.005 0.010 0.015 0.020 Heat Energy (1/2 K T ) (eV) 0.0 0.1 0.2 0.3 0.4 0.5 P ho to n E ne rg y (h f) (e V ) 0.000 0.005 0.010 0.015 0.020 Heat Energy (1/2 K T ) (eV) 0.0 0.1 0.2 0.3 0.4 0.5 P h o to n E n e rg y ( h f ) (e V ) 0.000 0.005 0.010 0.015 0.020 Heat Energy (1/2 K T ) (eV) 0.0 0.1 0.2 0.3 0.4 0.5 P h ot o n E n er gy ( h f) ( eV ) A: PbS , B: PbS e, C: PbTe. Fig.(1): Experime ntal transmi ttance versus wavelength for different temperatures. A: for InAs substrate coated with S iO. B: for Ge substrate coated with ZnS . C: for InS b substrate coated with ZnS . IBN AL- HAITHAM J . FO R PURE & APPL. SC I VO L. 23 (1) 2010 4.0 5.0 6.0 7.0 8.0 wavelemgth (um) 0 10 20 30 40 50 60 70 80 90 100 T ra n s m it ta n c e % Te mp.= 298 K Temp.=373 K Te mp.= 413 K Te mp.=4 43 K 7 8 9 10 11 12 13 14 Wavelength (um) 0 10 20 30 40 50 60 70 80 90 100 T ra n s m it ta n c e % Tem p.= 298 K Temp.=398 K Temp.= 473 K Temp.=523 K 7 8 9 10 11 12 13 14 15 Wavelength (um) 0 10 20 30 40 50 60 70 80 90 100 T ra n s m it ta n c e % Temp.= 298 K Temp.= 378 K Temp.= 408 K Temp.= 453 K Temp.= 498 K A B C Table (1): The important paramete rs values of some materials IBN AL- HAITHAM J . FO R PURE & APPL. SC I VO L. 23 (1) 2010 M aterial gE eV mT K γ Ω Germanium (Ge) 0.69 1210 0.15 1.0 Indium Arsenide (InAs) 0.39 1216 0.64 1.0 Indium Ant imonide (InSb) 0.23 808 0.20 1.0 A B C Fig. (3): Comparison between the present work and the experimental transmi ttance versus wavelength for different temperatures. A: In As substrate, B: InS b substrate, C: Ge Substrate The smooth line represents the theoretical results. The dotted line represents the experimental results 4.0 5.0 6.0 7.0 8.0 Wavelength (um) 0 10 20 30 40 50 60 70 80 90 100 T ra n s m it ta n c e % Temp .= 2 98 K Te mp.=373 K Te mp.= 410 K Temp.=443 K 7 8 9 1 0 11 1 2 13 14 15 Wavelength (um) 0 10 20 30 40 50 60 70 80 90 1 00 T ra n s m it ta n c e % Temp.=298 K T emp.= 378 K Temp.=408 K Temp.= 453 K Temp.= 498 K 7 8 9 10 11 12 13 14 wavelength (um ) 0 10 20 30 40 50 60 70 80 90 100 T ra n s m it ta n ce % T emp. =298 K Temp.=39 8 K Temp.=4 73 K Temp. =523 K