analysis of the drude model in metallic films

Analysis of the Drude model in metallic films

H. Y. Li, S. M. Zhou, J. Li, Y. L. Chen, S. Y. Wang, Z. C. Shen, L. Y. Chen, H. Liu, andX. X. Zhang

A method, believed to be new, to simulate Drude parameters for collective oscillation of the free carriersin metallic films is proposed. Plasma resonance frequency and relaxation were simulated simulta-neously from both the real and the imaginary parts of the dielectric function of a metallic film afterconsideration of their correlation in the Drude model. As examples, the contributions of the electrons inAg films and of the free carriers in metallic silicide, NbSi2 and TaSi2, films have been studied. © 2001Optical Society of America

OCIS codes: 310.0310, 300.0300.

1. Introduction

In the past few decades, the optical properties ofmetal films have been studied extensively. Amongoptical methods, such as reflectivity, absorption, andtransmission, scanning ellipsometry has become oneof the most powerful tools for studying the opticalproperties and electronic structures of solid materialsbecause it gives the dielectric function directly withoutthe Kramers–Kronig transformation. The dielectricfunction can be divided into contributions of intrabandand interband transitions from the free and the boundelectrons, respectively. The region far below the in-terband transition can be assumed to be nearly free-electron �or carrier� range. According to the Drudemodel, the dielectric function of free electrons, ε � ε1 �iε2, can be expressed as follows1:

ε1 � εb ��

2P

�2 � �2 , (1)

ε2 ��

2P�

�3 � ��2 , (2)

where εb refers to the dc dielectric function derivedfrom the core polarizability and an interband contri-

bution; �P ��Ne2�ε0m*�1�2� is the plasma frequency,where m* is the optical mass of free electrons; and �is the reciprocal relaxation time. Here we take �1. From Eqs. �1� and �2�, � can be expressed asfollows2,3:

� ��ε2

εb � ε1. (3)

The parameter εb is important to the frequency de-pendence of the reciprocal relaxation time. If εb isknown, the frequency dependence of � can be calcu-lated. In some experimental papers2–4 the recipro-cal relaxation was found to have the form � � �0 ��2 and tentatively accounted for the frequency de-pendence of � on the two-carrier effect2 or electron–electron collisions5 in the framework of the Fermi-liquid theory of Landau.5 When � �� , one can getapproximate equations:

ε1 � εb ��

2P

�2 , ε2 ��

2P�

�3 . (4)

In fact, the condition can hold for the noble metals Au,Ag, and Cu, and thus the above approximation is oftenused to describe the collective oscillation of conductionelectrons in noble metals in the low-energy range.According to Eqs. �4�, the real part ε1 will change lin-early with ��2. The slope yields the value of theplasma resonance frequency and the effective opticalmass. In optical studies of noble metals, Johnson andChristy6 calculated the plasma resonance fre-quency and the effective optical mass of free elec-trons from a plot of ε1 versus 2 and obtained thephoton-energy-independent reciprocal relaxation en-ergy from a plot of ε2� versus 2. Theye4 found thefrequency-dependent reciprocal relaxation of semi-transparent Au films by using Eq. �2�. The reciprocal

H. Y. Li, J. Li, Y. L. Chen, S. Y. Wang, Z. C. Shen, and L. Y. Chenare with the Department of Optic Science and Engineering, FudanUniversity, Shanghai 200433, China. S. M. Zhou �[email protected]� is with the Surface Physics Laboratory �National KeyLaboratory� and Department of Physics, Fudan University, Shang-hai 200433, China. H. Liu and X. X. Zhang are with the Depart-ment of Physics, Hong Kong University of Science and Technology,Clear Water Bay, Kowloon, Hong Kong, China.

Received 10 October 2000; revised manuscript received 30 May2001.

0003-6935�01�346307-05$15.00�0© 2001 Optical Society of America

1 December 2001 � Vol. 40, No. 34 � APPLIED OPTICS 6307

relaxation was neglected in the calculations of theplasma frequency. Obviously the plasma frequencyand the reciprocal relaxation were obtained separatelyfrom the real and the imaginary parts in the researchreported in Refs. 4 and 6. These methods have twoshortcomings. First, they are suitable only for sys-tems whose reciprocal relaxation is much smaller thanthe photon energy. One cannot employ them to ana-lyze the optical properties of transition metals, theiralloys, and metallic silicides, which have large recip-rocal relaxation. Second, this method is ambiguousin physics because the plasma frequency and the re-ciprocal relaxation should be correlated to each otherin Eqs. �1� and �2�. In studies of the collective oscil-lation of free electrons in metals, Nagel and Schnat-terly2 assumed that εb � 1, which is not always true inexperiments. Until now, we have seen no calculationof the plasma resonance frequency and the reciprocalrelaxation together from both the real and the imagi-nary parts. Thus a new, more reasonable physicalmethod method needs to be developed. In this paperwe propose a method for calculating plasma frequency,effective optical mass, and reciprocal relaxation that isa modification of the previous method. In our newmethod the plasma frequency and the effective massare assumed to be photon-energy independent and therelaxation energy is assumed to be a real function ofphoton energy.7,8

An �200-nm-thick Ag film was prepared upon aSi�100� substrate by electron evaporation. Thecharacteristic microstructure of these films was de-fined by x-ray diffraction in a wide angle range. TheAg film was found to be polycrystalline, with a latticeconstant of 0.409 nm. The dielectric function wasmeasured by scanning spectroscopic ellipsometry inthe visible and near-infrared ranges.9,10

2. New Analysis of the Drude Modeland Its Applications

It is obvious that both the real and the imaginaryparts of the dielectric function increase dramaticallywith decreasing photon energy in the infrared regionand that the dielectric function in the low-energyrange is more sensitive to the Drude parameters thanthat in the high-energy range. For comparison, datafor the dielectric function are displayed in Fig. 1,including measured data for the evaporated Ag filmprepared in the authors’ laboratory and data for an-other Ag film taken from Ref. 11. The difference inthe dielectric function between the two samples in-creases as the photon energy decreases. To obtainmore accurate values of the plasma frequency, theeffective optical mass of free electrons, and the recip-rocal relaxation, one might more profitably study theoptical properties of materials in the infrared region,far below the onset of the interband transition.Moreover, the penetration depth in the infrared re-gion becomes larger than that in the visible region,although extinction coefficient k increases dramati-cally with decreasing photon energy. In this case,the measured spectra can reflect the intrinsic physi-cal properties of the films because the effect of the

surface decreases with a further decrease of photonenergy. Normally the infrared region is far frominterband transitions and can be thought of as beingnearly the free-electron region, where one can use theDrude model to describe the collective oscillation offree carriers or free electrons in metallic materials.As is well known, the far-infrared region can also beused for study of the optical properties of carriers insemiconductors and semimetals.

The reciprocal relaxation of the noble metals Au,Ag, and Cu is �1 order of magnitude smaller than thephoton energy in both the visible and the infraredregions, and thus the real and the imaginary parts ofthe dielectric function are linear functions of ��2 and��3, respectively. In fact, we can use Eqs. �4� tocalculate the initial value of εb from the plot of ε1versus ��2 in the visible range. With the initialvalue of εb and the measured spectra of the dielectricfunction from the visible range to the infrared region,one can get the frequency-dependent reciprocal relax-ation energy from Eq. �3�.

Conversely, if the reciprocal relaxation is known,the plasma frequency and εb can be obtained from Eq.�1�. This process can be repeated. Finally, accu-rate values of the plasma frequency and the recipro-cal relaxation can be obtained. From abovediscussion, it is apparent that this process has over-come the shortcoming of previous methods, i.e., thatthe plasma frequency and the reciprocal relaxationare obtained separately from the real and the imag-

Fig. 1. Spectra of the dielectric function of evaporated Ag and thedata taken from Ref. 11. Inset, the absorption coefficient of Ag inthe infrared region.

6308 APPLIED OPTICS � Vol. 40, No. 34 � 1 December 2001

inary parts.4,6 Here the reciprocal relaxation andthe plasma frequency are correlated to each other.

Figure 2 shows the simulation results for the cor-responding data in Fig. 1. For both the evaporatedAg film and that described in Ref. 11, the real part ε1is a linear function of 1��2 � �2�, and ε2 is propor-tional to ��3 � ��2�. It is interesting that theplasma frequencies from the slopes of both the realand the imaginary parts are the same for both Agfilms. Plasma frequency �P is 7.9 and 7.6 eV, re-spectively, for these samples. In comparison, the in-set of Fig. 2 shows that the real part ε1 of theevaporated Ag film is a linear function of ��2 andthat �P � 8.4 eV. It is apparent that the plasmafrequency with the new method is smaller than thatobtained from the conventional method. Althoughε1 can also be a linear function of ��2, this does notmean that the corresponding value of the plasmafrequency in the inset is correct.

For transition metals or other metallic materials,such as the metal silicides NbSi2 and TaSi2, the re-ciprocal relaxation is large compared with that of thenoble metals and in some cases even larger than thatof the photon energy. In this case, reciprocal relax-ation cannot be neglected in Eqs. �1� and �2�, and thusthis simulation procedure cannot be used to simulatethe Drude parameter from the spectra of the dielec-tric function in metallic silicides. Here one can ad-just the value of εb to make the plasma frequenciesinferred from the real and the imaginary parts asnearly alike as possible. In this way, one can findthe plasma frequency and the frequency dependent

reciprocal relaxation. As an example, we shallstudy the optical properties of NbSi2 and TaSi2 films.This process is also suitable for the noble metals.

Figures 3�a� and 3�b� show the real and the imag-inary parts of the dielectric functions of NbSi2 andTaSi2 films versus 1��2 � �2� and ��3 � ��2�,respectively. Here the data of single-crystal NbSi2and TaSi2 were taken from Ref. 12, in which there areinterband transitions near 1.0 eV. In the above cal-culations the upper limit of the Drude model is set as0.1 eV, which is far from the onset of the interbandtransition. According to the slopes of the real andthe imaginary parts, one can find the plasma reso-nance frequencies �P � 2.3 eV for NbSi2 film and�P � 2.07 eV for TaSi2 film. Obviously the plasmafrequencies of metallic silicides are smaller thanthose of noble metals. This is so mainly because thedensity of the free carriers in the metallic silicides isless than that in noble metals, as one can see from thehigh resistivity of metallic silicides. εb is an impor-tant parameter in this simulation procedure. Forthese two materials it is large compared with that innoble metals, as was pointed out before.12 It shouldbe noted that, when � � 0.1 eV, real part ε1 is not alinear function of ��2 at all, as shown in the insets ofFig. 3. This means that compared with the photonenergy, the reciprocal relaxation cannot be neglected.After the reciprocal relaxation is taken into account,one can get the linear function.

As is shown in Fig. 4, reciprocal relaxation � inas-deposited Ag polycrystalline films and single-crystalline silicide films increases as a linear func-

Fig. 2. Real and imaginary parts of the �a�, �b� dielectric function of evaporated Ag film and of �c�, �d� the data taken from Ref. 11 versus1��2 � �2� and ��3 � ��2�, respectively. Inset, the real part ε1 of the dielectric function of evaporated Ag film versus ��2 in the visibleregion.


tion of �2, indicating that the electron–electroninteraction in these metallic films is the mainsource of the reciprocal relaxation energy.5 Thelinear dependence was also observed by Gugger etal.3 The reciprocal relaxation depends in fact onthe composition and preparation of the film. For

example, the reciprocal relaxation of metallic sili-cides is expected to be much larger than that of Agfilms at the same photon energy. Thesputtered-Ag film has a larger reciprocal relaxationthan the evaporated film. This result coincideswith the simulation results that the plasma fre-quencies of sputtered and evaporated Ag films aredifferent �Fig. 2�. Moreover, whereas � is a linearfunction of �2 in as-deposited Ag films �Fig. 4�, thelinear dependence in other cases can be found onlyafter a suitable annealing process.2

In summary, a new simulation method has beenproposed in which the plasma frequency and the re-ciprocal relaxation are inferred simultaneously fromboth the real and the imaginary parts of the dielectricfunction, thereby yielding a more reasonable physicalmethod for determining the optical properties ofmetal films. Differently from previous ones, thismethod is suitable not only for noble metals with lowreciprocal relaxation energy but also for metallicfilms with large reciprocal relaxation energy, such asmetallic silicides, transition metals, and their alloys.As examples, we have studied the collective oscilla-tion of carriers and electrons in Ag, NbSi2, and TaSi2films in the framework of this model. In the presentanalysis, εb is a crucial parameter.

This study was supported by the National ScienceFoundation of China, the Shanghai Research Centerfor Applied Physics �SRCAP�, the Science and Tech-nology Committee in Shanghai, and the EducationCommittee in Shanghai.

Fig. 3. Real and imaginary parts of the dielectric functions of �a�, �b� NbSi2 and �c�, �d� TaSi2 films versus 1��2 � �2� and ��3 � ��2�,respectively; the data were taken from Ref. 12. Insets, ε1 versus ��2 in the visible region.

Fig. 4. Reciprocal relaxation versus �2 for �a� the Ag films and �b�the NbSi2 and TaSi2 films.

6310 APPLIED OPTICS � Vol. 40, No. 34 � 1 December 2001

References1. G. Harbeke, “Optical properties of metals,” in Optical Proper-

ties of Solids, F. Abeles, ed. �North-Holland, Amsterdam,1972�, Chap. 3, p. 103.

2. S. R. Nagel and S. E. Schnatterly, “Frequency dependence ofthe Drude relaxation time in metal films,” Phys. Rev. B 9,1299–1303 �1974�.

3. H. Gugger, M. Jurich, and J. D. Swalen, “Observation of anindex-of-refraction-induced change in the Drude parameters ofAg films,” Phys. Rev. B 30, 4189–4195 �1984�.

4. M. L. Theye, “Investigation of the optical properties of Au bymeans of thin semitransparent films,” Phys. Rev. B 2, 3060–3078 �1970�.

5. R. N. Gruzhi, “Mutual electron correlations in metal optics,”Sov. Phys. JETP 8, 673–675 �1974�.

6. P. B. Johnson and R. W. Christy, “Optical constants of thenoble metals,” Phys. Rev. B 6, 4370–4379 �1972�.

7. P. E. Sulewski, A. J. Sievers, M. M. Maple, M. S. Torikachili,

J. L. Smith, and Z. Fisk, “Far-infrared absorptivity of UPt3,”Phys. Rev. 38, 5338–5352 �1988�.

8. J. W. Allen and J. J. C. Mikkelsen, “Optical properties of CrSb,MnSb, BiSb and NiAs,” Phys. Rev. B 15, 2952–2960 �1977�.

9. G. Q. Xia, R. J. Zhang, Y. L. Chen, H. B. Zhao, S. Y. Wang, S. M.Zhou, Y. X. Zheng, Y. M. Yang, L. Y. Chen, J. H. Chu, and Z. M.Wang, “New design of the variable angle infrared spectroscopicellipsometer using double Fourier transforms,” Rev. Sci. In-strum. 71, 2677–2683 �2000�.

10. L. Y. Chen, X. W. Feng, Y. Su, H. Z. Ma, and Y. H. Qian,“Improved rotating analyzer–polarizer type of scanning ellip-someter,” Thin Solid Films 234, 385–389 �1993�.

11. D. W. Lynch and W. R. Hunter, “Comments on the opticalconstants of metals,” in Handbook of Optical Constants ofSolids, E. D. Palik, ed. �Academic, New York, 1985�, pp. 350–357.

12. G. Guizetti and F. Marabelli, “Optical properties,” in Proper-ties of Metal Silicides, K. Maex and M. van Rossum, eds.�INSEC, London, 1995�, Chap. 6, p. 238.


analysis of the drude model in metallic films

Documents