Johnson - Optical Constants of Copper and Nickel as a Function of Temperature

7/28/2019 Johnson - Optical Constants of Copper and Nickel as a Function of Temperature

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P HYSICAL REVIEW B VOLU ME 11, NUMBER 15 FEBRUARY 1975Optical constants of copper and nickel as a function of temperatures

P. B. Johnson~ and R. W. ChristyDepartment of Physics and Astronomy, Dartmouth College, Hanover, eampshire 03755(Received 12 August 1974)

The optical constants were determined for copper and nickel from reflection and transmissionmeasurements on vacuum-evaporated thin films, in the spectral range 0.5-6.5 eV and at temperatures of78, 293, and 423 K. The imaginary part of the dielectric constant was nearly independent oftemperature for nickel, but for copper it increased with temperature in the intraband region below 2 eVand decreased above 4 eV in the interband region. Interpretation of the increase below 2 eV accordingto the Drude free-electron expression suggests a temperature and frequency dependence of the relaxationtime, which is not completely explained. The thermal behavior in the interband region can be largelyunderstood, however, if the zero-temperature theory of Williams, Janak, and Moruzzi is modified byincluding a Debye-%aller factor in transitions between nearly-free-electron-like bands.

I. INTRODUCTIONWe have previously measured the optical con-

stants of copper' and nickel as a function of photonenergy (0.5-6. 5 eV) at room temperature, andcompared the experimental values of &3, the imag-inary part of the dielectric constant, with theo-retical values from zero-temperature band-struc-ture calculations. ' Since the experimental peakswere generally lower and broader than the theo-retical ones, it is now important to have valuesfor the dielectric constants of copper and nickelas a function of temperature in order to completethe comparison with these calculations, which ne-glected atomic vibrations. In addition, accuratevalues will allow us to discuss the temperature-dependence mechanism; we shall propose that theobserved temperature dependence of e2 in the inter-band portion of the spectrum is mainly describedby the Debye-Wailer factor.Some mechanisms for explaining the tempera-ture dependence of &2 in metals have been summa-rized by Rosei and Lynch. ' Briefly, these mecha-nisms are volume thermal expansion (and, in thinfilms, shear strains resulting from different ther-mal-expansion rates in the substrate and sample),increased phonon population, Fermi-distributionbroadening, and shift of Fermi level. These ther-mal mechanisms cause small energy shifts in theband structure or Fermi level of the metals.These shifts in turn give small shifts in the posi-tion of peaks in the &2-vs-photon-energy curve,which are seen as changes in the magnitude of &2at, a given energy. In the case of copper, we havemeasured a relatively large decrease in && withincreasing temperature beyond 4 ev, and thesemechanisms are inadequate to account fully forthe observed changes.Therefore we consider another mechanism, 'the Debye-Wa11er factor, multiplying both the elec-

tron excitation energies (to account for energyshifts in &z) and the transition probabilities (to ac-count for changes in the magnitude of c,). TheDebye-%aller factor is a well-known thermal cor-rection for the potential-energy matrix elementswhich determine x-ray, neutron, or fast-electrondiffraction in a crystal, s and more recently forthe matrix elements of the pseudopotential used inthe calculation of band structures. ' In addition,however, the Debye-%aller factor will also appearas a correction for the momentum matrix elementswhich describe the probability of electron transi-tions between free-electron-like bands induced byphoton absorption. According to calculations byWilliams et aE. , a contribution to za of copperstarting at 4. 2 eV is the only one due to transi-tions between free-electron-like bands in copperor nickel. Therefore we will use the Debye-WaQerfactor applied to this transition to explain our mea-sured thermal decrease in the magnitude of a~ incopper, and the absence of a similar effect. innickel. This is, to our knowledge, the first evi-dence for the appearance of the Debye-%aller fac-tor in the magnitude of &3 through the momentummatrix elements, although it has previously beenrecognized as a possible cause of a shift in thepeak position through the potential-energy matrixelements and band gaps. 'Although earlier determinations of the dielec-tric constants as a function of temperature agreeas to the general behavior of the values with tem-perature, they do not agree as to the actual magni-tudes of the dielectric constants. These earlierdeterminations used the ellipsometric method;disagreement in the magnitudes of the dielectricconstants may be due to problems of sample prepa-ration, contamination (oxidation of the metal onheating or condensation of water vapor on coolingthe samples), or the cryostat windows, sincemethods which depend on the phase change on re-


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P. B . JOHNSON AND R. W. CHRIST Yflection are very sensitive to these factors. Inaddition, all of these investiga, tions were over amore limited spectral range.We have inverted reflection and transmissionmeasurements made on thin evaporated films inorder to determine the optical constants at 78,293, and 423 K. In the free-electron region of thespectrum, we wiG use the Drude theory to analyzeour measurements on copper films, and discussthe applicability of a frequency-dependent relaxa-tion time. In the interband region we will developthe Debye-%aller factor as an explanation for theobserved temperature changes in E2. Finally, wecompare our experimental values with other ex-periments and state why we believe our measure-ments to be more accurate.

II. EXPERIMENTAL CONSIDERATIONSOur preparation of evaporated metal films and

the optical instrumentation have been describedpreviously. ' The evaporation was in a vacuumof about 10 6 Torr, at a rate of about 100A/sec.The transmission and reflection measurementswere made with a Unicam SP 700 double-beamspectrophotometer from 0. 5 to 6. 5 eV. Now wehave made measurements on copper and nickelfilms at 78, 293, and 423 K in a cold-trapped evac-uated cryostat.A. Cryostat

The cryostat consists of an outer brass vacuumcan and an inner thin-waQed stainless-steel cylin-der which serves as a reservoir for either liquidnitrogen or hot oil. A copper bar extending fromthe brass bottom surface of the reservoir holds thesample. The temperature of the sample was moni-tored by a thermocouple screwed to the massivecopper s~ple holder about 1 em away from thesample itself. The spectrophotometer light beampasses through two $-in. -thick quartz windows andthe sample holder. The sample is surrounded bya copper shield which is in good thermal contactwith the copper-bar sample holder. There areholes in the shield which allow the light beamto pass through the sample, but the sample isshielded in line of sight from all surfaces exceptthe quartz windows. The vacuum-system pres-sure with the cryostat at room temperature was7x 10 7 Torr. When the sample was cooled to78 K by pouring llquld nltxogen into the resex'volx'in the cryostat, the pressure decreased to 4 x10 7Torr. To heat the cryostat to 423 K hot oil waspoured into the reservoir and maintained at a con-stant temperature with an immersion heater; thepressure rose to Qx10 7 Torr.The cryostat has two necessary functions be-side controlbng the sample temperature: At coldtemperatures it inhibits water vapor from freezing

onto the sample (for which purpose the coppershield is essential), and at hot temperatures itinhibits the sample from oxidizing. Nevertheless,these two effects were still a problem over longtime periods. Reflection measurements werefound to be much more sensitive to both condensa-tion and oxidation than transmission measurements,which were reproducible over a period of timewell within our estimated instrumental accuracy.The reflection measurements were observed tochange over long time periods, but from the ob-served rate of change we concluded that changesdue to contamination could be kept within our esti-mated error, and that the thermal change in theref lectivity could be reliably determined.The measurements made in the cryostat must benormalized to absolute values of reflectance andtransmittance, because the windows of the cryostatintroduce multiple reflections and are also slightlyabsorbing. We have accurately measured the re-flectance and transmittance of copper and nickeloutside the eryostat at room temperature. ' Sincethe changes in reflectance and transmittance insidethe cryostat between room temperature and theother measured temperatures are small, we canrenormalize our measurements in the cryostat ac-cording to the ratio of measured room-temperaturevalues outside and inside the cryostat.

B. Evaluation of optical constantsThe optical constants n and k of copper and nickelat the different temperatures were obtained by in-verting normal-incidence reflection measured from

opaque films together with normal-incidence trans-mission measured from thinner semitransparentfilms, both in the cryostat but renormah. zed as ex-plained. The details of inverting reflection andtransmission measurements have been describedpreviously. ~'2 In this study we were able to simpli- .fy our method of determining the thickness of thesemitransparent films. To determine the filmthickness, we inverted the normal-incidence room-temperature transmission using our previously de-termined room-temperature values~'~ for n and k ateach of the 48 measured wavelengths across our en-tire spectral range. This method gave va.lues forthe film thickness at each of the measured wave-lengths which agreed to within + 3 A. Using theaverage of these values as the known thickness ofthe semitransparent filmwe were able to invertthe normal-incidence reflection and transmissionmeasurements at different temperatures to deter-mine the optical constants. Thermal expansion ofthe film thickness was negligible.

III. RESULTS AND DISCUSSIONThe complex index of refra, ction n =n+R of copperat 78, 293, and 423 K is listed in Table I. The


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OPTICAL CONSTANTS OF COPPER AND NICKEL AS ATABLE I. Optical constants for copper at 78, 293,and 423 K.

78 K 293 K 423 K0.640.770. 891.021.141.261.391.51l. 64l. 761.882. 012. 132.262.382.502. 632. 752. 873.003.123.253.373.493.623.743.873.994.114.244.364, 494.614. 734. 864. 985. 115.235, 355.485.605.725. 855.976.106.226.346.476.59

0.690.490.410.340.250.230, 220.200.190.170.200, 290.6Vl. 09l.221.271.29l.28l.271.321.35l.361.40l.411.40l.381.411.41l.411.451.56l.58l.561.54l.52l.50l.46l.40l.351.28l.22l. 161.13l. 081.071.05l. 02l.010.99

13.2110.999.3588.1927.1846.4015.7305.1564.6504. 1833.7283.1822.6342.5002.5462.5072.4272.3422.2462. 1442. 0541.9851.916l. 857l. 8151.7761.7511.7041.6501.5811.5471.5601.5851.609l.664l. 724l. 7711.7891.780l.7571.7281.690l.6421.598l.5541.498l.446l.3961.345

l.090.760.600.480.360.320.300.260.240, 210.220.300.701.02l. 181.221.25l. 24l.251.28l.321.331.361.371.36l.341.381.381.40l.421.45l.461.45l.41l.411.371.34l. 281.23l. 181.131.081.041.010.990.980.970.950.94

13.4311.129.4398.245V.2176.4215.7685.1804.6654.2053.7473.2052.7042.5772.6082.5642.4832.3972.3052.2072. 1162. 045l.975l.916l. 864l. 821l. 783l. 729l.679l. 663l.6631.6461.668l.691l. 741l. 783l. 7991.8021.7921.768l.7371.6991.651l.599l.5501.4931.4401.388ly 337

1.541.040. 800.640, 470.410.380.320.280.240.250.320.71l. 021.181.23l.241.23l.23l.271.301.301.331,341.341.32l.35l.36l.38l.38l.40l.41l.391.37l.34l.31l.27l. 22l. 17l. 12l. 071.020.990.940.920.910.900.900.91

13.8511.379.5698.3797.3026.4775.9285.2034.6904.2083.7533.2052.7672. 6252. 6272.5822.5122.4222.3322.2372.1442, 0752.0021.9411.8831.842l.7961.7401.700l.6741.676l. 6881.7061.730l.770l.793l. 805l. 8071.7951.V74l. 743l. 706l.661l. 597l.548l.4851.4411.3851.336

corresponding values of &~ =2' are plotted in Fig.1. Figure 2 shows the results in differential form:e,(423)~,(VB)42378

The results were obtained by inverting reflectionmeasurements from opaque films together with0transmission measurements on a 347-A semitrans-pax'ent film. Reflection measurements on two sepa-rate sets of opaque films and transmission measure-ments on two different simultaneously evaporated347-A films both agreed well.The index of refraction of nickel at room tempera-ture is listed in Ref. 2. Reflection and transmissionmeasurements made on nickel films were indepen-

9 l io

8-l ie

I~7!Il, ilI

- 78K293K423K

I I I I3 4 5PHOTON ENERGY (eV)

FIG. 1. Imaginary part of the dielectric constant forcopper at 78, 293, and 423 K vs the photon energy.dent of temperature (within the accuracy of ourspectrophotometer) over the temperature range78423 K. Our dielectric constants for nickel are

l

P) pO

2 '3 4 5PHOTON ENERGY (eV)

I6FIG. 2. Difference between &2 at 423 and 78 K, divided

by the temperature difference, from Fig. I..


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1318 P ~ B ~ JOHNSON AND R. W. CHRIST~

175Cu

20,CLI

'j25OJ 10 0 e

25 l

0 8 ()jm )FIG. 3. Negative of the real part of the dielectric con-stant for copper at 78, 293, and 423 K vs the square ofthe wavelength.7 (pm )FIG. 4. Imaginary part of the dielectric constant forcopper at 78, 293, and 423 K divided by the wavelength,vs the wavelength squared.

therefore identical to the previously measuredroom-temperature values.A. Comparison with theory

taken to be frequency dependent. ' Accordingly, wehave determined different values for r at each of themeasured frequencies in the free-electron range,from the slopes of straight lines drawn from the or-The complex dielectric constant and the complexindex of refraction are related as follows: na= c=e~+ie~, so that e~=n -4 and e~=2nk. The Drudefree-electron theory predicts values for the dielec-tric constants of metals for photon energies belowthe onset of interband transitions. In copper thisoccurs near 2 eV; however, in nickel interbandtransitions are dominant throughout our entire mea-sured range and the free-electron effects are neg-ligible. We have therefore observed the tempera-ture dependence of the optical constants in the free-electron range only for copper, but for both copperand nickel in the interband region.j. Free-electron region

According to Drude's free-electron theorye~((o) =1 (

SO

1

!Cu!

where &o~~ =4vNe~/mo. If ape 1 then values for theoptical mass m, could be determined from the slopeof a plot, over the free-electron range, of cz ver-sus Xa (Fig. 3); and then values for the relaxationtime v from the slope of a plot of ea/X versus Xa(Fig. 4). In the latter plot, although the points ac-tually lie reasonably close to straight lines, thebest lines do not pass through the origin, as pre-dicted by the Drude theory with a constant relaxa-tion time. As a result the relaxation time might be

0 !22(eV)FIG. 5. Reciprocal of the relaxation times for copperat 78, 293, and 423 K vs the photon energy squared.Lines are least-squares fits.


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OPTICAL CONSTANTS OF COPP ER AND NICKEL AS A 1319

TABLE II. Free-electron parameters for copper cal-culated from data of Table I using the Drude theory, corn-pared with handbook values of electrical conductivity.

Extrapolateddc relaxationtime(sec)Optica]. mass(electron mass)Opticaldc conductivity(sec ~)Electricaldc conductivity(sec ~)Mean free path

78 K

11.4 x 10-151.561.56 x101745. 0 x 10~7200

293 K

6.99x10 151.49

1.OO x1O"5.32 x1O"

423 K

4.97 x 10"151,43

0.739 x 10~73.46 x 10~7

igin through each of the points shown in Fig. 4. Ifwe plot these values as 1/v versus photon energysquared (Fig. 5), we get acceptable straight linesat each temperature of measurement. [Plots of 1/vversus eV and v versus eV a3.so give equally straightlines; however, 1/v versus (eV)3 is related to theo-ry, as discussed below. j The resulting values of vfor different temperatures are nearly identical above2 eV, the upper end of the free-electron range. Theintercepts at zero frequency correspond to dc valuesfor v. The results for the optical masses and dcrelaxation times at V8, 293, and 423 K are shownin Table II.Using our extrapolated dc values for 7 one cancalculate values for the optical dc conductivity of cop-per, o'='v/mo. These values along with mea-sured electrical dc values" are also listed in TableII. Our extrapolated values at 293 and 423 K areboth abaut five times smaller than the correspond-

ing measured dc values. At V8 K, however, ourextrapolated value is about 30 times smaller thanthe measured value. The larger discrepancy atV8 K may be due to the limitation imposed on themean free path of the electrons by the thickness orgrain size of our metal films. The mean free pathis equal to the product of the Fermi velocity and therelaxation time, l =v&v. The resulting values for Iare presented in Table II. At 78 K the mean freepath is comparable to the sample thickness (possib-ly equal to the grain size in the film}. The resultinglimitation would cause a decreased relaxation timein comparison to the phonon-scattering limitationin bulk specimens.We have shown that, if we use the Drude free-electron theory with a single relaxation time, thenwe must assume that the relaxation time is frequen-cy dependent in order to account for the positive in-tercept in the plot of em/X versus X2. The contribu-tions to the relaxation times due to phonons~~ andstatic impurities are not expected to be frequency

dependent over our measured frequency and temper-

4-

0

~6-7~y&-636/$.Ml2 3 4 5 6

PHOTON ENERGY (eV}FIG. 6. Band-by-band decomposition (dashed lines) ofthe iInaginary part of the dielectric constant for copper

(full line) as calculated by Williams et al. (Ref. 3). Thedotted curve represents an 8% decrease in magnitude anda 4% shift (2% near 4 eV) in energy of the 6 7 transi-tion, to simulate the change in the Debye-Wailer factorbetween 0 and 387 K (or between 78 and 423 K).ature range. ~~ A functional form for a frequency-dependent relaxation time has been calculated byGurzhi, however, on the basis of the electron-electron interaction. Although his expression fairlyaccurately describes the frequency dependence ofour room-temperature relaxation time, it does notpredict the correct temperature dependence:Ourzhi's formula predicts that the rate of changeof 1/r with &u~ is independent of temperature; how-ever, the plot of our data (Fig. 5) clearly suggeststhat the rate of change is temperature dependent.More recently, Schnatterly and Nagel~ have cal-culated a functional form for a frequency-dependentrelaxation time based on scattering in crystallitesand grain boundaries in the sample. Their theoryalso predicts a linear rate of change of 1/r with &o~which is independent of temperature.Alternatively, the failure of the lines in Fig. 4to pass through the origin could be ascribed to anexcess contributiori to e& from interband effects(possibly indirect transitions, for example} belowthe sharp absorption edge at 2 eV.

2. Interband regionWilliams, Jan@]~, and Moruzzi~ have made self-consistent band-structure calculations for copper


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1320 P. B . JOHNSON AND R. W. CHRIST Yand nickel, neglecting atomic vibrations. In addi-tion, they calculated the transition-matrix elementsat each of 240 k points in ~of the Brillouin zoneand, using them, evaluated the interband contribu-tions to the imaginary part of the dielectric constantat T = 0. Their theoretical values for copper areshown in Fig. 6, together with their resolved con-tributions from transitions between different bands(numbered in order of increasing energy). We willdiscuss our results for copper and nickel in termsof these calculations, introducing into their calcu-lation for copper modifications to take into accountthe temperature dependence of the Debye-Wailerfactor for transitions between the free-electron-likebands (6- 7).The Fourier components V~ of the electronpseudopotential should be multiplied by the temper-ature-dependent factor e, where W=wF(T/8) jnthe Debye approximation to the phonon spectrum.The constant w and the function F(T/8) are definedas follows6:

8/ 2'dz e/(e' 1) +-,w 3h 2G3/2M@8,

where 2mb is Planck's constant, G is a reciprocal-lattice vector, 4' is Boltzmann's constant, 6 is theDebye temperature, and I is the mass of an atom.The above integral is approximately= (T/8)+-,', T O.

The temperature-dependent potential V~e ~ hastwo effects for free-electron-like bands. First, atthe zone boundary C the band gap is 2VGe ~, andso the transitions near this boundary will shift inenergy as a function of temperature. Secondly, thesame factor enters into the transition probabilitybetween the bands, i. e. , into the momentum matrixelements: The probability of finding an electron,initially in a state P;, in a higher energy state P&,due to the absorption of a photon, is22 approximatelyproportional to lgrzi V }g;) ~, where V is the gradi-ent operator. The wave functions g; and g& for elec-trons in free-electron-like states separated by anenergy 2&E near a zone boundary are linear com-binations of two plane waves~~ such that~4which is the result given by Harrison. 23 (Just onthe boundary &E= V~, but away from the boundaryAE is approximately independent of V~. ) Now whenV~ is multiplied by e ~, the probability of electrontransitions is proportional to the Debye-Wailer fac-tor e, and therefore so is the contribution to &&,which is due to transitions between the free-elec-

tron-like bands. The wave functions for electronsin localized states, on the other hand, are given inthe tight-binding approximation as Bloch functionswith atomic orbitals for a free atom. Since theatomic functions do not overlay significantly, theevaluation of (gq I &IP ~) removes" the plane-wavefunctions, leaving only the atomic functions whichare not affected by the Debye-Wailer factor. As aresult, we would roughly approximate the theoreticalvalues of e2 at temperature T as follows: For thecontributions from the free-electron-like bands(6- 7) we modify the static-ion calculation, de-creasing the band gay by e and decreasing thetransition probability by e ~~, but for the other con-tributions (involving d-band states) we make nochange. (Strictly speaking, the first effect occursonly not too far from the zone boundary and thesecond only not too near. )In the case of nickel, over our measured spectralrange no transitions between free-electon-like bandsare calculated. 3 If the Debye-Wailer factor isassumed to account for the main temperature de-pendence of e2, one would expect no appreciablechange in the dielectric constants with temperature.This prediction is in agreement with our experi-ments on nickel films. On the other hand, in thecase of copper there is an ene rgy range from 4. 2eV up, where transitions are calculated (Fig. 6)between two free-electron-like bands (6- 7). Theseoccur mainly in a volume of 0 space near the Lpoint. Over this same spectral range our measuredvalues of &~ show a relatively large temperature de-pendence (Figs. 1 and 2) which can be qualitativelyexplained by our use of the Debye-Wailer factor.The temperature dependence of the experimentalcurve of &~ for copper above 4. 2 eV is seen primar-ily as a progressive decrease in the magnitude ofthe 5-eV peak with increasing temperature, by about10%. Using as values for copper O=315 K, G= 2m& 3/a in the [111]direction, a =3.608 A, and M=1.063&&10 ' g, our Debye-Wailer factor e pre-dicts that there should be an 8% decrease in thecontribution to &~ from the 6- 7 transition with anincrease in temperature from 78 to 423 K. (In thecase of x-ray diffraction data on copper, this Debye-Waller factor exactly describes the measuredchangein the intensity. ) The 6- 7 transition accountsfor about 30% of the calculated total e~ value from4. 2 to 6. 6 eV (Fig. 6) and so this effect would beseen as a 3/0 decrease in the theoretical value of e~.(The other transitions are not between free-electron-like bands. In addition, in the case of nickel wherethe 1- 6 transition accounts for the entire high-en-ergy peak, there is no appreciable temperature de-pendence experimentally. ) Besides decreasing themagnitude of the contribution to &~ due to transitionsbetween tree-electron-like bands, the Debye-Wailerfactor also causes these transitions to shift in energy.


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OPTICAL CONSTANTS OF COPP ER AND NICKEI 1321At the L point the free-electron-like bands arebands 6 and 7, separated by 2V~. Here the factore will cause a decrease in the energy of band 7and an increase in the energy of band 6 so that thegap decreases by about 4/0 with a change in temper-ature from 78 to 423 K. As a result, the contribu-tion to &2 from 6- 7 must be shifted down in energyby 4%. But since the Fermi level remains un-changed, the low-energy edge of the 6- 7 contribu-tion, which starts with transitions at the Fermi level,is shifteddown in energyby only about2%%up. Aplot ofthe theoretical e~ with an 8% decrease in magnitudeand a 4/z shift in energy (2% on the low-energy side)of the 6-7 transition is included in Fig. 6, showingreasonable agreement with the experimental behav-ior in Fig. l. (The 0-K theoretical curve itself wasnot modified for the 0-K Debye-Wailer factor; in-clusion of this modification would further improvethe absolute agreement of the theoretical and experi-mental curves. ) From 4. 2 to 4. 5 eV the theoreticaltemperature dependence due to the shift of band 7 isincorrect. This effect may be cancelled because byraising band 6 in energy we actually decrease thenumber of electron states just below the Fermi lev-el, further reducing the contribution of the 6-7transition below 4. 5 eV. ~'

B. Comparison with other experimentsEarly studies of the temperature dependence innickel and copper were made by Roberts, "owhomeasured the optical properties at 90, 300, and500 K from 0.5 to 3 eV. The behavior of his valuesof E2 for copper as a function of temperature is

qualitatively similar to ours. Above 2 eV there isno temperature dependence up to 3 eV, and below2 eV the values of &2 increase significantly with in-creasing temperature. In his experiments on nickelRoberts, like us, determined that the opticalconstants were independent of temperature. Morerecent measurements were made by Pells andShiga, ' who used mechanically polished samplesin ultrahigh vacuum. They made measurements oncopper from 1.7 to 5. 9 eV and over the temperaturerange 77-920 K. Their values qualitatively agreewith ours in that &~ increases with temperature be-low 2 eV, does not change much with temperaturefrom 2 to 4. 2 eV, and decreases with temperatureabove 4. 2 eV. Nevertheless, their values for E,are as much as 501' larger than ours at some photonenergies. They also made measurements on nickelsamples from 0. 5 to 5. 9 eV. With a temperaturechange from 295 to 470 K they observed only avery small change in e2; inthis case their values wereasmuchas20'%%uolarger than ours. Finally, Stolid~de-termined the optical constants of copper from 0. 5 to3.3 eV over the temperature range 77-600 K.Stoll's values for c~ are in close agreement withours and a similar temperature dependence is ob-

served in his more limited energy range.The general behavior of the dielectric constantsfor copper and nickel as a function of temperatureis consistent in all of these ellipsometric studies,but the magnitudes differ. These differences maybe due to the surface condition of the samples,which depends on preparation techniques as wel. l asany oxidation or condensation which may occur. Inaddition, distortionsmay resultfrom the windowsofthe cryostat. Since our measurements are normalizedto our room-temperature measurements made out-side the cryostat, we believe that our values are themost accurate in magnitude. Only our measurementscovered the whole spectral range, including boththe infrared and ultraviolet regions. Knowledge of& over this wider range is essential to tie togetherthe behavior in the free-electron region and the in-terband region.Comparison can also be made with thermomodu-lation experiments' on copper. In the thermomodu-lation technique, the temperature of the sample ismodulated with about 10-K modulation amplitude.The resulting ac component in the reflectance, ~,and transmittance, &T, is measured. Values for&&& and 2 can be calculated from ~ and &T,provided that values for e, and &~ at the averagetemperature of the sample are known. This tech-nique is a useful tool for precisely locating struc-ture in the interbaiid absorption of solids, althoughits absolute accuracy is limited by knowledge of thethermomodulation amplitude and by the choice ofknown values for &, and e~. Rosei and Lynch usedtheir measurements of &R and ~T in conjunctionwith the e, and e2 values of Pells and Shiga" to de-termine values for &&2 from 1 to 5 eV. The struc-ture in our b,e~/'T curve (Fig. 2) agrees well withthat of Rosei and Lynch, considering that f rom 2 to4 eV our &~~ is smaller than our estimated absoluteerror. The differences in magnitude are not sur-prising, since Pells and Shiga's values for ~~ whichheyusedareasmuchas50%%uolarger th ours. Fur-thermore, their &e~ values were obtained as dif-ferences of two large quantities.Thermomodulation measurements of ~ havealso been made on nickel. 6 The magnitude of thestructure in these 4R measurements is muchsmaller than our estimated error, thereby con-firming that any change in the dielectric constantswith temperature would be very small in nickel.Thus our absolute measurements of &z as a func-tion of temperature for both nickel and copper arein satisfactory agreement with the thermomodula-tion experiments, which provide a sensitive testfor fine structure in z~.

IV. SUMMARY AND CONCLUSroNSWe have made reflection and transmission mea-surements on evaporated copper and nickel films


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1322 P. B. JOHNSON AND R. W. CHRIST Yat 78, 293, and 423 K from 0. 5 to 6. 5 eV and in-verted these measurements in order to determinevalues for the optical constants. The reflectionmeasurements were made on opaque films and thetransmission measurements on semitransparentfilms (-350 A). Since all of the measurementswere made in a cryostat where the windows mayintroduce distortions, we normalized our measure-ments to room-temperature measurements madeoutside the cryostat. In addition, we observed therates for condensation of water vapor on our sam-ples at 78 K and oxidation of the samples at 423 K,and determined that these effects were within theestimated error of our optical measurements.We detected some interesting changes in the di-electric constants for copper as a function of tem-perature, and found that by contrast they are ab-sent in nickel. In copper below 2 eV, where intra-band free-electron effects are thought to be domi-nant, we fit our results to Drude theory. Valuesfor the optical mass, which are determined by &&,decreased very slightly with temperature, an ef-fect which we believe to be real. The relaxationtime, which is determined from the values of theoptical mass and e&, was found to be frequency andtemperature dependent. Using the observed fre-quency dependence we extrapolated to the zero-fre-quency values for optical dc relaxation time, tocompare with the measured electrical dc conduc-tivity for copper. Our extrapolated values for theoptical dc conductivity were about five timessmaller than measured electrical values, exceptat V8 K, where a greater discrepancy can be ex-plained by the mean-free-path limitation due to oursample thickness and grain size. At room temper-ature the high-frequency dependence of the relaxa-tion time predicted from electron-electron orboundary-region scattering gives reasonable agree-ment with our experiments, but our indicatedchange in the frequency dependence with tempera-ture is not predicted. Possibly the grain boundarymechanism dominates at 78 K and the electron-electron scattering at higher temperatures. Acontribution to the frequency dependence calculatedfrom electron-phonon scattering theory ~ seems tobe much too small. Previous optical measure-ments on noble metals in the near infrared havelikewise exposed difficulties of interpretation interms of free-electron theory; these were sum-marized recently by Winsemius. ~ Precision re-flectance measurements on gold show a deviationabove 0.4 eV, which might a1.so be expected in cop-per. Similar effects are found in alkali metals3~ aswell. Perhaps it is not surprising that the assump-tion of a parabolic band fails to account for intra-band transitions at energies as large as 2 eV.(Another approach is to consider the discrepanciesas a low-energy tail of interband processes. ) No

intraband effects were expected or observed innickel.Above 2 eV interband processes are dominantin copper as well as nickel. We observed onlyvery small thermal changes in && from 2. 2 to 4. 2

eV, and the observed change above 5. 2 eV in cop-per can be reasonably well described by a Debye-Waller factor in the matrix elements for transi-tions between free-electron-like bands and for theelectron excitation energy. We modified the zero-temperature theoretical calculation of %.lliams etal. by incorporating (rather crudely) the Debye-Waller factor in order to explain this thermalchange in e~ of copper. The energies at which ex-perimental and theoretical changes occur agree ex-cellently, and the magnitudes of the changes agreereasonably well. This mechanism predicts nothermal change in nickel. Thus we conclude thatthe large negative values of 6&2/hT in copperaround 5. 2 eV are due mainly to the Debye-Wailerfactor, rather than to volume thermal expansion orchanges in the Fermi distribution. The experimen-tal effect of volume expansion can be estimatedfrom Gerhardt's volume-strain modulation results3~if they are multiplied by the thermal-expansion co-efficient of copper. The result agrees well withour positive peak at 2. 1 eV but is only one-fifth ofour negative peak at 5. 2 eV. The theoretical vol-ume effect which we expect on the basis of energy-band calculations33 is a downward shift of less than0. 1 eV (- l. 5%) for the 6- 7 and 1-6 peaks around5 eV. The predicted shifts of the 5- 6 peak at 2. 1eV and the 4, 5- 6 (X,- Z~. ) peak at 4.0 eV aremuch smaller yet, as is the effect of thermalchanges in the Fermi distribution. The Debye-Waller mechanism therefore seems to account forthe largest thermal interband changes. The Debye-Waller factor might also cause a temperaturechange in alkali metals, 3 but this has not beenverified to our knowledge.Our conclusion that the interband contribution isnearly independent of temperature from 2 to 4 eVin copper, as expected from the Debye-Wailermechanism, actually depends on the crossing ofthe intraband curves (Fig. 4) at 2 eV. If one sim-ply used the temperature-dependent (but frequency-independent) values of v calculated from the slopesto extrapolate the free-electron expression into theinterband region above 2 eV, a compensating tem-perature dependence would thereby be introducedinto the interband contribution to ez as well (be-cause experimentally the sum is almost indepen-dent of temperature). Since almost no tempera. -ture dependence is observed in the experimentalE~ from 2 to 4 eV, we assume that the intrabandcontribution remains independent of temperaturein the interband portion of the spectrum, or elsethat it decreases more rapidly with frequency at


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OP TIC AL CONST ANT S OF COPP ER AND NICKE I AS A, 1323all temperatures than is described by our Drudeexpression.Finally, we have compared our values for thedielectric constants with other experiments andfound qualitative agreement as to the thermal be-havior. The differences in the magnitudes of thedielectric constants we attribute to the sensitivityof the previous ellipsometric measurements to thecondition of the sample surface and to the windowsof the cryostat. We find fairly good agreement be-tween our thermal changes and the structure ob-

served by the thermomodulation technique in cop-per.ACKNOWLEDGMENTS

The Debye-Wailer factor as an explanation of theinterband temperature dependence was suggestedto us by Dr. A. H. Williams. We are also grate-ful to him and V. L. Moruzzi for sending us un-published results and permitting us to use themhere. We thank Professor W. E. Lawrence formany very helpful discussions.

*Partially supported by the NSF.fPresent address: Department of Physics and Astronomy,Louisiana State University, Baton Rouge, La. 70803.~P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370(1ev2).P. B. Johnson and R. W. Christy, Phys. Rev. B 9, 5056(19V4).A. R. Williams, J. F. Janak, and V. L. Moruzzi,Phys. Rev. Lett. 28, 671 (1972), and private communi-cations.4R. Rosei and D. W. Lynch, Phys. Rev. B 5, 3883 (1972).A. R. Williams (private communication).J. M. Ziman, Principles of the Theory of Solids (Cam-bridge U. P. , Cambridge, England, 1969), pp. 51-63.R. V. Kasowski, Phys. Rev. 187, 885 (1969); E.Christensen and B. O. Seraphin, Phys. Rev. B 4, 3321(19V1).A. G. Mathewson and H. P. Myers, J. Phys. F 2, 403(1ev2).S. Roberts, Phys. Rev. 114, 104 (1959).~ S. Roberts, Phys. Rev. 118, 1509 (1960).~~G. P. Pells and M. Shiga, J. Phys. C 2, 1835 (1969).M. Shiga and G. P. Pells, J. Phys. C 2, 1847 (1969).M. Ph. Roll, J. Appl. Phys. 40, 4533 (1969).~ P. B. Johnson, Ph. D. thesis (Dartmouth College, 1974)(unpublished).~5M. -L. Theye, Phys. Rev. B 2, 3060 (1970).CRC Handbook of Chemistry and Physics (ChemicalRubber Co. , Cleveland, Ohio, 1963), p. 2666.G. D. Mahan, J. Phys. Chem. Solids 31, 1477 (1970).i8B Caroli, J. Phys. (Paris) 23, 884 (1962).

R. Gurzhi, Zh. Eksp. Teoy. Fiz. 35, 965 (1958)[Sov. Phys. -JETP 8, 673 (1958)].20S. E. Schnatterly and S. R. Nagel, Phys. Rev. B 9,1299 (1974).~C. Kittel, Introduction to Solid State Physics, 4th ed.(Wiley, New York, 1971), p. 312,L. I. Schiff, Q~antgm Mechanics (McGraw-Hil]. , NewYork, 1970), pp. 397-405.3W. A. Harrison, Solid State Theory (McGraw-Hill,New York, 1970), pp. 317-326.4P. A. F].inn, G. M. McManus, and J. A. Rayne, Phys.Rev. 123, 809 (1961).2~W. E. Lawrence (private communication).J. Hanus, J. Feinleib, and W. J. Scouler, Phys. Rev.Lett. 19, 16 (1967).~T. Holstein, Ann. Phys. 29, 410 (1964).T. J. Wieting (private communication).9P. Winsemius, Temperatgre Dependence of the OpticalProperties of&u and Ag (Drukkerij J. H. Pasmans,'S-Gravenhage, 1973), p. 24.H. E. Bennett and J. M. Bennett, in Optical Propertiesand E'lectronic Structlre ofMetals and Alloys, editedby F. Abelbs (North-Holland, Amsterdam, 1966),pp. 175-186.3~N. V. Smith, Phys. Rev. 183, 634 (1969); Phys. Rev.B 2, 2840 (1970).U. Gerhardt, Phys. Rev. 172, 651 (1968).33H. L. Davis, J. S. Faulkner, and H. W. Joy, Phys.Rev. 167, 601 (1968).34C. -Y. Young, Phys. Rev. 183, 627 (1969).

Johnson - Optical Constants of Copper and Nickel as a Function of Temperature

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Transcript of Johnson - Optical Constants of Copper and Nickel as a Function of Temperature