Planck's Constant from a CdS Photoconductivity Cell J a m e s E. Sturm Lehigh University, Bethlehem, PA 18015 T o some, any suggestion of an elementary laboratory experiment involving determination of a fundamental constant may be categorized along with reinvention of the wheel. Such an exercise can nevertheless be instructive in our day if along with i t an augmented perspective is attained on characteristics of matter. Attention to the properties of currently available devices used in electronics can serve to communicate the relevance of basic principles. Some examples on this theme have appeared (1-3) but have been directed toward physics students. The traditional student laboratory method of measuring Planck's constant has been to exploit photoelectron emission in vacuum tubes. One update (4) presents more nearly current circuitry but still incorporates a vacuum phototube. Presented here is a description of how a light-sensitive semiconductor can be used t o obtain an estimate of Planck's constant. The same ideas offer a framework for understanding of semiconductors. ' The common CdS ~hotoconductivitvcell ( 5 . 6 ) is typical of the many doped se&conductors e x t k v e l y used in klectronic devices. Doped CdS is chosen because for it the population density of mobile charge carriers is rather low at room temperature as manifested by a large dark resistance. Absorption of light of sufficient photon energy to excite electrons into the system's conduction band results in an enhancement of the mobile charge carrier population. Under steady illumination, the rate of excitation to the conduction band is balanced by the electron-hole recombination rate. A constant population of carriers is manifested by a steady resistance somewhat lower than the dark resistance. Within saturation limits the conductivity is dependent on the intensity of the light. Available elertronics manuals (7)present the wavelength dependence of sensitivity of a CdS cell, but this property can also be determined with a spectrophotometer such as those often used in elementary chemistry laboratories. One can i n t e r ~ r e tthe lone-waveleneth end Xr." of this sensitivitv curve as indicating the minimum photon energy E ~ t excite o electrons from d o ~ e imnuritv d . " levels t o the svstem's conduction bald:
-
-
EG = helhc
One then has for n. Here k is the Boltzmann constant and E'Gis the Boltzmannaveraged energy of the conduction-band electrons. The dark conductivity a a t some temperature T depends further on electron mobility p., which is also temperature dependent (holescontributelittle to a i n thesematerials). As also in metals, p, is limited by electron-phonon scattering (81,which conveniently has a T-3'2dependence. One therefore has the simple relation a = exp (-E'clkT). The observable dark resistance R is likewise related exponentially to temperature since CT = 1/R.The logarithmic form resembles the van't Hoff isochore and is also applicable to common thermistors: In R = E'GlkT
+ constant
(4)
Equation 4 provides the hasis for an estimate of E'Gindependently of light-induced conductivity. One needs only a set of measured resistance R in darkness a t various T. A plot of in R vs. (11T) should be linear with a slope = E'clk. Finally, one equates EG and E'G, which should be nearly equal in the temperature range involved here. Planck's constant then is obtained from eq 1with EG= E'c:
The Laboratory Exercise Described below isa laboratory exercise carried out on this theme with a twical CdS Dhotoconductivitv cell and with apparatus read& available-in elementary chemistry laborahries. Determination of AG A Bausch & Lamb Speetronic 20 single-beam speetrophotameter was used. First, with only an empty cuvette in the sample chamber,
(1)
where h is Planck's constant and c is the speed of light in a vacuum. Attention is next turned t o the temperature dependence of the dark resistance of this same CdS cell. The thermally induced population density of electron-hole pairs follows Fermi-Dirac statistics (8). As noted earlier, practical photoconductors are chosen for their relatively large E c values. The Fermi-Dirac distribution a t these energies reduces t o the Boltzmann distribution. To express the number density n, of conduction-band electrons, one views the excitation as a dissociation from an impurity state (I) to the conduction band (cb) leaving behind a hole: e,
-
e,,
+ hole
Recourse is usually made to the electron-gas approximation to describe e+ for which the Dartition function Q, is that of a particle (electron, mass me)in a box: 1052
Journal of Chemical Education
Figure 1. Wavelengm dependence of rensitiviry of l a m p d e l m o r combma. lion. Bausch 8 Lomo Spemonic 20 spenraphotmter.
Figure 2. Wavelengm dependence of condvctlvity (= R - 1)of a CdS photoCOndUCtlvity cell exposed to light entering the sample comparrmem of a Specnonlc 20 spectrophotometer.
Figure 4. Temprature dependence of dark resistance of CdS photoconductivC ty cell: plot of In Rvs. 11T.
response to a unit intensity of light at the given wavelength. Above 900 nm the ordinate is low, independent of wavelength, and may very well represent a contribution by scattered light within the Spectronie 20 itself. (Laboratory ceiling lights were off and a thick felt cloth was draped over the euvette compartment.) There is some tailing of the curve above 760 nm. One can only assign to Ac a range of from 760-900 nm, or a value of 830 70 nm.
+
Figure 3. Wavelengm dependence of normalized response, (lIRSo),of a CdS photo~Dnd~Ctl~ity cell.
magnitudes of detector signal &(A) were determined relative to the maximum signal a t A = 520 nm by (1) setting % Tto 100 a t = 520 nm and (2) reading % T a t other wavelengths without changing the setting of the 100 WT knob. I t was, however, necessary to change scales, in effect, by resetting to 100 %T when the %T reading a t a given wavelength fell to 10%.Such resetting became necessary twice in scanning from 520 to 975 nm. I t should he noted that the normal detector in the Spectronic 20 is s vacuum phototuhe whose sensitivity is maximum near 500 nm and decrease cansiderahly toward 900 nm (7). Figure 1illustrates the wavelength dependence of this relative sensitivity So. (A wide-range phototuhe is available hut was not used in the present exercise.) The profile represents a combination of phototuhe sensitivity and hlaek-hody radiation intensity distribution (9). Next, a CdS photoconductivity cell was inserted into the cuvette, and its position was adjusted to give minimum resistance R at A = 520 nm as measured with a Keithley model 610B electrometer. Resistances ranging from 10" Q to lo8 Q were measured over the wavelength range from 520 to 975 nm; their reciprocals (o o: R-I) are shown on Figure 2. Already an indication of a long-wavelength limit can he seen by the sharp, 10-fold decrease in conductivity near 850 nm. Figure 3 shows a plot of the reciprocal product lI(RS0) as a function of wavelength. This ordinate is interpreted here to be proportional to the normalized response of the CdS cell; ie., its
Determination of ElG The CdS cell was wrapped in electrical insulator (glass wool or equivalent) and was placed in a ceramic cone wound on its outside with resistance heating wire. With this assembly covered with thick felt and with ceiling lights off, the heating wire was connected to a variable autotransformer. Several transformer settings were used to obtain steady-state temperatures between 132 OC (450 K) and 216 OC (489 K). Corresponding resistance measurements were made. Figure 4, a plot according to eq 4, was generated from the data. The slope of the least-squares line is (1.68 f 0.11) X lo4 K and correJ or 1.45 0.10 eV. As the sponds to E'o = (2.32 i 0.16) X intrinsic hand gap energy in CdS is 2.4 eV, somewhat larger than E'c, one can see that the photoconductivity cell is fabricated from doped CdS (5). Result and Assessment Use of the average Xc and the thermal E'c as determined above vields h = (6.4 f 0.6) X 10V4Js. a value within the oresent exoerimental un&ainty (if the much more precise accept& vnlue.'l:hose ill-disposed toward the theme of determmotion of Plnnrk's constant can treat h ar knctwnand thendrrect students'att~ntiontothaclose agreement between the two alternate estimates of the hand gap energy in this doped material. Either way, the exercise suggested here emphasizes two salient properties of s light-sensitive semiconductor. No specialized instruments are required, and the photomnductor itself is an inexpensive component available a t electronics stores. Interoretation of the observations confronts the student with the appropriateness ot further reading on semieondurm~sand thereby provides a framework for increased understanding of rhesp imporrant matrrinls. A taste is alw provided of the spectral charaeter of black body radiation ~~~~~
Lneratuie CRed 1. Kirkup, L.: Placido, F. Am. J. Phys. 1986.54,918420. 2. Cdlings, P. J. Am. J. Phys. 1980.48, L9FL99. 3. Devls,J. A.:Mualler. M. W . A m . J. Phys. 1977.45.770-771. 4. Boy8.D. W.;Cor. M. E.; Mykalslenko,W . M. Am. J.Phyr. 1978.46, 133-135. 5. Bube, R.H. PhofocanducliviiVo/Solidl; Wiley, 1967: pp 167ff. 6. Ambriziak, A. Somieonduetor Photoeleciiii Deuicrs: iliffe: ,968; p 257. 7. Technical Manual Pl-60. Photorubes and Phnlocdln; Radio Corp. ofAmerica: Lsncsster. PA. 1963: p 76. 8 Beeforth. T. H.; Goldsmid. H. J. Physics of SolidStateDeoiees: Pion: 1970:Chapter 1. 9. Maelwyn-Hughes, E. A. Physical Chemistry: Pcrpamon: 1957:Chapfer ill.
Volume 66
Number 12
December 1989
1053
