HAL Id: jpa-00213618
https://hal.archives-ouvertes.fr/jpa-00213618
Submitted on 1 Jan 1968
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
THE BAND STRUCTURE PARAMETERS OF PbTe FROM OPTICAL REFLECTIVITY, TRANSMISSION,
AND FARADAY EFFECT
Ju. Maltsev, I. Smirnov, Ju. Ukhanov, A. Veis
To cite this version:
Ju. Maltsev, I. Smirnov, Ju. Ukhanov, A. Veis. THE BAND STRUCTURE PARAMETERS OF PbTe FROM OPTICAL REFLECTIVITY, TRANSMISSION, AND FARADAY EFFECT. Journal de Physique Colloques, 1968, 29 (C4), pp.C4-99-C4-104. �10.1051/jphyscol:1968414�. �jpa-00213618�
JOURNAL DE PHYSIQUE CoZloque C 4 , supplkment au no 11-12, Tome 29, Nouembre-Dgcembre 1968, page C 4 - 99
THE BAND STRUCTURE PARAMETERS OF PbTe FROM OPTICAL REFLECTIVITY, TRANSMISSION, AND FARADAY EFFECT
by Ju. V. MALTSEV, I. K. SMIRNOV, Ju. I. UKHANOV, A. N. VEIS Department of Radioelectronics, Polytechnic Institute,
Leningrad, USSR
RBsumB. - Les spectres de rBflectivit6 et de transmission et d'effet Faraday ont BtB BtudiBs a
120 et 300 OKa des longueurs d'onde de 1 a 25 microns sur des cristaux de PbTe Bpitaxiques et massifs, de typep ou n et dans une gamme de concentrations allant de 4,4 x 10'7 a 1,2 x 1020 cm-3.
Les rksultats de ces mesures indiquent que les extrema de bandes prks du point L dans l'espace reciproque contribuent de faqon prkdominante aux effets magnetooptiques et optiques. Les masses effectives des Blectrons et des trous sont a peu prks Cgales, a la msme energie, dans les bandes, et la loi de dispersion n'est pas quadratique.
Abstract. - Reflection and transmission spectra and the Faraday effect were investigated at 120 and 300 OK at wavelengths from 1 to 25 p on epitaxial and bulk n-, and p-type crystals of PbTe with electron and hole concentration from 4.4 x 1017 to 1.2 x 1020 cm-3.
The results of these experimental studies indicate that the energy band extrerna placed near L-point in k-space give a predominant contribution in the magneto-optical and optical effects.
The effective masses of electrons and holes are approximately equal at the same energy in the bands and the energy dispersion law is not quadratic.
Lead telluride is a typical representative of IV-VI compounds and it was investigated lately by various methods [I], from which the useful data of band structure were determined. Optical and magneto- optical investigations of PbTe, especially of high doped one, are not enough [2-81, therefore it seemed advisable to carry out the supplementary measurements in wavelength region from 2 to 25 y by the carrier concen- tration up to loz0 ~ m - ~ .
Experimental methods. - OPTICAL SAMPLES. -
Three groups of samples were investigated. The crystal ingots of the first group were prepared by zone mixing method at slow cooling of the melt. The second group of bulk samples was prepared by the hot pressing with consequent heat treatment after which the sample structure contained small grains. These two groups of samples were prepared by E. D. Nensberg (Institute of Semiconductors, Leningrad) and R. B. Melnik (Polytechnic Institute, Leningrad) [9,10]. The samples of the third group were < 100 > oriented, epitaxial crystals grown by S. A. Semiletov (Institute of Crys- tallography, Moscow) on rock-salt substrates.
Electrical properties (Hall-effect, electroconducti- vity, thermoelectric power) were investigated by A. V. Petrov (Institute of Semiconductors, Leningrad) on the same samples before optical experiments.
The reflectivity was carried out on 10 x I0 x
20 min3 samples. One of the largest sample surfaces was polished mechanically. After reflectivity measure- ment the sample was mounted on a glass substrate with a resin and prepared by conventional grinding and polishing techniques till necessary thickness.
Then the sample was freed from the substrate, and freely mounted.
Reflectivity was measured on the great number of bulk samples but the thin samples were prepared only for two n-type bulk samples and three p-type samples. Electrical and other characteristics arelisted in the table only for the samples on which, except reflec- tivity, transmission, and Faraday rotation were mea- sured.
epita- 2100 300003.3;6;9
I I n I x i a l I 0.441 I I
Sam- ple Type
2 3 4 5 6
Sample thickness
(P)
Carrier density cm-3
Hall mobili- ty cm2/v.sec 293 oK] 77 OK
n n p p p
ingot ingot ingot ingot ingot
15.0 120
2.0 59.0 170
1400 300 1 150 410
7000 1 000 15 000 1 100
16 2.7 ; 3.0
30 5.0
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1968414
C 4 - 100 JU. V. MALTSEV, I. J. SMIRNOV, JU. I. UKHANOV, A. N . VEIS EXPERIMENTAL SYSTEMS. - Reflectivity, transmission
and Faraday rotation were investigated using infrared optical systems in conjunction with the electromagnet which is shown in figure 1. The monochromatic beam from exit slit of a prism monochromator was chopped by a 9-cps chopper and then was directed by flat and spherical mirrors in the space between the magnet poles where the sample was mounted in the holders of special construction for reflectivity (Fig. Ic, d) for transmission (Fig. la, b) or for Faraday rotation (Fig. lb). After passing through the sample, the radiation was directed by spherical and flat mirrors to the entrance window of the Golay detector (OAP-4).
Lineary polarized radiation was obtained with an eight-sheet selenium pile-of-plates polarizer placed in the beam before the sample. The Faraday rotation was measured with the same second polarizer placed after the sample.
ffom one side
FIG. 1. - A scheme of the optical system used for the expe- riments.
Since the magnet poles were used as side walls of vacuum chamber for the sample, the distance between poles was 9 mm and magnetic field was 25 kg in spite of a small electromagnet. The measurement chamber is shown in figure 1, where main peculiarities of its construction are seen. On a horizontal axis the chamber had two circular windows of KBr held in place with
putty. The third circular window with a lens was intented for visible control of the sample holder position between magnetic poles. When the sample temperature was close to the temperature of liquid nitrogen there was created a vacuum in the chamber.
If the sample was heated the chamber was filled with purified argon at the pressure of about 1,2 atm. for the preservation of the sample properties.
Measurement results and discussion. - Reflectivity R, and transmission t were measured by suitable methods : the sample - the aluminium mirror (for R), and with - without sample (for t). Faraday rotation was measured in most cases by the 450-method [l 11.
Reflectivity spectra. - Typical infrared reflectivity spectra for the bulk samples 2 - 4 are shown in figure 2. Reflectivity was measured at wavelength from 2 to 26 p, where the interband transition maxima were not observed as they were placed at shorter
FIG. 2. - The measured optical reflectivity spectra.
wavelength [12, 131. In the transmission region, in reflectivity spectra of epitaxial film samples there was a fringe structure (in Fig. 2 it is not shown) which arose because of interference.
In figure 2 one can see that in reflectivity spectra of high doped samples there is a plasma minimun which is
THE BAND STRUCTURE PARAMETERS OF PbTe C 4 - 101
shifted to shorter wavelength with the increase of the electron concentration in accordance with the formula
I-
cop 2 = Ne2/c0 E , m*, rn* is the effective mass at the Fermy energy, c, is the high frequency dielectric constant, c0 is the dielectric constant of vacuum in the International units systems.
For sample 4 the plasma minimum was placed in the region ;l > 25 p. Since the reflectivity is very weakly dependent on wavelength, the value c, = 34 + 2 was
determined from reflectivity (50 f 1) % in wave- length 2-5 p.
The carrier mobility is sufficiently high (see the Table) therefore, as was shown by Lyden [4], the approximate formula (1) is sufficiently exact for the calculation of the carrier effective mass. These masses determined by reflectivity method at 293 OK are equal : mz = (0.093 $. 0.008) mo, m3 = (0.30 0.02) mo, m5 = m6 = (0.23 f 0.02) mo for samples 2, 3, 5, 6, accordingly.
The plasma minimum was shifted to short wave- length and became deeper at decreasing temperature, but these minimum changings reduced with the increase of the carrier concentration. In figure 2 it is shown by dotted lines.
It is necessary to note that in the region A 5 lmi, reflectivity decreased more sharp by wavelength increasing than is expected from the formula
both for n- and p-type samples. Moreover the maximum was discovered before plasma minimum for a number of n-type crystals and pressed samples. This maximum shifted together with plasma minimum a t the increase of the electron concentration and became more distinct on cooling of the samples as it is shown in figure 2.
The origin of this maximum is not yet clear.
Absorption spectra. - The results of the transmis- sion measurements are shown for the n-type samples in figure 3 where absorption coefficient a is reproduced in logarithmic scale as a function of photon energy. The following peculiarities are seen here.
First of all, the absorption edge shifts to high energies a t the increase of the electron concentration in the conduction band (the Burstein-Moss effect). Since the electron concentration changed from -- 10"up to
-
lo2' ~ m - ~ , this absorption edge shift was observed both at low temperature and at room temperature.Second, upon cooling the samples to low temperature,
the absorption edge shift depends both in absolute value and in sign on the electron concentration. The third, the long wavelength absorption enlarges by the increasing of free carrier concentration and is propor- tional to wavelength in some exponent.
FIG. 3. - The measured optical absorption spectra.
Similar spectra were observed for p-type samples 4,6. The transmission of the sample 5 was unexpectedly negligible.
In epitaxial samples the eIectron concentration was low therefore their absorption edge was investigated carefully. The straight line portions were observed in the dependences ct2(m) and l/olo from which the energy gaps were determined a t 2930K for direct, allowed transitions E,, = (0.326 + 0.003) eV ; for indirect, allowed one E,, = (0.309 + 0.003) eV. The difference of these gaps is less than 0.032 ev determined by Scanlon [2]. This difference decreases upon cooling of free epitaxial samples which indicates on the nearness of main extrema of bands in k-space.
For determination of the point in k-space where the main band extrema are placed the experimental spectra of high doped samples 2, 3, 4, 6 were compared with
C 4 - 1 0 2 JU. V. MALTSEV, I. K. SMIRNOV, JU. I. IKHANOV, A. N. VEIS the theoretical dependences [14, 151 for the parabolic
energy bands
Here C, was supposed t o be independ on the frequency.
m, is the effective mass on the Fermy energy but not on the bottom of the conductivity band. Moreover the electron effective mass nz, is supposed to be equal to hole effective mass m,. M is the number of equiva- lent minima in the conductivity or valence bands, E, is the energy gap for direct transitions.
Formula (2) accounts only the effect of the filling of the conductivity band by electrons or of the valence band by holes.
The experimental data a t 12WK are shown in figure 4 by various points for the fundamental absorp-
I I I I I
42 83 0.4 0.5 06 kZ7 O photon energy , ev
FIG. 4. - Thed optical absorption edge spectra. Calculated curves for the absorption coefficient are shown assuming he free carriers to be in four equivalent nonparabolic and spherical bands located at L.
tion. These data were obtained by subtraction of free-carrier absorption from the experimental absorp- tion shown in figure 3. The results of calculation using formula (2) in M = 4 are shown by curves for samples 1-4, 6. As can be seen, the fit is poor for samples 2,4 and it is much better for the highest doped samples 3,6 both at room-temperature and at low temperature.
This permits to explain the absorption edge shift t o short wavelength upon cooling the samples 3,6 (but not to long wavelength as it is typical for lead salts with low free-carrier concentration and it is shown in figure 3 for the sample I), and the absorption edge shape change due to the essential redistribution of free carriers near the Fermy energy and due to increasing of the Fermi energy upon cooling the sample since the effective mass of free carriers essentially diminishes in lead salts [ I ] at decreasing temperature.
For the analysis of long wavelength absorption, the dependences In (n . a) = f (ln A) were represented, where n(A) - the index refraction dispersion was obtained for the samples 2-6 from reflectivity spectra. It was found that n.cr -- 1" for both n- and p-type samples, where x = 2 a t 293 OK and it increased feebly upon cooling the samples. In accordance with free-carrier theory [16,17], this testifies to the predominance of free-carrier scattering on acoustical phonons and ionized impurity centers.
Calculated from free-carrier absorption spectra, the effective masses were found at room temperature as following : m, = (0.094 + 0.008) mo ; m, = (0.099 +
0.006) mo ; m3 = (0.28 + 0.04) m o ; m6 = (0.31 k0.04) mo .
Faraday rotation. - The dependences 0(i12) are shown in figure 5 for the sample 2 at 130,293, 560 OK, and for the sample 4 a t 293 OK. The interband rotation is seen in the short wavelength region, the value of whiph increases at the cooling of the sample. The long wavelength rotation is in the main due to free carriers.
For quantitative analysis of the interband rotation, A0
Bd
(e)
= x2 is shown for the the dependence - -sample 2 at 130 OK in figure 6. The value A0 is equal to the difference between the experimental and free- carrier rotation extrapolated from the long wavelength
, En,,, = E, + EF = 0.45 eV region (see Fig. 6) ; on,, = --
i%
was determined from figure 3. Thus the interband rota- tion was supposed to be due to direct transitions not between extrema of the bands as the low levels of the conductivity band are filled by electron but between the valence band and the levels near the Fermi level in the conductivity band. The function f (Y) by Bos-
THE BAND STRUCTURE PARAMETERS OF PbTe C 4 - 103
FIG. 5. -The interband and free-carrier Faraday rotation for n-and p-type san~ples.
"=""'
/
- t4eory 8, - f , ( X ).
- experimentFIG. 6. - The interband Faraaay rotation for n-type sample.
warva et al. 1181 was fitted to experimental value A0
Bd - = 0.050 deg./G.cm at x2 = 0.36 and the effective factor was calculated at 130 OK : g,,, w - 40. In the same way, g,,, = f 2.5 was calculated for the sample 4 at 293 OK.
The effective electron mass was determined for the sample 2 from the dependence @(A2) in the region
w < ow, taking into consideration the dispersion of
refraction index. The value m 2 = (0.114 k 0.007) mo at 293 OKagrees to data [19, 201.
The decrease of the electron effective mass was calculated from the slope of the straight line portion of dependence @(A2) at heating of samples. For tempe- rature difference 130-560 OK- Am w 80 % ; here
m 1 3 0 OK
- - An - - 2.5.10M4 deg-' was supposed to be inde- n.AT
pendent on temperature and wavelength [6]. The strong temperature dependence of electron effective mass can be explained by positive value of forbidden band temperature coefficient dE, -- and by redistribution of
dT
free carriers near the Fermi level in the nonparabolic conductivity band.
Conclusions. - Experimental investigations of reflec- tivity, transmission, and Faraday rotation on epitaxial, synthetic n- and p-type crystals of PbTe indicated that the main extrema of bands are located near the same point of k-space, most likely at the < 11 1 > Brillouin zone boundaries. The increasing of the effective mass at the enlargement of free-carrier concentration indi- cated that both the conductivity and valence bands are nonparabolic.
Reference
[I] RAVICH (Ju. I.), EFIMOVA (B. A.), SMIRNOV (I. A.), The investigation methods of semiconductors applied to lead salts, Nauka, Moscow, 1968.
[2] Moss (T. S.), Optical properties of semiconductors, London, 1959.
[3] SCANLON:(W. W.), Phys. Rev., 1958,109,47.
141 LVDEN (H. A.), Phys. Rev., 1964, 134, A1106, 1964, 135, A514.
[5] KANAI (Y.), SHOHNO, J. Appl. Phys, Japan, 1963, 2, 6.
[6] ZEMEL (J. N.), JENSEN (J. D.), SCHOOLAR (R. B.) Phys. Rev., 1965,140, A330.
[7] RIEDL (H. R.), Phys. Rev., 1962,127, 162.
181 - - DIXON (J. R.), RIEDL (H. R.), Phys, Rev., 1965, 138 A873.
[9] NENSBERG (E. D.), IZV. ANSSSR, Neorg. materia&, 1965, no-6, 153.
C 4 - 104 JU. V. MALTSEV, I. K. SMIRNOV, JU. I. UKHANOV, A. N. VEIS MELNIK (R. B.), IZV. AN SSSR, Neorg. Materia&,
1966, 2, 12.
ALLEN (P. J.), RSI 1954,25, 394.
CARDONA (M.), GREENAWAY (D. L.), Phys. Rev., 1964,133, A1685.
BELLE (M. L.), FTT, 1965,7, 606.
PALIK (E. D.), MITCHELL (D. L.), ZEMEL (J. N.), Phys. Rev., 1964, 135, A763.
MALTSEV (Ju. V.), NENSBERG (E. D.), PETROV (A. V.), SEMILETOV (S. A.), UKHANOV (Ju. I.), FTT, 1966, 8, 2015.
[16] FAN (H. Y.), Rep. on Progu. in Physics, 1956, 19, 107.
[17] HAGA (E.), KIMURA (H. J.), Proc. Phys. Soc. Japan, 1964, 19, 1596.
1181 BOSWARVA (I. M.), LIDIARD (A. B.), Proc. Int. Conf., Exeter, 1962, 308.
[I91 Moss (T. S.), WALTON (A. K.), Physica, 1959,25,1142.
[2O] KURITA (S.), NAGASAWA (J.), TANAKA (K.), NISHINA (Y.), FURUROI (T.), Sci. Rep. RZTU, 1965, A, 17,
n o 1, 37.
