The optical properties of polycrystalline tellurium

J. Phys.

Chem. Solids

THE

Pergamon

OPTICAL

Press 1962. Vol. 23, pp. 1737-1742.

PROPERTIES

Printed in Great Britain.

OF POLYCRYSTALLINE

TELLURIUM J. N. HODGSON Physics Department,

The University,

(Received

Keele, Stafford&ire

22 June 1962)

AbstracL-The optical properties of evaporated films of tellurium have been investigated by several methods in the spectral range SOOO-27,000cm-1 (2.0 to 0.37 microns). A comparison of the experimental results with data on the infrared properties of tellurium crystals indicates that some films deposited on a glass prism had a random polycrystalline structure. The optical constants of these films are interpreted in terms of a tentative band structure for tellurium.

1. INTRODUCTION

reviewed by Moss.(s) In particular:

‘THE OPTICAL properties of tellurium in the infrared spectrum resemble those of the other semi,conducting elements. There is a region of low absorption in the infrared beyond an absorption edge at wave number k N 2700 cm-l, hv II 0.33 eV. A hole absorption band and absorption by free holes occur at lower wave numbers. The ,experimental data on the infrared absorption and refractive index of tellurium have been reviewed by FAN.(~) The experiments showed that the tellurium crystal is uniaxial with a high degree of anisotropy. Extrapolating the infrared measurements of CALDWELL and FAN(~) to zero frequency, one finds for the static dielectric constants e,,(O) = 38.6 and ~~(0) = 22.7 where the subscripts refer to an electric field parallel or perpendicular to the c-axis. These values ignore any contribution of holes to ~(0). The appropriate value for a random polycrystalline sample is ~(0) = &(G, +2eJ = 28-O. The optical properties of an isotropic material can be specified by the complex refractive index (n-i@ or alternatively, the di.electric constant e( = G-f@) and the conductivity c$ = n&z), as functions of the wave number k. The units of IJ are such that the conductivity in s1-l-cm-1 is o/30. c(k) and o(k) are related by the Kramers-Kronig integral relations, the application of which to optical measurements has been 1737

~(0) = 1+(4/r)

fmdX’a(X’), 0

where x’ is the wavelength. The major contribution to this integral must be due to the conductivity for wavelength less than 10-4 cm. Previous measurements of the optical constants of tellurium crystals in the visible and ultraviolet by VAN DYKJZ@) and MILLER(~) give values of 0 which seem too small to make the integral of the required size. In order to investigate this inconsistency, the optical constants of evaporated films of tellurium have been measured in the wave number of 5000 to 27,000 cm-l. The large anisotropy of tellurium crystals made it necessary to ensure that the evaporated films were polycrystalline without any preferred orientation. This was done by comparing the infrared value of E for films with the mean values from Caldwell and Fan. 2. EXF’ERIMENTAL

METHODS

The films used for optical measurements were prepared by vacuum evaporation of high purity (99.9 per cent) tellurium from a tantalum boat. The pressure in the evaporation chamber was held at N 10-b torr by a titanium evaporation

J.

1738

N. HODGSON

getter while the film was being deposited. Films were deposited on a glass slide or prism face which could be heated by a radiant heater and covered by a magnetic shutter. Films of thickness less than N 2000 il showed specular reflection of light with no scattering at both surfaces. Above this limit in thickness, the free surface of films had a cloudy appearance but the surface in contact with polished glass remained specular for all film thicknesses. The apparatus for optical measurements included two quartz Rochon polarizers and a quartz prism spectrometer and covered the spectrum 4000 to 27,000 cm-1 (2.5 to 0.37 microns). The optical Reflecting

face

B=fZ

FIG. 1. Glass or quartz prism.

entrance and exit prism faces. The azimuth I/Jand the phase difference A due to reflection in the prism were measured by the method described in a previous paper (HODGSON(@). If r, exp(iAp) and r, exp(iA,) are the complex reflection coefficients for the p and s waves, then: tan # = (T&~‘S) and A = (A,- A,). The expressions for E and G in, terms of #, A and 0, given in the previous paper, need to be multiplied by nt where no is the refractive index of the glass. Measurements were made first with the reflecting face uncoated to check for strain double refraction in the prism. The measured values of # were equal to the theoretical value (45”) within experimental error. The measured values of A are compared with theoretical values calculated from 710and 8 in Table 1. The differences have been used to correct the values of A measured with tellurium films on the reflecting face. The optical constants obtained from internal reflection by two tellurium films on glass prisms are shown in Fig. 2. Film Te-1 was deposited at a rate of N 10-s cm per set with the prism at N 100°C. The free surface of this film had a dark

Table 1. Values of A for uncoated glass prism, a = 75” k

4.7

6.7

10.9

16.9

A (exp.)

22.0

21.8

21.3

21.3”

A (theor.)

22.3

22.3

22.3

22.4”

constants of films on glass slides were measured by reflection of polarized radiation from the free surface. Films were deposited on the reflecting face of glass and quartz prisms (Fig. 1) for reflection measurements on tellurium-glass and tellurium-quartz surfaces. The effects of rate of deposition, film thickness and the temperature and nature of the substrate, on the optical properties were investigated. 3. EXPERIMENTAL RESULTS Two borosilicate crown glass prisms with CC= 75” were used in the first experiments. The angle of incidence 0 was made equal to the prism angle thus eliminating any polarization due to the

_

x lo3 cm-l

brown matt appearance. Film Te-2 was deposited at N 10-s cm per set with the prism at room temperature. The free surface of Te-2 was silvery matt in appearance. Other combinations of deposition rate and substrate temperature were tried but the optica! properties were intermediate between those of Te-1 and Te-2. All the films were sufficiently thick to be opaque for k > 5000 cm-l. The experimental errors in E and (Twere estimated from the errors in the measured angles and lie within the plotted circles and triangles of Fig. 2. The difference between the optical properties of Te-1 and Te-2 can be explained as an effect of different degrees of crystallite orientation in the two films. The values of E for Te-1 join up

THE

smoothly the mean constants typical of

OPTICAL

PROPERTIES

OF

POLYCRYSTALLINE

with the extrapolation of the curve for E from Caldwell and Fan. The optical of Te-1 have therefore been taken as random polycrystalline tellurium.

-3

0

1739

the glass prisms. The agreement between the calculated curves and the points for films (a) and (b) indicates that these films had nearly random structures. The occurrence of films with considerably different optical properties like film (c), may be explained by a greater tendency towards orientation of tellurium films on the regular lattice of crystalline quartz.

o’5I

-2 0

\

TELLURIUM

- IC1

“0 x

\o -0

0.1

I

01 5

I

I

10

15

k, 2

I k

3

cm-’ x104

FIG. 2. Optical properties of tellurium films on glass prism (a = 75”) by internal reflection. k = wave num0 = ber in cm-l; -c = dielectric constant = n2 42; conductivity = n/k Experimental points : 0 Te-1 ; 7 Te-2; 0 Moss(s); x CALDWELL and FAN.(~) Solid curve was drawn through points for Te-1 ; dotted curve represents *(6 il+2eJ.

A crystal quartz prism with M = 80” and the optic axis parallel to the prism faces was used in further optical measurements. Double refraction made measurement of A impossible but # was measured for several wave numbers. The p and s waves in the prism were the same as the 0 and E waves of the quartz crystal. Values of r, were calculated using the optical constants for Te-1 and Te-2 and no for quartz, ordinary index. A similar calculation was done for Y, and the calculated values of tan #( = rp/ys) were compared with measured values. The results of these experiments on three films are shown in Fig. 3. The influence of deposition rate and prism temperature on the optical properties seems to be greater than for

cm-’

x IO3

FIG. 3. Azimuth for internal reflection by tellurium films on crystal quartz prism (a = SO’). k = wave number in cm-l; tan $J = (Q/Q). Curves calculated for optical constants of Te-1 and T-2. Details of films: (b) (c) N ($ s N 10m6 N 10e5 cm/set Rate of deposition Temperature of prism Room N 1OO’C Room Experimental points x 0 V

Tellurium films were deposited on glass slides for optical reflection measurements on the free surfaces, tellurium-air. Films of three thicknesses were deposited under similar conditions, (i) 250 A, (ii) 500 A, (iii) 1600 A; thicknesses were measured by the interferometric method of TOLANSKY.(~) The optical constants were measured for K = 1.69 x 104 cm-l where the extinction coefficient has nearly its maximum value. This minimized the effect of film transmission on the reflection measurements. The intensity transmission coefficient of film (i) was 0.08, of films (ii) and (iii) < 0.01. The free surfaces of the fdms appeared completely specular without cloudiness. The values of # and A to be expected for reflection at 0 = 80” from a tellurium-air surface were calculated from the optical constants of Te-1 and Te-2. As shown

1740

J.

in Fig. 4, the measured lated values as the film values for film (i) were transmission. It seems

N.

HODGSON

values approach the calcuthickness is reduced. The affected by its appreciable that the free surface of a

in Table 2. The change of the apparent “constants” with 0 is further evidence of the non-ideal surface of tellurium films. It would seem difficult therefore to investigate the optical properties of tellurium by reflection from the free surface of an evaporated film. 4. BAND

813 -

‘\.

\ VTe-2 E-P,,

\

$ z D

‘qd’500A

\ \ \ \ \ \

q- 80-

STRUCTURE

The measured values of the dielectric constant E and the conductivity o of polycrystalline tellurium have been used to derive information about the electronic band structure. c and o are related by:

4d=250A

CC c(k)-

1 = (4/7r)

\

s

o(k’) dk’ ____ k’2

_

k2

(1)

0 \ \ \

\ !‘d=1600A

60

1

I 40

2 P, FIG. 4. Reflection

6C

50

degrees

from free surface of tellurium

films.

I/ = azimuth; (film thickness d); A = phase difference;

angle of incidence 0 = 80”; wave number k = 1.69 x 104 cm-l; 0 Experimental points; v Calculated points from optical constants of Te-1 and Te-2.

k and k’ are wave numbers. The graph of o vs. K consists of a series of peaks each corresponding to transitions from a filled to an empty band. The measured values of o form a single peak which can be completed by extrapolation at high k; this will be called peak (A). The contribution of peak (A) to the right hand side of (1) will be called EA. The remaining part of E is due to o for wave numbers above peak (A). All but a few per cent of E is due to transitions of the 535~4 valence electrons

Table 2. Apparent optical constants (ii) d = 500 _%

Film 0

80

72

(iii) d = 1600 A 80

72

-__ Te-1 75

Te-2 75”

n

3.81

3.84

2.93

3.48

4.37

4.31

R

3.72

3.59

262

1.87

368

3.88

-

tellurium film deviates from the ideal of a smooth plane bounding an isotropic medium, as assumed in the calculation of optical constants, to an extent which increases with film thickness. This deviation is evident in the optical measurements before it becomes sufficient to cause a visible cloudiness of the film surface. The apparent optical constants of films (ii) and (iii) were calculated from the measured values of # and A at 0 = 72 and 80”, as given

since the contribution of the core electrons to E is only = 0.1. The conductivity at wave numbers above peak (A) due to the valence electrons will be called peak (B), and the core electron contribution will be ignored. The corresponding EB can be estimated from the measured E and the EA calculated by means of (1). A convenient form of (1) for calculation is obtained by expressing o as a function of In k E s; with this transformation, (1)

THE

OPTICAL

PROPERTIES

OF POLYCRYSTALLINE

becomes : E/I(K) = E(k)---B(k)ds, s

PEAK (A)

1 = (2/nk)

1741

cm-1 are affected by the nature of the extrapolation of the cr vs. s curve. Adding 5.0, as an approximate value for e&O), to [l +EA(O)] leads to c(O) = 27.5 compared with the extrapolated value of 28.0 f or crystalline tellurium. The area on the CTvs. h graph needed to account for
1 t2)

c(s+s’)--Q(s--s’) sinh s’

1

TELLURIUM

A graph of o vs. s was drawn through the measured points for Te-1 and extrapolated linearly to zero through the last two points at the high s end. The results of graphical evaluations of the integral in (2) are given in Table 3. The curve of G vs. X, as

f = (4mc2/Nae2) j dk’o(k’)

Table 3. Dielectric constants k

0

4.9

8.1

13.4

e(k)

-

33.4

36.0

21.0

-

27.4

29.5

13.7

-11.0

5.0

5.5

6.3

CA(k) e&k)

(El

21.5 -

(A)

1,

cm x

10-4

FIG. 5. Conductivity (r vs. wavelength h for Te-1. The rectangle represents the area of peak (B).

shown in Fig. 5, was also integrated the equation : ~~(0) = E(O)-
1 = (4/z-)

and used in

j-

dh’o(X’)

PEA-iC (A)

The values of EB(k) for k = 22.0 and 26.9 x 104

22.0 6.8

3.2

26.9 x 103 cm-1 -

3%

-10.3 5.5

where Na is the number of atoms per cm3. Integration of the curve in Fig. 2 using the previously mentioned extrapolation, gives f N 1.8 per atom for peak (A). Band structure calculations by fiITZ(g) for a simplified model of the tellurium crystal lattice suggest that the 5p atomic level splits into three groups of bands separated by energy gaps. Each group of bands can hold two electrons per atom so the lowest and middle bands are filled and the top band is empty. Reitz considers that there is no mixing between the 5s and lowest 5p bands but HERMAN has suggested that mixing is probable. The calculations of REITZ do not show whether the lowest 5d band will overlap the 5p band so the identification of the conduction band is uncertain. The experiments of ROBINS on electron energy losses in tellurium films gave a plasmon energy of 17.0 eV compared with the free electron value for six valence electrons per atom, of 15 ~6 eV. The ultraviolet data of RUSTGI et aZ.(12) also show a rapid rise in the transmission of tellurium films for photon energies > 17 eV. The theory of NOZIERESand PINE@) shows that if the plasmon energy is nearly equal to the free electron value, as here, then the IS vs. k curve for the valence

J.

1742

N.

HODGSON

electrons has nearly all its area at photon energies considerably less than the plasmon energy. The main part of the conductivity peak (B) will therefore probably lie at k < 105 cm-l (hv < 12 eV). A tentative energy band diagram for tellurium which is consistent with the optical data is shown Y

I

I-

?

L,(A)

I

A81

1

Bottom

5p

and 55, posslblyl

wth gap between ( valence)

i i

0.33

I

is consistent with a conduction band width of N 2 eV. These tentative suggestions about the band structure of tellurium could be clarified by optical measurements at higher wave numbers in the ultraviolet and on single crystal specimens. Acknowledgement-1 would like to thank Dr. R. MCWEENY for advice about the tellurium band structure calculations.

i I I

I

eV

FIG. 6. Tentative

energy band diagram for telluriums. The arrows represent the limits of transitions for peak (A) and (B). in Fig. 6. The conduction band is identified with the top group of 5p bands and assumed equal in width to the middle group, as indicated by Reitz’s calculation. The soft X-ray absorption of tellurium, as measured by GIVENS et uZ.,(14) shows a double peak due to transitions from the N1v and Nv levels to the conduction band. The width of these peaks

REFERENCES 1. FAN H. Y., Rep. Progr. Phys. 19, 107 (1956). 2. CALDWELLR. S. and FAN H. Y., Phys. Rev. 114, 664 (1959). of semiconductors, 3. Moss T. S., Optical properties p. 22. Butterworths, London (1959). 6;919 (1922). 4. VA&DYKE G. D., J. opt. Sot. A&. 5. MILLER R. F., J. obt. Sot. Amer. 10. 621 (1925). 6. HODGSONJ. $, hoc. phys. Sot. iond. B68, 393 (1955). 7. TOLANSKYS., Multiple beam interferometry, p. 96. University Press, Oxford (1948). 8. SEITZ F., Modern theory of solids, p. 642. McGraw Hill, New York (1940). bITZ J. R., Phys. Rev. 105, 1233 (1957). Iz* HERMANF., Rev. mod. Phys. 30, 102 (1958). 11: ROBINSJ. L., PYOC.phys. Sot. Lond. 79, 119 (1962). 12. RUSTCI0. P., WALKERW. C. and WEISSLERG. L., J. opt. Sot. Amer. 51, 1357 (1961). NOZIERESP. and PINES D., Phys. Rev. 113, 1254 13. (1959). 14. GIVENS M. P., KOESTERC. J. and GOFFEW. L., Phys. Rev. 100, 1112 (1955).